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Abstract A model for oil-water coning in a partially perforated well has been developed and tested by perforated well has been developed and tested by comparison with numerical simulations. The effect of oil-water coning, including down-coning of oil, on field production is demonstrated by studying a small water drive reservoir whose complete production data arc known. production data arc known.The coning model is derived by assuming vertical equilibrium and segregated flow. A necessary correction for departure from vertical equilibrium in the immediate neighborhood of the well is developed The coning model is suitable for single-well studies or for inclusion in a reservoir simulator for two-dimensional, areal studies. Introduction The objective of this investigation of oil-water coning was to develop tools to evaluate operational problems for reservoirs with bottom water. Although problems for reservoirs with bottom water. Although any specific question can be answered (a least in principle) by finite-difference simulation, a practical principle) by finite-difference simulation, a practical problem occurs. Great detail may be necessary for problem occurs. Great detail may be necessary for a reservoir-wide simulation of problems involving coning. Two approaches are possible. One can use more accurate finite-difference equations (such as those derived by some type of Galerkin procedure) to solve the problem of insufficient accuracy. Or one can include in his simulator a "well model" that accurately predicts coning on the basis of near-well properties. The well model could be either another finite-difference subsystem or a formula theoretical or empirical (or both) in character. Our approach is to develop a theoretical model that can be installed in a finite-difference reservoir simulator. We feel that such a model, particularly if it is simple and widely applicable, has several advantages:(1)the assumptions made in the derivation aid in understanding coning; (2)the formula guides the engineer by indicating significant parameters and their relationships; (3)the existence parameters and their relationships; (3)the existence of a simple formula permits preliminary studies without a full simulation; and (4)the simple formula is easy to install in a reservoir simulator. This model for oil-water coning differs from others presented previously in two respects. First, presented previously in two respects. First, partial completion that does not necessarily extend partial completion that does not necessarily extend to the top of the formation is treated. Second, an effective radius that allows for vertical flow resistance is introduced. DESCRIPTION OF MODEL ASSUMPTIONS The geometric configuration for the coning model is a radially symmetric, homogeneous, anisotropic system with inflow at the outer boundary and with a partially perforated well. The fluid distribution is shown in Fig. 1. The presence of initial bottom water at 100-percent water saturation is considered. The perforated interval is assumed to be within the original oil column. The fluids are assumed to be incompressible. The model will be developed in steady-state flow. It is shown in Ref. 6 that the transient time for the start of flow is short for most practical problems and, thus, the rise of the cone can be represented as a succession of steady states. The fluids are assumed to flow in segregated regions, as shown in Fig. 1. The fractional flow into the perforated interval is assumed to be only a function of the fraction of the interval covered by each fluid and of the mobility ratio. The fluids are assumed to be in vertical equilibrium everywhere except near the wellbore. The departure from vertical equilibrium near the well caused by the vertical flow resistance is represented by an "effective radius." The expression for the effective radius represents the anisotropy through the vertical-to-horizontal permeability ratio. permeability ratio.The fluid flow equations are linearized by assuming that the average oil-column thickness over the drainage area can be used to compute the vertically averaged relative-permeability functions for the entire drainage area. SPEJ P. 65
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Introduction The semi-implicit reservoir simulator has become a very important part of the total simulation package necessary for the practicing reservoir engineer. Any multiphase simulation of a single well tom, problem (e.g., a study of water-oil coning) is very problem (e.g., a study of water-oil coning) is very expensive unless such a simulator is available. With a "standard" reservoir simulator, this difficulty arises because the relative permeabilities and capillary pressures, which depend upon saturations, lag one time step behind the pressure calculation. For this reason such a simulator is said to be implicit in pressure and explicit in saturation (abbreviated as IMPES). When "new" saturations are obtained, they are formed from "old" relative permeabilities and capillary pressures. The mathematical form of the equations is pressures. The mathematical form of the equations is such that an uncontrolled oscillation in the saturation values develops if the time step is too large. Only by taking smaller time steps can this oscillation be suppressed in an IMPES simulator, and very small time steps are then necessary for simulating coning behavior. The same problem can also appear in any production well model (in an IMPES simulator) that production well model (in an IMPES simulator) that distributes fluid production proportionally to phase mobilities. These saturation oscillations can be eliminated by making the well model "implicit in saturation." To overcome this instability, Blair and Weinaug developed a fully implicit simulator. All coefficients were updated iteratively until convergence occurred. It was necessary for them to stabilize their solution technique by the use of Newtonian iteration. Then it was found that a time-step limitation occurred because of nonlinearities, since the Newtonian iteration would not converge without a good initial estimate. Coats and MacDonald proposed an effective solution to this problem. They suggested estimating the relative permeabilities and capillary pressures by an extrapolation; e.g., pressures by an extrapolation; e.g., A mathematical investigation showed that since Sn + 1, the saturation at be new time step, is found simultaneously with the pressures as part of the solution, the mathematical time-step limitation inherent in the IMPES technique as a result of using "old" relative permeabilities would not occur. They also suggested that the equations be linearized by dropping products of (Sn+1 - Sn) and (pn+1 - pn). The equations are then more nearly linear. Hence, the difficulties in convergence of the solution technique are greatly reduced (Newtonian iteration is not needed). Nolen and Berry showed that linearization of the accumulation terms was not necessarily the best strategy (in problems that have solution gas), because material-balance errors would result. They felt that linearization of the flux terms made little difference. Many questions still remain unanswered by these papers. Nonlinearities remain in the equations, papers. Nonlinearities remain in the equations, particularly when a phase is near its immobile particularly when a phase is near its immobile saturation. Because of the use of upstream weighting of the relative permeabilities, another type of nonlinearity (potential reversal) can occur. Furthermore, the question of a practical procedure for selecting the time step must be settled. Finally, there are nonlinearities in the well model, which can cause slow convergence, or failure to converge, especially when dealing with a well completed in several layers or with a well that changes constraints. The problems just mentioned are all more severe if large time steps are used. Reducing time-step size is expensive, and in many cases difficult to automate. In this paper we present our experience in treating or circumventing these problems. We have felt that an important principle to follow is to eliminate time-step limitations due to mathematical instabilities. Thus, one should be able to run steady-state problems with essentially unlimited time-step size. For transient problems, it is expected that time truncation errors would normally govern the time-step size. One final, practical goal was to avoid running problems that would be annoying and mysterious to the field reservoir engineer. SPEJ P. 216