0720046

SIAM J. NUMER. ANAL.Vol. 20, No. 4, August 1983

1983 Society for Industrial and Applied Mathematics

0036-1429/83/2004-0003 $01.25/0

FINITE DIFFERENCE METHODS FOR TWO-PHASEINCOMPRESSIBLE FLOW IN POROUS MEDIA*

JIM DOUGLAS, JR.+

Abstract. Two-phase, incompressible flow in porous media is governed by a system of nonlinear partialdifferential equations. Convection physically dominates diffusion, and the object of this paper is to developa finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in

appearance, is discretized by a standard five-point difference method. The concentration equation is treatedby an implicit finite difference method that applies a form of the method of characteristics to the transport

terms. A convergence analysis is given for the method.

1. Introduction. There are two simple models used to describe the movementof two incompressible fluids in a porous medium. In one model the fluids are assumedto be completely miscible; one example is given by oil and a detergent solution. Inthe other model the fluids, such as water and oil, are considered to be completelyimmiscible. There are many other possible physical hypotheses that can be and areused to simulate more complicated flows; however, only the two simplest models willbe treated in this paper.

The differential systems for the two models can be put into quite similar forms,although the dependencies of the coefficients on the dependent variables are somewhatdifferent. In each case the dominant feature of the flow is the transport, rather thanthe diffusion, which provides only a small amount of parabolic regularization of thesolution. The differential system will consist of two equations, one elliptic in appearancethat effectively defines a pressure for the flow and the other an equation giving theconcentration or saturation of one of the fluids and formally being parabolic inappearance. It is this second equation for which the physical effects described by thefirst order spatial derivative terms dominate those described by the second order ones.

In recent years a variety of finite element methods have been introduced [3], [4],[7], [10], [11], [15], [16] and analyzed to treat the miscible displacement problem.Most of these methods would be more effective if the level of diffusion were higher,though the experimental results obtained with the self-adaptive, interior penaltyGalerkin procedure discussed in [4] are certainly more satisfying than those obtainedearlier. Quite recently, Russell [15] has adapted a method combining the method ofcharacteristics with a finite element procedure to the miscible displacement problem;this concept has been studied for both finite difference and finite element schemesfor scalar, transport-dominated diffusion problems by Russell and the author [8]. Theobject of this paper is to formulate a finite difference version of the characteristicprocedure for a differential system that contains both the miscible and the immiscibleproblems as special cases. Clearly, the motivation for including the method of charac-teristics is to follow the transport more accurately than the standard finite differenceor finite element methods do.

The differential systems will be discussed in the next section, and the finitedifference procedure based on the method of characteristics will be defined in thethird section. A convergence analysis for the method will be given for a periodicproblem in the fourth section; the treatment of the periodic case avoids some technicalquestions that are of very little interest in practical reservoir engineering problems,since the main interest in secondary and tertiary recovery processes is in the body of

* Received by the editors December 18, 1981.+ Department of Mathematics, University of Chicago, Chicago, Illinois 60637.

681

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682 JIM DOUGLAS, JR.

the reservoir, not at its edges. There will be certain simplifying assumptions made inthe analysis, mostly of a nature that lead to more regularity of the solution than isthe case when point sources and sinks (i.e., wells) are admitted. These assumptionsare similar to those made in all but one of the analyses of finite element methodsreferred to above and will be discussed in 2.

Experimental studies associated with the characteristic-finite difference methodwill be reported separately [9]. Refinements of the method, including certain self-adaptivity features, will be discussed later.

2. The differential systems. Let 12 be a domain in R2 (or in R3, in which casegravity terms, to be neglected in this presentation so as to concentrate attention onmore essential portions of the problems, should be included). The usual differentialsystem used to describe two-component, incompressible, miscible displacement isgiven by [3], [10], [13], [14]

(2.1)

V. u =-V. (k(X) Vp)=q,(c)

Oc--+u Vc-V’ (DVc) (?.-c)q,(x)at

O<t<-T,

xefl, O<t<=T,

where p is the pressure (unique to the system as miscibility implies no capillary forces),and k the porosity and the permeability of the rock, tz the concentration-dependent

viscosity of the fluid mixture, u the Darcy velocity of the fluid, c the concentrationof one of the basic components in the mixture, q the external flow rate, positive wherefluid is being injected. The function must be specified whenever injection is takingplace and is the concentration of the same component as measured by c in the injectedfluid. Where fluid is being produced, it will be assumed that c this is an assumptionabout the physics of the problem. The diffusion coefficient D arises from two sources,molecular diffusion, which is rather small for field-scale problems, and a velocity-dependent term called dispersion in the petroleum engineering literature. Thus, D isa matrix (i.e., V. (DVc)= (D%,)x,); its form is

(2.2) D =&(x)(dmoI +dlong[u[E +dran[u[E+/-),

where E represents projection along the velocity vector and is given by

(2.3) E (eli(U))= \[ulZ], u (Ul,""’, Un),

and E+/-=I-E is the complementary projection. The diffusion coefficient do,gmeasures the dispersion in the direction of the flow and dtr, that transverse to theflow. The pressure is determined up to an additive constant.

