A New Optimal Growth Constant for the Hele–Shaw Instability

ISSN： 0169-3913

Source： Transport in Porous Media, Vol.49, Iss.1, 2002-10, pp. : 27-40

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Abstract

We study the immiscible displacement of the oil from a homogeneous porous medium by using a less viscous fluid (water). We use the Hele–Shaw model, then a ‘sharp’ interface exists between the fluids. The ‘fingering’ phenomenon appears, first studied by Saffman and Taylor (1958). Gorell and Homsy (1983) consider an intermediate region (I. R.) between water and oil, containing a polymer mixture. The unknown viscosity in I. R. is a parameter which can improve the stability of the I. R.–oil interface. A numerical optimal viscosity profile in I. R. is given. Carasso and Paşa (1998) obtain an explicit formula for an optimal viscosity profile in I. R. An upper estimation of the growth constant is given. In this paper, a ‘very slow’ viscosity profile in I. R. is defined and an optimal formula for the growth constant is obtained, less than the previous estimation of Carasso and Paşa. Moreover, this formula is similar with the Saffman–Taylor result, only the water viscosity is replaced by the limit value of viscosity in I. R. on the interface with the oil. We explain the apparent contradiction between the previous results of Gorell and Homsy (1983) and Paşa and Polisevski (1992).