Skip to main content

Simulating fully 3D non-planar evolution of hydraulic fractures


Three-dimensional model of fracture propagation is proposed. The model simultaneously accounts rock deformation in the vicinity of a fracture and a cavity, fluid flow inside the fracture and its propagation in the direction that is selected by a growth criterion. The results of the sensitivity analysis of model solution to the variation of model parameters are presented.

This is a preview of subscription content, access via your institution.

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
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44


  • Abe H, Mura T, Keer LM (1976) Growth rate of a penny-shaped crack in hydraulic fracturing of rocks. J Geophys Res 81(29):5335–5340

    Article  Google Scholar 

  • Aidagulov G, Alekseenko O, Chang F, Bartko K, Cherny S, Esipov D, Kuranakov D, Lapin V (2015) Model of hydraulic fracture initiation from the notched open hole. In: Proceedings 2015 annual technical symposium & exhibition, Al Khobar, Saudi Arabia, April 21–23, SPE-178027-MS, pp 1–12

  • Alekseenko OP, Esipov DV, Kuranakov DS, Lapin VN, Cherny SG (2011) 2D step-by-step model of hydrofracturing Vestnik. Quart J Novosib State Univ Ser: Math Mech Inf 11(3):36–60 (in Russian)

    Google Scholar 

  • Alekseenko OP, Potapenko DI, Cherny SG, Esipov DV, Kuranakov DS, Lapin VN (2013) 3D Modeling of fracture initiation from perforated non-cemented wellbore. SPE J 18(3):589–600

    Article  Google Scholar 

  • Aliabadi MH (2002) The boundary element method: vol 2 (applications in solids and structures). Wiley, New York

    Google Scholar 

  • Barr DT (1991) Leading-edge analysis for correct simulation of interface separation and hydraulic fracturing. Massachusetts Institute of Technology, Department of Mechanical Engineering

  • Blandford GE, Ingraffea AR, Liggett JA (1981) Two-dimensional stress intensity factor computations using the boundary element method. Int J Numer Meth Eng 17(3):387–404

    Article  Google Scholar 

  • Briner A, Chavez JC, Nadezhdin S, Alekseenko O, Gurmen N, Cherny S, Kuranakov D, Lapin V (March 2015) Impact of perforation tunnel orientation and length in horizontal wellbores on fracture initiation pressure in maximum tensile stress criterion model for tight gas fields in the Sultanate of Oman SPE Middle East Oil & Gas Show and Conference, Manama, Bahrain, 8–11, pp 1–12, SPE 172663

  • Briner A, Florez JC, Nadezhdin S, Gurmen N, Cherny S, Kuranakov D, Lapin V (September 2015) Impact of wellbore orientation on fracture initiation pressure in maximum tensile stress criterion model for unconventional gas field in the Sultanate of Oman. In: Proceedings of North Africa technical conference and exhibition, Cairo, Egypt, 14–16, pp 1–13, SPE-175725-MS

  • Bueckner HF (1973) Field singularities and related integral representations. In: Sih GC (ed) Mechanics of fracture, vol 1: methods of analysis and solutions of crack problems. Nordhoff, Leyden, pp 239–314

    Chapter  Google Scholar 

  • Carbonell R, Desroches J, Detournay E (1999) A comparison between a semi-analytical and a numerical solution of a twodimensional hydraulic fracture. Int J Solids Struct 36(31–32):4869–4888

    Article  Google Scholar 

  • Carter BJ, Desroches J, Ingraffea AR, Wawrzynek PA (2000) Simulating fully 3D hydraulic fracturing. In: Zaman M, Booker J, Gioda G (eds) Modeling in Geomechanics. Wiley Publishers, New York, pp 525–557

  • Carter BJ (1992) Size and stress gradient effects on fracture around cavities. Rock Mech Rock Eng 25(3):167–186

    Article  Google Scholar 

  • Chang F, Bartko K, Dyer S, Aidagulov G, Suarez-Rivera R, Lung J (2014) Multiple fracture initiation in openhole without mechanical isolation: first step to fulfill an ambition. SPE-168638-MS, pp 1–18

  • Chen JT, Hong H-K (1999) Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl Mech Rev 52(1):17–33

