Nonlocal Diffusion in Porous Media: A Spatial Fractional Approach

Abstract

One dimensional diffusion problems in bounded porous media characterized by the presence of nonlocal interactions are investigated by a fractional calculus approach. Darcy's constitutive equation is assumed of convolution integral type and a power law attenuation function is implemented. Analogies and differences of the flow rate-pressure law with respect to other nonlocal and fractal models are outlined. By means of the continuity relationship, the fractional diffusion equation is then derived. It involves spatial Riemann-Liouville derivatives with non-integer order comprised between 1 and 2. The solution is obtained numerically via fractional finite differences and results are presented both in the transient and in the steady-state regimes. Eventually, the physical meaning of fractional operators is discussed and potential applications of the analysis are suggested.

Discrete mass-spring structure identification in nonlocal continuum space-fractional model

“…Consequently, in eqs. (24) and (30) we assume that: i) for i = 1: m = 1, and M = 0, and for all U in eqs. (22) and (28), a forward scheme is applied; ii) for i ∈ {2, 3, .…”

“…FC introduces a continuum-type description through the definition of the fractional derivative (FD) (derivative of an arbitrary order [13-16]), revealing certain hidden aspects of the discrete nature of the phenomena analysed. Furthermore, FC has an indisputable attribute, namely, that FC provides an infinite number of possible FD realizations [17,18]; thus we can identify a specific type of FD depending on the experimental evidence.An important family of FM concepts includes formulations where, in classic continuum mechanics (CCM) theory, the integer differential operators acting on a spatial variable are replaced by fractional operators; such models are called space-fractional continuum mechanics (s-FCM) [19][20][21][22][23][24][25]. The physical interpretation of s-FCM is that FD homogenizes, in a phenomenological sense, the underlying microstructure [26]; thus, the scale effect is included within FC, and the s-FCM model is nonlocal.…”

“…An important family of FM concepts includes formulations where, in classic continuum mechanics (CCM) theory, the integer differential operators acting on a spatial variable are replaced by fractional operators; such models are called space-fractional continuum mechanics (s-FCM) [19][20][21][22][23][24][25]. The physical interpretation of s-FCM is that FD homogenizes, in a phenomenological sense, the underlying microstructure [26]; thus, the scale effect is included within FC, and the s-FCM model is nonlocal.…”

Evaluating the Impact of the Covariance of the Friction and Potential Gradient on Describing the Infiltration Process

“…The memory effect is attributed to chemical reactions between the porous medium and the pore fluid and due to the transportation of particles through the channels (Caputo 2000 ; Sapora et al. 2017 ). Pachepsky et al.…”

Diffusion Process and Reaction on a Surface

“…Thus, species 1 reacts promoting the formation of species 2, in other words, following a first-order irreversible reaction 1 → 2. This reaction can be described by a kinetic equation [21,24,25]. In order to describe these processes, the model considered here is a set of coupled equations that may represent reaction, adsorption, and desorption processes Figure 1: Illustration of the interactions occurring between particles and, for example, a catalyst surface.…”