Modelling Physical Dispersion in Miscible Displacement-Part 2: Validation, Numerical Tests, and Applications
Authors: Shrivastava V.K. , Nghiem L.X. , Moore R.G. , Okazawa T. 
Published: March 2010

Abstract
A new method for the incorporation of physical dispersion into miscible displacement models was developed in an accompanying paper(1). This paper presents the numerical tests for validation of the proposed formulation, and applications for demonstration of the method in certain practical situations. The examples include miscible displacement in a linear system, miscible displacements in a quarter of a five-spot pattern, and the application of the proposed method in the simulation of vapour extraction (VAPEX) process.

Introduction
Molecular diffusion and mechanical dispersion are the main mechanisms responsible for gas-oil mixing that occurs in miscible displacements. Together, these mechanisms are called physical dispersion and appear as the coefficient of the second order derivative of concentration in the convection-diffusion equation. The dispersion coefficient is tensorial in nature and requires special treatment for inclusion in the molar conservation equations, especially for models with complex reservoir geometries. A new method based on a multi-point control-volume procedure was developed for incorporation of physical dispersion in miscible displacement modeling(1) on 3D hexahedron structured corner-point grids. The method was implemented in a compositional simulator that has a provision for a higher order scheme(2) to reduce numerical dispersion.
This paper seeks to address validation, testing, and application aspects of the implementation. These are done on few single and multi-dimensional miscible displacement problems. The methodology consists of using full tensor formulation of physical dispersion based on a flux continuous multi-point control-volume procedure in conjunction with a scheme for the reduction of numerical dispersion. For the latter, wherever necessary, a higher order scheme with a total variation diminishing (TVD) flux limiter(2) is used. The thermodynamic calculations are done with the Peng-Robinson equation-of-state(3).

Model Validation
Validation of the Proposed Formulation With Analytical Solutions
These runs compare numerical solution obtained from the proposed method with the analytical solution for miscible displacement in models with simplified geometry and well-defined boundary and initial conditions, and under assumptions necessary for arriving at the analytical solution. First, the analytical solutions are briefly presented followed by the model description and numerical solution with the proposed method.
The Analytical Solution to 1D Convection-Diffusion Equation
The flow of a component in a phase can be described by the convection- diffusion (C-D) equation in 1D, dimensionless form(4):
Equation 1–5 (available in full paper)
where C, CI, and CJ indicate current, initial, and injected concentrations, and L denotes the distance over which longitudinal dispersion, De, is measured. The dimensionless time, tD, corresponds physically to the number of cumulative pore volumes injected into the medium. The other variables are defined in the nomenclature.
Equation (1) assumes flow of a single phase incompressible fluid in permeable media and ideal mixing. The analytical solution of Equation (1) for miscible displacement of an initially uniform fluid distribution in a finite length L under the boundary conditions:
Equation 6a-6c (available in full paper)
has the form of an infinite series of error functions where successive terms arise from the superimposed reflections at the outlet.