Abstract
Hydraulic fracturing is a powerful technology, especially in stimulating fluid production from reservoirs. However, the problem of the intersection between hydraulic fractures and natural fractures is inevitable in engineering practice due to naturally fractured formations. This paper presents a new criterion for a toughness-dominated hydraulic fracture crossing a natural frictional interface through coupling the fluid flow and elastic deformation of the hydraulic fracture prior to intersecting with the natural frictional interface. The critical condition for the hydraulic fracture crossing the natural frictional interface is that the total superimposed stress does not satisfy the failure condition of the Mohr–Coulomb criterion. Simultaneously, the new criterion considers nonorthogonal intersection angles and six independent parameters relating to fluid flow (hydraulic fracture half-length, approaching distance and injection rate), rock mechanic properties (rock fracture toughness and Young’s modulus) and in situ stress. The prediction outcomes show good agreement with laboratory experiments as well as sufficient advantages compared with the analytical criteria of Blanton, extended Renshaw-Pollard and Llanos. Parameter sensitivity analysis is conducted using the control variable method. The parametric analysis results reveal that the influence sphere of different parameters is limited to a certain extent by the variations in the intersection angle except for Young’s modulus and the injection rate, which show slight effects on the intersection behaviors.
Similar content being viewed by others
Abbreviations
- HF:
-
Hydraulic fracture
- NF:
-
Natural fracture with frictional interface
- K I :
-
Stress intensify factor
- K IC :
-
Mode I rock fracture toughness (critical stress intensity factor)
- β :
-
Intersection angle between HF and NF
- P f :
-
Internal fluid pressure within HF
- σ h :
-
Minimum horizontal in situ stress
- σ H :
-
Maximum horizontal in situ stress
- ∆σ :
-
Stress difference between σH and σh
- t :
-
Injection time
- l(t):
-
Half-length of HF
- ξ :
-
Ratio of x and l(t) (ranging from 0 to 1)
- γ(ρ):
-
Dimensionless half-length of HF
- L(t):
-
Length scale, which has the dimension of length
- w(x, t):
-
Crack opening of HF
- Ω(ξ, t):
-
Dimensionless crack opening of HF
- P net(x, t):
-
Net fluid pressure inside HF
- Π(ξ, t):
-
Dimensionless net fluid pressure inside HF
- ε(t):
-
Small dimensionless parameter that guarantees the variation of Ω and Π from zero to infinity
- ρ(t):
-
Dimensionless evolution parameter
- \(E'\) :
-
Plane-strain elastic modulus
- v :
-
Poisson’s ratio
- Q 0 :
-
Injection rate
- \({\sigma _{xx}}\) :
-
Normal stress in the direction of x-axis induced by net pressure
- \({\sigma _{yy}}\) :
-
Normal stress in the direction of y-axis induced by net pressure
- \({\tau _{xy}}\) :
-
Shear stress induced by net pressure
- \({\sigma _{\beta x}}\) :
-
Total normal stress applied parallel to NF interface (along the direction of βx-axis)
- \({\sigma _{\beta y}}\) :
-
Total normal stress applied perpendicularly on NF interface (along the direction of βy-axis)
- \({\tau _{\beta xy}}\) :
-
Total shear stress applied on NF interface
- \({\sigma _{\gamma ,\beta x}}\) :
-
Normal stress component generated by in situ stress field along the direction of βx-axis on NF interface
- \({\sigma _{\gamma ,\beta y}}\) :
-
Normal stress component generated by in situ stress field along the direction of βy-axis on NF interface
- \({\tau _{\gamma ,\beta }}\) :
-
Shear stress component generated by in situ stress field on NF interface
- \({\sigma _{{P_{{\text{net}}}},\beta x}}\) :
-
Normal stress components along the direction of βx-axis generated by induced stress field on NF interface
- \({\sigma _{{P_{{\text{net}}}},\beta y}}\) :
-
Normal stress components along the direction of βy-axis generated by induced stress field on NF interface
- \({\tau _{{P_{{\text{net}}}},\beta }}\) :
-
Shear stress components generated by induced stress field on NF interface
- µ :
-
Friction coefficient of the NF
- c :
-
Cohesion of the NF
- T 0 :
-
Rock tensile strength
- r c :
-
Critical radius of nonlinear region where the stresses on the natural interface are maximized
- ∆l :
-
Hydraulic fracture approaching distance
- ∆:
-
Dimensionless stress difference
- F :
-
Slip function
- \(\overline {F}\) :
-
Dimensionless slip function
References
Adachi JI (2001) Fluid-driven fracture in permeable rock. PhD thesis, Minneapolis University of Minnesota
Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129
Beugelsdijk LJL, Pater CJd, Sato K (2000) Experimental hydraulic fracture propagation in a multi-fractured medium. In: SPE Asia Pacific conference on integrated modelling for asset management. Society of Petroleum Engineers
Blanton TL (1982) An experimental study of interaction between hydraulically induced and pre-existing fractures. In: Society of petroleum engineers unconventional gas technology symposium, Pittsburgh, May 16–18, SPE paper 10847, p 15
Blanton TL (1986) Propagation of hydraulically and dynamically induced fractures in naturally fractured reservoirs. In: Society of petroleum engineers unconventional gas technology symposium, Louisville, May 18–21, SPE paper 15261, p 15
Cai F, Liu ZG (2016) Simulation and experimental research on upward cross-seams hydraulic fracturing in deep and low-permeability coal seam. J China Coal Soc 41(1):113–119
Chuprakov D, Melchaeva O, Prioul R (2014) Injection-sensitive mechanics of hydraulic fracture interaction with discontinuities. Rock Mech Rock Eng 47(5):1625–1640
de Pater CJ, Beugelsdijk LJL (2005) Experiments and numerical simulation of hydraulic fracturing in naturally fractured rock. In: 40th U.S. rock mechanics symposium and 5th U.S.-Canada rock mechanics symposium, American Rock Society Association
Desroches J, Detournay E, Lenoach B, Papanastasiou P, Thiercelin C (1994) The crack tip region in hydraulic fracturing. Proc R Soc 447:39–48
Detournay E (1999) Fluid and solid singularities at the tip of a fluid-driven fracture. In: Durban D, Pearson J (Eds) Proceedings of the IUTAM symposium on non-linear singularities in deformation and flow, Haifa. Kluwer Academic, Dordrecht, pp 27–42
Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4(1):35–45
Detournay E (2016) Mechanics of hydraulic fractures. Annu Rev Fluid Mech 48(1):311–339
Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104
Garagash DI (1998) Near-tip processes of fluid-driven fractures. PhD thesis, Minneapolis University of Minnesota
Garagash DI (2000) Hydraulic fracture propagation in elastic rock with large toughness. In: Proceedings of 4th North American rock mechanics symposium, The Netherlands, pp 221–228. In: Girard J, Liebman M, Breeds C, Doe T (Eds) Pacific rocks 2000-proceedings of the 4th North American rock mechanics symposium. Balkema, Rotterdam, pp 221–228
Garagash DI, Detournay E (2002) Viscosity-dominated regime of a fluid-driven fracture in an elastic medium. In: IUTAM symposium on analytical and computational fracture mechanics of non-homogeneous materials, Cardiff. In: Karihaloo BL (Ed) Solid mechanics and its applications. Kluwer Academic, Dordrecht, pp 25–29. https://doi.org/10.1007/978-94-017-0081-8_4
Geertsma, de Klerk (1969) A rapid method of predicting width and extent of hydraulic induced fractures. J Petrol Technol 21(12):1571–1581
Gu H, Weng X (2010) Criterion for fractures crossing frictional interfaces at non-orthogonal angles. In: 44th US rock mechanics symposium and 5th US-Canada rock mechanics symposium, Salt Lake City, UT, June 27–30, ARMA Paper 10–198, p 6
Gu H, Weng X, Lund J, Mack M, Suarez-Rivera R (2011) Hydraulic fracture crossing natural fracture at non-orthogonal angles: a criterion and its validation. In: Society of petroleum engineers/hydraulic fracturing technology conference and exhibition, The Woodlands, January 24–26, SPE Paper 139984
