Page 399 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 334

2k h p − p wf,n )
π
( n
o
q = r (9.38)
n
µ ln h
oh
r w
which is a re-formulation of the steady state inflow equation presented in table 6.1. The
oil viscosity, µ oh is a function of the average temperature in the hot zone during the n th
time interval. The temperature will decline continuously during production as heat is
lost by conduction to the cap and base rock and by convection through the produced
fluids. A simple method for predicting the temperature decline allowing for both effects
9
has been presented by Boburg and Lantz . The pressure p in equ. (9.38) is the
n
average pressure during the time step at r h, the outer boundary of the hot zone, and
p wf,n the average wellbore pressure during the same period. If it is assumed that the
pressure declines at r h and r w can be approximated by a series of discrete pressure
steps then, in accordance with equ. (9.15), p and p wf,n can be expressed as
n

p = 1 2 (p n 1 + p n ) (9.39)
n
−
and

p wf,n = 1 2 (p wf,n 1 + p wf,n ) (9.40)
−

th
where p n and p wf,n are the pressures at r h and r w respectively at the end of the n time
step. The cumulative oil influx across the boundary r h by the end of the time step will be

N p,n = N p,n 1 + q . t
∆
n
−
th
in which N p,n-l is the known influx at the end of the (n - 1) time step. Using equs. (9.39)
and (9.40), the influx can be expressed as

α
N p,n = N p,n 1 + (p n 1 + p − p wf,n 1 − p wf,n ) (9.41)
n
−
2 − −
where

2k h t
π
∆
α = o r
µ oh ln r w h

An expression for the cumulative oil influx can also be obtained using the unsteady
state influx theory of Hurst and van Everdingen in the manner similar to that of
equ. (9.34)

n2
−
N p,n = U ∆ p W T − t j D ) + U p W T − t D n 1 ) (9.42)
∆
n 1
( D
D
( D
j
D
−
j0 −
=
in which T = n. ∆t and W D is the dimensionless influx function of Hurst and van
Everdingen, figs. 9.3-9.7. U is the aquifer constant defined in equ. (9.6), only in this
case applied to the oil reservoir for which f = 1, and the radius r o is replaced by r h.