The Evolution of Mathematical Models for Fluid Flow in Porous Media: From Darcy's Law to Fractional Continuum Models

Author(s)	Dr. Prahlad Singh
Country	India
Abstract	This paper provides an overview of the historical and mathematical progress of fluid flow models in porous media, starting from Darcy's empirical law to the potential contemporary generalized form of continuum mechanics models. The author studies aspects such as physical assumptions, governing equations, and limitations accompanying each generation of models in the discussion. Classical models such as Darcy's law, Brinkman equation, Forchheimer correction, and Richards equation are evaluated before introducing models of multiphase, heterogeneous, and deformable porous media. Trends of application of fractional calculus are described as powerful tools for modeling of abnormal diffusion, remote spatial interaction, and memory-based transport processes in fractured reservoirs, biological medicaments, geothermal systems, and non-conventional porous material. Several numerical methods are considered, emphasizing issues of computer modeling and its connection to machine learning technologies. Finally, the author offers an outlook for future research on hybrid fractional-physics-based methods.
Keywords	Porous media; Darcy's law; Brinkman equation; Forchheimer flow; Fractional calculus; Anomalous diffusion; Continuum mechanics; Mathematical modelling.
Published In	Volume 10, Issue 6, June 2019
Published On	2019-06-08
DOI	https://doi.org/10.71097/IJTAS.v10.i6.1388

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