Transport and Equilibrium Reactions - Environmental Modeling - page 160

and for the transformation matrix U results:

101 11 11

¼

U (8.38)

01 10110

Thus there are two totals which are a combination of the species concentrations:

¼

H þ þCa 2 þ OH þH 2 CO 3 CO 3 2 þCaHCO 3 þ
HCO 3 Ca 2 þ þH 2 CO 3 þCO 3 2

Tot 1
Tot 2

For each of the totals a differential equation is solved. The entire system is
reduced to the solution of the transport equation for two totals and the specification
calculations. The connection with transport was tested for a situation of fracture
within calcareous rock, entered by water which was under-saturated with respect to
calcite. Such a system was studied by Saaltink et al. ( 2001 ). Some parameter values
were adopted from that study.
A list of parameters is given in Table 8.3 . A 1D porous fracture of 100 m length
is modeled with water entering at a Darcy velocity of 2 m/a. Initial and input
concentrations for both species have to be specified as well.
The MATLAB

model for the problem is implemented as a combination of the
pdepe solver and the speciation calculations. pdepe is used for the usual transport
equation as described in Chap. 4 in detail. The speciation has to be performed for
each geochemical species set at each node and each time instance of interest.
The simulation using MATLAB ® shows the propagation of the front for all
species. Figure 8.2 depicts pH as an example. The pH rises, because carbon species,
entering the aqueous system by calcite dissolution, bind more of the available
H + -species. With pH

®

¼ 9.7 the inflowing front thus shows higher values than the

initial system with pH 8.75.

The given procedure can also be applied to systems in which both kinetic and
equilibrium reactions are expected. The mathematical formalism has to be slightly
extended, as it was demonstrated by Holzbecher ( 2005 ). Without going into details,
we demonstrate the procedure for a situation derived from the example above.

¼

Table 8.3 Parameter for calcite dissolution equilibrium test-case
Variable Value Unit

Length 100 m

Maximum simulated time 5 a

Porosity 0.1 -

Darcy velocity 2 m/a

m 2 /a Diffusivity 200

Initial concentration Tot1 4.019 log mol/l

Initial concentration Tot2 3.018 log mol/l

Input concentration Tot1 4.365 log mol/l

Input concentration Tot2 5.421 log mol/l