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Analytical Modeling of Particle Tracking for Dynamic Pumping Conditions

Water 2020, 12(9), 2469; https://doi.org/10.3390/w12092469

by Yuan Gao^1,*

and Thomas Sale²

Reviewer 1: Anonymous

Reviewer 2: Anonymous

Water 2020, 12(9), 2469; https://doi.org/10.3390/w12092469

Submission received: 29 July 2020 / Revised: 31 August 2020 / Accepted: 31 August 2020 / Published: 3 September 2020

(This article belongs to the Section Hydrology)

Round 1

Reviewer 1 Report

See attached file.

Comments for author File: Comments.pdf

Author Response

Reviewer #1:

General comments:

The manuscript describes an analytical method for tracking the movement of fluid particles under pumping and injection conditions in a well field. The method is based on superposition of the Theis solution. The work may be of interest to readers of WATER. However, some assumptions should be better specified, the results should be better presented and the discussion should be more focused.

The authors argue that numerical simulation methods may be insufficient to accurately track particles. Actually, the assumptions for the analytical model presented (however not completely specified in the manuscript) seem to be very simplified with respect to the potential of the flow and solute transport models available in the literature. Since the method is based on Theis solution, the assumptions and limits for applying the proposed method should be specified in the manuscript. In a well field, such as the one examined by the authors, it is possible that the aquifer is neither isotropic nor homogenous. In the work the hydraulic parameters (transmissivity, storativity, porosity and thickness of the aquifer) are considered to be constant, this is difficult to happen in a real aquifer of the type (bedrock sandstone) and size investigated by the authors. Precisely in these cases, numerical models can be more efficient. Furthermore, the results of the numerical models can be controlled through calibration and validation, also defining the uncertainty of the results. In the case of the analytical model presented, it is difficult to verify the validity of the results. I therefore suggest better specifying the assumptions of the method and the limits of the method in the discussion of results.

References to the literature appear to be limited and not always adequate. They actually derive from Theis's solution (1935), why not directly quote Theis? However, an extensive discussion of Theis solutions applied to pumping and injection wells can already be found in Bear (Hydraulics of groundwater, 1979).

In my opinion, the manuscript must undergo a moderate to major revision.

Dear reviewer #1:

Thank you for the comments which could greatly improve the quality of the manuscript. The basic idea of this study is to explore an analytical solution based on the Theis equation using dynamic water level data to track particles under dynamic pumping and injection conditions. This is a simple but a novel analytical method which has not been published in previous studies. Also, the focus of this study is the analytical method. Therefore, the geologic parameters are referred to the literatures on the study site, which were just used to testify the method. We understand your issues with the limitations of the method and totally agree with your comments:

The analytical solution developed in this study has limits to calculate particle tracking under dynamic pumping conditions.
Numerical models are more efficient to simulate particle tracking.
The results of the analytical method are difficulty to verify.
The cited literatures are limited.

Therefore, in the manuscript, the descriptions on the limitations of numerical models have been removed. Further, the assumptions of using the analytical solution developed in this study have been specified. Specifically, in the abstract, the limitations of numerical models have been removed, and the assumptions for using the analytical method have been mentioned in Lines 11-22:

“Movement of fluid particles about historic subsurface releases and through well fields is often governed by dynamic subsurface water levels. Motivations for tracking the movement of fluid particles include tracking the fate of subsurface contaminants and resolving the fate of water stored in subsurface aquifers. Based on superposition of the Theis solution in both space and time, this research explored an analytical solution based on Theis equation using dynamic pumping well data to resolve how fluid particles move around wells under dynamic pumping conditions. The results provide a relatively uniform capture zones for a pumping well. Further, the results show that even under continuous pumping and injection conditions, groundwater will not flow far away from the well. Accordingly, groundwater positions can be evaluated based on the research for dynamic pumping. Using the assumptions proposed by the Theis solution, the analytical solution developed in this study could provide a simple method to evaluate particle movement about well used to both store and recovery water.”

