V-27

ABSTRACT

In reservoir engineering, the upscaling of theabsolute permeability is now . a well-established technique. It is routinelyperformed in reservoir studies and recentadvances have extended its domain ofapplicability to unstructured grids .Nevertheless, there still exists unansweredquestions, some of which veere outlined byG. Matheron in the sixties . In this work, westudy the dependence of the upscaledpermeability on the choice of the boundaryconditions. The mere existence of differentmethod in everyday . practice (the so-calledno-flux, periodic or linear boundaryconditions) clearly indicates the relevance ofthis issue . In particular, we know that thisquestion is of a great interest when thecoarse grid is at an intermediate scale wittirespect to the finer one (i .e. the globalaggregation rate is love) . The analysisdeveloped here relies heavily onoptimization and control techniques, whichenable us to settle the question properly . Inthe new framework introduced herein weobtain interenting results which shéd lomelight on the issue . In addition we are able toestablish some connections witti thehomogenization theory. Preliminarynumerical experiments, support some of theresults presented here.

INTRODUCTION

As everybody knows, the Darcy's law formonophasic flow in porous media is aphenomenologicai law discovered in 1856 byD'Arcy in Dijon . In such a flow, we can definemacroscopic quantities such as filtrationvetocity and pressure . Then, in perfect analogywitti the Fourier's law for heat conduction, the

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Darcy's law establishes á proportionatityrelation between the flux -herein the filtrationvelocity- and the acting force -herein thepressure gradient . The coefficient ofproportionality leads to the definition of theabsolute permealiitity K

9=-K v P

where q is the filtration vetocity, P the pressure

and µ the fluid viscosity . For convenience we

suppose in the sequel that µ = l .

For a long time these have been attempts todeduce this law from first principles . -In the lastdecades many people . tried to recover theDarcy's law from .the description of movementof fluids at the pose level . One among the firstwas G. Matheron in the sixties witti his book•[7]. In . this work Matheron discuseed theexistence of different scales which make upporous media . He proved the emerging ofDarcy's law at the macroscopic level when theflow at the pose level is modelled by Stokesequations. Matheron gave also some resultsconcerning effectave pehmeability, i .e. thépermeability of a fictitious 'homogeneousformation for which the average flux is equal tothe one prevailing in the heterogeneousformation. The everyday practise of reservoirengineering teade to the related concept ofupscaled permeability . In niany applications,problems of flow through porous media aresolved numerically because of the complexity ofthe geological model . All numerical scheuresinvolve a discretization of the vaTiables and apartition of the flow domain into fanate siteelements, the so called large cells . In the case ofheterogeneous formation, assigning values ofthe permeabilities to the numerical cells is notstraightforward . This process . is known asupscaling, since it involves transition from thescale of variability of K to the lasges one of thenumerical cells. Several approaches have beendeveloped since the neventien to compote th e

PEPP I NG TERPOLILLI ' AND TH I ERRY HONTANS2

upscaled permeabilities . Most of them aredesc ribed in the review of Renard and DeMarsily [8] . It is common knowledge, that theupscaled permeabilities ' are proces depèndent :In pa rt icular they depend upon the boundaryconditions, which define the flows .The main goal of this work is to develop aframework enabling the study of thisdependency .We recall that homogenization theo ry [6] playsthis part with respect to effective property . Itprovides an adequate framework for proving theexistence of effective permeability . It is ' alsoproved in this framework that effectivepermeability is not process dependent : when theconditions required by homogenization theo ryare fulfilled it is shown that the effectavepermeability does not depend on bounda ryconditions. One of the results we present in thiswork proves that upscaled permeabilitiesconverge, in a certain sense, to the effectave per-meability when the aggregation rate increases .

1) A NEW APPROACH

Solving auxiliary problems, often 'Galled localproblems, is a basic step for most of the existingmethode for the computation of upscaledquantities . We introduce now the local problemsthat we consider.Let us consider a fanate piece (larger cell)co with boundary F, of a bounded reser-

voir 0 c IRd ,1 <_ d <_ 3 . We suppose to be 'given

the permeability field K(x) over co . Our goal is

to find a constant permeability tensor K inorder to affect it to o) . For this panpose, weintroduce the followirig pairs of local problems : .let m>0 be•a .given integer and i an index suchthat 1 <_ i S m

div(K(x) OP,. in [i)(11 .i)

