Predicting Gravity-Drainage Performance

Using a Three-Dimensional Model

H. N. HALLMEMBER AlME

PAN AMERICAN PEWtOIEUM CORP.TULSA, OKLA.

AbstractReservoir and prod[tcing Awictetlstics can :overn the

deci.riou 10 me either a one-, nvo- or tllvee-dittrensiotzalnwdcl for I)]aking prcdic(ions for gravity-drainage re.rer-wir.~. Examples of conditions rcqniring one-, IWO- andrhree-ditt tensional calc[dtuion r are given. In 196 I rhe au-thor prcsen!ed a me[hod for predicting t)tte.dit??el~.sior?algvavi[y-drainage performance. Titi.~ work iws been extendedm ohluin a [ilreeditnensional nmdel, wilici~ is desctvi~edand jor )ri~icil a saitlple proi)lenz is pre.rentrd.

IntroductionMarry pnpcrs on gravity drainage or gas-cap drive- have

been presented for conditions where the reservoir is treatedas a single arcaI segment so that, in effect, only one-dimen-sional flow is considered, Other authors- have given evi-dence showing that the one.dimensional concept would beunsatisfactory to predict reservoir performance for cerutin(ypcs of reservoirs. Experience has indicated that specificreservoir and producing conditions govern whether one-.two- or three-cfimensional concepts must be used as a basisfor reliable reservoir performance predictions,

Previously the author presented a paper outlining amethod for predicting the performatlce of gravity-drainagereservoirs? That paper treated the reservoir as a singleareal segment so that it was applicable for only one-dimen-sional flow. The work has been extended to obtain a three-dimensional model that can be used to predict performanceof gravity-drainage reservoirs. Although capable of beingthree-dimensional, it can be used equally weII for one. andtwo-dimensional predictions.

This paper discusses those reservoir and producing con-ditions governing the choice of model, and also describesa three-dimensional model suitable for predicting reservoirperformance in the more complex situations.

DiscussionThe Choice of Model

To predict reservoir performance, it is necessary to rep-resent the reservoir by a physical model. A mathematicalmodel, based on the assumed physical model, then is used

Original manuscript received in Society of Petroleum Engineers officeAUK.11, 1967. Revised manuscript received March S1. 1968. Paper (SPJ21878) was $wesentcd at SPIi 42nd Annuat Fall hleeting held in Houston,Tex, Ott: 1-4, 1!!67. 0 Copyright l!J68 A.nericwn Institute of Mininr!,Metallurgical, and Petroleum Engineers, Inc.

References 8iven at end of Paper.This paper will be printed in Tran8actioms volume 243, which will

cover 1968.

to predict the movement of oil and gas with~n the reser-voir. The word model is used here to connote the com-bination of both a physical model and a mathematicalmodel based on the physical model. Certain types ofreservoir situations may be predicted suitably with a one-dimensional model, whereas two- and three-dimensionalmodels may be required for other situations. This sectionillustrates reservoir conditions associated with the use ofa specific type of model.

Many types of reservoir structures are such that gravitydrainage possibly could be an important factor in oil pro-duction. Production practices, as well as the nature of thestructure, can influence the type of model that should beused. Obviously, it is impossible to discuss all combina-tions of reservoir structure and production practices thatmight be encountered in gravity-drainage reservoirs, Thefollowing arc typical examples of conditions where a choicemust be made among a one-, two- or three-dimensionalmodel to predict reservoir performance satisfactorily.

Conditions Requiriog aOne-Dinlensionttl Model

Fig. la shows contours for an asymmetrical anticline inwhich gravity drainage might bc expected to be important.Fig. Ib depicts a cross-section along Line A-A in Fig. 1a.Existence of a gas cap is a good clue that gravity drainagewill be an important factor in producing oil from thistype of reservoir. However, it is not essential since a gascap will form as oil is produced from the reservoi!, If theproper combination of reservoir permeability and with-drawal rates is present, the reservoir will behave like a

A

l-a l-bSTRUCTURAL CONTOUR SECTION A-A

Fig. lTypical reservoir, one-dimensional model.

