Page 153 - Fundamentals of Reservoir Engineering

MATERIAL BALANCE APPLIED TO OIL RESERVOIRS 92

or

W e = (c w+c f) W i ∆p (3.25)

in which the total aquifer compressibility is the direct sum of the water and pore
compressibilities since the pore space is entirely saturated with water. The sum of c w
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and c f is usually very small, say 10 /psi, therefore, unless the volume of water W i is
very large the influx into the reservoir will be relatively small and its influence as a drive
mechanism will be negligible. If the aquifer is large, however, equ. (3.25) will be
inadequate to describe the water influx. This is because the equation implies that the
pressure drop ∆p, which is in fact the pressure drop at the reservoir boundary, is
instantaneously transmitted throughout the aquifer. This will be a reasonable
assumption only if the dimensions of the aquifer are of the same order of magnitude as
the reservoir itself. For a very large aquifer there will be a time lag between the
pressure change in the reservoir and the full response of the aquifer. In this respect
natural water drive is time dependent. If the reservoir fluids are produced too quickly,
the aquifer will never have a chance to "catch up" and therefore the water influx, and
hence the degree of pressure maintenance, will be smaller than if the reservoir were
produced at a lower rate. To account for this time dependence in water influx
calculations requires a knowledge of fluid flow equations and the subject will therefore
be deferred until Chapter 9, in which a full description of the phenomenon is provided.
For the moment, the simple equation (3.25) will be used to illustrate the influence of
water influx in the material balance.

Using the technique of Havlena and Odeh (assuming that B w = 1), the full material
balance can be expressed as

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

in which the term E f,w, equ. (3.11), can frequently be neglected when dealing with a
water influx. This is not only for the usual reason that the water and pore
compressibilities are small but also because a water influx helps to maintain the
reservoir pressure and therefore, the ∆p appearing in the E f,w term is reduced.

This is a point which should be checked at the start of any material balance calculation
(refer exercise 9.2). If, in addition, the reservoir has no initial gascap then equ. (3.12)
can be reduced to

F = NE o + W e (3.26)
In attempting to use this equation to match the production and pressure history of a
reservoir, the greatest uncertainty is always the determination of the water influx W e. In
fact, in order to calculate the influx the engineer is confronted with what is inherently
the greatest uncertainty in the whole subject of reservoir engineering. The reason is
that the calculation of W e requires a mathematical model which itself relies on the
knowledge of aquifer properties. These, however, are seldom measured since wells
are not deliberately drilled into the aquifer to obtain such information. For instance,
suppose the influx could be described using the simple model presented as equ.
(3.25). Then, if the aquifer shape is radial, the water influx can be calculated as