SPE 22695

Modeling Geological Heterogeneities and its Impact onFlow SimulationV. Suro-P&ez, P, Ballin,* K. Aziz,* and A.G. Joyrrwl,* Stanford U,●SPE Members

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This praperwee prepared for prewntation at the ~th Annual Technical Conference and Exhlbl!’ n of the Soclety of Pefroleum Englnwr& held Irr Dellea,TX, October S-9, 19S1.P

}Thla paper wee aalec!ed for preaentatlon by an SPE Program Committee following review of inf rmalion contelned In en abstract submitted by the author(a).Contents of the paper,es preaanfed, have not been reviewed by tha society of Petrobum Englrtaersand awesubject t correction by the author(s). Tha maferial, as presented, dose not n6cewerlly reflecteny poaitlon of the Soclefy of Petroleum Englnwra, Ila officers, or membere.Papers presented at SPE meetings ere subject to publication review by Editorial Oommltteeaof the Sodetyof PetroleumEngineers,Permiaalon10copy Iereafrfcfadto en abstractof not mcwethan 390words. IIluafratfonsmaynot be copied. The abstractshouldcontainconspicuousacknowledgmentof whera and by whom the papw is prewnted, Write Publlcatlona Manager, SPE, P.O. Sox W3S3S, Richardson, TX 76083-3836 U.S.A. Telex, 730989 SPEDAL

Introduction

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Abstract

A single algorithm is proposed for the numerical modelingof geological facies and corresponding petrophyeical prop-erties. All reservoir parameters, whether categorical suchas Iithofaciea type or continuous such as permeability arecoded and processed aa a series of binary indicator vari-ables. Stochastic modeling of the reservoir is baaed onIndicator Principal Component Kriging and the sequen-tial simulation principle. The resulting alternative andequipru~able reservoir models honor the available infor-mation at wells and their statistics. These reservoir mod-els are then used to investigate the impact of geologicalheterogeneities on flow performance prediction in a water-flood scheme. Analysis of various production parameters(cumulative oil production, cumulative water oil ratio andbreakthrough times) indicates that they are more sensi-tive to the geological architecture of the reservoir than todetails of the statistical distributions of petrophysical vari-ables. Direct modeling of permeability across Iithofacies,i.e. ignorance of the geological architecture, may lead tosevere inaccuracy in reservoir performance prediction, par-ticularly after the first transient years of production.

This paper addresses the stochastic modeling of geologicalattributes, such as lithology or lithofacies; the generationof reservoir properties, such aa permeability, porosity andsaturation; and the prediction of reservoir performance,such as cumulative oil production, cumulative water oilratio and breakthrough times, The main goal is to evaluatehow performance prediction is affected by either explicitmodeling of geological heterogeneities or its ignorance,

A waterflood scheme is considered with one producerand one injector. Lithofaciea and reservoir petrophysicalproperties along these two wells are considered aJ+condi-tioning data. Reservoir description between wells is ob-tained through stochastic simulation [1, 2], Such stochas-tic modeling honors the available information at the wellsand allows reproduction of patterns of spatial variability.Some of the reservoir models contain explicit informationabout the lit hofacies distribution. Other models ignoresuch geological information with petrophysical propertiesbeing generated across lithofacies, The waterflood schemeis applied to each reservoiq model and the resulting produc-tion parameters are compared to evaluate the importanceof accounting for geological information in the modelingprocess,

References and figures at the end of the paper.

The stochastic approach used allows quantifying the un-certainty associated to the lack of information betweenwells. For each production parameter that uncertainty ischaracterized by the cumulative distribution function (calf)of the various outcomes corresponding to the alternativere,mrvoir models.

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2 Modeling Geological Heterogeneities and its hnpact on Flow Simulation SPE 22695

Modeling Geological Heterogeneities andFlow Properties

Recently, considerable effort haa been put on stochasticmodeling of reservoir parameters, suciI as permeability,porosity, lithofscies [2-7]. All algorithms attempt to pre-serve the spatial variability observed from the well data asreflected by Iiistograms and autocovariance functions.

