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Time-dependent crack propagation in a poroelastic medium using a fully coupled hydromechanical displacement discontinuity method

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Abstract

Many problems in subsurface rocks which are naturally filled with saturated cracks and pores (with one or more fluid phases) are better understood in a poroelastic framework. Displacement discontinuity method (DDM) is particularly ideal for problems involving fractures and discontinuities. However, the DDM in its original form is limited to elastic problems. The paper derives fundamental solutions of a poroelastic DDM. Then introduces a numerical formulation and implementation for the poroelastic DDM in a code named constant element poroelastic DDM (CEP-DDM). The accuracy and validity of the proposed solution and the newly developed code is verified by an analytical solution at short-time and long-time. Numerical results showed good agreement with analytical results at short time (undrained response) and long time (\(t=8000\) s) (drained response). A crack propagation scheme for crack propagation problems is introduced and demonstrated in an example which enables the code to follow crack propagation in time and space.

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References

  • Abdollahipour A (2015) Crack propagation mechanism in hydraulic fracturing procedure in oil reservoirs. University of Yazd, Yazd

    Google Scholar 

  • Abdollahipour A, Fatehi Marji M, Yarahmadi-Bafghi AR (2013) A fracture mechanics concept of in-situ stress measurement by hydraulic fracturing test. In: The 6th international symposium on in-situ rock stress. ISRM, Sendai, Japan

  • Adachi J, Detournay E (2008) Plane strain propagation of a hydraulic fracture in a permeable rock. Eng Fract Mech 75:4666–4694

    Article  Google Scholar 

  • Andrews LC (1985) Special functions for engineers and applied mathematicians. Macmillan, London

    Google Scholar 

  • Atkinson BK (1984) Subcritical crack growth in geological minerals. J Geophys Res 89:4077–4114

    Article  Google Scholar 

  • Behnia M, Goshtasbi K, Marji MF, Golshani A (2012) The effect of layers elastic parameters on hydraulic fracturing propagation utilizing displacement discontinuity method. Anal Numer Methods Min Eng 3:1–14

    Google Scholar 

  • Benedetti I, Aliabadi MH, Davi G (2008) A fast 3D dual boundary element method based on hierarchical martices. Int J Solids Struct 45:922–933

    Article  Google Scholar 

  • Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–164

    Article  Google Scholar 

  • Bobet A, Yu H (2015) Stress field near the tip of a crack in a poroelastic transversely anisotropic saturated rock. Eng Fract Mech 141:1–18

    Article  Google Scholar 

  • Boonei TJ, Ingrffea AR, Roegiers J-C (1991) Simulation of hydraulic fracture propagation in poroelastic rock with application to stress measurement techniques. Int J Rock Mech Min Sci Geomech abstr 28:1–14

    Article  Google Scholar 

  • Bush DD, Barton N (1989) Application of small-scale hydraulic fracturing for stress measurements in bedded salt. Int J Rock Mech Min Sci Geomech Abstr 26:629–635

    Article  Google Scholar 

  • Cisilino AP, Aliabadi MH (2004) Dual boundary element assessment of three-dimensional fatigue crack growth. Eng Anal Bound Elem 28:1157–1173. doi:10.1016/j.enganabound.2004.01.005

    Article  Google Scholar 

  • Crouch SL (1976) Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int J Numer Methods Eng 10:301–343

    Article  Google Scholar 

  • Crouch SL, Starfield AM (1983) Boundary element methods in solid mechanics. George allen & Unwin, London

    Google Scholar 

  • Davies R, Foulger G, Bindley A, Styles P (2013) Induced seismicity and hydraulic fracturing for the recovery of hydrocarbons. Mar Pet Geol 45:171–185

    Article  Google Scholar 

  • Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4:35–45

    Article  Google Scholar 

  • Detournay E, Cheng A (1987) Poroelastic solution of a plane strain point displacement discontinuity

  • Detournay E, Cheng AHD (1993) Fundamentals of poroelasticity. In: Fairhurst C (ed) Comprehensive rock engineering: principles, practice and projects, vol. ii, analysis and design method. Pergamon Press, Oxford

