Use of Irregular Grid in Cylindrical Coordinates
Authors: Settari A. , Aziz K. 
Published: December 2007

Abstract
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.

Introduction
Use of an irregular grid is often necessary or advantageous in the obtaining a finite-difference approximation to the solution of partial differential equations. In engineering applications, particularly in reservoir simulation, two methods of constructing an irregular grid are currently employed. In this first method, called the "block-centered grid" method, and domain is divided into rectangular "blocks" and a grid point is then located in the "center" of each block, as shown in Figure 1A for Cartesian coordinates. In this grid system, there are no grid points on the boundary, which is approximated by points on the boundary, which is approximated by boundaries of the blocks (dashed lines). The second method, which we referred to as the "grid-point" distribution" method in a previous paper, involves selecting grid points to approximate the actual boundary and then assigning blocks to grid points by locating block boundaries in the "middle" between the grid points (Fig. 1B). Consequently, grid points will not always be centered in the blocks. Points at the boundary are given by this grid system, whereas with the block-centered grid, it is not possible to obtain those points directly. Because the selection of grid spacings for each direction is independent, the investigation of grid system is a two-dimensional coning problem r-z coordinates. In such an application, the parts of the conservation equation representing the flow terms in the z-direction have the general form given by the operator, (1)d dU AU = (k(U, x)), (1) dx dx
and the terms in the r-direction have the general form given by the operator, (2)1 d dU CU = (x K(U, x)) (2) x dx dx
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.

FINITE-DIFFERENCE APPROXIMATIONS
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