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

domain is discretized along advective flow
paths of the Eulerian flow field by the travel
time � [s] of inert particles between an
injection plane and a control plane, both
oriented normal to the mean flow direction.
Each particle trajectory is regarded as a
separate one-dimensional stream tube of the
flow field with an infinitesimal cross
section. The probability density function
(pdf) g(�, x) of all particles travel times
completely reflects all hydraulic heterogeneities
of the model domain. Influences of
reactions (e.g. biodegradation, intraparticle
diffusion, sorption, etc.) are quantified by
means of the reaction function �(�, t),
which is evaluated by the BESSY model
(Jäger and Liedl, 2000) implemented in
SMART. With given g and � the normalized
breakthrough curve of a reactive
solute at a control plane is calculated by
(Finkel et al. 1998)
8
�
� �
0
�x, t�
g��,
x����t�d�
C ,
(24).
To overcome the limitations of both the
Eulerian and the Lagrangian concepts,
mixed Eulerian-Lagrangian methods can be
used, which take advantage of the particular
appropriateness of both concepts for modeling
advective and dispersive transport (e.g.
Neumann, 1981; Thorenz, 2001; Park et al.,
2006). In application 3 (section 3.3) a
combination of the software codes GeoSys /
Rockflow and SMART is used for a combined
application of the Eulerian and the
Lagrangian concepts. GeoSys / Rockflow is
used to model the Eulerian flow field in a
heterogeneous two dimensional domain and
to derive the representative pdf g(�, x). The
SMART model then utilizes the pdf for the
simulation of reactive transport in the
model domain.
Object- and process-oriented methods
The GeoSys / Rockflow code, which is used
for most of the numerical simulations
described in chapter 3, is written in the C++
language and thus allows the implementation
by object oriented programming (OOP)
methods. The OOP concept is especially
helpful for the development of complex
software in programmer teams, as encapsulation
and class-structures render the code
more stable and errors are easier to detect.
In GeoSys / Rockflow the OOP concept is
met by so called process orientation
(Kolditz and Bauer, 2004), which allows
the coupling of two-phase flow, heat transport,
mass transport, chemical reactions and
deformation in an efficient way (Wang et
al., 2006). The basic idea of process orientation
is that between each physical process
(e.g. single species transport) and its
numerical approximation by an algebraic
equation system (AES) exists a direct correspondence
(Fig. 4). The AES originates
from the temporal and spatial discretization
of the PDE on the computational grid. For
its solution the following steps are performed:
� AES assemblage and incorporation of
initial conditions
� determination of element matrices
� incorporation of boundary conditions
and source terms
� solving the AES by appropriate solvers
This procedure can be generalized for any
physical process regardless of its specific
type in an object oriented way by introducing
the process object (Kolditz and Bauer,
2004) (Fig. 4).
Transport of non-reactive
species in water
„PROCESS“
Multifield problem, if many
mobile species are
transported
Solution of a PDE
system
PROCESS-OBJECT
System of PDE
solved by
Multi-Process-Method
Fig. 4: Process analogy and process object,
shown for an instance of a transport
process (Kolditz and Bauer, 2004).