CHAPTER 11

SCALING ISSUES IN POROUS ANDFRACTURED MEDIA

VINCENT C. TIDWELLSandia National Laboratories, P. O. Box 5800, MS 0735, Albuquerque, NM 87185, USA

The continuum hypothesis of rational mechanics forms the basis of most subsurfaceflow and transport models. In this approach, the time-and-space dependence of statevariables is expressed in the form of differential balance equations formulated onthe principles of mass, momentum, and energy conservation. To achieve tractablesolutions for the resulting balance equations, simplifying assumptions are used thattypically introduce constitutive properties into the flow/transport equations. Theseproperties (e.g., permeability, dispersivity) account for the integrated effects of het-erogeneities and physical processes that occur at scales much smaller than the desiredscale of analysis. Constitutive properties are related not to a discrete point within theporous media but to a control volume or sample support and are assumed to varysmoothly enough in time and space so that the resulting balance equations can besolved by standard analytical/numerical methods of differential equations.

Because of technological and computational constraints, it is rarely possible tomeasure constitutive properties at the desired scale of analysis. For this reason, someaveraging or scaling model is required to transfer information from the scale of mea-surement to the desired scale of analysis. If the averaging process of the particularproperty under study were known, the problem would be alleviated. For example,the average porosity of a volume is simply the arithmetic average of the porosities ofall the samples that constitute it. The simple arithmetic averaging process holds truefor additive variables such as porosity and ore grade. Unfortunately, many constitu-tive properties (e.g., permeability) are not additive; that is, the scaling process notonly depends on the volume fraction present but other factors as well. These factorsinclude, but are not limited to, the heterogeneous characteristics (i.e., length scales,variance, spatial patterns) of the medium (e.g., Gelhar and Axness, 1983; Dagan,1984; Fogg, 1986), the nature (e.g., linear vs. convergent flow) of the flow field (e.g.,Desbarats, 1992a; Indelman and Abramovich, 1994; Tidwell et al., 1999), and scale

201C. Ho and S. Webb (eds.), Gas Transport in Porous Media, 201–212.© 2006 Springer.

202 Tidwell

dependent flow/transport processes (e.g., hydrodynamic dispersion is superceded bymacrodispersion at larger length scales).

Below we briefly consider the role of scaling in the modeling of fluid flow andmass transport in porous and fractured media. We begin by reviewing some basicapproaches to modeling material property scaling. We then present examples fromthe laboratory and field demonstrating scaling effects. The focus of this section is onthe scaling of permeability and dispersivity.

11.1 SCALING THEORIES

Simple averaging rules (arithmetic, geometric, and harmonic) represent the point ofentry for classical scaling theory. These averaging rules are only valid for a narrowrange of aquifer/reservoir conditions. Specifically, an infinitely stratified formationcomprised of individual layers with constant permeability that is subject to steady, lin-ear flow. Where flow is oriented parallel to the stratification the effective permeabilityis given by the arithmetic mean ka,

ka =n∑

i=1

kidi

d(11.1)

where n is the sample set size, d is the total thickness of the system, and ki and, di

are the permeability and thickness of each discrete bed, respectively. When flow isoriented normal to stratification the harmonic mean kh, given by

kh = dn∑

i=1di

/ki

(11.2)

represents the appropriate averaging rule. These two effective permeabilities arederived by solving the governing flow equation (Darcy’s Law) for each layer inparallel and in series, respectively. Where the formation lacks spatial correlation,Warren and Price (1961) found the geometric mean to provide a good estimate of theeffective permeability. The geometric mean kg is given by

kg = exp

[1/n

n∑

i=1

ln(ki)

](11.3)

It follows that, ka > kg > kh.Unfortunately, most natural systems lie somewhere in-between these extremes as

do the appropriate scaling rules. The power law average provides a convenient meansof handling this wide range of behavior. The power law average kp is given by:

kp =[

1/nn∑

i=1

kωi

]1/ω

(11.4)

