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Title: The simplest mixture-theory derivation of Darcy’s law
Abstract: Darcy’s law governs the velocity of a single fluid flowing in a porous medium. Originally deduced from experimental data in 1856, the law was extended to a three-dimensional form by the early 20th century and subsequently to flows with more complicated physics. While some people describe Darcy’s law as phenomenological, there are many derivations from more fundamental continuum mechanics. These derivations typically rest on such concepts as volume averaging, ensemble averaging, homogenization, or mixture theory. Published derivations based on mixture theory tend to be dauntingly technical. This lecture attempts to present the mixture-theoretic derivation as simply as possible. Starting with the basic balance laws, we adopt simple constitutive relations, apply thermodynamic constraints, then examine linear extensions from equilibrium that yield Darcy’s law in its most common form.