A Simple Approach to Optimization of Completion Interval in Oil/Water Coning Systems

Abstract

Summary An analytical solution is presented to estimate the maximum water-free production rate (critical oil rate) that considers the effect of limited wellbore penetration on the oil productivity of the well. To maximize the critical oil rate, an optimum wellbore penetration into an oil zone is determined theoretically. This analytical solution shows that the optimum wellbore penetration into a pay zone is less than one-third the total pay-zone thickness. Introduction Oil production through a well that partially penetrates an oil layer underlain by water causes the oil/water interface to deform into a cone shape. As the production rate is increased, the cone height above the original water/oil contact (WOC) also increases until, at a certain production rate, the cone becomes unstable and water is produced into the well. The maximum water-free production rate is the critical oil rate. Because oil production rate depends on wellbore penetration depth into the pay zone, perforation locations in the oil zone, and other factors, completion interval optimization and critical production rate calculation have been realistic research subjects for petroleum engineering investigators for many years. Muskat and Wyckoff1 presented their classic work on water-coning analysis in the 1940's. Because simultaneous determination of pressure distribution for the real water-coning system is effectively the same as that for the case with no cone and that the oil is flowing into a sand between two parallel, impermeable boundaries and into a well partially penetrating the sand. They presented a graphical procedure to determine critical oil rate. Meyer and Garder2 proposed an approximate equation that assumes radial flow and gave a simple equation describing the maximum theoretical oil flow rate with no water production while a water cone exists. This maximum theoretical flow rate is defined as the production rate at which the stable cone reaches the well bottom. Chaney et al.3 developed a set of curves based on Muskat and Wyckoff's1 mathematical analysis and potentiometric analyzer study by which maximum water-free production rates can be determined for wells having a gas/oil contact, WOC, or both. Henley et al.4 and Khan5 used scale models to study the water-coning problems. Karp et al.6 analyzed horizontal barriers for controlling water coning experimentally and theoretically using Muskat and Wyckoff's1 and Meyer and Garder's2 theories. Chierici and Ciucci7 presented an approach to determine maximum permissible oil rate without water/gas production using Muskat and Wyckoff's theory and potentiometric model technique. Welge and Weber8 presented a 2D method to calculate water-coning behavior numerically. Their results matched the coning behavior of the laboratory sandpacked model and of several producing wells that were experiencing water or free-gas production by coning. Sobocinski and Cornelius9 presented a correlation to predict water-cone behavior from static WOC to breakthrough conditions. Dagan and Bear10 investigated the problem of local interface upcoming in a coastal aquifer, and Blair and Weinaug,11 Letkman and Ridings,12 and Chappelear and Hirasaki13 studied the coning phenomonen by means of 2D numerical coning models. Bournalzel and Jeanson14 evaluated the breakthrough curve presented by Sobocinski and Cornelius9 and developed a simple expression to fit the curve. Schols15 developed a correlation to predict critical rates from experimental results, and Woods and Khurana16 and Trimble and McDonald17 developed 3D numerical coning models. Wheatley18 presented a 3D approximate theory of water coning where the effect of the existence of a water cone on the system pressure distribution was considered. Pressure distribution was approximately by superposing the uniform pressure distribution for a cylindrical case on several line/point sources along the wellbore. According to his theory, the critical flow rate can be obtained by a trial-and-error (iteration) procedure. Wheatley's approximation is good for dimensionless drainage-radius (ratio of drainage radius to thickness) values of 2 to 10. Chaperon19 compared critical rates in vertical and horizontal wells. Hoyland et al.20 presented two methods to predict critical oil rate in anisotropic formations having a well completed from the top of the formation; analytical and numerical solutions were reported. Their analytical solution is similar to that of Muskat and Wyckoff,1 except the assumption of uniform flux at the wellbore was replaced by the assumption of an infinitely conductive wellbore. They used Papatzacos'21 pressure distribution function and assumed that the potential distribution in the oil zone is unperturbed by the water conne. Piper and Gonzalez22 modified Wheatley's18 algorithm and extended it to three-phase problems. The effect of limited wellbore entry on the oil productivity of the well was not accounted for in these previous investigations. Therefore, the maximum water-free production rate would occur when wellbore penetration is zero, which is physically impossible. Abass and Bass23 studied the performance of water coning under different boundary conditions analytically, numerically, and experimentally. They used a fully implicit, strongly coupled mathematical model to handle rapid pressure/saturation changes, and they built a plexiglass mmodel to obtain qualitative and quantitative descriptions of the water-coning phenomenon. In their experimental study, no critical oil rate was observed for an unstable cone. Using an average pressure concept, they derived an analytical solution to calculate the water-free oil rate for steady-state and pseudo-steady-state flow conditions in a 2D radial-flow system. Although their 2D radial-flow assumption and average pressure concept may not be suitable for water-coning systems, they were the penetration on the maximum available water-free oil rate. According to their solution, the optimum fractional wellbore-penetration interval (completed from the top of the formation) should be 0.5 in oil/water coning systems. Recently, Yang and Wattenbarger24 presented two methods for water-coning calculations based on numerical simulation runs. Although many researchers have studied coning behavior to determine critical oil rate, their results frequently conflict because they make different assumptions to simplify the problem. In addition, while the existence of a critical oil rate associated with an unstable cone was proved analytically1,18 and was observed experimentally,4,5,9,10 it was not always observed.23 The purposes of this study were to investigate whether the unstable water cone and the critical oil rate associated with it exist, to reinvestigate the calculation of the critical oil rate with consideration of limited wellbore penetration effect, and to determie the optimum wellbore penetration where the maximum water-free oil rate can be obtained.