A new equation (Eq. 15, below) is presented for calculating the equivalent wellblock radii for all Nw wells in a reservoir. Arbitrary well rates and the interaction between wells are fully accounted for. The data required for the new equation may be obtained in a preprocessor by calculating Nw single-phase pressure distributions. Then an accurate equivalent wellblock radius can be calculated for each well each time step, even under conditions where well rates vary with time.
Previous work on handling of wells in reservoir simulators has, for the most part, focused on single wells and ignored the interaction between wells. For a single well, Peaceman defined the equivalent wellblock radius, r0, as that radius at which the single-well radial flow equation gives the same pressure as the wellblock pressure, p0. Then
Eq. 1 can be considered to be a definition of r0, but it can also be considered a well model because, if we know p0, q, and r0 for a wellblock, we can get the flowing well pressure, pwf. The usefulness of the well model therefore depends on the ability to calculate r0.
There is little in the literature that addresses the question of how r0 is affected by the interaction between wells. Palagi and Aziz show how r0 can be very different for one horizontal well (where the flow is primarily in the horizontal direction) and for two horizontal wells, one above the other, with one producing and one injecting (so the flow is primarily in the vertical direction). In another example, they examined the effect of rate, but only qualitatively.
The only work that quantitatively addresses the question of how r0 is affected by the interaction between wells for arbitrary well rates was provided in Ref. 3 (Eq. 23 there), for the case of multiple wells in a single isolated wellblock. (A wellblock is considered isolated when it is sufficiently far from other wellblocks and from the grid boundary that they have essentially no effect on the wells in this wellblock.) For uniform grid with uniform permeability, he showed that, for Nw wells in the block, the r0 for each well is given by
where rmk is the distance between wells m and k.
Up to now, there has been no work on how arbitrary flow rates affect r0 for wells in separate wellblock.