Modelling of fractal flow in dual media with fractional differentiation with power and generalized Mittag-Leffler laws kernels

Abstract

The paper has considered the fractal flow in a dual media with fractal properties, where the media could be elastic, heterogeneous and visco-elastic. We argued that, the fractal flow within a geological formation with elastic property cannot be accurately described with the concept of differentiation with local operator, as this operator is unable to include into mathematical formulation the effect of elasticity. Thus to include into mathematical formula the observed facts, we have modified the model by replacing the local derivative with the non-local operator with power. A more complex problem was considered where the geological formation is considered to have visco-elastic and heterogeneity properties. We argued that, the flow within a matrix rock with these two properties cannot either be described with local derivative nor a non-local derivative with power law. In this case two non-local operators were considered, an operator with Mittag-Leffler kernel and Mittag-Leffler-Power law kernel [F. Ali et al., J. Magn. Magn. Mater. 423, 327 (2017); F. Ali et al., Eur. Phys. J. Plus 131, 310 (2016); F. Ali et al., Eur. Phys. J. Plus 131, 377 (2016); F. Ali et al., Nonlinear Sci. Lett. A 8, 101 (2017); N.A. Sheikh et al., Neural Comput. Appl. (2016) https://doi.org/10.1007/s00521-016-2815-5]. For each model, a detailed study of existence and uniqueness of the system solutions was presented using the fixed point theorem. We solved numerically each model using a more acculturate numerical scheme known as Upwind. Some numerical simulations are presented to underpin the effect of the suggested fractional differentiation.