.- ~~#-

tank in that the gas-oil contact will move down-structureas a level discontinuity, and there will be no significantregional pressure gradient throughout the reservoir. Oildisplaced by movement of the gas-oil contact spreads outevenly and, in the case of operations not using pressuremaintenance, gas evolved throughout the structure willmove upward uniformly through the oil column into thegas cap. The dominant movement of gas is updip, and themovement of oil is downdip, therefore, it is one-dimen-sional. These conditions would be expected where proper-ties (such as porosity and permeability) are uniformthroughout the reservoir, where well spacing is uniformand where the producing rate from the field is Iow incomparison with the gravity reference rate.

Fig. 2a shows how the reservoir is separated into verticalblocks, (In this paper, %ertical blocks denotes blocksstacked in a vertical direction and horizontal blocksrefers to blocks lying in the same horizontal plane, ) Thefinal physical model of the reservoir to be used in one-dimensional calculations is shown in Fig. 2b. The reservoiris characterized by a series of stacked blocks. The area ofddch block is assigned in such a way that the pore volumeof each block corresponds to the actual portion of thereservoir it represents.

Conditions Requiring aTwo-Dimensional Model

Fig. 3a is a contour map for a slightly dipping reser-voin Fig. 3b is a cross-section view along Line A-A. Cookstated that even in reservoirs having low vertical perme.ability (less than 10 md ), gas liberated from solution cansegregate to the top of the sand and then flow through asecondary gas cap along the top of the sand into the mail;gas cap at the crest of the structure. He stated that thistype of performance was recognized as an importantmechanism active in massive sand reservoirs in westernVenezuela. Cook also concluded that gravity-segregationperformance could be influenced greatly by a downdipconcentration of withdrawal in this type of reservoir. Fig.

GAS-OIL CONTACT

2-aCROSS SECTION

2-bPERSPECTIVE

Fig. 2Physical model of reservoirmodel.

for one-ditnerrsional

4 shows the gas-oil contacts that could result. Gas sat-uration that can build up subsequently in downdip areaswill lead to early GOR problems, Other authoma haveinvestigated the tilting of a gas-oil contact for crestalgas-injection operations.

The general flow patterns that would exist under condi-tionsshown in Fig. 4precludethe use of a one-dimensionalmodel for predicting reservoir performance. If there areno regional lateral variations in reservoir properties (dip,permeability and porosity) and well concentration is lat-erally unform, then this type of performance could bepredicted with a two-dimensional model.

Fig. 5a depicts how the reservoir would be modeled bybreaking it up into areal segments that are further dividedinto vertical blocks, The reservoir model is shown in per-spective in Fig. 5b, with dashed lines showing the directionin which oil and gas are assumed to flow.

Conditions Requiring aThree-Din~eosiooal Model

Theasymmetrical anticline shown in Fig. 1 is used againfor illustrating reservoir and production ~~.ditions requir-ing a three-dimensional model. Instead of having equalwell spacing, it is specified that a certain production rateis to be maintained from the reservoir with a limited num.ber of wells. The main question is whether there will besufficient lateral migration of oil to achieve a conditionsimilar to that shown in Fig. 6a where the gas-oil contact

A--A

GAS-OIL CONTACT3-a

STRUCTURAL CONTOUR

3-bSECTION A-A

Fig. 3Typical reservoir, mwdimensionai model.

PRODUCTION

(

GAS-OIL CONTACT

Fig. 4-Gas-oil contact profi!e, segregated flowcross-section oj reservoir.

moves down uniformly. or whether there would be apronounced lowering of the gas-oil contact in the vicinityof the production wells as shown in Fig, 6b, If movementof the gas-oil contact is not uniform, it will be necessaryto consider the actual location of producing wells andregional changes in porosity, thickness and permeabilityto calculate this type of performance. Actual well produc-tion rates obviously cannot be estimated on average con-ditions; the true pressure in the vicinity of the well andlocation of the gas-oil contact at the well also must beconsidered.

Fig. 7 shows how a reservoir similar 10 that shown inFig. 1 worrId be modeled to make three-dimensional reser-voir predictions. The reservoir is separated into a largenumber of areal segments. each subdivided into verticalblocks. Each block is assigned a value of area, porosity,permeability and fluid content corresponding to that ac.[ually occurring in the reservoir. The three-dimensionalmathematical model constructed using this physical modelwould account for oil and gas fiow in two lateral direc.tions, and for the vertical direction. These are denoted inFig. 7 by arrows.

