Use of Fractional Derivatives for Fluid Flow in Heterogeneous Media

An original approach for modeling the dispersion of a chemical tracer through heterogeneous porous media is described in this paper. It is shown that, instead of introducing a time dependent dispersion coefficient, the presence of heterogeneities strongly modifies the structure of the transport equation. The presentation and the solution of this complicated equation are simplified by using the concept of fractional derivatives. Fractional derivatives are generalizations of the standard integration for a noninteger order q. In this study, the one-dimensional equation describing the transport of a tracer in a strongly heterogeneous medium (correlation at all the length’s scales) is demon strated. In such a medium, characterized by a parameter a (which can be related to a fractal dimension), the spreading width of a tracer varies as tα/², instead of t¹/² for standard dispersion, or t for pure convection in a layered medium. A gen eral transport equation for the concentration is derived using Laplace and Fourier transforms, which yield derivatives of a fractional order a in time and a second order in space. This form of transport equation can be reduced to the two limiting cases: (1) dispersive flow, which gives the standard convection-diffusion equation (a = 1), and (2), convective flow in layered media, which gives a wave equation (a = 2). The general equation represents a continuous crossover between dispersive and convective flow with only one tuning parameter (1 <a <2).