large-scale porous media and wavelet transformations

JULY/AUGUST 2003 Copublished by the IEEE CS and the AIP 1521-9615/03/$17.00 © 2003 IEEE 75

Editor: Dietrich Stauffer, [email protected]

SIMULATIONSC O M P U T E R S I M U L A T I O N S

heterogeneities of such porous media, whose linear sizes aretypically on the order of at least a few kilometers, manifestthemselves at four widely disparate length scales:

• microscopic scale—at the level of pores and grains (typicalsizes are up to tens of microns);

• mesoscopic scale—at the level of laboratory core plugs (typ-ical length scale on the order of a few centimeters);

• macroscopic scale—characteristic length scale of up to tens ofmeters for large blocks of a porous medium; and

• megascopic scale—the entire porous medium.

For two important reasons, LSPM are of immense inter-est. One reason is economical: oil and gas extracted from un-derground reservoirs are the world’s most important energysource. Much of our drinking water comes from ground-water aquifers. Landfills, and their effect on the surround-ing soil, are an unavoidable part of modern times. The sec-ond reason is political: much of the industrialized world doesnot produce enough oil and gas domestically and, therefore,relies on oil-exporting countries, most of which are politi-cally unstable. Many wars over the past 100 years, includingthe latest one in Iraq, have had at least partly to do with oilreservoirs and political control over them. In addition, theproblem of natural resource contamination—specifically,groundwater aquifers—has become so severe that no politi-cian who hopes to get elected can ignore it.

Regardless of whether we’re interested in extracting oilor gas from an underground reservoir, or wish to study howa contaminant may spread in an aquifer, we must be able tomodel flow and transport phenomena in LSPM. However,such models can be predictive and, thus, useful only if we

can characterize and model the LSPM’s morphology.

Working at Different LevelsIf it contains no fractures, we can model a porous medium atthe laboratory (or mesoscopic) length scale via a three-di-mensional (3D) network of interconnected pore throats—thenarrow channels that control a fluid’s passage through themedium—and pore bodies—the large chambers where mostof the medium’s porosity resides and where the pore throatsmeet. With the advent of sophisticated instruments and tech-niques over the past two decades, we can now get accurateand detailed images of a porous medium using 3D X-raycomputed tomography.4 Furthermore, we can compute theporous medium’s flow and transport properties based on the3D image with few (if any) approximations. For example, oneof a porous medium’s most important flow properties, whichis its effective permeability Ke, is defined through Darcy’s law,

, (1)

where v is the fluid’s velocity, µ its viscosity, and P the pres-sure, and can now be computed accurately.

The same level of knowledge and ability to model doesnot, however, exist when it comes to porous media at mega-scopic length scales. First, such porous media, unlike manymesoscopic or even macroscopic porous media, are not sta-tistically homogeneous. This means that the LSPM’s prop-erties have large-scale spatial variations and, therefore, de-pend on the length scale at which the porous medium isexamined. If we imagine an LSPM as consisting of a verylarge collection of mesoscopic blocks, the blocks’ effectivepermeability and porosity vary greatly, thus giving rise toscale-dependent properties. This scale-dependence impliesthe existence of extended correlations between the proper-ties at several length scales. Second, characterization of suchporous media is plagued by noisy and incomplete data. Evenwhen extensive data on LSPM do exist (a rare possibility),proper interpretation of the data and separation of noise anduncertainty from it still are challenging problems.

Even if all such problems were solved, we would still have

v = − ∇

KPe

µ

L ARGE-SCALE POROUS MEDIA (LSPM) SUCH

AS OIL, GAS, AND GEOTHERMAL RESER-

VOIRS, GROUNDWATER AQUIFERS,1,2 AND EVEN

LANDFILLS3 ARE HIGHLY HETEROGENEOUS. THE

LARGE-SCALE POROUS MEDIAAND WAVELET TRANSFORMATIONSBy Muhammad Sahimi

76 COMPUTING IN SCIENCE & ENGINEERING

to address the fundamental problem of scale-up—namely,the development of a proper LSPM model with two key at-tributes. One, the model should contain all the importantinformation, from microscopic to megascopic length scales,so that it accurately represents the porous medium’s truemorphology. This task is difficult because the disparity be-tween the smallest and largest length scales can be as largeas 10 orders of magnitude. Two, because a computationalgrid of the blocks (or nodes) to which we assign propertiessuch as porosity and permeability represents the porousmedium, the model must be computationally tractable. Thismeans that the grid cannot be too large: modeling fluid flowand transport in such a grid entails solving thousands of (dis-cretized) highly nonlinear equations thousands of times.

Despite the 40-year history of computer simulation ofprocesses in LSPM, it was only in the early 1980s that re-searchers recognized something big: the persistent deficien-cies and inaccuracies in predicting the performance of en-hanced oil-recovery (EOR) processes and the fate of pollutantsin aquifers had their roots in inadequate descriptions of theLSPM. A prime example is the recent discovery of an under-ground, plutonium-containing plume at a significant concen-tration roughly 1.3 km from the Nevada Test Site, where nu-clear testing began over 30 years ago.5 The site at YuccaMountain, which the United States Department of Energy re-cently approved as an underground repository for high-levelradioactive waste (a decision the state of Nevada is currentlychallenging) is on highly fractured and heterogeneous porousrock. It was predicted, based on conventional characterization,modeling, and simulation of flow and transport in the rock,that it would take tens of thousands of years for significantconcentrations of Pu to spread over distances on the order ofone kilometer or more from the test site. The recent discov-ery that this would not be the case has prompted serious ques-tions on the possible causes of this anomalous (and much fasterthan expected) contaminant transport.

These difficulties, together with other field observationsof surprisingly fast contaminant transport, suggest that theporous media’s complex structure is the main culprit for thebreakdown of the fluid-flow and transport models.

