Domenico Solution Is It Valid?

Transcription

1 Domenico Solution Is It Valid? by V. Sriniasan 1,T.P.Clement, and K.K. Lee 3 Abstract The Domenico solution is widely used in seeral analytical models for simulating ground water contaminant transport scenarios. Unfortunately, many tetbook as well as journal article treatments of this approimate solution are full of empirical statements that are deeloped without mathematical rigor. For this reason, a rigorous analysis of this solution is warranted. In this article, we present a mathematical method to derie the Domenico solution and eplore its its. Our analysis shows that the Domenico solution is a true analytical solution when the alue of longitudinal dispersiity is zero. For nonzero longitudinal dispersiity alues, the Domenico solution will introduce a finite amount of error. We use an eample problem to quantify the nature of this error and suggest some general guidelines for the appropriate use of this solution. Introduction Analytical solutions proide computationally efficient tools for modeling the fate and transport of ground water contaminant plumes (Aziz et al. 000; Clement et al. 00). In addition, they are also useful for testing comple numerical models (Clement et al. 1998; Clement 001; Quezada et al. 00). One of the most popular analytical solutions used for modeling ground water contaminant plumes is the Domenico (1987) solution. The Domenico solution is an approimate three-dimensional (3D) solution that describes the fate and transport of a decaying contaminant plume eoling from a finite patch source. This solution was based on an approach preiously published by Domenico and Robbins (1985) for modeling a nondecaying contaminant plume. Prior to this work, Cleary and Ungs (1978) presented an analytical solution to a similar 3D transport problem for a domain 1 Department of Ciil Engineering, Auburn Uniersity, Auburn, AL 389; (33) 8-70; fa: (33) 8-90; srinie@auburn.edu Corresponding author: Department of Ciil Engineering, Auburn Uniersity, Auburn, AL 3189; (33) 8-8; fa (33) 8-90; clement@auburn.edu 3 School of Earth and Enironmental Sciences, Seoul National Uniersity, Korea ; ; fa ; kklee@snu.ac.kr Receied May 00, accepted August 00. Copyright ª 007 The Author(s) Journal compilation ª 007 National Ground Water Association. doi: /j Vol. 5, No. GROUND WATER March April 007 (pages 13 1) finite in y and z directions. Later, Sagar (198) published an eact analytical solution to the transport problem considered by Domenico and Robbins (1985). Weler (199) etended the Sagar (198) solution to include the effects of reaction and presented an eact analytical solution to the transport problem considered by Domenico (1987). Howeer, these solutions are not closed form epressions since they inole numerical ealuation of a definite integral. This numerical integration step can be computationally demanding and can also introduce numerical errors. The key adantage of the Domenico and Robbins (1985) approach is that it proides a closed form solution without inoling numerical integration procedures. Due to this computational adantage, the Domenico solution has been widely used in seeral public domain design tools, including the U.S. EPA tools BIOCHLOR and BIO- SCREEN (Newell et al. 199; Aziz et al. 000). Although the Domenico solution is etensiely employed in seeral ground water transport models, its approimate nature has receied mied reiews oer the years. For eample, West and Kueper (00) compared the BIOCHLOR model against a more rigorous analytical solution and obsered considerable discrepancies. By comparing the near field concentration profiles, they concluded that the Domenico solution can produce errors up to 50%. Guyonnet and Neille (00) compared the Domenico solution against the Sagar (198) solution and presented the results in a nondimensional form. They concluded that for ground water flow regimes dominated by adection

2 and mechanical dispersion, the discrepancies between the two solutions can be considered negligible along the plume centerline. They further added that the errors may increase significantly outside the plume centerline. The preious reiew indicates that there are conflicting opinions regarding the performance of the Domenico solution. Furthermore, since the deelopment of the Domenico solution was based on a heuristic approach, researchers hae epressed skepticism regarding its alidity (West and Kueper 00). Currently, there are seeral unanswered issues related to the performance of this solution that include: Is there a mathematical basis for deriing the Domenico solution? If so, what are the approimations inoled in deriing the solution? What are the errors associated with these approimations? Finally, under what conditions are these approimations alid? To answer these questions, we need a fundamental understanding of the nature of the approimations inoled in the Domenico solution. The focus of this article is to perform a rigorous mathematical analysis on the origin and deelopment of the Domenico solution. The outcomes of this analysis are used to deelop some general guidelines for the appropriate use of the solution. Goerning