Flow and Transport in Porous Media and Fractured Rock (From Classical Methods to Modern Approaches)...

109

5Characterization of Field-Scale Porous Media:Geostatistical Concepts and Self-Affine Distributions

Introduction

In Chapter 4, we described various techniques for characterization of laboratory-scale (LS) porous media and interpreting the data on their properties. However,although characterization of the LS porous media provides valuable informationabout and insight into their morphology, it is not by itself nearly enough for de-veloping a realistic model of field-scale (FS) porous media, even if many coreplugsfrom such porous media are characterized. This is due to the fact that the FS porousmedia are heterogeneous over multiple and distinct length scales. However, charac-terization of the LS porous media provides information about the morphology overonly the two smallest length scales, namely, the pore and the sample size, but notlarger scales.

The characterization of a porous medium at length scales larger than the LS re-quires, in addition to data, use of special techniques and tools, and is collectivelyknown as geostatistics. The purpose of this chapter is to describe the most importantconcepts and ideas of modern geostatistics. Advances in the measurement and es-timation techniques, considerable progress in the development of theoretical andcomputational methods, and modern geostatistical techniques have made it possi-ble to develop highly-resolved models of the FS porous media. Such resolved mod-els, which usually contain a considerable amount of data on the spatial variationsof the porosity, permeability, and other important properties, are known as the ge-

ological models. They typically consist of a three-dimensional (3D) (or sometimes2D) computational grid with a few million grid blocks. The blocks’ linear dimen-sions are on the order of several (or more) meters in the vertical direction andtens of meters areally, with their effective permeabilities and porosities distributedthroughout according to the data and what the geostatistical analysis (see below)dictate. The grid blocks and their associated properties represent an intermediatescale between the LS and field scale.

However, due to computational limitations (mainly computation time), it is verydifficult to carry out simulation of fluid flow and transport using the highly-resolvedgeological models. It is, therefore, necessary to upscale the properties of the gridblocks in the geological models in order to develop coarsened grids that not onlycan they be used in computer simulations with an affordable amount of compu-

Flow and Transport in Porous Media and Fractured Rock, Second Edition. Muhammad Sahimi.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

110 5 Characterization of Field-Scale Porous Media: Geostatistical Concepts and Self-Affine Distributions

tation time, but also yield accurate results. The grid blocks in the upscaled gridsrepresent an intermediate step between the geological model and the field scale.Although, this is, in a sense, an artificial length scale introduced by computer sim-ulators, it is a very important practical length scale. Many techniques for upscalingthe geological models have been developed (see, for example, Rasaei and Sahimi,2008, 2009a,b; Sahimi et al., 2010).

To characterize and model the FS porous media, one must first analyze the avail-able data. We divide the data into two groups:

1. In one group are what we call the direct data. Two important properties of theFS porous media, namely, the spatial distributions of their porosity and per-meability, belong to this group. The spatial distribution of the porosity is mea-sured along, for example, wells during drilling or estimated relatively accuratelyby indirect methods, such as, measuring the resistivity of many samples. Thepermeability distribution can either be estimated by in-situ nuclear magneticresonance (NMR) (Mair et al., 1999; see Chapter 9), or by coring and laboratorymeasurements.Other types of logs, for example, the gamma ray and temperature logs, alsobelong to this class of data. If a porous medium is also fractured, the frac-tures’ permeability is usually estimated by relating the fracture aperture to thefracture excess conductivity measured on electrical image logs (Sibbit, 1995),critically-stressed fractures within the present-day stress field, or both. Howev-er, such methods yield, at best, a relative estimate of the fracture permeabilitythat must be calibrated against dynamic (flow) data. In any event, provided thatenough amounts of the direct data are available, they are combined with thegeostatistical methods (see below) to generate accurate spatial distributions ofthe porosity and permeability throughout a FS porous medium, and used inthe generation of the geological model.

2. In the second group are what we call the indirect data, the most important ofwhich are seismic recordings. Such data, when combined with a geostatisticalmethod, provide information about the porosity distribution. If a FS porousmedium is fractured, then, since the fractures are usually below the limit ofseismic resolution, they cannot be directly detected by seismic experiments.As a result, the static models of fractures and fracture networks are mainlyconstrained by the direct data. Seismic data do, however, yield insight into thelarge-scale structure of geological formations, such as, their stratification andfaults at the largest length scales. The static models include the mechanicalorigin (shear versus joint), the geometry (orientation, size, and frequency), andtopology (open or connected versus isolated or cemented) of the fractures andtheir network. These aspects will be described in Chapter 6. As described there,seismic data can also be combined with the well-log data for generating moreaccurate models of a FS porous medium.

Sometimes, it may be very difficult to obtain meaningful direct data for some FSporous media. Dehghani et al. (1999) provided an excellent example of such a situ-

5.1 Estimators of a Population of Data 111

ation by describing their work for a carbonate porous formation, though the prob-lem is general. The direct data are more useful if, in addition to being associatedwith specific depths and wells (hence, the region in which they are collected), theyare also associated with a certain formation type or facies. In addition, despite yearsof study, identification of the spatial distribution of fractures and faults remains alargely unsolved problem. Until very recently, it was also believed that porosity alone

cannot provide much insight into the whereabouts of the fractures. The permeabil-ity of fractures is much larger than that of the porous matrix in which they areembedded and, thus, data for the permeability provide insight into the spatial dis-tribution of fractures. There is, however, an insufficient volume of the permeabilitydata. Borehole-wall imaging is a very reliable method of mapping the intersectionsof the fractures and faults with the wells. Such imaging techniques are, however,expensive and are not always included in a logging run. In addition, they are notavailable for many of the older oil reservoirs and other FS porous formations.

Before starting the description and discussion of the various geostatistical con-cepts, let us point out that there are several excellent books on the principles ofgeostatistics, ranging from the pioneering books of Matheron (1967b) and Journeland Huijbregts (1978), to the more recent ones by Isaaks and Srivastava (1989),Cressie (1991), Wackernagel (1995), and Jensen et al. (2000). The book by Deutschand Journel (1998) contains some very useful computer programs for use in prac-tice. Thus, the discussions in the present chapter are not, and do not have to be,exhaustive. We provide a useful and understandable summary of the most impor-tant geostatistical concepts and techniques used in practice and invoked in the restof this book. It is assumed that the reader is familiar with the basic concepts of theprobability theory, although we recall the theory’s necessary elements whenevernecessary.

5.1Estimators of a Population of Data

Given a set or population of data points, how does one evaluate the property ofthe porous formation for which the population was obtained? If we had the prob-ability distribution function (PDF) of the population, then, using well-known lawsof probability theory, we could compute any quantity that we wish. However, forany given population or sample with a limited number of data points, the task isto extract as much information and insight as possible from the population with-out necessarily knowing the PDF. This task is commonly referred to as estimation,and any method for computing a characteristic quantity for the data in the popula-tion, for example, the arithmetic or geometric average of the property for which thepopulation is available, is called an estimator. As described by Jensen et al. (2000),estimation involves four essential ingredients that are as follows.

1. The characterizing statistic e to be evaluated, such as, various averages or thestandard deviation.


2. The estimator E .3. The estimate Oe.4. The standard error or confidence interval of Oe that enables one to evaluate the

accuracy of Oe. The standard error is σ( Oe), where σ2 is the variance.

How many parameters should one estimate? The number depends on the PDF andwhat one intends to do with the statistics. For example, a Gaussian distribution ischaracterized by two parameters, its mean and the variance. The next question is,what are the properties of a good estimator? In general, an accurate estimator musthave five properties: (1) small bias; (2) good efficiency; (3) robustness; (4) consis-tency, and (5) the ability to yield physically-meaningful estimates. An estimator’sbias b is defined by

b D E( Oe) � e , (5.1)

where E denotes the expected value, or the expectation, of Oe.1) In an ideal situation,b D 0, but, as usual, any practical situation is far from ideal and, therefore, b ¤ 0.The two sources that give rise to a nonzero b are the precision of the measurementsand the sampling.

The efficiency of an estimator is measured in terms of the variability of the es-timates that it provides. If the estimator’s variance is small, it is called an efficient

or a precise estimator. Note that the sample’s variance is not the same as the esti-mator’s. Thus, an efficient estimator has a small σ( Oe). It is very common (Jensen et

al., 2000) to have σ2( Oe) / N�1, where N is the number of data points. At the sametime, σ2( Oe) / e2 is quite reasonable.