Characterizing Large-Scale Porous MediaTo characterize LSPM, we divide all available data on theirmorphology into two groups. One group has what we call di-rect data—those data that provide direct and quantitative in-formation on the porous medium’s properties. An importantsource of such data is the logs (such as the porosity, resistivity,gamma ray, and temperature logs) collected during oil-well

drilling. With the advent of sophisticated instrumentation, it’sbecoming possible to measure the permeabilities in situ withnuclear magnetic resonance.

The second group has what we call indirect data—those thatprovide a picture of the porous medium at large length scales.Perhaps the most important of such data are 2D or 3D seis-mic recordings and, more recently, 4D seismic data—repeatedrecordings of the 3D data over a period of time.6 Indirect datado not provide quantitative information on an LSPM’s per-meability or porosity distribution. Nevertheless, given exten-sive coverage of the 3D (or 4D) data, and with proper cali-bration at the wells’ locations, indirect data provide us withthe opportunity for a more accurate characterization.

A critical aspect of an LSPM’s characterization is propertreatment of the data, ranging from denoising andsmoothing them (uncovering their special features anddiscovering their statistical distributions) to interpretingthem and understanding their implication for the charac-terization problem. Done properly, the data and theircharacteristics can help develop a realistic model for usein computer simulations.

Properly treating data, however, is fraught with complexi-ties. One important obstacle to achieving it is the fact that dif-ferent types of data are analyzed with different methods, oftenyielding conflicting results. Compounding these difficultiesare two important insights that have emerged over the pastdecade. One, many well logs and LSPM permeability distrib-utions follow fractal stochastic processes.1,2,7 In particular, am-ple data suggest that for many LSPM, the porosity logs in thedirection perpendicular to the bedding might obey the statis-tics of fractional Gaussian noise (fGn), while those that are par-allel to the bedding may follow a fractional Brownian motion(fBm), a stochastic process with a power spectrum S(ω) (thatis, the Fourier transform [FT] of its covariance) given by

, (2)

where H is the Hurst exponent, d is the system’s (or dataset’s) dimensionality, ad is a d−dependent constant, and ω =(ω1, …, ωd). Fractional Brownian motion is not differen-tiable, but in smoothing it numerically over a small interval,we get its numerical derivative, which is the fGn. For exam-ple, since the well logs are usually 1D data sets, the powerspectrum of fGn in 1D is given by

, (3) S

bH( )ω

ω= −

12 1

Sad

ii

d H d( ) /ωω

=

=

+

∑ 21

2

C O M P U T E R S I M U L A T I O N S

JULY/AUGUST 2003 77

where bd is a d−dependent constant; the corresponding 1Dpower spectrum for a fBm is given by S(ω) = a1/ω2H+1. Themost remarkable property of the fBm and fGn is that theygenerate correlations with an extent as large as the system’slinear size. Moreover, the type of the correlations dependson H. If H > 1/2, the data display persistence in the sense thata trend (for example, a property’s high or low value) at a pointx is likely to be followed by a similar trend at x + ∆x. If H< 1/2, the data contain antipersistence, in that a trend at x isnot likely to be followed by a similar trend at x + ∆x. For H= 1/2, the increments in the fBm-type data are uncorrelated.

The second important insight emerging over the past twodecades concerns the structure—in particular, the connectiv-ity—of an LSPM’s fracture networks, many of which containfractures on many distinct length scales. There is increasingevidence ranging from Monte Carlo simulations of fracturepropagation in rock8 to field observations9,10 that, one, thefracture network is very irregular at all the relevant lengthscales, and two, in many LSPM, the fracture network may bea fractal object.

An important implication of these results is that the frac-ture network’s connectivity bears no resemblance to the per-fectly ordered network of fractures used in traditional sim-ulations of fluid flow and transport in LSPM. In addition,due to its fractal nature, the fracture network’s spatial struc-ture is highly correlated.

After the data are denoised and interpreted properly, we gen-erate, via modern geostatistical methods,11 a highly detailed 3Dmodel of the LSPM that we refer to as the geological model,which typically contains several million grid blocks, or nodes.Such detailed models are important to managing LSPM (forexample, planning to keep an oil reservoir’s production on amore or less steady basis for several years), because fine-scalemorphology details dominate its flow and transport properties.Their use in the simulation of flow and transport in LSPM en-tails solving several million discretized equations thousands oftimes. Such simulations are currently not feasible, even withthe advent of massively parallel computational strategies andvector supercomputers.

To address the problem of making a transition from thegeological model to a computationally tractable one, re-searchers have developed methods to scale-up LSPM. Themain goal is to coarsen the highly detailed geological modelto levels suitable for computer simulation while avoiding los-ing important information. In a sense, the scale-up processis similar to the renormalization group methods in criticalphenomena whereby many degrees of freedom are elimi-nated from the system to develop a coarsened model whose

properties can be computed more easily than the original sys-tem’s. Indeed, past research has used renormalization groupmethods12–14 for scaling up geological models of LSPM.

Most of the scale-up methods proposed so far are too sim-plistic, not computationally efficient enough, or not applica-ble to fractured LSPM. Many of them average out the effectsof extreme values, such as those associated with thin commu-nicating layers in the geological formations, large flow barri-ers, and partially communicating faults. Most importantly,practically all the current scale-up methods are not applica-ble to fractured LSPM because the large differences betweenthe fracture and matrix permeabilities at the interface betweenthe two typically gives rise to singularities in the up-scalingscheme that cannot be resolved or removed. Despite consid-erable progress,15 much remains to be done, especially forfractured porous media.

This article describes a work in progress—a unified ap-proach to the characterization of LSPM, construction and up-scaling of a realistic model for them, that model’s dynamic up-dating, and proper interpretation of the results of simulationof flow and transport in such porous media. The approach de-scribed here is based on the use of wavelet transformations(WTs) at each of these steps. In what follows, we first intro-duce WTs and describe their properties that are relevant toour discussion. We’ll then look at how WTs can be used forcharacterizing and modeling LSPM by discussing their ap-plication to several aspects of the problems described earlier.

