26 Aug 2021
26 Aug 2021
Advances in the Hydraulic Interpretation of Water Wells Using Flowmeter Logs
 ^{1}Department of Energy and Fuels Systems, School of Mines and Energy, Universidad Politécnica de Madrid. Madrid, 28003, Spain
 ^{2}Gobierno de España – Ministerio de Ciencia e Innovación. Madrid, 28046, Spain
 ^{3}Department of Natural Resources and Environmental Engineering, School of Mining Engineering, Universidad de Vigo. Pontevedra, 36310, Spain
 ^{1}Department of Energy and Fuels Systems, School of Mines and Energy, Universidad Politécnica de Madrid. Madrid, 28003, Spain
 ^{2}Gobierno de España – Ministerio de Ciencia e Innovación. Madrid, 28046, Spain
 ^{3}Department of Natural Resources and Environmental Engineering, School of Mining Engineering, Universidad de Vigo. Pontevedra, 36310, Spain
Abstract. This paper reports on the methodology developed for a new hydraulic interpretation of flowmeter logs, allowing a better characterization of continental hydrological basins. In the course of a flowmeter log, different flow stretches are established mostly corresponding to permeable layers (aquifers), among which there are other stretches mainly corresponding to less permeable layers (aquitards). In such hydrological basins of sufficient thickness, these flow stretches may not have the same hydraulic head. This fact brings about the need for a new hydraulic interpretation that provides the actual distribution of horizontal permeability throughout the aquifer at depth. The modified hydraulic interpretation developed in this study focuses on the differences of the effective pressure gradient (considered as the difference between the hydraulic head in the well and the hydraulic head of each stretch) experienced by the different flow stretches along the well, due to the existence of different hydraulic heads. The methodology has been developed starting from a water well located in a multilayered aquifer within the sonamed Madrid Basin (the NW part of the continental basin of the Tajo River), located in the centre of the Iberian Peninsula. In this well, a stepdrawdown pumping test was conducted, in which the pumping rate versus drawdown and the specific capacity versus drawdown showed discrepancies with Darcian behaviour and an exponent of the Jacob equation of less than 1. Flowmeter logs were then recorded for different discharge rates and pump depths; the resulting water input from deeper permeable layers did not appear to show the expected relation with respect to drawdown. With the proposed methodology the results comply with the expected linearity and the cited discrepancies are solved.
Jesús DíazCuriel et al.
Status: final response (author comments only)

RC1: 'Comment on hess2021380', Frederick L Paillet, 01 Sep 2021
My review has been cursory as per your requet for rapid response. In looking at the mauscript the main thrust is given as including a correction for the effect of hydraulic head variation between otherwise siolated contributing aquifers in well production testing. The many factors acting to confound stepped drawdwn tests are cited such as nonlinearilty caused by turbulence and "skin" losses. Hydraulic head differences certainly contribute to this problem. However, the sbject of head differences in assessing the permeability of aquifers in multizone wells has been treated in elaborate detail by my USGS colleagues. Exactly 20 years ago I published an article (Paillet, Ground Water, v 39, no 5, p 667) that addresses just this problem. The theoretical background for this analysis was presented by Paillet, Water resources Research, v 34, no 5, p 997. Comparison of head interpretations were quantitatively compared to mutizone piezometer data by Paillet et al, 2000, Journal of Hydrology, v 234, p 208. Since then my colleagues and I have been advocating the use of flowmeters to determine hydraulic head differnces within hetreogeneous formations as being more indicative of largescale connections within fracture flow systems than the local transmissivity of specific flow zones where they intersect boreholes. Not long ago the USGS made a numerical code ackage (FLASH) available online for this analysis. The authors cite LeBorgne 2006 and that study uses the heads inferred from aquifer testing (using flowmeter data) in multizone wells to track the expansion and contraction of the cone of drawdown as a supply pump was cycled off and on.
That said, the topic is treated in the context of highresolution (EM and HP) flowmeters where other sources of nonlinearity are usually limited. Adding head difference considerations to the interpretation of impeller flowmeter logs in the presence of sources of nondarcyian flow would still be of interest. Just not as such a novel approach as implied here.

CC1: 'Reply on RC1', Jesús DíazCuriel, 06 Oct 2021
We thank you for your review and comments; we look forward to providing a satisfactory response.
In L423 it is explained why in this study turbulence is not considered a factor that justifies the anomalous behavior, as the flow in groundwater in granular media does not reach the turbulent regime. On the other hand, the subdivision of the well into stretches is done precisely to avoid the influence of differences in skin effects on the flowmeter results (L78).
In many of the papers involving USGS technicians that we had reviewed (Keys and Sullivan, 1979; Molz et al., 1990; Hess et al., 1991; Crowder et al., 1994; Gossell et al., 1999; Wilson et al., 2001; Johnson et al., 2005; Lane et al., 2002; Williams, 2008; Garcia et al., 2010; Paradis et al., 2011) we have not found that hydraulic heads differences in the assessment of aquifer permeability in multizone wells have been discussed in great detail.
