Richards' equation as a hydrodynamic limit: Chapman--Enskog reduction of the continuum kinetic equation for unsaturated soil water

Abstract

Richards' equation for unsaturated water flow is derived from the kinetic theory of the pore-filling distribution g(r,x,t) of a companion paper. It separates two limits that macroscopic theory usually conflates. The spatial limit is purely kinematic: contracting the representative elementary volume (REV) to a point yields the closed continuum kinetic equation (CKE) d_t g + div F = C[g], with F a pore-resolved pre-closure flux and C[g] the occupancy-gated redistribution operator. The dynamics lies in the temporal limit, the subject of this paper: a Chapman-Enskog (CE) reduction of the CKE controlled by the Damkohler number Da (redistribution time over forcing time), the structural analogue of the passage from Boltzmann to Navier-Stokes. The linearized redistribution operator J is self-adjoint and negative semidefinite in the mass inner product, with a one-dimensional kernel fixing a single invariant (water) and hence one macroscopic equation; the CE hierarchy inverts the same J at every order, only the source changing. Four results follow: the equilibrium step defines the retention curve; the linearized water budget plus the inter-REV source set the first-order equation; its solvability is mass conservation, i.e. Richards' equation; and the response function gives the macroscopic flux, identifying the conductivity K as a first-order transport coefficient, the counterpart of viscosity. Mean-field reduction recovers the standard integral formula; the serial-path correction gives Mualem's heterogeneity penalty. A bimodal pore-size distribution opens a gap in the relaxation spectrum; projecting onto the bands and applying CE within each, with cross-band relaxation surviving at O(1), derives the dual- and multiple-permeability models from first principles, the exchange coefficient set by inter-band connectivity. When Da is not small the expansion breaks down and the full CKE is needed.