The manuscript have been revised by two reviewers.
A significant number of major concerns have been raised during the review process. In particular one reviewer highlighted a fundamental inconsistency in the authors’ approach.
In their replies, the authors seems confident to be able to address the main issues raised during the reviewer process.
However, there is a crucial point in the authors’ replies that is not entirely convincing. The authors acknowledge that their approach entails a number of heuristic short cuts. They justify these on the basis of their ability to reproduce heads closely under some specific radial flow conditions. However, the purpose of pumping tests is to evaluate aquifer parameters which do not just reproduce observed head during a test, but allow modeling flow through the aquifer under other, arbitrary conditions.
Showing a good match between approximate and exact predictions of head during a test is in principle not enough to guarantee the latter, unless the model is physically (strongly) well founded. The latter requirement is not obviously the case in the proposed approach.
If the authors decide to revise and resubmit the manuscript to HESS they should carefully address this issue (together with the main criticisms pointed out by the two reviewers).

The manuscript has been revised by two reviews. One reviewer is satisfied with the paper, while and the second reviewer raises some issues that require some discussion.

The authors are encouraged to explain clearly the approximations associated with the upscaling technique they employ. I think the authors do so at the very beginning of Section 2.2. Here, they mention the idea of "spatial filtering of the flow equation", which is a key point raised by one Reviewer. However, I encourage them to be very clear about possible limitations and approximations introduced by the procedure, so that the reader does not have to become an expert of coarse graining to grasp the extent of the approximations involved. This is only in the spirit of having a self-contained contribution. Indeed, at about line 160 the authors state that they somehow localize the flow equation after coarse graining. However, the message is not crisp and making it clear would benefit the reader. Perhaps the message is diluted by the fact that the authors refer to "coarse-grained transmissivity" and "effective description of the transmissivity" instead of "effective description of the flow field expressed in terms of effective/representative parameters". All these comments are made in the interest of clarity and to maximize impact of the work.

An issue that I can see is that in Section 2.4 the authors state that they solve the integral in (3) by plugging into it the transmissivity (4). One can note that (3) is the solution of (2) which, in turn, is not an average flow equation. As such, it looks like the authors are not averaging a flow equation, but considering its solution and then plugging a coarse-grained parameter into it. This was also raised by the most critical Reviewer and the rationale underpinning this procedure must be clearly and convincingly explained.

As suggested by the Reviewer who worked on the revised manuscript, a few words should be added to address the way randomness of heads, as quantified for example by the variance of heads, can affect estimation errors. I consider this to be an element of importance to be included in a revised version of the manuscript. Additionally, it should be noted that the approach illustrated by Neuman et al. (2004) makes not only use of the (ensemble) average head/drawdown, but also of its variance, thus combining uncertainty within a unique methodology.


In summary, I do think the manuscript could still benefit from a set of revisions, which can be defined as minor, along the lines indicated.

A list of a few minor comments are following.

Abstract: the authors write "A radially depending description of transmissivity is required, including the parameters of log-transmissivity: mean, variance and correlation length". The way it is written is seems that the correlation length, mean and variance, depend on radial coordinate. I am not sure this is the case, so I suggest rephrasing.

Abstract: the sentence: "This is shown making use of virtual pumping tests, for both cases the ensemble mean drawdown and pumping tests at individual transmissivity fields." I am not sure what it means precisely. The authors are required to clarify.

Abstract: the authors write "Multiple pumping tests at an individual transmissivity fields, combined in a sampling strategy ..." this part is also not clear (how do the tests are combined in a sampling strategy? What kind of strategy are the authors referring to?).

Introduction. The authors state "Geostatistical distributions are generally used ...". How do they define a "geostatistical distribution"? I am not pointing out additional awkward terminology which the authors should revise.

Introduction. The authors state: "In the stochastic framework, the solution of the radial flow equation with a log-normal distributed transmissivity is also a random spatial function". What is the radial flow equation the authors are referring to? If they are considering a single realization, then the flow is not radial, but convergent. If the authors are referring to the equation satisfied by the (ensemble) average head, then it is fine. However, this needs to be clearly stated.

Line 55. I assume the authors are not referring to "a single constant value of transmissivity", but rather to a single constant effective/apparent value of transmissivity.

Line 75. What is exactly a "homogeneous substitute value"?

In the Introduction the authors refer to the solution derived by Neuman et al. (2004). Note that this solution was also applied in an actual field setting by Riva et al. (Riva, M., A. Guadagnini, J. Bodin, and F. Delay (2009), Characterization of the Hydrogeological Experimental Site of Poitiers (France) by stochastic well testing analysis, J. Hydrol., 369, 154-164.). I am not stating this to ask the authors to reference this work, but for completeness, since it appears from their phrasing that the solution of Neuman et al. (2004) has only been employed on the basis of synthetic numerical analyses. With reference to the solution of Neuman et al. (2004), note that it is not purely based on Monte Carlo numerical simulations. Otherwise, it draws its root on the analytical solution of the (ensemble) mean flow equation as well as the equation satisfied by the hydraulic head variance/covariance (with its approximations, of course). As such, the authors are encouraged to revise this part. In the context of purely radial (mean) flow, the analytical solutions upon which the type curves of Neuman et al. (2004) are based is that of Riva et al. (2001), which is then combined with the results of numerical Monte Carlo simulations for ease of use. This means that also the third point raised at line 105 is somehow misleading and/or incomplete. All these comments are not made in the spirit of asking the authors to reference the above mentioned manuscript. Otherwise, they are only made for the sake of clarity and complete depiction of the problem.

Additionally (and as I stated at the beginning of my assessment), it should be noted that the approach illustrated by Neuman et al. (2004) makes not only use of the (ensemble) average head/drawdown, but also of its variance, thus combining uncertainty within a unique methodology. I think this is also an important difference to be highlighted.

The authors concentrate on and refer to an "effective well flow solution". The key point they should explain is whether this solution derives from an effective equation of the kind which can be obtained by applying coarse graining to the flow equation governing the system behavior. This aspect is not unambiguously stated by the authors and I consider eliminating doubts about this part to be essential for final acceptance of the manuscript.

The authors consider transmissivity to be a function solely of the radial distance (and not of the angular coordinate) from the well in Section 2.1. I am not sure they comment on such an assumption and they are encouraged to do so. My reading of their (2) is that it is a stochastic differential equation (i.e., it should hold for a single realization). As such, I see no particular reason to drop the dependence of transmissivity from the angular coordinate. On the other hand, I see no critical problem in doing so for, say, an ensemble averaged flow equation. This is most likely done in their Section 2.2 but it should be made very clear.

I think the paper can be accepted after the minor modifications highlighted by the authors in their responses.