Describing and Predicting Breakthrough Curves for non · DESCRIBING AND PREDICTING BREAKTHROUGH...

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

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.

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TABLE OF CONTENTS

ABSTRACT.………………………………………………………………………..………ii

DEDICATION……………………………………………………………………………..iii

ACKNOWLEDGEMENTS.……………………………..………………………..…..……iv

TABLE OF CONTENTS …………………………………………………………….…….v

LIST OF TABLES ……………………………………………………………………….viii

LIST OF FIGURES ………………………………………………………………….……ix

CHAPTER 1. INTRODUCTION……..…………………………………………...……… 1

CHAPTER 2. MODELING SOLUTE TRANSPORT IN POROUS MEDIA…..….……....9

2.1 Introduction……………………………………………………………………..….…9

2.2 Solute Transport Model .…………………………………………………….….…10

2.2.1 Deterministic-Functional Models…………………………………………..…..11

2.2.2 Stochastic-Mechanistic Models………………………………………….….….11

2.2.3 Stochastic-Functional Models …………………………………………….……12

2.2.4 Deterministic-Mechanistic Models ……………………………………….……12

2.2.5 Solute Transport Models for Multi-Layered Porous media………………..…...14

2.2.6 Solutions of Governing Equations for Solute Transport Models……….….…...16

2.2.7 Parameter Estimation………………………………………………….…….….19

2.3 Modeling of Solute Dispersion……………………………………………..……….19

2.3.1 Dispersion Phenomena in Solute Transport……………………………..……...19

2.3.2 Models for Solute Dispersion at Various Scales...…………………………..….20

2.3.3 Estimation of Dispersion Coefficient…………………………………………...30

CHAPTER 3. DEVELOPMENT AND TESTING OF A LABORATORY COLUMN

SYSTEM FOR CONDUCTING MISCIBLE DISPLACEMENT EXPERIMENTS WITH

A FREE-INLET BOUNDARY……………………………………………………………33

3.1 Introduction…..……………………………………………………………………33

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3.2 Column System and Outlet Boundary Conditions ………………………….……...34

3.2.1 Column System……………………………………………………….………..34

3.2.2 Outlet Boundary Conditions………………………………………….………..37

3.3 Numerical Tests on the Effect of Injected Solute Distribution on BTCs…….…….44

3.3.1 Solute Distribution after Injection……………………………………….……44

3.3.2 Solute Transport Models……………………………………………….……...45

3.3.3 Procedures for Numerical Tests .……………………………………….……..47

3.3.4 Results of Numerical Tests……………………………………………………49

3.4 Experimental Tests on Effects of Injected Solute Distribution Width on BTCs..….61

3.4.1 Experimental Materials and Methods…………………………………………61

3.4.2 Experimental Results………………………………………………………….64

3.5 Conclusions………………………………………………………………………….68

CHAPTER 4. ANALYSIS OF BREAKTHROUGH CURVES USING NUMERICAL

SOLUTIONS OF THE CONVECTION DISPERSION EQUATION WITH SCALE-

DEPENDENT DISPERSIVITY …………………………………………………………69

4.1 Introduction……………………………………………………………….…………69

4.2 Time-Dependent and Distance-Dependent Dispersivity …………………………...78

4.3 Applicability of Scale-Dependent Dispersivity……………………………………..94

4.3.1 Solute Transport Problems………………………………………………….…..94

4.3.2 Scale-Dependent Dispersivity Functions..……………………………….……..98

4.3.3 Numerical Solution……………………………………………………………..99

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

VITA ……………………………………………………………………………………..190

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LIST OF TABLES

Table 2.1 Scale-dependent dispersivity functions…………………………………….25

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

1

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-

2

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.

6

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.

8

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

as: deterministic-mechanistic models, deterministic-functional models, stochastic-

mechanistic models, or stochastic-functional models.

10

2.2.1 Deterministic-functional models

The deterministic-functional models are based on a simplified continuum, mass balance

approach (De Smedt and Wierenga, 1978; Rose et al., 1982). This approach links the

solute and water balances using algebraic equations for mass conservation of solute and

water. It can be applied to solute transport in layered soil (Valocchi et al., 1981;

Addiscott et al., 1986), by dividing the soil profile into horizontal layers and taking a

mass balance for each layer. Different chemical, physical, and biological processes

occurring simultaneously within each layer can be taken into account.

2.2.2 Stochastic-mechanistic models

The stochastic-mechanistic models are mostly applicable to field situations. The field

heterogeneity is accounted for in the mechanistic model by considering the relevant soil

properties and transport parameters as random variables. These models can be used to

calculate the ensemble average solute concentration resulting from an input of a solute

over the entire field. These models have been used for describing conservative solute

transport (Sposito and Barry, 1987), transport including equilibrium adsorption of solutes

(Kabala and Sposito, 1991), transport including non-equilibrium adsorption of solutes

(Toride and Leij, 1996a, 1996b), and transport including nonlinear interactions between

the solute and the porous medium (Ginn et al., 1995).

The most popular and simplified stochastic-mechanistic model uses the stream tube

approach, in which the field is conceptualized as made up of independent vertical soil

columns (Toride and Leij, 1996a, 1996b; Stefano et al., 1999). The transport in each

stream tube is described by a deterministic model. Selected parameters in the

deterministic model for each tube are obtained by random sampling from some assumed

probability density function. Commonly used density functions are the lognormal

distribution (Dagan and Bresler, 1979), or a bivariate lognormal joint probability density

function (Toride and Leij, 1996a, 1996b). The field scale concentration at any depth in

the field at a given time is obtained as the ensemble average of the concentrations

calculated for the individual stream tubes.

11

The main steps in using the deterministic-stochastic model are:

1. Identify a suitable deterministic model to describe the solute transport in an individual

stream tube.

2. Assume that selected parameters such as hydraulic conductivity or pore water velocity

in the model are realizations of a stochastic process and can be described as some

probability density function or joint probability density function.

3. Determine the moments, such as the mean and standard deviation, of the probability

density function from field measurements.

4. Calculate the ensemble average concentration at any depth and time for the entire field.

The ensemble average concentration is obtained by substituting the probability density

functions for the selected parameter into the solution of the deterministic model. The

solution is then integrated over the entire domain of the selected parameter. One problem

in applying the stream tube approach is that it is difficult to identify a reliable statistical

distribution for each parameter (Toride et al., 1995)

2.2.3 Stochastic-functional models

Stochastic-functional models, which are also called transfer function models, were

developed by Jury and his co-workers (Jury, 1982; Jury et al., 1982, 1986, 1990). Jury

and Roth (1990) provided a review of these developments. In the transfer function

model, output is described as a function of input. The transfer function is taken to be the

travel time probability density function (PDF) or resident PDF describing the response of

the system to a Dirac delta function input. The transport process is assumed to be linear,

in order to apply the superposition principle to obtain the output as the convolution of the

input, for any boundary conditions. The transfer function can be a Fickian PDF,

lognormal PDF, or gamma PDF (Jury and Roth, 1990). The transfer function model can

be incorporated into the stream tube approach (Jury and Roth, 1990).

2.2.4 Deterministic-mechanistic models

Deterministic-mechanistic models constitute the largest group of models for describing

solute transport. This is because they are easily understood, they describe the

12

relationship among the different processes, and they can be directly combined with

models for water movement. They form the deterministic component in the stochastic

stream tube models. Deterministic-mechanistic models can be divided into four sub-

groups.

The first subgroup is the convection-dispersion equation (CDE) with or without

equilibrium adsorption. The CDE can be derived macroscopically based on continuum

mass conservation and Fick's law for diffusive mass transport (Bear, 1972). It can also be

derived using a microscopic approach based on Brownian motion. The CDE obtained

from the latter approach is also called the Fokker-Planck equation (Ito, 1951). When the

dispersion coefficient is distance independent, the CDE obtained macroscopically is

identical to the Fokker-Planck equation (Kinzelbach, 1988). When equilibrium

adsorption is considered to be an instantaneous and linear process, it can be easily

incorporated into the CDE. The resulting equation has the same form as the original

CDE but includes an additional parameter called the retardation factor to account for the

equilibrium adsorption. The equilibrium adsorption process can be also be described by

the non-linear Langmuir isotherm and the Freundlich isotherm (Leij and van Genuchten,

2000). In these cases, the retardation factor is a function of concentration and the CDE is

non-linear.

The second subgroup is comprised of transport models for solutes under physical non-

equilibrium conditions. Soil and subsurface porous materials are usually physically

heterogeneous, meaning that they are not uniform in structure and texture at different

locations in the subsurface. Physical heterogeneity at the pore scale results in non-

uniform pore water flow velocities and preferential water flow paths. In addition, not all

the water or solution may be mobile. Discrete portions of the fluid may be trapped in

dead end pores and pockets within the porous medium. This results in physical non-

equilibrium conditions. Conceptually, physical non-equilibrium for one-dimensional

transport implies that the solute concentration at a given time is not uniformly distributed

across a given cross-section of the porous medium. Physical non-equilibrium results in

excessive tailing and irregular distribution of the solute concentrations in the BTC.

13

Models that include physical non-equilibrium processes include the mobile-immobile

model (van Genuchten and Wierenga, 1976), the two-region model with a diffusion term

(Rao, 1980a, 1980b), the multi -flow domain model (Hutson and Wagenet, 1995; Ma and

Selim, 1995; Gwo et al., 1998), and the local equilibrium assumption model (Passioura,

1971; Parker and Valocchi, 1986).

The third subgroup consists of models that incorporate non-equilibrium chemical

interactions. These interactions are primarily kinetic adsorption processes that can be

easily included in the governing CDE. Generally, a two-site chemical adsorption

assumption is adopted. One site is considered as being in instantaneous equilibrium.

Solute interaction with the second site is a kinetic process. The kinetic process can be

described as first order (Nkedi-Kizza et al., 1983; Leij et al., 1993; Toride et al., 1995)

and second order (Ma and Selim, 1994; Selim et al., 1999).

The fourth subgroup is termed multi-process, non-equilibrium models. This subgroup

considers simultaneous physical and chemical non-equilibrium. Adsorption sites are sub-

divided into equilibrium and non-equilibrium sites, and the water phase is sub-divided

into mobile and immobile regions. This relatively complex deterministic model has been

shown to correctly describe solute transport in soils where both physical and chemical

non-equilibrium processes are common (Brusseau, 1989; Selim et al., 1999).

2.2.5 Solute transport models for multi-layered porous media

The models discussed in the previous section apply to single-layered media. This implies

that one-dimensional solute transport can be described using a mechanistic model with

one set of parameters. It is possible that the media characteristic may change sharply

along the direction of solute transport with the formation of interfaces, discontinuities,

and transition zones. The behavior of the solute will be different before and after it

crosses the interface. As a result, the whole medium has to be considered as multi-

layered. In this case, solute transport in each layer and in the transition zone between

layers will have to be included in the governing solute transport equations. Multi-layered

solute transport occurs commonly between the soil surface and ground water aquifers.

14

The vertical profile for most soils vary in texture or structure as evidenced by the

presence of distinct horizons. Butters et al. (1989), and Butters and Jury (1989), studied

solute transport in a 0.64 ha loamy-sand field soil. They found that the apparent

dispersion coefficient increased linearly through the top 3 meters soil. However, after a

narrow zone of fine texture at about 3 m, the apparent dispersion coefficient dropped

about 30% and then increased linearly.

The physical and mathematical analysis of solute transport across interfaces is very

important for prediction purposes. One approach assumed that the two adjacent layers

are independent meaning that transport in first layer is not affected by transport in the

second layer. This assumption implies that the output from one layer acts as the input

boundary condition for the next layer. This assumption was adopted by Jury and

Utermann (1992) in their zero correlation multi-layered transfer function model, by Delay

et al. (1997) in their elementary transport convolution and numerical particle tracking

model, and by Leij et al. (1991) in their semi-infinite first layer model.

Another approach assumed that the two adjacent layers are dependent, meaning that

transport of the solute in the first layer is affected by the transport in the second layer.

This assumption was made by Kreft and Zuber (1978), and by Barry and Parker (1987).

The latter authors assumed that both the solute flux and the resident concentration vary

continuously across any boundary plane. The solute flux concentration is the ratio of the

mass flux and the water flux across a given plane in the porous medium. It is different

from the resident concentration, which is the ratio of the solute mass in a given volume to

the water contained in the same volume. The same assumption was made by Leij et al.

(1991) in their finite first layer model, and by Jury and Utermann (1992) in their perfect

correlation, multi-layered transfer function model.

Mass conservation requires that the solute mass flux is continuous across an interface,

and that the flux concentration on either side of the interface is the same. The same

restriction does not necessarily apply to the resident concentration. Any difference in

resident concentration on either side of the interface is related to the hydrodynamic

15

characteristics of the adjacent layers i.e., pore water velocity and dispersion (Leij and

Dane, 1989). The discontinuity of resident concentration across an interface has

generated an interest to study the solute concentration distribution within the layered

profile in addition to the solute distribution in the breakthrough curve (Liu et al., 1998;

Jury and Utermann, 1992).

Discontinuities in pore water velocities or the effective porosity on either side of the

interface may result in discontinuity of the solute dispersion. When the random-walk

particle tracking method is used to simulate solute transport, resident concentration

discontinuities may occur if there is discontinuous dispersion (Labolle et al., 1996). Two

approaches have been developed to account for particle movement near an interface in

order to satisfy the condition for continuity in the resident concentration. One is the

reflection technique, which is based on the method of images (Feller, 1957; Uffink,

1985). The other is the velocity interpolation technique (Gelhar and Thompson, 1990).

The random-walk particle method, when modified by the reflection technique yields

resident concentration continuity near the interface for one-dimensional solute transport.

However, it is difficult to be apply this principle to 2- and 3-dimensional solute transport

problems (Labolle et al., 1996)

2.2.6 Solution of governing equations for solute transport models

The governing equations for a solute transport model represent a mathematical

description of the assumed transport mechanisms and processes. Most of the governing

equations are based on the convection-dispersion partial differential equation. In most

cases analytical solutions to the governing partial differential equation cannot be

obtained. For some well-defined initial and boundary conditions, analytical solutions can

be obtained by using transformed variables (Crank, 1975; Parker and van Genuchten,

1984), the Laplace transformation (van Genuchten and Alves, 1982), and the Fourier

transform and Bessel functions (Wexler, 1992). Analytical solutions of governing

equations for one-, two-, and three-dimensional solute transport under a variety of initial

and boundary conditions have been reviewed by Wexler (1992), Toride et al. (1995), and

Leij and van Genuchten (2000).

16

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 cannot be solved analytically. For solving these equations,

numerical approaches are used. In numerical approaches, spatial and temporal domains

of the problem are treated as a mesh of discrete points or cells. The concentration for

each discrete space node or cell is calculated at each discrete time step using numerical

techniques. Neuman (1984) classified the numerical approaches for solving CDE-based

models as Eulerian, Lagrangian, and mixed Lagrangian-Eulerian based on whether the

meshes are fixed or not. Numerical solutions can also be classified based on the

calculation methods. This results in three groups: finite difference, finite element, and

particle tracking.

The finite difference numerical method is simple and applicable for most situations.

Temporal and spatial derivatives are approximated using the Taylor series expansion. The

accuracy of approximation is based on the scheme selected and the mesh size of the

spatial and temporal domains. Finite difference methods are used for solution of many

solute transport problems (Rao et al. 1980b; Leij and Dane, 1989; Fong and Mulkey,

1990; Selim and Ma, 1997)

When the geometry of the spatial domain is complex or the boundaries are irregular, and

when the medium is heterogeneous or anisotropic, the precision of finite difference

methods will decrease. In these situations, the finite element method can be used

(Bedient et al., 1994). In the finite element method, selected piecewise functions are

defined in the domain which has been discretized into finite elements (Norrie and Vries,

1973). Generally, the Galerkin finite element method is used to solve solute transport

equations. In the Galerkin method, the domain of interest is divided into a set of finite

sub-domains called elements. Contaminant concentration is assumed to vary over these

sub-domains. The variation can be assumed as linear, quadratic, or cubic functions.

These variational functions are called the local node basis functions. A trial solution is

constructed as a linear combination of these basis functions. The Galerkin trial solution is

17

considered the weak-form solution of the partial differential equation when the test space

is the same as the trial space. In this trial process, the boundary conditions are

incorporated into the weak form when Green’s formula is applied to decrease the

derivative order (Braess, 1996). The finite element method has been successfully used

for solving the governing solute transport equations especially in the case of 2- and 3-

dimensional problems with complex auxiliary conditions (van Genuchten, 1980; Mishra

and Parker, 1990; Rabbani and Warner, 1997)

The finite difference and finite element numerical methods have been successfully

applied for solving dispersion-dominated transport problems. However, when convection

dominates the solute transport process, oscillation and numerical dispersion can occur in

the computational schemes. Some methods have been developed to overcome oscillation

and numerical dispersion, such as the upstream weighting method (Yeh, 1986; Haga et

al., 1999), the Langrangian method (Neuman, 1984), the characteristic-Galerkin method

(Peraire et al., 1986), and the Langrangian-Eulerian method using the zoomable hidden

fine-mesh approach (Yeh, 1990). In general, oscillation and numerical dispersion are

related and cannot decrease simultaneously (Gray and Pinder, 1976). One simple and

effective method is to decrease the local Peclet number by mesh refinement (Huyakorn

and Pinder, 1983).

The CDE with or without sink/source terms can also be solved by directly tracking a

large number of particles moving through the porous media. This is called the particle

tracking method or random-walk method. In this approach, the convection process is

simulated as the drift of these particles. The dispersion process is simulated by

superimposing a random-walk onto the particles in each time step. Conversion of the

particle cloud to a concentration distribution is realized by spatial discretization and

counting of the particles in each cell. The random-walk can be generated from a normal

distribution with mean zero and the standard deviation 2D t∆ , where D is dispersion

coefficient and ∆t is the discretized time step (Kinzelbach, 1988). It can also be

simulated by Uffink’s jump approach (Uffink, 1985; Delay et al., 1997). In Uffink’s

jump approach, at each advection-dispersion jump from cell i-1 to cell i, the particles in

18

cell i-1 are uniformly distributed around a point centered on i, along a distance

x∆± κ6 , where κ is the dispersivity, and ∆x is the length of a cell.

The concentrations calculated by the distribution generated using the random-walk

method tend to fluctuate over time. Increasing the number of tracked particles can reduce

the fluctuation. On the other hand, Uffink’s jump approach yields a deterministic

concentration at each time step. An advantage of the particle tracking method is that it

does not show any numerical dispersion and oscillation. In addition it can be used in

conjunction with any water flow model (Kinzelbach, 1988).

2.2.7 Parameter estimation

Some parameters in transport models can be independently determined by experimental

measurements. Other parameters, such as the dispersion coefficient, cannot be

determined directly. They are obtained by fitting the experimentally observed data to the

solution of the transport model. Parameter fitting is an inverse problem. Generally, the

solution to the inverse problem is obtained numerically using nonlinear least-squares

optimization based on the Levenberg-Marquardt method (Marquardt, 1963). General

algorithms for parameter fitting have been developed (van Genuchten, 1981; Toride et

al., 1995; Press et al., 1991)

2.3 Modeling of Solute Dispersion

2.3.1 Dispersion phenomena in solute transport

Water movement drives solute transport in porous media. However, the movements of

the solute and water are not identical. An important difference is that the solute spreads

as it is transported in the water. The spreading direction is both longitudinal, meaning

along the water movement direction, and transverse, meaning perpendicular to the water

movement direction. Spreading is either caused by non-sink/source processes or caused

by some sink/source processes. As an example of the latter, when an immobile region of

water exists in the porous media, the solute would exchange between the mobile and

immobile regions, and this results in spreading. The spreading caused by non-sink/source

processes are termed as dispersion.