The robustness of an estimator is measured in terms of its insensitivity to errorsin a small proportion of the data set. Hoaglin et al. (1983) and Barnett and Lewis(1984) provide detailed discussions of how to develop quantitative measures of ro-bustness. The so-called L-moments (Hosking and Wallis, 1997; Jensen et al., 2000)are used to measure the robustness of estimators. We define the moments

Mn D1Z

0

x F n dF , (5.2)

where F is the cumulative distribution function of the stochastic variable X (i.e., theintegral of its PDF). Then, the first four L-moments are given by L1 D M0 (which isjust the mean value of X); L2 D 2M1 � M0 (which is similar to the variance, exceptthat it attributes smaller weights to the extreme values of X); L3 D 6M2 �6M1 CM0

(which is a measure of the skewness of the distribution), and L4 D 20M3 �30M2 C12M1 � M0 (which is a measure of peakedness of the distribution). Note that theintegration in Eq. (5.2) is from zero to one because it is over all the probabilities,rather than all values of x. Because such moments attribute smaller weights tothe extreme values than the usual moments of the PDF, they can be accurately

1) Recall that if f (x ) is the PDF of the random variable X, E(X ) is defined by, E(X ) D R1

�1 x f (x )dx ,

while for a discrete variable X, E(X ) D PNnD1 pi Xi , where pi is the probability that X D Xi .

5.2 Heterogeneity of a Field-Scale Porous Medium 113

estimated using much smaller data bases than would otherwise be needed, whichis why they are valuable.

In general, quantitative assessment of an estimator robustness is difficult. If theerrors in the data are small, and if we have an analytical expression for the esti-mator E , then, based on the relation, Oe D E (X1, X2, . . . , XN ), the sensitivities arecalculated using a truncated Taylor series of E about the true values Xi ; that is,∆ Oe i D (@E/@Xi)∆ Xi , where ∆ Xi is the error in the measurement of the ith da-tum, assuming (implicitly) that the errors ∆ Xi are independent. The total erroris then the sum of the individual errors,

Pi ∆ Oe i . A similar analysis can be done

based on the variances, σ2(Xi).If the arithmetic average NX of a sample is the estimator, then its properties can

be analyzed. According to the central-limit theorem, for large N, the size of thesample or population (number of the data points) NX is normally distributed withthe mean and variance of Xi , regardless of the PDF of X. If X is itself normallydistributed, then NX is exactly normally distributed. The sample median, NX0.5, canbe another estimator that is more robust than the sample mean NX , but for whichthe central-limit theorem cannot be used for obtaining its distribution.

Note that the variability of NX decreases as N, the number of data points, increas-es. Also, note that if the Xi are distributed according to skewed distributions, suchas the log-normal distribution which give larger weights to the very small or verylarge values, then the sample mean is inefficient and inaccurate. For example, sincein many cases it is assumed that the permeabilities in a FS porous medium are dis-tributed according to a log-normal distribution, their arithmetic average is a poorestimate of the effective permeability of the porous formation.

Now, consider the sample variance s2 as the estimator (we use s2 to distinguish itfrom σ2, the variance of X). If X is distributed according to a Gaussian distributionwith mean µ and variance σ2, then, σ2

s (Os2) D σ2/N , where Os2 D s2/(N � 1). Moregenerally, regardless of the PDF of X, σ2

s (Os2) D fE [(X � µ)4] � σ2(X )g/[4(N � 1)σ2].

5.2Heterogeneity of a Field-Scale Porous Medium

Thus far in this book, we have repeatedly mentioned heterogeneous porous media,by which we have meant variations in the shapes, sizes, and connectivity of thepores. However, this type of heterogeneity is more suitable for describing the dis-order in the LS porous media. In case of the FS porous media, the heterogeneity isusually meant to be the spatial variations in the formation properties that affect fluid

flow at large length scales. The permeability variation is, of course, one key factorthat affects fluid flow and, therefore, the heterogeneity of the FS porous media maybe defined in terms of such variations. In general, however, one may define a het-erogeneity index in a variety of ways based either on a static property of a FS porousmedium or a dynamic one. Detailed discussions of both classes of heterogeneity arepresented by, among others, Jensen et al. (2000).


Static measures of the heterogeneity are based on the data taken from samplesof porous media. Then, a model of fluid flow in the samples helps quantifying theeffect on fluid flow of the variations (heterogeneity) in the property values. Thereare at least four static measures of the heterogeneity, and what follows is a briefdescription of each.

5.2.1The Dykstra–Parsons Heterogeneity Index

The index proposed by Dykstra and Parsons Dykstra and Parsons (1950) is perhapsthe most widely used measure of heterogeneity, and is defined by

HDP D 1 � K0.16

K0.5. (5.3)

Here, K0.5 is the median permeability, while K0.16 is the permeability at one stan-dard deviation below K0.5 in a log-probability plot. That is, K0.5 and K0.16 are thepermeabilities if, in a plot of log K versus the probability scale, the probabilitiesare 0.5 and 0.16. For a completely uniform porous medium, HDP D 0, whereasfor an “infinitely” heterogeneous porous formation, HDP D 1. The latter type ofporous formation is a hypothetical one with one layer of infinite permeability andnonzero thickness.

Given a set of permeability data, one orders them in increasing values. Witheach data point is associated a probability that represents the length of the intervalrepresented by the point. Although it was originally suggested that K0.5 and K0.16

should be obtained from the straight line that represents the best fit of the data ina log-probability plot, Jensen and Currie (1990) suggested to use the data directly.If ln K is normally distributed with a standard deviations σK, then Eq. (5.3) yields

HDP D 1 � exp(�σK) . (5.4)

In addition, it is important to recognize that HDP (or any other heterogeneity in-dex computed based on the permeability) varies significantly when it is computedbased on vertical well data or areally, both of which are distributed normally, evenif the permeabilities themselves are not.

If HDP is estimated from vertical wells, one finds that 0.65 � HDP � 0.99, butif it is estimated areally from arithmetically-averaged well permeabilities, one has0.12 � HDP � 0.93 (Lambert, 1981). Both estimates are distributed according toa Gaussian distribution, although most of the permeabilities are not distributedaccording to such a distribution. If one assumes that ln K is distributed accordingto a normal distribution, then the bias and standard error of HDP can be derivedanalytically (Jensen and Lake, 1988):

bDP D �0.749

�ln(1 � HDP)

�2 (1 � HDP)N

, (5.5)

5.2 Heterogeneity of a Field-Scale Porous Medium 115

σDP D �1.49(1 � HDP) ln(1 � HDP)p

N. (5.6)

Note that bDP < 0, which means that the Dykstra–Parsons index always underesti-

mates the heterogeneity, but the bias is typically small. Moreover, bDP / 1/N andattains its maximum for HDP ' 0.87.

However, HDP does have several weaknesses and, therefore, cannot be consid-ered as a universal measure of the heterogeneity. Lambert (1981) provided evidencethat HDP cannot differentiate between formation types. When oil recovery fromlaboratory-scale waterfloods was correlated with HDP (Lake and Jensen, 1991), itwas observed that the models are not sensitive to the heterogeneity if HDP < 0.5,but for broadly-distributed heterogeneities, the sensitivity was significant.

5.2.2The Lorenz Heterogeneity Index

A heterogeneity index more accurate than the DP index is the so-called Lorenzindex HL, which is computed based on the data for the porosity and permeability.In this method, the data are first ordered according to decreasing values of K/φ.One then computes the fraction of the total flow capacity defined by

FM D

MPmD1

Km hm

NPnD1

Kn hn

, (5.7)

and the fraction of total storage capacity defined by

CM D

MPmD1

φm hm

NPnD1

φn hn

, (5.8)

where N is the number of data points, hm the thickness of the layer m in which theporosity and permeability are φm and Km , and 1 � M � N . A plot of FM versusCM is then prepared (which should pass through the origin and the point (1, 1)).The area A between the curve and the diagonal straight line [passing through (0,0)and (1,1)] is then computed numerically. The Lorenz heterogeneity index HL isthen given by HL D 2A.

Similar to HDP, the Lorenz index is also zero for completely homogeneousporous media and approaches one for infinitely heterogeneous porous formations.Typical values of HL for field-measured values of the permeability and porosity are0.6 � HL � 0.9. If ln K is distributed according to a Gaussian distribution with astandard deviation σK, then the Lorenz and DP indeces are related:

HL D exp�

12

σK

�D erf

�� 1

2ln(1 � HDP)

�, (5.9)


where erf denotes the error function. In general, it can be shown that the errors inthe estimates of HL are smaller than those in HDP.

Although HL is somewhat more difficult to compute than HDP, it does haveseveral advantages over HDP that are as follows (Jensen et al., 2000).

1. It can be estimated accurately for any distribution.2. It does not rely on the best fits of the data, as does HDP.3. It depends on both the porosity φ and layer thickness h.4. If a porous medium consists of N uniformly stratified elements between wells

through which a fluid is flowing, then FM represents the fraction of the totalflow passing through a fraction CM of the medium’s volume. Therefore, in thisparticular case, the Lorenz coefficient has a clear physical interpretation. Otheruseful features of HL are described by Jensen et al. (2000).

The bias bL in estimating the true value of HL can be quite pronounced, particularlyif HL is large and the number of data points is relatively small. Similar to HDP, theLorenz index underestimates the heterogeneity. The standard error of HL is alsousually smaller than that of HDP.

