11.12.5 IJK Cartesian and Radial Cell Interpolation

RESQML provides support for radial cell interpolation as part of the IJK grid representation. Unlike reservoir simulation software, which provides special keywords to allow specification of grids directly in (r,,z) coordinates, RESQML continues to specify cell nodes as (x,y,z) points. This specification allows reasonable interchange with geologic model vendors, the majority of whom do not provide support for radial grids. However, when the “radial grid is complete” Boolean element is included in a RESQML grid description, together with the radial origin polyline geometry, it then implies that radial interpolation should be used to describe the cell shape.

• Specifically, for a Cartesian corner-point cell, the cell volume is defined by the tri-linear interpolant of (x,y,z) in () between the 8 corner nodes.

• In contrast, for a radial corner-point cell, the cell volume is defined by the tri-linear interpolant of (r2,,z).

This choice follows from the transformation of unit volumes: dx dy dz = r dr d dz = ½ dr2 d dz. In other words, the interpolation describes a radial (cylindrical) cell shape. The I (or ) coordinate (dimension: NI) is always in the radial direction, and the J (or ) coordinate (dimension: NJ) is always in the angular direction. For a fixed value of , we have a 2D radial coordinate system.

• To define r from (x,y,z), we need the coordinates of an origin at that same value of . The definition of the radial grid includes the specification of NKL origin points, which may be interpolated linearly in .

• To define  from (x,y,z), we may use the usual trigonometric functions.

Some care must be taken to ensure that the branch cut does not lie within the cell. For the special case of complete 360o grid cells with NJ=1, care must be taken to have strictly monotonic values for This construction is sufficiently general to represent simple radial grids with vertical coordinate lines, or more complex radial grids that may conform to a complex horizontal or undulating well trajectory.

Radial IJK grids may have a slightly different topology from Cartesian IJK grids. If a radial grid is complete, then it is periodic and covers 360o. Consequently the last J cell face is identical to the first, and NJL=NJ instead of the usual NJL=NJ+1.

Local radial grids use a modified interpolation method on the outermost ring(s) of cells (I=NI). To ensure geometric consistency with the parent grid, the  range is reduced so that the radial cell shape remains consistent with the cell volume interpolation method of the parent grid cell. Specifically, a circular arc between two node points on the local grid normally extends beyond a linear interpolant on the parent grid. The extent of the radial cell in the  direction is reduced so as not to exceed the linear interpolant.