simulation of permeability field conditioned to well test...

QSPE 49289

Simulation of Permeability Field Conditioned to Well Test DataSanjay Srinivasan, Andre G. Journel, SPE, Stanford University

mht 1~, ** ~ ~l~m Enginwrs, Inc.

~s paper was preparedfcfpressntstion at fhe 1SS6 SPE Annual Technical Ccmference andExhiMon held in New Orleans, Louisiana, 27-30 ~ptember 1998,

This papsr w selected for pressntatin by an SPE Program committee fo[l~ r~w ~infmmsfion contained in an absbact submitted by the author(s). Wteti d tie Pm, as

pn~, ~ve n@ *O r~”~ by tie *lety Of Petroleum Engineers and are subje b~ by the author(s). The matil, as pressmtsd, dw not necessarily reflect snypition & the Socii & Petroleum Engineers, ti offiirs, or members. Pspers presented atSPE m~ngs are subject to publication revisw by Ediirial timmti of ths ~Ieiy of~lwm Engineem. Elecbonic reproduction, ti~bution, or ~rage of any part of this psperfor cm-nmercial pu~ Woui the w?itten assnt cdthe %ely of Peboleurn Engineers ispmhibM, Permission to reproduce in priti is restricted b sn abstract of M more tian ~words; illustra~ns may nof k mpied. The atict must mntain mnspicuous~edgment of Mere and by Mom the paper ws presented. Writs Librarian, SPE, PO.Scu ~. Rmhsrdson, TX 7==, U.S.A., fax 01-972-952-9435,

Abstract

Well testing provides critical information about the effectivepermeability value around the well being tested. That infor-mation must be integrated with higher resolution, smaller scale,well log data. The difference of scale is handled by tilgingwith “block” averages, while the non-linear averaging of per-meability values is addressed by working on specific non-linearpower transform of the original permeability data. A case studyshows the simulated permeability fields to honor, in expectedvalue, the well test-derived effective permeability values in ad-dition to honoring the smaller scale well log data and statistics.Ignoring the well test information would result in permeabilityfields that are imprecise around the well being tested and whichdisplay too large uncertainty. The necessity of data integration,even when data we of widely different scales, is demonstrated.The paper also discusses cases when the well test-derived effec-tive permeability appears inconsistent with the well-log derivedsmaller scale permeability data. The paper points out the lim-itation of the linear power-averaging formulation for capturingthe dynamics of fluid flow.

Introduction

Well test data provide information about the average permeabil-ity around the well bore at a scale similar to that of the blocksused in flow simulation. That information, however, does nothave the resolution provided by other sources of data such aswell logs and core analysis. A single well test cannot providedetailed information about permeability anisotropy, also its area

Soeie~=f PetroleumEngineers

of reconnaissance is limited by well bore and boundary effects.Well test data must thus be integrated with other sources ofdata to provide the permeability field description required forflow simulation. One pioneering work in this area was that ofDeut3ch1 who suggested using simulated annealing to generatestochastic realizations of the permeability field conditioned toboth well log data and well test data. Many other authors havelater expanded on Deutsch’s original contribution2’3.

A prior calibration exercise allows approximating the welltest derived effective permeability ~tz as a power average ofthe permeability values k(u) at locations u within an annularvolume V centered at the well location:

(1)

where IV I is the volume of the drainage area V and w isthe power average parameter4, usually found in the interval[-1,+1].

In the presence of anisotropic 3D permeability fields, withvertical permeability differing from horizontal permeability, thecalibration procedure for the parameter w could be altered toaccount for the anisotropy3.

Stochastic modeling of the permeability field consists ofgenerating alternative equiprobable realizations {k(t)(u), u EA},l= l,..., L within the study area A (usually much largerthan V); t is the index of the realization number, L is the totalnumber of realizations. The simulated values k(t)(u) are de-fined on the smaller support volume of the well log data and areconditioned to the data available in the sense that:

1.

2.

- they should identify the well log derived data k(u~) attheir locations u~:

(2),(t)(ua) = k(ua), a = 1,... ,n, Vt

- the average permeability in the vicinity of the wellshould approximately match the well test-derived effec-tive permeability value:

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2 Sanjay Srinivasan, Andre G. Journel SPE 49289

In addition, the variogram T(t) (h) calculated from each realiza-tion {kfc) (u), u E A} should reproduce, in expected value, themodel V(h) inferred from well data and/or ancillary structuralinformation such as outcrop data:

or more precisely:

where h = u – u’ is the vector separating any two locationsu, u’ within the study area A.

The simulated annealing algorithm consists of iterativelyperturbing initial fields {k(to) (u), u E A} already verifyingthe exactitude condition (2) until conditions (3) and (4) are alsomet. The technique works well for reasonably small areas .45,but, typical of annealing algorithms, is slow and requires deli-cate tuning of the annealing schedule, One would wish to havea direct (mathematical) way to impose the well test constraint(3), for example through the well established sequential Gaus-sian simulation algorithm.

Sequential simulation revisited

Any sequential simulation algorithm calls for deriving at eachlocation u to be simulated, the cumulative conditional proba-bility distribution (ccdfl of the attribute value defined as:

Prob{K(u) < zldata} = F(u; zldata) (5)

where K(u) designates the random variable K at location umodeling the uncertainty about the unsampled permeabilityvalue k(u). z is any threshold value, here in permeability unit.The notation ]data indicates that the distribution is conditionedto all the data available in the neighborhood of u.

Actually, all that is needed are the mean and variance ofthat ccdf As long as that mean and variance identify the krig-ing estimate k* (u) and corresponding kriging variance a~ (u),it can be shown that both conditions (2) and (4) are honored,no matter the type of distribution retained for the ccdf (5)7’8.More precisely, that ccdf need not be Gaussian as in sequentialGaussian simulation.

There remains to introduce the well test constraint (3). Tothis purpose, a remarkable, albeit little known property of blockkriging will be revived9$10.

