integration of seismic attribute map into 3d facies modeling

Ž .Journal of Petroleum Science and Engineering 27 2000 69–84www.elsevier.nlrlocaterjpetscieng

Integration of seismic attribute map into 3D facies modeling

Tingting Yao a,), Anil Chopra b,1

a Department of Geological and EnÕironmental Sciences, Stanford UniÕersity, Stanford, CA 94305, USAb ARCO Exploration and Production Technology, 2300 West Plano Parkway, Plano, TX 75075, USA

Received 16 March 1999; accepted 3 March 2000

Abstract

Clastic reservoir characterization typically starts with the modeling of the facies distribution and geometry. Thearchitecture of the reservoir, governed by the facies geometry, is a major source of heterogeneity in such clastic systems.Seismic data potentially provide valuable information about the areal distribution of different facies. However, seismic dataare available only at coarse vertical resolution, more closely representing interval average rock properties, whereas well logfacies data more closely represent point rock properties. This scale difference or ‘‘volume support’’ difference between theseismic data and the facies data available along the wells makes direct integration difficult. A recently developed algorithmbased on the concept of cokriging with block average data is woven into probability field simulation for building faciesmodels. The seismic data at its coarse vertical scale, equivalent to attribute maps, can be fully accounted for without anyimplicit or explicit vertical duplication to match the fine vertical scale of geologic modeling. The cpu-speed advantage ofprobability field simulation is also retained. The algorithm is demonstrated on a clastic petroleum reservoir. The results arecompared with those obtained from facies indicator simulation without integrating seismic data and those using seismic dataduplicated along the vertical direction. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: reservoir; stochastic; facies; modeling; object; pixel

1. Introduction

The distribution of facies in a clastic reservoir isthe most important heterogeneity for reservoir char-

Žacterization and performance prediction Haldorsen.and Damsleth, 1990 . Reservoir modeling of clastic

systems starts with a modeling of facies geometry,

) Corresponding author. Present address: P.O. Box 2189, Hous-ton, TX 77252-2189, USA.

Ž .E-mail addresses: [email protected] T. Yao ,Ž [email protected] A. Chopra .

1 Tel.: q1-972-509-6081.

followed by the simulation of petrophysical proper-ties within each facies independently. There are manydifferent approaches to the stochastic modeling ofreservoir facies, the two main classes being object-based and pixel-based algorithms. Object-based algo-rithms can generate models displaying the crisp geo-metric shape of the facies, which are consistent withprior geological morphological interpretationsŽStoyan et.al., 1987; Omre et.al., 1990; Georgsen

.and Omre, 1993; Deutsch and Wang, 1996 . How-ever, the probabilistic distributions of the shape pa-rameters, required by the object-based algorithms,are difficult to infer from actual subsurface data. Inaddition, conditioning to dense local data can be

0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 00 00048-6

( )T. Yao, A. ChoprarJournal of Petroleum Science and Engineering 27 2000 69–8470

difficult. In this paper, we will focus on pixel-basedsimulation algorithms in integration of seismic data,addressing the problem of limited vertical resolutionof such data.

The commonly used pixel-based method is se-Žquential indicator simulation Journel and Alabert,

.1989, 1990 . This technique generates facies at eachpixel of the 3D model, one at a time, by drawingfrom the local conditional cumulative distribution

Ž .function ccdf estimated from the conditioning data.The previously simulated values are to be used asconditioning data for the pixels to be simulatedsubsequently, hence the spatial structure of the distri-bution of facies can be reproduced. Due to the sparsesampling of well data, it is important to integrateinformation from seismic data that deliver better

Žareal coverage Xu et al., 1992; Haas and Dubrule,.1994; Fournier, 1995; Tjolsen et al., 1995 . How-

ever, the utilization of seismic data for modeling 3Dfacies faces some severe problems, possibly the most

Žimportant being that of scale difference Doyen et. Žal., 1997 . The well data of categorical facies usu-

.ally considered as ‘‘hard’’ data are defined on amuch smaller volume support than the seismic data.They are distributed in the 3D space providing high

Ž .vertical resolution assuming vertical wells as re-quired by 3D geologic modeling, while seismic dataoften do not provide the same vertical resolution andrepresent only some vertically averaged informationover the layers being modeled. Integration of datadefined on such different scales is a difficult chal-

Ž .lenge. Behrens et al. 1996 and Deutsch et al.Ž .1996 offer reviews of algorithms presently used toaddress this challenge while building porosity mod-els. Most algorithms address the scale difference by,implicitly or explicitly, duplicating the seismic at-tribute map on each slice along the vertical directionthus creating artificially dense 3D seismic data. Sim-ulated annealing algorithms can bypass this problem,but they require delicate tuning of the annealingschedule parameters to reach convergence. In addi-tion, they may be cpu-intensive.

