  We report a local analysis of the flow of foams in two dimensional
heterogeneous model porous media. Pressure measurements are combined with
direct observation in order to determine simultaneously the effective viscosity
and the fraction of trapped gas. Experiments are conducted in the low-quality
regime, in a large range of capillary numbers and relative gas flow rates. The
effective viscosity is very high (up to a few thousands times that of water) at
low capillary numbers and rheology can be described by a decreasing power-law
function of the capillary number of exponent ranging from -1 to -0.75. The flow
is heterogeneous and the fraction of preferential paths increases with relative
gas flow rate. However, increase in fraction of preferential paths remains very
limited for increasing capillary number, thus failing to account for the
strongly shear thinning behavior observed. Additional experiments motivated by
this finding, carried out in a straight channel of varying cross-section, show
a deviation from Bretherton law at low capillary numbers in this geometry with
the bubble train behaving as a shear-thinning yield stress fluid. The pressure
threshold evidenced accounts quantitatively for the effective viscosity data in
the micromodel, indicating that it is the main mechanism underlying the
measured foam rheology.