Wavelet TransformationsHaar discovered the earliest wavelet function nearly a centuryago,16 but the Haar wavelet did not attract much attention atfirst. In the early 1980s, researchers developed a WT in prim-itive form for analyzing seismic data.17,18 Daubechies, Mallat,and others carried out the most significant work on the theo-retical development of the WTs in the late 1980s and early1990s.19–24 Their work provided rigorous foundations for themany applications that WTs have found.

The continuous WT of a function f(x), denoted for con-tinuous functions f(x) by f^(a, b), is defined by

, (4)

where

. (5) ψ ψa b

a a, ( )xx b

=−

1

ˆ ( , ) ( ) ( ),f a f da bb x x x=−∞

∞∫ ψ


Here, a > 0 is a dilation or rescaling parameter, b representsthe translation of the wavelet, and ψ(x) is called the motherwavelet. Keep in mind that ψa,b(x) has the same shape for allvalues of a; the wavelet function ψ(x) is not unique.

Depending on the intended application, we can use a varietyof wavelets. In fact, this nonuniqueness represents the great ad-vantage that a WT offers over the standard FT and Fourieranalysis. However, the choice of ψ(x) is not completely arbitrary.For example, in 1D, ψ(x) is a function with the property that

, (6)

so that ψ(x) has two important properties. One, it has a com-pact support (the support’s compactness is also a require-ment for 2D and 3D wavelets) to achieve localization inspace, and two, it has a zero mean—for instance,

, (7)

although higher moments of ψ(x) also can be zero. Equation7 implies that ψ(x) is wave-like, whereas Equation 6 ensuresthat ψ(x) is not a sustaining wave, hence the name wavelets(small waves). The compactness of the support of ψ(x) im-plies that computations with the WTs can be parallelized,because the wavelets are nonzero only over finite intervals.The inverse of a WT is given by

, (8)

where Cψ is a constant that depends on the wavelet, and d isthe system’s dimensionality.

Varying the parameter a dilates (a > 1) or contracts (a <1) f(x). Hence, the wavelet spreads out with increasing a andtakes into account only the large-scale features of f(x) (orthe long-time behavior, if a function f(t) is considered) andvice versa. Varying b, on the other hand, lets us analyze f(x)around different points b. These two fundamental proper-ties of the WTs make them an ideal tool for data process-ing. We can also view the WT as a microscope in which themagnification is given by 1/a and the optics by the choiceof the wavelet ψ(x).

So far, we’ve described continuous WTs. However, to an-alyze a discrete set of data, we must use a discrete WT. Forthe sake of clarity, let’s consider a 1D example to describe howa discrete WT comes about. To do this, we set the parameter

a = a0j, where j is an integer and a0 > 1 is a fixed dilation. We

also set b = kb0a0j, where b0 > 0 depends on the choice of ψ(x),

and k is an integer. Therefore, a0j plays the role of a magni-

fier, and thus, the WT of a function f(x) studies the functionat a particular location with the given magnification and thenmoves on to another (discrete) location. Hence, if we define

, (9)

the resulting WT is given by

. (10)

Daubechies discussed the conditions for choosing a0 andb0.22 They are fairly broad, so a0 and b0 admit very flexibleranges and values. A common choice is a0 = 2 and b0 = 1, inwhich case the resulting 1D WT of a discrete data array f(x),

, (11)

is usually called the data’s wavelet-detail coefficient, where k = 1,2, …, n, with n being the data array’s size, and the j’s being in-tegers. The resulting set of wavelets ψj,k for all j and k forman orthonormal basis. A remarkable property of these func-tions is that they also are orthonormal to their translates anddilates. Any square integrable function f(x) can be approxi-mated up to an arbitrarily selected high degree of precision by

. (12)

Such a representation of f(x) also implies that the coeffi-cients Dj(k) measure the contribution to f(x) of scale 2j at lo-cation 2jk. Thus, such series representation of f(x) is akin to aFourier series, except that the series in Equation 12 is double-indexed (indicating scale and location), and the basis functionshave a localization property. The Haar wavelet is the simplestof all orthogonal wavelets and is given by

. (13)ψ ( )/

/xx

x=≤ ≤

− ≤ <

1 0 1 21 1 2 1

0 otherwise

f x D k xj j k

kj

( ) ( ) ( ),==−∞

∞

=−∞

∞

∑∑ ψ

D k f x x k dxjj j( ) ( )/= −( )− −

−∞

∞∫2 22 ψ

ˆ ( , ) ( )/f j k a f x a x kb dxj j= −( )− −−∞

∞∫02

0 0ψ

ψ ψ ψj k j

j

jj jx

a

x kb a

aa a x kb,

/( ) =−

= −( )− −1

0

0 0

00

20 0

fC a

f a da dd a b( ) ˆ ( , ) ( ),x b x b= +∞

−∞

∞ ∫∫1 110ψ

ψ

ψ ( )x dx =

−∞

∞∫ 0

ψ ( )x dx2 1∫ =


JULY/AUGUST 2003 79

The detail coefficients and, more generally, a function’sWT, contain information only about the contrast be-tween two approximations of the same function at twosuccessive length scales. The most accurate approxi-mation of a function at a fixed or given scale is obtainedby using another function called the scale-function φ(x),which is orthogonal to ψ(x). Whereas the mean valueof ψ over the entire space is zero, the mean value of φis unity over the same space, implying that φ(x) < ψ(x),thus φ(x) provides us with complementary informationon the approximation to the function f(x). The waveletapproximate or scale coefficients are defined by

, (14)

where the definition of φj,k(x) is similar to that of ψj,k(x). As de-scribed later, the two functions are in fact interrelated.

Similar to the FT, the WTs decompose the object f intotwo separate components, because two different functions—a scaling function φ(x) and a wavelet ψ(x)—both act on f.However, as Equations 4, 11, and 14 indicate, the two func-tions do not separate the components into cosines and sines,but into averages and differences, with a wavelength equalto the window over which φ and ψ are nonzero.