    Article  Google Scholar 

  • Cherepanov GP (1979) Mechanics of brittle fracture (translated from the Russian by A. L. Peabody; ed. R. de Wit and W.C. Cooley). McGraw-Hill, London

  • Cherny S, Chirkov D, Lapin V, Muranov A, Bannikov D, Miller M, Willberg D (2009) Two-dimensional modeling of the near-wellbore fracture tortuosity effect. Int J Rock Mech Min Sci 36(6):992–1000

    Article  Google Scholar 

  • Cherny S, Esipov D, Kuranakov D, Lapin V, Chirkov D, Astrakova A (2015) Numerical method for solving a 3D problem of fracture initiation from a cavity in an elastic media presented in the international journal of fracture in 2015

  • Cooke ML, Pollard DD (1996) Fracture propagation paths under mixed mode loading within rectangular blocks of polymethyl methacrylate. J Geophys Res 101(B2):3387–3400

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

    Article  Google Scholar 

  • Cruse TA (1972) Numerical evaluation of elastic stress intensity factors by the boundary-integral equation method. Surfase cracks: physical problems and computational solutions, pp 153–170

  • Cruse TA (1973) Application of the boundary-integral equation method to three dimensional stress analysis. Comput Struct 3:509–527

    Article  Google Scholar 

  • Desroches J, Thiercelin M (1993) Modeling propagation and closure of micro-hydraulic fractures. Int J Rock Mech Min Sci 30:1231–1234

    Article  Google Scholar 

  • Dobroskok A, Ghassemi A, Linkov A (2005) Extended structural criterion for numerical simulation of crack propagation and coalescence under compressive loads. Int J Fract 133:223–246

    Article  Google Scholar 

  • Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85:519–525

    Article  Google Scholar 

  • Esipov DV, Cherny SG, Kuranakov DS, Lapin VN (2011) Multiple-zone boundary element method modeling of hydraulic fracture initiation from perforated cased wellbore. In: Proceedings of international conference “Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications”, devoted to the 90th anniversary of professor N. N. Yanenko (Novosibirsk, Russia, 30 May–4 June 2011). “Informregistr”. - Novosibirsk. -

  • Esipov DV, Kuranakov DS, Lapin VN, Cherny SG (2011) Multiple-zone boundary element method and its application to the problem of hydraulic fracture initiation from cased perforated wellbore. Comp Technol 16(6):13–26 (in Russian)

    Google Scholar 

  • Esipov DV, Kuranakov DS, Lapin VN, Cherny SG (2014) Mathematical models of hydraulic fracturing. Comp Technol 19(2):33–61 (in Russian)

    Google Scholar 

  • Germanovich LN, Cherepanov GP (1995) On some general properties of strength criteria. Int J Fract 71:37–56

    Article  Google Scholar 

  • Goldstein RV, Salganik RL (1974) Brittle fracture of solids with arbitrary cracks. Int J Fract 10:507–523

    Article  Google Scholar 

  • Gupta P, Duarte CAM (2014) Simulation of non-planar three dimensional hydraulic fracture propagation. Int J Nume Anal Methods Geomech 38(13):1397–1430

    Article  Google Scholar 

  • Hong H-K, Chen JT (1988) Derivation of integral equations in elasticity. J Eng Mech 114(6):1028–1044

    Article  Google Scholar 

  • Leblond J-B, Frelat J (2000) Crack kinking from an initially closed crack. Int J Solids Struct 37:1595–1614

    Article  Google Scholar 

  • Leblond J-B, Frelat J (2001) Crack kinking from an interface crack with initial contact between the crack lips. Eur J Mech A: Solids 20:937–951

    Article  Google Scholar 

  • Leblond JB, Frelat J (2004) Crack king from an initially closed, ordinary or interface crack, in the presence of friction. Eng Fract Mech 71:289–307

    Article  Google Scholar 

  • Liu YJ, Li YX (2014) Revisit of the equivalence of the displacement discontinuity method and boundary element method for solving crack problems. Eng Anal Bound Elem 47:64–67

    Article  Google Scholar 

  • Mi Y, Aliabadi MH (1992) Dual boundary element method for three-dimensional fracture mechanics analysis. Eng Anal 10(2):161–171

