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Propagation of a pre-existing turbulent fluid fracture

Institution:	1. College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China;2. State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China;1. Halliburton Energy Services, 2601 E. Belt Line Road, Carrollton, TX 75006, USA;2. University of Cincinnati, P.O. Box 210070, Cincinnati, OH 45221-0070, USA;1. IANIGLA, CCT Mendoza, CONICET, Parque San Martín s/n, 5500, Mendoza, Argentina;2. TEHEMA S.A., Subercaseaux 4100, Pirque, Chile;3. Universidad Nacional de San Luis, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina;4. Universidad de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas, Instituto de Estudios Andinos (IDEAN), Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina;5. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Argentina;1. Scientist-SE, ISRO Satellite Centre, Bangalore, 560017, India;2. Scientist-SG, ISRO Satellite Centre, Bangalore, 560017, India;3. ME Dept, Indian Institute of Science, Bangalore, 560012, India;1. School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, SiChuan 610500, China;2. HSE Quality Surveillance＆ Inspection Research Institute, CNPC Chuanqing Drilling Engineering Company Limited, Guanghan 613800, China;3. State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China

Abstract:	The propagation of rough and smooth wall pre-existing turbulent fluid fractures is investigated. The laminar fluid fracture is included as a special case for comparison. Lubrication theory is assumed to apply in the fracture and turbulence is introduced through the wall shear stress. The Perkins–Kern–Nordgren approximation is made in which the fluid pressure is proportional to the half-width of the fracture. The fracture half-width satisfies a non-linear diffusion equation. By using a linear combination of the Lie point symmetries of the non-linear diffusion equation a group invariant solution for the fracture length, volume and half-width is derived. The evolution of the length, half-width and mean flow velocity is analysed for a range of working conditions at the fracture entry. It is found that the mean flow velocity increases approximately linearly along the fracture.

Keywords:	Turbulent fluid fracture Wall shear stress Lubrication theory Perkins–Kern–Nordgren approximation Lie point symmetries Group invariant solution
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