Applied numerical modeling of saturated / unsaturated flow and ...

Applied numerical modeling of saturated / 

unsaturated flow and reactive contaminant transport 

- 

evaluation of site investigation strategies 

and assessment of environmental impact 

Dissertation 

zur Erlangung des Grades eines Doktors der Naturwissenschaften 

der Geowissenschaftlichen Fakultät 

der Eberhard-Karls-Universität Tübingen 

vorgelegt von 

Christof Beyer 

aus Braunschweig 

2007 

1

2 

Tag der mündlichen Prüfung: 23.02.2007 

Dekan: Prof. Dr. Peter Grathwohl 

1. Berichterstatter: Prof. Dr.-Ing. Olaf Kolditz 

2. Berichterstatter: Priv. Doz. Dr. rer. nat. Sebastian Bauer 

3. Berichterstatter: Prof. James F. Barker

Abstract 

Applied numerical modeling of saturated / 

unsaturated flow and reactive contaminant transport - 

evaluation of site investigation strategies 

and assessment of environmental impact 

CHRISTOF BEYER 

In this thesis numerical models of variably saturated flow and reactive transport 

processes in porous media are employed as assessment tools in two different fields of 

application. 

The first thematic complex studied is the computer based evaluation of investigation 

strategies for contaminated soils and aquifers. For this purpose the “Virtual Aquifer” 

(VA) concept is introduced and demonstrated by an assessment of the so called center 

line method for contaminant plume investigation. Errors and uncertainties in the 

estimation of contaminant degradation rates from center line data, which is often 

collected at field sites, are analysed. This is done by application at synthetic 

contaminated aquifers models, which are generated in the computer by numerical 

simulation of contaminant spreading. Monte Carlo simulations and sensitivity studies 

are performed to quantify the influences of sampling error magnitude, aquifer 

heterogeneity and model parameterisation on the estimated rate constants. In a second 

application example, the VA concept is used for the development and testing of a new 

approach for biodegradation parameter estimation. The performance of this method is 

studied in heterogeneous synthetic aquifers. Also, the propagation of errors and 

uncertainties from estimated rate parameters to a prognosis of the contaminant plume 

lengths is studied to obtain an indicator for the significance of the estimated degradation 

potential. The performance of the new parameter estimation method is assessed by 

comparison to frequently used approximations by first order kinetics. 

The second thematic complex addressed in this thesis is the evaluation of environmental 

impact of contaminant emissions from road constructions by type scenario modelling. 

Two general concepts for modelling of flow and transport, i.e. the Eulerian and the 

Lagrangian points of view, are combined in this study to assess the extent of leaching 

from contaminated demolition waste within road structures to the groundwater surface. 

Transport simulations are performed for a number of typical subsoil units of Germany 

to analyse the sensitivity of contaminant transport behaviour on subsoil properties. The 

relevant contaminant transport and attenuation processes in road constructions and the 

different subsoils are identified. The study allows to draw important conclusions on how 

these mechanisms could be used or enhanced to reduce contaminant leaching to groundwater. 

3

Angewandte numerische Modellierung der gesättigten / ungesättigten 

Strömung und des reaktiven Schadstofftransports- 

Evaluierung von Strategien der Standorterkundung 

und Umweltwirkungsprognose 

Kurzfassung 

CHRISTOF BEYER 

Diese Arbeit stellt die Verwendung numerischer Modelle gesättigt / ungesättigter 

Strömungs- und reaktiver Stofftransportprozesse in porösen Medien als 

Bewertungsinstrument in zwei verschiedenen Anwendungsfeldern vor. 

Der erste hier betrachtete Themenkomplex behandelt die computerbasierte Evaluierung 

von Erkundungsstrategien für kontaminierte Böden und Grundwasserleiter. Hierzu wird 

das Konzept der „Virtuellen Aquifere“ (VA) eingeführt und anhand einer Bewertung 

der sogenannten Center-Line Methodik zur Schadstofffahnenerkundung demonstriert. 

Die Center-Line Methode wird in der Praxis häufig zur Erhebung von Daten 

angewendet, auf deren Grundlage Schadstoff-Abbauratenkonstanten abgeschätzt 

werden können. Die Bewertung dieser Vorgehensweise wird durch eine Analyse der 

dabei auftretenden Fehler und Unsicherheiten vorgenommen, indem die Methode bei 

synthetischen kontaminierten Aquifermodellen angewendet wird, die am Computer 

durch numerische Simulation der Schadstoffausbreitung generierten wurden. Durch 

Monte-Carlo Simulationen und Sensitivitätsanalysen werden die Einflüsse von 

Messfehlern, Aquifer-Heterogenität und Modellparametrisierung quantifiziert. In einem 

zweiten Anwendungsbeispiel wird die VA-Methodik zur Ableitung und Prüfung eines 

neuen Ansatzes zur Schätzung von Parametern mikrobieller Abbaukinetiken 

angewendet. Die Eignung des Verfahrens wird in synthetischen heterogenen Aquiferen 

geprüft. Darüber hinaus wird die Fehler- und Unsicherheitspropagation von den 

geschätzten Abbauparametern zu mit diesen prognostizierte Schadstofffahnenlängen 

untersucht, um Indikatoren für die Aussagekraft der Parameter zu erhalten. Das 

Verfahren wird durch einen Vergleich mit häufig angewendeten Näherungen durch 

Kinetiken erster Ordnung bewertet. 

Der zweite im Rahmen dieser Arbeit behandelte Themenkomplex ist die Bewertung der 

Umweltauswirkungen von Schadstoffausträgen aus Straßenbauten durch Typszenarienmodellierung. 

Hier werden zwei allgemeine Konzepte zur Modellierung von Strömung 

und Transport, der Eulerische und der Larangesche Ansatz, miteinander kombiniert. 

Das Ausmaß des Schadstoffeintrags aus belastetem im Straßenbau eingesetztem 

Recycling-Bauschutt ins Grundwasser wird durch Einsatz der mit einander gekoppelten 

numerischen Modelle GeoSys/Rockflow und SMART bewertet. Die Transportsimulationen 

werden für eine Reihe typischer Unterbodeneinheiten Deutschlands 

durchgeführt, um die Sensitivität des Schadstofftransportverhaltens auf die 

Unterbodeneigenschaften zu untersuchen. Die relevanten Schadstofftransport- und 

Attenuierungsprozesse im Straßenaufbau und den Unterböden werden identifiziert. Die 

Studie ermöglicht wichtige Schlüsse im Hinblick eine Nutzung und Förderung dieser 

Mechanismen zu einer weiteren Reduktion der Schadstoffeinträge ins Grundwasser. 

5

Vorwort 

Die zurückliegenden drei Jahre am ZAG in Tübingen waren für mich eine sehr spannende Zeit 

und nicht nur im Hinblick auf meine Promotion sehr lehrreich. Aus diesem Grund möchte ich 

hier die Gelegenheit nutzen, ein paar persönliche Worte zu verlieren und einigen Personen zu 

danken, ohne deren Beteiligung diese Arbeit so sicher nicht hätte entstehen können. 

Die im Rahmen dieser Arbeit vorgestellten Ergebnisse wurden innerhalb der beiden Projekte 

“Virtueller Aquifer” (Förderkennzeichen 033 05 12/033 05 13) und “Übertragung der Ergebnisse 

des BMBF - Förderschwerpunktes „Sickerwasserprognose“ auf repräsentative Fallbeispiele” 

(Förderkennzeichen 02WP0517) erarbeitet. Beide Projekte wurden durch das Bundesministerium 

für Bildung und Forschung (BMBF) finanziell gefördert, wofür diesem hiermit 

gedankt sei. 

Besonders bedanken möchte ich mich bei den Betreuern dieser Arbeit. Herrn Prof. Dr.-Ing. Olaf 

Kolditz danke ich sehr für die Ermöglichung meiner Promotion, das in mich gesetztes Vertrauen 

und die stete Unterstützung. Seine immer optimistische Sicht der Dinge hat sich im Laufe der 

Zeit auch ein Wenig auf mich übertragen. Herr Dr. Sebastian Bauer ist mit Sicherheit die 

Person, die die Richtung meiner Arbeit am stärksten mitgeprägt hat. Sein aufrichtiges Interesse 

(auch abseits der fachlichen Dinge), viele ausführliche Diskussionen und immer konstruktive 

Kritik haben maßgeblich zu ihrem Gelingen beigetragen. Eine bessere Betreuung für seine 

Doktorarbeit kann man sich eigentlich kaum wünschen. 

Herrn Prof. Dr. James F. Barker danke ich für die Erstellung des dritten Gutachtens dieser 

Arbeit. 

Herzlich danken möchte ich Herrn Prof. Dr. Peter Grathwohl und Herrn Prof. Dr. Rudolf Liedl, 

die mir die Bearbeitung des Sickerwasserprognose-Projekts anvertrauten und mir so zu einem in 

jeder Hinsicht spannenden letzten Dreivierteljahr verhalfen. 

Ein herzlicher Dank geht an meine Kollegen Dr. Cui Chen, Dipl.-Geol. Jan Gronewold, Dr. 

Wilfried Konrad, Dr. Thomas Kalbacher, Dr. Chan Hee Park, Dr. Wenqing Wang, Dr. Chris 

McDermott, Dr. Martin Beinhorn, Robert Walsh, M.Sc., Dr. Dirk Schäfer von der Uni Kiel, Dr. 

Hermann Rügner und Dr. Peter Dietrich vom UFZ in Leipzig, sowie an zahlreiche weitere 

Kollegen für angenehm reibungslose Zusammenarbeit, viele hilfreiche Diskussionen und 

gemeinsame Pausen vom Zahlen hin und her schubsen. 

Mein größter Dank gilt meinen Eltern, da Ihr mir durch Eure Unterstützung diesen Weg erst 

ermöglicht habt, und Frauke für Deine große Geduld, Dein Verständnis, Deine Liebe und dafür, 

dass Du es trotz aller Umstände und Entfernungen gemeinsam mit mir bis hierhin geschafft 

hast. 

7

Table of Contents 

List of abbreviations and mathematical symbols 

1. Introduction 1 

2. Mathematical models 2 

2.1. Saturated / unsaturated flow 2 

2.2. Transport processes 3 

2.3. Reactive processes 4 

2.4. Numerics and software methods 6 

3. Modeling applications 9 

3.1. Evaluation of investigation strategies for contaminated aquifers using 

the Virtual Aquifer concept 9 

3.2. Development and testing of a new approach to estimating biodegradation 

parameters from field data 16 

3.3. Prognosis of long term contaminant leaching from recycling materials in 

road constructions 18 

4. Conclusions and outlook 21 

References 

Enclosed Publications 

23 

9

List of abbreviations and mathematical symbols 

a empirical sorption constant [-] 

b empirical sorption constant [-] 

C concentration [M L -3 ] 

C0 equilibrium concentration [M L -3 ] 

Cl liquid phase concentration [M L -3 ] 

Cs solid phase concentration [M M -1 ] 

Corg soil organic carbon content [%] 

Cw(�) water capacity function 

D tensor of hydrodynamic dispersion 

[L² T -1 ] 

Da aqueous molecular diffusion 

coefficient [L² T -1 ] 

Dae effective diffusion coefficient [L² T -1 ] 

Dap apparent diffusion coefficient [L² T -1 ] 

De tensor of effective hydrodynamic 

dispersion [L² T -1 ] 

Dm tensor of mechanical dispersion 

[L² T -1 ] 

DW demolition waste 

FE finite element 

FEM finite element method 

g travel time probability density 

function 

h hydraulic head [L] 

h´ erroneous head measurement [L] 

I identity tensor 

IC inhibition concentration [M L -3 ] 

K tensor of hydraulic conductivity 

[L T -1 ] 

kmax maximum degradation rate 

[M L -3 T -1 ] 

l pore connectivity parameter [-] 

m empirical Van Genuchten 

parameter [-] 

MC half-saturation concentration [M L -3 ] 

MM Michaelis-Menten 

n empirical Van Genuchten 

parameter [-] 

OOP object-oriented programming 

PDE partial differential equation 

pdf probability density function 

Q source or sink term [M L -3 T -1 ] 

r radial distance [L] 

REV representative elementary volume 

S specific storativity [L -1 ] 

Sr relative saturation [-] 

t time [T] 

V magnitude of velocity vector 

[L T -1 ] 

vmax maximum growth rate [T -1 ] 

v average linear velocity [L T -1 ] 

v vector of average linear velocity 

[L T -1 ] 

VA Virtual Aquifer 

X microbial population [M L -3 ] 

Y yield coefficient [-] 

z elevation [L] 

� empirical Van Genuchten 

parameter [L -1 ] 

�L 

�T 

longitudinal dispersivity [L] 

transverse dispersivity [L] 

� reaction function 

� tortuosity related coefficient [-] 

� random number 

� intraparticle porosity [-] 

��h maximum measurement error [L] 

� empirical sorption constant [-] 

�e effective porosity [-] 

� volumetric water content [-] 

�r residual water content [-] 

saturated water content [-] 

�s 

� empirical sorption constant 

[M 1-� L 3� M -1 ] 

� first order degradation rate 

constant [T -1 ] 

� microbial decay term 

� density [M L -3 ] 

2 

� Y ln(K) variance 

� travel time [T] 

tortuosity factor [-] 

�f 

� matric head [L] 

11

Applied numerical modeling of saturated / unsaturated 

flow and reactive contaminant transport – 

evaluation of site investigation strategies and assessment 

of environmental impact 

1. Introduction 

In the field of subsurface hydrology, mathematical 

models are used to simulate fluid 

flow and solute transport by translating 

physical and biogeochemical processes into 

mathematical equations, which can be 

solved by either analytical or numerical 

methods. Understanding of individual processes 

in domains of simple geometry is 

already a challenging task for itself. In large 

scale applications, however, we are faced 

with heterogeneous environments and interactions 

of many different types of spatially 

and temporally variable processes, which 

leave the numerical treatment of such complex 

coupled problems often as the only 

way to reach meaningful conclusions 

(Zheng and Bennett, 1995). As a benefit of 

the rapid development of computational capabilities 

(e.g. high performance parallel 

computing, specialized software implementation 

methods, data pre- and post-processing 

tools, graphical display routines) the 

numerical simulation of complex coupled 

problems in subsurface hydrology is continuously 

advanced. In general, numerical 

models of flow and transport in geosystems 

are used as tools for 

� qualitative and quantitative analysis of 

single or coupled processes 

� identification of relevant parameters 

� parameter estimation / inverse modeling 

� sensitivity and uncertainty analysis 

� prediction of system response to changes 

in initial or boundary conditions 

These capabilities as well as the spectrum 

of application would not have been 

achieved without the ever growing demand 

for groundwater resources and the concern 

about its quality. Groundwater is one of the 

main drinking water supplies and increasingly 

used for agricultural field irrigation 

(Morris et al., 2003). At the same time 

pollution from industrial activities, waste 

disposal, agricultural use of fertilizers or 

pesticides and urban waste waters (Scheidleder 

et al., 1999) poses a serious threat to 

our groundwater resources. 

In this thesis numerical modeling is used in 

three applications within the context of contaminant 

hydrology. The term applied numerical 

modeling here emphasizes the field 

scale application of computational methods 

to obtain solutions to systems of partial differential 

equations describing flow and 

reactive transport process interactions in 

porous media. Application 1 introduces the 

“Virtual Aquifer” (VA) concept, in which 

numerical modeling is used as a tool for the 

evaluation of investigation and remediation 

strategies for contaminated soils and aquifers. 

The concept is demonstrated by an 

assessment of the so called center line method 

for site investigation. In application 2 

the VA concept is applied for the development 

and testing of a new approach for biodegradation 

parameter estimation. Both 

applications make use of the GeoSys / 

Rockflow code (Kolditz et al., 2006) for the 

numerical simulations. In application 3 the 

Eulerian and Lagrangian concepts for contaminant 

transport modeling are combined. 

GeoSys / Rockflow is coupled with the 

SMART model (Finkel et al., 1998) and 

used for type scenario modeling to assess 

the environmental impact of recycling 

materials in road constructions. 

This synthesis is organized as follows: 

Chapter 2 presents the mathematical process 

models and numerical schemes used 

1

for the application studies outlined. The 

chapter is based on fundamental publications 

in the field of computational hydrology. 

However, it is not intended to 

serve as a comprehensive review of processes 

and model concepts, as this would be 

beyond the scope of this synthesis. Chapter 

3 and its subsections present the results 

and conclusions of the three application 

examples. Chapter 4 closes this synthesis 

with general conclusions and an outlook. 

2. Mathematical models 

The three-dimensional structure of natural 

porous media is manifested in its composition 

of three constituting phases, i.e. the 

solid (mineral or biophase), water and gaseous 

phases. At scales larger than the pore 

scale, a description of porous medium geometry 

becomes very complex and thus infeasible 

for modeling applications. To 

understand and formulate the dynamics of 

fluids in the subsurface the so called representative 

elementary volume (REV) concept 

is introduced (Bear, 1972): In the 

transition from the microscale to a larger 

macroscale material, parameters are averaged 

over a volume which is sufficiently 

large to describe the porous medium at that 

larger scale (see Fig. 1). 

Fig. 1: Representative elementary volume 

concept (Bear and Bachmat, 1990). 

This may also require a reformulation of the 

mathematical process descriptions. The 

derivation of representative or effective 

new material parameters and the corresponding 

governing equations is termed 

2 

upscaling. Within the REV the detailed 

structure of the medium is lost and becomes 

a continuous field. Parameters like porosity, 

permeability or dispersivity are considered 

constant over the averaging volume. In the 

following sections material parameters and 

governing equations are based on this 

continuum approach. 

2.1. Saturated / unsaturated flow 

The dynamics of the water in fully saturated 

three dimensional porous media can be described 

by the combination of the mass 

balance for the water phase (eq. (1)) and 

Darcy’s law (eq. (2)) as a constitutive equation 

(Bear, 1972) 

�h 

S � �� q 

� Q 

(1) 

�t 

q � �K�h 

(2) 

where S [m -1 ] is specific storativity, h [m] is 

the hydraulic head, given as the sum of elevation 

z [m] and the pressure head � [m], t 

[s] is time, q [m s -1 ] is the Darcy flux vector, 

K [m s -1 ] is the tensor of hydraulic conductivity 

and Q [s -1 ] is a source or sink 

term. The governing equation for groundwater 

flow under transient conditions is 

thus given by (Bear, 1972) 

�h 

S � � � �K �h��Q 

(3) 

�t 

which at steady state converts to 

� �h��Q 

� � K (4). 

This model of steady state flow in saturated 

porous media is employed in applications 1 

and 2 (sections 3.1, 3.2) of this synthesis. 

For unsaturated conditions, which are prevalent 

in application 3 (section 3.3), a more 

general form of eq. (2), the Buckingham- 

Darcy-law, can be used (Jury et al., 1991) 

q � �K( 

� ) �h 

(5) 

where K is a function of the pressure (or 

matric) head �, which itself depends on the 

volumetric water content � [-] of the porous 

medium. As for saturated conditions, the

Buckingham-Darcy law is combined with 

mass balance principles to yield the governing 

equation of flow for unsaturated 

conditions, i.e. the Richards equation. This 

equation exists in three main forms with �, 

� or both quantities as dependent variables 

(Jury et al., 1991). In GeoSys / Rockflow 

the �-based form (Freeze and Cherry, 

1979) is implemented (Du et al., 2005) 

C w 

�� 

�t 

�� K ( � ) �� 

� 

�K 

� 

�z 

�� Q 

(6) 

where Cw(�) is the water capacity function 

defined by d� /d� and with z positive in a 

downward direction. For the functional 

description of unsaturated hydraulic properties 

different mathematical formulations 

have been proposed in literature. A frequently 

used constitutive relation is the 

Van-Genuchten-Mualem model (Van Genuchten, 

1980) which is based on the 

statistical pore space model of Mualem 

(1976) and is given by 

� � � �2 �m 

m 

1� 

1� 

Sr 

l 

K( 

S ) � KS 

(7) 

Sr 

r 

� �� 

r � � 

� �� 

s 

r 

r 

1 

� � � �m n 

1� 

�� 

(8) 

m � 1� 1/ 

n 

(9) 

where Sr [-] is defined as the relative 

saturation, l [-] is a pore connectivity parameter, 

�, �r and �s are the actual, residual 

and saturated volumetric water contents, � 

[m -1 ], n [-], and m [-] are empirical parameters. 

Other constitutive relationships comprise 

approaches such as the Brooks-Corey 

model (Brooks and Corey, 1966), the Haverkamp 

model (Haverkamp et al., 1977), potential 

functions as introduced by Huyakorn and 

Pinder (1983) or the multimodal model of 

Durner (1994). Recently, also free form parameterizations 

were suggested (Bitterlich et al., 

2004). An overview on the prevalent approaches 

is given e.g. by Durner and Flühler (2005). 

2.2. Transport processes 

The most fundamental transport process of 

dissolved substances in saturated porous 

media is advection. Advection is passive 

with the flowing water. Purely advective 

transport of a solute plume is free of interference 

or mixing with the surrounding 

ambient water and is described with the advection 

equation (Zheng and Bennett, 1995) 

�C 

�t 

� �v 

�C 

� Q 

(10) 

where C [kg m -3 ] is the concentration of a 

dissolved species, v [m s -1 ] is the vector of 

average linear velocity which is given by 

division of q with the effective porosity �e 

[-], and Q [kg m -3 s -1 ] is a source or sink 

term for species C. 

In natural aquifers or soils, purely advective 

transport is practically not established as 

dissolved molecules migrate from high to 

low concentration regions by Brownian 

motion. This concentration gradient driven 

mass transport is termed molecular diffusion 

and occurs even when the fluid itself is 

stagnant. For transient systems the diffusion 

process in water can be described using 

Fick’s 2 nd law, (Fetter, 1993) 

2 

�C 

� C 

� �Da 

2 

�t 

�x 

(11) 

where Da is the molecular diffusion coefficient 

in water [m 2 s -1 ]. In porous media the 

diffusion process is hindered by the presence 

of the solid phase matrix and the tortous 

nature of the pores. Thus an effective 

diffusion coefficient Dae is derived as 

(Grathwohl, 1998) 

D 

� � 

e 

ae a 

f 

D � (12) 

� 

where � [-] is the constrictivity and �f [-] the 

tortuosity of the porous medium. Under 

unsaturated conditions Dae can also be 

related to � (e.g. Olsen and Kemper, 1968). 

As water moves through a porous medium, 

single streamline velocities can be greater 

or less than v. This effect is due to different 

3

path lengths of water molecules that bypass 

mineral grains of different size and shape, 

different pore diameters as well as inner 

pore friction which results in velocity contrasts 

along pore cross sections. The consequential 

divergence of transport velocities 

for dissolved solutes causes mixing with the 

ambient water along the flow path and thus 

results in solute spreading longitudinally 

and transversally to the main flow direction. 

This process is termed mechanical dispersion. 

For its mathematical description usually 

an analogy to the diffusion process is 

assumed. According to Bear (1972) the tensor 

of mechanical dispersion Dm is given by 

D m 

4 

� � I � � 

��L�� V 

TV 

T 

vv 

(13) 

where �L and �T [m] are longitudinal and 

transverse dispersivities, V is the magnitude 

of the velocity vector, I is the identity 

tensor and vv is the dyadic of the velocity 

vector. The tensor of hydrodynamic dispersion 

D [m 2 s -1 ] combines the dispersion and 

diffusion processes and is calculated by 

D m ae 

� D � D I 

(14). 

The mathematical formulation of advectivedispersive 

transport in fully saturated porous 

media assuming constant porosity is 

given by the sum of the advective and 

dispersive fluxes, i.e. the advection-dispersion-equation 

(Zheng and Bennett, 1995) 

�C 

� �� v C��D�C��Q 

(15). 

�t 

For unsaturated conditions the total solute 

flux in the water phase is described by 

��C 

� �� q�C��De�C��Q(16) �t 

where the effective hydrodynamic disper� 

sion tensor De is used, as besides Dae also 

�L and �T depend on � (Bear, 1979). 

2.3. Reactive processes 

The source or sink terms Q in eq. (15) and 

(16) represent a large variety of processes 

other than advection or hydrodynamic dispersion, 

which may cause temporal changes 

in the solute concentration C. Hence, Q 

may represent transfer of species between 

solid, water, gaseous or biophase (e.g. volatilization, 

non aqueous phase liquid dissolution, 

sorption), or equilibrium and kinetic 

reactions of (geo-)chemical or biochemical 

nature. According to Rubin (1983) (Fig. 2) 

reactive processes can be classified by 

� reaction velocity and reversibility (equilibrium 

or non-equilibrium; level A) 

� involvement of only a single or several 

phases (homogeneous / heterogeneous; 

level B) 

� reaction type: surface (e.g. sorption) or 

“classical” chemical reaction (level C) 

For the sake of brevity, here only process 

concepts relevant for the model applications 

of chapter 3 are explained in more detail. 

Fig. 2: Classification of reactions in porous 

media (Rubin, 1983). 

Kinetic sorption 

Transfer of dissolved species from the 

mobile phase to the solid matrix by 

physico-chemical processes is termed 

sorption. Sorbed species are immobilized 

and not transported with the water flux. 

Sorption is a reversible process, i.e. sorbed 

species can be remobilised by desorption. 

Temporary immobilisation by sorption results 

in lowered solute concentrations and 

retarded transport velocities. The manifold 

processes contributing to sorption phenomena 

include physical as well as chemical 

mechanisms (e.g. ion exchange and surface 

complexation through Coulomb or van der 

Waals forces, hydrogen-, hydrophobic- or

covalent bonding). To mathematically describe 

the sorption / desorption behaviour of 

a species so called sorption isotherms can 

be used, in which the sorbed amount of the 

species is a function of its dissolved 

concentration. A general nonlinear sorption 

model can be formulated as 

C 

s 

� �a�Cl� � � � abC � 

� 1 

(17) 

l 

where Cs [kg kg -1 ] is the sorbed solid phase 

and Cl [kg m -3 ] the liquid phase concentration, 

�, a,� b and � are empirical constants. 

For b = 0, a = 1 [-] and � = 1 [-] eq. 

(17) is the linear Henry isotherm, in this 

case � [m³ kg -1 ] is a simple equilibrium 

constant. For � = 1 and b = 1 eq. (17) is the 

Langmuir isotherm with � [kg kg -1 ] being 

the maximum amount of a species which 

can be sorbed to the solid phase and a 

[m³ kg -1 ] an adsorption constant. For b = 0, 

a = 1 [-] and � � 1 [-] (usually � < 1) the 

Freundlich isotherm is obtained, where � 

[kg 1-� m 3� kg -1 ] is the Freundlich coefficient 

and � [-] is the Freundlich exponent. 

In general, sorption processes may be treated 

as equilibrium reactions because sorption 

is fast compared to transport. However, 

there are exceptions to this rule because for 

some solute species as well as soils or 

aquifers, equilibration is a slow process. 

Possible mechanisms include slow diffusion 

into intraparticle pores accompanied by 

equilibrium sorption to surfaces within the 

pores or slow diffusion in organic matter 

(Ball and Roberts, 1991; Grathwohl, 1998; 

Rügner et al., 1999). Hence, from a 

macroscopic point of view sorption equilibrium 

may not be reached within the available 

contact time between mobile and solid 

phases. Assuming spherical particles, intraparticle 

diffusion kinetics can be described 

by Fick´s 2 nd law in radial coordinates (e.g. 

Grathwohl, 1998) 

2 

� C �� 

C 2 � C � 

� Dap� 

� 2 

� t 

� 

� � r r � r � 

(18) 

where r [m] is the radial distance from the 

particle center and Dap [m 2 s -1 ] is the appa- 

rent diffusion coefficient, which is calculated 

from Da by 

D 

D � 

a 

ap � (19) 

( � �� 

�) 

� f 

where � [-] is the intraparticle porosity, � 

the linear equilibrium sorption coefficient 

and � [kg m -3 ] the particle density. Other 

approaches to describe slow sorption / desorption 

kinetics comprise first- or secondorder, 

two-stage models (e.g. Brusseau and 

Rao, 1989; Ma and Selim, 1994; Streck et 

al., 1995). A comparison of first-order and 

diffusion limited approaches was recently 

published by Altfelder and Streck (2006). 

Kinetic degradation 

Degradation, whether biotic or abiotic, is 

the only process that reduces the overall 

mass of contaminants in natural porous 

media without transfer to other phases. 

Biological degradation mechanisms are numerous, 

complex and by far not completely 

understood nor even identified. The vast 

amount of different types of microorganisms 

in the subsurface provides many 

metabolic pathways for contaminant degradation 

under aerobic and anaerobic conditions. 

Through successive oxidation or 

reduction reactions contaminants can be 

transformed to innocuous compounds like 

methane, chloride, water or carbon dioxide 

(Wiedemeier et al. 1999). However, intermediate 

products can even be of significantly 

higher toxicity and persistence than 

their parent compounds (e.g., dechlorination 

of dichloromethane to vinyl chloride; 

Wiedemeier et al. 1999). 

Kinetic growth and decay of a microbial 

population X [kg m -3 ] can be described by a 

generalized Monod-type equation as given 

e.g. in Schäfer et al. (1998) 

�X 

� v 

�t 

� 

max 

X 

� 

I 

� 

I 

Ci 

M � C 

i Ci 

i 

C j 

� C 

j C j j 

�� 

�X� (20) 

5

where vmax [s -1 ] is a maximum growth rate, 

Ci [kg m -3 ] is the concentration of the i th 

substrate, MCi [kg m -3 ] is the corresponding 

half velocity concentration [kg m -3 ], Cj 

[kg m -3 ] is the concentration of the j th substance 

inhibiting microbial growth, ICj is the 

corresponding inhibition concentration 

[kg m -3 ] and �(X) is a microbial decay term, 

which is often modeled as being of first 

order. Consumption of substrates or production 

of k metabolites Ck [kg m -3 ] is both 

coupled to microbial growth via (Schäfer et 

al., 1998) 

�Ck �1 

��X 

� 

� 

�t 

Y � 

� �t 

� 

� 

6 

(21) 

k growth 

where Yk [-] is the yield coefficient for 

substrate or metabolite Ck, and [·]growth 

refers to the growth term only in eq. (20). 

From the generalized Monod-type equation 

different more simple kinetic formulations 

can be derived. For a temporally constant 

microbial population, i.e. growth and decay 

terms are constant and of equal magnitude, 

no inhibition and dependence on only a 

single substrate, eq. (20) and (21) can be 

combined to yield the Michaelis-Menten 

(MM) kinetics model (Simkins and 

Alexander, 1984) 

dC 

dt 

C 

� �kmax 

(22) 

C � M C 

where k max is the maximum degradation rate 

[kg m -3 s -1 ] and M C is the MM half-saturation 

concentration [kg m -3 ]. This approximation 

may be applicable e.g. when aquifer 

sediments have been exposed to contaminants 

for several years (Bekins et al., 1998) 

and is used in application 2 (section 3.2). 

Often, contaminant degradation is also 

described by simple first order kinetics, e.g. 

to simulate abiotic degradation reactions 

like hydrolysis and dehydrohalogenation of 

halogenated compounds (Wiedemeier et al., 

1999). First order kinetics can be derived 

from eq. (22) for C > MC eq. (22) 

approaches zero order kinetics. The first 

order model is used in application 1 and 3 

(sections 3.1 and 3.3). Extensive reviews on 

kinetic models of biodegradation can be 

found e.g. in Baveye and Valocchi (1989), 

Rittmann and VanBriesen (1996) or Islam 

et al. (2001). 

2.4. Numerics and software 

methods 

Numerical solution of the governing 

equations 

The governing equations for flow and reactive 

transport presented in sections 2.1 - 2.3 

belong to the group of partial differential 

equations (PDE), containing derivatives of 

first order in time and of first as well as 

second order in space. The classification of 

PDE can be based on mathematical aspects 

(highest order derivatives in the dependent 

variables) or on a physical point of view 

(problem type of physical process) (Kolditz, 

2002). Parabolic PDE are used for timedependent 

problems with dissipation process, 

such as diffusion (eq. (11)) or transient 

groundwater flow (eq. (3)), which 

convert to elliptic PDE for steady state 

conditions (eq. (4)). A third class of PDE 

are hyperbolic equations like the linear advection 

equation (eq. (10)), which are used 

to describe time-dependent problems without 

dissipation process. The transport equations 

(15) and (16) are of mixed type with a 

parabolic dispersion-diffusion term and a 

hyperbolic advection term. Their behaviour 

for a particular problem depends on the 

relative magnitudes of these flux components. 

In general the transport equations are 

of parabolic character which changes to 

hyperbolic for large ratios of v/�L as in this 

case the advective flux term is dominant 

(Kolditz, 2002). 

In general, PDE describing physical 

problems are well-posed when appropriate 

initial and boundary conditions are specified 

for the domain where a solution is 

required. While analytical solutions can be

found for a number of problems with 

simple geometries and boundary conditions 

(e.g. Bear, 1979; Van Genuchten and Alves, 

1982; Kinzelbach, 1983), for complex nonlinear 

problems exact solutions may not 

exist and thus approximate numerical 

solutions must be obtained. Given the 

governing equations with appropriate initial 

and boundary conditions for a specified 

problem, the general strategy for a numerical 

solution is to first convert the PDE into 

a system of discrete algebraic equations and 

then to find the exact solution of the latter, 

which is the approximate solution of the 

PDE. An overview of approximation 

methods for the solution of PDE is given in 

Fig. 3. 

Fig. 3: Overview of approximation methods 

for PDE (Kolditz, 2002). 

Among the many possible approaches for 

the numerical simulation of fluid dynamics 

and transport, finite difference, finite element 

and finite volume methods are the 

most frequently used. Their theory and implementation 

is content of numerous text 

books (e.g. Pinder and Gray, 1977; Baker, 

1983; Zienkiewicz and Taylor, 1993; 

Helmig, 1997; Kolditz, 2002). Here, a short 

description of the finite element method 

(FEM) only is given, as it is the numerical 

scheme which is implemented in the 

GeoSys / Rockflow code (Kolditz et al., 

2006) which is used in all applications 

presented in chapter 3. 

The concept of the FEM is based on the 

spatial discretization (“meshing”) of continuous 

structures into discrete substructures, 

i.e. the finite elements (FE). In comparison 

to finite differences, the big advantage of 

the FEM is its ability to also handle com- 

plex geometries by unstructured or arbitrarily 

shaped grids (Anderson and 

Woessner, 1992). The governing PDE are 

discretized by deriving integral formulations, 

e.g. by the method of weighted 

residuals. By introducing weighting functions 

the approximate solution is forced to 

satisfy the condition that the weighted 

average residual of the unknown true solution 

at the nodes over the computational 

domain is equal to zero. For any location 

within the domain the solution is obtained 

by linear combination of local interpolation 

functions and the solution at the nodes. 

With the standard Galerkin method 

(Huyakorn and Pinder, 1983) weighting and 

interpolation functions are selected 

identically. The FEM is globally mass 

conservative, locally, however, mass 

conservation problems may occur. Therefore 

mixed FEM approaches draw increasing 

attention (e.g. Starke, 2000; Knabner 

and Schneid, 2002; Korsawe et al., 2006). 

The governing PDE of flow and reactive 

transport in porous media outlined in the 

previous sections are formulated from the 

point of view of an fixed observer with the 

fluid and solute moving on a fixed spatial 

grid. This approach is termed the Eulerian 

concept and is able to handle dispersiondominated 

problems accurately and efficiently. 

For advection-dominated problems, 

however, the Eulerian concept is susceptible 

to numerical dispersion and artificial 

oscillations (Zheng and Bennett, 1995), For 

the price of increased computational effort 

this problem can be limited by sufficiently 

fine spatial and temporal discretization. 

By contrast, the Lagrangian concept is well 

suited for advection-dominated problems, 

but problematic when advection and dispersion 

must be solved together (Thorenz, 

2001). In the Lagrangian concept concentrations 

are not associated with fixed spatial 

points but rather with moving particles, that 

are transported with the prevailing flow 

velocity. The numerical model SMART 

(Finkel et al., 1998) is based on the 

Langrangian concept of Cvetkovic and 

Dagan (1996). In SMART the model 

7

domain is discretized along advective flow 

paths of the Eulerian flow field by the travel 

time � [s] of inert particles between an 

injection plane and a control plane, both 

oriented normal to the mean flow direction. 

Each particle trajectory is regarded as a 

separate one-dimensional stream tube of the 

flow field with an infinitesimal cross 

section. The probability density function 

(pdf) g(�, x) of all particles travel times 

completely reflects all hydraulic heterogeneities 

of the model domain. Influences of 

reactions (e.g. biodegradation, intraparticle 

diffusion, sorption, etc.) are quantified by 

means of the reaction function �(�, t), 

which is evaluated by the BESSY model 

(Jäger and Liedl, 2000) implemented in 

SMART. With given g and � the normalized 

breakthrough curve of a reactive 

solute at a control plane is calculated by 

(Finkel et al. 1998) 

8 

� 

� � 

0 

�x, t� 

g��, 

x��t�d� 

C , 

(24). 

To overcome the limitations of both the 

Eulerian and the Lagrangian concepts, 

mixed Eulerian-Lagrangian methods can be 

used, which take advantage of the particular 

appropriateness of both concepts for modeling 

advective and dispersive transport (e.g. 

Neumann, 1981; Thorenz, 2001; Park et al., 

2006). In application 3 (section 3.3) a 

combination of the software codes GeoSys / 

Rockflow and SMART is used for a combined 

application of the Eulerian and the 

Lagrangian concepts. GeoSys / Rockflow is 

used to model the Eulerian flow field in a 

heterogeneous two dimensional domain and 

to derive the representative pdf g(�, x). The 

SMART model then utilizes the pdf for the 

simulation of reactive transport in the 

model domain. 

Object- and process-oriented methods 

The GeoSys / Rockflow code, which is used 

for most of the numerical simulations 

described in chapter 3, is written in the C++ 

language and thus allows the implementation 

by object oriented programming (OOP) 

methods. The OOP concept is especially 

helpful for the development of complex 

software in programmer teams, as encapsulation 

and class-structures render the code 

more stable and errors are easier to detect. 

In GeoSys / Rockflow the OOP concept is 

met by so called process orientation 

(Kolditz and Bauer, 2004), which allows 

the coupling of two-phase flow, heat transport, 

mass transport, chemical reactions and 

deformation in an efficient way (Wang et 

al., 2006). The basic idea of process orientation 

is that between each physical process 

(e.g. single species transport) and its 

numerical approximation by an algebraic 

equation system (AES) exists a direct correspondence 

(Fig. 4). The AES originates 

from the temporal and spatial discretization 

of the PDE on the computational grid. For 

its solution the following steps are performed: 

� AES assemblage and incorporation of 

initial conditions 

� determination of element matrices 

� incorporation of boundary conditions 

and source terms 

� solving the AES by appropriate solvers 

This procedure can be generalized for any 

physical process regardless of its specific 

type in an object oriented way by introducing 

the process object (Kolditz and Bauer, 

2004) (Fig. 4). 

Transport of non-reactive 

species in water 

„PROCESS“ 

Multifield problem, if many 

mobile species are 

transported 

Solution of a PDE 

system 

PROCESS-OBJECT 

System of PDE 

solved by 

Multi-Process-Method 

Fig. 4: Process analogy and process object, 

shown for an instance of a transport 

process (Kolditz and Bauer, 2004).

The process object has access to all 

required data structures and functions, and 

thus is self configuring, executing and 

destructing. 

Contaminant transport problems usually 

involve a number of different processes as 

outlined in sections 2.1 – 2.3. The resulting 

multi-field problems can be approached by 

a multi-process algorithm, where one instance 

of the process object is created for 

each process considered. Solution of any 

number of flow or transport equations is 

fully automatic and encapsulated, guaranteeing 

high efficiency and flexibility. 

In GeoSys / Rockflow saturated and unsaturated 

flow as well as conservative transport 

are solved using standard Galerkin FE. A 

non-iterative operator splitting technique 

for the coupling of conservative transport 

and (bio-)chemical reaction processes is 

used (Xie et al., 2006; Bauer et al., 2006b). 

First, the flow field is solved followed by 

conservative transport for all species. In the 

third step the calculation of kinetic 

biochemical reactions is performed. Finally 

chemical equilibrium reactions are calculated. 

This approach allows an easy handling 

of any number of species and reaction 

processes as well as employing optimised 

mathematical methods for the solution of 

the corresponding equation systems. The 

non-iterative approach, however, is limited 

to small time steps in order to avoid 

numerical instabilities. It is also known not 

to converge necessarily to the exact solution 

(Carrayrou et al., 2004). These limitations 

can be overcome using a computationally 

more demanding iterative operator splitting 

approach (e.g. Kinzelbach et al., 1991). 

3. Modeling applications 

In this chapter three application examples 

of numerical models for flow and reactive 

transport as established in chapter 2 are 

presented. Section 3.1 introduces the Virtual 

Aquifer (VA) concept, in which numerical 

modeling is used for the evaluation of 

investigation strategies for contaminated 

sites. In section 3.2 the VA concept is 

applied to derive and test a novel method 

for the estimation of biodegradation kinetic 

parameters from measured field data. Section 

3.3 uses numerical modeling as a tool 

to predict the environmental impact of demolition 

waste used in road constructions 

3.1. Evaluation of investigation 

strategies for contaminated 

aquifers using the Virtual 

Aquifer concept 

Due to the limited accessibility of the 

subsurface, measurements of piezometric 

heads and pollutant concentrations at contaminated 

sites are sparse and may not be 

representative of the heterogeneous hydrogeologic 

conditions. Any site investigation 

is thus subject to uncertainty, reflecting the 

limited knowledge on aquifer properties 

and the extent of the contamination. Three 

main sources of uncertainty can be identified 

for site investigation, which are illustrated 

in Fig. 5. Conceptual model errors 

result from an incorrect identification of the 

governing processes at a site. Heterogeneity 

of the site causes an incomplete or wrong 

description of the relevant parameter distributions. 

For variables measured at observation 

wells like heads or concentrations 

measurement errors are inevitable. Due to 

this uncertainty, field investigation methods 

for plume screening and measuring of 

hydraulic conductivity or degradation rates 

can hardly be tested or verified in the field. 

The VA approach is particularly aimed to 

overcome this problem. Its basic idea is the 

computer based evaluation of the performance 

and reliability of field investigation 

methods by application in heterogeneous 

synthetic (i.e. virtual) aquifers. In this it 

resembles the concept of “virtual realities” 

(Schäfer et al., 2002) which are used e.g. in 

car industry (“virtual crash test”), education 

(flight simulators) or medicine (interactive 

operation planning). 

9

conceptional 

model error: 

wrong process 

description 

10 

measurement 

error 

resulting 

investigation error 

conceptual site model 

investigation 

virtual aquifer 

Fig. 5: Virtual aquifer concept and possible sources of investigation error. 

An application of the VA concept requires 

the definition of a synthetic site model and 

its translation into a numerical model for 

the simulation of the identified relevant processes. 

Synthetic site models are generated 

based on statistical properties of natural 

aquifers. A defined contaminant source is 

then introduced and the evolution of aquifer 

contamination is simulated by numerical 

modeling, thus generating a realistic 

contaminant distribution in the synthetic 

aquifer. In comparison to the "real world", 

the unique advantage of the synthetic aquifer 

is that the spatial distribution of all 

physical and geochemical properties and 

parameters as well as contaminant concentrations 

are exactly known. Once the 

synthetic contaminated aquifer is generated, 

it can be studied by standard monitoring 

and investigation techniques, e.g. by emplacement 

of observation wells. Although 

the parameter distribution of the synthetic 

aquifer is known a priori, only the data 

“measured” at wells (i.e. hydraulic heads or 

concentrations) are used and interpreted. 

This is done because in a real site investigation 

also only a limited amount of measured 

data would be available. Finally, the results 

are compared to the “true” parameter distribution 

known from the synthetic aquifer, 

allowing an evaluation of the accuracy of 

the investigation method used. Using the 

VA concept, sources of uncertainty or error 

can be considered individually and the 

heterogeneity: 

incomplete or 

wrong description 

of parameter 

distribution 

sensitivity of investigation results on these 

can be studied. Stochastic approaches like 

the Monte-Carlo method are applied to 

study the propagation of parameter variability 

and uncertainty into the investigation 

results. The VA has been first introduced 

by Schäfer et al. (2002) and was applied by 

Schäfer et al. (2004, 2006b), Bauer and 

Kolditz (2006), Bauer et al. (2005 [EP 1]; 

2006a [EP 2], 2007 [EP 4]) and Beyer et al. 

(2006 [EP 3], 2007a [EP 5]). An overview 

of VA applications is given in Bauer et al. 

(2006b) and Schäfer et al. (2006a). 

