Empirical and Conceptual Challenges in Hydraulic Fracturing with Special Reference to the Inflow
Publication: International Journal of Geomechanics
Volume 20, Issue 3
Abstract
In this paper, a systematic study was carried out to computationally explore the significance of different mechanisms involved in hydraulic fracturing treatments in practice, with special reference given to the hydrofracture inflow. For this reason, our well-established extended finite-element framework for modeling of hydraulic fracturing treatments in saturated porous media was employed. Accordingly, the formulation was adopted to develop the coupled set of equations governing the hydromechanical response of the fractured porous media. The spatial discretization was carried out by means of the extended finite-element method (X-FEM) in order to incorporate the strong discontinuity in the displacement field due to the fracture body as well as the weak discontinuity in the pressure field due to the leak-off flow. Important mechanisms and parameters studied include proppant settlement, fracture energy, leak-off flow, fluid-lag zone, fracture surface roughness and tortuosity, pressure fluctuations, and flow regimes. By benchmarking the results against the reference solution obtained in the absence of the mechanism under consideration, a beneficial criterion for the significance of each mechanism was deduced.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
The authors appreciate Prof. Amir R. Khoei for his valuable viewpoints on the X-FEM formulation of hydraulic fracturing treatments in saturated porous media.
References
Advani, S., T. Lee, R. Dean, C. Pak, and J. Avasthi. 1997. “Consequences of fluid lag in three-dimensional hydraulic fractures.” Int. J. Numer. Anal. Methods Geomech. 21 (4): 229–240. https://doi.org/10.1002/(SICI)1096-9853(199704)21:4%3C229::AID-NAG862%3E3.0.CO;2-V.
Bai, M., D. Elsworth, and J. C. Roegiers. 1993. “Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs.” Water Resour. Res. 29 (6): 1621–1633. https://doi.org/10.1029/92WR02746.
Beach, A. 1980. “Numerical models of hydraulic fracturing and the interpretation of syntectonic veins.” J. Struct. Geol. 2 (4): 425–438. https://doi.org/10.1016/0191-8141(80)90004-8.
Bear, J. 2013. Dynamics of fluids in porous media. New York: Dover Publications.
Belytschko, T., and T. Black. 1999. “Elastic crack growth in finite elements with minimal remeshing.” Int. J. Numer. Methods Eng. 45 (5): 601–620. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5%3C601::AID-NME598%3E3.0.CO;2-S.
Biot, M., and W. Medlin. 1985. “Theory of sand transport in thin fluids.” In Proc., SPE Annual Technical Conf. and Exhibition. Las Vegas: Society of Petroleum Engineers.
Boone, T. J., and A. R. Ingraffea. 1990. “A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media.” Int. J. Numer. Anal. Methods Geomech. 14 (1): 27–47. https://doi.org/10.1002/nag.1610140103.
Bowen, R. 1976. Theory of mixtures in continuum physics. 3rd ed. London: Academic.
Brown, E. T. 2007. Block caving geomechanics: International caving study 1997-2004. Brisbane, UK: Univ. of Queensland.
Cammarata, G., C. Fidelibus, M. Cravero, and G. Barla. 2007. “The hydro-mechanically coupled response of rock fractures.” Rock Mech. Rock Eng. 40 (1): 41–61. https://doi.org/10.1007/s00603-006-0081-z.
Carrier, B., and S. Granet. 2012. “Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model.” Eng. Fract. Mech. 79 (Jan): 312–328. https://doi.org/10.1016/j.engfracmech.2011.11.012.
Carter, B., J. Desroches, A. R. Ingraffea, and P. Wawrzynek. 2000. “Simulating fully 3D hydraulic fracturing.” In Modeling in geomechanics. New York: Wiley.
Carter, R. 1957. “Derivation of the general equation for estimating the extent of the fractured area. Appendix I of optimum fluid characteristics for fracture extension.” In Drilling and production practice, edited by G. C. Howard and C. R. Fast, 261–269. New York: American Petroleum Institute.
Chen, B., S. Cen, A. R. Barron, D. R. J. Owen, and C. Li. 2018. “Numerical investigation of the fluid lag during hydraulic fracturing.” Eng. Comput. 35 (5): 2050–2077. https://doi.org/10.1108/EC-02-2018-0087.
