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

46 CHAPTER 3. TRANSPORT UPSCALING USING MULTI- . . .
and strengths of the proposed approach, with an indication of avenues for
improvement.
3.2 Methodology
3.2.1 Background
Fine scale equations
At the fine scale, denoted herein by the superscript f, under steady-state
flow conditions and in the absence of sinks and sources, the flow equation of
an incompressible fluid in saturated porous media in a Cartesian coordinate
system can be obtained by combining the continuity equation and Darcy’s law
(Bear, 1972):
∇· [ K f (x f )∇h f (x f ) ] = 0 (3.1)
where h f [L] is the piezometric head; K f [LT −1 ] is a symmetric positive-definite
rank-two tensor; x f represents the fine scale coordinates.
Similarly, using the solute mass conservation equation and assuming that
Fick’s law is appropriate at the local scale, the three-dimensional advectivedispersive
equation (ADE) for solute transport is often written as (Freeze and
Cherry, 1979):
ϕ f ∂Cf (x f , t)
∂t
= −∇· [ q f (x f )C f (x f , t) ] + ∇· [ ϕ f D f ∇C f (x f , t) ]
(3.2)
where C f [ML −3 ] is the dissolved concentration of solute in the liquid phase;
ϕ f [dimensionless] is the porosity; q f [LT −1 ] is the Darcy velocity given by
q f (x) = −K f (x)∇h f (x); D f [L 2 T −1 ] is the local hydrodynamic dispersion
coefficient tensor with eigenvalues (associated with the principal axes, which
are parallel and perpendicular to the direction of flow) given by (Burnett and
Frind, 1987):
D f
i = Dm
|q
+ αi
f |
ϕf (3.3)
where αi are the local dispersivity coefficients, more specifically, αL, αH T and
αV T are, respectively, the longitudinal dispersivity coefficient and the transverse
dispersivity coefficient in the directions parallel and orthogonal to flow, and
Dm is the effective molecular diffusion coefficient.
The fine scale transport equation (3.2) is only valid if the Fickian assumption
is satisfied at the small scale. Here, we assume that the ADE is capable