Describing and Predicting Breakthrough Curves for non-Reactive Solute Transport in Statistically Homogeneous Porous Media Huaguo Wang Dissertation Submitted to the Faculty of the Virginia Polytechnic Institute and State University in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy In Crop and Soil Environmental Science Naraine Persaud, Chair Tao Lin Saied Mostaghimi Yakov Pachepsky Lucian Zelazny November, 2002 Blacksburg, Virginia Keywords: Solute Transport Modeling, Breakthrough Curves, Scale-Dependent Dispersivity, Statistically Homogeneous Porous Media, Column Experiments. Copyright 2002, Huaguo Wang
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Describing and Predicting Breakthrough Curves for non-Reactive
Solute Transport in Statistically Homogeneous Porous Media
Huaguo Wang
Dissertation Submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in Partial Fulfillment of the Requirement for the Degree of
Doctor of Philosophy
In
Crop and Soil Environmental Science
Naraine Persaud, Chair
Tao Lin
Saied Mostaghimi
Yakov Pachepsky
Lucian Zelazny
November, 2002
Blacksburg, Virginia
Keywords: Solute Transport Modeling, Breakthrough Curves, Scale-Dependent
DESCRIBING AND PREDICTING BREAKTHROUGH CURVES FOR NON-REACTIVE SOLUTE TRANSPORT IN STATISTICALLY
HOMOGENEOUS POROUS MEDIA by
Huaguo Wang
Crop and Soil Environmental Sciences
(ABSTRACT)
The applicability and adequacy of three modeling approaches to describe and predict breakthough curves (BTCs) for non-reactive solutes in statistically homogeneous porous media were numerically and experimentally investigated. Modeling approaches were: the convection-dispersion equation (CDE) with scale-dependent dispersivity, mobile-immobile model (MIM), and the fractional convection-dispersion equation (FCDE). In order to test these modeling approaches, a prototype laboratory column system was designed for conducting miscible displacement experiments with a free-inlet boundary. Its performance and operating conditions were rigorously evaluated. When the CDE with scale-dependent dispersivity is solved numerically for generating a BTC at a given location, the scale-dependent dispersivity can be specified in several ways namely, local time-dependent dispersivity, average time-dependent dispersivity, apparent time-dependent dispersivity, apparent distance-dependent dispersivity, and local distance-dependent dispersivity. Theoretical analysis showed that, when dispersion was assumed to be a diffusion-like process, the scale-dependent dispersivity was locally time-dependent. In this case, definitions of the other dispersivities and relationships between them were directly or indirectly derived from local time-dependent dispersivity. Making choice between these dispersivities and relationships depended on the solute transport problem, solute transport conditions, level of accuracy of the calculated BTC, and computational efficiency The distribution of these scale-dependent dispersivities over scales could be described as either as a power-law function, hyperbolic function, log-power function, or as a new scale-dependent dispersivity function (termed as the LIC). The hyperbolic function and the LIC were two potentially applicable functions to adequately describe the scale dependent dispersivity distribution in statistically homogeneous porous media. All of the three modeling approaches described observed BTCs very well. The MIM was the only model that could explain the tailing phenomenon in the experimental BTCs. However, all of them could not accurately predict BTCs at other scales using parameters determined at one observed scale. For the MIM and the FCDE, the predictions might be acceptable only when the scale for prediction was very close to the observed scale. When the distribution of the dispersivity over a range of scales could be reasonably well-defined by observations, the CDE might be the best choice for predicting non-reactive solute transport in statistically homogeneous porous media.
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DEDICATION
I dedicate this work to my parents and my wife for their love and support.
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ACKNOWLEDGEMENTS
I would like to begin by expressing my sincere appreciation to my advisor Dr. Naraine
Persaud for his patience, unreserved guidance, and invaluable support.
I would like also to express my sincere appreciation to Dr. T. Lin, Dr. S. Mostaghimi, Dr.
Y. Pachepsky, and Dr. L. Zelazny for serving on my research committee and for their
invaluable advice and suggestion.
I would like to especially thank Dr. Y. Pachepsky for his help and advice that were
indispensable for developing the content for this research.
Further thanks go to Ronnie Alls for making the experimental column system.
The financial support provided by the Department of Crop and Soil Environmental
Sciences is gratefully acknowledged.
Finally, words cannot express my deep and heartfelt gratitude to my parents and my wife
for their support throughout the course this study.
4.3.4 Results and Discussion..….………………………………………….…………99
4.4 Conclusions…………………………………………………………………………109
CHAPTER 5. A NEW FUNCTION FOR DESCRIBING SCALE-DEPENDENT
DISPERSIVITY IN STATISTICALLY HOMOGENEOUS POROUS MEDIA …….…111
5.1 Introduction ………………………………………………………………………111
5.2 Scale Dependent Heterogeneity and Statistically Homogeneous Porous Media …113
5.3 A New Function for Describing Scale-Dependent Dispersivity ..……..………….115
5.4 Materials and Methods ……………………………………………………………119
5.5 Results and Discussion …………………………………………..……………….122
5.6 Conclusions ……………………………………………………………………….130
CHAPTER 6. PREDICTION OF BREAKTHOUGH CURVES FOR NON-
REACTIVE SOLUTE TRANSPORT IN STATISTICALLY HOMOGENEOUS POROUS
MEDIA ……………..…………………………………….………………………..131
6.1 Introduction ……………………………………………………………………….131
6.2 Materials and Methods ……………………………………………………………134
6.3 Results and Discussion …………………………………………………………..138
6.3.1 BTC Prediction at Other Scales Using Parameters Observed at One Scale …138
6.3.2 BTC Prediction at Other Scales Using Parameters Observed at Two Scales ..153
6.4 Conclusions………………………………………………………………………...166
CHAPTER 7. SUMMARY AND CONCLUSIONS …………………………………….167 REFERENCES…………………………………………………………………………...174 APPENDIX 1. ANALYSIS OF INFLUENCE OF SOLUTE DISTRIBUTIONS
AFTER INJECTION INTO THE COLUMN ON BTCS GENERATED USING THE
ANALYTICAL SOLUTION OF THE CDE……………………………………………..186
APPENDIX 2. PROCEDURE USED TO PREPARE EXPERIMENTAL POROUS
MEDIA TO SIMULATE NATURE PORE-SCALE HETEROGENEITY ……………..189
Table 2.2 Comparison of the CDE and the FCDE…………………………………….30
Table 3.1 The number distribution of different sized glass beads used to pack experimental columns……………….……………………………………...62 Table 6.1 Parameters of three dispersivity functions obtained by analysis of the observed BTCs from three combinations of two length scales …..………..154
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LIST OF FIGURES
Figure 2.1 Available models for simulating solute dispersion at three scales based on the Gaussian and Lévy distributions...…………………………………….22
Figure 3.1 Schematic of the laboratory column system that was developed for conducting the miscible displacement experiments ………………..……..37
Figure 3.2 Computational scheme for the particle tracking method ………………….38
Figure 3.3 Algorithm used for solution of the CDE with zero concentration outlet boundary condition by the particle tracking method ….………………….40
Figure 3.4 Algorithm used for solution of the CDE with zero gradient finite outlet boundary condition by particle tracking method. ………………………..41
Figure 3.5 Algorithm used for solution of the CDE with zero gradient infinite outlet boundary condition by particle tracking method…………………………..42
Figure 3.6 Comparison of breakthrough curves calculated using the solution of the CDE by the particle tracking method for three outlet boundary conditions and four Peclet numbers …………………………………………………..44
Figure 3.7 BTCs calculated using the CDE for treatments where the injected solute was assumed to be normally distributed ………………………………….51
Figure 3.8 BTCs calculated using the CDE for treatments where the injected solute was assumed to be uniform distribution ………………………………….52
Figure 3.9 BTCs calculated using the CDE with distance-dependent dispersivity for treatments where the injected solute was assumed to be a normal distribution…………………………………………………………………54
Figure 3.10 BTCs calculated using the CDE with time-dependent dispersivity for treatments where the injected solute was assumed to be a normal distribution ………………………………………………………………..57
Figure 3.11 BTCs calculated using the MIM for treatments where the injected solute was assumed to be a normal distribution …………………………………58
Figure 3.12 BTCs calculated using the FCDE for treatments where the injected solute was assumed to be a normal distribut ion ………………………………….60
Figure 3.13 Dispersivities obtained by fitting experimental BTCs for different injection volumes, using the CDE, the CDE with distance-dependent dispersivity, the CDE with time-dependent dispersivity, and the MIM…………………….66
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Figure 3.14 Immobile porosity and mass transfer coefficient obtained by fitting experimental BTCs for different injection volumes, using the MIM.……..67
Figure 3.15 Fractional dispersion coefficient Df and the order of fractional differentiation α obtained by fitting experimental BTCs for different injection volumes, using the FCDE ……………………………………….67
Figure 4.1 BTC in a general space-time domain for one-dimensional solute transport ……………………………………………………………….….71
Figure 4.2 Specifying the distance-dependent dispersivity in the numerical schemes developed for solving the CDE with distance-dependent dispersivity ..….74
Figure 4.3 Specifying time-dependent dispersivity in the numerical scheme for solving the CDE with time-dependent dispersivity………………………………..77
Figure 4.4 Hypothetical spatial solute distributions at different times during one-dimensional transport for an initial solute source represented as a Dirac delta function ……………………………………………………………...83
Figure 4.5 Hypothetical spatial solute distributions during one-dimensional transport for an initial solute input represented as a Dirac delta function ……….…90
Figure 4.6 BTCs generated using the CDE with scale-dependent dispersivity functions based on the linear function for λT(t) for problem 1…………………….101
Figure 4.7 BTCs generated using the CDE with scale-dependent dispersivity functions based on the parabolic function for λT(t) for problem 1…………………102
Figure 4.8 Concentration distribution over spatial domain at time T/2 generated using the CDE with scale-dependent dispersivity functions based on the linear function for λT(t) for problem 1 ………………………………………….103
Figure 4.9 BTCs generated using the CDE with scale-dependent dispersivity functions based on the linear function for λT(t) for problem 2……………….…….106
Figure 4.10 BTCs generated using the CDE with local scale-dependent dispersivity functions based on the linear function for λT(t) for problem 3…………..108
Figure 4.11 BTCs generated using the CDE with different scale-dependent dispersivity functions based on the linear function for λT(t) for problem 3…………..108
Figure 5.1 Hypothetical local scale-dependent dispersivity and apparent scale-dependent dispersivity ………………………………………………….118
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Figure 5.2 Distribution of apparent dispersivities [αD(L)], observed at different column lengths ………………………………………………………………..…122
Figure 5.3 Comparison of the distribution of the observed apparent distance-dependent dispersivities αD(L), over length, with predicted distribution …….…….124
Figure 5.4 Comparison of the observed BTC at 141 cm and the BTCs predicted using the CDE with the three values of αD(L) calculated………………………125
Figure 5.5 Comparison of the observed BTC at 141cm with BTCs predicted using the numerical solution of the CDE with scale-dependent dispersivity given by the apparent dispersivity, local distance-dependent dispersivity and local time-dependent dispersivity ………………………………………….….126
Figure 5.6 Comparison of observed BTCs at different lengths for two sources input at two times but at one location in the space domain, with the corresponding BTCs predicted using the CDE with local distance-dependent dispersivity calculated ……………………………………………………………….128
Figure 5.7 Comparison of observed BTCs at different lengths for two simultaneous sources over the spatial domain, with the corresponding BTCs predicted using the CDE with local time-dependent dispersivity calculated ………129
Figure 6.1 Comparison of observed BTC, predicted BTCs, and fitted BTC at 141 cm. using the CCDE solute transport model …………………………………139
Figure 6.2 One set of fitted a and b at different column lengths obtained by fitting observed BTCs to the DC DE……………………………………………141
Figure 6.3 Comparison of the observed BTC at 59 cm to the BTCs fitted using the DCDE and the CCDE. ……………..…………………………………….142
Figure 6.4 Comparison of observed apparent dispersivity distribution with those calculated using Eq. (4.22) with a and b values obtained by fitting the DCDE to one observed BTC at 59 cm ………………………………….142
Figure 6.5 Comparison of the observed BTC at 141 cm and BTCs predicted for this length using the DCDE ……..…………………………………….….…..143
Figure 6.6 One set of fitted c and d at different column lengths obtained by fitting observed BTCs to the TC DE. …………………….………………….….145
Figure 6.7 Comparison of the observed BTC at 141 cm to the BTCs fitted using the TCDE and the CCDE ………………..………………………….…….…145
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Figure 6.8 Comparison of observed apparent dispersivity distribution with those calculated using Eq. (4.19) with c and d values obtained by fitting the TCDE to one observed BTC at 141 cm ………..…………………….….146
Figure 6.9 Comparison of the observed BTC at 141 cm and BTCs predicted for this length using the TCDE …………………………………………………..146
Figure 6.10 Distribution of (A) dispersivity κ, (B) first- order mass transfer coefficient beta and (C) immobile porosity θim , over different column lengths obtained by fitting experimental BTCs to the MIM…………………………….....149
Figure 6.11 Comparison of the observed BTC at 141 cm to the fitted BTCs and the predicted BTCs for this length using the MIM ………………….………150
Figure 6.12 Distribution of the parameters Df and α obtained by fitting observed BTCs for different column lengths to the FCDE ……………………………….152
Figure 6.13 Comparison of the observed BTC at 141 cm to the fitted BTCs and the predicted BTCs for this length using the FCDE …………………..……..152
Figure 6.14 Application of the parabolic function to predict the apparent scale-dependent dispersivity distributions over experimental column lengths, and to predict the BTC at 141 cm ………………………………….…..…….155
Figure 6.15 Application of the log-power function to predict the apparent scale-dependent dispersivity distributions over experimental column lengths, and to predict the BTC at 141 cm……………….. ……………………….….156
Figure 6.16 Application of the hyperbolic function to predict the apparent scale-dependent dispersivity distributions over experimental column lengths, and to predict the BTC at 141 cm ………………..…………………………..157
Figure 6.17 Hypothetical apparent dispersivity distribution in a fractal porous medium. The distributions were described using a parabolic dispersivity distribution Eq. (6.1)…………………………………………………………………..162
Figure 6.18 Sensitivity of the apparent dispersivity predicted for 3 length scales (150 cm, 300 cm and 600 cm) using four apparent distribution functions ……163
Figure 6.19 Predicted apparent scale dependent dispersivity using four apparent dispersivity functions ……………………………………………………164
Figure 6.20 Hyperbolic and LIC apparent dispersivity functions fitted to the observed apparent dispersivities reported by Zhang et al.(1994)…….…………….165
CHAPTER 1 INTRODUCTION
Describing and predicting solute transport behavior in porous media are essential to
optimally manage soils and subsurface aquifers and to address chemical pollution in these
resources. The variety and complexity of the physical, chemical and biological
interactions between the solute and the soil or subsurface aquifer medium often make it
very difficult to describe and predict solute transport behavior in these types of porous
media. Such porous media are naturally heterogeneous, meaning that their
physicochemical properties, such as texture, structure, chemical composition etc., vary
with space and time. This heterogeneity further complicates the interactions between the
solute and the medium, and increases the difficulty to predict solute transport behavior.
Description and prediction of solute transport behavior become even more difficult when
the dominant components of heterogeneity affecting solute transport are different at
differing spatial scales. At the laboratory column scale, the dominant component of
heterogeneity that affects solute transport, termed as the pore-scale heterogeneity, is
caused by micro-level variation in soil texture and soil structure. At field scales, the
dominant component of heterogeneity is due to variation in macro-level spatial
characteristics such as layering, presence of rocks and rock formations, solution channels
or channels formed by plant roots and earthworms, and disturbances caused by human
activities such as agriculture. Because it depends on the scale of observation,
heterogeneity of the transport medium has proven very difficult to characterize. A
medium made up of different particle sizes would be heterogeneous at the pore or micro
scale, but can be considered as homogeneous at the column or macro scale. The
homogeneity at the column scale is defined in a statistical sense, meaning that the micro
scale or pore scale heterogeneity of the porous medium is uniformly distributed within
the column. An observed property at this scale (macro scale) does not change
appreciably for some arbitrary change in the specified scale of the column. Similarly, at
the field scale, statistical homogeneity means that the components of macro scale
heterogeneity are uniformly distributed over the field, and that an observed property at
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the field scale does not change appreciably with some arbitrary change in the specified
scale for the field.
Subsurface solute transport is generally described and predicted using solute transport
models. Such models require quantitative descriptions of the mechanisms controlling
solute transport in subsurface media. Once developed and validated, these models
become cost-effective tools to assess the fate of solutes in the environment, since it is
generally not feasible to make such assessments by direct in-situ field sampling and
analysis over long periods of time. Most mechanistic transport models for solutes in
porous materials are based on the convection-dispersion equation (CDE). The CDE is a
partial differential equation representing mass continuity for movement of a given solute
in a porous medium by dispersion and convection under specified initial and boundary
conditions. Appropriate terms are incorporated into to the CDE to account for chemical
and/or physical sink/source interactions between the solute and porous medium. The
CDE can be developed microscopically based on Brownian motion, or macroscopically,
based on Fick’s law. CDE-based models have not been completely successful in
explaining observed solute transport breakthrough curves (BTCs), especially when the
porous media are heterogeneous at the scale of observations.
BTCs are extensively used to characterize the physicochemical processes involved in the
transport of solutes in porous media. These BTCs are usually obtained using packed or
undisturbed columns in laboratory experiments. Less frequently, they are determined in-
situ in the field. Even though it is very questionable, the parameters obtained from
column BTCs are commonly applied to field situations. The reason is that column
experiments are much more cost-effective and time-efficient. In addition, analysis of
column BTCs can provide useful estimates of the parameters for the physicochemical
processes involved in subsurface solute transport. Such parameters are essential for
developing and validating theoretical models for solute transport.
An important anomaly that has not been adequately explained is the enhanced spreading
or "tailing" of the BTCs observed in column experiments for the transport of non-
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reactive solutes. When the CDE is used to describe the observed BTCs, significant
differences appear between the observed and fitted BTCs especially for the tail portion of
the curve. In addition, significant errors occur when parameter values obtained by fitting
the CDE to the observed BTC at one scale, is used to predict the observed BTCs at other
scales. Such failure indicates that at least one parameter of the CDE may be scale-
dependent. Studies have shown that the scale-dependent parameter is dispersivity, which
is used to quantify dispersion in the CDE.
Three modeling approaches have been developed to account for the enhanced spreading
in the observed BTCs. These have been termed as: scale-dependent CDE, mobile-
immobile model (MIM), and the fractional convection-dispersion equation (FCDE, or
called as fractional advection-dispersion equation, FADE). In the scale-dependent CDE,
the dispersivity is included, not as a constant over all scales, but as a function of scale. In
the MIM approach, the fluid in the flow domain is described as a mobile phase and an
immobile phase. The solute exchanges between the two fluid phases as it moves in the
mobile phase. The FCDE assumes that the random movement of the solute particles
during transport obeys a non-Gaussian statistical distribution, called a α-stable
distribution. Scale-independence of the parameters is implicit in the development of the
MIM and FCDE.
Whether these three models can adequately describe and predict observed BTCs for
solute transport in statistically homogeneous porous media at column scales have not yet
been evaluated and compared by rigorous experiments. The results of such evaluation
and comparison can be directly useful for selecting the appropriate model to describe and
predict solute transport in soils and subsurface groundwater domains. In most situations,
these domains are considered as statistically homogeneous porous media at the solute
transport scales of interest. In this regard, the following questions need to be answered:
a. How well can these three models describe and predict BTCs for solute transport?
b. Are the parameters of the MIM and FCDE scale-independent? If they are scale-
independent, then the parameters identified at one scale can be directly used to predict
solute transport behavior at other scales.
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c. If the three models describe BTCs very well at one scale, how well can they predict the
BTCs at other scales? A related question is: what is accuracy of prediction for these
models when the prediction scale is different from the scale used for parameter
identification?
In applying the scale-dependent CDE, the scale dependence of the dispersivity needs to
be specified. Scale-dependent dispersivity can be expressed as time-dependent
dispersivity, or as distance-dependent dispersivity. The form of the expression used for
the scale-dependent dispersivity directly affects the choice of procedures for solving the
scale-dependent CDE. When the scale-dependent CDE with either time or distance-
dependent dispersivity is solved numerically, the dispersivity value must be updated as
appropriate for each discretized time or space step. Therefore, different algorithms are
required for solving the time-dependent CDE and distance-dependent CDE. When the
solute transport scenarios are complex, such as multidimensional and multiple-source
solute transport problems, the choice of time-dependent or distance-dependent
dispersivity becomes important for accurately and efficiently solving the scale-dependent
CDE.