5.2.3The Index of Variation

Another popular measure of the heterogeneity is the so-called index of variation,Hv, defined by

Hv D σK

E(K ), (5.10)

where E(K ) is the expected value of the permeability. In most cases, the mean andstandard deviations of the permeability data from different sources or populationsvary in the same direction so that Hv remains fairly constant. Hence, if two sampledata or populations are characterized by two very different values of Hv, the im-plication is that there must be fundamental structural differences between the twosamples.

As discussed by several authors (see, for example, Hald, 1952; Koopmans et al.,1964; Jensen et al., 2000), it is difficult to analyze the statistical properties of the esti-mators of Hv. However, if the permeability is distributed according to a log-normaldistribution, then an estimate of Hv can be derived. Suppose that KA and KH rep-resent, respectively, the arithmetic and harmonic averages of the permeability data.Then, an estimate of Hv is given by (Johnson and Kotz, 1970)

OHv D�

KA

KH� 1

� 12

. (5.11)

The effect of Hv on unstable miscible displacements has been extensively stud-ied, usually based on the assumption that Hv is a good measure of the heterogene-ity. Hv is also useful when one compares variabilities of different facies (Jensen et

5.3 Correlation Functions 117

al., 2000). Studies by Kittridge et al. (1990) and Goggin et al. (1992) indicated thatif one compares geologically-similar elements in outcrop and subsurface samples,the value of Hv remains largely unchanged even though there are large changes inthe average permeabilities of the two distinct samples.

5.2.4The Gelhar–Axness Heterogeneity Index

Gelhar and Axness (1983) introduced a heterogeneity index defined by

HGA D σ2ln K λD , (5.12)

where σ2ln K

is the variance of the distribution of ln K , and λD is the autocorrelationlength in the macroscopic direction of fluid flow. Physically, λD represents a lengthscale over which similar values of the permeability exist (see also below). In otherwords, λD is a correlation length for positive correlation between the permeabilityvalues. It is precisely due to λD that the Gelhar–Axness index is, in many ways,superior to the three heterogeneity indeces already defined. That is, whereas HDP,HL and Hv ignore such correlations, HGA does not. Indeed, studies by Kempers(1990) and Waggoner et al. (1992) suggested that HGA is a much better indicator offlow performance – and the effect of the heterogeneity on it – than the widely usedHDP; see also the discussion by Sorbie et al. (1994).

5.2.5The Koval Heterogeneity Index

The Koval index HK (Koval, 1963) is an empirical way of quantifying the effect ofthe heterogeneity on miscible displacements. It is a dynamic index and is definedby

HK D t�1BT , (5.13)

where tBT is the dimensionless time for the breakthrough – that is, when the dis-placing fluid first arrives at the outlet – in a miscible displacement in which themobility ratio is one (see Chapter 13 for the description of miscible displacements).The definition of HK also implies that the maximum oil recovery is obtained whenHK pore volumes of the displacing fluid has been injected. Thus, large values ofHK are detrimental to an efficient miscible displacement. The advantage of HK

over HDP and HL is that it is a more linear measure of the performance of a misci-ble displacement.

5.3Correlation Functions

The next important subject is the correlations between various properties of a FSporous medium. Many of such correlations are empirical, as there are no rigorous


theoretical foundations that relate one property, say the permeability, to anotherproperty such as the porosity. Thus, although important, we do not consider themin this chapter.

However, in order to generate a realistic model of a FS porous medium, onemust understand autocorrelation, that is, the correlation of a variable with itself. Tobetter understand this point, suppose that we are given the porosity logs along sev-eral widely-separated wells in an oil reservoir. To develop a model of the reservoir,we must be able to estimate the porosity in the interwell zones for which thereare no data. If the porosities were totally random (no correlation between them),then there would be no hope for developing any accurate model of the reservoir.The porosities are not, however, random, but are in fact highly correlated. Thistype of correlation is what is usually referred to as the autocorrelation, that is, thecorrelation between properties of the same sample space (in this example, the oilreservoir), but at different locations (between the porosities along the wells and thosein the interwell zones). Understanding such autocorrelations and being able to con-struct accurate models for them is an important task in any study of a FS porousmedium.

5.3.1Autocovariance

Suppose that Z(x) represents a property measured at x. In practice, the measure-ments are done at discrete points x1, x2, . . ., so that Zi D Z(xi) represents themeasured property at xi . The autocovariance R(Zi , Z j ) between data Zi and Z j isdefined by

R(Zi , Z j ) D E˚[Zi � E(Zi)][Z j � E(Z j )]

� D E(Zi Z j ) � E(Zi )E(Z j ) ,

(5.14)

where, as usual, E(x ) denotes the expected value of x. R(Zi , Z j ) describes the cor-relation between Zi and Z j as well as the effect of the heterogeneity. In practice,it is difficult to separate the two effects. Note that, R(Zi , Z j ) D R(Z j , Zi ) andR(Zi , Zi ) D σ2(Zi ), where σ2 is the variance. Given a set of data, an estimate OR ofthe true autocovariance is given by

OR(Zi , ZiCm) D 1N � m

N�mXiD1

(Zi � NZ )(ZiCm � NZ ) , (5.15)

where NZ is an estimate for the mean value of the variable Z.

5.3.2Autocorrelation

The autocorrelation coefficient CR(Zi , Z j ) is defined by

CR(Zi , Z j ) D R(Zi , Z j )σ(Zi)σ(Z j )

. (5.16)

5.3 Correlation Functions 119

Clearly, �1 � CR � C1. The extent of the autocorrelation is sometimes quantifiedby an integral scale �I defined by

�I D1Z0

CR(r)dr , (5.17)

which is defined for any autocorrelation with a finite correlation length. Note thatif the random variable Zi is independent, then, the variance σ2( NZ) of the samplemean is simply

σ2( NZ ) D σ2(Zi )N

. (5.18)

If, however, the random variable is dependent, then

σ2( NZ ) D 1N 2

NXiD1

NXj D1

R(Zi , Z j ) D σ2(Zi )N

241 C 2

N

NXiD1

NXj D1

CR(Zi , Z j )

35 .

(5.19)

For detailed analysis and discussions of cross-correlations between various types ofdata see Dashtian et al. (2011b).

5.3.3Semivariance and Semivariogram

The semivariance is defined by

γ (Zi , Z j ) D 12

Eh�

Zi � Z j

2i

. (5.20)

Clearly, γ � 0. A consequence of the nonnegativity of γ is that a semivariance canbe noncontinuous only at the origin. In that case, the height of the jump or discon-tinuity is usually called the nugget. If the autocovariance of a random function Z(x )exists, it is related to the semivariance by

2γ (x , y ) D R(x , x ) C R(y , y ) � 2R(x , y ) C E [Z(x ) � Z(y )]2 . (5.21)

One may also consider a normalized semivariance:

γn(Zi , Z j ) D γ (Zi , Z j )σ(Zi)σ(Z j )

. (5.22)

An estimate Oγ for the true semivariance is usually obtained from

2 Oγ D 1N � m

N�mXiD1

(Zi � ZiCm)2 . (5.23)

One may establish simple relations between the various quantities defined sofar if the data are stationary. A strictly stationary series is one for which all the


moments are independent of the position, that is, the moments are translation-ally invariant. Such a series is very rare. In practice, however, we usually requireup to second-order stationarity, that is, translational invariance of the first two mo-ments. Assuming that the data are second-order stationary, we have R(Zi , ZiCm) DR(m∆h) D R(h), where ∆h is the distance between two neighboring points in theseries and h D m∆h is the lag, or the lag distance. The second-order stationarity ofthe data also implies that the estimate OR(h) of the true autocovariance is given by

OR(h) D 1N � m

N�mXiD1

Zi ZiCm � NZ2 . (5.24)

Similarly, for second-order stationary data, we have CR(Zi , ZiCm) D CR(h), andγ (Zi , ZiCm) D γ (h). The plot of γ (h) versus h is called the semivariogram, whilethe plot of R(h) versus h is referred to as the autovariogram. More importantly, onehas a simple relation:

γ (h) D σ2 � R(h) , (5.25)

which holds so long as the variance σ2 is finite. Moreover, CR(h) and γ (h) followopposite trends: If CR(h) decreases, then γ (h) increases. Note that if a random fieldZ(x ) is stationary and ergodic (i.e., the volume average of the field is equal to itsensemble average), then

limh!1

γ (h) D σ2[Z(x )] , (5.26)

that is, the limit of the semivariogram is equal to the variance of the field. Thislimit is usually called the sill. The distance over which the difference between thesemivariogram γ (h) and its sill becomes negligible is called the range.

If the data contain no stochasticity (which, for any set of data for a disorderedporous medium, happens rarely, if ever), then the semivariogram increases mono-tonically with increasing lags. Step changes (increases) and quasi-periodic varia-tions are also sometimes obtained. If a porous medium is anisotropic, then onewill have distinct semivariograms in the horizontal and vertical directions. The op-posite is also true: distinct semivariograms in distinct directions imply anisotropy.

