Impact 
of 
Aquifer 
Heterogeneity 
on 
Subsurface 
Flow 
and 
Salt 
Transport 
at 
Different 
Scales 


From 
a 
Method 
to 
Determine 
Parameters 
of 
Heterogeneous 
Permeability 
at 
Local 
Scale 
to 
a 
Large-Scale 
Model 
for 
the 
Sedimentary 
Basin 
of 
Thuringia 


Dissertation 


zur 
Erlangung 
des 
akademischen 
Grades 
doctor 
rerum 
naturalium 
(Dr. 
rer. 
nat.) 


vorgelegt 
dem 
Rat 
der 
Chemisch-Geowissenschaftlichen 
Fakultät 
der 
Friedrich-Schiller-Universität 
Jena 


von 
Dipl.-Math. 
Alraune 
Zech 
geboren 
am 
02. 
Juni 
1983 
in 
Quedlinburg 



Gutachter: 


Prof.Dr. 
Sabine 
Attinger 
Friedrich-Schiller-Universität 
Jena 


Prof.Dr. 
Olaf 
Kolditz 
Technische 
Universität 
Dresden 


Tag 
der 
öffentlichen 
Verteidigung: 
20. 
September 
2013 



Abstract 


Groundwater 
is 
an 
important 
natural 
resource. 
More 
than 
two 
thirds 
of 
the 
German 
population 
rely 
on 
groundwater 
as 
the 
daily 
drinking 
water 
supply. 
Hence, 
the 
understanding 
of 
processes 
like 
fluid 
flow 
and 
solute 
transport 
in 
porous 
media 
are 
of 
broad 
relevance. 
A 
profound 
process 
understanding 
is 
not 
only 
important 
for 
rational 
management 
of 
water 
resources 
but 
in 
particular 
for 
the 
preservation 
of 
subsurface 
water 
quality. 
Further 
relevant 
aspects 
are 
the 
optimization 
of 
irrigation 
and 
drainage 
efficiency, 
safe 
and 
economic 
extraction 
of 
subsurface 
mineral 
and 
energy 
resources, 
as 
well 
as 
subsurface 
storage 
of 
energy 
and 
waste. 


Subsurface 
hydrology 
faces 
two 
problems. 
On 
the 
one 
hand, 
information 
about 
the 
character 
of 
the 
subsurface 
structure 
and 
hydraulic 
properties 
is 
scarce. 
Data 
of 
quantities, 
like 
hydraulic 
pressure, 
porosity, 
and 
permeability, 
are 
limited 
to 
a 
few 
locations. 
It 
results 
in 
a 
lack 
of 
spatial 
resolution 
of 
known 
aquifer 
parameters 
for 
large-scale 
flow 
and 
transport 
problems. 
This 
is 
aggravated 
by 
the 
fact 
that 
the 
processes 
are 
very 
slow. 
Thus, 
there 
is 
a 
deficiency 
in 
the 
temporal 
resolution 
of 
observations. 
On 
the 
other 
hand, 
many 
aquifer 
parameters 
are 
heterogeneously 
distributed 
in 
space. 
In 
particular, 
permeability 
shows 
a 
spatial 
variability 
that 
results 
from 
the 
complex 
geological 
processes 
through 
which 
aquifers 
evolve. 
Large 
differences 
in 
the 
permeability 
can 
be 
observed 
on 
local 
scale, 
i.e. 
in 
the 
range 
of 
meters, 
as 
well 
as 
on 
large 
scale, 
i.e. 
in 
the 
range 
of 
kilometers 
up 
to 
the 
scale 
of 
geological 
structures 
as 
the 
Thuringian 
Basin. 
However, 
permeability 
is 
a 
very 
important 
hydraulic 
property 
since 
it 
controls 
the 
groundwater 
flow 
velocity 
and 
hence 
flow 
and 
salt 
transport. 
In 
general, 
the 
scarcity 
of 
data 
does 
not 
allow 
to 
give 
a 
detailed 
spatial 
resolution 
of 
heterogeneously 
distributed 
permeability. 
The 
coincidence 
of 
both 
issues, 
scarcity 
of 
information 
and 
heterogeneity, 
inhibits 
a 
simple 
analysis 
of 
the 
most 
problems 
of 
subsurface 
hydrology. 


This 
thesis 
is 
dedicated 
to 
the 
question 
of 
how 
aquifer 
heterogeneity 
impacts 
on 
processes 
of 
flow 
and 
salt 
transport 
in 
the 
subsurface 
at 
different 
scales. 
We 
present 
a 
large-scale 
numerical 
model 
of 
the 
Thuringian 
basin 
in 
order 
to 
investigate 
the 
mechanisms 
of 
brine 
transport 
within 
the 
aquifers 
of 
a 
shallow 
sedimentary 
basin. 
The 
emphasis 
lies 
on 
the 
effects 
that 
heterogeneous 
permeability 
have 
on 
the 
flow 
and 
salt 
patterns. 
To 
perform 
numerical 
simulations, 
prior 
knowledge 
of 
hydraulic 
parameters 
is 
necessary. 
Of 
particular 
interest 
are 
those 
parameters 
that 
describe 
the 
heterogeneous 
structure 
of 
porous 
media. 
Therefore, 
the 
second 
major 
issue 
of 
this 
work 
is 
the 
development 
and 
discussion 
of 
a 
method 
to 
determine 
the 
statistical 
parameters 
which 
describe 
the 
spatial 
distribution 
of 
permeability. 


The 
first 
part 
of 
the 
work 
refers 
to 
the 
basic 
principles 
of 
modeling 
flow 
and 
salt 
transport 
in 
porous 
media. 
We 
introduce 
the 
physical 
equations 
which 
describe 
the 
relevant 
subsurface 
processes 
of 
fluid 
flow, 
salt 
transport 
and 
heat 
transport. 
We 



explain 
the 
basic 
concepts 
of 
geostatistics 
and 
upscaling 
theory. 
The 
focus 
lies 
on 
the 
statistical 
description 
of 
permeability, 
which 
is 
the 
basis 
for 
the 
upscaling 
methods 
developed 
in 
the 
second 
part. 
In 
addition, 
we 
present 
numerical 
benchmarks 
in 
order 
to 
indicate 
the 
usability 
of 
the 
simulation 
software 
OpenGeoSys. 
We 
use 
this 
numerical 
solver 
for 
the 
modeling 
of 
both, 
the 
complex 
large-scale 
model 
of 
the 
sedimentary 
basin 
of 
Thuringia 
in 
the 
second 
part 
and 
for 
the 
pumping 
test 
model 
in 
the 
third 
part. 


In 
the 
second 
part, 
we 
present 
a 
numerical 
modeling 
approach 
to 
investigate 
the 
fluid 
dynamics 
in 
the 
Thuringian 
basin. 
We 
focus 
on 
the 
impact 
of 
aquifer 
heterogeneity 
and 
fluid 
density 
differences 
on 
brine 
transport. 
Central 
questions 
are: 
How 
does 
the 
large-scale 
fluid 
dynamics 
look 
like? 
Does 
a 
coupling 
between 
thermallyinduced 
deep 
fluid 
convection 
and 
near-surface 
groundwater 
flow 
exist? 
How 
comes 
that 
saline 
groundwater 
reaches 
or 
comes 
close 
to 
the 
surface, 
which 
is 
a 
phenomenon 
that 
is 
observed 
in 
many 
places 
in 
the 
Thuringian 
Basin? 


We 
carry 
out 
numerical 
simulations 
of 
fluid 
flow, 
mass 
and 
heat 
transport 
in 
order 
to 
understand 
the 
role 
of 
geological 
features 
such 
as 
faults, 
aquifer 
heterogeneity, 
as 
well 
as 
fluid 
density 
differences 
caused 
by 
temperature 
and 
salt 
concentration 
gradients. 
For 
this 
purpose 
we 
construct 
a 
profile 
model 
that 
represents 
the 
geological 
setting 
of 
the 
Thuringian 
basin 
incorporating 
major 
hydraulic 
units 
and 
fault 
structures. 


The 
numerical 
results 
indicate 
that 
regional 
groundwater 
flow 
determines 
brine 
migration. 
The 
pattern 
of 
groundwater 
flow 
depends 
strongly 
on 
the 
local 
hydraulic 
parameters. 
A 
qualitative 
sensitivity 
analysis 
indicates 
that 
small 
variations 
in 
permeability 
can 
have 
significant 
influence 
on 
the 
flow 
and 
salt 
patterns. 
The 
local 
mean 
value, 
the 
degree 
of 
heterogeneity, 
and 
the 
local 
correlation 
structure 
of 
permeability 
impact 
on 
the 
location 
and 
amount 
of 
dissolved 
salt. 
This 
directly 
affects 
the 
amount 
of 
salt 
which 
is 
transported 
to 
near-surface 
regions. 
Also 
variation 
in 
fluid 
density 
due 
to 
salt 
concentration 
differences 
can 
cause 
significant 
changes 
in 
the 
flow 
pattern. 
Brine 
is 
heavier 
than 
fresh 
water. 
If 
saline 
groundwater 
is 
transported 
upwards, 
e.g. 
by 
pressure 
induced 
groundwater 
flow, 
then 
the 
higher 
fluid 
density 
of 
brine 
causes 
a 
counteracting 
downward 
movement. 
This 
effect 
leads 
to 
enhanced 
mixing. 
Increased 
mixing 
amplifies 
the 
salinization 
of 
the 
deep 
aquifers 
but 
prevents 
the 
upward 
movement 
of 
highly 
concentrated 
brine. 
Contrariwise, 
the 
simulations 
show 
that 
temperature 
can 
be 
neglected 
as 
driving 
mechanism 
for 
fluid 
flow. 
The 
shallow 
basin 
structure 
inhibits 
the 
developments 
of 
thermal 
convection 
on 
a 
regional 
scale, 
due 
to 
small 
temperature 
differences. 


The 
third 
part 
of 
the 
thesis 
refers 
to 
an 
upscaling 
method 
which 
we 
develop 
in 
order 
to 
determine 
the 
statistics 
of 
aquifer 
heterogeneity 
from 
pumping 
test 
data. 
Permeability 
is 
the 
key 
parameter 
to 
describe 
groundwater 
flow 
velocities 
and 
the 
salt 
distribution. 
However, 
the 
effects 
of 
heterogeneity 
cannot 
be 
captured 
by 
a 
single 



mean 
value. 
According 
to 
the 
statistical 
approach, 
permeability 
can 
be 
described 
by 
a 
log-normal 
distribution 
which 
provides 
additional 
parameters, 
like 
the 
variance 
and 
the 
correlation 
length. 
The 
variance 
represents 
the 
degree 
of 
heterogeneity, 
whereas 
the 
correlation 
length 
describes 
the 
distance 
up 
to 
which 
permeability 
values 
are 
correlated 
in 
space. 
Head 
measurements 
of 
pumping 
tests 
are 
commonly 
used 
to 
estimate 
hydraulic 
properties 
of 
porous 
media. 
Therefore 
it 
is 
reasonable 
to 
develop 
analytical 
tools, 
which 
allow 
the 
determination 
of 
parameters 
of 
aquifer 
heterogeneity 
from 
pumping 
test 
data. 


First, 
we 
derive 
a 
formula 
for 
the 
hydraulic 
head 
describing 
the 
mean 
drawdown 
of 
a 
three 
dimensional 
steady 
state 
pumping 
test 
in 
heterogeneous 
anisotropic 
porous 
media. 
The 
derivation 
is 
based 
on 
the 
Coarse 
Graining 
upscaling 
method. 
The 
closed 
form 
solution 
of 
the 
effective 
well 
flow 
hydraulic 
head 
can 
be 
understood 
as 
an 
extension 
of 
Thiem’s 
formula 
to 
heterogeneous 
porous 
media. 
The 
effective 
well 
flow 
solution 
is 
a 
function 
of 
the 
radial 
distance 
and 
accounts 
for 
the 
statistics 
of 
the 
permeability, 
namely 
geometric 
mean 
variance, 
horizontal 
correlation 
length 
and 
anisotropy 
ratio. 
We 
exploit 
the 
nature 
of 
the 
analytical 
solution 
to 
perform 
a 
sensitivity 
analysis 
on 
the 
parameters 
of 
the 
effective 
well 
flow 
head 
solution 
and 
implement 
an 
inverse 
estimation 
strategy. 
We 
analyze 
numerical 
pumping 
tests, 
both 
an 
ensemble 
of 
as 
well 
as 
single 
pumping 
tests, 
to 
show 
the 
applicability 
of 
the 
head 
solution 
to 
interpret 
drawdown 
data. 
The 
results 
of 
the 
inverse 
estimation 
procedure 
show 
excellent 
agreement 
of 
estimated 
statistical 
parameters 
with 
initial 
values. 


Second, 
we 
determine 
whether 
the 
analytical 
solution 
of 
effective 
well 
flow 
is 
capable 
of 
providing 
accurate 
and 
confident 
parameter 
estimates 
of 
a 
heterogeneous 
aquifer 
under 
limited 
data 
availability. 
This 
is 
of 
practical 
relevance 
since 
head 
measurements 
are 
limited 
in 
on-site 
pumping 
tests. 
We 
use 
simulated 
pumping 
tests 
to 
systematically 
reduce 
sampling 
size 
while 
also 
determining 
the 
accuracy 
and 
uncertainty 
of 
estimates 
at 
each 
level 
of 
data 
availability. 
Our 
findings 
indicate 
that 
the 
accuracy 
and 
uncertainty 
of 
estimated 
parameters 
are 
sensitive 
to 
the 
number 
and 
spatial 
distribution 
of 
head 
measurements. 
Piezometers 
are 
required 
at 
both 
locations, 
i.e. 
directly 
at 
the 
pumping 
well 
and 
at 
large 
distance, 
to 
reliably 
estimate 
the 
respective 
values 
of 
hydraulic 
conductivity. 
Likewise, 
several 
piezometers 
are 
needed 
in 
the 
vicinity 
of 
the 
well 
to 
increase 
accuracy 
and 
reduce 
uncertainty 
in 
estimates 
of 
horizontal 
correlation 
length. 
We 
then 
apply 
the 
same 
analytical 
solution 
to 
estimate 
the 
statistical 
parameters 
of 
a 
fluvial 
heterogeneous 
aquifer 
at 
the 
test 
site 
Horkheimer 
Insel, 
Germany. 
The 
estimated 
mean 
conductivity, 
variance 
and 
correlation 
length 
agree 
very 
good 
with 
results 
from 
laboratory 
measurements. 
Our 
work 
provides 
valuable 
implications 
regarding 
the 
conceptual 
design 
of 
ground 
water 
pumping 
tests 
and 
the 
predictive 
power 
of 
established 
pumping 
test 
sites. 



6 



Zusammenfassung 


Grundwasser 
ist 
eine 
wichtige 
natürliche 
Ressource. 
In 
Deutschland 
wird 
der 
tägliche 
Bedarf 
an 
Trinkwasser 
von 
mehr 
als 
zwei 
Drittel 
der 
deutschen 
Bevölkerung 
durch 
Grundwasser 
gedeckt. 
Aus 
diesem 
Grund 
ist 
das 
Verständnis 
für 
die 
Prozesse 
im 
Untergrund 
wie 
Fluidfluss 
und 
Stofftransport 
von 
weitreichender 
Relevanz. 
Ein 
profundes 
Prozessverständnis 
ist 
sowohl 
für 
ein 
vernünftiges 
Ressourcenmanagement 
als 
auch 
für 
die 
Sicherung 
der 
Grundwasserqualität 
wichtig. 
Weitere 
relevante 
Aspekte 
sind 
die 
Optimierung 
von 
Bewässerung 
und 
Entwässerung, 
die 
sichere 
und 
ökonomische 
Entnahme 
von 
Bodenschätzen 
und 
Energie, 
sowie 
die 
Einlagerung 
von 
Energie 
und 
Abfällen 
im 
Untergrund. 


Die 
Hydrogeologie 
sieht 
sich 
zwei 
grundlegenden 
Problemen 
gegenüber. 
Zum 
einen 
gibt 
es 
oft 
sehr 
wenige 
Informationen 
zur 
Beschaffenheit 
des 
Untergrunds 
sowie 
zu 
hydraulischen 
Eigenschaften. 
Messdaten 
von 
physikalischen 
Größen, 
wie 
piezometrischer 
Druck, 
Porosität 
und 
Leitfähigkeit 
sind 
beschränkt 
auf 
wenige 
Orte. 
Dadurch 
ergibt 
sich 
ein 
Mangel 
an 
räumlicher 
Auflösung 
von 
bekannten 
Daten 
für 
großskalige 
Fließ-und 
Transportprobleme. 
Hinzukommt, 
dass 
die 
Prozesse 
im 
Untergrund 
sehr 
langsam 
ablaufen, 
was 
eine 
geringe 
zeitliche 
Auflösung 
der 
Beobachtungsdaten 
zur 
Folge 
hat. 
Das 
zweite 
Problem 
bezieht 
sich 
auf 
die 
räumliche 
Heterogenität 
vieler 
Aquiferparameter. 
Insbesondere 
die 
Leitfähigkeit 
unterliegt 
sehr 
großen 
Schwankungen 
sowohl 
auf 
kleiner 
Skala, 
d.h. 
im 
Bereich 
von 
einigen 
Metern, 
als 
auch 
auf 
großer 
Skala, 
d.h. 
im 
Bereich 
von 
Kilometern, 
bis 
hin 
zu 
Ausdehnungen 
von 
großräumigen 
geologischen 
Strukturen 
wie 
dem 
Thüringer 
Sedimentbecken. 
Die 
starke 
räumliche 
Heterogenität 
resultiert 
aus 
den 
komplexen 
geologischen 
Entwicklungsprozessen. 
Die 
Permeabilität 
wiederum 
ist 
eine 
sehr 
wichtige 
hydraulische 
Eigenschaft 
des 
Bodens, 
denn 
sie 
kontrolliert 
die 
Grundwasserfließgeschwindigkeit 
und 
damit 
die 
Fließmuster 
und 
die 
daraus 
resultierende 
Salzverteilung. 
Im 
Allgemeinen 
erlaubt 
der 
Mangel 
an 
Daten 
keine 
detailgetreue 
räumliche 
Auflösung 
der 
heterogenen 
Leitfähigkeiten. 
Das 
Vorhandensein 
beider 
Aspekte, 
Informationsmangel 
und 
Heterogenität, 
erschweren 
eine 
einfache 
Analyse 
der 
meisten 
hydrogeologischen 
Fragestellung 
maßgeblich. 


Die 
vorliegende 
Arbeit 
ist 
der 
Frage 
gewidmet 
wie 
Aquiferheterogenität 
Fließ-und 
Transportprozesse 
im 
Untergrund 
auf 
verschiedenen 
Skalen 
beeinflusst. 
Zum 
einen 
wird 
ein 
großräumiges 
numerisches 
Untergrundmodell 
des 
Thüringer 
Beckens 
verwendet 
um 
die 
Mechanismen 
des 
Salztransports 
in 
Aquiferen 
eines 
flachen 
Sedimentbeckens 
zu 
untersuchen. 
Der 
Schwerpunkt 
liegt 
auf 
den 
Effekten 
die 
heterogene 
Leitfähigkeiten 
auf 
die 
Fließ-und 
Salzmuster 
haben. 
Um 
numerischen 
Simulationen 
durchführen 
zu 
können, 
werden 
Werte 
für 
hydrologische 
Parameter 
benötigt. 
Von 
besonderem 
Interesse 
sind 
dabei 
solche 
Parameter, 
die 
die 
heterogene 
Struktur 
des 
Untergrunds 
beschreiben. 
Aus 
diesem 
Grund 
liegt 
der 
zweite 
Schwerpunkt 
dieser 
Arbeit 
auf 
der 
Entwicklung 
und 
Diskussion 
einer 
Skalierungsmethode. 
Diese 



dient 
der 
Bestimmung 
von 
statistischen 
Parametern, 
mit 
denen 
sich 
die 
räumliche 


Verteilung 
von 
Leitfähigkeiten 
beschreiben 
lässt, 
aus 
Pumptestdaten. 
Im 
ersten 
Teil 
der 
Arbeit 
werden 
die 
Grundlagen 
der 
Modellierung 
von 
Fließ-
und 
Transportprozessen 
im 
Untergrund 
eingeführt. 
Beschrieben 
werden 
unter 
anderem 
die 
physikalischen 
Gleichungen 
der 
relevanten 
Prozesse 
Fluidfluss, 
Salz-und 
Wärmetransport. 
Weiterhin 
werden 
die 
grundlegenden 
Konzepte 
der 
Geostatistik 
sowie 
der 
Skalierungstheorie 
(Upscaling) 
erklärt. 
Der 
Fokus 
liegt 
auf 
der 
statistischen 
Beschreibung 
von 
heterogener 
Leitfähigkeit, 
welche 
Grundlage 
für 
die 
Methode 
ist, 
die 
im 
dritten 
Teil 
der 
Arbeit 
entwickelt 
wird. 
Zusätzlich 
werden 
numerische 
Benchmarks 
präsentiert 
um 
die 
Verwendbarkeit 
der 
Simulationssoftware 
OpenGeoSys 
zu 
zeigen. 
Sowohl 
für 
die 
Modellierung 
des 
Thüringer 
Beckens 
im 
zweiten 
als 
auch 
für 
die 
numerischen 
Pumptests 
im 
dritten 
Teil, 
wird 
dieser 
numerische 
Löser 
verwendet. 


Ein 
numerischer 
Modelansatz 
zur 
Untersuchung 
der 
Fluiddynamik 
des 
Thüringer 
Beckens 
wird 
im 
zweiten 
Teil 
der 
Arbeit 
präsentiert. 
Der 
Fokus 
liegt 
auf 
dem 
Einfluss 
von 
Aquiferheterogenitäten 
und 
der 
Fluiddichte 
auf 
den 
Salztransport. 
Zentrale 
Fragen 
sind: 
Wie 
sieht 
die 
großräumige 
tiefe 
und 
oberflächennahe 
Fluiddynamik 
aus? 
Existieren 
Kopplungen 
thermisch 
angetriebener 
tiefer 
Fluidkonvektion 
mit 
oberflächennahen 
Grundwasserströmungen? 
Wie 
ist 
das 
Phänomen 
zu 
erklären, 
dass 
salzige 
Grundwasser 
in 
oberflächennahen 
Schichten 
vorkommen, 
was 
an 
vielen 
Orten 
im 
Thüringer 
Becken 
zu 
beobachten 
ist? 


Numerische 
Simulationen 
zum 
Fluidfluss, 
Salztransport 
sowie 
zum 
Wärmetransport 
im 
Thüringer 
Becken 
werden 
durchgeführt 
um 
zu 
verstehen, 
welche 
Bedeutung 
geologische 
Besonderheiten 
wie 
Verwerfungen, 
Aquiferheterogenität 
und 
Dichteunter-
schiede 
haben. 
Schwankungen 
in 
der 
Fluiddichte 
können 
dabei 
durch 
Temperatur-
und 
Salzkonzentrationsgradienten 
entstehen. 
Zu 
diesem 
Zweck 
wird 
ein 
geologisches 
Profilmodell 
konstruiert, 
welches 
die 
geologischen 
Begebenheiten 
mit 
den 
Hauptgrundwasserleitern 
und 
-stauern 
sowie 
Verwerfungen 
im 
Thüringer 
Becken 
abbildet. 


Die 
Simulationsergebnisse 
zeigen, 
dass 
der 
regionale 
Grundwasserfluss 
die 
Haupttriebkraft 
für 
die 
Salzverteilung 
ist. 
Der 
Grundwasserfluss 
wiederum 
ist 
stark 
von 
den 
lokalen 
hydraulischen 
Eigenschaften 
abhängig. 
Eine 
qualitative 
Sensitivitätsanalyse 
zeigt, 
dass 
kleine 
Veränderungen 
in 
der 
Leitfähigkeit 
großen 
Einfluss 
auf 
die 
Fließmuster 
und 
die 
Salzverteilung 
haben 
können. 
Der 
lokale 
Mittelwert, 
die 
Stärke 
der 
Abweichungen 
vom 
Mittelwert 
sowie 
die 
räumliche 
Korrelationsstruktur 
der 
Leitfähigkeit 
beeinflussen 
die 
Verteilung 
sowie 
die 
Menge 
an 
gelöstem 
Salz 
im 
Grundwasser. 
Davon 
ist 
insbesondere 
die 
Menge 
an 
Salz 
abhängig, 
die 
in 
oberflächennahe 
Bereiche 
transportiert 
wird. 
Ebenso 
können 
salzkonzentrationsabhängige 
Variationen 
in 
der 
Fluiddichte 
die 
Fließmuster 
maßgeblich 
verändern. 
Salzige 
Grundwasser 
sind 
schwerer 
als 
frisches 
Grundwasser. 
Werden 
hochkonzentrierte 
Solen 
durch 
druckgetriebenen 
Grundwasserfluss 
aufwärts 
transportiert, 
dann 



bewirkt 
die 
höhere 
Fluiddichte 
eine 
entgegengesetzte 
Absinkbewegung 
die 
Durchmischung 
intensiviert. 
Die 
verstärkte 
Durchmischung 
wiederum 
erhöht 
die 
Versalzung 
der 
tiefen 
Grundwasserleiter, 
aber 
verhindert 
den 
Aufstieg 
von 
hochkonzentrierter 
Sole. 
Auf 
der 
anderen 
Seite 
zeigen 
die 
Simulationen 
auch, 
dass 
Temperatur 
als 
Triebkraft 
für 
konvektiven 
Fluidfluss 
vernachlässigt 
werden 
kann. 
Die 
flache 
Struktur 
des 
Beckens 
verhindert 
die 
Entwicklung 
von 
großräumiger 
thermischer 
Konvektion 
aufgrund 
der 
geringen 
Temperaturunterschiede. 


Der 
dritte 
Teil 
der 
Arbeit 
beschäftigt 
sich 
mit 
einer 
Skalierungsmethode, 
welche 
dazu 
dient 
die 
statistischen 
Parameter 
der 
heterogenen 
Leitfähigkeitsverteilung 
mittels 
Pumptestdaten 
zu 
bestimmen. 
Leitfähigkeit 
ist 
der 
Schlüsselparameter 
um 
Grundwasserfließgeschwindigkeiten 
und 
die 
Salzverteilung 
zu 
beschreiben. 
Allerdings 
benötigt 
es 
mehr 
als 
einen 
Mittelwert 
um 
die 
Effekte 
von 
Heterogenitäten 
zu 
beschreiben. 
Bei 
Verwendung 
eines 
statistischen 
Ansatzes 
lassen 
sich 
Leitfähigkeiten 
durch 
eine 
Log-Normal-Verteilung 
beschreiben. 
Zusätzliche 
statistische 
Parameter 
wie 
die 
Varianz 
und 
die 
Korrelationslänge 
erlauben 
eine 
detailliertere 
Charakterisierung. 
Dabei 
bestimmt 
die 
Varianz 
die 
Stärke 
der 
lokalen 
Schwankungen, 
d.h. 
den 
Grad 
der 
Heterogenität. 
Die 
Korrelationslänge 
beschreibt 
die 
Distanz 
über 
die 
Leitfähigkeitswerte 
räumlich 
korreliert 
sind. 
Pumptests 
werden 
häufig 
verwendet 
um 
die 
hydraulischen 
Eigenschaften 
eines 
Aquifers 
zu 
bestimmen. 
Deshalb 
ist 
es 
sinnvoll 
analytische 
Methoden 
zu 
entwickeln, 
welche 
es 
erlauben 
die 
statistischen 
Parameter 
der 
heterogenen 
Leitfähigkeitsverteilung 
aus 
Pumptestdaten 
zu 
bestimmen. 


Zunächst 
wird 
eine 
Formel 
für 
das 
hydraulische 
Druckpotential 
hergeleitet, 
die 
den 
mittleren 
Absenktrichter 
eines 
kleinskaligen 
stationären 
Pumptests 
in 
einem 
dreidimensionalen, 
anisotropen, 
heterogenen 
Medium 
beschreibt. 
Die 
Herleitung 
basiert 
auf 
der 
Skalierungsmethode 
Coarse 
Graining. 
Die 
analytische 
Lösung 
für 
den 
effektiven 
Brunnenfluss 
kann 
man 
verstehen 
als 
eine 
Erweiterung 
für 
Thiems 
Formel 
auf 
heterogene 
Medien. 
Die 
Lösung 
des 
effektiven 
Brunnenflusses 
ist 
einerseits 
eine 
Funktion 
vom 
Abstand 
zum 
Pumpbrunnen 
und 
anderseits 
abhängig 
von 
den 
statistischen 
Größen 
Mittelwert, 
Varianz, 
Korrelationslänge 
und 
Anisotropierate 
der 
Leitfähigkeit. 
Die 
geschlossen-analytische 
Form 
der 
Lösung 
erlaubt 
es 
eine 
Sensitivitätsanalyse 
bezüglich 
der 
statistischen 
Parameter 
durchzuführen 
sowie 
eine 
inverse 
Schätzmethode 
zu 
entwickeln. 
Die 
Analyse 
von 
numerischen 
Pumptests, 
sowohl 
einzelner 
als 
auch 
von 
Ensembles, 
zeigt, 
dass 
Absenktrichter 
aus 
Pumptests 
in 
heterogenen 
Medien 
sehr 
genau 
durch 
die 
Lösung 
des 
effektiven 
Brunnenflusses 
beschrieben 
werden 
können. 
Die 
Ergebnisse 
der 
Inversen 
Schätzung 
der 
statistischen 
Leitfähigkeitsparameter 
stimmen 
sehr 
gut 
mit 
den 
Anfangswerten 
überein. 


Es 
folgt 
die 
Untersuchung, 
ob 
die 
analytische 
Lösung 
des 
effektiven 
Brunnenflusses 
dazu 
in 
der 
Lage 
ist 
genaue 
und 
zuverlässige 
Schätzwerte 
für 
die 
statistischen 
Parameter 
unter 
reduzierter 
Datenverfügbarkeit 
zu 
liefern. 
Dies 
ist 
relevant, 
da 
in 
der 
Praxis 
meist 
nur 
wenige 
Messungen 
des 
hydraulischen 
Druckpotentials 
verfügbar 



sind. 
Numerische 
Pumptest 
werden 
generiert 
und 
ausgewertet 
mit 
Hinblick 
auf 
die 
Zuverlässigkeit 
und 
die 
Genauigkeit 
der 
Schätzergebnisse 
unter 
reduzierter 
Datenverfügbarkeit. 
Dazu 
wird 
die 
Anzahl 
von 
Messdatenpunkten 
systematisch 
reduziert 
und 
für 
jedes 
Level 
der 
Datenverfügbarkeit 
die 
statistischen 
Parameter 
und 
ihre 
Unsicherheiten 
invers 
geschätzt. 
Die 
Ergebnisse 
zeigen, 
dass 
die 
Genauigkeit 
und 
Zuverlässigkeit 
der 
geschätzten 
statistischen 
Parameter 
sehr 
sensitiv 
gegenüber 
Anzahl 
und 
räumlicher 
Verteilung 
der 
Druckpotentialmessungen 
sind. 
Untersuchungsbrunnen 
sind 
sowohl 
im 
Nahfeld, 
d.h. 
in 
direkter 
Umgebung 
des 
Pumpbrunnens, 
als 
auch 
im 
Fernfeld 
notwendig 
um 
eine 
sinnvolle 
Schätzung 
von 
Heterogenitätsparametern 
gewährleisten 
zu 
können. 
Insbesondere 
die 
Schätzbarkeit 
der 
Korrelationslänge 
erfordert 
eine 
gewisse 
Anzahl 
an 
Piezometern 
im 
Umfeld 
des 
Pumpbrunnens. 
Abschließend 
wird 
die 
Schätzmethode 
auf 
Messdaten 
von 
Pumptests 
auf 
der 
Horkheimer 
Insel, 
Deutschland 
angewendet. 
Es 
wird 
gezeigt, 
dass 
die 
Schätzwerte 
der 
Methode 
des 
effektiven 
Brunnenflusses 
für 
die 
mittlere 
Leitfähigkeit, 
Varianz 
und 
Korrelationlänge 
sehr 
gut 
mit 
den 
Messergebnissen 
aus 
Laboruntersuchungen 
übereinstimmen. 
Die 
Untersuchungen 
geben 
wertvolle 
Hinweise 
in 
Bezug 
auf 
das 
konzeptionelle 
Design 
von 
Pumptests 
als 
auf 
die 
Güte 
der 
Schätzung 
von 
Parametern 
aus 
vorliegenden 
Pumptestdaten. 



List 
of 
Symbols 
and 
Abbreviations 


Symbol 
Unit 
Quantity 


l 
[m] 
longitudinal 
solute 
dispersivity 


t 
[m] 
transversal 
solute 
dispersivity 


¯l 
[m] 
longitudinal 
thermal 
dispersivity 


¯t 
[m] 
transversal 
thermal 
dispersivity 


C 
[-] 
solute 
expansion 
coefficient 


T 
[K-1] 
thermal 
expansion 
coefficient 



(e) 
anisotropy 
function 


G 
exponential 
integral 
function 


d 
Kronecker 
delta 


. 
[-] 
proportionality 
factor 


. 
correlation 
function 


˜Fourier 
transform 
of 
correlation 
function 


. 
[mD] 
permeability 


. 
[m] 
filter 
constant 


s 
[kgm2 
s-3 
K-1] 
thermal 
conductivity 
of 
solid 


w 
[kgm2 
s-3 
K-1] 
thermal 
conductivity 
of 
water 


. 
[-] 
weighting 
factor 


µ 
mean 
of 
log-conductivity 


. 
[kgm-1 
s-1] 
viscosity 


f 
[-] 
porosity 


. 
[-] 
angular 
coordinate 


. 
[kgm-3] 
fluid 
density 


0 
[kgm-3] 
fresh 
water 
density 


s 
[kgm-3] 
density 
of 
solid 
rock 


2 
[-] 
variance 
of 
log-conductivity 


ˆ2 
[-] 
estimated 
variance 
of 
log-conductivity 


t 
[-] 
tortuosity 


. 
abbreviation 
for 
ln(Kefu/Kwell) 



w 
[m2] 
surface 
of 
the 
well 


V 
nabla 
operator 


A 
[-] 
relative 
space 
fraction 
of 
an 
element 
in 
a 
numerical 
mesh 
c 


[m2 
s-2 
K-1] 
thermal 
capacity 
of 
water 


cs 
[m2 
s-2 
K-1] 
thermal 
capacity 
of 
solid 
rock 


C 
[-] 
concentration 
C0, 
C1, 
C2 
integration 
constants 


¯ 


C 
[-] 
spatial 
mean 
concentration 

C 
[%] 
relative 
concentration 
difference 



Cr 
[-] 


CV 


d 
[-] 


D 
[m2 
s-1] 
Dm 
[m2 
s-1] 
Ds 
[m2 
s-1] 
DT 
[m2 
s-1] 


e 
[-] 
eˆ[-] 
f 
f˜ 


g/ 
g 
[ms-2] 
h 
[m] 
hˆ
[m] 


hefw(r) 


[m] 
hh(r)) 
[m] 
hh(r)) 
[m] 
hThiem(r) 
[m] 


H 
[-] 
Kf 
[ms-1] 
KA 
[ms-1] 
KG 
[ms-1] 
KH 
[ms-1] 
Keq 
[ms-1] 
Kef 
[ms-1] 


Kefu 
[ms-1] 
Kwell 
[ms-1] 
K(x) 
[ms-1] 


KCG(r) 


[ms-1] 
Kˆ
A 
[ms-1] 
Kˆ
G 
[ms-1] 
Kˆ
H 
[ms-1] 


Kˆ
well 
[ms-1] 
Kˆ
efu 
[ms-1] 
hKA) 
[ms-1] 
hKH) 
[ms-1] 
hKwell) 
[ms-1] 
i 
[m] 
`h 
[m] 
`v 
[m] 


`ˆ 
[m] 
L/Lz 
[m] 


Courant 
number 
covariance 
function 
number 
of 
spatial 
dimensions 
hydrodynamical 
dispersion 
tensor 
molecular 
diffusion 
coefficient 
mechanical 
dispersion 
tensor 
thermal 
diffusion/dispersion 
tensor 
anisotropy 
ratio 
estimated 
anisotropy 
ratio 
filter 
function 
Fourier 
transform 
of 
filter 
function 
gravitational 
acceleration 
/ 
vector 
hydraulic 
head 
estimated 
hydraulic 
head 
effective 
well 
flow 
hydraulic 
head 
simulated 
hydraulic 
head 
ensemble 
mean 
of 
simulated 
hydraulic 
heads 
hydraulic 
head 
of 
Thiem’s 
solution 
Hurst 
coefficient 
hydraulic 
conductivity 
arithmetic 
mean 
conductivity 
geometric 
mean 
conductivity 
harmonic 
mean 
conductivity 
equivalent 
conductivity 
effective 
conductivity 
effective 
conductivity 
for 
uniform 
flow 
effective 
conductivity 
at 
the 
well 
heterogeneous 
conductivity 
distribution 
radial 
depending 
mean 
Coarse 
Graining 
conductivity 
estimated 
arithmetic 
mean 
conductivity 
estimated 
geometric 
mean 
conductivity 
estimated 
harmonic 
mean 
conductivity 
estimated 
conductivity 
at 
the 
well 
estimated 
conductivity 
far 
from 
field 
’measured’ 
arithmetic 
mean 
conductivity 
’measured’ 
geometric 
mean 
conductivity 
local 
mean 
of 
conductivity 
values 
at 
the 
well 
correlation 
length 
horizontal 
correlation 
length 
vertical 
correlation 
length 
estimated 
correlation 
length 
aquifer 
thickness/vertical 
domain 
extend 



ODE 


p 
PD 


PDE 


pdf 


Peg 


q 


q 
Q 
Qw 
r 
rw 
R 


Rac 
RaS 
RaT 
REF 


s 


S 


SRF 


SV 


t 
T 


¯ 


T 

T 
Tf 


Z(x) 


[kgm-1 
s-2] 


[m] 
[-] 
[ms-1] 
[ms-1] 


[m3 
s-1] 


[m] 
[m] 
[m] 
[-] 
[-] 
[-] 
[m] 
[m3 
kg-1] 
[s] 
[C] 
or 
[K] 
[-] 
[%] 
[m2 
s-1] 
[ms-1] 
ordinary 
differential 
equation 
pressure 
penetration 
depth 
partial 
differential 
equation 
probability 
density 
function 
grid 
Peclet 
number 
Darcy 
velocity 
vector 
absolute 
Darcy 
velocity 
sink/source 
term 
pumping 
rate 
at 
the 
well 
radial 
distance 
radius 
of 
the 
well 
fixed 
radial 
distance 
from 
the 
pumping 
well 
critical 
Rayleigh 
number 
solute 
Rayleigh 
number 
thermal 
Rayleigh 
number 
representative 
elementary 
volume 
two-point 
separation 
distance 
specific 
storage 
coefficient 
spatial 
random 
function 
semivariogram 
function 
time 
temperature 
spatial 
mean 
temperature 
relative 
temperature 
difference 
transmissivity 
pore 
velocity 
vector 
spatial 
random 
function 



14 



Contents 


Abstract 
3 


Zusammenfassung 
7 


List 
of 
Symbols 
and 
Abbreviations 
11 


I. 
Modeling 
Fluid 
Flow 
and 
Salt 
Transport 
in 
a 
Sedimentary 
Basin 
17 


1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
19 


1.1. 
Introduction...................................... 
19 


1.2. 
PhysicsofPorousMedia............................... 
20 


1.3. 
GeostatisticsinSubsurfaceHydrology 
....................... 
25 


1.4. 
Upscaling 
....................................... 
29 


2. 
Numerical 
Benchmarks 
35 


2.1. 
Introduction...................................... 
35 


2.2. 
PumpingTestinHomogeneousPorousMedia 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
35 


2.3. 
UniformFlowinHeterogeneousPorousMedia. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
37 


2.4. 
TheElderBenchmark 
................................ 
40 


2.5. 
TheSaltdomeProblem................................ 
44 


II. 
Impact 
of 
Heterogeneity 
and 
Density 
Dependent 
Flow 
on 
Regional 
Scale 
51 


3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
53 


3.1. 
Introduction...................................... 
53 


3.2. 
Hydrogeological 
Characterization 
of 
the 
Study 
Area 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
55 


3.3. 
NumericalGroundwaterFlowModel 
........................ 
59 


3.4. 
DiscussionofSimulationResults 
.......................... 
64 


3.5. 
Conclusions 
...................................... 
84 


III. 
Determining 
Aquifer 
Heterogeneity 
on 
Local 
Scale 
87 


4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
89 


4.1. 
Introduction...................................... 
89 


4.2. 
StatementoftheProblem 
.............................. 
91 


4.3. 
MethodofCoarseGraining 
............................. 
93 


4.4. 
TheExtendedThiem’sSolution........................... 
95 


4.5. 
InterpretingNumericalPumpingTest 
....................... 
99 


4.6. 
SummaryandConclusions.............................. 
105 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
109 


5.1. 
Introduction...................................... 
109 


5.2. 
TheoreticalFramework................................ 
110 


5.3. 
Pumping 
Tests 
under 
Limited 
Data 
Availability 
-a 
Paradigm 
for 
Field 
Applications113 


5.4. 
FieldApplication................................... 
118 


5.5. 
Conclusions 
...................................... 
121 


6. 
Summary 
and 
Outlook 
123 


A. 
Appendix 
125 


A.1. 
The 
Coarse 
Graining 
Conductivity 
in 
Anisotropic 
Media 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
125 


A.2.TheAnisotropyFunction 
.............................. 
126 


A.3. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Three 
Dimensions 
. 
. 
. 
. 
. 
. 
. 
. 
127 


A.4. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Two 
Dimensions 
. 
. 
. 
. 
. 
. 
. 
. 
. 
128 


Bibliography 
131 



Part 
I. 
Modeling 
Fluid 
Flow 
and 
Salt 
Transport 
in 
a 
Sedimentary 
Basin 



18 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
1.1. 
Introduction 
During 
the 
last 
20 
years 
numerical 
modeling 
has 
gained 
significant 
importance 
in 
hydrogeology. 
The 
rapid 
evolution 
of 
computer 
technology 
allowed 
the 
development 
of 
complex 
software 
which 
reproduces 
the 
physical 
processes 
occurring 
in 
the 
subsurface 
with 
spatial 
and 
temporal 
resolution. 
Numerical 
models 
have 
several 
advantages. 
They 
can 
enhance 
the 
understanding 
of 
fluid 
dynamics 
due 
to 
the 
fact 
that 
they 
allow 
simultaneous 
evaluation 
of 
processes 
and 
thus 
investigation 
of 
coupling 
effects. 
Furthermore, 
models 
can 
serve 
quantitative 
aspects 
by 
providing 
estimates 
of 
quantities 
that 
can 
be 
used 
for 
engineering 
and 
management 
purposes. 


It 
requires 
several 
steps 
to 
establish 
a 
complex 
numerical 
model 
for 
subsurface 
flow 
and 
transport. 
Starting 
point 
is 
the 
formulation 
of 
a 
hydrogeological 
conceptual 
model. 
This 
aspect 
is 
significantly 
governed 
by 
the 
objective 
of 
the 
modeling 
project. 
The 
step 
contains 
the 
selection 
of 
the 
geological 
setup 
by 
identifying 
boundaries 
of 
the 
formation 
and 
significant 
geological 
features. 
The 
following 
step 
is 
to 
grasp 
the 
main 
mechanisms 
involved 
and 
to 
formulate 
physical 
equations 
describing 
the 
processes. 
In 
case 
existing 
numerical 
solvers 
are 
used, 
the 
third 
step 
contains 
the 
adaption 
of 
the 
hydrogeological 
conceptual 
model 
to 
the 
numerical 
framework. 
It 
requires 
the 
discretization 
of 
the 
domain 
in 
space 
and 
in 
time, 
to 
fix 
boundary 
conditions, 
initial 
conditions, 
as 
well 
as 
to 
set 
physical 
parameters. 


Developing 
a 
conceptual 
model 
is 
very 
specific 
to 
the 
problem 
under 
consideration. 
However, 
the 
physical 
equations 
which 
describe 
the 
major 
processes 
occurring 
in 
the 
subsurface 
are 
well 
known. 
We 
discuss 
them 
briefly 
in 
section 
1.2. 
These 
conservation 
laws 
form 
the 
basis 
of 
physically-based 
simulation 
software, 
like 
OpenGeoSys 
[Kolditz 
et 
al., 
2012b], 
FEFLOW 
[Fef, 
2012], 
MODFLOW 
[U.S. 
Geological 
Survey, 
2012], 
HydroGeoSphere 
[Brunner 
and 
Simmons, 
2012], 
SUTRA 
[Voss 
and 
Provost, 
2010], 
and 
many 
more. 
Complex 
numerical 
models 
with 
spatial 
and 
temporal 
resolution 
for 
thermal, 
hydraulic, 
and 
mechanical 
subsurface 
processes 
are 
important 
for 
the 
analysis 
of 
deep 
geological 
systems 
under 
high 
temperature, 
pressure 
and 
stress 
conditions. 
Application 
areas 
are 
e.g. 
reproduction 
and 
prediction 
of 
the 
solute 
plume 
development, 
geothermal 
energy 
utilization, 
nuclear 
waste 
disposal, 
and 
carbon 
dioxide 
storage 
in 
the 
deep 
geological 
formation. 


The 
hydraulic 
parameter 
of 
major 
interest 
within 
this 
work 
is 
the 
permeability. 
It 
shows 
a 
spatially 
heterogeneous 
distribution, 
which 
inhibits 
a 
simple 
analysis 
of 
many 
subsurface 
flow 
and 
transport 
problems. 
A 
detailed 
spatial 
resolution 
of 
the 
heterogeneous 
distribution 
is 
not 
possible 
in 
numerical 
models 
in 
general 
due 
to 
complexity. 
Therefore 
methods 
have 
been 
developed 
to 
handle 
aquifer 
heterogeneity 
in 
a 
stochastic 
manner 
which 
allows 
to 
incorporate 
uncertainty 
about 
the 
spatial 
distribution 
of 
permeability. 
We 
discuss 
the 
fundamental 
concepts 
of 
stochastic 
subsurface 
hydrology 
in 
section 
1.3. 
To 
capture 
the 
aspects 
of 
heterogeneous 
permeability 
in 
large 
scale 
hydrogeological 
models, 
it 
is 
necessary 
to 
reduce 
complexity 
by 
a 
certain 
averaging 
strategy. 
This 
procedure 
is 
in 
general 
denoted 
upscaling. 
Upscaling 
methods 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
aim 
to 
find 
representative 
parameters 
for 
spatially 
distributed 
quantities, 
which 
are 
capable 
of 
reproducing 
some 
average 
behavior 
of 
the 
system. 
We 
discuss 
the 
basics 
of 
upscaling 
in 
section 
1.4. 


1.2. 
Physics 
of 
Porous 
Media 
The 
simulation 
software 
of 
choice 
in 
this 
thesis 
is 
OpenGeoSys 
[Kolditz 
et 
al., 
2012a, 
b]. 
It 
follows 
the 
basic 
idea 
of 
continuum 
mechanics 
that 
the 
evolution 
of 
a 
physical 
system 
is 
completely 
determined 
by 
conservation 
laws. 
Basic 
properties 
such 
as 
mass, 
momentum 
and 
energy 
are 
conserved 
during 
the 
considered 
process 
at 
all 
times. 
Any 
physical 
system 
can 
be 
completely 
determined 
by 
these 
conservation 
properties. 
To 
reproduce 
processes 
in 
porous 
media 
numerically, 
additional 
information 
concerning 
the 
consistencies 
of 
the 
material 
(e.g. 
fluids, 
solids, 
porous 
medium) 
in 
the 
form 
of 
constitutive 
laws 
is 
necessary. 


In 
the 
following 
we 
shortly 
introduce 
the 
continuum 
concept 
of 
porous 
media 
and 
state 
the 
macroscopic 
balance 
equations 
of 
mass 
of 
fluid, 
mass 
of 
solute, 
fluid 
momentum 
and 
energy 
conservation. 
It 
is 
not 
aimed 
to 
give 
derivations 
nor 
to 
discuss 
or 
expand 
the 
fundamentals 
of 
porous 
media 
theory. 
A 
detailed 
description 
of 
the 
theoretical 
background 
with 
derivations 
can 
be 
found 
for 
instance 
in 
Bear 
[1972], 
Kinzelbach 
and 
Rausch 
[1995], 
as 
well 
as 
Nield 
and 
Bejan 
[1999]. 
Detailed 
discussion 
on 
density 
dependent 
flow 
effects 
can 
be 
found 
e.g. 
in 
Kolditz 
et 
al. 
[1998] 
and 
Diersch 
and 
Kolditz 
[2002]. 


1.2.1. 
Porous 
Media 
as 
Continuum 
Porous 
media 
is 
the 
geological 
material 
in 
the 
subsurface 
which 
consist 
of 
a 
solid 
matrix 
and 
pore 
space 
filled 
by 
water. 
The 
microscopic 
distribution 
of 
solid 
and 
pore 
space 
is 
complex 
and 
in 
general 
impossible 
to 
resolve. 
A 
description 
of 
the 
processes 
of 
flow 
and 
transport 
on 
microscopic 
level 
is 
neither 
feasible 
nor 
useful 
in 
most 
problems 
on 
macroscopic 
scale. 


The 
continuum 
approach 
targets 
at 
resolving 
these 
difficulties 
by 
a 
transition 
to 
the 
macroscopic 
level. 
At 
the 
macroscopic 
scale, 
porous 
media 
is 
regarded 
as 
a 
continuum 
that 
consists 
of 
an 
immobile 
solid 
phase 
and 
a 
mobile 
fluid 
phase. 
Both 
phases 
are 
characterized 
by 
macroscopic 
measurable 
quantities, 
which 
result 
from 
microscopic 
quantities 
by 
averaging 
over 
volumes. 
The 
transformation 
from 
microscopic 
to 
macroscopic 
level 
requires 
appropriate 
averaging 
strategies. 


The 
standard 
approach 
of 
averaging 
is 
to 
sample 
over 
a 
representative 
elementary 
volume 
(REV) 
[Bear, 
1972]. 
On 
the 
one 
hand, 
the 
REV 
has 
to 
be 
sufficiently 
large 
for 
fluctuations 
of 
spatially 
averaged 
properties 
to 
be 
negligible 
on 
microscopic 
scale. 
On 
the 
other 
hand, 
it 
has 
to 
be 
small 
enough 
to 
be 
regarded 
as 
a 
point 
at 
the 
macroscopic 
scale. 
In 
hydrogeology 
the 
REV 
is 
the 
smallest 
volume 
over 
which 
measurements 
can 
be 
made 
that 
yield 
a 
representative 
value 
for 
the 
entire 
domain 
under 
consideration. 
Quantities 
and 
parameters 
of 
porous 
media 
are 
only 
valid 
above 
the 
scale 
of 
the 
REV. 



1.2. 
Physics 
of 
Porous 
Media 
1.2.2. 
Continuity 
Equation 
of 
Flow 
The 
continuity 
equation 
is 
derived 
from 
the 
principle 
of 
mass 
conservation 
of 
the 
fluid. 
It 
describes 
the 
evolution 
of 
fluid 
flow 
in 
space 
and 
time. 
It 
relates 
the 
temporal 
change 
in 
fluid 
mass 
to 
the 
differences 
in 
spatial 
distribution 
in 
combination 
with 
the 
existence 
of 
a 
fluid 
sink 
or 
source. 


The 
macroscopic 
mass 
balance 
equation 
of 
the 
fluid, 
averaged 
over 
a 
representative 
elementary 
volume 
(REV), 
in 
a 
porous 
medium 
is 
given 
by 


. 
(S) 


+ 
r· 
(q)= 
Q. 
, 
(1.1)
@t 


where 
S 
is 
the 
specific 
storage 
coefficient, 
f 
is 
the 
porosity, 
. 
is 
the 
fluid 
density, 
t 
is 
the 
time, 
q 
is 
the 
Darcy 
velocity 
vector, 
and 
Q. 
is 
the 
sink/source 
term 
of 
the 
fluid 
mass 
in 
an 
aquifer. 


1.2.3. 
Darcy’s 
Law 
Darcy’s 
law 
characterizes 
the 
velocity 
vector 
q 
as 
a 
combination 
of 
porous 
medium 
characteristics 
and 
the 
hydraulic 
gradient. 
Darcy’s 
law 
can 
be 
understood 
as 
a 
momentum 
balance 
equation 
in 
porous 
media. 
In 
terms 
of 
hydraulic 
pressure 
p 
it 
is 
given 
by 


q 
= 
v 
= 
- 
. 
(rp 
+ 
g) 
, 
(1.2)

. 


where 
v 
is 
the 
fluid 
or 
pore 
velocity 
vector, 
. 
is 
the 
permeability 
of 
a 
porous 
medium, 
. 
is 
the 
viscosity, 
and 
g 
is 
the 
gravity 
vector. 


An 
alternative 
way 
to 
write 
Eq.(1.2) 
is 
the 
formulation 
of 
Darcy’s 
law 
with 
the 
freshwater 
hydraulic 
head 
h 
that 
directly 
relates 
the 
flow 
velocity 
to 
the 
driving 
forces 


q 
= 
-Kf 
(rh 
+(. 
- 
0)/0ez) 
, 
(1.3) 


where 
Kf 
is 
the 
hydraulic 
conductivity, 
0 
is 
the 
fresh 
water 
density, 
. 
is 
the 
actual 
fluid 
density 
and 
ez 
is 
the 
unit 
vector 
in 
the 
gravitational 
direction. 
The 
buoyancy 
term 
(. 
- 
0)/0ez 
is 
only 
present 
in 
processes 
where 
density 
differences 
are 
taken 
into 
account. 
In 
constant 
density 
processes 
the 
velocity 
is 
determined 
by 
the 
gradient 
in 
hydraulic 
head 
rh. 
The 
hydraulic 
conductivity 


0g

Kf 
= 
(1.4)

. 
is 
a 
porous 
medium 
property 
since 
it 
depends 
on 
permeability 
. 
as 
a 
property 
of 
the 
solid 
phase 
and 
on 
the 
fluid 
characteristics 
density 
0 
and 
viscosity 
. 


The 
relation 
between 
hydrostatic 
pressure 
p 
and 
hydraulic 
head 
h 
is 
given 
by 


p

h 
=+ 
zo,

0g 


where 
z0 
denotes 
the 
vertical 
distance 
to 
the 
zero 
pressure 
position. 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
1.2.4. 
Solute 
Transport 
Equation 
The 
transport 
of 
a 
solute 
like 
salt 
in 
groundwater 
is 
governed 
by 
the 
advection-dispersion 
equation 


. 
(C) 


+ 
r· 
(qC) 
-r· 
(D 
·rC)= 
QC 
, 
(1.5)
@t 
where 
C 
. 
[0, 
1] 
is 
the 
relative 
mass 
concentration, 
D 
hydrodynamic 
dispersion 
tensor, 
and 
QC 
is 
the 
mass 
supply. 


Several 
fluxes 
are 
involved 
in 
the 
mass 
transport 
process 
in 
Eq. 
(1.5): 
the 
first 
term 
describes 
the 
temporal 
change, 
the 
second 
term 
describes 
an 
advective 
flux 
and 
the 
third 
term 
includes 
diffusive 
and 
dispersive 
effects. 
The 
right 
hand 
side 
accounts 
for 
a 
source 
of 
solute 
mass. 


The 
dispersion 
tensor 
D 
in 
Eq.(1.5) 
incorporates 
dispersive 
and 
diffusive 
fluxes. 
According 
to 
Scheidegger’s 
law 
[Bear, 
1972] 
it 
is 
given 
in 
general 
tensor 
form 
by 


)qiqj

Dij 
= 
Dm 
+ 
Ds 
= 
Dm 
+ 
tqij 
+(l 
- 
t, 
(1.6) 


q 


where 
t 
is 
the 
tortuosity 
and 
Dm 
is 
the 
coefficient 
of 
molecular 
diffusion. 
Ds 
is 
the 
mechanical 
dispersion 
with 
the 
longitudinal 
and 
transversal 
dispersivities 
l 
and 
t, 
ij 
is 
the 
Kronecker 
delta, 
q 
= 
|q| 
is 
the 
absolute 
value 
of 
Darcy 
velocity, 
and 
qi 
and 
qj 
are 
the 
Darcy 
velocities 
in 
ith 
and 
jth 
principal 
directions, 
respectively. 


A 
simplified 
description 
can 
be 
given 
in 
a 
Cartesian 
coordinate 
system, 
which 
is 
suitable 
for 
instance 
for 
uniform 
flow. 
Choosing 
the 
x1-axis 
to 
be 
parallel 
to 
the 
average 
Darcy 
velocity 
q 


the 
tensor 
Ds 
simplifies 
to 


1

. 


BB
. 


lq 
00 


0 
tq 
0

CCA

Ds 
= 


. 


(1.7) 
00 
tq 


Thus 
the 
dispersion 
in 
main 
flow 
direction 
is 
given 
by 
the 
product 
of 
the 
average 
Darcy 
velocity 
q 
with 
the 
longitudinal 
dispersivity 
l. 
In 
the 
perpendicular 
directions 
the 
dispersion 
is 
proportional 
to 
the 
transversal 
dispersivity 
t, 
which 
is 
commonly 
assumed 
to 
be 
at 
least 
one 
order 
of 
magnitude 
smaller 
than 
l. 
According 
to 
Bear 
[1972], 
the 
axes 
of 
the 
coordinate 
system 
in 
which 
Ds 
is 
expressed 
in 
Eq. 
(1.7) 
are 
called 
the 
principal 
axes 
of 
dispersion. 


