Page 159 - gas transport in porous media

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of fluid provides a natural time variable
t ˆ
7 Sahimi et al.
0 Q t dt
τ = (8.15)
ψ
In the absence of dispersion the solution is simple: A concentration bank C = 1
(i.e., the pure fluid) displacing C = 0. In general, however, the concentrations
need not form a bank, since dispersion intervenes and develops a mixed zone. To
develop a probabilistic model that takes this effect into account, King and Scher (1987,
2
1990) interpreted (∂C/∂τ)d x as a two-dimensional probability density function for
concentration evolution. According to Eqs. (8.13) and (8.14) we can write
∂C 2 ∂C
d x =− dξ 1 dξ 2 =−dCdψ (8.16)
∂τ ∂ξ 1 ∂ξ 2
where ξ 1 and ξ 2 are local tangential and normal coordinates on the front.
Equation (8.16) can now be given a probabilistic interpretation: The probability of
2
concentration evolution at x [i.e., (∂C/∂τ)d x] is the product of the probability dψ
of fluid flow through x and the probability dC of a concentration gradient moving
through x. For a given realization, one samples the cumulants C and ψ, that is,
determines the flux contour C = r 1 and the streamline ψ = r 2 , and calculates their
intersection in the plane, which is also the point at which concentration is modified,
where r 1 and r 2 are two random numbers uniformly distributed in (0,1).
In such a simulation, one must employ a probabilistic interpretation of the FD
version of Eq. (8.13). We integrate this equation over a rectangular spatial region A ij
and time interval τ to obtain C, the change in the average concentration C ij . This
is given by
τ 8
δC ij =− Cdψ (8.17)
x y
∂A ij
Obviously, if δC ij / C is properly normalized, then it can be interpreted as the growth
probability at site ij (i.e., as the probability that the displacing fluid and the front
between the two fluids advance). One now has to fix C. According to Eq. (8.17),
the growth probability is non-zero only when the boundary integral overlaps the edge
of the cluster of the displacing fluid mixed with the displaced fluid. This method has
the advantage that finite values of the viscosity ratio can be used in the simulations.
We can now add the effect of dispersion. Consider first the static case, v = 0. The
2
evolution equation is simply the diffusion equation, ∂C/∂T − D L ∇ C = 0, where
T = D L t (King and Scher assumed that, D L = D T , which is, however, not justified).
In discrete form,
T 8
δC ij = ˆ n.∇Cdl (8.18)
x y
∂A ij