Modeling of Gravity-Imbibition and

Gravity-Drainage{ Prdcesses: Analytic

and Numerical SolutionsNl,eis Beth, SPE, i%@ Natl. Laboratory, Ole K. Jensen, SPE, Maersk Oil & Gss A/S, and Blrger

Nielsen, SPE, Cowiconsult.

SPE /gqJ g

Summery. A matrixifracwre exchange model for a fractured reservoir simditor is described. Ofl/wat~ imbibition is obtained froma @sion ~uation with water saturation as the dependent variable. Gas/oif gravity $frainage and imbibition are c$dcu@ted by ~ginto account the vertical saturation distniution in the matrix blocks.

Introduction

In most simulaton intended for natirally fractured resemoirs, thefractuw and matrix systems sre considered to be two overlappingmedia. Flow between tbe two is described in various ways by meansof source and sink terms. The description of the matrixffractuieinteraction is a key point in the modeling of dual-porosiiy~*m,l.14

In this paper, the modeling of oilhvater imbibition is based onthe diffusion equation approach of Becknei et al. 14 The effect ofgravity is inco~o&d &ugh a rncditication of the bouncktry con-ditions imposed. Analytical and .nunqical solutions are pcesent-ed, and computed results are compsred with experimental data.Gas/oil gravi~ drainage and imbibition are calculated by takinginto consideration the vertical satiation distribution in the matrix.Tbe principles for the implementation of the proposed methods ina reservoir simulator are described.

The folfowing limitations and assumptions apply.1. Tbe models presented are wdid ordy for two-phase oiUwater

and gas/oif systems.2. Matrix blocks within a grid cell am identical and box-shaped

with diinensions Lx, Ly, and L.z.3. For oillwater systems, capfhuy continuity exists inside a &d

cell between vertically stacked matrix blocks.4. The two ph~es ii the fFacpue system are gravity segregated.5. Analydcsd solutions cam be obtained only in the oif/water case

and onIy.if the water level in the fracture system rises with i mn-stant velocity and the diffision coefficient is constant.

& The mah-ix-bkwk gas and oil are at qilfarylg$+witational equi-librium. .

Flow Equatkms

Dual-porosity reservoirs ~e modeled by the continuum approach,where the fracture and matrix systems are considered to be twooverlapping continuous mqfia.

The basic equations for isothermal fluid flow in porous mediaare transformed to a system of ordinary tiifferential equations bymeans of the integral finite-difference method (see Pmess andBedvarssonls). In case of a dual-porosity, singls-permeabilityreservoir compxed of a congnuous fractuie system containing dis-continuous matrix blocks, the following .quations are obtained foreach component (1=0, g, or w) and the W grid cell in ~e reservoir.

Fracture e~on:

(d/dt)(mj)k= ~ {[qa~ac~ +(%o&ZJtili

-n4fPac&flk,}. . . . . . . . . . . . . . . . . . . . . . . . . . . ...(1)t

Manir eqtion:(d/dt)(mJj=- ~ (qapaC&)mjk, . . . . . . . . . . . . . . . . . . (2)

,fl

where mi=V@~p.JaC& . . . . . . . . . . . . . . . .. . . . . . . . . . ...(3).

Cowrlght 199T Society of Petmlern Engineers

and (u&f= &/Q.. ba,~Pa,t+@#h)kt]. . . . . . . . . . . .(4)

The summation is over all phases-i. e., .x=o, g, and w. The sumover Index P is over all grid C& adjacent to grid-cdl number k.Hence fndexktrefers to,tbe bou@qJ-between the gdd dfs k imdt.

The individual terms of@ 1 and 2 describe the transport ofComponent i through Phase a byvarious mechanisms. Fo

Fracture

_L,

Fig. l-bfatrlx block partially submerged In wat&. IIlustraflonof Imundaw conditions Imposed.

L

zwm

Ovl[ %

o

Fig. 2Mafrix block surrounded by fracture, 011/wafer system.

ever. analytical solutions caR be derived if the fracture water Ievelis specified as

z.#=v# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (lo)

where vw =velocity of the fracture water level. Analyti@ sOlu-tions to the diffusion equation arc given in the Appendix.

Contribution From Gravity. h a water/oil imbibition pmccss,the gravity effect is small and is nsgledcd during the derivationof Eq. 5. Itis bnpoztant, howaver, fordetetininin gthetitimateoil rccnvcry from makix blocks. The effect Pf gmv@ is ~clud~in the calculated water/oil imbibition by imposing the final or uki-IIIRtC average RWix-block water satu@ion as the boundary con-dition SW in Eq. 8.

