Numerical Simulation of Individual Wells in a Field Simulation Model
Authors: Akbar Ali M. , Arnold M.D. , Harvey A. Herbert 
Published: December 2007

Introduction
Pressures and fluid saturations in hydrocarbon reservoirs may be described at any point by differential equations involving reservoir rock and fluid properties. Numerical simulation of field performance is accomplished by establishing some performance is accomplished by establishing some type of reference grid, writing the appropriate equations for each mesh point, then solving the system of equations by a finite-difference technique. Since the number of mesh points must be finite, there is a necessary assumption that each mesh point is representative of a finite segment of the point is representative of a finite segment of the reservoir. Actually, however, pressures are not equal throughout such a segment of a producing field. This inequality of pressures within an element of the reference grid creates problems when the element contains a production or injection well. Since the finite-difference technique calculates a pressure that is representative of the entire element, pressure that is representative of the entire element, this pressure is not the bottom-hole pressure of the well. This situation will exist even though the well location may coincide with the grid point used to represent the element. Furthermore, this characteristic of the finite-difference approximation is not unique to the pressure calculation. Fluid saturations computed for a mesh point actually represent saturations of a finite segment of the reservoir. Fluid produced from the area of the wellbore is handled mathematically as if it were withdrawn from the entire area associated with a mesh point. Since the conventional finite-difference technique does not adequately describe reservoir conditions near a well, special mathematical techniques are required to handle the problem.

AVAILABLE TECHNIQUES
Several methods have been employed to predict well bottom-hole pressure for numerical simulation work. Attempts to use mesh-point pressure as bottom-hole pressure were generally unsatisfactory for reasons previously discussed. A more useful technique is to reduce permeability arbitrarily at mesh points corresponding to producing wells, thus obtaining mesh-point pressures that correspond to estimated bottom-hole pressures. It has also been suggested that it might be possible to represent pressure distributions by means of piecewise-polynomial pressure distributions by means of piecewise-polynomial approximations. The technique involves the use of high-order polynomials to represent the immediate vicinity of the wellbore, and lower-order polynomials to represent points more remote from polynomials to represent points more remote from the well.Another procedure that has been used with some success is to estimate bottom-hole pressure by extrapolating pressures from grid blocks adjacent to the block in which the well is located The extrapolation is based on Darcy's law written in radial form and integrated for steady - state conditions. The result of this integration may be written
(1)
where P is the average pressure on the edges of grid block, and r is the radius of a hypothetical circle with an area equal to that of the grid block.Although Eq. 1 is entirely adequate for estimating bottom-hole pressure in some instances, it can lead to erroneous results under unfavorable conditions. For example, a well with a large drawdown below bubble-point pressure may generate a high gas saturation in the vicinity of the borehole. This change in saturation reduces relative permeability to oil near the wellbore. Since the areal model does not account for this effect and the extrapolation technique uses the saturation computed by the areal model, this approach may predict a bottom-hole pressure that is too high. pressure that is too high. Another approach to the simulation of performance near a well has been described by MacDonald and Coats and by Letkeman and Ridings. Techniques described by those authors use radial coordinate grids to solve the problems of gas coning and water coning at individual wells.
SPEJ
P. 315