Analytical solutions for multiple species reactive transport in multiple dimensions

Many numerical computer codes used to simulate multi-species reactive transport and biodegradation have been developed in recent years. Such numerical codes must be validated by comparison of the numerical solutions with an analytical solution. In this paper, a method for deriving analytical solutions of the partial differential equations describing multiple species multi-dimensional transport with first-order sequential reactions is presented. Although others have developed specific solutions of multi-species transport equations, here a more general analytical approach, capable of describing any number of reactive species in multiple dimensions is derived. A substitution method is used to transform the multi-species reactive transport problem to one that can be solved using previously published single-species solutions for various initial and boundary conditions. One- and three-dimensional examples are presented to illustrate the steps involved in extending single-species solutions to a four-species system with sequential first-order reactions.     

Transport of a decay chain in homogenous porous media: analytical solutions

With the aid of integral transforms, analytical solutions for the transport of a decay chain in homogenous porous media are derived. Unidirectional steady-state flow and radial steady-state flow in single and multiple porosity media are considered. At least in Laplace domain, all solutions can be written in closed analytical formulae. Partly, the solutions can also be inverted analytically. If not, analytical calculation of the steady-state concentration distributions, evaluation of temporal moments and numerical inversion are still possible. Formulae for several simple boundary conditions are given and visualized in this paper. The derived novel solutions are widely applicable and are very useful for the validation of numerical transport codes.     

Semi-analytical solution of two-dimensional sharp interface LNAPL transport models

In this paper, we present semi-analytical solutions for two-dimensional equations governing transport of Light Non-Aqueous Phase Liquids (LNAPL) in unconfined aquifers. The proposed model is based on sharp interface displacement and steady groundwater flow assumptions, where both the water–LNAPL interface and the LNAPL–air interface are represented as sharp interfaces. In the case of steady groundwater flow, these equations can be reduced to a two-dimensional nonlinear solute transport equation, with the LNAPL thickness in the free product lens being the primary unknown variable. The linearized form of this solute transport equation falls into the category of two-dimensional transport equation with time-dependent dispersion coefficients. This equation can be solved analytically for an infinite domain region. In this paper, the general form of the analytical solution for the transport equation, as well as the solutions for some specific cases are presented. To demonstrate the utility of the proposed solution, numerical results obtained for two example problems are discussed and presented comparatively with a finite-element solution and other more restrictive solutions available in the literature. Although the solutions discussed in this paper have some simplifying assumptions, such as sharp-interfaces between fluid phases, steady groundwater flow and homogeneous aquifer properties, the semi-analytical solutions presented in this study may be used effectively as bench mark solutions in evaluating LNAPL migration in the subsurface. These solutions are simple and cost effective to implement and may be used in the calibration of other more complex numerical solutions that can be found in the literature.     

Kinematic analysis of solute mass flows in rock fractures with spatially random parameters

Field data of physical properties in heterogeneous crystalline bedrock, like porosity and fracture aperture, is associated with uncertainty that can have a significant impact on the analysis of solute transport in rock fractures. Solutions to the central temporal moments of the residence time probability density function (PDF) are derived in a closed form for a solute Dirac pulse. The solutions are based on a model that takes into account advection along the fracture plane, diffusion into the rock matrix and sorption kinetics in the rock matrix. The most relevant rock properties including fracture aperture and several matrix properties as well as flow velocity are assumed to be spatially random along transport pathways. The mass transport is first solved in a general form along one-dimensional pathways, but the results can be extended to multi-dimensional flows simply by substituting the expected travel time for inert water parcels. Based on data obtained with rock samples taken at Asp? Hard Rock Laboratory in Sweden, the solutions indicate that the heterogeneity of the rock properties contributes to increasing significantly both the variance and the skewness of the residence time probability density function for a pulse travelling in a fracture. The Asp? data suggests that the bias introduced in the variance of the residence time PDF by neglecting the effect of heterogeneity of the rock properties on the radionuclide migration is very large for fractures thinner than a few tenths of a millimetre.     

