Page 60 - Principles of Catalyst Development
P. 60
Part I: Reservoir Engineering Primer 41
In the limit as A/ -> 0 and Ax -* 0, we pass to the differential form of Eq. (5.5)
for the water phase:
dt -4(j) dx
A similar equation applies to the oil phase:
(5.7)
dt A$ dx
Since/, depends only on S w, we can write the derivative of fractional flow as
=
"a7 Ist "a7 (5 8)
'
Substituting dfjdx into dSJdx yields
dS w -q t df w
(5,9)
dt A dS dx
It is not possible to solve for the general distribution of water saturation S^x,
t) in most realistic cases because of the nonlinearity of the problem. For example,
water fractional flow is usually a nonlinear function of water saturation. It is
therefore necessary to consider a simplified approach to solving Eq. (5.9).
We begin by considering the total differential of S w(x, t):
w w QX w
(5.10)
dt dx dt dt
Equation (5.10) can be simplified by choosing x to coincide with a surface of
fixed S w so that dSJdt = 0 and
( dS.}
dx\
_____ I — _ ( \ dt / ) ff 1 1 \
.J / \ (5.H)
dx
Substituting Eqs. (5.8) and (5.9) into Eq. (5. II) gives the Buckley-Leverett
frontal advance equation:
