Solutions to Bentsen's equation with finite-element method

Abstract

Bentsen's equation is solved using the finite-element method. Based on the phenomena of variable inlet saturation observed in experiments, initial and boundary conditions are derived which can properly describe the inlet end effects. The inlet saturation has to be solved for as part of the solution to Bentsen's equation. A condition used to determine the inlet saturation is derived, and an iteration scheme is developed to evaluate the variation of inlet saturation. The present solution satisfies all the governing equations and the boundary conditions. The results reveal that the normalized inlet saturation is dependent on time and will rise to unity in finite time as observed in previous experiments. The behaviour of the inlet saturation can be fitted with an exponential function for a Corey medium ( n w= n w=2). Due to the inlet end effects, the solutions at early times are quite different from the Buckley-Leverett solution both in the fractional flow curve and in the saturation profile. For a large mobility ratio, a very long time is required for the normalized inlet saturation to reach unity. The gravity force has favorable influences by both increasing the breakthrough recovery efficiency and reducing the inlet end effects under the present simulated conditions. The breakthrough recovery efficiency from the Buckley-Leverett solution is a limiting value as the capillary number diminishes. The condition under which the capillary term can be neglected in evaluating the breakthrough recovery efficiency will be the product of C( S n) and N c ϖS n/ ϖξ, not the saturation gradient, ϖS n/ ϖξ, alone.