the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Jun 2024
Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function
Abstract. Groundwater storage in aquifers has become a vital water source due to water scarcity in recent years. However, aquifer systems are full of uncertainties, which inevitably propagate throughout the modeling computations, mainly reducing the reliability of the model output. This study develops a novel two-dimensional stochastic confined groundwater flow model. The proposed model is developed by linking the stochastic governing partial differential equations by means of their one-to- one correspondence to the nonlocal Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE). In the form of the LEFPE, the resulting deterministic governing equation describes the spatio-temporal evolution of the probability density function of the state variables in the confined groundwater flow process by one single numerical realization instead of requiring thousands of simulations in the Monte Carlo approach. Consequently, the ensemble groundwater flow process's mean and standard deviation behavior can be modeled under uncertainty in the transmissivity field and recharge and/or pumping conditions. In addition, an appropriate numerical method for LEFPE's solution is subsequently devised. Then, its solution is presented, discussed, and illustrated through a numerical example, which is compared against the results obtained by means of the Monte Carlo simulations. Results suggest that the proposed model appropriately characterizes the ensemble behavior in confined groundwater systems under uncertainty in the transmissivity field.
Status: closed
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RC1: 'Comment on egusphere-2024-1411', Anonymous Referee #1, 29 Aug 2024
The authors present an interesing approach for the modeling of the PDF of hydraulic
for 2-dimensional groundwater flow under spatially heterogeneous transmissivity. Formulating
the groundwater flow equation as an advection-diffusion equation, the latter rewritten in the
characteristic system. The characteristic equations represent stochastic differential
equation. Then a non-local Lagrangian-Eulerian equation for state variables is presented.
The authors then present a numerical scheme to solve this equation. The model is then
applied to a numerical experiment of two-dimensional groundwater flow. The authors state
that they propose a novel stochastic model. However, from the presentation of the model
in Sections 2 and 3 it is not entirely clear which model developments refer to the present
paper, and which refer to previous work for the second author. This needs to be clarified in
a revised version of the manuscript.Comments:
* Eq. 8-9: No partial derivative on x and y.
* Eq. 10: A different notation should be used for the coordinates in the
characteristic system. The expression for the diffusion terms in the characteristic
system should be given.* Eq. 14: Do the authors mean x_t as the x-coordinate at time t or the vector x at time t? Please
clarify.* Eqs. 19 and 21: These equations a linear and local in P. What is meant by non-locality?
* Eq. 40 and following: x refers to the x-ccordinate or vector x? Please clarify.
Citation: https://doi.org/10.5194/egusphere-2024-1411-RC1 -
AC1: 'Reply on RC1', Joaquin Meza, 23 Sep 2024
Comment on egusphere-2024-1411: Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head’s probability density function
Joaquín Meza (1) and Levent Kavvas (2)
(1) Departamento de Obras Civiles, Universidad Técnica Federico Santa María, Valparaiso, Chile
(2) Department of Civil and Environmental Engineering, University of California, Davis, CA, 95616, USA
Dear Editor and Reviewers,
We express our gratitude for your correspondence and the comprehensive feedback provided by the esteemed reviewers regarding the manuscript entitled "Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function.''
The constructive insights and comments offered by the reviewers have proven to be invaluable, significantly contributing to the enhancement of the manuscript. We have diligently reviewed each comment and incorporated necessary revisions to align the manuscript with the requisite standards. The amended sections have been distinctly highlighted in blue within the revised manuscript.
The ensuing sections delineate the principal corrections made in the paper, accompanied by our responses to the respective reviewer's comments:
Authors’ Response to Referee #1's Comments:
The authors present an interesing approach for the modeling of the PDF of hydraulic for 2-dimensional groundwater flow under spatially heterogeneous transmissivity. Formulating the groundwater flow equation as an advection-diffusion equation, the latter rewritten in the characteristic system. The characteristic equations represent stochastic differential equation. Then a non-local Lagrangian-Eulerian equation for state variables is presented. The authors then present a numerical scheme to solve this equation. The model is then applied to a numerical experiment of two-dimensional groundwater flow. The authors state that they propose a novel stochastic model. However, from the presentation of the model in Sections 2 and 3 it is not entirely clear which model developments refer to the present paper, and which refer to previous work for the second author. This needs to be clarified in a revised version of the manuscript.
