United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada OK 74820
Research and Development
EPA/600/S2-87/101 Feb. 1988
v°/EPA Project Summary
Physics of Immiscible Flow in
Porous Media
J. C. Parker, R. J. Lenhard, and T.Kuppusamy
This project addresses the
conceptual formulation, numerical
implementation and experimental
validation of a model for the
movement of organic chemicals
which are introduced into soils as
nonaqueous phase liquids via
surface spills or leakage from
subsurface containment facilities.
Numerical procedures were
investigated and implemented for
solution of the governing equations
for flow in three phase systems
under the assumption of constant
gas phase pressure, coupled with
the equations for three phase
component transport with equi-
librium phase partitioning. A two
dimensional finite element code was
developed which was used to
evaluate a proposed constitutive
model for properties governing
multiphase flow. The code was also
applied to investigate various
hypothetical problems involving
leakage from underground storage
facilities.
Continuum-based mathematical
models for fluid flow in porous media
require knowledge of relationships
between fluid pressures (P),
saturations (S) and permeabilities (k)
for the fluids and porous media of
concern to enable prediction of
convective velocities and
saturations. A concise parametric
description for S-P relations based
on a scaling procedure formulated to
separate porous medium-
dependent and fluid-dependent
effects has been developed. Relative
permeability-saturation relations are
predicted from the scaled S-P
functional, which is assumed to
reflect the pore size distribution of
the medium. The general form of the
model provides a unified theoretical
framework for the description of
three phase k-S-P relations subject
to arbitrary saturation paths,
including effects of hysteresis and
nonwetting fluid entrapment. Model
calibration is predicated on the
availability of two phase saturation-
pressure measurements alone. Data
requirements may be further reduced
by employing interfacial tension data
to determine scaling coefficients.
Experimental studies were carried
out to validate the constitutive model
for cases involving monotonically
draining water and total liquid
saturation paths. Procedures were
developed for the direct measure-
ment of fluid pressures and
saturations in three phase systems.
Measurements of S-P relations for
static two and three phase systems
provided model validation and
demonstrated the feasibility of using
interfacial tension data to simplify
model calibration. Model validation
under transient flow conditions was
investigated by comparison of
numerically simulated and
experimentally measured results.
Two phase investigations were
conducted involving measurement of
water displacement during constant
pressure NAPL infiltration and
redistribution event. These validation
studies indicated the k-S-P model
provides a reasonable description of
the system behavior under the
imposed experimental conditions and
may be satisfactorily calibrated using
relatively simple procedures.
This Project Summary was
developed by EPA's Robert S. Kerr
Environmental Research Laboratory,
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Ada, OK, 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
Contamination of groundwater by
organic chemicals has become a serious
threat to subsurface water resources.
Among the most widespread and
hazardous contaminants are organic
liquids of low water solubility introduced
into the subsurface environment via
surface spills, leaks from underground
storage facilities, or seepage from
improperly designed or managed landfills
or land disposal operations. Numerous
contamination incidents have been
documented resulting from petroleum
sources. Such contamination problems
generally involve complex mixtures of
multiple organic constituents which may
move in aqueous and nonaqueous liquid
phases and in the gas phase and which
may undergo chemically and biologically
mediated degradation with time.
Modeling of these systems requires
consideration of multiphase fluid flow and
multicomponent transport and reaction
within each phase. The development of
efficient numerical algorithms for such
problems presents many difficulties. An
even more fundamental impediment,
however, has been the substantial lack of
information concerning constitutive
relationships governing multiphase flow
and transport.
These studies focused attention on
the characterization of properties
governing multiphase flow with the
following specific goals:
• develop a parametric model for
hysteretic permeability-saturation-
pressure relationships in three phase
porous media systems,
• develop experimental methods and
specialized apparatii to directly
measure fluid pressures and
saturations in three phase systems
for purposes of model validation and
refinement,
• develop experimental and
computational procedures for routine
model calibration from two phase
system measurements,
• implement a two-dimensional finite
element model for liquid flow and
single component transport in a three
phase system at constant gas
pressure and experimentally validate
flow model for selected saturation
paths.
To model the transport of organic
contaminants introduced into the
subsurface as nonaqueous phase liquids
(NAPLs), it is first necessary to
accurately describe the convective
movement of coexisting fluid phases, i.e.,
water, air and NAPL. Continuum-based
mathematical models for fluid flow in
porous media require knowledge of
relationships between fluid pressures (P),
saturations (S) and permeabilities (k) for
the fluids and porous media of concern
to enable prediction of convective
velocities and saturations. Unfortunately,
such relationships are quite complex in
three phase (air-NAPL-water) systems
and very difficult to measure directly.
