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IssuesArchive of Issues2010-2pp.188-197

Archive of Issues

Total articles in the database: 1813
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A.Kh. Pergament and D.A. Ulkin, "Similarity and dimensional analysis in the plane problem of the propagation of a vertical hydraulic fracture crack in an elastic medium," J. Appl. Math. Mech. 74 (2), 188-197 (2010)
Year 2010 Volume 74 Issue 2 Pages 188-197
Title Similarity and dimensional analysis in the plane problem of the propagation of a vertical hydraulic fracture crack in an elastic medium
Author(s) A.Kh. Pergament (Moscow, Russia)
D.A. Ulkin (Moscow, Russia, dmulkin@yandex.ru)
Abstract The problem of the growth of a vertical hydraulic fracture crack in an unbounded elastic medium under the pressure produced by a viscous incompressible fluid is studied qualitatively and by numerical methods. The fluid motion is described in the approximation of lubrication theory. Near the crack tip a fluid-free domain may exist. To find the crack length, Irwins fracture criterion is used. The symmetry groups of the equations describing the hydraulic fracture process are studied for all physically meaningful cases of the degeneration of the problem with respect to the control parameters. The condition of symmetry of the system of equations under the group of scaling and time-shift transformations enables the self-similar variables and the form of the time dependence of the quantities involved in the problem to be found. It is established that at non-zero rock pressure the well-known solution of Spence and Sharp is an asymptotic form of the initial-value problem, whereas the solution of Zheltov and Khristianovich is a limiting self-similar solution of the problem. The problem of the formation of a hydraulic fracture crack taking into account initial data is solved using numerical methods, and the problem of arriving at asymptotic mode is investigated. It is shown that the solution has a self-similar asymptotic form for any initial conditions, and the convergence of the exact solutions to the asymptotic forms is non-uniform in space and time.
Received 26 June 2008
Link to Fulltext http://www.sciencedirect.com/science/journal/00218928
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