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

Abstract

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. INTRODUCTION At the end of primary recovery processes, when pressure in the reservoir has depleted, it is necessary to provide extra energy to the reservoir; one way to achieve this is by water injection. Waterflooding requires previous studies to forecast the performance and pattern of the displacement and to select the best recovery conditions.

Practical Mathematical Model for the Evaluation of Main Parameters in Polymer Flooding: Rheology, Adsorption, Permeability Reduction, and Effective Salinity

“…For fractional flow theory with Corey-type equations, one of the easiest expressions used is the end-point water–oil mobility ratio, M *. 14 , 63 , 64 …”

“…As a measure of mobility control for the displacement, the mobility ratio concept is used. For fractional flow theory with Corey-type equations, one of the easiest expressions used is the end-point water–oil mobility ratio, M *. ,, …”

“…For fractional flow theory with Corey-type equations, one of the easiest expressions used is the end-point water−oil mobility ratio, M*. 14,63,64 For waterflooding, one displacement front is obtained because viscosities and relative permeabilities are not affected during the process, so M* is constant in time and distance. Nevertheless, during polymer flooding, a water front and a polymer-rich front are presented because aqueous phase viscosity is a function of time and distance.…”

Modeling of Carbon Dioxide Leakage from Storage Aquifers

“…Equation 17is a nonlinear second order partial differential equation, which is also known as nonlinear convection-diffusion equation. A similar form of this equation has been developed before for the application of water flooding in oil reservoirs [32]. The first term in Equation 17is the accumulation term; the second term includes the contribution of capillary forces in the displacement process and has a diffusive nature; the third term contains the gravitational contributions to the flow and has an advective character; and the fourth term is the contribution of viscous forces and same as the third term has an advective character.…”