Equations The transport problem considered by Domenico (1987) assumes a patch source of constant concentration c o located at in a clean, semi-infinite aquifer. The contaminant is subjected to adection in the direction and dispersie miing in all three directions. Furthermore, it is assumed that the contaminant decays through a firstorder process. The goerning transport equation considered by Domenico @ 1 1 c The initial and boundary conditions are: 1 c z kc cð;y;z;0þ¼0; " 0,,N; N,y,N; N,z,N cð0;y;z;tþ¼c o ; Z,z,Z ;,y, ; " t>0 /N y/n ; otherwise @cð;y;z;tþ ðþ where c is the concentration of the contaminant [ML 3 ]; c o is the concentration at the source [ML 3 ]; and Z are the source dimensions in y and z directions, respectiely [L]; D, D y, and D z are the dispersion coefficients in, y, and z directions, respectiely [L T 1 ]; is the adection elocity in the direction [LT 1 ]; and k is the first-order decay coefficient [T 1 ]. Reiew of the Domenico Solution The Domenico (1987) solution was based on an approimate approach gien by Domenico and Robbins (1985). Therefore, we first present a detailed reiew of the deelopment of the Domenico and Robbins (1985) solution. Domenico and Robbins (1985) began their analysis by presenting the following eact analytical solution that describes the transport of an instantaneous pulse source in a 3D domain (Hunt 1978): cð;y;z;tþ¼ c o 8 f~ ð;tþf~ y ðy;tþf~ z ðz;tþ; t 1 X X >< >= >< t >= where f ~ ð;tþ¼ erf >: ðd tþ 1= erf 7 >: ðd tþ 1= 5 y1 9 8 y 93 >< >= >< >= f ~ y ðy;tþ¼ erf erf 7 ðd y tþ 1= >: ðd y tþ 1= 5 >: z1 Z 9 8 z Z 93 >< >= >< >= f ~ z ðz;tþ¼ erf erf 7 ðd z tþ 1= >: ðd z tþ 1= 5 >: ð3þ They then present the following one-dimensional (1D) analytical solution (Crank 1975): where cð; tþ ¼ c o f ð; tþ; t f ð; tþ ¼ erfc ðd tþ 1= ðþ Note that the preious epression is the solution to the standard 1D adection-dispersion equation for an instantaneous source etending from zero to negatie infinity (Bear 1979). To account for the transerse dispersion due to a finite-sized two-dimensional (D) source, they employed the following two analytical solutions (Crank 1975): cðy;tþ ¼ c o f yðy;tþ and cðz;tþ ¼ c o f zðz;tþ; y >< >= >< y >= where f y ðy;tþ ¼ erf >: ðd y tþ 1= erf 7 >: ðd y tþ 1= 5 z 1 Z Z >< >= >< z >= and f z ðz;tþ ¼ erf >: ðd z tþ 1= erf 7 >: ðd z tþ 1= 5 ð5þ Note that c(y,t) and c(z,t) are solutions to two independent 1D transient diffusion equations subjected to an instantaneous line source of widths and Z, respectiely. Further, it can be obsered that the terms f y (y,t) and f z (z,t) V. Sriniasan et al. GROUND WATER 5, no. :

3 in Equation 5 are identical to the terms f ~ y ðy; tþ and f~ zðz; tþ in the Hunt (1978) solution. Domenico and Robbins (1985) multiplied the 1D solution f (,t) with these transerse spreading terms f y (y,t) and f z (z,t) and presented the following epression: cð; y; z; tþ ¼ c o 8 f ð; tþf y ðy; tþf z ðz; tþ ðþ Howeer, the authors did not justify this superposition step. Note that the Hunt solution was neer used in this analysis. At this stage, Domenico and Robbins presented the following arguments: The product of these three integral solutions [Equation ] describes a semiinfinite contaminated parcel which moes in the positie direction with a 1D elocity but which continuously epands in size in directions transerse to throughout the whole domain of, i.e., in the positie and negatie regions. This is because the time t in the transerse spreading terms is interpreted as running time. Reinterpreting this time as / for a moing coordinate system, as is common in all transerse spreading models (Bruch and Street, 197; Ogata, 1970; Domenico and Palciauskas, 198) has the effect of maintaining the original source dimensions at so that the condition C ¼ Co is maintained at for t > 0. Using these arguments, they reinterpret the time term t in the transerse spreading terms f y (y,t) and f z (z,t) as/. Howeer, the authors did not proide a mathematical reasoning for this time reinterpretation step. Further, all the references cited in the preious tet sole fundamentally different problems and we will address this issue in a later section. Using this time reinterpretation step, Equation was modified as: cð;y;z;tþ ¼ c o 8 f ð;tþf y ðy;þf z ðz;þ; ( ) t where f ð;tþ ¼erfc ðd tþ 1= y >< >= >< y >= f y ðy;þ ¼ erf >: ðd y =Þ 1= erf 7 >: ðd y =Þ 1= 5 z 1 Z Z >< >= >< z >= f z ðz;þ ¼ erf >: ðd z =Þ 1= erf 7 >: ðd z =Þ 1= 5 ð7þ Equation 7 was presented as the final solution to the continuous finite patch source problem considered by Domenico and Robbins (1985). Domenico (1987) incorporated the effects of firstorder decay by replacing the f (,t) term with an analytical solution for the semi-infinite pulse source problem with a decay term presented by Bear (1979). The final solution was gien as (Domenico 1987): 138 V. Sriniasan et al. GROUND WATER 5, no. : 13 1 cð;y;z;tþ ¼ c o 8 f ð;tþf y ðy;þf z ðz;þ; where f ð;tþ ¼ ep ka #) 1= a 8 t 1 1 ka 91 1= >< >= C >: ða tþ 1= A y >< >= >< y >= f y ðy;þ ¼ erf >: ða y Þ 1= erf 7 >: ða y Þ 1= 5 z 1 Z Z >< >= >< z >= f z ðz;þ ¼ erf >: ða z Þ 1= erf 7 >: ða z Þ 1= 5 ð8þ where a ¼ D /, a y ¼ D y /, and a z ¼ D z / are the dispersiities in the, y, and z directions, respectiely [L]. Martyn-Hayden and Robbins (1997) later modified the Domenico (1987) solution, referred in this work as the modified-domenico solution, by incorporating the following 1D solution (which describes a constant sourceboundary) in the f (,t) term (Bear 1979): f ð; tþ ¼ ep ka 1= a 1= ( t 1 1 ka )! ða tþ 1= 1 ep ka #) 1= a ( 1 t 1 1 ka ða tþ 1= 1= )! ð9þ As pointed out by Bear (1979), if the alue of /a is sufficiently large, a condition usually satisfied in practice, the additional term in the preious equation can be safely ignored. The preious reiew shows that the deelopment of arious forms of the Domenico solution does not hae a rigorous mathematical basis. The empirical arguments proided by the authors are ague because the mathematical procedures implied by these arguments are ineplicit and nebulous. In the following section, we proide a more rigorous approach to derie the Domenico solution, clearly stating the approimations inoled. A Rigorous Approach to Derie the Domenico Solution The eact semi-analytical solution for the 3D transport problem described by Equations 1 and,

4 considered by Domenico (1987), was proided by Weler (199) as: Z s¼t cð;y;z;tþ¼ c o f 9ð;sÞf y 9ðy;sÞf z 9ðz;sÞds; 8 s¼0 where f 9ð;sÞ¼ ffiffiffiffiffiffiffiffi p ep pd D ep sks1 D 3 D s s 3= y >< >= >< y >= f y 9ðy;sÞ¼ erf >: ðd y sþ 1= erf 7 >: ðd y sþ 1= 5 z1 Z Z >< >= >< z >= f z 9ðz;sÞ¼ erf >: ðd z sþ 1= erf 7 >: ðd z sþ 1= 5 ð10þ To obtain the Domenico solution from the preious eact solution, we replace the alue of s in the transerse spreading terms f y 9ðy; sþ and f z 9ðz; sþ with / (the alidity of this substitution will be discussed later). This yields the following epression: Z s ¼ t cð;y;z;tþ ¼ c o 8 f yðy;þf z ðz;þ f 9 ð;sþds; s y >< >= >< y >= where f y ðy;þ ¼ erf >: ða y Þ 1= erf 7 >: ða y Þ 1= 5 z 1 Z Z >< >= >< z >= f z ðz;þ ¼ erf >: ða z Þ 1= erf 7 >: ða z Þ 1= 5 ð11þ where a y ¼ D y / and a z ¼ D z /. Note that by substituting s ¼ /, we hae made the transerse spreading terms f y (y,) and f z (z,) independent of time; hence, they will not participate in the integration process. Without the transerse terms, the definite integral can be ealuated analytically as shown in Appendi 1. Therefore, the preious equation can be simplified as: cð; y; z; tþ ¼ c o 8 f ð; tþf y ðy; Þf z ðz; Þ; where f ð; tþ ¼ ep ka #) 1= a ( t 1 1 ka 1= )! ða tþ 1= 1 ep ka #) 1= a ( 1 t 1 1 ka 1= )! ð1þ ða tþ 1= Equation 1 is identical to the modified-domenico solution shown in Equation 9. If we set the first-order decay coefficient k to zero, Equation 1 reduces to: cð; y; z; tþ ¼ c o 8 f ð; tþf y ðy; Þf z ðz; Þ; t where f ð; tþ ¼ erfc ða tþ 1= 1 t 1 ep erfc a ða tþ 1= ð13þ Equation 13 is similar to the Domenico and Robbins (1985) solution gien by Equation 7. The additional epression in the f (,t) term in Equations 1 and 13 is due to the use of the epanded form of the 1D solution that describes a constant concentration boundary condition instead of a semi-infinite pulse source boundary condition. The preious analysis shows that the only approimation required for rigorously deriing the Domenico solution is the time reinterpretation step, where s is replaced by / in the transerse dispersion terms. In the following section, we perform a detailed mathematical analysis to inestigate the alidity of this approimation. Mathematical Analysis of the Validity of the Approimation Inoled in the Domenico Solution Reiew of transport modeling literature indicates that it is common to replace s with / in the transerse dispersion terms when soling conection-dominated problems that hae low longitudinal miing. For eample, Bruch and Street (197) used a similar assumption to sole the adection-dispersion problem when the longitudinal miing was smaller than the transerse miing. Another eample of a conection-dominated problem that employs this approimation is the air pollution model used for predicting the fate and transport of smoke plumes eoling from chimneys (Wark and Warner 1981). Here, the transport is dominated by conection along the wind direction, and dispersie miing is restricted to the transerse directions only. Neglecting the effects of longitudinal dispersion in such problems simplifies the goerning transport @ 1 c 1 c z kc We consider solution to the preious transport problem subject to the following Domenico type initial and boundary conditions: cð;y;z;0þ¼0 " 0,,N; N,y,N; N,z,N cð0;y;z;tþ¼c o ; Z,z,Z ;,y, ; " t>0 otherwise; " y/n ð15þ V. Sriniasan et al. GROUND WATER 5, no. :

