Darcy, Forchheimer, Brinkman and Richards: classical hydromechanical equations and their significance in the light of th
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O R I G I NA L
Wolfgang Ehlers
Darcy, Forchheimer, Brinkman and Richards: classical hydromechanical equations and their significance in the light of the TPM
Received: 12 July 2020 / Accepted: 29 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Classical hydromechanics · Theory of Porous Media · Saturated and unsaturated porous materials
1 Introduction The present article aims at discussing the validity of classical hydromechanical equations describing porous media flow for saturated and unsaturated porous materials. Thus, the article is somehow in between a review article and an article presenting original research. As far as the author is aware, the famous Darcy law is widely used as a constitutive assumption for the description of mostly fully saturated pore flow conditions in porous media, rather than as the result of a continuum mechanical and thermodynamical investigation of a biphasic material of solid and fluid. In the same way, the Brinkman or Darcy–Brinkman equation as well as the Forchheimer or Darcy–Forchheimer equation is handled as extensions of Darcy’s law either by the introduction of frictional forces or by including inertia- or tortuosity-based terms added to the drag force. In contrast to these two equations, the Richards equation has been formulated for fluid flow in unsaturated media by combining the Darcy equation with the continuity equation of the pore fluid. In the following chapters, the classical approaches will be addressed, before biphasic and triphasic media are considered with the goal to find general relations that can be reduced to the classical equations. By this approach, it will easily be seen, what assumptions have to be made, such that the classical equations become evident. As the classical equations do not include thermal effects, the following investigations will be carried out under the assumption of isothermal conditions. W. Ehlers (B) Institute of Applied Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany E-mail: [email protected]
W. Ehlers
2 The classical equations 2.1 Darcy 1856 In 1856, Henry Darcy [1] published an extended treatise on the pu
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