Abstract
In a recent paper, van Poollen et al. presented equations relating observed field pressures to those calculated by a numerical simulator. The equations are applicable to steady- and semi steady-state flow for wells draining circular areas and using an equivalent block radius. They implicitly assume that wells are located at the center of the drainage area. In this note we present equations that generalize the previous method and relate field to model pressures for various shapes of drainage area, well pressures for various shapes of drainage area, well location and grid configuration. Using the generalized equations of flow of Brons and Miller, and following a method of derivation similar to that reported by van Poollen et al., we obtain:For steady-state flow: .................(1) For semisteady-state flow: ...................(2) For unsteady-state flow: .....................(3) where subscripts m and f refer to the model and field, respectively, and P D theta of Eq. 3 is the dimensionless initial pressure. For a circular system A pi r, and CA = 31.6 from Ref. 3. If we substitute these values in Eqs. 1 and 2, we obtain the results of van Poollen et al. as represented by their Eqs. 14a and 14b. In a manner analogous to that of van Poollen et al. the average reservoir pressure can be related to the dynamic pressure by using a dimensionless time based on the drainage area A rather than radius. Also, the relation between the average reservoir pressure and the simulator pressure in Eqs. 1 and 2 can be based on producing time as well as shut-in time, since it is possible to generate one method from the other as was shown by Ramey. NOMENCLATURE A = area in sq ftB = formation volume factor, res. bbl/STBCA = shape factorDD =c = compressibility, 1/psih = thickness, ftk = permeability, md PD, m, t = dimensionless model pressure, [141.2kb/ PD, m, t = dimensionless model pressure, [141.2kb/(q B)]P PD = dimensionless average reservoir pressure PD = dimensionless average reservoir pressure P = pressure, psiq = production rate, STB/Drw = well radius, ftt = flow time, days= porosity fraction= viscosity, cp P. 277