    Google Scholar 

  • Mi Y, Aliabadi MH (1994) Three-dimensional crack growth simulation using BEM. Comput Struct 52(5):871–878

    Article  Google Scholar 

  • Murakami Y (Editor-in-Chief) Stress intensity factors handbook. Pergamon Press, Oxford (1987)

  • Napier JAL, Detournay E (2013) Propagation of non-planar pressurized cracks from a borehole. In: Proceedings of the 5th international conference on structural engineering, mechanics and computation, SEMC 2013, pp 597–602

  • Neuber HK (1937) Verlag Julius Springer, Berlin

  • Novozhilov VV (1969) On a necessary and sufficient criterion for brittle strength. J Appl Math Mech 33(2):201–210

    Article  Google Scholar 

  • Nuismer RJ (1975) An energy release rate criterion for mixed mode fracture. Int J Fract 11:245–250

    Article  Google Scholar 

  • Paris A, Erdogan F (1963) A critical analysis of crack propagation law. J Basic Eng 85:528–534

    Article  Google Scholar 

  • Pereira JPA (2010) Generalized finite element methods for three-dimensional crack growth simulations. PhD Dissertation in Civil Engineering, University of Illinois

  • Portela A, Aliabadi MH, Rooke DP (1991) The dual boundary element method: efficient implementation for cracked problems. Int J Numer Methods Eng 32:445–470

    Article  Google Scholar 

  • Richard HA, Fulland M, Sander M (2005) Theoretical crack path prediction. Fatigue Fract Eng Mater Struct 28:3–12

    Article  Google Scholar 

  • Rizzo FJ (1967) An integral equation approach to boundary value problems of classical elastostatics. Quart J Appl Math 25:83–95

    Google Scholar 

  • Rungamornrat J (2004) A computational procedure for analysis of fractures in three dimensional anisotropic media. Ph.D. Dissertation, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin

  • Rungamornrat J, Wheeler MF, Mear ME (2005) A numerical technique for simulating nonplanar evolution of hydraulic fractures. Paper SPE 96968:1–9

    Google Scholar 

  • Savitski AA, Detournay E (2002) Propagation of a fluid-driven pennyshaped fracture in an impermeable rock: asymptotic solutions. Int J Solids Struct 39(26):6311–6337

    Article  Google Scholar 

  • Schollmann M, Richard HA, Kullmer G, Fulland M (2002) A new criterion for the prediction of crack development in multiaxially loaded structures. Int J Fract 117:129–141

    Article  Google Scholar 

  • Sedov LI (1997) Mechanics of continuous media. World Scientific, Singapore

    Book  Google Scholar 

  • Sneddon IN, Elliott HA (1946) The opening of a griffith crack under internal pressure. Quart Appl Math 4:262

    Google Scholar 

  • Snyder MD, Cruse TA (1975) Boundary-integral equation analysis of cracked anisotropic plates. Int J Fract 11(2):315–328

    Article  Google Scholar 

  • Sousa JL, Carter BJ, Ingraffea AR (1993) Numerical methods of 3D hydraulic fracture using Newtonian and power-law fluids. Int J Rock Mech Mining Sci Geomech Abst 30(7):1265–1271

    Article  Google Scholar 

  • Tada H, Paris P, Irwin G (2000) The stress analysis of cracks handbook, 3rd edn. ASME Press, New York

    Book  Google Scholar 

  • Vandamme L, Curran JH (1989) A three-dimensional hydraulic fracturing simulator. Int J Numer Methods Eng 28:909–927

    Article  Google Scholar 

  • Watson JO (1982) Hermitian cubic boundary elements for plane problems of fracture mechanics. Res Mechanica 4:23–42

    Google Scholar 

  • Watson JO (1986) Hermitian cubic and singular elements for plane strain. In: Banerjee PK, Watson JO (eds) Developments in boundary element methods - 4, Chapter 1. Elsevier Applied Science Publishers, London, pp 1–28

    Google Scholar 

  • Weber W, Kuhn G (2008) An optimized predictor–corrector scheme for fast 3d crack growth simulations. Eng Fract Mech 75:452–460. doi:10.1016/j.engfracmech.2007.01.005 International Conference of Crack Paths

    Article  Google Scholar 

Download references


Authors gratefully acknowledge the financial support of this research by the Russian Scientific Fund under grant number 14-11-00234.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Sergey Cherny.



