Peripheral water injection efficiency for material balance applications

Abstract The simple approach consisting of including injected water directly into the reservoir material balance equations is not appropriate to model peripheral injection, as it does not account for water lost to the aquifer as well as time required for pressure to diffuse to the reservoir boundary. Based on this observation, the authors have extended the van Everdingen and Hurst (1949) unsteady state edge aquifer model to account for peripheral sources. Taking advantage of the pressure diffusion equation linearity and problem symmetries, for simplified circular and linear geometries the problem can be cast as one-dimensional regardless of the number and position of peripheral injectors. Solutions are reported in the form of a tabulated cumulative efficiency function, defined as the amount of water having reached the reservoir boundary owing to the presence of a source injecting at unit rate, vs. time. Superposition principle can then be used to calculate time-dependent water influx for an arbitrary number of injectors and injection history. Solving the two-dimensional problem further provides the lateral influx distribution, and shows that pressure support efficiency as defined in this work, relevant to material balance applications, is conceptually different from transport efficiency provided by streamlines analysis. The latter is indeed unable to single out the individual contribution of a specific injector to reservoir voidage replacement from that of its neighbors and the aquifer itself.