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q
w
= (k
w
/
w
) P
w
/ x (9.1)
q
nw
= (k
nw
/
nw
) P
nw
/ x (9.2)
where
w
and
nw
are viscosities, and k
w
and k
nw
are relative permeabilities,
the subscripts w and nw here denoting wetting and nonwetting phases. For
mathematical simplicity, we assume zero capillary pressures P
c
, so that
P
nw
- P
w
= P
c
= 0 (9.3)
For water injection problems, this assumes that the displacement is fast (or,
inertia dominated), so that surface tension can be neglected; however, when
water breakthrough occurs, the assumption breaks down locally. In formation
invasion, this zero capillary pressure assumption may be valid during the early
periods of invasion near the well, when high filtrate influx rates are possible, as
the resistance offered by mudcakes is minimal. For slow flows, capillary
pressure is important; but generally, fast and slow must be characterized
dimensionlessly in the context of the model. Since P
nw
= P
w
holds, the pressure
gradient terms in Equations 9.1 and 9.2 are identical. If we divide Equation 9.2
by Equation 9.1, these cancel and we obtain
q
nw
=(k
nw w
/k
w nw
)q
w
(9.4)
At this point, we invoke mass conservation, and assume for simplicity a constant
density, incompressible flow. Then, it follows that
q
w
/ x = - S
w
/ t (9.5)
q
nw
/ x=- S
nw
/ t (9.6)
where S
w
and S
nw
are the wetting and non-wetting saturations. Since the fluid
is incompressible, these saturations must sum to unity; that is,
S
w
+ S
nw
= 1 (9.7)
Then, upon adding Equations 9.5 and 9.6, and simplifying with Equation 9.7, it
follows that
(q
w
+ q
nw
)/ x = 0 (9.8)
Thus, we conclude that a one-dimensional, lineal, constant density flow without
capillary pressure admits the general total velocity integral
q
w
+q
nw
= q(t) (9.9)
where an arbitrary functional dependence on time is permitted. We have not yet
stated what q(t) is, or how it is to be determined; this crucial issue is discussed in
detail later. It is convenient to define the fractional flow function f
w
for the
wetting phase by the quotient
f
w
= q
w
/q
(9.10)
Then, for the nonwetting phase, we obtain
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