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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