The usual form of the equations describing two-phase, immiscible, incompressibledisplacement in a gravity-free environment is given by [14]

(a) (So)-V’ k(x) kr-(s)Vpo =qo, x el2, O<t<=T,p.o

(2.4)

(b) (Sw)-V" k(x) krw(Sw)Vpw =qw, x eft, O<t<=T,p,w

where the subscripts "o" and "w" refer to oil and water, respectively, s the saturation,p the pressure, k,, the relative permeability,/x the viscosity, and q the external flow

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TWO-PHASE FLOW IN POROUS MEDIA 683

rate, each with respect to the th phase. The water and the oil will be assumed to fillthe pore space of the rock; i.e.,

(2.5) So+Sw=l.

Thus, one saturation can be eliminated trivially; let s So 1 Sw. The relative permea-bility functions are monotone functions of their respective saturations such that

(2.6) 0 kri(0) -< kri(Si) <= kri(1) 1, o or w.

These functions are also functions of position, kri(X, si); however, they will be takento be independent of position in this study. The pressures in the two fluids are relatedby the capillary pressure (with the sign reversed from its usual representation, so thatp’ _->0),

(2.7) p(s) =Po -Pw,

where Pc is assumed to depend on saturation. This function is often correlated to theporosity [12] as well; however, it will be taken in the form (2.7) here.

The system (2.4) is not in a form particularly similar to that of (2.1), but it canbe transformed using a different pressure variable introduced by Chavent [2]. Let

(2.8) A (s) ko (s__) + kw (s___)Id, t.l,

represent a total mobility of the two-phase fluid, and define relative mobilities by

k(s)(2.9) Zi(s) o, w.

lil (S

Chavent’s pressure variable is given by

po+Pw+ (Ao ()- Aw ()) d.(2.10) p=Let q qo +q be the total external flow rate. Then a simple calculation using the sumof equations (2.4a) and (2.4b) shows that

(2.11) -. (k(x)h(s)Vp)=q, x I), O<t<=T.

A second equation is derived using the difference of (2.4a) and (2.4b):

OsV. (k(x)A(s)(AoVpo-AwVpw))=qo-qw.24) 0----

Since Ao + Aw 1, it can be seen that

AoVpo X Vpw 2XoAwVpc + (Ao ;t )(Xo Vpo +wVpw

Thus,

2AoAwVpc + (Ao Aw)Vp.

Os 1 1dp--V (kAAoAwP Vs) -V. (kA (Ao Aw)Vp) -(qo qw ).

Now, set

(2.12) u =-k(x)h(s)Vp,

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and differentiate the last term on the left-hand side in the equation above and use(2.11). Since Ao Aw 2Ao 1,

Os6--- V (kAAoAwP Vs)

1+ A’ou" Vs -{(qo Aoq)- (qw Awq)}.

Make the assumptions that

(2.13)(a) qw=q and qo=O if q>-0,

(b) qw=3.wq and qo--Aoq if q<0,

which says that one does not inject oil as a matter of policy and that at a productionpoint the mixture produced is in the same proportions as those of the fluid residentat the point. Then, the differential system has assumed the form, for x 1 and 0 < <-_ T,

(2.14)(a) V.u=q,

Os [-Aoq, q >-_ O,(b) cD-:7+A’o(S)U Vs-V. (k(x)(,,owp’)(s)Vs)

O, q 0.

Thus, Chavent’s equations for immiscible displacement strongly resemble the usualones for miscible displacement; however, certain fundamental differences will bepointed out below.

Initial and boundary conditions must also be supplied. Initial values can bespecified only for the concentration in (2.1) and for saturation in (2.14), since thepressure variable is computable from the elliptic equation (2.7a) or (2.11) as soon asthe boundary conditions and the coefficients are determined. Note that the pressureis determined only up to an additive constant; thus, the conditions

(2.15) Iapdx =0, O<=t<=T,

can be applied to suppress the ambiguity. Quite a variety of boundary conditions canbe taken to represent different physical situations; for simplicity the domain fl willbe taken to be a square, scaled to have unit side length, and the solution will beassumed to be periodic of period one in each independent variable. In order that asolution be possible, the integral of q over 1" must vanish at all times.

