On the exactly solvable equation$S_t = [ ( \beta S + \gamma )^{ - 2} S_x ]_x + \alpha ( \beta S + \gamma )^{ - 2} S_x $ Occurring in Two-Phase Flow in Porous Media

Abstract

The symmetry approach of Fokas [J. Math. Phys. 21(1980), pp. 1318–1325] is used to show that the most general equation of the form $S_t = g( S )S_{xx} + f( S,S_x )$ , $dg/dS \ne 0$ which is exactly solvable is equivalent to the equation $S_t = [ ( \beta S + \gamma )^{ - 2} S_x ]_x + \alpha ( \beta S + \gamma )^{ - 2} S_x $. This equation is then mapped to the linear heat equation through a series of Bäcklund transformations. The above result is used to solve an initial-boundary value problem modeling the process of waterflooding, i.e., the displacement of oil, initially in a porous reservoir, by continuously injected water.