发表机构
University of Trento（特伦托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从孔隙填充分布的动力学理论推导非饱和水流的理查兹方程，通过查普曼 - 恩斯科格约化CKE，由达姆科勒数控制，得出相关结果，还能导出双渗透率和多渗透率模型，\(Da\)不小则需完整CKE。
AI 中文摘要
本文从伴随论文中孔隙填充分布\(g(r,x,t)\)的动力学理论推导出非饱和水流的理查兹方程。它区分了宏观理论通常混淆的两个极限。空间极限是纯运动学的：将代表性单元体积（REV）收缩到一点得到封闭的连续介质动力学方程（CKE）\(\partial_t g+\nabla\cdot F = C[g]\)，其中\(F\)是孔隙分辨的预封闭通量，\(C[g]\)是占有率门控再分布算子。动力学在于时间极限，即本文的主题：由达姆科勒数\(Da\)（再分布时间与强迫时间之比）控制的CKE的查普曼 - 恩斯科格（CE）约化，这是从玻尔兹曼方程到纳维 - 斯托克斯方程过渡的结构类似物。线性化再分布算子\(J\)在质量内积中是自伴且负半定的，其一维核确定单个不变量（水），从而得到一个宏观方程；CE层级在每个阶次都对相同的\(J\)求逆，只是源项不同。得出四个结果：平衡步定义了保持曲线；线性化水平衡加上REV间源项确定了一阶方程；其可解性是质量守恒，即理查兹方程；响应函数给出宏观通量，将传导率\(K\)识别为一阶输运系数，类似于粘度。平均场约化恢复了标准积分公式；串行路径校正给出了穆勒姆的非均质性惩罚。双峰孔径分布在弛豫谱中打开一个间隙；投影到能带并在每个能带内应用CE，跨能带弛豫在\(O(1)\)阶次存在，从第一原理导出双渗透率和多渗透率模型，交换系数由能带间连通性确定。当\(Da\)不小的时候，展开式失效，需要完整的CKE。
英文摘要
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.
