Critical Rate for Water Coning: Correlation and Analytical Solution

Abstract

Summary Two methods are presented for predicting critical oil rate for bottomwater coning in anisotropic, homogeneous formations with the well completed from the top of the formation. The first method is based on an analytical solution where Muskat's assumption of uniform flux at the wellbore has been replaced by that of an infinitely conductive wellbore. The potential distribution in the oil zone, however, is assumed unperturbed by the water cone. The method is derived from a general solution of the time-depedent diffusivity equation for compressible, single-phase flow in the steady-state limit. We show that very little difference exists between our solution and Muskat's. The deviation from simulation results is caused by the cone influence on potential distribution. The second method is based on a large number of simulation runs with a genral numerical reservoir model, with more than 50 critical rates determined. The results are combined in an equation for the isotropic case and in a single diagram for the anisotropic case. The correlation is valid for dimensionless radii between 0.5 and 50 and shows a rapid change in critical rate for values below five. Within the accuracy of numerical modeling results, Wheatley's theory is shown to predict the correct critical rates closely for all well penetrations in the dimensionless radius range from 2 to 50. Introduction Oil production from a well that partly penetrates an oil zone overlying water may cause the oil/water interface to deform into a bell shape. This deformation is usually called water coning and occurs when the vertical component of the viscous force exceeds the net gravity force. At a certain production rate, the water cone is stable with its apex at a distance below the bottom of the well, but an infinitesimal rate increase will cause cone instability and water breakthrough. This limiting rate is called the critical rate for water coning. Muskat and Wyckoff1 presented an approximate solution of the water-coning problem. For an istropic reservoir, the critical rate may be estimated from a graph in their work. Their solution is based on the following three assumptions:the single-phase (oil) potential distribution around the well at steady-state conditions is given by the solution of Laplace's equation for incompressible fluid;a uniform-flux boundary condition exists at the well, giving a varying well potential with depth; andthe potential contribution in the oil phase is not influenced by the cone shape. Meyer and Garder2 simplified the analytical derivation by assuming radial flow and that the critical rate is determined when the water cone touches the bottom of the well. Chaney et al.3 included completions at any depth in a homogeneous, isotropic reservoir. Their results are based on mathematical analysis and potentiometric model techniques. Chierici et al.4 used a potentiometric model and included both gas and water coning. The results are presented in dimensionless graphs that take into account reservoir anisotropy. Also, Muskat and Wyckoff's Assumption 2 is eliminated because the well was represented by an electric conductor. The graphs are developed for dimensionless radii down to five. For thick reservoirs with low ratios between vertical and horizontal permeability, however, dimensionless radii below five are required. Schols5 derived an empirical expression for the critical rate for water coning from experiments on Hele-Shaw models. Recently, Wheatley6 presented an approximate theory for oil/water coning of incompressible fluids in a stable cone situation. Through physical arguments, he postulated a potential function containing a linear combination of line and point sources with three adjustable parameters. The function satisfies Laplace's equation, and by properly adjusting the parameters. Wheatley was able to satisfy the boundary conditions closely, including that of constant well potential. Most important, his theory is the first to take into account the cone shape by requiring the cone surface---i.e., the oil/water interface---to be a streamle. Included in his paper is a fairly simple procedure for predicting critical rate as a function of dimensionless radius and well penetration for general anisotropic formations. Because of the scarcity of published data on correct critical rates, the precision of his theory is insufficiently documented. Although each practical well problem may be treated individually by numerical simulation, there is a need for correlations in large-gridblock simulators7 and for quick, reliable estimates of coning behavior.8 This paper presentsan analytical solution that removes Assumptions 1 and 2 in Muskat and Wyckoff's1 theory;practical correlations to predict critical rate for water coning based on a large number of simulation runs with a general numerical reservoir model; anda verification of the predictability of Wheatley's theory. All results are limited to a well perforated from the top of the formation. Analytical Solution The analytical solution presented in this paper is an extension of Muskat and Wyckoff's1 theory and is based on the work of Papatzacos.9,10 Papatzacos developed a general, time-dependent solution of the diffusivity equation for flow of a slightly compressible, single-phase fluid toward an infinitely conductive well in an infinite reservoir. In the steady-state limit, the solution takes a simple form and is combined with the method of images to give the boundary conditions, both vertically and laterally, as shown in Fig. 1 (See the appendix for details). To predict the critical rate, we superimpose the same criteria as those of Muskat and Wyckoff1 on the single-phase solution and therefore neglect the influence of cone shape on the potential distribution. A computer program was developed to give the critical rate in a constant-pressure square from Eqs. A-6 through A-13. The length of the square was transformed to an equivalent radius for a constant-pressure circle11 to conform with the geometry of Fig. 1 and the simulation cases. The results of the analytical solution are presented in Fig. 2, where critical rate, qcD, is plotted vs. dimensionless radius, rD, for five fractional well penetrations, Lp/ht, with the definitionsEquation (1)And Equation (2) Numerical Simulation The critical rate was determined for a wide range of reservoir and well parameters by a numerical reservoir model. The purpose was to check the validity of the analytical solutions and to develop separate pratical correlations valid to a low dimensionless radius. A summary is presented here; Ref. 12 gives the details. The numerical model used is a standard, three-phase, black-oil model with finite-difference formulation developed at Rogaland Research Inst. The validity of the model has been extensively tested. It is fully implicit with simultaneous and direct solution and therefore suitable for coning studies.