Abstract
This paper addresses the wave propagation problem within the context of the theory of materials with triple porosity under local thermal non-equilibrium. The time-harmonic wave solution is studied and the dispersion relation is explicitly established as an eighth-degree algebraic equation. It is shown that there are two shear waves undamped in time, that are non-dispersive and unaltered by the presence of the pore system or by thermal effects. Also, there are seven longitudinal wave solutions that are dispersive and damped in time: one longitudinal quasi-elastic wave, three longitudinal quasi-pore standing waves and three other longitudinal quasi-thermal standing waves. An illustrative simulation on a material like Berea sandstone shows that the effects of coupling triple porosity with local thermal imbalance and mechanical deformations leads to an increase in the propagation speed of the longitudinal quasi-elastic wave accompanied by an increase in its rate of decrease over time, as well as to a relaxation of the decrease in time for the quasi-pore and quasi-thermal standing waves.
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References
Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–165 (1941)
Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185 (1955)
Biot, M.A.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91–96 (1956)
Biot, M.A.: Theory of propagation of elastic waves in a fluid saturated porous solid. J. Acoust. Soc. Am. 28, 168–191 (1956)
Cheng, A.H.D.: Poroelasticity. Springer, Switzerland (2016)
Barenblatt, G.I., Zheltov, Y.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech. 24, 1286–1303 (1960)
Barenblatt, G.I.: On some boundary-value problems for the equation of filtration of fluid in fissurized rocks. Prikladnaya Matematika i Mekhanika, 27, No. 2, pp. 348–350 (1963) (in Russian); J. Appl. Math. Mech. (PMM) 27(2), 513–518 (1963) (English Translation). https://doi.org/10.1016/0021-8928(63)90017-0
Wilson, R.K., Aifantis, E.C.: On the theory of consolidation with double porosity. Int. J. Eng. Sci. 20(9), 1009–1035 (1982). https://doi.org/10.1016/0020-7225(82)90036-2
Wilson, R.K., Aifantis, E.C.: A double porosity model for acoustic wave propagation in fractured-porous rock. Int. J. Eng. Sci. 22(8–10), 1209–1217 (1984). https://doi.org/10.1016/0020-7225(84)90124-1
Beskos, D.E.: Dynamics of saturated rocks. I: equations of motion. J. Eng. Mech. ASCE 115(5), 982–995 (1989). https://doi.org/10.1061/(ASCE)0733-9399
Beskos, D.E., Vgenopoulou, I., Providakis, C.P.: Dynamics of saturated rocks. II: body waves. J. Eng. Mech. ASCE 115(5), 996–1016 (1989). https://doi.org/10.1061/(ASCE)0733-9399
Beskos, D.E., Papadakis, C.N.: Dynamics of saturated rocks. III: Rayleigh waves. J. Eng. Mech. ASCE 115(5), 1017–1033 (1989). https://doi.org/10.1061/(ASCE)0733-9399
Vgenopoulou, I., Beskos, D.E.: Dynamics of saturated rocks. IV: column and borehole problems. J. Eng. Mech. ASCE 118(9), 1795–1813 (1992). https://doi.org/10.1061/(ASCE)0733-9399(1992)118:9(1795)
Auriault, J.L., Boutin, C.: Deformable porous media with double porosity III: acoustics. Transp. Porous Media 14, 143–162 (1994). https://doi.org/10.1007/BF00615198
Olny, X., Boutin, C.: Acoustic wave propagation in double porosity media. J. Acoust. Soc. Am. 114(1), 73–89 (2003). https://doi.org/10.1121/1.1534607
Boutin, C., Royer, P.: On models of double porosity poroelastic media. Geophys. J. Int. 203, 1694–1725 (2015). https://doi.org/10.1093/gji/ggv378
