WATER RESOURCESRESEARCH,VOL. 27, NO. 5, PAGES699-708,MAY 1991
NumericalCalculationof EquivalentGrid Block PermeabilityTensors
for HeterogeneousPorousMedia
Louis J. DURLOFSKY
ChevronOil Field ResearchCompany,La Habra, Califi)rnia
A numericalprocedurefor the determination
of equivalentgrid block permeabilitytensorsfor
heterogeneous
porousmedia is presented.The methodentailssolutionof the fine scale pressure
equationsubjectto periodicboundaryconditions
to yield,uponappropriate
averagingof the finescale
velocityfield, the coarsescaleor equivalentgridblockpermeability.Whenthe regionover whichthis
coarsescale permeabilityis computedconstitutesa representativeelementaryvolume (REV), the
resultingequivalent permeabilitymay be interpretedas the effectivepermeabilityof the region.
Solutionof the pressureequationon the fine scaleis accomplished
throughthe applicationof an
accurate triangle-basedfinite element numericalprocedure,which allows for the modeling of
geometricallycomplexfeatures.The specification
of periodicboundaryconditionsis shownto yield
symmetric, positive definite equivalent permeabilitytensorsin all cases. The method is verified
throughapplicationto a periodicmodelproblemand is thenappliedto the scaleup of areal and cross
sectionswith fractally generatedpermeabilityfields.The applicabilityandlimitationsof the methodfor
these more general heterogeneityfields are discussed.
The determinationof the effective propertiesof heterogeneousporousmediahasbeen a problemof interestfor many
It is well establishedthat fine scale heterogeneitiesexist in years.The followingdiscussionof previouswork is by no
subsurface
formations, and methodsto approximatein detail means exhaustive; for more comprehensive literature rethe spatial distribution and variability of rock properties views, see Hewett and Behrens [1988] or Desbarats [!987].
suchas porosity and permeability are continually being Warren and Price [1961] considered three-dimensionalsysdeveloped
andimproved.It is not yet known,howeyer,how temscomprising
randompermeabilitieswith no spatialstrucbestto incorporatethis fine scale data into large scaleflow ture. They demonstratednumericallythat the geometric
simulationmodels. Specifically, averaging techniquesare mean providesa reasonableaverage for permeabilityfor
neededto scaleup the fine scalepermeabilitiesto the larger such systems.Dagan [1979] applied a self-consistentapscalesappropriatefor flow simulationand engineeringcal- proachto deriveboundson the effectivepermeabilityof
1.
INTRODUCTION
culations.
If the scale over which this "averaged" permeability is
isotropicsystems.His resultsprovide narrowerboundsthan
thosepreviouslyavailable.Beggand King [ 1985]considered
the effectivepermeabilityof regionswith discontinuous
flow
definedis large relative to the scaleof heterogeneitywithin
the porous medium, then the large scale permeability is
referredto as an effective permeability. This effectiveper-
barriers (i.e., low permeability shale streaks). They pre-
meability
is a propertyof the mediumanddoesnotvarywith
shale sizes and shale volume fractions. Gelhar and Axness
theflow conditionsto which the mediumis subjected.If the
largerscaledoesnot encompass
all the scalesof variationof
the permeabilityfield, by contrast, then the "averaged"
permeability
is referredto as an equivalentpermeability.If
theregionfor whichthe equivalentpermeabilityis computed
ischosento coincidewith a simulationgrid block, thenthis
permeability
is the appropriate"equivalentgrid blockpermeability"for use in a flow simulation.Unlike effective
sentedexpressions
for the effectivepermeabilityin termsof
[1983]applieda perturbationtechniqueto the determination
of effectivepermeabilityin porousmedia with statistically
anisotropicpermeabilityfields describedby a particular
covariancefunction. Their results are limited to small permeability variances,but indicate some general effects of
spatialcorrelation.Desbarats[1987]consideredsand-shale
sequences
with bothcorrelatedanduncorrelatedpermeability fields.His resultsextendthoseof Beggand King. Kasap
permeability,equivalentpermeabilityis not a constantprop- and Lake [1989] proposedan analytical techniquefor the
ertyof the medium.Rather, somevariationin the equivalent determination
of effectivepermeabilitythat is able to handle
permeability
under differentflow conditionsis typically locally tensorialpermeabilities.Their method essentially
expected.
provides
theabilityto calculate
thearithmetic
andharmonic
The purposeof this paper is to presenta numerical meansof layeredsystemscomprisingfull tensorpermeabil-
procedure
for computingthe effectiveor equivalentperme-
ities.
abilityof a region of a formationgiven the fine scale
None of the studies discussedabove provides a general
permeability
distributionwithin the region.The determina- framework for the numerical calculation of effective or
tionof the effectiveor equivalentpermeability
of heteroge- equivalent
permeability;
theyare all moreor lessapplicable
neous
rockregionsis importantin flow simulations
because to certainsituations.Specifically,all containeither implicit
thescaling
upprocess
allowsfor theuseof fewersimulation assumptions
regarding
the spatialdistribution
or correlation
gridblocks,resultingin computational
savings.
of permeability
heterogeneity
or areformulated
particularly
for a certaintypeof sequence(e.g., sand-shale)
or geometry
Copyright
1991by theAmericanGeophysical
Union.
(e.g., layeredsystems).Some attemptsto developmore
Papernumber91WR00107.
general
approaches
for the calculation
of effectiveor equiv0043-1397/91/91WR-00107505.00
699
700
DURLOFSKY:
CAI.CULATION
OF EQUIVALENT
GRID BLOCKPERMEABILITY
Blent permeability have recently been made. It is well
recognizedthat the boundaryconditionsimposedon the fine
scaleproblem, which must be solvedto determineeffective
permeability on the coarse scale, can significantlyinfluence
the coarse scale permeability results. Becausethe precise
flow field to which a region of the mediumwill be subjected
is not in general known a priori, the appropriateboundary
conditionsto be imposedon the fine scaleproblemcannotbe
specifieduniquely.Thus, attemptshave been madeto specify theseboundaryconditionsas generallyas possible,in the
hopethat the resultingeffectiveor equivalentpermeabilities
will predict flow phenomenain near agreementwith those
which would be predictedfrom a fine scalesimulationunder
a variety of different conditions.
proach,
similartothatdescribed
by Saezet al. [1989].Some
discussion
of the applicability
of the approach
to more
generalheterogeneity
fieldsis also presented.
