SPE 24742

hltegrating seismic Data in Reservoir Modeling:

The Collocated Cokriging AlternativeWenlong Xu and T.T. Tran,* Stanford U.; R.M. Srivastava, * FSS Intl.; andA.G. Jownel,* Stanford U,

●SPE Members

Copyright 1992, society of Petroleum Engineers km.

soobtyoflwoburnEnghaw8

This paper was prepared for prasemation al the 67th Annual Technical Conference and Exhibition of the Society of Petroleum I%gineera bald In Washington, DC, October 4-7, 1992.

Thk paper was aelscted for preaenlation by an SPE Program Committee following revkw of Information contained in an abstract $ubmlffed by the author(s). Contents of the paper,aa presen!ed, have not been reviewed by the Society of Petroleum Enginwm and are subject to correction by the author(e). The material, as prwamsd, dces not nscesaarliy roflecfany position of the Society of Pelrolaum Engineers, 11sofficers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Cot,tmlttwa of the SrscIelyof Petroleum Englnwra. Parmieelonto cow k reatricfedto an abstract of not morethan 300 words. IIlustrationemaynot be coplad. The abstract ehould mnlaln dczrnepkuousacknowledgmentof where and by whom tha paper Is preaentad. Write Llbrar:an, SPE, P,O, Box $33S36, Rlchardeen, TX 76083.3S3S U.S.A. Telex, 730389 SPEDAL,

Abstract

The two sources of information commonly available formodeling the top of a structure, depth data from wells andgeophysical measurements from seismic surveys, are often Mii-cult to integrate. W bile, the well data provide t he most accuratemeasurements of depths there are rarely enough wells to permitan accurate appraisal from well data alone. On the other hand,the seismic data are generally less precise but more abundant.Two geoatatistical methods, “external drift~ and “collocatedcokrigingfi, are proposed to integrate the two sources of infor-mation. A case study is used to document the strengths andweaknesses of both approaches for constructing contour mapscf the top structure and assessing the uncertainty on such mapsthrough stochastic simulations.

Introduction

A major contribution of geostatistics to reservoir modelinghas been the addressing of the general problem of data integra-tion, proposing algorithms for merging data of different types,reliabllit y, taken at different scales, into a more accurate reser-voir numerical model [I-5], As opposed to more traditionaldeterministic approaches, geostatistical algorithms provide anassessment of the resulting model uncertainty,

Typical of the genera! problem of data integration is theutilization of dense 3D seismic data atrd well test data for a bett-er characterization of reservoir heterogeneities, Traditionally,seismic data are used to map major reflective horizons. Due totheir increased resolution, these data are now being used” to im-prove the petrophysical characterization of the reservoir [6,7].As opposed to detecting major structural heterogeneities, the

References and illustrations at end of paper.

mapping of local petrophysical variability is a much more chal-lenging problem calling for algorithms that can handle informa-tion taken at different scales, Indeed, the highest resolution 3Dseismic data provide information still coarse when compared tocore samples; although, one can argue that this coarse infor-mation better matches the scale at which flow simulation andreservoir performance forecasts are made [8,9],

To provide a focus for discussion, consider the simplerproblem of mapping a specific horizon using dense 3D seismicdata (travel time) and sparse control well data (horizon depth),see Figure 1. The well data provide accurate local informationthat can be considered aa ‘hardn, The numerous seismic loca-tions provide a quaai exhaustive coverage, but each travel timerepresents some smooth local average of the depth around theCDP (Common Depth Point) location; also these seismic datashould be calibrated to the control wells. One wishes to mapthe horizon depth with its actual spatial variability aa shownby the well data, not the smoothed and possibly biased imageread directly from the seismic data. Reproduction of the actualvariability is much more critical when modeling permeabilitythan when mapping the depth of a horizon, Lastly, no matterthe algorithm used, the resulting estimated map is somewhatin error, and one would wish to have a meaaure of such poten-tial error, a meaimre that can be used to evaluate the impactof such error on reservoir performance prediction.

Algorithms for mapping a primary variable from both pri-mary (hard) and secondary (soft) data can be classified in twobroad categories:

● interpolation algorithms which yield a unique re-sponse (interpolated map), best in some sense. These in-terpolation algorithms are usually low-pass filters in thatthey tend to smooth out local dettifs of the spatial vari-abdity of the primary variable being mapped [2], In thebest case they provide a local mezaure of uncertainty, e.g.,

833

2

●

Integrating Seismic Data in Reservoir Modeling SPE 24742

a kriging variance, which falls short of being a measureof joint spatial uncertainty (one that involves many esti-mated points simultaneously).

