A Simple Method to Estimate Interwell Autocorrelation

8/9/2019 A Simple Method to Estimate Interwell Autocorrelation

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369

Chapter 26

A Simple Method to Estimate Interwell

AutocorrelationJorge Oscar de Sant Anna PizarroPetrobras

Rio de Janeiro, Brazil

Larry W. LakeCenter for Petroleum and Geosystems Engineering

The University of Texas at AustinAustin, Texas, U.S.A.

ABSTRACT

The estimation of autocorrelation in the lateral or interwell direction isimportant when performing reservoir characterization studies using stochas-tic modeling. This paper presents a new method to estimate the interwellautocorrelation based on parameters, such as the vertical range and the vari-ance, that can be estimated with commonly available data.

We used synthetic fields that were generated from stochastic simulations toprovide data to construct the estimation charts. These charts relate the ratio ofareal to vertical variance and the autocorrelation range (expressed variously)in two directions. Three different semivariogram models were considered:

spherical, exponential, and truncated fractal.The overall procedure is demonstrated using field data. We find that the

approach gives the most self-consistent results when it is applied to previouslyidentified facies; moreover, the autocorrelation trends follow the depositionalpattern of the reservoir, which gives confidence in the validity of the approach.

INTRODUCTION

The importance of reservoir characterization methodshas been established in the last decade by reservoirstudies that are based on stochastic models that accountfor the heterogeneity of the porous media.

A reliable study depends on prior quantification ofthe heterogeneity of the reservoir (Srivastava, 1994).The geostatistical approach describes the heterogeneitythrough averages, variances, and autocorrelation.Although it has several advantages, when comparedwith deterministic approaches the confidence in geo-statistical modeling will be a strong function of howwell the input data represent reality. The horizontal

autocorrelation, especially, is among the most criticalof the parameters to be estimated; we propose a newprocedure to estimate it in this paper. The methoduses serial data from several vertical wells, as wouldexist from wells in mature projects.

MOTIVATION

There are several works in the literature stressingthat the estimation of horizontal autocorrelation isimportant in achieving a good reservoir description.Lucia and Fogg (1989) stated that the principal diffi-culty in reservoir characterization is estimating the

de Sant Anna Pizzaro, J. O., L. W. Lake, A simplemethod to estimate interwell autocorrelation,1999, in R. Schatzinger and J. Jordan, eds.,Reservoir Characterization-Recent Advances,AAPG Memoir 71, p. 369380.


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370 Pizzarro and Lake

spatial distribution of petrophysical propertiesbetween vertical wel lbores. These points also havebeen highlighted (Hewett, 1986; Lemouzy et al., 1995;Jensen et al., 1996).

Lambert (1981) calculated and tabulated Dykstra-Parsons coefficients (measures of heterogeneity) inboth the horizontal and vertical directions from 689wells in 22 fields. In 90% of the cases, the ratio of the

coefficients in the horizontal and vertical directionswas less than 1. This result, governed by the deposi-tional trends observed in petroleum reservoirs, indi-cates a spatial dependence among these variables.

Figures 1 and 2 illustrate this idea in more detail.The figures show an idealized cross section (x,z) that isbeing penetrated by several vertical cored wells. Wecan calculate two types of variances for the cross sec-tion. The vertical variance is the arithmetic average ofthe variances for each individual well; the areal vari-ance is the variance of the well averages. A cross sec-tion that is strongly autocorrelated in the x-directionwill have a small areal variance; strong autocorrelationin thez-direction will lead to a small vertical variance.The ratio of these two is a measure of the extent ofautocorrelation in the respective directions.

If we quantify the relationship between the autocor-relation and the variances in both directions, we willbe able to est imate one parameter from the others;therefore, the main idea behind our method is that thehorizontal autocorrelation must depend on the ratio ofthe areal-to-vertical variances and the autocorrelationin the vertical direction. With a chart that expressesthis relationship, one can estimate autocorrelation inthe horizontal direction.

METHOD OF STUDY

This section states the approach used to estimateautocorrelation in the interwell region of a reservoir.In the following derivation, we focus on the estimation

of horizontal autocorrelation because, in most cases, ver-tical wells can provide information on autocorrelation inthe vertical direction. We also restrict attention to theestimation of horizontal autocorrelation in permeability.

