Calculation Of Flowing Well Pressures In Reservoir Simulation Using Nine-Point Differencing
Authors: Shiralkar Gautam S. 
Published: March 2010

Abstract
A quantity of considerable importance in flow rate predictions in reservoir simulation is the equivalent well block radius (r o) because it directly relates the numerically calculated grid block pressure to the actual well flowing pressure. Simulation of steam and miscible floods generally requires nine-point differencing. Previous work on approximating ro for nine-point differencing has either severe limitations or can give erroneous results and, hence, incorrect rate predictions. This paper presents, for the first time, a rigorous, closed form expression for ro for an isolated well, valid for any nine-point method. This expression is based on mathematical analysis and confirmed by careful numerical testing. It identically reduces to the expression for five-point differencing (obtained earlier by Peaceman) as a special case. Extension is made to the anisotropic case and confirmed numerically.
Results for some additional well-geometries are also presented for the case of nine-point differencing. These include: arbitrary location of a well in the well block, multiple wells within a single block, a well in an edge or comer block. Finally, effects of heterogeneity on predicting flow rates are considered.

Introduction
In the numerical simulation of reservoir flow, it is usually necessary to use grid blocks whose areal dimensions are much larger than the diameter of a well. Consequently, the numerically calculated well block pressure, po is substantially different from the actual flowing bottomhole pressure of the well, pwf) as shown in Figure 1. Well inflow models are used to relate the well block pressure to the actual wellbore pressure and flow rate through the concept of an equivalent well block radius(1) (ro)), defined by
Equation (1) Available In Full Paper.
Equation (1) can be expressed as a rate prediction equation:
Equation (2) Available In Full Paper.
Equation (1) indicates that ro, may be interpreted as the distance from the well at which the steady-state flowing pressure equals the numerically calculated well block pressure, as shown in Figure 1. Peaceman(1) showed that for an isolated vertical well centred in its grid block in a homogeneous and isotropic medium,
Equation (3) Available In Full Paper.
when conventional five-point differencing is employed over a square grid.
Peaceman obtained Equation (3) in two different ways. The first was by numerical experiments; the second by an approximate analytical approach that assumes that the pressures in the grid blocks immediately adjacent to the well block all satisfy the exact radial flow solution. For the square grid case, the first method yielded:
Equation (4) Available In Full Paper.
while the second method, which is henceforth referred to as the radial flow method, yielded
Equation (5) Available In Full Paper.
In a second paper, Peaceman(2) pointed out that the radial flow assumption is not generally valid and can result in significant error. For non-square (i.e. rectangular) grids, the radial flow method was found to predict:
Equation (6) Available In Full Paper.
whereas numerical experimentation showed that the correct expression is
Equation (7) Available In Full Paper.