integrating seismic attribute maps and well logs for porosity modeling in a west texas carbonate...

Ž .Journal of Petroleum Science and Engineering 28 2000 65–79www.elsevier.nlrlocaterjpetscieng

Integrating seismic attribute maps and well logs for porositymodeling in a west Texas carbonate reservoir: addressing the

scale and precision problem

Tingting Yao a,), Andre G. Journel b,1

a Department of Geological and EnÕironmental Sciences, Stanford UniÕersity, Stanford, CA 94305, USAb Department of Petroleum Engineering, Stanford UniÕersity, Stanford, CA 94305, USA

Received 7 February 2000; accepted 14 July 2000

Abstract

A 3-D porosity field in a west Texas carbonate reservoir is modeled conditional to both AhardB porosity data sampled bywells and 2-D seismic attribute map with less vertical resolution than the well log data. The difference-of-scales between thetwo sources of data is resolved by a prior 2-D estimation of vertically averaged porosity using well and seismic data. These2-D estimates are then used to condition the 3-D stochastic simulation of porosity. The algorithm used to merge 2-D averagevalues and 3-D data values, i.e., to solve the difference-of-scale problem, is a form of block kriging, which ensures thatvertical averages of the 3-D estimates reproduce exactly the 2-D average data values. The precision of the 2-D conditioningdata is also addressed. Several new geostatistical algorithms, such as automatic covariance modeling and direct sequentialsimulation algorithms, are weaved into the application. These new algorithms facilitate the process of integration of the softdata in petrophysical modeling. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: reservoir; carbonate; geostatistics; porosity; modeling; seismic

1. Introduction

Geostatistics is used in the petroleum industry tomodel the spatial distribution of petrophysical prop-erties, such as porosity. These 3-D numerical modelsof petrophysical properties provide the input into the

) Corresponding author. Current address: ExxonMobil Up-stream Research Company, ST 3209, PO Box 2189, Houston, TX77252-2189, USA. Tel.: q1-713-431-7174; fax: q1-713-431-6336.

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

1 Tel.: q1-650-723-1594.

flow simulator for reservoir performance prediction.Because sampling with well log data is sparse, it iscritical to integrate information from seismic data,

Žwhich delivers better areal coverage Abrahamsen et.al., 1996; Fournier, 1995; . However, the utilization

of seismic data for modeling 3-D petrophysical prop-erties faces some severe problems, possibly the most

Žimportant being that of scale difference Haas and.Dubrule, 1994; Gorell, 1995; Doyen et al., 1997 .

Ž .The well data usually considered as the AhardB dataare defined on a much smaller volume support thanthe seismic data. Well data are distributed in the 3-D

Žspace providing high vertical resolution assuming

0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 00 00068-1

( )T. Yao, A.G. JournelrJournal of Petroleum Science and Engineering 28 2000 65–7966

.vertical wells , while seismic data often do not pro-vide the same vertical resolution and represent onlysome vertically averaged information over the layerbeing modeled. Integration of data defined on suchdifferent scales is a difficult challenge. Behrens et al.Ž . Ž .1996 and Deutsch et al. 1996 offer reviews ofalgorithms presently used to address this challenge.Most algorithms address the scale difference by,implicitly or explicitly, repeating the 2-D seismicattribute map along each vertical slice, thus, creatingartificially dense 3-D seismic data and generatingvertical banding artifacts in the resulting petrophysi-cal model. Simulated annealing algorithms can by-pass this problem, but they require delicate tuning ofthe annealing schedule parameters to reach conver-gence. In addition, 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 data by first performing a 2-D cokriging of thevertically averaged porosity values using verticalaverage well data and the 2-D seismic attribute mapas a covariate. The resulting estimated verticallyaveraged porosity map is then used to condition a3-D stochastic simulation of the 3-D porosity field.In this second step, the authors solve the difference-of-scale problem with a Ablock krigingB procedure,whereby the previously AestimatedB vertically aver-aged porosity values are assimilated to AtrueB actualvertically averaged data values. Thus, the error ofestimation of the 2-D vertically averaged values isignored. In addition, the block kriging is performed

Žon the normal score transforms of the data both well.data and seismic data . The vertical linear averaging

is not preserved by such non-linear transform. Whatis needed is a simulation algorithm which can oper-ate directly on the original data without any priornon-linear transform. Nevertheless, the results shown

Ž .by Behrens et al. 1996 applied to two differentNigerian fields are remarkable, and their algorithm isworth revisiting and correcting.