Three-Dimensional ModelThe one-dimensional method previously referred to

is applicable for conditions of either complete or partialpressure maintenance. or for normal pressure depletion.Provisions were made to account for vertical variation inpermeability and fluid composition. The method was basedon representing the reservoir by a series of verticallystacked blocks, similar to that shown in Fig. 2b, Themathematical model that was developed accounts for theflow of oil and gas at all points ~hroughout the sectionwhile simultaneously satisfying material balance considera-tions. The numerical expressions developed consisted ofgas and oil material balance equations for each block inthe reservoir in conjunction with expressions describing theflow of oil and gas between blocks. These equations areimplicit; i.e.. they involve unknown values of pressure andfluid saturation at the end of a time step. An iterative

CROSS SECTION

t

t--+-DIRECTION OF OIL

c R GAS MOVEMENT5-b

PERSPECTIVE

Fig, 5Physical model of resert,[)ircalcltladons.

M jy 11 ${

process was described to solve these equations, and thuspredict gravity-drainage performance for one-dimensionalsystems,

Fig. 8 shows saturations obtained throughout a one-dimensional gravity-drainage model produced under pri-mary.type depletion (i.e., no gas injection). The productionrate was a low percentage of the gravity reference rate,and pressure was allowed to decline with time. Gas comingout of solution in the oil column migrated updip into thegas cap as oil flowed downward. The dynamic conditionsencountered were such that a near-constant average gassaturation was maintained in the oil column. Note that

S-OIL CONTACTVARIOUS TIMES

L 16-a

EQUAL MOVEMENT

L CONTACTIOUS TIMES

[ \6-b

UNEQUAL MOVEMENT

I PRODUCTIONFig. 7Physicrd model oj reservoir f,,r three-dimemionrd

crrlculrrtimz.s,

behind the front, especially near the top of the structure.most of the change in saturation occurs as a result of oilshrinkage rather than from flow.

A three-dimensional calculation method was developed,based on the one-dimensional method previously described.by considering a large number of one-dimensional (verti-cal) areal segments linked together by specifying themigration of oil and gas between adjacent blocks in thesegments. Fig. 9 shows a typical block in an areal segmentwith four surrounding blocks in adjacent areal segments.All these blocks are considered to be at the same eleva.tion. At the start of a time step, pressures and saturationsare known for each block, Actual migration that willoccur during a time step cannot be specified becausethe values of pressure and saturation in the surroundingblocks at the end of a particular time step will not bcknown. Therefore, it was decided to specify the migrationduring any time step by explicit methods. In other words.for calculating horizontal migration of oil and gas, pressureand saturations for the five horizontal blocks in Fig. 9were estimated at the mid-point of the next time step byextrapolating from previously calculated values at theprevious two time levels. For the first time step, pressureit time I 1 is assumed to be equal to pressure at time I.[n starting from static conditions, very little error is madeby this assumption since the system is in equilibrium andthe first time step is short. Subscripts a, j and i in Fig. 9correspond to vertical, x and y indicators, respectively.for designating blocks, Eq. 1 is an expression for pressureat any general point (designated as the point at spacelocation a, i, j) at a time midway between time I andtimet +1.

(P)l+i= W)f + Y2 [(i), (P)f.1l.,. . . . . . . (l]

Eq. 2 expresses the rate of oil migration at 14.7 psi and60F between blocks T (E,,,, ), , (.E*.),,, (5)

~hmc migration terms then are used as a pseudoproduc-tion term in the one.dimensional gravity-drainage predic-tion method, For example. Eq. 6 gives a materiaI balanceon the gas content of a block in the oil column (thisequation is the same as Eq. 13, in the Appendix of Ref. 7).

1(.s;),.,

[(R,),.,

1(

(1 .s ,)(P) 1.1(~)l+1 - (B:.,

[

(R,),R:-] ~ ($[p)~ $- ::::~:1-(K), - (B.,),.,

[ 1[At j (e.,), -t (e. ),., (R,-), -t (R, I,.,

( v+) ~ ~ (n,), i- (B, ),., 1

-[- 1[ 1(eC), -r (e.,),,, . (!

This equation is modified by Eqs. 4 and 5 to includehorizontal migration of oil and gas, and becomes Eq. 7.

1(s,,), , - - ( [-(R:),(R:) , ;,-(P,.),., -1- (q;), (e,,),+ (e,,-),., 1_:,(v+)(bV*j7+@}*jT A-i[[1 .$,.) (s,) !.! 1(l-.s,, ) (.s,,), [(N),l - (B,,), ,( (8;

.

