We consider the initial-boundary value problem for the two-dimensional generalized Benjamin-Bona-Mahony equation on upper half plane. This problem is closely related to the Buckley-Leverett equation for two-phase flow in a porous medium modified by including dependence of the capillary pressure on the rate of change of saturation. In this talk, we first introduce the background of this problem. Then we present our strategy for finding the classical solution of the equation.