We had not cited Paillet's publications, because we had understood that the methodology shown was largely based on the measurement of flowmeterlogs in ambient condition and with flow rates much lower (<5 l/min = 0.0014 l/s) than the case analyzed in this study. With reference to Paillet et al. (2000, Flowmetering of drainage wells in Kuwait city, Kuwait, Journal of Hydrology, v234, p208) and Paillet (2001, Hydraulic head applications of flow logs in the study of heterogeneous aquifers, Ground Water, v39, p667), the methodology to obtain the different hydraulic heads is not added in our new version, because we could not find the description of the procedure followed to obtain these values.
Following your comments we are going to make the following modification: In L53 "no methodology has been published to quantify its effects" to be more precise, we will add "in water wells in large continental detrital basins".
Paillet (1998, Flow modeling and permeability estimation using borehole flow logs in heterogeneous fractured formations, Water Resources Research, v34, p997) showed the results of two flowmeter logs obtained with a heatpulse flowmeter (lower limit of ~0.1 l/min and upper limit of ~20.0 l/min) in Waupun (Wisconsin, USA). These flowmeter logs was measured under ambient and injection conditions at about 4 l/min, and analyzed for pumping or injection rates typically 15 l/min. We think that the relationship used to estimate the transmissivity T_{k }of each fracture k, starting from the flow into the borehole q_{k} is: q_{k}^{b} – q_{k}^{a }= 2·π·T_{k · }(w^{a} – w^{b}) · ln(R_{0}/r_{w}) where a and b address the ambient and stressed conditions respectively, w^{a,b} is the water level in the borehole, R_{0} is the distance to the "outer edges" of the fracture, and r_{w} is the borehole radius. This relationship does not depend on the unknown value of the farfield head in the aquifer H_{k}. Later, in Paillet (2000, A field technique for estimating aquifer parameters using flow log data, Ground Water, v38, p510) ∑T_{k}·H_{k} = w^{a} · ∑T_{k} is used to determine T_{k}. In this work was stated that: "the results of high capacity tests, where the effects of ambient hydraulichead differences would not be significant", hydraulic head values (4.54, 4.91, 4.91 and 4.91 m below ground level) are presented for the four productive stretches in one of the boreholes analyzed, although the process followed is not reflected in this paper. In Paillet (2001, cited above) the hydraulic head estimates (cm above open hole water level) in the same borehole (+28, 11, 11, and 11 cm above open hole water level) are shown. Based on this methodology, DayLewis et al. (2011, A computer program for flowlog analysis of single holes (Flash), Ground Water, v49, pp926931) presented a computer program for flowlog analysis of single holes applicable up to 10 levels, in which the hydraulic head of each zone is determined by minimizing the differences between the flow rates obtained and those of the model, and between borehole's water level and farfield heads.
Following the reviewer's comments, in L222 we are going to add: “The main differences with the method used by Paillet (1998) are that we have chosen to use the Rehfeldt relationship (Eq. 2) for permeability instead of the Davis and DeWeist relationship (1966) relation for transmissivity, given that the thickness of the layers and the productive sections are taken into account. It has also been considered that the different hydraulic heads are below the static water level (the water level in ambient conditions from Paillet, 1998). The procedure developed is based on the linearity of the hydraulic behavior of the aquifer sections and each section is treated separately.”

AC5: 'Reply on RC1', Jesús DíazCuriel, 03 Dec 2021
We thank you for your review and comments; we look forward to providing a satisfactory response.
In L423 it is explained why in this study turbulence is not considered a factor that justifies the anomalous behavior, as the flow in groundwater in granular media does not reach the turbulent regime. On the other hand, the subdivision of the well into stretches is done precisely to avoid the influence of differences in skin effects on the flowmeter results (L78).
In many of the papers involving USGS technicians that we had reviewed (Keys and Sullivan, 1979; Molz et al., 1990; Hess et al., 1991; Crowder et al., 1994; Gossell et al., 1999; Wilson et al., 2001; Johnson et al., 2005; Lane et al., 2002; Williams, 2008; Garcia et al., 2010; Paradis et al., 2011) we have not found that hydraulic heads differences in the assessment of aquifer permeability in multizone wells have been discussed in great detail.
We had not cited Paillet's publications, because we had understood that the methodology shown was largely based on the measurement of flowmeterlogs in ambient condition and with flow rates much lower (<5 l/min = 0.0014 l/s) than the case analyzed in this study. With reference to Paillet et al. (2000, Flowmetering of drainage wells in Kuwait city, Kuwait, Journal of Hydrology, v234, p208) and Paillet (2001, Hydraulic head applications of flow logs in the study of heterogeneous aquifers, Ground Water, v39, p667), the methodology to obtain the different hydraulic heads is not added in our new version, because we could not find the description of the procedure followed to obtain these values.
Following your comments we are going to make the following modification: In L53 "no methodology has been published to quantify its effects" to be more precise, we will add "in water wells in large continental detrital basins".
