Non-Newtonian flow through porous media due to an instantaneous mass injection

none2Non-Newtonian fluid flow through porous media is of considerable interest in several fields, ranging from environmental sciences to chemical and petroleum engineering. In this paper we consider an infinite porous domain of uniform permeability k and porosity phi, saturated by a weakly compressible non-Newtonian fluid, and analyze the dynamics of the pressure variation generated within the domain by an instantaneous mass injection in its origin. The pressure is taken initially to be constant in the porous domain. The fluid is described by a rheological power-law model of parameters H (consistency index) and n (flow behavior index); n < 1 describes shear-thinning behavior, n > 1 shear-thickening behavior; for n = 1 , the Newtonian case is recovered. The law of motion for the fluid is a modified Darcy’s law based on the effective viscosity mu_eff, in turn a function of phi, H, n. Coupling the flow law with the mass balance equation yields the nonlinear partial differential equation governing the pressure field; an analytical solution is then derived as a function of a selfsimilar variable eta = rt^(-beta) (the exponent beta being a suitable function of n), combining spatial coordinate r and time t . We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution of the problem in d dimensions, valid for plane (d = 1), cylindrical (d = 2) and spherical (d = 3) geometry. When a shear-thinning fluid is considered, the analytical solution exhibits traveling wave characteristics, in variance with Newtonian fluids; no pressure front exists for shear-thickening fluids. For shear-thinning fluids, the front velocity is proportional to t^((n−2)/2) in plane geometry, t^((2*n-3)/(3-n)) in cylindrical geometry, and t^((3*n-4)/(4-1*n)) in spherical geometry.papers online http://www.lamc.ing.unibo.it/aimeta2011mixedDi Federico V.; Ciriello V.Di Federico V.; Ciriello V