Skip to main content
SpringerLink
Log in

A Cohesive Element-Based Numerical Manifold Method for Hydraulic Fracturing Modelling with Voronoi Grains

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

In this study, the cohesive element-based numerical manifold method with Voronoi grains is extended by incorporating a coupled hydro-mechanical (HM) model to investigate hydraulic fracturing of rock at micro-scale. The proposed hydraulic solving framework, which explicitly calculates the flow rate and fluid pressure of a compressible viscous fluid based on the cubic law and a linear fluid compressibility model, is first validated against analytical solutions for uncoupled transient and steady flow examples. Then the coupled HM procedure is further verified by two coupled examples, which respectively considers the elastic response of a pressurized fracture and hydraulic fracture (HF) propagation under different perforation inclinations and in situ stresses. Finally, the developed method is adopted to investigate the hydraulic fracture propagation in Augig granite possessing multi-fractures at micro-scale, based on which the effect of friction coefficient of natural fractures (NFs) on hydraulic fracture propagation is examined. The results show that the friction coefficient of the NFs has significant effects on the induced hydraulic fracture pattern. With increasing friction coefficient of the NFs, it becomes more difficult for the NFs to fail, which results in simpler HF patterns. This phenomenon is associated with the change in the type of interaction between HFs and NFs, i.e., from HFs being arrested by NFs to HFs crossing the NFs with offsets and then to HFs directly crossing NFs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

Abbreviations

\(E\) :

Young’s modulus of the granite sample

\(\nu\) :

Poisson’s ratio

\(\rho\) :

Bulk density of the rock material

\({M_i}\) :

Mathematical patch numbered i

\({P_i}\) :

Physical patch numbered i

\({E_i}\) :

Manifold element numbered i

\({G_{\text{I}}}\) :

Mode-I fracture energy

\({G_{{\text{II}}}}\) :

Mode-II fracture energy

\({\sigma _{\text{t}}}\) :

Tensile stress of cohesive element

\(\tau\) :

Shear stress of cohesive element

\(c\) :

Shear cohesion of cohesive element

\({\sigma _{\text{n}}}\) :

Compressive stress of cohesive element

\({D_{\text{o}}}\) :

Mode-I damage factor

\({D_{\text{s}}}\) :

Mode-II damage factor

\({f_{\text{s}}}\) :

Shear strength of the rock material

\({f_{\text{t}}}\) :

Tensile strength of the rock material

\({O_p}\) :

Critical opening displacement

\({O_{\text{r}}}\) :

Residual opening displacement

\({S_{\text{p}}}\) :

Critical sliding displacement

\({S_{\text{r}}}\) :

Residual sliding displacement

\(w\) :

Relative normal displacement of the contact pair

\(s\) :

Relative tangential displacement of the contact pair

\(\varphi\) :

Friction angle of the contact interface

\(V\) :

Volume of node

\(a\) :

Equivalent hydraulic aperture

\({q_{ij}}\) :

Flow rate from node i to j

\(\mu\) :

Dynamic viscosity of the fluid

\({\rho _{\text{f}}}\) :

Fluid density

\(g\) :

Acceleration of gravity

\({S_i}\) :

Saturation of the node i

\({V_{\text{f}}}\) :

Volume of fluid inside the node i

\({K_{\text{f}}}\) :

Bulk modulus of fluid

\(k\) :

Intrinsic hydraulic conductivity

\({{\text{N}}_i}\) :

Natural fracture numbered i

\({p_{{\text{fn}}}}\) :

Normal penalty parameters

\({p_{{\text{ft}}}}\) :

Tangential penalty parameters

\({p_{\text{n}}}\) :

Normal contact penalty

\({p_{\text{t}}}\) :

Tangential contact penalty

\({\varphi _{\text{f}}}\) :

Friction angle of fracture

References

  • Bai M, Roegiers J-C, Elsworth D (1995) Poromechanical response of fractured-porous rock masses. J Petrol Sci Eng 13:155–168

    Article  Google Scholar 

  • Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    Article  Google Scholar 

  • Biot M, Willis D (1957) The theory of consolidation. J Appl Elastic Coeff Mech 24:594–601

    Google Scholar 

  • Carslaw H, Jaeger J (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, Oxford

    Google Scholar 

  • Chen M, Jiang H, Zhang GQ, Jin Y (2010) The experimental investigation of fracture propagation behavior and fracture geometry in hydraulic fracturing through oriented perforations. Pet Sci Technol 28:1297–1306

