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Batu (this issue) has offered comments regarding the assessments of Srinivasan et al. (2007) and West et al. (2007) on the accuracy of the Domenico (1987) solution. In this reply, we respond only to comments directed at our paper. In general, Batu discussed (1) our method of presentation and (2) verification of the exact analytical solutions by Cleary and Ungs (1978) and Wexler (1992). We emphasize that nowhere does Batu assert that our work is incorrect, nor does he disagree with our final conclusions. His first comments pertain to our presentation of the mathematics (West et al. 2007, 128 to 130, equations 2 to 22). After a lengthy discussion of aspects of the work by Domenico and Robbins (1985) and Domenico (1987), Batu writes, “As a result, the Domenico and Robbins solution is not based on a systematic analytical solution as West et al. (2007)… expressed in their papers.” Our paper is entirely devoted to explaining the approximate nature of the Domenico (1987) and related solutions. At no time do we claim that they comprise a “systematic analytical solution.” On the contrary, we clearly distinguish the Domenico and the Domenico and Robbins approximations as “analytical expressions” and “approximate solutions” rather than “exact analytical solutions.” Furthermore, our equations 2 to 22 and the associated text explicitly discuss the various approximations (and their historical progression) underpinning Domenico (1987) and Domenico and Robbins (1985); we then use simulations to illustrate the influence of these approximations. Batu also comments, “In fact, the Domenico and Robbins solution was itself artificially developed. Thus, it is difficult to understand why West et al. (Eqs. (2) through (22)) … tried to derive the Domenico and Robbins (1985) and Domenico (1987) solutions.” We completely agree that these solutions were artificially developed, as we discuss in detail (West et al. 2007). Although the work by Domenico and Robbins (1985) and Domenico (1987) is approximate, it is methodical. These authors (and those of related solutions) should be credited with presenting their concepts clearly enough that we were able to reconstruct their derivation; this does not suggest that the expressions are exact but that they were obtained systematically. It is unclear what Batu is referring to because (1) we focused entirely on elucidating the approximate nature of the solutions; (2) we reconstructed the original methodologies for each expression; and (3) we conclusively stated that the solutions are approximate and their accuracy is parameter dependent. The second set of comments offered by Batu pertain to the validity of solutions by Cleary and Ungs (1978) and Wexler (1992). As we discussed (West et al. 2007, 127, paragraph 2), these are exact analytical solutions derived using different boundary conditions that result in different solution forms. Batu bases the bulk of his comments regarding verification and credibility on his assertion that “Wexler (1992, p. 55, equation 121a) used the same solution method as Cleary and Ungs (1978) in deriving the equation mentioned above.” When examining either Cleary and Ungs (1978) or the material presented on pages 64 to 75 of Wexler (1992), this claim is clearly incorrect, negating any criticism in this regard, because different boundary conditions and integral transforms were used. These two solutions are independent and thus verified as described in method 1 by Batu. To stress our point regarding verification, we further used method 1 and compared the results in West et al. (2007) with two additional exact analytical solutions. The first comparison was made with ATRANS by Neville (1998), which is an additional independent solution. The second comparison was made with the solution on pages 65 and 66 of Batu (1996). This is a particularly interesting comparison, because the Cleary and Ungs (1978) solution is in fact a special case of the more general solution presented by Batu (1996) (i.e., “one rectangular source having constant concentration” on pages 65 and 66). Figure 1 plots the three-dimensional transient comparison corresponding to figure 3B in West et al. (2007). Clearly, all solutions produce the same output. Batu comments that, “One immediate problem with the equation in Wexler, or equation 23 in West et al., is that when x = 0, the concentration C (x, y, z, t) becomes zero along the y-z plane where the rectangular source is located. This violates the assumption that the concentration at the source is equal to C0.” This statement has merit only for x values of exactly zero. Equation 121a of Wexler (1992) produces nonzero values for x > 0 (and C0 > 0), and the source boundary conditions are in fact maintained; the Wexler (1992) solution is rigorously derived and mathematically correct. The occurrence of C = 0 for x values of absolute zero is a mathematical consequence of the solution strategy. This can easily be demonstrated using x = 0.01 m, y =z = 0.0 m, and t > 0 with the parameter set in tables 2 through 4 of West et al. (2007). The Wexler (1992) solution gives C = 11.000 mg/L and the source boundary condition is honored. Building on the previously cited comment, Batu further writes, “The same situation also occurs in the corresponding solution of Cleary and Ungs (1978).” It can be demonstrated using both mathematics and simulations that this is incorrect, and C(x = 0, t > 0) is equal to C0 within the transverse bounds of the source zone when using either Cleary and Ungs (1978) or Batu (1996). Finally, to further verify our work, it should be noted that ATRANS by Neville (1998) has been compared to MT3D by Zheng (1990), and Batu (1996) has been compared to MODFLOWT by Duffield (2007) in Batu (2006, 171 to 174). Three-dimensional transient comparison of exact analytical solutions by Cleary and Ungs (1978), Wexler (1992), Batu (1996), and Neville (1998) with the approximate solution by Domenico (1987) for a monitoring well located 100 m downstream. This figure is adapted from figure 3B in West et al. (2007), and the solid line corresponds to the original output using both Cleary and Ungs (1978) and Wexler (1992). Input parameters are provided in tables 2 through 4 of West et al. (2007). Given the verification originally provided in West et al. (2007), and the additional verification provided here, it is clear that the work and conclusions provided in that paper are valid and complete. The authors extend their appreciation to Christopher Neville for sharing his insights and providing valuable feedback on this reply.

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