The two-dimensional advection-dispersion equations coupled with sequential first-order decay reactions involving arbitrary number of species in groundwater system is considered to predict the two-dimensional plume behavior of decaying contaminant such as radionuclide and dissolved chlorinated solvent. Generalized analytical solutions in compact format are derived through the sequential application of the Laplace, finite Fourier cosine, and generalized integral transform to reduce the coupled partial differential equation system to a set of linear algebraic equations. The system of algebraic equations is next solved for each species in the transformed domain, and the solutions in the original domain are then obtained through consecutive integral transform inversions. Explicit form solutions for a special case are derived using the generalized analytical solutions and are compared with the numerical solutions. The analytical results indicate that the analytical solutions are robust, accurate and useful for simulation or screening tools to assess plume behaviors of decaying contaminants.

Experimental and theoretical studies have been undertaken to understand the fate and transport of dissolved hazardous substances in subsurface environments because human health is threatened by a wide spectrum of contaminants in groundwater and soil. Analytical models are essential and efficient tools for understanding pollutants behavior in subsurface environments. Several analytical solutions for single-species transport problems have been reported for simulating the transport of various contaminants (Batu, 1989, 1993, 1996; Chen et al., 2008a, b, 2011; Gao et al., 2010, 2012, 2013; Leij et al., 1991, 1993; Park and Zhan, 2001; Pérez Guerrero and Skaggs, 2010; Pérez Guerrero et al., 2013; van Genuchten and Alves, 1982; Yeh, 1981; Zhan et al., 2009; Ziskind et al., 2011). Transport processes of some contaminants such as radionuclides, dissolved chlorinated solvents and nitrogen generally involve a series of first-order or pseudo first-order sequential decay chain reactions. During migrations of decaying contaminants, mobile and toxic successor products may sequentially form and move downstream with elevated concentrations. Single-species analytical models do not permit transport behaviors of successor species of these decaying contaminants to be evaluated. Analytical models for multispecies transport equations coupled with first-order sequential decay reactions are useful tools for synchronous determination of the fate and transport of the predecessor and successor species of decaying contaminants. However, there are few analytical solutions for coupled multispecies transport equations compared to a large body of analytical solutions in the literature pertaining to the single-species advective-dispersive transport subject to a wide spectrum of initial and boundary conditions.

Mathematical approaches have been proposed in the literature to derive a limited number of one-dimensional analytical solutions or semi-analytical solutions for multispecies advective–dispersive transport equations sequentially coupled with first-order decay reactions. These include direct integral transforms with sequential substitutions (Cho, 1971; Lunn et al., 1996; van Genuchten, 1985; Mieles and Zhan, 2012), decomposition by change-of-variables with the help of existing single-species analytical solutions (Sun and Clement, 1999; Sun et al., 1999a, b), Laplace transform combined with decomposition of matrix diagonization (Quezada et al., 2004; Srinivasan and Clememt, 2008a, b), decomposition by change-of-variables coupled with generalized integral transform (Pérez Guerrero et al., 2009, 2010), sequential integral transforms in association with algebraic decomposition (Chen et al., 2012a, b).

Multi-dimensional solutions are needed for real world applications, making them more attractive than one-dimensional solutions. Bauer et al. (2001) presented the first set of semi-analytical solutions for one-, two-, and three-dimensional coupled multispecies transport problem with distinct retardation coefficients. Explicit analytical solutions were derived by Montas (2003) for multi-dimensional advective-dispersive transport coupled with first-order reactions for a three-species transport system with distinct retardation coefficients of species. Quezada et al. (2004) extended the Clement (2001) strategy to obtain Laplace-domain solutions for an arbitrary decay chain length. Most recently, Sudicky et al. (2013) presented a set of semi-analytical solutions to simulate the three-dimensional multi-species transport subject to first-order chain-decay reactions involving up to seven species and four decay levels. Basically, their solutions were obtained species by species using recursion relations between target species and its predecessor species. For a straight decay chain, they derived solutions for up to four species and no generalized expressions with compact formats for any target species were obtained. Note that their solutions were derived for the first-type (Dirichlet) inlet conditions which generally bring about physically improper mass conservation and significant errors in predicting the concentration distributions especially for a transport system with a large longitudinal dispersion coefficient (Barry and Sposito, 1988; Parlange et al., 1992). Moreover, in addition to some special cases, the numerical Laplace transforms are required to obtain the original time domain solution. Besides the straight decay chain, the analytical model by Clement (2001) and Sudicky et al. (2013) can account for more complicated decay chain problems such as diverging, converging and branched decay chains.

