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This is to certify that the
thesis entitled !

EFFECTS OF HETEROGENEITY ON GROUNDWATER
FLOW AND CONTAMINANT TRANSPORT — A
VISUALIZATION LIBRARY

presented by

PRASANNA v SAMPATH ;

has been accepted towards fulfillment
of the requirements for the

MS -degree' in CIVIL ENGINEERING

 

 

(ftwajor wProfessor’f Signature/
‘1 [A]? Q I) 2m 5

ate

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EFFECTS OF HETEROGENEITY ON GROUNDWATER FLOW AND
CONTAMINANT TRANSPORT — A VISUALIZATION LIBRARY

By

Prasanna V Sampath

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Civil and Environmental Engineering

2006

Abstract

EFFECTS OF HETEROGENEITY ON GROUNDWATER FLOW AND
CONTAMINANT TRANSPORT — A VISUALIZATION LIBRARY

By
Prasanna V Sampath

It is well known that the subsurface environment is strongly heterogeneous with respect
to geological, hydrological, and chemical properties. These heterogeneities are often
“random” and can sometimes vary by several orders of magnitude over short distances.
Data limitations, in addition to the fact that the subsurface environment is invisible, make
the prediction of contaminant transport very difficult. This thesis presents a
comprehensive visualization library, consisting of approximately 50 high-resolution
numerical simulations and visualizations, which systematically explore the effects of
heterogeneity on groundwater flow and transport. In particular, this thesis studies the
effects of spatial heterogeneity, temporal variability, multi-scale heterogeneity and their
complex non-linear interactions. This thesis also investigates the applicability of effective
macrodispersion models for predicting plume transport, and the probabilistic structure of
parameters like head, concentration, seepage flux, and solute flux through high-resolution
Monte Carlo simulations. Several interesting conclusions emerge from these .
visualizations, such as the relative importance of porosity and partitioning coefficient
while considering the impact of interacting heterogeneities, the irreversibility of plume
transport due to interaction between temporal variability and spatial heterogeneity, the
effectiveness of macrodispersion models in the presence of multi-scale heterogeneity, and

the skewed, non-stationary nature of concentration uncertainty.

ACKNOWLEDGEMENTS

I owe my gratitude to all the people who have helped me during the course of my
graduate study. My heartfelt gratitude goes to Dr. Shu-Guang Li, my adviser and
committee chair, for his support and guidance. His advice and untiring help have played a

huge role in the successful completion of my thesis.

I thank Dr. Roger Wallace profusely for serving in my committee and for his expert
criticism of my writing. His invaluable comments made it possible for me to fine-tune the
thesis. I also thank Dr. Phanikumar Mantha for serving in my committee. I also wish to

thank Dr. Qun Liu for helping out with IGW-related issues.

I thank National Science Foundation (NSF) for funding my research. I thank all my

research group mates, Soheil Afshari, Andreanne Simard, David Ni, and Hassan Abbas

for their cooperation.

Finally, and most importantly, I want to thank my parents for encouraging me at each

step of this journey.

iii

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

LIST OF TABLES ix
LIST OF FIGURES xii
Introduction 1
1 REAL WORLD HETEROGENEITY 8
2 RANDOM FIELD REPRESENTATION OF HETEROGENEITY ................... 10
2.1 BORDEN AQUIFER, CANADA 10
2.1.1 Problem statement ........................................................................................... 10
2.1.2 Key observations .............................................................................................. 10
2.1.3 Additional observations and discussion ........................................................... 11
2.2 MULTIPLE REALIZATIONS - 14
2.2.] Problem statement ........................................................................................... 14
2.2.2 Key observations .............................................................................................. 14
2.2.3 Additional observations and discussion ........................................................... 14
2.3 EFFECT OF CORRELATION SCALES (A) 17
2.3.1 Problem statement ........................................................................................... I 7
2.3.2 Key observations .............................................................................................. 17
2.3.3 Additional observations and discussion ........................................................... 17
2.4 EFFECT OF In K VARIANCE 20
2.4.1 Problem statement ........................................................................................... 20
2.4.2 Key observations .............................................................................................. 20
2.4.3 Additional observations and discussion ........................................................... 20
2.5 RANDOM FIELD MODELS - - 23
2.5.1 Problem statement ........................................................................................... 23
2.5.2 Key observations .............................................................................................. 23
2.5.3 Additional observations and discussion ........................................................... 23
2.6 RANDOM FIELD GENERATION ALGORITHMS - 25
2.6.1 Problem statement ........................................................................................... 25
2.6.2 Key observations .............................................................................................. 25
2.6.3 Additional observations and discussion ........................................................... 25
2.7 UNCONDITIONAL AND CONDITIONAL SIMULATIONS 28
2.7.1 Problem statement ........................................................................................... 28

2. 7.2 Key observations .............................................................................................. 28
2.7.3 Additional observations and discussion ........................................................... 28

iv

3 SPATIAL HETEROGENEITY 31
3.1 EFFECTS OF HETEROGENEITY OF HYDRAULIC CONDUCTIVITY 31

 

 

 

 

 

 

 

 

 

3.1.1 Problem statement ........................................................................................... 31
3.1.2 Key observations .............................................................................................. 31
3. 1.3 Additional observations and discussion ........................................................... 32
3.1.4 Mathematical interpretation ............................................................................ 33
3.2 EFFECTS OF HETEROGENEITY OF HYDRAULIC CONDUCTIVITY -
WITH AND WITHOUT PORE-SCALE DISPERSION 38
3.2.1 Problem statement ..................................................................................... 38
3.2.2 Key observations .............................................................................................. 38
3.2.3 Additional observations and discussion ........................................................... 38
3.2.4 Mathematical interpretation ............................................................................ 39
3.3 INTERACTIONS BETWEEN CONDUCTIVITY AND POROSITY
HETEROGENEITIES 44
3.3.1 Problem statement ........................................................................................... 44
3.3.2 Key observations .............................................................................................. 44
3.3.3 Additional observations and discussion ........................................................... 44
3.3.4 Mathematical interpretation ............................................................................ 45
3.4 INTERACTIONS BETWEEN CONDUCTIVITY AND PARTITIONING
COEFFICIENT HETEROGENEITIES 49
3.4.1 Problem statement ........................................................................................... 49
3.4.2 Key observations .............................................................................................. 49
3.4.3 Additional observations and discussion ........................................................... 50
3.4.4 Mathematical interpretation ............................................................................ 50
3.5 INTERACTIONS BETWEEN CONDUCTIVITY, POROSITY AND
PARTITIONING COEFFICIENT HETEROGENEITIES 54
3.5.1 Problem statement ........................................................................................... 54
3.5.2 Key observations .............................................................................................. 54
3.5.3 Additional observations and discussion ........................................................... 55
3.5.4 Mathematical interpretation ............................................................................ 56
3.6 EFFECT OF CORRELATION SCALE (/1) OF CONDUCTIVITY
HETEROGENEITY 62
3.6.] Problem statement ........................................................................................... 62
3.6.2 Key observations .............................................................................................. 62
3.6.3 Additional observations and discussion ........................................................... 62
3.7 EFFECT OF CORRELATION SCALE (/1) OF CONDUCTIVITY
HETEROGENEITY 66
3.7.1 Problem statement ........................................................................................... 66
3.7.2 Key observations .............................................................................................. 66
3.7.3 Additional observations and discussion ........................................................... 66
3.8 EFFECTS OF ANISOTROPIC HETEROGENEITIES - - 70
3.8.1 Problem statement ........................................................................................... 70
3.8.2 Key observations .............................................................................................. 70
3.8.3 Additional observations and discussion ........................................................... 71
3.9 EFFECTS OF ANISOTROPIC HETEROGENEITIES 75
3.9.1 Problem statement ........................................................................................... 75

3.9.2 Key observations .............................................................................................. 75

 

 

 

 

 

 

3.9.3 Additional observations and discussion ........................................................... 75
4 TEMPORAL VARIABILITY 80
4.1 UNCONFINED AQUIFERS WITH INCREASING TRANSMISSIVITIES 80
4.1.1 Problem statement ........................................................................................... 80
4.1.2 Key observations .............................................................................................. 80
4.1.3 Additional observations and discussion ........................................................... 80
4.1.4 Mathematical interpretation ............................................................................ 81
4.2 UNCONFINED AQUIFERS WITH INCREASING SPECIFIC YIELDS 85
4.2.1 Problem statement ........................................................................................... 85
4.2.2 Key observations .............................................................................................. 85
4.2.3 Additional observations and discussion ........................................................... 85
4.2.4 Mathematical interpretation ............................................................................ 86
4.3 CONFINED AQUIFERS WITH INCREASING CONDUCTIVITIES ........ 90
4.3.1 Problem statement ........................................................................................... 90
4.3.2 Key observations .............................................................................................. 90
4.3.3 Additional observations and discussion ........................................................... 90
4.3.4 Mathematical interpretation ............................................................................ 91
4.4 CONFINED AQUIFERS WITH INCREASING STORAGE
COEFFICIENTS 95
4.4.] Problem statement ....................................................... '. ................................... 95
4.4.2 Key observations .............................................................................................. 95
4.4.3 Additional observations and discussion ........................................................... 95
4.4.4 Mathematical interpretation ............................................................................ 96
4.5 COMPARISON BETWEEN UNCONFINED AND CONFINED AQUIFERS
100
4.5.1 Problem statement ......................................................................................... 100
4.5.2 Key observations ............................................................................................ 100
4.5.3 Additional observations and discussion ......................................................... 100
3.5.4 Mathematical interpretation .......................................................................... 101
4.6 BOUNDARIES WITH DECREASING FREQUENCIES OF VARIABILITY
104
4.6.1 Problem statement ......................................................................................... 104
4.6.2 Key observations ............................................................................................ 104
4.6.3 Additional observations and discussion ......................................................... 104
4.6.4 Mathematical interpretation .......................................................................... 105
4.7 BOUNDARIES WITH INCREASING EXTENTS OF PENETRATION
INTO THE AQUIFER 109
4. 7.1 Problem statement ......................................................................................... 109
4.7.2 Key observations ............................................................................................ 109
4.7.3 Additional observations and discussion ......................................................... 109
4. 7.4 Mathematical interpretation .......................................................................... 110
4.8 PLUME TRANSPORT - COMPARISON BETWEEN HOMOGENEOUS
AND HETEROGENEOUS AQUIFERS 114
4.8.1 Problem statement ......................................................................................... 114

vi

4.8.2 Key observations ............................................................................................ 114

 

4.8.3 Additional observations and discussion ......................................................... 114
4.9 PLUME TRANSPORT - COMPARISON BETWEEN HETEROGENEOUS
AQUIFERS WITH AND WITHOUT EQUILIBRIUM SORPT ION ................ 118

4.9.] Problem statement ......................................................................................... 118

4.9.2 Key observations ............................................................................................ 118

4.9.3 Additional observations and discussion ......................................................... 118
4.10 PLUME TRANSPORT - COMPARISON BETWEEN AQUIFERS WITH
AND WITHOUT INTERACTING HETEROGENEITIES 122

4.10.1 Problem statement ....................................................................................... 122

4.10.2 Key observations .......................................................................................... 122

4.10.3 Additional observations and discussion ....................................................... 122

5 MACRODISPERSION 126

 

5.1 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
WEAKLY HETEROGENEOUS MEDIA (In K VARIANCE 0.5) WITH PORE-

 

SCALE DISPERSION 126
5.1.1 Problem statement ......................................................................................... 126
5.1.2 Key observations ............................................................................................ 126
5.1.3 Additional observations and discussion ......................................................... 127
5.1.4 Mathematical interpretation .......................................................................... 128

5.2 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
MODERATELY HETEROGENEOUS MEDIA (In K VARIANCE 1.0) WITH

 

PORE-SCALE DISPERSION 132
5.2.1 Problem statement ......................................................................................... 132
5.2.2 Key observations .............................................................................. . .............. 132
5.2.3 Additional observations and discussion ......................................................... 133
5.2.4 Mathematical interpretation .......................................................................... 134

5.3 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
STRONGLY HETEROGENEOUS MEDIA (In K VARIANCE 2.0) WITH

 

PORE-SCALE DISPERSION 139
5.3.1 Problem statement ......................................................................................... 139
5.3.2 Key observations ............................................................................................ 139
5.3.3 Additional observations and discussion ......................................................... 140
5.3.4 Mathematical interpretation .......................................................................... 141

5.4 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN VERY
STRONGLY HETEROGENEOUS NIEDIA (In K VARIANCE 4.0) WITH

 

PORE-SCALE DISPERSION 145
5.4.1 Problem statement ......................................................................................... 145
5.4.2 Key observations ............................................................................................ 145
5.4.3 Additional observations and discussion ......................................................... 146
5.4.4 Mathematical interpretation................................................... ....................... 147

5.5 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN

STRONGLY HETEROGENEOUS MEDIA (In K VARIANCE 2.0) WITHOUT

PORE-SCALE DISPERSION 151
5.5.1 Problem statement ......................................................................................... 151

 

vii

5.5.2 Key observations ............................................................................................ 151

 

 

 

 

 

 

 

 

5.5.3 Additional observations and discussion ......................................................... 152
5.5.4 Mathematical interpretation .......................................................................... 153
6 EFFECTS OF MULTI-SCALE HETEROGENEITY 157
6.1 EFFECTS OF MULTIPLE SCALES OF HETEROGENEITY .................. 157
6.1.1 Problem statement ......................................................................................... 15 7
6.1.2 Key observations ............................................................................................ 157
6.1.3 Additional observations and discussion ......................................................... 158
6.2 MACRODISPERSION IN THE PRESENCE OF MULTIPLE SCALES OF
HETEROGENEITY 161
6.2.1 Problem statement ......................................................................................... 161
6.2.2 Key observations ............................................................................................ 161
6.2.3 Additional observations and discussion ......................................................... 161
6.2.4 Mathematical interpretation .......................................................................... I63
7 MONTE CARLO SIMULATIONS 167
7.1 EFFECTS OF HYDRAULIC CONDUCTIVITY HETEROGENEITY —
DIFFERENT REALIZATIONS 167
7.1.1 Problem statement ......................................................................................... 167
7.1.2 Key observations ............................................................................................ 167
7.1.3 Additional observations and discussion ......................................................... 167
7.2 EFFECTS OF HYDRAULIC CONDUCTIVITY HETEROGENEITY -
DIFFERENT REALIZATIONS 170
7.2.1 Problem statement ......................................................................................... 170
7.2.2 Key observations ............................................................................................ I 70
7.2.3 Additional observations and discussion ......................................................... 170
7.3 MONTE CARLO SIMULATION — MEAN, VARIANCE AND
COVARIANCE 173
7.3.1 Problem statement ......................................................................................... 173
7.3.2 Key observations ............................................................................................ 1 73
7.3.3 Additional observations and discussion ......................................................... I 73
7.4 MONTE CARLO SIMULATION — PROBABILITY DISTRIBUTIONS
AND BREAKTHROUGH CURVES 177
7.4.1 Problem statement ......................................................................................... 177
7.4.2 Key observations ............................................................................................ I 77
7.4.3 Additional observations and discussion ......................................................... 1 78
8 CONCLUSIONS - 181
9 REFERENCES _ 185

 

viii

LIST OF TABLES

Table 2.1.1 Model parameters and inputs for cross-section A-A ’ ............................... 12
Table 2.1.1 Model parameters and inputs for cross-section B-B ' ............................... 12
Table 2.2 Model parameters and inputs ........................................................................ 15
Table 2.3 Model parameters and inputs ........................................................................ 18
Table 2.4 Model parameters and inputs ........................................................................ 21
Table 2.5 Model parameters and inputs ........................................................................ 23
Table 2.6 Model parameters and inputs ........................................................................ 25
Table 2.7 Model parameters and inputs ........................................................................ 29
Table 3.1 Model parameters and inputs ........................................................................ 35
Table 3.2 Model parameters and inputs ........................................................................ 41
Table 3.3 Model parameters and inputs ........................................................................ 46
Table 3.4 Model parameters and inputs ........................................................................ 51
Table 3.5 Model parameters and inputs ........................................................................ 57
Table 3.6 Model parameters and inputs ........................................................................ 64
Table 3.7 Model parameters and inputs ........................................................................ 68
Table 3.8 Model parameters and inputs ........................................................................ 72
Table 3.9 Model parameters and inputs ........................................................................ 77
Table 4.1 Model parameters and inputs ........................................................................ 82
Table 4.2 Model parameters and inputs ........................................................................ 87
Table 4.3 Model parameters and inputs ........................................................................ 92
Table 4.4 Model parameters and inputs ........................................................................ 97
Table 4.5 Model parameters and inputs ...................................................................... 102
Table 4.6 Model parameters and inputs ...................................................................... 106
Table 4.7 Model parameters and inputs ...................................................................... 111
Table 4.8.1 Model parameters and inputs ................................................................... 115
Table 4.8.2 Temporally varying boundary conditions ................................................ 115
Table 4.9.1 Model parameters and inputs ................................................................... 119

ix

Table 4.9.2 Temporally varying boundary conditions ................................................ 120

Table 4.10.1 Model parameters and inputs ................................................................. 123
Table 4.10.2 Temporally varying boundary conditions .............................................. 124
Table 5.1 Model parameters and inputs ...................................................................... 129
Table 5.2 Model parameters and inputs ...................................................................... 136
Table 5.3 Model parameters and inputs ...................................................................... 143
Table 5.4 Model parameters and inputs ...................................................................... 149
Table 5.5 Model parameters and inputs ...................................................................... 155
Table 6.1 Model parameters and inputs ...................................................................... 159
Table 6.2 Model parameters and inputs ...................................................................... 165
Table 7.1 Model parameters and inputs ...................................................................... 168
Table 7.2 Model parameters and inputs ...................................................................... 171
Table 7.3.1 Model parameters and inputs ................................................................... 174
Table 7.3.2 Maximum and minimum values for plotting ........................................... 175
Table 7.4 Model parameters and inputs ...................................................................... 179

LIST OF FIGURES

Figure 1.1 Gravel pit at Phoenix, Arizona showing natural variability consisting of
several layers of coarse and fine grained material of varying thickness (Modified from
Groundwater hydrology. Bouwer, 1978) ........................................................................ 8

Figure 1.2 "Permeability and porosity of cores collected at l-ft intervals from borehole
(11.056) in the Mt. Simon aquifer in Illinois (date from Bakr, 1976)." (Modified from

Stochastic Subsurface Hydrology, Gelhar, 1993) ........................................................... 9
Figure 2.1 Snapshot of the visualization ....................................................................... 13
Figure 2.2 Snapshot of the visualization ....................................................................... 16
Figure 2.3 Snapshot of the visualization ....................................................................... 19
Figure 2.4 Snapshot of the visualization ....................................................................... 22
Figure 2.4 Snapshot of the visualization ....................................................................... 24
Figure 2.6 Snapshot of the visualization ....................................................................... 27
Figure 2.7 Snapshot of the visualization ....................................................................... 30
Figure 3.1 Snapshot of the visualization ....................................................................... 37
Figure 3.2 Snapshot of the visualization ....................................................................... 43
Figure 3.3 Snapshot of the visualization showing spatial variation of n ...................... 48
Figure 3.4 Snapshot of the visualization showing spatial variation of K; .................... 53
Figure 3.5.1 Snapshot of the visualization .................................................................... 59
Figure 3.5.2 Spatial variation of n ................................................................................ 60
Figure 3.5.3 Spatial variation of K4 .............................................................................. 61
Figure 3.6 Snapshot of the visualization ....................................................................... 65
Figure 3.7 Snapshot of the visualization ....................................................................... 69
Figure 3.8 Snapshot of the visualization ....................................................................... 74
Figure 3.9 Snapshot of the visualization ....................................................................... 79
Figure 4.1 Snapshot of the visualization ....................................................................... 84
Figure 4.2 Snapshot of the visualization ....................................................................... 89
Figure 4.3 Snapshot of the visualization ....................................................................... 94
Figure 4.4 Snapshot of the visualization ....................................................................... 99

xi

Figure 4.5 Snapshot of the visualization ..................................................................... 103

Figure 4.6 Snapshot of the visualization ..................................................................... 108
Figure 4.7 Snapshot of the visualization ..................................................................... 113
Figure 4.8 Snapshot of the visualization ..................................................................... 117
Figure 4.9 Snapshot of the visualization ..................................................................... 121
Figure 4.10 Snapshot of the visualization ................................................................... 125
Figure 5.1 Snapshot of the visualization ..................................................................... 131
Figure 5.2 Snapshot of the visualization ..................................................................... 138
Figure 5.3 Snapshot of the visualization ..................................................................... 144
Figure 5.4 Snapshot of the visualization ..................................................................... 150
Figure 5.5 Snapshot of the visualization ..................................................................... 156
Figure 6.1 Snapshot of the visualization ..................................................................... 160
Figure 6.2 Snapshot of the visualization ..................................................................... 166
Figure 7.1 Snapshot of the visualization ..................................................................... 169
Figure 7.2 Snapshot of the visualization ..................................................................... 172
Figure 7.3 Snapshot of the visualization ..................................................................... 176
Figure 7.4 Snapshot of the visualization ..................................................................... 180

xii

Introduction

The study of the subsurface environment is both challenging and exciting, because of its
inherent complexities and heterogeneities. In the past, most research that investigated the
effects of heterogeneity employed theoretical analyses based on a number of highly
restrictive assumptions, such as stationarity, ergodicity, scale disparity, linearization, and
small perturbation. These assumptions severely limit our ability to understand the effects
of heterogeneity on groundwater flow and transport in real world scenarios. With the
advent of extremely efficient computers, and through the introduction of Interactive
Groundwater (IGW) — A real-time modeling paradigm (Li and Liu, 2003; Li and Liu,
2006), some of the limitations listed above are overcome. Taking advantage of these
recent developments, this thesis systematically explores the effects of heterogeneity on
groundwater flow and transport using hi gh-resolution numerical simulations and

visualizations.