Pride, S.R., Berryman, J.G.: Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation. Phys. Rev. E 68, 036603 (2003). https://doi.org/10.1103/PhysRevE.68.036603
Pride, S.R., Berryman, J.G.: Linear dynamics of double-porosity dual-permeability materials. II. Fluid transport equations. Phys. Rev. E 68, 036604 (2003). https://doi.org/10.1103/PhysRevE.68.036604
Al-Ahmadi, H.A.: A triple-porosity model for fractured horizontal wells. Master’s thesis, Texas A&M University (2010). http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8545. Accessed 1 April 2021
Al-Ahmadi, H.A., Wattenbarger, R.A.: Triple-porosity models: one further step towards capturing fractured reservoirs heterogeneity. In: SPE/DGS Saudi Arabia section technical symposium and exhibition. Society of Petroleum Engineers (2011). https://doi.org/10.2118/149054-MS. Accessed 1 April 2021
Tivayanonda, V., Apiwathanasorn, S., Economides, C., Wattenbarger, R.: Alternative interpretations of shale gas/oil rate behavior using a triple porosity model. SPE-159703-MS, Society of Petroleum Engineers (2012). https://doi.org/10.2118/159703-MS. Accessed 1 April 2021
Deng, J.H., Leguizamon, J.A., Aguilera, R.: Petrophysics of triple-porosity tight gas reservoirs with a link to gas productivity. SPE Reserv Eval Eng 14, 566–577 (2011). https://doi.org/10.2118/144590-PA
Shackelford, C.D.: Contaminant transport. In: Daniel, D.E. (ed.) Geotechnical Practice for Waste Disposal, pp. 33–65. Chapman and Hall, London (1993)
Gwo, J.P., Jardine, P.M., Wilson, G.V., Yeh, G.T.: A multiple-pore-region concept to modeling mass transfer in subsurface media. J. Hydrol. 164, 217–237 (1995). https://doi.org/10.1016/0022-1694(94)02555-P
Moutsopoulos, K.N., Konstantinidis, A.A., Meladiotis, I., Tzimopoulos, Ch.D., Aifantis, E.C.: Hydraulic behavior and contaminant transport in multiple porosity media. Transp. Porous Media 42, 265–292 (2001). https://doi.org/10.1023/A:1006745924508
Müller, T.M., Gurevich, B., Lebedev, M.: Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks - a review. Geophysics 75, 75A147-75A164 (2010). https://doi.org/10.1190/1.3463417
Pride, S.R.: Relationships between seismic and hydrological properties. In: Rubin, Y., Hubbard, S.S. (eds.) Hydrogeophysics. Water Science and Technology Library, vol. 50. Springer, Dordrecht (2005). https://doi.org/10.1007/1-4020-3102-5-9
Parotidis, M., Rothert, E., Shapiro, S.A.: Pore-pressure diffusion: A possible triggering mechanism for the earthquake swarms \(2000\) in Vogtland/NW/Bohemia. Central Europe. Geophys. Res. Lett. 30(20), 2075 (2003). https://doi.org/10.1029/2003GL018110
Abdassah, D., Ershaghi, I.: Triple-porosity systems for representing naturally fractured reservoirs. SPE Form. Eval. 1, 113–127 (1986). https://doi.org/10.2118/13409-PA
Bai, M., Elsworth, D., Roegiers, J.C.: Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resour. Res. 29, 1621–1633 (1993). https://doi.org/10.1029/92WR02746
Bai, M., Roegiers, J.C.: Triple-porosity analysis of solute transport. J. Contam. Hydrol. 28(3), 247–266 (1997). https://doi.org/10.1016/S0169-7722(96)00086-1
Aguilera, R.F., Aguilera, R.: A triple porosity model for petrophysical analysis of naturally fractured reservoirs. Petrophysics 45, 157–166 (2004)
Kuznetsov, A.V., Nield, D.A.: The onset of convection in a tridisperse medium. Int. J. Heat Mass Transf. 54, 3120–3127 (2011). https://doi.org/10.1016/j.ijheatmasstransfer.2011.04.021