In the introduction section, the limitations of numerical models have also been removed and the assumptions of the analytical method developed in this study have been mentioned in Lines 56-64 of the revised manuscript:

“To our knowledge, previous studies have not reported that using analytical models to track particles under dynamic pumping conditions. As such, the objective of this study is to investigate whether the analytical solution is able to capture dynamic aspects of groundwater flow for complex water surface with dynamic water levels, e.g., pumping conditions. By making assumptions proposed by [20], an analytical model was developed from the Theis equation [20,21] that can calculate head under dynamic pumping and injection conditions to evaluate particle tracking. The analytical solution is used to resolve the hydraulic gradient through a point of interest at a specified time and incremental time steps are used to resolve particle-flow path lines. Consideration is given to the prediction of flow under the influence of pumping and injection conditions.”

In Section 2 (Study site and dataset), the specific assumptions of the analytical method have been added in Lines 73-82 of the revised manuscript:

“In this study, several assumptions of the aquifer and flow conditions are made. The aquifer of the Meadows Pumping Center is assumed to be infinitely areal extent and homogeneous, isotropic, and of uniform thickness with 32.8 ft [22]. Transmissivity of the aquifer in the Meadows Pumping Center is 4000 gal/day/ft, storativity is 0.00005, and porosity is assumed to be 0.25 [22]. The wells in the center are assumed to be fully penetrating and flow to the wells is horizontal. Further, groundwater is assumed to be released instantaneously from storage with decline of hydraulic head, and the diameters of pumping wells in Meadows Pumping Center are very small so that storage in the well can be neglected. These assumptions of the aquifer and groundwater flow conditions also make the calculation of the drawdowns in the wells follow the Theis solution described Section 3.”

In Section 3 (Methodology), Theis (1935) and Bear (1979) have been cited instead of Davis (2013). These two references have been added in the reference section. The manuscript has also been revised in Lines 131-139:

“Theis superposition model under the assumptions can successfully predict drawdown produced by multiple wells in well fields that are cycled on and off [20,21]. Dynamic water-level data are through time with time-variant flow rates obtained using the Theis superposition model. [24] provides more than three years of hourly water levels and pumping rate data from operational well fields in Castle Rock, CO. Further [24] input well locations, pumping times associated with flow rates, as well as variables including transmissivity, storativity, natural slope of the potentiometric surface, and individual well loss constants into the Theis superposition model to calculate the drawdowns for all of its operational wells for more than a three-year period. The Theis equation is [20,21]:”

Theis, C.V. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Transactions of the American Geophysical Union 1935, 16, 519-524.
Bear, J. Hydraulics of Groundwater; McGraw-Hill series in water resources and environmental engineering, McGraw-Hill, New York, 1979.

In Section 5 (discussion section), the assumptions and limitations of the analytical method have been discussed in Lines 273-281 of the revised manuscript:

“The basic idea of this study is that using dynamic water level data and an analytical solution developed from the Theis equation to track particles under dynamic pumping conditions. Therefore, the focus of this study is the analytical method. The settings of geologic parameters were referred to the literatures on the study site and several assumptions of the aquifer and flow conditions were made. These assumptions were used to testify the method. This method can be used with a scope of assumptions; therefore, there are limitations existed for the method. For example, for heterogeneous and anisotropic conditions, the method is not applicable. Further, although several assumptions have been made, which could be applicable to the Theis equation, lab or field experiments can be conducted to verify the results.”

In Section 6 (conclusions), limitations of the method have been described and future research on the limitations have been discussed in Lines 300-306 of the revised manuscript:

“Limitations are also existed in this analytical method, since it was developed by making several assumptions of the aquifer and flow conditions. For tracking particles in heterogeneous and anisotropic conditions, or the aquifers are highly recharge affected (e.g., karst aquifers) under dynamic pumping conditions, the method may not be applicable. Further research may focus on developing simple analytical methods for tracking particles in complex aquifers, flow and hydrologic conditions. Moreover, field experiments or numerical solutions need to be conducted to verify the results derived from the analytical methods.”

The specific comments have also been responded in detail, which are shown below.

Specific comments:

Lines 14-15: …”numerical model are limited”… analytical models can be even more limited in tracking particles for heterogeneous and anisotropic aquifers (see previous general comments).

Thank you for the comment. We totally agree with your statement that the analytical models can be more limited in tracking particles for heterogeneous and anisotropic aquifers. Therefore, the relevant descriptions about “numerical model are limited…” in the manuscript have been removed. The manuscript has been revised in Lines 11-22:

Line 28: Reference [1] is not necessary, the statement is obvious.

Thank you for the suggestion. Reference [1] has been removed in the manuscript.

Lines 29-34: Cited references are limited, I suggest adding other more quoted references.