In this paper, the direct sequential simulationalgorithm is used to avoid the non-linear normalscore transform as in sequential Gaussian simulation,hence, the vertical linear averaging is preserved inthe final model. The precision problem of the previ-ously estimated 2-D conditioning data is addressedby associating those 2-D conditioning data with anerror item, which is represented by their estimation

variance. Some other new geostatistical algorithms,such as automatic covariance modeling approach, areweaved into this application.

2. Methodology

Ž .In a recent paper, Journel 1999 recalled a littleknown property of cokriging estimates, which allowsmatching, not only the hard data values at theirlocations, but also, any volume averaged data values,as long as the averaging process is linear.

In the following application, we used 2-D cokrig-ing to generate an AestimatedB 2-D field of verticallyaveraged porosity values conditioned to both seismicand well data. Instead of an analytical model for theŽ .cross covariances, we used the newly developed

Ž . Žtechnique of modeling cross spectrum tables Yao.and Journel, 1998 .

Next, we proceeded to a 3-D cokriging of poros-ity values conditioned to the 3-D well data and to thepreviously estimated 2-D vertically averaged poros-ity values. This 3-D kriging accounts for the errorvariance associated with the previous estimation ofthe 2-D vertically averaged porosity values: esti-mated vertical averages are not reproduced exactly,whereas actually measured vertical average values atwell locations are matched exactly. The 3-D porosityfield honors the sample cross-correlation between the2-D averaged porosity and the 2-D seismic attributemap.

Last, direct sequential simulation of 3-D porosityvalues is performed by drawing values from a posi-

Ž .tive distribution lognormal , whose mean is theestimated porosity value and the variance, the corre-

Ž .sponding kriging error variance. This results inseveral alternative, equiprobable, 3-D realizations ofthe porosity field which:

Ø match the hard 3-D porosity data along the wells;Ø reproduce the sample scattergram between the

vertically averaged porosity data and the collo-cated seismic data; and

Ø reproduce the sample 3-D porosity histogram andvariogram.

The realizations do not show any artifact ofsmoothing or vertical banding. The implementation

( )T. Yao, A.G. JournelrJournal of Petroleum Science and Engineering 28 2000 65–79 67

of the method is documented in detail in the follow-ing case study.

3. A west Texas carbonate reservoir

The case study developed here utilizes well logand seismic data from a carbonate reservoir in aPermian basin in west Texas. The production intervalis a shallowing upward, prograding, carbonate shelfsequence composed primarily of alternating

Ž .dolomites and siltstones Chambers et al., 1994 . Thedolomites have slightly poorer reservoir quality thanthe siltstones because of hydritic plugging of thepore space. Within the study area, there appears tobe a trend of decreasing siltstone proportion to thewest with a corresponding decrease in porosity. Thelocal complexity of the dolomitersiltstone geome-tries did not allow separating these two facies fromthe data available. In the following presentation, alldata and coordinate values were scaled to preserveconfidentiality.

The volume of study here is limited to a welldefined producing layer that has an area of 10 400=

Ž . Ž .10 400 ft 3170=3170 m and is 50 ft 15 m thick.This layer is seen in a seismic profile from a 3-Dsurvey as a single reflection event. Therefore, a 2-Dseismic attribute map represents the vertically aver-aged porosity rather than the individual 3-D porosityvalues. The corresponding estimationrsimulationgrid is discretized into 65=65 horizontal grid nodes

Ž .having 50 vertical layers, each 1 ft 0.305 m thick.Ž .The horizontal cell size is 160=160 ft 50=50 m ,

approximately that of the seismic data available.The study focuses on the integration of the 2-D

seismic attribute map for the estimationrsimulationof a 3-D volume of porosity values, each defined ona very small support volume assimilated to a quasiApointB support. The hard porosity data are log-de-rived from 62 wells within the study area A. Thelocation map and three of the well–log curves fromthe 62 wells are shown in Fig. 1: the greyscalerepresents the vertically averaged porosity valuesexpressed in percentage. The corresponding his-tograms of the 3-D log-derived porosity values and2-D vertically averaged porosity values are given inFig. 2. The porosity sample histogram shows an8.40% mean and positive skewness.