Stochastic hnaging (or simulation) techniques which

environment using field seismic data and actual well data, not

in a clean sterile environment of a lab. This is particularly im-portant when seismic data are to be used to map petrophysicalproperties: much too often the excellent correlation found be-

provide multiple possible realizations of the unknown sur-face or spatial distribution, yet all such realizations honorthe same original data. Typically, these stochastic al.gorithms are full-pass filters in that the simulated mapsreproduce the full spectrum of the data spatial variabil-ity, inasmuch aa that spectrum has been correctly in-ferred and modeled. Fluctuations between realizations(the stochastic images) provide a visuai and quantita-tive measure of the uncertainty about the underlying phe-nomenon,

This paper reviews algorithms in both categories, propos-ing adaptations and some innovative implementations, Thisreview is backed by comparative runs using a real data setfrom a reservoir overlying a salt dome in the Gulf of Mexico.AU runs were produced with the GSLIB software [10].

Notations:Let ZI(u) be the primary. variable of interest distributed

over a field A, with u c A being thi coordinate vector. In thefollowing case study, ZI represents the depth to the top of thereservoir at a 2D location u; integrals of S1(u) truncated bythe oil-water contact would provide the gross reservoir volume,In another application, Z1(u) could represent porosity at a 3DIocation u, and the integrals of Z1(u) wouid be associated tothe total pore volume.

.

Sparse hut accurate (hard) sample values Z1(Ua ), a =15as-lnl a~e available at well locations u~ ~ A. In addition, seis-mic data zz(u~),a = 1,..., 7ZZ,of different but related natureare available at a much larger number n2 of locations u: E A.Note that although the seismic CDP locations u: may be muchmore numerous, they may not provide enough resolution in thevertical dimension if the vertical extent of layer A is too small.Also, and typically, the seismic information z2(u~) relates toa fuzzy volume centered around location u~, much larger thanthe representative volume of the hard datum Z1(ua ). Yet itis assumed the seismic datum zz(u~) does carry informationabout the hard zl-values in the neighborhood of u:.

The goal of integration is to produce one or severalmaps for the distribution of ZI(u) over field A utilizing bothhard data {Z. (Ua), a = 1, ....nl} and soft data {Z2(U~), LY=1,t.., rzz}.

The first category of irAerpoIation algorithms aims atproviding a unique ‘best” estimate of Z1(u) at any node of thecontouring grid. The second category of stochastic slmu.-lation algorithms would provide a distribution of L images{zf~)(u), u G A], I = 1,..., L, each representing a possiblerepresentation of the underlying ‘tr.u@’ spatial phenomenon{ZI(U),U E A},

Note that for both categories the goal is to map the pri-mary variable Z1(u) making use of the secondary data zz (u:),it is not to map the secondary variable Z2(11). Hence, the stepof calibration of seismic data to control well data is essential:the greyscale map of travel time in Figure 1 is not directly a

map of the top of the structure as some geophysicists wouldlike to believe. Also, that calibration should be done in a real

tween sonic data and lab petrophysical measurements are notborne out by field data [7].

Regression algorithms

Most interpolation algorithms making use of secondary in-formation. to Map a primary variable are based on some formof regression of the unknown value ZI(u) cm the sample data,Z(ua), a = 1, .,,, N, the z-data being either of type ZI (primaryor well data) or of type 22 (secondary or seismic).

Kriging itself is but a generalized regression algorithmwhereby the unknown value is estimated by a linear combi-nation of the zl-data [11,12]:

(1)a51

The kriging surface z:(u) can be seen as a regression hy-perplane fitting a (nl + I)-dimensional calibration scattergramof values ZI(u) vs. Z1(u + ha) at distances h= = Ua - u away,

a = 1,,.., rat. In practice, this calibration is limited to inferenceof two-point statistics relating any two values ZI(u), ZI (U + h),such as the zl-variogram or covariance. The weights ~=’s deter-mining the regression (1) are then given by a system of normal(kriging) equations.

The apparently formidable cokriging is nothing more thanan extension of that regression to include data of type differentfrom ZI. For example, if n2 seismic data 22(u&) are avaiIablein addition to the nl well data Z1(u=) the cokriging estimatefor the primary variable at any unsampled location u is:

0%1 a=l

From a theoretical point of view, there is no differencebetween kriging and cokriging, Again, calibration is Iimit,cd.to inference of two-point statistics relating any two valt.esz(u), Z(U + h), where z can be of type Z1 or type 22, and the

(nl + nz) weights Ail) and .4~) are given by a system of normal(cokriging) equations. The only difference is of practical order,that of inference and consistent modeling of four covariancefunctions instead of a single one in the case of kriging

CII(h) = COV{ZI(U), Z,(U+ h)} (3)

C22(h) = COU{Z2(U), Z2(U + h)}

CIZ(h) = cou{/?fI(u), &(u + h)}

Czl(h) = COV{.ZZ(U),Z1(U + h)},

where C21(h) is usually assumed identical to C12(h).Modern desktop workstations can easily handle the in-

creased dimension of a cokriging system, (rJl + *2) instead ofnl for kriging. The hurdle to the wide spread use of cokrig-ing lies in the inference and, above all, the tedious modelingof the matrix of (cross) covariances (3). Although the task of

034

i .-,.

