On
the determination of a generalized Darcy equation for yield stress fluid in
porous media
Laurent Talon
Laboratoire Fluides, Automatique et Systemes Thermiques (FAST), Paris, France
Abstract
Flows of
Non-Newtonian fluids have different applications in porous media. Indeed,
slurries, colloidal suspensions, emulsions, foams or heavy oil which present
complex rheologies are commonly injected in porous
media for various applications (Oil recovery, hydraulic fracturing, soil
consolidation, etc.) Among the large number of different non-Newtonian fluids
an important class of behaviour is represented by the yield-stress fluids, viz.
fluids that require a minimum of stress to flow. Yield stress fluids are usually modelled as a
Bingham fluid or by the Herschel-Bulkley equation.
In the present
work, we initially use a Lattice-Boltzmann TRT scheme to determine the flow
structure in a synthetic porous medium. We then determined a generalized Darcy
equation by evaluating the flow for different applied pressure drops dP. As expected, one can determine a critical pressure drop
dPc below which there is no flow. Above this
threshold, we observe three different flowing regimes as function of the
distance to the critical pressure dP-dPc. Regime I corresponds to the situation where
the fluid is flowing in only one channel. Here, the relation between flow rate
and pressure drop is given by the non-Newtonian Poiseuille law. During Regime II, an increase in pressure
triggers the opening of new paths. The relation between flow rate and the
difference in pressure to the critical yield pressure becomes quadratic.
Finally, Regime III corresponds to the situation where all the fluid is
sheared. In which case one has a linear relationship between the flow rate and
the applied pressure.
In this
presentation, we will study the crucial importance of heterogeneities in these
flow regimes. In particular, we will show that even if these regimes are the
consequence of disorder, some statistical laws have exponents that are
independent of disorder. Finally, we will propose different models, based on the
pore network, in order to better understand the origin of theses exponents and
the influence of the disorder.
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