A homotopy series solution to a nonlinear partial differential equation arising from a mathematical model of the counter-current imbibition phenomenon in a heterogeneous porous medium

The mathematical model describing the counter-current imbibition phenomenon in a heterogeneous porous medium gives rise to a non-linear partial differential equation. This equation has been solved using ​homotopy analysis methods together with appropriate boundary and initial conditions. The solution represents the saturation of injected water during counter-current imbibition in a secondary oil recovery process, when water is injected. This saturation of injected ​water increases as the distance, X, from the imbibition phase ​increases, for a given time T>0. It is also deduced that the saturation of injected water in a homogeneous porous matrix is higher than the saturation of injected water in a heterogeneous porous matrix for the same distance X and time T>0. The numerical and graphical presentation of the saturation of injected water in heterogeneous as well as homogeneous porous ​matrices for distance X and time T>0 are obtained ​in Maple.