Page 132 - Fundamentals of Enhanced Oil and Gas Recovery

120 Pouria Behnoudfar et al.

The equation can be written in terms of density as follows [77]:
@ρ @ρ @ @ρ @ρ
5 D (4.27)
@t @c @x @x @c

Here, the dependency of density with respect to composition is assumed as it is
unknown. Hence, a linear approach is proposed base on the small composition gradi-
ents expected for the diffusion process of gases in heavily oil and bitumen. Thus,
Eq. (4.27) can be simplified as follows [77]:
@ρ @ @ρ
5 D (4.28)
@t @x @x
In the next step, the medium domain is discretized with mesh size Δx and time
step Δt, which is illustrated in Fig. 4.3.
It can also be assumed that concentration in the boundary A is constant, and there
is no density flux condition in the boundary B. The followings are obtained by discre-
tization of equations:
Internal points,
2
n
n
2 ρ 2 ρ n D n 1 ρ n 2 ρ D n 5 Δx ρ n11 2 ρ n (4.29)

i i21 i20:5 i11 i i10:5 i i
Δt
Boundary A,
2
2 2 ρ 2 ρ D 1 ρ n 2 ρ D n 5 Δx ρ n11 2 ρ n (4.30)

n
n
n

n
i A A i11 i i10:5 i i
Δt
Boundary B,
2
2 ρ 2 ρ n D n 1 ρ 2 ρ D n 5 Δx ρ n11 2 ρ n (4.31)
n
n
n

i10:5
i
i21
i20:5
i
B
Δt i i
Eqs. (4.29) (4.31) can be arranged in a matrix form as Ax 5 b. Matrix A can be
constructed from the discretization of governing equations. Vector b is made of the
density measurement at specific grid locations along the medium and boundary con-
ditions. x is also the unknown diffusion coefficients.
Figure 4.3 A medium domain with mesh size Δx and time step Δt.