Analytical Model for Vertical Oil/Water Displacement Under Combined Viscous, Capillary, and Gravity Effects

摘要

A mathematical model for water injection in vertical porous media initially saturated with oil and water is presented. The mathematical formulation takes the form of a nonlinear convection-diffusion equation. Its contribution comes from consideration of the three chief forces (viscous, capillary and gravity) in oil recovery processes. The model is general in that it can use any shape for relative permeability and capillary pressure functions, and it is developed to allow analysis of these forces individually or in a combined manner. By using Corey-type functions for relative permeability, logarithmic functions for capillary pressure, and Peclet and Gravity dimensionless numbers, the flow equation is written in a dimensionless form. In order to represent the physics of the oil-water displacement more accurately, variable saturation-dependent coefficients for the diffusive (capillary) and convection (viscous and gravity) terms were used. Thus, a nonlinear equation is obtained. A numerical model based on the finite-difference formulation with a fully implicit scheme was implemented to obtain the solution to this equation. The analytic solution for the diffusion-convection equation for the semi-infinite problem published elsewhere and the well-known Buckley-Leverett solution were used to validate the numerical algorithm. The numerical model allows evaluation of the influence of each of these three forces on the magnitude and direction of the dimensionless water velocity. Water velocity is defined as the sum of the velocity contribution of each force (viscous, capillary and gravity). This model also helps determine favorable scenarios for each force. For instance, the analytic equation and the numerical results show the cases in which one force dominates the others, under given petrophysical and fluid properties and oil or water injection velocity. Finally, by setting the proper boundary and initial conditions, this model can be used to simulate any displacement in which these three forces interact.