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

摘要

A mathematical model for water injection in vertical porousrnmedia initially saturated with oil and water is presented. Thernmathematical formulation takes the form of a nonlinearrnconvection-diffusion equation. Its contribution comes fromrnconsideration of the three chief forces (viscous, capillary andrngravity) in oil recovery processes. The model is general in thatrnit can use any shape for relative permeability and capillaryrnpressure functions, and it is developed to allow analysis ofrnthese forces individually or in a combined manner.rnBy using Corey-type functions for relative permeability,rnlogarithmic functions for capillary pressure, and Peclet andrnGravity dimensionless numbers, the flow equation is written inrna dimensionless form.rnIn order to represent the physics of the oil-water displacementrnmore accurately, variable saturation-dependent coefficients forrnthe diffusive (capillary) and convection (viscous and gravity)rnterms were used. Thus, a nonlinear equation is obtained. Arnnumerical model based on the finite-difference formulationrnwith a fully implicit scheme was implemented to obtain thernsolution to this equation. The analytic solution for therndiffusion-convection equation for the semi-infinite problemrnpublished elsewhere and the well-known Buckley-Leverettrnsolution were used to validate the numerical algorithm.rnThe numerical model allows evaluation of the influence ofrneach of these three forces on the magnitude and direction ofrnthe dimensionless water velocity. Water velocity is defined asrnthe sum of the velocity contribution of each force (viscous,rncapillary and gravity). This model also helps determinernfavorable scenarios for each force. For instance, the analytic equation and the numerical results show the cases in whichrnone force dominates the others, under given petrophysical andrnfluid properties and oil or water injection velocity.rnFinally, by setting the proper boundary and initial conditions,rnthis model can be used to simulate any displacement in whichrnthese three forces interact.