Page 220 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

194 Bin Yuan and Rouzbeh G. Moghanloo

@G @F 1 S w
1 5 0; G 5 ; F 52 (4.14)
@x D @ϕ f w f w

In the analytical solutions using MOC, the dynamic relations between
suspended nanoparticles and suspended fines concentration can be charac-
terized along two different characteristic directions, i.e., a slow path and a
fast path, as shown in Fig. 4.8. With the definition of initial and injection
(boundary) conditions, both slow path and fast path, where the nanopar-
ticles and fines concentrations vary, can be uniquely determined accord-
ing to the Rankine-Hugoniot condition and concept of discontinuity
(Lake, 1989; Hankins, 2004). After the suspended and retained fine parti-
cles concentrations are obtained from the above analysis, the characteristic
form of the lifting equation (Eq. (4.14)) can be expressed as follows:
dϕ @F f w S w ; σ a ; C FP Þ 2 S w
ð
@f w S w ;σ a ;C FP Þ
ð
5 σ 2 5 5 @S w (4.15a)
@G @f w S w ;σ a ;C FP Þ
ð
dx D 2
@S w

@f w @σ a 2 @σ a 1 @f w @C FP 2 @C FP
S w S w
dS w @σ a @ϕ @x D @C FP @ϕ @x D
5 (4.15b)
@f w
dx D
@S w

Figure 4.8 Composition diagram representing the relation between suspended
nanoparticles and fines concentrations along the slow path and fast path in new
coordinate of stream-function and distance (Yuan 2017a).