Local scale dependent dispersivity (either local time or local distance dependent) or
apparent scale dependent dispersivity (either apparent time or apparent distance
dependent) can be specified when the scale-dependent CDE is solved numerically. In
the former case, the local scale dependent dispersivity values depend only on the
position of the node in the discretized space/time domain. In this case, these dispersivity
values do not depend on the space location of the discretized domain at which the BTC
is to be evaluated. In the latter case, the scale dependent dispersivity values at a given
node in the discretized space/time domain represent an apparent or effective value,
which depends on the space location at which the BTC is to be evaluated. The
relationships between these dispersivities (local time-dependent dispersivity, apparent
time-dependent disperpsivity, local distance-dependent dispersivity, and apparent
distance-dependent dispersivity), and the conditions under which these four possibilities
are applicable need to be specified and tested experimentally.
4
To date, no special theoretical and experimental research has been conducted to answer
many important questions related to the use of scale-dependent dispersivity in describing
and predicting solute transport in porous media. Such questions are:
a. How are time-dependent dispersivity and distance-dependent dispersivity related, and
how are local scale dependent dispersivity and apparent scale dependent dispersivity
related? Some studies have indicated that they are related, but the relationship has not
been completely specified and explained.
b. If scale-dependent dispersivity is incorporated in the numerical solution of the scale-
dependent CDE for a given solute transport problem, such as a multi-source solute
transport problem, which choice for scale dependent dispersivity is more accurate and
efficient for the numerical solution? There are four possible choices: local time
dependent, local distance dependent, apparent time dependent, or apparent distance
dependent dispersivity.
c. Does the CDE incorporating common scale-dependent dispersivity functions
suggested in the literature, adequately describe solute transport in statistically
homogeneous laboratory scale porous media? If not, can a better alternative function be
developed?
d. Is it possible to completely specify the scale-dependent dispersivity function using
solute transport observations at only one scale? If not possible, what is the minimum
number of observation scales required to completely specify the function?
Experimentally, all the foregoing questions can be answered by generating replicated
BTCs for transport of a non-reacting solute in laboratory columns of varying length,
packed with porous media, that is statistically homogeneous at the column scale. These
BTCs can then be appropriately analyzed using the various possible forms of the scale-
dependent CDE, the MIM, and the FCDE. The simplest model that can adequately
describe and predict the experimental BTCs at all length scales will become apparent
from these experiments and analyses.
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The overall objective of the studies reported herein, was to attempt to answer all of the
foregoing questions using a series of carefully planned and rigorously conducted,
replicated laboratory column experiments for one-dimensional, single and multiple
source solute transport of a non-reactive solute. The specific objectives were:
a. To develop and test an innovative laboratory column system that would permit a free
inlet boundary, that can be used to implement different experimental solute transport
scenarios, including multiple source input conditions.
b. To seek a physical explanation for scale-dependent dispersivity based on scale-
dependent heterogeneity of porous media.
c. To detail and numerically examine the relationship between time-dependent
dispersivity and distance-dependent dispersivity.
d. To theoretically and experimentally determine the solute transport conditions under
which local time dependent, local distance dependent, apparent time dependent, or
apparent distance dependent dispersivity is appropriate for use in the numerical solution
of the scale-dependent CDE.
e. To elucidate the criteria for selecting the best of several choices for specifying scale-
dependent dispersivity for different solute transport problems, based on the accuracy and
efficiency in calculation of the numerical solution of the CDE with scale-dependent
dispersivity.
f. To develop a new scale-dependent dispersivity function specific to solute transport in
statistically homogeneous porous media, and to validate this new function by comparing
experimental BTCs to those predicted using the scale dependent CDE with this new
function, and with other scale-dependent dispersivity functions reported in the literature.
g. To determine whether parameters from experimental BTCs at a single observation
scale are sufficient for predicting BTCs at other scales using the MIM and the FCDE
models, in order to clarify whether the parameters of these models are scale-dependent or
scale-independent for non-reactive transport in statistically homogeneous porous.
h. To determine which of the three modeling approaches, namely, the scale-dependent
CDE, MIM, and the FCDE, is best for describing and predicting the anomalous tailing
behavior observed in BTCs for non-reactive solute transport in statistically homogeneous
media.
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The following presents a synopsis of each of the remaining six chapters of this
dissertation:
Chapter 2: Review of solute transport models in general with special emphasis on the
modeling of solute dispersion during transport in porous media. The review of solute
transport models includes a discussion of models for describing solute transport in single-
layered and multiple-layered porous media, the methods of solving the governing
equations of solute transport models, and techniques for estimation of model parameters.
The review of modeling of solute dispersion during transport introduces several concepts
and theories for explaining solute dispersion at the pore scale, laboratory scale, and field
scale. In addition, experimental methods for estimating solute dispersion parameters are
discussed.
Chapter 3: Description of the design and testing of the column system that was specially
developed for generating the experimental BTCs for one-dimensional, single and
multiple-source non-reactive solute transport. Tests on the effect of various column outlet
boundary conditions on BTCs calculated using the CDE are presented. Tests were also
made to determine the operating conditions under which the system can be used to obtain
reliable solute transport data for analysis using the CDE, the time-dependent CDE, the
distance-dependent CDE, the MIM and the FCDE. Algorithms developed for conducting
these tests are described.
Chapter 4: Details the algorithms for applying local time dependent, local distance
dependent, apparent time dependent, or apparent distance dependent dispersivity in
numerical solutions of the scale-dependent CDE. Explains why and how scale-dependent
heterogeneity is related to local time-dependent dispersivity, why and how local time-
dependent dispersivity is related to apparent time-dependent dispersivity, local distance-
dependent dispersivity, and apparent distance-dependent dispersivity. This chapter
includes numerical studies for testing the adequacy and applicability of applying the
relationships between these dispersivities for BTC analysis, and also includes numerical
7
studies on the criteria for selecting the scale-dependent dispersivity function for different
solute transport under different conditions.
Chapter 5: Reviews scale-dependent heterogeneity and the definition of statistically
homogeneous porous media. Explains the intrinsic connection between the local scale-
dependent dispersivity and the variance of hydraulic conductivity. Introduces a new
function for describing scale-dependent dispersivity in statistically homogeneous porous
media. The theory and application of this new function is discussed.
Chapter 6: Compares observed BTCs at different column scales with those described
and/or predicted using the CDE, the scale-dependent CDE with various scale dependent
dispersivity functions, the MIM, and the FCDE. The scale-dependent dispersivity
functions studied were the parabolic function, log-power function, hyperbolic function,
and the new function as developed in Chapter 5. Discusses whether the parameters in the
CDE, the MIM, and the FCDE are scale-dependent. Demonstrates that observed BTCs
for at least two-scales were needed to characterize the scale-dependent dispersivity
function, and demonstrates the advantage of using the new dispersivity function
developed in Chapter 5.
Chapter 7: Presents a summary of the principle finding s and conclusions of the studies
presented in the forgoing chapters.
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CHAPTER 2 MODELING SOLUTE TRANSPORT IN POROUS MEDIA
2.1 Introduction
This chapter reviews solute transport models. Special attention is focused on how
mechanical dispersion of solutes is incorporated in these models. Conceptually, the
components required in modeling of solute transport in porous media can be viewed as a
pyramid with three-layers. At the base of the pyramid, are concepts for modeling water
movement in porous media. Without accurate representation of the hydrological
processes controlling water movement, modeling solute transport would be impossible
since water movement drives the transport of solutes in porous media by convection. The
second layer represents concepts on the thermo-mechanical processes involved in solute
transport such as molecular diffusion and mechanical dispersion. The top layer includes
concepts related to modeling the effect of sinks/sources on solute transport. These
sink/source components of the solute transport model are used to describe physical,
chemical, and biological processes, other than convection and dispersion, which may
influence solute transport. A complete solute transport model is obtained when the
processes in each layer of the pyramid, modeled by themselves, are put together in a
reasonable manner. Not all the solute transport models would have sink/source
components, but all solute transport models must incorporate molecular diffusion and
mechanical dispersion effects into the water movement component.
Mechanical dispersion results primarily from the heterogeneous nature of porous media.
When the heterogeneity of porous media is complex and scale-dependent, modeling of
mechanical dispersion becomes difficult. In such cases, modeling of dispersion decides
the basic form of the solute transport model.
When water movement in a porous medium can be correctly described, modeling of
mechanical dispersion becomes sine qua non for further development of the solute
transport model. If the water movement is correctly described, the convection process is
9
defined. Molecular diffusion is relatively easy to define because it is a thermodynamic
characteristic of the solute at a given temperature. Sink/source components can be added
to the model by applying the mass conservation principle. However, such additions
would be meaningful only after mechanical dispersion of the solute has been adequately
described. Consequently, modeling of mechanical dispersion is a necessary step in the
construction of all models for non-reactive solute transport in porous media.
2.2 Solute Transport Models
As indicated in the above analogy, a large number of processes can be included in a
solute transport model. However, all the physical processes that are involved in solute
transport need not be explicitly included in the model. The form of the model is decided
by which physical processes one chooses to include, and by the assumptions one makes
regarding these processes. The specific processes that are included in the model may or
may not reflect the physical reality. A model is assumed to represent physical reality, if
results simulated with the model match the observations. Consequently, a large number
of solute transport models exist that vary in their complexity and sophistication.
In general, solute transport models can be categorized as deterministic and stochastic
models or as mechanistic and functional models (Addiscott and Wagenet, 1985). The
deterministic models assume that a given set of transport conditions lead to a uniquely
definable outcome. The stochastic models assume that a given set of transport conditions
lead to an uncertain outcome because of the spatial and temporal uncertainty of some
parameters that are included in the model. The mechanistic models are developed
directly and deductively from the fundamental mechanisms assumed for the process. The
functional models incorporate simplified, phenomenological treatments of water
movement and solute behavior in the subsurface, and make no assumptions of
fundamental mechanisms. Based on these distinctions, transport models can be classified
where ε∞ is an asymptotic dispersivity , β is a scale factor describing the linear growth of dispersivity process near the origin
--Finite element solution --Parameters are fitted by the BTC data at several distances. --Effective dispersivity, estimated by using BTCs, is the integral of local dispersivity.
Yates (1990)
--Distance-dependent --Linear α ( )
( ) ( )x a x
D x a x L b v== +
where: a is the slope of the distance-dispersivity relationship. b is a constant which characterizes the fluid diffusional processes and L is a characteristic distance
--Analytical solutions for initial concentration zero and constant concentration boundary condition, and for initial concentration zero and constant flux boundary condition were given
Jury and Roth (1990)
--Distance-dependent -- D
zl
Dz l=
-- Predict outflow concentration at distance z from the inlet by using the CDE transfer function gotten at the distance of l
Logan (1996)
--Distance-dependent --Exponential D x D L ee l
rx L( ) ( )/= + − −β 1 where De is the effective diffusion coefficient, L is a typical observation length scale in the problem, βL is the initial slope of the dispersion curve and r is the limiting value of the curve.
--Analytical solution with periodic boundary conditions and scale-dependent dispersion was given
25
Table 2.1 continued Reference Dispersion Function Application Pickens and Griska (1981)
--Time-dependent: represented as a function of mean travel distance from the input position. --Dispersivity is a constant for the entire system at any point in time but varies temporally. --Linear: α = ax --Parabolic: α = ax b --Asymptotic:
α = −+
AB
x B( )1
--Exponential α = − −E Fx[ exp( )]1 where x is mean travel distance, a, b and F are constants , A and E are asymptotic or maximum dispersivity value, B is characteristic half length(equals mean travel distance corresponding to A/2).
--Fit dispersivity equations by comparing variances of tracer distribution with finite element model results. --The scale-dependent dispersivity is necessary for prediction of solute transports for short mean travel distances, however the early scale-dependent dispersivity has little effect in simulating transports of long time. --All the practical examples used can be fitted by linear relationship within the scale-dependent region. --Finite element solution
Basha and El-Habel (1993)
--Time-dependent -- Linear D t D
tk
Dm( ) = +0
--Asymptotic D t D
tt k
Dm( ) =+
+0
--Exponential D t D t k Dm( ) [ exp( / )]= − − +0 1 where D0 is the maximum dispersivity, Dm is the molecular diffusion, and k is equal to the mean travel time corresponding to D0+Dm in the linear case, to 0.5D0+Dm in the asymptotic case, and 0.632D0+Dm for the exponential case.
--Transport in an infinite medium --Explicit analytical solution for the case of instantaneous injection. --Numerical integral analytical solution for the case of continuous injection.
Zou et al (1996)
--Time-dependent --Parabolic α ( )t dt b=where α is dispersivity d and b are constants --The dispersivity is constant for the entire system at any given time
--Parameters can be obtained by: 1. Spatial method I: Two times measurement of the concentration of the center of plume and application point 2. Spatial method II: Moment method of measuring spatial variance at two difference times 3. Temporal method: measuring Concentration -versus-time data from one location -- Analytical solution
26
A third type of laboratory scale models, called layer models (Figure 2.1), has been
developed from chromatography plate theory. In this theory, the chromatographic
column is visualized as a serious of plates with length l. The total number of the plates is
N = L/l, where L is the length of the column. Each plate consists of the mobile phase and
stationary phase. When the solute passes through the plate, it becomes distributed
between the two phases. The distribution is quantified by the partition coefficient. After
passing through a large number of plates, the solute distribution along the length of the
column is Gaussian. The principle and the results of the layer model are the same as
those of Bear’s one-D cell-channel array (Bear, 1988b).
In applying the layer model, the porous medium is divided into discontinuous layers with
depth. However, after the transport through the soil depth of concern, the solute
concentration distribution may not be Gaussian, because the number of layers is not large
enough. Thorburn et al. (1992) provided a detailed review of the application of layer
models to solute transport.
The laboratory scale models can also be applied in the field scale (Figure 2.1). For
example, if a porous medium at the field scale is assumed to be homogeneous, the CDE
model can be directly used for solute transport at the field scale. On the other hand, if the
medium is heterogeneous at the laboratory scale, the CDE may not be suitable for
describing solute dispersion at this scale, and the scale-dependent CDE or layer model is
applicable.
Field scale models: When applying deterministic models at the field scale, considerable
error may occur because some model parameters vary spatially and temporally in the
field. Usually, it is unrealistic to use one value for the dispersion coefficient to
completely describe solute dispersion in the field. To correctly simulate solute dispersion
in the field, the probabilistic nature of the dispersion coefficient has to be taken into
consideration in the model. Two approaches are used to treat the probabilistic
characteristics of the dispersion parameter at the field scale. One is termed as the
stochastic-mechanistic model or stream tube approach, in which the dispersion
27
coefficient is obtained as some statistical distribution. The other is the transfer function
approach. The underlying principles of these two modeling approaches have been
introduced in the foregoing paragraphs of this review of solute transport models.
Lévy distribution models: The framework for all of the dispersion models that have been
discussed above, are directly of indirectly related to the Gaussian distribution (Figure
2.1), meaning that their results are equivalent to that obtained with the solute dispersion
model directly derived from Brownian particle motion theory. There are two underlying
concepts of particle movement in Brownian motion theory. One is that the displacement
of the particles over any time element is Gaussian. The other is that the particle motion
is a Markov process in which the movement of a given particle depends only on its
current position. This means that the particle displacement at a given time is independent
of its history. Consequently, Brownian motion of particles can be conceptualized as the
resultant of a set of independent, identically distributed (iid) Gaussian random
displacements. The probability that a particle, which is undergoing the Brownian motion,
will be found in a particular spatial location at a particular time is governed by the
Fokker-Planck equation. In one dimension, the Fokker-Planck equation is:
xvP
xDP
tP
∂∂
−∂
∂=
∂∂ )()(
2
2
(2.1)
Where P is the probability, v is the mean particle instantaneous velocity, and D is the
parameter controlling the variance of the Gaussian random displacements of the particle
over time. If a large number of particles whose behavior is completely independent of
each other, are released to the system simultaneously (termed as an ensemble), then
according to the ergodic hypothesis, the probabilistic behavior of the ensemble will be the
same as that of an individual particle. If the ergodic hypothesis holds, then the ensemble
probability represents the fraction of the total mass present at a given time and location.
This can be converted to the solute concentration at this time and location.
However, the assumption that solutes are transported in soils and aquifers as Brownian
motion may be wrong. Soil and aquifer materials are deposited in continuous correlated
units. The hydraulic conductivity of these materials is generally auto-correlated spatially.
28
In such materials, a particle that is traveling faster (or slower) than some mean velocity at
a given time, is much more likely to be traveling faster (or slower) than the mean at later
time. This means that a particle's instantaneous movement is affected by its history.
Many observations in field and laboratory studies have shown that Brownian motion is
ill-suited to explain the solute transport in porous media. The breakthrough curves (BTC)
observed in these studies generally had heavier tails than predicted by the Gaussian based
models. These observations motivated research to find some new distribution instead of
the Gaussian, to explain the solute transport in porous media. To date, the alternative
distribution is the Lévy or α-stable distribution.
The Lévy distribution is a family of stable distributions, meaning that random variable
sums are distributed identically as the summands. The Gaussian distribution is therefore
a member of the Lévy distribution family. It follows from the Central Limit Theorem
that the scaled sum of n Lévy iid random variables (X), is distributed identically as X.
The scaled sum is:
α121 ...n
XXXS n
n+++
= (2.2)
where 0 2≤<α . When α=2, the Lévy distribution is Gaussian. The probability density
functions (pdf) of the symmetric α-stable distribution (SαSD) are given in Table 2.2. The
shape of the SαSD pdf is similar to that of the Gaussian pdf, except that the SαSD has a
longer tail than the Gaussian. One important property of Lévy distribution is that when
α<2 the second moment of the distribution is infinite. It is unreasonable to simulate the
variance of solute dispersion directly as the second moment of Lévy distribution, because
it is impossible for a solute particle to move an infinite distance in a single time interval
(called a “jump”) in soil and subsurface porous media. In order to obtain finite values for
the jumps to simulate the variance of solute dispersion, the αth moment of the Lévy
distribution is used. In this case, the αth moment is fractional moment for 1 2<<α . The
resulting partial differential equation for solute transport is the FCDE (Table 2.2). In the
FCDE, the heavy tails in the BTCs are explained by the α fractional derivatives in the
FCDE. The FCDE is usually incorrectly stated as being a Lévy distribution-based model;
actually it is the result of using the truncated Lévy distribution. Benson (1998), one of
29
the developers of the FCDE, said “a few truncated Lévy walks still look like a Lévy walk.
Only a large number of truncated Lévy walks looks like a Gaussian”.
Table 2.2 Comparison of the Gaussian based CDE
and the Lévy distribution-based FCDE
Item CDE FCDE
Partial Differential Equation (PDE):
xCv
xCD
xtC
∂∂
−
∂∂
∂∂
=∂∂
Symmetric FCDE
xCv
xC
xCD
tC
sf ∂∂
−
−∂∂
+∂∂
=∂∂
α
α
α
α
)(21
Standard Probability Density Function
Gaussian pdf
−=
2exp
21)(
2xxf N π
Symmetric α-stable pdf
φφφαα
ααα
α
α
α dUxUx
xf ∫
−
−= −
− 1
0
11
1
)(exp)(12
)(
πφφαπ
πφ
παφ
φ
αα
α21
1
cos)1(cos
2cos
2sin
)( −
=
−
U
Standardization
−
=σµ
σµσ xfxf NNS
1),,(
−
=σµ
σµσ αα
xfxf S1),,(
Analytical solution of PDE for initial value problem, Dirac delta δ input, and input mass m0. The porosity is n vt
Dt
xfn
mtxC NS
==
=
µσ
µσ
2
),,(),( 0
vt
tD
xfn
mtxC
sf
S
=
=
=
µ
πασ
µσ
α
α
1
0
2cos
),,(),(
2.3.3 Estimation of the dispersion coefficient
The dispersion coefficient D can be estimated from the soil physical characteristics by
using regression equations, or estimated from best-fitting of the experimental
breakthrough curves (BTCs). The methods of best-fitting of the BTCs have been
introduced in section 2.2.7.
The dispersion coefficient D is:
vDD κτ += 0 (2.3)
30
where D0 is the molecular diffusion coefficient, v is the average pore velocity, κ is
dispersivity, and τ is tortuosity. The tortuosity and λ can be estimated from the soil
physical characteristics as (Millington and Quirk, 1961):
34
n=τ (2.4)
where n is the porosity. κ can be estimated as (Thorburn et al., 1990):
5.48)ln(6.46 −= LFκ (2.5)
where LF(%) = leaching flux/infiltration, or as (Fried and Combarnous, 1971):
d)4.08.1( ±=κ (2.6)
where d is mean or effective grain diameter, or as (Xu and Eckstein, 1995):
(2.7) 414.210 )(log83.0 L=κ
where L is the distance from the source.