- Thank you for your valuable feedback. In order to clarify which model developments pertain to the present work and which refer to the previous contributions of the second author, we have provided a detailed breakdown below:
Novel Developments in the Present Paper: The Lagrangian-Eulerian Extension to the Fokker Planck Equation (LEFPE), that was developed in Kavvas (2003), was then applied to the modeling of stochastic one-dimensional (1D) groundwater flow in Cayar and Kavvas ( 2009a), and then to two-dimensional (2D) stochastic groundwater flow in Cayar and Kavvas (2009b). For modeling a real-life aquifer a 2D setting is necessary. However, in order to model the 2D stochastic groundwater flow by the symmetry method Cayar and Kavvas (2009b) had to restrict the 2D groundwater flow to a symmetric model domain with symmetric boundary conditions, and had to restrict the hydraulic conductivity, the main parameter, to a random constant in space (a random variable) while the hydraulic conductivity or its vertically-integrated form, transmissivity, are random functions of spatial location.In the current manuscript under review, co-authored by Joaquín Meza and M. Levent Kavvas, a new stochastic method of characteristics is applied to the 2D stochastic groundwater flow governing equation in order to render the resulting model applicable to the realistic setting of 2D groundwater flow with generally asymmetric boundary conditions and with the hydraulic conductivity or transmissivity as random functions of the spatial location over the model domain.
Secondly, instead of utilizing the conventional regular perturbation methods which usually have closure problems, for the first time in stochastic groundwater modeling, this study utilizes a one-to-one correspondence between the LEFPE and the stochastic ordinary differential equations, resulting from the application of the method of characteristics to the 2D stochastic groundwater flow governing equation to convert the stochastic modeling problem to the solution of a deterministic linear evolution equation for the time-space evolution of the probability density function (pdf) of the state variable, the hydraulic head. Once this pdf is solved from the LEFPE under specified initial and boundary conditions, then it is possible to determine the time-space varying ensemble mean and ensemble variance of the hydraulic head over the whole model domain.
Another key advancement in our work is the explicit incorporation of the spatial correlation in the hydraulic conductivity field. This development is crucial for capturing the realistic variability and interdependence of hydrological properties across the study area, which traditional models often oversimplify. Unlike previous works, which assumed that spatial correlation was non-existent (uncorrelated hydraulic conductivity field) in order to simplify the calculation of covariance from variance, our approach acknowledges and integrates these correlations directly.
Furthermore, our paper introduces an appropriate numerical method for solving the Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE), providing a practical tool for modeling confined groundwater systems under uncertainty. This capability is a significant extension beyond the studies of Kavvas (2003) and his co-workers (Cayar and Kavvas, 2009a,b), allowing for more accurate and robust predictions in hydrological modeling - We hope this clarifies the contributions and delineates the extensions made in the current manuscript relative to the previous studies of the second author. Additional details on specific equations and methodologies developed distinctly in this manuscript are outlined in Sections 2 and 3, which explicitly describe the integration of nonlocal stochastic elements into the modeling framework for confined aquifers.
Comments:
- Eq. 8-9: No partial derivative on x and y
- Thank you for your careful review and comment regarding the partial derivatives in equations 8 and 9. After revisiting the formulations and the intended mathematical representation, we realized that including partial derivatives was indeed a mistake in the manuscript. We have corrected this error in the revised manuscript by removing any incorrect indications of partial derivatives, ensuring the equations now accurately reflect the intended differentiated model.
- Eq. 10: A different notation should be used for the coordinates in the characteristic system. The expression for the diffusion terms in the characteristic system should be given.
- Thank you for your insightful comment regarding the notation used for the coordinates in the characteristic system. We appreciate your concern about the potential for confusion with the current notation. In the initial manuscript, we indeed used x, y, and t to denote the coordinates in the characteristic system. To address your concerns and to eliminate any ambiguity, we propose to introduce a clearer notation where λ explicitly denotes the parameter along the characteristic curves, which allows us to describe the dynamics more transparently.
