PHYSICAL REVIEW E 85, 066316 (2012) Transport in fractal media: An effective scale-invariant approach H. Hernandez-Coronado, M. Coronado, and E. C. Herrera-Hernandez Instituto Mexicano del Petr´ oleo, Eje central L´ azaro C´ ardenas 152, 07730, M´ exico D.F., Mexico (Received 16 February 2012; published 20 June 2012) In this paper an advective-dispersion equation with scale-dependent coefficients is proposed for describing transport through fractals. This equation is obtained by imposing scale invariance and assuming that the porosity, the dispersion coefficient, and the velocity follow fractional power laws on the scale. The model incorporates the empirically found trends in highly heterogeneous media, regarding the dependence of the dispersivity on the scale and the dispersion coefficient on the velocity. We conclude that the presence of nontrivial fractal parameters produces anomalous dispersion, as expected, and that the presence of convective processes induces a reescalation in the concentration and shifts the tracer velocity to different values with respect to the nonfractal case. DOI: 10.1103/PhysRevE.85.066316 PACS number(s): 47.53.+n, 05.60.Cd, 05.45.Df, 47.56.+r I. INTRODUCTION Transport through homogeneous media is described by the usual advection-dispersion equation (ADE) with constant coefficients [1,2]. For highly disordered heterogeneous media, whose properties exhibit variations over a wide range of length scales, the standard ADE does not describe transport phenomena properly (see [3–5] and references therein). In such cases, it has been found that dispersion [namely, the mean square displacement 〈r 2 (t )〉 of an initial deltalike plume] grows slower or faster than the classical t rate (see, e.g., [6,7]). Such behavior is referred to as anomalous transport and different approaches and tools have been used for describing it; they range from stochastic differential equations and random walks to percolation, scaling, and renormalization theory [4,5,8–12]. Most approaches giving rise to anomalous dispersion have been based on random walks in lattices. The reason for subdiffusion turns out to be that lattice sites bind the tracer particle very strongly, while superdispersion occurs due to the fact that very far distant regions result to be directly connected. Just as the probability to find the regular random walker in a position r at time t can be described effectively by the usual dispersion equation in the continuous limit, so the previous random walk process in the lattice can be described by probability distribution functions with long-tailed waiting times and jump lengths, obeying fractional dispersion equations as a continuous model [4]. This fractional dispersion equation is nonlocal and in general lacks a clear physical and geometrical interpretation [13,14]. Other studies that have led to anomalous behavior consist of ADE-like models with scale-dependent properties obeying power laws (see, e.g., [12,15]). In opposition to the fractional equations approach, these models are local and therefore they can be treated within the standard partial differential equations theory. The physical motivation for introducing power laws in the medium properties is based on heuristic arguments from scaling and percolation theory. Of particular interest is the previous work by O’Shaughnessy and Procaccia [12], where the dispersion equation for Euclidean lattices is generalized to the case of fractal lattices according to ∂P ∂  t − 1  r D−1 ∂ ∂ r   r D−θ −1 ∂P ∂ r  = 0, (1) where P is the probability density for a random walker on a fractal lattice,  r and  t are dimensionless variables, and D and θ are the fractal dimension and a permeability related connectivity index, respectively. In this work an advective-dispersion equation is proposed for describing transport through fractals, in the spirit of O’Shaughnessy and Procaccia’s approach. The model pro- posed here is obtained by demanding scale invariance, a transformation produced by a conformal group element (see, e.g., [16,17]). The fractal character of the medium is encoded in two fractal exponents: a mass fractal dimension D and a modified spectral dispersion coefficient α. Here α is used for convenience instead of the connectivity index θ ∗ , where θ ∗ is the connectivity index related to dispersion (as defined by Sahimi in Ref. [18]), which should not be confused with the connectivity index associated to permeability θ , as used in Eq. (1) [12]. In our case α = θ ∗ + 2. The fractal dimension provides the fraction of the Euclidean space that is occupied by the fractal system while α—together with D—determines the effective transversal flux area. We must supply D and α to describe the dynamic properties of the network because α contains information about the topology which is not in one-to-one correspondence with the mass scaling (D). We found that whereas the presence of both parameters produces anomalous transport, the effect of the spectral dispersion coefficient on the transport behavior is dominant. The rest of the paper is organized as follows. In Sec. II the scale-dependent properties are proposed following percolation arguments, and the model is defined. In Sec. III the general solution of the problem is found and two particular cases are considered. In Sec. IV the effects of the fractal parameters on the transport and its physical interpretation are discussed. Finally, we briefly present the conclusions in Sec. V. II. MODEL In nonporous media embedded in an Euclidean space, the concentration of a tracer C(x,t )—in the absence of sources— obeys the advection-dispersion equation given by ∂C ∂t + ∇ · (Cv − D dis ∇C) = 0, (2) where v and D dis are the tracer’s velocity and the dispersion tensor, respectively. When the tracer is transported (by a 066316-1 1539-3755/2012/85(6)/066316(8) ©2012 American Physical Society