摘要
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 < <em>n</em> < 5. The boundary value problem is solved numerically using a backward shooting method from the fracture tip, offset by 0 < <em>δ</em> <span style="white-space:nowrap;">≪</span> 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 < <em>n</em> < 5, to be finite for the Reynolds flow law <em>n</em> = 3 and to be zero for 2 < <em>n</em> < 3. The thin fluid film approximation breaks down at the fracture tip for 3 < <em>n</em> < 5 while it remains valid for increasingly tortuous fractures with 2 < <em>n</em> < 3. The effect of the touching asperities is to decrease the width averaged fluid velocity. An approximate analytical solution for the half-width, which was found to agree well with the numerical solution, is derived by making the approximation that the width averaged fluid velocity increases linearly with distance along the fracture.
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 < <em>n</em> < 5. The boundary value problem is solved numerically using a backward shooting method from the fracture tip, offset by 0 < <em>δ</em> <span style="white-space:nowrap;">≪</span> 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 < <em>n</em> < 5, to be finite for the Reynolds flow law <em>n</em> = 3 and to be zero for 2 < <em>n</em> < 3. The thin fluid film approximation breaks down at the fracture tip for 3 < <em>n</em> < 5 while it remains valid for increasingly tortuous fractures with 2 < <em>n</em> < 3. The effect of the touching asperities is to decrease the width averaged fluid velocity. An approximate analytical solution for the half-width, which was found to agree well with the numerical solution, is derived by making the approximation that the width averaged fluid velocity increases linearly with distance along the fracture.
作者
M. R. R. Kgatle-Maseko
D. P. Mason
M. R. R. Kgatle-Maseko;D. P. Mason(School of Computer Science and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, University of the Witwatersrand, Johannesburg, South Africa)
