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Propagation regimes of buoyancy-driven hydraulic fractures with solidification

Published online by Cambridge University Press:  16 May 2016

E. V. Dontsov*
Affiliation:
Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204, USA
*
Email address for correspondence: edontsov@central.uh.edu

Abstract

This study investigates the propagation of a semi-infinite buoyancy-driven hydraulic fracture in situations when the fluid is able to solidify along the crack walls. Such problems occur when hot magma ascends from a chamber due to buoyancy forces and solidifies by interacting with colder rock. In the model, the solidification rate is calculated assuming a one-dimensional heat transfer problem, in which case it becomes mathematically equivalent to Carter’s leak-off model, which is commonly used to describe the fluid leak-off from a hydraulic fracture into a porous rock formation. In order to construct a mathematical model for a buoyancy-driven hydraulic fracture with solidification, the aforementioned thermal problem is combined with (i) linear plane-strain elasticity to ensure equilibrium of the rock surrounding the fracture, (ii) linear elastic fracture mechanics to determine the fracture propagation, (iii) lubrication theory to capture the viscous fluid flow inside the crack and to account for the effect of buoyancy, and (iv) volume balance of the magma. To address the problem, the governing equations are first rewritten in terms of one integral equation with a non-singular kernel, which significantly simplifies the analysis and the procedure for obtaining a numerical solution. The latter solution is shown to obey a multiscale behaviour near the fracture tip that is fully resolved by the numerical scheme. In order to understand the structure of the solution and to quantify the regimes of propagation (and the associated transitions), a thorough analysis of the problem has been performed. Finally, the developments are applied to investigate the non-steady propagation of a buoyancy-driven fracture that is fed by a constant flux.