Chapter 11: Scaling Issues in Porous and Fractured Media 203

where ω is the power coefficient. With this approach the selection of an appropriatescaling rule is predicated on finding the proper power coefficient. Where flow isoriented parallel or normal to an infinitely stratified system ω is simply set to 1or −1 to yield an arithmetic or harmonic average, respectively. Alternatively, thegeometric mean is defined as ω = 0, and thus Equation (11.4) must undergo a limitedexpansion as currently defined. In general, empirical methods are used to determineω. Journel et al. (1986) and Deutsch (1989) provide algorithms for estimating ω forbimodal sand-shale sequences, while Desbarats (1992b) provides direction in the caseof log-normally distributed data.

Stochastic methods (e.g., Bakr et al., 1978; Gelhar, 1993) have been developed thatallow direct calculation of effective properties based on the heterogeneous character-istics of the aquifer/reservoir. Application of these methods requires an unboundeddomain, and uniform flow (i.e., the extent of the domain and the characteristic scaleof the flow nonuniformity are much larger than the correlation length scale of themedium). The permeability distribution must also be a weakly stationary and ergodicrandom variable with a relatively small variance (generally Var[lnk] < 1, how-ever under certain circumstances this limit can be exceeded). In this approach, theaquifer/reservoir is viewed as an ensemble of homogenous, isotropic blocks whosespatial distribution is fully characterized by its first two moments. Using a small per-turbation, first-order approximation of the governing stochastic differential equation,an expression for the effective permeability has been obtained. Gutjahr et al. (1978),extending the earlier work of Matheron (1967), found the effective permeability of aheterogeneous, isotropic medium to be:

keff = kg[1 − σ 2y /2] (11.5)

in one dimension,

keff = kg (11.6)

in two dimensions, and

keff = kg

[1 + σ 2

y

6

](11.7)

in three dimensions, where kg and σ 2y are the geometric mean and variance of the

natural log permeability distribution, respectively. An interesting result of this work isthe dependence of keff on the dimensionality of the flow domain. Gelhar and Axness(1983) extended this work to a 3-D statistically anisotropic medium. In this case, theeffective permeability is expressed as:

kii = kg[1 + σ 2y (0.5 − gii)] (11.8)

where gii is a geometric factor accounting for the degree of anisotropy and orientationof flow relative to the principal permeability axes and the subscript ii designates thetensor components. For the case of infinite stratification and flow parallel or normalto the bedding, Equation (11.8) reduces to that of an arithmetic and harmonic mean,

204 Tidwell

respectively. Indelman and Abramovich (1994) demonstrated gii to also be a functionof the shape of the covariance function.

Scaling has also been approached by purely numerical means. In this case (Ababouet al., 1989; Bachu and Cuthiell, 1990; Desbarats, 1987; Kossack et al., 1990) theeffective permeability of a spatially heterogeneous permeability field is determinedby numerically solving the steady-state flow equation for a prescribed domain. Real-izations of the heterogeneous permeability field are generated by statistical methods,based on a variety of principles ranging from sedimentological process models to geo-statistics (e.g., Koltermann and Gorelick, 1996). In formulating the flow model, caremust be taken to assign boundary conditions that reflect conditions the computationalelement will experience within the larger flow domain (Lasseter et al., 1986; Gomez-Hernandez and Journel, 1994). The statistical moments of the upscaled permeabilityfield can then be extracted though Monte Carlo simulation or from a single realizationprovided that the ergodicity requirements are satisfied. Use of numerical methods inapplied scaling problems is limited due to the excessive, sometimes insurmountablecomputational requirements.

