Modelling solute transport in porous media withspatially variable multiple reaction processes.

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MODELLING SOLUTE TRANSPORT IN POROUS

MEDIA WITH SPATIALLY VARIABLE

MULTIPLE REACTION PROCESSES

by

Linlin Xu

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF SOIL, WATER AND ENVIRONMENTAL SCIENCE

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

1995

1

OMI Number: 9603710

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UMI 300 North Zeeb Road Ann Arbor, MI 48103

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have

2

read the dissertation prepared by ___ L_I_N_L_I_N ___ X_U __________________________ _

entitled ____ M_O_D_E_L_L_I_N_G ___ SO_L_U_T_E ___ T_RA __ N_S_P_O_R_T ___ I_N ___ P_O_R_O_u_s __ ~_m_'D_I_A ___ W_I_T_f_I ____ __

SPATIALLY VARIABLE MULTIPLE REACTION ~ROCESSES

and recommend that it be accepted as fulfilling the dissertation

requirement for the Degree of DOCTOR OF :?HILOSOPHY

.-;1.. &' ::r ~ ;qC;:;-Date n:-rn Jfff

Arthur w. Warrick

1-( / 29 1 Cj\" ~~ ~\-rM}"2-Rajesh Srivastava Date

Date

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Dissertation Director

3

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements

for an advanced degree at The University of Arizona and is deposited in the Uni­

versity Library to be made available to borrowers under rules of the library.

Brief quotations from this dissertation are allowable without special per­

mission, provided that accurate acknowledgement of source is made. Requests for

permission for extended quotation from or reproduction of this manuscript in whole

or in part may be granted by the head of the major department or the Dean of the

Graduate College when in his or her judgment the proposed use of the material is

in the interests of scholarship. In all other instances, however, permission must be

obtained from the author.

SIGNED: ---o:::;.Y-:::-: ___ -=-~O_=} ___ ~_e:~~=:.:>~==_---_

4

ACKNOWLEDGEMENTS

I would like to thank my dissertation director Professor Mark L. Brusseau

for his instructions and helpful discussions during this study. Thanks are also due

to Professor Arthur W. Warrick, Dr. Rajesh Srivastava, Professor Shlomo Neuman

and Professor Dinshaw N. Contractor for their suggestions and their serving on my

committee.

I am indebted to my parents, my brother, and my sisters for their un­

derstanding and support. Special thanks are given to my wife, Jun Zhu, for her

patience and assistance, and to my son, Jim Jiayi Xu, for his innocent loving smile.

5

TABLE OF CONTENTS

page

ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER 1- INTRODUCTION AND LITERATURE REVIEW 8

1.1 Nwnerical Methods for the Advection-Dispersion Equation 8

Conventional Approach . . 8

Euler's Approach

Lagrange-Euler's Approach

1.2 Solute Transport in Heterogeneous Porous Media

Nonreactive Solute Transport

Reactive Solute Transport

1.3 Dissertation Format

CHAPTER 2 - PRESENT STUDY

2.1 Summary

REFERENCES

APPENDIX

A. A Combined Laplace Transform and Streamline Upwind

9

10

11

11

11

12

13

13

15

Approach for Nonideal Solute Transport in Porous Media. 21

B. Semi-Analytical Solution for Solute Transport in Porous Media

with Spatially Variable Multiple Reaction Processes . . . .. 49

C. FORTRAN Program for the Semi-Analytical Solution of Solute

Transport in Porous Media . . . . . . . . 78

6

ABSTRACT

In this dissertation, a new numerical method is developed for the simulation

of nonideal solute transport in porous media, and a first order semi-analytical

solution is derived for solute transport in porous media with spatially variable

multiple reaction processes. The Laplace transform is used to eliminate the time

dependency and the transformed transport equations are solved both numerically

and analytically. The transport solution is ultimately recovered by an efficient

quotient-difference inversion algorithm.

By introducing complex-valued artificial dispersion in the weighting func­

tions, characteristics of transport solutions have been successfully addressed. The

optimum of the complex-valued artificial dispersion has been derived for one di­

mensional problems. In multidimensional cases, the streamline upwind scheme

is modified by adding complex-valued artificial dispersion along the streamline.

Within this numerical scheme, the grid orientation problems have been success­

fully treated. The limitations on the cell Peclet number and on the Courant num­

ber were greatly relaxed. Both one dimensional and two dimensional numerical

examples are used to illustrate applications of this technique.

The analysis has been made for solute transport in systems with spatially

variable multiple reaction processes. Specific reaction processes include reversible

sorption and irreversible transformations (such as radioactive decay, hydrolysis re­

actions with fixed pH, and biodegradation). With the assumptions of solute trans­

port in a system with constant hydraulic conductivity and hydrodynamic dispersion

and spatially variable multiple reaction processes, a first-order semi-analytical so­

lution is derived for an arbitrary autocovariance function, which characterizes the

spatial variation of the multiple reaction processes. Results indicate that spatial

variation of the transformation constants for the solution phase and the sorbed

phase decreases the global rate of mass loss and enhances solute transport. If the

transformation constant for the sorbed phase is spatially uniform but not zero, a

similar effect is observed when there is spatial variation of the equilibrium sorption

coefficient. The global rate of mass loss and apparent retardation are decreased

7

when the spatial variability of the sorbed-phase transformation constant is posi­

tively correlated with the spatial variability of the equilibrium sorption coefficient,

and increased for a negative correlation. Spatial variation of the sorption rate

coefficient had minimal effect on transport.

8

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

1.1 Numerical Methods for the Advection-Dispersion Equation

A wide variety of mathematical models have been developed for the simu­

lation of contaminant transport in ground water. Most of them involve solution of

the advection-dispersion equation.

D82C _ q8C = 8C 8x2 8x at

where q is the rate of advection and D the dispersion coefficient. Analytical so­

lutions of this equation are available for some special cases [van Genuchten and

Wierenga, 1976; Hsieh, 1986; van Genuchten and Wagenet, 1989]. But many prob­

lems of practical interest (heterogeneous material properties and irregular domain

boundaries) must be solved numerically.

Conventional Approaches

Numerical solutions based on conventional finite-difference and/or finite­

element approaches are almost always plagued with either spurious oscillations or

excessive numerical dispersion when the advective term becomes dominant [Gresho

and Lee, 1979; Noorishad et al., 1992]. There are two parameters often used in

numerical analysis. One is the cell Peclet number, Pe, defined as q~;: where

~x is the characteristic length of the element and/or grid size. The other is the

Courant number, Cr, defined as ~r where ~t is the time difference between two

time steps. In most cases, conventional numerical schemes are subjected to the

constraints of Pe < 2 and Cr < 1. While these numerical difficulties can be easily

explained [Anderson et al., 1984] using, for example, general Fourier analysis,

the development of numerical schemes that overcome the problem is an ongoing

challenge. Although extremely fine mesh refinement is one possible solution, it is

often not a feasible alternative due to excessive computational requirements.

9

Euler's Approach

Through extensive studies of steady-state problems, it has been recognized

that apparent negative diffusion is inherent to conventional finite-difference and

finite-element schemes, and that artificial diffusion can be added to balance the

apparent negative diffusion [Christie et al., 1976]. The addition of artificial diffu­

sion has been accomplished through a variety of ways. In finite-difference schemes,

artificial diffusion can be implemented via an upwind difference expression for the

convective first derivative term, which essentially introduces a truncation error

having the same form as the physical diffusion term. In finite-element schemes,

artificial diffusion was added with the Petrov-Galerkin method, in which modi­

fied forms of interpolation functions were designed as weighting functions [Hein­

rich et al., 1977; Hughes, 1978]. The streamline upwind Petrov-Galerkin method

(SU /PG) [Brooks and Hughes, 1982] has demonstrated better performance in mul­

tidimensional problems.

As transient problems came under closer scrutiny, it was quickly recognized

that time discretization introduces new numerical difficulties, such as additional

numerical dispersion and the inaccurate representation of peak phase speed [Wes­

terink and Shea, 1989]. Artificial diffusion techniques developed for steady-state

problems generally produce overly diffusive solutions when applied to transient

problems. Extensions of the finite analytic [Chen and Chen, 1984] and optimal

test function [Celia et al., 1989] techniques to transient problems have typically ap­

proximated the time derivatives with finite difference approximation (time march­

ing). This has the effect of reducing the original transient problem to a series

of "steady state" problems defined as successive time steps. Unfortunately, the

resulting numerical solutions do not behave optimally [Li, et al., 1992]. In fact,

techniques which combine "optimal" spatial discretization with traditional tempo­

ral discretization suffer from the same problems as most other numerical methods

when advection dominates.

Many modellers have recently developed approaches that successfully bring

the time dependency into more active consideration. Tezduyar and Ganjoo [1986]

10

have modified the perturbation terms for SU /PG so that they are also dependent

upon the Courant number. Yu and Heinrich [1987] have considered a Petrov­

Galerkin approach that uses space-time finite elements and also includes the time

dependency in their upwind modifying functions. There are also higher order

schemes based more rigorously on the increased order of accuracy in the approx­

imation algorithms. Leonard [1979] presented a third-order space-accurate finite

difference algorithm for the transport equation, which performs accurately for the

entire range of Peclet numbers. Donea et al. 's [1987] Crank-Nicholson Taylor­

Galerkin finite element scheme is fourth-order accurate in both space and time.

Laplace transform is another alternative for addressing time constraints.

Recently, the Laplace transform/finite element method has been applied to ground­

water flow problems [Moridis and Reddell, 1991] and to solute transport problems

[Yu et al., 1987]. In Laplace domain, solutions are generally smoother than in the

time domain [Sudicky, 1989 and Sudicky 1990]. In addition, the Courant number

constraint is eliminated. The solutions for different times are independent (or par­

allel), without numerical error accumulations. However, the numerical solutions

are still subject to the limitations of a small cell Peclet number. The close in­

teraction between space and time dependence in the advective-dispersive equation

requires that both the temporal and spacial derivatives be accurately approximated

in order to obtain satisfactory solutions [Celia et al., 1990].

Lagrange-Euler's Approaches

The structure of the advective-dispersive equation suggests the use of "op­

erator splitting" techniques [Valocchi and Malmstead, 1992] which treat the ad­

vective term and the dispersive term differently. This approach leads to numerical

algorithms that combine Eulerian and Lagrangian solution methods. When the ad­

vective portion of the equation is solved with Lagrangian (or characteristic-based)

methods, Courant and cell Peclet number constraints can be significantly relaxed.

The performances of such methods are, however, critically dependent on the ac­

curacy of the backward trajectory-tracking and concentration interpolation [Holly

and Preissmann, 1977; Li and Venkataraman, 1991]. Linear interpolation [Neuman,

11

1981; Neuman and Sorek, 1982; Wheeler and Dawson, 1988; Chiang et ai., 1989]

can lead to excessive numerical dispersion [Roache, 1972; Huffenus and Khaletzky,

1981]. Quadratic interpolation [Douglas and Russell, 1982; Baptista, 1987] reduces

numerical damping somewhat but introduces spurious oscillations. Higher-order

interpolation techniques based on more interpolation nodes can reduce numerical

dispersion further but may introduce even more severe oscillations. The Hermite

interpolation technique [Fisher, 1977] achieves good accuracy at the expense of

increased computational effort, particularly for multidimensional problems. The

Lagrangian cubic spline interpolation technique [Thomson et ai., 1984] was rec­

ommended by Lindstrom [1992]. This technique uses data from across the entire

domain of definition, whereas the low-degree polynomials use only local data. Re­

cently, the Petrov-Galerkin method has been applied with the operator-splitting

technique [Miller and Rabideau, 1993].

1.2 Solute Transport in Heterogeneous Porous Media

Nonreactive Solute Transport

Field experiments indicate that field-scale dispersion is orders of magnitude

larger than that observed in the laboratory [Mackay et ai., 1986; Leblanc et ai.,

1991 and Gelhar, et ai, 1992]. The research about these scale effects motivated

the development of stochastic models for solute transport [Dagan, 1989]. Many

stochastic theories of flow and transport in heterogeneous porous media have been

developed for nonreactive solutes [Gelhar, 1986 and Neuman et ai., 1987]. The

stochastic analyses showed that transport of nonreactive solute is dominated by

spatial variations in hydraulic conductivity, which result in heterogeneous advec­

tion fields.

