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<ul><li><p>Basic Transport Properties in NaturalPorous Media</p><p>Continuum Percolation Theory and Fractal Model</p><p>A. G. HUNT</p><p>Received September 9, 2004; revision accepted December 7, 2004; accepted December 7, 2004</p><p>1. INTRODUCTION</p><p>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</p><p>to flow through, are soils and rocks.</p><p>This article tells how it has finally been possible to predict</p><p>the flow of air and water (and several other fundamental</p><p>transport properties) in such natural porous media.</p><p>It has not been easy for scientists to agree on the appro-</p><p>priate description of such media. These porous media are</p><p>disordered on many scales, all the way from the individual</p><p>grains or pores, through nominally homogenous laboratory-</p><p>sized samples and then field or plot sizes to formation sizes</p><p>and larger, altogether over more than 12 orders of magni-</p><p>tude of size. Fractal, multifractal, Gaussian, and log-normal</p><p>models have all been proposed, probably in every scale</p><p>range. The verification of a consistent theoretical framework</p><p>for calculation of transport properties, at least at some</p><p>scales, has the potential to eliminate much confusion re-</p><p>garding both the appropriate theoretical techniques to use</p><p>as well as the appropriate model to choose.</p><p>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</p><p>[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.</p><p>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-</p><p>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.</p><p>The first question asked bypercolation theorists is</p><p>essentially, can particles or gasmolecules fit through the porespace and arrive at the other</p><p>side of the system .</p><p>22 C O M P L E X I T Y 2005 Wiley Periodicals, Inc., Vol. 10, No. 3DOI 10.1002/cplx.20067</p></li><li><p>mated from calculations [26]) and are not fit parameters.The understanding that is developed is universal.</p><p>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 .</p><p>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.</p><p>The distribution of pore sizes is defined by the followingprobability density function [11]:</p><p>Wr 3 Dp</p><p>rm3Dp r</p><p>1Dp r0 r rm. (1)</p><p>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</p><p>3,for a pore of radius r. The result for the total porosityderived from equation (1) is [11]</p><p> 3 Dp</p><p>rm3Dp </p><p>r0</p><p>rm</p><p>r3r1Dpdr 1 r0rm3Dp</p><p>, (2)</p><p>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.</p><p>Using the substitutions, 31 , and Dp3Ds the resultis obtained [12]:</p><p> r0rm3Ds</p><p>. (3)</p><p>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!</p><p>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],</p><p>A/V r0</p><p>rm r2r1Dsdr</p><p>r0rm r3r1Dsdr</p><p> 3 Ds2 Ds 1rm rmr0 Ds2 11 </p><p> .(4)</p><p>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.</p><p>3. BASIC RELEVANCE OF PERCOLATION</p><p>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.</p><p>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</p><p> 2005 Wiley Periodicals, Inc. C O M P L E X I T Y 23</p></li><li><p>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).</p><p>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?</p><p>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-</p><p>tion) approaches for geologic scale up-</p><p>scaling problems [35]. As will be seen here all basic transport</p><p>phenomena will be described by some combination of ideas</p><p>from percolation theory. The combination will vary according</p><p>to the property.</p><p>3.2. Percolation PerspectiveThe first question asked by percolation theorists is essen-</p><p>tially, can particles or gas molecules fit through the pore</p><p>space and arrive at the other side of the system [6]? The</p><p>answer is based on whether the pore space is continuously</p><p>connected. In case the system is below the percolation</p><p>threshold and the pore space is not continuously con-</p><p>nected, the answer is no; if above the percolation threshold,</p><p>one can also ask how the transport property varies with the</p><p>proximity of the system to the percolation threshold (i.e.,</p><p>how much gas, water, or solute arrives in a given time</p><p>frame). This perspective of percolation theory presumes its</p><p>usefulness only for systems near the percolation threshold</p><p>[35, 36]. This perspective is too narrow, and has contributed</p><p>to an underestimation of the value of percolation theory,</p><p>particularly in porous media. In order to develop a full</p><p>understanding of transport in porous media it is necessary</p><p>to be able to apply percolation theory to systems both nearand far from the percolation threshold.