Colebrook, C., and C. White. 1937 “Experiments with fluid friction in roughened pipes.” Proc. R. Soc. London Ser. A: Math. Phys. Sci. 161 (906): 367–381.
Coussy, O. 2004. Poromechanics. New York: Wiley.
Daneshy, A. 1978. “Numerical solution of sand transport in hydraulic fracturing.” J. Petrol. Tech. 30 (01): 132–140. https://doi.org/10.2118/5636-PA.
de Borst, R., J. Réthoré, and M. A. Abellan. 2006. “A numerical approach for arbitrary cracks in a fluid-saturated medium.” Arch. Appl. Mech. 75 (1–12): 595–606. https://doi.org/10.1007/s00419-006-0023-y.
Dehghan, A. N., K. Goshtasbi, K. Ahangari, Y. Jin, and A. Bahmani. 2017. “3D numerical modeling of the propagation of hydraulic fracture at its intersection with natural (pre-existing) fracture.” Rock Mech. Rock Eng. 50 (2): 367–386. https://doi.org/10.1007/s00603-016-1097-7.
Desroches, J., E. Detournay, B. Lenoach, P. Papanastasiou, J. Pearson, M. Thiercelin, and A. Cheng. 1994. “The crack tip region in hydraulic fracturing.” Proc. R. Soc. London: Math. Phys. Eng. Sci. R. Soc. 447 (1929): 39–48. https://doi.org/10.1098/rspa.1994.0127.
Detournay, E. 2004. “Propagation regimes of fluid-driven fractures in impermeable rocks.” Int. J. Geomech. 4 (1): 35–45. https://doi.org/10.1061/(ASCE)1532-3641(2004)4:1(35).
Detournay, E., and A. H. D. Cheng. 1993. “Fundamentals of poroelasticity.” Chap. 5 in Vol. 2 of Comprehensive rock engineering: Principles, practice and projects, II, edited by C. Fairhurst, 113–171. Oxford, UK: Pergamon Press.
Detournay, E., and A. Peirce. 2014. “On the moving boundary conditions for a hydraulic fracture.” Int. J. Eng. Sci. 84 (Nov): 147–155. https://doi.org/10.1016/j.ijengsci.2014.06.010.
Elsworth, D. 1987. “A boundary element-finite element procedure for porous and fractured media flow.” Water Resour. Res. 23 (4): 551–560. https://doi.org/10.1029/WR023i004p00551.
Ferté, G., P. Massin, and N. Moës. 2016. “3D crack propagation with cohesive elements in the extended finite element method.” Comput. Methods Appl. Mech. Eng. 300 (Mar): 347–374. https://doi.org/10.1016/j.cma.2015.11.018.
Forchheimer, P. 1914. Hydraulik. Leipzig, Germany: BG Teubner.
Frank, U., and N. Barkley. 1995. “Remediation of low permeability subsurface formations by fracturing enhancement of soil vapor extraction.” J. Hazard. Mater. 40 (2): 191–201. https://doi.org/10.1016/0304-3894(94)00069-S.
Frankel, N. A., and A. Acrivos. 1967. “On the viscosity of a concentrated suspension of solid spheres.” Chem. Eng. Sci. 22 (6): 847–853. https://doi.org/10.1016/0009-2509(67)80149-0.
Geertsma, J., and F. de Klerk. 1969. “A rapid method of predicting width and extent of hydraulically induced fractures.” J. Pet. Tech. 21 (12): 1571–1581. https://doi.org/10.2118/2458-PA.
Gordeliy, E., and A. Peirce. 2013. “Coupling schemes for modeling hydraulic fracture propagation using the XFEM.” Comput. Methods Appl. Mech. Eng. 253 (Jan): 305–322. https://doi.org/10.1016/j.cma.2012.08.017.
Gordeliy, E., and A. Peirce. 2015. “Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems.” Comput. Methods Appl. Mech. Eng. 283 (Jan): 474–502 https://doi.org/10.1016/j.cma.2014.09.004.
Gu, M., P. M. Kulkarni, M. Rafiee, E. Ivarrud, and K. K. Mohanty. 2014. “Understanding the optimum fracture conductivity for naturally fractured shale and tight reservoirs.” In Proc., SPE/CSUR Unconventional Resources Conf.–Canada. Richardson, TX: Society of Petroleum Engineers.
Gudmundsson, A., and S. L. Brenner. 2001. “How hydrofractures become arrested.” Terra Nova 13 (6): 456–462. https://doi.org/10.1046/j.1365-3121.2001.00380.x.