Ž .Following an original lead by Xu et al. 1992 ,Ž .Behrens et al. 1996 suggested integrating the seis-

mic attribute map by considering them as ‘‘block’’data through a ‘‘block cokriging’’ paradigm. Theapproach was demonstrated on a case study of build-ing 3D porosity models integrating seismic attribute

Ž .map. In a recent paper, Journel 1999 recalled theproperty of ‘‘block cokriging’’ that allows matchingnot only the hard data values at their locations butalso any volume-averaged data values, as long as theaveraging process is linear and all the values withinthe same volumes are estimated using the sameconditioning data configuration. Yao and JournelŽ .1999 applied it in a more theoretically rigorousway to build 3D porosity models in a West Texascarbonate reservoir. The results from both Behrens et

Ž . Ž .al. 1996 and Yao and Journel 1999 illustrate theapproach to build 3D models of a continuous vari-able, i.e., porosity, using the improved sequentialGaussian simulation algorithm with block cokrigingoption.

Intuitively, this block cokriging approach can alsobe incorporated into a sequential indicator simulationalgorithm to integrate the seismic information whilebuilding categorical facies models, provided the pro-cessed seismic data contain information about the

Žfacies proportion along the vertical dimension i.e.,the seismic data are the vertical averages of the

.indicators of each facies . However, under the se-quential indicator simulation paradigm, adding previ-ously simulated values as conditioning data for sub-sequent nodes will downplay the influence of seis-mic data, especially when the channel sand facieshas a long range of continuity along some horizontaldirection. Therefore, the information conveyed bythe seismic data cannot be fully accounted for. Thiscan be seen in the following case study in Section 4.

In this paper, we propose to weave the newapproach of ‘‘block cokriging’’ into the probability

Ž .field p-field simulation algorithm. For the p-fieldsimulation, the local ccdf’s are kriged from theoriginal conditioning data only, while the probabilityvalues used to draw from the local ccdf’s are spa-tially correlated. The reproduction of spatial structure

Ž .of facies distribution or variogram model isachieved through the spatially correlated p-field,rather than through using the previously simulatedvalues as conditioning data for subsequent simula-tion as in sequential indicator simulation. Whilebuilding the local ccdf’s at each pixel, the seismicdata from attribute map are integrated as block data

Ž .as proposed by Behrens et al. 1996 . Since onlyoriginal data are used as conditioning data, same dataconfiguration can be retained while estimating the

( )T. Yao, A. ChoprarJournal of Petroleum Science and Engineering 27 2000 69–84 71

local ccdf’s at all the pixels along the same verticalline. There will be no previously simulated values tobe used to downplay the effect of the seismic datafor building local ccdf, hence the information fromseismic data can be fully accounted for. That is themain advantage of the improved p-field simulationover the sequential indicator simulation. In addition,the p-field simulation only requires building the localccdf’s once at each node, using all the informationfrom original conditioning well data and seismicdata. To generate multiple realizations, it only re-quires generating different correlated probabilityfields used to draw from the saved local ccdf’s.Therefore, the cpu advantage is gained in the im-proved p-field simulation algorithm.

In the following sections, we will first recall thetraditional indicator simulation algorithms, followedby the improved sequential indicator simulation aswell as improved p-field simulation. The proposednew algorithms will be illustrated by a case study ina real clastic reservoir.

2. Sequential indicator simulation: recall

The traditional sequential indicator simulation al-gorithm builds on the sequential simulation paradigmŽ .see Goovaerts, 1997, p. 376 . More precisely, at

Ž .each node of the simulation grid, it proceeds with: ia sequential determination of the local facies proba-bility distribution conditional to both original data

Ž .and previously simulated values, followed by iidrawing the facies type from that local distribution.

Assuming there are altogether K facies, at anysample location u , each original facies sample isa

w Ž .recoded as a vector of binary indicator i u ; k ,a

xks1, . . . K with:

i u ; k s1 if u g facies k ,Ž .a a

s0 otherwise 1Ž .