Fig. 6: Virtual investigation of a heterogeneous 

contaminant plume by the center 

line method (Bauer et al., 2005 [EP 1]). 

The VA concept is used here to study errors 

and uncertainties in degradation rate constants 

estimated from data typically collected 

by site investigation with the so called

center line method (Fig. 6). This method is 

frequently used in field studies when natural 

attenuation is considered as a remediation 

alternative and is based on observation 

wells that are placed along the presumed 

center line of the contaminant plume. 

Influence of measurement errors 

The first aspect studied here is the influence 

of measurement errors in hydraulic heads 

on degradation rate constant estimates 

(Bauer et al., 2007 [EP 4]). The VA concept 

used in this study is based on a two 

dimensional conceptual model of the 

groundwater body using a homogeneous 

distribution of hydraulic conductivity K. 

The development of the contaminant plume 

originating from a rectangular source zone 

is simulated until steady state conditions are 

established. The numerical simulations are 

performed using the GeoSys / Rockflow 

code, which was introduced in the previous 

chapter. The contaminant is subject to a 

first order kinetics (eq. (23)) degradation 

reaction using a uniform degradation rate 

constant �. Additionally, a conservative 

tracer is emitted from the source zone. The 

contaminant plume thus generated is then 

investigated by the center line method. 

From the hydraulic heads measured at three 

initial observation wells (one being located 

directly in the center of the source zone) 

first the direction of groundwater flow is 

estimated by construction of a hydrogeological 

triangle. Head measurements are 

obtained by reading the model output at the 

respective well locations and adding a 

random measurement error by 

h � � h � �� 

(25) 

h 

where h´ and h are the erroneous and exact 

(i.e. simulated) heads, respectively, � is an 

evenly distributed random number from the 

interval [-1, 1] and ��h is the maximum 

measurement error. Along the estimated 

(and potentially erroneous) flow direction 

three new observation wells are installed, 

one at every 10 m. These wells were then 

used to measure local (erroneous) heads, 

contaminant concentrations and hydraulic 

conductivities along the presumed plume 

center line. From the hydraulic head difference, 

true porosity and well positions 

groundwater flow velocities are calculated. 

Together with the concentration data this 

allows the determination of � using any of 

the analytical models presented in Tab. 1. 

As hydraulic conductivity K is distributed 

homogeneously and concentrations are 

assumed to be measured precisely, the only 

source of error here is the measured head. 

Tab. 1: Analytical models for the estimation of the first order degradation rate constant �. 

method formula description reference 

1 

2 

3 

4 

v � � 

a C ( x) 

� � � � 

� 

� 

� 

1 ln 

�x 

� C0 

� 

v � � � 

a � C ( x) 

C0 

� � � 

� 

2 ln 

�x 

� C � � 

� 0 C ( x) 

� 

with 

�C( x) 

C � 

2 

v � 

� 

a �� 

ln 

0 � 

� 

� � 

3 � 

� 

�1 

� 2� 

L 

� 1 

4� 

� 

L �� 

�x 

� � 

�C( x) 

( C � ) � 

2 

v � 

� 

a �� 

ln 

0 � 

� 

� � 

4 � 

� 

�1 

� 2� 

L 

� 1 

4� 

� 

L �� 

�x 

� � 

� � 

� 

W S 

� � erf 

� 

� � 

� 4 � T � x � 

analyt. solution of 1D advection 

equation with first order 

degradation 

same as method 1; concentrations 

normalized by conservative tracer 

analyt. solution of 1D advectiondispersion 

equation with first 

order degradation 

analyt. solution of 2D advectiondispersion 

equation with first 

order degradation and accounting 

for the source area width . 

Newell et al. 

(2002) 

Wiedemeyer 

et al. (1996) 

Buscheck 

and Alcantar 

(1995) 

Zhang and 

Heathcote 

(2003) 

11

The conceptual model used is a rigorous 

simplification of the processes observed in 

natural aquifer systems, as K varies in space 

and contaminant degradation usually follows 

more complicated laws, depends on 

microorganism growth and may be limited 

or inhibited by other substances. The 

simplifications assumed here are however 

necessary to study the sole effects of measurement 

error on rate constant estimation 

under otherwise ideal conditions for the 

application of the center line method and 

the analytical models of Tab. 1. 

In this synthesis only results for method 1 

of Tab. 1 are presented. The degradation 

rate estimated from evaluation of measured 

heads and concentrations is divided by the 

true rate constant used in the numerical 

model yielding normalized overestimation 

factors. To assess the range of uncertainty 

resulting from the random measurement 

error a Monte Carlo analysis is conducted 

for five increasing values of ��h, each with 

a sample size of 100. Fig. 7 presents the 

rate constants thus estimated versus ��h. 

Without measurement error the correct rate 

constant is obtained. However, already for 

very small ��h < 1 cm, a significant 

overestimation can be observed. This has 

two main reasons: Firstly, an erroneous 

head yields an incorrect transport velocity 

and thus results in rate constant 

overestimation for too high velocities and in 

underestimation for too low velocities. 

Secondly, erroneous heads result in an incorrect 

derivation of the flow direction 

using the hydrogeologic triangle. Thus the 

observation wells installed downgradient of 

the source may be placed off the true center 

line position (compare Fig. 6). Hence, concentrations 

measured in off center line wells 

are too low, indicating an overly high rate 

of degradation. Overestimation of the 

degradation rate increases with the maximum 

head error and reaches factors of more 

than 20 in the worst cases. 

In Bauer et al. (2007 [EP 4]) also the 

influence of concentration measurement 

errors was studied. It was found that also 

12 

this type of error results in overestimation 

of the rate constant on average. Erroneous 

heads, however, were found to have a larger 

impact on the rate constant estimates. For 

real field applications therefore much care 

has to be taken when measuring hydraulic 

heads for the derivation of the plume center 

line position. 

Fig. 7: Influence of head measurement 

error on rate constant estimates (Bauer et 

al., 2007 [EP 4]). The reference rate 

constant is indicated by the horizontal line, 

small symbols represent single estimates, 

big symbols are ensemble averages with 

standard deviation as error bars. 

Influence of aquifer heterogeneity 

The second aspect studied using the VA 

concept is the influence of spatially heterogeneous 

hydraulic conductivity distributions 

on the accuracy of rate constant 

estimation methods 1 – 4 of Tab. 1. For this 

end the conceptual model used so far is 

modified by assuming all head, concentration 

and K measurements to be free of 

measurement error and regarding K as a 

spatial random variable. This is done to 

study the sole influence of heterogeneous 

conductivity. Multiple realizations of heterogeneous 

K fields for four degrees of 

aquifer heterogeneity characterized by the 

ln-conductivity variance 2 

� Y were generated, 

using a Monte Carlo approach to study 

the range of investigation result uncertainty. 

� at least 100 different reali- 

For each 2 

Y

zations were generated. For all realizations 

the spreading of a conservative and a 

reactive contaminant plume subject to first 

order degradation was simulated. The resulting 

heterogeneous plumes were investigated 

as explained above. Fig. 8 presents 

normalized estimated rate constants for 

2 

methods 1 – 4 (Tab. 1) versus � Y (Bauer et 

al., 2005 [EP 1]). Clearly, most rate 

constants are larger than 1, i.e. the rate 

constant is generally overestimated. Single 

realizations show overestimation by several 

orders of magnitude. It is obvious that an 

increase in K heterogeneity causes higher 

overestimation. Also the spread of the 100 

realizations and the resulting ensemble 

standard deviations increase significantly, 

causing higher uncertainty in the rate constant 

estimate. The main reasons for the 

observed overestimation are identified as 

deviation of observation wells from the true 

plume center line position, an incorrect 

approximation of the transport velocity and 

no or inadequate consideration of concen- 

normalized deg. rate constant [-] 

normalized deg. rate constant [-] 

1000 

100 

10 

1 

0.1 

0.01 

1000 

100 

10 

1 

0.1 

0.01 

(a) method 1 (b) method 2 

(c) method 3 (d) method 4 

0 1 2 3 4 5 

tration reduction by longitudinal and 

transverse dispersion. Comparing the performance 

of methods 1 – 4 yields that 

method 2 is the most accurate and reliable 

among the four approaches. The superiority 

of method 2 follows from the correction of 

contaminant concentrations by normalization 

to the concentrations of a conservative 

tracer, which is spread from the same 

source. The tracer correction successfully 

accounts for the effects of dispersion and 

measuring off the center line. Method 3 explicitly 

accounts for longitudinal, method 4 

for both, longitudinal and transverse dispersion. 

However, for each realization investigated, 

method 3 yields a higher rate constant 

estimate than methods 1 or 2. Longitudinal 

dispersion of a degrading contaminant 

results in a stronger spreading of the solute 

downstream and thus in higher concentrations 

along the center line of a steady state 

plume. To model an observed concentration 

reduction with a one-dimensional model 

like method 3 which accounts for �L only 

0 1 2 3 4 5 

�2 �2 Y Y 

Fig. 8: Estimated degradation rate constants versus aquifer heterogeneity for methods 1 (a), 

2 (b), 3 (c) and 4 (d) of Tab. 1 (Bauer et al., 2005 [EP 1]). Small symbols represent single 

realization results, large symbols ensemble averages of 100 realizations. 

13

equires therefore a higher degradation rate 

constant. Method 4 yields significantly 

lower rate constant estimates than 

method 1, when aquifer heterogeneity is 

low, but almost the same results as method 

3 for high heterogeneity. The rate constant 

underestimation for low heterogeneity is 

due to an “over correction” for transverse 

dispersion by the error function term � in 

the rate equation. Hence, both methods fail 

to yield closer rate constant estimates than 

the simpler approach of method 1, which 

completely neglects the dispersion process. 

Influence of dispersivity parameterization 

Since the results of methods 3 and 4 depend 

on longitudinal and transverse dispersivities, 

an adequate parameterization is crucial 

for their success. In this study, parameterization 

of �L as well as �T is based on the 

scale of the contamination problem, as 

common with many field applications. 

Clearly from the results in Fig. 8 derivation 

of �L and �T from the assumed length of the 

contaminant plume is not appropriate. As 

known from stochastic hydrogeology (e.g. 

Dagan, 1989) �L as well as �T strongly 

depend on travel time and distance as well 

as on the correlation structure of hydraulic 

conductivity and flow velocity. Therefore 

the influence of dispersivity parameterization 

on the performance of methods 3 and 4 

was analyzed (Bauer et al., 2006a [EP 2]). 

From the often very limited amount of data 

on the degree of aquifer heterogeneity and 

the spatial correlation structure of hydraulic 

conductivity available for real field sites, 

the inference of dispersivities by methods 

of stochastic theory is rarely possible or at 

least afflicted by high uncertainty. 

Therefore a sensitivity analysis is performed 

to cover a wide range of values for 

�L and �T. Varying both parameters the 

heterogeneous plume realizations were reevaluated 

using the different parameterizations 

of method 4. Fig. 9 shows results of 

method 4 in terms of ensemble medians of 

the 100 estimated rate constants for each of 

the four different degrees of heterogeneity 

14 

and all combinations of �L and �T considered. 

As method 4 converges with 

method 3 for �T = 0, results for �T = 0 are 

also representative for method 3. 

For all degrees of heterogeneity, decreasing 

median degradation rates are found with 

increasing �T. For low and medium heterogeneity 

(Fig. 9 (a) and (b)) combinations of 

�L and �T can be found, which allow for an 

optimal estimation of the degradation rate 

constant, i.e. the results are of comparable 

accuracy as those of methods 1 or 2. For the 

highest degree of heterogeneity (Fig. 9 (d)), 

no such parameter combination is found 

within the range of dispersivities considered 

here. However, with an inadequate parameterization 

also for low heterogeneities 

severe over- as well as underestimation is 

possible. Moreover, with the exception of 

the low heterogeneity case, only unphysical 

combinations of low �L and high �T yield 

close estimates of the rate constant. These 

parameters do not represent actual dispersivities, 

but must be considered as mere 

lumped fitting parameters, as they are used 

to correct for off center line measurements 

and not the dispersion process only. As the 

magnitude of bias caused by off center line 

measurements is not known and the K 

correlation structure and degree of heterogeneity 

are usually not properly characterized 

at many field sites, choosing 

dispersivities for an application of method 4 

would be highly uncertain and arbitrary. 

From the results presented here, it can be 

concluded that for contaminated sites, 

where the assumption of first order degradation 

kinetics can be justified, method 2 

should be preferably applied, if the degradation 

rate constant is to be derived with 

the center line approach. In situations, 

where a correction of concentrations by a 

conservative tracer is not possible, 

alternatively method 1 should be used, as 

its application implies the least amount of 

parameterization uncertainty.

Fig. 9: Influence of dispersivity parameterization on estimated degradation rate constants for 

2 

method 4 (and method 3 with �T = 0) of Tab. 1 and four degrees of aquifer heterogeneity � Y 

(Bauer et al., 2006a [EP 2]). Symbols represent ensemble medians of 100 realizations. 

Extensive monitoring networks 

For the previous studies, linearly arranged 

center line observation wells have been 

used for the VA investigation (compare Fig. 

6). Contaminant plumes in natural heterogeneous 

aquifers may however show a 

substantial amount of meandering (Wilson 

et al., 2004) resulting in non-linear center 

line orientations. As documented, an 

inappropriate assumption of a linear center 

line may result in misplacement of the 

observation wells and in significant overestimation, 

if degradation rate constants are to 

be derived. Moreover, only measurements 

in observation wells located on the plume 

axis are evaluated and additional information 

possibly at hand (i.e. well data downgradient 

from the source but not on the 

center line) is not explicitly accounted for 

in the rate estimation. Therefore the performance 

of methods 1, 3 and 4 of Tab. 1 was 

Fig. 10: Site investigation for degradation 

rate constant evaluation by (A) non-linearly 

positioned center line wells and (B) using 

all observation wells of the monitoring 

network (Beyer et al., 2007a [EP 5]). Small 

dark squares show observation well 

locations, larger squares show presumed 

center line well locations, contour lines are 

concentrations calculated analytically with 

the approach of Stenback et al. (2004). 

15

studied for heterogeneous sites with extensive 

monitoring networks allowing the estimation 

of the degradation rate constant 

based on freely positioned observation 

wells from which a center line is constructed 

(strategy A, Fig. 10 (a)). Results are 

compared to a two-dimensional inverse 

modeling approach of Stenback et al. 

(2004), which accounts for information 

from all observation wells of a monitoring 

network (strategy B, Fig. 10 (b)). Both rate 

constant estimation strategies are applied to 

a set of synthetic contaminated sites with 

independently designed extensive monitoring 

networks (Beyer et al., 2007a [EP 5]). 

a) 

b) 

16 

norm. degradation rate constant [-] 


100 

10 

1 

0.1 

100 

10 

1 

0.1 

SW = 4 m 

SW = 16 m 

meth. 1 

(A) 

meth. 3 

(A) 

meth. 4 

(A) 

investigation strategy 

Stenback 

(B) 

Fig. 11: Rate constant estimates obtained 

for investigation strategy A and methods 1, 

3 and 4 as well as strategy (B) for a source 

width WS of 4 (a) and 16 m (b) (Beyer et al., 

2007a [EP 5]). Small symbols represent 

single realization results, large symbols 

ensemble averages. 

Two different types of plumes are considered 

in this comparison: narrow contaminant 

plumes with a small source area width 

WS = 4 m (Fig. 11 (a)) and wider plumes 

with WS = 16 m (Fig. 11 (b)). For small 

source widths overestimation by strategy B 

is comparable to or at best slightly less than 

for approach A. For large source widths and 

wider plumes, however, strategy B yields 

closer estimates of the degradation rate 

constant than strategy A on average. The 

results of this study suggest that incorporation 

of off center line information for the 

estimation of the degradation rate constant 

can improve results of the plume investigation 

significantly. 

3.2. Development and testing of a 

new approach to estimating 

biodegradation parameters 

from field data 

In the previous section, the VA method was 

applied to evaluate the performance of 

different analytical models for the derivation 

of first order degradation rate constants 

from center line investigation data in heterogeneous 

aquifers. From the literature, 

however, it is well known that the use of 

first order kinetics may be problematic in 

some situations, as it is an inaccurate representation 

of the processes occurring in 

contaminated aquifers. Usage of a first 

order model outside its range of validity 

may result either in significant under- or 

overestimation of the attenuation potential 

at a site (Bekins et al., 1998). In an 

extension to this study, therefore a new 

approach for the estimation of degradation 

parameters kmax and MC for Michaelis- 

Menten (MM) kinetics (eq. 22) from the 

same plume investigation strategy was 

developed and tested in Beyer et al. (2006 

[EP 3]) using the VA method. 

In its integral form eq. (22) can be 

rearranged to

v 

� 

�C�Cx) � 

� C0 

� 

ln�� 

C�x�� 

M � � 1 

� 

� 

( k C � C( 

x) 

k 

x C 

(25). 

a 0 

max 0 

max 

With the same type of center line 

investigation data used for the estimation of 

the first order degradation rate constant (i.e. 

local concentrations, heads, hydraulic conductivities), 

eq. (25) can be utilized to 

estimate the MM parameters kmax and MC 

by linear regression. 

The Monte Carlo scenario definition and 

site investigation procedure for this study 

are similar to those explained in section 3.1. 

Numerical simulations of plume development 

in homogeneous and heterogeneous 

aquifers were performed with the GeoSys / 

Rockflow code. Here, however, the contaminant 

plumes investigated were generated 

using MM instead of first order degradation 

kinetics. The parameters k max and M C 

estimated with eq. (25) for the different 

plume realizations were normalized to the 

true values used in the numerical 

simulations and are shown in a 3D-scatterplot 

versus aquifer heterogeneity (Fig. 12). 

Fig. 12: Normalized Michaelis-Menten parameters 

(given as overestimation factors) 

versus aquifer heterogeneity. 

In general an overestimation of both k max 

and M C is observed, which increases with 

heterogeneity. An overestimation of k max increases 

the velocity of contaminant degradation 

as long as concentrations are much 

higher than MC. The simultaneous overestimation 

of MC counterbalances this effect 

because the concentration threshold is 

raised at which the kinetic begins to show a 

dependence on concentration and transits 

from zero to first order and hence decreases 

the rate of degradation. 

To obtain an indicator for the significance 

of the estimated degradation potential, the 

MM parameters determined were used in an 

analytical transport model to estimate the 

contaminant plume lengths. These then 

were compared to the respective true plume 

lengths from the numerical simulations 

(Fig. 13 (a)). As a consequence of overestimating 

the degradation parameters, calculated 

plume lengths for high heterogeneities 

are estimated to about 75 % of the true 

length on average and thus are not conservative. 

For low heterogeneities, however, 

the suggested regression approach on average 

yields good estimates of the plume 

length and the degradation potential. 

In addition to the effect of aquifer heterogeneity 

on estimated MM parameters and the 

resultant plume length estimates, also the 

effect of a wrong process identification 

(compare Fig. 5) is studied in Beyer et al. 

(2006 [EP 3]). Although it is well known 

that contaminant degradation in natural 

aquifers is governed by complex processes 

and kinetic laws, simple first order models 

are routinely used at many field sites. This 

study therefore highlights some of the 

problems that result from an insufficient 

wrong process identification. For this end 

investigation of the plumes following MM 

degradation kinetics is repeated, assuming 

the appropriateness of a first order rate law 

to approximate the contaminant degradation 

behaviour. Hence the methods of Tab. 1 

were used to derive first order rate constants 

for the multiple plume realizations. 

As for the estimated MM parameters the 

estimated first order rate constants were 

evaluated by analytical transport models to 

yield estimates of the contaminant plume 

length. Results for method 1 (Tab. 1) are 

presented in Fig. 13 (b). 

17

Fig. 13: Plume length overestimation factors 

versus aquifer heterogeneity (Beyer et 

al., 2006 [EP 3]). Plume lengths were estimated 

for plumes following Michaelis-Menten 

degradation kinetics estimated using the 

Michaelis-Menten model (a) and assuming 

validity of a first order rate law (b). 

In comparison with Fig. 13 (a) an additional 

error is introduced which stems from the 

first order approximation. Uncertainty as 

well as bias increase significantly, as can be 

seen by the wider spread of single realization 

results around the ensemble medians. 

Estimated plume lengths here are found to 

be less than 40 % of the true length on 

average even for mildly heterogeneous 

aquifers. Plume lengths calculated using the 

MM parameters in general are significantly 

closer to the correct length compared to 

those obtained by a first order approximation. 

This approach is therefore recom- 

18 

mended, if field data collected along the 

center line of a plume give evidence of MM 

type degradation kinetics. 

3.3. Prognosis of long term contaminant 

leaching from recycling 

materials in road 

constructions 

The third application of the numerical 

models and methods presented in section 2 

is focussed on the prognosis of contaminant 

leaching and transport by seepage water 

from pollutant loaded recycling materials, 

which are used in earthworks or road 

constructions. According to the German 

federal soil protection decree (BBodSchV, 

1999) such a prognosis is required for 

contaminated sites as well as for constructions 

or depositions of contaminated materials 

in order to assess the extent and environmental 

impact of potential contaminant 

leaching through the vadose zone to the 

groundwater. In such a prognosis, the 

relevant attenuation processes need to be 

considered and quantified, as significant 

contaminant attenuation could result in less 

restrictive utilization criteria without 

compromising the protection of groundwater 

resources. For this end, the application 

of process based numerical transport 

models is favorable, as complex geometries 

of the model scenarios and possible process 

interactions limit the applicability of analytical 

models or expertise founded “verbalargumentative” 

assessments. 

In this study, process based type scenario 

modeling is used as a tool to assess 

contaminant leaching from recycled demolition 

waste (DW) material. The type 

scenarios are based on three different case 

studies for the utilization of the DW, i.e. 

recycling as base and subbase layers of a 

parking lot, a noise protection dam and a 

road dam (Fig. 14) (Beyer et al., 2007b 

[EP 6]). Instead of regarding the full spectrum 

of contaminants typically embodied in 

DW three model substances are considered 

in the type scenarios: a conservative tracer

1.3m 

0.3m 

as a representative for highly soluble salts, 

naphthalene for moderately sorbing and 

phenanthrene for strongly sorbing organic 

compounds. Contaminant leaching from the 

DW to the groundwater surface is studied 

with six different characteristic subsoil 

units of Germany (BGR, 2006) to be able to 

compare the influence of hydraulic and 

basic physico-chemical soil properties on 

contaminant attenuation. Fig. 14 presents 

the conceptual model for the road dam. 

Here coarse grained DW is used as unbound 

base / subbase layers below the 

asphalt surface of the road. According to 

German road construction regulations, the 

base / subbase layers are covered by low 

and high permeable soil layers along the 

embankment (Fig. 14). 

RCB 

Körnung 0/32 

b horizon 

c horizon 

symmetry axis 

10m 1.5m 

asphalt layer 

impermeable low permeable soil 

3% 

4% 

12% 

1:1. 5 

2.3m 

10 cm high 

permeable soil 

2m 

groundwater surface 

(point of compliance) 

Fig. 14: Road construction with demolition 

waste recycled in base / subbase layers 

(Beyer et al., 2007b [EP 6]). 

The simulation strategy for this study 

combines the Eulerian and Lagrangian 

frameworks of transport modeling. Unsaturated 

flow, i.e. the hydraulics of the constructions, 

is modeled with GeoSys / 

Rockflow using standard FEM. Fig. 15 

shows that the two-dimensional scenarios 

exhibit complex flow patterns under unsaturated 

conditions. The velocity vectors at 

the element nodes of the FEM mesh for the 

road dam presented in Fig. 15 indicate 

unhindered infiltration of water from the 

low permeable soil on top of the embankment 

into the DW material. Along the 

sloped material boundary with high permeable 

soil on top of the coarse DW, however, 

strong capillary barrier effects are observed. 

These cause a concentration of the water 

flux on top of the DW and generation of 

lateral runoff, almost completely bypassing 

the DW. 

Fig. 15: Element nodes and velocity vectors 

of unsaturated water flow in a section of the 

model domain (Beyer et al., 2007b [EP 6]). 

Reactive transport of the model compounds 

is simulated using the stream tube concept 

of the SMART code, to take full advantage 

of the reactive process models implemented 

in SMART (degradation, intraparticle 

diffusion kinetics, sorption, etc.). Coupling 

between GeoSys / Rockflow and SMART is 

achieved through the travel time pdf. These 

are generated by simulation of conservative 

tracer breakthrough curves by GeoSys / 

Rockflow using the FEM. The pdf are used 

as input for the reactive transport simulations 

with SMART. The model output of 

SMART for the three model substances at 

the groundwater surface represents concentrations 

integrated along the cross-section 

of the contaminant transport path only. 

These breakthrough curves for the tracer in 

the road dam are displayed in Fig. 16 as 

grey curves. The three black curves 

represent the same tracer breakthrough 

concentrations but are integrated along the 

groundwater surface of the overall model 

domain. Hence, they account for dilution by 

uncontaminated seepage water which 

bypasses the actual transport path due to the 

capillary barrier. 

Breakthrough curves in relative concentrations 

C/C0 [-] are given here for three of 

the six soils regarded in this study, i.e. for a 

cambisol, a podzol and a chernozem. Comparing 

the concentration breakthrough for 

the transport path (grey curves), it is found 

that the earliest breakthrough time for the 

19

concentration maximum is for the cambisol, 

followed by the podzol and the chernozem. 

In general, however, breakthrough times are 

very similar for the three soils. Also the 

maximum concentrations observed are of 

comparable magnitude. As the tracer is conservative, 

concentration reductions of about 

70 % can be attributed to the dispersion 

process. Integration of breakthrough concentrations 

along the overall lower model 

boundary (black curves) yields a further 

reduction of concentrations, as about 30 % 

of the infiltration bypasses the contaminant 

transport path. 

Fig. 16: Tracer breakthrough curves at the 

groundwater surface below the road dam 

for three different subsoil types (Beyer et 

al., 2007b [EP 6]). 

For naphthalene and phenanthrene, contaminant 

transport is regarded with and 

without degradation. For the cases with 

degradation, a simple first order kinetics is 

used. Sorption of both compounds is 

modeled by a linear isotherm, where the 

equilibrium sorption coefficient is derived 

from the soil organic carbon content and the 

distribution coefficient between organic 

carbon and the aqueous phase (Beyer et al., 

2007b [EP 6]). Sorption kinetics are quantified 

using the intraparticle diffusion model 

(eq. 18). Concentration breakthroughs are 

presented in Fig. 17. For the moderately 

sorbing naphthalene without decay the influence 

of soil organic carbon Corg is clearly 

20 

visible. The cambisol, which is almost free 

of Corg (0.01 %), shows the earliest breakthrough 

of the three soils. For the podzol 

with a little higher Corg (0.21 %), the maximum 

concentration breakthrough is slightly 

retarded. The latest breakthrough time is 

observed for the chernozem, which is the 

soil with the highest Corg (0.88 %) regarded 

here. For the strongly sorbing phenanthrene 

this behaviour is even more characteristic. 

For the chernozem, the maximum concentration 

breakthrough is not yet observed 

within 275 a of simulated contaminant leaching. 

In contrast to the tracer, for which 

source concentrations are depleted within a 

few years, retardation within the DW causes 

naphthalene and phenanthrene leachate 

concentrations to stay on an almost 

unreduced level throughout the simulation 

period of 275 a (Beyer et al., 2007b [EP 6]). 

As high contaminant concentrations are 

constantly delivered from the source 

material, dispersive concentration reductions 

remain ineffective. Hence for the 

transport path maximum concentration 

breakthrough between 70 and 90 % of C0 is 

observed. As for the tracer, these are 

reduced by about 30 % when integrated 

along the overall lower model boundary. 

In comparison to the other type scenarios 

(parking lot, noise protection dam) studied 

in Beyer et al. (2007b [EP 6]), the contaminant 

residence times in the road dam are 

rather short, as high amounts of runoff 

water from the road asphalt which infiltrate 

along the embankment result in comparably 

high flow velocities. The short residence 

times reduce the effectiveness of degradation. 

Hence, naphthalene and phenanthrene 

concentrations are reduced by factors 

between 2.5 and 5, respectively, while for 

the other two type scenarios concentration 

reductions by factors between 10 and 150 

were observed. From this type scenario 

modeling study the relevant transport and 

attenuation processes for contaminant 

leachate from DW used in road constructions 

could be identified and quantified for 

the assumed model structure.

C/C 0 [-] 

C/C 0 [-] 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

1 

0.1 

0.01 

0.001 

cambisol (dil.) 

podzol (dil.) 

chernozem (dil.) 

cambisol 

podzol 

chernozem 

0 50 100 150 200 250 

time [a] 

a) naphthalene b) phenanthrene 

c) naphthalene 

with degradation 

d) phenanthrene 

with degradation 

0 50 100 150 200 250 

time [a] 

Fig. 17: Concentration breakthrough curves at the groundwater surface below the road dam 

for three different subsoil types showing naphthalene without (a) and with degradation (c), 

and phenanthrene without (b) and with degradation (d) (Beyer et al., 2007b [EP 6]). 

The results allow important conclusions on 

mechanisms (e.g. capillary barrier effects) 

and complementary design criteria for road 

constructions, which can be used to reduce 

contaminant leaching to groundwater. 

4. Conclusions and outlook 

In this thesis, the utilization of process 

based numerical models of saturated / unsaturated 

flow and reactive contaminant transport 

is demonstrated for two different fields 

of application. 

The Virtual Aquifer (VA) concept is 

introduced, which uses numerical modeling 

as a tool for the computer based evaluation 

of investigation and remediation strategies 

for contaminated soils and aquifers. In the 

application examples presented, the VA 

concept proves its usefulness for assessing 

the uncertainty involved in the investigation 

of heterogeneous sites and the parameterization 

of degradation process models. 

Moreover, the VA concept is successfully 

applied to test a newly developed approach 

for the inference of biodegradation parameters 

from data typically collected during site 

investigation. Both applications exemplify 

the importance but also the pitfalls of a 

careful and accurate collection of investigation 

data, if these are to be used for 

prognosis of the contaminant behaviour at a 

site. The main advantage of the VA is, that 

individual factors, such as hydraulic heterogeneity 

or conceptual model errors can be 

studied in detail at low costs and without 

high effort, either individually or in combination 

and under otherwise ideal and 

controlled conditions. As the sum of these 

21

possibilities can not be provided neither by 

large scale field experiments nor in the 

laboratory, the VA concept can be 

considered a valuable contribution complementing 

state of the art experimental 

methods. Future applications of the VA 

concept will incorporate more realistic degradation 

kinetics and structures of aquifer 

heterogeneity, in order to study the 

influence of flow and transport channelling 

on the effectiveness of natural contaminant 

attenuation processes. 

In the second field of interest regarded here, 

numerical models are applied for an 

assessment of environmental impact of 

recycling materials used in road constructions. 

In this study, the reactive streamtube 

model SMART is used for the first time in 

combination with the finite element model 

GeoSys / Rockflow to simulate the extent 

of contaminant leaching from contaminated 

demolition waste within road structures to 

the groundwater surface. The coupling 

between GeoSys / Rockflow and SMART 

combines the Eulerian and Langrangian 

frameworks of flow and transport. In so 

doing, the complex hydraulic behaviour of 

the two-dimensional model geometries are 

successfully modeled by travel time 

probability density functions, which allows 

to take full advantage of the process models 

implemented in the SMART code. For the 

assessment of contaminant leaching consequences 

on groundwater recharge quality, 

type scenario modeling is used. The 

relevant transport and attenuation processes 

in road constructions are identified. The 

study allows important conclusions on how 

these mechanisms could be used or 

enhanced to further reduce contaminant 

leaching to groundwater. As an example, 

hydraulic processes like capillary barrier 

formation from layered composition of 

granular base / subbase materials in a road 

dam could be exploited to reduce water 

fluxes through contaminant loaded recycling 

materials. This study demonstrates 

that process based numerical models can 

provide valuable tools for optimizing 

22 

design and construction with regard to both 

functionality and environmental impacts. 

In as much as the application of numerical 

models provides preeminent and cost 

effective means of assessment and 

prognosis, their value and credibility relies 

on a thorough collection and preparation of 

the required parameters and input data. 

Only if utmost care is taken in the stage of 

setting up the model, reliable results can be 

expected. As the data and parameter acquisition 

is of fundamental importance to a 

successful modeling of hydrogeosystems - 

be it by direct measurements in laboratory 

and field experiments, research in literature 

and databases or utilization of expert knowledge 

- both processes are closely related 

and therefore should, if possible, go hand in 

hand.

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26

Enclosed publications 

[EP 1] Bauer, S., Beyer, C., Kolditz, O., (2005): Assessing measurements of first order 

degradation rates by using the Virtual Aquifer approach. In: Thomson, N.R. (Ed.), GQ2004, 

Bringing Groundwater Quality Research to the Watershed Scale. Proceedings of a 

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[EP 4] 

10.1016/j.jconhyd.2006.04.006. 




degradation rates by us1ing the virtual aquifer approach.), Grundwasser, 12, 3–14, 

10.1007/s00767-007-0019-8. 

[EP 5] Beyer, C., Chen, C., Gronewold, J., Kolditz, O., Bauer, S. (2007a): Determination of first 

order degradation rate constants from monitoring networks. (accepted by Ground Water.). 

[EP 6] Beyer, C., Konrad, W., Park, C.H., Bauer, S., Rügner, H., Liedl, R., Grathwohl, P. (2007b): 




via SpringerLink), 10.1007/s00767-007-0025-x.

Enclosed Publication 1 

Bauer, S., Beyer, C., Kolditz, O. (2005): Assessing measurements of first order degradation 

rates by using the Virtual Aquifer approach. In: Thomson, N.R. (Ed.), GQ2004, Bringing 

Groundwater Quality Research to the Watershed Scale. Waterloo, Canada, July 2004. IAHS 

Publication 297, IAHS Press, Wallingford, 274-281. 

The enclosed article was reproduced and is made available with the permission of IAHS 

Press. 

It can be obtained from IAHS Press at http://www.cig.ensmp.fr/~iahs/redbooks/297.htm.

274 

INTRODUCTION 

Bringing Groundwater Quality Research to the Watershed Scale (Proceedings of GQ2004, the 4th International 

Groundwater Quality Conference, held at Waterloo, Canada, July 2004). IAHS Publ. 297, 2005. 

Assessing measurements of first-order degradation 

rates through the virtual aquifer approach 

SEBASTIAN BAUER, CHRISTOF BEYER & OLAF KOLDITZ 

Center for Applied Geoscience, University of Tübingen, Sigwartstrasse 10, D 72076 Tübingen, 

Germany 

sebastian.bauer@uni-tuebingen.de 

Abstract The principal idea behind the “virtual aquifer” is to simulate and 

evaluate investigation strategies for contaminated sites by modelling typical 

contamination scenarios. In this paper, first-order degradation rates using 

various methods were the focus of study. A virtual reality of a contaminated 

aquifer was generated by simulating the spreading of a plume, originating 

from a defined source zone, subject to first-order degradation. This plume was 

investigated through monitoring wells placed along the plume centre-line. 

Using information such as head measurements, concentration and hydraulic 

conductivity, first-order degradation rates were calculated and compared to the 

true predefined value. This comparison was conducted for varying degrees of 

heterogeneity, represented by ln(KF), randomly distributed conductivity fields. 

It was found that when heterogeneity was increased, “measured” degradation 

rates overestimated the true degradation rate by several orders of magnitude. 

The range of degradation rates obtained roughly corresponds to the range 

stated in literature values. 

Key words first-order degradation; modelling; natural attenuation; virtual reality 

At a real contaminated site, the true hydrogeological properties (e.g. conductivity, 

porosity, recharge rates, source position, degradation rates, etc.) are generally unknown 

(spatially). The basic idea behind the “virtual aquifer” is to produce a “virtual” contaminated 

site, where the spatial distribution of parameters is exactly known. By using a 

process-based flow and transport model, the fate of contaminants in the subsurface can 

be simulated, including plume development. The second step for creating a virtually 

contaminated site is to examine the virtual aquifer properties using standard investigative 

procedures (e.g. interpolating hydraulic head and contaminant concentrations 

measured at monitoring wells). The result of this virtual investigation can be compared 

to the true hydrogeological property distribution of the virtual aquifer because, contrary 

to an actual contaminated site, the exact distribution is known. The investigation 

techniques used can thus be tested and evaluated with respect to certain influences, i.e. 

sensitivity to aquifer heterogeneity or variation of other parameters. When the virtual 

plume is examined, only the data obtained by these investigation techniques are used, 

i.e. only the data that would also be measured at a real field site. Data such as hydraulic 

head and contaminant concentrations are “measured” in the virtual aquifer by “reading” 

the model output. The “virtual aquifer” approach offers the ability to test and evaluate 

site investigation techniques, which cannot be performed in the real world. In this 

paper, four methods for determining first-order degradation rate constants, all based on 

the plume centre-line method, are examined by the “Virtual Aquifer” approach.

Assessing measurements of first-order degradation rates through the virtual aquifer approach 275 

METHODS 

Virtual aquifers were produced by generating random fields of hydraulic conductivity 

for the model area (dimensions: 184 × 64 m, Fig. 1). Flow direction was from left to 

right, with a mean hydraulic gradient of 0.003. For this application, a mean hydraulic 

conductivity of 7.2 × 10 -5 m s -1 was assumed, with ln(KF) variances of 0.38, 1.71, 2.7 

and 4.5; the variances were chosen to simulate different degrees of heterogeneity. An 

exponential variogram model, with an integral scale of 2.33 m, was used to describe 

spatial correlation. A virtual contaminant source zone of widths 4, 8 and 16 m was 

introduced into the aquifer, emitting a contaminant which was subject to a first-order 

degradation constant, λ, of 1 year -1 . A conservative tracer was also released. By using 

first order degradation kinetics for the reactive contaminant, the plume evolving 

corresponds to the methods used to estimate the first order degradation rate. This is 

certainly not true in reality, where the degradation of a contaminant follows changing 

and more complex kinetics. The plume was simulated using a process-based numerical 

flow and transport model, assuming longitudinal and transverse dispersivities of 0.25 

and 0.05, respectively. Thus, the virtual contaminated aquifer was generated. In the 

second step, the plume’s properties were examined using the centre-line approach. For 

this investigation, not the full data of the model is used but only the values which are 

obtained by the centreline approach. There were three initial observation wells; one 

was directly in the source zone, while the other two were outside of the source (Fig. 1). 

At these three wells, the hydraulic heads were measured by reading the model output. 

A hydrogeological triangle was constructed and the direction of groundwater flow was 

thus determined. Along the estimated direction of groundwater flow, new observation 

wells were installed at every 10 m. These wells were then used to measure (arrows in 

Fig. 1) hydraulic head, contaminant concentrations, tracer concentrations and local 

hydraulic conductivity. From the hydraulic head difference, the true porosity and the 

well positions the respective groundwater flow velocities are calculated. Together with 

the concentration data, this allows the determination of the degradation rate constant 

Fig. 1 Method used to obtain plume centreline concentrations. The upper graph 

depicts the investigation procedure and the lower graph is the virtual site. 

virtual reality investigation

276 

S. Bauer et al. 

Table 1 Methods used for calculating first-order rate constants λ. va is the transport velocity, ∆x is the 

observation well distance, C(x) is the downstream concentration and C0 the source concentration, while 

αL and αT are the longitudinal and transverse dispersivities, WS is the source width and erf is the error 

function. 

Method Formula for degradation rate Description 

1 

va � C( 

x) 

� 

λ1 

= − ln� � 

∆x 

� � 

� C0 

� 

Analytical solution to 0-D transport equation 

(batch reactor) 

2 

v � ∗ � 

a � C( 

x) 

C0 

λ = − 

� 

2 ln 

∆x 

� C ∗ � 

� 0 C( 

x) 

� 

Concentration normalized to a non-degrading cocontaminant, 

thus accounting for dilution and 

dispersion 

3 

� 

2 

v 

( ) � 

a �� 

ln C( 

x) 

C0 

� 

λ = � − α 

� −1� 

3 1 2 L 

4α 

� 

� 

L �� 

∆x 

� � 

Analytical solution to the 1-D transport equation. 

Accounts for longitudinal dispersion. 

4 

� 

2 

v 

( ) � 

a �� 

ln C( 

x) 

( C0β) 

� 

λ = � − α 

� −1� 

3 1 2 L 

4α 

� 

� 

L �� 

∆x 

� � 

Analytical solution to the 2-D transport equation. 

Accounts for longitudinal as well as transverse 

dispersion and a finite source width. 

� � 

with: � WS 

β = erf � 

� � 

� 4 αT 

∆x 

� 

first-order degradation; modelling; natural attenuation; virtual reality. The setup is thus 

designed to resemble ideal conditions for the application of the four methods for 

estimating the degradation rate constant. The only uncertainty and variability is 

introduced by the aquifer heterogeneity. The data used for the centreline method is 

thus only the data which can be measured at the three initial and the three downstream 

wells. Thus neither the mean hydraulic conductivity nor the variance or correlation 

length is known. 

The four centre-line methods used are provided in Table 1. Method 1 (Wiedemeier 

et al., 1996) is the batch solution to a first-order degradation (i.e. no transport is 

included). Method 2 was proposed by Wilson et al. (1994) (see also Wiedemeier et al., 

1996) and is similar to Method 1, except amended concentrations are used. The 

measured concentrations for the reactive contaminant are corrected by using the ratio 

of the conservative co-contaminant at the observation well. This method corrects for 

dispersion and measurements taken outside of the plume centre-line at the three new 

observation wells. Method 3 was proposed by Buscheck & Alcantar (1995) and is 

based on the solution of a one-dimensional (1-D) transport equation with a first-order 

decay constant. This method accounts explicitly for longitudinal dispersion of the 

plume. To account for transverse dispersion (Method 4), Stenback et al. (2004) suggest 

using the analytical solution for a 2-D transport equation with first-order decay. In 

order to carry out Methods 3 and 4, it is required that the longitudinal and the transverse 

dispersivities be known. The following dispersivities were used according to 

Wiedemeier et al. (1999): 0.1 of the plume length for the longitudinal dispersivity, and 

0.33 of longitudinal dispersivity for the transverse dispersivity. For each of the four 

approaches, first-order degradation rates were calculated and compared to the value 

used in generating the plume. For each degree of heterogeneity, 100 realizations were 

evaluated to obtain a statistical measure of the error introduced by the heterogeneity of


the hydraulic conductivity. For each realization, the procedures described above were 

followed and a degradation rate was calculated for each method, at each downstream 

well and for every source width. 

RESULTS AND DISCUSSION 

Figure 2 illustrates the results for the calculated first-order rate constants. Calculated rate 

constants were reported as normalized rate constants (i.e. the calculated rate constant 

was divided by the true rate constant used in the numerical simulation). The normalized 

rate constant can thus be interpreted as an overestimated factor or an underestimated 

factor. Inspection of Fig. 2 yields that most calculated rates are higher than one (i.e. the 

degradation rate is overestimated). This conclusion is quite concerning for single 

realizations, where overestimations can be of several orders of magnitude. On the lefthand 

side of Fig. 2(a), the variation of the calculated normalized rate with the source 

zone width is illustrated. It is clear that for Method 1, the calculated rates improve when 

the source zone width is increased; this is because Method 1 does not account for 

dilution, dispersion or measurements outside of the plume. These factors become less 

relevant with increasing source width since the basic assumptions inherent in Method 1 

are better fulfilled, and the overestimation factor drops accordingly. On the right-hand 

side of Fig. 2(a), the dependence of the calculated rate on the degree of heterogeneity 

(given as variance) is shown. It is obvious that an increase of σ²ln(KF) leads to an 

overestimation of the calculated degradation rate. Furthermore, the standard deviation of 

the mean calculated degradation rate increases, leading to greater uncertainty in the 

calculated rate. For the smallest degree of heterogeneity the mean overestimation is a 

factor a bit smaller than 2, which increases to values between 3 and 5 for medium to 

high heterogeneity, and 10 for very high heterogeneity. 

Figure 2(b) shows the results for Method 2. Degradation rates for this method were 

also overestimated. However, when comparing this method to Method 1, the overestimation 

factor and standard deviation are generally smaller (i.e. both the error and 

the uncertainty are lower compared to Method 1). When examining the left-hand side 

of Fig. 2(b), the calculated rates show no dependence on source width. This effect is 

inherent to the method, since Method 2 accounts for dispersion, dilution and 

measurements taken outside of the plume. Method 3 depicts results similar to Method 

1 regarding the rate dependence on source width and on the degree of heterogeneity 

(Fig. 2(c)). However, the normalized degradation rates for Method 3 are higher than 

for Method 1 (and also higher than Method 2); this is due to the dispersivity term. In 

comparison to Method 1, a portion of the concentration reduction from the source 

observation well to the downstream observation well is attributed to dispersion and 

corrected for, and thus a higher degradation rate is estimated. Method 4 (Fig. 2(d)) 

displays behaviour similar to Method 3, except that the rate values are slightly lower. 