In the convergence analysis to be given below certain liberties will be taken withthe physics. As in all but one [11] of the analyses of finite element methods for themiscible problem so far published [7], 10], 11 ], 15], 16], the external flow functionq will be assumed to be smoothly distributed; clearly, on any reasonable scale, wellsare point sources and sinks. The capillary term is degenerate in the immiscible problem,since AAoAwP’c vanishes at s 0 and s 1; however, the diffusion coefficient will beassumed bounded below by a positive constant in the analysis. In a sense, this isequivalent to treating problems in which 0 < e s(x, t)<-1-e for some positive e.This is not much of a loss physically, but it may be a bit misleading practically. Thecapillary diffusion coefficient is small enough that at a practical level of spatialdiscretization it cannot be safely considered to give a sufficient level of parabolicregularization. Thus, it may be advisable to add an artificial diffusivity to the real,physical one. This will be discussed in the formulation of the difference equations.There will also be some assumptions, implied or explicit, on the smoothness of thecoefficients 4 and k; these are minor in mathematical importance relative to the otherhypotheses introduced above.

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3. The finite difference procedure. Consider the differential system

(3.1)(a) V u g(x, t),

Os(b) &-:-7+b(s)u Vs-V" (DVs)=f(x,t,s), x f, O<t<=T,

dt

where 12 is the unit square,

(3.2) u -k(x, s)Vp

and D -d(x, s)I +Dx(u), with Dl(U) positive definite. Assume that the integral of qover f vanishes and that the coefficients, the data, and the solution are periodic ofperiod one in each of the independent variables. Let h =N-x, xit--(ih, ]h), At >0,

nat, and w(xi, ) wij.

The difference method will consist of two parts: first, if the approximate saturationSi]. is known at time ", (3.1a) and (3.2) will be discretized to obtain an approximationP’} to the pressure at time t", and an approximate velocity U’. will be evaluated usingS" and P". Then, a new saturation, S "+a, will be calculated, with s, + bu Vs beingreplaced by a difference in the characteristic direction. The diffusion term will behandled somewhat differently in the miscible and immiscible cases, and the specificswill be given later.

Let

(3.3) K in+l/2,j [k (xij, Sin,j) q- k (Xi+l,j, Sin+l,t)],

and let K,i+l/2 be defined analogously; the symbol kin+l/2,i will denote the evaluationabove with s in place of S. Let

(a)(3.4)

(b)

6(K6xP)i’ h-Z[K "i+x/2,i (Pi+I" ,i -P,")-K"i-/2, (P-Pi"--x.)],

V(KVP)i 8(KS,P) i+ 8(KSP) i;

then set

(a) -Vh(KVP),"t=G,.=h- f q(x,t")dx,+Qh

(3.5)N

(b) (P", 1)= Pi].h2=0,i,j=

where Qh is the square of side length h centered at the origin and the discrete innerproduct (., is taken over a fundamental period for the periodic problem (3.1). Thedefinition of Gi]. is such that (G", 1)=0; thus, under the assumption (which is validfor both the miscible and immiscible problems if the permeability k (x) is nonsingular)that

(3.6) k(x,s)>-k.>O,

there exists a unique periodic solution P" of (3.5).Compute the approximate velocity U (V, W) as follows:

1[ P" Pi.-PiCx,t](3 7) V] -- K" i+l,t-P,t+K.i-1/2,ti+1/2,j h h

with Wi’ being the corresponding average in the other direction. Next, set

(3.8) Xit Xi] b (S i]) Uit At/qit,

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and let

(3.9) S, S (X, ),

where S" (x) is the extension of the knot values {S ’.) by piecewise, bilinear interpolation.The choices (3.8) and (3.9) will be the ones considered in the analysis below; however,since it is the characteristic through the point (xi, "+) that is desired, it can be

n--1advantageous to extrapolate the velocity to time + by replacing Ugi by 2U0. U0and to change the value S. to the value $0 given in (3.8) and (3.9) in b(s) to find analternate evaluation of S0"

Ui]At/bil(3.8’)

Xq x, b(S"(x,i

U6 2Uii U’-1,(3.9’) (Xi]).