Characterization of the autocorrelations in natural porous media is not as clean-cut. As described by Jensen et al. (2000), autocorrelations in clastic formationsare typically scale-dependent. Moreover, many natural porous formations areanisotropic, with the anisotropy caused by stratification. For such porous me-dia, the horizontal and vertical autocorrelations may be completely distinct. Thevertical autocorrelations in clastic porous formations are usually weak, but they arestrong in the horizontal direction. One may even have anisotropy in a horizontalplane that is caused by a unidirectional deposition process, or by a unidirectionalflow, as in fluvial porous formations. Characterizing the autocorrelations is moredifficult if there exist multiple distinct and relevant length scales in the porousformation. The stratification may not be so clear in carbonate porous formations,but the anisotropy does usually exist.

5.4 Models of Semivariogram 121

5.4Models of Semivariogram

Empirical variograms cannot be computed at every lag distance h,simply becausethere are rarely enough data to do so. Moreover, due to variations in the estima-tion, it is not even guaranteed that one obtains a valid semivariogram – valid in thesense defined above. On the other hand, some geostatistical methods that are usedfor generating models of the FS porous media, such as Kriging and co-Krigingmethods (see below), require valid semivariograms. Thus, the experimental semi-variograms are often approximated by models that guarantee the validity of thesemivariogram.

5.4.1The Exponential Model

The model is given by

γ (h) D �σ2 � σ2

0

�1 � exp

�� αh

�e

��C σ2

0δ(0,1)(h) . (5.27)

The parameter �e is the range that, physically, represents an autocorrelation length.α is a purely numerical parameter. σ2 is the sill of the semivariogram, while σ2

0 isthe nugget.

5.4.2The Spherical Model

The spherical variogram is defined by

γ (h) D �σ2 � σ2

0

("32

�h

�s

�� 1

2

�h

�s

�3#

δ(0,�s)(h) C δ(�s,1)

)C σ2

0δ(0,1)(h) .

(5.28)

The spherical model is used extensively in modeling of the FS porous media. Theparameters σ2, σ2

0 and �s represent, respectively, the sill (which is approximatelythe sample variance), the nugget (which can be measured directly), and the rangeor the autocorrelation length. If the data indicate that nugget � sill, the semivari-ogram is said to exhibit a pure nugget effect.

5.4.3The Gaussian Model

The Gaussian model of a semivariogram is given by

γ (h) D �σ2 � σ2

0

"1 � exp

� αh2

� 2g

!#δ(0,�g)(h) C σ2

0δ(�g,1)(h) . (5.29)


The physical meanings of σ2, σ20, and �g are the same as before. In all the three

models introduced so far, the function δ(x ,y )(h) is defined by

δ(x ,y )(h) D(

1 if h 2 (x , y ) ,

0 otherwise .(5.30)

5.4.4The Periodic Model

Some empirical semivariograms exhibit a degree of periodicity or quasi-periodicity.Therefore, it is important to have a model that can accurately describe such semi-variograms. One such model is given by

γ (h) D σ20 C �

σ2 � σ20

�1 C

��p

h

�sin

�h

�p

��. (5.31)

The model contains peaks and valleys that are the solutions of the equation, h/�p Dtan(h/�p), with the first peak and valley being at h/�p D 3π/2 and h/�p D 5π/2,respectively. In many sedimentary formations, deeper valleys in a particular di-rection are common, implying the need for an anisotropic semivariogram model.Then, one can use a model such as

γ (h) D σ20 C �

σ2 � σ20

�1 � cos

�h

�p

��, (5.32)

in the direction in which the deeper valleys are seen in the semivariogram.

5.5Infinite Correlation Length: Self-Affine Distributions

There is increasing evidence that many FS porous media do not have a finite corre-lation length, but an infinitely long one. While the concept of an infinite correlationlength may seem abstract, it does have a clear physical interpretation: the correla-tion length is as large as the linear size of the porous formation. The larger the size,the longer is the correlation length.

To describe such geological formations, a nonstationary stochastic process calledthe fractional Brownian motion (FBM) has been used. We first consider the 1D case,and define the FBM process BH(x ) by (Mandelbrot and van Ness, 1968)

BH(x ) � BH(0) D 1Γ�H C 1

2

24 tZ

�1

K(x � s)dB(s)

35 , (5.33)

where x can be a spatial or temporal variable, Γ (x ) is the gamma function, and H

is called the Hurst exponent. As described in Chapter 6, the FBM is also used to

5.5 Infinite Correlation Length: Self-Affine Distributions 123

describe and model rough surfaces, in which case it is referred to as the roughness

exponent. The kernel K(x � s) is given by

K(x � s) D(

(x � s)H� 12 0 � s � t

(x � s)H� 12 � (�s)H� 1

2 s < 0 .(5.34)

It is not difficult to show that

BH(bx ) � BH(0) � bH�BH(x ) � BH(0)

�, (5.35)

where “�” means “statistically equivalent to.” Equation (5.35) indicates that theFBM defines a self-affine stochastic distribution (Feder, 1988). Self-affinity is sim-ilar to self-similarity described in Chapters 3 and 4, except that it is direction de-pendent. That is, the self-similarity of the object or the distribution in differentdirections is by different scale factors (see also Chapter 6). To demonstrate that theFBM generates stochastic series with an infinite correlation length (or time), con-sider the correlation function C(x ) of “future” increments BH(x ) with the “past”increments �BH(�x ), defined by

C(x ) D h�BH(�x )BH(x )ihBH(x )2i . (5.36)

It is straightforward to show that C(x ) D 2(22H�1 � 1) independent of x; that is, thecorrelations persist everywhere.

The type of the correlations can be tuned by varying the Hurst exponent H. IfH > 1/2, then the FBM displays persistence or positive correlations, that is, a trend(for example, a high or a low value) at x is likely to be followed by a similar trendat x C ∆x , whereas with H < 1/2, the FBM generates antipersistence or negative

correlations, that is, a trend at x is not likely to be followed by a similar one at xC∆x .For H D 1/2, the past and future are not correlated and, thus, the increments inBH(x ) are completely random and uncorrelated.

The 1D FBM may be generalized to 2D or 3D. Hence, if we consider two arbitrarypoints x and x0 in 2D or 3D space, the FBM is defined byD�

BH(x ) � BH(x0)�2E � jx � x0j2H . (5.37)

Figure 5.1 presents 1D profiles generated by the FBM. The FBM is not a stationaryprocess, though its increments are, although they are not ergodic. The variance ofa FBM for a large enough array is divergent, implying that the variance increaseswith the size of the series without bounds (see below). The FBM trace in d dimen-sions is a self-affine fractal with a local fractal dimension Df D d C1� H . The FBMis not differentiable at any point, but by smoothing it over an interval one obtainsits approximate numerical derivative which is called the fractional Gaussian noise

(FGN). The semivariogram of a FBM is given by

γ (h) D γ0h2H , (5.38)


Figure 5.1 Examples of the 1D FBM: (a) H = 0.8; (b) H = 0.3 (courtesy of Dr. Fatemeh Ebrahi-mi).

where γ0 D γ (h D 1). Due to its simplicity and elegance, and the fact that theFBM does seem to describe accurately considerable amount of data (see below),the use of a FBM in the simulation of the FS porous media has become popular.The semivariogram of a 1D FGN is given by (Bruining et al., 1997)

γ (h) D γ0 s2H � γ1

2s2

�(h C s)2H � 2h2H C jh � sj2H

�, (5.39)

where s is a smoothing parameter, and γ0 and γ1 are two constants.In a pioneering work, Hewett (1986) presented evidence that the porosity logs of

the FS porous media may follow the statistics of a FBM or FGN. More precisely,Hewett provided the first concrete evidence that the porosity logs in the directionperpendicular to the bedding may follow the statistics of a FGN, while those par-allel to the bedding may follow a FBM. Since Hewett’s original work, extensivestudies by several research groups have provided compelling evidence for the va-lidity of Hewett’s proposal (Crane and Tubman, 1990; Sahimi and Yortsos, 1990;Taggart and Salisch, 1991; Aasum et al., 1991; Hardy, 1992; Sahimi and Mehrabi,1999; Sahimi et al., 1995). For example, Crane and Tubman (1990) analyzed threehorizontal wells and four vertical wells in a carbonate reservoir, and found that theporosity logs are described well by a FGN with H ' 0.88. Likewise, Hardy (1992)analyzed 240 porosity logs from a carbonate formation and found them to be de-scribed well by a FGN with H ' 0.82. The analysis of a single horizontal wellin the same formation also indicated the FGN statistics with a slightly higher H.The analysis of porosity logs of a major Iranian oil reservoir (Sahimi and Mehra-bi, 1999) indicated that at least some of the logs are described reasonably well by aFGN (most Iranian oil reservoirs are of carbonate-type). In all such studies, the low-er limit of self-affine fractal behavior was below the resolution of the instrumentsused for measuring the data. A surprising discovery of most of the studies was thatthe same value of H was found for the porosity and resistivity logs that, in general,is not expected to be the case.

