Geology Reference
In-Depth Information
f
nw
= q
nw
/q = (q - q
w
)/q = 1 - f
w
(9.11)
where we used Equation 9.9. Equations 9.10 and 9.11 can be rewritten as
q
w
= q f
w
(9.12)
q
nw
= q ( 1 - f
w
)
(9.13)
Substituting into Equation 9.4, the function q(t) drops out, so that
1 - f
w
= ( k
nw
w
/k
w nw
) f
w
(9.14)
w
/
nw
) = 1/{1 + (k
nw
w
/k
w nw
)}
(9.15)
f
w
(S
w
,
The function f
w
(S
w
,
w
/
nw
) in Equation 9.15, we emphasize, is a function of
the
constant
viscosity ratio
w
/
nw
and the saturation
function
S
w
itself.
According to Equation 9.12, q
w
must likewise be a function of S
w
. Thus, we
can rewrite Equation 9.5 with the more informed nomenclature
S
w
/ t=-
-1
q
w
/ x
-1
q
=-
f
w
(S
w
,
w
/
nw
)/ x
-1
qdf
w
(S
w
,
=-
w
/
nw
)/S
w
S
w
/ x
(9.16)
or
S
w
/ t + {q(t)/ }df
w
(S
w
,
w
/
nw
)/dS
w
S
w
/ x = 0 (9.17)
Equation 9.17 is a first-order nonlinear partial differential equation for the
saturation S
w
(x,t). Its general solution can be easily constructed using concepts
from elementary calculus. The total differential dS
w
for the function S
w
(x,t)
can be written in the form
dS
w
=
S
w
/ t dt + S
w
/ x dx
(9.18)
If we divide Equation 9.18 by dt, we find that
dS
w
/dt =
S
w
/ t + dx/dt S
w
(9.19)
Comparison with Equation 9.17 shows that we can certainly set
dS
w
/dt = 0
(9.20)
provided
dx/dt = {q(t)/ }df
w
(S
w
,
w
/
nw
)/dS
w
(9.21)
Equation 9.20 states that the saturation S
w
is constant along a trajectory whose
speed is defined by Equation 9.21. (This constant may vary from trajectory to
trajectory.) In two-phase immiscible flows, we conclude that it is the
characteristic velocity dx/dt = {q(t)/
}df
w
(S
w
,
w
/
nw
)/dS
w
that is important,
and not the simple dx/dt = q(t)/ obtained for single-phase flow. But when
shocks form, it turns out that Equation 9.39 applies.
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