The autocorrelation will be a function of the type ofmodel chosen to represent it. Three different theoreti-cal semivariogram models were tested: spherical,exponential, and truncated fractal (power-law). The

fractal model usually does not have a finite autocorre-lation, but we will truncate at some upper cutoff;hence, the use of a truncated fractal.

The way we express autocorrelation depends on thesemivariogram model being used. For the sphericalsemivariogram model,

(1)

autocorrelation is expressed by the range . For thevertical direction, =

z; for the lateral or horizontal,

= x. For the exponential model,

(2)

we use the autocorrelation length. As before, for thevertical direction, = z; for the horizontal, = x. Forthe truncated fractal fBm model,

(3)

we use the cut-off length, l. For the vertical direction,l = lz; for the horizontal, l = lx. We use the wordrangeto generically express all three autocorrelation

h

h h

h

H

( )= ( )

cov 0

0

1

2

h eh

( )= ( )

cov 0

h

h hh

h

( )= ( )

cov 0

3

2

1

20

1

3

Figure 1. An idealized crosssection (x,z) penetrated byvertical wells.


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A Simple Method to Estimate Interwell Autocorrelation 371

measures. Because a log-normal distribution isassumed for the permeability, all variances and aver-ages are calculated on the logarithm of the perme-ability (lnk). In equations 13, cov(0) stands for thesill, h the lag distance, and Hthe Hurst coefficient.Note that the parameter expresses the autocorrela-tion of the permeability in all three models; however,because the models are different, is also differentfor each; for the spherical model is the range, forthe exponential model is the autocorrelationlength, and for the fractal model is the cutofflength. We seek, in this paper, to estimate zD givencertain variances and zD.

Formally, the vertical variance is the expectation ofthe conditional variances of Y(x,z) = lnk(x,z):

(4)

where var [Y(x0,z) ] indicates az-direction variance con-ditioned to a fixed location x0. As suggested in Figure 2,2vert is estimated as

(5)sN

Y i j j N vertw i

N

l

w2

1

11= ( )[ ]( ) =

= var , , ,

vert E Y x z2

0= ( )[ ]( )var ,

where i andj are the indices on the x- andz-directions,respectively. The areal variance is the variance of theconditional expectation of Y:

(6)

It is estimated as

(7)

See Jensen et al. (1996) for details on conditional vari-ances and expectations.

Our approach consists of the generation of synthetic,equiprobable permeability fields using stochasticsimulation and a statistical treatment of the obtaineddata. There are several methods that can generatestochastic reservoir images (Srivastava, 1994). One isthe matrix decomposition method (MDM). Thismethod is an averaging technique based on the decom-position of the autocovariance matrix. The presentstudy uses a program developed by Yang (1990) thatperforms simulations based on MDM.

sN

Y i j I N areall j

N

w

l2

1

11= ( )

= =var , , ,

areal E Y x z2

0= ( )[ ]( )var ,

Figure 2. Schematic of theprocedure to calculate theareal and vertical variances.


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The MDM can generate permeability fields forreservoir modeling, but several statistics must beknown as input data. MDM requires that the geo-logic knowledge be quantified in terms of a fewparameters that can characterize the static propertiesof the reservoir. In doing this, some assumptions arenecessary. The permeability distribution must be bya single log-normal population that obeys second-

order stationarity. The log-normal assumption wascorroborated by the work of several authors, theresults of which are summarized in a paper byJensen et al . (1987).

The generation of synthetic permeability fields pro-vided the necessary data to describe the relationamong all investigated parameters. Averaging resultsfrom several realizations will provide a good estimatefor the actual value of x.

TYPE-CURVE PROCEDURE

To perform the numerical experiments, we estab-lished the following procedure. First, we chose a rec-tangular cross section with 4000 blocks, 200 blocks inthe x-direction and 20 in thez-direction. The samplingprocedure represents a reservoir with 11 vertical wellsaligned in one direction having a constant spacingbetween them. We later concluded that the differencein results caused by staggered wells is small (Pizarro,1998); therefore, the analysis can be used for irregu-larly spaced wells, provided the average spacing isused as the reference distance.

The ranges input to MDM were converted todimensionless form to provide more generality to theresults. The dimensionless horizontal autocorrelationwas normalized by the interwell spacing; the totalthickness is the reference distance in the vertical direc-tion. We sample the generated fields at a constantinterval to generate the estimated variances.