Ž .Fig. 1. A Location map of the 62 wells. The grey scale repre-sents the value of the vertical averaged porosity at each well

Ž .location. B Porosity curves from the three wells.

A number of 2-D seismic attribute maps wereavailable to crossplot with the vertically averaged


Fig. 2. Histograms of log-derived 3-D porosity data and verticallyaveraged 2-D porosity data.

porosity data. The seismic attribute, which had thehighest collocated correlation with the vertically av-eraged porosity, was the low-frequency reflectionenergy. There is no specific rock physics considera-tion relating this seismic attribute to vertically aver-aged porosity. The low-frequency reflection energywas retained because it has a higher correlation withthe vertically averaged porosity based on pure statis-tical grounds. Fig. 3 shows the greyscale map of theseismic data. Fig. 4 shows the scattergram of the 62collocated pairs of seismic vs. vertically averagedporosity values. The higher seismic values in the NEcorner are consistent with the higher sample porosityvalues of Fig. 1. The 2-D correlation coefficient issignificant at 0.60.

3.1. Cokriging Õertically aÕeraged porosity

The first step of the study consists of generating a2-D field of vertically averaged porosity values that

will be used to condition the final 3-D simulation ofpoint-support porosity. Cokriging was used whereby,at each location x of the 65=65 seismic grid, thea

vertically averaged porosity is estimated from a lin-ear combination of up to 12 neighboring vertically

Ž .averaged porosity data f from wells and 12 seis-mic data:

12)f x ym s l f x ymŽ . Ž .Ýf a a f1 1

a s11

12

q h s x ym 1Ž . Ž .Ý a a s2 2a s12

Ž .where x is the set of horizontal coordinates, f xa1

is the vertically averaged porosity data at the wellŽ .location x , and s x is the seismic data ata a1 2

location x . The m and m are the correspondinga f s2

porosity and seismic mean values, l and h area a1 2

the cokriging weights.Such cokriging requires three covariance models,

the porosity covariance, the seismic covariance and

Fig. 3. Map and histogram of the seismic data.


Fig. 4. Calibration cross-plot between the vertically averagedporosity and the collocated seismic data.

the cross-covariance. In this study, instead of follow-ing the traditional route of calculating experimentalcovariances and modeling them with a linear model

Ž .of coregionalization Goovaerts, 1997 , the directcovariance table approach proposed by Yao and

Ž .Journel 1998 was retained. More precisely,

1. the three experimental correlogram tables are cal-culated and the results are presented in Fig. 5Ž .Deutsch and Journel, 1998 ,

2. these correlogram tables are smoothed and filled-in for missing entries by a preliminary smoothing,

3. the completed correlogram tables are transformedinto quasi-spectrum tables through fast Fourier

Ž .transform FFT ,4. the quasi-spectrum tables are corrected into per-

Žmissible spectrum tables positive definiteness.condition , and

5. they are inverse Fourier transformed into permis-sible, jointly positive definite correlogram tablesŽ .Fig. 6 .

All correlogram and cross-correlogram values re-quired by the cokriging process are read directlyfrom the tables of Fig. 6. This approach, in additionto being fast and easy to implement, allows a moreaccurate modeling of the original experimental cor-relogram values, and it is not constrained by anyclosed-form analytical expression.

The greyscale map of the cokriging estimates ofthe vertically averaged porosity and their histogramis given in Fig. 7. Note that the estimated values are

Fig. 5. Experimental auto- and cross-correlograms of the verticallyaveraged porosity and seismic attribute.


Fig. 6. Final smoothed auto- and cross-correlograms of the verti-cally averaged porosity and seismic attribute.

Fig. 7. Grey-scale map and histogram of estimated average poros-ity using cokriging.

high in the NE corner due to the influence of theseismic data. Also, the influence of the conditioningwell data is reflected in the central lower half of thestudy area. The overall estimated mean is fs8.25,a value slightly less than the sample mean of 8.40.This smaller value can be explained by the decluster-ing effect of kriging, which underweights the clusterof wells with high porosity values in the NE corner,see location map of Fig. 1.

