Document Type : Original Research Paper


1 Department of Mining and Metallurgical Engineering, Faculty of Engineering, Yazd University, Yazd, Iran

2 School of Mining Engineering, College of Engineering, University of Tehran, Tehran, Iran


The presence of pores and cracks in porous and fractured rocks is mostly accompanied by fluid flow. Poroelasticity can be used for the accurate modeling of many rock structures in the petroleum industry. The approach of the stress to the value of the fracture stress and the effect of pore pressure on the deformation of rock are among the effects of fluid on the mechanical behavior of the medium. Due to the deformation-diffusion property of porous media, governing equations, strain-displacement, and stress-strain relationships can be changed to each other. In this study, constitutive equations and relationships necessary to investigate the behavior and reaction of rock in a porous environment are stated. Independent and time-dependent differential equations for an impulse and point fluid source are used to obtain the fundamental solutions. Influence functions are obtained by using the shape functions in the formulation of the fundamental solutions and integrating them. To check the validity and correctness of provided formulation, several examples are mentioned. In the first two examples, numerical application and analytical solution are used at different times and in undrained and drained conditions. In times 0 (undrained response of medium) and 4500 seconds (drained response of medium), there is good coordination and agreement between the numerical and analytical results. In the third example, using the numerical application, a crack propagation path in the wellbore wall is obtained, which is naturally in the direction of maximum horizontal stress.


[1]. Crouch, SL. and Starfield, AM. (1983). Boundary Element Methods in Solid Mechanics: With Applications in Rock Mechanics and Geological Engineering. Allen & Unwin.
[2]. Crouch, SL. (1976). Engineering U of MD of C and M, Program NSF (U.S). RA to NN. Analysis of Stresses and Displacements Around Underground Excavations.
[3]. Crouch, SL. (1976). Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int J Numer Methods Eng. 10 (2): 301–343.
[4]. Chaoxi, L. and Suaris, W. (1991). Hadamard’s principle for displacement discontinuity modeling of cracks. Eng FractMech; 39:141–5.
[5]. Fatehi-Marji, M. (2011). On the crack propagation mechanism of brittle rocks under various loading conditions.
[6]. Haeri, H. Shahriar, K. Fatehi-Marji, M. and Moarefvand, P. (2014). Experimental and numerical study of crack propagation and coalescence in pre-cracked rock-like disks. Int J Rock Mech Min Sci; 67:20–8.
[7]. Fatehi-Marji, M. (2014). Rock fracture mechanics with displacement discontinuity method. Ger L Lambert Acad Publ.
[8]. Crawford, A.M. and Curran, J.H. (1982). Higher-order functional variation displacement discontinuity elements. Int J Rock Mech Min Sci Geomech Abstr ;19: 143–8.
[9]. Napier, J.A.L. and Malan, D.F. (1997). A viscoplastic discontinuum model of time-dependent fracture and seismicity effects in brittle rock. Int J Rock Mech Min Sci; 34 (7): 1075–89.
[10]. Fatehi-Marji, M. (2015). Simulation of crack coalescence mechanism underneath single and double disc cutters by higher order displacement discontinuity method. J Cent South Univ; 22 (3):1045–54.
[11]. Abdollahipour, A. and Fatehi-Marji, M. (2020). A thermo-hydromechanical displacement discontinuity method to model fractures in high-pressure, high-temperature environments. RenewEnergy.
[12]. Fatehi-Marji, M. (1996). Modeling of cracks in rock fragmentation with a higher order displacement discontinuity method. Ankara, Turkey: Middle East Technical University.
[13]. Abdollahipour, A. Fatehi-Marji, M. Yarahmadi-Bafghi, A. and Gholamnejad, J. (2016). On the accuracy of higher order displacement discontinuity method (HODDM) in the solution of linear elastic fracture mechanics problems. J Cent South Univ; 23 (11):2941–50.
[14]. Exadaktylos, G. and Xiroudakis, G. (2010). The G2 constant displacement discontinuity method–Part I: Solution of plane crack problems. Int J Solids Struct; 47(18-19): 2568–77.
[15]. Dehghani-Firoozabadi, M.H. Fatehi-Marji, M. Abdollahipour, A. Yarahamdi-Bafghi, A. and Mirzaeian, Y. (2022). Simulation of Crack Propagation Mechanism in Porous Media using Modified linear Element Displacement Discontinuity Method. Volume 13, Issue 3, July 2022, Pages 903-927.
[16]. Exadaktylos, G. and Xiroudakis G. (2010). A G2 constant displacement discontinuity element for analysis of crack problems. Comput Mech; 45 (4):245–61.
[17]. Fatehi-Marji, M. Hosseini-Nasab, H. and Kohsary, A.H. (2007). A new cubic element formulation of the displacement discontinuity method using three special crack tip elements for crack analysis. JP J Solids Struct; 1:61–91.
[18]. Yan, X. (2005). An efficient and accurate numerical method of stress intensity factors calculation of a branched crack. J Appl Mech; 72:330–40.
[19]. Yan, X. (2006). Multiple crack fatigue growth modeling by displacement discontinuity method with crack-tip elements. Appl Math Model; 30 (6):489–508.
[20]. Li, J. Sladek, J. Sladek, V. and Wen, P.H. (2020). Hybrid meshless displacement discontinuity method (MDDM) in fracture mechanics: static and dynamic. Eur J Mech
[21]. Naredran, V.M. and Cleary, M.P. (1983). Analysis of growth and interaction of multiple hydraulic fractures. Reserv. Stimul. Symp., San Francisco.
[22]. Adachi, J.I. and Detournay, E. (2008). Plane strain propagation of a hydraulic fracture in a permeable rock. Engng Fract Mech 2008; 75:4666–94. mech.04.006.
[23]. Ito, T. (2008). Effect of pore pressure gradient on fracture initiation in fluid saturated porous media: Rock. Eng Fract Mech; 75:1753–62.
[24]. Huang, J. Griffiths, D.V-V and Wong, S. (2012). Initiation pressure, location and orientation of hydraulic fracture. Int J Rock Mech Min Sci; 49: 59–67.2011.11.014.
[25]. Yu, W. Luo, Z. Javadpour, F. Varavei, A. and Sepehrnoori, K. (2014). Sensitivity analysis of hydraulic fracture geometry in shale gas reservoirs. JPetSciEng; 113:1–7.
[26]. Bush, D.D. and Barton, N. (1989). Application of small-scale hydraulic fracturing for stress measurements in bedded salt. Int J Rock Mech Min Sci Geomech Abstr; 26:629–35.
[27]. Schmitt, D.R. and Zoback, M.D. (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.
[28]. Abdollahipour, A. Fatehi Marji, M. and Yarahmadi-Bafghi, A.R. (2013). A fracture mechanics concept of in-situ stress measurement by hydraulic fracturing test. 6th Int. Symp. In-situ Rock Stress, Sendai, Japan: ISRM.
[29]. Legarth, B. Huenges, E. and Zimmermann, G. (2005). Hydraulic fracturing in a sedimentary geothermal reservoir: Results and implications. Int J Rock Mech Min Sci ;42:1028–1041.
[30]. Reinicke, A. Zimmermann, G. (2010).  Hydraulic stimulation of a deep sandstone reservoir to develop an enhanced geothermal system:laboratory and field experiments. Geothermics39:70–77.
[31]. Davis, R. and Carter, L. (2013). Fracking Doesn’t Cause Significant Earthquakes. Durham Univesity.
[32]. Hofmann, H. Babadagli, T. and 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.
[33]. Jaeger, J. Cook, N. and Zimmerman, R. (2009). Fundamentals of rock mechanics. Wiley, New York.