Schematic representation of two-dimensional transport of decaying contaminants in a uniform flow field with flux boundary source located at of the inlet boundary.

Based on the aforementioned reviews, this study presents a parsimonious explicit analytical model for two-dimensional multispecies transport coupled by a series of first-order decay reactions involving an arbitrary number of species in groundwater system. The derived analytical solutions have four salient features. First, the third-type (Robin) inlet boundary conditions which satisfy mass conservation are considered. Second, the solution is explicit, thus solution can be easily evaluated without invoking the numerical Laplace inversion. Third, the generalized solutions with parsimonious mathematical structures are obtained and valid for any species of a decay chain. The parsimonious mathematical structures of the generalized solutions are easy to code into a computer program for implementing the solution computations for arbitrary target species. Fourth, the derived solutions can account for any decay chain length. The explicit analytical solutions have applications for evaluation of concentration distribution of arbitrary target species of the real-world decaying contaminants. The developed parsimonious model is robustly verified with three example problems and applied to simulate the multispecies plume migration of dissolved radionuclides and chlorinated solvent.

This study considers the problem of decaying contaminant plume migration. The
source zone is located in the upstream of groundwater flow. The source zone
can represent leaching of radionuclide from a radioactive waste disposal
facility or release of chlorinated solvent from the residual NAPL phase into
the aqueous phase. After these decaying contaminants enter the aqueous
phase, they migrate by one-dimensional advection with flowing groundwater
and by simultaneously longitudinal and transverse dispersion processes.
While migrating in the groundwater system, the contaminants undergo linear
isothermal equilibrium sorption and a series of sequential first-order
decaying reactions. Sudicky et al. (2013) provided the detailed modeling
scenario. The scenario considered in this study can be ideally described as
shown in Fig. 1. A steady and uniform velocity in the

The initial and boundary conditions for solving Eqs. (1a) and (1b) are

Comparison of spatial concentration profiles of four species along
the longitudinal direction (

Equation (3) means that a third-type boundary condition satisfying mass
conservation at the inlet boundary is considered. Equation (4) considers the
concentration gradient to be zero at the exit boundary based on the mass
conservation principle. Such a boundary condition has been widely used for
simulating solute transport in a finite-length system. Equations (5) and (6)
assume no solute flux across the lower and upper boundaries. It is noted
that in Eq. (3), we assume arbitrary time-dependent sources of species

Comparison of spatial concentration profiles of four species along
the transverse direction (

Transport parameters used for convergence test example 1 involving the four-species radionuclide decay chain problem used by van Genuchten (1985).

Values for coefficients of Bateman-type boundary source for four-species transport problem used by van Genuchten (1985).

Equations (1)–(6) can be expressed in dimensionless form as

Comparison of spatial concentration profiles of four species along
the transverse direction (

Following Chen et al. (2012a, b), the generalized analytical solutions
in compact formats can be obtained as follows (with detailed derivation
provided in Appendix A)

Concise expressions for arbitrary target species such as described in Eqs. (13) to (15) facilitate the development of a computer code for implementing the computations of the analytical solutions.

Comparison of spatial concentration profiles of four species along
the longitudinal direction (

The generalized solutions of Eq. (13) accompanied by two corresponding
auxiliary functions

Comparison of spatial concentration profiles of four species along
the transverse direction (

Solution convergence of each species concentration at transect of
inlet boundary (

Solution convergence of each species concentration at transect of

By substituting Eq. (16b) into Eqs. (13)–(15), we obtain

Comparison of spatial concentration profiles of four species along
the transverse direction (

Comparison of spatial concentration profiles of ten-species along

Based on the special-case analytical solutions in Eq. (17) supported by two
auxiliary functions, defined in Eqs. (18) and (19), a computer code was
developed in FORTRAN 90 language with double precision. The details of the
FORTRAN computer code are described in Supplement. The derived analytical
solutions in Eqs. (17)–(19) consist of summations of double infinite series
expansions for the finite Fourier cosine and generalized integral transform
inversions, respectively. It is straightforward to sum up these two infinite
series expansions term by term. To avoid time-consuming summations of these
infinite series expansions, the convergence tests should be routinely
executed to determine the optimal number of the required terms for evaluating
analytical solutions to the desired accuracies. Two-dimensional four-member
radionuclide decay chain