The finite
elementnumerical
solutionprocedure
is thendescribed
in
somedetail.Examples
verifyingthemethodanddemonstrating its advantages
over more simplisticapproaches
are
presented
in section3. Then,the applicability
of themethod
to the scaleup of moregeneralsystems,fractallygenerated
arealandcrosssections,
will beassessed.
Furtherdiscussion
of the advantages
andlimitationsof the methodconstitutes
section 4.
2.
CALCULATIONOF EFFECTIVE PERMEABILITY
The generalnumericalprocedurefor computingtheeffecWhite and Horne [ 1987]presenteda numericaltechnique
for the determinationof equivalentpermeabilitytensorsthat tiveor equivalent
gridblockpermeability
of a heterogeneous
entails the use of several different sets of boundary condi- regionof a porousmedium is describedin this section.The
tions and the subsequentaveragingof the coarse scale methodis basedon a two-scaleapproachand is therefore
permeability results. This method is, however, limited in
that the resulting(averaged)equivalentpermeabilityis not in
generala symmetrictensor(the requirementthat permeability be a symmetric tensor will be discussedin section 2).
Further, becausetheir techniqueis basedon a finite difference method, it is not sufficientlygeneralto treat geometrically complex cross-bedding.
Bourgeat [1984] applied homogenizationtheory to the
determinationof effectivepermeabilityand effectiverelative
permeability.This approachis strictly applicableonly for
rigorouslyapplicableonly to somecases;e.g., periodic
or
nearly periodic systems or regions in which the scaleof
heterogeneityis small comparedto the scale over whichthe
effective permeability is computed. However, for reasons
discussed
in section2.2, it can be expectedto be somewhat
applicableto morege•ral systems,as is in fact verifiedin
section3. The developmentin this paper is for a twodimensional system. Extension to the three-dimensional
case will be discussed below.
periodicsystems,though,withinthisframework,it provides 2.1. GoverningEquationsand the Two-ScaleApproach
a means to compute effective permeability for complex
Single-phase,
incompressible
flow througha heterogesystemsin a consistent
manner,alwaysyieldinga symmetric neousporousmedium,in the absenceof sourcesandsinks,
effective permeabilitytensor. Though it is a powerful is describedby Darcy's law and continuity:
methodfor the determination
of effectivepermeability,this
methodhas,to date,onlybeenapplied
to extremely
simple
1
u= ---k
- Vp,
(1)
periodicheterogeneity
fields.Its applicabilityto moregenera[heterogeneity
fieldshasnotbeenassessed,
evenempirically. Bourgeat'stheoreticalapproachhas recentlybeen
V- u = O,
(2)
extendedby Saezet al. [1989]andMei andAuriault[1989].
In a related effort, Kitanidis [1990]appliedgeneralized whereu is the localfluidvelocityvector,p the localpressure
Taylor dispersion theory to the calculation of effective (pressurecan be modified in the usual way to include
permeability.This approachyields equationsidenticalto gravitationalpotential),tx the fluid viscosityand k the local
thosethatresultfromthe homogenization
methodology. permeability tensor. In two dimensions, in the x-z coordiThe mathematicallyanalogous
problemof the determina- nate system, k is represented as
tionof theeffective
therm',d
conductivity
of a heterogeneous
materialhas been treatedby a varietyof investigators.
k=kkzx
kz .
(3)
Periodic,two-dimensional
systems
werestudied
by, among
others,Sanganiand Yao [1988]andDurandand Ungar Certainpropertiescan be ascribedto k. From a macroscopic
[1988].Theseformulations
are applicable
to the determina- pointof view [Bear, 1972]k can be shownto be symmetric
tion of effectivepermeabilityof heterogeneous
media,
(kxz= kzx)andpositive
definite
(kxxkzz
> k]z,kxx> 0,kzz
thoughneitherof the numerical
solution
procedures
is ap- > 0). These results can also be derived frdm Stokes flow
plicableto the heterogeneous
systemswe wishto consider.
considerations
[Durlofsk7and Brady, 1987].
Our intenthereis to developandimplement
a numerical To proceed with our two-scale formulation, we must
methodfor the determination
of the effectiveor equivalent identifytwo distinctscalesof permeabilityvariation,a fine
permeabilityof heterogeneous
porousmedia. The method scale y with fast variation and a coarse scale x with slow
entailssolution
ofthefinescalepressure
equation
subject
to variation.This nomenclatureis that of Saez et al. [1989].The
periodicboundaryconditionsand will be seento overcome
simplestexampleof sucha mediumis a spatiallyperiodic
someof the limitations
of previousmethods
in thatit (1) is medium,depictedschematically
in Figure !. In thiscasek
ableto modelcomplex
geometries,
suchasthosetypicalof variesonly in y. Now, combiningDarcy's law with incomcross-bedded
strataand (2) handlesboundaryconditions pressibilityyields the one-phasepressureequationfor
consistently,
alwaysyielding
symmetric
effective
permeabil- steady,incompressibleflow (all variables from here on are
ities.Towardthisgoal,the paperproceeds
as follows.In considereddimensionless):
section2, the problemto be solvedfor the determination
of
effectivepermeability
is formulated
via a two-scale
ap-
V-[k(x, y). Vp] = O,
{4)
DURLOFSKY:
CALCULATION
OFEQUIVALENT
GRIDBLOCKPERMEABILITY
subject
to prescribed
pressure
andfluxdataat thedomain
701
n2
boundaries.
We havewrittenk ask(x, y) to emphasize
thatk
varieson both scales.The intenthereis to replace(4) with a
homogenized
(averaged)
equationcharacterized
by aneffective permeability,designatedk*, that variesonly on the
coarsescalex; i.e.,
%. [k*(x) - V.p] = 0,
(5)
wherethe x subscripton the gradient operator indicatesthat
it operates
on the x scale.In the idealizedcasewherethe
• n4
n34-----
mediumis periodic,k doesnot vary in x and (5) simplifiesto
k*: VxVxp = 0.
(6)
Anequationof the form of (5) representsa greatsimplification over (4) because, in solving (5) numerically, the com-
putational
gridneednotresolvedetailsonthey scale,which
clearlyresultsin considerablecomputationalsavings.
That the simplification accomplished from (4) to (5) is
nl
c•
D1
possible,
and that the resultinghomogenized
equationis
indeedof the form of (5), was rigorously demonstratedby
Fig. 2.
Unit cell for effective permeability calculation.
