application of homotopy perturbation method to nonlinear equations describing cocurrent and countercurrent imbibition in fractured porous media

in oil industry, spontaneous imbibition is an important phenomenon in recovery from fractured reservoirs which can be defined as spontaneous uptake of a wetting fluid into a porous solid. spontaneous imbibition involves both cocurrent and countercurrent flows. when a matrix block is partially covered by water, oil recovery is dominated by cocurrent imbibition i.e. the production of non wetting phase has the same direction of flow as the wetting phase. however if the matrix block is completely covered by water then countercurrent flow takes place, and the production of non wetting phase has an opposite direction of flow to that of the imbibing wetting phase. each of these processes can be described by a nonlinear partial differential equation (pde). in this paper, the homotopy perturbation method (hpm) which is a powerful series-based analytical tool, is used to approximate the solutions of cocurrent and countercurrent equations. hpm decomposes a complex partial differential equation under study to a series of simple ordinary differential equations that are easy to be solved. the solutions obtained by hpm are compared with that found using a common numerical method applied by matlab software. the difference between the two is seemed to be virtually negligible. a good agreement is also achieved from the comparison of the solutions obtained by hpm with those of a numerical method (nm).