Page 185 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

160 Thomas Russell et al.

pressure P W . The equation for water pressure separates from the system.
As particle detachment is modeled in equilibrium conditions here, the
detachment rate will be zero; hence, S a will only provide initial and
boundary conditions, as in Section 3.5. Thus, the system has four
Eqs. (3.211, 3.212, 3.214, and 3.215) with four unknowns, s, γ, S s , and c.
The initial conditions of this system are:

8
0 X w , X , X mI
>
0 1
>
>
q
<
T 5 0:S a 5 S cr @ p ffiffiffiffi ; γ I A X mI , X , X crl ; (3.217)
2πr e X
>
>
>
:
S aI X . X crl
8
S aI X w ,X ,X mI
>
0 0 11
>
>
q
<
T 50:cX;0Þ5 @ S aI 2S cr @ p ffiffiffiffi;γ I AA X mI ,X ,X crI ; (3.218)
ð
2πr e X
>
>
>
:
0 X .X crI
T 50:γ5γ ;s5s I ; (3.219)
I
X 5X w :c50;γ5γ ;f 51: (3.220)
J
3.7.2 Splitting method for integration of two-phase systems
Based on works by Pires et al. (2006), Shen (2016),and Borazjani et al. (2017),
this section presents the splitting technique for a hyperbolic system of equations.
In order to split, the stream-function ϕ(X,T) is introduced, assuming
that the solution of the parameters s(X,T), c(X,T), S s (X,T), S a (X,T), and
γ(X,T) is already known:
ð X;TÞ
ð
ϕ X; TÞ 5 fdT 2 sdX: (3.221)
ð
ð 0;0Þ
It follows from Eq. (3.211) that s and f are the partial derivatives of
the stream-function ϕ(X,T):
@ϕ @ϕ
s 52 ; f 5 ; (3.222)
@X @T
This function is independent of the integration path that links point
(X,T) with the origin (0,0) (Courant and Friedrichs, 1976).