Open Access
Issue
Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles
Volume 75, 2020
Article Number 68
Number of page(s) 20
DOI https://doi.org/10.2516/ogst/2020062
Published online 09 October 2020
  • Albinali A., Holy R., Sarak H., Ozkan E. (2016) Modeling of 1D anomalous diffusion in fractured nanoporous media, presented at the low permeability media and nanoporous materials from characterization to modeling, Can we do better? Rueil-Malmaison, France, Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles 71, 4, 56. https://doi.org/10.2516/ogst/2016008. [CrossRef] [Google Scholar]
  • Angulo J.M., Ruiz-Medina M.D., Anh V.V., Grecksch W. (2000) Fractional diffusion and fractional heat equation, Adv. Appl. Probab. 32, 4, 1077–1099. [Google Scholar]
  • Artus V. (2020) Numerical upscaling of discrete fracture networks for transient analysis URTEC-2020-3087-MS, in: Presentation at the Unconventional Resources Technology Conference held in Austin, Texas, USA, pp. 20–22. [Google Scholar]
  • Barenblatt G.I., Zheltov Iu.P., Kochina I.N. (1960) Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech. 24, 5, 1286–1303. [CrossRef] [Google Scholar]
  • Barker J.A. (1988) A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rock, Water Resour. Res. 24, 10, 1796–1804. [Google Scholar]
  • Beier R.A. (1994) Pressure-transient model for a vertically fractured well in a fractal reservoir, SPE Form. Eval. 9, 2, 122–128. https://doi.org/10.2118/20582-PA. [CrossRef] [Google Scholar]
  • Belayneh M., Masihi M., Matthäi S.K., King P.R. (2006) Prediction of vein connectivity using the percolation approach: model test with field data, J. Geophys. Eng. 33, 219–229. https://doi.org/10.1088/1742-2132/3/3/003. [CrossRef] [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]
  • Benson D.A., Meerschaert M.M., Revielle J. (2013) Fractional calculus in hydrologic modeling: A numerical perspective, Adv. Water Resour. 51, 479–497. [CrossRef] [PubMed] [Google Scholar]
  • Bisdom K., Bertotti G., Nick H.M. (2016) The impact of different aperture distribution models and critical stress criteria on equivalent permeability in fractured rocks, J. Geophys. Res. Solid Earth 121, 5, 2169–9356. https://doi.org/10.1002/2015JB012657. [Google Scholar]
  • Cacas M.C., Ledoux E., de Marsily G., Tillie B., Barbreau A., Durand E., Feuga B., Peaudecerf P. (1990a) Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The flow model, Water Resour. Res. 26, 3, 479–489. https://doi.org/10.1029/WR026i003p00479. [Google Scholar]
  • Cacas M.C., Ledoux E., de Marsily G., Barbreau A., Calmels P., Gaillard B., Margritta R. (1990b) Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation: 2. The transport model, Water Resour. Res. 26, 3, 491–500. https://doi.org/10.1029/WR026i003p00491. [Google Scholar]
  • Cacas M.C., Daniel J.M., Letouzey J. (2001) Nested geological modelling of naturally fractured reservoirs, Petrol. Geosci. 7, S43–S52. https://doi.org/10.1144/petgeo.7.S.S43. [CrossRef] [Google Scholar]
  • Caine J.S., Evans J.P., Forster C.B. (1996) Fault zone architecture and permeability structure, Geology 24, 11, 1025–1028. [Google Scholar]
  • Camacho-Velázquez R.G. (1984) Response of wells producing commingled reservoirs: Unequal fracture length, Master’s thesis, University of Tulsa, Tulsa, OK. [Google Scholar]
  • Camacho-Velázquez R.G., Raghavan R., Reynolds A.C. (1987) Response of wells producing layered reservoirs: unequal fracture length, SPE Form. Eval. 2, 1, 9–28. https://doi.org/10.2118/12844-PA. [CrossRef] [Google Scholar]
  • Camacho-Velázquez R.G., Raghavan R. (1989) Boundary-dominated flow in solutions-gas-drive reservoirs, SPE Reserv. Eng. 4, 4, 503–512. https://doi.org/10.2118/18562-PA. [CrossRef] [Google Scholar]
  • Caputo M. (1967) Linear models of dissipation whose Q is almost Frequency Independent-II, Geophys. J. Roy. Astron. Soc. 13, 5, 529–539. [NASA ADS] [CrossRef] [Google Scholar]
