Moments of the Flow Variables,Part I :The Flow Equation and the Hydraulic Head

Abstract

The parameters and boundary conditions which control the flow processes vary greatly over short distances and over time, and it is inevitable that flow variables such as the hydraulic head and the fluid’s velocity are spatially variable. It is reasonable to model them as SRFs, not only because of the spatial variability, but also due to our inability to model them in detail with only the limited number of measurements usually available. This chapter presents the basic principles and a few methods for modeling flow variables as SRFs, while showing how these moments reflect the physics of the flow and how to use them in applications. The starting point in our derivations will be the moments that characterize the medium variability, such as σ2γ and I γ, and our goal is to develop the SRF models of flow variables such as the head and the velocity in terms of the media parameters and boundary conditions. To drive this point home, let us consider Darcy’s law for isotropic hydraulic conductivity: where qi is the specific flux in the ith direction, K is the isotropic hydraulic conductivity, and H is the hydraulic head. For a spatially variable K, the flux qi is also spatially variable (with the exception of the one-dimensional flow case). An SRF model for K can be used to construct an SRF for qi through (4.1), with the aid of the flow equation. SRFs can be characterized through their low-order statistical moments, but preferably through their pdfs. For example, the hydraulic head will be defined by its expected value (H) and its variance σ2 H = ((H - ( H ) ) 2 ) , but also through its pdf fH(h) where fH(h) dh is the probability of having H in the vicinity dh of h. SRFs may or may not be stationary depending on the nature of the hydrogeological variables such as K and on the boundary conditions. For example, the head is always nonstationary, except when there is no flow.