modeling geological heterogeneities and its impact on flow simulation

7/30/2019 Modeling Geological Heterogeneities and Its Impact on Flow Simulation

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SPE 22695Modeling Geological Heterogeneities and its Impact onFlow SimulationV. Suro-P&ez, P, Ballin,* K. Aziz,* and A.G. Joyrrwl,* Stanford U,qSPE Members ; ).,,, :..

1 ,,.,.. . . . . . T .-:--.;-(,{ {Copyright 1S91,society of Pdroiium Engineers Inc, :( / /,/, /,) )1 1 l /This praperweepreparedfor prewntation at the ~th Annual TechnicalConferenceand Exhlbl! n of the Soclety of PefroleumEnglnwr& held IrrDellea,TX, October S-9, 19S1.P

}hlapaper wee aalec!edfor preaentatlon by an SPE ProgramCommitteefollowing reviewof inf rmalion contelned In en abstract submittedby the author(a).Contentsof the paper,es preaanfed,havenot been reviewedby tha society of Petrobum Englrtaersand awesubject t correctionby the author(s).Tha maferial,as presented, dose not n6cewerlly reflectenypoaitlonof the Soclefyof PetroleumEnglnwra, Ilaofficers, or membere.Paperspresentedat SPEmeetingsere subjectto publicationreviewby EditorialOommltteeaof theSodetyof PetroleumEngineers,Permiaalon10copy Iereafrfcfadtoen abstractof notmcwehan 390words.IIluafratfonsmaynotbecopied.Theabstractshouldcontainconspicuousacknowledgmentof whera and by whom the papw is prewnted, Write PubllcatlonaManager, SPE, P.O. Sox W3S3S,Richardson, TX 76083-3836U.S.A. Telex, 730989SPEDAL

I n t r oduc t i on

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

Mode lin g Geologic a l He t e rogen e it ie s an dF low Pr op er t ie sRecently, 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.Geo logi ca l H et e r o ge n e it i e sConsider 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 K

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 PropertiesAs 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) ~ .%kO 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) , s sI(X; ZK )]T (6)

However, as opposed to the indicator vector ( 2), the defi-nition ( 5) entails a vector with a series of Osand 1s. TheS h@. ? t r LUIS h iO n from O t O 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 alinear combination of indicator data:

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


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q

q

e

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 KrigingAs 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)

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

n

a= lfollowed by the back transfo~m:

I*(x) = AY* (X)with

Prob* {Z(x) S .Zk {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 zkI{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 inaccurate


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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 of

each 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 ipcs 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 to

horizontal absolute permeabilities.

Dy n am i c Pr op er t ie sDynamic 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-

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.F low P er fo rm a n ceA 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:

q

q

q

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.


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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.

q 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.

q 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 ResultsTables 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 no

flow 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 from

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) and

D (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 A

which 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.


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.6 Modeling Geological Heterogeneities and its Impact on Flow Simulation SPE 22695the reservoir modeling process yields uncertainty modelswhich may underestimate severely the actual uncertainty,ConclusionsThis 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.Acknow l edgemen t sThis research has been supported by the Stanford Center

[5]

[6]

[7]

[8]

[9]

[10]

[11]

multiphsse fluid flow for 3D reservoir studies, in 2nd,European Conference on the Mathematics of Oil Re-covery, eds, D. Gu6rillot and O, Guillon, Ed. Technip,p. 11-20,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.


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

AppendixFive 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 .) 7c~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. a re the coordinates in the horizontal andvertical direction. Exp~ is the exponential covariance with

practical range a, defined by:

For the absolute permeabili ty field 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 proportionin 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.U nit s of C a re va ria nce unit s, a a re ma p unit s a nd T uni ts are d imensionless .


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Ta ble 4: S t ruct ura l P a ra met ers of the IP C Autocova ria nce Models.Th e va r ia n ces h ave been st a nd ar diz ed t o 1.

IPC C(0) CO 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

Tab le 5: Phase End P oint Sa tura tions Ta ble 6: P ha se E nd P oin t Rela tive P ermea bilit iesFIJ SWi(,frac.) %(frac.)1 0.20 0.20

FU 1


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Ta ble 9: Cumula t ive Wa ter Oil Ra t io for t he 2nd. Yea r . U nit s a redimensionless

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: Cumula t ive Wa ter Oil Ra t io for the 4th. Yea r.

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


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q Figure 1: on e s toch a st ic lit hof acies s im ula t ion . Th er e a r e six lihofaci~t ha t la t er

nAis

a re summa rized in to four flow un it s.70 ~60 : :---- ; ~-------50 : ii ;...............

0.0 0.2 0.4 0.6 0.8 1.0WATER SATURATION (fr a c.)

e Figure2: Ca pillar ypressurecurvescorresponding o the four flow unitsco&id er ed in t his s tu dy .

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modeling geological heterogeneities and its impact on flow simulation

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