A Mathematical Model for Dispersion in the Direction Of Flow in Porous Media

Abstract

Introduction The problem of multicomponent single-phase flow through porous media is encountered in the study of petroleum reservoirs, gas chromatographic and ion-exchange columns, industrial fixed-bed contacters and elsewhere. These particular examples span the wide range of flow conditions possible; Reynolds numbers of less than 10 are not unusual in oil-production problems, while values in excess of 10 are common in large fixed-bed operations. The study of flow-dependent transport phenomena is complicated both by the changes in the character of the flow over this range, and by the irregularity of the flow boundaries inherent to porous media.Dispersion is one of the important phenomena known to depend fundamentally on flow conditions as well as on fluid and medium properties. As used in this paper, the term "dispersion" refers to the observed mixing of fluid elements of different composition which occurs in flow systems. The actual mechanism may be one or more of a number listed below. Only dispersion in the direction of the mean flow (referred to as axial or longitudinal dispersion (or mixing) is considered here, although lateral dispersion arises as part of certain coupled mechanisms. MECHANISMS FOR LONGITUDINAL DISPERSION A number of distinct mechanisms are known to contribute to the phenomenon of longitudinal dispersion. The more important of these are as follows.Molecular diffusion in the flow direction.Turbulent (cell) mixing.Lateral transport processes coupled with velocity and/or residence time distributions, including:"Taylor" diffusion caused by the interaction of velocity profiles in individual voids with lateral molecular diffusion;separation and remixing or interdiffusion of streams having different velocities around particles; andthe coupling of gross velocity profiles, caused by viscous instability or inhomogeneous porosity, with lateral dispersion.Finite mass-transfer rate between a porous matrix and the flowing phase, and finite diffusion rate inside elements of the porous matrix. It is a well documented fact that these mechanisms, whether acting individually or in combinations, produce essentially similar integral effects. The observed dispersion can be described approximately by solutions of the diffusion equation with a properly chosen dispersion coefficient.Necessary conditions for equivalence between dispersion and diffusion are discussed by Klinkenberg and Sjenitzer. Apparently these conditions are not satisfied exactly in many real systems. "Best fit" solutions of the diffusion equation often depart appreciably from measured breakthrough curves for the usual pulse- or step-forced experimental systems. In particular, pulse response curves usually show some degree of asymmetry and "tailing" which cannot be reproduced by the one-parameter diffusion model.The three-parameter model proposed herein attempts to produce these effects in a realistic manner. A symmetry and tailing are predicted for certain sets of parameter values, while for other sets the new model agrees with the diffusion model. VELOCITY DEPENDENCE OF DISPERSION Molecular diffusion apparently controls at sufficiently low Reynolds number in both gas and liquid systems. At Re greater than 20, turbulent cell mixing is dominant for gases flowing at low pressure; the data of McHenry and Wilhelms show that the mixing cell is approximately 1 particle diameter long in random sphere packs. SPEJ P. 49^