1.2.5. 
Heat 
Transport 
Equation 
Energy 
conservation 
has 
to 
be 
taken 
into 
account 
when 
describing 
temperature 
dependent 
processes. 
The 
heat 
balance 
equation 
in 
porous 
media 
is 
based 
on 
a 
thermal 
equilibrium 
between 
solid 
and 
liquid 
phase, 
which 
can 
be 
assumed 
due 
to 
the 
low 
velocities. 
The 
equation 
is 
given 
by 


ss)@T 


(c. 
+ 
(1 
- 
)c 
+ 
cq 
·rT 
-r· 
(cDT 
·rT 
)= 
QT 
, 
(1.8)

@t 
where 
c. 
+ 
(1 
- 
)css 
is 
the 
heat 
capacity 
of 
the 
porous 
medium, 
with 
porosity 
, 
density 
. 
and 
specific 
heat 
capacity 
c 
of 
water 
as 
well 
as 
density 
s 
and 
specific 
heat 
capacity 
cs 
of 



1.2. 
Physics 
of 
Porous 
Media 
solid. 
T 
is 
the 
temperature, 
t 
it 
the 
time, 
q 
is 
the 
Darcy 
velocity, 
and 
DT 
is 
the 
thermal 
diffusion/dispersion 
tensor. 
The 
heat 
source 
term 
is 
given 
in 
detail 
by 
QT 
= 
Qw 
+ 
(1 
-

T 
)sQs 
, 
where 
Qw 
is 
the 
heat 
production 
of 
water 
and 
Qs 
is 
the 
heat 
production 
of 
solid. 


TT 
T 


The 
thermal 
diffusion-dispersion 
tensor 
DT 
is 
given 
in 
analogy 
to 
the 
hydrodynamic 
dispersion 
tensor 
D 
in 
Eq.(1.6), 


DT 
)qiqj 


ij 
= 
DT 
ij 
+¯tqij 
+(¯l 
- 
¯t, 
(1.9) 


q 


w+(1-)s

where 
DT 
= 
is 
the 
thermal 
diffusivity 
with 
heat 
conductivity 
w 
and 
s 
of 
water 


c. 


and 
solid, 
respectively. 
The 
second 
term 
and 
third 
term 
are 
part 
of 
the 
thermal 
dispersion 
tensor 
with 
heat 
dispersion 
coefficients 
¯t 
and 
¯l 
in 
transversal 
and 
longitudinal 
directions. 
ij 
is 
the 
Kronecker-delta, 
q 
= 
|q| 
is 
the 
absolute 
value 
of 
Darcy 
velocity, 
qi 
and 
qj 
denote 
the 
Darcy 
velocities 
in 
ith 
and 
jth 
principle 
directions, 
respectively. 


Due 
to 
the 
low 
velocities 
q 
in 
porous 
medium, 
thermal 
diffusion 
dominates 
the 
process 
of 
thermal 
dispersion. 
Therefore 
the 
thermal 
dispersion 
term 
in 
Eq. 
(1.9) 
is 
mostly 
neglected, 
resulting 
in 
cDT 
= 
w 
+ 
(1 
- 
)s 
in 
Eq.(1.8). 


1.2.6. 
Equation 
of 
State 
for 
Fluid 
Density 
The 
description 
of 
processes 
where 
changes 
in 
fluid 
density 
take 
place 
requires 
a 
constitutive 
fluid 
density 
model. 
Based 
on 
experiments, 
equations 
are 
formulated 
which 
describe 
the 
relationship 
between 
fluid 
density 
and 
other 
relevant 
quantities 
like 
water 
pressure, 
solute 
concentration 
and 
temperature. 


Different 
descriptions 
exist, 
like 
an 
exponential 
function 
as 
given 
by 
Kolditz 
et 
al. 
[1998] 
or 
the 
additive 
description 
of 
Herbert 
et 
al. 
[1988]. 
The 
most 
common 
approximate 
relationship 
is 
the 
linearized 
equation 
of 
the 
bulk 
fluid 
density 
[Kolditz 
et 
al., 
1998; 
Diersch 
and 
Kolditz, 
2002; 
Nield 
and 
Bejan, 
1999; 
Magri, 
2005] 
accounting 
for 
thermal 
and 
solute 
mass 
effects 


. 
= 
0 
(1 
+ 
C 

C 
- 
T 

T 
) 
, 
(1.10) 


where 
. 
denotes 
the 
fluid 
density, 
0 
is 
the 
reference 
density 
of 
freshwater, 

C 
. 
[0, 
1] 
is 
the 


T 
-T0

change 
in 
relative 
solute 
concentration, 
C 
is 
the 
solute 
expansion 
coefficient, 

T 
= 
2

T0 


[0, 
1] 
is 
the 
change 
in 
temperature 
relative 
to 
the 
reference 
temperature 
T0, 
and 
T 
is 
the 
thermal 
expansion 
coefficient. 


A 
similar 
model 
could 
be 
assumed 
for 
fluid 
viscosity 
since 
this 
property 
also 
depends 
on 
temperature 
and 
solute 
mass. 
However, 
investigations 
of 
Magri 
[2005] 
and 
Kaiser 
et 
al. 
[2011] 
show 
that 
the 
effects 
of 
fluid 
viscosity 
on 
flow 
pattern 
at 
basin 
scale 
is 
much 
less 
pronounced 
in 
comparison 
to 
density 
effects. 
Thus 
a 
constant 
viscosity 
model 
is 
appropriate 
for 
moderate 
temperature 
and 
solute 
mass 
gradients. 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
1.2.7. 
Stability 
Criteria 
for 
Density 
Dependent 
Flow 
The 
Rayleigh 
number 
is 
a 
classical 
instrument 
to 
characterize 
density 
dependent 
flow 
regimes. 
The 
value 
of 
the 
Rayleigh 
number 
characterizes 
the 
stability 
of 
the 
system 
and 
is 
used 
to 
predict 
the 
onset 
of 
convection. 
It 
results 
from 
a 
dimensional 
analysis 
of 
Darcy’s 
law 
(1.3) 
in 
combination 
with 
the 
equation 
of 
state 
for 
fluid 
density 
(1.10) 
and 
the 
transport 
equations 
for 
solutes 
(1.5) 
or 
heat 
(1.8). 


The 
Rayleigh 
number 
is 
defined 
with 
reference 
to 
the 
quantity 
of 
influence 
on 
the 
fluid 
density 
as 
Rayleigh 
number 
of 
solute 
RaS 
and 
thermal 
Rayleigh 
number 
RaT 
by 


Kf 
LzC 

CKf 
Lz|T 
|
T

RaS 
= 
and 
RaT 
= 
, 
(1.11)

D 
DT 


where 
Kf 
denotes 
the 
hydraulic 
conductivity, 
Lz 
is 
the 
vertical 
domain 
extent, 
f 
is 
the 
porosity, 

C 
is 
change 
in 
solute 
concentration, 
C 
is 
the 
solute 
expansion 
coefficient, 
D 
is 
the 
hydrodynamic 
dispersion 
in 
the 
direction 
of 
interest, 

T 
is 
the 
change 
in 
temperature, 
T 
is 
the 
thermal 
expansion 
coefficient, 
and 
DT 
is 
thermal 
diffusion-dispersion. 


In 
double 
diffusive 
phenomena, 
where 
both 
concentration 
and 
temperature 
differences 
cause 
buoyancy-driven 
transport, 
the 
Rayleigh 
number 
results 
as 
sum 
or 
difference 
of 
RaS 
and 
RaT, 
corresponding 
to 
the 
direction 
of 
impact 
[Nield 
and 
Bejan, 
1999]. 
Perturbation 
theory 
identified 
two 
important 
critical 
Rayleigh 
numbers: 
Ra(1) 
=4p 
which 
characterizes 
the 
onset 
of 


c 


convection 
in 
stable 
stationary 
rolls 
and 
Ra(2) 
˜ 
400 
marking 
the 
limit 
between 
stable 
and 


c 


unstable 
patterns. 


1.2.8. 
Quantitative 
Indicators 
Characteristic 
measures 
can 
help 
to 
quantify 
differences 
in 
simulation 
results. 
This 
issue 
is 
in 
particular 
important 
for 
benchmarking 
and 
for 
sensitivity 
analysis 
of 
parameters. 
Indicators 
can 
underline 
visually 
observed 
results 
or 
quantify 
discrepancies 
between 
simulations 
which 
cannot 
be 
detected 
by 
pure 
visual 
comparison. 
A 
quantitative 
indicator 
has 
to 
be 
chosen 
according 
to 
the 
problem 
of 
interest. 
For 
instance 
Prasad 
and 
Simmons 
[2005] 
presented 
a 
number 
of 
methods 
to 
compare 
simulation 
results 
for 
variable 
density 
flow 
problems. 


We 
distinguish 
between 
indicators 
referring 
to 
a 
single 
simulation 
and 
comparative 
measures 
which 
quantify 
the 
difference 
between 
two 
simulations. 
Typical 
representatives 
for 
the 
first 
group 
are 
spatial 
and/or 
temporal 
means 
as 
integrated 
measures. 
Further 
characteristics 
are 
the 
penetration 
depth 
of 
concentration 
isolines, 
the 
center 
of 
mass, 
minimal 
or 
maximal 
velocities, 
breakthrough 
curves, 
and 
so 
forth. 
Indicatiors 
of 
the 
second 
group 
compare 
results 
which 
requires 
the 
analysis 
of 
the 
spatial 
structure 
and 
the 
distribution 
of 
quantities 
for 
of 
different 
simulations. 
This 
can 
be 
a 
quite 
challenging 
task. 
Various 
disciplines 
use 
pattern 
matching 
algorithms, 
like 
image 
analysis, 
meteorology 
or 
geographical 
information 
science. 
For 
a 
broad 
literature 
scan 
see 
e.g. 
Hagen-Zanker 
[2006]. 
We 
focus 
on 
two 
integrated 
measures. 



1.3. 
Geostatistics 
in 
Subsurface 
Hydrology 
Mean 
Concentration 


We 
define 
the 
mean 
for 
the 
concentration 
C 
as 
the 
spatial 
average 
over 
the 
entire 
domain 
of 
a 
simulation 
k 
with 
an 
arbitrary 
shape 
of 
the 
numerical 
mesh 
by 


N

X

¯ 


Ck 
= 
A(i)Ck(i) 
and 
(1.12) 


i=1 


where 
N 
is 
the 
number 
of 
elements 
of 
the 
mesh, 
A(i) 
. 
[0;1] 
is 
the 
relative 
space 
fraction 
of 
¯

the 
element 
i 
withPNi=1 
A(i) 
= 
1; 
C(i) 
is 
the 
concentration 
value 
of 
the 
element 
i. 
C 
is 
a 
dimensionless 
quantity 
due 
to 
the 
fact 
that 
the 
concentration 
is 
given 
in 
this 
work 
in 
relative 


¯

amounts 
C 
. 
[0;1]. 
The 
mean 
concentration 
C 
corresponds 
directly 
to 
the 
amount 
of 
salt 
being 
solved 
within 
the 
domain. 


Relative 
Concentration 
Difference 
and 
Relative 
Temperature 
Difference 


To 
perform 
a 
comparison 
between 
two 
simulations 
k 
and 
l 
on 
the 
same 
numerical 
grid, 
we 
define 
the 
relative 
difference 
in 
concentration 
and 
temperature 
according 
to 


N

XP


Ckl 
= 
100% 
A(i) 
|Ck(i) 
- 
Cl(i)| 
and 
i=1 
(1.13)

N

100% 
XP


T 
kl 
= 
A(i) 
|Tk(i) 
- 
Tl(i)| 
,
Tmax 
- 
Tmin 


i=1 


where 
N 
is 
the 
number 
of 
elements 
of 
the 
mesh, 
A(i) 
. 
[0;1] 
is 
the 
relative 
space 
fraction 
of 
the 
element 
i 
withPNi=1 
A(i) 
= 
1. 
Ck(i) 
and 
Cl(i) 
are 
the 
concentration 
values 
and 
Tk(i) 
and 
Tl(i) 
are 
the 
temperature 
of 
element 
i 
of 
the 
simulations 
k 
and 
l, 
respectively. 
The 
division 
by 
the 
difference 
between 
the 
minimal 
temperature 
Tmin 
and 
maximal 
temperature 
Tmax 
is 
necessary 
to 
gain 
a 
relative 
quantity 

T 
. 
The 
relative 
difference 
measures 
the 
accordance 
of 
spatial 
pattern 
between 
two 
simulations. 
A 
value 
of 
zero 
corresponds 
to 
similarity. 


1.3. 
Geostatistics 
in 
Subsurface 
Hydrology 
Subsurface 
hydrology 
faces 
two 
problems 
concerning 
the 
characterization 
of 
hydraulic 
properties 
of 
porous 
media: 
a 
strong 
variability 
of 
properties 
in 
space 
and 
the 
scarcity 
of 
information. 
The 
coincidence 
of 
both 
issues 
complicates 
the 
description 
and 
prediction 
of 
processes 
like 
the 
fluid, 
mass 
and 
heat 
transport 
in 
porous 
media. 
To 
account 
for 
the 
heterogeneity 
of 
subsurface 
characteristics, 
the 
theory 
of 
geostatistics 
has 
been 
developed 
in 
the 
context 
of 
hydrogeology 
in 
the 
1960s. 
One 
of 
the 
first 
scientists 
using 
this 
concept 
was 
Matheron 
[1967]. 
Geostatistics 
attempts 
to 
find 
statistical 
characterizations 
which 
capture 
the 
patterns 
of 
spatial 
variability 
and 
to 
explain 
field 
observations. 
It 
further 
aims 
to 
recognize 
and 
quantify 
the 
impact 
of 
subsurface 
heterogeneity 
on 
flow 
and 
transport 
processes. 
Variables 
of 
interest 
are 
the 
hydraulic 
conductivity, 
transmissivity, 
hydraulic 
head, 
concentration, 
fracture 
density, 
and 
dispersivity. 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
A 
detailed 
discussion 
on 
multiple 
geostatistical 
aspects 
are 
given 
e.g. 
in 
Dagan 
[1989]; 
Gelhar 
[1993] 
and 
Rubin 
[2003]. 
In 
the 
following 
we 
focus 
on 
hydraulic 
conductivity 
Kf 
. 
All 
statements 
and 
results 
are 
equally 
valid 
for 
the 
porous 
media 
quantity 
permeability, 
which 
is 
in 
one 
to 
one 
correspondence 
to 
the 
hydraulic 
conductivity 
trough 
the 
relation 
given 
in 
Eq. 
(1.4). 


The 
spatial 
variability 
of 
porous 
media 
is 
a 
result 
of 
complex 
geological 
processes 
through 
which 
aquifers 
evolve. 
Physical 
and 
chemical 
processes, 
like 
structural 
deformation, 
deposition 
and 
diagenesis 
control 
the 
geometry 
and 
texture 
of 
sedimentary 
deposits 
[Miall, 
1996]. 
Field 
studies 
indicate 
that 
hydraulic 
conductivity 
can 
vary 
by 
orders 
of 
magnitude 
over 
short 
distances 
of 
the 
order 
of 
meters 
[Sudicky, 
1986; 
Gelhar, 
1993]. 
The 
spatial 
variability 
of 
hydraulic 
properties 
influences 
fluid 
flow 
and 
transport, 
due 
to 
the 
fact 
that 
the 
conductivity 
controls 
direction 
and 
magnitude 
of 
flow 
by 
Darcy’s 
law. 
Furthermore, 
heterogeneity 
triggers 
mechanical 
dispersion 
and 
dominates 
the 
movement 
of 
a 
solute 
plume. 
However, 
it 
is 
of 
practical 
significance 
to 
understand 
the 
actual 
movement 
of 
contaminants 
for 
their 
removal 
by 
remediation 
technologies. 
A 
deterministic 
approach 
to 
characterize 
complex 
spatial 
patterns 
is 
inappropriate 
because 
it 
requires 
the 
estimation 
of 
a 
large 
number 
of 
parameters 
which 
is 
in 
contrast 
to 
the 
rather 
small 
number 
of 
measurements 
available. 
The 
lack 
of 
perfect 
or 
complete 
measurements 
makes 
our 
knowledge 
of 
the 
variable 
uncertain 
, 
thus 
justifying 
the 
probabilistic 
approach. 


1.3.1. 
Concept 
of 
Spatial 
Random 
Function 
The 
geostatistical 
description 
of 
a 
spatially 
distributed 
quantity, 
like 
hydraulic 
conductivity, 
uses 
the 
concept 
of 
spatial 
random 
functions 
(SRF). 
These 
are 
random 
variables 
that 
depend 
on 
the 
location 
and 
exhibit 
a 
stochastic 
spatial 
structure. 
The 
goal 
of 
constructing 
a 
subsurface 
quantity 
as 
SRF 
is 
to 
reduce 
the 
ensemble 
of 
measurements 
to 
a 
few 
useful 
statistics 
which 
capture 
mathematically 
the 
pattern 
of 
spatial 
variability. 


A 
SRF 
Z(x) 
can 
be 
characterized 
by 
a 
certain 
set 
of 
statistical 
parameters, 
in 
general 
its 
statistical 
moments. 
The 
spatial 
variation 
of 
a 
SRF 
often 
shows 
an 
overall 
trend 
and 
a 
random 
component 
associated 
with 
erratic 
fluctuations. 
These 
characteristics 
can 
be 
captured 
by 
the 
first 
two 
moments 
of 
Z(x). 
The 
first 
order 
moment 
is 
the 
expected 
value 
or 
arithmetic 
mean 
ZA(x), 
marking 
the 
trend. 
The 
second 
order 
moments 
include 
variance 
Var[Z(x)], 
the 
auto-covariance 
CV[Z(x)] 
and 
the 
semivariogram 
SV[Z(x)]. 
The 
interdependency 
of 
conductivity 
values 
in 
space 
is 
covered 
by 
the 
model 
of 
spatial 
correlation, 
which 
is 
expressed 
by 
the 
covariance 
function 
or 
semivariogram. 
In 
linear 
geostatistics 
higher 
moments 
are 
neglected. 


For 
hydraulic 
conductivity 
the 
property 
of 
stationarity 
is 
often 
assumed, 
meaning 
that 
the 
statistics 
do 
not 
change 
over 
space. 
Thus 
the 
mean 
is 
constant 
and 
the 
auto-correlation 
does 
not 
depend 
on 
x, 
but 
on 
the 
separation 
distance 
of 
two 
points 
s. 
Additionally, 
there 
exists 
a 
one-on-one 
correspondence 
between 
auto-covariance 
CV 
and 
semivariogram 
SV: 


ZA 
= 
E[Z(x)], 
CV(s) 
= 
Cov[Z(x 
+ 
s),Z(x)] 
= 
E[Z(x 
+ 
s)Z(x)] 
- 
ZA2 
, 
SV(s) 
= 
Var[Z(x)] 
- 
CV(s). 



1.3. 
Geostatistics 
in 
Subsurface 
Hydrology 
1.3.2. 
Hydraulic 
Conductivity 
as 
Log-normally 
Distributed 
SFR 
It 
is 
common 
to 
model 
the 
hydraulic 
conductivity 
K(x) 
conceptually 
as 
log-normally 
distributed 
SRF, 
referring 
to 
the 
summary 
of 
several 
field 
studies 
given 
by 
Gelhar 
[1993]. 
It 
means 
that 
Y 
(x) 
= 
ln 
K 
x) 
is 
normally 
distributed 
with 
a 
Gaussian 
probability 
density 
func


( 
(


1(x-µ)2

tion 
pdfY 
(x)= 
22 
exp 
-in 
uni-variate 
form 
with 
µ 
and 
2 
being 
the 
mean 
and 
the

22

variance 
of 
Y 
, 
respectively. 
The 
probability 
density 
function 
of 
K(x) 
is 
given 
by 




1 
ln(x) 
- 
µ

pdfK(x)= 
v 
exp-. 
(1.14)

22

x 
22 


The 
moments 
of 
K(x) 
can 
be 
calculated 
using 
the 
statistical 
parameters 
µ 
and 
2. 
The 
arithmetic 
mean 
KA 
(first 
moment), 
the 
geometric 
mean 
KG 
as 
well 
as 
the 
harmonic 
mean 
KH 
are 
important 
for 
the 
purpose 
of 
upscaling 
(section 
1.4), 


KG 
= 
exp 
(µ), 
(1.15) 




1

KA 
= 
expµ 
+ 
21 
2= 
KG 
exp2 
2, 
(1.16) 




11

KH 
= 
expµ 
- 
2 
2= 
KG 
exp-2 
2. 
(1.17) 


The 
second 
moment 
of 
K(x) 
is 
given 
by 






Var(K) 
= 
exp(2) 
- 
1exp2µ 
+ 
12 
2. 


The 
correlation 
structure 
of 
conductivity 
in 
space 
is 
captured 
by 
the 
covariance 
model. 
It 
can 
be 
expressed 
using 
Y 
’s 
covariance 
model 




Cov 
[K(x 
+ 
s),K(x)]=exp 
2µ 
+ 
2 
+ 
CVY 
(s). 


The 
statistical 
description 
of 
spatial 
correlation 
is 
based 
on 
the 
assumed 
covariance 
model. 
A 
bunch 
of 
models 
exist, 
which 
all 
exhibit 
properties 
of 
spatial 
correlation 
that 
can 
be 
observed 
in 
practice. 
Finding 
an 
appropriate 
model 
of 
correlation 
structure 
from 
field 
data 
includes 
an 
estimation 
procedure, 
see 
e.g 
Samper 
and 
Neuman 
[1989a, 
b]. 
In 
general, 
spatially 
distributed 
measurements 
are 
log-transformed 
and 
interpreted 
to 
gain 
an 
experimental 
semivariogram 
or 
a 
covariance 
function, 
respectively. 
The 
data 
is 
then 
fitted 
to 
the 
covariance 
model. 
At 
this 
point 
different 
models 
might 
lead 
to 
equivalent 
good 
results 
describing 
the 
correlation 
structure. 


The 
exponential, 
Gaussian 
and 
spherical 
models 
are 
typical 
representatives 
for 
covariance 
models 
used 
in 
practice 
[Rubin, 
2003]. 
They 
differ 
in 
the 
near-origin 
behavior 
and 
the 
rate 
of 
decay 
of 
correlation 
as 
function 
of 
distance. 
The 
exponential 
model 
is 
indicative 
of 
sharp 
transition 
occurring 
between 
blocks 
of 
different 
values 
and 
the 
spatial 
correlation 
decays 
at 
a 
very 
low 
rate 
with 
large 
distances. 
The 
Gaussian 
covariance 
model 
describes 
a 
gradual 
transition 
between 
blocks 
of 
different 
conductivity 
values 
and 
shows 
a 
faster 
decay 
of 
correlation. 
The 
spherical 
model 
describes 
sharp 
transitions 
and 
discontinuities 
between 
zones 
of 
different 
conductivity 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
values. 
In 
practice 
it 
is 
often 
very 
difficult 
to 
determine 
which 
of 
the 
models 
fit 
best 
to 
the 
observed 
data, 
e.g. 
Turcke 
and 
Kueper 
[1996] 
. 
Riva 
and 
Willmann 
[2009] 
showed 
that 
mean 
values 
of 
quantities 
like 
the 
hydraulic 
head 
are 
not 
considerably 
influenced 
by 
the 
choice 
of 
one 
of 
these 
three 
variogram 
types. 


We 
focus 
on 
a 
3D-Gaussian 
model 
with 
the 
auto-covariance 
function 
CovG(s)= 
2G(s), 
where 
G(s) 
is 
the 
correlation 
function 


X

G(s) 
= 
exp 
-(si/`i)2 
. 
(1.18) 


i 


The 
model 
relies 
on 
stationarity 
and 
hence 
can 
be 
expressed 
as 
function 
of 
the 
separation 
distance 
s 
=(s1,s2,s3). 
The 
Gaussian 
model 
refers 
to 
a 
finite 
correlation 
structure, 
which 
is 
expressed 
by 
the 
correlation 
length 
. 
=(`1,`2,`3) 
> 
0. 
It 
characterizes 
the 
degree 
of 
continuity 
in 
every 
direction. 
It 
is 
most 
likely 
to 
assume 
that 
the 
correlation 
length 
is 
different 
for 
different 
directions. 
In 
case 
of 
equality 
`1 
= 
`2 
= 
`3 
the 
medium 
is 
said 
to 
be 
isotropic. 
Otherwise 
it 
is 
called 
anisotropic, 
where 
the 
anisotropy 
between 
the 
two 
directions 
i 
and 
j 
is 
expressed 
by 
the 
ratio 
e 
= 
`i/`j. 


Another 
quantity 
commonly 
used 
to 
describe 
the 
spatial 
persistence 
of 
K(x) 
is 
the 
integral 
scale 
defined 
as 
Ii 
=1/2R8 
C(si)dsi, 
in 
ith 
direction. 
The 
larger 
the 
integral 
scale, 
the 


0 


longer 
is 
the 
spatial 
persistence 
of 
the 
correlation. 
The 
integral 
scale 
is 
proportional 
to 
the 
correlation 
length 
and 
can 
be 
transformed 
according 
to 
the 
model 
of 
choice. 
For 
the 
Gaussian 
model 
the 
relation 
is 
given 
by 
Ii 
= 
v 
/2`i. 


These 
mentioned 
models 
all 
assume 
a 
finite 
correlation 
scale 
continuously 
distributed 
over 
space. 
However, 
geological 
media 
exhibit 
correlation 
on 
multiple 
scales 
depending 
on 
the 
problem 
of 
consideration 
and 
observation 
[Gelhar, 
1986; 
Dagan, 
1986]. 
A 
fractal 
model 
allows 
to 
account 
for 
an 
evolving 
heterogeneity 
structure. 
Neuman 
et 
al. 
[2008] 
showed 
that, 
even 
though 
sample 
data 
appear 
to 
fit 
either 
Gaussian 
or 
Exponential 
variograms, 
they 
could 
be 
represented 
equally 
well 
in 
terms 
of 
a 
truncated 
power 
law 
variogram 
model. 


The 
semivariogram 
of 
the 
power 
law 
model 
is 
given 
by 


SVPL(r)= 
C0r 
2H 
, 
(1.19) 


with 
Hurst 
coefficient 
0 
<H< 
1 
and 
constant 
C0. 
The 
model 
is 
also 
referred 
to 
as 
self-similar. 
The 
power 
law 
model 
does 
not 
have 
a 
finite 
variance 
nor 
a 
finite 
correlation 
length, 
capturing 
the 
effect 
of 
correlation 
on 
multiple 
scales 
[Neuman, 
1990]. 
The 
shape 
of 
a 
conductivity 
field 
depends 
on 
the 
choice 
of 
the 
Hurst 
coefficient: 
for 
H> 
0.5 
the 
probability 
of 
observing 
increments 
to 
the 
same 
sign 
over 
large 
distances 
is 
high 
and 
fields 
becoming 
smooth 
in 
appearance. 
For 
H< 
0.5 
larger 
small-scale 
variability 
can 
be 
observed 
and 
the 
appearance 
is 
more 
rugged. 


The 
unbounded 
growth 
of 
variability 
is 
a 
theoretical 
hypothesis 
which 
cannot 
be 
captured 
in 
numerical 
studies. 
For 
generating 
fields 
of 
hydraulic 
conductivity 
numerically 
a 
truncated 
power 
law 
is 
used, 
where 
a 
cutoff 
length 
is 
chosen 
larger 
than 
the 
domain 
size. 



1.4. 
Upscaling 
1.3.3. 
Quantifying 
Variables 
of 
Flow 
and 
Transport 
in 
Stochastic 
Media 
Two 
approaches 
are 
commonly 
used 
to 
solve 
a 
problem 
in 
stochastic 
subsurface 
hydrology: 
the 
analytical 
and 
the 
numerical 
approach. 
The 
idea 
of 
the 
analytical 
approach 
is 
to 
derive 
explicit 
expressions 
for 
effective 
quantities 
from 
stochastic 
differential 
equations. 
The 
numerical 
approach 
uses 
random 
field 
generators 
to 
produce 
hydraulic 
conductivity 
fields. 
They 
serve 
as 
input 
to 
numerical 
flow 
and 
transport 
models 
which 
provide 
numerical 
solutions 
for 
the 
flow 
field 
and/or 
concentration 
patterns 
in 
heterogeneous 
media. 


Since 
hydraulic 
parameters 
may 
vary 
over 
distance 
and 
time, 
variables 
of 
flow 
and 
transport 
like 
hydraulic 
head 
or 
the 
concentration 
distribution 
of 
a 
tracer 
may 
be 
spatially 
dependent, 
too. 
Introducing 
K(x) 
as 
SRF 
into 
Darcy 
law, 
Eq.(1.2), 
and 
the 
mass 
conservation 
equation, 
in 
Eq.(1.5), 
results 
in 
stochastic 
PDE’s. 
Hence, 
flow 
variables 
are 
also 
SRF. 
It 
is 
often 
aimed 
to 
express 
these 
SRF 
with 
the 
statistical 
parameters 
of 
K(x), 
which 
are 
KG, 
2 
and 
f 
for 
a 
Gaussian 
covariance 
model. 
In 
general, 
the 
hydraulic 
head 
is 
less 
variable 
than 
the 
log 
conductivity 
due 
to 
physical 
constrains, 
given 
by 
the 
PDE’s 
[Rubin, 
2003]. 


Using 
the 
Monte 
Carlo 
approach 
to 
solve 
the 
flow 
equation 
with 
heterogeneous 
conductivity 
distributions 
numerically 
is 
conceptually 
simple 
and 
needs 
no 
particular 
assumption. 
It 
allows 
the 
analysis 
of 
complex 
scenarios 
which 
are 
not 
accessible 
to 
analytical 
methods. 
The 
main 
steps 
of 
such 
a 
procedure 
are 
(i) 
assuming 
a 
probability 
distribution 
of 
the 
known 
input 
variable, 
which 
is 
in 
general 
the 
hydraulic 
conductivity 
K(x); 
(ii) 
generating 
realizations 
synthetically; 


(iii) 
solving 
the 
flow 
equation 
with 
appropriate 
boundary 
conditions 
for 
every 
realization; 
(iv) 
computing 
the 
statistics 
of 
the 
model 
output 
variables. 
An 
advantage 
of 
this 
approach 
is 
the 
flexibility 
in 
the 
problem 
description. 
A 
disadvantage 
is 
that 
the 
calculation 
of 
huge 
ensembles 
of 
realizations 
might 
be 
necessary, 
which 
can 
be 
quite 
consuming 
in 
time 
and 
computational 
power. 
Furthermore, 
it 
is 
often 
difficult 
to 
predict 
how 
many 
realizations 
and 
thus 
simulations 
are 
necessary 
to 
achieve 
convergence 
of 
solutions. 
1.4. 
Upscaling 
One 
of 
the 
questions 
arising 
in 
groundwater 
flow 
problems 
is 
how 
to 
treat 
heterogeneity 
in 
aquifer 
models. 
In 
order 
to 
reduce 
complexity 
it 
is 
important 
to 
find 
representative 
parameters, 
which 
are 
capable 
of 
reproducing 
some 
average 
behavior 
of 
the 
system 
by 
upscaling. 
Upscaling 
basically 
aims 
to 
connect 
scales, 
either 
to 
connect 
conceptual 
model 
scale 
to 
observation 
scale 
or 
to 
connect 
different 
observation 
scales. 


The 
emergence 
of 
upscaling 
in 
hydrogeology 
is 
directly 
linked 
to 
the 
application 
of 
geostatistical 
methods 
on 
subsurface 
flow 
and 
transport 
problems. 
Matheron 
[1967] 
was 
the 
first 
who 
described 
heterogeneously 
distributed 
hydraulic 
conductivity 
as 
SRF 
and 
directly 
asked 
for 
representative 
values 
and 
bounds. 
Since 
then 
a 
huge 
amount 
of 
literature 
emerged, 
concerning 
upscaling 
of 
hydraulic 
conductivity. 
For 
a 
detailed 
review 
see 
e.g. 
Dagan 
[1989]; 
Renard 
and 
de 
Marsily 
[1997]; 
Sánchez-Vila 
et 
al. 
[2006]. 


In 
the 
context 
of 
stochastic 
subsurface 
hydrology, 
numerical 
models 
can 
serve 
for 
validating 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
theoretical 
upscaling 
approaches. 
It 
is 
often 
very 
complicated 
to 
define 
upscaled 
hydraulic 
parameters 
for 
complex 
flow 
regimes. 
Due 
to 
the 
lack 
of 
data 
it 
is 
very 
difficult 
to 
validate 
averaged 
values 
obtained 
from 
upscaling 
approaches 
directly. 
Numerical 
models 
can 
be 
used 
to 
fill 
the 
gap 
of 
missing 
measurements. 
Based 
on 
the 
physical 
process 
equations 
we 
can 
perform 
numerical 
experiments 
and 
generate 
virtual 
data. 
Results 
from 
upscaling 
theory 
can 
be 
validated 
by 
comparing 
analytical 
solutions 
with 
numerical 
experiment 
data. 
The 
first 
following 
this 
strategy 
was 
Freeze 
[1975]. 
We 
discuss 
the 
combination 
of 
theoretical 
framework 
and 
numerical 
methods 
to 
find 
upscaled 
descriptions 
of 
hydraulic 
conductivity 
in 
the 
flow 
conditions 
of 
uniform 
flow 
and 
well 
flow 
in 
detail 
in 
the 
sections 
2.3 
and 
4, 
respectively. 


1.4.1. 
Scale 
Hierarchy 
Heterogeneity 
in 
hydraulic 
properties 
of 
porous 
media 
can 
be 
observed 
on 
multiple 
scales. 
The 
characterization 
of 
the 
spatial 
correlation 
scale 
of 
the 
property 
of 
interest 
coincides 
with 
the 
definition 
of 
spatial 
scales 
of 
flow 
domains. 
Dagan 
[1986] 
defined 
three 
major 
scales: 
the 
laboratory, 
the 
local, 
and 
the 
regional 
scale 
with 
corresponding 
correlation 
on 
pore 
scale, 
correlation 
of 
hydraulic 
conductivity, 
and 
correlation 
of 
transmissivity. 


Heterogeneity 
on 
pore 
scale 
is 
not 
considered 
here. 
For 
flow 
on 
local 
scale 
the 
hydraulic 
conductivity 
K(x)= 
K(x, 
y, 
z) 
is 
the 
variable 
of 
interest. 
The 
vertical 
component 
of 
the 
flow 
has 
to 
be 
taken 
into 
account, 
since 
the 
flow 
patterns 
are 
strongly 
influenced 
by 
the 
three 
dimensional 
nature 
of 
the 
porous 
media. 
Furthermore, 
anisotropy 
effects 
have 
to 
be 
considered. 


For 
flow 
on 
regional 
scale 
the 
relation 
between 
the 
horizontal 
extent 
and 
the 
aquifer 
thickness 
becomes 
very 
small. 
Thus, 
it 
is 
assumed 
that 
vertical 
flow 
can 
be 
neglected 
and 
the 
flow 
field 
can 
be 
seen 
as 
horizontal. 
The 
porous 
media 
can 
be 
described 
by 
the 
two 
dimensional 
transmissivity 
T 
(x), 
which 
is 
the 
vertical 
average 
of 
the 
hydraulic 
conductivity 
K(x)= 
K(x, 
y, 
z) 
over 
the 
domain 
thickness 
L, 


ZL 


T 
(x, 
y)=K(x, 
y, 
z)dz. 


0 


1.4.2. 
Effective 
and 
Equivalent 
Conductivity 
Two 
main 
approaches 
have 
evolved 
in 
literature 
to 
find 
representative 
parameters. 
The 
first 
is 
the 
approach 
of 
effective 
quantities 
which 
are 
gained 
by 
performing 
ensemble 
averages. 
The 
second 
approach 
uses 
spatial 
averaging 
to 
find 
equivalent 
parameters. 
However, 
determining 
upscaled 
parameters 
for 
a 
certain 
problem 
of 
consideration 
can 
be 
quite 
challenging. 
Effective 
values 
might 
not 
exist 
or 
spatial 
averages 
can 
strongly 
depend 
on 
the 
scale 
of 
the 
averaging 
volume. 


Effective 
hydraulic 
conductivity 
is 
defined 
as 
the 
value 
Kef 
(or 
Keff) 
which 
satisfies 
Darcy’s 
law 
hq(x)) 
= 
-Kefhrh(x)iwith 
ensemble 
averaged 
quantities 
h.iof 
flux 
q 
and 
hydraulic 
head 
h. 
The 
effective 
conductivity 
relates 
expected 
values 
of 
specific 
discharge 
and 
head 
gradient. 
Kef 
is 
defined 
as 
a 
function 
of 
aquifer’s 
material 
properties 
and 
must 
be 
valid 
throughout 
the 
entire 



1.4. 
Upscaling 
domain; 
it 
is 
a 
characteristic 
property 
of 
the 
medium 
but 
must 
not 
be 
influenced 
by 
flow 
conditions 
[Rubin, 
2003]. 


Closed-form 
results 
for 
an 
effective 
conductivity 
Kef 
are 
available 
in 
steady 
uniform 
flow 
for 
cases 
of 
low 
variability 
and 
simple 
flow 
regimes, 
discussed 
in 
more 
detail 
in 
section 
1.4.3. 
In 
complex 
flow 
configurations 
or 
for 
large 
variances 
the 
effective 
conductivity 
can 
be 
bracketed 
by 
bounds 
of 
arithmetic 
and 
harmonic 
mean 
KA 
and 
KH. 
These 
are 
given 
in 
Eq.(1.16) 
and 
Eq.(1.17) 
for 
log-normally 
distributed 
hydraulic 
conductivity 
fields 
[Matheron, 
1967; 
Sánchez-
Vila 
et 
al., 
2006]. 
Non-uniform 
flow 
regimes 
like 
in 
pumping 
tests 
lead 
to 
Kef 
depending 
on 
space 
coordinates 
and/or 
boundary 
conditions. 
Thus, 
the 
common 
definition 
of 
effective 
conductivity 
is 
not 
appropriate 
to 
describe 
representative 
conductivity 
in 
these 
flow 
regimes. 


Another 
approach 
to 
define 
an 
upscaled 
conductivity 
refers 
to 
volume 
averaging. 
The 
equivalent 


¯¯r-= 
Kq 
eq 


h 
with 
the 
spatial 
average 
of 
the 
Equivalent 
parameters 
are 
associated 
with 


conductivity 
Keq 
is 
defined 
to 
fulfill 
Darcy’s 
law 


R

and 
of 
the 
head 
gradient 

h¯
.


flux


q 
=
a 
particular 
geometry 
and 
boundary 
conditions 
and 
can 
thus 
be 
defined 
for 
non-uniform 
flow 
conditions. 


In 
case 
of 
ergodicity, 
ensemble 
averages 
can 
be 
replaced 
by 
spatial 
averages. 
Hence, 
effective 
and 
equivalent 
conductivity 
are 
identical. 
In 
stationary 
conductivity 
fields 
and 
when 
flow 
is 
induced 
by 
a 
linear 
pressure 
gradient, 
ergodicity 
can 
be 
assumed 
throughout 
the 
entire 
domain. 
The 
mean 
flux 
is 
equal 
to 
the 
spatial 
average 
of 
the 
flux 
[Rubin, 
2003] 
and 
thus 
Kef 
= 
Keq. 
For 
non-uniform 
flow 
this 
relation 
is 
not 
valid, 
due 
to 
the 
ergodicity 
breakdown. 


Beyond 
the 
two 
approaches 
of 
effective 
and 
equivalent 
parameters 
a 
lot 
of 
work 
has 
been 
devoted 
to 
finding 
representative 
descriptions 
of 
conductivity 
for 
more 
complex 
flow 
regimes, 
as 
discussed 
in 
detail 
e.g. 
in 
Sánchez-Vila 
et 
al. 
[2006]. 


¯ 


1.4.3. 
Upscaled 
Conductivity 
for 
Uniform 
Flow 
Steady 
state 
parallel 
flow 
in 
stationary 
media 
is 
one 
of 
the 
particular 
situations 
where 
effective 
conductivity, 
denoted 
by 
Kefu, 
exists. 
The 
theoretical 
approach 
provides 
analytical 
solutions 
for 
the 
effective 
conductivity 
of 
a 
log-normally 
distributed 
isotropic 
conductivity 
K(x) 
in 
dependence 
on 
the 
dimensionality 
d 
of 
the 
considered 
flow 
field, 


11

Kefu 
= 
KG 
exp 
22 
- 
, 
(1.20)

d 


with 
the 
statistical 
quantities 
geometric 
mean 
KG 
(Eq.(1.15)) 
and 
variance 
2 
of 
K(x). 
For 
the 
more 
complex 
solution 
in 
anisotropic 
media, 
see 
section 
4.2.2 
or 
[Gelhar 
and 
Axness, 
1983; 
Dagan, 
1989]. 


For 
all 
dimensions 
Kefu 
is 
bounded 
between 
KA 
and 
KH, 
regardless 
of 
the 
assumed 
correlation 


(1d)

structure. 
In 
one 
dimension 
the 
effective 
conductivity 
is 
the 
smallest 
with 
K= 
KH 
= 


efu



V 
q 
dV

1 


V

KG 
exp-

2, 
due 
to 
the 
fact 
that 
in 
one 
direction 
a 
serial 
flow 
pattern 
dominates, 
where 


the 
harmonic 
mean 
is 
the 
average 
of 
choice. 
In 
two 
and 
even 
stronger 
in 
three 
dimensions, 
flow 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
(2d)

can 
circumvent 
areas 
of 
low 
conductivity, 
thus 
the 
effective 
values 
are 
given 
by 
K= 
KG

efu



(3d) 


= 
KG 
exp

16

2

.


(or 
transmissivity 
Tefu 
= 
TG, 
respectively) 
and 
K

efu 


In 
section 
2.3 
we 
show 
that 
the 
analytical 
description 
of 
Kefu 
in 
Eq. 
(1.20) 
can 
be 
reproduced 
by 
the 
effective 
hydraulic 
conductivity 
calculated 
from 
an 
ensemble 
of 
simulations 
with 
uniform 
flow 
through 
two-dimensional 
heterogeneous 
log-normally 
distributed 
conductivity 
fields. 


1.4.4. 
Upscaled 
Conductivity 
for 
Well 
Flow 
Most 
field 
methods, 
which 
rely 
on 
well 
flow 
scenarios 
to 
obtain 
hydraulic 
conductivity 
values, 
assume 
homogeneity 
[Kruseman 
and 
de 
Ridder, 
2000]. 
In 
homogeneous 
media 
well 
flow 
can 
be 
described 
by 
Thiem’s 
solution 
[Thiem, 
1906] 
for 
steady 
state 


h(r)= 
- 
Qw 
ln 
r 
+ 
hR 
(1.21)

2LK 
R 


or 
Theis’ 
solution 
[Theis, 
1935] 
for 
transient 
flow, 
where 
h(r) 
is 
the 
hydraulic 
head, 
depending 
on 
the 
radial 
distance 
r 
from 
the 
well. 
Qw 
is 
the 
discharge 
at 
the 
well, 
L 
is 
the 
aquifer 
thickness, 
K 
is 
the 
homogeneous 
conductivity, 
and 
hR 
= 
h(R) 
is 
the 
known 
hydraulic 
head 
at 
a 
certain 
distance 
R 
from 
the 
well. 
The 
analytical 
solution 
of 
Thiem 
(1.21) 
can 
be 
used 
to 
benchmark 
numerical 
codes 
for 
convergent 
flow 
regimes, 
see 
section 
2.2. 


Well 
flow 
in 
heterogeneous 
media 
reveals 
a 
much 
more 
complex 
behavior 
than 
in 
homogeneous 
media. 
The 
constitutive 
equation, 
which 
is 
the 
average 
of 
Darcy’s 
law, 
has 
a 
non-local 
structure 
for 
non-uniform 
flow 
and 
it 
is 
in 
general 
not 
possible 
to 
find 
a 
single 
upscaled 
conductivity 
[Matheron, 
1967; 
Indelman 
and 
Abramovich, 
1994; 
Indelman 
et 
al., 
1996]. 
The 
steep 
pressure 
gradients 
which 
occur 
at 
the 
well, 
lead 
to 
a 
breakdown 
of 
the 
ergodicity 
assumption. 
An 
effective 
conductivity 
value 
Kef, 
which 
is 
valid 
throughout 
the 
entire 
range, 
does 
not 
exist 
and 
thus 
the 
definition 
of 
effective 
parameters 
is 
not 
applicable 
to 
convergent 
flow. 


Although 
a 
single 
representative 
conductivity 
for 
well 
flow 
does 
not 
exist, 
the 
asymptotic 
behavior 
is 
well 
known 
[Indelman 
and 
Abramovich, 
1994]. 
Flow 
can 
be 
assumed 
to 
be 
uniform 




in 
average 
far 
from 
the 
well, 
thus 
the 
representative 
value 
is 
Kefu 


= 
KG 
exp

16

2

in 
three 


dimensions 
and 
Tefu 
= 
TG 
in 
two 
dimensions. 
The 
near-well 
representative 
value 
Kwell 
or 
Twell 
is 
stronger 
impacted 
by 
heterogeneity 
due 
to 
the 
high 
pressure 
gradients. 
The 
value 
depends 
on 
the 
boundary 
condition 
assumed 
at 
the 
well. 
It 
is 
either 
the 
arithmetic 
mean 
KA 
or 
TA 
for 
a 
constant 
head 
or 
the 
harmonic 
mean 
KH 
or 
TH 
for 
a 
constant 
flux 
boundary 
condition, 


e.g. 
[Shvidler, 
1966; 
Dagan, 
1989; 
Indelman 
et 
al., 
1996; 
Indelman 
and 
Dagan, 
2004]. 
A 
detailed 
discussion 
on 
that 
is 
given 
in 
section 
4.2.2. 
Since 
ensemble 
averaging 
is 
not 
applicable, 
the 
concept 
of 
spatial 
averaging 
can 
be 
used 
to 
obtain 
an 
equivalent 
conductivity, 
which 
depends 
on 
the 
distance 
to 
the 
well. 
Indelman 
et 
al. 
[1996] 
defined 
the 
equivalent 
conductivity 
Keq(r) 
based 
on 
Thiem’s 
formula 
(1.21) 
as 
a 
spatial 



1.4. 
Upscaling 
average 
over 
a 
circle 
of 
radius 
r 
around 
the 
well 
in 
the 
sense 
of 
Matheron 
[1967], 


Qw(ln 
r 
- 
ln 
rw)

Keq(r)= 


2L(h(r) 
- 
h(rw)), 


where 
Qw 
is 
the 
flow 
rate, 
rw 
is 
the 
radius 
of 
the 
well, 
h(rw) 
is 
the 
head 
at 
the 
well 
and 
L 
is 
the 
aquifer 
thickness. 
Keq(r) 
can 
be 
interpreted 
as 
the 
conductivity 
of 
a 
cylinder 
of 
radius 
r 
in 
homogeneous 
media 
which 
conveys 
the 
same 
discharge 
as 
the 
actual 
medium 
for 
the 
same 
mean 
head 
difference. 
In 
contrast 
to 
Kefu 
in 
uniform 
flow 
Keq(r) 
depends 
on 
the 
flow 
geometry. 
Firmani 
et 
al. 
[2006] 
showed 
by 
the 
use 
of 
numerical 
simulations 
that 
Keq(r) 
is 
appropriate 
to 
describe 
the 
conductivity 
of 
well 
flow 
only 
for 
small 
variances 
and 
high 
anisotropy 
ratios. 


A 
large 
amount 
of 
work 
is 
focused 
on 
finding 
upscaled 
descriptions 
of 
the 
conductivity 
and 
transmissivity 
in 
well 
flow 
regimes 
expressed 
by 
a 
radial 
dependency. 
Some 
studies 
are 
based 
on 
the 
definition 
of 
Keq(r) 
or 
Teq(r), 
respectively; 
e.g. 
Fiori 
et 
al. 
[1998]; 
Dagan 
and 
Lessoff 
[2007]. 
But 
also 
other 
approaches 
have 
emerged, 
e.g. 
Neuman 
and 
Orr 
[1993]; 
Sánchez-Vila 
[1997]; 
Sánchez-Vila 
et 
al. 
[1999]; 
Guadagnini 
et 
al. 
[2003]; 
Neuman 
et 
al. 
[2004]. 


1.4.5. 
Well 
Flow 
Adapted 
Filtering 
Schneider 
and 
Attinger 
[2008] 
presented 
a 
promising 
alternative 
approach 
of 
upscaling 
well 
flow 
in 
two 
dimensions. 
They 
presented 
a 
radially 
dependent 
upscaled 
transmissivity 
T 
CG(r), 
which 
allows 
a 
transition 
from 
near 
field 
transmissivity 
TH 
to 
far 
field 
value 
Tefu 
= 
TG 
controlled 
by 
the 
correlation 
length 
`, 


. 
1 
2 
. 


T 
CG

(r)= 
TG 
exp 
- 
, 
(1.22)
H 
2 
(1 
+ 
2r2/`2) 


where 
r 
is 
the 
radial 
distance 
from 
the 
well, 
TG 
is 
the 
geometric 
mean, 
2 
is 
the 
variance 
and 
. 
is 
the 
correlation 
length 
of 
the 
log-normally 
distributed 
transmissivity 
T 
(x). 
. 
is 
a 
factor 
of 
proportionality. 


The 
formula 
of 
T 
CG(r) 
in 
Eq(1.22) 
was 
derived 
making 
use 
of 
the 
upscaling 
procedure 
Coarse 


H 


Graining, 
introduced 
for 
uniform 
flow 
by 
Attinger 
[2003]. 
The 
procedure 
of 
Coarse 
Graining 
and 
its 
application 
on 
two-dimensional 
well 
flow 
can 
be 
found 
in 
Schneider 
and 
Attinger 
[2008]. 
We 
discuss 
it 
for 
three-dimensional 
well 
flow 
in 
section 
4.3. 
The 
fundamental 
idea 
is 
to 
perform 
a 
spatial 
filtering 
on 
the 
flow 
equation 
which 
is 
appropriate 
to 
the 
non-uniform 
character 
of 
a 
pumping 
test. 
The 
filter 
is 
proportional 
to 
the 
radial 
distance 
from 
the 
well. 
Hence 
near 
the 
well 
the 
filter 
length 
is 
very 
small, 
so 
nearly 
no 
filtering 
is 
applied 
and 
the 
heterogeneity 
of 
the 
local 
transmissivities 
is 
still 
resolved. 
Far 
away 
from 
the 
well 
the 
filter 
volumes 
are 
very 
large 
and 
the 
local 
heterogeneous 
transmissivity 
values 
are 
replaced 
by 
the 
effective 
value 
Tefu. 


The 
analytical 
expression 
of 
T 
CG(r) 
in 
Eq.(1.22) 
allows 
a 
direct 
calculation 
of 
the 
corresponding 


H 
upscaled 
hydraulic 
head 
hewf(2d)(r) 
by 
solving 
the 
groundwater 
flow 
equation, 
as 
done 
in 
the 
Appendix 
A.4. 
The 
solution 
hewf(2d)(r), 
called 
the 
effective 
well 
flow 
head, 
provides 
an 
effective 



1. 
Principles 
of 
Fluid 
Flow 
and 
Salt 
Transport 
in 
Heterogeneous 
Porous 
Media 
description 
of 
the 
hydraulic 
head 
for 
large-scale 
pumping 
tests 
in 
a 
two-dimensional 
flow 
regime 




hewf(2d)(r)= 
- 
Qw 
..(z(r)) 
- 
..(z(R)) 
- 
e 

22 
..z(r) 
- 
2 
+e 

22 
..z(R) 
- 
2 
+ 
hR 
,

4TG22

(1.23) 
where 
r 
is 
the 
radial 
distance 
from 
the 
well, 
Qw 
is 
the 
pumping 
rate, 
TG 
is 
the 
geometric 
mean, 
and 
2 
is 
the 
variance 
of 
the 
log-transmissivity. 
..(z) 
is 
the 
Theis 
function 
or 
exponential 
integral 


R

z 
exp(z0) 
01

function 
..(z)=0dz. 
z(r) 
is 
an 
abbreviation 
for 
z(r)= 
2 
(1+(r/`)2), 
likewise 
z(R)= 


-8 
z2 


2 
1 


(1+(R/`)2), 
with 
f 
being 
the 
correlation 
length 
and 
. 
being 
the 
factor 
of 
proportionality. 
R 


is 
an 
arbitrary 
distance 
from 
the 
well, 
where 
the 
hydraulic 
head 
h(R)= 
hR 
is 
known. 
Numerical 
simulations 
of 
pumping 
tests 
in 
a 
two-dimensional 
flow 
regime 
showed 
that 
ensemble 
averaged 
drawdowns 
can 
be 
reproduced 
very 
well 
by 
making 
use 
of 
T 
CG(r) 
as 
shown 
by 


H 


Schneider 
and 
Attinger 
[2008]. 



2. 
Numerical 
Benchmarks 
2.1. 
Introduction 
Complicated 
hydrogeological 
problems 
can 
be 
analyzed 
using 
modern 
numerical 
codes. 
Major 
challenges 
for 
scientists 
are 
the 
understanding 
of 
the 
coupled 
processes, 
the 
implementation 
of 
algorithms, 
and 
the 
integration 
of 
the 
available 
experimental 
data. 
To 
cope 
with 
the 
problems, 
interdisciplinary 
cooperation 
and 
an 
interactive 
development 
procedure 
are 
required. 


Benchmarking 
is 
a 
common 
procedure 
for 
model 
verification. 
Complex 
numerical 
models 
rely 
on 
a 
properly 
and 
efficiently 
implemented 
code. 
Therefore, 
the 
numerical 
software 
is 
tested 
on 
examples. 
In 
case 
analytical 
solutions 
exist, 
benchmarking 
comprises 
of 
the 
comparison 
with 
the 
numerical 
solution. 
Another 
approach 
is 
to 
compare 
simulated 
results 
with 
observational 
data. 
In 
these 
cases 
caution 
has 
to 
be 
given 
to 
the 
fact 
that 
measurements 
can 
be 
biased. 
For 
models 
without 
analytical 
solutions 
or 
observational 
evidence 
a 
cross 
comparison 
with 
other 
numerical 
codes 
is 
often 
done. 
For 
the 
simulation 
software 
OpenGeoSys 
an 
extensive 
benchmarking 
documentation 
is 
given 
in 
Kolditz 
et 
al. 
[2012a]. 


The 
scope 
of 
this 
section 
is 
to 
introduce 
four 
specific 
benchmarks 
that 
refer 
to 
the 
complex 
problems 
discussed 
in 
parts 
II 
and 
III. 
On 
the 
one 
hand, 
it 
serves 
as 
code 
testing. 
On 
the 
other 
hand, 
we 
outline 
important 
features 
of 
numerical 
simulations 
dealing 
with 
heterogeneous 
parameter 
distributions 
and 
density 
dependent 
flow. 


The 
four 
benchmarks 
can 
be 
understood 
as 
examples 
for 
the 
development 
of 
simple 
hydrogeological 
models 
with 
reference 
to 
the 
steps 
of 
model 
evolution, 
as 
described 
in 
section 
1.1. 
We 
introduce 
the 
conceptual 
model 
and 
state 
the 
process 
equations 
of 
relevance 
which 
are 
discussed 
in 
detail 
in 
section 
1.2. 
We 
further 
adapt 
the 
problems 
to 
the 
software 
OpenGeoSys 
[Kolditz 
et 
al., 
2012b]. 
This 
includes 
defining 
the 
model 
domain 
with 
a 
spatial 
grid 
resolution, 
fixing 
a 
temporal 
resolution, 
defining 
initial 
and 
boundary 
conditions 
as 
well 
as 
specifying 
physical 
parameters. 
In 
addition, 
the 
discussion 
of 
each 
of 
the 
four 
benchmarks 
includes 
a 
short 
literature 
review. 


2.2. 
Pumping 
Test 
in 
Homogeneous 
Porous 
Media 
Pumping 
tests 
are 
a 
widely 
used 
tool 
to 
identify 
hydraulic 
properties. 
In 
a 
pumping 
test 
water 
is 
extracted 
from 
a 
single 
location 
which 
causes 
a 
radial 
flow 
character. 
From 
the 
mathematical 
point 
of 
view 
this 
point 
source 
creates 
a 
singularity 
at 
the 
pumping 
well, 
which 
complicates 
the 
investigation 
of 
the 
flow 
equation 
in 
particular 
in 
heterogeneous 
porous 
media. 
For 
homogeneous 
media 
an 
analytical 
solution 
exists, 
introduced 
by 
Thiem 
[1906]. 
We 
benchmark 
the 
numerical 
solver 
OpenGeoSys 
for 
convergent 
flow 
by 
comparing 
a 
simulated 
pumping 
test 
drawdown 
with 
Thiem’s 
solution. 



2. 
Numerical 
Benchmarks 
Figure 
1: 
Radial 
refinement 
of 
the 
numerical 
mesh 
in 
the 
area 
of 
1m 
around 
the 
pumping 
well. 


2.2.1. 
Thiem’s 
Solution 
We 
give 
a 
short 
derivation 
of 
Thiem’s 
solution. 
The 
starting 
point 
is 
the 
equation 
for 
saturated 
groundwater 
flow, 
which 
is 
a 
combination 
of 
the 
continuity 
equation 
of 
flow 
(1.1) 
and 
Darcy’s 
law 
(1.3), 


@h(x,t)

S 
-r· 
(K(x)rh(x,t)) 
= 
Q(x,t) 
, 
(2.1)

@t 
where 
S 
is 
the 
storage 
capacity, 
h(x,t) 
is 
the 
hydraulic 
head 
in 
space 
x 
=(x1,x2) 
and 
time 
t, 
K(x) 
is 
the 
hydraulic 
conductivity 
and 
Q(x,t) 
represents 
a 
source 
or 
sink 
term. 