Paillet (1998, Flow modeling and permeability estimation using borehole flow logs in heterogeneous fractured formations, Water Resources Research, v34, p997) showed the results of two flowmeter logs obtained with a heatpulse flowmeter (lower limit of ~0.1 l/min and upper limit of ~20.0 l/min) in Waupun (Wisconsin, USA). These flowmeter logs was measured under ambient and injection conditions at about 4 l/min, and analyzed for pumping or injection rates typically 15 l/min. We think that the relationship used to estimate the transmissivity T_{k }of each fracture k, starting from the flow into the borehole q_{k} is: q_{k}^{b} – q_{k}^{a }= 2·π·T_{k · }(w^{a} – w^{b}) · ln(R_{0}/r_{w}) where a and b address the ambient and stressed conditions respectively, w^{a,b} is the water level in the borehole, R_{0} is the distance to the "outer edges" of the fracture, and r_{w} is the borehole radius. This relationship does not depend on the unknown value of the farfield head in the aquifer H_{k}. Later, in Paillet (2000, A field technique for estimating aquifer parameters using flow log data, Ground Water, v38, p510) ∑T_{k}·H_{k} = w^{a} · ∑T_{k} is used to determine T_{k}. In this work was stated that: "the results of high capacity tests, where the effects of ambient hydraulichead differences would not be significant", hydraulic head values (4.54, 4.91, 4.91 and 4.91 m below ground level) are presented for the four productive stretches in one of the boreholes analyzed, although the process followed is not reflected in this paper. In Paillet (2001, cited above) the hydraulic head estimates (cm above open hole water level) in the same borehole (+28, 11, 11, and 11 cm above open hole water level) are shown. Based on this methodology, DayLewis et al. (2011, A computer program for flowlog analysis of single holes (Flash), Ground Water, v49, pp926931) presented a computer program for flowlog analysis of single holes applicable up to 10 levels, in which the hydraulic head of each zone is determined by minimizing the differences between the flow rates obtained and those of the model, and between borehole's water level and farfield heads.
Following the reviewer's comments, in L222 we are going to add: “The main differences with the method used by Paillet (1998) are that we have chosen to use the Rehfeldt relationship (Eq. 2) for permeability instead of the Davis and DeWeist relationship (1966) relation for transmissivity, given that the thickness of the layers and the productive sections are taken into account. It has also been considered that the different hydraulic heads are below the static water level (the water level in ambient conditions from Paillet, 1998). The procedure developed is based on the linearity of the hydraulic behavior of the aquifer sections and each section is treated separately.”

CC1: 'Reply on RC1', Jesús DíazCuriel, 06 Oct 2021

CC2: 'Comment on hess2021380  Advances in the Hydraulic Interpretation of Water Wells Using Flowmeter Logs', Teresa Valente, 16 Oct 2021
Very interesting work.
I believe this methodology is novel, comparing with that which is classically adopted. The procedure considers that, vertically, the multilayer basins can be modeled, as a whole, as an equivalent porous model.
Teresa
 AC2: 'Reply on CC2', Jesús DíazCuriel, 18 Nov 2021

RC2: 'Comment on hess2021380', Frederick L Paillet, 17 Oct 2021
The author’s response to my rather cursory review (as per the request for quick turnaround) has been responded to with a note that much of the earlier work on using flowmeter profiles to quantify hydraulic head differences in aquifer layers relates to aquifer tests made at low pumping rate in lowcapacity wells. Actually, these flowmeters can be adapted to measure substantial flow rates in some cases. The general method is applicable to stratified granular aquifers, and has been used in that application, even in most of the referenced works were for fractured bedrock or karst. The generalized method has already been described as applicable to any pair of quasisteady flow conditions, which could be static and pumped, a pair of pumped conditions or even injection. Some authors have extended that to consider several such states, and then fit the flow model to the profiles. The general scheme is the same: measure flow under two conditions to get two sets of inflow/outflow data, and then “invert” the model fit to solve for transmissivity and head simultaneously. In general, this is like solving any set of coupled equations for two variables. If more than two data sets, the problem is treated as a standard overdetermined inversion.
That said, the technique has not been applied to impeller flowmeter logs and that has real practical applications. Other flowmeter work has demonstrated that the technique works effectively within screened wells. The analysis gives relative head difference between quasi steady open hole water level during pumping and the farfield head in each producing zone. But that can be converted to actual head by reference to the static water level if known. But keep in mind that the corrections to inferred interval transmissivity still involve skin and turbulent inflow contributions, negligible at ultralow pumping rates. But a more important point is that the transmissivity value applies to the immediate vicinity of well bore or screen. This is imposed by the convergent flow regime. Largescale aquifer properties are, however, indicated by the hydraulic head estimates. This application is especially relevant in contamination studies where one wants to know if one contaminated aquifer is isolated from another.