Lower rate values are attributed to the additional term in the rate equation, which 

accounts for transverse dispersion. It should also be noted that at the lowest degree of 

heterogeneity, the normalized degradation rates were actually underestimated; this is 

due to an “over correction” of the effects for transverse dispersion. To effectively 

illustrate the over and underestimation of the degradation rates for all four methods, 

the degradation rates where calculated (for a homogeneous hydraulic conductivity) and

278 

(a) Method 1 

(b) Method 2 

(c) Method 3 

(d) Method 4 


Fig. 2 “Measured” first-order degradation rate constants normalized to the true 

degradation rate constant vs source width (left) and degree of heterogeneity (right) for 

(a) Method 1, (b) Method 2, (c) Method 3 and (d) Method 4. All figures show results 

for all observations (small symbols) as well as their mean value (large symbols) and 

the corresponding standard deviations (error bars).


plotted as small horizontal bars for σ²ln(KF) of 0 on the right-hand side graphs of 

Fig. 2. It is obvious that for Method 1 and Method 2, the normalized degradation rate 

is exactly 1 for the homogeneous case (i.e. these methods yield the correct result). 

Method 3 shows a slight overestimation, while Method 4 yields very small, normalized 

degradation rates. The smallest rate, from Method 4, was obtained for the smallest 

source width, as then the correction is largest (compare Method 4 in Table 1). For large 

source widths, the argument of the error function of method 4 approaches 1. 

As mean and standard deviations are true for the ensemble mean, but not for single 

observations, an alternative method was chosen for comparing the four methods. The 

four methods were contrasted by plotting the probability of success against the error 

factor; the results are illustrated in Fig. 3. An error factor of 10 corresponds to an 

interval of 0.1 to 10 for normalized degradation rates, i.e. the interval that is obtained 

by multiplying 1 with the error factor and dividing 1 by the error factor (“within one 

order of magnitude”). An error factor of 5 thus corresponds to an interval of 0.2 to 5. 

These plots illustrate the probability that the measured degradation rate is within the 

interval of the corresponding error factor. Figure 3(a) represents the lowest degree of 

heterogeneity and shows that the probability of calculating the degradation rate with an 

error factor of less than 2 (i.e. “within a factor of 2”) is about 0.7 for Method 1, 0.9 for 

Method 2, 0.55 for Method 3 and 0.3 for Method 4. When increasing the error factor 

(a) (b) 

(c) (d) 

Fig. 3 Probability plots for all methods for σ²ln(KF) of: (a) 0.38, (b) 1.71, (c) 2.7 and 

(d) 4.5, resembling the four degrees of heterogeneity, shown for a source width of 

4 m. The probability of method success is plotted against the error factor.

280 


to 5, Methods 1, 2 and 3 show a success probability of about 1, while Method 4 yields 

a probability of 0.7. Figure 3 also illustrates that all methods exhibit a decreasing 

probability of success when the degree of heterogeneity is increased. An error factor 

of 5 yields a success probability of about 1 for the lowest degree of heterogeneity 

(0.38) for Method 1; this probability decreases to 0.7, 0.5 and 0.35 for a σ²ln(KF) of 

1.71, 2.7 and 4.5, respectively. For Method 2 the corresponding success probabilities 

are 1.0, 0.9, 0.8 and 0.6; Method 3 yields values of 1.0, 0.55, 0.35 and 0.25; and 

Method 4 0.7, 0.7, 0.6 and 0.4. 

To achieve the degradation rate within a factor of 10 of the correct degradation 

rate (for high hydraulic heterogeneity, Fig. 3(c)), the success probabilities were found 

to be 0.8, 0.95, 0.8 and 0.6 for Methods 1 through 4, respectively. For all degrees of 

heterogeneity, Method 2 yielded the highest probability for achieving the correct 

degradation rate. For medium to very high heterogeneity, Method 4 was second best, 

and was the worst for aquifers that were only slightly heterogeneous (Fig. 3(a)). Method 

1, although the simplest method, yielded similar probabilities as Method 4, except 

Method 1 works well for aquifers of low heterogeneity. Method 3 yielded the lowest 

success probabilities, with the exception of an aquifer of low heterogeneity. 

CONCLUSIONS 

From the study presented, it can be concluded that the four methods for examining 

decay coefficients behave differently when heterogeneity is increased. All methods 

show a decrease in success probability with increasing heterogeneity. Figure 2 

illustrated that the decrease in success probability is attributed to an overestimation of 

the degradation rate constant. The overestimation was largest for Method 3, yielding 

the lowest success probability. Method 2 was the least affected by an increase in 

heterogeneity and was also the method that depicted the lowest overestimation of 

degradation rates. Methods 3 and 4, the most realistic since they are based on the 1-D 

and 2-D transport equations, illustrated low success probabilities and high overestimation 

of degradation rates, Method 3 being the worst. Methods 3 and 4 were 

prone to errors due to the introduction of longitudinal and transverse dispersivities, 

which were required to calculate the degradation rate constant. Method 1, the simplest 

method, yielded results comparable to Method 4. 

It can be concluded from this study, that Method 2 (using a non-reactive cocontaminant) 

is the preferred method for field investigations, where first-order 

degradation rates are to be estimated. If this co-contaminant is not available, then 

Method 1 is preferred as then no uncertainty regarding the dispersivities is introduced. 

Although widely used and published in the literature, the method of Buscheck & 

Alcantar (1995), Method 3, yielded the worst results in this study. 

Acknowledgements This work is funded by the German Ministry of Education and 

Research as part of the KORA priority programme, sub-project 7.2.


REFERENCES 

Buscheck, T. E. & Alcantar, C. M. (1995) Regression techniques and analytical solutions to demonstrate intrinsic 

bioremediation, In: Intrinsic Bioremediation (ed. by R. E. Hinchee, T. J. Wilson, & D. Downey), 109–116. Batelle 

Press, Columbus, Ohio, USA. 

Stenback, G. A., Ong, S. K., Rogers, S. W. & Kjartonson, B. H. (2004) Impact of transverse and longitudinal dispersion on 

first-order degradation rate constant estimation. J. Contam. Hydrol. 73, 3–14. 

Wiedemeier, T. H., Swanson, M. A., Wilson, J. T., Kampbell, D. H., Miller, R. N. & Hansen, J. E. (1996) Approximation 

of biodegradation rate constants for monoaromatic hydrocarbons (BTEX) in ground water. Ground Water 

Monitoring Remed. 16(3), 186–194. 

Wiedemeier, T. H., Rifai, H. S., Wilson, J. T. & Newell, C. (1999) Natural Attenuation of Fuels and Chlorinated Solvents 

in the Subsurface. Wiley, New York, USA. 

Wilson, J. T., Pfeffer, F. M., Weaver, J. W., Kampbell, D. H., Wiedemeier, T. H., Hansen, J. E., & Miller, R. N. (1994) 

Intrinsic bioremediation of JP-4 jet fuel. In: Symposium on Intrinsic Bioremediation of Ground Water (Denver, 

Colorado, USA), 60–72. US-EPA/540 R-94/515, Washington DC, USA.


Bauer, S., Beyer, C., Kolditz, O. (2006a): Assessing measurement uncertainty of first-order 

degradation rates in heterogeneous aquifers, Water Resour. Res., 42, W01420, 

doi:10.1029/2004WR003878. Copyright 2006 American Geophysical Union. 

Reproduced by permission of American Geophysical Union. 

The enclosed article can be obtained online from AGU at 

http://www.agu.org/pubs/crossref/2006.../2004WR003878.shtml.

WATER RESOURCES RESEARCH, VOL. 42, W01420, doi:10.1029/2004WR003878, 2006 

Assessing measurement uncertainty of first-order 

degradation rates in heterogeneous aquifers 

Sebastian Bauer, Christof Beyer, and Olaf Kolditz 

Center for Applied Geoscience, University of Tübingen, Tübingen, Germany 

Received 7 December 2004; revised 7 October 2005; accepted 18 October 2005; published 31 January 2006. 

[1] The principal idea of this paper is to simulate and evaluate the determination of 

first-order degradation rate constants at heterogeneous contaminated sites under realistic 

conditions. First, a set of heterogeneous and contaminated synthetic aquifers is generated; 

second, the spreading of a solute plume subject to first-order degradation is simulated. 

Third, this plume is investigated using ‘‘monitoring wells’’ placed along the presumed 

plume center line. Using only piezometric heads, concentrations and hydraulic 

conductivities obtained at these monitoring wells, first-order degradation rate constants are 

calculated by methods typically used in field applications. The estimated rate constants 

are compared to the ‘‘real’’ value known from the simulations. This comparison is 

conducted for different degrees of heterogeneity, represented by lognormally distributed 

random conductivity fields. The results indicate that, with increasing degree of 

heterogeneity, ‘‘measured’’ degradation rate constants become uncertain with a high 

variability around the true constant. Measured rate constants tend to overestimate the true 

constant by up to one order of magnitude. A sensitivity analysis of the influences of source 

width, transport velocity, and dispersivity shows that (1) with increasing source width, 

measured rate constants decrease their relative error and increase their accuracy; (2) the 

choice of dispersivity can produce both over- and under-estimation of the true rate 

constant; and (3) that large-scale measurements of hydraulic conductivity yield better 

estimates of flow velocities as compared to local scale measurements. These results 

explain in part the high variability of field measured degradation rate constants reported in 

the literature. 

Citation: Bauer, S., C. Beyer, and O. Kolditz (2006), Assessing measurement uncertainty of first-order degradation rates in 

heterogeneous aquifers, Water Resour. Res., 42, W01420, doi:10.1029/2004WR003878. 


[2] This work studies the uncertainty involved in estimating 

first order degradation rate constants by the plume 

center line method for the assessment of natural attenuation 

at contaminated groundwater sites. Natural attenuation, also 

known as intrinsic bioremediation, refers to the observed 

reduction in contaminant concentration via natural processes 

as contaminants migrate from the source into environmental 

media [U.S. Environmental Protection Agency (EPA), 1999; 

Wiedemeier et al., 1999]. The processes contributing to 

natural attenuation include dilution, dispersion, sorption, 

volatilization and biodegradation, where biodegradation is 

the only process that decreases the total contaminant mass. 

The relative efficiencies of the attenuation processes active 

at a contaminated site must be carefully assessed before 

natural attenuation can be adopted as a cleanup remedy or 

risk reduction strategy. Thus degradation rates of the contaminants 

under consideration may play an important role in 

decision making and site management, when natural attenuation 

is considered as a remedial alternative or a remedial 

step in contaminated site management. Degradation rate 

constants can be used to estimate (1) the total overall natural 

Copyright 2006 by the American Geophysical Union. 

0043-1397/06/2004WR003878$09.00 

W01420 

attenuation potential of an aquifer, (2) contaminant plume 

lengths and (3) downstream concentrations. They can also be 

used for identifying potential receptors and exposure levels 

in case of a risk analysis. 

[3] Several approaches for estimating biodegradation 

rates in ground water in the field are commonly used, 

including mass balances, in situ microcosm studies and 

the use of concentration-distance relations obtained along 

the plume center line [Chapelle et al., 1996; Wiedemeier et 

al., 1999]. The latter include a batch-reaction solution 

[Wiedemeier et al., 1996], normalization to a recalcitrant 

co-contaminant [Wiedemeier et al., 1996, 1999] and the 

method of Buscheck and Alcantar [1995]. The method of 

Buscheck and Alcantar [1995] utilizes contaminant concentrations 

measured along the plume center line, which are 

evaluated by an analytical solution to the one-dimensional 

transport equation with first-order degradation. The firstorder 

degradation rate is calculated from the concentrations 

and an assumed longitudinal dispersivity. An 

additional requirement is, that the plume has reached 

steady state. This approach has been used by a number 

of authors, e.g., Chapelle et al. [1996], Wiedemeier et al. 

[1996], Zamfirescu and Grathwohl [2001], Suarez and Rifai 

[2002] or Bockelmann et al. [2003]. Recently, two- and 

three-dimensional approaches were suggested [Zhang and 

Heathcote, 2003; Stenback et al., 2004] as extensions to the 

1of14

W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES W01420 

method by Buscheck and Alcantar [1995], which are based 

on analytical solutions for transport in two and three dimensions 

and account for finite source widths as well as 

transverse dispersion. By a method comparison with the 

original data Zhang and Heathcote [2003] showed that the 

method of Buscheck and Alcantar [1995] overestimates 

the degradation rate by 21% and 65% in case of a twoand 

three-dimensional plume, respectively. McNab and 

Dooher [1998] reported that the method by Buscheck and 

Alcantar [1995] is easily subject to misinterpretation, as 

transverse dispersivities and temporal effects can produce 

center line concentration profiles which resemble a degrading 

contaminant, even in the absence of degradation. 

[4] The spatial variability of aquifer properties has a 

significant influence on the distribution of contaminants 

and plume development. As a consequence, the methods for 

the estimation of degradation rates presented above are 

prone to effects of hydraulic heterogeneity, as they rely on 

concentration samples along the (presumed) plume center 

line as well as on estimations of site specific dispersivity. As 

Wilson et al. [2004] point out, the center line of a plume can 

easily be missed by monitoring wells installed based on 

assumed, but incorrect, groundwater flow directions. Moreover, 

contaminant plumes may wander in all three dimensions 

due to macroscale heterogeneities [Wilson et al., 

2004]. However, so far no study has been reported in 

literature which investigates these effects. Aim of this work 

is therefore to assess the influence of spatially heterogeneous 

hydraulic conductivities on the determination of firstorder 

degradation rates using sets of synthetic aquifer 

models. 

[5] Owing to the limited accessibility of the subsurface, 

measurements of piezometric heads and contaminant concentrations 

at contaminated sites are sparse and may not be 

representative of the heterogeneous hydrogeologic conditions. 

Therefore site investigation is subject to uncertainty, 

reflecting the limited knowledge on the aquifer properties 

and the extent of the contamination. Owing to this uncertainty, 

field investigation methods for plume screening or 

measuring hydraulic conductivity or degradation rates can 

neither be tested nor verified in the field. The only way of 

assessing the performance and reliability of field investigation 

methods is by studying them in synthetic aquifers 

within a Monte Carlo framework. By applying the investigation 

method under consideration in the synthetic contaminated 

and heterogeneous aquifer, the method results can be 

compared to the true values. These are known from the 

synthetic aquifer, unlike in reality, where the true values are 

unknown. 

[6] This approach uses synthetic aquifer models, which 

are generated as the first step based on statistical properties 

of real aquifers and have a defined source of contamination. 

A reactive transport model is then used to simulate the 

spreading of the plume, resulting in realistic concentration 

distributions in the synthetic aquifer. In comparison to the 

‘‘real world,’’ the unique advantage of the synthetic aquifer 

is that the spatial distribution of all physical and geochemical 

properties and parameters as well as the contaminant 

concentrations are exactly known. In the second step, the 

synthetic aquifer is investigated by standard monitoring and 

investigation techniques. In this step, only the data obtained 

by the investigation methods, i.e., heads and concentrations 

2of14 

at the observation wells, is used, because in case of a real 

site investigation the true parameter distribution is unknown. 

In the third step, the results from the investigation 

are compared to the true values, which allows to test and 

evaluate the investigation method used. Using synthetic 

aquifers offers furthermore the possibility to single out the 

influence of different parameters, such that sources of 

uncertainty and error for the investigation method can be 

studied individually. Owing to this possibility of extensive 

and detailed scenario analysis and visualization, this approach 

is well suited to explore the uncertainty involved in 

hydrogeologic investigation and management. It has been 

applied under the term ‘‘virtual aquifer’’ by Schäfer et al. 

[2002, 2004], Bauer et al. [2005] and Bauer and Kolditz 

[2006]. 

[7] This paper uses synthetic heterogeneous and contaminated 

aquifers in a Monte Carlo approach to assess for the 

first time the influence of spatially heterogeneous hydraulic 

conductivities on the determination of first-order degradation 

rates. To this end, plumes formed by contaminants 

degrading according to a first-order degradation rate in 

aquifers of different degrees of heterogeneity are investigated 

by the center line approach. By comparison of the 

estimated degradation rate constant with the true degradation 

rate constant the methods are tested and evaluated. This 

is performed by individually studying the influence of 

aquifer heterogeneity, source width, flow velocity and 

dispersivity on the estimated rate constant. 

2. Methods 

2.1. Model Domain 

[8] The model domain used for the numerical investigation 

is a two-dimensional aquifer with 184 m length and 

64 m width (Figure 1). Flow is from left to right, with a 

mean hydraulic gradient of 0.003, which is induced by 

fixed head boundary conditions on the left and the right 

hand side of the model domain. No flow boundary 

conditions are assigned to all other sides of the model 

domain. The model domain is discretized with a grid 

density of 0.5 m in both directions. A contaminant source 

is emplaced 11.5 m downstream of the inflow boundary in 

the center of the aquifer, emitting a contaminant subject to 

first-order degradation with a degradation rate constant l 

of 1 a 1 (one per year). The contaminant source is 

represented by a fixed concentration boundary condition 

at the source position. Neither sorption, i.e., retardation, 

nor volatilization or dilution by recharge are accounted for. 

Additionally, a conservative compound is emitted from the 

source. The model setup is thus designed to provide ideal 

conditions for the application of the four center line 

methods to be studied. This is certainly not the case in 

nature, where the reaction kinetics will follow more 

complicated laws and may be spatially dependent, or 

influences from sorption and dilution have to be accounted 

for. However, these assumptions are used here to be able 

to study the standard methods closely and evaluate individually 

the influence of heterogeneity of the hydraulic 

conductivity. Further studies will use model setups which 

incorporate, e.g., different degradation kinetics. 

[9] A plume is generated using a process based numerical 

flow and reactive transport model. The simulation code

W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES 

Figure 1. Model area of the synthetic aquifer and 

boundary conditions applied. 

GeoSys/RockFlow [Kolditz, 2002; Kolditz et al., 2004] is 

used here, which solves the flow and transport equations by 

standard Galerkin finite element methods [e.g., Huyakorn 

and Pinder, 1983] and using implicit Euler time stepping. 

The governing equations are given as [e.g., Bear, 1972]: 

and 

S @h 

@t 

¼rðKrhÞþq ð1Þ 

@C 

@t ¼ varC þrðDrCÞ lC ð2Þ 

where S is the storage coefficient, h is the piezometric head, 

K is the tensor of hydraulic conductivity, q are sources and 

sinks of water, C is concentration, v a is the transport 

velocity, D is the dispersion tensor, l is the first order 

degradation rate constant and t is time. The model 

parameters used in this study are given in Table 1. Details 

on numerical and software issues can be found in the work 

of Kolditz [2002] and Kolditz and Bauer [2004]. The 

simulation code has been used for ground water flow and 

transport simulations by Kolditz et al. [1998], Diersch and 

Kolditz [1998, 2002], Thorenz et al. [2002] and Beinhorn et 

al. [2005]. 

[10] To study the effects of spatially variable hydraulic 

conductivity, K is regarded as a random variable following a 

lognormal distribution with an expected value of E[Y = 

ln(K)] = 9.54. This corresponds to an effective hydraulic 

conductivity K ef of 7.2 10 5 ms 1 using the geometric 

mean [Rubin, 2003]. Using a porosity n of 0.33, the mean 

transport velocity is given by 6.5 10 7 ms 1 . The spatial 

correlation structure is characterized by an isotropic exponential 

covariance function C Y = s Y 2 exp( Dh/lY), with an 

integral scale of l Y = 2.67 m and the variance s Y 2 . Four 

different cases of increasing heterogeneity with ln(K) variances 

s Y 2 of 0.38, 1.71, 2.70 and 4.50 are considered, 

representing mildly to highly heterogeneous conductivity 

fields. The value of s Y 2 = 0.38 as well as the integral scale lY 

is taken from the Borden field site [Sudicky, 1986]. The 

value of 1.71 stems from an alluvial valley aquifer in 

southern Germany [Herfort, 2000]. The values of 2.70 

and 4.50 were reported for the Columbus Air Force Base 

site [Rehfeldt et al., 1992]. The geostatistical software tool 

gstat2.4 [Pebesma and Wesseling, 1998] is used to generate 

100 realizations of the random field for each value of s Y 2 by 

unconditional sequential Gaussian simulation. The random 


K values are generated over a two-dimensional grid of 

density 0.5 m, exactly matching the numerical grid. Thus, 

following a rule of thumb of Ababou et al. [1989], a 

sufficient resolution of 5.33 > 1 + sY 2 grid nodes per integral 

scale is ensured. 

[11] To generate steady state plumes, as required by the 

methods under consideration, a stationary flow field is 

assumed. The time development of the plume is calculated, 

until the plume has reached steady state. A local longitudinal 

dispersivity aL = 0.25 m and a local transversal 

dispersivity of aT = 0.05 m are used for the numerical 

simulations (compare Table 1). 

2.2. Center Line Method 

[12] Four methods for the determination of first-order 

degradation rate constants are investigated here, which are 

all based on the plume center line method. Method 1 is 

based on the one-dimensional transport equation, considering 

advection and first-order degradation only. The steady 

state solution for the concentration profile can be rearranged 

to yield the first-order degradation rate constant for method 

1, i.e., l 1 [T 1 ]as: 

l1 ¼ va 

Dx 

Cx 

ln ðÞ 

C0 

where va [L T 1 ] is the transport velocity, Dx [L] is the 

distance between the observation wells, and C0 and C(x) 

[M L 3 ] are the upstream and downstream contaminant 

concentrations at the observation wells. In this formulation, 

all concentration changes resulting from processes other 

than degradation, i.e., diffusion, dispersion and dilution, 

are attributed to degradation. Therefore the rate constant 

l1 determined with method 1 can be considered rather 

an overall (or bulk) attenuation rate than a degradation 

rate constant [Newell et al., 2002]. Also, if the 

downstream observation well is not placed on the plume 

center line, the measured concentration is smaller than on 

the plume center line and the degradation rate constant is 

overestimated. 

[13] Method 2 was proposed by Wiedemeier et al. [1996] 

and is based on the same transport equation as method 1. 

However, to overcome the above mentioned drawbacks, 

amended concentrations are used: The measured concentrations 

of the reactive contaminant are corrected by the ratio 

of upgradient concentration C* 0 to downgradient concentration 

C(x)* [M L 3 ] of a nondegrading co-contaminant at the 

same observation wells. Thus the method corrects for 

dispersion of the plume or for the effects of unintended 

measurements off the plume center line. The degradation 

Table 1. Model Parameters Used in the Simulations 

Parameter Value 

Kef 7.2 10 5 ms 1 

lY sy 

2.67 m 

2 

n 

0, 0.38, 1.71, 2.7, 4.5 

0.33 

l 1a 1 

S, q 0 

aL 0.25 m 

0.05 m 

a T 

W01420 

ð3Þ


rate constant for method 2 is then calculated as [Wiedemeier 

et al., 1996]: 

l2 ¼ va 

Dx 

Cx 

ln ðÞ 

C0 

C* 0 

C* ðÞ x 

A prerequisite for the application of method 2 is that 

physicochemical properties of degradable and recalcitrant 

compounds like Henry’s Law constants and sorption 

coefficients must be comparable. A group of substances 

that has been proven to be well suited for the normalization 

of downgradient concentrations in BTEX plumes under 

anaerobic conditions are several trimethylbenzene (TMB) 

isomers [EPA, 1998; Wiedemeier et al., 1996, 1999]. When 

TMB is subject to biodegradation the estimated rate 

constant will be less than the actual value. For chlorinated 

solvent plumes, inorganic compounds like chloride may be 

appropriate substances for the normalization. Reductive 

dechlorination results in the production of chloride along 

the flow path, which by means of a mass balance can be 

used to derive the correction factor [EPA, 1998]. 

[14] The third method investigated here was proposed by 

Buscheck and Alcantar [1995]. It is based on the steady 

state solution to the one-dimensional transport equation, 

accounting for advection, dispersion and first-order degradation. 

In comparison to method 1, method 3 accounts 

additionally for effects of longitudinal dispersion and thus 

requires an estimate of the longitudinal dispersivity aL [m]. 

The degradation rate constant for method 3 is given by 

Buscheck and Alcantar [1995]: 

l3 ¼ 

va 

ln 

1 2aL 

4aL 

Cx ðÞ 

00 

B 

C0 

@@ 

Dx 

1 

A 

2 

1 

ð4Þ 

C 

1A 

ð5Þ 

Method 4 used here is the modified method of Buscheck and 

Alcantar [1995], as proposed by Zhang and Heathcote 

[2003]. Since in this study a two-dimensional synthetic 

aquifer is used to assess the different approaches, for method 

4 the analytical solution to the two-dimensional transport 

equation including first order decay [Domenico, 1987] is 

adopted. Method 4 accounts for a finite source width as well 

as longitudinal and transverse dispersion. Therefore longitudinal 

and transverse dispersivities aL [m] and aT [m] as 

well as the source width WS [m] perpendicular to the average 

flow direction have to be known or estimated. Given these 

prerequisites, the degradation rate constant is given as 

[Zhang and Heathcote, 2003]: 

l4 ¼ 

va 

1 

4aL 

ln 

2aL 

Cx ðÞ 

00 

B 

@@ 

C0b 

Dx 

b ¼ erf 

WS 

1 

A 

2 

1 

C 

1A 

4 ffiffiffiffiffiffiffiffiffiffiffi p ð6Þ 

aT Dx 

2.3. Investigation Scenario 

[15] The site investigation mimicked in this study is 

depicted in Figure 2, where a steady state plume has 

evolved from a contaminant source. This plume is investigated 

by the center line approach. Figure 2a shows the 

initial situation, where three observation wells are present in 


Figure 2. Representation of the investigation method to 

obtain plume center line concentrations: (a) initial 

situation, (b) estimation of flow direction by application 

of a hydrogeologic triangle, (c) measured concentrations on 

the inferred center line, and (d) comparison to true 

concentrations and heads. 

the aquifer. One of these wells (solid circle) is in the source 

and concentrations are high, while the other two show no 

concentration. This situation is the starting point for the 

investigation scenario for all realizations and represents the 

initial knowledge on the site. In the first investigation step, 

the flow direction is determined (Figure 2b). Hydraulic 

heads are measured in the three wells (by reading the model 

output at the well positions), a hydrogeologic triangle is 

constructed and the hydraulic gradient is calculated. In the 

second investigation step, three new observation wells are 

installed every 10 m along the estimated direction of flow 

(Figure 2c). These wells are then used to obtain hydraulic 

heads and concentrations of the contaminant and the nonreactive 

co-contaminant as well as local hydraulic conductivities 

at the observation wells by using the model input for 

hydraulic conductivity at the corresponding location. From 

the head difference, the true porosity and the well positions 

the respective groundwater flow velocities are calculated. 

An effective conductivity Kef between each upstream and 

downstream well is estimated by calculating the geometric 

mean of the local conductivities. Together with the concentration 

data, this information allows for the determination of


the first-order degradation rate constants by the four methods 

presented above. The investigation setup is designed to 

resemble ideal conditions for the application of the four 

methods for estimating the degradation rate constant. All 

measurements are assumed to be exact, which means that 

there is no measurement error involved. The only uncertainty 

and variability is introduced by the heterogeneity of 

hydraulic conductivity. For methods 3 and 4, additionally 

aL and aT have to be known. These are estimated following 

Wiedemeier et al. [1999] as 0.1 of the plume length for aL, 

with aT being about 0.33 of the longitudinal dispersivity. As 

plume length the maximum distance covered by the observation 

wells, i.e., 30 m, is used. As the plumes are generally 

longer, this assumption yields rather low dispersivities. 

Thus aL and aT are estimated to be 3.0 and 1.0 m, 

respectively. These estimates of dispersivities are not optimal, 

as they are not based on the heterogeneity of the 

hydraulic conductivity. However, dispersivities based on 

results from stochastic hydrogeology are difficult to obtain, 

as for most field sites structure and degree of heterogeneity 

are not well known. Also, the four samples taken in this 

investigation scenario do not allow for an estimation of the 

correlation length or the ln(K) variances. Both the correct 

source width and the correct porosity are used. In the last 

step, by each of the four approaches the corresponding 

first-order degradation rate constants l 1 through l 4 are 

calculated. These values can be compared to the value used 

to generate the plume (l =1a 1 ). For each realization, the 

investigation procedure described above is followed and a 

degradation rate is calculated for each method, each downstream 

well and for each source width. For each of the four 

classes of heterogeneity used in this study (ln(K) variances 

s Y 2 of 0.38, 1.71, 2.70 and 4.50) a minimum of 100 realizations 

is evaluated. Thus statistical measures of the errors 

and uncertainties introduced by the heterogeneity of the 

hydraulic conductivity are obtained. Additionally, also the 

impact of the width of the source zone is studied. Here it 

is expected, that for increasing source width the onedimensional 

methods yield better results, as then the 

investigated situation corresponds better to the assumptions 

of the method. Source widths W S of 4 m, 8 m and 16 m are 

used, corresponding to 1.5, 3 and 6 integral scales l Y. Then 

methods for estimating the flow velocity are elucidated for 

the different degrees of heterogeneity. This is because the 

goodness of the calculated value for lambda is directly 

related to estimated transport velocity accuracy. Finally the 

influence of estimated longitudinal and transversal dispersivities 

on results by methods 3 and 4 is studied in a 

sensitivity analysis. 

2.4. Numerical Tests 

[16] Convergence of the Monte Carlo simulation with 

regard to the sample size N of estimated degradation rate 

constants was tested by a procedure following Goovaerts 

[1999]. The test is only conducted for the highest degree of 

heterogeneity used in this study (sY 2 = 4.5) and the smallest 

source width of 4 m, as this is the case of highest variability. 

A total of 1000 realizations of the random conductivity field 

was generated. For each realization, plume development 

was simulated and the degradation rate constant l1 was 

calculated using method 1. The resulting set of 1000 

degradation rates is assumed to be sufficiently large to 

Figure 3. Influence of Monte Carlo sample size N on the 

average subset mean (l1), the standard deviation of subset 

means (sl1 ), and the average standard deviation of the 

subset population (sl1 ) for the highest degree of heterogeneity 

(sY 2 ). 

represent the global population. The global population of 

l1 was randomly sampled with a sample size of N =2, 

yielding a subset of two l1. For this subset the mean 

degradation rate as well as the standard deviation were 

calculated. Random sampling was repeated 999 times, resulting 

in 1000 subsets of N = 2. From these subsets, the average 

subset mean l1, the standard deviation of subset means sl1 and the average standard deviation of the subset population 

sl1 are calculated. Random sampling was repeated with 

increasing subset sizes N =3,4,..., 1000, resulting in 999 

triplets of the statistics, one for each subset size. Dependence 

of the three statistics on N is shown in Figure 3. 

[17] The middle curve in Figure 3 displays the average 

subset mean l1, which shows almost no dependence on N 

and yields values very close to the global mean of 8.9. The 

standard deviation of the subset means (s , lower curve) 

l1 

shows a strong decrease from 12.3 (N = 2) to 2.2 (N = 50) 

and 1.5 (N = 100), with a significantly reduced decrease for 

larger subset sizes. In relation to the global mean of 8.9, the 

variation among the subsets is therefore small for N 100. 

The upper curve in Figure 3 shows the average standard 

deviation of the random sample subsets sl1 , which strongly 

increases with subset size for small N, but with a much 

smaller increase for N > 50. For N = 100 a value of 15.5 is 

found, which is 95% of the standard deviation of the global 

population, as obtained for N = 1000. The observed reduction 

in increase of sl1 with N indicates the redundancy of 

additional realizations with regard to the subset variability. 

As the rate of decrease of s as well as the rate of increase 

l1 

of sl1 becomes small for more than 100 realizations, we feel 

confident that a sample size of N = 100 is sufficient to yield 

stable ensemble averaged rate coefficients. Since the analysis 

was conducted for the largest degree of heterogeneity, 


W01420


Figure 4. Estimated first-order degradation rate constants Li (normalized to the true rate constant l) 

versus degree of heterogeneity sY 2 for (a) method 1, (b) method 2, (c) method 3, (d) and method 4. All 

figures show results for all single realizations (small symbols) as well as their ensemble means (large 

crosses) with their corresponding standard deviations (error bars) and ensemble medians (large diamonds). 

The reference rate constant used in the numerical simulations is indicated by the horizontal line. 

for lower values of s Y 2 convergence can be expected already 

at lower sample sizes. 

[18] Another check was performed concerning the mean 

flux over the flow domain. Although deviations between 

single realizations are quite distinct and increase with s Y 2 , 

the ensemble averages for each s Y 2 match the theoretical 

value with less than a 1% error. Furthermore, the correct 

operation of the investigation methods was verified by 

applying the investigation procedure for a source of infinite 

width, i.e., a width equal to the model area, and by 

assuming the aquifer is homogeneous. Then all methods 

reduce to the one-dimensional method and yield the correct 

degradation rate constant. This result was obtained and thus 

the correct operation verified. 

3. Results and Discussion 

3.1. Influence of Heterogeneous Conductivity 

[19] To examine the influence of heterogeneity on the 

estimation of the rate constants, a contaminant source of 


width W S = 4 m perpendicular to the average flow direction 

is emplaced in the synthetic aquifer and flow as well as 

reactive transport are simulated. With the investigation 

scenario described in section 2.3 applied to a single realization, 

each of the four methods yields differing rate 

constants, l i, for each of the three center line observation 

wells (10 m, 20 m, and 30 m distance from the source). For 

the assessment of the four methods, these rate constants 

l i,10, l i,20 and l i,30 are averaged to yield one single 

estimated l i for each method. This procedure is repeated 

for all realizations. Figure 4 presents results of the calculated 

rate constants l i versus the degree of heterogeneity 

(given as ln conductivity variance s Y 2 ). Calculated rates li 

are reported as normalized rates L i, i.e., the calculated rate 

l i is divided by the true rate l used in the numerical 

simulation. The normalized rate constants L i thus can be 

interpreted as over- or under-estimation factors. The homogeneous 

case (s Y 2 = 0) is included for reference. 

[20] Figure 4a presents the results for method 1, i.e., based 

on the one-dimensional advection-degradation solution


(equation (3)). It can be clearly seen, that an increase of 

sY 2 leads to an increase in spread of the single realizations. 

Quite a number of realizations exhibit normalized rate 

constants L1 of more than 10 up to about 100, i.e., the 

true rate constant is severely overestimated by a factor of 

10 to 100. Also, L1 increases with increasing sY 2 as well 

as the standard deviations of the ensemble means. This 

points to an increase in uncertainty and reflects the spread 

of the single realizations. Mean overestimation increases 

from a factor of about 1.6 for sY 2 = 0.38 to about 10.2 in 

the case of sY 2 = 4.50. In the homogeneous case (sY 2 =0), 

method 1 yields the correct result. Although on average l 

is overestimated, even for high values of sY 2 in single 

realizations l may actually be underestimated by L1. For 

the highest degree of heterogeneity, the L1 of the single 

realizations span about three orders of magnitude. Comparison 

of ensemble means with the corresponding 

medians shows that in all cases the medians are significantly 

lower. The populations of estimated l1 are positively 

skewed as some exceedingly large values of l1 

shift the means to high values. However, the general trend 

of increasing overestimation with heterogeneity is also 

distinct for the medians. 

[21] Figure 4b shows the corresponding results obtained 

with method 2, i.e., using the one dimensional advectiondegradation 

equation with normalization to a recalcitrant cocontaminant 

(equation (4)) [Wiedemeier et al., 1996]. As for 

method 1, the spread of the single realizations increases 

with increasing s Y 2 . Compared to method 1, however, the 

spread is smaller and more equally distributed about L 2 =1. 

Therefore the average overestimation factors as well as the 

standard deviations are much smaller than for method 1 and 

both the error and the uncertainty are lower. Average rate 

constants L 2 are 1.1 for s Y 2 = 0.38, increasing to 3.3 for sY 2 = 

4.50. For homogeneous conditions, also method 2 yields the 

correct result, i.e., L 2 = 1 for s Y 2 = 0. Ensemble medians are 

just slightly above the true l, i.e., deviating less than a 

factor of 2. 

[22] Degradation rate constants calculated with method 

3, i.e., the one-dimensional method introduced by 

Buscheck and Alcantar [1995] (equation (5)), are displayed 

in Figure 4c. They exhibit a similar general 

behavior as found with method 1, i.e., increasing spread 

and increasing overestimation of the true rate constant for 

higher s Y 2 . However, mean L3 are significantly higher 

than the corresponding L 1, with ensemble means of 2.0 

for s Y 2 = 0.38 increasing to 29.4 for sY 2 = 4.50. In the 

homogeneous case, the true rate constant is slightly 

overestimated, i.e., L 3 =1.25fors Y 2 = 0. Spread in the 

single realizations is higher compared to method 1, now 

spanning nearly four orders of magnitude for the largest 

variance value of s Y 2 . 

[23] Results for method 4, i.e., the two-dimensional 

solution (equation (6)) suggested by Zhang and Heathcote 

[2003], are depicted in Figure 4d. Compared to the other 

methods, method 4 displays the largest spread of calculated 

L 4 around the mean values. For the highest degree of 

heterogeneity, the spread of the single realizations covers 

nearly five orders of magnitude. Ensemble means of L 4 

increase from 0.6 for s Y 2 = 0.38 to about 23.0 for sY 2 = 4.5, 

i.e., for low s Y 2 the normalized rate constant L4 is actually 

underestimated in most realizations, while for larger varian- 


W01420 

ces the ensemble averages approach the results of method 3. 

In the homogeneous case (sY 2 = 0), the estimated rate 

constant l4 is about two orders of magnitude lower than 

the true rate constant l. Ensemble medians for method 4 

show a similar behavior as the ensemble means, with their 

values closer to the true rate constant than for methods 1 

and 3 for large heterogeneities (sY 2 = 1.71). This reflects the 

fact, that the spread of the single realizations is distributed 

symmetrically around L4 = 1, however, the spread of the 

populations and thus the uncertainty is significantly larger 

than for methods 1 and 3. 

[24] Method 1 is based on the one-dimensional solution 

to first-order biodegradation and advection. Therefore it is 

expected that rate constants estimated with method 1 

overestimate the true rate constant due to two effects. 

Firstly, method 1 does not account for measuring off the 

center line. So if an observation well is placed off the plume 

center line, concentrations sampled there will be smaller 

than on the center line and therefore the degradation rate 

constant will be estimated too high. Secondly, as method 1 

is based on a one-dimensional solution of the transport 

equation, it does not account for transverse dispersion, 

which lowers concentrations on the plume center line. This 

second effect also causes an overestimation of the rate 

constant. Both effects together cause the overestimation 

of the rate constant as shown in Figure 4a. Method 2 

tries to overcome these two problems by normalization 

to a conservative tracer. Both above effects also determine 

the concentration of the nonreactive component, and are 

thus corrected for by the normalization. As is shown in 

Figure 4b, results of method 2 are considerably better than 

of method 1, both considering spread and ensemble averages. 

However, as can be seen from Figure 4b, method 2 

does not correct for all effects, as overestimation is observed 

with increasing s Y 2 . Effects of dispersion and measuring off 

the center line are accounted for by method 2, so the 

deviation seen for method 2 has to have a hydraulic cause. 

This deviation is introduced by the determination of the 

average flow velocity between the observation wells, which 

is calculated using an averaged value of the hydraulic 

conductivity at the two observation wells. This averaged 

value may not be representative of the flow path between the 

two wells and bias may be introduced into the calculation of 

degradation rate constants. This effects is studied closely 

below in section 3.3. 

[25] Method 3 is based on the one-dimensional transport 

equation including advection, degradation and longitudinal 

dispersion. Results from method 3, as shown in Figure 4c, 

display a higher spread and higher ensemble averages 

compared to method 1. Because method 3 includes longitudinal 

dispersion, it should be closer to reality and advantageous 

over method 1. The differences in estimated rate 

constants between method 1 and 3 are therefore due to the 

longitudinal dispersivity a L in method 3. With the onedimensional 

transport model used, pronounced longitudinal 

dispersion of a degrading contaminant results in a stronger 

spreading of the solute downstream and thus in higher 

concentrations along the plume center line compared to an 

advection only case. Therefore a larger rate constant is 

calculated to accomplish a given concentration decrease 

between the upgradient and the downgradient observation 

well. l 3 grows linearly with a L and is always larger than l 1,


as can be seen by expanding the squared brackets in 

equation (3) and using C(x) C0: 

l3 ¼ va aL 

va 

lnð Cx ðÞ=C0 

Dx 

lnð Cx ðÞ=C0 

Dx 

2 

¼ l1 

! 

lnðCx ðÞ=C0Þ 

Dx 

[26] If a L = 0, method 3 reduces to the advection only 

case, i.e., method 1. Method 3 still does not account for 

transverse dispersion, which is the process causing smaller 

concentrations on the plume center line. Method 4 is based 

on a two-dimensional solution to the transport equation 

including advection, longitudinal and transverse dispersion 

and first-order degradation. Results from method 4 

(Figure 4d) show an underestimation of the true rate 

constant for homogeneous or slightly heterogeneous conditions 

(sY 2 

1.71), while for high degrees of heterogeneity 

(sY 2 

2.7), the ensemble averages of the estimated rate 

constants approach the respective values obtained with 

method 3. The underestimation for low heterogeneities is 

a consequence of the correction for transverse dispersion, 

represented by the error function b in equation (4). The 

effect of lower concentrations along the plume center line 

due to transverse dispersion is strong for small source 

widths, large transversal dispersivities and large well spacings. 

b is always less than 1 and asymptotically approaches 

unity for arguments of the error function larger than 2, i.e., 

l4 converges toward l3 for small aT or large WS. For 

arguments


Figure 5. Estimated first-order degradation rate constants versus degree of heterogeneity sY 2 for 

(a) method 1, (b) method 2, (c) method 3, and (d) method 4 and source widths WS of 4, 8, and 16 m, 

respectively. Solid lines show ensemble means normalized to the true degradation rate constant (left 

axis), dashed lines show the corresponding standard deviations (right axis). 

[29] In the following, the influence of different approaches 

for estimating K (and thus va) on the determination of li 

is analyzed. As the plume samples only part of the full 

domain, the hydraulic conductivity needed is not the effective 

conductivity for the complete domain, but an equivalent 

hydraulic conductivity valid for the flow path along the 

center line wells. This equivalent hydraulic conductivity 

may differ from the global effective value. During the field 

investigation by the center line method, only the local 

hydraulic conductivities measured at the observation wells 

placed along the inferred plume center line or only the 

global effective hydraulic conductivity are known. Neither 

the underlying statistical parameters nor the complete conductivity 

field are known. The first approach represents the 

situation, where local hydraulic conductivities have been 

obtained in the observation wells, e.g., by slug tests or sieve 

analysis, and are averaged to obtain an estimate of the 

equivalent hydraulic conductivity between the two observation 

wells used for rate determination. In addition to the 

geometric mean Kg, as used so far, also the arithmetic mean 

Ka and the harmonic mean Kh are used, as they present 


W01420 

upper and lower limits for Kef [e.g., Renard and de Marsily, 

1997]. As hydraulic conductivity data is available only at 

four wells, a full characterization of the conductivity field is 

not possible. Often at a real site, values of hydraulic 

conductivity are not available at the locations of the 

observation wells at the plume center line, but only from 

a single well not on the plume center line and thus not used 

for rate determination. This case is simulated by using a 

value KS of hydraulic conductivity from a well at point 

[55.10 m, 55.20 m] (see Figure 1). In the last approach, the 

true global effective conductivity KG of the synthetic 

aquifer is known, i.e., from a large-scale pumping test, 

and KG is used as an estimator for the local equivalent 

hydraulic conductivity at the observation wells. 

[30] Figure 6 shows the ensemble averages (Figure 6a) 

and medians (Figure 6b) of the degradation rate constants 

estimated using the above five approaches. Only results for 

method 2 and WS = 16 m are presented here, because 

method 2 is only affected by the hydraulic error due to a 

wrong estimation of flow velocity. Inspection of Figure 6a 

shows that all approaches lead to an overestimation of the


Figure 6. Ensemble (a) means and (b) medians of 

calculated L 2 for the different approaches of average flow 

velocity estimation and (c) the corresponding coefficients of 

variation. 

ensemble averaged L2, which increases with sY 2 . This 

increase is most pronounced for Ka and KS. For sY 2 = 4.50 

the average L2,K a is about 4.3 times larger than L2,K g , 

whereas for single realizations the difference may reach a 

10 of 14 

factor of 70 (not shown here). If only the single KS value is 

used, overestimation is even larger for all degrees of 

heterogeneity. In comparison to the geometric average, 

using the harmonic average Kh or the true global effective 

conductivity KG reduces the overestimation of L2 considerably. 