Olusola, B.K., Yu, G., Aguilera, R.: The use of electromagnetic mixing rules for petrophysical evaluation of dual-and triple-porosity reservoirs. SPE Reserv. Eval. Eng. 16, 378–389 (2013). https://doi.org/10.2118/162772-PA
Zou, M., Wei, C., Yu, H., Song, L.: Modelling and application of coalbed methane recovery performance based on a triple porosity/dual permeability model. J. Nat. Gas Sci. Eng. 22, 679–688 (2015). https://doi.org/10.1016/j.jngse.2015.01.019
Desbois, G., Urai, J.L., Hemes, S., Schröppel, B., Schwarz, J.O., Mac, M., Weiel, D.: Multi-scale analysis of porosity in diagenetically altered reservoir sandstone from the Permian Rotliegend (Germany). J. Pet. Sci. Eng. 140, 128–148 (2016). https://doi.org/10.1016/j.petrol.2016.01.019
Said, B., Grandjean, A., Barre, Y., Tancret, F., Fajula, F., Galameau, A.: LTA zeolite monoliths with hierarchical trimodal porosity as highly efficient microreactors for strontium capture in continuous flow. Microporous Mesoporous Mater. 232, 39–52 (2016). https://doi.org/10.1016/j.micromeso.2016.05.036
Liu, C., Abousleiman, Y.N.: N-porosity and N-permeability generalized wellbore stability analytical solutions and applications. In: 50th U.S. Rock Mechanics/Geomechanics Symposium, 26-29 June, Houston, Texas, American Rock Mechanics Association ARMA-2016-417 [9 pages]
Liu, C., Abousleiman, Y.N.: Multiporosity/multipermeability inclined-wellbore solutions with mudcake effects. SPE J. 23, 1723–1747 (2018). https://doi.org/10.2118/191135-PA
Mehrabian, A., Abousleiman, Y.N.: Multiple-porosity and multiple-permeability poroelasticity: theory and benchmark analytical solution. In: 6th Biot Conference on Poromechanics, Poromechanics 2017—Paris, France, July 9 2017–July 13 2017. pp. 262–271. Paris, France: American Society of Civil Engineers (ASCE) (2017). https://doi.org/10.1061/9780784480779.032
Svanadze, M.: Fundamental solutions in the theory of elasticity for triple porosity materials. Meccanica 51(8), 1825–1837 (2015). https://doi.org/10.1007/s11012-015-0334-6
Svanadze, M.: Potential method in the theory of elasticity for triple porosity materials. J. Elast. 130(1), 1–24 (2018). https://doi.org/10.1007/s10659-017-9629-2
Svanadze, M.: Potential method in the coupled theory of elastic double-porosity materials. Acta Mech. (2021). https://doi.org/10.1007/s00707-020-02921-2
Straughan, B.: Waves and uniqueness in multi-porosity elasticity. J. Therm. Stresses 39(6), 704–721 (2016). https://doi.org/10.1080/01495739.2016.1169136
Straughan, B.: Modelling questions in multi-porosity elasticity. Meccanica 51(12), 2957–2966 (2016). https://doi.org/10.1007/s11012-016-0556-2
Straughan, B.: Uniqueness and stability in triple porosity thermoelasticity. Rend. Lincei-Mat. Appl. 28(2), 191–208 (2017). https://doi.org/10.4171/RLM/758
Galeş, C., Chiriţă, S.: Wave propagation in materials with double porosity. Mech. Mater. (2020). https://doi.org/10.1016/j.mechmat.2020.103558
Chiriţă, S.: Attenuation of an external signal in a thermoelastic material with triple porosity in local thermal non-equilibrium. J. Therm. Stresses 44(6), 768–783 (2021). https://doi.org/10.1080/01495739.2021.1914529
Straughan, B.: Mathematical Aspects of Multi-Porosity Continua. Advances in Mechanics and Mathematics, Springer, New York (2017)
Svanadze, M.: Potential Method in Mathematical Theories of Multi-porosity Media. Series: Interdisciplinary Applied Mathematics, vol. 51. Springer, Cham (2019)
