Applied numerical modeling of saturated / unsaturated flow and ...

for the application studies outlined. The
chapter is based on fundamental publications
in the field of computational hydrology.
However, it is not intended to
serve as a comprehensive review of processes
and model concepts, as this would be
beyond the scope of this synthesis. Chapter
3 and its subsections present the results
and conclusions of the three application
examples. Chapter 4 closes this synthesis
with general conclusions and an outlook.
2. Mathematical models
The three-dimensional structure of natural
porous media is manifested in its composition
of three constituting phases, i.e. the
solid (mineral or biophase), water and gaseous
phases. At scales larger than the pore
scale, a description of porous medium geometry
becomes very complex and thus infeasible
for modeling applications. To
understand and formulate the dynamics of
fluids in the subsurface the so called representative
elementary volume (REV) concept
is introduced (Bear, 1972): In the
transition from the microscale to a larger
macroscale material, parameters are averaged
over a volume which is sufficiently
large to describe the porous medium at that
larger scale (see Fig. 1).
Fig. 1: Representative elementary volume
concept (Bear and Bachmat, 1990).
This may also require a reformulation of the
mathematical process descriptions. The
derivation of representative or effective
new material parameters and the corresponding
governing equations is termed
2
upscaling. Within the REV the detailed
structure of the medium is lost and becomes
a continuous field. Parameters like porosity,
permeability or dispersivity are considered
constant over the averaging volume. In the
following sections material parameters and
governing equations are based on this
continuum approach.
2.1. Saturated / unsaturated flow
The dynamics of the water in fully saturated
three dimensional porous media can be described
by the combination of the mass
balance for the water phase (eq. (1)) and
Darcy’s law (eq. (2)) as a constitutive equation
(Bear, 1972)
�h
S � �� �q
� Q
(1)
�t
q � �K�h
(2)
where S [m -1 ] is specific storativity, h [m] is
the hydraulic head, given as the sum of elevation
z [m] and the pressure head � [m], t
[s] is time, q [m s -1 ] is the Darcy flux vector,
K [m s -1 ] is the tensor of hydraulic conductivity
and Q [s -1 ] is a source or sink
term. The governing equation for groundwater
flow under transient conditions is
thus given by (Bear, 1972)
�h
S � � � �K �h��Q
(3)
�t
which at steady state converts to
� �h���Q
� � K (4).
This model of steady state flow in saturated
porous media is employed in applications 1
and 2 (sections 3.1, 3.2) of this synthesis.
For unsaturated conditions, which are prevalent
in application 3 (section 3.3), a more
general form of eq. (2), the Buckingham-
Darcy-law, can be used (Jury et al., 1991)
q � �K(
� ) �h
(5)
where K is a function of the pressure (or
matric) head �, which itself depends on the
volumetric water content � [-] of the porous
medium. As for saturated conditions, the