On efficient simulation of hydraulic fracturing in terms of particle velocity

Hydraulic fracture has been a subject of many numerical simulations. They usually employed the opening and pressure as unknowns. In this paper, we demonstrate the advantages of using, instead, the particle velocity and the lubrication equation reformulated in its terms. It reduces the problem to finding functions, which are analytical near the fluid front. This overcomes difficulties caused by strong non-linearity of the equations, moving boundary and high stiffness of equations resulting from spatial discretization. The efficiency of the approach is illustrated on the classical problems by Nordgren, 1972 and Spence and Sharp, 1985. Simple analytical solutions of these problems are obtained. It appears that when leak-off is negligible, the particle velocity is almost constant along the entire flow region what elucidates the conditions of the proppant movement in low permeability rocks.