Modelling Physical Dispersion in Miscible Displacement-Part 1: Theory and the Proposed Numerical Scheme
Authors: Shrivastava V.K. , Nghiem L.X. , Moore R.G. , Okazawa T. 
Published: March 2010

Abstract
Physical dispersion, comprising molecular diffusion and mechanical dispersion, is one of the primary fluid mixing mechanisms in reservoir processes dominated by compositional change. Its effect controls the characteristics and magnitude of oil recovery by miscible displacement. Standard compositional simulators used to model miscible displacement generally do not include physical dispersion effects and solve the governing equations by first order finite-difference scheme with singlepoint upstream weighting of mobilities. This approach leads to unphysical smearing of fronts (known as numerical dispersion), which is assumed to compensate for the physical dispersion. Unfortunately, this assumption is valid only for one-dimensional problems under very restrictive conditions and can lead to erroneous results in multiple dimensions.
The incorporation of physical dispersion in geologically complex models, such as the ones described by non-orthogonal corner-point grids, requires the use of advanced techniques of flux approximation to retain both physical and numerical accuracy. The use of the tensorial form of the permeability or dispersion coefficient becomes a necessity for convective or dispersive transport when flows are not aligned to the principal coordinate axes, which is almost always the case in practical reservoir simulation.
In this paper, a new dispersive flux-continuous scheme based on a multi-point control volume procedure is developed to allow the inclusion of the full tensor form of physical dispersion into compositional simulation of miscible displacement on 3-dimensional hexahedron structured corner-point grids.

Introduction
Gas injection into oil reservoirs results in a number of physical mechanisms that help in mobilizing and extracting the oil. Depending on the pressure, temperature, and the compositions of reservoir oil and the injected gas, immiscible or miscible displacements occur. The mobility ratio of displacing to displaced fluids and the gravity and capillary forces determine the extent of viscous fingering that would take place in a gas displacement process. Often the mass transport is affected by dispersion in different directions due to varying velocity gradients.
The oil recovery in a miscible displacement process depends on the size of the mixing zone between the injected fluid and the reservoir oil. For maximum oil recovery at breakthrough to occur, the mixing zone should remain small compared to the reservoir volume so that the oil produced is not diluted by the injected fluid. Ideally in a reservoir with a small mixing zone, for complete oil recovery, slightly more than one reservoir pore volume of injection fluid is required. However, if the mixing zone is large, several reservoir pore volumes of injection fluid may be needed to achieve complete recovery(1). On the other hand, the mixing due to diffusion and dispersion can dampen out viscous fingers in an unstable displacement, leading to increased sweep efficiency.
Dispersive mixing is caused by molecular diffusion and mechanical dispersion and is the main part of the mixing in miscible displacements(2). Molecular diffusion is a phenomenon whereby the transport of mass of a species (component) occurs within a single fluid phase from one point to the other in the direction of decreasing concentration.