Haddad, M., and K. Sepehrnoori. 2015. “Simulation of hydraulic fracturing in quasi-brittle shale formations using characterized cohesive layer: Stimulation controlling factors.” J. Unconventional Oil Gas Resour. 9 (Mar): 65–83. https://doi.org/10.1016/j.juogr.2014.10.001.
Haddad, M., and K. Sepehrnoori. 2016. “XFEM-based CZM for the simulation of 3D multiple-cluster hydraulic fracturing in quasi-brittle shale formations.” Rock Mech. Rock Eng. 49 (12): 4731–4748. https://doi.org/10.1007/s00603-016-1057-2.
Jeffrey, R. G., X. Zhang, and M. J. Thiercelin. 2009. “Hydraulic fracture offsetting in naturally fractures reservoirs: Quantifying a long-recognized process.” In Proc., SPE Hydraulic Fracturing Technology Conf. Richardson, TX: Society of Petroleum Engineers.
Kern, L., T. Perkin, and R. Wyant. 1959. “The mechanics of sand movement in fracturing.” J. Pet. Tech. 11 (07): 55–57. https://doi.org/10.2118/1108-G.
Khoei, A. R. 2014. Extended finite element method: Theory and applications. New York: Wiley.
Khoei, A. R., and B. Bahmani. 2018. “Application of an enriched FEM technique in thermo-mechanical contact problems.” Comput. Mech. 62 (5): 1127–1154.
Khoei, A. R., M. Vahab, E. Haghighat, and S. Moallemi. 2014. “A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique.” Int. J. Fract. 188 (1): 79–108. https://doi.org/10.1007/s10704-014-9948-2.
Khoei, A. R., M. Vahab, and M. Hirmand. 2018. “An enriched–FEM technique for numerical simulation of interacting discontinuities in naturally fractured porous media.” Comput. Methods Appl. Mech. Eng. 331 (Apr): 197–231. https://doi.org/10.1016/j.cma.2017.11.016.
Khristianovic, S., and Y. Zheltov. 1955. “Formation of vertical fractures by means of highly viscous fluids.” In Proc., 4th World Petroleum Congress, 579–586. London: World Petroleum Congress.
Lecampion, B. 2009. “An extended finite element method for hydraulic fracture problems.” Int. J. Numer. Method. Biomed. Eng. 25 (2): 121–133. https://doi.org/10.1002/cnm.1111.
Lecampion, B., and E. Detournay. 2007. “An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag.” Comput. Methods Appl. Mech. Eng. 196 (49): 4863–4880. https://doi.org/10.1016/j.cma.2007.06.011.
Lhomme, T., C. De Pater, and P. Helfferich. 2002. “Experimental study of hydraulic fracture initiation in Colton sandstone.” In Proc., SPE/ISRM Rock Mechanics Conf. Society of Petroleum Engineers. Richardson, TX: Society of Petroleum Engineers.
Lomize, G. 1951. Flow in fractured rocks [In Russian.], 127–197. Moscow: Gosenergoizdat.
Louis, C. 1967. “Strömungsvorgänge in klüftigen Medien und ihre Wirkung auf die Standsicherheit von Bauwerken und Böschungen im Fels.” Ph.D. thesis, Institut für Bodenmechanik und Felsmechanik, Technical Univ. Karlsruhe.
Mikhailov, D. N., M. J. Economide, and V. N. Nikolaevskiy. 2011. “Fluid leakoff determines hydraulic fracture dimensions: Approximate solution for non-Newtonian fracturing fluid.” Int. J. Eng. Sci. 49 (9): 809–822. https://doi.org/10.1016/j.ijengsci.2011.03.021.
Milanese, E., P. Rizzato, F. Pesavento, S. Secchi, and B. Schrefler. 2016. “An explanation for the intermittent crack tip advancement and pressure fluctuations in hydraulic fracturing.” Hydraul. Fract. J. 3 (2): 30–43.
Moës, N., J. Dolbow, and T. Belytschko. 1999. “A finite element method for crack growth without remeshing.” Int. J. Numer. Methods Eng. 46 (1): 131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1%3C131::AID-NME726%3E3.0.CO;2-J.
Mohammadi, S. 2012. XFEM fracture analysis of composites. New York: Wiley.