For 
a 
steady 
state 
pumping 
test 
the 
time 
derivative 
is 
zero. 
Furthermore, 
we 
assume 
homogeneity 
in 
hydraulic 
conductivity, 
thus 
K(x)= 
K. 
Transforming 
Eq.(2.1) 
from 
Cartesian 
to 
polar 
coordinates 
(x1,x2) 
. 
(r, 
), 
with 
radius 
r 
and 
angular 
coordinate 
, 
results 
in 


1dh(r)d2h(r)

0= 
K 
+ 
K 
2 
. 
(2.2) 


r 
dr 
dr

According 
to 
the 
nature 
of 
radial 
flow 
the 
solution 
h(r) 
does 
not 
depend 
on 
the 
angular 
coordinate 
. 
In 
Eq.(2.2) 
the 
sink 
term 
is 
not 
included, 
which 
is 
instead 
used 
as 
boundary 
condition. 
A 
constant 
discharge 
Qw 
at 
the 
well 
corresponds 
to 
a 
Neumann 
boundary 
condition 
and 
is 
calculated 
as 
the 
flux 
over 
the 
surface 

w 
of 
the 
pumping 
well, 


I

Qw 
=q(rw)d
w 
= 
-2rwLKh0(rw), 



w 


where 
rw 
is 
the 
radius 
of 
the 
well, 
L 
is 
the 
aquifer 
thickness, 
and 
q(r) 
is 
the 
flux, 
which 
is 
given 
by 
Darcy’s 
law 
in 
radial 
coordinates 
q(r)= 
-Kh0(r)er 
with 
er 
being 
the 
unit 
vector 
in 
radial 
direction. 
The 
second 
boundary 
condition 
for 
the 
ODE 
(2.2) 
is 
given 
by 
a 
fixed 
hydraulic 
head 
h(R) 
at 
the 
radial 
distance 
R. 



2.3. 
Uniform 
Flow 
in 
Heterogeneous 
Porous 
Media 
Figure 
2: 
Drawdowns 
of 
simulated 
pumping 
test 
(dots) 
and 
analytical 
Thiem’s 
solution 
(solid 
line) 
in 
dependence 
on 
the 
radial 
distance 
r 
from 
the 
well. 


The 
solution 
of 
the 
ODE 
(2.2) 
is 
given 
by 
h(r)= 
C1 
ln 
r 
+ 
C2. 
The 
integration 
constants 
C1 


and 
C2 
are 
determined 
using 
the 
previously 
discussed 
boundary 
conditions, 
resulting 
in 
C1 
= 
dh(r) 
Qw 
Qw

r 
= 
- 
and 
C2 
= 
ln 
R 
+ 
h(R). 
Thus 
Thiem’s 
solution 
reads 


dr 
2LK 
2LK 


h(r)= 
- 
Qw 
ln 
r 
+ 
h(R) 
. 
(2.3)

2LK 
R 


2.2.2. 
Simulation 
Results 
We 
carried 
out 
simulations 
on 
a 
two 
dimensional 
numerical 
mesh 
ranging 
from 
-6.4m 
to 
6.4m 
in 
both 
directions. 
We 
established 
a 
radial 
refinement 
at 
the 
well 
in 
the 
range 
of 
-1m 
to 
1m 
as 
visualized 
in 
Figure 
1. 
It 
provides 
high 
resolution 
of 
the 
hydraulic 
head 
drawdown 
around 
the 
well. 
The 
well 
is 
not 
included 
as 
a 
point 
source 
but 
as 
a 
hollow 
cylinder 
with 
radius 
rw 
=0.01m. 
We 
chose 
a 
constant 
head 
of 
zero 
at 
the 
outer 
radial 
distance 
R 
=6.4m 
as 
boundary 
conditions. 
At 
the 
well 
we 
used 
a 
constant 
total 
pumping 
rate 
of 
Qw 
= 
-1e-3m3/s. 


Figure 
2 
shows 
the 
comparison 
of 
the 
drawdowns 
from 
a 
simulated 
pumping 
test 
and 
the 
analytical 
Thiem’s 
solution, 
Eq. 
(2.3). 
Theoretical 
and 
numerical 
drawdowns 
are 
in 
very 
good 
agreement. 
Especially 
at 
the 
well 
the 
refined 
mesh 
provides 
good 
resolution. 
Thus, 
the 
simulation 
software 
OpenGeoSys 
can 
reproduce 
very 
well 
the 
radial 
dependent 
groundwater 
flow 
with 
steep 
gradients 
at 
a 
pumping 
well. 


2.3. 
Uniform 
Flow 
in 
Heterogeneous 
Porous 
Media 
The 
existence 
of 
an 
analytical 
expression 
for 
the 
effective 
hydraulic 
conductivity 
Kefu 
makes 
the 
uniform 
flow 
regime 
favorable 
for 
numerical 
benchmarking. 
We 
performed 
simulations 
in 
heterogeneous 
porous 
media 
with 
a 
uniform 
flow 
field 
in 
a 
simple 
domain 
design. 
It 
allows 
the 



2. 
Numerical 
Benchmarks 
application 
of 
the 
Monte 
Carlo 
approach 
to 
determine 
the 
statistics 
of 
flow 
in 
stochastic 
media, 
as 
described 
in 
section 
1.3.3. 
From 
the 
simulated 
hydraulic 
head 
distribution 
a 
simulated 
effective 
conductivity 
Kˆ
efu 
can 
be 
inferred 
using 
Darcy’s 
law 
and 
averaging 
strategies. 
The 
benchmark 
focuses 
on 
the 
comparison 
of 
the 
analytical 
expression 
of 
Kefu 
with 
the 
simulated 
effective 
conductivity 
Kˆ
efu. 


Analytical 
and 
Simulated 
Effective 
Hydraulic 
Conductivity 


The 
evaluation 
of 
an 
effective 
conductivity 
for 
uniform 
flow 
in 
heterogeneous 
porous 
media 
has 
been 
subject 
of 
several 
studies 
since 
the 
beginning 
of 
geostatistics 
in 
hydrogeology. 
Matheron 
[1967] 
started 
by 
obtaining 
the 
upper 
and 
lower 
bounds, 
later 
Gelhar 
and 
Axness 
[1983]; 
Dagan 
[1989] 
presented 
a 
general 
analytical 
expression, 
called 
Kefu. 
The 
expression 
for 
all 
dimensions 
d 
=1, 
2, 
3 
can 
be 
written 
according 
to 
Eq. 
(1.20) 
as 


21 
1

Kefu 
= 
KG 
exp 
2 
- 
,

d 


with 
KG 
being 
the 
geometric 
mean 
and 
2 
the 
variance 
of 
the 
log-normal 
distributed 
conductivity 
K(x). 


Several 
authors 
addressed 
the 
issue 
of 
comparing 
the 
analytical 
expression 
for 
Kefu 
with 
effective 
mean 
values 
obtained 
from 
simulations. 
Although 
the 
analytical 
expression 
of 
Kefu 
was 
derived 
for 
unbounded 
domains, 
it 
can 
be 
assumed 
that 
it 
is 
valid 
in 
bounded 
domains 
for 
a 
sufficiently 
large 
domain 
extent, 
discussed 
in 
detail 
in 
Sánchez-Vila 
et 
al. 
[2006]. 
Freeze 
[1975] 
was 
the 
first 
who 
performed 
numerical 
investigations 
of 
one 
dimensional 
flow 
in 
heterogeneous 
media. 
Follin 
[1992] 
gave 
a 
numerical 
confirmation 
of 
Kˆ
efu 
= 
Kefu 
in 
two 
dimensions 
(up 
to 
2 
= 
16) 
and 
Dykaar 
and 
Kitanidis 
[1992b] 
in 
three 
dimensions 
(up 
to 
2 
= 
6). 
Dykaar 
and 
Kitanidis 
[1992b] 
also 
investigated 
the 
relation 
of 
domain 
extent 
and 
convergence 
of 
the 
effective 
conductivity 
toward 
an 
asymptotic 
value. 
Their 
results 
confirm 
that 
an 
analysis 
with 
Kefu 
is 
valid 
in 
case 
of 
large 
domain 
extent 
with 
respect 
to 
the 
corresponding 
directional 
correlation 
length. 


To 
obtain 
an 
effective 
conductivity 
Kˆ
efu 
from 
simulations, 
we 
averaged 
the 
simulated 
hydraulic 


Rˆ

head 
hˆ
in 
the 
direction 
perpendicular 
to 
the 
flow 
direction, 
hˆ
(x)=h(x, 
y)dy. 
In 
the 
second 
step, 
we 
inserted 
hˆ
(x) 
into 
an 
inverted 
Darcy’s 
law, 
Eq.(1.3), 


Z

x2 


(Lx 
- 
x)

Kˆ
efu 
=Qw 
dx, 
x1 
hˆ
(x) 


where 
Qw 
denotes 
the 
flow 
rate 
and 
Lx 
the 
horizontal 
domain 
extent. 
The 
integration 
limits 
x1 
and 
x2 
were 
chosen 
sufficiently 
far 
from 
the 
boundaries. 


Numerical 
Simulation 
Setup 


We 
carried 
out 
simulations 
in 
a 
two-dimensional 
domain 
with 
a 
spatial 
extension 
of 
Lx 
= 
200 
m 
and 
Ly 
= 
50 
m 
and 
a 
regular 
grid 
spacing 
of 
1m. 
We 
established 
a 
constant 
flow 
rate 
of 
Q 
= 



2.3. 
Uniform 
Flow 
in 
Heterogeneous 
Porous 
Media 
Figure 
3: 
Realization 
of 
a 
generated 
normally 
distributed 
log-conductivity 
field 
Y 
= 
ln 
K(x) 
with 
Gaussian 
correlation 
function. 
Light 
patches 
indicate 
high 
conductivity, 
whereas 
dark 
patches 
indicate 
low 
conductivity. 
Statistical 
parameters: 
KG 
= 
1e-4m/s, 
2 
= 
1, 
f 
= 
5 
m, 
and 
e 
= 
1. 


5·e-7m3/s 
at 
the 
left 
boundary 
(x 
= 
0m) 
as 
inflow. 
The 
outflow 
area 
is 
at 
the 
right 
side 
(x 
= 
200 
m), 
with 
a 
constant 
head 
of 
h(x 
= 
200) 
= 
0 
m. 
The 
top 
and 
bottom 
boundaries 
are 
impervious 
to 
the 
flow. 
The 
setting 
resembles 
a 
section 
of 
a 
confined 
aquifer 
with 
a 
mean 
head 
gradient 
of 
1m 
over 
a 
distance 
of 
200m. 


We 
generated 
an 
ensemble 
of 
20 
heterogeneously 
distributed 
conductivity 
fields. 
Fields 
are 
of 
log-normal 
distribution 
with 
a 
Gaussian 
correlation 
structure, 
as 
defined 
in 
Eq. 
(1.14) 
and 
Eq.(1.18). 
The 
statistical 
parameter 
are 
the 
following: 
mean 
conductivity 
of 
KG 
= 
1e-4m/s, 
variance 
of 
2 
= 
1, 
correlation 
length 
of 
f 
= 
5 
m, 
and 
anisotropy 
ratio 
of 
e 
= 
1. 
We 
visualized 
a 
snapshot 
of 
one 
realization 
in 
Figure 
3. 


One 
correlation 
length 
is 
resolved 
over 
five 
elements 
of 
the 
numerical 
grid. 
The 
relation 
of 
domain 
size 
to 
correlation 
length 
is 
Lx/f 
= 
20 
which 
is 
sufficient 
for 
Gaussian 
correlated 
fields 
to 
neglect 
boundary 
effects 
according 
to 
Dykaar 
and 
Kitanidis 
[1992b]. 


Results 


We 
plotted 
the 
simulated 
drawdowns 
of 
20 
realizations 
and 
the 
ensemble 
mean 
drawdown 
in 
comparison 
to 
the 
drawdown 
in 
homogeneous 
media 
with 
Kefu 
= 
KG 
as 
substitute 
value 
in 
Figure 
4. 
The 
effective 
conductivities 
Kˆ
efu 
of 
single 
realizations 
deviate 
up 
to 
10% 
from 
the 
analytical 
solution 
Kefu 
= 
KG. 
The 
ensemble 
mean 
of 
the 
20 
simulation 
runs 
deviates 
only 
less 
than 
1% 
from 
KG. 


In 
general, 
it 
was 
possible 
to 
reproduce 
the 
effective 
conductivity 
for 
uniform 
flow 
Kefu 
with 
numerical 
simulations 
of 
flow 
in 
heterogeneous 
media 
quite 
well. 
However, 
the 
domain 
size 
is 
too 
small 
to 
guarantee 
convergence 
to 
Kefu 
within 
a 
single 
realization. 
Convergence 
to 
the 
analytical 
solution 
can 
be 
found 
by 
taking 
the 
mean 
of 
several 
simulations 
runs. 
These 
results 
underline 
the 
fact 
that 
the 
sample 
size 
and 
the 
number 
of 
realizations 
is 
crucial 
in 
simulations 
with 
heterogeneous 
conductivity 
distributions. 



2. 
Numerical 
Benchmarks 
Figure 
4: 
Simulated 
drawdowns 
of 
hydraulic 
head: 
drawdowns 
of 
20 
simulation 
runs 
(dotted 


lines), 
ensemble 
mean 
drawdown 
(dashed 
line) 
and 
drawdown 
in 
homogeneous 
media 


with 
Kefu 
= 
KG 
as 
homogeneous 
substitute 
value 
(solid 
line). 


2.4. 
The 
Elder 
Benchmark 
The 
Elder 
benchmark 
is 
one 
of 
the 
commonly 
used 
benchmarks 
to 
verify 
density-dependent 
flow 
in 
hydrogeological 
simulation 
software 
[Voss 
and 
Souza, 
1987; 
Oldenburg 
and 
Pruess, 
1995; 
Kolditz 
et 
al., 
1998; 
Frolkovic 
and 
Schepper, 
2001; 
Diersch 
and 
Kolditz, 
2002; 
Prasad 
and 
Simmons, 
2003; 
Park 
and 
Aral, 
2007]. 
The 
wide 
use 
of 
this 
benchmark 
is 
due 
to 
the 
fact 
that 
it 
is 
a 
problem 
of 
free 
convection 
where 
flow 
is 
purely 
driven 
by 
density 
differences. 


Background 


The 
benchmark 
is 
based 
on 
numerical 
studies 
of 
the 
"short 
heater 
problem" 
introduced 
by 
Elder 
[1967a, 
b] 
in 
combination 
with 
experimental 
measurements 
of 
thermally 
driven 
convection 
in 
porous 
media. 
Voss 
and 
Souza 
[1987] 
adapted 
and 
reformulated 
the 
problem 
in 
terms 
of 
density 
dependent 
flow 
induced 
by 
salt 
concentration 
differences 
for 
code 
testing 
purposes. 
Resulting 
salt 
concentration 
distribution 
patterns 
of 
Elder 
[1967b] 
and 
Voss 
and 
Souza 
[1987] 
matched 
quite 
well. 
In 
subsequent 
numerical 
studies 
different 
steady 
state 
solutions, 
varying 
in 
number 
and 
development 
of 
salt 
plumes, 
have 
been 
observed 
e.g. 
Oldenburg 
and 
Pruess 
[1995]; 
Frolkovic 
and 
Schepper 
[2001]; 
Diersch 
and 
Kolditz 
[2002]; 
Park 
and 
Aral 
[2007]. 
At 
first, 
these 
discrepancies 
have 
been 
mainly 
attributed 
to 
mesh 
resolution 
[Frolkovic 
and 
Schepper, 
2001; 
Diersch 
and 
Kolditz, 
2002]. 
By 
making 
use 
of 
bifurcation 
analysis 
Johannsen 
[2003] 
showed 
that 
(at 
least) 
three 
stable 
and 
a 
further 
eight 
unstable 
steady 
state 
solutions 
co-exist. 
Different 
domain 
discretizations, 
numerical 
schemes 
and 
initial 
conditions 
can 
lead 
to 
different 
solutions 



2.4. 
The 
Elder 
Benchmark 
Figure 
5: 
Geometry 
and 
boundary 
conditions 
of 
the 
Elder 
benchmark 
for 
the 
left 
half 
domain. 


in 
number 
of 
down-welling 
plumes, 
shown 
by 
Park 
and 
Aral 
[2007]. 
However, 
the 
existence 
of 
multiple 
steady 
state 
solutions 
is 
an 
intrinsic 
characteristic 
of 
the 
Elder 
problem. 


The 
lack 
of 
a 
unique 
steady 
state 
solution 
makes 
the 
Elder 
problem 
questionable 
for 
benchmarking 
purposes. 
By 
making 
use 
of 
a 
pseudo-spectral 
approach, 
van 
Reeuwijk 
et 
al. 
[2009] 
confirmed 
the 
results 
of 
Johannsen 
[2003] 
and 
underlined 
the 
fact 
that 
bifurcation 
depends 
on 
the 
Rayleigh 
number 
as 
defined 
in 
Eq. 
(1.11). 
According 
to 
their 
analysis 
only 
one 
steady 
state 
solution 
exists 
for 
lower 
Rayleigh 
numbers 
than 
originally 
used 
in 
the 
Elder 
benchmark. 
Consequently, 
they 
established 
a 
low 
Rayleigh 
number 
Elder 
benchmark 
with 
slightly 
changed 
setting. 
We 
adopting 
that 
idea 
and 
present 
results 
for 
both, 
the 
classical 
Elder 
benchmark 
as 
well 
as 
the 
low 
Rayleigh 
number 
Elder 
benchmark. 


Problem 
Definition 


The 
Elder 
domain 
is 
a 
vertical 
cross 
section 
of 
x 
= 
600 
m 
length 
and 
z 
= 
150 
m 
height 
(Figure 
5). 
Half 
of 
the 
surface 
contains 
a 
salt 
source 
with 
a 
constant 
concentration 
of 
C 
=1. 
The 
bottom 
has 
a 
fixed 
concentration 
of 
zero. 
The 
hydraulic 
head 
is 
constant 
in 
the 
entire 
domain 
and 
all 
boundaries 
are 
impervious 
to 
flow 
and 
mass 
transport. 
The 
hydraulic 
parameters 
are 
listed 
in 
Table 
2. 
Voss 
and 
Souza 
[1987] 
proposed 
to 
use 
these 
values 
to 
obtain 
a 
Rayleigh 
number 
of 
Ra 
= 
400, 
following 
the 
lines 
of 
the 
"short 
heater 
problem" 
of 
Elder 
[1967b]. 


Following 
the 
lines 
of 
van 
Reeuwijk 
et 
al. 
[2009] 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
remains 
with 
the 
geometry 
and 
the 
parameter 
setting 
of 
the 
classical 
Elder 
benchmark 
with 
the 
exceptions 
of: 
(i) 
a 
reduced 
value 
of 
maximal 
density 
of 
1030 
kg/m3 
instead 
of 
1200 
kg/m3, 
resulting 
in 
a 
Rayleigh 
number 
of 
Ra 
= 
60;(ii) 
a 
change 
in 
boundary 
condition 
at 
the 
surface: 
the 
concentration 
it 
fixed 
to 
zero 
at 
x 
=0 
- 
150 
m 
and 
x 
= 
450 
- 
600m; 
and 
(iii) 
a 
coarser 
time 
stepping. 
Parameters 
different 
for 
both 
simulation 
settings 
are 
listed 
in 
Table 
3. 



2. 
Numerical 
Benchmarks 
symbol 
[unit] 
quantity 
value 


f 
[-] 
porosity 
0.1 
. 
[m2] 
permeability 
4.845e-13 

C 
[-] 
relative 
concentration 
difference 
1 
0 
[kgm-3] 
fresh 
water 
density 
1000 
. 
[kgm-1 
s-1] 
viscosity 
0.001 
Dm 
[m2 
s-1] 
molecular 
diffusion 
coefficient 
3.565 
e-6 
l,t 
[m] 
solute 
dispersivities 
0 


Table 
2: 
Parameters 
valid 
for 
classical 
and 
low 
Rayleigh 
number 
Elder 
benchmark. 


symbol 
[unit] 
quantity 
Ra 
= 
400 
Ra 
= 
60 


C 
[-] 
volumetric 
expansion 
0.2 
0.03 

t 
[s] 
time 
stepping 
1314000 
7884000 
vmax 
[m/s] 
maximal 
velocity 
2e-6 
3.0e-7 
Peg 
grid 
Peclet 
number 
1.3 
0.2 
Cr 
Courant 
number 
1.1 
1 


Table 
3: 
Parameters 
and 
numerical 
stability 
numbers 
for 
classical 
and 
the 
low 
Rayleigh 
number 
Elder 
benchmark. 


We 
reproduced 
the 
classical 
and 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
with 
OpenGeoSys 
[Kolditz 
et 
al., 
2012b]. 
We 
performed 
all 
simulations 
on 
the 
half 
domain, 
which 
is 
sufficient 
due 
to 
the 
symmetry 
of 
the 
problem. 
The 
mesh 
contains 
128 
· 
64 
identical 
square 
elements, 
corresponding 
to 
a 
mesh 
refinement 
level 
of 
6, 
according 
to 
Frolkovic 
and 
Schepper 
[2001]; 
Diersch 
and 
Kolditz 
[2002]. 


The 
governing 
equations 
used 
to 
solve 
variable 
density 
flow 
consist 
of 
the 
three 
fundamental 
conservation 
equations: 
the 
continuity 
equation 
of 
flow, 
given 
in 
Eq. 
(1.1), 
the 
momentum 
equation 
(1.2), 
and 
the 
mass 
transport 
equation 
(1.5). 
The 
equations 
are 
coupled 
to 
the 
equation 
of 
fluid 
density 
(1.10) 
and 
the 
hydrodynamic 
dispersion 
(1.6). 
We 
used 
no 
numerical 
stabilization 
schemes. 
Calculating 
grid 
stability 
criteria 
like 
the 
grid 
Peclet 
number 
and 
Courant 
number, 
as 
e.g. 
defined 
in 
Kolditz 
[2012] 
result 
in 
values 
satisfying 
the 
criteria 
for 
numerical 
stability 
in 
spatial 
and 
temporal 
discretization 
(Table 
3). 


Simulation 
Results 


We 
visualized 
the 
results 
of 
the 
concentration 
patterns 
in 
combination 
with 
the 
flow 
velocity 
field 
for 
both 
Elder 
Benchmarks 
in 
Figure 
6. 
The 
first 
time 
step 
is 
at 
an 
early 
stage 
of 
plume 
evolution, 
the 
second 
step 
is 
at 
a 
later 
stage 
when 
simulations 
reached 
the 
steady 
state. 
For 
the 
Elder 
benchmark 
this 
is 
at 
t 
= 
3 
year 
and 
t 
= 
30 
years 
and 
for 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
at 
t 
= 
20 
year 
and 
t 
= 
200 
years. 
The 
differences 
in 
evolution 
time 
corresponds 
to 
the 
difference 
in 
the 
Rayleigh 
number. 



2.4. 
The 
Elder 
Benchmark 
Figure 
6: 
Isolines 
of 
simulated 
salt 
concentration 
C 
= 
[0.1,...,0.9] 
of 
the 
classical 
and 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
for 
two 
time 
steps: 
top 
left 
Ra=400, 
t=3years; 
down 
left 
Ra=400, 
t=30years; 
top 
right 
Ra=60, 
t=20years; 
down 
right 
Ra=60, 
t=200 
years. 
The 
arrows 
mark 
the 
flow 
directions, 
their 
color 
the 
flow 
velocity. 


The 
evolving 
concentration 
pattern 
for 
the 
classical 
Elder 
benchmark 
shows 
several 
downwelling 
fingers 
at 
an 
early 
stage 
and 
finally 
one 
central 
down-welling 
finger 
at 
the 
steady 
state 
resembling 
the 
results 
of 
Frolkovic 
and 
Schepper 
[2001]; 
Diersch 
and 
Kolditz 
[2002]; 
Park 
and 
Aral 
[2007]. 
The 
results 
for 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
correspond 
perfectly 
to 
the 
distribution 
of 
isolines 
presented 
in 
van 
Reeuwijk 
et 
al. 
[2009]. 


For 
a 
quantitative 
comparison 
we 
calculated 
the 
maximal 
penetration 
depth 
of 
several 
concentration 
isolines, 
listed 
in 
Table 
4. 
For 
the 
penetration 
depth 
of 
the 
isoline 
of 
C 
=0.6 
the 
value 
of 
the 
classical 
Elder 
benchmark 
corresponds 
perfectly 
to 
the 
calculated 
value 
given 
in 
Prasad 
and 
Simmons 
[2005]. 
The 
values 
for 
the 
low 
Rayleigh 
number 
Elder 
benchmark 
match 
the 
location 
of 
the 
isolines 
given 
in 
van 
Reeuwijk 
et 
al. 
[2009]. 


C 
PD 
(RaS 
= 
400) 
PD 
(RaS 
= 
60) 
0.2 
-145 
m 
-138 
m 
0.4 
-141 
m 
-127 
m 
0.6 
-136 
m 
-113 
m 
0.8 
-56.3 
m 
-84.4 
m 


Table 
4: 
Maximal 
penetration 
depth 
PD[m] 
at 
steady 
state 
for 
different 
concentration 
isolines 
for 
the 
classical 
and 
the 
low 
Rayleigh 
number 
Elder 
benchmark. 



2. 
Numerical 
Benchmarks 
Figure 
7: 
Geometrical 
setting 
and 
boundary 
conditions 
of 
the 
Saltdome 
benchmark. 


2.5. 
The 
Saltdome 
Problem 
The 
Saltdome 
benchmark 
reflects 
the 
idealized 
groundwater 
flow 
over 
a 
salt 
dome, 
taking 
significant 
variations 
of 
fluid 
density 
due 
to 
salt 
dissolution 
into 
account. 
It 
was 
originally 
formulated 
in 
the 
frame 
of 
the 
international 
HYDROCOIN 
groundwater 
modeling 
project 
(Case 
5, 
Level 
1 
problem). 
The 
setting 
is 
in 
reference 
to 
the 
salt 
dome 
nuclear 
waste 
disposal 
site 
at 
Gorleben, 
Germany 
[Herbert 
et 
al., 
1988; 
Oldenburg 
and 
Pruess, 
1995; 
Konikow 
et 
al., 
1997]. 
The 
model 
aims 
to 
identify 
how 
convective 
flow, 
which 
is 
induced 
by 
fluid 
density 
differences, 
interacts 
with 
the 
advective 
flow 
process 
initiated 
by 
pressure 
differences. 
We 
examine 
the 
Saltdome 
benchmark 
more 
closely 
since 
the 
idealized 
setting 
is 
of 
similar 
character 
as 
the 
cross 
section 
model 
of 
the 
Thuringian 
Basin, 
which 
we 
discuss 
in 
section 
3. 


2.5.1. 
Problem 
Definition 
The 
benchmark 
consists 
of 
three 
processes. 
Continuity 
equation 
of 
flow 
(1.1) 
and 
Darcy’s 
law 
(1.2) 
determine 
the 
groundwater 
flow. 
Eq. 
(1.5) 
describes 
the 
salt 
transport. 
The 
heat 
transport 
process 
requires 
the 
evolution 
of 
the 
energy 
balance 
equation 
in 
the 
porous 
media, 
Eq.(1.8). 
These 
processes 
are 
successfully 
implemented 
into 
the 
physical 
based 
simulation 
software 
OpenGeoSys 
[Kolditz 
et 
al., 
2012b], 
which 
we 
used 
to 
evaluate 
the 
Saltdome 
benchmark. 


The 
model 
region 
is 
a 
two 
dimensional 
vertical 
cross 
section 
of 
900m 
length 
and 
300m 
height, 
representing 
a 
homogeneous 
and 
isotropic 
aquifer. 
The 
geometrical 
setting 
and 
the 
boundary 
conditions 
are 
visualized 
in 
Figure 
7. 
We 
applied 
a 
linear 
pressure 
gradient 
of 
1e5Pa 
at 
the 
top 
from 
left 
to 
right. 
We 
established 
a 
constant 
salt 
concentration 
boundary 
at 
the 
central 
third 
of 
the 
bottom, 
representing 
a 
salt 
source 
exposed 
to 
the 
aquifer. 
The 
boundary 
conditions 
of 
the 
heat 
transport 
process 
are 
given 
by 
a 
constant 
temperature 
of 
T 
=8 
C 
at 
the 
surface 
and 
of 
T 
= 
35 
C 
at 
the 
bottom 
of 
the 
basin. 
It 
implies 
a 
constant 
temperature 
gradient 



2.5. 
The 
Saltdome 
Problem 
Figure 
8: 
Numerical 
mesh 
used 
for 
the 
Saltdome 
benchmark 
simulations. 


symbol 
[unit] 
quantity 
value 


f 
[-] 
porosity 
0.2 
. 
[mD] 
permeability 
1000 
. 
[kgm-1 
s-1] 
viscosity 
0.001 
0 
[kgm-3] 
fresh 
water 
density 
1000 
C 
[-] 
normalized 
salt 
concentration 
[0, 
1] 
C 
[-] 
solute 
expansion 
coefficient 
0.2 
Dm 
[m2 
s-1] 
molecular 
diffusion 
5e-08 
l,t 
[m] 
solute 
dispersivities 
20;2 


T 
[ 
C] 
temperature 
[8, 
35] 
T 
[K-1] 
thermal 
expansion 
coefficient 
-3e-04 
w 
[kgm2 
s-3 
K-1] 
thermal 
conductivity 
of 
water 
0.6 
s 
[kgm2 
s-3 
K-1] 
thermal 
conductivity 
of 
solid 
2.0 
cw 
[m2 
s-2 
K-1] 
thermal 
capacity 
of 
water 
4200 
cs 
[m2 
s-2 
K-1] 
thermal 
capacity 
of 
solid 
rock 
1000 
s 
[kgm-3] 
rock 
density 
2700 


Table 
5: 
Parameters 
for 
the 
Saltdome 
benchmark 
in 
line 
with 
the 
problem 
definition 
of 
Herbert 
et 
al. 
[1988]; 
Kolditz 
et 
al. 
[1998]. 


of 
30 
C/km. 
The 
lateral 
and 
bottom 
boundaries 
are 
impervious 
to 
flow, 
salt, 
and 
heat. 
The 
initial 
concentration 
in 
the 
domain 
is 
C 
= 
0. 
The 
initial 
temperature 
is 
T 
=8 
C. 
The 
specific 
hydraulic 
parameters 
used 
in 
this 
application 
are 
given 
in 
Table 
5. 


For 
numerical 
simulations, 
we 
use 
an 
unstructured 
grid 
consisting 
of 
23581 
triangular 
elements 
with 
gradual 
refinements 
in 
the 
area 
of 
the 
salt 
dome. 
The 
edge 
lengths 
are 
between 
1m 
and 
30 
m. 
Figure 
8 
gives 
a 
visualization 
of 
the 
mesh. 
We 
adapt 
the 
time 
stepping 
to 
the 
choice 
of 
the 
mesh 
and 
the 
processes 
incorporated. 
Generally, 
a 
time 
step 
is 
in 
the 
range 
of 
one 
month. 
Simulations 
reached 
the 
steady 
state 
after 
1000 
years. 


We 
performed 
numerical 
simulations 
with 
two 
different 
approaches 
of 
transport. 
First, 
we 
assume 
conservative 
transport 
of 
salt 
and 
heat. 
We 
neglect 
their 
impact 
on 
fluid 
density 
by 
using 
a 
constant 
density 
model 
. 
= 
0. 
The 
system 
is 
not 
coupled: 
mass 
and 
heat 
transport 
depend 
on 
fluid 
flow 
but 
not 
vice 
versa. 
Second, 
fluid 
density 
effects 
are 
taken 
into 
account. 



2. 
Numerical 
Benchmarks 
Figure 
9: 
Isolines 
of 
calculated 
salt 
concentration 
C 
= 
[0.1,..., 
0.9] 
in 
the 
Saltdome 
benchmark 
domain 
after 
1000 
years 
simulation 
time 
for 
(a) 
the 
constant 
density 
model 
0 
and 


(b) 
for 
the 
concentration 
and 
temperature 
dependent 
density 
model 
(C, 
T 
). 
The 
arrows 
mark 
the 
flow 
directions, 
the 
color 
of 
the 
arrows 
indicate 
the 
flow 
velocity. 
Differences 
in 
salt 
concentration 
and 
temperature 
affect 
the 
fluid 
density 
via 
a 
linear 
density 
model 
. 
= 
(C, 
T 
), 
as 
described 
in 
Eq.(1.10). 
As 
a 
consequence, 
the 
fluid 
flow 
equation 
depends 
on 
the 
salt 
mass 
and 
heat 
transport 
resulting 
in 
a 
non-linear 
coupling 
of 
equations. 


2.5.2. 
Results 
Simulation 
results 
of 
the 
salt 
concentration 
distribution 
for 
the 
constant 
fluid 
density 
model 
. 
= 
0 
are 
given 
in 
Figure 
9a. 
A 
uniform 
regional 
groundwater 
flow 
regime 
develops 
from 
left 
to 
right 
due 
to 
the 
imposed 
pressure 
gradient. 
Mass 
and 
heat 
are 
transported 
conservatively 
with 
the 
groundwater 
according 
to 
the 
flow 
pattern. 
High 
concentrated 
solute 
(up 
to 
C 
=0.4) 
rises 
to 
the 
surface 
on 
the 
top 
right 
edge. 
We 
observe 
no 
salt 
at 
the 
left 
side 
of 
the 
salt 
dome 
area. 
The 
temperature 
pattern 
is 
similar, 
as 
it 
can 
be 
seen 
in 
Figure 
10a. 
A 
stable 
temperature 
stratification 
is 
suppressed 
by 
inflowing 
cold 
water 
at 
the 
left 
side 
of 
the 
domain, 
whereas 
warmer 
water 
(up 
to 
T 
= 
24 
C) 
from 
the 
bottom 
region 
is 
transported 
to 
the 
surface 
in 
the 
discharge 
area 
at 
the 
right 
side. 


Figure 
9b 
shows 
the 
simulated 
salt 
concentration 
distribution 
for 
the 
non-constant 
fluid 
density 
model 
. 
= 
(C, 
T 
). 
In 
contrast 
to 
the 
results 
of 
the 
constant 
density 
model 
(Figure 
9a), 
salt 
can 
be 
observed 
in 
at 
the 
left 
side 
of 
the 
salt 
dome. 
The 
entire 
bottom 
region 
of 
the 
domain 
is 
filled 
with 
highly 
concentrated 
brine. 
Though 
the 
concentration 
of 
brine 
which 
reaches 
the 
surface 
at 
the 
right 
edge 
is 
less 
than 
C 
=0.2. 



2.5. 
The 
Saltdome 
Problem 
Figure 
10: 
Calculated 
isolines 
of 
temperature 
T 
= 
[9, 
12,..., 
33] 
C 
in 
the 
Saltdome 
benchmark 
domain 
after 
1 
000 
years 
simulation 
time 
for 
(a) 
the 
constant 
density 
model 
0 
and 


(b) 
for 
the 
concentration 
and 
temperature 
dependent 
density 
model 
(C, 
T 
). 
The 
arrows 
mark 
the 
flow 
directions, 
the 
color 
of 
the 
arrows 
indicate 
the 
flow 
velocity. 
The 
regional 
groundwater 
flow 
is 
mainly 
present 
in 
the 
shallow 
basin 
region, 
whereas 
in 
deep 
basin 
regions 
the 
impact 
of 
the 
pressure 
gradient 
is 
significantly 
reduced. 
Flow 
occurs 
in 
vertical 
direction, 
visible 
at 
the 
flow 
field 
arrows. 
The 
temperature 
distribution 
resembles 
the 
salt 
distribution. 
The 
reduced 
impact 
of 
advective 
flow 
results 
in 
a 
smoother 
temperature 
stratification, 
as 
visible 
in 
Figure 
10b. 


2.5.3. 
Discussion 
A 
comparison 
of 
the 
simulation 
results 
for 
the 
different 
density 
models 
reveals 
that 
the 
evolving 
salt 
concentration 
and 
temperature 
patterns 
differ 
significantly 
from 
each 
other. 
The 
salt 
concentration 
is 
higher 
in 
the 
lower 
basin 
region 
for 
the 
fully 
coupled 
simulation 
with 
. 
= 
(C, 
T 
), 
but 
less 
highly 
concentrated 
saline 
groundwater 
is 
transported 
to 
the 
surface 
in 
contrast 
to 
the 
uncoupled 
simulation 
with 
. 
= 
0. 


The 
total 
amount 
of 
dissolved 
salt 
is 
higher 
for 
the 
concentration 
dependent 
density 
model, 
see 
Table 
6. 
To 
emphasize 
on 
that 
the 
horizontal 
distribution 
of 
salt 
concentration 
at 
z 
= 
-200m 
depth 
C(x, 
z 
= 
-200m)forconstantandnon-constantdensitymodel 
isvisualizedinFigure11. 
It 
can 
be 
clearly 
seen 
that 
the 
slope 
of 
the 
salt 
plume 
is 
steeper 
for 
the 
constant 
density 
model, 
indicating 
a 
much 
stronger 
impact 
of 
advective 
flow 
in 
deep 
aquifer 
regions. 


In 
contrast 
to 
the 
advection 
dominated 
transport 
processes 
in 
the 
constant 
density 
simulations, 



2. 
Numerical 
Benchmarks 
Figure 
11: 
Salt 
concentration 
after 
1000 
years 
at 
z 
= 
-200m 
depth 
for 
the 
constant 
density 
model 
0 
(solid 
line) 
and 
the 
non-constant 
density 
model 
(C, 
T 
) 
(crosses). 


0 
(C, 
T 
) 
(C) 
(T 
) 
C 
¯0.097 
0.229 
0.224 
0.086 


Table 
6: 
Mean 
concentration 
C 
¯ 
for 
different 
density 
models, 
as 
defined 
in 
Eq.(1.12). 


the 
flow 
regime 
for 
salt 
and 
temperature 
dependent 
density 
is 
of 
mixed 
convection 
[Nield 
and 
Bejan, 
1999]. 
The 
main 
groundwater 
flow 
is 
driven 
by 
a 
pressure 
gradient, 
but 
density 
effects 
can 
give 
rise 
to 
free 
convection 
at 
the 
bottom 
of 
the 
aquifer. 
This 
is 
supported 
by 
the 
high 
Rayleigh 
number 
of 
solute 
RaS 
˜ 
9600, 
calculated 
according 
to 
Eq.(1.11). 
The 
horizontal 
density 
differences 
provoke 
the 
formation 
of 
eddies, 
resulting 
in 
an 
increased 
velocity 
in 
vertical 
direction. 
Higher 
velocities 
in 
combination 
with 
dispersion 
lead 
to 
a 
stronger 
spread 
of 
salt 
and 
thus 
the 
effect 
of 
recirculation 
provokes 
stronger 
upconing. 
An 
increased 
density 
inhibits 
the 
upward 
transport 
of 
highly 
concentrated 
brine 
into 
shallow 
region 
by 
the 
groundwater 
flow. 
The 
evolving 
flow 
pattern 
represents 
a 
balance 
among 
buoyancy 
effects, 
advection 
and 
dispersion. 


Several 
studies 
[Herbert 
et 
al., 
1988; 
Oldenburg 
and 
Pruess, 
1995; 
Konikow 
et 
al., 
1997; 
Kolditz 
et 
al., 
1998; 
Diersch 
and 
Kolditz, 
1998; 
Younes 
et 
al., 
1999] 
investigated 
the 
effects 
of 
boundary 
and 
initial 
conditions, 
spatial 
discretization, 
dispersion 
and 
diffusion 
coefficients 
as 
well 
as 
other 
parameters 
for 
the 
Saltdome 
benchmark 
with 
concentration 
dependent 
fluid 
density. 
Of 
particular 
interest 
is 
the 
setting 
of 
diffusion 
and 
dispersion 
coefficients. 
Coarse 
numerical 
meshes 
require 
a 
certain 
numerical 
and/or 
physical 
diffusion 
or 
sufficiently 
large 
dispersion 
to 
prevent 
numerical 
errors 
and 
guarantees 
convergence. 
Large 
values 
of 
diffusion 
or 
dispersion 
results 
in 
stronger 
upconing 
of 
salt. 
The 
dependency 
of 
the 
amount 
of 
dissolved 
salt 
on 
the 
refinement 
of 
the 
mesh 
is 
due 
to 
the 
synergetic 
feedback 
effect 
of 
dispersive 
transport. 
It 
is 
more 
pronounced 
for 
coarser 
meshes 
since 
the 
numerically 
calculated 
advective 
flux 
depends 
on 



2.5. 
The 
Saltdome 
Problem 

C 
[%] 
0 
(C, 
T 
) 
(C) 
(C) 
0 
0 
15.1 
15.7 
1.1 
(C, 
T 
) 
15.1 
0 
0.6 
15.2 
(C) 
15.7 
0.6 
0 
15.7 
(T 
) 
1.1 
15.2 
15.7 
0 


Table 
7: 
Relative 
concentration 
difference 

C 
[%] 
between 
density 
models, 
as 
defined 
in 
Eq.(1.13). 



T 
[%] 
0 
(C, 
T 
) 
(C) 
(T 
) 
0 
0 
18.8 
19.5 
1.9 
(C, 
T 
) 
18.8 
0 
0.7 
20.1 
(C) 
19.5 
0.7 
0 
20.8 
(T 
) 
1.9 
20.1 
20.8 
0 


Table 
8: 
Relative 
temperature 
difference 

T 
[%] 
between 
density 
models, 
as 
defined 
in 
Eq.(1.13). 


the 
scale 
of 
discretization 
[Konikow 
et 
al., 
1997]. 
Refined 
meshes 
allow 
for 
more 
realistic, 
lower 
values 
of 
diffusion 
and 
dispersion 
resulting 
in 
less 
salt 
spreading 
as 
shown 
by 
Younes 
et 
al. 
[1999]. 
In 
this 
line 
the 
discretization 
of 
the 
constant 
salt 
boundary 
condition 
at 
the 
bottom 
is 
of 
importance, 
since 
it 
regulates 
the 
amount 
of 
brine 
entering 
from 
the 
bottom 
source 
region. 
This 
in 
turn 
is 
responsible 
for 
the 
development 
of 
a 
gravity-driven 
clockwise 
circulation 
against 
the 
overall 
counterclockwise 
flow 
imposed 
by 
the 
pressure 
boundary 
conditions 
[Oldenburg 
and 
Pruess, 
1995; 
Konikow 
et 
al., 
1997; 
Kolditz 
et 
al., 
1998]. 
The 
more 
brine 
is 
dissolved, 
the 
stronger 
is 
the 
recirculation 
effect. 


The 
impact 
of 
the 
temperature 
process 
on 
the 
evolving 
flow 
pattern 
is 
small. 
Simulation 
results 
for 
purely 
temperature 
dependent 
density 
. 
= 
(T 
) 
(not 
visualized) 
and 
constant 
density 
. 
= 
0 
are 
very 
similar. 
The 
same 
is 
valid 
for 
simulation 
results 
for 
the 
variable 
density 
model 
without 
and 
with 
temperature 
. 
= 
(C) 
(not 
visualized) 
and 
. 
= 
(C, 
T 
), 
respectively. 
These 
findings 
are 
underlined 
by 
the 
similar 
relative 
mean 
salt 
concentration 
and 
mean 
temperature, 
listed 
in 
Table 
6 
and 
the 
small 
relative 
difference 
in 
salt 
concentration 
and 
temperature 
distribution, 
listed 
in 
Tables 
7 
and 
8. 
These 
findings 
are 
supported 
by 
the 
relatively 
low 
thermal 
Rayleigh 
number 
of 
RaT 
˜ 
170. 
It 
is 
above 
the 
critical 
value 
of 
Rac 
=4, 
which 
marks 
the 
onset 
of 
convection. 
However, 
advective 
flow 
suppresses 
the 
development 
of 
convection 
cells 
and 
thus 
the 
upward 
movement 
of 
fluid 
due 
to 
smaller 
density. 
This 
is 
in 
line 
with 
the 
findings 
of 
Diersch 
and 
Kolditz 
[1998]. 
Thus, 
temperature 
differences 
present 
in 
this 
setting 
are 
too 
small 
to 
significantly 
impact 
on 
the 
flow 
pattern. 


More 
recently, 
Holzbecher 
et 
al. 
[2010] 
investigated 
the 
sensitivity 
of 
double 
diffusive 
convection 
above 
a 
salt 
dome, 
in 
a 
setting 
comparable 
with 
the 
Saltdome 
benchmark. 
Holzbecher 
et 
al. 
[2010] 
confirmed 
that 
the 
formation 
of 
eddies 
is 
an 
effect 
of 
salinity 
differences. 
The 
smaller 
the 
buoyancy 
effect, 
the 
less 
pronounced 
is 
the 
upconing 
of 
salt. 
Moreover 
a 
stronger 
potential 



2. 
Numerical 
Benchmarks 
flow 
reduces 
the 
tendency 
of 
eddy 
formation. 
Simulations 
accounting 
for 
temperature 
effects 
on 
fluid 
density 
underlined 
the 
fact 
that 
for 
gradients 
of 
25 
C/km, 
typically 
found 
in 
sedimentary 
basins 
in 
Germany, 
the 
effect 
of 
temperature 
on 
the 
evolving 
flow 
and 
salt 
pattern 
is 
marginal. 
Furthermore, 
Holzbecher 
et 
al. 
[2010] 
re-evaluated 
parameters 
aiming 
to 
use 
characteristic 
values 
most 
likely 
to 
be 
found 
in 
a 
real 
aquifer, 
confirming 
most 
values 
used 
in 
the 
Saltdome 
benchmark. 


Finally, 
we 
can 
state 
that 
the 
results 
for 
the 
Saltdome 
benchmark 
calculated 
with 
Open-
GeoSys 
are 
in 
line 
with 
previous 
findings 
of 
Herbert 
et 
al. 
[1988]; 
Kolditz 
et 
al. 
[1998]; 
Diersch 
and 
Kolditz 
[1998]. 
Although 
the 
setting 
is 
highly 
idealized, 
the 
Saltdome 
benchmark 
is 
of 
general 
interest 
for 
studying 
strongly 
coupled 
groundwater 
flow 
and 
solute 
transport 
due 
to 
highly 
concentrated 
brine. 
The 
results 
show 
that 
flow 
patterns 
look 
significantly 
different 
when 
taking 
into 
account 
density 
effects. 
Even 
though 
the 
effects 
of 
mixed 
convection 
are 
different 
from 
a 
purely 
advective 
regime, 
the 
results 
point 
toward 
the 
fact 
the 
saline 
groundwater 
can 
be 
transported 
to 
shallow 
aquifer 
regions 
by 
an 
interplay 
of 
regional 
groundwater 
flow 
and 
dispersive 
flux 
in 
deep 
aquifer 
regions. 



Part 
II. 
Impact 
of 
Heterogeneity 
and 
Density 
Dependent 
Flow 
on 
Regional 
Scale 



52 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
3.1. 
Introduction 
Saline 
groundwater, 
which 
comes 
close 
to 
or 
reaches 
the 
surface, 
is 
a 
phenomenon 
that 
can 
be 
observed 
in 
many 
places 
in 
the 
Thuringian 
Basin 
[Seidel, 
2003]. 
However, 
it 
is 
not 
obvious 
why 
denser 
brine 
overlays 
lighter 
fresh 
water 
in 
this 
region. 
The 
hydrogeological 
processes, 
which 
cause 
the 
rising 
of 
saltwater 
plumes 
from 
deeper 
geological 
layers 
to 
the 
surface, 
are 
not 
yet 
fully 
understood. 
The 
aim 
of 
this 
numerical 
study 
is 
to 
investigate 
the 
mechanism 
of 
brine 
transport 
within 
the 
aquifers 
of 
shallow 
sedimentary 
basins 
in 
general, 
and 
of 
the 
Thuringian 
Basin 
in 
particular. 


Numerous 
mechanism 
can 
be 
responsible 
for 
fluid 
migration 
within 
sedimentary 
basins 
like 
gradients 
in 
hydraulic 
head 
driven 
by 
topography 
and 
fluid 
density 
variations, 
but 
also 
tectonic 
loading, 
seismogenic 
pumping 
and 
the 
production 
of 
diagenetic 
fluids 
[Person 
et 
al., 
1996]. 
The 
relative 
importance 
and 
the 
degree 
of 
interaction 
of 
these 
driving 
forces 
on 
fluid 
flow 
varies, 
depending 
on 
the 
local 
hydraulic 
and 
tectonic 
conditions 
(e.g. 
permeability, 
porosity 
and 
geological 
structure). 
The 
role 
of 
specific 
driving 
mechanisms 
on 
fluid 
flow 
within 
different 
tectonic 
environments 
is 
examined 
in 
several 
studies, 
e.g. 
Magri 
et 
al. 
[2005, 
2008] 
and 
Lampe 
and 
Person 
[2002]. 


The 
understanding 
of 
basin-scale 
flow 
and 
transport 
phenomena 
is 
important 
for 
geothermal 
applications, 
hydrocarbon 
migration 
and 
in 
particular 
for 
assessing 
long-term 
behavior 
of 
pollutants. 
Results 
from 
numerical 
simulations 
provide 
essential 
information 
about 
active 
processes 
in 
the 
subsurface 
on 
the 
basis 
of 
physical 
principles. 
Mathematical 
models 
complement 
field 
or 
laboratory-based 
investigations 
because 
they 
represent 
geologic 
processes 
that 
occur 
at 
very 
slow 
rates 
and 
over 
large 
length 
scales. 
Furthermore 
they 
take 
into 
account 
the 
simultaneity 
of 
flow, 
heat 
and 
mass 
transport 
and 
can 
consider 
coupling 
effects 
[Nield 
and 
Bejan, 
1999; 
Simmons 
et 
al., 
2001; 
Diersch 
and 
Kolditz, 
2002]. 


Regional 
groundwater 
flow 
develops 
due 
to 
hydrostatic 
pressure 
gradients 
caused 
by 
differences 
in 
topography. 
Permeability 
is 
the 
property 
of 
porous 
media, 
which 
mainly 
determines 
the 
flow 
velocities 
and 
thus 
the 
groundwater 
flow 
pattern 
by 
Darcy’s 
law. 
However, 
the 
distribution 
of 
permeabilities 
is 
subject 
to 
a 
high 
degree 
of 
uncertainty. 
On 
the 
one 
hand, 
the 
amount 
of 
available 
measured 
data 
is 
very 
low. 
On 
the 
other 
hand, 
permeability 
is 
heterogeneously 
distributed 
in 
space. 
Variations 
over 
orders 
of 
magnitude 
are 
common 
[Aehnelt 
et 
al., 
2011]. 
An 
additional 
factor 
of 
uncertainty 
are 
the 
permeabilities 
in 
fault 
zones. 
These 
regions 
often 
show 
very 
distinct 
hydraulic 
behavior 
compared 
to 
undisturbed 
basin 
compartments: 
sometimes 
they 
act 
as 
conduit, 
other 
times 
as 
barrier 
to 
groundwater 
flow 
[Lampe 
and 
Person, 
2002; 
Bense 
and 
Person, 
2006]. 


Due 
to 
the 
strong 
heterogeneity 
of 
permeability 
values, 
estimates 
of 
mean 
permeability 
on 
basin 
scale 
can 
hardly 
be 
representative 
for 
the 
entire 
area 
of 
interest. 
Generally, 
averages 
can 
only 
be 
given 
within 
ranges. 
Thus, 
large 
scale 
groundwater 
flow 
simulations 
using 
one 
set 
of 
homogeneous 
permeabilities 
for 
all 
sedimentary 
layers 
might 
not 
be 
appropriate 
to 
describe 
the 
fluid 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
12: 
Geological 
surface 
map 
of 
Thuringia 
with 
focus 
on 
the 
Thuringian 
Basin 
(modified 
according 
to 
Thüringer 
Landesamt 
für 
Umwelt 
und 
Geologie 
(TLUG) 
[2002]). 
The 
bold 
lines 
indicate 
locations 
of 
cross 
sections, 
where 
A-A’ 
marks 
the 
location 
of 
the 
southwest-northeast 
cross 
section 
used 
for 
numerical 
modeling. 


dynamics 
satisfactorily. 
Furthermore, 
averaged 
permeability 
values 
for 
large 
sedimentary 
units 
do 
not 
take 
into 
account 
the 
effects 
of 
local 
heterogeneity 
on 
the 
flow 
pattern. 
In 
combination 
with 
density 
dependent 
flow, 
caused 
by 
salt 
concentration 
and/or 
temperature 
gradients, 
they 
can 
play 
an 
import 
role. 
Several 
studies 
like 
Simmons 
et 
al. 
[2001] 
point 
towards 
the 
fact 
that 
heterogeneity 
and 
disturbances 
in 
the 
geological 
layering 
can 
have 
quite 
diverse 
impacts 
on 
the 
onset 
and 
development 
of 
convection 
cells. 
Local 
differences 
in 
the 
permeability 
can 
cause 
the 
onset 
of 
convection. 
Contrariwise, 
heterogeneity 
can 
enhance 
mixing 
and 
thus 
lead 
to 
the 
breakdown 
of 
convection. 
Classical 
measures 
like 
the 
Rayleigh 
number 
are 
probably 
not 
appropriate 
to 
predict 
the 
onset 
of 
convection 
in 
heterogeneous 
porous 
media. 


The 
importance 
of 
temperature 
as 
source 
for 
density 
instabilities 
that 
can 
create 
free 
convection 
in 
deep 
permeable 
sedimentary 
rocks 
is 
well 
known, 
e.g. 
Elder 
[1967a]; 
Diersch 
and 
Kolditz 
[1998]; 
Nield 
and 
Bejan 
[1999]. 
Magri 
et 
al. 
[2005, 
2008] 
showed 
for 
the 
North 
East 
Ger-
man 
Basin 
that 
salinization 
is 
probably 
caused 
by 
the 
interaction 
of 
hydrostatic 
and 
thermal 
forces. 
The 
simulation 
results 
point 
out 
that 
the 
interaction 
between 
temperature 
induced 
deep 
thermohaline 
convection 
and 
topography 
driven 
flow 
in 
shallow 
aquifers 
can 
transport 
saline 
groundwater 
over 
large 
distances 
into 
near-surface 
regions. 
More 
recent 
studies 
by 
Kaiser 
et 
al. 
[2011] 
suggest 
that 
convection, 
which 
is 
triggered 
by 
temperature 
dependent 
fluid 
density 
variations, 
affects 
the 
flow 
pattern 
of 
the 
North 
East 
German 
Basin 
only 
locally. 



3.2. 
Hydrogeological 
Characterization 
of 
the 
Study 
Area 
From 
Rayleigh 
theory 
it 
is 
known 
that 
the 
development 
of 
convection 
requires 
highly 
permeable 
sedimentary 
units 
with 
large 
vertical 
expansion 
and 
large 
differences 
in 
temperature 
[Nield 
and 
Bejan, 
1999]. 
Comparing 
the 
flat 
and 
shallow 
geological 
setting 
of 
the 
Thuringian 
basin 
with 
the 
much 
deeper 
and 
strongly 
structured 
North 
East 
German 
Basin, 
it 
cannot 
be 
expected 
that 
thermal 
convection 
is 
the 
major 
driving 
force 
within 
the 
Thuringian 
basin. 
It 
is 
rather 
assumed 
that 
regional 
groundwater 
flow 
interacting 
with 
the 
fault 
structure 
is 
the 
major 
force 
driving 
deep 
saline 
water 
to 
shallow 
aquifers. 


To 
give 
numerical 
proof 
of 
the 
hypothesized 
fluid 
dynamics 
of 
the 
Thuringian 
basin, 
we 
performed 
transient 
coupled 
simulations 
of 
hydraulic, 
thermal 
and 
mass 
transport 
processes 
on 
a 
selected 
profile. 
The 
aim 
of 
the 
numerical 
study 
is 
to 
analyze 
the 
major 
mechanisms 
influencing 
the 
flow 
pattern: 
topography 
driven 
flow 
in 
a 
geological 
setting 
with 
faults, 
impact 
of 
hydraulic 
properties 
and 
heterogeneity, 
as 
well 
as 
fluid 
density 
differences 
caused 
by 
temperature 
and 
salt 
concentration 
gradients. 


3.2. 
Hydrogeological 
Characterization 
of 
the 
Study 
Area 
3.2.1. 
Geological 
Overview 
The 
geologically 
closed 
unit 
of 
the 
Thuringian 
Basin 
is 
located 
in 
the 
center 
of 
Germany. 
It 
stretches 
in 
an 
oval 
shape 
approximately 
150km 
from 
northwest 
to 
southeast 
and 
80km 
from 
northeast 
to 
southwest, 
see 
Figure 
12. 
In 
the 
north, 
it 
borders 
on 
the 
Harz 
and 
Kyffhäuser 
mountains, 
in 
the 
west 
on 
the 
Eichsfeld 
Swell, 
and 
in 
the 
south 
on 
the 
Thuringian 
Forest 
mountains 
as 
well 
as 
Thuringian 
shale 
mountains 
[Hoppe, 
1959; 
Jordan 
and 
Weder, 
1995; 
Seidel, 
2003; 
Kober, 
2008]. 


Over 
much 
of 
its 
history 
the 
Thuringian 
Basin 
was 
part 
of 
the 
North 
German 
Basin. 
It 
became 
separated 
in 
the 
Late 
Cretaceous 
by 
the 
nearly 
100km 
long 
Finne 
fault 
zone, 
which 
symbolizes 
the 
morphological 
and 
geological 
northeast 
border 
of 
the 
Thuringian 
Basin 
[Malz 
and 
Kley, 
2012]. 


