United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada, OK 74820
Research and Development
EPA/600/S2-89/035 Sept. 1989
&EPA Project Summary
Kinematic Modeling of
Multiphase Solute Transport in
the Vadose Zone
R. J. Charbeneau, J. W. Weaver and V. J. Smith
The goal of this research was the
development of a computationally
efficient simulation model for
multiphase flow of organic hazardous
waste constituents in the shallow soil
environment. Such a model is
appropriate for investigation of fate
and transport of organic chemicals
introduced to the soil through spills
on the ground surface, leakage from
surface impoundments or under-
ground storage tanks, or land treat-
ment of hazardous wastes. During the
initial phases of a site investigation
there usually does not exist sufficient
data to support the application of
comprehensive, computationally ex-
pensive numerical models. Simplified
physically based models which can
address the transport of an organic
constituent experiencing volatiliza-
tion, multiphase partitioning, biodeg-
radation and migration may be
preferred. Two models based on the
kinematic theory of multiphase flow
are developed and presented, along
with a number of illustrative
examples. The Kinematic Oily
Pollutant Transport (KOPT) model
assumes steady infiltration of water
based on the expected annual
infiltration rate; the Kinematic Rainfall
and Oily Pollutant Transport (KROPT)
model includes transient hydrologic
phenomena (evaporation and infiltra-
tion) along with a model of stochastic
generation of rainfall. The examples
presented suggest that the KOPT
model may be preferred for most
applications.
This Project Summary was
developed by EPA's Environmental
Research Laboratory, Ada, Oklahoma,
to announce key findings of the
research project that is fully
documented in a separate report of
the same title (see Project Report
ordering information at back).
Introduction
The focus of this research has been
on the fate and transport of organic
hazardous waste constituents in the
shallow soil environment. Organic
hazardous waste constituents may be
introduced into the soil by spills on the
ground surface, leakage from surface
impoundments or underground storage
tanks, or land treatment of hazardous
wastes. Oftentimes in these circum-
stances, an "oil" phase is present in
addition to the water, air, and soil of the
natural media, and the fate and transport
characteristics are determined by the
mobility of the water and oil, as well as
by multiphase partitioning of the constit-
uent between the air, water, oil and soil
matrix. The emphasis of the research has
been on development of simplified
models for fate and transport of organic
contaminants which may be appropriate
for the initial screening or investigation of
a site when the available data base does
not warrant the application of more
general models.
The primary objectives of this cooper-
ative agreement have been to support
RSKERL in their modeling studies of
subsurface contamination from organic
wastes, including hazardous waste land
treatment operations, and to investigate
the application of kinematic models for
multiphase processes including the flow
of water and oil, as well as multiphase-
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partitioning, volatilization, and bio-
degradation.
The Kinematic Oily Pollutant Transport
(KOPT) and Kinematic Rainfall and Oily
Pollutant Transport (KROPT) programs
are implementations of a kinematic
multiphase transport model that was
developed as a part of this research. The
KOPT program is a simplified
implementation of the kinematic multi-
phase transport model, intended to be
used as a screening tool for hydrocarbon
spills or near-surface releases. It
addresses the questions of how far an oil
release might go into the soil and how
soon it might get there. In the KOPT
implementation, steady water infiltration
is assumed to occur, and the specified
water saturation is taken as represen-
tative of climatic conditions. The KROPT
program is the full implementation of the
kinematic multiphase transport model
and includes modeling of transient
hydrologic phenomena (evaporation and
infiltration), along with a model for the
stochastic generation of rainfall. This
model handles multiple loadings or
releases of oily wastes at or near the
ground surface, multiple rainfall events,
potential oil migration, sorption, volatiliza-
tion, and biodegradation.
Research Background and
Scope
The objective of this research was to
apply kinematic modeling theory to fate
and transport of hydrocarbons in the
vadose zone for oil spills and land
treatment sites. The purpose of the work
was to provide a way to estimate the
gross movement of pollutants in a
multiphase system resulting from spills,
leaks and at land treatment sites. In the
case of land treatment systems, the long-
term behavior of the site is of interest, so
computationally efficient simulations are
needed. This was the reason for using a
simplified approach.