Mohammadnejad, T., and A. R. Khoei. 2013. “An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model.” Finite Elem. Anal. Des. 73 (Oct): 77–95. https://doi.org/10.1016/j.finel.2013.05.005.
Narsilio, G., I. W. Johnston, A. Bidarmaghz, S. Colls, O. Mikhaylovaa, A. Kivi, and R. Aditya. 2014. “Geothermal energy: Introducing an emerging technology.” In Proc., Int. Conf. on Advances in Civil Engineering for Sustainable Development, 141–154. Amsterdam, Netherlands: Elsevier.
Nguyen, T., and A. Selvadurai. 1998. “A model for coupled mechanical and hydraulic behaviour of a rock joint.” Int. J. Numer. Anal. Methods Geomech. 22 (1): 29–48. https://doi.org/10.1002/(SICI)1096-9853(199801)22:1%3C29::AID-NAG907%3E3.0.CO;2-N.
Nicodemo, L., L. Nicolais, and R. F. Landel. 1974. “Shear rate dependent viscosity of suspensions in Newtonian and non-Newtonian liquids.” Chem. Eng. Sci. 29 (3): 729–735. https://doi.org/10.1016/0009-2509(74)80189-2.
Nicolas, A., and M. Jackson. 1982. “High temperature dikes in peridotites: Origin by hydraulic fracturing.” J. Petrol. 23 (4): 568–582. https://doi.org/10.1093/petrology/23.4.568.
Noorishad, J., C. Tsang, and P. Witherspoon. 1984. “Coupled thermal-hydraulic-mechanical phenomena in saturated fractured porous rocks: Numerical approach.” J. Geophys. Res. 89 (B12): 10365–10373. https://doi.org/10.1029/JB089iB12p10365.
Nordgren, R. 1972. “Propagation of a vertical hydraulic fracture.” Soc. Petrol. Eng. J. 12 (4): 306–314. https://doi.org/10.2118/3009-PA.
Novotny, E. 1977. “Proppant transport.” In Proc., SPE Annual Fall Technical Conf. and Exhibition. Richardson, TX: Society of Petroleum Engineers.
Oda, M. 1986. “An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock masses.” Water Resour. Res. 22 (13): 1845–1856. https://doi.org/10.1029/WR022i013p01845.
Oka, F., B. Shahbodagh, and S. Kimoto. 2019. “A computational model for dynamic strain localization in unsaturated elasto-viscoplastic soils.” Int. J. Numer. Anal. Methods Geomech. 43 (1): 138–165. https://doi.org/10.1002/nag.2857.
Oliaei, M. N., A. Pak, and K. Soga. 2014. “A coupled hydro-mechanical analysis for prediction of hydraulic fracture propagation in saturated porous media using EFG mesh-less method.” Comput. Geotech. 55 (Jan): 254–266. https://doi.org/10.1016/j.compgeo.2013.09.001.
Palmer, I. 1993. “Induced stresses due to propped hydraulic fracture in coalbed methane wells.” In Proc., Low Permeability Reservoirs Symp. Richardson, TX: Society of Petroleum Engineers.
Parchei Esfahani, M., and R. Gracie. 2019. “On the undrained and drained hydraulic fracture splits.” Int. J. Numer. Method. Biomed. Eng. 118 (12): 741–763. https://doi.org/10.1002/nme.6036.
Perkins, T., and L. Kern. 1961. “Widths of hydraulic fractures.” J. Pet. Tech. 13 (9): 937–949. https://doi.org/10.2118/89-PA.
Rahman, M. M., and S. S. Rahman. 2013. “Studies of hydraulic fracture-propagation behavior in presence of natural fractures: Fully coupled fractured-reservoir modeling in poroelastic environments.” Int. J. Geomech. 13 (6): 809–826. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000274.
Remij, E., J. Remmers, J. Huyghe, and D. Smeulders. 2015. “The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials.” Comput. Methods Appl. Mech. Eng. 286 (Apr): 293–312. https://doi.org/10.1016/j.cma.2014.12.025.
Réthoré, J., R. de Borst, and M. A. Abellan. 2007. “A two-scale approach for fluid flow in fractured porous media.” Int. J. Numer. Methods Eng. 71 (7): 780–800. https://doi.org/10.1002/nme.1962.
