1) Scientific Significance

The manuscript presents a model for radial dispersion of solutes injected in wells considering the effect of mixing in the wellbore, the influence of the surrounding skin zone, as well as mobile and immobile regions. The latter, conceptually modeled as two continuums with spatially uniform parameters which co-exist over the entire aquifer, allow simulations of early arrivals and long tailing of the breakthrough curves specific to spatially heterogeneous aquifers. The first-order reactive transport is governed by a system of coupled equations with constant coefficients which can be solved analytically. The analytical solutions derived by the authors in Laplace domain are tested against finite-element numerical solutions and experimental data. It is shown that the new model performs better than partial modes which do not consider simultaneously the mixing, skin, and heterogeneity effects.

As an overall evaluation, the manuscript contributes to the scientific progress in the research filed and within the scope of the HESS journal.

2) Scientific Quality

The authors present only the solutions in Laplace domain. At page 14 it is mentioned that “the de Hoog method will be employed to conduct the inverse Laplace transform”. A section in the Supplementary Materials with the computation of the inverse Laplace transform or, at least, references for the method and the software used in their study should be included.

At page 22 it is mentioned the “genetic algorithm (GA) … employed to search the optimal parameter values”, again without any details in Supplementary Materials or references for the algorithm and codes used. These should be included as well.

Apart from these missing details, the applied methods are valid and the results are discussed in with consideration of related work.

3) Presentation Quality

The results and conclusions are presented in a clear way and in a good English language. The figures and tables included are appropriate and the manuscript contains the relevant references to the literature.

The paper proposes a new analytical solutions based on the mobile-immobile framework for redial dispersion within a wellbore that considers mixing effect, skin effect, scale effect, aquitard effect and limited media heterogeneity, as this is considered only in the context of the mobile-immobile as a ratio of conductivities in the aquifer, and not as a spatially varying heterogeneity which is a more realistic pattern. The paper is hard to follow, and generally lacks real clarity, specifically there is no in depth explanation on the “skin-effect” as they previously did in [Li et al., 2019], and it is hard to understand how the derivation differ from their [Wang et al., 2020] paper which focuses on the transport. Moreover, it is not clear how the model is better then existing models? In line 100 the authors claim that other models, namely MRMT, CTRW, and fADE, are “usually unavailable or difficult to develop” yet a quick search show that there are models that cope with that problem well in CTRW [Dentz et al., 2015; Hansen et al., 2016], fADE [Chen et al., 2017; Soltanpour Moghadam et al., 2022], and even a combination of MRMT and CTRW [Kang et al., 2015]. Also, specifically for reactive transport in radial conditions there are experimental evidence for the scaling of dispersion, mixing, and reaction [Edery et al., 2015; Leitão et al., 1996], which are similar to the scaling in this study. The authors should refer to this literature and explain how their analytical solution differ and why is it better as they claim.

Line 135-137 needs to be clarified

Are we defining the asymptotical value for the model in line 137-139, please clarify.

Line 157-162 defines reaction rate (or radioactive decay, or biodegradation), and retardation factor yet there is no example to using these parameters in the results since R=1, μ is so small it is negligeable, so the sensitivity to these parameters must be small. Can the author comment on the choice of parameters? Also, why is this part in the supplementary and not in the text?

As the comsol solution was based on equation 14, which is the basis for the analytical solution equation 20-23, it is not surprising that the match between them in figure S4 is good, yet why do they differ so much from the observation in Chao et al 1999? Moreover, can the authors supply an R-square or quantify how well the analytical solution performs for all figure, and not just figure 3, where the comsol solution is very different? Please, add the error to the figure caption as it is confusing to switch between the figure to the table?

Another point is that there is no explanation as to why the error is so big, and why the analytical solution is better than the numerical one with respect to the error.

To summarize, the paper seems like an important contribution as it considers many physical aspects for radial dispersion (reaction, retardation, conductivity change in the skin area), and provides an analytical solution that considers these aspects. However, at the moment the advantage of the analytical solution, when compared to experimental data and even to the numerical solution is not clear enough. The paper is not approachable, as the figures need to be combined with the error while all the details of the modeling and results need to be ordered and clarify. Lastly, there is a bulk of literature that need to be added to put this work in the right context. I believe that addressing these comments will make the paper more approachable, provide the right context and make a stronger case for the analytical solution presented here.

Chen, K., H. Zhan, and Q. Yang (2017), Fractional models simulating non‐Fickian behavior in four‐stage single‐well push‐pull tests, Water Resources Research, 53(11), 9528-9545.

Dentz, M., P. K. Kang, and T. Le Borgne (2015), Continuous time random walks for non-local radial solute transport, Advances in water resources, 82, 16-26.

Edery, Y., I. Dror, H. Scher, and B. Berkowitz (2015), Anomalous reactive transport in porous media: Experiments and modeling, Physical Review E, 91(5), 052130.

Hansen, S. K., B. Berkowitz, V. V. Vesselinov, D. O'Malley, and S. Karra (2016), Push‐pull tracer tests: Their information content and use for characterizing non‐F ickian, mobile‐immobile behavior, Water Resources Research, 52(12), 9565-9585.

Kang, P. K., T. Le Borgne, M. Dentz, O. Bour, and R. Juanes (2015), Impact of velocity correlation and distribution on transport in fractured media: Field evidence and theoretical model, Water Resources Research, 51(2), 940-959.

Leitão, T. E., J. Lobo-Ferreira, and A. J. Valocchi (1996), Application of a reactive transport model for interpreting non-conservative tracer experiments: The Rio Maior case-study, Journal of contaminant hydrology, 24(2), 167-181.

Li, X., Z. Wen, H. Zhan, and Q. Zhu (2019), Skin effect on single-well push-pull tests with the presence of regional groundwater flow, Journal of Hydrology, 577, 123931.

Soltanpour Moghadam, A., M. Arabameri, and M. Barfeie (2022), Numerical solution of space-time variable fractional order advection-dispersion equation using radial basis functions, Journal of Mathematical Modeling, 1-14.

Wang, Q., J. Wang, H. Zhan, and W. Shi (2020), New model of reactive transport in a single-well push–pull test with aquitard effect and wellbore storage, Hydrology and Earth System Sciences, 24(8), 3983-4000.