The need to visualize

More than a couple of decades of research have gone into understanding the influence of
heterogeneities on groundwater flow and transport. However, data limitations, in addition
to the fact that the subsurface environment is invisible to the human eye, have made it
difficult to physically understand groundwater dynamics. The objective of this research is
to create a comprehensive library covering different aspects of stochastic subsurface
hydrology, using a set of high-resolution numerical simulations and visualizations. This
library systematically explores the effects of heterogeneities, interactions between aquifer

parameters and temporal variability, and multi-scale effects. Macrodispersion models and

Monte Carlo simulations are also visualized in this thesis. Sharing information in the
form of a visualization library has been made possible by the rapid development of

computers and the Internet.

Chapter 1 of this thesis illustrates the variability of aquifers in the real world, while
Chapter 2 deals with the representation of heterogeneity using random field models. The
effects of spatial heterogeneity and interactions with other aquifer parameters, such as
porosity and partitioning coefficient are explored in Chapter 3. The effects of temporal
variability and its interaction with spatial heterogeneity are dealt with in Chapter 4.
Chapter 5 is devoted towards the use of macrodispersion models, while Chapter 6
illustrates the effects of multiple scales of heterogeneity. Stochastic methods like Monte

Carlo simulations are discussed in Chapter 7.

Factors influencing groundwater flow and transport

There are spatial and temporal factors that influence plume transport in the subsurface
environment. Spatial factors are those that vary in the spatial domain of the aquifer
matrix, such as conductivity, porosity, and partitioning coefficient. Temporal factors are
those that vary with time, such as the frequency of oscillation of a time-variable boundary
condition. This thesis explores, in detail, these heterogeneities and their effect on
groundwater flow and transport. The application of macrodispersion models and Monte

Carlo simulations are also studied.

Random field representation of heterogeneity

In the real world, the subsurface environment is extremely complex and variable. In order
to create an “accurate” model, it is necessary to obtain data samples, which can
characterize the type and extent of the variability. The type of variability that has the
most influence is the conductivity of the aquifer, although there are other variabilities,
such as porosity and partitioning coefficient. The accuracy of the model depends on the
resolution of the data. Higher resolution data, in general, will provide better accuracy.
Hi gh-resolution data is not available at all times since data acquisition is costly and time
consuming. However, it is possible to predict the internal structure of the aquifer using
statistical methods on available data. The most common method adopted is to obtain the
mean, variance, covariance, and correlation scale of the heterogeneity of the aquifer
properties. Using this information, a random field can be generated that has the same
statistical properties as the data set. Chapter 2 deals exclusively with the use of random
field models, algorithms, and unconditional and conditional simulations to represent real

world heterogeneity.

Effects of spatial heterogeneity

The main parameter of interest in groundwater hydrology is the hydraulic conductivity of
the aquifer. The effect of heterogeneity of this parameter is studied in Section 3.1.
Another parameter of interest in a groundwater model is the pore-scale dispersivity. This
parameter is needed in order to compensate for spatial averaging. However, the

introduction of pore-scale dispersivity into the model lowers the maximum concentration

of the plume. Section 3.2 deals with the effect of pore-scale dispersivity on plume

transport.

Generally, the conductivity in a domain is modeled as a spatially varying parameter,
while the values of porosity and partitioning coefficient used in models are assumed to be
constant over the modeling domain. This assumption can be questioned since it is known
that porosity and partitioning coefficient are correlated with the conductivity. Porosity
and partitioning coefficient are, in general, positively and negatively correlated with
conductivity respectively. These correlations affect plume spreading. The effects of these

correlations are explored in Sections 3.3, 3.4, and 3.5.

The correlation scale of heterogeneity plays a very important role in the spreading of
plumes. Depending on whether the correlation scale is smaller, of the same size, or larger
than the initial size of the plume, the plume can spread in drastically different ways. This
difference can be attributed to the effect of correlation scales on the formation of
“preferential” paths for plume transport. Further complications can arise if the
heterogeneity is anisotropic. Strongly anisotropic heterogeneity can result in varied
degrees of plume spreading, depending on the direction of flow. Sections 3.6, 3,7, 3.8,

and 3.9 elucidate the effect of correlation scales on plume transport.

Effects of temporal variability
While spatial heterogeneity is very important, temporal variability can also result in

complex groundwater flow situations. A sinusoidally oscillating boundary condition can

affect the head in the aquifer, and hence the direction of flow. The effect of such a
boundary on an aquifer is dependent on the type of aquifer, as well as its transmissivity
and storage properties. Unconfined aquifers are usually affected less than a confined
aquifer of similar transmissivity, due to the drastically different storage properties of the
two aquifers. Unconfined aquifers release water by draining of the inter—granular pore
space. On the other hand, confined aquifers release water by elastic expansion of the
aquifer matrix and decompression of water. Therefore, confined aquifers have to drain a
larger area to obtain the same amount of water as an unconfined aquifer. The sinusoidally
oscillating boundary condition is characterized by an amplitude and a time period. The
magnitude of the amplitude and time period has an impact on the area of aquifer affected.

Chapter 4 explores the effects of temporal variability on groundwater flow and transport.

Macrodispersion model

In order to characterize the spatial heterogeneity in an aquifer, data is required. Data
acquisition is economically not feasible in all situations. Therefore, it becomes essential
to characterize the aquifer using other means. One of the statistical means to characterize
the aquifer is Gelhar’s asymptotic macrodispersion model (Gelhar and Axness, 1983a).
This model approximates the complex heterogeneous field with pore-scale dispersion
coefficient to a simpler homogenous field with an effective macrodispersion coefficient.
One of the basic assumptions of this model is that the heterogeneous field has a relatively
small In K variance. The advantage of this model is that the complex heterogeneities are
“lumped” together into an equivalent coefficient, making the process of modeling much

simpler. However, this model is only able to predict the average concentration over a

large spatial area. In reality, the concentration fluctuates, which is extremely important
when the maximum concentration is to be estimated. Chapter 5 elucidates the application
of macrodispersion models to predict plume transport in heterogeneous media and

evaluates the effectiveness of these models.

Effects of multi-scale heterogeneity

Uniquely characterizing spatial heterogeneity is extremely difficult, since it requires

hi gh-resolution data. It becomes even more difficult because spatial heterogeneity exists
on many scales. Modeling all the scales of heterogeneity is impossible. Smaller scales of
heterogeneity within larger scales have a significant impact on plume spreading. Usually,
available data is “coarse,” allowing modeling of larger scales. While the larger scales
provide the “trend” or the general path of transport, it is the smaller scales that determine
the amount of spreading. Therefore, the ability to characterize these smaller scales is
important. This can be achieved by using Gelhar’s macrodispersion model. Although
Gelhar’s model assumes a homogeneous field with macrodispersion, a moderately
heterogeneous field with a large scale of heterogeneity is a close approximation. Chapter

6 deals with effects of multiple scales of heterogeneity on plume transport.

Monte Carlo simulations

Despite the use of various methods to characterize the aquifer structure, due to the
inherent complexity of the medium, there is an element of uncertainty in the results
obtained. It is in this context that stochastic methods are needed. Stochastic methods

make it possible to quantify the uncertainty, using techniques like Monte Carlo

simulations. In spite of characterizing aquifer structure using a random field model, each
realization of this random field is only one of an infinite number of possible situations.
Therefore, the results obtained from one realization are not the “truth.” Neither is it
possible to determine what the truth is. A Monte Carlo simulation, which consecutively
generates infinite realizations, is a technique used to obtain the probability of occurrence
of an event. The results obtained from each realization are used to update the statistics of
the probability distribution. After a sufficiently large number of realizations, the
probability distribution “stabilizes.” This result gives not only the probability of
occurrence, but also the standard deviation. This technique is very useful for probabilistic

analysis and risk estimation, and is discussed in Chapter 7.

Each Chapter in this thesis, with the exception of Chapter 1, consists of a number of
Sections, which are further divided into sub-sections. Each Section is devoted to a
visualization, which attempts to explain certain aspects of flow and transport. Each
Section is divided into the following sub-sections: Problem statement, Key observations,
Additional observations and discussion, Mathematical interpretation (if applicable), and
Model parameters and inputs. A snapshot of the visualization and a URL to view the

visualization on the Internet are also provided in every Section.

It is the intention of this visualization library to provide a systematic “reference manual”
for students, researchers, and practitioners of groundwater modeling. Images in this thesis
are presented in color. The visualization library can be accessed on the Internet at

http://www.egr.msu.edu/igw/DL/SSH

1 REAL WORLD HETEROGENEITY

The subsurface environment is inherently heterogeneous with respect to geologic,
hydrologic and chemical properties. Aquifer properties such as hydraulic conductivity
and partitioning coefficient can sometimes vary by several orders of magnitude over

surprisingly short distances.

Figure 1.1 shows a typical aquifer outcrop in a gravel pit at Phoenix, Arizona

(Groundwater hydrology, Bouwer, 1978). Note that there are several layers of various

aquifer materials of varying thickness in the picture.

    

, , ,, .1;- 1, _.- _. ., ’T '3- : ‘5 is ‘
Figure 1.1 Gravel pit at Phoenix, Arizona showing natural variability consisting of
several layers of coarse and fine grained material of varying thickness (Modified
from Groundwater hydrology, Bouwer, 1978)

   

Figure 1.2 shows the range of porosity and permeability values collected at 1 feet
intervals fi'om a borehole in Mt. Simon aquifer, Illinois (Stochastic Subsurface
Hydrology, Gelhar, 1993). While porosity varies over a wide range, permeability varies

by several orders of magnitude.

 

 

 

 

 

 

 

 

 

porosity (percent) penneabliwmd)
0 an 1 10 100 1000
0 u -
100 - ‘ A "
a D
E , -="_‘
8 _ -
.0 -—-—-——- ....:.'
_—
200 - H -—- '7‘
I ‘_'.L
300 P 3 _‘

 

 

Figure 1.2 "Permeability and porosity of cores collected at l-ft intervals from
borehole (IL056) in the Mt. Simon aquifer in Illinois (date from Bakr, 1976)."
(Modified from Stochastic Subsurface Hydrology, Gelhar, 1993)

From Figures 1.1 and 1.2 it is clear that natural variability in the horizontal and vertical
directions is significant. A set of pictures on other real world aquifers can be accessed at

the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/RWH

2 RANDOM FIELD REPRESENTATION OF

HETEROGENEITY

2.1 BORDEN AQUIFER, CANADA

2.1.1 Problem statement

This video demonstrates the multiple realizations for the statistical parameters of the

Borden aquifer, Canada (Sudicky, 1986). Details are provided in Tables 2.1.1 and 2.1.2.

2.1.2 Key observations

The following observations can be made from the video:

A random field is characterized by a set of statistical parameters, such as mean,
variance and correlation scale.

Multiple realizations of the Borden aquifer data share the same “texture” as the
data.

In spite of having the same structure, each realization is different from other
realizations.

An infinite number of realizations exist that satisfy the same statistical
parameters; hence, uniquely characterizing aquifer structure with limited data is
impossible.

All the realizations statistically represent the truth. None of them can

deterrninistically represent the truth.

10

2.1.3 Additional observations and discussion

Any aquifer can be statistically represented using a mean, ln K variance and a correlation

scale. A brief description of these parameters is given below:

0 Mean — Is the geometric mean of the data, which represents the
average conditions that exist in the aquifer

0 Correlation scale — Is a measure of the spatial distance across which
the aquifer properties are supposed to be correlated. This parameter

can be different in all three directions making the aquifer anisotropic.

However, in most cases, the horizontal correlation scales (xix and 11,)

are assumed to be constant. Vertical correlation scale (22) is in general

smaller than the horizontal correlation scales. Correlation scales give
an idea about the “texture” or “pattern” of variability.

o Variance — Is a measure of how variable the aquifer properties are with
respect to the mean. A higher variance means higher variability of

aquifer properties and vice versa.

The Borden aquifer is a test site, which generated a large amount of hydraulic
conductivity data with extremely fine resolution. Using this data, Sudicky (1986)
performed a statistical analysis to obtain the mean, variance and correlation scale for this
site. Given these parameters, it is possible to create a random field that statistically

represents the Borden aquifer.

11

Table 2.1.1 Model parameters and inputs for cross-section A-A '

 

 

 

 

 

 

 

 

 

Parameter All realizations
Geometric mean hydraulic conductivity, Kg (cm/s) 9.75 x 10-3

In K variance 0.38
Correlation scale, [ix (m) 2-8

Correlation scale, 2.2 (m) 0-12

Size of model (m) 19 x 1.75
Covariance function Exponential
Grid 761 x 71

Cell size, Ax (m) 0.025 x 0.025

 

 

 

 

Table 2.1.1 Model parameters and inputs for cross-section B-B ’

 

 

 

 

 

 

 

 

 

 

 

 

Parameter All realizations
Geometric mean hydraulic conductivity, Kfi (cm/s) 9.75 x 10.3

In K variance 0.38
Correlation scale, 11x (m) 2.8

Correlation scale, ’12 (m) 0-12

Size of model (m) 12 x 1.75
Covariance function Exponential
Grid 481 x 71

Cell size, Ax (m) 0.025 x 0.025

 

The model parameters and inputs are explained below:

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, In K
variance and covariance.

The geometric mean, In K variance, and correlation scale is constant for all cases.
The covariance function for all realizations is exponential.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

12

 

Figure 2.1 Snapshot of the visualization

For a detailed visualization, access the following addresses:
URL: http://wwwegrmsu.edu/igw/DL/SSH/RFRH/RFRHOO.htm

URL: http://wwwegr.msu.cdu/igw/DL/SSH/RFRH/RFRHO.htm

2.2 MULTIPLE REALIZATIONS

2.2.1 Problem statement

This video demonstrates the multiple realizations that are possible from a given set of

statistical parameters. Details are provided in Table 2.2.

2.2.2 Key observations

The following observations can be made from the video:
0 The statistical properties of all realizations are identical; each realization has the
same underlying statistical “structure.”
0 In spite of having the same structure, each realization is different from other
realizations.
0 An infinite number of realizations exist that satisfy the same statistical
parameters; hence, uniquely characterizing aquifer structure with limited data is

impossible.

2.2.3 Additional observations and discussion

Since aquifer properties vary dramatically within very short distances, a point-to-point
detailed description of the aquifer structure is impossible. With available data, statistical
parameters such as mean, variance, and correlation scale can be obtained. Using these
parameters, a random field model can be created. Infinite realizations of the random field,
which satisfy the statistical parameters, can be created. Although each of these
realizations is created from the same underlying statistical structure, they are different

from each other.

14

Table 2.2 Model parameters and inputs

 

 

 

 

 

 

 

 

 

Parameter All realizations
Geometric mean hydraulic conductivity, Kg (m/day) 10

In K variance 1.0
Correlation scale, A (m) 5

Size of model (m) 100 x 100
Covariance function Exponential
Grid 203 x 203

Cell size, Ax (m) 1 x 1

 

 

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

0 The geometric mean, In K variance, and correlation scale is constant for all cases.

0 The covariance function for all realizations is exponential.

o The cell size, Ax, is selected such that the correlation scale, A, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

15

 

 

 

Figure 2.2 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/RFRH/RFRH1.htm

 

2.3 EFFECT OF CORRELATION SCALES (2.)

2.3.1 Problem statement

This video demonstrates the effect of correlation scales on the aquifer texture. Details are

provided in Table 2.3.

2.3.2 Key observations

The following observations can be made from the video:

0 Correlation scales are an important statistical property of the aquifer, which
determine the texture of the aquifer’s heterogeneity.

0 Depending on the magnitude of the correlation scales, the texture of the
heterogeneity can be drastically different.

0 Depending on the ratio of the correlation scales, aquifers can be isotropic,

anisotropic or strongly anisotropic in the horizontal or vertical planes.

2.3.3 Additional observations and discussion

Given the statistical parameters of an aquifer, the aquifer structure can be determined.

The parameter that controls the “texture” of the aquifer is the correlation scale.

Case 1: The correlation scales, 2,, and 11),, are 5 m in both directions for this aquifer. This

isotropic structure is most commonly found in the horizontal plane.

Case 2: The correlation scales, 2, and 23., are 30 m and 5 m respectively. This aquifer

structure is anisotropic, and is most commonly found in a vertical cross-section.

17

Case 3: The correlation scales, xix and 2,, are 100 m and 5 m respectively. This aquifer

structure is strongly anisotropic, and is most commonly found in a vertical cross-section.

Such an aquifer is called “perfectly” stratified.

Case 4: The correlation scales, xix and 2,, are 5 m and 100 m respectively. This aquifer

structure is strongly anisotropic and not very commonly found.

Table 2.3 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic conductivity,
10 10 10 10

Kg (m/day)

In K variance 1.0 1.0 1.0 1.0
Correlation scale, 21,; (m) 5 30 100 5
Correlation scale, 11y (m) 5 5 5 100

Size of model (m) 100 x 100 100 x 100 100 x 100 100 x 100
Covariance function Exponential Exponential Exponential Exponential
Grid 203 x 203 203 x 203 203 x 203 203 x 203
Cellsize,Ax(m) 1x1 1x1 1x1 1x1

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following

set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance

and covariance.

I The geometric mean and ln K variance is constant for all cases.

0 The correlation scales, 2,, and 11,, are 5m and 5m, 30 m and 5 m, 100 m and 5 m,

and 5 m and 100 m for cases 1, 2, 3, and 4 respectively.

18

 

0 The covariance function for all realizations is exponential.
o The cell size, Ax, is selected such that the correlation scale, xi, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

 

 

 

Figure 2.3 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/RFRH/RFRH2.htm

2.4 EFFECT OF In K VARIANCE

2.4.1 Problem statement

This video demonstrates the effect of In K variance on the degree of variability in the

aquifer. Details are provided in Table 2.4.

2.4.2 Key observations

The following observations can be made from the video:
0 The In K variance is an important statistical property of the aquifer, which
determines the degree of variability in the aquifer.
0 Depending on the magnitude of the In K variance, aquifers can be weakly,
moderately, strongly or very strongly heterogeneous.
o The In K variance does not affect the pattern of heterogeneity in the aquifer, it

only controls the extent of variability.

2.4.3 Additional observations and discussion

Given the statistical parameters of an aquifer, the aquifer structure can be determined.

The parameter that controls the variability of the aquifer is the In K variance.

Case 1: The In K variance is 0.5 for this aquifer. This aquifer is weakly heterogeneous.

Case 2: The In K variance is 1.0 for this aquifer. This aquifer is moderately

heterogeneous.

20

Case 3: The In K variance is 2.0 for this aquifer. This aquifer is moderately to strongly

heterogeneous.

Case 4: In K variance is 4.0 for this aquifer. This aquifer is strongly heterogeneous.

Table 2.4 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic conductivity, K g 10 10 10 10

(tn/day)

In K variance 0.5 1.0 1.5 2.0
Correlation scale, xix (m) 5 5 5 5

Size of model (m) 100 x 100 100 x 100 100 x 100 100 x 100
Covariance function Exponential Exponential Exponential Exponential
Grid 203 x 203 203 x 203 203 x 203 203 x 203
Cellsize,Ax(m) 1x1 1x1 1x1 1x1

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following

set of statistical parameters: geometric mean (K3) of K, correlation scale, variance

and covariance.

0 The geometric mean and correlation scale is constant for all cases.

0 The In K variance is 0.5, 1.0, 2.0, and 4.0 for cases 1, 2, 3, and 4 respectively.

0 The covariance function for all realizations is exponential.

o The cell size, Ax, is selected such that the correlation scale, A, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

21

 

 

-anvar=l.0

 

 

For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/RFRH/RFRH3.htm

 

22

2.5 RANDOM FIELD MODELS

2.5.1 Problem statement
This video demonstrates the different random field models, such as Gaussian,

Exponential, Whittle, and Mizell-A, which can be used to represent the heterogeneity in

aquifer structure. Details are provided in Table 2.5.

2.5.2 Key observations

The following observations can be made from the video:
0 Random field models, such as Gaussian, Exponential, Whittle, and Mizell-A,
define the covariance structure of the aquifer.
o The different random field models produce aquifer structures that are very similar

to each other for a particular realization.

2.5.3 Additional observations and discussion

Given the statistical parameters of an aquifer, the aquifer structure can be modeled using
a number of random field models. Depending on the type of model, the data interpolation
is done using different sets of equations. Expectedly, the aquifer structures generated by

each model, for a particular realization, are very similar to each other.

Table 2.5 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

. , 10 10 10 10
conductrvrty, Kg (m/day)
In K variance 1.0 1.0 1.0 1.0
Correlation scale, xix (m) 5 5 5 5
Size of model (m) 100 x 50 100 x 50 100 x 50 100 x 50
Covariance function Gaussian Exponential Whittle Mizell~A
Grid 203 x 102 203 x 102 203 x 102 203 x 102

 

23

 

 

lCellsize,Ax(m) Tlxl [1x1 ]1x1 |1x1 j

The model parameters and inputs are explained below:

 

 

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

The geometric mean, In K variance, and correlation scale is constant for all cases.
The covariance function is Gaussian, Exponential, Whittle, and Mizell-A for
cases 1, 2, 3, and 4 respectively.

The cell size, Ax, is selected such that the correlation scale, 2., is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

 

Figure 2.4 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/RFRH/RFRHS.htm

24

2.6 RANDOM FIELD GENERATION ALGORITHMS

2.6.1 Problem statement

This video demonstrates the different random field generation algorithms, such as
Spectral algorithm, Sequential Gaussian simulation, Tumings Bands algorithm, and
Simulated Annealing, which can be used to represent the heterogeneity in aquifer

structure. Details are provided in Table 2.6.

2.6.2 Key observations

The following observations can be made from the video:
0 The different random field generation algorithms produce aquifer structures that

are very similar to each other for a particular realization.

2.6.3 Additional observations and discussion

Given the statistical parameters of an aquifer, the aquifer structure can be modeled using
a number of random field generation algorithms. These algorithms in turn use a random
field model to generate the random field. Depending on the type of algorithm, the data
interpolation is done using different sets of equations. Expectedly, the aquifer structures

generated by each algorithm, for a particular realization, are very similar to each other.