Reactive Solute Transport

The transport of reactive solutes may be influenced additionally by spatial

variations of chemical and microbiological properties of the porous media, and by

processes such as rate-limited sorption, rate-limited mass diffusion and transfonna­

tion [see Brusseau, 1994 for a review]. Smith and Schwartz [1980] and Garabedian

12

[1987] examined the effects of heterogeneities of both porosity and the distribu­

tion coefficient, as well as hydraulic conductivity, on the solute transport. They

showed that solute spreading is enhanced when a negative correlation exists be­

tween the hydraulic conductivity and the distribution coefficient. Valocchi [1989]

studied the transport of kinetically sorbing solute where the pore water velocity,

dispersion coefficient, distribution coefficient, and adsorption rate coefficient are

spatially variable. This study also found that a negative correlation between pore

water velocity and retardation factor increases solute dispersion. Chrysikopou­

los et ai. [1990] derived a first-order analytical solution for solute transport in a

one-dimensional homogeneous advection field in which retardation factor due to

instantaneous sorption is spatially variable. His results indicated that the spatially

variable retardation factor yields an increase in spreading of the solute front as

well as a decrease in the velocity of the center of mass. Bellin and Rinaldo [1995]

considered different degrees of correlation between the retardation factor and the

permeability of the porous media.

The spatial variability of irreversible reactions such as biotransformation

and its impact on transport has rarely been addressed. FUrther more, the spatial

variability of multiple reaction processes has rarely been addressed.

1.4 Dissertation Format

This dissertation consists of two manuscripts prepared for publication in

scientific journals. The manuscript attached as Appendix A has been accepted by

Water Resources Research. The other one has been submitted for peer review.

My advisor, Professor Mark L. Brusseau, helped select this subject as my

dissertation research. This research is based on the conceptual models developed

by Professor Brusseau and colleagues. I developed the new numerical method

presented in Appendix A for simulations of nonideal solute transport, and am

responsible for the solutions and conclusion in Appendix B about the effects of

spatial variable chemical and microbiological properties of porous media on the

transport and fate of contaminants. The encouragement and many suggestions

from Professor Brusseau contributed to the completion of this dissertation.

CHAPTER 2

PRESENT STUDY

13

The methods, results and conclusions of this study are presented in the

papers appended to this dissertation. The following is a summary of the most

important findings in these papers.

2.1 Summary

In this dissertation, a new numerical method is developed for the simula­

tion of solute transport in porous media with linear multiprocesses, and a first­

order semi-analytical solution is derived for solute transport in porous media with

spatially variable multiple reaction processes. The Laplace transform is used to

eliminate the time dependency and the transformed transport equations are solved

both numerically and analytically. The transport solution is ultimately recovered

by an efficient quotient-difference inversion algorithm.

By introducing complex-valued artificial dispersion in the weighting func­

tions, characteristics of transport solutions have been successfully addressed. The

optimum of the complex-valued artificial dispersion has been derived for one di­

mensional problems. In multidimensional cases, the streamline upwind scheme

is modified by adding complex-valued artificial dispersion along the streamline.

Within this numerical scheme, the grid orientation problems have been success­

fully treated. The limitations on the cell Peclet number and on the Courant num­

ber were greatly relaxed. Both one dimensional and two dimensional numerical

examples are used to illustrate applications of this technique.

The analysis has been made for solute transport in systems with spatially

variable multiple reaction processes. Specific reaction processes include reversible

sorption and irreversible transformations (such as radioactive decay, hydrolysis re­

actions with fixed pH, and biodegradation). With the assumptions of solute trans­

port in a system with constant hydraulic conductivity and hydrodynamic dispersion

and spatially variable multiple reaction processes, a first-order semi-analytical so­

lution is derived for an arbitrary auto covariance function, which characterizes the

14

spatial variation of the multiple reaction processes. Results indicate that spatial

variation of the transformation constants for the solution phase and the sorbed

phase decreases the global rate of mass loss and enhances solute transport. If the

transformation constant for the sorbed phase is spatially uniform but not zero, a

similar effect is observed when there is spatial variation of the equilibrium sorption

coefficient. The global rate of mass loss and apparent retardation are decreased

when the spatial variability of the sorbed-phase transformation constant is posi­

tively correlated with the spatial variability of the equilibrium sorption coefficient,

and increased for a negative correlation. Spatial variation of the sorption rate

coefficient had minimal effect on transport.

15

REFERENCES

[1] Anderson, D. A., J. C. Tannehill, and R. H. Pletcher, Computational Fluid

Mechanics and Heat Transfer, Hemisphere, Washington, 1984

[2] Baptista A. M. de M., Solution of Advection Dominated Transport by

Eulerian-Lagrangian Methods Using the Backward Method of Character­

istics, Ph.D. thesis, Dept. of Civil Eng., Mass. Inst. of Techno!', Cambridge,

1987

[3] Bellin, A. and A. Rinaldo, Analytical solutions for transport of linearly ad­

sorbing solutes in heterogeneous formations, Water Resour. Res., 31, 1505-

1511, 1995

[4] Brusseau, M. L., R. E. Jessup, and P. S. C. Rao, Modeling the transport

of solutes influenced by multiprocess nonequilibrium, Water Resources Re­

search, 25, 1971-1988, 1989

[5] Brusseau, M. L., R. E. Jessup, and P. S. c. Rao, Modeling solute trans­

port influenced by multiprocess nonequilibrium and transformation reac­

tions, Water Resources Research, 28, 175-182, 1992

[6] Brusseau, M. L., Transport of rate-limited sorbing solutes in heterogeneous

porous media: Application of a one-dimensional multifactor nonideality

model to field data, Water Resources Research, 28, 2485-2497, 1992

[7] Brusseau, M. L., Transport of reactive contaminants in heterogeneous

porous media, Reviews of Geophysics, 32(3), 285-313, 1994

[8] Brooks, A. N., and T. J. R. Hughes, Streamline upwind Petrov-Galerkin

formulations for convection dominated flows with particular emphasis on

the incompressible Navier-Stokes equations, Comput. Methods App!. Mech.

Eng., 32, 199-259, 1982

[9] Celia, M. A., J. S. Kindred, and 1. Herrera, Contaminant transport and

biodegradation in porous media, Water Resour. Res., 25, 1141-1148, 1989

[10] Celia, M. A., T. F. Russell,!. Herrera, and R. E. Ewing, An Eulerian­

Lagrangian localized adjoint method for the advection-diffusion equation,

Adv. Water Resources, 13(4), 187, 1990

16

[11] Chen, C. J., and H. C. Chen, Finite-analytic numerical method for unsteady

two-dimensional Navier-Stokes equations, J. Comput. Phys., 53, 209-226,

1984

[12] Chiang, C. Y., M. F. Wheeler, and P. B. Bedient, A modified method of

characteristics technique and mixed finite-element method for simulation of

groundwater solute transport, Water Resour. Res., 25(7), 1541-1549, 1989

[13] Christie, I., D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz, Finite

element methods for second order differential equations with significant first

derivatives, Int. J. Numer. Methods Eng., 10, 1389-1396, 1976

[14] Chrysikopoulos, C. V., P. K. Kitanidis and P. V. Roberts, Analysis of one­

dimensional solute transport through porous media with spatially variable

retardation factor, Water Resour. Res., 26, 437-446, 1990

[15] Dagan, G., Flow and transport in porous formations, Springe-Verlag, New

York, 1989

[16] De Hook, F. R., J. H. Knight, and A. N. Stokes, An improved method for

numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput., 3,

357-367, 1982

[7) Donea, J., L. Quarlapelle, and V. Selmin, An analysis of Time Discretization

in the Finite Element Solution of Hyperbolic Problems, J. Comput. Phys.,

70, 463-499, 1987

[18) Douglas, J., and T. F. Russell, Numerical methods for convection-dominated

diffusion problems based on combing the method of characteristics with

finite element or finite difference procedure, SIAM J. Numer. Anal., 19,

1-67, 1982

[19) Fisher, K., Convective difference schemes and hermite interpolation, in Pro­

ceedings International Symposium Innovative Numerical Analysis in Ap­

plied Engineering Science, Versailles, France, p.3.31, 1977

[20] Garabedian, S. P., Large-Scale dispersive transport in aquifers: field ex­

periments and reactive transport theory, Ph.D. dissertation, Mass. Inst. of

Technol., Cambridge, Mass., 1987

17

[21] Gelhar, L. M., Stochastic subsurface hydrology from theory to applications,

Water Resour. Res., 22, 135S-145S, 1986

[22] Gelhar, L. M., C. Welty and K. R. Rehfeldt, A critical review of data on

field-scale dispersion in aquifers, Water Resour. Res., 28, 1955-1974, 1992

[23] Gresho, P. M., and R. J. Lee, Don't suppress the wiggles -- They are

telling something !, in Finite Element Methods for Convection Dominated

Flows, edited by T. J. R. Hughes, AMD, 34, 37-61, 1979

[24] Heinrich, J. C., P. S. Huyakorn, O. c. Zienkiewicz, and A. R. Mitchell,

An upwind finite element scheme for two-dimensional convective transport

equation, Int. J. Numer. Methods Eng., 11, 131-143, 1977

[25] Holly, F. M., and A. Preissmann, Accurate calculation of transport in two

dimensions, J. Hydraul. Div. Am. Soc. Civ. Eng., 103(HYll), 1259-1277,

1977

[26] Hsieh, J. A., A new formula for the analytical solution of the radial disper­

sion problem, Water Resour. Res., 22(11), 1597-1605, 1986

[27] Huffenus, J. P., and D. Khaletzky, The Lagrangian approach of advective

term treatment and its application to the solution of N avier-Stokes equa­

tions, Int. J. Numer. Methods Fluids, 1, 365-387, 1981

[28] Hughes, T. J. R., A simple scheme for developing upwind finite elements,

Int. J. Numer. Methods Eng., 12, 1359-1365, 1978

[29] Leblanc, D. R., S. P. Garabedian, K. M. Hess, L. W. Gelhar, R. D. Quadri,

K. G. Stollenwerk and W. W. Wood, Large-scale natural gradient tracer

test in sand and gravel, Cape Cod, Massachusetts, 1, Experimental design

and observed tracer movement, Water Resour. Res., 27(9), 895-910, 1991

[30] Leonard, B. P., A survey of finite difference of opinion on numerical mod­

elling of incomprehensible defective confusion equation, in Finite Element

Methods for Convection Dominated Flows, edited by T. J. R. Hughes, AMD,

34, 1-16, 1979

18

[31] Li, S. G., F. Ruan, and D. McLaughlin, A space-time accurate method for

solving solute transport problems, Water Resour. Res., 28(9), 2297-2306,

1992

[32} Li, S. G., and L. Venkataraman, Discussion on time interpolation for the

solution of the dispersion equation by J. C. Yang and E. L. Hsu, J. Hydraul.

Res., 29(4), 568-571,1991

[33] Lindstrom, F. T., A mathematical model for the one-dimensional transport

and fate of oxygen and substrate in a water-saturated sorbing homogenous

porous medium, Water Resour. Res., 28(9), 2499-2511, 1992

[34] Mackay, D. M., D. L. Freyberg, P. V. Roberts and J. A. Cherry, A natural

gradient experiment on solute transport in a sand aquifer, 1, Approach on

overview and plume movement, Water Resour. Res., 22, 2017-2029, 1986

[35} Miller, C. T., and A. J. Rabideau, Development of split-operator, Petrov­

Galerkin methods to simulate transport and diffusion problems, Water Re­

sour. Res., 29, 2227-2240, 1993

[36] Moridis, G. J., and D. L. Reddell, The Laplace transform finite difference

method for simulation of flow through porous media, Water Resour. Res.,

27, 1873-1884, 1991

[37] Neuman, S. P., An Eulerian-Lagrangian numerical scheme for the dispersion­

convection equation using conjugate space-time grids, J. Compo Phys., 41,

270-294, 1981

[38} Neuman, S. P., and S. Sorek, Eulerian-Lagrangian methods for advection­

dispersion, in Finite Elements in Water Resources, vol 4, edited by K. P.