    Article  Google Scholar 

  • Chen W, Konietzky H, Liu C, Tan X (2018) Hydraulic fracturing simulation for heterogeneous granite by discrete element method. Comput Geotech 95:1–15

    Article  Google Scholar 

  • Cheng W, Wang R, Jiang G, Xie J (2017) Modelling hydraulic fracturing in a complex-fracture-network reservoir with the DDM and graph theory. J Nat Gas Sci Eng 47:73–82

    Article  Google Scholar 

  • Choo LQ, Zhao Z, Chen H, Tian Q (2016) Hydraulic fracturing modeling using the discontinuous deformation analysis (DDA) method. Comput Geotech 76:12–22

    Article  Google Scholar 

  • Damjanac B, Cundall P (2016) Application of distinct element methods to simulation of hydraulic fracturing in naturally fractured reservoirs. Comput Geotech 71:283–294

    Article  Google Scholar 

  • Detournay E, Cheng AH-D (1988) Poroelastic response of a borehole in a non-hydrostatic stress field. Int J Rock Mech Min Sci Geomech Abstr 25:171–182

    Article  Google Scholar 

  • Geertsma J, de Klerck F (1969) A rapid method of predicting width and extent of hydraulically induced fractures. J Petrol Technol 21:1571–1581

    Article  Google Scholar 

  • Goodman R (1976) Methods of geological engineering in discontinuous rocks

  • Harr ME (1962) Groundwater and seepage. McGraw-Hill Book Co., Inc., New York

    Google Scholar 

  • Hu Y, Chen G, Cheng W, Yang Z (2014) Simulation of hydraulic fracturing in rock mass using a smeared crack model. Comput Struct 137:72–77

    Article  Google Scholar 

  • Hu M, Rutqvist J, Wang Y (2017a) A numerical manifold method model for analyzing fully coupled hydro-mechanical processes in porous rock masses with discrete fractures. Adv Water Resour 102:111–126

    Article  Google Scholar 

  • Hu MS, Wang Y, Rutqvist J (2017b) Fully coupled hydro-mechanical numerical manifold modeling of porous rock with dominant fractures. Acta Geotech 12:231–252

    Article  Google Scholar 

  • Itasca (2005) UDEC version 4.0 user’s manuals. Itasca Consulting Group Inc, Minnesota

    Google Scholar 

  • Jiao Y-Y, Zhang H-Q, Zhang X-L, Li H-B, Jiang Q-H (2015a) A two-dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing. Int J Numer Anal Methods Geomech 39:457–481

    Article  Google Scholar 

  • Jiao YY, Zhang XL, Zhang HQ, Li HB, Yang SQ, Li JC (2015b) A coupled thermo-mechanical discontinuum model for simulating rock cracking induced by temperature stresses. Comput Geotech 67:142–149

    Article  Google Scholar 

  • Jing L, Ma Y, Fang Z (2001) Modeling of fluid flow and solid deformation for fractured rocks with discontinuous deformation analysis (DDA) method. Int J Rock Mech Min Sci 38:343–355

    Article  Google Scholar 

  • Khristianovic S, Zheltov Y (1955) Formation of vertical fractures by means of highly viscous liquid. Paper presented at the 4th World petroleum congress, Rome, Italy

  • Li Y, Deng JG, Liu W, Feng Y (2017) Modeling hydraulic fracture propagation using cohesive zone model equipped with frictional contact capability. Comput Geotech 91:58–70

    Article  Google Scholar 

  • Li X, Zhang QB, Li JC, Zhao J (2018) A numerical study of rock scratch tests using the particle-based numerical manifold method. Tunn Undergr Sp Technol 78:106–114

    Article  Google Scholar 

  • Lin JS (2003) A mesh-based partition of unity method for discontinuity modeling. Comput Methods Appl Mech Eng 192:1515–1532

    Article  Google Scholar 

  • Lisjak A, Liu Q, Zhao Q, Mahabadi OK, Grasselli G (2013) Numerical simulation of acoustic emission in brittle rocks by two-dimensional finite-discrete element analysis. Geophys J Int 195:423–443

    Article  Google Scholar 

  • Lisjak A, Kaifosh P, He L, Tatone BSA, Mahabadi OK, Grasselli G (2017) A 2D, fully-coupled, hydro-mechanical, FDEM formulation for modelling fracturing processes in discontinuous, porous rock masses. Comput Geotech 81:1–18