Table 2.6 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

, , 10 10 10 10
conductivrty, [is (m/day)
In K variance 0.5 0.5 0.5 0.5
Correlation scale, 2x (m) 5 5 5 5
Size of model (m) 100 x 50 100 x 50 100 x 50 100 x 50
Covariance function Exponential Exponential Exponential Exponential
Random field algorithm Spectral Sequential Gaussian Turning Bands Simulated

 

25

 

 

 

I algorithm algorithm algorithm Annealing I
IGrid 203 x 102 203 x 102 203 x 102 203 x 102 |
ICellsize,Ax(m) 1x1 1x1 1x1 1x1 I

 

 

 

 

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

o The geometric mean, In K variance, correlation scale, and covariance function is
constant for all cases.

0 The random field algorithm is Spectral algorithm, Sequential Gaussian
simulation, and Tumings Bands algorithm, and Simulated Annealing for cases 1,
2, 3, and 4 respectively.

0 The cell size, Ax, is selected such that the correlation scale, A, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

26

 

 

Figure 2.6 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/RFRH/RFRH6.htm

27

 

2.7 UNCONDITIONAL AND CONDITIONAL SIMULATIONS

2.7.1 Problem statement

This video demonstrates the effect of conditioning of data on the aquifer texture. The

effect of extent of conditioning is also shown. Details are provided in Table 2.7.

2.7.2 Key observations

The following observations can be made from the video:
0 Unconditional simulations can predict the extent and texture of variability in the
aquifer, but do not satisfy the data at the data points.
0 Conditional simulations can predict the extent and texture of variability in the
aquifer, and also satisfy the data at the data points.
0 By increasing the number of data points, conditional simulations become

increasingly better representations of “reality.”

2.7.3 Additional observations and discussion

Given a set of data for representing the variability in an aquifer, two methods can be used
to represent the aquifer structure: a) Unconditional simulations, b) Conditional
simulations. Unconditional simulations can predict the variability in an aquifer, but are
not constrained by the data at the data points. On the other hand, conditional simulations
can predict the variability, and are also constrained by the data. This makes conditional
simulations a better representation of the “real” aquifer. Obviously, the effectiveness of

conditional simulations increases by increasing the number of data points.

28

Table 2.7 Model parameters and inputs

 

 

 

 

 

 

 

 

Parameter “Truth”
Geometric mean hydraulic conductivity, K & (m/day) 10

In K variance 2.0
Correlation scale, 11(m) 10

Size of model (m) 100 x 75
Covariance function Exponential
Grid 201 x 151
Cell size, Ax (m) 0.5 x 0.5

 

 

 

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

0 The geometric mean, In K variance, correlation scale, and covariance function is

constant for all cases.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved;

typically, Ax is at least 3 to 4 times smaller than the correlation scale.

29

Unconditional Simulations

Conditional - 100 Ioints Conditional - 20 points

 

 

Figure 2.7 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: Mpzflwwwegr.msu.edu/igw/DL/SSH/RFRH/RFRH4htm

30

3 SPATIAL HETEROGENEITY

3.] EFFECTS OF HETEROGENEITY OF HYDRAULIC CONDUCTIVITY

3.1.1 Problem statement

This video demonstrates the effect of heterogeneous hydraulic conductivity media on the

transport of a conservative solute in the absence of pore-scale dispersion. The initial size

of the plume is much larger than the scale of heterogeneity. The modeling domain

consists of constant head boundaries on the left and right extremes, and no-flow

boundaries at the top and bottom. Case 1 has a homogeneous field; and cases 2 through 4

show increasing degrees of heterogeneity. Details are provided in Table 3.1.

3.1.2 Key observations

The following observations can be made from the video:

Although there is no pore-scale dispersion, the plume spreads because of
heterogeneity. This large-scale spreading is called macrodispersion.

As long as the plume encounters heterogeneity, it continues to spread.

Plume spreading increases with increasing ln K variance.

Longitudinal spreading is significantly greater than transverse spreading.

Mean displacement remains largely unaffected, despite the heterogeneity.
Transport through heterogeneous media results in complex irregular plumes; in

homogeneous media the plume remains regular.

31

3.1.3 Additional observations and discussion

The pattern of plume spreading shows a clear trend. The plume does not spread at all
when the In K variance is 0.0, and spreads the most when the In K variance is 3.0. Even
though the plume spreading is enhanced, the mean plume displacement is largely
unaffected. The fingers of the plume contain the maximum mass of the plume, but are
concentrated over a small spatial area. The tails have less mass, but are elongated.
Consequently, the center of mass, and thus mean displacement, remains the same,

irrespective of the In K variance.

The effect of the symmetric In K field on plume spreading is asymmetric. The fingers
(leading edge) of the plume are shorter than the tails (trailing edge) of the plume, which
results in a skewed plume. This can be explained very simply. High K zones are
convergent zones, through which the plume prefers to travel, and in which it converges.
Convergence is the opposite of spreading. Therefore, shorter fingers are found. In
contrast, low K zones are zones of divergence, which impede the movement of the plume.
When a low K zone is encountered, the plume travels around it rather than through it and
this results in spreading. If a part of the plume reaches the low K zones, it gets ‘trapped’
due to the extremely low velocities in that zone. The bulk of the plume continues moving,
while the portion trapped in the low K zone is released slowly; thus contributing to tail

elongation.

32

3.1.4 Mathematical interpretation

Mathematically, these processes can be explained using the following equations. In a
heterogeneous field, in the absence of pore-scale dispersion, the conservative solute

concentration at any point for any irregular plume realization is given by:

ac auiC
__ + :
at 8x,-

 

0, (3.1.1)

where:
C - Concentration
u,- - Seepage velocity.
To determine large-scale behavior, the method of perturbation is used, and on performing

local spatial averaging, we get the following:

85 8175 a T

 

where:

C - Mean concentration
C ' - Perturbation to mean concentration
5 - Mean seepage velocity

u',- - Perturbation to mean seepage velocity.

This term, (— u ,.-C j , reflects the effects of heterogeneity.

It can be shown (Gelhar and Axness, 1983a) that:

—u,-'C'== rah-£92. (3.1.3a)
ax-

 

The coefficient A; is called macrodispersivity and is given by (Gelhar, 1993):

33

 

02 K
AU— .. ,1; A22- _ (aL +617), (3.1.3b)

0.12“ K

where:

0]an - In K variance

r1 - Correlation scale

A u — Longitudinal macrodispersivity

A22 - Transverse macrodispersivity

aL - Longitudinal pore-scale dispersivity
a‘r - Transverse pore-scale dispersivity

Substituting (3.1.3a) into (3.1.2) yields:

8C duC_ -3C
. 3.1.4
at _+ ax, arc—TIA” a a.) ( )

 

This is the classical advection-dispersion equation. The additional term, Barri/iii; g—ic],
x

in (3.1.4) represents macrodispersion. From (3.1.3b), it is evident that with increasing In
K variance, the longitudinal component of macrodispersion will increase. However, there

will be no effect on transverse macrodispersion, since no pore-scale dispersion is present.

The mean velocity (Z) controls the mean displacement of the plume. Darcy’s equation

gives the seepage velocity of the flow as:

=£§£

,
n ax,-
where:

K - Mean conductivity

n - Porosity.

34

The mean velocity is given by:

-,; 3am
n 3x,- n ax,‘

Gelhar (1993, equation 4.1.48) showed that:

2 ' 2 "
7(ng ”Eng; K'aiz £11193,
2 3x,- 2 3x,-

where:
K - Perturbation to mean conductivity

h - Mean hydraulic head

h. - Perturbation to mean hydraulic head
Kg - Geometric mean of hydraulic conductivity.

Substituting (3.1.6) into (3.1.5), we get:

_ Kg 3;,"
u=——-.
n ax,-

(3.1.5)

(3.1.6)

(3.1.7)

Since the geometric mean hydraulic conductivity and mean hydraulic gradient are the

same for all cases, u is the same. As a result, the mean plume displacement remains

largely unaffected.

Table 3.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

. _ 10 10 10 10
conductrvrty. Kg (m/day)
In K variance 0.0 0.5 1.0 3.0
Correlation scale, ,1 (m) NA 2 2 2

 

 

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Porosity, n 0.3 0.3 0.3 0.3

Local pore-scale dispersivity O 0 0 0

Head difference (m) 1 1 1 1

Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function NA Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10

Grid 40] x 407 401 x 407 401 x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step, At (days) 1 1 l 1

 

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.
0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

o The geometric mean is constant for all cases, while the In K variance increases

from 0 to 3.

0 The correlation scale does not apply to case 1 and is a constant for cases 2, 3 and

4.

o The covariance function for cases 2, 3 and 4 is exponential.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved;
typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than /i in one time step.

36

 

 

., ‘
Figure 3.1 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/EHST/EHSTl.htm

 

37

3.2 EFFECTS OF HETEROGENEITY OF HYDRAULIC CONDUCTIVITY -

WITH AND WITHOUT PORE-SCALE DISPERSION

3.2.1 Problem statement

This video demonstrates the effect of heterogeneity on transport of a conservative solute
plume in the presence of pore-scale dispersion. The initial size of the plume is larger than
the scale of heterogeneity. The modeling domain consists of constant head boundaries on
the left and right extremes, and no-flow boundaries at the top and bottom. Cases 1 and 2
consist of heterogeneous fields with and without pore-scale dispersion respectively.

Details are provided in Table 3.2.

3.2.2 Key observations

The following observations can be made from the video:

0 Transport through heterogeneous media causes large scale spreading.

Pore-scale dispersion does not have a significant impact on overall spreading.
0 Pore-scale dispersion has little impact on longitudinal macrodispersion, while it
has a relatively greater impact on transverse macrodispersion.

0 Pore-scale dispersion significantly reduces the concentration variability, and thus

enhances the degree of dilution.

3.2.3 Additional observations and discussion

Since the two models have the same realization, the degree of macrodispersion is the
same. The difference lies in the ‘smearing’ of the plume. When there is pore-scale

dispersion, the plume is slightly larger (or smeared) than when there is none. This

38

smearing is both longitudinal and transverse. The absence of pore-scale dispersion results
in numerous fingers and tails; hence, the variance of concentration is higher.

Pore-scale dispersion is represented numerically by the longitudinal and transverse
dispersivities. The amount of dispersion is obtained by multiplying the dispersivity by the
velocity in a particular cell in the grid. Since the ln K field is fixed for a particular

realization, the velocities in each cell of the grid are also fixed.

Physically, pore-scale dispersion can be understood as a measure of our uncertainty or
ignorance about the complex pore geometry. If it is possible to map out the microscopic
pore structure, and resolve it numerically, a purely advective model can predict plume
spreading accurately. This is, however, impossible. Moreover, we do not care about that
level of detail, making it necessary to have a numerical correction factor called pore-scale

dispersion.

3.2.4 Mathematical interpretation

The similarity in spreading between the two situations can be explained mathematically.
Sirrrilarly, a mathematical argument can be posed for the need to incorporate pore-scale
dispersion if risk based assessment is to be performed.

In a heterogeneous field, in the presence of pore-scale dispersion, the conservative solute

concentration at any point for any irregular plume realization is given by:

0C au-C 3 BC
I =— D-- — , 3.2.1

 

where:

C - Concentration

39

u,- - Seepage velocity

Di}- - Dispersion coefficient.

3.2.4.1 Impact of pore-scale dispersion on macrodispersion

For a heterogeneous K field with pore-scale dispersion, applying the perturbation theory,
(3.2.1) is modified as follows:
8C auC a BC —.-r
—+ — D.. ——+ —u.C . 3.2.2
at 8x, =ax,[ U Bx. ( ' )J ( )

J

 

Substituting (3.1.3a) into (3.2.2), we get:

65 BEE a §_c —ac
— =— — . .2.
at + 0x, 8x, [D d dx +[A'd u x]] (3 3)

 

XI

The additional term, 59—:(A, Ii BC], in (3.2.3) represents macrodispersion. From (3.1.3b),

it is apparent that the local dispersivities do not have any impact on longitudinal
macrodisperion. However, they have some effect on transverse macrodispersion. The
magnitude of macrodispersion is larger than pore-scale dispersion. Therefore, the plume
in case 2 has the same longitudinal extent as in case 1, while it is slightly smeared in the

transverse direction.

3.2.4.2 Impact of pore-scale dispersion on concentration variance

Kapoor and Gelhar (1994) showed that:

 

(3.2.4)

40

 

80% -80% — 320%;
——+u—=u --+a~ — 2u
at Bx,- (AU U ) dxixj
211a,):
IE .
(H

where:

0% - Concentration variance

2’ - Variance decay coefficient

A? - Concentration microscale.

- .. an 3.2
J 3x,- axj C,

(3.2.5)

(3.2.6)

The concentration variance without and with pore-scale dispersion is given in (3.2.4) and

(3.2.5) respectively. These equations are necessary in order to perform a risk-based

assessment of the plume concentration. The sharp peaks and valleys of concentration in

the plume are smoothed when pore-scale dispersion is present, resulting in a decrease in

the concentration variance. This can be mathematically proven using (3.2.5).

Table 3.2 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

[Parameter Case 1 Case 2
Geometric mean hydraulic conductivity.
Kg (m/day) 10 10

In K variance 3.0 3.0
Correlation scale, A (m) 2 2
Porosity, n 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0 0.01
Transverse pore-sclae dispersivity (m) 0 0.001
Head difference (m) 1 1

Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial plume size (at) 10 10

Grid 40] x 407 401 x 407

 

 

 

 

 

41

 

ICell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 I
ITime step, At (days) 1 1 I

 

 

 

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.
0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance

and covariance.

0 The geometric mean, In K and correlation scale are constant for all cases.

0 The covariance function for both cases is exponential.

0 Case 1 has no longitudinal and transverse dispersivities, while the corresponding
values for case 2 are 0.01 m and 0.001 m.

o The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than A in one time step.

42

In K var = 3.0

InKvar= 3.0

 

Figure 3.2 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSl-I/EHST/EHSTthm

43

3.3 INTERACTIONS BETWEEN CONDUCTIVITY AND POROSITY

HETEROGENEITIES

3.3.1 Problem statement

This video demonstrates the impact of interactions between conductivity and porosity
heterogeneities on the spreading of a conservative solute plume. The modeling domain
consists of constant head boundaries on the left and right extremes, and no-flow

boundaries at the top and bottom. Details are provided in Table 3.3.

3.3.2 Key observations

The following observations can be made from the video:
0 Interaction between conductivity (K) and porosity (n) heterogeneities has a
significant impact on plume spreading.
0 In most situations n is positively correlated with K, which results in less plume
spreading compared to when there is no heterogeneity in n.
0 When n is negatively correlated with K, plume spreading is greater compared to
when there is no heterogeneity in n.

0 When n and K are independent of each other, plume spreading is less than case 1,

although in general, this might not be the case.

3.3.3 Additional observations and discussion

Case 1: Even though K varies continuously over space, n remains constant throughout the
field. This is not a commonly encountered situation. In the absence of sufficient data, n is

assumed to be constant throughout the field.

44

Case 2: In a more common situation, n interacts positively with K. When n increases, the .
seepage velocity decreases. The seepage velocity is given by q/n, where q is the Darcy
velocity. When n increases, the seepage velocity decreases, resulting in less plume

spreading. Hence, the plume spreads lesser than in case 1.

Case 3: In this case, n interacts negatively with K. This means that it decreases when K
increases. This relatively uncommon situation results in enhanced spreading of the

plume. The eventual size of the plume is greater in comparison to case 1.

Case 4: When n varies with no relationship to K, velocity variability is random. The

plume spreading is dependent on 1n n realization.

Giving n a single value ignores the variability, results in increasing or reducing plume

spreading. The video shows that the effect on plume spreading is significant.

3.3.4 Mathematical interpretation

The interaction between it and K is modeled using the following equations (Paulson,

2002):

n .
—= (emf +W), (3.3.1)

"m
where:
n - Porosity

nm - Mean value of porosity

a- Correlation factor

45

f- In K

w - White noise.
The positive sign in the formula is applicable for positive correlation (case 2) and the
negative sign is applicable for negative correlation (case 3). The correlation factor is
computed by using the coefficient of variation, which is defined as the ratio of standard

deviation of porosity to the Mean porosity. The value of acontrols the variation in n. The

value of aat any point is given by the ratio of coefficient of variation to f '. w is the

random white noise for normalized n.

Table 3.3 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

, . 10 10 10 10
conductrvrty, 5g (m/day)
In K variance 2.0 2.0 2.0 2.0
Correlation scale, xi (m) 2 2 2 2
Mean porosity, "m 0.25 0.25 0.25 0.25
Coefficient of variation — n and K NA 0.27 0.25 0'19 a“ ”

variance)

Correlation - n and K NA Positive Negative None
White noise, w NA 0 0 0
Local pore—scale dispersivity 0 0 0 0
Head difference (m) 1 1 l 1
Size of model (at) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function Exponential Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10
Grid 401 x 407 401 x 407 401x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step, At (days) 1 1 1 1

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

46

 

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

The geometric mean, In K and correlation scale are constant for all cases.

The covariance function for all cases is exponential.

The interactions between n and K are characterized by a coefficient of variation
and the type of correlation.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

47

The video shows only the variation ofK over space. The variation ofn over space is
shown below. Notice that, even though the cases look very different from each other, the
head contours are the same. The head is dependent only on K. Since the In K realization

is the same for all cases, the head contours are identical.

hr 11 var = 0.0

 

 

. or ' . ' o
Figure 3.3 Snapshot of the visualization showing spatial variation

 

   

of n
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/NIT/NlT3.htm

 

48

3.4 INTERACTIONS BETWEEN CONDUCTIVITY AND PARTITIONING

COEFFICIENT HETEROGENEITIES

3.4.1 Problem statement

This video demonstrates the impact of interactions between conductivity and partitioning
coefficient (linear equilibrium sorption) heterogeneities on the spreading of a
conservative solute plume. The modeling domain consists of constant head boundaries on
the left and right extremes, and no-flow boundaries at the top and bottom. Details are

provided in Table 3.4.

3.4.2 Key observations
The following observations can be made from the video:
0 Interaction between conductivity (K) and partitioning coefficient (Kd)
heterogeneities has a significant impact on plume spreading.
0 When Kd is positively correlated with K, plume spreading is reduced than when

there is no heterogeneity in K0,.

0 In most situations K4 is negatively correlated with K, which results in
significantly greater plume spreading compared to when there is no heterogeneity
in Kd.

0 When Kd and K are independent of each other, plume spreading is less than case
1, although in general, this might not be the case.

0 Chemical heterogeneity (Kd) causes physical plume dispersion.

49

3.4.3 Additional observations and discussion

Case 1: Even though K varies continuously over space, Kd remains constant throughout

the field. This is not a commonly encountered situation. In the absence of sufficient data,

K1 is assumed to be constant throughout the field.

Case 2: In this case, Kd interacts positively with K. This means that Kd increases when K

increases. This relatively uncommon situation results in a decrease in plume spreading.

The eventual size of the plume is underestimated in comparison to case 1.

Case 3: In a more common situation, Kd interacts negatively with K. When Kd decreases,

the retardation factor decreases and hence the contaminant velocity increases and vice
versa, which results in increased spreading. The actual size of the plume is larger than in

comparison to case 1. The tails, in particular, are longer due to higher retardation factors.

Case 4: When Kd varies with no relationship to the K, velocity variability is random. The

plume spreading is dependent on ln Kd realization.

Giving Kd a single value ignores the variability, results in increasing or reducing plume

spreading. The video illustrates that the effect of variability in K; can be important.

3.4.4 Mathematical interpretation

The interaction between partitioning coefficient and conductivity are modeled using the

following equations (Paulson, 2002):

50

(2.4.1)

where:
Kd — Partitioning coefficient
Kdm — Mean value of Partitioning coefficient

_0t_ — Correlation factor

f ’ - In K

w — White noise.
The positive sign in the formula is applicable for positive correlation (case 2) and the
negative sign is applicable for negative correlation (case 3). The correlation factor is
computed by using the coefficient of variation, which is defined as the ratio of standard

deviation of partitioning coefficient to the mean partitioning coefficient. The value of

acontrols the variation in Kd. The value of aat any point is given by the ratio of

coefficient of variation to f ‘. w is the random white noise for normalized Kd.

Table 3.4 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

_ , 10 10 10 10
conductrvrty, Kg (m/day)
In K variance 2.0 2.0 2.0 2.0
Correlation scale, xi (m) 2 2 2 2
Porosity, n 0.3 0.3 0.3 0.3
Mean artitionin coefficient, K - - - -

3 P 8 dm 10 7 10 7 10 7 10 7
(m lg)
Coefficient of variation - Kd and K NA 1.65 1-47 1'59 (In Kd
variance)

Correlation _ K d and K NA Positive Negative None
White noise, w NA 0 0 0
Local pore-scale dispersivity 0 0 0 0

 

51

 

 

 

 

 

 

 

 

 

 

 

 

 

Head difference (m) l l l 1

Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function Exponential Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10

Grid 401 x 407 401 x 407 401 x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step. At (days) 2 2 2 2

 

The model parameters and inputs are explained below:

 

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

The geometric mean, In K and correlation scale are constant for all cases.

The covariance function for all cases is exponential.

The interactions between Kd and K are characterized by a coefficient of variation
and the type of correlation.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

52

The video shows only the variation ofK over space. The variation ode over space is
shown below. Notice that, even though the cases look very different from each other, the
head contours are the same. The head is dependent only on K. Since the In K realization

is the same for all cases, the head contours are identical.

 

 

Negative Correlation with In K, Var Coeff = 1.47

"HUT. 1‘1}. .
a? Lit-ti}.

a v . ‘
_ no .- .
. a
'4 . -1 “‘5'“ 1- 1.219111,
_ . .

V

No Correlation with In K. In Krl var = 1.5 s '

mo

 

'3 on ~~".",‘ 0

Figure 3.4 Snapshot of the visualization showing spatial variation of K,

 

For a detailed visualization, access the following address:

URL: htt ://www.e r.msu.edu/i w/DL/SSH/NlT/NIT4.htm

 

53

3.5 INTERACTIONS BETWEEN CONDUCTIVITY, POROSITY AND

PARTITION IN G COEFFICIENT HETEROGENEITIES

3.5.1 Problem statement

This video demonstrates the impact of interactions between conductivity, porosity and
partitioning coefficient (linear equilibrium sorption) heterogeneities on the spreading of a
conservative solute plume. The modeling domain consists of constant head boundaries on
the left and right extremes, and no-flow boundaries at the top and bottom. Details are

provided in Table 3.5.