The conditional probability of occurrence of fa-cies k at any unsampled location u isestimated by

Ž .indicator kriging as Goovaerts, 1997, p. 293 :

) < ) <Prob Z u sk n sE I u; k n� 4 � 4Ž . Ž . Ž . Ž .

s i ) u; k , ks1, . . . , KŽ .

with:n

)i u; k yp s l u; k i u ; k yp 2Ž . Ž . Ž . Ž .Ýk a a kas1

� Ž .4where p sE I u; k is the global proportion ofk<Ž .facies k; the notation n means ‘‘conditioning to

the n neighborhood data available’’; these includeboth original data and previously simulated values.The l terms are the weights determined by solvinga

the kriging system.The indicator kriging process is repeated for the

K facies, and the resulting conditional probabilities) Ž .i u; k are then assembled into a cumulative condi-k

tional distribution function, or ccdf:

kX

) w xP u; k s i u; k g 0,1Ž . Ž .Ý kXk s1

The simulated facies type at u is obtained byMonte Carlo drawing from that local ccdf.

Ž .The simple indicator kriging expression 2 canalso be provided with smoothly varying local proba-

Ž .bility p u :k

n)i u; k yp u s l u; kŽ . Ž . Ž .Ýk a

as1

= i u ;k yp u 3Ž . Ž . Ž .a k

One possibility is to determine the locally varyingŽ .prior probability p u from seismic data. Note thatk

Ž .expression 3 requires knowledge of the prior prob-Ž .ability p u at every grid location u, that is at thek

resolution of the 3D simulation grid. But, typically,seismic data do not have the vertical resolution ofthe simulated grid. The traditional way is to assumethat a seismic measurement carries the same infor-mation for all the nodes along the same vertical line,which amounts to duplicating the seismic data verti-

Ž .cally Gorell, 1995; Goovaerts, 1997, p. 190 .

3. Improved algorithms with block cokriging ap-proach

Seismic data have limited vertical resolution,hence can only provide vertical average of rockproperties, such as facies proportion map. In the

Ž .kriging expression 2 , the seismic data of facies


Ž .proportion s x; k from the seismic attribute map atŽ .the horizontal location xs x, y of the 3D grid

Ž .location us x, z is added as a ‘‘block average’’Žvalue Journel and Huijbregts, 1978, p. 305; Abra-

hamsen et al., 1996; Haas and Natinger, 1996;.Behrens et al., 1996; Journel, 1999 , where z repre-

sents the vertical coordinate of the location u. TheŽ .seismic value s x; k from the attribute map is

Ž .common to all facies indicator variables i u; zalong the vertical line at x but its correlation withŽ .i u; k is specific to the facies k and the coordinate

Ž .z of us x; z . Its relation to the vertical average ofŽ .the indicator variable i u; k is modeled as:

1s x; k s i x, z ; k d tqe u 4Ž . Ž . Ž . Ž .H

Z Z

Ž .where Z is the vertical thickness and e u is an errorthat is assumed to be independent of the integral.Although the raw seismic data from survey, such asthe seismic volume of amplitude, are not directlyrelated with the facies indicator variables as in theabove equation, the geophysicist can process the rawseismic data and come up with an attribute map

informing the facies proportion areally, based on thecross validation between seismic data and the faciesproportion at well locations. The example is given inthe case study in Section 4.

The improved kriging estimate is then written as:n

)i u; k yp s l u; k i u ;k ypŽ . Ž . Ž .Ýk a a kas1

ql s x; k yp 5Ž . Ž .0 k

Ž .The seismic data s x; k are introduced through anadditional term in the kriging expression. Due to the

Ž .linearity of expression 4 and the errorrsignal inde-Žpendence assumption, the correlation between s x;

. Ž .k and any value i u; k is simply:

C u ,x; k sCov I u; k ,S x; k� 4Ž . Ž . Ž .I a

Z1s C I x, z ; k ,S x; k� 4Ž . Ž .Ý IZ zs1

where Z is the vertical dimension at horizontal loca-tion x.

The corresponding kriging system is:

n°l C u yu ; k ql C u ,x; k sC uyu ; k ,as1, . . . ,nŽ . Ž .Ž .Ý b I b a 0 I a I a

bs1~ 6Ž .n

l C u ,x; k ql Var S x; k sC u,x; k� 4Ž . Ž .Ž .Ý b I b 0 I¢bs1

where:Z Z1

Var S x; k s C� 4Ž . Ý Ý I2XZ zs1 z s1

= I x, z ; k , I x, zX ; k� 4Ž . Ž .yVar e x� 4Ž .

