Modeling liquid–vapor fronts in porous media using time-fractional derivatives: An innovative framework

This study presents an innovative framework for modeling the liquid–vapor phase change interface in porous media by employing time-fractional derivatives. Traditional heat transfer models often rely on integer-order derivatives, assuming local and instantaneous diffusion processes, which fail to fully capture the memory effects and nonlocal dynamics inherent in many real-world phase transition processes. To address this limitation, we incorporate time-fractional derivatives into the energy balance for both the liquid and vapor phases and the dynamics of the phase-change front. Using the Caputo fractional derivative, we model the nonlocal temporal behavior, offering a more accurate and comprehensive representation of heat transfer and phase transition dynamics. The study focuses on the time-fractional dynamics of liquid–vapor front in porous media in a geothermal context, but the methodological approach is broadly applicable to systems exhibiting anomalous diffusion and memory effects, particularly those involving phase transitions. Numerical solutions are computed using a finite difference method with fourth-order differentiation matrices, ensuring high accuracy and stability. Simulations reveal that increasing the fractional order parameter α slows the phase-change front, indicating sub-diffusive behavior characteristic of porous structures. Governing parameters such as heat generation, heat absorption, density ratio, porosity, temperature contrast, and fractional order parameter are analyzed, demonstrating combined impact on heat transfer and liquid–vapor front dynamics. These findings provide critical insights for optimizing energy extraction and environmental engineering applications, offering a fresh perspective on phase transition modeling in complex systems.