Similarly, permeability measurements on outcrop surfaces have provided addi-tional evidence that the FBM- and FGN-type behavior in oil reservoirs and other


natural formations is not the exception, but the rule. For example, the analysis ofGoggin et al. (1992) on sandstone outcrop data indicated that the logarithm of thepermeabilities along the lateral trace followed the FGN statistics with H ' 0.85.

There is also evidence that some properties of oil reservoirs follow the statisticsof a FBM. In the original analysis of Hewett (1986), a FBM was used for generat-ing the horizontal properties of the reservoir. This was justified based on the dataobtained from groundwater plumes (see, for example, Pickens and Grisak, 1981).Emanuel et al. (1989), Mathews et al. (1989), and Hewett and Behrens (1990) all ob-tained accurate descriptions of oil reservoirs using a FBM. Neuman (1994) provid-ed further evidence that the permeability distributions of many aquifers follows thestatistics of a FBM with H < 0.5. Makse et al. (1996b) analyzed data for two sand-stone samples from distinct environments, and found the permeability distributionto be described by a FBM with H ' 0.82�0.9. Moreover, analyzing extensive da-ta for the elastic moduli, the density, and seismic wave speeds in eight offshoreand onshore oil and gas reservoirs, Sahimi and Tajer (2005) showed that even suchproperties follow the statistics of the FBM. Dashtian et al. (2011a) showed, in fact,that practically all type of well logs can be described by self-affine distributions.

Note that, as pointed out earlier, a FBM is a nonstationary stochastic process.However, if the self-affine fractal behavior indicated by the FBM is only manifestedbelow a low-wave number cutoff 1/L, then the permeability or porosity field iseffectively stationary at length scales larger than L, with the variance given by

σ2 D c

2HL2H , (5.40)

where c is a constant. Equation (5.40) demonstrates an earlier assertion that thevariance of a FBM increases with the size L of the system without bound. A com-prehensive discussion of such issues is given by Molz et al. (2004).

In general, the description of a FS porous medium by a FBM does not differmuch from that by a FGN if the wells are closely spaced. Moreover, both the FBMand FGN descriptions approach a stratified medium as the variations in the porousmedium properties between the wells decrease with decreasing well spacing. Weshould, however, point out that under practical circumstances, one encounters withthe FS porous media, a FGN with 0 < H < 1, producing results that are verysimilar to those obtained with the FBM with 1 < H < 2, provided that one usesH C 1 instead of H in Eq. (5.38).

If the data contain long tails, then an FBM or FGN may not accurately describethem. For example, if an oil reservoir or aquifer is fractured, then one must detectlarge jumps in the permeability distribution, indicating the presence of the frac-tures since as a probe moves along a well and passes from the porous matrix tothe fractures, the permeability should significantly increase. Compared with thenumber of pores of any geological formation, the fractures are relatively rare and,therefore, their presence represents rare events manifested by large jumps in thepermeabilities and giving rise to long tails in their distribution that are not seenin data that follow the statistics of a FBM or FGN. In this case, a fractional Lévymotion (a generalization of a FBM) and the associated Lévy-stable distribution may


provide a more accurate and complete characterization of the data (Mehrabi et al.,1997). We will come back to this point in the next chapter.

To give the reader some idea about the complexities that are involved in the logdata, we show in Figure 5.2 a vertical porosity log that was collected along a verticalwell in an oil reservoir in southern Iran (Sahimi and Mehrabi, 1999). The well’sdepth is about 1900 m and the porosity φ was estimated or measured every 20 cmso that over 600 data points were collected. The question then is: given a porositylog, or any other log or the permeabilities along the same well of a FS porous medi-um, how can one accurately analyze it to uncover its mathematical structure? Inparticular, if such data follow the statistics of a FBM or FGN, how can one estimatethe Hurst exponent H that characterizes the correlations in the data?

A closely related issue is the efficient and accurate numerical simulation andgeneration of an array that follows the statistics of a FBM or a FGN if they describethe porosity logs, permeability distributions, and other properties of a FS porousmedium. In particular, such numerical simulations are needed when one uses theconditional simulations described in Section 5.7. Efficient and accurate simulationof a FBM or FGN is not straightforward at all. There are several techniques for nu-merical simulation and generation of an FBM array with a given Hurst exponent H

(Mehrabi et al., 1997). For example, Hamzehpour and Sahimi (2006a) developed amethod by which a FBM array is generated by an optimization method using sim-ulated annealing (see Chapters 6–8). We now describe four efficient methods forgeneration of a FBM or FGN.

Figure 5.2 A porosity log along a well in a fractured oil reservoir (after Sahimi and Mehrabi,1999).


5.5.1The Spectral Density Method

A convenient way of representing a stochastic function is through its spectraldensity Sd(ω), the Fourier transform of its two-point correlation function in d-dimensions. Bruining et al. (1997) derived the spectral density of the FBM. Theresult for a 1D FBM is given by

S1(ω) D H γ0

Γ (1 � 2H ) cos(Hπ)1

ω2HC1 . (5.41)

Equation (5.41) indicates why a FBM is not differentiable because in order to beso, its spectral density should decay faster than ω�3, which is not the case even forH D 1. Using a formula due to Tsuji (1955),

S2(ω) D 1π

1Z0

d

dω1

�1

ω1

d

dω1S1(ω1)

�qω2

1 � ω2 dω1 , (5.42)

where ω denotes the magnitude of ω, one obtains the corresponding 2D density:

S2(ω) D 22H γ0H2

π2

Γ (H )2 sin(Hπ)ω2

x C ω2y

�HC1. (5.43)

Tsuji (1955) also showed that

S3(ω) D � 12πω

�dS1(ω1)

dω1

�ω1Dω

, (5.44)

and, therefore,

S3(ω) D γ0H

2πΓ (1 � 2H ) cos(Hπ)2H C 1

ω2x C ω2

y C ω2z

�HC 32

. (5.45)

Note that the spectral representation of a FBM also allows one to introduce a cutofflength scale `c D 1/ωc such that

Sd(ω) D adω2

c C ω2x C ω2

y C ω2z

�HC d2

, (5.46)

where ad is a d-dependent constant. The cutoff `c allows one to control the lengthscale over which the spatial properties of a porous formation are correlated, or anti-correlated. Thus, for length scales L < `c, the properties preserve their correlationsor anticorrelations, whereas for L > `c, they become random and uncorrelated.

The spectral density for a 1D FGN is given by

S1(ω) D 2π s2 γ0H Γ (2H ) sin(πH )[1 � cos(sω)]

1ω2HC1 . (5.47)


In the limit s ! 0, Eq. (5.47) reduces to

S1(ω) D 1π

γ0H Γ (2H ) sin(πH )1

ω2H�1 . (5.48)

The spectral density of an FBM or FGN provides a convenient method for theirnumerical generation using a fast Fourier transformation (FFT) technique. In thismethod, one first generates random numbers, distributed either uniformly in [0, 1],or according to a Gaussian distribution with random phases, and assigns them tothe sites of a d-dimensional lattice. In most cases, the linear size L of the lattice is apower of two, but the only requirement is that L can be partitioned into small primenumbers so that the FFT algorithm can be used. One must also keep in mind thatsince the variance of a FBM increases with the size L of the array or lattice, gener-ating a FBM array with a given variance requires selecting an appropriate size L. Inany case, the Fourier transform of the resulting d-dimensional array is calculatednumerically and then multiplied by

pSd(ω), and the inverse Fourier transform of

the results is then computed numerically. The array so obtained follows the statis-tics of a FBM. To avoid the problem associated with the periodicity of the numbersarising as a result of their Fourier transforming, one must generate the array usinga much larger lattice size than the actual size that is to be used in the analysis,and use the central part of the array. Clearly, a similar method can be used forgenerating a FGN. It should be emphasized that, as Hough (1989) pointed out, in-terpreting a power-law spectral density as an indication of fractality is not withoutdifficulties and, thus, one must be careful in using such an analysis. In particular,a power-law spectral density might also be the characteristic of a nonstationary, butalso nonfractal system.

The two-point correlation function C2(r) of a FBM array is given by

C2(r) D C0r2H , (5.49)

which is similar to its semivariogram, Eq. (5.38), where, C0 D C(r D 1). Tomake the generation of a FBM array more efficient and accurate, Pang et al. (1995)and Makse et al. (1996c) modified the FFT method. Since the correlation functionEq. (5.49) has a singularity at r D 0, they considered a slightly different correlationfunction,

C2(r) D

1 CdX

iD1

r2i

!�2

, (5.50)

which, in the limit r ! 1, has the same qualitative behavior as Eq. (5.49). The FTof the correlation function Eq. (5.50) can be determined analytically, with the resultbeing,

S(ω) D 2πd2

Γ (� C 1)

ω2

��K�(ω) . (5.51)

Here, � D 1/2( � d), ω D jωj, and K� is the the modified Bessel functionof order �. For ω � 1, one has the asymptotic relation that K�(ω) � ω� . The


correlated array generated based on Eqs. (5.50) and (5.51) corresponds to a FBMwith the Hurst exponent, H D 1 � 1/2 . Let us mention that Mehrabi and Sahimi(2009) extended the FFT method significantly in order to generate the FBM or FGNin a curved space, for example, along the curved strata in a FS porous medium.