19

The existence of dispersion can be demonstrated by performing a miscible displacement

experiment in which a step input of a perfectly inert tracer is applied at one end of a

uniformly packed soil column, and then displaced by tracer free solution under steady

conditions. The tracer concentration over time appears at the other end of the column,

not as a step output, but as a sigmoid-shaped curve. When the tracer is input as a Dirac

delta function, the output concentration over time is a bell-shaped curve (Bear, 1988a).

Two processes contribute to solute dispersion. One is molecular diffusion, and the other

is mechanical dispersion. Molecular diffusion is the spontaneous process that occurs

whenever a chemical potential gradient exists in the transition zone between the tracer

solution and the resident solution that it displaces, or vice-versa. Mechanical dispersion

in laboratory columns is merely caused by local variations of water flow at the

microscopic scale level. At this level, these variations include differences in water flow

velocity within an individual pore (or channel) and between pores, or variations in the

direction of mean flow velocity in different pores. In individual pores the maximum

flow velocity occurs in the center, and decreases to zero at the solid surface because of

fluid viscosity effects. Between pores, the mean flow velocity is higher in the pores with

bigger pore diameters as occurs, for example, in preferential flow in soil.

2.3.2 Models for solute dispersion at various scales

Mechanical dispersion is primarily a consequence of the local variations of the water

flow velocity, and these are produced by the intrinsic heterogeneity of the porous

medium. Heterogeneity can be defined at various scales of observation. The following

scales of observation for the variations of water flow velocity in porous media are

adapted from the hierarchical classification of these scales developed by Weber (1986).

Microscopic scale or pore scale: At this scale of observation, heterogeneity is defined

over distances in the order from mm to cm, corresponding to the size of the grains and

pores. The heterogeneity at this scale is mainly caused by variability of soil texture and

soil structure.

20

Macroscopic scale or laboratory scale: The heterogeneity at this scale is defined over

distances in the order from cm to m corresponding to the size of cores and other

reasonably sized samples. At this scale of observation, the porous medium consists of

many grains and pores. The water flow velocities observed at this scale represent the

effect of the pore scale heterogeneity on water flow averaged over the macroscopic scale.

Darcy’s law was developed as the governing equation for water flow at this scale. Porous

media properties, such as porosity, permeability, and dispersivity, are commonly defined

in this scale. This scale corresponds to the scale termed as the representative elementary

volume (REV) proposed by Bear (1988) for defining macroscopic characteristics of

natural porous media.

Megascopic scale or field scale: The heterogeneity at this scale is defined over distances

in the order from m to 100m, corresponding a landscape unit, or in the vertical direction,

the whole thickness of vadose zone. 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.

Dispersion is directly caused by the heterogeneity, and since heterogeneity can be defined

at various scales, solute dispersion can be incorporated in solute transport models at

various scales of heterogeneity. This results in several general types of models for

simulating solute dispersion in porous media, namely, pore scale models, laboratory scale

models, and field scale models. Some models, such as the layer model and the scale-

dependent dispersivity CDE model, may be applicable over two consecutive scales in the

hierarchy. Some, such as the fractional convection-dispersion equation (FCDE) model,

may be applicable over all three scales. For the most part, pore scale models were

developed to explain dispersion theoretically. Laboratory scale models were developed

to identify solute transport parameters and explain transport mechanisms. Field models

were designed mainly for practical description and prediction of solute transport in fields.

21

Another classification of models for solute dispersion is based on the assumption of

statistical behavior of the moving tracer particles. These models can be divided into

those based on the Gaussian distribution, and those based on the Levy distribution. To

date, only one model, termed as the fractional advection-dispersion equation (FADE)

Model (Pachepsky, 2000) is based on the Levy distribution. Like the advection-

dispersion equation (ADE) is also termed as the convection-dispersion equation (CDE),

the FADE will be termed as the fractional convection-dispersion equation (FCDE) in the

dissertation. Models based on the Gaussian distribution, either explicitly or implicitly,

have been developed for all three scales of observation. Solute transport models in these

two groups are listed in Figure 2.1. Models that span more than one scale are indicated

in Figure 2.1. The transfer function models are shown as partially belonging to the

Gaussian-based category, since some of them are also based on the lognormal or gamma

probability density functions. A more detailed discussion of these dispersion models

follows.

Gaussian DistributioniGaussian Distributioni Levy Distribution

Pore scale

Laboratory

scale

Field -scale

Exact mathematical models based on simplegeometric assumption(1) Taylor’s capillary tube(2) Bear’s cell-channel array

Statistical models(1) 1-D random walk(2) Scheidegger’s random walk model(3) de Josselin, de Jong’s network (4) Saffman’s network

Mass conservation and Fick’s Law:Convection-dispersion equation (CDE)

Scale-dependent CDE(1) Distance-dependent CDE (2) Time-dependent CDE

Layer model

Stochastic models:(1) Mechanistic model: Stream tube

Factional convection-dispersion equation (FC

DE)

(2) Transfer function

Figure 2.1 Available models for simulating solute dispersion at three scales based on the

Gaussian and Lévy distributions.

22

Pore scale models: Two main approaches are used in dispersion models at the pore scale

(Figure 2.1). One is based on exact mathematical methods, the other on statistical

methods (Bear, 1988). In the former approach, the porous media are represented as

simple, idealized geometric units such as capillary tubes or a series of cells, for which

solute transport behavior can be well-defined physically and mathematically. Taylor

(1953) developed the theory for solute dispersion in a single capillary tube based on the

parabolic fluid velocity distribution within the tube. Bear (1988) modeled the porous

medium as an array of small cells with short interconnecting channels, which he termed

as a "one-D cell-channel array". Solute transport was modeled as the sequential mixing

of the tracer solution between consecutive cells in the array.

The complex nature of discrete porous media makes it very difficult, if not impossible, to

realistically represent them as a collection of simple geometrical units. When the size

and shape of the grains and pores occur randomly in the discrete porous medium, the

transport of tracer molecules in the pores can be treated as a statistical process. Statistical

distributions and methods can be used to average the microscopic motions of the solute in

order to obtain a macroscopic description. The observed concentration distribution

represents the ensemble average of the microscopic solute transport behavior in the pores.

Even though it is not possible to exactly describe the behavior of an individual tracer

particle in the porous medium, statistical theory permits predicting the spatial distribution

of a cloud of tracer particles undergoing random movements over time. Statistically

based models include random walk models and pore network models, such as

Scheidegger’s random walk model, de Josselin de Jong’s network model, and Saffman’s

network model. A detailed review of these pore scale models has been provided by Bear

(1988b).

Laboratory scale models: Most experimental research on solute transport and dispersion

in porous media has been carried out at the laboratory scale. The operational procedures

for conducting laboratory scale solute transport experiments are relatively easy. The

breakthrough curves (BTCs) obtained from such experiments can provide sufficient

23

information for identifying the physicochemical processes involved in solute transport.

Parameters obtained at this scale can be used for comparison of the physicochemical

characteristics of various media. These parameters may be applied in field scale models,

if the media at the field scale are homogeneous.

A variety of laboratory scale models have been developed (Figure 2.1). A main source of

this variety is how dispersion is accounted for in these models. In one type of models,

termed as CDE models, a single constant is used to account for dispersion. When

dispersion is scale-dependent due to the heterogeneity of the media, more complex

functions are needed to describe dispersion. The functions may be either space or time

dependent. When these functions are used in place of the dispersion constant in the CDE,

the resulting models are commonly called scale-dependent CDE models. Such models

would include both distance-dependent CDE and time-dependent CDE categories

depending on whether space or time dependent functions were used to describe

dispersion. Some reported scale-dependent dispersivity functions used in this type of

models are listed in Table 2.1.

24

Table 2.1 Scale-dependent dispersivity functions

Reference Dispersion Function Application

Mishra and Parker (1990)

--Distance dependent --Local hyperbolic dispersivity :

ε

ε β

( )x

x

=+

∞

11 1

--Effective dispersivity:

ε εβ ε

β ε( ) [

ln ( / )/

]xx

x= −

+∞

∞

∞

11

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

entire procedure is iterated.

.. .. .. .... .. .. .. .. .... .. .. .. .. .. ..I-7 N(I-7,E) N(I-7,F) N(I-7,F) N(I-7,F) N(I-7,F) N(I-7,F) N(I-7,E)I-6 N(I-6,E) N(I-6,F) N(I-6,F) N(I-6,F) N(I-6,F) N(I-6,F) N(I-6,E)I-5 N(I-5,E) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,E)I-4 N(I-4,E) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,E)I-3 N(I-3,E) N(I-3,F) N(I-3,F) N(I-3,F) N(I-3,F) N(I-3,F)I-2 N(I-2,E) N(I-2,F) N(I-2,F) N(I-2,F) N(I-2,F)I-1 N(I-1,E) N(I-1,F) N(I-1,F) N(I-1,F)I

N(I-7) N(I-6) N(I-5) N(I-4) N(I-3) N(I-2) N(I-1) N(I)

At time t:

L+

N'(I-7) N'(I-6) N'(I-5) N'(I-4) N'(I-3) N'(I-2) N'(I-1) N'(I)

Convection and dispersion within ∆ t:

At time t+ ∆ t:SumSumSumSumSumSumSumSum

Noutlet

Sum

N(I-3,E)N(I-2,F) N(I-2,E)N(I-1,F) N(I-1,F) N(I-1,E)

N(I)

Index

Figure 3.3 Algorithm used for solution of the CDE with zero concentration outlet

boundary condition by the particle tracking method. See text for explanation.

40

Physically, the zero gradient finite outlet boundary condition, specifies that there is no

concentration gradient at x = L. Consequently, at the dispersion step in particle tracking

procedure, there are no net particles crossing the outlet boundary at x = L by dispersion.

This means that that if one particle disperses out of the column at x = L, one outside

particle disperses back into the column in the same time. The algorithm for the particle

tracking method using the Zero gradient finite outlet boundary condition is illustrated in

Figure 3.4. The notation for this algorithm is identical to that already described for the

zero concentration outlet boundary condition. However, as discussed, the no-flux

boundary condition specifies that there are no net particles crossing the outlet boundary at

x = L by dispersion. As a result, the redistribution of the particles for the discretized cells

close to the boundary (indexed from I – (K+1) to I), during each convection-dispersion

step is different, as shown in the corresponding rows of the matrix in Figure 3.4.

N(I-7) N(I-6) N(I-5) N(I-4) N(I-3) N(I-2) N(I-1) N(I)

At time t:

L+

N'(I-7) N'(I-6) N'(I-5) N'(I-4) N'(I-3) N'(I-2) N'(I-1) N'(I)

Convection and dispersion within ∆ t:

At time t+ ∆ t:SumSumSumSumSumSumSumSum

.. .. .. .. .. .. ..

I-5 N(I-5,E) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,F) N(I-5,E)

I-4 N(I-4,E) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,F) N(I-4,E)N(I-3,E)

I-3 N(I-3,E) N(I-3,F) N(I-3,F) N(I-3,F) N(I-3,F) N(I-3,F)N(I-2,E) N(I-2,F)

I-2 N(I-2,E) N(I-2,F) N(I-2,F) N(I-2,F) N(I-2,F)N(I-1,E) N(I-1,F) N(I-1,F)

I-1 N(I-1,E) N(I-1,F) N(I-1,F) N(I-1,F)

I N(I)

Noutlet

Index

Figure 3.4 Algorithm used for solution of the CDE with zero gradient finite outlet

boundary condition by particle tracking method. See text for explanation

It is generally impossible to simulate an infinite condition in a finite domain such as a

computer memory. However, if the finite domain is large enough, some infinite

conditions can be approximated in the finite domain. The infinite condition specified in

41

Eq. (3.4) was simulated in an appropriately large finite domain. The infinite domain was

approximated as an imaginary extension of the finite column with an imaginary outlet

boundary at x = 50L, as illustrated in Figure 3.5.

The boundary condition at the imaginary outlet was taken to be the no-flux finite

boundary condition as described in Eq. (3.3). The discharge point was treated as the

midpoint of a cell just outside the end of the column at x = L as shown in Figure 3.5.

Because the imaginary outlet boundary was very far away from x = L, the convection and

dispersion of any particles in a cell near the imaginary outlet boundary would not affect

the particle number distribution in a cell near the discharge point at x = L. Therefore, the

imaginary outlet boundary behaves as if it were at some infinite location.

Injection Outlet

50LL

Observation Cell(BTC determined)

Finite condition

Figure 3.5 Algorithm used for solution of the CDE with zero gradient infinite outlet

boundary condition by particle tracking method. The infinite boundary condition was

approximated using the finite condition for an appropriately large domain. See text for

explanation.

For all the three calculation schemes described above, one zero concentration outlet

boundary condition was specified for the end of the column at which the tracer free

background solution was pumped in (the left end of the column in Figure 3.1). In the

numerical tests, the distance from the injection point to the left end was assumed to be

30L in order that no particles could disperse out from the left end of the column. This

would hold even for a very low value of the solute transport Peclet number (< 5). For all

calculation of BTC generating and BTC fitting, the mass of solute recovery at the outlet

42

boundary was calculated in order to observe whether all the injected solute was

recovered.

BTCs calculated using the solution to the CDE by the particle tracking method with these

three outlet boundary conditions, for different column Peclet numbers are presented in

Figure 3.6. The dimensionless Peclet number is defined as vL/D (= L/κ), where v is the

average pore water velocity, L is the column length, and D (= vκ) is the dispersion

coefficient. BTCs were calculated for Peclet numbers of 5, 20, 60, and 100. The results

showed that the BTCs for the three boundary conditions were observably different for

Peclet numbers equal to 5 and 20. BTCs for Peclet numbers above 60 were

indistinguishable. Low Peclet numbers imply that the solute transport is dominated by

the dispersion process. High Peclet numbers imply that convection is the dominant

process controlling solute transport. The results in Figure 3.6 indicate that when

convection is the dominant process in the solute transport, the outlet boundary condition

specified for the free-inlet initial value problem does not significantly affect the

calculated BTCs. van Genuchten and Alves (1982) reported the same behavior of their

calculated BTCs at high Peclet numbers for finite columns with both inlet and outlet

boundary conditions.

43

Peclet number=5

0

0.005

0.01

0.015

0 1 2 3 4

Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 )Peclet number=5

0

0.005

0.01

0.015

0 1 2 3 4


Con

cent

ratio

n (m

mol

e/cm

3 )

Zero concentration Zero-gradient infinite conditionZero-gradient finite condition

Peclet number=20

0

0.006

0.012

0.018

0.024

0 1 2


Con

cent

ratio

n (m

mol

e/cm

3 )

Zero concentration Zero-gradient infinite conditionZero-gradient finite condition

Peclet number=60

0

0.01

0.02

0.03

0.04

0 0.5 1 1.5 2 2.5


Con

cent

ratio

n (m

mol

e/cm

3 ) Zero concentration Zero-gradient infinite condition

Zero-gradient finite condition

Peclet number=100

0

0.01

0.02

0.03

0.04

0.05

0 0.5 1 1.5 2 2.5Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) Zero concentration Zero-gradient infinite conditionZero-gradient finite condition

3

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. In all cases, the total mass of solute injected = 1 mmole, porosity = 0.33, and L

= 37.5 cm.

.

3.3 Numerical Tests on the Effect of Injected Solute Distribution on BTCs

Numerical tests were made in order to determine how the initial solute distribution after

injection influences solute transport, and how critical is this effect of the initial

distribution on the observed BTCs.

3.3.1 Solute distribution after injection

After solute is injected into the column, the solute is assumed to mix completely and

instantaneously with the background solution, and forms some solute concentration

distribution along the length of column, centered at the plane of injection in which the

needles are located. In order to test the effect of solute distribution after injection on the

44

BTCs, the distribution after injection was assumed to be either a normal distribution with

a standard deviation of σ, or a uniform distribution with a width of w. A normal

distribution approaches a Dirac delta function as σ → 0. If the solution of a solute

transport problem is known when the initial condition is specified as a Dirac delta

function, the solution for all other initial solute distribution can be obtained using the

convolution theorem (Toride et al. 1995).

3.3.2. Solute transport models

Five non-reactive solute transport models were used for numerical testing of the

dependence of BTCs on the initial solute distribution. These models were the CDE, the

CDE with distance-dependent dispersivity, the CDE with time-dependent dispersivity,

the MIM, and the FCDE. These models are detailed below.

Model 1. The convection-dispersion equation (CDE) with constant dispersivity: The

CDE for one-dimensional conservative solute transport under steady, saturated flow

conditions is:

xCv

xCD

tC

∂∂

−∂∂

=∂∂

2

2

(3.5)

where C is the concentration (ML-3), v is average pore-water velocity (Lt-1), x is distance

(L), and t is time (t). D is the hydro-dynamic dispersion coefficient (L2t-1) and represents

a quasi-diffusion coefficient that accounts for both mechanical dispersion and molecular

diffusion. D = κv + Do [or Eq. (2.3) ], where κ is the dispersivity (L), and Do is the

molecular diffusion coefficient.

Model 2. The CDE with distance-dependent dispersivity (distance-dependent CDE):

Based on fractal theory, Wheatcraft and Tyler (1988) incorporated D in the CDE as κv

using κ (dispersivity) as a power-law distance-dependent equation. Expressing D in this

way, implies that molecular diffusion is neglected. The CDE becomes:

xCv

xCvax

xtC b

∂∂

∂∂

∂∂

∂∂

−= ])[( (3.6)

45

where a and b are two constants related to the porous medium structure.

Model 3. The CDE with time-dependent dispersivity (time-dependent CDE): The

dispersivity can also be described using a power-law time-dependent equation (Zou et al.,

1996), and when the molecular diffusion is neglected, the corresponding CDE is:

xCv

xCct

tC d

∂∂

∂∂

∂∂

−= 2

2

(3.7)

where c and d are two constants.

Model 4. The mobile-immobile model (MIM): The mobile-immobile model (MIM)

assumes that the liquid phase can be partitioned into a mobile (flowing) region and an

immobile (stagnant) region (Coats and Smith, 1964; van Genuchten and Wierenga,

1976). Solute transport in mobile region obeys the CDE with a sink/source term

describing solute exchange between two regions. Solute exchange is modeled as a first-

order process. For saturated, steady water flow conditions, the MIM is written as:

θ∂

∂θ

∂∂

∂∂

βmm

mm

m im

Ct

DC

xq

Cx

C C= − − −2

2 ( ) (3.8a)

θ∂

∂βim

imm im

Ct

C C= −( ) (3.8b)

where the subscripts m and im represent the mobile and immobile regions, respectively,

θm and θim represent the volumetric water content, q is the discharge velocity or the Darcy

flux (Lt-1), and β is a first-order mass transfer coefficient (t-1) controlling solute

exchange between the two regions. For saturated conditions and θ = θm +θim is total

porosity. The meaning of the other variables and parameters of the MIM are the same as

those already defined in the CDE.