Consider a matrix blcck containing oif and water par!ially zub-merged in wdtcr, as sbmvn in Fig. 2. The oil and water inside thematrix block are assumed to be in capiIlary/grnvitationaf equilibri-um. fR the fracture. cauillaw forces are assumed zero, ond the twophases are completel~ seg~egatcd.

The system in Fig. 2 is in capilkwy andgravitaticmal equilibriumwhen

g@wPo)(zw-zm)+Pcm(zm) =o$ . . . . . . . . . . . . . . . (11)

assuming equal phase densities in fractnrc and matrix.This condition @q. 11) cnrrcspon& to or dgfmcs m ultimate aver-

age water saturation for the matrix block, Sw, which is US~ ~the expression for the iqmscd boundary saturations (@ 8).

130

- Lz

Sg = 1- s,. -%__ -_-_. -

- 20..

gas ~___ - _ ---_--

-%*..

oil

Fig. 3-Matrix block surrounded by fraqure, gas/oil system.

GaslOIl GrawlW Drabmfe and Irrrblbltlorr

higa.doil systems, thegravity ef@t is of primary importance fnthis sccdonon expression is derived for the mati-block oil drainagerate wbcn gravity and c.apilbuy forces arc the driving mechanisms.

Consider a mimix block containing connate water, oil, and gasthat is sln?ounded bya d-actwc system filled Widl gmvity-.%grcgatcdoil and gas (sze Ffg. 3). Tbe oil and gas tilde the matrix blinkare assumed to bc in capillary/gmvbadonal eqdfibrium. Belowz~m, tied blink OJmins no gas. Above z.. ~d if ZOIII

rABLE lDATA FROM KLEPPE AND MORSE EXPERIMEIAND COMPUTED DIFFUSION COEFFICIENT

side bf ~. .13, qc,o;ti, is @e oil ilow.mmed mainly by oil/watercapilkuy pressure, but modified to include the contdbution fromoiUwater gravity forces. The third term is the gas/oil gravi~drainage and imbMtion term.

The matrix/fracmre gas-exchange rate contains two terms, thestandard convection term and the gjsloii drainage term. Tbe latterwas given in Eq. 12. The flow is seen to be driven by gravitationalas. well as capilkwy forces.

The matrix{ fracbme exchange rate of wafer contains a convec-tion term and a capillary pressure term. TIE COIIVW6011 term isdrivti by. differences between matrix and fracture in reference prez-sure. FIOW tamed by oiUwater capillary pressure is contaioed inthe second terni, which is determined from the diffusion equationas explai@ previously for a single ma@x blc.sk. The mti-blockboundary condition is modified to take into account the contib..tion from gravity, ~ shown previously.

It is.assumed that a!J matrix blocks within a grid cell are identi-cal as far as dimeosiom and pbysicalproperties are concerned. fbisis not necessarily so with regard to fluid content and environment.Tbe 0i3/water and gaslol situations are txeated differently, as ex-olained below... ..--... ,,

l@ number of matrix blocks within a grid cell is

.ntiB=vfL&yLz. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . (16)

ne number of ma~x blocks stacked vertically witbin i grid cell is

n~=tifLz. . . . .. . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . .. (17)

Tbe number of matix blocks in each matrix block layer within agrid cell is

L=nMBk7K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(1.3)

OdiWatez System. fn the OiUwiter case, capilhy continuity isassumed to exist betw%n the matrix blocks io the s~k. fn fact,@e dfision equation is solved for just one rrmhi.x block with theheight Az. This approximation has been found acceptable by com-paring results from two simulations. In the fti case, five grid cel$of height AZ, each with one matrix block having Lz =&, werestacked ve@ally. The system initially bmained oil and COooa@water. Oil and water were produced from the top, and wa~r wasinjected from the bottom. The second case was identical except that.there was only om grid celj of height 5Az containing oe matrix

Permeability, mdPorosityWater viscosity, CPOil viscosity, cp

SW0.30000.31740.32470.3521

0.3695

0.36660.40420.42160.43690.45630.47370.4911

0.50340.52580.54320.56050.57760.5953

0.61260.0300

km k

0.7500 =0.6979 0.00030.6477 0.00070.5591 0.0010

0.5506 0.00140.50420.45620.40960.36090.3146

0:27320.21660.16640.11810.07640.04320.0163

0.00760.00290.0000

0.00170.00220.00290.0036

.0.00480.00660.0089

0.01170.01540.02370.02460.04730.0612

0.0797.0.1000

Pm.