P = g; on I'

div(H V Ui f, in o)

U.=g1 , on I,

Problems (l .l .i) model 'a inónophasic flowthrough Cv with perrneability field K(x) ; f,- is

a source term and g ; the condition imposed on

the boundary F . We denote by P, the unique

solution of (l .l .i) and g; =-K(x) OP,. the

corresponding filtration velocity . The sameapplies to problems (1 .2.i) where a constantfield H is substituted to the heterogeneousgaven field K(x) . Let U . devote the unique

solution of (1 .2.i) and v; _ -H D U . the

corresponding filtration velócity .We. give , now lome precisions about thefunctional .framework we adopt to solveproblem (l .l .i) . Very often, the heterogeneouspermeability field K(x) is given in a fine grid

using geostatistical tools . So we must be ab[e toconsider discontinuous coefficients in (l .l .i)and then consider weak solutions which belong

to Sobolev space H' ( w ) . We assume the field

K(x) satisfying the following assumptions :

(i) for each x e w , K(x) is symmetrie

(ii) there exist positive constante a, and

such that 0 < a <_ Q and b'~ E IR"

a~~ ~ ~ .< K(x )g • ~:5

Qg2

a . e . x E coWe introduce; the s et .M(h , a , w) of constant

uniformely positive definite symmetrie tensoron co, where h denotes the harmonie average

i . e . h= 1 f K -' {x}c x and a is the

arithmetic average defined by :

a= . ƒ K(x)dx : the constant fields H in

M(h,a,w) satisfy : hg : ~ <_ Hg . g <_ ag. g

V~ E Iltd . This set is a bóunded closed convex

set in the vector space generated by syminetric 'tensor s of order 3 :In this p aper wë pre sent our method only withDirichlet boundary conditions in order . tolighten the maan ideas . But our theo ry applies toNeumann , periodic or :mixed conditions too.Th is fact is important to consider no flux .conditions for example . We will . gave thecorresponding results when appropriate . Weintroduce now the . basic ideas which set up ourapproach .

1.1) DISSIPATED ENERGYThe basic idea is to look for the tensor H'which minimizes the discrepancy between thedissipated energy in problems (1 . Li) and (1 .2 .i) .This is consistent with classical methodeconsidered in Matheron [7], Bamberger [l],Corre [3] where the dissipated energy in theupscaled field is equa l to that dissipated in theoriginal one.For each index i, j such that 1 <_ r, j m we

consider the following quantities :

E~ (x(x)) ='1 f x{x}vP. .vPJdx - J f; P, dx2

E,; ( I-i) - 1 J H vU v U ,dx- j f U;dx .2

For i = j these quantities correspond exactly to

the energy dissipated for the correspondingflows.To compute the upscaled permeabili ty weconsider the following optimization problem:

m

(P) min I(H) _ IIE,(H) - E, (K(x )) ] zHe M(h,a,0)

ij= 1

This optimization problem is finite d innensional,this is the main point since the existence of aminimizer is then simpler to prove. Actually asM(h,a,w) is a closed bounded set it suffices toprove the continuity of I to obtain the existenceof a minimizer using Weierstrass theorem . Wehave the following result :Theorem 11 : The functional 1 is continuousánd differentiable .Corollary 1 .1: The problem (P) hos at leastone rninimizer in M(h,a,co) .Definition 1.1 : We say that H* is an upscaledtensor lor the energy if H * is sólutio❑ of theproblem (P) .

1.2) AVERAGED VELDCITY FIELD

We consider now for each local problem theaveraged velocity fields defined by :

1 -~and .T;.(H) = ƒ v; chic .

. ull m

We consider then the following optimizationprablem :

2

~ P ~ He M (min

h o,m) J(H)

- ~ I V ~ (~ - v iK (x} }r_ j

where ~.~ is the Euclidien norm. The remarksgiven, for problem (P) apply agáin ánd we

easily obtain the existence of a minimizer for

(P) .

Theorem 1 .2 : The functional J is continuousand differentiable .

Corollary 1 .2 : The problem (P) hos at least

one minimizer in M(h,a,[o) .

Definition 1 .2: We say that H* is an upscaled

tensor for the velócty if H` is solution of the

probíem (P) .

If the existence of a minimizer is readilyabtained, the problem of uniqueness is far frombeing obyious . In the next section, we show thatseveral classical approaches to upscaling fallinfo our framework and for the corresponding

optimization problems we . have uniqueness ofsolutian . We come back to this issue in section3 .