Both approaches result in nearly identical ensemble 

averages. However, inspecting ensemble medians as shown 

in Figure 6b, harmonic averaging of K measurements results 

in an underestimation of L2 in more than 50% of all 

realizations. Taking this into account, usage of the global 

KG seems to be the best approach for the estimation of the 

local average flow velocity between two observation wells 

along the plume center line. This finding is supported by 

Figure 6c, which shows coefficients of variation (CV) ofL2 

as a measure of uncertainty. Using KG yields the smallest 

spread. Comparison of all five approaches yields: CVK G < 

CVK g CVK h < CVK a < CVK S for all sY 2 . These findings 

indicate that for the conditions given in this work the best 

estimate of local flow velocities and of the degradation rate 

constant is given by the global geometric average of the 

hydraulic conductivity. This result is expected for large well 

distances, as then ergodic conditions have been reached. 

Our results show that also for nonergodic conditions due to 

small well distances of just a few integral scales the global 

geometric average yields better estimates of local flow 

velocity compared to using locally measured hydraulic 

conductivities. The spread observed in Figure 6 is thus 

due to the nonergodic conditions of single realizations, i.e., 

the fact that the plume has sampled only a part of the 

domain. Thus for a field case, a single large-scale pumping 

test would be preferable to small-scale local information 

obtained directly at the sampling wells. 

3.4. Influence of Dispersivity Parameterization 

[31] As demonstrated in the previous sections, the parameterization 

of macrodispersivities a L and a T solely 

based on the scale of the contaminant plume according to 

Wiedemeier et al. [1999] using 0.1 of the assumed plume 

length for a L and a T = a L/3 introduces a significant 

additional error when rate constants are estimated with 

methods 3 or 4. For method 3, this error is always toward 

higher rate constants, while for method 4 errors in both 

directions may occur. It is known that aL and aT do not only 

depend on plume scale, because the variability of flow 

velocity resulting from the heterogeneity of hydraulic conductivity 

is also important for the spreading during solute 

transport. If additional information on the spatial distribution 

and variability of hydraulic conductivity is available, 

i.e., from an geologically analogous aquifer, where these 

parameters have been determined, this information can be 

used to obtain estimates of the dispersivity values based on 

aquifer heterogeneity. So far in the manuscript, only estimates 

based on plume length have been used. 

[32] In a two-dimensional isotropic domain with zero 

local dispersivity and for ergodic conditions, the asymptotic 

large time longitudinal dispersivity is given by aL = sY 2 lY, 

whereas the asymptotic limit of aT is 0 [Dagan, 1989; 

Rubin, 2003]. For the two-dimensional model setup investigated 

in this manuscript and the values of sY 2 (0.38, 1,71, 

2.7, 4.5) and lY (2.67) m, the corresponding values for the 

asymptotic limit of aL are 1.02, 4.57, 7.21 and 12.02 m, 

respectively. Close to the source, i.e., in the preasymptotic


Figure 7. Ensemble medians of estimated first-order rate constants L 4 for (a) s Y 2 = 0.38, (b) 1.71, 

(c) 2.7, and (d) 4.5 for different values of a L versus a T. Values for method 3 are obtained using 

L 3(a L)=L 4(a L, a T = 0). Medians for L 1 and L 2 are depicted in each diagram by dashed and dash-dotted 

horizontal lines for reference. 

region, aL will show lower values. aT grows with travel 

distance until aT reaches peak values at about 2.5 lY of 0.1, 

0.44, 0.69 and 1.15 m, respectively. The asymptotic limit 

of aT is 0. For nonzero but small local dispersivities (aL,T 

l 1 

Y 1), aL is unaffected or slightly reduced, while aT 

increases and a non zero long time limit will establish 

[Zhang and Neuman, 1996]. As the observation wells used 

for the investigation of the plumes are located within 3.8 to 

11.2 lY from the contaminant source, it is expected, that the 

asymptotic limits of aL and aT are not yet reached. Thus 

for methods 3 and 4, aL values below those listed above 

should be used. Similarly, aT values in the range between 

the peak values and the asymptotic limit should be used 

for method 4. As this rough estimation already shows, 

applying the stochastically derived values in a real field 

case may introduce uncertainty, as the local conditions are 

generally not known exactly. 

[33] To study the influence of dispersivity parameterization 

on rate constants estimated with methods 3 and 4, the 

sensitivity of L3 and L4 on different values of aL and aT is 

investigated. Because in a real field application these values 


W01420 

will be always uncertain, this investigation is performed as a 

sensitivity analysis, which allows to cover a wider range of 

values for aL and aT. For aL values of 12, 9, 6, 3, 0.8 and 

0.25 m are used, while for aT values of 2, 1.15, 1, 0.75, 0.5, 

0.3, 0.07 and 0 m are used. Thus the range of reasonable 

values of aL and aT up to the maximum values for each sY 2 

is well represented. Figure 7 presents for each sY 2 the 

medians of estimated rate constants L3, using L3(aL) = 

L4(aL, aT = 0), and L4(aL, aT) in dependence on aT for the 

different values of aL. For reference, also medians for L1 

and L2 (compare Figure 4) are included. Because method 2 

corrects for dispersion as well as for measuring off the 

plume center line, but not for errors in the estimated 

transport velocity, and the same value of the transport 

velocity is used for all four methods, L2 can be seen as 

the optimal lower limit for methods 3 and 4. Method 1 

involves no correction for longitudinal and transverse dispersion 

or measuring off the center line. Thus a good 

parameterization of aL and aT for method 4 should result 

in rate constant estimates L4 < L1. However, L4 < L2 would 

indicate over correction for transverse dispersion.


[34] Results of the sensitivity study are presented in 

Figure 7. For all degrees of heterogeneity, decreasing 

median degradation rates are found with increasing aT. 

Differences between the different aL are most distinct for 

low values of aT. Considering method 3, it is clearly shown 

that L3 = L1 only for aL = 0, and L3 > L1 for aL >0. 

This demonstrates again that method 3 always yields 

higher estimated degradation rate constants as compared 

to method 1 (compare equation (7)). Considering that 

generally l is overestimated in our study, accounting only 

for aL by using method 3 aggravates this problem. 

[35] For method 4 it is found that for low and medium 

heterogeneity values for aL and aT exist, which allow for 

an optimal estimation of the degradation rate constant, i.e., 

L4 = 1, and thus L4 L2. For sY 2 = 2.70, L4 > 1.0 

always, but L4 < L2 can be achieved for large values of 

aT and small values of aL. However, for the highest 

degree of heterogeneity, L4 > L2 > 1 always. For small 

values of aT, the degradation rate is overestimated for all 

degrees of heterogeneity, while for low and medium 

heterogeneity L4 < 1 is possible for large values of aT. 

[36] For sY 2 = 0.38 and aT = 0.07 m, all medians are larger 

than L 1 (when a L > 0.8), at a T = 0.3 all medians are 0.5 m (Figure 7c). However, even 

for an unrealistically large a T of 2 m, L 4 is still significantly 

larger than L 2. A similar behavior is found for s Y 2 =4.5 

(Figure 7d), where only using a L = 9 m and a T >1.15m 

will result in L 4 being closer to the true rate constant than 

the corresponding L 1. 

[37] These results show, that a wide range of dispersivities 

can and must be used to obtain better estimates using 

method 4 than using method 1 or method 2. It is also found 

that the theoretical values (as given above) lead only for the 

case of low heterogeneity to considerably better estimates 

than using method 1. Therefore a large fraction of the 

observed overestimation must result from off center line 

measurements, as it cannot be corrected for by reasonable 

values of a T. Especially for the cases of high heterogeneity, 

the effects of dispersion seem to be minor in comparison to 

the effect of missing the center line. 

[38] As shown above, method 4 could yield estimated 

rate coefficients that are as close or even closer to the true 

rate constant than rate coefficients estimated with method 2, 

regardless of the degree of heterogeneity. However, this 

requires unreasonably low values for a L and very high 

values for a T, as these parameters would have to correct for 

the off center line measurement errors. In this case, a L 

and a T would no longer represent the actual dispersivities, 

but are lumped fitting parameters. As the magnitude of 

bias introduced by missing the center line as well as the 

exact values of s Y 2 or lY are usually not known at a real 

field site, choosing dispersivities is highly uncertain and 

may cause over- as well as under-estimation of the 

degradation rate constant. Thus estimation of dispersivity 


introduces an additional error into the estimation of degradation 

rate constants using methods 3 or 4. Only for aquifers 

of low heterogeneity method 4 yields better estimates than 

method 1. Better estimates than using method 2 are only 

possible by assuming unphysical dispersivity values. 

4. Summary and Conclusions 

[39] In this paper the performance of four different 

methods for the estimation of degradation rate constants 

in an aquifer with a heterogeneous distribution of the 

hydraulic conductivity is studied. All four methods are 

based on the center line approach. The results demonstrate 

that a heterogeneous distribution of the hydraulic conductivity 

may lead to severe overestimation of the ensemble 

averaged degradation rate constant. Furthermore, the single 

realizations show a large spread and a large standard 

deviation, indicating that results obtained from any one 

estimation are highly uncertain. Mean overestimation as 

well as spread increase with degree of aquifer heterogeneity. 

By method comparison, the main reasons are identified as 

‘‘measuring off the plume center line’’ and effects of 

transverse dispersion. Best method performance is observed 

for method 2, which is based on the one-dimensional 

transport equation including advection and first order degradation. 

By normalizing the measured concentrations of the 

degrading contaminant to a nonreactive co-contaminant 

emitted by the same source, the above mentioned effects 

are corrected for. Method 2 thus shows the lowest spread 

and the lowest overestimation of estimated degradation rate 

constants of all four methods. However, the presence of the 

recalcitrant co-contaminant needed for the normalization 

approach may not always be given at a site. Second best 

performance is observed for method 1, which yields consistently 

higher spread and degradation rate constant overestimation 

as compared to method 2. Methods 3 and 4, 

although more realistic in the sense that they base on the 

one-dimensional and two-dimensional transport equation, 

respectively, show higher spread and larger overestimation. 

Both methods are prone to errors introduced by estimating 

longitudinal and transverse dispersivities. The choice of 

these values introduces additional uncertainty without yielding 

substantially better results than methods 1 or 2. The 

ensemble averaged degradation rate constant is highest for 

method 3, due to the longitudinal dispersivity term in 

equation (3), while performance of method 4 crucially 

depends on the choice of an appropriate transverse dispersivity 

value a T.Ifa T is chosen too small with respect to the 

real macrodispersivity at the field site under consideration, 

the degradation rate constant may be overestimated. If a T is 

chosen too large, the degradation rate may be underestimated. 

For all methods, a high spread of the results from the 

single realizations is found, causing a high uncertainty of 

the estimated degradation rate constant for all methods. For 

a single realization, the estimated degradation rate may 

deviate by one or even two orders of magnitude from the 

correct value. This deviation is caused only by the heterogeneity 

of the hydraulic conductivity. The spread observed 

here may contribute to the spread observed in degradation 

rate constants observed in the field. Wiedemeier et al. [1999] 

and Aronson and Howard [1997] report measured degradation 

rate constants for PCE ranging over two orders of


magnitude from 0.07 a 1 to 1.2 a 1 and for TCE ranging 

from 0.05 a 1 to 4.75 a 1 . For benzene, toluene and xylene 

rate constants of 0.07–3.0 a 1 , 0.36–21.0 a 1 and 0.32– 

76.0 a 1 , respectively, are reported [Wiedemeier et al., 

1999]. Method performance increases with increasing 

source width for all methods. For sources very wide with 

respect to the integral scale of the hydraulic conductivity 

field, all methods yield reasonable results. In reality, however, 

when sources are heterogeneous or formed by a 

complex combination of a number of zones, the total source 

width may be difficult to estimate. If degradation rates are 

used for assessing the NA potential at a contaminated site, 

overestimation of the degradation rates is a critical point. 

Overestimation of the degradation rate constant leads to an 

overestimation of the overall natural attenuation potential. If 

plume lengths are calculated with too high degradation rate 

constants, then estimated plume lengths are too short. 

Remediation times as well as downgradient concentrations 

may be underestimated. The results presented show that 

determination of degradation rate constants suffers from two 

main sources of error, i.e., sampling off the plume center 

line and an incorrect estimate of the average transport 

velocity. The first can be overcome by using method 2, 

the second can be resolved by conducting tracer tests or 

additional measurements of the hydraulic conductivity. A 

tracer test would furthermore prove, that the observation 

wells under consideration are sampling the same flow path. 

Further work on this subject will include the effects of 

measurement error on the estimated degradation rates, both 

in measuring hydraulic head and contaminant concentration. 

Also effects of different formulations of the kinetic reactions 

used to simulate the plume will be investigated. 

[40] Acknowledgments. This work is funded by the German Ministry 

of Education and Research as part of the KORA priority program, 

subproject 7.1 Virtual Aquifer. We would like to acknowledge the thoughtful 

reviews of three anonymous reviewers. Their comments have greatly 

improved this manuscript. 

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S. Bauer, C. Beyer, and O. Kolditz, Center for Applied Geoscience, 

University of Tübingen, Sigwartstr. 10, D-72076 Tübingen, Germany. 

(sebastian.bauer@uni-tuebingen.de; christof.beyer@uni-tuebingen.de; 

kolditz@uni-tuebingen.de)


Beyer, C., Bauer, S., Kolditz, O. (2006): Uncertainty Assessment of Contaminant Plume 

Length Estimates in Heterogeneous Aquifers. J. Contam. Hydrol., 87, 73-95, doi: 

10.1016/j.jconhyd.2006.04.006. 

Reprinted from Journal of Contaminant Hydrology, 87, Beyer, Christof, Bauer, Sebastian and 

Olaf Kolditz, Uncertainty Assessment of Contaminant Plume Length Estimates in 

Heterogeneous Aquifers, 73-95, Copyright 2006, with permission from Elsevier. 

The enclosed article can be obtained online via ScienceDirect at 

http://www.sciencedirect.com/science/journal/01697722.

Uncertainty assessment of contaminant plume length 

estimates in heterogeneous aquifers 

Abstract 

Journal of Contaminant Hydrology 87 (2006) 73–95 

Christof Beyer ⁎ , Sebastian Bauer, Olaf Kolditz 

www.elsevier.com/locate/jconhyd 

Center for Applied Geoscience, University of Tübingen, Sigwartstraße 10, D 72076 Tübingen, Germany 

Received 4 November 2005; received in revised form 14 April 2006; accepted 25 April 2006 

Available online 16 June 2006 

The Virtual Aquifer approach is used in this study to assess the uncertainty involved in the estimation of 

contaminant plume lengths in heterogeneous aquifers. Contaminant plumes in heterogeneous twodimensional 

conductivity fields and subject to first order and Michaelis–Menten (MM) degradation kinetics 

are investigated by the center line method. First order degradation rates and plume lengths are estimated 

from point information obtained along the plume center line. Results from a Monte-Carlo investigation 

show that the estimated rate constant is highly uncertain and biased towards overly high values. Uncertainty 

and bias amplify with increasing heterogeneity up to maximum values of one order of magnitude. 

Calculated plume lengths reflect this uncertainty and bias. On average, plume lengths are estimated to about 

50% of the true plume length. When plumes subject to MM degradation kinetics are investigated by using a 

first order rate law, an additional error is introduced and uncertainty as well as bias increase, causing plume 

length estimates to be less than 40% of the true length. For plumes with MM degradation kinetics, 

therefore, a regression approach is used which allows the determination of the MM parameters from center 

line data. Rate parameters are overestimated by a factor of two on average, while plume length estimates are 

about 80% of the true length. Plume lengths calculated using the MM parameters are thus closer to the 

correct length, as compared to the first order approximation. This approach is therefore recommended if 

field data collected along the center line of a plume give evidence of MM kinetics. 

© 2006 Elsevier B.V. All rights reserved. 

Keywords: Natural Attenuation; Heterogeneity; Plume length; Center line method; Uncertainty analysis; Monte-Carlo; 

Numerical modelling 

⁎ Corresponding author. Tel.: +49 7071 29 73176; fax: +49 7071 5059. 

E-mail address: christof.beyer@uni-tuebingen.de (C. Beyer). 

0169-7722/$ - see front matter © 2006 Elsevier B.V. All rights reserved. 

doi:10.1016/j.jconhyd.2006.04.006

74 C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95 


One major requirement for the implementation of natural attenuation (NA) as a remedial and risk 

reduction strategy for contaminated aquifers is an assessment of the dimensions of contaminant 

plumes and to predict their fate. Down gradient contaminant concentrations, i.e. plume lengths, must 

be calculated or estimated to identify potential receptors and predict exposure levels. For this purpose 

analytical and numerical solute transport models (e.g. Bioscreen (Newell et al., 1996), Biochlor 

(Aziz et al., 2000), Bioplume III (Rifai et al., 1998)) are routinely used. The rate of contaminant 

removal through biodegradation is a key parameter, as concentrations and the modeled plume 

lengths are highly sensitive to the degradation rate (McNab, 2001; Suarez and Rifai, 2004). Although 

very detailed mathematical descriptions of contaminant degradation in the subsurface are available 

(Baveye and Valocchi, 1989; Rittmann and VanBriesen, 1996; Wiedemeier et al., 1999; Islam et al., 

2001), for applications in the field, usually simplified approaches are used because the identification 

of a large number of parameters and processes from field data is often impossible. Due to its 

mathematical simplicity, its ease of implementation into transport models and the necessity of 

determining only a single parameter, the most frequently used degradation model is first order 

kinetics (Wiedemeier et al., 1999). Field methods for the determination of biodegradation rates in 

ground water include mass balance calculations, in-situ microcosm studies and the center line 

method (Chapelle et al., 1996; Wiedemeier et al., 1999). The latter is frequently used for plume 

monitoring and degradation rate evaluation (e.g. Chapelle et al., 1996; Wiedemeier et al., 1996; 

Zamfirescu and Grathwohl, 2001; Suarez and Rifai, 2002; Wilson and Kolhatkar, 2002; Bockelmann 

et al., 2003), and is based on contaminant concentrations measured in observation wells 

installed along the presumed center line of a plume. This approach, however, is only applicable for 

plumes that have reached a (quasi-) steady state, i.e. the plume is neither shrinking nor expanding and 

the measured concentrations do not change with time. The concentration-distance relations thus 

obtained for a steady state plume can be used to estimate the first order rate constant λ. This 

parameter can then be used with an appropriate transport model to estimate the contaminant 

distribution in the aquifer. However, as the spatial variability of aquifer properties has a substantial 

influence on the distribution of contaminants and plume development, also the results of such an 

assessment are affected. Wilson et al. (2004) point out that the approach is prone to errors because the 

center line of a plume may be missed by monitoring wells if the inferred ground water flow direction 

is incorrect or the contaminant plume meanders in all three dimensions due to macro-scale 

heterogeneities. McNab and Dooher (1998) demonstrated that, even in a homogeneous aquifer, 

transverse dispersion can produce center line concentration profiles of recalcitrant compounds that 

exhibit characteristics consistent with first order degradation; this can easily lead to misinterpretation 

of the monitoring data. The result of these complicating factors is that the degradation potential may 

be severely overestimated, causing underestimation of plume length or contaminant mass and an 

over optimistic prognosis of down gradient concentrations and exposure levels. Moreover, it is well 

known that the use of first order kinetics may be problematic in some situations, as it is a poor 

representation of the processes occurring in contaminated aquifers. Usage of a first order model 

outside its range of validity may result either in significant under- or overestimation of the 

attenuation potential at a site (Bekins et al., 1998). In a numerical experiment, Schäfer et al. (2004a) 

demonstrated that for specific points in time, first order kinetics may be able to approximately 

reproduce mass and dimensions of contaminant plumes that follow from a far more complex 

degradation model. For a long term prognosis, however, the first order approximation proved 

inappropriate, resulting in an underestimation of plume length and contaminant mass. Recently, 

Bauer et al. (2006) performed a sensitivity study on the influences of aquifer heterogeneity on first

order degradation rate constants estimated from using the center line method. This study demonstrated 

that aquifer heterogeneity introduces significant uncertainty in the estimated rate constants 

and may cause a severe overestimation of the degradation potential. 

Since the determination of degradation rates is usually only an intermediate step for the 

characterization of contaminated sites, the present paper takes the approach of Bauer et al. (2006) one 

step further. Here, the uncertainty involved in the estimation of contaminant plume lengths in 

heterogeneous aquifers is evaluated using the Virtual Aquifer concept. Three different scenarios are 

studied in detail. In case A, synthetic contaminant plumes following first order degradation kinetics 

are investigated. First order rate constants are estimated by methods typically used in field 

applications. The rate constants are then used to calculate the corresponding contaminant plume 

lengths with analytical transport models. As the first order degradation model results in theoretically 

infinite plumes, a relative concentration contour line is defined as the plume length here. Results are 

analysed with regard to errors and uncertainty in the rate constants and their propagation to the plume 

length estimates. In case B, the additional error is studied that arises when a first order approximation 

is used although degradation kinetics deviate from a first order rate law. Here the attenuation 

potential for plumes following Michaelis–Menten (MM) degradation kinetics is assessed using the 

same first order methods employed in case A. In case C, a regression approach is used to estimate the 

MM parameters and plume lengths for the plumes with MM degradation kinetics. Results are 

compared to cases A and B to allow conclusions about potentials and limitations of this approach. 

2. Virtual Aquifer concept 

C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95 

Due to the limited accessibility of the subsurface, measurements of piezometric heads and 

pollutant concentrations at contaminated sites are sparse and may not be representative of the 

heterogeneous hydrogeologic conditions. Any site investigation is thus subject to uncertainty, 

reflecting the limited knowledge of the aquifer properties and the extent of the contamination. Due 

to this uncertainty, field investigation methods for plume screening and measuring of hydraulic 

conductivity or degradation rates can neither be tested nor verified in the field. One appropriate 

method of assessing the performance and reliability of field investigation methods is by studying 

them in heterogeneous synthetic (virtual) aquifers. With this approach the results of a particular 

method can be compared to the “true” values, as these values are known from the synthetic aquifer. 

The “Virtual Aquifer” concept is a combination of different methodologies, tools and techniques, 

particularly aimed at this type of problem. Its two key components are (1) a flexible and efficient 

modelling system, allowing the numerical simulation of reactive multi-component transport in the 

subsurface, and (2) an extensive database, containing statistical information, physical and (bio-) 

geochemical data from a large number of well investigated sites and aquifers. Moreover, the concept 

comprises a collection of analytical and numerical methods, that are commonly used for the 

investigation of contaminated sites and aquifers or the interpretation of measured data. The synthesis 

of both, the database and the simulation system allows a proper definition and computer based 

evaluation of scenarios and case studies, focussed on investigation strategies, redevelopment and 

monitoring at contaminated sites. 

As a first step for such an analysis, synthetic aquifer models are generated based on the statistical 

properties of real aquifers. Thus, to a certain degree, these aquifer models represent realistic analogues 

of existing sites. A defined source of contamination is then introduced and a reactive transport model 

is used to simulate the evolution of the plume, resulting in realistic concentration distributions in the 

synthetic aquifer. In comparison to the “real world”, the unique advantage of the synthetic aquifer is 

that the spatial distribution of all physical and geochemical properties and parameters as well as the 

75


contaminant concentrations are exactly known. In the second step, the synthetic aquifer is investigated 

by standard monitoring and investigation techniques. Although the parameter distribution of the 

synthetic aquifer is known a priori, only the data “measured” at the observation wells (i.e. hydraulic 

heads and concentrations) are used and interpreted. This is done because in a real site investigation 

also only a limited amount of measured data would be available, with the amount of information 

depending on investigation intensity and project finances. In the third step, the investigation results are 

compared to the “true” values, allowing an evaluation of the accuracy of the investigation method. In 

addition, the use of synthetic aquifers offers the possibility to assess the influence of different 

parameters, such that sources of uncertainty and error for the investigation method can be considered 

individually and the sensitivity of investigation results on these can be studied. Stochastic simulation 

techniques like the Monte-Carlo method are applied to study the propagation of parameter variability 

and uncertainty into the investigation results. Due to the ability to perform extensive and detailed 

scenario analysis and visualization, this approach is well suited to the exploration of the uncertainty 

involved in hydrogeologic investigation and management. The methodology has been applied under 

the term “Virtual Aquifer” by Schäfer et al. (2002, 2004b), Bauer and Kolditz (2005) and Bauer et al. 

(2005, 2006). 

3. Scenario definition 

In this study, the Virtual Aquifer concept is used in a Monte-Carlo framework to assess the 

influence of spatially heterogeneous hydraulic conductivities on the estimation of degradation 

rates and contaminant plume lengths. Multiple plume realizations of contaminants degrading 

according to a first order degradation kinetics or Michaelis–Menten kinetics in aquifers with 

different degrees of heterogeneity are investigated using the center line approach. By comparison 

of the estimated degradation rates and plume lengths with the respective virtual reality data the 

investigation methods are tested and evaluated. Three different cases are studied in detail: 

In case A, four different standard methods for the determination of the first order rate constant λ 

are applied to concentration vs. distance data obtained from investigation of synthetic contaminant 

plumes following first order degradation kinetics. Accordingly, four different rate constants are 

estimated for each plume realization. For each λ the length of the contaminant plume is estimated 

using an analytical transport model. The four methods and the corresponding analytical transport 

models are introduced in Sections 4.1 and 4.2. The main objectives of case A are to test the 

applicability and performance of the four different methods of determining the first order degradation 

rate constant in heterogeneous aquifers and to analyse the propagation of errors and 

uncertainty from the rate constant to the plume length estimate. 

In case B, the same four methods are evaluated with regard to their ability to approximate the 

degradation potential and estimate the plume length, when the true degradation kinetics deviate 

from first order. Here plumes following MM degradation kinetics are investigated in an analogous 

manner to case A. The additional error that arises from the first order approximation is studied. The 

motivation behind this scenario is that although it is well known that contaminant degradation in 

natural aquifers may follow far more complicated processes and kinetic laws than a simple first 

order model, the latter is routinely used at many field sites. Therefore, this scenario highlights some 

of the problems that result from this discrepance. 

In case C, a regression approach is studied which allows the estimation of MM kinetic parameters 

from plume center line data. This method is developed in Section 4.1 and tested under the influence 

of aquifer heterogeneity in case C. Here the plumes following MM degradation kinetics are 

investigated. As for cases A and B the propagation of errors and uncertainty from the estimated MM

Table 1 

Overview of scenarios studied 

Case Contaminant plume 

following 

parameters to the plume lengths is analysed. Results are set in relation to cases A and B to discuss 

advantages and disadvantages, potentials and limitations of the MM parameter estimation approach. 

An overview of the three different cases is given in Table 1. 

4. Methods 


Degradation rate 

determined with 

4.1. Estimation of degradation rate constants 

Plume length 

determined with 

A First order kinetics First order kinetics (Eqs. (1)–(4)) First order kinetics (Eqs. (7)–(9)) 6.1 

B Michaelis–Menten 

kinetics 

First order kinetics (Eqs. (1)–(4)) First order kinetics (Eqs. (7)–(9)) 6.2 

C Michaelis–Menten 

kinetics 

Michaelis–Menten kinetics (Eq. (5)) Michaelis–Menten kinetics (Eq. (10)) 6.3 

Results in 

section 

Four different standard methods for the determination of the first order degradation rate 

constant λ (Table 2, Eqs. (1)–(4)) are compared in this study. Each of the four methods requires 

concentration-distance relations obtained by measuring contaminant concentrations in several 

observation wells along the center line of a steady state plume. The degradation rate constant λ is 

estimated by fitting a linear function to the logarithms of concentration vs. distance from the source 

by linear regression. Methods 1–4 are introduced only briefly here, more details are given in Bauer 

et al. (2006). Furthermore, a regression approach that allows the estimation of MM kinetics 

parameters from center line data is developed and tested as method 5 (Table 2, equation (5)). 

Method 1 (Table 2, equation (1)) is based on the one dimensional transport equation, considering 

advection and first order degradation only. Rate constants determined with method 1 can 

be considered rather an overall or bulk attenuation rate than a degradation rate constant (Newell et 

al., 2002) as all concentration changes that result from processes other than degradation, such as 

diffusion, dispersion, volatilization and dilution, are attributed to the degradation process. Method 

2(Table 2, equation (2); Buscheck and Alcantar, 1995) is based on the steady state solution to the 

one dimensional transport equation considering advection, longitudinal dispersion and first order 

degradation. Method 2 thus requires an estimate of longitudinal dispersivity αL [m]. Method 3 

(Table 2, equation (3)) was proposed by Zhang and Heathcote (2003) and represents a twodimensional 

modification of method 2. A correction term derived from the analytical solution of 

the two-dimensional transport equation including first order decay (Domenico, 1987) is used to 

account for lateral spreading and the width of the source zone. A similar approach, which also 

allows the inclusion of measurements off the plume center line was presented by Stenback et al. 

(2004). Method 4 (Table 2, equation (4); Wilson et al., 1994; Wiedemeier et al., 1996) is based on 

the same transport equation as method 1. However, measured concentrations of the reactive 

contaminant are scaled by up and down gradient concentration C0 ⁎ and C(x)⁎ [M L − 3 ] of a nondegrading 

conservative solute spreading from the same source. Since dispersion and measuring off 

the center line also affect the measured concentrations of non-reactive solutes, this procedure 

allows a correction for both effects. Method 5 (Table 2, equation (5)) is valid for contaminant 

plumes following Michaelis–Menten (MM) degradation kinetics (Beyer et al., 2005). When the 

77


Table 2 

Estimation of first order degradation rate constants (methods 1–4) and Michaelis–Menten degradation kinetics parameters (method 5) from center line data 

Formula Description Parameter 

estimated 

Method 

(equation) 

λ 

1D transport equation with advection and 

first order degradation; advective velocity, 

va, source concentration C0, down gradient 

concentration, C(x), first order degradation 

rate constant, λ, distance, x 

(1) k1 ¼ − va CðxÞ 

ln 

Dx C0 

! 

2 

λ 

1D transport equation with advection, 

longitudinal dispersion and first order 

degradation; longitudinal dispersivity, αL 

−1 

lnðCðxÞ=C0Þ 

Dx 

1−2aL 

(2) k2 ¼ va 

4aL 

λ 

! 

2D transport equation with advection, 

longitudinal and transverse dispersion, 

source width and first order degradation; 

transverse dispersivity, αT, source area 

width, WS 2 

−1 

lnðCðxÞ=ðC0bÞÞ Dx 

1−2aL 

k3 ¼ va 

4aL 

(3) 

4 ffiffiffiffiffiffiffiffiffiffi p 

aTDx 

C * 0 

CðxÞ * 

! 

WS 

with b ¼ erf 

λ 

Same as method 1; contaminant 

concentrations normalized with 

regard to conservative solute 

concentrations, C0⁎, C(x)⁎ 

CðxÞ 

ln 

C0 

(4) k4 ¼ − va 

Dx 

k max, M C 

1D transport equation with advection and 

Michaelis–Menten degradation kinetics; 

maximum degradation rate, kmax, 

half saturation concentration, MC 

1 

þ 

kmax 

lnðC0=CðxÞÞ C0−CðxÞ 

MC 

¼ 

kmax 

Dx 

ð Þ 

(5) 

va C0−CðxÞ

Table 3 

Calculation of contaminant plume lengths for first order and Michaelis–Menten degradation kinetics 

Method Formula Equation 

1 L1 ¼ Dx ¼ − va 

k lnðCðxÞ=C0Þ (7) 

2 

lnðCðxÞ=C0Þ 

L2 ¼ Dx ¼ 2aL 

1−ð1 þ 4kaL=vaÞ 0:5 

( " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 

) 

(8) 

3, 4 L3;4 ¼ Dxi; for CðxÞ 

5 LMM ¼ Dx ¼ va 

degradation of a contaminant is not limited by electron acceptor availability and the microbial 

density is assumed to be constant with time, Eq. (6) applies (Simkins and Alexander, 1984): 

dC 

dt 

C 

¼ −kmax 

C þ MC 


where kmax is the maximum degradation rate [M L − 3 T − 1 ] and MC is the MM half-saturation 

concentration [ML − 3 ]. This approximation may be applicable when aquifer sediments have been 

exposed to contaminants for several years (Bekins et al., 1998). The integral form of Eq. (6) can be 

rearranged to yield equation (5) of Table 2. According to Robinson (1985), equation (5) is the most 

reliable of several different formulations of the integrated MM model for estimation of kmax and 

MC. Both parameters are estimated by a linear least squares fit of Δx/[va(C0−C(x))] vs. ln(C0/C 

(x))/(C0−C(x)). Thus kmax is obtained as the reciprocal of the intercept of the linear function and 

MC as its slope multiplied by kmax. Application of method 5 assumes advective transport only (see 

also Parlange et al., 1984). As Robinson (1985) points out, application of a linearized integrated 

MM model for least squares estimation of its parameters may be problematic, because measured 

concentrations C(x) appear in the dependent as well as in the independent variable. A preliminary 

examination of several nonlinear least squares approaches for fitting the MM parameters to the 

concentration vs. distance data showed that the parameters determined by method 5 were on 

average more accurate than those obtained by other methods. 

4.2. Estimation of contaminant plume lengths 

C0 

kmax 

−exp Dxi 

2aL 

1− 

1 þ 4kaL 

va 

Given a first order degradation rate constant λ (by estimation with one of the four center line 

methods presented above), equations (1)–(3) (Table 2) can be rearranged to calculate the length of 

the steady state contaminant plume (see Table 3). The plume length here is defined as the largest 

distance between the source and the concentration isoline for concentration CPL [ML − 3 ]. Equation 

(7) gives this distance for the purely advective case of method 1, equations (8) and (9) correspond 

with methods 2 and 3, respectively. As the rate constant estimated with method 4 is corrected for 

dispersion, this process has to be accounted for when the plume length is calculated. Therefore 

equation (9) is also used for calculation of plume lengths based on λ4. A steady state plume length 

can also be calculated using the MM parameters estimated using method 5. Rearrangement of 

equation (5) of Table 2 yields equation (10) and gives the distance L MM at which concentrations fall 

erf 

WS 

4 ffiffiffiffiffiffiffiffiffiffiffi p ¼ 0 (9) 

aTDxi 

MCln C0 

CðxÞ þ C0−CðxÞ (10) 

79 

ð6Þ


below CPL. To define the plume length for this study, the 1% relative concentration contour line of 

the contaminants is used, i.e. CPL=C(x)/C0=0.01. 

4.3. Numerical Monte-Carlo simulations 

To study the influence of heterogeneous hydraulic conductivity on the investigation results 

two-dimensional virtual aquifers are used. The model domain has dimensions of 184 m length and 

64 m width (Fig. 1). A mean hydraulic gradient I of 0.053 is induced by fixed head boundary 

conditions on the left and the right hand side of the model domain. No flow boundary conditions 

are assigned to all other sides. Flow conditions are at steady state. 

The model domain is discretized with a grid density of 0.5 m in both directions. A contaminant 

source of 3 m×8 m, represented by a fixed concentration boundary condition, is centered at [11.5 m; 

32.0 m] down stream of the inflow boundary. The source emits two reactive contaminants and a 

conservative compound, each with a unit concentration of 1. The first reactive contaminant is 

degraded by first order kinetics with a rate constant λ=5.87·10 −7 s −1 . Degradation of the second 

reactive contaminant follows MM kinetics. MM parameters are taken from Bekins et al. (1998) 

(kmax=3.9·10 −9 g L −1 s −1 and MC=1.33·10 −3 g L). Using the source concentration of 2.68·10 −2 g 

L −1 given in Bekins et al. (1998) these parameters were scaled to a dimensionless source 

concentration of 1.0, as used here, yielding relative values (in normalized units) of 

kmax=1.45·10 −7 s −1 and MC=4.97·10 −2 . Thus the first order and MM plume lengths for both 

compounds are equal in a two-dimensional homogeneous aquifer for CPL=C(x)/C0=0.01. Neither 

growth of microorganisms nor limitation or inhibition of degradation by other substances is 

considered here. All compounds are not retarded and show no volatilization. The conceptual model 

used in this study is a rigorous simplification of the processes observed in natural aquifer systems, 

where degradation follows more complicated laws and is spatially dependent. The model setup is thus 

designed to provide ideal conditions for the application of the center line methods to be studied. This 

is certainly not the case in nature, where the reaction kinetics will follow more complicated laws, may 

be spatially dependent, be steered by the availability of electron donors and acceptors, or additional 

influences from transient effects and dilution have to be accounted for. However, these simplifying 

assumptions are used here to be able to study the standard methods closely and evaluate individually 

the influence of heterogeneity of the hydraulic conductivity and the influence of degradation kinetics 

on the performance of the methods under otherwise ideal conditions. Case B (Table 1) is the case were 

we study the combination of errors stemming from hydraulics and from reaction kinetics. 

The hydraulic conductivity K of the virtual aquifers is regarded as a spatial random variable, 

following a lognormal distribution with an expected value of E[Y=ln(K)]=−9.54, which corresponds 

to an effective conductivity Kef of 7.19·10 −5 ms −1 using the geometric mean. An isotropic 

exponential covariance function with an integral scale lYof 2.67 m is used for the spatial correlation 

Fig. 1. Virtual Aquifer model domain and boundary conditions.

structure. Four different cases with increasing heterogeneity, i.e. ln(K) variances σY 2 of 0.38, 1.71, 2.7 

and 4.5, respectively, are considered in this study, representing mildly to highly heterogeneous 

conductivity fields. The parameters lY and σY 2 =0.38 are taken from the Borden field site (Sudicky, 

1986); the value of 1.71 was found at the Testfeld Süd in southern Germany (Herfort, 2000); the 

values of 2.7 and 4.5 were reported for the Columbus Air Force Base site (Rehfeldt et al., 1992). A 

constant porosity n of 0.33 is used, resulting in a mean flow velocity va of 1.16·10 −5 ms −1 . For each 

of the four degrees of heterogeneity, ensembles of 100 realizations of the random field are generated 

by unconditional Gaussian simulation. The Monte-Carlo strategy is chosen in order to obtain 

statistical measures of the errors and uncertainties introduced by the heterogeneity of K. Plumes of 

the three compounds are generated in each virtual aquifer using a process based numerical flow and 

reactive transport model. The GeoSys/RockFlow simulation code (Kolditz et al., 2004)isusedhere, 

which solves the flow and transport equations by finite element methods. The governing equations 

are given as (e.g. Bear, 1972; Kolditz, 2002): 

jðKjhÞ ¼0 ð11Þ 

∂C 

∂t ¼ −vajC þ jðDjCÞ−C ð12Þ 

with h as the piezometric height, K the tensor of hydraulic conductivity, C concentration, D the 

dispersion tensor, t time and Γ a sink term representing first order or MM degradation kinetics. For 

local dispersivities αL and αT values of 0.25 m and 0.05 m were used. Details of numerical and 

software issues can be found in Kolditz (2002) and Kolditz and Bauer (2004). All model parameters 

are summarized in Table 4. 

5. Investigation of the synthetic plumes 

The steady state contaminant plumes in each virtual aquifer are investigated by the center line 

method (see Fig. 2). Initially three observation wells are present in the aquifer (Fig. 2 (a)), one being 

located directly in the source in the center of the aquifer (full circle) at [13.0 m; 32.0 m], showing 

high concentrations. This setup is the starting point for the investigation of all realizations. The initial 

knowledge on the site comprises only the hydraulic heads at the three wells. The full concentration, 

Table 4 

Model parameters used in the numerical simulations 

Parameter Value 

Kef Effective conductivity 7.19·10 − 5 − 1 

ms 

σY 2 

ln(K)-variance 0, 0.38, 1.71, 2.7, 4.5 

lY Integral scale 2.67 m 

n Porosity 0.33 

I Hydraulic gradient 0.053 

αL Longitudinal dispersivity 0.25 m 

α T Transverse dispersivity 0.05 m 

λ First order degradation rate constant 5.87·10 − 7 − 1 

s 

kmax 

Maximum degradation velocity a 

1.45·10 − 7 − 1 

s 

MC 


Half saturation concentration a 

a In normalized units, see explanation in the text. 

0.0497 

81


Fig. 2. Investigation of a virtual plume by the center line approach: (a) initial situation, (b) estimation of flow direction by 

application of a hydrogeologic triangle, (c) observation wells on the inferred center line, (d) comparison to virtual reality 

(concentration and heads). 

head and conductivity distributions of the virtual sites are assumed unknown. In the first investigation 

step, the flow direction is determined (Fig. 2 (b)). A hydrogeologic triangle is constructed 

and the hydraulic gradient is calculated using the heads measured at the three wells. After this, five 

new observation wells are installed along the estimated flow direction with distances of 15, 50, 75, 

100 and 125 m from the source (Fig. 2 (c)). At these and at the well at the source hydraulic heads and 

concentrations of the three compounds are measured. Additionally, local hydraulic conductivities are 

determined (e.g. by a slug test). The geometric mean of these six K values is used as an estimator for 

the effective conductivity Kef along the flow path. From the head difference, the true porosity and Kef 

an average va is calculated. For methods 3 and 4, estimates of dispersivities αL and αT are required. 

Following Wiedemeier et al. (1999) αL is taken as 0.1 of the plume length and αT is assumed to be 0.1 

of αL. As the true plume length is unknown at this stage of the site investigation, the maximum 

distance covered by the observation wells, i.e. 125 m, is used. Consequently, α L and α Tare estimated 

to be 12.5 and 1.25 m, respectively. Of course, such rather rough estimates of α L and α T are not 

optimal, as they are not based on the heterogeneity structure of the aquifer. In practice, however, 

dispersivities based on results from stochastic hydrogeology are difficult to obtain, as for most field 

sites structure and degree of heterogeneity are not well characterized. Also with this scenario, the

samples taken at all eight wells do not allow for an estimation of lYand σY 2 , which would be required 

to derive αL and αT. A detailed study on the effects of dispersivity parameterization on the 

performance of methods 2 and 3 is presented in Bauer et al. (2006). 

The information obtained by the site investigation allows the application of the five methods for 

estimation of the degradation kinetic parameters and the subsequent calculation of contaminant 

plume lengths as presented in Sections 4.1 and 4.2. The investigation setup is designed to resemble 

ideal conditions for this purpose. All measurements are assumed to be exact, i.e. without measurement 

error, and are obtained by reading the model output at the respective well positions. The 

only uncertainty and variability is introduced by the heterogeneity of the hydraulic conductivity. 

6. Results and discussion 


As outlined in Section 3, three different cases A, B and C are investigated here (see Table 1). The 

next three Sections 6.1, 6.2 and 6.3 present detailed results and discussions for each case studied. A 

detailed comparison and discussion of the performance of the different approaches is given in 

Section 6.4. 

6.1. Case A: estimation of first order degradation rate constants and plume lengths for plumes 

following a first order degradation kinetics 

6.1.1. Estimation of rate constants 

In case A methods 1–4 are tested and compared based on their ability to estimate the first order 

degradation rate constant λ and the contaminant plume length in heterogeneous aquifers. 

Therefore, plumes following first order degradation kinetics are investigated here. The four 

estimated rate constants λi are divided by the true rate constant to yield corresponding normalized 

Λi, which can be interpreted as an over- or underestimation factor. Fig. 3 presents ensemble means 

with corresponding standard deviations as error bars, medians, coefficients of variation and the 

single realization results for methods 1–4 against the aquifer heterogeneity σY 2 . 

Results for method 1 (Fig. 3 (a)) show that Λ1=1 in the homogeneous case (σY 2 =0), thus in this case 

the true rate is obtained. However, already for the lowest degree of heterogeneity (σY 2 =0.38), about 

77% of the rate constants estimated fall above the reference line which indicates the true rate constant, 

overestimating λ up to factors of 4.79 in the worst case. In the remaining realizations λ is 

underestimated. The spread of the ensemble of 100 realizations covers about one order of magnitude. 

When σY 2 is further increased, spread and standard deviation also increase. For σY 2 =4.5 the Λ1 cover 

almost two orders of magnitude with single realizations showing overestimation factors N10. Mean Λ1 

is 3.3 with standard deviation and coefficient of variation of 3.65 and 1.11, respectively. Increasing the 

heterogeneity of the aquifer thus results in a higher uncertainty of λ and increases the probability of a 

significant overestimation. For method 2 normalized rate constants Λ2 are shown in Fig. 3 (b). As seen 

for method 1 a systematic overestimation of λ can be observed, increasing with σ Y 2 . However, the Λ2 

are significantly larger than the corresponding Λ 1. In the homogeneous case method 2 yields Λ 2=1.7, 

increasing to 9.38 for σY 2 =4.50. In extreme cases, overestimation of λ is larger than a factor of 50. Also 

spread and standard deviations are larger than for Λ1. Estimated rate constants Λ3 obtained by method 

3(Fig. 3 (c)) are only slightly lower than those of method 2, ranging from 1.17 in the homogeneous 

case to 7.39 for σY 2 =4.5. Method 4 (Fig. 3(d)) yields the true rate for homogeneous conditions, i.e. 