When placed in porous media,
immiscible pollutants retain their unique
properties. Although these pollutants are
largely immiscible in water, they are
capable of causing ground-water contam-
ination by their dissolved constituents.
Immiscibility leads to distinct bodies of
pollutant in close contact with both air
and water, where present. Above the
water table, the addition of an immiscible
pollutant causes the original two-phase
water/air system to become a three-
phase water/pollutant/air system. When
flowing, the transport and fate of
immiscible fluids are governed by a set
of relationships which are, basically,
expanded forms of the single phase
governing equations. The former provides
the theoretical framework for solving
immiscible (or multiphase) flow problems
in general. The specific application of the
theory depends on the initial and
boundary conditions imposed during
pollutant migration as well as other
features unique to pollution incidents.
Currently available analytic and semi-
analytic multiphase flow models share
the limitation that they rely on an
assumed shape of the oil profile that is
not based upon solution of the governing
physically based equations. After an oil
release ends, the oil is gradually replaced
by air as it drains. A drainage profile,
similar to those observed for water is
expected, since the same forces that act
on the water also act on the oil. Other
phenomena outside the scope of these
models are biodegradation, which also
causes variable oil saturations, and
simultaneous unsteady flow of the water.
The latter is typical of actual time-varying
conditions in the field.
Only a few numeric solutions of the
multiphase flow equations have been
presented for the specific problem of
shallow aquifer contamination. A limita-
tion of the numeric approach is that the
models are computationally very inten-
sive. To exploit fully their capacity for
modeling geologic variability, a large
amount of data is required. Because data
is usually sparse, the full capacity of a
numeric model may never be reached
when applied to specific site inves-
tigations.
The analytic and semi-analytic flow
models illustrate that if a constant amount
of water is present in the pore space,
only the mass conservation equation for
the oil needs to be solved. A solution,
that has not been used for this purpose,
is the kinematic model for soil moisture
transport. In addition to a solution for the
duration of water application, this model
gives a solution for the unsteady flow
occurring in soils undergoing drainage
(replacement of soil moisture by air). The
solution is obtained by solving an
approximate governing equation for the
water by the method of characteristics
(MOC). With the proper constitutive
relationships for two-phase flow, the
water solution is analytic. Adapting this
solution for the oil phase gives a more
realistic model than the analytic and
semi-analytic models mentioned above,
since the oil drainage profile is deter-
mined by solution of the governing
equation. An analytic solution is not
possible for the oil because the relative
permeability function is too complicated
and depends on the amount of water in
the pore space. However, a semi-ana
solution for the oil phase can
achieved, allowing relatively low c
putation times to be achieved.
The following assumptions are i
for the model formation. The m<
equations are solved for one-dimensi
flow. This is one of the neces:
assumptions to extend kinematic th
to multiphase flow. Lateral migra
however, reduces the amount
pollutants and water that m
downward. The one-dimensional forn
tion is conservative as it will deten
the maximum amount of substances
will reach a given depth. A
component pollutant is assumed. Th
acts as an inert carrier for the dissc
pollutant. Various transformations, (
than biodegradation, that the oil
undergo are not modeled. Adve<
transport is used to model the tran
of the constituent by oil and w
Equilibrium, linear partitioning rela
ships are used for sorption, dissol
into the water and for volatilization
the air phase. Volatilization is modele
applying the wet zone/dry zone m
To extend kinematic modeling to tr
phase, three more assumptions are i
First, the flow of the air does not im
the flow of the liquids. Second, the
phase transport is dominated by gr
Lastly, the medium is assumed t
strongly wetted so that water will r
in the smallest pores, air in the lai
and oil in the intermediate sized pore
Under the conditions discussed
kinematic model can be applied 1) I
unsteady flow of both the oil am
water and 2) to the fate and transp
the dissolved constituent. If onh
advective motion of the constitue
modeled, the resulting equation;
compatible with the numerical app
taken for the oil equations.
