# 3D model of transversal fracture propagation from a cavity caused by Herschel–Bulkley fluid injection

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## Abstract

The paper presents an extension of authors’ previous model for a 3D hydraulic fracture with Newtonian fluid, which aims to account for the Herschel–Bulkley fluid rheology and to study the associated effects. This fluid rheology model is the most suitable for description of modern complex fracturing fluids, in particular, for description of foamed fluids that have been successfully utilized recently as fracturing fluids in tight and ultra-tight unconventional formations with high clay contents. Another advantage of using Herschel–Bulkley rheological law in the hydraulic fracture model consists in its generality as its particular cases allow describing the behavior of the majority of non-Newtonian fluids employed in hydraulic fracturing. Except the Herschel–Bulkley fluid flow model the considered model of hydraulic fracturing includes the model of the rock stress state. It is based on the elastic equilibrium equations that are solved by the dual boundary element method. Also the hydraulic fracturing model contains the new mixed mode propagation criterion, which states that the fracture should propagate in the direction in which mode \({{\mathrm{\mathrm {II}}}}\) and mode \({{\mathrm{\mathrm {III}}}}\) stress intensity factors both vanish. Since it is not possible to make both modes zero simultaneously the criterion proposes a functional that depends on both modes and is minimized along the fracture front in order to obtain the direction of propagation. Solution for Herschel–Bulkley fluid flow in a channel is presented in detail, and the numerical algorithm is described. The developed model has been verified against some reference solutions and sensitivity of fracture geometry to rheological fluid parameters has been studied to some extent.

## Keywords

3D model of hydraulic fracture propagation Herschel–Bulkley fluid flow in fracture Dual boundary element method 3D mixed mode crack front deflection criterion## Notes

### Acknowledgements

The authors acknowledge the financial support of this research by the Russian Science Foundation (Grant No. 17-71-20139).

