- Ali I., Malik N., Chanane B. (2014) Fractional diffusion model for transport through porous media, in 5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry, K. Vafai, A. Bejan, A. Nakayama, O. Manca (eds). ECI Symposium Series. [Google Scholar]
- Angulo J.M., Ruiz-Medina M.D., Anh V.V., Grecksch W. (2000) Fractional diffusion and fractional heat equation, Adv. Appl. Prob. 32, 4, 1077–1099. [CrossRef] [Google Scholar]
- Balakrishnan A.V. (1960) Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 2, 419–437. https://doi.org/10.2140/pjm.1960.10.419. [CrossRef] [MathSciNet] [Google Scholar]
- Bazhlekova E. (2019) Subordination principle for space-time fractional evolution equations and some applications, Integr. Transform. Spec. Funct., 431–452 https://doi.org/10.1080/10652469.2019.1581186. [CrossRef] [Google Scholar]
- Bazhlekova E., Bazhlekov I. (2019) Subordination approach to space-time fractional diffusion, Mathematics 2019, 7, 415. https://doi.org/10.3390/math7050415. [CrossRef] [Google Scholar]
- Beier R.A. (1990) Pressure Transient Field Data Showing Fractal Reservoir Structure, in: Paper 90–04 presented at the Annual Technical Meeting, Calgary, Alberta Petroleum Society of Canada. https://doi.org/10.2118/90-04. [Google Scholar]
- Benson D.A., Wheatcraft S.W., Meerschaert M.M. (2000) Application of a fractional advection-dispersion equation, Water Resour. Res. 36, 6, 1403–1412. [Google Scholar]
- Benson D.A., Tadjeran C., Meerschaert M.M., Farnham I., Pohll G. (2004) Radial fractional-order dispersion through fractured rock, Water Resour. Res. 40, W12416. https://doi.org/10.1029/2004WR003314. [Google Scholar]
- Bernard S., Delay F., Porel G. (2006) A new method of data inversion for the identification of fractal characteristics and homogenization scale from hydraulic pumping tests in fractured aquifers, J. Hydrol. 328, 3–4, 647–658. https://doi.org/10.1016/j.jhydrol.2006.01.008. [CrossRef] [Google Scholar]
- Bourdet D., Whittle T.M., Douglas A.A., Pirad Y.M. (1983) A new set of type curves simplifies well test analysis, World Oil 95–106. [Google Scholar]
- Boyadjiev L., Luchko Y. (2017) Mellin integral transform approach to analyze the multidimensional diffusion-wave equations, Chaos Solit. Fract. 102, 127–134. https://doi.org/10.1016/j.chaos.2017.03.050. [CrossRef] [Google Scholar]
- Cai M., Li C. (2019) On Riesz derivative, Fract. Calc. Appl. Anal. 22, 2, 287–301. https://doi.org/10.1515/fca-2019-0019. [CrossRef] [MathSciNet] [Google Scholar]
- Caputo M. (1967) Linear models of dissipation whose Q is almost Frequency Independent-II, Geophys. J. R. Astron. Soc. 13, 5, 529–539. [CrossRef] [Google Scholar]
- Chang J., Yortsos Y.C. (1990) Pressure-transient analysis of fractal reservoirs, SPE Form. Eval. 5, 1, 31–39. [CrossRef] [Google Scholar]
- Chang A., Sun H., Zhang Y., Zheng C., Min F. (2019) Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs, Phys. A Stat. Mech. Its Appl., 519, 119–126. [Google Scholar]
- Chen C., Raghavan R. (2015) Transient flow in a linear reservoir for space-time fractional diffusion, J. Pet. Sci. Eng. 128, 194–202. https://doi.org/10.1016/j.petrol.2015.02.021. [Google Scholar]
- Chow V.T. (1952) On the determination of transmissibility and storage coefficients from pumping test data, Trans. Am. Geophys. Un. 33, 397–404. [CrossRef] [Google Scholar]
- Chu W., Pandya N., Flumerfelt R.W., Chen C. (2019a) Rate-transient analysis based on power-law behavior for permian wells, SPE Res. Eval. Eng. 22, 4, 1360–1370. https://doi.org/10.2118/187180-PA. [CrossRef] [Google Scholar]
- Chu W., Scott K., Flumerfelt R.W., Chen C. (2019b) A new technique for quantifying pressure interference in fractured horizontal shale wells, SPE Res. Eval. Eng. 23, 01, 143–157. https://doi.org/10.2118/191407-PA. [Google Scholar]
- Cloot A., Botha J.F. (2006) A generalised groundwater flow equation using the concept of non-integer order derivatives, Water SA 32, 1, 1–7. www.wrc.org.za. [Google Scholar]
- Cooper H.H., Jacob C.E. (1946) A generalized graphical method for evaluating formation constants and summarizing well-field history, Trans. AGU 27, 526–534. [CrossRef] [Google Scholar]
