Effective Rheology of Bi-viscous Non-newtonian Fluids in Porous Media

We model the flow of bi-viscous non-Newtonian fluids in porous media by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime, where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of them being the correlation length exponent. We find that both critical exponents depend on the parameters of the model.