    Article  Google Scholar 

  • Ma GW, An XM, Zhang HH, Li LX (2009) Modeling complex crack problems using the numerical manifold method. Int J Fract 156:21–35

    Article  Google Scholar 

  • Ma G, An X, He LEI (2010) The numerical manifold method: a review. Int J Comput Methods 07:1–32

    Article  Google Scholar 

  • Mahabadi OK, Lisjak A, Munjiza A, Grasselli G (2012) Y-Geo: new combined finite-discrete element numerical code for geomechanical applications. Int J Geomech 12:676–688

    Article  Google Scholar 

  • Malan D, Napier J (1995) Computer modeling of granular material microfracturing. Tectonophysics 248:21–37

    Article  Google Scholar 

  • Morgan WE, Aral MM (2015) An implicitly coupled hydro-geomechanical model for hydraulic fracture simulation with the discontinuous deformation analysis. Int J Rock Mech Min Sci 73:82–94

    Article  Google Scholar 

  • Ning YJ, Zhao ZY (2013) A detailed investigation of block dynamic sliding by the discontinuous deformation analysis. Int J Numer Anal Methods Geomech 37:2373–2393

    Article  Google Scholar 

  • Ning YJ, Yang Z, Wei B, Gu B (2017) Advances in two-dimensional discontinuous deformation analysis for rock-mass dynamics. Int J Geomech 17:E6016001

    Article  Google Scholar 

  • Nordgren R (1972) Propagation of vertical hydraulic fracture. Soc Petrol Eng J 12:306–314

    Article  Google Scholar 

  • Parker AP (1981) Mechanics of fracture and fatigue: an introduction. E & F N Spon, London

    Google Scholar 

  • Perkins T, Kern L (1961) Widths of hydraulic fractures. J Petrol Technol 13:937–949

    Article  Google Scholar 

  • Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61:2316–2343

    Article  Google Scholar 

  • Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108:1451–1476

    Article  Google Scholar 

  • Rudnicki JW (1985) Effect of pore fluid diffusion on deformation and failure of rock. In: Bazant ZP (ed) Mechanics of geomaterials. Wiley, New York, pp 315–347

    Google Scholar 

  • Shi GH, Goodman RE (1985) 2 Dimensional discontinuous deformational analysis. Int J Numer Anal Methods Geomech 9:541–556

    Article  Google Scholar 

  • Shimizu H, Murata S, Ishida T (2011) The distinct element analysis for hydraulic fracturing in hard rock considering fluid viscosity and particle size distribution. Int J Rock Mech Min Sci 48:712–727

    Article  Google Scholar 

  • Shimizu H, Ito T, Tamagawa T, Tezuka K (2018) A study of the effect of brittleness on hydraulic fracture complexity using a flow-coupled discrete element method. J Petrol Sci Eng 160:372–383

    Article  Google Scholar 

  • Terzaghi K (1951) Theoretical soil mechanics. Chapman and Hall, Limited, London

    Google Scholar 

  • Wang YT, Zhou XP, Kou MM (2018) Peridynamic investigation on thermal fracturing behavior of ceramic nuclear fuel pellets under power cycles. Ceram Int 44:11512–11542

    Article  Google Scholar 

  • Weng L, Huang LQ, Taheri A, Li XB (2017) Rockburst characteristics and numerical simulation based on a strain energy density index: a case study of a roadway in Linglong gold mine, China. Tunn Undergr Sp Technol 69:223–232

    Article  Google Scholar 

  • Wu Z, Wong LNY (2014) Extension of numerical manifold method for coupled fluid flow and fracturing problems. Int J Numer Anal Methods Geomech 38:1990–2008

    Article  Google Scholar 

  • Wu Z, Fan L, Liu Q, Ma G (2017) Micro-mechanical modeling of the macro-mechanical response and fracture behavior of rock using the numerical manifold method. Eng Geol 225:49–60

    Article  Google Scholar 

  • Wu Z, Xu X, Liu Q, Yang Y (2018) A zero-thickness cohesive element-based numerical manifold method for rock mechanical behavior with micro-Voronoi grains. Eng Anal Bound Elem 96:94–108

