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  • Basic Transport Properties in NaturalPorous Media

    Continuum Percolation Theory and Fractal Model

    A. G. HUNT

    Received September 9, 2004; revision accepted December 7, 2004; accepted December 7, 2004


    P orous media include many man-made as well as natural materials.In fact, all solid substances are po-rous either to some degree, or at somelength scale. But the most familiar nat-ural media, which are porous enoughfor significant amounts of water or air

    to flow through, are soils and rocks.

    This article tells how it has finally been possible to predict

    the flow of air and water (and several other fundamental

    transport properties) in such natural porous media.

    It has not been easy for scientists to agree on the appro-

    priate description of such media. These porous media are

    disordered on many scales, all the way from the individual

    grains or pores, through nominally homogenous laboratory-

    sized samples and then field or plot sizes to formation sizes

    and larger, altogether over more than 12 orders of magni-

    tude of size. Fractal, multifractal, Gaussian, and log-normal

    models have all been proposed, probably in every scale

    range. The verification of a consistent theoretical framework

    for calculation of transport properties, at least at some

    scales, has the potential to eliminate much confusion re-

    garding both the appropriate theoretical techniques to use

    as well as the appropriate model to choose.

    The present description has two the-oretical inputs. The first is that power-law distributions of pore sizes best de-scribe the pore space and that suchdistributions are consistent with theself-similarity associated with fractalmodels [15]. The second is that theappropriate theoretical description offlow is based on percolation theory

    [6 9]. The choice to apply continuum percolation theory,rather than the site or bond variations, is required by thechoice of the fractal model. But the limitations of the fractalmodel are then predicted by continuum percolation theory.

    Using continuum percolation theory on the probabilisticfractal model, it is possible to predict the ratio of the unsat-urated to the saturated hydraulic conductivities [10, 11], thepressure-saturation relationships [12, 13] (including hyster-etic aspects [14]), solute and gas diffusion relationships [15]that are observed in experiment [16 21], the observed [22,23] air permeability [24] and the known [2] electrical con-ductivity relationships [25]. All parameters are consistentfrom one property to another. The values of the parametersthat describe the hydraulic properties can be predicted [11,12] from physical measurements [16, 19, 20] (or approxi-

    Correspondence to: Allen G. Hunt. E-mail: allenghunt@msn.comA. G. Hunt is at the Department of Physics and Departmentof Geology, Wright State University, Dayton, OH 45435.

    The first question asked bypercolation theorists is

    essentially, can particles or gasmolecules fit through the porespace and arrive at the other

    side of the system .

    22 C O M P L E X I T Y 2005 Wiley Periodicals, Inc., Vol. 10, No. 3DOI 10.1002/cplx.20067

  • mated from calculations [26]) and are not fit parameters.The understanding that is developed is universal.

    2. FRACTAL PORE-SPACE MODELIn porous media fractal models have been used to describethe solid volume, the pore volume, or the interface betweenthe two. Fractal models of pore space were developed ini-tially in the mid-1980s [1, 2] and used in the petroleumphysics and engineering communities. Some applicationswere to the pressure dependence of the saturation, some tohydraulic or electrical conductivity. Turcotte [3] proposed afractal fragmentation model, which identified a physicalbasis for the existence of fractal soils in the scale invarianceof the fragmentation of soil particles. Fragmentation can beviewed as the chief mechanism of physical weathering. Hereit will not be necessary to account for the present complex-ity in fractal modeling. In particular, it will be necessary todiscuss only two different fractal dimensionalities [11, 12].Ds, describes the solid space; Dp, describes the pore space.No particular geometrical assumptions need to be made.The values of Dp and Ds are related by symmetry [12], i.e.,particle quantities are obtained from pore quantities bysubstitution everywhere in an equation of 31 .

    The particular model used is of a probabilistic, trun-cated, continuous (rather than discrete) fractal. Model char-acteristics are, however, defined so that the porosity andwater retention functions are identical to those of the dis-crete and explicit fractal model of Rieu and Sposito [4](called hereafter the RS model). Moreover, integration overthe continuous pore size distribution between qr and r,where q 1 is an arbitrary factor, yields the contribution tothe porosity from each size class obtained by RS. Thus thepresent model is just a continuous version of RS.