In the stochastic simulations, the dimensionlessranges ranged between 0.1 and 100 in the x-directionand between 0.1 and 2.0 in the z-direction. This rangeis believed to describe the autocorrelation existing inmost petroleum reservoirs.

To make the results represent the expectations inequations 5 and 7 accurately, several realizations wereperformed. For each realization, Yvalues in each wellwere collected in a table in which there were 11columns in the horizontal direction (each correspond-ing to one well) and 20 lines in the vertical directionrepresenting layer values. For each well the arithmeticaverage of Ywas calculated in the vertical direction aswell as the variance in the vertical direction.

Once both areal and vertical variances are known,the calculation of the ratio between them is straight-forward. Because all of the variances used in equa-tions 5 and 7 are directly proportional to thevariance, the ratio will be a function only of xD, zD ,and the semivariogram model; therefore, if s

2

areal ands

2

vert are calculated, and zD is estimated from fitting aspecific semivariogram model to the vertical data,xD can be estimated.

RESULTS

Here, we present the simulation results used tobuild the type-curve charts. The procedure previ-ously stated was implemented, and several simula-tions were performed.

All fields were generated for 2 = 1. This popula-tion variance is equivalent to a Dykstra-Parsons coef-

ficient of 0.63, a value that is within the range of thevertical VDP values tabulated by Lambert (1981) fromcore. Because both areal and vertical variances arederived from the autocovariance, their ratio is inde-pendent of 2 (Pizarro, 1998). We used the minimalnumber of realizations (NR) that would provide stableresults, NR = 50 (Pizarro, 1998).

The results obtained are expressed in Figures 35for the spherical, exponential, and truncated fractalsemivariogram models, respectively. Each figure is aplot of s

2

areal/s2

vert versus zD

with xD

as a parameter.The truncated fractal fBm plot uses a Hurst coefficientof 0.25 because this seems to fit various types of fielddata (Neuman, 1994). Each figure contains results

from 3500 MDM simulations because we calculate theaverage of 50 realizations.

Validation

The overall procedure cannot be validated becausewe have only analytical expressions for some limitingcases. We would need an extraordinary amount offield data to cover all the possible situations asidefrom the limiting cases; however, some insight intothe results can be given by analyzing the availablesolutions.

The first result that can be compared is for the casewhen we have no autocorrelation in either the hori-

zontal or vertical directions. This means that the per-meability values are completely random, with nospatial correlation. The central limit theorem statesthat no matter what distribution a group of indepen-dent random variables are from, the sample mean ofthese variables is approximately normally distributed.So, if Y1,Y2,Y3,,YNdenote independent random vari-ables each having the same mean, , and variance, 2,and Y

equals the mean of Nof these random variables,

we will obtain

(8)

Applying the property that the variance of a sum ofindependent variables will be equal to the sum of thevariance of each one, we obtain

(9)

According to this procedure, the variance of the meancan be approximated by the areal variance ( s

2

areal).

VarYN

( )=2

Y Y

N

Y

N

Y

N NYN i

i

N

= + + ==1 2

1

1


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Hence, the above relation states that, for the case of anuncorrelated field (xD = zD = 0), the areal varianceshould be equal to

(10)

When the number of layers (Nl) equals 20, the aboveexpression gives

2

areal = 0.050. In the numerical experi-ments with MDM, we obtained s

2

areal = 0.047, a goodapproximation.

areallN

22

=

Another way to validate the results is to make useof an analytical expression that describes the relation-ship between variances of properties measured at dif-ferent scales. Several authors, including Neuman(1994), Lasseter et al. (1986), and Haldorsen (1986),have reported the effect of the scale on the measure-ment of heterogeneity. One of the strong points of geo-statistics is its ability to represent this behavior. Forinstance, the permeability of a well can be measuredfrom a series of core measurements representingblocks of a certain size, shape, and orientation. As thesize of these blocks increases, the variance of the mean

0.0

0.4

0.8

1.2

1.6

0.1 1 10 100

Dimensionless Horizontal Range

ArealVari

ance/VerticalVariance

1.0

0.8

0.5

0.3

0.1

Spherical Model

zD

=

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 . 1 1 10 100

Exponential Model

ArealVariance/Ve

rticalVariance

zD

= 1 . 0

0 . 8

0 . 5

0. 3

0. 1

Dimensionless Horizontal Correlation Length

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

0. 1 1 10 100

Dimensionless Horizontal Autocorrelation Upper Cutoff

ArealVariance/VerticalV

ariance

Fractal Model (H = 0.25)

0. 8

0.5

0. 3

0. 1

z D

= 1. 0

Figure 3. Autocorrelationchart for spherical semivari-ogram model obtained from50 realizations.