    Google Scholar 

  • Dirgantara T, Aliabadi MH (2001) Dual boundary element formulation for fracture mechanics analysis of shear deformable shells. Int J Solids Struct 38:7769–7800. doi:10.1016/S0020-7683(01)00097-X

    Article  Google Scholar 

  • Erdogan F, Sih GC (1963) On the crack extension in plates under plate loading and transverse shear. J Basic Eng 85:519–27

    Article  Google Scholar 

  • Fatehi Marji M (2014) Rock fracture mechanics with displacement discontinuity method. LAP Lambert Academic publishing, Saarbrucken

    Google Scholar 

  • Fatehi Marji M (2015) Higher order displacement discontinuity method in rock fracture mechanics. Yazd University, Yazd

    Google Scholar 

  • Fatehi Marji M (1997) Modelling of cracks in rock fragmentation with a higher order displacement discontinuity method. Middle East Technical University, Ankara

    Google Scholar 

  • Fatehi Marji M, Gholamnejad J, Eghbal M (2011) On the crack propagation mechanism of brittle substances under various loading conditions. In: 11th international multidisciplinary scientific geo-conference. Albena, Bulgaria

  • Fatehi Marji M, Hosseini Nasab H, Kohsary AH (2006) On the uses of special crack tip elements in numerical rock fracture mechanics. Int J Solids Struct 43:1669–1692. doi:10.1016/j.ijsolstr.2005.04.042

    Article  Google Scholar 

  • Fedelinski P, Aliabadi MH, Rooke DP (1993) The dual boundary element method in dynamic fracture mechanics. Eng Anal Bound Elem 12:203–210. doi:10.1016/0955-7997(93)90016-E

    Article  Google Scholar 

  • Garagash D (2007) Plane-strain propagation of a fluid-driven fracture during injection and shut-in: asymptotics of large toughness. Engng Fract Mech 74:456–481

    Article  Google Scholar 

  • Greene WH (2003) Econometric analysis. Pearson Education, Gurgaon

    Google Scholar 

  • Greetesma J, de Klerk F (1969) A rapid method of predicting width and extent of hydraulic induced fractures. J Pet Tech 21:1571–1581

    Article  Google Scholar 

  • Haeri H, Shahriar K, Fatehi Marji M, Moaref Vand P (2013a) Simulating the bluntness of TBM disc cutters in rocks using displacement discontinuity method. In: 13th international conference on fracture. China

  • Haeri H, Shahriar K, Fatehi Marji M, Moarefvand P (2013b) An experimental and numerical study of crack propagation and cracks coalescence in the pre-cracked rock-like disc specimens under compression. Int J Rock Mech Min Sci Geomech Abstr 67:20–28

    Google Scholar 

  • Hofmann H, Babadagli T, Zimmermann G (2014) Hot water generation for oil sands processing from enhanced geothermal systems: process simulation for different hydraulic fracturing scenarios. Appl Energy 113:524–547

    Article  Google Scholar 

  • Hu J, Garagash D (2010) Plane-strain propagation of a fluid-driven crack in a permeable rock with fracture toughness. J Eng Mech ASCE 136:1152–1166

    Article  Google Scholar 

  • Huang B, Liu C, Fu J, Guan H (2011) Hydraulic fracturing after water pressure control blasting for increased fracturing. Int J Rock Mech Min Sci 48:976–983

    Article  Google Scholar 

  • Ito T (2008) Effect of pore pressure gradient on fracture initiation in fluid saturated porous media: rock. Eng Fract Mech 75:1753–1762. doi:10.1016/j.engfracmech.2007.03.028

    Article  Google Scholar 

  • Jaeger JC, Cook NGW, Zimmerman R (2009) Fundamentals of rock mechanics. Wiley, New York

  • Ji L (2013) Geomechanical aspects of fracture growth in a poroelastic, chemically reactive environment. The University of Texas at Austin, Austin

