Page 137 - Fundamentals of Reservoir Engineering

MATERIAL BALANCE APPLIED TO OIL RESERVOIRS 76

reservoir. In the case of an oil reservoir, however, the situation is generally more
complex since below the bubble point two phases, oil and gas, will co-exist and,
due to the gravity difference between the phases, will tend to segregate. As a
result, the point at which the average pressure should be determined will vary
with time. Precisely how the volume averaged reservoir pressure can be
determined from the analysis of pressure tests in wells will be detailed in
Chapter 7.

3.3 THE MATERIAL BALANCE EXPRESSED AS A LINEAR EQUATION

Since the advent of sophisticated numerical reservoir simulation techniques, the
Schilthuis material balance equation has been regarded by many engineers as being of
historical interest only; a technique used back in the nineteen forties and fifties when
people still used slide-rules. It is therefore interesting to note that as late as 1963-4,
Havlena and Odeh presented two of the most interesting papers ever published on the
subject of applying the material balance equation and interpreting the results. Their
4,5
papers, described the technique of interpreting the material balance as the equation
of a straight line, the first paper describing the technique and the second illustrating the
application to reservoir case histories.

To express equ. (3.7) in the way presented by Havlena and Odeh requires the
definition of the following terms

F = N p (B o + (R p R s) B g) + W p B w (rb) (3.8)

which is the underground withdrawal;

E o = (B o − B oi) + (R si − R s) B g (rb/stb) (3.9)

which is the term describing the expansion of the oil and its originally dissolved gas;

B
E = B oi g − 1 (rb / stb) (3.10)
g
B gi

describing the expansion of the gascap gas, and

cS + c
E f,w = (1 m) B oi w wc f ∆ p (rb / stb) (3.11)
+
1 − S wc
for the expansion of the connate water and reduction in the pore volume. Using these
terms the material balance equation can be written as

(3.12)
F = N(E o+mE g+E f,w) + W eB w

Havlena and Odeh have shown that in many cases equ. (3.12) can be interpreted as a
linear function. For instance, in the case of a reservoir which has no initial gascap,
negligible water influx and for which the connate water and rock compressibility term
may be neglected; the equation can be reduced to