Another important difference between the WTs and FTis that the WTs are recursive, so we can apply them in suc-cession to any set of averages produced using the waveletsto produce another level of averages and another level of de-tails. Another important property of the wavelets is that aspatially localized function, such as a finite impulse function,cannot be represented by a few terms of its Fourier series.Instead, a very large number of terms may be needed toachieve the convergence. In the wavelet space, however, thesame function can be completely described by a few of itswavelet coefficients.

Consider now the 1D wavelets. Orthonormal waveletsthat are compactly supported—those that are nonzero overonly small intervals of x—are formed by rescaling and trans-lating ψ(x) and φ(x) via ψk

j(x) = 2–j/2ψ(2–jx – k) and φkj(x)

= 2–j/2φ(2–jx – k). One important family of such wavelets con-tains the Daubechies wavelets of order M19,22 (usually re-ferred to as DBM). The first M moments of the DBMwavelets are zero. The scaling function φ(x) is related tothose at the finer length scales by

. (15)

Equation 15 is usually referred to as the two-scale relation.Similarly,

, (16)

where L = 2M. Here, hk and mk, the filter coefficients, are re-lated by mk = (–1)k hL–k–1, with k = 0, 1, …, L – 1; they areusually nonzero for only a few values of k.

The Haar wavelet’s filter coefficients wavelet are (h0, h1)= ( )(1, 1), thus (m0, m1) = ( )(–1, 1) = (–h1, h0).Actually, the extra factor (which is often not in-cluded) is included here to ensure orthonormality betweenφ and ψ. For the DB2 wavelet, we have (h0, h1, h2, h3) =( )(1 + , 3 + , 3 – , 1 – ), and (m0, m1, m2,m3) = (–h3, h2, –h1, h0). Again, the extra factor is in-cluded for the orthonormality of φ and ψ. Figure 1 showsthree wavelets (DB2, DB6, and DB10) in 1D.

Cohen-Daubechies separable wavelets, which are built bytensor products of the 1D wavelets, are one of the manymethods suggested for constructing 2D wavelets. In thismethod, there is one scaling function and three wavelets,

, (17)

, (18)

, (19)

. (20)

The extension to 3D is straightforward. Another method,

ψ φ ψj k k k

jkjx y x y, ,

( ) ( , ) ( ) ( )1 2 1 2

3 =

ψ φ ψj k k k

jkjx y x y, ,

( ) ( , ) ( ) ( )1 2 1 2

2 =

ψ φ ψj k k k

jkjx y x y, ,

( ) ( , ) ( ) ( )1 2 1 2

1 =

φ φ φj k k kj

kjx y x y, , ( , ) ( ) ( )

1 2 1 2=

1 2/33 331 2/

1 2/ 1 2/ 1 2/

ψ φ( ) ( )x m x kk

k

L

= −=

−

∑2 20

1

φ φ( ) ( )x h x kk

k

L

= −=

−

∑2 20

1

S k f dj j k( ) ( ) ( ),=

−∞

+∞∫ φ x x x

0

1.5

1

0.5

0

–0.5

–1

–1.5

500 1,000

ψ χ(

)

0

1.5

1

0.5

0

–0.5

–1

–1.5

2,0003,0001,000 0

1.5

1

0.5

0

–0.5

–1

–1.5

2,000 4,000

(a) (b) (c)

Figure 1. Three one-dimensional Daubechies wavelet functions ψ(x):(a) DB2, (b) DB6, and (c) DB10. These three functions show the wave-like structure and locality of wavelets.


suggested by Sweldens,25 divides the WT into two steps: thefirst computes the wavelet detail coefficients, and the seconduses these coefficients to speed up the computation of thescaling coefficients.

Wavelet Characterization of Large-Scale Porous MediaHaving introduced WTs and discussed their main proper-ties, let’s look at their application to LSPM characterization.

Data Smoothing and Relevant-Scale IdentificationAs discussed earlier, an important aspect of analyzing director indirect data in LSPM is the ability to separate noisycomponents from actual data. The same problem arises inthe analysis of various well logs. In the context of fluid-flowproblems, we must address the same problem in isolation ofturbulent flows’ coherent structures. Several ways of sepa-rating noise from the data’s real part exist. For example, sim-

ilar to the power spectrum (the FT of a statistical distribu-tion’s covariance), we can define a wavelet spectrum26,27 as

, (21)

which, for orthonormal wavelets, is given by

. (22)

This wavelet spectrum successfully identifies the dominantscales or modes of variations in turbulent flows by condens-ing the phase plane (also called the scalogram) into a singlefunction of scale. The wavelet spectrum provides a bettermeasure of variance in a data set than the power spectrumdoes because in the WTs, the coefficients are influenced bylocal events or variations, whereas data over the entire do-main affect the Fourier coefficients in the power spectrum.Local maxima in the wavelet spectrum provide clues about

S j D kjk

( ) ( )= ∑2

S a f a d( ) ˆ ( , )=

−∞

∞∫ b b


Figure 2. A noisy data set (a time series) and three of its smoothed versions. We get these three versions by using a wavelettransformation and three levels of smoothing. (a) Original noisy schedule, (b) one-level denoising, (c) two-level denoising, and(d) three-level denoising.

10

8

6

4

2

0

–2

–3

–6

–8

–10

10

8

6

4

2

0

–2

–3

–6

–8

–10

Original noisy signal 1-level denoising

8

6

4

2

0

–2

–3

–6

–8

8

6

4

2

0

–2

–3

–6

–8

2-level denoising 3-level denoising

(a) (b)

(c) (d)

0 200 400 600 800 1,000 1,200

0 200 400 600 800 1,000 1,200

0 200 400 600 800 1,000 1,200

0 200 400 600 800 1,000 1,200

JULY/AUGUST 2003 81

the scales at which features provide significant contributions,which they can make in one of two ways: by one feature witha large contribution or by a series of smaller features.