46

Model 5. The fractional convection dispersion equation (FCDE): The FCDE (Benson,

1998) assumes that solute dispersion obeys a Levy distribution instead of a Gaussian

distribution as in the CDE. The corresponding governing solute transport equation is:

xCv

xC

xCD

tC

f ∂∂

−

−∂

∂+

∂∂

=∂∂

α

α

α

α

)(21 (3.9)

where Df is the fractional dispersion coefficient (Lαt-1), α is the order of fractional

differentiation and 0 < α ≤ 2.

3.3.3. Procedures for numerical tests

At time zero, the solute mass injected was assumed as a normal distribution or a uniform

distribution over some region along the length of the column. This region was discretized

into 1000 cells, and in each cell, the solute distribution was assumed to be a Dirac delta

function. The cells covered the entire region for the uniform distribution. For the normal

distribution, the cells covered a region taken as 8σ representing more than 99.999% of

the injected mass. The width w of the uniform distribution, and the interval 2σ of the

normal distribution, was assumed as 5%, 10%, 20%, and 40% of the column length. The

column length for this purpose was defined as the distance from the injection point to

discharge point and was denoted as L. The different values of w or 2σ were taken as

treatments for the numerical tests using the 5 solute transport models. An additional

treatment was included with the injected solute distribution taken as a Dirac delta

function. In this case, w = 2σ ≈ 0% of the column length, and this treatment was denoted

as ~0%. The infinite outlet boundary condition was used in all numerical tests.

The BTCs for the CDE, the CDE with time dependent dispersivity, and the FCDE were

generated using the convolution theorem. For these models, the solute concentration at

the outlet was calculated as:

(3.10) ),()(),(500

500tlLfimtLC i

i−= ∑

−=

Where L is column length from injection point to outlet point, m(i) is the solute mass in

cell i at the initial time, and li is the distance from the injection point to the center of cell

47

i. If the cell i is located between the injection and discharge points, the li is positive,

otherwise, it is negative. f(L-li, t) is an auxiliary function for the convolution. For the

~0% treatment, this auxiliary function represents the analytical solution.

For the CDE, f(L-li, t) is:

−−−=−

DtvtlL

DttlLf i

i 4)(

exp41),(

2

πθ (3.11)

For the time-dependent CDE, f(L-li, t) is (Zou et al., 1996):

−−−=−

λλπθ~4

)(exp~4

1),(2

vvtlL

vtlLf i

i (3.12)

Where 1

1~ +

+= dt

dcλ

For the FCDE, f(L-li, t) is defined as (McCulloch and Panton, 1997):

∫

−

−= −

− 1

0

11

1

)(exp)(12

)( φφφαα

αα

α

ε

α

α dUyUy

yf (3.13a)

( )πφ

φαππφ

παφ

φ

αα

α21

21

1

cos)1(cos

2cos

2sin

−

=

−

U (3.13b)

απασ

1

'

2cos

= tDsf (3.13c)

−−

=− ''

1),(σθσ α

vtlLftlLf i

i (3.13d)

The BTCs for the distance-dependent CDE and the MIM were generated using the

particle tracking method developed by Uffink (1985) and Delay et al. (1997). For the

~0% treatment, all the particles were assumed to be injected in a single cell centered at

the plane of injection.

48

The analytical solutions of the CDE with the initial solute distribution as a normal

distribution or uniform distribution are given in Appendix 1.

3.3.4 Results of the numerical tests

The BTCs generated by the 5 models using the different injection treatments described

above were compared in the numerical tests.

Results using the CDE: In the CDE, the dimensionless column Peclet number is

generally used to express the relative contribution of dispersion and convection to the

solute transport. The Peclet number of the CDE is:

DvLnumberPeclet = (3.14)

When the Peclet number is large (>100), convection is the dominant solute transport

process. When the Peclet number is small (< 10), dispersion is dominant. The BTCs are

calculated using dimensionless time that equals to tv/L, where L/v is the time taken for

displacement of one pore volume through the column.

Results of the numerical tests to determine the effect of the different injection treatments

on the BTCs obtained using the CDE are presented in Figure 3.7 for the normal

distribution assumption, and in Figure 3.8 for the uniform distribution assumption. BTCs

were generated for column Peclet numbers of 1, 10, 100, and 1000. Comparison of the

results of Figure 3.7 and Figure 3.8 showed that the BTCs for the two initial solute

distribution assumptions behaved similarly. In both cases, when the Peclect number was

equal to 1, the BTCs for the treatments of ~0%, 5%, 10 %, 20%, and 40% were almost

identical. As the Peclet number was increased to 10, the BTC for the treatment of 40%

was somewhat different from the BTCs of other four treatments. The BTCs of the

treatments ~0%, 5%, 10% and 20% were identical. For Peclet number = 100, the BTCs

of the treatments of ~0%, 5%, and 10% were same, but were different from those of the

treatments of 20% and 40%. Also, the BTCs for the latter two treatments were markedly

different from each other. At the highest value of 1000 for the Peclet number, the BTCs

49

of all five treatments were clearly different each other. The peakedness of the BTCs

decreased with increasing Peclet number.

These results show that if the column Pelect number is <100, and the value of w or 2σ is

no more than 10% of L, the analytical solution of the CDE developed for solute transport

with initial conditions represented as a Dirac delta function (the ~0% treatment), is

adequate. When using this analytical solution, the calculated BTCs will not be different

from corresponding BTCs calculated by applying the convolution procedures when the

injected solute is uniformly or normally distributed. This finding may also be

approximately applicable, when the Peclet number is not much larger than 100 and w or

2σ is less than 5% of L.

These results also showed that the assumed distribution of the solute mass after injection,

was not important in the calculation of the BTCs. There was minimal, if any, effect on

the calculated BTCs between the normal distribution and the uniform distribution.

However, the injection is not instantaneous since there are four injection units, and

therefore there is random mixing of the tracer solution with the background solution in

the column. When these factors are taken into consideration, the normal distribution may

physically more closely represent the distribution of the solute mass after injection, than

the uniform distribution. Consequently, only the normal distribution for the injected

solute was tested for the other four models.

50

Peclet number=1

0.000

0.001

0.002

0.003

0.004

0.005

0 1 2 3 4 5Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

Peclect number=10

0.000

0.005

0.010

0.015

0 1 2 3

Dimensionless time TC

once

ntra

tion

(mm

ole/

cm3 ) 5%

10%20%40%~0%

Peclet number=100

0.000

0.010

0.020

0.030

0.040

0 0.5 1

Peclet number=100

0.000

0.010

0.020

0.030

0.040


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

Peclet number=1000

0.000

0.040

0.080

0.120

0 0.5 1 1.5Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

2

4

Figure 3.7 BTCs calculated using the CDE for treatments where the injected solute was

assumed to be normally distributed with 2σ of 5%, 10%, 20%, and 40% of L . Treatment

denoted as ~0% represents an injected solute distribution described as a Dirac delta

function. In all cases, the total mass of solute injected = 1 mmole, porosity = 0.33.

51

Peclet number=1

0.000

0.001

0.002

0.003

0.004

0.005

0 1 2 3 4 5Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

Peclet number=10

0.000

0.003

0.006

0.009

0.012

0.015

0 1 2 3 4Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

Peclet number=100

0.00

0.01

0.02

0.03

0.04

0 0.5 1

Peclet number=100

0.00

0.01

0.02

0.03

0.04


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

Peclet number=1000

0.000

0.040

0.080

0.120

0 0.5 1 1.5Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

2

Figure 3.8 BTCs calculated using the CDE for treatments where the injected solute was

assumed to be uniform distribution with w of 5%, 10%, 20%, and 40% of L . Treatment



Results using the CDE with distance-dependent dispersivity: BTCs calculated using the

CDE with distance-dependent dispersivity are presented in Figure 3.9. Four parameters

were used in these calculations: the parameters a and b, velocity v, and average Peclet

number. The average Peclet number was defined as:

DvLnumberPecletAve =. (3.15a)

52

vL

dxaxD

Lb∫

= 0 (3.15b)

Eq. (3.15b) represents the dispersivity (κ = axb) averaged over the length of the column.

In order to examine the effects of the parameters, the BTCs in Figure 3.9 were calculated

for different combinations of the parameters a, b, and v, using a fixed length L, as shown

below.

BTC I II III IV V VI a 0.01 0.01 0.1 0.1 0.1 0.01 b 1 1 1.5 1 1 1.5 v 0.06 0.6 0.6 0.06 0.6 0.6 Ave. Peclet number 200 200 4 20 20 40

Out of the four parameters, the average Peclet number had the greatest effect on the

calculated BTCs. When the average Peclet number was smaller than 40, there was no

discernible difference in the BTCs using the numerical solution of the distance-dependent

CDE with initial conditions represented as a Dirac delta function (the ~0% treatment),

and those when 2σ was <10% of L. However, when the average Peclet number was

increased to 200, the BTCs were markedly different from those using initial conditions as

Dirac delta function, except when 2σ = 5% of L.

53

a=0.01, b=1, v=0.06Ave. Peclet number=200

0.00

0.02

0.04

0.06

0 0.5 1 1.5 2Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%


0

0.005

0.01

0.015

0.02

0.025


0

0.005

0.01

0.015

0.02

0.025


Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%


0.00

0.02

0.04

0.06

0 0.5 1 1.5 2


Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%


0.00

0.02

0.04

0.06

0 0.5 1 1.5 2


Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%


0.000

0.005

0.010

0.015

0.020

0.025a=0.1, b=1, v=0.6

Ave. Peclet number=20

0.000

0.005

0.010

0.015

0.020

0.025

0 0.5 1 1.5 2 2.5


Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%

a=0.1, b=1.5, v=0.6Ave. Peclet number=4

0

0.005

0.01

0.015

0.02

0 1 2 3Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%

a=0.01, b=1.5, v=0.6Ave.Peclet number=40

0

0.01

0.02

0.03


Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%

I

II

III

IV

V

VI

Figure 3.9 BTCs calculated using the CDE with distance-dependent dispersivity for

treatments where the injected solute was assumed to be normal distribution with 2σ of

5%, 10%, 20%, and 40% of L. Treatment denoted as ~0% represents an injected solute

distribution described as a Dirac delta function. In all cases, the total mass of solute

injected = 1 mmole, porosity = 0.33.

54

Results using the CDE with time-dependent dispersivity: BTCs calculated using the CDE

with time-dependent dispersivity are presented in Figure 3.10. Four parameters were

used in these calculations: the parameters c and d, velocity v, and average Peclet number.

In this case, the average Peclet number was defined as:

DvLnumberPecletAve ~. = (3.16a)

vT

dtct~

'

T

0

d'

∫=D (3.16b)

where T' = L/v is the time taken for displacement of one pore volume through the

column.

In order to examine the effects of the parameters, the BTCs in Figure 3.10, were

calculated for different combinations of the parameters c, d, and v, using a fixed length L,

as shown below.

BTC I II III IV V VI c 0.01 0.1 0.01 0.1 0.01 0.02 d 1 1.5 1.5 1 1 1 v 0.6 0.6 0.6 0.6 0.06 0.6 Ave. Peclet number 120 2 18 12 12 60

The BTCs in Figure 3.10, show similar findings to those reported in Figure 3.9 for the

CDE with distance dependent dispersivity. When the average Peclet number was smaller

than 120, the BTCs calculated using analytical solution to the CDE with time dependent

dispersivity, for initial solute distribution represented as a Dirac delta function (~0%

treatment), were almost identical to those calculated by applying the convolution theorem

when 2σ = 5% of L.

Intuitively, one would expect that the width of the initial solute distribution would be

very important for both the CDE with distance dependent or time dependent dispersivity,

because the dispersivity is not a constant in space or time domain. However, the

foregoing results showed that, it was the average Peclet number rather than the width of

the initial distribution, that was critical for selection of the method used for solving the

injected solute transport problem described in Eq.(3.6) and Eq.(3.7). In one method, the

55

initial injected solute distribution can be represented as a Dirac delta function. The other

method makes use of the convolution theorem, provided that the initial solute distribution

function is known. When it is unknown, the initial solute distribution can be represented

as a Dirac delta function provided that the average Peclet number is not very large.

Results of the MIM: As discussed above, the MIM model includes a number of

parameters. The effect of varying three of these parameters on the BTCs generated for

the five treatments were analyzed. The three parameters were the immobile phase

porosity θim, mass transfer coefficient β, and the Peclet number for solute transport in

mobile phase = (qL) / (θm D). The immobile phase porosity θim was assumed as either

10% or 30% of total porosity θ, β as either 0.001 or 0.01 min-1, and the Peclet number as

10 and 100. The results are presented in Figure 3.11. As shown, the Peclet number was

the most important factor controlling the BTCs for the different treatments. For Peclet

number = 10, the initial solute distribution could be represented as a Dirac delta function

provided 2σ was 20% of L. However, for Peclet number = 100, the BTCs calculated

assuming a Dirac delta function (~0% treatment), reasonably matched those for the

normal initial solute distribution only when 2σ was smaller than 10% of L.

56

c=0.01, d=1, v=0.6 Ave. Peclet number=120

0.00

0.01

0.02

0.03

0.04

0.05

0 1 2DImensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

c=0.1, d=1, v=0.6, Ave. Peclet number=12

0

0.005

0.01

0.015

0 1 2 3Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

c=0.1, d=1.5, v=0.6Ave. Peclet number=2

0.000

0.002

0.004

0.006

0.008


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

c=0.01, d=1, v=0.06, Ave. Peclet number=12

0.000

0.005

0.010

0.015

0 1 2Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 )

5%10%20%40%~0%

c=0.01, d=1.5, v=0.6Ave. Peclect number=18

0.000

0.005

0.010

0.015

0.020

0 1 2 3

c=0.01, d=1.5, v=0.6Ave. Peclect number=18

0.000

0.005

0.010

0.015

0.020

0 1 2 3


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

c=0.02, d=1, v=0.6Ave. Peclet number=60

0.00

0.01

0.02

0.03


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

I

II

III

IV

V

VI

3

Figure 3.10 BTCs calculated using the CDE with time-dependent dispersivity for

treatments where the injected solute was assumed to be normal distribution with 2σ of

5%, 10%, 20%, and 40% of L. Treatment denoted as ~0% represents an injected solute

distribution described as a Dirac delta function. In all cases, the total mass of solute

injected = 1 mmole, porosity = 0.33.

57

=0.1 , =0.001,Peclet number=100

0.00

0.01

0.02

0.03

0.04


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

=0.3 , =0.001,Peclet number=100

0

0.01

0.02

0.03

0.04

0.05

0 1 2

=0.3 , =0.001,Peclet number=100

0

0.01

0.02

0.03

0.04

0.05


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%

10%20%

40%~0%

=0.1 , =0.01, Peclet number=100

0

0.01

0.02

0.03

0.04


Con

cent

ratio

n (m

mol

e/cm

3 )

=0.1 , =0.01, Peclet number=100

0

0.01

0.02

0.03

0.04


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

=0.1 , =0.001, Peclet number=10

0.000

0.005

0.010

0.015

0 1Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

=0.3 , =0.001,Peclet number=10

0.000

0.005

0.010

0.015

0.020

0 1


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

=0.1 , =0.01, Peclet number=10

0.000

0.005

0.010

0.015

0 1


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%

20%40%

~0%

θimθim

θim θim

θim θimθ θ

θ θ

θ θ

β β

β β

β β

2

2

2

Figure 3.11 BTCs calculated using the MIM for treatments where the injected solute was

assumed to be normal distribution with 2σ of 5%, 10%, 20%, and 40% of L. Treatment



.

58

Results of the FCDE: Three parameters, namely Df , α, and v, must be specified for

calculating solute BTCs with the FCDE model . Because the dimension of Df is Lαt-1, it

was difficult to find a dimensionless parameter such as the Peclet number as defined as

the Eq.(3.14), which would similarly encapsulate the relative contributions of convection

and fractional dispersion on solute transport. The dimensionless number of fD

vL 1−α

did not

give meaningful results when used in numerical calculations for evaluating the effect of

the width of injection distribution on the predicted BTCs. Consequently, a parameter

with dimension of L2-α , termed as the equivalent Peclet number, was defined as:

fD

vLnumberPeclet. =Equiv (3.17)

Application of this equivalent Peclet number for the FCDE model was equivalent to

applying the Peclet number in the CDE. BTCs were calculated for the 5 treatments using

the FCDE with values of α = 1.2 and 1.8, and equivalent Peclet numbers of 10 and 100.

These BTCs are presented in Figure 3.13. The results showed that when the equivalent

Peclet number = 10, regardless whether α = 1.2 or 1.8, the BTCs for treatments 5%, 10%

and 20% of L, were almost identical to the BTC obtained using a Dirac delta function for

the initial injected solute distribution (~0% treatment). However, when the equivalent

Peclet number was 100, only the BTCs for treatment 5% of L, matched the BTC obtained

using the Dirac delta function.

59

=1.2, Equiv. Peclet number=100

0.00

0.01

0.02

0.03

0.04

0.05

0 0.5 1 1.5 2


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%


0.00

0.01

0.02

0.03

0.04

0.0 0.5 1.0 1.5 2.0Dimensionless time T

Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

=1.2, Equiv. Peclect number=10

0

0.002

0.004

0.006

0.008

0 1 2 3 4


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%


0.000

0.002

0.004

0.006

0.008

0.010


Con

cent

ratio

n (m

mol

e/cm

3 ) 5%10%20%40%~0%

α α

α α

Figure 3.12 BTCs calculated using the FCDE for treatments where the injected solute

was assumed to be normal distribution with 2σ of 5%, 10%, 20%, and 40% of L.

Treatment denoted as ~0% represents an injected solute distribution described as a Dirac

delta function. In all cases, the total mass of solute injected = 1 mmole, porosity = 0.33.

In summary, the results of the numerical tests showed that, for all 5 models, if the width

of the solute distribution immediately after injection, was not larger than 5% of the

column length, the solute transport BTCs were almost identical to that obtained assuming

an initial distribution represented as a Dirac delta function. This result applies assuming

the injected solute is distributed normally or uniformly, and centered on the column

cross-section coincident with the plane of injection, and for Peclet numbers smaller than

200. The width is defined as 2σ of this normal distribution or the width of the uniform

distribution, and the column length as the distance from the injection point to the

60

discharge point. This finding is important, since in real experiments, the injected solute

distribution is generally unknown. In such situations, these results make it possible to

define the experimental conditions under which the injected solute distribution can be

approximated as a Dirac delta function.

3.4 Experimental Tests on Effects of Injected Solute Distribution Width on BTCs

The foregoing numerical tests were made in order to test the behavior of solutions to a set

of initial value problems posed by use of the injection method in one-dimensional solute

transport experiments. Analysis of this problem was an important step in order to

determine operational conditions for application of the column system. As discussed,

solutions to the initial value problem where the injected solute distribution is assumed to

be a Dirac delta function is adequate when w or 2σ of the injected solute distribution was

smaller than 5% of L. The numerical tests also showed that the important factor affecting

the BTCs was the width of initial distribution, rather than the assumption of the form of

the initial solute distribution.

These results of the numerical tests needed to be validated by experiments, since the

width and form of the injected solute distribution cannot be quantified in real situations,

and therefore the convolution theorem cannot be applied. Practically, it is impossible to

inject solute to obtain an initial distribution as a Dirac delta function. However, the

numerical tests suggested that the BTCs generated using the Dirac delta function for the

initial distribution were not different from those generated using the solution to the initial

value problem with very small w or 2σ. In the inlet free column system described above,

differing widths of injected solute can be easily obtained by adjusting the injection

volume. Experiments were designed to test the effect of solute injection width as

indicated by these results of the numerical tests.

