Improved Treatment of Dispersion in Numerical Calculation of Multidimensional Miscible Displacement

In previous papers by this author on numerical calculations of multidimensional miscible displacement, some simplifying assumptions were made in writing the dispersion term of the differential equation. It was assumed that the flow vector were essentially parallel to the x-axis. However, in most multidimensional miscible displacements, the flow vectors are not so simply oriented with respect to the coordinate axes. This paper derives the correct dispersion term for the more general case; gives a difference approximation for the dispersion term; and derives the stability criterion for the corresponding explicit difference equation. Introduction The differential equation for solvent concentration in miscible displacement is: .........(1) where v is the bulk flow velocity, C is concentration and D is the dispersion coefficient. For simple isotropic dispersion, D is a scalar quantity. however, dispersion in porous media is not generally isotropic since it is usually greater in the direction parallel to flow than in the direction transverse to flow. Hence, D must be treated as a tensor. Scheidegger has shown for an isotropic porous medium that, for the dispersion tensor to be invariant under coordinate transformations, there can be no more than two independent dispersivity factors; these are the longitudinal dispersivity D1, which acts in the direction of flow, and the transverse dispersivity Dt, which acts in the direction perpendicular to flow, In general, both D1 and Dt are functions of the magnitude of the flow velocity. In previous papers, the author made some simplifying assumptions in writing the dispersion term of the differential equation. It was assumed that the flow vectors were essentially parallel to the x-axis and, therefore, that the dispersion term. D C could be replaced by the sum: ...... (2) However, in most multidimensional miscible displacements, the flow vectors are not so simply oriented with respect to the coordinate axes. The purpose of this paper is to derive, without using tensor notation, the correct dispersion term for the more general case, to give a difference approximation for the dispersion term, and to derive the stability criterion for the corresponding explicit difference equation. DERIVATION OF DISPERSION TERM Let x, y be a fixed coordinate system and at any point let v be the velocity vector with magnitude v and angle, measured counter-clockwise from the x-axis. Let q be the vector which describes the rate and direction of flow of solvent due to dispersion. Consider a rotated coordinate system r, s where r is in the direction parallel to v, and s is in the perpendicular direction. Then, for an isotropic medium: SPEJ P. 213ˆ