SPE ‘24742 W, Xu, T. Tran, R.M, %ivastava, and A,G. Jzmrnel 3

such modeling can be made much easier using interactive andintelligent graphics software, lt.Mt. algorithmic developments indata integration have consisted in shortcutting the modeling ofcross-covariances.

The external drifl model [10,13]

This model consists of assuming that the secondary datumZZ(W)reflects, up to a linear resealing, the behavior of somelocal average of the primary .zl-vidues around location U:

/+ v(u)Z1(U’)du’ N a + bz2(u) (4)

where V(u) represents some volume/area centered at u,with measure I V 1.

In terms of a random function model, the secondary vari-able 22(u) is interpreted as a linear resealing of the locallyvariable, hence non-stationary, expected value J?{ZI (u)} of theprimary variable:

E{ Zl(u)} = a + bz2(u) (5)

In words, the spatial variabdity of the secondary variablez2(u) is assumed to be related to local trends in the primaryvariable Z1(u).

Kriging with an external drift model consists of estimat-ing by regression from the collocated Z1-ZZdata the coefficientsa and b, then using these estimates a“, b* to perform krigingfrom the sole primary residual data Z1{us)-[a*+h*z2(u~)], @=1. .... m. This two-step procedure is collapsed into a single %ni-versal” kriging-type system of equations [12] with the estimatebeing a linear combination of the zl-data alone:

[~l(U)]&z~.&~f* -ml = ~va[zl(~a) - t~~] (6)a=l

{

~pi, v@G (w - u.)+ #l -t /1222(u.) = Cl(u-us),

X;i, u~ = 1

m = 1,...jnl

ZL3:, %9’2(W) = %2(U)

where m 1 = E{ZI (u) is the stationary mean.

~

● The algorithm is extremely easy to implement. As com-pared to cokriging, it does not require inference of thecovariances C2 (h) and C;z (h), the system is of dimension(rr~ + 2) instead of (n, + nz).

● Itprovides, by construction, Z1-maps that closely resem-ble the secondary z2-map, see constitutive hypothesis re-lation (5),

U

●

●

●

It provides zs-maps that closely resemble zz-maps,whether hypothesis (5) is correct or not.

It does not capture the full Z1-ZZspatial cross-correlationas does cokriging,

It requires that zz secondary data be available at all lo-cations um, a = 1~..., m of the primary data and at aunodee u being estimated, see system (6).

● Theory requires that the covariance Cl (h) used in system(6) be that of the residual 21(u) -[a i- 622(u)],not thatof Z1(u) aa used moat often, nor that of the expmimentalresiduals Zl(u) - [a”+ b*Za(u)].

Ideally, the assumption (4) should be based on the physicsof the problem. For example, a t we-way travel time 22(u) canbe reasonably associated to some local average of the reflectinghorizon depth ZI(u). However, it would be more ditlicult tojustify from basic principles a relation of type (4) between anyseismic parameter Z2(u), whether travel time or seismic ampli-tude, and a petrophysical property zi [u), say, porosity or direc-tional permeability y. In practice, there is never enough denselydrilled well data ZI(u’), u’ E V(u), to statistically check rela-tions (4) or (5).

The collocated cokriging model

One implementation problem associated with a full cokrig-ing approach to integration of seismic data, or of any denselysampIed secondazy data, is matrix instability. Indeed, the ex-treme proximity and the large auto-correlation of contiguousseismic zz-data, as opposed to the large separation distancesand poor auto-correlation between primary Z1 data taken atdifferent wells, create unstable cokriging matrices (close to sin-gularity), In addition, the secondary datum ZZ(U) collocatedwith the value Z1(u) to be estimated tends to screen the influ-ence of further away secondary data.

A arduticm to this problem consists simply of retaining ateach location u to be estimated only the collocated secondarydatum zz(u), thus makhg nz = 1 in expression (2). The es-timate z;(u) and corresponding simple cokriging system arewritten:

z;(u) - ml = ~ ~ [ (u ) - m~] + A(2)[z2(u) - m2] (7)“* ~(l) 21 ~

Q=l

{

~;~l Aj%(uls - u.)+ A(2)C21(U - u.) = Cl(u - %),a=l ~-~ m

sp:, &c,2(uB - u)+ A(2)C2(o) = C12(0)

where ml = E{ Z1(U)), mz = E{ ZZ(U)} are the two stationarymeans, Cl(h), Cz (h), CM(h) = C21 (h) are the (Cro$a) Covari-ances defined in (3).

