Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs

Fernando Alcántara-López, Carlos Fuentes, Rodolfo G. Camacho-Velázquez, Fernando Brambila-Paz and Carlos Chávez
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Fernando Alcántara-López: Department of Mathematics, Faculty of Science, National Autonomous University of Mexico, Circuito Exterior S/N, Mexico City 04510, Mexico
Carlos Fuentes: Mexican Institute of Water Technology, Paseo Cuauhnáhuac Num. 8532, Jiutepec 62550, Mexico
Rodolfo G. Camacho-Velázquez: Engineering Faculty, National Autonomous University of Mexico, Circuito Exterior, Ciudad Universitaria, Mexico City 04510, Mexico
Fernando Brambila-Paz: Department of Mathematics, Faculty of Science, National Autonomous University of Mexico, Circuito Exterior S/N, Mexico City 04510, Mexico
Carlos Chávez: Water Research Center, Department of Irrigation and Drainage Engineering, Autonomous University of Querétaro, Cerro de las Campanas S/N, Col. Las Campanas, Querétaro 76010, Mexico

Energies, 2022, vol. 15, issue 13, 1-11

Abstract: Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools have been implemented, from fractal geometry to characterize the structure of the porous medium to fractional calculus to include the memory effect in the fluid flow. Considering infinite naturally fractured reservoirs (Type I system of Nelson), a spatial fractional Darcy’s law is proposed, where the spatial derivative is replaced by the Weyl fractional derivative, and the resulting flow model also considers Caputo’s fractional derivative in time. The proposed model maintains its dimensional balance and is solved numerically. The results of analyzing the effect of the spatial fractional Darcy’s law on the pressure drop and its Bourdet derivative are shown, proving that two definitions of fractional derivatives are compatible. Finally, the results of the proposed model are compared with models that consider fractal geometry showing a good agreement. It is shown that modified Darcy’s law, which considers the dependency of the fluid flow path, includes the intrinsic geometry of the porous medium, thus recovering the heterogeneity at the phenomenological level.

Keywords: Weyl fractional derivative; Caputo fractional derivative; fractal porous media; naturally fractured reservoir (search for similar items in EconPapers)
JEL-codes: Q Q0 Q4 Q40 Q41 Q42 Q43 Q47 Q48 Q49 (search for similar items in EconPapers)
Date: 2022
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