    Google Scholar 

  • Kamali Yazdi A, Omidvar B, Rahimian M (2011) Improving the stability of time domain dual boundary element method for three dimensional fracture problems: A time weighting approach. Eng Anal Bound Elem 35:1142–1148. doi:10.1016/j.enganabound.2011.05.003

    Article  Google Scholar 

  • Kim K, Pereira JP (1997) Rolling friction and shear behaviour of rock discontinuities filled with sand. Int J Rock Mech Min Sci 34:244.e1-–244.e17

    Google Scholar 

  • Koegl M, Gaul L (2001) Dual Reciprocity BEM for free vibration analysis of anisotropic solids. In: Twenty-third international conference on the boundary element method. Lemnos, Greece, pp 289–298

  • Lawn R, Wilshaw R (1975) Review indentation fracture: principles and applications. J Mater Sci 10:1049–1081

    Article  Google Scholar 

  • Legarth B, Huenges E, Zimmermann G (2005) Hydraulic fracturing in a sedimentary geothermal reservoir: results and implications. Int J Rock Mech Min Sci 42:1028–1041

    Article  Google Scholar 

  • Lobao M, Eve R, Owen DRJ et al (2010) Modelling of hydro-fracture flow in porous media. Engng Comput 27:129–154. doi:10.1108/02644401011008568

    Article  Google Scholar 

  • Mitchell S, Kuske R, Peirce A (2007) An asymptotic framework for the analysis of hydraulic fractures: the impermeable case. J Appl Mech Trans ASME 74:365–372

    Article  Google Scholar 

  • Mitchell S, Kuske R, Peirce A (2006) An asymptotic framework for finite hydraulic fractures including leak-off. SIAM J Appl Math 67:364–386

    Article  Google Scholar 

  • Morozov VA, Savenkov GG (2013) Limiting velocity of crack propagation in dynamically fractured materials. J Appl Mech Tech Phys 54:142–147

    Article  Google Scholar 

  • Naredran VM, Cleary MP (1983) Analysis of growth and interaction of multiple hydraulic fractures. In: Reservoir stimulation symposium. San Francisco

  • Natarajan S, Mahapatra DR, Bordas SPA (2010) Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework. Int J Numer Methods Eng 83:269–294

    Google Scholar 

  • Pollard DD, Aydin A (1988) Progress in understanding jointing over the past century. Geol Soc Am Bull 100:1181–1204

    Article  Google Scholar 

  • Portela A, Aliabadi MH, Rooke DP (1992) The dual boundary element method, effective implementation for crack problems. Int J Numer Methods Eng 22:1269–1287

    Article  Google Scholar 

  • Reinicke A, Zimmermann G (2010) Hydraulic stimulation of a deep sandstone reservoir to develop an enhanced geothermal system: laboratory and field experiments. Geothermics 39:70–77. doi:10.1016/j.geothermics.2009.12.003

    Article  Google Scholar 

  • Rice JR, Cleary MP (1976) Some basic stress diffusion solutions for fluid saturated elastic porous media with compressible constituents. Rev Geophys 14:227–241

    Article  Google Scholar 

  • Romlay FRM, Ouyang H, Ariffin AK, Mohamed NAN (2010) Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method. Eng Anal Bound Elem 34:297–305. doi:10.1016/j.enganabound.2009.09.006

  • Sanford RJ (2003) Principles of fracture mechanics. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Schmitt DR, Zoback MD (1989) Poroelastic effects in the determination of the maximum horizontal principal stress in hydraulic fracturing tests–A proposed breakdown equation employing a modified effective stress relation for tensile failure. Int J Rock Mech Min Sci Geomech Abstr 26:499–506

    Article  Google Scholar 

  • Shou KJ, Crouch SL (1995) A higher order displacement discontinuity method for analysis of crack problems. Int J Rock Mech Min Sci Geomech Abstr 32:49–55

    Article  Google Scholar 

  • Shou K-J, Napier JALAL (1999) A two-dimensional linear variation displacement discontinuity method for three-layered elastic media. Int J Rock Mech Min Sci 36:719–729