Thank you for the suggestion. More references have been added in the text and the reference section. The manuscript has been updated in Lines 27-32 and 314-330:

Lines 27-32:

“The contamination of groundwater, however, is also a widespread problem and requires solid techniques for its remediation [1,2,3,4,5]. Although many contaminants can be naturally attenuated in subsurface via microorganism activities [6,7], residual chemicals can persist in for a long period, which has posed substantial harms to natural groundwater resources, particularly in public water supply well fields which are located at sites that were historically impacted by releases [8,9,10,11].”

Lines 314-330:

“2. Lu, H.; Li, J.; Chen, Y.; Lu, J. A multi-level method for groundwater remediation management accommodating non-competitive objectives. Journal of Hydrology 2019, 570, 531-543.

Roy, J.W.; Bickerton, G. Proactive screening approach for detecting groundwater contaminants along urban streams at the reach-scale. Sci. Technol. 2010, 44(16), 6088-6094.
Schipper, L.A.; Robertson, W.D.; Gold, A.J.; Jaunes, D.B.; Cameron, S.C. Denitrifying bioreactors-an approach for reducing nitrate loads to receiving water. Eng. 2010, 36(11), 1532-1543.
Wiafe, S.; Ofosu, S.A.; Ahima, J. The quality of groundwater resources around auto-mechanic workshop enclaves in Ghana. Sci. Technol. 2013, 1(1), 38-49.
Aelion, C.M.; Bradley, P.M. Aerobic biodegradation potential of subsurface microorganisms from a jet fuel-contaminated aquifer. Applied and Environmental Microbiology 1991, 57(1), 57-63.
Nevin, K.P.; Finneran, K.T.; Lovley, D.R. Microorganisms associated with uranium bioremediation in a high-salinity subsurface sediment. Applied and Environmental Microbiology 2003, 69(6), 3672-3675.
Bayer-Raich, M.; Jarsjo, J.; Liedl, R.; Ptak, T.; Teutsch, G. Integral pumping test analyses of linearly sorbed groundwater contaminants using multiple wells: inferring mass flows and natural attenuation rates. Water Resour. Res. 2006, 42, W08411.
Guo, Z.; Bruseau, M. The impact of well-field configuration and permeability heterogeneity on contaminant mass removal and plume persistence. Journal of Hazardous Materials 2017, 333, 109-115.”

Lines 57-63: Again the role of numerical modeling in tracking the movement of fluid particles is diminished. Advective, dispersive and diffusive flow can be better analyzed through numerical models even in complex hydrogeological situations. I suggest better specifying under which conditions and assumptions the analytical solutions can be used.

Thank you for the suggestions. This section has been updated by removing the statement of limitations of numerical models. The assumptions for the analytical models in this study has been added, in particular Section 2. The updates are shown in Lines 56-64 and 73-82 of the revised manuscript:

Lines 56-64:

Lines 73-82:

Line 66: Please refer to Theis (1935) or Bear (1979), rather than [12].

Thank you for the suggestion. Theis (1935) and Bear (1979) have been both cited in the text instead of Davis (2013). The manuscript has been updated in Lines 65-67, and the two references have been added in the reference section in Lines 353-357.

Lines 57-61:

“As such, the objective of this study is to investigate whether the analytical solution is able to capture dynamic aspects of groundwater flow for complex water surface with dynamic water levels, e.g., pumping conditions. By making assumptions proposed by [20], an analytical model was developed from the Theis equation [20,21] that can calculate head under dynamic pumping and injection conditions to evaluate particle tracking.”

Lines 342-346:

“20. Theis, C.V. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Transactions of the American Geophysical Union 1935, 16, 519-524.

21.Bear, J. Hydraulics of Groundwater; McGraw-Hill series in water resources and environmental engineering, McGraw-Hill, New York, 1979.”

Lines 70-72: …”The paper is divided……..Section 6” is not necessary.

Thank you for the comment. This sentence has been removed from the text.

Lines 82-83: Please, better detail these assumptions and the approximations of the results that can derive from them, as the aquifer does not seem to me to be homogeneous in terms of thickness, transmissivity and storativity from Davis (2013).

Thank you for the comment. The assumptions of the aquifer and flow conditions have been described in the section of study site in Lines 73-82 of the revised manuscript:

Fig. 1: The figure is of poor quality and difficult to read. I would suggest reporting only the location of the study area and the detail of the Meadows Center well field with the graphical scale.

