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. Plain Language Summary This technical note introduced a framework aimed at unifying and enhancing existing theories/models for interpreting various soil‐water processes in soils. A unified physical law, termed the “tempered stable density law,” was assumed for soil‐water properties, leading to a unified model which well captures the core soil water processes that have been modeled (many times inefficiently) by tens of different and sometimes competing theories/models over the past century. This unified theory/model has the potential for expansion to other land‐surface equilibrium and kinetic processes that cover a wide range of spatiotemporal scales and share a similar dynamic nature as water in soils. Key Points A tempered stable law with subordination built a unified model governing water equilibrium and kinetics in saturated/unsaturated soils Fractional‐derivative equations fitted soil‐water data sets, such as non‐Darcian flow, as well as or better than classical models The assumed tempered stable density distribution for so