.=

3.ioo2.s442.1171.8041.5691.4111.255 ~1.1091.0020.90970.626S0.74290.6697030720.54460.4s050.44190.39330.24630.3000

2900.225,

2.:

(ftz%c). 0.0000

-0.9261 x 10 8.-0.1396 X 10 5-0.14S6X10-6-o.1483xli-5.-0.1576 x 1.0.-5.-0.1,731 X.lo -$-0.1967 x10-5-0.1925 x10-5

-0.2270x 10 5

-O.31OOX1O-5-0.3600X IO+-0.4312 xI0-5-0.5227X 10 -5-. 0.7479 X.10-5:0.9654x 10 5-0.1251 X 10-*

-0.1590 XI0-4-0.2021 XI0-!-0.2576 X 10=4

block with L= =543 In this example, we fourid that calailated ofi,recovery was almost identical for the MO cases.

At present, the cahd@on of qd,o,mf(E!q. 13) and qc,W,@@q.15) is semiexplicit. ,~at k, the fracture water level it the previoustimestep is used for the calculation of tbe boundary conditions.

Gas/Oil System. h the gasloil case, the matrix bl@s in a gridcell are &vi@ into tbr.x groups, as shown @Fig. fl. Matrix bloc@in Group 1 are surrounded by gas and re.sid.d fracture oif, if any.Matrix blocks in Group 2 facefhe fracture. gas/.oiI contact. Matrixbltiks io Group 3 are ddIy submerged in oil (~d water, if soy).It is assomed that only matrix blocks belonging to Groups. 1 and2 contain s.9s. If tbe grid-cell matrix gas saturation is S-, thwthe average gas saturations of niatri.x blccks of Types 1 Wd 2 are

Vsgm ..58,1=

Vdnml +%211 b+,)]}.. . . . . . . . . . .. ..(19)

and ~gm2=[l (zOfLz)]~g,l if nMBl> O..... .:: .,. . ..tiOa)

or ~gav2=(A?/Lz)Sgm if:nMB1=O, . . . . . . . . . . ..... ..:(206)

where nB1=nMBnw2nMB3, . . . . . . . . . . . . . . . . . .. ..(21)

MB2=nL lfO

Comparison with Kleppe and Morse

LOW Velocity

: s..,,,,.

---: .Um. rk.l

/

i!Liz_-_:010203040 $J6970n090100

Cumulative Wer Jnjectiofi% pv

Fig. 5-Comparison between expedmerit,al, analytic, and iw.meqlcal results on the low-rate Kleppe and Morse xperlment.

sion equation-(Eq:5$ with a diflisio~;oefticient m. 6) was usedwith the boundary condition in Eq. 7.

The corresponding es flow rate is

qG,#naf=-qG,O,n@ . . . . . . . . . . . . . . . . . . . . . . . . . . ..~~

Validation of Approach-Exampk Problem+

10 this section we comp6retbe diffusion approach with two &pesof laboratory experiments snd the gadoil gravity dmioage modelwith a fin~grid iiiulation.

Comparison With Kleppe aod Morse Experiment. The Kleppeand Morse 17 expaiment was previously modeled by Beckner etal. 14 with a fine-grid simulation using the diffi!aion equ6tion. We

present our results with the diffusion approach using a numeric61solution tecbniaue to solve the diffusion eauation with a nordiieardiffusion coeff]~ient and an analytic soMi& &.th a constant diffu-sion coefficient.

fn modeline the extmiments of Klemx ad Morse, the 2D diffu-

TABLE 2-RELATIVE pERMmBILITIES AND CAPILLARYPRESSURE DATATOGETHER WITH THE COMPUTED

DIFFUSION COEFFICIENT

Permeability, mdPorosilvWater iiscosity, CDOil viscosity, ip

0.160.197

0.4s3.3

sA0.0620

0.09270.12340.15410.12470.21540.2461

0.27680.30750.33620.36660.39950.4s02

0.46090.49160.52230.55290.56360.6143

0.6450

kJ1.0000

0.93360.66730.60090.734s0.6631

0.60180.53540.46900.40270,33630.26990.20360.13720.06620.05710.03090.01190.00360.0000