The 
recent 
tectonic 
structure 
of 
the 
Thuringian 
Basin 
developed 
mainly 
during 
the 
Late 
Cretaceous 
and 
the 
Cenozoic. 
The 
sedimentary 
succession 
of 
the 
Thuringian 
Basin 
started 
with 
the 
Upper 
Permian 
Zechstein. 
Polyphase 
deformations 
of 
the 
basements 
and 
of 
the 
sedimentary 
cover 
have 
created 
the 
complex 
tectonics 
of 
the 
Thuringian 
Basin 
from 
Triassic 
and 
latest 
Cretaceous 
time. 
Several 
northwest-southeast 
striking 
faults 
subdivide 
the 
synclinar 
structure 
[Seidel, 
2003; 
Malz 
and 
Kley, 
2012]. 
Their 
influence 
on 
the 
groundwater 
flow 
pattern 
can 
be 
significant 
on 
a 
local 
up 
to 
a 
regional 
scale. 
Depending 
on 
the 
size, 
location, 
direction 
and 
their 
petrophysical 
characteristics, 
they 
can 
act 
as 
barriers 
or 
preferential 
pathways. 


The 
basin 
is 
filled 
with 
Permian 
and 
Triassic 
sediments, 
while 
the 
overlying 
layers 
of 
Jurassic 
and 
Cretaceous 
age 
are 
mainly 
eroded. 
Tertiary 
and 
Quaternary 
deposits 
are 
restricted 
to 
recent 
river 
valley. 
The 
internal 
structure 
of 
the 
basin 
is 
strongly 
influenced 
by 
the 
Upper 
Permian 
Zechstein 
salt 
deposits. 
In 
the 
center 
of 
the 
basin 
the 
Zechstein 
is 
located 
in 
1000 
m 
depth, 
whereas 
the 
outcrops 
of 
the 
Zechstein 
deposits 
mark 
the 
edge 
of 
the 
basin 
[Seidel, 
2003]. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
sedimentary 
unit 
sedimentary 
subunit 
abbreviation 
thickness 
hydrogeol. 
unit 
Trias 
Keuper 
middle 
km 
0 
- 
220 
m 
shallow 
aquifer 
lower 
ku 
0 
- 
65 
m 
Muschelkalk 
upper 
mo 
45 
- 
75 
m 
middle 
mm 
45 
- 
115 
m 
lower 
mu 
85 
- 
120 
m 
Buntsandstein 
upper 
so 
100 
- 
190 
m 
aquiclude 
middle 
sm 
140 
- 
240 
m 
deep 
aquiferlower 
Bernburg 
suB 
150 
- 
170 
m 
Calvörde 
suC 
170 
- 
190 
m 


Table 
9: 
Diagrammatic 
plan 
of 
the 
sedimentary 
units 
used 
in 
the 
Thuringian 
Basin 
model 
with 
subunits, 
abbreviations, 
and 
thicknesses 
according 
to 
Gaupp 
et 
al. 
[1998] 
and 
Seidel 
[2003]. 
The 
rightmost 
column 
shows 
the 
hydrogeological 
classification 
which 
is 
based 
on 
the 
permeabilities 
of 
the 
sedimentary 
units, 
given 
in 
Table 
10. 


3.2.2. 
Hydrogeological 
Characterization 
of 
Sedimentary 
Units 
We 
reduced 
the 
hydrogeological 
characterization 
of 
the 
Thuringian 
Basin 
to 
the 
sedimentary 
units 
overlaying 
the 
Zechstein 
salt. 
It 
can 
be 
assumed 
that 
the 
thick 
sequence 
of 
Zechstein 
salt 
decouples 
mechanically 
and 
hydraulically 
the 
sub-salt 
Permian 
deposits 
from 
the 
Trias-
sic 
sediments 
[Rödiger, 
2005]. 
A 
diagrammatic 
plan 
with 
thickness 
and 
general 
hydrological 
characterization 
is 
given 
in 
Table 
9. 


The 
dominating 
basin 
filling 
sediments 
are 
the 
Buntsandstein 
(Lower 
Triassic) 
and 
the 
Muschelkalk 
(Middle 
Triassic), 
which 
form 
the 
two 
main 
aquifer 
systems. 
Small 
areas 
of 
Keuper 
deposits 
operate 
as 
permeable 
shallow 
aquifers, 
mainly 
in 
the 
center 
of 
the 
Thuringian 
Basin. 
They 
have 
minor 
influence 
on 
the 
flow 
pattern 
due 
to 
their 
spatially 
limited 
occurrence. 
Younger 
remained 
deposits 
-from 
Tertiary 
and 
Quaternary 
-are 
not 
relevant 
for 
the 
large 
scale 
hydrogeology 
of 
the 
basin. 
To 
allow 
a 
differentiation 
of 
the 
lithology 
dependent 
hydraulic 
and 
thermal 
properties 
the 
Triassic 
units 
Buntsandstein 
and 
Muschelkalk 
were 
further 
divided 
into 
subunits 
(Table 
9). 


Buntsandstein 


The 
Buntsandstein 
sediments 
in 
the 
Thuringian 
Basin 
have 
an 
overall 
thickness 
of 
500- 
720m 
[Puff 
et 
al., 
2003]. 
They 
are 
structured 
into 
Upper, 
Middle 
and 
Lower 
Buntsandstein 
(so, 
sm 
and 
su). 
Due 
to 
significant 
differences 
in 
the 
hydraulic 
properties, 
the 
Lower 
Buntsandstein 
is 
further 
divided 
into 
Bernburg 
formation 
(suB) 
and 
Calvörde 
formation 
(suC), 
see 
Table 
9. 


The 
Röt-formation 
(Upper 
Buntsandstein, 
so) 
contains 
large 
contents 
of 
clay, 
forming 
the 
regional 
most 
important 
aquiclude 
of 
the 
Thuringian 
Basin. 
The 
Röt 
separates 
the 
aquifer 
of 
Lower 
and 
Middle 
Buntsandstein 
from 
the 
Muschelkalk 
aquifer 
system 
[Jordan 
and 
Weder, 
1995; 
Hecht, 
2003; 
Rödiger, 
2005], 
see 
Table 
9. 
Only 
few 
values 
for 
measurements 
of 
permeabil



3.2. 
Hydrogeological 
Characterization 
of 
the 
Study 
Area 
57 
(1) 
[mD] 
(2) 
[mD] 
(3) 
[mD] 
(4) 
[mD] 
km 
1 
000 
---
ku 
1 
000 
---
mo 
50 
- 
10 
000 
10 
- 
100 
--
mm 
50 
- 
10 
000 
10 
- 
10 
000 
--
mu 
50 
- 
10 
000 
20 
- 
200 
-2.5 
so 
--10 
0.025 
- 
2.5 
sm 
100 
- 
300 
3 
- 
3 
000 
100 
- 
5 
000 
25 
- 
2 
500 
suB 
100 
- 
300 
3 
- 
3 
000 
100 
- 
5 
000 
25 
- 
250 
suC 
20 
- 
40 
-100 
- 
5 
000 
2.5 


Table 
10: 
Permeability 
values 
. 
for 
the 
sedimentary 
units 
from 
literature: 
(1)Jordan 
and 
Weder 
[1995], 
(2)Hecht 
[2003] 
and 
Merz 
[1987], 
(3)Rödiger 
[2005], 
and 
(4)Aehnelt 
et 
al. 
[2011]. 
Transmissivities 
Tf 
from 
pumping 
tests 
in 
Jordan 
and 
Weder 
[1995]; 
Hecht 
[2003]; 
Merz 
[1987], 
and 
Rödiger 
[2005] 
are 
converted 
to 
permeability 
by 
. 
= 
Tf 
/(Lg). 
Permeability 
measurements 
by 
Aehnelt 
et 
al. 
[2011] 
from 
bore 
logs 
are 
multiplied 
by 
the 
factor 
25 
to 
account 
for 
effects 
of 
fractures 
according 
to 
Hauthal 
[1967]. 


ities 
can 
be 
found 
in 
literature, 
see 
Table 
10. 
In 
his 
hydrogeological 
model 
of 
the 
Buntsandstein 
aquifer 
system 
in 
the 
Eastern 
Part 
of 
the 
Thuringian 
Basin, 
Rödiger 
[2005] 
used 
a 
conductivity 
of 
Kf 
(so)= 
1e-7m/s. 
Bore-log 
permeability 
measurements 
[Aehnelt 
et 
al., 
2011] 
point 
towards 
much 
smaller 
values 
of 
. 
=0.001 
- 
0.1mD, 
which 
corresponds 
to 
Kf 
= 
1e-11-1e-9m/s. 


The 
facies 
of 
Lower 
and 
Middle 
Buntsandstein 
fluctuate 
between 
fluvial 
sandstones 
and 
lacustrine 
deposits 
[Gaupp 
et 
al., 
1998]. 
The 
units 
of 
the 
Middle 
Buntsandstein 
(sm) 
and 
the 
upper 
part 
of 
the 
Lower 
Buntsandstein, 
the 
Bernburg 
formation 
(suB), 
can 
be 
assumed 
as 
one 
hydraulic 
unit 
[Jordan 
and 
Weder, 
1995; 
Rödiger, 
2005]. 
They 
form 
an 
aquifer 
with 
doubleporosity 
characteristic 
[Jordan 
and 
Weder, 
1995]. 
The 
effects 
of 
the 
flow 
in 
the 
pore 
matrix 
is 
overlaid 
by 
the 
much 
faster 
reacting 
groundwater 
flow 
in 
the 
fissures 
and 
fractures. 
Comparisons 
of 
permeability 
measurements 
and 
pumping 
test 
results 
point 
toward 
a 
20 
- 
30 
times 
higher 
velocity 
in 
the 
fissures 
than 
in 
the 
pore 
matrix 
[Hauthal, 
1967]. 


The 
fissured 
character 
of 
the 
sm/suB-aquifer 
is 
recognizable 
in 
large 
variations 
of 
permeability. 
Hecht 
[2003] 
stated 
a 
spread 
of 
permeabilities 
over 
four 
orders 
of 
magnitudes, 
depending 
on 
their 
position, 
from 
undisturbed 
rock 
formation 
up 
to 
strongly 
shattered 
sediments 
in 
crossings 
of 
fault 
zones. 
Recent 
investigations 
on 
Buntsandstein 
aquifer 
characteristics 
by 
Aehnelt 
et 
al. 
[2011] 
support 
these 
findings. 
In 
general, 
the 
range 
of 
permeability 
is 
very 
large, 
whereas 
values 
tend 
to 
decrease 
with 
depth. 
Typical 
values 
found 
in 
literature 
are 
listed 
in 
Table 
10. 


The 
Calvörde 
formation 
(suC) 
of 
the 
Lower 
Buntsandstein 
has 
a 
mixed 
hydraulic 
character. 
The 
upper 
part, 
connecting 
to 
the 
Bernburg 
formation 
(suB) 
is 
acting 
as 
an 
aquiclude. 
Lower 
units, 
which 
connect 
to 
the 
Zechstein, 
consist 
of 
permeable 
deposits 
with 
aquifer 
characteristics 
[Jordan 
and 
Weder, 
1995; 
Rödiger, 
2005]. 
Thus, 
different 
mean 
values 
for 
the 
permeability 
emerge 
in 
literature, 
see 
Table 
10. 
Permeability 
values 
of 
the 
suC 
are 
assumed 
to 
be 
smaller 
than 
those 
of 
the 
sm/suB-aquifer 
[Aehnelt 
et 
al., 
2011; 
Jordan 
and 
Weder, 
1995]. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Muschelkalk 


The 
Muschelkalk 
is 
marked 
by 
a 
prevailing 
marine 
environment 
and 
is 
subdivided 
into 
three 
units 
which 
reach 
on 
average 
an 
overall 
thickness 
of 
about 
250m 
[Gaupp 
et 
al., 
1998]. 
The 
Lower 
Muschelkalk 
(mu) 
consists 
mainly 
of 
massive 
limestone 
with 
varying 
composition. 
The 
Middle 
Muschelkalk 
(mm) 
deposits 
are 
evaporites, 
such 
as 
dolomit 
marls, 
gypsum 
and 
even 
rock 
salt 
and 
dolomit 
limestone. 
The 
Upper 
Muschelkalk 
(mo) 
is 
characterized 
by 
limestones 
and 
marl 
[Gaupp 
et 
al., 
1998]. 
The 
thicknesses 
of 
the 
subunits 
are 
listed 
in 
Table 
9. 


The 
Muschelkalk 
forms 
a 
connected 
aquifer 
system, 
although 
it 
is 
marked 
by 
an 
alternating 
layering 
of 
aquifers 
and 
aquicludes 
[Hecht, 
2003; 
Merz, 
1987]. 
A 
closed 
thick 
aquiclude 
layer, 
like 
the 
Röt 
formation, 
is 
not 
present 
above 
or 
within 
the 
Muschelkalk. 
Since 
the 
Muschelkalk 
is 
mainly 
formed 
by 
limestone 
sediments, 
it 
is 
a 
Karst 
aquifer 
were 
flow 
takes 
place 
in 
faults 
and 
fissures. 
Depending 
on 
the 
degree 
of 
subrosion, 
large 
variations 
in 
permeability 
can 
be 
observed, 
which 
result 
in 
a 
strong 
heterogeneity 
of 
the 
aquifer 
[Merz, 
1987; 
Hecht, 
2003; 
Rödiger, 
2005]. 
Higher 
permeabilities 
are 
often 
present 
where 
the 
rock 
formation 
shows 
a 
looser 
structure 
due 
to 
tectonic 
faults. 
Especially 
the 
low 
permeable 
limestone 
from 
the 
Muschelkalk 
can 
have 
increased 
permeable 
character 
in 
fault 
zones 
and 
change 
its 
character 
from 
aquiclude 
to 
aquifer 
[Jordan 
and 
Weder, 
1995]. 
Mean 
permeabilities 
from 
literature 
are 
given 
in 
Table 
10. 


3.2.3. 
Hydrodynamic 
The 
hydrodynamic 
of 
the 
Thuringian 
Basin 
is 
dominated 
by 
the 
two 
large 
aquifer 
systems 
of 
the 
Muschelkalk 
and 
Buntsandstein. 
The 
aquiclude 
Röt 
separates 
both 
hydraulic 
units, 
allowing 
only 
little 
communication, 
except 
in 
fault 
zones 
or 
graben 
structures 
[Rödiger, 
2005]. 


For 
the 
Buntsandstein 
areas 
the 
groundwater 
catchments 
coincide 
with 
the 
surface 
water 
catchments, 
giving 
that 
the 
groundwater 
is 
flowing 
mainly 
according 
to 
the 
morphological 
conditions 
[Jordan 
and 
Weder, 
1995]. 
This 
can 
be 
assumed 
for 
the 
Muschelkalk 
aquifer 
system 
as 
well. 
The 
central 
part 
of 
the 
Thuringian 
Basin 
belongs 
to 
the 
Unstrut-catchment, 
whereas 
the 
eastern 
part 
belongs 
to 
the 
Saale-catchment. 


Areas 
of 
groundwater 
recharge 
are 
at 
the 
southwest, 
west 
and 
southeast 
margins 
of 
the 
basin. 
The 
main 
discharge 
takes 
place 
at 
the 
northeast 
margin, 
which 
coincides 
with 
the 
outflow 
direction 
of 
the 
rivers 
Unstrut 
and 
Saale. 
The 
study 
of 
Meincke 
[1967] 
points 
towards 
higher 
flow 
velocities 
at 
the 
recharge 
areas 
than 
in 
the 
basin 
center 
and 
in 
the 
discharge 
areas. 
Even 
a 
stagnating 
flow 
regime 
can 
be 
assumed 
for 
parts 
of 
the 
basin 
center. 
The 
different 
flow 
velocities 
indicate 
the 
existence 
of 
local 
spots 
of 
discharge. 


The 
general 
direction 
of 
groundwater 
flow 
is 
from 
southwest 
to 
northeast 
[Meincke, 
1967]. 
The 
main 
fault 
zones 
striking 
from 
northwest 
to 
southeast 
can 
cause 
significant 
modifications 
of 
the 
flow 
pattern. 
A 
fault 
can 
act 
as 
barrier 
or 
preferential 
pathway 
depending 
on 
size, 
location, 
direction, 
and 
petrophysical 
characteristics. 
Thus, 
faults 
can 
connect 
or 
disconnect 
permeable 
layers 
and 
accordingly 
they 
can 
operate 
as 
areas 
of 
recharge 
or 
discharge 
[Meincke, 
1967; 
Jordan 
and 
Weder, 
1995; 
Hecht, 
2003]. 



3.3. 
Numerical 
Groundwater 
Flow 
Model 
Figure 
13: 
Geological 
structure 
of 
the 
southwest-northeast 
cross 
section 
with 
fault 
zones 
and 
sedimentary 
units: 
km 
-Middle 
Keuper, 
ku 
-Lower 
Keuper, 
mo 
-Upper 
Muschelkalk, 
mm 
-Middle 
Muschelkalk, 
mu 
-Lower 
Muschelkalk, 
so 
-Upper 
Buntsandstein, 
sm 
-Middle 
Buntsandstein, 
suB 
-Lower 
Buntsandstein 
(Bernburg 
formation), 
suC 
-Lower 
Buntsandstein 
(Calvörde 
formation) 
. 
Superelevation 
factor 
of 
10. 


Salinization 
of 
ground 
water 
due 
to 
subrosion 
of 
Zechstein 
salt 
was 
observed 
especially 
in 
the 
Saale-Valley 
[Hecht, 
2003; 
Rödiger, 
2005]. 
The 
effect 
of 
salinization 
is 
particularly 
strong 
in 
fault 
zones. 


3.2.4. 
Temperature 
Distribution 
Hurtig 
and 
Meincke 
[1969] 
investigated 
the 
temperature 
distribution, 
thermal 
conductivities 
and 
heat 
flow 
in 
the 
Thuringian 
Basin. 
They 
showed 
that 
only 
small 
thermal 
anomalies 
are 
present 
in 
the 
Thuringian 
Basin. 
Unusual 
or 
disturbing 
heat 
sources 
are 
most 
likely 
not 
present. 
Observed 
heat 
flow 
anomalies 
are 
due 
to 
the 
relief 
and 
the 
rock 
composition 
of 
the 
basement. 


According 
to 
Meincke 
et 
al. 
[1967], 
inflowing 
meteoric 
fresh 
water 
influences 
the 
temperature 
distributionupto 
400mdepth. 
Especiallyatthebasinmarginsacoolingeffectcanbeobserved. 
The 
thermal 
gradient 
amounts 
in 
average 
to 
30 
C/km. 
Higher 
gradients 
up 
to 
33 
C/km36 
C/km 
were 
observed 
in 
elevated 
regions, 
like 
the 
Eichsfeld 
Swell. 
In 
flat 
areas 
like 
the 
basin 
center 
the 
gradients 
decrease 
to 
values 
of 
21 
C/km-27 
C/km 
[Meincke 
et 
al., 
1967]. 
The 
relatively 
low 
thermal 
gradients 
correspond 
to 
the 
fact 
that 
thermal 
water 
springs 
are 
not 
known 
in 
the 
Thuringian 
Basin 
[Hecht, 
2003]. 


3.3. 
Numerical 
Groundwater 
Flow 
Model 
3.3.1. 
Cross 
Section 
Model 
We 
performed 
simulations 
on 
a 
southwest-northwest 
cross 
section 
of 
approximately 
70km 
length 
and 
maximal 
1000 
m 
depth, 
shown 
in 
Figure 
13. 
The 
profile 
ranges 
from 
the 
northern 
rim 
of 
the 
Thuringian 
Forest 
in 
the 
southeast, 
cutting 
through 
the 
region 
of 
Erfurt 
in 
the 
basin 
center, 
to 
the 
Finne 
fault 
in 
the 
northeast. 
The 
location 
of 
the 
cross 
section 
in 
the 
Thuringian 
Basin 
is 
depicted 
in 
Figure 
12 
as 
line 
A-A’. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
14: 
Detail 
of 
the 
numerical 
mesh 
in 
the 
environment 
of 
the 
Eichenberg-Gotha-Saalfeld 


fault 
zone. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Without 


superelevation. 


We 
chose 
the 
transect 
perpendicular 
to 
the 
northwest-southeast-striking 
faults. 
The 
choice 
is 
based 
on 
the 
one 
hand, 
on 
the 
general 
flow 
direction 
from 
southwest 
to 
northeast. 
On 
the 
other 
hand, 
we 
aim 
to 
investigate 
the 
effects 
faults 
have 
on 
the 
flow 
pattern, 
by 
incorporating 
the 
main 
fault 
structures, 
as 
given 
in 
Figure 
13. 


The 
cross 
section 
model 
is 
based 
on 
the 
basin 
stratigraphy 
as 
described 
in 
Jordan 
and 
Weder 
[1995]; 
Gaupp 
et 
al. 
[1998]; 
Seidel 
[2003]; 
Rödiger 
[2005]. 
It 
contains 
the 
Triassic 
sediments 
of 
the 
Buntsandstein, 
Muschelkalk 
and 
Keuper 
as 
discussed 
in 
detail 
in 
section 
3.2. 
The 
Zechstein 
deposits 
form 
a 
natural 
boundary 
in 
depth 
for 
the 
study 
area 
due 
to 
their 
impermeability 
to 
fluid 
flow. 
We 
did 
not 
incorporate 
the 
Zechstein 
layer 
into 
the 
cross 
section 
model. 
However, 
at 
the 
bottom 
of 
the 
basin, 
where 
the 
Triassic 
sediments 
are 
in 
contact 
with 
the 
Zechstein, 
dissolved 
salt 
can 
be 
transported 
by 
groundwater 
flow. 


We 
developed 
a 
work 
flow 
to 
gain 
a 
spatially 
discretized 
mesh, 
which 
is 
appropriate 
for 
numerical 
transport 
simulations 
from 
geostratigraphical 
input 
data. 
We 
generated 
a 
balanced 
crosssectionsmodelusingthesoftware2D-
MoveonthebasisofgeologicalsurfacemapsGK 
1 
: 
25000 
of 
Thuringia 
with 
stratigraphical 
information 
from 
a 
digital 
underground 
model 
of 
Kober 
[2008]. 
We 
incorporated 
faults 
structures 
according 
to 
Malz 
and 
Kley 
[2012]. 
For 
generating 
the 
finite 
element 
mesh, 
we 
transformed 
and 
loaded 
the 
profile 
into 
the 
software 
Gocad. 
Subsequently 
we 
used 
Gocad 
and 
the 
interfaces 
described 
in 
Zehner 
[2011] 
to 
generate 
the 
numerical 
grid. 
A 
part 
of 
the 
numerical 
mesh 
in 
the 
environment 
of 
the 
Eichenberg-Gotha-Saalfeld 
fault 
zone 
is 
visualized 
in 
Figure 
14. 
The 
unstructured 
finite 
element 
mesh 
consists 
of 
121983 
triangular 
elements. 
The 
average 
side 
length 
of 
a 
triangle 
is 
35-40m. 
Themaximumsidelengthamounts 
to 
50m. 
Locally 
refined 
triangles 
can 
have 
a 
side 
length 
of 
about 
5m. 
The 
mesh 
satisfies 
the 
regularity 
conditions 
and 
the 
resolution 
allows 
to 
model 
variations 
in 
fluid 
density. 
We 
assume 
mesh 
convergence 
due 
to 
the 
fact 
that 
additional 
simulations, 
which 
we 
performed 
on 
finer 
meshes 
give 
the 
same 
results 
for 
all 
simulation 
scenarios. 


We 
incorporated 
discontinuities 
in 
the 
layering, 
which 
result 
from 
major 
fault 
structures, 
to 
the 
mesh. 
This 
results 
in 
a 
natural 
break 
up 
of 
the 
horizontal 
stratification 
into 
compartments 
of 
flat, 
undisturbed 
basin 
fill, 
interrupted 
by 
faults 
zones 
of 
regional 
importance 
(Figure 
13). 
Partly 
the 
fault 
zones 
have 
a 
horizontal 
extension 
(Eichenberg-Gotha-Saalfeld 
fault 
zone, 
Erfurter 
fault 
zone 
and 
Finne 
fault 
zones), 
which 
allows 
for 
a 
separate 
assignment 
of 
parameters. 



3.3. 
Numerical 
Groundwater 
Flow 
Model 
3.3.2. 
Processes 
and 
Numerical 
Framework 
We 
carried 
out 
all 
simulations 
using 
the 
software 
OpenGeoSys. 
This 
finite 
element 
based 
simulator 
provides 
algorithms 
for 
solving 
coupled 
fluid 
flow, 
mass 
and 
heat 
transport 
processes 
in 
porous 
media. 
The 
software 
was 
successfully 
tested 
against 
a 
wide 
range 
of 
benchmarks 
for 
fluid, 
heat 
and 
mass 
transport 
by 
Kolditz 
et 
al. 
[2012a], 
and 
for 
density 
dependent 
flow 
in 
particular 
in 
section 
2.4 
and 
2.5. 


Three 
processes 
have 
been 
established 
on 
the 
cross 
section 
model: 
a 
hydraulic 
process 
calculates 
the 
groundwater 
flow; 
a 
mass 
transport 
process 
gives 
the 
distribution 
of 
salt; 
and 
a 
heat 
transport 
process 
calculates 
the 
temperature 
distribution. 
Several 
partial 
differential 
equations 
describe 
these 
processes 
of 
flow 
and 
transport 
in 
saturated 
porous 
media: 
Darcy’s 
law 
given 
in 
Eq.(1.2), 
the 
continuity 
equation 
of 
flow 
(1.1), 
the 
equation 
of 
solute 
mass 
conservation 
(1.5) 
as 
well 
as 
the 
energy 
balance 
equation 
of 
the 
fluid 
and 
the 
porous 
media 
(1.8). 
The 
resulting 
system 
is 
fully 
implemented 
in 
OpenGeoSys. 


We 
conducted 
several 
simulation 
scenarios 
to 
investigate 
the 
impact 
of 
two 
main 
transport 
mechanisms. 
First, 
we 
performed 
simulations 
with 
different 
mean 
hydraulic 
permeability 
distributions. 
The 
parametrization 
of 
these 
scenarios 
is 
given 
in 
section 
3.3.4. 
Thereby 
we 
used 
a 
constant 
density 
model 
of 
conservative 
mass 
and 
heat 
transport. 
Second, 
we 
investigated 
the 
role 
of 
density 
dependent 
flow. 
This 
required 
a 
coupling 
of 
all 
equations 
through 
a 
mass 
and 
temperature 
dependent 
fluid 
density 
model. 
In 
OpenGeoSys 
a 
linear 
relation 
for 
mass 
and 
temperature 
dependent 
fluid 
density 
is 
implemented, 
according 
to 
Eq.(1.10). 


3.3.3. 
Boundary 
and 
Initial 
Conditions 
We 
set 
the 
fluid 
pressure 
at 
the 
surface 
boundary 
to 
a 
constant 
value 
of 
zero 
to 
establish 
a 
regional 
groundwater 
flow. 
This 
corresponds 
to 
a 
hydraulic 
head 
following 
the 
local 
topographical 
elevation. 
It 
induces 
a 
steady 
regional 
flow 
with 
recharge 
areas 
in 
the 
highlands 
and 
discharge 
areas 
in 
the 
lowlands. 


We 
implemented 
a 
mass 
transport 
in 
all 
simulations 
by 
fixing 
a 
constant 
salt 
concentration 
at 
the 
bottom 
of 
the 
basin 
model, 
where 
the 
Buntsandstein 
is 
in 
contact 
with 
the 
Zechstein 
salt. 
We 
fixed 
a 
zero 
mass 
concentration 
at 
the 
surface, 
representing 
the 
recharge 
with 
meteoric 
fresh 
water. 
Although 
this 
condition 
does 
not 
allow 
saline 
water 
to 
enter 
the 
surface, 
it 
is 
sufficient 
to 
localize 
zones 
where 
deep 
groundwater 
rises 
to 
near-surface 
levels. 


We 
applied 
a 
constant 
surface 
temperature 
of 
8 
C 
for 
the 
heat 
transport 
process. 
It 
corresponds 
to 
the 
current 
average 
annual 
temperature 
for 
the 
area 
under 
consideration. 
We 
fixed 
the 
temperature 
values 
at 
the 
lower 
boundaries 
corresponding 
to 
a 
linear 
vertical 
gradient 
of 
30 
C/km 
[Meincke 
et 
al., 
1967]. 
Thus, 
the 
temperature 
at 
the 
bottom 
of 
the 
basin 
varies 
between 
30 
C 
and 
38.4 
C, 
depending 
on 
the 
depth 
and 
distance 
to 
the 
surface. 


We 
defined 
the 
lateral 
boundaries 
impermeable 
for 
fluid, 
heat 
and 
mass 
transport. 
The 
initial 
conditions 
are 
given 
by 
a 
zero 
pressure 
and 
pure 
fresh 
water 
distribution 
in 
the 
whole 
simulation 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
symbol 
[unit] 
quantity 
value 


C 
- 
normalized 
salt 
concentration 
[0, 
1] 
T 
[C] 
temperature 
[8, 
38.4] 
T 
K-1 
thermal 
expansion 
coefficient 
-0.0003 
C 
- 
solute 
expansion 
coefficient 
0.2 
Dm 
[m2 
s-1] 
molecular 
diffusion 
coefficient 
1e-8 
l, 
t 
[m] 
longitudinal 
and 
transversal 
dispersion 
20; 
2 
0 
[kgm-3] 
density 
of 
fresh 
water 
1 
000 


. 
[kg 
m-1 
s-1] 
viscosity 
0.001 
w 
[kgm2 
s-3 
K-1] 
thermal 
conductivity 
of 
water 
0.6 
cw 
[m2 
s-2 
K-1] 
thermal 
capacity 
of 
water 
4 
200 
cs 
[m2 
s-2 
K-1] 
thermal 
capacity 
of 
solid 
rock 
1 
000 


Table 
11: 
Simulation 
parameters 
for 
the 
entire 
model 
domain. 


domain. 
The 
initial 
temperature 
was 
8C. 
We 
performed 
all 
simulations 
with 
a 
computing 
time 
of 
200000 
years 
to 
assure 
that 
the 
system 
reaches 
steady 
state. 


3.3.4. 
Parametrization 
Table 
11 
contains 
all 
model 
parameters 
for 
fluid, 
mass 
and 
heat 
transport 
processes, 
which 
are 
valid 
for 
the 
entire 
cross 
section 
model. 
Physical 
properties, 
which 
can 
be 
distinguished 
for 
the 
stratigraphical 
units, 
are 
listed 
in 
Table 
12. 
We 
grouped 
the 
nine 
sedimentary 
layers 
to 
three 
main 
hydraulic 
units: 
a 
shallow 
aquifer 
consisting 
of 
Keuper 
and 
Muschelkalk 
(km/ 
ku/mo/mm/mu), 
the 
aquiclude 
unit 
Röt 
(so), 
and 
the 
deep 
aquifer 
formed 
by 
sediments 
from 
Middle 
and 
Lower 
Buntsandstein 
(sm/suB/suC), 
see 
also 
Table 
9. 
Reasonable 
ranges 
and 
representative 
mean 
values 
of 
permeability 
are 
given 
in 
Table 
12 
for 
every 
sedimentary 
layer, 
based 
on 
the 
findings 
from 
literature 
as 
discussed 
in 
section 
3.2 
(Table 
10). 
We 
considered 
each 
layer 
homogeneous 
and 
isotropic 
in 
density, 
thermal 
properties 
and 
porosity. 


We 
applied 
three 
reasonable 
mean 
permeabilities 
for 
every 
sedimentary 
layer: 
a 
lower 
(l), 
a 
reference 
(r) 
and 
a 
higher 
(h) 
permeability, 
given 
in 
Table 
12. 
Having 
three 
permeability 
distributions 
for 
every 
of 
the 
three 
hydraulic 
units 
(shallow 
aquifer, 
aquiclude 
and 
deep 
aquifer) 
results 
in 
27 
possible 
simulation 
scenarios 
with 
different 
mean 
permeabilities. 
Additionally, 
we 
performed 
simulations 
with 
varied 
permeability 
values 
within 
the 
fault 
zones 
(EichenbergGotha-
Saalfeld 
fault 
zone, 
Erfurter 
fault 
zone 
and 
Finne 
fault 
zone). 
While 
we 
kept 
the 
permeabilities 
in 
the 
undisturbed 
basin 
units 
constant 
at 
the 
reference 
value 
(r), 
we 
varied 
the 
permeability 
values 
within 
the 
faults 
over 
several 
orders 
of 
magnitude 
by 
multiplying 
with 
the 
factors 
10, 
100, 
0.1 
and 
0.01. 
This 
is 
in 
line 
with 
findings 
of 
permeability 
values 
in 
fault 
zone 
in 
the 
Thuringian 
basin 
[Hecht, 
2003; 
Aehnelt 
et 
al., 
2011]. 


We 
generated 
ensembles 
of 
heterogeneous 
permeability 
distributions 
(x) 
according 
to 
a 
log 
normal 
distribution 
(x) 
. 
LN 
(µ, 
2), 
as 
described 
in 
section 
1.3.2. 
The 
statistical 
distribution 
is 
characterized 
by 
the 
geometric 
mean 
. 
= 
exp 
µ 
and 
variance 
2. 
The 
investigation 



3.3. 
Numerical 
Groundwater 
Flow 
Model 
layer 
L 
s 
s 
f 
range 
of 
. 
(l) 
(r) 
(h) 
km 
0 
- 
45 
2 
600 
2.0 
0.30 
100 
- 
1 
000 
100 
100 
1 
000 
ku 
0 
- 
55 
2 
600 
2.0 
0.25 
100 
- 
1 
000 
100 
100 
1 
000 
mo 
80 
2 
800 
2.0 
0.20 
5 
- 
200 
5 
20 
200 
mm 
50 
2 
800 
2.0 
0.20 
20 
- 
1 
000 
20 
100 
1 
000 
mu 
100 
2 
800 
2.0 
0.20 
5 
- 
200 
5 
20 
200 
so 
100 
2 
400 
2.7 
0.05 
0.02 
- 
2 
0.02 
0.2 
2 
sm 
240 
2 
400 
2.1 
0.15 
30 
- 
3 
000 
30 
300 
3 
000 
suB 
160 
2 
400 
2.1 
0.13 
30 
- 
300 
30 
100 
300 
suC 
185 
2 
400 
1.7 
0.08 
2 
- 
100 
2 
20 
100 


Table 
12: 
Simulation 
parameters 
for 
sedimentary 
layers: 
mean 
thickness 
L 
[m], 
rock 
density 
s 
[kg 
m-3], 
thermal 
conductivity 
s 
[kgm2 
s-3 
K-1], 
porosity 
f 
[-], 
and 
permeability 
. 
[mD] 
(=[1e-15m2]). 
The 
ranges 
of 
. 
correspond 
to 
results 
found 
in 
the 
literature 
(Table 
10.) 
(l), 
(r), 
and 
(h) 
are 
the 
three 
mean 
permeability 
values 
for 
every 
sedimentary 
layer, 
with 
abbreviations 
(l) 
for 
low, 
(r) 
for 
reference, 
and 
(h) 
for 
high. 


included 
three 
degrees 
of 
heterogeneity: 
mildly 
heterogeneous 
media, 
expressed 
by 
a 
variances 
of 
2 
=0.2, 
medium 
heterogeneous 
media 
with 
2 
= 
1 
and 
strongly 
heterogeneous 
media 
with 
2 
= 
4. 
We 
used 
two 
different 
spatial 
correlation 
structure 
models. 
First, 
we 
applied 
a 
Gaussian 
covariance 
model, 
accordingtoEq.(1.18), 
characterizedby 
a 
horizontal 
and 
a 
vertical 
correlation 
length 
`h 
and 
`v. 
We 
generated 
heterogeneous 
fields 
with 
correlation 
lengths 
over 
four 
orders 
of 
magnitude: 
from 
local 
scale 
`h 
= 
20 
m 
(`v 
= 
2 
m) 
corresponding 
to 
the 
size 
of 
a 
grid 
cell, 
over 
regional 
scale 
`h 
= 
200 
m 
(`v 
= 
20 
m) 
and 
`h 
= 
2000 
m 
(`v 
= 
50 
m) 
up 
to 
basin 
scale 
`h 
= 
20000 
m 
(`v 
= 
50 
m) 
which 
corresponds 
to 
the 
maximal 
size 
of 
an 
undisturbed 
basin 
compartment. 
Second, 
we 
used 
the 
truncated 
power 
law 
covariance 
model, 
according 
to 
Eq.(1.19). 
In 
this 
model 
there 
is 
no 
finite 
correlation 
length, 
but 
correlation 
exists 
at 
any 
scale. 
We 
used 
an 
upper 
cut-off 
value 
of 
`max 
= 
25000 
m 
(`max 
= 
50 
m) 
which 
is 
above 
the 


hv 


maximal 
size 
of 
an 
undisturbed 
basin 
compartment 
in 
the 
Thuringian 
Basin. 
Every 
ensemble 
consists 
of 
200 
realizations 
with 
identical 
choice 
of 
statistical 
parameters. 


We 
performed 
simulations 
to 
investigate 
density 
dependent 
flow, 
where 
the 
fluid 
density 
can 
change 
due 
to 
dissolved 
salt 
or 
temperature 
variations. 
We 
applied 
a 
linear 
density 
model, 
according 
to 
Eq.(1.10): 
. 
= 
0(1 
+ 
C 

C 
+ 
T 

T 
). 
We 
set 
the 
solute 
expansion 
coefficient 
to 
C 
=0.2, 
which 
refers 
to 
a 
maximal 
saturated 
brine 
density 
of 
C 
= 
1200 
kg/m3 
for 
a 
normalized 
salt 
concentration 
C 
. 
[0, 
1]. 
The 
thermal 
expansion 
coefficient 
is 
T 
= 
-0.0003 
corresponding 
to 
minimal 
density 
of 
T 
= 
991 
kg/m3 
for 
the 
maximal 
temperature 
of 
38 
C 
and 
a 
fresh 
water 
density 
of 
0 
= 
1000 
kg/m3 
for 
8 
C 
(Table 
11). 


We 
made 
some 
simplifying 
assumptions. 
We 
assumed 
the 
brine 
to 
be 
pure 
NaCl 
solution 
and 
neglected 
viscosity 
variations 
due 
to 
temperature 
and 
salt 
concentration. 
Furthermore, 
the 
simulation 
results 
cannot 
reproduce 
the 
real 
groundwater 
flow, 
which 
is 
naturally 
three 
dimensional. 
Nevertheless, 
the 
two 
dimensional 
approach 
allowed 
to 
investigate 
brine 
migration 
in 
shallow 
sedimentary 
basins 
and 
to 
gain 
insight 
into 
the 
role 
of 
geological 
fault 
structures, 
hydraulic 
parameters 
and 
fluid 
density 
variation 
on 
the 
flow, 
salt 
and 
temperature 
pattern. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
3.4. 
Discussion 
of 
Simulation 
Results 
In 
the 
following 
we 
discuss, 
compare, 
and 
interpret 
the 
results 
for 
velocity 
distribution, 
salt 
concentration, 
and 
temperature 
pattern 
of 
different 
simulation 
scenarios.The 
results 
of 
the 
reference 
simulation 
serve 
as 
basis 
for 
comparison 
with 
all 
other 
simulation 
scenarios. 
We 
then 
present 
the 
results 
for 
changes 
in 
the 
mean 
permeability 
of 
different 
sedimentary 
layers 
and 
in 
the 
fault 
zones. 
It 
is 
followed 
by 
an 
interpretation 
of 
the 
simulation 
results 
with 
heterogeneous 
permeability 
distributions 
and 
an 
analysis 
of 
the 
impact 
of 
the 
statistical 
quantities. 
The 
final 
step 
involves 
the 
discussion 
of 
simulation 
results 
for 
density 
dependent 
flow. 


Besides 
visual 
comparison, 
we 
use 
quantitative 
characteristics 
to 
support 
interpretation 
results. 


¯

For 
every 
simulation 
we 
computed 
the 
mean 
of 
the 
salt 
concentration 
C 
. 
[0, 
1], 
as 
given 
in 
Eq.(1.12) 
which 
gives 
a 
relative 
value 
of 
the 
total 
amount 
of 
dissolved 
salt 
in 
in 
the 
entire 
domain. 
For 
every 
simulation 
scenario, 
we 
calculated 
the 
relative 
concentration 
difference 

C 
[%] 
and 
the 
relative 
temperature 
difference 

T 
[%] 
to 
the 
reference 
simulation, 
as 
given 
in 
Eq.(1.13). 
Solute 
and 
thermal 
Rayleigh 
numbers 
RaS 
and 
RaT 
are 
calculated 
according 
to 
Eq.(1.11) 
for 
density 
dependent 
flow. 


3.4.1. 
Reference 
Simulation 
The 
initial 
point 
for 
all 
comparisons 
of 
different 
simulation 
scenarios 
is 
the 
reference 
simulation. 
Boundary 
and 
initial 
conditions 
as 
well 
as 
domain 
parameters 
are 
defined 
in 
section 
3.3.3 
and 
section 
3.3.4. 
We 
assumed 
the 
permeability 
to 
be 
homogeneously 
distributed 
with 
mean 
values 
of 
all 
layers 
according 
to 
the 
reference 
values 
(r), 
defined 
in 
Table 
12. 
The 
transport 
of 
salt 
and 
heat 
is 
conservative, 
meaning 
that 
a 
constant 
fluid 
density 
model 
is 
used, 
where 
salt 
and 
temperature 
do 
not 
impact 
on 
the 
flow 
pattern. 


Simulation 
results 
of 
velocity, 
normalized 
salt 
concentration 
distribution 
and 
the 
temperature 
distribution 
after 
100000 
years 
are 
visualized 
in 
Figure 
15. 
We 
assume 
that 
these 
results 
represent 
the 
steady 
state 
solution 
because 
for 
all 
processes 
the 
changes 
to 
subsequent 
simulation 
time 
steps 
are 
less 
than 
0.05%. 
The 
time 
needed 
to 
establish 
the 
regional 
flow 
pattern 
from 
the 
initial 
conditions 
can 
be 
neglected 
compared 
to 
the 
time 
needed 
to 
reach 
a 
steady 
state 
in 
temperature 
and 
salt 
concentration. 
The 
temperature 
distribution 
reaches 
steady 
state 
after 
30000 
years, 
whereas 
the 
salt 
requires 
approximately 
100000 
years 
to 
reach 
a 
stadium 
of 
stable 
salt 
stratification. 
The 
faster 
development 
of 
thermal 
steady 
state 
is 
due 
to 
the 
higher 
diffusivity 
of 
the 
heat 
transport 
process. 


The 
topography 
determines 
the 
flow 
pattern 
(Figure 
15a), 
according 
to 
the 
choice 
of 
the 
constant 
hydraulic 
head 
flow 
boundary 
condition 
at 
the 
basin 
surface. 
Flow 
takes 
place 
from 
elevated 
regions 
to 
lower 
basin 
parts. 
The 
main 
area 
of 
recharge 
is 
the 
southwest 
basin 
margin 
at 
the 
Thuringian 
forest 
border 
fault. 
Discharge 
takes 
place 
locally, 
especially 
in 
flat 
basin 
regions 
and 
near 
the 
fault 
zones. 
We 
observe 
high 
velocities 
in 
the 
shallow 
Keuper 
layer 
(km/ku) 
and 
the 
Middle 
Buntsandstein 
(sm) 
aquifer 
due 
to 
the 
high 
permeabilities. 
We 
find 
very 
low 
velocity 
in 
the 
Röt 
aquiclude 
(so) 
and 
in 
the 
Lower 
Buntsandstein 
layers 
(suB/suC). 
Especially 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
15: 
Simulation 
results 
for 
reference 
simulation: 
(a) 
Velocity 
distribution 
with 
arrows 
marking 
the 
flow 
direction, 
(b) 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
and 
(c) 
Temperature 
distribution 
with 
isolines 
at 
distance 
of 
2C. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


in 
the 
flat 
basin 
center 
flow 
is 
nearly 
stagnant, 
which 
corresponds 
to 
the 
findings 
of 
Meincke 
[1967]. 
The 
fault 
geometry 
has 
distinct 
effects 
on 
the 
flow 
pattern. 
Where 
the 
EichenbergGotha-
Saalfelder 
fault 
zone 
and 
the 
Erfurter 
fault 
zone 
serve 
as 
transit 
areas 
for 
flow, 
the 
Sömmerdaer 
fault 
has 
a 
barrier 
effect 
on 
the 
flow 
causing 
the 
groundwater 
to 
rise. 


The 
salt 
concentration 
pattern 
(Figure 
15b) 
corresponds 
to 
the 
distribution 
of 
velocities. 
The 
choice 
of 
permeabilities 
allow 
the 
development 
of 
large 
scale 
groundwater 
advection 
cell. 
They 
reach 
the 
deep 
aquifer 
units 
of 
the 
Buntsandstein 
and 
transport 
saline 
groundwater 
to 
nearsurface 
regions. 
Steep 
plumes 
of 
salt 
water 
develop 
in 
the 
two 
central 
undisturbed 
basin 
compartments 
and 
one 
in 
the 
vicinity 
of 
the 
Sömmerdaer 
fault. 
The 
mean 
concentration 
in 
the 


¯¯

entire 
domain 
is 
C(k/m/s) 
=0.159, 
where 
the 
amount 
of 
salt 
in 
the 
shallow 
aquifer 
C(k/m) 
=0.016 
is 
very 
low 
compared 
to 
the 
amount 
in 
the 
deep 
aquifer 
C¯(sm/su) 
=0.224 
(Table 
13). 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
(k/m) 
(so) 
(sm/su) 
¯ 
C(k/m/s) 
¯ 
C(k/m) 
¯ 
C(so) 
¯ 
C(sm/su) 

C 
[%] 
r 
r 
r 
0.157 
0.014 
0.030 
0.224 
0 
h 
r 
r 
0.154 
0.004 
0.025 
0.221 
2.66 
l 
r 
r 
0.171 
0.022 
0.028 
0.239 
3.73 
r 
h 
r 
0.133 
0.010 
0.013 
0.190 
6.16 
r 
l 
r 
0.194 
0.012 
0.050 
0.274 
7.88 
r 
r 
h 
0.189 
0.023 
0.031 
0.266 
7.82 
r 
r 
l 
0.142 
0.009 
0.022 
0.202 
5.22 
r 
h 
h 
0.163 
0.014 
0.013 
0.231 
4.44 
r 
h 
l 
0.124 
0.007 
0.012 
0.177 
6.44 
r 
l 
h 
0.194 
0.016 
0.050 
0.273 
8.73 
r 
l 
l 
0.179 
0.008 
0.046 
0.254 
4.70 
h 
r 
h 
0.172 
0.006 
0.023 
0.247 
6.56 
h 
r 
l 
0.140 
0.002 
0.019 
0.203 
5.16 
l 
r 
h 
0.174 
0.028 
0.027 
0.257 
7.51 
l 
r 
l 
0.143 
0.014 
0.023 
0.202 
5.26 
h 
h 
r 
0.133 
0.005 
0.012 
0.191 
6.05 
l 
h 
r 
0.139 
0.013 
0.015 
0.197 
5.92 
h 
l 
r 
0.175 
0.002 
0.033 
0.266 
7.35 
l 
l 
r 
0.218 
0.032 
0.075 
0.299 
10.03 
h 
h 
h 
0.149 
0.007 
0.013 
0.215 
3.31 
h 
h 
l 
0.122 
0.002 
0.001 
0.177 
6.44 
h 
l 
h 
0.172 
0.002 
0.026 
0.248 
7.16 
h 
l 
l 
0.178 
0.001 
0.037 
0.256 
5.22 
l 
h 
h 
0.165 
0.019 
0.016 
0.233 
4.55 
l 
h 
l 
0.126 
0.011 
0.013 
0.180 
6.36 
l 
l 
h 
0.216 
0.044 
0.073 
0.292 
10.80 
l 
l 
l 
0.178 
0.018 
0.054 
0.262 
5.33 


¯

Table 
13: 
Mean 
concentration 
C, 
as 
defined 
in 
Eq.(1.12) 
and 
relative 
concentration 
difference 

C 
to 
the 
reference 
simulation, 
as 
defined 
in 
Eq.(1.13) 
for 
combinations 
of 
homogeneous 
permeabilities 
of 
the 
three 
hydraulic 
units 
shallow 
aquifer 
Keuper/Muschelkalk 
(k/m), 
aquiclude 
Röt 
(so) 
and 
deep 
aquifer 
Middle 
and 
Lower 
Buntsandstein 
(sm/su). 
The 
first 
three 
columns 
denote 
the 
choice 
of 
permeability 
: 
l=low, 
r=reference, 
h=high, 
according 
to 
the 
values 
given 
in 
Ta


¯ 
¯¯

ble 
12; 
C(k/m/s) 
is 
the 
mean 
concentration 
in 
the 
entire 
domain; 
C(k/m), 
C(so), 
and 
C¯(sm/su) 
are 
the 
mean 
concentrations 
in 
the 
corresponding 
hydraulic 
units. 



3.4. 
Discussion 
of 
Simulation 
Results 
The 
distribution 
of 
temperature 
(Figure 
15c) 
develops 
according 
to 
the 
thermal 
gradient 
of 
30 
Cper 
1kmdepth: 
from 
8 
C 
at 
the 
surface 
to 
maximal 
38.4 
C 
at 
the 
bottom 
of 
the 
basin. 
The 
stable 
temperature 
stratification, 
which 
develops 
by 
conductive 
heat 
transport 
trough 
the 
solid 
and 
fluid 
phase, 
is 
disturbed 
by 
regional 
groundwater 
flow. 
Surface-near 
regions 
of 
lower 
and 
higher 
temperatures 
coincide 
with 
the 
areas 
of 
recharge 
and 
discharge: 
cold 
meteoric 
fresh 
water 
enters 
the 
system 
and 
is 
transported 
with 
the 
regional 
groundwater 
flow 
into 
deeper 
aquifer 
units. 
This 
can 
be 
observed 
especially 
in 
the 
environment 
of 
fault 
zones. 
In 
discharge 
areas 
the 
warmer 
groundwater 
from 
deeper 
regions 
is 
transported 
to 
the 
surface. 
This 
can 
be 
observed 
mainly 
in 
the 
flat 
undisturbed 
basin 
units, 
where 
also 
salt 
water 
plumes 
develop. 
Thus, 
conduction 
is 
the 
main 
transport 
mechanism 
in 
deep 
aquifer 
units, 
which 
is 
visible 
at 
the 
flat 
isolines. 
Whereas 
advection 
dominates 
the 
temperature 
distribution 
in 
the 
shallow 
aquifer 
units. 


3.4.2. 
Scenarios 
of 
Homogeneous 
Permeability 
The 
combination 
of 
the 
three 
choices 
of 
permeability 
, 
as 
listed 
in 
Table 
12, 
for 
every 
of 
the 
three 
hydraulic 
units 
results 
in 
27 
possible 
scenarios 
of 
mean 
permeability 
distribution. 
In 
Table 
13 
we 
give 
the 
calculated 
characteristic 
numbers 
of 
mean 
concentration 
C 
¯ 
in 
the 
entire 
domain 
(k/m/s) 
and 
the 
hydraulic 
units 
Keuper/Muschelkalk 
(k/m), 
Röt 
(so) 
and 
Middle 
and 
Lower 
Buntsandstein 
(sm/su) 
as 
well 
as 
the 
relative 
difference 
to 
the 
reference 
simulation 

C 
for 
all 
simulations. 
To 
distinguish 
between 
the 
simulation 
scenarios, 
we 
use 
subscripts 


¯

symbolizing 
the 
choice 
of 
. 
in 
the 
corresponding 
hydraulic 
unit; 
for 
instance 
Clrh 
describes 
the 
mean 
concentration 
for 
the 
scenario 
with 
low 
permeability 
in 
the 
shallow 
aquifer 
Keuper/
Muschelkalk 
(k/m)= 
l, 
reference 
permeability 
in 
the 
aquiclude 
Röt 
(so)= 
r 
and 
high 
permeability 
in 
the 
deep 
aquifer 
Middle/Lower 
Buntsandstein 
(sm/su)= 
h. 


The 
discussion 
and 
visualization 
of 
simulation 
results 
focus 
on 
the 
salt 
distribution 
of 
scenarios 
where 
permeabilities 
of 
only 
one 
hydraulic 
unit 
were 
modified 
compared 
to 
the 
reference 
simulation. 
For 
brevity, 
simulation 
results 
for 
the 
flow 
field 
are 
not 
visualized 
separately, 
since 
it 
can 
be 
inferred 
from 
the 
salt 
concentration 
distribution. 
Furthermore, 
the 
temperature 
distribution 
is 
not 
visualized 
and 
discussed, 
because 
changes 
to 
the 
reference 
temperature 
distribution 
are 
small. 


Variation 
of 
Permeability 
in 
the 
Shallow 
Aquifer 


The 
simulated 
concentration 
distribution 
for 
the 
scenarios 
with 
increased 
and 
decreased 
permeabilities 
in 
the 
Keuper 
and 
Muschelkalk 
layers 
are 
visualized 
in 
Figure 
16. 
The 
overall 
flow 
pattern 
only 
changes 
little 
compared 
to 
the 
reference 
simulation. 
Number 
and 
location 
of 
salt 
plumes 
are 
similar, 
which 
is 
emphasized 
by 
the 
small 
concentration 
difference 
to 
the 
reference 
simulation 
of 

Chrr 
=2.66% 
and 

Clrr 
=3.73% 
(Table 
13). 


We 
observe 
differences 
in 
the 
mean 
concentration 
of 
the 
entire 
domain 
which 
is 
higher 
for 
lower 
permeabilities 
and 
vice 
versa: 
C¯ 
lrr(k/m/s) 
= 
0.171 
and 
C¯ 
hrr(k/m/s) 
= 
0.154 
compared 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
16: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
(a) 
increased 
and 
(b) 
decreased 
permeability 
in 
shallow 
aquifer 
Keuper/Muschelkalk 
(k/m). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


to 
C¯ 
rrr(k/m/s) 
=0.159. 
Significant 
differences 
occur 
in 
the 
shallow 
aquifer 
units 
Keuper 
and 
Muschelkalk 
(Table 
13). 
The 
spreading 
of 
salt 
within 
the 
Muschelkalk 
is 
apparently 
damped 
for 
higher 
permeabilities. 
This 
effect 
can 
be 
explained 
by 
deeper 
intruding 
fresh 
water 
due 
to 
higher 
velocities. 
However, 
a 
change 
in 
permeability 
of 
the 
shallow 
aquifer 
has 
little 
impact 
on 
the 
flow 
pattern 
in 
the 
aquiclude 
Röt 
or 
the 
Buntsandstein 
aquifer. 


Variation 
of 
Permeability 
in 
the 
Aquiclude 


Simulated 
concentration 
distribution 
for 
varied 
permeabilities 
in 
the 
aquiclude 
Röt 
(so) 
are 
given 
in 
Figure 
17. 
The 
changes 
compared 
to 
the 
reference 
simulation 
are 
significant, 
visible 
in 

Crhr 
=6.16% 
and 

Crlr 
=7.88% 
(Table 
13). 


An 
increased 
mean 
permeability 
leads 
to 
more 
salt 
plumes 
with 
slightly 
changed 
position 
in 
comparison 
to 
those 
of 
the 
reference 
simulation. 
The 
salt 
plumes 
are 
steeper 
and 
have 
sharper 
interfaces. 
Furthermore, 
less 
salt 
is 
dissolved 
( 
C¯ 
rhr(k/m/s) 
=0.133, 
Table 
13). 
An 
increased 
permeability 
in 
the 
Röt 
allows 
a 
stronger 
communication 
between 
the 
shallow 
aquifer 
and 
the 
deep 
aquifer. 
Thus, 
the 
regional 
groundwater 
flow 
has 
a 
stronger 
impact 
in 
deeper 
basin 
units, 
resulting 
in 
more 
pronounced 
large 
scale 
advection 
cells. 


The 
flow 
and 
salt 
patterns 
change 
completely 
when 
reducing 
the 
permeability 
of 
the 
Röt. 
An 
impermeable 
Röt 
layer 
disconnects 
Muschelkalk 
and 
Middle/Lower 
Buntsandstein 
layers, 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
17: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 


simulations 
with 
(a) 
increased 
and 
(b) 
decreased 
permeability 
in 
aquiclude 
Röt 
(so). 


Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


which 
results 
in 
a 
break 
down 
of 
large 
scale 
advection 
cells. 
In 
the 
flat 
basin 
center 
diffusive 
and 
dispersive 
salt 
transport 
dominates 
due 
to 
the 
very 
low 
groundwater 
flow 
velocities 
in 
this 
region. 
A 
strong 
salinization 
of 
the 
deep 
aquifer 
(sm/su) 
can 
be 
observed 
(Figure 
17b). 


¯

Although 
the 
mean 
salt 
concentration 
increased 
to 
Crlr(k/m/s) 
=0.194, 
the 
amount 
of 
salt 
reaching 
near-surface 
areas 
is 
reduced. 


Variation 
of 
Permeability 
in 
the 
Deep 
Aquifer 


Changes 
in 
the 
permeabilities 
of 
the 
deep 
aquifer 
units 
have 
strong 
impact 
on 
the 
flow 
pattern, 
visualized 
in 
in 
Figure 
18. 
The 
concentration 
distribution 
for 
varied 
permeabilities 
in 
the 
deep 
aquifer 
differs 
significantly 
from 
the 
results 
of 
the 
reference 
simulation 
(Figure 
15b). 
This 
is 
underlined 
by 
the 
high 
relative 
differences 
of 

Crrh 
=7.82% 
and 

Crrl 
=5.22% 
(Table 
13). 