A more general method of denoising or smoothing a data setis via threshold partitioning. In this method, we calculate the dataset’s wavelet-detail coefficients, set a threshold ∈d (as a fractionof the coefficients’ maximum value), and eliminate (that is, setthem to zero) all the coefficients that are less than ∈d. Then,the denoised data are reconstructed again with the zero andnonzero detail coefficients, meaning we compute the inverseof the denoised data’s discrete WT. If we chose the thresholdappropriately, the method is very powerful (details about choos-ing the threshold appear elsewhere28). Figure 2 presents a noisydata set (time-series data) and the three smoothed versions weget via the threshold-partitioning method.

We can also use wavelets for denoising and interpreting in-direct information—in particular, seismic data, which moti-vated WTs’ development in the first place. The analyses ofseismic data and their proper interpretation have, of course,a long history. However, it became increasingly clear thatconventional methods for doing so couldn’t provide deep in-sight into what the data imply for LSPM’s morphology. In-stead, it became apparent that the most significant advancesin seismic data processing and analysis come from uncon-ventional methods of describing wave propagation in LSPM.Using fractal distributions to analyze direct data and WTs toanalyze these and indirect data have proved very practical.

However, the problem of processing and analyzing seis-mic data poses a significant computational challenge. Dueto their large volume, especially when repeated measure-ments are made, using an effective means of processing andanalyzing seismic data is critical. Wavelet transformationshave already been suggested for compressing, transmitting,and decompressing such data.29,30

Analysis of Fractal Data Sets and Their Synthetic ReconstructionThe direct data for LSPM that are typically available, suchas porosity, resistivity, and gamma-ray logs, usually have acomplex structure that makes it seemingly impossible to un-cover their mathematical structures (even more so if they fol-low a fractal stochastic process). Figure 3 presents a verticalporosity log collected alongside an oil well. Researchers mea-sured porosity every 25 cm and collected several thousandsdata points. However, these data do not constitute a randomset; they contain correlations with an extent that can be verylarge. Given such a complex data set, the question is, how canwe accurately analyze the set and investigate whether it con-

Figure 3. A vertical porosity log alongside an oil well in an oilfield. Depths are in meters, whereas porosities are inpercentages.

30

25

20

20

15

10

5

0

Depth (m)1,780 1,800 1,820 1,840 1,860 1,880 1,900 1,920

Poro

sity

Figure 4. A synthetic porosity log generated with a one-dimensional fractional Brownian motion and its analysis bythe wavelet method. The input Hurst exponent H was 0.2,whereas the wavelet method found it to be about 0.21.

0.6

0.5

0.4

0.3

0.2

0.1

j

0 50 100 150 200 250

Poro

sity

10

8

6

4

2

0 2.5 3 4 4.5 5

Log 2

(σdj

)

1.5 2 3.5

Figure 5. A wavelet analysis of Figure 3’s porosity log. Fromthe slope of the straight line, we can compute the Hurstexponent H.

–5

–10

–15

–20

–25

–30

j2 3 4 5 6 7 8

9

1

Log 2

(σdj

)


tains long-range correlations? In the case of fBm- or fGn-type data, previous work31 has shown that using WTs canproduce a highly efficient and accurate analysis.

In this method, we calculate the wavelet-detail coefficientsof the data defined by Equation 11 by fixing j, varying k, andcomputing Dj(k). For each j, we determine n such numbersand calculate their variance σ2(j). We can then show that re-gardless of the type of wavelet function ψ(x) we use, we have

log2[σ2(j)] = (2H + 1)j + constant. (23)

Thus, plotting log2[σ2(j)] versus j yields the Hurst exponentH. Figure 4 presents such an analysis for a synthetic datagenerated by a fBm with H = 0.2. Figure 5 shows the waveletanalysis of Figure 3’s porosity log.

Mallat20 showed that for stochastic processes possessinga power-law spectrum (for example, the fBm and fGn), thewavelet spectrum S(j) satisfies the following equation,

S(j) = 22HS(j + 1), (24)

so on a logarithmic scale, the wavelet spectra at differentscales have a linear relationship. Note that we could inter-pret the spectrum S(j) as a sort of “energy” of the system,so Mallat’s relation has a particularly attractive physical in-terpretation for self-similar stochastic processes. Flandrin32

showed that although fBm itself is not a stationary process(even though a fBm’s increments are stationary), its WT is.Finally, Wornell33 showed that we could use orthonormalwavelets to construct self-similar stochastic processes X(x)by writing

, (25)

where the coefficients dj(k) are uncorrelated for any distinctpair j and k and have a power-law variance. Although certainconditions regarding the wavelets used in Equation 25 mustbe met, the class of wavelets that do satisfy these conditionsis fairly broad and includes the Daubechies wavelets, ofwhich the Haar wavelet is the simplest member.

Essentially, not only can WTs analyze fractal data sets ac-curately and efficiently, we can also use them for construct-ing synthetic fractal data (as in Equation 25), which are ofgreat use in developing the geological model of LSPM. Ifan LSPM’s heterogeneities follow fractal distributions(which is often true), we must use fractal interpolation forthe interwell zones (those regions between the wells for

which no data are available to generate realistic estimatesof their properties).

Identification of Rock Fractures’ Spatial DistributionMost geological formations contain fractures at some lengthscale. In particular, many oil reservoirs and groundwater aquifersare fractured; such cracks are critical to fluid flow in porous me-dia. For example, carbonate oil reservoirs (typical of Iranian oilreservoirs) typically are characterized by a spatial distribution ofvery low porosities; they can’t produce large volumes of oil ifthey do not contain a network of interconnected fractures con-nected to the producing wells. Fractures can have a detrimen-tal effect on groundwater aquifers, however. They can act as fastconduits for dispersal of contaminants in low permeability soils.Fractures and faults also affect our understanding of tectonicmotions and earthquakes’ hypocenters.8–10,34,35

In spite of the need for it, precise identification of frac-tures’ and faults’ spatial distribution has remained a largelyunsolved problem. Porosity distribution is not usually usedfor identifying this distribution; most researchers believethat porosity alone cannot provide insight into fractures’whereabouts. Although a fracture’s permeabilities are muchlarger than those of the porous matrix (which means the per-meability data can, in principle, provide insight into spatialdistribution), such data are not used routinely for this pur-pose, because we usually don’t have enough of such data.