Ž .If all the estimated nodes us x, t along thesame vertical at x utilize the same data configuration,

) Ž .then the resulting 3D estimated values i u; k areexact in the sense that:

Ž .1. they honor the well data i u ; k at their na

locations u :a

i) u ; k s i u ; k ,Ž . Ž .a a

;as1, . . . ,n; ;ks1, . . . , K

2. their vertical averages identify the conditioning� Ž .4seismic data from attribute map, if Var e x s0

Ž .usually unknown :

Z1)i x, z ; k ss x; k , ;x 7Ž . Ž . Ž .Ý

Z zs1

The proof for this exactitude property is providedŽ .by Journel 1999 . If the data configuration for all

the nodes along the vertical dimension at x varies,Ž .the exactitude relation 7 will not hold anymore. In

a sequential simulation paradigm where previouslysimulated values are added as conditioning data forthe nodes to be simulated later, it is impossible tokeep the same data configuration for the nodes alongthe same vertical line. Therefore, depending on the


number of previously simulated nodes to be used, theŽ . Z ) Ž .estimated average 1rZ Ý i x, z; k might notzs1

Žbe well correlated with the input seismic data s x;.k . This can be shown in the case study in Section 4.

In order to fully account for the seismic data thatrepresent the vertical averages of the facies indica-

Ž Ž ..tors Eq. 4 , the ccdf’s for the nodes along thesame vertical line should be estimated using only theoriginal data with the same data configuration. Prob-

Žability field simulation algorithm Srivastava, 1992;.Xu, 1995; Deutsch and Journel, 1998, p. 181 pro-

) Ž . ) Ž X .vides a solution. Assuming i u; k , i u ; k arelocal probabilities of prevailing facies k at location u

X Ž .and u estimated from Eq. 5 using only the originalindicator data from well logs and seismic data asblock data. The simulated facies are then drawn fromthese local distributions using spatially correlated

Ž l .Ž . Ž l .Ž X. Ž .probability values p u and p u , where the ldenotes the lth realization. The probability valuesŽ l .Ž . Ž l .Ž X.p u , p u are spatially correlated in that they

Ž .are from the same realization l of a ‘‘p-field’’,w xwith stationary uniform distribution in 0,1 and co-

variance modeled from the sample covariance of theuniform transforms of the data. Hence, even thoughthe local ccdf’s are estimated without using thepreviously simulated values, the spatial structure ispreserved in the generated model through the corre-lated probability field. This p-field simulation withlocal ccdf’s derived from block cokriging approachcan take full account of the seismic data since nopreviously simulated values are used as conditioningdata that would downplay the influence of the seis-mic data when estimating the local ccdf.

The improved algorithms of sequential indicatorsimulation and p-field simulation with block cokrig-

Ž .ing are now applied to the modeling of a sand payfacies in a clastic reservoir. The results are compared

Ž .to: a modeling that ignores the seismic information,Ž .and b using seismic information with vertical

duplication. For the improved sequential indicatorsimulation algorithm, the sensitivity of the results tothe number of previously simulated values used isalso studied.

4. Case study

The data for this case study are from a fluvialreservoir, which has an area of 5000 ft=5000 ft.

Fig. 1. Example of facies logs along four wells: 0 indicates shaleand 1 indicates sand.

Ž .The pay facies channel sand consist of amalga-mated channels. From the well data, the volume


proportion of distributary channels is estimated atŽ20%. Mudstone and other facies distributary mouth

.bar, shale, coal and limestone constitute the remain-

ing 80%. The average gross thickness is 66X, withfacies description in cores available at 1-ft intervals.Thus, vertical resolution for the simulation grid was

Ž . Ž . Ž .Fig. 2. a Sand proportion map from seismic data; b histogram of sand proportions from seismic data; c scatterplot of sand proportionsŽ .from seismic data vs. those from well data after rescaling to the well average proportion 0.20 .


set at 1 ft. The grid includes 100=100 horizontalnodes with 50X spacing. The available data include17 wells with facies indicator data and seismic data

that were interpreted for the areal proportion ofŽ .channel sands attribute map . The original seismic

data from 3D survey were first inverted into a seis-

Fig. 3. Six realizations of the scatterplot between seismic-derived and well-derived sand proportions from bootstrap sampling of the 17 dataŽ .points bottom of Fig. 2 .


mic impedance volume. This impedance volume wasthen interpreted for a seismic attribute map of sand

Fig. 4. Experimental indicator variogram of sand facies calculatedfrom the well data and the models adopted. Model for horizontalvariogram is based on geologic interpretation.