For 
higher 
permeabilities 
(sm/su)= 
h 
less 
salt 
plumes 
develop 
(Figure 
18a). 
We 
observe 
only 
one 
large 
plume 
in 
the 
flat 
basin 
center. 
It 
shows 
that 
the 
occurrence 
of 
large 
scale 
groundwater 
advection 
is 
reduced. 
The 
salinization 
is 
stronger 
in 
the 
Lower 
Buntsandstein 
layers, 
visible 
in 
a 
higher 
amount 
of 
dissolve 
salt, 
C¯ 
rrh(k/m/s) 
=0.189 
(Table 
13). 


A 
reduction 
of 
permeability 
in 
the 
Buntsandstein 
aquifer 
has 
contrary 
effects. 
The 
concentration 
distribution 
shows 
more 
and 
smaller 
salt 
plumes 
than 
the 
reference 
simulation 
(Figure 
18b 
compared 
to 
Figure 
15b). 
The 
total 
amount 
of 
dissolved 
salt 
is 
reduced, 
C¯ 
rrl(k/m/s) 
=0.142, 
as 
well 
as 
the 
mean 
concentration 
for 
all 
hydrological 
units 
(Table 
13). 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
18: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
(a) 
increased 
and 
(b) 
decreased 
permeability 
in 
deep 
aquifer 
Middle 
and 
Lower 
Buntsandstein 
(sm/su). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


Variation 
of 
Permeabilities 
in 
the 
Faults 


The 
impact 
of 
the 
permeability 
distribution 
in 
the 
fault 
zones 
is 
investigated 
by 
multiplying 
the 
mean 
permeabilities 
of 
all 
sedimentary 
layers 
in 
the 
fault 
zones 
with 
a 
certain 
factor, 
while 
keeping 
the 
mean 
permeability 
distribution 
in 
the 
undisturbed 
basin 
compartments 
constant. 
Simulation 
results 
are 
visualized 
in 
Figures 
19 
and 
20. 
Table 
14 
contains 
the 
calculated 
mean 


¯

concentration 
C 
in 
the 
entire 
domain 
and 
in 
the 
sedimentary 
units, 
as 
well 
as 
the 
relative 
difference 
to 
the 
reference 
simulation 

C. 


The 
changes 
in 
flow 
and 
salt 
patterns 
are 
recognizable 
but 
overall 
small. 
The 
number 
of 
salt 
plumes 
and 
the 
total 
amount 
of 
dissolved 
salt 
is 
constant. 
This 
is 
underlined 
by 
the 
small 
concentration 
differences 
to 
the 
reference 
simulation 

C 
for 
all 
scenarios, 
which 
are 
between 
1.4% 
and 
4% 
(Table 
14). 


We 
observe 
differences 
to 
the 
reference 
simulation 
locally 
in 
the 
shape 
and 
positions 
of 
the 
salt 
plumes. 
Most 
significant 
are 
the 
changes 
in 
flow 
pattern 
for 
highly 
impermeable 
faults, 
Figure 
20b. 
Differences 
can 
be 
seen 
in 
the 
location 
of 
the 
salt 
plume 
at 
the 
Sömmerdaer 
fault 
and 
of 
the 
plume 
in 
the 
vicinity 
of 
the 
Eichenberg-Gotha-Saalfeld 
fault 
zone. 
Furthermore, 
the 
salt 
distribution 
in 
the 
shallow 
aquifer 
of 
the 
central 
salt 
plume 
is 
modified. 
These 
changes 
can 
be 
explained 
by 
differences 
in 
the 
velocity 
of 
inflowing 
fresh 
water 
in 
the 
fault. 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
19: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
increased 
permeability 
. 
in 
fault 
zones: 
(a) 
10 
· 
, 
(b) 
100 
· 
. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 



Figure 
20: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
decreased 
permeability 
. 
in 
fault 
zones: 
(a) 
0.1 
· 
, 
(b) 
0.01 
· 
. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
* 
¯ 
C(k/m/s) 
¯ 
C(k/m) 
¯ 
C(so) 
¯ 
C(sm/su) 

C 
[%] 
1 
0.159 
0.016 
0.030 
0.224 
0 
10 
0.158 
0.016 
0.033 
0.222 
1.43 
100 
0.154 
0.016 
0.031 
0.217 
1.65 
0.1 
0.159 
0.013 
0.029 
0.225 
1.97 
0.01 
0.156 
0.013 
0.025 
0.221 
4.04 


¯

Table 
14: 
Mean 
concentration 
C, 
as 
defined 
in 
Eq.(1.12) 
and 
relative 
concentration 
difference 

C 
to 
the 
reference 
simulation, 
as 
defined 
in 
Eq.(1.13) 
for 
different 
permeabilities 
in 
fault 
zones: 
* 
denotes 
the 
factor 
of 
modified 
permeability 
in 
fault 
zones, 


¯

as 
defined 
in 
section 
3.3.4; 
C(k/m/s) 
is 
the 
mean 
concentration 
in 
the 
entire 
domain, 
C¯(k/m), 
C¯(so) 
and 
C¯(sm/su) 
are 
the 
mean 
concentrations 
in 
the 
corresponding 
hydraulic 
units. 


2 
`h 
[m] 
`v 
[m] 
( 
¯ 
C(k/m/s)) 
( 
¯ 
C(k/m)) 
( 
¯ 
C(so)) 
( 
¯ 
C(sm/su)) 
h
C) 

hC) 
0 
0 
0 
0 
0.159 
0.014 
0.030 
0.224 
0 
0 
10 
0.2 
20 
2 
0.153 
0.014 
0.028 
0.217 
1.15 
0.84 
20 
0.2 
200 
20 
0.156 
0.014 
0.030 
0.220 
2.27 
1.39 
30 
0.2 
2 
000 
50 
0.157 
0.014 
0.028 
0.222 
3.59 
1.96 
40 
0.2 
20 
000 
50 
0.159 
0.014 
0.027 
0.225 
3.63 
2.26 
50 
0.2 
25 
000 
50 
0.158 
0.014 
0.027 
0.223 
3.08 
1.86 
11 
1 
20 
2 
0.156 
0.017 
0.031 
0.219 
2.39 
1.77 
21 
1 
200 
20 
0.165 
0.020 
0.037 
0.231 
4.79 
3.45 
31 
1 
2 
000 
50 
0.170 
0.020 
0.036 
0.238 
6.73 
4.26 
41 
1 
20 
000 
50 
0.170 
0.017 
0.032 
0.240 
6.32 
4.20 
51 
1 
25 
000 
50 
0.168 
0.017 
0.031 
0.236 
5.52 
3.61 
12 
4 
20 
2 
0.158 
0.022 
0.038 
0.220 
5.11 
3.93 
22 
4 
200 
20 
0.169 
0.026 
0.049 
0.232 
7.41 
5.64 
32 
4 
2 
000 
50 
0.188 
0.030 
0.058 
0.259 
10.49 
7.39 
42 
4 
20 
000 
50 
0.190 
0.028 
0.054 
0.262 
9.91 
6.95 
52 
4 
25 
000 
50 
0.181 
0.025 
0.048 
0.251 
8.84 
6.20 


Table 
15: 
Ensemble 
mean 
concentrations 
and 
relative 
concentration 
differences 
for 
all 
ensembles 
of 
heterogeneous 
permeability 
distributions: 
The 
first 
column 
denotes 
the 
ID 
of 
the 
ensemble. 
The 
choice 
of 
statistical 
parameters 
is 
given 
by 
the 
variance 
2 
and 
the 
horizontal 
and 
vertical 
correlation 
lengths 
`h 
and 
`v, 
respectively. 
hC¯(k/m/s)) 
is 
the 
ensemble 
mean 
concentration 
in 
the 
entire 
domain, 
hC¯(k/m)i, 
hC¯(so)i, 
and 
hC¯(sm/su)) 
are 
the 
ensemble 
mean 
concentrations 
in 
the 
corresponding 
hydraulic 
units. 
They 
are 
calculated 
as 
ensemble 
average 
over 
the 
mean 
concentrations 
C 
¯ 
of 
the 
200 
realization; 
h
C) 
[%] 
is 
the 
ensemble 
average 
of 
the 
relative 
differences 
of 
concentration 

Ci, 
i 
=1,..., 
200 
between 
single 
realization 
i 
and 
the 
reference 
simulation. 
Whereas 

hC) 
[%] 
is 
the 
relative 
difference 
of 
concentration 
between 
the 
ensemble 
mean 
hC) 
and 
the 
reference 
simulation. 



3.4. 
Discussion 
of 
Simulation 
Results 
3.4.3. 
Scenarios 
of 
Heterogeneous 
Permeability 
Distribution 
For 
every 
ensemble, 
we 
calculated 
the 
spatially 
distributed 
ensemble 
concentration 
hC) 
as 
the 
mean 
over 
the 
concentration 
distribution 
of 
all 
realizations. 
Table 
15 
contains 
the 
calculated 
quantitative 
measures 
for 
the 
simulated 
ensembles 
of 
heterogeneous 
permeability 
distributions. 
The 
ensemble 
mean 
concentration 
hC¯) 
is 
the 
ensemble 
average 
of 
the 
mean 
concentrations 
C 
¯ 
of 
all 
realizations. 
The 
relative 
ensemble 
difference 
h
C) 
is 
the 
ensemble 
average 
of 
the 
relative 
difference 

C 
between 
every 
single 
realization 
and 
the 
reference 
simulation, 
whereas 
the 
ensemble 
relative 
difference 
in 
concentration 

hC) 
denotes 
the 
difference 
between 
the 
ensemble 
mean 
hC) 
and 
the 
reference 
simulation. 


Impact 
of 
the 
Correlation 
Structure 


We 
compare 
the 
simulation 
results 
for 
different 
correlation 
structures 
while 
keeping 
the 
variance 
at 
a 
medium 
level 
of 
2 
= 
1. 
First, 
we 
investigate 
the 
impact 
of 
a 
Gaussian 
correlation 
structure 
for 
four 
cases 
of 
correlation 
length 
(`h 
= 
20 
m, 
200m, 
2000 
m 
and 
20000 
m) 
ranging 
from 
local 
to 
basin 
scale, 
as 
defined 
in 
section 
3.3.4. 
Second, 
we 
analyze 
an 
ensemble 
with 
a 
truncated 
powerlawcovariancestructurewithcutoff 
alengthof 
25000 
m. 


Figure 
21 
shows 
the 
simulated 
salt 
concentration 
distributions 
for 
two 
realizations 
and 
the 
ensemble 
mean 
of 
ensemble 
11. 
The 
correlation 
length 
of 
`h 
= 
20 
m 
ranges 
in 
the 
size 
of 
one 
grid 
cell. 
It 
can 
be 
seen 
that 
a 
small 
correlation 
length 
causes 
local 
fluctuations 
in 
the 
salt 
concentration, 
but 
does 
not 
impact 
on 
the 
shape 
and 
the 
position 
of 
the 
salt 
plumes 
compared 
to 
the 
homogeneous 
reference 
simulation 
(Figure 
15b). 
This 
is 
underlined 
by 
the 
small 
mean 
difference 
of 
h
C11) 
=2.39% 
(Table 
15). 
The 
differences 
between 
realizations 
(Figure 
21a 
and 
21b) 
are 
little. 
Thus, 
the 
ensemble 
mean 
concentration 
pattern 
hC11) 
is 
nearly 
identical 
to 
the 
concentration 
distribution 
with 
homogeneous 
permeabilities 
(Figure 
21c 
compared 
to 
Figure 
15b). 
This 
fact 
is 
underlined 
by 
the 
small 
relative 
difference 

hC11) 
=1.52% 
and 
a 
similar 
amount 
of 
dissolved 
salt 
hC¯ 
11(k/m/s) 
) 
=0.156 
compared 
to 
the 
reference 
simulation. 


We 
visualize 
the 
simulation 
results 
for 
a 
medium 
scale 
correlation 
length 
of 
`h 
= 
200 
m 
(ensemble 
12) 
in 
Figure 
22. 
Small 
changes 
exist 
in 
position 
and 
shape 
of 
salt 
plumes 
for 
the 
heterogeneous 
realizations 
(Figures 
22a 
and 
b) 
compared 
to 
the 
homogeneous 
reference 
simulation 
(Figure 
15b). 
Local 
fluctuations 
are 
more 
pronounced 
in 
comparison 
to 
the 
results 
for 
a 
small 
correlation 
length 
of 
20 
m. 
This 
is 
supported 
by 
a 
higher 
relative 
concentration 
difference 
of 
h
C21) 
=4.79%. 
An 
increased 
spreading 
of 
salt 
is 
visible 
in 
a 
higher 
amount 
of 
dissolved 
salt 
hC¯ 
21(k/m/s) 
) 
=0.165 
(Table 
15) 
compared 
to 
the 
reference 
simulation. 
The 
ensemble 
mean 
of 
heterogeneous 
simulations 
hC21) 
in 
Figure 
22c 
mirrors 
changes 
in 
the 
concentration 
pattern: 
salt 
plumes 
are 
damped 
and 
there 
exists 
a 
certain 
range 
in 
which 
salt 
plumes 
develop. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
21: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution: 
(a) 
and 
(b) 
are 
single 
realizations 
and 
(c) 
is 
the 
mean 
over 
200 
realizations 
of 
ensemble 
11 
with 
small 
correlation 
length 
(. 
= 
20 
m). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
22: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution: 
(a) 
and 
(b) 
are 
single 
realizations 
and 
(c) 
is 
the 
mean 
over 
200 
realizations 
of 
ensemble 
21 
with 
medium 
correlation 
length 
(. 
= 
200 
m). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
23: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution: 
(a) 
and 
(b) 
are 
single 
realizations 
and 
(c) 
is 
the 
mean 
over 
200 
realizations 
of 
ensemble 
31 
with 
large 
correlation 
length 
(. 
= 
2000 
m). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


Figure 
23 
shows 
the 
simulation 
results 
for 
large 
scale 
correlation 
length 
of 
`h 
= 
2000 
m. 
Local 
fluctuation 
are 
of 
larger 
scale 
and 
can 
lead 
to 
significantly 
different 
salt 
patterns 
for 
different 
realizations 
(Figure 
23a 
and 
b). 
We 
observe 
differences 
in 
number, 
position 
and 
shape 
of 
salt 
plumes. 
The 
differences 
to 
the 
reference 
simulation 
(Figure 
15b) 
are 
large, 
supported 
by 
the 
relative 
ensemble 
difference 
of 
h
C31) 
=6.73%. 
A 
higher 
amount 
of 
dissolved 
salt 
hC¯ 
31(k/m/s)) 
=0.17 
indicates 
a 
wider 
spread 
of 
salt 
plumes. 
The 
ensemble 
concentration 
pattern 
hC31) 
(Figure 
23c) 
shows 
a 
flat 
salt 
distribution 
without 
clear 
salt 
plumes. 
It 
displays 
the 
range 
of 
positions 
of 
salt 
plumes. 


In 
Figure 
24, 
we 
visualize 
the 
simulation 
results 
for 
a 
correlation 
length 
of 
`h 
= 
20000 
m, 
which 
corresponds 
to 
the 
maximum 
distance 
of 
an 
undisturbed 
basin 
compartment. 
The 
results 
show 
significant 
differences 
in 
number, 
position, 
and 
shape 
of 
salt 
plumes. 
Realizations 
differ 
strongly 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
24: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution: 
(a) 
and 
(b) 
are 
single 
realizations 
and 
(c) 
is 
the 
mean 
over 
200 
realizations 
of 
ensemble 
41 
with 
correlation 
length 
at 
basin 
scale 
(. 
= 
20000 
m). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


among 
each 
other 
and 
from 
the 
homogeneous 
reference 
simulation, 
visible 
at 
h
C41) 
=6.32% 
(Table 
15). 
Due 
to 
the 
very 
large 
correlation 
length, 
the 
results 
can 
be 
similar 
to 
salt 
patterns 
of 
simulations 
with 
homogeneous 
permeabilities, 
as 
discussed 
in 
section 
3.4.2. 
The 
amount 
of 
dissolved 
salt 
is 
higher 
(hC¯ 
41(k/m/s)) 
=0.17) 
compared 
to 
homogeneous 
simulations. 
The 
pattern 
of 
the 
ensemble 
mean 
hC41) 
(Figure 
24c) 
is 
similar 
to 
that 
of 
the 
large 
scale 
correlation 
length 
`h 
= 
2000 
m 
(Figure 
23c) 
mirroring 
the 
areas 
of 
salt 
plumes. 


Figure 
25 
shows 
the 
simulation 
results 
with 
the 
truncated 
power 
law 
correlation 
model, 
displaying 
a 
heterogeneous 
medium 
with 
correlation 
on 
all 
scales. 
In 
the 
resulting 
concentration 
patterns, 
we 
observe 
local 
fluctuations, 
as 
in 
simulations 
with 
small 
correlation 
lengths 
and 
changes 
in 
position 
and 
shape 
of 
salt 
plumes, 
as 
observed 
in 
simulations 
with 
high 
and 
basin 
scale 
correlation 
lengths. 
The 
concentration 
distributions 
differ 
in 
average 
h
C51) 
=5.52%. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
25: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution: 
(a) 
and 
(b) 
are 
single 
realizations 
and 
(c) 
is 
the 
mean 
over 
200 
realizations 
of 
ensemble 
51 
with 
infinite 
correlation 
length 
(truncated 
power 
law 
model). 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


The 
mean 
concentration 
pattern 
hC91) 
(Figure 
25c) 
mirrors 
changes 
compared 
to 
the 
homogeneous 
reference 
simulation, 
as 
for 
the 
large 
scale 
correlation 
length. 
Generally, 
the 
resulting 
salt 
concentration 
patterns 
are 
very 
similar 
to 
those 
with 
correlation 
length 
f 
= 
2000 
m 
and 
`= 
20000 
m, 
which 
shows 
that 
the 
effect 
of 
large 
correlation 
lengths 
is 
dominant. 


Impact 
of 
the 
Variance 


We 
analyze 
simulations 
with 
three 
choices 
of 
variance: 
mildly 
heterogeneous 
media 
(2 
= 
0.2), 
medium 
heterogeneous 
media 
(2 
=1), 
and 
strongly 
heterogeneous 
media 
(2 
=4). 
Figure 
26 
shows 
the 
simulated 
salt 
distributions 
for 
three 
realization 
with 
different 
variances 
and 
a 
Gaussian 
correlation 
length 
of 
`h 
= 
2000 
m. 
We 
chose 
the 
spatial 
correlation 
structure 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
26: 
Distribution 
of 
normalized 
salt 
concentration 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
of 
simulations 
with 
heterogeneous 
permeability 
distribution 
for 
different 
variances: 
realizations 
from 
ensembles 
30, 
31 
and 
32 
with 
identical 
correlation 
structure 
(`h 
= 
2000 
m) 
and 
(a) 
2 
=0.2, 
(b) 
2 
=1 
and 
(c) 
2 
=4. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


identically 
for 
all 
variances, 
which 
allows 
a 
direct 
comparison 
of 
realizations. 
It 
can 
be 
seen 
that 
a 
higher 
variance 
goes 
along 
with 
a 
stronger 
salinization. 
This 
is 
valid 
for 
all 
choices 
of 
correlation 
structure, 
which 
is 
visible 
in 
increasing 
values 
of 
hC¯) 
for 
increasing 
variance 
for 
all 
correlation 
lengths 
(Table 
15). 
However, 
the 
variance 
does 
not 
directly 
impact 
on 
number 
and 
position 
of 
salt 
plumes, 
only 
on 
the 
shape 
due 
to 
the 
higher 
amount 
of 
dissolved 
salt. 
This 
is 
in 
particular 
valid 
in 
combination 
with 
larger 
correlation 
lengths. 


Heterogeneity 
contributes 
to 
the 
mechanical 
dispersion 
[Bear, 
1972]. 
The 
variance 
controls 
the 
range 
of 
permeability 
values, 
amplifying 
differences 
between 
areas 
of 
higher 
and 
lower 
permeability. 
As 
a 
consequence, 
the 
larger 
the 
variance, 
the 
stronger 
are 
the 
variations 
in 
velocity. 
Higher 
velocities 
result 
in 
increased 
mechanical 
dispersion 
and 
thus 
in 
higher 
amounts 
of 
dissolved 
salt. 
This 
effect 
can 
be 
observed 
in 
particular 
in 
the 
shallow 
aquifer 
units. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
density 
model 
¯ 
C(k/m/s) 
¯ 
C(k/m) 
¯ 
C(so) 
¯ 
C(sm/su) 

C 
[%] 

T 
[%] 
0 
0.157 
0.014 
0.030 
0.224 
0 
0 
(T 
) 
0.157 
0.014 
0.029 
0.222 
1.00 
0.17 
(C) 
0.309 
0.079 
0.149 
0.408 
15.87 
2.17 
(T, 
C) 
0.308 
0.078 
0.146 
0.406 
15.71 
2.14 


Table 
16: 
Mean 
concentration 
C¯, 
as 
defined 
in 
Eq.(1.12), 
relative 
difference 
in 
concentration 

C 
and 
relative 
difference 
in 
temperature 

T 
to 
the 
reference 
simulation, 
as 
defined 
in 
Eq.(1.13) 
for 
different 
fluid 
density 
models: 
0 
is 
the 
constant 
density 
model, 
(T 
) 
is 
the 
purely 
temperature 
dependent 
density 
model, 
(C) 
is 
the 
purely 
salt 
concentration 
dependent 
density 
model, 
and 
(T,C) 
is 
the 
concentration 
and 
temperature 
dependent 
density 
model 
according 
to 
Eq. 
(1.10). 
C¯(k/m/s) 
is 
the 
mean 
concentration 
in 
the 
entire 
domain, 
C¯(k/m), 
C¯(so) 
and 
C¯(sm/su) 
are 
the 
mean 
concentrations 
in 
the 
corresponding 
hydraulic 
units. 


3.4.4. 
Density 
Dependent 
Flow 
Impact 
of 
the 
Temperature 


The 
simulation 
results 
for 
a 
temperature 
dependent 
fluid 
density 
model 
(T 
) 
are 
given 
in 
Figure 
27. 
The 
salt 
and 
temperature 
distributions 
resemble 
those 
of 
the 
constant 
density 
reference 
simulation 
in 
Figure 
15. 
We 
observe 
no 
differences 
in 
the 
temperature 
distribution 
visually. 
Small 
changes 
in 
the 
salt 
pattern 
can 
be 
found 
in 
the 
form 
of 
the 
central 
salt 
plume. 
The 
marginal 
differences 
are 
underlined 
by 
the 
nearly 
identical 
values 
of 
the 
mean 
concentration 
(Table 
16) 
and 
the 
small 
values 
of 
the 
relative 
difference 
in 
temperature 

T 
=0.17% 
and 
salt 
concentration 

C 
=1.0%. 


The 
marginal 
changes 
for 
a 
temperature 
dependent 
density 
model 
are 
due 
to 
the 
fact 
that 
the 
temperature 
differences 
are 
too 
small 
to 
initiate 
convective 
flow. 
According 
to 
the 
linear 
density 
model 
described 
in 
Eq.(1.10), 
the 
density 
of 
water 
can 
vary 
between 
1000 
kg/m3 
and 
991kg/m3, 
which 
corresponds 
to 
the 
thermal 
expansion 
coefficient 
of 
T 
= 
-0.0003 
and 
the 
minimal 
and 
maximal 
temperature 
of 
8 
C 
and 
38.4 
C 
respectively. 
The 
maximal 
thermal 
Rayleigh 
number 
RaT, 
as 
defined 
in 
Eq.(1.11) 
for 
all 
sedimentary 
units 
can 
be 
found 
in 
the 
Middle 
Buntsandstein 
aquifer 
with 
RaT 
= 
11.7. 
This 
is 
beneath 
the 
critical 
value 
of 
4p 
which 
characterizes 
the 
onset 
of 
convection 
[Nield 
and 
Bejan, 
1999]. 


Even 
for 
higher 
permeabilities, 
thus 
higher 
thermal 
Rayleigh 
numbers, 
regional 
groundwater 
flow 
would 
superpose 
convective 
processes 
due 
to 
higher 
flow 
velocities. 
From 
the 
Rayleigh 
theory 
we 
know 
that 
temperature 
induced 
convection 
requires 
highly 
permeable 
media 
of 
sufficiently 
large 
vertical 
extent 
and 
which 
is 
undisturbed 
from 
other 
flow 
sources. 
Since 
this 
is 
not 
present 
in 
the 
Thuringian 
Basin, 
conduction 
and 
advection 
dominate 
the 
heat 
transport 
process. 
Thus, 
temperature 
dependent 
density 
driven 
flow 
is 
negligible 
in 
the 
Thuringian 
Basin 
due 
to 
the 
significant 
regional 
flow. 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
27: 
Simulated 
temperature 
and 
salt 
distribution 
for 
flow 
impacted 
by 
temperature 
induced 
density 
variations 
and 
regional 
groundwater 
flow: 
(a) 
salt 
concentration 
distribution 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
and 
(b) 
temperature 
distribution 
with 
isolines 
at 
distance 
of 
2 
C. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
Figure 
28: 
Results 
of 
temperature 
and 
salt 
distribution 
for 
flow 
impacted 
by 
density 
variations 
due 
to 
salt 
concentration 
differences 
and 
regional 
groundwater 
flow: 
(a) 
salt 
concentration 
distribution 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
and 
(b) 
temperature 
distribution 
with 
isolines 
at 
distance 
of 
2 
C. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


Impact 
of 
Salt 
Concentration 


Effects 
of 
dissolved 
salt 
on 
the 
fluid 
density 
result 
in 
salt 
concentration 
and 
temperature 
patterns 
given 
in 
Figure 
28. 
Significant 
differences 
are 
visually 
recognizable 
compared 
to 
the 
reference 
simulation 
in 
Figure 
15b. 
More 
salt 
is 
dissolved 
for 
a 
salt 
dependent 
density 
model 
(C) 
than 
for 
the 
constant 
density 
model 
0: 
C¯(k/m/s) 
=0.309 
compared 
to 
C¯(k/m/s) 
=0.159, 
respectively 
(Table 
16). 
The 
salt 
plume 
pattern 
is 
less 
pronounced. 
High 
concentration 
brine 
(C> 
0.7) 
does 
not 
move 
upwards. 
Salt 
is 
stronger 
laterally 
distributed, 
in 
particular 
in 
the 
central 
basin 
compartments. 


Changes 
in 
temperature 
distribution 
(Figure 
28b) 
are 
less 
pronounced 
but 
observable 
compared 
to 
the 
reference 
simulation 
(Figure 
15c). 
The 
temperature 
isolines 
are 
smoother, 
which 
results 
from 
changed 
velocity 
fields 
in 
the 
deep 
aquifer, 
due 
to 
salt 
mass 
effects. 
The 
relative 
difference 
in 
temperature 
pattern 
amounts 
to 

T 
=2.17%. 


The 
strong 
impact 
of 
salt 
on 
the 
flow 
and 
salt 
pattern 
can 
be 
explained 
by 
the 
interplay 
of 
density 
differences 
and 
the 
regional 
groundwater 
flow. 
Full 
salt 
saturation 
can 
lead 
to 
a 
fluid 
density 
of 
up 
to 
1200 
kg/m3. 
This 
results 
in 
large 
solute 
Rayleigh 
numbers 
RaS 
in 
all 
sedimentary 
layers. 
In 
the 
deep 
aquifer 
the 
values 
range 
between 
RaS 
= 
800 
- 
6000, 
where 
the 
highest 
values 
occur 
in 
the 
Middle 
Buntsandstein 
of 
the 
undisturbed 
central 
basin 
compartment. 
In 
areas 
of 
discharge 
water 
pressure 
gradients 
cause 
an 
upward 
movement 
of 



3.4. 
Discussion 
of 
Simulation 
Results 
Figure 
29: 
Results 
of 
temperature 
and 
salt 
distribution 
for 
flow 
impacted 
by 
density 
variations 
due 
to 
temperature 
and 
salt 
concentration 
differences 
and 
regional 
groundwater 
flow: 
(a) 
salt 
concentration 
distribution 
with 
isolines 
at 
C 
= 
[0.1,...,0.9] 
and 
(b) 
temperature 
distribution 
with 
isolines 
at 
distance 
of 
2 
C. 
Sedimentary 
units 
are 
separated 
by 
bold 
black 
lines. 
Superelevation 
factor 
of 
10. 


brine. 
The 
effect 
of 
an 
increased 
fluid 
density 
due 
to 
solved 
salt 
leads 
to 
the 
counteracting 
effect 
of 
downward 
convection. 
It 
results 
in 
smoothed 
pressure 
gradients 
in 
horizontal 
direction 
but 
increased 
vertical 
velocities. 
Higher 
velocities 
cause 
an 
increase 
in 
dispersion. 
Furthermore, 
local 
eddies 
develop 
as 
described 
in 
Holzbecher 
et 
al. 
[2010], 
which 
enhance 
local 
mixing. 
Both 
effects 
result 
in 
a 
stronger 
spreading 
of 
salt. 


Impact 
of 
Temperature 
and 
Salt 
Concentration 


Simulation 
results 
for 
a 
salt 
and 
temperature 
dependent 
density 
model 
(C,T) 
are 
given 
in 
Figure 
29. 
The 
salt 
and 
temperature 
distributions 
resemble 
the 
patterns 
with 
a 
purely 
salt 
dependent 
density 
(C) 
(Figure 
28). 
Changes 
in 
the 
characteristic 
numbers 
are 
small: 
the 
total 
amount 
of 
salt 
and 
the 
relative 
difference 
to 
the 
reference 
simulation 
are 
marginally 
lower 
than 
for 
purely 
salt 
dependent 
density 
(Table 
16). 
In 
general, 
the 
effect 
of 
salt 
is 
slightly 
reduced, 
but 
still 
dominant. 
The 
reduction 
can 
be 
explained 
by 
the 
opposite 
character 
of 
salt 
and 
of 
temperature 
on 
fluid 
density. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 
3.5. 
Conclusions 
The 
aim 
of 
the 
study 
was 
to 
investigate 
the 
main 
fluid 
dynamics 
within 
the 
Thuringian 
Basin. 
We 
performed 
transient 
coupled 
simulations 
of 
hydraulic, 
thermal, 
and 
mass 
transport 
processes 
on 
a 
selected 
profile 
to 
quantify 
the 
impact 
of 
the 
permeability 
distribution 
and 
density 
differences 
on 
the 
flow 
pattern, 
the 
salt 
concentration 
and 
the 
temperature 
distribution. 
To 
capture 
the 
uncertainty 
in 
permeability, 
which 
determines 
regional 
groundwater 
flow, 
we 
performed 
simulations 
with 
different 
mean 
permeabilities 
in 
the 
main 
hydraulic 
units 
as 
well 
as 
in 
the 
fault 
zones. 
We 
further 
analyzed 
effects 
of 
heterogeneity. 
Thereby 
permeability 
was 
characterized 
by 
three 
parameters: 
mean, 
variance 
and 
correlation 
structure. 
In 
a 
first 
step, 
we 
examined 
the 
effect 
of 
mean 
permeability 
varying 
within 
a 
realistic 
range 
by 
conducting 
simulation 
with 
different 
distributions 
of 
homogeneous 
permeability 
within 
the 
three 
hydraulic 
units 
shallow 
aquifer 
(Keuper/Muschelkalk), 
aquiclude 
(Röt), 
and 
deep 
aquifer 
(Middle 
and 
Lower 
Buntsandstein). 
In 
a 
second 
step, 
we 
investigated 
the 
impact 
of 
heterogeneity 
using 
ensembles 
of 
heterogeneous 
permeability 
distributions 
with 
different 
values 
of 
variance 
and 
correlation 
structure. 
Thirdly, 
we 
examined 
the 
impact 
of 
salt 
mass 
and 
temperature 
on 
the 
groundwater 
flow. 
We 
performed 
simulations 
with 
a 
non-constant 
fluid 
density 
model, 
depending 
on 
salt 
concentration 
and 
temperature 
differences. 


The 
main 
driving 
force 
of 
fluid 
flow 
in 
the 
Thuringian 
Basin 
is 
the 
water 
pressure 
gradient, 
due 
to 
differences 
in 
the 
topography. 
The 
shallow 
basin 
structure 
causes 
the 
regional 
groundwater 
flow 
to 
impact 
on 
flow 
directions 
and 
velocities 
down 
to 
the 
deep 
aquifer 
units, 
connecting 
to 
the 
Zechstein 
salt 
in 
up 
to 
1000 
m 
depth. 
Especially 
in 
vicinity 
of 
faults, 
meteoric 
fresh 
water 
can 
infiltrate 
deeply 
into 
the 
basin. 
It 
results 
in 
high 
flow 
velocities 
and 
causes 
anomalies 
in 
the 
temperature 
stratification. 
Areas 
of 
discharge 
are 
the 
flat 
undisturbed 
basin 
compartments 
located 
in 
the 
basin 
center. 


Since 
flow 
velocities 
are 
proportional 
to 
the 
permeabilities 
of 
the 
sedimentary 
layers, 
their 
distribution 
significantly 
determines 
flow 
and 
salt 
concentration 
pattern. 
Simulation 
results 
with 
permeability 
variations 
in 
all 
hydraulic 
units 
indicate 
that 
variations 
in 
the 
permeability 
of 
the 
shallow 
aquifer 
(Keuper/Muschelkalk) 
have 
little 
impact 
on 
the 
salt 
distribution 
in 
the 
deep 
aquifer 
(Middle 
and 
Lower 
Buntsandstein). 
On 
the 
contrary, 
the 
permeability 
of 
the 
aquiclude 
(Röt) 
determines 
the 
amount 
of 
communication 
between 
deep 
and 
shallow 
aquifers 
and 
thus 
the 
flow 
pattern: 
the 
lower 
the 
permeability, 
the 
lower 
is 
the 
interaction 
and 
the 
more 
salt 
is 
dissolved 
in 
the 
deep 
aquifer, 
but 
the 
less 
salt 
is 
transported 
to 
surface 
near 
regions. 
Changes 
in 
permeability 
of 
the 
deep 
aquifer 
have 
significant 
impact 
on 
the 
amount 
of 
dissolved 
salt: 
the 
higher 
the 
permeability, 
the 
more 
salt 
is 
dissolve, 
but 
the 
salt 
plume 
pattern 
is 
less 
pronounced, 
thus 
less 
salt 
is 
transported 
to 
the 
surface. 


Heterogeneity 
in 
permeability 
can 
have 
significant 
impact 
on 
the 
developing 
flow 
pattern 
and 
thus 
the 
salt 
distribution. 
In 
general, 
an 
increase 
in 
correlation 
length 
and 
variance 
goes 
along 
with 
a 
deviation 
of 
the 
flow 
and 
salt 
pattern 
from 
results 
for 
a 
homogeneous 
permeability 
distribution. 
Fluctuations 
in 
the 
salt 
concentration 
pattern 
correspond 
to 
the 
choice 
of 
correlation 
length: 
small 
correlation 
lengths 
in 
the 
range 
of 
20m 
cause 
local 
fluctuations 
with 
similar 
salt 



3.5. 
Conclusions 
patternsashomogeneoussimulation. 
Largecorrelationlengthsintherangeof 
2000 
mcontrol 
large 
ranges 
of 
hydraulic 
conductivity. 
Thus 
heterogeneous 
simulation 
results 
can 
vary 
significantly 
in 
number 
and 
position 
of 
salt 
plumes, 
as 
it 
was 
the 
case 
for 
homogenous 
simulations 
with 
increased 
and 
decreased 
conductivity 
in 
the 
entire 
sedimentary 
layers. 
The 
truncated 
power 
law 
correlation 
structure 
could 
capture 
the 
effects 
of 
correlation 
lengths 
on 
all 
scales. 
A 
comparison 
with 
the 
Gaussian 
correlation 
structure 
for 
different 
correlation 
lengths 
showed 
that 
effects 
of 
large 
scale 
correlation 
lengths 
dominate 
over 
those 
of 
small 
scale 
correlation 
lengths. 


The 
variance 
as 
degree 
of 
heterogeneity 
has 
a 
different 
effect. 
Differences 
in 
permeability 
are 
stronger 
the 
larger 
the 
variance 
is. 
The 
resulting 
spread 
in 
local 
flow 
velocity 
enhances 
mechanical 
dispersion. 
Thus, 
higher 
variances 
only 
slightly 
change 
the 
overall 
flow 
and 
salt 
pattern 
but 
strongly 
impact 
on 
the 
amount 
of 
dissolved 
salt. 
The 
effect 
of 
an 
increased 
amount 
of 
dissolved 
salt 
is 
also 
given 
for 
increasing 
correlation 
length, 
although 
it 
is 
not 
that 
strong 
as 
for 
the 
variance. 
In 
cases 
of 
large 
correlation 
lengths 
and 
large 
variances, 
the 
deviation 
from 
salt 
distributions 
for 
homogeneous 
permeabilities 
is 
the 
strongest. 


Effects 
of 
a 
non-constant 
fluid 
density 
can 
impact 
on 
the 
flow 
pattern 
and 
thus 
the 
distribution 
of 
salt 
and 
temperature 
significantly. 
This 
is 
in 
particular 
the 
case 
for 
density 
variations 
due 
to 
salt 
concentration 
differences. 
High 
amounts 
of 
dissolved 
salt 
result 
in 
an 
increased 
density 
which 
causes 
downward 
movement 
of 
fluid. 
The 
counteracting 
effect 
of 
regional 
groundwater 
flow 
in 
discharge 
areas 
changes 
the 
velocity 
field 
and 
leads 
to 
the 
development 
of 
local 
eddies. 
It 
results 
in 
an 
enhanced 
mixing 
and 
thus 
an 
increase 
in 
the 
amount 
of 
dissolved 
salt. 
Solute 
Rayleigh 
numbers 
above 
the 
critical 
value 
of 
4p 
in 
all 
aquifer 
units 
confirm 
these 
findings. 
On 
the 
contrary, 
the 
impact 
of 
temperature 
is 
little. 
Thermal 
induced 
convection 
is 
not 
present 
due 
to 
low 
temperature 
gradients 
and 
high 
regional 
flow 
velocities. 
These 
results 
from 
simulations 
are 
supported 
by 
Rayleigh 
theory: 
calculated 
thermal 
Rayleigh 
number 
RaT 
are 
lower 
than 
the 
critical 
value 
for 
onset 
of 
convection. 


The 
numerical 
study 
showed 
that 
in 
general 
two 
main 
transport 
mechanism 
interact 
in 
the 
Thuringian 
Basin: 
topography 
driven 
regional 
groundwater 
flow 
and 
convection 
due 
to 
fluid 
density 
differences. 
The 
interaction 
of 
both 
processes 
results 
in 
a 
wide 
spread 
of 
salt 
in 
the 
deep 
aquifer 
(Middle 
and 
Lower 
Buntsandstein) 
in 
the 
central 
basin. 
Salt 
transport 
to 
near-surface 
regions 
can 
be 
observed 
locally, 
especially 
in 
the 
basin 
center. 


In 
contrast 
to 
the 
dynamics 
of 
deep 
sedimentary 
basins 
like 
the 
North 
East 
German 
Basin, 
the 
shallow 
character 
of 
the 
Thuringian 
Basin 
inhibits 
the 
evolution 
of 
large 
thermal 
convection 
cells. 
From 
Rayleigh 
theory 
it 
is 
known 
that 
onset 
of 
convection 
requires 
thick, 
high 
permeable 
aquifers 
which 
are 
undisturbed 
from 
external 
flow. 
Those 
units 
are 
not 
present 
in 
the 
Thuringian 
Basin. 
However, 
other 
mechanism 
are 
responsible 
for 
brine 
transport 
to 
near-surface 
regions, 
which 
we 
showed 
by 
this 
numerical 
study. 



3. 
Mechanisms 
of 
Salt 
Transport 
in 
the 
Thuringian 
Basin 

Part 
III. 
Determining 
Aquifer 
Heterogeneity 
on 
Local 
Scale 



88 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
4.1. 
Introduction 
Determining 
hydraulic 
properties 
of 
an 
aquifer 
has 
been 
a 
matter 
of 
research 
for 
several 
decades. 
Not 
only 
for 
characterizing 
groundwater 
flow 
but 
also 
for 
describing 
transport 
processes 
in 
the 
subsurface 
a 
good 
perception 
of 
the 
heterogeneous 
structure 
of 
porous 
media 
is 
necessary, 
see 
for 
e.g. 
Dagan 
[1989]; 
Gelhar 
[1993]; 
Rubin 
[2003]. 


Pumping 
tests 
are 
a 
widely 
used 
tool 
to 
identify 
hydraulic 
parameters 
which 
effect 
the 
groundwater 
flow 
pattern. 
Analyzing 
Darcy’s 
Law 
in 
combination 
with 
the 
continuity 
equation 
for 
steady 
state 
pumping 
tests 
in 
homogeneous 
porous 
media 
leads 
to 
the 
well 
known 
Thiem’s 
solution 


hThiem(r)= 
- 
Qw 
ln 
r 
+ 
h(R) 
. 
(4.1)

2LK 
R 
It 
describes 
the 
drawdown 
of 
the 
hydraulic 
head 
h(r) 
depending 
on 
the 
radial 
distance 
from 
the 
well 
r 
for 
homogeneous 
hydraulic 
conductivity 
K. 
Thiem’s 
solution 
is 
valid 
in 
a 
confined 
aquifer 
of 
thickness 
L 
with 
fully 
penetrating 
well 
and 
the 
constant 
discharge 
Qw; 
h(R) 
is 
a 
known 
reference 
head 
at 
the 
distance 
R 
from 
the 
well. 


The 
applicability 
of 
Thiem’s 
solution 
to 
pumping 
tests 
in 
heterogeneous 
media 
is 
limited. 
It 
requires 
a 
representative 
conductivity 
value 
K 
for 
the 
whole 
range 
of 
the 
depression 
cone. 
As 
stated 
by 
Matheron 
[1967], 
a 
single 
representative 
K-value 
for 
well 
flow 
does 
not 
exist, 
due 
to 
the 
emergence 
of 
different 
mean 
conductivities 
characterizing 
the 
behavior 
near 
and 
far 
from 
the 
well. 
Since 
then 
an 
enormous 
amount 
of 
work 
has 
been 
devoted 
to 
find 
representative 
conductivity 
descriptions 
for 
well 
flow 
and 
to 
estimate 
statistical 
parameters 
of 
the 
hydraulic 
conductivity 
from 
drawdown 
data. 
For 
a 
detailed 
review 
see 
Sánchez-Vila 
et 
al. 
[2006]. 
Most 
of 
the 
studies 
are 
limited 
to 
a 
two 
dimensional 
analysis, 
like 
Shvidler 
[1966]; 
Desbarats 
[1992]; 
Sánchez-Vila 
et 
al. 
[1999]; 
Copty 
and 
Findikakis 
[2004]; 
Neuman 
et 
al. 
[2004, 
2007]; 
Dagan 
and 
Lessoff 
[2007]; 
Schneider 
and 
Attinger 
[2008] 
and 
many 
more. 
Only 
a 
few 
authors 
addressed 
the 
impact 
of 
modeling 
the 
conductivity 
in 
three 
dimensions 
upon 
radial 
flow 
[Indelman 
and 
Abramovich, 
1994; 
Indelman 
et 
al., 
1996; 
Guadagnini 
et 
al., 
2003]. 
In 
particular, 
only 
Firmani 
et 
al. 
[2006] 
presented 
a 
fully 
three 
dimensional 
numerical 
investigation. 


In 
order 
to 
find 
a 
description 
of 
the 
hydraulic 
head 
field 
in 
pumping 
tests 
for 
heterogeneous 
media, 
the 
conductivity 
K(x) 
is 
commonly 
modeled 
as 
a 
log-normal 
distributed 
spatial 
random 
function. 
Based 
on 
this 
assumption 
Indelman 
and 
Abramovich 
[1994] 
solved 
an 
averaged 
Darcy’s 
Law 
and 
presented 
a 
fundamental 
solution 
for 
the 
mean 
head 
distribution 
for 
arbitrary 
boundary 
conditions. 
Since 
it 
is 
given 
in 
Fourier 
space, 
only 
approximate 
solutions 
in 
real 
space 
are 
available. 
In 
this 
line 
Indelman 
et 
al. 
[1996] 
performed 
a 
perturbation 
expansion 
in 
the 
variance 
2 
of 
ln 
K(x) 
to 
present 
a 
first 
order 
solution 
in 
the 
hydraulic 
head 
for 
well 
flow. 
The 
result 
has 
been 
expanded 
by 
several 
authors 
to 
higher 
orders, 
e.g. 
Fiori 
et 
al. 
[1998]; 
Indelman 
[2001]. 
Additionally, 
Guadagnini 
et 
al. 
[2003] 
presented 
a 
three-dimensional 
steady 
state 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
solution 
for 
mean 
flow 
based 
on 
recursive 
approximations 
of 
exact 
non-local 
moment 
equations. 


The 
implicit 
character 
of 
these 
head 
solutions 
inhibits 
the 
application 
to 
analyze 
pumping 
test 
drawdowns 
directly. 
Furthermore, 
the 
reliability 
of 
a 
perturbation 
approach 
to 
describe 
well 
flow 
is 
questionable 
due 
to 
a 
ergodicity 
breakdown 
near 
the 
well 
which 
is 
mathematically 
a 
singularity. 
None 
of 
the 
solutions 
could 
reproduce 
the 
head 
drawdown 
of 
a 
pumping 
test 
exactly 
as 
numerical 
investigation 
showed 
[Guadagnini 
et 
al., 
2003; 
Firmani 
et 
al., 
2006] 
nor 
allowed 
an 
inverse 
estimation 
of 
the 
parameters 
of 
K(x) 
in 
highly 
heterogeneous 
porous 
media. 


Making 
use 
of 
the 
equivalent 
conductivity 
Keq 
as 
defined 
by 
Matheron 
[1967] 
and 
their 
first 
order 
solution, 
Indelman 
et 
al. 
[1996] 
derived 
the 
expression 
Keq(r)= 
Kwell(1-(r))+(r)Kefu. 
It 
relates 
the 
near 
well 
representative 
conductivity 
Kwell 
to 
the 
far 
field 
value 
Kefu 
(in 
detail 
discussed 
in 
section 
4.2.2) 
by 
a 
weighting 
factor 
(r) 
which 
depends 
on 
the 
statistical 
parameters 
of 
K(x) 
and 
the 
radius 
r. 
This 
description 
was 
used 
by 
Firmani 
et 
al. 
[2006] 
for 
inverse 
parameter 
estimation 
from 
numerical 
pumping 
tests. 
But 
as 
their 
simulations 
showed, 
the 
description 
of 
Keq(r) 
is 
only 
valid 
for 
small 
variances 
2 
up 
to 
0.5. 
Furthermore, 
the 
estimation 
of 
the 
parameters 
is 
of 
high 
uncertainty. 


To 
overcome 
the 
above 
mentioned 
limitations 
Schneider 
and 
Attinger 
[2008] 
showed 
that 
in 
two 
dimensions 
another 
description 
for 
the 
conductivity, 
respectively 
transmissivity, 
is 
appropriate 
to 
describe 
well 
flow 
effectively. 
They 
introduced 
a 
new 
approach 
by 
applying 
an 
upscaling 
technique 
to 
the 
flow 
equation 
to 
derive 
their 
representative 
description 
for 
the 
transmissivity 
T 
CG(r), 
depending 
on 
the 
radial 
distance 
and 
the 
statistical 
parameters. 
Based 
on 
that 
they 
performed 
forward 
simulations 
to 
achieve 
a 
head 
drawdown 
from 
T 
CG(r) 
and 
compared 
it 
to 
ensemble 
averages 
of 
simulated 
two 
dimensional 
pumping 
tests 
in 
heterogeneous 
media. 
They 
stated 
that 
this 
method 
allows 
a 
much 
better 
parameter 
estimation 
for 
T 
(x) 
than 
existing 
methods 
do. 


In 
this 
study 
we 
do 
not 
only 
extend 
the 
results 
of 
Schneider 
and 
Attinger 
[2008] 
to 
three 
dimensions 
but 
will 
go 
one 
step 
further 
by 
introducing 
a 
closed 
form 
solution 
for 
the 
effective 
well 
flow 
hydraulic 
head 
hefw(r). 
It 
describes 
the 
depression 
cone 
of 
a 
three 
dimensional 
pumping 
test 
in 
heterogeneous 
media 
effectively. 
This 
new 
solution 
hefw(r) 
can 
be 
understood 
as 
an 
extension 
of 
Thiem’s 
formula 
(4.1) 
to 
heterogeneous 
media. 
It 
accounts 
for 
the 
statistical 
parameters 
of 
K(x) 
and 
reproduces 
the 
vertical 
mean 
hydraulic 
head 
field 
at 
every 
radial 
distance 
r 
from 
the 
well 
preserving 
the 
flow 
rates. 


In 
contrast 
to 
existing 
solutions, 
hefw(r) 
does 
not 
result 
from 
a 
perturbation 
analysis 
of 
the 
mean 
head 
by 
expansion 
on 
the 
variance 
2. 
Therefore, 
it 
is 
also 
valid 
for 
highly 
heterogeneous 
media. 
Furthermore, 
hefw(r) 
directly 
allows 
to 
estimate 
parameters 
of 
K(x) 
without 
the 
detour 
to 
a 
representative 
description 
of 
the 
conductivity 
as 
done 
by 
Indelman 
et 
al. 
[1996] 
and 
Firmani 
et 
al. 
[2006]. 


After 
stating 
the 
problem, 
we 
will 
shortly 
sum 
up 
known 
results 
for 
near 
and 
far 
field 
representative 
conductivities 
of 
pumping 
tests 
in 
section 
4.2. 
In 
part 
4.3, 
we 
introduce 
the 
upscaling 
method 
Coarse 
Graining 
and 
apply 
it 
to 
three 
dimensional 
well 
flow 
resulting 
in 
the 
representative 
conductivity 
description 
KCG(r). 
In 
section 
4.4, 
we 
derive 
hefw(r) 
by 
solving 
the 
radial 



4.2. 
Statement 
of 
the 
Problem 
flow 
equation 
with 
KCG(r) 
and 
perform 
a 
sensitivity 
analysis 
for 
the 
parameters 
of 
K(x) 
on 
the 
drawdown 
hefw(r). 
We 
finally 
prove 
the 
applicability 
of 
hefw(r) 
by 
analyzing 
three 
dimensional 
numerical 
pumping 
tests 
in 
highly 
heterogeneous 
media. 
Moreover, 
we 
implement 
an 
inverse 
estimation 
procedure 
to 
infer 
on 
the 
statistics 
of 
K(x) 
in 
part 
4.5. 
The 
results 
presented 
within 
this 
section 
refer 
to 
the 
publication 
Zech 
et 
al. 
[2012]. 


4.2. 
Statement 
of 
the 
Problem 
4.2.1. 
Flow 
and 
Conductivity 
Model 
In 
this 
study 
we 
focus 
on 
steady 
state 
pumping 
tests 
with 
fully 
penetrating 
well 
in 
a 
confined 
aquifer, 
where 
water 
is 
extracted 
at 
a 
constant 
pumping 
rate 
Qw. 
The 
three 
dimensional 
flow 
is 
characterized 
by 
Darcy’s 
Law 
in 
combination 
with 
the 
continuity 
equation 


-r(K(x)rh(x)) 
= 
Q(x) 
, 
(4.2) 


with 
x 
=(x1,x2,x3). 
The 
sink/source 
term 
Q(x) 
can 
be 
written 
as 
Qw(xw 
- 
x), 
using 
the 
delta 
distribution 
function 
, 
where 
xw 
is 
the 
position 
of 
the 
well. 


To 
cover 
the 
spatial 
structure 
of 
the 
hydraulic 
conductivity 
K(x) 
we 
model 
it 
as 
log-normal 
distributed 
field 
K(x) 
. 
LN 
(µ, 
2), 
meaning 
that 
ln 
K(x)/K0 
is 
a 
normal 
distributed 
quantity 
with 
mean 
µ 
and 
variance 
2. 
The 
spatial 
correlation 
structure 
described 
by 
a 
Gaussian 
shaped 
covariance 
model 
CV(x 
- 
x0)= 
2(x 
- 
x0) 
includes 
the 
correlation 
function 


 !

222

x1 
x2 
x3

(x) 
= 
exp-- 
- 
, 
(4.3)

`2`2`2

123

where 
`i 
is 
the 
correlation 
length 
in 
ith 
direction. 
To 
reflect 
the 
anisotropic 
structure 
of 
three 
dimensional 
heterogeneous 
media, 
we 
presume 
the 
correlation 
length 
in 
both 
horizontal 
directions 
to 
be 
identical 
`1 
= 
`2 
= 
g 
whereas 
in 
vertical 
direction 
we 
assume 
a 
smaller 
correlation 
length 
`3 
= 
e`, 
with 
e 
. 
[0, 
1] 
denoting 
the 
anisotropy 
ratio. 


4.2.2. 
Far 
and 
Near 
Field 
Conductivities 
The 
issue 
of 
representative 
conductivity 
values 
for 
well 
flow 
has 
been 
discussed 
intensively 
over 
several 
years, 
starting 
with 
Shvidler 
[1966] 
and 
Matheron 
[1967]. 
However, 
most 
of 
the 
studies 
are 
limited 
to 
a 
two 
dimensional 
analysis. 
For 
three 
dimensional 
convergent 
flow 
Indelman 
and 
Abramovich 
[1994] 
concluded 
that 
the 
far 
field 
behavior 
is 
best 
covered 
by 
the 
effective 
hydraulic 
conductivity 
for 
uniform 
flow 
in 
three 
dimensions 


1

Kefu 
= 
KG 
exp 
22 
- 

(e) 
, 
(4.4) 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Figure 
30: 
Relationship 
between 
Kefu, 
KA 
and 
KH. 
The 
values 
on 
the 
left 
refer 
to 
a 
variance 
of 
2 
=1 
and 
KG 
=1m/s 
and 
are 
drawn 
to 
scale. 


where 

(e) 
is 
the 
anisotropy 
function, 
known 
from 
Dagan 
[1989], 
e 
1 
q


(e)= 
parctan( 
1/e2 
- 
1) 
- 
e. 
(4.5)

2(1 
- 
e2)1 
- 
e2 


Depending 
on 
the 
degree 
of 
anisotropy, 

(e) 
varies 
between



13

(e=1) 
and 
zero 
(e=0), 
thus 


causing 
Kiso 


efu 
= 
KG 
exp



16

2to 
be 
the 
limit 
for 
isotropic 
media 
and 
the 
arithmetic 
mean 
KA 
= 




2to 
be 
the 
limit 
for 
stratified 
media. 


KG 
exp

12

For 
a 
representative 
description 
of 
the 
conductivity 
near 
the 
well, 
in 
the 
following 
denoted 
by 
Kwell, 
different 
results 
emerge 
in 
literature 
(e.g. 
Indelman 
et 
al. 
[1996]; 
Indelman 
and 
Dagan 
[2004]), 
depending 
on 
the 
description 
of 
the 
discharge 
at 
the 
well. 
Theoretically, 
either 
a 
constant 
head 
hw 
(corresponding 
to 
a 
Dirichlet 
boundary 
condition) 
or 
a 
constant 
pumping 
rate 
Qw 
(Neumann 
boundary 
condition) 
can 
be 
applied. 
A 
constant 
pumping 
rate, 
in 
the 
following 
denoted 
as 
BCH, 
leads 
to 
a 
varying 
head 
along 
the 
well 
resulting 
is 
the 
harmonic 
mean 
of 
the 
hydraulic 
conductivity 
values 
at 
the 
well. 
On 
the 
contrary, 
a 
constant 
head 
gives 
the 
arithmetic 
mean 
as 
representative 
near 
well 
conductivity 
value. 
If 
ergodicity 
is 
fulfilled 
the 




means 
are 
given 
by 
KA 
= 
KG 
exp

12

2

and 
KH 
= 
KG 
exp-

12

2

. 


An 
additional 
approach 
was 
introduced 
by 
Indelman 
et 
al. 
[1996] 
where 
a 
constant 
discharge 
was 
subdivided 
into 
fluxes 
proportional 
to 
the 
local 
conductivities 
along 
the 
well. 
This 
assumption 
leads 
to 
similar 
conditions 
as 
the 
constant 
head 
boundary 
condition, 
giving 
Kwell 
= 
KA, 
we 
will 
therefore 
call 
this 
BCA. 
All 
of 
these 
results 
were 
confirmed 
by 
numerical 
simulations 
by 
Firmani 
et 
al. 
[2006]. 


An 
important 
feature 
of 
three 
dimensional 
well 
flow 
is 
the 
asymmetric 
relation 
between 
KA, 
KH 
and 
Kefu, 
especially 
when 
taking 
anisotropy 
into 
account. 
As 
shown 
in 
Figure 
30 
the 
distance 
between 
Kefu 
and 
KA 
is 
much 
smaller 
than 
between 
Kefu 
and 
KH, 
even 
becoming 
zero 
for 
stratified 
media. 
Unlike 
in 
two 
dimension 
where 
we 
have 
K2d 
= 
KG 
and 
thus 


efu 


23

| 
ln(K2d 
efu/KH)| 
=efu/KA)| 
= 
| 
ln(K2d 
3d(iso)

whereas 
| 
ln(K/KH)|

efu 


=


2

. 


12

2, 
we 
find 
in 
three 
dimensions 
| 
ln(KThis 
fact 
becomes 
important 
when 
comparing 
results 
for 


3d(iso)
efu 


/KA)| 
=

2 


the 
different 
boundary 
conditions 
BCA 
and 
BCH 
at 
the 
well. 



