A Model of Oil-Water Coning for Two-Dimensional, Areal Reservoir Simulation
Authors: Chappelear J.E. , Hirasaki G.J. 
Published: December 2007

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