3.4.1 Experimental Materials and Methods

In the laboratory column system already described, the distance between the injection

point and the discharge outlet of the column was 37.5 cm. The column was uniformly

packed with different sizes of soda-glass beads to obtain a specified fractal dimension in

61

order to simulate aggregate distribution in natural soils. The required number distribution

of the glass beads needed to obtain the fractal dimension of 2.02 was obtained using

fragmentation theory (Reiu and Sposito, 1991). The number distribution of different

sized glass beads, based on a total of 1000 beads, was calculated using Eq. (A2.5) and

Eq. (A2.6) in Appendix 2, and is listed in Table 3.1. This number distribution can be

easily converted to the weight distribution because of the same bead density for all the

different sized glass beads.

Table 3.1. The number distribution based on a total of 1000 beads

of different sized glass beads used to pack experimental columns

Diameter (mm) 1.1 1.6 1.9 2.2 2.6 3.1 3.7 5.1

Number 498 150 103 70 52 37 41 45

Fluorescein solution (3 mM fluoescein + 7 mM NaCl) was used as the non-reactive

tracer. The background or displacement solution was 10 mM NaCl. Treatments of

injection volume were 0.5 ml, 2 ml, 4 ml, and 8 ml, respectively. Each treatment was

replicated four times in the same column. The solute mass was close to 0.0015 millimole

for all injections. Light was excluded from the entire column system in order to

minimize photobleaching of the fluorescein. The outflow samples collected in the

fraction collector, were kept in the dark and analyzed within 24 hours. The concentration

of fluorescein in the samples was determined using a spectrophotometer. Preliminary

tests were made to ensure that light was adequately excluded to ensure that there was not

photobleaching under the experimental conditions.

An attractive feature of using fluorescein as the tracer and glass beads as the porous

medium, was that the width of the solute distribution after injection could be visually

estimated. Tests showed that it was very easy to visually distinguish the color difference

between fluorescein-free background solution and a 0.001 µM fluorescein solution. The

0.001 µM fluorescein solution was 3000 times lower than the concentration of the

fluorescein solution injected for the 0.5 ml treatment. The injection widths for the 0.5, 2,

4, and 8 ml treatments were visually estimated as 1 cm, 2 cm, 3 cm, and 6 cm,

62

respectively. The estimated width was taken as equivalent to w if a uniform distribution

was assumed, or equivalent to 2σ if a normal distribution was assumed. The four widths

were about 2.6%, 5.3%, 8%, and 16% of the L. The numerical tests discussed above

showed that, for a given value of w or 2σ, if the solution that was developed using the

Dirac delta function for the initial solute distribution after injection was applicable for a

high Peclet number, the same solution would also be applicable for a lower Peclet

number. Consequently, only the results of the numerical tests for Peclet numbers

between 100 and 200 were experimentally verified.

The solutions developed for the 5 models already discussed with injected solute

distribution represented as a Dirac delta function, was fitted to the experimental BTCs.

The fitted parameters of the solute transport models were estimated using a nonlinear

least-squares optimization method based on the Levenber Marquardt procedure

(Marquardt, 1963). The algorithm used for fitting was taken from Toride et al. (1995),

and coded using C++. From the results of the numerical tests, it was reasonable to

assume that the treatment of 0.5 ml would result in an initial distribution that was most

closely equivalent to a Dirac delta function. Therefore, if the fitted parameters of one of

the other 3 treatments were not significantly different from the corresponding parameters

of the 0.5 ml treatment, then the injected solute distribution for this treatment was also

closely equivalent to a Dirac delta function.

For the CDE, the fitted parameter used for evaluation of the injection treatments was the

dispersivity κ. Because convection was dominant in the experiments, the molecular

diffusion was neglected and κ was calculated as:

vD /=κ (3.18)

In the CDE with distance dependent dispersivity, neither parameter a or b taken

individually was suitable for evaluating the injection treatments since they depend on

each other. However, their interdependence is controlled in the model by specifying the

apparent dispersivity κ as:

63

b

Lb

Lb

aL

dxax

10

+==

∫κ (3.19)

The apparent dispersivity κ , was therefore used for evaluating the different injection

treatments for the CDE with distance dependent dispersivity.

Parameters of c and d in the time-dependent CDE were also dependent each other, and

the apparent dispersivity κ~ defined by the following equation was used for evaluation of

the injection treatments:

d

Td

Td

cT

dtct'

'0

1~

'

+==

∫κ (3.19)

Both fitted parameters Df and α in the FCDE were used for evaluating the injection

treatments. In the experiments, the volume discharge rate was kept same for all the

treatments, so that Df and α for different treatments could be directly compared. In the

MIM, the fitted parameters of θim, β, and the dispersivity κm defined for the solute

transport in the mobile region were used for evaluation. The dispersivity κm is:

κm = D/(q/θm) (3.20)

Statistical differences between the means of the fitted parameters for the different

injection treatments, were determined using multiple comparisons based on the LSD at

the 5% level of significance.

3.4.2 Experimental Results

The fitted dispersivity parameters, defined above as κ [Eq. (3.18)], κ [Eq. (3.19)], κ~ [Eq.

(3.20)], and κm [Eq. (3.20)], obtained by fitting the experimental BTCs using the CDE,

the CDE with distance-dependent dispersivity, the CDE with time-dependent

dispersivity, and the MIM are presented in Figure 3.13. The Peclet number was about 130

64

for the CDE, the CDE with distance-dependent dispersivity, and the CDE with time-

dependent dispersivity. For the MIM, the Peclet number was about 175.

Figure 3.13 shows the mean value of the fitted dispersivity parameter for the four

injection treatments. The fitted dispersivities for the 2 ml and 4 ml injection treatments

were not significantly different from those of the 0.5 ml treatment for the CDE, the

distance-dependent CDE, and the CDE with time-dependent dispersivity. These

experimental results supported those of the numerical tests for the CDE (Figure 3.7,

Peclet number =100), for the CDE with distance-dependent (Figure 3.9, Peclet number =

200), and for the CDE with time-dependent dispersivity (Figure 3.10, Peclet number =

120). For the MIM, the fitted dispersivity parameter for the 2 ml treatment was not

significantly different from that of the 0.5 ml. This was also indicated in the numerical

tests with the MIM (Figure 3.11, Peclet number = 100).

Analyses for the fitted immobile porosity θim and the fitted mass transfer coefficient β,

for the MIM are presented in Figure 3.14. The mean fitted β and θim values for the 2 ml

treatment were not significantly different from those of 0.5 ml treatment. The mean fitted

θim value was about 1% of the total porosity, and accounted for the tailing present in the

experimental BTCs. Although the porous medium consisted of packed solid glass beads,

it was possible that immobile fluid existed in the column. This immobile fluid would

represent water present as surface boundary layer films, trapped in the neighborhood of

contact between particles, directly adsorbed on the surface of the beads, or trapped in

crevices formed by microscopic imperfections on the bead surfaces.

Analysis of the fitted results for Df and α using the FCDE are presented in Figure 3.15.

The mean values of Df and α for the 2 ml treatment, were not significantly different from

that for 0.5 ml treatment. The equivalent Peclet number for the FCDE was about 160.

These experimental results were in accordance with those obtained in the numerical tests

with the FCDE (Figure 3.12, equivalent Peclet number = 100).

65

For all the five models, the experimental results showed that the parameters fitted for the

2 ml treatment were not significantly different from those of the 0.5 ml treatment. The

width of initial solute distribution formed by the 2 ml injection was visually estimated to

be about 5% of the column length (L), which, as stated previously, was the distance from

the injection point to the outlet point. The experimental results confirmed the findings

obtained in the numerical tests. If the initial solute distribution width was no more than

5% of L, and the column Peclet number was smaller than 200, the solute distribution after

a single injection could be assumed as a Dirac delta function for solving the initial value

problem posed by the five models.

Volume of injection(ml)

8.04.02.0.5

95%

CID

ispe

rsiv

ity(c

m)

.34

.30

.26

.22


8.04.02.0.5

Dis

pers

ivity

(cm

)

.34

.30

.26

.22

bb

b

a


8.04.02.0.5

95%

CI A

vera

gedi

sper

sivi

ty(cm

).36

.32

.28

.24


8.04.02.0.5

Appa

rent

dis

pers

ivity

(cm

).36

.32

.28

.24

bb

b

a


8.04.02.0.5

95%

CI A

vera

gedi

sper

sivi

ty(cm

)

.34

.30

.26

.22


8.04.02.0.5

.34

.30

.26

.22

b b b

a


8.04.02.0.5

95%

CI d

ispe

rsiv

ity(c

m)

.26

.22

.18

.14


8.04.02.0.5

Dis

pers

ivity(

cm)

.26

.22

.18

.14

c cb

a

I I

III IV


8.04.02.0.5

95%

CI d

ispe

rsiv

ity(c

m)

.26

.22

.18

.14


8.04.02.0.5

Dis

pers

ivity(

cm)

.26

.22

.18

.14

c cb

a

I I

III IV

Appa

rent

dis

pers

ivity

(cm

)

II

Figure 3.13 Dispersivities obtained by fitting experimental BTCs for different injection

volumes, using the CDE (I) for κ , the CDE with distance-dependent dispersivity(II)

forκ , the CDE with time-dependent dispersivity(III) for κ~ , and the MIM (IV) for κm.

66

Values represent the treatment means and 95% confidence level. Means with differing

letters are significantly different at P≤0.05.


8.04.02.0.5

95%

CI m

ass

trans

fer c

oeffi

cien

t(1/m

in)

.0014

.0010

.0006

.0002


8.04.02.0.5

Mas

s tra

nsfe

r coe

ffici

ent(1

/min

)

.0014

.0010

.0006

.0002

abab

b

a


8.04.02.0.5

95%

CI P

oros

ity o

f im

mob

ile p

hase

.020

.015

.010

.005

0.000


8.04.02.0.5

Poro

sity

of i

mm

obile

pha

se

.020

.015

.010

.005

0.000

a ab b

a


8.04.02.0.5

95%

CI P

oros

ity o

f im

mob

ile p

hase

.020

.015

.010

.005

0.000


8.04.02.0.5

Poro

sity

of i

mm

obile

pha

se

.020

.015

.010

.005

0.000

a ab b

a

Figure 3.14 Immobile porosity and mass transfer coefficient obtained by fitting

experimental BTCs for different injection volumes, using the MIM. Values represent the

treatment means and 95% confidence level. Means with differing letters are significantly

different at P≤0.05.

volume of injection(ml)

8.04.02.0.5

95%

CI T

he o

rder

of f

ract

iona

l diff

eren

tiatio

n

1.9

1.8

1.7

1.6


8.04.02.0.5

95%

CI T

he o

rder

of f

ract

iona

l diff

eren

tiatio

n

1.9

1.8

1.7

1.6

acab

a


8.04.02.0.5

95%

CI T

he o

rder

of f

ract

iona

l diff

eren

tiatio

n

1.9

1.8

1.7

1.6


8.04.02.0.5

The

orde

r of f

ract

iona

l diff

eren

tiatio

n

1.9

1.8

1.7

1.6

acab

a


8.04.02.0.5

95%

CID

f

.32

.28

.24

.20


8.04.02.0.5

95%

CID

f

.32

.28

.24

.20

c bc

b

a


8.04.02.0.5

95%

CID

f

.32

.28

.24

.20


8.04.02.0.5

Df

.32

.28

.24

.20

c bc

b

a

c


8.04.02.0.5

95%

CID

f

.32

.28

.24

.20


8.04.02.0.5

Df

.32

.28

.24

.20

c bc

b

a

c

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. Values represent the treatment means and 95% confidence level. Means with

differing letters are significantly different at P≤0.05.

67

3.5 Conclusions

A laboratory column system was developed for studying one-dimensional solute transport

with a free inlet boundary. The system utilized an injection assembly to inject the tracer

solution directly into the column. This injection assembly was designed so that solute

transport in the column can be treated as a 1-D problem for all times after injection, and

the mass injected can be easily and accurately determined, and the column system can be

easily used for multiple injections over time and space. The most attractive feature of the

column system, is that it can be easily adapted to study problems for one-dimensional

transport of solutes from multiple sources distributed over the space domain. These

problems cannot be studied using a column system with tracer applied at one end.

When using the system, the initial distribution of the injected solute over the length of the

column is very difficult to quantify. Numerical simulations and experimental tests were

performed to determine the operating conditions under which the system can be used to

obtain reliable BTCs for analysis of the transport behavior of injected solutes. These

tests showed that, if a single injection of solute is distributed over a distance no larger

than 5% of the column length and the column Peclet number is smaller than 200, then 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 column length

was taken as the distance from the injection point to the discharge point. The five models

were: the CDE, the CDE with distance-dependent dispersivity, the CDE with time-

dependent dispersivity, the MIM, and the FCDE. The system will be used to generate

BTCs in the experiments for the research covered in following chapters.

68

69

CHAPTER 4

ANALYSIS OF BREAKTHROUGH CURVES USING NUMERICAL

SOLUTIONS OF THE CONVECTION DISPERSION EQUATION

WITH SCALE-DEPENDENT DISPERSIVITY

4.1 Introduction

One of most important mechanistic models for non-reactive solute transport in porous

media is the convection-dispersion equation (CDE). For one-dimensional solute

transport, the CDE is given as:

xC

vxC

Dxt

C∂∂

∂∂

∂∂

∂∂

−

= (4.1)

where C is solute concentration (ML-3), D is the hydraulic dispersion coefficient (L2t-1), v

is average pore water velocity (Lt-1), x is distance (L), and t is time (t).

The parameter D in the CDE represents a quasi-diffusion coefficient (termed as the

dispersion coefficient) used to describe the solute spreading caused by mechanical

dispersion and molecular diffusion, and is defined as:

0DDD m += (4.2a)

vDm κ= (4.2b)

where Dm is the mechanical dispersion coefficient, D0 is the molecular diffusion

coefficient (L2t-1), κ is the dispersivity (L).

For a given domain in space or time, such as a solute transport column, if κ is a constant

over the domain, the CDE can be written as:

xC

vxC

DvtC

∂∂

κ∂∂

−∂∂

+= 2

2

0 )( (4.3)

70

If κ is not a constant, the dispersivity is scale-dependent over the space or time domain.

When the scale-dependent dispersivity is expressed as a distance-dependent function, the

CDE becomes:

xC

vxC

Dvxxt

C∂∂

∂∂

κ∂∂

∂∂

−

+= ])([ 0 (4.4)

where κ(x) is the distance-dependent dispersivity function.

When the scale-dependent dispersivity is expressed as a time-dependent function, the

CDE becomes:

xC

vxC

DvttC

∂∂

κ∂∂

−∂∂

+= 2

2

0 ])([ (4.5)

where κ(t) is the time-dependent dispersivity function. When κ(x) is constant, Eq. (4.4)

reduces to Eq. (4.3), and when κ(t) is constant, Eq. (4.5) becomes Eq. (4.3).

Many authors have reported that dispersivity is scale-dependent rather than constant, both

at the column scales (Han et al., 1985; Porro et al., 1993; Zhang et al., 1995), and at the

field scale (Yasuda et al., 1994; Snow et al., 1995; Ellworth et al., 1996). Gelhar et al.

(1992) reviewed dispersivity observations from 59 field sites. His results indicated that

there was a tendency for the longitudinal dispersivity to increase with increasing scale.

At most laboratory column scales (Han et al., 1985), and at some field scales (Sudicky et

al., 1983), the dispersivity approached a maximum value after some distance or time.

Usually, the scale-dependence of the dispersivity over solute transport domain is

explained by the heterogeneity of the porous medium. Soil and subsurface ground water

domains are heterogeneous systems, and their heterogeneity is scale-dependent (Weber,

1986; Wheatcraft and Tyler 1988; Neuman, 1990). Scale-dependent dispersivity is a

consequence of the scale-dependent heterogeneity of porous media properties such as

structure and texture (Wheatcraft and Tyler, 1988), or scale-dependent variation of the

hydraulic conductivity (Neuman, 1990). Using carefully prepared artificial porous media

with periodically repeated heterogeneity, Irwin et al. (1996) showed that the scale-

dependent dispersivity was directly related to the scale of the heterogeneity, and not the

absolute size of solute transport domain.

71

There are several ways of specifying the scale dependent dispersivity function, depending

on whether Eq. (4.4) and Eq. (4.5) are solved ana lytically or numerically to obtain the

BTC at a specified length L. Figure 4.1 presents a generalized space-time domain for

one-dimensional solute transport. If an analytical solution of Eq. (4.4) is used to generate

the five points of C(L1,t1), C(L1,t2), C(L1,t3), C(L1,t4), and C(L1,t5) on BTC1 at L1 in Figure

4.1, one dispersivity value, namely κ(L1) needs to be substituted in the analytical solution

of Eq. (4.4) for generating these five points. Similarly, another value, namely κ(L2) is

needed to generate the five points of C(L2,t1), C(L2,t2), C(L2,t3), C(L2,t4), and C(L2,t5) on

BTC2 at L2 in Figure 4.1 using the analytical solution of Eq. (4.4). However, if an

analytical solution of Eq. (4.5) is applied to generate these five points on the BTC1, it

would require explicitly or implicitly defining five values of κ(t), namely, κ(t1) for

C(L1,t1), κ(t2) for C(L1,t2) and so on. These values of κ(t), namely κ(t1), κ(t2), κ(t3), κ(t4)

and κ(t5), are also needed to generate the five points on the BTC2 using the analytical

solution of the Eq. (4.5).

Figure 4.1 BTC in a general space-time domain for one-dimensional solute transport. At

L1, BTC1 is observed (or generated), and at L2, BTC2 is observed (or generated). Each

BTC shown has five concentration points obtained at time t1, t2, t3, t4 and t5. For BTC1,

these five concentration points are C(L1, t1), C(L1, t2), C(L1, t3), C(L1, t4) and C(L1, t5),

and for BTC2, the five concentration points are C(L2, t1), C(L2, t2), C(L2, t3), C(L2, t4) and

C(L2, t5).

DistanceL 10

C(L1, t5 )

t1

t2

t3

t4

t5

Tim

e

C(L 1,t2)

C(L 1,t1)

C(L 1,t3)

C(L 1 ,t4)

L 2

C(L 2 ,t5)

C(L 2, t2 )

C(L 2, t1 )

C(L 2, t3 )

C(L2,t4 )

72

On the other hand, generating a BTC at L using a numerical solution to Eq. (4.4) and Eq.

(4.5) would require setting a dispersivity value for each discretized unit in space at each

discretized time step. The discretized unit in space can be a discretized node in the finite

difference method, a discretized element in the finite element method, or a discretized

cell in the particle tracking method. Without loss of generality, the discretized unit

indexed as i in space and j in time, can be denoted as cell (i, j). Using κ(i, j) to express the

dispersivity value in cell (i, j), Eq. (4.4) for cell (i, j) is :

)(

),()(

),(]),([

)()(),(

0 ixjiC

vix

jiCDvi

ixjtjiC

∂∂

∂∂

κ∂

∂∂

∂−

+= (4.6)

and Eq. (4.5) in the cell(i, j) is:

)(

),()(

),(])(,[)(

),(2

2

0 ixjiCv

ixjiCDvj

jtjiC

∂∂

∂∂κ

∂∂ −+= (4.7)

where C(i ,j) denotes solute concentration in cell (i, j), i.e. the concentration at x=i∆x, and

t=j∆t. ∆x is the space length of a discretized cell, and ∆t is the time length of a discretized

time step.