    The distribution of pore sizes is defined by the followingprobability density function [11]:

    Wr 3 Dp

    rm3Dp r

    1Dp r0 r rm. (1)

    The power law distribution of pore sizes is bounded by amaximum radius, rm, and truncated at the minimum radius,r0. Equation (1), as written, is compatible with a volume, r

    3,for a pore of radius r. The result for the total porosityderived from equation (1) is [11]

    3 Dp




    r3r1Dpdr 1 r0rm3Dp

    , (2)

    exactly as in RS. If a particular geometry for the pore shapeis envisioned, it is possible to change the normalizationfactor to maintain the result for the porosity, and also main-tain the correspondence to RS.

    Using the substitutions, 31 , and Dp3Ds the resultis obtained [12]:


    . (3)

    As a reminder, in equation (2) r0 and rm refer explicitly tothe minimum and maximum pore sizes, and in equation (3)to the minimum and maximum particle sizes, respectively.However, I make the assumption, made long before [2729]that pore and particle radii are proportional to each other.Thus it must also be assumed that the ratio of r0 to rm is thesame for both the pores and the particles. Equations (2) and(3) also imply directly a property pointed out in RS, thattypically Dp Ds. But the basis for this relationship is thatusually 0.5. If 0.5, the relationship is reversed. Notethat in the fractal treatment of [2] and [30] the porosity isgiven by equation (3), but using Dp as the fractal dimen-sionality!

    It is possible, aside from such complications as individ-ual particle geometry, (including, especially the tendency ofclay particles to be flat, i.e., two dimensional) to develop anexpression for the surface area to volume ratio, A/V, of themedium, assuming that it is related to the ratio of thesurface area (r2) of all the particles to the volume (r3) of allthe particles [12],

    A/V r0

    rm r2r1Dsdr

    r0rm r3r1Dsdr

    3 Ds2 Ds 1rm rmr0 Ds2 11


    Equation (4) must be multiplied by 1 to give a surface areaper unit volume of the porous medium [12]. Though equation(4) is used only for later comparisons, it is given now becauseof its connection with the power-law pore size distribution andits absence of explicit connection with flow.


    3.1. The Upscaling ProblemIn heterogeneous or disordered media, the difficulty oftransporting mass or energy may vary greatly from place toplace. Thus the hydraulic conductivity may be a strongfunction of the position. The porous media communitiesinvestigate what they call upscaling for flow and trans-port. In physics terms this would be formulated as findingan effective transport coefficient in terms of its microscopic(or local) variability.

    In the hydrology and soil physics literature the mostcommon suggestion is that the upscaled K will lie some-where between the harmonic and the arithmetic mean ofthe individual K values. The basis of this statement is in two

    2005 Wiley Periodicals, Inc. C O M P L E X I T Y 23

  • obvious extreme results. One is that a collection of resistors(electrical or hydraulic) arranged in series has an equivalentresistance equal to the sum of all the resistances (the con-ductivity given by the harmonic mean conductivity, as longas all the resistances are of equal length). On the other hand,a collection of the same resistances configured in parallelhas an equivalent conductance equal to the sum of all theconductances (the conductivity given by the arithmeticmean conductivity).

    The question is: what fraction of the distribution of re-sistance values in a real medium should be considered asbeing configured in series and what remaining fractionshould be considered as in parallel?

    This question has been addressed for the electrical conduc-tivity [7, 8] of disordered solids, and the unambiguous answerobtained [31, 32] that the appropriate configuration is de-scribed by percolation theory. What percolation theory statesis that all resistances smaller than or equal to some criticalvalue must be considered to be in series with each other, butin parallel with the remainder of the distribution. The quantileof this critical resistance in the distribution of resistance valuesis equal to the percolation probability. The same method [33]was later validated [34] in network simulations of the saturatedhydraulic conductivity. Because OhmsLaw for electrical conduction and Dar-cys Law for fluid flow are mathemati-cally identical, this statement of topolog-ical and analytical equivalence is notsurprising. Percolation theory also per-formed better than stochastic (perturba-

    tion) approaches for geologic scale up-

    scaling problems [35]. As will be seen here all basic transport

    phenomena will be des


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