The simple kriging (SK) used to determine the mean k* (u)and variance o~ (u) of the ccdf(5) typically retains as data onlythe well log ‘hard’ data k(ue ). The idea is to also consider in

the kriging, the block average data value ~Y,, as derived fromwell test interpretation and the power u calibration process, seeexpression (1).

If that block average ~L, is a linear average (U =“-1), thenblock kriging with the additional datum value %J ensures thatthe estimated values k*(u) within block 1’ average out into ~~,that is: .,.

-1 k* (u)du = ~T., exactly1;[ “

(6)

However, most often the calibration of well test data results ina power average which is not linear; w # 1. To capitalize onthe previous average value identification one should work noton the permeability value k(u) itself but on its w-power trans-form kti(u). Then kriging would provide estimates [k@(u)]*

that would identify the power transformed datum value ~,:

~ ~ [k”(u)]”du = ~v, exactly

Consequently:

Thus, instead of simulating permeability values kt(u), onewould simulate values [kP(u)]W by drawing from a ccd~with:

● mean equal to the kriging estimate [k@(u)]* using the ad-

ditional block data ~j

● variance equal to the corresponding kriging variance

The resulting simulated values [kP(u)]’”, ? = 1,..., L wouldhonor, in expected value, the well test constraint in the sensethat:

The inverse power (~) backtransform then provides the sim-

ulated values kt(u). This direct sequential simulation processhonors exactly the constraint (2) per exactitude of kriging andhonors, in expected value, the constraint (3). Per simulation,the input variogram model for [k(u)]ti is reproduced, thus itis the rank order variogram of the permeability k(u) which isreproduced.

Finally, the sample (or any target) histogram of k(u) can bereproduced by a general backtransform similar to the back nor-mal score transform included in the sequential Gaussian simula-tion algorithm, This general backtransform is performed usingthe program trans 11.

Note that the direct sequential simulation8’12 algorithm pro-

posed never calls for any Gaussian model, instead it replaces theGaussian (normal score) transform by an w-power transform tocapitalize on the kriging exactitude property (6).

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SPE49289 permeability simulation conditioned to well test 3

Case Study

To demonstrate and check the algorithm proposed to integratewell test data, an exhaustively known permeability data setwould bc convenient. One would (sparsely) sample such ref-erence data for hard well data of the type (2), forward simulatea WCIItest to derive the well test datum of type (3), [hen try toreconstruct the reference permeability image using that sparseinformation. Short of (ever) having such reference data set onecould synthesize one, using an established stochastic simulationalgorithm but at the risk of a recursive demonstration. Insteadwe have retained a “real” CT scan data13 set resealed to identifya “real” permeability distribution.

The reference data set Computer Tomography (CT) mea-surement consists of generating a beam of high energy pho-tons, transmitting the photons through the object and recordingthem on an array of detectors placed diametrically opposite thesource. The resultant photon intensity at the detectors is usedto reconstruct the CT characteristics of the object. The scannedcrosssection is discretized into voxels and a CT number is at-tributed to each voxel. The CT number is proportional to thedensity of material within the scan plane.

For the purpose of this research, a core taken from a cleanblock of Stanford sandstone was scanned and the resulting CTvalues were resealed to identify the histogram of actual well-Iog derived permeability values from a deep water turbiditicreservoir14. The resultant permeability greyscale map and cor-responding histogram is given in Figure 1. This reference 2-D data set comprises 50 x 50 gridded values. It is assumedrepresentative of a reservoir layer of lateral dimensions 2500ftx 2500ft, the grid size is thus 50ft x 50 ft. The permeability val-ues define the horizontal transmissivities of the layer assumedof constant 30ft \lertical thickness.

The histogram of Figure 1 if put in arithmetic scale woulddisplay a positively skewed shape (although not lognormal) typ-ical of core scale permeability distributions. The permeabilityvalues range from 12md to 2201md. with a coefficient of vari-ation of O.82.

Figure 2 gives four directional semivariograms calculatedfrom the reference data set : no clear anisotropy is apparent.All four variograms were standardized by the reference vari-ance (321)2.

Sample data set Ten “hard” well data k(u=), a = 1, . . . . rz =10 were retained at locations indicated on Figure 3. The sam-ple mean is 620md significantly larger than the reference mean390md.

The most central well is assumed to have been tested. Thewell test has been forward simulated using the reference per-meability data set. The reservoir is assumed initially saturated

with oil. The oil production rate is set constant at 500 STB/day.The oil viscosity /~0 is assumed to bc 1.0 cps and the formationvolume factor BO is 1.0 rb/STB. The pressure at the bottom ofthe well is monitored and consequently wellbore storage effectsare considered non-existent. The test is run for a duration of 10days.

The semi-log plot of bottom hole pressure versus time isshown in Figure 4. The straight line at early times is followedby a sharper decline in pressure at later times indicative ofboundary effects. The slope of the straight line is IO! = 4.35psi/cycle, The effective permeability xl is thus 623 rnd. Theproducer is located in a region of high permeabilities, see Fig-ure 1, consequently this relatively high effective permeabilityvalue appears reasonable. The simulated permeability real-izations kt(u), u c .4 are to be constrained to this well test-derived datum.

Calculation of the drainage area (horizontal section ofblock 1‘) and corresponding power average parameter u waspcrformed~15. The best fit parameter was found to be ~ = 0.9

and the radii of the annular shtiped drainage area are Tnin =26.85 ft and rmfl. = 210.03 ft. The producer is located in anisolated region of high permeabilities, the permeability at theproducer location is 1068 md, see location map in Figure 3.Since the extent of that high permeability region is limited, theduration of the infinite acting flow regime is limited and thedrainage volume informed by the well test is restricted. Alsosince the well is situated in a region of high permeabilities thevalue u close to 1 (1inear average) is reasonable,

Permeability simulation The direct sequential simulation al-gorithm previously outlined was applied to generate L = 20realizations of the permeability field o\Jer the 50 x 50 studyarea. The two first realizations are displayed in Figure 5.