In summary, the authors have added a note that the idea of inferring hydraulic head in situ in multilevel aquifers (both fractured bedrock and granular sedimentary layers) has been around for a while, but almost none of this has been worked out for impeller flowmeters in highdischarge wells. Hence their manuscript provides new results. Even more interest could be added by addressing the specific complications produced under such well test conditions. But even more relevance can be added by citing the need to understand the largescale structure of aquifers concerning recharge and contaminant communication rather than just a correction to standard ump test evaluations of transmissivity based on the assumption of a single aquifer. The ability to infer head differences in situ for multilevel aquifers has a lot more application than just correcting measured transmissivity for the presence of those head differences.

AC3: 'Reply on RC2', Jesús DíazCuriel, 18 Nov 2021
Dear Reviewer;
We thank you for your review and comments; we attach the response indicating the changes we have made. We are confident that we have given a satisfactory response to your suggestions.
A document has been attached in which the proposed changes are differentiated in red text. In addition, a new figure is included.
−) But keep in mind that the corrections to inferred interval transmissivity still involve skin and turbulent inflow contributions, negligible at ultralow pumping rates. …
In the manuscript, the "skin" effects are mentioned and it is specifically stated that the determination of stretches is made precisely in order not to take into account the different values that these effects may take in each screen.
As mentioned in the manuscript, the groundwater flow does not become turbulent even for the maximum flow rate used (for which the groundwater velocity has been calculated in those vicinities, in particular, for a radius equal to that of the well).
−) The ability to infer head differences in situ for multilevel aquifers has a lot more application than just correcting measured transmissivity for the presence of those head differences.
−) But even more relevance can be added by citing the need to understand the largescale structure of aquifers concerning recharge …
In discussion section, L476, we have added:
“This study has allowed to carry out the hydrological and hydraulic division of the studied basin that had not been done before, and such division involve a more precise obtaining of the permeability values in each stretch (and hence in its corresponding aquifer) which was neither been before. Certainly, the new procedure developed to obtain the hydraulic head differences in heterogeneous granular basins and the results obtained for the first time in the Madrid basin may allow hydrogeological hypotheses to understand the largescale structure of aquifers concerning recharge. According to the results obtained, the fact that the Madrid Basin is considered a single aquifer should be replaced, at least from a depth of 200 m, by a sequence of stretches aquifers differentiated by their different permeability values. From 345 m depth (the one of stretch 4), it was also found that the aquifers corresponding to stretches 4, 5 and 6 have different "hydraulic heads" than the upper aquifers. One hypothesis would be that this means different "recharge pathways". So that it could be deduced that above 345 m the Madrid Basin can be considered a single heterogeneous aquifer (with different subaquifers of different permeability), and below 345 m, the Madrid Basin consists of a sequence of confined aquifers (the last three coarsegrained ones shown in the welllogs, see Fig. 4) that are hydraulically separated from the rest of the aquifers.
New figure Fig. 10 (PLEASE SEE ATTACHED DOCUMENT)
Figure 10. Largescale scheme of NW arc of the Madrid basin
It should be emphasized that the hydrogeological hypotheses that can be made as the previous scheme must be contrasted with results in more wells within the NW arc of the Madrid Basin.”
−) … and contaminant communication rather than just a correction to standard ump test evaluations of transmissivity based on the assumption of a single aquifer.
Just next to above text, we have added:
“The division of the studied well also allows proposing a strategy regarding the arsenic propagation in the Madrid basin. The obtained results indicate the stretch of the studied well that is "activated" when the dynamic level exceeds the "hydraulic head" of the aquifer to which it corresponds, is the rather connected to a point or zone where the arsenic focus is. As the exploitation of that stretch in different points of the basin will cause the contaminant to move towards those points, that critical dynamic level should be not allowed.”

AC3: 'Reply on RC2', Jesús DíazCuriel, 18 Nov 2021

CC3: 'Comment on hess2021380', Teresa Albuquerque, 18 Oct 2021
The authors introduce an interesting approach for a new hydraulic interpretation of flowmeter logs. The methodology was implemented using a case study a well located in the Madrid Basin. In my opinion, it would be advisable to validate the results using other and different case study examples.

AC4: 'Reply on CC3', Jesús DíazCuriel, 18 Nov 2021
Thank you very much for your comments. We agree with the interest of conducting similar studies in other aquifers with these characteristics, although at this time we do not have such data. In fact, the manuscript mentions a future line of action to determine the spatial extension of this behavior in this particular aquifer.

AC4: 'Reply on CC3', Jesús DíazCuriel, 18 Nov 2021

RC3: 'Comment on hess2021380', Anonymous Referee #2, 26 Oct 2021
Comments on the paper HESS2021380 entitled: “Advances in the hydraulic interpretation of water wells using flowmeter logs”, by J. DiazCuriel et al.
This contribution revisits the concepts of flowmeter logs (i.e. measuring water velocities within the casing of a well) to discuss on the ability of the technique to identify various hydraulic behaviors (mainly hydraulic conductivity) of flowing water bearing bodies crosscut by a pumping well. These water bearing bodies are referred to as stretches by the authors, some kind of subseries of neighbor superimposed sedimentary layers with presumably homogeneous hydraulic behavior.