Thank you for the suggestion. Figure 1 has been updated to make the figures more clearly. The figure for Meadows Pumping Center has been updated with graphical scale. The revised figures are shown as below:

Figure 1. (a) Location map for the Denver Basin [23]. (b) Castle Rock well locations. (c) Wells locations at the Meadows Pumping Center in Castle Rock, CO.

Figs. 2-3: I suggest pairing Figures 2 and 3 into a single figure, so that the reader can see pumping rates and water levels at the same time.

Thank you for the suggestion. Figure 2 and 3 have been paired together as one figure in the revised manuscript:

Figure 2. Pumping rates ((a)-(h)) and water levels ((i)-(p)) in eight wells at the Meadows Pumping Center, CO.

Line 122: Why 130 and 6000 days? and not, for example, 180 and 365 days?

Thank you for the comment. The 130 days and 6000 days employed in the study are just assumed periods. The reason that the 6000 days was used is because we want to observe more flow path of fluid particles under dynamic injection conditions. Also, these periods are flexible, researchers use this method can define their specific period based on the local conditions. The manuscript has also been updated in Lines 116-117:

“The reason that a longer period used for the injection conditions is because a clear flow path of particle tracking can be observed.”

Lines 124-125: How were pumping rates and injection rates chosen? Based on the operational functioning of the well field? Are these pumping and injection rates the same for all eight wells?

Thank you for the comment. In this study, particle movement around the well 82 was evaluated. According to Figure 2d, we selected the pumping rate of 244 gal/min as the pumping rate in this study. The injection rates for 130-day and 6000-day periods are both assumed values. The manuscript as also been updated in Lines 117-122:

“In this study, particle movement around the well 82 was evaluated. The pumping and injection rates of the well 82 employed correspond to their periods are shown in Figure 3. Figure 3a shows a period for 130 days. The pumping rate of 244 gal/min is selected (Figure 2d) and a -144 gal/min is assumed to be the injection rate. Figure 3b shows a period for 6000 days. The pumping rate of 244 gal/min is selected and injection rate is assumed to be -244 gal/min.”

Line 137: I suggest referring to the most quoted works of Theis (1935) or Bear (1979), rather than [12].

Thank you for the suggestion. The reference [12] provides water level and pumping data for this study, therefore, it was cited in some parts of the text. Both of Theis (1935) and Bear (1979) have been cited in the revised manuscript, in particular in the methodology section in Lines 131-133 and 139:

Lines 131-133:

“Theis superposition model under the assumptions can successfully predict drawdown produced by multiple wells in well fields that are cycled on and off [20,21].”

Line 139:

“The Theis equation is [20,21]:”

Line 142: See the previous comment, the equation 1,2 …. were first proposed by Theis, the reference to [12] is wrong.

Thank you for the comment. The equations proposed by Theis (1935) have been cited with Theis (1935) and Bear (1979) in the revised manuscript in Lines 139, 142-143, and 150-152:

Line 139:

“The Theis equation is [20,21]:”

Line 142-143:

“ is the well function that can be expressed as the infinite series [20,21]:”

Line 150-152:

“For a well field with wells, associated pumping rates of , ,…, , and radial distance from each well , ,…, , the following equation is used [20,21]:”

Line 160: as above.

Thank you for the suggestion. Theis (1935) and Bear (1979) have been cited instead of [12]. The manuscript has been revised in Lines 155-157:

“Based on the Theis superposition model [20,21], this research developed a new analytical model to track particle under dynamic pumping conditions.”

Line 164: …”of a regionally sloping potentiometric surface”, I do not find in the rest of the work how the regionally sloping potentiometric surface was considered. In 6000 days it will change, has this been taken into account in the definition of the water circles around the well?

Thank you for the comments. A regionally sloping potentiometric surfaced was calculated using a function () without dynamic pumping conditions, and a fluid particle can be considered on this surface. Under dynamic pumping conditions, equation (7) was employed to calculate the head of particles on the surface. The time step (t) has been considered in the calculation which is shown in both equation (6) and (7). Therefore, the head of the particle changes with time as the regional surface changed and the water circles around the well can be simulated in 6000 days. The manuscript has been revised in Lines 163-169 to make the description clear:

“A regression is performed to obtain a solution for the potentiometric surface (, [L]) [28]:

(6)

where, and is a position of interest, is the gradient of head in the direction (dimensionless), is the gradient of head in the direction [dimensionless], is a constant defined as the elevation of the water table at (0,0) (L), and is the time interval.

