geostatistics for reservoir characterization

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SPE 20750Geostatistics for Reservoir CharacterizationA.GSPE

. Journel,Member

Stanford

SPE-of~m

Copyright 19S0, Socie!y of Petroleum Engineers Inc.This paper was prepared for presentation atthe 65th Annual Technical Conference and Exhibition ofthe Society of Petroleum Engineers held in NewOrleans, LA, Septemb er23-2S, 1S90.Thle paper was selected for presentat ion by an SPE Program Committee fol lowing review of information contained in an abatracl eubmitted by the aulhor(e). Conteme of the paper,aa preeenled, have not been reviewed by the Society of Petroleum Engineera and are subject to correct ion by the author(s), The malerial, as presented, does not necessari ly rel leclany poaltion of the Society of Petroleum Engineers, its officers, or members. Papere presented at SPE meetings are subject to publication review by Editorial Committees O!the Societyof Petroleum Engineere. Permieelon 10copy is restricted to an abetract of notmore than S&3words. Illuelrations maynot be copied. Theabstract should contain conspicuous acknowledgmentOfwhere and by whom the paper is presented. Write Publications Manager, SPE, PO. Box 83383S, Richardson, TX 75083.383S. Telex, 730989 SPEDAL.

Abstract calling for better numerical models of a reservoir rock and fluidGe osta tistic s a nd , m ore sp ec ific ally, sto cha stic m od - propcrties, geostatistics is receiving renewed attention,Until recently, geostatistics was often associated with onlyelirig of reservoir heterogeneities are being increasingly

considered by reservoir ana ly sts and eag irte ers for their one of its important contributions and was used as a synonympotential in generating more accurate reservoir models for kriging, a multiple regression technique6-s that has beentogether with usable measures of spatial uncertainty. most commonly used as a gridding procedure, By reducingGeostatistics provides a probabilistic framework and a geostatistics to another canned gridding softwsreg, early userstoolbox for data analysis with early integration of infer. failed to realize its full potential as a set of spatial data analysismation. The uncertainty about the epatial distribution tools, as a probabilistic language to be shared by geologist slO,of critical reservoir parameters is modeled and trans- geophysicists and reservoir engineers23, and as a vehicle forferred all the way to a risk-conscious reservoir manage- integrating various sources of uncertain informat ion2, Over-ment, The stochastic imaging (modeling) algorithms al.low the generation ofmultiple, equiprobable, unsoothed

sold as canned software with w regtird for prior education,reservoir models yet all honoring the data available, geostatistics generated overcxpectations and the ensuing disrtp-pointment. Fortunately, rwccnt experience has brought forward

In rod uctiol~l more reasonable expectations and has broadened the scope ofNumerical models of reservoirs often fails to capture the applications,Spatial Data Analysisheterogeneities that are critically imPortantl-4 for reservoir .

performance. With production always being the primary func- Geostatistics begins with an emphasis on describing andtioq of w well, the available data cue typically biased toward modelling the spatial variability of reservoir properties rmd thethe more productive regions in a resevvoir and are regrettably spatial correlation between related properties such as porositysparse, The interpolation and griclding algorithms commonly and scisrnic velocity, These models can then be used in theused by industry further cxaccrbatc the problem since they construction of numerical mod& for a variety of purposesare low-pass filters that tend to smcwth out the little spatird interpolations for a property W11OSCverage is critically impor.variability that the sparse data reveal. While core plugs and tant, stochastic simulations for a property whose extremes arcwell logs are not the only sourcee of information, other data, critically important,such as geophysical information, arc often difficult to integrate Whether onc needs to transfer information from one rescr-since they have different levels of reliability4$ and are reprcsen- voir to rtnother, or between different units within the sametativc of very different voh.rtnes of rock, History matching on reservoir, or whether one needs to transfer information fromhistorical production information and well test data does not one discipline to another, a quantitative vchic]c is necessary,guarantee reliable forcca.sts of a reservoirs future performance. Gcostutistical models of spatial variability and dependence pro.