For the kriging process needed to calculate the estimatedvalues [kd(u)]”, a model with relative nugget effect 0.01 anda spherical structure with isotropic range 430 ft and sill equalto the variance of the 10 sample values kd (ua ) was retained.This model is based on the reference variograms of Figure 2,in real practice the variogram model will have to be inferredfrom whatever structural information is available such as out-crop data, geological interpretation or seismic data since 10data are insufficient to even attempt at inferring an horizontalvariogram. The parameter value w = 0.9 was utilized. Thekriging did not yield any negative estimate. Because krigingis a non-convex estimation algorithm, negative kriging weightsapplied to a large datum value kti (uO ) could generate negativeestimated values [kti(u)]” which then could not be backtrans-formed with the inverse power ~. The correctiorrl” whereby allkriging weights are shifted by the modulus of the most negativeweight (if any) could then have been used.

Lognormal distributions (ccd~s) are used to draw the sim-ulated realizations [kt(u)]ti, ensuring that all such valuesare positive hence could be backtransformed into realizations


kt(u). A final backtransform can identify any permeability tar-get histogram. For this application we consider for target dis-tribution a Iognormal distribution with parameters identified tothe 10 well data sample mean and variance.

Results All 20 realizations generated are checked to reproduceexactly the 7 “hard” well data, refer to constraint (2).

Reproduction, in average, of the constraint (8) with ~,g =(623)09 is excellent as shown by the histogram of the 20 poweraverages of the simulated permeability values k(t) (u) withinthe annular shaped drainge area IT, see Figure 6.

Reproduction in average, of the well test-derived datum

TV = 623 md by the 20 effective values ~~, / = 1,...,20,each obtained by well testing the corresponding simulated per-

meability field is shown in Figure 7. The 20 values ~~ fluctu-ate around a mean value 571md close to the target value 623md. The reproduction of the target value is poorer on Figure 7than in Figure 6: this can be explained by uncertainty associ-ated to the 20 well test interpretations and to boundary effectsdue to the proximity of the well being tested to the Northeast-ern boundary of the field. The algorithm only ensures good re-production of power averages as seen on Figure 6; similarly, asimulated annealing approach would ensure only reproductionof power averages.

The cloud of 20 omni-directional semivariograms calcu-lated from the 20 simulated permeability fields is given in Fig-ure 8: they appear reasonably well clustered around the targetvariogram model with nugget effect 0.01 and range 430 feet;refer to constraint (4). All simulated variograms were standard-ized by their respective variances.

In practice, good reproduction of both the target histogramand variogram model is mixed blessing because these modelsare extremely uncertain due to sparsity of well data (here only10). However, this good reproduction indicates that relevantsensitivity analyses could be conducted using the proposed sim-ulation algorithm.

Figure 5 gives the results specific to the first two realiza-tions drawn, {k(t) (u), u G A}, ~ = 1,2. All realizations ap-pear different thus providing an assessment of the underlyinguncertainty. The permeability in the region around the wellis high as would be expected due to conditioning by the rela-tively high effective permeability datum 1068md. The effec-tive permeability computed by simulating the well test resultsin xv = 542 md for the second realization, a value significantlylower than the target 623 md. This lower effective permeabil-ity can be attributed to the cluster of high permeability pixelsstretching up to the NE boundary as observed in Figure 5 (sec-ond realization). This causes the producer to see the effects ofthe boundary much earlier and causes the computed effectivepermeability to be lower. Figure 5 also shows the variograrnreproduction of the two realizations.

The importance of conditioning the realizations to well testeffective permeability can be assessed by comparing the uncer-tainty in xv as shown in Figure 7 against a similar distributioncomputed over a suite of sequential Gaussian simulated real-

11 These realizations are condi-izations using program sgsim .tioned to only the hard data at the ten wells, Figure 9 shows theresultant distribution of effective permeability value computedon 20 sgsim realizations. The sgsim realizations yield a widerrange of effective permeabilities with a mean 925 md as com-pared to the target 623 md. As opposed to the previous case andFigure 7, the high datum value 1068 md at the producing wellis not balanced by the well test lower value 623 md.

Simulated Annealing For comparison purposes, integrationof the well test information using simulated annealing andprogram sasimi 1 was also attempted, A specific objectivefunction component accounting for the deviation of the aver-

age value Xfi) at iteration number i from the target value XVwas included:

Ow,eff = [1:) – ZV]2

The computation of ~t and ~~) utilize the same parameters J,rmaz and rmin as calculated before.

The histogram of the final average values ~~ over 20 real-izations is shown in Figure 10. Since the deviation from thetarget value 623 md is explicitly minimized in the sasim pro-

cedure, the resultant deviation of ~~. is virtually nil. However.the ultimate objective of the permeability simulation is to repro-duce the target well test derived effective permeability. For thisreason, the forward well test was simulated on the 20 sasimrealizations. The uncertainty in the effective permeability over20 realizations is shown in Figure 11. The distribution is biasedtowards low effective permeability values. The mean effectivepermeability is 433 md. compared to the target 623 md. Com-parison of Figure 7 and Figure 11 indicates that for this casestudy, the simulated annealing approach does not perform bet-ter than our proposed direct approach.

In order to understand the reason for the bias in computedeffective permeabilities, the pixel image of two specific sasimrealizations are shown in Figures 12a and b. Figure 12a cor-responds to a low (340 md) effective permeability realization,whereas Figure 12b corresponds to the high (592 md) realiza-tion, It can be seen in Figure 12a that the high permeabilityin the vicinity of the well stretches to the boundary. Well testpressure response on this realization is thus influenced by theboundary, specifically, the boundary hastens the pressure de-cline, which in turn results in a low effective permeability in-terpretation. Permeability realizations generated using the sim-ulated annealing technique are characterized by minimum en-tropy. ~is implies that the corresponding simulated permeabil-ity have larger clusters of connected high permeability pixels.Because of that lower entropy, the occurence of connected highpermeability streak extending to the boundary is more frequent

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SPE 49289 permeability simulation conditioned to well test 5

in simulated annealing permeability realizations. This in turnresults in the bias towards lower effective permeability values.