A numerical model for transient-hysteretic flow and solute transport in unsaturated porous media

A two-dimensional flow and transport model was developed for simulating transient water flow and nonreactive solute transport in heterogeneous, unsaturated porous media containing air and water. The model is composed of a unique combination of robust and accurate numerical algorithms for solving the Richards', Darcy flux, and advection-dispersion equations. The mixed form of Richards' equation is solved using a finite-element formulation and a modified Picard iteration scheme. Mass lumping is employed to improve solution convergence and stability behavior. The flow algorithm accounts for hysteresis in the pressure head-water content relationship. Darcy fluxes are approximated with a Galerkin and Petrov-Galerkin finite-element method developed for random heterogeneous porous media. The transport equation is solved using an Eulerian-Lagrangian method. A multi-step, fourth-order Runge-Kutta, reverse particle tracking technique and a quadratic-linear interpolation scheme are shown to be superior for determining the advective concentration. A Galerkin finite-element method is used for approximating the dispersive flux. The unsaturated flow and transport model was applied to a variety of rigorous problems and was found to produce accurate, mass-conserving solutions when compared to analytical solutions and published numerical results.     

Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform media

A new reactive transport modelling approach and examples of its application are presented, dealing with the impact of sorption/desorption kinetics on the spreading of solutes, e.g. organic contaminants, in groundwater. Slow sorption/desorption is known from the literature to be strongly responsible for the retardation of organic contaminants. The modelling concept applied in this paper quantifies sorption/desorption kinetics by an intra-particle diffusion approach. According to this idea, solute uptake by or release from the aquifer material is modelled at small scale by a "slow" diffusion process where the diffusion coefficient is reduced as compared to the aqueous diffusion coefficient due to (i) the size and shape of intra-particle pores and (ii) retarded transport of solutes within intra-particle pores governed by a nonlinear sorption isotherm. This process-based concept has the advantage of requiring only measurable model parameters, thus avoiding fitting parameters like first-order rate coefficients.In addition, the approach presented here allows for modelling of slow sorption/desorption in lithologically nonuniform media. Therefore, it accounts for well-known experimental findings indicating that sorptive properties depend on (i) the grain size distribution of the aquifer material and (ii) the lithological composition (e.g. percentage of quartz, sandstone, limestone, etc.) of each grain size fraction. The small-scale physico-chemical model describing sorption/desorption is coupled to a large-scale model of groundwater flow and solute transport. Consequently, hydraulic heterogeneities may also be considered by the overall model. This coupling is regarded as an essential prerequisite for simulating field-scale scenarios which will be addressed by a forthcoming publication.This paper focuses on mathematical model formulation, implementation of the numerical code and lab-scale model applications highlighting the sorption and desorption behavior of an organic contaminant (Phenanthrene) with regard to three lithocomponents exhibiting different sorptive properties. In particular, it is shown that breakthrough curves (BTCs) for lithologically nonuniform media cannot be obtained via simple arithmetic averaging of breakthrough curves for lithologically uniform media. In addition, as no analytical solutions are available for model validation purposes, simulation results are compared to measurements from lab-scale column experiments. The model results indicate that the new code can be regarded as a valuable tool for predicting long-term contaminant uptake or release, which may last for several hundreds of years for some lithocomponents. In particular, breakthrough curves simulated by pure forward modelling reproduce experimental data much better than a calibrated standard first-order kinetics reactive transport model, thus indicating that the new approach is of high quality and may be advantageously used for supporting the design of remediation strategies at contaminated sites where some lithocomponents and/or grain size classes may provide a long-term pollutant source.     

Multidimensional, multicomponent, subsurface reactive transport in nonuniform velocity fields: code verification using an advective reactive streamtube approach

High performance computing has made possible the development of high resolution, multidimensional, multicomponent reactive transport models that can be used to analyze complex geochemical environments. However, as increasingly complex processes are included in these models, the accuracy of the numerical formulation coupling the nonlinear processes becomes difficult to verify. Analytical solutions are not available for realistically complex problems and benchmark solutions are not generally available for specific problems. We present an advective reactive streamtube (ARS) transport technique that efficiently provides accurate solutions of nonlinear multicomponent reactive transport in nonuniform multidimensional velocity fields. These solutions can be compared with results from Eulerian-based advection-dispersion-reaction models to evaluate the accuracy of the numerical formulation used. The ARS technique includes mixed equilibrium and kinetic complexation and precipitation-dissolution reactions subject to the following assumptions: (1) transport is purely advective (i.e., no explicit diffusion or dispersion), and (2) chemistry is described by a canonical system of reactions that evolves with time and is unaffected by position in space. Results from the ARS technique are compared with results from the massively parallel, multicomponent reactive transport model MCTRACKER on a test problem involving irreversible oxidation of organic carbon and reaction of the oxidation products with two immobile mineral phases, gypsum and calcite, and fifteen aqueous complexes. Truncation error, operator splitting error, and the nonlinear transformation of these errors in the high-resolution reactive transport model are identified for this problem.     

Successive identification of biodegradation rates for multiple sequentially reactive contaminants in groundwater.

At the field scale, the biodegradation rate is usually estimated from analytical solutions to single species transport with first-order reactions, using measured data as input. Because many contaminants, e.g., chlorinated solvents, are degraded in a sequential pattern, with degradation products further reacting to produce new species, it is of great interest to quantify the transformation rate of every reaction. The conventional inverse solutions for identifying the transformation rates are limited to single species problems. In the present study, we propose a successive optimization approach to identify the biodegradation rate for each species by using a previously developed analytical solution to multi-species first-order reactive transport using data obtained at the field scale. By specifying a link between analytical solutions to sequentially reactive transport problems and optimization methods and assuming constant transport parameters (velocity, dispersivities, and retardation factors), the first-order transformation rates are optimized successively from parent species to its daughter species.     

A legislative view on science and predictive models

Preparation of European environmental policy needs detailed and consistent assessments. All possible and available means have to be applied to get the optimal solution for current and upcoming problems in the environment. As there will be a recognition of the increasing importance of linkages between the various aspects in the environment, a sound and appropriate method for tackling the integration problem is needed. One of the main features of Integrated Environmental Assessment (IEA) deals with the linkage between societal activities and the relevant environmental issues. The European Environment Agency (EEA) uses a chain of linkages between the driving forces within society (D), the pressure on the environment (P), the state of the environment itself (S), the impact on people and nature (I) and the desirable response (R). This approach is called the DPSIR chain. For each environmental issue a chain, including the linkages, could be developed by combining available knowledge and data. In fact it is a complex multi-dimensional system of relations between the chains belonging to the various aspects. Modelling the integration of all available scientific knowledge and data is the only way to deliver adequate information for the policy preparation process. The EEA made use of this approach for its assessments of Europe's environment. Every three years the EEA will publish an "Assessment" and a "State of the Environment". The latest publication was "Europe's Environment: The Second Assessment", in June 1998 (EEA,1998). The next will be the "State of the Environment 1998" (EEA,1998), to be published in 1999. A prospective analysis forms part of the preparation of this report, in which modelling, based on the same DPSIR-approach is essential. The above-mentioned publications contribute to the preparation of new policy lines within the European Community.     

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities (υ) and retardation factors (R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities (υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.     