Therefore, much effort was addressed to
development of methods to describe k-
S-P relations with maximum accuracy,
while keeping procedures for model
calibration to a minimum to facilitate
practical application to real problems
given reasonable resources. The
approach to this problem was to develop
a concise parametric description for S-
P relations based on a scaling procedure
formulated so as to separate porous
medium-dependent and fluid-
dependent effects. Relative perme-
ability-saturation relations are predicted
from the scaled function which is
regarded as an index of the pore size
distribution of the medium. The
formulation permits calibration to be
performed by measuring only two phase
system S-P relations. Furthermore, it
was found that scaling coefficients may
be estimated with good accuracy from
fluid interfacial tension data, thus
enabling model calibration from only
air-water S-P data and interfacial
measurements. Validation of such
calibration procedures is demonstrated
for static two phase systems and for
transient and three phase systems.
A critical assumption in the k-S-P
model is that two phase S-P data can
be used to predict three phase S-P
relations. Although such assumptions
have been commonly invoked in the field
of petroleum engineering for many years,
no direct validation has been reported
due to a lack of experimental procedures
for measuring three phase S-P
relations. The investigators, therefore,
developed a new experimental device
expressly for the purpose of measuring
three phase S-P relations and have
utilized this apparatus to test the three
phase S-P submodel for cases of
monotonically draining water and total
liquid saturation paths.
In lieu of evaluating the k-S sub-
model by direct measurement of three
phase k-S-P relations, which is a v<
tedious and arduous task, t
investigators validated the model
comparing results of transient fl
experiments with numerical simulatic
based on the parametric mod
Numerical procedures were mvestiga
and implemented for the solution of
governing equations for flow in two £
three phase systems under t
assumption of constant gas pha
pressure in two dimensional spal
domains using finite element methc
which have been verified numerically *,
employed to investigate a number
hypothetical problems. Expenmen
validation of the multiphase flow moi
was achieved first for transient two phc
experiments involving measurement
water displacement by TCE and
benzene-derivative hydrocarbon.
In order to validate the k-S-P moi
for transient flow conditions in thr
phase systems, procedures had to
developed for the measurement of fli
saturations and pressures in three pha
porous media systems. To accompli
this, a dual energy gamma attenuati
device was designed, built and install
in a laboratory dedicated to this functk
The apparatus enables simultaneo
determinations of air, water and NA
saturations to be made with spat
resolution of less than 1 mm on s
columns or flumes. Soil columns we
designed and constructed for use in t
dual energy gamma system and we
equipped with specially f.abricat<
sensors capable of measuring NAPL a
water phase pressures concurrent
Instruments for measuring NAI
pressures were adopted from the thr
phase S-P apparatus design utilizing
special technique for producing stat
hydrophobic sensors by silanization
ceramic tensiometers Experiments we
performed involving simultaneoi
measurement of liquid pressures ai
saturations during transient flow in thr
phase air-NAPL-water systems whii
provided successful validation of tl
three phase k-S-P model and h.
demonstrated the feasibility of employii
model parameters calibrated using or
two phase air-water S-P relations ai
interfacial tension data
The studies mentioned to this poi
have all been constrained to cases
which hysteresis and nonwetting flu
entrapment is ignored or eliminaU
experimentally by considering on
situations involving monotonic water ai
total liquid drainage. In field situatior
effects of hysteresis and especially flu
entrapment may be expected to ha1
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significant effects on flow and mass
ransport. The volume that a NAPL
plume will eventually occupy before
becoming effectively immobilized due to
saturation path reversals will exert a
major influence on the long term
behavior of aqueous and gaseous phase
transport from a contamination event,
and this behavior cannot be modeled
without explicitly considering entrapment
effects on the k-S-P relations.
Previous analyses of hysteresis in three
phase systems have been incapable of
dealing with arbitrary saturation paths
and have generally considered
hysteresis in k-S relations only.
To deal with the problem of hysteresis
and fluid entrapment.the investigators
developed a unified theoretical
framework for the description of three
phase k-S-P relations subject to
arbitrary saturation paths. The theory
extends their previous scaled parametric
model, which it includes as a special
case. Model calibration is still predicated
on the availability of two phase data
alone and requires only slightly more
information than the nonhysteretic model.
The investigators experimentally
investigated calibration procedures for
the hysteretic S-P submodel which also
provided model validation for two phase
main drainage and primary imbibition
paths.
Numerical studies were initiated in the
final stages of the project dealing with
the analysis of coupled three phase flow
and single component transport
controlled by local phase equilibrium.
