Page 226 - Fundamentals of Enhanced Oil and Gas Recovery

214 Mohammad Ali Ahmadi

Therefore, the correct solution is
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tq t df w
r f 5 r e 2 r 2 (7.39)
2
e
πhφ dS w f
In field unit, the expression is
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5:615tq t df w
r f 5 r e 2 r 2 (7.40)
2
e πhφ dS w f
There are other extensions, generalizations, and improvements to
Buckley Leverett theory developed to obtain and enhance understandings of the
complicated flow behavior of multiple phases in porous media. In particular, the
Buckley Leverett fractional flow theory has been generalized and applied by various
researchers to study enhanced oil recovery (EOR) [7], surfactant flooding [8], polymer
flooding [9], mechanisms of chemical methods [10], and alkaline flooding [11].
More recently, studies have extended the Buckley Leverett solution to flow in a
composite, one-dimensional heterogeneous, composite-reservoir system [12], to non-
Newtonian fluid flow [13 17] and the non-Darcy displacement of immiscible fluids
in porous media [18 22]. Fundamentals of the physics of flow of multiphase fluids in
porous media have been understood through laboratory experiments, theoretical anal-
ysis, mathematical modeling, and field studies [23 25]. Analysis of porous medium
flow processes relies traditionally on Darcy’s law-based approaches, and application of
such analysis has provided quantitative methodologies and modeling tools for many
related scientific and engineering disciplines.
Fayers and Sheldon [26] described the Frontal advance theory as an application of
the law of conservation of mass. Flow through a small volume element with length
Δx and cross-sectional area “A” can be expressed in terms of total flow rate q t as
(7.41)
q t 5 q w 1 q o
q w 5 q t 3 f w (7.42)

q o 5 q t 3 f w 5 q t 3 ð1 2 f w Þ (7.43)
where q denotes volumetric flow rate at reservoir conditions and subscripts {o,w,t}
refer to oil, water, and total rate, respectively, and f w and f o are fractional flow to
water and oil (or water cut and oil cut), respectively.

kk ro @ AP o Þ
ð
q o 5 (7.44)
μ @r
o
kk rw @ AP w Þ
ð
q w 5 (7.45)
μ @r
w