applications of simulated annealing on actual but atypical permeability...

SPE 35345Societv of PetrolevrnEndneers

Applications of Simulated Annealing on Actual but Atypical Permeability Data

G. B. Savioli, SPE, and E. D. Falcigno, SPE, and M. S. Bidner, SPE, University of Buenos Airesand L. W. Lake, SPE, University of Texas at Austin.

Copyright 19S6 Scclety of Petroleum Eng, neers

This paper was prepared for presentatmn at the Intl Petroleum Conference& Exhlbrt!on of Mex$coheld m Wlahermosa, Mexco, 5.7 March, 1696

This paper was selected for presentation by the SPE Program Commtttee follwwmg review ofmformat)on contained I. an abstract submtted by the author(s) Contents of the paper aspresented, have not been rwewed by the Somety of Petroleum Engmews and are subject tocorfectmn by the authors(s) The material, as presented, does not necessw!y reflect any Powtlonof the Souety of Petroleum Engineers of Ik members Papers presented at SPE meebngs aresubject to publ!catlon rev!ew by EdNor#al Commtttee of the Scaety of Petroleum Eng{neersPermlswon to copy IS resfncted to an abstract of not more tfwn W words Illustrations may notbe copted The abstract should contain conspicuous acknowledgement G4where and by whomthe paper was presented VJrIte Llbrarmn, Spe P O Box 83333836, Richardson, TX ?5083-3838U S A faX 01-214-952-9435

AbstractThe usc of geostatistics is becoming recognized as a standardmeans of representing reservoir heterogeneity. Geostatistics hasenjoyed an extensive use and a fairly well developed theoreticalbase. This is a little less true of simulated annealing (SA), theform of geostatistics tested here, but it is also a maturetechnology.

Yet there remains a need to exercise these procedures underactual conditions of nonuniformly sampled data, non-Gaussiandistributions and truncated data sets, Providing insights intohow to deal with these nonidealitics is the objective of this work.

We find that SA estimates are improved when the originaldata sets are power-transformed. However, SA estimates tend todeviate from the input cumulative distribution function (CDF)bccausc of cxccssive rejections, This deviation can be correctedby including the CDF into the SA objective function.

IntroductionStochastic reservoir modeling refers to the generation ofsynthetic reservoir properties that are conditioned toobservations. Ideally, the generated ‘image” of reservoirproperties should honor all available data; seismic traces,geological description, core measurements, well logs, pressuretest analysis, ctc

Various methods have been applied to the stochastic modelingof resctvoir heterogeneities. One of them, which has beenrcccntly introduced is simulated annealing (SA) 1‘2. This is a

combinatorial optimization technique that involves a two-stepprocedure: first, an objective fimction (OF) is built, and second,the OF is minimized by an appropriate algorithm,

The main advantage of SA is that it can combine data fromdifferent sources by simply adding extra information into theOF.

In this work, the objective tirnction is minimized by theMetropolis algorithm, as described by Kirkpatrick et al, 1 and Senct al. 3, After analyzing the performance of the three algorithms,Sen et aL3 concluded that the Metropolis algorithm is the fastestwhen solving small problems. And this is our case.

Our purpose is to test the ability of SA to generate syntheticpermeability fields that represent actual heterogeneity. With thataim, the generated image of permeabilitics is compared withcore measurements, Our data consist of three sets ofpermeability measurements as functions of depth. Theycorrespond to three wells from different reservoirs: well A, Band C.

It has been generally accepted4 that in a rock type unit,permeabilitics follow a log-normal distribution, However, thereare no theoretical background for that thoughts. Jensen et al 6proposed a power transformation of permeability data, whichdepends on one parameter p, Any random series-parallel

arrangement of permeability elements yields a p-normaldistribution such that –1 < ps 1. The normal distribution has

p = I and the log-normal distribution has p = O. Jensen6

showed several data sets which fall within these limits.None of the three sets of permeability data follow log-normal

distributions. Therefore, we applied the Jensen p-transformationto them. Permeabilities from well B do follow a normaldistribution after p-transformation. But those from well A showan exponential distribution On the other hand, permcabilitiesfrom WCI1C could not be arranged in a known distribution eitherbefore or after the p-transformation.

The goal of this work is to analyze the performance of the SAmethod in conjunction with the Metropolis algorithm on thesethree atypical sets of data.

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2 APPLICATIONS OF SIMULATED ANNEALING ON ACTUAL BUT ATYPICAL PERMEABILITY DATA SPE 35345

The Metropolis algorithm departs from an initial permeabilityfield randomly drawn from the PDF: this is the generationprobability, Other parts of the algorithm are the OF, aperturbation mechanism, an acceptance probability (Metropoliscriterion) which depends on a control parameter, and aprocedure to update the OF,

Commonly, the objective function has been built to reproduceonly the autocorrelation by matching the semivariogram of theimage to the semivariogram of the data. The PDF was expectedto be automatically adjusted.