System (7) does not carry any more the small covariancevalues associated to highly redundant secondary zz-data, butit still requires inference of the cross-covariance ClZ(h).

A further approximation consists of retaining for Cl~(h) aMarkov-type model, see Appendix to this paper.

E

i.eda

The Markov modelConsider the Markov-type screening hypothesis:

22(u) I 21(U), 21(U+ h)} = ~{zz(l.t) I Z1(U)}, Vz!(u+h)

(8)the primary datum Z1(u) screens the influence of any other:m Z1(u + h) on the secondary collocated variable i72(u).Then it can be shown, see Appendix, that the croes-

covariance C12(h) = C21(h) takes the congenial form:

c12(0)Cl(h), Vhc12(h) = ~ (9)

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4 Integrating Seismic Data in Reservoir Modeling SPE 24742

or equivalently,

P12(~) = IX2(0)P1 (h), Vh

745)c h being the 21 -correlogram, pl? (h) =with pi(h) =

*being the Z1-ZZ cross-correlogram, and pM(0) be-

i&-the ~raditiona.1 (collocated) coefficient of correlation be-tween %I(u) and Z2(u).

The Markov model (9) is particularly congenial in that itprovides the cross-covariartce model as a resealed version of theprimary covariance Cl(h), Thk model can easily be checkedby running experimental covariances and cross-covariances andcomparing their resealed plots:

C12(h) God yfim~—p c, (o)

Collocated cokrigiug under a Markov modelWith the Markov model, the collocated cokriging estimate

and system (7) are rewritten in their standardized form:

{

~:1 Aj%l(ufl - rla] + N2)/?12(o)pi(u - Ua) = pl(u - Us),a = 1,..., nl

up:, ~$?lw(o)dw -u)+ .4(2)= PM(O)

where u; = Var{Z~(u)}, u? = Var{Z2(u)} are the twostationary variances.

Remarks

If PIz(0) = O, then ~@ = O, and the collocated secondarydatum ZQ(U)is ignored.

If 012(0) = 1, the system (10) is identical to a simp!e

As in kriging with an external drift, the system (10) re-quires the secondary variable Z2(U) be sampled at allnodes where ZI is to be estimated,

The algorithm ignores the information brought by non-collocated secon~ary data beyond that of th~ coUocateddatum ZZ(U).

Another major difference between the external drift-basedestimate [6) and the collocated cokriging estimate (10) is thatthe former is not directly related to the secondary variable val-ues, whereas the latter is. The secondary data values z2(u~)

play a role only in informing about the shape of the Z1 trend,hence the name “external drift model”. Whereas, in the bona-fide cokriging expression (10) the secondary datum ZZ(U) influ-ences directly the estimated zl-value,

However, in our opinion, the major advantage of the collo-cated cokriging model is that it relies on a calibration (tuning)parameter: the correlation coefficient P12(0), and it can be in-validated by checking the Markov model (9). There is no suchtuning parameter in the external drift model, nor can it beinvaEdated a priori from the data,

All regression algorithms, including kriging, full cokriging,kriging with an external drift model or collocated cokrigingwith a Markov-type crosz-covariance model, are low-pass filtersthat tend to yield an over-smoothed. image of the actual spatialvariability of the primary attribute Z1, This smoothing may bedesirable for applications involving static volumetric calculwtions such as mapping the top of a reservoir. It may be harm-ful for applications invoIving dynamic flow simulations suchas the modeling of permeability spatial distributions: smooth-ing would, lead to an under-representation of extreme values(conditional biaa) and an erasing of p~tterns of spatial connec-tivity of such extreme values (flow barriers or flow paths if theprimary variable is permeability y), see [2,14]. The solution isto consider a full-pass mapping algorithm that reproduces thefull spectrum (i.e., covariance) of spatial variability. Stochasticsimulations are such algorithms.

kri&g system with ‘(nl + 1) primary data ZI(u=) i&dzz(u). The exactitude of kriging then yields the solution:

Jg) = O,VcN= 1,..,, nl, and A(2J = 1. Therefore, the esti-mate (10) identifies the standardized collocated secondarydatum, as expected.

&

o The algorithm is easy to implement, as easy as krigingwith an external drift model, compare system (6) and(10),

Q As opposed to the constitutive hypothesis (5) of the ex-ternal drift approach, the Markov model [9) can easily bechecked from data.

h

● unless I p12(0) I is large, the resulting collocated cokrigingZI-map may not look alike the secondary z2-map. Thismay be considered a safeguard to overconfidence in thesecondary map,

Stochastic sirnulaticm algorithms(Gaussian-based)

This section is limited to a review of stochastic simu-lation algorithms involving the use of secondary data, morespecifically seismic data complementing ‘hardn well data inthe framework of reservoir modeling. These include two broadcategories:

1.

