A Mechanical Picture of Fractal Darcy’s Law

• Published in: Fractal and fractional 2023-09, Vol.7 (9), p.639
• Main Authors: Damián Adame, Lucero, Gutiérrez-Torres, Claudia del Carmen, Figueroa-Espinoza, Bernardo, Barbosa-Saldaña, Juan Gabriel, Jiménez-Bernal, José Alfredo
• Format: Article
• Language: English
• Subjects: Anisotropy; Calculus; Cartesian coordinates; Differential calculus; Differential equations; Domains; Euclidean space; Flow velocity; Fluid flow; fractal continuum calculus; fractal Darcy law; Fractal geometry; Fractal models; Fractals; Fractured reservoirs; Geometry; Hausdorff dimension; Heterogeneity; Hydraulic gradient; Hydraulics; Mathematical morphology; naturally fractured reservoir; Operators (mathematics); Parameters; Permeability; Porous materials; Porous media; pressure gradient; Reservoirs; Topology
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract7090639

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0