But this is not always tree. During the minimization process,the PDF matching is sometimes lost, To overcome this situationwe analyze two possible solutions: (1) estimating thesemivariance with the p-transformed permeabilities, (2) buildinga more complex OF that accounts for the PDF as well as thesemivariogram.

TheoryIn order to characterize permeability values as a function ofdepth, different statistical measures are computed. We aremainly interested in estimating two of those measures: the

frequency or probability distribution function (PDF) and thesemivariogram, because they are input to the SA technique.Besides, the common measures, such as the arithmetic mean(measure of central tendeney) and the standard deviation(measure of dispersion) are obtained.Frequency or Probability Dktribution Function (PDF). ThePDF is a descriptive device which let us realize how a certainproperty varies. It assigns to each value of the property a specificprobability of occurrence. In many cases it is possible to tit atheoretical PDF to the experimental values. For the permeabilitydata analyzed here, only two theoretical functions are used: thenorma[ or Gaussian distribution and the exponentialdisfribufion. One parameter, the mean, determines the latterfunction, while the former requires also the standard deviationto be completely defined’.

In order to find a suitable theoretical PDF, the commonpractice is to plot a histogram from a frequency table ofmeasured data, The histogram shows the behavior of thevariable, In fact, if wc deal with a continuous variable, as thenumber of observations increases the histogram becomes thePDF. Therefore, we use the histogram to search a suitabletheoretical PDF, Then statistical tests for that distribution, likethe Lilliefors test for the exponential distribution or the Shapiro-Wilk test for the normal distribution?, are applied to validate theproposed PDF.

Frequently, it is not possible to fit measured permeabilitiesinto a known PDF, so, a transformation is performed. Jensen etal, (1987) suggest the following transformation,

,(1)

[ink p=()

Equation (1) is a power transformation that is defined by oneparameter, p. The value of p modifies the shape of thecorresponding histogram, The optimal p-value that best fitstransformed data into a known PDF is estimated by minimizing(the Lilliefors test) or maximizing (the Shapiro-Wilk test) thestatistic or indicator defined in the corresponding statistical testfor that distribution.

Nevertheless, in the SA technique, the cumulative distributionfunction (CDF) is more useful than the PDF to generate randompermeability values. The CDF is defined as,

CDF(x)= prob(X~x),.,..,..,,,,,,.,,.,,.,,.,,.,.,..,,,,,...,.,,.,....,,,.(2)

i,c, is the probability of finding a value of a random variable Xwhich is less than or equal to x. If the random variable X iscontinous, the relationship between CDF and PDF is,

1CDF(X)= PDF(u) du ....... ......... ......... ......... ...............(3)–m

CDF’S cannot be obtained directly from measured data. First,a suitable PDF is approximated as described above; then thecorresponding CDF is estimated by Eq. 3.Autocorrelation -Semivariance Estimator. The autocorrelationis the degree of similarity between spatia[ly separated data, It iscommonly quantified by a statistic, the semivariogram. Thesemivariogram is a plot of the semivariance of N samplesmeasured a distance h apart as a fimction of h, which is calledthe separation or lag distance,

The classical semivariance estimator is defined as9

y(h) = &N#)[Z(xi)_Z(xi +h)]z . . . . . .. . .. ,(4)

where Z is the variable under consideration (i.e., permeability),N(h) is the number of data pairs and x, denotes the spatial

location of data.The main drawback of this estimator is its imprecision at

moderate to large lag distances. This imprecision arises becausethe number of data pairs involved in the calculation decreases aslag distance increases, Therefore, only the first few points of thesemivariogram are really significant. Common practice is todisregard any estimate of ?’ for lag distances greater than onehalf of the sampled interval,

The estimator defined in (4) has an important Iimltation: itneeds equally spaced measurements, which are not alwaysavailable in practice. In order to handle these cases, Samper andCarrera10 proposed an alternative algorithm, Their algorithm isapplied in this work.

The techniques mentioned above compute the experimentalsemivariogram, because they only use measured data.Sometimes, a theoretical model is fit to the experimentalsemivariogram in order to obtain an equation. But importantfcalurcs of the spatial behavior may be lost using an inadequate

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SPE 35345 G. B. SAVIOLI, E.D.FALCIGNO, M. S BIDNER, L.W LAKE 3

equation, Thcrcforc, in this paper, only the experimentalscmivariogram is used.Simulated Annealing. Simulated Annealing is a techniquebased on a combinatorial optimization scheme for generatingstochastic permeability fields. The actual heterogeneity of theformation is simulated honoring the available information. Oncpossible approach to optimization by SA, the Metropolisalgorithm, is briefly dcscribcd here,

1

2.

3.

4,

5.

