  Steady-state upscaling of relative permeability is studied for a range of
reservoir models. Both rate-dependent upscaling and upscaling in the capillary
and viscous limits are considered. In particular, we study fluvial depositional
systems, which represent a large and important class of reservoirs. Numerical
examples show that steady-state upscaling is rate dependent, in accordance with
previous work. In this respect we introduce a scale-dependent capillary number
to estimate the balance between viscous and capillary forces. The difference
between the limit solutions can be large, and we show that the intermediate
flow rates can span several orders of magnitude. This substantiate the need for
rate-dependent steady-state upscaling in a range of flow scenarios. We
demonstrate that steady-state upscaling converges from the capillary to the
viscous limit solution as the flow rate increases, and we identify a simple
synthetic model where the convergence fails to be monotone. Two different sets
of boundary conditions were tested, but had only minor effects on the presented
reservoir models. Finally, we demonstrate the applicability of steady-state
upscaling by performing dynamic flow simulation at the reservoir scale, both on
fine-scaled and on upscaled models. The considered model is viscous dominated
for realistic flow rates, and the simulation results indicate that viscous
limit upscaling is appropriate.