- Cortis A., Knudby C. (2006) A continuous time random walk approach to transient flow in heterogeneous porous media, LBNL-59885, Water Resour. Res. 42, W10201. [Google Scholar]
- Dassas Y., Duby Y. (1995) Diffusion toward fractal interfaces, potentiostatic, galvanostatic, and linear sweep voltammetric techniques, J. Electrochem. Soc. 142, 12, 4175–4180. [Google Scholar]
- Deng C.-S., Schilling R.L. (2019) Exact asymptotic formulas for the heat kernels of space and time-fractional equations, Fract. Calc. Appl. Anal. 22, 4, 968–989. https://doi.org/10.1515/fca-2019-0052. [CrossRef] [MathSciNet] [Google Scholar]
- Doe T., Shi C., Knitter C., Enachescu C. (2014) Discrete fracture network simulation of production data from unconventional wells, paper URTeC 1923802, in: Proceedings of The Unconventional Resources Technology Conference, Denver CO. [Google Scholar]
- Erdelyi A., Magnus W.F., Oberhettinger F., Tricomi F.G. (1955) Higher transcendental functions. Chapter 18: Miscellaneous Functions, Vol. 3, McGraw-Hill, New York, pp. 206–227. [MathSciNet] [Google Scholar]
- Estrada-Rodriguez G., Gimperlein H., Painter K.J., Stocek J. (2018) Space-time fractional diffusion in cell movement models with delay, Math. Models Methods Appl. Sci. 29, 01, 1–2. https://doi.org/10.1142/S0218202519500039. [Google Scholar]
- Flamenco-Lopez F., Camacho-Velazquez R. (2001) Fractal transient pressure behavior of naturally fractured reservoirs, in: Paper 71591 presented at the Annual Technical Conference and Exhibition, Soc. Pet. Eng., New Orleans, LA. https://doi.org/10.2118/71591-MS. [Google Scholar]
- Fomin S., Chugunov V., Hashida T. (2011) Mathematical modeling of Anomalous Diffusion in Porous Media, Fractional Differential Calculus 1, 1–28. [CrossRef] [MathSciNet] [Google Scholar]
- Garcia-Rivera J., Raghavan R. (1979) Analysis of short time pressure transient data dominated by wellbore storage and skin, J. Pet. Tech. 31, 5, 623–631. [CrossRef] [Google Scholar]
- Garra R., Salusti E. (2013) Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media, Phys. D Nonlinear Phenom. 250, 52–57. https://doi.org/10.1016/j.physd.2013.01.014. [CrossRef] [Google Scholar]
- Gehlhausen A.N. (2009) Evaluation of the fractional Theis solution, Master of Science Thesis, University of Nevada, Reno. http://hdl.handle.net/11714/4109. [Google Scholar]
- Gorenflo R., Loutchko J., Luchko Yu (2002) Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal. 5, 491–518. [MathSciNet] [Google Scholar]
- Gorenflo R., Mainardi F. (1998) Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal. 1, 2, 167–191. http://www.math.bas.bg/fcaa/. [MathSciNet] [Google Scholar]
- Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. (2014) Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin Heidelberg, p. 443. [Google Scholar]
- Gurtin M.E., Pipkin A.C. (1968) A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 2, 113–126. https://doi.org/10.1007/BF00281373. [CrossRef] [MathSciNet] [Google Scholar]
- Henry B.I., Langlands T.A.M., Straka P. (2010) An introduction to fractional diffusion, in: Presented at the Conference: Complex Physical, Biophysical and Econophysical Systems – Proceedings of the 22nd Canberra International Physics Summer School, pp. 37–89. [Google Scholar]
- Huang F., Liu F. (2005) The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput. 18, 1–2, 339–350. https://doi.org/10.1007/BF02936577. [CrossRef] [MathSciNet] [Google Scholar]
- Kim S., Kavvas M.L., Ercan A. (2015) Fractional ensemble average governing equations of transport by time-space nonstationary stochastic fractional advective velocity and fractional dispersion. II: Numerical investigation, J. Hydrol. Eng. 20, 2, 04014040. [CrossRef] [Google Scholar]
- Kwaśnicki M. (2017) Ten equivalent definitions of the fractional Laplace operator, Fract Calc Appl Anal. 20, 1, 7–51. [MathSciNet] [Google Scholar]
- Le Mehaute A., Crepy G. (1983) Introduction to transfer and motion in fractal media: The geometry of kinetics, Solid State Ionics 1, 9–10, 17–30. [CrossRef] [Google Scholar]
- Lenormand R. (1992) On use of fractional derivatives for fluid flow in heterogeneous media, in: Proceedings 3rd European Conference on the Mathematics of Oil Recovery, Delft The Netherlands. [Google Scholar]
