A Unified Phenomenological Model Captures Water Equilibrium and Kinetic Processes in Soil

Soil water sustains life on Earth, and how to quantify water equilibrium and kinetics in soil remains a challenge for over a century despite significant efforts. For example, various models were proposed to interpret non-Darcian flow in saturated soils, but none of them can capture the full range of non-Darcian flow. To unify the different models into one overall framework and improve them if needed, this technical note proposes a theory based on the tempered stable density (TSD) assumption for the soil-hydraulic property distribution, recognizing that the underlying hydrologic processes all occur in the same, albeit very complex and not measurable at all the relevant scales, soil-water system. The TSD assumption forms a unified fractional-derivative equation (FDE) using subordination. Preliminary applications show that simplified FDEs, with proposed hydrological interpretations and TSD distributed properties, effectively capture core equilibrium and kinetic water processes, spanning non-Darcian flow, water retention, moisture movement, infiltration, and wetting/drying, in the soil-water system with various degrees and scales of system heterogeneity. Model comparisons and evaluations suggest that the TSD may serve as a unified density for the properties of a broad range of soil-water systems, driving multi-rate mass, momentum, and energy equilibrium/kinetic processes often oversimplified by classical models as single-rate processes.