While the position Xi and the function Si could be iterated to convergence in (3.8’)and (3.9’), no essential improvement should be expected beyond the predictor-corrector formulation (3.8’), (3.9’). For the miscible problem only the trivialmodification from Uii to Uii is needed, as b (s)= 1.

The flow is essentially along the characteristic for the transport; so, considerdifferentiation in the characteristic direction’

(a) 4(x,s,u)=[&(x)Z+b(s)Zlul]/2(3.10)

+bu(b)Or 0

& Ot

Thus,

(3.11)OS OS--+ bu Vs O--.&Ot Or

Note that the direction r is dependent on both space and time. If

tnn+l n+I At/ii and (g/i,Xi] Xij --b(sii )Uij Sij S ),(3.12)

then

(3.13) 00s (Xii, tn+)=ii sii --Sii+o A,rAt OT2

The function gi] differs from i’ in two ways, as it is the evaluation of s(., t") andnot that of an interpolant and it is evaluated at g., which does not coincide withX0. However, (3.13) motivates the choice

(3.14) & --+ b(s)u Vs (Xii, n+l) i] S’+1At

The discretization of the diffusion term is of a different nature for the twoproblems. In the case of immiscible displacement, D d (x, s)I and the term V (DVs)can be approximated by

n+l(3 15) (Ds)ii Th(D(Sn)hS) n+l"ij

n+lhere, it is intended that, e.g., Di+l/2d is given by

/- +1 __lfl-t’Xi,j2"[1,-’n(3.16) i+1/2,i Sii) q-D(Xi+l,j, Si+l,i)}.

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Since S ij must be computed in any event, there is no extra work involved in (3.16)over basing D in+l/2,i on the knot values S i’); clearly, there are advantages in havingthe capillary terms evaluated more nearby as they have affected the fluid resident at(xii, "/1) during the time step.

At a practical level of spatial discretization for the immiscible problem, thecapillary term can be insufficient to provide adequate smoothing for the numericalmethod, and it may be advisable to add an additional, artificial diffusive term, as hasbeen discussed in many articles concerning the capillarity-free, single space variableproblem. In particular, a generalization of the techniques discussed in [5], [6] will beconsidered. Since the artificial diffusion will be taken in a form resembling the tensorialdiffusion in the miscible problem, a specification of this term will be given after themiscible diffusion term has been treated.

For the .miscible case, the diffusion splits into a molecular diffusion,V. (dmol(X)t (x)Vs), which can be approximated by Vh (dmolt VhS)i"+ 1, and the velocity-dependent term

(3.17) V. (D(u)Vs)= E Dk,l 10Xk

which will be approximated by the sum

,,+ [(Dh(D(U )hS), ’8,,S)+8(D228S)(3.18) +1/4(Ox +O)(D(Oy +Og)S)+1/4(Oy +3)(D21(Ox +O)S)].+,where O, and 3 are the forward and backward difference quotients in the first spacevariable and uX2=u2, U 1, and D22 are evaluated at U", as indicated on theleft-hand side and as will be analyzed. Alternately, these coefficients could be evaluatedat ". Note that the values of D2 are required only at the node points xx,i andxi,ia in (3.18).

The artificial diffusivity term for the immiscible problem should be taken in thesame form as (3.18), with the coefficients do.g and dtr, of (2.2) and (2.3) that definethe Dx(u) matrix to be of the form (now dependent on x and t)

(3.19) dlong

where dl should be larger than d, if (3.8’) is used in place of (3.8), the correspondingmodification should be made here. The constant d, should not be taken to be zero.The asymptotic error analysis to be given below cannot give any information thatcould be used to develop criteria for the selection of d and d,, since the real capillaritywill eventually dominate the artificial.

Letn+lh(OhS)ii

denote the discrete diffusion term arising from (3.15) or (3.18) or their combination.Then, the difference equation associated with the saturation or concentration (i.e.,(3. b))is

(3.20) Oijt

either S-+x occurs linearly in f or f should be linearized about S., so that the algebraicequations remain linear.

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Thus, the finite difference method consists of the two equations (3.5) and (3.20),which hold for all x0. These equations have a periodic solution (Pi’, $i’) for n >-1,assuming that $. is periodic.

4. Convergence analysis. The demonstration of the convergence of the finitedifference solution to that of the underlying differential system (3.1) has some pointsin common with that given by Ewing and Wheeler 10] for standard Galerkin methods.In particular, the error in the gradient of the pressure or, equivalently, in the Darcyvelocity will be related to the error in the concentration variable. The argument isalso based on that given by Russell and the author [8] for the treatment of scalar,convection-dominated parabolic problems by characteristic finite difference or finiteelement methods.