In general, each of the aforementioned theories assume the porous media exhibits adiscrete hierarchy of scales (i.e., a finite correlation length scale exists). However, notall porous media behave in such manner, but rather exhibit a continuous hierarchy ofscales or continuous evolving heterogeneity (i.e., infinite length scale). A convenientmeans of modeling evolving heterogeneities is with fractals. In fact, a number ofresearchers have found geologic materials to display fractal characteristics. Fractalbehavior was noted for the case of sandstone porosity (Katz and Thompson, 1985),soil properties (Burrough, 1983), fracture networks (Barton and Larson, 1985), andreservoir porosity (Hewett, 1986). Upon examining transmissivity and permeabilitydata measured over scales of 10 cm to 45 km, Neuman (1994) argued that the datascale according to a power-law semivariogram (i.e., infinite length scale). Usingthe stochastic framework of Gelhar and the apparent power-law scaling, Neumanoffers a model for the effective permeability. The model predicts that the effectiveisotropic permeability will decrease with increasing sample support (i.e., samplevolume) in one-dimensional media, increase in three-dimensional media, and showno systematic variation within two-dimensional media. The concept of multifractalscaling of hydraulic conductivity distributions has also been developed to deal withdata sets where conductivity variations are more heterogeneous at smaller scales thanat larger scales (Liu and Molz, 1997).

Similar approaches have been used to predict effective transport parameters suchas the dispersion coefficient or macrodispersivity. The macrodispersivity tensor is ameasure of the influence hydraulic conductivity imposes on large-scale solute mixing.Calculations based on stochastic theory and assuming a finite correlation length scalepredict that the longitudinal macrodispersivity, Aij increases directly with the varianceof log hydraulic conductivity in the isotropic case (Gelhar and Axness, 1983)

A11 = σ 2y λ

γ 2(11.9)

Chapter 11: Scaling Issues in Porous and Fractured Media 205

where λ is the correlation length scale and γ = q1 / KgJ1 (where q1 and J1 arethe flux and average gradient in the longitudinal direction, respectively). In strati-fied media, the transverse mixing process is highly anisotropic; that is, the horizontaltransverse macrodispersivity in the plane of bedding is much larger than vertical trans-verse macrodispersivity associated with a direction perpendicular to that of bedding.Assuming finite correlation length scales, Dagan (1988) shows that the longitudinalmacrodispersivity grows with travel time to an asymptotic value that is independentof the anisotropy ratio. These travel distances are on the order of tens of horizontallog-conductivity correlation length scales to reach asymptotic mixing conditions. Onelimitation of these and similar theories is the assumption of mild heterogeneity (i.e.,σ 2

y < 1). For more heterogeneous materials, Neuman and Zhang (1990) employedquasi-linear theory to describe the evolution of the dispersion process.

Fractal concepts have also been used to explain the scaling of basic transport pro-cesses like diffusion (Spoval et al., 1985) and macrodispersivity (Wheatcraft andTyler, 1988). For porous media exhibiting evolving scales of heterogeneity, classicalFickian dispersive and diffusive concepts fail; that is the macrodispersivity continuesto grow with scale rather than reaching an asymptotic limit. Wheatcraft and Tyler(1988) developed a Lagrangian model for dispersion in a set of fractal streamtubes.They found the dispersivity is proportional to the straight-line travel distance raisedto a power of 2D − 1 where D is the fractal dimension. This evolving scale depen-dent behavior is seen when one plots dispersivity versus experimental scale (e.g.,Gelhar et al., 1992). Assuming a hierarchy of fractal media and quasi-linear the-ory, Neuman (1990) offers a theoretical basis for interpreting this long recognizedbehavior.

11.2 RESULTS FROM LABORATORY AND FIELD TESTSDOCUMENTING SCALING EFFECTS

Field and laboratory studies have been performed to gain a better understanding ofscaling processes. In general, these studies involve the measurement of some propertyover a range of different scales. However, the manner in which the scaling studies areconducted can vary significantly in terms of the measurement techniques employed,the range in sample support (i.e., sample volume) investigated, and the geologicmedia interrogated. Laboratory tests have the advantage that greater control over themeasurement process can be maintained, while field studies allow a larger range ofscales to be investigated.