Λ4=1. As for the other methods, spread and uncertainty increase with σY 2 . Compared to methods 1–3 

the spread is smaller and balanced around the true rate constant. For σY 2 ≤2.7 the ensemble averages 

and medians match the true rate constant well. For σ Y 2 =4.50 the average Λ4 reaches 1.73. 

83


The overestimation observed for the different Λi results from a combination of several effects. 

Method 1, which is the simplest approach used in this study, is based on the one dimensional transport 

equation only accounting for advection and degradation. Concentration reductions on the plume 

center line caused by transverse dispersion therefore are incorrectly attributed to the degradation 

process and Λ1 is overestimated. Moreover, when the inferred center line deviates from the true center 

line, concentrations measured are too low, which further increases rate overestimation. A third source 

of error is the estimate of v a (see Section 5), which may not be representative for the flow path. 

Overestimation is caused if v a is estimated too high, while a too low value of v a causes underestimation 

of the rate constant. All three error types are also relevant for method 2. The additional bias of method 

2 towards too large rate constants in comparison with method 1 is a consequence of accounting for αL 

in equation (2) of Table 2. Longitudinal dispersion of a degrading contaminant results in a stronger 

spreading of the solute down stream and consequently in higher concentrations along the center line of 

a steady state plume. Equation (2) can be rearranged to show that λ2=λ1+αLva(ln(C(x)/C0)/Δx) 2 , i.e. 

λ2 grows linearly with αL and is always larger than λ1 for αLN0.Method3isaffectedbyerrorsinva 

and off center line measurements. By accounting for transverse dispersion, rate constant estimates are 

improved compared to method 2, because βb1.0 in equation (4) and thus Λ 3bΛ 2 always. Because β 

approaches unity for arguments N2, Λ3 converges with Λ2 for small αT, short transport distances Δx 

Fig. 3. Estimated first order degradation rate constants Λi (normalized to the true rate constant λ, indicated by the 

horizontal line) vs. heterogeneity of the aquifer σ Y 2 for methods 1 (a), 2 (b), 3 (c) and 4 (d) for case A.


and large source widths WS. A surprising result is that method 1, despite its simplicity, yields closer 

estimates of the true rate constant than the more comprehensive description by method 3. Since 

method 3 depends on longitudinal and transverse dispersivities, an adequate parameterization is 

crucial for its success. From stochastic hydrogeology it is known that αL as well as αT strongly 

depends on travel time and distance as well as on the correlation structure of hydraulic conductivity 

and flow velocity (e.g. Dagan, 1989). Consequently, a uniform parameterization solely based on the 

scale of the contaminant problem as used in this study (and with many field applications) is not 

adequate. A detailed sensitivity study on the influence of dispersivity parameterization on the 

performance of methods 2 and 3 is presented in Bauer et al. (2006). It is found that for method 2 no 

value of αL and for method 3 only very high and thus unphysical values of αT yield the correct 

degradation rate constant. The required values, however, cannot be deduced from aquifer heterogeneity 

σY 2 alone, as the other errors also influence the estimated degradation rate constant. 

Method 4 circumvents this problem and corrects for transverse dispersion as well as for measuring 

off the center line by normalizing concentrations to a conservative tracer. The bias towards too large 

degradation rate constants observed for the other methods is significantly reduced yielding the closest 

estimates of λ of the four methods. The remaining deviation from the true rate constant is due to the 

hydraulic error introduced by the approximation of va. For low heterogeneities there is no evidence for 

a systematic bias towards either too high or too low rate constants. A prerequisite which may limit the 

applicability of method 4 is the presence of a suited normalization compound. A discussion of potential 

normalization compounds is provided in U.S. EPA (1998) and Wiedemeier et al. (1996, 1999). 

6.1.2. Estimation of plume lengths 

During site characterization, estimating degradation rate constants rarely is a goal per se. Here, the 

kinetics of contaminant degradation are quantified to be used for prediction of the steady state length 

of the plumes. In site assessment, such information could be used to identify potential receptors and 

exposure levels. Rate constants λ1, λ2 and λ3 are evaluated using the respective corresponding 

equations (7), (8) and (9) of Table 3 yielding plume length estimates L1, L2 and L3. Forλ4 also 

equation (9) is used yielding L4. To be able to compare the results of all realizations, the Li are 

normalized by the respective true length L read from the model output. Resulting over- respectively 

underestimation factors against aquifer heterogeneity σY 2 are presented in Fig. 4 (single realization 

results, ensemble means with standard deviations as error bars, medians, coefficients of variation). 

Plume lengths L 1 and L 2, calculated from λ 1 and λ 2, show exactly identical results, although 

the λ2 show a stronger overestimation than the corresponding λ1 for all realizations. The reason 

for the equivalence of L1 and L2 is that the bias introduced by estimating λ2 with a one 

dimensional model accounting for longitudinal dispersion only is reversed by using the same 

transport equation to calculate the plume length. While for homogeneous conditions the true 

plume length is obtained, L is underestimated in most realizations for all degrees of heterogeneity. 

Mean L1 and L2 decrease to 0.59 for σY 2 =4.5. As for λ, spread and uncertainty of L increase with 

σ Y 2 . The overestimation of the degradation rate constant is thus reflected in an underestimation of 

the plume length. 

Plume lengths L3 calculated with the two-dimensional transport equation on average are lower 

than the corresponding L1 and L2. This is a consequence of the β term in equation (5) (Table 2), 

which is used to correct down gradient concentrations for transverse dispersion. When estimating 

the rate constant with method 3, each down gradient concentration C(x) is scaled by a different 

value β, as the correction factor is dependent on the distance from the source. This scaling is not 

fully reversed when the plume length is calculated using equation (9) (Table 3), as then only one 

single Δx is used. 

85


As shown before, degradation rate constants λ4 in general are significantly better estimated 

than those obtained by the other three approaches. This is also reflected in plume lengths L4 (Fig. 

4 (c)). For all σY 2 the ensemble means almost perfectly match the true L. Of all four approaches the 

L4 consistently show the lowest coefficients of variation, indicating the lowest spread and thus the 

lowest uncertainty. 

The plume length results presented for case A demonstrate that the determination of 

degradation rate constants using the center line method as well as the subsequent calculation of 

contaminant plume lengths both are subject to uncertainty induced by the heterogeneity of the 

medium. However, the uncertainty observed for the rate constants is only partially transferred to 

the calculated plume lengths. For all approaches the observed spread of the ensembles of estimated 

plume lengths is smaller than for the corresponding rate constants. This is most obvious for the L4, 

which also show the best agreement with the true plume lengths. Moreover, the L4 are unbiased, 

showing equal amounts of under- as well as overestimation. However, both types of error are 

undesirable. Underestimation of the contaminant plume length, i.e. a “non-conservative” result, 

may pose a threat to down gradient receptors. On the other hand, a “conservative” result, i.e. an 

overestimation of plume dimensions, might result in wrong decisions regarding the necessity and 

dimensioning of engineered remediation measures with unnecessary financial expenses. In 

Fig. 4. Plume lengths Li calculated with degradation rate constants λ1 through λ4. The Li are normalized by the true plume 

length L (indicated by the horizontal line) for case A.


contrast to L4, plume lengths L1, L2 and L3 show a clear tendency of underestimation, demonstrating 

the non-conservativeness of too large rate constants. Comparing methods 1, 2 and 3, 

differences in calculated plume lengths are not as pronounced as for the rate constants. Thus it can 

be concluded, that the same underlying equation should be used for rate constant and plume length 

estimation. However, if e.g. a rate constant estimated with method 2 is used in a two- or threedimensional 

(numerical) transport model, results are susceptible to the bias in method 2, resulting 

in a significantly stronger underestimation of the plume length. 

6.2. Case B: estimation of first order degradation rate constants and plume lengths for plumes 

following Michaelis–Menten degradation kinetics 

In case B the additional error is studied, that arises when the four methods for the estimation of 

first order degradation rate constants (Table 2, equations (1)–(4)) and the respective equations for 

plume lengths (Table 3, equations (7)–(9)) are used for plumes with a degradation kinetics 

deviating from first order but following MM degradation kinetics instead (see Table 1). 

Since a direct comparison of estimated rate constants λ 1 through λ 4 with the MM parameters k max 

and MC used in the numerical simulations is not possible, the evaluation here is based on calculated 

Fig. 5. Normalized plume lengths Li calculated with degradation rate constants λ1 through λ4 for the contaminant plumes 

following Michaelis–Menten degradation kinetics. 

87


plume lengths only. The estimated contaminant plume lengths Li are displayed in Fig. 5 (single 

realization results, ensemble means with standard deviations as error bars, medians, coefficients of 

variation). As for case A, L1 and L2 (Fig. 5 (a)) yield very similar results. However, in comparison to 

case A, underestimation is clearly increased. This is most obvious for homogeneous conditions, where 

L1 and L2 are only about 50% of the true length and thus no longer yield the correct result. 

Underestimation of L increases with heterogeneity, yielding mean values of L1 and L2 of 0.45 for 

σ Y 2 =4.5. Spread is about 0.75 orders of magnitude for low heterogeneity and one order of magnitude 

for high heterogeneities, which is similar to case A. Plume lengths L 3 (Fig. 5 (b)) show the same 

general behaviour as the L1 and L2, with mean values being slightly lower. In contrast to this, L is 

overestimated for most realizations by L4 (Fig. 5 (c)). While for homogeneous conditions a too short 

L4 of 0.65 is obtained, L4 increases to values larger than one for higher degrees of heterogeneity. 

Spread of single realizations is significantly increased in comparison to case A (compare Figs. 4 (c) 

and 5(c)), as a spread of about one order of magnitude can be observed for all degrees of heterogeneity. 

The additional error introduced by using methods 1 through 4 for plumes following Michaelis– 

Menten kinetics degradation is most pronounced for low heterogeneities. Compared to case A, 

plume lengths are underestimated to a larger extent, as can be seen by the lower mean values and 

Fig. 6. Normalized Michaelis–Menten kinetics parameters kmax vs. MC estimated for σY 2 =0.38 (a), 1.71 (b), 2.7 (c) and 4.5 

(d), respectively.


medians. Only for L4 plume length overestimation is observed and uncertainty is increased in 

comparison to case A. 

6.3. Case C: estimation of Michaelis–Menten kinetics parameters and plume lengths for plumes 

following Michaelis–Menten degradation kinetics 

6.3.1. Estimation of rate parameters 

As demonstrated in case B, using a first order kinetics approximation for plumes following MM 

degradation kinetics introduces an additional error, yielding less conservative estimates of plume 

lengths (L1−L3) as well as higher uncertainty (L4). Therefore, method 5 for the estimation of the 

MM parameters is tested by application to the contaminant plumes following MM degradation 

kinetics (see Table 1). Using the same concentration vs. distance data as in Section 6.2, MM 

parameters kmax and MC are estimated by a linear fit as explained in Section 4.1. These parameters 

then are used to calculate the length of the contaminant plumes LMM by equation (10) (Table 3). 

Fig. 6 presents the results of the parameter estimation in single diagrams for each σ Y 2 (results 

for single realizations, ensemble means with corresponding standard deviations as error bars, 

medians). Since for each realization the maximum degradation rate kmax and half-saturation 

concentration MC are estimated simultaneously, the parameters are shown in scatter plots. For the 

homogeneous case (not shown), kmax is slightly overestimated with a value of 1.06, while for MC 

an underestimation is observed with MC=0.91. This error results from neglecting the dispersion 

process. Fig. 6 shows that normalized kmax and MC are increasingly overestimated with increasing 

σY 2 . Also uncertainty increases with σY 2 . This is a similar behaviour as found for the first order rate 

constants in case A. Approximate orientation of the data points along a diagonal axis with positive 

slope gives evidence of a weak positive correlation between k max and M C. An overestimation of 

kmax increases the degradation rate as long as concentrations are higher than MC. However, 

overestimation of MC raises the concentration threshold at which kinetic degradation transits from 

zeroth to slower first order, which counter-balances the effects of the kmax overestimation. 

Fig. 7. Normalized plume lengths LMM calculated with Eq. (10) based on estimated Michaelis–Menten kinetics parameters 

k max and M C. 

89


6.3.2. Estimation of plume lengths 

For the determination of contaminant plume lengths, the estimated parameters kmax and MC are 

evaluated with equation (10) (Table 3). Normalized plume lengths LMM are presented in Fig. 7 

(single realization results, ensemble means with corresponding standard deviations as error bars, 

medians, coefficients of variation). For the homogeneous case the estimated and true plume 

lengths agree well with LMM=0.97. Also for σY 2 =0.38 the mean yields the correct result with 

L MM=1.0 and spread is about 0.5 orders of magnitude. For higher heterogeneities, however, 

plume lengths again tend to be underestimated with L MM decreasing to 0.80 for σ Y 2 =4.5. Spread 

for all degrees of heterogeneity is smaller than one order of magnitude. When the spread of 

estimated kmax and MC values is compared to the spread of calculated MM plume lengths, it is 

found – as for case A – that the uncertainty in MM parameters is not fully propagated to the 

plume lengths. Comparing MM plume lengths with those determined for the plumes subject to 

first order degradation in case A (see Fig. 4), the LMM outperform L1, L2 or L3 with regard to 

accuracy as well as uncertainty. Only the L4 are closer to the true lengths on average, while the 

spread of single realizations is comparable. It is thus found in this study that the proposed method 

of estimating MM kinetics parameters is adequate and performs better than the methods for 

estimating first order rate constants. 

6.4. Comparison and discussion of cases B and C 

In this section the plume lengths LMM calculated in case C using the MM model (Table 3, 

equation (10)) are compared to the L1 −L4 of case B, where plume lengths were calculated using 

the first order kinetics approximations of equations (7) through (9) of Table 3. Table 5 compares 

the accuracy of calculated plume lengths using MM parameters estimated by method 5 with the 

four different first order approaches. Given are the percentages of realizations for which LMM 

constitutes a closer estimate of L than L1, L2, L3, orL4. In comparison to L1, L2 and L3 it is found 

that LMM is the closer estimate for 93 up to 99% of all realizations, depending on the degree of 

heterogeneity and the first order method used. In comparison with L4 LMM constitutes the closer 

estimate still for 53 up to 72% of all realizations. 

As a more quantitative criterion, plume length error factors EF are calculated for each 

realization and all five estimation methods: 

EF ¼ L * 

L 

a 

with 

a ¼ −1jL * bL 

a ¼þ1jL * zL 

where L ⁎ and L are the estimated and true plume lengths, respectively, and the exponent a is used to 

obtain a standardized measure for over- as well as for underestimation. Hence, EF gives the degree of 

accuracy of the plume length estimate L ⁎. For each ensemble of Li and LMM cumulative empirical 

Table 5 

Percentage of realizations for which LMM is a closer estimate of L than L1−L4 

LMM vs. L1, L2 % LMM vs. L3 % LMM vs. L4 % 

0.38 97 99 72 

1.71 99 99 57 

2.7 98 99 59 

4.5 94 93 53 

σ Y 2 

ð13Þ


Fig. 8. Cumulative empirical distribution functions of plume length error factors for estimated plume lengths L1 through L4 

and LMM and aquifer heterogeneities σY 2 of 0.38 (a), 1.71 (b), 2.7 (c) and 4.5 (d). The diagrams yield the probability of 

obtaining an estimate of the plume length L with an accuracy (EF) as given on the abscissa. 

distribution functions (edf) of the EF were calculated. These are presented in Fig. 8 (a) through (d) for 

each degree of heterogeneity. The edf give the probability of obtaining an estimate of the plume 

length with an EF less than or equal to the associated quantile on the abscissa. In Fig. 8 (a) with 

σY 2 =0.38, for example, the probability of estimating L by LMM, given an accuracy of factor 2, i.e. 

allowing a maximum underestimation by 50% or overestimation by 100%, is about 0.98. In contrast 

to this, the probability of obtaining L1 (=L2)orL3 as accurate as a factor of 2 is only 0.68 and 0.55, 

respectively, while for L4 the probability is approximately 0.89. When aquifer heterogeneity 

increases, the probabilities are reduced, as can be seen by the shift towards higher EF and the 

flattening of the slopes (note the logarithmic scale of the abscissa). Moreover, differences between 

LMM and L4 diminish. Thus for σY 2 =4.5 (Fig. 8 (d)) edf for both approaches almost coincide and 

show the same degree of accuracy. However, LMM still yields significantly higher probabilities for a 

given EF than L1, L2 or L3. 

The results obtained in this comparison clearly support that for plumes following MM degradation 

kinetics usage of the MM parameter estimation approach allows a distinct improvement of 

plume length estimates over those obtained using a first order approximation, especially when 

aquifer heterogeneity is not too high. For the majority of realizations investigated in this study, L MM 

91


constitutes a more accurate estimate of the true plume length than any of the first order methods used. 

Only when aquifer heterogeneity is very high, L4 is able to yield plume length estimates of almost 

comparable accuracy. This is mainly, because increased transverse dispersion produces concentration 

profiles that allow a log-linear fit of a first order rate constant. 

7. Summary and conclusions 

In this study the Virtual Aquifer concept is used to assess the uncertainty involved in estimating 

down stream contaminant concentrations and plume lengths in heterogeneous aquifers. For such 

an analysis a key element is the quantification of the degradation rate at the site under study. 

Therefore, the main focus of this work is on the influence of this parameter on the estimated plume 

lengths. Three different scenarios are analysed. 

In case A, four different standard field methods based on the center line investigation strategy 

are tested and compared with regard to their capability of estimating first order degradation rate 

constants and the contaminant plume length in heterogeneous aquifers. The four methods are 

applied to plumes following first order degradation kinetics. It is found that both, the estimated 

degradation rate constants and the calculated plume lengths, are subject to high uncertainty. On 

average, estimated rate constants exceed the true degradation rate constant, causing calculated 

plume lengths that are too short. Both bias and uncertainty of estimated degradation rate constants 

increase with the degree of heterogeneity to about a factor in the order of a magnitude, respectively. 

However, the uncertainty observed in the rate constants does not fully propagate to the plume length 

estimates. On average plume lengths are underestimated by about 50% of the true plume length, and 

up to a factor of ten in the worst cases. Of the four different methods, the approach of Wiedemeier et 

al. (1996) using concentrations normalized to a conservative tracer yields the best results for the rate 

constants and for the plume lengths. Consequently, this method should preferably be used. 

However, the presence of a suitable recalcitrant compound may not always be given. In this case, 

the most simple of the four approaches, which is based on the advection equation and neglects 

dispersive processes should be used to determine the degradation potential, as this method yields 

closer estimates of the first order rate constant than the approaches using the one- (Buscheck and 

Alcantar, 1995) and two-dimensional advection dispersion equations (Zhang and Heathcote, 2003). 

The non-conservative plume length estimates might cause threats to down stream receptors, as the 

risk of contamination could be underrated. The high uncertainty when estimating plume dimensions 

could result in incorrect decisions regarding the necessity and dimensioning of engineered remediation 

measures or when considering the applicability of natural attenuation. All four methods are only 

applicable to steady state plumes. In reality, however, contaminant plumes often show temporal 

variations in extent and orientation as the plume expands, the source slowly depletes, or the flow 

regime changes over time. For expanding plumes, application of methods 1–4 wouldresultinan 

overestimation of the rate constant, and thus in underestimation of the plume length at steady state. This 

is because the down stream transient contaminant concentrations are lower than those at steady state, 

causing a higher concentration decrease along the center line, which would be falsely attributed to the 

degradation process. For the same reason rate constants would also be overestimated for shrinking 

plumes. An examination of the current state of the plume is necessary, when one of the four methods is 

to be applied at a site. An overview of techniques for this purpose is given by Newell et al. (2002). 

Case B analyses the additional error that results from application of the four methods, although 

the true degradation kinetics deviate from first order. Here plumes following Michaelis–Menten 

degradation kinetics are investigated. Results show that for the three approaches without 

correction of concentrations to a conservative tracer, an additional underestimation occurs which

is largest for low heterogeneities. For the approach using the tracer correction, the estimated 

plume lengths still match the true lengths on average. However, the uncertainty of the plume 

lengths is significantly increased as compared to case A. 

In case C, a regression approach is introduced to estimate the parameters of the MM kinetics and to 

calculate the plume lengths for the same plumes as investigated in case B. Since longitudinal and 

transverse dispersion are neglected in this method, the MM parameters are overestimated on average. 

Overestimation increases from less than a factor of two for low degrees of heterogeneity to a factor of 

roughly four for high heterogeneity. Consequently an underestimation of the corresponding plume 

lengths is observed for most realizations. For low heterogeneity, the plume length is estimated 

precisely with low uncertainty, for high heterogeneity the average estimated plume length is about 

80% of the true plume length, with estimates as low as 40–35% in the worst cases. However, in 

comparison with the first order approximation investigated in case B the error resulting from this 

approach is significantly reduced, with the uncertainty of the plume length estimates being reduced by 

a factor of three. Therefore, if field data collected along the center line of a plume gives evidence of 

MM kinetics (linear behaviour of concentrations vs. distance in a linear plot, concave down behaviour 

in a semi-logarithmic plot (Bekins et al., 1998)), this approach is recommended. 

Generally, the Virtual Aquifer concept has proven useful in assessing the performance of 

methods for investigating first order rate constants and plume lengths from plume center line 

measurements. The main advantage is, that individual factors, as here the hydraulic heterogeneity 

and the assumption of a wrong plume kinetics, can be studied either individually in detail or in 

combination and under otherwise ideal conditions. However, the restrictions from the simplified 

setup of the model scenario have to be kept in mind when drawing conclusions for field 

applications. In reality, contaminant degradation follows more complicated rate laws, depending 

e.g. on electron acceptor and donor availability, may include transient effects, or dilution and 

phase changes to the unsaturated zone. Further studies will thus incorporate more realistic 

degradation kinetics as well as influences from e.g. measurement errors. 

Acknowledgements 

This work is funded by the German Ministry of Education and Research (BMBF) under grant 

033 05 12/033 05 13 as part of the KORA priority program, sub-project 7.2. We wish to thank 

Robert Walsh for his helpful comments on the manuscript. We also wish to thank our project 

partners at the Christian-Albrechts-University Kiel Andreas Dahmke and Dirk Schäfer for their 

support in our research. We acknowledge Uwe Wittmann, Iris Bernhardt and Ludwig Luckner for 

coordination of the project work. Furthermore we would like to acknowledge the thoughtful 

reviews of the anonymous reviewers and the Editor. Their comments have greatly improved the 

manuscript. 

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degradation rates by us1ing the virtual aquifer approach.), Grundwasser, 12, 3–14, doi: 

10.1007/s00767-007-0019-8 

The enclosed article is made available with the permission of Springer and was published in 

the journal Grundwasser, 12, 3-14 (2007). Copyright © 2007 Springer. 

The article can be obtained online via SpringerLink at 

http://www.springerlink.com/openurl.asp?genre=journal&eissn=1432-1165.

Einfluss von Heterogenität und Messfehler auf die 

Bestimmung von Abbauraten erster Ordnung – eine 

Virtueller Aquifer Szenarioanalyse ∗ 

Influence of heterogeneity and measurement error on the determination 

of first order degradation rates - a virtual aquifer scenario analysis 

Sebastian Bauer, Christof Beyer, Olaf Kolditz 

Zentrum für Angewandte Geowissenschaften, Universität Tübingen, Sigwartstraße 10, 

D 72076 Tübingen. Tel: 07071-2973171. Fax: 07071 5059 

e-mail: sebastian.bauer@uni-tuebingen.de, christof.beyer@uni-tuebingen.de, kolditz@uni-tuebingen.de 

Header: Heterogenität und Messfehler bei Abbauratenbestimmung 

Kurzfassung 

Die grundlegende Idee des Virtuellen Aquifers ist, durch numerische Modellierung von typischen 

Schadensfällen Erkundungsstrategien zu simulieren und zu bewerten. In diesem Beitrag wird die 

Bestimmung von Abbauraten erster Ordnung untersucht. Eine Schadensquelle wird dabei in einen 

virtuellen Aquifer eingebracht und die stationäre Schadstofffahne unter der Annahme einer 

Abbaukinetik erster Ordnung simuliert. Diese Fahne wird dann durch Beobachtungspegel entlang der 

Zentrallinie der Fahne untersucht. Anhand von vier typischen Methoden werden Abbauraten erster 

Ordnung berechnet und mit dem vorgegebenen Wert verglichen. Dieser Vergleich wird für 

unterschiedlich stark ausgeprägte hydraulische Heterogenitäten durchgeführt. Dabei zeigt sich, dass 

mit zunehmender Heterogenität die ermittelten Abbauraten die tatsächliche Abbaurate um Größenordnungen 

überschätzen können und sie somit sehr unsicher sind. Bei der Untersuchung der 

Messfehler wurden Abweichungen bei der Bestimmung der Piezometerhöhe und der Konzentration 

angenommen. Hierbei ergibt sich, dass Messfehler ebenfalls zu einer hohen Unsicherheit der 

Ratenkonstante führen können, wobei Messfehler der Piezometerhöhe einen stärkeren Einfluss 

haben. 

Abstract 

The principal idea of the Virtual Aquifer is to simulate and evaluate monitoring strategies and 

remediation options for contaminated sites by modelling of typical contamination scenarios. Here the 

determination of first order degradation rates is studied. A virtual reality is generated by simulating the 

spreading of a plume, originating from a defined source and subject to first order degradation. This 

plume is investigated using monitoring wells placed along the plume center line. From the information 

thus obtained first order degradation rates are calculated by methods typically used and are then 

compared to the predefined value. This comparison is conducted for different degrees of 

heterogeneity. It is found that with increasing heterogeneity degradation rates overestimate the real 

degradation rate by up to orders of magnitude and show a high uncertainty. Then measurement errors 

are introduced for piezometric head and concentration measurements. It is found that deviations of the 

estimated first order rate constant from the true one of up to orders of magnitude can occur, with 

errors of the piezometric head measurement causing the dominant uncertainty. 

Keywords: natural attenuation, heterogeneity, first-order degradation rate, virtual aquifer, scenario 

analysis 

∗ Bauer, S., Beyer, C., Kolditz, O. (2007): Einfluss von Heterogenität und Messfehler auf die Bestimmung von Abbauraten erster 

Ordnung - eine Virtueller Aquifer Szenarioanalyse. (Influence of heterogeneity and measurement error on the determination of 

first order degradation rates by us1ing the virtual aquifer approach.), Grundwasser, 12, 3–14, doi: 10.1007/s00767-007-0019-8. 

Der Artikel wurde in der Zeitschrift Grundwasser publiziert und mit Erlaubnis von Springer reproduziert. Copyright © 2007 Springer. 

Der Artikel ist online abrufbar via SpringerLink: http://www.springerlink.com/openurl.asp?genre=journal&eissn=1432-1165. 

1

Einführung 

In dieser Arbeit wird eine „Virtueller Aquifer“ Szenarioanalyse verwendet, um die Unsicherheit bei der 

Bestimmung von Abbauratenkonstanten erster Ordnung anhand von Messstellen auf der Zentrallinie 

der Schadstofffahne („Center line method“) im Kontext von Natural Attenuation zu untersuchen und zu 

quantifizieren. Natural Attenuation (NA), auch als Bioremediation bekannt, bezieht sich auf die Abnahme 

von Schadstoffkonzentrationen durch natürliche Abbauprozesse mit zunehmendem Abstand von 

der Quelle (US-EPA, 1999; WIEDEMEIER et al., 1999). Dabei werden Dispersion, Verdünnung, 

Sorption, Ausgasung und Bioabbau betrachtet, wobei der Bioabbau der einzige Prozess ist, der zu 

einer Verringerung der Schadstoffmasse führt. Die an einem Standort ablaufenden Prozesse müssen 

sorgfältig charakterisiert werden, um Aussagen über NA treffen zu können. Hierbei können 

insbesondere die Abbauraten der betrachteten Schadstoffe für mögliche Sanierungen und das 

Standortmanagement eine Rolle spielen. Die Abbauraten werden verwendet, um das gesamte NA- 

Potential am Standort zu charakterisieren, um die Länge von Schadstofffahnen in der Zukunft zu 

prognostizieren und um unterstromige Konzentrationen zu berechnen, die für eine Auswirkungsprognose 

benötigt werden (WIEDEMEIER et al., 1999). 

Zur Bestimmung von Abbauraten im Feld stehen derzeit mehrere Methoden zur Verfügung, wie z.B. 

Massenbilanzen, in-situ Mikrokosmosstudien oder die Verwendung von Konzentrations-Abstands- 

Beziehungen, die auf der Fahnenzentrallinie einer stationären Schadstofffahne ermittelt wurden 

(CHAPELLE, 1996). Für letztere sind in der Literatur vier unterschiedliche Methoden beschrieben. Die 

erste beruht auf der eindimensionalen Transportgleichung mit Advektion und Abbau erster Ordnung 

(WIEDEMEIER et al., 1996). Die zweite Methode ist eine Erweiterung der ersten, indem die 

Konzentrationen des betrachteten Schadstoffes auf die Konzentrationen eines nichtreaktiven Mitkontaminanden 

normiert werden (WIEDEMEIER et al., 1996, 1999; WILSON et. al., 1994). Die dritte 

Methode, von BUSCHECK & ALCANTAR (1995) vorgeschlagen, beruht auf der eindimensionalen 

Transportgleichung mit Advektion, Dispersion und Abbau erster Ordnung und wurde bereits an einigen 

Standorten eingesetzt (CHAPELLE et al., 1996; WIEDEMEIER et al., 1996; ZAMFIRESCU & GRATHWOHL, 

2001; SUAREZ & RIFAI, 2002; BOCKELMANN et al., 2003). In letzter Zeit wurden als Erweiterung zur 

Methode von BUSCHECK & ALCANTAR (1995) zwei- und dreidimensionale Methoden entwickelt (ZHANG 

& HEATHCOTE, 2003; STENBACK et al., 2004). ZHANG & HEATHCOTE (2003) konnten zeigen, dass die 

eindimensionale Methode durch Vernachlässigung der Querdispersion die Abbauraten um 21% im 

Vergleich zum zweidimensionalen und um 65% im Vergleich zum dreidimensionalen Fall überschätzt. 

MCNAB Jr. & DOOHER (1998) beschrieben, wie transversale Dispersion und instationäre Strömungszustände 

Konzentrationsverteilungen erzeugen können, die durch die Methode von BUSCHECK & 

ALCANTAR (1995) dann fälschlich als Bioabbau klassifiziert werden können, obwohl am Standort kein 

Abbau stattfindet. 

Aufgrund der eingeschränkten Zugänglichkeit des Untergrundes sind Beobachtungen an 

kontaminierten Standorten nur an einzelnen räumlichen Punkten möglich. Die aus einzelnen 

Messpunkten abgeleiteten Ergebnisse und Aussagen sind daher immer mit Unsicherheit behaftet, die 

das eingeschränkte und punktuelle Wissen über den Standort widerspiegelt. Eine „Virtuelle Aquifer“ 

Szenarioanalyse kann verwendet werden, um diese Unsicherheit näher zu betrachten und zu quantifizieren 

(SCHÄFER et al., 2002; SCHÄFER et al., 2004; BAUER et al., 2006; BEYER et al., 2005a; BAUER & 

KOLDITZ, 2005). Eine Untersuchung des Einflusses der Parametrisierung der Abbaukinetik auf die 

prognostizierte Fahnenlänge wird in SCHÄFER et al. (2005) durchgeführt. Dabei werden numerische 

synthetische Modelle typischer Aquifere generiert. Mithilfe eines reaktiven Transportmodells können 

dann typische Schadensszenarien, wie beispielsweise die Ausbreitung einer Schadstofffahne ausgehend 

von einer Schadstoffquelle, simuliert werden. Der große Vorteil der Virtuellen Aquifere ist, dass 

die so erhaltene realistische räumliche Verteilung aller Parameter, wie z.B. Piezometerhöhe oder 

Konzentration, exakt bekannt ist. Die virtuelle Schadstoffahne wird dann in einem zweiten Schritt 

durch typische Erkundungsstrategien (hier: Zentrallinienmethode) untersucht. Eine Messung im 

virtuellen Aquifer bedeutet, die entsprechenden Werte aus der Modellausgabedatei zu lesen. Bei 

dieser Erkundung der virtuellen Schadstofffahne werden nur die Messwerte (hier: Piezometerhöhe, 

Konzentration) berücksichtigt, die auch an einem echten Standort erhalten werden können. 

Ausgehend von diesen (virtuellen) Messwerten werden dann weitere Parameter ermittelt (hier: 

hydraulische Durchlässigkeiten, Abbauraten). Bei diesem zweiten Schritt ist also die richtige 

Parameterverteilung nicht bekannt. Indem das Ergebnis der virtuellen Erkundung im dritten Schritt mit 

der wahren Parameterverteilung (hier: Abbauratenkoeffizient) verglichen wird, können die 

verwendeten Untersuchungsmethoden getestet und bewertet werden. 

Die räumliche Heterogenität von Aquiferparametern hat einen erheblichen Einfluss auf die 

Fahnenentwicklung und die resultierende Konzentrationsverteilung eines Schadstoffes im Aquifer. Die 

2

oben erwähnten Methoden zur Bestimmung von Abbauratenkonstanten erster Ordnung unterliegen 

daher dem Einfluss der Heterogenität, da sie auf Messdaten entlang der vermuteten Fahnenzentrallinie 

und auf Abschätzungen der Dispersivität beruhen. Wird die Grundwasserfließrichtung 

falsch abgeschätzt, kann die Fahnenzentrallinie leicht verfehlt werden (WILSON et al., 2004). Bisher 

existiert in der Literatur keine Studie, die den Einfluss der Heterogenität auf die Ermittlung von 

Abbauratenkonstanten erster Ordnung untersucht. Bei der Probenahme im realen Fall können darüber 

hinaus noch Messfehler auftreten, die ebenfalls die Ratenkonstanten beeinflussen. Diese können 

sowohl bei der Pegeleinmessung, bei der Bestimmung der hydraulischen Leitfähigkeit, bei Wasserstandsmessungen 

und insbesondere bei Konzentrationsmessungen auftreten. Auch hierzu existieren 

noch keine Untersuchungen in der Literatur. Ziel dieser Arbeit ist daher a) die Genauigkeit von 

Abbauraten, wie sie aus Feldmessungen gewonnen werden, zu ermitteln, und b) die verschiedenen 

Methoden zu Bestimmung von Abbauraten zu bewerten. Dazu wird der Einfluss von hydraulischer 

Heterogenität und Messfehlern anhand einer „Virtueller Aquifer“ Szenarioanalyse untersucht. Hierzu 

werden in einem synthetischen Aquifer mit unterschiedlich stark ausgeprägter Heterogenität virtuelle 

stationäre Fahnen simuliert, die einer Abbaukinetik erster Ordnung unterliegen. Die anhand der oben 

genannten vier Methoden bestimmten Abbauratenkonstanten erster Ordnung werden mit dem wahren 

Wert verglichen und so Aussagen über die Zuverlässigkeit und Unsicherheit der Methoden und der 

resultierenden Ratenkonstanten abgeleitet. Bei der Untersuchung des Einflusses des Messfehlers auf 

die Ratenkonstante wird anschließend von einem homogenen Aquifer ausgegangen. Der Einfluss von 

Messfehlern bei der Bestimmung der Piezometerhöhe und bei der Konzentrationsmessung wird 

separat untersucht und quantifiziert. 

Methodik 

Das verwendete Modellgebiet ist zweidimensional und misst 184 m auf 64 m (vergl. Abb. 1). Das 

Grundwasser strömt von links nach rechts, der mittlere hydraulische Gradient beträgt 0.003 und wird 

durch Festpotentialrandbedingungen am linken und rechten Modellrand erzeugt. Alle anderen Ränder 

sind undurchlässig. Das Modellgebiet ist mit einer Gitterweite von 0.5 m regelmäßig diskretisiert. 11.5 

m vom oberstromigen Rand ist eine Schadstoffquelle, durch eine Festkonzentrationsrandbedingung 

dargestellt, eingebracht, die einen Schadstoff emittiert, der einer Abbaukinetik erster Ordnung folgend 

mit einer Ratenkonstanten λ abgebaut wird. Zusätzlich wird ein nichtreaktiver Stoff aus der Quelle 

freigesetzt. Es werden Quellbreiten von 4, 8 und 16 m betrachtet. Weder Sorption noch Ausgasung 

werden berücksichtigt. Der Schadstoff verhält sich also genau so, wie es von den Methoden zur 

Bestimmung der Abbaurate angenommen wird – eine Annahme, die in der Realität nur annähernd 

erfüllt ist, da die Abbauraten komplexeren Gesetzmäßigkeiten folgen. Diese Abstraktion ist hier 

notwendig, um die vier Methoden möglichst genau untersuchen zu können. Als longitudinale und 

transversale Dispersionslängen werden im Modell 0.25 m und 0.05 m angenommen. Diese stellen die 

lokale Dispersion dar, in die effektive Dispersion geht noch die Heterogenität der hydraulischen 

Durchlässigkeit ein, die hier im Modell explizit dargestellt wird. Mit Hilfe des Programms GeoSys 

(KOLDITZ et al., 2005) wird dann eine stationäre Schadstofffahne erzeugt. Um den Einfluss räumlicher 

Heterogenität untersuchen zu können, wird die hydraulische Durchlässigkeit K als eine ln – 

normalverteilte Zufallsvariable aufgefasst mit einem Erwartungswert von ln(K) = -9.54, was einer 

mittleren hydraulischen Durchlässigkeit von 7.2 10 -5 m s -1 entspricht. Mit der verwendeten Porosität 

von 0.33 ergibt sich so eine mittlere Transportgeschwindigkeit von 6.5·10 -7 m s -1 . Die räumliche 

Struktur wird durch ein isotropes exponentielles Covarianzmodell Cln(K) = σ 2 exp (-∆h/l) abgebildet, 

wobei eine integrale Länge von l = 2.67 m verwendet wird, was einer Korrelationslänge von 8.0 m 

entspricht (RUBIN, 2003). Vier Klassen von hydraulischer Heterogenität werden betrachtet, die durch 

ln(K)-Varianzen σ 2 von 0.38, 1.71, 2.70 und 4.50 gegeben sind und das Spektrum von wenig bis stark 

heterogenen Bedingungen abdecken. Der Wert σ 2 = 0.38, das Covarianzmodell, die mittlere 

hydraulische Durchlässigkeit und Porosität als auch die integrale Länge l entsprechen dem Standort 

Borden (SUDICKY, 1986). Der Wert 1.71 wurde in einem alluvialen Aquifer in Süddeutschland bestimmt 

(HERFORT, 2000), während die Werte 2.70 und 4.50 von Untersuchungen der Columbus Air Force 

Base stammen (REHFELDT et al., 1992). Die geostatistische Software gstats2.4 (PEBESMA & 

WESSELING, 1998) wurde zur Generierung der Zufallsverteilungen der hydraulischen Durchlässigkeit 

durch unkonditionierte Gauss’sche Simulation verwendet. 

Die so erzeugten Fahnen werden nun anhand der Zentrallinien-Methode untersucht. Dabei sind 

anfänglich drei Beobachtungsbrunnen vorgegeben, von denen einer direkt in der Schadstoffquelle 

liegt, während die anderen keine Kontamination zeigen (Abb.1 a). An diesen drei Brunnen werden nun 

die Piezometerhöhen „gemessen“, indem die Modellausgabe gelesen wird. Durch Konstruktion eines 

hydrogeologischen Dreiecks wird die lokale Fließrichtung an der Schadensquelle bestimmt (Abb.1b). 

3

Entlang der so bestimmten Grundwasserfließrichtung werden nun drei neue Messstellen im Abstand 

von je 10 m installiert. An diesen Messstellen werden dann Piezometerhöhen, Konzentrationen des 

reaktiven und des nichtreaktiven Schadstoffes und die lokale hydraulische Durchlässigkeit „gemessen“ 

(Abb. 1c). Aus der Differenz der Piezometerhöhen, den hydraulischen Durchlässigkeiten und der 

wahren, d.h. der auch im Modell verwendeten, Porosität wird die Grundwasserfließgeschwindigkeit 

bestimmt. Zusammen mit den gemessenen Schadstoffkonzentrationen können dann Abbauraten 

erster Ordnung anhand von vier Methoden berechnet werden. Der Virtuelle Aquifer und das Messszenario 

sind so ausgelegt, dass optimale Bedingungen für eine Bestimmung der Abbauraten 

vorliegen. Die einzige Unsicherheit wird durch die heterogene Verteilung der hydraulischen 

Durchlässigkeit erzeugt. 

4 

Anfangssituation 

Erkundungsschritt 1 

Erkundungsschritt 2 

Virtuelle Realität 

Abb. 1 Verwendete Methodik, um Konzentrationen von der Zentrallinie der Fahne zu erhalten: a) 

Anfangszustand, b) Abschätzung der Fließrichtung anhand des hydrogeologischen Dreiecks, c) 

„gemessene“ Konzentrationen auf der ermittelten Zentrallinie, d) Vergleich mit der Virtuellen Realität 

(Piezometerhöhen und Fahne). 

Für den zweiten Teil dieses Beitrags, in dem der Einfluss von Messfehlern untersucht wird, wird der 

Aquifer als homogen angenommen. Die Unsicherheit wird nun also nicht durch das heterogene 

Fließfeld erzeugt, sondern durch fehlerhaftes „Messen“ von Piezometerhöhe und Schadstoffkonzentration. 

Im Falle der Piezometerhöhe lautet das Fehlermodell: 

[1] h′ = h + z∆hmax 

wobei h’ die fehlerbehaftete, hh die wahre Piezometerhöhe und ∆hmax der maximale Messfehler für die 

Piezometerhöhe ist. z ist eine gleichverteilte Zufallszahl aus dem Intervall [-1, 1], mit der ∆hmax

multipliziert wird, um einen fehlerbehafteten Messwert für die Piezometerhöhe aus dem Intervall [h- 

∆hmax, h+∆hmax ] zu erhalten. Die fehlerbehafteten Messwerte streuen also symmetrisch um den 

wahren Wert. ∆hmax wird zwischen 0 und 5 cm variiert. Bei der Ermittlung der Ratenkonstante wird nun 

bei der Probenahme für die Piezometerhöhe ein Messfehler gemäß des vorgestellten Fehlermodells 

berücksichtigt. Die gesamte Auswertung wird einhundert mal durchgeführt, um statistisch 

repräsentative Aussagen über den Einfluss des Messfehlers auf die Ratenkonstante zu erlangen. 

Da für die Messung von Konzentrationen ein höherer Messfehlerbereich zu erwarten ist, wurde das 

Fehlermodell angepasst: 

a 

[2] c ′ = c z∆c 

) 

5 

( max 

wobei c’ die fehlerbehaftete, ch die wahre Konzentration und ∆cmax der maximale Messfehlerfaktor für 

die Konzentration ist. Der Exponent a ist –1 oder 1 und wird zufällig bestimmt, z ist eine gleichverteilte 

Zufallszahl aus dem Intervall [0, 1], mit der ∆cmax multipliziert wird. Man erhält so einen fehlerbehaftet 

Messwert für die Konzentration aus dem Intervall [c/∆cmax, c∆cmax ]. ∆cmax wird zwischen 1 und 100 

variiert. Dieses Verfahren ist ähnlich zu dem für die Piezometerhöhe angewendeten Verfahren und ein 

Messfehlerfaktor von 2 entspricht dem umgangsprachlichen „auf einen Faktor 2 genau“. Für die 

Bestimmung des Einflusses von Konzentrationsmessfehlern wurden ein homogenes Strömungsfeld 

und keine Messfehler für die Piezometerhöhe angenommen. Für den konservativen und den reaktiven 

Stoff wird dasselbe Messfehlermodell angenommen. 

Die vier Methoden, anhand derer die Abbauraten bestimmt werden, sind in Tabelle 1 aufgeführt. 

Methode 1 stellt die Lösung zur eindimensionalen Advektionsgleichung mit Abbau erster Ordnung dar. 

Methode 2 wurde von WIEDEMEIER et al. (1996) vorgeschlagen, und baut auf Methode 1 auf. 

Allerdings werden die Konzentrationen des reaktiven Stoffes auf die Konzentrationen des nichtreaktiven 

Stoffes bezogen. Dadurch berücksichtigt Methode 2 Dispersion, Verdünnung und „Aus der 

Fahne Messen“ an Beobachtungspegeln, die nicht genau auf der Zentrallinie der Fahne liegen. Die 

dritte Methode wurde von BUSCHECK & ALCANTAR (1995) vorgestellt und basiert auf der eindimensionalen 

Transportgleichung mit Advektion, Dispersion und Abbau erster Ordnung. Diese Methode 

beinhaltet somit explizit die longitudinale Dispersion. ZHANG & HEATHCOTE (2003) beschrieben die 

vierte Methode, die auf der analytischen Lösung zur zweidimensionalen Transportgleichung beruht 

und zusätzlich zur longitudinalen auch die transversale Dispersion berücksichtigt. 

