Abstract
When non-Newtonian fluids flow through porous media, the topology of the pore space leads to a broad range of flow velocities and shear rates. Consequently, the local viscosity of the fluid also varies in space with a non-linear dependence on the Darcy velocity. Therefore, an effective viscosity is usually used to describe the flow at the Darcy scale. For most non-Newtonian flows the rheology of the fluid can be described by a (non linear) function of the shear rate. Current approaches estimate the effective viscosity by first calculating an effective shear rate mainly by adopting a power-law model for the rheology and including an empirical correction factor. In a second step this averaged shear rate is used together with the rheology of the fluid to calculate. In this work, we derive a semi-analytical expression for the local viscosity profile using a Carreau type fluid, which is a more broadly applicable model than the power-law model. By solving the flow equations in a circular cross section of a capillary we are able to calculate the average viscous resistance directly as a spatial average of the local viscosity. This approach circumvents the use of classical capillary bundle models and allows to upscale the viscosity distribution in a pore with a mean pore size to the Darcy scale. Different from commonly used capillary bundle models, the presented approach does neither require tortuosity nor permeability as input parameters. Consequently, our model only uses the characteristic length scale of the porous media and does not require empirical coefficients. The comparison of the proposed model with flow cell experiments conducted in a packed bed of monodisperse spherical beads shows, that our approach performs well by only using the physical rheology of the fluid, the porosity and the estimated mean pore size, without the need to determine an effective shear rate. The good agreement of our model with flow experiments and existing models suggests that mean viscosity is a good estimate for the effective Darcy viscosity providing physical insight into upscaling of non-Newtonian flows in porous media.
Highlights
Flow through porous media is ubiquitous in many natural and industrial systems
This suggests that the above assumptions are questionable. Most of these models predict a linear relationship between the effective shear rate and the Darcy velocity. In this manuscript we show that for a Carreau fluid [27], the local viscosity can be derived directly from the fluid’s constitutive law and the velocity profile in a mean pore size, using a circular capillary to mimic the flow at pore scale
In combination with the Carreau model (Equation 4), the shear rate can be used to calculate the local viscosity μ(r) in a cross section of the capillary from which we can infer the spatial average of μ as μ(r)dA. Based on this formalism, we propose that the effective viscosity can be appropriately estimated directly from the average viscosity μ without using an effective shear rate γeff
Summary
Flow through porous media is ubiquitous in many natural and industrial systems. Examples include flow through biological tissues, blood vessels and bones [1,2,3] or through soils, sediments and rocks, with long-standing interest in hydrology [4, 5], petroleum [6], and chemical engineering [7,8,9]. While Darcy’s law is a sound description for the bulk behavior of a fluid whose viscosity μ is constant, many relevant fluids in e.g., in food [11,12,13] and petroleum [14, 15] industry, show a much more complex constitutive law. For most of these socalled non-Newtonian fluids, the viscosity can be described by a nonlinear function of the stress-strain its first principal invariant γrate =
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