The key question motivating the revisiting is that each stretch could be under the influence of its own hydraulic head gradient (which is logical, if those stretches are weakly or not connected within the subsurface aquifer system), a feature usually not accounted for in classical interpretations.
As the application targets a deep and largediameter well crosscutting several stretches under forced flow (from large extraction rates in the monitored well above the depth investigated via flow logs), the interpretation of logs needs for a preprocessing of the raw data. My understanding is that flow within the blind or the screened casing of the well stands between inertial and turbulent flow. In any case, the water velocity measured by the impeller of the flow log probe is not the mean velocity through the whole section. This raw velocity is corrected by relying upon classical formulas of turbulent flow in pipes to provide actual velocities at various depths along the cased well. These profiles also allow for calculating the local head loss (Deltah) along the well by relying upon the DarcyWeisbach relationship stating that the head loss (or head gradient) is proportional the squared velocity of water. It is worth noting that the authors are very picky on the way they preprocess the data. It is not clear in the writing if this procedure is mandatory for getting reliable information. I could also figure out that this procedure is specific to the application of log interpretations over wide wells and under large extraction rates. One can doubt on the usefulness of all that stuff in small wells. Do we become overgunned for example, when passing through fractured (karstified) layers, with smalldiameter wells weakly stressed by the extraction rates of a few liters per second (a classical investigation in that context). Anyway, who can do more, can also do less! I keep still in mind the fact that head losses within the widediameter wells (as stated by the authors with a different phrasing, lines 341) are negligible compared with hydraulic head loss in the diverse water bearing bodies crosscut by the well. Stated differently reinterpreting inflows from stretches in a well could also consider that the head within the well is uniform over its whole depth… Good news!
Another interesting feature of the study is that it investigates a single well for water supply (of 475 m depth) passing through several separated and confined aquifers. This is completely prohibited by the legislation of many countries! In these countries a single well should be completely blindcased and cemented until it reaches the targeted aquifer. Reaching and monitoring another aquifer above or beneath would need for another distant well, also blind cased over its wellbore through the above formations. It goes without saying it that a single well passing through several stretches is an opportunity to record continuous flow logs, with less errors than those of reconstituting a synthetic log from different distant wells. Another way to say that is to consider that the well investigated by the authors is probably an outlier (at least an exception) in the Literature, which deserves exposure in a scientific paper of large audience.
The diverse flow stretches are analyzed under the forced flow conditions that allow for the interpretation of their hydraulic property. My understanding is that the authors employ the classical formula of the drawdown in the well in the form of the Jacob equation: s = aQ + bQ^p with s the drawdown, a and b constants, Q the extracted flow rate from a stretch, p a power law factor. With several extracted rates Qi over the diverse stretches (i), for the different sum of extracted rates (sum Qi) above the log, the authors force the Jacob relationship with a factor p=1… Then, (a+b)i is proportional to the hydraulic conductivity of the stretch and identified (via the Thiem equation) if the static head far from the pumped well is known… Not that much tricky but absolutely not clearly explained at all in the paper… I would urge the authors to put dots on I and cross on T, especially in Section 4.6, on the way they derive conductivities from local measurements of flow rates (from flow logs) and an overall head drawdown between the monitored well and a distant location. It took me time and a few head scratch, to conjecture how the authors did it. For the rest, everything is clear…
Finally, I thing that the paper could be published almost as it appeared at its first release. A few imprecisions could be cured with cosmetics adjustments.
 Line 96. Not well said. A single scalar value (that of a conductivity in a layer) is always proportional to another one (that of the whole wellbore) up to a multiplying constant: a = (a/b)*b!
 Fig. 1left (or Fig. 6). I would have swapped in one of the figs the horizontal and vertical axis, so they can read exactly the same way without leaning the head at 90° in Fig 1left, to find the same plot as in Fig6. By the way, in Fig. 1left the coefficient “A1” = 0.6 should read “A4” = 0.6.
 Line 180. Specify which terms are involved in the Reynolds number, especially the “length” that I guess to be the diameter (radius) of the well.
 P. 7, Fig. 2. Specify that k in the notations Rek is the iteration index of the convergence algorithm, and not anything else.
 Line 200, Eq. 7. Please also remind the form employed for the DarcyWeisbach equation. Several form exist, even if one can guess that in here the form is: Gradh (or Deltah) = (f/2g)*(V^2/D) (D effective diameter of the well, g gravity, V water velocity, and f friction factor).
 P. 13, Table 2. Not clear how the deltah in the table are calculated. Is that a mean from bottom to top handling a mean friction factor and a mean velocity over the whole depth investigated? Or is that (what I think better) the cumulated Delta h adding the successive local delta values relying upon local friction factors and local water velocities?
 P. 15, Fig.8a (the three left plots). It I unclear to me what mean the alternating grey and white bars beneath (left to) the three curves. They do not seem to be the alternation of geological layers, as they are not the stretches (#1 to #6) reported in Fig. 8B (the three right plots). Do they correspond to intervals where the monitored velocities in the flow logs are quite uniform?