Holz et al., Springer-Verlag, New York, p. 14.41-14.68, 1982

[39] Neuman, S. P., C. L. Winter and C. M. Newman, Stochastic theory of field­

scale Fickian dispersion in anisotropic porous media, Water Resour. Res.,

23, 453-466, 1987

[40] Noorishad, J., C. F. Tsang, P. Perrochet, and A. Musy, A perspective on

the numerical solution of convection-dominated transport problems: A price

to pay for the easy way out, Water Resour. Res., 28(2), 551-561, 1992

19

[41] Parker, J. C. and M. Th. van Genuchten, Flux-averaged and volume­

averaged concentrations in continuum approaches to solute transport, Water

Resour. Res., 20, 866-872, 1984

[42] Roache, P. J., Computational Fluid Dynamics, Hermosa, Albuquerque, N.

M.,1972

[43] Smith, L. and F. W. Schwartz, Mass transport, 2, Analysis of uncertainty

in prediction, Water Resour. Res., 17, 351-369, 1981

[44] Sudicky, E. A., The Laplace transform Galerkin technique: A time continu­

ous finite element theory and application to mass transport in groundwater,

Water Resour. Res., 25(8), 1833-1846, 1989

[45] Sudicky, E. A., The Laplace transform Galerkin technique for efficient time­

continuous solution of solute transport in double-porosity media, Geoderma,

46, 209-232, 1990

[46] Tezduyar, T. E., and D. K. Ganjoo, Petrov-Galerkin formulations with

weighting functions dependent upon spatial and temporal discretization:

Applications to transient convection-diffusion problems, Compo Method

Appl. Mech. Eng., 59, 49-71, 1986

[47] Thomson, N. R., J. F. Sykes, and W. C. Lennox, A Langrangian porous

media mass transport model, Water Resour. Res., 20(3), 391-399, 1984

[48] Valocchi, A. J., Spatial moment analysis of the transport of kinetically ad­

sorbing solutes through stratified aquifers Water Resour. Res., 22, 273-279,

1989

[49] Valocchi, A. J., and M. Malmstead, Accuracy of operator splitting for

advection-dispersion-reaction problems, Water Resour. Res., 28(5), 1471-

1476, 1992

[50] van Genuchten, M. T., and P. J. Wierenga, Mass transfer studies in sorbing

porous media: analytical solutions, Soil Sci. Soc. Am. J., 40, 473-479, 1976

[51] van Genuchten, M. T., and R. J. Wagenet, Two-site/Two-region models for

pesticide transport and degradation: theoretical development and analytical

solutions, Soil Sci. Soc. Am. J., 53, 1303-1309, 1989

20

[52] Westerink, J. J., and D. Shea, Consistent higher degree Petrov-Galerkin

methods for the solution of the transient convection-diffusion equation, Int.

J. Numer. Methods Eng., 28, 1077-1101, 1989

[53] Wheeler, M. F., and C. N. Dawson, An operator-splitting method for

advection-difFusion-reaction problems, in MAFELAP Proceedings, vol. VI,

edited by J. A. Whiteman, Academic, San Diego, Calif., 463-482, 1988

[54] Yu, C. C., and J. C. Heinrich, Petrov-Galerkin methods for multidimen­

sional, time dependent convection-diffusion equations, Int. J. Numer. Meth­

ods Eng., 24, 2201-2215, 1987

[55] Yu, Y. S., M. Heidari, and W. T. Wang, A semi-analytical method for

solving unsteady flow equations, AS ME National Conference on Hydraulic

Engineering, Am. Soc. Mech. Eng., Williamsburg, Va., Aug., 3-7, 1987

APPENDIX A

A COMBINED LAPLACE TRANSFORM AND STREAMLINE

UPWIND APPROACH FOR NONIDEAL TRANSPORT OF

SOLUTES IN POROUS MEDIA

Linlin X u and Mark L. Brusseau

Departments of Soil and Water Science and

Hydrology and Water Resources

University of Arizona, Tucson, Arizona 85721

Has been accepted for publication by:

WATER RESOURCES RESEARCH

21

22

ABSTRACT

A new numerical method for nonideal transport of solutes in porous media

is developed by first using the Laplace transform to eliminate the time dependency

and then the streamline upwind scheme to solve the spatial equations. The transient

solution is ultimately recovered by an efficient quotient-difference inversion algo­

rithm. By introducing complex-valued artificial dispersion in the weighting func­

tions, characteristics of the transient solutions have been successfully addressed.

The optimum of the complex-valued artificial dispersion has been derived for one

dimensional problems. In multidimensional cases, the streamline upwind scheme

is modified by adding complex-valued artificial dispersion along the streamline.

Within this numerical scheme, the grid orientation problems have been success­

fully treated. The limitations on the cell Peclet number and on the Courant num­

ber were greatly relaxed. Both one dimensional and two dimensional numerical

examples are used to illustrate applications of this technique.

23

INTRODUCTION

Most transport models involve nwnerical solution of the advection disper­

sion equation. Through extensive studies of steady-state problems, it has been rec­

ognized that apparent negative dispersion is inherent to the Galerkin finite-element

schemes, and that artificial dispersion can be added to balance the apparent nega­

tive dispersion [Christie et ai., 1976]. The addition of artificial dispersion has been

accomplished through the Petrov-Galerkin method, in which modified fonns of in­

terpolation functions were designed as weighting functions [Heinrich et al., 1977;

Hughes, 1978]. In the applications to multidimensional problems, the streamline

upwind Petrov-Galerkin method (SU /PG) [Brooks and Hughes, 1982] has demon­

strated better perfonnance. As transient problems came under closer scrutiny, it

was quickly recognized that time discretization introduces new numerical difficul­

ties, such as additional numerical dispersion and the inaccurate representation of

peak phase speed [Westerink and Shea, 1989]. Artificial dispersion techniques devel­

oped for steady-state problems generally produce overly dispersive solutions when

applied to transient problems.

Recently, the Laplace transform/Galerkin finite element method has been

applied to solute transport problems [Sudicky, 1989]. In Laplace domain, solutions

are generally smoother than in the time domain. In addition, the solutions for dif­

ferent times are independent (or parallel), without numerical error accumulations.

The constraint of small Courant number Cr (Cr = 2i~) is eliminated. However,

the numerical solutions are still subject to the limitations of a small cell Peclet

number Pe (Pe = q~x). The close interaction between space and time dependence

in the advective-dispersive equation requires that both the temporal and spatial

derivatives be accurately approximated in order to obtain satisfactory solutions

[Celia et al., 1990].

We present an Eulerian numerical scheme which combines both Laplace

transform and streamline-upwind methods. The time derivative is eliminated

by taking the Laplace transfonn of the original equation and the resulting

24

"steady-state" equation is solved by a special streamline-upwind scheme contain­

ing complex-valued parameters. The solution in the physical domain is ultimately

recovered by using an efficient quotient-difference inversion algorithm [De Hook

et al., 1982]. As the Laplace transform/Galerkin method, this technique is devel­

oped for linear transport problems with steady flow fields and time-independent

transport parameters. The use of this technique is illustrated for selected cases.

MATHEMATICAL AND NUMERICAL APPROACH

Conceptual Model

The conceptual framework upon which the numerical model is based is that

associated with the multi-process models presented by Brusseau and colleagues

[1989, 1992]. Processes responsible for nonideal transport of solutes include rate­

limited diffusive mass transfer, rate-limited sorption, and transformation. Trans­

port is expressed by the following set of differential equations:

8 (1) - F. K C a - a a a (la)

8 (1) - F. K e n - n n n (lb)

8 (2) ~ = k ((1- F. )K C - 8(2») - ,,(2)8(2) at a a a a a r-a a (lc)

8 (2) ~ = k ((1 - F. )K C - 8(2») - 1/(2) 8(2) 8t n n n n n r-n n (ld)

8e 88(1) On( atn + J1.nCn) + (1- J)p( ;; + J1.~1)8~1»)

88(2) +(1 - f)p 8; = a(Ca - Cn) (Ie)

25

(If)

with the following initial and boundary conditions:

At t=O:

Ca = <p, 8 (1) - K F. In a - a aT'

8(2) - 0 a -

(2)

Cn = 0, 8(2) = 0 n

At boundary r:

( aCa) n· ~q·C - "D··- - n·q··I. I '" I a 'f IJ ax. - I 1'1-'

J

(3)

where lower index represents advective and nonadvective regions, and upper index

indicates different sorption domains; C is liquid solute concentration; 8 the mass

of sorbate in each domain divided by the mass of sorbent in the corresponding

region; F the fraction of sorbent in each region for which sorption is instantaneous;

K is equilibrium sorption constant; k the first order sorption constant; J.L the first

order degradation constant; () the fractional volumetric water content; p the bulk

density of soil; f the mass fraction of sorbent constituting the advective region; q

the Darcy flux; D the hydrodynamic dispersion coefficient; a the first order mass

transfer constant between advective and nonadvective regions; ni the out normal

of the boundary; Cs the source function; And e, TJ, cp and tP are given parameters

and/or functions to represent initial and boundary conditions,

26

Laplace Transform and Its Numerical Inversion

The Laplace transform of real-valued function G(t) with G(t)=O for t < 0

is defined as

(4)

where r = v+zw is the complex-valued. Laplace transform variable; C(t) is assumed

to be at least piecewise continuous and of exponential order 'Y (i.e., IG(t)1 :5

M e"'tt ,M being some arbitrary but finite constant), in which case the transform

function G ( r) is well defined for v > 'Y.

The inversion of the Laplace transform G( r) can be carried out analytically

in the following

1 111

+100

G(t) = -2 eTtG(r)dr 7rZ 11-100

(5a)

or, ellt 100

_ _

G(t) = - (~G(v + zw) coswt - 9G(v + zw)sinwt)dw 7r 0

(5b)

which can be approximated using a trapizoidal rule with a step size of ~ as the

following

The error E of approximation (6) has been derived for NT -400,

E = 2::=1 e-2mIlTG(2mT + t), 0< t < 2T (7)

The larger the vT, the smaller the error of the approximation with the

trapizoidal rule. It is known that the summation converges very slowly with this

direct approach. Many numerical algorithms have been developed to accelerate the

numerical convergence. Among them, the quotient-difference scheme demonstrated

the best performance [De Hook et ai., 1982].

To outline the quotient-difference scheme, we follow De Hook et al. [1982].

Given a power series 2::=0 amzm, we generate a sequence of quotient differences

27

AB2M as successive approximations to the power series, A2M and B2M are evaluated 2M

by the following recurrence:

m=1, .. ,,2M (8)

with the initial values A-I = 0, B-1 = 1, Ao = do, Bo = 1. In equation (8), dm

are constants,

Upon defining

e(i) = q(i+l) _ q(i) + e(i+l) r r r r-l

(i+l) (i) _ (i+l) er - l

qr - qr-l (i) er -l

{ r = 1, .. " M i = 0, .. ,,2M - 2r

(9a)

{r = 2, .. " M i = 0, .. ,,2M - 2r - 1 (9b)

and the initial values e~i) = ° for i = 0, .. ,,2M and q~i) = a~:l for i _

0, .. ,,2M -1, then dm are given by

do = ao, d - q(O) 2m-l - - m , m=1, ... , M (10)

A Combined Laplace Transform and Streamline Upwind Method

By taking the Laplace transform of the nonideal transport equations for

transient problems, the following equations are derived:

(11a)

at r (11b)

where

(12a)

f p(1 - Fa)KakaT OlG + +-~ T + J.L~2) + ka Ol + G

G = (On + (1 - f)pFnKn)T + (OnJ.Ln + (1- f)pFnKnJ.L~l»

(1 - J)p(1 - Fn)KnknT

+ T + J.L~) + kn

28

(12b)

(12c)

C a and C 8 represents the Laplace transforms of the solute concentration in the

advective region and the source function.

By weighting the Laplace transformed equations (11) with some test func­

tion Wand p, and integrating by parts, the following weighted residual equations

are derived:

2- -

1 a Ca aCa -p( -D;j a a + q;-a + RCa - Q)dA

A Xi Xj Xi

1 aCa - aw + (Dij- - qiCa)-dA A aXj aX;

(13)

1 - i - aCa + (RCa - Q)WdA + (qiCa - Dij -a

)niWdr = 0 A r ~

where W is the interpolation function used to approximate the unknown C a at

each element. Sudicky [1989J used only W as the test function in his Laplace

Transform/Galerkin approach, with p = 0 without any addition of artificial dis-

perslOn.