Under dynamic pumping conditions, head can be calculated by employing the static water surface elevations (, [L]) minus the drawdown at any time [24],”

Line 195: what does the symbol Φ mean in equations (12)?

Thank you for the comment. Simbol is effective porosity (dimensionless). The manuscript has also been revised in Lines 193:

“where, is effective porosity (dimensionless), and is aquifer thickness [L].”

Line 198: In the results or in Sections 2 or 3 it is necessary to specify whether the drawdown in the wells or observation wells actually follows the Theis model. In addition, it is necessary to specify whether the different simulations shown in Figures 5, 6, 7, 8 and 9 concern the same pumping and injection rates in all eight wells or otherwise.

Thank you for the comments. In Section 2, the assumptions of the aquifer and groundwater flow conditions have been added to make the calculation of drawdown in the wells meet follow the Theis model in Lines 73-82 of the revised manuscript. The revised manuscript has also described that the results shown in the figures concern the same pumping and injections rates that were presented in Figure 3. These improvements are shown in the Lines 117-122 and 194-196 of the revised manuscript:

Lines 73-82:

Lines 117-122:

Lines 194-196:

“Based on the assumptions of the aquifer and groundwater flow conditions made in Section 2, the pumping and injection rates described in Figure 3 were used in the analytical solution to track particles under dynamic pumping and injection conditions in different periods.”

Lines

Line 209: 21 or 22 days? as shown in Fig. 5.

Thank you for the comment. There are totally 21 days. The figure has been revised in the manuscript:

Figure 4. Movement of particles around one representative of eight wells for 21 days under (a) pumping and (b) injection conditions at the Meadows Pumping Center, CO.

Lines 219-220: Reference [18] is not necessary, the statement is obvious.

Thank you for the comment. Reference [18] has been removed from both the text and reference sections in the revised manuscript.

Fig. 6: Please explain the meaning of the blue arrows and numbers in the caption.

Thank you for the comment. The blue arrows mean the particle flow directions in each pumping and injection process. The numbers mean the particle position at the beginning or ending of each pumping and injection step. For example, the initial particle position is at the number 0. 21 days later, particle position is at number 1. The pumping condition continuously occurred during these 21 days, therefore, particle flow direction is from 0 to 1. The description has also been updated in the figure caption in the revised manuscript:

Figure 5. Movement of particles around one representative of eight wells for 130 days under continuous pumping and injection conditions at the Meadows Pumping Center, CO. The blue arrows represent the particle flow directions in pumping and injection processes. The numbers are used to illustrate particle positions at the beginning or ending of pumping and injection processes.

Fig. 8: The figure is not understandable, see the previous comment.

Thank you for the comment. The blue arrows in this figure represent the particle flow directions under pumping processes, while the red arrows represent the particle flow directions under injection processes. The numbers in this figure mean particle positions at the beginning or ending of pumping and injection processes. For example, the number 0 mean the initial particle position. Number 1 represent the particle position at the end of the pumping process, which is the 21th day. The meanings of arrows and numbers have also been added in the figure caption:

Figure 8. Movement of particles around one representative of eight wells for 6000 days under continuous pumping and injection conditions at the Meadows Pumping Center, CO. The blue and red arrows represent the particle flow directions under pumping and injection processes, respectively. The numbers represent particle positions at the beginning or end of pumping and injection processes.

Fig. 9a: I cannot understand the position of the water circle in this figure when I compare it with figures 9c and 9e.

Thank you for the comment. The water circle represents the initial water position under each pumping or injection process. For example, for the step when particle flows from 0 to 1 (Figure 9a), initial water position is at number 0. Therefore, water circle is far away from the well. For the step when particle flows from 1 to 2 (Figure 9b), initial water position is at number 1. Water circle at this time is accordingly close to the well. For Figure 9c and 9e, initial water positions are actually both away from the well and particles flows from the far-end to the near-end due to the pumping, which are similar to the water circle in Figure 9a. Also, it can be observed that for pumping conditions, water circles are larger at the far-end than they are at the near-end.