In laboratory experiments, BTCs of a step input solute transport are commonly used for
D estimation. Examples are (Fried and Combarnous, 1971):
2
2
1
1
2)()(
81
−−
−=
tvtx
tvtxD (2.8)
where t1 at the point of C/C0=0.16, and t2 at the point of C/C0=0.84 on the BTC.
In the field, a single well injection-recovery test can be used (Mercado, 1966), in this
case, κ is given as:
[ ]
p
p
I
VV
V
bnVV
∆=
=
*
5.1
2*5.0
323π
κ (2.9)
where b is the thickness of injection aquifer layer, n is the porosity, VI is the total
injection volume, Vp is withdrawal volume, ∆Vp is withdrawal volume.
A field test may also involve a pulse injection of tracer in one well and observing the
BTCs at an observation well. In this case, κ is given as (Mercado, 1966):
31
2
max643
∆=
ttrowκ (2.10)
where row is the distance between injection well and observation well, tmax is the time
from the injection to peak concentration occurrence in the observation well, ∆t is the time
interval of Cmax/e on the BTC. Here e is the base of the natural logarithm.
Dsf and α in the FCDE (Table 2.2) can be estimated by best fitting of the BTC to the
FCDE (Pachepsky et al., 2000). In addition, α can also be estimated from dispersivities
of CDE at several scales as (Benson, 1998):
( ) αλ 12
12
1)( tttt m ∝∝ , and
12+
=m
α (2.11)
32
CHAPTER 3 DEVELOPMENT AND TESTING OF A LABORATORY COLUMN
SYSTEM FOR CONDUCTING MISCIBLE DISPLACEMENT
EXPERIMENTS WITH A FREE-INLET BOUNDARY
3.1 Introduction
Most laboratory column systems used for researching one-dimensional solute transport in
porous media are designed to apply tracer from one end of the column, displace the
applied tracer with a tracer-free solution (termed miscible displacement), and collect the
effluent at the other end (Rao, 1980a; Rao, 1980b; Porro et al., 1993; Ma and Selim,
1994; Zhang et al., 1994). These systems physically simulate the initial-boundary value
problem of one-dimensional transport. However, such column systems do not simulate
solute transport scenarios with multiple simultaneous inputs of the same (or different)
solute at several locations in the transport domain. Additionally, analyses of the BTCs
obtained in such systems are affected by the assumed inlet boundary conditions (van
Genuchten and Alves, 1982; Parker and van Genuchten, 1984; Barry and Sposito, 1988)
and by the manner in which the solute is applied at the column inlet (Horton and
Kluitenberg, 1990).
In their laboratory experiments, Delay et al. (1997) used a syringe to directly inject
concentrated tracer at the center of the cross-section at an arbitrary location along the
length of a Plexiglas column packed with an artificial porous medium. They used this
system to simulate a one-dimensional initial value solute transport problem with a free
inlet boundary. The one-dimensional solute transport domain was discretized into cells
and a particle tracking method was developed for solving the convection-dispersion
equation (CDE) and the mobile-immobile (MIM) solute transport equation. Their results
showed that the injection method was a useful alternative to simulate 1-D solute transport
in laboratory columns. In addition, their injection method was extended to simulate
multiple source solute transport over the space or time domain.
33
However, several questions remained to be answered when applying the injection method
of Delay et al. (1997) to column experiments. These are:
a. How to describe the solute distribution in the column at the initial time, or just after the
injection?
b. How does the initial solute distribution after injection influence solute transport, and
how critical is this effect of the initial distribution on the observed BTCs?
c. What is the best technique to inject the solute in order that one-dimensional solute
transport conditions are maintained for all times after injection?
d. Under what injection conditions can the initial solute distribution be treated as a Dirac
delta function?
e. How should the outlet boundary conditions be specified for the free-inlet initial value
problem, and how do the specified outlet boundary conditions affect the analysis of the
observed BTCs?
This chapter attempts to answer the above questions by numerical simulations and
supporting laboratory column experiments. The specific objectives included:
a. Development of a column system that can be used to physically simulate one-
dimensional solute transport with a free inlet boundary
b. Testing of the system to determine the operating conditions under which the system
can be used to obtain reliable BTCs for analysis of solute transport behavior.
Specifically:
(i) Identify the influence of outlet boundary assumptions on the analysis of the BTCs
(ii) Quantify the influence of the initial solute distributions on the observed BTCs
3.2 Column System and Outlet Boundary Conditions
3.2.1 Column system
Figure 3.1 is a schematic of the column system that was developed for the miscible
displacement experiments. It consisted of a 5.1-cm diameter plexiglass column
containing the porous medium. The column was saturated and subjected to a constant
fluid flow rate by the means of the peristaltic pump. A length of tubing (termed as a
balance tubing) was connected to the inlet end of the column. The other end of the
34
balance tubing was open at the same elevation as that of discharge end of the column. A
clip was applied on the balance tubing.
An injection assembly was developed to directly inject tracer at an arbitrary location
along the column length. The assembly consisted of four injection units. Each injection
unit consisted of two syringes and one needle. The needles penetrated into the interior of
the column, and were aligned along radii of the column so that adjacent needles were
perpendicular to each other. The needle was connected to the syringes with tubing. In
order to inject the solute as uniformly as possible over the cross section of the column,
two rows of tiny openings were made on opposite sides along the length of the needles.
The needles were oriented in the column such that they were always parallel to the plane
of the cross-section. The objective of designing the injection assembly in this way, was
to ensure that the solute transport remained strictly one-dimensional for all times after
injection.
One syringe of each injection unit was filled with tracer solution, and the other was filled
with tracer–free background solution. As shown in Figure 3.1, an injection unit consisted
of syringe A, syringe B, tubing A, tubing B, tubing C, a clip, and a needle. Assuming
syringe A was filled with known amount of the tracer solution, then syringe B was filled
with the background solution.
Before injection of the tracer, the background solution was pumped through the column
until the required fluid flow velocity was achieved and the flow was steady. During this
time the background solution was allowed to fill the balance tubing, which was then
closed with a clip. At the time of injection, the pump was stopped, and the clip on the
balance tubing was removed. The clip on tubing C was removed, and tracer solution was
injected into the column passing through the tubing A and C. Some of the background
solution in syringe B was then injected into the column passing through tubing B and C,
in order to flush any tracer solution remaining in tubing C and in the needle, into the
column. Tubing C was then closed with the clip. This procedure was repeated for all the
other three injection units. After injection, the balance tube was once again closed, and
35
the pump was restarted. The time when the pump was restarted was taken as zero time for
analyses of the initial value solute transport problem.
Any residual tracer solution in tubing A was flushed back into syringe A using the
background solution remaining in syringe B. Syringe A was then removed and the
contents transferred into a volumetric flask. Any residual tracer solution in syringe A
was removed by repeated washings into the volumetric flask using the tracer-free
background solution. The solution in the flask was then diluted to the known volume
with tracer-free background solution. A sample of the solution in the flask was then
taken and analyzed in order to quantify the mass of residual tracer in the syringe A. The
solute mass difference in syringe A between before and after injection gave the solute
mass injected. The total mass injected was the sum of the masses injected by each of the
four injection units.
Keeping both the outlet and the balance tubing open during injection was very important
for keeping the injected tracer solution symmetrically distributed on either side of the
cross-sectional plane containing the needles. This was necessary in order to take the
injection location as the origin of the space coordinate for simulating one-dimensional
solute transport. Without use of the balance tubing, the injected solute peak would
deviate to the outlet side of the injection location, regardless of whether the discharge
tubing was left open or closed during injection. If this happened, the origin of the space
coordinate could not be specified.
The column system in Figure 3.1 can be easily extended to simulate multiple-source
solute transport over the time domain by multiple injections at one location but at
different times, or over the space domain by putting multiple injection units on one
column with arbitrary interval distance between the units, and injecting solute into the
column at same time from these injection units.
36
Peristalticpump
Backgroundsolution tank
Two rows of operating on each needle. Needle werePerpendicular to columnWall.
Fraction collector
Ceramic porous plate
20cmLength based on experimental design
The column was set vertically in experiments.
Holes
Stoppers
ClipTubing B
Tubing CTubing A
Syringe BSyringe A
Needle
Column wall
Design of injection assembly
Balance tubingClip
I
II
III
Figure 3.1 Schematic of the laboratory column system that was developed for conducting
the miscible displacement experiments with a free-inlet boundary, showing I.
experimental column system, II. injection assembly, and III. detail of a needle.
3.2.2 Outlet boundary conditions
The influence of boundary conditions on BTCs generated using columns, in which the
tracer was applied from one end and collected on the other end, has been researched for
many years (van Genuchten and Alves, 1982; van Genuchten and Parker, 1984). The
column system for this study was designed to physically simulate an inlet boundary free
problem with an outlet boundary. However, there were no similar studies that focused on
the influence of the outlet boundary conditions on BTCs generated when using a column
with an outlet boundary but no inlet boundary.
The influence of various outlet boundary conditions on the BTCs for this case was
therefore investigated numerically in this study. These investigations were carried out
using a particle tracking method (Uffink, 1985; Delay et al., 1997) to solve the
convective-dispersive equation (CDE) for solute transport. The detailed numerical
37
procedure was given by Delay et al. (1997). The influence of outlet boundary conditions
on the BTCs was determined by comparing BTCs generated using this solution of the
CDE subject to three different outlet boundary conditions. The computational scheme of
Delay et al. (1997) is illustrated in Figure 3.2.
i - 4 i - 3 i - 2 i - 1 i i + 2 i + 3i + 1 i + 4
i - 4 i - 3 i - 2 i + 3 i + 4i - 2 i - 1 i i + 1 i + 2
S S
K ∆ x K ∆ x
C o n v e c t i o n f r o m c e l l i - 1 t o i
D i s p e r s i o n
Figure 3.2 Computational scheme for the particle tracking method (adapted from Delay
et al., 1997).
In the Figure 3.2, the experimental domain is discretized into continuous cells of equal
size of ∆x. Time is discretized into time steps ∆t =∆x/v, which is equal to the time for
convection of all particles from cell i-1 to i. Here v represents the average pore water
velocity (Lt-1). Between time t and t+∆t, all particles in cell i-1 are moved by
convection to cell i (top portion of Figure 3.2). After the convection step, dispersion
distributes the particles in cell i, uniformly over a length 2S, centered in i (bottom
portion of Figure 3.2). Thus, for each time step ∆t, the Ni-1 particles in cell i-1 will drift
to i and become uniformly distributed around a point centered on i. The length S for
each convection-dispersion step is given by (Delay et al., 1997) as:
xS ∆= κ6 (3.1)
where κ is dispersivity.
38
Because the Ni-1 particles are assumed to be uniformly distributed around a point
centered on i, each cell completely covered by segment 2S receives Ni-1∆x/2S particles.
The two end cells not completely covered by segment 2S receive {Ni-1[S-(K+0.5)
∆x]}/2S particles, where K is the integer number of cells excepting cell I, covered by
the segment S. Starting with a given number of particles N in a given cell at time t = 0,
iteration of this method would generate a normal distribution of these particles along a
point x = vn∆t after n convection-dispersion steps.
The three outlet boundary conditions were:
a. The Zero concentration boundary condition (Barry and Sposito, 1988) given as:
C (3.2) 0)t,L( =+
where C is the concentration, L+ indicates a location just outside the outlet boundary of
the column of length L.
b. The zero gradient finite outlet boundary condition:
0),( ==∂∂ tLx
xC (3.3)
where L indicates the outlet boundary of the column of length x = L, and x is length.
c. The zero gradient infinite boundary condition:
0)t,x(xC
=∞→∂∂ (3.4)
The algorithm for the particle tracking method using the zero concentration outlet
boundary condition is illustrated in Figure 3.3. The zero concentration outlet boundary
condition means physically that there are no particles outside the column outlet boundary
at x = L that can disperse back into the column. It also implies that a particle is
discharged once it crosses the column outlet boundary by convection or dispersion.
In Figure 3.3, it is assumed that K = 2 (K and S are defined in Figure 3.2). Discretized
cells are indexed as I, I-1, I-2, .... The letters N and N' denote the particle number in the
discretized cells at time t and t + ∆t respectively. The matrix in Figure 3.3 illustrates how
39
the particle number for each discretized cell are calculated from N at initial time t, to N' at
time t + ∆t. In the matrix, F indicates that the cell is completely covered by S, and E
indicates an end cell that is partially covered by S, at completion of a convection-
dispersion step. As shown, each row of the matrix is indexed identically as the
discretized cells, and represents the redistribution by convection and dispersion of the N
particles initially present in the cell at time t. For example, during the convection-
dispersion step, the particles in discretized cell I-2 will redistribute as shown in row I-2 of
the matrix. In this row, N(I-2,F) = N(I-2)/(2*S), N(I-2,E)=[N(I-2)/(2*S)]*[S-(2+0.5)∆x],
and N(I-2) = 5*N(I-2,F)+2*N(I-2,E). The particles in the elements of each column of the
matrix are summed to obtain N'. Noutlet represent the sum of all particles discharged
outside of x = L by convection and dispersion. Noutlet is converted to a discharge solute
concentration as [Noutlet / (θ∆x)] * (mo/No) to obtain the BTC. Here θ = porosity, ∆x is the
discretized cell length, and mo / No is some arbitrary initial mass to number ratio of the
injected solute. For the next convection-dispersion step, N' is substituted for N and the
field scale) (Weber, 1986). At the microscopic scale, the heterogeneity is caused by
micro-level variation in soil texture and soil structure. At the macroscopic scale, the
heterogeneity is due to variation of pore scale heterogeneity, and variation in pebble and
cobble size. At the megascopic scale or field scale, the heterogeneity is due to variation
in macro-scale spatial characteristics such as layering, presence of rocks and rock
formations, solution channels or channels formed by plant roots or earthworms, and
disturbances caused by human activities such as agriculture.
Pore size heterogeneity can directly related to heterogeneity of solid particles in porous
media (Whitaker, 1972). The heterogeneity of solid particles can be expressed as some
size distribution. For example, natural soil aggregate size distribution can be described as
a mass-based lognormal distribution (Gardner, 1956), number-based fractal distribution
(Rieu and Sposito, 1991a), and mass-based fractal distribution (Rasiah et al., 1993).
A medium made up of different particle sizes would be heterogeneous at the pore or
micro scale, but can be considered as homogeneous at the column or macro-scale. In this
case, homogeneity at the column scale is defined in a statistical sense, meaning that the
113
micro scale heterogeneity of the porous medium is uniformly distributed within the
column. An observed property at this scale (macro-scale), does not change appreciably
for some arbitrary change in the specified scale of the column. Similarly, at the field
scale, statistical homogeneity means that the components of macro-scale heterogeneity
are uniformly distributed over the field, and that an observed property at the field scale
does not change appreciably with some arbitrary change in the specified scale for the
field.
Saying that porous media are heterogeneous does not mean that the spatial variation of
hydraulic conductivity of the porous media is completely random. In fact, the hydraulic
conductivity is spatially auto-correlated (Mulla and McBratney, 2000), and the auto-
correlation range is related to the scale of heterogeneity. For example, in a porous
medium with heterogeneity at the microscopic scale, the auto-correlation range should be
at the same scale of heterogeneity. A porous medium for a given observation scale is
termed as statistically homogeneous, if the variance of the natural logarithm of the
hydraulic conductivity [denoted as ln (K)] and the correlation tensor are fixed and finite
(Neuman, 1990). Also, a porous medium is statistically homogeneous if the spatial auto-
correlation of the field comprising ln(K) vanishes over the given observation scale
provided the field is stationary (Smith and Schwartz, 1980). The porous medium is not
statistically homogeneous, if the variance of ln(K) and its spatial auto-correlation scale
are infinite, such as in fractal heterogeneous media (Wheatcraft and Tyler, 1988), or in
hierarchical porous medium with universal scaling of hydraulic conductivity (Neuman,
1990). Also, if the spatial auto-correlation of the field comprising ln(K) does not vanish
over a given observation scale, and the field is stationary, the porous medium is not
statistically homogeneous at the given observation scale.
As shown in Chapter 4, the mechanical dispersion of a solute during transport in a porous
medium is caused by the fluctuations of the pore-water velocity, which are assumed to be
normally distributed in a given spatial scale. Mechanical dispersion is quantified by
dispersivity. The fluctuations in the pore-water velocity are caused by the heterogeneity
of the porous medium. Therefore, when the heterogeneity of the porous medium is scale-
114
dependent, the dispersivity is also scale-dependent. The fluctuations of velocity at a
given scale are determined primarily by the variation in hydraulic conductivity
observations at that scale. The variation in hydraulic conductivity can be quantified by
taking the variance of ln(K) at the given scale. For solute transport in a porous medium,
the solute is distributed over some spatial scale at a given time t. At time t, the change in
the variance of the solute distribution, which is quantified by the local time-dependent
dispersivity λT(t), is determined by the fluctuation of the pore-water velocities at this
scale (Figure 4.4). Therefore, if the variance of ln(K) at scale L1 is larger than that at
scale L2, where L2 > L1, the magnitude of pore-water velocity fluctuations in L2 will be
larger than that in L1. As a result, the local time-dependent dispersivity [λT(t1)] at time t1
will be larger than that [λT(t2)] at t2, where t1 and t2 are the times at which the solute is
distributed over spatial scales L1 and L2 respectively. When the porous medium is
statistically homogeneous at a given scale, the spatial variation of ln(K) observations at
this scale is completely random, and the variance of ln(K) is constant. Therefore, at this
scale, the fluctuation of pore-water velocities will be constant, and the corresponding
local time-dependent dispersivity will also be constant.
5.3 A New Function for Describing Scale-Dependent Dispersivity
As discussed in the previous section, it would be logical to expect that the local time-
dependent dispersivity function λT(t), could be directly derived from the distribution of
the variances of ln(K) over spatial scales. Some aspects of the general behavior of the
spatial dependence of the variances of ln(K) can be inferred from the semi-variogram of
ln(K). The semi-variogram is a plot of half the value of the mean squared differences
(called the semi-variance) between values of a set of observations in space, versus
adjacent values of the same set that are separated by a specified distance called the lag
separation. The semivariances of ln(K) increases with the lag separation over a range of
lag separation, beyond which it becomes constant. The increase over the range is
commonly assumed to be linear, spherical, or exponential (Mulla and McBratney, 2000).
Consequently, the local time-dependent dispersivity function λT(t), was assumed to be of
the same form as the semi-variogram of ln(K). Thus λT(t) was assumed to increase
linearly within time 0< t < tl, and become constant thereafter. tl is the time at which the
115
solute becomes spatially distributed over a spatial scale equal to the range of the semi-
variogram of ln(K). Many researchers (Pickens and Griska, 1981; Han et al., 1985;
Yate, 1990; Basha and El-habel, 1993) have suggested that the scale-dependent
dispersivity function increases linearly over some scale.
This new function was abbreviated as LIC, meaning that local scale-dependent
dispersivity linearly increases within some scale, after which it becomes constant. The
LIC is expressed as:
(5.1)
≥=<=
lT
lT
ttkvttttkvtt
)()(
λλ
(5.2)
≥=<=
lxklxlxkxx
D
D
)()(
λλ
where x is the mean solute travel distance, t is the mean solute travel time, k is the slope
of the linear portion of λT(t), v is the average pore-water velocity, tl is a transition time
after which the local time-dependent dispersivity λT(t) becomes constant, and l is the
transition region after which the local distance-dependent dispersivity λD(x) becomes
constant. Here l = vtl and is not equal to the range in the semivariogram of ln(K).
When a BTC is generated at L, the corresponding apparent scale-dependent dispersivities
based on Eq. (5.1) and Eq. (5.2), can be calculated using Eq. (4.18) and Eq. (4.21) as:
≥−+
=
<=
∫
∫
l
ll
t
T
l
T
T
tTT
tTkvtdkvT
tTT
dkvT
l
)()('
)('
0
0
ττα
ττα
(5.3)
where τ is a dummy integration variable and T is an arbitrarily defined time which is
assumed to represent the expected value of time distribution of the BTC at L (i.e. the
mean travel time of the BTC at L).
116
Similarly:
≥−+
=
<=
∫
∫
lLL
lLkldkL
lLL
dkL
l
D
L
D
)()(
)(
0
0
ζζα
ζζα (5.4)
where ζ is a dummy integration variable.