We apply the streamline upwind method (p t- 0) to our transformed equa­

tions (11). This approach guarantees the addition of artificial dispersion only along

the streamline if p takes the following form [Brooks and Hughes, 1982]

{3qi aw p=-­

qjqj aXi

However, the artificial dispersion {3 is allowed to be complex-valued here.

(14)

Equation (13) needs to be solved NT +l times according to the different

values of T used in equation (6). The numerical solutions at a given time could

29

be obtained directly with the quotient-difference inversion algorithm. Preliminary

analysis showed that NT = 14 (M = 7) is large enough for most transport problems.

In the finite difference scheme, temporal discretization of (1) may lead to

similar spatial equations as (11) and (13). To obtain the numerical solutions at

a given time, however, it is necessary to use a time-marching process, in which

equation (13) has to be solved as many times as the time steps in the time-marching

process.

One Dimensional Case

Using a linear finite element with element length ~, we derive the following

element matrix from equation (13):

kll = D + (3 + !(q _ (3R) + AR A 2 q 3

(15a)

k12 = D + (3 + !(q _ (3R) + AR ~ 2 q 6

(15b)

k21 = _ D + (3 _ !(q _ (3R) + AR A 2 q 6

(15c)

k22 = D + (3 _! (q _ (3R) + AR A 2 q 3

(15d)

For steady state problems with R=O. Christie et al. [1976] showed that if (3

is chosen as Aq Pe 2

(3 = -(coth- --) 2 2 Pe

(16)

the finite element matrix (15) is exact and the finite element formulation gives

exact solutions at all nodes. In equation (16) Pe = ~, which is defined as the cell

Peclet nwnber.

For transient problems (R #0), however, the finite element matrix can not

be exact with the use of only one parameter in the weighting functions. In order

to achieve optimal approximation, we choose to keep the following identity [see

Appendix]:

30

k k -Pe 12 = 21 e (17)

From equation (17) we see that k12 is smaller than k21 and that k12 ap­

proaches 0 as Pe increases. Given the following:

- BCa1 - -qCa1 -D--a;- = k ll Ca1 +k12 Ca2

- BCa2 - -qCa2 - D--a;- = -k21 Ca1 - k 22 Ca2

(18)

we see that equation (17) implies that flux is weighted more at upstream locations

than at downstream locations. The combination of (15) and (17) yields

6.q (3 =-0'

2 (19a)

coth Pe _ ..l... + AR 2 Pe 3q ( b)

0' = 1 + AR coth Pe 19 2q 2

For steady state problems with R = 0, equation (19) gives the same value

as that given by Christie et ai's formula (16). But when R ;l:0, equation (19) gives

different complex values.

Multidimensional Cases

In the application to multidimensional problems, the complex-valued artifi­

cial dispersion is added only along the streamline through the weighting function

p. we employ ad hoc generalizations [Brooks and Hughes, 1982] for the optimal

value of parameter (3 in the function p:

(20a)

where coth E.£ - ..l... + A iqjf

2 Pe 3 q 0' = ..

1 + AJ 91 R coth E.£ 21ql2 2

(20b)

(20c)

31

(20d)

in which ei are unit vectors and ~i the characteristic lengths of the element. D is

the scalar dispersion coefficient in the direction of flow.

A two-dimensional element is presented in Figure 1, with A, A', B and B'

as midpoints. We define ~ 1 = AA', ~ 2 = B B', and ;r = ~~:, e2 = ~~:. A

three-dimensional element is shown in Figure 2, with A, A', B, B', C and C' as

midpoints. We define ~l = AA', ~2 = BB', ~3 = CC' and ;r = ~~:, e2 = ~~:, e3 = gg:. In the calculations of the finite element matrix, the two dimensional

element is further divided into two triangular sub elements where linear interpola­

tion functions are introduced, and the three dimensional element is further divided

into six linear tetrahedral subelements. The use of linear subelements eliminates

the second derivatives in the integrals in equation (13).

32

NUMERICAL EXAMPLES AND DISCUSSION

In this section, two problems are presented to demonstrate the effective­

ness of the combined Laplace transform and streamline upwind scheme. The first

problem involves uniform flow with solute mass injected through a point at a con­

stant rate. The advection rate is assumed to be much larger than the dispersion

rate. Solute mass can not spread upstream near the point source and the solute

concentration varies sharply upstream near the point source and downstream near

the propagation front. The numerical solution was compared to that obtained by

the Laplace transform Galerkin finite element approach. This problem is used to

demonstrate the accuracy of the numerical solution. In the second problem, flow is

uniform and non-orthogonal to the mesh. This example is used to demonstrate that

the mesh orientation problem can be solved with the combined Laplace transform

and streamline upwind method.

In both problems, we take v = ~, T=6000 days and N r =15 in the quotient­

difference Laplace transform inversion algorithm. For multidimensional problems,

the tensor dispersion coefficients are defined as:

(21)

where a, and at are longitudinal and transverse dispersivities, respectively. In both

numerical examples, we use the parameter data (Table 1) suggested by Brusseau

[1992] for the Borden field experiment.

Example 1. Constant Point Flux

In this problem, we simulate the transport and propagation of the solute

concentration front along a line. It is assumed that flow is uniform at a rate of

q=O.OI and that a constant point flux of qOo is given at x = O. Both initial and

boundary values are taken as zero (cp = 0 and 'Ij; = 0). With these assumptions,

Q = q~O 8( x). Spatial discretization are over a segment of 20 meters ( -2 < x < 18)

with four different element lengths: ~=0.4, 0.04, 0.01 and 0.0008 meter. For these

33

spatial discretization, the corresponding cell Peclet numbers are: Pe=476, 47.6,

12, and 0.95, respectively.

The exact solution was obtained by first solving analytically the transformed

equation (11) and then performing the quotient-difference inversion algorithm. The

calculated spatial distributions of solute concentration at time t=300 days are

shown in Figure 3. For small cell Peclet number of 0.95, a good match is observed

between the Laplace transform Galerl{in finite element approach and our combined

Laplace transform and streamline upwind method. However, as the cell Peclet

number increases, the numerical solutions based on the Laplace transform Galerkin

finite element approach quickly break down. They are oscillatory at the upstream

domain for intermediate cell Peclet numbers of 12 and 47.6 and divergent at both

upstream and downstream locations for a large cell Peclet number of 476. Clearly,

the combined Laplace transform and streamline upwind method gave satisfactory

solutions for all the four cell Peclet numbers.

Example 2. Advection Skew to the Mesh

In this problem, the flow is uniform at the rate of q=0.03 and skew to

the mesh. Solute was injected at a constant rate of qCo along a line segment:

x = 0, 0 < y < 2. There is no mass transport across the boundary. The initial

condition is Ca = 0 and there are no interior sources. We used a 20 by 20 rectangular

finite element mesh over an area of 10 by 10 square meters. This example involves

discontinuities both along and across the streamline.

Numerical results were obtained for four different advection directions:

B = 0°,22.5°,45° and 67.5° (qz = qcosB and qy = qsinB). We define directional

Peclet numbers: Pez = qb~£ and Pey = qb~l!. With Table 1, it is derived that

Pez =594, 554, 433, 240 and Pey=O, 240,433, 554, corresponding to the four dif­

ferent advection directions. The spatial distributions of the calculated solute con­

centration at time t=300 days are shown in Figure 4. From these data, it is clear

that the solute front spreads smoothly along the streamline. The discontinuities

across the streamline are also represented well, with small oscillations across the

34

streamline near the discontinuities. When this problem is solved with the Laplace

transform/Galerkin finite method, strongly oscillatory solutions are obtained.

CONCLUSION

A new method for the nonideal transport of solutes in porous media is de­

veloped by first using Laplace transform to eliminate the time dependency and

then the streamline upwind scheme to solve the spatial equations. The transient

solution is ultimately recovered by performing an efficient Laplace inversion algo­

rithm. By introducing complex-valued artificial dispersion in the weighting func­

tions, characteristics of the transient solutions have been successfully addressed.

The optimum of the complex-valued artificial dispersion has been derived for one

dimensional problems. In multidimensional cases, the streamline upwind scheme

is modified by adding complex-valued artificial dispersion along the streamline.

Within this numerical scheme, grid orientation problems have been successfully

treated. The limitations on the cell Peclet number and on the Courant number

were greatly relaxed. Both one dimensional and two dimensional numerical exam­

ples have demonstrated the effectiveness of the combined Laplace transform and

streamline upwind approach.

Acknowledgments

This research was supported in part by a grant provided by the Water

Quality Research Program of the US Department of Agriculture.

35

REFERENCES

[1] Brusseau, M. L., R. E. Jessup, and P. S. C. Rao, Modeling the transport

of solutes influenced by multiprocess nonequilibrium, Water Resources Re­

search, 25, 1971-1988, 1989

[2] Brusseau, M. L., R. E. Jessup, and P. S. C. Rao, Modeling solute trans­

port influenced by multiprocess nonequilibrium and transformation reac­

tions, Water Resources Research, 28, 175-182, 1992

[3] Brusseau, M. L., Transport of rate-limited sorbing solutes in heterogeneous

porous media: Application of a one-dimensional multifactor nonideality

model to field data, Water Resources Research, 28, 2485-2497, 1992

[4] Brooks, A. N., and T. J. R. Hughes, Streamline upwind Petrov-Galerkin

formulations for convection dominated flows with particular emphasis on

the incompressible N avier-Stokes equations, Comput. Methods Appl. Mech.

Eng., 32, 199-259, 1982

[5] Celia, M. A., T. F. Russell, I. Herrera, and R. E. Ewing, An Eulerian­

Lagrangian localized adjoint method for the advection-diffusion equation,

Adv. Water Resources, 13(4), 187, 1990

[6] Christie, I., D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz, Finite

element methods for second order differential equations with significant first

derivatives, Int. J. Numer. Methods Eng., 10, 1389-1396, 1976

[7] De Hook, F. R., J. H. Knight, and A. N. Stokes, An improved method for

numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput., 3,

357-367, 1982

[8] Heinrich, J. C., P. S. Huyakorn, O. c. Zienkiewicz, and A. R. Mitchell,

An upwind finite element scheme for two-dimensional convective transport

equation, Int. J. Numer. Methods Eng., 11, 131-143, 1977

[9] Hughes, T. J. R., A simple scheme for developing upwind finite elements,

Int. J. Numer. Methods Eng., 12, 1359-1365, 1978

36

[10] Sudicky, E. A., The Laplace transform Galerkin technique: A time continu­

ous finite element theory and application to mass transport in groundwater,

Water Resour. Res., 25(8), 1833-1846, 1989

[11] Westerink, J. J., and D. Shea, Consistent higher degree Petrov-Galerkin

methods for the solution of the transient convection-diffusion equation, Int.

J. Numer. Methods Eng., 28, 1077-1101, 1989

37

APPENDIX

At the element: Xl ~ X ~ X2, assuming constant D, q, R and Q = 0, then

equation (lla) can be simply written as

(A -1)

The analytical solution of (A-I) is

q - Jq2 +4DR >'1 = 2D (A - 2)

>. _ q+ Jq2 +4DR 2 - 2D

From equation (A-2), we could derive the flux expression:

(A-3)

From equation (A-3), we could obtain the finite element equation:

(A - 4a)

(A - 4b)

From equation (A-4b) we see that K12 = K 21 e-Pe

38

Table 1. Input Parameter Data

Ka = 0.76 Fa = 0.20 ka = 0.36

Kn = 0.76 Fn = 0.20 kn = 0.36

f = 0.72 p = 1.81 a = 0.013

J1.a = 0 J1.~1) = 0 J1.~2) = 0

J1.n = 0 J1.~1) = 0 J1.<;> = 0

(}a = 0.238 (}n = 0.092 q = 0.03

a, = 5 X 10-4 at = 5 X 10-5 Do = 4.3 X 10-5

39

a'

, , I

A -+ - - A' , I I

Fig. 1 Typical four-node 2-dimensional element geometry

40

Fig. 2 Typical eight-node 3-dimensional element geometry

0 0 ........ 0

7

6

I

Exact

5 0

Combined

A

4

I Galerkin

3 I • i ! •

I

• 2 • • • • 1 • • • • 0

-2 -1 0 1 2 3 4 5 6 7 X

Fig. 3a Solute concentration profile at t=300 days due to

a point fiux. 6=0.4 and Pe=476

41

8

0 0 --0

1.2

1 .. 0.8 .. .. 0.6

.. .. 0.4

0.2

0 .. -0.2 ..