Dispersive—diffusive transport of non-sorbed solute in multicomponent solutions

The composition of fuels, mixed-solvent wastes and other contaminants that find their way into the subsurface are frequently chemically complex. The dispersion and diffusion characteristics of multicomponent solutions in soil have rarely been compared to equivalent single-solute systems. The purpose of this work was to examine the diffusive and dispersive transport of single- and multi-component solutions in homogeneous porous media. The miscible displacement technique was used to investigate the transport behavior of ¹⁴C-labelled 2,4-dichlorophenoxyacetic acid (2,4-D) in two materials for which sorption of 2,4-D was minimal. Comparison of breakthrough curves collected for 2,4-D in single- and multi-component solutions shows that there is little, if any, difference in transport behavior over a wide range of pore-water velocities (70, 7, 0.66 and 0.06 cm h⁻¹). Thus, dispersivities measured with a non-sorbing single-solute solution should be applicable to multicomponent systems.     

Transport and Diffusion Calculations during Time- and Space-Varying Meteorological Conditions Using a Microcomputer

Small computers can now solve complex transport and diffusion problems. Making such calculations rapidly and on the spot would be very useful for decision makers during emergencies. This paper describes a microcomputer model that can simulate nonsteady state transport and diffusion and calculate mass-consistent flow fields in uneven terrain. Transport and diffusion calculations use a Puff model approach to simulate dispersion from a continuous point source. Three-dimensional wind fields are generated from linear combinations of solutions that are obtained ahead of time on a larger machine. A computer code has been written in BASIC and successfully run on an Apple II personal computer.     

Influence of soluble copper on the electrokinetic properties and transport of copper oxychloride-based fungicide particles

This article describes the influence of dissolved copper on the electrokinetic properties and transport of a copper oxychloride-based fungicide (COF) in porous media. The Zeta potential (ζ) of COF particles increases (viz. becomes less negative) with increasing concentration of Cu(2+) in the bulk solution. ζ decreases for COF when the electrolyte (NaNO(3)) concentration is raised from 1 to 10mM. This can be ascribed to ion correlation of Cu(2+) in the electrical double layer (EDL). COF transport tests in quartz sand columns showed the addition of Cu(2+) to the bulk solution to result in increased retention of the metal. Modelling particle deposition dynamics provided results consistent with kinetic attachment. Based on the effect of soluble Cu on colloid mobility, the transport of particulate and soluble forms of copper is coupled via the chemistry of pore water and colloid interactions. Mutual effects between cations and colloids should thus be considered when determining the environmental fate of particulate and soluble forms of copper in soil and groundwater (especially at copper-contaminated sites).     

Modeling radium and radon transport through soil and vegetation

A one-dimensional flow and transport model was developed to describe the movement of two fluid phases, gas and water, within a porous medium and the transport of 226Ra and 222Rn within and between these two phases. Included in this model is the vegetative uptake of water and aqueous 226Ra and 222Rn that can be extracted from the soil via the transpiration stream. The mathematical model is formulated through a set of phase balance equations and a set of species balance equations. Mass exchange, sink terms and the dependence of physical properties upon phase composition couple the two sets of equations. Numerical solution of each set, with iteration between the sets, is carried out leading to a set-iterative compositional model. The Petrov-Galerkin finite element approach is used to allow for upstream weighting if required for a given simulation. Mass lumping improves solution convergence and stability behavior. The resulting numerical model was applied to four problems and was found to produce accurate, mass conservative solutions when compared to published experimental and numerical results and theoretical column experiments. Preliminary results suggest that the model can be used as an investigative tool to determine the feasibility of phytoremediating radium and radon-contaminated soil.     

Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations

In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovic and de Schepper [Frolkovic, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference in obtaining a unique stationary solution for this problem. The results of our study also indicate that the density differences are critical in the solution of the Elder problem and that high density differences lead to the physical instability that is inherent with this problem. Other than the physical instability associated with the level of density differences used in the Elder problem, the following two points should be considered in solving the Elder problem in a consistent manner: (i) strict attention should be paid to the vertical grid Peclet number in developing the criteria for convergent grid selection; and, (ii) with a globally continuous velocity calculation stable solutions can be obtained at lower grid densities.     