The notation κ(i, ) in Eq. (4.6) is used to specify that the distance-dependent (but time-

independent) dispersivity values vary spatially, but for a given cell, it is unvarying with

time, meaning that for index k and h, when k and h ∈ j and k≠h, κ(i, k) =κ(i, h).

Similarly,κ( ,j) in Eq. (4.7) is used to specify that the dispersivity values are constant over

entire spatial domain at a given time, but changes with time, meaning that for index k

and h, when k and h∈ i and k≠h, κ(k, j) =κ(h, j).

The distance-dependent dispersivity κ(i, ) can be specified as a function of the distance

from the coordinate origin to the point where the BTC is observed (or generated), or as a

function of i∆x for cell (i, j). In the former case, the independent variable of the

73

function κ(i, ) is identical to the independent variable of the function κ(x) in the

analytical solution of Eq. (4.4). When a location for calculating a BTC is given, the

dispersivity values in all cells related to the calculation are set to a single value, which is

a function only of the location where the BTC is calculated. When a BTC at another

location is calculated, the dispersivities in all cells related to this BTC calculation have to

be reset to a new dispersivity value, which is now determined by the new location where

the BTC is calculated. When calculated in this manner, the distance-dependent

dispersivity κ(i, ) in Eq. (4.6) is termed as the apparent distance-dependent dispersivity

(αD), and is specified as αD(L) in the numerical solution of the CDE with distance-

dependent dispersivity. The apparent distance-dependent dispersivity has also been

called the effective dispersivity (Mishra and Parker, 1990).

In the latter case, where κ(i, ) is specified as a function of i∆x for cell (i, j), once the

spatial location of a cell is given, the dispersivity value in this cell is defined, regardless

of where the BTC is calculated. This value does not necessarily have to be reset in the

numerical scheme when the location for calculating the BTC is changed. In this case, the

κ(i, ) in Eq. (4.6) is termed as the local distance-dependent dispersivity(λD), and

specified as λD(x) in the numerical solution.

Specifying the apparent distance-dependent dispersivity αD(L) and local distance-

dependent dispersivity λD(x) in the numerical schemes that were developed for solving

the CDE with distance-dependent dispersivity given in Eq. (4.6), is illustrated in Figure

4.2. As shown, the space-time domain in Figure 4.2 is discretized into cells with length

∆x and time step ∆t. BTC1 and BTC2 are calculated at different locations L1 and L2 in the

same one dimensional space domain with the same origin. Five columns of discretized

cells are involved in the calculation of BTC1 at L1, and 7 columns of discretized cells are

involved in the calculation of BTC2 at L2.

74

Figure 4.2 Specifying the distance-dependent dispersivity in the numerical schemes

developed for solving the CDE with distance-dependent dispersivity given by Eq. (4.6).

I. and III. BTC1 calculated at L1 using apparent distance-dependent dispersivity αD (I),

and local distance-dependent dispersivity λD (III). II and IV. BTC2 calculated at L2

using apparent distance-dependent dispersivity αD (II), and local distance-dependent

dispersivity λD (IV).

The coordinate system of the 4 parts of Figure 4.2 is identical. Calculation of the five

concentration points for the BTC in each part of Figure 4.2 can be carried out in one

iterative computation, starting at time = 0 and proceeding sequentially to time = t1, t2, ....

In part I of Figure 4.2, BTC1 at L1 is calculated using the apparent distance-dependent

dispersivity, so the dispersivity values in all cells are identical and given as αD(L1).

When BTC2 at L2 is calculated using the apparent distance-dependent dispersivity (part

II), the dispersivity values in the cells related to this calculation have to be set to αD(L2).

Parts III and IV show the specification of the local-distance dependent dispersivity λD(x)

in the numerical scheme for calculating BTC1 at L1 (part III) and for calculating BTC2 at

L2 (part IV). As shown, the dispersivity values in parts III and IV depend only on the

L10

C(L1,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5

Tim

e

αD(L1) αD(L1) αD(L1) αD(L1) αD(L1)





C(L1,t2)

C(L1,t1)

C(L1,t3)

C(L1,t4)

x5 L20

C(L2,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5






C(L2,t2)

C(L2,t1)

C(L2,t3)

C(L2, t4)

x5 x6 x7

αD(L2) αD(L2)

αD(L2) αD(L2)

αD(L2) αD(L2)

αD(L2) αD(L2)

αD(L2) αD(L2)

DistanceL10

C(L1,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5

Tim

e

λD(x1) λD(x2 ) λD(x3 ) λD(x4 ) λD(x5)

λD(x1) λD(x2 ) λ D(x3 ) λD(x4 ) λD (x5)

λD(x1) λD(x2 ) λ D(x3 ) λD (x4) λD (x5)

λD(x1) λD(x2 ) λD(x3 ) λD (x4) λD (x5)

λD (x1) λD(x2 ) λD(x3 ) λD (x4) λD(x5)

C(L1 ,t2)

C(L1 ,t1)

C(L1 ,t3)

C(L1,t4)

x5

DistanceL20

C(L2,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

C(L2,t2)

C(L2,t1)

C(L2

C(L2,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

λD(x1) λ

D(x2) λD(x3) λ

D(x4) λD(x5)

C(L2,t2)

C(L2,t1)

C(L2,t3)

C(L2,t4)

x5 x6

λD(x6)

λD(x6)

λD(x6)

λ D(x6)

λD(x6)

λD(x7)

λD(x7)

λD(x7)

λD(x7)

λD(x7)

I II

III IV

x7

75

location of cells, and the dispersivity values in the cells at x1, x2, x3, x4 and x5 for parts III

and IV are identical.

In the same manner as the distance dependent dispersivity, the time-dependent

dispersivity κ( , j) in Eq. (4.7) can be a function of the time coordinate starting from time

= 0 to the time at which a concentration point on the BTC is calculated, or a function of

j∆t for cell (i, j).

In the former case, the κ( , j) is termed as the average time-dependent dispersivity (αT),

and specified as αT(t) in the numerical solution. In order to generate a concentration

point C(L, t) on a BTC, all the dispersivity values in the cells with time index j for which

j∆t<t, would be set as αT(t), in the iterative calculation starting from t=0. If another

concentration point C(L, t’) on this BTC needs to be generated, all the dispersivity values

in the cells with time index j for which j∆t<t’ , would have to be reset as αT(t’), and the

iterative calculation restarted from t = 0. Consequently, if 100 concentration points on a

BTC need to be generated, it would require that the iterative calculation be repeated 100

times, with each iterative calculation starting from t=0

In the latter case, where κ( , j) is specified as a function of j∆t for cell(i, j), κ( , j) is called

the local time-dependent dispersivity (λT), and specified as λT(t) in the numerical

solution. The λT(t) value is a function of the time directly related to the cell ( , j) in

which j∆t = t, regardless of the time at which a given concentration point on the BTC is

calculated. A BTC can be generated in one iterative computation, starting at time = 0,

and proceeding sequent ially to time t1 = ∆t, t2 = 2∆t, .... At time step j, one dispersivity

value, which is a function of the time j∆t, would be required for the calculation at this

time step. Consequently, if a BTC with 100 concentration points needs to be calculated,

only one iterative computation starting at time t=0 would be required.

Specifying the average time-dependent dispersivity αT(t) and local time-dependent

dispersivity λT(t) in the numerical scheme for solving the CDE with the time-dependent

76

dispersivity (Eq. (4.7) is illustrated in Figure 4.3. As shown, the space-time domain in

Figure 4.3 is discretized into cells with length of ∆x and time step ∆t. There are four

parts in Figure 4.3. Each part uses the same dimensional space-time domain and the

same coordinate origin. Parts I and II show how αT(t) is specified to calculate C(L, t3)

and C(L, t5), where L represents L1 or L2. Parts III and IV show how λT(t) is specified to

calculate C(L, t3) and C(L,t5) where, as before, L represents L1 or L2. In part I,

calculating concentration C(L, t3) requires that the dispersivity values in all cells( , j) for

which j∆t<t3, be set to αT(t3), in the iterative calculation starting from t=0. When the

concentration point C(L, t5) is calculated in the part II, the dispersivity values in all cells (

, j) for which j∆t<t5, are set to be αT(t5) in the iterative calculation, which is restarted

from t= 0. However, when the local time-dependent dispersivity is used in the numerical

scheme (parts III and IV in Figure 4.3), the dispersivity value for a given cell is

determined by the time coordinate of that cell. Consequently, the dispersivity values in

all cell located at t1, t2, and t3 in part III for calculating C(L, t3), and in part IV for

calculating C(L, t5) are identical. In this case, concentration points C(L, t3) and C(L, t5)

can be generated sequentially starting from t = 0, and a complete BTC can be generated

by only one iterative computation.

77

Figure 4.3 Specifying time-dependent dispersivity in the numerical scheme for solving

the CDE with time-dependent dispersivity. I and III. Concentration points C(L1, t3) and

C(L2, t3) calculated using average time-dependent dispersivity αT (I), and using local

time-dependent dispersivity λT (III). II and IV. Concent ration points C(L1, t5) and C(L2,

t5) calculated using average time-dependent dispersivity αT. (II), and using local time-

dependent dispersivity λT (IV).

Several questions remain to be answered when using the numerical solutions of the CDE

with scale-dependent dispersivity for generating the BTCs. These are:

a. How are αD(L), αT(t), λD(x) and λT(t) defined theoretically?

b. How can λD(x) and λT(t) be set for each cell when αD(L) or αT(t) are known, and vice

versa? This requires specifying relationships between αD(L), αT(t), λD(x) and λT(t).

c. Which of the functions αD(L), αT(t), λD(x) or λT(t), should be applied for defining

scale dependent dispersivity in order to accurately and efficiently compute BTCs for a

given solute transport problem.

The overall objective of the studies reported in this chapter, was to attempt to answer the

above questions by theoretical research and numerical experiments. Specific objectives

were to:

L20 x1 x2 x3 x4

t1

t2

t3

αT(t3) αT(t3) αT(t3) αT(t3) αT(t3)



C(L2,t 3)

x5 x6 x7

αT(t3) αT( t3)

αT(t3) αT(t3)

αT(t3) αT( t3)

L1

C(L1,t3)

L 20

C(L 2,t5)

x1 x2 x3 x4

t1

t2

t3

t4

t5

αT(t5) α T( t5) αT(t5) αT(t5) αT(t5)

αT(t5) α T( t5) αT(t5) αT(t5) αT(t5)

αT(t5) α T( t5) αT(t5) αT(t5) α T( t5)

αT(t5) α T( t5) α T( t5) αT(t5) α T( t5)

αT(t5 ) α T( t5) α T( t5) αT(t5) α T( t5)

x 5 x6 x 7

αT(t5) αT( t5 )

αT(t5) α T(t5)

αT(t5) α T(t5)

αT(t5) α T(t5)

αT(t5) α T(t5)

C(L 1,t5)

L 1

L20 x1 x2 x3 x 4

t1

t2

t3

λ T (t1) λT (t1) λT (t1) λT (t1) λ T (t1)

λ T (t2) λT (t2) λ T (t2) λT (t2) λ T (t2)

λ T (t3) λT (t3) λ T (t3) λ T( t3) λ T (t3)

C(L 2,t3 )

x5 x 6 x7

λT (t3) λT (t3)

λT (t2) λ T(t2 )

λ T(1 λT (t1)

L1

C(L1 ,t3 )

L20

C(L 2,t5)

x1 x2 x 3 x4

t1

t2

t3

t4

t5

λT(t1) λT(t1) λT(t1) λT(t1) λT(t1)

λT(t2) λT(t2) λT(t2) λT(t2) λT(t2)

λT(t3) λT(t3) λT(t3) λT( t3 ) λT(t3)

λT(t4) λT(t4) λT( t4) λT( t4 ) λT(t4)

λT(t5) λT(t5) λT( t5) λT( t5 ) λT(t5)

x 5 x 6 x 7

λT(t5) λT(t5)

λT(t4) λT(t4)

λT(t3) λT(t3 )

λT(t2) λT(t2 )

λT(t1) λT(t1 )

C(L1,t 5)

L1

Tim

eTi

me

Distance Distance

I

II

III

IV

78

a. Use scale-dependent heterogeneity in porous media to propose explanations of scale-

dependent dispersivity, and define αD(L), αT(t), λD(x), or λT(t) theoretically.

b. Describe and explain possible relationships between αD(L), αT(t), λD(x), or λT(t), and

identify the applicability of these relationships for different solute transport conditions.

c. Compare the accuracy and computation efficiency of applying αD(L), αT(t), λD(x), or

λT(t) for calculation of several classical solute transport problems in porous media. The

solute transport problems included single source solute transport, multiple source solute

transport over the space domain, and multiple source solute transport over the time

domain. Results were used to decide whether αD(L), αT(t), λD(x), or λT(t), should be used

to generate BTCs numerically for these three transport problems in porous media.

4.2 Time-Dependent and Distance-Dependent Dispersivity

In the CDE, mechanical dispersion is assumed to be a diffusion- like, quasi-Fickian

process. Conceptually, mechanical dispersion can be understood as follows (Jury, 1988).

Solute spreading during transport in a very “tiny” volume, such as the volume of a tiny

drop in the water phase of the porous medium, is caused by molecular diffusion. As the

volume increases to V consisting of many such “tiny” volumes, water velocities in these

tiny volumes in V are different from each other because of the existence of medium

heterogeneity in V. These water velocities fluctuate around an average pore water

velocity, which can be understood as an observable characteristic on V. Consequently,

when an average pore water velocity is applied to describe convection of solute in V, the

differences between the average velocity and the velocity in these “tiny” volumes (termed

as the residual part of the velocities) appear as the observed mechanical solute spreading

in V. When the residual part of velocities is normally distributed, then the mechanical

solute spreading is diffusion- like or quasi-Fickian. This effect of the velocity fluctuations

around the average pore water velocity on solute transport in V is termed as mechanical

dispersion of the solute in the porous medium.

At a given scale of V, the residual part of velocities determines the change in the solute

spreading between two times at this scale. When the fluctuation is quantified as the

variance of the normally distributed residual part of velocities, the larger the value of this

79

variance, the larger would be the change in solute spreading between two times at this

scale.

Because the mechanical dispersion of solute in porous media is assumed to be diffusion-

like or quasi-Fickian, it can be quantified using the theories and methods developed to

describe molecular diffusion in liquids. For an initial solute input represented as a Dirac

delta function at time t = 0, molecular diffusion in a liquid can be quantified using the

diffusion coefficient (Einstein, 1905; Pickens and Grisak, 1987), which is defined as:

dt

)]t([d21

)t(D2

00

σ= (4.8a)

where [σ0(t)] 2 is the variance of the spatial distribution of the solute molecules in the

liquid after time t. Consequently, the mechanical dispersion coefficient can be written as:

dt

)]t([d21

)t(D2

mm

σ= (4.8b)

where [σm(t)] 2 is the variance of the spatial distribution of the solute molecules in the

porous medium at time t, for an initial solute input represented as a Dirac delta function.

When solute transport in porous media is under steady water flow conditions, the average

pore water velocity, v, is constant. From Eq. (4.2b) Dm = κ(t)v. Combining this with Eq.

(4.8b) gives:

dt

)]t([dv21

)t(2

mσ=κ (4.9a)

and ∫ ττκ=σt

0

2m d)(v2)]t([ (4.9b)

where τ is a dummy variable for integration.

When [σm(t)]2 is specified in this way, κ(t) in Eq.(4.9a) and Eq. (4.9b) represents the

parameter used to quantify the change of the spatial variance of solute distribution in

80

porous medium with time. As the variance of the solute distribution in the porous media

increases, the solute spreading in the porous media increases. In the numerical scheme of

the CDE (Figure 4.3), the local time dependent dispersivity λT(t) accounts for the change

of the variance of the solute distribution at time t. This implies that the value of λT(j∆t)

accounts for the change of solute spreading from time j∆t to (j+1)∆t (or from time (j-1)∆t

to j∆t). When ∆t→0, λT(j∆t) approaches κ(t) in Eq. (4.9a, b). Consequently, the local

time dependent dispersivity λT(t) in the numerical scheme (Figure 4.3) has the same

definition as the κ(t) in Eq. (4.9a) and Eq. (4.9b), and therefore:

dt

tdv

t mT

2)]([21

)(σ

λ = (4.10a)

∫=t

Tm dvt0

2 )(2)]([ ττλσ (4.10b)

For a given time t, [σm(t)] 2 is traditionally given by:

vtttm )('2)]([ 2 κσ = (4.11)

When [σm(t)]2 is specified in this way, κ'(t) represents the parameter used to quantify

the spatial variance of the solute distribution in the porous media at time t.

Combining Eq. (4.10b) and Eq. (4.11) gives:

t

dt

t

T∫= 0

)()('

ττλκ (4.12)

As shown, κ'(t) in Eq(4.12), represents a parameter that quantifies the average change of

the spatial variance of solute distribution in porous media from time zero to time t.

Consequently, κ'(t) is the average time-dependent dispersivity αT(t) defined in Figure 4.3

and represents the averaged integral effect of local time-dependent dispersivity λT(t). Eq.

(4.12) can therefore be written as:

t

dt

t

T

T

∫= 0

)()(

ττλα (4.13a)

81

As a result, when a given concentration point C(L, k∆t) on a BTC in Figure 4.3 is

calculated, the average time dependent dispersivities [αT(k∆t)] related to this iterative

calculation can be obtained from the local time dependent dispersivities [λT(j∆t), j =

1,2,…,k] by:

tk

tjtk

k

jT

T ∆

∆=∆

∑=1

)()(

λα (4.13b)

Mechanical dispersion is directly caused by heterogeneity of porous media. The

heterogeneity of porous media is scale-dependent, meaning that the dominant

components of heterogeneity affecting solute transport are different at differing scales.

At the laboratory column scale, the dominant component of heterogeneity that affects

non-reactive solute transport is caused by micro-scale heterogeneity. Consequently,

when solute transport in porous media is observed at laboratory column scales, and the

average pore water velocity is applied to describe convection of solute transport at this

scale, the mechanical dispersion is caused by the residual part of the velocities, which is

determined by the micro-scale heterogeneity.

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. Consequently, when solute transport in

porous media is observed at a field scale, and the average pore water velocity is applied

to describe convection of solute transport at this scale, the mechanical dispersion is

caused by the residual part of the velocities, which is determined by the macro-scale

heterogeneity.

It would be logical to expect that the larger the scale of solute transport in porous media,

the larger would be the value of the variance of the distribution of the residual part of

velocities. Consequently, as the scale of solute transport in porous media increases, the

change in the spatial variance of the solute distribution in the porous media over a given

82

time interval would increase. Therefore, the change in the spatial variance of solute

distribution in porous media would also be scale-dependent. However, such continuous

increase in the change of the spatial variance with increasing scale would theoretically

occur only in fractal heterogeneous porous media (Wheatcraft and Tyler, 1988).

However, if the porous medium were considered as statistically homogeneous, it would

imply that no new heterogeneity components are introduced after some scale of

observation of solute transport. In this case, the variance of the distribution of the

residual part of velocities is constant after such scale of observation of solute transport,

and consequently, the change of the spatial variance of solute distribution in porous

media becomes constant after this scale.

In Eq. (4.10a), the change of the spatial variance of solute distribution at time t is

quantified by the local time dependent dispersivity, λT(t), at the time t. As shown above,

the change of the spatial variance of solute distribution is expected to be scale-dependent.