3.5.2 Key observations
The following observations can be made from the video:
0 Interaction between conductivity (K), porosity (n) and partitioning coefficient

(Kd) heterogeneities has a drastic impact on plume spreading.

0 Plume spreading decreases when variability of n (positive correlation with K) is

considered.

0 Plume spreading increases when variability of K; (negative correlation with K) is
considered.

0 Heterogeneity in Kd has a greater influence on plume spreading than

heterogeneity in n.

54

3.5.3 Additional observations and discussion

Case 1: Even though K varies continuously over space, n and K4 remain constant
throughout the field. This is not a commonly encountered situation. In the absence of
sufficient data, n and Kd is assumed to be constant throughout the field.

Case 2: In this case, n interacts positively with K, while Kd remains constant throughout.

This means that n increases when K increases. This results in less extensive spreading. In

comparison to case 1, the plume is slightly smaller.

Case 3: In a more common situation, it remains constant throughout, while Kd interacts

negatively with K, meaning thath will decrease with increase in K. When Kd decreases,

the retardation factor decreases and hence the contaminant velocity increases, resulting in

enhanced spreading. The size of the plume is significantly larger than in case 1, and the
tails, in particular, are longer due to higher retardation factors. This indicates that K,

heterogeneity has a significant impact on plume spreading.

Case 4: This is the most commonly encountered situation. Porosity interacts positively

with K, while Kd interacts negatively with K. The resulting plume is considerably larger

than in case 1, which is the result of heterogeneity in Kd. However, in comparison to case

3, the plume is slightly smaller, due to the variability in n.

A comparison of the four situations shows that the effect of the interaction between K, n

and K4 is worth considering. Wherever possible, the effect of variability in all three must

55

be considered. In most cases, only the variability in K is considered. If the variability in n
is ignored, spreading of the plume is enhanced. If the variability in K4 is ignored,
spreading of the plume is reduced. If both are ignored, spreading of the plume is reduced,

indicating that the effect of heterogeneity in Kd is much greater than heterogeneity in n.

3.5.4 Mathematical interpretation

The interactions between porosity, partitioning coefficient and conductivity are modeled

using the following equations (Paulson, 2002):

n i ’
—=(e 09‘ +W), (3.5.1)
"m
K + '
-—d- = (e‘af + W), (3.5.2)
Kdm
where:
n —- Porosity

nm — Mean value of porosity

Kd — Partitioning coefficient

Kdm — Mean value of Partitioning coefficient
a— Correlation factor

f ' — In K

w — White noise.

Porosity is positively correlated with conductivity (cases 2, 4), while partitioning

coefficient is negatively correlated (cases 3, 4). The correlation factor is computed by

56

using the coefficient of variation, which is defined as the ratio of standard deviation of

porosity (or partitioning coefficient) to the Mean porosity (or partitioning coefficient).
The value of acontrols the variation in n (or Kd). The value of aat any point is given by

the ratio of coefficient of variation to f ‘. w is the random white noise for normalized n (or

Kd).

Table 3.5 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

. , 10 10 10 10
conductrvrty, Kg (m/day)
In K Variance 2.0 2.0 2.0 2.0
Correlation scale, xi (m) 2 2 2 2
Mean porosity, n 0.25 0.25 0.25 0.25
Coefficient of variation - n and K 0.0 0.27 0.0 0.27
Correlation - n and K NA Positive NA Positive
Partitioning coefficient. Kd (mg 10-7 10.7 10.7 10-75
Coefficient of variation - Kd and K 0-0 0-0 1-47 1-47
Correlation _ K d and K NA NA Negative Negative
Local pore-scale dispersivity O 0 0 0
Head difference (m) 1 1 l 1
Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function Exponential Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10
Grid 401 x 407 401 x 407 401 x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step, At (days) 2 2 2 2

 

0 The In K field is a normally distributed random field. Therefore, there is an equal

The model parameters and inputs are explained below:

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

 

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance

and covariance.

57

The geometric mean, In K and correlation scale are constant for all cases.

The covariance function for all cases is exponential.

The interactions between n and K, and between Kd and K are characterized by a
coefficient of variation and the type of correlation. Only 4 of the 16 possible

situations are modeled, since those are the most commonly encountered

situations.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than x1 in one time step.

58

The video shows only the variation of K over space. The variations of n and Kai over
space are shown below. Notice that, even though the cases look very different from each
other, the head contours are the same. The head is dependent only on K. Since the In K

realization is the same for all cases, the head contours are identical.

 

In 11 var = 0.0, In Kd var = 0.0
4’ O u 9‘ g
. O

O

i

z.‘ 152‘. .1._ -

2'") x...-
! ‘KJKL‘
' ‘15

Figure 3.5.1 Snapshot of the visualization

 

For a detailed visualization, access the following address:

59

URL: http://wwwegr.msu.edu/igw/DL/SSH/NIT/NITS.htm

In 11 var = 0.0, In Krl var = 0.0

 

 

 

60

In 11 var = 0.0. In Krl var = 0.0

 

 

61

3.6 EFFECT OF CORRELATION SCALE (xi) OF CONDUCTIVITY

HETEROGENEITY

3.6.1 Problem statement

This video demonstrates the impact of correlation scale (xi) of conductivity heterogeneity
on the transport of a conservative solute plume. The modeling domain consists of
constant head boundaries on the left and right extremes, and no-flow boundaries at the

top and bottom. Details are provided in Table 3.6.

3.6.2 Key observations

The following observations can be made from the video:
0 Correlation scale, xi, has a significant impact on plume transport.

0 When the correlation scale is much smaller than the plume source size, the plume
spreads in longitudinal and transverse directions. The center of mass of the plume
moves, in general, along a straight line. The plume boundary displays small scale
irregularities.

0 When the correlation scale is of the order of the plume source size, the
longitudinal spreading is greater than in case 1, transverse spreading is lesser than
in case 1 and the mean plume displacement is greater than in case 1. The center of
mass moves in general, along a meandering path. The irregularities of the plume

boundary are of a larger scale than in case 1.

3.6.3 Additional observations and discussion

The two cases differ only in the magnitude of their correlation scales. However, the

plumes look completely different. In case 1, the plume spreading is apparent in both

62

directions. Since the scale of heterogeneity is much smaller than the plume source size,
small preferential channels are formed, which ‘shred’ the plume into numerous fingers
and tails. Since the plume travels a distance equal to at least 50xi, the plume has the
opportunity to encounter the full range of conductivities. This is reflected in the mean

velocity of the plume and the mean plume displacement.

In case 2, where the correlation scale is about the size of the plume source, the plume
moves predominantly along a preferential channel (higher K regions). In comparison to
case 1, the width of the preferential channel is larger, and the plume is ‘sucked’ into it. In
any preferential channel, the streamlines of flow tend to converge, resulting in
longitudinal stretching and transverse compression of the plume. Therefore, we see that
the longitudinal spreading is greater than in case 1, and the transverse spreading is lesser
than in case I. Since the plume encounters only the higher range of conductivities, the
mean velocity of the plume will be higher than in case 1; therefore, the mean plume

displacement is also much higher.

Data limitation can seriously impair the ability to accurately characterize the aquifer.
Correlation scales are determined from field data using standard statistical methods.
When data is limited, the correlation scale will be overestimated. This results in
overestimation of the mean plume displacement and longitudinal spreading and

underestimation of transverse spreading.

63

Table 3.6 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2

Geometric mean hydraulic conductivity, K g 10 10
(m/day)

In K variance 2.0 2.0

Correlation scale, xi (m) l 5

Porosity, n 0.3 0.3

Local pore-scale dispersivity 0 0

Head difference (m) l 1

Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10

Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 l

 

The model parameters and inputs are explained below:

 

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

The geometric mean and In K variance are constant for all cases..

The correlation scale is 1 m for case 1 and 5 m for Case 2.

The covariance function for both cases is exponential.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

64

 

Figure 3.6 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: Mpd/www.egr.msu.edu/igw/DL/SSH/EHST/EHST7htm

65

3.7 EFFECT OF CORRELATION SCALE (xi) OF CONDUCTIVITY

HETEROGENEITY

3.7.1 Problem statement

This video demonstrates the impact of correlation scale (xi) of conductivity heterogeneity
on the spreading of a conservative solute plume. The modeling domain consists of
constant head boundaries on the left and right extremes, and no-flow boundaries at the

top and bottom. Details are provided in Table 3.7.

3.7.2 Key observations

The following observations can be made from the video:

0 Correlation scale, xi, has a significant impact on plume spreading.

0 When the correlation scale is much smaller than the source size of the plume, the
plume encounters a significant amount of heterogeneity and spreads as a result.

0 When the correlation scale is comparable to or larger than the plume source size,
the plume moves predominantly through preferential channels resulting in
enhanced longitudinal spreading and increased mean plume displacement.

0 Enhanced longitudinal spreading causes ‘negative’ transverse dispersion, which
means that the eventual size of the plume is smaller than the source size of the

plume, in the transverse direction.

3.7.3 Additional observations and discussion

The two models differ only in their correlation scale. However, the plumes look
completely different. In case I, the plume exhibits more detailed spreading in both

directions while in case 2, the plume spreading is prominent in the longitudinal direction

66

and minimal in the transverse direction. When the correlation scale is smaller than the
source size, the plume encounters more variability. As a result, the plume exhibits
numerous fingers and tails. The mean displacement is lower than in case 2, because the
chance of the plume encountering a low K zone is as high as that of encountering a high

K zone. Hence, the mean velocity of the plume is lower.

When the correlation scale is larger than the size of the plume, plume migration is highly
dependent on the In K realization. If the plume source is located in a zone of low K, the
plume will remain trapped for a long time. Conversely, if the source is located in a zone
of high K, the plume moves rapidly. Plume spreading also depends on how large the
correlation scale is, in comparison to the source size. Predicting the plume migration is
error-prone if xi is many orders of magnitude larger than the source. In case 2, xi is
approximately the same size as the size of the plume source. Hence, the chance of getting
trapped in a low K zone is not drastically increased. In this particular realization, the high
K zones are well connected, forming hi gh-velocity preferential channels. The plume
travels almost completely through these channels. The chance of encountering a low K
zone is minimized and as a result, the transverse spreading is considerably reduced. It is
obvious that the mean velocity of the plume will be higher than in case 1, because it
travels mostly through preferential channels. Therefore, the mean displacement is also

much higher.

Correlation scales are determined from field data using statistical methods. Data

limitation can seriously impair the ability to characterize the aquifer. In the absence of

67

data, the correlation scale of the variability can be overestimated. This results in serious

overestimation of the mean plume displacement and underestimation of plume spreading

in the transverse direction.

Table 3.7 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Geometric mean hydraulic conductivity, Kg (m/day) 10 10

In K variance 2.0 2.0
Correlation scale, xi (m) 1 10
Porosity, n 0.3 0.3

Local pore-scale dispersivity 0 0

Head difference (m) 1 1

Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10

Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 1

 

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance. The geometric mean and In K variance are constant for all cases.

0 The correlation scale is 1 m for case 1 and 10 m for case 2.

0 The covariance function for both cases is exponential.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved.

Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

68

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

 

Figure 3.7 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://wwwengsu.edu/igw/DL/SSH/EHST/EHST8htm

69

3.8 EFFECTS OF ANISOTROPIC HETEROGENEITIES

3.8.1 Problem statement

This video demonstrates the effects of anisotropic heterogeneities on the spreading of a
conservative solute plume. This type of anisotropic heterogeneity is most commonly
found in vertical cross-sections. The modeling domain consists of constant head
boundaries on the left and right extremes, and no-flow boundaries at the top and bottom.

Details are provided in Table 3.8.

3.8.2 Key observations

The following observations can be made from the video:

0 Anisotropic heterogeneities have a drastic impact on plume spreading.

Increasing horizontal correlation scales correspond to increasingly higher degrees

of stratification, resulting in less plume spreading in the transverse direction.

0 Longitudinal spreading is dependent on the In K realization.

o The vertical correlation scale is of the same order as the plume source size, but the
horizontal correlation scale controls plume spreading.

0 Plume spreading is most pronounced for the smallest horizontal correlation scale
(100 m), since the high K zones are not well connected. The longitudinal
spreading is prominent.

o For the largest horizontal correlation scale (5000 m), the plume rrrigration is

controlled, for the most part, by the preferential channels, which extend

longitudinally for about 5000 m. This implies that, once a preferential channel is

70

entered, the plume can travel for a very long distance within that channel, thus

minimizing transverse spreading.

3.8.3 Additional observations and discussion

Case 1: The correlation scale is 100 m in the horizontal direction and 10 m in the vertical
direction. The plume spreads to a great extent in the longitudinal direction, since the high
K zones are not well connected. The mass of the plume is concentrated in the leading

edges, and the plume tails are elongated.

Case 2: The correlation scale is 500 min the horizontal direction and 10 m in the vertical

direction. The plume spreads to a lesser extent in the longitudinal direction, since the high
K zones are connected better than in case 1. The mass of the plume is not concentrated as

much in the leading edges as in case 1. The highly skewed plume spreading seen in case

1 is not observed.

Case 3: The correlation scale is 1000 m in the horizontal direction and 10 m in the
vertical direction. The plume spreads to a lesser extent in the longitudinal direction, since
a longer high K zone (1000 m) is present than in case 1. The mass of the plume is not
concentrated as much in the leading edges as in case 1. The highly skewed plume
spreading seen in case 1 is not observed. In comparison to case 2, there is little difference

in longitudinal and transverse spreading.

Case 4: The correlation scale is 5000 m in the horizontal direction and 10 m in the

vertical direction. The plume spreads the least in the longitudinal direction, since an

71

extremely long high K zone (5000 m) is present. The plume spreads approximately
equally about its mean. This is evidenced by the presence of a number of small fingers
and tails that are not elongated. In comparison to cases 2 and 3, the plume appears

different, especially in the longitudinal direction.

A comparison of the four situations shows that the effect of anisotropic heterogeneities
on plume spreading is drastic. The most common places where such heterogeneity is
encountered are vertical cross-sections. It is interesting to note that, with increasing
stratification, the head contours approach a uniform distribution, as though the field were
homogeneous. However, the spreading of the plume indicates that the velocity vectors
differ greatly. The differential rates of advection are responsible for the spreading of the

plume.

Table 3.8 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 ase 3 Case 4
Geometric mean hydraulic conductivity,

10 10 10 10
K (m/day)
In K Variance 2.0 2.0 2.0 2.0
Correlation scale, xix (m) 100 500 1000 5000
Correlation scale, 42 (m) 10 10 10 10
Porosity, n 0.3 0.3 0.3 0.3
Pore-scale dispersivity O 0 0 0
Head difference (m) I 1 1 1
Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function Exponential Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10
Grid 401 x 407 401 x 407 401 x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step, At (days) 1 l I l

 

 

72

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

o The geometric mean and In K variance are constant for all cases.

0 The horizontal correlation scales vary from 100 m to 5000 m, while the vertical
correlation scales remain constant at 10 m.

0 The covariance function for all cases is exponential.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

73

 

Correlation lengths - 100 m, 10 m

 

5.3—2: ._._._-_---_———--I—

—-——_—_—-_-~-—:_—-‘

 

_ ___._ _._..__.____._i-_.—-—-‘_

i 7: i ‘ Correlation Iengths- 5000 m. 10 m

o—u—-—————--———
V... : ' ‘ -L-.:=t R-“4_.l.nnh‘:
.‘ .- ., ....-_ ..-. .......- . - w“, __.

.--.-. V 4-..“... _...

.— —-~— urn-u! W “- ---
«‘m— W ..a n».

..7.- _-..— _... __;._.-

 

 

Figure 3.8 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/EHST/EHST14.htm

74

3.9 EFFECTS OF ANISOTROPIC HETEROGENEITIES

3.9.1 Problem statement

This video demonstrates the effects of anisotropic heterogeneities on the spreading of a

conservative solute plume. The modeling domain consists of constant head boundaries on

the left and right extremes, and no-flow boundaries at the top and bottom. Details are

provided in Table 3.9.

3.9.2 Key observations

The following observations can be made from the video:

Anisotropic heterogeneities have a drastic impact on plume spreading.

When the sources size of the plume is greater than both horizontal and vertical
correlation scales, plume spreading is enhanced.

Increasing vertical correlation scales result in less plume spreading, in both
longitudinal and transverse directions.

Plume spreading is most pronounced for the smallest vertical correlation scale (2
m), since the high K zones are not well connected in the transverse direction,
resulting in significant spreading.

For the largest horizontal correlation scale (25 m), the plume migration is
controlled, for the most part, by the In K realization, which in turn, governs the

location of preferential channels.

3.9.3 Additional observations and discussion

Case 1: The correlation scale is 20 m in the horizontal direction and 2 m in the vertical

direction. This correlation structure is similar to what is found in a vertical cross—section.

75

However, it is not perfectly stratified, resulting in elongated plumes. The plume spreads
to a great extent in the longitudinal direction, since the high K zones are not well
connected in the transverse direction. The magnitude of transverse spreading is less in

comparison to that of longitudinal spreading.

Case 2: The correlation scale is 20 min the horizontal direction and 5 min the vertical
direction. This correlation structure is similar to what is found in a vertical cross-section.
However, it is not perfectly stratified, resulting in elongated plumes, although less
elongated than in case 1. The plume spreads to a less extent in the longitudinal direction,

since the high K zones are connected better than in case 1.

Case 3: The correlation scale is 20 min the horizontal direction and 10 min the vertical
direction. This correlation structure is commonly found in a horizontal view. Plume
migration is controlled by the In K realization. If a low K zone is encountered, the plume
is trapped, and is released slowly at later points in time. The horizontal correlation scale
is such that both low K and high K zones have to be encountered. This means that the
different parts of the plume move at significantly different velocities, which means that

the longitudinal spreading will be enhanced.

Case 4: The correlation scale is 20 min the horizontal direction and 25 min the vertical
direction. This correlation structure is commonly found in a horizontal view. Plume
migration is controlled by the In K realization. If a low K zone is encountered, the plume

is trapped, and is released slowly at later points in time. The horizontal correlation scale

76

is such that both low K and high K zones have to be encountered. This means that the

different parts of the plume move at significantly different velocities, and longitudinal

spreading will be enhanced. In comparison to case 3, the spreading is less extensive,

since the plume does not encounter the different heterogeneities.

A comparison of the four situations shows that the effect of anisotropic heterogeneities

on plume spreading is drastic. With increasing correlation scales, plume spreading is

reduced in the transverse direction, since it becomes difficult to encounter the

heterogeneities. The differential rates of advection are responsible for the spreading of

the plume.

Table 3.9 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3 Case 4
Geometric mean hydraulic

, , 10 10 10 10
conductrvrty, Kg (m/day)
In K Variance 2.0 2.0 2.0 2.0
Correlation scale, xix (m) 20 20 20 20
Correlation scale, xiZ (m) 2 10 25
Porosity, n 0.3 0.3 0.3 0.3
Pore-scale dispersivity 0 O 0 0
Head difference (m) 1 1 l 1
Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50
Covariance function Exponential Exponential Exponential Exponential
Approximate initial plume size (m) 10 10 10 10
Grid 401 x 407 401 x 407 401 x 407 401 x 407
Cell size, Ax (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5
Time step, At (days) I 1 l l

 

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

77

 

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

The geometric mean and In K variance are constant for all cases.

The vertical correlation scales vary from 2 m to 25 m, while the horizontal

correlation scales remain constant at 20 m.

The covariance function for all cases is exponential.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

78

 

 

Correlation lengths - 20 m. 2 m ‘
c, __'- ‘~
-_ .

a..-
'-

 

C'on‘elation lengths - 20 m. 25 m
. . ( '
,4 ' ~ 0' '
.. “ a %
Figure 3.9 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/EHST/EHSTIS.htm

 

79

4 TEMPORAL VARIABILITY

4.1 UNCONFINED AQUIFERS WITH INCREASING TRANSMISSIVITIES

4.1.1 Problem statement
This video demonstrates the effect of a sinusoidally oscillating boundary condition on the
hydraulic head in unconfined aquifers with increasing transrnissivities. The modeling

domain consists of a time-variable head boundary on the left extreme, and a no-flow

boundary on the right. Details are provided in Table 4.1.

4.1.2 Key observations

The following observations can be made from the video:

0 Sinusoidal oscillation at the boundary affects the head in the aquifer sinusoidally.

0 As distance from the boundary increases, the effect on the head decreases.

o The area that is influenced by the boundary is called the “response zone”. The
response zone depends on several parameters, such as transmissivity, storage
coefficient, frequency of oscillations, amplitude of oscillations, and distance from
the boundary.

0 Lower transmissivity values result in a smaller response zone.

4.1.3 Additional observations and discussion

Case 1: The average transmissivity is 2 m2/day and the specific yield is 0.1. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. Since the aquifer material is not highly transmissive, the resulting

transient head distribution in the aquifer is concentrated in a small zone very close to the

80

river. Beyond approximately 20 m from the river, the oscillating head in the river

produces no visible effect on the head in the aquifer.

Case 2: The average transmissivity is 20 m2/day and the specific yield is 0.1. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. Since the aquifer material is more transmissive than in case 1, the
resulting transient head distribution in the aquifer is concentrated in a larger zone

compared to case 1. Beyond approximately 50 m from the river, the oscillating head in

the river produces no visible effect on the head in the aquifer.

Case 3: The average transnrissivity is 200 mzlday and the specific yield is 0.1. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. Since the aquifer material is more transmissive than in cases 1 and
2, the resulting transient head distribution in the aquifer is concentrated in a larger zone
compared to cases 1 and 2. Beyond approximately 100 m from the river, the oscillating

head in the river produces no visible effect on the head in the aquifer.