Let

(4.1) rr =p-P, =s-S.Recall the notation defined in (3.3) and the remark immediately following (3.3). Then,

(4.2) --Vh (k hP)ij-- Gii + tij,

where

(4.3) 18The symbol M(.,., ..) will indicate a generic function of its arguments; the explicitdependence of these functions on the coefficients will not usually be spelled out. Thenorms I1" II,.o are those for the periodic Sobolev space Wt’ (.f); the notation

I1 11=< . ’/=

will denote the norm on the periodic discrete space 12(fD. Also, (gVh’lr, hTr) denotesthe square of the weighted semi-norm on the discrete space h l(f) corresponding toH’(lq)=

Subtract (3.5a) from (4.2) to obtain the relation

(4.4) --Vh (KVhTr’) ii t ii "4- Vh (k K)VhP ii.

Test (4.4) against 7r" and sum by parts:

(4.5) (KnVhTr

hence by (3.6.),(4.6)

Recall that the standard Riemann sum over a uniform partition is accurate to arbitraryorder as a quadrature rule for periodic functions. Thus, since (p", 1)= 0,

(4.7) (p", 1) M(llp"]14.)h 4.Set

(4.8)

and apply the discrete Poincar6 inequality for periodic functions,

(4.9) lift" 7"/’ave liX7hrn II;thus,

(4.10) Ilv, " /M(llp"l]4,)h 4.

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As a consequence of (4.3), (4.6), and (4.10), it follows that

(4.11) IlVhzr"ll<--M(llPll4,oo, IIs"ll,oo){ll"ll / hE}.It can be seen, by noting that

(4.12) 18,1--< M(IIP"II3,o, IIs" II=.oo)h,that Vhrr" can also be bounded by the inequality

(4.13) IlV.r" _-< M(llp" 113.oo, IIs 112.o)(11" / h).

Consider the saturation equation next. The arguments for the two forms of thediffusion coefficient are somewhat different, and the case D d (x, s)I will be treatedfirst. Let (3.7)-(3.9) hold, and recall that

.+1 At/qbi].(4.14) Z i] Xij b(s .+1 )u i]

dn+l n+lIf --i+I/E,’ represents the evaluation of (3.16) with sit replacing Sij, then

:i+1 __(sn(g)__Si]) n+l-V(DVh).+ =f(s.+)-f(S )+Vh((d-D)Vs)’.++e,i] At

(4.15)

where

(4.16) le0’l-<-M s I1,,L=(J" ;to)) (h + At),

with J" [t", "+a] and r 7"i+1 the characteristic direction at (xii, t"+a). Interpret " (x)as the piecewise bilinear interpolant of the values {’.}, " ", and let o " (Xo).Then,

(4.17) ,+a (s (Xi]) --n n+l --n --n --nn-n-Si])-’(ij -i!.)-(s (,i])-s (Xii))-((1-)s (Xq))

Note that

(4.18) I((1-N)s")(.,i’})l<_MllsllE,oomin (h E hat)-1with the hat alternative arising when Ib(Si)Uii[ii At is less than .5h. Next,

Sn ( inl S (’ )l Mils" I1,1 i. i[

(4.19) =Mils I[1,[b(si] +1 )ua -b(Sij)Ui]

M([[P[[wI.( ;wl.), IIsllwl.(.w,))(t + I, +where the inequality ]ui uilg(llp[lx,)(lil + [v[) has been used. The difference

n+lf(s.+l)-f(Sii )is bounded by Mli].+ [. Now, let

(4.20) = maxM([[sn+ll3,, =sll ’llsllw(’";w*)’ [IpII( ;wl’))2 L(J ;Lm)

Thus, (4.15)-(4.20) show that

i] i’+ (i]-- Vh (DVh).+ < (li.l + l.+ + lvh i.l + h +At)At

(4.21)+Vh((d-D)Vs)+a.

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(4.22)

Let

Test the relation against .,+a. Then by (4.21)

1 n+l) (t(n, +(Dn+ h{(’"+ -+)} v-+

(ll"+ll2 + II"ll2 + IlVh"ll= + h + (at)2}+((on+l-dn+l)hsn+l hn+l)

(4.23) M =M+maxM,L(

Then, (4.a3) aows the elimination of [IVhZr"l[z from (4.22) by replacing/r by/r. Sincen+l -n

SS i] Si] (s n+ -n, s (x,].))+(s"(x) (,,;))+(( )s")(x,)+,,it follows that

I((Dn+a--d"+a)Vhs "+’, Vhn+l)[(4.24) - V .+x-< zd.II , = +r{(At)= + I1"11z / Ilx7,r 2 / h’ / II("llZ}.where d, is a lower bound for d(x, s). Thus, since

(4.25)Ar{ll" 2 + II"+a]l2 + h 2 + (At)2}.