All of M is not NCWS;these problems have come to the vide a quantitative summary of geological observatiotd, andforefrrmt as industry focuses on enhancing rccovcry from known can therefore serve as a such a vehicle. Gcostatistical nlod-rwmrvoirs. With performance prediction for EOR processes CISmake it eaaicr to compare data from different sedimentary

I]tcferellces and [Ihtstratioas at ctd of paper basins, from different formations, and from different horizons,and are therefore valuable aids to any attempt at buildh~g aw


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tGEOSTATISTICS FOR RESERVOIRCHARACTERIZATION en-

classification system. They aiso enable geologists , for example,o put their vaiuable information in a format that can be used~y reservoir engineers,

Geologica.i interpretation of the direction of maximum con-tinuity can be corroborated through the use of directional ccm-elogrtuns and the plot of correlat ion ranges, see figure 1. In theibsence of sufficient data in a particular reservoir, correlogramsmd other statistical characteristics can be borrowed from other;elevant data sets such as originating from more mature fields13n similar geological environments, outcrop studies14 or geolog-cai cross-sections and maps. By capturing the criticai het-erogeneities in a quantitative form, directional correlogramsmable the use of this important information that often goesgnored since it commonly has only a quaikative expression,

Geostatisticians commonly use variograms to describe spa-;ial continuity. While a correlogram, such as that shown in fig-lre 1, shows the decay in the correlation between sample vaiuesw a function of increasing separation distance, the variogram6;hows the increase in dissimilarity between sample vaiues ver-ms increasing separation distance.

Though variograms and correlograms are commonly used;O study spatiai continuity of a rvticular variable, the same;OOIScan be applied to the study J the cross-continuity of dif-krent variables at different locations. One could, for example,:ompare porosity at a particular location to travel time at aocation nearby. Once modelled, this spatiai cross-correlation:an be used in a muitivariate regression procedure known as:okrigingll for building a porosit y map not only from the avail-ible porosity data but aiso from the more abundant seismicinformation,

Non-iinear transformations of the data vaiues can also bevery useful in spatiai data anaiysis. The spatiai continuity ofVery skewed data, such as permeability vaiues, is usually betterunderstood through anaiysis of the logarithms of these data,Another example is the non-iinear binary indicator transform,which can be defi;ed as 1 if the permeability sample vaiueis greater than a given threshold and O o therwise, The vari-~grarn of such an indicator transformation provides a measureof spatial connectivity of the high permeability values that areresponsible for flow pat hs and the low permeability y values thatserve as barriers to flow16,

In addition to variograms and correlograms, a geostatisti-cians toolkit contains a wide variety of plots and graphs fordetecting, corroborating and modelling patterns of spatial vari-ability and interdependence.Generalized Regression (Kriging - CokrigingO-6J1 )

Any unsampled vaiue, a porosity for example, can be es-timated by generalized regression from surrounding measure-ments of the same value once the statistical relationship be-tween the unknown being estimated and the available samplevalues hus been defined, This is exactly what the correlograxnprovides: a prior model of the statistical similarity betweendata vaiues, As was pointed out in the previous section, thisgeneralized regression can also include nearby measurementsof .somcdifferent variable-seismic travel time, for example, ora facies code, When using such secondary information, oneaiso necd6 a prior model of the cross-corrclogram, which pro.vides information on the statistical similarity between different

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variables at d~fferent locations. These generalized regressgonthms are collectively known as kriging and cokrigThey generdlze tradtionai regression as appiied in weanalysis in two senses:(i) The data used (the independent variables in tradhi

statistical jargon) need not be independent one froother, This allows redundancy to be taken into accan important factor when using several related wesimultaneously.

(ii) The sample correlation is modelled before being ato the regression anaiysis. This ailows for filtering caspects (frequencies) of the sample correlation. Fample, white noise and more generaily the high frequcomponents of the data can be filtered out, leavingterpolated map that reflects only the large scaie trenthese datas. Conversely, a particular trend knownindirect information can be built into the map ina way that it honors as closely as possible the avadata17.