From the derivation in the file attached, it follows that, aligning the characteristic system with the representation used in the manuscript. This approach ensures that each variable clearly corresponds to its role in the characteristic framework, separating them distinctly from their usage in the general field equations.We have revised the notation in the manuscript accordingly to reflect this clarification, ensuring that the representation is consistent and clear across all sections. We appreciate your guidance on this matter and hope that the revised notation and the inclusion of the diffusion terms will meet the rigorous standards of clarity and precision required for our study.
- Thank you for your insightful comment regarding the notation used for the coordinates in the characteristic system. We appreciate your concern about the potential for confusion with the current notation. In the initial manuscript, we indeed used x, y, and t to denote the coordinates in the characteristic system. To address your concerns and to eliminate any ambiguity, we propose to introduce a clearer notation where λ explicitly denotes the parameter along the characteristic curves, which allows us to describe the dynamics more transparently.
- Eq. 14: Do the authors mean xt as the x-coordinate at time t or the vector x at time t? Please clarify.
- We appreciate the valuable feedback from the reviewer. In response, we have significantly improved the notation along the document to clearly differentiate the coordinate “x” and the "vector x". In the case of equation 14 it effectively refers to "vector x” at time “t”, which was already modified in the whole document to ensure clarity in the use of the variables.
- Eqs. 19 and 21: These equations a linear and local in P. What is meant by non-locality
- We value the reviewer’s suggestion and have addressed it by extending the explanation about "non-locality". This concept refers to the influence of conditions or parameters at a specific point that depend on information from distant points in space or earlier moments in time. In contrast to purely local equations, where the evolution of a variable in time or space only depends on the conditions at the same location, non-local processes involve the dependence of variables on their interaction with conditions at other locations or past times.
This study involves covariances over space and time, indicating that the evolution of variables at a given location is influenced by their interaction with other variables at different time-space locations. For example, the covariance terms in equation (15) described in the document imply that to understand the evolution of a system at a particular location, one must account for the system’s past states or its state at different spatial positions.Thus, the concept of non-locality emphasizes that the system's behavior is not just a function of local conditions but is interconnected with the system's behavior across space and time. In summary, non-locality refers to the fact that the behavior of a system at a specific point location is not independent but is influenced by the broader spatial and temporal context, as reflected in the integral terms and covariances in the equation (21).
We have expanded the discussion around these equations and explained the concept of non-locality as it applies to our model. The revisions are intended to clarify this concept, emphasizing how the probability density function "P" is influenced across a range of spatial interactions.
- We value the reviewer’s suggestion and have addressed it by extending the explanation about "non-locality". This concept refers to the influence of conditions or parameters at a specific point that depend on information from distant points in space or earlier moments in time. In contrast to purely local equations, where the evolution of a variable in time or space only depends on the conditions at the same location, non-local processes involve the dependence of variables on their interaction with conditions at other locations or past times.
- Eq. 40 and following: x refers to the x-ccordinate or vector x? Please clarify.
- Thank you for your observation concerning the clarification needed in Equation 40 and the subsequent equations regarding whether "x" refers to the spatial coordinate or a vector.
In the initial manuscript, the notation "x" was indeed ambiguous, and we appreciate your pointing this out. To clarify, in Equation 40 and the following equations, "x" and "y" refer specifically to the spatial coordinates in the x-direction and y-direction, respectively, not to vectors. These equations model the stochastic dynamics along each spatial dimension independently but within the context of the two-dimensional field.To eliminate any potential confusion in the revised manuscript, we have adjusted the notation as follows:
- We use xi to denote the spatial coordinates, where i = {1, 2} corresponds to the x- and y-directions, respectively.
- We have ensured that all instances of the coordinate "x" and vector "\underline{x}" in the manuscript are now clearly defined, with an explicit definition provided in Section 2.2.
The corrected equations are presented with clear differentiation between the handling of the x- and y-directions, ensuring that the interpretation aligns with the physical processes being modeled. This notation not only enhances readability but also aligns with the mathematical precision required for stochastic modeling.
- Thank you for your observation concerning the clarification needed in Equation 40 and the subsequent equations regarding whether "x" refers to the spatial coordinate or a vector.