This section presents an integrated approach to the nu-merical mddeling of oil reservoirs. The algorithm of indica-tor principal component kriging (IPCK) [8] is used, first tosimulate the reservoir geological architecture, then for thesimulation of flow properties specific to each of the previ-ously simuhkted lithofacies. The underlying assumption ofsuch a two-steps .approaoh is that flow is largely controlledby the major geological structures.

Geological Heterogeneities

Consider that K exclusive geological categories are ob-served at the wells. At any location x, an indicator randomvariable is defined as:

1(x; k) ={

1 if L(x)= kO otherwise

(1)

with k being one of the K geological categories found inthe reservoir and L(x) being the actual category observedat x, These K indicator variabhx define the vector, ‘

I(x) = [1(x; 1) . . . I(x; K)]T (2)

Note that one and only one element of vector ( 2) is equalto 1 since the geological. categories are exclusive. Indica-tor auto(crces)covariances inferred from the correspondingindicator data characterize the, relative geometry of the Kcategories:

C~(h; k, k’) = E[l(x; k)l(x + h; k’)] – p~pkl (3)

with pk being the proportion of category k in the reservoirand h, a separation vector. There are K* such indicatorauto(cross)covariances. Thus, when K is greater than 3that inference may become tedious.

Indicator auto(croas)covariances can be used to estimatethe likelihood of occurrence of a particular category at anyunsampled location. Indeed, it can be shown [1] that amodel of conditional distribution can be obtained by aweighted linear combination of indicator data.

-\

Pro6”{L(x) = k’lL(xa), a = 1,..s, n} = pkt +

The weights & are determined by solving a linear systemof normal equations [1].

Conditional probability models of type ( 4) provideinformation about heterogeneities distribution betweenwells. They can be used to yiehJ images of the reser-voir geometric architecture [2]. This is done by drawing.the category m lithof&iea prevailing at any unsampled I*cation from he corresponding probability distribution oftype ( 4).

F1OW Properties

As done above for categorical variables, determination ofa conditional probability model of type ( 4) provides thelikelihood of occurrence of a certain class of permeabilityor porosity value at any unsampled location. As before, acontinuous property, say Z(x), can be coded into a seriesof indicators:

“Z(x; %k)={

1 if Z(X) ~ .%k

O otherwise(5)

with %kbeing anyone of K threshold values discretizingthe range of variability of Z(x), The definition ( 5) yieldsa vector of indicator variables similar to ( 2):

I(x) = [1(X; ZI) , ssI(X; ZK)]T (6)

However, as opposed to the indicator vector ( 2), the defi-nition ( 5) entails a vector with a series of O’sand 1’s. TheSh@.? trLU’IShiOn from O tO 1 (~(x; Zk) = (), 1(X; %k+l) = 1)

indicates that Z(x) belongs to the interval (zk, %k+l].A conditional distribution model is agtin provided by a

linear combination of indicator data:

Prob*{Z(x) ~ zk!ll~(xa), a = 1,,., ,n} = Pkf +

(7),

with pp being the proportion of values Z(x) below thethreshold value %kl.As before, the weights AOare obtainedby solving the corresponding system of normal equations.

Stochastic Imaging

Expressions ( 4) and ( 7) represent uncertainty models ateach unsampled location. They provide quantitative infor-mation about how much is ( or is not) known about theattribute Z(x) or category L(x) at location x, The sequen-tial simulation algorithm [1] provides a way to interpolatebetween sample locations using the uncertainty models oftype ( 4) or ( 7).

The sequential simulation algorithm is here briefly re-called:

● Define a random path over the domain to be simu-lated.

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●

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At-each location x, determine the conditional proba-bility ( 4) or ( 7) for all K indicator variables. Use aaconditioning data {n} the original sample data as wellaa the previously simulated values within a predefineneighborhood.

