{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17358",
  "author_total": 1,
  "authors": [
    {
      "name": "Riccardo Rigon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/796a5bg86fxnm.bib",
  "cited_by": 1,
  "doi": "10.48550/arxiv.2607.17358",
  "fulltext_url": null,
  "id": "796a5bg86fxnm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17358",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "physics.geo-ph"
    ],
    [
      "openalex",
      "2205|Soil and Unsaturated Flow"
    ]
  ],
  "openalex": "W7169886305",
  "pdf_url": "https://arxiv.org/pdf/2607.17358",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/796a5bg86fxnm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2607.17358",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17358"
    }
  ],
  "subjects": [
    {
      "field": "earth",
      "origin": "arxiv:physics.geo-ph",
      "subfield": "geophysics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Richards' equation as a hydrodynamic limit: Chapman--Enskog reduction of the continuum kinetic equation for unsaturated soil water",
  "title_text": "Richards' equation as a hydrodynamic limit: Chapman--Enskog reduction of the continuum kinetic equation for unsaturated soil water",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:14Z",
  "url": "https://research.draconitic.com/paper/796a5bg86fxnm",
  "venue": "arXiv",
  "versions": []
}