Spatial Variability of Soil Moisture and the Scale Issue: A Geostatistical Approach

We study the standard deviation of water saturation SDS as function of the mean saturation (Formula presented.) by a stochastic model of unsaturated flow, which is based on the first-order solution of the three-dimensional Richards equation. The model assumes spatially variable soil properties, following a given geostatistical description, and it explicitly accounts for the different scales involved in the determination of the spatial properties of saturation: the extent L, i.e., the domain size, the spacing Δ among measurements, and the dimension (Formula presented.) associated to the sampling measurement. It is found that the interplay between those scales and the correlation scale I of the hydraulic properties rules the spatial variability of saturation. A “scale effect” manifests for small to intermediate L/I, for which SDS increase with the extent L. This nonergodic effect depends on the structural and hydraulic parameters as well as the scales of the problem, and it is consistent with a similar effect found in field experiments. In turn, the influence of the scale (Formula presented.) is to decrease the saturation variability and increase its spatial correlation. Although the solution focuses on the medium heterogeneity as the main driver for the spatial variability of saturation, neglecting other important components, it explicitly links the spatial variation of saturation to the hydraulic properties of the soil, their spatial variability, and the sampling schemes; it can provide a useful tool to assess the impact of scales on the saturation variability, also in view of the several applications that involve the saturation variability.