On some numerical methods for solving 2D radial flow towards an oil well

In this paper we study a family of finite difference schemes in two dimensions to model the single phase flow of oil through heterogeneous porous media. That family depends on one parameter θ, 0 ≤ θ ≤ 1. Using a suitable order of equations and unknowns, a linear system of equations, with a particular structure, is obtained. The corresponding matrix, excluding the first row and column, has up to five elements in each row, arranged in five diagonals. The system of linear equations is solved by a method based on Taylor series of matrix functions (TSMF). The convergence conditions for this technique are established and the most convenient θ is selected to increase the time step Δt. Besides, TSMF is compared with two iterative methods, ADI and block-SOR, usually applied to solve multidimensional equations. Both methods, ADI and block-SOR, are adapted to this particular problem. We conclude that TSMF is the fastest technique using adequate values of θ and Δt, but the time increment Δt must remain small because of the convergence condition. On the other hand, block-SOR converges using large values of Δt, but it uses a large amount of CPU time. ADI is discarded for not presenting advantages over the other two techniques. Therefore, TSMF is recommended when a short period of time must be simulated, and block-SOR is suitable for long simulations and applying a variable time increment.