The automatic gridding of a well with any trajectory has been the most challenging numerical development for KAPPA to date.
Developing a numerical model for a simple slanted well was already a challenge. Naturally, analytical well index calculations are easy to implement, but this works for coarse grids only and with commensurate large time steps. It also means the characteristics around the well, particularly in the vertical direction, bring little value. However, if one wants to model the shut-in for such a well, with logarithmic time sampling, it is necessary to create a series of grids that will handle the very early time radial or elliptical flow towards the wellbore, the vertical diffusion when the well is of limited entry, the transition towards horizontal flow and finally the horizontal flow itself.
With our experience to date of using the Voronoi grid it was natural to follow this road in the development of this particular well model. But the path was nearly as convoluted as some of the well trajectories we were being asked to model. Many technical barriers prevented us from following a straight road to get to the 3D solution. These included honouring horizons with a reasonable number of cells, connecting to other grid modules and handling anisotropy etc. These barriers wereprogressively removed on the way to the solution offered today and it is interesting to share this journey.
Attempt 1: Use of a locally refined cartesian grid
We started with a simple octree procedure to successively refine the grid around the wellbore. Since it was built in a stratigraphic space, its main advantage was to rigorously honor any succession of complex horizons, while offering full flexibility at the level of refinement. It was also extremely fast. But this solution meant the infamous cell connections could be in real contradiction to the classical k-orthogonality assumption. However, this octree approach enabled us to develop new internal transmissivities, that proved to be extremely robust and the transient response obtained with the octree matched the analytical perfectly on a loglog plot.
Attempt 2: Using a 3-D Delaunay grid
As a consequence of the work done in attempt 1 and thanks to these transmissivities, nothing was preventing us from trying 3-D Delaunay grids, since these, although non-orthogonal, are extremely efficient in ensuring both local refinement and cell conformity everywhere it is needed. This Delaunay grid was used as a sort of ‘cement’ to fill the gaps between the large coarse well grid and the main Voronoi grids. It worked, however far too many cells were required in order to get reasonable aspect ratios, and the gridding process was too slow.
Attempt 3: Using a 3-D Voronoi grid
This was the right one (so far): We filled the surroundings of the wellbore with points, ensuring a high density close to the drain, decimating as we radiate outwards from the well. The grid was then built using a 3D Voronoi grid builder from this collection of points. The result is a 3D unstructured grid that honors the well drain, and the surrounding horizons, with reasonable aspect ratios and orthogonal connections. In order to keep an acceptable grid size, even for long wells, the first ring of cells surrounding the drain was kept voluntarily coarser than what we normally use with other well types. Experience gained from the octree and the Delaunay approaches provided us with the right way to handle complex transmissivity derivations and internally provide 'virtual cells' to the solver within this coarse ring in order to capture early transient flow. We ended up with a fully transient numerical model, validated by the analytical solution at every time scale.
Crossing horizons
Once this fundamental problem had been solved the next challenge was to allow this slanted well to cross several layers, honoring both flow towards the well and the horizons. This was done by first gridding the vicinity of the drain, then carefully positioning additional points and honoring the horizons by construction. The current procedure is still iterative but a direct construction is now under development.
The Wriggly well
The next step was to combine multiple segments, with any number of segments in any direction. The well is no longer enclosed in a single plane. Voronoi sites around the segments are positioned so that the corners are honored with no interference between segments. Multiple perforations along the drain are handled.