Fig. 5. Simulated 3D cubes of sand facies, using different algo-Ž .rithms: a improved sisim with seismic data used as blockŽ . Ž .average; b sisim with no seismic data used; c sisim with

seismic data used as local mean.

proportion or prior probability; this interpretationwas based on the sample cross validation betweenseismic impedance and sand thickness at well loca-tions.

Fig. 1 shows four typical sand facies logs at a 1-ftvertical resolution. The sand proportion map inter-preted from the seismic data and the correspondinghistogram are shown in Fig. 2a and b. The average


Ž .Fig. 6. Vertically averaged simulated sand proportion maps and their scatterplots with the seismic-derived prior proportions: a improvedŽ . Ž .sisim with seismic data used as block average; b sisim with no seismic data used; c sisim with seismic data used as local mean.


sand proportion obtained from the seismic data is0.18, a value that is slightly smaller than the 0.20proportion obtained from well data. To provide con-sistency with the well data, the seismic proportionmap was rescaled to match the well-derived estimate.The scatterplot between the prior seismic-derivedsand proportion data and the co-located 17 wellmeasurements is shown in Fig. 2c; the linear correla-tion is significant at 0.68. To validate this correlation

Žcoefficient, we applied bootstrap Efron and Tibshi-

.rani, 1993 on the 17 data and the additional sixscatterplots given in Fig. 3 does support the similarcorrelation. One objective of this facies simulationexercise is to reproduce that correlation.

Fig. 4 shows the experimental indicator vari-ograms for the channel sands, calculated from the 3Dwell indicator data. The variogram model adoptedhas a range of 2500 ft in the NS direction, and 1000ft in the EW direction. Due to the sparsity of lateral

Ž .data only 17 well locations , the horizontal ranges

Ž . Ž .Fig. 7. Ensemble averages of 20 realizations using different algorithms: a improved sisim with seismic data used as block average; bŽ .sisim with no seismic data used; c sisim with seismic data used as local mean.


were obtained from geological interpretation abouthow continuous the channel sands are in the princi-

pal directions. The range in the vertical direction is0.5 using stratigraphic coordinates varying from 0 to

Ž .Fig. 8. Ensemble average sand proportion maps and their scatterplots with the seismic-derived prior proportions: a improved sisim withŽ . Ž .seismic data used as block average; b sisim with no seismic data used; c sisim with seismic data used as local mean.


X Ž .1, or an average of 33 in true distance units . Thesemivariogram model is:

g h s0.16SphŽ .

=

2 2 2h h hx y zq q(ž / ž / ž /ž /1000 2500 0.5

Fig. 5a–c shows the simulated realizations of theŽ .sand facies black pixels , obtained by each of the

following three algorithms:

1. the improved indicator simulation algorithm, inte-grating the seismic attribute map at its proper

Ž .scale through expression 5

2. a traditional indicator simulation ignoring theseismic information

3. indicator simulation using the seismic data as aŽ .local mean through expression 3

To check how well each algorithm reproduces thescatterplot between averaged sand proportion andinput seismic data, the vertical average of the simu-lated sand indicators was retrieved at each horizontallocation. The sand vertical average maps correspond-ing to the three realizations are shown on the leftside of Fig. 6. The right side of that figure gives thescatterplots between these vertical averages and the

Fig. 9. Simulated 3D sand facies cube using a maximum of 24 original data and an increasing maximum number n of previously simulatedŽ . Ž . Ž . Ž . Ž .values: a ns1 prev. simul. value; b ns3 prev. simul. values; c ns6 prev. simul. values; d ns12 prev. simul. values; e ns24

prev. simul. values.


input proportion values as derived from seismic data.The improved sequential indicator simulation algo-

Ž .rithm upper row of Fig. 6 produces a 0.27 correla-

tion; the other two algorithms show no correlation.The explanation for the low correlation coefficientfor the improved sequential indicator simulation is

Ž .Fig. 10. Scatterplots between simulated sand proportions calculated from Fig. 11 vs. the input seismic-derived prior proportions: a ns1Ž . Ž . Ž . Ž .prev. simul. value; b ns3 prev. simul. values; c ns6 prev. simul. values; d ns12 prev. simul. values; e ns24 prev. simul.

values.