5.5.2Successive Random Additions

In the successive random addition method (Voss, 1985), one begins with the twoend points in the interval [0, 1], and assigns a zero value to them. Then, Gaussianrandom numbers ∆0 with a zero mean and unit variance are added to these values.In the next stage, new points are added at a fraction r of the previous stage byinterpolating between the old points (by either linear or spline interpolation), andGaussian random numbers ∆1 with a zero mean and variance r2H are added to thenew points. Thus, given a sample of Ni points at stage i with resolution λ, stagei C1 with resolution r λ is generated by first interpolating NiC1 D (Ni �1)(1/r �1)new points from the old ones, and then adding Gaussian random numbers ∆ i

to all of the new points. At stage i with r < 1, the Gaussian random numbershave a variance, σ2

i � r2 i H , consistent with the FBM. For example, with r D 1/3and Ni D 5, the old (o) and new (n) points are in the order, (onnonnonnonno),so that there are NiC1 D 8 new points in the array. The process is continueduntil the desired length of the data array is reached. The method can easily beextended to any dimension (Lu et al., 2003). It is not, however, very efficient if large3D correlated arrays are to be generated. In addition, extending the method forgenerating anisotropic 2D or 3D arrays is difficult.

The problem with the method is that the points that are generated in earlier gen-erations are not statistically equivalent to those generated later. To remedy this, onecan add a random Gaussian displacement with a variance r2(n�1)H to all the pointsduring the nth stage of the process, though doing so increases the computationtime – it roughly doubles it. Moreover, if one is interested in generating an FBMarray with a very wide range, one may start the process by assigning a Gaussianrandom number with a variance 22H to one end of the [0, 1] interval. The general-ization of this method to higher dimensions is straightforward.

5.5.3The Wavelet Decomposition Method

The wavelet decomposition method is a suitable tool for analyzing the FBM-typedata (Flandrin, 1992). For such data, one computes the following quantity,

D j (k) D 2�j

2

1Z�1

BH(x )ψ(2� j x � k)dx , (5.52)

where D j (k) is called the wavelet-detail coefficient of the FBM, and represents thewavelet transformation of the data set. Here, ψ is the wavelet function (see Press et


al., 2007) for an introduction to wavelet functions) k D 1, 2, . . . , N , where N is thesize of the data array, and the js are integer numbers. Thus, in this method, onefixes j and varies k to calculate D j (k). For each j, one determines N such numbersand calculates their variance σ2( j ). Then, it can be shown that regardless of thewavelet ψ, one has

log2[σ2( j )] D (2H C 1) j C constant . (5.53)

Thus, plotting log2[σ2( j )] versus j yields H.The analysis of the porosity log of Figure 5.2 by the wavelet decomposition

method is shown in Figure 5.3 (Sahimi and Mehrabi, 1999), which indicates thatthere are actually two distinct scaling regimes corresponding, respectively, to small(small j) and large (large j) length scales. For small length scales, one obtainsH ' 0.8, indicating positive correlations. Such correlations are expected, as twopoints that are at short distances from each other presumably fall (at least in thiscase) within a single stratum, hence exhibiting positive correlations. On the otherhand, for large length scales, one obtains H ' 0.33, indicating negative correla-tions, which is again expected as two points that are widely separated presumablybelong to two strata with contrasting properties, hence exhibiting negative correla-tions.

Figure 5.3 Wavelet decomposition analysis of the porosity log of Figure 5.2 (after Sahimi andMehrabi, 1999).


5.5.4The Maximum Entropy Method

The maximum entropy method computes the spectral density of the data without

using the FFT, hence avoiding the pitfalls that one encounters in using the FFT. Inthis method, the spectral density is approximated by

Sd(ω) ' a0ˇˇ1 C

MPkD1

ak z k

ˇˇ2 , (5.54)

where the coefficients ak are calculated such that Eq. (5.54) matches the Laurentseries, Sd(ω) D PM

�M bi zi . Here, z is the frequency in the z-transform plane,

z � exp(2π i ω∆), and ∆ is the sampling interval in the real space. In practice, tocalculate the coefficients ai , one first computes the correlations functions

C j D hdi diC j i ' 1N � j

N� jXiD1

vi viC j , (5.55)

where N is the number of data points, and di is the datum at point i. The coeffi-cients ai are then calculated from

MXj D1

a j Cj j �kj D Ck , k D 1, 2, . . . , M . (5.56)

The advantage of Eq. (5.54) over the Laurent series is that if Sd(ω) contains sharppeaks, it detects them easily because they manifest themselves as the poles of theequation, whereas one may have to use a very large number of terms in the Laurentseries to detect the same peaks.

The analysis of the porosity log of Figure 5.2 by the maximum entropy methodis shown in Figure 5.4 (Sahimi and Mehrabi, 1999). Except for large values of ω,that is, very short distances, the spectral density of the porosity log exhibits clearpower-law scaling with the frequency. The estimated value of H, which is about0.35, agrees with what one obtains with the wavelet decomposition method forlarge length scales. However, the spectral density method (and, hence, the max-imum entropy method by which one computes the spectral density) is often notcapable of detecting a scaling regime for short length scales, if one exists, becausethe measurement resolution is not high enough for distinguishing the points thatare in the same neighborhood and, therefore, the data might be too noisy for thespectral density method.

While the four methods described are all highly accurate in terms of the mini-mum required size of the data array for reliable characterization of the long-rangecorrelations, the maximum entropy method is the most reliable method (Mehrabiet al., 1997) since it yields valuable information and accurate estimates of H, evenwhen one has a small data array. This is a great advantage of the method sincethe development of a large data base for the FS porous media is costly and timeconsuming.


Figure 5.4 Spectral density S1(ω) of the porosity log of Figure 5.2 computed by the maximumentropy method (after Sahimi and Mehrabi, 1999).

5.6Interpolating the Data: Kriging

The analysis of the various well logs and other types of data is an important steptoward characterization and modeling of the FS porous media. An even more im-portant issue is how to interpolate and extrapolate the known data which are usuallyfor the areas along the wells in, for example, oil reservoirs, to the interwell zones.Aside from some information for the large-scale structure of porous media thatone gleans from seismic recordings, there are usually very little quantitative datafor the interwell zones. In addition, even the data measured along the wells maycontain significant gaps and, therefore, one must interpolate the existing data inorder to fill the gaps.

This issue is particularly important because in order to develop the geologicalmodel of a FS porous medium, one must have quantitative information for everygrid block of the model. However, the vast majority of such blocks represent theinterwell zones and those representing along the wells where there are gaps in theexisting data and, therefore, one must find an accurate way of interpolating andextrapolating the existing data to such zones. However, the problem sounds easi-er than it actually is. The problem is uncertainty, namely, given that a FS porousmedium is typically highly heterogeneous, one must find a way of not only extrap-olating the existing data to the zones for which there are no data, but also takinginto account the effect of the uncertainty in any property value that is attributed toany grid block.

5.6 Interpolating the Data: Kriging 133

This becomes clearer once we recognize that every point in a FS porous mediumas well as every random variable associated with the point (that represents a proper-ty value) has a PDF. If a property value at a given point is already measured – and,therefore, is known deterministically – then the PDF associated with it is simplya δ function or a spike because, aside from the possible measurement error, thereis no uncertainty in the property value. If, however, a property value at that pointhas not been measured because, for example, it is impossible to do so, then a PDFis associated with it with a nonzero variance. If there are no physical constraints,for example, some qualitative information about the property, then the PDF can, inprinciple, take on any shape.

The best known technique for extrapolation and interpolation of the data is theKriging method,2)which is essentially a statistical method for estimating the propertyvalues for certain points, given some data for other points in the space. Krige (1951)developed a geological point of view that is based on the assumption of continuedmineralization between measured values. Assuming, that is, prior knowledge en-capsulates how minerals co-occur as a function of space. Then, given an orderedset of measured grades, interpolation by Kriging predicts mineral concentrationsat unobserved points. Kriging has been used in mining, hydrogeology, modelingof fluid flow in oil reservoirs, natural resources, environmental science, remotesensing, and even the so-called black box modeling in computer experiments.

The theory of Kriging was developed by Matheron (1963, 1967a)3), based on theKrige’s work. Matheron had already derived an estimate K� for the permeabilitywhich was the “estimateur” in his work and a precursor to the kriged estimate orkriged estimator. However, what he had failed to derive was σ2(K�), the varianceof his estimateur. In fact, he had computed the length-weighted average grade ofeach “panneau”, but did not compute the variance of its central value.