k~

0.0000

0.00290.00590.00660.011s0.01470.0177

0.02060.02360.02650.02950.03240.03540,03230.04340.05260.06370.07930.06500.0860

:P3

141.756.5037.5431.2726,5121.76.19.91

16.4016.8915.3814.0013.1712.3311.46

10.66

9.S206.9346.147

,7.3110,2125

~ti::ec)0.0000

-0.6762x 10 6-O.3I9OX1O-6-O.2O2SX1O-6-0.2251 XlO-e-0.1 S76X10-6-0.1096 xIO-6-0.1064 x10-8-o.ii5sxlo-6-0.11 .tsxlo-e-0.87s1 X1 O-7-0. W66x10-7-0.571 SXI0-7-0.4598 x 10-7- 0.2S67 X 10 7-0.2461 X10-7-0.1428 x10-7-0.5731 x 10 -a-0.6407x 10 a-0.0000

Comparison with Kleppe and Morse

64. High Velocity

z .,.,,.,,..,w*- : ..W.

2---, ..,.,,..1

:40..

:

:,.

=

,:m -

~

~io -

C$

,0 m ,0 0 50 60 n so 90 !04Cumulative Water Injection % Pv

Fig. 6-Comparison between experimental, analytic, and nu-merical results on the high.rst6 Kleupe and Mors6 experiment.

In Table 1, the relative pemmabilities,and WLP~W Pr=mre ~

the KIeppe and Morse experiments are given. The two water in-jexfion 16tes of 3.3 aod 35 cm3 huh [0.03 and 0.32 B/D] corre-spond to a water velocity in fracture of 3.403 snd 36.076 @d[11.2aod 118 fUD], respectively. The 6odytical solution derived in theAppendix MS been applied, with a di~sion const60t correspond-ingto maximum water saturation. A comparison between the ana-lytic, numerical, amd experimental restits is shown in Figs. 560d6. In both cases, gcdagreementis obsqw?d. The Iowervaluefromthe samlytic solution in the high-rate case can be explained by thefact dmt no contribution from the top face is added when the waterfront reaches the top. The calculated results for the high-rate cased~agree with the results reported by Beclmer st al 14

The cfitical water injection rate, introduced by Matt6x and@S18 mthcmtiat wtichtie water advmcemtektie~~block equals the water advance rate in the fracture, correspondsinthediftilon approach tothewater velocity rsoge ir,thefrac-tore, where a steady-state solution of the diffmion. equation isachieved. From the derived .3mdytic solution in the Appendm, themmsient p6rt is seen to be negligible when

DItiI,,(Z#@)>l, . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(28)

where z@ =height of the water froot in the fracture. LIIus, the ,ait-

icalrate isanonooique oumber. ffwedefme tbecriticd rate astie m.te at which the transient pat of the solution conhibutes

TABLE 3RELATIVE PERM2ABILITV AND CAPILLARYPRESSURE DATA FROM REF. 20 I

Reservoir pressure, PSIMatrix permeability, mdPorosityWater visoosity, cp

Grid dimensionAx, nAy, ftAZ, ft

4,5001

0.290.35

SX8X8

0.01, 1, 2,2, 2,2, 1, 0.010.1, 1, 2,2, 2, 2, 1,0.1

s_.J!-0.2000

0.25000.30000.35000.40000.4600

0.5000

0.60000.70000.7500

k40.0000

0.00500.01000.02000.03000.0450

0.0600

0.11000.18000.2200

Sq kv

0.0000 ~ 0.00000.1000 0.01500.2000 0.0500

0.3000 0.1030

0.4000 0.1900

0.5000 0.31000.5500 0.4200.

0.3040 -1.2000.1540 -4.0000.0420 -10.000.0000 -40.00

Pcwk (psi)

.~ _

1.0000 0.39330.7000 0.44ss0.4500 0.49820.2500 0.60310.1100 0.7604

0.0280 1 .s3730.0000 2.0243

Morse data. l%is is in a!qeement with the observation of Beckneret al. 14

Comptiison With Field hnb~ition Data. ma typical laboratoryimbibition experiment, a core plug saturated with oil is immerselcompletely in water and the uptake is recorded against time. fImwfeling such 2n experiment with a difision process, coontercu2-ront oil flow has to be taken ioto account. This baz been done byB12ir.,19 who derived a diffusion equation with tb~ miencOefflcient

D2(SW)=[kkOkW/(@O+pOkW)l(dPJdSJ. . . . . . . . . . . ..(29)

At low rate, D2 =D1, while D2 goes to zero for SW =SW,- andD, attains its maxirnom value.

