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Chemical Engineering: Dept. of Nuclear Engg., MIT
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The University of Arizona W. Wunderlich
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Ruhr-Universitat Bochum
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G. Pinder D-4639 Bochum-Ouerenburg (FRG)
School of Engineering, Dept. of Civil Engg.
Prinecton University Structural Engineering, Fluids and
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Laser Fusion - Plasma: Inst. fur Statik und Dynamik der
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Lecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
8
Linda M. Abriola
Multiphase Migration
of Organic Compounds
in a Porous Medium
A Mathematical Model
Spri nger-Verlag
Berlin Heidelberg New York Tokvo 1984
Series Editors
C. A. Brebbia . S. A. Orszag
Consulting Editors
J. Argyris . K.-J. Bathe' A. S. Connor' J. Connor' R. McCrory
C. S. Desai' K.-P. Holz . F. A. Leckie' L. G. Pinder' A. R. S. Pont
J. H. Seinfeld . P. Silvester' P. Spanos· W. Wunderlich' S. Yip
Author
Linda M. Abriola
Department of Civil Engineering
University of Michigan
Ann Arbor, Michigan 48109-2125
USA
ISBN-13:978-3-540-13694-1 e-ISBN-13:978-3-642-82343-5
001: 10.1007/978-3-642-82343-5
Library of Congress Cataloging in Publication Data
Abriola, Linda M.
Multiphase migration of organic compounds in a porous medium.
(Lecture notes in engineering; 8)
1. Water, Underground - Pollution - Mathematical models.
2. Organic water pollutants - Mathematical models.
3. Porous materials - Mathematical models.
I. Title. II. Series.
TD426.A27 1984 628.1'68 84-13970
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payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin, Heidelberg 1984
To my mother and father
ABSTRACT
In recent years, great attention has focused on the contamination
of the subsurface by organic chemicals. This problem is geographically
widespread and persistent. A literature review of contamination case
histories and present modeling techniques reveals the need for a more
comprehensive approach to the modeling of the chemical contamination pro
cess. This approach should be capable of tracing the muZtiphase migration
of a pollutant (i.e. its migration as a solute, a gas, and a non-aqueous
phase). In this thesis, such an approach is developed and implemented in
the construction of a numerical simulator.
Four separate phases are included in the development: solid (soil),
water, gas, and contaminant. The contaminant phase may be composed of two
distinct components - one volatile and the other non-volatile. Transfer of
the volatile component to the water and/or gas phases is permitted. As a
starting point in the analysis, the microscopic mass balance law of con
tinuum mechanics is averaged over a representative elementary volume to
produce a macroscopic mass balance equation for each system componento
Based on the physical characteristics of these components, various con
stitutive relations and approximations can be introduced. Incorporation
of these relations into the balance laws yields a system of three non
linear partial differential equationso
This system of equations is not amenable to solution by analytical
means. Approximate solutions to these equations, however, can be sought
at specific points by replacing the differential operators by finite dif
ference operators. The system of equations is thereby reduced to a system
VI
of implicit nonlinear algebraic equations in discreet unknowns. A
Newton-Raphson iteration scheme provides an effective technique for
the solution of these· equations. Development of a one-dimensional
computer model proceeds along these lines.
In order to apply this finite difference model to a specific
problem, a number of equation parameters must be evaluated. These
parameters include three-phase relative permeabilities, saturations,
partition coefficients, and mixture densities and viscosities. Once
expressions for these parameters are obtained, the numerical model
may be used to simulate various one-d~mensional conta~ination sce
narios. Pollution of an unsaturated soil column by a petroleum mix
ture and migration ofTCE in a water-saturated column are considered.
Convergence and mass balance properties of the scheme are examined
for each of these problems.
The numerical model can also be extended to handle two
dimensional problems. Implementation of a 04 ordering scheme reduces
the computer solution time and storage requirements of the model. Simu
lation of the migration of TCE in a confined aquifer demonstrates the
applicability of the model to a field problem.
ACKNOWLEDGEMENTS
Many individuals have been instrumental in the development of this
work. Thanks must be expressed to George F. Pinder who first encouraged
me to pursue this project and has contributed much of his knowledge and
ideas to its completion. My gratitude must also be extended to William
G. Gray for his support and constructive criticisms of the text. Portions
of this work have taken shape as a result of discussions with Michael A.