A central doctrine of the geostatistics that Matheron developed is that the spatialdependence (of some properties) need not be verified, but may be assumed to ex-ist between two or more points, determined in samples selected at positions withdifferent coordinates. This doctrine of assumed causality was the quintessence ofMatheron’s new geostatistics. The question remains, however, as to whether as-sumed causality makes sense in any other scientific discipline. For an account ofthe history of Kriging, see Cressie (1990).

The general idea that forms the basis for Kriging is as follows. Suppose thatsome data for a property, Zi D Z(xi) with i D 1, 2, . . . , N , are known and one isinterested in obtaining an estimate OZ of the same property at a point for which

2) The method was first proposed by miningengineer Daniel Gerhardus Krige who wasthe pioneer plotter of distance-weightedaverage gold grades at the Witwatersrandreefs complex in South Africa. He developedthe method as part of his Master’s thesis, butnever used the word Kriging.

3) Georges Francois Paul Marie Matheron

(1930–2000) was a French mathematicianand geologist who made importantcontributions to geostatistics and flowthrough porous media. The method wascalled krigeage for the first time in his 1960Krigeage un Panneau Rectangulaire par sa

Priphrie.


there are no data. A common interpolator is given by

OZ DNX

iD1

wi Zi . (5.57)

Since the data points are spatial positions x1, x2, . . ., one may think of the parame-ters or parameters wi as depending on the spatial position xi as well. Formulatingthe estimator OZ as in Eq. (5.57), the main problem is then determining the “opti-mal” values of the weights wi , such that Eq. (5.57) provides an accurate estimate OZ .Note that Eq. (5.57) provides the estimate as a linear combination of all the knowndata Zi . If the estimate is obtained by an unbiased method, namely, by one suchthat the expectation value of the property value is equal to its true value, then OZis called a BLUE (best linear unbiased estimate) estimate. We note that one mayalso use a log-normal Kriging that interpolates positive data by means of their log-arithms. However, in what follows, we only work with the data themselves, ratherthan their logarithms.

Krige developed a method for estimating wi , which is why the parameters arereferred to as the Kriging weights (KWs). Depending on the stochastic propertiesof the random fields that represent the properties of FS porous medium, severaltypes of Kriging have been developed. The type of Kriging used determines thelinear constraint on the weights wi implied by Eq. (5.57) and, hence, the methodfor calculating the weights depends upon the type of Kriging used.

5.6.1Biased Kriging

This method is also called simple Kriging. Mathematically, it is the simplest of allthe Kriging methods, but it is also the least general. It assumes that the expecta-tion of the random field is known, and relies on a covariance function in order todetermine the KWs wi . However, in most applications, neither the expectation northe covariance are known a priori. The method is built upon three fundamentalassumptions: (1) the wide sense stationarity of the random field Z(x ); (2) the expec-tation of the field is zero everywhere, and (3) a covariance function is known. Thus,the method first defines a sensible variance. Let us define

σ2BK D E [(Z � OZ )2] . (5.58)

Note that the Kriging variance defined by Eq. (5.58) (as well as in what follows) mustnot be confused with the variance of the Kriging predictor itself. Equation (5.58),after substituting Eq. (5.57) for Z, becomes

σ2BK D σ2

Z � 2NX

iD1

wi R(Z, Zi ) CNX

iD1

NXj D1

wi w j R(Zi , Z j ) , (5.59)

where R(x , y ) is the usual covariance of x and y. Note that the true value of thevariance σ2

Z is not known. It is estimated from the sample variance or from the


semivariogram of the data (see Eq. (5.40)). To determine the KWs wi , the varianceis minimized with respect to wi , that is, one sets @σ2

BK/@wi D 0 for i D 1, 2, . . . , N .After taking the derivatives and some rearrangements, one obtains

NXiD1

R(Zi , Z j )wi D R(Z, Z j ) , j D 1, 2, . . . , N (5.60)

which represents a set of N simultaneous linear equations for the N unknownsw1, . . . , wN . As the equations are linear, the minimum value of σ2

BK is actually itstrue global minimum, and not a local one.

Suppose that after determining the KWs wi , we wish to obtain the Kriging esti-mates OZ D [ OZ1, . . . , OZn ]T at n grid points (T denotes the transpose operation). Theestimates are given by the following vector equation:

OZ D MTZ , (5.61)

where M is an N � n matrix of the KWs wi j given by

MT D

264

w11 w1n

......

...wN1 wM n

375 . (5.62)

Note that wi j now has two indeces which refer to the measurement (data) at point i

and estimates at point j.The resulting estimates are biased because in determining the KWs wi , we did

not impose the constraint that the true average value at the unsampled locations beproduced.

5.6.2Unbiased Kriging

Unbiased Kriging is also called ordinary Kriging, and is the most commonly usedtype of Kriging. It assumes a constant but unknown mean. Typical assumptionsfor using the ordinary or unbiased Kriging are (1) intrinsic or wide sense station-arity of the field Z(x ); (2) there are enough observations or data to estimate thevariogram, and (3) the mean of the random field is unknown but constant. Undersuch assumptions, the KWs fulfill the unbiasedness condition. To remove the biasfrom the simple biased Kriging, we first recall Eq. (5.1) for the bias b( OZ):

b( OZ) D E( OZ) � E(Z) D E(Z)

NX

iD1

wi � 1

!. (5.63)

Thus, if E(Z) is itself unbiased, a vanishing bias b requires setting

NXiD1

wi D 1 . (5.64)


In other words, the KWs wi must be determined subject to the constraint that theirsum must add to one. As such, determining the KWs for an unbiased estimateis a constraint optimization problem, a bit more difficult than determining theKWs wi in the biased Kriging problem. To impose the constraints, one introducesthe Lagrange multiplier in the variance to be minimized:

σ2UK D σ2

BK C 2

NX

iD1

wi � 1

!, (5.65)

so that in addition to the KWs wi , the Lagrange multiplier must also be deter-mined. Use of the Lagrange multiplier ensures that the true minimum of σ2

UK isobtained. After minimizing σ2

UK with respect to both wi and , one obtains

NXiD1

R(Zi , Z j )wi D R(Z, Z j ) C , j D 1, . . . , N (5.66)

which is only slightly more complex than Eq. (5.59) and gives rise to a matrix equa-tion similar to Eq. (5.60), but with one additional unknown, .

The main properties of any unbiased Kriging are as follows (Chilés and Delfiner,1999; Wackernagel, 1995). (1) The Kriging estimates are unbiased. (2) The Krig-ing estimates honor the data, assuming that no measurement error exists. (3) TheKriging estimates are the best (most accurate) linear unbiased estimate if all theassumptions hold. If, however (Cressie, 1991), the assumptions do not hold, Krig-ing could result in poor estimates. (4) There might be better nonlinear and/or bi-ased methods than Kriging. (5) No accurate estimate is guaranteed if the wrongvariogram is used. However, one may still obtain a “good” interpolation. (6) Thebest estimate is not necessarily an accurate one. For example, if there are no spa-tial correlations, the Kriging interpolation is only as good as the arithmetic mean.(7) Kriging provides a measure of precision. The measure relies, however, on thecorrectness of the variogram.

5.6.3Kriging with Constraints for Nonadditive Properties

The Kriging procedure described so far is usually accurate for obtaining estimatesfor those properties of a FS porous medium that are additive, such as the porosity.For nonadditive properties such as the permeability, however, even the unbiasedconstraint is weak. That is because an estimate of the permeability at any point inspace is not normally a linear combination of the known (measured) permeabil-ities. Thus, one must impose more stringent constraints that result in relativelyaccurate estimates.

It is known that the average or overall permeability of a region that consists ofseveral subregions with permeabilities K1, K2, . . . satisfy certain upper and lower


bounds. For example, K satisfies the following bounds

NXiD1

K�1i � K �

NXiD1

Ki . (5.67)

Much tighter bounds are described by Sahimi (2003a). Thus, one may impose theconstraint that the bounds Eq. (5.67) must be satisfied. Alternatively, one can usethe results of well testing to constrain the permeabilities. In well testing, one mea-sures the time-dependence of the variations in the pressure at the well by imposingchanges on the volume flow rate of the well, from which the permeability of theporous formation around the well is estimated. Thus, one can constrain the esti-mate of the permeability based on well-testing results, although such a constraintis unable to provide information on the spatial variations of the permeability in theinterwell zone. We should keep in mind that any nonlinear constraint comes at theprice of more intensive computations.

5.6.4Universal Kriging

Universal Kriging assumes a general linear-trend model. When there are long-range correlations in the data, then, in order to produce reasonable estimates byKriging, one first subjects the data to a preprocessing step. For example, givensome insight into the structure of a porous formation, one may divide the data in-to several segments, with each segment corresponding to a formation’s zone, andthen carry out separate Kriging for each zone.