Bourgeat[1984],Saezet at. [1989],Mei and Aurialdt[1989]
and Kitanidis [1990] in the context of porous media flow.
This in itself is a significant finding, as the homogenized
versionsof variable coefficient partial differential equations
arenot necessarilyof the same form as the original equation
(e.g.,the homogenizedversion of the Stokesequationsfor
flow in dilute porous media is the Brinkman equation).
Further,as shown by Mei and Auriault, k* is itself symmetricandpositivedefinite,just as is the fine scalepermeability.
All these derivations, however, are for periodic systems.
Theyfurther require that the ratio of the microlengthscaleto
the macrolengthscale be much less than unity and that the
regionin question be far removed from boundariesand
sources and sinks.
2.2.
Effective Permeability.'of Periodic Systems
We now consider the determination of the effective permeability of a porousmedium comprisingperiodically distributed heterogeneities,as illustrated in Figure 1. If large
scaleflow is to occur, pressurevariation on the large scale
(x) must exist. Then, on the x scale, pressure can be
expressedlocally as
P=P0 +G
.(x-x0),
(7)
wherex0 representsthe centerof the regionunderstudy,p 0
is the pressureat x0 andG(x0) is the localpressuregradient,
on the scaleof x, at x0. The actual numericalvaluesof the
componentsof G are arbitrary. We wish to relate the average
flow through a periodic cell (i.e., the average over the y
scale), designated(u), to this x scale pressuregradient. The
proportionalityconstantis the effective permeabilityk*,
defined via
(u)=-k*-G.
(8)
To determinek*, we need to solve the pressure equation
over the unit cell, i.e.,
Vy. [k(y). Vyp]= 0,
(9)
where the y subscripton the gradient operator indicates
operationon the y (unit cell) scale, and then computethe
velocity (u) (definedexplicitly below) throughthe cell. The
boundaryconditionsfor (9) are derivedfrom the periodicity
of the systemand the impositionof the pressuregradientG.
Thesespecifications
requirethat both the pressurefield and
the local flux, or equivalently the local velocity field, be
themselvesperiodic over the unit cell. However, becausea
pressuregradientis imposedon the system,the pressure
field undergoesa jump 'from one boundary to the corre-
sponding
boundaryof magnitude]n ßG[!, wheren is the
outwardpointingnormalat either of the boundariesandl is
the distanceseparatingthe two boundaries.
x1
The boundaryconditionscan be specifiedexplicitly with
reference
to Figure 2. The unit cell is here a squareof side
Fig.1. Spatially
periodic
porous
medium.
Here,k variesrapidly
length 1. We resolveG into its two componentsin the y
in y and slowly (or not at all) in x.
702
DURLOFSKY:CALCULATIONOF EQUIVALENTGRID BLOCKPERMEABILITY
coordinatesystem;G = G•il + G2i2, whereil andi2 are the tionof homogenization
theory.
In thehomogenization
apunit coordinate directions. To compute the full effective proaches,
theequations
whichmustbe solved
aresimilar
to
permeabilitytensor, two problemsmust be solved. In the (9)subject
to(10).However,
instead
of specifying
ajumpin
first, we take G2 -- 0. Now boundaryspecifications
are as the pressurelike
variable(referredto as S by Mei and
follows:
Auriault[1989])fromoneboundary
to another,
theequation
(10a)
P(Yl, Y2 = O)= P(Yl, Y2= 1)
on ODi and OD2,
u(yl, y2=0)'
n l=-u(y•,y2=
1). n2
on ODi
(10b)
and OD2,
P(Yl = 0, Y2)= P(Yl = 1, Y2)- G1
(10c)
isdriven
byasource
termoftheformV. k. Strictperiodicity
(nojump)is imposed
ontheS field.The effectivepermeability isthencomputed
by anappropriate
volumeaverage,
over
they scale,of thevelocityfield,in a procedure
analogous
to
that specifiedby (11).
Thedevelopment
aboveis strictlyvalidonlyfor mediafor
whichtwo distinctlengthscalesof heterogeneity
existorin
moregenerallyheterogeneous
porousmediawhenthe scale
overwhichtheeffective
permeability
is computed
is large
relativeto the correlationscaleof heterogeneitywithinthe
on OD3 and OD4,
u(y 1 = 0, Y2)' n3 = --u(Yl = 1, y2)- n4
(10d)
on 0D 3 and OD4.
Note that the actual values of pressure and flux are not
specifiedat any point on the domain boundaries.Rather,
correspondencesbetween pressuresand fluxes on opposite
boundaries are stated. With these boundary conditions,
alongwith the specificationof pressureat any singlepoint
insidethe domain,the problemis well posedandthe solution
is unique.
Upon solutionof (9) subjectto (10), the averagevelocity
through the unit cell can be determined as follows:
{Ul)
=-•0ußn3dy2,
(11a)
(u2)=-/o
u-
(11b)
D3
Dt
porous
medium(i.e., theeffectivepermeability
is computed
for a representativeelementaryvolume or REV). Some
natural porous media do in fact approximate two-scale
behavior.Examplesincludemany cross-bedded
systems,
whichcontainnearlyregular,thoughgeometrically
complex,
layerings.For the moregeneralcaseof a porousmediumfor
whichneitherof thetwo criteriaaboveis satisfied,
however,
the boundaryconditionsthat must be imposedon the fine
scaleproblemto determinethe equivalentgridblockpermeabilityof the regioncan no longerbe specifiedunambiguously, as they result from the local flow field, which is not
generallyknowna priori. However, theseboundaryconditionsmuststillbe formulatedso as to assurethe symmetry
and positive definitenessof the resulting equivalentgrid
blockpermeabilitytensors.Thesetwo propertiesare essential if the resultingequivalentpermeabilitytensoris to have
any physical meaning. Only if the permeability is everywhere symmetricare its eigenvalues(i.e., principalvalues)
guaranteedto be real; otherwise complex principalvalues
canresult. Positivedefinitenessof the equivalentgridblock
permeabilitytensorassuresthat energyis alwaysdissipated
duringflow; violationof this can yield flow "up" a pressure
Note that, dueto the periodicity,the firstintegralcouldalso gradient.
be evaluatedalongOD4 and the secondalongOD2 (aftera
Boundaryconditionson the fine scale problemmust,
changein sign),with identicalresults.Writing out the two therefore, be specifiedin a manner that maintainsthe symcomponentsof (8) givesexplicitexpressions
for k* in terms
metry and positivedefinitenessof k* (equivalentpermeabilof (ul) and (u2), as follows:
ity is also designatedas k*). It is also desirableto specify
these
boundaryconditionsin as generala way as possible.