Exact .. -0.4

.. 0 .. Combined

-0.6 A Galerkin

-0. -2 -1 0 1 2 3 4 5 6 7

X

Fig. 3b Solute concentration profile at t=300 days due to

a point flux. 6=0.04 and Pe=47.6

42

8

0 0 '""-0

1

0.8

0.6

0.4

0.2

0

-0.2 Exact

0

Combined -0.4

-0.6 A Galerkin

-0.8 -2 -1 0 1 2 3 4 5 6 7

X

Fig. 3c Solute concentration profile at t=300 days due to

a point fiux. 6'=0.01 and Pe=12

43

8

0 () '""-()

1

0.9 Exact 0

0.8 Combined

0.7 ~

Galerkin

0.6

0.5

0.4

0.3

0.2

0.1

0 -2 -1 0 1 2 3 4 5 6 7

X

Fig. 3d Solute concentration profile at t=300 days due to

a point flux. 6=0.0008 and Pe=0.95

44

8

Fig. 48 Two dimensional spatial distribution of solute at t:=300 days. e = 00

45

(} = 22.5°

Fig. 4b Two dimensional SPatial distribution of solute at t:::800 days. (} == 22.50

46

Fig. 4c Two dimensional spatial distribution of solute at t==300 days. (J ~ 450

47

Fig. 4d Two dimensional Spatial distribution of solute at t::::300 days. fI == 67.50

48

APPENDIX B

SEMI-ANALYTICAL SOLUTION FOR SOLUTE

TRANSPORT IN POROUS MEDIA WITH MULTIPLE

SPATIALLY VARIABLE REACTION PROCESSES

Linlin Xu and Mark L. Brusseau

Departments of Soil and Water Science and

Hydrology and Water Resources

University of Arizona, Tucson, Arizona 85721

PREPARED FOR:

WATER RESOURCES RESEARCH

June 20, 1995

49

50

ABSTRACT

A first-order semi-analytical solution is derived for solute transport in porous

media with multiple spatially variable reaction processes. Specific reactions of in­

terest include reversible sorption, reversible mass transfer, and irreversible trans­

formation (such as radioactive decay, hydrolysis reactions with fixed pH, and

biodegradation). Laplace transform is employed to eliminate the time derivatives

in the linear transport equations, and the transformed equations are solved an­

alytically. The transient solution is ultimately obtained by use of an efficient

quotient-difference inversion algorithm. Results indicate that spatial variation of

transformation constants for the solution phase and the sorbed phase decreases the

global rate of mass loss and enhances solute transport. If the sorbed-phase trans­

formation constant is spatially unifonn but not zero, a similar effect is observed

when there is spatial variation of the equilibrium sorption coefficient. The global

rate of mass loss and apparent retardation are decreased when the spatial vari­

ability of the sorbed-phase transformation constant is positively correlated with

the spatial variability of the equilibrium sorption coefficient, and increased for a

negative correlation. Spatial variation of the sorption rate coefficient had minimal

effect on transport.

51

INTRODUCTION

Many stochastic theories of flow and transport in heterogeneous porous me­

dia have been developed for nonreactive solutes [Gelhar, 1986; Neuman et ai., 1987;

Dagan, 1989]. These analyses indicate that transport of nonreactive solute is domi­

nated by spatial variations in hydraulic conductivity, which result in heterogeneous

advection fields. The transport of reactive solutes, however, may be influenced ad­

ditionally by spatial variations of chemical and microbiological properties of the

porous media. These chemical and microbiological properties are inherently associ­

ated with processes such as rate-limited sorption, rate-limited mass diffusion, and

transformation.

Smith and Schwartz [1981] and Garabedian [1987] examined the effects of

spatially variable porosity and sorption, as well as hydraulic conductivity, on so­

lute transport. They showed that solute spreading is enhanced when a negative

correlation exists between the hydraulic conductivity and the distribution coef­

ficient. Valocchi [1989] studied the transport of kinetically sorbing solute where

the pore water velocity, dispersion coefficient, distribution coefficient, and adsorp­

tion rate coefficient are spatially variable. This study also found that a negative

correlation between pore water velocity and retardation factor increases solute dis­

persion. Chrysikopoulos et ai. [1990] derived a first-order analytical solution for

solute transport in a one-dimensional homogeneous advection field in which re­

tardation due to instantaneous sorption is spatially variable. His results indicated

that the spatially variable retardation factor yields an increase in spreading of the

solute front as well as a decrease in the velocity of the center of mass.

The spatial variability of irreversible reactions such as transformation and

its impact on transport has rarely been addressed. Furthermore, the spatial vari­

ability of multiple reaction processes has rarely been addressed. We investigate the

one-dimensional transport of reactive solutes in a uniform advection field wherein

multiple reaction processes are spatially variable. Analytical procedures are used

52

to solve the linear multiprocess transport equations proposed by Brusseau and col­

leagues [1989 and 1992]. The time derivatives in the transport equations are elimi­

nated by taking the Laplace transform. A first-order analytical solution is derived

for an arbitrary autocovariance function, which describes the spatial variability of

the multiple reaction processes. The trarisient solution is ultimately recovered by

use of an efficient quotient-different inversion algorithm [De Hook et al., 1982].

Selected examples are used for illustration.

GOVERNING EQUATIONS

The conceptual framework upon which the analysis is based is that associ­

ated with the multi-process models presented by Brusseau and colleagues [1989,

1992]. Processes responsible for nonideal transport of solute include rate-limited

mass diffusion, rate-limited sorption, and transformation. Transport is expressed

by the following set of differential equations:

(la)

(lb)

8 (2) ~ = k ((1 - F. )K C - S(2») - 1l(2)S(2) at a a a a a r-a a (Ie)

8S(2) _n _ _ k ((1 - F. )K C _ S(2») _ 1l(2) S(2) at - n n n n n r-n n (ld)

8e 8S(1) On( atn +J.lnCn)+(l-f)p( ;; +J.l~)S~l»)

8S(2) +(1 - f)p 8; = a(Ca - Cn) (Ie)

82Ca 8Ca (8Ca ) D 8x2 - q 8x + C, = Oa 7ft + J.laCa

8S(1) 8S(2) +fp( ~ + J.l~l)S~l») + fp ~ + a(Ca - Cn) (If)

with the following initial and boundary conditions:

53

At t=O:

Ca = <p, si1) = KaFa<P, si2) = 0

(2)

Cn = 0, S~l) = 0, S(2) = 0 n

At x=O: C aCa

q a - D ax = q"p (3a)

At x = xo:

Daca

= 0 ax

(3b)

where the lower index represents advective (a) and nonadvective (n) regions, and

the upper index indicates different sorption domains (1 = instantaneous and 2

= rate limited); C is liquid solute concentration; S is the mass of sorbate in each

domain divided by the mass of sorbent in the corresponding region; F is the fraction

of sorbent in each region for which sorption is instantaneous; K is the equilibrium

sorption coefficient; k is the first-order sorption rate coefficient; JL is the first-order

transformation constant; () is the fractional volumetric water content; p is the bulk

density of soil; f is the mass fraction of sorbent constituting the advective region; q

is the Darcy flux; D is the hydrodynamic dispersion coefficient; a is the first-order

mass transfer coefficient between advective and nonadvective regions; Cs is the

source function; And l/J, "p represent initial and boundary conditions.

We are using first-order kinetics to represent irreversible transformation

reactions at a generic level. However, first-order kinetics accurately describes ra­

dioactive decay and, under certain conditions (e.g., fixed pH), hydrolysis reac­

tions. In addition, biodegradation often can be adequately simulated as first-order

with respect to solute concentration, especially if other factors such as sorption

and transport are accurately represented. First-order kinetics strictly applies to

biodegradation for conditions of temporally constant biomass and no nutrient or

electron-acceptor limitations. Biodegradation is often specified to occur only in the

solution phase. However, this assumption is not valid for radioactive decay and

54

often will not be valid for hydrolysis reactions. Therefore, we allow transformation

to occur in any relevant domain.

The flux-type condition (3a) implies concentration discontinuity at the inlet

and leads to material balance conservation [Parker and van Genuchten, 1984]. The

boundary condition (3b) preserves concentration continuity at the outlet. In this

work, all reaction-related parameters in the above transport equations (1-3) are

allowed to be spatially variable.

By taking the Laplace transform [see Appendix] of equations (1-3), the

following equations are derived:

where

2- -D a C a _ q ac a _ RCa + Q = 0

ax2 aX

- aCa -qCa - D ax = qt/J, x = 0

DaCa

= 0, ax x = Xo

R = (Oal-'a + fpFaKal-'~l» + (Oa + fpFaKa)T

f p(1 - Fa)KakaT aG + +-~ T + 1-'~2) + ka a + G

G = (On + (1- f)pFnKn)T + (Onl-'n + (1- J)pFnKnl-'~l»

+ (1 - f)p(1 - Fn)KnknT

T + 1-'~2) + kn

(4a)

(4b)

(4c)

(5a)

(5b)

(5c)

C a and C IJ represent, respectively, the Laplace transforms of the solute concentra­

tion in the advective region and the source function.

55

SEMI-ANALYTICAL SOLUTION

We assume homogeneous hydrodynamic processes, such that the dispersion

coefficient D and the flow rate q are constants. However, the coefficient R describing

the properties of the multiple reaction processes is considered to be a stationary

stochastic variable. The coefficient R and the transformed solute concentration C a

are expressed as

R = (R) + {R}

Ca = (Ca) + {Cal

(6a)

(6b)

where { .} signifies fluctuation, and (.) represents ensemble mean or expected values.

The substitution of (6) into (4) yields

x = xo (7e)

Taking the ensemble mean of all terms in (7), we obtain the following

stochastic equation and boundary conditions:

Da~~a) _ qa~a) _ (RHCa) - ({RHCa}) + Q = 0 (8a)

(C ) - Da(Ca ) - .1. X = 0 q a ax - q'f/, (8b)

x = xo (8e)

Similar equations for the fluctuation {Cal are attained by subtracting (8)

from (7):

56

(9b)

x = Xo (ge)

Comparing (8) to (4), we see that (8) contains an additional tenn

({RHCa }), which accounts for the effect of the spatial variability of the multi­

ple reaction processes on the transfonned concentration (Ca ).

Assuming that the fluctuation {R} is small, we ignore the last two terms in

equation (9a) and obtain the following first-order solution for {Cal:

(10)

where W(x, y) = {WI (X, y) + W2(X, y), 0:5 y < x (Ua)

WI(X,y), X<y:5xo

A2eA2(XO- Y) - Al eA1 (XO-Y) Al eA1X - A2 eA2X

W1(X,y) = D(A2 _ AI) A~eA2XO _ A~eA1XO (Ub)

eA2(X-Y) _ eA1 (X-Y)

W2(x, y) = D(A2 _ Ad (Ue)

A - q - Jq2 + 4q(R} (Ud) 1 - 2D

). _ q + Jq2 + 4q(R} (Ue) 2 - 2D

The substitution of (10) into (8) yields

QR(X) = 1xO

({R(x)HR(y)})(Ca(y)}W(x,y)dy (12b)

(c ) - DO(Ca ) - .1. X = 0 q a ox - qoy, (12e)

x = Xo (12d)

57

In equation (12), there exists an integral-differential equation that is difficult

to solve analytically. As we make a first-order approximation in the derivation of

the fluctuation {Ca }, we can consistently replace (Ca ) in the integral in (12b) by

(Ca)o, which is determined by the following equations:

D82~~;)O _ q 8(~:)o _ (R)(Ca)o + Q = 0

(c ) - D 8(Ca)o - .1. X = 0 q a 0 8x - q'f',

D 8(Ca )o) = 0, 8x x = Xo

(13a)

(13b)

(13c)

Thus, equation (12) simplifies to an ordinary differential equation that can be

solved analytically. The solution is written as:

(14a)

or

(Ca(X)) = (Ca(x))o + 1xO 1xO

W(x,y)W(y,z)(Ca(z))o({R(y)}{R(z)})dydz

(14b)

Equation (14) represents the general result for the transformed solute con­

centration with which any autocovariance function of {R} can be employed. Ex­

amples are provided in the following section.