- We hope that this adjustment resolves the ambiguities and thank you for helping us improve the clarity and accuracy of our manuscript. Your meticulous attention to our manuscript is sincerely appreciated,
- Thank you for your valuable feedback. In order to clarify which model developments pertain to the present work and which refer to the previous contributions of the second author, we have provided a detailed breakdown below:
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AC1: 'Reply on RC1', Joaquin Meza, 23 Sep 2024
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RC2: 'Comment on egusphere-2024-1411', Anonymous Referee #2, 06 Sep 2024
I have several suggestions for the authors to consider.
- The paper is presenting (what seems to me) a tedious math derivation that may not be easy for many readers to follow. I am not against tedious derivations in principle, but the authors should convince the reader that the effort is worthwhile.
- The open science questions this paper attempts to address should be stated clearly. And then answered. As it is now, it seems that the paper is trying to show computational speed under (what sems to me) a very limited set of conditions and limiting assumptions.
- The title of the paper states “probability density function”. However, the text deals with mean and standard deviations. There is equivalence here only if the probability density function is Gaussian. But there is no proof provided for this assumed/implied equivalence.
- Page 4 (last paragraph) states: “upscales the governing stochastic differential equations from a point-scale (at which they are valid) to a field scale.” There are also no explanation of what upscaling means or how it is carried out.
- The authors need to define “point-scale” and “field local-scale”.
- The follow-up paragraph (page 5) suggests (through references to previous works) that upscaling and ensemble averaging are equivalent. This can be true under strict conditions which need to be stated. These conditions, once met, pose severe limitations on the applicability of the proposed approach.
- The paper attempts to show the superiority of the proposed method compared to Monte-Carlo simulations. This could be true under a very limited set of conditions which casts doubts on the value and generality of this approach:
- The method is demonstrated for a 2D, confined aquifer.
- The assumption stated following equation 6 is not defended (except for the case study). There is a suggestion to test it using Monte Carlo simulations, which defeats the entire purpose of the study.
- The authors should list the models of spatial variability that can be accommodated: (1) does the method requires stationarity of some sort (must be, because of the assumed equivalence between upscaling and ensemble averaging, but it is not stated)? (2) what types of spatial covariances can be accommodated? (3) there is no mention of the type of spatial covariance used for the demo.
- How to condition the simulations on data (multiple types, multiple scales, varying data quality)? The authors should take a deep look at this topic and show how it can be done. Such an example could start with a reasonable scenario of a real-life aquifer, proceed through model calibration and testing against data (validation)
- Top of p.26: “Therefore, this governing equation needs to be upscaled to the corresponding field scale to predict its behavior correctly”: again, what does that mean? And how does this upscaling affect the possibility to condition the simulations on local data at its locations.
Citation: https://doi.org/10.5194/egusphere-2024-1411-RC2 -
AC2: 'Reply on RC2', Joaquin Meza, 27 Sep 2024
Comment on egusphere-2024-1411: Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head’s probability density function
Joaquín Meza (1) and Levent Kavvas (2)
(1) Departamento de Obras Civiles, Universidad Técnica Federico Santa María, Valparaiso, Chile
(2) Department of Civil and Environmental Engineering, University of California, Davis, CA, 95616, USA
Dear Editor and Reviewers,
We express our gratitude for your correspondence and the comprehensive feedback provided by the esteemed reviewers regarding the manuscript entitled "Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function.''
The constructive insights and comments offered by the reviewers have proven to be invaluable, significantly contributing to the enhancement of the manuscript. We have diligently reviewed each comment and incorporated necessary revisions to align the manuscript with the requisite standards. The amended sections have been distinctly highlighted in blue within the revised manuscript.
The ensuing sections delineate the principal corrections made in the paper, accompanied by our responses to the respective reviewer's comments:
Authors’ Response to Referee #2's Comments:
1) The paper is presenting (what seems to me) a tedious math derivation that may not be easy for many readers to follow. I am not against tedious derivations in principle, but the authors should convince the reader that the effort is worthwhile.
- Thank you for your thoughtful comment. We appreciate your perspective on the mathematical derivations presented in our paper. We acknowledge that the derivations are extensive and may be challenging to follow. However, we believe that this mathematical rigor is essential and forms a significant part of our contribution of this study.
