Abstract
In this paper, a fully coupled 3D. numerical simulation of hydraulic fracture propagation in saturated deformable porous media is presented in the context of the extrinsically enriched element free Galerkin (EFG) method. By exploiting the partition of unity property of moving least square shape functions, weak and strong discontinuities are simulated using the Ridge and the Heaviside enrichment functions, respectively. The cohesive crack model is used to describe the nonlinear fracture processes developing in the area in front of the crack tip where the energy dissipation takes place. The fracturing fluid flow within the fracture is modeled using Darcy’s law and the fracture permeability is considered to follow the cubic law. The developed fully coupled numerical framework can simulate the fluid leak-off phenomenon and formation of the fluid-lag zone. For verification of the developed computational algorithm, a problem with an analytical solution was simulated and a good agreement was seen between numerical and analytical results. The numerical simulations and the parametric studies results show that the proposed numerical framework can successfully simulate various aspects of the complicated process of the hydraulic fracturing treatment.
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References
Li, Q., Xing, H., Liu, J., Liu, X.: A review on hydraulic fracturing of unconventional reservoir. Petroleum. 1, 8–15 (2015)
Nilsson, B., Tzovolou, D., Jeczalik, M., Kasela, T., Slack, W., Klint, K.E., Haeseler, F., Tsakiroglou, C.D.: Combining steam injection with hydraulic fracturing for the in situ remediation of the unsaturated zone of a fractured soil polluted by jet fuel. J. Environ. Manag. 92, 695–707 (2011). https://doi.org/10.1016/j.jenvman.2010.10.004
Kumari WGP, Ranjith PG, Perera MSA, Li X, Li LH, Chen BK, et al. Hydraulic fracturing under high temperature and pressure conditions with micro CT applications : Geothermal energy from hot dry rocks 2018;230:138–154. https://doi.org/10.1016/j.fuel.2018.05.040
Khristianovic, S., Zheltov, Y.: Formation of vertical fractures by means of highly viscous liquid. Proc Fourth World Pet Congr Rome. 579–586 (1955)
Geertsma, J., De Klerk, F.: Rapid method of predicting width and extent of hydraulically induced fractures. J. Pet. Technol. 21, 1571–1581 (1969). https://doi.org/10.2118/2458-pa
Perkins, T.K., Kern, L.R.: Widths of hydraulic fractures. J. Pet. Technol. 13, 937–949 (1961). https://doi.org/10.2118/89-pa
Nordgren, R.P.: Propagation of a vertical hydraulic fracture. Soc. Pet. Eng. J. 12, 306–314 (1972). https://doi.org/10.2118/3009-pa
Settari, A.: Simulation of hydraulic fracturing processes. Soc. Pet. Eng. J. 20, 487–500 (1980). https://doi.org/10.2118/7693-PA
Adachi, J., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44, 739–757 (2007). https://doi.org/10.1016/j.ijrmms.2006.11.006
Yamamoto K, Shimamoto T, Maezumi S. Development of a true 3D hydraulic fracturing simulator. Soc Pet Eng - SPE Asia Pacific Oil Gas Conf Exhib 1999, APOGCE 1999 1999. https://doi.org/10.2523/54265-ms.
Yamamoto, K., Shimamoto, T., Sukemura, S.: Multiple fracture propagation model for a three-dimensional hydraulic fracturing simulator. Int J Geomech. 4, 46–57 (2004). https://doi.org/10.1061/(ASCE)1532-3641(2004)4:1(46)
Boone, T.J., Ingraffea, A.R.: A numerical procedure for simulation of hydraulically-driven fracture propagation in POROELASTIC media. Int. J. Numer. Anal. Methods Geomech. 14, 27–47 (1990)
Desroches, J., Thiercelin, M.: Modelling the propagation and closure of Micro-hydraulic fractures. Int. J. Rock Mech. Min. Sci. 30, 1231–1234 (1993)
Pak A. Numerical modeling of hydraulic fracturing. PhD Thesis, Univ Alberta, Canada 1997
Simoni, L., Secchi, S.: Cohesive fracture mechanics for a multi-phase porous medium. Eng. Comput. 20, 678–698 (2003)
Secchi, S., Simoni, L., Schrefler, B.A.: Mesh adaptation and transfer schemes for discrete fracture propagation in porous materials. Int. J. Numer. Anal. Methods Geomech. 31, 331–345 (2007). https://doi.org/10.1002/nag
Réthoré, J., De Borst, R., Abellan, M.A.: A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks. Comput. Mech. 42, 227–238 (2007). https://doi.org/10.1007/s00466-007-0178-6
Réthoré, J., De Borst, R., Abellan, M.A.: A two-scale approach for fluid flow in fractured porous media. Int. J. Numer. Methods Eng. 71, 780–800 (2007). https://doi.org/10.1002/nme
Lobão, M.C., Eve, R., Owen, D.R., Souza Neto, E.A.: Modelling of hydro-fracture flow in porous media. Eng. Comput. 27, 129–154 (2010). https://doi.org/10.1108/02644401011008568