58

EXAMPLES AND DISCUSSION

Solutions

In this section, the expected value of the solution is given for a semi-infinite

medium (xo = 00) under constant flux of qCo at x = O. The initial condition

is ¢; = 0, and there is no interior source (Cs = 0). For the expected values of

parameters, we use those data (Table 1) suggested by Brusseau [1992] for the

Borden field experiment. The dispersion coefficient is defined as: D = ctl q + BaDo.

With these assumptions, the solution of (13) is written as:

(C) - qCO A1% a 0 - DA2 e

and as Xo -l- 00, the kernel function W(x,y) is simplified as:

(15)

(16)

Four parameters are allowed to be spatially varied in this section. These

are the first-order transformation constant for the solution phase, J1.a, and for the

sorbed phase, J1.~I), the equilibrium sorption coefficient Ka (Kn = K a), and the

sorption rate coefficient ka (kn = ka). In the remainder of this section, we define

WI = J1.a, W2 = J1.~I), W3 = K a, W4 = ka, and assume exponential autocovariance

functions for the fluctuation {wd:

(17)

From (5b) and (5c), we obtain the following first-order approximation:

(18)

Thus,

(19)

59

The substitution of (19) and (15) into (14b) yields

4 4

(Ca(x» = (Ca(x)}o + L LO"ijhij(X) (20a) i=1 j=1

- 8R 8R qCo 100 100 -l.i.=..!J.+~lZ hij(X) = {Wi}{W;}---- W(x,y)W(y,z)e >'ij dydz

8Wi 8Wj DA2 0 0 (20b)

With these assumptions, we carry out the integral in (20b) and obtain the

expected value of {Ca }. The transient solutions for heterogeneous {Ca } and ho-

mogeneous {Ca}o systems are obtained by use of the quotient-difference inversion

algorithm [De Hook et ai., 1982]. The effect of the spatial variability of each pro­

cess on the transport solution {Ca } is explicitly expressed in (20) in terms of the

coefficients O"ij and the error functions hij. In the following examples, a correlation

length of 10 is assumed for Aij in all cases. In discussion of results, we only present

the error functions hi;, which are independent of the coefficient 0" ij.

Single-Parameter Variability

We assign a value of 0.01 to {Pa} and zero to other transformation constants.

The spatial distributions of the error functions hii (i=I,3) at t=800 are given in

Figure la, and the temporal distributions of the error functions hii (i=I,3) at X=lO

are shown in Figure lb.

In this case, 0"11 h11 accounts for the spatial variability of the solution-phase

transformation constant with uniform sorption. Because 0"11 is positive, the positive

value of hll indicates a positive contribution to {Ca } from the spatial variability of

the solution-phase transformation constant. In other words, more solute exists in

the solution phase for this system. Thus, the spatial variation of the solution-phase

transformation constant decreases the global rate of mass loss and enhances solute

movement.

The case of spatially variable equilibrium sorption coefficient with uniform

solution-phase transformation is represented by 0"33h33' The function h33 has neg­

ative values in the upstream region and positive values in the downstream region

for the spatial distribution, and positive values at earlier times and negative values

60

at later times for the temporal distribution. With a positive 0'33, this indicates that

the spatial variation of the equilibrium sorption coefficient increases the spreading

of the solute front, as reported by previous investigators. Furthermore, it is clear

that spatially variable sorption does not affect the global rate of solution-phase

transformation. We are not aware of this finding being previously reported.

A value of 0.01 is assigned to (/L~l)} and zero to other transformation con­

stants when transformation occurs only in the sorbed phase. The spatial distribu­

tions of the error functions hjj (j=2,3) at t=800 are given in Figure 2a, and the

temporal distributions of the error functions hjj (j=2,3) at X =10 are shown in

Figure 2b.

For the case of uniform sorption, the effect of spatially variable sorbed-phase

transformation (0'22 h 22 ) is similar to that of spatially variable solution-phase trans­

formation (0'11 h11 ). However, the error function accounting for the spatial variabil­

ity of the equilibrium sorption coefficient with uniform sorbed-phase transformation

(h33) exhibits a nonzero positive value at later times. This indicates that the global

rate of mass loss is reduced not only by the spatial variability of the sorbed-phase

transformation coefficient, but also by the spatial variability of sorption.

The value of 0.01 is assigned to {/La} and (/L~l)}, and other transformation

constants are assumed to be zero for the third case. The spatial distributions of the

error functions hkk (k=1,2,3) at t=800 are given in Figure 3a, and the temporal

distributions of the error functions hkk (k=1,2,3) at X =10 are shown in Figure 3b.

In this case 0'33h33 accounts for the spatial variability of the equilibrium sorption

coefficient with uniform transformation in both solution and sorbed phases. It

has similar spatial and temporal distributions as did the case with only sorbed­

phase transformation. For the case of uniform sorption, 0'11 h11 + 0'22 h22 represents

the contribution of spatially variable transformation in both solution and sorbed

phases. It has the same characteristics as 0'11 h11 for solution-phase transformation

and 0'22h22 for sorbed-phase transformation.

The error function h44 , accounting for the contribution of spatial variability

of sorption kinetics is much smaller in magnitude for all the cases discussed above,

61

and is not presented in this work. Solute transport is insensitive to spatial variation

of the sorption-rate coefficient for the conditions employed in this analysis.

Multi-Parameter Variability

When more than one property is spatially variable, a question of major in­

terest is whether the variabilities of the properties are correlated, and if they are,

in what manner and to what degree .. In addition, the potential synergistic or an­

tagonistic interactions created by cross correlations and their impact on transport

is of great interest. In this work, we will illustrate this concept by examining cross

correlation between sorption and transformation. Results are obtained for the fol­

lowing cases: 1) solution-phase transformation and sorption, and 2) sorbed-phase

transformation and sorption.

The coefficient (1'i3 (i=l, 2) is positive for a positive correlation, and nega­

tive for a negative correlation. (1'13hI3 accounts for correlated sorption and solution­

phase transformation, and (1'23h23 for correlated sorption and sorbed-phase trans­

formation. The spatial distributions for error function hi3 (i=1,2) are shown in

Figure 4a, and the temporal distributions for error function hi3 (i=1,2) are shown

in Figure 4b.

The positive value of hi3 indicates that a positive correlation between sorp­

tion and transformation in the solution phase (+(1'13) or in the sorbed phase ( +(1'23)

results in a positive (1'i3hi3 and thus a decrease in apparent retardation. Conversely,

a negative correlation between sorption and transformation in the solution phase

( -(1'13) or in the sorbed phase (-(1'23) yields a negative (1' i3 hi3 and thus an increase

in apparent retardation.

The global rate of mass loss is altered as indicated by the nonzero h23

values at later times. Specifically, the global rate of mass loss is increased when

the equilibrium sorption coefficient is negatively correlated with the sorbed-phase

transformation constant (-(1'23) and decreased for a positive correlation (+(1'23)' The

spatial and temporal distributions of solute concentration for the case of correlated

sorption and sorbed-phase transformation are presented in Figure 5a and Figure

5b [specific variances used are: (1'22=0.16, (1'33=0.04, and (1'23 = ±0.12].

62

CONCLUSION

A first-order semi-analytical solution is derived for solute transport in porous

media with multiple spatially variable reaction processes. Specific reactions of in­

terest include reversible sorption, reversible mass transfer, and irreversible trans­

formation (such as radioactive decay, hydrolysis reactions with fixed pH, and

biodegradation). Laplace transform is employed to eliminate the time derivatives

in the linear transport equations, and the transformed equations are solved an­

alytically. The transient solution is ultimately obtained by use of an efficient

quotient-difference inversion algorithm. Results indicate that spatial variation of

transformation constants for the solution phase and the sorbed phase decreases the

global rate of mass loss and enhances solute transport. If the sorbed-phase trans­

formation constant is spatially uniform but not zero, a similar effect is observed

when there is spatial variation of the equilibrium sorption coefficient. The global

rate of mass loss and apparent retardation are decreased when the spatial vari­

ability of the sorbed-phase transformation constant is positively correlated with

the spatial variability of the equilibrium sorption coefficient, and increased for a

negative correlation. Spatial variation of the sorption rate coefficient had minimal

effect on transport.

Acknowledgments

This research was supported in part by a grant provided by the Water

Quality Research Program of the US Department of Agriculture.

63

REFERENCES

[1] Brusseau, M. L., R. E. Jessup, and P. S. C. Rao, Modeling the transport of

solutes influenced by multiprocess nonequilibrium, Water Resour. Res., 25,

1971-1988, 1989

[2] Brusseau, M. L., R. E. Jessup, and P. S. C. Rao, Modeling solute trans­

port influenced by multiprocess nonequilibrium and transformation reac­

tions, Water Resour. Res., 28, 175-182, 1992

[3] Brusseau, M. L., Transport of rate-limited sorbing solutes in heterogeneous

porous media: Application of a one-dimensional multifactor nonideality

model to field data, Water Resour. Res., 28, 2485-2497, 1992

[4] Chrysikopoulos, C. V., P. K. Kitanidis and P. V. Roberts, Analysis of one­

dimensional solute transport through porous media with spatially variable

retardation factor, Water Resour. Res., 26, 437-446, 1990

[5] Dagan, G., Flow and transport in porous formations, Springe-Verlag, New

York, 1989

[6] De Hook, F. R., J. H. Knight, and A. N. Stokes, An improved method for

numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput., 3,

357-367, 1982

[7] Gelhar, L. M., Stochastic subsurface hydrology from theory to applications,

Water Resour. Res., 22, 135S-145S, 1986

[8] Garabedian, S. P., Large-Scale dispersive transport in aquifers: field ex­

periments and reactive transport theory, Ph.D. dissertation, Mass. Inst. of

Techno!., Cambridge, Mass., 1987

[9] Neuman, S. P., C. L. Winter and C. M. Newman, Stochastic theory of field­

scale Fickian dispersion in anisotropic porous media, Water Resour. Res.,

23, 453-466, 1987

[10] Parker, J. C. and M. Th. van Genuchten, Flux-averaged and volume­

averaged concentrations in continuum approaches to solute transport, Water

Resour. Res., 20, 866-872, 1984

64

[11] Smith, L. and F. W. Schwartz, Mass transport, 2, Analysis of uncertainty

in prediction, Water Resour. Res., 17, 351-369, 1981

[12] Valocchi, A. J., Spatial moment analysis of the transport of kinetically ad­

sorbing solutes through stratified aquifers Water Resour. Res., 25, 273-279,

1989

65

Appendix

The Laplace transfonn of real-valued function G(t) with G(t)=O for t < 0

is defined as

(A -1)

where r = v+~w is the complex-valued Laplace transfonn variable; C(t) is assumed

to be at least piecewise continuous and of exponential order, (i.e., IG(t)1 < M e"Yt ,M being some arbitrary but finite constant), in which case the transform

function G(r) is well defined for v > ,. The inversion of the Laplace transform G( r) can be carried out analytically

in t.he following

1 111

+100

G(t) = -2 eTtG(r)dr 7r~ V-IOO

(A - 2a)

or,

evt 100

G(t) = -~ G(v + ~w)elwtdw 7r 0

(A - 2b)

which can be approximated using a trapizoidal rule with a step size of ¥ as the

following evt 1- "NT - m7r J.%lU.

G(t) = -;-{ "2 G(v) + ~ L-m=l G(v + ~T)e T } (A - 3)

The error E of approximation (A-3) has been derived for NT -+ 00,

0< t < 2T (A -4)

The larger the vT, the smaller the error of the approximation with the

trapizoidal rule. It is known that the summation converges very slowly with this

direct approach. Many numerical algorithms have been developed to accelerate the

numerical convergence. Among them, the quotient-difference scheme demonstrated

the best performance [De Hook et al., 1982].

To outline the quotient-difference scheme, we follow De Hook et al. [1982].