Consequently, the dispersivity should also be scale-dependent. The above concepts can

be better understood with reference to Figure 4.4.

Figure 4.4 shows hypothetical solute distributions at three times t1 <t2< t3 during

transport in a one-dimensional porous medium under steady water flow conditions. The

initial solute distribution was represented as a Dirac delta function. At time t1, the spatial

scale of the solute distribution over the transport domain is L1. Similarly, L2 is the spatial

scale of the solute distribution at t2, and L3 is the spatial scale of the solute distribution at

t3. Since t3 > t2 > t1, it implies that L3 > L2 > L1.

At a given spatial scale, the variance of the distribution of the residual part of velocities

(denoted as VR) is an observable characteristic for this scale. When L3 > L2 > L1, it

implies that VR3 ≥ VR2 ≥ VR1, where VR1 is the variance of the distribution of the

residual part of velocities for the scale L1, VR2 for scale L2, and VR3 for scale L3. When

VR3 ≥ VR2 ≥ VR1, it implies that d[σm(t3)] 2 /dt ≥ d[σm(t3)]2/dt ≥ d[σm(t1)] 2/dt, and

consequently, λT(t3)≥ λT(t2)≥ λT(t3) and αT(t3)≥αT(t2)≥αT(t1).

83

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.

L1, L2 and L3 are the spatial scales of the solute distribution at the time t1, t2 and t3,

respectively. Average pore water velocity is assumed to be constant.

As discussed above, the application of an average time-dependent dispersivity in the

numerical solution shown in Figure 4.3 results in a computationally inefficient algorithm.

Computational efficiency is quantified using the time exponent of a time complexity

function O(ta). Compared to the most efficient linear time complexity O(t) for applying

local time-dependent dispersivity λT(t) to generate a BTC, the time complexity of

applying average time-dependent dispersivity αT(t) to generate this BTC is Ο(t2).

Consequently, it is impractical to numerically generate the BTC using the CDE with

average time dependent dispersivity for large-scale solute transport problems or multi-

dimensional solute transport problems.

One challenge of using λT(t) in the numerical solution of the CDE, is that this function

may not be directly obtained from analysis of the solute concentration distribution curve.

Generally, only the average time dependent dispersivity αT(t) can be obtained from a

concentration distribution curve using Eq. (4.11). Either graphical analysis or parameter

fitting procedures can be used to obtain the average time dependent dispersivity. For

most laboratory and field situations, only the BTC is usually observed. The BTC

t = t1t = t2t = t3

t = t1t = t2t = t3

L1

L2

L3 Water flow direction

84

represents concentration distribution over time at a given location, rather than the

concentration distribution over space a given time as illustrated in Figure 4.4. Therefore,

if the time-dependent dispersivity values λT(t) can be directly related to the location at

which the BTCs is observed or generated, analysis of the BTCs using the CDE with time-

dependent dispersivity [Eq. (4.5)] will be facilitated. A potential approach to relate the

time-dependent dispersivity to the distance is by defining the mean solute travel time.

For generating the BTC at a given distance L, the mean solute travel time T', which is the

expected value of the time distribution of the BTC at L, may be used to describe the time

distribution of the BTC at L. The mean solute travel time T' of an observed BTC for

solute transport is defined as:

∫∞

=0

),(' dttLtfT (4.14)

where f(L,t) is a concentration distribution function of the BTC generated at L. T' is the

expected value of the time distribution of the BTC at L.

Substituting the definition of T' into Eq. (4.13a) for t = T' gives:

'

)()'(

'

0

T

dT

T

T

T

∫=

ττλα (4.15)

For a BTC generated at L, αT(T') is a function of L and can be denoted as α'T(T').

For this condition, Eq. (4.15) can be rewritten as:

'

)()'('

'

0

T

dT

T

T

T

∫=

ττλα (4.16)

In the numerical solution of the CDE in Eq. (4.5), α'T(T') is used to generate a BTC at L.

However, the average time-dependent dispersivity αT(T') is used to generate a

concentration point at time T' on the BTC at L. When applying α'T(T') for a given L, all

the dispersivity values in the cells involved in calculating the BTC at L are set to be

85

α'T(T'). In this case, α'T(T') is termed as apparent time-dependent dispersivity. The time

complexity of applying α'T(T') to generate a BTC is Ο(t).

As shown in Figure 4.4, the change of the spatial variance of solute distribution at a time

t is determined by the variance of the distribution of the residual part of the pore water

velocities (VR). VR is an observable characteristic for the spatial scale of the solute

distribution at time t. For example, d[σm(t2)]2/dt is determined by VR (denoted as VR2)

for spatial scale L2. Dispersion can also be modeled using the Brownian particle motion

theory discussed in section 2.3.6. In this theory, the movement of a solute particle by

dispersion between time t and t + ∆t is modeled as the random displacement of the

particle. The random displacement of the particle may or may not be dependent on the

time (t) or on the position of the particle at time t. The change in the variance of the

spatial solute distribution over the time interval ∆t embodies the change in the random

displacements for all the solute particles in the porous medium. The statement that

d[σm(t2)]2/dt is determined by VR2, can now be restated in terms of Brownian particle

motion. The resultant is that the random displacement for all particles at time t2,

regardless where they are in the spatial domain, is determined by the VR2, which is an

observable characteristic for the spatial scale L2. However, Eq. (4.16) implies that the

spreading movement (or random displacement) for all the solute particles undergoing

transport, is determined by an apparent VR regardless how long a solute particle remains

in the spatial domain covered by L. This apparent VR represents the average effect of the

VR (a function of time) from t = 0 to t = T' on the solute dispersion observed as a BTC at

L. The spatial scale of solute distribution at time t = T' (denoted as LT) is not the L at

which the BTC is observed. However, LT is related to L, since T' is defined at L (Eq.

4.14).

As shown in Eq. (4.14), α'T(T') can be directly defined as a function of L and denoted as

αD(L). In this case, the dispersivity is called as apparent distance-dependent dispersivity.

Application of the dispersivity defined as αD(L) in the numerical scheme for solution of

the CDE [Eq. (4.4)] is presented in Parts I and II of Figure 4.2.

86

The apparent distance-dependent dispersivity, αD(L), is a function of L. However, αD(L)

is not related to the variance of the distribution of the residual part of velocities VR for

the scale L. The αD(L) has the same meaning as α'T(T'), which represents the average

effect of VR from t = 0 to t = T' on the solute dispersion observed as a BTC at L.

Therefore:

)'(')( TL TD αα = (4.17)

For describing the BTC at L, it implies that an apparent distance-dependent dispersivity

αD(L) could be adequately defined from the time-dependent dispersivity λT(t) using Eq.

(4.14), Eq. (4.15) , Eq. (4.16), and Eq. (4.17). When the CDE with constant dispersivity κ

[Eq. (4.3)], the CDE with apparent time-dependent dispersivity α'T(T'), and the CDE with

apparent distance-dependent dispersivity αD(L) are applied to fit an observed BTC at L,

the fitted values of κ, α'T(T'), and αD(L) should be identical. The mean travel time T' can

be determined from the observed BTC using Eq. (4.14). The local time-dependent

dispersivity λT(t) can be specified using Eq. (4.16) when α'T(T') and T' are known.

The definition of the T' [Eq. (4.14)] is dependent on f(L, t) which is the concentration

distribution function of the BTC generated at L. Therefore, α'T(T') and αD(L) are

dependent on the BTC at L. When the objective of applying scale-dependent dispersivity

is to predict the BTC at L, it is redundant and impractical to define a scale-dependent

dispersivity, which is dependent on a BTC, in order to generate this BTC at L.

The mean solute travel time T' represents the expected value of the time distribution of

the BTC at L. However, as discussed in the previous paragraph, using T' to relate the

time-dependent dispersivity to distance is an improper approach for the practical

purposes of prediction. In order to relate the time-dependent dispersivity to the distance

for the practical purposes of prediction, a time defined as a function of L, has to be

independent of the BTC at L. For the solute transport problem with an initial solute

source distribution represented as a Dirac delta function, such a time can be arbitrarily

87

defined as L/v and denoted as T. In this situation, T is assumed to represent the expected

value of the time distribution of the BTC at L. Therefore, Eq. (4.16) becomes:

T

dT

T

T

T

∫= 0

)()('

ττλα (4.18)

And Eq. (4.17) becomes:

)(')( TL TD αα = (4.19)

Substituting T = L/v and t = x/v , where x is the physical spatial distance in the solute

transport domain uniquely defined by x=vt, into Eq. (4.18) and using the relationship of

Eq. (4.19) gives:

L

dxvxL

L

T

D

∫= 0

)/()(

λα (4.20)

In Eq. (4.20), λT(x/v) is an explicit function of distance rather than time. For a given

time t, x is uniquely defined by x = vt. Consequently, λT(x/v) can be directly defined as a

function of distance x. In this case, λT(x/v) is called local distance-dependent dispersivity

and denoted as λD(x). The application of λD(x) in the numerical solution of Eq. (4.4) has

been introduced in Parts III and IV of Figure 4.2. The relationship of the local time

dependent dispersivity λT(t) and the local distance-dependent dispersivity λD(x) can be

expressed as:

)()/()( xvxt DTT λλλ == (4.21)

For a BTC observed at L, Eq. (4.20) can be rewritten as:

L

dxxL

L

D

D

∫= 0

)()(

λα (4.22)

88

Therefore, the apparent distance-dependent dispersivity αD(L) represents the averaged

integral effect of the local distance-dependent dispersivity λD(x) on the BTC, which is

observed at L. Eq. (4.22) was also given by Mishra and Parker(1990).

As shown in Eq. (4.20), λT(x/v) or λD(x) is defined from Eq. (4.18) by using the

relationships T = L/v and t = x/v. However, λT(x/v) or λD(x) cannot be defined from Eq.

(4.16) when T' ≠ L/v. For example, for analyzing an observed BTC, when T' ≠ L/v but t

= x/v , the value of λD(x) directly obtained from Eq. (4.22) by using αD(L) = α'T(T') [Eq.

(4.17)] will be different from that of λT(t), which is obtained from Eq. (4.16). Such

difference between the values of λT(t) and λD(x) is denoted as ∆. Therefore, when the

observed BTC is analyzed for determining λD(x) and αD(L), Eq. (4.21) and Eq. (4.22)

can be simultaneously correct only when T = T', T = L/v and t = x/v . However, when T ≈

T', T = L/v and t = x/v, Eq.(4.21) and (4.22) might simultaneously be applied for practical

purposes, if the error caused by simultaneously using these two relationships were

acceptable. If ∆ were small, λT(t) ≈ λD(x).

For a given solute transport problem under given solute transport conditions, α'T(T),

αD(L) and λD(x) can be defined from λT(t) using the relationships between of them [Eq.

(4.18), Eq. (4.19), Eq. (4.21) and Eq. (4.22)]. If BTCs generated at L using the numerical

solution of the CDE with these λT(t), α'T(T), αD(L), and λD(x) were indistinguishable

from each other, these relationships would be applicable for describing or predicting the

BTCs at L for the given solute problem under the given solute transport condition.

Otherwise, some of these relationships (or maybe all of them) would be inapplicable.

The local distance-dependent dispersivity λD(x) can be understood as a parameter used to

quantify the change with time of the variance of the spatial solute distribution at the

location x. The spatial solute distribution is defined using the CDE with local distance-

dependent dispersivity λD(x). If the dispersion were modeled using Brownian particle

motion theory as discussed above (also see section 2.3.6), the spreading movement (or

random displacement) for the particles at x, regardless of time, would be determined by

89

the variance of the distribution of the residual part of velocities, which is also a function

of x. Therefore, this variance of the distribution of the residual part of velocities is

dependent on the distance x and denoted as VR(x). However, even though VR(x) is a

function of x, it is not determined by the heterogeneity of the porous medium at x. It is

determined by the heterogeneity of the porous medium for the spatial scale covered by

the solute distribution at time t = x/v . VR(x) is a characteristic for the spatial scale of the

solute distribution at time t, which is defined using the CDE with local-time dependent

dispersivity given in Eq. (4.5). The above concepts can be better understood with

reference to Figure 4.5.

Figure 4.5 shows hypothetical spatial distributions during one-dimensional transport of

an initial solute input represented as a Dirac delta function. Assume that spatial solute

distribution I was obtained using the CDE with local distance-dependent dispersivity

λD(x) at time t. Solute particles at any point on this spatial solute distribution will be

randomly displaced during time t to time t+∆t. The change in the variance of the spatial

solute distribution over the time interval ∆t, would embody the change in the random

displacements for all the solute particles in the porous medium.

Without loss of generality, only the random displacements of solute particles at points A

and B as shown in Figure 4.5 will be considered. The random displacement of the

particles at any point x for the solute distribution I, can be quantified using local distance-

dependent dispersivity λD(x). Therefore, the random displacement of the particles at A

can be quantified using λD(xA), and the random displacement of the particles at B can be

quantified using λD(xB).

At xA and xB, the values of the local time dependent dispersivities λT(xA /v) and λT(xB /v)

are determined by VR(xA) and VR(xB), which are the variances of the distribution of the

residual parts of velocities over spatial scale LA and LB, respectively. LA is the spatial

scale of the solute distribution over the transport domain at time tA = xA /v as shown in

solute distribution II of Figure 4.5. This distribution has to be defined using the CDE

with the local time-dependent dispersivity. Similarly, LB is the spatial scale of the solute

90

distribution over the transport domain (solute distribution III in Figure 4.5) at time tB = xB

/v, which also has to be defined using the CDE with the local time-dependent

dispersivity. Therefore, λD(xA) is determined by VR(xA), which is a characteristic for the

scale of LA, and λD(xB) is determined by VR(xB), which is a characteristic for the scale of

LB. This implies that, for a given solute transport porous medium, the random

displacement over time ∆t of the particles at A is determined by the heterogeneity within

the scale of LA, and the random displacement of the particles at B is determined by the

heterogeneity within the scale of LB.

Figure 4.5 Hypothetical spatial solute distributions during one-dimensional transport for

an initial solute input represented as a Dirac delta function. See text for explanation.

Combining Eq. (4.11), Eq. (4.12), Eq. (4.13a), Eq. (4.15), and Eq. (4.16) gives:

vttt Tm )('2)]([ 2 ασ = (4.23)

For any arbitrary time t, there is a mean solute travel distance x' uniquely defined for the

spatia l solute distribution over the solute transport domain at time t. Substituting Eq.

(4.19) into Eq. (4.20) and using the assumption that the mean travel distance x' = x, in

which x = vt, gives the traditionally used function for describing the variance of the

spatial solute distribution in the porous medium. This function is:

')'(2)]'([ 2 xxx Dm ασ = (4.24)

Solute distribution I

Solute distribution II

Solute distribution III

A

L A

B

xA xB

L B

Water flow direction



Solute distribution I:

Solute distribution II:

Solute distribution III:

91

Wheatcraft and Tyler (1988) substituted fractal distance into Eq. (4.24) in order explain

the distance-dependence of dispersivity. Their result implied that, for a given solute

transport problem, only distance should be considered in defining scale dependent

dispersivity. Their explanation and result has become popularly accepted. However, in

Eq. (4.24) above, the mean travel distance x' is a function of time rather than the physical

spatial location x in the solute transport domain. Therefore, their result may need to be

reconsidered when x ≠ x', such as for the solute transport problem with multiple source

input over time.

As shown in this section, the fact that mechanical dispersion in the CDE is assumed to be

a diffusion- like process, and the heterogeneity in porous medium is scale-dependent,

directly indicates that the scale-dependent dispersivity is locally time-dependent. The

apparent time-dependent dispersivity α'T (T), the apparent distance-dependent

dispersivity αD(L), and the local distance-dependent dispersivity λD(x) are defined from

the local time-dependent dispersivity λT(t). The reasons of defining the functions α'T (T),

αD(L), and λD(x) are:

a. The BTC represents solute concentration distribution over time at a given location, and

it is easy to use scale-dependent dispersivity, which is directly or indirectly defined as a

function of distance.

b. The function λT(t) may not be directly obtained from analysis of a BTC. However, it

may be easily obtained from α'T (T), αD(L), and λD(x) by using the relationships given

above. These relationships include Eq. (4.18) in which the α'T (T) represents the

averaged integral effect of the λT(t) on the BTC, Eq. (4.19) in which the value of α'T (T)

equals the value of αD(L), Eq. (4.21) in which the value of λT(t) equals the value of

λD(x) when x = vt, and Eq. (4.22) in which αD(L) represents the averaged integral effect

of the λD(x) on the BTC. Among the three possibilities namely, α'T (T), αD(L), and λD(x),

the apparent distance-dependent dispersivity αD(L) may be most easily obtained by

analysis of a BTC. At a given L, the BTC generated using the CDE with αD(L) [Eq.

(4.4)] is identical to the BTC generated using the CDE with constant dispersivity [Eq.

92

(4.3)]. Therefore, the constant dispersivity in Eq. (4.3) is the value of the αD(L) at L.

Most of the current methods used to determine dispersivity from the observed BTCs are

developed for the CDE with constant dispersivity [Eq. (4.3)], either by graphical analysis

(Bear, 1988b), or by parameter fitting procedures (Toride et al., 1995). When the values

of αD(L) at several locations are obtained by analyzing the BTCs at these locations, an

explicit function for αD(L) can be obtained from these values. Once the explicit function

for αD(L) is obtained, λT(t), α'T(T), and λD(x) can be obtained using the relationships

between them.

c. The local time dependent dispersivity λT(t) is difficult to apply when the solute input is

not instantaneous. In this case, the convolution theorem has to be used when the BTC is

calculated using the CDE with local time-dependent dispersivity. However, it would be

expected that the convolution calculation would be a computationally inefficient

algorithm.

All the relationships [Eq. (4.18), Eq. (4.19), Eq. (4.21) and Eq. (4.22)] between these

scale-dependent dispersivities, and the definition of α'T(T), αD(L), and λD(x) were directly

or indirectly obtained from λT(t), using the assumption of that T ( =L/v ) could represent

the expected value of the time distribution of the BTC at L for the solute transport

problem with an initial source represented as a Dirac delta function. For generating the

BTC at a given distance L, the time T may or may not represent the expected value of the

time distribution of the BTC at L under all solute transport conditions (e.g. high Peclet

number or low Peclet number). This means that the BTCs generated using the CDE with

α'T(T), αD(L), or λD(x) obtained from λT(t) and the relationships between them (based on

T = L/v) may or may not be identical to the BTCs generated using the CDE with λT(t).

Therefore, the applicability and adequacy of using α'T(T), αD(L), and λD(x), or the

relationships between them, in this solute transport problem under differing solute

transport conditions has to be carefully checked.

When the solute transport problem involves multiple source input over time but at one

location, the time t is difficult to define for practical purposes because it is difficult to

define the origin of the time coordinate. Therefore, it is difficult to define a λT(t) to

93

quantify the solute dispersion of solute transport with multiple sources over time. In

order to apply λT(t) in this case, several λT(t) have to be defined for each source

separately, and then BTCs generated for every source are superimposed (using the

convolution theorem) to form the BTC for such solute transport problems. The

advantage of applying the convolution theorem is that the generated BTC is accurate. The

disadvantage is that the application of the convolution theorem in numerical solution of

the CDE is computationally inefficient.