2) CLASSICAL METHODS AS SPECIALINSTANCES

In this section we give results which show theclose relationship of most classical methodswith our framework . Classical methods differmainly by the choice of boundary conditions forthe auxiliary problems they solve . But, for thesemethods a more secret feature wilt appear in thefollowing analysis : a physical quantity is preser-ved when we go Erom the fine . grid to the largerone. In our opinion this fact is not suff}cientlyemphasized in the literature, even if there existscattered restilts covering special instances inMatheron [7], Gallouët-Guérillot [4] .We illustrate our point of view mainly in thesituation of Diricnlet boundary value problemsand the use of dissipated energy. But we give atable of results which could be useful toenlighten the relationship . between manyclassical approaches for the computing ofupscaled permeabitities .

2 .1) UPSCALING WITH CINEAR BOU iv-DARY CONDITIaNS . .

We consider problems (1 .1 .i) and (1 .2.i) withf; = 0 and g; = x, for i =1, . . .,d , d being thedimension of the considered space . We considerthen :

-(2 . 1 .

div (K(x ) OP 0 in col~

P,. = x ; . on t

1<_i <_ d

- div (H D Uj = 0 in w(2 . 2 . i)

U; = x; on t

l<i<d .The ,energy dissipated is given by :

E ij (K(x)} = ' J K(x)VP . OP~dr2

l:5 i, j :5

d

E'' (H) fr 2 HOU ; .VUjdx 1<_i,j<_ d

and the averaged velocity fields :

ƒ~~(x)VP, dxf

1<_i<d

V, (H) = ~ fHVUdx~

1<_i5dAs U; = xr is the unique~ solution of problems

(2 .2 .i) for 1 <_ i < d we obtain :

2

Vi (H) = H. 19i 1 :5 i :5 dWe consider the optimization problem (P) in

this context which is denoted (P. )

d w H.. 1 Z

ij= l

It appears that 1 is a convex quadratic functionand a straightforward computation gives :

d I(H)

.Hij 2

We obtain the following equivalence :

dI(h) =Or~H~* K(x)oP .oP, dzr

It is easíly Been that the upscaled tensor H+ with

elements H* belongs to M{h,a,w} and so is a

solution of (PL) . As Í is strictly convex, we

obtain the uniqueness of the solution . Moreover

we immediately see • that 1(H*) = 0 which

means that the dissipation energy in the coarsegrid is equal the one in the fine grid. It is

interesting to outline that H* is actually solutionof the following optimization problem withrelaxed constraints :

min I (H)trFsI

where S+ is the vector space generated bysymmetrie definite positive matrices .

We consider problem (P) in the same contextand we obtain :

2( _ ->

min J(H) I H j (K (x)) . el 1NeM(I,,u,~) ,,111

As the function J is quadratic, strictly convex,we usa the same step as previously to obtain :

Hij = II ( r K(x) 0P dxl . e, 1 :5 i, j5 d1

With no difficulty veeshow that :

K(x) VP,. .OP~ dx = qP,. dx~ . e~

and we recover the same solution as for problem

(PL) . By strict convexity of J we have

uniqueness . ,The previous results show that a classicalmethod first proposed by Hamberger [1] andCorte [3] in reservoir engineering applicationfans into vut framework. A nice feature of out

approach is to enable easy proof for knownresults such as conservation of dissipatedenergy . We outline the fact that we obtain thesame upscaléd Censor if we minimize thediscrepancy' between dissipated energy oraveraged velocity.For other boundary conditions, such as no-fluxconditions or periodic conditions we obtain thesame result . We do nnt .give details here but wepresent a table which rums up some of them andgives comments . Al] results and proofs could befound in [5] . To cover the result of Gallouët-Guérillot [4] we briefly recall their approach .Considering problems (2 .1 .i) and (2 .2.i) theyintroduce the following optimization prablern :

