Page 220 - Fundamentals of Enhanced Oil and Gas Recovery

208 Mohammad Ali Ahmadi

water from left to right. The rate the water enters to the medium element from
left-hand side is
q t 3 f w 5 water flow rate entering the element
The rate of water leaving element from the right-hand side is

q t 3 f w 1 Δf w 5 water flow rate leaving the element
The change in water flow rate across the element is found by performing a mass
balance. The movement of mass for an immiscible, incompressible system gives

Change in water flow rate 5 water entering 2 water leaving
(7.1)

5 q t 3 f w 2 q t f w 1 Δf w 52 q t 3 Δf w
This is equal to the change in element water content per unit time (see Fig. 7.1).
Let S w be the water saturation of the element at time t. Then, if oil is being displaced
from the element, at the time (t 1 Δt), the water saturation will be (S w 1 ΔS w ).
Therefore, water accumulation in the element per unit time is
ΔS w 3 A 3 φ 3 Δx
water accummulation per unit time 5 (7.2)
Δt
where φ is porosity; equating Eqs. (7.1) and (7.2) results
ΔS w 3 A 3 φ 3 Δx ΔS w 2 q t 3 Δf w
52 q t 3 Δf w - 5 (7.3)
Δt Δt A 3 φ 3 Δx

In the limit as Δt - 0 and Δx - 0 (for the water phase):

ΔS w 2 q t df w
5 (7.4)
Δt A 3 φ dx
x t
The subscript x on the derivative indicates that this derivative is different for each
element. It is not possible to solve for the general distribution of water saturation
S w (x, t) in most realistic cases because of the nonlinearity of the problem. For exam-
ple, water fractional flow is usually a nonlinear function of water saturation. It is,
therefore, necessary to consider a simplified approach to solving Eq. (7.4).

Figure 7.1 Horizontal bed containing oil and water.