Assumptions:

We seek a coarse grid that

To coarsen the grid, we manipulate a set of partition vectors according to the following coarsening principles:

The general framework is implemented as a set of algorithmic primitives that create partition vectors (sources) and a set of primitives that manipulate them (filters):

References:

The first example of the amalgamation framework published by our group was the non-uniform coarsening method proposed by Aarnes, Efendiev, and Hauge [1]. The method consists of four steps, as illustrated in the figure to the right:

Later, the algorithm has been improved in several directions by using different neighbour definitions to grow blocks, combining flow adaption with a priori partitions, etc, as discussed by Hauge et al. [2].

Flow-based coarsening generally gives improved accuracy in the computation of fluid transport, as shown in the figure below. We consider an extruded PEBI model with petrophysical parameters sampled from Layers 50-60 of the SPE10 benchmark and a diagonal displacement crated by placing an injector and a producer in opposite corners. The figure compares the saturation distribution after one pore-volume of water has been injected, simulated on three different grids. We observe that the METIS grid has too large blocks in the flow channel to capture the flow well, while the time-of-flight grid to a large degree matches the flow pattern from the original fine grid.

Original fine grid, 11 864 cells.

Coarsening based on time-of-flight, 127 cells.

Topological coarsening using METIS, 175 blocks.

The amalgamation framework is quite general and contains a large number of different coarsening methods as special cases. Each particular method can be expressed by combining the generic filters for merging and refining blocks with different a priori partitions and indicator functions. In the figure below, we show grids that are generated using a combination of structured topological and flow-based partitions.

Hybrid grid: topological partition combined with a flow-based partition with velocity as indicator.

Hybrid grid: topological partition combined with a flow-based partition with time-of-flight as indicator.

The coarsening can also be constrained to various geological features. In the figure below we use information of facies to constrain each coarse block so that it contains only one facies.

The distribution of facies in the domain which is considered as a static partition that is intersected with a standard flow-based partition based on e.g., flow magnitude or time-of-flight.

Cartesian fine grid.

PEBI fine grid.

Because the all coarse grids created by agglomeration of cells from a fine grid are represented using a partition vector, it is (almost) straightforward to add dynamical adaptivity. We generate a coarse grid based on the time-of-flight indicator intersected with a uniform topological partitioning, as shown in the figure above. Local refinement, down to the resolution of the original grid, is added in regions near the displacement front. Cells that have time-of-flight values a certain fraction larger than those near the displacement front will likely belong to the unswept zone and are therefore merged into a single coarse block. Moreover, a band of blocks with intermediate resolution is kept ahead of the displacement front (measured in time-of-flight) to localise the search for regions that need to be refined in the next step and as a precaution when multiple time steps are computed without updating the grid. The figure below shows examples the adaptive grids for Layer 22 from the SPE10 model. For comparison, we also show the saturation profiles on the original and on a corresponding static coarse grid.

Examples of locally adapted grids for Layer 22 of the SPE10 model. The three plots to the left show saturation profiles after 0.1 pore volumes have been injected, whereas the rightmost plots show saturations at 0.5 PVI.

Published October 21, 2010

A portfolio of strategic research projects funded by the Research Council of Norway

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