The scattergram of estimated, vertically averagedporosity values vs. the collocated seismic value isgiven in Fig. 8: the reproduction of the samplescattergram of Fig. 4 is good, although the correla-

Ž .tion coefficient is too high 0.71 instead of 0.60 .This is typical of cokriging.

An alternative to the full cokriging approach ofŽ . ŽEq. 1 could have been collocated cokriging Xu et

.al., 1992 , retaining the single seismic datum valueŽ .s x at the location x being estimated. We chose to


Fig. 8. Scattergram of estimated vertically averaged porosity vs.the seismic low-frequency attribute.

Ž .present the more general formulation as in Eq. 1 ,which would be required if the simulation grid ismuch finer than the seismic grid.

As expected from any regression-type estimationalgorithm, cokriging yields a smoothed image. The

Ž .2variance of the estimated values in Fig. 7 is 1.62 ,a value smaller than the corresponding variance of

Ž .2 Ž .the well data, i.e. 1.91 Fig. 2 .

3.2. 3-D cokriging of porosity

A 3-D estimated map of quasi point-supportporosity values is generated conditional to the 3-D

Ž .porosity data from wells and the previously ob-tained 2-D vertically averaged porosity values whichcarry the seismic information. Consider the collo-cated cokriging estimate:

n)f u ym s l f u ymŽ . Ž .Ýf a a f

as1

)ql f x ym 2Ž . Ž .0 f

Ž .where us x, z is the set of 3-D coordinates: x isthe vector of horizontal coordinates and z is the

) Ž .depth coordinate, f u is the estimated 3-D poros-)Ž . Ž .ity, f u are the 3-D porosity data, f x is thea

previously estimated vertically averaged porosity athorizontal location x, l and l are the correspond-a 0

ing kriging weights, and m is the global meanf

value of porosity.

The corresponding cokriging system is given inAppendix A.

If this cokriging process is applied, the resulting) Ž .3-D estimated values f u are fully exact in the

sense that:

Ž .1. they match the well data values f u at their na

3-D locations u :a

f ) u sf u , ;as1, . . . ,n , and 3Ž . Ž . Ž .a a

2. their vertical averages match the 2-D conditioningdata:

Nz1) )f x , z sf x , ;x 4Ž . Ž . Ž .Ý kNz ks1

where k is the index of the vertical level z and Nk zŽis the number of vertical layers N s50 in this casez

.study .This 3-D cokriging process requires inference of

only the 3-D porosity variogram model.The analytical semivariogram model retained is

Ž .sill standardized to 1 :

g hŽ .2 2 2h h hx y z

s0.6Sph q q(ž / ž / ž /ž /3000 1000 12

q0.4Sph

=

2 2 2h h hx y zq q(ž / ž / ž /ž /6000 30 000 50

5Ž .

where Sph designates a spherical model of unit-rangeŽ .and sill, hs h , h , h are the coordinates of ax y z

separation vector h in the EW, NS and verticaldirections, respectively.

This anisotropic model is displayed in Fig. 9,together with the corresponding sample semivari-ograms.

) Ž .In order for the 3-D cokriging estimates f u tomatch the vertically averaged data, the same data


Ž .Fig. 9. Experimental variograms of 3-D porosity and the fitted model in three directions sill standardized to 1 .


Ž .configuration must be used for all locations x, zk

along the same vertical at x. In this study, the 50Ž .locations x, z , ks1, . . . , 50 are estimated fromk

the same 33 data, more precisely:

Ø four data points taken at regular intervals z X ,k

kX s10, 20, 30, 40 from the eight wells closest tolocation x; and

) Ž .Ø the collocated, vertically averaged value f x ,as obtained from the previous 2-D cokriging us-ing the seismic data.

Fig. 10 gives two vertical sections, NS at xs30and EW at ys30. The vertical bands seen in Fig.10 are the consequences of conditioning the esti-mates of porosity along a vertical column to thesame vertically averaged datum value. This bandingis actually a by-product of the smoothing effect ofkriging and will be corrected by the process of

Ž .simulation; see later Fig. 16. Behrens et al. 1996noted the same banding artifact with a 3-D cokrigingapproach, which was also corrected in their simula-tion results.