    Article  Google Scholar 

  • Xia L, Yvonnet J, Ghabezloo S (2017) Phase field modeling of hydraulic fracturing with interfacial damage in highly heterogeneous fluid-saturated porous media. Eng Fract Mech 186:158–180

    Article  Google Scholar 

  • Xie L, Min K-B, Shen B (2016) Simulation of hydraulic fracturing and its interactions with a pre-existing fracture using displacement discontinuity method. J Nat Gas Sci Eng 36:1284–1294

    Article  Google Scholar 

  • Xing P, Yoshioka K, Adachi J, El-Fayoumi A, Damjanac B, Bunger AP (2018) Lattice simulation of laboratory hydraulic fracture containment in layered reservoirs. Comput Geotech 100:62–75

    Article  Google Scholar 

  • Yan C, Jiao Y-Y (2018) A 2D fully coupled hydro-mechanical finite-discrete element model with real pore seepage for simulating the deformation and fracture of porous medium driven by fluid. Comput Struct 196:311–326

    Article  Google Scholar 

  • Yan C, Zheng H, Sun G, Ge X (2015) Combined finite-discrete element method for simulation of hydraulic fracturing. Rock Mech Rock Eng 49:1389–1410

    Article  Google Scholar 

  • Yang Y, Tang X, Zheng H, Liu Q, Liu Z (2018) Hydraulic fracturing modeling using the enriched numerical manifold method. Appl Math Model 53:462–486

    Article  Google Scholar 

  • Zhang Y, Wong LNY (2018) A review of numerical techniques approaching microstructures of crystalline rocks. Comput Geosci 115:167–187

    Article  Google Scholar 

  • Zhang HH, Li LX, An XM, Ma GW (2010) Numerical analysis of 2-D crack propagation problems using the numerical manifold method. Eng Anal Bound Elem 34:41–50

    Article  Google Scholar 

  • Zhao GF, Zhao XB, Zhu JB (2014) Application of the numerical manifold method for stress wave propagation across rock masses. Int J Numer Anal Methods Geomech 38:92–110

    Article  Google Scholar 

  • Zhao G, Kazerani T, Man K, Gao M, Zhao J (2015) Numerical study of the semi-circular bend dynamic fracture toughness test using discrete element models. Sci China Technol Sci 58:1587–1595

    Article  Google Scholar 

  • Zheng H, Xu DD (2014) New strategies for some issues of numerical manifold method in simulation of crack propagation. Int J Numer Methods Eng 97:986–1010

    Article  Google Scholar 

  • Zhou XP, Cheng H (2017) Multidimensional space method for geometrically nonlinear problems under total lagrangian formulation based on the extended finite-element method. J Eng Mech 143:04017036

    Article  Google Scholar 

  • Zhou J, Zhang L, Pan Z, Han Z (2016) Numerical investigation of fluid-driven near-borehole fracture propagation in laminated reservoir rock using PFC 2D. J Nat Gas Sci Eng 36:719–733

    Article  Google Scholar 

  • Zhou J, Zhang L, Pan Z, Han Z (2017) Numerical studies of interactions between hydraulic and natural fractures by smooth joint model. J Nat Gas Sci Eng 46:592–602

    Article  Google Scholar 

  • Zhou XP, Wang YT, Shou YD, Kou MM (2018) A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads. Eng Fract Mech 188:151–183

    Article  Google Scholar 

  • Zhuang X, Augarde CE, Mathisen KM (2012) Fracture modeling using meshless methods and level sets in 3D: framework and modeling. Int J Numer Methods Eng 92:969–998

    Article  Google Scholar 

Download references

Acknowledgements

The research work is supported by the National Natural Science Foundation of China (Grant nos. 41502283, 41772309) and the Sino-British Fellowship Trust Visitorship of the University of Hong Kong.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Wuhan University, Wuhan, 430072, China

    Zhijun Wu & Hao Sun

  2. Department of Earth Sciences, The University of Hong Kong, Pokfulam, Hong Kong

    Louis Ngai Yuen Wong

Authors
  1. Zhijun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louis Ngai Yuen Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louis Ngai Yuen Wong.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Z., Sun, H. & Wong, L.N.Y. A Cohesive Element-Based Numerical Manifold Method for Hydraulic Fracturing Modelling with Voronoi Grains. Rock Mech Rock Eng 52, 2335–2359 (2019). https://doi.org/10.1007/s00603-018-1717-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-018-1717-5

Keywords

Search

Navigation