Celia, whose contributions are also gratefully acknowledged.
Thanks should go to Elizabeth Kaminski for her excellent and cre
ative layout and typing of this text and to Thomas Agans for his fine
draftsmanship on a number of the figures.
This work was supported, in part, by the U.S. Department of Energy
under contract and the Industrial Support Group of
#DE~AC02-79EV10257
Princeton University. Other funding was supplied by fellowshi~ from the
DuPont Corporation, the Shell Oil Foundation, and WAPORA, Inc.
TABLE OF CONTENTS
Chapter I - Introducti on •••.•••••.•••.•.•.•.•.•..•.•.••••••.•.•.••
Chapter II - Equation Development ••.••••.••••••••••..•••••.••••••• 14
2.1-Presentation of the Balance Laws ••••••••.•.•.•••••.• 14
2.2-Soi1 Species Equation ••••••••••••••••.••••••..••..•• 19
2.3-Water Equation •.•••••••••.•.•.••••••..•.•..••••••.•. 22
2.4-Inert Chemical Species Equation •.•........•.....•.•. 32
2.5-Air Species Equation •••.••...•.•.•......•......••... 37
2.6-Species 2 Equation.................................. 38
2.7-Partitioning of Mass .•.....•.••••.••..•••.•.•.••..•• 46
2.8-Equation Summary.................................... 49
Chapter III - Development of the 1-0 Simulator :................... 53
3.l-Background ••••••.••.••••.•••....•.•••...•.•.•.•••..• 53
3.2-Formation of the Difference Equations ••••••••.•••••• 57
3.3-Incorporation of Boundary and Initial Conditions..... 67
3.4-Evaluation of Coefficients .•••••.••..•...••.•••••••• 72
3.5-Solution of the Nonlinear Matrix Equations ...••.•... 82
Chapter IV - Computer Simulations in One Dimension ••.•..••...•..•• 88
4.1-0il Contamination Simulations ...•.••....•.•••••••••. 88
4.2-TCE Simulations ..••••••••••••.•.•.•••••••.•...•••.•• 112
Chapter V - The Two-Dimensional Simulator .•........•.•.....••...•• 143
5.1-Extension to Two Space Dimensions ••••.•••••••.•••••• 143
5.2-Matrix Equation Structure and Solution ••.••.•.•.•.•• 151
5.3-Examp1e Simulations .•••••••••••••••...•••.•.•••••••. 158
x
Summary and Conclusions 175
References 178
Appendices
Appendix A - Derivation of the Macroscopic Mass
Balance Equation •........ ~ ....••••..... 188
Appendix B.l - Properties of the Difference
Operator and its Solutions 195
Appendix B.2 - Analysis of Truncation Terms 198
Appendix B.3 - The Newton-Raphson Iteration Method ... 203
Appendix C.l - Coefficients for the l-D 206
~4atrix ~4odel
Appendix C.2 - Newton-Raphson Matrix Coefficients
for the 1-0 Model .••..•.••.••..•....•. 214
CHAPTER I
INTRODUCTION
Groundwater has long been one of the world's most important
resources. It accounts for approximately 96% of all fresh water in the
United States and supplies more than 50% of the population with potable
water. Historically, this water source has generally been regarded as
pristine. However, in recent years, contamination of ground water by
industrial products has become a problem of growing concern.
During the past four decades, the variety and quantity of organic
chemicals produced in the U.S. has steadily increased. Currently, more
than 40,000 different organic compounds are being manufactured, trans
ported, used and eventually disposed of in the environment (Wilson, et
!l (1981». Production and consumption of petroleum products has also
risen in this same time period. Many of these industrial compounds
are highly toxic and slightly water soluble. Thus, they pose a poten
tial threat to large volumes of groundwater if they are somehow intro
duced into the subsurface. Increased production of chemicals implies
the increased risk of accidental spills or leakage to the soil, and
indeed, the literature abounds with contamination case histories.
Description:Groundwater has long been one of the world's most important resources. It accounts for approximately 96% of all fresh water in the United States and supplies more than 50% of the population with potable water. Historically, this water source has generally been regarded as pristine. However, in recen