Svanadze, M.: On the linear theory of double porosity thermoelasticity under local thermal nonequilibrium. J. Therm. Stresses 42(7), 890–913 (2019). https://doi.org/10.1080/01495739.2019.1571973
Franchi, F., Lazzari, B., Nibbi, R., Straughan, B.: Uniqueness and decay in local thermal non-equilibrium double porosity thermoelasticity. Math. Methods Appl. Sci. 41(16), 6763–6771 (2018). https://doi.org/10.1002/mma.5190
Chiriţă, S.: Modeling triple porosity under local thermal nonequilibrium. J. Therm. Stresses 43(2), 210–224 (2020). https://doi.org/10.1080/01495739.2019.1679057
Nield, D.A.: A note on modelling of local thermal non-equilibrium in a structured porous medium. Int. J. Heat Mass Transf. 45(21), 4367–4368 (2002). https://doi.org/10.1016/S0017-9310(02)00138-2
Rees, D.A.S., Bassom, A.P., Siddheshwar, P.G.: Local thermal non-equilibrium effects arising from the injection of a hot fluid into a porous medium. J. Fluid Mech. 594, 379–398 (2008). https://doi.org/10.1017/S0022112007008890
Straughan, B.: Convection with Local Thermal Non-equilibrium and Microfluidic Effects. Advances in Mechanics and Mathematics, vol. 32. Springer, New York (2015)
Jaeger, J.C., Cook, N.G.W., Zimmerman, R.W.: Fundamentals of Rock Mechanics, 4th edn. Blackwell Publishing, Malden (2007)
Khaled, M.Y., Beskos, D.E., Aifantis, E.C.: On the theory of consolidation with double porosity—III A finite element formulation. Int. J. Numer. Anal. Meth. Geomech. 8(2), 101–123 (1984). https://doi.org/10.1002/nag.1610080202
Coyner, K.B.: Effects of stress, pore pressure, and pore fluids on bulk strain, velocity, and permeability of rocks. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridg1e (1984). https://dspace.mit.edu/handle/1721.1/15367. Accessed 01 April 2021
Shankland, T.J., Johnson, P.A., Hopson, T.M.: Elastic wave attenuation and velocity of Berea sandstone measured in the frequency domain. Geophys. Res. Lett. 20(5), 391–394 (1993). https://doi.org/10.1029/92GL02758
Davis, E.S., Sturtevant, B.T., Sinha, D.N., Pantea, C.: Resonant ultrasound spectroscopy studies of Berea sandstone at high temperature. J. Geophys. Res. Solid Earth 121(9), 6401–6410 (2016). https://doi.org/10.1002/2016jb013410
Davis, E.S.: Anomalous Elastic Behavior in Berea Sandstone. Ph.D. Thesis, University of Houston, Houston, USA (2018). https://uh-ir.tdl.org/uh-ir/bitstream/handle/10657/3638/DAVIS-DISSERTATION-2018.pdf?sequence=1&isAllowed=y Accessed 01 April 2021
Ikeda, K., Goldfarb, E., Tisato, N.: Calculating effective elastic properties of Berea sandstone using the segmentation less method without targets. J. Geophys. Res. Solid Earth 125(6), e2019JB018680 (2020). https://doi.org/10.1029/2019JB018680
Chiriţă, S., Arusoaie, A.: Thermoelastic waves in double porosity materials. Eur. J. Mech. A/Solids (2021). https://doi.org/10.1016/j.euromechsol.2020.104177
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The authors acknowledge support from the Project STARDUST-R H2020-MSCA-ITN-2018, Grant Agreement No. 813644, Alexandru Ioan Cuza University of Iasi.
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Chiriţă, S., Galeş, C. Wave propagation and attenuation in time in local thermal non-equilibrium triple porosity thermoelastic medium. Acta Mech 232, 4217–4233 (2021). https://doi.org/10.1007/s00707-021-03044-y
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DOI: https://doi.org/10.1007/s00707-021-03044-y