A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium

摘要

In the present paper is proposed a new analytical solution to simulate the motion of the interface of two immiscible fluids when a gravitationally segregated displacement is occurring in a homogeneous tilted bed. The theory presented applies to an oil reservoir driven by water whereas it could be equally applied to a gas drive system or to a gas reservoir with aquifer support. The equation of motion has been obtained from the mass and linear momentum conservation laws and has been expressed in terms of interface local tilt angle. Through the development of the equation have been neglected the contribution from the gravity terms associated to the curvature of the interface but retained those dependent by interface inclination. The analytical solution is presented in terms of tilt angle and from it has been obtained, by proper integration, the formulations of: the interface position, the interface velocity and the fractional flow of water. The proposed solution proves to be in complete agreement with the classical segregated flow theory from Dietz since it predicts the same long term behavior of the interface in case of stable flow and has its equation for unstable flow as special case. Also the approach of averaging saturation by Dake can be regarded as a special case of the presented solution. To check the validity of the solution during the transient, it has been tested in various conditions against the solution of the complete equation of motion obtained through a numerical simulator which employs an explicit finite difference approximation. It proved to be in good agreement with the numerical solution particularly for unstable and conditionally stable flow but also for unconditionally stable condition, provided that the mobility ratio is not too small. In present days the problems of phase displacements are solved mainly by numerical simulator based on finite difference scheme which allows dealing with the highly complex environment of a hydrocarbon reservoir. In such schemes, due to the discrete form given to the domain representing the reservoir and to the equation of motion, the simulation carries numerical dispersion which can ruin the accuracy of the solution. In this respect, when the assumption of full pressure support, one dimensional displacement (i.e. line drive pattern) and segregated fluids are applicable to the problem under study, the proposed analytical solution can represent a useful tool to benchmark more sophisticated approach such as numerical simulation.