Conclusions: I suggest specifying the limitations of this approach (for example, what happens if the drawdown does not follow Theis' model? What happens if the aquifer is not homogeneous? How much can the recharge affect the forecast at 6000 days? .... etc.), rather than focusing on the limitation of the numerical models. The proposed method seems more limited to me than a calibrated and validated numerical model. How can the reader or user of the proposed method verify the validity and uncertainty of the results?

Thank you for the comments. The conclusions section has been entirely revised and improved. The descriptions about the limitation of numerical models have been removed, and more descriptions on the limitations for the analytical solution developed in this study have been added. The manuscript has been revised in Lines 283-306:

“Groundwater at well fields has potential to become contaminated by organic or inorganic compounds from releases. For the field site with dynamic pumping conditions, this research explored an analytical solution developed from the Theis superposition model [20,21] to solve particle tracking under dynamic pumping conditions with space and time. A well field was employed in this study to predict the movement of groundwater by tracking particles under dynamic pumping and injection conditions, relying on dynamic water-level data and the analytical solution.

The Theis superposition model [20,21] provides exact solutions for gradients about pumping wells under dynamic pumping conditions. Based on the Theis superposition model and the analytical solution, flow path lines of fluid particles under dynamic pumping and injection conditions at well fields were obtained. Under dynamic pumping conditions, the results of this study provide relatively uniform capture zones. The results show that although groundwater may flow away from the well to the aquifer during the pumping process and flow toward to the well from the aquifer during the injection process, positions of the groundwater may change following each process but does not flow far away from the well. Accordingly, groundwater positions can be evaluated based on the research for dynamic pumping. Under specific assumptions, the analytical solution developed in this study provides a clue or even a simple method to evaluate particle movement about well fields used to both store and recovery water.

Limitations are also existed in this analytical method, since it was developed by making several assumptions of the aquifer and flow conditions. For tracking particles in heterogeneous and anisotropic conditions, or the aquifers are highly recharge affected (e.g., karst aquifers) under dynamic pumping conditions, the method may not be applicable. Further research may focus on developing simple analytical methods for tracking particles in complex aquifers, flow and hydrologic conditions. Moreover, field experiments or numerical solutions need to be conducted to verify the results derived from the analytical methods.”

Lines 296-297: Please avoid unnecessary reference to [12].

Thank you for the suggestion. The reference [12] has been removed. Theis (1935) and Bear (1979) have been both cited in Lines 284-286 and 289-290:

Lines 284-286:

“For the field site with dynamic pumping conditions, this research explored an analytical solution developed from the Theis superposition model [20,21] to solve particle tracking under dynamic pumping conditions with space and time.”

Lines 289-290:

“The Theis superposition model [20,21] provides exact solutions for gradients about pumping wells under dynamic pumping conditions.”

Lines 304-306: For the requested data, I don't know if the method and the conditions presented "help people". In other words, it could probably also help, but how can I check the validity of the results?

Thank you for the comment. The description has been revised and the limitations of the analytical solution developed in this study have been added in the discussion and conclusion sections. The manuscript has been updated in Lines273-281 and 297-306:

Lines 273-281:

Lines 297-306:

“Under specific assumptions, the analytical solution developed in this study provides a clue or even a simple method to evaluate particle movement about well fields used to both store and recovery water.

Author Response File: Author Response.pdf

Reviewer 2 Report

The model developed in the paper is available to such prediction.

The draft is, however, not completed as a final material and the following further consideration is preferable;

Line 38; particle-tracking schemes >>> In the text, any particles are not described. Why are you define the analytical model as the particle tracking scheme?
Line 72; Section 4 and 5 >>> There is not section 5 in the text.
Figure 1 : All figures are not clear. Especially, the characters in the figure C are too small. The number of wells should be clearly indicated.
Figure 4 ; Please express the date clearly; 7/17/08 means July 17^th,2008 ?, 3/17/24 means March 17^th, 2024 ?
Line 133; What is “Theis equation” ?
Line 198; Discussion should be in a separate session.
Line 212; How we can recognize “the current particle position”?
Line 336; Reference [12] is very important in the text. But it is a master thesis. You need a simple introduction of the reference [12].

Author Response

Reviewer #2:

The model developed in the paper is available to such prediction.

The draft is, however, not completed as a final material and the following further consideration is preferable:

Line 38: particle-tracking schemes >>> In the text, any particles are not described. Why are you define the analytical model as the particle tracking scheme?

