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

Buckingham-Darcy law is combined with
mass balance principles to yield the governing
equation of flow for unsaturated
conditions, i.e. the Richards equation. This
equation exists in three main forms with �,
� or both quantities as dependent variables
(Jury et al., 1991). In GeoSys / Rockflow
the �-based form (Freeze and Cherry,
1979) is implemented (Du et al., 2005)
C w
��
�t
��� � � � �K ( � ) ��
�
�K
�
�z
��� � Q
(6)
where Cw(�) is the water capacity function
defined by d� /d� and with z positive in a
downward direction. For the functional
description of unsaturated hydraulic properties
different mathematical formulations
have been proposed in literature. A frequently
used constitutive relation is the
Van-Genuchten-Mualem model (Van Genuchten,
1980) which is based on the
statistical pore space model of Mualem
(1976) and is given by
� � � �2 �m
m
1�
1�
Sr
l
K(
S ) � KS
(7)
Sr
r
� ��
r � �
� ��
s
r
r
1
� � � �m n
1�
��
(8)
m � 1� 1/
n
(9)
where Sr [-] is defined as the relative
saturation, l [-] is a pore connectivity parameter,
�, �r and �s are the actual, residual
and saturated volumetric water contents, �
[m -1 ], n [-], and m [-] are empirical parameters.
Other constitutive relationships comprise
approaches such as the Brooks-Corey
model (Brooks and Corey, 1966), the Haverkamp
model (Haverkamp et al., 1977), potential
functions as introduced by Huyakorn and
Pinder (1983) or the multimodal model of
Durner (1994). Recently, also free form parameterizations
were suggested (Bitterlich et al.,
2004). An overview on the prevalent approaches
is given e.g. by Durner and Flühler (2005).
2.2. Transport processes
The most fundamental transport process of
dissolved substances in saturated porous
media is advection. Advection is passive
with the flowing water. Purely advective
transport of a solute plume is free of interference
or mixing with the surrounding
ambient water and is described with the advection
equation (Zheng and Bennett, 1995)
�C
�t
� �v
�C
� Q
(10)
where C [kg m -3 ] is the concentration of a
dissolved species, v [m s -1 ] is the vector of
average linear velocity which is given by
division of q with the effective porosity �e
[-], and Q [kg m -3 s -1 ] is a source or sink
term for species C.
In natural aquifers or soils, purely advective
transport is practically not established as
dissolved molecules migrate from high to
low concentration regions by Brownian
motion. This concentration gradient driven
mass transport is termed molecular diffusion
and occurs even when the fluid itself is
stagnant. For transient systems the diffusion
process in water can be described using
Fick’s 2 nd law, (Fetter, 1993)
2
�C
� C
� �Da
2
�t
�x
(11)
where Da is the molecular diffusion coefficient
in water [m 2 s -1 ]. In porous media the
diffusion process is hindered by the presence
of the solid phase matrix and the tortous
nature of the pores. Thus an effective
diffusion coefficient Dae is derived as
(Grathwohl, 1998)
D
� �
e
ae a
f
D � (12)
�
where � [-] is the constrictivity and �f [-] the
tortuosity of the porous medium. Under
unsaturated conditions Dae can also be
related to � (e.g. Olsen and Kemper, 1968).
As water moves through a porous medium,
single streamline velocities can be greater
or less than v. This effect is due to different
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