- Luchko Y. (2016) A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis, Math. Model. Nat. Phenom. 11, 3, 1–17. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- Luchko Y. (2017) On some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation, Mathematics 5, 4, 76. https://doi.org/10.3390/math5040076. [CrossRef] [Google Scholar]
- Luchko Y. (2019) Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation, Theory Probab. Math. Stat. 98, 121–141. [Google Scholar]
- Magin R.L., Ingo C., Colon-Perez L., Triplett W., Mareci T.H. (2013) Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy, Microporous Mesoporous Mater. 178, 15, 39–43. https://doi.org/10.1016/j.micromeso.2013.02.054. [CrossRef] [Google Scholar]
- Mainardi F., Luchko Y., Pagnini G. (2001) The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4, 2, 153–192. [MathSciNet] [Google Scholar]
- Mainardi F. (2010) Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, p. 344. [Google Scholar]
- Mattax C.C., Dalton R.L. (eds) (1990) Reservoir simulation, SPE Monograph Series 13, 187 pp. [Google Scholar]
- Meerschaert M.M., Zhang Y., Baeumer B. (2008) Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett. 35, L17403. https://doi.org/10.1029/2008GL034899. [CrossRef] [Google Scholar]
- Metzler R., Glockle W.G., Nonnenmacher T.F. (1994) Fractional model equation for anomalous diffusion, Physica A 211, 1, 13–24. [Google Scholar]
- Molz F.J. III, Fix G.J. III, Lu S.S. (2002) A physical interpretation for the fractional derivative in Levy diffusion, Appl. Math. Lett. 15, 7, 907–911. [Google Scholar]
- Moodie T.B., Tait R. (1983) On thermal transients with finite wave speeds, J. Acta Mechanica 50, 1–2, 97–104. https://doi.org/10.1007/BF01170443. [CrossRef] [Google Scholar]
- Nigmatullin R. (1984) To the theoretical explanation of the universal response, Phys. Status Solidi B Basic Res. 123, 2, 739–745. [CrossRef] [Google Scholar]
- Noetinger B., Estebenet T. (2000) Up-scaling of double porosity fractured media using continuous-time random walks methods, Transp. Porous Med. 39, 3, 315–337. [Google Scholar]
- Noetinger B., Roubinet D., Russian A., Le Borgne T., Delay F., Dentz M., Gouze P. (2016) Random walk methods for modeling hydrodynamic transport in porous and fractured media from pore to reservoir scale, Transp. Porous Med. 115, 2, 345–385. https://doi.org/10.1007/s11242-016-0693-z. [CrossRef] [MathSciNet] [Google Scholar]
- Norwood F.R. (1972) Transient thermal waves in the general theory of heat conduction with finite wave speeds, ASME. J. Appl. Mech. 39, 3, 673–676. https://doi.org/10.1115/1.3422771. [CrossRef] [Google Scholar]
- Oldham K.B., Spanier J. (1974) The fractional calculus; theory and applications of differentiation and integration to arbitrary order, Academic Press, New York, p. 234. [MathSciNet] [Google Scholar]
- Park H.W., Choe J., Kang J.M. (2000) Pressure behavior of transport in fractal porous media using a fractional calculus approach, Energy Sources, Part A: Recovery, Utilization, and Environmental Effects 22, 10, 881–890. [Google Scholar]
- Podlubny I. (1998) Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, some methods of their solution and some of their applications, Academic Press, New York, p. 340. [Google Scholar]
- Podlubny I. (2005) Mittag-Leffler function. www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function. [Google Scholar]
- Raghavan R., Ozkan E. (1994) A method for computing unsteady flows in porous media, Pitman Research Notes in Mathematics Series (318), Longman Scientific & Technical, Harlow, UK, p. 188. [Google Scholar]
- Raghavan R., Chen C. (2015) Anomalous subdiffusion to a horizontal well by a subordinator, Transp. Porous Med. 107, 387–401. https://doi.org/10.1007/s11242-014-0444-y. [CrossRef] [MathSciNet] [Google Scholar]
- Raghavan R., Chen C. (2018) A conceptual structure to evaluate wells producing fractured rocks of the Permian Basin, in: Paper SPE-191484-MS, Presented at the Annual Technical Conference and Exhibition, 24–28 September, Dallas, TX, USA. [Google Scholar]