Fig. 11 is a crossplot to verify that the verticalaverages of the 3-D estimated porosity values matchexactly the conditioning 2-D vertical averages.

The characteristic smoothing of kriging is re-vealed by the histogram and q–q plot of Fig. 12.The estimation process is exact and unbiased: itreproduces exactly the mean 8.25 of the 2-D condi-tioning data of Fig. 7, but the variance of the 3-D

Ž .2estimated values is only 2.0 , much lower than the

Fig. 10. Two vertical sections of the estimated 3-D porosity.Artifact banding will be corrected by simulation.

Fig. 11. Scatterplot of vertical averages of the 3-D estimatedporosity vs. the previously estimated 2-D average porosity.

Ž .2sample variance 3.37 given in Fig. 2. Again, theprocess of simulation will correct that smoothing.

Fig. 12. Histogram of the estimated 3-D porosity values and theirquantile plot with the well-porosity data.


Fig. 13. Scatterplot of vertical averages of the 3-D estimatedporosity vs. the previously estimated 2-D average porosity, ac-counting for precision of the 2-D conditioning data.

3.3. Addressing the precision problem

In the AblockB kriging system, providing the esti-) Ž . Ž .mate f u defined in Eq. 2 , if the covariances

) Ž .associated with the estimates f x are made equalto the covariances associated with the true vertical

Ž .averages f x , the exactitude property of krigingŽwould apply, resulting in the exactitude relation Eq.

)Ž .. Ž .4 . The data f x are, however, estimated val-ues, hence, they should be reproduced only up to

Ž . 2 Ž .their estimation kriging variances s x whichSK

are generated as by-products of the 2-D krigingprocess. If the covariances associated with the esti-

) Ž .mates, f x , account for the corresponding estima-2 Ž . Žtion variance s x , the exactitude relation Eq.SK

Ž .. Ž4 prevails only at well locations x see Appendixa

. ) Ž .A . More precisely, the 3-D estimated porosity f uvalues are now such that:

Ž .1. they match the 3-D well data f u at their na

3-D locations u :a

f ) u sf u , ;as1, . . . ,n , and 6Ž . Ž . Ž .a a

2. their vertical averages are equal to the 2-D mea-sured values, only at the well locations x :a

Nz1)f x ,t sf x 7Ž . Ž . Ž .Ý a z aNz ks1

The scatterplot between the vertical averages ofthe estimated 3-D porosity values and the condition-ing 2-D vertical averages shown in Fig. 13 indicatesthat the estimated vertical averages are not repro-duced exactly. This is due to the consideration of theprecision of the 2-D conditioning data. For partial-

Ž .penetrating wells, the f x could be approximateda

by partial vertical average, associated with someerror to take account for the precision problem.

3.4. 3-D simulation of porosity

To preserve the linear relation between the 2-Dvertically averaged porosity and the 3-D quasi-pointporosity, direct sequential simulation is performedwithout any normal score transform. The theory ofdirect sequential simulation calls for drawing simu-

Ž l .Ž .lated values ff u of the 3-D porosity from a

Fig. 14. Histogram of simulated 3-D porosity and quantile plot vs.well data.


Fig. 15. Two different horizontal sections of the simulated 3-Dporosity values.

Fig. 16. Two vertical sections of the simulated 3-D porosityvalues.

Fig. 17. Histogram of vertical averages calculated from the 3-Dsimulated realization and scatterplot with the conditioning 2-Daverage porosity of Fig. 7.

Fig. 18. Scattergram of simulated vertically averaged porosity vs.collocated seismic data.