Haimson BC (1975) Deep in-situ stress measurements by hydrofracturing. Tectonophysics 29(1–4):41–47
Jeffrey RG, Weber CR, Vlahovic W, Enever JR (1994) Hydraulic fracturing experiments in the great Northern Coal Seam. In: SPE Asia Pacific oil & gas conference, Melbourne, Australia, pp 7–10
Jeffrey RG, Bunger AP, Lecampion B et al (2009) Measuring hydraulic fracture growth in naturally fractured rock. In: SPE annual technical conference and exhibition, New Orleans, Louisiana, USA, pp 4–7
Khristianovic SA, Zheltov YP (1955) Formation of vertical fractures by means of highly viscous liquid. In: Proceedings of the fourth world petroleum congress, Rome, pp 579–586
Llanos EM, Jeffrey RG, Hillis R, Zhang X (2017) Hydraulic fracture propagation through an orthogonal discontinuity: a laboratory, analytical and numerical study. Rock Mech Rock Eng 6:1–18
Meng ZP, Tian YD, Li GF (2010) Characteristics of in-situ stress field in Southern Qinshui Basin and its research significance. J China Coal Soc 35(6):975–981
Nordgren R (1972) Propagation of vertical hydraulic fractures. J Pet Tech 253:306–314 (SPE3009)
Ouyang ZH, Qi QX, Zhang Y, Zhao SK (2011) Mechanism and experiment of hydraulic fracturing in rock burst prevention. J China Coal Soc 36(2):321–325
Perkins TK, Kern LR (1961) Widths of hydraulic fractures. Soc Pet Eng 13(9):937–949
Renshaw CE, Pollard DD (1995) An experimentally verified criterion for propagation across unbounded frictional interfaces in brittle, linear elastic-materials. Int J Rock Mech Min Sci Geo-Mech Abstr 32(3):237–249
Rice JR (1968) Mathematical analysis in the mechanics of fracture. Fracture, an advanced treatise. In: Liebowitz H (ed) Fracture. Academic, New York, pp 191–311
Rossmanith HP (1983) Rock fracture mechanics. Springer, Berlin
Sarmadivaleh M, Rasouli V (2014) Modified Renshaw and Pollard criteria for a non-orthogonal cohesive natural interface intersected by an induced fracture. Rock Mech Rock Eng 47(6):2107–2115
Spence T (1985) Magma-driven propagation of crack. J Geophys Res Solid Earth 90(B1):575–580
Sun RJ (1969) Theoretical size of hydraulically induced horizontal fractures and corresponding surface uplift in an idealized medium. J Geophys Res 74(25):5995–6011
Valko P, Economides MJ (1995) Hydraulic fracture mechanics. Wiley, Chichester
Wang YF, He XQ, Wang EY, Li YZ (2014) Research progress and development tendency of the hydraulic technology for increasing the permeability of coal seams. J China Coal Soc 39(10):1945–1955
Warpinski NR, Teufel LW (1987) Influence of geologic discontinuities on hydraulic fracture propagation. SPE J Pet Technol 39(2):209–220. https://doi.org/10.2118/13224-pa
Warpinski NR, Lorenz JC, Branagan PT, Myal FR, Gall BL (1993) Examination of a cored hydraulic fracture in a deep gas well. SPE Prod Facil 8(8):150–164. https://doi.org/10.1016/0148-9062(94)92893-2
Wu YZ, Kang HP (2017) Pressure relief mechanism and experiment of directional hydraulic fracturing in reused coal pillar road way. J China Coal Soc 42(5):1130–1137
Xu YQ, Zhang KN, Wang Y (2012) Numerical investigation for enhancing injectivity of CO2 storage in saline aquifers. Rock Soil Mech 33(12):3825–3832
Yoon JS, Zimmermann G, Zang A (2015) Discrete element modeling of cyclic rate fluid injection at multiple locations in naturally fractured reservoirs. Int J Rock Mech Min Sci 74:15–23
Zhou J, Chen M, Jin Y, Zhang GQ (2008) Analysis of fracture propagation behavior and fracture geometry using tri-axial fracturing system in naturally fractured reservoirs. Int J Rock Mech Min Sci Geo-Mech Abstr 45:1143–1152
Acknowledgements
This work is supported by the National Natural Science Foundation of China (nos. 51374257, 50804060).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhao, Y., He, P., Zhang, Y. et al. A New Criterion for a Toughness-Dominated Hydraulic Fracture Crossing a Natural Frictional Interface. Rock Mech Rock Eng 52, 2617–2629 (2019). https://doi.org/10.1007/s00603-018-1683-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00603-018-1683-y