Page 219 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Using Nanofluids to Control Fines Migration in Porous Systems 193

onto pore surfaces, and water component flowing through 1D permeable
media can be written as:
Water component:

@f w @S w
1 5 0 (4.11a)
@x D @t D
Fine particles component:

ð
ð
@ C FP f w Þ @ S w C FP Þ 1 @σ a σ a 5 σ cr ; σ cr , σ cr;max
1 1 5 0; (4.11b)
@x D @t D φ @t D σ a 5 0; σ cr 5 σ cr;max
Nanoparticles component:
^
8
^ > ^ C NP;max K NP C NP ^ ^
@ C NP f w Þ @ S w C NP Þ 1 @C NP < C NP 5 ; C NP ,C NP;max
ð
ð
1 1 50; 11K NP C NP
@x D @t D φ @t D >
: ^ ^ ^
C NP 50;C NP ,C NP;max
(4.11c)
The relative permeability of the wetting water-phase behaves accord-
ing to a function of the retained fine particles concentration onto pore
surfaces expressed as Eq. (4.12). The viscosity of flowing water is a func-
tion of fines concentration, which is modeled using the Flory-Huggins
equation (Pope and Nelson, 1978):

ð
k rw S w ; σ a Þ k rw S w ; σ a;initial
5 (4.12)
ð
ð
μ C FP Þ ð 1 1 βσ FP Þμ 1 1 aC FP Þ
w w
where, β; a are constant coefficents.
Yuan (2017a) and Yuan and Moghanloo (2018c) introduced the
splitting method (Borazjani et al., 2016, Borazjani and Bedrikovetsky,
2016) to reduce the above governing system (Eq. (4.9)) from the 3 3 3
system to two sub-systems, including, a nanoparticles and fines retention-
kinetics auxiliary subsystem (Eq. (4.13)) and a lifting system for unknown
water saturation (Eq. (4.14)). The procedure to solve the problem
includes: (1) transformation of Eq. (4.11) using a stream-function and
splitting technique; (2) analytical solutions of both the auxiliary system
and lifting system using method of characteristics (MOC) (Appendix A);
and (3) inversion of solutions by transforming the coordinates:

@C FP 1 @σ a
1 5 0 (4.13a)
@x D φ @ϕ
^
@C NP 1 @C NP
1 5 0 (4.13b)
@x D φ @ϕ