Abstract
A widely used approach for upscaling relative permeability is based on a steady-state 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 semi-analytical method to find the low-rate limit solution, while the high-rate limit solution is found by running a simulation without gravity and capillary pressure. These limit solutions allow us to understand when rate-dependent 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 two-dimensional model. The rate-dependent simulations show smooth transitions between the low- and high-rate limits, and these transitions are in general non-trivial. 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 rate-dependence of the force balance, we evaluate when different steady-state upscaling procedures are applicable for an unsteady flooding process. We observe that the capillary-limit upscaling, which also takes gravity into account, reproduces the low-rate limit fine-scale simulations. Such capillary-limit upscaling is also able to reproduce the transition to capillary equilibrium normal to the flow direction. As already known, the viscous-limit upscaling is only applicable when we have close to constant fractional flow within each coarse grid block.