(ul) = -(kTiG• + k•2G2),
(12a)
To accomplishthis, considerthe pressureand flow fieldsin
a heterogeneous
porousmedium, away from boundariesand
(u2) = -(k•lG 1+ k•2G2).
awayfrom sourcesor sinks.The local pressurefield, onthe
Now, becauseG2 = 0, k•l andk•l are easilydetermined coarse scale, may often be reasonably approximatedas
from knowledgeof G1 and(u).
linearover the regionin question.In suchcases,the boundTo determinethe remainingtwo componentsof k*, a ary conditionsfor pressurespecifiedin (10) will approximate
secondproblemmustbe solved.By specifying
G• = 0 and the localbehaviorof the pressurefield. The periodicity
of
G2 •: 0, solvingthe pressure
equation(theboundary
condi- flux specifiedin (10) can no longerbe expectedto hold
tionsin thiscasedifferfromthosein (10)in thatthejumpin exactly.To computek* consistently,however,we stillmust
pressure
is between
ODi andOD2)andevaluating
(u), k•2 requirethatthetotalfluxesthroughopposite
boundaries
of
andk•2 canbe determined
analogously.
In all cases,the theregionbe equalandopposite.Otherwise,calculation
of
resulting
values
fork•2 andk•i areidentical,
asrequired
by k* becomesambiguous.It is not sufficientmathematically,
thesymmetry
of thepermeability
tensor,andk* is positive however,to simplyspecifythat the total fluxesthrough
definite.Both of thesepropertiesresultfrom the solutionof oppositeboundaries
be equal and opposite;rather,flux
the well-posedboundaryvalueproblem(9) andthe consis- relationships
at eachpointmustbe specified.In theabsence
tent handlingof the boundaryconditions(10).
of anya prioriknowledge
of the globalflowfield,specifica-
Wehavenowcompleted
thespecification
oftheboundary tion of periodicityof fluxesis a reasonablemeansto accomplish this. We note that the use of periodicboundary
Thoughthe resulting
equations
areof a differentform,this conditions
is common
for a varietyof problems
involving
formulation
is analogous
to thatresulting
fromtheapplica- effectivemedia calculationsand flow simulationsevenwhen
value problem to be solved for the determination of k*.
DURLOFSKY:
CALCULATION
OF EQUIVALENT
GRID BLOCKPERMEABILITY
703
thesystem
is notstrictlyperiodic.Foranexample
oftheuse
In the nonconformingfinite element method, the starting
ofperiodic
boundary
conditions
in anapplication
involving point for the numericalprocedureis the pressureequation
flowthroughporousmedia,seeDurlofst,3•
andBrady[1987].
Fromthe abovediscussion,
the useof periodicboundary
conditions
in the calculationof equivalentgrid blockpermeabilitiesappearsreasonablein somecases.This boundary
(9) approximated over the unit cell. The solution domain is
discretizedinto a set of M triangular elements. Pressureis
linearly approximatedover each element via the following
expansion:
specification
cannotbe expected
to be completely
general,
however,as the equivalentgrid block permeabilityis not an
intrinsicpropertyof the medium.It may lead to an inaccuratedescriptionof the large scaleflow in regionswhere the
N
P(Yl,3'2)=Z ¾i•i(Yl,Y2),
(13)
i=1
pressure
fieldvariesespecially
rapidly(e.g.,in regions
with
highheterogeneity
relativeto neighboring
regions).Never- where N is the total number of element edgesand y• are the
theless,the specificationof periodicity avoids the need to
(unknown) pressuresat the edge midpoints. The noncon-
specify
a particular
pressure
field(ora sequence
ofparticular forming(I)i basisfunctionsare definedlinearlyover each
pressure
fields),assures
the symmetry
andpositivedefinite- element
suchthat•i = 1 at the midpointof edgei andzero
nessof the resulting equivalent permeability tensor and
at the midpointsof the other two edgesof the element (these
yields
thecorrecteffective
permeability
tensorin cases
when elements are referred to as P1 nonconforming triangular
(1)twodistinctscalesof heterogeneityexistor (2) the region elements).Becausethesebasisfunctions interpolate through
for whichk* is computedis a valid REV. The applicability edge midpoints,rather than through the triangle vertices as
and limitations of these boundary conditions for the calculationof equivalentpermeability will be further discussedin
section 4.
in standard finite element methods, the pressure is not
continuousat all points along the edges between adjacent
elements(thus "nonconforming" finite elements). However,
We now turn to a discussionof the numericalprocedure
the fluxesthroughthe element edges, obtained via differenusedto solve the pressure equation over the unit cell (9)
tiation of the pressurefield at the edge midpoint and applisubjectto boundaryconditions(10).
cation of Darcy's law, are continuous between adjacent
edges. Further, this differentiation does not degrade the
2.3.
Numerical Solution Procedure
accuracy of the resulting fluxes relative to the pressure
solution.
A variety of numerical procedurescould be used to solve
To be specific, the finite element procedure based on
thepressureequation(9) subjectto periodicboundaryconlinear
nonconformingbasisfunctions yields a second-order
ditions(10). However, because we are interested in systems
accurate
solution for the pressure field; i.e., pressureis
that can involve complex cross-bedding,it is essentialthat
whereh is the lengthof a typicalelement
the numerical method be capable of resolving complex O(h2) accurate,
geometry.This suggeststhe use of a finite elementmethod. edge. A standard Galerkin finite element procedure with
Equations(9) and (10) indicate that the determinationof k* linear basis functions also yields a second-order accurate
involvesintegrationof fluxes throughthe boundariesof the pressuresolution,but the fluxesobtainedvia differentiation
unit cell. Thus, accurate fluxes are required. Mixed finite of this pressuresolution are only first-order accurate. For
elementmethodsare, therefore, well suitedfor this problem, triangle-based,nonconformingfinite elementmethods,howasthey(1) allow for quitegeneralgeometriesand(2) provide ever, the fluxes through the element edges, computedby
fluxesto a higher degree of accuracythan standardfinite differentiationof the pressure solution, do not degrade in
elementmethods.Another family of finite elementmethods, accuracy. Further, Marini [1985] demonstratedthat the
so-called"nonconforming"finite elements,are also appli- fluxesobtainedfrom the triangle-basednonconformingfinite
cableto this problem.This is becausethesemethodsyield,
element method are identical
undersome circumstances,fluxes identical to those of the
morecostlylowestordermixedfinite elementmethod.Thus,
solutionswith the accuracy of the mixed method can be
obtained via a less costly nonconforming finite element
mixedfiniteelementmethod(this pointwill be discussed
in
moredetail below). In addition, the nonconformingfinite
elementmethodusedin this study is more computationally
efficientthan the analogousmixed method becausethe
mixedmethodrequiresmoreunknownsandresultsin a set
of linearequationsthat is lessamenableto iterativesolution
techniques
than in the equationset resultingfrom the nonconformingfinite element discretization.Becausethey assurecontinuous
fluxesbetweenadjacentelements,boththe
mixedandnonconforming
finiteelementmethodscaneasily
to those obtained
from the
method. For this reason, we proceed with the development
of the nonconformingfinite element method.
