The Operational Significance of the Continuum Hypothesis in the Theory of Water Movement Through Soils and Aquifers

Abstract

The operational meaning of the representative elementary volume (REV) concept, on which current foundational theories of water movement through porous media are based, is analyzed critically. It is concluded that the REV concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable. In its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments. The effects of resolution and precision differences among experimental measuring devices, now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers, can be incorporated naturally within the relativist concept but are excluded a priori in the REV concept. The relativist point of view, unlike the REV concept, has a clear operational meaning and, in principle, extends theoretical validity to all local measurements made on porous media. It is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated, anisotropic, deformable soil or aquifer.

What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports

“…Baveye and Sposito [1984] recognized that the value of a macroscopic field variable and its accuracy are inherently related to the method of measurement and to the physical features of the porous media. Accordingly, they proposed the weighting function as a convenient means for relating characteristics of an instrument's measurement to the measured macroscopic field variable.…”

Il Comportamento Di Suoli Agrari Rigonfiabili

“…Secondo le proposte di Stirk [42] a seguito di perdita graduale dell'umidità a partire da un campione di terreno saturo nelle condizioni in situ ha inizio un campo di variazione, che indicheremo come dominio (2) di umidità strutturale, durante il quale la perdita di acqua in volume, spesso irregolare, è in genere superiore alla riduzione del volume del suolo (dw > dV map ; ciò in generale implica pendenza media della curva di contrazione <1). Il contributo di questo dominio al contenuto idrico del terreno può raggiungere anche il 60-80% di quello totale ( fig.…”

“…Ciò significa che alla contrazione di e, che accompagna la riduzione di w, corrisponde una riduzione non lineare di ψ pm,int < ψ pm,mis , che rende conto dell'inconciliabile discordanza tra WRC e ShC per questo dominio (a parte possibilità di accettabile approssimazione nelle condizioni pratiche). Forse più accurate determinazioni potrebbero scaturire dalle variazioni di scala [2].…”

Technical Notes on Volume Averaging in Porous Media I: How to Choose a Spatial Averaging Operator for Periodic and Quasiperiodic Structures

“…This method is used in image processing (Buades et al 2005), signal processing (Witkin 1984) or in fluid mechanics (e.g., turbulence modeling (Sagaut 2006) and smoothed particle hydrodynamics (Monaghan 2005)). For the VAT in porous media, the idea dates back to the seminal works of C. Marle in Marle (1965, 1967, who anticipated difficulties associated with the standard definition that we thoroughly present in this paper, and is also discussed in Baveye and Sposito (1984), Hassanizadeh and Gray (1979), Mls (1987), Quintard and Whitaker (1994a, b), Baelmans (2015a, b, 2016). With this generalized definition, we can readily recover the more standard averaging operators, for example using a rectangular function for the kernel.…”