After integration, Eq. (5.3) can be written as:
≥−=
<=
ll
lT
lT
tTT
vktkvtT
tTkvTT
2)('
2)('
2
α
α (5.5)
And Eq. (5.4) can be written as:
≥−=
<=
lLL
klklL
lLkLL
D
D
2)(
2)(
2
α
α (5.6)
The relationships between the local scale-dependent dispersivities [Eq. (5.1) and Eq.
(5.2)] and the corresponding apparent scale-dependent dispersivities [Eq. (5.5) and Eq.
(5.6)] are illustrated in Figure 5.1 generated using assumed values for the parameters.
In Figure 5.1, it was assumed that k = 0.1, tl = 20, average pore water velocity v (Lt-1) =
1. These values imply that l (L) is also 20. As shown, the local scale-dependent
dispersivity increases linearly for L < l or T < tl, and becomes constant thereafter. The
apparent scale-dependent dispersivity (either apparent time-dependent dispersivity or
apparent distance-dependent dispersivity) curve shows three features:
a. it is linear or quasi-linear, when L<<l or T<<tl
117
b. there is not a sharp transition between the varying value region and the constant value
region of the curve.
c. the dispersivity becomes asymptotically constant, when L>>l or T>>tl.
These three features were observed in most laboratory column experiments (Han et al.,
1985; Irwin et al., 1996), and some field experiments (Sudicky et al., 1983; Mishra and
Parker, 1990).
0
1
2
3
0 20 40 60 80
Dis tance or Tim e
Dis
pers
ivity
Local scale-dependent dispersiv ityApparent scale-dependent dispersiv ity
100
Figure 5.1 Hypothetical local scale-dependent dispersivity and apparent scale-dependent
dispersivity generated using Eq. (5.1) through Eq.(5.6). See text for explanation.
There are two parameters (k, and tl or l) in Eq. (5.5) and Eq. (5.6). Therefore, they can be
determined if at least two apparent dispersivities are known for a statistically
homogeneous medium. These two apparent dispersivities can be used for estimating k
and l (or tl) provided that at least one of them is obtained at a scale larger than l (or tl).
Once the k and l (or tl) are obtained, the scale-dependent dispersivity distributions [Eq.
(5.1), Eq. (5.2), Eq. (5.4), and Eq. (5.5)] are completely specified. The scale-dependent
dispersivity in the scale-dependent CDE [Eq.(4.4) and Eq. (4.5)] is then quantified, and
therefore, a numerical solution of the scale-dependent CDE can be used to describe the
BTC at any location in the statistically homogeneous medium.
Eq. (5.5) and Eq (5.6) can also be applied, when the analytical solution of the CDE with
constant dispersivity [Eq. (4.3)] is used to describe solute transport in a statistically
118
homogeneous porous medium. This is because the value of the apparent-scale
dispersivity at a given scale L (or T) should also be the value of dispersivity in the CDE
with constant dispersivity [Eq. (4.3)], which is used to describing the BTC at L (or T),
provided that T (or L =Tv) would represent the expected value of the time distribution of
the BTC at scale L. When the BTC at a given scale needs to be described using the
analytical solution of the CDE with constant dispersivity [Eq. (4.3)], the apparent scale-
dependent dispersivity is first calculated at this scale using Eq.(5.5) or Eq. (5.6), and then
the CDE is solved by substituting the apparent dispersivity into the analytical solution.
As discussed above, tl is the time at which the solute is distributed over a spatial scale
equaling the range of the semivariogram of ln(K). It is possible to identify the range of
the semivariogram of ln(K) for a statistically homogeneous porous medium, by analyzing
BTCs for non-reactive solute transport with an initial solute input represented as a Dirac
delta function. In this case, the solute, at a given time, is normally distributed over some
spatial scale in the porous medium. When tl is obtained from the BTC analysis, the
variance σ2 of solute distribution over space at time tl, can be calculated using Eq. (4.11),
to obtain the standard deviation (σ) of the solute distribution. σ can be used to quantify
the scale covered by the spatial distribution of the solute at time tl , and this scale may
approximate the range in the semi-variogram of ln(K).
5.4 Materials and Methods
The laboratory column system detailed in Chapter 3 was used in the solute transport
experiments. Fluorescein was used as the non-reactive tracer, and the volume of solution
injected for one source at an injection assembly (four injection units) did not exceed 1 ml.
As discussed in Chapter 3, this volume would result in an initial injected solute
distribution that can be very closely represented as a Dirac delta function in the analyses
of the experimental BTCs. The column was uniformly packed with glass beads. Pore
scale heterogeneity of the glass beads was obtained by combining different sizes of the
glass beads as already presented in Chapter 3. This artificial medium was expected to be
heterogeneous at pore scale, but statistically homogeneous at the column scale.
119
The three solute transport problems described in Chapter 4 were simulated in the
experiments. For the problem 1, only one source was applied. Column lengths (distance
between source and outlet) were 16 cm, 38 cm, 59 cm, 83 cm, 103 cm, and 141 cm.
Average pore velocity was about 0.8 cm min-1. Four observed BTCs (replications) were
generated for each length. Apparent distance-dependent dispersivities were obtained by
fitting the observed BTCs to the analytical solution of the CDE [Eq. (3.10)]. Apparent
distance-dependent dispersivities obtained from BTC analyses for two lengths were used
to determine l and k in Eq. (5.5). The combination of two lengths selected for this
purpose was 38 cm and 59 cm, 38 cm and 83 cm, and 38 cm, and 103 cm. The values of
l and k obtained were used to predict the apparent scale-dependent dispersivity [Eq. (5.6)]
at 141 cm. The accuracy of the prediction of the apparent scale-dependent dispersivity
was evaluated by comparing the BTC generated with the CDE using the predicted
apparent scale-dependent dispersivity to the fitted BTC at 141cm. The fitted BTC was
generated with the CDE using the fitted apparent scale-dependent dispersivity obtained at
141 cm. If there was no statistically significant difference between the predicted BTC and
the fitted BTC, the predicted value of the apparent distance-dependent dispersivity was
accurate. The statistical significance of differences between the predicted BTC and the
fitted BTC was assessed using Fisher’s F statistic, with F = (sr,predicted)2/ (sr,fitted)2, where
sr2 is the lack-of-fit-square (Whitemore, 1991). The hypothesis of equality between
sr,predicted2 and sr,fitted
2 was rejected when the Fisher’s test statistic was bigger than the
critical value of F0.025, N-2, N-2 or smaller than F0.975, N-2, N-2, where N is the number of
samples of the BTC.
For problem 2, two sources, separated by a time interval, were applied at one location in
the transport domain. The distance L between the location of injection and the outlet was
98 cm, 77 cm and 55 cm, respectively. The time interval between the two sources was
decided by the value of L. It was about 15 min when L = 98 cm, 18 min when L = 77
cm, and 15 min when L = 55 cm.
Local distance-dependent dispersivity [Eq. (5.2)] was used in the numerical solution of
the CDE with scale-dependent dispersivity for predicting the BTCs at L. The
120
applicability of using the local distance-dependent dispersivity λD(x) was tested by
comparing the observed BTCs to the predicted BTCs. The predicted BTCs were
numerically generated using the CDE with local distance-dependent dispersivity [Eq.
(5.2)], defined by the values of k and l obtained from problem 1.
In order to simulate problem 3 physically, two injection assemblies were installed at
separate locations on one column. The distance between these two assemblies was 22
cm. The sources were simultaneously applied using these two assemblies. The location of
one assembly, which was close to the outlet, was defined as the origin of the space
coordinate. The location of the other assembly was taken as –22 cm from its origin.
Column lengths were 98 cm, 77 cm, and 55 cm.
The applicability of applying the local time-dependent dispersivity λT(t) to the numerical
scheme to solve the scale-dependent CDE for solute transport with simultaneous
multiple-source inputs over space, was tested by comparing the observed BTCs to the
predicted BTCs. The predicted BTCs were numerically generated using the CDE with
local time-dependent dispersivity [Eq. (5.1)], specified by the values of k and l obtained
from problem 1.
The particle tracking method (explained in Chapter 3) was used to numerically solve the
scale-dependent CDE for problem 2 and problem 3. The algorithms for applying local-
time dependent dispersivity and local distance-dependent dispersivity in the numerical
scheme for solving the scale dependent CDE have been already detailed in Section 4.1 of
Chapter 4.
For problem 1, the apparent distance-dependent dispersivity at a given length was
estimated by fitting the analytical solution of the CDE [Eq. (4.3)] to the BTC observed at
that length. The fitting was conducted using a nonlinear least-squares optimization
method, which were introduced in Chapter 3. l and k were obtained by fitting Eq. (5.6)
to the apparent distance-dependent dispersivity values obtained at the two lengths
121
specified above, using the NLIN procedure (non-linear least squares regression
procedure) in SAS.
5.5 Results and Discussion
As discussed in Section 4.2 of Chapter 4, for solute transport with a single solute input
source with initial distribution represented as a Dirac Delta function, T = L/v can be used
to represent the expected value of the time-distribution of the BTC at L under
experimental conditions specified by the Peclet number. Values of the Peclet number (Pe)
were about 40 when L = 16 cm; 95 when L = 38 cm; 145 when L = 59 cm; 215 when L =
83cm; 250 when L = 103 cm; and 320 when L = 141 cm. For these values of Pe, the
dispersivity obtained at a given L by fitting the observed BTCs to the CDE with a
constant dispersivity, was the same as the apparent scale-dependent dispersivity [αD(L)]
at L. BTCs observed at different column lengths were fitted using Eq. (3.10). The
apparent distance-dependent dispersivity αD(L) values obtained from these fits were
plotted versus column length as presented in Figure 5.2.
0.15
0.25
0.35
0.45
0 20 40 60 80 100 120 140 160
Length (cm)
Dis
pers
ivity
(cm
)
a
bbc
cd cd d
Figure 5.2 Distribution of apparent dispersivities [αD(L)], observed at different column
lengths. Plotted points represent the mean of 4 observations. Means with differing letters
were significantly different at p ≤ 0.05. The 95% confidence intervals are indicated on
the error bars.
122
As shown in the Figure 5.2, the apparent distance-dependent dispersivity αD(L),
increases as column length increases. However, the rate of this increase is reduced after
80 cm. This implied that the apparent distance-dependent dispersivity in the
experimental porous medium could asymptotically approach a constant after some length
scale. The experimental columns were uniformly packed with glass beads so that the
medium was heterogeneous at the pore-scale. This porous medium was expected to be
statistically homogeneous when the column was long enough. The apparent dispersivity
distribution with distance in Figure 5.2, indicated that the experimental medium could be
treated as statistically homogeneous after some column scale.
Statistical comparisons using the least significant difference (LSD) test at 5%
significance level showed that αD(L) at L = 16 cm was significantly different from αD(L)
at all other column lengths. αD(L) at L = 81 cm, 103 cm, and 141 cm were not
significantly different from each other.
The parameters k and l of the new scale-dependent dispersivity function were determined
by fitting Eq. (5.6) to the observed αD(L) at two column lengths. When the combination
of these two lengths was selected as 38 cm and 59 cm, values of k = 0.0281 and l =15.21
cm were obtained. The corresponding values for lengths 38 cm and 83 cm were k =
0.0208 and l = 21.13 cm. For lengths 38 cm and 103 cm, k = 0.0221 and l = 19.10 cm
were obtained.
The three sets of values obtained for k and l were substituted into Eq. (5.6) for predicting
the apparent distance-dependent dispersivity αD(L) distribution over distance. The three
predicted αD(L) distributions with distance were compared to the observed αD(L)
distribution, and the results are presented in Figure 5.3. As shown in Figure 5.3, all three
predicted αD(L) distribution curves were very close to the observed distribution. These
results indicated that the new function, in which the parameters were identified by
analyzing the BTCs observed at two length scales, could be used to predict the
distribution of scale-dependent dispersivity over the entire length scale for a statistically
homogeneous porous medium.
123
0
0.2
0.4
0.6
0 40 80 120 160Dis tance (cm )
Dis
pers
ivity
(cm
)
O b s e rv e d D is p e rs iv it ie sP re d ic te d a p p a re n t d is p e rs iv ity d is trib u t io n (k =0 .0 2 6 1 , l=1 5 .1 2 c m )P re d ic te d a p p a re n t d is p e rs iv ity d is trib u t io n (k =0 .0 2 0 8 , l=2 1 .1 3 c m )P re d ic te d a p p a re n t d is p e rs iv ity d is trib u t io n (k =0 .0 2 2 1 , l=1 9 .1 0 c m )
Figure 5.3 Comparison of the distribution of the observed apparent distance-dependent
dispersivities αD(L), over length, with predicted distribution using Eq. (5.6) and the three
sets of experimental values for k and l.
The accuracy of the prediction of the apparent scale-dependent dispersivity using Eq.
(5.6), was evaluated by comparing the BTC generated using the CDE with the predicted
αD(L) to the fitted BTC at 141 cm. The result of these comparisons is presented in
Figure 5.4. The comparison between the predicted BTC and the fitted BTC (Figure 5.4)
indicated that the predicted BTCs were not significantly different from the fitted BTC at
the 5% significance level. This result showed that the apparent dispersivity estimated
using Eq. (5.6) could be used in the scale-dependent CDE for accurately predicting the
BTCs for solute transport problem 1.
The result of numerical tests reported in Chapter 4 showed that, for the problem 1, the
BTCs estimated using apparent time and distance dependent dispersivity α'T(T) and
αD(L), local distance-dependent dispersivity λD(x), and local time-dependent dispersivity
λD(t) were indistinguishable for solute transport Peclet number >100. This numerical
result (summarized as principal concept (c) in section 5.1 for applying the scale-
dependent dispersivity) was verified in these experiments for the BTC at 141cm (Figure
5.5), for which the solute transport Peclet number was about 300. The numerical solution
124
of the CDE with apparent dispersivity [αD(L) or α'T(T)] , λD(t), or λD(x), was used to
generate the BTCs at 141 cm for comparison with the observed BTC for this column
Figure 5.4 Comparison of the observed BTC at 141 cm and the BTCs predicted using
the CDE with the three values of αD(L) calculated with Eq. (5.6). Critical values for the F
test statistic were F(0.025, 34,34) = 1.98, and F(0.975, 34,34) = 0.505, and values used were:
sample number N = 36, number of parameters = 2.
125
k = 0 . 0 2 6 1 , l = 1 5 . 1 2 c m , A p p a r e n t d i s p e r s i v i t y = 0 . 3 7 3 4 c m
0
0 .0 0 3
0 .0 0 6
0 .0 0 9
0 .0 1 2
1 1 0 1 3 0 1 5 0 1 7 0 1 9 0T im e ( m in )
Con
cent
ratio
n(m
M)
O b s e rv e d B T CB T C p re d ic t e d u s in g a p p a re n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l d is t a n c e -d e p e n d e n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l t im e -d e p e n d e n t d is p e rs iv it y
k = 0 . 0 2 0 8 , l = 2 1 . 1 3 c m , A p p a re n t d i s p e rs i v i t y = 0 . 4 0 6 6
0
0 .0 0 3
0 .0 0 6
0 .0 0 9
0 .0 1 2
1 1 0 1 3 0 1 5 0 1 7 0 1 9 0T im e (m in )
Con
cent
ratio
n(m
M) O b s e rv e d B T C
B T C p re d ic t e d u s in g a p p a re n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l d is t a n c e -d e p e n d e n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l t im e -d e p e n d e n t d is p e rs iv it y
k = 0 . 2 2 1 , l = 1 9 . 1 0 c m , A p p a r e n t d i s p e r s i v i t y = 0 . 3 9 3 5 c m
0
0 .0 0 3
0 .0 0 6
0 .0 0 9
0 .0 1 2
1 1 0 1 3 0 1 5 0 1 7 0 1 9 0T im e (m in )
Con
cent
ratio
n(m
M)
O b s e rv e d B T CB T C p re d ic t e d u s in g a p p a re n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l d is t a n c e -d e p e n d e n t d is p e rs iv it yB T C p re d ic t e d u s in g lo c a l t im e -d e p e n d e n t d is p e rs iv it y
Figure 5.5 Comparison of the observed BTC at 141 cm with BTCs predicted using the
numerical solution of the CDE with scale-dependent dispersivity given by the apparent
dispersivity [αT'(T) or αD(L)], local distance-dependent dispersivity λD(x) and local time-
dependent dispersivity λD(t)
126
For problem 2, two sources separated by a time interval, were applied at one location in
the transport domain. BTCs predicted using the CDE with local distance-dependent
dispersivitiy λD(x) [Eq. (5.2)] specified by the three sets of values of k and l obtained in
the problem 1, and the corresponding BTCs observed at different lengths are presented in
Figure 5.6. As shown in Figure 5.6, the predicted BTCs at 77 cm and 55 cm were
virtually indistinguishable from the observed BTCs. The predicted BTC at 98 cm were
somewhat distinguishable from the observed BTC; however, the error of the prediction
was acceptable.
For problem 3, solute was simultaneously injected input at two locations in the transport
domain. The local time-dependent dispersivity [Eq. (5.1)], specified by the three sets of
values of k and l obtained in the problem 1, was used to simulate scale-dependent
dispersivity in the numerical solution of the scale-dependent CDE. The predicted BTCs
and the observed BTC at different lengths are presented in Figure 5.7. The results in
Figure 5.7 show that for all lengths, the predicted BTCs were indistinguishable from the
corresponding observed BTCs.
The experimental results verified the findings of Chapter 4 (summarized as principal
concepts (d) and (e) in section 5.1 for applying the scale-dependent dispersivity), that the
local time dependent dispersivity λT(t) should be used when the numerical solution of
scale-dependent CDE is used to describe solute transport with simultaneous multiple
sources over the spatial domain. Further, they verified that the local distance-dependent
dispersivity λD(x) should be used when the numerical solution of the scale-dependent
CDE is applied to describe solute transport for multiple sources over time, but input at
one spatial location. In both cases, the numerical accuracy is acceptable and the time
Figure 5.6 Comparison of observed BTCs at different lengths for two sources input at
two times but at one location in the space domain, with the corresponding BTCs
predicted using the CDE with local distance-dependent dispersivities [λD(x)] calculated
using Eq. (5.6) and the three sets of experimental values for k and l.
128
L e n g t h = 9 8 c m a n d 1 2 0 c m
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2
5 0 7 0 9 0 1 1 0 1 3 0 1 5 0T im e ( m in )
Con
cent
ratio
n(m
M)
O b s e r v e d B T CP r e d ic t e d B T C : k = 0 .0 2 6 1 , l= 1 5 .1 2 c mP r e d ic t e d B T C : k = 0 .0 2 0 8 , l= 2 1 .1 3 c mP r e d ic t e d B T C : k = 0 .0 2 2 1 , l= 1 9 .1 0 c m
L e n g t h = 7 7 c m a n d 9 9 c m
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2
3 0 6 0 9 0 1 2 0T im e ( m in )
Con
cent
ratio
n (m
M)
O b s e r v e d B T CP r e d ic t e d B T C : k = 0 .0 2 6 1 , l= 1 5 .1 2 c mP r e d ic t e d B T C : k = 0 .0 2 0 8 , l= 2 1 .1 3 c mP r e d ic t e d B T C : k = 0 .0 2 2 1 , l= 1 9 .1 0 c m
L e n g t h = 5 5 c m a n d 7 7 c m
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2
0 .0 2 5
2 0 4 0 6 0 8 0 1 0 0T im e ( m in )
Con
cent
ratio
n(m
M)
O b s e r v e d B T CP r e d ic t e d B T C : k = 0 .0 2 6 1 , l= 1 5 .1 2 c mP r e d ic t e d B T C : k = 0 .0 2 0 8 , l= 2 1 .1 3 c mP r e d ic t e d B T C : k = 0 .0 2 2 1 , l= 1 9 .1 0 c m
Figure 5.7 Comparison of observed BTCs at different lengths for two simultaneous
sources over the spatial domain, with the corresponding BTCs predicted using the CDE
with local time-dependent dispersivities [λT(t)] calculated using Eq. (5.6) and the three
sets of experimental values for k and l.
129
5.6 Conclusions
At a given scale in a porous medium, if the porous medium is statistically homogeneous,
the local time-dispersivity λT(t) can attain a constant value after some transport time tl.
