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 symmet...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.展开更多
The mechanism of grain fracturing in a zirconia metering nozzle used in the continuous casting process was studied. The phase composition, microstructure, and chemical composition of the residual samples were studied ...The mechanism of grain fracturing in a zirconia metering nozzle used in the continuous casting process was studied. The phase composition, microstructure, and chemical composition of the residual samples were studied using an X-ray fluorescence analyzer, scanning electron microscope, and electron probe. Results revealed that the composition, structure, and mineral phase of the original layer, transition layer, and affected layer of the metering nozzle differed because of stabilizer precipitation and steel slag permeation. The stabilizer MgO formed low-melting phases with steel slag and impure SiO2 on the boundaries（pores） of zirconia grains; consequently, grain fracturing occurred and accelerated damage to the metering nozzle was observed.展开更多
文摘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.
基金funded by the National Natural Science Foundation of China(51372193)the Natural Science Basic Research Plan in the Shaanxi Province of China(2014JM6224)
文摘The mechanism of grain fracturing in a zirconia metering nozzle used in the continuous casting process was studied. The phase composition, microstructure, and chemical composition of the residual samples were studied using an X-ray fluorescence analyzer, scanning electron microscope, and electron probe. Results revealed that the composition, structure, and mineral phase of the original layer, transition layer, and affected layer of the metering nozzle differed because of stabilizer precipitation and steel slag permeation. The stabilizer MgO formed low-melting phases with steel slag and impure SiO2 on the boundaries（pores） of zirconia grains; consequently, grain fracturing occurred and accelerated damage to the metering nozzle was observed.
