Numerical simulation of general hydrodynamic dispersion in porous medium

Abstract

A general two-dimensional equation of dispersion in a porous medium is presented. The second order linear partial differential equation describing the transient concentration distribution has mixed partial derivatives which is the result of treating the dispersion coefficients as second order symmetric tensors. Using the principles of calculus of variations a "functional" is developed for the dispersion equation that has mixed partial derivatives. The two-dimensional region is divided into triangular finite elements of arbitrary size and shape. The concentration is assumed to vary linearly over each triangular finite element. Minimization of the functional in combination with the finite element method leads to a system of simultaneous, first order, linear, ordinary differential equations. The matrix differential equation is numerically integrated using the fourth order Runge-Kutta and Adams-Moulton multistep predictor-corrector methods. Before proceeding with the use of the new functional, solutions were obtained for the dispersion equation with mixed partial derivatives in a rotated coordinate system. The numerical solutions using the new functional for one- and two-dimensional problems compared favourably with the available analytic solutions and the results obtained by finite element method that use a different functional. It was shown that the new functional can handle different ratios of lateral to longitudinal dispersion. A general stability criteria for the resulting matrix equation is developed. Stability dependent on the data is discussed in detail with examples. A brief description of the numerical instability is also given.