Improved Distance Based Upgridding and Diffuse Source Upscaling for High Resolution Geologic Models
Authors: Syed Imroj , Liu Ching-Hsien , Kelkar Mohan G. , King Michael J. 
Published: October 2020

Abstract
Challenges remain in the upscaling and flow simulation of reservoir models from strong heterogeneity that may arise when representing complex patterns of connectivity and barriers. This is especially true in high contrast systems, e.g. for carbonate reservoirs, where statistical upgridding and upscaling approaches developed for clastic reservoirs perform less well. This has led to the development of a novel "Distance Based" upgridding technique which we combine with "Diffuse Source" upscaling to successfully simulate such models.
We replace previously developed variance-based statistical sequential layer grouping reservoir coarsening analyses, with a novel distance-based calculation. It relies on the local errors in the interstitial velocity and the time-of-flight introduced when layers are grouped, as a measure of distance between reservoir models. The use of a distance measure allows for the inclusion of flow capacity in this calculation, and avoids the strong biases that arise from the previous variance-based approaches, especially with high contrast systems.
We utilize the "Diffuse Source" (DS) upscaling approach to obtain the intercell transmissibility and well indices for the upscaled reservoir model. The DS approach is an extension of pseudo-steady-state (PSS) flow-based upscaling that utilizes the diffusive time of flight to identify well-connected sub-volumes in each adjacent pair of coarsened reservoir grid cells. DS upscaling retains the same localization advantage as the PSS approach. Unlike steady state upscaling, there is no coupling to a global flow field and local-global iterations are not required. DS upscaling was previously developed for more general reservoir coarsening, but the description now includes an extremely simple implementation for 1×1×N layer coarsening that naturally avoids the issues that arise in the use of the harmonic average for the vertical transmissibility.
Starting from a high-resolution fine scale 3D geologic model, sequential layer grouping provides us with a series of increasingly coarsened reservoir models. Each model in turn minimizes the integrated distance between the fine and coarsened models, which is used as a measure of heterogeneity lost during coarsening. From these layer designs we apply a combined cost and heterogeneity objective function to determine an optimal layer design, which is then used for upscaling and flow simulation.
We show that this distance-based optimal layer design does not experience the over-grouping of layers that arose from the previous variance-based approach. The new approach has been able to integrate the flow capacity similar to a Lorenz plot into the calculation of distance between reservoir models to include the impact of reservoir quality. This replaces the simple use of a net-to-gross cutoff utilized in previous work. The distance calculation uses a hyper-volume Lebesgue measure which provides a consistent means of combining different physical attributes: in this case interstitial velocity and the time-of-flight. The generalization of the hyper-volume to multiple properties, e.g., anisotropic permeability, is straight-forward.
1×1xN layer grouping is examined specifically to show the improvement in the optimal layer design compared to the previous statistical analyses. Areal coarsening is increased beyond 1×1 to show the general applicability of the DS upscaling approach.
The combination of distance-based upgridding and DS upscaling is tested and performs extremely well on a series of sector, outcrop and full-field 3D reservoir models using a research simulation code and a commercial finite difference simulator. These models include the SPE10 reference model, the Amellago carbonate model, and additional full field examples.
A novel distance-based measure of reservoir heterogeneity has been developed and applied to the design of an optimal reservoir simulation layering scheme, given a prior 3D geologic model. The distance measure combines elements of the Lorenz plot and previous variance-based analyses to avoid the strong biases and collapse of layering seen in the earlier approaches. Flow simulation based on Diffuse Source upscaled properties are shown to perform extremely well compared to the fine scale model.