</p><p>As a basis to consider both uses of percolation theoryconsider an infinite square grid in 2D (like a very largewindow screen). Imagine cutting single wires (individualbonds) at random. What fraction, p, of bonds do you have tobreak before the structure, or screen, falls apart? That valueof p is called pc, and in this particular case pc 0.5. Theprobability, P, that an infinite continuous path of brokenbonds exists, is zero for p pc, but 1 for p pc. Similarly,emplacement of a fraction, pc, of the wires into their screenpositions will just construct an infinitely large connectedwire screen. This screen will have lots of holes, in fact holesof all sizes (meaning a fractal structure) but it will be con-tinuous. If, in such a system, all the local grid connectionsare identical in size, then the electrical conductivity, , ofthe screen can be accurately predicted as a function of pusing scaling concepts of percolation theory. Because theconducting portion of the medium near the percolationthreshold is self-similar, must vanish according to a pow-er-law in (p pc). The result is, (p pc)</p><p>t with t 1.88(1.27) in 3D (2D) [9, 37]. Although pc turns out to depend onthe particulars of the mesh geometry, i.e., whether it is</p><p>square, triangular, or hexagonal, thevalue of t turns out to be independentof nearly everything except the dimen-sionality of the system. If all the wires ofthe grid are present, but of very differ-ent radii, however, the optimal meansto calculate the electrical conductivityin terms of the variability of the wires</p><p>uses a more abstract application of percolation theory.Consider now that a large number of resistances with a</p><p>wide range of values are connected between random pairsof nearest-neighbor sites on such a regular grid. Such a gridcan also be used to represent an idealized problem in the(unsaturated) hydraulic conductivity and is known as anetwork model [38]. In particular one can imagine connect-ing cylindrical tubes with a wide range of tube radii betweenthe sites. Calculating the conductivity of the electrical ana-logue is somewhat more straightforward because in a realsituation, the right-angle junctions of different radius water-filled tubes could contribute to flow nonlinearities, but aslong as such complications are ignored, the fundamentalidea is identical for both cases. In this case pc is a moreimportant quantity than t above. If one knows the pc for anarbitrary regular grid representing a network model with aspread of pore sizes and thus connecting conductances, onecan find the upscaled K [39] or electrical conductivity. Val-ues of pc are tabulated for many specific cases [40].</p><p>Percolation theory used in the form of critical path anal-ysis (CPA) quantifies the characteristic resistance of thepath of least resistance [40, 41], and the critical, orbottleneck resistances on this path control the entire field</p><p>Percolation theory in the form ofcritical path analysis quantifiesthe characteristic resistance of</p><p>the path of least resistance.</p><p>24 C O M P L E X I T Y 2005 Wiley Periodicals, Inc.</p></li><li><p>of potential drops [34]. Consider a problem of a log uniformdistribution of resistance values with a distribution width 10orders of magnitude, e.g., from 100 to 1010 in arbitrary units.Place each resistor at random on a square lattice. Hydrolo-gists intuitively suspect that the conductivity of this ar-rangement is given in terms of the middle value, 105, orgeometric mean resistance [42] and this is correct becausepc 0.5. What this means is that, for the square lattice, onecan put 1/2 the resistors into grid positions at random, andit will be possible to find an interconnected path of theseresistances, which is guaranteed to reach infinite size (in aninfinite system). If one has chosen that half of the resistancedistribution with the smallest resistances, one has mini-mized the total resistance. In summary, a continuous pathfor current flow can just be found across an infinite squarelattice, which completely avoids resistances larger than themedian value.</p><p>For a square grid, the value of pc 0.5 is obtainedprecisely by the approximate formula, Zpc d/(d 1), withd the Euclidean dimensionality [43] and Z the number ofnearest neighbors for a given site. In two dimensions...</p></li></ul>


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