Two separate implementations
created. The first (KOPT) was
kinematic model for the oil mig
only, with a constant amount of
present in the pore space. This me
analogous to the simplified analyti
semi-analytic flow models noted <
but with the capability to sin-
degrading oil and the transport (
dissolved constituent. In related re:
this model was coupled with a me
oil spread along the water table, wl
turn supplied input for a model of
transport in the saturated zone. He
overall model was intended
screening tool for oil spills. A
Guide for the KOPT model is pre
in Appendix A while the listing
code is presented in Appendix B.
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The second implementation (KROPT)
vas a model to investigate the
nteractions between incoming rainfall
and oil moving within the profile. This
model is suited for studying the effect of
•3 series of rainfall events, irrigations and
cultivations on a land treatment site.
There are practical limitations to
ipplication of the KROPT model, and it is
:onsidered a research code at this time.
Discussion
This work has shown that kinematic
heory can be applied to multiphase flow
)roblems. Kinematic models have an
idvantage over traditional numeric mod-
»ls in that the kinematic models are well-
tiited for studying the hyperbolic or
/ave behavior of the multiphase flow
iquations. Inclusion of sharp fronts is
atural to the kinematic models, while
inite difference or finite element models
ncounter difficulties when derivatives in
he governing equations tend to become
ndefined. Solution of the kinematic
lodel shows how the oil and water fronts
nteract. Thus, by simplifying the gov-
rning equations, some of the funda-
lental behavior of the solution is clearly
evealed. The effect of the neglected
apillary pressure gradients are to
crease the infiltration capacity of the
oil and smear the fronts. Much of the
mportant physics is retained, however, in
he kinematic model.
Computationally, by reducing the
nginal system of coupled partial
ifferential equations to a system of
rdinary differential equations, the
ifficulty of finding a numerical solution is
educed greatly. Resulting computer
^odes are well suited to solving problems
where there is a constant amount of
water in the pore space. Such a situation
corresponds to a soil at the so-called field
:apacity and can represent typical
:onditions of the profile. The best per-
ormance of the KOPT model is obtained
vhen the oil is nondegrading and the oil
iharacteristics are straight. Curving char-
ictenstics, due to biodegradation, in-
rease the computational effort.
The implementation of the more
eneral model (KROPT) reveals the
iteractions between the oil and water
onts. The model is limited to situations
'here the initial water profile is uniform.
ince again the best performance is
btained when the oil characteristics are
raight, since the characteristic equa-
Dns are not solved explicitly. Per-
imance is degraded where rainfalls
cceed the kinematic capacity of the soil.
ince the infiltration capacity is infinite
initially, very large water fluxes are
possible until runoff is produced. During
this period small time steps are required.
This modeling work enhances
understanding of multiphase flow through
the results obtained from the example
problems presented. Most important is
the material concerning the oil banks. If
incoming rainfalls displace oil, then the
banks are created. Although this is a
result of simplified modeling, the notion
that incoming water displaces oil into a
bank moving ahead of the water front is a
more general result. In situations where
the flow of the fluids is not kinematic,
banks will also exist. In certain cases, the
banks may be very pronounced, although
short-lived. In others, the banks may
persist for longer periods but would be
virtually undetectable in the laboratory
and the field or by numerical methods
that are not based upon the kinematic
model.
The kinematic model results indicate
that the water displaces all of the mobile
oil. This result was contrary to what was
expected but was established through
conservation of mass. Some displace-
ment of oil would be expected, along with
the creation of an oil bank, but the model
result may or may not be confirmed by
experiment.
The value of the residual oil saturation
is critical for the results obtained from the
model. This parameter could be easily
manipulated to show that oil would
spread over a smaller depth.
Biodegradation of the oil has only a
small effect upon the ultimate depth
attained by the oil. It does affect the
mobility of the oil (reducing it). An
interesting effect on the constituent is
caused by biodegradation of the oil.
Concentrations increase as the oil
degrades because of the loss of solvent.
If the constituent is also biodegradable,
the shorter half life determines if the
concentration will increase or not.
