WaterSubject

Transport problems involving sequentially decaying contaminants are frequently analyzed by groundwater hydrologists to assess water quality issues associated with environmental and health hazards. Examples of sequentially decaying contaminants include radioactive waste materials, chlorinated solvents, and nitrogenous species. Several types of models, using both analytical and numerical procedures, have been formulated for solving these sequentially coupled reactive transport problems and . Although numerical models are capable of solving complex and heterogeneous problems, their performance often needs to be tested against experimental datasets or analytical models. Experimental simulations of complex reactive transport problems are not only time consuming but can also be expensive. Therefore, analytical models provide a convenient, cost-effective alternative to test and validate numerical formulations . Furthermore, analytical models also provide computationally efficient screening tools for simulating the fate and transport of reactive contaminants in groundwater systems and.

The analytical solution given by McLaren and McLaren , which describes the steady-state, one-dimensional transport of a five species nitrogen chain, is one of the first multi-species solutions derived for solving sequentially coupled reactive transport problems. This work assumed that the transport was governed only by advection, and the effects of dispersion and sorption were ignored. Cho developed explicit analytical solutions to a three species transport problem that was subjected to advection, dispersion, linear equilibrium sorption and coupled through sequential first-order reactions. Explicit analytical solutions were obtained using Laplace transform procedures for the Dirichlet boundary condition. Misra et al. derived semi-analytical solutions to a problem similar to the one solved by Cho using a pulse source boundary. One of the limitations of these solutions is that only the first species in the chain was subjected to sorption. Burkholder and Rosinger and Lester et al. developed solutions for the advective dispersive transport of radionuclide chains subjected to linear equilibrium sorption. Explicit analytical solutions were presented for a three species problem involving distinct retardation factors for each species, for both impulse and decaying-band release boundary conditions. In addition, they also provided solutions for the case of no dispersion and for the case of identical retardation factors.

Harada et al. published a research report presenting a general format for obtaining semi-analytical solutions to sequentially coupled one-dimensional reactive transport problems of arbitrary chain lengths subjected to arbitrary source release modes. However, one of the major limitations of the solution strategy was that, the semi-analytical solution for a given species in the chain required the computation of its entire predecessor species. This would result in computationally inefficient algorithms especially when analyzing transport problems involving long reactive chains. Harada et al. and Higashi and Pigford also provided explicit closed-form solutions for a set of purely advective (no dispersion) transport problems with various types of boundary conditions.

Gurehian and Jansen presented an analytical solution to a transport problem involving a three member, first-order decay chain in a multi-layered system, subjected to advection, dispersion and linear equilibrium sorption processes for both continuous and band source release conditions. Convolution theorems and Laplace transform techniques were used to obtain semi-analytical solutions for the case involving both advective and dispersive transport and explicit closed-form analytical solutions for the case involving non-dispersive transport. van Genuchten developed explicit analytical solutions to model a sequentially coupled four species transport problem governed by advection, dispersion and linear equilibrium sorption processes involving, first-order reactions. It was assumed that all the species had distinct retardation factors. One of the key contributions of this work is that it considered both Dirichlet and Cauchy boundary conditions. Furthermore, van Genuchten developed a robust computer code (CHAIN) for implementing his analytical solution.

Angelakis et al. developed a semi-analytical solution to a sequentially coupled two-species reactive transport problem governed by advection, dispersion and linear equilibrium sorption subjected to Dirichlet boundary condition. The transport problem assumed that the reactions were first-order and each of the species had different dispersion coefficients and distinct retardation factors. The authors also demonstrated that when the dispersion coefficients of both the species were equal, their solution reduced to the closed-form solution similar to the solutions presented by Cho and Misra et al. Furthermore, the authors also provided solutions for the no dispersion (pure advection) case. Angelakis et al. developed an interesting semi-analytical solution for a problem involving the coupled transport of two solutes and a gaseous product in soils. The solute migration was governed by advection, dispersion, linear equilibrium sorption and sequential first-order reaction, whereas the gas migration was governed by diffusive transport coupled with reversible linear equilibrium dissolution.

Lunn et al. solved a three-species transport problem, which was similar to the Cho problem, using the Fourier transform method. The authors demonstrated that the use of Fourier transforms enabled them to solve problems having non-zero initial conditions by solving two special case problems. Khandelwal and Rabideaudeveloped semi-analytical solutions for a three species, sequentially-coupled, first-order reactive transport problem. The key contribution of this work was that they addressed cases involving linear, non-equilibrium sorption mechanisms. Eykholt and Li developed a solution method based on kinetic response functions to solve a linearly coupled non-sequential reactive transport problem having different retardation factors. Although, there was no restriction on the number of species in the system, this method required numerical procedures to evaluate the final solution. Furthermore, for the case of the non-ideal plug flow scenarios (advective dispersive transport), the accuracy of this method appears to decrease with decrease in Peclet number.

Sun et al. developed a method that can solve multi-species advective dispersive transport equations coupled with sequential first-order reactions involving arbitrary number of species for different types of initial and boundary conditions. Their method was based on the use of a transformation format to uncouple the system of equations, which could then be solved analytically in the transformed domain. The final solutions are obtained by retransforming the solutions to the original domain. Later, Sun et al. extended the transformation format to solve problems involving a combination of serial and parallel reactions. Clement presented a more general and fundamental approach to derive the Sun et al. solution by employing the similarity transformation method. The approach presented by Clement can solve problems involving serial, parallel, converging, diverging and/or reversible first-order reaction network. However, all of these methods are only applicable for solving problems involving identical retardation factors.

Bauer et al. presented a method to solve one-, two-, and three-dimensional sequentially coupled reactive transport problems with distinct retardation factors. This method was based on transforming the system of equations to a Laplace domain and then obtaining a set of fundamental solutions to each of the equations in the transformed domain. The specific solutions in the Laplace domain can then be obtained through a linear combination of the fundamental solutions. However, in order to accomplish this, the fundamental solutions must be linearly independent. Finally, the Laplace domain solutions can be transformed back to the time domain using the inverse Laplace transform procedure, which could be accomplished either analytically or numerically. Although this method can be applied to solve different types of boundary conditions, the solution procedure is mathematically tedious; specifically obtaining analytical inverse transform expressions for long chain lengths can be a challenge.

Montas developed an analytical procedure to solve a three species, multi-dimensional transport problem coupled by a first-order, non-sequential reaction network subject to a pulse type boundary condition. This procedure involved obtaining a basis solution (of a convoluted form) for the transport equation and then evaluating the basis solution using Laplace transforms. One of the key advantages of this procedure is that it can model transport problems with distinct retardation factors. However, as mentioned earlier, this solution was limited to a three species system.

Quezada et al. extended the approach given by Clement and developed a method that can solve multi-species transport equations coupled with a network of first-order reactions involving distinct retardation factors. This method involves transforming the system of governing equations to a Laplace domain and then solving the transformed system of equations using the Clement pproach. The solutions in the Laplace domain are then retransformed to the time domain using an inverse Laplace transform procedure. One of the key limitations of this approach is that, except for a simple two species transport problem, the solutions presented by Quezada et al. are in general semi-analytical since they require a numerical inverse Laplace transform routine to evaluate them.

Our literature review indicates that one-dimensional reactive transport equations coupled through sorption and sequential first-order reactions, have explicit closed-form analytical solutions only for short chains, up to four species. To model transport problems involving longer reaction chains, one has to either use semi-analytical solutions or purely numerical solutions. In this paper, we develop closed-form analytical solutions for the sequential decay problem involving arbitrary number of species subjected to a generic exponentially decaying Bateman-type source boundary, and spatially varying initial conditions.