The next objective in the analysis is to relate <4(", ("> and <&", "). Write (" inthe form

’. " (xii b (s .)u

(4.26)

+ [j" (xij b (s .)U i"" At/&it) J" (xii b (s i]’)u i’} At/cbii)].

Represent the first term on the right-hand side by (and recall the periodicity)

Nijn.n(4.27) (xii-b(sq)uiiAt/tii) E Olklgi+k,i+l,

k,/=l

where

(4.28) a7 =>0, Y a 1,k,l

and at most four of the weights are nonzero for each triple (i,/’, n). Also, eta? 0unless IXkl--Xiil<--lAt. Since, e.g.,

b(Si+la)Ui+a’iAt/&i+a’J -b(sii)uiiAt/cbii =-O-x -] hat,

it follows that

(4.29) E [Ol. + l,j,n ijn, -c,l-<M(lls IIx,,llp IIz,)At,k,l

i-k,i-k,n(4.30) ( i-k,j-IOl kl [1 + O(At)]tij,k,l

and from this and (4.28) it can be seen thatDow

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with the constant having the same form as that of (4.29). Then, using (4.28),2

E ij" (xo b (s o)u ’At/cbq)2h 2 E qbo a ki+kd+ hi] 7 kl

iin 2 2

(4.31) i kl

i-k,j-l,n 2E E $i-kd-takt (0)2hq kl

The remaining terms on the right-hand side of (4.26) will be shown (after a ratherlengthy argument) to generate inner products that are asymptotically negligible. First,estimate the second term on the right-hand side of (4.26) as follows"

x-b S)Uat/.V" Uo d(4.32)

,,-b()u,a,/,, "Iu,lMIU,311,lat max {IG.I" Ix.o

Thus, with Iz I Ilzll and Iz 11.2 Ilzll’,

IIn II:----<M(At):E [Uij[ Iii max{lVhoq lx. -xlh +MIUoIAt}h 2

ij

(4.33)

Under slightly different circumstances but with a proof easily modifiable to the currentsituation, Bramble [1] has demonstrated that

(4.34) IvlMlvl.= log

for any mesh function. Consequently,

(4.35) I1 "I1= MIg(=+MIgI a) I’1.=(at) log ,1and

1(4.36) (&r/", :"+’> -< MII"+’II=At +MIU"I3I"I.=At log

under the hypothesis that

(4.37) AtMh

this hypothesis is very reasonable, since even in the case of a scalar parabolic equationall that can be expected of the basic method is an O(h + At) error estimate.

The third expression in (4.26) can be bounded in a similar fashion:

" (xo b (s,)UAt/&o) " (xo b (s,.)u .At/o)

(4.38) V" Uii -uo,,-s).,/.,, Iu. -uld

MIU" u"lAt max {IV.Go I’ Ixoo -x,l h +M max (Iu, I, lu.I)At}.

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Thus, (4.37) and (4.38) imply that

(4.39) [["II=<MIU-and

(4.40)

Inequalities (4.31), (4.36), and (4.40) can be applied to (4.25) to obtain the relation

2At

(4.41)

,,3 ,, 1U" U" ]/ MIU I[’ II.:zlog’+/lr(l+[ [)1 -u"[ 1’"[1,.

In order to complete the analysis it is appropriate to estimate U" and U"-u" interms of " and 7r. It is easy to see from the definitions of Ui’ and u. that

(4.42)

log +h = /MlVhzr"[o.

Next, if (4.4) is differenced in a spatial direction and then tested against that spatialdifference of r" and the results added over the spatial directions, it can be shown that

where the obvious inequality [n[1,oo<=h-lln[1,2 has been used. Now, apply (4.34) to

VhZr" to see from (4.42) and (4.43) that

(4.44) IU"-u"l<_-El"l,=(x +h-ll"ll)+h] log

From this it follows that

(4.45) [U"I<-Ar X /["l,2(X /h-ll"ll) log-Let

(4.46) O" max [MIUI[I2 1 Uk k ]O=kn1,2 log -+ll,(1

Then, (4.41) implies that there exists {M"},

(4.47) M" <- lffI et"t" <- iffI eY4rT M*,

such that

(4.48)k=l

02--<o l’[,2at+M"{ll’ +h-+(+/-t)=}.

