Equation for Oil and Gas Two-Phase Flow into Vertical Well- A Theoretical Derivation

Generally reservoirs are classified into five categories according to the reservoir fluids. Single phase flow in reservoir is expected during the production life for wet gas and dry gas reservoirs. But for black oil, volatile oil, and gas condensate reservoirs, two-phase flow will occur after pressure declines to saturated pressure and the second phase saturation reaches the critical saturation. Theoretical derivation of single-phase flow into vertical well had been done by former investigators. The solutions to different boundaries at different flow status were provided through different simplifications and assumptions. These solutions provide powerful tools for reservoir study. But when it comes to oil and gas two-phase flow, they cannot be applied. To the best of our knowledge there is no rigorous and theoretical derivation for the oil and gas two-phase flow into the well. In this study, rigorous and theoretical derivation of oil and gas two-phase flows from reservoir into the well was conducted. Starting from the famous Darcy’s equation, the flows of oil and gas are controlled by the rock/fluid properties, pressure, and temperature condition. The continuity equation was applied. With the combination of equations of state and the concept of compressibility, the governing equation of oil and gas two-phase flow was constructed. Even numerical approach is required to solve the equation to get gas and oil flow rates, the new equation still has its significance due to its theoretical and rigorous derivation. It can be used as a unique tool to solve the saturated oil and gas condensate reservoirs that experience two-phase flows. The proposed equation addresses the rigorous derivation of two-phase flow in a radial reservoir. It eliminates the uncertainty in the empirical equation and has the advantage of quickness and simplicity comparing with reservoir simulation method. It becomes a powerful tool for petroleum engineers.