Reactive transport

Many times in geochemical modeling we want to understand not only what reactions proceed in an open chemical system, but where they take place (e.g., Steefel et al., 2005). In a problem of groundwater contamination, for example, we may wish to know not only the extent to which a contaminant might sorb, precipitate, or degrade, but how far it will migrate before doing so. 

To construct models of this sort, we combine reaction analysis with transport modeling, the description of the movement of chemical species within flowing groundwater, as discussed in the previous chapter (Chapter 20). The combination is known as reactive transport modeling, or, in contaminant hydrology, fate and transport modeling. 

A lysimeter filled with sediments was equilibrated with the following water (concentrations in mmol/L)  

At a time Tl, an acid mine drainage of the following composition is added (concentration in mmol/L)  

Calculate the distribution of the concentrations within the lysimeter column taking into account the cation exchange (discretisation and time steps as in the example in chapter 2.2.2.3). Selectivity coefficients are taken from the exemplary data of WATEQ4F.dat data set and an exchange capacity of 0.0011 mol per kg water is assumed. Neither diffusion nor dispersion is considered. Present your results graphically. 

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Beginning in the late 1980s, a number of groups have worked to develop reactive transport models of geochemical reaction in systems open to groundwater flow. As models of this class have become more sophisticated, reliable, and accessible, they have assumed increased importance in the geosciences (e.g., Steefel et al., 2005). The models are a natural marriage (Rubin, 1983 Bahr and Rubin, 1987) of the local equilibrium and kinetic models already discussed with the mass transport... 

In a reactive transport model, the domain of interest is divided into nodal blocks, as shown in Figure 2.11. Fluid enters the domain across one boundary, reacts with the medium, and discharges at another boundary. In many cases, reaction occurs along fronts that migrate through the medium until they either traverse it or assume a steady-state position (Lichtner, 1988). As noted by Lichtner (1988), models of this nature predict that reactions occur in the same sequence in space and time as they do in simple reaction path models. The reactive transport models, however, predict how the positions of reaction fronts migrate through time, provided that reliable input is available about flow rates, the permeability and dispersivity of the medium, and reaction rate constants. 

Reactive transport models are, naturally, more challenging to set up and compute... 

Fig. 2.11. Configurations of reactive transport models of water-rock interaction in a system open to groundwater flow (a) linear domain in one dimension, (b) radial domain in one dimension, and (c) linear domain in two dimensions. Domains are divided into nodal blocks, within each of which the model solves for the distribution of chemical mass as it changes over time, in response to transport by the flowing groundwater. In each case, unreacted fluid enters the domain and reacted fluid leaves it.

Since a valid reaction model is a prerequisite for a continuum model, the first step in any case is to construct a successful reaction model for the problem of interest. The reaction model provides the modeler with an understanding of the nature of the chemical process in the system. Armed with this information, he is prepared to undertake more complex calculations. Chapters 20 and 21 of this book treat in detail the construction of reactive transport models. 

Such models are known as reactive transport models and are the subject of the next chapter (Chapter 21). We treat the preliminaries in this chapter, introducing the subjects of groundwater flow and mass transport, how flow and transport are described mathematically, and how transport can be modeled in a quantitative sense. We formalize our discussion for the most part in two dimensions, keeping in mind the equations we use can be simplified quickly to account for transport in one dimension, or generalized to three dimensions. 

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

Equation 20.24 can be solved analytically to give closed-formed answers to simple problems. Many mass transport problems, however, including all but the most straightforward in reactive transport, require the equation to be evaluated numerically (e.g., Phillips, 1991). There are a variety of methods for doing so, including... 

Each nodal block represents a distinct system, as we have defined it (Fig. 2.1). Conceptually, the properties of the entire block are projected onto a nodal point at the block s center (Fig. 20.2). A single value for any variable is carried per node in a transport or reactive transport simulation. There is one Ca++ concentration, one pH, one porosity, and so on. In other words, there is no accounting in the finite difference method for the extent to which the properties of a groundwater or the... 

A reactive transport model, as the name implies, is reaction modeling implemented within a transport simulation. It may be thought of as a reaction model distributed over a groundwater flow. In other words, we seek to trace the chemical reactions that occur at each point in space, accounting for the movement of reactants to that point, and reaction products away from it. 

Substituting these relations into the reactive transport equation (Eqn. 21.2), and taking porosity to be constant, gives... 

Retardation also arises when a fluid undersaturated or supersaturated with respect to a mineral invades an aquifer, if the mineral dissolves or precipitates according to a kinetic rate law. When the fluid enters the aquifer, a reaction front, which may be sharp or diffuse, develops and passes along the aquifer at a rate less than the average groundwater velocity. Lichtner (1988) has derived equations describing the retardation arising from dissolution and precipitation for a variety of reactive transport problems of this sort. 

In the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

A reactive transport model in a more general sense treats a multicomponent system in which a number of equilibrium and perhaps kinetic reactions occur at the same time. This problem requires more specialized solution techniques, a variety of which have been proposed and implemented (e.g., Yeh and Tripathi, 1989 Steefel and MacQuarrie, 1996). Of the techniques, the operator splitting method is best known and most commonly used. 