For solute transport with an initial source represented as a Dirac delta function, at such
time tl, the solute is spatially distributed over a scale which is approximately equal to the
range of semi-variogram of ln(K) in the porous medium. A local time-dependent
dispersivity function (LIC) was developed by assuming that λT(t) linearly increases over
time from 0 to tl, and then becomes constant thereafter. The local distance-dependent
dispersivity λD(x), and the apparent scale-dependent dispersivity αD(L) and α'T(T') can be
directly or indirectly obtained from λT(t) using the relationships between them. The LIC
can be defined by analyzing the BTCs observed at two length scales, where at least one
these BTCs is obtained at a scale larger than vtl. Miscible displacement experiments
conducted in specially designed columns showed that the LIC could accurately predict
the scale-dependent dispersivity distribution over increasing scales in statistically
homogeneous porous media. The analyses of experimentally observed BTCs using the
LIC also verified the principal concepts (c), (d) and (e) stated in the section 5.1 for
applying scale-dependent dispersivity in numerical solutions of the scale-dependent CDE.
130
CHAPTER 6 PREDICTION OF BREAKTHOUGH CURVES FOR
NON-REACTIVE SOLUTE TRANSPORT IN STATISTICALLY
HOMOGENEOUS POROUS MEDIA
6.1 Introduction
It is generally not practical feasible to make assessments of solute transport behavior in
soils and aquifers by in-situ field sampling and analysis over long periods of time.
Consequently, solute transport models have been developed to permit describing and
predicting solute transport behavior in the subsurface environment. The ability of a
model to describe and predict solute transport is determined by its parameters, which are
determined by the properties of the solutes and the subsurface porous media, and the
solute transport conditions.
Description of observed solute transport behavior, such as BTCs, by a solute transport
model is generally conducted by fitting the solute transport model to the observed data.
The adequacy of different models to describe observed solute transport behavior is
generally evaluated by comparing how well the models fit the observed data (Rao et al.,
1980a,b; Ma and Selim, 1994; Pachepsky et al., 2000). The fitting procedure optimizes
the values of parameters of the model in order to minimize the error of fitting. The
model with the least error is judged to be the most adequate. Consequently, when
applying such procedures, the value of the error becomes more important for evaluating
the adequacy of different models, rather than the values of the fitted parameters for the
model.
However, the values of the parameters of a model are very important when the model is
subsequently applied for purposes of predicting solute transport behavior. In such
situations, the accuracy of prediction of a solute transport model is mostly determined by
the values of applied parameters. When a parameter in the model is scale-dependent, its
value would change with different solute transport scales. The parameter value identified
at one scale represents the solute transport characteristics at this scale, and does not
131
necessarily represent the solute transport characteristics at other scales. Consequently,
the parameter value identified at one scale cannot be directly used to predict solute
transport behavior at other scales.
When the scale for predicting solute transport is different from the scale that was used to
experimentally identify the values of the model parameters, two different modeling
approaches (termed as approach I and approach II) for prediction are possible. In
approach I, a solute transport model with scale-independent parameters can be used. In
approach II, a solute transport model with scale-dependent parameters is used if such
models exist. However, the latter approach would entail explicitly or implicitly
specifying the value of the scale-dependent parameter at the scale of prediction (or
specifying the scale-dependent parameter distribution over scales) before the scale-
dependent solute transport model can be used for prediction. If this step is possible, the
specified parameter values (or specified distribution) are incorporated in the model for
use at the given scale of prediction.
As discussed in Chapters 4 and 5, for non-reactive solute transport in statistically
homogeneous porous media, the variance of the solute distribution grows non-linearly
with time. When the CDE is used to describe solute transport in these porous media, the
dispersivity parameter of the CDE is scale-dependent. Therefore, in this case, the CDE
with a constant dispersivity over scales [Eq. (4.3)] cannot be directly used for predicting
solute transport when the scale of parameter identification is different from the scale of
prediction. This means that the dispersivity identified at one scale cannot be directly
incorporated into the CDE [Eq. (4.3)] for predicting solute transport behavior (e.g. BTCs)
at other scales. Consequently, when non-reactive solute transport behavior (e.g. BTCs) in
statistically homogeneous porous media needs to be predicted, and when the scale of
prediction and the scale of parameter identification are different, either approach I or
approach II have to be applied.
For approach I, two potential models are the mobile-immobile model (MIM) (van
Genuchten and Wierenga, 1976) and the fractional convection-dispersion function,
132
abbreviated as the FCDE (Benson, 1998; Pachepsky et al., 2000). In the MIM, the flow
domain is described as a mobile region and an immobile region. The non-linear growth of
the variance of the solute distribution over time is explained by the solute exchange
between the two regions as it moves in the mobile phase. The FCDE assumes that the
random movements of solute particles obey a non-Gaussian statistical distribution, called
an α-stable distribution. Random movements of the solute particles according to an α-
stable distribution accounts for the non-linear growth of the variance of the solute
distribution over time. Because of these assumptions in the development of these two
models, their parameters are expected to be scale-independent. It implies that, for these
two models, their parameters identified at one observation scale could be directly used for
predicting BTCs at other scales. However, a search of the available literature, did not
show that rigorous experimental tests were conducted to determine whether the
parameters of these two models identified at one scale, can be directly using for
predicting BTCs at other scales in statistically homogeneous porous media.
In approach II, the scale dependent dispersivity in the CDE is described as a function of
time (Pickens and Griska, 1981; Basha and El-Hebel, 1993; Zou et al., 1996) or a
function of distance (Mishra and Parker, 1990; Yates, 1990; Logan, 1996). The physical
meaning of the scale-dependent dispersivity has been detailed in Chapter 4 and Chapter
5. Instead of using the dispersivity as a constant over scales, approach II treats the
dispersivity as some distribution function over scales. Therefore, if this scale-dependent
dispersivity distribution function can be identified over scales by observations, the
prediction of solute transport at other scales using the CDE becomes possible. Two
questions then need to be addressed, namely, how to identify the scale-dependent
dispersivity distribution function by observations and, what is the minimum number (one
or more than one) of observation scales required for such identification.
The objectives of this chapter were to:
a. Test whether the CDE with constant dispersivity [Eq. (3.5) or Eq. (4.3)], the MIM [Eq.
(3.8)], and the FCDE [Eq. (3.9)] can be used to predict BTCs for solute transport in
statistically homogeneous porous media at other scales using parameters determined from
133
observations at one scale.
b. Test whether observations at one scale would be sufficient to define the scale-
dependent dispersivity distribution function assuming the function to be power-law, and
whether this defined power-law scale-dependent dispersivity function can be used in the
CDE [Eq. (3.6) and Eq. (3.7)] to accurately predict experimental solute transport BTCs .
c. Test whether observations at two scales would be sufficient to define the scale-
dependent dispersivity distribution function assuming the function to be one of four types
namely, power-law, log-power, hyperbolic, and the LIC (see Chapter 5). Also, whether
these defined scale-dependent dispersivity functions can be used in the scale dependent
CDE to accurately predict experimental solute BTCs.
d. Analyze the applicability of the above four functions to adequately describe the
dispersivity distribution function over scales for predicting solute transport in statistically
homogeneous porous media.
6.2 Materials and Methods
BTCs obtained from miscible displacement column experiments with single source
injection (see Chapter 5) were analyzed. Two separate sets of analyses were conducted on
these BTCs. Firstly, BTCs at other scales were predicted using model parameters
identified from observations at one scale. Secondly, BTCs at other scales were predicted
using model parameters identified from observations at two scales.
In the first set of analyses, five models with parameters identified at a given observation
scale were used to predict the BTC at 141 cm. Observation scales of 16 cm, 38 cm, 59
cm, 83 cm, or 103 cm were used to identify the model parameters. The accuracy with
which the BTCs at 141 cm were predicted using these parameters in the five models was
evaluated by comparing the predicted BTC and the fitted experimental BTC. The fitted
BTC was generated with the models using the fitted parameters obtained 141 cm. The
statistical significance of differences between the predicted BTC and the fitted BTC was
assessed using Fisher's F statistic (see Chapter 5).
134
The five models were, the CDE with constant dispersivity denoted as “CCDE” [Eq. (3.4)
or Eq. (4.3)], the MIM [Eq. (3.7)], the FCDE [Eq. (3.8)], the CDE with power-law
distance-dependent dispersivity denoted as DCDE [Eq. (3.5)], and the CDE with power-
law time-dependent dispersivity denoted as TCDE [Eq. (3.6)]. The analyses using the
CCDE, the MIM, and the FCDE were conducted to satisfy objective (a) above. Analyses
using the DCDE and the TCDE were conducted for achieving objective (b) above. The
analytical solutions presented as Eq. (3.10), Eq. (3.11), and Eq. (3.12) were used to solve
the CCDE, the TCDE, and the FCDE respectively. Particle tracking methods were used
to numerically solve the DCDE and the MIM. The parameters of these five models were
estimated using the nonlinear least-squares optimization algorithm that was introduced in
Chapter 3.
In the second set of analyses, the CDE with scale-dependent dispersivity was used to
describe and predict the experimental BTCs. The scale-dependent dispersivity
distribution function over scales was determined from BTCs observed at two scales.
Determination of the dispersivity distribution functions over scales, meant specifying the
values of the function parameters. A given scale-dependent dispersivity distribution
function can be expressed in four different ways namely, as apparent distance-dependent
dispersivity αD(L), apparent time-dependent dispersivity α'T(T)], local time-dependent
dispersivity λT(t), or local distance-dependent dispersivity λD(x). However, because
αD(L) and α'T(T) were the same for the experimental solute transport problem [see Eq.
(4.19)] used in these analyses, they were both termed as apparent dispersivity, and
hereafter denoted only as αD(L). BTCs at 141 cm were predicted using the numerical
solution of the CDE with scale-dependent dispersivity. As already detailed in Chapter 4,
the specified dispersivity distribution functions were used to set the dispersivity value for
each cell at each time step in this numerical solution.
Four scale-dependent dispersivity distribution functions were used in this set of analyses
namely, the power-law function, log-power function, hyperbolic function, and LIC. The
LIC was detailed in Chapter 5. Analysis of the BTCs for these four scale-dependent
dispersivity functions was conducted to address objective (c) above.
135
The power-law functions for αD(L), λD(x), and λT(t) used in these analyses were based on
the function for αD(L) reported by Neuman (1990), Pachepsky et al. (2000), Wheatcraft
and Tyler (1988), and Su (1997) as :
(6.1a) bD aLL =)(α
from which
(6.1.b) bD xbax )1()( +=λ
(6.1c) bT vtbat ))(1()( +=λ
can be obtained using Eq. (4.18), Eq. (4.19), Eq. (4.21), and Eq. (4.22). Here v is the
average pore-water velocity, a and b are constants.
The log-power functions for αD(L), λD(x), and λT(t) were based on the function for αD(L)
reported by (Xu and Eckstein, 1995) as:
(6.2a) nD LmL )(log)( 10=α
from which
+= 1
)ln()(log)( 10 x
nxmx nDλ (6.2b)
+= 1
)ln()(log)( 10 vt
nvtmt nTλ (6.2c)
can be obtained using Eq. (4.18), Eq. (4.19), Eq. (4.21), and Eq. (4.22). Here m and n
are constants.
The hyperbolic functions for αD(L), λD(x), and λT(t) were based on the functions for
αD(L) and λD(x) reported by Mishra and Parker (1990) as:
( )
+−=
pqLpqLpLD
1ln1)(α (6.3a)
qxp
xD 111)(+
=λ (6.3b)
136
from which
qvtp
tT 111)(+
=λ (6.3c)
can be obtained using Eq. (4.21). Here p and q are constants
The constants of the dispersivity distribution functions were determined by fitting Eq.
(6.1a), Eq. (6.2a), and Eq.(6.3a) to the apparent dispersivities obtained at two selected
column lengths (Figure 5.2). For each column length (treatment) there were four
replications. Therefore 8 values of the apparent dispersivity were used to determine the
dispersivity distribution function for each pair of selected column lengths. The
combination of two column lengths selected for this purpose was 38 cm and 59 cm, 38
cm and 83 cm, and 38 cm and 103 cm. The fitting was conducted using the NLIN
procedure in the SAS software. When the specified apparent dispersivity distribution
functions were applied for prediction, the value of the apparent dispersivity at 141 cm
was first calculated. Then this value was incorporated into the analytical solution [Eq.
(3.11)] of the CDE with constant dispersivity for generating the predicted BTC at 141
cm. When the specified λD(x) and λT(t) functions were applied for prediction, the particle
tracking solution of the CDE with scale-dependent dispersivity [Eq. (4.4) and Eq. (4.5)]
was used for generating the predicted BTC at 141 cm. The particle tracking method was
introduced and explained in Chapter 3. The free draining outlet boundary condition [Eq.
(3.2)] was used for all BTCs generated. The algorithms for incorporating λD(x) and λT(t)
in the numerical scheme for solving the scale dependent CDE have been detailed in
Section 4.1 of Chapter 4. The accuracy with which these fitted scale-dependent
dispersivity functions were able to predict solute transport BTCs at other scales, after
incorporation in the scale dependent CDE, was evaluated by comparing the predicted
BTC and the fitted BTC. The F-statistic used for this comparison was introduced in
Chapter 5.
Objective (d) above was realized by comparing calculated apparent dispersivity
distributions over a large scale (>> the scale of parameter identification) based on the
137
four types of functions [Eq. (6.1a), Eq. (6.2a), Eq. (6.3a), and Eq. (5.6)]. Also, by
comparing the sensitivity of apparent dispersivity distributions predicted using these four
functions to changes in the observed apparent dispersivity values at one of the two scale
lengths used to determine these distribution functions. At a given prediction scale, the
sensitivity can be defined as:
−
−=
b
ba
b
ba
ODODOD
PDPDPD
S (6.4)
where S is the sensitivity of the apparent dispersivity distribution calculated by one of the
four functions at the given prediction scale length, PD is the predicted apparent
dispersivity at that scale, OD is one of the two observed apparent dispersivity values
used to determine the apparent dispersivity distribution. The subscript a represents the
value after the change, and the subscript b represents the value before the change. The
change means addition or subtraction of some arbitrary number taken to represent the
absolute error in measuring this observed value.
6.3 Results and Discussion
6.3.1 BTC Prediction at Other Scales Using Parameters Observed at One Scale
The CDE with Constant Dispersivty (CCDE): The observed apparent dispersivity
distribution over differing column lengths was presented in Figure 5.2 of Chapter 5.
Apparent dispersivity values obtained for column lengths of 16 cm, 38 cm, 59 cm, 83 cm,
and 103 cm were directly substituted into the CCDE [Eq. (4.3)] to predict the BTC at 141
cm. Comparison between the observed BTC, the predicted BTCs, and the fitted BTC at
141 cm are presented in Figure 6.1. The F-test indicated that the BTCs predicted using
the apparent dispersivities obtained at 16 cm and 38 cm were significantly different at the
5% level from the fitted BTC at 142 cm. However, the BTCs predicted using the apparent
dispersivities obtained at 59 cm, 83 cm, and 103 cm were not significantly different from
the fitted BTC at 141 cm.
As discussed in Chapter 5, the apparent dispersivity distribution in the experimental
porous media could be explained using the LIC [Eq. (5.6)]. The parameter l in the LIC
138
for the experimental porous media was close to 20 cm. In the LIC, the apparent
dispersivities are not so different from each other for scales >> l. Therefore, the apparent
dispersivity obtained by observation at one scale may be directly used for prediction of
the BTC at other scales, when the observed scale and predicted scale are both >> l. This
was the case for prediction at 141 cm using values obtained at 59 cm, 83 cm and 103 cm.
In the LIC, if the observed scale is not >> l when the predicted scale is >>l, using the
CCDE may cause serious error in the prediction. This was the case for the prediction at
141 cm using values obtained at 16 cm and 38 cm.
0.000
0.003
0.006
0.009
0.012
0.015
110 130 150 170 190Time(min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with ave.disp at 16cm (disp=0.2316cm, F=6.7334)
0.000
0.003
0.006
0.009
0.012
0.015
110 130 150 170 190Time(min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with ave.disp at 16cm (disp=0.2316cm, F=6.7334)BTC predicted with ave.disp at 38cm (disp=0.3124cm, F=2.351)BTC predicted with ave.disp at 59cm (disp=0.3360cm, F=1.786)BTC predicted with ave.disp at 83cm (disp=0.3834cm, F=1.136)BTC predicted with ave.disp at 103cm (disp=0.3845cm, F=1.128)Fitted: Dispersivity=0.4227cm, R2=0.97
Figure 6.1 Comparison of observed BTC, predicted BTCs, and fitted BTC at 141 cm.
using the CCDE solute transport model. The test statistic F = sr,predicted2/sr,fitted
2 where sr =
the lack-of-fit mean square. Critical values for the F statistic were F(0.025, 34, 34) =1.98 and
F(0.0975, 34, 34) = 0.505, and values used were : sample number N = 36, number of
parameters = 2. Ave. disp = average of the observed apparent dispersivities of four
replications.
The CDE with Power-law Distance-dependent Dispersivity (DCDE): Observed BTCs at
different column lengths were fitted using the DCDE. For a given observed BTC, the
non-linear least squares fitted values of the two parameters a and b in the local distance-
dependent dispersivity function λD(x) varied for different initial trial values of a and b.
Therefore the values of a and b were not uniquely defined for any observed BTC. One
139
set of fitted a and b values for the four replications of each of the six column lengths
(treatments) are presented in Figure 6.2. Because the a and b values could not be
uniquely defined, the results in Figure 6.2 could not be used to evaluate whether a and b
were scale-dependent or scale-independent.
Further analysis showed that a and b were not independent each other. The DCDE was
fitted to an observed BTC at 59 cm using three arbitrary initial trial values of a and b.
This resulted in fitted a and b values of a = 0.00183 and b = 1.48, a = 0.00253 and b =
1.392, and a = 0.00324 and b = 1.321. Even though the fitted values of a and b were
different, the fitted BTCs for these three combinations of a and b were not different. In
addition, all three fitted BTCs using the DCDE were the same as the BTC fitted for the
same observed BTC at 59 cm using the CCDE. Comparison of the observed BTC at 59
cm, the three fitted BTCs using different combinations of a and b in the DCDE, and the
BTC fitted using the CCDE is presented in Figure 6.3.
The fitted apparent dispersivity αD(L) obtained at 59 cm using the CCDE was 0.3041 cm.
As shown in Eq. (4.22), the apparent dispersivity αD(L) at 59 cm could also be obtained
from local distance-dependent dispersivity λD(x), which is specified by the fitted values
of a and b. The calculated αD(L) for these three a and b combinations were 0.3082 cm (a
= 0.00183 and b = 1.48), 0.3086 cm (a = 0.00253 and b = 1.392), and 0.3047 cm ( a =
0.00324 and b = 1.321). These results implied that the procedure of fitting the DCDE to
an observed BTC was equivalent to determining an apparent dispersivity at the observed
scale rather than determining unique values for a and b at this scale.
Using Eq. (4.22) and the above values of a and b, three calculated apparent dispersivity
distributions were obtained. These distributions are presented in Figure 6.4. As shown in
Figure 6.4, the fitted a and b values defined the apparent dispersivity at the scale of fitting
that matched the observed value. However, the apparent dispersivities at other scales
defined using these fitted a and b values did not match the observed values. The results in
Figure 6.4 also showed that the three calculated apparent dispersivity distributions were
different from each other. This fact implied that, even when a porous medium is
140
completely fractal, dispersivity distribution over scales could not be uniquely defined by
BTC analysis at one observed scale.
Consequently, it would be logical to expect that directly applying fitted values of a and b,
which are specified by BTC analysis at one observed scale, to predict BTCs at other
scales in statistically homogeneous porous media might produce serious error. The
expectation was verified by predicting the BTC at 141 cm using a and b, which were
determined by fitting the DCDE to the observed BTCs at 16 cm, 38 cm, 59 cm, 83 cm
and 103 cm. Comparison of the observed BTC and predicted BTCs is presented in
Figure 6.5. No statistical analysis was carried out to evaluate the accuracy of prediction,
because the differences between the predicted and observed BTCs were obvious. The
results in Figure 6.5 further confirmed that the dispersivity parameters a and b cannot be
completely nor uniquely determined by fitting of the observed BTC at one scale.
Therefore, such fitted values cannot be directly applied for predicting BTCs at other
scales in statistically homogeneous porous media.
Fitted 'a'
0
0.01
0.02
0 50 100 150Distance(cm)
a va
lue
Fitted 'b'
0
1
2
0 50 100 150Distance(cm)
b va
lue
Figure 6.2 One set of fitted a and b at different column lengths obtained by fitting
Figure 6.3 Comparison of the observed BTC at 59 cm to the BTCs fitted using the
DCDE and the CCDE. The DCDE was fitted using three arbitrary initial trial values of a
and b, which resulted in different fitted a and b values with the DCDE. The calculated
apparent dispersivity were obtained using Eq. (4.22) with these a and b values.