2.

836

Gaussian-baaed simulations: the variable to be simulatedis the normal score transform of the primary variableZI (u), The Gaussian conditional distributions are estab-lished either with a full cokriging using all secondary dataor with the less demanding collocated cokrighg.

In case the Gaussian modeI is rwoven inadequate, a non-parametricl non-Gaussian, ind-cater appro~h should beconsidered, This indicator approach, although more gen-eral, more powerfui and also much more demandhrg, isnot discussed in this paper, see [2,12].

I

SPE 24742 W. Xu, T. Tran, R,M. Srivastava, at~d A.G. Journel 5

Gaussian simulation with full cokriging

Provided a rnult~variate Gaussian model is accepted, thisia the most straightforward approach. It does require though afull cokriging in the normal score transform space, with ail theshortcomings attached to such full cokriging (matrix instabilityand tedious modeling of cowmiance function matrix).

More precisely, consider the normal score transformsYj(u) = ~1(ZI (u)) and Yz(u) = 42(22(u)) of the primary andsecondary variables, 21(u) and 22(u) respectively. The covari-ances and cross-covariances of Y1(u) and Yz(u) are inferred andmodeled from the corresponding normal score datw, let thembe:

Sl(h) = ~{ YI(u)YI(u + h)} (11)

S,(h) = ~{ fi(U)fi(U+ h)]

S,.(h) = SQ;(h) = E{ YI(u]Y2(u+ h)}

Recall that E{ M(u)} == E{ fi(u)] =. 0, by definition ofthe normal score transform.

The full simple cokriging of YI(u) using both primary datath(ua), cu = 1, .... w and secondary data ~z(u~), a = 1, ..., tt2

determines fully the Gaussian conditional cumulative distribu-tion (ccdf) of Y1(u). The ccdf mean is the simple cokrigingestimate ~;(u) and the ccdf variance is the corresponding sim-ple kriging variance:

The (nl -+ n2 ) weights J$i) and Ag) are given by a simplecokriging (normal) system.

J@iJ

o Straightforward Gaussian theory.<*?iw...

l“● The ccdf of any YI(u) ii ad~conditionaJ to all secondary

data yQ(u&) available i the n~ighborhood of u.

Cons:.—

●

$

9

All the limitations of a very stringent joint multivariateGaussian model, includlng maximum entropy, and lack ofstructure of extreme values [14],

Tedious inference and modeling of the normal score co-variance matrix (11),

Instability of the cokriging matrix if the secondary dataare numerous and smoothly variable in space, as are seis-mic data,

Realizations of the Gaussian random field Z1 (u) are ob-tained using the sequential simulation partitgm aa follows[10,12]:

1. Define a random path visiting all nodes ui to be simulated,i=l,..., N.

2.

9.

4.

5.

At node Ui, determine the Gaussian ccdf of Y1(Ui) fromsimple cokriging uuing atl original normal score data (oJboth types PI. and y2) and al! previously uimulated values

yf’)(uj), j < i falling into a neighborhood of Ui.

Draw a value ~~’)(ui) from that ccdf and add it to the fileof primar~ Vi-data.

Return to step (!?) until all N nodes have been simulated,

Back-transform the realization {~~a)(ui), i = 1,.,., N}into the original Z1-space with Zf’)(u) = drl (~~’)(u)).

Repeat the entire procem with another random path to gen.crate another realization {Z\ ’)(Ui), i = 1,,.., N}.

Gaussian simulation with collocated cokrlging

This approach is similar to the previous one except thatthe full cokriging to determine the ccdf’s is replaced by a lessdemanding collocated cokriging of type (10) using a Markovmodel of type (9) for the covariimce S12(h). The mean andvariance of the Gaussian ccdf of Y1*u) are

ml

v;(u) = ~ Xl% (u. ) 1- A(24/2(u) (13)a=l

nl

C&@) = 1 _ ~ A8)SI(U - U.) - A(2)S12(0)Q=l

~

● Straightforward Gaussian theory.

● No need to model the cross-covariance S12(h).

● Stable kriging matrices.

@l&

c Dependence on stringent joint multivariate Gaussianmodel with all its limitations.

● Dependence on the Markov model of type (9) for SIQ(h).However, this model can be checked using the originalnormai score data vl (u-) and w (u&).

o Secondary data information is not retained beyondcollocated datum y2(u).