There is another aspect of simulated annealing which alsoinfluences the forward simulation of well tests. Since the blockaverage constraint is explicitly included in the objective func-tion, the resultant permeability images exactly honor the tar-get IV value. In this particular case study, since the targetkv = 623 md whereas the hard permeability datum at the wellis 1068 md, the resultant permeability images have an abrupttransition from 1068 md at the well to values close to 623 mdin the immediate vicinity of the well, in order to honor the blockaverage constraint. This results in a pseudo-skin effect whichin turn results in lower effective permeability values calculatedon the sasim realizations.

A possible reason for the large fluctuations in the simulatedwell test responses observed in both the dssim and sasim re-alizations could be the limited volume of influence of the welltest, here r max = 210 ft. The volume of influence is deter-mined so as to minimize the difference between the target keff

and the power average TV. The pair of values [u, V] is obtainedjointly by minimizing the aforementioned difference. The com-putation of V in addition utilizes information on the duration ofthe radial flow period. The small rm.z found here implies thatonly a small region around the well bore is constrained by thewell test datum. Regions away from the well are unconstrained(if there are no nearby hard data) and consequently high perme-ability streaks extending to the boundary may be generated insome realizations.

Power Averaging

The results discussed previously point to the limitations of thepresent power average formulation in capturing the dynamicsof fluid flow in the reservoir. This section explores some alter-nate formulations for the power average that attempt to accountfor the physics of fluid flow,

Traditional power averaging assumes that every locationwithin the drainage volume V contributes equally to the av-erage. The consequence of this assumption is that permeabilityvalues simulated at locations away from the test well, yet withinits drainage area, influence equally the block average. Since inthis case study, the test well is located off center and the targetxv is high (623 red), equal weighting may result in realizationswhich connect to the boundary. This in turn yields lower esti-mates of effective permeability. Hence the idea to express theblock average permeability as a weighted linear combination ofthe w transformed permeabilities k(u):

The weights p(u) could vary from one location to the otherand account for the distance from location u to the test well UO.Various techniques for computing the weights p(u) have been

735

proposed 217. The application of two such weighting schemesfor estimating the weights is discussed below.

Oliver’s Weight Function The effective permeability is inter-preted from the pressure decline at the test well. Since thepower average permeability Iv is expected to be a proxy for theeffective permeability interpreted from transient pressure data,it is appropriate that the weights p(u) be related to the solutionof the radial diffusivity equation that models the flow of fluidsaround the test well. Oliver derived the solution to the radialdiffusivity for the case of spatially variable permeability, re-sulting in an analytical expression for the slope of the semi-logpressure plot, which is at the basis for transient well testing:

where tD is a dimensionless time, PD is the dimensionlesspressure, K(rD, tD) is the weighting function, ~(~D, tD) is thepermeability perturbation (deviation from the mean) and rD isa dimensionless distance. The equation expresses the effec-tive permeability, given by the slope of the semilog pressureplot, as the weighted sum of permeability within annular re-gions whose radii propagates with time. The behavior of theweight function at different times is shown in Figure 13. Oliverpoints out that at a given tD, only the annular region boundedby (0.12fi, 2.34~ contributes to the power average. Theanalytical formulation for the weight function K(rD, tD) andan overview of Oliver’s derivation is included in the Appendix.

Oliver’s derivation of the weight function assumes:

● small isotropic permeability perturbations about its mean

. ideal radial flow towards the wellbore. This is necessaryfor the analytical solution to hold.

In many practical situations, the above assumptions will beviolated. In order to extend Oliver’s formulation to practicalreservoir applications, a calibration procedure15 is necessaryin order to scale the dimensionless time by a factor A specificto the reservoir under study. The underlying argument behindthis correction is that, while radial assumptions may not holdin reservoirs with strong permeability anisotropies, the analyt-ical development of Oliver may be extended to such reservoirsby suitably scaling the annular regions over which the weightfunctions are defined. The parameter A obtained by calibration,therefore, injects the specifics of the reservoir under study intoOliver’s formulation derived for ideal flow.

For the case study, the block average was calculated usingOliver’s weight function calibrated for this specific case study.The procedure for calculating A and w was as follows:

● The approximate duration of the infinite acting flow pe-riod is estimated from the well test plot. Assume a num-ber of (A, u) pairs, For each A calculate tD. Then foreach tD the smaller and larger radii of the annular regionare calculated.


● The weight function is calculated at each location u usingthe analytical formulation of Oliver.

● The weights are applied on the w transformed permeabil-ity values to compute the power average ~v.

. The pair of values (.4 and U) that minimizes the deviationof~~ from the target lc,ff = 623 md is retained.

For this case study, the optimal values were found to beu = (),4, rmin = 28 ft. and rm.z = 541 ft. The correspond-ing spatial variation in weights as well as the weight profile inthe radial direction is shown in Figure 14. The figure depictsan essential feature of Oliver’s kernel function. The maximumweight is attributed not to the pixel immediately adjacent to thewell but to the pixel located at 0.92fi distance away fromthe well. For the calculation above, the duration of the infiniteacting flow period was assumed to be 0.2 hrs., estimated fromthe semi-log plot of pressure vs. time.

The parameters calculated above were used in the dssimsimulation. The simulation procedure is the same except thatthe block-point covariances are now calculated utilizing theweights. Thus and for example:

.7(U, v)”= I p(u)c(u, U’)du’

u’Gt’(II)

The results of the dssim simulation are shown in Fig-ure 15. As seen from the lower right histogram, incorporat-ing the weight function does improve the accuracy of the k~f f

distribution. The mean k~ff over 20 realizations is 599 md.very close to the target value of 623 md. This result is to becompared against the mean value of 571 md. obtained by con-sidering equal weights, see Figure 7. The spread of the k~f

idistribution remains substantial. The realizations correspon -ing to the two extreme keff values is also shown in Figure 15.The realization corresponding to the low value appears to havepermeability discontinuity in the vicinity of the well. This re-sults in a pseudo-skin effect, in turn leading to lower interpretedk,f f .