Although several methods can help identify the fractures’spatial distribution, most of them suffer from shortcomings.One method6 combines seismic data, various types of avail-able direct data, and semiempirical laws governing the speedof seismic waves in LSPM to guess the fractures’ where-abouts and orientations. Although this method is used heav-ily, it could be subject to uncertainty. We might need tosomehow calibrate the method with knowledge of the frac-tures’ distribution in some zones in the rock to ensure thatthe method’s inferred distribution of the fracture is realistic.Unfortunately, reliable data for such a calibration do not of-ten exist. Borehole-wall imaging is a reliable way of mappingdiscontinuities (meaning fractures and faults) within bore-holes. However, such imaging techniques are expensive andnot always included in a logging run. In addition, the data-base for many geological formations that have been in usefor many decades do not contain such images at all.

My research group has created a novel method36 for iden-tifying rock fracture (or fault) spatial distribution that usesdirect data, based on WTs. The method uses the simplesttype of direct data for LSPM—namely, the porosity logs—

X x d k xj j k

kj

( ) ( ) ( ),= ∑∑ ψ


JULY/AUGUST 2003 83

although the technique can be used with any other type ofwire logs. This advantage’s flexibility is that the amount ofporosity data for many reservoirs is relatively large and theircollection is economical.

The method is based on computing the porosity logs’wavelet detail coefficients (WDCs). The WDCs separatethe data into different length scales and have three key at-tributes that we can exploit for this problem.

One, they can be quite large even if the data’s value at agiven depth x is small, and vice versa. Thus, each piece ofthe data is given its proper weight, which for our problemimplies that although the fractures’ porosity is typicallysmaller than that of the porous matrix, its significance anddistinction from that of the matrix is recognized by theWT and reflected in the corresponding WDCs. As ex-plained earlier, local maxima in the WDC spectrum giveclues about the scales at which important features providesignificant contributions: either one feature (a single frac-ture) with a large contribution or a series of smaller fea-tures (several microfractures).

Two, if the log is composed of two (or more) distinct seg-ments (types of data), each having its own statistical distrib-

ution or characters, which the WDCs distinguish one fromanother, implying for our problem that because the frac-tures’ porosity distributions and the porous matrix differ, theWT can differentiate between the two types, with the dif-ference reflected in their corresponding WDCs.

Three, as already mentioned, the WDCs are influenced bylocal events or features, which contrasts with the power spec-trum of the same data. The implication for the problem ofidentifying the fractures’ spatial distribution is clear: since pas-sage from the porous matrix to a fracture (and vice versa) is alocal event, the fractures’ presence should be reflected in theWDCs’ distribution.

Consider an unfractured porous medium for which a discreteset of data (such as a porosity log) is given. A plot of the data’sWDCs versus the depths at which the data were collected (forexample, along the wells) then produces a featureless diagram.The WDCs fluctuate around a well-defined mean (typicallyzero), exhibiting no particular structure, because all the datahave the same distribution. However, if the data are for a frac-tured zone, the plot of their WDCs versus their locations isagain featureless, except where the fractures intersect the wells,at which point the WDCs exhibit local maxima.36

Figure 6. A synthetic porosity log generated by (a) an fBm with H = 0.2 and (b) its wavelet-detail coefficients. (c) The same log butmodified at a several points. (d) The corresponding wavelet-detail coefficients. The modifications indicate the presence of fractures.

40

35

30

25

20

(a) 3,000 3,500 4,500 5,000

Poro

sity

(p

erce

ntag

e)

40

35

30

25

20

(b) 3,000 3,500 4,500 5,000

Poro

sity

(p

erce

ntag

e)

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.03,000 3,500 4,500 5,000

A–

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.03,000 3,500 4,500 5,000

A–

Depth, ft Depth, ft

(c) (d)Depth, ft Depth, ft


To demonstrate the method’s utility, we use an fBm-gen-erated synthetic porosity log. Its WDCs fluctuate aroundthe same mean value and exhibit no distinct features (seeFigure 6), because the data all follow the same distribution.However, if the log contains two types of data with distinctdistributions, the WDCs distinguish one from the otherwith the difference appearing as local maxima in the plot ofthe WDCs versus the depths.

As an example, we remove a few of the data in Figure 6’ssynthetic array and replace them with other data distrib-uted uniformly. We chose the inserted data’s values delib-erately such that we cannot distinguish them from the restof the original data array. A plot of the new composite dataarray’s WDCs exhibits sharp spikes at precisely the depthsat which we replaced the fBm data with the uniformly dis-tributed data. However, the emergence of the sharp spikeshas nothing to do with the inserted data’s magnitude, but,rather, the inserted data’s statistical distribution, whichcompletely differs from that of the rest of the data array.We describe the test of this method with real porosity logs(and how it can be used to map out the fractures’ spatialdistribution in the interwell zones for which no data areavailable) elsewhere.36

Researchers also have used wavelets for detecting themost important features of a given rock sample’s fracturepattern.37,38 For example, one method enables transfor-mation to large-scale maps from smaller scale ones andquantification of the multiscale behavior of fracturinganisotropy.37 The technique is based on finding an opti-mum wavelet that can reveal the local structure at eachpoint of the fracture map and, thus, help us detect themost important fractures in a complex network of frac-tures. This is very useful for simulating flow and transportin fractured LSPM.

Scale-Up of the Geological ModelWe can also use WTs39–43 for scaling up an LSPM’s geo-logical model and generating a coarsened grid for large-scale simulation of multiphase flows. Suppose, for example,that the geological model is represented by its distributionf[K(x)] of the permeabilities assigned to the grid points(nodes) or blocks of the geological model. We assume thatthe distribution f[K(x)] is broad and contains correlations atall the relevant length scales. We’ll describe the method fora square grid, but the method can be used with equal facil-ity for a 3D grid.