Ž . Ž . Ž .Fig. 19. Sample variogram dots , input variogram model continuous line and the variogram of simulated values dash line in threedirections.


sequence of conditional distributions whose meansand variances are identified by the kriging means

) Ž . 2Ž .f u and variances s u . For this kriging, weK

retain the algorithm that accounts for the precision ofthe estimated 2-D vertical average data. The condi-tional distribution type can be anything. For thisapplication, we retain a lognormal distribution type,which yields only positive simulated values. Thelognormal parameters a and b 2 of each lognormaldistribution at location u are given by the relation:

a s lnf ) u yb 2r2Ž .b 2 s ln 1qs 2 u rf ) 2 u 8Ž . Ž . Ž .Ž .K

A random number g is then drawn from a stan-dard normal distribution, the simulated porosity valueat u is:

Ž l . w xff u sexp aqb g 9Ž . Ž .Ž .The superscript l indicates that the value relates

� Ž l .Ž . 4to the l th simulated realization ff u , ugA .Ž l .Ž .That simulated value ff u is immediately

stored as a datum value to be used for cokriging of) Ž X. X

f u at any subsequent and neighboring node u ;recall the sequential simulation paradigm, e.g. in

Ž . ) Ž .Goovaerts 1997 . The cokriging of f u considersfor data the previously obtained, 2-D estimated, ver-tically averaged porosity of Fig. 7 in addition to the3-D original f data and the neighboring, previouslysimulated ff Ž l . values.

The last step of the simulation algorithm consists� Ž l .Ž . 4of transforming each realization ff u , ugA

to approximate the sample porosity histogram of Fig.2, or any other target distribution deemed appropri-ate. The rank order-preserving transform algorithmand program trans have been used for this purposeŽ .Deutsch and Journel, 1998 . This algorithm pre-

Ž l .Ž .serves the exactitude of the simulation ff u vs.the hard 3-D well data, i.e., the final simulated

Ž l .Ž .values f u are such that:

f Ž l . u sff Ž l . u sf u , ;as1, . . . ,nŽ . Ž . Ž .a a a

10Ž .

Fig. 14 shows that the target sample histogramŽ .well data has been well approximated by the simu-

� Ž l .Ž . 4lated values f u , ugA of the first realizationŽ .ls1 produced; compare to the sample statistics ofthe well data in Fig. 2.

Figs. 15 and 16 show several horizontal andvertical sections from the first simulated realization.When compared to estimation as depicted in Fig. 10,it appears that the process of simulation has cor-rected the artifact smoothing and vertical banding ofestimation.

To check the impact of conditioning the simula-tion to vertical averages, the 3-D simulated porosityvalues were vertically averaged. The histogram ofthese simulated vertical averages and their scatter-gram with the corresponding 2-D vertical averagesof Fig. 7 are given in Fig. 17. Although the simula-tion does not reproduce exactly at each location x

Ž .the conditioning estimated vertical average value,the statistics are closely reproduced and the collo-cated 2-D correlation is 0.90. Note that since the

) Ž .vertically averaged values f x are only esti-Ž .mated, they need not should not be reproduced

exactly.Reproduction of the sample correlation of verti-

cally averaged porosity vs. seismic data is checkedthrough Fig. 18 and compared to Fig. 4. Both the

Ž .non-linear shape and the correlation value 0.60 ofthe sample cloud are well reproduced by the simula-tion. The exact match of the correlation value is acoincidence.

Fig. 19 provides a final check, plotting the three-directional variograms calculated from the simulatedrealizations vs. the input variogram model and thesample variogram calculated from the original welldata.

The variograms from the simulated realization arereasonably close to the sample variograms, actuallycloser to them than to the input model. In thepresence of dense conditioning data, as is the case

Ž .here 62 wells , the sample statistics prevail overwhatever model is given as input: this is a positiveconsequence of exact conditioning to the well datavalues.

4. Conclusions

The block kriging procedure initially proposed byŽ .Behrens et al. 1996 is improved to provide an

algorithm for simulating 3-D fields of a petrophysi-cal property conditioned to 3-D hard data and 2-D

Žseismic attribute maps with lower vertical resolu-


.tion . The difference-of-scale problem is solved bygenerating first a 2-D estimated field of verticallyaveraged values conditioned to the 2-D seismic at-tribute map, followed by a 3-D AblockB kriging-typeestimation to generate alternative simulated 3-D real-izations by direct-sequential simulation.

This paper weaves together several recently de-veloped geostatistical algorithms for data processingand integration, more precisely:

1. A fast modeling of covariance and cross-covari-ance tables with FFT, which does not requirechoosing a parametric analytical model and cir-cumvents completely the tedious fitting of a linearmodel of coregionalization.

