Natural length scales define the range of applicability of the Richards equation for capillary flows

Abstract

The rapid expansion of remotely sensed spatial information and enhanced computational capabilities fuel raising scientific and public expectations for reliable hydrologic predictions across time and spatial scales. Process-based hydrologic models often rely on the Richards equation (RE) formalism to represent unsaturated flow processes at multiple scales which raises the much debated question: does the underlying physics in the RE formulation apply at large scales of practical interest? The study analyses recent findings from different unsaturated flow processes (soil evaporation, internal redistribution, and capillary flow from point sources) revealing inherent characteristic length scales that delineate the spatial range of applicability of the RE. These length scales reflect the role of intrinsic porous medium properties that shape liquid phase continuity and interplay of forces that drive and resist unsaturated flow. The study revisits some of the key assumptions in the RE and their ramifications for numerical discretization. An intrinsic length scale for hydraulic continuity deduced from pore size distribution has been shown to control soil evaporation dynamics (i.e., stage 1 to stage 2 transition), to provide upper bounds for regional evaporative losses, and governs the dynamics of internal redistribution toward field capacity. For large-scale hydrologic applications, we show that the spatial extent of lateral flow interactions under most natural capillary gradients rarely exceed a few meters. The study provides a framework for guiding numerical and mathematical models for capillary flows across different scales considering the conditions for coexistence of stationarity, hydraulic continuity, and capillary gradients - essential ingredients for physically consistent application of the RE.