However, when solute sources occur at one location, which can be treated as the origin of

the space coordinate, it is easier to use the distance-dependent dispersivity λD(x) and

αD(L) for practical purposes. This situation implies that the arbitrarily defined time T

=L/v could represent the expected value of time distribution of the BTC at L for this

multiple source solute transport problem. It also implies that the relationships between

λT(t), α'T(T), λD(x), and αD(L), which were developed for solute transport with an initial

source represented as a Dirac delta function, could also be directly used to define λD(x)

and αD(L) for the solute transport problem with multiple source inputs over time. These

implications need to be carefully checked experimentally.

Similarly, when the solute transport problem involves simultaneous multiple source

inputs over the spatial domain, it is difficult to uniquely define the values of distance x

and L for practical purposes, because it is difficult to uniquely define the origin of the

space coordinate. Consequently, it is difficult to uniquely define a time T to represent the

expected value of the time distribution of the BTC at L. As discussed above, α'T(T),

λD(x), and αD(L) are derived from λT(t) using the arbitrarily defined time T. When T is

not uniquely defined, α'T(T), λD(x), and αD(L) will also not be uniquely defined.

Therefore, in this situation, only the time-dependent dispersivity λT(t) should be applied

to the numerical scheme for solving the scale-dependent CDE. However, the statement

that it is difficult to uniquely define a time T to represent the expected value of the time

distribution of the BTC at L has to be critically examined for practical purposes.

94

In the remaining sections of this chapter, numerical tests will be introduced to test the

applicability and adequacy of using λT(t), α'T(T), λD(x), and αD(L) or the relationships

between them, provided in Eq. (4.18), Eq. (4.19), Eq. (4. 21), and Eq. (4.22) in generating

BTCs using the CDE for differing solute transport problems under differing solute

transport conditions.

4.3 Applicability of Scale-Dependent Dispersivity

Three typical solute transport problems under steady flow condition with different Peclet

numbers were used to test the applicability and adequacy of using λT(t), α'T(T), λD(x), and

αD(L) to numerically generate BTCs using the CDE with these scale-dependent

dispersivity functions. These functions were assumed to be linear or power-law. The

one-dimensional particle tracking method was used to solve the CDE with scale-

dependent dispersivity to numerically generate the BTCs.

4.3.1 Solute transport problems

Three infinite domains, one-dimensional, non-reactive solute transport problems were

used for testing. Average pore water velocity v was assumed to be constant in the

governing CDE for all solute transport conditions. BTCs were numerically generated at a

distance L for these solute transport problems under different flow conditions as defined

by the Peclet number. These solute transport problems were as follows:

Problem 1: This solute transport problem is described by the CDE [Eq. (4.1)] with the

following auxiliary conditions:

0),(lim

0),(lim)()0,0(

0,0)0,(

=

==

≠=

+→

−∞→

txC

txCxC

xxC

Lx

x

δ (4.23)

These conditions define an initially instantaneous and narrow-pulse solute input, which

can be represented as a Dirac delta function, applied at location zero and time zero.

95

In this problem, the objective was to numerically test the applicability and adequacy of

α'T(T), λD(x), and αD(L) in generating BTCs using the scale-dependent CDE under

differing solute transport conditions. All these three dispersivity functions are directly or

indirectly defined from local time-dependent dispersivity λT(t) by using the arbitrarily

defined time T = L/v, in which T is assumed to represent the time distribution of the BTC

at L. These numerical tests were carried by comparing the BTCs generated numerically

using the CDE with λD(x), α'T(T), and αD(L) to those generated using the CDE with λT(t).

λD(x), α'T(T), and αD(L) were obtained from λT(t) using the relationships between them,

provided in Eq. (4.18), Eq. (4.19), Eq. (4. 21), and Eq. (4.22). Solute transport conditions

were defined by Peclet numbers of about 20, 50, 100, and 200 in each case.

Problem 2: This solute transport problem is described by the CDE (Eq. 4.1) with the


0),(lim

0),(lim)()2,0(

)()0,0(0,0)0,(

=

==

=≠=

+→

−∞→

txC

txCxTC

xCxxC

Lx

x

δδ

(4.24)

These conditions define two instantaneous, narrow-pulse solute inputs, each of which can

be represented by a Dirac delta function, applied at location zero. One is applied at time

zero, and the other is applied at time T/2, in which vLT = . This solute transport

problem falls into the general category of multiple source input over the time domain.

The objective of this solute transport problem was to test the two practical implications

when the distance-dependent dispersivity λD(x) and αD(L) are used in the scale-dependent

CDE to numerically generate the BTC at L. In this situation λD(x) and αD(L) are defined

from the local time-dependent dispersivity λT(t) using the arbitrary defined time T =L/v.

One implication is that the arbitrarily defined time T =L/v could represent the expected

value of the time distribution of the BTC at L for this multiple source solute transport

96

problem. The other is that the relationships between λT(t), α'T(T), λD(x), and αD(L),

which were developed for the solute transport with an initial source represented as a

Dirac delta function, could also be directly used to define λD(x) and αD(L) for the solute

transport problem with multiple source input over time. For problem 2 under a given

solute transport condition (e.g a given Peclet number), if both implications were

applicable and adequate for practical purposes (or it were possible to directly apply αD(L)

and λD(x) to the scale-dependent CDE), the BTCs generated using the CDE with λD(x)

and αD(L) would be indistinguishable to those generated using the CDE with λT(t). For

the case of applying λT(t), the convolution theorem has to be applied to superimpose the

two BTCs generated for the two sources separately.

The test was carried out by comparing BTCs numerically generated at L using the CDE

with αD(L) and λD(x), to those obtained by superimposing the BTCs, generated

separately using the CDE with λT(t) for both inputs, using the convolution theorem.

Solute transport conditions for generating BTCs in problem 2 were defined by Peclet

numbers of 20, 40 60, and 100.

Problem 3: This solute transport problem is described by the CDE (Eq. 4.1) with the


0),(lim

0),(lim)()0,2(

)()0,0(0,0)0,(

=

==−

=≠=

+→

−∞→

txC

txCxLC

xCxxC

Lx

x

δδ

(4.25)

These conditions define two instantaneous, narrow-pulse solute inputs applied

simultaneously at time zero. Each of these narrow-pulse inputs can be represented as a

Dirac delta function. One is applied at location zero, and the other is applied at location

minus L/2. This solute transport problem falls into the general category of multiple source

over the space domain.

97

For this solute problem, the objective was to test the statement that it is difficult to

uniquely define a time T to represent the expected value of the time distribution of the

BTC at L for practical purposes, and that only the time-dependent dispersivity λT(t)

should be applied into the numerical scheme of the scale-dependent CDE. The test was

carried out by comparing BTCs numerically generated at L using the CDE with λT(t) to

those obtained using the CDE with three possible values of T. Even though it is difficult

to uniquely define a T to represent the expected value of the time distribution of the BTC

at L, it would be logical to expect that T would fall in the range from L/v to 1.5L/v. Here,

L is the distance from the location where the BTC is generated to the location of the

closer solute input. For analysis of this problem, the three possible values for T were

assumed to be L/v, 1.25L/v, and 1.5L/v.

The auxiliary conditions of Eq. (4.25) are not unique for defining the problem 3, because

the origin of the space coordinate is not uniquely defined for this problem, for example,

Eq. (4.25) can also be expressed as:

0),(lim

0),(lim)()0,2(

)()0,0(0,0)0,(

=

==

=≠=

+→

−∞→

txC

txCxLC

xCxxC

Lx

x

δδ

(4.26)

or

0),(lim

0),(lim)()0,4/(

)()0,4/(0,0)0,(

=

==

=−≠=

+→

−∞→

txC

txCxLC

xLCxxC

Lx

x

δδ

(4.27)

Therefore, the local distance-dependent dispersivity λD(x) is not unique. If λD(x) were

used in the numerical scheme of the scale-dependent CDE, the BTC would be obtained

by superimposing the BTCs generated separately using the CDE with λD(x) for both

inputs using the convolution theorem. In this case, the computation is inefficient. The

98

comparison of the BTCs generated using the CDE with λT(t) to those obtained by

superimposing the BTCs generated separately using the CDE with λD(x) for both inputs

using the convolution theorem were also conducted for this problem.

Solute transport conditions for generating BTCs in problem 3 were also defined by Peclet

numbers of 20 and 100.

4.3.2 Scale-dependent dispersivity functions

Two classes of scale-dependent dispersivity functions were used in the CDE for

numerically generating the BTCs. One class was based on defining λT(t) as a linear

function:

attT =)(λ (4.28a)

The other functions α'T(T), αD(L), and λD(x) were then defined as:

2

)()(' 0 aT

T

dttT

T

T

T ==∫λ

α (4.28b)

bxxD =)(λ (4.28c)

2

)()( 0 bL

L

dxxL

L

D

D ==∫λ

α (4.28d)

where a and b are two constants, and b = a/v.

The other class was based on defining λT(t) as a power- law function:

dctt =)(λ (4.29a)

with the other functions α'T(T), αD(L), and λD(x) defined as:

1

)()(' 0

+==

∫dcT

T

dttT

d

T

T

λα (4.29b)

dD gxx =)(λ (4.29c)

99

1

)()( 0

+==

∫dgL

L

dxxL

d

L

D

λα (4.29d)

where c, d, and g are constant, and g = c/vd.

For those cases where the set of scale-dependent dispersivity functions was based on the

linear function for defining λT(t), the spatial solute distribution at time t = T/2 was also

numerically generated for comparison.

4.3.3 Numerical solution

The one-dimensional particle tracking method was used to generate the BTCs using the

CDE with the different scale-dependent dispersivity functions. The algorithms for

applying distance-dependent dispersivity (Figure 4.3) and time dependent dispersivity

(Figure 4.4) in the numerical schemes for solving the scale dependent CDE [Eq. (4.4) and

Eq. (4.5)] have been detailed in section 4.1. The particle tracking method was

introduced and explained in Chapter 3. The free drainage outlet boundary condition [Eq.

(3.2)] was used for all BTCs generated. In comparison to using finite difference and

finite element methods to numerically solve the CDE, an important and attractive feature

of using the particle tracking method is that there is no possibility of numerical

dispersion. In addition, finite difference and finite element methods were developed for

solving boundary value problems.

4.4. Results and Discussion

Problem 1: BTCs generated using the CDE with scale-dependent dispersivity functions

based on the linear function for defining λT(t), are presented in Figure 4.6 for different

Peclet numbers of about 20, 50, 100, and 200. Corresponding BTCs generated using the

CDE with scale-dependent dispersivity functions based on the power- law function for

defining λT(t), are presented in Figure 4.7. The response of the BTCs to increasing Peclet

number was similar as shown in Figure 4.6 and Figure 4.7. These results indicated that

BTCs generated using the CDE with α'T(T), αD(L), and λD(x) were observably different

to the BTC generated using the CDE with λT(t) for solute transport under conditions

100

defined by Peclet numbers of 20 and 50. For Peclet numbers of 100 and 200, meaning

that solute transport is dominated by convection, BTCs generated using the CDE with

α'T(T), αD(L), and λD(x) were indistinguishable from the BTC generated using the CDE

with λT(t).

The results indicated, that when the Peclet number is larger than 100, application of the

dispersivity functions α'T(T), αD(L), and λD(x) to the numerical solution of the scale-

dependent CDE [Eq. (4.4) and Eq. (4.5)], will not cause observable errors in the BTCs

generated for solute transport problem 1. Therefore, the arbitrarily defined T = L/v can

be used to represent the expected value of the time distribution of the BTC at L for this

problem when the Peclet number is larger than 100. Application of α'T(T), αD(L), and

λD(x) will not cause an unacceptable error when the Peclet number is larger than 50.

However, when dispersion dominates the solute transport process and the Peclet number

is small (e.g. 20), the application of α'T(T), αD(L), and λD(x) will cause observable errors

in the generated BTCs. Therefore, the arbitrarily defined T = L/v cannot be used to

represent the expected value of the time distribution of the BTC at L for this problem

when the Peclet number is small (e.g. 20). The results also indicated that the accuracy of

applying α'T(T), αD(L), and λD(x) is determined primarily by the solute transport

conditions (Peclet number), rather than by the form (linear or power- law) of the function

for λT(t), on which they are based.

The spatial concentration distribution generated at time T/2 using the CDE with scale-

dependent dispersivity functions based on the linear function for defining λT(t) are

presented in Figure 4.8. As shown, the solute distribution curves over the spatial

transport domain at t=T/2, generated using the CDE with α'T(T) and αD(L) are observably

different from that generated using the CDE with λT(t) for all the given solute transport

conditions. This result implied that α'T(T) and αD(L), which were developed to define the

average integrated effect of λT(t) and λD(x) on solute dispersion as observed in a BTC at

L, cannot be used to analyze the spatial solute distribution at a given time. However, the

spatial solute distribution curves generated using λD(x) were not observably different

101

from those generated using λT(t) for the solute transport under conditions specified by

Peclet numbers 100 and 200. They are different for the solute transport under conditions

prescribed by Peclet number 20 and 50. Therefore, the CDE with dispersivity function

λD(x) can be applied to numerically generate the spatial solute distribution curves for

solute transport under conditions specified by high Peclet numbers ( e.g. >100). It

cannot be used for low Peclet numbers (e.g. <50).

Figure 4.6 BTCs generated using the CDE with scale-dependent dispersivity functions

based on the linear function for λT(t) for problem 1. In all cases, the mass of solute

injected = 1 mmole, porosity = 0.33, average pore water velocity v = 1 cm/min, and L =

120cm. a = 0.1 and b = 0.1 when Peclet number (Pe) = 20, a = 0.04 and b =0.04 when

Peclet number = 50, a = 0.02 and b =0.02 when Peclet number = 100, and a = 0.01 and b

= 0.01 when Peclet number = 200.

Pe = 20

0

0.002

0.004

0.006

0.008

0 1 2 3Dimensionless time

Con

cent

ratio

n (m

mo

le/c

m3 )

Pe = 50

0

0.002

0.004

0.006

0.008

0.01


Con

cent

ratio

n (m

mo

le/c

m3 )

Pe = 100

0

0.003

0.006

0.009

0.012


Co

nce

ntr

atio

n (m

mol

e/c

m3 )

Pe = 200

0

0.003

0.006

0.009

0.012

0.015

0.018

0 1 2 3

Dimensionless time

Con

cent

ratio

n (m

mo

le/c

m3 )

λT(t) λT(t)

λT(t)λT(t)

α’T(T) α’T(T)

α’T(T)α’T(T)

λD((x) λD((x)

λD((x)λD((x)

αD(L) αD(L)

αD(L)αD(L)

102


based on the power- law function for λT(t) for problem 1. In all cases, the mass of solute

injected = 1 mmole, porosity = 0.33, average pore water velocity v = 1.5 cm/min, and L =

120cm. c = 0.065, d = 1.2, g = 0.04 when Peclet number (Pe) = 21, c = 0.0244, d = 1.2,

g = 0.015 when Peclet number (Pe) = 56, c = 0.013, d = 1.2, g = 0.008 when Peclet

number (Pe) = 106, and c = 0.0065, d = 1.2, g = 0.04 when Peclet number (Pe) = 211.

Pe = 21

0

0.002

0.004

0.006

0.008


Co

nce

ntr

atio

n (m

mol

e/c

m3)

λT(t)

α’T(T)

λD((x)

αD(L)

Pe = 56

0

0.002

0.004

0.006

0.008

0.01


Co

nce

ntr

atio

n (m

mol

e/cm

3)

λT(t)

α’T(T)

λD((x)

αD(L)

Pe = 211

0

0.003

0.006

0.009

0.012

0.015

0.018

0 1 2 3

Dimensionless time

Co

nce

ntr

atio

n (m

mo

le/c

m3 )λT(t)

α’T(T)

λD((x)

αD(L)

Pe = 106

0

0.003

0.006

0.009

0.012

0 1 2 3

Dimensionless time

Co

nce

ntr

atio

n (m

mol

e/cm

3)

λT(t)

α’T(T)

λD((x)

Pe = 106

0

0.003

0.006

0.009

0.012

0 1 2 3

Dimensionless time

Co

nce

ntr

atio

n (m

mol

e/cm

3)

λT(t)

α’T(T)

λD((x)

αD(L)

103

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. In all cases, the mass of solute injected = 1 mmole, porosity = 0.33,

average pore water velocity v = 1 cm/min, and L = 120cm. a = 0.1 and b = 0.1 when

Peclet number (Pe) = 20, a = 0.04 and b =0.04 when Peclet number = 50, a = 0.02 and b

=0.02 when Peclet number = 100, and a = 0.01 and b = 0.01 when Peclet number = 200.

Pe = 20

0

0.003

0.006

0.009

0.012

0.015

0 40 80 120

Distance (cm)

Pe = 20

0

0.003

0.006

0.009

0.012

0.015

0 40 80 120

Distance (cm)

Con

cent

ratio

n (m

mo

le/c

m3 ) λT(t) α T(T)

λD((x) αD(L)

Pe = 50

0

0.006

0.012

0.018

0.024

0 40 80 120

Distance (cm)

Co

nce

ntr

atio

n (

mm

ole

/cm

3) λT(t) α T(T)

λD((x) αD(L)

Pe = 100

0

0.007

0.014

0.021

0.028

0.035

0 40 80 120Distance (cm)

Con

cent

ratio

n (m

mol

e/c

m3) λT(t) α T(T)

λD((x) αD(L)

Pe = 200

0

0.007

0.014

0.021

0.028

0.035

0.042


Con

cent

ratio

n (m

mol

e/c

m3) λT(t) α T(T)

λD((x) αD(L)

104

Problem 2: The results for problem 1 showed that the effect of the scale-dependent

dispersivity functions were determined primarily by the Peclet number, rather than by the

form (linear or power- law) of the function for λT(t). Consequently, only the dispersivity

functions based on the linear definition for λT(t), were used in the CDE to generate the

BTCs for this problem. For this solute transport problem, the objective was to test the

possibility of directly applying αD(L) and λD(x) to the scale-dependent CDE. Therefore,

the BTCs generated using the CDE with αD(L) and λD(x) were compared to the BTCs

generated using the CDE with λT(t). The BTCs generated for Peclet numbers of 20, 40,

60, and 100 are presented Figure 4.9.

As shown in Figure 4.9, the BTCs generated using the CDE with αD(L), λD(x) and λT(t)

were different from each other when Pelect number was 20. When Peclet number was

40, the BTC generated using the CDE with λD(x) closely matched to that generated using

the CDE with λT(t), however, the BTC generated using the CDE with αD(L) was

distinguishable from that generated using the CDE with λT(t). When the Peclet number

was 60, the BTC generated using the CDE with λD(x) was indistinguishable to that

generated using the CDE with λT(t), and the BTC generated using the CDE with αD(L)

was closely matched to that generated using the CDE with λT(t). When Peclet became

100, three BTCs generated using the CDE with D(L), λD(x) and λT(t) were

indistinguishable from each other.

These results indicated that it was possible to directly apply αD(L) and λD(x) in the

numerical scheme for solving the scale-dependent CDE for solute transport problems in

which multiple time input of solute are made at one location when Peclet number is large

(e.g. >60), and the error is virtually unobservable or acceptable. When solute transport

occurs under conditions specified by small Peclet numbers (e.g. 20), the error of the BTC

generated using the CDE with αD(L) and λD(x) may be unacceptable.

