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

where vmax [s -1 ] is a maximum growth rate,
Ci [kg m -3 ] is the concentration of the i th
substrate, MCi [kg m -3 ] is the corresponding
half velocity concentration [kg m -3 ], Cj
[kg m -3 ] is the concentration of the j th substance
inhibiting microbial growth, ICj is the
corresponding inhibition concentration
[kg m -3 ] and �(X) is a microbial decay term,
which is often modeled as being of first
order. Consumption of substrates or production
of k metabolites Ck [kg m -3 ] is both
coupled to microbial growth via (Schäfer et
al., 1998)
�Ck �1
��X
�
�
�t
Y �
� �t
�
�
6
(21)
k growth
where Yk [-] is the yield coefficient for
substrate or metabolite Ck, and [·]growth
refers to the growth term only in eq. (20).
From the generalized Monod-type equation
different more simple kinetic formulations
can be derived. For a temporally constant
microbial population, i.e. growth and decay
terms are constant and of equal magnitude,
no inhibition and dependence on only a
single substrate, eq. (20) and (21) can be
combined to yield the Michaelis-Menten
(MM) kinetics model (Simkins and
Alexander, 1984)
dC
dt
C
� �kmax
(22)
C � M C
where k max is the maximum degradation rate
[kg m -3 s -1 ] and M C is the MM half-saturation
concentration [kg m -3 ]. This approximation
may be applicable e.g. when aquifer
sediments have been exposed to contaminants
for several years (Bekins et al., 1998)
and is used in application 2 (section 3.2).
Often, contaminant degradation is also
described by simple first order kinetics, e.g.
to simulate abiotic degradation reactions
like hydrolysis and dehydrohalogenation of
halogenated compounds (Wiedemeier et al.,
1999). First order kinetics can be derived
from eq. (22) for C > MC eq. (22)
approaches zero order kinetics. The first
order model is used in application 1 and 3
(sections 3.1 and 3.3). Extensive reviews on
kinetic models of biodegradation can be
found e.g. in Baveye and Valocchi (1989),
Rittmann and VanBriesen (1996) or Islam
et al. (2001).
2.4. Numerics and software
methods
Numerical solution of the governing
equations
The governing equations for flow and reactive
transport presented in sections 2.1 - 2.3
belong to the group of partial differential
equations (PDE), containing derivatives of
first order in time and of first as well as
second order in space. The classification of
PDE can be based on mathematical aspects
(highest order derivatives in the dependent
variables) or on a physical point of view
(problem type of physical process) (Kolditz,
2002). Parabolic PDE are used for timedependent
problems with dissipation process,
such as diffusion (eq. (11)) or transient
groundwater flow (eq. (3)), which
convert to elliptic PDE for steady state
conditions (eq. (4)). A third class of PDE
are hyperbolic equations like the linear advection
equation (eq. (10)), which are used
to describe time-dependent problems without
dissipation process. The transport equations
(15) and (16) are of mixed type with a
parabolic dispersion-diffusion term and a
hyperbolic advection term. Their behaviour
for a particular problem depends on the
relative magnitudes of these flux components.
In general the transport equations are
of parabolic character which changes to
hyperbolic for large ratios of v/�L as in this
case the advective flux term is dominant
(Kolditz, 2002).
In general, PDE describing physical
problems are well-posed when appropriate
initial and boundary conditions are specified
for the domain where a solution is
required. While analytical solutions can be