  • Carrera J., Sánchez-Vila X., Benet I., Medina A., Galarza G., Guimerá J. (1998) On matrix diffusion: formulations, solution methods and qualitative effects, Hydrogeol. J. 6, 1, 178–190. [Google Scholar]
  • Carslaw H.S., Jaeger J.C. (1959) Conduction of heat in solids, 2nd edn., Clarendon Press, Oxford, p. 510. [Google Scholar]
  • Chang J., Yortsos Y.C. (1990) Pressure-transient analysis of Fractal Reservoirs, SPE Form. Eval. 5, 1, 31–39. [CrossRef] [Google Scholar]
  • Chen C. (1982) A study of naturally fractured reservoirs, MS Thesis, University of Tulsa, Tulsa, OK. [Google Scholar]
  • Chen C., Raghavan R. (2013) On some characteristic features of fractured-horizontal wells and conclusions drawn thereof, SPE Reserv. Eval. Eng. 16, 1, 19–28. https://doi.org/10.2118/163104-PA. [CrossRef] [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. [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. (2019) 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. (2020) A new technique for quantifying pressure interference in fractured horizontal shale wells. SPE Res. Eval. Eng. 23, 1, 143–157 https://doi.org/10.2118/191407-PA. [CrossRef] [Google Scholar]
  • Chu W. (2018) Personal Communication. [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]
  • Cinco-Ley H., Meng H.-Z. (1988) Pressure transient analysis of wells with finite conductivity vertical fractures in double porosity reservoirs, in: Presented at the SPE Annual Technical Conference and Exhibition, 2–5 October, Houston, Texas. https://doi.org/10.2118/18172-MS. [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]
  • de Swaan-O A. (1976) Analytic solutions for determining naturally fractured reservoir properties by well testing, Soc. Pet. Eng. J. 16, 3, 117–122. https://doi.org/10.2118/5346-PA. [CrossRef] [Google Scholar]
  • Defterli O., D’Elia M., Du Q., Gunzburger M., Lehoucq R., Meerschaert M.M. (2015) Fractional Diffusion on Bounded Domains, Fract. Calc. Appl. Anal. 18, 2, 342–360. [Google Scholar]
  • Dershowitz W., Klise K., Fox A., Takeuchi S., Uchida M. (2002) Channel network and discrete fracture network modeling of hydraulic interference and tracer experiments and the prediction of phase C experiments, SKB Report IPR-02-29, SKB, Stockholm. [Google Scholar]
  • Doe T.W. (1991) Fractional dimension analysis of constant-pressure well tests, in: Paper SPE-22702-MS, Presented at the SPE Annual Technical Conference and Exhibition, 6–9 October, Dallas, Texas. https://doi.org/10.2118/22702-MS. [Google Scholar]
  • Doe T., Shi C., Knitter C., Enachescu C. (2014) Discrete fracture network simulation of production data from unconventional wells, in: Paper URTeC 1923802, Proceedings of The Unconventional Resources Technology Conference, Denver, CO. [Google Scholar]
  • Dong Y., Fu Y., Yeh T.-C.J., Wang Y.-L., Zha Y., Wang L., Hao Y. (2019) Equivalence of discrete fracture network and porous media models by hydraulic tomography, Water Resour. Res. 55, 4, 3234–3247. https://doi.org/10.1029/2018WR024290. [Google Scholar]
  • Dontsov K.M., Boyrchuk B.T. (1971) Effect of characteristics of fractured media on pressure buildup behavior, Izvestia VUZ, Oil and Gas N1, 42–46 (in Russian). [Google Scholar]
  • Doyle P.G., Snell J.L. (1984) Random walks and electric networks, Carus Mathematical Monographs, Vol. 22, Mathematical Association of America, 174 pp. https://www.jstor.org/stable/10.4169/j.ctt5hh804. [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. [Google Scholar]
  • Evans J.P. (1988) Deformation mechanisms in granitic rocks at shallow crustal levels, J. Struct. Geol. 10, 5, 437–443. [Google Scholar]
  • Fomin S., Chugunov V., Hashida T. (2011) Mathematical modeling of anomalous diffusion in porous media, Fract. Differ. Calc. 1, 1–28. [Google Scholar]
  • Gale J.F.W., Laubach S., Olson J.E., Eichhuble P., Fall A. (2014) Natural Fractures in shale: A review and new observations, AAPG Bull. 98, 11, 2165–2216. [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]
