Mathematical modelling of flow towards an oil well

We describe the numerical approximations and applications of a mathematical model that governs the #ow of oil towards a well. The #ow of a single-phase #uid in a porous medium is governed by a parabolic equation obtained by combining the Darcyʹs and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial}boundary value problem is numerically solved by a "nite-di!erence family of numerical schemes, which depends on a parameter h. The stability*conditional or unconditional, depending on h*and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative e$ciencies are carefully assessed. TSMF is the fastest technique given that adequate values of h and time step *t are used*but *t must remain small. A combination of TSMF and block-SOR with variable *t seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments