Modeling flow of Carreau fluids in porous media

Carreau fluids occur routinely in porous medium systems for a range of applications, and the dependence of the viscosity for such fluids on the rate of strain tensor poses challenges to modeling at an averaged macroscale. Traditional approaches for macroscale modeling such flows have relied upon experimental observations of flows for generalized Newtonian fluids (GNFs) and a phenomenological approach referred to herein as the shift factor. A recently developed approach based upon averaging conservation and thermodynamic equations from the microscale for Cross model GNFs is extended to the case of Carreau fluids and shown to predict the flow through both isotropic and anisotropic media accurately without the need for GNF-flow experiments. The model is formulated in terms of rheological properties, a standard Newtonian resistance tensor, and a length-scale tensor, which does require estimation. An approach based upon measures of the morphology and topology of the pore space is developed to approximate this length-scale tensor. Thus, this work provides the missing components needed to predict Carreau GNF macroscale flow with only rheological information for the fluid and analysis of the pore morphology and topology independent of any fluid flow experiments. Accuracy of predictions based upon this approach is quantified, and extension to other GNFs is straightforward.

Figure 1

Porous medium systems used: (a) a BCC array of spheres; (b) a BCE array of ellipsoids; (c) random sphere packs with 25 spheres; (d) random ellipsoid packs with 25 ellipsoids; and (e) random sphere packs with 1000 spheres. In all figures the blue box represents the domain.

Figure 2

Comparison of predicted vs observed resistance for: (a) a BCC array of spheres; (b) a random sphere packing with 25 spheres (${\epsilon }^{\overline{\overline{w}}}=0.4,$ variance = 0.1); (c) a BCE system in the $x$ dimension; and (d) a BCE system in the $z$ dimension.

Figure 3

Comparison of predicted vs observed values for $\overline{{\mathsf{L}}^w}$ resulting from nonlinear least squares applied using (a) Function 2 and (b) Function 5.

Figure 4

Comparison of observed resistance and resistance calculated from Eq. (15) using ${\widehat{\mathsf{L}}}^w$ predicted from model function 2.