B#

ALILCT

d

A similar procedure is used for modifying all other oiland gas material balance equations listed in the Appendixof the paper describing the one-dimensional method forpredicting gravity-drainage performance.

By using this technique to specify horizontal migrationfor gas and oil flow, the reservoir can be divided into alarge number of areal segments and the implicit approachfor a single segment can be utilized for each segment topredict three-dimensicmal performance.

Esmnplc of Predicted ReservoirPerformance Using a Three-DimensionalGravity-Drainage Model

Fig. IO is a contour map of a hypothetical reservoir.Origin~l oil in place was 650 million bbl, and originalreservoir pressure was 4,100 psia at a datum of 7,200 ftsubsea, Fig 11 represents values of permeability timesthickness, and Fig. 12 shows porosity times thicknessvalues used. Connate water saturation used was 18 percrmt.A gas-oil contact existed at 7,070 ft subsea cm discovery.Production wells are located as indicated. Total oil produc-tion was maintained constant at 20,000 STB/D with wellsbeing produced al equal rates. All oil production camefrom blocks below the gas-oil contacl. ~-he section open intiny well was at least 10 ft below the location of the gas-oilcontac!. No coning was assumed in this example during atime step.

Fluid properties were selected in such a way that theyvaried with structure in the reservoir. Fluid at the gas-oil

Fig. 1 lPermeabifi/y fhickness cm~lwws, sclmpkreservoir (darcy-feef).Fig. 10SlrucVurol cotllwr~, .wnple problew.

TABLE 1 FLUID PROPERTIESSAMPLE PROBLEMTop of Bottom of

Reservoir Reservoir. ..

Saturation pressure (psig) 4,300 2,150Oil viscosity (cp) 0.26Oil reservoir volume factor

0.71,84 1.26

Gas in solution (scf/STB of oil) 1,720 700..-

contact had a saturation pressure of 4.300 psi and de-creased downdip to a value of 2,150 psi. Fluid propertiesare given in Table 1. Variations in saturation pressure andmsociated fluid properties were not linear,

The reservoir was represented by the type of physicalmodel shown in Fig. 7, and 22 rectangular, areal segmentswere used. In the j direction (along the major axis of thereservoir). blocks were five times the length in the idirection. Each block was25 ft high.

Fig. 13 shows pressures and GORs calculated for 25years of production from this reservoir. The solid line inthe pressure. vs-time curve in Fig. 13 represents the calcu-lated, volumetrically weighted average reservoir pressureat a datum of 7,200 ft subsea. The circles represent anaverage of reservoir pressuie in the vicinity of producingwells at various times, Note that the average of the pres-sures in the vicinity of the wells is lower than the averagereservoir pressure. This is emphasized further by Fig. 14.which represents pressure contours throughout the reser-voir after 20 ytars of production history. The shape ofthe reservoir pressure-vs-tirne curve is characteristic of

\

undersaturated conditions existing in the reservoir at thestart of production. Pressure declines rapidly to the satura-tion pressure, and then declines at a slower rate due toevolution of gas in the oil column. The fact that satura-tion pressure changes with structure makes this a verygradual change as compared with predicted performanceusing standard material balance calculations where an oilwith a single saturation pressure is considered.

The produced GOR shown in Fig. 13 increased slightlyfrom an initial value of 970 scflbbl during the first 5years of production. This reflects the fact that all wellswere producing from the oil column only, and due topermeability variation the largest amount of oil came from

4000

g 3000

u

: 2000

mw~ 1000

0 5 10 15 20 25YEARS OF PRODUCTION

2000

u12500 l PRODUCINGWELLSFfg. 14Presswe conrours, sample problem (20 ye[m ,Ij

ptductirm).

low in [he structilrc. MOSI of the produccri oil was stillundcrsaturatwi duc to the variation in saturation pressurewith elevation. The decline in GOR with time indicatesthat cmmtcrcurren[ flow existed in the oil column belowthe gas.l. il contact, As rcwrvoir prcssuiw declined, producedGORs were only slightly higher than solution GOR, sincemost of the gas evolved in the reservoir migrated updipinto the gas cap. The liquid saturation at wrrious timeschangcxi in a manner similar to that shown in Fig. 8 whereu iow gas saturation is observed heiow the gas.oii contac[at all times.