4.1.4 Mathematical interpretation

The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:

2 015' 2

yx ~
)srn(at -

 

s(x,t) = s, exp(—

 

), (4.1)

where:

s - Head change in the aquifer

81

s, - Amplitude of stage change in the surface water body

ar- Frequency of stage change
x - Distance from the boundary
T - Transmissivity of aquifer

t - Time

Sy - Storage coefficient (specific yield) of aquifer

Equation 4.1 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The amplitude of
variation is damped exponentially with distance away from the boundary. The amplitude
is dependent on aquifer properties such as transmissivity and storage coefficient. When
transmissivity increases, amplitude also increases, and when storage coefficient increases,

amplitude decreases.

Table 4.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Average transmissivity, T (mzlday) 2 20 200
Specific yield, 5), 0.1 0.1 0.1
Storage coefficient, S NA NA NA
Porosity, n 0.3 0.3 0.3
Starting head - aquifer (m) 5 5 5
Mean head — river (m) 5 5 5
Amplitude, s, (m) 5 5 5

Time period (days) 1 1 1

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cell size,Ax(m) 2x2 2x2 2x2
Time step, At (days) 0.1 0.1 0.1

 

 

 

 

 

The model parameters and inputs are explained below:

82

o The average transmissivity values are 2 m2/d, 20 mZ/d, and 200 m2/d for cases 1,

2, and 3 respectively, and specific yields are 0.1 for all cases.

0 The sinusoidal oscillation in the river is characterized by the following
parameters: amplitude of 5 m and a time period of 1 day.

0 The time step, At, is selected such that the temporal variability of head in the river

is resolved. The head in the river varies sinusoidally over a time period of 1 day,

and hence a time step of 0.1 days is selected.

83

 

*.",."’«‘.-n’ .-;- raw... .j' " "(,_’,' ' ' '
44.1'.(" .' " ‘3! 317:5}: an. ,zl ' ' I .

 

' ‘ ‘. . ’artrestrnaamrmt
:

 

Figure 4.1 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://wwwegr.msu.edu/igw/DL/SSH/TV/TV9.htm

84

4.2 UNCONFINED AQUIFERS WITH INCREASING SPECIFIC YIELDS

4.2.1 Problem statement

This video demonstrates the effect of a sinusoidally oscillating boundary condition on the
hydraulic head in unconfined aquifers with increasing specific yields. The modeling
domain consists of a time-variable head boundary on the left extreme, and a no-flow

boundary on the right. Details are provided in Table 4.2.

4.2.2 Key observations

The following observations can be made from the video:

0 Sinusoidal oscillation at the boundary affects the head in the aquifer sinusoidally.

0 As distance from the boundary increases, the effect on the head decreases.

0 The area that is influenced by the boundary is called the “response zone”. The
response zone depends on several parameters, such as transmissivity, storage
coefficient, frequency of oscillations, amplitude of oscillations, and distance from
the boundary.

0 Higher specific yield values result in a smaller response zone.

4.2.3 Additional observations and discussion

Case 1: The average transmissivity is 200 mzld and the specific yield is 0.01. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. The resulting transient head distribution in the aquifer is
concentrated in a zone close to the river. Beyond approximately 250 m from the river, the

oscillating head in the river produces no visible effect on the head in the aquifer.

85

Case 2: The average transmissivity is 200 m2/d and the specific yield is 0.1. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. The resulting transient head distribution in the aquifer looks very
similar to case 1. The higher specific yield compared to case 1, decreases the area
influenced by the river. Beyond approximately 100 m from the river, the oscillating head

in the river produces no visible effect on the head in the aquifer.

Case 3: The average transmissivity is 200 m2/d and the specific yield is 0.3. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. The resulting transient head distribution in the aquifer looks very
similar to cases I and 2. The higher specific yield of 0.3 compared to cases 1 and 2,
decreases the area influenced by the river. Beyond approximately 60 m from the river, the

oscillating head in the river produces no visible effect on the head in the aquifer.

4.2.4 Mathematical interpretation

The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:

2 (US 2

x
y )sin(at -

 

 

s(x,t) = s, exp(—

), (4.2)

where:

s - Head change in the aquifer

s, - Amplitude of stage change in the surface water body

60- Frequency of stage change

86

x - Distance from the boundary
T - Transmissivity of aquifer
t - Time

Sy - Storage coefficient (specific yield) of aquifer

Equation 4.2 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The amplitude of
variation is dampened exponentially with distance away from the boundary. The
amplitude is dependent on aquifer properties such as transmissivity and storage
coefficient. When transmissivity increases, amplitude also increases, and when storage

coefficient increases, amplitude decreases.

Table 4.2 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Average transmissivity, T (mz/day) 200 200 200
Specific yield, 81, 0.01 0.1 0.3
Storage coefficient, S NA NA NA
Porosity, n 0.3 0.3 0.3
Starting head - aquifer (m) 5 5 5

Mean head - river (m) 5 5 5
Amplitude, s, (m) 5 5 5

Time period (days) I l 1

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cell size,Ax(m) 2x2 2x2 2x2
Time step, At (days) 0.1 0.1 0.1

 

The model parameters and inputs are explained below:
0 The transmissivity is 200 m2/d for all cases; specific yield is 0.01, 0.1, and 0.3 for

cases 1, 2, and 3 respectively.

87

0 The sinusoidal oscillation in the river is characterized by the following
parameters: amplitude of 5 m and a time period of 1 day.

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 1 day,

and hence a time step of 0.1 days is selected.

88

 

 

' i' 1. . vi: ’2 tsfltfikufla

. Wat-angst. arm”...

I
l
.

 

Figure 4.2 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/TV/TV10.htm

89

4.3 CONFINED AQUIFERS WITH INCREASING CONDUCTIVITIES

4.3.] Problem statement

This video demonstrates the effect of a sinusoidally oscillating boundary condition on the
hydraulic head in confined aquifers with increasing conductivities. The modeling domain
consists of a time-variable head boundary on the left extreme, and a no-flow boundary on

the right. Details are provided in Table 4.3.

4.3.2 Key observations

The following observations can be made from the video:

0 Sinusoidal oscillation at the boundary affects the head in the aquifer sinusoidally.

0 As distance from the boundary increases, the effect on the head decreases.

0 The area that is influenced by the boundary is called the “response zone”. The
response zone depends on several parameters, such as transmissivity, storage
coefficient, frequency of oscillations, amplitude of oscillations, and distance from
the boundary.

0 Lower transmissivity values result in a smaller response zone.

4.3.3 Additional observations and discussion

Case 1: The transmissivity is 0.2 m2/d and the storage coefficient is 2 x 104. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. Since the aquifer material is not highly transmissive, the resulting
transient head distribution in the aquifer is concentrated in a small zone very close to the
river. Beyond approximately 75 m from the river, the oscillating head in the river

produces no visible effect on the head in the aquifer.

90

Case 2: The transmissivity is 2 mz/day and the storage coefficient is 2 x 104. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a
time period of 1 day. Since the aquifer material is more transmissive than in case 1, the
resulting transient head distribution in the aquifer is concentrated in a larger zone
compared to case 1. Beyond approximately 200 m from the river, the oscillating head in

the river produces no visible effect on the head in the aquifer.

Case 3: The transmissivity is 20 mz/day and the storage coefficient is 2 x 104. The

sinusoidal oscillation of head in the river is characterized by an amplitude of 5 m and a

time period of 1 day. Since the aquifer material is more transmissive than in cases 1 and
2, the resulting transient head distribution in the aquifer is concentrated in a larger zone
compared to cases 1 and 2. The variable head in the river affects the head in the entire

modeled area (450 m).

4.3.4 Mathematical interpretation
The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:

tuS'x2 . tztS'x2
)srn(a)t —
T 2T

 

 

), (4.3)

s(x,t) = s, exp(—

where:

s - Head change in the aquifer

s, - Amplitude of stage change in the surface water body

91

60- Frequency of stage change
x - Distance from the boundary
T - Transmissivity of aquifer

t - Time

5 - Storage coefficient of aquifer

Equation 4.3 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The amplitude of
variation is dampened exponentially with distance away from the boundary. The
amplitude is dependent on aquifer properties such as transmissivity and storage
coefficient. When transmissivity increases, amplitude also increases, and when storage

coefficient increases, amplitude decreases.

Table 4.3 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Transmissivity, T(m2/day) 0-2 2 20
Storage coefficient, S 2 x [0‘4 2 x 10‘4 2 x 10'4
Porosity, n 0.3 0.3 0.3
Startini head - aquifer (m) 5 5 5

Mean head - river (m) 5 5 5
Amplitude, s, (m) 5 5 5

Time period (days) 1 1 1

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cell size,Ax(m) 2x2 2x2 2x2
Time step, At (days) 0.1 0.1 0.1

 

 

 

 

 

The model parameters and inputs are explained below:

92

0 The transmissivity values are 0.2 m2/d, 2 mzld, and 20 m2/d for cases 1, 2, and 3
respectively, and storage coefficients are 2 x 10'4 for all cases.

0 The sinusoidal oscillation in the river is characterized by the following
parameters: amplitude of 5 m and a time period of 1 day.

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 1 day,

and hence a time step of 0.1 days is selected.

93

 

 

 

 

 

 

 

Figure 4.3 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/TV/TVl l.htm

94

4.4 CONFINED AQUIFERS WITH INCREASING STORAGE COEFFICIENTS

4.4.] Problem statement

This video demonstrates the effect of a sinusoidally oscillating boundary condition on the
hydraulic head in unconfined aquifers with increasing storage coefficients. The modeling
domain consists of a time-variable head boundary on the left extreme, and a no-flow

boundary at the right. Details are provided in Table 4.4.

4.4.2 Key observations

The following observations can be made from the video:

0 Sinusoidal oscillation at the boundary affects the head in the aquifer sinusoidally.

0 As distance from the boundary increases, the effect on the head decreases. The
area that is influenced by the boundary is called the “response zone”.

0 The area that is influenced by the boundary is called the “response zone”. The
response zone depends on several parameters, such as transmissivity, storage
coefficient, frequency of oscillations, amplitude of oscillations, and distance from
the boundary.

0 Lower storage coefficient values result in a larger response zone.

4.4.3 Additional observations and discussion

Case 1: The transmissivity is 2 m2/d and storage coefficient is 2 x 103. The sinusoidal

oscillation of head in the river is characterized by an amplitude of 5 m and a time period
of 1 day. The resulting transient head distribution in the aquifer is spread over a large
area. Since the aquifer material can store “less” water, the head in the aquifer is affected

in the whole of the modeled area.

95

Case 2: The transmissivity is 2 mZ/d and storage coefficient is 2 x 10-4. The sinusoidal

oscillation of head in the river is characterized by an amplitude of 5 m and a time period
of 1 day. The resulting transient head distribution in the aquifer looks very different from
case 1. The higher storage coefficient compared to case 1, means that the aquifer can
store more water. Therefore, a smaller area is affected. Beyond approximately 150 m
from the river, the oscillating head in the river produces no visible effect on the head in

the aquifer.

Case 3: The transmissivity is 2 mzld and storage coefficient is 2 x 10-5. The sinusoidal

oscillation of head in the river is characterized by an amplitude of 5 m and a time period
of 1 day. The resulting transient head distribution in the aquifer is concentrated in a small
zone very close to the river. The higher storage coefficient compared to cases 1 and 2, has
a significant impact on the area influenced by the river. Since the aquifer can store more
water than in cases 1 and 2, beyond approximately 150 m from the river, the oscillating

head in the river produces no visible effect on the head in the aquifer.

4.4.4 Mathematical interpretation

The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:
2 2
0th (1th
s x,t :5 ex -— sin (ot- , 4.4
()rp(‘i2T)( 2T) ()

s - Head change in the aquifer

 

 

where:

96

s, - Amplitude of stage change in the surface water body

01- Frequency of stage change

x - Distance from the boundary

T - Transmissivity of aquifer

t - Time

S - Storage coefficient of aquifer
Equation 4.4 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The amplitude of
variation is dampened exponentially with distance away from the boundary. The
amplitude is dependent on aquifer properties such as transmissivity and storage
coefficient. When transmissivity increases, amplitude also increases, and when storage

coefficient increases, amplitude decreases.

Table 4.4 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Transmissivity, T (m2/day) 2 2 2
Storage coefficient, S 2 x 10'3 2 x 10'4 2 x 1075
Porosity, n 0.3 0.3 0.3
Starting head - aquifer (m) 5 5 5

Mean head - river (m) 5 5 5
Amplitude, s, (m) 5 5 5

Time period (days) 1 1 1

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cell size,Ax(m) 2x2 2x2 2x2
Time step, At (days) 0.1 0.1 0.1

 

 

 

 

 

97

The model parameters and inputs are explained below:

0 The transmissivity is 2 m2/d for all cases; storage coefficient is 2 x 103, 2 x 10'4
and 2 x 10'5 for cases 1, 2 and 3 respectively.

0 The sinusoidal oscillation in the river is characterized by the following
parameters: amplitude of 5 m and a time period of 1 day.

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 1 day,

and hence a time step of 0.1 days is selected.

98

 

 

 

 

 

 

 

Figure 4.4 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://wwwegrmsu.edu/igw/DL/SSH/TV/TVl2.htm

99

4.5 COMPARISON BETWEEN UNCONFINED AND CONFINED AQUIFERS

4.5.1 Problem statement

This video compares the effect of a sinusoidally oscillating boundary condition on the
hydraulic head in an unconfined aquifer with that in a confined aquifer. The two aquifers
differ only in the magnitude of their storage properties (specific yield and storage
coefficient). The modeling domain consists of a time-variable head boundary on the left

extreme, and a no-flow boundary on the right. Details are provided in Table 4.5.

4.5.2 Key observations

The following observations can be made from the video:

0 In comparison to a confined aquifer, a similar unconfined aquifer has a smaller
response zone.

0 Unconfined aquifers release water by draining water-filled pores, while confined
aquifers release water by elastic expansion of the aquifer matrix and
decompression of water. Therefore, unconfined aquifers are able to drain more
water from a unit area of aquifer material per unit decline in head; hence, the

smaller response zones.

4.5.3 Additional observations and discussion

Case 1: The average transmissivity of the unconfined aquifer is 20 m2/d and the specific

yield is 0.1. The sinusoidal oscillation of head in the river is characterized by an
amplitude of 5 m and a time period of 1 day. The response zone is approximately 50 m in

length.

100

Case 2: The transmissivity of the confined aquifer is 20 mz/d and the storage coefficient
is 2 x 10.4. The sinusoidal oscillation of head in the river is characterized by an amplitude

of 5 m and a time period of 1 day. The response zone is larger than in case 1, since the

confined aquifer has a much lower storage coefficient compared to the unconfined

aquifer.

Since storage properties of confined aquifers are smaller than those of unconfined
aquifers by several orders of magnitude (typically 2 or more), the response zones are

proportionally larger.

3.5.4 Mathematical interpretation
The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:

2 2
s(x,t) = s, exp(—‘/ (:9; )sin(wt - “ZS; ) , (4.5)

s - Head change in the aquifer

 

 

Where:

s, - Amplitude of stage change in the surface water body

60- Frequency of stage change
x - Distance from the boundary
T - Transmissivity of aquifer

t - Time

S - Storage coefficient of aquifer

101

Equation 4.5 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The amplitude of
variation is dampened exponentially with distance away from the boundary. The
amplitude is dependent on aquifer properties such as transmissivity and storage
coefficient. When all pr0perties of the two aquifers are similar, a decrease in storage

coefficient causes an increase in amplitude and size of the response zone.

Table 4.5 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Average transmissivity, T (m2/day) 20 20
Specific yield. Sy 0.1 NA
Storage coefficient, S NA 2 x “)4
Porosity, n 0.3 0.3
Starting head - aquifer (m) 5 5

Mean head - river (m) 5 5
Amplitude, s, (m) 5 5

Time period (days) 1 1

Size of model (m) 500 x 10 500 x 10
Grid 251 x 6 251 x 6
Cell size, Ax (m) 2 x 2 2 x 2
Time step, At (days) 0.1 0.1

 

The model parameters and inputs are explained below:

0 The transmissivity values are 20 mzld for both cases; specific yield is 0.1 for case
1 and storage coefficient is 2 x 104 for case 2.

o The sinusoidal oscillation in the river is characterized by the following
parameters: amplitude of 5 m and a time period of 1 day.

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 1 day,

and hence a time step of 0.1 days is selected.

102

 

 

 

    

Figure 4.5 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/TV/TVI3htm

 

103

4.6 BOUNDARIES WITH DECREASING FREQUENCIES OF VARIABILITY

4.6.1 Problem statement

This video demonstrates the effect of decreasing the frequency of a sinusoidally
oscillating boundary condition on the hydraulic heads in confined aquifers. The modeling
domain consists of a time-variable head boundary on the left extreme, and a no-flow

boundary on the right. Details are provided in Table 4.6.

4.6.2 Key observations

The following observations can be made from the video:
0 Decreasing the frequency of Sinusoidal oscillation at the boundary decreases the
frequency of change in head in the aquifer.
o The area influenced by the boundary is called the “response zone”. The response
zone depends on several parameters, such as transmissivity, storage coefficient,
frequency of oscillations, amplitude of oscillations, and distance from the

boundary.

0 Smaller values of frequency of oscillation result in a larger response zone.

4.6.3 Additional observations and discussion

Case 1: The transmissivity is 2 mzld and the storage coefficient is 2 x 104. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of
0.25 days. Since the frequency of oscillations is very high, the resulting transient head
distribution in the aquifer is concentrated in a small zone close to the river. Beyond
approximately 100 m from the river, the oscillating head in the river produces no visible

effect on the head in the aquifer.

104

Case 2: The transmissivity is 2 mz/d and the storage coefficient is 2 x 10-4. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of 1
day. Since the frequency of oscillations is smaller than in case 1, the resulting transient
head distribution in the aquifer is concentrated in a larger zone compared to case 1.
Beyond approximately 200 m from the river, the oscillating head in the river produces no

visible effect on the head in the aquifer.

Case 3: The transmissivity is 2 mz/d and the storage coefficient is 2 x 10-4. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of
10 days. Since the frequency of oscillations is smaller than in cases 1 and 2, the resulting
transient head distribution in the aquifer is concentrated in a larger zone compared to
cases 1 and 2. The variable head in the river affects the head in the entire modeled area

(450 m).

4.6.4 Mathematical interpretation

The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:
2 2
atS'x . aer
s x,t =s ex - srn ax— , 4.6
()rp(‘i2T)( 2T) ()

s - Head change in the aquifer

 

 

where:

s, - Amplitude of stage change in the surface water body

105

60- Frequency of stage change
x - Distance from the boundary
T - Transmissivity of aquifer

t - Time

S - Storage coefficient of aquifer

Equation 4.6 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The size of the
response zone is dependent on aquifer properties, as well as the frequency. When the

frequency of variability decreases, the size of the response zone increases.

Table 4.6 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Transmissivity, T (mz/day) 2 2 2
Storage coefficient, S 2 x 10" 2 x 10'4 2 x “)4
Porosity, n 0.3 0.3 0.3
Starting head - aquifer (m) 5 5 5

Mean head - river (m) 5 5 5
Amplitude, s, (m) 10 1° 10
Time period (days) 0.25 l 10

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cellsize,Ax(m) 2x2 2x2 2x2
Time step. At (days) 0.05 0.05 0.05

 

The model parameters and inputs are explained below:
0 The transmissivity and storage coefficient values are 2 m2/d and 2 x 10'4

respectively for all cases.

0 The sinusoidal oscillation in the river is characterized by amplitude of 10 m for all

cases, and time periods of 0.25, 1, and 10 days for cases 1, 2, and 3 respectively.

106

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 0.25

days, and hence a time step of 0.05 days is selected.

107

   

ll

0- -.--

 

 

 

Figure 4.6 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://wwwegrmsu.edu/igw/DL/SSH/TV/TVl4.htm

108

4.7 BOUNDARIES WITH INCREASING EXTENTS OF PENETRATION INTO

THE AQUIFER

4.7.1 Problem statement

This video demonstrates the effect of increasing extents of penetration into the aquifer on
the hydraulic heads in confined aquifers. The modeling domain consists of a time-
variable head boundary on the left extreme, and a no-flow boundary on the right. Details

are provided in Table 4.7.

4.7.2 Key observations

The following observations can be made from the video:
0 Increasing the extent of penetration into the aquifer has little impact on the head
in the aquifer.
o The area influenced by the boundary is called the “response zone”. When the
extent of penetration increases, the size of the response zone increases, although

the increase is not drastic.

4.7.3 Additional observations and discussion

Case 1: The transrrrissivity is 2 m2/d and the storage coefficient is 2 x 10-4. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of 1
day. The surface water body penetrates only 5 m of the 20 m thickness of the aquifer.

Beyond approximately 230 m from the river, the oscillating head in the river produces no
visible effect on the head in the aquifer. There is considerable head loss under the surface

water boundary.

109

Case 2: The transmissivity is 2 m2/d and the storage coefficient is 2 x 10-4. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of 1
day. The surface water body penetrates only 10 m of the 20 m thickness of the aquifer.
Beyond approximately 240 m from the river, the oscillating head in the river produces no
visible effect on the head in the aquifer. The small increase in the size of the response

zone is due to the lower head loss under the surface water boundary.

Case 3: The transmissivity is 2 mz/d and the storage coefficient is 2 x 104. The sinusoidal

oscillation of head in the river is characterized by amplitude of 10 m and time period of 1
day. The surface water body penetrates the entire 20 m thickness of the aquifer. Beyond
approximately 250 m from the river, the oscillating head in the river produces no visible
effect on the head in the aquifer. Since the surface water body penetrates the entire
aquifer, there is no head loss under the boundary. Therefore, to achieve the same head

loss as cases 1 and 2, water in a larger area has to be displaced.

4.7.4 Mathematical interpretation

The head change in an aquifer connected to a surface water body with a sinusoidally

oscillating stage was studied by Ferris (1951), and is given by:
2 2
trer 0er
' ,t = ex — sin art— , 3.7
AU)» p("2T)( 2T) ()

s - Head change in the aquifer

 

 

where:

s, - Amplitude of stage change in the surface water body

110

60- Frequency of stage change
x - Distance from the boundary
T - Transmissivity of aquifer

t - Time

S - Storage coefficient of aquifer

Equation 4.7 shows that the head change in the aquifer varies sinusoidally with the same
frequency as the stage in the surface water body, but with a time lag. The size of the
response zone is dependent on aquifer properties, as well as the frequency. When the

frequency of variability decreases, the size of the response zone increases.

