Page 154 - Fundamentals of Reservoir Engineering

MATERIAL BALANCE APPLIED TO OIL RESERVOIRS 93

2
2
W = (c + c ) (r − r )fhφ ∆ p (3.27)
π
e
f
e
o
w
in which r e and r o are the radii of the aquifer and reservoir, respectively, and f is the
fractional encroachment angle which is either Θ/2π or Θ/360°, depending on whether Θ
is expressed in radians or degrees. It should be realised that the only term in
equ. (3.27) which is known with any degree of certainty is π! The remaining terms all
carry a high degree of uncertainty. For instance, what is the correct value of r e? Is the
aquifer continuous for 20 kilometers or is it truncated by faulting? What is the correct
value of h, the average thickness of the aquifer or φ, the porosity? These can only be
estimated, based on the values determined in the oil reservoir. For such reasons,
building a correct aquifer model to match the production and pressure data of the
reservoir is always done on a "try it and see" basis and even when a satisfactory model
has been achieved it is seldom if ever, unique. Therefore, the most appropriate way of
applying equ. (3.26) is by expressing it as

F W
= N + e (3.28)
E o E o

and plotting F/E o, corresponding to the observed production, versus W e/E o, where We
is calculated using an aquifer model such as equ. (3.27).

w - too small
e
w - correct
e
incorrect geometry
F
w - too large
E o e
(stb)

45°

N

w / E (stb)
e
o
Fig. 3.9 Trial and error method of determining the correct aquifer model
(Havlena and Odeh)

This model is linked to the reservoir by the pressure drop term ∆p which is interpreted
as the pressure drop at the original reservoir-aquifer boundary, and is normally
assumed to be equal to the average pressure drop in the reservoir due to the
production of fluids. If the aquifer model is incorrect, the plotted data points will deviate
from the theoretical straight line which has a slope of 45° and intercept N, when
W e/E o = 0, as shown in fig. 3.9.

The deviation labelled as being due to using the wrong geometry means that radial
geometry has been assumed whereas linear geometry would probably be more
appropriate. With radial geometry there is a larger body of water in close proximity to