A modified Greenʹs function to estimate the interface stability in oil recovery Original Research Article

In C. Carasso and G. Paşa [Math. Modell. Numer. Anal. (M2AN) 32 (1998) 211–221] a problem concerning the secondary recovery was considered: the oil contained in a porous medium is obtained by pushing it with a second fluid (water). If the second fluid is less viscous than the oil, the “fingering” phenomenon appears, first studied in P.G. Saffman and G. Taylor [Proc. Roy. Soc. A 245 (1958) 312–329]. To minimise this phenomenon, an intermediate polymer-solute region, with a variable viscosity μ, is considered between water and oil, see S.B. Gorell and G.M. Homsy [SIAM J. Appl. Math. 43 (1983) 79–98]. The stability of the interface between the intermediate region and oil is governed by a Sturm–Liouville problem; the eigenvalues are present in the boundary conditions. A finite-difference procedure is used in C. Carasso and G. Paşa [Math. Modell. Numer. Anal. (M2AN) 32 (1998) 211–221] to solve this problem and to obtain an “optimal” viscosity μ in the intermediate region. In this paper, we prove the convergence of the finite-difference method for the above Sturm–Liouville problem. For this, we define a “modified” Greenʹs function.