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 mu(eff) 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 real rheology of the
fluid to calculate mu(eff). 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 <mu > 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 the mean viscosity <mu > is a good estimate for the
effective Darcy viscosity mu(eff) providing physical insight into
upscaling of non-Newtonian flows in porous media.
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