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120 140 16
Dis tance (cm )
Dis
pers
ivity
(cm
)
A ppa re n t d is pe rs iv ity c a lc u la te d us ing a =0 .00183 a nd b=1 .480A ppa re n t d is pe rs iv ity c a lc u la te d us ing a =0 .00253 a nd b=1 .392A ppa re n t d is pe rs iv ity c a lc u la te d us ing a =0 .00324 a nd b=1 .321O bs e rve d a ppa re n t d is pe rs iv ity
0
Figure 6.4 Comparison of observed apparent dispersivity distribution with those
calculated using Eq. (4.22) with a and b values obtained by fitting the DCDE to one
observed BTC at 59 cm.
142
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average a and b at 16cm ( a=0.01673 and b=1.2198)BTC predicted with the average a and b at 38cm ( a=0.00679 and b=1.297)BTC predicted with the average a and b at 59cm ( a=0.00387 and b=1.3093)BTC predicted with the average a and b at 83cm ( a=0.00186 and b=1.4085)
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average a and b at 16cm ( a=0.01673 and b=1.2198)BTC predicted with the average a and b at 38cm ( a=0.00679 and b=1.297)BTC predicted with the average a and b at 59cm ( a=0.00387 and b=1.3093)BTC predicted with the average a and b at 83cm ( a=0.00186 and b=1.4085)BTC predicted with the average a and b at 103cm ( a=0.00117 and b=1.4446)Fitted BTC at 141cm (a=0.000537, b=1.539 and R2=0.975)
Figure 6.5 Comparison of the observed BTC at 141 cm and BTCs predicted for this
length using the DCDE. Each combination of the parameters a and b used for prediction
was determined by fitting the DCDE to the observed BTCs at one column length (16 cm,
38 cm, 59 cm, 83 cm or 103 cm)
The CDE with Power-law Time-Dependent Dispersivity (TCDE): As discussed in
Chapter 4, the dispersivity distribution function for the TCDE was physically equivalent
to that for the DCDE. It was shown that the local time-dependent dispersivity λT(t) is
equal to the local distance dependent dispersivity λD(x) when the Peclet number >50 and
x = vt [Eq. (4.21) and Figure (4.6) and Figure (4.7)] . Therefore, fitting the TCDE to an
observed BTC at a given scale was equivalent to determining the apparent dispersivity at
the given scale rather than uniquely specifying the parameters c and d for describing the
dispersivity distribution over scales. Therefore, it would be expected that the results of
analyses of BTCs using the TCDE would not be any different from those obtained using
the DCDE.
143
One set of the fitted values of c and d at different column lengths is presented is Figure
6.6. Because the c and d values could not be uniquely defined for the TCDE, the results
in Figure 6.6 could not be used to evaluate whether c and d were scale-dependent or
scale-independent. Fitted values of c and d for a given BTC at an observed scale were
affected by the initial trial values of c and d. The fitted values were related to each other
by the apparent dispersivity at this observed scale [Eq. (4.18)]. For example, when one
observed BTC at 141 cm was fitted to the TCDE, an infinite combination of c and d
values were possible. For purposes of illustration and argument, three arbitrary
combinations of fitted values of c and d were obtained from fitting the TCDE to one of
the observed BTCs at 141 cm. These values were c = 0.000174 and d = 1.755; c =
0.0000527 and d = 2.008; and c = 0.00000247 and d = 2.655. The calculated apparent
dispersivities using these values were 0.4244 cm, 0.419 cm, and 0.4168 respectively and
were not different each other. These calculated apparent dispersivities were also not
different from the apparent dispersivity of 0.4277 cm obtained by fitting the observed
BTC at 141 cm to the CCDE. These results are presented in Figure 6.7 and demonstrated
that fitting the TCDE to an observed BTC at a given scale was equivalent to determining
the apparent dispersivity at the given scale rather than uniquely specifying the parameters
c and d for describing the dispersivity distribution over scales.
Using Eq. (4.18), Eq. (4.19), and the above values of c and d, three calculated apparent
dispersivity distributions were obtained, and these are presented in Figure 6.8.
Comparison of these calculated apparent dispersivity distributions to the observed
distribution, showed that the apparent dispersivity defined by the fitted c and d values at
the scale of fitting matched the observed value. However, the apparent dispersivities at
other scales defined using these fitted c and d values did not match the observed values.
The results in Figure 6.8 also showed that the three calculated apparent dispersivity
distributions were different from each other.
As discussed above, values of c and d obtained from analysis of the observed BTC at one
scale, cannot be used to completely and uniquely define the dispersivity distribution over
scales. Consequently, and if they were directly used to predict the BTCs at other scales,
144
the prediction may be grossly wrong. This expectation was verified by the results
presented in Figure 6.9. In Figure 6.9, the BTC at 141 cm was predicted by directly
using the values of c and d, which were determined by fitting the TCDE to the observed
BTCs at 16 cm, 38 cm, or 59 cm. When these predicted BTCs were compared to the
observed BTC at 141 cm, there were serious discrepancies.
Fitted 'c'
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0 50 100 150Distance(cm)
c va
lue
Fitted 'd'
0.0
1.0
2.0
3.0
0 50 100Distance(cm)
d va
lue
150
Figure 6.6 One set of fitted c and d at different column lengths obtained by fitting
Figure 6.7 Comparison of the observed BTC at 141 cm to the BTCs fitted using the
TCDE and the CCDE. The TCDE was fitted using three arbitrary initial trial values of c
and d, which resulted in different fitted c and d values with the TCDE. The calculated
apparent dispersivity were obtained using Eq. (4.19) with these c and d values.
145
0
0.1
0.2
0.3
0.4
0.5
0.6
0 20 40 60 80 100 120 140 160
Mean travel time (min)
Dis
pers
ivity
(cm
)Apparent dis pers ivity ca lcula ted us ing c=0.000174 and d=1.755Apparent dis pers ivity ca lcula ted us ing c=0.0000527 and d=2.008Apparent dis pers ivity ca lcula ted us ing c=0.00000247 and d=2.655Obs erved apparent dis pers ivity
Figure 6.8 Comparison of observed apparent dispersivity distribution with those
calculated using Eq. (4.19) with c and d values obtained by fitting the TCDE to one
observed BTC at 141 cm.
0
0 .005
0 .01
0 .015
0 .02
120 140 160 180 200
T im e (m in )
Con
cent
ratio
n(m
M)
O b s e rv e d B T C .
B T C p re d ic t e d wit h t h e a v e ra g e c a n d d a t1 6 c m ( c = 0 .0 3 7 7 1 a n d d = 0 .8 1 2 9 )B T C p re d ic t e d wit h t h e a v e ra g e c a n d d a t3 8 c m ( c = 0 .0 2 3 4 3 a n d d = 0 .8 3 1 7 )
B T C p re d ic t e d wit h t h e a v e ra g e c a n d d a t5 9 c m ( c = 0 .0 1 9 3 7 a n d d = 0 .8 0 7 2 )B T C fit t e d a t 1 4 1 c m (c = 2 .4 6 8 a n dd = 2 .6 5 5 2 )
Figure 6.9 Comparison of the observed BTC at 141 cm and BTCs predicted for this
length using the TCDE. Each combination of the parameters c and d used for prediction
was determined by fitting the TCDE to the observed BTCs at one column length (16 cm,
38 cm, or 59 cm)
146
Mobile Immobile Model (MIM): Four parameters are obtained when an observed BTC is
fitted to the MIM. These four parameters are average pore water velocity v, dispersivity
κ [= D/v in Eq. (3.8a)], first–order mass transfer coefficient β (or beta), and immobile
porosity θim. The value of velocity was determined by the water discharge flux for each
replication (or run) in the miscible displacement experiments. However, the other three
parameters were expected to depend only on the characteristics of the porous medium
and the solute. Consequently, their values were expected to be the same for all
treatments and replications for the experimental medium and solute. Distributions of
these three fitted parameters over observed lengths are presented in Figure 6.10. As
shown in Part A of Figure 6.10, the dispersivity κ was not constant with increasing
column lengths. The results in Part B of Figure 6.10 showed that only the value of beta at
16 cm was significantly higher than the values at other column lengths. The values of
beta were not significantly different from each other for column lengths of 38 cm, 59 cm,
83 cm, 103 cm, and 141 cm. Similarly, values of immobile porosity θim (Part C of Figure
6.10), were not significantly different each other for the column lengths of 59 cm, 83 cm,
103 cm, and 141 cm. However, these values were significantly lower than the
corresponding values for the 16 cm and 38 cm column lengths. These results implied that
the assumption that κ, beta, and θim were scale-independent might not be adequate and
applicable for the experimental column lengths. However, it might be adequate and
applicable to assume that θim and beta were constant when the column length >38 cm.
The average fitted values of κ, beta, and θim for the BTCs obtained at each column length
(16 cm, 38 cm, 59 cm, 83 cm or 103 cm ) for the four replications were directly used in
the MIM to predict the BTC at 141 cm. Comparison of the observed BTC, predicted
BTCs, and the BTC fitted using the MIM at 141 cm, are presented Figure 6.11. As
shown in Figure 6.11, all the predicted BTCs were significantly different from the fitted
BTC at 141 cm based on the F-test at the 5% significance level.
The MIM was developed for explaining non-symmetric BTCs with long tails for non-
reactive solute transport in porous media. This advantage of the MIM was verified in
these experiments. The tails of the observed BTCs could be accurately explained (or
147
fitted) using the MIM such as the BTC at 141cm in Figure 6.11. However, all the other
models used in these analyses were not able to accurately describe the tailing effect. This
can be seen by comparing the fitted BTC at 141 cm in Figure 6.11 with those fitted using
the CCDE (Figure 6.1), the DCDE (Figure 6.3), the TCDE (Figure 6.7), and the CDE
with scale-dependent apparent dispersivity functions (Figure 5.4). As will be shown
later, the same was true for the FCDE (Figure 6.13). Even through the experimental
porous media consisted of packed solid glass beads, it was still possible that 1 to 2 % of
immobile fluid existed in the column. This possibility was discussed in Chapter 3.
The results implied that directly applying the parameters identified by fitting the MIM to
an observed BTC might cause some error in predicting the BTCs for non-reactive solute
transport in statistically homogeneous porous media at other scales. These results also
implied that, although the MIM provided reasonable description (or explanation) of a
given experimental BTC, this did not necessarily mean that the model parameters
obtained could be used for prediction purposes. This may be more likely when the
observed scale for identifying the model parameters is different from the scale for the
prediction.
148
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a
b b
b
b b
A
B
C
Figure 6.10 Distribution of (A) dispersivity κ, (B) first- order mass transfer coefficient
beta and (C) immobile porosity θim , over different column lengths obtained by fitting
experimental BTCs to the MIM. Values represent the treatment means and 95%
confidence level (CL) for multiple comparisons. Means with differing letters are
significantly different at P≤ 0.05.
149
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time (min)
Con
cent
ratio
n(m
M)
Observed BTCBTC predicted with ave. value at 16cm (Disp.=0.1714cm, n_im=0.02413, beta=0.004305. R2=0.659, F=122.34)BTC predicted with ave. value at 38cm (Disp.=0.2386cm, n_im=0.0168, beta=0.001392. R2=0.988, F=4.505)BTC predicted with ave. value at 59cm (Disp.=0.2666cm, n_im=0.01088, beta=0.009095. R2=0.995, F=2.141)BTC predicted with ave. value at 83cm (Disp.=0.3027cm, n_im=0.009925, beta=0.009808. R2=0.991,F=3.322)BTC predicted with ave. value at 103cm (Disp.=0.2683cm, n_im=0.0114, beta=0.009443. R2=0.996, F=1.749)
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time (min)
Con
cent
ratio
n(m
M)
Observed BTCBTC predicted with ave. value at 16cm (Disp.=0.1714cm, n_im=0.02413, beta=0.004305. R2=0.659, F=122.34)BTC predicted with ave. value at 38cm (Disp.=0.2386cm, n_im=0.0168, beta=0.001392. R2=0.988, F=4.505)BTC predicted with ave. value at 59cm (Disp.=0.2666cm, n_im=0.01088, beta=0.009095. R2=0.995, F=2.141)BTC predicted with ave. value at 83cm (Disp.=0.3027cm, n_im=0.009925, beta=0.009808. R2=0.991,F=3.322)BTC predicted with ave. value at 103cm (Disp.=0.2683cm, n_im=0.0114, beta=0.009443. R2=0.996, F=1.749)Fitted BTC ( Disp.=0.2487, n_im=0.0141, and beta=0.0008847, R2=0.998)
Figure 6.11 Comparison of the observed BTC at 141 cm to the fitted BTCs and the
predicted BTCs for this length using the MIM. Each combination of the parameters for
dispersivity κ (disp), first-order mass transfer coefficient (beta), and immobile porosity
θim (im) was determined by fitting the MIM to the observed BTCs at one column length
(16 cm, 38 cm, 59 cm, 83 cm or 103 cm). The test statistic F = sr,predicted2/sr,fitted
2 where sr
= the lack-of-fit mean square. Critical values for the F statistic were F(0.025, 32, 32) = 2.025
and F(0.0975, 32, 32) = 0.494, and values used were: sample number N = 36, number of
parameters = 4. Ave.= average of observed or fitted values for 4 replications.
The Factional Convection Dispersion Equation (FCDE): Distributions of the fractional
differentiation order α, and the fractional dispersion coefficient Df obtained by fitting the
BTCs at different column lengths using the FCDE are presented in Figure 6.12. The fitted
values of α ranged from 1.5 to 1.8, and were lower than 2, which is the derivative order
in the CDE. Except for the Df value fitted for the BTC at 16 cm, the Df values at all the
other lengths (38, 59, 81, 103, and 141 cm) were not significantly different from each
other at 5% level of significance (Figure 6.12). The results for the fitted α values at
different column lengths were more complex than the corresponding results for Df. The
fitted values of α were not significantly different from each other for the following
column lengths :16 cm, 59 cm, and 103 cm; 38 cm and 103 cm; 38 cm and 141 cm; and
150
59 cm and 83 cm. Overall, the results in Figure 6.12 implied that the assumption that Df
and α were scale-independent might be inadequate and inapplicable for the experimental
porous media used in this study.
Figure 6.13 presents the comparison of the observed BTC, the BTC fitted using the
FCDE at 141 cm, and the predicted BTCs at this length calculated using the FCDE with
the values of Df and α averaged over the four replications for each column length. As
shown in Figure 6.13, only the BTCs predicted using the average Df and values α
obtained at 83 cm or 103 cm were not significantly different to the BTC fitted at 141cm
at the 5% significance level. When compared to the results using the CCDE (Figure 6.1),
it becomes apparent that the accuracy of the BTC predictions using the FCDE was no
better than those obtained using the CCDE for the porous media used in these
experiments. In both cases good prediction were obtained only when the fitted
parameters for the BTCs at 83 cm and 103 cm were used for predicting the BTC at 141
cm. This implied that, for these two models, good predictions would be obtained if the
scale of the predicted BTC were not markedly different from the observation scale for the
BTCs used for parameter identification.
In the FCDE, Df is expected to be constant over scales because the fractional
differentiation order α is introduced to account for observed long-tailed solute dispersion
effects. The order α is expected to be constant also, since it is determined by the
characteristics of porous media. The FCDE was developed by assuming that the
movement of solute particles in porous media obeys a symmetric Levy distribution. The
symmetric Levy distribution is similar to the Gaussian distribution, except that it has
fatter tails than the Gaussian. Therefore, the FCDE is expected to explain long-tailed
BTCs. However, in these experiments, the long tail of the BTC could not be explained
by the FCDE. The tails of the fitted BTCs and observed BTCs were obviously different
(Figure 6.13). Among the CCDE, DCDE, TCDE, MIM, and FCDE, only the MIM
could explain the long tails of observed BTCs in these experiments (Figure 6.11).
151
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ab
aba
ab
ab
Figure 6.12 Distribution of the parameters Df and α obtained by fitting observed BTCs
for different column lengths to the FCDE. Values represent the treatment means and 95%
confidence level (CL) for multiple comparisons. Means with differing letters are
significantly different at P≤ 0.05
for different column lengths to the FCDE. Values represent the treatment means and 95%
confidence level (CL) for multiple comparisons. Means with differing letters are
significantly different at P≤ 0.05
0
0.003
0.006
0.009
0.012
100 110 120 130 140 150 160 170 180 190 200
Tim e (min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average value at 16cm (Df =0.1721 and α =1.6307. R2 =0.947, F=4.467)BTC predicted with the average value at 38cm (Df =0.1909 and α =1.6223. R2=0.975, F=2.095) BTC predicted with the average value at 59cm (Df =0.1987 and α =1.6903. R2 =0.968, F=2.725)
0
0.003
0.006
0.009
0.012
100 110 120 130 140 150 160 170 180 190 200
Tim e (min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average value at 16cm (BTC predicted with the average value at 38cm (DBTC predicted with the average value at 59cm (DBTC predicted with the average value at 83cm (Df =0.2196 and α =1.7121. R2 =0.978, F=1.807)BTC predicted with the average value at 103cm (Df =0.2071 and α =1.6365. R2 =0.979, F=1.738)Fitted BTC (Df =0.2017 d α =1.5129. R2=0.988
0
0.003
0.006
0.009
0.012
100 110 120 130 140 150 160 170 180 190 200
Tim e (min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average value at 16cm (Df =0.1721 and α =1.6307. R2 =0.947, F=4.467)BTC predicted with the average value at 38cm (Df =0.1909 and α =1.6223. R2=0.975, F=2.095) BTC predicted with the average value at 59cm (Df =0.1987 and α =1.6903. R2 =0.968, F=2.725)
0
0.003
0.006
0.009
0.012
100 110 120 130 140 150 160 170 180 190 200
Tim e (min)
Con
cent
ratio
n (m
M)
Observed BTCBTC predicted with the average value at 16cm (BTC predicted with the average value at 38cm (DBTC predicted with the average value at 59cm (DBTC predicted with the average value at 83cm (Df =0.2196 and α =1.7121. R2 =0.978, F=1.807)BTC predicted with the average value at 103cm (Df =0.2071 and α =1.6365. R2 =0.979, F=1.738)Fitted BTC (Df =0.2017 d α =1.5129. R2=0.988
Figure 6.13 Comparison of the observed BTC at 141 cm to the fitted BTCs and the
predicted BTCs for this length using the FCDE. Each combination of the parameters Df
and α was determined by fitting the FCDE to the observed BTCs at one column length
(16 cm, 38 cm, 59 cm, 83 cm or 103 cm). The test statistic F = sr,predicted22/sr,fitted
22 where sr =
the lack-of-fit mean square. Critical values for the F statistic were F(0.025, 33, 33) = 2.002,
F(0.0975, 33, 33) = 0.499 and values used were: sample number N = 36, number of parameters
=3. Average value = average of observed or fitted values for 4 replications for a given
column length.
Figure 6.13 Comparison of the observed BTC at 141 cm to the fitted BTCs and the
predicted BTCs for this length using the FCDE. Each combination of the parameters Df
and α was determined by fitting the FCDE to the observed BTCs at one column length
(16 cm, 38 cm, 59 cm, 83 cm or 103 cm). The test statistic F = sr,predicted /sr,fitted where sr =
the lack-of-fit mean square. Critical values for the F statistic were F(0.025, 33, 33) = 2.002,
F(0.0975, 33, 33) = 0.499 and values used were: sample number N = 36, number of parameters
=3. Average value = average of observed or fitted values for 4 replications for a given
column length.
152
6.3.2 BTC Prediction at Other Scales by Parameters Observed at Two Scales
As discussed in Chapters 4 and 5, one procedure of applying scale-dependent dispersivity
involves the following steps:
a. Determine apparent dispersivity values at observed scales by fitting the CCDE to the
observed BTCs.
b. Determine the apparent dispersivity distribution by fitting a selected distribution
function to these apparent dispersivity values
c. Apply the distribution function determined in this way for predicting the BTCs at other
scales. If the CCDE is used for prediction, the apparent dispersivity values at the scales
for prediction are estimated first by the distribution function. These estimated apparent
dispersivities are then substituted into the CCDE for prediction. If the CDE with local
time-dependent dispersivity λT(t), or local distance-dependent dispersivity λD(x), is
applied for prediction, first the λT(t) or λD(x) has to be derived from the apparent
dispersivity function using the relationships between them [Eq. (4.18), Eq. (4.19), Eq.
(4.21) and Eq. (4.22)]. Then the λT(t) or λD(x) obtained in this manner is applied into the
numerical solution of the CDE for predicting the BTCs.
