A New Mathematical Model For 1D Miscible Displacement
Authors: Tan J. , Bentsen R.G. 
Published: March 2010

Abstract
This paper presents a new model for one-dimensional miscible displacement. The model was constructed using Darcy's law, Fick's dispersion law and the continuity equation. The effects of viscosity contrast between the displaced and displacing fluids and the heterogeneity of a porous medium can be investigated using the model. An approximate analytical solution to the model was developed and it indicates that the transition zone grows with the square root of time at early times when dispersion dominates. As the displacement goes on, the transition zone grows linearly with time when the effect of dispersion is negligible.

Introduction
When two fluids mix in all proportions and their mixtures remain a single phase, they are called first-contact miscible. When one such fluid is injected into a porous medium, which is pre-saturated with the other miscible fluid, a transition zone is developed wherein the fluid compositions change continuously from the injected to the displaced fluid. If the porous medium is homogeneous, and if the miscible fluids have the same viscosity, the performance of the process can be described by the traditional convection- dispersion equation(1). The solution to this equation for an infinitely long, one-dimensional system indicates that the concentration of the displaced fluid in the effluent versus time (effluent curve) is sigmoid and the length of the transition zone increases with the square root of time(2). If the injected fluid has a lower viscosity than that of the displaced fluid and/or if the porous medium is not homogeneous, the effluent curve may demonstrate a skewed "S" shape and the transition zone grows linearly with time(3–7).
A significant effort has been made to develop mathematical models capable of describing adequately the miscible displacement process(3, 6–10). If the porous medium is homogeneous, and if the fluid viscosities are equal, Fick's diffusion law may be combined with the continuity equation to obtain the standard dispersion- convection equation which may be expressed as follows:
Equation 1 (available in full paper)
As noted by Mannhardt and Nasr-El-Din(11), this equation predicts that the transition-zone length grows with the square root of time, and that the effluent concentration curve is symmetrically "S" shaped under certain boundary conditions. If the assumptions concerning homogeneity of the formation and equality of the fluid viscosities are violated, Equation (1) is no longer able to describe adequately the miscible displacement process. In particular, for inhomogeneous formations and/or unmatched fluid viscosities, the effluent concentration curve may no longer have a symmetrical "S" shape, and the length of the transition-zone no longer grows with the square root of time(2, 7). To enable the prediction of unfavourable viscosity ratio displacements in inhomogeneous formations, some researchers(6, 10) have suggested using immiscible, two-phase flow theory to describe the miscible displacement process. When this approach is taken, dispersion is neglected and the transition zone grows linearly with time. However, it can be inferred from recent experimental work(7) that, when the porous medium is inhomogeneous, the transition zone grows at a rate intermediate between that predicted by the standard dispersionconvection equation and that predicted by immiscible two-phase flow theory.