Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection

Non-Newtonian fluid flow through porous media is of considerable interest in several fields, ranging from environmental sciences to chemical and petroleum engineering. In this article, we consider an infinite porous domain of uniform permeability k and porosity φ, 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 given consistency index H and flow behavior index n; 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 μ_ef, in turn a function of φ, 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 self-similar variable η = r/t^β (the exponent β 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 to the problem depending on a geometrical parameter d, valid for plane (d = 1), cylindrical (d = 2), and semi-spherical (d = 3) geometry. When a shear thinning fluid is considered, the analytical solution exhibits traveling wave characteristics, in variance with Newtonian fluids; the front velocity is proportional to t^[(n−2)/2] in plane geometry, t^[(2n−3)/(3−n)] in cylindrical geometry, and t^[(3n−4)/[2(2−n)] in semi-spherical geometry. To reflect the uncertainty inherent in the value of the problem parameters, we consider selected properties of fluid and matrix as independent random variables with an associated probability distribution. The influence of the uncertain parameters on the front position and the pressure field is investigated via a global sensitivity analysis evaluating the associated Sobol’ indices. The analysis reveals that compressibility coefficient and flow behavior index are the most influential variables affecting the front position; when the excess pressure is considered, compressibility and permeability coefficients contribute most to the total response variance. For both output variables the influence of the uncertainty in the porosity is decidedly lower.