  • Gorenflo R., Loutchko J., Luchko Yu. (2002) Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal. 5, 491–518. [Google Scholar]
  • Gradshteyn S., Ryzhik I.M. (1965) Table of integrals, series, and products, 5th edn, in: Jeffrey A. (eds.), Academic Press, New York. [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]
  • Haggerty R., Gorelick S.M. (1995) Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res. 31, 10, 2383–2400. https://doi.org/10.1029/95WR10583. [Google Scholar]
  • Haubold H.J., Mathai A.M., Saxena R.K. (2011) Mittag-Leffler functions and their applications, J. Appl. Math. 2011, 298628. https://doi.org/10.1155/2011/298628. [Google Scholar]
  • Henry B.I., Langlands T.A.M., Straka P. (2010) An Introduction to Fractional Diffusion, in: Dewar R.L., Detering F. (eds), Complex Physical, Biophysical and Econophysical Systems, World Scientific, Hackensack, NJ, p. 400. [Google Scholar]
  • Hilfer R., Anton L. (1995) Fractional master equations and fractal time random walks, Phys. Rev. E 51, 2, R848–R851. [Google Scholar]
  • Holy R.W., Ozkan E. (2016) A Practical and Rigorous Approach for Production Data Analysis in Unconventional Wells, in: Paper 181662-MS presented at the SPE Low Perm Symposium, 5–6 May, Denver, Colorado, USA. [Google Scholar]
  • Houze O.P., Horne R.N., Ramey H.J. (1988) Pressure-transient response of an infinite-conductivity vertical fracture in a reservoir with double-porosity behavior, SPE Form. Eval. 3, 3, 510–518. https://doi.org/10.2118/12778-PA. [CrossRef] [Google Scholar]
  • Jourde H., Pistrea S., Perrochet P., Droguea C. (2002) Origin of fractional flow dimension to a partially penetrating well in stratified fractured reservoirs. New results based on the study of synthetic fracture networks, Adv. Water Resour. 25, 4, 371–387. [Google Scholar]
  • Kang P.K., Le Borgne T., Dentz M., Bour O., Juanes R. (2015) Impact of velocity correlation and distribution on transport in fractured media: Field evidence and theoretical model, Water Resour. Res. 51, 2, 940–959. https://doi.org/10.1002/2014WR015799. [Google Scholar]
  • Karimi-Fard M., Durlofsky L.J., Aziz K. (2004) An efficient discrete-fracture model applicable for general-purpose reservoir simulators, SPE J. 9, 2, 227–236. https://doi.org/10.2118/88812-PA. [CrossRef] [Google Scholar]
  • Kazemi H. (1969) Pressure transient analysis of naturally fractured reservoirs, Trans. AIME 256, 451–461. [Google Scholar]
  • Kenkre V.M., Montroll E.W., Shlesinger M.F. (1973) Generalized master equations for continuous-time random walks, J. Stat. Phys. 9, 1, 45–50. [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. [Google Scholar]
  • Klafter J., Sokolov I.M. (2011) First steps in random walks, Oxford University Press, p. 152. [Google Scholar]
  • Larsen L., Hegre T.M. (1991) Pressure-transient behavior of horizontal wells with finite-conductivity vertical fractures, in: Paper SPE 22076, Presented at the International Arctic Technology Conference, May 29–31, Anchorage, Alaska, USA. [Google Scholar]
  • Le Mẽhautẽ A., Crepy G. (1983) Introduction to transfer and motion in fractal media: The geometry of kinetics, Solid State Ion. 1, 9–10, 17–30. [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, Micropor. Mesopor. Mater. 178, 15, 39–43. https://doi.org/10.1016/j.micromeso.2013.02.054. [CrossRef] [PubMed] [Google Scholar]
  • Mainardi F. (2010) Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, p. 344. [Google Scholar]
  • Metzler R., Glockle W.G., Nonnenmacher T.F. (1994) Fractional model equation for anomalous diffusion, Phys. A 211, 1, 13–24. [CrossRef] [Google Scholar]
  • Metzler R., Chechkin A.V., Goncharb V.Yu., Klafter J. (2007) Some fundamental aspects of Lévy flights, Chaos Soliton. Fract. 34, 1, 129–142. [CrossRef] [Google Scholar]
  • Metzler R., Jeon J.-H., Cherstvy A.G., Barkai E. (2014) Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16, 24128–24164. [CrossRef] [PubMed] [Google Scholar]