~Where n is o~ffd.normal to the boundary F ,da the measure on C' and M thé -sèt ofconstant tensor. The function F measures thediscrepancy between the flux , through Fcorresponding to (2 .],i) and (2 .2 .i) .We eau I the energy function, J the velocityfunction and F the flux function, respectively .Table 1 shows that most of the classicalmethods are_ special instances of ourframewark . At the same time wé ' give morephysical foundation for such classical methods .For example simpte inspeetion shows that foreach boundary condition we obtain the sameclassical result using energy function or velocityfunction. This gives clear answer to those whowondered about the telavance of conservationof dissipated energy to compute upscaledpermeabilities . In Bach case the cast function isquadratic and strictly convex. This propertygives us uniqueness for the optimizationproblem associated with each method. Anotherimportant fact is the nullity of the colt functionat the solution . In the case of energy functionthis can be interpretsd . as conservation ofdissipated énergy at thé large and * fine scala .This property was known for linear and periodicconditions only [1], [2] . In out frámework weeasily obtain the result for all situations :particularly for no-flux boundary, conditionswhich seems to be a new result. A specialmeetion for periodic boundary cónditions forwhich we obtain the same solution whatever thefunction we minimize . Actually we alwaysobtain the result Bivan by periodic homogeni-zation. The fact that this minimizes the fluxfunction gives ' an unexpected relation withGallauët-GuérilloE's method even' if,Brie [2]has shown conservation of the flux in that case .

3) UNICITY AND STABILITY

We give here a short presentation of twoimportant properties. The first one is a result

about uniqueness for problems (P) and (P) ..

The second one relates to the convergente ofupscaled permeabilities to effective permea-bility when the aggregation rate goes to infinity.

3.1) UN ICITY

It can be proved (sec [5] ) that if (g, )I<f<d is a

set of boundary conditions such that we have

uniqueness for the problem (P) [resp . (P)]

then uniqueness is kept for boundary conditions

g; in a neighborhood of g; , tor each index - i ,

3.2) STABILITY

Convergente of upsdcaled permeabilitiestoward effective permeability is oftenmentioned in the literature [8] . But it is not easyto Eind a forma) statement, even lens to End aproof. This fact is likely related to the difficultyto define' convergente of tensor field KE ,

consistent with convergente of solutions of flow

equations - div(KE (x)OP)= f and flexible

enough to cover realistic situation .Spagnolo' [9] introduces such a notion whichbecame the cornerstone for the development of

homogenization theory: this was G-convergentewhere G stands for Green function . Basically,homogenization theory can bé uséd to prove the

G-convergence of permeabiliry fields KE

toward the effectivè permeability field K * . If

for Bach E > 0 we consider KÉ the upscaled

tensor of K£.' (Def. 1 .1), we can prove [5] thatG

K€ --> K * , which is the G-convergence of

upscaled permeabilities 'toward the effectivepermeability .

4) NUMERICAL SIMULATION S

The approach we advocate here enable one tocompute upscaled permeabi l ity using genera)boundary conditions . Theorem 1 .1 tells us thatthe energy function I is differentiable .Moreover control theory allows us to computethe derivative of I at any given tensor H . Weare then able to design an optimizationalggrithm for salving problem (P), The

numerical results in this section are aimed atillustrating this scheme . The domgin cu used is

a square defined by : 5 :5 x, <_ 10 and 5 :5 x 2 S 10

(figure 1) . Note that permeabili ty at the center is1 D and in the rest of tee cell is K D . In thefollowing we consider two cases with K=10 andK-100. We want to compute the upscaledpermeabili ty using problem (P) with local

problems (1 . IJ) and (1 . 2 . i) 1 :5 i <_ 2 subjectedto thé following boundary conditions

g1 (x 1 , x2) =5x,

92 (X1 1X2) 1~

These non-standard boundary conditions areplotted in the figure 2, and , may be comparedwith classical linear boundary condition . Localproblems are solved by a mixed finite elementmethod and an optimization routine is used tosolve the problem (P) .

For K=10, after 11 iterátions (figure 3) the tost

function is I(H*) =1.31 E - 04 and the

upscaled permeabili ty tensor is :

6.192 IE.-051

IE-05 7 .623

In the case K=100 we' obtain the followingupscaled tensor:

[54.564 0 .H

0. 71,087

We remark that the diagonal coefficients for thetensors obtained by minimization are not thesame even if the permeábility fields aresymetric . It is a consequente of the non-standards bounda ry conditions ' taken intoaccount . . Other resuits, obtained by classicalupscal ing methods are presented in the table 2for comparison .

