Two-Dimensional Radial Treatment of Wells Within a Three-Dimensional Reservoir Model
Authors: Mrosovsky Ivan , Ridings R.L. 
Published: December 2007

Abstract
A reservoir simulator is described in which two-dimensional cylindrical models are used for individual wells within the framework of a three-dimensional, rectangular-grid, reservoir model. This approach to simulation should give more realistic results than those incorporating simpler well production routines. Application should be production routines. Application should be particularly pertinent to cases in which gas and particularly pertinent to cases in which gas and water coning are important. Computation cost is increased by this sophisticated treatment of wells, but total running costs are nevertheless low enough that judicious use of the model is warranted.

Introduction
Changes in pressure and fluid saturations occurring in a reservoir on production may be described by certain well known differential equations. These equations can be solved by finite-difference techniques. To this end a reservoir may be divided, more or less regularly, into a number of blocks. We consider here a three-dimensional model. A rectangular grid is superimposed on the reservoir, which is further divided vertically into several layers. This kind of subdivision is generally adequate for treating most of a reservoir. However, difficulty arises in the neighborhood of a well. Fluid saturations and pressures typically exhibit steep gradients in the neighborhood of a producing well. Moreover, these properties have values near the well that are markedly different from those prevailing over most of the well's drainage area. To prevailing over most of the well's drainage area. To model this situation using finite-difference methods, a fine grid would be necessary in the vicinity of the well. To extend such a fine grid throughout a reservoir, with many wells, would result in an impracticably large number of blocks. Instead, it is customary to simplify and approximate the calculation of well performance. A common recourse is to use a radial flow formula such as(1)
Here the external radius, re, is set so that the area of a circle of this radius should equal the area of the block in which the well is situated. Obviously, this formula takes no account of the radial variation of fluid saturations. In this paper a different approach to the calculation of well performance is described. Each well in the three-dimensional (3D) reservoir model is also assigned a two-dimensional (2D) radial model covering its immediate vicinity. These radial models are solved simultaneously with the 3D reservoir model. Thus, precision is concentrated where it is most needed - near the wells; yet the 3D reservoir model is not blown up unduly. The equations and method of solution for the 3D model are briefly sketched in the Appendix. The 2D radial models are basically the same as that described by Letkeman and Ridings. We now discuss some details of the radial well models. In particular we shall be concerned with the interface particular we shall be concerned with the interface between the 2D radial well models and the 3D reservoir model.

RADIAL WELL MODEL
To make the discussion simpler and more concrete we take, as an example, a reservoir model we have run. This 3D reservoir model had 15 layers. Consider a well situated in one vertical column of 15 blocks. The 2D radial model for this well also has 15 layers. Each layer in the 2D model is further split radially into six concentric blocks. The relationship between the two models is depicted in Fig. 1. The total pore volume in any layer of the 2D model must equal the pore volume of the corresponding block in the same pore volume of the corresponding block in the same layer of the 3D model; that is, the block through which the well passes. Consider now the calculation of one time step, say 30 days, for the 3D reservoir model. To make this calculation, we require a well production rate. Actually we need three rates: one each for oil, water, and gas. These rates are obtained by solving the 2D model over the same 30-day period.
SPEJ
P. 127