Permeability Sensitivity Coefficients The effective perme-ability interpreted from well test is a summary of the pressuresignature at an observation well due to production at a constantrate. The block average permeability xv is a fast proxy to theeffective permeability datum, a proxy that can be convenientlyutilized within a sequential simulation or simulating annealingmode to condition permeability realizations to the large scalek,f f. The goal is to derive a block average that correctly cap-tures the flow behavior such that forward simulation on the per-meability simulation correctly retrieve the target k~f f value.

Given the dynamic nature of effective permeability, a suit-able measure of the relative importance of the permeability at aparticular location could be its contribution to the pressure re-sponse at the well. This contribution, termed as the permeabil-ity sensitivity coefficient, should account for both the proximity

of the pixel to the well and its connectivity to the well. Math-ematically the permeability sensitivity coefficient is defined asthe partial derivative ~A~U, ~, i.e the change in pressure at the

well p due to an incremental change in permeability Ak(ui) ata location uz 18.

The derivation of the permeability sensitivity coefficientsutilizing the discretized form of the radial diffusivity equationsis discussed in the Appendix. Since the pressure response atthe well is impacted most by the permeability variation in theregion adjacent to the well, the sensitivity coefficient is calcu-lated over a restricted area, thus saving on computation timeand cost.

The sensitivity coefficient approach is numerical and henceit can handle any realization, with any pattern of hetero-geneities. There is no restriction on isotropy or small varianceof the permeability fie}d. However, it is important to understandthe similarities between this approach and Oliver’s weightingfunctions. Oliver’s analytical formulation is based on the radialdiffusivity equation. The sensitivity coefficients are calculatedusing the discretized form of the same equation. Both Oliver’sweighting function and the sensitivity coefficient are based onthe relative contribution to the well pressure decline due to thepermeability at an arbitrary location. The numerical approachhas the additional advantage that forward well tests on perme-ability realizations are performed using similar discretized flowsimulators.

The weight distribution based on the sensitivity coefficientsis shown in Figure 16. The weights have been standardizedsuch that they add up to 1, Note that the weights distribution isnot strictly radial as when using Oliver’s kernel. This is due tothe fact that unlike in Oliver’s approach, the sensitivity coeffi-cients are calculated based on the actual permeability variationsin the reference image. The optimal u which minimizes the de-viation of the weighted average from the target 623 md is 0.2.This optimal w is significantly different from that calculatedusing Oliver’s weighting function (w = 0.4). This is becausein the sensitivity coefficient approach a larger area around thewell is assigned substantial weight. The lower ti value implieslesser contribution of high permeability locations to the poweraverage, The corresponding results of the dssim simulation areshown in Figure 17. The results indicate that the distributionof effective permeability obtained by forward flow simulationof 20 permeability realizations is accurate. The block averageis reproduced in an expected value sense. The uncertainty ofthe k$~f distribution remains wide. The two extreme realiza-tions are also shown in Figure 17. Again the low effective per-meability realization appears more discontinuous near the wellresulting in a pseudo skin effect.

Permeability simulation was also performed using the simu-lated annealing procedure (program sasim) modified to acc(ountfor weighted power average. The weights computed using thesensitivity coefficient approach were utilized. The results areshown in Figure 18. The resulting distribution of effective per-meabilities is improved in accuracy (compared to earlier resultsusing equal weights, see Figure I I). The block average datum

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is honored exactly using the sasim procedure. The low and higheffective permeability realizations are shown Figure 18. As ex-pected, sasim realizations show the least entropy and since theyhonor the block datum exactly, the simulated permeability fieldjumps instantaneously from the 1068 md value at the well tothe 623 md block average value in the vicinity of the well. Thisartifact discontinuity results in a pseudo-skin effect.

A comparison of the various k,f f distributions presented inthis section reveals the following:

. The distribution corresponding to Oliver’s weightingfunction is the least precise of the three. The coefficientof variation of this distribution is 0.27 compared to 0.22for the dssim realizations using sensitivity coefficientsand 0.20 for the sasim realizations using sensitivity co-efficients. The mean k,f f of realizations generated usingOliver’s kerne[ is 599 md compared to the target 623 md.

. The sasim realizations using sensitivity coefficients havethe least coefficient of variation. However, the distribu-tion is biased with a mean k,f f = 514 compared to target623 md. The reduced coefficient of variation is due to thelow entropy character of sasim realizations.

. The dssim realizations using sensitivity coefficients arethe most accurate, mean k,f f = 649 md compared to thetarget 623 md.

The improvement in accuracy of the effective permeabilitydistribution using the weighted average, indicates that the mod-ifications go part of the way in improving the estimated perme-ability maps. However serious deficiencies in well test interpre-tation and the power average formulation for capturing the welltest information persist, Indeed all previous weighting systemsutilize an averaging framework where permeability pixels aretaken one at a time, independent of their connectivity/similaritywith values at nearby pixels. Instead, flow response is depen-dent on multi-point connectivity. Improved power averagingaccounting for multi point connectivity of permeability may benecessary. Incorporating non-linear and multi-point power av-erages into geostatistical simulations would require significanttheoretical development perhaps in the form of extended nor-mal equations and multi- point connectivity functionslg.

Implementation notes

. In order to ensure fair comparison of the methods andtechniques presented in this paper, all calibration wasdone on the exhaustive reference data. This includes thecomputation of o and the volume of influence I” as wellas the permeability sensitivity coefficients. In practice,such calibration would have to bc performed on uncondi-tional simulations (or on simulations conditioned only tohard data).