  • Mitchell T.M., Faulkner D.R. (2009) The nature and origin of off-fault damage surrounding strike-slip fault zones with a wide range of displacements: A field study from the Atacama fault system, northern Chile, J. Struct. Geol. 31, 8, 802–816. https://doi.org/10.1016/j.jsg.2009.05.002. [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]
  • Montroll E.W., Weiss G.H. (1965) Random walks on lattices II, J. Math. Phys. 6, 167–181. [Google Scholar]
  • Moodie T.B., Tait R. (1983) On thermal transients with finite wave speeds, J. Acta Mech. 50, 1–2, 97–104. https://doi.org/10.1007/BF01170443. [CrossRef] [Google Scholar]
  • Nigmatullin R.R. (1984) To the theoretical explanation of the universal response, Phys. Status Solidi B Basic Res. 123, 2, 739–745. [CrossRef] [Google Scholar]
  • Nigmatullin R.R. (1986) The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi B Basic Res. 133, 1, 425–430. [CrossRef] [Google Scholar]
  • Noetinger B. (2015) A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow, J. Comput. Phys. 283, 205–223. [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. [CrossRef] [Google Scholar]
  • Noetinger B., Estebenet T., Landereau P. (2001) A direct determination of the transient exchange term of fractured media using a continuous time random walk method, Transp. Porous Med. 44, 3, 539–557. https://doi.org/10.1023/A:1010647108341. [CrossRef] [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] [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]
  • O’Shaughnessy B., Procaccia I. (1985a) Analytical solutions for diffusion on fractal objects, Phys. Rev. Lett. 54, 5, 455–458. [Google Scholar]
  • O’Shaughnessy B., Procaccia I. (1985b) Diffusion on fractals, Phys. Rev. A 32, 5, 3073–3083. [Google Scholar]
  • Ozkan E., Raghavan R. (1991) Some new solutions to solve problems in well test analysis: I-analytical considerations, SPE Form. Eval. 3, 359–368. https://doi.org/10.2118/18615-PA. [CrossRef] [Google Scholar]
  • Palacio J.C., Blasingame T.A. (1993) Decline-curve analysis with type curves – analysis of gas well production data, Presented at the SPE Low Permeability Reservoirs Symposium, 26–28 April, Denver. https://doi.org/10.2118/25909-MS. [Google Scholar]
  • Povstenko Y. (2015) Linear fractional diffusion-wave equation for scientists and engineers, Birkhäuser, p. 460. [Google Scholar]
  • Prats M. (1961) Effect of vertical fractures on reservoir behavior – incompressible fluid case, Soc. Pet. Eng. J. 1, 2, 105–118. https://doi.org/10.2118/1575-G. [CrossRef] [Google Scholar]
  • Pruess K., Narasimhan T. (1985) A practical method for modeling fluid and heat flow in fractured porous media, SPE J. 25(1), 14–26. https://doi.org/10.2118/10509-PA. [Google Scholar]
  • Raghavan R. (1980) The effect of producing time on type curve analysis, J. Petrol. Technol. 32, 6, 1053–1064. https://doi.org/10.2118/6997-PA. [CrossRef] [Google Scholar]
  • Raghavan R. (2004) A review of applications to constrain pumping test responses to improve on geological description and uncertainty, Rev. Geophys. 42, RG4001. https://doi.org/10.1029/2003RG000142. [Google Scholar]
  • Raghavan R. (2008) A note on the drawdown, diffusive behavior of fractured rocks, Water Resour. Res. 45, 2, W02502. [Google Scholar]
  • Raghavan R. (2011) Fractional derivatives: Application to transient flow, J. Pet. Sci. Eng. 801, 7–13. https://doi.org/10.1016/j.petrol.2011.10.003. [Google Scholar]
  • Raghavan R., Ohaeri C.U. (1981) Unsteady Flow to a Well Produced at Constant Pressure in a Fractured Reservoir, in: Paper SPE-9902MS, Presented at the SPE California Regional Meeting, 25–27 March, Bakersfield, California. https://doi.org/10.2118/9902-MS. [Google Scholar]
  • Raghavan R., Chen C. (2017) Addressing the influence of a heterogeneous matrix on well performance in fractured rocks, Transp. Porous Med. 117, 1, 69–102. https://doi.org/10.1007/s11242-017-0820-5. [CrossRef] [Google Scholar]
  • Raghavan R., Chen C. (2018a) Time and space fractional diffusion in finite systems, Transp. Porous Med. 1231, 173–193. https://doi.org/10.1007/s11242-018-1031-4. [CrossRef] [Google Scholar]