    Article  Google Scholar 

  • Simpson R, Trevelyan J (2011) A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics. Comput Methods Appl Mech Eng 200:1–10

    Article  Google Scholar 

  • Sneddon IN (1951) Fourier transforms. McGraw-Hill Book Company, New York

    Google Scholar 

  • Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipes in Fortran 77 and Fortran 90. Cambridge University Press, Cambridge

    Google Scholar 

  • Thanh TuB, Popov V (2008) Boundary element dual reciprocity method with overlapping sub-domains. Bound Elem Other Mesh Reduct Methods XXX I:179–187. doi:10.2495/BE080181

    Article  Google Scholar 

  • Verruijt A (1969) Elastic storage in aquifers. Flow through porous media. Academic Press, New York, pp 331–376

    Google Scholar 

  • Yin S, Dusseault MB, Rothenburg L (2009) Multiphase poroelastic modeling in semi-space for deformable reservoirs. J Pet Sci Eng 64:45–54

    Article  Google Scholar 

  • Yin S, Dusseault MB, Rothenburg L (2007) Analytical and numerical analysis of pressure drawdown in a poroelastic reservoir with complete overburden effect considered. Adv Water Resour 30:1160–1167

    Article  Google Scholar 

  • Yu W, Luo Z, Javadpour F et al (2014) Sensitivity analysis of hydraulic fracture geometry in shale gas reservoirs. J Pet Sci Eng 113:1–7. doi:10.1016/j.petrol.2013.12.005

    Article  Google Scholar 

  • Yun BI, Ang WT (2010) A dual-reciprocity boundary element approach for axisymmetric nonlinear time-dependent heat conduction in a nonhomogeneous solid. Eng Anal Bound Elem 34:697–706. doi:10.1016/j.enganabound.2010.03.013