A case-study: Mapping of a sak.kme

This case-study relates to a reservoir in the Gulf of lMexico,where upthrust of a saltdome has deformed the overlying strataincluding the top of the reservoir. The data set comprises, seeFigure 1,

●

●

depths to the top of the reservoir as measured from 20wells; these are considered as hard data z?(u*), u =1,..,, nl = 20.

the two-way travel times from 3D seismic, recorded at15,753 CDP locations; these are considered as soft datazz(u~)ja = 1, ....n-z = 15,753 informing the primary vari-able ZI(u).

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6 Integrating Seismic Data in Reservoir .Modeling SPE 24742

The gross reservoir volume is that defined between the topsurface and the oil-water contact at a depth of 5130 m.

Of the 20 wells, only 7 are located on the upthrust struc-ture: lighter grey areas on Figure la corresponding to the righttail of the histogram of well data and the left tail of the his-togram of travel times. The scattergram of well data vs. collo-cated travel times show two populations, see Figure Id. The ex-cellent correlation, p = -0.96, is not representative of the mainstructure (dome area). If only the seven wells of that dome areaare retained, the correlation coefficient drops to p = -0.60.

If travel time is indeed a good indicator of the depth ofthe structure, any mapping of that structure using only thewell data would fail to reveal the details of the dome structure,see Figure 2b.

Ordinary KrigingOrdinary kriging was performed at each of tire 15,753, CDP

locations. At each location, ali 20 well data were used. Theisotropic semi-wmiogram model was inferred from the omnidi-rectional experimental semi-variogram of the 20 normal scoredata, see Figure 2a: ‘

~, (h)+ A’[.05 + .95GwMss35(I h l)] ,(14)

with Gaussa(h) = 1 - ezp(-h2/a2) being a Gaussianmodel with unit sill; parameter a and effective range am. Themultiplicative factor K plays no roIe on the resulting krigingestimates; it could be identified to the variance of the 20 welldepth data, i.e., h“ = 314m2, see the h~stogram of Figure lb,The small 5% relative nugget effect was added to avoid krigingmatrix instability iw{sociated with very continuous models suchas Gaussian with zero nugget effect. The Gaussian model waschosen because one ,expects extreme spatial continuity whenmapping the top of a sedimentary structure. The experimentalvariogram of the original well data was too noisy to be usefulrThe effective range am= 1100rn was borrowed from the lessnoisy variogram of the normal score transform of Figure 2a.

Ordinary kriging results in the smooth map of Figure 2bwhich fails to reveal the sharp boundaries of the dome structureseerr on the travel time map of Figure la. Compare the his.togram of the kriging estimates of Figure 2C to that of the welldata of Figure lb, Total gross reservoir volume is estimatedfrom that kriging to be 84.21 x 106m3.

Kriging with external driftA first alternative to integrate the seismic data is kriging

with a trend model linearly resealed from the travel time data,see relation (5).

The resulting kriging map is shown on. Figure 3a, with thecorresponding histogram of 15,753 kriging estimates shown onFigure 3b. The variogram model used is that considered forordinary kriging, see expression (14) and Figure 2a, All 20well data and, correspondingly, the 20 collocated seismic datazz(u~) were retained for each kriging, see system (6), :“

The resulting &timai&d map delineates better the domearea shown on the seismic map of Figur& la. Note also $he sig-nitlca,ntly larger standard deviation of estimates (less s,rnooth-ing effect): u = 16.4 from Figure 3b, when compared to that,.of the ordinary ~riging: estimates: u = 12.2 from Figure 2c.

The rtisulting total, gross re~rvoir volume @72,79 x’10!rn3.,, #

Collocated cokriging ~A second alternative to integrate the seismic data is cok- ~

~iging retaining only the travel time datum collocated with the :point being estimated, and using a hfarkov model for the’ cross- “covariance, see system (10).

As mentioned before, the Markov model (9) can be jchecked. Figure 4a shows the omnidirectional experimen-tal cross-varir?gram versus tlie variogram obtained from theMarkov model. The excellent match validates the Maikov hy-pothesis for this case,

The resulting cokriging map is shown on Figure 4b, withthe corresponding histogram of 15,753 estimated values shownon Figure 4c. The pri~ary variogram model used is that re- ;tained for ordinary kriging, see expression (14) and Figure 2a.The correlation coefficient PIZ(0) used for the Markov model (9)is that based on the %ven wells intersecting the dome strtw ‘.tute, i,e., gMI(0) = -0,6, As for the other krigings, all 20 welldata were retained for each cokriging fit = 20 in system (10).

The resulting estimated map delineates the dome area aswell as the external drift map of Figure 3a although with alarger smoothing effect: u = 11.73 for collocated cokriging, u =‘16,44 for kriging with an external drift.” Recall the standarddeviation a = 17.72 of the original 20 well data.

