Page 64 - Principles of Applied Reservoir Simulation 2E

Part I: Reservoir Engineering Primer 49

3
f = 0, x f < x < L (62)
ox

Equations (6.1) and (6.2) apply to those parts of the medium containing water
and oil respectively. They assume that the fluids are incompressible, and that
the oil-water interface is a piston-like displacement in the ^-direction. The piston-
like displacement assumption implies a discontinuous change from mobile oil
to mobile water at the displacement front. This concept differs from the Buckley-
Leverett analysis presented in Chapter 5. Buckley-Leverett theory with Welge's
method shows the existence of a transition zone as saturations grade from mobile
oil to mobile water. The saturation profile at the interface between the immiscible
phases depends on the fractional flow characteristics of the system. The present
method has less structure in the saturation profile, but is more readily suited for
analyzing the stability of the displacement front.
Boundary conditions at the displacement front are given by continuity of
phase pressure
x
P 0 = P w at = x f ( t ) (6.3)

and continuity of phase velocity

(64)

where A.J is the mobility of phase 0. Equation (6.3) is valid when we neglect
capillary pressure, and the effect of gravity has been excluded from Eq. (6.4).
The exclusion of gravity corresponds physically to flow in a horizontal medium.
Boundary conditions at the edges of the porous medium are
P w=P latx=0 (6.5)
and
= at*= L (6.6)
P 0 P 2
Equations (6.1) through (6.6) may be solved analytically. We begin by
integrating Eqs. (6.1) and (6.2) to find the general solutions
P w = A wx+B w (6.7)
and