Page 400 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 335

Values of the pressure drop ∆p j at the hot zone boundary can be evaluated using
equ. (9.16) as

∆ p = 1 2 (p j 1 − p j 1 ) (9.16)
i
+
−
and in particular

∆ p n1 = 1 2 (p n 2 − p n )
−
−
therefore, equ. (9.42) can be written as

n2 β
−
N p,n = U ∆ p W T − t j D ) + (p n 2 − p n ) (9.43)
D
j
( D
−
j0 2
=
)
where β = U W D (T D−t D
n -1
Equations (9.41) and (9.43) can now be equated and solved explicitly for the boundary
th
pressure p n at r h at the end of the n time step, i.e.
1 n2
−
p ∆ p W T − t ) − 2N + β p + α (p + p − p ) (9.44)
2U
−
−
−
−
n= j D ( D j D p,n 1 n 2 wf,n 1 wf,n n 1
(α + β ) j0
=
Providing that the manner in which the bottom hole flowing pressure will be allowed to
decline is specified, then everything on the right hand side of equ. (9.44) is determined,
thus p n can be calculated and subsequently p using equ. (9.39). Finally, substituting
n
this value of p in equ.(9.38) will lead to the determination of q . In fact, it is necessary
n
n
to iterate on this oil rate to correctly model the heat loss due to convection, since, as
mentioned already, the temperature in the hot zone, which determines the oil viscosity
in equ. (9.38), is reduced due to the removal of heat by the produced oil. This
convective heat loss is directly proportional to the production rate and therefore, the oil
rate and viscosity are dependent upon one another necessitating the iterative solution
technique.
10
Bentsen and Donohue have reported the use of the above technique incorporated in
a dynamic programming model for optimising steam soak operations.

REFERENCES

1) van Everdingen, A.F. and Hurst, W., 1949.TheApplication of the Laplace
Transformation to Flow Problems in Reservoirs. Trans. AIME. 186: 305-324.

2) Fetkovitch, M.J., 1971. A Simplified Approach to Water Influx Calculations-Finite
Aquifer Systems. J.Pet.Tech., July: 814-828.

3) van Everdingen, A.F., Timmerman, E.H. and McMahon, J.J., 1953. Application of
the Material Balance Equation to a Partial Water-Drive Reservoir. Trans. AIME.
198:51.

4) Havlena, D. and Odeh, A.S., 1963. The Material Balance as an Equation of a
Straight Line. J.Pet.Techn, August: 896-900. Trans. AIME.