Transport and Sorption - Environmental Modeling - page 125

The idea of two function variables in one vector has to be adopted to understand
the functions of the M-file. Where a single value was sufficient in former programs,
now two values need to be given; the first for the single phase differential equation,
the second for the solid phase differential equation. The functions read as follows:
The coefficient for the time derivative term is 1 in both differential equations.
Thus, both components in the column-vector c are equal to 1. The flux term f
contains the negative of Fickian diffusion in the first component and zero in the
second (as there is no diffusion in the solid phase). Note that the variable DuDx
contains the spatial derivatives of the concentrations and is a two-component
column vector within the function.
In a similar manner the variable s contains the contributions from advection,
decay and sorption. The first component of the first term -v*DuDx(1) contains the
advection term already known from the other M-files. The second component in
the first term is zero, as there is no advection in the solid phase. 6 The second term of
the s ¼ ... assignment includes decay terms, which are allowed to be phase-depen-
dent. If they are phase dependent, the variables lambdaf and lambdas have to be
chosen differently. The last term denotes the interphase exchange. In the coefficient
term (in round brackets) the amount of exchange is computed. The exchange needs
to be included with a positive sign in the first differential equation and with
a negative in the second differential equation. In order to achieve that, one has to
multiply with the [1/theta;-1/rhob] vector.
For both phases initial conditions are specified in the slowsorpic function. Note
that c0 is a two component vector, which contains the specified initial
concentrations for both phases. For the boundary conditions the concentration of
6 In most applications on porous media, the solid phase or porous matrix is assumed to be fixed in
the chosen spatial coordinate system. However, in some cases this may not be true. In sediments
the solids move with respect to a fixed level in space. If the interface between the sediments and the
overlying region of free flow is taken as a reference, one may obtain a situation with no flux of
solids. However, this trick works doesn't work if there are temporal changes in sedimentation, or
even in steady state, when compaction has to be taken into account.