A Model of Slurry Flow in a Fracture

Abstract

Abstract An accurate calculation of slurry flow in a fracture is an important issue for fracture design. A model taking into account particle dynamics on a micro-level has been developed. The model shows that the slurry dynamics, to a significant extent, is governed by particle fluctuations generated in a high shear-rate flow. Particles migrate from high shear-rate zones at the fracture walls towards the fracture centre. Thus, a slurry flow is characterized by non-uniform concentration across a fracture. Low solids concentration near the walls leads to reduction of slurry-wall friction compared to that predicted by a model not taking particle migration into account. Reduction in the friction leads to a significant decrease in the pressure gradient and the net pressure, respectively. Introduction Hydraulic fracturing is a complicated but well-studied technology for well stimulation(1). There are a number of computational codes for calculating technological parameters of a fracturing procedure. Such codes solve equations describing slurry hydrodynamics coupled with equations modelling formation fracturing. Since the percentage of fracturing job failures is high (above 30%), it is important to enhance the accuracy of modelling of different components of hydraulic fracturing to make this procedure a more reliable operation. In the current research, we suggest a model of slurry flow in a fracture taking into account an important effect of proppant migration to the fracture centre. Model Slurry flow moving in a fracture is usually characterized by a high mean shear rate (up to γm = um/w= 200 1/s), where um is the mean (superficial) slurry velocity and w is the fracture width. In our analysis, we will consider a steady-state slurry flow in a channel of a constant width. We also neglect impact of fluid leakoff into the formation on slurry dynamics. Many fracturing fluids used in practice are characterized by the power-law rheology. Engineers usually employ a known solution of the Navier-Stokes equation for power-law slurry(1) obtained by assuming that particles have the same velocity as a fluid and that the concentration of solids is uniformly distributed across a fracture. Lattice-Bolzmann computations have shown that in a high shear rate slurry, flow particles fluctuate(2). For the present study, the kinetic theory of granular flows(3) was employed for flow modelling. The kinetic theory assumes that particles collide with each other moving like ideal gas molecules and the velocity distribution of particle fluctuations is Maxwellian. The key parameter of this approach is the granular temperature determining intensity of particle fluctuations: Equation 1 Available in Full Paper. where < υ2s> is the mean-square fluctuation velocity of a particle. An equation of a granular energy balance for a flat channel is written as(4): Equation 2 Available in Full Paper.