  • Raghavan R., Chen C. (2018b) 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) Evaluation of Fractured Rocks through Anomalous Diffusion, in: Paper SPE-195305-MS, Presented at the SPE Western Regional Meeting, 23–26 April, San Jose, California, USA. [Google Scholar]
  • Raghavan R., Chen C., Agarwal B. (1997) An analysis of horizontal wells intercepted by multiple fractures, SPE J. 2, 3, 235–245. https://doi.org/10.2118/27652-PA. [CrossRef] [Google Scholar]
  • Raghavan R., Dixon T.N., Phan V.Q., Robinson S.W. (2001) Integration of geology, geophysics, and numerical simulation in the interpretation of a well test in a fluvial reservoir, SPE Reserv. Eval. Eng. Soc. Petrol. Eng. 4, 3, 201–208. https://doi.org/10.2118/72097-PA. [CrossRef] [Google Scholar]
  • Savage H.M., Brodsky E.E. (2011) Collateral damage: Evolution with displacement of fracture distribution and secondary fault strands in fault damage zones, J. Geophys. Res. 116, B03405. https://doi.org/10.1029/2010JB007665. [Google Scholar]
  • Saxena R.K., Mathai A.M., Haubold H.J. (2006) Fractional reaction-diffusion equations, Astrophys. Space Sci. 305, 3, 289–296. [Google Scholar]
  • Scholz C.H., Dawers N.H., Yu J.Z., Anders M.H., Cowie P.A. (1993) Fault growth and fault scaling laws: Preliminary results, J. Geophys. Res. 98, B12, 21951–21961. [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, in: Paper URTEC-2154675, Proceedings of Unconventional Resources Technology Conference, San Antonio, Texas. https://doi.org/10.15530/URTEC-2015-2154675. [Google Scholar]
  • Spath J.B., Ozkan E., Raghavan R. (1994) An efficient algorithm for computation of well responses in commingled reservoirs, SPE Form. Eval. 9, 2, 115–121. https://doi.org/10.2118/21550-PA. [CrossRef] [Google Scholar]
  • Stehfest H. (1970a) Algorithm 368: Numerical inversion of Laplace transforms [D5], Commun. ACM 13, 1, 47–49. [Google Scholar]
  • Stehfest H. (1970b) Remark on algorithm 368: Numerical inversion of Laplace transforms, Commun. ACM 13, 10, 624. [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., Hashida T., Li K., 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. [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. Nat. 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. American Geophysical Union Transactions 16, 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 Res. Eval. Eng. 8, 3, 248–254. https://doi.org/10.2118/77452-PA. [CrossRef] [Google Scholar]
  • Uchaikin V.V. (2013) Fractional derivatives for physicists and engineers, Volume I: Background and theory, Springer, New York, p. 384. [Google Scholar]
  • van Everdingen A.F., Hurst W. (1949) The application of the LaPlace transformation to flow problems in reservoirs, Trans. AIME 186, 305–324. [Google Scholar]
  • Warren J.E., Root P.J. (1963) The Behavior of Naturally Fractured Reservoirs, Soc. Pet. Eng. J. 3, 3, 245–255. https://doi.org/10.2118/426-PA. [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. [Google Scholar]
  • Yeh N.S., Davison M.J., Raghavan R. (1986) Fractured well responses in heterogeneous systems-application to Devonian Shale and Austin Chalk reservoirs, ASME. J. Energy Resour. Technol. 108, 2, 120–130. https://doi.org/10.1115/1.3231251. [Google Scholar]
  • Yeh T.-C.J., Mao D., Zha Y., Wen J., Wan L., Hsu K., Lee C. (2015) Uniqueness, scale, and resolution issues in groundwater model parameter identification, Water Sci. Eng. 8, 3, 175–194. https://doi.org/10.1016/j.wse.2015.08.002. [Google Scholar]
  • Zhang Y., Benson D.A., Reeves D.M. (2009) Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Res. 32, 561–581. [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. [Google Scholar]
  • Zhokh A., Strizhak P. (2019) Investigation of the anomalous diffusion in the porous media: a spatiotemporal scaling, Heat Mass Trans. 55, 1–10. https://doi.org/10.1007/s00231-019-02602-4. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.