Authors: Sindre T Hilden Carl Fredrik Berg
Publish Date: 2016/08/16
Volume: 115, Issue: 1, Pages: 125-152
Abstract
A widely used approach for upscaling relative permeability is based on a steadystate assumption For small time intervals and at small scales the flooding process can be approximated as being in a steady state However at large scales with large time steps water flooding of a reservoir is an unsteady process In this article we first investigate the balance of viscous capillary and gravity forces on the fine scale during the water flooding of a reservoir at different flow velocities We introduce a semianalytical method to find the lowrate limit solution while the highrate limit solution is found by running a simulation without gravity and capillary pressure These limit solutions allow us to understand when ratedependent simulations approach a point where some forces become negligible We perform a series of numerical simulations on the fine scale to construct solution transitions between the established outer limits Simulations are run both on homogeneous models on different layered models and on a more complex twodimensional model The ratedependent simulations show smooth transitions between the low and highrate limits and these transitions are in general nontrivial In all our example cases one of the limit solutions gives a lower bound for the rate dependent results while they do not in general provide an upper bound Based on the ratedependence of the force balance we evaluate when different steadystate upscaling procedures are applicable for an unsteady flooding process We observe that the capillarylimit upscaling which also takes gravity into account reproduces the lowrate limit finescale simulations Such capillarylimit upscaling is also able to reproduce the transition to capillary equilibrium normal to the flow direction As already known the viscouslimit upscaling is only applicable when we have close to constant fractional flow within each coarse grid block
Keywords: