On the stochastic foundations of the theory of water flow through unsaturated soil

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Abstract
The parabolic differential equation that describes the isothermal isohaline transport of water through an unsaturated soil is shown to be the mathematically rigorous result of a fundamental stochastic hypothesis: that the trajectory of a water molecule is a nonhomogeneous Markov process characterized by space‐ and time‐dependent coefficients of drift and diffusion. The demonstration is valid in general for heterogeneous anisotropic soils and provides for three principal results in the theory of water flow through unsaturated media: (1) a derivation of the Buckingham‐Darcy flux law that does not rely directly on experiment, (2) a new theoretical interpretation of the soil water diffusivity and the hydraulic conductivity in molecular terms, and (3) a proof that the soil water diffusivity for anisotropic soil is a symmetric tensor of the second rank. A dynamic argument at the molecular level is developed to show that the fundamental Markovian hypothesis is physically reasonable in the case of water movement through an unsaturated soil.

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