Draw from these conditional distributions model Ksimulated indicator values i’ (x; k) or i’ (x; Zk), k =1,... , K. The upperscript s refers to a simulatedvalue. Add these simulated values to the set of con-ditioning data.

Loop over all locations of the simulation grid.

It can be shown that this algorithm allows honor-ing original data at their locations and reproducingglobal statistics such aa proportions p~ and indicatorauto(cross)covariances.

Indicator Principal Component Kriging

As mentioned before, the derivation of the conditionalprobabilities ( 4) or ( 7) requires the inference of K2 in-dicator auto(croas]covariances. IPCK allows shortcuttingthat inference by working with on a limited number of in-dicator prinqipal components (ipc) [2, 7, 8]. Consider theindicator covariance matrix for a given separation vectorh’:

~I(h’) = Chv{I(x), I(x + h’)} (8)

with I(x) being the indicator vector ( 2) or ( 6). Nextconsider the unique orthogonal decomposition of that co-variance matrix [9]:

X1(h’) = ADAT (9)

with AT being a K x K orthogonal matrix, defining thevector of ipc:

Y(x) = ATI(x) (lo)

with.,

Y(x) =[YI(X) . . . YK(X)]T

By definition, the K principal components Yk(x) andYkl(x + h’) are orthogonal to each other, i.e., their covari-ance matrix at separation h’ is diagonal.

In practice the separation vector h’, at which the de-composition ( 9) is done, ia chosen small or zero. Thischoice ensures in most cases [2, 8, 10], that the principalcomponent covariatice matrix is orthogonal, to an excellentdegree of approximation, “atall separation vectors h.

This property allows reduction of the modeling effortfrom K2 indicator auto(croas)covariancea for I(x) to onlyK ipc autocovariances for Y(x).

The linear combinations (4) or(7) are replaced by leastsquare regressions of the Yk(x), i.e.,

n

a=l

followed by the back transfo~m:

I*(x) = AY* (X)

with

Prob* {Z(x) S .ZkI{n]} =

Integrating Flow PropertiesAttributes

(11)

Z“(X; Zk)

and Geological

The integration of geological and petrophysical informa-tion is accomplished by making the simulation of flowproperties conditional to a previously simulated geologicalimage. The probability distribution for the petrophysicalproperty Z(x) is made specfic to each category k’, andwould consider aa conditioning data only samples pertain-ing to that category k’, i.e.,—

Prob*{Z(x) S zk I{n) E k’]

Thus, inference of statistical parameters, histograms andautocovariances, is done separately for each geological cat-egory. A shortsighted decision would be to avoid suchcomplex inference by pooling together all available dataregardless of geological facies. The fact that inference ismade easier is no justification for ignoring essential. char-acteristics in the reservoir, Ignorance of major geologicalheterogeneities in reservoir modeling may yield inaccurateperformance forecast as shown hereafter.

A Heterogeneous Oil Reservoir

A number of alternative reservoir models (2D sections)have been simulated using the sequential simulation al-gorithm and IPCK. The conditioning data is constitutedby lithofacies, porosities and permeabilities sampled at twowells. Each reservoir image comprises 2650 cells, 50 in thehorizontal and 53 in the vertical direction.

Static Properties .. .

Six different Iithofacies are initially considered and later,lumped into four different flow units. Table 1 gives thevolume proportion of each Iithofacies and its wisigned flowunit. Figure 1 illustrates one stochastic image of the reser-voir. It shows a layer-type reservoir whith each layer cor-responding to a different lithofacies, itself constituted bydifferent Iithologies. Also, note that there is no perfect

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4 Modeling Geological Heterogeneities and its Impact on Flow Simulation SPE 22695

continuity between the two wells, Table 2 gives the pa-rameters of the first five ipc autocovariances. The sixthipc is a constant value [10]. Additional information aboutthese ipc autocovariances is given in Appendix.