5 In Appendi, we use Laplace transform techniques to sole the preious problem and the resulting eact analytical solution is: cð; y; z; tþ ¼ c o 8 f o ð; tþf yð; yþf z ð; zþ; where f oð; tþ ¼ ep k n u t o n where u t o is the step function gien by; n u t o 0 if t ¼ 1 if t > ð1þ where f y (,y) and f z (,z) are identical to the epressions gien in Equation 11. Since the Domenico approach approimates s as / in the transerse dispersion terms, we hypothesize that the Domenico approimation must be alid when a is zero. To test this hypothesis, we perform a iting analysis of the modified-domenico solution by forcing a to zero; this is epressed as: cð; y; z; tþ ¼ c o a /0 8 f ð; tþf y ðy; Þf z ðz; Þ ð17þ The mathematical details of this iting analysis are gien in the supplementary material. The analysis shows that when a approaches zero, the modified-domenico solution relaes to the eact analytical solution gien by Equation 1. This proes that the Domenico approimation indeed yields an eact analytical solution when a is equal to zero. Analysis of the Error Associated with the Domenico Solution The mathematical analysis presented in the preious section demonstrates that the time reinterpretation step, where s is replaced with /, is eactly alid when a. From these results, one could also infer that this time reinterpretation process proides a reasonable approimation when longitudinal dispersion plays an insignificant role in the oerall transport. Hence, the Domenico solution can be epected to produce reasonable estimates for adection-dominated problems; howeer, it can introduce significant errors for longitudinal dispersion-dominated problems. Another important feature of the time reinterpretation step is that it forces a quasi-steady-state condition along the transerse directions at all times. In other words, the conceptual residence time (/ alue) associated with a point located at the centerline to disperse contaminant mass in the transerse directions is independent of the simulation time. Further, this residence time is also assumed to increase linearly with respect to. These unrealistic assumptions regarding residence times will lead to erroneous predictions, especially beyond the adectie front. For eample, consider a problem where ¼ 50 m/year, and we are interested in predicting the concentration distribution of a -year-old plume (t ¼ years) at a location ¼ 00 m. The Domenico solution will estimate the residence time s for our location of interest ¼ 00 m as s ¼ / ¼ years; this in fact is greater than the total simulation time itself. This is an unrealistic assumption since a -year-old plume simply cannot hae the time to disperse for years. For a particle located at the adectie front, the residence time assumed by the Domenico solution is years (the simulation time), and for all the particles located behind the adectie front, the residence time assumed by the Domenico solution will be equal to / (which will be < years); these seem to be reasonable estimates. Howeer, for all points beyond the adectie front, i.e., > 100 m, the Domenico solution will assign unrealistic conceptual residence times, which will be greater than the simulation time t ¼ years. It must be noted that this incorrect behaior will anish when a is zero because, for this case, the plume will abruptly end at the adectie front, and the residence time for each particle located at or behind the adectie front will in fact be equal to /. When soling steady-state problems, the assumption related to residence time should be a reasonable approimation. This is because, at steady state, the theoretical adectie front will be at infinity. Therefore, the time reinterpretation should be reasonable for any finite domain. Hence, the performance of the Domenico solution under steady-state conditions can be epected to be better. Howeer, it is important to note that een under steady-state conditions, the solution will not be eact because it will still ignore the transport due to longitudinal miing. In general, it can be concluded that the Domenico solution can be epected to perform better behind the adectie front. In the following section, we use an eample problem to illustrate the implication of these theoretical results. Eample Problem The eample problem presented by Domenico and Robbins (1985) is considered in this analysis. The transport parameters used in the problem are summarized in Table 1. The performance of the modified-domenico solution was tested by comparing its results against those generated using the eact solution gien by Weler (199). It has been established in the preious sections that the Domenico approimation makes unreasonable assumptions regarding the residence time beyond the adectie front and reasonable assumptions behind the Table 1 Parameters Used in the Eample Problem Parameter Longitudinal dispersiity (a ) Transerse dispersiity (a y ) Transerse dispersiity (a z ) Velocity () Source width in direction () Source width in Z direction (Z) Source concentration (C) Simulation Time (t) Value.58 m 8.3 m 0.00 m m/d 0.0 m 5.0 m 850 mg/l 5110 d 10 V. Sriniasan et al. GROUND WATER 5, no. : 13 1

