Computer Simulation of Water Coning By a Sequential Solution Method
Authors: Ko S.C.M. , Aziz K. , Settari A. 
Published: March 2010

Abstract
A new two-phase (oil-water) coning model using a sequential solution (SEQ) approach was developed. The model employed the paint-distributed grid system and also incorporated both the outlet effect and compatibility condition in handling the wellbore/reservoir boundaries. The SEQ approach proposed in this study requires that oil pressure and water saturation be computed sequentially during a time step calculation.
Analysis of various treatments of the nonlinearities in the saturation equation showed that stability of the SEQ model was improved by using a linearized implicit chord slope method for the transmissibility and a linearized implicit tangent slope method for the capillary pressure. In addition, the nonlinear term containing the product of the linearized implicit transmissibility and capillary pressure was also dropped to enhance stability.
The new coning simulator referred to here as the SEQ model is stable with relatively small time step sensitivity and low grid size sensitivity in the radial direction. Simulation of coning problems, even with large capillary pressures, is possible with this model. Mathematical details of the SEQ model are also presented in this paper.

Introduction
Numerical simulation of well coning behaviour is probably one of the more difficult problems in reservoir simulation. This problem is characterized by the high flow velocities and high pore volume throughput in blocks near the wellbore resulting from the convergent flow to the well.
The limitation of using explicit transmissibilities in the difference equations for coning studies was recognized as early as 1964 by Welge and Weber. Shortly thereafter, fully implicit(2), linearized implicit (3,4) and semi-implicit(5) treatments of transmissibilities were proposed and successfully applied in coning simulations. In 1973, Settari and Aziz(6) showed that correct representation of wellbore/reservoir boundary conditions by including the outlet effect and the condition of compatibility could improve the stability of a coning model as well as reduce discretization errors.
All the aforementioned research efforts were mainly concerned with improvements in the stability of a numerical model. The resulting flow equations remained coupled and therefore were solved by the simultaneous solution (SS) method(1, 8). More recently, Spillette et al.(9) have reported the development of a high-stability sequential solution approach (SEQ) to solve the flow equations. This new technique was reported to yield the calculation stability previously associated only with the SS method. Furthermore, coning, gas percolation and other high-flow velocity problems have been claimed tractable through the use of semi-implicit saturation functions. No mathematical details were given in this paper.
The object of this study was to develop a suitable SEQ technique for two-phase, two-dimensional coning simulation. A model has been developed that incorporates the physically correct wellbore/reservoir boundary conditions which account for both the outlet effect and condition of compatibility. The mathematical and finite-difference formulation of the SEQ method are presented in this paper. The effect of various methods for the treatment of nonlinearities in the flow equations on the model behaviour is also discussed.

Mathematical Formulation

Governing Equations
The most suitable coordinate system for a coning model is the cylindrical (r, z, θ) system.