Tab. 1 Methoden zur Bestimmung von Abbauratenkonstanten erster Ordnung λ. va ist die 

Transportgeschwindigkeit, ∆x ist der Abstand der verwendeten Beobachtungspegel, C(x) ist die unterstromige 

und C0 die Quellkonzentration. αL und αT sind die longitudinale und transversale Dispersivität, WS ist die 

Quellbreite und erf ist die Fehlerfunktion. 

Methode Formel für Abbauratenkonstante Beschreibung 

1 ⎟ v ⎛ ⎞ 

a C( 

x) 

λ = − 

⎜ 

1 ln 

∆x 

⎝ C0 

⎠ 

2 

3 

4 

v ⎛ ∗ ⎞ 

a ⎜ C( 

x) 

C0 

λ = − 

⎟ 

2 ln 

∆x 

⎜ C ∗ ⎟ 

⎝ 0 C( 

x) 

⎠ 

( C( 

x) 

C ) 

2 

v ⎛ 

⎞ 

a ⎜⎛ 

ln 

0 ⎞ 

λ 

− ⎟ 

3 = 

⎜ 

⎜1− 

2α 

L 

⎟ 1 

4α 

⎟ 

L ⎝⎝ 

∆x 

⎠ ⎠ 

( C( 

x) 

( C β ) ) 

2 

v ⎛ 

⎞ 

a ⎜⎛ 

ln 

0 ⎞ 

λ 

− ⎟ 

3 = 

⎜ 

⎜1− 

2α 

L 

⎟ 1 

4α 

⎟ 

L ⎝⎝ 

∆x 

⎠ ⎠ 

mit: 

⎛ ⎞ 

⎜ 

WS 

β = erf ⎟ 

⎜ ⎟ 

⎝ 4 αT 

∆x 

⎠ 

Analytische Lösung der 1D 

Advektionsgleichung mit Abbau erster 

Ordnung 

Wie Methode 1, aber Konzentration normiert 

auf einen nichtreaktiven Mitkontaminand, 

berücksichtigt Verdünnung und Dispersion 

Analytische Lösung der 1D Transportgleichung 

mit longitudinaler Dispersion 

Analytische Lösung der 2D Transport- 

gleichung. Berücksichtigt sind longitudinale 

und transversale Dispersion und die 

Quellbreite

Für die letztgenannten Methoden müssen die longitudinale und die transversale Dispersivität bekannt 

sein. Gemäß einem Ansatz in WIEDEMEIER et al. (1999) wurde die longitudinale Dispersivität als 10% 

der Fahnenlänge angenommen, die transversale Dispersivität beträgt 33% der longitudinalen 

Dispersivität. Als Fahnenlänge wurde der Abstand von der Quelle zur entferntesten Messstelle 

angenommen. Dieser Abstand beträgt 30 m, die longitudinale Dispersivität somit 3 m und die 

Transversale Dispersivität 1 m. Die Dispersivitäten sind damit recht gering geschätzt, eine detaillierte 

Untersuchung des Einflusses der angenommenen Dispersivitäten ist in BAUER et al. (2005) zu finden. 

Mithilfe der vier vorgestellten Ansätze werden Abbauraten erster Ordnung ermittelt und mit dem 

Modelleingabewert verglichen. Für jede Klasse der hydraulischen Heterogenität werden je 100 

Realisierungen betrachtet, um ein statistisches Maß für die Unsicherheit zu erhalten, die durch die 

hydraulische Heterogenität erzeugt wird. Für jede Realisierung wurde die oben beschriebene 

Auswertung für jede Methode und jede Quellbreite durchgeführt. Dabei wurden jeweils die Abbauraten 

erster Ordnung ausgehend von der Quelle für die unterstromigen Brunnen im Abstand von 10 m, 20 m 

und 30 m bestimmt und anschließend arithmetisch gemittelt. Für Methode 4 wird für die Ermittlung der 

Abbaurate die wahre Quellbreite angenommen. 

Ergebnisse und Diskussion 

Einfluss der Heterogenität 

Abb. 2 zeigt die Ergebnisse der Berechnung der Abbauratenkonstante erster Ordnung. Die Raten 

werden normalisiert dargestellt, d.h. die berechnete Ratenkonstante wird durch die wahre Ratenkonstante 

geteilt. Die normalisierte Rate kann so als Überschätzungs- bzw. Unterschätzungsfaktor 

interpretiert werden. Abb. 2 zeigt, dass die meisten berechneten Ratenkonstanten größer als 1.0 sind, 

d.h. dass die Ratenkonstante überschätzt wird. Dieser Effekt ist in einzelnen Realisierungen sehr 

stark, wo Überschätzungen der Ratenkonstante von einigen Größenordnungen auftreten können. 

Abb. 2a zeigt links die Variation der berechneten normalisierten Ratenkonstante mit der Quellbreite 

der emittierenden Schadstoffquelle. Es ist deutlich, dass mit zunehmender Quellbreite die berechnete 

Ratenkonstante sich der wahren Ratenkonstante annähert, d.h. sich dem Wert 1 annähert, was am 

deutlichsten für große Heterogenitäten sichtbar ist. Dies ist in Methode 1 begründet, die weder 

Verdünnung noch Dispersion oder „Aus der Fahne messen“ berücksichtigt. Diese Effekte werden mit 

zunehmender Quellbreite weniger signifikant, da dann die Annahmen zur Anwendung von Methode 1 

besser erfüllt sind. Auf der rechten Seite der Abb. 2 ist die Abhängigkeit der berechneten Ratenkonstante 

von der verwendeten Heterogenitätsklasse, durch die zugehörige Varianz σ²ln(K) bezeichnet, 

dargestellt. Es ist zu erkennen, dass eine Erhöhung von σ²ln(K) im Mittel zu einer Überschätzung der 

Ratenkonstante führt. Zusätzlich zu diesem Trend nimmt auch die Standardabweichung der 

berechneten Ratenkonstanten zu, was die Zunahme der Spanne der ermittelten Ratenkonstanten 

wiederspiegelt. Diese Zunahme der Spanne kann als Zunahme der Unsicherheit der berechneten 

Ratenkonstante verstanden werden. Die Überschätzung beträgt im Mittel ca. 2 für geringe Heterogenität, 

steigt jedoch auf Werte von 4 bis 10 für mittlere und hohe Heterogenität und erreicht einen 

Wert von ca. 100 im Falle der sehr hohen hydraulischen Heterogenität. 

Abb. 2b stellt die Ergebnisse für Methode 2 dar. Auch anhand von Methode 2 werden die 

Ratenkonstanten überschätzt, jedoch sind, verglichen mit Methode 1, die Überschätzungen als auch 

die Standardabweichung generell geringer. Methode 2 liefert also bessere und mit weniger 

Unsicherheit behaftete Ergebnisse. Wie man an der linken Grafik in Abb. 2b erkennen kann, ergibt 

sich für Methode 2 keine Abhängigkeit von der Quellbreite. Dies ist in der Methode begründet, da sie 

Effekte von Dispersion, Verdünnung und „Aus der Fahne messen“ durch die Normierung explizit 

berücksichtigt. Methode 3 zeigt ein ähnliches Verhalten wie Methode 1, sowohl in Abhängigkeit von 

der Quellbreite als auch in Abhängigkeit von der Heterogenität (Abb. 2c). Jedoch sind die 

normalisierten Abbauratenkonstanten höher als für Methode 1, was an der Berücksichtigung der 

Dispersion in Methode 3 liegt. 

Im eindimensionalen Aquifer mit einer Festkonzentration als Randbedingung führt die 

Berücksichtigung der longitudinalen Dispersion zu höheren Konzentrationen entlang der stationären 

Fahne, verglichen mit dem Fall ohne Dispersion. Dies wird durch den zusätzlichen dispersiven 

Massenaustrag aus der Schadstoffquelle verursacht. Um eine gemessene Konzentrationsabnahme 

zwischen Quelle und unterstromigen Brunnen mit Methode 3 erklären zu können ist also eine höhere 

Abbauratenkonstante notwendig als mit Methode 1. Methode 4 (Abb. 2d) schließlich zeigt ein sehr 

ähnliches Bild wie Methode 3. Aufgrund der in Methode 4 berücksichtigten Querdispersivität sind die 

bestimmten Abbauratenkonstanten etwas geringer als bei Methode 3. Für geringe Heterogenität wird 

die Ratenkonstante durch Methode 4 sogar unterschätzt. Dies liegt an einer zu hohen Korrektur durch 

6

den Querdispersionsterm, der die Effekte der Querdispersion für geringe Heterogenität überschätzt. 

Für den Fall geringer oder keiner Heterogenität sind die hier gewählten und in Methode 4 

verwendeten Dispersivitäten zu groß. 

a) 

b) 

c) 

d) 

normalisierte Ratenkonstante [-] 




1000 

100 

10 

1 

0.1 

0.01 

0 4 8 12 16 20 

1000 

100 

10 

1 

0.1 

0.01 

0 4 8 12 16 20 

1000 

100 

10 

1 

0.1 

0.01 

0 4 8 12 16 20 

1000 

100 

10 

1 

0.1 

0.01 

0 4 8 12 16 20 

Quellbreite [m] 

7 

0 1 2 3 4 5 

0 1 2 3 4 5 

0 1 2 3 4 5 

0 1 2 3 4 5 

σ 2 ln(K) 

Abb. 2 “Gemessene” Abbauratenkonstanten erster Ordnung, normalisiert auf die wahre 

Abbauratenkonstante, aufgetragen gegen Quellbreite (links) und Heterogenität (rechts). a) Methode 1, 

b) Methode 2, c) Methode 3 und d) Methode 4. Alle Abbildungen zeigen die Einzelergebnisse aller 

Realisierungen (kleine Symbole), ihren Mittelwert (große Symbole) und die zugehörige 

Standardabweichung (Fehlerbalken). Unterschiedliche Grautöne geben die Quellbreite, 

unterschiedliche Symbole die verwendeten σ²ln(K) an.

Um diesen Effekt zu verdeutlichen sind in Abb. 2 für alle Methoden noch die für den homogenen Fall 

bestimmten Ratenkonstanten eingetragen. Die Ergebnisse sind für σ²ln(K) = 0 in den Abbildungen 

rechter Hand als kleine horizontale Balken dargestellt. Für Methode 1 und Methode 2 beträgt die 

normalisierte Ratenkonstante im homogenen Fall genau 1.0, d.h. diese Methoden liefern exakt die 

wahre Ratenkonstante. Mit Methode 3 ergibt sich eine geringe Überschätzung, mit Methode 4 eine 

deutliche Unterschätzung der wahren Ratenkonstante. Die geringste Ratenkonstante wird für die 

geringste Quellbreite ermittelt, da hier die Korrektur durch β am größten ist (vergl. Tab 1). 

Da die bisher gezeigten Mittelwerte und Standardabweichungen repräsentativ für das Ensemblemittel 

sind, nicht jedoch für die einzelnen Realisierungen, wurde ein weiteres Maß zum Vergleich der 

anhand der vier Methoden bestimmten Ratenkonstanten entwickelt. Abb. 3 zeigt die Wahrscheinlichkeit, 

mit der eine Methode zum Erfolg führen kann, worunter hier verstanden wird, dass die Abbaurate 

anhand der Methode mit einer gewünschten Genauigkeit ermittelt wird. Die gewünschte Genauigkeit 

wird als sogenannten Fehlerfaktor angegeben. Ein Fehlerfaktor von 10 entspricht dem Intervall 0.1 bis 

10 der normierten Ratenkonstanten, wobei dieses Intervall durch Division und Multiplikation von 1.0 

mit dem Fehlerfaktor (10) ermittelt wird. Dies entspricht somit der umgangssprachlichen Formulierung 

„... innerhalb einer Größenordnung ...“. Ein Fehlerfaktor von 5 entspricht somit dem Intervall 0.2 bis 5 

der normierten Ratenkonstanten. Abb. 3 gibt daher die Wahrscheinlichkeit dafür an, dass eine 

„gemessene“ Ratenkonstante innerhalb des durch den Fehlerfaktor aufgespannten Intervalls liegt. 

Abb. 3a zeigt für die geringste Heterogenität, dass die Wahrscheinlichkeit, die Abbauratenkonstante 

mit einem Fehlerfaktor kleiner als 2.0 zu bestimmen („ ... bis auf einen Faktor zwei ...“), für Methode 1 

ca. 70 %, für Methode 2 ca. 90%, für Methode 3 ca. 55% und für Methode 4 ca. 30% beträgt. Wird der 

Fehlerfaktor auf 5 erhöht, dann erhält man mit Methoden 1 bis 3 eine Erfolgswahrscheinlichkeit von 

100%, nur mit Methode 4 beträgt die Erfolgswahrscheinlichkeit ca. 70%. Je geringer die Werte einer 

Methode in Abb. 3 sind, desto geringer ist die Wahrscheinlichkeit, die Abbauratenkonstante mit der 

gewünschten Genauigkeit bestimmen zu können. Wie Abb. 3 zeigt, sinkt für alle Methoden die Erfolgswahrscheinlichkeit 

mit zunehmender Heterogenität. Beträgt die Erfolgswahrscheinlichkeit von Methode 

1 für einen Fehlerfaktor von 5 noch ca. 100% für die geringste Heterogenität, so sinkt diese 

Wahrscheinlichkeit auf 70%, 50% und schließlich 35% für die höheren Heterogenitätsklassen. Für 

Methoden 2 bis 4 sind die Werte analog Abbildung 3 zu entnehmen. 

Abb. 3 erlaubt daher einen direkten Vergleich der in dieser Arbeit untersuchten vier Methoden. Wird 

die Ratenkonstante in einem stark heterogenen Aquifer beispielsweise auf einen Faktor zehn genau 

benötigt, betragen die Erfolgswahrscheinlichkeiten ca. 80%, 95%, 80% und 60% für Methoden 1 bis 4 

(Abb. 3c). Für alle Heterogenitätsklassen liefert Methode 2 die höchste Wahrscheinlichkeit, das 

richtige Ergebnis zu bekommen. Für mittlere bis hohe Heterogenitäten folgt Methode 4 an zweiter 

Stelle, während diese Methode für geringe Heterogenität die schlechtesten Erfolgswahrscheinlichkeiten 

aufweist. Obwohl Methode 1 die einfachste Methode ist, liefert sie sehr ähnliche Ergebnisse wie 

Methode 4. Für geringe Heterogenitäten ergeben sich mit Methode 1 sogar die besseren 

Abschätzungen der Ratenkonstante. Methode 3 zeigt – bis auf geringe Heterogenität – immer die 

geringste Erfolgswahrscheinlichkeit. 

Für größere Quellbreiten ergeben sich qualitativ dieselben Ergebnisse. Methode 2 ist unabhängig von 

der Quellbreite, daher ist auch für Quellbreiten von 8 m oder 16 m die Erfolgswahrscheinlichkeit von 

Methode 2 dieselbe wie im Falle von 4 m. Für die anderen Methoden ergeben sich mit zunehmender 

Quellbreite höhere Erfolgswahrscheinlichkeiten. So steigt z.B. die Erfolgswahrscheinlichkeit von 

Methode 3 für σ²ln(K) = 2.7 von 35% für eine Quellbreite von 4 m (Abb. 3c) auf 40% für eine 

Quellbreite von 8 m und ca. 50% für eine Quellbreite von 16 m. Somit steigt die Erfolgswahrscheinlichkeit 

für Methoden 1, 3 und 4 mit der Quellbreite an, erreichen aber dennoch nicht die 

Erfolgswahrscheinlichkeit von Methode 2. 

8

1 a) b) 1 

c) 

0.8 

0.6 

0.4 

0.2 

Methode 1 

Methode 2 

Methode 3 

Methode 4 

0 

1 10 100 1000 

1 

0.8 

0.6 

0.4 

0.2 

0 

Methode 1 

Methode 2 

Methode 3 

Methode 4 

1 10 100 1000 

Fehlerfaktor 

9 

0.8 

0.6 

0.4 

0.2 

d) 

Methode 1 

Methode 2 

Methode 3 

Methode 4 

0 

1 10 100 1000 

1 

0.8 

0.6 

0.4 

0.2 

0 

Methode 1 

Methode 2 

Methode 3 

Methode 4 

1 10 100 1000 

Fehlerfaktor 

Abb. 3 Erfolgswahrscheinlichkeit für alle vier Methoden gegen den Fehlerfaktor für σ²ln(K) a) 0.38, b) 

1.71, c) 2.7 and d) 4.5. Die Quellbreite beträgt 4 m. 

Einfluss des Messfehlers 

Bei der Untersuchung des Einflusses des Messfehlers wird von einem homogenen Aquifer ausgegangen. 

Die Unsicherheit wird nun nicht durch das heterogene Fließfeld erzeugt, sondern durch 

falsches „Messen“ von Piezometerhöhe und Schadstoffkonzentration. 

Da in der vorigen Untersuchungen gezeigt wurde, dass Methode 4 mit der gewählten transversalen 

Dispersivität im homogenen Fall zu kleine Abbauratenkonstanten liefert, wurde die transversale 

Dispersivität, die für die Auswertung mit Methode 4 (Tabelle 1) verwendet wird, auf 0.15 m verringert. 

Damit ergibt sich im homogenen Fall die richtige Abbauratenkonstante. 

In Abb. 4 ist die auf die wahre Ratenkonstante normierte fehlerbehaftete Ratenkonstante gegen den 

maximal möglichen Fehler bei der Bestimmung der Piezometerhöhe, ∆hmax, gesondert für die vier 

Auswertemethoden aufgetragen. In Abb. 4a erkennt man ein deutliches Ansteigen der normierten 

Ratenkonstanten mit zunehmendem ∆hmax für die Auswertung mit Methode 1. Bereits für einen 

maximalen Messfehler von 1 cm ergibt sich eine mittlere Überschätzung von ca. 2, der für ein ∆hmax 

von 5 cm auf ca. 5 steigt. Außerdem ist zu erkennen, dass für einzelne Messungen Überschätzungen 

der Ratenkonstante von mehr als 10 möglich sind. Generell liegt eine Überschätzung vor, da eine 

fehlerhaft bestimmte Piezometerhöhe zu einer falschen Abschätzung der Fließrichtung führt. Die neu 

platzierten unterstromigen Brunnen liegen dann nicht mehr auf der Zentrallinie, d.h. die gemessenen 

Konzentrationen sind kleiner als auf der Zentrallinie und dies führt zu einer Überschätzung der 

Ratenkonstanten. Es wird jedoch nicht nur die Fließrichtung falsch ermittelt, sondern auch die 

Fließgeschwindigkeit entlang der (angenommenen) Zentrallinie, da diese aus den gemessenen 

Piezometerhöhen, Brunnenabständen und hydraulischen Durchlässigkeiten berechnet wird. Der 

hieraus resultierende Fehler kann sowohl zu große als auch zu kleine Ratenkonstanten produzieren. 

Für Methode 2 (Abb. 4b) ergibt sich ein anderes Bild. Die mittlere Ratenkonstante ist auch für große 

∆hmax sehr nahe bei 1, d.h. es liegt kein Trend zu Über- oder Unterschätzung vor. Die Unsicherheit der 

bestimmten Abbaurate ist kleiner als bei Methode 1, wie an den Fehlerbalken zu erkennen ist. Da 

Methode 2 das „aus der Fahne messen“ korrigiert, lässt sich dieser Fehler für Methode 2 nicht 

beobachten (Abb. 4b). Es kommt sowohl zu einer Über- als auch Unterschätzung der Ratenkonstante, 

also einem eher symmetrischen Fehler ohne generelle Tendenz, der auf die falsche Abschätzung der 

Fließgeschwindigkeit zurückzuführen ist. Mit zunehmendem ∆hmax nimmt auch für Methode 2 die

Unsicherheit zu. Für Methode 3 ( Abb. 4c) und Methode 4 (Abb. 4d) ergibt sich – wie schon für die 

hydraulische Heterogenität – ein sehr ähnliches Bild wie für Methode 1, jedoch mit höheren 

Überschätzungen und Unsicherheiten als für Methode 1. Für Methode 3 und 4 beträgt die mittlere 

Überschätzung der Ratenkonstanten für ∆hmax = 1 cm bereits ca. 5, jeweils verbunden mit einer hohen 

Unsicherheit. 

normalisierte Abbauratenkonstante [-] 


100 

10 

1 

0.1 

100 

10 

1 

0.1 

0 1 2 3 4 5 6 

10 

100 

a) b) 

0 1 2 3 4 5 6 

maximaler Messfehler Piezometerhöhe [cm] 

10 

1 

0.1 

100 

c) d) 

10 

1 

0.1 

0 1 2 3 4 5 6 

0 1 2 3 4 5 6 

maximaler Messfehler Piezometerhöhe [cm] 

Abb. 4 Normalisierte Abbauratenkonstante, aufgetragen gegen den maximalen Messfehler bei der 

Bestimmung der Piezometerhöhe für a) Methode 1, b) Methode 2, c) Methode 3 und d) Methode 4. 

Dargestellt sind die Einzelergebnisse (kleine Symbole), die Mittelwerte (durch Linie verbundene 

Punkte) und die zugehörige Standardabweichung (Fehlerbalken). 

Für Konzentrationsmessungen wurde ein höherer Messfehlerfaktor von 100 und eine exakte Messung 

der Piezometerhöhe angenommen (siehe Kapitel Methodik). Ein Messfehlerfaktor von 100 ist 

sicherlich sehr hoch gegriffen und wird hier nur zu Demonstrationszwecken eingesetzt. Das Ergebnis 

für Methode 1 zeigt Abb. 5a, in der die normierte Ratenkonstanten gegen den Fehlerfaktor 

aufgetragen sind. Man erkennt, dass für geringe Messfehlerfaktoren die Abbaukonstante sehr gut 

bestimmt werden kann, für einen Messfehlerfaktor von 2 erhält man im Mittel noch das richtige 

Ergebnis, für einen Messfehlerfaktor von 5 erhält man eine mittlere Überschätzung der Ratenkonstante 

von ca. 3. Die maximale mittlere Überschätzung der Ratenkonstante von ca. 5 wird bei einem 

Messfehlerfaktor von ca. 10 erreicht und steigt auch für größere Fehlerfaktoren nicht weiter an. 

Allerdings nimmt die Unsicherheit, dargestellt als Standardabweichung, mit dem Messfehlerfaktor 

stark zu und man kann sowohl zu große als auch zu kleine Ratenkonstanten erhalten. Für Methode 2 

(Abb. 5b) erhält man generell eine geringere Überschätzung der Ratenkonstante von maximal ca. drei, 

jedoch eine ähnlich hohe Unsicherheit wie mit Methode 1. Generell werden bei Methode 2 eher 

kleinere Ratenkonstanten erzeugt als bei Methode 1. Methoden 3 und 4 (Abb 5.c und Abb. 5d) zeigen 

einen deutlichen Anstieg der ermittelten Überschätzung mit dem Messfehlerfaktor, maximale Werte 

liegen hier bei etwa 10. Ebenso ist die Unsicherheit dieser Werte sehr hoch.



1000 

100 

10 

1 

0.1 

0.01 

1000 

100 

10 

1 

0.1 

0.01 

a) 

1 10 100 

c) 

1 10 100 

maximaler Messfehlerfaktor Konzentration [-] 

11 

1000 

100 

10 

1 

0.1 

0.01 

1000 

100 

10 

1 

0.1 

0.01 

b) 

1 10 100 

d) 

1 10 100 

maximaler Messfehlerfaktor Konzentration [-] 

Abb. 5 Normalisierte Abbauratenkonstante, aufgetragen gegen den maximalen Messfehlerfaktor bei 

der Bestimmung der Konzentration für a) Methode 1, b) Methode 2, c) Methode 3 und d) Methode 4. 

Dargestellt sind die Einzelergebnisse (kleine Symbole), die Mittelwerte (durch Linie verbundene 

Punkte) und die zugehörige Standardabweichung (Fehlerbalken). 

Schlussfolgerungen 

Die vorgestellten Ergebnisse zeigen, dass sich die anhand der vier untersuchten Methoden ermittelten 

Abbauratenkonstanten erster Ordnung sowohl für zunehmende hydraulische Heterogenität als auch 

für zunehmende Messfehler bezüglich der Piezometerhöhe und der Konzentration unterscheiden. Alle 

Methoden zeigen geringere Erfolgswahrscheinlichkeiten mit zunehmender Heterogenität (Abb.3) 

sowie eine erhöhte Unsicherheit (Abb. 2). Aus Abb. 2 ist ersichtlich, dass diese Abnahme der 

Erfolgswahrscheinlichkeit durch eine generelle Überschätzung der Ratenkonstante verursacht wird. 

Diese Überschätzung ist am größten für Methode 3. Methode 2 wird dagegen am wenigsten von der 

hydraulischen Heterogenität beeinflusst und zeigt die geringsten Überschätzungen (Abb. 2) und 

Unsicherheiten. Obwohl Methoden 3 und 4 realitätsnäher sind, da sie auf der eindimensionalen bzw. 

zweidimensionalen Transportgleichung beruhen, zeigen sie eine geringere Erfolgswahrscheinlichkeit 

und eine größere Überschätzung der Ratenkonstanten als Methoden 1 und 2. Beide Methoden sind 

anfällig für Fehler, die durch die Abschätzung der longitudinalen und transversalen Dispersivitäten 

erzeugt werden können. Methode 1 als die einfachste Methode, da sie weder die Abschätzung der 

Dispersivität noch einen nichtreaktiven Mitkontaminanden benötigt, zeigt bessere oder ähnlich gute 

Ergebnisse wie Methode 4. Sowohl mittlere Überschätzung als auch Unsicherheit steigen mit 

zunehmender Größe des Messfehlers. Die Untersuchung des Messfehlers der Piezometerhöhe zeigt, 

dass bereits geringe Messfehler von 1 cm zu deutlicher Überschätzung der Ratenkonstanten und 

einer erhöhten Unsicherheit führen können. Der Einfluss des Messfehlers der Konzentration auf die 

Ratenkonstante ist generell geringer. Auch bei der Untersuchung der Messfehler erhält man, wie 

schon im Falle von hydraulischer Heterogenität, die besten Ergebnisse mit Methode 2. Methoden 3 

und 4 zeigen sehr ähnliche Ergebnisse und weisen größere Überschätzungen auf als Methode 2 oder 

Methode 1. 

Insgesamt ergibt sich, dass die Verwendung von Methode 2, die auf der Normierung der 

Schadstoffkonzentration mit einem nichtreaktiven Mitkontaminanden beruht, zu den besten 

Ergebnissen bei der Bestimmung von Abbauratenkonstanten erster Ordnung führt. Daher sollte diese 

Methode, wenn möglich, angewendet werden. Ist kein nichtreaktiver Mitkontaminand vorhanden, sollte 

Methode 1 verwendet werden, da sie die Abbauratenkonstante besser als Methoden 3 und 4 vorher-

sagt sowie die zusätzliche Problematik des Abschätzens der Dispersivitäten umgeht. In BAUER et al. 

(2005) wird der Einfluss der geschätzten Dispersivitäten auf die Abbauratenkonstante sowohl für die 

longitudinale als auch transversale Dispersivität anhand einer Sensitivitätsstudie eingehend untersucht. 

Es zeigt sich dabei, dass mit Methode 3 nie, mit Methode 4 nur unter Annahme sehr hoher 

transversaler Dispersivitäten, die nicht mit der Heterogenität (σ²ln(K)) des Aquifers begründet werden 

können, die korrekten Abbauraten ermittelt werden. In diesem Aufsatz werden weiterhin der Einfluss 

der Quellbreite sowie verschiedene Methoden zur Berechnung der Fliessgeschwindigkeit untersucht. 

Es zeigt sich, dass die Anordnung der Messpegel, abgeleitet aus dem hydraulischen Dreieck, großen 

Einfluss auf die geschätzte Abbaurate hat, wie er sich auch in Abbildung 1 widerspiegelt. Ebenso ist 

die Mittelung der an den einzelnen Messpegeln ermittelten hydraulischen Durchlässigkeiten von 

Bedeutung. Hier ergibt sich, dass eine geometrische Mittelung zu bevorzugen ist (BAUER et al., 2006). 

Die gefundene generelle Überschätzung der Ratenkonstante ist im Hinblick auf eine Prognose der 

Fahnenlänge problematisch, da eine zu hohe Ratenkonstante den Abbau überschätzt und daher zu 

geringe prognostizierte Fahnenlängen verursacht. Dieser Fehler ist daher nicht konservativ und kann 

zu einer zu optimistischen Einschätzung der Standortverhältnisse führen. Im Folgenden ist die 

Fahnenlänge definiert als die Strecke zwischen Schadensquelle und dem Ort auf der 

Fahnenzentralachse, an dem das Verhältnis C/C0 einen vorgegebenen Wert erreicht hat 

(beispielsweise 10 -3 ), wobei C die Konzentration in der Fahne und C0 die Konzentration in der Quelle 

ist. Für eindimensionale Lösungen ohne Dispersion ergibt sich, dass Abbaurate und Fahnenlänge 

umgekehrt proportional zueinander sind und die Fahnenlänge linear mit dem Kehrwert der Abbaurate 

wächst (Vergleiche Methoden 1 und 2 in Tabelle 1). Eine Überschätzung der Abbaurate um den 

Faktor 5 verursacht also eine Unterschätzung der Fahnenlänge ebenfalls um den Faktor 5. Im 

eindimensionalen Fall mit Dispersion sowie im zweidimensionalen Fall ist die Fahnenlänge annähernd 

umgekehrt proportional zum Kehrwert der Quadratwurzel der Abbaurate, die genauen Werte hängen 

von den gegebenen Dispersivitäten, der Quellbreite und dem Verhältnis von Abstandsgeschwindigkeit 

zu Abbaurate ab (vergleiche Methoden 3 und 4 in Tabelle 1). Eine Überschätzung der Fahnenlänge 

um den Faktor 5 bzw. 10 kann beispielsweise eine Unterschätzung der Fahnenlänge um den Faktor 3 

bzw. 5 verursachen. Der Einfluss der Dispersion verringert die Sensitivität der Fahnenlänge auf die 

Abbaurate. Eine genauere Untersuchung des Einflusses der ermittelten Abbaurate auf die 

prognostizierte Fahnenlänge wurde von BEYER et al., (2005b) durchgeführt. Dabei zeigte sich eine 

durchschnittliche Unterschätzung der Fahnenlängen um 50 %. Für einzelne Realisierungen ergaben 

sich zum Teil jedoch erheblich größere Fehler, sodass in ungünstigen Fällen die geschätzte 

Fahnenlänge lediglich 10% des tatsächlichen Wertes betrug. Dabei zeigt sich, dass die mit Methoden 

1, 3 und 4 berechneten Abbauraten prinzipiell zu ähnlichen Abschätzungen der Fahnenlänge führen, 

wenn diese mit einem mit der Abbaurate korrespondierenden Transportmodell berechnet wird. Die 

durch Vernachlässigung bzw. falsche Parametrisierung der Längs- und Querdispersion verursachten 

Fehler in der Abbaurate werden bei der Fahnenlängenprognose teilweise wieder aufgehoben. Als 

problematisch stellte es sich jedoch heraus, Abbauraten, die mit der eindimensionalen Lösung 

berechnet wurden, in einem zwei- oder dreidimensionalen Transportmodell zur Prognose der 

Fahnenlänge zu verwenden (BEYER et al., 2005b). 

Danksagung: Der Beitrag entstand im Rahmen des Projekts TV 7.1 „Modellierung und 

Prognose“ des BMBF- Förderschwerpunktes “Kontrollierter natürlicher Rückhalt und Abbau 

von Schadstoffen bei der Sanierung kontaminierter Grundwässer und Böden (KORA)“ an den 

Universitäten Kiel und Tübingen. Den Projektträgern und dem BMBF sei für die hierbei 

gewährte Unterstützung gedankt. 

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13


Beyer, C., Chen, C., Gronewold, J., Kolditz, O., Bauer, S. (2007a): Determination of first 

order degradation rate constants from monitoring networks. (accepted by Ground Water.). 

The enclosed article was accepted for publication in the journal Ground Water. Copyright © 

2007 Blackwell Publishing. 

The definite version will be available at www.blackwell-synergy.com.

Determination of first order degradation rate constants from 

monitoring networks ∗ 

Christof Beyer, Cui Chen, Jan Gronewold, Olaf Kolditz, Sebastian Bauer 

Center for Applied Geoscience, Eberhard-Karls-University of Tübingen, 

Sigwartstraße 10, D 72076 Tübingen, Germany 

Tel.: +49 7071 29 73176; Fax: +49 7071 5059. E-mail: christof.beyer@uni-tuebingen.de 

Abstract 

In this paper different strategies for estimating first order degradation rate constants from measured field data are 

compared by application to multiple synthetic contaminant plumes. The plumes were generated by numerical simulation 

of contaminant transport and degradation in virtual heterogeneous aquifers. These sites then were individually and 

independently investigated on the computer by installation of extensive networks of observation wells. From the data 

measured at the wells, i.e. contaminant concentrations, hydraulic conductivities and heads, first order degradation rates 

were estimated by three one-dimensional center line methods, which use only measurements located on the plume axis, 

and a two-dimensional method, which uses all concentration measurements available downgradient from the 

contaminant source. Results for both strategies show that the true rate constant used for the numerical simulation of the 

plumes in general tends to be overestimated. Overestimation is stronger for narrow plumes from small source zones with 

an average overestimation factor of about 5 and single values ranging from 0.5 to 20, decreasing for wider plumes, with 

an average overestimation factor of about 2 and similar spread. Reasons for this overestimation are identified in the 

velocity calculation, the dispersivity parameterization and off-center-line measurements. For narrow plumes the onedimensional 

and the two-dimensional strategies show approximately the same amount of overestimation. For wide 

plumes, however, incorporation of all measurements in the two-dimensional approach reduces the estimation error. No 

significant relation between the number of observation wells in the monitoring network and the quality of the estimated 

rate constant is found for the two-dimensional approach. 

Introduction 

Although detailed mathematical descriptions of contaminant degradation in the subsurface are available (e.g. Rittmann 

and VanBriesen 1996; Wiedemeier et al. 1999; Islam et al. 2001), in field studies often simplified approaches are used for 

contaminant transport modeling. This is mainly because the identification of a large number of parameters and processes 

from field data often is impossible. Due to its mathematical simplicity, its easy implementation into transport models and 

the necessity of determining only a single parameter, the biodegradation model most frequently used is first order kinetics 

(Wiedemeier et al. 1999). Field methods for the determination of biodegradation rates in ground water comprise a variety 

of approaches, including mass balance calculations, in-situ microcosm studies and the so called center line method 

(Chapelle et al. 1996). With the center line method, only concentration measurements in observation wells located on the 

plume axis are evaluated and additional information possibly at hand (e.g. well data downgradient from the source but 

not on the center line) is not explicitly accounted for in the rate estimation. Thus, the center line method is an essentially 

one-dimensional approach to rate constant estimation, as flow, transport and degradation are only evaluated for a single 

streamline. Assuming that contaminant biodegradation can be approximated by a first order kinetics, the degradation rate 

constant can be calculated with analytical transport models to yield the sought for first order rate constant. An overview 

of common methods for estimating degradation rate constants from field data within the context of monitored natural 

attenuation is given by Newel et al. (2002). The authors discuss approaches based on (point-) concentration over time 

data and different concentration vs. distance relations, where dispersion is completely neglected as well as accounted for. 

Important to note is that the different approaches yield different types of degradation rates that should be used for 

destined purposes only. Rate constants estimated from concentration vs. time data for single monitoring wells for 

example represent point decay rates that are typically used to assess source decay or the time required to reach defined 

remediation goals at a particular location (Newell et al. 2002). Complete neglect of dispersion in the rate estimation 

procedure for concentration vs. distance data along the center line of the plume yields bulk attenuation rates, which 

quantify the reduction of contaminant concentrations with distance from the source due to the combined effects of 

dispersion, dilution (e.g. by recharge) and degradation processes. The frequently used method of Buscheck and Alcantar 

∗ 

Beyer, C., Chen, C., Gronewold, J., Kolditz, O., Bauer, S. (2007): Determination of first order degradation rate constants from monitoring networks. 

(accepted by Ground Water.). 

The manuscript was accepted for publication in the Journal Ground Water. Copyright © 2007 Blackwell Publishing. The definite version will be 

available at www.blackwell-synergy.com. 

1

(1995) yields a “hybrid” rate constant between a bulk attenuation and a pure biodegradation rate constant, as it accounts 

for longitudinal dispersion whereas the effects of transverse dispersion are still reflected in the rate constant. A pure 

biodegradation rate constant can be estimated analytically by normalizing contaminant concentrations to concentrations 

of a recalcitrant tracer, if such a compound is emitted from the same source zone as the contaminant of concern 

(Wiedemeier et al. 1996). If this is not the case, an approach of Zhang and Heathcote (2003) may be used, in which the 

Buscheck and Alcantar (1995) method is extended to account for dispersion in two or three dimensions. Pure 

biodegradation rate constants exclusive of dispersion or other attenuation processes (and only those) may be used in 

contaminant transport models for prognoses of plume trends. It is important to note that these center line based rate 

constant estimation approaches are only applicable to contaminant plumes that have reached steady state conditions, as 

for still expanding plumes the rate constant would be overestimated. 

Although it is known, that biodegradation rate estimates obtained from an investigation of the plume center line are 

subject to substantial uncertainty, this strategy is frequently used in practice. Even in homogeneous aquifers vertical and 

horizontal transverse dispersion can produce center line concentration profiles of recalcitrant compounds that could be 

mistaken as following from first order degradation (McNab Jr. and Dooher 1998). Moreover, the plume axis may easily 

be missed by monitoring wells when the inferred ground water flow direction is incorrect, changes over time due to 

transient flow behavior or when the contaminant plume shows large scale meandering due to aquifer heterogeneity 

(Newell et al. 2002; Wilson et al. 2004). Therefore, biodegradation rate constants obtained from such field data should be 

taken as rough estimates only (Chapelle et al. 2003). As first order rates calculated from center line data include all 

effects and processes that lower local contaminant concentrations, the precarious result of the different sources of 

uncertainty is that the degradation potential may be severely overestimated (Rittmann 2004), causing underestimation of 

plume length and contaminant mass as well as a too optimistic prognosis of down gradient concentrations and exposure 

levels. These aspects were recently studied in-depth by Bauer et al. (2006a) and Beyer et al. (2006) by means of 

numerical experiments in two-dimensional synthetic heterogeneous contaminated aquifers. The numerical experiments 

were based on coarse monitoring networks with 6 to 8 wells, which were all placed along an inferred plume center line. 

In reality, however, monitoring networks typically are designed to suit multiple and sometimes conflicting requirements 

and objectives, which may also change with time, depending on the stage of the site investigation. Objectives in this 

context are the detection of ground water contamination (e.g. Storck et al. 1997), site characterization and spatial 

delineation of the contamination (e.g. McGrath and Pinder 2003) or the long term monitoring of the plume behavior (e.g. 

Wu et al. 2006). These aims require spatially more extensive monitoring networks with larger numbers of wells 

compared to the relatively simple center line well configurations necessary to estimate a degradation rate constant. 

Recently, Stenback et al. (2004) demonstrated that additional off center line measurements can be incorporated in the 

estimation of the degradation rate, when a two-dimensional analytical transport model is fitted to contaminant 

concentrations of all monitoring wells of an extensive monitoring network downgradient from the source. In a field 

application example Stenback et al. (2004) showed, that accounting for the additional information on contaminant 

concentrations and distribution significantly reduced rate constant estimates obtained from the conventional center line 

approach to about 50 %, pointing out the well known problem of rate constant overestimation with the 1D center line 

method. 

This paper therefore studies the performance of several rate constant estimation approaches based on center line 

investigation data for sites with extensive monitoring networks and compares the results to the two-dimensional approach 

of Stenback et al. (2004). Adopting the terminology of Newell et al. (2002) the types of degradation rate constants 

regarded here comprise bulk attenuation, biodegradation and “hybrid” rate constants. Point decay rates are not addressed 

in this paper. Both strategies for rate constant estimation, i.e. the investigation of the plume center line and the approach 

of Stenback et al. (2004), are applied to a set of synthetic sites with extensive monitoring networks, which were 

independently designed and installed by individual test persons engaged in hydrogeological research and consulting. The 

networks were installed for a general characterization and quantification of the contaminant plume (Chen et al. 2005; 

Bauer et al. 2006b), and are used here as a basis for degradation rate estimation. Estimated rate constants for both 

strategies are compared with regard to magnitude of errors and variability to draw conclusions on their limitations in 

view of the monitoring network used. Thereby the studies of Bauer et al. (2006a) and Beyer et al. (2006) are considerably 

extended, as the monitoring networks used in this paper are representative of real field situations by using all installed 

observation wells, and are not restricted to an one-dimensional plume center line. Moreover, this analysis allows an 

evaluation of the “human factor” on estimated rate constants resulting from individual notions of “sufficient accuracy” in 

plume investigation. 

Background and Scope 

This study is based on a set of synthetic contaminated two-dimensional aquifers, generated by multiple stochastic 

simulations of heterogeneous hydraulic conductivity fields and subsequent numerical simulation of contaminant 

spreading from a defined source zone in the synthetic aquifers. The evolved virtual plumes were independently 

investigated by a number of German scientists engaged in hydrogeological and environmental research using an 

2

interactive plume investigation and mapping software (Chen et al. 2005) with any number of wells considered necessary 

for a characterization of the synthetic sites and the delineation of the contaminant plume. The plume investigations were 

performed within a related project (Bauer et al. 2006b) and are described in detail below. The results of these virtual 

plume investigations yield individually and realistically investigated plumes. The “measured” concentrations, hydraulic 

heads and conductivities at the observation well networks of the different plumes investigated are used as a basis for the 

determination of the degradation rate constants. Two different strategies are compared here: 

• Strategy A - One-dimensional center line approach: From all observation wells installed the concentration 

distribution is analyzed and wells along the approximate center line of the contaminant plume are identified (see 

Figure 1(a)). Concentrations measured in these wells are evaluated using three different methods for the 

inference of a degradation rate constant (Newell et al. 2002; Buscheck and Alcantar 1995; Zhang and Heathcote 

2003). Thus, here only a subset of concentrations measured in the original monitoring network is used for rate 

constant estimation. 

• Strategy B - Two-dimensional evaluation: An approximate solution for the two-dimensional steady state 

concentration distribution is fitted to concentrations measured in all observation wells of the monitoring network 

downgradient of the source zone using a residual least squares criterion (Stenback et al. 2004) (Figure 1(b)). 

Fitting parameters are the first order rate constant and the source concentration. 

Figure 1: a) Strategy A and b) strategy B for the inference of the degradation rate constant using an existing monitoring 

network. Shown is the true virtual plume (unknown to the investigator), the monitoring network (small squares) and the 

identified center line wells (larger squares) (strategy A) as well as an example of a fitted 2D analytical plume (contour 

lines) (strategy B). 

Generation of synthetic sites 

The data basis of this study consists of 20 different realizations of heterogeneous steady state contaminant plumes. The 

plumes were generated by numerical simulation of ground water flow and contaminant transport in 20 different twodimensional 

horizontal aquifer realizations with heterogeneous hydraulic conductivity distributions. Into ten of these 

conductivity fields a source with a width of 4 m perpendicular to the mean flow direction was introduced, for the other 10 

aquifer realizations, a source width of 16 m was used. For the numerical simulations the GeoSys/Rockflow code (Kolditz 

et al. 2006; Kolditz and Bauer 2004) was used, which solves the flow and transport equations by finite element methods. 

