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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.V. Theis, Eos Trans. Am. Geophys. Union 16, 519 (1935)
M.S. Hantush, J. Hydraul. Div. 87, 83 (1961)
M.S. Hantush, J. Hydraul. Div. 87, 171 (1961)
H.H. Cooper, C.E. Jacob, Eos Trans. Am. Geophys. Union 27, 526 (1946)
J.E. Warren, P.J. Root, SPE J. 3, 245 (1963)
G.E. Barenblatt, I.P. Zheltov, I.N. Kochina, J. Appl. Math. Mech. 24, 1286 (1960)
K. Serra, A.C. Reynolds, R. Raghavan, J. Pet. Technol. 35, 2271 (1983)
A. de Swaan O, SPE J. 16, 117 (1976)
D. Bourdet et al., New Type Curves Aid Analysis of Fissured Zone Well Tests (World Oil, 1984)
B. O’Shaughnessy, I. Procaccia, Phys. Rev. A 32, 3073 (1985)
J. Leveinen, J. Hydrol. 234, 116 (2000)
T. Hirata, Pure Appl. Geophys. 121, 157 (1989)
K.B. Oldham, J. Spanier, in The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering (Academic Press, 1974), Vol. V
B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, in Fractional Calculus and Its Applications, Lecture Notes in Mathematics (1975), Vol. 457, pp. 1–36
L. Debnath. Int. J. Math. Educ. Sci. Technol. 35, 487 (2004)
B.S.T. Alkahtani, Chaos Solitons Fractals 89, 547 (2016)
A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)
R. Courant, E. Isaacson, M. Rees, Commun. Pure Appl. Math. 5, 243 (1952)
H.K. Versteeg, W. Malalasekera, in An Introduction to Computational Fluid Dynamics (1995), Chap. 5, p. 104
H.K. Versteeg, W. Malalasekera, in An Introduction to Computational Fluid Dynamics (1995), Chap. 5, p. 105
F. Ali, N.A. Sheikh, I. Khan, M. Saqib, J. Magn. Magn. Mater. 423, 327 (2017)
F. Ali, S.A.A. Jan, I. Khan, M. Gohar, N.A. Sheikh, Eur. Phys. J. Plus 131, 310 (2016)
F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)
F. Ali, N.A. Sheikh, M. Saqib, A. Khan, Nonlinear Sci. Lett. A 8, 101 (2017)
N.A. Sheikh, F. Ali, I. Khan, M. Saqib, Neural Comput. Appl. (2016) https://doi.org/10.1007/s00521-016-2815-5
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ahokpossi, D.P., Atangana, A. & Vermeulen, D.P. Modelling of fractal flow in dual media with fractional differentiation with power and generalized Mittag-Leffler laws kernels. Eur. Phys. J. Spec. Top. 226, 3705–3727 (2017). https://doi.org/10.1140/epjst/e2018-00002-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2018-00002-4