Page 194 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 132

p , V 3
3
4
p , V 4
q 3
q
4

p , V 1
1
q 2
q
1
2
p , V 2

Fig. 5.3 Reservoir depletion under semi-steady state conditions.

(5.12)
q i ∝ V i
and hence the volume average in equ. (5.11) can be replaced by a rate average, as
follows

pq i
i
p res = i (5.13)
q i
i

and, whereas the V i's are difficult to determine in practice, the q i's are measured on a
routine basis throughout the lifetime of the field thus facilitating the calculation of p ,
res
which is the pressure at which the reservoir material balance is evaluated. The method
by which the individual p 's can be determined will be detailed in Chapter 7. sec. 7.
i
c) Steady State condition

q = constant

Pressure ∂ p
∂ t = 0 p = constant
e
fluid index
p wf
r r
r w e
Fig. 5.4 Radial flow under steady state conditions

The steady state condition applies, after the transient period, to a well draining a cell
which has a completely open outer boundary. It is assumed that, for a constant rate of
production, fluid withdrawal from the cell will be exactly balanced by fluid entry across
the open boundary and therefore,

p = p e = constant, at r = r e (5.14)

∂ p
and = 0 for all r and t (5.15)
t ∂