In this paper, a numerical method for solving the problem of steam injection in the heavy oil reservoirs is presented. We consider a three-dimensional model of three-phase flow, oil, water, and steam with the effect of three-phase relative permeability. Interphase mass transfer of water and steam is considered, oil is assumed nonvolatile. We apply the simultaneous solution approach to solve the corresponding nonlinear discretized partial differential equation in the fully implicit form. The convergence of finite difference scheme is proved by the Rosinger theorem. The HJFNK (Heuristic Jacobian-Free-Newton-Krylov (JFNK)) method is proposed for solving the system of algebraic equations. The result of this proposed numerical method is well compared with some experimental results. Our experimental observations show that the first iteration of the full approximation scheme (FAS) provides a good initial guess for the Newton method. Therefore we propose a new Hybrid-FAS-HJFNK method while there is no steam in the reservoir. The experimental results show that the Hybrid-FAS-HJFNK method converges faster than the HJFNK method. Evaluation studies of cyclic steam injection in the heavy oil reservoir show the important roles of steam injection rate and steam injection period and the role of the soaking period is not noticeable.

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