The governing equation of a hydrologic process is derived by conserving mass and momentum/energy over a differential control volume. This derivation over a differential control volume represents the process at “point scale” since it is at the scale of a differential control volume. When the governing equation for a specific process is applied over a field or a region of several square kilometers, this “point-scale” equation becomes uncertain at any location over the field or regional model domain due to the heterogeneity of the material of the model domain, rendering the equation’s parameters uncertain, due to the uncertainty in the generally time-space varying sources and sinks into and from the model domain, and due to the spatially varying boundary conditions over the field or regional model domain.
A general modeling framework in terms of the probability density function (pdf) of the state variable of the process, which, in general, may vary with time and spatial locations, would be a useful approach for deriving the ensemble mean and ensemble variance of the process over the model domain since the process mean and variance are the most often addressed process moments in hydrologic practice that attempts to address process uncertainty in a physically-based framework. One fundamental advantage of this approach is that the resulting pdf of the process is obtained directly from the one-to-one correspondence between a Lagrangian-Eulerian extension to the conventional Fokker-Planck Equation (LEFPE) of statistical physics and the subject governing equation of a particular hydrologic process. Although the LEFPE has been developed about two decades ago (Kavvas, 2003) and could be useful in determining the generally time-space varying pdf of the state variable of any particular hydrologic process under specified initial and boundary conditions, it has been largely unknown to the hydrologic community. Accordingly, in this article we tried to provide as much information as permitted by the paper length limit imposed by the publisher, on the LEFPE in terms of the confined aquifer flow process, and its application toward the derivation of the time-space varying ensemble mean and ensemble variance of the process under spatially varying process parameter (transmissivity) and boundary conditions around the model domain. An important aspect of this methodology is that each hydrologic process will have a one-to-one correspondence to a LEFPE. As such, this approach provides a general framework for determining the generally time-space varying pdf of a hydrologic process under specified initial and boundary conditions without needing any assumption about the underlying distribution of the state variable’s pdf. As explained in detail in Kavvas (2003), the fundamental approximation of this methodology is that it is valid to the order of the covariance time of a specified hydrologic process (to the order of the second cumulant of the process). - Justification of the Mathematical Derivations
- Necessity for Addressing Fundamental Challenges: The heterogeneity of geological formations and the uncertainties in hydrological parameters present significant challenges in accurately modeling groundwater flow. Our detailed mathematical derivations are necessary to develop a robust framework that can address these challenges and provide accurate simulations.
- Contribution to Hydrological Modeling: Our paper introduces a novel methodology that establishes a one-to-one correspondence between the stochastic governing equations of confined groundwater flow and a Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE). This approach allows us to derive the time-space evolution of the probability density function (pdf) of the hydraulic head without making assumptions about its underlying distribution, which is a significant advancement in stochastic hydrology.
- Mathematical Rigor as a Core Contribution: The rigorous mathematical development is integral to our contribution. It ensures the validity and applicability of our methodology. By thoroughly presenting these derivations, we enable other researchers and practitioners to understand the foundations of our approach and potentially extend it to other hydrological processes.
- We believe that the mathematical rigor presented in our paper is justified and worthwhile, as it addresses fundamental uncertainties in hydrological modeling and offers a substantial contribution to the field. Our work fits well within the scope of HESS by providing innovative research that enhances the understanding of hydrological systems and processes.
We appreciate your feedback, which has prompted us to performed a major revision of our paper to clarify the whole methodology, which hopefully will describe its value to the readerAnd to better communicate the importance and applicability of our work.
2) “The open science questions this paper attempts to address should be stated clearly. And then answered. As it is now, it seems that the paper is trying to show computational speed under (what sems to me) a very limited set of conditions and limiting assumptions.”
- We appreciate the reviewer’s insightful comment and the opportunity to clarify the open scientific questions our paper aims to address. We recognize the importance of explicitly stating these questions and demonstrating how our work contributes to advancing hydrological science.
- Open Scientific Questions Addressed:
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- How can we determine the ensemble mean and variance of a hydrological process without making assumptions about the underlying probability distribution of the state variable or assuming stationarity/ergodicity? Traditional stochastic modeling approaches often rely on specific assumptions about the probability distributions and stationarity of hydrological processes. However, these assumptions may not hold in heterogeneous media with spatially varying parameters and boundary conditions. Our study seeks to develop a methodology that overcomes these limitations by directly deriving the time-space evolution of the probability density function (pdf) of the hydraulic head from the governing equations.
