The lectures in Topic A of this course focused on some of the fundamental concepts that would serve us when  considering groundwater geochemical problems. In Topic B, we will be shifting gears a bit and devote our attention to the specific modeling tools and approaches that are available for solving geochemical speciation problems. This lecture will serve as an introduction to the topic. Subsequent lectures will introduce the PHREEQC model and describe how to formulate problems and solve a few examples. We will finish Topic B with a discussion of the coupling of geochemical speciation and transport models, the theoretical crux of this course.

Itís fairly straightforward to create a relatively simple geochemical speciation model capable of at least solving the aqueous phase speciation problem for a multi-component system. There is no doubt that a large number of such models have been created for specialized applications all over the world that have never been used by anyone other than the author(s) of the model. However, there are several models (or, more accurately, model families) that have come into wide use. The more familiar of these include:

 WATEQ. WATEQ (and its later derivates), first developed by Truesdell and Jones in 1974, was one of the first geochemical equilibrium models to be widely used. WATEQ relies on a relatively simple iteration scheme to solve aqueous speciation problems. While it does not explicitly handle heterogeneous reactions, it does compute mineral saturation indices.

 MINEQL. MINEQL was developed by Westall et al. in 1976 after the earlier REDEQL model of Morel and Morgan. MINEQL emphasizes a basis species approach in its computation (see discussion below), that reduces the number of unknowns and equations to solve in the iteration routine. In addition, MINEQL relies on the Newton-Raphson iteration scheme (see discussion below) that improves on the rate of convergence offered by the simpler substitution method employed by WATEQ. MINEQL is also more advanced than WATEQ in that it can handle simple heterogeneous reactions (mineral precipitation and dissolution), although complex reaction path modeling with incongruent dissolution reactions (see Lecture 4) are beyond its capabilities.

 MINTEQA2. MINTEQA2 (Allison et al., 1990), was developed from MINEQL. Major improvements over MINEQL include a large thermodynamic database (originally based on the WATEQ database) with many aqueous complexes as well as mineral phases. Complexation data for organic ligands, a topic we will explore in an example problem in a later lecture, are particularly important. MINTEQA2 also offers a number of surface complexation models. Another useful capability is the ease of modeling the system with different redox couples turned on or off.

 PHREEQE. The PHREEQE program and its derivatives were developed by the U.S. Geological Survey with an eye toward groundwater chemistry. The current version of PHREEQE, known as PHREEQC (a reference to the conversion of the code from the Fortran programming language to the C language) shares many of the same capabilities as EQ3/6, although it lacks some of the reaction path capabilities, has a smaller database, and cannot simulate reactions under geothermal conditions. Nevertheless, PHREEQC offers a number of advantages for most users as an all-purpose geochemical model for groundwater systems:

1. Itís a well-established model with a long history. A large number of papers have been published in the literature over the years that have used PHREEQE models to solve geochemical problems.

2. The current manifestation, PHREEQC, is versatile in that it can solve a wide range of problems, including those entailing surface chemistry phenomena and reaction kinetics.

3. Itís based upon a robust numerical engine that rarely crashes.

4. It can access thermodynamic data from large, well-established databases or rely on user-provided data, depending on user specifications.

7. An especially important point for this course: PHREEQC is public domain software that can be easily and freely downloaded from the web.

All of the common geochemical speciation models listed above, and any other equilibrium model, depend on their respective thermodynamic databases to produce accurate predictions.

 Continuing Fraction

To see how the continuing fraction method works, consider a generic mass action expression of the following form (neglecting activity coefficient corrections in the mass action expression):

Now consider a mass balance equation for the component represented by master species m:

By combining Eqs. 1 and 2, we see that we can develop an expression for the concentration of the

Using Eq.-3, we can estimate the concentration of a master species based on the concentrations of other master species that occur with it in various reactions. Of course, we donít know the concentrations of any of the master species at the beginning of a simulation. Thus, we make an educated guess, such as assuming that the master species concentration equals the total concentration of that element, and start our calculations using Eq.-3. Each time we use Eq.-3 for each of the master species, we improve our guesses. The iterations will eventually converge, giving us self-consistent solutions for the master species equations. Calculating the concentrations of non-master species is then a simple matter of substitution of the master species concentrations in the mass balance expressions (i.e., using Eq.-1). Well go through an example of how to use this approach at the end of this lecture.

 Newton-Raphson Iteration

The continuing fraction method is easy to program and converges fairly rapidly for small, simple systems that do not exhibit strong complexation. However, in other cases, convergence may be relatively slow. Moreover, the iteration scheme becomes much more difficult and cumbersome when charge balance, operational valance balance, and mineral precipitation and dissolution reactions are considered. Therefore, most "heavy duty" geochemical models use the Newton-Raphson iteration technique to arrive at a solution.

Newton-Raphson iteration is a multi-variate analog to the well-known Newtonís method that is used to estimate numerical solutions to equations that are otherwise intractable to closed-form solution (such as non-linear equations). Briefly, the method involves defining a Jacobian matrix for the all of the variables and algebraic equations that describe the geochemistry of the system. The Jacobian is of the form:

            (Eq.-4)

Letís take a look at an example of how the continuous fraction method can be used for a simple speciation calculation. Consider a water samples containing 1 x 10^-3 mol/L of total inorganic carbon and 1 x 10^-3 mol/L of calcium at pH 7 ([H⁺] = 1 x 10^-7 mol/L). If we define our master species as CO₃^2- and Ca²⁺, respectively, then we can write the following reactions:

From Eq.-3, we can write out two mass balance equations on the master species as:

If we postulate as our initial guesses that [CO₃^2-] = 1 x 10^-3 mol/L and that [Ca²⁺] = 1 x 10^-3 mol/L (i.e., the same as the total concentrations of each respective element), then, neglecting activity coefficient corrections, iteration on these two equations gives us:

As you can see, the iteration scheme converges rapidly to a solution for this simple problem. The mass action expressions can now be used to calculate concentrations of the other (i.e., non-master) species. For example, for CaHCO₃⁺,

we can calculate [CaHCO₃⁺] ~ 1 x 10^-5 mol/L. The interested student is invited to calculate the remaining concentrations and to verify that the resulting total set of concentrations yields the proper mass balance for both elements.

You are now ready to continue to

TOPIC B: MODELING APPROACHES.

LECTURE 2: Introduction to Geochemical Speciation Modeling with PHREEQC: Model Structure and Model Definition.

You may e-mail me questions and comments.

Walt W. McNab
E-mail address: Walt McNab <WaltMcNab@prodigy.net>

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