Effects of physical processes and chemical reactions on the
concentration contours of four-species at

In order to determine the optimal term number of series expansions for the finite Fourier cosine transform inverse to achieve accurate numerical evaluation, we specify a sufficiently large number of series expansions for the generalized transform inverse so that the influence of the number of series expansions for the generalized integral transform inverse on convergence of series expansion for finite Fourier cosine transform inverse can be excluded. A similar concept is used when investigating the required number of terms in the series expansions for the generalized integral transform inverse. An alternative approach is conducted by simultaneously varying the term numbers of series expansions for the generalized integral transform inverse and the finite Fourier cosine transform inverse.

Tables 3, 4 and 5 give results of the convergence tests up to 3 decimal
digits of the solution computations along the three transects (inlet
boundary at

Spatial concentration contours of five-species at

To further examine the series convergence behavior, example 2 considers a
transport system of large aspect ratio
(

Solution convergence of each species concentration at transect of
exit boundary (

Using the required numbers determined from the convergence test, the computational time for evaluation of the solutions at 50 different observations only takes 3.782, 11.325, 23.95 and 67.23 s computer clock time on an Intel Core i7-2600 3.40 MHz PC for species 1, 2, 3, and 4 in the comparison of example 1.

Solution convergence of each species concentration at transect of
inlet boundary (

Three comparison examples are considered to examine the correctness and robustness of the analytical solutions and the accuracy of the computer code. The first comparison example is the four-member radionuclide transport problem used in the convergence test example 1. The second comparison example considers the four-member radionuclide transport problem used in the convergence test example 2. The third comparison example is used to test the accuracy of the computer code for simulating the reactive contaminant transport of a long decay chain. The three comparison examples are executed by comparing the simulated results of the derived analytical solutions with the numerical solutions obtained using the Laplace transformed finite difference (LTFD) technique first developed by Moridis and Reddell (1991). A computer code for the LTFD solution is written in FORTTRAN language with double precision. The details of the FORTRAN computer code are described in Supplement.

Figures 2, 3 and 4 depicts the spatial concentration distribution along one
longitudinal direction (

The third example involves a 10 species decay chain previously presented by
Srinivasan and Clement (2008a) to evaluate the performance of their
one-dimensional analytical solutions. The relevant model parameters are
summarized in Tables 8 and 9. Our computer code is also compared against the
LTFD solutions for this example. Figure 8 depicts the spatial concentration
distribution at

Physical processes and chemical reactions affect the extent of contaminant
plumes, as well as concentration levels. To illustrate how the physical
processes and chemical reactions affect multispecies plume development, we
consider the four-member radionuclide decay chain used in the previous
convergence test and solution verification. The model parameters are the
same, except that the longitudinal (

Solution convergence of each species concentration at transect of

Transport parameters used for verification example 2 involving the ten-species transport problem used by Srinivasan and Clement (2008b).

Coefficients of Bateman-type boundary source for ten-species transport problem used by Srinivasan and Clement (2008b).

Figure 9 illustrates the spatial concentration of four species at

Transport parameters used for example application involving the five-species dissolved chlorinated solvent problem used by BIOCHLOR.

Natural attenuation is the reduction in concentration and mass of the
contaminant due to naturally occurring processes in the subsurface
environment. The process is monitored for regulatory purposes to demonstrate
continuing attenuation of the contaminant reaching the site-specific
regulatory goals within reasonable time, hence, the use of the term
monitored natural attenuation (MNA). MNA has been widely accepted as a
suitable management option for chlorinated solvent contaminated groundwater.
Mathematical models are widely used to evaluate the natural attenuation of
plumes at chlorinated solvent sites. The multispecies transport analytical
model developed in this study provides an effective tool for evaluating
performance of the monitoring natural attenuation of plumes at a chlorinated
solvent site because a series of daughter products produced during
biodegradation of chlorinated solvent such as PCE

A study of 45 chlorinated solvent sites by McGuire et al. (2004) found that
mathematical models were used at 60 % of these sites and that the public
domain model BIOCHLOR (Aziz et al., 2000) provided by the Center for
Subsurface Modeling Support (CSMoS) of USEPA was the most commonly used
model. An illustrated example from BIOCHLOR manual (Aziz et al., 2000) is
considered to demonstrate the application of the developed analytical model.
This example application demonstrated that BIOCHLOR can reproduce plume
movement from 1965 to 1998 at the contaminated site of Cape Canaveral Air
Station, Florida. The simulation conditions and transport parameters for
this example application are summarized in Table 10. Constant source
concentrations rather than exponentially declining source concentration of
five-species chlorinated solvents are specified in the

Coefficients of Bateman-type boundary source used for example application involving the five-species dissolved chlorinated solvent problem used by BIOCHLOR.