Khoei, A.R., Barani, O.R., Mofid, M.: Modeling of dynamic cohesive fracture propagation in porous saturated media. Int. J. Numer. Anal. Methods Geomech. 35, 1160–1184 (2011). https://doi.org/10.1002/nag
Barani OR, Khoei AR, Mofid M. Modeling of cohesive crack growth in partially saturated porous media ; a study on the permeability of cohesive fracture. Int. J. Fract. 2011;167:15–31. https://doi.org/10.1007/s10704-010-9513-6
Mohammadnejad, T., Khoei, A.R.: An extended finite element method for fluid flow in partially saturated porous media with weak discontinuities; the convergence analysis of local enrichment strategies. Comput. Mech. 51, 327–345 (2012). https://doi.org/10.1007/s00466-012-0732-8
Mohammadnejad, T.: Khoei a. R. an extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model. Finite Elem. Anal. Des. 73, 77–95 (2013). https://doi.org/10.1016/j.finel.2013.05.005
Salimzadeh, S., Khalili, N.: A three-phase XFEM model for hydraulic fracturing with cohesive crack propagation. Comput. Geotech. 69, 82–92 (2015). https://doi.org/10.1016/j.compgeo.2015.05.001
Khoei, A.R., Hirmand, M., Vahab, M., Bazargan, M.: An enriched FEM technique for modeling hydraulically driven cohesive fracture propagation in impermeable media with frictional natural faults : numerical and experimental investigations. Int. J. Numer. Methods Eng. 104, 439–468 (2015). https://doi.org/10.1002/nme
Khoei, A.R., Vahab, M., Hirmand, M.: Modeling the interaction between fluid-driven fracture and natural fault using an enriched-FEM technique. Int. J. Fract. 197, 1–24 (2015). https://doi.org/10.1007/s10704-015-0051-0
Vahab, M., Akhondzadeh, S., Khoei, A.R., Khalili, N.: An X-FEM investigation of hydro-fracture evolution in naturally-layered domains. Eng. Fract. Mech. 191, 187–204 (2018). https://doi.org/10.1016/j.engfracmech.2018.01.025
Vahab M, Khalili N. Computational Algorithm for the Anticipation of the Fluid-Lag Zone in Hydraulic Fracturing Treatments 2018;18:1–15. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001273
Vahab, M., Khalili, N.: A super-convergent staggered algorithm for the simulation of hydraulic fracturing treatments. Int. J. Fract. 217, 1–16 (2019). https://doi.org/10.1007/s10704-019-00362-0
Mortazavi MS, Pirmoradi P, Khoei AR. Numerical simulation of cold and hot water injection into naturally fractured porous media using the extended – FEM and an equivalent continuum model 2022:0–39. https://doi.org/10.1002/nag.3314
Shi X, Qin Y, Xu H, Feng Q, Wang S, Xu P. Numerical simulation of hydraulic fracture propagation in conglomerate reservoirs 2021. https://doi.org/10.1016/j.engfracmech.2021.107738
Cong, Z., Li, Y., Tang, J., Martyushev, D.A.: Numerical simulation of hydraulic fracture height layer-through propagation based on three-dimensional lattice method. Eng. Fract. Mech. 264, 108331 (2022). https://doi.org/10.1016/j.engfracmech.2022.108331
Rabczuk, T., Zi, G.: A Meshfree method based on the local partition of Unity for cohesive cracks. Comput. Mech. 39, 743–760 (2006). https://doi.org/10.1007/s00466-006-0067-4
Khoshghalb, A., Khalili, N.: A meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media. Int. J. Numer. Anal. Methods Geomech. 37, 716–743 (2012). https://doi.org/10.1002/nag
Iranmanesh, M.A., Pak, A., Samimi, S.: Non-isothermal simulation of the behavior of unsaturated soils using a novel EFG-based three dimensional model. Comput. Geotech. 99, 93–103 (2018). https://doi.org/10.1016/j.compgeo.2018.02.024
Liu, G.R.: Meshfree methods: moving beyond the finite element method. Boca Raton: CRC Press. (2003). https://doi.org/10.1115/1.1553432
Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and their Programming. Springer, Dordrecht, The Netherlands (2005). https://doi.org/10.1007/1-4020-3468-7
Modaressi, H., Aubert, P.: Element-free Galerkin method for deforming multiphase porous media. Int. J. Numer. Methods Eng. 340, 313–340 (1998)
Oliaei, M.N., Soga, K., Pak, A.: Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems. Int. J. Numer. Anal. Methods Geomech. 33, 915–938 (2009). https://doi.org/10.1002/nag.747
Khoshghalb, A., Khalili, N.: A stable meshfree method for fully coupled flow-deformation analysis of saturated porous media. Comput. Geotech. 37, 789–795 (2010). https://doi.org/10.1016/j.compgeo.2010.06.005
Soares Jr., D.: Iterative dynamic analysis of linear and nonlinear fully saturated porous media considering edge-based smoothed meshfree techniques. Comput. Methods Appl. Mech. Eng. 253, 73–88 (2013). https://doi.org/10.1016/j.cma.2012.10.010
Tootoonchi, A., Khoshghalb, A., Liu, G.R., Khalili, N.: A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media. Comput. Geotech. 75, 159–173 (2016). https://doi.org/10.1016/j.compgeo.2016.01.027
