2.1. Percolation Theory
 A system of pores that have low interconnectivity is best approached as a percolation problem [Sahimi, 1994; Hunt and Ewing, 2009]. Unfortunately, few hydrologists and soil scientists (the authors included) learned percolation theory as students, so the topic often seems a confusing morass from which we do not know how to extract the relationships we want. Here we present some basic concepts of percolation theory, as background for applying it to diffusive exchange between a porous sphere and the surrounding fluid.
 Percolation theory [Stauffer and Aharony, 1994] is a mathematical framework for describing the macroscopic properties that emerge in a system composed of many roughly equivalent parts, given the local degree of connection between those parts. In the context of this study, the parts are the individual pores within a single particle (grain, cobble, etc.), and the macroscopic properties of chief interest are the accessible porosity and the intragranular diffusion coefficient. Ewing et al.  discussed why it is reasonable to expect that many geological media have low pore connectivity; here we will briefly discuss how some properties emerge as a consequence of low connectivity. Extensive presentations of percolation theory in the context of porous materials are given in the work of Berkowitz and Balberg , Sahimi , and Hunt and Ewing .
 Consider a large two-dimensional lattice, for example a brick wall, and suppose that individual sites (e.g., bricks) are colored black with probability p, or white with probability 1 − p. As p increases from 0, black sites change from being rare and isolated, to occurring in clusters of black sites. Further increases in p increase the mean size of the black clusters, and eventually a threshold is crossed: the black clusters merge to form an “infinite cluster” that spans the entire brick wall, however large it may be. The value of p at which the percolation threshold is reached (the infinite cluster forms) is called pc, the critical probability. When p > pc, the white sites occur as disconnected clusters, and further increasing p will decrease the mean size of the white clusters until they become rare and isolated. Much of percolation theory can be developed from this simple sketch.
 When p = 1.0, all sites are black, so all sites are on the infinite cluster. When p < pc, no black site are on the infinite cluster because there is no infinite cluster. The fraction of black sites that belong to the infinite cluster (denoted P, the “power” of the infinite cluster) therefore goes from 0 for p < pc, to 1 at p = 1. As it happens, P is described by a simple power law: when p > pc, P ∼ (p – pc)β. This behavior is independent of (most) details of the lattice: even though square, triangular, cubic, and irregular lattices have different values for pc, the same power law behavior is seen in all of them, a phenomenon known as universality. The value of β is different in different dimensions, but values are known for both two and three dimensions (Table 1). A consequence of universality is that relationships developed for an easy case, say a square two-dimensional lattice, can then be applied (with some care) to irregular three-dimensional problems.
Table 1. Fundamental Percolation Exponents Discussed in This Study
|Exponent||Value in 2-D||Value in 3-D||Meaning|
|β||5/36a||0.41||Defines the fraction of the medium that is on the infinite cluster (the “power” of the infinite cluster).|
|ν||4/3||0.88||Defines χ, the fundamental length scale near pc. Indicates size-scaling.|
| ||1.13||1.34||Fractal dimension of the chemical path between two points on the same cluster; related to tortuosity.|
|μ||1.31||2.0||Describes conductivity near pc in an infinite medium.|
 A useful parameter is the correlation length, χ. For p < pc, χ is essentially the mean radius of the finite clusters, while for p > pc, χ is the mean radius of the holes in the infinite cluster. This has practical implications: if only the black phase is conducting, then pathways through the black infinite cluster are on average separated by a distance χ; this is why χ is also called the “mean separation of paths.” χ is known to scale as χ ∼ |p – pc|−ν, and the presence of the percolation exponent ν (Table 1) generally indicates some form of size scaling. χ functions as a characteristic length, and we find that systems behave homogeneously at length scales L ≫ χ, while behavior at scales less than χ is typically scale dependent. Because χ ∞ as p pc, the only operational length scale at pc is the system size.
 Suppose a system is near pc, and we want to know the length of a path between two given sites on the same cluster. The straight line or Euclidean distance, l, can be calculated trivially, but the connecting pathway, proceeding from site to connected site, is more complex. The length of this path is called the chemical distance, λ, because in a porous medium, with sites corresponding to pores, this is the shortest path available to a molecule traveling between the two sites. The path is fractal at scales greater than the scale of individual sites but less than χ; for these distances λ is related to the Euclidean distance as (Table 1). The ratio λ/l is therefore the tortuosity τ, the value of which must be scale dependent at scales greater than a single pore and less than χ [Ewing et al., 2010].
 As a final illustration, consider conduction through the infinite cluster. If p = 1, the system's conductivity, g, is clearly determined by the distribution of conductivities. But as p decreases toward pc, the individual conducting pathways become more tortuous (with ), while the separation between individual pathways becomes greater (with ν). At some point [Hunt, 2004; Ewing and Hunt, 2006] the topological considerations become more important than the conductivity distribution, and the system's conductivity is found to scale with proximity to the percolation threshold as g ∼ (p – pc)μ (Table 1).
 Instead of having sites be black with probability p (“site percolation”), we could have had bonds (the connections between adjacent sites) being active with probability p (“bond percolation”). The values of the critical probabilities, and of the prefactors needed to convert the scaling relationships to equalities, are specific to the lattice used (square, honeycomb, etc.), and whether it is site or bond percolation that obtains. However, the scaling relationships we have shown hold generally across these as well as other considerations. Our RW model uses bond percolation on a simple cubic lattice, so we know that pc ≈ 0.2488 [Stauffer and Aharony, 1994]. In a physical system (e.g., Borden sand), we do not know either p or pc, so different methods must be found to work with percolation concepts. Both conceptual and physical systems are considered in this study.