4.3. 
Method 
of 
Coarse 
Graining 
4.3. 
Method 
of 
Coarse 
Graining 
We 
use 
the 
upscaling 
technique 
Coarse 
Graining 
as 
introduced 
by 
Attinger 
[2003] 
to 
gain 
a 
representative 
description 
for 
the 
well 
flow 
conductivity. 
Schneider 
and 
Attinger 
[2008] 
already 
applied 
Coarse 
Graining 
to 
radial 
convergent 
flow 
in 
two 
dimensions, 
focusing 
on 
regional 
scale 
flow. 
The 
following 
procedure 
of 
deriving 
KCG(r) 
will 
be 
similar, 
but 
we 
focus 
on 
three 
dimensional 
media, 
additionally 
incorporating 
anisotropy. 


4.3.1. 
Coarse 
Graining 
as 
Spatial 
Filtering 
Procedure 
The 
idea 
of 
the 
Coarse 
Graining 
method 
is 
to 
apply 
a 
spatial 
filter 
of 
variable 
volume 
size 
on 
the 
flow 
equation 
(4.2) 
to 
transform 
it 
to 
a 
coarser 
scale. 
The 
approach 
was 
originally 
developed 
for 
Large 
Eddy 
Simulations, 
see 
Layton 
[2002]. 
The 
background 
of 
using 
Coarse 
Graining 
in 
hydrogeological 
modeling 
is 
to 
find 
a 
representative 
conductivity 
on 
a 
coarser 
scale 
still 
considering 
sub-scale 
effects. 


The 
procedure 
is 
based 
on 
averaging 
the 
flow 
equation 
over 
a 
filter 
volume 
controlled 
by 
the 
filter 
length 
scale 
. 
Mathematically 
this 
is 
expressed 
by 
a 
convolution 
with 
a 
smoothing 
function 
f(x). 
It 
results 
in 
a 
flow 
equation 
on 
coarser 
scale 
for 
the 
filtered 
heads 
h(x)= 


R

1

R

0R3 
f(x0)h(x' 
+ 
x)d3x' 
where 
the 
effects 
of 
head 
fluctuations 
smaller 
than 
. 
are 


f(x0)dx

covered 
by 
the 
scale 
dependent 
-filtered 
conductivity 
KCG(x). 
Recapitulating 
the 
method 


. 


derived 
in 
Attinger 
[2003], 
the 
head 
equation 
is 
transformed 
to 
Fourier 
space, 
then 
the 
filtering 
procedure 
is 
applied 
to 
the 
transformed 
head 
equation. 
After 
some 
mathematical 
treatment 
and 
certain 
approximations 
the 
equation 
is 
transformed 
back 
and 
we 
result 
in 
the 
filtered 
head 




KCG

-r(x)rh(x)= 
Q(x) 
, 
(4.6)

. 


where 
Q(x) 
is 
the 
filtered 
sink 
term. 
KCG(x)= 
KCG()+ 
K˜
CG(x,) 
denotes 
the 
-filtered

. 
A 


hydraulic 
conductivity, 
which 
is 
still 
a 
spatial 
random 
function 
on 
a 
coarser 
scale. 
It 
is 
composed 
of 
the 
filtered 
mean 
conductivity 
KCG(), 
depending 
on 
, 
but 
not 
on 
x 
and 
the 
filtered 


A 


fluctuation 
part 
K˜
CG(x,). 


The 
result 
for 
KCG() 
depends 
on 
the 
geostatistical 
model 
and 
on 
the 
choice 
of 
the 
filter 


A 




x

function 
f(x). 
Using 
a 
Gaussian 
shaped 
filter 
function 
f(x) 
/rexp22 
and 
making 
use 
of 
renormalization 
theory, 
Attinger 
[2003] 
presented 
a 
closed 
form 
solution 
for 
isotropic 
media 




by 
KCG()= 
Kiso 
321+ 
2 
- 
3 
efu 
is 
defined 
in 
Eq.(4.4) 
with 
e 
= 
1.

A 
efu 
exp 
14`22 
, 
where 
Kiso 


4.3.2. 
Coarse 
Graining 
for 
Well 
Flow 
Coarse 
Graining 
comprises 
the 
possibility 
of 
applying 
a 
filter 
of 
variable 
volume 
size 
on 
the 
flow 
equation 
which 
is 
of 
significant 
importance 
when 
dealing 
with 
well 
flow. 
For 
pumping 
tests 
the 
singular 
character 
of 
the 
source 
causes 
a 
strong 
influence 
of 
local 
heterogeneities 
on 
the 
flow 
pattern 
near 
the 
well, 
where 
the 
impact 
decreases 
with 
distance 
from 
the 
well. 
We 
therefore 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
think 
that 
an 
adaptive 
filter 
with 
high 
resolution, 
thus 
small 
filter 
volumes 
near 
the 
well 
and 
increasing 
filter 
volumes 
towards 
the 
far 
field, 
applies 
best 
to 
the 
flow 
equation. 
It 
still 
fulfills 
the 
physics 
of 
this 
strongly 
nonuniform 
flow 
pattern. 


As 
proposed 
by 
Attinger 
[2003] 
and 
Schneider 
and 
Attinger 
[2008] 
we 
use 
a 
Gaussian 
shaped 
filter 
function. 
Based 
on 
the 
nonuniform 
flow 
pattern 
for 
radial 
convergent 
flow 
we 
consider 
the 
filter 
length 
scale 
`to 
be 
proportional 
to 
the 
radius 
r`as 
described 
in 
Schneider 
and 
Attinger 
[2008], 
covering 
the 
idea 
of 
nearly 
no 
smoothing 
near 
the 
well 
and 
large 
smoothing 
in 
the 
far 
field. 
For 
isotropic 
media 
we 
choose 
a 
filter 
function 
of 
the 
form 


2 


fiso 
1 
x

(x)= 
3 
exp 
-6,
r 


3/232

r2r

with 
x 
26R3, 
r`the 
radial 
distance 
from 
the 
well, 
treated 
as 
parameter 
and 
`a 
constant 
of 
proportionality, 
determining 
the 
width 
of 
the 
Gaussian 
filter. 
Note 
that 
fiso(x) 
is 
normalized. 


r 


In 
case 
of 
anisotropy 
we 
modify 
the 
filter 
in 
that 
way 
that 
in 
horizontal 
direction 
the 
filter 
width 
is 
1,2 
/6r, 
whereas 
in 
vertical 
direction 
the 
filter 
is 
weighted 
with 
the 
anisotropy 
ratio 
3 
/6e`·6r. 
At 
the 
same 
time 
the 
filter 
volume 
stays 
constant 
and 
independent 
of 
e, 


-

thus 
1 
·62 
·63 
/6r3. 
Hence 
we 
find 
1 
= 
2 
= 
e`

13

r`and 
3 
= 
e`

23

r`which 
results 
in 
a 
more 


general 
filter 
function 
for 
anisotropic 
media 


22-22

faniso 
1 
x1 
+ 
x2 
+ 
ex3

(x1,x2,x3)= 
3 
exp 
-6.`(4.7)

r 


-2/32r2

3/23re

What 
is 
the 
physical 
meaning 
of 
this 
spatial 
averaging? 
Imagine 
the 
filter 
in 
the 
original 
Coarse 
Graining 
procedure 
in 
three 
dimensions 
to 
be 
a 
block 
of 
volume 
3 
around 
a 
point 
x, 
where 
all 
spatial 
heterogeneities 
are 
averaged 
within 
this 
block, 
leading 
to 
a 
filtered 
head 
h(x). 
Then 
the 
radial 
depending 
filter 
can 
be 
seen 
as 
a 
cube 
around 
x 
growing 
with 
distance 
from 
the 
well. 
Near 
the 
well, 
the 
filter 
is 
very 
small 
leaving 
nearly 
all 
heterogeneity 
of 
K(x) 
unchanged, 
whereas 
far 
away 
the 
local 
conductivities 
are 
replaced 
by 
an 
averaged 
value. 
In 
the 
same 
way 
heads 
and 
fluxes 
are 
stepwise 
filtered 
with 
distance 
to 
the 
well. 
The 
influence 
of 
the 
anisotropy 
is 
covered 
by 
an 
adapted 
filter 
in 
vertical 
direction, 
giving 
that 
the 
filtering 
is 
still 
proportional 
to 
the 
correlation 
length 
in 
all 
directions. 
For 
e6

= 
1 
the 
fictitious 
cube 
filter 
around 
a 
point 
x 
is 
deformed 
to 
a 
cuboid 
with 
the 
same 
volume, 
reflecting 
the 
fact 
that 
the 
vertical 
compensation 
is 
reduced 
due 
to 
the 
stratification. 


Following 
the 
line 
of 
derivation 
in 
Attinger 
[2003] 
and 
Schneider 
and 
Attinger 
[2008] 
we 
evaluate 
KCG(r) 
with 
the 
correlation 
function 
in 
Eq.(4.3) 
and 
the 
adapted 
filter 
function 
in 
Eq.(4.7). 


A 


Details 
on 
the 
derivation 
are 
presented 
in 
the 
appendix 
A.1. 
We 
result 
in 
the 
filtered 
conductivity 
KCG(x)= 
KCG(r)+ 
K˜
CG(x,r), 
where 
K˜
CG(x,r) 
is 
the 
fluctuating 
part 
and 
KCG(r) 
is

r 
AA 


the 
scale 
dependent 
mean 
hydraulic 
conductivity 
given 
by 


. 


. 


KCG 


A 
(r)= 
Kefu 
exp 
@2
(e) 
1+ 


3

. 
22 
!-

r

p6

32 


2

`2 


. 


,`(4.8) 


e 



4.4. 
The 
Extended 
Thiem’s 
Solution 
with 
the 
anisotropy 
function 

(e) 
and 
Kefu 
given 
in 
Eq.(4.5) 
and 
Eq.(4.4). 


4.3.3. 
Representative 
Conductivity 
for 
Well 
Flow 
The 
Coarse 
Graining 
conductivity 
KCG(x)= 
KCG(r)+ 
K˜
CG(x,r) 
still 
contains 
local 
fluctu


r 
A 


ations. 
Since 
we 
are 
interested 
in 
a 
representative 
conductivity 
for 
well 
flow 
which 
depends 
only 
on 
the 
radius, 
we 
perform 
a 
vertical 
average 
over 
a 
sufficiently 
thick 
aquifer. 
This 
reflects 
the 
fact 
that 
heads 
measured 
in 
real 
pumping 
tests 
can 
be 
regarded 
as 
means 
over 
the 
vertical 
extension 
of 
the 
well. 
Since 
the 
average 
of 
the 
fluctuating 
part 
K˜
CG(x,r) 
is 
zero 
by 
definition, 
KCG(r) 
remains 
as 
representative 
conductivity 
value 
for 
well 
flow. 
It 
can 
be 
seen 
in 


A 
Eq.(4.8) 
that 
the 
asymptotic 
behavior 
of 
KCG(r) 
covers 
KA 
as 
near 
field 
and 
Kefu 
as 
far 
field 


A 


representative 
conductivity 
values. 


As 
discussed 
in 
section 
4.2.2 
different 
boundary 
conditions 
at 
the 
well 
result 
in 
different 
near 
field 
representative 
conductivities. 
We 
therefore 
generalize 
the 
result 
to 
both 
possibilities 
of 
Kwell. 
For 
the 
BCA 
with 
Kwell 
= 
KA 
we 
use 
the 
abbreviation 
A 
= 
2
(e) 
and 
for 
BCH 
with 
Kwell 
= 
KH 
we 
write 
H 
= 
2(
(e) 
- 
1). 
We 
result 
in 
a 
general 
expression 
for 
the 
representative 
well 
flow 
conductivity 
for 
both 
boundary 
conditions 


. 


KCG(r)= 
Kefu 
exp 
@(2 
,e) 
1+ 


3

. 
22 
!-

r

v 


32 


. 


2

. 


.`(4.9) 


e 


`2 


A 
universal 
definition 
of 
`is 
given 
by 
`= 
ln(Kwell/Kefu). 
It 
allows 
KCG(r) 
to 
act 
as 
a 
general 
interpolating 
function 
between 
the 
near 
and 
far 
field 
representative 
conductivity 
values 
Kwell 
and 
Kefu, 
respectively. 


4.4. 
The 
Extended 
Thiem’s 
Solution 
4.4.1. 
Derivation 
of 
hefw(r) 
The 
Coarse 
Graining 
conductivity 
KCG(r), 
as 
given 
in 
Eq.(4.9) 
reflects 
the 
impact 
of 
heterogeneity 
on 
well 
flow 
in 
porous 
media. 
It 
is 
the 
representative 
conductivity 
value 
for 
which 
the 
solution 
of 
the 
flow 
equation 
best 
fits 
the 
drawdown 
of 
pumping 
tests. 
More 
precisely 
inserting 
KCG(r) 
to 
the 
flow 
equation 
(4.2) 
delivers 
a 
head 
which 
reproduces 
the 
vertically 
averaged 
drawdown 
of 
a 
three 
dimensional 
pumping 
test 
at 
every 
radial 
distance 
r`from 
the 
well 
in 
dependence 
on 
the 
parameters 
KG, 
2,`and 
e`of 
K(x). 


The 
final 
step 
in 
our 
approach 
is 
to 
derive 
the 
effective 
well 
flow 
head 
hefw(r) 
by 
solving 
the 
radial 
flow 
equation 
with 
KCG(r). 
We 
transform 
Eq. 
(4.2) 
to 
polar 
coordinates, 
evaluate 
the 
vertical 
component 
and 
result 
in 


hefw(r) 


= 
C1 
exp(-) 
ln 
r`+ 
C1 
sinh()U1(r) 
(4.10)

R`
+C1 
(1 
- 
cosh()) 
U2(r)+ 
h(R) 
,`


4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Figure 
31: 
Comparison 
of 
hefw(r) 
and 
Thiem’s 
solution 
for 
two 
conductivity 
fields 
(KG 
= 


10-4 
m/s, 
2 
=1, 
e 
=1, 
f 
=5m, 
and 
f 
=10m) 
and 
both 
boundary 
conditions, 


BCA 
and 
BCH. 
Reference 
head 
is 
h(R 
=32m)=0m. 


with 
C1 
=- 
Qw 
and 


2LKefu 


r)+11 
1 


U1(r)=lnu(+ 


u(R)+1- 
u(r)u(R) 
, 
r)1 
1 
1 
1 


U2(r)=lnu(
, 


u(R)- 
2u(r)2 
+2u(R)2 
- 
4u(r)4 
+4u(R)4 


q

p

where 
u(r)serves 
as 
abbreviation 
for 
u(r)=1+(r)2/(3e`)2 
and 
. 
=ln(Kwell/Kefu). 
L 
is 
the 
aquifer 
thickness 
and 
Qw 
the 
pumping 
rate. 
h(R) 
is 
a 
reference 
head 
measured 
at 
the 
arbitrary 
distance 
R. 
This 
can 
be 
e.g. 
the 
head 
at 
the 
well 
h(R 
=rw)or 
a 
measured 
head 
value 
h(R)=hR 
in 
the 
far 
field, 
within 
the 
radius 
of 
influence 
of 
the 
pumping 
test. 
Details 
on 
the 
mathematical 
derivation 
can 
be 
found 
in 
the 
appendix 
A.3. 


The 
solution 
hefw(r)is 
a 
general 
expression 
for 
both 
boundary 
conditions 
at 
the 
well 
with 
hefw(r)

A 


hefw

for 
Kwell 
=KA 
(BCA) 
and 
hefw(r)for 
Kwell 
=KH 
(BCH). 
hefw(r)and 
(r)differ 
in 
the 


H 
AH 


parameter 
. 
For 
BCA 
we 
fix 
A 
=0.8. 
For 
BCH 
we 
set 
H 
=1.6. 
The 
relation 
A 
=0.5* 
H 
results 
from 
the 
fact 
that 
the 
transition 
zone 
from 
KA 
to 
Kefu 
is 
half 
the 
size 
of 
the 
transition 
from 
KH 
to 
Kefu. 
It 
results 
from 
the 
asymmetric 
relation 
between 
Kefu, 
KA 
and 
KH 
as 
discussed 
in 
section 
4.2.2, 
see 
Figure 
30. 


Analyzing 
hefw(r)in 
Eq.(4.10), 
we 
see 
that 
the 
first 
term 
resultsin 
Thiem’s 
Formula 
with 
Kwell 
as 
homogeneous 
substitute 
value. 
The 
terms 
U1 
and 
U2 
operate 
as 
correction 
terms, 
gaining 
in 
influence 
with 
increasing 
distance 
from 
the 
well. 
For 
homogeneous 
media 
. 
becomes 
zero 
and 
Eq.(4.10) 
reduces 
to 
Thiem’s 
solution, 
Eq.(4.1). 



4.4. 
The 
Extended 
Thiem’s 
Solution 
Figure 
32: 
Derivation 
of 
hefw(r)and 
hefw(r)with 
respect 
to 
parameters 
2 
and 
R 
in 
dimen-

AH 


sionless 
scale 
r/`. 
The 
left 
plot 
shows 
the 
derivatives 
for 
2 
=0.5, 
the 
right 
one 
for 
2 
=2. 
Used 
setting: 
KG 
=10-4 
m/s, 
e 
=1and 
h(r/R 
=6.4)=0m. 


As 
shown 
in 
Figure 
31, 
hefw(r)interpolates 
between 
the 
drawdowns 
of 
Thiem’s 
solution 
with 
Kwell 
and 
Kefu 
as 
homogeneous 
substitute 
values, 
where 
the 
transition 
is 
determined 
by 
the 
correlation 
length 
`. 
In 
Figure 
31 
also 
the 
different 
sizes 
of 
the 
transition 
zone 
for 
BCA 
and 
BCH 
can 
be 
seen. 


4.4.2. 
Sensitivity 
Analysis 
Formula 
(4.10) 
for 
the 
effective 
well 
flow 
head 
hefw(r) 
allows 
for 
a 
detailed 
analysis 
of 
the 
influence 
of 
the 
parameters 
of 
K(x) 
on 
the 
drawdown. 
Depending 
on 
Kwell 
meaning 
the 
applied 
boundary 
condition 
(BCA 
or 
BCH), 
the 
effects 
can 
be 
quite 
different. 
However, 
for 
both 
boundary 
conditions 
we 
state 
from 
our 
findings 
that 
the 
geometric 
mean 
KG 
influences 
the 
entire 
curve. 
The 
variance 
2 
mainly 
impacts 
on 
the 
drawdown 
at 
the 
well. 
And 
the 
correlation 
length 
R 
determines 
the 
’velocity’ 
of 
transition 
from 
near 
to 
far 
field 
behavior. 


The 
variance 
2 
determines 
the 
magnitude 
of 
the 
drawdown 
in 
the 
vicinity 
of 
the 
well. 
For 
BCA 
a 
higher 
variance 
2 
causes 
larger 
values 
for 
KA 
and 
hence 
flattens 
the 
depression 
cone. 
For 
BCH 
the 
opposite 
effect 
appears. 
The 
larger 
2, 
the 
steeper 
becomes 
the 
depression 
cone. 
This 
effect 
can 
be 
seen 
in 
Figure 
32, 
where 
we 
plotted 
the 
absolute 
values 
of 
the 
derivative 


of 
hefw(

r)with 
respect 
to 
the 
variance 
2 
for 
both 
near 
well 
conductivity 
values 
Kwell 
=KA 
and 
Kwell 
=KH 
and 
two 
choices 
of 
2. 
For 
hefw(r), 
we 
can 
see 
that 
the 
influence 
of 
the 
variance 


A 
near 
the 
well 
is 
very 
strong 
and 
smooths 
out 
in 
the 
far 
field, 
but 
is 
still 
present 
through 
Kefu. 
For 
hefw 


H 
(r) 
the 
effect 
of 
2 
on 
the 
drawdown 
reverses 
because 
of 
the 
different 
signs 
in 
the 
exponent 
of 
KH 
and 
Kefu. 
In 
particular, 
within 
the 
distance 
of 
the 
first 
correlation 
length 
there 
exists 
an 
area 
where 
a 
change 
in 
2 
does 
not 
influence 
the 
depression 
of 
hefw(r)for 
BCH. 


The 
effect 
of 
a 
change 
in 
correlation 
length 
R 
on 
the 
drawdown 
can 
be 
seen 
in 
Figure 
31 
where 
we 
plotted 
hefw(r)for 
two 
different 
correlation 
lengths. 
The 
larger 
`, 
the 
longer 
takes 
the 
transition 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Figure 
33: 
Contour 
plot 
of 
anisotropy 
ratio 
e: 
the 
black 
lines 
show 
the 
isolines 
for 
the 
head 
hefw(r)in 
dependence 
on 
the 
dimensionless 
distance 
r/f 
and 
the 
anisotropy 


A 


ratio 
e. 
Used 
setting: 
KG 
=10-4 
m/s, 
2 
=1and 
h(r/f 
=6.4)=0m. 


from 
Kwell 
to 
Kefu. 
In 
Figure 
32 
we 
plotted 
the 
dimensionless 
version 
of 
the 
derivative 
of 
hefw(r) 
with 
respect 
to 
the 
correlation 
length 
f 
for 
two 
different 
values 
of 
the 
variance 
2. 
Noticeable 
is 
the 
increasing 
influence 
of 
f 
with 
increasing 
variance 
2 
for 
both 
boundary 
conditions 
BCA 
and 
BCH. 
This 
is 
caused 
by 
a 
larger 
distance 
between 
Kwell 
and 
Kefu 
for 
larger 
variances. 
Furthermore, 
it 
can 
be 
seen 
that 
the 
influence 
of 
f 
on 
hefw(r)vanishes 
quickly 
with 
increasing 
distance 
to 
the 
well. 
This 
is 
caused 
by 
the 
fact 
that 
the 
drawdown 
reaches 
the 
far 
field 
behavior 
after 
approximately 
two 
correlation 
lengths, 
meaning 
that 
hefw(r> 
2`)= 
hThiem(r> 
2`) 
with 
Kefu 
as 
homogeneous 
substitute 
value 
in 
Eq. 
(4.1). 
This 
is 
in 
line 
with 
the 
findings 
of 
Neuman 
et 
al. 
[2004, 
2007]. 


If 
anisotropy 
(e6

=1) 
is 
assumed 
an 
additional 
quantity 
impacts 
the 
hydraulic 
head 
drawdown. 
The 
anisotropy 
rate 
e 
influences 
the 
far 
field 
behavior 
by 
its 
impact 
on 
Kefu 
in 
Eq.(4.4). 
Additionally, 
it 
is 
present 
as 
a 
scaling 
factor 
to 
the 
horizontal 
correlation 
length 
(visible 
in 
the 
expression 
of 
u(r)), 
since 
the 
relation 
between 
Kefu 
and 
Kwell 
also 
manipulates 
the 
transition 
zone. 
The 
impact 
of 
a 
change 
in 
e 
on 
the 
drawdown 
is 
plotted 
in 
Figure 
33 
for 
BCA, 
where 
we 
see 
that 
the 
sensitivity 
of 
hefw(r)towards 
e 
is 
very 
low. 
The 
same 
can 
be 
observed 
for 
BCH. 


A 


This 
can 
be 
explained 
by 
the 
fact 
that 
a 
stronger 
stratification 
does 
not 
impact 
very 
much 
on 
the 
flow 
pattern 
of 
pumping 
tests, 
which 
is 
mainly 
determined 
by 
horizontal 
flow. 


4.4.3. 
The 
Head 
Solution 
as 
an 
Inverse 
Estimation 
Tool 
The 
analytical 
solution 
hefw(r)provides 
a 
useful 
tool 
to 
analyze 
pumping 
test 
data. 
Depending 
on 
the 
available 
amount 
of 
data, 
hefw(r)enables 
the 
inverse 
estimation 
of 
statistical 
properties 
under 
the 
assumption 
of 
a 
log-normal, 
Gaussian 
correlated 
hydraulic 
conductivity 
field. 



4.5. 
Interpreting 
Numerical 
Pumping 
Test 
Examining 
Eq.(4.10) 
more 
closely 
we 
see 
that 
KG 
and 
2 
are 
both 
incorporated 
in 
Kwell 




and 
Kefu, 
where 
the 
relation 
Kwell 
= 
KG 
exp±12 
2only 
holds 
if 
ergodicity 
is 
fulfilled 
at 
the 
well. 
Generalizing 
hefw(r) 
to 
non-ergodic 
conditions, 
we 
shift 
the 
input 
variables 
from 
KG 
and 
2 
to 
Kwell 
and 
Kefu 
by 
using 
. 
= 
ln(Kwell/Kefu). 


However, 
from 
the 
discussion 
in 
section 
4.4.2 
it 
can 
be 
seen 
that 
Kwell, 
Kefu 
(respectively 
2 
and 
KG 
under 
ergodic 
conditions) 
and 
f 
all 
have 
a 
unique 
influence 
on 
the 
drawdown, 
which 
serves 
as 
a 
good 
basis 
for 
estimating 
them 
through 
a 
regression. 
Furthermore, 
Figure 
32 
shows 
that 
the 
estimation 
of 
f 
becomes 
even 
more 
certain 
the 
higher 
the 
variance 
2 
is. 
In 
contrast, 
e 
cannot 
be 
treated 
as 
an 
independent 
parameter, 
because 
of 
its 
low 
influence 
on 
hefw(r), 
see 
Figure 
33. 
We 
therefore 
support 
the 
statement 
of 
Firmani 
et 
al. 
[2006] 
that 
an 
estimation 
of 
the 
anisotropy 
ratio 
e 
through 
pumping 
test 
data 
is 
very 
error 
prone. 


What 
is 
not 
discussed 
until 
now 
is 
the 
influence 
of 
the 
boundary 
condition 
assumed 
at 
the 
well 
on 
the 
parameter 
estimation. 
The 
certainty 
in 
estimating 
Kwell 
and 
Kefu 
(respectively 
KG 
and 
2) 
is 
independent 
on 
the 
choice 
of 
BCA 
or 
BCH. 
But 
to 
infer 
on 
f 
the 
crucial 
area 
is 
the 
transition 
zone 
from 
Kwell 
to 
Kefu. 
This 
zone 
is 
smaller 
for 
BCA 
than 
for 
BCH, 
independent 
of 
2, 
because 
of 
the 
relation 
of 
Kefu 
to 
KA 
and 
KH, 
see 
Figure 
30. 
In 
case 
of 
anisotropy 
the 
transition 
zone 
for 
BCA 
becomes 
even 
smaller 
and 
vanishes 
for 
stratified 
media 
since 
Kefu(e 
= 
0) 
= 
KA. 
This 
makes 
a 
parameter 
estimation 
with 
BCA 
more 
difficult 
and 
less 
reliable. 
For 
BCH 
the 
opposite 
effect 
occurs. 
A 
convergence 
of 
Kefu 
to 
KA 
enlarges 
the 
transition 
zone, 
thus 
improves 
the 
ability 
of 
estimating 
`. 


4.5. 
Interpreting 
Numerical 
Pumping 
Test 
In 
this 
section 
we 
analyze 
three 
dimensional 
numerical 
pumping 
tests 
in 
highly 
heterogeneous 
porous 
media 
to 
confirm 
the 
validity 
of 
hefw(r) 
to 
describe 
well 
flow 
effectively. 
We 
examine 
the 
drawdown 
results 
and 
develop 
a 
method 
to 
infer 
on 
the 
statistical 
parameters 
of 
the 
conductivity 
distribution 
K(x). 
With 
our 
simulations 
we 
characterize 
the 
transition 
zone 
of 
near 
well 
to 
far 
field 
behavior 
being 
the 
main 
area 
of 
influence 
of 
heterogeneity. 
Furthermore, 
we 
discuss 
the 
question 
if 
a 
single 
pumping 
test 
realization 
is 
sufficient 
to 
infer 
on 
all 
parameters 
of 
K(x) 
and 
present 
several 
ways 
to 
cope 
with 
a 
lack 
of 
ergodicity. 


4.5.1. 
Simulation 
Setup 
To 
examine 
the 
question 
of 
ergodicity 
and 
the 
influence 
of 
domain 
size 
on 
a 
numerical 
pumping 
test 
drawdown, 
we 
perform 
simulations 
on 
several 
meshes 
of 
different 
horizontal 
and 
vertical 
extension. 
Starting 
from 
the 
findings 
of 
Firmani 
et 
al. 
[2006], 
we 
generate 
a 
large 
domain 
G3 
with 
horizontal 
mesh 
size 
of 
R3 
= 
40`, 
a 
medium 
sized 
domain 
G2 
with 
R2 
= 
25.6f 
and 
a 
small 
one, 
G1 
with 
R1 
=6.4`, 
all 
of 
them 
having 
a 
vertical 
extension 
of 
64`v. 
Adapting 
to 
the 
radial 
flow 
system, 
we 
establish 
a 
mesh 
refinement 
in 
the 
range 
of 
-f 
to 
`. 
It 
provides 
a 
high 
resolution 
of 
the 
head 
drawdown 
near 
the 
well. 
Additionally, 
the 
well 
is 
not 
included 
as 
a 
point 
source 
but 
as 
a 
hollow 
cylinder 
with 
radius 
rw 
=0.01m. 
All 
meshes 
refer 
to 
a 
correlation 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Ensemble 


KG 
[10-4 
m/s] 
2 
i 
[m] 
e 


E1 


1.0 
1.0 
5.0 
1 
E2 


1.0 
2.0 
5.0 
1 
E3 


1.0 
1.0 
10.0 
1 
E4 


1.0 
1.0 
10.0 
0.5 
Table 
17: 
Input 
parameters 
of 
generated 
ensembles. 


length 
of 
i 
= 
5 
m 
and 
a 
resolution 
of 
5 
cells 
per 
correlation 
length, 
resulting 
in 
a 
cell 
size 
of 
1m, 
being 
in 
the 
range 
of 
an 
idealized 
small 
scale 
pumping 
test. 
Note 
that 
for 
the 
ensembles 
with 
a 
correlation 
length 
of 
i 
= 
10 
m 
these 
ratios 
change 
while 
keeping 
the 
meshes 
unchanged. 


The 
boundary 
conditions 
of 
the 
simulations 
are 
the 
following: 
The 
upper 
and 
lower 
horizontal 
planes 
delimiting 
the 
domain 
are 
set 
impervious 
which 
reflects 
a 
confined 
aquifer. 
At 
the 
radial 
distance 
Ri 
according 
to 
Gi 
a 
constant 
head 
h(Ri) 
= 
0 
is 
applied, 
giving 
a 
circular 
outer 
horizontal 
boundary 
condition. 
At 
the 
well 
we 
use 
a 
constant 
total 
pumping 
rate 
of 
Qw 
= 
-10-3 
m3/s 
with 
two 
different 
boundary 
conditions: 
(i) 
constant 
flux 
(BCH), 
where 
we 
assign 
the 
same 
pumping 
rate 
Qi 
at 
every 
grid 
cell, 
resulting 
in 
Kwell 
= 
KH 
and 
(ii) 
proportional 
flux 
(BCA), 
where 
the 
assigned 
rate 
is 
proportional 
to 
the 
local 
conductivity 
of 
the 
grid 
cell 
Qi 
. 
Ki, 
giving 
Kwell 
= 
KA, 
as 
stated 
by 
Indelman 
et 
al. 
[1996] 
and 
Firmani 
et 
al. 
[2006]. 


All 
simulations 
are 
performed 
using 
the 
finite 
element 
software 
OpenGeoSys 
developed 
by 
Kolditz 
et 
al. 
[2012a]. 
The 
code 
was 
tested 
against 
a 
steady 
state 
pumping 
test 
with 
homogeneous 
conductivity 
and 
the 
results 
in 
two 
and 
three 
dimension 
are 
in 
very 
good 
agreement 
with 
the 
analytical 
solution 
of 
Thiem 
(4.1). 


To 
generate 
heterogeneous, 
log-normal 
distributed, 
Gaussian 
correlated 
conductivity 
fields 
we 
make 
use 
of 
the 
statistical 
field 
generator 
randomfield 
provided 
by 
Cirpka 
[2010]. 
Hydraulic 
conductivity 
fields 
are 
created 
by 
a 
given 
deterministic 
power 
spectra, 
as 
described 
in 
Dykaar 
and 
Kitanidis 
[1992a]. 
We 
generate 
several 
realizations 
of 
hydraulic 
conductivity 
fields 
in 
three 
dimensions 
of 
the 
same 
statistical 
parameters, 
i.e. 
geometric 
mean 
KG, 
variance 
2, 
correlation 
length 
i 
and 
anisotropy 
ratio 
e 
all 
forming 
one 
ensemble. 
To 
investigate 
the 
influence 
of 
parameters 
on 
the 
drawdown 
numerically, 
we 
generate 
several 
ensembles 
with 
varying 
parameter 
setups, 
listed 
in 
Table 
17. 
In 
this 
study 
every 
ensemble 
consists 
of 
20 
realizations. 


RL

We 
focus 
on 
the 
vertical 
average 
of 
the 
simulated 
head 
hh(x1,x2)) 
=1/L0 
h(x1,x2,x3)dx3. 
We 
take 
the 
mean 
of 
hh(x1,x2)) 
on 
the 
x1 
and 
x2 
axes 
as 
representative 
drawdown 
hh(r)) 
for 
a 
realization, 
only 
depending 
on 
the 
distance 
to 
the 
well 
r. 
In 
a 
first 
step, 
we 
compare 
the 
drawdown 
hh(r)) 
with 
hefw(r) 
and 
Thiem’s 
solution 
hThiem(r), 
using 
Kefu 
and 
Kwell 
as 
homogeneous 
substitutes. 
In 
a 
second 
step, 
we 
use 
hefw(r) 
to 
infer 
on 
the 
statistical 
parameters 
of 
K(x) 
by 
applying 
a 
nonlinear 
regression 
to 
find 
estimates 
Kˆ
G,ˆ2 
,`ˆwhich 
minimize 
the 
mean 
square 
error 
between 
the 
numerical 
drawdown 
data 
hh(r)) 
and 
hefw(r). 


In 
this 
estimation 
procedure 
we 
include 
all 
available 
points 
ri 
in 
the 
range 
of 
5i 
(corresponds 
to 
25m) 
of 
the 
drawdown 
hh(r)i. 
The 
question 
of 
the 
applicability 
of 
hefw(r) 
on 
limited 
head 
data 
is 
of 
quite 
a 
complex 
nature. 
Answering 
it 
from 
the 
perspective 
of 
numerical 
pumping 
tests 



4.5. 
Interpreting 
Numerical 
Pumping 
Test 
Figure 
34: 
Comparison 
of 
20 
simulated 
drawdowns 
(BCH 
on 
G1) 
from 
ensemble 
E2 
(KG 
= 
10-4 
m/s, 
2 
=2, 
g 
=5m, 
e 
=1) 
in 
log 
scale. 


with 
full 
data 
availability 
results 
in 
a 
complicated 
selection 
criteria. 
The 
necessary 
detailed 
statistical 
analysis 
is 
given 
in 
section 
5. 
Though 
in 
reality 
a 
full 
range 
of 
head 
data 
will 
not 
be 
available, 
we 
first 
focus 
on 
examining 
the 
validity 
of 
hefw(r) 
more 
theoretically 
in 
order 
to 
describe 
three 
dimensional 
well 
flow 
effectively. 


4.5.2. 
Influence 
of 
Domain 
size 
We 
test 
different 
horizontally 
extended 
domains 
to 
investigate 
how 
the 
position 
of 
the 
outer 
boundary 
condition 
influences 
the 
simulated 
head. 
Comparing 
the 
numerical 
drawdown 
results 
for 
identical 
heterogeneity 
fields 
on 
all 
three 
domains 
G1, 
G2 
and 
G3 
shows 
negligible 
differ




ences 
in 
the 
qualitative 
behavior, 
valid 
for 
all 
realizations 
and 
all 
ensembles. 
A 
quantitative 


hhG1(r)i-hhG3 
(r)) 


between 
the 


hhG1(r)i

comparison 
proves 
that 
the 
maximum 
relative 
difference 
maxr





drawdowns 
on 
G1 
and 
G3 
is 
less 
than 
0.5%. 
Furthermore, 
the 
inverse 
estimation 
results 
are 


nearly 
identical 
(not 
shown 
here). 
The 
numerical 
simulations 
-independent 
of 
the 
used 
domain 
-confirmed 
the 
findings 
of 
the 
sensitivity 
analysis 
in 
4.4.2 
that 
the 
mean 
drawdown 
reaches 
the 
far 
field 
representative 
behavior 
after 
less 
than 
two 
correlation 
lengths. 
It 
shows 
that 
reducing 
the 
horizontal 
extension 
does 
not 
impact 
on 
the 
simulated 
drawdown, 
as 
long 
as 
the 
outer 
boundary 
applies 
at 
a 
distance 
of 
more 
than 
tree 
to 
four 
correlation 
length. 
In 
the 
following 
we 
will 
thus 
reduce 
the 
discussion 
on 
numerical 
results 
to 
domain 
setup 
G1, 
since 
we 
believe 
that 
it 
contains 
all 
information 
necessary 
to 
determine 
the 
impact 
of 
heterogeneity 
on 
the 
drawdown. 


In 
the 
next 
step, 
we 
test 
the 
influence 
of 
the 
vertical 
domain 
extent 
on 
the 
drawdown 
results. 
We 
observe 
significant 
differences 
in 
the 
mean 
drawdown 
when 
reducing 
the 
vertical 
domain 
size, 
being 
in 
the 
line 
with 
the 
findings 
of 
Firmani 
et 
al. 
[2006]. 
They 
state 
that 
a 
vertical 
extension 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Figure 
35: 
Plot 
of 
simulated 
drawdowns 
(BCA 
and 
BCH 
on 
G1) 
of 
a 
single 
realization 
of 


ensemble 
E1 
(KG 
= 
10-4 
m/s, 
2 
=1, 
f 
=5m, 
e 
=1) 
versus 
hefw(r) 
with 
local 


and 
theoretical 
value 
for 
near 
well 
conductivity: 
hKAi=1.422 
·10-4 
m/s, 
KA 
= 


1.649 
·10-4 
m/s, 
hKHi=0.523 
·10-4 
m/s, 
KH 
=0.607 
·10-4 
m/s. 
of 
L 
60`z 
ensures 
ergodicity. 
To 
check 
whether 
this 
also 
holds 
true 
for 
our 
simulation 
setup 
with 
L 
being 
even 
64`z 
we 
compare 
the 
resulting 
drawdowns 
hh(r)ifor 
one 
ensemble 
and 
find 
large 
differences 
between 
the 
20 
realizations. 
We 
find 
that 
behavior 
in 
particular 
near 
the 
well, 
as 
shown 
in 
Figure 
34 
for 
ensemble 
E2, 
which 
is 
in 
contradiction 
to 
the 
findings 
of 
Firmani 
et 
al. 
[2006]. 


The 
reason 
for 
the 
spreading 
within 
one 
ensemble 
becomes 
evident 
when 
relating 
the 
drawdown 
hh(r)iof 
every 
realization 
to 
its 
local 
hydraulic 
conductivity 
value 
at 
the 
well 
hKwelli. 


P

1 
m

According 
to 
both 
boundary 
conditions, 
we 
calculate 
the 
arithmetic 
mean 
hKAi= 
mi=1 
Ki

-1

P

1 
m 
1

and 
the 
harmonic 
mean 
hKHi=i=1 
of 
the 
conductivity 
values 
Ki 
of 
the 
m 
= 
320 


mKi

cells 
along 
the 
well. 
As 
predicted 
by 
theory, 
the 
local 
mean 
hKwellidetermines 
the 
depression 
in 
the 
vicinity 
of 
the 
well. 
But 
in 
contrast 
to 
Firmani 
et 
al. 
[2006], 
who 
found 
differences 
of 




less 
than 
1.4% 
between 
hKwelliand 
the 
theoretical 
value 
Kwell 


= 
KG 
exp±

12

2

, 
we 
found 


differences 
up 
to 
20%. 




The 
discrepancies 
between 
local 
and 
theoretical 
expected 
means, 
hKwelli6= 
KG 
exp±

12

2

, 
we 


trace 
back 
on 
a 
too 
small 
sample 
size. 
A 
mean 
over 
320 
K-values, 
moreover 
spatially 
correlated, 
is 
not 
representative 
for 
a 
full 
K-field 
of 
over 
one 
million 
values, 
ensuring 
the 
convergence 
to 
the 
theoretical 
expected 
mean. 
We 
therefore 
state 
that 
a 
single 
three 
dimensional 
pumping 
test, 
with 
a 
randomly 
chosen 
well 
position, 
does 
not 
fulfill 
ergodic 
conditions, 
even 
for 
very 
large 
vertical 
domain 
extensions. 



4.5. 
Interpreting 
Numerical 
Pumping 
Test 
Figure 
36: 
Statistics 
on 
results 
of 
ensemble 
E1 
(n 
= 
20): 
The 
upper 
boxplots 
compare 
the 
local 
means 
at 
the 
well 
hKwell) 
with 
the 
inverse 
estimated 
values 
Kˆ
well. 
The 
lower 
boxplots 
compare 
the 
estimation 
results 
of 
Kˆ
efu 
and 
`ˆfor 
both 
boundary 
conditions 
(BCA 
and 
BCH). 
The 
appropriate 
theoretical 
value 
is 
marked 
on 
the 
vertical 
axis. 


4.5.3. 
Analysis 
of 
a 
Single 
Pumping 
Test 
To 
overcome 
the 
lack 
of 
ergodicity, 
we 
analyze 
single 
pumping 
tests 
with 
hefw(r), 
accounting 
for 
local 
variations 
at 
the 
well. 
We 
use 
the 
universal 
definition 
of 
. 
= 
ln(hKwelli/Kefu) 
as 
introduced 
and 
discussed 
in 
section 
4.3.3 
and 
4.4.3 
with 
the 
local 
mean 
hKwell) 
instead 
of 
the 
theoretical 
value 
Kwell 
and 
find 
very 
high 
accordance 
between 
hh(r)) 
and 
hefw(r). 
Figure 
35 
shows 
the 
impact 
of 
hKwell) 
on 
the 
simulated 
drawdown 
of 
a 
single 
realization 
for 
ensemble 
E1 
and 
the 
large 
differences 
between 
hefw(r) 
when 
using 
hKwell) 
compared 
to 
Kwell. 
It 
can 
be 
seen 
quite 
well 
that 
hefw(r) 
with 
hKwell) 
matches 
the 
simulated 
drawdown 
for 
both 
boundary 
conditions. 
This 
is 
clearly 
not 
the 
case 
for 
hefw(r) 
with 
Kwell, 
because 
of 
the 
significant 
differences 
of 
13.8% 
between 
Kwell 
and 
hKwell) 
for 
both 
boundary 
conditions. 


It 
should 
be 
mentioned 
that 
a 
modification 
of 
hefw(r) 
goes 
along 
with 
a 
shift 
of 
input 
parameters 
from 
KG 
and 
2 
to 
the 
far 
and 
near 
field 
representative 
values 
hKwell) 
and 
Kefu. 
In 
terms 
of 
analyzing 
single 
realizations, 
it 
can 
be 
interpreted 
as 
a 
decoupling 
of 
the 
full 
field 
variance 
2 
incorporated 
in 
Kefu 
and 
a 
local 
variance 
at 
the 
well 
h2) 
incorporated 
in 
hKwelli. 
Both 
values, 
local 
and 
full 
field 
variance, 
might 
differ 
significantly 
from 
each 
other. 


We 
performed 
a 
parameter 
estimation 
by 
using 
Eq. 
(4.10) 
with 
. 
= 
ln 
hKwell) 
on 
every 
realiza-

Kefu 


tion 
of 
ensemble 
E1. 
A 
statistical 
analysis 
of 
the 
results 
is 
shown 
in 
the 
box 
plots 
of 
Figure 
36. 
For 
both 
boundary 
conditions, 
we 
find 
a 
very 
high 
accordance 
between 
the 
local 
mean 
values 
at 
the 
well 
hKwell) 
(hKA) 
and 
hKHi, 
respectively) 
and 
the 
estimated 
values 
Kˆ
well 
(Kˆ
A 
and 
Kˆ
H), 
plotted 
in 
the 
upper 
box 
plots. 
The 
estimation 
results 
for 
the 
far 
field 
conductivity 
Kˆ
efu 
are 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
ˆKG 
[10-4 
m/s] 
ˆ2 
ˆKefu 
[10-4 
m/s] 
ˆKA 
[10-4 
m/s] 
`ˆ 
[m] 
E1 
1.0 
1.0 
1.181 
1.649 
5.0 
0.972 
1.064 
1.16 
1.654 
4.99 
(±0.014) 
(±0.028) 
(±0.013) 
(±0.008) 
(±0.42) 
E2 
1.0 
2.0 
1.396 
2.718 
5.0 
0.946 
2.083 
1.338 
2.68 
5.05 
(±0.023) 
(±0.05) 
(±0.023) 
(±0.028) 
(±0.41) 
E3 
1.0 
1.0 
1.181 
1.649 
10.0 
0.954 
1.079 
1.142 
1.636 
10.12 
(±0.005) 
(±0.011) 
(±0.004) 
(±0.005) 
(±0.55) 


Table 
18: 
Parameter 
estimation 
results 
of 
simulated 
mean 
head 
hh(r)) 
for 
three 
ensembles 


with 
BCA. 
Expected 
ensemble 
parameters 
in 
italic, 
the 
inverse 
estimates 
in 
bold 
and 


the 
95% 
confidence 
intervalls 
in 
brackets. 


shown 
in 
the 
lower 
left 
box 
plot. 
We 
can 
see 
that 
BCA 
underestimates 
and 
BCH 
overestimates 
the 
theoretical 
expected 
value 
Kefu, 
but 
only 
by 
small 
deviations. 
The 
estimated 
values 
for 
the 
correlation 
length 
`ˆ 
match 
the 
theoretical 
value 
f 
very 
well, 
visible 
in 
the 
lower 
right 
box 
plot. 
However, 
the 
variability 
is 
quite 
large, 
especially 
for 
BCA. 
This 
corresponds 
to 
the 
large 
spread 
of 
hKA) 
from 
KA, 
because 
the 
estimation 
of 
the 
correlation 
length 
depends 
strongly 
on 
the 
transition 
zone 
and 
is 
therefore 
triggered 
by 
the 
local 
discrepancies 
at 
the 
well. 


We 
conclude 
from 
the 
analysis 
of 
a 
single 
realization 
that 
hefw(r) 
allows 
a 
good 
estimation 
of 
the 
statistical 
parameters 
Kˆ
well, 
Kˆ
efu 
and 
`ˆ. 
Since 
Kˆ
well 
is 
very 
much 
influenced 
by 
the 
local 
distribution 
of 
the 
conductivity 
near 
the 
well, 
it 
does 
not 
necessarily 
correspond 
to 
the 




theoretical 
value 
Kwell 
= 
KG 
exp±21
2and 
we 
thus 
recommend 
to 
be 
careful 
when 
tracing 
back 
KG 
and 
2 
from 
Kˆ
well 
and 
Kˆ
efu 
for 
a 
single 
pumping 
test. 


4.5.4. 
Analysis 
of 
an 
Ensemble 
of 
Pumping 
Tests 
Since 
a 
vertical 
extension 
of 
L 
= 
64`z 
does 
not 
ensure 
ergodicity 
sufficiently, 
we 
state 
that 
even 
in 
three 
dimensions 
it 
is 
necessary 
to 
investigate 
an 
ensemble 
of 
pumping 
tests 
to 
infer 
on 
the 
statistics 
of 
K(x). 
A 
further 
extension 
in 
vertical 
direction 
would 
also 
be 
a 
possibility 
to 
ameliorate 
the 
results 
in 
theory. 
However, 
even 
assuming 
high 
anisotropy 
rates 
these 
conditions 
will 
hardly 
be 
fulfilled 
in 
reality. 
On 
the 
other 
hand, 
a 
number 
of 
pumping 
tests 
within 
an 
observation 
area 
will 
give 
much 
better 
insights 
into 
the 
heterogeneous 
structure 
of 
the 
subsurface. 


Within 
our 
setting 
we 
tested 
several 
ensemble 
sizes 
up 
to 
n 
= 
20 
realizations 
and 
estimate 


P

n

the 
statistical 
parameters 
for 
the 
ensemble 
mean 
hh(r)i=i=1hhi(r)i. 
We 
found 
a 
quick 


n 


convergence 
of 
Kˆ
efu 
and 
`ˆto 
the 
theoretical 
assigned 
values: 
less 
than 
10 
realizations 
are 
sufficient 
to 
be 
in 
a 
range 
of 
95% 
accuracy. 
For 
the 
near 
well 
conductivity 
about 
15-20 
realizations 
are 
necessary 
to 
ensure 
an 
acceptable 
good 
agreement 
between 
Kˆ
well 
and 
Kwell. 
Table 
18 
and 
Table 
19 
show 
results 
for 
the 
estimated 
parameters 
Kˆ
well, 
Kˆ
efu 
and 
`ˆfrom 
hh(r)in=20 
for 
three 
different 
ensembles 
and 
both 
boundary 
conditions. 
The 
values 
for 
Kˆ
G 
and 
ˆ2 
can 
either 
be 



4.6. 
Summary 
and 
Conclusions 
ˆKG 
[10-4 
m/s] 
ˆ2 
ˆKefu 
[10-4 
m/s] 
ˆKH 
[10-4 
m/s] 
`ˆ 
[m] 
E1 
1.0 
1.0 
1.181 
0.607 
5.0 
1.008 
0.972 
1.186 
0.62 
5.01 
(±0.003) 
(±0.005) 
(±0.004) 
(±0.001) 
(±0.09) 
E2 
1.0 
2.0 
1.396 
0.368 
5.0 
0.986 
2.025 
1.383 
0.358 
5.07 
(±0.007) 
(±0.013) 
(±0.012) 
(±0.001) 
(±0.08) 
E3 
1.0 
1.0 
1.181 
0.607 
10.0 
1.031 
0.94 
1.206 
0.644 
10.26 
(±0.006) 
(±0.011) 
(±0.009) 
(±0.001) 
(±0.27) 


Table 
19: 
Parameter 
estimation 
results 
of 
simulated 
mean 
head 
hh(r)) 
for 
three 
ensembles 
with 
BCH. 
Expected 
ensemble 
parameters 
in 
italic, 
the 
inverse 
estimates 
in 
bold 
and 
the 
95% 
confidence 
intervalls 
in 
brackets. 


evaluated 
from 
Kˆ
well 
and 
Kˆ
efu 
or 
directly 
estimated 
by 
using 
A 
= 

(e) 
and 
H 
= 
(
(e)-1); 
the 
estimation 
results 
are 
also 
listed 
in 
Table 
18 
and 
Table 
19. 


Again 
we 
state 
that 
the 
good 
estimation 
results 
for 
Kˆ
well 
and 
ˆ2 
are 
mainly 
caused 
by 
the 
fact 
that 
for 
a 
sufficiently 
large 
number 
of 
realizations, 
the 
local 
mean 
at 
the 
well 
hKwell) 
converges 
to 
the 
theoretical 
assigned 
value 
Kwell. 
A 
precise 
number 
of 
pumping 
tests 
needed 
to 
ensure 
ergodicity 
can 
be 
traced 
back 
on 
the 
question: 
What 
sample 
size 
N 
is 
necessary 
to 
guarantee 
the 
convergence 
of 
the 
mean 
of 
a 
sample 
of 
spatially 
correlated 
log-normal 
distributed 




1 
PN 
1

values 
hKiN 
= 
i=1 
Ki 
to 
the 
theoretical 
expected 
mean 
KA 
= 
KG 
exp2 
2. 
Answering 


N

this 
question 
is 
out 
of 
the 
scope 
of 
this 
study. 
Most 
likely, 
there 
does 
not 
exist 
a 
single 
number 
being 
appropriate 
for 
all 
possible 
statistical 
and 
geometrical 
settings. 


An 
item 
which 
is 
not 
discussed 
until 
now 
is 
the 
interpretation 
of 
drawdowns 
in 
anisotropic 
media. 
As 
discussed 
in 
section 
4.4.2 
we 
do 
not 
incorporate 
e 
into 
our 
estimation 
procedure 
due 
to 
the 
low 
sensitivity 
of 
hefw(r) 
towards 
e. 
However, 
applying 
hefw(r) 
on 
drawdown 
data 
in 
anisotropic 
media 
is 
possible 
and 
useful, 
as 
shown 
in 
Figure 
37. 
Assuming 
a 
reasonable 
value 
for 
e 
leads 
to 
very 
good 
accordance 
of 
hh(r)in=20 
and 
hefw(r) 
and 
further 
allows 
the 
estimation 
of 
Kˆ
efu, 
Kˆ
well 
and 
`ˆ. 
For 
application 
on 
real 
pumping 
tests 
data 
we 
would 
recommend 
to 
fix 
e 
to 
a 
reasonable 
value 
or 
to 
carry 
out 
several 
estimations 
with 
various 
ratios 
for 
e 
like 
0.01, 
0.1, 
0.5 
and 
1.0 
and 
interpret 
the 
results 
with 
respect 
to 
the 
accordance 
of 
Kˆ
efu, 
Kˆ
well 
and 
`ˆ 
to 
the 
drawdown. 


We 
conclude 
by 
stating 
that 
our 
numerical 
results 
show 
that 
hefw(r) 
is 
a 
promising 
tool 
to 
characterize 
aquifer 
properties 
like 
mean 
conductivity, 
variance, 
and 
spatial 
correlation 
at 
a 
very 
local 
scale 
by 
interpreting 
the 
near 
well 
behavior 
of 
steady 
state 
pumping 
tests. 


4.6. 
Summary 
and 
Conclusions 
In 
this 
study 
we 
introduced 
a 
representative 
description 
of 
the 
hydraulic 
head 
drawdown 
for 
a 
steady 
state 
pumping 
test 
with 
fully 
penetrating 
well 
for 
highly 
heterogeneous 
media. 
By 
making 
use 
of 
the 
upscaling 
technique 
Coarse 
Graining, 
we 
derived 
a 
radial 
depending 
con



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 
Figure 
37: 
Plot 
of 
ensemble 
drawdown 
hh(r)in=20 
for 
ensemble 
E4 
(KG 
= 
10-4 
m/s, 
2 
= 


1, 
f 
= 
10 
m, 
e 
=0.5) 
versus 
hefw(r) 
in 
anisotropic 
media 
for 
BCA 
and 
BCH. 
The 


inset 
shows 
the 
near 
well 
behavior 
in 
log 
scale. 


ductivity 
KCG(r). 
It 
interpolates 
between 
the 
known 
near 
and 
far 
field 
representative 
conductivities 
for 
well 
flow. 
Therefore, 
we 
deduced 
the 
effective 
well 
flow 
head 
solution 
hefw(r) 
which 
reproduces 
the 
mean 
drawdown 
of 
a 
pumping 
test 
adequately. 
We 
understand 
hefw(r) 
as 
an 
extension 
of 
Thiem’s 
Formula 
incorporating 
the 
effects 
of 
the 
statistical 
parameters 
of 
the 
underlying 
log-normal 
distributed 
conductivity 
field 
K(x) 
on 
the 
flow 
pattern. 


The 
analytical 
character 
of 
hefw(r) 
allowed 
us 
to 
perform 
a 
sensitivity 
analysis 
for 
the 
parameters 
of 
K(x) 
on 
the 
drawdown. 
We 
found 
that 
the 
variance 
2 
has 
the 
strongest 
impact 
on 
the 
hydraulic 
head 
directly 
at 
the 
well. 
The 
horizontal 
correlation 
length 
f 
determines 
the 
transition 
from 
near 
to 
far 
field 
behavior. 
In 
particular 
the 
impact 
of 
f 
increases 
with 
increasing 
variance 
2, 
which 
makes 
a 
prediction 
of 
f 
easier 
for 
highly 
heterogeneous 
media. 
The 
anisotropy 
ratio 
e 
has 
only 
little 
influence 
on 
the 
drawdown, 
giving 
that 
hefw(r) 
shows 
very 
low 
sensitivity 
towards 
changes 
in 
e. 


To 
validate 
the 
applicability 
of 
hefw(r) 
we 
performed 
steady 
state 
numerical 
pumping 
tests 
in 
three 
dimensional 
highly 
heterogeneous 
anisotropic 
media, 
with 
variances 
up 
to 
2 
=2. 
Our 
investigations 
confirmed 
the 
findings 
of 
Indelman 
et 
al. 
[1996] 
that 
the 
far 
field 
behavior 




is 
covered 
by 
Kefu 
= 
KG 
exp2(

12

- 

(e)). 
We 
also 
found 
the 
near 
well 
representative 


conductivity 
Kwell 
to 
be 
the 
arithmetic 
KA 
or 
harmonic 
mean 
KH, 
depending 
on 
the 
assigned 


Dirichlet 
or 
Neumann 
boundary 
condition. 
However, 
the 
means 
at 
the 
well 
have 
to 
be 
considered 
very 
locally. 
Investigations 
on 
the 
local 
distribution 
of 
K(x) 
showed 
that 
the 
arithmetic 
and 
harmonic 
mean 
of 
the 
conductivity 
values 
directly 
along 
the 
well 
hKA) 
and 
hKH) 
are 
not 
representative 
for 
the 
theoretically 
expected 




means 
of 
the 
full 
field 
KA 
= 
KG 
exp

12

2

and 
KH 
= 
KG 
exp

-


12

2

. 
We 
found 
discrepancies 


up 
to 
20% 
between 
KA 
and 
hKA) 
as 
well 
as 
between 
KH 
and 
hKH) 
for 
all 
tested 
variances. 
We 
therefore 
conclude 
that 
a 
single 
pumping 
test 
realization 
does 
not 
fulfill 
ergodic 
conditions 
in 
the 
vicinity 
of 
the 
well 
even 
for 
a 
large 
vertical 
extension 
of 
more 
than 
60 
correlation 
lengths. 