Gcncratc an initial permeability field on a desired griddra~ving values from the corresponding CDF, Computethe objective function OF,”,Introduce a perturbation by selecting a grid CC1l atrandom and replacing its permeability value by anothervalue also drawn from the CDF.Compute OFnewand AOF = OFn,W- OF,

If AOF<0 accept the perturbation.Otherwise, apply the Metropolis criterion: scleet arandom number z, 0 s z <1. If cxp(–AOF/ T) > z accept

the change, else reject it. T is a convergence parameter.Repeat s(cps 2-4 until a specified number of perturbationsarc accepted.Lower T and repeat steps 2-5 until a convergencecriterion is satisfied or a specified number ofperturbations cxcecded. In our case, the process isstopped when the objective function remains roughlyconstant during 5000 iterations.

The common approach only includes in the objective functionthe semivariogram, based on the idea that the CDF adjustmentwill not be distorted despite the accepting/rejecting criteria, Thisis not always true because permeability values are changed oneby one, Frequently, most of the values accepted in step 4 belongto a narrow permeability range, As a conscquencc, the CDFobtained from the optimum permeability field may be differentfrom the original CDF. This problem is solved by 1) includingthe permeability transformation in the scmivariancc estimationor/and 2) building a more complex objective function thataccounts for the cumulative distribution fimction (CDF) as WCIIas the scmivariogram Thcrcforc, several objective fimctions arcproposed.Objective Functions. Wc built four objective functions: OF1,OF2, OF3 and 0F4.

OF1 is the common approach,

OFl=~(yac, (h)-ymn(h))2 . . .(5)h

where Y(h) is the classical semivariance estimator. Thesubscripts ‘act’ and ‘sire’ mean actual (obtained frommcasurcmcnts) and simulated (obtained from generated data),rcspcctivcly.

0F2 also tits only the scmivariogram, but the scmivarianccestimator is computed from the transformed permcabilities.Therefore,

0F2=~(YaCt(h)- j&(h))2 ....... ... ......................... ......(6)h

where ~(h) is the semivariance obtained from transformed

data.0F3 is a more complex objective function, It adds a ncw term

to OF1, in order to force a fit to the CDF,

01’3 = ~(y,c, (fr)- y,,#d)2 +w~@lF(Y),C( -C@Y),,m )2 ..(7)h 1

where CDF(Y) is the cumulative distribution fimction of thetransformed permeabilities, y. The weighting factor, w, must becarefully selected to maintain the initial PDF adjustment. Up tonow, we do it by trial and error.

0F4 is obtained applying both ideas simultaneously, i.e.,adding the controlling term to 0F2,

0F4 = ~(;act(h) - j,tm(h))2 + W~ c~~(y)ac, -CDF(Y),lm 2 ...(8)h 1

Definition of Errors. In order to analyze the behavior of SAwith the four objective functions mentioned above, a suitableerror function has to be defined.

Synthetic permeabilities arc compared with measured data totest the ability of SA to represent the actual heterogeneity.Therefore, in each location where a measurement is available,wc detine an absolute error,

ci=k –kCxpi gen, . . . (9)

where kCXP, is the experimental value and k ~,n, is the

gcncratcd one.The most general way to quantify the c, is by generating their

CDF, This way the median, the central tendency measure Icastsusceptible to outlicrs, can be gotten by inspection as well as canthe prcpondcrancc of extreme values, Extreme values seem toaffect fluid flow more than central ones

DataThe theo~ already described is applied to sets of permeabilitydata measured at three wells. They are located in differentreservoirs of the Neuquen Basin of Argentina. These wells arenamed wells A, B and C throughout this paper. The analyzeddata consist of permeability mcasurcmcnts as a function of depthobtained through whole core laboratory tests: 65 values for wellA, 139 values for B and 112 values for C. They are representedby points and crosses in Figure 3 for well A, in Figure 7 for Band in Figure 10 for C.

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Table 1 shows the lowest and highest permeability values, thearithmetic average, the standard deviation, the variance and thecoefficient of variation of the three wells.

Permeability data are characterized by their probabilitydistribution function (PDF) and autocorrelation(semivariogram).PDF, Figures la- lC represent permeability histograms for wellsA, B and C, Well A histogram lacks symmetry. Therefore,power transformations (Eq. 1) are applied to seek a knowndistribution. Data from well A fit an exponential typedistribution with the following transformation:

~~33_l

yA =T+2.64s E(0,28) ,,,,,,,,..,.,,,..,.,.,.,,.,,,.,,.,,., ,(10)

where 0,28 is the parameter of the exponential distribution,

pDF(y*) =0,28e~(-0,28y* ),,,,,.,..,.,,,,.,........,.............(n)

The Lilliefors testE for exponential distribution accepts thishypothesis, The transformed histogram and the correspondingPDF can be seen in Figure ld. This is an atypical behavior,Therefore, the assumption ofp-normality is not always correctc.

