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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Fernando Alcántara-López & Carlos Fuentes & Rodolfo G. Camacho-Velázquez & Fernando Brambila-Paz & Carlos Chávez, 2022. "Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs," Energies, MDPI, vol. 15(13), pages 1-11, July.
  • Handle: RePEc:gam:jeners:v:15:y:2022:i:13:p:4837-:d:853801
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    References listed on IDEAS

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    1. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
    2. Chang, Ailian & Sun, HongGuang & Zhang, Yong & Zheng, Chunmiao & Min, Fanlu, 2019. "Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 519(C), pages 119-126.
    3. Razminia, Kambiz & Razminia, Abolhassan & Torres, Delfim F.M., 2015. "Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 374-380.
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