Numerical simulation of three-phase coning in petroleum reservoirs

Abstract

A study of computer simulation of three-phase flow in petroleum reservoirs with particular reference to coning problems is the topic of the present investigation. The mathematical formulation of flow equations is treated in detail and the implications of physically correct as opposed to conventional boundary conditions are clarified. The proper construction of finite difference approximations is discussed. A new finite difference formulation of the boundary conditions is developed, which satisfies the outlet effect and the condition of compatibility between the flow in the reservoir and in the wellbore. Several mathematical properties of the system of difference equations are proved and the methods for solving the resulting system of algebraic equations are surveyed. The investigation of various methods for handling the nonlinearities shows that Newton's method can be used to obtain a fully implicit solution in few iterations. The stability and truncation errors of Newton's method are compared with the linearized and semi-implicit method on test problems . A new method is proposed for handling the nonlinearity due to accumulation terms. The numerical coning model developed on the basis of these theoretical investigations is highly stable, simulates correctly the boundary effects and has a small time-step sensitivity. These features are demonstrated on test problems of two-phase coning and three-phase coning with gas percolation. The investigation of the fea sibility of Newton's iteration shows that best results are obtained if only one iteration per time step is performed and smaller time steps are selected, instead of performing several Newton's iterations with extremely large time steps. The numerical model is used to study the influence of some variables on coning, including capillary pressure and wettability, and to investigate some possibilities of controlling coning.