As detailed in Chapter 5, this procedure was very efficient and accurate for predicting the
BTCs when the two-parameter LIC function was used to describe the scale-dependent
dispersivity distribution over the experimental length scales, and the observed BTCs at
two scales were used to determine this LIC. In this section, three other two-parameter
apparent dispersivity functions [Eq (6.1a), Eq. (6.2a), Eq. (6.3a)] were selected, and the
procedure outlined above and previously implemented using the LIC, was repeated with
these three other functions.
Analysis of the observed BTCs from three combinations of two length scales was used to
determine the apparent scale-dependent dispersivity distribution functions specified by
Eq. (6.1a), Eq. (6.2a) and Eq. (6.3a). The parameters obtained for these functions are
listed in Table 6.1. The results of applying these functions namely, the power-law
function, log-power function, and hyperbolic function, to describe the dispersivity
153
distribution over experimental column lengths, and to predict the BTC at 141 cm are
presented in Figure 6.14 (power-law), Figure 6.15 (log-power), and Figure 6.16
(hyperbolic). Each of these figures consists of four parts. Part (A) presents the
comparison of the observed dispersivity values and the predicted apparent scale-
dependent dispersivity distribution [αD(L)] over the experimental length scales. Part (B)
presents the comparison of the observed BTC at 141 cm, the fitted BTC at 141 cm using
the CCDE, and the BTCs predicted using the CDE with αD(L) for this length. Part (C)
presents the comparison of the observed BTC at 141 cm, the fitted BTC at 141 cm using
the CCDE, and the BTCs predicted using the CDE with λD(x). Part (D) presents the
comparison of the observed BTC at 141 cm, the fitted BTC at 141 cm using the CCDE,
and the BTCs predicted using the CDE with λT(t).
As shown in these three sets of figures, observations at two length scales were sufficient
to determine the scale-dependent dispersivity distribution in the experimental porous
media. The predicted BTCs at 141 cm were not significantly different from the fitted
BTC at this length in all cases analyzed. Applying αD(L), λD(x), or λT(t) did not
markedly affect the predicted BTCs at 141 cm regardless of the three functional forms
used for their description. This finding, taken in conjuction with the results of Chapter 5,
implied that the four scale-dependent dispersivity functions (power-law, log-power,
hyperbolic, and LIC) had the same accuracy for describing the scale dependent
dispersivity distribution and for predicting the BTCs at the experimental column scales
used in this study.
.
Table 6.1 Parameters of three dispersivity functions obtained by analysis of the
observed BTCs from three combinations of two length scales. Parameter At 38 cm and 59 cm At 38 cm and 83 cm At 38 cm and 103 cm
Power-law function: A 0.1596 0.1312 0.1573 B 0.189 0.2428 0.1928
Log-power function: M 0.2273 0.2031 0.2207 N 0.7289 0.9751 0.7936
Hyperbolic function: P 0.4317 0.4989 0.4624 Q 0.0964 0.0637 0.0774
154
A0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150Distance (cm)
Disp
ersi
vity
(cm
)
Observed Predicted: a=0.1596 and b=0.1890Predicted: a=0.1312 and b=0.2428Predicted: a=0.1573 and b=0.1928
A0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150Distance (cm)
Disp
ersi
vity
(cm
)
Observed Predicted: a=0.1596 and b=0.1890Predicted: a=0.1312 and b=0.2428Predicted: a=0.1573 and b=0.1928
B
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Conc
entr
atio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, disp=0.4066cm, F=1.024Predicted:a=0.1312, b=0.2428, disp=0.4328cm, F=1.006Predicted: a=0.1573, b=0.1928, disp=0.4084cm, F=1.018Fitted: Dispersivity=0.4227cm and R 2=0.97
C
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Con
cnet
ratio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, F=1.301.Predicted:a=0.1312, b=0.2428, F=1.338.Predicted: a=0.1573, b=0.1928, F=1.299Fitted: Dispersivity=0.4227cm and R 2=0.97
D
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Conc
entr
atio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, F=1.316Predicted: a=0.1312, b=0.2428, F=1.373Predicted: a=0.1573, b=0.1928, F=1.316Fitted: Dispersivity=0.4227cm and R 2=0.97
D
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Conc
entr
atio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, F=1.316Predicted: a=0.1312, b=0.2428, F=1.373Predicted: a=0.1573, b=0.1928, F=1.316Fitted: Dispersivity=0.4227cm and R 2=0.97
D
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Conc
entr
atio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, F=1.316Predicted: a=0.1312, b=0.2428, F=1.373Predicted: a=0.1573, b=0.1928, F=1.316Fitted: Dispersivity=0.4227cm and R 2=0.97
D
0
0.003
0.006
0.009
0.012
100 120 140 160 180 200Time(min)
Conc
entr
atio
n(m
M)
Observed Predicted: a=0.1596, b=0.1890, F=1.316Predicted: a=0.1312, b=0.2428, F=1.373Predicted: a=0.1573, b=0.1928, F=1.316Fitted: Dispersivity=0.4227cm and R 2=0.97
Figure 6.14 Application of the power-law function to predict the apparent scale-
dependent dispersivity distributions over experimental column lengths, and to predict the
BTC at 141 cm. (A) Comparison of observed and predicted apparent dispersivity
distributions. (B) Comparison of observed BTC, fitted BTC, and BTCs predicted using
the CDE with αD(L). (C) Comparison of observed BTC, fitted BTC, and BTCs predicted
using the CDE with λD(x). (D) Comparison of observed BTC, fitted BTC, and BTCs
predicted using the CDE with λT(t). The test statistic F = sr,predicted2/sr,fitted
2 where sr = the
lack-of-fit mean square. Critical values for the F statistic were F(0.025, 33, 33) = 2.002,
F(0.0975, 33, 33) = 0.499 and values used were: sample number N = 36, number of parameters
=3.
155
A
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150Distance(cm)
Dis
pers
ivity
(cm
)Obs erved P redic ted: m=0.2273 and n=0.7289P redic ted: m=0.2031 and n=0.9751P redic ted: m=0.2207 and n=0.7936
Figure 6.16 Application of the hyperbolic function to predict the apparent scale-
dependent dispersivity distributions over experimental column lengths, and to predict the
BTC at 141 cm. (A) Comparison of observed and predicted apparent dispersivity
distributions. (B) Comparison of observed BTC, fitted BTC, and BTCs predicted using
the CDE with αD(L). (C) Comparison of observed BTC, fitted BTC, and BTCs predicted
using the CDE with λD(x). (D) Comparison of observed BTC, fitted BTC, and BTCs
predicted using the CDE with λT(t). The test statistic F = sr,predicted2/sr,fitted
2 where sr = the
lack-of-fit mean square. Critical values for the F statistic were F(0.025, 33, 33) = 2.002,
F(0.0975, 33, 33) = 0.499 and values used were: sample number N = 36, number of parameters
=3.
These four scale-dependent dispersivity functions had the same accuracy for predicting
the dispersivity distribution, and the BTCs at 141 cm over the range of the experimental
length scales. However, this does not mean that this result can be generalized for solute
transport at all scales for a given statistically homogeneous porous medium. In these
157
experiments, the observed scales for determining the dispersivity parameters, and the
predicted scales were not too much different. However, in practical situations, the
predicted scale may be much larger than the observed scale used for parameter
identification. Therefore, when the predicted scale >> the observed scale, the
applicability of the above four scale-dependent dispersivity functions to adequately
predict the dispersivity distribution over scales, and to predict BTCs for solute transport
in statistically homogeneous porous media has to be checked carefully.
These four functions are all two-parameter functions. Therefore, mathematically
observations at any two scales could be used to uniquely determine the two parameters.
However, physically, this might not be true for accurately determining the scale-
dependent dispersivity distribution for prediction of BTCs in statistically homogeneous
porous media. In order to accurately predict the solute transport BTCs using the CDE
with a scale-dependent dispersivity in a statistically homogeneous porous medium,
several factors have to be taken into consideration when two-scale observations are used
to determine the dispersivity distribution. These factors are: error of the observed
dispersivity values, difference between observed scales and predicted scale,
characteristics of the selected distribution function, and the sensitivity of the apparent
dispersivity predicted using these four functions to change in the observed apparent
dispersivity values. The error of the observed dispersivity values is determined by both
the random properties of porous media and the accuracy of the analysis.
In order to show the effect of these factors on the prediction, the case for a hypothetical
fractal porous medium was first analyzed in order to better understand their effects before
extending the analysis to statistically homogeneous media. In a fractal porous medium, it
would be logical to assume that the scale-dependent dispersivity distribution would be a
power-law function (Wheatcraft and Tyler, 1988). It is assumed that there must be some
power-law function that would describe the scale dependent dispersivity for the
hypothetical porous medium. Three distribution curves calculated using the power-law
function with different parameters, and a set of discrete points taken to represent the
hypothetical observed distribution, are presented in Figure 6.17. The three calculated
158
curves in Figure 6.17 were named as distribution 1, distribution 2, and distribution 3. As
shown in Figure 6.17, these three calculated curves were indistinguishable from each
other when distance was < 50 (arbitrary units). At < 50 distance the difference of the
values between these three curves was at the second decimal place. However, they were
obviously different from each other when distance was >80.
If two distances < 50 were selected for making observations to determine the power-law
distribution, and the numerical precision in making the dispersivity observation were one
decimal place, any one of these three (or more) curves might be obtained by fitting the
observed dispersivities to the power-law function. In this case, one decimal place
indicates a lower numerical precision than the difference of the values between these
three curves, which was at the second decimal place when the distance < 50. It is
obvious that the prediction would be wrong when either distribution 1 or distribution 3
were applied to predict the scale dependent dispersivity value at a distance > 80.
However, the prediction might be considered acceptable when the prediction distance <
80.
If one of the observed length scales in Figure 6.17 were larger than 100, and the error of
observed dispersivities were still at the first decimal place, neither distribution 1 nor
distribution 3 would be the result of fitting. Distribution 2 (or some other distribution
close to the distribution 2) would be the result of fitting. In this situation, predictions can
be made at scales larger than 100. Therefore, if the error in the observed dispersivities
could significantly affect the departure of the fitted dispersivity distribution functions
from the real dispersivity distribution, the fitted functions might not accurately predict the
scale dependent dispersivity when the scale for prediction >> observed scale.
The foregoing discussion illustrated the sensitivity as defined by Eq. (6.4) of the apparent
scale dependent dispersivity function. When experimental BTCs are observed in a given
porous medium the error of observation is fixed by the accuracy of the experimental
procedures and methods of measurement. This means that addition or subtraction of
some arbitrary number, taken to represent the absolute error in measuring an observed
159
value to be used in determining the apparent dispersivity distribution, is fixed. Therefore,
the sensitivity of a dispersivity distribution function can significantly limit the range of
the scales over which the function can be used for making accurate predictions.
In order to evaluate the sensitivity of the four apparent dispersivity distribution functions
[Eq. (6.1), Eq. (6.2), Eq. (6.3), and Eq. (5.6)], observations at two length scales namely
38 cm and 83 cm were used to fit the apparent scale dependent distribution functions.
The dispersivity value at 83 cm was then assumed to change ±10%. The sensitivity
calculated using Eq. (6.4) of the four apparent dispersivity distribution functions for
predicted scale lengths of 150 cm, 300 cm, and 600 cm are presented in Figure 6.18.
Predicted apparent scale dependent dispersivity distributions are shown in Parts A
through C of Figure 6.19. The distributions in Part A were calculated using parameters
obtained by fitting the four functions using the observed dispersivity values of 0.3124 cm
and 0.3834 cm obtained for the 38 cm and 83 cm length scales respectively. In Part B,
the observed dispersivity at 83 cm was arbitrarily increased by 10% to 0.4217 cm. In
Part C it was arbitrarily decreased by 10% to 0.3451 cm.
The results in Figure 6.18, and in Part B of Figure 6.19 showed that the power-law
function was the most sensitive to the changes in the observed dispersivity value, and the
LIC was the least sensitive. The power-law function was almost twice as sensitive as the
LIC at 600 cm. For example, in Figure 6.19, the predicted dispersivities at 500 cm by the
power-law function [Eq. (6.1a)], log-power function [Eq. (6.2a)], hyperbolic function
[Eq. (6.3a)], and the LIC [Eq. (5.6)] were 0.5918 cm, 0.5339 cm, 0.4661 cm, and 0.4301
cm, respectively (Part A). When the observed dispersivity at 83 cm was arbitrarily
increased by 10% (Part B of Figure 6.19), the four predicted dispersivities at 500 cm
became 0.8095 cm, 0.6937 cm, 0.5772 cm, and 0.4953 cm, respectively. When the
observed dispersivity at 83 cm was decreased by 10% (Part C of Figure 6.19), the
predicted dispersivity distributions for these four functions were very close to each other
at large scales.
160
The results in Part A of Figure 6.19 showed that the difference between the dispersivity
distributions predicted by these four functions increased as the scale of prediction was
increased. At some scales, such as 141 cm the distributions described by the four
functions were not markedly different. This result is in accordance with those already
discussed for Figure 6.14, Figure 6.15, and Figure 6.16. However, as the scale increases,
dispersivities predicted by the four functions became obviously different. These results
showed that the dispersivity predicted by the power-law function and log-power function
continuously increased as the predicted scale increased. For this reason, these two
functions might not be adequate and applicable to describe the dispersivity distribution in
the statistically homogeneous porous media when the observed scales >> predicted. As
defined Chapter 5, the apparent dispersivity in statistically homogeneous porous media
would asymptotically approach a constant value after some length scale (l).
Overall, the above analysis implied that the LIC and the hyperbolic function would be the
better choice to describe the scale-dependent dispersivity distribution, especially when
the predicted scale >> the observed scales. As discussed in Chapter 5, when two-scale
observations are used to determine the LIC, at least one observed scale has to be larger
than the l (or tl ) after which the local dispersivity becomes constant [see Eq. (5.6)]. This
may also be a necessary condition for identifying the parameters for the hyperbolic
function.
An indication that this may be the case was obtained by analysis of the experimental data
of Zhang et al. (1994). The hyperbolic function and the LIC were fitted to the apparent
dispersivities observed in column experiments using homogeneous and heterogeneous
porous media conducted by Zhang et al. (1994). The results are presented in Figure 6.20.
For the homogeneous porous media (Part A of Figure 6.20), both functions could
describe dispersivity distribution very well (R2 > 0.8 for both fitted functions). The value
of l was about 1000 cm. However, for the heterogeneous porous media, the fitted results
were not so good (Parts B and C of Figure 6.20). The largest R2 was 0.625 and the value
of l was larger than the longest observed column length of 1200 cm. The reason for this
might be that the experimental length scales were not large enough for estimating l. The
161
fitted results might be improved if the heterogeneous porous media used by Zhang et al.
(1994) extended to a large length (e.g. 30 m). In this case, the medium might be treated
as statistically homogeneous at the new scale, provided that the structural heterogeneity
of the medium remained the same.
0
1
2
3
4
0 30 600
1
2
3
4
0 30 60 90 120 150 180Distance
Dis
pers
ivity
Real distributionDistribution1: a=0.00183, b=1.48Distribution 2: a=0.00253, b=1.392Distribution 3: a=0.00324, b=1.323
Figure 6.17 Hypothetical apparent dispersivity distribution in a fractal porous medium.
The distributions were described using a power-law dispersivity distribution Eq. (6.1)
162
At 300cm
0
1
2
3
Parabolic Log-power Hyperbolic LIC
Sens
itivi
ty
At 600cm
0
1
2
3
4
Parabolic Log-power Hyperbolic LIC
Sens
itivi
tyAt 150cm
0
0.5
1
1.5
2
Parabolic Log-power Hyperbolic LIC
Sens
tivity
-10% +10%
-10% +10%
-10% +10%
Figure 6.18 Sensitivity of the apparent dispersivity predicted for 3 length scales (150 cm,
300 cm and 600 cm) using four apparent distribution functions. These functions were
determined by analysis of experimental results at 38 cm and 83 cm. The sensitivity values
[Eq. (6.4)] were obtained by increasing or decreasing the observed dispersivity at 83 cm
was by ±10%.
163
A
0
0.3
0.6
0.9
0 100 200 300 400 500Distance(cm)
Dis
pers
ivity
(cm
)Observed ParabolicLog-power HyperbolicLIC
B
0
0.3
0.6
0.9
0 100 200 300 400 500Distance(cm)
Dis
pers
ivity
(cm
)
LIC (Reference) ParabolicLog-pow er HyperbolicLIC
C
0
0.3
0.6
0.9
0 100 200 300 400 500
Distance(cm)
Dis
pers
ivity
(cm
)
LIC (Reference) ParabolicLog-pow er HyperbolicLIC
Figure 6.19 Predicted apparent scale dependent dispersivity using four apparent
dispersivity functions. (A) calculated using parameters obtained by fitting the functions to
observed dispersivities at the 38 cm and 83 cm length scales. (B) and (C) calculated in
the same manner after arbitrarily changing the observed dispersivity at 83 cm by ± 10%
respectively. The distribution for the LIC in (A) was repeated in (B) and (C) as a
reference curve to emphasize the differences caused by this change.
164
A
0
1
2
3
4
5
6
0 200 400 600 800 1000 1200 1400Length(cm)
Dis
pers
ivity
(cm
)
Observed Fitted using hyperbolic equation: p=25525074, q=0.00735.R2=0.887Fitted using LIC equation: k=0.007, l=1032cm, R2=0.876
B
0
50
100
150
200
250
0 200 400 600 800 1000 1200 1400Length(cm)
Dis
pers
ivity
(cm
)
ObservedFitted using hyperbolic equation: p=1043, q=0.2295, R2=0.610Fitted using linear equation: k=0.1013, R2=0.625
C0
100
200
300
0 200 400 600 800 1000 1200 1400Length(cm)
Dis
pers
ivity
(cm
)
ObservedFitted using hyperbolic equation: p=367.8, q=0.5588, R2=0.512Fitted using linear equation: k=0.151, R2=0.418
Figure 6.20 Hyperbolic and LIC apparent dispersivity functions fitted to the observed
apparent dispersivities reported by Zhang et al.(1994) for (A) homogeneous soil column
(v = 0.6 cm/min); (B), (C) heterogeneous soil column (v = 1.12 cm/min and v = 0.882
cm/min respectively).
165
6.4 Conclusions An experimental porous medium was specially developed to simulate statistically homogeneous porous media at the column scale. Five solute transport models were evaluated for their applicability and adequacy to predict solute transport BTCs over increasing scales using parameters determined at one observed scale. All of the five models namely, the CCDE, DCDE, TCDE, MIM, and FCDE could not accurately predict BTCs as the scale was increased. The predictions might be acceptable only when the scale for prediction was very close to the observed scale for the CCDE, MIM, and FCDE. The applicability of the DCDE and the TCDE were limited, since the dispersion parameters determined from one scale observation can define only the apparent dispersivity values at the observed scale but not a dispersivity distribution over scales. The MIM was the only model that could explain the long tail of the observed BTCs. Observations at two length scales could be used to define a scale dependent dispersivity distribution over scales as a power-law function, log-power function, hyperbolic function, and the LIC function. BTCs at a given scale were predicted using the CDE with these four dispersivity distribution functions specified from analysis of experimental results at two different length scales. Predicted BTCs were not significantly different from the BTC fitted to the experimental data at the given scale used for prediction. Error and sensitivity analyses showed that the power-law and log-power functions might not be applicable or adequate to describe the dispersivity distribution in statistically homogeneous porous media when the scale of parameter determination << the scale of prediction. The reason was that predicted dispersivities were highly sensitive (relative to the other two functions) to changes in the observed dispersivity values used to define these functions. Also, the predicted dispersivities using these two distribution functions increased monotonically with increasing scale. This was not the case for the hyperbolic function and LIC function. Consequently, the hyperbolic function and LIC function were two potentially applicable functions to adequately describe the scale dependent dispersivity distribution in statistically homogeneous porous media. The condition that at least one of the observations for fitting the LIC has to be larger than l (or tl), may also be usefully applied for defining the hyperbolic dispersivity function in statistically homogeneous porous media.
166
CHAPTER 7 SUMMARY AND CONCLUSIONS
Solute transport models are developed to describe and predict solute transport behavior in
porous media such as soils and subsurface aquifers, since it is generally not practically
feasible to make such assessments by direct in-situ field sampling and analysis over long
periods of time. The form of the model is decided by the physical processes included, and
by the assumptions made regarding these processes. A model is assumed to represent
physical reality, if results simulated with the model match the observations. However,
different processes may match the same set of observations equally well when included
in the model. Consequently, the specific processes that are included in a model may or
may not necessarily reflect the physical reality.