Let us notice that Lambert5 also obtained exponentialdistribution functions for 102 wells over the 689 she studied,though she analized untransformed data.

On the other hand, well B histogram show more symmetry.After p-transformation, permeability data fit a normal typedistribution, as the Shapiro-Wilk test8 conllrrns. Thecorresponding transformation is:

y~=k#3-1

-0.3-0,59 zN(1,34,0.44),.,,,,,,.,,,.,,,.,,.,,,,,,,,, .(12)

where N( 1.34,0.44) means the following normal distribution,

PDF(yB) = ~10,44c~{-#y’1:34)2]

.(13)

Figure lC shows the transformed permeability histogram and theadjusted PDF.

The data from WCI1C are unusual in that, owing to themeasurement procedure used, there are no values below 1mD.Such data sets are said to be truncated; in the case of the well Cdata, the truncation is more than half of the entire set. Toaccount for this, we set all the unmeasured values to 1 rnD andused a logarithmic (p=O) transformation of the data in the SA.See Sinclair” for a discussion of how to correct a data set fortruncation.Semivariogram. The classical scmivariance estimators areplotted in Figure 2, They are calculated from the rawpermeability data (Figures 2a, 2b and 2c) or from the power-transformed data (Figures 2d, 2e and 2f) for wells A, B and Crespectively. According to the common practice, we consider

significant only the first half of the classical semivariograrn(heavy solid line),

As it can be seen from the plots, one of the advantages ofincluding the transformation is the reduction of thesemivariance scale. Besides, in many cases, using thistransformation the semivariograms show strongerautoeorrelation3, as happens in well B,

Whichever semivariance estimation is performed, the well Adata show the greater autocorrelation. The approximate rangesare 7 m for well A and 3 m for wells B and C. They have beencalculated from the spherical theoretical model, as described byGoggin et al,’ 2 and Falcigno et al, 13

ResultsIn this section, we compare the behavior of the SA method usingthe four objectwe functions defined in Eqs, 5-8 on wells A, Band C. In each case, we randomly selected 10’%oof the actualdata to be conditioning points and estimated the remaining go~o,

The latter 90% were used in the error estimate,Well A. Figure 3 shows the results obtained minimizing thefour objective functions, In each case, the points and the solidline represent the measured and generated permeability valuesfor every grid block, The measurements selected as conditioningpoints are shown with crosses. Figures 4 and 5 show theagreements between the imposed and calculated semivariogramsand CDFS, respectively,

OF1 (Figure 3) corresponds to the common approach - itminimizes the difference between the actual semivariogram andthe simulated semivariogram, And the actual semivariogram isestimated from the raw permeability data, Nevertheless, itsresults are clearly the worst, Figure 5 shows the main cause: thedesired CDF is lost after the annealing process,

The semivariogram is more appropriately estimated includingthe permeability transformation in the classical semivarianceestimator, as it is done in 0F2 (Eq. 6). Applying 0F2, thematching of the CDF improves (Figure 5), And so does thematching of the semivariogram (Figure 4), As a consequence,the image of the permeability field is better adjusted to themeasurements for 0F2 than for OF1 (Figure 3).

The CDF is included in both 0F3 and 0F4 defined by Eqs, 7and 8. Figure 5 shows the improvement of the CDF match for0F3 and 0F4, The simulated permeability fields arc closer tothe actual fields, as it can be seen in Figure 3 although thematch of the semivariogram is somewhat worse for 0F3 (Figure4).Well B. The main feature of well B is that its permeability data

follow a normal distribution after power transformation (Figurele). Figure 7 represents the actual and simulated permeabilityfields obtained applying OF 1, 0F2 and 0F3, OF1 provides theworst match. The adjustment of the semivariogram (estimatedfrom the raw data) is poor, as it can be seen in Figure 8, And itdoes not honor the CDF, as it is shown in Figure 9.

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SPE 35345 G. B. SAVIOLI, E,D,FALCIGNO, M, S. BIDNER, L.W.LAKE 5

The image obtained by 0F2, is a better representation of theactual permeability field (Figure 7), The adjustment of thesemivariogram (obtained from power transformed data, Figure8) is excellent, And the CDF is honored (Figure 9),

Figure 7 also shows that the image determined by 0F3 isequivalent to the image dctcrmincd by 0F2, 0F3 adds the termwhich takes into account the CDF, Because of that, it tits theCDF very well (Figure 9) despite some loss in matching of thesemivariogram (Figure 8),

Results, for 0F4 are not shown in Figures 7-9, Thescmivariogram, the CDF and the permeability field simulated by0F4 are equivalent to those simulated by 0F2. Well Bpermcabilities show a Gaussian CDF after p-transformation.There is no need to include the CDF into the objective function(as in 0F4), because the CDF is automatically honored (as in0F2).Well C. As it was already said, the permeability data set for WICIIC is truncated for values less than 1mD. Figure 10 shows actualand generated permeability fields obtained applying the four OF.