The residualequationsfor the nonconformingfinite element methodare formed by weighting the pressureequation
withtheß k basisfunctions
andintegrating
overtheunitcell
D:
[v.(k.vv)]as
=0.
handle
highlydiscontinuous
permeability
fields,whichcan
cause
problemsfor standardfiniteelementprocedures.
Both
(14)
ofthesemethods
wereappliedto the solutionof (9) subject Integratingby parts and applying the divergencetheorem
to (10). However, becausethe formulationbasedon noncon-
forming
finiteelements
is morestraightforward
andappears
superior,
only its applicationto the problemat handis
described
in detail. For backgroundon both mixedand
nonconforming
finite element methods,see Thomasset
[198•1.
gives
(kßVp).
V•kdA+
•o(ußn)•
t'dl=0,
D
(15)
704
DURkOFSKY:
CALCU1LATION
OFEQUIVALENT
GradBLOCK
PERMEABILITY
equationset wouldbe costlyfor large
wherewe havemadeuseof Darcy'slaw in theintegralover of the resulting
OD.Now, introducing
the expansion
for p (equation
(13)) problems;for simplegeometries,low ordermixedmethods
usingrectangularblock elementswould probablybe more
gives
computationally
efficient.
This completesthe descriptionof our effectivepermeabil.
ity calculation. We now present some numericalresults
which displaythe capabilitiesof the methods.
'YifD(k'•7•i)'
VcI)kdA+fo
(u-n)•tCdl=0,
D
(16)
wherethe appearance
of a repeated
indexindicates
summation over that index.
3.
NUMERICAL
RESULTS
In this sectionwe present resultsfor the calculationof the
Theimposition
of periodic
boundary
conditions
requires effective and equivalent permeabilities of heterogeneous
thatthe equations
for all •'i fallingon ODbe replaced
by regions. The first example, a model problem, servesto
periodicity
conditions,
i.e., correspondences
betweenpres- demonstratethe advantagesof the method comparedto
sureandfluxon opposite
boundaries.
The correspondence
more simplistic approaches. The next examples involve
between
pressure
on opposite
boundaries
is specified
via calculationsfor fractally generatedareal and crosssections.
equations
relating
thepressures
onthetwoboundaries
tothe
imposed
pressure
gradient.
Forexample,
relationships
ofthe 3.1. Model Problem
form
The examplepresentedin this sectionis a verificationof
the method for a model problem that can be solvedanalytically. Thoughthis exampleis quite simple,the geometryis
whereYaand•,• areedgesopposite
oneanother
onbound- somewhatrepresentativeof somecross-beddedsystems.At
ariesOD4 andOD3respectively
(seeFigure2) andG1 is the the same time, the results of an overly simplistic approach,
imposed
pressure
gradient
in they• direction,
arespecified. which gives results in substantialdisagreementwith the
Similarequations
relateoppositeedgeson OD• and OD2 to analytical solution, are presented and discussed.We first
G,. The flux relationshipsspecifythat the flux into D at an considerthis simplifiedapproach,and then definethe examelementedge on one boundary(say OD3) is equal and ple problemand presentnumericalresults.
opposite
to thefluxthroughthecorresponding
elementedge
The most simple numericalprocedurefor the determinaon the oppositeboundary(0D4). Thus, theserelationships tion of effectiveor equivalentpermeability, which hasbeen
are of the form
and continues to be used, involves the solution of the
Ya- •/0 = G•,
(17)
--
In ß (k . Vp)] a+[n
-(k ß Vp)] 0=0,
(18)
pressureequation(9) subjectto the constantpressure
andno
flux boundary conditions:
whereagaina and b indicateedgeson oppositeboundaries.
p = 0 on OD3,
(19a)
By writing the Vp termsout via the expansion(13), (18) can
be written in terms of Tip = 1 on OD4,
(19b)
The equationsetto be solvedis definedby (16)(applicable
for edgemidpointsnot lying on OD) andby (17) and (18) for
u ß n•=0
on OD•,
(19c)
boundaryedges.Upon specificationof a value for pressure
u ß n2 = 0 on OD2.
(19d)
at one point in the domain,the equationset can be solved,
fluxes can be evaluated through differentiationof (13) and
Uponsolution,thefluxthrougheitherof they 1 facesofthe
effectivepermeabilitycan be computedas describedin
section2.2. The equation set is of a form that is very similar
to that of standard(conforming)finite elementmethods.The
matrix is sparseand is banded for regular discretizations.
Additional couplingsdue to the periodic boundary conditions,which relateterms on oppositeboundaries,complicate
the structure and render the matrix nonsymmetric. Solution
can be accomplished through either direct or iterative techniques. The iterative method used for solution here is a
GMRES (generalized minimal residual) solution technique
[Saad and Schultz, 1986] with a diagonal scalingpreconditioner. Other preconditionersand/or solvers were not tried,
though it is possible that faster convergence could be
achieved with a different iterative
method.
Extension of the approach described in this section to
three-dimensionalsystems is reasonably straightforward
conceptually. The governing equations developed in sections 2.1 and 2.2 generalize immediately. The numerical
approachused in three dimensionswould be analogousto
that describedhere. Three-dimensionalspacecould be discretized quite generally via tetrahedra; the nonconforming
finite element basis functions for tetrahedral
elements are
analogousto those for trianglesin two dimensions.Solution
regioniscomputed
andpermeability
is calculated
asin(12a),
yielding,for G1 = 1, k•l = -(ul). An analogous
procedure
givesk'•2. Note that the crosstermsof k* cannotbe
determinedthroughthis procedure,as no flux boundary
conditions
are prescribed
on the boundaries
with normal
perpendicular
totheimposed
pressure
gradient.