Given a power series 1::'=0 amzm, we generate a sequence of quotient differences

66

AB2M as successive approximations to the power series, A2M and B2M are evaluated 2M

by the following recurrence:

Am = A m- 1 + zdmAm-2

m=1, .. ,,2M (A-5)

with the initial values A-I = 0, B-1 = 1, Ao = do, Bo = 1. In equation (A-5), dm

are constants,

U pan defining

e(i) = q(i+1) _ q(i) + e(i+l) r r r r-l {

r = 1, .. " M i = 0, .. ,,2M - 2r

{r = 2, .. " M i = 0, .. ,,2M - 2r - 1

and the initial values e~i) = ° for i _ 0, .. ,,2M and q~i) _

0, .. ,,2M - 1, then dm are given by

do = ao, d - q(O) 2m-l - - m , m = 1, .. " M

(A - 6a)

(A - 6b)

(A -7)

In our applications, z = e¥, am = C(v+z '}11'), V = ~, T=6000, and N r=15,

67

Table 1. Expected Values of Model Parameters

(Ka) = 0.76 Fa = 0.20 (ka) = 0.36

(Kn) = 0.76 Fn = 0.20 (kn) = 0.36

f = 0.72 p = 1.81 a = 0.013

()a = 0.238 ()n = 0.092 q = 0.03

az = 5 x 10-4 at = 5 X 10-5 Do = 4.3 X 10-5

68

0.8~------------------------------------~

0.6

0.4

0.2

-0.2

-0.4

-e­h33

x

Fig. 1a The spatial distributions (t=800) of error functions

hii (i=1,3) for spatially variable sorption and

solution-phase transformation

r3 ........ '. ' . ..0::

69

1~--------------------------------------------~

0.8 -+-h11

0.6 -a-h33

0.4

0.2

0

-0.2

-0.4

-0.6~~~~~~~~lIlTlrllrnIlITTrrrrMnI~TT~ 300 500 700 900 1100

t

Fig. Ib The temporal distributions (X=lO) of error functions

hii (i=1,3) for spatially variable sorption and

sol ution-phase transformation

~ ........ .... .... ..c:

1.2

1 -+-h22

0.8 -e-h33

0.6

0.4

0.2

0

-0.2

-0.4 0 5 10 15 20

X

Fig. 2a The spatial distributions (t=800) of error functions

hjj (j=2,3) for spatially variable sorption and

sorbed-phase transformation

70

25

tJ ........ .... .... ..:::

1.4

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

-e­haa

71

-0.4+------~--~----------~1-----~----------~------.-----~----------r-~ aoo 400 500 600 700 800 900 1000 1100

t

Fig. 2b The temporal distributions (X=lO) of error functions

hjj (j=2,3) for spatially variable sorption and

sorbed-phase transformation

r:3 ....... ole ole

..t::

0.35

0.3 --+-h11

0.25 -.-h22

0.2 -e-hS3

0.15

0.1

0.05

0

-0.05

-0.1 0 5 10 15 20

X

Fig. 3a The spatial distributions (t=800) of error functions

hkk (j=1,2,3) for spatially variable sorption and

transformation in solution and sorbed phases

72

25

tJ ....... .., .., ~

73

0.6~--------------------------------------~

0.5 -+-h11

---.-0.4 h22

-e-

0.3 h33

0.2

0.1

-0. 1 ~-r-r-.,.-,--,....,...,...,.........,......,...,~..,.....,..~ee~,....,...,.-r-r-..,...,...,~..,........"-'-'--r-T""""""'..-.J 300 500 700 900 1100

t

Fig. 3b The temporal distributions (X=10) of error functions

h/r:k (j=1,2,3) for spatially variable sorption and

transformation in solution and sorbed phases

74

0.3,---------------------.

0.25 --+-h13

........ 0.2 h23

tJ ....... 0.15 .~

..c:!

0.1

0.05

5 10 15 20 x

Fig. 4a The spatial distributions (t=800) of error functions

hi3 (i=1,2) for correlated spatially variable sorption

and transformation

25

r3 ........ .., .. ~

0.25~------------------------------------~

-+-

0.2 h13 ......-

h23

0.15

0.1

0.05

o~~~~~~~~~~~~~~~~~ 300 500 700 900 1100

t

Fig. 4b The temporal distributions (X=10) of error functions

hi3 (i=1,2) for correlated spatially variable sorption

and transformation

75

~ "'" 1\ C[j

0 V

76

1

0.9 Homogeneous

0.8 ~

+ Correlation 0.7 --..-0.6

- Correlation

0.5

0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20

X

Fig. Sa Spatial distribution (t=800) of solute concentration for

correlated spatially variable sorption and sorbed-phase

transformation (0'22=0.16, 0'23 = ±O.12 and 0'33=0.04)

~ -A <U (.) V

77

0.6~--------------------------------------~

0.5

0.4

0.3 Homogeneous

-a-0.2 + Correlation

~

- Correlation 0.1

O~~~--~----~--~--~--~----~--~ 300 400 500 600 700 800 900 1000 1100

t

Fig. 5b Temporal distribution (X=10) of solute concentration for

correlated spatially variable sorption and sorbed-phase

transformation (0"22=0.16, 0"23 = ±O.12 and 0"33=0.04)

APPENDIX C

FORTRAN PROGRAM FOR THE SEMI-ANALYTICAL

SOLUTION OF SOLUTE TRANSPORT IN POROUS MEDIA

Linlin Xu

Department of Soil and Water Science

University of Arizona, Tucson, Arizona 85721

Developed for Dissertation Research

78

PROGRAM HETEROGENOUS IMPLICIT NONE REAL 0, V common 0, V REAL rho, f, alpha,

& theta a, F a, Kd a, k2 a, mu a, mu aI, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu-nl, mu:n2

common /mlb/rho~ f, alpha, - - -& theta a, F a, Kd a, k2 a, mu a, mu aI, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu:nl, mu:n2

REAL max time, time, t-min, t max - -REAL X, X_min, X_max, sigma_LAMD(10,2), Cases(10,2,lO) INTEGER I, K, iI, i2, check, nod_tm, NOD_sp common /hetero/sigma LAMD DOUBLE PRECISION inversion, WOW, WOW htr(lO) DOUBLE COMPLEX conc_lap(95) ,conc_lap:htr(95) OPEN (55,FILE='HTRG.DAT ' ,STATUS='OLD ' ) OPEN (66,FILE='x.dat',STATUS='new') OPEN (77,FILE='t.dat ' ,STATUS=' new ') READ (55, *) READ (55,*) 0, V READ (55, *) READ (55,*) rho, f, alpha READ (55, 1c) READ (55,*) theta_a, F_a, Kd_a, k2_a, mu_a, mu_al, mu_a2 READ (55, *) READ (55,*) theta n, F_n, Kd_n, k2_n, mu_n, mu_nl, mu_n2 READ (55,*,ERR=90) READ (55,*,ERR=90) DOK=l,10

READ (55,*,ERR=90) ( Cases (I, l,K), 1=1,10) READ (55,*,ERR=90) ( Cases(I,2,K), 1=1,10) wr i te (*,12,ERR=90) ( Cases (l,l,K), 1=1,10) write(*,12,ERR=90) (Cases(I,2,K), 1=1,10)

12 format ( lx, 10 (f6.2,lx) ) END DO READ (55,*,err=90) READ(55,*,err=90)max time

C PERFORM THE INVERSION OF THE LAPLACE TRANSFORM 30 READ (55,*,ERR-90)

&

READ (55,*,ERR=90) CHECK IF (CHECK.EQ.l) THEN

read (55,*,err=30) read (55,*,err=30) time WRITE (66,*) write(66,*) It=?I, TIME WRITE (66,*) WRITE (66,*) x=

IC a= read (55,*,err=30) read(55,*,err=30)X_min, X_max, NOD_sp DO 1=1, NOD sp

x=X_min+(I-l)*(X_max-X_min)/(NOD_sp-l) do i 1=1, 10

do i2=1, 2 sigma_lamd(il,i2)=0

79

end do end do call lap(x, conc_lap, conc_lap_htr, max_time) wow=inversion(conc lap,time,max time)

do k=l, 10 - -do i 1=1, 10

do i 2=1, 2 sigma_I amd (i I , i 2) =Cases (i I , i 2, k)

end do end do call lap(x, conc_lap, conc_lap_htr, max_time) wow_htr(k)=inversion(conc_lap_htr,time,max_time)

end do

80

WRITE (66,33)x,wow,wow htr(I)*mu a**2,wow htr(2)*mu al**2, & wow-htr (3)*Kd-a**2,wow-htr (6)*mu-a*Kd a, & wow-htr(8)*mu-al*Kd a - --

33 FORMAT(IX,f6.I,f6.3, S(lx,EIO.3f) -

C 90

&

& &

END DO ELSE IF (CHECK.EQ.2) THEN

read (55,*,err=30) read (55,*,err=30)x WR I TE (77 , *) wr i te (77 , *) I x=? I, X WR I TE (77 , *) WRITE (77,*) t=

IC a= read (55,*,err;30) read(55,*,err=30)t min, t_max, nod_tm do i=l, nod tm -

time=t mTn+(i-I)*(t max-t min)/(nod tm-l) do i l=T, 10 - - -

do i2=1, 2 sigma_lamd(il,i2)=0

end do end do call lap(x, conc_lap, conc_lap_htr, max_time) wow=inversion(conc_lap,time,max_time) do k=l, 10

do i 1 = 1, 10 do i2=1, 2

s i gma_l amd (i 1 • i 2)=Cases (i 1 , i 2. k) end do

end do call lap(x, conc_lap, conc_lap_htr, max_time) wow_htr(k)=inversion(conc_lap_htr.time.max_time)

end do WRITE (77.33)time,wow,wow_htr (1)*mu_a**2.wow_htr (2)*mu_ al**2,

end do END IF go to 30

wow_htr(3)*Kd_a**2,wow_htr(6)*mu_a*Kd_a, wow_htr(8)*mu_al*Kd_a

THE END OF THE INVERSION STOP END

subroutine lap (x, conc_lap, conc_lap_htr, max_time) implicit none REAL 0, V common 0, V real x, max_time, pO double complex p_i, conc_lap(95) , conc_lap_htr(95) , R, C_a double complex LAMD_I, LAMD_2 integer I po=8.634694/max_time DO 1=1, 95

p_i=dcmplx( pO, (i-l)*3.141592654/(O.8*max_time) LAMD_l=(V-sqrt (V**2+4*D*R (p_i»)/(2*D) LAMD_2=(V+sqrt(V**2+4*D*R(p i»)/(2*D) conc_lap(i)=V/p_i/(D*LAMD_2f*EXP(LAMD_l*X) conc_lap_htr(i)=C_a(x, p_i)

END DO return end

DOUBLE COMPLEX FUNCTION R(tau) IMPLICIT NONE REAL 0, V common 0, V REAL rho, f, alpha,

& theta a, F a, Kd a, k2_a, mu_a, mu_al, mu a2, & theta-n, F-n, Kd-n, k2 n, mu n, mu nl. mu:n2

common /mlb/rho~ f, alpha, - - -& theta a, F a, Kd a, k2 a, mu a, mu aI, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu:nl, mu:n2

DOUBLE COMPLEX tau, GG- - - -GG= theta n*mu n+(l-f)*rho*F n*Kd n*mu nl

& +(theta-n+(l=f)*rho*F n*Kd-n)*tau -& +(l-f)*rho*(l-F n)*Kd-n*k2-n*tau/(tau+k2 n+mu n2)

IF (GG.NE.O) GG=alpha*GG/(alpha+GG) --R= theta a*mu a+f*rho*F a*Kd a*mu al

& +(theta a+f*rho*F a*Kd a)*tau -& +f*rho*11-F a)*Kd-a*k2-a*tau/(tau+k2 a+mu a2) & +GG - - - --

RETURN END

DOUBLE COMPLEX FUNCTION R_mu(tau) IMPLICIT NONE REAL 0, V common 0, V REAL rho, f, alpha,

& theta a, F a, Kd a, k2 a, mu a, mu ai, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu-nl, mu:n2

common /mlb/rho~ f, alpha, - - -& theta a, F a, Kd a, k2 a, mu a, mu aI, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu:nl, mu:n2

DOUBLE COMPLEX tau - - - -R mu= theta a RETURN -END

81

DOUBLE COMPLEX FUNCTION R_mul (tau) IMPLICIT NONE REAL D. V common D. V REAL rho. f. alpha.

& theta a. F a. Kd a. k2_a. mu_a. mu_al. mu a2. & theta-no F-n. Kd-n. k2 n. mu n. mu nl. mu:n2

common /mlb/rho7 f. alpha. - - -& theta a. F a. Kd a. k2 a. mu_a. mu al. mu a2. & theta-no F-n. Kd-n. k2:n. mu_n. mu:nl. mu:n2

DOUBLE COMPLEX tau - -R mul= f*rho*F a*Kd a RETURN --END

DOUBLE COMPLEX FUNCTION R_Kd(tau) IMPLICIT NONE REAL D. V common D. V REAL rho. f. alpha.

