Analytical solutions and parametric uncertainty of non-Newtonian fluid flow in porous media

Transient 1-D motion of power-law non-Newtonian fluids through porous domains may be investigated via analytical techniques. Coupling a modified Darcy’s law accounting for the fluid rheology with the mass balance equation yields a nonlinear partial differential equation, whose solution is derived in terms of a self-similar variable. Two case studies are presented. In the first one, we consider an infinite porous domain of assigned geometry (plane, radial or spherical) and uniform characteristics, initially at constant pressure and saturated by a weakly compressible fluid, and analyze the dynamics of the pressure variation generated within the domain by fluid injection or withdrawal in the origin, for different boundary conditions. The second case is the motion of a shallow viscous gravity current in a uniform porous layer above a horizontal impermeable boundary, driven by the instantaneous or maintained injection of a volume of fluid. To reflect the uncertainty inherent in the value of the problem parameters, we then consider selected properties of fluid and matrix as independent random variables with an associated probability distribution. The relative influence of the uncertain parameters on variables of interest is investigated via Global Sensitivity Analysis, evaluating the associated Sobol’ indices.