The mathematical formulation for the
problem was derived and implemented
numerically for a two dimensional spatial
domain.
Discussions and Conclusions
Model for Nonhysteretic
Multiphase Flow
The investigators have presented a
parametric model for relative
permeability saturation-pressure
relationships in porous media which
contain up to three coexisting fluid
phases following monotonic saturation
paths. The model provides simple closed
form expressions for fluid saturation-
pressure functions and their derivatives
and for relative permeability-saturation
functions.
By interpreting the scaled saturation-
capillary head function as an index of the
pore size distribution of the medium,
expressions for the relative perme-
abilities of air, water, and NAPL in a three
phase system are derived which also
degenerate to appropriate functions for
any subset two phase case. No
additional parameters are introduced
over those involved in describing the
saturation-capillary head with the
exception of a gas slippage correction
factor for gas permeabilities. In fact.
several parameters in the saturation-
capillary pressure function do not arise in
the permeability-saturation functions.
Direct verification of the relative
permeability model will require
comparison of model-predicted
permeabilities with experimentally
observed values on two phase and three
phase systems
The proposed parametric model for
constitutive relationships governing
multiphase convection is expected to
provide an adequate representation of
fluid-porous media systems subject to
their meeting certain criteria implied in
deriving the model. One difficulty to
contend with is the assumption of a rigid
porous medium with no significant fluid-
solid phase interactions The data shown
for a clayey soil, albeit of relatively low
swelling potential, indicates that the
proposed scaling procedure may not
deteriorate seriously m fine-grained
materials. However, even rather small
changes in the pore size distribution,
which in certain instances may not be
clearly evident in the saturation-
pressure relations (e g , development of
small fissures due to clay shrinkage in
the presence of low dielectic organic
fluids), may markedly affect the
permeability functions. In some instances
such problems may be a substantial
concern and research is warranted to
investigate methods of dealing with their
effects quantitatively
Another proviso on the model which
should be emphasized is that the
investigators have assumed saturation
paths corresponding to monotonically
decreasing wetting phase saturations in
all instances, and for three phase flow
that total liquid saturation also
monotonically decreases. When these
conditions are not met, hysteresis in the
constitutive properties is expected to be
of importance, particularly when imbibing
and draining saturation-capillary head
curves do not close at zero capillary
head. For example, following an initial
injection of oil into a water-saturated
core, reflooding with water will commonly
fail to achieve full water saturation due to
entrapment of some oil.
Measurement and Estimation of
Static Two Phase Saturation-
Pressure Relations
The format proposed to scale
saturation-capillary pressure relations of
two phase air-water, air-organic and
organic-water porous media systems
was evaluated by applying the procedure
to relations measured for four organic
liquid systems in a sandy porous
medium. Multiphase versions of the
Brooks-Corey and van Genuchten
retention functions were fit to the
experimental data using nonlinear
regression analyses
A procedure was outlined to obtain
parameters of retention functions which
are used to predict saturation-pressure
relations m three phase systems that
preserves a unique pore size distribution
for rigid porous media. In addition, it was
shown that fluid-dependent scaling
factors used to scale saturation-capillary
pressure relations of different two phase
systems in the sandy porous medium
investigated are well approximated from
ratios of interfacial tensions provided the
latter are measuied using fluids subject
to any impurities occurring in the actual
porous medium system. Using scaling
factors from interfacial tension data
permits prediction of saturation-capillary
pressure relations for any two phase fluid
system in a porous medium from direct
measurements of single two phase
system and appropriate interfacial
tension data Three phase system
behavior may likewise be predicted
/Measurement and Estimation of
Three Phase Saturation-
Pressure Relations
An experimental apparatus was
developed to measure three phase air-
oil-water S-h relations in un-
consolidated porous media The
apparatus is also capable of measuring
S-h relations of two phase air-water
and air-oil systems Measurements of
three phase S-h relations were
conducted and compared to two phase
S-h measurements for three porous
media to directly test assumptions
involved m the prediction of three phase
S-h relations from two phase
measurement Good agreement between
total liquid saturations in three phase
air-oil-water systems and oil
saturations of two phase air-oil systems
as functions of air-oil water systems and
oil saturations of two phase air-oil
systems as functions of air-oil capillary
head was observed Close agreement
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was also observed between water
saturation versus oil-water capillary
head relations measured in two and three
phase systems.
Agreement between two and three
phase S-h relationships for monotonic
saturation paths imply that researchers
modeling flow of continuous oil and water
phases in the vadose zone may employ
more readily available two phase S-h
measurements to predict flow
phenomena in three phase air-oil-
water systems if fluid saturations are
assumed to be unique functions of
capillary head (i.e., no hysteresis).