Table 4.7 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Transmissivity, T(m2/day) 2 2 2
Storage coefficient, 5 2 x 10‘4 2 x 10’4 2 x 10'4
Porosity, n 0.3 0.3 0.3
Starting head - aquifer (m) 5 5 5

Mean head - river (m) 5 5 5
Amplitude, s, (m) 10 10 10
Time period (days) 1 1 1

Size of model (m) 500 x 10 500 x 10 500 x 10
Grid 251x6 251x6 251x6
Cellsize,Ax(m) 2x2 2x2 2x2
Time step, At (days) 0.05 0.05 0.05

 

The model parameters and inputs are explained below:

 

0 The transmissivt and storage coefficient values are 2 mz/day and 2 x 10.4

respectively for all cases.

0 The sinusoidal oscillation in the river is characterized by amplitude of 10 m for all

cases, and time periods of 0.25, l, and 10 days for cases I, 2, and 3 respectively.

111

0 The time step, At, is selected such that the temporal variability of head in the river
is resolved. The head in the river varies sinusoidally over a time period of 0.25

days, and hence a time step of 0.05 days is selected.

112

 

 

 

 

 

 

Figure 4.7 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: Mpfl/www.eer.msu.edu/igw/DL/SSH/TV/TVl5.htm

113

4.8 PLUME TRANSPORT - COMPARISON BETWEEN HOMOGENEOUS AND

HETEROGENEOUS AQUIFERS

4.8.1 Problem statement

This video compares the effect of temporally varying boundary conditions on plume
transport in homogeneous aquifers with heterogeneous aquifers. The modeling domain
consists of a time-variable head boundary on both left and right extremes. Details are

provided in Table 4.8.

4.8.2 Key observations

The following observations can be made from the video:
0 Temporally varying boundary conditions cause to and fro motion of the plume in
both homogeneous and heterogeneous aquifers.
o Plume transport is predominantly reversible in homogeneous aquifers, while it is
irreversible in heterogeneous aquifers.
o The plume spreads significantly due to the interaction between spatial
heterogeneity and temporal variability. In the absence of spatial heterogeneity, the

plume does not spread, but undergoes slight distortion in shape.

4.8.3 Additional observations and discussion

Case 1: The average conductivity of the aquifer is 5 m/day. The temporally varying
boundary conditions cause the head in the aquifer to fluctuate. As a result, the flow of
groundwater changes direction, resulting in to and fro motion of the plume. However, due
to the homogeneous properties of the aquifer, there is no spreading of the plume, apart

from a slight distortion in the shape of the plume.

114

Case 2: The geometric mean conductivity of the aquifer is 5 m/day. The temporally
varying boundary conditions cause the head in the aquifer to fluctuate. As a result, the
flow of groundwater changes direction, resulting in to and fro motion of the plume. Since
the aquifer is heterogeneous, the to and fro motion of the plume is irreversible. When a
part of the plume enters a low K zone, it is “trapped” and releases the contaminant
slowly. As a result, the plume spreads significantly. Compared to case 1, the plume’s
shape is lost and its position in the aquifer is drastically different. Since the aquifer is
anisotropic, plume spreading is predominantly in the horizontal plane, and very little in

the vertical plane.

Table 4.8.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Geometric mean conductivity, K&(m/day) 5 5
MKNMMMe 00 20
Correlation scale, xix (m) NA 10
Correlation scale, xiz (m) NA 3
Porosity, n 0.3 0.3

Size of model (m) 200 x 100 200 x 100
Grid 251x126 251x126
Cell size, Ax (m) 0.8 x 0.8 0.8 x 0.8
Time step, At (days) 10 10

 

Table 4.8.2 Temporally varying boundary conditions

Time da Left
0 73
30 71.5
60 70
90 70

120 69
150
180
210
240
270

 

115

 

Q00 70.5 71
| 330 72 73
| 360 73 75

 

 

 

 

 

The model parameters and inputs are explained below:

The mean conductivity values are 5 m/d for both cases.

The In K variance for case 1 is 0.0 and for case 2 it is 2.0.

The horizontal correlation scale (xix), and vertical correlation scale (xiz) is 10 m
and 3 m respectively for case 2.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that the temporal variability of head in the river

is resolved. Since the variability is 30 days, a time step of 10 days is used.

116

 

 

(‘onfimng Layer

In K var = 0.0

(‘onlinmg Layer

InK var = 2.0

Figure 4.8 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://wwwegr.msu.edu/igw/DL/SSH/TV/TVl .htm

117

4.9 PLUME TRANSPORT - COMPARISON BETWEEN HETEROGENEOUS

AQUIFERS WITH AND WITHOUT EQUILIBRIUM SORPT ION

4.9.1 Problem statement

This video compares the effect of temporally varying boundary conditions on plume
transport in heterogeneous aquifers with and without equilibrium sorption. The modeling
domain consists of a time—variable head boundary on both left and right extremes. Details

are provided in Table 4.9.

4.9.2 Key observations

The following observations can be made from the video:

0 Temporally varying boundary conditions cause to and fro motion of the plume in
the aquifers.

o In the absence of equilibrium sorption, the plume spreads significantly due to the
interaction between spatial heterogeneity and temporal variability. The spreading
of the plume is irreversible.

o In the presence of equilibrium sorption, the plume spreading is similar as above.

However, the extent of the plume is smaller because of retardation.

4.9.3 Additional observations and discussion

Case 1: The geometric mean conductivity of the aquifer is 5 m/day. The temporally
varying boundary conditions cause the head in the aquifer to fluctuate. As a result, the
flow of groundwater changes direction, resulting in to and fro motion of the plume. Since
the aquifer is heterogeneous, the to and fro motion of the plume is irreversible. When a

part of the plume enters a low K zone, it is “trapped” and releases the contaminant

118

slowly. As a result, the plume spreads significantly. Since the aquifer is anisotropic,
plume spreading is predominantly in the horizontal plane, and very little in the vertical

plane.

Case 2: The geometric mean conductivity of the aquifer is 5 m/day. The temporally
varying boundary conditions cause the head in the aquifer to fluctuate. As a result, the
flow of groundwater changes direction, resulting in to and fro motion of the plume. Since
the aquifer is heterogeneous, the to and fro motion of the plume is irreversible. Due to the
presence of retardation in the form of sorption, whena part of the plume enters a low K
zone, it is “trapped” and releases the contaminant extremely slowly. However, sorption is
uniform throughout the aquifer; therefore, the velocity in the aquifer is lowered
throughout. As a result, the plume spreading is lesser than in comparison to case 1. Since
the aquifer is anisotropic, plume spreading is predominantly in the horizontal plane, and

very little in the vertical plane. Compared to case 1, the plume’s size is slightly smaller.

Table 4.9.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Geometric mean conductivity, Kg (m/day) 5 5

In K variance 2.0 2.0
Correlation scale, it (m) 10 10
Correlation scale, xiZ (m) 3 3
Partitioning coefficient, Kd (kg/m3) 0-0 10-7
Porosity, n 0.3 0.3

Size of model (m) 200 x 100 200 x 100
Grid 251x126 251x126
Cell size, Ax (m) 0.8 x 0.8 0.8 x 0.8
Time step, At (days) 10 10

 

 

 

 

 

119

 

Table 4.9.2 Temporally varying boundary conditions

Time da Left
0 73
30 71.5
60 70
90 7O
69

 

The model parameters and inputs are explained below:

The average conductivity values are 5 m/d for both cases.

The In K variance is 2.0 and for both cases.

The horizontal correlation scale (xix) and vertical correlation scale (AZ), is 10 m
and 3 m respectively for both cases.

The partitioning coefficient is 0.0 for case 1 and 10'7 kg/m3 for case 2.

The cell size, Ax, is selected such that the correlation scale, 2., is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that the temporal variability of head in the river

is resolved. Since the variability is 30 days, a time step of 10 days is used.

120

 

 

('onftnrng Layer

('unlimng Layer

In Kd var = 0.0

Figure 4.9 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: mpzflwww.egr.msu.edu/igw/DL/SSH/TV/TVthm

121

 

4.10 PLUNIE TRANSPORT - COMPARISON BETWEEN AQUIFERS WITH

AND WITHOUT INTERACTING HETEROGENEITIES

4.10.1 Problem statement

This video compares the effect of temporally varying boundary conditions on plume
transport in aquifers with and without interacting heterogeneities. The modeling domain
consists of a time-variable head boundary on both left and right extremes. Details are

provided in Table 4.10.

4.10.2 Key observations

The following observations can be made from the video:
0 Temporally varying boundary conditions cause to and fro motion of the plume in
the aquifers.
0 Interaction between conductivity and partitioning coefficient heterogeneity in the
presence of temporal variability does not have a significant impact on plume

transport.

4.10.3 Additional observations and discussion

Case 1: The geometric mean conductivity of the aquifer is 5 m/day. The temporally
varying boundary conditions cause the head in the aquifer to fluctuate. As a result, the
flow of groundwater changes direction, resulting in to and fro motion of the plume. Since
the aquifer is heterogeneous, the to and fro motion of the plume is irreversible. Due to the
presence of retardation in the form of sorption, when a part of the plume enters a low K

zone, it is “trapped” and releases the contaminant extremely slowly. However, sorption is

122

uniform throughout the aquifer; therefore, the velocity in the aquifer is lowered

throughout.

Case 2: The geometric mean conductivity of the aquifer is 5 m/day. The temporally
varying boundary conditions cause the head in the aquifer to fluctuate. As a result, the
flow of groundwater changes direction, resulting in to and fro motion of the plume. Since
the aquifer is heterogeneous, the to and fro motion of the plume is irreversible. Due to the
presence of retardation in the form of sorption, when a part of the plume enters a low K
zone, it is “trapped” and releases the contaminant extremely slowly. Since partitioning
coefficient is negatively correlated with conductivity, the velocity in the low conductivity
zones is lowered drastically. However, in the high conductivity zones, velocity is lowered
by much less. As a result, velocity variability is effectively lesser in comparison to case

1; hence, the plume size is marginally larger in case 2.

Table 4.10.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Geometric mean conductivity, K g (m/day) 5 5

In K variance 2.0 2.0
Correlation scale, xix (m) 10 10
Correlation scale, xi2 (m) 3 3
Partitioning coefficient, Kd (kg/m3) 10.7 10-7
Coefficient of variation — Kai and K NA 2
Correlation - Kd and K NA Negative
Porosity, n 0.3 0.3

Size of model (m) 200 x 100 200 x 100
Grid 251x126 251x126
Cell size, Ax (m) 0.8 x 0.8 0.8 x 0.8
Time step, At (days) 10 10

 

123

Table 4.10.2 Temporally varying boundary conditions

Time da Left Ri

0 73 75

30 71.5 74

60 7O

9O 70 7O
69 68

67

68
70
71
73
75

 

The model parameters and inputs are explained below:
0 The average conductivity values are 5 m/d for both cases.
0 The In K variance is 2.0 and for both cases.
0 The horizontal correlation scale (2,) and vertical correlation scale (xiz), is 10 m
and 3 m respectively for both cases.

0 The mean partitioning coefficient is 10‘7 kg/m3 for both cases.

0 The interaction between K; and K is characterized by a coefficient of variation of
2.0 and a negative correlation.

0 The cell size, Ax, is selected such that the correlation scale, A, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that the temporal variability of head in the river

is resolved. Since the variability is 30 days, a time step of 10 days is used.

124

 

('onfinmg Law-r

('onlinmg Layer

In Kd var = 2.0

 

Figure 4.10 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/TV/TV3.htm

125

5 MACRODISPERSION

5.1 A MACRODISPERSION MODEL FOR PLUNIE MIGRATION IN WEAKLY
HETEROGENEOUS MEDIA (In K VARIANCE 0.5) WITH PORE-SCALE

DISPERSION

5.1.1 Problem statement

This video compares two different modeling techniques to predict macrodispersion of a
conservative solute plume in a weakly heterogeneous medium. The first technique
employs a detailed description of spatial variability of conductivity using statistical
parameters, and a pore-scale dispersion coefficient. The second technique uses Gelhar’s
asymptotic macrodispersion model (Gelhar and Axness, 1983a) that consists of a
homogeneous medium and a macrodispersion coefficient. This video demonstrates the
effectiveness of Gelhar’s model in predicting plume spreading. The modeling domain
consists of constant head boundaries on the left and right extremes, and no-flow
boundaries at the top and bottom; the initial size of the plume is significantly larger than

the correlation scale of heterogeneity. Details are provided in Table 5.1.

5.1.2 Key observations

The following observations can be made from the video:

0 The macrodispersion model approximately predicts plume location and amount of

spreading in both longitudinal and transverse directions.

0 The macrodispersion model is unable to predict the irregular shape of the plume,

and hence, the maximum concentration and degree of dilution.

126

Similar experiments with larger perturbations (higher In K variance) are shown in

subsequent videos.

5.1.3 Additional observations and discussion

Gelhar’s macrodispersion model attempts to predict the large-scale spreading that occurs
in a heterogeneous field. The underlying assumption for the macrodispersion model is
that the conductivity perturbations in the heterogeneous field are small. These
perturbations control plume spreading in the heterogeneous field. Longitudinal spreading
in the heterogeneous field is greater than transverse spreading. The macrodispersion
model predicts that the longitudinal spreading is directly proportional to the correlation
scale of heterogeneity and the In K variance, and transverse spreading is directly
proportional to the pore-scale dispersivity and the In K variance. Since the correlation
scale is at least two orders of magnitudes larger than pore-scale dispersivity, longitudinal
spreading is significantly larger than its transverse counterpart. Therefore, the
macrodispersion model is able to predict the overall spreading of the plume

approximately.

Since the macrodisperion model consists of a homogeneous medium, the plume obtained
is always regular. As a result, the irregular shape of the plume seen in the heterogeneous
field is not predicted. The irregular shape of the plume translates to irregular variations of
concentration, which are also not predicted. Consequently, the macrodispersion model is
unable to accurately predict the maximum concentration and the degree of dilution of the

plume.

127

5.1.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

macrodispersion:
59. au,c___i 01.1.39 , (5.1.1)
a: 3x,- Bxi dxj
a? 355 3 ac ( —)
—+ =—— D--—+ —u-C , 5.1.2
at 3x,- 3x,- [ U GXJ' l J ( )
where:
C — Concentration
u,- — Seepage velocity
D,-j — Dispersion coefficient.
It can be shown (Gelhar and Axness, 1983a) that:
—u,'C'zA,-,It'2§-. (5.1.3a)
3x,-

The coefficient A,, is called macrodispersivity and is given by (Gelhar, 1993):

2
an

 

All :UlitK/l’ A22 = (63L +07), (5.1.3b)

where:

01%, K — In K variance

xi — Correlation scale

A 1 1 — Longitudinal macrodispersivity

128

 

A22 —- Transverse macrodispersivity
aL — Longitudinal dispersivity

aT — Transverse dispersivity.

Equation (5.1.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of

longitudinal macrodispersion is many orders of magnitude larger than the transverse

macrodispersion. Substituting (5. 1.3a) into (5.1.2):

 

85 8175 a
— + = — D,
a: 8x,- Bx,-
a: a: 5

The pore-scale dispersivity, as], is typically one-hundredth of a meter, while the

macrodispersivity, Aij, is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.
Equation (5.1.5) simplifies the complex heterogeneous field, represented by (5.1.2). A
simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

Table 5.1 Model parameters and inputs

6C
7a7j+i5‘"5c§

3C

 

 

 

 

arameter Case 1 Case 2
Mean hydraulic conductivity, Kg 10 10
(m/day)
In K variance 0.5 0.0
Correlation scale, xi (m) 1 NA

 

 

 

 

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Porosity, n . 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0.01 NA
Transverse pore-scale dispersivity (m) 0.001 NA
Longitudinal macrodispersivity (m) NA 0.5
Transverse macrodispersivity (m) NA 0.0008125
Head difference (m) l 1

Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10

Grid 801 x 201 801 x 201
Cell size. Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 1

 

 

 

 

The model parameters and inputs are explained below:

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

The geometric mean, In K variance and correlation scale are constant for both

cases.

The covariance function for both cases is exponential.

Case 1 has longitudinal and transverse dispersivities of 0.01 m and 0.001 m
respectively, while they do not apply to case 2. The corresponding
macrodispersivities for case 2 are 0.5 m and 0.0008125 m.

The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

130

 

Figure 5.1 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://www.egr.msu.edu/igw/DL/SSH/SMM/SMM9.htm

I31

5.2 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
MODERATELY HETEROGENEOUS MEDIA (In K VARIANCE 1.0) WITH

PORE-SCALE DISPERSION

5.2.1 Problem statement

This video compares two different modeling techniques to predict macrodispersion of a
conservative solute plume in a moderately heterogeneous medium. The first technique
employs a detailed description of spatial variability of conductivity using statistical
parameters, and a pore-scale dispersion coefficient. The second technique uses Gelhar’s
asymptotic macrodispersion model (Gelhar and Axness, 1983a) that consists of a
homogeneous medium and a macrodispersion coefficient. This video demonstrates the
effectiveness of Gelhar’s model in predicting plume spreading. The modeling domain
consists of constant head boundaries on the left and right extremes, and no-flow
boundaries at the top and bottom; the initial size of the plume is significantly larger than

the correlation scale of heterogeneity. Details are provided in Table 5.2.

5.2.2 Key observations

The following observations can be made from the video:

0 The macrodispersion model approximately predicts plume location and amount of

spreading in both longitudinal and transverse directions.

0 The macrodispersion model is unable to predict the irregular shape of the plume,

and hence the maximum concentration and degree of dilution.

132

o In comparison to the previous experiment with weakly heterogeneous media, the
effectiveness of the macrodispersion model in predicting plume spreading

decreases.

Similar experiments with larger perturbations (higher In K variance) are shown in

subsequent videos.

5.2.3 Additional observations and discussion

Gelhar’s macrodispersion model attempts to predict the large-scale spreading that occurs
in a heterogeneous field. The underlying assumption for the macrodispersion model is
that the conductivity perturbations in the heterogeneous field are small. These
perturbations control plume spreading in the heterogeneous field. Longitudinal spreading
in the heterogeneous field is greater than transverse spreading. The macrodispersion
model predicts that the longitudinal spreading is directly proportional to the correlation
scale of heterogeneity and the In K variance, and transverse spreading is directly
proportional to the pore—scale dispersivity and the In K variance. Since the correlation
scale is at least two orders of magnitudes larger than pore-scale dispersivity, longitudinal
spreading is significantly larger than its transverse counterpart. Therefore, the
macrodispersion model is able to predict the overall spreading of the plume

approximately.

Since the macrodisperion model consists of a homogeneous medium, the plume obtained
is always regular. As a result, the irregular shape of the plume seen in the heterogeneous

field is not predicted. The irregular shape of the plume translates to irregular variations in

133

concentration, which are also not predicted. Consequently, the macrodispersion model is
unable to accurately predict the maximum concentration and the degree of dilution of the

plume.

In comparison to the previous video, where the In K variance was 0.5, this
macrodispersion model is not as accurate. Although the model captures the overall extent
of the plume, the plume is more irregular, which is not captured by the macrodispersion
model. Since a In K variance of 1.0 can still be considered only moderately

heterogeneous, the predictive ability of Gelhar’s model is not seriously hampered.

5.2.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

macrodispersion:
pg au.C:_3_ DUE. , (5.2.1)
at Bx,- ax, Bx]-
35 335 3 ac —.—-
__ =_ D~— _ -C , 5.2.2
at + ax,- dxi[ 'dej+( “' l] ( )
where:

C — Concentration
u,- — Seepage velocity
Di]- — Dispersion coefficient.

It can be shown (Gelhar and Axness, l983a) that:

134

 

-u,-'C' = 4,529. (5.2.3a)
3x-

 

The coefficient A,, is called macrodispersivity and is given by (Gelhar, 1993):

 

A, 1 = 012“ th; A22 = 0'; K (aL + a1 ), (5.2.3b)

where:

0?}, K — In K variance

xi — Correlation scale

A 1 1 — Longitudinal macrodispersivity

A22 — Transverse macrodispersivity

aL — Longitudinal dispersivity

ar — Transverse dispersivity.

Equation (5.2.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of
longitudinal macrodispersion is many orders of magnitude larger than the transverse

macrodispersion. Substituting (5.2.3a) into (5.2.2):

 

at? are a ac -aE

— =—— D~— .- — 5.2.4

at + 8x,- 8x, U ax], +[Auuaxi] ( )
35 35?: ac
— —= .- .- —. 4.2.5
at + a: (“'1 l A" ax,- ( )

135

The pore-scale dispersivity, cry, is typically one-hundredth of a meter, while the

macrodispersivity, Aij, is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.

Equation (5.2.5) simplifies the complex heterogeneous field, represented by (5.2.2). A
simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

Table 5.2 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IParameter Case 1 Case 2
IMean hydraulic conductivity, Kg (in/day) 10 10
Iln K variance 1.0 0.0
ICorrelation scale, xi (m) 1 NA
Porosity, n 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0.01 NA
Transverse pore-scale dispersivity (m) 0.001 NA
Longitudinal macrodispersivity (m) NA 1.0
Transverse macrodispersivity (m) NA 0.001625
Head difference (m) 1 1
Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10
Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 1

 

 

 

 

The model parameters and inputs are explained below:

 

o The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

136

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

The geometric mean, In K variance and correlation scale are constant for both
cases.

The covariance function for both cases is exponential.

Case 1 has longitudinal and transverse dispersivities of 0.01 m and 0.001 m
respectively, while they do not apply to case 2. The corresponding
macrodispersivities for case 2 are 1.0 m and 0.001625 m.