6 front. Therefore, we analyze the results of this comparison in two parts one behind the adectie front and the other beyond the adectie front (note that for our base case the front is at ¼ 1100 m). Plume Comparison Analysis behind the Adectie Front Figures 1a and 1b compare the D concentration contours of both solutions on the X- and X-Z planes. (Note: an aspect ratio of.:1 was maintained for the X- plots, and an aspect ratio of 55:1 was maintained for the X-Z plots.) Since the problem is symmetric about the X- ais, only half of the plume is presented. It can be obsered from Figure 1 that the modified-domenico solution is reasonably close to the true solution, though there are some noticeable discrepancies. To eplore the its of these discrepancies, we performed a series of sensitiity simulations. In the first set of sensitiity simulations, we aried the alue of the longitudinal dispersiity (a ) by an order of magnitude aboe and below the assumed baseline alue. Figures a and b compare the D concentration contours of the solutions for both cases. Comparison of the data shown in Figures 1a and indicates that the discrepancies between the two solutions were large when the alue of longitudinal dispersiity was high. Also, as Figure. Sensitiity results for ariations in the longitudinal dispersiity alue: solutions behind the adectie front for (a) a h 10 and (b) a /10. Figure 1. Concentration contours predicted by the Domenico and Weler solutions for the base case: solutions behind the adectie front for (a) X- plane and (b) X-Z plane. epected, when the longitudinal dispersiity was low, there was an ecellent match between the solutions. Similar trends were also obsered in the concentration contours predicted on the X-Z plane. Since the spreading terms in the y and z directions are identical in structure, the contours in the X- and X-Z planes will ehibit identical trends. Therefore, from this point onward, our analysis will be restricted to X- contours. In the second set of sensitiity simulations, we aried the alue of the transerse dispersiity (a y ) by an order of magnitude aboe and below the baseline alue. These results (Figures S1 and S) indicated that the transerse dispersiity in the y direction does not play a significant role in influencing the error associated with the modified- Domenico solution. Similar sensitiity analysis performed on other transport parameters, including the transerse dispersiity a z and the source dimensions and Z, also showed minimal sensitiity. Although the original problem considered by Domenico and Robbins (1985) does not inole reactions, in order to test the performance of the modified-domenico solution in the presence of first-order decay, a third set of sensitiity simulations were completed for a decaying contaminant plume by assuming arious first-order rate coefficients (k). Comparison of the concentration contours for k alues of and 0.001/d (Figures S1 and S) indicated that the presence of a decay term does not introduce any significant additional error. V. Sriniasan et al. GROUND WATER 5, no. :

7 Figure. Concentration contours predicted by the Domenico and Weler solutions for the base case: solutions include concentration contours beyond the adectie front. Figure 3. Sensitiity results for ariations in the transport elocity: solutions behind the adectie front for (a) h 10 and (b) /10. to the unrealistic assumptions made by the Domenico solution when computing the conceptual residence times beyond the adectie front. The results of a sensitiity analysis performed on the parameter a are summarized in Figures 5a and 5b. These figures indicate a trend similar to those present for regions within the adectie front. The higher the alue of the longitudinal dispersiity, the greater the error associated with the modified-domenico solution. Further, it can be obsered that the error systematically increases when the contaminant is transported beyond the adectie front. The fourth set of sensitiity simulations inoled arying the adection elocity () by an order of magnitude aboe and below the baseline alue. Figures 3a and 3b compare the concentration contours of the solutions for the two cases. From these figures, it can be concluded that the adection elocity has ery little effect in determining the accuracy of the solution. Note that in the absence of first-order decay, arying the adection elocity will hae the same effect as arying the total simulation time (t). Since decay does not play any significant role in determining the accuracy of the modified-domenico solution, it can be safely concluded that ariations in the total simulation time will hae little sensitiity on its accuracy. The preious results indicate that within the adectie front, the longitudinal dispersiity plays a ery important role in determining the accuracy of the modified-domenico solution. All the other transport parameters hae negligible effect on the accuracy of the solution. Plume Comparison Analysis beyond the Adectie Front Figure compares the concentration contours of the two solutions in the X- plane for the base case parameters. (Note: here an aspect ratio of :1 is maintained for the X: plane to capture the plume beyond the adectie front; also, the location of the adectie front is indicated by an arrow on the -ais.) It can be obsered from Figure that, as we moe beyond the adectie front, the accuracy of the modified-domenico solution reduces rapidly. As pointed out in the earlier sections, this is due Figure 5. Sensitiity results for ariations in the longitudinal dispersiity alue: solutions include concentration contours beyond the adectie front for (a) a h 10 and (b) a /10. 1 V. Sriniasan et al. GROUND WATER 5, no. : 13 1

