Geodesic
A solution to the hydraulic fracture problem in the traveling wave form
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2009), pp. 46-49

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Solutions to the system of equations describing the propagation of hydraulic fracture cracks in a porous medium are obtained in the travelling wave form. The only sought solution is the separatrix of integral curves on the “penetration depth–crack width” plane. Some necessary dependencies that should be given at the crack inlet are found for the fluid flow rate and the fluid pressure. The crack width and the penetration depth are related by power laws in the limiting cases when the crack propagation processes or the penetration processes are dominant. 
@article{VMUMM_2009_4_a7,
     author = {V. R. Tagirova},
     title = {A solution to the hydraulic fracture problem in the traveling wave form},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {46--49},
     publisher = {mathdoc},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2009_4_a7/}
}
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V. R. Tagirova. A solution to the hydraulic fracture problem in the traveling wave form. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2009), pp. 46-49. http://geodesic.mathdoc.fr/item/VMUMM_2009_4_a7/