Figure 4. Autocorrelationchart for exponential semi-variogram model obtainedfrom 50 realizations.

Figure 5. Autocorrelationchart for fractal semivari-ogram model obtained from50 realizations.


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value within the blocks gets smaller, although theymay have the same mean.

Kriges relationship (Journel and Huijbregts, 1978)can be adapted to the present application. If a miningdeposit (D) is split up into blocks (V), the variance ofthe properties of these blocks will be called the blockvariance and be denoted by

2

(V/D). The variance ofpoint grades (O) within a block is expressed by

2

(O/V).

This relationship states that these variances can berelated to the variance of point grades within adeposit (

2

(O/D))

(11)

In our application, the block represents a well, thepoint grades will be core samples within each well,and the deposit will correspond to the reservoir. Withthis analogy, Kriges relationship will be

(12)

Figure 6 illustrates how each variance behaves withxD increasing from 0.1 to 100. When

2 equals 1, anexponential model is used and zD = 0.8. The sum ofs

2

areal and s2

vert is constant and equals the value expectedfrom Kriges relationship for all values of xD. Theagreement between equation 12 and the numericalresponse is very good. The good agreement betweenanalytical (equation 11) and numerical (equation 12)results in Figure 6 indicates that the dependence of ourresults on sample size is small. Equation 11 applies foressentially infinite sampling, whereas the terms in

equation 12 must be estimated from finite samples.The chart in Figure 6 reveals, also, how both s2

areal ands

2

vertbehave with increasing xD. As described, for smallxD, s2

areal reaches a maximum and s2

vert a minimum. AsxDbecomes greater than 1, s

2

areal declines, while s2

vert

rises. For very large values of xD, the horizontal auto-correlation is so large that s

2

areal tends to zero, while s2

vert

approaches the population variance (2).Kriges equation also provides an alternative way to

develop the relationships expressed in the charts ofFigures 35. Knudsen and Kim (1978) showed that,

s sareal vert2 2 2+ =

V D O D O V ( ) ( ) ( )

= 2 2 2

using the definition of the autocovariogram, cov(h),the areal variance could be calculated by

(13)

where h stands for the distance between any two

points in the volume V, and the integrals represent anintegration over a volume; however, the amount ofeffort to be spent in these integrations is excessive,which motivated us to adopt the numerical approachusing MDM.

Discussion

Here, we discuss the results and interpret somefeatures of the type charts. The differences in resultsamong the three semivariogram models are relatedwith the degree of autocorrelation that each modelincorporates. As s

2

areal/s2

vert represents a ratio betweentwo variances, the effect of the autocorrelation

model will depend on which of the variances con-trols the final result. One way to illustrate this effectis by plotting the dependence of

2

areal/2

vert on theHurst coefficient for the truncated fractal model.This parameter represents the degree of autocorrelationamong data.

Figure 7 shows how s2

areal/s2

vert can vary withHandxD for zD = 1. For instance, considering a reservoirstrongly autocorrelated in the horizontal direction (xD= 100), the larger the H, the smaller will be s

2

areal/s2

vert.When dealing with smaller values of xD, however, thebehavior of the vertical variance will control the ratios

2

areal/s2

vert; therefore, as Hincreases, s2

areal/s2

vert alsoincreases.

The charts in Figures 35 also show that s2

areal/s2

vertincreases when zD is greater than 0.1; however, in thehorizontal direction, s

2

areal/s2

vert decreases very slowlywith the vertical autocorrelation until xD reaches 1.0.This behavior is the same for all three semivariogrammodels and shows that a field with xD = 1 will behavesimilarly to an uncorrelated field in this direction. Thisresult is a consequence of the fact that s

2

areal dependsstrongly on horizontal autocorrelation only for rangesgreater than the interwell spacing. Ranges smaller than1 have only a slight effect on s

2

areal. This observation also

V D

vvVh dvdv( )= ( )

22

1cov

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.1 1 10 100


Variance

Numerical Result

Analytical Solution

sareal

2svert

2

+

svert

2

sareal

2

Figure 6. Analytical valida-

tion (exponential model, zD= 0.8, and 50 realizations).


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illustrates that, as expected, it is impossible to estimateranges smaller than the well spacing (xD < 1).

