A theory of dispersion in a porous medium

This paper is concerned with the dispersion of a dynamically neutral material quantity in a fluid flowing through a porous medium. The medium is regarded as an assemblage of randomly orientated straight pores, and it is assumed that the path of a marked element of the material quantity consists of a sequence of statistically independent steps whose direction and duration vary in some random manner. The probability density function for the displacement of a single marked element is calculated and values for the dispersion of a cloud of marked elements then follow. The case is examined in which the flow satisfies Darcy's law (i.e. the mean velocity is linearly proportional to the mean pressure gradient), and the molecular diffusivity is sufficiently small for the dispersion to be primarily due to the randomness of the streamlines, but it is not assumed that effects of molecular diffusion can be altogether neglected. It is shown that the longitudinal dispersion in the direction of the mean flow may be described asymptotically by an effective diffusivity which is a function of U, l, a, K and T. (U denotes the average velocity, l the pore length, a the pore radius which is shown to be related to the permeability, k the molecular diffusivity, and T the time from the initial instant.) Expressions for the longitudinal diffusivity k_l are obtained according to the relative values of l/U, T, t_0 = (l^2)/2k and t_1 = (a^2)/8k. These are given in §4, equations (4.3), (4.4) and (4.5). Speaking roughly, when t0 [Gt ] T [Gt ] l/U,kl/Ul is alogarithmic function of UT/l and increases with T; when T [Gt ] t_0 [Gt ] l/U, which must eventually be the case however small k,k_l/Ul is alogarithmic function of Ul/k and independent of T. The theoretical results are compared with experimental data reported in the literature and approximate agreement is obtained when E is put equal to the average diameter of the particles composing the porous medium. The lateral dispersion in the direction perpendicular to that of the mean velocity is found to be governed asymptotically by an effective diffusivity k_t = 3/16 Ul. However, it is pointed out that some of the assumptions, namely that successive steps are statistically independent and that the dispersion of a cloud follows immediately from the statistical properties of the displacement of a single marked element, may not be valid for the lateral dispersion and this result is therefore suspect. Remarks are made in §5 on the dispersion for high values of the Reynolds number Ul/v (v = kinematic viscosity) when Darcy's law is not obeyed, and it is argued that k_l/Ul should decrease as Ul/v increases