Page 187 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

162 Thomas Russell et al.

So the transformation of the original system (Eqs. (3.211 3.215))
into (X,ϕ)-coordinates has the form (Eqs. (3.226, 3.229, 3.230, and
3.231). System (Eqs. (3.229 3.231)) and Eq. (3.226) are called the auxil-
iary system and the lifting equation, respectively. The auxiliary system
(Eqs. (3.229 3.231)) splits from the lifting Eq. (3.226). A detailed deriva-
tion of the splitting technique and its application to colloidal flow in
porous media is provided by Borazjani and Bedrikovetsky (2017).
Substituting a trajectory ϕ(X) and T(X) into the flux (Eq. (3.223)) and
taking the corresponding derivatives show the relationship between rare-
faction and shock wave speeds in planes (X,ϕ) and (X,T):

V 5 fD 2 s: (3.232)
Here, V and D are the wave speeds in (X,ϕ) and (X,T) coordinates,
respectively.
The initial and boundary conditions translate to:
0 X w , X , X mI
8
>
0 1
>
>
q
<
ϕ 52 s I X:S a 5 S cr @ p ffiffiffiffi ; γ I A X mI , X , X crI ; (3.233)
2πr e X
>
>
>
:
S aI X . X crI
8
S aI X w , X , X mI
>
0 0 11
>
>
q
<
ϕ 52 s I X:c 5 @ S aI 2 S cr @ p ffiffiffiffi ; γ I AA X mI , X , X crI ;
2πr e X
>
>
>
:
0 X . X crI
(3.234)
ϕ 52 s I X:γ 5 γ ; F 52N; (3.235)
I
X 5 X w :c 5 0; γ 5 γ ; U 5 1: (3.236)
J
By using the splitting procedure, the original (4) 3 (4) system of
quasi-linear hyperbolic Eqs. (3.211, 3.212, 3.214, and 3.215) is reduced
to a single equation for the saturation (Eq. (3.226)), which separates from
the remaining three Eqs. (3.229 3.231) for the salinity and the suspended
and strained particle concentrations. This new (3) 3 (3) system is called
the auxiliary system.
3.7.3 Exact solution for the auxiliary system
As performed in Section 3.5, the salt and suspended particle concentra-
tion can be solved from Eqs. (3.230 and 3.231) using the method of