Though
this
limitationis obvious,the approachcontinues
to be used
underthetacitassumption
thatthediagonal
termsofthek*
computedin thismannerare correctandthatthe crossterms
are unimportant.
Consider
thelayeredmedium
depicted
in Figure3. This
mediumis considered
to be infiniteand periodic;therefore
k* doesnotvaryin x. Thepermeabilities
of thetworegions,
designatedk l and k 2, are as follows:
k•= 2 '
k2=
0.0! '
Thethickness
ofregion
1istakentobe7 times
thatofregion
2. Forthisperiodic
system,
theeffective
permeability
is
DURLOFSKY:
CALCULATION
OFEQUIVALENT
Grad BLOCKPERMEABILITY
easilycalculated
throughweightedarithmetic(k•) and
harmonic
(k'22)meansas
k.=(1.875
0)
0
0.0772947
705
x2
(21)
'
Region2
Thisresultis recoverednumericallyby both the solutionof
•
(9)subject
to (10) (periodicboundary
conditions)
andby
solutionof (9) subjectto (19) (pressure-fluxboundaryconditions).The unit cell for these calculations,designated
by
thedashedbox, is shownin Figure 3.
Region
1
We now rotate the layered system relative to the coordi45"
nateaxesan angle of -45 ø, as shownin Figure 4, where the
unitcellis againdesignatedby the dashedbox. The effective Fig. 4. Laminated porousmedium with laminationsnot aligned
permeability
isnowtheresultpresented
in (21)rotated-45 ø,
with coordinate axes. Dashed box indicates unit cell.
accomplished
throughthe use of a tensorrotation[Bear,
1972]:
strictly valid. We now turn to more general permeability
fields, with permeabilityvariation on all scales. For such
-0.898853 0.976147,]
systems,.theconceptof effective permeability is no longer
This result is recovered exactly through solution of (9) applicable;rather, we computeequivalentgrid block permesubjectto (10), i.e., through use of periodic boundary abilities.The applicabilityof the method for these casesis
conditions.Pressure-flux boundary conditions (19), how- not known a priori. Therefore, to verify the scale up procedurefor thesecases,we shall comparefine and coarsescale
ever,give a very different result:
numericalresults. The level of agreementattained between
0.976147
-0.898853
h
(22)
k* =
0.193757
0)
0
0.193757
'
(23)
Thus, it is apparentthat the pressure-fluxboundaryconditionapproach can give incorrect values for the diagonal
terms as well as for the cross terms in some cases. The error
arisesthrough the tacit assumption that the coordinate
directionscoincidewith the principal directionsof the effective permeabilitytensor. When this is indeedthe case,k*
computed
throughuse of pressure-fluxboundaryconditions
will agree with that computed through the more general
periodicboundaryconditionapproach.Unfortunately,it is
notin generalknown a priori when this is the case.Periodic
boundary
conditions,by contrast,are lessrestrictive,anddo
notrequireknowledgeof the principaldirectionsof k* to
the fine and coarse scale results for the different cases will
allow us to assessthe applicability and limitations of the
scaleup procedurefor generalpermeabilityfields.
We now considerthe solution of the pressureequationon
heterogeneousregions with permeability synthesized
througha fractalgenerationtechni.que[Hewett and Behrens,
1988].This generatingtechniqueis basedon the WeierstrassMandelbrotfractal function (discussedin detail by Berry and
Lewis [1980]), an approximation of fractional Brownian
motionon a finitegrid. For areal sections,the methodentails
the generationof a porosity field via a two-dimensional
generalizationof the Weierstrass-Mandelbrotfunction.
Giventhe porosityfield, permeability, consideredisotropic,
is modeledthroughan exponentialmapping;i.e.,
yield correct results.
3.2.
k = a10/'4
Fractally Generated Areal and CrossSections
In the exampleconsideredin section3.1, the systemwas
periodicand k* thereforedisplayedno variationin x. For
(24)
where the choiceof a and b scalesthe resultingpermeabil-
ity. The Weierstrass-Mandelbrot
functioncontainstwo im-
portantparameters,
thefractaldimension
Df andthe scale
ratior. ThevalueofDf determines
therelationship
between
suchsystems,
the conceptof effectivepermeability
is appli- the amplitudeof variationsat every scale of the porosity
cableandthe scaleup procedurepresentedin thispaperis trace while r determinesits texture or "lacunarity." For the
arealporosityfieldsusedin this study,Df = 2.2 andr =
0.707.
Region2
'
I
,,I
....
,i,
I
I
....
RegionI
Figure 5 depictsa porosity field on a 50 x 50 grid
computedvia the methoddescribedabove (note that the
globalcoordinatesystemis now designated(x, y)). The
porosityhererangesfrom 0.23 to 0.38, with darkershades
indicatinghigherporosity.The constantsa andb are chosen
to givea permeabilityfield with the minimumpermeability
(kmi
n) of 0.05andthe maximumpermeability
(kma
x) of 20,
givinga permeabilityvariation (kmax/kmi
n) of about 2.5
ordersof magnitude.The assessment
of the scaleup procedure for this problemproceedsas follows. We solve the
pressure
equationwitha uniformsourceof unit magnitude
over a domain with fixed pressure boundaries'
Fig.3. Larninated
porous
medium
withlaminations
aligned
with
coordinate axes. Dashed box indicates unit cell.
V-(kVp)
= -1,
(25a)
706
DURLOFSKY.'
CALCULATION
OFEQUIVALENT
GRIDBLOCK
PERMEABILITY
0.10
0.08
o.o
0.02
0
0
:j
Fig. 5.
Fractally generatedporosity map for areal section.
p=0
on OD.
0.2
0.4
0.6
0.8
1.0
Fig. 6. Comparisonof fine scale (solid curves) and coarsescale
{dashedcurves) pressure profiles at various values of y for flow
through fractally generated areal section.
