Dynamic Diffuse-Source Upscaling in High-Contrast Systems
Authors: Nunna Krishna , King Michael J. 
Published: September 2019

Traditional upscaling methods are dependent on steady-state (SS) concepts of flow, whereas flow simulation itself is used for the calculation of pressure and saturation transients, which can be considered as a sequence of pseudosteady-state (PSS) solutions. In high-contrast or low-permeability systems, neither the SS nor the PSS limits need to be reached within each coarse-cell volume during a simulation timestep, introducing a potentially significant bias into an upscaling or downscaling calculation. We use an asymptotic pressure analysis for transient flow, dependent on the diffusive time of flight, to improve the resolution of these dynamic effects.

We introduce a novel upscaling approach with two major differences from SS upscaling. First, we transition from SS- to PSS-flow solutions. This has been shown to provide identical results to SS upscaling in one dimension, but to have improved localization for upscaling in two and three dimensions. Specifically, there is no longer an explicit dependence upon global pressure boundary conditions. Development of this PSS upscaling approach has also required the introduction of a new transmissibility-weighted pressure-averaging definition instead of the pore-volume (PV) -weighted pressure average used for SS flow. The second difference is in using pressure-transient concepts to identify well-connected subvolumes within a coarse-cell volume. The local source/sink terms during the transient are no longer solely proportional to porosity, as in the PSS limit. Instead, these terms now include a spatial dependence obtained from the asymptotic transient pressure approximation. This dependence is especially important for high-contrast or low-permeability systems. The methodology we have developed is an application of the concepts of the diffusive time of flight and transient drainage volume to obtain source functions that capture both the early- and late-time limits of the transient-flow patterns. Diffuse-source (DS) functions are introduced within each fine cell of a coarse-cell pair, consistent with the transients and with a specified total flux between the coarse cells. The ratio of this flux to the averaged pressure drop is used to obtain the effective transmissibility between the cell pair.

The application of pressure-transient concepts has allowed us to develop completely local upscaling and downscaling calculations. A characteristic time is determined for which a well-connected subvolume for each coarse-cell pair is sufficiently close to PSS. This enables us to distinguish between well-connected and weakly connected pay while upscaling. Unlike SS upscaling calculations, which explicitly impose flow on the boundaries of an upscaling region and implicitly couple the local problem to a global flow field, these calculations are completely local. The methodology is tested on SPE10 (Christie and Blunt 2001) with permeability variations over eight orders of magnitude, making it a high-contrast example. We also test the method on a low-net/gross onshore tight gas reservoir consisting of thin fluvial channels undergoing primary depletion. The comparisons of performance prediction with fine-scale numerical simulation and SS upscaling demonstrate the accuracy of the proposed approach.

NOTE: Supplement available in Supporting Information section.