Porosity and permeability have been modeled as lognor-mal distributions. Table 3 shows the corresponding statis-tics for each flow unit. Note the heterogeneous behaviorbetween flow units, going from the highly permeable flowunit 1 to an almost impermeable barrier (flow unit 4). Re-garding spatial variability, porosity within each flow unitis considered uncorrelated whereas absolute permeabilityis modeled with spatial correlation.

For the simulation of the absolute permeability field,nine threshold values (zk ) corresponding to the deciles ofeach lognormal distribution were selected. Table 4 showsthe parameters of the three ipc autocovariances retainedfor the evaluation of the conditional distribution of type( 11), The other ipc’s are considered uncorrelated. Also,it was assumed that the permeability spatial variabilityis the same for all flow units. However, the geometry ofeach lithofacies is different. Also, note that the correla-tion ranges of permeability are smaller than the correlationranges of lithofacies geometry.

A small anisotropy ratio of 2:3 is used for vertical tohorizontal absolute permeabilities.

Dynamic Properties

Dynamic fluid properties are assumed constant within eachof the four flow units. Fluid saturation values are basedon capillary-gravity equilibrium, which is is achieved ifthe capillary pressure and relative permeability relations.are parametrized by absolute permeability [11]. The di-mensionless capillary pressure function group known as J-function, is used to parametrize the imbibition capillarypresure curve-a as function of the absolute permeability.Further, it is assumed that interracial tension and contactangle changes between flow units are negligible,

The connate water saturation (SWi) and residual oil sat-uration (SOr) are based on the correlations of Kocberberand Collins [11], Table 5 shows the critical saturationsused for each flow unit, These values assume a water wetrock system, thus it is expected that the residual oil satu-ration is not much tiected by the pore geometry as givenby the correlation [11].

The capillary pressure curve of the second SPE compar-ison problem [12] is normalized and transformed to repre-sent an imbibition system with SOr = 0.2. It is used asreference for the parametrization process. The computedcapillary presure curves are shown in Figure 2.

The relative permeability curves considered in this studydepend on the phase end point values and an exponent[13] whose value depends on the rock type. The phase endpoint relative permeability values are based on the corre-

lations [14], while the experimental results [15] are used todefine the curvatures. Table 6 shows the end points andFigure 3 presents the resulting water oil relative perme-ability curves for the four flow units.

Flow Performance

A waterflood exercise was designed to measure the impactof geological heterogeneities on flow performance. A two-phsse, tw-dimensional black-oil simulator [16] is used toderive cumulative oil production, cumulative water oil ra-tio and breakthrough times under dfierent scenarios. Thewaterflood scheme consists of one injector and one pro-ducer, with water being injected at constant rate. Theinjection period was five years.

Seven different cases are considered in this study, Someof the cases include an explicit modeling of the geologicalarchitecture w obtained from simulation of the lithofacies,Other cases consist of stochastic simulation of the absolutepermeability across all lithofacies. For all cases, the samewell data are used as conditioning data. The total numberof alternative reservoir models considered is 50 for eachcase. Description of the seven cases follows:

●

●

●

●

●

402

Case A: ‘The geological architecture of Figure 1 isconsidered identically for all 50 realizations, Eachflow unit of each realization is informed with anuncorrelated permeability field with the correct his-togram. Relative permeabilities are made specific toeach flow unit.

Case B: This case is in all points identical to caseA, except for the permeability fields that now presentspecific autocorrelation within each flow unit.

Case C: No lithofacies differentidion is considered.The spatially correlated permeability field was gener-ated using the average (over all flow units) statistics(histograms and autocovariances) of c,ase B. The rel-ative permeability curves are the same as in cases Aand B. Therefore, even though no lithofacies is con-sidered, there is a partial knowledge of flow unit typesthrough using different relative permeabilities.

Case D: Same as case C but now a single relativepermeability curve is used. The phase end pointsare computed as a weighted average of the phase endpoints of the four flow units. The weights are the pr~portions of the tiow units in the reservoir. The curva-ture exponent is the same considered in the originalrelative permeability curves. Figure 4 gives the capil-lary pr&sure and relative permeability curve used forthis case.

