Page 184 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 122

k/ µ w = M ≤ 1
′
rw
′
k/ µ o
ro
where M is known as the end point mobility ratio and, since both and k′ are k′ the end
rw
ro
point relative permeabilities, is a constant. If M ≤ 1 it means that, under an imposed
pressure differential, the oil is capable of travelling with a velocity equal to, or greater
than, that of the water. Since it is the water which is pushing the oil, there is therefore,
no tendency for the oil to be by-passed which results in the sharp interface between
the fluids.

The displacement shown in fig. 4.10(a) is, for obvious reasons, called "piston-like
displacement". Its most attractive feature is that the total amount of oil that can be
recovered from a linear reservoir block will be obtained by the injection of the same
volume of water. This is called the movable oil volume where,

1 (MOV) = PV(1 – S or − S wc)

The non-ideal displacement depicted in fig. 4.10(b), which unfortunately is more
common in nature, occurs when M > 1. In this case, the water is capable of travelling
faster than the oil and, as the water pushes the oil through the reservoir, the latter will
be by-passed. Water tongues develop leading to the unfavourable water saturation
profile.

Ahead of the water front oil is again flowing in the presence of connate water. This is
followed, in many cases, by a waterflood front, or shock front, in which there is a
discontinuity in the water saturation. There is then a gradual transition between the
shock front saturation and the maximum saturation S w = 1−S or. The dashed line in
fig. 4.10(b) depicts the saturation distribution at the time when the shock front breaks
through into the producing well (breakthrough). In contrast to the piston-like
displacement, not all of the movable oil will have been recovered at this time. As more
water is injected, the plane of maximum water saturation (S w = 1−S or) will move slowly
through the reservoir until it reaches the producing well at which time the movable oil
volume has been recovered. Unfortunately, in typical cases it may take five or six
MOV's of injected water to displace the one MOV of oil (as will be demonstrated in
exercises 10.2 and 10.3 of Chapter 10). At a constant rate of water injection, the fact
that much more water must be injected, in the unfavourable case, protracts the time
scale attached to the oil recovery and this is economically unfavourable. In addition,
pockets of by-passed oil are created which may never be recovered.

Mobility control

If the end point mobility ratio for water displacing oil is unfavourable, the injection
project can be engineered to overcome this difficulty. The manner in which this is done
can be appreciated by considering the general expression

Mobility of the displacing fluid k/ µ
′
M = = rd d (4.36)
Mobility of the displaced fluid k/ µ o
′
ro