Computational Mechanics

, Volume 61, Issue 1–2, pp 137–155

# Fully coupled simulation of multiple hydraulic fractures to propagate simultaneously from a perforated horizontal wellbore

• Qinglei Zeng
• Zhanli Liu
• Tao Wang
• Yue Gao
• Zhuo Zhuang
Original Paper

## Abstract

In hydraulic fracturing process in shale rock, multiple fractures perpendicular to a horizontal wellbore are usually driven to propagate simultaneously by the pumping operation. In this paper, a numerical method is developed for the propagation of multiple hydraulic fractures (HFs) by fully coupling the deformation and fracturing of solid formation, fluid flow in fractures, fluid partitioning through a horizontal wellbore and perforation entry loss effect. The extended finite element method (XFEM) is adopted to model arbitrary growth of the fractures. Newton’s iteration is proposed to solve these fully coupled nonlinear equations, which is more efficient comparing to the widely adopted fixed-point iteration in the literatures and avoids the need to impose fluid pressure boundary condition when solving flow equations. A secant iterative method based on the stress intensity factor (SIF) is proposed to capture different propagation velocities of multiple fractures. The numerical results are compared with theoretical solutions in literatures to verify the accuracy of the method. The simultaneous propagation of multiple HFs is simulated by the newly proposed algorithm. The coupled influences of propagation regime, stress interaction, wellbore pressure loss and perforation entry loss on simultaneous propagation of multiple HFs are investigated.

## Keywords

Multiple hydraulic fractures Fluid partitioning XFEM Fully coupled simulation Perforation entry loss

## Notes

### Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11532008 and 11372157), the Special Research Grant for Doctor Discipline by Ministry of Education, China (Grant No. 20120002110075), the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201326). Tsinghua University Initiative Scientific Research Program is also acknowledged.