The anomalous shape is shown again in Figure 12 of CDFversus log k. The CDF is the initial state for SA. In this case wecould not find a theoretical representation of the CDF,Therefore, wc have approximated the empirical function shownin Figure 12 by two straight lines,

Once more the traditional OF 1 gives the worst match (Figures10 -12). It gets slightly better for 0F2, In this case OF2 is builtby using log k in the estimation of the semivariogram. A greaterimprovement appears when the CDF is added in the OF, as inOF3 and OF4. Visually, 0F4 gives the best match for thesemivariogram (Figure 11) and also for the CDF (Figures 12).

DiscussionUp to now, comparison among results obtained by applying thefour objective functions has been done qualitatively. So as to

quantify the goodness of the results, the absolute errors detincdby Eq 9 arc calculated for each cell. Then, the CDF of thoseerrors is gcncratcd for each run, corresponding to a givenobjective function. Finally, the median of errors is estimated,

Figure 6 is a plot of the CDF of the absolute errors for well A,From it, the following information can be extracted, 50% ofgenerated values have an error which is less than 27 mD withOF1, 2 mD with 0F2, 0,8 mD with 0F3 and 0.6 mD with 0F4.Clearly, in all cases, the improvement of the autocorrelationestimation by the power transformed permeabilities and theinclusion of the CDF in the objective function are both helpful.The best results arc obtained with 0F4,

These errors can be compared with the statistics of Table 1,The difference between the highest permeability value and theIowcst is 116 mD. The arithmetic average is 10.3 mD. Thestandard deviation is 23.8 mD and the coefficient of variation 1s2.3 mD.

The median of errors can be seen in Table 2 for WCIISA, Band C.

For well B, SOY. of the generated values have an error lessthan 57 mD with OF1, 17 mD with 0F2, And the same medianof the errors, 17 mD, is found when applying 0F3 and 0F4.Well B shows a difference of 1093 mD between the highest andthe lowest permeability values. The arithmetic average is 58mD.Other statistics ean bc seen in Table 1.

For well C, 50% of permeability values have an error which isless than 2mD with OF1, 0.3 mD with 0F2, 0,09 mD with 0F3and 0.08 mD for 0F4 (Table 2), Arithmetic average ofpermeability is very low: 4.6 mD. The difference betweenmaximum and minimum permeability values is narrow: 61 mD(Table 1),

In order to compare across the wells the median of errors aredivided by the arithmetic mean permeability, Results are shownin Table 3,

It is interesting to notice that the worst results correspond towell B. In spite that WCIIB is the only onc showing a normaldistribution of permeability after transformation. But well B hasa smaller correlation length than well A. The former is 3m andthe latter is 7m,

Well C behavior is very atypical because permeability data aretruncated and have very low values. Because of that, it showsthe lowest errors in Tables 2 and 3.

ConclusionsSimulated annealing (SA) in conjunction with the Metropolisalgorithm is applied to stochastic modeling of permeabilityfields, The aim is to verify the potential of SA to generateimages of atypical heterogeneities. Atypical data are three sets ofcore measurements which were taken from three wells: A, B andC. None of the three sets of data show a Gaussian PDF ofpermeability logarithms, After applying power transformation topermeability measurements from WCIIS A and B, the latterexhibits a normal PDF and the former exhibits an exponentialPDF. Well C histogram cannot bc transformed to give a knowndistribution, because there are no measurements of permeabilityvalues less than 1 mD. They are just mentioned as non-permeable zones, which are present at different depths,

In order to handle our problematic data several new objectivefunctions have been proposed and tested, The conclusions are:1, The experimental semivariogram, instead of an analytical

model, is included in the objective function. But, in somecases, the traditional estimator of the experimentalsemivariogram is too noisy, Besides, it needs equally spacedmeasurements, In order to overcome this limitation and tosmooth the fluctuations, an algorithm suggested by Samperand Carrera’0 is successfidly applied

2, Best images arc obtained considering the power transformeddata in the scmivariance estimation, for wells A and B. Inthis way, the semivariance estimation is consistent with theproposed power transformation of the PDF,

.3. For our data, when the OF is built to match only thesemivariogram, the PDF is automatically honored only for

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4

well B, The main feature of well B is that itspermeabilities,after transformation, show a normal distribution.Thus, it is important to control the PDF adjustment duringthe annealing process. Otherwise, the initial fitness may belost, We propose to add a new term to the objective timction,which compares the calculated and desired CDF. Thesynthetic permeability fields are best adjusted to the actualmeasurements when this new OF is applied.

AcknowledgmentsWe are indebted to the University of Buenos Aires and the oilcompanies PlusPetrol, Amoco Argentina and Astra forsupporting this work. M. S.Bidner is a Research Fellow at theConsejo Naciona[ de Investigaciones Cientljicas y Tecnicas deArgentina.

