Page 67 - Principles of Applied Reservoir Simulation 2E

52 Principles of Applied Reservoir Simulation

The boundary conditions for the phase potentials are
Q (6.22)

and
0> 0 = 0 2 atx = L (6.23)

Capillary pressure is still neglected in this formulation. Equation (6.20) is the
analog of Eq. (6.3).
The solutions of the second-order ordinary differential equations Eqs,
(6.18) and (6.19) are the linear relationships
= (6.24)
® w A' wx+B' w
= A' 0x+Bi (6.25)
<S> 0
The coefficients are evaluated by substituting Eqs. (6.24) and (6.25) into Eqs.
(6.18) and (6.19) and applying the boundary conditions. The coefficients are

AL = - (6.26)
ML + (1 - M)x f

= MAI (6-27)
A' 0

*; = 0, (6.28)

(6.29)
The Darcy velocity of the water phase is

= =
v w ~d w ;~" ~A A (6.30)
dx
The velocity of frontal advance in a dipping reservoir is found by substituting
Eq. (6.30) into Eq. (6.15) to find

dt W-S or-S wc) ML+(l-M)x f ( ' }