Réthoré, J., R. de Borst, and M. A. Abellan. 2008. “A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks.” Comput. Mech. 42 (2): 227–238. https://doi.org/10.1007/s00466-007-0178-6.
Rubin, A. M. 1995. “Propagation of magma-filled cracks.” Annu. Rev. Earth Planet. Sci. 23 (1): 287–336. https://doi.org/10.1146/annurev.ea.23.050195.001443.
Saad, Y. 2003. Iterative methods for sparse linear systems. Philadelphia: Society for Industrial and Applied Mathematics.
Salimzadeh, S., and N. Khalili. 2015. “Fully coupled XFEM model for flow and deformation in fractured porous media with explicit fracture flow.” Int. J. Geomech. 16 (4): 04015091. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000623.
Samimi, S., and A. Pak. 2016. “A fully coupled element-free Galerkin model for hydro-mechanical analysis of advancement of fluid-driven fractures in porous media.” Int. J. Numer. Anal. Methods Geomech. 40 (16): 2178–2206. https://doi.org/10.1002/nag.2525.
Sarris, E., and P. Papanastasiou. 2012. “Modeling of hydraulic fracturing in a poroelastic cohesive formation.” Int. J. Geomech. 12 (2): 160–167. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000121.
Schrefler, B., S. Secchi, and L. Simoni. 2006. “On adaptive refinement techniques in multi-field problems including cohesive fracture.” Comput. Methods Appl. Mech. Eng. 195 (4): 444–461. https://doi.org/10.1016/j.cma.2004.10.014.
Secchi, S., and B. Schrefler. 2012. “A method for 3-D hydraulic fracturing simulation.” Int. J. Fract. 178 (1–2): 245–258. https://doi.org/10.1007/s10704-012-9742-y.
Shahbodagh, B., M. Mirjalili, S. Kimoto, and F. Oka. 2014. “Dynamic analysis of strain localization in water-saturated clay using a cyclic elasto-viscoplastic model.” Int. J. Numer. Anal. Methods Geomech. 38 (8): 771–793. https://doi.org/10.1002/nag.2221.
Sobhaniaragh, B., M. Haddad, W. J. Mansur, and F. C. Peters. 2019. “Computational modelling of multi-stage hydraulic fractures under stress shadowing and intersecting with pre-existing natural fractures.” Acta Mech. 230 (3): 1037–1059. https://doi.org/10.1007/s00707-018-2335-8.
Soliman, M., M. Wigwe, A. Alzahabi, E. Pirayesh, and N. Stegent. 2014. “Analysis of fracturing pressure data in heterogeneous shale formations.” Hydraul. Fracturing J. 1 (2): 8–12.
Souley, M., P. Lopez, M. Boulon, and A. Thoraval. 2015. “Experimental hydromechanical characterization and numerical modelling of a fractured and porous sandstone.” Rock Mech. Rock Eng. 48 (3): 1143–1161. https://doi.org/10.1007/s00603-014-0626-5.
Spence, D., and P. Sharp. 1985. “Self-similar solutions for elastohydrodynamic cavity flow.” Proc. R. Soc. London: Math. Phys. Eng. Sci. 400 (1819): 289–313. https://doi.org/10.1098/rspa.1985.0081.
Taleghani, A. D. 2009. “Analysis of hydraulic fracture propagation in fractured reservoirs: An improved model for the interaction between induced and natural fractures.” Ph.D. dissertation, Hildebrand Dept. of Petroleum and Geosystems Engineering, Univ. of Texas at Austin.
Taleghani, A. D., and J. E. Olson. 2011. “Numerical modeling of multistranded-hydraulic-fracture propagation: Accounting for the interaction between induced and natural fractures.” SPE J. 16 (3): 575–581.
Tsang, Y. W., and P. Witherspoon. 1981. “Hydromechanical behavior of a deformable rock fracture subject to normal stress.” J. Geophys. Res. 86 (B10): 9287–9298. https://doi.org/10.1029/JB086iB10p09287.
Tzschichholz, F., and H. Herrmann. 1995. “Simulations of pressure fluctuations and acoustic emission in hydraulic fracturing.” Phys. Rev. E. 51 (3): 1961. https://doi.org/10.1103/PhysRevE.51.1961.
Vahab, M., S. Akhondzadeh, A. R. Khoei, and N. Khalili. 2018. “An X-FEM investigation of hydro-fracture evolution in naturally-layered domains.” Eng. Fract. Mech. 191 (Mar): 187–204. https://doi.org/10.1016/j.engfracmech.2018.01.025.