4.6. 
Summary 
and 
Conclusions 
This 
is 
in 
contrast 
to 
a 
previous 
work 
published 
by 
Firmani 
et 
al. 
[2006]. 
In 
order 
to 
make 
predictions 
for 
the 
overall 
statistics, 
we 
analyzed 
ensembles 
of 
pumping 
tests 
and 
showed 
that 
hefw(r) 
does 
not 
only 
reproduce 
the 
ensemble 
drawdown 
but 
enables 
the 
estimation 
of 
the 
statistical 
parameters 
with 
very 
high 
accuracy. 
In 
isotropic 
media 
the 
estimated 
results 
Kˆ
G, 
ˆ2 
and 
`ˆ 
differ 
less 
than 
6% 
from 
the 
expected 
ones 
for 
all 
ensembles 
with 
very 
high 
confidence. 
Solely 
the 
anisotropy 
ratio 
e 
is 
difficult 
to 
infer. 
We 
agree 
with 
Firmani 
et 
al. 
[2006] 
that 
the 
estimation 
of 
e 
from 
pumping 
test 
data 
is 
very 
error 
prone. 


Nonetheless, 
hefw(r) 
also 
allows 
the 
interpretation 
of 
drawdowns 
in 
anisotropic 
media 
(e< 
1) 
by 
assuming 
a 
reasonable 
ratio 
e 
and 
then 
estimating 
the 
parameters 
Kˆ
G, 
ˆ2 
and 
`ˆ. 
However, 
being 
limited 
to 
ensemble 
averages 
of 
multiple 
pumping 
tests 
is 
clearly 
a 
limitation 


for 
interpreting 
real 
drawdown 
data. 
To 
overcome 
the 
lack 
of 
ergodicity 
at 
the 
well 
in 
a 
single 
realization 
we 
adapted 
our 
proposed 
formula 
hefw(r) 
on 
local 
statistics 
of 
the 
conductivity, 
incorporating 
hKA) 
and 
hKHi, 
which 
gives 
a 
much 
better 
reproduction 
of 
the 
simulated 
depression 
cone 
than 
with 
theoretically 
values 
KA 
and 
KH. 
This 
modification 
allows 
to 
estimate 
Kˆ
A 
and 
Kˆ
H 
respectively, 
Kˆ
efu 
and 
`ˆ 
for 
single 
drawdown 
data. 
If 
several 
pumping 
tests 
in 
one 
area 
are 
available 
each 
can 
be 
interpreted 
with 
hefw(r) 
and 
afterwards 
a 
statistical 
analysis 
can 
be 
applied 
to 
infer 
on 
ˆ2 
and 
Kˆ
G. 
Thus 
hefw(r) 
can 
serve 
as 
a 
helpful 
tool 
to 
interpret 
real 
drawdown 
data 
for 
an 
arbitrary 
number 
of 
steady 
state 
pumping 
tests. 


Exploiting 
our 
results 
with 
respect 
to 
predictions 
on 
a 
real 
pumping 
test 
sampling 
design 
we 
suppose 
that 
the 
quality 
of 
the 
parameter 
estimation 
mainly 
depend 
on 
the 
position 
of 
the 
observation 
wells. 
A 
good 
estimation 
of 
the 
variance 
2 
requires 
measurements 
directly 
at 
the 
well. 
To 
infer 
on 
the 
correlation 
length 
f 
the 
vicinity 
of 
the 
well, 
meaning 
the 
area 
within 
two 
correlation 
lengths 
has 
to 
be 
investigated. 
Measurements 
far 
from 
the 
well 
allow 
to 
infer 
on 
Kefu. 
The 
larger 
the 
number 
of 
head 
data 
in 
the 
corresponding 
area 
of 
influence 
of 
a 
parameter, 
the 
more 
reliable 
are 
its 
estimation 
result. 
Thus 
we 
can 
use 
hefw(r) 
not 
only 
to 
infer 
on 
the 
statistics 
but 
it 
also 
allows 
to 
judge 
the 
usefulness 
of 
measurements 
with 
respect 
to 
the 
estimation 
of 
the 
parameters 
for 
the 
underlying 
hydraulic 
conductivity 
field. 



4. 
The 
Extended 
Thiem’s 
Solution 
-Including 
the 
Impact 
of 
Heterogeneity 

5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
-a 
Paradigm 
for 
Field 
Applications 
5.1. 
Introduction 
Groundwater 
pumping 
tests 
are 
applied 
as 
common 
and 
well-established 
tool 
to 
estimate 
hydraulic 
properties 
of 
porous 
media. 
Estimating 
parameters 
such 
as 
the 
saturated 
hydraulic 
conductivity 
of 
an 
aquifer 
is 
based 
on 
measurements 
of 
the 
hydraulic 
head 
across 
the 
area 
of 
drawdown. 
Classical 
approaches 
to 
estimate 
hydraulic 
conductivity 
from 
pumping 
tests 
are 
based 
on 
Thiem’s 
formula 
[Thiem, 
1906] 
or 
Theis’ 
solution 
[Theis, 
1935] 
for 
steady 
state 
or 
transient 
flow 
conditions, 
respectively; 
the 
latter 
having 
been 
simplified 
by 
Cooper 
and 
Jacob 
[1946]. 
Detailed 
information 
on 
the 
application 
of 
those 
approaches 
under 
numerous 
boundary 
conditions 
was 
provided 
by 
Kruseman 
and 
de 
Ridder 
[2000]. 


With 
regard 
to 
natural 
aquifers, 
a 
critical 
shortcoming 
of 
these 
methods 
is 
the 
simplified 
assumption 
of 
aquifers 
being 
homogeneous. 
The 
vast 
majority 
of 
natural 
aquifers 
are 
characterized 
by 
discontinuities 
evolved 
from 
geomorphologic 
processes 
which 
affect 
the 
drawdown 
curve 
derived 
from 
pumping 
tests 
[Dagan 
and 
Neuman, 
1997]. 
Estimating 
the 
heterogeneous 
structure 
of 
aquifers 
is 
crucial 
for 
characterizing 
the 
groundwater 
flow 
toward 
the 
pumping 
well 
and 
for 
predicting 
solute 
transport, 
particularly 
in 
regard 
to 
contaminant 
migration 
and 
in-situ 
remediation. 
For 
example, 
assuming 
a 
log-normal 
hydraulic 
conductivity 
model, 
the 
horizontal 
dispersivity 
of 
an 
aquifer 
is 
proportional 
to 
the 
product 
of 
the 
variance 
2 
and 
the 
horizontal 
correlation 
length 
f 
[Gelhar, 
1993]. 


Taking 
heterogeneity 
into 
account, 
the 
hydraulic 
conductivity 
is 
assumed 
to 
be 
represented 
by 
a 
spatial 
random 
function 
K(x) 
with 
log-normal 
distributed 
values 
K(x) 
. 
LN 
(µ, 
2), 
where 
µ 
and 
2 
denote 
the 
mean 
and 
variance 
of 
the 
normal 
distribution 
ln 
K(x), 
respectively. 
However, 
using 
a 
single 
representative 
mean 
value 
of 
K(x) 
to 
describe 
the 
water 
flow 
toward 
the 
pumping 
well 
is 
inadequate 
for 
steady 
state 
conditions 
[Matheron, 
1967], 
because 
distinct 
representative 
values 
emerge 
for 
the 
flow 
near 
(Kwell) 
and 
far 
from 
the 
pumping 
well 
(Kefu). 
The 
mathematical 
expressions 
of 
Kwell 
and 
Kefu 
depend 
on 
µ 
and 
2 
and 
are 
related 
to 
the 
dimensionality 
of 
the 
water 
flow, 
see 
Sánchez-Vila 
et 
al. 
[2006] 
for 
details. 


Most 
studies 
concerning 
well 
flow 
in 
heterogeneous 
porous 
media 
are 
limited 
to 
a 
two 
dimensional 
flow 
model 
and 
therefore 
are 
representative 
for 
large 
scale 
pumping 
tests, 
e.g. 
Desbarats 
[1992]; 
Sánchez-Vila 
et 
al. 
[1999]; 
Copty 
and 
Findikakis 
[2004]; 
Neuman 
et 
al. 
[2004]; 
Leven 
and 
Dietrich 
[2006]; 
Neuman 
et 
al. 
[2007]; 
Dagan 
and 
Lessoff 
[2007]; 
Schneider 
and 
Attinger 
[2008]. 
However, 
for 
small 
scale 
pumping 
tests 
vertical 
flow 
in 
the 
vicinity 
of 
the 
pumping 
well 
is 
a 
critical 
component 
that 
influences 
the 
drawdown. 
Those 
considerations 
indicate 
the 
need 
to 
reliably 
estimate 
statistical 
parameters 
of 
K(x) 
using 
data 
from 
pumping 
tests 
and 
applying 
simple 
analytical 
solutions 
for 
three 
dimensional 
water 
flow 
[Indelman 
and 
Abramovich, 
1994; 
Indelman 
et 
al., 
1996; 
Indelman, 
2001; 
Guadagnini 
et 
al., 
2003; 
Firmani 
et 
al., 
2006; 
Zech 
et 
al., 
2012], 
thereby 
reducing 
time 
and 
cost-load 
otherwise 
involved 
with 
methods 
such 
as 
laboratory 
trials. 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
By 
deriving 
the 
vertical 
mean 
drawdown 
of 
a 
steady 
state 
pumping 
test 
in 
three 
dimensional 
heterogeneous 
porous 
media, 
we 
presented 
in 
the 
previous 
section 
4.4 
an 
analytical 
tool 
to 
analyze 
statistical 
parameters 
of 
K(x). 
The 
developed 
formula 
– 
referring 
to 
the 
effective 
well 
flow 
head 
hefw(r) 
– 
depends 
on 
the 
radial 
distance 
from 
the 
pumping 
well 
r 
and 
the 
statistical 
properties 
KG, 
2, 
c 
and 
e 
of 
K(x). 
In 
contrast 
to 
Indelman 
and 
Abramovich 
[1994] 
and 
subsequent 
works, 
this 
method 
is 
based 
on 
an 
upscaling 
procedure 
called 
Coarse 
Graining 
[Attinger, 
2003] 
rather 
than 
perturbation 
theory. 
Numerical 
simulations 
of 
pumping 
tests 
showed 
that 
hefw(r) 
reliably 
reproduces 
the 
radial 
depression 
cone 
of 
a 
steady 
state 
pumping 
test 
for 
a 
range 
of 
statistical 
parameter 
values. 


In 
contrast 
to 
simulated 
pumping 
tests, 
the 
underlying 
K(x) 
distribution 
of 
on 
site 
aquifers 
is 
unknown, 
which 
hampers 
the 
qualitative 
assessment 
of 
parameter 
estimates 
when 
using 
empirical 
field-site 
data. 
Moreover, 
sample 
size 
of 
head 
measurements 
is 
limited, 
which 
involves 
uncertainty 
in 
parameter 
estimates, 
e.g. 
if 
pumping 
tests 
are 
by 
chance 
conducted 
in 
areas 
of 
extraordinary 
low 
or 
high 
permeability. 
Thus 
the 
question 
remains 
whether 
hefw(r) 
is 
capable 
to 
estimate 
the 
statistical 
parameters 
of 
an 
natural 
aquifer 
using 
field-site 
pumping 
test 
data. 


The 
aim 
of 
this 
study 
is 
to 
close 
the 
gap 
between 
theoretical 
and 
field 
application 
of 
hefw(r) 
on 
pumping 
test 
data. 
We 
examined 
the 
capability 
and 
predictive 
power 
of 
hefw(r) 
to 
provide 
estimates 
of 
statistical 
parameters 
of 
K(x) 
from 
pumping 
test 
data 
such 
as 
under 
field-site 
conditions. 
As 
a 
first 
step 
(section 
5.3), 
we 
analyzed 
simulated 
pumping 
tests 
by 
reducing 
the 
sample 
size 
of 
head 
measurements 
and 
evaluating 
the 
quality 
of 
parameter 
estimation. 
As 
a 
second 
step 
(section 
5.4), 
we 
applied 
hefw(r) 
on 
empirical 
groundwater 
pumping 
test 
data 
from 
the 
field 
site 
Horkheimer 
Insel, 
Germany 
[Schad 
and 
Teutsch, 
1994] 
and 
compared 
the 
hefw(r) 
estimates 
with 
the 
findings 
from 
laboratory 
investigations 
[Schad, 
1997]. 
Based 
on 
these 
findings, 
we 
provide 
consideration 
regarding 
the 
conceptual 
design 
of 
groundwater 
pumping 
tests 
and 
the 
predictive 
power 
of 
established 
pumping 
tests 
sites. 


5.2. 
Theoretical 
Framework 
This 
study 
is 
based 
on 
the 
effective 
well 
flow 
head 
hefw(r) 
introduced 
in 
section 
4 
and 
published 
in 
Zech 
et 
al. 
[2012]. 
The 
relation 
between 
the 
effective 
well 
flow 
head 
and 
the 
hydraulic 
head 
for 
pumping 
tests 
conducted 
in 
heterogeneous 
porous 
media 
is 
depicted 
in 
Figure 
38. 
Moreover, 
the 
fundamentally 
distinct 
method 
of 
predicting 
the 
hydraulic 
head 
of 
homogeneous 
aquifers 
using 
Thiem’s 
solution 
is 
illustrated. 
Starting 
point 
is 
a 
spatially 
distributed 
heterogeneous 
conductivity 
field 
K(x,KG,2, 
`, 
e). 
Conducting 
pumping 
tests 
on 
such 
an 
aquifer 
results 
in 
a 
spatially 
distributed 
hydraulic 
head 
field 
h(r, 
, 
KG,2, 
`, 
e), 
also 
depending 
on 
the 
statistics 
of 
K(x). 
Tracing 
back 
the 
parameters 
of 
K(x) 
from 
h(r, 
, 
KG,2, 
`, 
e) 
is 
the 
fundamental 
goal 
of 
estimating 
aquifer 
parameters 
inversely. 


The 
approach 
of 
Zech 
et 
al. 
[2012] 
is 
based 
on 
adapted 
spatial 
averaging 
of 
K(x,KG,2, 
`, 
e) 
according 
to 
the 
conditions 
of 
pumping 
tests. 
Using 
the 
upscaling 
procedure 
Coarse 
Graining 
[Attinger, 
2003] 
a 
representative 
conductivity 
KCG(r, 
KG,2, 
`, 
e) 
is 
derived, 
which 
is 
not 
fully 
homogenized, 
but 
still 
depends 
on 
the 
statistical 
parameter 
of 
K(x). 



5.2. 
Theoretical 
Framework 
Figure 
38: 
Relation 
between 
the 
deduction 
of 
hefw(r)and 
processes 
occurring 
during 
field 
site 
pumping 
tests. 


Coarse 
Graining 
can 
be 
best 
explained 
as 
a 
spatial 
filtering 
procedure 
that 
averages 
over 
volumes 
of 
variable 
filter 
size. 
Applied 
to 
pumping 
tests, 
Coarse 
Graining 
allows 
to 
account 
for 
the 
character 
of 
radial 
convergent 
flow. 
Near 
the 
pumping 
well, 
where 
the 
impact 
of 
heterogeneity 
is 
large, 
the 
filter 
size 
is 
adjusted 
such 
as 
that 
small 
volumes 
are 
captured, 
thereby 
leaving 
the 
heterogeneity 
unchanged. 
Whereas 
far 
from 
the 
pumping 
well, 
the 
filter 
size 
is 
larger 
resulting 
in 
averaged 
values. 


Performing 
a 
pumping 
test 
on 
KCG(r)results 
in 
the 
effective 
well 
flow 
head 
hefw(r, 
KG,2, 
`, 
e) 
(Figure 
38). 
Numerical 
simulation 
in 
section 
4.5 
showed 
that 
hefw 
reproduces 
the 
vertical 
averaged 
hydraulic 
head 
h(r, 
, 
KG,2, 
`, 
e)of 
a 
pumping 
test 
in 
randomly 
heterogeneous 
medium, 
due 
to 
its 
ability 
to 
capture 
effects 
of 
heterogeneity 
in 
contrast 
to 
Thiem’s 
solution 
hT(r, 
KG). 


5.2.1. 
The 
Effective 
Well 
Flow 
Head 
The 
effective 
well 
flow 
head 
hefw(r) 
can 
be 
considered 
an 
extension 
of 
Thiem’s 
solution 
to 
heterogeneous 
media. 
It 
reproduces 
the 
vertical 
mean 
drawdown 
of 
a 
steady 
state 
pumping 
test 
in 
relation 
to 
the 
radial 
distance 
r 
and 
the 
statistical 
parameters 
of 
K(x): 


hefw(

r)=C1 
exp(-)ln 
r 
+C1 
sinh()U1(r)+C1 
(1- 
cosh())U2(r)+h(R) 
, 
(5.1)

R 


r

Qw 
, 
. 
=lnKwell 
r

with 
the 
abbreviations 
C1 
=- 
, 
u(r)=1+pand 
the 
terms 


3

2LKefu 
Kefu 
e`2 


r)+11 
1 


U1(r)=ln 
u(+ 


u(R)+1- 
u(r)u(R) 
r)1 
1 
1 
1 


U2(r)=ln 
u(
, 


u(R)- 
2u(r)2 
+2u(R)2 
- 
4u(r)4 
+4u(R)4 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
where 
Qw 
is 
the 
pumping 
rate, 
L 
is 
the 
aquifer 
thickness, 
. 
is 
a 
constant 
of 
proportionality, 
f 
is 
the 
horizontal 
correlation 
length 
and 
e 
. 
[0, 
1] 
is 
the 
anisotropy 
ratio 
assuming 
a 
Gaussian 
shaped 
spatial 
correlation 
structure 
in 
horizontally 
isotropic 
and 
vertically 
anisotropic 
media. 
The 
reference 
head 
h(R) 
is 
measured 
at 
the 
radial 
distance 
R, 
which 
can 
either 
be 
at 
the 
pumping 
well 
or 
in 
the 
far 
field. 


The 
hefw(r)-solution 
interpolates 
between 
the 
representative 
hydraulic 
conductivities 
at 
the 
pumping 
well 
and 
in 
the 
far 
field 
– 
Kwell 
and 
Kefu, 
respectively. 
They 
differ 
remarkably 
for 
steady 
state 
flow 
conditions, 
thereby 
affecting 
the 
flow 
characteristics 
of 
each 
radial 
section 
of 
the 
depression 
cone. 


For 
log-normal 
distributed 
conductivity 
the 
asymptotic 
drawdown 
behavior 
in 
the 
far 
field 
of 
a 
steady 
state 
pumping 
test 
in 
three 
dimensions 
is 
characterized 
by 
the 
effective 
hydraulic 
conductivity 
for 
uniform 
flow 
[Dagan, 
1989; 
Indelman 
and 
Abramovich, 
1994; 
Sánchez-Vila 
et 
al., 
2006]: 




1

Kefu 
= 
KG 
exp 
22 
- 

(e), 
(5.2) 


where 
KG 
denotes 
the 
geometric 
mean, 
2 
the 
variance 
and 

(e) 
is 
the 
anisotropy 
function 


p

e 
v 
1


(e)= 
2 
arctan1/e2 
- 
1- 
e.

2(1-e2)1-e

Estimates 
of 
Kwell 
depend 
on 
the 
assigned 
boundary 
condition, 
[Indelman 
et 
al., 
1996; 
Indelman 
and 
Dagan, 
2004]: 
(i) 
the 
Neumann 
boundary 
condition 
(hereafter 
referred 
to 
as 
BCH) 
is 
based 
on 
constant 
flux 
and 
corresponds 
to 
the 
harmonic 
mean 
Kwell 
= 
KH, 
whereas 
(ii) 
the 
Dirichlet 
boundary 
condition 
(BCA) 
is 
based 
on 
a 
constant 
head 
and 
corresponds 
to 
the 
arithmetic 
mean 
Kwell 
= 
KA. 
If 
ergodicity 
applies 
at 
the 
pumping 
well 
the 
values 
for 
Kwell 
can 
be 




calculated 
as 
KH 
= 
KG 
exp

-


12

2

for 
BCH, 
and 
KA 
= 
KG 
exp

12

2

for 
BCA. 
Both 
bound


ary 
conditions 
are 
addressed 
in 
Eq.(5.1) 
through 
Kwell 
in 
. 
= 
ln 
Kwell 
and 
the 
parameter 
,

Kefu 


which 
is 
BCH 
=1.6 
and 
BCA 
=0.8, 
see 
section 
4.4.1. 
Notably, 
hefw(r) 
is 
a 
deterministic 
head 
solution 
governed 
by 
the 
statistical 
characteristics 
of 
K(x) 
rather 
than 
a 
spatial 
random 
function. 
The 
solution 
presented 
in 
Eq.(5.1) 
is 
based 
on 
the 
four 
parameters 
Kefu 
and 
Kwell, 
f 
and 
e. 
If 
ergodic 
conditions 
apply 
2 
and 
KG 
can 
be 
calculated 
via 


2 
= 
2 
(ln 
Kwell 
- 
ln 
Kefu) 
2
(e) 
- 
1 
± 
1 
, 
(5.3) 


with 
+ 
for 
BCA 
and 
- 
for 
BCH 
and 
KG 
= 
Kwell 
exp

=


12

2-2

= 
Kefu 
exp

12

- 

(e)


. 


However, 
for 
single 
pumping 
tests 
ergodicity 
cannot 
be 
presumed 
in 
the 
vicinity 
of 
the 
pumping 
well 
and 
Kefu 
and 
Kwell 
are 
required 
for 
Eq.(5.1) 
to 
be 
independent 
from 
any 
ergodicity 
assumption. 
This 
modification 
is 
particularly 
useful 
for 
single 
pumping 
tests 
where 
Kwell 
is 
not 
related 
to 
the 
theoretical 
values 
of 
KH 
or 
KA 
(section 
4.5.2). 
Both 
parameterizations, 
KG 
and 
2 
for 
ergodic 
conditions, 
and 
Kefu 
and 
Kwell 
for 
non-ergodic 
conditions, 
are 
able 
to 
reproduce 
ensemble 
head 
means 
of 
simulated 
pumping 
tests. 
Since 
the 
aim 
of 
this 
study 
is 
to 
examine 
the 
capability 
and 
predictive 
power 
of 
hefw(r) 
to 
estimate 
statistical 
parameters 
using 
single 
pumping 
test 
data 
from 
field 
campaigns, 
we 
focus 
on 
non-ergodic 
conditions 
hereafter. 



5.3. 
Pumping 
Tests 
under 
Limited 
Data 
Availability 
-a 
Paradigm 
for 
Field 
Applications 
113 
5.2.2. 
Inverse 
Parameter 
Estimation 
While 
the 
analytical 
character 
of 
hefw(r) 
allows 
for 
examining 
the 
influence 
of 
statistical 
parameters 
of 
K(x) 
on 
the 
depression 
cone, 
it 
also 
can 
be 
utilized 
to 
estimate 
those 
parameters 
inversely 
from 
pumping 
tests. 
Based 
on 
this 
premise 
nonlinear 
regression 
is 
used 
to 
estimate 
best 
values 
of 
Kˆ
efu,Kˆ
well 
and 
`ˆto 
minimize 
the 
mean 
square 
error 
of 
the 
difference 
between 
hefw(r) 
and 
the 
measured 
drawdown 
hh(r)i. 


However, 
to 
determine 
the 
accuracy 
and 
certainty 
of 
those 
estimates 
it 
is 
essential 
to 
ascertain 
the 
sensitivity 
of 
hefw(r) 
to 
each 
of 
the 
parameters. 
A 
sensitivity 
analysis, 
performed 
in 
section 
4.4.2, 
revealed 
that 
Kwell 
and 
Kefu 
influence 
hefw(r) 
in 
the 
vicinity 
of 
the 
pumping 
well 
and 
the 
far 
field, 
respectively, 
whereas 
the 
correlation 
length 
f 
controls 
the 
transition 
between 
Kwell-driven 
and 
Kefu-driven 
drawdown. 
The 
larger 
f 
the 
larger 
is 
the 
transition 
zone 
around 
the 
pumping 
well. 


Moreover, 
hefw(r) 
is 
highly 
sensitive 
to 
changes 
in 
values 
of 
Kefu, 
Kwell 
and 
`, 
whereas 
the 
sensitivity 
of 
hefw(r) 
towards 
changes 
in 
the 
anisotropy 
ratio 
e 
is 
very 
low. 
Hence, 
estimating 
e 
from 
pumping 
tests 
is 
hardly 
possible 
[Firmani 
et 
al., 
2006; 
Zech 
et 
al., 
2012] 
and 
reliable 
estimates 
of 
Kˆ
efu, 
Kˆ
well 
and 
`ˆ 
can 
only 
be 
achieved 
if 
e 
is 
considered 
constant. 


Comparing 
the 
confidence 
intervals, 
denoted 
by 
CI, 
of 
the 
three 
parameters 
reveals 
that 
CI`ˆ 
is 
much 
greater 
than 
CI 
ˆand 
CI 
ˆ. 
This 
is 
caused 
by 
the 
distinct 
areas 
that 
effect 
Kˆ
efu

Kefu 
Kwell

and 
Kˆ
well 
– 
far 
field 
and 
vicinity 
of 
pumping 
well, 
respectively 
– 
whereas 
the 
accuracy 
of 
`ˆalso 
depends 
on 
the 
estimates 
of 
Kˆ
efu 
and 
Kˆ
well. 
Alternatively, 
to 
eliminate 
this 
intrinsic 
uncertainty 
and 
thereby 
decreasing 
the 
confidence 
interval, 
the 
correlation 
length 
`˜ 
rather 
than 
f 
can 
be 
estimated 
using 
Kˆ
efu 
and 
Kˆ
well 
as 
constant 
parameters. 


5.3. 
Pumping 
Tests 
under 
Limited 
Data 
Availability 
-a 
Paradigm 
for 
Field 
Applications 
We 
examined 
the 
capability 
of 
hefw(r) 
to 
estimate 
statistical 
parameters 
of 
a 
heterogeneity 
under 
limited 
data 
availability 
as 
found 
under 
conditions 
of 
on-site 
pumping 
tests. 
In 
order 
to 
characterize 
the 
impact 
of 
the 
number 
and 
location 
of 
head 
measurements 
on 
the 
estimation 
quality, 
represented 
by 
accuracy 
and 
uncertainty, 
we 
performed 
simulated 
numerical 
experiments. 
The 
simulations 
were 
based 
on 
head 
data 
of 
pumping 
tests 
conducted 
in 
three 
dimensional 
aquifers 
with 
randomly 
generated 
hydraulic 
conductivity 
fields. 
Sample 
size 
of 
head 
measurements 
was 
systematically 
reduced 
with 
regard 
to 
their 
horizontal 
distribution 
and 
their 
radial 
distance 
relative 
to 
the 
pumping 
well. 


5.3.1. 
Pumping 
Test 
Model 
The 
finite 
element 
software 
OpenGeoSys 
[Kolditz 
et 
al., 
2012a] 
was 
used 
to 
simulate 
pumping 
tests 
on 
randomly 
generated 
realizations 
of 
three 
dimensional 
hydraulic 
conductivity 
fields. 
The 


-1

statistical 
parameters 
of 
the 
aquifer 
were 
set 
as 
KG 
= 
10-4ms, 
2 
= 
1, 
f 
=8m, 
and 
e 
= 
1. 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
r 
i 
in[m] 
#r 
i 
characteristics 


S1 
r 
w, 
1,..,32 
33 
full 
range, 
equidistant, 
1m 
interval 
S2 
r 
w, 
2,..,32 
17 
full 
range, 
equidistant, 
2m 
interval 
S3 
r 
w, 
4,..,32 
9 
full 
range, 
equidistant, 
4m 
interval 
S4 
r 
w, 
8,..,32 
5 
full 
range, 
equidistant, 
8m 
interval 
S5 
r 
w, 
1,..,8 
9 
vicinity 
of 
the 
well 
S6 
9,..,32 
24 
far 
field 
S7 
r 
w, 
2, 
4, 
6, 
8, 
16, 
24, 
32 
8 
full 
range, 
focus 
on 
vicinity 
of 
the 
well 
S8 
r 
w, 
8, 
16, 
20, 
24, 
28, 
32 
7 
full 
range, 
focus 
on 
far 
field 


Table 
20: 
Scenarios 
S1-S8 
of 
limited 
radial 
head 
data 
r 
i. 
The 
total 
number 
of 
measurements 
is 
denoted 
as 
#r 
i. 
Head 
measurements 
were 
sampled 
from 
all 
four 
axial 
directions. 


The 
numerical 
grid 
was 
similar 
to 
the 
one 
used 
previously 
in 
section 
4.5, 
with 
small 
adaption 
to 
conditions 
of 
short 
term 
pumping 
tests 
as 
conducted 
on-site 
at 
the 
Horkheimer 
Insel 
by 
Schad 


[1997], 
(section 
5.4). 
The 
horizontal 
grid 
size 
was 
8H 
(64m) 
with 
a 
uniform 
grid 
cell 
size 
of


18

H 
(1m) 
except 
for 
cells 
in 
the 
vicinity 
of 
the 
pumping 
well, 
which 
were 
required 
to 
be 
smaller 
for 
the 
purpose 
of 
integrating 
steep 
head 
gradients. 
The 
vertical 
grid 
size 
was 
15eH 
with 
a 
uniform 


grid 
cell 
size 
of


18

eH 
. 


Impermeable 
horizontal 
layers 
formed 
the 
base 
and 
top 
of 
the 
aquifer. 
The 
outer 
boundary 
condition 
was 
set 
as 
fixed 
head 
h 
(R 
=4H 
) 
= 
0. 
At 
the 
pumping 
well 
both 
BCH 
and 
BCA 


3-1

were 
applied, 
respectively 
(section 
5.2). 
For 
the 
first 
the 
pumping 
rate 
of 
Q 
w 
= 
-10-3ms 
was 
equally 
distributed 
over 
all 
vertical 
grid 
cells 
at 
the 
pumping 
well, 
whereas 
for 
the 
latter 
the 
pumping 
rate 
was 
proportional 
to 
the 
vertically 
distributed 
hydraulic 
conductivity 
values 
at 
the 
pumping 
well 
Q 
i 
. 
K 
i, 
which 
corresponds 
to 
the 
constant 
head 
boundary 
condition 
[Firmani 
et 
al., 
2006; 
Zech 
et 
al., 
2012]. 


Simulated 
heads 
were 
measured 
at 
the 
four 
axial 
directions 
. 
1 
=0,. 
2 
=1/ 
2, 
. 
3 
= 
, 
. 
4 
=3/ 
2p 
with 
a 
resolution 
of 
1m. 
We 
used 
vertically 
averaged 
heads 
to 
reflect 
on 
head 
measurements 
as 
derived 
under 
field-site 
conditions, 
eventually 
only 
depending 
on 
the 
horizontal 
position 
de


RL

scribed 
by 
the 
radial 
distance 
r 
i 
and 
the 
angular 
position 
. 
j: 
hh 
(r 
i,. 
j)) 
=1/L0 
h 
(r 
i,. 
j,z 
)dz 
. 


5.3.2. 
Methods 
of 
Sample 
Size 
Reduction 
We 
reduced 
the 
sample 
size 
of 
simulated 
head 
measurements 
stepwise 
according 
to 
(i) 
their 
angular 
position, 
(ii) 
radial 
distance 
relative 
to 
the 
pumping 
well, 
and 
(iii) 
a 
combination 
of 
both. 
For 
each 
step 
we 
estimated 
the 
statistical 
parameters 
Kˆ
efu, 
Kˆ
well 
and 
`ˆaccording 
to 
the 
procedure 
described 
in 
section 
5.2.2. 
The 
average 
hydraulic 
head 
hh 
(r 
)) 
derived 
as 
the 
mean 
of 
the 
four 
axial 
directions 
. 
1 
= 
0, 
. 
2 
=1/ 
2p 
, 
. 
3 
= 
p 
, 
. 
4 
=3/ 
2p 
and 
measured 
at 
1m 
intervals 
serves 
as 
reference 
scenario 
S1 
(Table 
20) 
with 
full 
data 
availability. 


Firstly, 
we 
controlled 
the 
sample 
size 
of 
head 
measurements 
with 
respect 
to 
their 
angular 
arrangement, 
while 
remaining 
the 
full 
sample 
size 
with 
regard 
to 
their 
radial 
distance 
from 
the 
pumping 
well. 
The 
statistical 
parameters 
were 
estimated 
for 
each 
of 
the 
four 
axial 
directions, 



5.3. 
Pumping 
Tests 
under 
Limited 
Data 
Availability 
-a 
Paradigm 
for 
Field 
Applications 
115 
ˆKwell 
[10-4 
m/s] 
ˆKefu 
[10-4 
m/s] 
`ˆ 
[m] 
reference 
parameter 
0.648 
1.181 
8.0 
hh(r)) 
0.665 
(±0.003) 
1.165 
(±0.012) 
8.86 
(±0.54) 
hh(r, 
1)) 
hh(r, 
2)) 
hh(r, 
3)) 
hh(r, 
4)) 
0.657 
(±0.006) 
0.663 
(±0.006) 
0.677 
(±0.009) 
0.664 
(±0.005) 
1.176 
(±0.020) 
1.207 
(±0.027) 
1.199 
(±0.054) 
1.106 
(±0.016) 
7.58 
(±0.83) 
8.95 
(±1.07) 
12.52 
(±2.42) 
7.60 
(±0.82) 
h˜h(r, 
C1)) 
h˜h(r, 
C2)) 
h˜h(r, 
C3)) 
h˜h(r, 
C4)) 
h˜h(r, 
C5)) 
0.660 
(±0.011) 
0.659 
(±0.010) 
0.661 
(±0.012) 
0.672 
(±0.011) 
0.660 
(±0.010) 
1.155 
(±0.039) 
1.152 
(±0.034) 
1.179 
(±0.049) 
1.173 
(±0.048) 
1.134 
(±0.030) 
8.04 
(±1.73) 
7.72 
(±1.52) 
8.73 
(±2.08) 
9.65 
(±2.17) 
7.12 
(±1.40) 


Table 
21: 
Estimates 
of 
Kˆ
well, 
Kˆ
efu 
and 
`ˆ 
for 
the 
mean 
of 
all 
angular 
directions 
hh(r)i, 
single 
transects 
hh(r, 
j)) 
and 
five 
distinct 
combinations 
of 
heads 
with 
randomly 
chosen 
angular 
position 
hh˜
(r, 
Ck)ik=1,...,5. 
In 
brackets 
are 
given 
the 
95% 
confidence 
intervals. 
The 
sampling 
size 
with 
regard 
to 
the 
radial 
distance 
remained 
constant 
at 
1m 
intervals. 


hereafter 
referred 
to 
as 
transects 
hh(r, 
1)i,..., 
hh(r, 
4)) 
(Table 
21). 
Then 
the 
drawdown 
curves 
were 
analyzed 
with 
a 
random 
combination 
of 
head 
measurements 
from 
any 
of 
the 
four 
axial 
directions 
hh˜
(r, 
C1)i,. 
.., 
hh˜
(r, 
C5)i, 
listed 
in 
Table 
21. 


Secondly, 
we 
controlled 
the 
sample 
size 
regarding 
the 
radial 
distance 
to 
the 
pumping 
well, 
while 
using 
the 
average 
hydraulic 
head 
of 
all 
transects. 
The 
sample 
size 
was 
reduced 
with 
respect 
to 
equidistant 
sampling 
intervals 
for 
scenarios 
S2-S4 
(Table 
20). 
For 
scenarios 
S5 
and 
S6 
sampling 
intervals 
remained 
at 
1m, 
but 
focused 
only 
on 
head 
measurements 
in 
the 
vicinity 
of 
the 
pumping 
well 
or 
the 
far 
field, 
respectively 
(notably, 
head 
measurement 
at 
the 
pumping 
well 
was 
excluded 
for 
S6). 
Likewise, 
head 
measurements 
focused 
on 
the 
vicinity 
of 
the 
pumping 
well 
or 
the 
far 
field 
for 
scenarios 
S7 
and 
S8, 
respectively, 
but 
heads 
were 
sampled 
across 
the 
entire 
range 
of 
the 
aquifer 
(logarithmic 
scaled 
arrangement). 


Finally, 
we 
combined 
the 
two 
controlled 
experiments 
and 
randomly 
sampled 
head 
measurements 
from 
any 
of 
the 
four 
transects 
for 
all 
scenarios 
as 
defined 
in 
Table 
20. 


5.3.3. 
Parameter 
Estimation 
under 
Angular 
Limitation 
The 
drawdown 
curves 
in 
Figure 
39 
– 
angular 
mean, 
the 
four 
transects 
and 
one 
random 
angular 
combination 
of 
head 
measurements 
– 
were 
in 
good 
accordance 
in 
the 
far 
field 
and 
directly 
at 
the 
pumping 
well. 
Hence, 
estimates 
of 
Kˆ
well 
and 
Kˆ
efu 
were 
accurate 
and 
only 
little 
uncertainty 
was 
involved 
for 
both, 
the 
four 
transects 
hh(r, 
i)) 
and 
the 
random 
combinations 
hh(r, 
Ci)) 
(Table 
21). 
The 
hydraulic 
conditions 
of 
both 
locations, 
pumping 
well 
and 
far 
field, 
were 
the 
same 
for 
all 
j 
and 
Ck, 
thereby 
resulting 
in 
similar 
estimates 
for 
Kˆ
well 
and 
Kˆ
efu. 


At 
a 
radial 
distance 
from 
the 
pumping 
well 
corresponding 
to 
one 
correlation 
length 
(i 
= 
8m) 
the 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
Figure 
39: 
Simulated 
drawdowns 
of 
a 
single 
realization 
(with 
statistics 
KG 
=10-4 
m/s, 
2 
= 
1, 
e 
=1, 
i 
=8m 
and 
h(R 
=32m)=0m). 
Black 
line 
denotes 
the 
angular 
mean 
hh(r)) 
of 
the 
four 
transects 
hh(r,1)i,...,hh(r,4)) 
(grey 
dashed 
lines). 
The 
black 
dots 
mark 
one 
random 
combination 
of 
head 
measurements 
hh(r,C1)) 
. 


drawdowns 
of 
the 
four 
transects 
hh(r,i)) 
(dashed 
lines 
in 
Figure 
39) 
deviated 
from 
the 
angular 
mean 
hh(r)i, 
indicating 
that 
the 
estimated 
correlation 
length 
`ˆwas 
sensitive 
to 
the 
selected 
transect. 
Likewise, 
estimates 
of 
`ˆwere 
less 
accurate 
for 
selected 
transects, 
e.g., 
`ˆof 
hh(r,3)) 
= 


12.52 
in 
Table 
21. 
A 
possible 
explanation 
for 
this 
is 
the 
occurrence 
of 
patches 
of 
much 
higher 
or 
lower 
than 
the 
average 
hydraulic 
conductivity 
in 
the 
environment 
of 
the 
pumping 
well, 
which 
critically 
influence 
hydraulic 
heads 
of 
the 
transition 
zone 
between 
Kwell-driven 
and 
Kefu-driven 
depression. 
For 
randomly 
combined 
head 
measurements 
from 
all 
four 
transects 
hh(r,Ci)) 
(black 
dots 
in 
Figure 
39) 
the 
estimates 
of 
`ˆwere 
more 
accurate 
but 
less 
certain, 
as 
indicated 
by 
larger 
confidence 
intervals 
(Table 
21). 
The 
random 
distribution 
of 
measurements 
around 
the 
pumping 
well 
buffers 
the 
effect 
of 
estimating 
local 
values 
as 
for 
single 
transects, 
but 
also 
results 
in 
a 
larger 
deviation 
from 
the 
mean 
head 
hh(r)) 
(Figure 
39). 
Based 
on 
these 
findings, 
we 
conclude 
that 
for 
heterogeneous 
aquifers 
the 
angular 
arrangement 
of 
head 
measurements 
is 
critical 
to 
estimate 
sound 
and 
certain 
values 
of 
`ˆ. 


5.3.4. 
Parameter 
Estimation 
under 
Radial 
Limitation 
For 
each 
scenario 
S1-S8 
(Table 
20) 
the 
estimates 
of 
Kˆ
efu, 
Kˆ
well 
and 
`ˆand 
the 
corresponding 
confidence 
intervals 
CI 
ˆ, 
CI 
ˆand 
CI`ˆ 
– 
all 
normalized 
by 
the 
initial 
parameters 
Kefu, 
Kwell 


KefuKwell 


and 
i 
– 
are 
depicted 
in 
Figure 
40, 
where 
values 
close 
to 
one 
indicate 
high 
accuracy. 
Likewise, 
`˜
is 
plotted 
as 
alternative 
estimate 
to 
`ˆto 
depict 
estimates 
of 
the 
correlation 
length 
while 
Kˆ
efu 
and 
Kˆ
well 
being 
constant. 
For 
each 
scenario 
the 
approach 
of 
estimating 
`˜separately 
from 
Kˆ
efu 
and 
Kˆ
well 
was 
found 
to 
be 
very 
suitable 
to 
reduce 
the 
confidence 
interval 
CI`˜dramatically 
(Figure 
40). 



5.3. 
Pumping 
Tests 
under 
Limited 
Data 
Availability 
-a 
Paradigm 
for 
Field 
Applications 
117 
Figure 
40: 
Normalized 
estimation 
results 
for 
eight 
scenarios 
of 
radially 
limited 
head 
data 
(Ta


ble 
20) 
with 
error 
bars 
marking 
the 
normalized 
95% 
confidence 
intervals. 
Circles 


denote 
results 
for 
Kˆ
efu/Kefu, 
squares 
for 
ˆ`/i 
and 
diamonds 


Kwell/hKwelli, 
stars 
for 
ˆ
for 
˜

`/`. 


Parameter 
estimates 
of 
Kˆ
well 
and 
Kˆ
efu 
were 
nearly 
insensitive 
to 
any 
sampling 
arrangement 
with 
regard 
to 
the 
radial 
distance 
except 
for 
S6 
where 
the 
estimation 
of 
Kˆ
well 
failed 
due 
to 
a 
lack 
of 
head 
measurement 
at 
the 
pumping 
well. 
As 
a 
consequence 
also 
estimates 
of 
correlation 
length 
were 
unreliable 
for 
S6. 
Otherwise, 
estimates 
of 
the 
correlation 
length 
were 
sensitive 
to 
increasing 
sampling 
intervals 
(S2-S4) 
and 
the 
distance 
of 
the 
head 
measurements 
from 
the 
pumping 
well 
(S5, 
S7 
and 
S8). 


These 
findings 
emphasize 
that 
at 
least 
one 
(if 
measured 
at 
the 
pumping 
well) 
to 
three 
(if 
measured 
in 
the 
vicinity 
of 
the 
pumping 
well) 
head 
measurements 
are 
required 
to 
facilitate 
reliable 
estimates 
of 
Kˆ
well. 
Likewise, 
four 
to 
five 
head 
measurements 
located 
in 
the 
far 
field 
of 
the 
depression 
cone 
enhance 
reliable 
estimates 
of 
Kˆ
efu. 


Based 
on 
this 
findings, 
we 
conclude 
that 
limited 
sample 
size 
with 
regard 
to 
the 
radial 
distance 
from 
the 
pumping 
well 
affects 
the 
uncertainty 
of 
the 
estimates 
Kˆ
efu 
and 
Kˆ
well, 
represented 
by 
larger 
confidence 
intervals 
rather 
than 
their 
accuracy. 
This 
uncertainty 
also 
influences 
the 
uncertainty 
of 
the 
correlation 
length 
`ˆ, 
whereas 
the 
accuracy 
of 
`ˆ 
critically 
depends 
on 
the 
number 
of 
head 
measurements 
located 
at 
the 
transition 
zone 
(0 
- 
2`). 
This 
result 
is 
critical 
for 
field 
applications 
where 
the 
correlation 
length 
is 
the 
target 
rather 
than 
the 
input 
parameter. 


The 
combination 
of 
both 
approaches, 
sample 
size 
reduction 
with 
regard 
to 
angular 
and 
radial 
head 
limitations, 
even 
amplified 
the 
effects: 
the 
accuracy 
of 
parameter 
estimates 
decreased 
while 
confidence 
intervals 
increased, 
particularly 
for 
the 
correlation 
length. 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
Figure 
41: 
Distribution 
of 
pumping 
wells 
(black 
dots) 
and 
observation 
wells 
(circles) 
in 
the 
area 
of 
investigation 
at 
the 
field 
site 
Horkheimer 
Insel 
(according 
to 
Schad 
[1997]). 


5.4. 
Field 
Application 
Given 
our 
findings 
from 
the 
simulated 
pumping 
tests 
in 
section 
5.3, 
we 
applied 
hefw(r) 
using 
empirical 
data 
from 
on-site 
pumping 
tests 
to 
estimate 
the 
parameters 
Kˆ
G, 
ˆ2 
and 
`ˆof 
a 
natural 
aquifer. 


5.4.1. 
Field 
Site 
Horkheimer 
Insel 
We 
used 
empirical 
pumping 
test 
data 
from 
the 
Horkheimer 
Insel 
[Schad, 
1997], 
located 
in 
the 
Neckar 
Valley 
in 
southern 
Germany. 
The 
aquifer 
consisted 
of 
unconsolidated 
fluvial 
sediments 
with 
poorly 
sorted 
sand 
and 
gravel 
and 
can 
be 
considered 
heterogeneous 
with 
a 
log-normal 
conductivity 
distribution 
[Schad 
and 
Teutsch, 
1994]. 
On 
average 
the 
saturated 
thickness 
L 
was 
3m, 
overlain 
by 
impermeable 
flood 
deposits 
and 
underlain 
by 
limestone 
formations. 
The 
dominating 
general 
hydraulic 
gradient 
was 
0.001 
towards 
the 
Neckar 
River. 
The 
infiltration 
rate 
from 
bedrock 
into 
the 
aquifer 
was 
negligible. 
According 
to 
the 
findings 
of 
Schad 
[1997] 
the 
average 
hydraulic 
conductivity 
derived 
from 
flowmeter 
measurements 
was 
KG 
=4.57e-3m/s. 
The 
average 
transmissivity 
was 
estimated 
as 
T 
=3.1e-2m2/s 
using 
Theis’ 
analytical 
solution, 
which 
is 
equivalent 
to 
Kefu 
= 
T/L 
=1.0e-2m/s. 
Detailed 
variogram 
analysis 
revealed 
that 
the 
vertical 
and 
horizontal 
correlation 
length 
ranged 
between 
`v 
=0.15 
- 
0.25m 
and 
f 
=6 
- 
10m, 
respectively, 
resulting 
in 
an 
anisotropy 
ratio 
of 
e 
= 
`v/f 
=0.015 
- 
0.04. 
The 
variance 
of 
hydraulic 
conductivity 
ranged 
from 
2 
=3.17 
for 
permeameter 
measurements 
and 
2 
=2.32 
for 
grain 
size 
analyzes 
to 
2 
=1.57 
for 
flowmeter 
measurements. 
More 
details 
regarding 
the 
field 
site, 
wells, 
pumping 
tests, 
and 
parameter 
estimates 
of 
K(x) 
from 
laboratory 
trials 
are 
provided 
in 
Schad 
[1997]. 


For 
the 
purpose 
of 
this 
study 
we 
used 
data 
derived 
from 
small 
scale 
pumping 
tests 
(Figure 
41). 
At 
each 
of 
the 
three 
pumping 
wells 
W40, 
W42, 
and 
W44, 
two 
pumping 
tests 
were 
performed. 



5.4. 
Field 
Application 
For 
each 
of 
the 
pumping 
tests 
the 
drawdown 
was 
measured 
at 
four 
observation 
wells. 
The 
tests 
continued 
for 
two 
hours 
with 
pumping 
rates 
varying 
between 
2l/s 
and 
5.5l/s 
depending 
on 
the 
well 
yield. 


The 
pumping 
test 
duration 
of 
two 
hours 
is 
assumed 
to 
be 
ideal 
(neither 
too 
short 
nor 
too 
long) 
because 
(i) 
transient 
drawdown 
data 
at 
later 
times 
corresponds 
to 
a 
quasi 
steady 
state 
flow 
regime 
[Neuman 
et 
al., 
2007], 
(ii) 
the 
influence 
of 
field-site 
dependent 
boundaries 
can 
be 
neglected 
until 
three 
hours 
of 
pumping 
[Schad 
and 
Teutsch, 
1994], 
(iii) 
storativity, 
well 
bore 
storage 
and 
well 
loss 
have 
a 
significant 
influence 
to 
drawdown 
at 
early 
times 
which 
can 
be 
eliminated 
by 
using 
drawdown 
data 
at 
later 
times 
[Kruseman 
and 
de 
Ridder, 
2000]; 
(iv) 
the 
influence 
of 
areas 
with 
low 
permeability 
do 
not 
affect 
effective 
conductivity 
until 
after 
two 
hours 
of 
pumping 
[Schad 
and 
Teutsch, 
1994]; 
(v) 
after 
two 
hours 
the 
flow 
is 
essentially 
horizontal, 
whereas 
vertical 
flow 
becomes 
negligible 
and, 
hence, 
this 
phase 
is 
interpreted 
as 
influenced 
by 
inherent 
aquifer 
properties 
only 
[Schad 
and 
Teutsch, 
1994]. 


5.4.2. 
Analysis 
of 
Empirical 
Data 
The 
measured 
drawdown 
data 
hh(r)) 
of 
pumping 
tests 
PT40, 
PT42 
and 
PT44 
was 
compared 
to 
Thiem’s 
solution 
(Figure 
42), 
such 
that 
it 
matched 
the 
drawdown 
near 
and 
at 
a 
large 
distance 
from 
the 
pumping 
well, 
eventually 
resulting 
in 
representative 
conductivities 
that 
can 
be 
considered 
first 
estimates 
of 
Kwell 
and 
Kefu, 
respectively. 


Then, 
hefw(r) 
was 
applied 
using 
nonlinear 
regression 
to 
estimate 
the 
parameters 
of 
K(x), 
as 
described 
in 
section 
5.2.2. 
The 
boundary 
conditions 
at 
the 
pumping 
well 
were 
assumed 
to 
be 
BCH 
because 
pumping 
tests 
were 
performed 
at 
constant 
pumping 
rates 
and 
Kwell 
<Kefu. 
Thus, 
the 
representative 
hydraulic 
conductivity 
at 
the 
pumping 
well 
was 
set 
to 
a 
value 
corresponding 
to 
the 
harmonic 
mean 
Kwell 
= 
KH 
and 
H 
=0.8 
in 
Eq.(5.1). 
The 
anisotropy 
ratio 
was 
set 
to 
e 
=0.025. 
The 
reference 
head 
h(R) 
corresponded 
to 
the 
observation 
well 
with 
the 
largest 
distance 
to 
the 
pumping 
well. 


Pumping 
rates 
were 
normalized 
to 
a 
value 
of 
Qw 
=0.001 
m3/s 
and 
combined 
pumping 
tests 
sharing 
the 
same 
pumping 
well, 
eventually 
resulting 
in 
three 
quasi-steady 
state 
tests 
with 
eight 
observational 
points 
hh(ri)ii=1,...,8 
(with 
h(r8)= 
h(R) 
= 
0), 
including 
the 
drawdown 
at 
the 
pumping 
well. 
Further, 
we 
calculated 
Kˆ
G 
and 
ˆ2 
by 
substituting 
the 
estimates 
of 
Kˆ
efu 
and 
Kˆ
well 
into 
Eq.(5.3). 


The 
set-up 
of 
the 
three 
pumping 
tests 
is 
well 
reflected 
by 
scenario 
S8 
(Table 
20), 
with 
observational 
wells 
being 
located 
at 
large 
distance 
rather 
than 
in 
the 
vicinity 
of 
the 
pumping 
well. 
Hence, 
head 
measurements 
in 
distance 
of 
one 
correlation 
length 
from 
the 
pumping 
well 
(l 
=6 
- 
10m 
[Schad, 
1997]) 
are 
rare, 
eventually 
biasing 
the 
accuracy 
and 
certainty 
of 
estimates 
of 
`ˆ(section 
5.3.4). 
Therefore, 
we 
subdivided 
the 
parameter 
estimation: 
firstly, 
we 
estimated 
Kˆ
efu 
and 
Kˆ
well, 
and 
then 
estimated 
`˜while 
keeping 
Kˆ
efu 
and 
Kˆ
well 
fixed 
(section 
5.2.2). 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
Figure 
42: 
Drawdowns 
of 
the 
three 
pumping 
tests 
with 
fitted 
analytical 
solution 
hefw(r) 
(bold 
line) 
and 
Thiem’s 
solution 
for 
Kˆ
efu 
(dashed 
line) 
and 
Kˆ
well 
(dashed 
dotted 
line) 
as 
well 
as 
hydraulic 
head 
measurements 
of 
the 
pumping 
tests 
PT40, 
PT42 
and 
PT44. 



5.5. 
Conclusions 
121 
PT40 
PT42 
PT44 
mean 
ˆ
6.62 
Kefu 
[10-3m/s] 
4.19 
(±1.23) 
12.25 
(±1.4) 
3.42 
(±0.90) 
ˆKwell 
[10-3m/s] 
1.16 
(±0.18) 
2.36 
(±0.13) 
0.67 
(±0.07) 
1.40 
ˆKG 
[10-3m/s] 
2.23 
(±0.18) 
5.47 
(±0.21) 
1.53 
(±0.14) 
3.08 
ˆ2 
1.32 
(±0.46) 
1.68 
(±0.17) 
1.66 
(±0.37) 
1.55 
`˜ 
[m] 
8.22 
(±1.38) 
7.89 
(±0.46) 
9.36 
(±1.27) 
8.49 


Table 
22: 
Estimation 
results 
for 
PT40, 
PT42, 
PT44 
and 
the 
mean 
of 
the 
three 
pumping 
tests 
with 
95% 
confidence 
intervals 
in 
brackets. 


5.4.3. 
Estimated 
Field 
Parameters 
For 
a 
selected 
range 
of 
pumping 
tests 
at 
the 
Horkheimer 
Insel, 
the 
effective 
well 
flow 
head 
hefw(r) 
provided 
reliable 
predictions 
of 
drawdown 
for 
the 
entire 
range 
of 
the 
depression 
cone 
(Figure 
42). 
Whereas 
Thiem’s 
solution 
predicted 
the 
drawdown 
accurately 
only 
in 
the 
vicinity 
or 
at 
large 
distance 
from 
the 
pumping 
well. 
This 
is 
due 
to 
the 
additional 
parameters 
2 
and 
l 
integrated 
into 
hefw(r), 
considering 
the 
influence 
of 
heterogeneity, 
and 
hence, 
capturing 
the 
transition 
between 
Kwell-driven 
and 
Kefu-driven 
drawdown 
(section 
5.2.2). 


The 
estimated 
parameters 
of 
K(x) 
for 
PT40, 
PT42 
and 
PT44 
(Table 
22) 
were 
in 
good 
accordance 
with 
those 
derived 
by 
Schad 
[1997] 
(section 
5.4.1). 
Regarding 
the 
diverse 
analyzing 
methods 
used 
by 
Schad 
[1997] 
the 
parameters 
from 
flowmeter 
measurements 
corresponded 
best 
to 
our 
estimates 
using 
hefw(r). 
This 
is 
no 
surprise 
as 
measurements 
from 
both 
flowmeter 
and 
pumping 
tests 
represent 
data 
that 
is 
averaged 
across 
large 
volumes 
of 
the 
aquifer. 


The 
estimated 
correlation 
length 
was 
similar 
across 
the 
three 
pumping 
tests, 
whereas 
the 
estimated 
hydraulic 
conductivities 
Kˆ
well 
and 
Kˆ
efu 
were 
considerably 
different. 
Substituting 
those 
estimates 
into 
Eq.(5.3)) 
clarified 
that 
the 
variability 
of 
Kˆ
well 
and 
Kˆ
efu 
was 
caused 
by 
different 
values 
of 
Kˆ
G 
rather 
than 
ˆ2, 
indicating 
that 
the 
local 
permeability 
was 
considerably 
different 
across 
the 
three 
pumping 
wells 
but 
the 
overall 
heterogeneity 
structure 
of 
the 
aquifer 
regarding 
variance 
ˆ2 
=1.55 
and 
correlation 
length 
`˜=8.5m 
was 
the 
same 
for 
the 
area 
influenced 
by 
the 
pumping 
tests 
(Figure 
41). 


The 
uncertainty 
involved 
with 
the 
parameter 
estimation 
was 
low 
for 
the 
estimates 
of 
variance 
and 
the 
conductivity 
values. 
Since 
there 
was 
only 
a 
limited 
number 
of 
observation 
wells 
located 
in 
the 
vicinity 
of 
the 
well, 
CI`˜was 
higher 
compared 
to 
CI 
ˆand 
CI 
ˆ. 
However, 
the 
method 


Kefu 
Kwell

of 
estimating 
`˜rather 
than 
`ˆestablished 
as 
valid 
procedure 
to 
decrease 
the 
confidence 
interval, 
thereby 
increasing 
the 
certainty 
of 
the 
estimates 
of 
the 
correlation 
length. 


