Gridding and Upscaling*

Also see:

• SPE10 and Upscaling

• a correct solution to the Unidirectional Flow Problem - a simple test for K-orthogonality and numerical dispersion using any grid (a solution has never been demonstrated on other than a Cartesian grid)

Single-phase flow-based upscaling on Cartesian grids

In a single coarse Cartesian gridblock of size DX DY DZ, Darcy’s Law directly gives the permeability in each direction Kx, Ky, and Kz (using a unit viscosity single phase incompressible fluid) as Qx DX/( DY DZ DPx), Qy DY/(DX DZ DPy), and Qz DZ/(DX DY DPz), respectively. The Q/DP rate/pressure difference ratios come from separate solutions in each direction on the corresponding (nx,ny,nz) fine grid system, and they may be considered as boundary condition/solution or as solution/boundary condition – these definitions and boundary condition values make no difference. In that fine grid system, the equivalent boundary conditions are effectively applied using planar wells on the edges, for an exactly equivalent physical system.

This method is not [has not been] generally applicable to non-orthogonal grids such as corner point, SKUA, or unstructured, since Darcy's Law does not give the coarse block solution for any discretization other than Cartesian.

So, how are the grids in use today upscaled?  We suspect that usually they are not.  It seems that most users today simply generate their grids at the desired scale without being able to determine the optimal one. Grid size sensitivity is usually skipped, but if done consists only of comparing results obtained on different scale grids, which are almost guaranteed to be different without any of the conditioning of upscaling that makes it effective. That leads to the wrong conclusion that simulation at the geological scale is required, or to severe numerical dispersion (model error) in coarse grids, which causes error in input/output relationships, severely inhibiting our abilities in optimization and forecasting.

* taken from an SPE Reservoir Simulation Technical Interest Group discussion, October 28, 2011 (deleted by SPE, 2014)

New (April 2016)

We have solved the single-phase flow-based upscaling problem for non-orthogonal grids, including corner point and SKUA, by adapting our Cartesian upscaling program.  Significant usability and workflow improvements are provided by integration with Sensor.  Application is restricted to regular grids with no fault throws represented in the grid (no non-neighbor connections), single-porosity reservoirs, and to the matrix portion of dual permeability reservoirs.  Non-orthogonal grids require input of grid block pore volumes and transmissibilities that are calculated by the gridding program/preprocessor.

For fractured dual porosity/permeability reservoirs,  upscale as a single porosity model to the scale indicated by your manual dual porosity/permeability (dp) upscaling - comparing discrete sp vs. dp results - then add DUAL option and fracture data.

See or ask about the new Sensor UPSCALE feature.  Options include stopping the run following upscaling and writing the coarse grid file, or continuing the run with either the upscaled coarse grid or the original fine grid.

The implications are obvious, not only for the solution of the (single-phase) upscaling problem for non-orthogonal systems, but for potentially achieving the goal of quantifying uncertainty in probabilistic forecasts and optimizations, even for field-scale models.  Examples are coming soon.

For optimal upscaling, the geological model should be created on a regular (Cartesian) grid.  Block size uniformity is often critical to reservoir performance (the smallest cells limit performance),  Simply lay the grid over the reservoir and discretize the properties and features (boundaries. fault planes, and well trajectories).  There is no need for flexible grids in systems that have been discretized, i.e. properties are assumed fixed within each discrete volume (gridblock). Like a television picture, good resolution does not require the use of non-rectangular pixels or a non-orthogonal grid.  This allows for robust application of single-phase flow-based upscaling for permeability.

No matter what hardware systems you have, when simulation is required to answer the question(s) and where Sensor is applicable, nothing is more efficient or effective than running the optimal number of tuned serial Sensor realizations in parallel (see Parallel Reservoir Simulation?)  That optimal number is usually equal to the number of effective cores.