The cell size, Ax, is selected such that the correlation scale, 2, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

137

 

Figure 5.2 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://wwwengsu.edu/igw/DL/SSH/SMM/SMMlO.htm

138

5.3 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
STRONGLY HETEROGENEOUS MEDIA (In K VARIANCE 2.0) WITH PORE-

SCALE DISPERSION

5.3.1 Problem statement

This video compares two different modeling techniques to predict macrodispersion of a
conservative solute plume in a strongly heterogeneous medium. The first technique
employs a detailed description of spatial variability of conductivity using statistical
parameters, and a pore-scale dispersion coefficient. The second technique uses Gelhar’s
asymptotic macrodispersion model (Gelhar and Axness, 1983a) that consists of a
homogeneous medium and a macrodispersion coefficient. This video demonstrates the
effectiveness of Gelhar’s model in predicting plume spreading. The modeling domain
consists of constant head boundaries on the left and right extremes, and no-flow
boundaries at the top and bottom; the initial size of the plume is significantly larger than

the correlation scale of heterogeneity. Details are provided in Table 5.3.

5.3.2 Key observations

The following observations can be made from the video:
0 The macrodispersion model approximately predicts plume location and
underestimates the amount of spreading in both longitudinal and transverse

directions.

0 The macrodispersion model is unable to predict the irregular shape of the plume,

and hence the maximum concentration and degree of dilution.

139

 

0 In comparison to previous experiments with weakly and moderately
heterogeneous media, the effectiveness of the macrodispersion model in

predicting plume spreading decreases drastically.

5.3.3 Additional observations and discussion

Gelhar’s macrodispersion model attempts to predict the large-scale spreading that occurs
in a heterogeneous field. The underlying assumption for the macrodispersion model is
that the conductivity perturbations in the heterogeneous field are small. These
perturbations control plume spreading in the heterogeneous field. Longitudinal spreading
in the heterogeneous field is greater than transverse spreading. The macrodispersion
model predicts that the longitudinal spreading is directly proportional to the correlation
scale of heterogeneity and the In K variance, and transverse spreading is directly
proportional to the pore-scale dispersivity and the In K variance. Since the correlation
scale is at least two orders of magnitudes larger than pore-scale dispersivity, longitudinal
spreading is significantly larger than its transverse counterpart. Therefore, the
macrodispersion model is able to predict the overall spreading of the plume

approximately.

Since the macrodisperion model consists of a homogeneous medium, the plume obtained
is always regular. As a result, the irregular shape of the plume seen in the heterogeneous
field is not predicted. The irregular shape of the plume translates to irregular variations in
concentration, which are also not predicted. Consequently, the macrodispersion model is
unable to accurately predict the maximum concentration and the degree of dilution of the

plume.

140

 

In comparison to the previous video, where the In K variance was 1.0, this
macrodispersion model is not as accurate. Although the model captures the overall extent
of the plume, the plume is more irregular, which is not captured by the macrodispersion
model. The In K variance of 2.0 is considered strongly heterogeneous; the predictive

ability of Gelhar’s model is hampered.

5.3.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

 

macrodispersion:
5_€ BuiC =_3_ Dad—C , (5.3.1)
a: 6x,- 3x,- ax}-
36 355 3 ac ( —)
-——+ =—-—- D--—+ —u-C , 5.3.2
at 3x,- 8x,- [ U fix} I J ( )
where:
C — Concentration
u,- — Seepage velocity
Di} — Dispersion coefficient.
It can be shown (Gelhar and Axness, 1983a) that:
"lli'C.z AliHE. (5.3.33)
fix,-
The coefficient A,-,- is called macrodispersivity and is given by (Gelhar, 1993):
2 02
All = 01“ K2; 422 = '2 K (aL + a1)” (53.315)

141

 

where:
0'12" K - In K variance
xi — Correlation scale
A 1 1 — Longitudinal macrodispersivity
A22 — Transverse macrodispersivity
aL — Longitudinal dispersivity

ctr — Transverse dispersivity.

Equation (5.3.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of
longitudinal macrodispersion is many orders of magnitude larger than the transverse

macrodispersion. Substituting (5.3.3a) into (5.3.2):

 

ac are 3 ac ~a'c'

— =— o-.— -- — , 5.3.4

at + ax, ax, " ax]. {Aware} ( )
a? 355 aE
—— —= -- -. —. 5.3.5
at + a: (55" + A" ax,- ( )

The pore-scale dispersivity, at], is typically one-hundredth of a meter, while the

macrodispersivity, Aij’ is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.

142

 

 

[F'

 

Equation (5.3.5) simplifies the complex heterogeneous field, represented by (5.3.2). A
simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

Table 5.3 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Parameter Case 1 Case 2
Mean hydraulic conductivity, Kg (m/day) 10 10
In K variance 2.0 0.0
Correlation scale, xi (m) 1 NA
Porosity, n 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0.01 NA
Transverse pore-scale dispersivity (m) 0.001 NA
Longitudinal macrodispersivity (m) NA 2.0
Transverse macrodispersivity (m) NA 0.00325
Head difference (m) l 1
Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10
Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 l

 

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘Iow’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

o The geometric mean, In K variance and correlation scale are constant for both

C3363.

- The covariance function for both cases is exponential.

143

0 Case I has longitudinal and transverse dispersivities of 0.01 m and 0.001 m
respectively, while they do not apply to case 2. The corresponding
macrodispersivities for case 2 are 2.0 m and 0.00325 m.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

 

Figure 5.3 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://www.eg1.msu.edu/igw/DL/SSH/SMM/SMMlI.htm

5.4 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN VERY
STRONGLY HETEROGENEOUS MEDIA (In K VARIANCE 4.0) WITH PORE-

SCALE DISPERSION

5.4.1 Problem statement

This video compares two different modeling techniques to predict macrodispersion of a
conservative solute plume in a very strongly heterogeneous medium. The first technique
employs a detailed description of spatial variability of conductivity using statistical
parameters, and a pore-scale dispersion coefficient. The second technique uses Gelhar’s
asymptotic macrodispersion model (Gelhar and Axness, 1983a) that consists of a
homogeneous medium and a macrodispersion coefficient. This video demonstrates the
effectiveness of Gelhar’s model in predicting plume spreading. The modeling domain
consists of constant head boundaries on the left and right extremes, and no-flow
boundaries at the top and bottom; the initial size of the plume is significantly larger than

the correlation scale of heterogeneity. Details are provided in Table 5.4.

5.4.2 Key observations

The following observations can be made from the video:
0 The macrodispersion model approximately predicts plume location and
significantly underestimates the amount of spreading in both longitudinal and
transverse directions.

0 The macrodispersion model is unable to predict the irregular shape of the plume,

and hence the maximum concentration and degree of dilution.

145

o In comparison to previous experiments with weakly, moderately and strongly
heterogeneous media, the effectiveness of the macrodispersion model in

predicting plume spreading decreases drastically.

5.4.3 Additional observations and discussion

Gelhar’s macrodispersion model attempts to predict the large-scale spreading that occurs
in a heterogeneous field. The underlying assumption for the macrodispersion model is
that the conductivity perturbations in the heterogeneous field are small. These
perturbations control plume spreading in the heterogeneous field. Longitudinal spreading
in the heterogeneous field is greater than transverse spreading. The macrodispersion
model predicts that the longitudinal spreading is directly proportional to the correlation
scale of heterogeneity and the In K variance, and transverse spreading is directly
proportional to the pore-scale dispersivity and the In K variance. Since the correlation
scale is at least two orders of magnitudes larger than pore-scale dispersivity, longitudinal
spreading is significantly larger than its transverse counterpart. Therefore, the
macrodispersion model is able to predict the overall spreading of the plume

approximately.

Since the macrodisperion model consists of a homogeneous medium, the plume obtained
is always regular. As a result, the irregular shape of the plume seen in the heterogeneous
field is not predicted. The irregular shape of the plume translates to irregular variations in
concentration, which are also not predicted. Consequently, the macrodispersion model is
unable to accurately predict the maximum concentration and the degree of dilution of the

plume.

146

 

In comparison to the previous video, where the In K variance was 2.0, this
macrodispersion model is not as accurate. Although the model captures the overall extent
of the plume, the plume is more irregular, which is not captured by the macrodispersion
model. The In K variance of 4.0 is considered very strongly heterogeneous; the predictive

ability of Gelhar’s model is seriously hampered.

5.4.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

macrodispersion:
§p+_au,-c =1 Dij 5_C , (5.4.1)
at dxi Bx,- ax}-
35 325 3 ac ( —)
—+ =— D--—+ -u-C , 5.4.2
at 0x,- dxi[ 5 dxj ' J ( )
where:
C — Concentration
u,- — Seepage velocity
Di]- — Dispersion coefficient.
It can be shown (Gelhar and Axness, 1983a) that:
-—u,-'C' = 4,1729. (5.4.3a)
3x-

I

The coefficient A,, is called macrodispersivity and is given by (Gelhar, I993):

147

 

All = O'lan/i; A22 2 2K (0L +aT ),, (5H43b)

where:
0'13, K — In K variance
2 — Correlation scale
A11 — Longitudinal macrodispersivity
A22 — Transverse macrodispersivity
aL - Longitudinal dispersivity

aT - Transverse dispersivity.

Equation (5.4.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of
longitudinal macrodispersion is many orders of magnitude larger than the transverse
macrodispersion. Substituting (5.4.3a) into (5.4.2):

at? are 3 ac —a'E
— =— D-— -- — , 5.4.4
at + 8x,- 8x,- 5 dxj +[Auudx-J ( )

 

at" 32.? ac
— —— -- .- —. 5.4.
a: + at (“'1 + A" ax, ( 5)

The pore-scale dispersivity, aij, is typically one-hundredth of a meter, while the

macrodispersivity, Aij, is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.

148

Equation (5.4.5) simplifies the complex heterogeneous field, represented by (5.4.2). A
simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

Table 5.4 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IParameter Case 1 Case 2
Mean hydraulic conductivity, K g (m/day) 10 10

ln K variance 4.0 0.0
Correlation scale, 2 (m) 1 NA
Porosity, n 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0.01 NA
Transverse pore-scale dispersivity (m) 0.001 NA
Longitudinal macrodispersivity (m) NA 4.0
Transverse macrodispersivity (m) NA 0.0065
Head difference (m) 1 1

Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10

Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) I 1

 

 

 

 

The model parameters and inputs are explained below:

 

- The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

0 The geometric mean, In K variance and correlation scale are constant for both

C ases.

o The covariance function for both cases is exponential.

149

0 Case I has longitudinal and transverse dispersivities of 0.01 m and 0.001 m
respectively, while they do not apply to case 2. The corresponding
macrodispersivities for case 2 are 4.0 m and 0.0065 m.

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

 

Figure 5.4 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: httD://www.egr.msu.edu/igw/DL/SSH/SMM/SMM12.htm

150

5.5 A MACRODISPERSION MODEL FOR PLUME MIGRATION IN
STRONGLY HETEROGENEOUS MEDIA (In K VARIANCE 2.0) WITHOUT

PORE-SCALE DISPERSION

5.5.1 Problem statement

This video compares two different modeling techniques to predict macrodispersion of a
conservative solute plume in a strongly heterogeneous medium without pore—scale
dispersion. The first technique employs a detailed description of spatial variability of
conductivity using statistical parameters. The second technique uses Gelhar’s asymptotic
macrodispersion model (Gelhar and Axness, 1983a) that consists of a homogeneous
medium and a macrodispersion coefficient. This video demonstrates the effectiveness of
Gelhar’s model in predicting plume spreading. The modeling domain consists of constant
head boundaries on the left and right extremes, and no-flow boundaries at the top and
bottom; the initial size of the plume is significantly larger than the correlation scale of

heterogeneity. Details are provided in Table 5.5.

5.5.2 Key observations

The following observations can be made from the video:
0 The macrodispersion model approximately predicts plume location but
underestimates the amount of spreading in the longitudinal direction.
0 The macrodispersion model predicts that there will be no transverse spreading,
contrary to the heterogeneous model.

0 The macrodispersion model is unable to predict the irregular shape of the plume,

and hence the maximum concentration and degree of dilution.

151

 

0 In comparison to the macrodispersion model in 4.3, the effectiveness of this

macrodispersion model in predicting transverse spreading decreases.

5.5.3 Additional observations and discussion

Gelhar’s macrodispersion model attempts to predict the large-scale spreading that occurs
in a heterogeneous field. The underlying assumption for the macrodispersion model is
that the conductivity perturbations in the heterogeneous field are small. These
perturbations control plume spreading in the heterogeneous field. Longitudinal spreading
in the heterogeneous field is greater than transverse spreading. The macrodispersion
model predicts that the longitudinal spreading is directly proportional to the correlation
scale of heterogeneity and the In K variance, and transverse spreading is directly
proportional to the pore-scale dispersivity and the In K variance. Since there is no pore-
scale dispersivity, the macrodispersion model predicts that there will be no transverse
spreading. Since the transverse spreading in the heterogeneous model is not significant,
compared to longitudinal spreading, the macrodispersion model is able to predict the

overall spreading of the plume approximately.

Since the macrodisperion model consists of a homogeneous medium, the plume obtained
is always regular. As a result, the irregular shape of the plume seen in the heterogeneous
field is not predicted. The irregular shape of the plume translates to irregular variations in
concentration, which are also not predicted. Consequently, the macrodispersion model is
unable to accurately predict the maximum concentration and the degree of dilution of the

plume.

152

 

Although the model captures the overall extent of the plume, the peaks and valleys are
sharp, which is not accurately captured by the macrodispersion model. In the absence of
pore—scale dispersion, Gelhar’s model loses the ability to predict transverse spreading.

Despite this limitation, the overall predictive ability is not seriously hampered.

5.5.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

 

macrodispersion:
59. BuiC =_a_ 01.1.29 , (55.1)
a: 8x,- ax, ax].
32‘ 3276 3 ac ( —)
—+ =— D--—+ —u-C , 5.5.2
3t ax,- axi[ 5 3x} I J ( )
where:
C — Concentration
u,- - Seepage velocity
Di,- — Dispersion coefficient.
It can be shown (Gelhar and Axness, 1983a) that:
-u,-'C'= AiiH'a—C. (5..53a)
3x,-
The coefficient A,-,- is called macrodispersivity and is given by (Gelhar, 1993):
2 ‘72
All = 0'1“ K/i; A22 2 ng (UL + a7 ), (553D)

where:

153

0'12" K — In K variance

2 — Correlation scale

A 1 1 — Longitudinal macrodispersivity
A 22 — Transverse macrodispersivity
aL — Longitudinal dispersivity

aT — Transverse dispersivity.

Equation (5.5.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of
longitudinal macrodispersion is many orders of magnitude larger than the transverse

macrodispersion. Substituting (45.5.3a) into (5.5.2):

 

ac are a ea —32:'
—— =—— D~——-—— -. — , 5.5.4
at + 0x,- 3x,- 5 ax], +[Auu a») ( )

at" 355 36

— —= -- .. —. 5.5.5
at + at (55’ + A” ax, ( )

The pore—scale dispersivity, (1,3, is typically one-hundredth of a meter, while the

macrodispersivity, Aij, is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.
Equation (5.5.5) simplifies the complex heterogeneous field, represented by (5.5.2). A
simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

154

Table 5.5 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

arameter Case 1 Case 2
Mean hydraulic conductivity, K g (m/day) 10 10
In K variance 2.0 0.0
Correlation scale, 2 (m) 1 NA
Porosity, n 0.3 0.3
Longitudinal pore-scale dispersivity (m) 0 NA
Transverse pore-scale dispersivity (m) 0 NA
Longitudinal macrodispersivity (m) NA 2.0
Transverse macrodispersivity (m) NA 0.0
Head difference (m) 1 1
Size of model (m) 200 x 50 200 x 50
Covariance function Exponential Exponential
Approximate initial size of plume (m) 10 10
Grid 801 x 201 801 x 201
Cell size, Ax (m) 0.25 x 0.25 0.25 x 0.25
Time step, At (days) 1 l

 

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal

probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

0 The geometric mean, In K variance and correlation scale are constant for both

cases.
0 The covariance function for both cases is exponential.

- Cases 1 and 2 do not have longitudinal and transverse pore-scale dispersivities.

The macrodispersivities for case 2 are 2.0 m and 0.0 m.

155

0 The cell size, Ax, is selected such that the correlation scale, xi, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.
0 The time step, AI, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

 

Figure 5.5 Snapshot of the visualization

For a detailed visualization, access the following address:

URL: http://wwwegrmsu.edu/igw/DL/SSH/SMM/SMMl3.htm

156

6 EFFECTS OF MULTI-SCALE HETEROGENEITY

6.1 EFFECTS OF MULTIPLE SCALES OF HETEROGENEITY

6.1.1 Problem statement

This video demonstrates the effects of multiple scales of heterogeneity, in the absence of
pore—scale dispersion, on the spreading of a conservative solute plume. The modeling
domain consists of constant head boundaries on the left and right extremes, and no-flow

boundaries at the top and bottom. Details are provided in Table 6.1.

6.1.2 Key observations

The following observations can be made from the video:

0 Multiple scales of heterogeneity have a drastic impact on plume spreading.

0 In case 1, the scale of heterogeneity is much larger than the initial size of the
plume. As a result, the plume does not encounter all the heterogeneity, and hence
does not spread greatly.

- In case 2, two large scales of heterogeneity are modeled, the smallest scale of
heterogeneity is larger than the initial size of the plume. Plume spreading is
enhanced in comparison to case 1, though not significantly, because the plume
does not encounter all of the heterogeneity.

0 In case 3, three scales of heterogeneity are modeled, plume spreading is
considerably enhanced. Since the smallest scale of heterogeneity is smaller than
the initial size of the plume, the plume encounters all the heterogeneity and hence

the spreading is increased.

157

0 Mean plume displacement, for this particular realization, increased as more scales

of heterogeneity were modeled.

6.1.3 Additional observations and discussion

Case 1: The correlation scale of heterogeneity is 500 m, which is much larger than the
initial size of the plume; therefore the plume does not spread greatly. In such cases, the In

K realization is an important factor that controls plume migration.

Case 2: The correlation scales of heterogeneity in the first and second scales are 500 m
and 100 m respectively. The initial size of the plume is of the order of 100 m. The plume
does not exhibit detailed fingers and tails. However, in comparison to case 1, spreading is
increased. The mean plume displacement is greater than in case 1, because the plume

encounters more preferential paths (high velocity channels).

Case 3: The correlation scales of heterogeneity in the first, second and third scales are
500 m, 100 m, and 10 m respectively. The third scale is smaller than the initial size of the
plume; most of the heterogeneity is encountered, and thus plume spreading is greatly
enhanced. The mean displacement is greater than in cases 1 and 2, because the plume
encounters more preferential paths (high velocity channels).

A comparison of the three cases shows that increasing the number of scales of
heterogeneity increases plume spreading. Eliminating smaller scales of heterogeneity
from the model underestimates plume spreading. Incorporating smaller scales requires

hi gher-resolution data, which is costly, and perhaps infeasible. When only larger scales

are modeled, the detailed structure of the aquifer represented by the smaller scales of

158

heterogeneity is averaged. In this process of averaging, considerable amount of detail is
lost. These details are critical, and make a huge difference in the prediction of plume

spreading.

Table 6.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2 Case 3
Geometric mean hydraulic

, , 10 10 10
conductrvrty, Kg (m/day)
In K variance (for all scales) 1.0 1.0 1.0
Scale I correlation scale, 2 (m) 500 500 500
Scale 2 correlation scale, xi (m) - 100 100
Scale 3 correlation scale, 2 (m) - - 10
Porosity, n 0.3 0.3 0.3
Pore-scale dispersivity 0 0 0
Head difference (m) l 1 1
Size of model (m) 2000 x 500 2000 x 500 2000 x 500
Covariance function Gaussian Gaussian Gaussian
Approximate initial plume size (m) 100 100 100
Grid 801 x 201 801 x 201 801 x 201
Cell size, Ax (m) 2.5 x 2.5 2.5 x 2.5 2.5 x 2.5
Time step, At (days) 100 100 100

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

0 The geometric mean and In K variance are constant for all cases.

0 The first scale of heterogeneity is common to all cases, and has a correlation scale

of 500 m.

159

0 The second scale of heterogeneity is applicable only to cases 2 and 3, and has a
correlation scale of 100 m.

0 The third scale of heterogeneity is applicable only to cases 2 and 3, and has a
correlation scale of 10 m.

0 The covariance firnction for all cases is Gaussian.

0 The cell size, Ax, is selected such that the correlation scale, 2, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

'

Three Scales

 

Figure 6.1 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egg.msu.edu/igw/DL/SSH/SDT/SDTl .htm

I60

6.2 MACRODISPERSION IN THE PRESENCE OF MULTIPLE SCALES OF

HETEROGENEITY

6.2.1 Problem statement

This video compares the spreading of a conservative solute plume in In K fields where
two different methods of representing the small-scale heterogeneity have been employed;
the large scale heterogeneities in both fields are identical. In case 1, the small-scale
heterogeneity is represented using equivalent macrodispersivities; in case 2, the small-
scale heterogeneity is modeled using a random field with pore-scale dispersivities. The
modeling domain consists of constant head boundaries on the left and right extremes, and

no-flow boundaries at the top and bottom. Details are provided in Table 6.2.

6.2.2 Key observations

The following observations can be made from the video:

0 The position and extent of the plume predicted in case 1 (macrodispersion) is
approximately the same as that observed in case 2 (random field with pore-scale
dispersion).

- The macrodispersion model smoothes the detailed interior structure of the plume

observed in case 2.

6.2.3 Additional observations and discussion
Case 1: The small scale of heterogeneity is represented by equivalent macrodispersivities

(A 1 1 = 10 m, A22 2 0.001625 m) using Gelhar’s asymptotic macrodispersion model that is

discussed in detail in the next section. The correlation scales (2) of heterogeneity of the

161

large scales are 500 m and 100 m respectively. The initial size of the plume is of the

order of 100 m.

Case 2: The small scale of heterogeneity is represented by a random field model (In K
variance = 1.0, 2 = 10 m, aL = 0.01 m, a1: 0.001 m). The correlation scales (2) of

heterogeneity of the large scales are 500 m and 100 m respectively. The third scale of

heterogeneity is smaller than the initial size of the plume.