AC1: 'Reply on RC3', Jesús DíazCuriel, 18 Nov 2021
Dear Reviewer;
We thank you for your review and comments; we attach a detailed response indicating the changes we have made considering all suggestions. We are confident that we have given a satisfactory response to your suggestions, and we are grateful for the exhaustive review of the manuscript, which has allowed us to correct some mistakes and clarify some issues.
A document differentiating the proposed changes in red text has been attached. In addition, the modified and new figures, which could not be included in the interactive response, are included.
In response to the issues raised by the reviewer general comments:
− It is not clear in the writing if this procedure is mandatory for getting reliable information.
The need for an exhaustive treatment of the flowmeter logs arose initially to avoid doubts on observed anomalies in the characteristic curves of the stepdrawdown test could stem from the reliability of the flowmeter log results. Thus, it had to be shown that such effects were not due to head losses along the well. In addition, considering that the flow velocity used in the DarcyWeisbach equation is raised to a power of two, the differences between the head losses resulting from considering the actual flow velocity instead of the velocity directly measured by the sonde is greatly amplified.
− One can doubt on the usefulness of all that stuff in small wells.
Applying the rigorous formulation presented to process the flowmeter logs (Eqs. 3 to 7) and considering that the sonde has a significant diameter (r_{D}), the values of <V>/V(r_{D}) vary between 0.85 and 0.94. However, if the well diameter is smaller (close to the diameter of the sonde, (V(r_{D}) approaches Vmax, resulting that the <V>/V(r_{D}) ratio presents a greater variation (from 0.50 to 0.83) for the range of Re found, than if the diameter of the well analyzed is used.
− Not that much tricky but absolutely not clearly explained at all in the paper… especially in Section 4.6, on the way they derive conductivities from local measurements of flow rates (from flow logs) and an overall head drawdown between the monitored well and a distant location.
Following the reviewer's recommendation, several changes have been introduced in sections 3, 4.3, 4.4, 4.5 and 4.6.
1) In section 3, L215, the following changes have been made:
“In most flowmeter logging with several pumping steps, the drawdown used in the Thiem (1906) equation is the same for all of the aquifer stretches in a well, d_{0}(s)= h_{DL}(s)−H_{SL} where h_{DL}(s) is the dynamic level for the ‘s’ pumping step and H_{SL} is the dynamic level of the entire well. However, under the hypothesis presented in this work, the hydraulic head of each stretch, and therefore the corresponding drawdown, can be different. Numerically, the drawdown of each flow stretch т_{N} will be given by the following relation:
d_{N}(s)= h_{DL}(s)  h_{SL}(s) (8)
where h_{SL}(n) is the static level for flow stretch т_{N}. In short, the proposed method consists of replacing the single drawdown d in Eq. (2) from Rehfeldt by a drawdown for each stretch.”
2) In L225, we have corrected:
"The proposed method for obtaining the hydraulic head of each flow stretch is to 1) correct the drawdown values of the total head loss due to flow along the pipeline and 2) modify the height of the hydraulic head for each flow stretch until the straight line fitted to the data, q_{N}(s) versus d_{N}(s), reaches the maximum regression coefficient (where q_{N}(s) is the water input in flow stretch N for the s pumping step). "
3) In section 4.3, L307, we have added:
"The static level H_{SL} was measured at a depth of 157 m before the beginning of flowmeter logging. Flowmeter logs were obtained for pumping rates of 20 l/s (measured dynamic level at 172 m), 30 l/s (dynamic level at 178 m), and 70 l/s (dynamic level at 205 m). The drawdowns of the entire well for each pumping rate, without including the head losses, hence are 15 m, 21 and 58 for 20 l/s, 30, and 70 respectively."
4) In section 4.4, L334, we have added:
“The obtained values of the head loss ∆h(s) for each pumping rate are shown in Table 2, which will be used in the calculation of the effective drawdown produced.”
In L370, we have added:
Table 2. Head loss values for each pumping rate in the case study
(PLEASE SEE THE PDF ATTACHED)
5) In section 4.5, table 3, we have the next changes:
Table 3. Water inputs of flow stretches for different pumping rates and fractions over the total flow rate Q_{T} in the case study.
(PLEASE SEE THE PDF ATTACHED)
6) In section 4.6, the following changes have been made:
Before L385 we have added the explanation of the calculation of permeability, which, as the reviewer rightly points out, had not been specified.
“The permeability of each stretch has been calculated using Eq. (2). Instead of the contribution of each layer q_{j}, the sum total of the contributions of each stretch q_{N}(s) is considered (see table 3). The unique initial drawdown d considered in Eq. (2) has been modified by the drawdown of the entire well d_{0}(s)= h_{DL}(s)−H_{SL}− Δh(s) for each pumping rate (s) (Δh(s) being the head losses showed in table 2). The static level H_{SL} is 157 m (as determined before the flowmeter logging was conducted) and the dynamic levels h_{DL}(s) are 172 m for pumping rate of 20 l/s, 178 m for pumping rate of 30 l/s y 205 m for pumping rate of 70 l/s. The thickness of each layer Δz_{j} has been replaced by the thickness of each stretch Δz(т_{N}) (depth intervals in Table 3). The radius of influence (R_{0}) considered is 950 m (as in the previous calculations), and the well radius (r_{w}) is 0.404/2=0.202 mm. The characteristic curves of each stretch are shown in Fig. 9.a.”