One obvious approach to investigating scaling processes is to dissect a heteroge-neous medium and then reconstruct its effective properties from its component parts.Laboratory-based studies conducted by Henriette et al. (1989) reconstructed the effec-tive permeability of two 15 × 15 × 50 cm blocks of rock, one a sandstone and theother a limestone. The reconstruction was based first on permeability measurementsmade on 15 × 15 × 15 cm intermediate-scale blocks cut from the former and then300 core samples cut from them. Scaling was manifest as trends in the statisticalproperties of the measured permeability values. Specifically, the sample variance of

206 Tidwell

the permeability decreased with increasing sample volume. This is a common findingreflecting the fact that larger sample volumes integrate over more heterogeneity. Theinvestigators also found the sample mean of the permeability decreased slightly withincreasing sample volume. This is consistent with theory that suggests the effectivepermeability of a medium with short-range spatial correlation, as was the case here,approaches that of the geometric mean of the smaller scale measurements (e.g., Gelharand Axness, 1983).

An alternative approach to scaling investigations is to collect samples from the fieldthat are subsequently analyzed in the laboratory. For example, Parker and Albrecht(1987) acquired soil cores of three different volumes (92, 471, and 1770 cm3) takenfrom two different soil layers along closely spaced transects. Saturated hydraulic con-ductivities and solute dispersivities were then measured in the laboratory.As expected,the variance of the natural-log conductivity and dispersivity decreased with increasingcore volume. However, in this case the mean permeability was found to increase withincreasing sample volume, which was inconsistent with the short-range spatial cor-relation and linear flow imposed in the test. Upon closer inspection the investigatorsconcluded that the integrity of the smaller core samples had been compromised duringcollection. This highlights but one of the difficulties with conducting scaling experi-ments. That is, care must be taken to avoid introducing bias into the experiment dueto changes in sample integrity, sample density, or measurement precision/accuracyfor different scales of measurement.

In an effort to avoid such biasing, Tidwell and Wilson (1997, 1999a, 1999b, 2000)employed a consistent measurement device and sampling strategy to acquire per-meability data over a range of different scales. Specifically, they used a computerautomated minipermeameter test system with six different size tip seals, each provid-ing approximately an order of magnitude larger sample volume than the next smaller.Over 150,000 permeability values were collected from three, meter-scale blocks ofrock; including, two cross-bedded sandstones and one volcanic tuff. Characterizationof each block face involved high-resolution mapping of the heterogeneous permeabil-ity field with each of five different size tip seals, plus the collection of a single large tipseal measurement designed to interrogate most of the sampling domain. These exhaus-tive data sets, measured under consistent experimental conditions yielded empiricalevidence of permeability scaling (e.g., Figure 11.1). Specifically, as the sample sup-port increased the sample variance decreased, the semivariogram range increasedlinearly, while the small-scale (i.e., smaller than the tip seal) spatial structure waspreferentially filtered from the permeability maps and semivariograms. Although allthree-rock samples exhibit similar qualitative scaling trends, distinct differences werealso noted. These differences were most evident in the quantitative characteristics ofthe aforementioned trends and in the scaling of the mean permeability. These differ-ences in part can be explained on the basis of the spatial characteristics of the threerock samples and the divergent flow geometry imposed by the minipermeametertip seal.

Beyond the work of Tidwell and Wilson, there is a growing body of evidencedemonstrating the role of flow geometry on scaling processes. One example is the

Chapter 11: Scaling Issues in Porous and Fractured Media 207

-36 -32 -28 -24

ln(k)

-33

-32

-31

-30

-29

-28

0.6

0.6

0.40.20 0.8 1 1.2 1.4 0.60.40.20 0.8 1 1.2 1.4

E[l

n(k)

]

Inner tip seal radius (cm)

20

10

0

0 10 20 10

10

0

10

20

0

20

200 10 200

Digital image

Z-a

xis

(cm

)Y

-axis

(cm

)

X-axis (cm) Y-axis (cm) Z-axis (cm)