& theta a. F a. Kd a. k2 a. mu a, mu al. mu a2. & theta-n, F-n. Kd-n, k2-n. mu-n, mu-nl. mu:n2

common /mlb/rho7 f. aTpha. - - -& theta a. F a. Kd a, k2 a. mu a. mu al. mu a2. & theta-n, F-n. Kd-n, k2-n. mu-n. mu:nl. mu:n2

DOUBLE COMPLEX tau. GG- - - -GG= theta n*mu n+(I-f)*rho*F n*Kd n*mu nl

& +(theta-n+(I=f)*rho*F n*Kd-n)*tau -& +(I-f)*rho*(I-F n)*Kd-n*k2-n*tau/(tau+k2 n+mu n2) R Kd= ( f*rho*F aimu al+f*rho*F a*tau - -

& - +f*rho*(T-F af*k2 a*tau/{tau+k2 a+mu a2) ) & +( alpha/(alpha+GG)-)**2 --& *( (l-f)*rho*F n*mu nl+(I-f)*rho*F n*tau & +(I-f)*rho*(T-F nf*k2 n*tau/(tau+k2 n+mu n2) )

RETURN - - --END

DOUBLE COMPLEX FUNCTION R k2(tau) IMPLICIT NONE -REAL D. V common D. V REAL rho. f. alpha,

& theta a. F a. Kd a. k2 a. mu a. mu al. mu a2. & theta-no F-n. Kd-n, k2-n, mu-n, mu-nl, mu:n2

common /mlb/rho7 f, alpha, - - -& theta a, F a, Kd a, k2 a, mu a, mu aI, mu a2, & theta-n, F-n, Kd-n, k2-n, mu-n, mu:nl, mu:n2

DOUBLE COMPLEX tau, GG- - - -GG= theta n*mu n+(I-f)*rho*F n*Kd n*mu nl

& + (theta - n+ (1:'f) *rho*F n*Kd-n) *tau -& +(I-f)*rho*(l-F n)*Kd-n*k2-n*tau/(tau+k2 n+mu n2)

R k2=( f*rho*(I-F-a)*Kd-a*tau*(tau+mu a2)/{tau+k2 a+mu a2)**2 ) & - +( alpha/(alpha+GG) f**2 _ - --& *«1-f)*rho*(1-F_n)*Kd_n*tau*(tau+mu_n2)/(tau+k2_n+mu_n2)**2)

RETURN END

82

DOUBLE COMPLEX FUNCTION C_sigma(X, tau, LAMD) IMPLICIT NONE REAL D, V common D, V REAL X, LAMD DOUBLE COMPLEX tau, R DOUBLE COMPLEX LAMD 1, LAMD 2, SUM 1, SUM 2, SUM 3, SUM_4 LAMD 1=( V-sqrt(V*V+4*D*R(tau» )/(2*D) - -LAMD-2=V/D-LAMD 1 SUM T=o -SUM-1=SUM l+LAMD 1**2*(EXP(i*(-LAMD 2+LAMD l)*X)-l)

& - - /(-LAMD 2+LAMD 1+1/LAMD)72/(-LAMD 2+LAMD 1) SUM l=SUM l-LAMD l*LAMD 2*(EXP«-LAMD 2+LAMD-l)*X)-T)

& - - / (-LAMD 2+LAMD 1+1/LAMD)/ (:'LAMD 2+LAMD 1) SUM l=SUM 1-LAMD l*LAMD 2*(EXP«-LAMD 2+LAMD l)*xf-l)

& - - *LAMD7(-LAMD-2+LAMD 1) - -SUM l=SUM l+LAMD 2**2*LAMD*X -SUM-l=SUM-l+(LAMD 2*LAMD-LAMD l/(-LAMD 2+LAMD l+l/LAMD»

& - - *( LAMD l*(EXP«-LAMD 2+LAMD l-l/LAMD)*X)-l) & /(-LAMD 2+LAMD 1-1/LAMD) -& +LAMD 2*LAMD*(EXP(-X/LAMD)-1) )

SUM l=SUM l*EXP(LAMD l*X) SUM-2=0 - -SUM-2=SUM 2+LAMD 1**2*(EXP(2*(-LAMD 2+LAMD l)*X)-l)

& - - /2/ (-LAMD 2+LAMD 1) - -SUM 2=SUM 2+LAMD 2**2*X -SUM-2=SUM-2+2*LAMD l*LAMD 2*(EXP«-LAMD 2+LAMD l)*X)-l)

& - - / (LAMD I-LAMD T) --SUM 2=SUM 2*EXP(LAMD l*xf/(LAMD 2-LAMD 1+1/LAMD) SUM-3=0 - - --SUM-3=SUM 3+LAMD 1*EXP«2*LAMD l-LAMD 2)*X)

& - - /(LAMD 2-LAMD 1-1/LAMD)/2/(-LAMD 2+LAMD 1) SUM 3=SUM 3-LAMD 2*EXP(LAMD l*X) --

& - - *LAMD7(LAMD 2-LAMD 1) SUM 3=SUM 3+(LAMD l/(LAMD 2-LAMD l-l/LAMD)+LAMD 2*LAMD)

& - - *EXP«LAMD 1-17LAMD)*X)/(LAMD 2-LAMD-1+1/LAMD) SUM 3=SUM 3*(LAMD l*EXP«LAMD l-LAMD 2)*X)-LAMD-2) SUM-4=0 - - - - -SUM-4=SUM 4-LAMD 1*EXP«2*LAMD l-LAMD 2)*X)/2/(LAMD 1-LAMD 2) SUM-4=SUM-4+LAMD-2*EXP(LAMD l*X)/(LAMD 1-LAMD 2) - -SUM-4=SUM-4*(LAMD l*EXP«LAMD 1-LAMD 2f*X)-LAMD 2)

& - - /(LAMD-2-LAMD 1+1/LAMD) - -C_sigma=(SUM_1+SUM_2+SUM:3+SUM_4)

& /(D*LAMD 2)**2/(LAMD 2-LAMD 1)**2 & *V/tau/(D*LAMD 2) - -

RETURN -END

83

DOUBLE COMPLEX FUNCTION C a(X, tau} IMPLICIT NONE -REAL X, sigma_LAMD(10,2} common /hetero/sigma LAMD DOUBLE COMPLEX tau, C_sigma, SUM DOUBLE COMPLEX R mu, R mul. R Kd. R_k2 SUM=O - - -IF (sigma_LAMD(1.1) .NE.O} THEN

SUM=SUM + sigma_LAMD(l. 1) *C_sigma (X.tau.sigma_LAMD (1.2}) & *R mu (tau) **2

END IF -IF (sigma_LAMD(2.1).NE.0) THEN

SUM=SUM + sigma LAMD(2, l)*C_sigma (X.tau.sigma_LAMD (2.2» & *R mul1tau) **2

END IF -IF (s i gma_LAMD <3. 1) • NE. 0) THEN

SUM=SUM + sigma_LAMD(3. l)*C_sigma (X.tau.sigma_LAMD (3.2» & *R Kd(tau)**2

END IF -IF (sigma_LAMD(4.1) .NE.O) THEN

SUM=SUM + sigma_LAMD(4. 1)*C_sigma(X.tau.sigma_LAMD(4.2» & *R k2(tau)**2

END IF -IF (sigma_LAMD(S.]) .NE.O) THEN

SUM=SUM + sigma_LAMD(S. l)*C_sigma(X.tau.sigma_LAMD(S.2» & *R mu(tau)*R mul (tau)

END IF - -IF (sigma_LAMD(6.1).NE.0) THEN

SUM=SUM + sigma_LAMD (6. l)*C_sigma (X.tau.sigma_LAMD (6.2» & *R mu (tau) *R Kd(tau)

END IF - -IF (s i gma_LAMD (7.1) .NE.O) THEN

SUM-SUM + sigma_LAMD(7. 1)*C_sigma(X.tau.sigma_LAMD(7,2» & *R mu(tau)*R k2(tau)

END IF - -IF (sigma_LAMD(8.1) .NE.O) THEN

SUM=SUM + sigma LAMD(8.1)*C_sigma(X.tau.sigma_LAMD(8.2» & *R mul1tau)*R Kd(tau)

END IF - -IF (sigma_LAMD(9.1) .NE.O) THEN

SUM-SUM + sigma LAMD(9.1)*C_sigma(X,tau,sigma_LAMD(9.2» & *R mul1tau)*R k2(tau)

END IF - -

84

IF (s i gma_LAMD (10. 1) .NE .0) THEN SUM=SUM + s i gma_LAMD (10. 1) *C sigma (X. tau, s i gma_LAMD (10.2) ) ..

& *R Kd(tau)*R k2(tauf END IF - -C a=SUM RETURN END

double precision function inversion(conc_lap,time,max_time) imp I i cit none integer max_lap,i parameter (max_lap=95) real pi, max_time, pO parameter (pi=3.141592654) real time double complex record (max_lap) , conc(max_lap) ,conc_lap(max_lap) common /transform/conc double complex temp_time, AM, BM po=8.634694/max_time do i=l, max_lap

conc (i) =conc_l ap (i) end do call coeff_QD(record) temp_time=dcmplx(O.O, pi*time/0.8/max_time) temp_time=exp(temp_time) call ab(am,bm,record,temp_time) if ( (am.eq.O) .and. (bm.eq.O) ) then

inversion=O else if (bm.eq.O) then

write(*,*) 'ERRORS EXIST IN THE INVERSION CALCULATIONS' stop' INVERSION ERROR'

else inversion= exp(pO*time)/0.8/max_time*Dreal (AM/BM)

end if return end

subroutine coeff_QD(record) implicit none integer max_lap parameter (max_lap=95) double complex record (max_lap) integer count integer i, j, trafic common /trafic/trafic double complex epsilon(max_lap, 2), Term count=1 do j=I,2

do i=l,max lap eps i lon1i ,j)'" (0.0,0.0)

end do end do record (count)=Term(O) do while (count.lt.max_lap)

call add (count, epsilon) count=count+l record (count) =-eps i Ion (l , 1)

end do return end

85

subroutine ab(am,bm,record,z) implicit none integer max_lap, i parameter (max_Iap=95) double complex record(max_lap) double complex z, am, bm double complex temp_l (max_lap), temp_2(max_Iap), h temp_l (1) =record (1) temp_l (2) =record (1) temp_2 (1) = (1.0,0.0) temp_2(2)=I+record(2)*z do i=3, max_lap

if (i.lt.max lap) then temp_l (j r =temp_l (j -1) +record (j) *temp_l (i - 2) *z temp 2 (i)=temp_2(i-l)+record(i)*temp_2(i-2)*z

else if (i.eq.max_Iap) then h=0.5+0.5*z*( record(i-l) - record(i) ) h=-h*( 1 - sqrt(l+z*record(i)/h**2) ) temp_l (i)=temp_l (i-1)+h*temp_1 (i-2)*z temp_2 (i)=temp_2 (i-l)+h*temp_2 (i-2)*z

end if end do am=temp_l (max_l ap) bm=temp_2(max_Iap) return end

subroutine add (count, epsilon) implicit none integer max_I ap parameter (max_Iap=95) double complex epsilon(max_lap, 2) integer count integer trafic, i, switch common /trafic/trafic double complex Term if (trafic.eq.1) then epsilon(count,2)-Term(count)/Term(count-l) switch-l do i=count-l,l,-l

if (switch.gt.O) then eps i 1 on (i ,2) =eps i lon (j + 1 , 1) +eps i lon (i+ 1 ,2) -eps i 1 on (j , 1)

else if (swi tch. 1 t .0) then eps i Ion (j ,2) =eps i Ion (i + 1 , 1) 1ceps i Ion (i + 1 ,2) / eps i Ion (i , 1)

end if switch=-switch

end do do i-l, count

eps i Ion (i , 1) -eps i Ion (i ,2) end do end if return end

86

double precision function inversion(conc_lap,time,max_time) implicit none double complex function Term(l) implicit none integer max_lap parameter (max_lap=95) double complex conc(max_lap) common /transform/conc integer 1, trafic common /trafic/trafic if (1. eq. 0) then

Term= 0.5*Conc(1) else

Term=Conc (1+1) end if if ( abs(Term) .It.exp(-50.0) ) then

trafic=O else

trafic=l end if return end

87