Integrating Data of Different TypesMultivcuiate regression, or coknging as a geostatisti

would tail it, is usuaily not a convenient framework fintegration of too widely different types of data such asitative geological information, which is usuaily oniy indicin nature, and direct laboratory measurements.

At a particular location where one does not yet hporosity measurement, a consideration of the iithofaciesmation might provide a reasonable intervai within whicunknown value should faills, If we are certain that we amparticular type of sandstone, for example, we might knowthe porosity must be somewhere in the intervai from 130%. If, in addition, we also have enough core plug mements within that type of sandstone to build a histogramcould go further than simply stating the previous cm-istinterval. We could use that histogram to provide a probadistribution that might, for example, tell us that the unkporosity is more likely to be on the low end of our 109$ trange than on the high end,

The indicator framework of geostatistics and the sofing algorithm ailow an updating of such prior distributionnearby data that may be either soft or hard, To cowith our previous example, we could locaily update theous probability y distribution obtained from lithofacies conations with more specific local information, This couldinformation, such as the fact that the location in questionthe upper haif of a fining-upwards sequence, or hard inftion, such as the fact that a full core measurement takewell only 50 feet away had a porosity of 15,670,

The result of this updating is a posterior probabilitytribution that provides the probability for the unsampledto belong within any given class of values, say, betweenand 1570 porosity. From such a dist ribut ion, any optestimate for the unmrnplcd vaiue can be derived once anmaiity criterion has been specifiedie, The optimaiity crpreferred by most statisticians is the least square errorrion; for this definition of optimaiity, the mean of the po


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. -1SPE 20750 A. G. Journel 3probability distribution provides the best estimate, One could Just as one can draw a series of outcomes from a univariate:hose instead to use the absolute er:or, rather th~ the squ~ed probability distributional, one can also draw a series of out-:rror, es the yardstick by which to measure the goodness of an comes from a multivariate probability distribution47~ 13. With:stimate; for this optimtdity criterion, the median of the poste- a (K . JV)-variate probability distribution representing the un-rior probability distribution turns out to be the best estimate, certainties in K rock and fluid properties at N locations, eachRather than chosing some convenient, but arbitrary, defi- drawing from such a multivariate probability distribution rep-nition of optimality, one could also analyze the intended use of resents a stochastic image cjf the reservoir. If IV is very large,my estimate and come up with a specific {loss function lg120 several million nodes on a, 3-dimensioned grid, each outcomethat describes the penalty associated to any particular level of provides a high resolution, image of the reservoir rock and fluiderror, Such a loss function would likely be asymmetric since properties, Such stochastic images honor all of the availablethe impact of an underestimation oi a particular magnitude information, both hard and soft, at all locations and all of theis rarely the same as the i:npact of an overestimation of the structural information entered through the correlograms andsame magnitude, Thk project-specific 10SSfunction can also cross-correlograms, yet they are different one from each other,be minimized, yielding a best estimate that aims specifically see figure 2.at minimizing the impact of error on each specific project. The differences between these stochastic images provide a

Indicator geostatistics encodes all amenable information directly usable visualization of the uncertainty about the reser-into a series of indicator (binary) dat a718. Hard data result voir rock and fluid properties. Where all the different outcomesin a coding that consists entirely of Os and 1 s; soft data re- agree, there is little or no uncertainty; where they differ mostsuit in a coding that consists of intermediate values between there is maximum uncertainty. For example, a streak of highD and 1. These indicator data are then interpreted as prior permeability values that persists on 90% of the images mayprobability for any unsampled variable to take upon any par- be considered as reliable, whereas a streak that is present ontitular value. No matter what their origin, be it well logs, only 50% of the images is less likely to exist. If the uncertainseismic data or geological interpretation, all indicator data are streaks happen to be in areas that are highly consequential topooled together. Experimental indicator variograms are then reservoir performance, the need for additional data has beencalculated to reveal the patterns of spatial continuity in these established together with the locations at which these data areindicator data, Once the pattern of spatial continuity is well seeded. Stochastic imaging might, for example, reveal thatunderstood, it can be modelled and used in a multiple indica- there is a lot of uncertainty about whether or not a particulartor regression to yield the posterior probability distribution of EOR injector/producer pattern contains a high permeabilityany unsampled value, streak near the top of the producing unit, !,f the nttture of the

