A modified Green's function to estimate the interface stability in oil recovery

In C. Carasso and G. PaSa [Math. Modell. Numer. Anal. (M2AN) 32 (1998) 211- 221] a problem concerning the secondary recovery was considered: the oil co ntained in a porous medium is obtained by pushing it with a second fluid (w ater). If the second fluid is less viscous than the oil, the "fingering" ph enomenon appears, first studied in P.G. Saffman and G. Taylor [Proc. Roy. S ec. A 245 (1958) 312-329]. To minimise this phenomenon, an intermediate pol ymer-solute region, with a variable viscosity mu, is considered between wat er and oil, see S.B. Gorell and G.M. Homsy [SIAM J. Appl. Math. 43 (1983) 7 9-98]. The stability of the interface between the intermediate region and o il is governed by a Sturm-Liouville problem; the eigenvalues are present in the boundary conditions. A finite-difference procedure is used in C. Caras so and G. PaSa [Math. Modell. Numer. Anal. (M2AN) 32 (1998)211-221] to solv e this problem and to obtain an "optimal" viscosity mu in the intermediate region. In this paper, we prove the convergence of the finite-difference me thod for the above Sturm-Liouville problem. For this, we define a "modified " Green's function. (C) 2000 Elsevier Science S.A. All rights reserved.