Case E: The same unique lithofacies geometry usedin cases A and B is considered for all 50 realizationsof this case, but the permeability field is modified

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SP.E 22695 V. Suro-P&ez, P. Ballin, K. Aziz and A. G. Journel 5

such that the correlation range within each flow unitis increased by about 2570.

● Case F: There are now 50 different reservoir geome-tries, i.e. uncertainty about the geological architec-ture is accounted for. The absolute permeability fieldis generated using the parrimeters of case E. The rel-ative permeabilities of Figure 3 were assigned to thecorresponding flow unit.

● Case G: The same 50 realizations of Iithofacies geom-etry generated for case F are used again. The perme-ability fields are simulated anew using the statisticsconsidered for case B.

All cases consider the same oil in place and total porevolume, This constraint ensures a fair comparison betweenthe production parameters derived from each case.

Discussion of Results

Tables 7 and 8 show cumulative oil production for thesecond and fourth years ; Tables 9 and 10 present cumu-lative water oil ratios for the same two production years.In all cases, each statistical parameter is derived from thecorresponding 50 realizations.

Case A, for both production parameters, presents thesmaller spread or variance. This is a consequence of thespatially uncorrelated absolute permeability. There are noflow paths or flow barriers since tb.e absolute permeabilityfield lacks any spatial pattern ~f correlation. The flowis entirely controlled by the geological architecture. If themean (3?)or median (qO.s) are retained as estimators, thereis no dramatic difference between this case A and the othercases at least for the first three years. For longer times (>3 years), the production parameters begin to deviate fromthe other cases.

Caaes B and E share the same geological architecture,differing only by the permeability correlation ranges. Forcase E the ranges are 25% larger than for B. This causesa reduction of the cumulative oil production and an in-crease of the cumulative water oil ratio, from case E to B.However, these differences are not significant, and reser-voir management decisions based in either case would beequivalent.

C&s C and D are very different. Case C presents theminimum cumulative oil production and the maximum cu-mulative water oil ratio. Its performance appears shiftedwith respect to the other cases. This is due to using theoriginal relative permeabilities without association to thegeometrical architecture. This observation is verified incake D corresponding to totaI ignorance of geometry. Av-erage values and medians are now closer to the other casesfor times lesser than 3 years. Howeverj observe that thevariance in case D is small ss compared to the other cases

and that, as time increases (> 3 years), the production pa-rameters begin to deviate from the cases that have accessto the geological architecture, This shows that charac-terization of reservoir geometry is an important factor foraccurate forecast of reservoir performance.

Finally, cases F and G show the highest variances be-cause the geometry of the reservoir is now changing fromone realization to another. There is no practical differencebetween these two cases, supporting the argument thatin heterogeneous reservoirs flow performance is primarilycontrolled by the geometry of such heterogeneities. Re-call that the 50 realizations of lithofacies geometry are thesame for both cases F and G.

Figures 5a to 5C illustrate this point with water satu-ration maps at the end of the first year. Each map isone realization of cases B (Figure 5a), C (Figure 5b) andD (Figure 5c). Note that for cases B and C water chainnelling follows the geological structure whereas in case Dno such behavior is observed. Cases C and D show a morepronounced gravity effect: the water is coming down astime progresses. Case B shows a less important gravityeffect due to the geometrical architecture, Note on Figure1 below the region of high permeability at the top of thereservoir, there is a layer that acts as a flow barrier. Thisgeometric feature is accounted for in case B and ignored,partly, in case C and totally in case D,

Statistics of breakthrough times are illustrated in T&ble 11. The maximum average corresponds to case Awhich presents also, the minimum variance. Case C de-parts considerably from the other cases. Its median corre-sponds approximately to the minimum values of the CMWwhich use explicit knowledge of reservoir geometry. It isobserved that larger variances are associated to cases wherethe reservoir geometry is accounted for.