5.5. 
Conclusions 
We 
showed 
that 
the 
analytical 
effective 
well 
flow 
solution 
hefw(r) 
is 
capable 
to 
provide 
reliable 
parameter 
estimates 
of 
a 
heterogeneous 
aquifer 
under 
field-site 
conditions. 
By 
using 
simulated 
pumping 
tests 
to 
systematically 
reduce 
the 
sampling 
size, 
we 
assessed 
the 
accuracy 
and 
uncertainty 
of 
estimates 
at 
each 
level 
of 
data 
availability. 
Then 
we 
applied 
hefw(r) 
using 
empirical 



5. 
Estimating 
Parameters 
of 
Aquifer 
Heterogeneity 
Using 
Pumping 
Tests 
pumping 
test 
data 
from 
the 
field 
site 
Horkheimer 
Insel. 
The 
estimated 
parameters 
of 
the 
fluvial 
heterogeneous 
aquifer 
are 
in 
agreement 
with 
estimates 
derived 
from 
laboratory 
measurements, 
published 
in 
Schad 
[1997]. 
From 
these 
findings, 
we 
have 
elucidated 
how 
the 
quality 
of 
parameter 
estimates 
is 
affected 
by 
number 
and 
spatial 
distribution 
of 
measurement 
locations 
having 
implications 
for: 
(i) 
the 
conceptual 
design 
of 
field 
site 
pumping 
tests, 
and 
(ii) 
the 
potential 
power 
to 
estimate 
reliable 
parameters 
at 
established 
pumping 
test 
sites. 


Regarding 
the 
former 
(i), 
it 
is 
recommended 
to 
optimize 
the 
spatial 
distribution 
of 
piezometers 
relative 
to 
the 
pumping 
well. 
A 
large 
number 
of 
piezometers 
facilitate 
accurate 
and 
confident 
parameter 
estimation, 
but 
also 
provoke 
high 
cost-load. 
Likewise, 
limited 
sample 
size 
of 
head 
measurements 
or 
poorly 
conducted 
piezometers 
with 
regard 
to 
their 
relative 
location 
to 
the 
pumping 
well 
provoke 
poor 
parameter 
estimates, 
and 
high 
work-and 
cost-load. 


To 
achieve 
reliable 
parameter 
estimates, 
it 
is 
critical 
to 
capture 
the 
entire 
range 
of 
the 
depression 
cone. 
Our 
findings 
emphasize 
that 
at 
least 
three 
to 
five 
head 
measurements 
located 
in 
the 
far 
field 
are 
required 
to 
facilitate 
accurate 
estimates 
of 
Kefu. 
To 
gain 
any 
estimates 
of 
Kˆ
well, 
drawdown 
data 
at 
the 
pumping 
well 
is 
a 
prerequisite. 
For 
sound 
and 
confident 
estimates 
of 
the 
correlation 
length 
`ˆ, 
several 
piezometers 
located 
in 
the 
transition 
zone 
are 
required, 
i.e. 
between 
pumping 
well 
and 
the 
distance 
of 
few 
correlation 
lengths. 
This 
is 
particularly 
challenging 
since 
this 
parameter 
is 
usually 
unknown 
a 
priori. 
However, 
simple 
laboratory 
tests, 
such 
as 
grain 
size 
analyses 
at 
random 
site 
locations, 
can 
be 
used 
as 
indicators 
for 
pre-estimating 
both 
the 
vertical 
and 
horizontal 
correlation 
length. 


Regarding 
the 
spatial 
distribution 
of 
piezometers 
our 
findings 
point 
toward 
two 
strategies. 
On 
the 
one 
hand, 
piezometers 
located 
along 
one 
transect 
from 
the 
pumping 
well 
provide 
high 
confidence 
but 
low 
accuracy 
in 
parameter 
estimates 
because 
each 
transect 
represents 
the 
statistical 
properties 
along 
one 
angular 
direction 
only. 
Whereas 
spatially 
random 
distributed 
piezometers 
around 
the 
pumping 
well 
better 
represent 
the 
statistics 
of 
the 
overall 
conductivity 
field, 
thereby 
providing 
more 
accurate 
parameter 
estimates 
with 
lower 
confidence. 


Regarding 
the 
latter 
(ii) 
examining 
the 
pumping 
test 
design 
in 
relation 
to 
the 
number 
and 
spatial 
distribution 
of 
the 
piezometers 
is 
important 
to 
clarify 
what 
accuracy 
and 
uncertainty 
can 
be 
expected. 
Likewise, 
the 
predictive 
power 
of 
the 
established 
pumping 
test 
design 
depends 
on 
the 
parameter 
of 
interest. 
Having 
drawdown 
data 
at 
the 
pumping 
well 
and 
at 
a 
distance 
of 
several 
correlation 
lengths 
from 
the 
pumping 
well 
enables 
accurate 
estimates 
of 
hydraulic 
conductivity 
at 
both 
locations 
( 
Kˆ
well 
and 
Kˆ
efu). 
These 
parameters 
can 
then 
be 
used 
to 
reduce 
the 
uncertainty 
in 
estimates 
of 
the 
correlation 
length. 
Likewise, 
piezometers 
located 
in 
the 
vicinity 
of 
the 
well 
increase 
the 
accuracy 
of 
correlation 
length 
estimates. 


Based 
on 
the 
high 
accuracy 
and 
relatively 
low 
uncertainty 
involved 
with 
the 
inverse 
estimation 
of 
aquifer 
parameters, 
we 
conclude 
that 
hefw(r) 
is 
a 
simple 
alternative 
to 
laboratory 
trials 
to 
estimate 
the 
statistical 
heterogeneity 
parameter 
KG, 
2 
and 
1 
more 
cost-efficiently. 



6. 
Summary 
and 
Outlook 
The 
current 
work 
provided 
insights 
into 
the 
effects 
of 
aquifer 
heterogeneity 
on 
subsurface 
flow 
and 
salt 
transport. 
The 
hydraulic 
property 
of 
permeability 
is 
a 
key 
parameter 
to 
describe 
groundwater 
flow. 
However, 
spatial 
heterogeneity 
is 
an 
intrinsic 
property 
of 
permeability 
and 
a 
large 
range 
of 
values 
can 
be 
found 
in 
measurement 
data. 
By 
capturing 
the 
spatial 
variability 
with 
a 
stochastic 
approach 
we 
examined 
how 
permeability 
impacts 
on 
subsurface 
processes 
at 
large 
scale 
and 
on 
local 
scale. 


We 
first 
performed 
a 
numerical 
study 
on 
regional 
scale 
to 
investigate 
the 
fluid 
dynamics 
of 
the 
Thuringian 
Basin. 
The 
results 
underline 
that 
the 
permeability 
is 
a 
decisive 
factor 
for 
the 
pattern 
of 
regional 
groundwater 
flow 
and 
salt 
distribution. 
The 
impact 
of 
heterogeneity 
is 
strong. 
Amongst 
others 
we 
performed 
simulations 
with 
different 
mean 
permeability 
values 
in 
all 
sedimentary 
layers, 
with 
permeability 
distributions 
of 
different 
variances 
and 
of 
different 
correlation 
structures. 
We 
saw 
that 
all 
these 
parameters 
impact 
on 
the 
evolving 
fluid 
circulation 
pattern 
and 
thus, 
the 
amount 
of 
dissolved 
salt 
and 
the 
location 
where 
salt 
reaches 
near-surface 
regions. 
Additionally, 
we 
examined 
how 
density 
differences 
caused 
by 
temperature 
and 
salt 
concentration 
gradients 
impact 
on 
the 
fluid 
dynamics. 
We 
found 
that 
thermally 
induced 
convection 
is 
not 
present 
in 
the 
shallow 
sedimentary 
basin 
of 
Thuringia. 
However, 
the 
changes 
in 
fluid 
density 
due 
to 
dissolved 
salt 
can 
lead 
to 
significant 
changes 
in 
the 
distribution 
and 
amount 
of 
salt 
in 
the 
Thuringian 
Basin. 


In 
a 
second 
step, 
we 
showed 
that 
heterogeneity 
in 
permeability 
values 
also 
effects 
flow 
on 
local 
scale. 
Of 
interest 
are 
well 
flow 
regimes, 
because 
pumping 
tests 
are 
a 
widely 
used 
tool 
to 
infer 
on 
porous 
medium 
parameters 
in 
particular 
in 
shallow 
aquifer 
units. 
However, 
pumping 
tests 
in 
heterogeneous 
aquifers 
show 
a 
distinct 
drawdown 
behavior 
compared 
to 
well 
flow 
in 
homogeneous 
media. 
We 
used 
this 
specific 
characteristic 
of 
well 
flow 
to 
determine 
the 
parameters 
of 
aquifer 
heterogeneity. 
We 
derived 
an 
analytical 
solution 
to 
describe 
the 
mean 
drawdown 
of 
a 
steady 
state 
pumping 
test 
in 
three 
dimensional 
heterogeneous 
anisotropic 
media. 
By 
combining 
the 
analytical 
solution 
with 
an 
inverse 
estimation 
strategy 
we 
determined 
parameters 
of 
aquifer 
heterogeneity, 
like 
mean 
permeability, 
variance, 
and 
a 
quantity 
capturing 
the 
correlation 
structure 
from 
on-site 
pumping 
tests. 


The 
method 
to 
describe 
mean 
well 
flows 
is 
currently 
limited 
to 
steady 
state 
pumping 
tests. 
In 
future 
work, 
we 
aim 
to 
expand 
the 
theory 
and 
its 
practical 
application 
to 
transient 
pumping 
tests. 
This 
kind 
of 
pumping 
tests 
is 
of 
high 
practical 
relevance. 
The 
temporal 
resolution 
of 
data 
can 
provide 
more 
information 
of 
the 
area 
of 
investigation. 
Further 
fields 
of 
expansion 
are 
pumping 
tests 
in 
heterogeneous 
porous 
media 
with 
advanced 
descriptions 
of 
the 
spatial 
correlation 
structure 
like 
stochastic 
media 
with 
facies 
structure 
or 
media 
with 
infinite 
correlation 
structure. 


The 
numerical 
study 
of 
the 
Thuringian 
Basin 
showed 
that 
the 
knowledge 
about 
the 
permeability 
distribution 
is 
the 
key 
to 
determine 
the 
flow 
pattern 
and 
salt 
distribution. 
Additional 
data 
is 
necessary 
to 
reduce 
uncertainty 
and 
enhance 
the 
reproduction 
and 
prediction 
of 
the 
fluid 



6. 
Summary 
and 
Outlook 
dynamics 
in 
the 
Thuringian 
Basin. 
The 
application 
of 
the 
upscaling 
method 
to 
pumping 
test 
performed 
in 
the 
Thuringian 
Basin 
can 
help 
to 
gain 
more 
information 
about 
the 
heterogeneous 
structure 
of 
the 
shallow 
aquifers. 
The 
analysis 
of 
existing 
and 
the 
performance 
of 
additional 
pumping 
test 
in 
the 
Thuringian 
Basin 
will 
we 
be 
a 
goal 
of 
future 
research. 


Deep 
drilling 
projects 
are 
necessary 
to 
gain 
information 
of 
geological 
units 
in 
several 
hundreds 
of 
meters 
depth. 
As 
part 
of 
the 
research 
project 
INFLUINS 
a 
scientific 
drilling 
is 
currently 
(July 
2013) 
performed 
in 
the 
center 
of 
the 
Thuringian 
Basin 
near 
Erfurt. 
Numerous 
experiments 
concerning 
fluid 
flow 
between 
the 
different 
sedimentary 
units 
are 
done 
as 
part 
of 
the 
drilling. 
Furthermore, 
bore-log 
data 
can 
provide 
further 
information 
about 
the 
sedimentary 
and 
hydrogeological 
character 
of 
the 
sediments. 
This 
drilling 
project 
can 
enhance 
the 
data 
availability 
for 
numerical 
basin 
models 
and 
thus 
lead 
to 
more 
precise 
reproductions 
and 
predictions 
of 
the 
fluid 
dynamics 
of 
the 
Thuringian 
Basin. 


With 
this 
work 
we 
contribute 
to 
the 
understanding 
of 
flow 
and 
transport 
processes 
in 
the 
subsurface 
that 
are 
influenced 
by 
density 
differences 
and 
local 
geological 
characteristics. 
We 
observed 
the 
mechanisms 
controlling 
the 
fluid 
dynamics 
in 
the 
Thuringian 
Basin, 
but 
they 
can 
be 
representative 
for 
any 
shallow 
sedimentary 
basin 
with 
conditions 
comparable 
to 
those 
in 
Thuringia. 



A. 
Appendix 
A.1. 
The 
Coarse 
Graining 
Conductivity 
in 
Anisotropic 
Media 
As 
presented 
in 
Attinger 
[2003] 
the 
filtered 
conductivity, 
gained 
from 
Coarse 
Graining, 
covering 
the 
small 
scale 
effects 
in 
the 
filtered 
flow 
equation 
(4.6) 
is 
of 
the 
form 
KCG(x)= 
KCG()+ 


. 
A 


K˜
CG(x,), 
with 
a 
filtered 
fluctuation 
part 
K˜
CG(x,) 
and 
scale 
dependent 
mean 
conductivity 
KCG

()= 
KA 
+KCG(). 
The 
latter 
one 
is 
composed 
of 
the 
arithmetic 
mean 
of 
the 
unfiltered 


A 
conductivity 
KA 
and 
the 
scale 
dependent 
partial 
mean 
conductivity 
KCG(), 
given 
by 
the 
integral 


2 
Z2

..


1

KCG()= 
-KA 
˜(y)y1 
- 
f˜ 
(y)dd 
y 
,`(A.1)

(2)dRd 
y2

where 
d`is 
the 
dimension, 
2 
is 
the 
variance 
of 
the 
log 
conductivity, 
˜(y) 
is 
the 
Fourier 
transformed 
of 
the 
correlation 
function, 
and 
f˜ 
(y) 
is 
the 
filter 
function 
in 
Fourier 
space. 


Adapting 
the 
integral 
(A.1) 
to 
Coarse 
Graining 
for 
well 
flow 
in 
three 
dimensions 
we 
replace 
the 
scaling 
factor 
`by 
x,y 
. 
r`and 
z 
. 
e`· 
r. 
The 
Fourier 
transforms 
of 
the 
adapted 
filter 
function 
fr, 
as 
given 
in 
Eq. 
(4.7), 
and 
the 
correlation 
function 
, 
as 
given 
in 
Eq.(4.3), 
are 




1 
2222

˜(y)= 
3/2e`3 
exp-4`2(y1 
+ 
y2 
+ 
ey3),`(A.2) 




1 
-2/32 
2222

f˜ 
r(y) 
= 
exp-4 
er`2(y1 
+ 
y2 
+ 
ey3),`(A.3) 


where 
r`is 
the 
radial 
distance 
from 
the 
well,`is 
the 
correlation 
length 
in 
horizontal 
direction, 
e`is 
the 
anisotropy 
ratio 
and 
`is 
the 
factor 
of 
proportionality. 


Applying 
Eq.(A.2) 
and 
Eq.(A.3) 
to 
Eq.(A.1) 
results 
in 


Z

2 
y2y2

11

KCG(r)= 
-KA2 
˜(y)dy 
-2 
˜(y)f˜ 
r(y)dy 
(A.4)

(2)3 
R3 
yR3 
y

0!-3/21

. 
2

2

r

= 
-KA2 
@
(e) 
- 

(e) 
v 
A

1+ 
3,`(A.5) 
e

2`2 


where 

(e) 
is 
the 
anisotropy 
function 
as 
presented 
in 
Eq.(4.5). 
We 
give 
the 
solution 
of 
the 




2 
-3/2

integrals 
in 
(A.4) 
to 
gain 

(e) 
and 

˜(e)= 

(e)1+ 
p
2er22separately 
in 
section 
A.2. 


3

Following 
the 
line 
of 
procedure 
in 
Attinger 
[2003] 
we 
find 
the 
solution 
for 
the 
upscaled 
mean 
conductivity 
KCG(r)= 
KA 
+ 
KCG(r) 
by 
interpreting 
the 
terms 
in 
Eq.(A.5) 
as 
first 
terms 
of 


A 


an 
exponential 
series, 


0. 
!-3/2. 
22

r

KCG 
A

(r)= 
Kefu 
exp 
@2
(e) 
1+ 
v 
,`(A.6)
A 
3

2`2

e

..


where 
Kefu 
= 
KA 
exp-2
(e)is 
the 
effective 
conductivity 
for 
uniform 
flow 
as 
given 
in 
Eq.(4.4). 
Attinger 
[2003] 
showed 
the 
validity 
of 
this 
procedure 
using 
renormalization 
theory. 



126 
A. 
Appendix 


A.2. 
The 
Anisotropy 
Function 
The 
function 

(e), 
as 
given 
in 
Eq. 
(4.5), 
expresses 
the 
impact 
of 
anisotropy 
e 
on 
the 
effective 
conductivity 
for 
uniform 
flow 
in 
three 
dimensions. 
It 
results 
as 
solution 
of 
the 
integral 


Z

1 
y12


(e)= 
2 
˜(y)dy 
. 
(A.7)

(2)3
R3 
y

Using 
an 
Gaussian 
correlation 
function 
, 
as 
given 
in 
Eq. 
(4.3) 
with 
the 
Fourier 
transformed 
˜
as 
given 
in 
Eq.(A.2), 
the 
integral 
(A.7) 
results 
in 


Z


(e)= 
23e`
33 
/22y1222exp-
14 
`2(y12 
+ 
y22)exp-
14 
e 
2`2 
y32 
dy1 
dy2 
dy3 
, 
(A.8) 


R3 
y1 
+ 
y2 
+ 
y

3 


We 
use 
a 
short 
cut 
to 
solve 
Eq.(A.8). 
We 
substitute 




2Z1

y1 
1 
2 
z 
222

= 
y1 
exp- 
(y1 
+ 
y2 
+ 
y3)dz. 
(A.9) 


y22244

1 
+ 
y2 
+ 
y30 


p

..

R8 
2


into 
Eq.(A.8), 
resort 
the 
terms 
and 
solve 
the 
integral 
in 
y 
by 
usingexp-axdx 
= 
p

-8 


..

R8 
2
v 

a 


and-8 
x2 
exp-axdx 
= 
2a3/2 
: 


Z8 
Z

e`3 


2 
122 
12


(e)= 
y1 
exp-4 
(y1 
+ 
y2)(`2 
+ 
z)exp-4 
y3(e 
2`2 
+ 
z)dy1 
dy2 
dy3 
dz

253/2

0R3 


v 
pp

Z1

e`3 
p 
p 


= 
ppdz

253/20 
2((`2 
+ 
z)/4)3/2 
(`2 
+ 
z)/4(e2`2 
+ 
z)/4 


Z1

e`3 
11 


= 
v 
dz. 


20 
`2 
+ 
ze2`2 
+ 
z 


We 
solve 
the 
integral 
in 
z 
by 
substitution 
and 
results 
in 


2s3

v 
8 


4Z
1 
z 
+ 
e2`2 
e2`2 
+ 
z 
5


(e)= 
e`32)3/2 
arctan

2 
`3(1 
- 
e`2 
- 
e2`2 
+ 
`2(1 
- 
e2)(`2 
+ 
z) 


0

 s!

e 
11 
e 


= 
2)3/2 
arctan2 
- 
1 
- 
2,

2(1 
- 
ee1 
- 
e

whichisthesolutionofthegammafunctionasgiveninEq.(4.5). 
Theresultcanbetransformed 
to 
the 
solution 
presented 
in 
Dagan 
[1989]. 


The 
integral 
for 

˜(e,`,r) 
as 
given 
in 
Eq. 
(A.4) 
can 
be 
solved 
in 
a 
similar 
way. 
We 
evaluate 

˜(e,`,r) 
using 
Eq.(A.2) 
and 
Eq.(A.3) 
with 
the 
abbreviation 
!2 
= 
e-2/32r2 
as 
well 
as 
Eq.(A.9): 


Z

1 
y12


˜(e, 
`, 
r)= 
2 
˜(y)f˜ 
r(y)dy

(2)3
R3 
ye`3 
Zy2Z

1 
1 
2222`21 
2222

= 
2 
exp-4 
`2 
y1 
+ 
y2 
+ 
y3e 
exp-4 
!2 
y1 
+ 
y2 
+ 
ey 
dy

233/2R3 
y3 



A.3. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Three 
Dimensions 
Z1

e`3 
11 


= 
v 
dz
20 
(`2 
+ 
!2 
+ 
z)2 
e2`2 
+ 
e2!2 
+ 
z 


2Ps3

8 


e`3 
4P
1 
e2`2 
+ 
e2!2 
+ 
z
5

= 
2!2)3/2 
arctan

2(`2 
+ 
!2 
- 
e2`2 
- 
e`2 
- 
e2`2 
+ 
!2 
- 
e2!2 
0

"Pv 
#8 


e`3 
e2`2 
+ 
e2!2 
+ 
z

+2 
(`2 
- 
e2`2 
+ 
!2 
- 
e2!2)(`2 
+ 
!2 
+ 
z) 
0 
e`3 
1 
(1 
- 
e2)(`2 
+ 
!2) 
e`

= 
2)2/3(`2 
+ 
!2)2/3 
arctan 
-

2(1 
- 
ee2(`2 
+ 
!2) 
(1 
- 
e2)(`2 
+ 
!2)3/2

0sP1P

e`1 
(1 
- 
e2) 
e``3 


@A

= 
2)3/2 
arctan 
-

2

2 
(1 
- 
ee(1 
- 
e2)(`3 
+ 
!2)3/2 



(e)
= 
3/2
1+ 
2r2 


2/3`2

e

A.3. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Three 
Dimensions 
In 
order 
to 
derive 
the 
effective 
well 
flow 
head 
hefw(r) 
we 
solve 
the 
steady 
state 
flow 
equation 
(4.2) 


with 
KCG(r) 
as 
defined 
in 
Eq.(4.9). 
We 
transform 
the 
flow 
equation 
to 
polar 
coordinates 


and 
evaluate 
the 
vertical 
component. 
We 
result 
in 
the 
ODE 
for 
the 
hydraulic 
head 
h(r)=

RL 


h˜
(r,`z)dz,
0 
 1dh(r)d2h(r) 
!dKCG(r)dh(r)


0= 
KCG(r)++ 
.`(A.10)

2

r`dr`drdr`dr`

q

P

v 




For 
brevity, 
we 
write 
KCG(r)= 
Kefu 
exp 
with 
u(r)=1+ 
2r2/( 
3e2`2) 
and 
`= 


u3(r) 


ln 
(Kwell/Kefu). 
Solving 
the 
ODE 
(A.10) 
by 
separation 
of 
variables 
using 
d 
KCG(r)= 
KCG(r)· 


Pdr 
`· 
d1 
, 
we 
result 
in 


dru3(r) 


Z

r2 


1 
-`

h(r2) 
- 
h(r1)= 
C0expdr`

ru3(r)

r1 


1Z

XP(-)ir2 
1 
= 
C0 
dr,`(A.11) 


i=0 
i!r1 
ru3i(r)

where 
we 
perform 
a 
series 
expansion 
of 
the 
exponential 
function. 
C0 
is 
the 
integration 
constant. 


For 
every 
i`the 
solution 
of 
the 
integral 
in 
Eq.(A.11) 
is 
given 
by

Z

1 


dr`= 
ln 
r`for 
i=0 


ru3i(r)
11


= 
ln 
r`- 
ln(u(r)+1)+ 
+ 
...`+ 
for 
i`= 
1 
odd 


u(r) 
(3i`- 
2)u3i-2(r) 
11

= 
ln 
r`- 
ln 
u(r)+ 
+ 
...`+ 
for 
i`= 
2 
even. 


2u2(r) 
(3i`- 
2)u3i-2(r) 


We 
insert 
this 
result 
to 
Eq.(A.11) 
and 
resort 
the 
sum 
in 
terms 
of 
r. 
Furthermore, 
we 
use 


(-)j

the 
definition 
of 
the 
exponential 
functionP8 
j=0 
j! 
=e-. 
and 
of 
the 
hyperbolic 
sine 
and 
(-)2j+1 
P8 
(-)2jcosine 
functionsP8 
j=0 
(2j+1)! 
= 
- 
sinh(),j=0 
(2j)! 
= 
cosh(). 
We 
neglect 
all 
terms 
of 



128 
A. 
Appendix 


1

the 
form 
juj 
(r) 
with 
j`= 
3 
and 
result 
in 


h(r2) 
- 
h(r1)= 
C0 
exp 
(-)ln 
r2 
+ 
C0 
sinh()U1 
+ 
C0(1 
- 
cosh())U2 
,`(A.12) 
r1 


with 


U1 
= 
ln 
u(r2)+1 
1 
1 
u(r1)+1 
- 
u(r2)+ 
u(r1) 
,`
1111 


U2 
= 
ln 
u(r2)+ 
+ 
.`

u(r1) 
- 
2u2(r2)2u2(r1) 
- 
4u4(r2)4u4(r1) 


The 
final 
result 
for 
hefw(r) 
as 
presented 
in 
Eq. 
(4.10) 
results 
by 
inserting 
the 
boundary 
conditions 
r2 
= 
r, 
r1 
= 
R`and 
C0 
= 
- 
Qw 
. 
We 
derive 
the 
constant 
C0 
from 
the 
relations 


2LKefu 
LKCGh0(rwKefu


Qw 
= 
-2rw) 
and 
h0(rw)= 
C0 
KCG 
, 
with 
Qw 
being 
the 
pumping 
rate 
and 
L`the 


rw 


aquifer 
thickness. 


The 
solution 
in 
Eq.(A.12) 
is 
nearly 
exact. 
The 
logarithmic 
terms 
dominate 
the 
drawdown. 


1

The 
truncated 
parts 
contain 
terms 
of 
the 
form 
O(k)juj 
(r), 
with 
j,`k`= 
3 
impacting 
on 
the 
drawdown 
for 
very 
small 
r`and 
large 
variances 
2 
and 
can 
therefore 
be 
neglected 
without 
changing 
the 
character 
of 
the 
solution. 


A.4. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Two 
Dimensions 
We 
apply 
the 
same 
procedure 
as 
presented 
in 
section 
A.3 
to 
find 
the 
closed 
form 
solution 
for 
the 
upscaled 
hydraulic 
head 
hewf(2d)(r) 
as 
given 
in 
Eq.(1.23). 
The 
steady 
state 
groundwater 
flow 
equation 
in 
radial 
coordinates 
with 
the 
upscaled 
Coarse 
Graining 
transmissivity 
T`CG(r)

H 


given 
in 
Eq.(1.22) 
reads 


 !

T`CG

1dh`d2h`(r)dh

H

0= 
THCG(r)+ 
2+ 
.`(A.13) 


r`dr`drdr`dr`

Performing 
the 
derivation 
in 
T`CG(r) 
and 
solving 
the 
ODE 
(A.13) 
by 
separation 
of 
variables 


H 


results 
in 
 !

Z

r2 
2

11 


h(r2) 
- 
h(r1)= 
C1expdr,`(A.14) 


r1 
r`2 
(1+ 
2r2/`2)

with 
integration 
constants 
C1 
and 
C2. 


We 
evaluate 
the 
integral 
(A.14) 
analytically 
by 
making 
use 
of 
the 
exponential 
integral 
function 


R

x 
exp(z)1

..(x) 
- 
..(X)=Xz 
dz. 
It 
requires 
a 
substitution 
of 
the 
argument 
z`= 

2 
2 
(1+2r2/`2) 
with 


dr`= 
- 
`2 
2 
- 
1- 
21 
dz. 
Eq. 
(A.14) 
results 
in 


4z22z 


Z

2 
z(r2) 
exp(z)

h(r2) 
- 
h(r1)= 
C1 
dz

4z(r1) 
z(z`- 

2 
2 
) 


Z

C1 
z(r2) 
exp(z) 
exp(z)

= 
- 
dz

2z(r1) 
z`- 

2 
2 
z`


A.4. 
Derivation 
of 
the 
Effective 
Well 
Flow 
Head 
in 
Two 
Dimensions 
ZZ

C1 
z(r2)- 
22 
exp(z 
+ 

2 
2 
) 
C1 
z(r2) 
exp 
(z)

=dz 
- 
dz

2z(r1)- 
2 
z 
2z

2 
z(r1) 




C1 
2 
C1

= 
2exp..z(r2) 
- 
2 
- 
..z(r1) 
- 
2 
- 
(..(z(r2)) 
- 
..(z(r1))) 
.

222

2 


We 
derive 
the 
constant 
C1 
= 
- 
Qw 
from 
the 
boundary 
condition 
of 
constant 
flux 
at 
the 


2TG 


T 
CG(rwT 
CG(rw)C1 
TG/T 
CG(rw

well 
Qw 
= 
-2rw)h0(rw)= 
-2rw). 
Inserting 
r1 
= 
r, 
r2 
= 
R 


rw 


1

with 
h(R)= 
hR 
and 
resubstituting 
z(r)= 
22 
(1+2r2/`2) 
gives 
the 
effective 
well 
flow 
head 
for 
large 
scale 
pumping 
tests 


  ! !!

hefw(2d)(r)= 
- 
Qw 
2 
1 
2 
1

G 
- 
..

4TG 
2 
1+ 
2r2/`2 
2 
1+ 
2R2/`2 


  ! !!



Qw2 
2 
-2r2/`2 
2 
-2R2/`2 


+ 
exp2G 
- 
..+ 
hR 
.
4TG 
2 
1+ 
2r2/`2 
2 
1+ 
2R2/`2 


(A.15) 

130 
A. 
Appendix 



References 
131 


References 


FEFLOW 
6.1: 
Finite 
Element 
Subsurface 
Flow 
and 
Transport 
Simulation 
System 
User 
Manual, 
DHI-WASY, 
2012. 


Aehnelt, 
M., 
D. 
Beyer, 
C. 
Kunkel, 
and 
R. 
Gaupp, 
PoroPerm-Daten 
im 
Thüringer 
Becken 
aus 
Bohrlochuntersuchungen, 
unpublished 
material, 
2011. 


Attinger, 
S., 
Generalized 
coarse 
graining 
procedures 
for 
flow 
in 
porous 
media, 
Comput. 
Geosci., 
7(4), 
253–273, 
2003. 


Bear, 
J., 
Dynamics 
of 
Fluids 
in 
Porous 
Media, 
Elsevier, 
New 
York, 
1972. 


Bense, 
V. 
F., 
and 
M. 
A. 
Person, 
Faults 
as 
conduit-barrier 
systems 
to 
fluid 
flow 
in 
siliciclastic 
sedimentary 
aquifers, 
Water 
Resour. 
Res., 
42, 
W05421, 
2006. 


Brunner, 
P., 
and 
C. 
T. 
Simmons, 
Hydrogeosphere: 
A 
fully 
integrated, 
physically 
based 
hydrological 
model, 
Ground 
Water, 
50(2), 
2012. 


Cirpka, 
O. 
A., 
randomfield.m, 
http://m2matlabdb.ma.tum.de/download.jsp?MC_ID=6&MP 
_ID=31, 
2010. 


Cooper, 
H. 
H. 
J., 
and 
C. 
E. 
Jacob, 
A 
generalized 
graphical 
method 
for 
evaluating 
formation 
constants 
and 
summarizing 
well 
field 
history, 
Eos 
Trans. 
AGU, 
27, 
526–534, 
1946. 


Copty, 
N. 
K., 
and 
A. 
N. 
Findikakis, 
Stochastic 
analysis 
of 
pumping 
test 
drawdown 
data 
in 
heterogeneous 
geologic 
formations, 
J. 
Hydraul. 
Res., 
42, 
EXTRA 
ISSUE, 
59–67, 
2004. 


Dagan, 
G., 
Statistical 
theory 
of 
groundwater 
flow 
and 
transport: 
Pore 
to 
laboratory, 
laboratory 
to 
formation, 
and 
formation 
to 
regional 
scale, 
Water 
Resour. 
Res., 
22, 
120S–134S, 
1986. 


Dagan, 
G., 
Flow 
and 
Transport 
on 
Porous 
Formations, 
Springer 
Verlag, 
New 
York, 
1989. 


Dagan, 
G., 
and 
S. 
C. 
Lessoff, 
Transmissivity 
upscaling 
in 
numerical 
aquifer 
models 
of 
steady 
well 
flow: 
Unconditional 
statistics, 
Water 
Resour. 
Res., 
43, 
W05431, 
2007. 


Dagan, 
G., 
and 
S. 
Neuman 
(Eds.), 
Subsurface 
Flow 
and 
Transport: 
A 
Stochastic 
Approach, 
Cambridge 
University 
Press, 
Cambridge, 
1997. 


Desbarats, 
A., 
Spatial 
averaging 
of 
transmissivity 
in 
heterogeneous 
fields 
with 
flow 
toward 
a 
well, 
Water 
Resour. 
Res., 
28(3), 
757–767, 
1992. 


Diersch, 
H., 
and 
O. 
Kolditz, 
Coupled 
groundwater 
flow 
and 
transport: 
2. 
Thermohaline 
and 
3D 
convection 
systems, 
Adv. 
Water 
Resour., 
21, 
401–425, 
1998. 


Diersch, 
H., 
and 
O. 
Kolditz, 
Variable-density 
flow 
and 
transport 
in 
porous 
media: 
Approaches 
and 
challenges, 
Adv. 
Water 
Resour., 
25(3), 
899–944, 
2002. 



132 
References 


Dykaar, 
B., 
and 
P. 
K. 
Kitanidis, 
Determination 
of 
the 
effective 
hydraulic 
conductivity 
for 
heterogeneous 
porous 
media 
using 
a 
numerical 
spectral 
approach: 
1. 
Method, 
Water 
Resour. 
Res., 
28(4), 
1155–1166, 
1992a. 


Dykaar, 
B., 
and 
P. 
K. 
Kitanidis, 
Determination 
of 
the 
effective 
hydraulic 
conductivity 
for 
heterogeneous 
porous 
media 
using 
a 
numerical 
spectral 
approach: 
2. 
Results, 
Water 
Resour. 
Res., 
28(4), 
1167–1178, 
1992b. 


Elder, 
J., 
Steady 
free 
convection 
in 
porous 
medium 
heated 
from 
below, 
J. 
Fluid 
Mech., 
27(1), 
29–48, 
1967a. 


Elder, 
J., 
Transient 
convection 
in 
porous 
medium, 
J. 
Fluid 
Mech., 
27(3), 
609—-623, 
1967b. 


Fiori, 
A., 
P. 
Indelman, 
and 
G. 
Dagan, 
Correlation 
structure 
of 
flow 
variables 
for 
steady 
flow 
toward 
a 
well 
with 
application 
to 
highly 
anisotropic 
heterogeneous 
formations, 
Water 
Resour. 
Res., 
34, 
699–708, 
1998. 


Firmani, 
G., 
A. 
Fiori, 
and 
A. 
Bellin, 
Three-dimensional 
numerical 
analysis 
of 
steady 
state 
pumping 
tests 
in 
heterogeneous 
confined 
aquifers, 
Water 
Resour. 
Res., 
42, 
W03422, 
2006. 


Follin, 
S., 
Numerical 
calculations 
of 
heterogeneity 
of 
groundwater 
flow, 
Ph.D. 
thesis, 
Royal 
Institute 
of 
Technology, 
Stockholm, 
1992. 


Freeze, 
R., 
A 
stochastic-conceptual 
analysis 
of 
the 
one-dimensional 
groundwater 
flow 
in 
nonuniform 
homogeneous 
media, 
Water 
Resour. 
Res., 
11(5), 
725–742, 
1975. 


Frolkovic, 
P., 
and 
H. 
D. 
Schepper, 
Numerical 
modeling 
of 
convection 
dominated 
transport 
coupled 
with 
density 
driven 
flow 
in 
porous 
media, 
Adv. 
Water 
Resour., 
24, 
63–72, 
2001. 


Gaupp, 
R., 
T. 
Voigt, 
and 
H. 
Lützner, 
Stratigraphy 
and 
sedimentological 
evolution 
of 
the 
Lower 
and 
Middle 
Triassic 
deposits 
in 
the 
SE 
part 
of 
the 
German 
Triassic 
Basin, 
Hallesches 
Jahrb. 
Geowiss., 
Reihe 
B, 
Beiheft, 
6, 
99–120, 
1998. 


Gelhar, 
L., 
Stochastic 
Subsurface 
Hydrology, 
Prentice 
Hall, 
Englewood 
Cliffs, 
New 
York, 
1993. 


Gelhar, 
L. 
W., 
Sochastic 
subsurface 
hydrology 
from 
theory 
to 
applications, 
Water 
Resour. 
Res., 
22(9), 
135S–145S, 
1986. 


Gelhar, 
L. 
W., 
and 
C. 
L. 
Axness, 
Three-dimensional 
stochastic 
analysis 
of 
macrodispersion 
in 
aquifers, 
Water 
Resour. 
Res., 
19(1), 
161–180, 
1983. 


Guadagnini, 
A., 
M. 
Riva, 
and 
S. 
Neuman, 
Three-dimensional 
steady 
state 
flow 
to 
a 
well 
in 
a 
randomly 
heterogeneous 
bounded 
aquifer, 
Water 
Resour. 
Res., 
39(3), 
405–419, 
2003. 


Hagen-Zanker, 
A., 
Comparing 
continuous 
valued 
raster 
data 
-a 
cross 
disciplinary 
literature 
scan, 
Tech. 
rep., 
Research 
Institute 
for 
Knowledge 
Systems 
(RIKS 
bv), 
Maastricht, 
The 
Netherlands, 
2006. 



References 
133 


Hauthal, 
U., 
Zum 
Wasserleitvermögen 
von 
Gesteinen 
des 
Mittleren 
Buntsandsteins, 
Zeitschrift 
für 
angewandte 
Geologie, 
13, 
405 
– 
407, 
1967. 


Hecht, 
G., 
Grundwässer 
-In 
G. 
Seidel, 
Geologie 
von 
Thüringen, 
Schweizerbart, 
Stuttgart, 
2003. 


Herbert, 
A., 
C. 
Jackson, 
and 
D. 
Lever, 
Coupled 
groundwater 
flow 
and 
solute 
transport 
with 
fluid 
density 
stronly 
dependent 
upon 
concentration, 
Water 
Resour. 
Res., 
24(10), 
1781–1795, 
1988. 


Holzbecher, 
E., 
C. 
Kohfahl, 
M. 
Mazurowski, 
A. 
Bacik, 
M. 
Dobies, 
and 
M. 
Schneider, 
The 
sensitivity 
of 
thermohaline 
groundwater 
circulation 
to 
flow 
and 
transport 
parameters: 
A 
numerical 
study 
based 
on 
double-diffusive 
convection 
above 
a 
salt 
dome, 
Transp 
Porous 
Med, 
83, 
771–791, 
2010. 


Hoppe, 
W., 
Exkursionsführer 
Thüringer 
Becken, 
Akademie-Verlag, 
Berlin, 
1959. 


Hurtig, 
E., 
and 
W. 
Meincke, 
Subsurface 
temperature, 
heat 
conductivity 
and 
heat 
flow 
in 
the 
Thuringia 
Basin 
and 
surrounding 
areas, 
GDR, 
Bulletin 
Volcanologique, 
33, 
229 
– 
242, 
1969. 


Indelman, 
P., 
Steady-state 
source 
flow 
in 
heterogeneous 
porous 
media, 
Transport 
in 
Porous 
Media, 
45(1), 
105–127, 
2001. 


Indelman, 
P., 
and 
B. 
Abramovich, 
Nonlocal 
properties 
on 
nonuniform 
averaged 
flows 
in 
heterogeneous 
media, 
Water 
Resour. 
Res., 
30(12), 
3385–3393, 
1994. 


Indelman, 
P., 
and 
G. 
Dagan, 
A 
note 
on 
well 
boundary 
condition 
for 
flow 
through 
heterogeneous 
formations, 
Water 
Resour. 
Res., 
40, 
W03601, 
2004. 


Indelman, 
P., 
A. 
Fiori, 
and 
G. 
Dagan, 
Steady 
flow 
toward 
wells 
in 
heterogeneous 
formations: 
Mean 
head 
and 
equivalent 
conductivity, 
Water 
Resour. 
Res., 
32(7), 
1975–1984, 
1996. 


Johannsen, 
K., 
On 
the 
validity 
of 
the 
Boussinesq 
approximaion 
for 
the 
Elder 
problem, 
Comp. 
Geoscience, 
7, 
169–182, 
2003. 


Jordan, 
H., 
and 
H. 
Weder, 
Hydrogeologie, 
Enke, 
Stuttgart, 
1995. 


Kaiser, 
B. 
O., 
M. 
Cacace, 
M. 
Scheck-Wenderoth, 
and 
B. 
Lewerenz, 
Characterization 
of 
main 
transport 
processes 
in 
the 
Northeast 
German 
Basin: 
Constraints 
from 
3-D 
numerical 
models, 
Geochem. 
Geophys. 
Geosyst., 
12, 
Q07011, 
2011. 


Kinzelbach, 
W., 
and 
R. 
Rausch, 
Grundwassermodellierung, 
Gebrüder 
Borntraeger 
Berlin 
Stuttgart, 
1995. 


Kober, 
M., 
Erstellung 
eines 
ArcGis-und 
Petrel-basierten 
digitalen 
Untergrundmodells 
der 
Thüringer 
Mulde, 
Master’s 
thesis, 
Friedrich-Schiller-University 
of 
Jena, 
2008. 


Kolditz, 
O. 
(Ed.), 
Computational 
Methods 
in 
Environmental 
Fluid 
Mechanics, 
Springer 
Science 
Publisher, 
2012. 



134 
References 


Kolditz, 
O., 
R. 
Ratke, 
H. 
Diersch, 
and 
W. 
Zielke, 
Coupled 
groundwater 
flow 
and 
transport: 
1. 
Verification 
of 
variable 
density 
flow 
and 
transport 
models, 
Adv. 
Water 
Resour., 
21, 
27–46, 
1998. 


Kolditz, 
O., 
U. 
Goerke, 
H. 
Shao, 
and 
W. 
Wang 
(Eds.), 
Thermo-Hydro-Mechanical-Chemical 
Processes 
in 
Porous 
Media, 
Springer 
Verlag, 
2012a. 


Kolditz, 
O., 
et 
al., 
OpenGeoSys: 
An 
open-source 
initiative 
for 
numerical 
simulation 
of 
thermohydro-
mechanical/chemical 
(THM/C) 
processes 
in 
porous 
media, 
Environ 
Earth 
Sci, 
67, 
589–599, 
2012b. 


Konikow, 
L. 
F., 
W. 
E. 
Sanford, 
and 
P. 
J. 
Campbell, 
Constant-concentration 
boundary 
condition: 
Lessons 
from 
the 
HYDROCOIN 
variable-density 
groundwater 
benchmark 
problem, 
Water 
Resour. 
Res., 
33, 
2253–2261, 
1997. 


Kruseman, 
G., 
and 
N. 
de 
Ridder 
(Eds.), 
Analysis 
and 
Evaluation 
of 
Pumping 
Test 
Data, 
International 
Institute 
for 
Land 
Reclamation 
and 
Improvement 
(ILRI), 
Wageningen, 
2000. 


Lampe, 
C., 
and 
M. 
Person, 
Advective 
cooling 
within 
sedimentary 
rift 
basins 
– 
application 
to 
the 
Upper 
Rhinegraben 
(Germany), 
Mar. 
Pet. 
Geol., 
19, 
361–375, 
2002. 


Layton, 
W. 
J., 
A 
mathematical 
introduction 
to 
large 
eddy 
simulation, 
Research 
note, 
Department 
of 
Mathematics, 
University 
of 
Pittsburgh, 
Pittsburgh, 
PA 
15260, 
U.S.A., 
2002. 


Leven, 
C., 
and 
P. 
Dietrich, 
What 
information 
can 
we 
get 
from 
pumping 
tests?-comparing 
pumping 
test 
configurations 
using 
sensitivity 
coefficients, 
J. 
of 
Hydrology, 
319, 
199–215, 
2006. 


Magri, 
F., 
Mechanisms 
and 
fluid-dynamics 
driving 
saline 
waters 
within 
the 
North 
East 
German 
Basin: 
Results 
from 
thermohaline 
numerical 
simulations, 
Ph.D. 
thesis, 
Freie 
Universität 
Berlin, 
2005. 


Magri, 
F., 
U. 
Bayer, 
M. 
Tesmer, 
P. 
Möller, 
and 
A. 
Pekdeger, 
Salinization 
problems 
in 
the 
NEGB: 
Results 
from 
thermohaline 
simulations, 
Int. 
J. 
Earth 
Sci., 
97, 
1075–1085, 
2008. 


Magri, 
F., 
et 
al., 
Fluid-dynamics 
driving 
saline 
water 
in 
the 
Norht 
East 
German 
Basin, 
Int. 
J. 
Earth 
Sci., 
94, 
1056–1069, 
2005. 


Malz, 
A., 
and 
J. 
Kley, 
The 
Finne 
fault 
zone 
(central 
Germany): 
structural 
analysis 
of 
a 
partially 
inverted 
extensional 
fault 
zone 
balanced 
cross-sections, 
Int. 
J. 
Earth 
Sci., 
101, 
2167–2182, 
2012. 


Matheron, 
G., 
Elements 
pour 
une 
theorie 
des 
milieux 
poreux, 
Maisson 
et 
Cie, 
Paris, 
1967. 


Meincke, 
W., 
Hydrodynamik 
im 
Solling-Sandstein 
(Mittleren 
Buntsandstein) 
des 
Thüringer 
Beckens, 
Zeitschrift 
für 
angewandte 
Geologie, 
13, 
399–405, 
1967. 


Meincke, 
W., 
E. 
Hurtig, 
and 
J. 
Weiner, 
Temperaturverteilung, 
Wärmeleitfähigkeit 
und 
Wärmefluss 
im 
Thüringer 
Becken, 
Geophysik 
und 
Geologie, 
11, 
40–71, 
1967. 



References 
135 


Merz, 
G., 
Zur 
Petrographie, 
Stratigraphie, 
Paläogeographie 
und 
Hydrogeologie 
des 
Muschelkalks 
(Trias) 
im 
Thüringer 
Becken, 
Z. 
geo. 
Wiss, 
15, 
457–473, 
1987. 


Miall, 
A., 
The 
Geology 
of 
Fluvial 
Deposits: 
Sedimentary 
Facies, 
Basin 
Analysis, 
and 
Petroleum 
Geology, 
Springer, 
Berlin, 
Heidelberg, 
1996. 


Neuman, 
S. 
P., 
Universal 
scaling 
of 
hydraulic 
conductivities 
and 
dispersivities 
in 
geologic 
media, 
Water 
Resour. 
Res., 
26(8), 
1749–1758, 
1990. 


Neuman, 
S. 
P., 
and 
S. 
Orr, 
Prediction 
of 
steady 
state 
flow 
in 
nonuniform 
geologic 
media 
by 
conditional 
moments: 
Extract 
nonlocal 
formalism, 
effective 
conductivities 
and 
weak 
approximation, 
Water 
Resour. 
Res., 
29(2), 
341–364, 
1993. 


Neuman, 
S. 
P., 
A. 
Guadagnini, 
and 
M. 
Riva, 
Type-curve 
estimation 
of 
statistical 
heterogeneity, 
Water 
Resour. 
Res., 
40, 
W04201, 
2004. 


Neuman, 
S. 
P., 
A. 
Blattstein, 
M. 
Riva, 
D. 
Tartakovsky, 
A. 
Guadagnini, 
and 
T. 
Ptak, 
Type 
curve 
interpretation 
of 
late-time 
pumping 
test 
data 
in 
randomly 
heterogeneous 
aquifers, 
Water 
Resour. 
Res., 
43, 
W10421, 
2007. 


Neuman, 
S. 
P., 
M. 
Riva, 
and 
A. 
Guadagnini, 
On 
the 
geostatistical 
characterization 
of 
hierarchical 
media, 
Water 
Resour. 
Res., 
44, 
W02403, 
2008. 


Nield, 
D., 
and 
A. 
Bejan, 
Convection 
in 
Porous 
Media, 
Springer, 
New 
York, 
1999. 


Oldenburg, 
C., 
and 
K. 
Pruess, 
Dispersive 
transport 
dynamics 
in 
a 
strongly 
coupled 
groundwater-brine 
flow 
system, 
Water 
Resour. 
Res., 
31(2), 
289–302, 
1995. 


Park, 
C.-H., 
and 
M. 
Aral, 
Sensitivity 
of 
the 
solution 
of 
the 
Elder 
problem 
to 
density, 
velocity 
and 
numerical 
perturbations, 
J. 
Contam. 
Hydrol., 
92, 
33–49, 
2007. 


Person, 
M., 
J. 
Raffensperger, 
S. 
Ge, 
and 
G. 
Garven, 
Basin-scale 
hydrogeological 
modeling, 
Reviews 
of 
Geophysics, 
34(1), 
61–87, 
1996. 


Prasad, 
A., 
and 
C. 
T. 
Simmons, 
Unstable 
density-driven 
flow 
in 
heterogeneous 
porous 
media: 
A 
stochastic 
study 
of 
the 
Elder 
[1967b] 
"short 
heater" 
problem, 
Water 
Resour. 
Res., 
39(1), 
1007, 
2003. 


Prasad, 
A., 
and 
C. 
T. 
Simmons, 
Using 
quantitave 
indicators 
to 
evaluate 
results 
form 
variabledensity 
groundwater 
flow 
models, 
Hydrogeol. 
J., 
13, 
905–914, 
2005. 


Puff, 
P., 
R. 
Langbein, 
and 
G. 
Seidel, 
Mesozoikum 
-In 
G. 
Seidel, 
Geologie 
von 
Thüringen, 
Schweizerbart, 
Stuttgart, 
2003. 


Rödiger, 
T., 
Characterisierung 
und 
Modellierung 
des 
Buntsandsteinfließsystems 
im 
Osten 
des 
Thüringer 
Beckens, 
Ph.D. 
thesis, 
Friedrich-Schiller-University 
of 
Jena, 
2005. 


Renard, 
P., 
and 
G. 
de 
Marsily, 
Calculating 
equivalent 
permeability: 
A 
review, 
Adv. 
Water 
Resour., 
20(5-6), 
253–278, 
1997. 



136 
References 


Riva, 
M., 
and 
M. 
Willmann, 
Impact 
of 
log-transmissivity 
variogram 
structure 
on 
groundwater 
flow 
and 
transport 
predictions, 
Adv. 
Water 
Resour., 
32, 
1311–1322, 
2009. 


Rubin, 
Y., 
Applied 
Stochastic 
Hydrogeology, 
Oxford 
Univ. 
Press, 
New 
York, 
2003. 


Samper, 
F. 
J., 
and 
S. 
Neuman, 
Estimation 
of 
spatial 
covariance 
structure 
by 
adjoint 
state 
maximum 
likelihood 
cross 
validation: 
1. 
Theory, 
Water 
Resour. 
Res., 
25(3), 
351–362, 
1989a. 


Samper, 
F. 
J., 
and 
S. 
Neuman, 
Estimation 
of 
spatial 
covariance 
structure 
by 
adjoint 
state 
maximum 
likelihood 
cross 
validation: 
2. 
Synthetic 
experiments, 
Water 
Resour. 
Res., 
25(3), 
363–371, 
1989b. 


Sánchez-Vila, 
X., 
Radially 
convergent 
flow 
in 
heterogeneous 
porous 
media, 
Water 
Resour. 
Res., 
33(7), 
1633–1641, 
1997. 


Sánchez-Vila, 
X., 
C. 
L. 
Axness, 
and 
J. 
Carrera, 
Upscaling 
transmissivity 
under 
radially 
convergent 
flow 
in 
heterogeneous 
media, 
Water 
Resour. 
Res., 
35(3), 
613–621, 
1999. 


Sánchez-Vila, 
X., 
A. 
Guadagnini, 
and 
J. 
Carrera, 
Representative 
hydraulic 
conductivities 
in 
saturated 
groundwater 
flow, 
Rev. 
Geophys., 
44, 
RG3002, 
2006. 


Schad, 
H., 
Variability 
of 
hydraulic 
parameters 
in 
non-uniform 
porous 
media: 
Experiments 
and 
stochastic 
modeling 
at 
different 
scales, 
Ph.D. 
thesis, 
Universität 
Tübingen, 
1997. 


Schad, 
H., 
and 
G. 
Teutsch, 
Effects 
of 
the 
investigation 
scale 
on 
pumping 
test 
results 
in 
heterogeneous 
porous 
aquifers, 
J. 
of 
Hydrology, 
159, 
61–77, 
1994. 


Schneider, 
C. 
L., 
and 
S. 
Attinger, 
Beyond 
Thiem 
-a 
new 
method 
for 
interpreting 
large 
scale 
pumping 
tests 
in 
heterogeneous 
aquifers, 
Water 
Resour. 
Res., 
44, 
W04427, 
2008. 


Seidel, 
G., 
Geologie 
von 
Thüringen, 
Schweizerbart, 
Stuttgart, 
2003. 


Shvidler, 
M. 
I., 
The 
source-type 
solution 
of 
the 
problem 
of 
unsteady 
flow 
in 
random 
porous 
media, 
Akad. 
Nauk 
USSR 
Mekh. 
Zhidk. 
Gaza, 
4, 
137–141, 
1966. 


Simmons, 
C., 
T. 
Fenstemaker, 
and 
J. 
Sharp, 
Variable-density 
groundwater 
flow 
and 
solute 
transport 
in 
heterogeneous 
porous 
media: 
Approaches, 
resolutions 
and 
future 
chalanges, 
J. 
Contam. 
Hydrol., 
52, 
245–275, 
2001. 


Sudicky, 
E., 
A 
natural 
gradient 
experiment 
on 
solute 
transport 
in 
a 
sand 
aquifer: 
Spatial 
variability 
of 
hydraulic 
conductivity 
and 
its 
role 
in 
the 
disperion 
process, 
Water 
Resour. 
Res., 
22(13), 
2069–2082, 
1986. 


Theis, 
C., 
The 
relation 
between 
the 
lowering 
of 
the 
piezometric 
surface 
and 
the 
rate 
and 
duration 
of 
discharge 
of 
a 
well 
using 
groundwater 
storage, 
Trans. 
Am. 
Geophys. 
Union, 
16, 
519–524, 
1935. 


Thiem, 
G., 
Hydrologische 
Methoden, 
J.M. 
Gebhardt, 
Leipzig, 
1906. 



References 
137 


Thüringer 
Landesamt 
für 
Umwelt 
und 
Geologie 
(TLUG), 
Geologie 
und 
Hydrogeologie 
im 
Überblick 
1:200000, 
CD-ROM, 
2002. 


Turcke, 
M., 
and 
B. 
Kueper, 
Geostatistical 
analysis 
of 
the 
borden 
aquifer 
hydraulic 
conductivity 
field, 
J. 
of 
Hydrology, 
178, 
223–240, 
1996. 


U.S. 
Geological 
Survey, 
MODFLOW: 
3D 
finite-difference 
groundwater 
flow 
model, 
http://water.usgs.gov/nrp/gwsoftware/modflow.html, 
2012. 
van 
Reeuwijk, 
M., 
S. 
Mathias, 
C. 
Simmons, 
and 
J. 
Ward, 
Insights 
from 
a 
pseudospectral 
approach 
to 
the 
Elder 
problem, 
Water 
Resour. 
Res., 
45, 
W04416, 
2009. 


Voss, 
C., 
and 
W. 
Souza, 
Variable 
density 
flow 
and 
solute 
transport 
simulation 
of 
regional 
aquifers 
containing 
a 
narrow 
freshwater-saltwater 
transition 
zone, 
Water 
Resour. 
Res., 
23(10), 
1851–1866, 
1987. 


Voss, 
C. 
I., 
and 
A. 
M. 
Provost, 
Sutra, 
a 
model 
for 
saturated-unsaturated 
variable-density 
ground-water 
flow 
with 
solute 
or 
energy 
transport, 
Water-resources 
investigations 
report 
024231, 
U.S. 
Geological 
Survey, 
version 
of 
September 
22, 
2010, 
2010. 


Younes, 
A., 
P. 
Ackerer, 
and 
R. 
Mose, 
Modeling 
variable 
density 
flow 
and 
solute 
transport 
in 
porous 
medium: 
2. 
Re-evaluation 
of 
the 
salt 
dome 
flow 
problem, 
Transp 
Porous 
Med, 
35, 
375–394, 
1999. 


Zech, 
A., 
C. 
L. 
Schneider, 
and 
S. 
Attinger, 
The 
extended 
Thiem’s 
solution 
-Including 
the 
impact 
of 
heterogeneity, 
Water 
Resour. 
Res., 
48(10), 
W10535, 
2012. 


Zehner, 
B., 
Constructing 
geometric 
models 
of 
the 
subsurface 
for 
finite 
element 
simulation, 


Conference 
of 
the 
International 
Association 
of 
Mathematical 
Geosciences 
(IAMG 
2011), 
Salzburg, 
Austria, 
2011. 



Danksagung 


Mein 
erster 
Dank 
geht 
an 
meine 
Betreuerin 
Prof. 
Sabine 
Attinger, 
dafür 
dass 
sie 
mir 
die 
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hat 
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Durch 
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Ich 
bin 
dankbar, 
dass 
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auf 
diesem 
Weg 
begleitet 
und 
gefördert 
hat. 
Ebenso 
möchte 
ich 
Prof. 
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für 
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in 
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letzten 
drei 
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sowie 
für 
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Begutachtung 
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Arbeit 
danken. 


Besonderen 
Dank 
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Sven 
Arnold, 
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Ich 
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EnfInf 
gedankt. 


Zum 
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danke 
ich 
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und 
meinen 
Freunden. 
Ihr 
habt 
mir 
das 
Selbstvertrauen 
geschenkt, 
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schaffen. 
Danke 
für 
eure 
Unterstützung. 
Ein 
ganz 
besonderer 
Dank 
geht 
an 
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und 
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weil 
ihr 
in 
meiner 
dunkelsten 
Stunde 
bei 
mir 
wart. 



Selbstständigkeitserklärung 


Ich 
erkläre, 
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