Nomenclaturee=

h=

k=

OF =

P=T=

Y=z=

Greek letters

1’=?=

Subscriptsact =

exp =

gen =

i=

sim =

References

absolute error, rnD.lag distance, m,permeability, mD.objective function used in SA.power transformation parameter, Eq, 1.convergence parameter used in 5A.p-transformed permeability.random number used in SA.

semivariance computed from raw data.semivariance computed from transformeddata,

actualexperimentalgeneratediti measurementsimulated

1. Kirkpatrick, S., Gelatt, C. D., Jr. and Vecchi, M. P.: “Optimizationby Simulated Annealing”, Science ( 1983), 220,671-680,

I TABLE 1- PERMEABILITY DATA SAMPLE ISTATISTICS

well A well B well C

Number of data 64 139 112. . . . . . . . . . .... ........ .. .. ..... .. .... .. .. . . .Maximum, kmu (mD) 116.0 1093,0 61,6....... .. .. .. ... . . . . . .. . .. .. .. . . . .. . . . ............ .. ...Minimum k (mD) 0.002 1.92 1 O*

. . . . . .. .. .. ....~.....!w .. .. .... ................ .. . .............. . . ......................IArithmetic mean ka,~(rnD) 10,3 5s.0 4.6..... . .. .. . . ...........?. ... .... . . . . .. . . . . . . . . .Standard deviation a (rnD) 23,8 144.8 9,5........ . .............. . . ........... . . . . ..... .......... .. . . .. . . ....... ... .. . . . .

Variance, 02 (mD ) 567 20,973 91. . . . . . .. .. . . . .. .. .. . . . . .. ......... . . .. ..... ..Coeftlcient of variation. c., 2,3 2.5 2.1

2.

3.

4.

5.

6.

7.

8.

9.

10

13,

Deutsch, C. V. and Journel, A. G.: “The Application of SimulatedAnnealing to Stochastic Reservoir Modeling”, SPE AdvancedTechnology Series (1994), 222-227.Sen, M, K,; Datta Gupta, A,; Stoffa, P,L.; Lake, L.W. and Pope,G.A. :“Stochastic Reservoir Modeling Using Simulated Annealingand Genetic Algorithm”, paper SPE 24754 presented at the 1992SPE Annual Technical Conference and Exhibition, Washington,DC, Oct. 4-7.Law, J.: “Statistical Approach to the Interstitial Heterogeneity ofSand Reservoirs”, Tram. AIIUE ( 1944) Vol. 155,202.Lambert, M. E.: “A Statistical Study of Reservoir Heterogeneity”,MS thesis, U. of Texas, Austin (1 981 ).Jensen, J. L.; Hinkley, D. V. and Lake, L. W.: “A Statistical Studyof Reservoir Permeability: Distributions, Correlations, andAverages”, SPEFE ( 1987), 2, No. 4,461468.Savioli, G,B.; Bidner, M.S. and Jacovkis, P.M.: “The Influence ofHeterogeneities on Well Test Pressure Response - A SensitivityAnalysis”, paper SPE 26985, proceedings of the 3rd SPE LatinAmerican and Caribbean Petroleum Engineering Conference(1 994), Vol. III, 1107-1117, Buenos Aires, April 27-29Conover, W.J,: Practical Nonparametric Statistics, 2ed, JohnWiley & Sons, New York ( 1980),Li, Dachang and Lake, Lany W.: “A moving window semivarianceestimator”, Wafer Resources Research (1994), 30, No 5, 1479-14!39.Samper Calvete, F y Carrera Ramirez, J:Geoestadistica:Aplicaciones a la Hidrologia Subterranean, CentroIntemacional de Metodos Numericos en Jrrgenieria, Barcelona(1 990),Sinclair, Alastair J.: “Applications of Probability Graphs in MineralExploration”, Association of Exploration Geologists SpecialVofume No. 1, Richmond, British Columbia ( 1976).Goggin, D. J.; Chandler, M. A.; Kocurek, G., and Lake, L. W.:“Patterns of Permeability in Eolian Deposits: Page Sandstone(Jurassic), Northeastern Arizona”, SPEFE ( 1988), 3, No. 2, 297-306,Falcieno. ED.: Savioli. G.B. and Bidner. MS,: “CaracterizacionGeoe~tadistica’ de Medicines de Poro’sidad y PermeabilidadHorizontal y Vertical”, proceedings of the 3eras Jomadas deInformatica Aplicada a la Production de Hidrocarburos ( 1994),L4P, 277-298. Buenos Aires, Nov. 9-11.