5.6.5Co-Kriging

If more than one set of data are available, then, in order to obtain Kriging estimates,one carries out a co-Kriging procedure. For example, suppose that in addition to adata set for the permeability along a certain path, additional information on theresistivity is also available, and we wish to obtain estimates of the permeability atthe points for which no data are available. The estimate is then written as

OZ DNX

iD1

wi Zi CNX

iD1

λ i X i , (5.68)

where Xi D X(xi) represent the additional data at xi . Clearly, one may use exactlythe same procedure that was described for the unbiased Kriging, except that onemust determine two sets of co-Kriging weights, wi and λ i .


5.7Conditional Simulation

Despite its many desirable features, Kriging is also not without problems. For onething, the FS porous media are highly heterogeneous and, therefore, the spatialvariations of such properties as the permeability are great – often by several ordersof magnitude. However, the estimates that Kriging provides give rise to permeabil-ity fields that are too smooth. Sharp changes are almost never produced by Krigingalone. The second shortcoming of Kriging is that it is a completely deterministicmethod. In other words, the same data Z(xi ) always produce the same KWs wi

and, hence, the same estimates OZ. The third significant feature of Kriging that canbe bothersome is that all the estimates that it provides are Gaussian. This is easilyseen by considering Eq. (5.57) and recalling the central-limit theorem.

The Gaussianity of the Kriging estimates is dealt with through a procedure calleddisjunctive Kriging. In this method, the data are preprocessed to be rendered Gaus-sian by a suitable transformation. The questions of smoothness of the fields gen-erated by Kriging and their deterministic nature are addressed by conditional simu-

lation. There are actually three variations on the conditional simulation, and whatfollows is a description of each.

5.7.1Sequential Gaussian Simulation

As its name suggests, sequential Gaussian simulation (SGS) is a Kriging-basedmethod that generates a Gaussian field for the entire system. The entire procedurecan be summarized as follows (Jensen et al., 2000). Suppose that we have N datapoints, Z(xi ) with i D 1, 2, . . . , N , and wish to generate property values for M gridblocks, numbered j D 1, 2, . . . , M , including those at which the data are available.Thus, there are M – N grid blocks that are unconditioned, that is, no data for themare available. We also assume that the semivariogram model is available as it canbe computed using the N data points. The data are also assumed to be error free.The property values of the conditioned blocks, that is, those for which the data areavailable, do not change during the SGS.

1. The data are first transformed to be Gaussian.2. The M – N unconditioned grid blocks are assigned property values equal to

those of the nearest conditioned blocks.3. A random walk through the unconditioned blocks is constructed such that each

of such blocks is visited only once.4. Each time an unconditioned block is visited, the number and locations of the

local conditioned blocks around it are identified. To do so, one must specifythe shape and boundary of the region that constitutes the local neighborhoodof the visited unconditioned block. Typically, the local neighborhood is definedin such a way that it has roughly the ellipse range identified by the semivar-iogram model (recall that most FS porous media are anisotropic). The local

5.7 Conditional Simulation 139

neighborhood may contain the blocks that have already been simulated, that is,a property value has already been assigned to them.

5. Unbiased Kriging (or co-Kriging if needed) is carried out for the local neigh-borhood using the data (actual as well as the already simulated that may bein the local neighborhood) as the conditioning points. Once this is done, themean and variance σ2

UK of the local region are computed. The estimate OZ has aGaussian PDF with the computed mean and variance. Local Kriging is used inorder to avoid solving the large-scale problem indicated by Eq. (5.65). However,one may also use a single Kriging for all the unconditioned grid blocks at once,thereby including all the trends at all the data points.

6. A random number r, uniformly distributed in [0, 1], is generated and used topick a property value p from the Gaussian PDF constructed in step (v) by solvingthe equation

r DpZ

pmin

f (x )dx ,

where f (x ) is the Gaussian PDF, and pmin is the minimum value of the prop-erty. The value is attributed to the unconditioned grid block. Thus, the numberof the data points (measured as well as simulated) increases by one.

7. The next step of the random walk to the next unconditioned grid block is takenand steps 4–6 are repeated.

8. Steps 6 and 7 are repeated until all the unconditioned grid blocks are visitedand property values are attributed to them. The final result is a realization ofthe model conditioned upon the available data.

Clearly, multiple realizations of the model can also be generated using this pro-cedure. Each realization is a valid representation of the porous formation to bemodeled.

5.7.2Random Residual Additions

The random residual additions (RRA) method is, in some sense, more realisticthan the SGS method because it no longer uses the concept of local neighborhoodaround an unconditioned grid block. What follows is a step-by-step description ofthe method.

1. An unbiased Kriging is carried out for the entire grid using the available dataas the conditioning points.

2. An unconditioned grid is also generated using the semivariogram. Because itis unconditioned, it does not honor the data at the specified point. However, itdoes have all the required statistical properties, including the second momentof the PDF of the data because the semivariogram is used. The unconditionedsystem is not deterministic, but represents a single realization of the ensemble


of all the possible realizations with the required statistical properties. The un-conditioned grid is generated by any appropriate method. For example, if thedata follow a fractional Brownian motion (FBM), any of the methods of gener-ating a FBM described earlier may be used.

3. Use the result of step 2 at the blocks for which the actual data are availableas the conditioning points in order to carry out a second unbiased Kriging (orco-Kriging).

4. The unconditioned grid values of step 2 are subtracted from those generatedby the second conditioned Kriging (or co-Kriging). Clearly, since the secondKriging was carried conditioned upon the unconditioned grid, the residuals forthe blocks at which actual data are available are zero.

5. The resulting residuals obtained in step 4 are then added to the first Krigingcarried out in step 1. Thus, the data are honored at the specified points – zeroresiduals are attributed to such grid blocks – and the nonzero residuals repre-sent the uncertainty for the property values at the unconditioned grid blocks.

The resulting distribution of the property values in the computational grid hasmany desirable features: (1) it honors the data at the specified points; (2) it has thecorrect statistical properties; (3) it is unbiased, and (4) it includes the large-scaleheterogeneity that is expected to exist in the geological formation and its model.

5.7.3Sequential Indicator Simulation

This is a method that classifies each grid block into a facies category. The categoryis built up using a variety of data, such as, seismic-amplitude map extractions thatare calibrated using core and log data and include, for example, channels, overbankdeposits, and others.

It is assumed that no two facies can exist in the same grid block. Thus, themethod is sometimes referred to as a simulation for generating the facies. Aftereach grid block is assigned to its facie category, its property value is attributed toit from the PDF of the facie. Thus, the method is also called sequential indicator

simulation-probability distribution function (SIS-PDF). The overall PDF of the faciesrepresents the pattern of their occurrence at the scale of the porous medium to bemodeled. The PDF is obtained from the usual sources, that is, either log data or amap of the facies. The overall procedure for the SIS-PDF is as follows (Jordan andGoggin, 1995).

1. As in the SGS method, a random walk is taken through the computational gridthat represents the FS porous medium such that the unconditioned blocks arevisited once and only once.

2. For each visited unconditioned grid block, the prespecified number of condi-tioning facies data from the wells and already simulated blocks, as well as anyother sources of data, is identified.

5.7 Conditional Simulation 141

3. An indicator Kriging is carried out in order to estimate the conditional prob-ability for each facies category. An indicator Kriging uses indicator functionsinstead of the process itself in order to estimate the transition probabilities. Itproceeds exactly like the usual Kriging, except that an indicator semivariogramγI is used in place of the usual semivariogram, and the data values Zi are re-placed by their respective indicators. To construct γI, a threshold value Zc isintroduced that varies between a minimum and maximum value and, hence,can take on several values Zci

. Then, the indicator variable I(Zci) is defined by

I(Zci) D

(1 Zi � Zci

,

0 Zi > Zci.

The cumulative distribution function OF (Zc) is then constructed by

OF (Zc) D 1M

MXiD1

I(Zci) ,

where M < N , with N being the number of the data points. The indicatorsemivariogram is constructed based on I(Zci

), and the cumulative probabilitydistribution is used for estimating the conditional probabilities.

4. Each facie’s probability is then normalized by the sum of the probabilities ofall facies. The result is then used to construct a local cumulative probabilitydistribution.

5. A random number r, distributed uniformly in [0, 1], is generated and used to-gether with the local cumulative distribution function in order to determine thesimulated facies category in the visited unconditioned grid block.

6. For each unconditioned block in the random path, steps 2–5 are repeated. Thefinal result is the facies distribution.

5.7.4Optimization-Reconstruction Methods

In addition to Kriging and conditional simulation methods, there are other tech-niques that have been used for developing geological models of the FS porous me-dia. They do not, however, represent geostatistical methods and have been bor-rowed from other disciplines, for example, the statistical mechanics of thermalsystems. Chief among them, are the simulated annealing method and the geneticalgorithm, both of which represent optimization methods. That is, given a limit-ed amount of data, the two methods are used for determining an optimal structure(spatial distributions of the porosity and permeability) of a porous medium that notonly honors the data, but can also provide accurate predictions for other propertiesfor which no data are available. This method is also referred to as reconstruction,as one attempts to reconstruct a model of a porous medium for which limitedamounts of data are available. The two optimization methods will be described indetail in Chapters 6–8.

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