Vahab, M., and N. Khalili. 2017. “Numerical investigation of the flow regimes through hydraulic fractures using the X-FEM technique.” Eng. Fract. Mech. 169 (Jan): 146–162. https://doi.org/10.1016/j.engfracmech.2016.11.017.
Vahab, M., and N. Khalili. 2018a. “Computational algorithm for the anticipation of the fluid-lag zone in hydraulic fracturing treatments.” Int. J. Geomech. 18 (10): 04018139. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001273.
Vahab, M., and N. Khalili. 2018b. “An X-FEM formulation for the optimized graded proppant injection into hydro-fractures within saturated porous media.” Transp. Porous. Med. 121 (2): 289–314. https://doi.org/10.1007/s11242-017-0959-0.
Vahab, M., and N. Khalili. 2018c. “X-FEM modeling of multizone hydraulic fracturing treatments within saturated porous media.” Rock Mech. Rock Eng. 51 (10): 3219–3239. https://doi.org/10.1007/s00603-018-1419-z.
Vahab, M., A. R. Khoei, and N. Khalili. 2019. “An X-FEM technique in modeling hydro-fracture interaction with naturally-cemented faults.” Eng. Fract. Mech. 212 (May): 269–290. https://doi.org/10.1016/j.engfracmech.2019.03.020.
van Der Veen, C. J. 2007. “Fracture propagation as means of rapidly transferring surface meltwater to the base of glaciers.” Geophys. Res. Lett. 34 (1): L01501. https://doi.org/10.1029/2006GL028385.
Wang, J., Y. Zhang, J. Liu, and B. Zhang. 2010. “Numerical simulation of geofluid focusing and penetration due to hydraulic fracture.” J. Geochem. Explor. 106 (1–3): 211–218. https://doi.org/10.1016/j.gexplo.2009.11.009.
Warpinski, N. 1985. “Measurement of width and pressure in a propagating hydraulic fracture.” Soc. Petrol. Eng. J. 25 (1): 46–54. https://doi.org/10.2118/11648-PA.
Wells, G., and L. Sluys. 2001. “A new method for modelling cohesive cracks using finite elements.” Int. J. Numer. Methods Eng. 50 (12): 2667–2682. https://doi.org/10.1002/nme.143.
Witherspoon, P. A., J. Wang, K. Iwai, and J. Gale. 1980. “Validity of cubic law for fluid flow in a deformable rock fracture.” Water Resour. Res. 16 (6): 1016–1024. https://doi.org/10.1029/WR016i006p01016.
Yao, Y. 2012. “Linear elastic and cohesive fracture analysis to model hydraulic fracture in brittle and ductile rocks.” Rock. Mech. Rock. En. 45 (3): 375–387. https://doi.org/10.1007/s00603-011-0211-0.
Yarushina, V., D. Bercovici, and M. Oristaglio. 2012. “Effect of rock rheology on fluid leak-off during hydraulic fracturing.” In Proc., EGU General Assembly Conf. Abstracts, 3812. Munich, Germany: European Geosciences Union.
Zhang, G. Q., and M. Chen. 2010. “Dynamic fracture propagation in hydraulic re-fracturing.” J. Pet. Sci. Eng. 70 (3–4): 266–272. https://doi.org/10.1016/j.petrol.2009.11.019.
Zhang, X., R. G. Jeffrey, and M. Thiercelin. 2007. “Deflection and propagation of fluid-driven fractures at frictional bedding interfaces: A numerical investigation.” J. Struct. Geol. 29 (3): 396–410. https://doi.org/10.1016/j.jsg.2006.09.013.
Zienkiewicz, O. C., A. Chan, M. Pastor, B. Schrefler, and T. Shiomi. 1999. Computational geomechanics. Chichester, UK: Wiley.
Zimmerman, R. W., and G. S. Bodvarsson. 1996. “Hydraulic conductivity of rock fractures.” Transport. Porous. Med. 23 (1): 1–30. https://doi.org/10.1007/BF00145263.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 20 • Issue 3 • March 2020
Copyright
©2019 American Society of Civil Engineers.
History
Received: Dec 9, 2018
Accepted: May 30, 2019
Published online: Dec 18, 2019
Published in print: Mar 1, 2020
Discussion open until: May 18, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.