On the exactly solvable equationSₜ = [ ( β S + γ )- 2 Sₓ ]ₓ + α ( β S + γ )- 2 Sₓ Occurring in Two-Phase Flow in Porous Media

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 Sₜ = g( S )Sₓₓ + f( S,Sₓ ) , dg/dS ≠ 0 which is exactly solvable is equivalent to the equation Sₜ = [ ( β S + γ )- 2 Sₓ ]ₓ + α ( β S + γ )- 2 Sₓ. 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.