One limitation of this method is the variability ofs

2

areal/s2

vert among the 50 realizations. Figure 8 expresseshow the results for zD = 0.1 and 0.5, obtained from the

exponential model, vary with xD. The figure shows theaverage of 50 realizations and also curves representingplus or minus one standard deviation about the mean.The standard deviation band shrinks slightly when lat-eral autocorrelation (as expressed by zD) is large. Theband is considerably smaller at zD = 0.1 than at zD =0.5. A fact that is equally important because the curvesbecome more horizontal for smaller zD is that the errorat a fixed variance ratio becomes quite large when lat-eral autocorrelation is small.

APPLICATION

We chose data from a particular field to demon-strate how the procedure works. The results representan illustration of the method rather than a comprehen-sive analysis of the reservoir. A complete analysisrequires additional effort and integration among geol-ogists and reservoir engineers to interpret the resultsin the light of all the knowledge about the field.

We applied the procedure to the El Mar field,located in the Delaware basin of west Texas and New

Mexico (Figure 9). This unit is currently operated byBurlington Resources Company.

The El Mar

This field is located 30 km north of Mentone, Texas.The entire field covers approximately 40 km2, with two-thirds of the field located in the western portion of Lov-ing County, Texas, and the remainder in southwest LeaCounty, New Mexico. In this study, we investigatedonly the data from the El Mar (Delaware) unit, whichcovers an area of 5 8 km and contains 175 wells.

The primary producing layer is the Ramsey sand,which is composed of an upper Asand and a lowerBsand separated by shale laminae. These sandstoneswere deposited in deep water, probably by submarine-fan complexes formed by turbidity-current depositionduring lowstands of sea level. Dutton et al. (1996) dis-cussed this and other alternative models of Delaware

sandstone deposition. For the purposes of this paper,the most important detail in Figure 9 is that the sourceof the turbiditic sands lies approximately north-north-east of the El Mar unit. The Ramsey formation lies at anaverage producing depth of 1500 m, and the forma-tions weighted average porosity and permeability are,respectively, 23% and 22 md. A description of otherpetrophysical properties of the Delaware formation canbe found in Jenkins (1961). The difficulty in estimating

0. 0

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0. 7

0. 0 0 0. 0 5 0 . 10 0 . 15 0. 2 0 0. 2 5 0. 3 0 0 . 35 0 . 4

Hurst Coefficient

ArealVariance/VerticalVariance

1 0

1.0

100

xD = ( )z D = 1Fractal ModelFigure 7. Estimation ofs2

areal/s2

vert sensitivity to theHurst coefficient for differenttruncated ranges.

Figure 8. Variability(expressed as 1 standarddeviation) of s

2

areal/s2

vert

among the 50 realizations.

0. 0

0. 4

0. 8

1. 2

1. 6

2. 0

0. 1 1 10



0. 1

+

+

Exponential Model

zD

0. 5


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the water saturation caused a large number of coredwells to be taken. This made it possible to find severalcored wells aligned in the orthogonal directions of thefield, leaving it in a favorable condition to perform thepresent analysis. We selected six wells aligned in thenorth-south direction and two more in the east-westdirection, for a total of three east-west wells.

We selected the data from the eight wells and builta vertical semivariogram of Yfor each. Because the

core sampled unit A indiscriminately, it is highlylikely that the data are from multiple turbidite facies.Most of the data analyzed were from unit A because ofthe predominance of this unit in this part of the field.Our approach was to separate the data from both unitsbecause they seem to have different petrophysicalproperties. We performed the analysis only for Ram-sey A because the amount of data for Ramsey B wasless; only two wells, 1814 and 1824, have data thatinclude core samples from both units A and B.

The goal here was simply a practical demonstrationof the method; therefore, we assumed that there were no

problems concerning data acquisition, and the furtheranalysis was based on the semivariograms constructedwith primary data. We also did not consider the possi-bility of nugget effects in the semivariogram fitting.

The procedure outlined was performed to calcu-late the average values and the s

2

areal and s2

vert for eachset of data. To calculate zD, the experimental semi-variograms must be fitted by a theoretical model. Thedifficulties and the importance of this step are well

described in the literature by several authors, includ-ing Journel and Huijbregts (1978), Isaaks and Srivas-tava (1989), and Olea (1994).

Estimating xDWe tested three different semivariogram models,

trying to find the one that best described the experi-mental data.

