A Mechanical Picture of Fractal Darcy’s Law

Abstract

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 0α11 and 1α12, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

Keywords

• fractal Darcy law
• fractal continuum calculus
• Hausdorff dimension
• pressure gradient
• naturally fractured reservoir