Upscaling and Inverse Modeling of Groundwater Flow and Mass ...

48 CHAPTER 3. TRANSPORT UPSCALING USING MULTI- . . .
transfer processes taking place within the coarse block and largely associated
with the within-block heterogeneity (e.g., Zinn and Harvey, 2003; Fernàndez-
Garcia and Gómez-Hernández, 2007; Willmann et al., 2008; Fernàndez-Garcia
et al., 2009).
We have chosen the multi-rate mass transfer model (MRMT) (Haggerty
and Gorelick, 1995; Carrera et al., 1998) as the mass exchange expression to
be used at the coarse scale. Alternative models such as the continuous time
random walk (Berkowitz and Scher, 1998) or fractional derivatives (Benson
et al., 2000) could be used as well. Fernàndez-Garcia et al. (2009) discussed
the use of the MRMT for upscaling purposes in 2D, and its versatility to
treat complex heterogeneities; furthermore, it was successfully applied at the
MADE aquifer by Feehley et al. (2000) and at the Lauswiesen site by Riva
et al. (2008). Many transport codes based on the MRMT model (e.g., Zheng
and Wang, 1999; Carrera et al., 1998; Salamon et al., 2006a; Silva et al., 2009)
indicate the great potential of this approach.
The upscaled transport equation, including the MRMT model, can be
described by the following governing equation (Haggerty and Gorelick, 1995;
Carrera et al., 1998):
ϕ c ∂C
m
c m(xc , t)
= −∇·
∂t
[ q c (x c )C c m(x c , t) ] + ∇· [ ϕ c mD c ∇C c m(x c , t) ] − ϕ c mΓ(x c , t)
(3.5)
where ϕc m [dimensionless] defines the pore volume fraction of the mobile domain;
Cc m[ML−3 ] is the solute concentration in the mobile region of the coarse
block; qc [LT −1 ] is the Darcy velocity derived from the upscaled hydraulic conductivity;
Dc [L2T −1 ] is an enhanced block dispersion tensor, which includes
the fine scale local hydrodynamic dispersion (αi) and a macrodispersivity term
(Ai) (Fernàndez-Garcia and Gómez-Hernández, 2007; Fernàndez-Garcia et al.,
2009):
D c i = Dm + (αi + Ai) |qc |
ϕc (3.6)
m
where Dc i are the eigenvalues of Dc associated with the principal axes, which
are parallel and perpendicular to the flow direction. The additional mass
exchange term Γ(xc , t) [ML−3T −1 ] can be expressed in terms of mobile concentrations
by using a convolution product with a memory function g(xc , t)
[T −1 ] (Carrera et al., 1998; Haggerty et al., 2000):
Γ(x c , t) = β(x c ∫ t
)
0
g(x c ∫ ∞
, t) =
0
g(x c , τ) ∂Cc m(xc , t − τ)
dτ
∂τ
αf(x c , α)e −αt dα
(3.7)