Transport and Equilibrium Reactions - Environmental Modeling - page 158

It is appropriate to multiply the equation from the left by a matrix U with the
property U S T ¼ 0. The matrix U has N s -N r rows and N s columns and is not

unique. The result is:

@
@t

S T r Uc ¼ U r j ð c Þþ U (8.34)

S T 0. If the

transport terms are linear, multiplication with matrix U and differentiation can be
exchanged. From ( 8.34 ) one directly obtains a differential equation for the total
concentrations u

The last term on the right side can be omitted because of U ¼

:¼ U c:

@
@t

¼r j ð Þ

u u (8.35)

Equation ( 8.35 ) represents a system of N s -N r transport equations for the N s -N r
components of u. The problem is thus significantly simpler than the original system
with N s coupled equations. The gain is paid by the need for speciation calculations.
When the vector u is calculated, the N s species have to be computed in a second step
with the help of ( 8.22 ).
The formulation of the entire problem, given by ( 8.35 ) and ( 8.22 ), offers various
advantages. The given formulation has the following properties:

1. Transport and reaction modeling are not coupled, because transport can be
solved independently from the speciation; solving the transport problem, the
knowledge of the c-vector is not required
2. The transport problem consists of N s -N r linear differential equations
3. The differential equations of the transport problem are independent from each
other
4. The reaction problem consists of a nonlinear system of N s equations which have
to solved for each node

For the solution of such a system it is justified to apply a sequential non-iterative
approach (SNIA, see: Steefel and MacQuarrie 1996 ): transport is solved first in
order to obtain u, from which c is determined in a second step. In the first step N s -N r
independent linear transport equations have to be solved.
In the case of reactive transport with equilibrium equations, the SNIA approach
causes no additional errors in contrast to other approaches. If the type of boundary
conditions is the same for all total concentrations, it is even sufficient to solve the
transport equation only once. When the transport step is completed, the speciation
has to be calculated for each block (or node). These computations are independent
from each other, i.e. the speciation in one block does not depend on the speciation in
any other block of the model region.
The de-coupled solution procedure is only possible, because within the formu-
lation itself ( 8.35 ) and ( 8.22 ) transport and equilibrium geochemistry are not
coupled. The sequential (de-coupled) treatment of transport and speciation is not