Traditionally, data of different types have been processed EOR process makes gravity override a concern, the exerciseseparately, leading to several different modelsa geological of stochastic imaging will have identified the neeu for moremodel, a geophysical model, a production model, . ..which accurate information on the upper portion of the pattern inare difficult, if not impossible, to merge. Indicator geostat is- question.tics takes a quite different route by first merging all of the Stochastic images represent alternative, equiprobable nu-relevant information through a common coding of that infor- merical models of the reservoir rock acd fluid properties and,mation and, then, producing reservoir models consistent with as such, can be used as input to flow simulators for sensitivitythat information. 2i122.Since a complex flow simulator could be very ex -nalyslsStochastic Imaging of Reservoir Heterogeneities pensive to run, it is often necessary to consider only a limited

The thrust of modern geostatistics is not least-squares spa- number of reservoir models representative of the range of un-tial regression but the building of probability distributions that certainty: some unfavorable ones, some intermediate ones andsome favorable ones. This selection can be done by runningcharacterize the present uncertainty about a reservoir rock andfluid properties, These probability distributions should ac- a stripped-down simulator that is faat and efficient yet alsocaptures the relevant feature of the flow problem under con-count for all relevant information through models of the spa-tial dependence between each piece of information and the un- sideration, A simple particle tracking algorithm, for example,known variable of interest. As more information is collected, might be useful in identifying the main flow paths23, therebythe uncertainty about the unknown variable of interest lessens allowing a trained engineer quickly to assess whether a par-ticular reservoir model will produce favora$le or unfavorableand the spread of the posterior probability distribution d > results.creases.

The unknown can be a single particular unsarnple~ value, ~eference~say porosity, at a single location, or it can be the unsampledvalues of a particular variable at many locations (the nodes of 1, Weber, K, J ,: How heterogcneit y affects oil recovery, ina regular grid, for example). It can even be the unsampled Rest rvoir Characterizntio~ (1986), Lake and Caroll cd,,values of all relevant variables-porosity, permeability, satura- Academic Press, 487-584,tions, pressures,... -at many locations. With a single variableat a single location, one hss a l-variate problem; with a sin- 2. Hewett, T. A,: Fractal distr ibutions of reservoir hetero-gle variable at JV locations, the problem becomes an .V-variate geneity and their influence on fluid transport, SPE pa-problmn; with K interdependent variables at N locations, the pcr 15386 presented at the 1986 SPE Annual Conference,problem becoms a (K oN)-variate problem. New Orleans (Oct. 5-8, 1986),


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1-4 GEOSTATISTICS FOR RESERVOIRCHARACTERIZATION SPE 203. Tang, R.W,, Behrens, R. A,and Emanuel, A.S; Reser-

voir studies using geostatistics to forecast performance,SPE paper 18432 presented at the 1989 SPE Symposiumon Reservoir Simulation, Houston (Feb. 6-8, 1989).

4, Haldorsen, H.H and Damsleth, E.: Stochastic model-ing, JPT (Apr. 1990),

14, Go@n, D. J., Chandler, M. A., Lake, L.W. and KocurekG.: Permeability transects in eolian sands arid their usin generating random permeability fields: SPE pape19586 presented at the 1989 SPE Annual Conference, SaAntonio (Oct. 8-11, 1989).

