This paper presents the development and application of semianalytical solutions to model multi-phase flow of oil and water. Our solutions are modifications of single-phase solutions, with multi-phase flow introduced using the Perrine-Marin "total mobility" to replace single-phase mobility in the standard diffusivity equation. This modification led to very satisfactory agreement between calculated and simulated results. Our results include new solutions for an outer boundary condition of "specified flux." With water influx, this "specified flux" will vary with time and is calculated using an aquifer model coupled with the reservoir model.
The importance of this work is in its possible implementation in software designed for production data and well test data analysis. Most commercial software is limited to single-phase flow, with multi-phase water-oil flow modeled (if at all) by highly simplified models.
Our model assumes a producing well centered in its drainage area surrounded by a cylindrical aquifer or injected water bank. Cumulative water influx is related to the well production and reservoir flow properties by material balance. Any standard water influx model can be used to relate cumulative water influx and the reservoir-aquifer interface pressure.
We relate the average reservoir pressure to well performance using solutions first derived to model single-phase, slightly compressible liquid flow. Oil and water production rates are obtained from the total rate using Darcy's law. Fluid saturations at a given time are computed volumetrically, and this technique does not require knowledge of saturation distribution in the reservoir. Pre- and post-breakthrough oil and water rates are based on fractional flow theory. Although we considered only the case of constant-rate production, our method can be extended readily to the case of constant-pressure production.