Thank you for the comment. In this study, we employed an analytical method to track a fluid particle, e.g., a water particle in groundwater to track its movement under dynamic pumping conditions. Also, the particle can be also considered as a solute particle or a sediment in groundwater if water is contaminated. In the study site, we consider a fluid particle in groundwater and its movement was successfully obtained using the analytical solution in either pumping or injection process by making assumptions proposed by Theis (1935). The particles have been described in methodology, results and discussion sections. The reason that we use particle tracking schemes is to describe the importance of particle tracking in the area of groundwater and aquifer contamination. It can be simulated using both numerical and analytical method.

Line 72: Section 4 and 5 >>> There is not section 5 in the text.

Thank you for the comment. The manuscript has been updated and Section 5 is the discussion section in the revised manuscript.

Figure 1: All figures are not clear. Especially, the characters in the figure C are too small. The number of wells should be clearly indicated.

Thank you for your suggestions. All the figures in Figure 1 have been updated. The characters in Figure 1c have also been improved:

Figure 1. (a) Location map for the Denver Basin [14]. (b) Castle Rock well locations. (c) Wells locations at the Meadows Pumping Center in Castle Rock, CO.

Figure 4: Please express the date clearly; 7/17/08 means July 17th, 2008?, 3/17/24 means March 17th, 2024?

Thank you for the suggestions. In Figure 4, the date format has been updated, which is mm/dd/yyyy. The figures have been updated:

Figure 3. Pumping and injection rates employed for particle tracking over (a) 130 days and (b) 6000 days under continuous pumping and injection conditions at the Meadows Pumping Center, CO.

Line 133: What is “Theis equation”?

Thank you for the comment. The Theis equation (Theis, 1935) describes radial confined groundwater flow in a uniformly thick horizontal, homogeneous, isotropic aquifer of infinite areal extent:

(1)

The relevant reference has been added both in the manuscript and reference. In particular, the Theis equation was originally from Theis (1935) instead of Davis (2013). Therefore, reference [12] or Davis (2013) has been replaced with Theis (1935) and Bear (1979) throughout the manuscript. The manuscript has been updated in Lines 131-139:

Line 198: Discussion should be in a separate session.

Thank you for the suggestion. The discussion section has been put in a separate session from the Lines 246-281 in the revised manuscript:

“5. Discussion

Particle tracking under continuous pumping and injection conditions was also studied for a longer period of time, in order to see if the similar results can be obtained. Figure 7 shows the movement of water for 6000 days under continuous pumping and injection conditions. Also, uninterrupted drawdowns and recoveries occurred successively because of continuous pumping and injection. The associated pumping stress is shown in Figure 3b. The movement of water under each pumping and injection process is shown in Figure 8. The red circle represents as the water position under each pumping and injection process.

Figure 7. Movement of particles around one representative of eight wells for 6000 days under continuous pumping and injection conditions at the Meadows Pumping Center, CO. The blue and red arrows represent the particle flow directions under pumping and injection processes, respectively. The numbers represent particle positions at the beginning or end of pumping and injection processes.

Figure 8. Water circles around a well for 6000 days under each pumping and injection conditions at the Meadows Pumping Center, CO.

For a longer period of time, 6000 days in this case, for a single well, water also does not flow far away from the well under continuous pumping and injection. For the first 1000 days (Figure 8a), water moves 861.12 ft from the aquifer to the well during the pumping. In Figure 8b, water is injected into the aquifer and moves 183.1 ft back to the aquifer from the well. In the following processes (Figure 8c and 8d), water moves 513.81 ft from the aquifer to the well and 176.41 ft from the well back to the aquifer, respectively. Finally, water again moves 659.45 ft from the aquifer to the well and 180.84 ft from the well back to the aquifer (Figure 8d and 8f). Based on the results, the analytical method could provide the movement of particles under pumping conditions.

The basic idea of this study is that using dynamic water level data and an analytical solution developed from the Theis equation to track particles under dynamic pumping conditions. Therefore, the focus of this study is the analytical method. The settings of geologic parameters were referred to the literatures on the study site and several assumptions of the aquifer and flow conditions were made. These assumptions were used to testify the method. This method can be used with a scope of assumptions; therefore, there are limitations existed for the method. For example, for heterogeneous and anisotropic conditions, the method is not applicable. Further, although several assumptions have been made, which could be applicable to the Theis equation, lab or field experiments can be conducted to verify the results.”