## References

1. 1.
Zhang X, Bunger AP, Jeffrey RG (2014) Mechanics of two interacting magma-driven fractures: a numerical study. J Geophys Res Solid Earth 119:8047–8063. doi:
2. 2.
Lucantonio A, Noselli G, Trepat X, DeSimone A, Arroyo M (2015) Hydraulic fracture and toughening of a brittle layer bonded to a hydrogel. Phys Rev Lett 115:188105. doi:
3. 3.
Khristianovic SA, Zheltov YP (1955) Formation of vertical fractures by means of highly viscous liquid. Paper presented at the proceedings of 4th world petroleum congress, Rome, ItalyGoogle Scholar
4. 4.
Geertsma J, De Klerk F (1969) A rapid method of predicting width and extent of hydraulically induced fractures. J Pet Technol 21:1571–1581. doi:
5. 5.
Perkins TK, Kern LR (1961) Widths of hydraulic fractures. J Pet Technol 13:937–949. doi:
6. 6.
Nordgren RP (1972) Propagation of a vertical hydraulic fracture. Soc Pet Eng J 12:306–314. doi:
7. 7.
Spence DA, Sharp P (1985) Self-similar solutions for elastohydrodynamic cavity flow. In: Proceedings of the royal society of london series A, mathematical and physical sciences, vol 400, pp 289–313Google Scholar
8. 8.
Adachi J (2001) Fluid-driven fracture in permeable rock. Dissertation, University of MinnesotaGoogle Scholar
9. 9.
Savitski AA, Detournay E (2002) Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Int J Solids Struct 39:6311–6337. doi:
10. 10.
Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4:35–45. doi:
11. 11.
Garagash DI (2006) Propagation of a plane-strain hydraulic fracture with a fluid lag: early-time solution. Int J Solids Struct 43:5811–5835. doi:
12. 12.
Bažant ZP, Ohtsubo H, Aoh K (1979) Stability and post-critical growth of a system of cooling or shrinkage cracks. Int J Fracture 15:443–456. doi:
13. 13.
Nemat-Nasser S, Keer LM, Parihar KS (1978) Unstable growth of thermally induced interacting cracks in brittle solids. Int J Solids Struct 14:409–430. doi:
14. 14.
Nemat-Nasser S, Sumi Y, Keer LM (1980) Unstable growth of tension cracks in brittle solids: stable and unstable bifurcations, snap-through, and imperfection sensitivity. Int J Solids Struct 16:1017–1035. doi:
15. 15.
Bažant ZP, Tabbara MR (1992) Bifurcation and stability of structures with interacting propagating cracks. Int J Fracture 53:273–289. doi: Google Scholar
16. 16.
Olson JE, Taleghani AD (2009) Modeling simultaneous growth of multiple hydraulic fractures and their interaction with natural fractures. Paper presented at the SPE hydraulic fracturing technology conference society of petroleum engineers, The Woodlands, TexasGoogle Scholar
17. 17.
Taleghani AD (2011) Modeling simultaneous growth of multi-branch hydraulic fractures. Paper presented at the 45th US rock mechanics/geomechanics symposium American rock mechanics association, San Francisco, CaliforniaGoogle Scholar
18. 18.
Mack MG, Elbel JL, Piggott AR (1992) Numerical representation of multilayer hydraulic fracturing. Paper presented at the 33th US symposium on rock mechanics (USRMS) American rock mechanics association, Santa Fe, New MexicoGoogle Scholar
19. 19.
Bunger AP (2013) Analysis of the power input needed to propagate multiple hydraulic fractures. Int J Solids Struct 50:1538–1549. doi:
20. 20.
Lecampion B, Desroches J (2015) Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore. J Mech Phys Solids 82:235–258. doi:
21. 21.
Wu K, Olson JE (2015) Mechanisms of simultaneous hydraulic-fracture propagation from multiple perforation clusters in horizontal wells. SPE J 21:1–9. doi: Google Scholar
22. 22.
Gordeliy E, Peirce A (2013) Coupling schemes for modeling hydraulic fracture propagation using the XFEM. Comput Methods Appl Mech Eng 253:305–322. doi:
23. 23.
Gordeliy E, Peirce A (2013) Implicit level set schemes for modeling hydraulic fractures using the XFEM. Comput Methods Appl Mech Eng 266:125–143. doi:
24. 24.
Khoei AR, Hirmand M, Vahab M, Bazargan M (2015) 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. doi:
25. 25.
Gupta P, Duarte CA (2016) Coupled formulation and algorithms for the simulation of non-planar three-dimensional hydraulic fractures using the generalized finite element method. Int J Numer Anal Methods Geomech 40:1402–1437. doi:
26. 26.
Faivre M, Paul B, Golfier F, Giot R, Massin P, Colombo D (2016) 2D coupled HM-XFEM modeling with cohesive zone model and applications to fluid-driven fracture network. Eng Fract Mech 159:115–143. doi:
27. 27.
Mohammadnejad T, Khoei AR (2013) 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. doi:
28. 28.
Wu K (2014) Numerical modeling of complex hydraulic fracture development in unconventional reservoirs. Dissertation, The University of Texas at AustinGoogle Scholar
29. 29.
Gordeliy E, Detournay E (2011) A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag. Int J Numer Anal Methods Geomech 35:602–629. doi:
30. 30.
Adachi J, Siebrits E, Peirce A, Desroches J (2007) Computer simulation of hydraulic fractures. Int J Rock Mech Min Sci 44:739–757. doi:
31. 31.
Peirce A, Detournay E (2008) An implicit level set method for modeling hydraulically driven fractures. Comput Methods Appl Mech Eng 197:2858–2885. doi:
32. 32.
Xu DD, Liu ZL, Liu XM, Zeng QL, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54:489–502. doi:
33. 33.
Fox RW, McDonald AT, Pritchard PJ (2010) Introduction to fluid mechanics. Wiley, New York
34. 34.
Moody LF (1944) Friction factors for pipe flow. Trans ASME 66:671–684Google Scholar
35. 35.
Colebrook CF (1939) Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J Inst Civ Eng 11:133–156. doi:
36. 36.
Swamee PK, Jain AK (1976) Explicit equations for pipe-flow problems. J Hydraul Div Asce 102:657–664Google Scholar
37. 37.
Haaland SE (1983) Simple and explicit formulas for the friction factor in turbulent pipe flow. J Fluids Eng 105:89–90. doi:
38. 38.
Crump JB, Conway MW (1988) Effects of perforation-entry friction on bottomhole treating analysis. J Pet Technol 40:1041–1048. doi:
39. 39.
Cramer DD (1987) The application of limited-entry techniques in massive hydraulic fracturing treatments. Paper presented at the SPE production operations symposium, Oklahoma City, OklahomaGoogle Scholar
40. 40.
McDaniel BW, Willett RM, Underwood PJ (1999) Limited-entry frac applications on long intervals of highly deviated or horizontal wells. Paper presented at the SPE annual technical conference and exhibition, Houston, TexaGoogle Scholar
41. 41.
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150. doi:
42. 42.
Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304. doi:
43. 43.
Zeng QL, Liu ZL, Xu DD, Wang H, Zhuang Z (2016) Modeling arbitrary crack propagation in coupled shell/solid structures with X-FEM. Int J Numer Methods Eng 106:1018–1040. doi:
44. 44.
Gordeliy E, Peirce A (2015) Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems. Comput Methods Appl Mech Eng 283:474–502. doi:
45. 45.
Zhuang Z, Liu Z, Cheng B, Liao J (2014) Extended finite element method. Elsevier, Tsinghua University Press, OxfordGoogle Scholar
46. 46.
Dolbow J, Moës N, Belytschko T (2001) An extended finite element method for modeling crack growth with frictional contact. Comput Methods Appl Mech Eng 190:6825–6846. doi:
47. 47.
Gosz M, Moran B (2002) An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Eng Fract Mech 69:299–319. doi:
48. 48.
Walters MC, Paulino GH, Dodds RH (2005) Interaction integral procedures for 3-D curved cracks including surface tractions. Eng Fract Mech 72:1635–1663. doi: