Stability analysis and numerical simulation of 1-D and 2-D radial flow towards an oil well

The radial flow of oil towards a well in one and two dimensions is modeled by a family of finite difference schemes. This family depends on one parameter θ, 0 ≤ θ ≤ 1. The stability of the proposed schemes is analyzed applying the matrix method, which takes into account boundary conditions. Particularly, in the 2-D case, an “almost pentadiagonal” matrix is obtained choosing an appropriate order of equations and unknowns. We prove that this matrix may be symmetrized by a similarity transformation. Therefore, studying bounds for the corresponding eigenvalues, unconditional stability is found for θ ≥ 1/2 and stability restrictions are established for θ < 1/2. Numerical simulations are presented using the BSOR (Block Successive Over Relaxation) method to solve the resulting system of linear equations. The finite difference solution has perfectly reproduced the analytical solution of a simplified 1-D model.