A Numerical Model of Multiphase Flow Around a Well
Authors: Sonier F. , Ombret O. 
Published: December 2007

Abstract
This paper describes a two-dimensional three-phase numerical model for simulating two- or three-phase coning behavior. The model is fully implicit with respect to all variables and uses the simultaneous solution of the different equations describing multiphase flow. For determining well flow rates from all blocks communicating with the well, particular attention has been paid to the well boundary condition, which is considered to be a physical boundary. The mathematical expression of these well conditions enables flow rates to be calculated in a perfectly implicit manner and thus makes the model very stable so that the computational error in time is very small. The model described is appreciably different in this respect from previous models in which the well is represented by source points and in which the flour terms are calculated by using various simplifications. The results of several tests are presented. The model was checked by the simulation of several water coning cases that had previously been studied on a physical model. Four examples are given here. In these examples, the boundary influx conditions and fluid mobility ratio are made to vary. One of them illustrates the care that must be taken when using simplified solution schemes for the boundary conditions.

Introduction
Multiphase numerical models have usually employed finite-difference approximations in which relative permeabilities are evaluated explicitly at the beginning of each time step. But simulators of this type are not capable of solving problems characterized by high flow velocities and such phenomena as well coning, except perhaps at phenomena as well coning, except perhaps at extremely high cost. Recently, some papers were published describing a method that employs semi-implicit relative permeabilities and uses the simultaneous solution of multiphase equations. This method is very efficient. In these simulators, the well is represented by source points, and flow rate terms are calculated by using various simplifications (mobility or potential methods). potential methods). This paper describes a new numerical coning model. The numerical part of the model is similar to that in the latest models, but its representation of wellbore conditions is quite different and more nearly expresses physical phenomena caused by end effects. The well is represented full-scale and not by source points. Furthermore, so as not to partially screen out wellbore conditions, the partially screen out wellbore conditions, the producing interval, even if it is small, may be producing interval, even if it is small, may be advantageously represented by several layers. Any condition may be specified for the external boundaries. All the leading physical parameters are treated semi-implicitly. When a flow rate is imposed on the well, taking into account the well-wall boundary conditions, the calculation of production terms is fully implicit. This calculation is iterative, but at almost each time step a simple algorithm enables a direct solution to be obtained. The results of numerous simulations are presented. Studies on physical models have demonstrated the full validity of the numerical model. The simulation of actual field cases shows that the model is very efficient.