Deterministic and Stochastic Approaches to Modeling Water Filtration Through Saturated and Unsaturated Porous Media: A Review

Water flow through saturated and unsaturated porous media underlies groundwater assessment, contaminant transport, agricultural drainage and dam-seepage analysis, yet existing reviews typically address only one link in this chain: a single governing equation, a single flow regime, or a single modeling paradigm. This review connects them. It revisits the hydrodynamic foundations of filtration, from Darcy’s original experiments to the steady-state and transient saturated-flow equations, and extends the treatment to variably saturated conditions through the Richards equation and soil-water-retention models. Deterministic and stochastic modeling paradigms are then compared directly, and the comparison is made concrete through an original numerical case study: the Richards equation is solved for ponded infiltration into a sand filter column, first deterministically and then over a convergence-checked Monte Carlo ensemble (up to 500 realizations) of spatially correlated hydraulic-conductivity fields, showing that realistic heterogeneity alone produces a more than three-fold spread in the predicted wetting-front arrival time. Recent developments in numerical solvers, high-performance computing and machine-learning surrogate models are reviewed to show how the two paradigms are converging into hybrid, physics-informed frameworks, and the review is positioned explicitly against seven related studies published between 2017 and 2025. Persistent challenges in characterizing heterogeneity and validating hybrid models are identified, and directions for future research are outlined.