Uncertainty Models

The very reason for using stochastic reservoir models is toprovide an appreciation of the spatial uncertainty left be-yond the availabIe information. Uncertainty exists becauseof incomplete knowledge of geology and flow properties inthe reservoir.

A good uncertainty model needs to account for all im-portant features of the reservoir. Modeling of geologicalheterogeneities or equivalently flow units does impact theresulting uncertainty measure, For example, consider thecumulative oil production for the fifth year, Figure 6 showsthe cumulative distribution function for cases A, B, D andG. The maximum difference between minimum and mmci-mum cumulative oil productions is obtained for the case G,which accounts for the uncertainty about lithofacies. CaseA which freezes the geological architecture and considersan uncorrelated permeability field, shows the smaller vari-ation. This figure indicates that excessive simplification of

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6 Modeling Geological Heterogeneities and its Impact on Flow Simulation SPE 22695

the reservoir modeling process yields uncertainty modelswhich may underestimate severely the actual uncertainty,

Conclusions

This paper has presented the application of stochasticimaging for modeling heterogeneous oil reservoirs. Flowproperties, such as absolute permeability and geologicalattributes such as lithofacies, are the variables consideredin that modeling. It has been shown that the geologi-cal architecture of the reservoir plays an important role influid flow, hence in performance prediction. The modelingof petrophysical properties such as porosity/permeabilityshould be made conditional to previously modeled geolog-ical architecture.

XIIpretsence of a heterogeneous porous media, flow ismainly controlled by the high permeability contrast be-tween some flow units. It appears that knowledge of sp-tial variability of flow properties within each flow unit isnot critical. Models with different permeability correlationranges but sharing the same geological architecture, es-sentially give the sa,me flow performance. Future researchshould consider systematic variation in the spatial variabil-ity of geometric characteristics and measure their impacton flow.

This study has also shown that larger uncertainty re-sults from the consideration of geometric architecture. Anon-conservative underestimation of actual uncertainty inproduction forecast may result by ignoring either the in-fluence of geometric architecture or that of the spatial cor-relation of the petrophysical variables.

Acknowledgements

This research has been supported by the Stanford Centerfor k.eservoir Forecasting and SUPRI-B,

References

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Haldorsen H, H., Brand, P, J. and Macdonald, C. J,,1988, “Review of the stochastic nature of reservoirs”,in Mathematics in Oil Production, eds. S. Edwardsand P. R. King, Oxford Science Publications.

Hoiberg, J., Omre, H. and Tjelmeland, H., 1990,“Large scale barriers in extensively drilled reservoirs”,in 2nd. European Conference on the Mathematics ofOil Recovery, eds, D, Guc%illot and O. Guillon, Ed,Technip, p. 31-41.

Journel, A. G. and Alabert, F., 1988, “Focus-ing on spatial connectivity of extreme-valued at-tribut~’: stochastic indicator models of reservoir het-erogeneities”, SPE paper No, 18324.

Suro-P&ez, V. and Journel, A. G., 1991, “IndicatorPrincipal Component Kriging”, Math. GeoL, V. 23,No. 5, in press,

Golub, G. and Van Loan, Ch., 1983, Matr~z Compu-tations, The John Hopkins University Press, 476 p.

Suro-P&ez, V., 1991, “Stochastic simulation of cate-gorical variables for reservoir description”, submittedto Math. Geol,.

Kocberber, S, and Collins, R, E., 1990, ‘Impact ofreservoir heterogeneityy on initial distributions of hy-drocarbons”, SPE paper No. 20547.

Chappelear, J. E. and Nolen, J. S., “Second compar-ative solution project: a three-phase coning study”,SPE paper No. 10489.

Corey, A. T., Rathjens, C. H., Henderson, J. H. andWyllie, M. R. J., 1956, “Three-phase relative perme-ability”, Trans. AIME, V, 207, p. 349-351.