●

●

●

A highly negative power (U + – 1) would correspond toharmonic average of quasi-annu!ar rings of different per-meability values around the well being tested. Harmonicaverage tends to weight more low permeability values.Thus if a negative power u (w j –1) is coupled with arelatively high well test value k},, the algorithm proposedmay yield unreasonably high simulated permeability val-ues kt~l (u) within the well test volume IT. These highvalues are needed to meet (he high well test constraint ifu is negative. Abnormal permeability values obtainedunder such conditions is actually a nice feature of thesimulation algorithm: it points to the inconsistency be-tween the u and ku- specified.

Suppose the hard data k(u~) at the well test location ismuch higher than the well test-derived effective perme-ability, say 2000 md versus 200 md. The simulated re-alization would display a large drop in permeability be-tween the well test location and the surrounding pixels.Subsequent well test performed on a simulation with suchcontrasting permeabilities would reveal a high pressuredrop followed by a region with more gradual pressurechange and finally a region under boundary influence.The initial high pressure drop is akin to a skin effect andshould be ignored in the derivation of the effective per-meability. However, it must be noted that the skin effectmay not be clearly discernible in the pressure responsewhen the permeability contrast is not severe, as in thiscase study. In such cases the slope of the semilog pres-sure plot may be erroneously estimated, leading to im-precise keff values.

Correcting the kriging weights to ensure positivity ofthe estimated value-[k~(u)] “-entails departure from strictkriging using block averages and hence from the strictexactitude property (7). Practice will determine if suchdeparture is severe enough to justify further correction..

Conclusions

A little known property of block kriging indicates that krig-ing estimate reproduce exactly any linear average value suchas a “block” datum value. A well test-derived effective per-meability can he interpreted as such linear average of smallscale permeability values taken to a certain power w, usuallyw c (–0.1, +1). Kriging on the w-power transformed perme-ability values, followed by an inverse ~ power transform, thusallows generating estimated permeability fields which identifythe target power average, hence whose effective values over thewell test drainage area approximate the well test effective per-meability value, In a stochastic simulation mode, the alternativesimulated permeability values will honor that well test data inexpected value, that is in average over many realizations. Acase study based on a CT scan data set of a real sandstone coreillustrates the proposed algorithm. Similar stochastic model-ing ignoring the well test data results in imprecise permeability

737


fields around the well being tested. also the realizations gen-erated display much larger uncertainty. This demonstrates theimportance of integrating data, even if they are of widely dif-ferent scales (well log vs. well test), in any reservoir modelingendeavor.

Power averages of single permeability values (ignoring theirspatial connectivity) are but proxies to the actual effective per-meabilities, thus reproduction of well test values is presentlylimited, sometimes severely, by this simplistic averaging for-mulation.

Nomenclature

u =u =h =

‘Y =e=k =

1=v=k.ff =K=

power average parameterlocation coordinate vectorlag separation vectorvariogram functionrealization indexpermeability

block average permeabilityVolume for permeability averagingwell test effective permeabilitypermeability weight kernel

References

1.

2.

3.

4.

5.

6.

Deutsch, C. V., “Annealing techniques applied to reser-voir modeling and the integration of geological and engi-neering (well test) data”, Ph.D. thesis, Stanford Univer-sity, 1992.

Campozano, F.P., Lake, Larry W. and Sepehnoori, K., “Reservoir modelling constrained to multiple well test per-meabilities”, SPE 36569, Society of Petroleum EngineersATCE, Denver, Colorado, 1996.

Hu, L.Y., Blanc, G., Noetinger, B., “Estimation of litho-facies proportions using well and well test data’’,SPE36571, Society of Petroleum Engineers ATCE, Denver,Colorado, 1996.

Journel, A. G., Deutsch, C.V. and Desbarats, A., “Poweraveraging for block effective permeability”, 56*h Califor-nia Regional Meeting, Society of Petroleum Engineers,p.329-334, April 1986.

Deutsch, C.V. and Journel, A. G., “The Application ofSimulated Annealing to Stochastic Reservoir Modeling”,SPE Advanced Technology Series, Vol. 2, No. 2, April1994.

Goovaerts, P., Geostatistics for natural resources evalua-tion, Oxford University Press, New York, 1997.

7.

8.

9.

10.

11.

12.

13.

14,

15.

Journel, A.G., “Modeling uncertainty: Some conceptualthoughts”, R. Dimitrakopoulos, editor, Geostatistics forthe Next Century, p.30-43, Kluwer, Dordrecht, Holland,1993.

Bourgault, G., “Using non-Gaussian distributions in geo-statistical simulations”, ,Wathematical Geology, Vol. 29,No. 3, April 1997.

Journel, A.G. and Huijbregts, Mining Geostatistics, Aca-demic Press, New York, 1978.

Journel, A. G., “ Conditioning geostatistical operationsto non-linear volume averages, Part I: Theory and PartII: Applications”, .submitted to Mathematical Geology,1998.

Deutsch, C.V. and Journel, A.G., GSLIB: GeostatisticalSoftware LibraV and User’s Guide, Second Edition, Ox-ford University Press, New York, 1997.

Xu, W. and Journel, A.G., “DSSIM: A general sequen-tial simulation algorithm”, Stanford Center for ReservoirForecasting - Report 7,1994.

Srinivasan. S., Deutsch, C.V. and Bertin, H., ‘6Collectionand analysis of porosity data from Stanford sandstonefor uncertainty analysis”, Stanford Center for ReservoirForecasting Report -10, 1997.

Scholle, PA. and Spearing, D., Sandstone DepositionalEnvironments, The American Association of PetroleumGeologists, Tulsa, Oklahoma, 1992.

Wen, X-H, Cullick, A.S. and Deutsch, C,V., ” Constraintson the spatial distribution of permeability due to a singlewell pressure transient test”, Stanford Center for Reser-voir Forecasting - Report 10,1997.

