A general description of the mathematical and numerical formulations used in modern numerical reactive transport codes relevant for subsurface environmental simulations is presented. The formulations are followed by short descriptions of commonly used and available subsurface simulators that consider continuum representations of flow, transport, and reactions in porous media. These formulations are applicable to most of the subsurface environmental benchmark problems included in this special issue. The list of codes described briefly here includes PHREEQC, HPx, PHT3D, OpenGeoSys (OGS), HYTEC, ORCHESTRA, TOUGHREACT, eSTOMP, HYDROGEOCHEM, CrunchFlow, MIN3P, and PFLOTRAN. The descriptions include a C. I. Steefel ( ) · B. Arora · S. Molins · N. Spycher

Many hydrogeochemical models have appeared in recent years for simulating subsurface solute transport. The hydrological transport of solutes can be described by a set of linear partial differential equations, and the chemical equilibria are described by a set of nonlinear algebraic equations. Three approaches are currently used to formulate the problem: (1) the mixed differential and algebraic equation (DAE) approach, (2) the direct substitution approach (DSA), and (3) the sequential iteration approach (SIA). An extremely important consideration in any approach is the choice of primary dependent variables (PDVs). Six types of PDVs have been employed in the existing models: (1) concentrations of all species, (2) concentrations of all component species and precipitated species, (3) total analytical concentrations of aqueous components, (4) total dissolved concentrations of aqueous components, (5) concentrations of aqueous component species, and (6) hybrid concentrations. Because of many possible combinations of PDVs and approaches, many hydrogeochemical transport models for multicomponent systems have been developed. This paper critically evaluates and discusses these models. The discussion and evaluation are conducted in terms of (1) how severe can the constraints be that a model imposes on computer resources, (2) which equilibrium geochemical processes can a model include, and (3) how easily can a model be modified to deal with mixed kinetic and equilibrium reactions. The use of SIA models leads to the fewest constraints on computer resources in terms of central processing unit (CPU) memory and CPU time; both DAE and DSA models require excessive CPU memory and CPU time for realistic two‐ and three‐dimensional problems. Only those models that use the first three types of PDVs can treat the full complement of equilibrium reactions simultaneously. DAE and SIA models can be modified with reasonable ease to handle mixed chemical kinetics and equilibria. DSA models require strenuous efforts to modify for treating mixed chemical kinetics and equilibria. Therefore SIA models using the third type of PDVs are recommended for their practicality and flexibility. DSA and DAE models should remain research tools for one‐dimensional investigations.

This paper presents the development and demonstration of a two‐dimensional finite‐element hydrogeochemical transport model, HYDROGEOCHEM, for simulating transport of reactive multispecies solutes. The model is designed for application to heterogeneous, anisotropic, saturated‐unsaturated media under transient or steady state flow conditions. It simulates the chemical processes of complexation, dissolution‐precipitation, adsorption‐desorption, ion exchange, redox, and acid‐base reaction, simultaneously. A set of four example problems are presented. The examples illustrate the model's ability to simulate a variety of reactive transport problems. Important results presented include a depiction of the propagation of multiple precipitation‐dissolution fronts, a display of the large errors in model response if the number of iterations between the hydrologic transport and chemical equilibrium modules is limited to one, an illustration of the development of greater concentration of contaminants In groundwater away from a waste site than near the source, and a demonstration of the variation in distribution coefficients of more than 6 orders of magnitude.

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[1] This paper presents the development and illustration of a numerical model of reactionbased geochemical and biochemical processes with mixed equilibrium and kinetic reactions. The objective is to provide a general paradigm for modeling reactive chemicals in batch systems, with expectations that it is applicable to reactive chemical transport problems. The unique aspects of the paradigm are to simultaneously (1) facilitate the segregation (isolation) of linearly independent kinetic reactions and thus enable the formulation and parameterization of individual rates one reaction by one reaction when linearly dependent kinetic reactions are absent, (2) enable the inclusion of virtually any type of equilibrium expressions and kinetic rates users want to specify, (3) reduce problem stiffness by eliminating all fast reactions from the set of ordinary differential equations governing the evolution of kinetic variables, (4) perform systematic operations to remove redundant fast reactions and irrelevant kinetic reactions, (5) systematically define chemical components and explicitly enforce mass conservation, (6) accomplish automation in decoupling fast reactions from slow reactions, and (7) increase the robustness of numerical integration of the governing equations with species switching schemes. None of the existing models to our knowledge has included these scopes simultaneously. This model (BIOGEOCHEM) is a general computer code to simulate biogeochemical processes in batch systems from a reaction-based mechanistic standpoint, and is designed to be easily coupled with transport models. To make the model applicable to a wide range of problems, programmed reaction types include aqueous complexation, adsorption-desorption, ionexchange, oxidation-reduction, precipitation-dissolution, acid-base reactions, and microbial mediated reactions. In addition, user-specified reaction types can be programmed into the model. Any reaction can be treated as fast/equilibrium or slow/ kinetic reaction. An equilibrium reaction is modeled with an infinite rate governed by a mass action equilibrium equation or by a user-specified algebraic equation. Programmed kinetic reaction rates include multiple Monod kinetics, nth order empirical, and elementary formulations. In addition, user-specified rate formulations can be programmed into the model. No existing models to our knowledge offer these simultaneous features. Furthermore, most available reaction-based models assume chemical components a priori so that reactions can be written in basic (canonical) forms and implicitly assume that fast equilibrium reactions occur only for homogeneous reactions. The decoupling of fast reactions from slow reactions lessens the stiffness typical of these systems. The explicit enforcement of mass conservation overcomes the mass conservation error due to numerical integration errors. The removal of redundant fast reactions alleviates the problem of singularity. The exclusion of irrelevant slow reactions eliminates the issue of exporting their problematic rate ...