The governing equations for steady state flow conditions are given as (e.g. Bear 1972): 

∇( K ∇h) 

= 0 

(1) 

∂C 

= −va∇C 

+ ∇( 

D∇C) 

− λC 

(2) 

∂t 

with h [L] the hydraulic head, K [L T -1 ] the tensor of hydraulic conductivity, C [M L -3 ] concentration, D [L 2 T -1 ] the 

dispersion tensor which is calculated acording to Bear (1972), va [L T -1 ] the transport velocity, t [T] time and λ [T -1 ] 

representing the first order degradation rate constant. Equation (2) was solved over a two-dimensional numerical grid of 

184 m length by 64 m width with node spacing of 0.5 m in both directions and setting the left hand side of equation (2) to 

0. This guarantees that the simulated plumes are at steady state. For local dispersivities αL and αT values of 0.25 m and 

0.05 m are used. A mean hydraulic gradient I of 0.003 is induced by fixed head boundary conditions on the left and the 

right hand side of the model domain (Figure 2). Flow conditions are at steady state. Hydraulic conductivity K of the 

virtual aquifers is regarded as a spatial random variable (Figure 2), following a lognormal distribution with an expected 

value of E[Y = ln(K)] = -9.54, which corresponds to an effective conductivity Kef of 7.19·10 -5 m s -1 using the geometric 

mean. As the aquifer models are two-dimensional and horizontal, an isotropic exponential covariance function with an 

3

2 

integral scale lY of 2.67 m and ln-conductivity variance σ Y = 2.7 is used for the spatial correlation structure. Kef and lY 

2 

are taken from the Borden field site (Sudicky 1986), whereas the degree of heterogeneity σ Y was reported for the 

Columbus Air Force Base site (Rehfeldt et al. 1992). Porosity n is set to 0.33, resulting in va = 6.54 10 -7 m s -1 . For local 

dispersivities αL and αT values of 0.25 m and 0.05 m are used. The contaminant source introduced in the aquifers is of 

rectangular shape centered at [11.5 m; 32.0 m] downstream of the inflow boundary and has an area of either 3 * 4 m or 3 

* 16 m, corresponding to a source width WS of 1.5 or 6 correlation lengths transverse to the mean flow direction. It emits 

a contaminant with the contaminant concentration fixed in the source area at C / C0 = 1.0. The dissolved contaminant is 

subject to a first order kinetics degradation reaction with a rate constant λ of 1.59·10 -8 s -1 (0.5 a -1 = 0.5 yr -1 ). This value is 

well within the range of reported / recommended first order degradation rate constants for chlorinated solvents as well as 

petroleum hydrocarbons under anaerobic conditions listed in Aronson and Howard (1997). The model parameters used 

are summarized in Table 1. 

contaminant source: C = 1.0 

0 

fixed head 

h = 0.552 m 

4 

no flow 

no flow 

fixed head 

h = 0.0 m 

-1 

K [m s ] 

Figure 2: A single realization of the spatially correlated random hydraulic conductivity field and model boundary 

conditions. 

The conceptual model used here is a rigorous simplification of contaminant transport and degradation observed in natural 

aquifer systems, where biodegradation is a function of electron acceptor and donor availability and the aquifer structure 

and thus reaction kinetics follow more complicated laws, show spatial dependence and may include transient effects, 

dilution or phase changes to the unsaturated zone. Furthermore, the contaminant is not retarded and shows no 

volatilization. However, as the methods assume a uniform rate constant to be valid, only using such simplified 

representations of reality as virtual test sites allows for a detailed analysis of the influence of heterogeneous hydraulic 

conductivity on the different methods under otherwise ideal conditions. 

1.0*10 -2 

1.0*10 -3 

1.0*10 -4 

1.0*10 -5 

1.0*10 -6 

1.0*10 -7 

1.0*10 -8 

1.0*10 -9 

Table 1: Model parameters used in the numerical simulations. 

parameter description value 

Kef effective conductivity 7.19·10 -5 m s -1 

2 

σ Y 

ln(K)-variance 2.7 

lY integral scale 2.67 m 

n porosity 0.33 

αL longitudinal dispersivity 0.25 m 

αT transverse dispersivity 0.05 m 

I hydraulic gradient 0.003 

λ first order degradation rate constant 1.59·10 -8 s -1 

Investigation of synthetic sites 

The plumes were independently investigated by 85 different test persons, engaged in hydrogeological and environmental 

research, consulting or administration. These investigators were confronted with the scenario of contaminant migration 

downstream from a source zone in a virtual aquifer. A two-dimensional top view of the site, the mean ground water flow 

direction and the approximate location of the contaminant source were given as the only prior information for the site 

investigation. Neither the plume nor the hydraulic heads calculated are known at this stage. The task of the investigators 

was an as exact as possible characterization of the contaminated aquifer by the following procedure: 

• Step 1) Emplacing observation wells into the virtual aquifer: Using an interactive graphical user interface, the 

investigator positions observation wells on the virtual site. At the wells local contaminant concentrations and 

hydraulic heads are measured. 

• Step 2) Regionalization of local measurements: Using different interpolation schemes (e.g. Kriging or Inverse 

Distance Weighting) the investigator interpolates contaminant concentrations and hydraulic heads measured at 

the observation wells to the virtual aquifer.

Steps 1 and 2 could be repeated as often as desired by the investigator, until the interpolated contaminant plume was 

deemed to be investigated accurately enough to properly characterize the contaminant distribution. Thus the interactive 

site investigation is an iterative procedure and is evaluated by a comparison of the “true” plume with the investigation 

result. More details on this procedure can be found in Chen et al. (2005). 

Using this methodology, a total number of 85 individual investigation results were obtained. 47 of these investigations 

were conducted for plumes originating from a source of width 4 m, while the plumes with a source width of 16 m were 

investigated 38 times. Accordingly, the majority of the 20 different plumes were investigated three or four times by 

different investigators. 

The configuration and the number of monitoring wells varies substantially between the different realizations and even for 

the same realization but different investigators (12 – 93 wells), as the decision about how many wells were needed for an 

accurate characterization of the site was left to the individual investigators. For two contaminant plumes (one for each 

source width) the investigation was repeated 13 times. Each investigation was performed by a different investigator. In 

addition to the general comparison of strategies A and B for the inference of the degradation rate constant, this subset of 

the whole data set allows for an analysis of estimated rate constant variability for a single site, resulting from different 

notions of “sufficient accuracy” of the virtual plume investigation. 

Rate constant estimation 

Using the monitoring networks installed by the investigators of the virtual plumes, first order rate constants were 

estimated following strategies A and B. Methods A1, A2 and A3 are used in strategy A and are described in detail in 

Bauer et al. (2006a). Therefore, only a brief description is given here: 

Method A1 (equation (3)) is based on the one-dimensional transport equation, considering advection and first order 

degradation only. The steady state solution for the concentration profile is rearranged to yield λA1, i.e. the first order 

degradation rate constant for method A1: 

va ⎛ C( 

x) 

⎞ 

λ A1 

= − ln⎜ ⎟ 

∆x 

⎜ ⎟ 

(3) 

⎝ C0 

⎠ 

va [L T -1 ] is the transport velocity, ∆x [L] the distance between the observation wells and C0 and C(x) [M L -3 ] are 

upstream and downstream contaminant concentrations at the observation wells, respectively. λA1 can be considered rather 

an overall or bulk attenuation rate than a degradation rate constant (Newell et al. 2002), as all concentration changes due 

to diffusion, dispersion, volatilization or dilution are attributed to the degradation process. 

The method introduced by Buscheck and Alcantar (1995) is the second approach applied in this study (equation (4)). It is 

based on the steady state solution of the one-dimensional transport equation considering advection, longitudinal 

dispersion and first order degradation. Method A2 requires an estimate of longitudinal dispersivity αL [L] and yields a 

“hybrid” rate constant between a bulk attenuation and a pure biodegradation rate constant, as the effects of transverse 

dispersion are still reflected in the rate constant estimate λA2. 

⎛ 

2 

v 

( ) ⎞ 

a ⎜⎛ 

ln C( 

x) 

C0 

⎞ 

λ = ⎜1 

− 2 

⎟ −1⎟ 

A2 

α L 

4α 

⎜ 

⎟ 

L ⎝⎝ 

∆x 

⎠ ⎠ 

(4) 

Zhang and Heathcote (2003) proposed modifications to the method of Buscheck and Alcantar (1995) to improve the 

estimation of λ with regard to transverse dispersion. Correction terms derived from analytical solutions to the two- and 

three-dimensional advection dispersion equations including first order decay are used to account for lateral spreading and 

the width of the source zone WS [L] in two and three dimensions, respectively. Therefore information about WS, αL and 

αT are required for this approach. Method A3 (equation (5)) is the two-dimensional form of the method by Zhang and 

Heathcote (2003) and yields the biodegradation rate constant estimate λA3: 

2 

v ⎛ ( ) ⎞ 

a ⎜⎛ 

ln C( 

x) 

( C0β 

) ⎞ 

⎛ ⎞ 

λ = − ⎟ 

A3 

⎜ 

⎜1− 

2α 

L 

⎟ 1 with 

⎜ WS 

β = erf ⎟ (5) 

4α 

⎟ 

L ⎝⎝ 

∆x 

⎠ 

⎜ ⎟ 

⎠ 

⎝ 4 αT 

∆x 

⎠ 

For evaluation strategy B, method B (equation (6)) is used, which corresponds to the approach of Stenback et al. (2004). 

An approximate solution for the steady state concentration distribution derived from the two-dimensional advectiondispersion 

equation with first order degradation (Domenico and Schwartz 1990) is fitted to measured concentrations: 

C ⎪⎧ 

⎛ 

⎞⎪⎫ 

⎪ 

⎧ ⎛ ⎞ ⎛ ⎞⎪ 

⎫ 

0 ⎛ x ⎞ 

⎨ 

⎜ + ⎟ ⎜ − 

⎨ ⎜ 

4λ 

⎟⎬ 

− 

⎟ 

⎜ 

⎟ 

Bα 

L 2y 

WS 

2y 

WS 

C( 

x, 

y) 

= exp 1− 

1+ 

erf 

erf ⎬ (6) 

2 ⎪⎩ ⎝ 2α 

⎠ 

⎜ 

⎟ 

⎪⎭ ⎪⎩ 

⎜ ⎟ ⎜ ⎟ 

L ⎝ 

va 

⎠ ⎝ 4 αT 

x ⎠ ⎝ 4 αT 

x ⎠⎪⎭ 

5

The approximate solution employed has the same basis as method 3. Here, however, a real two-dimensional approach is 

used, as equation (6) is fitted to the concentrations of all observation wells and not only to those measured along the 

center line. 

In strategy A the first step is an analysis of the concentration distribution based on all measurements made at the 

observation wells of the monitoring network. Starting at the well with the highest measured concentration, those 

downgradient wells are identified, that best represent the center line of the plume. These then are used to estimate the rate 

constant. Only locally measured quantities are used here, i.e. local hydraulic conductivities, hydraulic heads and 

contaminant concentrations are “measured” at these wells by reading the model data at the respective nodes of the 

numerical grid. The transport velocity va between each pair of center line wells is approximated by: 

∆h 

va = K ef 

(7) 

n∆x 

with Kef the effective conductivity of local hydraulic conductivities at up and down gradient wells, n the porosity, ∆h the 

head difference and ∆x the distance between the wells. According to Rubin (2003) in stationary isotropic twodimensional 

domains with gaussian probability density functions of Y = ln(K) the effective conductivity Kef can be 

calculated as the geometric mean (cf. Bauer et al. 2006a). The porosity is assumed to be known correctly. With va, λ can 

then be calculated for each pair of center line wells using methods A1 – A3. For a set of k center line wells thus k-1 rate 

constants are calculated for one method. These are averaged to yield an estimate of the mean degradation rate constant λ. 

As an alternative to using only the locally measured conductivities, rate constant estimation is also performed using a 

global estimate of Kef, which is obtained from the geometric mean value of all hydraulic conductivities measured at all 

observation wells. 

In strategy B, method B (equation (6)) is fitted to measured concentrations of all observation wells of the monitoring 

network. Both the biodegradation rate constant λ and the source concentration C0 are varied simultaneously to achieve 

correspondence of measured and calculated concentrations, as a preliminary analysis (in agreement with results of 

Stenback et al. (2004)) showed that this procedure on average yields closer estimates of the true rate constant than fitting 

only λ with a single fixed estimate of the source concentration. A least squares criterion for the concentration residuals is 

used in the fitting procedure. As in strategy A the flow velocity va is approximated using equation (7). Kef is calculated as 

the geometric mean of hydraulic conductivities measured at all wells of the network. The average hydraulic gradient over 

the entire site is approximated by fitting a linear trend surface to all head measurements by ordinary least squares 

regression. 

For methods A2, A3 and B estimates of longitudinal and transverse dispersivities are required. Practical guidance on 

estimating these parameters at the field scale is given e.g. by Wiedemeier et al. (1999), where one suggestion is to use 0.1 

times the plume length for αL and αT as 0.1 αL. Here, however, an alternative strategy is employed: Macrodispersivities 

2 

αL and αT are derived from correlation scale, aquifer heterogeneity σ Y and travel distance (Dagan 1984; Hsu 2003; 

Rubin et al. 2003). Thus, αL is taken as 7 m, which roughly corresponds to the large time asymptotic limit for the given 

conductivity distribution, while αT is taken as the approximate peak value of transverse macrodispersivity, calculated as 

0.7 m (which thus also corresponds to the frequently used relationship as αT ≈ 0.1 αL). These values are well within the 

ranges of dispersivities commonly used for the field scale modeling of contaminant transport. The true value of the 

source width WS is assumed to be known from the site investigation. These approximations and assumptions were made 

to ensure that the error introduced in estimated rate constants due to the parameterization of methods A2, A3 and B is as 

small as possible. 

Results and Discussion 

Strategy A - One-dimensional center line approach 

The rate constants λA1 - λA3 estimated with equations (3) – (5) are divided by the “true” value used in the numerical 

simulations to yield normalized rate constants ΛA1 - ΛA3, which can directly be interpreted as overestimation or 

underestimation factors. Results in terms of mean values, medians, standard deviations and coefficients of variation (cv) 

as well as the number of realizations (N) used for strategy A are presented in Table 2 and Figure 3. Comparing methods 

A1 - A3 for the small source width WS = 4 m yields that all approaches on average result in a distinct overestimation of λ. 

For method A1 λ is overestimated on average by a factor of 6.88, while for A2 an mean ΛA2 = 8.24 is observed. Hence, 

method A1 performs better than A2. Method A3 yields a slightly lower mean of ΛA3 = 6.82, while the spread of results 

for A3 is noticeably larger, as can also be seen by the higher cv. Three main error sources for the observed 

overestimation exist: 

• neglect of dispersion by method A1, which thus is attributed to the degradation process (λA1 represents a bulk 

attenuation rate constant) 

6

• deviation of sampling well locations from the true center line position, which causes the sampling of too low 

concentrations 

• unrepresentative estimates of the local va along the flow path, as the estimated λ increases and decreases with va 

(cf. equations (3) – (5)). 

These effects are discussed in detail in Bauer et al. (2006a). For method A2 the additional bias towards too large rate 

constants is a consequence of accounting only for αL in equation (4), as with a one-dimensional transport model 

longitudinal dispersion of a degrading contaminant results in a stronger spreading of the solute downstream and 

consequently in higher concentrations along the center line of a steady state plume compared to an advection only case. 

Therefore a larger rate constant is needed to fit a given concentration decrease and the “hybrid” rate constant estimate ΛA2 

is always larger than the bulk attenuation rate constant ΛA1 (cf. Bauer et al. 2006a), which appears contradictory. 

Table 2: Normalized degradation rate constants estimated 

with strategy A. 

WS = 4 m 

method A1 method A2 method A3 

mean 6.88 8.24 6.82 

median 4.43 5.50 2.65 

stdv. 6.36 7.41 10.09 

cv 0.92 0.90 1.48 

N 47 47 47 

WS = 16 m 


mean 4.46 5.33 4.99 

median 2.66 3.23 2.45 

stdv. 4.81 6.29 9.59 

cv 1.08 1.18 1.92 

N 38 38 36 

For the source width WS = 16 m all methods improve (Table 2, Figure 3). Still, method A1 shows the closest estimates of 

the rate constant and the lowest variability of single realization results in comparison to methods A2 and A3. With 

method A3 for two out of 38 plumes no reasonable rate constant estimate is obtained. This effect is due to an overcorrection 

for transverse dispersion in the β term of equation (5), which can cause the corrected C(x) to be very close to 

or even larger than the respective upgradient concentration C0, yielding very small or even negative λA3. 


100 

10 

1 

W S = 4 m W S = 16 m 

single realization result 

ensemble mean 

0.1 

A1 1 A2 2 A3 3 

A1 1 A2 2 A3 3 

method 

method 

Figure 3: Normalized degradation rate constants estimated with strategy A and methods A1, A2 and A3 for source widths 

of 4 m (left diagram) and 16 m (right diagram). Small symbols represent results of individual realizations, large symbols 

the mean of all realizations. Kef used for rate estimation is calculated from measurements at center line wells only. 

7

As Bauer et al. (2006a) demonstrated, the local flow velocity estimate is a crucial parameter for the estimation of the 

degradation rate constant. In a sensitivity study the authors found that inclusion of field scale information on hydraulic 

conductivity, which is more representative for the entire site, can improve the accuracy of estimated rate constants over 

usage of local information only. Therefore the center line data obtained in strategy A are re-evaluated using a field scale 

estimate of the effective conductivity at the investigated sites: Kef is calculated as the geometric mean of local 

conductivity values measured at all observation wells of the site and not only using those measured on the center line. 

Results for this approximation are presented in Table 3 and Figure 4. It is found that consideration of all conductivity 

measurements available improves the rate constants estimated for all three methods and both source widths. On average 

all estimates are closer to the true rate than those estimated using only local conductivity information. The average 

improvement is between a factor of 1.5 for WS = 4 m and a factor of two for WS = 16 m. Also the standard deviations 

from the mean values and the cv are lower. This finding is true for the settings investigated here, i.e. a correlation length 

smaller than the average distance between observation wells and a stationary K distribution. If the correlation length is on 

the order or longer than the average observation well distance or K is not stationary, this finding is probably not 

transferable. 


100 

10 

1 

0.1 

Table 3: Normalized degradation rate constants estimated 

with strategy A, re-evaluated using a field scale estimate of 

Kef. 

WS = 4 m 


mean 4.75 5.73 4.14 

median 4.04 4.67 2.66 

stdv. 3.75 4.33 4.06 

cv 0.79 0.76 0.98 

N 47 47 47 

WS = 16 m 


mean 2.16 2.53 2.42 

median 1.14 1.50 0.90 

stdv. 2.08 2.45 3.98 

cv 0.96 0.97 1.64 

N 38 38 36 

W S= 4 m W S = 16 m 


ensemble mean 

A1 1 A2 2 A3 3 

A1 1 A2 2 A3 3 

method 

method 

Figure 4: Normalized degradation rate constants estimated with strategy A and methods A1, A2 and A3 for source widths 

of 4 m (left diagram) and 16 m (right diagram), respectively. Kef used for rate estimation is calculated as a field scale 

estimate from all monitoring wells available for each investigated plume. 

8

An interesting observation is that the differences between the three methods of strategy A are not as distinct as observed 

in Bauer et al. (2006a). One explanation for this finding is that due to the larger number of observation wells used for the 

site investigation in this study, the plume center line positions are better identified on average and concentration samples 

are taken closer to the true plume axis. Methods A1, A2 and A3 show a different sensitivity on deviations of 

measurement locations from the plume center line. This is because in method A1, the rate constant estimate is linearly 

related to ln( C( x) 

/ C0 

) / ∆x 

, while in method A2 this term appears in linear as well as squared form after rearrangement 

of equation (4). As a consequence, increasing deviations of observation well locations from the plume center line and 

thus lower measured contaminant concentrations will result in increasingly stronger overestimation of λ by A2 relative to 

A1. For method A3, the correction factor β has to be taken into account additionally (cf. equation (5)). For significant 

deviations from the center line, however, the same effect as for A2 can be shown. 

In Figure 5 the degradation rate constants ΛA1 estimated with method A1 were plotted against the number of observation 

wells with relative concentrations C/C0 > 0.001 in the respective monitoring networks. A clear relationship between the 

number of wells and the accuracy of ΛA1 is neither observed for WS = 4 m nor for WS = 16 m. It seems, however, that the 

spread of estimated rate constants decreases slightly with increasing numbers of wells, but a larger number of samples, 

especially for numbers of monitoring wells > 30 would be required to derive more meaningful results. For the other 

methods A2 and A3, the similar observations are made (not shown here). 

norm. deg. rate constant Λ A1 [-] 

50 

10 

1 

0.1 

Y = -0.074 * X + 6.531; R 

0 20 40 60 80 

no. of wells with C/C0 > 0.001 

2 = 0.063 

Y = -0.043 * X + 3.174; R2 linear fits 

= 0.072 

Figure 5: Degradation rate constants ΛA1 for source widths of 4 m (grey diamonds) and 16 m (black crosses) versus the 

number of wells showing relative concentrations C/C0 > 0.001. 

9 

W S = 4 m 

W S = 16 m 

Strategy B - Two-dimensional evaluation 

Applying method B it is not always possible to obtain a rate constant λB > 0 when minimizing the sum of squared 

residuals of concentration. The concentration distribution calculated with the analytical model indicates absence of 

contaminant degradation for the closest fit to the measured data for six out of 47 plumes when WS = 4 m. For the 

remaining 41 plumes, usage of method B on average yields ΛB = 4.51 (Table 4, Figure 6), which is considerably closer to 

the true rate constant than for methods A1 - A3 with using only local measurements of hydraulic conductivity along the 

center line (cf. Table 2). The standard deviation for method B, however, is slightly larger than for A1 and A2. With WS = 

16 m for five out of 38 plumes no ΛB > 0 is found. Here, the improvement over methods A1 – A3 is even more distinct 

with the mean ΛB = 1.92. Also the spread of single realizations is reduced for the larger source width. However, no 

definite improvement of ΛB over those obtained by strategy A using the field scale estimate of Kef (cf. Table 3) is 

observed. For WS = 4 m the mean ΛB is slightly lower than the mean ΛA1, but slightly larger than ΛA3. Also the variability 

of estimated ΛB is larger than for methods A1 – A3. For WS = 16 m the mean ΛB is slightly lower than for all methods of 

strategy A, while the observed spread is only lower in comparison to A3. 

In the estimation of λ with method B the source concentration is included as a fitting parameter. On average the true 

source concentration is overestimated by a factor of three for WS = 4m, while for WS = 16m deviations are lower than 5 

%. As for strategy A a distinct relationship between the number of observation wells in the monitoring network and the 

quality of the degradation rate estimates is not observed.

Table 4: Normalized degradation rate 

constants estimated with method B. 

WS = 4 m WS = 16 m 

mean 4.51 1.92 

median 1.96 1.14 

stdv. 8.34 2.97 

cv 1.85 1.55 

N 41 33 


100 

10 

1 

0.1 

WS = 4 m WS = 16 m 

Figure 6: Normalized degradation rate constants estimated with method B for source widths of 4 m and 16 m. 

10 


ensemble mean 

Variability of estimated rate constants for a single site 

For each source width, one single plume was repeatedly investigated 13 times by different investigators. For this subset 

of plumes the variability of estimated rate constants for a single site, resulting from different notions on “sufficient 

accuracy” of the plume investigation, is studied. As the incorporation of conductivity measurements from all observation 

wells of the monitoring network significantly improves rate constant estimation with methods A1, A2 and A3, this 

approach is used in this comparison. Results for strategies A and B are summarized in Table 5 and Figure 7. 

For source width WS = 4 m a slightly stronger overestimation of λ can be observed for strategy A (methods A1 - A3) in 

comparison to the results for the complete data set. Here, the mean ΛA1 resulting from 13 investigations of the single site 

is 5.26, while ΛA1 for the complete data set is 4.75. For methods A2 and A3 the same tendency of increased 

overestimation is found. The variability among the sets of 13 estimated ΛA1, ΛA2 and ΛA3, however, is much smaller. The 

cv here are only 0.3, 0.34 and 0.68, respectively. With strategy B, i.e. method B, a slightly smaller mean ΛB = 4.26 is 

found than for the complete data set and the cv of 0.78 here also is reduced. For a source width of 16 m, all four methods 

show a much lower mean value of normalized rate constants than for the complete data set. On average, method A1 and 

A3 yield the correct result for the single investigated site, while method A2 and B show a slight overestimation. 

Variability of the estimated rate constants in terms of cv is lower than for the complete data set of WS = 16 m. For 

strategy B no clear dependence between the precision of the rate constant estimate and the absolute number of wells in 

the monitoring network nor the number of respective wells showing considerable contaminant concentrations (e.g. 

C/C0 > 0.001) could be observed. The absence of such an obvious relationship may be due to the low sample size of only 

13 investigations of both plume realizations. As described before, it seems, however, that the spread of estimated rate 

constants decreases slightly with increasing numbers of wells (not shown here). 

The stronger overestimation observed for WS = 4 m as well as the more precise estimation of λ for WS = 16 m do not 

allow for any general conclusions regarding the absolute magnitude of errors, as this might probably be an effect of the 

two individual realizations investigated and their respective plume geometry. The variability observed among the 13 

results for each of the two single plumes however shows, that for a single realization of a contaminant plume, 

substantially different investigation results, in this case degradation rate constants, may be obtained, when investigated 

by individual persons. For both, the narrow plume with a comparatively small source zone of 4 m width as well as for the 

wider plume with WS = 16 m, the variability of investigation results here has a magnitude of 25 % up to 69 % of the 

variability observed among different realizations of the plume, depending on the method used for rate constant 

estimation. This clearly shows, that a substantial part of rate constant estimation uncertainty is due to the individual 

configuration of the monitoring network.


100 

10 

1 

0.1 

W S = 4 m W S = 16 m 


ensemble mean 

A1 1 A2 2 A3 3 4B 

A1 1 A2 2 A3 3 4B 

method 

method 

Figure 7: Comparison of normalized degradation rate constants estimated with methods A1 – A3 and B for two single 

plume realizations with different source widths (Ws = 4 m: left diagram; Ws = 16 m: right diagram). Both plume 

realizations were repeatedly and independently investigated by 13 individual persons each. 

Summary and Conclusions 

Table 5: Normalized degradation rate constants estimated with methods 

A1 – A3 and B for two individual plume realizations with different source 

widths (Ws = 4 m and 16 m). Both plume realizations were independently 

investigated by 13 individual persons each. 

WS = 4 m 

method A1 method A2 method A3 method B 

mean 5.26 6.25 4.56 4.26 

median 5.02 5.88 2.94 3.47 

stdv. 1.59 2.11 3.10 3.33 

cv 0.30 0.34 0.68 0.78 

N 13 13 13 13 

WS = 16 m 

method A1 method A2 method A3 method B 

mean 1.01 1.28 1.00 1.21 

median 0.97 1.15 0.87 1.20 

stdv. 0.31 0.51 0.48 0.51 

cv 0.31 0.40 0.48 0.42 

N 13 13 13 13 

In this study, the frequently used center line approach for estimation of degradation rate constants is compared to a 

methodology suggested by Stenback et al. (2004), which uses all concentration measurements downstream of the 

contaminant source zone to estimate the degradation rate. In a numerical experiment, both strategies are applied to a set 

of 85 synthetic contaminant plumes, subject to first order degradation, and investigated using extensive monitoring 

networks. Rate constants are estimated using concentrations, hydraulic heads and conductivities locally measured at the 

observation wells of the monitoring network. 

Results show that rate constants tend to be overestimated in general. Using the center line approach (strategy A) and a 

simplified observation well set up causes a general overestimation due to three factors: 

• biased estimates of the mean flow velocity, 

• inadequate parameterizations of longitudinal and transverse dispersivities and 

• off center line measurements. 

11

Comparing results for source widths of 4 and 16 m yields, that rate constant estimation improves significantly for wider 

plumes, as it is less likely, that observation wells are placed off the center line. Estimated rate constants also become 

more accurate when more than only local information on hydraulic conductivity is used to approximate the effective 

conductivity along the flow path: Taking Kef as the geometric mean of all measured K values from the monitoring 

network reduces rate constant overestimation, although wells are not necessarily located on the flow path or even within 

the contaminant plume. 

For the two-dimensional approach (strategy B), sources of error are inadequate parameterizations of longitudinal and 

transverse dispersivities and an erroneous approximation of the mean flow velocity. As all observation wells 

downgradient of the source zone are incorporated in the estimation procedure, off center line measurements are not 

relevant for this strategy. However, when the contaminant plume shows significant meandering, the applicability of 

strategy B is hampered, as the analytical model which is fitted to the concentration measurements presumes a linear 

plume axis. This problem is observed for the small source width of 4 m, where overestimation by strategy B is 

comparable to the one-dimensional approaches using the global estimate of Kef. For larger source widths like the 16 m 

used in this study, however, strategy B yields closer estimates of the degradation rate constant than strategy A on 

average. 

One drawback of the two-dimensional approach is that source concentration and degradation rate constant have to be 

fitted simultaneously to obtain accurate estimates of the degradation rate constant. This proves problematic especially for 

the small source width of 4 m where for some realizations the source concentration is overestimated by up to 300 %. For 

the source width of 16 m, however, the source concentration is fitted precisely within 5% of the “true” source 

concentration. These results suggest that incorporation of off center line information in the estimation of the degradation 

rate can improve results of the plume investigation considerably. Especially for wide plumes, the two-dimensional 

approach of Stenback et al. (2004) proves superior to the one-dimensional center-line investigation strategy, resulting in 

more precise estimates of the degradation rate constant. 

Studying the variability of estimated degradation rates due to different monitoring network configurations, i.e. the 

personal factor due to different “site investigators”, it is found that for both source widths the variability of investigation 

results of one realization ranges from 25 % up to 69 % of the variability observed among different realizations. This 

demonstrates that the configuration of the monitoring network following from the investigators individual site 

investigation approach represents a substantial part of the uncertainty of the estimated rate constant. 

Overestimation of the degradation rate at a contaminated site is a critical point if monitored natural attenuation is 

considered as an alternative to conventional engineered remediation measures because the overall NA potential is 

assessed too positive. As demonstrated by Beyer et al. (2006) plume lengths calculated with biased degradation rates will 

result in an underestimation of the contaminated regions of the aquifer. Such an application, however, might not be of 

primary concern, when monitoring networks of sufficient density have already been established over a site. Nonetheless, 

a too high rate constant could falsely lead to the conclusion, that a plume is at steady state, when the present day length 

observed fits the calculated steady state plume length. In the assessment of contaminated sites, indication for a steady 

state plume is an important result, e.g. for the acceptance of NA as a remediation scheme. Also for well investigated sites, 

a reliable determination of contaminant degradation rates is important, when the rates are to be used as modeling 

parameters, for comparison with other sites and for comparing and optimizing alternatives of remediation measures. 

Acknowledgements: This work is funded by the German Ministry of Education and Research (BMBF) under grant 

033 05 12 / 033 05 13 as part of the KORA priority program, sub-project 7.2. We wish to thank our project partners at the 

Christian-Albrechts-University Kiel Andreas Dahmke and Dirk Schäfer for their support in our research. We 

acknowledge the support of Uwe Wittmann, Iris Bernhardt and Ludwig Luckner in coordination of the project work. Last 

but not least we are grateful to Bernie Kueper and two anonymous reviewers for their thoughtful comments and 

suggestions which considerably improved this paper. 

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13


Beyer, C., Konrad, W., Park, C.H., Bauer, S., Rügner, H., Liedl, R., Grathwohl, P. (2007b): 




via SpringerLink), doi:10.1007/s00767-007-0025-x 

The enclosed article is made available with the permission of Springer and was published in 

the journal Grundwasser, online. Copyright © 2007 Springer. 

The article can be obtained online via SpringerLink at 

http://www.springerlink.com/openurl.asp?genre=journal&eissn=1432-1165.

Modellbasierte Sickerwasserprognose für die Verwertung von 

Recycling-Baustoff in technischen Bauwerken * 

Model based prognosis of contaminant leaching for reuse of demolition waste 

in construction projects 

Christof Beyer 1 , Wilfried Konrad 1 , Hermann Rügner 2 , Sebastian Bauer 1 , Park Chan Hee 1 , Rudolf 

Liedl 3 , Peter Grathwohl 1 

1 Eberhard-Karls-Universität Tübingen, Zentrum für Angewandte Geowissenschaften (ZAG), Sigwartstraße 10, 72076 

Tübingen; Telefon: 07071-29 73176, Telefax: 07071-5059, E-mail: christof.beyer@uni-tuebingen.de 

2 

Umweltforschungszentrum Leipzig-Halle (UFZ), Permoserstraße 15, 

04318 Leipzig 

3 

TU Dresden, Institut für Grundwasserwirtschaft, 01062 Dresden 

Header: Sickerwasserprognose für Recycling-Baustoff 

Kurzfassung: 

In dieser Studie wird die in der BBodSchV rechtlich etablierte Sickerwasserprognose in Bezug auf die 

Beurteilung der von Recycling-Baustoff-Verwertungen im Straßenbau ausgehenden Schadstoffeinträge ins 

Grundwasser weiterentwickelt. Anhand numerischer reaktiver Stofftransportsimulationen für drei 

praxisrelevante Verwertungsszenarien (Parkplatz, Lärmschutzwall, Straßendamm) sowie eine Auswahl 

regionaltypischer Unterbodeneinheiten Deutschlands werden zeitliche Konzentrationsverläufe verschiedener 

Stoffklassen an der Grundwasseroberfläche berechnet. Der Durchbruchszeitpunkt konservativer Tracer wird 

allein von den hydraulischen Eigenschaften der Unterböden gesteuert, für organische Schadstoffe sind vor 

allem deren KOC-Werte und die Corg-Gehalte der Unterböden ausschlaggebend. Signifikante dispersive 

Konzentrationsverminderungen ergeben sich nur bei deutlicher Abnahme der Quellstärke vor dem 

Durchbruch der Konzentrationspeaks. Bei lang anhaltend hohen Quellkonzentrationen relativ zur 

Transportzeit bleiben die Konzentrationsdurchbrüche unvermindert. Biologischer Schadstoffabbau führt zu 

deutlich reduzierten Durchbruchskonzentrationen. Für die Szenarien Lärmschutzwall und Straßendamm 

werden Kapillarsperreneffekte beobachtet, die zu einem teilweisen Umfließen der Schadstoffquelle führen. 

Bei Berücksichtigung des am Recyclingmaterial vorbeiströmenden Sickerwassers durch Konzentrationsmittelung 

über die gesamte Bauwerksbreite ergeben sich Konzentrationsminderungen um 30-40%. 

Abstract: 

In this study contaminant leaching from recycling materials in road constructions to groundwater is assessed 

by the “Sickerwasserprognose”. Numerical transport simulations for three scenarios (parking lot, noise 

protection dam, road) and a number of characteristic subsoils of Germany are performed to estimate the 

breakthrough of different contaminant classes at the groundwater table. Conservative tracer breakthrough 

times (BTT) primarily depend on subsoil hydraulic properties, for organic pollutants Koc and subsoil OC are 

the controlling parameters. Significant concentration reductions from dispersion only occur when source 

concentrations decrease prior to contaminant breakthrough. If source concentrations remain high for long 

periods relative to peak BTT, concentration breakthrough is undamped. Accounting for biodegradation 

reduces breakthrough concentrations significantly. For the scenarios "noise protection dam" and "road" capillary 

barrier effects cause the seepage water to partially bypass the recycling material. Accounting for this 

bypass flow and averaging spatially across the constructions reduces concentrations by about 30-40%. 

Keywords: ground water risk assessment; reuse; demolition waste; type-scenarios; road construction; 

modelling 

* Beyer, C., Konrad, W., Park, C.H., Bauer, S., Rügner, H., Liedl, R., Grathwohl, P. (2007b): Modellbasierte Sickerwasserprognose für 

die Verwertung von Recycling-Baustoff in technischen Bauwerken. (Model based prognosis of contaminant leaching for reuse of demolition 

waste in construction projects.) (accepted by Grundwasser), doi:10.1007/s00767-007-0025-x 

Der Artikel wurde von der Zeitschrift Grundwasser zur Veröffentlichung angenommen, online publiziert und mit Erlaubnis von Springer 

reproduziert. Copyright © 2007 Springer. Der Artikel ist online abrufbar via SpringerLink: 

http://www.springerlink.com/openurl.asp?genre=journal&eissn=1432-1165 

1

Einleitung und Zielsetzung 

In Deutschland fallen jährlich etwa 250 Mio. t mineralischer Abfälle an, die in erheblichem Umfang im Erd-, 

Straßen- und Verkehrsflächenbau verwertet werden. Recycling-Baustoffe und Rückstände aus der 

industriellen Produktion (z. B. Hochofenschlacken) oder aus der Abfallbehandlung (z. B. Hausmüllverbrennungsaschen) 

finden unter anderem in Trag- und Frostschutzschichten beim Straßenbau, zum Bau von 

Lärmschutzwällen oder als Verfüllmaterial zunehmende Verwendung (Krass et al. 2004a; 2004b). Die 

Verwertung von Recycling-Baustoffen (RCB) in technischen Bauwerken wird in Deutschland in 

Übereinstimmung mit der diesbezüglichen Politik der Europäischen Union gegenüber einer Deponierung 

prinzipiell vorgezogen (KrW-/AbfG 1994). Aufgrund der häufig vorliegenden Belastung von RCB durch 

organische und anorganische Schadstoffe (z.B. PAK, Salze, Schwermetalle) muss bei der Verwertung in 

technischen Bauwerken jedoch berücksichtigt werden, dass Inhaltsstoffe durch das Sickerwasser 

ausgewaschen werden und das Grund- und Oberflächenwasser belasten können. Aus diesem Grund ist 

eine Bewertung von Verwertungsmaßnahmen hinsichtlich ihrer Umweltauswirkungen notwendig. Das 

Schadstoffaustragsverhalten technischer Bauwerke ist deshalb ein sowohl auf nationaler als auch auf 

internationaler Ebene aktuelles und intensiv bearbeitetes Forschungsgebiet. So führten Hjelmar et al. (2007) 

großskalige Feldstudien an einem Straßenabschnitt in Dänemark durch und beobachteten ein zeitliches 

Abklingen der Schadstoffquellstärke, welches bei der Umweltwirkungsprognose berücksichtigt werden sollte. 

Für einige Schadstoffe ergäben sich so weniger konservativen Prognosen, die weniger restriktive 

Grenzwerte erlauben würden, ohne den Schutz des Grundwassers zu gefährden (Hjelmar et al. 2007). Die 

Autoren wiesen zudem das Auftreten von Kapillarsperren in Straßendämmen nach, die zu einem Umfließen 

des Verwertungsmaterials und somit zu reduzierten Wasserflüssen durch die Schadstoffquelle führten. 

Kapillarsperreneffekte wurden auch von Hansson et al. (2006) bei numerischen Simulationen der 

Wasserströmung in Straßendämmen beobachtet. Susset (2007) führte Freilandlysimeteruntersuchungen mit 

RCB über Löss- und geringsorptiven Sandbodenmonolithen durch und stellte einen Rückhalt von PAK über 

bisher 4 Jahre nach, wobei die Wasserdurchsatzraten übertragen auf Feldbedingungen mehreren 

Jahrzehnten entsprechen. Sulfatkonzentrationen an der Unterkante des RCB gingen im 

Beobachtungszeitraum kaum zurück und brachen in voller Höhe durch. Für Chlorid hingegen konnte eine 

dispersive Verdünnung der Maximalkonzentration beobachtet werden, da die „Lebensdauer“ der 

Chloridquelle deutlich kleiner als die Transportzeit durch die Lysimeter war. In umfangreichen Säulenexperimenten 

mit RCB über Böden unter feldnahen Bedingungen zeigten Stieber et al. (2006), dass 

organische Kontaminanten wie PAK in der ungesättigten Zone effektiv durch mikrobiellen Abbau eliminierbar 

sind, sofern die dafür physiologisch notwendigen Milieubedingungen nicht durch das Sickerwasser nachteilig 

verändert werden. 

Die Vielzahl der zu berücksichtigenden hydraulischen, geochemischen und mikrobiologischen Prozesse 

sowie die Komplexität ihrer Interaktionen macht deutlich, dass die Anwendung von prozessbasierten 

Modellen bei der Bewertung von umweltoffen verwerteten Recyclingmaterialien große Vorteile bietet. Vor 

diesem Hintergrund untersucht diese Arbeit in Anlehnung an die Methodik der Sickerwasserprognose nach 

BBodSchV (1999) anhand numerischer reaktiver Stofftransportsimulationen, welche Schadstoffeinträge von 

im Straßenbau eingesetztem RCB über die ungesättigte Zone ins Grundwasser unter Berücksichtigung 

möglicher zeitabhängiger Festlegungs- und Abbauprozesse (Sorption, Intrapartikeldiffusion, Bioabbau) 

erfolgen können. Dazu wird hier erstmals eine Kopplung des Stromröhrenmodells SMART (Finkel 1998, 

Finkel et al. 1998) mit dem Finite-Elemente-Modell GeoSys/Rockflow (Kolditz & Bauer 2004; Kolditz et al. 

2006) eingesetzt. Die Kombination verschiedener repräsentativer Verwertungsszenarien (Parkplatz, 

Lärmschutzwall, Straßendamm), RCB-typischer Schadstoffklassen und regionaltypischer Unterböden führt 

zu einem umfangreichen Satz von Typ-Szenarien-Simulationen, deren Ergebnisse es erlauben, 

• den aus den Verwertungsszenarien zu erwartenden charakteristischen zeitlichen Verlauf der 

Schadstoffeinträge in Boden und Grundwasser darzustellen, 

• die während des Transports für Schadstoffrückhalt und Konzentrationsminderung relevanten 

Prozesse zu identifizieren, 

• Einflüsse der Unterbodeneigenschaften hinsichtlich des Filter- und Puffervermögens herauszustellen 

und 

• das Potential der modellbasierten Sickerwasserprognose praxisnah zu demonstrieren. 

Modellkonzept 

Die Berechnung der Schadstoffeinträge aus dem RCB über die ungesättigte Zone ins Grundwasser wird in 

dieser Studie erstmalig mit Hilfe einer Kopplung zwischen dem Finite-Elemente-Modell GeoSys/Rockflow 

2

und dem stochastischen Stromröhrenmodell SMART durchgeführt. Die Simulationsaufgabe wird hier in die 

zwei Teilprobleme Hydraulik/Strömung und reaktiver Transport zerlegt. Diese Vorgehensweise wurde 

gewählt, um die Auswirkung der räumlich variablen Sickerwasserströmung in den Verwertungsszenarien auf 

den Schadstofftransport berücksichtigen sowie physikalisch realistische Schadstofffreisetzungsprozesse in 

der Quelle und die durch Intrapartikeldiffusion zeitabhängige Sorption beim Transport quantifizieren zu 

können. 

Die Kopplung zwischen der Strömungssimulation mit GeoSys/Rockflow und der reaktiven 

Transportsimulation mit SMART erfolgt über Verteilungsfunktionen (pdf, "probability density function") der 

Verweilzeiten des Sickerwassers, welche die Einflüsse der hydraulischen Heterogenität auf das 

Strömungsfeld beschreiben (vgl. Utermann et al. 1990; Bold 2004). Die pdf sind Modellergebnis der 

Strömungssimulation und werden von SMART als Modelleingabe benötigt, um in der reaktiven 

Transportsimulation Massendurchbruchskurven an einer unterstromigen Kontrollebene (hier die 

Grundwasseroberfläche) zu berechnen. Da das SMART-Konzept zwar Transportsimulationen in heterogen 

zusammengesetzten Materialien erlaubt (Lithokomponenten-Ansatz nach Kleineidam et al. 1999a), entlang 

der Stromröhre jedoch eine homogene Materialverteilung vorausgesetzt wird, muss für jede Material- bzw. 

Modellschicht der aus mehreren Baumaterialien und Unterbodenhorizonten bestehenden Typ-Szenarien 

eine separate SMART-Simulation gerechnet werden. Das Modell ist somit als eine Reihe von hintereinander 

geschalteten 1D-Stromröhren konzipiert (siehe Abb. 1). 

GeoSys/Rockflow SMART 

35cm 

Charakterisierung 

der Strömung 

konst. Infiltration 

Tragschicht 

RCB 

Körnung 

0/32 

Unterboden 

Untergrund 

Grundwasseroberfläche 

(GWO) 

Tracereingabe 

BTC Tragschicht 

Tracereingabe 

BTC Unterboden 

Tracereingabe 

BTC Untergrund 

pdf Tragschicht 

pdf Unterboden 

pdf Untergrund 

3 

reaktiver 

Transport 

Stromröhre 

Tragschicht 

Stromröhre 

Untergrund 

Modellergebnis 

BTC an GWO 

Abb. 1: Veranschaulichung des Modellkonzepts anhand des Parkplatzszenarios: Simulation der 

Tracerdurchbruchskurven zur Ableitung von Laufzeitverteilungen (pdf) der einzelnen Modellschichten mit 

GeoSys/Rockflow; reaktive Transportsimulationen mit SMART (BTC = Durchbruchskurve); Kopplung beider 

Modelle durch die pdf. 

Die modellierte Massendurchbruchskurve an der unterstromigen Kontrollebene jeder Stromröhre 

repräsentiert jeweils die oberstromige zeitabhängige Konzentrationsrandbedingung der nachfolgenden 

Stromröhre bzw. im Fall der untersten Modellschicht den Schadstoffeintrag ins Grundwasser. 

