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

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 implement...

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Bibliographic Details
Published in:Energies (Basel) 2022-07, Vol.15 (13), p.4837
Main Authors: Alcántara-López, Fernando, Fuentes, Carlos, Camacho-Velázquez, Rodolfo G., Brambila-Paz, Fernando, Chávez, Carlos
Format: Article
Language:English
Subjects:
Caputo fractional derivative
Diffusion
Fluid dynamics
Fluid flow
Fractal geometry
Fractal models
fractal porous media
Fractals
Fractional calculus
Fractured reservoirs
Heterogeneity
naturally fractured reservoir
Permeability
Porosity
Porous media
Pressure drop
Reservoirs
Weyl fractional derivative
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Summary: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.
ISSN:1996-1073
1996-1073
DOI:10.3390/en15134837