A Diffusion Model to Explain Mixing of Flowing Miscible Fluids in Porous Media

Abstract

Published in Petroleum Transactions, AIME, Volume 210, 1957, pages 345–349. Abstract This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. Introduction A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media. The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod and of von Rosenberg are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara, and recently in this country by Day. Yuhara arrives at his conclusions through analogy with turbulent diffusion, describing the microscopic velocities in terms of "eddy motion" although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger's concept of the dispersion of velocities. After a very clear description of Scheidegger's theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1.