(25b)
6%). However, it must be emphasized that geometric means
This equation modelsflow in an oil reservoir or aquifer with
are applicable to a very limited range of problems; specifiuniform injection (or analogouslyproduction) at all points
cally those with nearly random heterogeneity and local
within the domain and outflow (or inflow) at the boundaries.
isotropy. Equivalent permeabilities computed through geoFor the case of constant permeability, the maximum presmetric means cannot be expected to apply to more general
sure occurs at the center of the region. Equation (25) is first cases.
solved on the fine scale (50 x 50) via a standard finite
The next case considered is a cross section based on a
difference technique. Next, a 10 x 10 coarse grid is superfractal interpolation between well logs. The resulting porosimposed on the 50 x 50 fine grid. Each coarse scale grid
ity field, which can be used to approximate the permeability
block now corresponds to 25 fine scale grid blocks. The
field through use of (24), is shown in Figure 7 (darker shades
coarse scale equivalent permeability in each grid block is
againindicate higher porosity). This particular porosityfield
computed via the method described in section 2, using the
is representative of a portion of an oil reservoir in the
permeabilities of the 25 fine scale grid blocks that compose
Western overthrustbelt. The vertical variation of porosityis
the coarse scale grid block. Then, given the coarse grid and
based on a fractional Gaussian noise model, in contrast to
an equivalent permeability tensor for each coarse scale grid
the horizontal variation, which is again based on a fractional
block, (25) is again solved (via a finite element method, in
which each grid block is divided into two fight triangles)and
the resulting pressure field compared with the fine scale
result.
This comparisonis displayedin Figure 6, where pressure
profilesin x, at variousy locations,are displayed.The fine
and coarse scaleresults are in clear agreement,indicating
the applicabilityof the scaleup procedureto this problem.
Comparableagreementis alsoobservedfor pressureprofiles
in y at various x locations. The grid block equivalent
permeabilities,however, showlittle anisotropy.On average,
the crossterms of the permeabilitytensor are only about
0.5% of the diagonalterms, with the maximumbeingabout
2%.The average
magnitude
of thedifference
between
the
two diagonalterms is only about 10% of their magnitude;
.
.,
"• ......•':..ish.:x.•.:::..:..:.;;:::.i::.:!:..:.-i::;::::i.:i•i.
•-:•r)•?..:s:
•.•-:•.•:s:-:,•
.
<i ß
'• - -'
:•.................
?: ..:•:.•}{•;;'5•;::•iO?•{•:•:=!•g:
:,•'i•i•:•:ii
..........
s:•::•:•:}•::•::-':.-½!;-'
...........
'•:,.•liii:•?:?z::•i'•}•:•g•&::•,•C•g•,•,i.•:'d.-,•;,..'•.'.....'
:..-
::.:...::....•..::..:::•:'
:i.:.:.:.:.':ii-:...
•
:
•:•<•;?:•i•e.:::$:..•..---.... .........
.:•
........
;i•;;½:,½:•½,-,,-•.•&4,•,
........) •-............
.................................
..::•.:•:.::•:•.
....................
..•::-..
........ ß-"..;:.i•}•:{•i'.-:..:.:.:.{•i{!{•i•4•5:5•{•Si•5'iiiE•;:•-•Z•':
'•-:., , ..-..
.:,-.,<.•.,..
i.e., 2[k*x.•
- k•?l/(k*
xx + k'*yy) -" 0.1. Further,the local
variation of the permeability field appears somewhatrandom. Therefore, becausethe fine scale permeabilityis isotropic and more or less random, the equivalentpermeabilities can be reasonablywell describedby computingthe
geometric mean of the fine scale permeabilities. Indeed,
coarse scale permeabilities computed via the geometric
meanare in goodagreementwith k* computedthroughour
procedure (variation between the geometricmean perme-
abilityandtheaverageof k"*
xxandk*yyis typicallyonlyabout
,•,&,.,,;
....
. ........
ß
:.:...... :, ..:::::..:......:,•.,::•::::.:•:•.,>,::,.....,:v.
:.....:...........-..-:;.,..:.-.:.:.::
.-.-...-...k-.......ß....
.:,.;:,.......
....
:...:.
.....
Fig. 7. Fractallygeneratedporositymap for crosssection.
DURLOFSKY:
CALCULATION
OFEQUIVALENT
GRIDBLOCK
PERMEABILITY
707
1.0
z
0.5
z
0
0.5
1.•
0.5
0
0.5
x
Fig. 8. Streamlinesfor fine scalesimulationof flow throughfractally generated cross section.
1.0
x
Fig. 9.
Streamlines for coarse scale simulation of flow through
fractally generated cross section.
There is considerableanisotropy in the equivalent grid
Brownian
motionmodel.The fractaldimension
Df of the
porositytrace in the horizontaldirectionis 1.2; this corre- block permeabilitiesin this case, in contrastto the resultsfor
sponds
to a fractal dimensionof 2.2 areally, as usedfor the the areal section.Here, though the fine scale permeabilities
arealsectionabove. This porosity field was generatedas a
sequenceof one-dimensional interpolations between well
logs.The correct vertical structure is maintained approximatelybecausethe variation horizontally is much less than
the variation vertically. This is the case because fractional
Brownianmotion is roughly the integral of fractional Gaussiannoise and therefore displays less variation. Both the
horizontaland vertical permeabilities vary over a range of 6
ordersof magnitude.At all locations,vertical permeabilityis
1
taken
tobe; of thehorizontal
permeability.
The assessmentof the scale up techniquefor this case is
similarto the assessmentperformed above, thoughhere we
solvea problem more applicable to a cross-sectionalmodel.
displayonly moderateanisotropy(kz,,/kxx= 0.25), the
coarse scale anisotropies are much greater; on average,
k zz/kxx"• 0.01,a factorof 25 lessthanfor the fine scale.This
is due to the fact that the system is highly layered, as is
evident from Figure 7. The cross terms of the equivalent
permeabilitiesare still rather small though they are greater
thanin the arealcase;on average,]
• 0.1 Their
relative unimportancein this case is due to the fact that the
orientation of the layers coincides very nearly with the
orientation of the coordinate system.
4.
DISCUSSION
The scenario considered here is two vertical wells located at
Effective permeabilityprovides a valid descriptionof large
scale flow through heterogeneousporous media when the
region for which the effective permeability is computed is
boundaries.
Solutionof this problementailssolutionof the largerelative to the scaleof heterogeneity. In suchcases,the
pressureequation (4) or (5) subject to the boundary condi- generalnumericalmethod presented in section 2 provides a
tionsp = 1 onx =0, p = 0onx = 1 andu-n=0onz
= meansto accuratelycomputethis effective permeability. For
0 andz = 1. Again, fine scaleand coarsescaleproblemsare more general heterogeneities,equivalent grid block permesolved,with the scaleup procedureidenticalto that for the abilities,which are not unvarying properties of the medium
arealproblem described above. The fine scale grid in this but rather are processdependent, at least to some extent, are
caseis 30 x 55 and the coarsescalegrid 6 x 11. Streamlines requiredfor scale up. Equivalent grid block permeabilities
for thesetwo simulationsare shownin Figures8 and 9. computedthroughapplicationof the method describedhere
Therearemoresignificantdifferences
betweenthefinescale can still provide reasonable large scale pressure and flow
and coarse scale results in this case than in the areal section
results in many cases.
