The analytical investigation of hydraulic fracture dynamics according to the incomplete coupling principle

M. M. Ramazanov ; A. V. Karakin ; V. E. Borisov

Matematičeskoe modelirovanie, Tome 29 (2017) no. 11, pp. 3-18.

Résumé

In the paper we present a self-similar solution of the coupling problem about slow movement in the hydraulic fracture and corresponding deformation and fluid percolation in the external medium. These movements are generated with fluid uploading into well. The flow in a crack is described with the hydrodynamics Stokes equations in the approach of the lubricating layer. The outer problem is described with the poroelasticity equations. We consider an option of the heterogeneous pressure into a crack in three- and two-dimensions. In the second case one can have the self-similar solution in the analytical form.

Détail

Keywords: hydraulic fracture problem, self-similar solution, incomplete coupling principle, equilibrium crack.

Bibliographie

[1] Yu.N. Gordeev, “Self-similar solution of the problem of pseudo-three-dimension vertical hydraulic fracture in an impermeable stratum”, Fluid Dynamics, 30:6 (1995) | DOI | Zbl

[2] D.A. Spence, P. Sharp, “Self-similar solutions for elastohydrodynamic cavity flow”, Proceedings of the Royal Society of London. Ser. A, 400 (1985), 289–313 | DOI | MR | Zbl

[3] Yu.N. Gordeev, A.A. Zazovskiy, “Tochnoe reshenie problemy rasprostraneniya treshchiny gidrorazryva bolshoy vysoty i bolshoy dliny v nepronitsaemoy srede”, Izv. AN SSSR. Mekhanika tverdogo tela, 1992, no. 1, 94

[4] J.I. Adachi, E. Detournay, R.S. Carbonell, “Plane strain propagation of a fluid-driven fracture II: Zero-toughness self similar solution”, Proceedings of the Royal Society of London, 2002

[5] J.I. Adachi, E. Detournay, “Self-similar solution of a plane-strain fracture driven by a power-law fluid”, Int. J. Numer. Anal. Meth. Geomech., 26 (2002), 579–604 | DOI | Zbl

[6] E. Detournay, D.I. Garagash, “The near-tip region of a fluid-driven fracture propagating in a permeable elastic solid”, J. Fluid Mech., 494 (2003), 1–32 | DOI | Zbl

[7] A.P. Bunger, E. Detournay, D.I. Garagash, “Toughness-dominated hydraulic fracture with leak-off”, Int. J. Fract., 134:2 (2005), 175–190 | DOI | Zbl

[8] J.I. Adachi, E. Detournay, “Plane strain propagation of a hydraulic fracture in a permeable rock”, Engng. Fract. Mech., 75 (2008), 4666–4694 | DOI

[9] E.D. Carter, “Optimum fluid characteristics for fracture extension”, Drilling and production practices, eds. Howard G. C., Fast C. R., American Petroleum Institute, Tulsa, 1957, 261–270

[10] M.A. Biot, “General solutions of the equations of elasticity and consolidation for a porous material”, J. Appl. Mech. Trans. ASME, 78 (1956), 91–96 | MR

[11] A.V. Karakin, “Printsip nepolnoj svyazannosti v modelyah porouprugih sred”, Matematicheskoe modelirovanie, 18:2 (2006), 24–42

[12] A.V. Karakin, “Hydraulic fracturing in the upeer crust”, Izvestiya, Physics of the Solid Earth, 42:8 (2006), 652–667 | DOI

[13] A.V. Karakin, M.M. Ramazanov, V.E. Borisov, “Problema nepolnoj svyazannosti uravnenij gidrorazryva”, Matematicheskoe modelirovanie, 29:6 (2017), 115–134 | Zbl

[14] J.R. Rice, “The Mechanics of Earthquake Rupture”, Proc. International School of Physics ‘Enrico Fermi’, Course 78 (1979), Physics of the Earth's Interior, eds. A.M. Dziewonski, E. Boschi, Italian Physical Society and North-Holland Publ. Co., 1980, 555–649

[15] M. Kachanov, B. Shafiro, Ig. Tsukrov, Handbook of Elasticity Solutions, Springer Science+Bisiness Media, B.V., 2003 | MR

[16] L.I. Sedov, Mechanics of Continuous Media, In 2 Vol., 4th ed., World Scientific Publishing, 1997 | MR | Zbl