In a previous paper, the authors investigated various finite-difference approximations of [] and showed the advantages of the grid-point distribution method over the block-centered grid. In this paper, the proper finite-difference approximation with an irregular grid for is discussed. Terms of this type are associated with problems in cylindrical coordinate systems, like the coning problems in petroleum engineering work. The results of the truncation error analysis presented here confirm the conclusions reached in the previous work. It is shown that for the definition of block boundaries the choice of a logarithmic mean radius in r instead of in r yields an approximation that satisfies certain desirable integral properties of the differential operator in cylindrical coordinates. This scheme has been successfully implemented in two- and three-phase coning models.
The finite-difference approximations for the operator, associated with both block-centered and point-distributed grids, have already been treated point-distributed grids, have already been treated in detail. Our purpose here is to carry out a similar analysis for the operator . In view of the practical importance of operator, it is surprising that no such analysis has appeared in the literature before.
In one dimension, the geometrical quantities involved in the construction of an irregular grid system for block-centered and point-distributed grid are shown in Figs. 2A and 2B. When x is a linear coordinate, the grid points in the block-centered grid are placed halfway between the boundaries (= =) and the boundaries in the point-distributed grid are placed halfway between every two grid points (= =, ).
SPEJ
p. 396