Crack propagation algorithm

Quasi-static crack growth

Let us consider the initial fracture with the front defined by vertices \({{\mathrm{\mathbf x}}}_i^0, i=1,\ldots ,N_{fr}\). The case of an unloaded fracture in a stretched media and the case of a loaded fracture in a compressed media are combined in the algorithm by a generalized loading pressure p. A step-by-step fracture propagation is indicated with the superscript n. The general scheme of fracture propagation algorithm is displayed in Fig. 43. The iterative process

$$\begin{aligned} p^{m+1} = \mathbb {P}(p^m) \end{aligned}$$

is introduced to achieve the fulfillment of the condition

$$\begin{aligned} \max \limits _i K_I({{\mathrm{\mathbf x}}}_i^{n+1 \ s}, p^m) = K_{Ic}. \end{aligned}$$
Fig. 45
figure 45

Flow chart of dynamic fracture growth algorithm, derived viscous fluid flow

Fig. 46
figure 46

Flow chart of the algorithm for hydrodynamics-elasticity problem solution

The iterations

$$\begin{aligned} L_i^{s+1} = \mathbb {L}(L_i^s), \quad \theta _i^{s+1} = \mathbb {Q}(\theta _i^s) \end{aligned}$$

provide the fulfillment of conditions

$$\begin{aligned} K_I({{\mathrm{\mathbf x}}}_i^{n+1 \ s}, p^m) = K_{Ic}, \quad K_{II}({{\mathrm{\mathbf x}}}_i^{n+1 \ s}, p^m) = 0 \end{aligned}$$

at each vertex of the crack front at propagation step \(n+1\). The iterative schemes (73) and (75) are based on the methods of solving equations (74) and (76) respectively. The criteria (19) and (17) are implemented iteratively with the desired accuracy, at each vertex of the crack front, at every step of propagation algorithm shown in Fig. 43. If the scaling law of crack incrimination is taken as a propagation criterion, and the direction of propagation is defined by the formula (16) itself, then the algorithm becomes essentially simpler (Fig. 44). The crack trajectories calculated using the algorithms in Fig. 43 and Fig. 44 are compared in the section 4.2.2 “Fatigue crack growth under cyclic loading – scaling law for crack front increment”.

Viscous fluid crack growth

Let the fracture be loaded by the pressure of viscous flow. The fluid front (labeled with its vertices \({{\mathrm{\mathbf x}}}_{f \ i}^n\)), the fracture front with vertices \({{\mathrm{\mathbf x}}}_{r \ i}^n\), and the lag between fluid and fracture front \(L_{r \ i}\) are included into the algorithm. In the algorithm there is a fluid volume \(V^n\) calculated from the fracture width

$$\begin{aligned} V^n = \int \limits _{S^+} W^n dS. \end{aligned}$$

The hydrodynamics-elasticity problem in the algorithm in Fig. 45 provides the relation between the fracture width \(W^{n+1 \ s}\) and the pressure \(p^{n+1 \ s}\) which is produced by the fluid flow in the fracture in the crack front position \({{\mathrm{\mathbf x}}}_{r \ i}^{n+1 \ s}\) and the fluid front \({{\mathrm{\mathbf x}}}_{f \ i}^n\). The scheme of the solution algorithm for the hydrodynamics-elasticity problem is shown in Fig. 46. The iterative process \(\Delta t^{k+1} = \mathbb {T}(\Delta t^k)\) is implemented to fulfill the condition

$$\begin{aligned} \max \limits _i \left| {{\mathrm{\mathbf v}}}_i^{m+1 \ k} \right| = v_f, \end{aligned}$$

which provides the equivalence of the fluid velocity and the fracture front velocity \(v_f = L_f^0 / \Delta t\), where \(\Delta t\) is calculated using the fracture volume dynamics.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cherny, S., Lapin, V., Esipov, D. et al. Simulating fully 3D non-planar evolution of hydraulic fractures. Int J Fract 201, 181–211 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • 3D boundary element method
  • Fracture initiation
  • 2D fluid flow
  • Hydraulic fracture propagation
  • Numerical simulation
  • Fully coupled