Fig. 11. Simulated 3D sand facies cube using improved probabil-ity field simulation with seismic data as block average.

that adding more previously simulated values asconditioning data has downplayed the effect of seis-mic data. This can be seen in the sensitivity analysis.

Since simulation honors the block data only in anexpected value sense, the previous analysis should beconducted using the results of many realizations. Assuch, 20 realizations from each algorithm were simu-lated and three ensemble averages computed overeach of the three sets of 20 realizations are shown inFig. 7. The grey scale represents the sand proportionat each 3D node obtained by averaging the 20 simu-lated sand indicator values. Fig. 8 provides a checksimilar to that performed in Fig. 6, and it can be seenthat the proposed algorithm yields the best correla-

Ž .tion 0.42 , whereas the other two algorithms yieldagain almost zero correlation. However, 0.42 is still

significantly lower than the sample correlation 0.68.Again, one explanation is that by using the previ-ously simulated node, the sequential indicator simu-lation downplays the influence of the block average

Ž .data seismic data from attribute map . Indeed,checking the simulation debug file confirms that thekriging weight attributed to the block average datadecreases as the sequential simulation progresses andmore previously simulated data are being generated.The sensitivity of the results to the number of previ-ously simulated values used can be seen in thefollowing study.

4.1. SensitiÕity analysis

To investigate the impact of the number of previ-ously simulated nodes on reproduction of the scatter-plot of Fig. 2, the proposed indicator simulationalgorithm was repeated retaining different numbersof previously simulated values. All other simulationparameters were kept constant.

Fig. 9 shows five realizations, all use a maximumof 24 original well indicator data plus a maximum ofn previously simulated values, with n increasingfrom 1, 3, 6, 12 to 24.

Fig. 10 shows the correlation between the result-ing simulated sand proportions with the seismic-de-rived prior sand proportions. When the number of

Ž .previously simulated nodes used is small ns1 , thesimulated facies cube is very noisy as expected.

Fig. 12. Areal proportion map calculated from the improved probability field simulation realization and its scatterplot vs. the inputseismic-derived prior proportions.


Conversely, the correlation with the prior seismic-de-rived proportion is high. As the number of previ-ously simulated nodes used increases, the simulatedimages are less noisy but the correlation with theseismic data decreases. The kriging solution at somenodes was checked extensively and this effort con-firms that the weight attributed to the seismic blockaverage data decreases with the number of previ-ously simulated nodes used. This explains the lowercorrelation with the seismic data.

The problem of decreasing influence of seismicdata with increasing number of previously simulatedvalues, associated with sequential simulation, can becircumvented by using the proposed improved prob-ability field simulation algorithm. Fig. 11 shows onesimulated realization of the sand facies using theimproved p-field simulation algorithm. Fig. 12 showsthat the resulting simulated vertical sand proportionmap is well correlated with the input seismic map ofFig. 2 with a correlation coefficient 0.67 that is closeto the sample correlation of 0.68. Since the localccdf’s are built from only the original data in p-fieldsimulation algorithm, adequate weight is assigned tothe seismic block average data. Also the original data

Ž .configuration used is the same for all nodes x; talong the same vertical line at x.

5. Conclusions and discussions

We have proposed a modification to the sequen-tial indicator simulation and probability field simula-tion in order to account for block average data thatcovers the seismic information, as initially proposed

Ž .by Behrens et al. 1996 . The seismic-derived sandproportions from the interpreted seismic data areconsidered as vertically averaged ‘‘block’’ data andare integrated through a cokriging framework in the3D facies indicator simulation. The results are com-pared with those from the traditional algorithms, inregards to honoring the prior cross validation be-tween averaged facies proportion from wells andsand proportion from seismic attribute map. Utiliza-tion of previously simulated facies indicator valuesin the improved sequential indicator simulation algo-rithm results in poor reproduction of the samplecorrelation between simulated facies vertical propor-tions and seismic-derived prior proportions. One al-

ternative is to use the proposed improved probabilityfield simulation algorithm where the facies condi-tional probabilities are determined only from theoriginal well log data and seismic data.

Acknowledgements

The authors thank Godo Perez and David Tambefrom ARCO for providing the detailed geologicalbackground of the reservoir, and Sanjay Srinivasanfrom Stanford for his careful peer review of thepaper. The authors acknowledge many helpful andconstructive comments from the reviewers, Dr. Clay-ton Deutsch, Dr. Bruce Hart and Dr. Alain-YvesHuc.

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