Toble 2 gives relative permeabtities and capihry pressure datafor am example. In modeliig the corresponding imbibition data, a3D diftision equation was used with the boundary condition

SW(x,y,z,$=(l -,Sor)(le-&) . . . . . . . . . . . . . . . . . ., . . ..(30)

at the faces of a 3D block with side lengths Lx, & and L,, Here

6 is an inVe15e time constit reflecting a delayed wetting of theblock stices. If@=m, imtanbmeous imbibition occum at the blccksurfaces. fn modeling the imbibition &ta, a constant diffosion ineff-icient of m2x(lD2 [) was used in the mmlydc solution of the totaluptake derived in the Appendix.

As seen from !be analytic expression, the following dbnension-less number is naturally introduced

b=@l#(l/Lx2+l/Ly 2)Dsl. . . . . . . . . . . . . . . . . . . . . .. (31)

Because imbibition occurs tbmugb all faces, we should rather de-fine this dimensionless gzoup as

b=--$min(

1 1 1

)llLx~ + ULY2 ULX2 + l\LZ2 ULJ + 11LZ2

. . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . ..(32)

This provides a way to scale the inverse time Conztant .8, deter-mined from laboratory experiments, to a tield value. A compari-

Gas/Oil gravity Drainage

Fine Grid - Single Bloci Dual Simulation61

;.5 Pcgof = 0L

g .4-

~

----------

>,3-.

----: ,!,!. ,,..,

? : ,,. WI.~ ,2-

g

.1-

,00! 234 567

Years

Fig. 8Comparison between single-block dual simulation andfine-grid simulation. Data from Ref. 20 with no coplllarypre30ure in fracture.

son of both sptaneous and delayed imbibition with the laborato~imbibition data is shown in Fig. 7. This shows tiat a delayed imbi-bition with b=O.75 gives a very goad agreement with the experi-ment, whereas a spontaneous imbibition does not.

Comparison Gas/OiI With Fim-Grfd .%mdation. Single-blockstudies of gadoil gravity drainage from a 3,o5 x 3.o5 x 3 .05-in[1OX IOX 10-ti] block, with data given by Fwoozabadi andTboma220 and listed in Table 3, have been compared to an6X8 x8 tine-grid simulation using ECLIPSE. The grid data are afsogiven in Table 3. The rezukz are shown in Figs. 8 ond 9, and verygood agreement is observed both foz zero capillary pressure in thefractures and for Pczo =0.69 kPa [0.1 psi] in the fractures.

SPE Compsmtive solution Pzoject Problem. fbe reservoir simu-lator with the described tecboique has alzo be.$n used on the SixthSPE Comparadve Solution Reject problem. Descripdon oftkpmb-Lem and results are given in Ref. 20.

cOncIusIOrIs

1. Special core analysis data can be tested for consistency. Rep-resentative inhibition rates can be expressed with a few parameters.

2. The matrix/fracmre exchange term takes into account satum-tion gradients within mahix blocks and the imbibition through oewlycontacted matrix-block su@ces. Furthermore, wettabilby proper-tie$ cm be modeled by a simple akeration of the surface diffusionrate.

3. Calculation of gas/oil drainage takes saturation distribution invertical matrix blocks into account.

Nomenclature

,4 = 2rea, mz [W]C = maw fmctionD = ditlizion coefficient, m2/s [t12/see]f= matrix block fraction

8 = gmvit2ti0n2f constmt, dsz [ft21xclAk = vertical distance betweeo grid cell centers, m [fi]

k = permeability, m2 [red]L = distance between grid ceU centers, m [tl]m = mas, kg flbm]

L = number Of IMtri.x blocks in each matrix block layerwitbin a grid celf

nm = number of matrix blocks in @d UII

n= = n~ber Of maw bl~~ SWM vezdially witbin agrid cell

r1.

[

I

Gas/Oil gravity Drainage

Fine Grid - Single Block Dual Simulation,61

.5->0

Pcgof

!Lo

~.4-

m.

J._-----: single block

>.3-C : Fine gridu

>

0 .2-U

alx.. .

