Stochastic analysis of unsaturated steady flows above the water table

Abstract

Steady flow takes place into a three‐dimensional partially saturated porous medium where, due to their spatial variability, the saturated conductivity Ks, and the relative conductivity Kr are modeled as random space functions (RSF)s. As a consequence, the flow variables (FVs), i.e., pressure‐head and specific flux, are also RSFs. The focus of the present paper consists into quantifying the uncertainty of the FVs above the water table. The simple expressions (most of which in closed form) of the second‐order moments pertaining to the FVs allow one to follow the transitional behavior from the zone close to the water table (where the FVs are nonstationary), till to their far‐field limit (where the FVs become stationary RSFs). In particular, it is shown how the stationary limits (and the distance from the water table at which stationarity is attained) depend upon the statistical structure of the RSFs Ks, Kr, and the infiltrating rate. The mean pressure head true〈normalΨtrue〉 has been also computed, and it is expressed as 〈Ψ〉=Ψ0(1+ψ), being ψ a characteristic heterogeneity function which modifies the zero‐order approximation Ψ0 of the pressure head (valid for a vadose zone of uniform soil properties) to account for the spatial variability of Ks and Kr. Two asymptotic limits, i.e., close (near field) and away (far field) from the water table, are derived into a very general manner, whereas the transitional behavior of ψ between the near/far field can be determined after specifying the shape of the various input soil properties. Besides the theoretical interest, results of the present paper are useful for practical purposes, as well. Indeed, the model is tested against to real data, and in particular it is shown how it is possible for the specific case study to grasp the behavior of the FVs within an environment (i.e., the vadose zone close to the water table) which is generally very difficult to access by direct inspection.

A scale‐invariant property of the water retention curve in weakly heterogeneous vadose zones

“…We wish also to recall here that, in order to be ergodic (thus allowing to use the ensemble average to compute the effective water retention curve), a flow domain has to be large enough compared with the integral scales of the random space functions. This has been assessed previously in a series of studies (Severino et al, , ), and therefore, such a requirement is taken for granted in the present study.…”

“…In words, for extremely dry conditions, the heterogeneity of the vadose zone does not impact the behaviour of the retention curve (irrespective of n ). A similar conclusion was drawn by Severino, Scarfato, and Comegna (), who analysed the spatial distribution of the mean head ⟨Ψ⟩ = Ψ 0 + ⟨Ψ 2 ⟩ in nonstationary unsaturated steady flows.…”

On a Class of Integrals Useful to Solve Well‐Type Flows in Heterogeneous Porous Formations

“…At aquifer scale the spatial distribution of the hydraulic conductivity K (and concurrently of Y ) is largely irregular (see, e.g., the data sets in Fallico et al, ; Severino et al, , ), such that it is unrealistic (due to the many logistic/economic limitations) to model its spatial variations within a deterministic framework. The uncertainty in the spatial distribution of K is accounted for by regarding Y as a stationary, normally distributed, random space function of 2‐point autocorrelation ρ ≡ ρ ( x ) and variance (Dagan, ; Rubin, ).…”

A fractal analysis of the water retention curve

“…The study of flow (Severino et al ) and transport (Severino et al ) phenomena in unsaturated porous media (soils) requires the knowledge of the hydraulic conductivity, which in turn is a function of the water content ϑ (or alternatively of the matric potential ψ ). The hydraulic conductivity may attain values within a wide range of ψ (see, e.g., Comegna et al, ), with variations of orders of magnitude.…”