15. Journel, A.G. and Alabert, F. G.: Non-Gaussian dat5. Krause, F.F. and Collins, H.N.: Pembina Cardium, re- expansion in the Earth Sciences, Terra Nova, 1(2), 12134 (1989).

covery efficiency study: A geological and engineering syn-thesis, vol. I and II, Petroleum Recovery Institute, Cal- 16. Sandjivy, L,: The factorial kriging analysis of regionalgary, 1984. ized data: Its application to geochemical prospecting,

6. Hohn, M.E .: Geostatistics and Petroleum Geology, (1988), in Geostatistics for Natural. .Resources Charact~

Van Nostrand Reinhold, 264 p. Part 1, 571-579, (1984), Verly et al. cd., Reidel publ.

7. Journel, A.G,: Fundamentals of Ge 17, Henning, O. and Halvorsen, K.B.: The Bayesian bridgostatistics in Five Les- between simple and universal kriging, Math Geology~, Short Course in Geology 8, (1989), AGU publ. 40 21(7), 767-786, (1989).P.8. Isaaks, E.H. and Srivastava, R.M .: An Introduction to App- 18. Kostov, C, and Journel, A. G.: Coding and extrap~

lied Geostatistics, (1989), Oxford Press, 561 p. lating expert information for reservoir description,~ Character izatio~ (1986), Lake and Caroll c9. Bluepack-3D Users Manual, Centre de Geostatistique, I Academic Press, 249-264.

Fontainebleau, l?ra~;e, 209p.0. Fogg, G.E. . - ucia, J. F.: Reservoir modelling in the

Dune field, Crane County, TX: Geologic/geostatisticalcharacterization and analysis of heterogeneity, recoveryefficiency and in-fill drilling: a Report of the Bureau ofEconomic Geology, U. of Texas at Austin, 64 p. (1989).

1. Doyen, P.: Porosity from seismic data A geostatis~icalapproach, Geophysics, 53(10), 1263-1275 (1988).

12, Journel, A.G. and Alabert, F.: New method for reser-\oir mapping, JPT (Feb. 1990).

l?,. Journel, A.G. and Gomez-Hernandez, J. J.: Stochasticimaging of the Wilmington clastic seq~ence, SPE paper19857 presented at the 1989 SPE Annual Conference, SanAntonio (Oct. 8-11, 1989).

19. Berger, J. O.: Statistical Decision Theory and BayesianAnalysis (1980), Springer Verlag, 617p,

20. Srivastava, R, M,: Minimum variance or maximum proitability, CIM Bulletin (May 1987), 80(901), 63-68.

21. Genrich, J,)7mand ~ommer, Fos,: L~Novelapproach to sesitivity analysis, JPT (Sept, 1989), 930-932 and 98985.

22. Matthews, J. L., Emanuel, A,S, and Edwards, K,A .: Etal methods improve Mitsue miscible predictions, JP(NOV. 1989) 1136-1142.

23. Giordano, R., Salter, S, and Mohanty, K.: The effeof permeability y variations on flow in porous media, Spaper 14365, 1985.


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,., , bSPE20750

Correlogramsfww)1 1

model L

Rose of correlation rangesxperimental value

o r 6a2 - al(direction X2) ( direction Ml)Ihl

Figure 1: Typical spatial dependence analysisThe variable is porosity as sampled (after calibration) from sonic logs. A directional correlo-

gram p(l~l, a) is a plot versus 1~1of the linear correlation coefficient between all pairs of porositydata O(Z), c#(g+k) separated by approximately the same distance l~i in approximately the samedirection a. The correlogram p(l~, a), measures the loss of correlation aa the separation distance1~1between two sample values increases, The distance at which the correlation vanishes is calledthe range aa, The polar plot, aa vs. al, of these range values provides a rneamre of anisotropy.with the largest range corresponding to the direction of best spatial continuity (correlation),

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X$nr xWR .!

maIL2bK9nr xW* z

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Figure 2: Four equiprobakle realizations of the sand-shale sequence over a verticalsection, (The vertical exaggeration is 10:1)

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geostatistics for reservoir characterization

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