TABLE 2- MEDIAN OF ERRORS (mD)

well A I well B well COF 1 27 57 2

* truncated

332

SPE 35345 G. 8. SAVIOLI, E.D.FALCIGNO, M. S. BIDNER, L.WLAKE 7

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0000 06 12 1

log (k)

Fig. 1 - Permeability Histograms:(a) Well A; (b) Well B; (c) Well C.PDF and Histograms after p-transformation: (d) Well A, p=0,33;(e) Well B, P=-0.3

Semivariance. mD2~oo➤..__—I (c)

lCOL

o ~-—...o 10 20 30 4

Distance, m

Semivariance

06(f)

04 .

,,. . ...7 —: —. —..

02 -

i

I00 ~_J_ ..—

10 20 30 4

Distance, m

)

8

Fig. 2- Semivariograrns computed from raw permeability data: (a) Well A, (b) Well B and (c) Well C. Scmivariograms computed

from transformed data: (d) Well A, p-transformation (p=O.33); (e) Well B, p-transformation (p=-O.3); (f) Well C, log(k)-

Iransformation

333


Permeability, mD0.01 1 100

,.— .mmrm.-—

2515

2520

E 2525I

r-= 2530

:

2535 !

2540

.

OF1

Permeability, mD Permeability, mD0,01 1 100 001 1 100—

r >. .>.! ,,, ,!,,1 \ ,, !!>!! ,?!!1,,

OF2

X-–==-=?..ZI

“:-......‘2‘~--%

. __-g5--“.+==- .,

.52..>

>,,,,.,,.,,,,

0F3

.

Permeability, mD0.01 1 100

[—..0F4

.—— —.$-$..

1 .~==l

1“ .>..” “s.,t ~.. ;%-.. “ _y5”-t ..>

~ -..

~7---

.===..__.. _ --.-–=-----x .c—<—_

“=+==$

Fi!z, 3- Well A: Actual (points) and simulated (solid line) permeability fields obtained using the four objective functions. OF] fits theraw data semivariogram~OF2 its the transformed data sernivariograrn, OF3 fits the raw data semivariogram and the CDF and OF4tits the transformed data scmivariogram and the CDF Crosses represent conditioning points.

Semivariance, mD22500

2000

1500

1000

500

0

OFIrawdata . .

. .

.“

2.5 5.0 7.5 10,0 12

r–— -- -.– -,2500 ___

I 0F3zooo / mwdata

$. “.”’1

1500. .

1:1<,

0 ;:”.-———i~25 75 10,0 12.5

Distance, m

40

30

20

10

0

Semivariance

OF2~

p-transformed data . . .

. . . . . .

2.5

40 OF4 I

30

20

10

0

p-transformed data . . .I

I,.J

2.5 5.0 7.5 10.0 12.5

Distance, m

I

Fig. 4- Well A: Comparison between actual (points) and simulated (solid line) semivariograms. The images arc generated bySA using the four objective functions. OF I fits the raw data semivariogram, 0F2 fits the transformed data scmivariogram, 0F3fits the raw data semivariogram and the CDF and 0F4 fits the transformed data semivariograrn and the CDF.

334

SPE 35345 G. B. SAVIOLI, E.D.FALCIGNO, M, S. BIDNER, L.W.LAKE 9

1.00

0.75

050

025

0.00

Cumulative Distribution Function Cumulative Distribution Function

OF1

— actual

—. simulated

——– J

5 10 1; 20

1.00

0.75

0.50

025

0.00

0F2

— actual

—. simulated

o 5 10 15 I

1.00 : -—-— 1.00: 0F3 0F4

0,75 ~ 0.75

050 050

0.25 ~ — actual0.25

— actual

simulated II — simulatedIJ,oo Y___ .—-1

o 5 10 15 2000001 .-. —–. —____

5 10 15 20

p-Transformed Permeability pTransformed Permeability

Fig. 5- Well A: Comparison between actual (obtained by p-transformation) and simulated CDF. The images are generated bySA using the four objective functions. OF1 fits the raw data semivariogram, OF2 fits the transformed data semivariogram,0F3 tits the raw data scmivariogram and the CDF, and 0F4 tits the transformed data semivariogram and the CDF,

Cumulative Distribution Function,.00 .-.. ..—.

OF1

075

050 iI

median = 27 mD

0.00 oL_ ...._. ..L_L_—_.. ---J20 40 60

1.00

075

050

025

000(

100

0.75

0.50

0.25

Cumulative Distribution Function

0F2

.Oo!l!v!!wo 20

100

0.75

0.50

0.25

40 60

OF4

l,~dian=06mll I_————..-

il 200.00

40 60 0 20 40 60

median = 0,6 mD

Absolute Errors, mD Absolute Errors, mD

Fig. 6- Well A: Cumulative distribution of errors obtained with the four objective functions, OF 1 fits the raw data semivariogram,OF2 fits the transformed data semivariogram, OF3 fits the raw data semivariogram and the CDF, and OF4 fits the transformeddata semivariogram and the CDF.

335


2405

2410

E 2415

~- 2420n

:2425

2430

2435

Permeability, mD10 100 1000

3.

