Rate Dependency in Steady-State Upscaling

TL;DR

This paper investigates how steady-state upscaling of relative permeability depends on flow rate in reservoir models, emphasizing the importance of rate-dependent effects and demonstrating the applicability of viscous limit upscaling for realistic flow scenarios.

Contribution

It introduces a scale-dependent capillary number to quantify rate effects and shows the convergence from capillary to viscous limits in steady-state upscaling.

Findings

Steady-state upscaling is rate dependent.

Convergence from capillary to viscous limit as flow rate increases.

Viscous limit upscaling is suitable for realistic flow rates.

Abstract

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.