Type
Papers
Copyright
© 2016 Cambridge University Press 

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References

Adachi, J., Siebrits, E., Peirce, A. & Desroches, J. 2007 Computer simulation of hydraulic fractures. Intl J. Rock Mech. Min. Sci. 44, 739757.CrossRefGoogle Scholar
Adachi, J. I., Detournay, E. & Peirce, A. P. 2010 An analysis of classical pseudo-3D model for hydraulic fracture with equilibrium height growth across stress barriers. Intl J. Rock Mech. Min. Sci. 47, 625639.Google Scholar
Bolchover, P. & Lister, J. R. 1999 The effect of solidification on fluid-driven fracture, with application to bladed dykes. Proc. R. Soc. Lond. A 455, 23892409.CrossRefGoogle Scholar
Broberg, K. B. 1999 Cracks and Fracture. Academic Press.Google Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.Google Scholar
Carter, E. D. 1957 Optimum fluid characteristics for fracture extension. In Drilling and Production Practices (ed. Howard, G. C. & Fast, C. R.), pp. 261270. American Petroleum Institute.Google Scholar
Dahm, T. 2000 On the shape and velocity of fluid-filled fractures in the Earth. Geophys. J. Intl 142, 181192.Google Scholar
Detournay, E. 2004 Propagation regimes of fluid-driven fractures in impermeable rocks. Intl J. Geomech. 4, 3545.Google Scholar
Dontsov, E. & Peirce, A. 2015 A non-singular integral equation formulation to analyse multiscale behaviour in semi-infinite hydraulic fractures. J. Fluid. Mech. 781, R1.CrossRefGoogle Scholar
Economides, M. J. & Nolte, K. G.(Eds) 2000 Reservoir Stimulation, 3rd edn. Wiley.Google Scholar
Garagash, D. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. Trans. ASME J. Appl. Mech. 67, 183192.CrossRefGoogle Scholar
Garagash, D. I., Detournay, E. & Adachi, J. I. 2011 Multiscale tip asymptotics in hydraulic fracture with leak-off. J. Fluid Mech. 669, 260297.Google Scholar
Irwin, G. R. 1957 Analysis of stresses and strains near the end of a crack traversing a plate. Trans. ASME J. Appl. Mech. 29, 361364.CrossRefGoogle Scholar
Khristianovic, S. A. & Zheltov, Y. P. 1955 Formation of vertical fractures by means of highly viscous fluids. In Proceedings of the 4th World Petroleum Congress, vol. 2, pp. 579586. World Petroleum Congress.Google Scholar
Lister, J. R. 1990 Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263280.CrossRefGoogle Scholar
Lister, J. R. 1994a The solidification of buoyancy-driven flow in a flexible-walled channel. Part 1. Constant-volume release. J. Fluid Mech. 272, 2144.Google Scholar
Lister, J. R. 1994b The solidification of buoyancy-driven flow in a flexible-walled channel. Part 2. Continual release. J. Fluid Mech. 272, 4565.CrossRefGoogle Scholar
Lister, J. R. & Kerr, R. C. 1991 Fluid-mechanical models of crack propagation and their application to magma transport in dykes. J. Geophys. Res. 96, 1004910077.Google Scholar
Menand, T. & Tait, S. R. 2002 The propagation of a buoyant liquid-filled fissure from a source under constant pressure: an experimental approach. J. Geophys. Res. 107, ECV 16.Google Scholar
Peirce, A. 2010 A Hermite cubic collocation scheme for plane strain hydraulic fractures. Comput. Meth. Appl. Mech. Engng 199, 19491962.Google Scholar
Peirce, A. P. 2015 Modeling multi-scale processes in hydraulic fracture propagation using the implicit level set algorithm. Comput. Meth. Appl. Mech. Engng 283, 881908.Google Scholar
Revalta, E., Bottinger, M. & Dahm, T. 2005 Buoyancy-driven fracture ascent: experiments in layered gelatine. J. Volcanol. Geotherm. Res. 144, 273285.CrossRefGoogle Scholar
Revalta, E. & Dahm, T. 2006 Acceleration of buoyancy-driven fractures and magmatic dikes beneath the free surface. Geophys. J. Intl 166, 14241439.Google Scholar
Rice, J. R. 1968 Mathematical analysis in the mechanics of fracture. In Fracture: An Advanced Treatise (ed. Liebowitz, H.), pp. 191311. Academic Press.Google Scholar
Roper, S. M. & Lister, J. R. 2005 Buoyancy-driven crack propagation from an over-pressured source. J. Fluid Mech. 536, 7998.CrossRefGoogle Scholar
Roper, S. M. & Lister, J. R. 2007 Buoyancy-driven crack propagation: the limit of large fracture toughness. J. Fluid Mech. 580, 359380.CrossRefGoogle Scholar
Rubin, A. M. 1995 Propagation of magma-filled cracks. Annu. Rev. Earth Planet. Sci. 23, 287336.Google Scholar
Settari, A. & Cleary, M. P. 1986 Development and testing of a pseudo-three-dimensional model of hydraulic fracture geometry (P3DH). In Proceedings of the 6th SPE Symposium on Reservoir Simulation (SPE 10505), pp. 185214. Society of Petroleum Engineers.Google Scholar
Spence, D., Sharp, P. & Turcotte, D. 1987 Buoyancy-driven crack propagation: a mechanism for magma migration. J. Fluid Mech. 174, 135153.Google Scholar
Spence, D. & Turcotte, D. 1985 Magma-driven propagation of cracks. J. Geophys. Res. 90, 575580.CrossRefGoogle Scholar
Taisne, B. & Tait, S. 2011 Effect of solidification on a propagating dike. J. Geophys. Res. 116, B01206.Google Scholar
Tsai, V. C. & Rice, J. R. 2010 A model for turbulent hydraulic fracture and application to crack propagation at glacier beds. J. Geophys. Res. 115, F03007.Google Scholar
Wilson, L. & Head, J. W. 2002 Heat transfer and melting in subglacial basaltic volcanic eruptions: implications for volcanic deposit morphology and meltwater volumes. In Volcano–Ice Interaction on Earth and Mars (ed. Smellie, J. L. & Chapman, M. G.), Geological Society of London.Google Scholar