Each node or square block of the grid is assigned a perme-ability K selected from the distribution f[K(x)]. The grid is as-

sumed to have the finest possible resolution—a more detailedgrid cannot be built because no information on f[K(x)] at finerlength scales is available. We then apply a one-level discrete WT(DWT) to f[K(x)]; we attempt to coarsen the grid by a factor of2, but the coarsening is done “intelligently” instead of uniformly.Associated with the DWT at every block with its center at x = (k1, k2)—or at x = (k1, k2, k3) in 3D—are four wavelet coeffi-cients (eight in 3D). Following Equations 11 and 14, the fourwavelet coefficients are given in more precise forms by

, (26)

, (27)

where j is the level of coarsening (the original geological modelrepresents the j = 1 level), and Ω is the problem’s domain. Asmentioned earlier, Sj(k1, k2) contains information about thepermeability K(x) at x in the coarser grid. The coefficientsDj

(d)(k1, k2) measure the contrast between K(x) at point x in thecoarser scale and those of its neighbors in the previous finerscale, with d = 1,2, and 3 (d = 1, …, 7, in 3D) corresponding tothe y, x, and diagonal directions, respectively.

In practice, we can calculate the wavelet coefficients at anylevel j from the scale coefficients at the previous finer level j –1 in two steps (three in 3D). First, we apply a 1D Mallat algo-rithm20,21 to the scale coefficients associated with the blocks(or nodes) in a given direction—the x−direction. The Mallatalgorithm for a DWT is given by

, (29)

, (30)

where hl and ml are the filter coefficients defined earlier, andSj–1(l + 2k1) is the scale coefficient at level j – 1. The scale co-efficients at the j = 0 level are simply the values of K(x) at thegrid block centered at x. We then apply the 1D Mallat algo-rithm to the calculated Sj(k1) and Dj

(d)(k1) in the y-direction.Each application results in two coefficients, thus we get thefour wavelet coefficients (in 2D).

To carry out the coarsening process, two thresholds ∈sand ∈d are introduced, where ∈s is a measure of the per-

D k m S l kj

dl j

l

L( ) ( ) ( )1 1 1

0

1

2= +−=

−

∑

S k h S l kj l j

l

L

( ) ( )1 1 10

1

2= +−=

−

∑

D k k f K dj

dj k kd( ), ,

( )( , ) [ ( )] ( )1 2 1 2= ∫ x x x

Ωψ

S k k f K dj j k k( , ) [ ( )] ( ), ,1 2 1 2

= ∫ x x xΩ

φ


JULY/AUGUST 2003 85

meability at a given block (or the cor-responding wavelet scale coefficient)and is set as a fraction of the largestscale coefficient of the system. ∈d, onthe other hand, measures the contrastbetween the neighboring blocks’ per-meabilities (or the associated WDCsand is also set as a fraction of the sys-tem’s largest detail coefficient. Eachblock or node’s scale coefficient isthen examined. If it is higher than ∈s,implying that its permeability is largeenough, it is left intact, and the nextblock or node is examined. If, how-ever, the examined scale coefficient issmaller than ∈s, the associated detailcoefficients are examined and set to zero if they are smallerthan ∈d. Setting Dj

(d)(k1, k2) = 0 means that the neighbor of(k1, k2) corresponding to the direction (d), which in thefine-scale model is only one block away from (k1, k2), is re-moved—the two blocks merge and form a larger block. Inthis way, the coarsening method resembles the thresholdpartitioning described earlier.

Therefore, depending on the structure of the distributionf[K(x)], several blocks (or nodes) in the fine-scale model arecoarsened. If the distribution f[K(x)] is relatively narrow andcontains no correlations, the blocks are coarsened more orless uniformly throughout the model, whereas with a broadand correlated f[K(x)], the coarsened blocks are scatteredthroughout the system.

The resulting grid or network is coarsened again by ap-plying the DWT to the scale coefficients obtained at the pre-vious level (which contain information about the permeabil-ities K(x) in the current coarsened model) and by calculatinga new set of four coefficients for each block of the coarsenedgrid. The resulting new detail coefficients are again set tozero if they are smaller than the threshold ∈d, and the corre-sponding blocks (or nodes) in the current network or grid arecoarsened. In effect, at coarsening level j, each block is com-pared with those at a distance 2j–1 from it, where the distanceis measured in units of the blocks’ length in the fine-gridmodel. This process is repeated until no significant numberof blocks or nodes is coarsened (or removed). Typically, afterthree or four levels, the grid is no longer effectively coars-ened—no significant reduction in the number of the nodesor blocks is obtained—thus efficiently yielding the finalcoarsened model for fixed ∈s and ∈d. In practice, the numberof coarsening levels depends on the original grid’s size and

the permeability distribution’s broadness. The extent of re-duction in the number of nodes or blocks depends on thethresholds ∈s and ∈d. Clearly, the higher the two thresholds,the larger the number of nodes or blocks is removed. Thethresholds’ numerical values are fixed by the level of detailwe would like to include in the final coarsened model and theamount of computation time we can afford.

After coarsening the blocks, the next important issue is theassignment of effective permeability of the larger coarsenedblocks (or the corresponding nodes). This can be done viaseveral methods, but the most straightforward one involvesreconstructing the coarsened model’s permeability distrib-ution (by computing the permeability distribution’ inverseDWT after each coarsening) and assigning the local per-meabilities based on the reconstructed distribution. This ap-proach’s key attribute is that it preserves all the importantfeatures of the original permeability distribution f(K).

In general, starting with a square network or grid, thecoarsened grid will contain three types of nodes:

• those that have four neighbors (six in 3D);• boundary nodes, for which the boundary conditions are

discretized in the usual way; and• those that are at the interface between coarsened and fine

blocks, or two blocks with different levels of coarseningor sizes.