The results implied that the arbitrarily defined T = L/v could be used to approximate the

expected value of the time distribution of a BTC at L for problem 2 when Peclet number

105

is not very small (e.g. 40 and 60). And it could be used to represent the expected value of

the time distribution of a BTC at L when Peclet number is large (e.g. 100). The results

also implied that the relationships between λT(t), α'T(T), λD(x), and αD(L), which were

developed for the solute transport with an initial source represented as a Dirac delta

function, could be directly used to define the λD(x) and αD(L) for this kind of multiple-

source solute transport problems without causing unacceptable error when Pelect number

is not very small(e.g. >60).

When multiple solute sources are applied at one spatial location, in order to generate the

BTC using the CDE with local time-dependent dispersivity λT(t), the convolution

theorem has to be applied. As pointed out, it would be expected that the algorithm for

applying the convolution theorem to numerically solve the CDE with λT(t) would be

computationally inefficient. In this situation, αD(L) and λD(x) could be used to achieve

acceptable accuracy and computational efficiency when Peclet number is not very small.

106


based on the linear function for λT(t) for problem 2. In both cases, the mass of each

solute injected = 1 mmole, porosity = 0.33, average pore water velocity v = 1 cm/min and

L = 120 cm. a = 0.1 and b = 0.1 when Peclet number (Pe) = 20, a = 0.05 and b =0.05

when Peclet number = 40, a = 0.033 and b =0.033 when Peclet number = 60, and a =

0.02 and b = 0.02 when Peclet number = 200.

Problem 3: The BTCs for multiple spatial sources generated using the CDE with linear

functions for λT(t) and λD(x) are presented in Figure 4.10. As shown, the BTCs generated

using the CDE with λT(t) and λD(x) were not obviously different from each other for

solute transport conditions defined by Peclet numbers of 20 and 100. However, the

algorithm when applying λD(x) is computationally inefficient because of applying the

convolution theorem.

Pe = 100

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (

mm

ole

/cm

3 )

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

λT(t)

λD(x)

αD(L)

Pe = 60

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (m

mo

le/c

m3 )

0.000

0.002

0.004

0.006

0.008

0.010

0.012

λT(t)

λD(x)

αD(L)

Pe = 40

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (

mm

ole

/cm

3 )

0.000

0.002

0.004

0.006

0.008

0.010

λT(t)

λD(x)

αD(L)

Pe = 20

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (m

mo

le/c

m3 )

0.000

0.002

0.004

0.006

0.008

0.010

λT(t)

λD(x)

αD(L)

107

As pointed out for this problem, the possible T, which might represent the time

distribution of the BTC at L for practical purposes, would fall in the range from L/v to

1.5L/v. Three possible values of T were arbitrarily defined as L/v, 1.25L/v, and 1.5L/v.

The corresponding α'T(T) were applied to test the possibility of directly applying α'T(T)

in the numerical scheme to generate the BTC for this problem using the scale-dependent

dispersivity. The results are presented in Figure 4.11

As shown in these Figures, no matter how T were defined (T = L/v, 1.25L/v or 1.5L/v),

the BTCs generated using α'T(T) were different from those generated using local time

dependent dispersivity λT(t), for solute transport conditions defined by Peclet numbers 20

and 100.

Overall, the results of numerical tests conducted for problem 3 indicated that only local

time-dependent dispersivity λT(t) should be applied in the numerical scheme for solving

the scale-dependent CDE for the solute transport problems with multiple source input

over space. The accuracy of applying the local distance-dependent dispersivity λD(x) is

acceptable, however, the algorithm is computationally inefficient because the convolution

theorem has to be used for calculation. Application of αT'(T) may cause serious error

because it is difficult to define a value of T to represent the time distribution of the BTC

at L for this problem.

108

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. In both cases, the mass of

each solute injected = 1 mmole, porosity = 0.33, average pore water velocity v = 1

cm/min and L = 120cm. a = 0.1 and b = 0.1 when Peclet number (Pe) = 20, and a = 0.04

and b =0.04 when Peclet number = 100.

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. In both cases, the mass of

each solute injected = 1 mmole, porosity = 0.33, average pore water velocity v = 1

cm/min and L = 120cm. a = 0.1 and b = 0.1 when Peclet number (Pe) = 20, and a = 0.04

and b =0.04 when Peclet number = 100.

Pe = 20

0

0.002

0.004

0.006

0.008


Co

nce

ntr

atio

n (

mm

ole

/cm

3 ) λT(t)

λD((x)

Pe = 100

0

0.003

0.006

0.009

0.012

0 1 2 3

Dimensionless time

Con

cent

ratio

n (m

mol

e/c

m3 ) λT(t)

λD((x)

Pe = 100

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (m

mo

le/c

m3 )

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

λT (t)

α'T(T): T = L/v

α'T(T): T = 1.25L/v

α'T(T): T = 1.5L/T

Pe = 20

Dimensionless time

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Co

nce

ntr

atio

n (m

mo

le/c

m3 )

0.000

0.002

0.004

0.006

0.008

0.010

λT(t)

α'T(T): T = L/v

α'T(T): T = 1.25L/v

α'T(T): T = 1.5L/v

109

4.4 Conclusions

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. As a result, the scale-dependent dispersivity can be

specified as local time-dependent dispersivity λT(t), average time-dependent dispersivity

αT(t), apparent time-dependent dispersivity α'T (T') or α'T (T), apparent distance-

dependent dispersivity αD(L), and local distance-dependent dispersivity λD(x). The

algorithm for applying the average time-dependent dispersivity αT(t) to generate the BTC

is computationally inefficient, consequently, it is not useful for practical purposes.

The fact that mechanical dispersion in the CDE is assumed to be a diffusion- like process,

and the heterogeneity in porous medium is scale-dependent, directly indicate that the

scale-dependent dispersivity is locally time-dependent. The relationships [Eq. (4.16), Eq.

(4.17), Eq. (4.18), Eq. (4.19), Eq. (4.21) and Eq. (4.22)] between these scale-dependent

dispersivities, and the definition of α'T(T') or α'T(T'), αD(L), and λD(x) were directly or

indirectly obtained from λT(t), using the concept of mean solute transport time T' for a

BTC at L, or from the assumption that an arbitrarily defined time T (=L/v) could

represent the expected value of the time distribution (= T') of the BTC at L for the solute

transport problem with an initial source represented as a Dirac delta function. α'T(T') can

not be used for predicting a BTC, since T' can only be defined when the BTC is known.

For a given solute transport problem, when a BTC at L is predicted using the numerical

solution of the scale-dependent CDE, the choice between using λT(t), α'T(T), αD(L) or

λD(x) depended on the solute transport problem, solute transport conditions, level of

accuracy of the calculated BTC, and computational efficiency. For the solute transport

problem with a single source (Problem 1, T=L/v) under high Peclet number solute

transport conditions (e.g. >50), any one of these four dispersivities could be use to

generate the BTCs at L. However, when the Peclet number is small (e.g. <50), only λT(t)

should be used. For the solute transport problem with multiple source input over time but

at one spatial location (Problem 2, T = L/v), λD(x) or αD(L) has to be used in order to

110

achieve computational efficiency for generating the BTC when the solute transport Peclet

number is not too small(e.g. >60). However, when the Peclet number is small (e.g. <40),

only λT(t) should be used. For the solute transport problem with simultaneously multiple-

source inputs over the space domain (Problem 3), it is difficult to arbitrarily define a T to

represent the expected value of time distribution of the BTC at L. Therefore, only λT(t)

should be used.

CHAPTER 5 A NEW FUNCTION FOR DESCRIBING SCALE-

DEPENDENT DISPERSIVITY IN STATISTICALLY

HOMOGENEOUS POROUS MEDIA

5.1 Introduction

As discussed in Chapter 4, when the CDE with scale-dependent dispersivity is solved

numerically, the scale-dependent dispersivity has to be specified either as local time-

dependent dispersivity λT(t), apparent time-dependent dispersivity α'T(T), local distance-

dependent dispersivity λD(x), or apparent distance-dependent dispersivity αD(L). The

principal concepts regarding scale dependent dispersivity that have been explained and

developed thus far, can be summarized as follows:

a. Scale-dependent dispersivity results from scale-dependent heterogeneity of a porous

medium. Therefore, the scale-dependent dispersivity can be assumed to be a

characteristic of the porous medium. This assumption enables definition of explicit

functions to describe the scale-dependent dispersivity for solute transport in the porous

medium.

b. Scale-dependent dispersivity can be explicitly expressed as some function of physical

distance or time. These functions may be directly derived theoretically such as the fractal

approach of Wheatcraft and Tyler (1988), or developed empirically (Xu and Eckstein,

1995). In the numerical scheme for solving the scale-dependent CDE, these functions

can be expressed as local time-dependent dispersivity λT(t), apparent time-dependent

dispersivity α'T(T), local distance-dependent dispersivity λD(x), or apparent distance-

dependent dispersivity αD(L). They can be directly or indirectly related to each other

[Eq. (4.18), Eq. (4.19), Eq. (4.21), Eq. (4.22)]. These functions can be identified by the

analysis of solute BTCs, either by graphical analysis or by parameter fitting.

111

c. For predicting the BTC of solute transport problem with a single solute input over the

time or space domain, it is mplied that T = L/v. This T can represent the expected value of

time distribution of the BTC at L when the solute transport Peclet number is larger than

50. BTCs numerically generated using the CDE with α'T(T), αD(L), and λD(x) are close

(e.g. Peclet number = 50) or indistinguishable (e.g. Peclet number = 100 and 200) to

those generated using the CDE with λT(t). Therefore, for this solute transport problem,

the λT(t) should be applied. However, any of the other three dispersivities can also be

applied if the solute transport Peclet number were larger than 50.

d. For predicting the BTC of the solute transport problem with multiple source input over

time but at one spatial location, it is implied that T = L/v, and this T can approximate or

represent the expected value of time distribution of the BTC at L when the solute

transport Peclet number is not too small (e.g. >40). Therefore, λD(x) or αD(L) has to be

used in order to achieve computational efficiency for generating the BTCs for these

solute transport conditions.

e. For predicting the BTC of the solute transport problem with simultaneous multiple

source input over the space domain, it is difficult to define a value for T to represent the

expected value of time distribution of the BTC at L, therefore, only λT(t) should be used.

In this chapter an attempt is made to develop a new function to describe the scale-

dependent dispersivity in statistically homogeneous porous media. Most porous media,

such as natural geologic media or experimental column media, are statistically

homogeneous after some scale. Development of the new function was based on the

spatial auto-correlation of hydraulic conductivity in statistically homogeneous porous

media. It was used to define λT(t), αT(T), λD(x'), or αD(L'). Specially designed

experimental columns, in which the porous media can be considered as statistically

homogeneous, were used to simulate the three solute transport problems described in

section 4.3.1. The applicability of the concepts summarized in (c), (d) and (e) above for

the new function was tested using the BTCs obtained from these column experiments.

112

This chapter will:

a. Introduce some concepts on scale-dependent heterogeneity and the definition of

statistically homogeneous porous media.

b. Detail development of a new function for describing scale-dependent dispersivity in

statistically homogeneous porous media

c. Present results of tests of the appropriateness of incorporating the new dispersivity

function into the scale dependent CDE to describe BTCs from column experiments.

d. Present results of experimental tests of the applicability of the concepts summarized in

(c), (d), and (e) above using λT(t), α'T(T), λD(x), or αD(L) defined for the new function.

5.2 Scale Dependent Heterogeneity and Statistically Homogeneous Porous Media

In natural geologic media, the scale of heterogeneity is hierarchical. The hierarchy can be

treated as a continuous structure with an infinite range of scales (Neuman 1990;

Wheatcraft and Tyler, 1988), or treated as discrete hierarchical units, such as microscopic

scale (or pore scale), macroscopic scale (or laboratory scale), and megascopic scale (or

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

length (Figure 5.5).

0

0.002

0.004

0.006

0.008

0.01

0.012

110 120 130 140 150 160 170 180 190 200Time(min)

Con

cent

ratio

n(m

M)

Obs erved BTCP redic ted BTC I: Dis pe rs ivity=0.3734cm, k=0.0261, l=15.12cm, F=1.22P redic ted BTC II: Dis pers ivity=0.4066cm, k=0.0208, l=21.13cm, F=1.02P redic ted BTC III: Dis pe rs ivity=0.3935cm, k=0.0221, l=19.10cm, F=1.07Fitted BTC: Dis pers ivity=0.4227cm, r-Square=0.97

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

complexity of computation is linear [O(t)].

127

Length = 98 cmInterva = 22.5 min

0

0.005

0.01

0.015

0.02

50 70 90 110 130 150Time(min)

Con

cent

ratio

n(m

M)

Observed BTCPredicted BTC: k=0.026, l=15.12cmPredicted BTC: k=0.0208, l=21.13cmPredicted BTC: k=0.0221, l=19.10cm

Length = 98 cmInterva = 22.5 min

0

0.005

0.01

0.015

0.02

50 70 90 110 130 150Time(min)

Con

cent

ratio

n(m

M)


Length = 77 cmInterval = 18 min

0

0.005

0.01

0.015

0.02

30 50 70 90 110 130

Time(min)

Con

cent

ratio

n(m

M)


Length = 77 cmInterval = 18 min

0

0.005

0.01

0.015

0.02

30 50 70 90 110 130

Time(min)

Con

cent

ratio

n(m

M)


Length = 55 cm Interval =15 min

0

0.005

0.01

0.015

0.02

20 40 60 80 100Time(min)

Con

cent

ratio

n(m

M)


Length = 55 cm Interval =15 min

0

0.005

0.01

0.015

0.02

20 40 60 80 100Time(min)

Con

cent

ratio

n(m

M)


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)


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)


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


b va

lue

Figure 6.2 One set of fitted a and b at different column lengths obtained by fitting

observed BTCs to the DCDE.

141

0

0.003

0.006

0.009

0.012

50 70 90 110 130Time(min)

Con

cent

ratio

n(m

M)

Observed BTCFitted BTC: a=0.00183, b=1.48, R2=0.991, Calculated apparent dispersivity=0.3082cm

0

0.003

0.006

0.009

0.012

50 70 90 110 130Time(min)

Con

cent

ratio

n(m

M)

Observed BTCFitted BTC: a=0.00183, b=1.48, R2=0.991, Calculated apparent dispersivity=0.3082cmFitted BTC: a=0.00253, b=1.392, R2=0.991, Calculated apparent dispersivity=0.3086cmFitted BTC: a=0.00324, b=1.321, R2=0.992, Calculated apparent dispersivity=0.3047cmBTC fitted using the CCDE. R2=0.989, Fitted apparent dispersivity=0.3041cm

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


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

observed BTCs to the TCDE.

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

Observed BTCFitted BTC: c=0.000174, d=1.755, R2=0.984, Calculated apparent dispersivity=0.4244cmFitted BTC: c=0.0000527, d=2.008, R2=0.985, Calculated apparent dispersivity=0.4194cmFitted BTC: c=0.00000247, d=2.655, R2=0.986,Calculated apparent dispersivity=0.4168cmBTC fitted using the CCDE, R2=0.97, Fitted apparent dispersivity=0.4227cm

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

Distance(cm)

14110383593816

95%

CID

ispe

rsiv

ity(c

m)

.4

.3

.2

.1

Distance(cm)

14110383593816

Dis

pers

ivity

(cm

).4

.3

.2

.1

d

aab bc

ac a

Distance(cm)

14110383593816

Imm

obile

por

osity

.04

.03

.02

.01

0.00

a

b

c c cc

Distance(cm)

14110383593816

Imm

obile

por

osity

.04

.03

.02

.01

0.00

Distance(cm)

14110383593816

Imm

obile

por

osity

.04

.03

.02

.01

0.00

a

b

c c cc

Distance(cm)

14110383593816

Beta

.006

.005

.004

.003

.002

.001

0.000

-.001

a

b b

b

b b

Distance(cm)

14110383593816

Beta

.006

.005

.004

.003

.002

.001

0.000

-.001

Distance(cm)

14110383593816

Beta

.006

.005

.004

.003

.002

.001

0.000

-.001

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

Distance(cm )

14110383593816

95%

CI a

lpha

1 .9

1.8

1.7

1.6

1.5

1.4

D istance(cm )

14110383593816

Alph

a1 .9

1.8

1.7

1.6

1.5

1.4

bd bcad

a

bd

c

Distance(cm )

14110383593816

95%

CID

f

.28

.26

.24

.22

.20

.18

.16

.14

.12

D istance(cm )

14110383593816

Df

.28

.26

.24

.22

.20

.18

.16

.14

.12

b

ab

aba

ab

ab

Distance(cm )

14110383593816

95%

CID

f

.28

.26

.24

.22

.20

.18

.16

.14

.12

D istance(cm )

14110383593816

Df

.28

.26

.24

.22

.20

.18

.16

.14

.12

b

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%


significantly different at P≤ 0.05

for different column lengths to the FCDE. Values represent the treatment means and 95%


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


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.


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,


=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


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


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)


D

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Conc

entr

atio

n(m

M)


D

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Conc

entr

atio

n(m

M)


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,


=3.

155

A

0

0.1

0.2

0.3

0.4

0.5

0.6


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

B

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

ObservedPredicted: m=0.2273, n=0.7289, disp=0.4066cm, F=1.024Predicted: m=0.2031, n=0.9751, disp=0.4328cm, F=1.006Predicted: m=0.2207, n=0.7936, disp=0.4084cm, F=1.018Fitted: Dispersivity=0.4227cm, R2=0.97

C

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

Observed Predicted: m=0.2273, n=0.7289, F=1.321Predicted: m=0.2031, n=0.9751, F=1.313Predicted: m=0.2207, n=0.7936, F=1.307Fitted: Dispersivity=0.4227cm, R2=0.97

D

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

ObservedPredicted: m=0.2273, n=0.7289, F=1.331Predicted: m=0.2031, n=0.9751, F=1.338Predicted: m=0.2207, n=0.7936, F=1.318Fitted: Dispersivity=0.4227cm, R2=0.97

Figure 6.15 Application of the log-power function to predict the apparent scale-







2 where sr = the



=3.

156

A

0

0.1

0.2

0.3

0.4

0.5

0.6


Dis

pers

ivity

(cm

)Obs erved P redic ted: p=0.4317 and q=0.0964P redic ted: p=0.4989 and q=0.0637P redic ted: p=0.4624 and q=0.0774

B

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

ObservedPredicted: p=0.4317, q=0.0964, disp=0.3839cm, F=1.133Predicted: p=0.4989, q=0.0637, disp=0.4174cm, F=1.003Predicted: p=0.4624, q=0.0774, disp=0.3997cm, F=1.046Fitted: Dispersivity=0.4227cm, R2=0.97

C

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

ObservedPredicted: p=0.4317, q=0.0964, F=1.357Predicted: p=0.4989, q=0.0637, F=1.302Predicted: p=0.4624, q=0.0774, F=1.313Fitted: Dispersivity=0.4227cm, R2=0.97

D

0

0.003

0.006

0.009

0.012

100 120 140 160 180 200Time(min)

Con

cent

ratio

n(m

M)

ObservedPredicted: p=0.4317, q=0.0964, F=1.61Predicted: p=0.4989, q=0.0637, F=1.314Predicted: p=0.4624, q=0.0774, F=1.320Fitted: Dispersivity=0.4227cm, R2=0.97

Figure 6.16 Application of the hyperbolic function to predict the apparent scale-







2 where sr = the



=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


Sens

itivi

tyAt 150cm

0

0.5

1

1.5

2


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


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


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


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


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


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.

190

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