TITLE:
Mathematical Model of a Hyperbolic Hydraulic Fracture with Tortuosity
AUTHORS:
M. R. R. Kgatle-Maseko, D. P. Mason
KEYWORDS:
Hyperbolic Crack Law, Partially Open Fracture with Tortuosity, Modified Reynolds Flow Law, Backward Shooting Method, Approximate Analytical Solution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.9 No.5,
May
31,
2021
ABSTRACT: The aim of the research is to study the propagation of a hydraulic fracture with tortuosity due to contact areas between touching asperities on opposite crack walls. The tortuous fracture is replaced by a model symmetric partially open fracture with a hyperbolic crack law and a modified Reynolds flow law. The normal stress at the crack walls is assumed to be proportional to the half-width of the model fracture. The Lie point symmetry of the nonlinear diffusion equation for the fracture half-width is derived and the general form of the group invariant solution is obtained. It was found that the fluid flux at the fracture entry cannot be prescribed arbitrarily, because it is determined by the group invariant solution and that the exponent n in the modified Reynolds flow power law must lie in the range 2 n δ ≪ 1 to avoid singularities, to the fracture entry. The numerical results showed that the tortuosity and the pressure due to the contact regions both have the effect of increasing the fracture length. The spatial gradient of the half-width was found to be singular at the fracture tip for 3 n n = 3 and to be zero for 2 n n n