- How does the heterogeneity of aquifer materials and uncertainty in source/sink conditions affect the time-space varying probabilistic structure of confined groundwater flow, and how can we model this behavior in a physically-based framework? Understanding the influence of spatial variability and uncertainty on groundwater flow is crucial for accurate modeling and prediction. We aim to explore how these factors impact the ensemble behavior of the hydraulic head and to develop a modeling framework that captures these effects without oversimplifying assumptions.
- How Our Paper Addresses These Questions:
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- We introduce a novel methodology that establishes a one-to-one correspondence between the stochastic governing equations of confined groundwater flow and a Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE). This approach allows us to determine the time-space evolution of the pdf of the hydraulic head under specified initial and boundary conditions, without assuming any specific underlying distribution or stationarity.
- By applying this methodology, we derive the ensemble mean and variance of the hydraulic head directly from the governing equations, accounting for the heterogeneity of the aquifer medium and the uncertainty in source/sink terms.
- Our numerical application demonstrates how the ensemble mean and variance evolve over time and space in a two-dimensional, asymmetric flow domain with spatially varying transmissivity and boundary conditions.
- Addressing Concerns About Limited Conditions and Assumptions: While our numerical example focuses on a specific scenario to illustrate the methodology, the approach itself is general and not limited to particular conditions. The methodology does not rely on restrictive assumptions about the system’s properties, such as homogeneity or stationary processes. Instead, it provides a framework that can handle time-space varying parameters and conditions, making it applicable to a wide range of hydrological processes and settings.
Our study aims to move beyond traditional methods that may equate ensemble means with spatial averages or require ergodicity assumptions. By not making such assumptions, we address the complexities inherent in natural systems, where parameters and conditions can vary significantly across space and time. - Revisions Made to the Manuscript: In response to your valuable suggestion, we have substantially revised the manuscript to clearly state the open scientific questions our paper addresses. Specifically:
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- Abstract and Introduction: We have explicitly outlined the fundamental scientific questions at the beginning of the manuscript, providing a clear context for our study.
- Summary and conclusions: We have expanded these sections to directly answer the stated questions, highlighting the significance of our findings and their implications for hydrological modeling.
- Clarification of Methodology: We have enhanced the description of our methodology to emphasize its generality and applicability beyond the specific example presented.
- We believe that these revisions improve the clarity of our manuscript and better communicate the importance and scope of our work. We are grateful for your thoughtful feedback, which has helped us strengthen our manuscript. We hope that the revisions address your concerns.
3) “The title of the paper states “probability density function”. However, the text deals with mean and standard deviations. There is equivalence here only if the probability density function is Gaussian. But there is no proof provided for this assumed/implied equivalence.”
- Clarification on the Use of the Probability Density Function: Our paper focuses on determining the time-space evolution of the probability density function (pdf) of the hydraulic head in a confined groundwater flow system under uncertainty. The pdf is obtained directly from the solution of the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE) under specified initial and boundary conditions. This approach does not assume any specific form for the underlying probability distribution of the hydraulic head; in particular, it does not assume that the distribution is Gaussian.
Relationship Between the PDF, Mean, and Standard Deviation: While the ensemble mean and standard deviation are statistical moments commonly used to describe a distribution, they do not uniquely define the distribution unless additional information or assumptions are made (such as normality). In our study, we derive the pdf of the hydraulic head without assuming it follows a Gaussian distribution. From this derived pdf, we calculate the ensemble mean and variance (and hence the standard deviation) of the hydraulic head using the standard definitions:
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- Ensemble Mean: \langle h(x, y, t) \rangle = \int h(x, y, t) P(h, x, y; t) \, dh
- Ensemble Variance: \text{Var}[h(x, y, t)] = \int h^2(x, y, t) P(h, x, y; t) \, dh - \langle h(x, y, t) \rangle^2
These expressions allow us to compute the mean and variance directly from the pdf obtained through the LEFPE, regardless of the distribution’s shape.
- No Assumption of Gaussianity: Our methodology does not assume or imply that the hydraulic head follows a Gaussian distribution. The LEFPE provides a framework to determine the pdf of the hydraulic head based on the stochastic properties of the transmissivity field and source/sink terms, without specifying the form of the distribution a priori.
