A Fully Implicit Three-Dimensional Model in Curvilinear Coordinates
Authors: Sonier F. , Chaumet P. 
Published: December 2007

Abstract
Because of saturation instabilities, conventional multiphase three-dimensional simulators cannot use large time-step sizes if grid blocks are small. Therefore, to be economical, such models do not use a fine grid near wells, and they are inadequate for accurately describing reservoir phenomena around producing wells. Only radial circular models can be used inexpensively to describe coning behavior. However, they are not always reliable because there usually is no axial symmetry around producing wells. producing wells. We propose a three-dimensional reservoir model capable of simultaneously describing flows at a distance from and around wells in a part of an oil reservoir subject to lateral drive from an aquifer or a gas cap. The model uses an asymmetrical curvilinear grid whose mesh sizes increase with distance from the wells. It is a fully implicit model that can use large time-step sizes. A convergence acceleration device is used for equation solving. The results of this model have been checked against those obtained with a large laboratory model. Wells were in a line pattern and in a staggered one. pattern and in a staggered one

Introduction
In an oil reservoir subject to lateral drive from an aquifer or a gas cap, no axial symmetry exists around the producing wells. As a result of such production and of lateral drive, the transition zone becomes deformed and, in particular, the gas moves along the top of the layer toward the wells. Pressures (especially if the problem is with imposed Pressures (especially if the problem is with imposed bottom-hole pressures) and coning phenomena near the wells in such transition zones is very difficult to describe with existing numerical reservoir models. The grids and numerical methods usually used for this are not properly adapted to solve the problem.

FROM THE STANDPOINT OF PATTERN
A circular radial coning model is not representative. Also, a conventional three-dimensional model with parallelepipedic grids has the disadvantage of using large blocks corresponding to producing wells (this will be called well blocks in the remainder of the paper). The average pressure and saturation values are quite different from the actual well values since it is in the vicinity of these wells that pressures and saturation vary the most in space. One way of solving this would be to adopt a highly irregular grid with small cells in the vicinity of the wells. The cost of simulation with such a system would be very high because of the excessive number of grids, including some perfectly useless ones in distant zones with low gradients.

FROM THE NUMERICAL STANDPOINT
From the numerical standpoint a conventional model of the type described above using the finite-difference technique and "implicit-pressure explicit-saturation" concept requires time steps that become smaller as mesh size decreases. This is an additional reason for trying to avoid any tightening up of the grid. As a result, while maintaining a relatively loose grid, some authors have tried to find special processing methods for the individual points made up by the wells:arbitrarily reducing permeability in the well blocks to adjust bottom-hole pressures,approximating pressures by polynomial functions of a high order pressures by polynomial functions of a high order in the vicinity of wells and of a lower order elsewhere,estimating bottom-hole pressure by extrapolation (based on Darcy's law written in radial form and integrated for steady-state conditions) from grid blocks adjacent to the well block, andinserting a radial-circular coning model in well blocks. (This last solution is not a good one because of the unsatisfied symmetry conditions and assumptions on the linking of both models.)
We propose here a three-dimensional reservoir model capable of simultaneously describing flows at a distance and in the vicinity of wells. This model is discretized according to a curvilinear grid in the plane of the layer, enabling small meshes to be formed around the well and large meshes in zones farther away without creating any difficulties of linking between meshes.
SPEJ
P. 361