8 A similar set of sensitiity simulations were performed on different transport parameters, including a y, a z,, Z, and K for regions beyond the adectie front as well. As epected, the results indicated that these parameters had negligible contribution in determining the accuracy of the solution. A final set of sensitiity simulations were performed by arying the alue of the adection elocity () by an order of magnitude aboe and below the baseline alue. The results of this sensitiity analysis are summarized in Figures a and b. Initial obserations of these figures may indicate that at higher elocities the modified- Domenico solution appears to perform better. Howeer, a closer analysis of these figures with respect to their respectie adectie front locations indicates that at higher elocities a greater portion of the plume is behind the adectie front, whereas at lower elocities a relatiely lesser portion of the plume is behind the adectie front. Comparison of these figures made in the light of their respectie adectie fronts reeals that the adection elocity has little effect in determining the accuracy of the solution. Howeer, it must be noted that the adection elocity itself plays an important role in determining the location of the adectie front, which is one of the key parameters that affects the performance of the solution. Variations in the total simulation time (t) will hae a similar effect as that of the adection elocity. From the results of these sensitiity simulations, it can be safely concluded that the two most important factors that affect the accuracy of the modified-domenico solution are the alue of the longitudinal dispersiity (a ) and the position of the adectie front (t). The solution will hae minimal errors when the alue of a is low and when the adectie front is farther away from the source. It must be noted that the conclusions obtained for the modified-domenico solution apply to the Domenico solution as well, proided the alue of /a is sufficiently large (Bear 1979). General Discussions Regarding the Accuracy of the Domenico Solution Since the original Domenico solution lacked a theoretical basis, seeral misconceptions regarding its performance hae eoled oer the years. One of the common misconceptions is that the error will be a minimum along the plume centerline. For eample, Guyonnet and Neille (00) compared the Domenico solution against the Sagar solution and concluded that the results of the ealuation confirm that along the plume centerline, and for ground water flow regimes dominated by adection and mechanical dispersion rather than by molecular diffusion, discrepancies between the two solutions (namely the Domenico solution and the Sagar solution) can be considered negligible for all practical purposes. Howeer the errors in the Domenico (1987) solution may increase significantly outside the plume centerline. Howeer, our simulation results indicate that this conclusion might not be true for all cases. To illustrate this, we compare the y and z concentration transects predicted by the two solutions for our base case scenario. Figure 7a compares the concentration profiles along the y direction at ¼ 1000 and 1500 m, and similar results for the z direction are Figure. Sensitiity results for ariations in the transport elocity: solutions include concentration contours beyond the adectie front for (a) h 10 and (b) /10. shown in Figure 7b. It is eident from these figures that the error is not minimal along the centerline, but rather at a point, which will always be away from the centerline. This error pattern can also be obsered in all the D contours. Further, it can be obsered from Figures 7a and 7b that the absolute error is, in fact, maimum along the plume centerline. Another important issue that we would like to address here is the nature in which the error associated with the Domenico approimation is propagated spatially. Our results show that the plumes predicted by the modified-domenico solution are always wider than the actual plumes. This phenomenon can easily be obsered in all the figures presented in this study. This can be attributed to the fact that the Domenico approimation oerpredicts the conceptual residence times of all particles along the centerline (hence allows more time to disperse in the transerse directions). This oerprediction would lead to a decrease in the centerline concentrations; therefore, solutions that employ the Domenico approimation will always underpredict the oerall etent of the plume in the longitudinal direction. An important transport parameter not addressed so far is the retardation factor (R). Retardation affects the adection elocity and possibly the decay constant (depending on the phase where the decay occurs). Since the presence of a decay term does not introduce any significant additional error to the Domenico solution, its effect can be ignored. Howeer, retardation changes the location of the adectie front by changing the adection elocity V. Sriniasan et al. GROUND WATER 5, no. :