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To complete the convergence argument it is sufficient to show that t" tends to zeroas h tends to zero. This will be done using an induction argument. Assume that, forconstants M of the form (4.23) and some positive B, which can be taken to be lessthan one,

(4.49) (1 + h-l[ItSkll)lk [1.2 --</roh , 0 < k < n.

For n k 0, the inequality is trivial, as tS 0. Next, if (4.49) holds,

(4.50) Iul, [Uk --ukloo<=fC2h log 0_<-k <=n,

and

(4.51) 1 d.0 -< <

for h (and At, by (4.37)) sufficiently small. Thus, by (4.48), with 4, =inf4(x) andagain noting that 5= 0,

(4.52) ,11/11= + d,l/al,=At <-_M*(h 2 + (At)2).In particular,

(4.53) II/llqh,

and, if (4.37) is strengthened to read

(4.54)

then

(4.55)

Mlh TM <- At <--_ M2h for some 3’ < 2,

Thus, (4.49) is satisfied for 0-<_k =<n + 1 when h is sufficiently small and (4.54) isimposed. So, the induction hypothesis has been established, and it follows that

T/At /,)1/2

(4.56) max II:"ll + ,ko( <M’h,

where (with L’(X)=L’(O, T;X))

(4.57) M’=M’ Ilsll,*,w., Ilsllwl.oo,, 0, ,%

This completes the analysis of the slightly regularized immiscible case for which thediffusion coefficient has the form D =d(x,s)L since (4.13) and a version of (4.10)with the O(h4)-term replaced by M(l[p[ILoo(w2.%)h 2 provided error bounds for thepressure and its gradient. Note that one implication of (4.13) is that the Darcy velocityconverges at an/2-rate of O(h).

Consider next the case when D has the form

(4.58) D (u {dmo,/ + d,oneJu lE (u + dtra,,lu lE+/-(u )}cb (x ),

where E(u) is defined by (2.3). The analysis of the approximate pressure needs noalteration, and, in particular, (4.13) remains valid. In the actual miscible problem, thecoefficient b (s) is a constant; thus, the treatment of the characteristic derivative termgiven above is more than adequate. The two terms in (4.22) depending upon the

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diffusion must be reconsidered. Note first that

(4.59) (dmollVhn+l, hn+l) --dmol(hn+l, hn+l)

and

(4.60) ((dmol-dmol[)VhS n+x, Vhljn+a) O.

Thus, only the velocity dependent terms need to be analyzed. Observe first that, ifthe vector Vh" (0x, 0y:) is decomposed into components c and/3 with c parallelto some vector u and/3 orthogonal to u, then

(4.61) (dlongE(u)h + dtranE-l-(bl)h, h)R dlongltx 12 + dtranlfll2 >-dtran[h]2.

This implies that, with u being the Darcy velocity at xii at time for the solutionof the differential problem,

(4.62)

I’/ .n+l n+l 2E (i](drno[ +donglUi]lE(ui)+dtranlUii[E+/-(uq))Vhi Vhi] )R2hii

e Y’. (tii(dmol’k-dtranlUin’[)Vh"+1, Vh/+1 )RZh 2

ii

(6 (dmol + dtranlunl)vhjn+x, VhS"+).Next, note that the (k,/)-entry of the matrix IU}E(U)-IuIE(u) satisfies the

inequality

(4.63)](U(Ut-ut)-(u-U)UlI+(lul-lUI)IUllu[

<--(1/ 2llul)In-uI.The symmetry in U and u in the left-hand side implies that the roles of U and u canbe reversed in the inequality. Hence, the minimum of IUIlul and lullU1-1 can beused, pointwise, and it follows that

(4.64) UkUtll __< 3[UIu[ lulA coercivity inequality can be derived for the term (--hDl(U)hj, j) as follows.