The macrodispersion model in case 1 is able to predict spreading of the plume in case 2.
In the absence of hi gh-resolution data, a “coarse” heterogeneous field that incorporates
macrodispersion can be used to predict the position and extent of plume spreading
approximately. However, the macrodispersion model only predicts the general shape of

the plume, and not the peaks and valleys of the plume.

Lack of data limits the ability to model small scales of heterogeneity. In such cases, an
effective macrodispersion model with a large-scale trend can be used to obtain better
results. From Section 4.1 it is obvious that eliminating smaller scales of heterogeneity
from the model underestimates plume spreading. Incorporating smaller scales requires
higher-resolution data, which is costly, and perhaps infeasible. When only larger scales
are modeled, the detailed structure of the aquifer represented by the smaller scales of
heterogeneity is averaged. In this process of averaging, considerable amount of detail is

lost. These details are critical, and make a huge difference in the prediction of plume

I62

spreading. However, it is possible to “retrieve” some of the detail using models such as

Gelhar’s macrodispersion model.

6.2.4 Mathematical interpretation

The following equations show the relationship between pore-scale dispersion and

 

 

macrodispersion:
BC du-C a 3C
__ I =__ D--—— , 6.2.1
at 8x,- ax, " ax]. ( )
a‘c' du—C 3 ac —.—-
_ =_ D--— _ -C , 6.2.2
at + 3x, axi[ 53x}- +( ul )] ( )
where:

C — Concentration
u,- — Seepage velocity

Di]- — Dispersion coefficient.

It can be shown (Gelhar and Axness, 1983a) that:

—u_,-’C_' z 2,1229. (6.2.3a)
ax,-

The coefficient A,-,- is called macrodispersivity and is given by (Gelhar, 1993):

2
0'
A” = 012“ K2; 422 = 'g" (aL +617), (6.2.3b)

 

where:

0'12“ K - In K variance

163

2 — Correlation scale

A 1 1 — Longitudinal macrodispersivity
A22 — Transverse macrodispersivity
a1, — Longitudinal dispersivity

ar — Transverse dispersivity.

Equation (6.2.3b) shows that the longitudinal macrodispersion is unaffected by pore-scale
dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of
longitudinal macrodispersion is many orders of magnitude larger than the transverse

macrodispersion. Substituting (6.2.3a) into (6.2.2):

 

35 are a are -aE

—— =—— D--— -. — 6.2.4

at + 3x,- 3x,- ” ij +[Auudxii ( )
35 are 35
_ —: e' " —. 6.205
at + a: (a5 + A" ax, ( )

The pore-scale dispersivity, as}, is typically one-hundredth of a meter, while the

macrodispersivity, Aij, is approximately one meter. Therefore, macrodispersion is the

dominant process and controls the total spreading of the plume.

Equation (6.2.5) simplifies the complex heterogeneous field, represented by (6.2.2). A

simple mean macrodispersion model is obtained by replacing the dispersion coefficient

with a macrodispersivity value. This model can predict the mean spreading of the plume.

164

Table 6.2 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Case 1 Case 2
Mean hydraulic conductivity, K g (m/day) 10 10

In K variance (for all scales) 1.0 1.0

Scale 1 correlation scale, xi (m) 500 500
Scale 2 correlation scale, xi (m) 100 100
Scale 3 correlation scale, 2 (m) - 10
Porosii n 0.3 0.3
Longitudinal pore-scale dispersivity (m) NA 0.01
Transverse pore-scale dispersivity (m) NA 0.001
Longitudinal macrodispersivity (m) 10 NA
Transverse macrodispersivity (m) 0.001625 NA

Head difference (m) 1 1

Size of model (m) 2000 x 500 2000 x 500
Covariance function Gaussian Gaussian
Approximate initial plume size (m) 100 100

Grid 801 x 201 801 x 201
Cell size, Ax (m) 2.5 x 2.5 2.5 x 2.5
Time step, At (dpys) 100 100

 

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘Iow’ K zone or a ‘high’ K zone.
0 The heterogeneous conductivity field is uniquely characterized by the following

set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance

and covariance.
o The geometric mean and In K variance are constant for both cases.

0 The first and second scales of heterogeneity are common to both cases, and have a

correlation scale of 500 m and 100 m respectively.

0 The third scale of heterogeneity is applicable only to cases 1 and 2, and has a

correlation scale of 10 m.

o The covariance function for all cases is Gaussian.

I65

o The cell size, Ax, is selected such that the correlation scale, 2, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.
0 The time step, AI, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

Three Scales

 

Figure 6.2 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: http://www.egg.msu.edu/igv_v/DL/SSH/SDT/SDT2.htm

I66

7 MONTE CARLO SIMULATIONS

 

7.1 EFFECTS OF HYDRAULIC CONDUCTIVITY HETEROGENEITY —

DIFFERENT REALIZATIONS

7.1.1 Problem statement

This video demonstrates 4 different realizations of hydraulic conductivity heterogeneity.
The spreading of a conservative solute plume through these realizations is shown. The
modeling domain consists of constant head boundaries on the left and right extremes, and

no-flow boundaries at the top and bottom. Details are provided in Table 7.1.

 

7.1.2 Key observations

The following observations can be made from the video:
0 Despite the plume being larger than the scale of heterogeneity, different
realizations cause significantly different plume spreading.

0 The different plumes vary in all aspects, such as size, shape, mean displacement,

and overall extent of spreading.

7.1.3 Additional observations and discussion

Even though all the realizations have the same set of statistical parameters, the eventual
patterns of spreading exhibited by the plumes are significantly different. The plumes
differ not only in their eventual extents of spreading, but also in their shape, size, and
mean displacement. There is no way to compare the patterns of spreading of the different
realizations using a generalization. However, if a large number of such realizations are

considered, the mean of all those realizations will be a Gaussian distribution.

I67

Uniquely characterizing the plume’s behavior from a single realization is difficult. This
uncertainty in plume behavior is a result of uncertainty in aquifer properties. Therefore,
stochastic methods such as Monte Carlo simulations need to be used in order to perform a

systematic probabilistic analysis that can be used for risk-estimation.

Table 7.1 Model parameters and inputs

Realizations

mean conducti , K (m/da ) 10
In K variance 0
' 2 m
.n 0.3
Local d' ' ' 0

Head difference m 1
Covariance function
initial size of 10

Grid
Cell size,Ax m
Time At da

 

The model parameters and inputs are explained below:
0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.
0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

o The geometric mean, In K variance, correlation scale (2), and covariance function

are constant for all cases.
0 The cell size, Ax, is selected such that the correlation scale, 2, is resolved.

Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

168

 

0 The time step, A1, is selected such that a particle ofthe plume travels a distance

that is less than xi in one time step.

 

‘
' .r sd!141.-41-a- A

’. wl.’
I c.

‘,‘.o ‘ "'o
1 '44“)

Figure 7.1 Snapshot of the visualization

 

For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/IMCFTS/IMCFTSI.htm

 

I69

7.2 EFFECTS OF HYDRAULIC CONDUCTIVITY HETEROGENEITY -

DIFFERENT REALIZATIONS

7.2.1 Problem statement

This video demonstrates 4 different realizations of hydraulic conductivity heterogeneity.
The spreading of a conservative solute plume through these realizations is shown. The
modeling domain consists of constant head boundaries on the left and right extremes, and

no-flow boundaries at the top and bottom. Details are provided in Table 7.2.

7.2.2 Key observations

The following observations can be made from the video:
0 The plume is of the same order as the scale of heterogeneity, different realizations
cause drastically different plume spreading.
0 The different plumes vary in all aspects, such as size, shape, mean displacement,

and overall extent of spreading.

7.2.3 Additional observations and discussion

Even though all the realizations have the same set of statistical parameters, the eventual
patterns of spreading exhibited by the plumes are drastically different. The plumes differ
not only in their eventual extents of spreading, but also in their shape, size, and mean
displacement. There is no way to compare the patterns of spreading of the different
realizations using a generalization. However, if a large number of such realizations are
considered, the mean of all those realizations will be a Gaussian distribution. Comparing
the plume behavior with Section 7.1, it is obvious that the plumes become increasingly

different with increasing correlation scale.

170

Uniquely characterizing the plume’s behavior from a single realization is difficult. This

uncertainty in plume behavior is a result of uncertainty in aquifer properties. Therefore,

stochastic methods such as Monte Carlo simulations need to be used in order to perform a

systematic probabilistic analysis that can be used for risk-estimation.

Table 7.2 Model parameters and inputs

 

Realizations

Geometric mean h conducti ' , K (m/da ) 10

In K variance 0
' 2 m 10

- ,n 0.3
Local ' ° ' 0

Head difference m I
Covariance function
initial size of 10

Grid
Cell size, Ax at
Time At

The model parameters and inputs are explained below:

The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

The geometric mean, In K variance, correlation scale (2), and covariance function
are constant for all cases.

The cell size, Ax, is selected such that the correlation scale, 2, is resolved.

Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

171

0 The time step, AI, is selected such that a particle of the plume travels a distance

that is less than xi in one time step.

 

Re ailiza-fion .1 _\

sph’r or
ya

7
_____________ 47 x,t." '
as
. :7

 

Figure 7. 2 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: htt ://www.e .msu.edu/i w/DL/SSH/IMCFTS/IMCFTSZ.htm

 

 

7.3 MONTE CARLO SIMULATION - MEAN, VARIANCE AND COVARIANCE

7.3.1 Problem statement

This video illustrates the statistical results (mean concentration, concentration variance,
and the covariance structure of concentration at a point) obtained from a set of 500
realizations of a Monte Carlo simulation. The Monte Carlo simulation is generated for a
random K field. The modeling domain consists of constant head boundaries on the left
and right extremes, and no-flow boundaries at the top and bottom. Details are provided in

Table 7.3.

7.3.2 Key observations

The following observations can be made from the video:

0 Each realization is very different from other realizations; yet, the mean
concentration after 500 realizations is predominantly uniform.

- The concentration variance obtained over 500 realizations is non-stationary,
exhibits the “double peak” structure (Graham and McLaughlin, 1989).
Concentration variance increases with concentration gradient.

0 The covariance structures are strongly “non-stationary,” since they are very

different from each other at different points.

7.3.3 Additional observations and discussion

Mean concentration: Although each realization is dramatically different from other
realizations, the statistical best estimate of concentration is given by the mean
concentration obtained from all realizations. During the first few realizations, the mean

concentration is strongly controlled by the concentration obtained from the realizations.

173

 

After ~50 realizations, the mean concentration approaches a uniform distribution.
Additional realizations cause little change to the mean concentration. At the end of 500

realizations, the mean concentration is predominantly uniform.

Concentration variance: The concentration variance during the initial realizations changes
drastically with each realization, since the concentration for that realization has a
significant impact on the variance. With increasing number of realizations, the variance
begins to exhibit the “double-peak” structure (Graham and McLaughlin, 1989).
Downstream of the concentration source, the chance of obtaining a mean concentration
away from the centerline of the source is low compared to along the centerline of the
source. Therefore, the variance along the centerline is low, and higher away from the

centerline, resulting in the double peak.

Covariance structure: Typically, a single covariance structure is used for a site. Such a
procedure can lead to inaccuracies because the covariance is strongly non-stationary; i.e.
it is dependent on the location. This can be seen from the covariance structure which is

shown for 6 different points: (50,15), (50,25), (50,35), (70,15), (70,25), and (70,35).

Table 7.3.1 Model parameters and inputs

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter All realizations
Geometric mean hydraulic conductivity, K g (m/day) 10

In K variance 2.0
Correlation scale, 2 (m) 4

Porosity, n 0.3
Longitudinal pore-scale dispersivity (m) 0.01
Transverse pore-scale dispersivity (m) 0.001

Head difference (m) 1

Size of model (m) 150 x 50
Covariance function Gaussian

 

I74

 

initial size in 10
Grid 301 x 101

Cell size, Ax m 0.5 x 0.5
Time At da 2

 

The model parameters and inputs are explained below:

0 The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (K g) of K, correlation scale, variance
and covariance.

o The geometric mean and In K variance are constant for all realizations.

0 The correlation scale of heterogeneity is common for all realizations.

0 The covariance function for all cases is Gaussian.

o The cell size, Ax, is selected such that the correlation scale, 2, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than 2 in one time step.

Table 7.3.2 Maximum and minimum values for plotting

 

 

 

 

Uarameter Minimum Maximum I
I Concentration covariance (ppmz) ~350 1400 I
LConcentration variance (ppmz) 0 2000 I
I Transmissivity (ml/day) 1.5 5000 I

 

 

 

175

 

 

Figure 7.3 Snapshot of the visualization
For a detailed visualization, access the following address:

URL: Mptflwwwegnmsuedu/igw/DL/SSH/IMCFTS/IMCFTS3.htm

I76

7.4 MONTE CARLO SIMULATION - PROBABILITY DISTRIBUTIONS AND

BREAKTHROUGH CURVES

7.4.1 Problem statement

This video illustrates the statistical results (probability distributions of In K, hydraulic
head and concentration at specified locations, probability distribution of the seepage flux
across a cross-section of the aquifer, breakthrough curves of solute flux across a cross-
section and concentration at a point) obtained from a set of 500 realizations of a Monte
Carlo simulation. The Monte Carlo simulation is generated for a random K field. The
modeling domain consists of constant head boundaries on the left and right extremes, and

no-flow boundaries at the top and bottom. Details are provided in Table 7.4.

7 .4.2 Key observations

The following observations can be made from the video:

0 Although the In K probability distribution function (PDF) over the modeling
domain is Gaussian, the hydraulic head PDF at MW 6 is approximately Gaussian,
though it is skewed.

0 The concentration PDFs at MW 1, MW 2, MW 3, and MW 5 are very different
from each other. All of them are skewed in different directions to varying degrees.
While the PDF at MW 1, MW 3, and MW 5 are skewed towards low
concentrations, MW 2 is skewed towards the higher concentration. This drastic
difference means that MW 2 has a greater probability of a high concentration
compared to MW 1, MW 3, and MW 5. It is more likely that the concentrations at

MW 1 and MW 3 are low compared to MW 5.

177

 

0 The seepage flux PDF at cross-section AA' obtained over 500 realizations is
predominantly Gaussian.

o The solute flux breakthrough curve at cross-section AA’ obtained over 500
realizations indicates that the average solute flux across AA’ is ~ 2000 g/day, with
a standard deviation of ~ 1000 g/day.

0 The concentration breakthrough curve at Monitoring Well (MW 2) obtained over
500 realizations indicates that the average concentration at MW 2 is ~ 60 ppm,

with a standard deviation of ~ 40 ppm.

7.4.3 Additional observations and discussion

Hydraulic head PDF: Since hydraulic head is almost linearly related to In K, hydraulic

head PDF is also Gaussian.

Concentration PDFs: Concentration PDFs are strongly skewed, in either direction (low or
high concentrations). Concentration is neither Gaussian nor lognorrnal. A normal
distribution is characterized by a mean and a variance. Therefore, a mean and variance
cannot be used to characterize uncertainty of concentration. In addition, concentration

PDF is non-stationary, in the sense that it varies with respect to location.

Seepage flux PDF: Seepage flux PDF is almost Gaussian, though it appears to have a

“flat peak.”

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Solute flux breakthrough curve: The solute flux breakthrough curve obtained from 500
realizations, predicts a mean solute flux of ~ 2000 g/day with a standard deviation of ~

1000 g/day.

Concentration breakthrough curve: The concentration breakthrough curve obtained from
500 realizations, predicts a mean concentration ~ 60 ppm with a standard deviation of ~

40 ppm.

Table 7.4 Model parameters and inputs

 

Parameter All realizations

 

Geometric mean hydraulic conductivity, Kg (m/day) 10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In K variance 2.0
Correlation scale, 1 (m) 4
Porosity, n 0.3
Longitudinal pore-scale dispersivity (m) 0.01
Transverse pore-scale dispersivity (m) 0.001
Head difference (m) 1

Size of model (m) 150 x 50
Covariance function Gaussian
Approximate initial plume size (m) 10

Grid 301 x 101
Cell size, Ax (m) 0.5 x 0.5
Time step, At (days) 2

 

The model parameters and inputs are explained below:

 

o The In K field is a normally distributed random field. Therefore, there is an equal
probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone.

0 The heterogeneous conductivity field is uniquely characterized by the following
set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance
and covariance.

0 The geometric mean and In K variance are constant for all realizations.

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o The correlation scale of heterogeneity is common for all realizations.

0 The covariance function for all cases is Gaussian.

0 The cell size, Ax, is selected such that the correlation scale, ,1, is resolved.
Typically, Ax is at least 3 to 4 times smaller than the correlation scale.

0 The time step, At, is selected such that a particle of the plume travels a distance

that is less than A in one time step.

Probability at MW 1 3 Probability at MW 2 » probabililv d1 MW i

"‘1-f(

3‘ .
a,
I

 

For a detailed visualization, access the following address:

URL: http://www.egg.msu.edu/igw/DL/SSH/IMCFTS/IMCFTS4.htm

180

8 CONCLUSIONS

1.

The subsurface environment is inherently heterogeneous, exhibiting dramatic
variations over surprisingly small distances. Since it is practically impossible to
measure these detailed point-to-point variations, heterogeneity translates into
uncertainty. A random field representation characterized by statistics such as
mean, variance, and covariance derived from data provides a systematic way to

describe the heterogeneous aquifer properties.

Small-scale heterogeneity in conductivity causes large scale spreading of plume
resulting in numerous fingers and tails. Even though In K distribution in the field
is Gaussian, the variation of plume concentration with time at any point is non-

Gaussian.

When a heterogeneous conductivity field is coupled with a heterogeneous
porosity field, the effect on plume spreading is not drastic. Since porosity varies
over a small range of values compared to conductivity, the impact on velocity
variability is minimal. When n and K are positively/negatively correlated, the

plume is slightly smaller/larger compared to when n and K are not correlated.

When a heterogeneous conductivity field is coupled with partitioning coefficient,
the effect on plume spreading is drastic. Since partitioning coefficient variability

is comparable to that of conductivity, the impact on velocity variability is

significant. When K, and K are positively/negatively correlated, the plume is

significantly smaller/larger compared to when Kd and K are not correlated.

181

 

5.

6.

Depending on the scale of the heterogeneity with respect to the initial size of the
plume, the type of plume transport is significantly different. Usually, when the
plume is larger than the scale of heterogeneity, plume spreading is considerably
higher. However, when the plume is much smaller than the scale of heterogeneity,
the plume travels predominantly along preferential channels, while not spreading
by a great extent. When the plume is extremely small compared to the scale of

heterogeneity, the plume undergoes “negative transverse dispersion.”

Temporal variability caused by seasonal fluctuations, in the absence of spatial
heterogeneity, causes to and fro motion of the plume, which is reversible.
Generally, the plume does not undergo extensive spreading. When temporal
variability interacts with spatial heterogeneity, the effect on plume transport is
drastic as the plume is shred into fingers and tails. In addition, the process cannot
be reversed because the plume may get trapped in a low conductivity zone, which
causes enhanced spreading of the plume. When additional heterogeneities, such as
partitioning coefficient, are coupled together with temporal variability and spatial

heterogeneity, the effect on plume spreading is compounded.

Heterogeneity in the real world is practically not deterministic. In order to model
this highly complex variable conductivity field, a stochastic approach is used. In
this approach, the aquifer structure is represented using a set of statistical
parameters, such as mean, correlation scales, and variance. These statistical

parameters are obtained by analysis of the data.

182

0 Mean - Is the geometric mean of the data, which represents the
average conditions that exist in the aquifer

- Correlation scales — Are a measure of the spatial distance across which
the aquifer properties are supposed to be correlated. This parameter

can be different in all three directions making the aquifer anisotropic.
However, in most cases, the horizontal correlation scales (xix and Ay)

are assumed to be constant.

0 Variance — Is a measure of how variable the aquifer properties are with
respect to the mean. A higher variance means higher variability of
aquifer properties and vice versa.

Using these prOperties, the aquifer can be described stochastically, which allows

systematic probabilistic analysis.

Macrodispersion models predict the mean plume spreading and the overall extent
of the plume. However, these models do not provide information on the maximum

concentration, concentration variance and the degree of dilution.

Multiple scales of heterogeneity have a significant impact on plume spreading.
Large scales of heterogeneity provide only a “trend” for plume transport, while
the smaller scales spread the plume. If the smaller scales are not modeled, the
spreading of the plume is significantly less. When the smaller scales are modeled
using an effective macrodispersion model the model is able to predict the overall
extent of the plume. However, this model does not provide information on the

maximum concentration, concentration variance and the degree of dilution.

183

 

10. Stochastic methods such as Monte Carlo simulations provide information on the
mean plume behavior by statistically analyzing the plume characteristics obtained
from an infinite number of realizations. Monte Carlo simulations can also provide
information on the probability distribution function (PDF) of any parameter of
interest, such as head, In K, concentration at a monitoring well, and seepage and

solute flux across a cross-section of the aquifer.

11. Although a Gaussian distribution of In K characterizes the random field, the
concentration PDF and solute flux PDF are strongly skewed. However, the head
and seepage flux PDF across a cross-section are predominantly Gaussian. In the
vicinity of stresses such as pumping wells, head and seepage flux PDF may not be

Gaussian.

12. If the concentration PDF is assumed to be Gaussian, a mean and variance will be
sufficient to estimate the probability distribution. Since concentration PDF is
strongly skewed, characterizing the uncertainty is difficult. If the concentration
PDF is lognormal, it can be characterized using a mean and a variance. However,
concentration is strongly location dependent, and a lognorrnal distribution does
not hold good always. Therefore, variance is not a good measure of uncertainty
for concentration PDF. Hence, the actual concentration PDF is needed to

determine the variability at each point of interest.

184

 

9 REFERENCES

I.

10.

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Freeze, R.A., and J .A.Cherry, Groundwater, Prentice-Hall, Englewood Cliffs, NJ,
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Gelhar, L.W., Stochastic subsurface hydrology, Prentice-Hall, Englewood Cliffs,
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Gelhar, L.W., and CL. Axness, Three-dimensional stochastic analysis of .
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Graham, W. and D. McLaughlin, Stochastic analysis of nonstationary subsurface
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Sudicky, E. A., A natural gradient experiment on solute transport in a sand

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