Generalized solution for 1-D non-Newtonian flow in a porous domain due to an instantaneous mass injection

Non-Newtonian fluid flow in porous media is of considerable Interest in hydrology, chemical and petroleum engineering, and biofluid mechanics. We consider an infinite porous domain of plane ( d =1), cylindrical ( d = 2 ) or semi-spherical ( d = 3 ) geometry, having uniform permeability k and porosity φ , initially at uniform pressure and 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 0 m in its origin. The fluid is described by a rheological power-law model of given consistency index H and flow behavior index n ; the flow law is a modified Darcy’s law depending on φ , H , n . Coupling flow law and mass balance equations yields the nonlinear partial differential equation governing the pressure field; an analytical solution is derived in space r and time t as a function of a self-similar variable η = r/t^β. We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution to the problem for d = 1,2,3 . When a shear-thinning fluid ( n <1) is considered, the analytical solution exhibits traveling wave characteristics, in variance with Newtonian fluids; 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. The front position is a markedly increasing function of n and is inversely dependent on d ; the pressure front advances at a slower rate for larger values of compressibility, higher injected mass and lower porosity. When pressure is considered, it is seen that an increase in d from 1 to 3 brings about an order of magnitude reduction. An increase in compressibility implies a significant decrease in pressure, especially at early times. To reflect the uncertainty inherent in values of the problem parameters, we then consider selected properties of fluid (flow behavior index n ) and porous domain (permeability k , porosity φ , and medium compressibility c_p as independent random variables with a uniform probability distribution. The influence of the uncertain parameters on front position and pressure field is investigated via a Global Sensitivity Analysis, evaluating the associated Sobol’ indices with the Polynomial Chaos Expansion technique. The analysis reveals that compressibility and flow behavior index are the most influential variables affecting the front position; when pressure is considered, compressibility and permeability contribute most to the total response variance. The influence of the uncertainty in the porosity is decidedly lower.