9 using a better approimation for the time reinterpretation step. This improed approimation should include the effects of transport due to longitudinal dispersion (currently ignored by the Domenico solution). Furthermore, it will be useful to deelop some quantitatie guidelines for the use of the Domenico solution. This could be accomplished by using a set of nondimensional parameters. Acknowledgments This research was supported in part by the Office of Science (BER), U.S. Department of Energy Grant No. DE-FG0-0ER13, and by a project from the DOE Westinghouse Saannah Rier Laboratory. This joint effort with the Seoul National Uniersity was supported by the Frontier Project of the Korea Ministry of Science and Technology. We would like to thank the Ground Water reiewers and editors for their thoughtful comments. Figure 7. Concentration profiles predicted by the Domenico solution compared with the Weler solution at ¼ 1000 m and 1500 m (a) along -ais and (b) along Z-ais. and hence would influence the oerall accuracy of the Domenico solution. Conclusions Based on the theoretical results presented in this study, we conclude that the key assumption used to derie the Domenico solution is the time reinterpretation step, where the time s in the transerse dispersion terms is replaced with /. Our deriations proe that this substitution process is alid only when the longitudinal dispersiity is zero. For all nonzero longitudinal dispersiity alues, the solution will hae a finite error. The spatial distribution of this error is highly sensitie to the alue of a and the position of the adectie front (t), and is relatiely less sensitie to other transport parameters. Based on the results of this study, we conclude that the error in the Domenico solution will be low when soling transport problems that hae low longitudinal dispersiity alues, high adection elocities, and large simulation times. Despite its itations, the Domenico approimation offers a simple alternatie for etending 1D analytical solutions to 3D analytical solutions. This approach is useful for deeloping approimate solutions for unsoled, 3D, multispecies reactie transport problems that hae eplicit 1D solutions. Howeer, such solutions should be used carefully after understanding the itations identified in this study. Finally, it is of the authors opinion that the performance of the Domenico solution can be improed by Supplementary Material The following supplementary material aailable for this article: Appendi S1. Limiting Analysis of the Domenico solution (a ). Figure S1. Sensitiity results for ariations in the transerse dispersiity alue; solutions behind the adectie front for (a) a y *10 and (b) a y /10. Figure S. Sensitiity results for ariations in the decay rate; solutions behind the adectie front for (a) K.0001 day 1 and (b) K.001 day 1. This material is aailable as part of the online article from: j (This link will take you to the article abstract). Please note: Blackwell Publishing is not responsible for the content or functionality of any supplementary materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. Appendi 1 The integral in Equation 11 can be ealuated as: Z s ¼ t I ¼ f 9ds ¼ pffiffiffiffiffiffiffiffi ep s pd D Z s ¼ t ep s ks 1 D D s 3 ds ða1þ s s 3= Applying Laplace transform to Equation A1, we get: l½iš ¼ ffiffiffiffiffiffiffiffi 1 p ep pd D s n ep s ks 1 o 3 D 3 h l D s 7 5 ðaþ s 3= 1 V. Sriniasan et al. GROUND WATER 5, no. : 13 1

10 where l is the Laplace transform operator and s is the Laplace ariable. Equation A can be epressed as: l½iš ¼ ffiffiffiffiffiffiffiffi 1 p pd s ep D 3 ep 3 h l ep D s 1 ks 7 D 5 ða3þ s 3= The second term within the Laplace operator can be ealuated by using Selby (1971, Equation 8, 97) as: n o 3 ep D l s " p ffiffiffiffiffiffiffiffi 7 5 ¼ pd s 3= ep p ffiffiffiffiffi D pffiffi # s ðaþ The entire epression within the Laplace operator can be ealuated by using Selby (1971, Equation 11, 91) and Equation A as: l½iš ¼ " s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ep( s ep )# pffiffiffiffiffi s 1 1 k D D D y; z; tþ y; z; ðbþ Applying Laplace transform to c(,y,z,t) in Equation p where a y ¼ D y p ¼ and a z ¼ D z ðþ p; ðb3þ where s is the Laplace ariable and p is the concentration in the Laplace domain. The boundary conditions get modified as: pð0; y; zþ ¼ c o s ; Z, z, Z ;, y, ; y; zþ y; ðbþ Inerse Laplace transform of the preious equation yields (Bear 1979): I ¼ ep 1 a 1 1 ka #) 1= 1= ( t 1 1 ka ) ða tþ 1= 1 ep ka #) 1= a ( 1 t 1 1 ka ða tþ 1= where a ¼ D 1= Appendi The goerning transport @ c )! ; ðaþ c ¼ kc ðb1þ The initial and boundary conditions are: cð;y;z;0þ¼0; " 0,,N; N,y,N; N,z,N cð0;y;z;tþ¼c o ; Z, z, Z ;,y, ; " t > 0 ; otherwise " t>0 Equation B3 can be interpreted as a transient D diffusie reactie transport problem. Its boundary conditions gien by Equation B represent an instantaneous pulse of a plane source. The solution to this problem without the decay term can be readily deduced from Hunt (1978) by ignoring the adection term and reducing the problem to two dimensions. Thus, the solution to the preious problem without the reaction term is gien by: p9ð;y;zþ¼ c o s f yð;yþf z ð;zþ; y >< >= >< y >= where f y ð;yþ¼ erf >: ða y Þ 1= erf 7 >: ða y Þ 1= 5 ( Z ) ( Z z1 z f z ð;zþ¼erf erf ða z Þ 1= ða z Þ 1= 3 ) 7 5 ðb5þ Now, we make use of a method similar to the Danckwert s method described by Crank (1975) to include the reaction term. If p9 is the solution for the diffusion problem without reaction; the solution for the same problem with a first-order reaction, with a rate constant, for the same initial and boundary condition is gien as: p ¼ p9ep ðbþ V. Sriniasan et al. GROUND WATER 5, no. :