- Revisions Made to the Manuscript: To address this concern and enhance clarity, we have made the following revisions:
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- Emphasized the Role of the PDF: We have updated the manuscript to highlight that the pdf of the hydraulic head is a primary outcome of our methodology, and the mean and standard deviation are calculated from this pdf without assuming any specific distribution.
- Clarified the Absence of Gaussian Assumption: We have explicitly stated in the text that no assumption is made regarding the Gaussianity of the hydraulic head distribution. This clarification is included in the methodology section and reinforced in the discussion of results.
- Included Discussion on the Shape of the PDF: Where appropriate, we have included discussions on the potential shape and properties of the pdf obtained from the LEFPE, acknowledging that it may not be Gaussian and that higher-order moments could be calculated if needed.
- We hope that this explanation clarifies that our study does not assume an equivalence between the pdf and its mean and standard deviation based on Gaussianity. Instead, we derive the pdf directly from the LEFPE and compute the statistical moments from this pdf. This approach allows us to capture the full probabilistic behavior of the hydraulic head under uncertainty, without restrictive assumptions about the form of its distribution.
4) “Page 4 (last paragraph) states: “upscales the governing stochastic differential equations from a point-scale (at which they are valid) to a field scale.” There are also no explanation of what upscaling means or how it is carried out.”
- Clarification on the Use of “Upscaling”: In the original manuscript, we used the term “upscaling” to describe the process of developing a stochastic model that determines the probabilistic behavior of a hydrological process at the field or regional scale, starting from its governing equations derived at the point scale (i.e., over a differential control volume). Specifically:
- How Upscaling Was Addressed: In our initial manuscript, we intended to convey that our methodology transitions from the deterministic point-scale equations to a stochastic representation suitable for field-scale applications. However, we recognize that the term “upscaling” was not adequately explained and may have caused confusion.
- Revisions Made to Address This Issue: In response to your valuable feedback, we have made the following revisions to the manuscript:
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- Removed the Term “Upscaling”: To avoid ambiguity and enhance clarity, we have removed the term “upscaling” from the manuscript. We acknowledge that without proper explanation, the term can be misleading or confusing.
- Provided Detailed Explanation of the Methodology: We have revised the text to provide a detailed explanation of how we develop the stochastic model for the confined groundwater flow process, starting from the point-scale governing equations. This includes:
•Derivation of the LEFPE from Point-Scale Equations: We explain how the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE) is derived directly from the point-scale governing equations of confined groundwater flow. This derivation is presented step-by-step to ensure that readers understand the transition from deterministic to stochastic modeling.
•Accounting for Field-Scale Heterogeneity and Uncertainty: We elaborate on how heterogeneity in the transmissivity field, uncertainties in source/sink terms, and spatially varying boundary conditions are incorporated into the stochastic framework. This clarifies how the model represents the probabilistic behavior of the hydraulic head at the field scale. - Clarified Terminology and Concepts: Throughout the manuscript, we have ensured that all terms and concepts are clearly defined and explained. We avoid using technical jargon without proper context and provide explanations where necessary.
- By removing the term “upscaling” and providing a more detailed explanation of our methodology, we aim to enhance the clarity and accessibility of our manuscript. Our goal is to ensure that readers can follow the development of our stochastic modeling approach from the point-scale governing equations to the field-scale representation without confusion.
5) The authors need to define “point-scale” and “field local-scale”.
- The “point-scale” is the scale of the differential control volume within which the governing equation of the particular process is derived. What was meant by field or regional scale is the area size of the modeling domain which, in the case of the reported confined aquifer flow modeling study, corresponds to several square kilometers.
6) The follow-up paragraph (page 5) suggests (through references to previous works) that upscaling and ensemble averaging are equivalent. This can be true under strict conditions which need to be stated. These conditions, once met, pose severe limitations on the applicability of the proposed approach.
- Thank you for your insightful comment. We appreciate the opportunity to clarify the relationship between upscaling and ensemble averaging in our methodology. Due to the time-space variation of the pdf and all of the moments of the confined aquifer flow process (in terms of the hydraulic head’s pdf), the time-space varying ensemble mean (ensemble average) of the process is not equivalent to the areal average of the process. In the reported methodology there is no stationarity or ergodicity assumption, or any assumption about the underlying distribution of the process state variable, the hydraulic head. The fundamental assumption of the reported modeling methodology is the approximation of the process to the covariance time of the process (second-order cumulant expansion). This approximation is stated in the revised version of the paper. The derivation of the LEFPE that corresponds to any particular governing equation of a process is given in Kavvas (2003).
