Analytical Analysis of the Impact of Capillary Pressure Nonlinearity on Counter-Current Imbibition

The modelling of counter-current spontaneous imbibition in porous media is primarily driven by nonlinear partial differential equations. The formulation for the capillary pressure plays a crucial role in fluid transport dynamics. This study employs the reduced differential transform method to derive semi-analytical solutions for the one-dimensional imbibition equation. To assess how capillary pressure nonlinearity affects the water saturation profiles and the propagation of the imbibition front, we examine three different constitutive models: a baseline linear model, a logarithmic model, and an exponential model. The reduced differential transform method is applied to convert the governing nonlinear partial differential equations into algebraic recurrence relations, producing power-series solutions that maintain the system’s nonlinear characteristics. A comparative analysis shows that the logarithmic formulation leads to aggressive, singularity-driven displacement rates. Conversely, the exponential model mathematically captures the vanishing mobility of the non-wetting phase as the system approaches maximum wetting saturation, ensuring physical consistency and mathematical stability. Convergence analysis confirms that the truncated reduced differential transform method series is reliable across all tested nonlinearities. These findings demonstrate the effectiveness of the reduced differential transform method in addressing strongly nonlinear multiphase flow phenomena, making it a robust semi-analytical tool for analyzing displacement dynamics in porous media.