Development and analysis of an adaptive transport scheme

An approach to solving the advection dominated atmospheric mass transport problem which adaptively constructs and updates the computational mesh is implemented and analyzed. The formulation of the mesh adaptation algorithm allows for information from other model processes as well as transport to influence mesh refinement. Comparisons to other methods are limited to a pure advection test problem. The scheme is based on a Petrov–Galerkin finite element method using quadratic interpolating polynomials over triangular elements. The temporal component of the transport equation is discretized using the Crank–Nicolson method. A single time step is applied to the entire domain regardless of mesh refinement levels. Two means of upwind biasing the solution via modification of the finite element weighting functions are investigated. Criteria for refining and unrefining the mesh based on advective flux and an estimate of the error are proposed and compared. The effects of varying the number of time steps taken between mesh refinements are analyzed. Results for adapted mesh solutions of a test problem are shown to be superior to corresponding uniform mesh finite element solutions. Comparisons to several other popular uniform mesh advection schemes indicate either equal or improved accuracy for the adapted mesh solutions.     

Differences in uptake and translocation of selenate and selenite by the weeping willow and hybrid willow

BACKGROUND, AIM, AND SCOPE: Due to its essentiality, deficiency, and toxicity to living organisms and the extensive use in industrial activities, selenium (Se) has become an element of global environmental and health concern. Se removal from contaminated sites using physical, chemical, and engineering techniques is quite complicated and expensive. The goal of this study was to investigate uptake and translocation of Se in willows and to provide quantitative information for field application whether Se phytoremediation is feasible and ecologically safe. MATERIALS AND METHODS: Intact pre-rooted plants of hybrid willows (Salix matsudana Koidz x alba L.) and weeping willows (Salix babylonica L.) were grown hydroponically and treated with selenite or selenate at 24.0 +/- 1 degrees C for 144 h. Removal of leaves was also performed as a treatment to quantify the effect of transpiration on translocation and volatilization of Se. At the end of the study, total Se in the hydroponic solution and in different parts of plant tissues was analyzed quantitatively by hydride generation-atomic fluorescence spectrometry. The capacity of willows to assimilate both chemical forms of Se was also evaluated using detached leaves and roots in sealed glass vessels in vivo. Translocation efficiency of Se in both plants was estimated. RESULTS: Significant amounts of the applied selenite and selenate were eliminated from plant growth media by willows during the period of incubation. Both willows showed a significantly higher removal rate for selenate than for selenite (p < 0.05). Substantial differences existed in the distribution of both chemical forms of Se in plant materials: lower stems and roots were the major sites for accumulation of selenite and selenate, respectively. Translocation efficiency for selenite was significantly higher than that for selenate in both willow species (p < 0.01). Compared to the intact trees, remarkable decrease in the removal rate of both chemical forms of Se was found for willows without any leaves (p < 0.01). Volatilization of Se by plant leaves was estimated to be approximately 10% of the total applied selenite or selenate. Significant reduction (>20%) of selenate was observed in the sealed vessel with excised roots of willows, whereas trace amounts of selenite were eliminated from the hydroponic solution in the presence of roots. Detached leaves from neither of them reduced the concentration of selenite or selenate in the solution. DISCUSSION: Due to the significant difference in the removal rate and the distribution of the two chemical forms of Se in plant materials, the conversion of selenate to selenite in hydroponic solution prior to uptake and within plant tissues is unlikely. An independent uptake and translocation mechanisms are likely to exist for each Se chemical species. Uptake of selenate is mediated possibly through an active transport mechanism, whereas that of selenite may possibly depend on plant transpiration. Uptake velocities of selenite are linear (zero-order kinetics), while selenate removal processes obey first-order kinetics. In experiments with detached leaves in closed bottles, the cuticle of leaves was the major obstacle to extract both chemical forms of Se from the hydroponic solution. Phytovolatilization is a biological process playing an important role in Se removal. CONCLUSIONS: Although faster removal rates of selenate than selenite from plant growth media were observed by both willow species, selenite in plant materials was more mobile than selenate. Significant decrease in removal rates of both chemical forms of Se was detected for willows without any leaves. Significant differences in extraction, assimilation and transport pathways for selenite and selenate exist in willow trees. RECOMMENDATIONS AND PERSPECTIVES: Phytoremediation of Se is an attractive approach of cleaning up Se contaminated environmental sites. More detailed investigation on the assimilation of Se in plant roots and transport in tissues will provide further biochemical evidence to explain the differences in uptake and translocation mechanisms between selenite and selenate in willows. A relevant phytoremediation scheme can then be designed to clean up Se contaminated sites. Willows show a great potential for uptake, assimilation and translocation of both selenite and selenate. Phytotreatment of Se is potentially an efficient and practical technology for cleaning up contaminated environmental sites.     

