Darcy, Forchheimer, Brinkman and Richards: classical hydromechanical equations and their significance in the light of the TPM

Abstract

In hydromechanical applications, Darcy, Brinkman, Forchheimer and Richards equations play a central role when porous media flow under saturated and unsaturated conditions has to be investigated. While Darcy, Brinkman, Forchheimer and Richards found their equations mainly on the basis of flow observations in field and laboratory experiments, the modern Theory of Porous Media allows for a scientific view at these equations on the basis of precise continuum mechanical and thermodynamical investigations. The present article aims at commenting the classical equations and at deriving their counterparts by the use of the thermodynamical consistent Theory of Porous Media. This procedure will prove that the classical equations are valid under certain restrictions and that extended equations exist valid for arbitrary cases in their field.

Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

“…The singular source term was chosen so that the CEM solution would require additional basis functions, as the source term contains subgrid features. In all numerical examples, the coarse mesh size was 1 10 and the fine mesh size was 1 100 . For the time discretization, we consider the final time T = 0.05.…”

“…We would like to explore the case in which the diffusion operator is nonlinear. This, in Equation (1), we set f (u) = −∇ • (κα(u)∇u).…”

“…In many applications, these problems have multiscale nature and contain multiple scales and high contrast. Examples include nonlinear porous media flows (Richards' equations, Forchheimer flow and so on; see [1,2]), where the media properties contain many spatial scales and high contrast. Due to high contrast in the media properties, these processes also occur on multiple time scales.…”

A historical review on porous‐media research

“…From a modern point of view, Darcy's law can be found by combination of a constitutive equation for the direct momentum production term and the momentum balance of the liquid component included in a binary system of solid and fluid. Furthermore, Darcy's law only proves to hold in case of quasi-static situations, when a creeping hydraulic flow motivates a neglect of acceleration terms and when, furthermore, the frictional fluid stresses can be neglected by arguments of dimensional analysis in comparison with the friction included in the constitutive assumption for the momentum production, compare [11]. However, Darcy's law is widely used today in hydraulic engineering as a given constitutive equation without the consideration of its domain of validity.…”

Geometric Perturbation Theory and Travelling Waves profiles analysis in a Darcy–Forchheimer fluid model

“…Along the presented analysis, the relevant nonlinear flow equations are given by the following set of equations (refer to [20] for a modern derivation of typical fluids models, as Darcy and Forchheimer ones, under the scope of the Theory of Porous Media):…”