2.2. Model Development
 When accessible porosity is not uniform, a FD model needs local values of accessible porosity in order to convert between concentration (the driving force) and solute mass. Our FD model therefore requires equations giving local values of both accessible porosity, φa, and the diffusion coefficient, pm. When the sphere has low pore connectivity, these parameters vary as functions of both the proximity to the percolation threshold (i.e., with p – pc), and the intragranular distance l to the sphere's exterior. We address the accessible porosity first. Recall that the percolation exponent β gives the fraction of active sites that are on the infinite cluster, and in an infinite system, only sites on the infinite cluster would be accessible over great distances. Meanwhile, scaling with distance occurs for distances l < χ. The value of χ for a system above the percolation threshold is given by χ = a (p – pc)−ν, with a being a system-specific prefactor. The β and ν exponents combine to give an exponent –β/ν, which governs how accessible porosity varies with distance from the exterior. If p and pc are known (e.g., when comparing the FD model with the RW model) then a is a fitting parameter; for a physical system (e.g., the Borden sand) we instead consider χ to be a fitting parameter.
 At distances l > χ, φa takes a constant plateau value which we designate φp. Closer to the exterior, φa decreases with an exponent –β/ν. Equating these at l = χ defines a prefactor b ≡ φpχβ/ν, so
where the scaling with distance shows up through both the exponent ν, and the crossover in scaling at l = χ (Figure 1). With these relationships, and given χ and φp for a sphere of radius R, then φa for the whole sphere can be calculated as
and given any three of [R, χ, φp, φa], we can calculate the fourth.
 When modeling the sphere without distinguishing between the infinite and finite clusters (the “merged” model), equation (2) suffices to describe the porosity distribution. But when modeling the infinite and finite clusters as separate systems operating in parallel (the “split” model), their accessible porosities must be treated separately. All accessible porosity must be on either the infinite cluster (φi) or those finite clusters intersecting the edge (φf): φa = φi + φf. Accessible porosity on the infinite cluster varies with intragranular distance l to the grain's exterior as
 By definition, all accessible pores at distances l > χ belong to the infinite cluster (Figure 1). Consequently, accessible porosity due to the finite clusters, φf, is nonzero only for l < χ, and is given by the difference between equations (2) and (4):
When the sphere is small relative to the correlation length (i.e., when R < χ), there is no infinite cluster. In this case we consider that the split and merged models are identical.
 The FD model also requires a percolation-based estimate of the diffusion coefficient, pm. As with φa, two distinct scaling issues apply: scaling with distance l from the sphere's exterior, and scaling with p – pc, the proximity to the percolation threshold. As with porosity, diffusion scaling with distance has a crossover at l = χ, and is known to take the form [Stauffer and Aharony, 1994]
where the exponent ϑ ≡ (μ – β)/ν ≈ 1.807 in 3-D. As we would expect, this distance-scaling exponent has the exponent ν in the denominator.
 The second kind of scaling, scaling with p – pc for distances l > χ, was seen by Ewing et al.  to follow
They gave the exponent ψ ≈ 1.42 in 3-D, and conjectured that ψ ≡ (2 + ϑ)/2 . On the other hand, Havlin and Ben-Avraham  and Stauffer and Aharony  give the exponent ψ ≡ 2ν – β (≈1.35 in 3-D) for p < pc, which is to say, for finite clusters near criticality. This exponent can be interpreted as meaning that diffusion is restricted to a single pathway (on average) through each area χ2 (because the mean distance between paths scales with the exponent –ν), and is directly proportional to the fraction of the porosity that is on the infinite cluster (controlled by the β exponent). For comparisons against the RW model, we ran the FD model with both values of ψ.
 Because diffusion and conductivity are related, one might expect that the relevant exponent would be μ rather than ψ. The short explanation is that in our porous spheres we have diffusion on both the infinite cluster and the finite clusters; when diffusion is restricted to the infinite cluster, the μ exponent is appropriate. A fuller explanation [e.g., Stauffer and Aharony, 1994] is beyond the scope of this paper.
 Equation (7) is problematic in terms of actual implementation, most obviously because the endpoints are wrong. For example, we may want pm at p = 1.0 to take some known value (denoted k), and can easily normalize to obtain it. But equation (7) would also give pm = 0 at p = pc, which is only asymptotically true for infinite systems, in contrast to the finite spheres we are considering. Specifically, if we are working with a grain of size R ≤ χ, all intragranular pores lie within the region of decreasing porosity and increasing tortuosity (Figure 1). Within this region, all values of p between pc at the low end, and pc + (R/a)−1/ν at the high end, should give identical values of the diffusion coefficient (combining both the infinite and finite clusters): all of these systems behave essentially identically within a distance l < χ of the exterior [Ewing et al., 2010, Figure 10]. We therefore modify equation (7) to
This modification is further justified in section 5. Combining the two kinds of scaling (equations (6) and (8)) and normalizing to the known k at p = 1.0 gives
 The above discussion involves an artificial system for which p – pc is known. In contrast, a physical system's proximity to the percolation threshold is not generally known. In this case we combine the several unknowns, defining
which allows us to treat 0 as our second fitting parameter (after χ). The local diffusion coefficient is then simply given as