Most of the governing equations for these models are based on the convection-dispersion
partial differential equation (CDE). For some well-defined initial and boundary
conditions, analytical solutions can be developed to solve the CDE . When unsteady
water flow conditions, spatial and temporal variability of soil properties, or complicated
initial and boundary conditions are considered in the model, the partial differential
equations have to be solved numerically, using methods based on finite differences, finite
elements, or particle tracking. Some parameters in transport models can be
independently determined by experimental measurements. Other parameters have to be
obtained by fitting the experimentally observed data to the analytical or numerical
solution of the transport model.
When water movement in a porous medium can be correctly described, modeling of
mechanical dispersion becomes sine qua non for further development of the solute
transport model. Mechanical dispersion results primarily from the heterogeneous nature
of porous media, which can be defined at various scales of observation, such as pore
scale, laboratory scale, and field scale. Therefore, mechanical dispersion can be modeled
at various scales, resulting in pore scale models, laboratory scale models, and field scale
models.
167
Conceptually, mechanical dispersion is the physical manifestation of velocity fluctuations
around the average pore water velocity on solute transport in the porous medium. The
differences between the average velocity and the microscopic velocities (termed as the
residual part of the velocities) appear as the observed solute spreading.
These velocity fluctuations are directly related to spatial variation in the hydraulic
conductivity (K) of the porous medium. However, the hydraulic conductivity is typically
spatially auto-correlated, and the auto-correlation range is related to the scale of
heterogeneity. A porous medium for a given observation scale can be considered as
statistically homogeneous, if the variance of the natural logarithm of the hydraulic
conductivity [ln (K)] and the correlation tensor of are fixed and finite. In a statistically
homogeneous porous medium, the variance of ln(K) is scale-dependent. Its value
increases as the scale increases, then attains a constant value after some scale, which is a
characteristic of the porous medium. Mechanical dispersion is quantified by the
dispersivity parameter in the CDE. It is determined by the velocity fluctuations, which
are directly caused by the variance of the ln (K). Therefore, when the variance of ln (K)
is scale dependent the dispersivity is scale-dependent.
Most mechanical dispersion models are directly of indirectly related to the Gaussian
distribution. This means that their results are equivalent to those obtained with solute
dispersion models derived directly from Brownian particle motion theory, in which the
displacement of a solute particle at a given time depends only on its current position
rather than on its history. When the particle displacement at a given time depends not
only on its current position, but also on its history, some other distribution instead of the
Gaussian, has to be applied to explain mechanical dispersion of the solute in porous
media. An alternative distribution that has been proposed is the Lévy or α-stable
distribution. The corresponding governing solute transport equation is called fractional
convection dispersion equation (FCDE).
168
A medium made up of different particle sizes would be heterogeneous at the pore or
micro scale, but can be considered as homogeneous at the column or macro scale.
Homogeneity at the column scale is defined in a statistical sense, meaning that the micro
scale heterogeneity of the porous medium is uniformly distributed within the column. An
observed property at this scale (macro scale) does not change appreciably for some
arbitrary change in the specified scale of the column. Similarly, at the field scale,
statistical homogeneity means that the components of macro scale heterogeneity are
uniformly distributed over the field, and that an observed property at the field scale does
not change appreciably with some arbitrary change in the specified scale for the field.
Few, if any, experimental investigations have been made to describe and predict solute
transport in statically homogeneous porous media. When the CDE is used to describe the
observed BTCs for solute transport in statistically homogeneous media, differences
appear between the observed and fitted BTCs especially for the tail portion of the curve.
In addition, significant errors occur when dispersivity values obtained by fitting the CDE
to the observed BTC at one scale, is directly used to predict the observed BTCs at other
scales.
In this study, the applicability and adequacy of three modeling approaches to describe
and predict BTCs of solute transport in statistically homogeneous porous media were
numerically and experimentally investigated. These approaches were: the scale-
dependent CDE, mobile-immobile model (MIM), and the fractional convection-
dispersion equation (FCDE).
In applying the scale-dependent CDE, scale-dependent dispersivity was described as a
power-law function, hyperbolic function, log-power function, or LIC. The LIC was a new
scale-dependent dispersivity function, which was developed for describing the scale-
dependent dispersivity distribution in statistically homogeneous media. In developing the
LIC, it was assumed that local scale-dependent dispersivity linearly increases within
some scale, after which it becomes constant.
169
When the CDE with scale-dependent dispersivity is solved numerically for generating a
BTC at L, the value of scale-dependent dispersivity has to be set for each discretized unit
in space at each discretized time step. Therefore, the scale-dependent dispersivity can be
specified in several ways namely, as local time-dependent dispersivity λT(t), average
time-dependent dispersivity αT(t), the apparent time-dependent dispersivity α'T (T), the
apparent distance-dependent dispersivity αD(L), and the local distance-dependent
dispersivity λD(x).
A prototype laboratory column system for conducting miscible displacement experiments
with a free-inlet boundary was designed to generate the experimental BTCs for non-
reactive single or multiple source solute transport. Since it was a prototype, the
performance and operating conditions of this column system was rigorously evaluated.
Special attention was given to testing of the injection assembly used to input tracer
solution directly into the column. BTCs were generated at different column lengths, and
were used to verify the results of numerical tests on applying the scale-dependent
dispersivity functions in numerical solutions of CDE. The BTCs were also used to test
the applicability and adequacy of the three modeling approaches for describing and
predicting the solute BTCs in the statistically homogeneous media. The principal results
and conclusions of this study were:
a. Tests of the experimental column system showed that it could be used to generate
accurate and reliable BTCs needed for this study. The experimental artificial porous
media used could be considered as statistically homogeneous when the column was long
enough. When a single injection of the tracer solution was distributed over a distance no
larger than 5% of the column length, and the column Peclet number was smaller than
200, the solute distribution could be assumed as a Dirac delta function for solving the
initial value problem posed by five different non-reactive solute transport models. The
five models were: the CDE, the CDE with power-law distance-dependent dispersivity,
the CDE with power-law time-dependent dispersivity, the MIM, and the FCDE.
170
b. When mechanical dispersion in the CDE is assumed to be a diffusion-like process, and
the heterogeneity in porous medium is scale-dependent, it implies that the scale-
dependent dispersivity is locally time-dependent. In this case, the definition of αT(t),
α'T(T), αD(L), and λD(x), and relationships between them can be directly or indirectly
obtained from the definition of λT(t). These definitions and relationships were based on
either of two concepts regarding solute transport with an initial solute distribution
represented as a Dirac delta function. The first was the concept of mean solute travel
time for a BTC observed at some distance L. The second was the assumption that an
arbitrarily defined time could be used to represent the expected value of the time
distribution of the BTC at L.
c. The algorithm of applying the average time-dependent dispersivity αT(t) to generate
the BTC was computationally inefficient, and consequently, it was difficult to use for
practical purposes. When a BTC at L is to be predicted using the numerical solution of
the scale-dependent CDE, the choice between using λT(t), α'T(T), αD(L), and λD(x) would
depend on the solute transport problem, solute transport conditions, level of accuracy
required of the calculated BTC, and computational efficiency. For the solute transport
problem with a single source (T = L/v), under solute transport conditions specified by
high Peclet numbers ( >50), any one of these four dispersivities can be use to generate the
BTCs at L. However, when the solute transport Peclet numbers are small (<50), only
λT(t) should be used. Although defined differently, α'T (T) and αD(L) are no different
when applied in the numerical algorithms.
For solute transport problems with multiple source solute input over time but at one
spatial location (T = L/v), either λD(x) or αD(L) has to be used in order to achieve
computational efficiency when the solute transport Peclet number is not too small ( >20).
For solute transport problems with simultaneous multiple source solute input over the
space domain, it is difficult to arbitrarily define a value of time (T) to represent the
expected value of the time distribution of the BTC at L. Therefore, only λT(t) should be
used in this case.
171
d. The LIC dispersivity distribution function can be defined by analyzing the BTCs
observed at two length scales, in which at least one of them is obtained at a scale larger
than l = vtl. Miscible displacement experiments showed that the LIC could accurately
predict the scale-dependent dispersivity distribution over increasing scales in statistically
homogeneous porous media.
e. Solute transport models varied considerably in their applicability and adequacy to
predict solute transport BTCs over increasing scales using parameters determined at one
observed scale. The five models evaluated namely, the CCDE (CDE with constant
dispersivity), DCDE (CDE with power-law distance-dependent dispersivity), TCDE
(CDE with power-law time dependent dispersivity), MIM (mobile-immobile model), and
FCDE (fractional convection-dispersion equation), could not accurately predict the
experimental BTCs as the scale was increased. The predictions might be acceptable only
when the scale for prediction was very close to the observed scale for the CCDE, MIM,
and FCDE. The applicability of the DCDE and the TCDE were limited since the
dispersion parameters determined from one scale observation can define only the
apparent dispersivity values at the observed scale but not a dispersivity distribution over
scales. The MIM was the only model that could explain the long tail in the experimental
BTCs.
f. BTCs observed at two length scales could be used to define a scale dependent
dispersivity distribution over scales either as a power-law function, log-power function,
hyperbolic function, or as the new function termed as the LIC. The predicted
dispersivities with the power-law and log-power functions were highly sensitive (relative
to the other two functions) to changes in the observed dispersivity values used to define
these functions. Also, the predicted dispersivities using these two distribution functions
increased monotonically with increasing scale. Because of their high sensitivity to such
measurement error the power-law and log-power functions might not be applicable or
adequate to describe the dispersivity distribution in statistically homogeneous porous
media when the scale of parameter determination << the scale of prediction. This was
not the case for the hyperbolic function and LIC. Consequently, the hyperbolic function
172
and LIC were two potentially applicable functions to adequately describe the scale
dependent dispersivity distribution in statistically homogeneous porous media.
Taken in their entirety, the results of the numerical and laboratory experiments
demonstrated that several models could be applied to describe experimental observations
of non-reactive solute transport behavior in porous media. However, although these
models performed reasonably well for description, this did not directly imply that they
could be used for prediction of solute transport at different scales and under different
solute transport conditions. The reason was that the ability of the model to predict solute
transport behavior not only depended on how well the model described the experimental
observations, but also depended on whether parameters identified at given scales and
transport conditions could applied for other scales or transport conditions, or on the
manner in which these parameters had to be extrapolated for prediction purposes. The
assumption that solute dispersion was quasi-Fickian directly implies that scale-dependent
dispersivity should be specified as local time-dependent dispersivity when the CDE is
solved numerically for calculating BTCs. However, several other ways could be used for
specifying scale dependent dispersivity, because they were comparatively easier to be
defined, or were more computationally efficient. The feasibility of applying these other
ways for BTC prediction could be evaluated by comparing between BTCs predicted by
these other ways to the corresponding predictions using the local time-dependent
dispersivity for a given solute transport problem under specified conditions.
173
174
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186
APPENDIX 1
ANALYSIS OF INFLUENCE OF SOLUTE DISTRIBUTIONS AFTER
INJECTION INTO THE COLUMN ON BTCS GENERATED USING
THE ANALYTICAL SOLUTION OF THE CDE
The experimental column system physically represents a solute transport problem with a free
inlet boundary. If the outlet boundary condition is assumed to be an infinite boundary condition
[Eq. (3.4)], solute transport in the experimental column system can be mathematically described
as an initial value solute transport problem in a one-dimensional infinite domain. A BTC at the
outlet of column can be written as:
C L t C l f L l t dl( , ) ( ) ( , )= −−∞
+∞
∫ 0 (A1.1)
where L is the distance from the injection location to the outlet, C0(l) is the initial concentration
at the location of l, and f(L-l, t) is an auxiliary function for the convolution. f(L-l, t) is the BTC
for a solute transport problem with an initial solute concentration distribution represented as a
Dirac delta function δ at (L-l) or:
C L lL l
0 ( )( )
− =−δ
θ (A1.2)
where θ is the effective porosity of the porous media. When the analytical solution of the CDE
with constant dispersivity is used to solve the problem describing solute transport in the column
system, the f(L-l, t) is:
−−−=−
DtvtlL
Dt
mtlLf
4)(
exp4
),(2
πθ (A1.3)
where m is solute mass.
If the solute distribution after injection is assumed to be a Dirac delta function at the injection
location or at the location of l = 0, Eq. (A1.1) becomes:
C L tm
DtL vt
Dt( , ) exp
( )= −
−
0
2
4 4θ π (A1.4)
187
where m0 is the solute mass injected.
However, the solute distribution after injection is not a Dirac delta function. The distribution has
some width w. In the experiment, fluorescein was used as tracer. Therefore, the w could be
visually estimated. The value of the w was observed to be about 1 cm for the column with L =
16 cm in the experiments.
If the injection location is used as the coordinate origin and the solute distribution at t = 0 is
assumed to be a uniform distribution over the interval [-w/2, w/2]. Eq (A1.1) becomes:
C L tm
w DtL vt l
Dtdl
w
w
( , ) exp( )
/
/
= −− −
−∫ 0
2
2
2
4 4θ π
=− +
−− +
mw
erfL vt w
Dterf
L vt wDt
0
22
22
2θ/ /
(A1.5)
If the solute distribution at t = 0 is assumed to be a normal distribution with a standard deviation
of σ, Eq. (A1.1) becomes:
C L tm
Dt
l L vt lDt
dl( , ) exp exp( )
= −
−
− −
−∞
∞
∫ 0
2
2
2
2
8 2 4θσ π σ
=+
−−+
m
Dt
L vtDt
0
2
2
22 4 4 2θ πσ π σ
exp( )
(A1.6)
Differences in the BTCs generated using Eq. (A.1.4), Eq. (A1.5), and Eq. (A1.6) was
estimated for solute transport in the column with length of 16 cm. The column with a length of
16 cm was the shortest column used in the experiments. If the difference was indistinguishable
for the column with a length of 16 cm, the difference will be indistinguishable for columns with
length >16 cm, provided that the solute transport conditions are the same (see detail in Chapter
3). Comparison of the BTC generated using Eq. (A1.4), and BTCs generated using Eq. (A1.5)
with w = 1 cm, 2 cm and 6 cm, is presented in Figure A1.1 (Part A). Comparison of the BTC
generated using Eq. (A1.4) and BTCs generated using Eq. (A1.6), in which σ = 0.5 cm, 1 cm,
and 2 cm, is presented in figure A1.1 (Part B). As shown in the figure A1.1 , the BTC
188
generated using Eq. (A1.4) was indistinguishable to the BTC generated using Eq. (A1.5) with w
= 1 cm, and it was indistinguishable to the BTC generated using Eq. (A1.6) with σ = 0.5 cm.
When the initial distribution width w was 2 cm or the standard deviation (σ) of the initial
distribution was 1 cm, the error caused by using Eq. (A1.4) instead of using Eq. (A1.5) or Eq,
(1.6) might be acceptable. Marked differences existed when w = 6 cm or σ = 2 cm. These
results indicated that when w or σ was small, the assumed distribution of the solute mass after
injection was not important in calculation of the BTCs under the experimental solute transport
conditions. Also that it was feasible to directly use Eq. (A1.4) to generate the BTCs in the
experiments instead of using Eq. (A1.5) and Eq. (A1.6). This result was very useful for
analyzing the experimental BTCs, because it was impossible to directly determine the exact
initial solute mass distribution and the exact width of the initial solute mass distribution in the
experiments.
Figure A1.1 (A) Comparison of BTC generated using Eq. (A1.4) to the BTCs generated using
the Eq. (A1.5), and (B) Comparison of BTC generated using Eq. (A1.4) to the BTCs
generated using the Eq. (A1.6) (Parameter values were L = 16 cm, D = 2 cm2/min, v = 0.8
min, effective porosity = 0.33. In legend, std denotes σ).
0
0.1
0.2
0.3
0.4
0.5
0 5 10 15 20 25 30 35 40
Time (min)
C/C
0
BTC calculated using the E q.(A1.4)BTC calculated using the Eq . (A1.5) with w=1 cmBTC calculated using the Eq . (A1.5) with w=2 cmBTC calculated using the Eq . (A1.5) with w=6 cm
0
0.1
0.2
0.3
0.4
0.5
0 5 10 15 20 25 30 35 40
Time (min)
C/C
0
BTC calculated using the E q.(A1.4)BTC calculated using the Eq . (A1.5) with w=1 cmBTC calculated using the Eq . (A1.5) with w=2 cmBTC calculated using the Eq . (A1.5) with w=6 cm
0.00
0.10
0.00
0.10
0.20
0.30
0.40
0.50
0 5 10 15 20 25 30 35 40
Time (min)
C/C
o
BTC calculated using theEq .(A1.4)BTC calculated using the Eq .(A1.6) with std=0.5 cmBTC calculated using the Eq .(A1.6) with std=1.0 cmBTC calculated using the Eq .(A1.6) with std=2.0 cm
A
B
0
0.1
0.2
0.3
0.4
0.5
0 5 10 15 20 25 30 35 40
Time (min)
C/C
0
BTC calculated using the E q.(A1.4)BTC calculated using the Eq . (A1.5) with w=1 cmBTC calculated using the Eq . (A1.5) with w=2 cmBTC calculated using the Eq . (A1.5) with w=6 cm
0
0.1
0.2
0.3
0.4
0.5
0 5 10 15 20 25 30 35 40
Time (min)
C/C
0
BTC calculated using the E q.(A1.4)BTC calculated using the Eq . (A1.5) with w=1 cmBTC calculated using the Eq . (A1.5) with w=2 cmBTC calculated using the Eq . (A1.5) with w=6 cm
0.00
0.10
0.00
0.10
0.20
0.30
0.40
0.50
0 5 10 15 20 25 30 35 40
Time (min)
C/C
o
BTC calculated using theEq .(A1.4)BTC calculated using the Eq .(A1.6) with std=0.5 cmBTC calculated using the Eq .(A1.6) with std=1.0 cmBTC calculated using the Eq .(A1.6) with std=2.0 cm
A
B
0.00
0.10
0.00
0.10
0.20
0.30
0.40
0.50
0 5 10 15 20 25 30 35 40
Time (min)
C/C
o
BTC calculated using theEq .(A1.4)BTC calculated using the Eq .(A1.6) with std=0.5 cmBTC calculated using the Eq .(A1.6) with std=1.0 cmBTC calculated using the Eq .(A1.6) with std=2.0 cm
A
B
APPENDIX 2 PROCEDURE USED TO PREPARE EXPERIMENTAL POROUS
MEDIA TO SIMULATE NATURE PORE-SCALE HETEROGENEITY
The particle (or glass beads) size fractal distribution using in the experiments was
developed using the fractal fragmentation theory (Rieu and Sposito, 1991a; Rieu and
Sposito, 1991b). The particle size fractal distribution can be expressed as:
crDrN fk +−= 2log)(log (A2.1)
where N(rk) is the number of particles with radii ≥ rk. Df is the bulk fractal dimension,
and c is a constant.
For a given soil or medium, the smallest radius of particles is assumed to be rg.
When rk equals rg, the Eq. (A2.1) becomes:
crDrN gfg +−= 2log)(log (A2.2)
where N(rg) is the total number of particles. Eq. (A2.2) can be rewritten as:
gfg rDrNc 2log)(log += (A2.3)
Substituting Eq.(A2.3) into Eq.(A2.1) gives:
fD
k
ggk r
rrNrN
= )()( (A2.4)
If the particles are divided into “g” classes according their radii, the corresponding
number of particles in these classes in decreasing order of size n1, n2, …,ng-1, and ng, is
given as
fD
gg r
rrN
=
11 )(n (A2.5)
ff D
k
gg
D
k
ggk r
rrN
rr
rN
−
=
−1
)()(n , k=2,3, …, g (A2.6)
189
VITA
The author was born in Inner Mongolia, China on January 28, 1969. He is the son of
Wang Diangui and Guan Shuzhong. He received his B.S. in Applied Physics from
Beijing Agricultural University in July 1990. Then he was employed by the Laboratory of
Application of Nuclear Techniques in Beijing Agricultural University, where he did the
research on irrigation and fertilizer use efficiency for winter wheat. In September of
1993, he enrolled in the M.S. program of Biophysics at Beijing Agricultural University,
and did pesticide environmental toxicology research. He received his M.S. in Biophysics
from Beijing Agricultural University in July 1996. He remained at the Laboratory of
Application of Nuclear Techniques in Beijing Agricultural University, where he did the
research on pesticide environmental toxicology and taught two undergraduate courses for
another two years. He enrolled in the Ph.D. program in Crop and Soil Environmental
Sciences at Virginia Polytechnic Institute and State University in August of 1998.