An accurate calculation of a slurry flow in a fracture is important for fracture designing. A model taking into account particle dynamics on 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 center. Thus, a slurry flow is characterized by non-uniform concentration across the fracture. Low solids concentration near the walls leads to reduction of a slurry-wall friction, which in turn causes a significant decrease in the pressure gradient. (Figure) (Available in full paper)
Slurry flows in fractures during hydraulic fracturing are characterized by high mean shear rates (Available in full paper). For practical applications engineers usually employ a known solution of the Navier-Stokes equation for the power-law slurry rheology [1] 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]. Power spent on slurry pumping dissipates through viscous friction caused by these fluctuations [2]. Wylie et al [2] approximated a mean particle drag coefficient in a slurry shear flow obtained by the Lattice-Bolzmann computations as a function of both the particle Reynolds number in the fluctuation motion and the solids volume concentration.
For this study, the kinetic theory of granular flows [3] was used to develop the model of slurry flow in a flat channel (fracture). The key parameter of this approach is the granular temperature determining an intensity of particle fluctuations: (equation (1)) (Available in full paper)
where (Available in full paper) is the mean-square fluctuation velocity of a particle. Note that the kinetic theory was primarily developed for gas/solids systems. However, it could be applied for liquid/solids flows if the ratio of the velocity relaxation time for a particle fluctuating in a liquid to the time of a particle free path between particle-particle collisions is much higher than unity. Based on the Lattice-Bolzmann computations Wylie et al [3] showed that the kinetic theory provides high accuracy if the aforementioned ratio is larger than 2. An equation of a granular energy balance for a flat channel is written as [4]: (equation (1)) (Available in full paper)
where q is the granular energy flux that is a function of the particle size, density, concentration and the granular temperature [3]; (Available in full paper)? is the shear stress in a slurry flow, (Available in full paper) is the non-Newtonian slurry viscosity, u is the slurry velocity; (Available in full paper) are the coefficients determining fracturing fluid rheology; (Available in full paper) is the function of solids concentration determining increase in the slurry viscosity with increase in the solids loading [5]; (Available in full paper) is the energy dissipation rate caused by viscous friction of fluctuating particles with a fluid (one can find the correlation in [2]);