## References

- Abass HH, Brumley JL, Venditto JJ et al (1994) Oriented perforations-a rock mechanics view. In: SPE annual technical conference and exhibitionGoogle Scholar
- Adachi JI, Detournay E (2002) Self-similar solution of a plane-strain fracture driven by a power-law fluid. Int J Numer Anal Methods Geomech 26:579–604CrossRefGoogle Scholar
- Adachi JI, Detournay E (2008) Plane strain propagation of a hydraulic fracture in a permeable rock. Eng Fract Mech 75(16):4666–4694CrossRefGoogle Scholar
- Aliabadi MH (2002) The boundary element method, applications in solids and structures, vol 2. Wiley, ChichesterGoogle Scholar
- Aud WW, Wright TB, Cipolla CL, Harkrider JD (1994) The effect of viscosity on near-wellbore tortuosity and premature screenouts. In: SPE annual technical conference and exhibition, New Orleans, Louisiana. Society of Petroleum EngineersGoogle Scholar
- Barati R, Liang J-T (2014) A review of fracturing fluid systems used for hydraulic fracturing of oil and gas wells. J Appl Polym Sci 131(16):1–11CrossRefGoogle Scholar
- Barati R, Hutchins RD, Friedel T, Ayoub JA, Dessinges M-N, England KW (2009) Fracture impact of yield stress and fracture-face damage on production with a three-phase 2D model. SPE Product Oper 24(02):336–345 (SPE-111457-PA)CrossRefGoogle Scholar
- Bunger AP, Detournay E (2007) Early-time solution for a radial hydraulic fracture. J Eng Mech 133(5):534–540CrossRefGoogle Scholar
- Bunger AP, Detournay E, Garagash DI (2005) Toughness-dominated hydraulic fracture with leak-off. Int J Fract 134(2):175–190CrossRefGoogle Scholar
- Chen J-T, Hong H-K (1999) Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl Mech Rev 52(1):17–33CrossRefGoogle Scholar
- Cherny SG, Lapin VN (2016) 3D model of hydraulic fracture with Herschel–Bulkley compressible fluid pumping. Procedia Struct Integr 2:2479–2486CrossRefGoogle Scholar
- Cherny S, Chirkov D, Lapin V, Muranov A, Bannikov D, Miller M, Willberg D, Medvedev O, Alekseenko O (2009) Two-dimensional modeling of the near-wellbore fracture tortuosity effect. Int J Rock Mech Min Sci 46(6):992–1000CrossRefGoogle Scholar
- Cherny S, Lapin V, Esipov D, Kuranakov D, Avdyushenko A, Lyutov A, Karnakov P (2016) Simulating fully 3D non-planar evolution of hydraulic fractures. Int J Fract 201(2):181–211. https://doi.org/10.1007/s10704-016-0122-x CrossRefGoogle Scholar
- Cleary MP, Johnson DE, Kogsbøll H-H, Owens KA, Perry KF, de Pater CJ, Stachel A, Schmidt H, Mauro T (1993) Field implementation of proppant slugs to avoid premature screen-out of hydraulic fractures with adequate proppant concentration. In: Low permeability reservoirs symposium, Denver, Colorado. Society of Petroleum EngineersGoogle Scholar
- Cooke ML, Pollard DD (1996) Fracture propagation paths under mixed mode loading within rectangular blocks of polymethyl methacrylate. J Geophys Res 101(B2):3387–3400CrossRefGoogle Scholar
- Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4(1):35–45CrossRefGoogle Scholar
- Detournay E (2016) Mechanics of hydraulic fractures. Annu Rev Fluid Mech 48(1):311–339CrossRefGoogle Scholar
- Dontsov EV (2016) An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off. R Soc Open Sci 3(12):160737CrossRefGoogle Scholar
- Dontsov EV, Kresse O (2018) A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid. J Fluid Mech 837:210–229CrossRefGoogle Scholar
- Economides MJ, Nolte KG (2000) Reservoir stimulation, 3rd edn. Wiley, ChichesterGoogle Scholar
- Esipov DV, Kuranakov DS, Lapin VN, Cherny SG (2014) Mathematical models of hydraulic fracturing of a reservoir. Comput Technol 19(2):33–61 (in Russian)Google Scholar
- Garagash DI (2006) Transient solution for a plane-strain fracture driven by a shear-thinning, power-law fluid. Int J Numer Anal Methods Geomech 30(14):1439–1475CrossRefGoogle Scholar
- Garagash D, Detournay E (2000) The tip region of a fluid-driven fracture in an elastic medium. J Appl Mech 67(1):183–192CrossRefGoogle Scholar
- Herschel WH, Bulkley R (1926) Konsistenzmessungen von gummi-benzollösungen. Kolloid-Zeitschrift 39(4):291–300CrossRefGoogle Scholar
- Hong H-K, Chen J-T (1988) Derivations of integral equations of elasticity. J Eng Mech 114(6):1028–1044CrossRefGoogle Scholar
- Kauzlarich JJ, Greenwood JA (1972) Elastohydrodynamic lubrication with Herschel–Bulkley model greases. ASLE Trans 15(4):269–277CrossRefGoogle Scholar
- Kuranakov DS, Esipov DV, Lapin VN, Cherny SG (2016) Modification of the boundary element method for computation of three-dimensional fields of strain-stress state of cavities with cracks. Eng Fract Mech 153:302–318CrossRefGoogle Scholar
- Linkov A (2015) Bench-mark solution for a penny-shaped hydraulic fracture driven by a thinning fluid. ArXiv e-prints arXiv:1508.07968
- Mi Y, Aliabadi MH (1992) Dual boundary element method for three-dimensional fracture mechanics analysis. Eng Anal Bound Elem 10(2):161–171CrossRefGoogle Scholar
- Mi Y, Aliabadi MH (1994) Three-dimensional crack growth simulation using BEM. Comput Struct 52(5):871–878CrossRefGoogle Scholar
- Mitsoulis E (2007) Flows of viscoplastic materials: models and computations. In: Rheology Reviews 2007. British Society of RheologyGoogle Scholar
- Nuismer RJ (1975) An energy release rate criterion for mixed mode fracture. Int J Fract 11(2):245–250CrossRefGoogle Scholar
- Ouyang S, Carey GF, Yew CH (1997) An adaptive finite element scheme for hydraulic fracturing with proppant transport. Int J Numer Methods Fluids 24:645–670CrossRefGoogle Scholar
- Pereira JPA (2010) Generalized finite element methods for three-dimensional crack growth simulations. Ph.D. thesis, Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, p 221Google Scholar
- Rungamornrat J, Wheeler MF, Mear MF (2005) Coupling of fracture/non-newtonian flow for simulating nonplanar evolution of hydraulic fractures. In: SPE annual technical conference and exhibition, SPE-96968-MSGoogle Scholar
- Savitski AA, Detournay E (2002) Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Int J Solids Struct 39(26):6311–6337CrossRefGoogle Scholar
- Shokin Yu, Cherny S, Esipov D, Lapin V, Lyutov A, Kuranakov D (2015) Three-dimensional model of fracture propagation from the cavity caused by quasi-static load or viscous fluid pumping. Commun Comput Inf Sci. https://doi.org/10.1007/978-3-319-25058-8-15 Google Scholar
- Sousa JL, Carter BJ, Ingraffea AR (1993) Numerical simulation of 3D hydraulic fracture using newtonian and power-law fluids. Int J Rock Mech Min Sci Geomech Abstr 30(7):1265–1271CrossRefGoogle Scholar