CONCLUSION S

We have developed a new approach to upscalepermeability field which can manage genera)boundary conditions . Much of the classicá lmethods are particu lar instances of this newapproach . In the ncw, framework developedherein we veere able to prove new results forclassical methods and to give a formal ' statementabout the convergente of upscaled permeabi-l ities . Numerica l experiments show theeffectiveness of the approach .ACKNOWLEDGMENTS

The second au thor is grateful to E lf ExplorationProduction for financial support tb conduct thisre s earch .

kt EA T I t E \ C E S

I- A . Bambcrgcr, Approximation des Coef=

icieTits d'Opérateurs Elliptiques Stablespour la G-Co»vert*cnce, Piiblications dit

Centre de Mathématiques Appliquées de

I'Eco le Po lytechnique, N° MAP/[ 5 . 1977 .

~- O . Que , Analysis of art Upscaling Method

13ased cm Conservation of Dissipaticzn,Trans . in Porous Media, 17, pp . 77-86, 1994 .

_~- 13 . Corre, Techniques d"Homagêné i sation .

llomiogénéisatian des Pennéab ilités Aó-

solues - Composition des Perniéabilités

Relatiees, Rapport I nterne Elf Ayu i taine

Production, 1 99 1 .

4- T . Gallouët and D. Guérillat, Average

1 letero~eneous Poreus Media hy Minimiza-

tion of the error in the flow rate, ProceedingsECMOR 1V, 1994 .

~- f . Hontans, Hamogénéisation Numérique de

Naramètres Pétrophysiques jour des Mailla-(,es Déstructurés en 5irnu lation de Réservoir,

Ilicsis of Pau University, 2000 .

[,- I . Ilornung, Ho ►nogenizatian and PorousMedia, I ntc;rd i sciplinaty Applied Mathema-

tics, Springier, 1997 .

G. Matheron, Flémcnts puur une Théorie desMilieux Poreus, Masson Paris, 1967 .

S- P . Renard and G. De Marsily, Calculating

Equivalent PerEineability : A Rev i ew, Water .

Elesour ., Vol . 20, N° 5-6, pp . 253-278, 1 997 .

>- S . Spagnolo, Sulla Convergenza d i So l uzioni

di Equazioni Paraboliche cd E l littiche, Ann .

Scuola Norm. Stip . Pisa (3) 22 pp . 577-597,1)(,~

Funckons

Enagy Veluaty Flux

ri i Classical Classtral l.allnuët

(Corre) (Corre) 1--, um ílot

na fltu; Clasgcal Classical hde%v2uundaty (no-flux) (no-flux} Result

canrLtaons penodic Penodic Petiodic Classiwl

baroog. hom o g i C?urlufsky}

;u,rhw[u- Stodi StaltlHomog . Ham o g

Tahle I : Sunmary of'iipscaled tensor obtain bvniiiitni /at uit ~~ith cl .itisical boundary condition s

H 11)

K n~ .

e,o

so

d~

3a

2 0

,o

~ hnea r

~~ 5 n 5g S S r '~ 9 5 10

F i gure 2 : Boundary condition s

1 ,O UIE-i ~+

1 onE . ~~;

ooe .u-.

1ooE,o3

i oo e +o i

i ,ooe -r~;

1 2 ., 4 5 6 ~ - iu 1 1

Figure 3 : Convergence ofcast funct i or7 I

~ F~ HHa ~~x H 'ra ~ ~ca VI' 4~:rc V-{ HP H~.rr~ Hyc ~ I: h

~ 7 D 3.0 8 E,fl2 6 .1 7 E .46 6 47 6A8 6&Z 6T 0 7 .29 T .bl 7 .75

2rr~ h.BA ~ 3 ] .62 50. .26 53Ai1 58 E.4 ~B Y G4 .23 fi1 .T3 692D 6 4 .72 762 5

Tab l e 2 : Upscaled tensors obta ined by c lass i ca lizlelhods .

Nn.-r [resp . f/ax~ : Harmonic

[res p . Geometrie, A r ithrnetic ] average

fl R i, : Renormalization roetbod

HG, : Cardwel l-Parsons roe t bod

H tirc[resp . H,irc ] : No-flux method witti

CFE [resp . MFEI

H1If, : Perindi c hamogeni7at ion

H ,f,; :M ult igrid method

Hff [resp . H,,,f ] : Linear rnc thod with

CFE [resp. MFE ]

CFE [resp . MFF.] indicates that l ocals p rob l ems

are solved witti a Conform ing [resp. Mixed ]

Finite Element roetbod .

Fioure 1 : Dnrnain c9 to upscale