  In this work, we study non-Newtonian fluid flow in heterogeneous porous
media. We are interested in fluids presenting a specific change in rheology:
Newtonian below a certain shear rate and power law above. Since porous media
generally exhibit strong spatial heterogeneity at large geological scales, we
study the interaction between such inhomogeneity and the nonlinear rheology of
the fluid. The coupling between permeability heterogeneity and nonlinear
rheology significantly affects the flow. We are particularly in the statistical
properties of the velocity field (mean, variance, correlation, etc). Depending
on the imposed mean pressure gradient, three macroscopic flow regimes are
identified. For a low or high average pressure gradient, the average flow rate
increases linearly or according to a power law, respectively. In the latter
regime, we observe that the velocity field is more heterogeneous for
shear-thinning fluids than for shear-thickening fluids. This is corresponding
to a channeling effect of shear-thinning fluids. The intermediate regime
corresponds to a progressive and inhomogeneous change of the local rheology.
This transient regime is then characterized in terms of pressure gradient
range. The flow field is also analyzed statistically. The spatial distribution
of the regions above the rheology threshold shows interesting statistical
properties. For instance, they exhibit multiscale characteristics (fractal),
similar to other critical systems (percolation, avalanches, etc.).
  More surprisingly, even though some statistical properties are independent of
the parameters, an interesting abrupt rotation of the correlations is found for
a particular set of parameters. This is explained by using some symmetries of
the problem.