In addition to the proviso that results
have only been presented for
monotonically draining water and total
liquid saturation paths, the procedures
for predicting three phase S-h relations
are expected to be valid only for strongly
water-wet and rigid porous media. Soils
and aquifer materials are generally
expected to meet the criterion of being
strongly water-wet, and at least in
relatively coarse grained materials the
rigid medium constraint may be
sufficiently closely met as well. In finer
textured porous media, the rigid medium
assumption is most prone to cause
deviations from theory and more
extensive investigations for materials with
a wide compositional range is warranted
in the future.
The scaling procedure that we have
advanced combined with a suitable
parametric representation of the scaled
retention function enables development
of a multiphase retention model which
can be used to describe two and three
phase S-h relations. The results
indicate that calibration of the model by
fitting to two phase air-water, air-oil
and oil-water S-h data yield an
accurate description of two and three
phase drainage S-h relations. If two
phase air-oil and oil-water are not
available, model parameters can be
estimated from two phase air-water S-
h data and relatively simple
measurements of air-water, air-oil and
oil-water interfacial tensions with only
moderate deterioration in model
precision.
Finite Element Model for
Multiphase Flow
A finite element formulation based on
the variational approach has been
successfully employed to solve the
coupled immiscible flow problem
involving a three fluid-phase system of
air, water, and NAPL in soil with air at
constant pressure. For two hypothetical
fluid storage tank problems, the leakage
of organic fluids was analyzed and the
NAPL plume movement predicted The
results show the potential of applying the
finite element method developed here to
the design of monitoring systems and
remediation schemes for leaking
underground storage tanks.
Model Validation and
Calibration for Two Phase
Transient Flow
Three methods of calibrating the k-
S-P model developed in previous
chapters were investigated. (1) direct
measurement of S-P relations for air-
water, air-oil and oil-water systems, (2)
direct measurement of air-water S-P
relations with scaling coefficients
estimated from interfacial tension data,
and (3) numerical inversion of transient
oil-water displacement experiments
using a nonlinear optimization procedure
The methods were evaluated by analysis
of results for two NAPL-porous media
systems.
Of the methods investigated for
calibrating the proposed multiphase k-
S-P model, all have yielded reasonably
good descriptions of both static S-P
relations and of transient two phase
displacement experiments, although the
equilibrium-based procedures (Methods
1 and 2) naturally describe S-P data
with greatest accuracy, while the
transient inversion procedure (Method 3)
predicts transient flow behavior more
closely. The substantial agreement
between various calibration methods not
only facilitates the selection of calibration
procedures to ease experimental effort
involved, but provides validation of the
form of the proposed k-S-P model for
the conditions involved in the present
study
Model Validation and
Calibration for Three Phase
Transient Flow
A three phase air-oil-water flow
experiment was designed and conducted
involving simultaneous measurements of
liquid saturations and pressures during
monotonic water and total liquid
drainage. Oil and water saturations were
measured with a dual gamma radiation
apparatus after doping the liquid phases
to enhance separation of gamma
attenuation coefficients. Oil and water
pressures were measured with
hydrophobic (treated) and hydrophilic
(untreated) ceramic tensiometers,
respectively.
Measured water saturation vs c
water capillary head data during the thr
phase flow experiment agreed well w
predictions using the proposed k-S
model calibrated from static air-wa
S-P relations and interfacial tension d,
(Method 1). Measured total liqi
saturation vs air-oil capillary head d,
deviated more severely from the c
water data predictions and exhibit
marked scatter, which was inferred
reflect, in part, errors in expenmen
measurements or perhaps due
deviations from the assumption
constant gas phase pressure Moc
parameters were also evaluated
directly fitting to the observed thr
phase S-P data (Method 2)
Good agreement was found betwe
measured liquid saturation distributions
time and space during the transient fli
experiment and those predicted by t
finite element multiphase flow code usi
the proposed k-S-P model calibrat
using either Method 1 or Method
Measured liquid pressures agreed me
closely with simulations employu
Method 2 parameters Neverthele:
considering the practical advantages
using Method 1 for model calibration, t
accuracy of the simulations based
these estimates was relatively good
Although the experimental regir
considered in the present paper w
limited to monotonically decreasing wa
and total liquid saturation paths, t
methodology is applicable to the study
arbitrary saturation histories Hystere:
and nonwettmg fluid entrapme
associated with complex saturation pat
may substantially affect three phase
S-P relations, and studies of the effe(
of such phenomenal will be necessary
obtain confidence in our ability
evaluate field scale multiphase fk
behavior
Model for Hysteretic
Permeability-Saturation-
Pressure Relations
A model is presented f
permeabihty-saturation-pressu
relationships in porous media syster
containing up to three immiscible fli
phases which accounts for hystere:
and fluid entrapment effects The moc
was formulated by proposing a sine
hysteretic scaled saturation-capilla
head functional which describes wal
saturation vs air-water capillary head
two phase systems and water saturati
vs. oil-water capillary head or total hqi
saturation vs air-oil water capillary he
m three phase systems and can I
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employed to predict fluid permeabilities
via a parametric model. The scaling of
heads is obtained by a linear
transformation and saturations are scaled
by introducing the concept of apparent
fluid saturations which include entrapped
nonwetting fluid contents with those of
the fluid under consideration.
