Peaceman defined the well-block pressure obtained from reservoir simulation/or equally spaced grids as the steady state flowing pressure at the equivalent radius and he found that an equivalent radius occurring at a distance of 0.2 times the grid spacing from the center of the well was required to obtain the correct steady-state, Muskat analytical rate. This pressure does not correspond to the average pressure of the well block and generally requires changes to pressures obtained from pressure transient analysis for use in the simulator. In the method described here, it is found that a distance of 0.342 times the grid spacing is needed to obtain the correct steady-state, Muskat analytical rate. This corresponds to the average pressure of the well block and, therefore, no adjustment to pressure transient analysis data is required when the method described here is employed. A procedure is provided for implementation of the method into an existing simulator. Applicability of the method to large, effective wellbore radii such as could be encountered in wells which are extensively and vertically fractured is also considered. In such cases, with qualification, wellbore radii nearly as large as the grid spacing can be accommodated in models employing the method.
A method of dealing with wells in grid blocks of reservoir simulators was developed by Peaceman in 19781. This method continues to be very popular today2,3. A very recent rehash of the method can be found in the technical paper by Sharpe and Ramesh3. In the development for uniform grid spacing by Peaceman, a new definition of well-block pressure appeared along with a rather enigmatic constant, 0.2 (or more precisely, 0.1985064), which when multiplied by the grid size gave an equivalent radius of thewell block. Peaceman defined this well-block pressure as the steady-state flowing pressure at the equivalent radius.
Yet, there were others (van Poolen et al.5 and Coats et al.6 are examples) who maintained that the well-block pressure should be the average pressure which should occur at a radius of 0.342 (or more precisely, 0.342198, to give itthe same numbers of digits as Peaceman's constant) times the grid spacing. Because Peaceman's well-block pressure is neither the average pressure of the well block, nor is it located at the appropriate position for the average pressure, corrections are generally required to average pressure data obtained from pressure transient analysis before input to the simulator to accommodate Peaceman's method7.
In this paper, this controversy is more fully explored with the ultimate conclusion that the more appropriate constant should indeed be 0.342198 when used with the procedure described below.
Before focusing on the main issue of this paper, a brief review of point-centering of grids is in order. A more complete review can be found in Aziz and Settari8. Figure 1 shows two examples of point-centered grids for both a one-dimensional system such as might be used in modeling a laboratory core flood and a two-dimensional system such as might be used in modeling a symmetry element of a pattern study (e.g., quarter of a five-spot). It is for these types of systems, where the outer boundary corresponds to an injection or production face or well, that point-centered grids are most advantageous.