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

Yong Zhang*, Martinus Th van Genuchten, Dongbao Zhou, Golden J. Zhang, Hong Guang Sun*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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.

Original languageEnglish
Article numbere2023WR035782
Number of pages14
JournalWater Resources Research
Issue number3
Publication statusPublished - Mar 2024


  • fractional-derivative model
  • soil
  • tempered stable law
  • water equilibrium and kinetics


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