Adopt for the moment the notation

where

and

II 1Di+1/2,i,n-O (xi+1/2,], Ui+l/2,j

Ui+l/2jn (Vin+l/2j, (W + Wi+l,j))n

Vi+ 1/2,/’ -K in+ 1/2,/’ (P+1," P’)h

and W0. correspond to (3.7) and let d a:l+l/2,],n be the evaluation of the Da:-coefficient

kl klat the corresponding values of p Similarly, define the other values of D, and d,.Then, by (4.64) it is easy to see that

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-<TaD(U")Ta_E{Dll )2 22 .+1)2i+l/2,i,n(Ox!.+1 -I--Di,i+x/2,n(Oyii

i,i

(4.65) +(D 12 21 n+l n+l n+l en+l

+1/2,.. (ax7+ +i,i

, +d,)(a.+1 +a_.)(a +a._l)}h

11 22Next, the argumentsin d+/2j,, and d,+1/2,, can be shifted to (x, u) by inducingan additional term of the form O(p"[[2,h[[Vh"+[[2). Hence, after another shift by hin the evaluation of the coecients,

,i k,l

n+l )2 22,. + (a.+’)i,l

(4.66) +(d, +d,)wg-,g, +a g,_)}h

+ o(I." u"l + llp"ll.a )llv" +’II.The second term on the right above is nonnegative, modulo anotherO (llp ll2,h ll"+ll)- term, since,Combine this remark with (4.62) and (4.66); it follows that

-<VD(U")V(4.67)

-{[u"and coercivity holds for small h, provided that ]u"-U"] tends to zero with h.

The term on the right-hand side of (4.22) can be estimated using (4.13) and (4.64)"

(4.68)amo,<Ov

If (4.67) and (4.68) are applied to (4.22), the remainder of the argument between(4.22) and (4.48) can be repeated to give (4.48) again with p" replaced by

(4.69) max l(IU-uloo+h)+p".O<__k<-n

The remainder of the argument is essentially unchanged, and the final inequality (4.56)for the error in the concentration (or saturation variable) is again valid, with theconsequence through (4.10) and (4.13) of an O(h) convergence rate in/2(f) for thepressure and for the Darcy velocity.

REFERENCES

J. H. BRAMBLE, A second order finite difference analog of the first biharmonic boundary value problem,Numer. Math., 4 (1966), pp. 236-249.

[2] G. CHAVENT, A new formulation of diphasic incompressible flows in porous media, Lecture Notes inMathematics 503, Springer-Verlag, New York, 1976.

[3] J. DOUGLAS, JR., The numerical solution of miscible displacement in porous media, ComputationalMethods in Nonlinear Mechanics, J. T. Oden, ed., North-Holland, Amsterdam, 1980.

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[4] J. DOUGLAS, JR., M. F. WHEELER, B. L. DARLOW AND R. P. KENDALL, Self-adaptive finiteelement simulation of miscible displacement in porous media, SIAM J. Sci. Stat. Comp., to appear.

[5] J. DOUGLAS, JR., B. L. DARLOW, M. F. WHEELER AND R. P. KENDALL, Self-adaptive finiteelement and finite difference methods for one-dimensional, two-phase, immiscible flow, to appear.

[6] J. DOUGLAS, JR., Simulation ofa linear waterflood, Free Boundary Problems, vol. II, Istituto Nazionaledi Alta Matematica "Francesco Severi", Roma, 1980.

[7] J. DOUGLAS, JR., Recent results concerning simulation of miscible flow in porous media, Seminar onNumerical Analysis and its Application to Continuum Physics, Coleco Atas, vol. 12, Rio deJaneiro, 1980.

[8] J. DOUGLAS, JR. AND T. F. RUSSELL, Numerical methods for convection-dominated diffusion problemsbased on combining the method of characteristics with finite element or finite difference procedures,this Journal, 9 (1982), pp. 871-875.

[9] J. DOUGLAS, JR. AND J. E. ROBERTS, Comparison of some numerical methods for miscible displace-ment, to appear.

[10] R. E. EWING AND M. F. WHEELER, Galerkin methods for miscible displacement problems in porousmedia, this Journal, 17 (1980), pp. 351-365.

11 ., Galerkin methods for miscible displacement problems with point sources and sinksuunit mobilitycase, to appear.

[12] T. C. FRICK, Petroleum Production Handbook, vol. II, McGraw-Hill, New York, 1962.[13] D. W. PEACEMEN, Improved treatment of dispersion in numerical calculation of multidimensional

miscible displacement, Soc. Pet. Eng. J. (1966), pp. 213-216.[14] ., Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.[15] T. F. RUSSELL, An incompletely iterated characteristic finite element method for a miscible displacement

problem, Thesis, Univ. of Chicago, Chicago, 1980.[16] M. F. WHEELER AND B. L. DARLOW, Interior penalty Galerkin methods for miscible displacement

problems in porous media, Computational Methods in Nonlinear Mechanics, J. T. Oden, ed.,North-Holland, Amsterdam, 1980.

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