The resulting gross reservoir volume is 79.37 x 106t?a3i a - jvalue between that given by ordinary kriging 84.21,x 10sm3 andkriging with an exteqnal drift 72.79 x 108m3. Some evaluationof the potential for errtir of these estimation values is in order. ~

Conditional sirmdation with collocated coldgingGaussian simulation conditional to both normal score well

data and seismic data was implemented to generate 100 real- ~izations of the structure top. 1

The vanogram model fitted from the normal score trans-forms of the 20 well data is that of expression (14) and shotiin Figure 2a. The correlation coefficient used for the Markovmodel of cross-covariance is P12(0) = -0.6 corresponding tothe seven wells intersecting’ the dome structure. Brews thatcorrelation value -0.6 is mediocre and because the primary var.iogram model (14) is so continuous at the origin, far away ppi-rnary (well) data will carry mor~ weight than the collocated .secondary (seismic) datum. Indeed, the primary correlogram

PI(h) drops to under 0.6 only for distances I h I greater than430 m. This has for consequence to dilute the influence of theseismic data and tlieir associated smooth spatial variabWy,

Figure 5a and 5b gives the realizations with maximumand minimum gross reservoir volumes. Figure 5C gives the E-.type estimated map, i.e., the map obtained by averaging ateach of the 15,753 locations all 100 simulated values. Figure6 gives the histogram of the 100 simulated gross reservoir VOLumes: the mean is the E-type estimated gross reservou volume,84.07 x 10srn3; the spread of that histogram provides a measureof uncertainty about that volume, e.g., the 95% probability in- /terval for the gross reservoir volume V is: I

,; I~ob{V c [62.73, 102.44] X 10%n3} = 0,95

,’,,,1:

Not! tlie similarity in shapd of the &tjpe estimated ma! ~ !of Figure 6C to the travel time rnarr of. Figure la.

,,!,,

SW%“24742 W. Xu, T. Tran, R.M. Srivastava, and A.G. Journel 7

ValidationAfter this case study has been done, an additional 25 wells

were made available to us over the study area, see dots onFigure 7. The collocated cokriging (not simulation) algorithm(10) was repeated using all 45 wells, resulting in the map ofFigure 7a to be compared to that of Figures 3a and 4b.

The updated gross reservoir volume 75.59 x 106m3 is seento fall within the 95% probability interval provided by simula-tion.

Figure 8a gives the scattergram of the 25 depth estimatesprovided by kriging with external drift (using only 20 well data)us. the actual values at these locations. F@re 811gives thescattergram with, now, the 25 estimates provided by collocatedcokriging (again using only 20 well data), Collocated cokrigingis seen to perform slightly better.

Conclusions

In the category of Gaussian model-based algorithms, wherenormal score transforms of primary and secondary data are as-sumed to be jointly Gaussian-distributed, the collocated cok-riging approximation represents an interesting practical alter-native in that

1. it is fast, robust and easy to implement

2. it does not require inference of a cross-cevariance model(under an additional Markov-type hypothesis that can bechecked from data)

3. it can be applied in either an estimation or a simulationmode

4. it is firmly grounded in Gaussian theory,

A comparative study using we~l and two-way travel timeseismic data for mapping of a saltdome structure indicates thatcollocated cokriging performs well in both

● the estimation mode, as compared to ordinary kriging andkriging with an external drift, and

o the simulation mode.

Presently available code for Gaussian sequential simula-tion, such as sgsim of GSLIB [10], can be easily adapted tohandle the single additional equation required by collocatedcokriging. ‘“

NomenclatureT(h) =A,u =

p(h) =UI,U2 =

u:~(u) =Ci,j(h) =

q.} =E{ Z21Z1 = z} =

semi-variogramkriging weightscorrelogram function E [-1, +1]standard deviationkriging (estimation) variancecovariance between any tworandom variables i and jexpected valueconditional expection of 22

given the value Zi = z

a,b =ae$b” =

h =TIJi =

n, nl ==

:.#i{U) =

y;(u) =

Z:(U) =

Zi(U) =

z:(u) =

coefficientsestimates of a and bseparation vectorstationary means of va:iable inumber of datalocation vectorvolumenormal scored data at location ukriging estimate in normal spacerandom variable at location udata at location uan estimate of value 21(u)

References

1.

2.

3.

4.

5.

6.

7,

8.

9.

100

11.

12.

13.

14.

Haldorsen,H.H. and Damsleth,E. (1990) “Stochastic mod-elingn in Journal of Pet, Technology, p. 404-412, April1990.

Journel,A.G. and Alabert,F, (1988) “Focusing on spatialconnectivity of extreme-valued attributes: Stochastic in-dicator models of reservoir heterogeneities”, SPE paperno. 18324.

Doyen, Ph. (198$) ‘Porosity from seismic data: A geosta-tistical approachn, in Geophysics, 53{10), p.1263-1275.