2. The concept of Ablock krigingB, which allowsexact reproduction of large-scale data values ex-pressed as linear averages of small-scale attributevalues. This algorithm is critical for integratingdata defined on different volume supports.

3. The utilization of an estimation variance to ac-count for the precision of the AdataB, which arenot actually measured, but are the result of a prior

Ž .estimation process see Appendix A .4. The direct sequential simulation algorithm, which

generalizes the well established sequential Gauss-ian simulation algorithm. No prior normal scoretransform is required; simulation is performed onthe original data values, which allow preservingthe linear average characteristic of some data and

Ž Ž .the related exactitude property see point 2 just.above .

Any of the four previous algorithms may not befamiliar to the reader. The original contribution ofthis paper, however, is the combination of these fouralgorithms into a general methodology for the inte-gration of data of different sources and resolutionaccounting for their reliability.

Application to seismic data integration for 3-Dporosity mapping within a west Texas carbonatefield has yielded excellent results with none of theartifact smoothing associated to using data withwidely different resolutions.

Nomenclaturea mean of the lognormal distributionb 2 variance of the lognormal distribution

g variogram functions 2 kriging varianceK

m mean porosity valuef

m mean seismic values

s seismic valuez the k th vertical level in 3-D modelk

Ž .u x, z , 3-D coordinates vectork

A interested study areaf porosity valuef ) estimated porosity valueff Ž l l . simulated porosity valuef Ž l . transformed porosity value to identify tar-

get histogram( .f x vertically averaged porosity value at 2-D

location x; for allg within

Appendix A. Block cokriging accounting for scaleand precision

Ž .Recall Eq. 2 of the 3-D cokriging estimate:n

)f u ym s l f u ymŽ . Ž .Ýf a a f

as1

)ql f x ym A-1Ž . Ž .0 f

) Ž .where f x is the estimated 2-D vertical averageporosity at horizontal location x using both 3-D welldata and 2-D estimates of vertically averaged poros-ity values. These estimates carry the seismic infor-

Ž .mation. The corresponding true vertical average f xcan be written as:

)f x sf x qe x A-2Ž . Ž . Ž . Ž .Ž .where the error e x is, by definition of kriging

Ž .Journel and Huijbregts, 1978 , orthogonal to the) Ž .estimator f x , in which case:

) 2Var f x sVar f x ys x G0 A-3Ž . Ž . Ž . Ž .� 4 � 4 SK

2 Ž . � Ž .4 Ž .where s x sVar e x is the kriging errorSK

variance at x.The previous relation expresses the smoothing

) Ž .effect of kriging: the estimated value f x displaysa variance deficiency equal to the kriging variance.


Ž .Assuming independence of the error e x at xŽ X.with any variable f u at a location with horizontal

coordinates xX/x, the following covariance is writ-

ten:X X

)Cov f u ,f x sCov f u ,f x q0Ž . Ž . Ž . Ž .� 4 � 4A-4Ž .

with for the point-to-vertical average covariance:

XCov f u ,f xŽ . Ž .� 4Nz1

X Xs Cov f u ,f x , z sC u , x� 4Ž . Ž . Ž .Ý kNz ks1

A-5Ž .XŽ .Thus, the covariance value C u , x is merely

Ž Xaveraged from the point 3-D covariance model C u.yu .

The Ablock krigingB system corresponding to the3-D kriging estimate can nowbe developed asŽ .Journel and Huijbregts, 1978 :

n

l C u yu ql C u , x sC uyu ,Ž . Ž . Ž .Ý b b a 0 a a

bs1

as1, . . . ,n A-6Ž .n

)l C u , x ql Var f x sC u , xŽ . Ž . Ž .� 4Ý b b 0bs1

) 2� Ž .4 � Ž .4 Ž .where Var f x sVar f x ys x .SK2 Ž .If s x s0, that is, if the estimated data valueSK

) Ž . Ž .f x is assimilated to the true value f x , thenthe exactitude property of kriging would apply, en-

Ž . 2 Ž .tailing relation 3 . If s x /0, then the dataSK) Ž .value f x is not any more reproduced exactly.

) Ž .The weight l given to the data value f x in the0

cokriging expression is controlled by its estimation2 Ž .variance s x .SK

References

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