    Article  Google Scholar 

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Appendix

Appendix

Time-independent part of influence functions

$$\begin{aligned} \left( {\sigma _{xx}^{ds,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{111} \left( {x-\eta ,y} \right) } d\eta =\frac{G}{2\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^{+a} {\left[ {\frac{8y\left( {x-\eta } \right) ^{3}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}-\frac{2y\left( {x-\eta } \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right] } d\eta \nonumber \\= & {} \left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{y\left( {3\left( {x-\eta } \right) ^{2}+y^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a}\nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{y\left( {3\left( {x-a} \right) ^{2}+y^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{y\left( {3\left( {x+a} \right) ^{2}+y^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(51)
$$\begin{aligned} \left( {\sigma _{yy}^{ds,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{221} \left( {x-\eta ,y} \right) } d\eta =\frac{G}{2\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^{+a} {\left[ {\frac{8\left( {x-\eta } \right) y^{3}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}-\frac{2y\left( {x-\eta } \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right] } d\eta \nonumber \\= & {} \left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{y\left( {y^{2}-\left( {x-\eta } \right) ^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a}\nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{y\left( {y^{2}-\left( {x-a} \right) ^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{y\left( {y^{2}-\left( {x+a} \right) ^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(52)
$$\begin{aligned} \left( {\sigma _{yx}^{ds,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{211} \left( {x-\eta ,y} \right) } d\eta =\frac{G}{2\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^{+a} {\left[ {\frac{8\left( {x-\eta } \right) ^{2}y^{2}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) }} \right] } d\eta \nonumber \\= & {} \left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{\left( {x-\eta } \right) \left( {y^{2}-\left( {x-\eta } \right) ^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{\left( {x-a} \right) \left( {y^{2}-\left( {x-a} \right) ^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{\left( {x+a} \right) \left( {y^{2}-\left( {x+a} \right) ^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(53)
$$\begin{aligned} \left( {\sigma _{xx}^{dn,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{112} \left( {x-\eta ,y} \right) } d\eta =\frac{G}{2\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^{+a} {\left[ {\frac{8\left( {x-\eta } \right) ^{2}y^{2}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) }} \right] } d\eta \nonumber \\= & {} \left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{\left( {x-\eta } \right) \left( {y^{2}-\left( {x-\eta } \right) ^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{\left( {x-a} \right) \left( {y^{2}-\left( {x-a} \right) ^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{\left( {x+a} \right) \left( {y^{2}-\left( {x+a} \right) ^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(54)
$$\begin{aligned} \left( {\sigma _{yy}^{dn,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{222} \left( {x-\eta ,y} \right) } d\eta \nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\int \limits _{-a}^{+a} \left[ \frac{8y^{4}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\right. \nonumber \\&\left. -\frac{4y^{2}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) }-\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) } \right] d\eta \nonumber \\= & {} -\left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{\left( {x-\eta } \right) \left( {\left( {x-\eta } \right) ^{2}+3y^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} -\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{\left( {x-a} \right) \left( {\left( {x-a} \right) ^{2}+3y^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{\left( {x+a} \right) \left( {\left( {x+a} \right) ^{2}+y^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(55)
$$\begin{aligned} \left( {\sigma _{yx}^{dn,\lambda } } \right) ^{0}= & {} \int \limits _{-a}^{+a} {\sigma _{212} \left( {x-\eta ,y} \right) } d\eta =\frac{G}{2\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^{+a} {\left[ {\frac{8\left( {x-\eta } \right) y^{3}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}-\frac{2\left( {x-\eta } \right) y}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right] } d\eta \nonumber \\= & {} \left. {\frac{G}{2\pi \left( {1-\nu _u } \right) }\left( {\frac{y\left( {y^{2}-\left( {x-\eta } \right) ^{2}} \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right) } \right| _{\eta =-a}^{\eta =+a}\nonumber \\= & {} \frac{G}{2\pi \left( {1-\nu _u } \right) }\left( \frac{y\left( {y^{2}-\left( {x-a} \right) ^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}\right. \nonumber \\&\left. -\frac{y\left( {y^{2}-\left( {x+a} \right) ^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}} \right) \end{aligned}$$
(56)
$$\begin{aligned} \left( {P_x^\lambda } \right) ^{0}= & {} \int \limits _{-a}^a {p_1 (x-\eta ,y)d\eta } =-\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\nonumber \\&\times \int \limits _{-a}^a {\left[ {\frac{2\left( {x-\eta } \right) y}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right] d\eta } \nonumber \\= & {} \left. {-\,\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( {\frac{y}{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) } \right| _{x=-a}^{x=+a}\nonumber \\= & {} -\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( \frac{y}{\left( {x-a} \right) ^{2}+y^{2}}-\frac{y}{\left( {x+a} \right) ^{2}+y^{2}} \right) \nonumber \\ \end{aligned}$$
(57)
$$\begin{aligned} \left( {P_y^\lambda } \right) ^{0}= & {} \int \limits _{-a}^a {p_2 (x-\eta ,y)d\eta }\nonumber \\= & {} \frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\int \limits _{-a}^a {\left[ {\frac{\left( {x-\eta } \right) ^{2}-y^{2}}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}} \right] d\eta } \nonumber \\= & {} \left. {\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( {\frac{\left( {x-\eta } \right) }{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) } \right| _{x=-a}^{x=+a}\nonumber \\= & {} \frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( {\frac{\left( {x-a} \right) }{\left( {x-a} \right) ^{2}+y^{2}}-\frac{\left( {x+a} \right) }{\left( {x+a} \right) ^{2}+y^{2}}} \right) \nonumber \\ \end{aligned}$$
(58)