11 This can be easily erified by checking if the solution p satisfies the goerning equation and the initial and boundary conditions. Since p9 is the solution to Equation B3 without the reaction term, it p9 a ðb7þ Also, differentiating p with respect to, y, and z to the respectie orders yields: dp d p ep dp9 1 ep p9 ¼ ep p9 ðb8þ From Equations B through B8, we get Equation B3. This proes that solution p satisfies the goerning differential equation. To check for the initial condition, we substitute in Equation B. When, the eponential term becomes unity and hence Equation B reduces to p ¼ p9; thus, the initial condition is satisfied. In order to check for the boundary condition in the y direction, we need to take the deriatie of the solution p with respect to y. This is gien as: dp dy ¼ ep dp9 dy ðb9þ In the iting case, when y approaches N, the deriatie of p9 with respect to y becomes 0. From Equation B9, we can conclude that the deriatie of p with respect to y also becomes 0. This proes that the boundary condition in the y direction is satisfied. On similar lines, we see that the boundary condition is also satisfied in the z direction. Hence, it is proed that Equation B is the solution for the system of equations described by Equations B3 and B. Equation B can be written as: pð; y; zþ ¼ c o s ep f y ð; yþf z ð; zþ ðb10þ Inerse Laplace transform of Equation B10 gies the final solution as (Selby 1971, Equation 1, 95): cð; y; z; tþ ¼ c o 8 f o ð; tþf yð; yþf z ð; zþ f oð; tþ ¼ ep k n u t o ; n where u t o is the step function gien by; 8 n u t o 0 if t 9 < = ¼ : 1 if t > ; ðb11þ References Aziz, C.E., C.J. Newell, J.R. Gonzales, P. Haas, T.P. Clement, and. Sun BIOCHLOR: Natural attenuation decision support system. 1.0, User s manual. U.S. EPA Report EPA/00/R-00/008. Cincinnati, Ohio: U.S. Enironmental Protection Agency. Bear, J Hydraulics of Groundwater. New ork: McGraw-Hill. Bruch, J.C., and R.L. Street Two-dimensional dispersion. Journal of the Sanitary Engineering Diision, Proceedings of the American Society of Ciil Engineers, 93, no. SA: Cleary, R., and M.J. Ungs Analytical models for groundwater pollution and hydrology. Report No. 78-WR-15. Princeton, New Jersey: Water Resources Program, Princeton Uniersity. Clement, T.P Generalized solution to multispecies transport equations coupled with a first-order reaction network. Water Resources Research 37, no. 1: Clement, T.P.,. Sun., B.S. Hooker, and J.N. Petersen Modeling multispecies reactie transport in ground water aquifers. Ground water Monitoring & Remediation Journal 18, no. : Clement, T.P., M.J. True, and P. Lee. 00. A case study for demonstrating the application of U.S. EPA s monitored natural attenuation screening protocol at a hazardous waste site. Journal of Contaminant Hydrology 59, no. 1 : Crank, J The Mathematics of Diffusion, nd ed. New ork: Oford Uniersity press. Domenico, P.A An analytical model for multidimensional transport of a decaying contaminant species. Journal of Hydrology 91, no. 1 : Domenico, P.A., and G.A. Robbins A new method of contaminant plume analysis. Ground Water 3, no. : Domenico, P.A., and V.V. Paliciauskas Alternatie boundaries in solid waste management. Ground Water 0, no. 3: Guyonnet, D., and C. Neille. 00. Dimensionless analysis of two analytical solutions for 3-D solute transport in groundwater. Journal of Contaminant Hydrology 75, no. 1 : Hunt, B Dispersie sources in uniform ground-water flow. Journal of the Hydraulic Diision 10 no. H1: Martyn-Hayden, J., and G.ARobbins Plume distortion and apparent attenuation due to concentration aeraging in monitoring wells. Ground Water 35 no. : Newell, C.J., R.K. McLeod, and J.R. Gonzales BIO- SCREEN: Natural attenuation decision support system user s manual. EPA/00/R-9/087. Ada, Oklahoma: Robert S. Kerr Enironmental Research Center. Ogata, A Theory of dispersion in a granular medium. USGS Professional Paper Reston, Virginia: USGS. Quezada, C.R., T.P. Clement, and K.K. Lee. 00. Generalised solution to multi-dimensional multi-species transport equations coupled with a first order reaction network inoling distinct retardation factors. Adances in Water Resources 7, no. 5: Sagar, B Dispersion in three dimensions: Approimate analytical solutions. ASCE Journal of Hydraulic Diision 108, no. H1: 7. Selby, S.M Standard Mathematical Tables, 19th ed. Cleeland, Ohio: The Chemical Rubber Co. Wark, K., and C.F. Warner Air Pollution, Its Origin and Control, nd ed. New ork: Addison Wesley Longman Inc. West, M., and B.H. Kueper. 00. Natural attenuation of solute plumes in bedded fractured rock. In Proceedings of USEPA/NGWA Fractured Rock Conference, Portland, Maine: National Ground Water Association. Weler, E.J Analytical solutions for one-, two-, and threedimensional solute transport in ground-water systems with uniform flow. USGS TWRI Book 3, Chapter B7. Reston, Virginia: USGS. 1 V. Sriniasan et al. GROUND WATER 5, no. : 13 1