The spherical semivariogram fit was consideredpoor. The second model tested was the truncated fBmfractal or power-law model. For the fitting, a value of

Figure 9. Map of the Delaware basin (from Dutton et al., 1996).


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H= 0.15 gave a better adjustment than the value ofH= 0.25; therefore, a new set of curves for the fractalmodel, analogous to those in Figure 5, was generatedfor H= 0.15. The final result was better than thespherical fitting for most wells. The exponentialmodel gave a fit close to that obtained with the frac-tal for most wells, and reproduced a similar responsefor the ones in which the spherical model worked

better. The results are summarized in Table 1 for theeast-west section and in Table 2 for the north-southsection.

Estimating zDUsing the zD values given in Table 1 for the appro-

priate model and the ratio s2

areal/s2

vert for each direction,we enter the charts and interpolate s

2

areal/s2

vert as shownin Figure 10. This represents the autocorrelation in thex or lateral direction, the autocorrelation beingexpressed through an exponential semivariogram

model. The results shown in Table 3 for the spherical,exponential, and fractal are, respectively, zD|EW= 1.2,1, and 3 and xD|NS = 4, 2, and 6.

The results show anisotropy in the autocorrelationpattern of the El Mar field, at least along the directionsindicated. The permeability range in the north-southdirection is at least twice that in the east-west direc-tion. This behavior roughly coincides with the deposi-

tional characteristic of the field because the major axisof the turbidite channel is aligned with the northeast-southwest direction (Figure 9).

A less rigorous approach would be to group thedata from both units A and B together, instead of per-forming the analysis only for unit A. Additional datafrom two wells (1814 and 1824) must be considered.Because the wells in the east-west section do not con-tain the Ramsey B sand, only results from the north-south section will change. The main difference is thatthe value of s

2

areal changes from 0.23 to 0.36, making

Table 1. Summary of Results for the East-West Cross Section of the El Mar Field

Wells1514 1513 1512 Average

Average of ln(k) in vertical direction 1.15 2.25 2.17 1.86Variance of ln(k) in vertical direction 1.96 2.44 2.81 2.40Vertical correlation range with spherical model 0.21 0.18 0.17 0.19Vertical correlation length with exponential model 0.09 0.07 0.08 0.08Vertical truncated upper cutoff with fractal model (H= 0.15) 0.35 0.26 0.34 0.32

Table 2. Summary of Results for the North-South Cross Section of the El Mar Field

Wells

1514 1524 1534 1532 1814 1824 Average

Average of ln(k) in vertical direction 1.15 1.89 2.37 2.40 1.85 1.58 1.87Variance of ln(k) in vertical direction 1.96 1.47 0.46 2.09 4.00 1.71 1.95Vertical correlation range with spherical model 0.21 0.20 0.23 0.20 0.22 0.24 0.22Vertical correlation length with exponential model 0.09 0.08 0.12 0.09 0.11 0.11 0.10Vertical truncated upper cutoff with fractal model (H= 0.15) 0.35 0.28 0.30 0.34 0.42 0.30 0.33

0.00

0.05

0.10

0.15

0.20

0.25

0.1 1 10 100

Exponential Model


zD =0.1


2

East-West

North-South

xD

=

0.08

Figure 10. Determining xDfor the north-south and theeast-west sections of the ElMar field.


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s2

areal/s2

vert = 0.18 instead of 0.12. This change leads to theconclusion that the ranges are similar in both direc-tions, in conflict with the depositional pattern. Thisillustrates the importance of prior geological analysisof the data to be used in making the estimation.

Final Remarks

The autocorrelation analysis of the El Mar field showsthat results obtained by using different semivariogrammodels can be different. These results were expectedbecause each model expresses the autocorrelation in a

slightly different manner. For example, the sphericalmodel (Figure 3) cannot be used to estimate zD valuesof less than 1, meaning that the spherical model containsno information about autocorrelation if the range is lessthan the well spacing. The exponential model, however,is discriminating to zD values less than the well spac-ing (Figure 4). Although this seems a little strange atfirst, it may not have a great effect on a simulated flow

response. Indeed, Fogg et al. (1991) referred to a study inwhich two ranges of 120 and 390 m caused significantlydifferent permeability patterns, but practically identicalcumulative oil recoveries and water-oil rates, whenthese values were used in simulation.

