Analysis of Hydraulic Fracture Propagation in Toughness Dominant with Considering Fluid Viscosity and Inertia Parameters Interaction: Higher Order Terms

Document Type : Research Article


1 Assistant Professor of Geotechnical Engineering, Faculty of Engineering and Technology, University of Mazandaran, Babolsar, Iran.

2 استادیار دانشکده عمران و محیط زیست


In the process of hydraulic fracture, various physical parameters such as; viscosity, inertia of fluid and toughness of rock do not influence the fracture propagation identically, and it is probable that one or more of the parameters be more pronounced. Therefore, it may persuade one special regime which is named base on the dissipation of energy. In an impermeable rock, the two limiting regimes can be identified with the dominance of one or the other of the two energy dissipation mechanisms corresponding to extending the fracture in the rock and to flow of viscous fluid in the fracture, respectively. In the viscosity-dominated regime, dissipation in extending the fracture in the rock is negligible compared to the dissipation in the viscous fluid flow, and in the toughness-dominated regime, the opposite holds. It is supposed that the flow of incompressible fluid in the fracture is unidirectional and laminar. Besides, the fracture is fully fluid-filled at all times and fracture propagation is described in the framework of linear elastic fracture mechanics (LEFM). The contribution of this research is a detailed study of the evaluation parameters’ effects on the propagation of hydraulic fracture an impermeable brittle rock. Here, the modified perturbation method suggested for evaluating fluid viscosity and inertia parameters interaction (FVII). The proposed method provides a good estimate of the solution in the wide range of the viscosity/inertia parameters because of the coexistence of both small parameters in the governing equations. The results showed that considering the FVII reduce the length of the crack, and the crack length decreases with increasing viscosity parameter, and the decreasing trend will be intensified by increasing the inertia of fluid. On the other hand, the effects of fluid viscosity in the hydraulic fracture injection process are more pronounced than the effects of the inertia parameter on the assumption of a laminar flow. Neglecting the effect of the FVII result in a significant error. These errors continue to increase with the increase, and may reach about 300%. At last, the results are compared with the available references, which confirms the logical process. 


Main Subjects

[1] M.J. Economides, K.G. Nolte, U. Ahmed, Reservoir stimulation, Wiley Chichester, 2000.
[2]  J.L. Gidley, Recent advances in hydraulic fracturing,  (1989).
[3] R.J. Clifton, A.S. Abou-Sayed, A variational approach to the prediction of the three-dimensional geometry of hydraulic fractures, in:  SPE/DOE Low Permeability Gas Reservoirs Symposium, Society of Petroleum Engineers, 1981.
[4] A. Ingraffea, T. Boone, Simulation of hydraulic fracture in poroelastic rock, Numerical Methods in Geomechanics (Innsbruck 1988), Balkema, Rotterdam,  (1988) 95-105.
[5] S.H. Advani, T. Lee, J. Lee, Three-dimensional modeling of hydraulic fractures in layered media: part I-finite element formulations, Journal of energy resources technology, 112(1) (1990) 1-9.
[6] J. Sousa, B. Carter, A. Ingraffea, Numerical simulation of 3D hydraulic fracture using Newtonian and power-law fluids, in:  International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1993, pp. 1265-1271.
[7] K. Shah, B. Carter, A. Ingraffea, Hydraulic fracturing simulation in parallel computing environments, in:  International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1997, pp. 474.
[8] D.I. Garagash, Plane-strain propagation of a fluid-driven fracture during injection and shutin: Asymptotics of large toughness, Engineering Fracture Mechanics, 73(4) (2006) 456-481.
[9] R. Nilson, Gas-driven fracture propagation, J. Appl. Mech.;(United States), 48 (1981).
[10]   D. Spence, D. Turcotte, Magma driven propagation of cracks, Journal of Geophysical Research: Solid Earth (1978-2012), 90(B1) (1985) 575-580.
[11]   E. Detournay, D. Garagash, The near-tip region of a fluid-driven fracture propagating in a permeable elastic solid, Journal of Fluid Mechanics, 494 (2003) .23-1
[12]   D. Spence, P. Sharp, Self-similar solutions for elastohydrodynamic cavity flow, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1819) (1985) 289-313.
[13]   J.R. Lister, Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors, J. Fluid Mech, 210 (1990) 263-280.
[14]   R. Carbonell, J. Desroches, E. Detournay, A comparison between a semi-analytical and a numerical solution of a two-dimensional hydraulic fracture, International journal of solids and structures, 36(31) (1999) 4869-4888.
[15] A. Savitski, E. Detournay, Propagation of a pennyshaped fluid-driven fracture in an impermeable rock: asymptotic solutions, International journal of solids and structures, 39(26) (2002) 6311-6337.
[16] D.I. Garagash, E. Detournay, Plane-strain propagation of a fluid-driven fracture: small toughness solution, Journal of Applied Mechanics, 72 (2005) 916.
[17]  D. Garagash, E. Detournay, Viscosity-dominated regime of a fluid-driven fracture in an elastic medium, in:  IUTAM Symposium on Analytical and Computational Fracture Mechanics of NonHomogeneous Materials, Springer, 2002, pp. 25-29.
[18]   D. Garagash, E. Detournay, An analysis of the influence of the pressurization rate on the borehole breakdown pressure, International journal of solids and structures, 34(24) (1997) 3099-3118.
[19] J.I. Adachi, Fluid-driven fracture in permeable rock, University of Minnesota, 2001.
[20] J. Adachi, E. Detournay, Self similar solution of a plane strain fracture driven by a power law fluid, International Journal for Numerical and Analytical Methods in Geomechanics, 26(6) (2002) 579-604.
[21] J.I. Adachi, E. Detournay, Plane strain propagation of a hydraulic fracture in a permeable rock, Engineering Fracture Mechanics, 75(16) (2008) 4666-4694.
[22] P.A. Charlez, Rock mechanics: petroleum applications, Editions Technip, 1997.
[23] D. Mendelsohn, A review of hydraulic fracture modeling-part I: general concepts, 2D models, motivation for 3D modeling, Journal of energy resources technology, 106(3) (1984) 369-376.
[24] P. Valko, M. Economides, Hydraulic Fracturing Mechanics, in, John Wiley and Sons, New York, USA, .5991
[25] A. A s g a r i, A. G o l s h a n i, H ydraulic Fracture Propagation in Impermeable Elastic Rock With Large Toughness: Considering Fluid Inertia Parameter Sharif Journal of Civil Engineering, 31.2(3.2) (2015) 23-29.
[26]  A. A s g a r i, A. G o l s h a n i, A. L a k i r o u h a n i, Hydraulic Fracture Propagation in Brittle Rock: C onsidering Interaction Term Between Fluid Inertia and Viscosity Parameters, Sharif Journal of Civil Engineering, 32.2(2.1) (2016) 59-66.
