Multi-well analytical solution for coning under simultaneous steady-state flow of three phases

Unwanted water or gas production from upward or downward coning can significantly Erode profits in oil recovery processes. Simulation estimates of the magnitude and timing of coning can be erroneous owing to unknown reservoir heterogeneity, large grid blocks near the wells, and inaccuracies in simulation well models, such as that from Peaceman. This paper presents a multi-well steady-state analytical solution for coning of three phases (oil, water, and gas) flowing simultaneously. The solution for multiple wells is developed using superposition with a coning potential function that includes capillary pressure and relative permeability. The assumption of vertical equilibrium (VE) is made, which gives maximum crossflow vertically and therefore the largest possible coning. Any model for relative permeability and capillary pressure can be used, although we used Stone 2 for relative permeability and Brooks-Corey for capillary pressure. The solution is important to benchmark numerical solutions and potentially improve the accuracy of Peaceman's well model. Further, we give inflow performance windows (IPW) to show the allowable physical window of three-phase rates and the maximum oil rate as a function of the water and gas rates. The new potential functions are also used to demonstrate superposition for two production wells in a reservoir with perpendicular no flow boundaries.