By employing an adaptation of van
Genuchten's retention function to
characterize the pore size distribution of
the system in conjunction with Mualem's
relative permeability model, closed form
expressions for relative permeabilities
are obtained. Calculations for a
hypothetical problem indicate that the
proposed model predicts behavior that is
in accordance with general behavior that
has been reported in the literature for
such systems. It may be noted that other
parametric models could have been
utilized to obtain similar ends,
however,the evaluation of possible
advantages in various various formations
must be left to future investigations.
Detailed measurements of three phase
permeability-saturation-pressure
relations will be necessary to rigorously
validate the model.
Calibration of the proposed hysteretic
k-S-h model was investigated for a
sandy porous medium with p-cymene
as the NAPL Two methods were studied
Method 1 in which model parameters
were fit to a combined set of two phase
air-water, air-oil and oil-water
drainage and imbibition saturation-
capillary head data, and Method 2
utilizing air-water drainage path data,
single imbibition S-h points for air-
water air-oil and oil-water systems and
interfacial tension data. Predicted and
observed drainage and imbibition data
for all three of the two phase systems
were described very closely by Method 1
parameters and only slightly less well by
those using Method 2. This is a very
encouraging result since the use of
Method 2 greatly simplifies model
calibration. The most tedious aspect of
calibration in Method 2 and probably the
most crucial in practice, is the
determination of maximum residual
nonwetting phase saturations. The
development of simpler experimental
procedures or empirical estimation
methods for the latter would be of
substantial value
Model parameters estimated by the
two methods were employed to predict
three phase k-S-h relations during a
hypothetical NAPL contamination
scenario Differences in paths predicted
by the parameters obtained for the two
calibration methods were generally small
again providing encouragement for use
of the simpler procedure. Fluid
entrapment effects were provided to lead
to greater hysteresis in three phase S-h
relations than were evident in the two
phase data. Effects of fluid entrapment
on water relative permeability was not
great for the saturation paths simulated,
while effects on air relative permeability
were large at high total liquid saturations.
The results indicate that hysteresis
and nonwetting fluid entrapment effects
in k-S-h relations of three phase
systems may be quite substantial.
However, since the proposed model is
based on a number of hypotheses
regarding the extension of two phase
behavior to three phase systems,
validation will require direct comparisons
between model predictions and three
phase system behavior.
Numerical Model for Multiphase
Component Transport
A model is described for multiphase
contaminant transport and its numerical
implementation for a simple problem
involving a single contaminant moving
simultaneously in water, gas and
nonaqueous liquid phases is
demonstrated. A computational scheme
involving decoupling of the solutions for
flow and mass transport equations was
implemented and found to provide
satisfactory results. For future extensions
to multicomponent transport, this
procedure is expected to provide very
substantial savings in computer execution
costs. Much work remains to be done
pertaining to the development of
numerical solutions for multiphase
transport to consider transport of
complex NAPL mixtures, constituent
interactions (e.g., cosolvent effects) and
biochemical transformations,
nonequihbrium phase partitioning, gas
phase convection, hysteresis and other
effects, as well as to improve and
develop algorithms and computational
procedures which lead to improved
computational efficiency. Experimental
studies should be pursued concurrently
to further develop and refine constitutive
models for multiphase flow and transport
and to validate numerical codes wtiich
implement these.
J. C. Parker, R. J. Lenhard, and T. Kuppusamy are with Virginia Polytechnic
Institute and State University, Blacksburg. VA 24061.
Thomas Short is the EPA Project Officer (see below).
The complete report, entitled "Physics of Immiscible Flow in Porous Media,"
(Order No. PB 88-131 008/AS; Cost: $19.95) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA2216J
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Robert S. Kerr Environmental Research Laboratory
U.S. Environmental Protection Agency
P.O. Box 1198
Ada, OK 74820
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Environmental Protection Information POSTAGE & FEES PAID
Agency Cincinnati OH 45268 EPA
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