Hydraulic fracturing is a commonly used technique to enhance oil recovery and involves the use of a non-Newtonian fluid to pressurize a wellbore until a fracture develops in the rock formation. The fluid Bows into the fracture, extending it further into the rock. Sand particles, added to the fluid, prevent the fracture from closing completely at the end of the process. The fluid at the inlet to the fracture is cooler than the rock, so free and forced (or mixed) convection effects occur. This paper examines steady two-dimensional laminar mixed convection Bow of Newtonian and non-Newtonian (power-law) fluids in a vertical fracture of constant width and constant wall temperatures. Approximate solutions of the problem are obtained by making certain assumptions and simplifications, the main assumption being that the fracture width is small compared to its height and length. This assumption leads to approximations similar to these of lubrication theory, and solutions can be found provided that epsilon Re << 1 and Gr/Re << 1, where epsilon (<< 1) is the ratio of fracture width to height, Re is the Reynolds number and Gr is the Grashof number. The problem reduces to finding solutions of a parabolic equation in one space dimension (the other space dimension being treated as a time-like variable) and a second-order nonlinear partial differential equation. For a Newtonian fluid an analytical solution is obtained; for a power-law fluid the equations are solved numerically using finite difference methods. The buoyancy-driven secondary flow velocities, which are normal to the horizontal primary Bow are computed at various distances along the fracture for typical oilfield parameter values. The secondary Bow speeds are found to be of the same order of magnitude as a typical Stokes' settling velocity, which means that at the centre of the fracture particle settling is enhanced. This enhancement could be detrimental to the fracturing process.