Following the reviewer's comments on Fig. 6, the axes of the graph in Figure 9 have been inverted, being now d_{N}(s) versus q_{N}(s). A new figure has also been added (Fig. 9.a) showing the resulting curves considering a unique hydraulic head for all stretches. It should be noted that by inverting the axes (d_{N}(s) versus q_{N}(s) instead of q_{N}(s) versus d_{N}(s)), the coefficients of the fitting curves are the inverse of those shown in the old figure.
7) In the paragraph of L385 we have made the following changes:
“Analyzing the specific capacities of different flow stretches, т_{1} and т_{3} show the expected proportionality for a confined aquifer. However, this is not the case for flow stretches т_{4}, т_{5} and т_{6}, whose d_{N}(s) versus q_{N}(s) data fit to a power function with exponents of 0.22, 0.37 and 0.67, respectively (see Fig. 9.a). Not only does this not reflect Darcian behavior, but it also indicates an exponent p in the Jacob equation of less than 1, as is the case with the well as a whole (see Fig. 6).”
8) In the paragraph of L390 we have made the following changes:
“However, if it is considered that flow stretches т_{4}, т_{5} and т_{6} have different hydraulic heads, the results vary. Through an iterative process, the value of the static level (hydraulic head) of each flow stretch for which the total water input of the flow stretch versus the drawdown acquires greater alignment can be determined. This means that when the data are fitted to a straight line, the regression coefficient is maximum. In other words, the resulting exponent in the Jacob equation when the data are fitted to a power function is p=1. Thus, for flow stretch т_{6}, the static level for which inputs versus drawdown acquire greater alignment occurs at a depth of 165 m. Similarly, the resulting static level for flow stretch т_{5} is located at a depth of 175 m. For a pumping rate of 70 l/s, flow stretch т_{4} undergoes an “activation” effect (even higher than flow stretch т_{5}) when the dynamic level exceeds the true static level of т_{4}, which is computed at a depth of 177.5 m. Summarizing, the hydraulic heads h_{SL}(N) obtained with this criterion are 157 m for т_{1} and т_{3}; 177.5 m for т_{4}; 175 m for т_{5}; and 165 m for т_{6}.”
9) In L398 we have added:
“Figure 9.b shows the regression lines of water inputs versus drawdown for each stretch, with the corresponding relationships and R^{2} coefficients”
New Fig. 9 and its caption:
(PLEASE SEE THE PDF ATTACHED)
Figure 9. Drawdown versus water inputs for different flow stretches in the case study. a) d_{N}(s) # q_{N}(s) with a unique hydraulic head for all the stretches. b) d_{N}(s) # q_{N}(s) with the modified hydraulic heads for each stretch obtained considering that p is at least equal to one in the Rorabough equation).
10) In L401, the next changes have been added:
With these differentiated static levels, the hydraulic conductivities of each flow stretch were obtained using a next change of Eq. (2) (Rehfeldt et al. 1989) replacing d_{0}(s) by d_{N}(s)= h_{DL}(s)−h_{SL}(N)−Δh(s), which values are presented in Table 4.
Next, we have added:
The successive relationships used to arrive to the actual permeability with depth have been:
(PLEASE SEE THE PDF ATTACHED)
It must point out that the k_{N} is the same for the different (s) because the ratio q_{N}(s)/d_{N}(s) is the same for any pumping rate (d_{N}(s) versus q_{N}(s) are fitted to a straight line).
11) In L403, table 4, the next changes have been made:
Table 4. Specific capacities and permeabilities of flow stretches for the static level determined in the case study
(PLEASE SEE THE PDF ATTACHED)
12) Sentence in L406 has been modified as follows:
“The average hydraulic conductivities of the stretches in the studied part of the well (200 to 470 m) have values between 2·10^{−4} and 1.3·10^{−3} Darcy, providing a geometric mean value of 5·10^{−4} Darcy, which is close to the average hydraulic conductivity obtained with the pumping tests.”
Specific Comments
 Line 96. Not well said. A single scalar value (that of a conductivity in a layer) is always proportional to another one (that of the whole wellbore) up to a multiplying constant: a = (a/b)*b!
We appreciate the reviewer's recommendation; we have added the following in L97:
“To achieve hydraulic interpretation from flowmeter logs, most authors (Molz et al., 1989; Rehfeldt et al., 1992; Ruud and Kabala, 1996; Zlotnik and Zurbuchen, 2003a; BarahonaPalomo, et al. 2011; Riva et al., 2012) start from the basis that hydraulic conductivity values for each permeable layer (from each screen) are proportional to the hydraulic conductivity of the entire well up to a multiplying constant.”
