Analysis of subdiffusion in disordered and fractured media using a Grünwald-Letnikov fractional calculus model

• Published in: Computational geosciences 2018-10, Vol.22 (5), p.1231-1250
• Main Authors: Obembe, Abiola D., Abu-Khamsin, Sidqi A., Hossain, M. Enamul, Mustapha, Kassem
• Format: Article
• Language: English
• Subjects: Computer simulation; Constitutive equations; Constitutive relationships; Diffusion; Discretization; Dye dispersion; Earth and Environmental Science; Earth Sciences; Finite difference method; Flow rates; Flow velocity; Fractional calculus; Fractures; Geotechnical Engineering & Applied Earth Sciences; Hydrogeology; Material balance; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematical models; Oil reservoirs; Original Paper; Pore pressure; Porous media; Pressure drop; Reservoirs; Soil Science & Conservation; Stability; Stability analysis; Stability criteria
• ISSN: 1420-0597; 1573-1499
• DOI: 10.1007/s10596-018-9749-1

The increasing applications of fractional calculus in simulating the anomalous transport behavior in disordered and fractured heterogeneous porous media has grown rapidly over the past decade. In the present study, a temporal fractional flux relationship is employed as a constitutive equation to relate the volumetric flow rate to the gradient of the pore pressure. The novelty of this paper entails interpreting the time fractional derivative operator in the flux relationship by the Grünwald-Letnikov (G-L) definition as opposed to the Caputo interpretation which has been widely considered. Subsequently, a numerical scheme based on the block-centered finite-difference discretization is formulated to handle the resulting non-linear fractional diffusion model. In addition, a linear stability analysis is successfully performed to establish the stability criterion of the developed numerical scheme. An expression for the modified incremental material balance index was derived to assess the effectiveness of the numerical discretization process. Finally, numerical experiments were performed to provide qualitative insights into the nature of pressure evolution in a hydrocarbon reservoir under the influence subdiffusion. In summary, the results establish that subdiffusion regime results in the development of higher pressure drop in the reservoir. This paper will provide a strong foundation for researchers interested in investigating anomalous diffusion phenomena in porous media.