The latter such nodes have only three (or fewer) neigh-bors and, therefore, the governing flow equations writtenfor such points cannot be discretized in the usual way. In thelanguage of the network models, some of the bonds con-nected to such nodes are missing, implying that their per-

Figure 7. An anisotropic (layered) permeability field and the resulting coarsenedgrid generated with the thresholds εd = 0.7 and εs = 0.8. Lightest and darkest regionscorrespond, respectively, to the highest and lowest permeabilities.


meabilities are zero. In reality, this is not the case; the zonesaround such nodes do have finite permeabilities.

To overcome this difficulty, we can use one of the followingtwo methods. One, we can resort to the method of finite vol-umes (FVs) for solving the flow equations in the coarsenedgrid. In this case, the coarsened grid still has a regular struc-ture, except that the block sizes are not all the same. Details ofthe blocks’ internal structure are irrelevant in the FV method.

Two, we can add an extra node or grid point to the modelat the center of the neighboring coarsened (larger) block,use the block’s corner points as neighbors of the added node,and hence increase the number of neighbors to four. Thegoverning equations at the added nodes are written in theusual manner, except that because the corner points are usedas neighboring nodes, the coordinates (x, y) must be rotatedat 45ο to the new coordinates (ξ, η) related to the (x, y) co-ordinates by the following equations:

, . (30)

We can use a similar technique in 3D. In such cases, thegoverning equations at the nodes are transformed to thisnew coordinate system. The resulting coarsened networkusually is completely irregular. Note, though, that the num-ber of added grid points is typically small. Figure 7 presentsthe grid structure we get with this method.

Both methods have their advantages and disadvantages.The first method’s advantage is that it is somewhat simpler

than the second. It generates a completely regular grid, al-beit with blocks of various sizes, and as such it easily can becoupled to any other simulator for computing other quan-tities of interest (for example, the rock’s elastic properties).This method’s disadvantage is that because we must use theFV method to solve for the governing equations, its accu-racy will be limited if the size of the coarsened blocks be-comes too large. In addition, it may not be obvious how todevelop a FV formulation of a problem that is more com-plex than the flow problem.

The second method’s advantage is its very high accuracy.Its possible disadvantage is that due to the irregular grid itgenerates, its use for solving problems more complex thanthe flow process might need a more detailed formulation,although use of structureless grids in numerical computa-tions has become quite common.

Figure 8 compares the effective permeability Ke of theoriginal fine-scale model with that of the coarsened grid Kcfor a range of H. The fine-scale model is a 1,024 × 1,024square grid in which the permeabilities of the blocks ornodes are distributed according to a fractional Brownianmotion. Hence, we must solve nearly 1.1 million equationsto determine the pressure distribution in the grid and itspermeability using Darcy’s law (Equation 1). In contrast, thecoarsened grid contains only about 4,500 nodes, which rep-resents a reduction by a factor of more than 200 in the totalnumber of equations to be solved (creating at least over twoorders of magnitude less computation time). The agreementbetween the two is excellent, with the difference betweenthem being less than 1 percent.

AcknowledgmentsI thank Dietrich Stauffer who insisted that I should writethis article, and should do so before the deadline that hehad set for me. I am also grateful to Bahram Dabir, Fate-meh Ebrahimi, Mehrdad Hashemi, Amir Heidarinasab,and Ali Reza Mehrabi who have worked with me on vari-ous applications of the wavelet transformations. Thepreparation of this article was supported in part by the Pe-troleum Research Fund, administered by the AmericanChemical Society.

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Fractals, Percolation, Cellular Automata, and Simulated Annealing,” Rev.Modern Phys., vol. 65, no. 4, 1993, pp. 1393–1534.

2. M. Sahimi, Flow and Transport in Porous Media and Fractured Rock, Wiley-VCH, 1995.

3. M. Hashemi et al., “Computer Simulation of Gas Generation and Trans-

η = −

22

( )x y ξ = +

22

( )x y


Figure 8. Dependence of Kc/Ke, the ratio of the effectivepermeability Kc of the coarsened grid using the wavelets, and Ke,the effective permeability of the fine-scale grid, on the Hurstexponent H and the threshold εd for εd = 0.9 (◊), εd = 0.8 (ο), εd =0.7 (υ), and εd = 0.5 (ν). For all cases, εs = 0.9. The fact that Kc/Keis extremely close to 1 indicates that the wavelet methodsuccessfully coarsened the original fine-scale model.

0

1.006

1.006

1.002

1.000

0.998

0.996

0.994

0.9920.1

H1

K c/K

e

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

JULY/AUGUST 2003 87

port in Landfills––I: Quasi-Steady-State Condition, Chemical Eng. Science,vol. 57, no. 13, 2002, pp. 2475–2501.

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36. M. Sahimi and M. Hashemi, “Wavelet Identification of the Spatial Distri-bution of Fractures,” Geophysical Research Letters, vol. 28, no. 4, 2001,pp. 611–614.

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39. A.R. Mehrabi and M. Sahimi, “Coarsening of Heterogeneous Media: Ap-plication of Wavelets,” Physical Rev. Letters, vol. 79, no. 22, 1997, pp.4385–4388.

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41. F. Ebrahimi and M. Sahimi, “Multiresolution Wavelet Coarsening andAnalysis of Transport in Heterogeneous Media, Physica A, vol. 316, no.4, 2002, pp. 160–188.

42. F. Ebrahimi and M. Sahimi, “Multiresolution Wavelet Coarsening andAnalysis of Miscible Displacements in Flow Through HeterogeneousPorous Media,” to be published in Transport in Porous Media, 2003.

43. M. Sahimi, A. Heidarinasab, and B. Dabir, “Computer Simulation ofConduction in Heterogeneous Materials: Application of Wavelet Trans-formations,” to be published in Chemical Eng. Science, 2003.

Muhammad Sahimi is professor and chairman of chemical and petro-

leum engineering at the University of Southern California. One of his

main areas of research interests is porous media. He received his BS from

the University of Tehran and his PhD from the University of Minnesota,

both in chemical engineering. He is a member of the American Institute

of Chemical Engineers, the American Physical Society, and the Society of

Petroleum Engineers. Contact him at [email protected].

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large-scale porous media and wavelet transformations

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