Time-dependent part of influence functions

$$\begin{aligned} \Delta \sigma _{xx}^{ds,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left. \left( y\left( {y^{2}-3\left( {x-\eta } \right) ^{2}} \right) \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right. \right. \nonumber \\&\left. \left. -2y^{3}\xi ^{4}e^{-\xi ^{2}} \right) \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\nonumber \\&\times \left\{ \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}-2y^{3}\xi ^{4}e^{-\xi ^{2}}} \right] \right. \nonumber \\&\left. \times \frac{y\left( {y^{2}-3\left( {x-a} \right) ^{2}} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}-\frac{y\left( {y^{2}-3\left( {x+a} \right) ^{2}} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}} \right. \nonumber \\&\left. {-\,\frac{2y^{3}\xi ^{4}e^{-\xi ^{2}}}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}+\frac{2y^{3}\xi ^{4}e^{-\xi ^{2}}}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}}} \right\} \end{aligned}$$
(59)
$$\begin{aligned} \Delta \sigma _{yy}^{ds,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left. \left[ y\left( {3\left( {x-\eta } \right) ^{2}-y^{2}} \right) \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right. \right. \nonumber \\&\left. \left. -\,2\left( {x-\eta } \right) ^{2}y\xi ^{4}e^{-\xi ^{2}} \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\nonumber \\&\times \left( {\frac{1}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}}} \right) \nonumber \\&\times \left[ \left( y\left( {3\left( {x-a} \right) ^{2}-y^{2}} \right) \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right. \right. \nonumber \\&\left. -\,2\left( {x-a} \right) ^{2}y\xi ^{4}e^{-\xi ^{2}} \right) -\left( y\left( {3\left( {x+a} \right) ^{2}-y^{2}} \right) \right. \nonumber \\&\left. \left. {\left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] -2\left( {x+a} \right) ^{2}y\xi ^{4}e^{-\xi ^{2}}} \right) \right] \nonumber \\ \end{aligned}$$
(60)
$$\begin{aligned} \Delta \sigma _{yx}^{ds,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left. \left[ \left( {x-\eta } \right) \left( {\left( {x-\eta } \right) ^{2}-3y^{2}} \right) \right. \right. \nonumber \\&\times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \nonumber \\&\left. \left. +\,2\left( {x-\eta } \right) y^{2}\xi ^{4}e^{-\xi ^{2}} \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\left( \frac{1}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}\right. \nonumber \\&\left. -\frac{1}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}} \right) \left( \left[ \left( {x-a} \right) \left( {\left( {x-a} \right) ^{2}-3y^{2}} \right) \right. \right. \nonumber \\&\times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \left. +2\left( {x-a} \right) y^{2}\xi ^{4}e^{-\xi ^{2}} \right) \nonumber \\&-\left( \left( {x+a} \right) \left( {\left( {x+a} \right) ^{2}-3y^{2}} \right) \right. \nonumber \\&\left. \left. \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right) +2\left( {x+a} \right) y^{2}\xi ^{4}e^{-\xi ^{2}} \right) \nonumber \\\end{aligned}$$
(61)
$$\begin{aligned} \Delta \sigma _{xx}^{dn,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left[ \left( {x-\eta } \right) \left( \left( {x-\eta } \right) ^{2}-3y^{2} \right) \right. \nonumber \\&\times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \nonumber \\&+ \left. \left. \,2\left( {x-\eta } \right) y^{2}\xi ^{4}e^{-\xi ^{2}} \right] {} \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\nonumber \\&\times \left( {\frac{1}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}}} \right) \nonumber \\&\times \left( \left[ \left( {x-a} \right) \left( {\left( {x-a} \right) ^{2}-3y^{2}} \right) \right. \right. \nonumber \\&\left. \times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] +2\left( {x-a} \right) y^{2}\xi ^{4}e^{-\xi ^{2}}\right) \nonumber \\&-\left( \left( {x+a} \right) \left( {\left( {x+a} \right) ^{2}-3y^{2}} \right) \right. \nonumber \\&\left. \left. \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right) +2\left( {x+a} \right) y^{2}\xi ^{4}e^{-\xi ^{2}} \right) \nonumber \\ \end{aligned}$$