●.-.......”. .

,. .

iiFi!Ei“. .

●. . . .

...

-.

. .

OF1

I

Permeability, mD10 ICHI 1000

%

0F2. .. .

=. .. . .

““e?

~.

... . . ...

. .. ,,

%!5.,

Permeability, mD10 100 1000

l!!i5-OF3

;. .

. .

. .1 ..

....,.. .

z

. . ..... . .,

..%. .

,: &_: .. *<–_

Fig. - Well B: Actual (points) anh simu Ited (solid line) permeability folds obtaifunctions, OF1 fits the raw data semivariogram, OF2 fits the transformed data semivariograrn, and OF3 its theraw data semivariogram and the CDF. Crosses represent conditioning points.

Semivariance, mD 240000

30000 . . . .

2oc00 .

. . . -

10000 ~OFIraw data

o.0 4 8 12 16

Distance, m

Semivariance, mD 240000

30000. .

20000. . .

I. . ..

10000 .0F3 i

I raw data I

ed using the three objective

Semivariance

0“25~T0.20

0.10

0.05

0.00 Io 4 8 12 16

Distance, m

Fig. 8- Well B: Comparison between actual(points) and simulated (solid line)semivariograms. The images are generated bySA using the three objective functions. OF 1fits the raw data semivariogram, OF2 fits thetransformed data semivariogram, and OF3 fitsthe raw data semivariogram and the CDF.

o ~..-o 4 8 12 16

Dktance, m

336

SPE 35345 G. B. SAVIOLI, E.D.FALCIGNO, M. S. BIONER, L.W.LAKE 11

Cumulative Distribution Function Cumulative Distribution Function100

0.75

0.50

0.25

OFl

— actual

— simulated

1.00

0.75

0.50

0.25

3F2

— actual

0.0000 05 1,0 1.5 2.0 2

0000.0 05 1.0 1.5 2.0 2.5

p-Transformed Permeability

Cumulative Distribution Function


‘“” &I

Fig. 9- Well B: Comparison between actual (obtained0,75 by p-transformation) and simulated CDF. The images

are generated by SA using the three objective timctions.

050 OF 1 tits the raw data semivariogram, OF2 fits theItransformed data semivariogram, and 0F3 tits the rawdata semivariogram and the CDF,

O25— actual

— simulated

000 -0.0 0.5 10 1.5 2.0 25


Permeahlity, mD1 10

——

Permeability, mD1 10

Permeability mD1 10

Permeability, mD1 10

2F4 ~—~===&-

S“. .

2390

OF1 c..= ..*_— 3F2 3F3

I2400

E

K- 2410g

o

2420

C..I

t

I

F.:I

2430l–-

Fig. 10- Well C: Actual (points) and simulated (solid line) perm~udity fields obtained using the four objectthe raw data semivariograrn, 0F2 fits the transformed data semivariogram, OF3 fits the raw data semivariogram and the CDF, and

‘e functions. OF 1 fits

OF4 fits the transformed data scmivariogram and the CDF.

337


Semivariance, mD 2 Semivariance150 ~~ 04~

‘w.....””. .

loot . . . ...!.. ” “.

.-. .. .

.. .50 .“ . ..

.“ OFI..raw data

I “ ‘“””

\ ... .“,, ..

I Iptranaformed data

01~ 0010 5 10 20 0 5 10 15 2(

150 r~ 04~

.“- ..4.“

. . .. ,. . .

100 -..

- ..... i : “M].“.’

R?d01 . OF4..

p-t ra naformed data

o0 5 10 15 20

000 5 10 15 – 20

Distance, m Distance, m

Fig. 11- Well C: Comparison between actual (points) and simulated (solid line) semivariograms. The images aregenerated by SA using the four objective fimctions. OF1 fits the raw data semivariogram, OF2 fits the transformeddata semivariogram, OF3 fits the raw data semivariogram and the CDF, and OF4 fits the transformed datasemivariogram and the CDF.

100

075

050

Cumulative

i OFI

Distribution Function

7

,,

>-. .

_.

::1-.----.-2!!3H!o- 1 10 100

Cumulative Distribution Functionlm

I oF2075

050

O 25— actual

_ simulated

o mo~100

Permeab#l~, mD Permeability, mD

Cumulative Distribution Function Cumulative Distribution Function

XL ~ :rlo 1 10 100 0- 1 10 lC

Permeability, mD Permeability, mD

1

Fig. 12- Well C: Comparison between actual (obtained by p-transformation) and simulated CDF. Theimages are generated by SA using the four objective functions. OF 1 fits the raw data semivariogram, 0F2fits the transformed data scmivariogram, 0F3 fits the raw data semivariogram and the CDF, and 0F4 fits thetransformed data semivariogram and the CDF.

338

applications of simulated annealing on actual but atypical permeability...

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