We present an analytical model with a parsimonious mathematical format for two-dimensional multispecies advective-dispersive transport of decaying contaminants such as radionuclides, chlorinated solvents and nitrogen. The developed model is capable of accounting for the temporal and spatial development of an arbitrary number of sequential first-order decay reactions. The solution procedures involve applying a series of Laplace, finite Fourier cosine and generalized integral transforms to reduce a partial differential equation system to an algebraic system, solving for the algebraic system for each species, and then inversely transforming the concentration of each species in transformed domain into the original domain. Explicit special solutions for Bateman type source problems are derived via the generalized analytical solutions. The convergence of the series expansion of the generalized analytical solution is robust and accurate. These explicit solutions and the computer code are compared with the results computed by the numerical solutions. The two solutions agree well for a wide spectrum of concentration variations for three test examples. The analytical model is applied to assess the plume development of radionuclide and dissolved chlorinated solvent decay chain. The results show that dispersion only moderately modifies the size of the plumes, without altering the relative order of the plume sizes of different contaminant. It is suggested that retardation coefficients, decay rate constants and the predecessor species plume distribution mainly govern the order of plume size in groundwater. Although there are a number of numerical reactive transport models that can account for multispecies advective-dispersive transport, our analytical model with a computer code that can directly evaluate the two-dimensional temporal-spatial concentration distribution of arbitrary target species without involving the computation of other species. The analytical model developed in this study effectively and accurately predicts the two-dimensional radionuclide and dissolved chlorinated plume migration. It is a useful tool for assessing the ecological and environmental impact of the accidental radionuclide releases such as the Fukushima nuclear disaster where multiple radionuclides leaked through the reactor, subsequently contaminating the local groundwater and ocean seawater in the vicinity of the nuclear plant. It is also a screening model that simulates remediation by natural attenuation of dissolved solvents at chlorinated solvent release sites. It should be noted the derived analytical model still has its application limitations. The model cannot handle the site with non-uniform groundwater flow or with multiple distinct zones. Furthermore, the developed model cannot simulate the more complicated decay chain problems such as diverging, converging and branched decay chains. The analytical model for more complicated decay chain problems can be pursued in the near future.

In this appendix, we elaborate on the mathematical procedures for deriving the analytical solutions.

The Laplace transforms of Eqs. (7a), (7b), (9)–(12) yield

The finite Fourier cosine transform is used here because it satisfies the
transformed governing equations in Eqs. (A1a) and (A2b) and their
corresponding boundary conditions in Eqs. (A4) and (A5). Application of the
finite Fourier cosine transform on Eqs. (A1)–(A3) leads to

Using changes-of-variables, similar to those applied by Chen and Liu (2011),
the advective terms in Eqs. (A8a) and (A8b) as well as nonhomogeneous terms
in Eq. (A9) can be easily removed. Thus, substitutions of the
change-of-variable into Eqs. (A8a), (A8b), (A9) and (A10) result in
diffusive-type equations associated with homogeneous boundary conditions

As detailed in Ozisik (1989), the generalized integral transform pairs for
Eqs. (A13a) and (A13b) and its associated boundary conditions Eqs. (A14) and
(A15) are defined as

The generalized integral transforms of Eqs. (13a) and
(13b) give

Solving for Eqs. (A20) and (A21) algebraically for each species,

where

The solutions in the original domain are obtained by a series of integral transform inversions in combination with changes-of-variables.

The inverse generalized integral transform of Eq. (A26) gives

The finite Fourier cosine inverse transform of Eq. (A28) results in

The analytical solutions in the original domain will be completed by taking
the Laplace inverse transform of Eq. (A29).

Thus, the Laplace inverse of

Putting Eq. (A34) into Eq. (A2) we can obtain the following form:

Note that Eq. (A33) is invalid for some of

The generalized formulae for the cases with some of

The authors are grateful to the Ministry of Science and Technology, Republic of China, for financial support of this research under contract MOST 103-2221-E-0008-100. The authors thanks three anonymous referees for their helpful comments and suggestions. Edited by: M. Giudici