As in field cases where constituent
concentrations in water are measured, a
large mass of constituent may reside in
the oil phase and not be detected. In one
of the examples, concentrations in the oil
are 50 times greater than in the water. If
only the water phase concentrations are
measured, most of mass in the soil is
missed. This is particularly a problem if
the oil is immobilized, since the oil
cannot flow into an observation well
(phenomena akin to end effects may
prevent oil flow into wells even at higher
saturations). The presence of the oil
material is not detected by the
observation wells; the core material itself
must be analyzed.
Using a differential equation solver is
well suited to solving the kinematic
model, since the model equations can be
written as a system of ordinary
differential equations. The speed of the
solver is degraded as more and more
equations are added. For these
situations, it might be advantageous to
solve the method of characteristics
portion of the solution by finite difference
techniques instead of tracking the
characteristics. The overhead associated
with using the Runge-Kutta-Fehlberg
solver may be comparable to that for the
discretization methods. Also, the
necessity to track sets of oil bank
characteristics makes the program ex-
ceedingly complex Extension of the
model may not be practical because of
this.
Conclusions and
Recommendations
The results of this research found that
kinematic theory was successfully ap-
plied to multiphase flow of water and oil.
Kinematic models have an advantage
over traditional numeric models in that
the kinematic models are well-suited foi
studying the hyperbolic or wave behavioi
of the multiphase flow equations.
Inclusion of sharp fronts is natural to the
kinematic models, while finite difference
or finite element models encounter
difficulties when derivatives in the
governing equations tend to become
undefined. The kinematic model simpli-
fies the governing equations revealing
some of the fundamental behavior of the
solution and showing how the oil and
water fronts interact. Solution of the
kinematic model indicates that the
incoming water must displace all of the
mobile oil for mass to be conserved.
Another important result is that oil banks
are formed when the incoming water
displaces the oil that is present. The main
limitation to the kinematic model is that
displacement of oil by water may be
unstable depending on the fluid prop-
erties of the oil and the speed of the
incoming water.
The two computer codes written to
implement two versions of the kinematic
model, KOPT and KROPT, run in a
computationally efficient manner and
may be used to estimate the vertical
migration of oil and a dissolved con-
stituent within the soil profile, including
the effects of volatilization and
biodegradation. Running these codes
showed that the amount of residual oil
saturation strongly affects the total depth
over which the oil will be predicted to
spread.
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In light of the success of kinematic
models for multiphase flow problems,
further development of the KOPT and
KROPT programs is recommended.
Specifically, the following recommenda-
tions are suggested:
• Generalize KOPT and KROPT models
for use in simulation of facilitated
transport. Such an application could
look at the use of solvents (such as
heptane) for flushing a soil profile to
remove low mobility organic
hazardous wastes and then model the
much more rapid volatilization and
degradation of the solvent.
Development and implementation of
user-friendly versions of the KOPT
and KROPT programs.
Expand KROPT program to handle
unstable displacements of oil by
water, thereby eliminating the main
limitation of the model and enabling
the use of randomly generated
rainfalls that may exceed the kine-
matic rate. This program enhancement
would also handle cases where oil is
the displacing fluid, instead of water,
such as for when the oil loading rate
exceeds the kinematic capacity of an
initially nonuniform water profile.
Expand KROPT to handle lay
systems, where the effects of
fluids crossing into media of diff<
intrinsic permeabilities and pore
distributions can have an impo
effect on their migration.
Modify KROPT to have the optk
use other oil relative permea!
models, in addition to the Br<
Corey approach.
Modify KROPT to handle depth
time variability in oil resi
saturations.
R. J. Charbeneau, J. W. Weaver, and V. J. Smith are with the University of Texas
at Austin, Austin, TX 78712.
John E. Matthews is the EPA Project Officer (see below).
The complete report, entitled "Kinematic Modeling of Multiphase Solute
Transport in the Vadose Zone," (Order No. PB 89-207 948/AS; Cos?:
527.95, subject to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Robert S. Kerr Environmental Research Laboratory
U.S. Environmental Protection Agency
Ada, OK 74820
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Official Business
Penalty for Private Use $300
EPA/600/S2-89/035
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