The examples presented in section 3 demonstrate the
consideredabove, though the qualitativebehaviorof the
coarsescale solution is correct; e.g., the bulk of the flow applicabilityandlimitationsof the methodfor the calculation
occursin the same region of the domainin the two cases. of equivalentgrid block permeabilities. From these exam-
theedgesof the crosssectionand prescribedto have a fixed
pressuredifference, with no flow at the upper and lower
Thetotalfluxesthroughthesystemin thetwocasesarealso ples,as well as from the discussionin section2.2, the scale
up procedurebasedon solutionof (9) subjectto (10) can be
expectedto yieldaccuratelargescaleresultsfor caseswhere
procedure
is not asprecisein this caseasit wasfor the areal the permeabilityvariationin the regionunder studyis not
sectionconsideredabove because,for the crosssection,a excessive relative to its variation in neighboring regions.
comparable;9.69 in the fine scale simulationand 8.96 in the
coarsescale simulation,a differenceof 7.5%. The scaleup
smallnumberof highpermeabilitylayersdominatetheflow. When this variation is excessive, some inaccuracy may
In suchcases,scaleup mustbe donewithcaresoasnotto occur.The applicationof the boundaryconditionsdescribed
losetheimportant
effectsof a fewfinescaledetails.
Thiswas in (!0) to generally heterogeneoussystemsfor the calculanot a difficultyfor the areal sectionbecauseno dominant tion of equivalentgrid block permeability was basedon the
assumption
that the pressurevariationwas locally approxismallscaleregionsexist.
708
DURLOFSKY:
CALCULATION
OFEQUIVALENT
GRIDBLOCK
PERMEABILITY
A. J., Numerical
estimation
of effective
permeability
in
matelylinear.This requirement
can readilybe checkeda Desbarats,
sand-shale
formations,WaterResour.Res., 23, 273-286,1987.
posterJori,
andrefinement
addedto regionswherepressure
Durand,
P.P., andL.H. Ungar,Application
of theboundary
variation is noticeably nonlinear.
elementmethodto densedispersions,
Int. J. Numer.Methods
Eng., 26, 2487-2501,1988.
The scaleup procedure
presented
in thispaper,like any
equation
averaging
procedure,
cannotcapture
all thedetailof a fine Durlofsky,L., andJ. F. Brady,Analysisof theBrinkman
asa model
forflowin porous
media,Phys.Fluids,30,3329-3341,
scalemodel.Therefore,if a few fine scalefeatures,suchas
1987.
highpermeability
streaks,havea dominanteffecton the Durlofsky,L. J., andE. Y. Chung,Effectivepermeabilityof heter.
resultunderstudy,they mustbe includedexplicitlyin the
ogeneous
reservoirregions,in 2nd EuropeanConference
onthe
flow model.Thus,only someregionsof the domainshould Mathematicsof Oil Recovery,edited by D. Guerillotand0.
be homogenized;
othersmustbe modeledin detail.Because Guillon, pp. 57-64, Editions Technips, Paris, 1990.
thefractallygenerated
crosssectionmodeledin section3.2 Gelhar, L. W., and C. L. Axness, Three-dimensionalstochastic
analysis
of macrodispersion
in aquifers,WaterResour.Res.,19,
contains several dominant fine scale features (high perme-
161-180, 1983.
abilitystreaks),itsscaleupis notasaccurateasthescaleup Hewett, T. A., and R. A. Behrens, Conditional simulationof reserof the areal section,which containsfew if any especially voir heterogeneity with fractals, paper presented at the SPE
dominant fine scale details.
The use of a triangle-based
nonconforming
finite element
techniquefor the calculationof k* allowsfor the accurate
modelingof complexgeometries.The geometricflexibility
offeredby thismethodwasof usein griddingthe45ølayered
systemdepictedin Figure 4, thoughthe true potentialof
triangularfinite elementsfor griddingirregularfeatureswas
notexploited.Thisabilityis essentialin the modelingof flow
throughsome heterogeneous
cross-beddedstrata because
these strata typically displaygeometricallycomplexbeddingsorientedat variousinclinations.The methodis applied
to sucha systemby Durlofsky and Chung [1990],where the
scaleup of a regionof an eolianoutcropis considered.
AnnualTechnical
Conference
andExhibition,Soc.of Pet.Eng.,
Houston, Tex., 1988.
Kasap, E., and L. W. Lake, An analytical methodto calculatethe
effectivepermeabilitytensorof a grid block and its application
in
an outcrop study, paper presented at the SPE Symposiumon
Reservoir Simulation, Soc. of Pet. Eng., Houston, Tex., 1989.
Kitanidis, P.K., Effective hydraulic conductivityfor gradually
varyingflow, Water Resour. Res., 26, 1197-1208, 1990.
Mar•.ni,L. D., An inexpensivemethodfor the evaluationof the
solution of the lowest order Raviart-Thomas mixed method,
SIAM J. Numer. Anal., 22,493-496, 1985.
Mei, C. C., andJ.-L. Auriault, Mechanicsof heterogeneous
porous
media with several spatial scales, Proc. R. $oc. London, Ser.A,
426, 391-423, 1989.
Saad, Y., and M.H. Schultz, GMRES: A generalized minimal
residual algorithm for solving nonsymmetric linear systems,
SIAM J. Sci. Stat. Cornput., 7, 856-869, 1986.
Saez, A. E., C. J. Otero, and I. Rusinek, The effective homogeneousbehavior of heterogeneousporous media, Transp. Porous
Acknowledgments.i am gratefulto J. M. Rutledgefor his help
Media, 4, 213-238, 1989.
with the implementationof the GMRES solutiontechnique,to T. A.
Hewett for discussionsregardingthe fractal generationtechniques Sangani,A. S., and C. Yao, Transport processesin randomarrays
of cylinders,!, Thermal conduction,Phys. Fluids, 31, 2426-2434,
and to anonymousreviewersfor insightfulcommentsand sugges1988.
tions.
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(Received April 17, 1990;
revised October 4, 1990;
accepted January 10, 1991.)