[27]   A. A s g a r i, A. G o l s h a n i, Mathematical Modeling of Hydraulic Fracture Propagation in Elastic Medium: Viscosity-Toughness-Dominated, Sharif Journal of Civil Engineering,  (Accepted In 2017).
[28]  S. Khristianovic, Y. Zheltov, Formation of vertical fractures by means of highly viscous fluids, in:  Proc. 4th world petroleum congress, Rome, 1955, pp. 579586.
[29] G.I. Barenblatt, The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, Journal of Applied Mathematics and Mechanics, 23(3) (1959) 622-636.
[30] T. Perkins, L. Kern, Widths of hydraulic fractures, Journal of Petroleum Technology, 13(09) (1961) 937949.
[31] J. Geertsma, F. De Klerk, A rapid method of predicting width and extent of hydraulically induced fractures, Journal of Petroleum Technology, 21(12) (1969) 1571-1581.
[32]   R. Nordgren, Propagation of a vertical hydraulic fracture, Society of Petroleum Engineers Journal, 12(04) (1972) 306-314.
[33]   J. Geertsma, R. Haafkens, A comparison of the theories for predicting width and extent of vertical hydraulically induced fractures, Journal of energy resources technology, 101(1) (1979) 8-19.
[34]   J. Geertsma, Two-dimensional fracture propagation models, in:  Recent Advances in Hydraulic Fracturing, SPE Richardson, TX, 1989, pp. 81-94.
[35]   K.B. Naceur, M. Economides, Production from naturally fissured reservoirs intercepted by a vertical hydraulic fracture, SPE formation evaluation, 4(04) (1989) 550-558.
[36]   M.G. Mack, N.R. Warpinski, Mechanics of hydraulic fracturing, Reservoir stimulation,  (2000) 6-1.
[37]   M. Biot, L. Masse, W. Medlin, A two-dimensional theory of fracture propagation, SPE Production Engineering, 1(01) (1986) 17-30.
[38]   R. Nilson, Similarity solutions for wedge-shaped hydraulic fractures driven into a permeable medium by a constant inlet pressure, International Journal for Numerical and Analytical Methods in Geomechanics, 12(5) (1988) 477-495.
[39]   R.S. Carbonell, Self-similar solution of a fluid-driven fracture in a zero toughness elastic solid, Proc .Roy. Soc. London. Ser, A submitted for publication, 1996.
[40]   J. Desroches, E. Detournay, B. Lenoach, P. Papanastasiou, J. Pearson, M. Thiercelin, A. Cheng,
The crack tip region in hydraulic fracturing, in:  Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 1994, pp. 39-48.
[41]   N. Huang, A. Szewczyk, Y. Li, Self-similar solution in problems of hydraulic fracturing, Journal of Applied Mechanics, 57 (1990) 877.
[42]   D. Garagash, E. Detournay, The tip region of a fluiddriven fracture in an elastic medium, Journal of Applied Mechanics, 67(1) (2000) 183-192.
[43]   G. Batchelor, An Introduction to Fluid Dynamics Cambridge Univ, Press, Bentley House, London,  (1967).
[44]   E.D. Dmitry Garagash, Similarity solution of a semiinfinite fluid-driven fracture in a linear elastic solid, Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics-Physics-Chemistry-Astronomy, .292-582 )8991( )5(623
[45]   D. Garagash, Hydraulic fracture propagation in elastic rock with large toughness, in:  4th North American Rock Mechanics Symposium, American Rock Mechanics Association, 2000.
[46]   D. Garagash, Transient solution for a plane-strain fracture driven by a shear-thinning, power-law fluid, International Journal for Numerical and Analytical Methods in Geomechanics, 30(14) (2006) 1439-1475.
[47]   A.P. Bunger, E. Detournay, R.G. Jeffrey, Crack tip behavior in near-surface fluid-driven fracture experiments, Comptes Rendus Mecanique, 333(4) (2005, c) 299-304.
[48]   D. Garagash, E. Detournay, Erratum:”Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution [Journal of Applied Mechanics, 2005, 72 (6), pp. 916-928], Journal of Applied Mechanics, 74(4) (2007) 832-832.
[49]   R.A. Shapiro, The dynamics and thermodynamics of compressible fluid flow, New York: Ronald Press, 2(1) (1954).
[50]   I.N. Sneddon, M. Lowengrub, P. Mathematician, Crack problems in the classical theory of elasticity, Wiley New York, 1969.
[51]   J.R. Rice, Mathematical analysis in the mechanics of fracture, Fracture: an advanced treatise, 2 (1968) 191-311.
[52]   A.P. Bunger, R.G. Jeffrey, E. Detournay, Toughnessdominated near-surface hydraulic fracture experiments, in:  Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), American Rock Mechanics Association, 2004.
[53]   D.I. Garagash, Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution, International journal of solids and structures, 43(1819) (2006) 5811-5835.
[54]   M.D. Van Dyke, Perturbation methods in fluid dynamics, Stanford: Parabolic Press, 1975.