Mathematical Model of a Hyperbolic Hydraulic Fracture with Tortuosity

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 < 5. The boundary value problem is solved numerically using a backward shooting method from the fracture tip, offset by 0 < 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 < 5, to be finite for the Reynolds flow law n = 3 and to be zero for 2 < n < 3. The thin fluid film approximation breaks down at the fracture tip for 3 < n < 5 while it remains valid for increasingly tortuous fractures with 2 < n < 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.

Effect of the number of irradiation holes on rock breaking under constant laser energy

The use of mechanical drilling in accessing energy resources stored in deep and hard rock formations is becoming increasingly challenging.Thus,laser irradiation has emerged as a novel drilling method with considerable in this context.This study examines the variation of rock fracture length,fracture tortuosity,hole size,and rock breaking efficiency for a different number of holes and laser power,based on the constant total energy of laser irradiation.As indicated by the results,increasing the laser power increases the laser intensity,which helps increase the hole diameter and depth.Moreover,for the same laser power,increasing the number of irradiated holes reduces the laser energy absorbed by each hole,which is not conducive to increasing the hole depth.As the number of holes increases,the mass loss of the rock also increases,while both specific energy(SE)and modified specific energy(MSE)decrease.When the number of holes remains the same,the mass of the shale removed by low power is less than that removed by high power,while SE and MSE have an inverse relation with power.Therefore,high laser power and multiple-hole irradiation are more conducive to rock breaking.Besides,the fracture length and fracture tortuosity of the rock irradiated by the low laser power will increase first and then decrease with the increase in the number of holes,and reach the peak value when the irradiation takes place through three holes.When a high-power laser irradiates the rock,the fracture length and tortuosity will increase with the increase in the number of irradiation holes.This is because a rock irradiated by low power dissipates more energy,with the result that the energy absorbed by the sample with four irradiation holes is not enough to break the rock quickly.This study is expected to provide some guidance to break rock for drilling deep reservoirs and hard rock formations using laser irradiation.