Journel,A.G. (1991) “Geostatistics for reservoir charac-terization”, SPE paper no. 20750.

Hohn,M. (1988) Geostatistics and Petroleum Geohw,Van Nostrand, 264p.

Doyen, Ph. and Guidish,T. (1990) “Seismic discrimina-tion of lithology: a Bayesian approach”, in Proc. of G-statistics Symposium, Calgary AB, May 1990.

Han, D., Nur,A., and Morgan,D. (1986) “Effects of poros-ity and clay content on wave velocities in sandstone” inGeoph~sics, 51, p.2093-2107.

Ballin,P., Journel,A.G., and Aziz,K. (1991) “Predictionof uncertainty in reservoir performance forecaatingm, inProc. of CIM/AOSTRA Tech. Conf,, Banff AB, April1991.

Aziz,K. and Settari,A. (1979) Petroleum Reservoir Sim-ulation, Applied Science Publ.} N.Y., 476p.

Deutsch,C.V. and Journel,A.G. (1992) CSLH3: Geosta-tistical Sojtware Librar~ and User% Guide, Oxford Univ.Press, in press.

Goldberger,A, (1962) “Best linear unbiased prediction inthe generalized linear regression model”, JASA 57, p.369-375.

Journel,A.G. (1989) Fundamentals of Geostatistics inFive Lessons, 8, Short Course in Geology, American Geo-physical Union publ., Washington D.C., 40P.

Marechal,A, (1984) “Kriging seismic data in presence offaults”, in G.Verly et al. cd., Geostatistics jor NaturalResources Characterization, 1, p.271-294, RAdel publ.,Dordrecht, Holland.

Journel,A.G. ‘and Alabert,F. (1989) “Non-Gauasian dataexpansion in the earth sciences” Term Noua, 1, p.123-134

839

8 Integrating Seismic Data in Reservoir Modeling. !WE 24742

Appendix: The M~rlcov rnorlel for cr~ss-

CovarianCe

Consider two stationary random functions Z;(u) andZ2(U). Without 10ss of generality, these random functions canbe assumed to have zero means and unit variances; their co-variances are then:

PI(w = E{ Z,(U)ZI(U + h)}@?(h) = E{Z2(U)ZZ(U + h)} (15)

mi(h) = P21(h)= E{Z2(U)ZI(U + h)}

Next consider the two conditions:

E{ Z2(U) I 21[22) ==21} = p12(o)zl (16]

i.e., the regression of 22 on 21 is linear (note that (16) isverified if 21 (u) and Zz(u) are jointly Gaussian-distributed),and

E{ Z2(U) / 21 (U) = z, ZI(U + h) = 2’)

(17)= E{ Z2(U) I 21(u) = Z}, Vh, %’

i.e., the collocated datum 21(u) = z screens the influenceof any other zl-data on 22(u), then

Theorem: Conditions (16) and (17) entail the followingexpression for the cross-covariance:

w(h) = m(o)m(h), vh (18)

Proof Let fh(z, z’) be the bivariate pdf of the tworandom variables 21 (u) and Z1(u + h), the cross-covarianceplz(h) is written:

P12(h) = E{ Z2(U)Z1 (U + h)}

p~2(h) = ff~{z2(u)z1@ +h) ~ZI(U) ==z,&(u + h) = Z’}

jh(z, Z’)dzdz’

= ffz’.I5{Z2(u) I ZI(U) = z, Z,(u- I-h)= Z’}fh(z,Z’)dZ~Z’

= ~~z’E{Zz(U) I z,(u) = z}~~(z,z’)dzsfz’, using (17)

= P12(~] f f zz’fh(z, Z’)dZtfZ’, using (16)

= p12(0)pI (h), by sfe~inition oj PI(h).

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a - using two-w?.y travel time (rns). The dots indicate well locations

b - histogram of depths measured at the wells

c - histogram of travel time data

d - scattergram of travel time and depth at 20 well locations

1 840

SPE 24742 W. Xu, ‘I’. Tran, R.M. Srivastava, and A.G. Journ.el 9

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a - 6reYsc~@ maP of kriging estimates with external drift Ib - histogram of kriging estimates with external d! ift

Gross Reservoir Volume = 7937x 10’ m3

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841 ‘1

10 Integrating Seismic Data in Reservoir Modeling

GrossReservoir Volume = 109.77x10’ m’

GrossRestwoh-Vokme = 6132 x 10’ m’

GrossReservoir Volume= 2407x 106ms

Figure 5: Simulation with collocated cokriging

.a - realization with maximum gross reservoir volume

b . realization with minimum gross reservoir volume

c - E-type estimation map

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a - kriging with an extermd drift using 20 original WCIIS

b - collocated cokriging using 20 original wells