16. Journel, A.G. and Rae, S.E., “ Deriving conditional dis-tributions from ordinary kriging”, Geostatistics Wol/on-gong ’96, Baafi, E.Y. and Schofield, N.A. editors, Vol-ume 1, Kluwer Academic Publishers, 1997.

17. Oliver, D.S. “The averaging process in permeability esti-mation from well test data”, SPE Formation Evaluation,p. 319-324, September 1990.

18. Chu, L., Reynolds, A.C. and Oliver, D.S.,” Computationof sensitivity coefficients for conditioning the permeabil-ity field to well-test pressure data”, In Situ, 19(2), p. 179-223, 1995.

19. Journel, A.G. and Alabert, G. A., “New method for reser-voir mapping”, Journal of Petroleum Technology, Febru-ary 1990, p. 212-218.

20. Aziz, K. and Settari, A., Petroleum Reservoir Simulation,Elsevier Applied Sciences, Publishers, London, 1986.

738


Reference Image9GI-U-IM--- 2500.

1000.0

l..

2000.

SW0Im,

*+,

100.0

0.

H indicateslocation of producing well

~.~ Reference DataNumber of Data 2500

mean 3S0std. dev. 327

coef. of var 0.82maximum 2201

upper ~:~~& :~~

g 0200:Iwer quartile 192

minimum 12

~k O.lm-

O,m

1 10 100 1am 1woo

Permeability

Figure 1: Reference permeability greyscale map and histogram

,,20 Reference variograma

/r

/ WI 35E. - East /,.,

10Q/..= _4. . .

.? -. /--------. -> -X,ti

.?p. N45E

0.80

~ 0.60 “ ~

,“0.40

0.20

~

/

0.00 izdo, qdo. 6do, Gdo, Iobo,

Figure 2: Reference directional standardized semivariograms(45° azimuth increments starting from North)

Location of Well Data

,lb.

I2obQ 2:

1000.

100

...

10.

),

,,00 Histogram of Well Data

i /080

0.60

0.40

0.20 1 /

Number of Data 10mean IS20

std. dev. 637coef, of var 102

maximum 2201

“p~r$’%!% n!lower quartile 174

mln[mum 23

/0.00 ‘

lb “To Iw 1Obo

Figure 3: Ten well locations and corresponding permeabilitycumulative histogram. ~e central well indicated by a crosshas been tested yielding kl. = 623 md

‘050-

750t

Figure 4: Well test pressure decline (reference data)

739


000+(

2b. 4b. 6&I.

,mo120 .

! If

..

iw

.

080.

~, ‘y 0,0 ●

.0,40:.

0201000

Omi’0 21m 4b &

D,stawe

Figure 5: Two simulated realizations of the permeability fieldand their variograms (dots) plotted against the variogram model(isotropic). The two well test effective permeabilities are: ~} =542 red., ~v = 650 md.vs. ~VtanQet = 623 md.

,Histogrsm of Block Avg. Permeabilities4

i

Number of Data 20mean 595.

std.dev. 39.coef. of var. 0.07

maximum 675,upper quamle 619.

median 600.loweri:y:fl~ ;Jg,

Block Average

Figure 6: Histogram of the 20 simulated power averages (w =0.9) over the drainage volume V. The target well test referencevalue 623 md is indicated by the dot ●

Histogram of DSSIM Effective Permeabilities0.300J . Number of Data 20

mean 571.std. dev. 124.

coef. of var 0.22m.”im, ,fi Rfi7

3*L,@l.

‘O’i- - Mn’mum’o’

k.eft

Figure 7: Histogram of effective permeability values derivedby well test performed on the 20 simulated permeability fields.The target well test reference value 623 md is indicated by thedot ●

Y

1,20

1.00

0.60

060

0.40

I

Figure 8: Cloud of standardized omnidirectional semivari-ograms calculated from the 20 simulated permeability fields.The target model is given by the thick continuous curve

740


Histogram of SGSIM Effective Permeabilities

02

0.1

~~o,l

2L

0.0

0.0

k_eff

Figure 9: Histogram of effective permeability values calculatedfrom 20 sgsim realizations not accounting for the well test data(623md indicated by the dot ●) : note the bias

,,00 Histogram of Block Avg.

0.80

a 0.60g

z~ 0.40

0.20

II

PermeebilitiesNumber of Data 20

meen 623.1std. dev. 0.6

coet, of var 0.0max!mum 625

““er% ”&?% 823lower quatille 623

mlnlmum 622

0,00 i II340. 5~o. ● 7io.

Block Average

Figure 10: Histogram of 20 simulated power averages(U = 0.9) using a simulated annealing algorithm.The target well test value 623 md. is indicated by the dot ●

025033.

0209.,2195

j~30.15: 85g

:010

0.05

000

K.eff

Figure 11: Histogram of effective permeability values derivedby well test performed on the 20 simulated (annealing) perme-ability fields. The target well test reference value 623 md. isindicated by the dot.

Figure 12: Two simulated realizations (annealing) of the per-meability field

741


,>,CO t.b?cta ,.C-?C.XM0.4.

~

1::0.2

0,1.

0,0

0,0 0.5 10 15 2.0 2,5 30

1.09(1~)

Figure 13: Oliver’s weight function evaluated at different times.The figure shows the spatial distribution of weights as time pro-gresses.

Weights computed using Oliver’s Kernel15%,0

0.00 East m,

Weights profile radially away from the well

1 ,m.1

1.b.2

1

1 W-3

i C9-4

1 W.5

1 OR.6

- 1 oe.7

$

I

.

i Oe-8

1.-.9 .

1.C9.10

i.oz.11.

1.w-t2

. ...