- Raghavan R., Chen C. (2019) The Theis solution for subdiffusive flow in rocks, Oil Gas Sci. Technol. - Rev. IFP Energies nouvelles 74, 6. https://doi.org/10.2516/ogst/2018081. [CrossRef] [Google Scholar]
- Raghavan R., Chen C. (2020) A study in fractional diffusion: Fractured rocks produced through horizontal wells with multiple, hydraulic fractures, Oil Gas Sci. Technol. - Rev. IFP Energies nouvelles 75, 68. https://doi.org/10.2516/ogst/2020062. [Google Scholar]
- Sapora A., Cornetti P., Chiaia B., Lenzi E.K. (2017) Nonlocal diffusion in porous media: a spatial fractional approach, J. Eng. Mech. 143, 5, D4016007-1–D4016007-7. [CrossRef] [Google Scholar]
- Scott K.D., Chu W.-C., Flumerfelt R.W. (2015) Application of real-time bottom-hole pressure to improve field development strategies in the Midland Basin Wolfcamp Shale, paper URTEC-2154675, in: Proceedings of Unconventional Resources Technology Conference, San Antonio, Texas. https://doi.org/10.15530/URTEC-2015-2154675. [Google Scholar]
- Sinkov K., Weng X., Kresse O. (2021) Modeling of proppant distribution during fracturing of multiple perforation clusters in horizontal wells, paper SPE-204207-MS, in: Presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition. https://doi.org/10.2118/204207-MS. [Google Scholar]
- Stanislavsky A., Weron K., Weron A. (2014) Anomalous diffusion with transient subordinators: A link to compound relaxation laws, J. Chem. Phys. 140, 5, 054113. https://doi.org/10.1063/1.4863995. [CrossRef] [PubMed] [Google Scholar]
- Su N. (2014) Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration, J. Hydrol. 519, B, 1792–1803. [CrossRef] [Google Scholar]
- Su N., Nelson P.N., Connor S. (2015) The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests, J. Hydrol. 529, 1262–1273. https://doi.org/10.1016/j.jhydrol.2015.09.033. [CrossRef] [Google Scholar]
- Suzuki A.T., Hashida K.Li, Horne R.N. (2016) Experimental tests of truncated diffusion in fault damage zones. Water Resour. Res. 52, 8578–8589. https://doi.org/10.1002/2016WR019017. [CrossRef] [Google Scholar]
- Tao S., Gao X., Li C., Zeng J., Zhang X., Yang C., Zhang J., Gong Y. (2016) The experimental modeling of gas percolation mechanisms in a coal-measure tight sandstone reservoir: A case study on the coal-measure tight sandstone gas in the Upper Triassic Xujiahe Formation, Sichuan Basin, China. J. Natural Gas Geosci. 1, 6, 445–455. https://doi.org/10.1016/j.jnggs.2016.11.009. [CrossRef] [Google Scholar]
- Theis C.V. (1935) The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans. AGU 16, 2, 519–524. [CrossRef] [Google Scholar]
- Thomas O.O., Raghavan R., Dixon T.N. (2005) Effect of scaleup and aggregation on the use of well tests to identify geological properties, SPE Reserv. Eval. Eng. 8, 3, 248–254. https://doi.org/10.2118/77452-PA. [CrossRef] [Google Scholar]
- Uchaikin V.V. (2012) Fractional derivatives for physicists and engineers; background and theory, Vol. 1, Springer-Verlag, New York, NY, p. 385. [Google Scholar]
- Wang Y. (2013) Anomalous transport in weakly heterogeneous geological porous media, Phys. Rev. E 87, 032144-1-032144-8. https://doi.org/10.1103/PhysRevE.87.032144. [Google Scholar]
- Yanga S., Zhoua H.W., Zhang S.Q., Ren W.G. (2019) A fractional derivative perspective on transient pulse test for determining the permeability of rocks, Int. J. Rock Mech. Min. Sci. 113, 92–98. [CrossRef] [Google Scholar]
- Zhokh A., Strizhak P. (2018) Non-Fickian transport in porous media: Always temporally anomalous?, Transp. Porous Med. 124, 2, 309–323. https://doi.org/10.1007/s11242-018-1066-6. [CrossRef] [Google Scholar]
- Zhokh A., Strizhak P. (2019) Investigation of the anomalous diffusion in the porous media: a spatiotemporal scaling. Heat Mass Transf. 55, 2693–2702. https://link.springer.com/article/10.1007/s00231-019-02602-4. [CrossRef] [Google Scholar]
Open Access
| Issue |
Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles
Volume 76, 2021
|
|
|---|---|---|
| Article Number | 78 | |
| Number of page(s) | 10 | |
| DOI | https://doi.org/10.2516/ogst/2021058 | |
| Published online | 20 December 2021 | |
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