In this method, we write the governing equation (Eqn. 21.1 or 21.2) for reactive transport in the conceptual form,... 

Fig. 21.1. Operator splitting method for tracing a reactive transport simulation. To step forward from t = 0, the initial condition, to t = At, evaluate transport of the chemical components into and out of each nodal block, using the current distribution of mass. The net transport is the amount of component mass accumulating in a block over the time step. Once the updated component masses are known, evaluate the chemical equations to give a revised distribution of mass at each block. Repeat procedure, stepping to t = 2At, t = 3 At, and so on, until the simulation endpoint is reached.

In this chapter, we build on applications in the previous chapter (Chapter 26), where we considered the kinetics of mineral dissolution and precipitation. Here, we construct simple reactive transport models of the chemical weathering of minerals, as it might occur in shallow aquifers and soils. 

In a first reactive transport model (Bethke, 1997), we consider the reaction of silica as rainwater infiltrates an aquifer containing quartz (SiC>2) as the only mineral. Initially, groundwater is in equilibrium with the aquifer, giving a SiC>2(aq) concentration of 6 mg kg-1. The rainwater contains only 1 mg kg-1 Si02(aq), so as it enters the aquifer, quartz there begins to dissolve,... 

Only for the intermediate cases - those with velocities in the range of about 100 m yr-1 to 1000 m yr-1 - does silica concentration and reaction rate vary greatly across the main part of the domain. Significantly, only these cases benefit from the extra effort of calculating a reactive transport model. For more rapid flows, the same result is given by a lumped parameter simulation, or box model, as we could construct in REACT. And for slower flow, a local equilibrium model suffices. 

In the previous two chapters (Chapters 26 and 27), we showed how kinetic laws describing the rates at which minerals dissolve and precipitate can be integrated into reaction path and reactive transport simulations. The purpose of this chapter is to consider how we can trace the reaction paths that arise when redox reactions proceed according to kinetic rate laws. 

Groundwater remediation is the often expensive process of restoring an aquifer after it has been contaminated, or at least limiting the ability of contaminants there to spread. In this chapter, we consider the widespread problem of the contamination of groundwater flows with heavy metals. We use reactive transport modeling to look at the reactions that occur as contaminated water enters a pristine aquifer, and those accompanying remediation efforts. 

Fig. 33.3. Steady-state distribution of microbial activity and groundwater composition in an aquifer hosting acetotrophic sulfate reduction and acetoclastic methanogenesis, obtained as the long-term solution to a reactive transport model.

Lichtner, P. C., 1996, Continuum formulation of multicomponent-multiphase reactive transport. Reviews in Mineralogy 34, 1-81. 

Steefel, C. I., 2001, Gimrt, Version 1.2 Software for modeling multicomponent, multidimensional reactive transport, User s Guide. Report UCRL-MA-143182, Lawrence Livermore National Laboratory, Livermore, California. 

Steefel, C. I., D. J. DePaolo and P. C. Lichtner, 2005, Reactive transport modeling An essential tool and a new research approach for the Earth sciences. Earth and Planetary Science Letters 240, 539-558. 

Steefel, C.I. and K.T. B. MacQuarrie, 1996, Approaches to modeling of reactive transport in porous media. In PC. Lichtner, C.I. Steefel and E.H. Oelkers (eds.), Reactive Transport in Porous Media, Reviews in Mineralogy 34, 85-129. 

Steefel, C.I. and S.B. Yabusaki, 1996, Os3d/GIMRT, Software for multicomponent-multidimensional reactive transport User s Manual and Programmer s Guide. Report PNL-11166, Pacific Northwest National Laboratory, Richland, Washington. 

Yabusaki, S. B., C. I. Steefel and B. D. Wood, 1998, Multidimensional, multicomponent subsurface reactive transport in non-uniform velocity fields code verification using an advective reactive streamtube approach. Journal of Contaminant Hydrology 30,299-331. 

Since publication of the first edition, the held of reaction modeling has continued to grow and hnd increasingly broad application. In particular, the description of microbial activity, surface chemistry, and redox chemistry within reaction models has become broader and more rigorous. Reaction models are commonly coupled to numerical models of mass and heat transport, producing a classification now known as reactive transport modeling. These areas are covered in detail in this new edihon. 

At the same time, reaction modeling is now commonly coupled to the problem of mass transport in groundwater flows, producing a subfield known as reactive transport modeling. Whereas a decade ago such modeling was the domain of specialists, improvements in mathematical formulations and the development of more accessible software codes have thrust it squarely into the mainstream. 

I expand treatment of sorption, ion exchange, and surface complexation, in terms of the various descriptions in use today in environmental chemistry. And I integrate all the above with the principals of mass transport, to produce reactive transport models of the geochemistry and biogeochemistry of the Earth s shallow crust. As in the first edition, I try to juxtapose derivation of modeling principles with fully worked examples that illustrate how the principles can be applied in practice. 

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