1.27 cm tip seal0.31 cm tip seal

Berea sandstone

Topopah tuff

0.01

0.1

1

10

Var

[ln(

k)]

Inner tip seal radius (cm)

Berea sandstone

Topopah tuff

(c)

(a)

(b)

(d)

–30.5 –30.0 –29.5 –29.0 –28.5

ln(k)

30

20

10

0

30

20

10

0

30

30

20

20

10

10

0

0 3020100 3020100

Figure 11.1. Permeability scaling measured on a block of Berea Sandstone and Topopah Spring Tuff. (a)Digital image of the Berea Sandstone sample and corresponding natural log permeability fields measuredwith two different size tip seals. (b) Same as (a) but for a block of Topopah Spring Tuff. (c) Scaling ofthe mean of the natural log permeability as measured on the two rock samples shown above (note thatpermeability maps for two of the tip seals are not shown). (d) Scaling of the variance of the natural logpermeability

208 Tidwell

tracer studies conducted by Chao and others (2000). Experiments involved a two-dimensional horizontal laboratory tank measuring 244 by 122 by 6.35 cm filledheterogeneously with different silica sands. Both uniform flow and convergent flowtracer tests were performed using a conservative tracer. Tests involving uniform flowyielded results consistent with classical stochastic theory that is the dispersivitiesapproached a constant value at large displacements. The convergent flow tests gavevery different results; specifically the dispersivities exhibited sustained scale depen-dence across a wide range of displacements. For the convergent tests, the dispersivitywas also noted to be dependent on both the spatial extent of the source term and itspoint of injection (even when the radial distance from the pumping well was constant).

To investigate transport scaling processes over a larger range of scales, experimentshave been conducted in the field under controlled conditions. These tests generallyincluded detailed descriptions of the subsurface geology followed by a natural and/orforced gradient tracer experiment. The advection, dispersion and reaction of thecontaminant plume are then monitored over time through the sampling of a densenetwork of boreholes. Efforts are then made to interpret the measured tracer behaviorwith that predicted by stochastic transport models.

One of the best-known field tests was conducted at the Borden Site in Ontario,Canada. Natural gradient experiments were conducted using a variety of conserva-tive and nonconservative tracers in this relatively homogeneous (Var[lnk] = 0.29)sand aquifer (Freyberg, 1986; Sudicky, 1986). In much the same manner, field testswere conducted in Cape Cod, Massachusetts (Garabedian, 1987; Hess et al., 1991).Natural-gradient tracer experiments conducted in this gravel aquifer extended overa travel distance of about 275m. Additionally, Moltyaner and Killey (1988a, 1988b)describe a 40 m long tracer test conducted in a fluvial sand aquifer at Chalk River,Ontario, termed the Twin Lakes site.

In each of these tests, macrodispersivities calculated from the field data werecompared with that predicted by the stochastic theory of Gelhar and Axness (1983)subject to the spatial distributions of conductivity characterizing each aquifer. Foreach of the field tests, favorable comparisons were found. As predicted by theory,longitudinal spreading dominated the evolution of the tracer plume with the transversedispersion playing a relatively weak role, particularly in the vertical direction.

More recently several natural-gradient tracer tests were conducted at the Colum-busAir Force Base in Mississippi, commonly known as the MADE (MacrodispersionExperiment) site. The MADE site is associated with a shallow alluvial aquifer that isat least one order of magnitude more heterogeneous than the aquifers noted above.Owing to the considerable heterogeneity in the measured hydraulic conductivity dis-tribution the classical Fickian advection-dispersion model had difficulty reproducingthe measured tracer behavior (Zheng and Jiao, 1998; Eggleston and Rojstaczer, 1998).Part of the problem was believed to be due to transport along preferential flow path-ways. Similar studies of tracer dispersion and diffusion in fractured dolomite requiredhigher-order models to explain mass-transfer between the fractures and host matrix(Haggerty et al., 2001).

Chapter 11: Scaling Issues in Porous and Fractured Media 209

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