[34]. Boonei, T.J. Ingraffea, A.R. 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.
[35]. Yin, S. Dusseault, M.B. and 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. 
[36]. Ji, L. (2013). Geomechanical aspects of fracture growth in a poroelastic, chemically reactive environment. The University of Texas at Austin.
[37]. Bobet, A. and Yu, H. (2015). Stress field near the tip ofa crack in a poroelastic transversely anisotropic saturated rock. Eng Fract Mech 141:1–18.
[38]. Greetesma, J. and de Klerk, F. (1969). A rapid method of predicting width and extent of hydraulic induced fractures. J Pet Tech 21:1571–1581.
[39]. Detournay, E. (2004). Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4:35–45.
[40]. Mitchell, S, Kuske, R. and Peirce, A. (2006). An asymptotic framework for finite hydraulic fractures including leak-off. SIAM J Appl Math 67:364–386
[41]. 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.
[42]. Mitchell, S. Kuske, R. and Peirce, A. (2007). An asymptotic framework for the analysis of hydraulic fractures: the impermeable case. J Appl Mech Trans ASME 74:365–372.
[43]. Hu, J. and 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.
[44]. Lobao, M. Eve, R. Owen, DRJ. et al. (2010). Modelling of hydrofracture flow in porous media. EngngComput 27:129–154.
[45]. Behnia, M. Goshtasbi, K. Fatehi Marji, M. and Golshani, A. (2012). On the crack propagation modeling of hydraulic fracturing by a hybridized displacement discontinuity/ boundary collocation method. J Min Environ; 2.
[46]. Darvish, H. Nouri-Taleghani, M. Shokrollahi, A. and Tatar, A. (2015). Geo-mechanical modeling and selection of suitable layer for hydraulic fracturing operation in an oil reservoir (south west of Iran). J African Earth Sci; 111:409–20.
[47]. Yaylaci, M. Merve, A. Ecren, U. Yaylaci H. (2022). The contact problem of the functionally graded layer resting on rigid foundation pressed via rigid punch, Steel and Composite Structures. 43 661–672.
[48]. M. Yaylacı, M. Abanoz, E.U. Yaylacı, H. Ölmez, D.M. Sekban, A. Birinci, Evaluation of the contact problem of functionally graded layer resting on rigid foundation pressed via rigid punch by analytical and numerical (FEM and MLP) methods, Archive of Applied Mechanics. 92 (2022) 1953–1971.
[49]. M. Turan, E. Uzun Yaylacı, M. Yaylacı, Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods, Archive of Applied Mechanics. 93 (2023) 1351–1372.
[50]. Öner, E. Şengül Şabano, B. Uzun Yaylacı, E. Adıyaman, G. Yaylacı, M. Birinci, A. (2022) On the plane receding contact between two functionally graded layers using computational, finite element and artificial neural network methods, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik. 102.
[51]. Shou, K. J. and Crouch, S. L. (1995). ‘A Higher Order Displacement Discontinuity Method for Analysis of Crack Problems’; Int. J. Rock Mech. Min. Sci. and Geomech. Abstr, 32, pp. 49-55.
[52] Biot, MA. (1941). General theory of three-dimensional consolidation. J Appl Phys 12:155–164.
[53]. Verruijt, A. (1969). Elastic storage in aquifers. Flow through porous media. Academic Press, New York, pp 331–376.
[54]. Rice, J.R. and Cleary, M.P. (1976). Some basic stress diffusion solutions for fluid saturated elastic. Rev Geophys 14:227–241.
[55]. Detournay, E. and Cheng, A. (1987). Poroelastic solution of a plane strain point displacement discontinuity.
[56]. Abdollahipour, A. (2015). Crack propagation mechanism in hydraulic fracturing procedure in oil reservoirs. University of Yazd, Yazd.
[57]. Sneddon, I.N. (1951). Fourier transforms. McGraw-Hill Book Company, New York.