A unified fractal analytical model for spontaneous imbibition in rough porous media considering dynamic wettability and gravitational effects

Dynamic wetting on rough walls is crucial to the mass transfer mechanism of gas-liquid spontaneous imbibition in multiscale porous energy systems. However, obtaining analytical solutions for dynamic wetting in rough porous media under gravity remains challenging. In this work, we propose a novel unified mathematical model that explicitly couples the static Wenzel thermodynamic correction with the kinetic energy dissipation of the moving contact line, while systematically incorporating gravitational effects. Specifically, by introducing the Lambert W function, we transform the complex nonlinear ordinary differential equation (ODE) arising from the coupling process into a closed-form explicit analytical solution. In this model, fractal theory is applied to characterize wall roughness and facilitate macroscopic upscaling. The accuracy of the proposed model is validated against existing experimental data. Through comprehensive simulations, pore-scale control mechanisms and dimensionless scaling are investigated. The results indicate that surface roughness statically suppresses capillary forces and the equilibrium distance in micropores. Furthermore, dynamic wetting nonlinearly hinders imbibition, exponentially prolonging the equilibrium time in large pores. This effect enables intermediate pores to maintain the longest high-speed stage and the maximum dynamic contact angle. Temporally, spontaneous imbibition is sequentially governed by fractal roughness, dynamic wetting, and the Bond number. Macroscopically, multi-parameter sensitivity analysis demonstrates that neglecting dynamic wetting severely overestimates early imbibition in high-permeability media. Conversely, intrinsic viscous resistance dominates ultra-low permeability systems. Overall, this model provides an efficient analytical tool and novel physical insights into complex mass transfer processes within rough porous media.