.i -

.0 I I 8 I f t I

0 1 2 3 5 7

Yeari 6

lg.9-Comparison between single-block dual simulation and fins-grid simulation. D?fsfromIef. 20 with capillary pressure In frscfure equal to 0.1 psi.

n. = number of latersl mesh points in mstrix blcck @dnz = number of vertical nmzh points in matrix block grid

p = pressure, Pa bsilq = volume flow rate, msh IS/D]

q, = flow caused by oilh?ater capillsry pressurs, m3 /s

S=satistion

~~ = sm&zt average mshix-block gas saturation at whichmaximum or top mahix-blcck gas saturation equals

Stm, gss saNrstiOn at rssidual oilSiW = connsts water saturationS$ =,wafer saturation atw@choil/water capilkuypressure

is zero

t = time, seconds,,

%f(o = ~ewpenfimre watezlevilequals2, secondsu = phase velocity, m/[fbD1v = velocity, M/s [ft/D]Y= volume, m3 mbl]x = akial (literal) ccordmate, m [ft]z = vertical coordinate, m [tl] ,

.Z.f= fmctureofl Ievel rehtivet omatiblcckbottom, Fig.4, m [ft]

z~fti =grid W-fracNre oil level, Fig. 4,m[ft]Z.MQ = -e~t kVelabJve matrix-block bottmn w,here

matn.x-blpck gas saturation is maximum, Fig. 3, m[n].

&U lsrg*tl evelabovem atrix-blockb ottOm~hcre~~.blcxkgsz saturation is smallest, Fig. 3,m

zd= frsctme Watsrlevel relative tomstrix.block bottom,

i3 = invezse time constant, seconds1~ . mentier

A=mObilhy,Pa:s-l fpsihec]

o =denshy, kg/m3 @brrJgall,J = shape factor, llmz [1/ftZ]

@ = porosity

Subscriptsb = boundary~ . api!.lsg

f= t%wture

g= gas

G == gravityi,j = mesh-paint index

.?,( = @d-cell indexnm = matrix

mm = maximumfi=am

n = mode numbero=~s

~ = r~idau . ddmab

x = x directiony = y direoionz = z direction

a = phaze index

sufH3cript3

i = cnmpment index

n = timeztep index- = average

...

Fig. 1, m [ft]Z- = sm~e~t be] above matrix-block bottom wbe~ - Acknowledgment

mstnx-blwk oil saturation is maximum, Fig. 2, m Pardsl funding of this work by the Dsrdsh Mmistrjof Energy is[ft] gratefully acknowledged.

134 SPE R i E i i F b 1991

References

1. Kazmd, H. et d. , Nmwicd Skmdadrm cd WaterfOfi F!OW .U Nmu-tiy Fractured Reservoirs,,, SPEJ @cc. 1976) 317-26; T-., AIME,261.

2. Oilman, J.R. and Kazemi, H.: Jmprovementz in Sinndation of Naluc-ally Fmcfored Re.SeIVC,kS,S, SPEJ (Aug. 1983) 695-707.

3. Thomas, L. K., Dixon, T, W., and Pierson, R. G.: F.-a@red Reser-voir Simulation, SPEJ (Feb. 19S3) 42-52.

4. Rossen, R.H. and Ctm, E.: Simulation of Gas/Oil Dra@ge andWakr/Oil Jmbibitim in NmmJIY Fractured Reservoirs. v, SPERK (Nov.198P) 4d4-7& Trm.. AJME.2S7

5. Dml JL3L aid Lo, L:L~~ Skdations of NamrdJy Fnwtvsed Resez-voics, WERE (f&y 1988) 638-48.

6. Litvak, B.L.: SimuJatio and Ch?.mdmizmion of Namrdly FrmturedReservoirs, Proc., Reservoir Cbmctmization Technical Conference,DaJtas (A@t 29-MrIy 1, 1983).

7. Sonier, F,, SouiUard, P., and Blakwich, F. T.:

ISeth

*

f,: .

--i--Jensen Nielsen

Nlels Beth has worked for the last tO yeacz in the appllcttion and development of resewolr simulators at RisO NalLaboratory In Roskilde, Denmark. He holdsan MS degrem Ielectronic engineering and numerical mathematics from tiTechnical U. of Denmark. Ole K. .knsen Is chief Produtik

engineer with Maersk Oil&Gas A/Sin Copenhagen. His maiinterests are reservoir development planning and resewnsimulation development. He holds an MS degree from thTechnical U. of Denmark and a PhO degree from the U. .3-

~ ,rb.o ,ru

5.2-2

,1-

.0 r01 2 3 56 >

Yea&.

Fig.9-Co2np8rixonbetween Wxwetoryimbibitbn de@mdepente22m24endd.14yetdlmbiMtion modol.

Fig.11-Compszleon botwooneingle-block dud-aimuletionend flnx.grid xImulaUon. Detx hum SPE-10 compsretlve solutionpm-. w~ apill~w pmesofe in fmcturo equai to 0.1pd.

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