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SPE 22695 V. Suro-P&ez, P. Ballin, K, Aziz and A. G. Journel 7

Appendix

Five ipc autocovariances were considered for the simulationof the six lithofacies. Two nested exponential structureswith a large geometrical anisotropy are used for all five ipcautocovariances. The model is:

Cyk (h=, h.) 7= c~hpal( h= +r~hz)+

with Cyh(hZ, h~) being the k:h ipc autocovariance, Cl thesill and aI the horizontal cor *elation range for the l:h struc-ture; h= and h. are the coordinates in the horizontal andvertical direction. Exp~ is the exponential covariance with

practical range a, defined by:

For the absolutepermeabilityfield three ipc autocovari-ances are considered. The general expression of such au-tocovariancea is:

with C(0) being the variance and COthe nugget effect. Thespherical model Spha(.) is:

3h lh~, Spha(h) = 1 –Z;+ ~(z)

/’ i(\

Table 1: Equivalence between Lit hofacies and Flow Units. LF corre-sponds to Jithofacies, FU to flow units and VOL to the volume proportion

in percent.

LF FU VOL11 0.4624 0.2232 0.044 3 0.0952 0.1162 0.05

Table 2: Structural Parameters of the IPC Autocovariance Models.Units of C are variance units, a are map units and T units are dimensionless.

IPC c1 C21 0.1 0.15 1::0 5:0 4;;8 ;;2 0.036 0.05 11.25 162.5 2.81 14.13 0.037 0.04 62.5 12,5 15.62 4.164 0.025 0.027 50.0 16.25 15.62 5.415 0,017 0.02 25.0 16.25 7.14 8.12

Table 3: Porosity and Permeability Statistics. ~i and u: are themean or variance of either porosity (~) or permeability (k) in md. Theseparameters define a lognormal distribution.

?Tik u: m~ u;

1 300.0 360000.0 0.2 0.012 50.0 22500.0 0.15 0,00563 5.0 225.0 0.10 0.0025

—

1

4 0.5 1.0 0.05 0.00062

.z..—-

—

Table 4: Structural Parameters of the IPC Autocovariance Models.The variances have been standardized to 1.

IPC C(0) C’O Cl1 1.0 0.10 0.8 3~0 JO2 1.0 0.05 0,95 20.0 4.03 1.0 0.0 1.0 10.0 4.0

Table 5: Phase End Point Saturations Table 6: Phase End Point Relative Permeabilities

FIJ SWi(,frac.) %(frac.)1 0.20 0.20

FU 1<.W~a%.(frac.) K,O ~o=,(frac,)1 0.50 1.00,

2 0.30 0.225 2 0.35 0.903 0.45 0.25 3 0.18 0.784 0.60 0,275 4 0.02 0.65

Table 7: Cumulative Oil Production for the 2nd. Year. Case refersto the different situations studied in the text; z is the arithmetic average; S2is the variance; qo.25, qo,5 and qo.T5 are the first quartile, median and thirdquartile respectively; min and max are the minimum and maximum values.

The units are STB in all cases, except the variance which is in STB2

B 7314.26 53980.39 7172.93 7348.41 7487.10 6553.86 7914.66c 6824.48 14925.25 6742.88 6812.83 6892.65 6593.86 7396.13D 7101.49 9288.49 7030.68 7081.62 7164.51 6940.01 7530.79E 7154.93 96604.41 6958.35 7154.60 7413.05 6406.39 7598.67F 7261.16 79047.86 7087.71 7323.21 7468,24 6266.91 7599.93G 7272.63 81616,80 7192.33 7319.89 7456.95 6408.24 7615.71

Table 8: Cumulative Oil Production for the 4nd, Year.