When calculating the data to construct the charts inFigures 35, we also derived a chart to correct the ver-tical variance measured at the wells to the value thatshould be used in conditional simulation studies.Recall that the vertical variance measured at the wellswould be equal to the population variance only whenthere was no autocorrelation. Figure 11 shows hows

2

areal/2 varies with zD and xD. The more verticallyautocorrelated the permeability field is, the more the

s2

vert is different from 2.Figure 12 shows an analogous result, but it is

expressed as the Dykstra-Parsons coefficient instead ofthe variance. To express the variance in terms of theVDP, the chart is no longer general and depends on thepopulation VDP. Figure 12 considered a VDPpop of 0.9.

Table 3. Summary of the Results fromAutocorrelation Analysis of the El Mar Field

Summary North-South East-West

Areal variance 0.23 0.37VDP areal 0.46 0.46Vertical variance 1.95 2.40

VDP vertical 0.75 0.79Ratio areal/vertical variances 0.12 0.15xD (spherical model) 4 1.2xD (exponential model) 2 1xD (fractal model) 6 3

0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 1 0 100


Ratio

ofVertica

lto

Population

Variance

Exponential Model

2= 1

zD= 0.1

0 .3

0 .5

0 .8

0.7

0.8

0.9

0.1 1 10 100


VerticalDykstra-P

arsons

Coefficient

Exponential Model

zD=0.1

0.3

0.8

0.5

VDPpop. = 0.9

Figure 11. Differencesbetween the vertical (sample)and population variancecaused by autocorrelation.

Figure 12. Differencesbetween the vertical (sample)and populationV

DPcaused

by autocorrelation.


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These charts can be used to correct data for inputinto stochastic simulation. For instance, from Figure 12we can conclude that a sample VDP of 0.85 (s

2

vert = 3.6)calculated from a reservoir with zD = 0.3 and xD = 1.0,given by the exponential model, will have a popula-tion VDPpop = 0.9 (

2 = 5.3).

SUMMARY AND CONCLUSIONSWe developed a set of charts for estimating autocor-

relation in the horizontal direction based on the ratio ofareal-to-vertical variances, the vertical autocorrelation,and the type of semivariogram model used.

The procedure was partially validated through acomparison of results; analytical solutions wereavailable for some limiting situations. A field casewas analyzed to demonstrate how the method couldbe applied. Data from eight wells in the El Mar field,Ramsey A sand, were used to estimate horizontalautocorrelation for two directions of the field.

Results from the El Mar analysis point out the

anisotropy in the permeability distribution of this field.The autocorrelation ranges in the vertical directionwere found to be between 0.1 and 0.3 of the sampledinterval, depending on the type of model used. Thehorizontal range in the north-south direction rangedbetween two and six t imes the mean well spacing,being at least twice the autocorrelation in the east-westdirection. These results can be used in conditional sim-ulations to generate the expected permeability patternof this field.

The present approach seems to be an effective toolto be used, along with the geological knowledge of thefacies continuity, for estimating the autocorrelation forthe interwell region between vertical wells.

NOMENCLATURE

a = semivariogram rangecov = covarianceD = mining depositE = expectation valueh = distance lagH = Hurst coefficientk = permeabilityNl = number of layersNr = number of realizationsNw = number of wells

O = point gradess2 = estimation of the varianceV = volume of a blockVDP = Dykstra-Parsons coefficientvar = varianceY = natural logarithmic of the permeability = autocorrelation range (general) = range (spherical model) = correlation length (exponential model) = truncated upper cutoff (fractal model) = population mean

= population standard deviation2 = population variance2areal = areal variance2vert = vertical variance

Subscripts

D = dimensionless variable

O/D = point grades within a mining depositO/V = point grades within a blockpop = population valuesV/D = blocks within a mining depositx,y,z = Cartesian coordinate directions

ACKNOWLEDGMENTS

The authors acknowledge the Enhanced Oil Recov-ery Research Program of the Center for Petroleum andGeosystems Engineering at The University of Texas atAustin for partial support of this work. Jorge Pizarrothanks Petrobras S.A. for financial support during hisstay at The University of Texas. Larry W. Lake holds

the W.A. (Monty) Moncrief Centennial EndowedChair. We thank I.H. Silberberg for his editorial com-ments and John Barnes with Burlington Resources forsupplying the core data on the El Mar (Delaware) unit.

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A Simple Method to Estimate Interwell Autocorrelation

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