Scale Dependent Fluid Momentum and Solute Mass Macroscopic Balance Equations: Theory and Observations

Abstract

A mathematical development is presented concerning extensions to the macroscopic momentum balance equation for compressible Newtonian fluids flowing through saturated porous matrices, and the macroscopic mass balance equation of solutes transported with the fluids. It is shown that each of these balance equations is composed of a dominant macroscopic equation associated with a larger spatial scale, coupled with a secondary macroscopic balance equation valid at a smaller spatial scale. The dominant fluid momentum balance equation can govern the propagation of shock waves, conform to Forchheimer's law or to Darcy's law when friction at the solid-fluid interface is dominant. Concurrently, the secondary momentum balance equation is governed by inertia flow that conforms to a wave equation propagating the intensive momentum and the dispersive momentum flux, both deviating from their corresponding dominant average terms. The dominant macroscopic solute mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic solute mass balance equation describes pure advection of the product of deviations from the average solute mass fraction and the average fluid density. Field observations under natural gradient flow conditions show excessive high concentration of colloids (average of 50 mg/L) under land irrigated by sewage effluents. The high concentration of colloids in a macroscopic flow field where specific discharge varies between 4 to 16 m/yr, is suggested to be due to the enhancement of colloidal mobility resulting from the secondary fluid momentum equation governed by inertia and the secondary solute mass equation of pure advection, both proven to be valid at a scale smaller then the one considered for spatial averaging.