Line 212: How we can recognize “the current particle position”?

Thank you for the comment. The current particle position mentioned in the manuscript can be considered as a initial position. For example, the initial position shown in Figure 4 (in the revised manuscript). If we know this initial or current position, we can use particle tracking method developed in this study to determine where the particles were, for example, 21 days later, which is also shown in Figure 5.

Line 336: Reference [12] is very important in the text. But it is a master thesis. You need a simple introduction of the reference [12].

Thank you for the suggestion. We noticed that the Theis equation is not originally from reference [12], therefore, we have replaced reference [12] with Theis (1935) and Bear (1979) where the Theis equation has been described in detail. Especially in the methodology section, the Theis equation is described in detail in Lines 139-157 and the two references have been added in the references sections:

“The Theis equation is [20,21]:

(1)

where, is drawdown [L] at a particular radial distance from the pumped well and time since the start of pumping; is pumping rate [L³T^-1]; is transmissivity [L²T^-1]; is the well function that can be expressed as the infinite series [20,21]:

(2)

with being defined as:

(3)

where is the storativity of the aquifer.

In a multiple well system, aquifer drawdown is influenced by more than one pumping well. Appling superposition of the Theis equation, the drawdown at any point in the aquifer can be calculated as the sum of the drawdown created by each well individually. For a well field with wells, associated pumping rates of , ,…, , and radial distance from each well , ,…, , the following equation is used [20,21]:

(4)

(5)

with defined as time from the start of pumping for well with . Based on the Theis superposition model [20,21], this research developed a new analytical model to track particle under dynamic pumping conditions.”

“ 20. Theis, C.V. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Transactions of the American Geophysical Union 1935, 16, 519-524.

21.Bear, J. Hydraulics of Groundwater; McGraw-Hill series in water resources and environmental engineering, McGraw-Hill, New York, 1979.”

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

See the attached file.

Comments for author File: Comments.pdf

Author Response

Reviewer #1

The manuscript has been improved. There are still some issues to clarify or improve:

Figures 1a and 1b are hard to read (for example, many toponyms cannot be read)

Thank you for the comment. Figure 1a has been improved and the toponyms in the figure are much clearer. Figure 1b has been removed since it is not important. Figure 1 has two subfigures in the revised manuscript:

Figure 1. (a) Location map for the Denver Basin [23]. (b) Wells locations at the Meadows Pumping Center in Castle Rock, CO.

the formatting of the text is not entirely consistent with the rules of the journal (the equations are displaced with respect to their numbering)

Thank you for the comment. The equations and their numbers have been improved using the journal format in the revised manuscript.

3) it is not yet clear to me what are the necessary input data to apply the method:

- is the natural hydraulic head required before pumping application?

Thank you for the comment. The hydraulic head is required before pumping application. The hydraulic head is used for calculating water surface elevations using equation (6) in Lines 161-162 of the manuscript:

“A regression is performed to obtain a solution for the potentiometric surface (, [L])

[28]:

(6)”

- is changes in the hydraulic head over 6000 days due to only to pumping? So there is no recharge (this seems to me an unlikely hypothesis)

Thank you for the comment. The hydraulic heads over 6000 days employed in this study are the observed data sets from Davis (2013), so the influence of recharge on the changes in hydraulic head is considered.

- only the effect on one of the wells is examined, do the other wells work with the same pumping and injection rate of the well examined?

Thank you for the comment. The pumping and injection rates used for other wells are different with the well (CR 82). Davis (2013) provides more than three years of hourly water level and pumping rate data from these wells. The pumping and injection rates used for other wells were based on the observed data from Davis (2013) (Figure 2).

Author Response File: Author Response.pdf

Reviewer 2 Report

The revised version is well considered and the figures are better arranged.

But some new sentences include mistake. For example; Line 360; What do you indicate by "which"?

So, please check the sentences carefully.

Author Response

Reviewer #2

The revised version is well considered and the figures are better arranged.

But some new sentences include mistake. For example; Line 360; What do you indicate by "which"?

So, please check the sentences carefully.

Thank you for the suggestion. The sentence has been revised in the manuscript in Lines 278-280:

“Further, although several assumptions that are applicable to the Theis equation have been made, lab or field experiments can be conducted to verify the results.”

Further, all the other texts have also been carefully checked and revised.

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Analytical Modeling of Particle Tracking for Dynamic Pumping Conditions

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