  Immiscible two-phase flow in porous media with mixed wet conditions was
examined using a capillary fiber bundle model, which is analytically solvable,
and a dynamic pore network model. The mixed wettability was implemented in the
models by allowing each tube or link to have a different wetting angle chosen
randomly from a given distribution. Both models showed that mixed wettability
can have significant influence on the rheology in terms of the dependence of
the global volumetric flow rate on the global pressure drop. In the capillary
fiber bundle model, for small pressure drops when only a small fraction of the
tubes were open, it was found that the volumetric flow rate depended on the
excess pressure drop as a power law with an exponent equal to 3/2 or 2
depending on the minimum pressure drop necessary for flow. When all the tubes
were open due to a high pressure drop, the volumetric flow rate depended
linearly on the pressure drop, independent of the wettability. In the
transition region in between where most of the tubes opened, the volumetric
flow depended more sensitively on the wetting angle distribution function and
was in general not a simple power law. The dynamic pore network model results
also showed a linear dependence of the flow rate on the pressure drop when the
pressure drop is large. However, out of this limit the dynamic pore network
model demonstrated a more complicated behaviour that depended on the mixed
wettability condition and the saturation. In particular, the exponent relating
volumetric flow rate to the excess pressure drop could take on values anywhere
between 1.0 and 1.8. The values of the exponent were highest for saturations
approaching 0.5, also, the exponent generally increased when the difference in
wettability of the two fluids were larger and when this difference was present
for a larger fraction of the porous network.
