Anuj Pankajbhai Raval, Mitesh Sureshbhai Joshi

Analytical Solution of Co-Current Capillary Imbibition in Homogeneous Porous Media Using the Homotopy Analysis Method

Introduction

Analytical solution of co-current capillary imbibition in homogeneous porous media using the homotopy analysis method. Analytical solution for co-current capillary imbibition in homogeneous porous media using Homotopy Analysis Method (HAM). Solves nonlinear diffusion, revealing spatio-temporal dynamics and capillary rates.

Abstract

Co-current imbibition is a basic mechanism that controls multiphase flow in porous media. In this mechanism, the non-wetting phase is pushed by the wetting phase in the same direction due to capillary forces. Here, we provide an analytical study of co-current capillary-induced imbibition in a homogeneous porous medium. We neglect the gravitational and viscous effects of the non-wetting phase to focus on the pure capillary effect. The nonlinear governing partial differential equation is derived from Darcy’s law, mass conservation, and the capillary pressure-saturation relationship. This yields a Boussinesq-type nonlinear diffusion equation for the wetting phase saturation. To obtain an accurate semi-analytical solution, the Homotopy Analysis Method (HAM) is employed. This method provides a fast-convergent analytical series without the need for linearization or small-parameter assumptions. The symbolic computation and formulation are performed in Mathematica using the BVPH 2.0 package, which automatically handles boundary conditions and higher-order derivatives. The obtained solution reproduces the spatio-temporal dynamics of wetting phase saturation and correctly predicts the capillary imbibition rate in the porous system. The graphical analysis shows that the saturation front moves faster in co-current imbibition than in counter-current flow, which verifies the predominance of capillary forces in enhancing fluid displacement. The presented analytical framework proves the universality of the HAM Mathematica (BVPH 2.0) method for solving nonlinear transport problems in porous media and sets a standard for future research on capillary-driven infiltration and fluid recovery processes.

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