Dementsprechend muss für jede Modellschicht eine eigene pdf abgeleitet werden. Aus Gründen der 

Modellvereinfachung werden die einzelnen pdf als unabhängig voneinander betrachtet (vgl. Vanderborght et 

al. 2007). Zur Ableitung der pdf wird in GeoSys/Rockflow für jede einzelne Materialschicht an der 

Schichtobergrenze ein konservativer Tracer mit konstanter Konzentration C0 in das Strömungsfeld 

eingegeben und dessen Durchbruch C(t) an der Schichtuntergrenze zeitlich registriert (siehe Abb. 1). Die pdf 

der Modellschicht ergibt sich als Ableitung der Durchbruchskurve dC/dt aufgetragen über die Zeit t. 

Die Kopplung beider Simulationsmodelle wurde als sogenannte lose Kopplung realisiert, d.h. die Quellcodes 

wurden nicht kombiniert sondern kommunizieren über Batch-Aufrufe und automatisch generierte Eingabe- 

/Ausgabedateien während bzw. nach der Programmausführung. Diese Art der Einbindung hat im Hinblick auf

die Anzahl Simulationsläufe den Vorteil, dass pro Kombination aus Verwertungsmaßnahme und 

Unterbodentyp nur eine Simulation des Strömungsfeldes zur Bestimmung aller pdf notwendig ist und diese 

jeweils für die darauf folgenden Transportsimulationen der einzelnen Schadstoffklassen mit SMART 

verwendet werden können. 

Numerische Simulation der Strömung und des reaktiven Transports 

Die Simulation der ungesättigten Strömung mit GeoSys/Rockflow zur Ableitung der pdf erfolgt auf Grundlage 

der Richards-Gleichung, zu deren Lösung das Van-Genuchten-Mualem-Modell (van Genuchten 1980; 

Mualem 1976) verwendet wird (Du et al. 2005). Auf eine umfassende Ausführung der allgemein bekannten 

Modellgleichung wird hier und im Folgenden verzichtet, eine ausführliche Erläuterung der verwendeten 

mathematischen Modelle ist in Grathwohl et al. (2006) zu finden. Die Berechnung der Tracerbewegung im 

ungesättigten Strömungsfeld zur Ableitung der pdf erfolgt auf Grundlage der Konvektions- 

Dispersionsgleichung. Die Beschreibung des reaktiven Transports mit SMART erfolgt dagegen wie oben 

beschrieben nicht entlang von Raumkoordinaten sondern in Abhängigkeit der Verweilzeit entlang einer 

Stromröhre (Finkel, 1998). Die Rückhalteprozesse beim Transport werden durch das in SMART integrierte 

numerische Modell BESSY (Jäger & Liedl 2000) quantifiziert. Die Berücksichtigung der den 

Sorptionsprozess zeitlich limitierenden Intrapartikel-Porendiffusion ist insbesondere für Böden mit hohem 

Kies- oder Skelettanteil notwendig (Grathwohl 1998; Rügner et al. 1997; 1999). Die Sorptionskinetik wird 

unter Annahme von sphärischen Partikeln durch das 2. Fick´sche Gesetz in Radialkoordinaten beschrieben: 

2 

∂ w ⎡∂ 

Cw 

2 ∂ C 

= Da⎢ 

+ 2 

∂ t ⎣ ∂ r r ∂ r 

C w 

r [m] bezeichnet den radialen Abstand vom Kornmittelpunkt, CW [kg m -3 ] die Konzentration in der wässrigen 

Phase der Intrapartikelporen und Da [cm 2 s -1 ] den scheinbaren Diffusionskoeffizienten, der gegenüber der 

Diffusion in Wasser (Daq) vermindert ist: 

D 

D 

ε 

4 

⎤ 

⎥ 

⎦ 

(1) 

aq 

a = (2) 

( ε + Kd 

ρk 

) τ f 

ε [-] bezeichnet die Intrapartikelporosität, τf [-] den Tortuositätsfaktor, Kd [m -3 kg -1 ] den Gleichgewichts- 

Verteilungskoeffizienten und ρk [kg m -3 ] die Partikeldichte gemäß ρk = ( 1− 

ε )ρ mit ρ [kg m -3 ] der 

Mineraldichte. 

Die Verteilung der Schadstoffe zwischen dem in den Intrapartikelporen gelösten und dem sorbierten Anteil 

wird durch eine lineare Sorptionsisotherme beschrieben, für den Schadstoffabbau im mobilen Porenwasser 

wird eine Kinetik erster Ordnung angenommen. Somit ergibt sich unter stationären Fließbedingungen ein 

insgesamt lineares mathematisches Modell für die im Sickerwasser auftretenden Konzentrationen C und 

eine Proportionalität zwischen C und den anfänglich aus der Schadstoffquelle eluierenden Schadstoffkonzentrationen 

C0, bzw. den anfanglich sorbierten Stoffmengen. 

Parametrisierung der Typ-Szenarien 

Im Rahmen dieser Studie werden drei repräsentative Verwertungsszenarien betrachtet: Parkplatz (PP), 

Lärmschutzwall (LSW) und Straßendamm (SD). Das in diesen zum Einsatz kommende Verwertungsmaterial 

besteht aus granularem RCB. Als Schadstoffe werden vier Modellsubstanzen verwendet, die in ihren 

Eigenschaften für RCB typische Schadstoffe repräsentieren: Naphthalin (NAP) und Phenanthren (PHE) als 

schwach bzw. mäßig sorbierende organische Schadstoffe, ein stark sorbierender für das durchschnittliche 

Verhalten der 15 EPA-PAK repräsentativer Summenparameter (Σ15 EPA-PAK) sowie ein konservativer 

Tracer als Beispiel leicht löslicher Salze wie Chlorid. Für die Transportstrecken unterhalb der Quellterme 

werden sechs regionaltypische Unterbodeneinheiten Deutschlands berücksichtigt, um Charakteristika und 

Unterschiede hinsichtlich ihrer Filter- und Pufferkapazitäten herauszustellen. Die folgenden Abschnitte 

erläutern die Parametrisierung der Typ-Szenarien (für detailliertere Ausführungen siehe Grathwohl et al. 

2006).

Verwertungsszenarien und -materialien 

Das PP-Szenario wird als vertikales 1D-Modell betrachtet und orientiert sich am Querschnitt für 

Verkehrsflächen der Bauklasse VI mit Pflasterdecke und ungebundener Tragschicht auf einer 

Frostschutzschicht (FSS) nach FGSV (2001). Für Tragschicht und FSS, die im Modell als einheitliche 

Schicht von 0.35 m betrachtet werden, wird RCB mit einer Körnung der Sieblinie 0/32 (FGSV 2004) 

eingesetzt. Das Pflaster selbst wird nicht berücksichtigt, da die geforderte Versickerungsleistung von 2.7*10 -5 

m s -1 (FGSV 1998) deutlich über den hier angenommenen Infiltrationsraten liegt (s.u.). Abb. 1 zeigt neben 

dem Modellkonzept auch den für die Modellierung vereinfachten PP-Aufbau. 

Die Szenarien LSW und SD werden als 2D-Vertikalschnitte betrachtet, bei denen die Simulationen aus 

Symmetriegründen nur für jeweils eine Hälfte des Querschnitts durchgeführt wurden. Das LSW-Modell (Abb. 

2) orientiert sich an einem von Mesters (1993) experimentell untersuchten LSW. Als Bodenabdeckung des 

Wallkerns aus RCB wird ein Lehmboden verwendet. 

0.5m 

3.5m 

1m 6m 

2m 

RCB 

Körnung 0/32 

Unterboden 

Untergrund 

Symmetrieachse 

1:1. 5 

Planum 

Abb. 2: Lärmschutzwall mit Wallkern aus Recycling-Baustoff (RCB). 

0.5m 

5 

Bodenabdeckung 

kulturfähiger 

Lehmboden 


(GWO) 

Das SD-Modell (Abb. 3) orientiert sich am Regelquerschnitt RQ 26 für vierstreifige Autobahnen (FGSV 

1996). Das Bankett wird nach FGSV (2005) als schwach durchlässiger Boden SŪ* ausgeführt (entspr. 

Bodenart Su3), die Bodenabdeckung der Böschung als stark durchlässiger Sand (entspr. Bodenart Ss). Die 

hydraulischen Eigenschaften der für Bankett und Böschung verwendeten Böden sind in Tab. 1 aufgeführt. 

Unterhalb der Asphaltdeckschicht schließt sich eine ungebundene Tragschicht an, für die wie für den 

darunter liegenden Dammkern RCB der Sieblinie 0/32 angenommen wird. 

1.3m 

0.3m 

RCB 

Körnung 0/32 

Unterboden 

Untergrund 

Symmetrieachse 

10m 1.5m 

Asphaltdeckschicht 

undurchlässig 

3% 

Planum 4% 

12% 

schwach durchlässiger 

Boden 

1:1. 5 

2.3m 

10 cm stark durchlässiger 

Boden 

2m 


(GWO) 

Abb. 3: Straßendamm mit Frostschutzschicht/Tragschicht aus Recycling-Baustoff (RCB). 

Zu den ungesättigten hydraulischen Eigenschaften von Recyclingmaterialien im Straßenbau finden sich in 

der Literatur kaum experimentelle Angaben. Deshalb wurden die Van-Genuchten-Parameter für den RCB in 

den einzelnen Verwertungsszenarien mit einem Ansatz von Arya & Paris (1981) bzw. Mishra et al. (1989) auf

Grundlage des Korngrößenspektrums der 0/32-Siebline, der Lagerungsdichte ρb = ( V ⋅ ρ p ) und der 

Porosität η = ( 1− 

ρb 

/ ρ) 

abgeleitet. V [-] ist der Verdichtungsgrad und ρp die Proctordichte [g cm -3 ] (siehe 

Tab. 1). Dieser Ansatz wurde für natürliche Böden entwickelt, sollte nach Hansson et al. (2006) jedoch auch 

für relativ grobkörnige Materialien wie Tragschichtschotter im Straßenbau geeignet sein. Für die gesättigte 

Leitfähigkeit Ks von Tragschichten wird ein Mindestwert von 5.4*10 -5 m s -1 gefordert (FGSV 1998). Aufgrund 

der großen Spannbreite experimentell in Labor und in-situ bestimmter Durchlässigkeiten (z.B. Wörner et al. 

2001; Stoppka 2002; Kellermann 2003) wird dieser Wert hier in allen Verwertungsszenarien für den RCB 

angenommen. Abb. 4 zeigt die θ-ψ- sowie die θ-K-Beziehungen des RCB für PP, LSW und SD. Die 

Kurvenverläufe sind durchweg sehr ähnlich und durch sehr geringe Kapillarität gekennzeichnet, was zu einer 

schnellen Drainage der Tragschichten erforderlich ist. 

Tab. 1: Hydraulische Eigenschaften der Baumaterialien für Parkplatz, Lärmschutzwall 

und Straßendamm 

Parkplatz Lärmschutzwall 

RCB RCB Böschung ‡ 

6 

Straßendamm 

RCB Bankett † Böschung † 

θs 0.27 0.31 0.43 0.25 0.36 0.37 

θr 0.00 0.00 0.08 0.00 0.00 0.04 

n 1.29 1.29 1.56 1.29 1.28 1.57 

α [m -1 ] 166 154 3.60 146 2.64 8.74 

l 0.50 0.50 0.50 0.50 0.50 0.50 

ρ [g cm -3 ] ¶ 2.66 2.66 2.65 2.66 2.65 2.65 

ρb [g cm -3 ] 1.94 1.84 1.51 2.00 1.69 1.67 

ρp [g cm -3 ] ¶ 1.94 1.94 - 1.94 - - 

V [-] § 1 0.95 - 1.03 - - 

Ks [m s -1 ] 5.40*10 -5 5.40*10 -5 1.00*10 -6 5.40*10 -5 1.00*10 -6 2.90*10 -6 

‡ : Carsell & Parish 1988; † : Hennings 2000; 

§ : FGSV 2002 (PP), 1997 (LSW), 2004 (SD); ¶ : Kellermann 2003 

α, n, l: empirische Van-Genuchten-Parameter 

θr, θs: residualer und gesättigter Wassergehalt 

ρ, ρb, ρp: Mineral-, Lagerungs- und Proctordichte 

V: Verdichtungsgrad; Ks: gesättigte hydraulische Leitfähigkeit 

ψ [m] 

10 2 

10 1 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

Parkplatz 

Lärmschutzwall 

Straßendamm 

0 0.1 0.2 0.3 0.4 

θ [-] 

Krel [-] 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

10 -5 

10 -6 

10 -7 

10 -8 

10 -9 

10 -10 

0 0.1 0.2 0.3 0.4 

θ [-] 

Abb. 4: Wassergehalts-SAugspannungs- (links) sowie Wassergehalts-Leitfähigkeits-Beziehungen (rechts) für 

Recycling-Baustoff in den drei Verwertungsszenarien (ψ = Matrixpotential, Krel = relative Leitfähigkeit, θ = 

Wassergehalt).

Die Kd-Werte von NAP, PHE und Σ15 EPA-PAK im RCB von 106, 496 bzw. 1333 l kg -1 sowie die 

Intrapartikelporosität ε = 0.015 wurden von Henzler (2004) experimentell bestimmt. Daq für PHE und NAP 

wurde mit 7.86*10 -10 bzw. 9.15*10 -10 m² s -1 nach Hayduk & Laudie (1974) abgeschätzt. Die heterogene 

Zusammensetzung des RCB aus verschiedenen Korngrößenfraktionen wurde durch den Lithokomponenten- 

Ansatz nach Kleineidam et al. (1999a) berücksichtigt. Wegen sehr langer Rechenzeiten wird der RCB durch 

zwei Korngrößenklassen modelliert (siehe auch Rügner et al. 2005), für die die kinetische Sorption jeweils 

separat berechnet wird. Auf die Fein- und Grobfraktion entfallen Anteile von 32.8 % bzw. 67.2 % mit 

„effektiven“ Kornradien von a = 0.25 mm bzw. a = 8 mm (siehe Henzler (2004)). 

Auswahl und Klassifizierung der Unterbodenprofile 

Die Charakterisierung der Unterbodenprofile erfolgte auf Basis der nutzungsdifferenzierten Bodenübersichtskarte 

1:1.000.000 (BÜK1000; BGR 2006). Aus deren 672 Referenzprofilen wurden sechs Profile nach 

Flächenrepräsentanz und dem zu erwartenden charakteristischen Transportverhalten für die numerischen 

Simulationen des reaktiven Stofftransports ausgewählt (Braunerde und Podsol aus Sand, Fahlerde aus 

Geschiebelehm, Schwarzerde und Parabraunerde aus Löss, Pelosol aus verwittertem Mergel- und 

Tonstein). Zur Vereinfachung der Simulationen wurde eine Reduktion der vertikalen Profildifferenzierung 

durch Zusammenfassung mehrerer Horizonte vorgenommen, soweit eine einheitliche Betrachtung nach 

bodenkundlichen und geologischen Aspekten möglich schien. Bei allen Profilen wurden somit Daten aus 

verschiedenen Horizonten nach deren Mächtigkeit gewichtet gemittelt (Tab. 2). Als zu berechnende 

Profiltiefe wurde für die Böden die in der BÜK1000 (BGR 2006) beschriebene Profiltiefe ohne den 

Oberboden angesetzt, da dieser bei der Bebauung abgetragen wird. Die resultierenden Profiltiefen (Tab. 2) 

sind im Sinne von Mindesttiefen zu verstehen, da die in der Praxis relevanten Grundwasserflurabstände 

häufig deutlich größer sein dürften. 

Zur Festlegung der Ks-Werte und Van-Genuchten-Parameter wurden Pedotransferfunktionen von Wösten et 

al. (1998) angewendet. Der Sättigungswassergehalt θs wurde für alle Bodenschichten der Porosität 

gleichgesetzt. Ks bezieht sich auf den Feinbodenanteil (< 2 mm) und wurden bei höheren Skelettgehalten um 

die Reduktion der Leitfähigkeit durch den Skelettanteil mit einem Verfahren von Brakensiek & Rawls (1994) 

korrigiert. Für die Simulation der Tracerversuche in GeoSys/Rockflow zur Ableitung der pdf wurden die 

Dispersivitäten mit αL = 0.1 m und αT = 0.01 m festgelegt. 

Die Kd–Werte wurden aus dem Gehalt an organischem Kohlenstoff foc [-] (= 0.01 Corg; vgl. Tab. 2) und dem 

K = K f abgeschätzt. Für 

auf den Corg-Gehalt normierten Verteilungskoeffizienten KOC [l kg -1 ] durch d OC oc 

Horizonte, die nach BÜK1000 (BGR 2006) als „humusfrei“ ausgewiesen sind, wurde als konservative 

Annahme für den Feinboden ein minimaler Corg von 0.01 % angenommen. Zur Abschätzung des KOC wurde 

die auf der Wasserlöslichkeit S [mol l -1 ] beruhende Korrelation von Seth et al. (1999) verwendet (für 

vergleichbare Ansätze siehe auch Allen-King et al. 2002): 

log KOC = − 0. 

88 log S + 0. 

07 (3) 

S beträgt für PHE 3.46*10 -5 , für NAP 8.74*10 -4 mol l -1 . Für Σ15 EPA-PAK wurde S über ein mittleres 

Molgewicht von 202 g mol −1 und einer effektiven Wasserlöslichkeit von 2.5 g l -1 (Grathwohl 2004) mit 1.14 

*10 -5 mol l -1 abgeschätzt. Für die verschiedenen Böden ergeben sich so Kd-Werte für NAP zwischen 0.06 

und 5.10 l kg -1 , für PHE zwischen 0.99 und 87.2 l kg -1 sowie für Σ15 EPA-PAK zwischen 2.45 und 217 l kg -1 

Für die Skelettanteile wurden KOC-Werte von 4.0 l kg -1 (NAP), 5.2 l kg -1 (PHE) und 5.7 l kg -1 (Σ15 EPA-PAK) 

angenommen (Rügner et al. 2005). Mit diesen vergleichsweise hohen Werten wird berücksichtigt, dass es 

sich beim Corg dieser Komponenten idR. um gealtertes Material mit höherer Sorptionskapazität handelt 

(Kleineidam et al. 1999b). Der foc wurde mit 0.0005 angesetzt (Kleineidam et al. 1999b). 

Die Sorptionskapazität und -kinetik für den Feinboden wird vor allem durch das partikuläre organische 

Material bestimmt. Es wurden folgende effektive Parameter zugrunde gelegt: a = 11.7 µm, ε = 0.00175, 

ρ = 2.65 g cm -3 und die Tortuosität τ f = 1/ 

ε (Grathwohl, 1992; Rügner et al. 1999). Für die Grobfraktion (> 2 

mm) wurden a = 1 cm und ε = 0.01 angenommen (Rügner et al. 1999; Kleineidam et al. 1999a). 

Die Transportsimulationen für NAP, PHE und Σ15 EPA-PAK wurden jeweils mit und ohne Berücksichtigung 

von Bioabbau durchgeführt, wobei nur die in Lösung vorliegenden Anteile als abbaubar betrachtet wurden. 

Zur Beurteilung der langfristigen Filterwirkung des Bodens bedarf es repräsentativer Langzeit- 

Ratenkonstanten (Henzler et al. 2006), deren quantitative Abschätzung in der ungesättigten Zone jedoch 

durch die Komplexität der Wechselwirkungen zwischen schadstoffspezifischen Eigenschaften, klimatischen 

und geochemischen Randbedingungen (Grathwohl et al. 2003, Höhener et al. 2006) bisher kaum möglich 

ist. Aus diesem Grund wurde ein relativ niedriger Wert von 1.15*10 -7 s -1 (Halbwertzeit = 70 d) angenommen. 

7

Dies ist als konservative Abschätzung zu betrachten, da viele Schadstoffe in der ungesättigten Zone unter 

Feldbedingungen deutlich schneller abgebaut werden können (Maier & Grathwohl 2005, Rügner et al. 2005). 

Tab. 2: Bodeneigenschaften der sechs betrachteten Unterböden aus der BÜK1000 (BGR 2006). 

Für vertikal aggregierte Horizonte wurden die Parameter über die Mächtigkeit der einzelnen 

Horizonte gewichtet gemittelt. 

Bodentyp Braunerde Podsol Fahlerde 

8 

Schwarzerde 

Parabraunerde 

Pelosol 

Fläche [km²] 1672 7237 6933 3455 14428 5614 

Einteilung Unterboden Unterboden Unterboden Untergrund Unterboden Unterboden Unterboden 

von - bis [m] 0.00-1.70 0.00-1.70 0.00-0.90 0.90-1.70 0.00-1.70 0.00-1.70 0.00-1.75 

Ton [%] 3.82 10.40 31.85 31.85 19.69 19.78 39.88 

Schluff [%] 14.54 20.48 20.48 20.48 69.40 70.28 37.00 

Sand [%] 57.41 63.21 38.68 38.68 8.44 8.43 16.48 

Skelett [%] 24.24 5.91 9.00 9.00 2.47 1.50 6.64 

Corg [%] 0.01 ‡ 0.21 0.01 ‡ 0.01 ‡ 0.88 0.01 ‡ 0.03 

ρb [g cm -3 ] 1.71 1.61 1.62 1.75 1.42 1.63 1.60 

θs 0.29 0.36 0.35 0.34 0.45 0.38 0.39 

† 

θr 

0.01 0.01 0.01 0.01 0.01 0.01 0.01 

n † 1.34 1.26 1.17 1.08 1.15 1.13 1.16 

α † [m -1 ] 5.93 5.98 6.21 4.95 1.76 1.24 2.73 

l † 1.77 -0.30 -0.66 -3.57 -1.91 -0.32 -2.75 

Ks † [m s -1 ] 1.52*10 -6 3.37*10 -6 3.09*10 -6 8.46*10 -7 3.83*10 -6 1.38*10 -6 9.48*10 -7 

‡ : 0.00 nach BÜK1000; Annahme eines minimalen Corg-Gehaltes des Feinbodens von 0.01% 

† : abgeleitet nach Wösten et al. (1998); Ks korrigiert um Skelettanteil nach Brakensieck & Rawls (1994) 

θr, θs: residualer und gesättigter Wassergehalt; ρb: Lagerungsdichte 

α, n, l: empirische Van-Genuchten-Parameter; Ks: gesättigte hydraulische Leitfähigkeit 

Abschätzung der Sickerwasserrate 

Für eine langfristige Beurteilung der Schadstoffverlagerung ist es sinnvoll und zulässig von vereinfachten 

Fließverhältnissen auszugehen (Grathwohl & Susset 2001; Henzler et al. 2006). Aus diesem Grund wird eine 

stationäre ungesättigte Strömung angenommen. Die jährlichen mittleren Infiltrationsraten I [mm a -1 ] der 

Verwertungsszenarien wurden als Mittelwerte für Deutschland abgeschätzt. Hierzu wurde das BAGLUVA- 

Verfahren (Glugla et al. 2003) angewendet. Als mittlerer korrigierter Niederschlag Njahr wurde ein Wert von 

859 mm a −1 angenommen (BMU 2000). I ergibt sich als Differenz von Njahr und der Jahressumme der 

Evapotranspiration ET. Diese wird als Funktion der maximalen ET und standortspezifischer Parameter 

bestimmt. Für den PP wurde I so mit 583 mm a -1 abgeschätzt, was im oberen Bereich für sickerfähige 

Pflasterdecken liegt (Richter 2003). Die Annahme wird jedoch durch Freilandlysimeter-Untersuchungen von 

Flöter (2006) gestützt, in denen der Anteil von I am Niederschlag über 4 a im Mittel rund 80 % betrug. Für 

den LSW wurde I mit 313 mm a -1 abgeschätzt. Für den SD ergibt sich die zu infiltrierende Wassermenge als 

Summe des Oberflächenabflusses der Asphaltdecke (Abflussbeiwert = 0.9; FGSV 2005) und des auf 

Bankett und Böschung fallenden Niederschlags abzüglich der ET. Für die räumliche Verteilung der Infiltration 

mussten vereinfachende Annahmen getroffen werden, da der SD-Querschnitt eine neue Bauweise nach 

FGSV (2005) darstellt, bei der der Straßenabfluss zu größeren Teilen über die Böschung infiltrieren soll, 

bisher jedoch keine experimentellen Daten dazu vorliegen. Aus diesem Grund wurde angenommen, dass die 

Versickerung gleichmäßig über Bankett und Böschung erfolgt. Auf den Querschnitt bezogen ergibt sich I mit 

2318 mm a -1 . Für den Böschungsfuß wurden wie für den LSW I = 313 mm a -1 angenommen.

Ergebnisse und Diskussion 

Die im Folgenden vorgestellten zeitlichen Konzentrationsverläufe der Modellsubstanzen an der GWO stellen 

über den unteren Modellrand integrierte Durchbruchskurven dar. Dabei ist für das LSW- und SD-Szenario 

zwischen den mit SMART berechneten, auf den Transportpfad bezogenen (graue dicke Kurven, Abb. 5, 8, 9) 

und den über den gesamten Querschnitt gemittelten Konzentrationen (schwarze dünne Kurven, Abb. 5, 8, 9) 

zu unterscheiden (siehe auch Abb. 6). 

Die Tracer-Durchbruchskurven an der GWO für die drei Szenarien PP, LSW und SD sind in Abb. 5 (a)-(c) 

dargestellt. Für den PP erfolgt der Durchbruch des Tracers am frühesten bei Braunerde und Podsol, später 

bei Pelosol, Parabraunerde und Schwarzerde (Abb. 5 (a)), die eine geringere 

Porenwasserfließgeschwindigkeit als die Sandböden aufweisen. Dispersion führt zu Verminderungen der 

Durchbruchskonzentrationen auf C/C0 zwischen 0.23 (Braunerde) und 0.16 (Schwarzerde). Die hier 

betrachteten Transportstrecken von bis zu 1.75 m ab der RCB-Unterkante dürften idR. kürzer als die in 

vielen Praxisfällen relevanten Grundwasserflurabstände sein, wodurch unter Umständen eine stärkere 

dispersive Konzentrationsreduktion möglich wäre. 

C/C0 [-] 

0.3 

0.2 

0.1 

a) Parkplatz 

Braunerde 

Podsol 

Schwarzerde 

Parabraunerde 

Fahlerde 

Pelosol 

0 

0 1 2 3 

Zeit [a] 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

0 5 10 15 

Zeit [a] 

9 

b) Lärmschutzwall 

Legende siehe 

Straßendamm 

0.3 

0.2 

0.1 

0 

c) Straßendamm 

Braunerde (verd.) 

Podsol (verd.) 

Schwarzerde (verd.) 

Braunerde 

Podsol 

Schwarzerde 

0 1 2 3 

Zeit [a] 

Abb. 5: Durchbruchskurven des Tracers an der Grundwasseroberfläche für (a) Parkplatz-, (b) 

Lärmschutzwall- und (c) Straßendamm-Szenario. Gezeigt werden Durchbruchskurven für den 

Schadstofftransportpfad ((b) und (c), graue dicke Kurven) bzw. auf das Gesamtbauwerk bezogene 

Durchbruchskurven unter Berücksichtigung der Verdünnung durch nicht belastetes Sickerwasser ((b) und 

(c), schwarze dünne Kurven). 

Die über den unteren Modellrand integrierten Tracer-Durchbruchskurven für das LSW-Szenario (Abb. 5 (b)) 

zeigen im Vergleich zum PP ein stärkeres Tailing und über längere Zeiträume anhaltende hohe 

Konzentrationen C/C0. Aus Anschaulichkeitsgründen werden die Ergebnisse hier nur für Braunerde, Podsol 

und Schwarzerde vorgestellt. Das ausgeprägte Tailing resultiert aus der geringeren Infiltrationsrate sowie 

aus der Geometrie des LSW. Die Überdeckung des grobkörnigen RCB mit einem feinkörnigen Lehm führt an 

der Böschung zu einer Kapillarsperre, welche die Infiltration in den RCB vermindert und ein Umströmen des 

Wallkerns verursacht (siehe Abb. 6 (a)). 

Die Kapillarsperre bildet sich aus, da der RCB aufgrund geringer Kapillarität bereits bei geringen 

Saugspannungen einen Großteil des Porenwassers verliert (siehe Abb. 4) und die hydraulische Leitfähigkeit 

gegenüber dem Lehm deutlich stärker abnimmt. Das Sickerwasser kann so entlang der geneigten 

Materialgrenze in dem nun besser wasserleitenden Lehmboden abfließen. Mit zunehmender Distanz von der 

Wallkrone erhöhen sich die Wasserflüsse im Lehm, sodass in Abhängigkeit der Druckverhältnisse 

zunehmend Wasser infiltrieren kann (Ross 1990). Die räumliche Variabilität der Sickerwasserströmung führt 

zu Bereichen mit stark reduzierten Fließgeschwindigkeiten und so zu einem langsameren Auswaschen des 

Tracers aus dem RCB. Die Maximalkonzentrationen erreichen die GWO nach etwa 2 (Braunerde) bis 3.5 a 

(Schwarzerde) (Abb. 5 (b)). Im Vergleich zum PP kommt es zudem zu höheren Peakkonzentrationen C/C0 

für den Transportpfad (Braunerde: 0.43, Schwarzerde: 0.35), da zum Zeitpunkt des Peak-Durchbruchs an 

der GWO die Eluatkonzentrationen der Quelle noch deutlich höher als beim PP sind (siehe Abb. 7 (a)) und 

so die dispersive Konzentrationsminderung weniger effektiv ist. Der länger anhaltende hohen 

Eluatkonzentrationen resultieren aus den geringeren Sickerwassermengen, der Reduktion der Infiltration in 

den RCB durch Kapillarsperren und der größere Mächtigkeit des RCB gegenüber dem PP. Auf das gesamte 

Bauwerk bzw. auf die gesamte infiltrierende Wassermenge bezogen reduzieren sich die 

Maximalkonzentrationen durch kleinräumige Mittelung jeweils um ca. 40 %. Zu beachten ist in diesem 

Zusammenhang, dass die Ausprägung und Effektivität einer Kapillarsperre von einer Reihe von

Randbedingungen abhängig ist. Aus diesem Grund sind die berechneten Verdünnungsfaktoren nur für die 

hier betrachteten Geometrien, Infiltrationsraten und Materialkombinationen quantitativ gültig. 

(a) Lärmschutzwall (b) Straßendamm 

Modellergebnisse bezogen auf den 

Transportpfad (SMART-Output) 

über Gesamtbreite gemittelte Modellergebnisse über Gesamtbreite gemittelte Modellergebnisse 

10 

Modellergebnisse bezogen auf den 

Transportpfad (SMART-Output) 

Abb. 6: Geschwindigkeitsvektoren der ungesättigten Strömung an allen Elementknoten im Lärmschutzwall 

(a) und in einem Ausschnitt des Straßendammes (b). Die Konzentration des Flusses in den 

Bodendeckschichten ist Resultat des Kapillarsperreneffektes, wird durch die feinere Netzdiskretisierung 

jedoch überzeichnet. 

C/C0 [-] 

1 

0.1 

0.01 

a) Tracer 

Parkplatz 

Lärmschutzwall 

Straßendamm 

0.01 0.1 1 10 

Zeit [a] 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

b) Parkplatz 

Naphthalin 

Phenanthren 

Σ15 EPA PAK 

0.1 1 10 100 

Zeit [a] 

Abb. 7: Zeitliche Entwicklung der aus der Schadstoffquelle eluierenden Konzentrationen an der Recycling- 

Baustoff-Unterkante (RCB) für ausgewählte Schadstoff-Verwertungsszenario-Kombinationen: (a) 

konservativer Tracer für Parkplatz-, Lärmschutzwall- und Straßendammszenario; (b) Naphthalin, 

Phenanthren und Σ15 EPA-PAK für das Parkplatzszenario. 

Auch für das SD-Szenario ist ein Kapillarsperreneffekt zu beobachten (Abb. 6(b)). Während sich im 

Bankettbereich eine relativ gleichförmige Verteilung der Sickerwasserströmung zeigt, strömt in der 

Bodenabdeckung der Böschung ein Teil des Sickerwassers am RCB vorbei und infiltriert konzentriert am 

Böschungsfuß. Bei kleinräumiger Mittelung der Durchbruchskonzentrationen ergibt sich eine Verdünnung um 

ca. 31 %. Die Durchbruchskurven des Tracers (Abb. 5 (c)) zeigen ein steileres Ansteigen als beim LSW und 

fallen innerhalb von fünf Jahren auch deutlich schneller ab. Die maximalen Durchbruchskonzentrationen 

C/C0 liegen im kleinräumigen Mittel zwischen 0.20 (Braunerde) und 0.18 (Schwarzerde). 

Abb. 8 zeigt die Durchbruchskurven für NAP, PHE und Σ15 EPA-PAK jedoch ohne Abbau. Für das PP- 

Szenario zeigt sich in Abb. 8 (a)-(c) deutlich der Effekt der vom Corg des Unterbodens abhängigen Sorption. 

Während für die Corg-armen Böden NAP bereits nach kurzer Zeit durchbricht, wird es beim mäßig Corghaltigen 

Podsol leicht, bei der Corg-reichen Schwarzerde stärker retardiert (Abb. 8 (a)). Für PHE und 

Σ15 EPA-PAK (Abb. 8 (b) und (c)) zeigen auch die Corg-armen Böden eine deutliche Retardation. Der 

flachere Verlauf der Durchbruchskurven für Braunerde im oberen Konzentrationsbereich deutet zudem auf 

einen Einfluss der langsamen Sorptionskinetik des Skelettanteils hin. Für alle Böden bis auf die Schwarzerde 

wird innerhalb der Simulationszeit der Durchbruch der PHE-Peaks mit C/C0 zwischen 0.70 und 0.96 und eine 

500

anschließende Konzentrationsabnahme auf Grund der zeitlichen Abnahme der Quellstärke (siehe Abb. 7 (b)) 

beobachtet. Für die Schwarzerde steigt C/C0 nach 500 a noch deutlich an. 

C/C0 [-] 

C/C0 [-] 

C/C0 [-] 

1 

0.1 

0.01 

1 

0.1 

0.01 

1 

0.1 

0.01 

a) Parkplatz 

Naphthalin 

d) Lärmschutzwall 

Naphthalin 

g) Straßendamm 

Naphthalin 

0.1 1 10 100 

Zeit [a] 

b) Parkplatz 

Phenanthren 

e) Lärmschutzwall 

Phenanthren 

h) Straßendamm 

Phenanthren 

0.1 1 10 100 

Zeit [a] 

11 

c) Parkplatz 

Σ15 EPA PAK 

Braunerde 

Podsol 

Schwarzerde 

Parabraunerde 

Fahlerde 

Pelosol 

f) Lärmschutzwall 

Σ15 EPA PAK 




Braunerde 

Podsol 

Schwarzerde 

i) Straßendamm 

Σ15 EPA PAK 




Braunerde 

Podsol 

Schwarzerde 

0.1 1 10 100 

Zeit [a] 

Abb. 8: Durchbruchskurven der sorbierbaren Stoffe Naphthalin (schwach sorptiv, links) und Phenanthren 

(mäßig gut sorptiv, Mitte) und des Summenparameters Σ15 EPA-PAK (stark sorptiv, rechts) für die drei 

Szenarien Parkplatz ((a)-(c)), Lärmschutzwall ((d)-(f)) und Straßendamm ((g)-(i)). Gezeigt werden für 

Lärmschutzwall und Straßendamm jeweils Durchbruchskurven für den Schadstofftransportpfad (graue dicke 

Kurven) bzw. auf das Gesamtbauwerk bezogene Durchbruchskurven unter Berücksichtigung der 

Verdünnung durch nicht belastetes Sickerwasser (schwarze dünne Kurven). 

Abb. 8 (d)-(f) zeigt die NAP-, PHE- und Σ15 EPA-PAK- Durchbruchskurven des LSW-Szenarios. Ein 

deutlicher Unterschied zum PP ist das längere Anhalten hoher Konzentrationen, welches aus den im 

Durchschnitt niedrigeren Strömungsgeschwindigkeiten resultiert. Für NAP werden unter Berücksichtigung 

der Verdünnung durch am RCB vorbeiströmendes Sickerwasser Maximaldurchbrüche von C/C0 = 0.56 

erreicht (Abb. 8 (d)). Für PHE und Σ15 EPA-PAK zeigen sich bei allen betrachteten Böden nach 500 a noch 

deutlich ansteigende Konzentrationen (Abb. 8 (e) und (f)). 

Im Vergleich dazu zeigen sich für das SD-Szenario aufgrund der stark erhöhten Infiltrationsraten steilere 

Durchbruchskurven und somit ein früheres Erreichen der Konzentrationspeaks (Abb. 8 (g)-(i)). NAP zeigt 

seine Maximaldurchbrüche bei der Braunerde bereits nach 19 a (C/C0 = 0.59), bei der Schwarzerde nach 46 

a (C/C0 = 0.60). 

Ohne Abbau erfolgt langfristig der Eintrag der gesamten Schadstoffmasse des RCB ins Grundwasser. Wird 

Abbau bei der Simulation mit berücksichtigt (Abb. 9 (a)– (i)), ist die Massenreduktion bei gegebener 

Abbauratenkonstante nur von der Verweilzeit der Substanzen im Bodenwasser abhängig. Retardation hat

keinen Einfluss auf die abgebaute Masse, da Mikroorganismen idR. nur zum Abbau in Lösung vorliegender 

Schadstoffe fähig sind. So zeigen Abb. 9 (a)-(c) für den PP, dass der Abbau bei der Parabraunerde trotz 

früherer Durchbruchszeiten effektiver ist, als z.B. bei der Braunerde oder dem Podsol. Dies liegt an der 

geringeren Abstandsgeschwindigkeit va = q / θ (= Transportgeschwindigkeit des Tracers) bzw. dem höheren 

Wassergehalt der feinkörnigen gegenüber den sandigen Böden bei vorgegebener Infiltrationsrate (= 

Darcyfluss q), woraus sich eine längere effektive Aufenthaltszeit der Substanzen im Bodenwasser des Profils 

ergibt. Insgesamt ist bei der angenommenen Halbwertzeit von 70 d eine Reduktion der 

Durchbruchkonzentrationen für den PP um einen Faktor von bis zu 30 möglich. 

C/C0 [-] 

C/C0 [-] 

C/C0 [-] 

1 

0.1 

0.01 

0.001 

1 

0.1 

0.01 

0.001 

1 

0.1 

0.01 

0.001 

a) Parkplatz 

Naphthalin 

d) Lärmschutzwall 

Naphthalin 




Braunerde 

Podsol 

Schwarzerde 

g) Straßendamm 

Naphthalin 




Braunerde 

Podsol 

Schwarzerde 

0.1 1 10 100 

Zeit [a] 

b) Parkplatz 

Phenanthren 

Braunerde 

Podsol 

Schwarzerde 

Parabraunerde 

Fahlerde 

Pelosol 

e) Lärmschutzwall 

Phenanthren 

h) Straßendamm 

Phenanthren 

0.1 1 10 100 

Zeit [a] 

12 

c) Parkplatz 

Σ15 EPA PAK 

f) Lärmschutzwall 

Σ15 EPA PAK 

i) Straßendamm 

Σ15 EPA PAK 

0.1 1 10 100 

Zeit [a] 

Abb. 9: Durchbruchskurven der sorbierbaren Stoffe Naphthalin (schwach sorptiv, links) und Phenanthren 

(mäßig gut sorptiv, Mitte) und des Summenparameters Σ15 EPA-PAK (stark sorptiv, rechts) unter 

Berücksichtigung des biologischen Abbaus für die drei Szenarien Parkplatz ((a)-(c)), Lärmschutzwall ((d)-(f)) 

und Straßendamm ((g)-(i)). Gezeigt werden für Lärmschutzwall und Straßendamm jeweils 

Durchbruchskurven für den Schadstofftransportpfad (graue dicke Kurven) bzw. auf das Gesamtbauwerk 

bezogene Durchbruchskurven unter Berücksichtigung der Verdünnung durch nicht belastetes Sickerwasser 

(schwarze dünne Kurven). 

Aufgrund der langsameren Strömungsgeschwindigkeiten ist beim LSW die durchschnittliche Verweilzeit der 

gelösten Schadstoffe im Boden höher als beim PP. Dies führt zu einer stärkeren Konzentrationsreduktion bei 

Abbau (Minderungsfaktoren zwischen 25 und 150; Abb. 9 (d)-(f)). Umgekehrt zeigen sich beim SD auf Grund 

höherer Strömungsgeschwindigkeiten deutlich geringere Aufenthaltszeiten, sodass sich trotz des Abbaus nur 

geringe Minderungsfaktoren zwischen 2.5 und 5 ergeben (Abb. 9 (g)–(i)).

Schlussfolgerungen 

Die in dieser Studie durchgeführten umfangreichen Typ-Szenarien-Simulationen ergänzen die im BMBF- 

Verbundprojekt „Sickerwasserprognose“ erarbeiteten wissenschaftlichen Grundlagen und methodischen 

Instrumentarien durch die Anwendung auf praxisrelevante Fallbeispiele. Aus den dabei gewonnenen 

Ergebnissen lassen sich folgende Schlussfolgerungen ableiten: 

• Dispersive Konzentrationsminderung ist nur bedingt wirksam. Eine nennenswerte Abschwächung der 

Durchbruchskonzentrationen ist nur für Fälle möglich, in denen das Abklingen der Quellkonzentrationen 

deutlich vor dem Durchbruch des Schadstoffpeaks an der GWO erfolgt. Bei länger anhaltenden 

Quellstärken (z.B. für Salze wie Sulfat) ist dagegen mit weitgehend unvermindertem 

Konzentrationsdurchbruch zu rechnen. 

• Für stärker sorbierende Schadstoffe (PHE, Σ15 EPA-PAK) zeigt sich schon bei geringem Corg der 

Unterböden eine Retardation der Peak-Durchbruchszeitpunkte um viele Jahrzehnte bis Jahrhunderte. 

Mit signifikanten Stoffeinträgen (in Abhängigkeit der Quellstärke) ins Grundwasser ist so bei „natürlichen“ 

Grundwasserneubildungsverhältnissen erst nach sehr langen Zeiträumen zu rechnen. 

• Auch bei relativ niedrigen Ratenkonstanten ist eine deutliche Reduktion der Konzentrationen organischer 

Schadstoffe durch mikrobiellen Abbau möglich. Bezüglich der Effektivität des Abbaus ergeben sich 

jedoch erhebliche Unterschiede zwischen den drei betrachteten Verwertungsszenarien, die aus den 

unterschiedlichen Strömungs- bzw. Transportgeschwindigkeiten resultieren. 

• Die Stoffverlagerung in den in zwei Raumdimensionen betrachteten Szenarien Lärmschutzwall und 

Straßendamm zeigt darüber hinaus einen ausgeprägten Einfluss der räumlich variablen 

Sickerwassermenge. Bereiche hoher Strömungsgeschwindigkeit führen zu früheren Ankunftszeiten der 

Schadstoffe im Grundwasser. Transportprognosen unter Vernachlässigung dieser Effekte können 

deshalb zu einer Unterschätzung der Schadstoffverlagerung führen und sind nicht konservativ. 

• Die Strömungsbilanzen für Lärmschutzwall und Straßendamm legen nahe, dass mit geeigneten 

Bauwerksgeometrien und Materialien eine Reduktionen der Infiltration in das Verwertungsmaterial und 

des Austrags von Schadstoffen ins Grundwasser durch das Ausnutzen von Kapillarsperreneffekten zu 

erzielen ist. Der hier vorgestellte Modellansatz schafft grundsätzlich die Möglichkeit, das Design von 

Verwertungsszenarien ohne großen Mehraufwand in der Umsetzung durch numerische Simulationen in 

Richtung einer möglichst geringen Umweltbelastung bei hoher Verwertungsquote optimieren zu können. 

Danksagung: Die Untersuchungen wurden im Rahmen des BMBF-Förderschwerpunkts 

„Sickerwasserprognose“ (Projektnummer 02WP0517) durchgeführt. Dem BMBF sei für die Förderung 

gedankt. Des Weiteren danken wir Herrn Dr. Utermann und Herrn Dr. Duijnisveld (BGR, Hannover), Herrn 

Dr. Susset und Herrn Dr. Leuchs (LANUV, Recklinghausen), Herrn Dr. Henzler (UFZ, Leipzig) und Frau Dr. 

Kocher (BaST, Bergisch Gladbach) für ausführliche und hilfreiche Diskussionen sowie Frau Dr. Kouznetsova 

und Herrn Duran für Hilfe bei der Durchführung der Simulationen. 

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Applied numerical modeling of saturated / unsaturated flow and ...

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