(62)
$$\begin{aligned} \Delta \sigma _{yy}^{dn,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left[ \left( {x-\eta } \right) \left( {3y^{2}-\left( {x-\eta } \right) ^{2}} \right) \right. \nonumber \\&\times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \nonumber \\&\left. \left. +2\left( {x-\eta } \right) ^{3}\xi ^{4}e^{-\xi ^{2}} \right] {} \right| _{\eta =-a}^{\eta =+a}\nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\nonumber \\&\times \left( {\frac{1}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}}} \right) \nonumber \\&\times \left( \left[ \left( {x-a} \right) \left( {3y^{2}-\left( {x-a} \right) ^{2}} \right) \right. \right. \nonumber \\&\left. \times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] +2\left( {x-a} \right) ^{3}\xi ^{4}e^{-\xi ^{2}}\right] \nonumber \\&-\left[ \left( {x+a} \right) \left( {3y^{2}-\left( {x+a} \right) ^{2}} \right) \right. \nonumber \\&\left. \left. \times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] +2\left( {x+a} \right) ^{3}\xi ^{4}e^{-\xi ^{2}} \right] \right) \nonumber \\\end{aligned}$$
(63)
$$\begin{aligned} \Delta \sigma _{yx}^{dn,\lambda }= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\frac{1}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{3}}\nonumber \\&\times \left[ y\left( {3\left( {x-\eta } \right) ^{2}-y^{2}} \right) \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] \right. \nonumber \\&\left. \left. -2\left( {x-\eta } \right) ^{2}y\xi ^{4}e^{-\xi ^{2}} \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2Gc\left( {\nu _u -\nu } \right) }{\pi \left( {1-\nu _u } \right) \left( {1-\nu } \right) }\nonumber \\&\left( {\frac{1}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{3}}-\frac{1}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{3}}} \right) \nonumber \\&\times \left( {\left[ {y\left( {3\left( {x-a} \right) ^{2}-y^{2}} \right) \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] } \right. } \right. \nonumber \\&\left. -\,2\left( {x-a} \right) ^{2}y\xi ^{4}e^{-\xi ^{2}} \right] -\left[ y\left( {3\left( {x+a} \right) ^{2}-y^{2}} \right) \right. \nonumber \\&\left. \left. \times \left[ {1-\left( {1+\xi ^{2}} \right) e^{-\xi ^{2}}} \right] -2\left( {x+a} \right) ^{2}y\xi ^{4}e^{-\xi ^{2}}\right] \right) \nonumber \\ \end{aligned}$$
(64)
$$\begin{aligned} \Delta \sigma _{xx}^{df,\lambda }= & {} \frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left[ \sqrt{\frac{\pi }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \nonumber \\&\left. \times \left( {\frac{\left( {x-\eta } \right) \xi }{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) } \right. \nonumber \\&-\,\left. {\left. {\frac{\left( {x-\eta } \right) }{\left( {x-\eta } \right) ^{2}+y^{2}}\left( {1-e^{-\xi ^{2}}} \right) } \right] } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left\{ \left[ \sqrt{\frac{\pi }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \right. \nonumber \\&\times \left( {\frac{\left( {x-a} \right) \xi }{\left( {x-a} \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x-a} \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) } \nonumber \\&-\,\left. {\frac{\left( {x-a} \right) }{\left( {x-a} \right) ^{2}+y^{2}}\left( {1-e^{-\xi ^{2}}} \right) } \right] \nonumber \\&-\,\left. \left[ \sqrt{\frac{\pi }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \right. \nonumber \\&\times \left( {\frac{\left( {x+a} \right) \xi }{\left( {x+a} \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x+a} \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) }\nonumber \\&\left. \left. -\frac{\left( {x+a} \right) }{\left( {x+a} \right) ^{2}+y^{2}}\left( {1-e^{-\xi ^{2}}} \right) \right] \right\} \end{aligned}$$
(65)