Case - s’

A 918t.72 1007.62 9%%0 9;;;:5 9;:t;9 91!;87 92~~4B 9014.76 38757.57 8922.55 9033.88 9139.96 8312.82 9299.57c 8331.09 18691.42 8238.97 8319.45 8401.62 8082.17 8944.26D 8730.63 9751.76 8665.96 8724.81 8782.49 8528.31 9229.44~ 8873.20 65190,74 8653.91 8875.84 9076.87 8247,06 9243.257F 8920.46 52771.23 8773.49 8940,20 9124.57 8281.72 9235.89G 8919.72 56302.08 8776.09 8946.92 9102.43 8016.04 9278.93

..,

Table 9: Cumulative Water Oil Ratio for the 2nd. Year. Units aredimensionless

B 0.041 0.0011 0.013 0.032 0.054 0.0003 0.157c 0.113 0.0003 0.099 0.114 0.125 0.029 0.150D 0.070 0.0002 0.060 0.073 0.080 0.0111 0.094E 0.065 0.0021 0.024 0.060 0.088 0.002 0,1721’ ().049 0,0017 0.016 0.038 0.060 0.!)02 0.209G 0.047 0.0018 0.016 0.039 0.054 0,0001 0,182

Table 10: Cumulative Water Oil Ratio for the 4th. Year.

B 0.670 0.0013 0.643 0.663 0.679 0,619 “0.807c 0.804 0.0008 0,785 0.805 0.820 0.682 0.858D 0.722 0.0003 0,711 0.723 0.731 0.631 0.762E 0.696 0.0023 0,655 0,690 0,733 0.628 0.822F 0.687 0.0019 0.647 0.681 0.710 0,630 0.814G ‘0.687 0.0021 0.651 0.681 0.708 0.622 0.873

Table 11: Breakthrough Times. Units are days, except for S* for whichare days2

Case - a

A 64;70 26;.84 6&5 6?+;;7 6?~;i0 6;2~5 6;3:;0B 607.87 2515.83 565.75 611.37 642.75 492.75 693.50C’ 460i35 1258.21 428.87 456.25 474.5 401,50 %02.25E 577.24 4643.61 520.12 574.88 629.63 428.88 702.63F 590.17 4162.69 547.50 602.25 642.75 419.75 693.50G 583.69 5301.83 548.50 602.25 642.75 410.62 711.75

vroduux inlectw

1 50

● Figure 1: one stochastic lithofacies simulation. There are six lihofaci~

that later

n

Ais

are summarized into four flow units.

70~

60 ::

---- ; ~-------

50 :ii

;...............

0.0 0.2 0.4 0.6 0.8 1.0

WATER SATURATION (frac.)

e Figure2: Capillarypressurecurvescorrespondingto thefourflowunitsco&idered in this study.

..

1,0

0.8

0.6

0.4

0.2

0.0

L \

~—1

f

l’0.0 0.2 0.4 0.6 0.8 1

WATER SATURATION (frac.).1

‘1,,i

● Figure 3: Relative permeability curves for the four flow units.

408.. . . .

,.

,.

GrJ

al

1.0

0.8

0.6

0.4

0.2

0.0

— Case - D

4

t * , , , t , i,nfi.,,t,’

O.O 1.0;ATER i&uRA#oN (fiit~j

(a)

18

16

14

12

10

8

6

4

2

0

i’ — Case - D

I

O.O 0.2 0.4 0.6 0.8 l.O

WATER SATURATION (frac.) ---’-

(b)

. Figure 4: Dynamic properties used in case D. (a) Relative permeability.

(b) Capillary pressure.

409

Sw Distribution Sw Distribution

50

40

30N

20

10

0

*o

50

40

30N

20

10

0

0 10 20 30 40 50x

(a)

t “1 0.2 0

Sw Distribution

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

0 10 20 30 40 50x

(c)

o 10 20 30 40 50

(U)

Cumulative 011 Production (5th. Year)

0.9 : ,

0.8 : *#

o.? *1*

0.6 1

R1

0.5 ‘“D

K #*

0.4 9

0.3

0.2

0.1

z [STBI

● Figure 6: Uncertainty models for the cumulative oil production at t~e&-

. Figure 5: Sw distribution at the end of the lst. year. (a) case B. (b) end of 5th. year. (—) case A; (*) case E (– –) case D and (+) we ~

case C. (c) caae D. G. Ml.

.1