On efficient numerical solution of one-dimensional convection–diffusion equations in modelling atmospheric processes

A new approach is proposed to the numerical solution of one-dimensional convection–diffusion equations that arise in modelling atmospheric processes and air pollution modelling. The technique is based on upstream-type difference approximations for first-order derivatives and non-standard difference approximations for second-order derivatives of convection–diffusion equations. This approach leads to the significant qualitative improvements in the numerical solutions behaviour. The relative contribution of convection and diffusion is directly incorporated into the corresponding numerical scheme in such a way that large spatial grids can be taken without affecting solution stability. The method is compared with the contemporary computational schemes for solving problems with severe internal and boundary gradients and is shown to be stable and computationally efficient. The results of a numerical experiment are given.     

A hybrid micro-scale model for transport in connected macro-pores in porous media

This paper presents a hybrid model for transport in connected macro-pores in porous media. A pore-scale model is used to parameterize the hybrid model. The hybrid model explicitly models the advection and diffusion of species in the connected macro-pores and treats the porous media around the connected macro-pores as a continuum with effective transport properties. The pore-scale model is used to calculate the effective transport properties of the porous continuum. This approach negates the need to calibrate the hybrid model against experimental data, which is common for continuum-scale models of porous media, and allows an arbitrary microstructure to be considered. The paper presents the multi-scale modeling approach along with the details of the hybrid and pore-scale models. Validation of the model is also presented along with several case studies investigating the applicability of the multi-scale modeling approach to different geometries and transport conditions. The case studies show that the multi-scale modeling approach is accurate for various connected macro-pore geometries given that the porosity of the porous medium around the connected macro-pores is sufficiently small. The accuracy of the hybrid model decreases with increasing porosity of the matrix.     

Modeling hydrocarbon biodegradation in tidal aquifers with water-saturation and heat inhibition effects.

A model is developed for hydrocarbon biodegradation, which includes saturated and unsaturated flow, multi-species transport, heat transport, and bacterial growth processes. Numerical accuracy of the model was tested against analytical solutions. The model was also verified against laboratory results for a saturated-flow problem and reasonable match was obtained. Expressions are proposed for inhibition due to water content and temperature fluctuations. Bioactivities under cyclic water content variation were studied under no-flow conditions. A quantitative approach was used to reconcile some of the apparent contradictory conclusions regarding the efficiency of biodegradation of soils under wetting and drying conditions. The efficiency depends on the nature of the oxygenation process. For cases involving the presence of dissolved oxygen and the absence of O2 vapor, subjecting the soil to constant water content close to its optimal value for degradation is most efficient. However, wetting and drying can enhance degradation if O2 is only provided through aeration or direct contact between air and the medium. Also presented are the results of a typical field application of the model and a discussion of the effects of tides, saturation inhibition, and heat inhibition. Other inhibition factors, such as pH or salinity, can be easily incorporated in the formulation. The quantitative approach developed here can be used in assessing bioremediation not only in tidal aquifers but also in areas where water-table or temperature effects are of significance. The approach can be useful in the design of remediation strategies under water-flow or no-flow conditions involving water content and temperature fluctuations.