..

l,oe.13

1.W.14 , Imo Woo 4A.O 2m.o 00

Radial dishnce froinwell

Histogram of B/ock Avg. K Histogram of Effective K

Figure 15: Results of the dssim simulation using weights basedon Oliver’s kernel. The realization to the left corresponds to thelow k.ff = 352 md. The realization to the right corresponds tothe high k=f f = 949 md. The distributions of block averagesand effective permeability values derived by forward simulatingwell tests on 20 realizations are shown. The target well testreference value 623 md. is indicated by the dot ●

Figure 14: Weight distribution calculated using the referenceexhaustive data and Oliver’s weight kernel. Note that the max-imum weight is calculated at 0.92& distance away from thewell

742


,=, Weights tiom sens/t/v/ty coeff/c/ente

H[stogrem of DSSIM Block Avg. Histogrem of DSSIM Eff. K

I Weighte pmflle (EW-dn.)

1

I Weighte profile (NS-dn.)

3

L$ ,M.1

~.-.

i

..

.

* 1.M.3 ..

..

1.ti.4 ..

? *52DJ,0 2W.O 4W.O .0 0.0

1 b.1

i

1,0+2

?

L

..

.3 1,0..3 .

..

.1,0.-4 .

..

.t .0..5

2W,0 W.o W.o m,o an

Figure 16: Distribution of weights obtained using the perme-ability sensitivity coefficients. The radial profile in the EWand NS directions are shown.

MUM ofDa!n20 0.2%

mean S28std. dav. 78

cmf ofvar 012 ~mmm”. 7@6

m $~q;ll ggInveigullll, 574

mpm”m 514 $ 0.1601[

Bw Av.fap EflelIve K

Figure 17: Results of dssim simulation using weights calcu-lated based on permeability sensitivity coefficients. The real-ization to the left corresponds to the low keff = 393 md whilethe realization to the right corresponds to the high keff = 922md. The distributions of block averages and forward simulatedeffective permeabilities calculated on 20 realizations are alsoshown. The target reference effective permeability value, 623is indicated by the dot ●

743


Hlstograrnof r3/ock Avg. K ffistogrsm of Effective K

0,200

0,150

i

Nutie, 0( Datamean

std. dev..cel ofvar

wT;Jy~;

lower qu.flllenun(m.m

20514104020678:9:

44Q244

Bkxk Aveca.a. K Ken

Figure 18: Results ofsasim simulation using weights calcu-Iatedbased onpermeability sensitivity coefficients. The real-ization to the left corresponds to the low k~ff = 244 md whilethe realization to the right corresponds to the high keff = 678md. The distribution of block averages and forward simulatedeffective permeabilities calculated on 20 realizations are alsoshown. The target reference effective permeability value, 623is indicated by the dot ●

744


Appendix can be incorporated into Oliver’s solution by applying suitablescaling factors:

Oliver’s Weight Kernel 17 The general solution to the radialdiffusivity equation corresponding to a continuously varying tD=~.

0.000264ket

permeability field is written as: ~pctr~

where:

pD=pDO+L/“

r

/[

~(TD,~D) 1 –1

27r ~ kD(rD,6) 1d6drD—r kt. : effective permeability from well test

@ : porositywhere: P : viscosity

kDCf : total compressibility

: an arbitrary dimensionless permeability rwdistribution function

: wellbore radius

rD : the dimensionless radial distancefrom the well bore

tD :the dimensionless time

PD : dimensionless pressure

PD. : the pressure response of an equivalenthomogeneous reservoir

K(r~,t~) : weighting function accounting forvariable permeability

O indicates that the equation is expressed in cylindrical co-ordinates. The spatially varying permeability is assumed to bethe sum of a constant “average” permeability and a small per-turbation component. Thus:

kD(rD,o) = [1 – Cf(rD,6)]-1

where ~(rD, O) is a first order perturbation function. Forsuch perturbation expression, Oliver derives the slope of thesemilog pressure plot to be:

For tD >100, the weight function K(TD, tD) is then:

where lf71/,,1/, (z) is the Whittaker’s function, a confluenthypergeometric function. The behavior of the weight func-tion at different times is shown in Figure 13. Oliver pointsout that at a given tD, only the annular region bounded by(0.12fi, 2.34~ contributes to the power average.

Oliver’s derivation of the weight function assumes small,isotropic permeability perturbations about the mean, Since thederivation is based on the radial diffusivity equation, ideal ra-dial tlow towards the wellbore is implicitly assumed. Since inpractical reservoir characterization these assumptions are vio-lated, a calibration15 procedure is proposed to scale the dimen-sionless time tL) by a factor A to account fOr departures frO.rnOliver’s assumptions. The underlying argument being that fac-tors such as permeability anisotropy cause the annular region tostretch preferentially in the direction of the anisotropy and this

Permeability Sensitivity Coefficients The discretized form ofthe radial diffusivity equation is written as:

[T]{p} ‘+’ = {B}

where the time derivative of pressure and the Laplace operatorhave been approximated by the corresponding forwarddifferences20. The matrix T is termed the transmissibility ma-trix accounting for the spatial and time discretizations as wellas the boundary conditions. The matrix B is the right hand sidematrix accounting for the time discretizations and the boundaryconditions. The flow simulation consists of inverting the matrixT and finding the pressure solution at time t + 1 from:

{p} ’+1 = [T]-l{B)

The sensitivity coefficients express the influence of the per-meability at any location within the reservoir on the weIl pres-sure response. They are obtained by differentiating the equationabove:

a{p}t+l8Ak(ui)

= [T]-l aB _ [~]-1dAk(uz) f3fiuz){p}f+1

for all locations Uz within the reservoir.

Thus knowing the transmissibility matrix T, the right handmatrix B and the inverse [T]–l, the sensitivity coefficient canbe calculated easily. Inverting the transmissibility matrix or theJacobian is an essential step called for in flow simulation. Uti-lizing that inverse matrix to calculate the sensitivity coefficientis consequently inexpensive. Furthermore, because the pres-sure response at the well is most influenced by the permeabilitywithin a limited region around the vJell, the flow equation andconsequently the Jacobian can be calculated over a restricted re-gion, This further reduces the cost of calculating the sensitivitycoefficients.

745

simulation of permeability field conditioned to well test...

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