A Class of Viscosity Profiles for Oil Displacement in Porous Media or Hele-Shaw Cell

Abstract

This paper is devoted to the study of the stability of oil displacement in porous media. Results are applied to the secondary oil recovery process: the oil contained in a porous medium is obtained by pushing it with a second fluid (usually water). As in Saffman and Taylor (1958) and Gorell and Homsy (1983) the porous medium will be modelized by a Hele-Shaw cell. If the second fluid is less viscous, the fingering phenomenon appears, first studied by Saffman and Taylor (1959). In order to minimize this instability, we consider, as in Gorell and Homsy (1983), an intermediate polymer-solute region (i.r.), with a variable viscosity \mu , between water and oil. This viscosity increases from water to oil. The linear stability of the interfaces is governed by a Sturm–Liouville problem which contains eigenvalues in the boundary conditions. Its characteristic values are the growth constants of the perturbations. The stability can be improved by choosing a ‘‘minimizing’’ viscosity profile \mu which gives us an arbitrary small positive growth constant. In this paper, we suggest a class of minimizing profiles. This main result is obtained by considering the Rayleigh quotient to estimate – without any discretization – the characteristic values of the above Sturm–Liouville problem. A finite-difference procedure and Gerschgorin’s localization theorem were used by Carasso and Pasa (1998) to solve the above problem. A formula of an exponential viscosity profile in (i.r.) was obtained. The new class of minimizing viscosity profiles described in this paper includes linear and exponential profiles. The corresponding total amount of polymer and the (i.r.) length are estimated in terms of the limit value of \mu on the (i.r.) – oil interface. Our results are compared with the previous theoretical viscosity profiles. We show that the linear case is more favorable compared with the exponential profile. We give lower estimates of the total amount of polymer and of the (i.r.) length for a given improvement of the stability , compared with the Saffman–Taylor case.