 1left (or Fig. 6). I would have swapped in one of the figs the horizontal and vertical axis, so they can read exactly the same way without leaning the head at 90° in Fig 1left, to find the same plot as in Fig6.
By the way, in Fig. 1left the coefficient “A1” = 0.6 should read “A4” = 0.6.
Many thanks for the correction; we have replaced A1 by A4.
(PLEASE SEE THE PDF ATTACHED)
We appreciate the reviewer's suggestion. We have opted to modify the Fig. 6 (instead Fig 1left as proposed by reviewer) because it is easier to recognize the Jacob (1947) or Rorabaugh, (1953) behavior (d=A·Q+B·Q^{p}). We have divided figure 6 in two graphs, one with the drawdown in ordinates versus pumping rate, and the other with the drawdown in abscissas versus specific capacity.
(PLEASE SEE THE PDF ATTACHED)
Figure 6. a) Drawdown versus the pumping rate, b) and the specific capacity versus drawdown in the case study.
 Line 180. Specify which terms are involved in the Reynolds number, especially the “length” that I guess to be the diameter (radius) of the well.
Following the reviewer's recommendation, we have added in L182:
“It begins by taking the measured velocity V_{meas} at a given depth as the initial flow velocity and the initial Reynolds number Re_{ini} according to its definition, that is, Re= ρ·<V>·D/μ, where ρ is the water density, D the well diameter and μ the dynamic viscosity.”
 7, Fig. 2. Specify that k in the notations Re_{k} is the iteration index of the convergence algorithm, and not anything else.
We appreciate the reviewer's recommendation. We have modified the subscript of Re according to the one used in Figure 7 and we have added the following clarification in L187:
“Then, using the relationship for the velocity factor F_{vel}(τ), defined as the ratio between Vmax and the flow velocity <v>, the first flow velocity is obtained with the corresponding Reynolds number Re_{ini}, which is closer to the actual value. Applying τ(Re), V(r_{D}), and F_{vel}(τ), a new Re value Re_{k} is obtained (k being the iteration index of the convergence algorithm). This process is repeated until a given convergence criterion c_{CR} is reached, following the flow chart in Fig. 2 (adapted from DíazCuriel et al., 2020), to obtain Re(z).”
 Line 200, Eq. 7. Please also remind the form employed for the DarcyWeisbach equation. Several form exist, even if one can guess that in here the form is: Gradh (or Deltah) = (f/2g)*(V^2/D) (D effective diameter of the well, g gravity, V water velocity, and f friction factor).
Following the reviewer's recommendation, we have added in L198199:
“Once the Reynolds number at each depth is known, the head loss can be obtained by the DarcyWeisbach equation (Darcy 1857; Weisbach 1845), given by Δh=f·(ℓ/D)·(<V>^{2}/2g), where g is the gravity acceleration (m·s^{−2}), <V> is the average flow velocity (m·s^{1}), D is inner diameter of the well (m), ℓ is the length of each considered pipe element (m), and f the friction factor (dimensionless) for smooth pipes given by Eq. (7):”
 13, Table 2. Not clear how the deltah in the table are calculated. Is that a mean from bottom to top handling a mean friction factor and a mean velocity over the whole depth investigated? Or is that (what I think better) the cumulated Delta h adding the successive local delta values relying upon local friction factors and local water velocities?
The reviewer is right in his assessment, and to make it clearer we have added in L333:
“The total head loss below the pump is obtained by integrating the head loss throughout the well based on the flow velocity obtained at each depth (see Fig. 7), that is, the cumulated Δh adding the successive local head loses values relying upon local friction factors and local water velocities.”
Reviewing the manuscript, we have noticed an error in L340:
“In this case, the friction factor reaches values six times higher at the bottom of the well than at the initially recorded depth, and the value of the head loss is low (0.006 m) because …”
would be:
“In this case, the friction factor reaches values six times higher at the bottom of the well than at the initially recorded depth, and the value of the head loss is low (0.06 m) because …”
 15, Fig.8a (the three left plots). It I unclear to me what mean the alternating grey and white bars beneath (left to) the three curves. They do not seem to be the alternation of geological layers, as they are not the stretches (#1 to #6) reported in Fig. 8B (the three right plots). Do they correspond to intervals where the monitored velocities in the flow logs are quite uniform?
This was not its purpose. The bars only reflect the depth intervals of each screen (we segmented the measured continuous logs with vertical segments thinking that it reveals that flow only increases in screens, and it better shows the flow steps between them, but if you think we should return the bottom and top points of the flow rate in each screen to its center, please let us know). To clarify this we have added in the caption of Figure 8:
“Figure 8. Flowmeter results in the case study (grey horizontal bars reflect the depth intervals of each screen). a) upward flow rate versus depth; b) water inputs from each screen.”
Jesús DíazCuriel et al.
Data sets
Spinner Flowmeter JESÚS DÍAZ CURIEL https://data.mendeley.com/datasets/gx8dwgvygn/1
Jesús DíazCuriel et al.
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