Ghaffaripour, O., Khoshghalb, A., Khalili, N.: An edge-based smoothed point interpolation method for elasto-plastic coupled hydro-mechanical analysis of saturated porous media. Comput. Geotech. 82, 99–109 (2017). https://doi.org/10.1016/j.compgeo.2016.10.002
Samimi, S., Pak, A.: Three-dimensional simulation of fully coupled hydro-mechanical behavior of saturated porous media using element free Galerkin (EFG) method. Comput. Geotech. 46, 75–83 (2012). https://doi.org/10.1016/j.compgeo.2012.06.004
Samimi, S., Pak, A.: A novel three-dimensional element free Galerkin (EFG) code for simulating two-phase fluid flow in porous materials. Eng Anal Bound Elem. 39, 53–63 (2014). https://doi.org/10.1016/j.enganabound.2013.10.011
Samimi, S., Pak, A.: A three-dimensional mesh-free model for analyzing multi-phase flow in deforming porous media. Meccanica. 51, 517–536 (2015). https://doi.org/10.1007/s11012-015-0231-z
Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., Snm Liu, W.K.: Smoothing and accelerated computations in the element free Galerkin method. J. Comput. Appl. Math. 74, 111–126 (1996). https://doi.org/10.1016/0377-0427(96)00020-9
Organ, D., Fleming, M., Terry, T., Belytschko, T.: Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput. Mech. 18, 225–235 (1996). https://doi.org/10.1007/BF00369940
Fleming, M., Chuu, Y.A., Moran, B., Belytschko, T.: Enriched element-free Galerkin methods for crack tip fields. Int. J. Numer. Methods Eng. 40, 1483–1504 (1997). https://doi.org/10.1002/(SICI)1097-0207(19970430)40:8<1483::AID-NME123>3.0.CO;2-6
Ventura, G., Xu, J.X., Belytschko, T.: A vector level set method and new discontinuity approximations for crack growth by EFG. Int. J. Numer. Methods Eng. 54, 923–944 (2002). https://doi.org/10.1002/nme.471
Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61, 2316–2343 (2004). https://doi.org/10.1002/nme.1151
Cordes, L.W.W., Moran, B.: Treatment of material discontinuity in the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139, 75–89 (1996). https://doi.org/10.1016/S0045-7825(96)01080-8
Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008). https://doi.org/10.1016/j.matcom.2008.01.003
Iranmanesh, M.A., Pak, A.: Extrinsically enriched element free Galerkin method for heat and fluid flow in deformable porous media involving weak and strong discontinuities. Comput. Geotech. 103, 179–192 (2018). https://doi.org/10.1016/j.compgeo.2018.07.013
Oliaei, M., Pak, A., Soga, K.: A coupled hydro-mechanical analysis for prediction of hydraulic fracture propagation in saturated porous media using EFG mesh-less method. Comput. Geotech. 55, 254–266 (2014)
Samimi, S., Pak, A.: 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, 2178–2206 (2016). https://doi.org/10.1002/nag
Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941). https://doi.org/10.1063/1.1712886
Khoei, A.R.: Extended finite element method. Wiley. (2014). https://doi.org/10.1016/C2012-0-01326-9
Lewis, R.W., Schrefler, B.A.: The finite element method in the static and dynamic deformation and consolidation of porous media. Chichester: Wiley. (1998). https://doi.org/10.1137/1031039
Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res. 16, 1016–1024 (1980)
Schrefler, B.A., Zhan, X., Simoni, L.: A coupled model for water flow, airflow and heat flow in deformable porous media. Int J Numer Methods Heat Fluid Flow. 5, 531–547 (1995)
Zi, G., Rabczuk, T., Wall, W.: Extended meshfree methods without branch enrichment for cohesive cracks. Comput. Mech. 40, 367–382 (2007). https://doi.org/10.1007/s00466-006-0115-0
Moës, N., Cloirec, M., Cartraud, P., Remacle, J.F.: A computational approach to handle complex microstructure geometries. Comput. Methods Appl. Mech. Eng. 192, 3163–3177 (2003). https://doi.org/10.1016/S0045-7825(03)00346-3
Barenblatt, G.I.: The formation of equilibrium cracks during brittle fracture . general ideas and hypotheses. Axially-symmetric cracks. J. Appl. Math. Mech. 23, 622–636 (1959)
Sarris, E., Papanastasiou, P.: The influence of the cohesive process zone in hydraulic fracturing modelling. Int. J. Fract. 167, 33–45 (2011). https://doi.org/10.1007/s10704-010-9515-4
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Iranmanesh, M.A., Pak, A. Three-dimensional numerical simulation of hydraulically driven cohesive fracture propagation in deformable reservoir rock using enriched EFG method. Comput Geosci 27, 317–335 (2023). https://doi.org/10.1007/s10596-023-10198-2
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DOI: https://doi.org/10.1007/s10596-023-10198-2