Figs. 15a and 15b represent the location of the gas-oilcontact aiong Lines A-A and B-B of Fig. 10 after 20years of production. It is obvious that horizontal migrationhas been ineffective in maintaining a uniform rate of ad-vance of the gas-oil contact throughout the reser joir. Oilhas migrateci from the edges of the reservoir; however,most of the produced oii has come from movement ofthe gas-oii contact in the vicinity of the producing wells.Aiong the iine A-A only the section at the extreme edgesis underwturated, whiie along B-B, a large portion of thesouthern end of the field was under.saturated.

A I(hough constant production rates for each wcii wereconsidered in the sampie problem, actuai ikid practicesmay be such !hat individual weli.producing rates wouhi heaiiocated on tnc basis of equal drawdown, equal wellheatillowing pressures or some other criteria. The uneven move-ment of gas-oil contact with time, and regional variationsin pressure throughout the reservoir, indicate that individ-ual weli productivity wili not rermin constant throughoutthe iife of a reservoir, anti that it cannot he cstimate[iaccurately by considering average reservoir conditions.ro obtain the best estimate possibie regarding future per-formance of a reservoir of this type, it is essential toconsider actual Ioctition of wells and regional changes inporosity, thickness anti structure.

B,, = oil formation volume factor (bbl at reser-voir temperature and pressure) /(bbl at60F and 14.7 psi)

e. = oii influx rate in verticaI dirwtion, cu ft/day at reservoir temperature and pressure

Cfl= gas influx rate in verticai direction, cu ft/day at reservoir temperature and pressure

E,. = horizontal oii migration rate between aci-jacent blocks at the same elevation, cuft/day at 14.7 psi and 60F

E,, = horizontal gas migration rate between ad-jacent blocks at the same elevation, cuft/day at 14,7 psi and 60F

F = eievation difference between points in thereservoir (used to determine gravity gra-dient between blocks ), ft

k = absolute permeability, darciesk, = relative permeability to oiik,, = relative permeability to gas

L = distance from mid-point of block to edgeof other block, ft

M = 6,33 X absolute permeability X cross-sec-tional area, da rcics X sq ft

P = pressure, psiq = oii producing rate, cu ft/{iay (measured al

60F and 14.7 psi)

SECTION B-B

R=

R. =

#/a, i.t.

t = vaiue at time It+l = value at time (t+l)

r-l-% = value at mid-point of time interval

References

1, Stewart, F. M., ffarthwaite, D. L. and Krebil, F, K.: Pres-sure Maintenance by Inert Gas Injection in the High ReliefElk Basin Field, Trans., AIME ( 195S) Vol. 204, 49-55.

2. SI_sreve,D. R, and Welch, L. W., Jr.: Gas Drive and GravityDrainage Analysis for Pressure Maintenance Operations,Trans., AIME (1956) Vol. 207, 136-143.

3. Kirby, J. E., Jr., Stamm, H. E. IIJ and Schnitz, L. Il.: Cal-culation of the Depletion History and Future Performance ofa Gas-Cap-Drive Reservoir, Tram., AIME (1957) Vol. 210,218.226.

4. Martin, John C.: Reservoir Analysis for Pressure Mainte-nance Operations Based on Complete Segregation of MobileFluids, Trans., AlME (1958) Vol. 213, 220-227.

5, Cook, Robert E.: Analysis of Gravity Segregation Perform-

. .

anee During Natural Depletion, SOC. Pet, Eng. J, (Sept,,1962) 261-274.

6, Sheldon, J, W. and Fayer~, F. J,: The Motion of an Inter-face Between Two Fluids m a Slightly Dippi~lg Porous Medi-um, SOc, Pel. .Errg.J. (Sept., 1962) 275-282.

7, Hall, H, N,: Analysis of Gravity Drainage, J. Pet, Tech.(Sept., 1961) 927-936, **

Ii. N, Hall is u .sfafl reseurch en~inrwworking in ihe Prrxlwtion ResearchDiv. cjf P(IIIA tnericu)t Peir,>leuttt C[>rp,,vRe.wircil Cm ter, IuI.Yu. HiIll receiveda BS degree in chcinical tw.gineerittg}ro)u okkihot)u: Stuie U. atrd UN MSdrgrw itl tmi IIwt }I(ifirs frf utl Tk, V. 0/Tul.wl.

S24 JfJURNA1. OF PRTROI.F.VM TECHNOLOGY