$$\begin{aligned} \Delta \sigma _{yy}^{ds,\lambda }= & {} \left. {\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left[ {\frac{\left( {x-\eta } \right) }{\left( {x-\eta } \right) ^{2}+y^{2}}\left( {1-e^{-\xi ^{2}}} \right) } \right] } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( {1-e^{-\xi ^{2}}} \right) \nonumber \\&\times \left[ {\frac{\left( {x-a} \right) }{\left( {x-a} \right) ^{2}+y^{2}}-\frac{\left( {x+a} \right) }{\left( {x+a} \right) ^{2}+y^{2}}} \right] \end{aligned}$$
(66)
$$\begin{aligned} \Delta \sigma _{yx}^{ds,\lambda }= & {} \left. {-\,\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left[ {\frac{y}{\left( {x-\eta } \right) ^{2}+y^{2}}\left( {1-e^{-\xi ^{2}}} \right) } \right] } \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} -\frac{BG\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left( {1-e^{-\xi ^{2}}} \right) \nonumber \\&\times \left[ {\frac{y}{\left( {x-a} \right) ^{2}+y^{2}}-\frac{y}{\left( {x+a} \right) ^{2}+y^{2}}} \right] \end{aligned}$$
(67)
$$\begin{aligned} \Delta P_x^\lambda= & {} -\frac{4BGc\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left. \left[ {\frac{y}{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}\xi ^{4}e^{-\xi ^{2}}} \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} -\frac{4BGc\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\xi ^{4}e^{-\xi ^{2}}\nonumber \\&\times \left[ {\frac{y}{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}-\frac{y}{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}}} \right] \end{aligned}$$
(68)
$$\begin{aligned} \Delta P_y^\lambda= & {} -\frac{4BGc\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\left. \left[ {\frac{\left( {x-\eta } \right) }{\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) ^{2}}\xi ^{4}e^{-\xi ^{2}}} \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} -\frac{4BGc\left( {1+\nu _u } \right) }{3\pi \left( {1-\nu _u } \right) }\xi ^{4}e^{-\xi ^{2}}\nonumber \\&\times \left[ {\frac{\left( {x-a} \right) }{\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) ^{2}}-\frac{\left( {x+a} \right) }{\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) ^{2}}} \right] \end{aligned}$$
(69)
$$\begin{aligned} \Delta P_f^\lambda= & {} \frac{2B^{2}G\left( {1-\nu } \right) \left( {1+\nu _u } \right) ^{2}}{9\pi \left( {1-\nu _u } \right) \left( {\nu _u -\nu } \right) }\left[ \sqrt{\frac{\pi }{2\left( {\left( {x-\eta } \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \nonumber \\&\times \left. \left. \left( {\frac{\left( {x-\eta } \right) \xi }{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x-\eta } \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) } \right] \right| _{\eta =-a}^{\eta =+a} \nonumber \\= & {} \frac{2B^{2}G\left( {1-\nu } \right) \left( {1+\nu _u } \right) ^{2}}{9\pi \left( {1-\nu _u } \right) \left( {\nu _u -\nu } \right) }\left\{ \left[ \sqrt{\frac{\pi }{2\left( {\left( {x-a} \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \right. \nonumber \\&\left. \times \left( {\frac{\left( {x-a} \right) \xi }{\left( {x-a} \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x-a} \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) } \right] \nonumber \\&-\,\left[ \sqrt{\frac{\pi }{2\left( {\left( {x+a} \right) ^{2}+y^{2}} \right) }}\xi {erf}\right. \nonumber \\&\left. \left. \times \left( {\frac{\left( {x+a} \right) \xi }{\left( {x+a} \right) ^{2}+y^{2}}} \right) e^{\left( {-\left( {{y^{2}}/{\left( {x+a} \right) ^{2}+y^{2}}} \right) \xi ^{2}} \right) } \right] \right\} \end{aligned}$$
(70)

where \(\xi =\sqrt{\frac{r^{2}}{4ct}}\).

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Abdollahipour, A., Marji, M.F., Bafghi, A.Y. et al. Time-dependent crack propagation in a poroelastic medium using a fully coupled hydromechanical displacement discontinuity method. Int J Fract 199, 71–87 (2016). https://doi.org/10.1007/s10704-016-0095-9

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