Page 198 - Fundamentals of Reservoir Engineering

CHAPTER 6

WELL INFLOW EQUATIONS FOR STABILIZED FLOW CONDITIONS

6.1 INTRODUCTION

In this chapter solutions of the radial diffusivity equation, for liquid flow, will be sought
under stabilized flow conditions. These have already been defined in the previous
chapter as semi-steady state and steady state for which the time derivative in
equ. (5.20) is constant and zero, respectively. The solution technique for semi-steady
state flow is set out in some detail since the method is a perfectly general one which
can be applied for a variety of radial flow problems. Finally, the constraint that the outer
boundary of the cell must be radial is removed by the introduction of Dietz shape
factors. This allows a general form of inflow equation to be developed for a wide range
of geometries of the drainage area and positions of the well within the boundary.

6.2 SEMI-STEADY STATE SOLUTION

The radial diffusivity equation, (5.20), will be solved under semi-steady state flow
conditions for the geometry and radial pressure distribution shown in fig. 6.1.

q = constant

Pressure p e

h
p wf

r
r w r e

Fig. 6.1 Pressure distribution and geometry appropriate for the solution of the radial
diffusivity equation under semi-state conditions

At the time when the solution is being sought the volume averaged pressure within the
cell is p which can be calculated from the following simple material balance

cV (p i − p) = qt (6.1)

in which V is the pore volume of the radial cell, q is the constant production rate and t
the total flowing time. The corresponding boundary pressures at the time of solution
are p e at r e and p wf at r w. For the drainage of a radial volume cell, the semi-steady state
condition was derived in the previous chapter as

∂ p q
=− (5.10)
2
π
r ∂ cr hφ
e
which, when substituted in the radial diffusivity equation, (5.20), gives