Steady-State Coning Solutions With Multiple Wells and Reservoir Boundaries

Abstract

Abstract The critical oil (or gas) rate to avoid coning of unwanted fluids into production wells is an important design parameter. Simulation methods are useful to predict critical rates in reservoirs with complex heterogeneities and boundaries, but they are manpower intensive and prone to errors when large grid blocks are used. Current analytical methods are quick and easy to use, but their assumptions are too restrictive. Thus, there is a need for improved analytical methods that can account for well patterns and more complex boundaries, and also serve as further benchmarks for simulation. This paper makes analytical solutions more realistic by extending existing single-well analytical solutions to account for multiple wells and common no-flow and constant pressure boundary conditions. A potential function is derived to superimpose existing single well coning solutions for single- or simultaneous two-phase flow. Capillary pressure and relative permeability effects on coning are included. The only limiting assumptions are vertical equilibrium (VE) and steady-state flow. Comparisons with simulation show good agreement in predicted critical oil rates when steady state and VE are approached. VE and steady-state are approximately achieved when aspect ratios are greater than about 10. Even when aspect ratios are less than 10, the predicted critical rates are useful in that they are always conservative. The proposed analytical solutions are quick and easy to use compared to reservoir simulation and can be used in the development of downhole water-sink technology (DWS). Introduction Water or gas coning can adversely affect oil production in oil reservoirs and gas production in gas reservoirs. In oil reservoirs, a large oil rate can cause upward coning of water or downward coning of gas into the well perforations. Once gas or water is produced, the oil rate decreases and the cost of water and/or gas handling is increased. It is a common industry practice to reduce water coning in oil reservoirs by perforating vertical wells as far above the oil-water contact (OWC) as possible and to produce the wells at or below the critical oil rate. Similarly, wells are often perforated low in the oil column away from the gas-oil contact (GOC) in gas-oil reservoirs. The benefits of this practice are mixed in that limited perforations may increase the pressure gradient (the drawdown) near the well, which can exacerbate coning. There has also been success in reducing coning with polymers and gels.[1] A more recent and novel approach is to use downhole water-sink technology (DWS) where water is produced separately from the oil using dual packers.[2] The water production below the OWC may reduce upward water coning so that the oil rate can be increased. The DWS technology, however, requires a good understanding of how fluid rates affect coning, which is one of the goals of this paper. Dupuit[3] published one of the first papers on the down coning of air into aquifers. The Dupuit equation, which assumes vertical equilibrium and segregated flow, gives the steady-state relationship between the water production rate and water table elevation in the vicinity of a single wellbore. The Dupuit equation is still used today to determine the elevation of the water table when producing water to a well.[4,5] Muskat and Wyckoff,[6] who coined the term "water-coning," derived an approximate steady-state solution for two-dimensional water coning in an oil reservoir. Pirson[7] extended the Dupuit approach to estimation of critical oil rates for oil flow in a segregated gas-oil-water reservoir. Numerical simulation has also been used to estimate critical rates to avoid coning.[8–12] Although numerical simulation methods are useful, numerical dispersion can affect simulation results leading to errors in predicted critical rates. Analytical methods can provide quick first estimates of critical rates, and are independent of the grid-block size. Johns et al.[13] derived new analytical solutions of "Dupuit form" that allow for both single- and simultaneous two-phase flow that include the effect of capillary pressure and relative permeability on fluid interfaces. Their solution, however, assumed a single well in an infinite acting reservoir.