* corresponding author:
jerome.carrayrou@unistra.fr
Keywords
Geochemical modelling; instantaneous equilibrium chemistry; activity
correction, outer fixed-point algorithm, Davis activity, Debye-Hückel
activity, B-dot model.
Abstract
Reactive transport codes are very useful elements of environmental
research. They now contain multiphysics with very complex algorithms,
including flow, transport, chemical and sometimes heat transport,
mechanical and/or biological algorithms. Because of this complexity,
some parts of these algorithms still have not been sufficiently studied.
Here, we present a comparison of 3 algorithms for activity correction, a
specific subset of equilibrium chemistry algorithms. We show that the
most used algorithm (the inner fixed-point algorithm) or the most
rigorous algorithm (the full Newton) might not be the most efficient,
and we propose the outer fixed-point algorithm, which is more robust and
faster than other algorithms.
Introduction
The problem of groundwater management has received increasing attention,
and many tools have been developed to address this issue. One of these
tools, reactive transport modeling, was first limited to laboratory
experiments [1] and then extended to the comprehension of problems
in various fields [2]. Reactive transport modeling is actually a
mature research field that has produced important results in many
environmental domains, such as water management, sea water intrusion
[3], long-term nuclear waste storage [4] and heavy metal
contamination [5]. Numerous reactive transport codes are available,
and some review articles [6–12] propose an overview of them.
Examining these articles, it can be seen that all these codes include
one or more activity correction models. Even though the different models
of activity correction are usually well-detailed, the algorithmic method
used to compute the activity coefficients and to incorporate these
calculations into the entire chemistry algorithm is not given.
We show here that different algorithms can handle activity corrections
and that they are not all equivalent. It seems that the most used
algorithm, named the inner fixed-point algorithm, leads to numerical
instabilities when handling highly-charged chemical species. We then
recommend the new algorithm presented in this work: the outer
fixed-point algorithm. It is more robust, faster and less sensitive to
the initial conditions.
General concepts
A general formulation of a chemical reaction leading to the formation of
one of the Nc species (Ci) from the number Nx of chosen
components Xj is written as:
Instantaneous equilibrium chemistry is usually described using two
fundamental concepts: mass conservation equations and mass action laws.
According to the classical formulation stated by Morel and Morgan, mass
conservation equations describe the conservation of the total
concentrations of the components (Tj), and mass action
laws describe the formation of each chemical species as a combination of
the Nx chosen components.
Mass conservation equations are written using the species concentrations
[Ci]:
On the other hand, mass action laws are written using the species
{Ci} and components {Xj} of the
activities:
To ensure the closure of the system, an activity model is used. The
activity coefficient (i) is less than one and determines
the species activity from its concentration.
Several activity models have been developed, all of which use the ionic
strength of the solution (I):