- We appreciate your feedback, which has prompted us to clarify the assumptions and limitations of our approach. By explicitly stating the conditions under which our methodology operates and ensuring that our explanations are clear and accurate, we aim to provide a robust and transparent framework for modeling stochastic groundwater flow.
7) The paper attempts to show the superiority of the proposed method compared to Monte-Carlo simulations. This could be true under a very limited set of conditions which casts doubts on the value and generality of this approach:
- Thank you for your insightful comment. We appreciate the opportunity to clarify the value and generality of our proposed method in comparison to Monte Carlo simulations.
- Clarification on the Purpose of Our Method: Our paper does not aim to demonstrate the superiority of our proposed method over Monte Carlo simulations. Instead, we present an alternative approach for modeling stochastic groundwater flow that offers certain characteristics making it a valuable tool for evaluating temporal and spatial uncertainties in parameters and their influence on the system’s response.
- Value and Generality of Our Approach:
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- Alternative Framework for Uncertainty Analysis: Our methodology provides a different perspective on handling uncertainties in hydrological modeling. By directly deriving the time-space evolution of the probability density function (pdf) of the hydraulic head from the governing equations, we offer a framework that can complement existing methods like MC simulations. This approach can be particularly useful in scenarios where assessing the influence of parameter uncertainties on the system’s behavior is essential.
- Direct Evaluation of Temporal and Spatial Uncertainties: The method allows for explicit incorporation of temporal and spatial uncertainties in model parameters, such as transmissivity and source/sink terms. By solving the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE), we can directly observe how these uncertainties propagate through the system and affect the hydraulic head’s response over time and space.
- Applicability to Realistic Conditions: The proposed approach does not rely on restrictive assumptions like stationarity, ergodicity, or specific probability distributions of the state variables. This generality enhances its applicability to a wide range of hydrological systems, including those with heterogeneous aquifer properties and complex boundary conditions.
- Context of Monte Carlo Simulations in Our Study: Due to the lack of comprehensive publicly available datasets for confined aquifers with sufficient spatial and temporal resolution, we employed Monte Carlo simulations to validate our developed stochastic model. The MC simulations served as a benchmark to compare the results obtained from our method under realistic initial and asymmetric boundary conditions in a two-dimensional confined aquifer flow over a 4 sq. km region.
- Addressing the Concern About Limited Conditions: While our numerical example focuses on a specific scenario, the methodology itself is general and not limited to particular conditions. Our intention is to provide an alternative tool that can be applied to various hydrological problems, offering insights into the effects of parameter uncertainties on groundwater flow without the need for extensive computational resources often associated with MC simulations.
- Our proposed method offers a complementary approach to existing techniques for modeling stochastic groundwater flow. By providing a framework that can effectively evaluate temporal and spatial uncertainties and their influence on the system’s response, we contribute to a more comprehensive understanding of groundwater flow under uncertainty.
We appreciate your feedback, which has helped us clarify the purpose and scope of our work. We have revised the manuscript to emphasize that our method is an alternative approach rather than a replacement for Monte Carlo simulations, highlighting its potential benefits and applications without overstating its superiority.
(a) The method is demonstrated for a 2D, confined aquifer.
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- Relevance of the 2D Confined Aquifer Scenario:
- Practical Significance: Modeling a 2D confined aquifer allows us to capture essential features of groundwater flow in a manner that is both computationally manageable and practically relevant. Many real-world groundwater problems involve large-scale aquifers where horizontal flow dominates, making a 2D approach appropriate for initial investigations.
- Illustrative Purposes: The 2D scenario provides a clear and focused context to demonstrate the capabilities of our proposed method. It enables us to effectively illustrate how the methodology handles spatial heterogeneity, uncertain parameters, and complex boundary conditions without the added computational complexity of a three-dimensional model.
- Relevance of the 2D Confined Aquifer Scenario:
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- Generality and Applicability of the Method:
- Method Generality: While we have demonstrated the method using a 2D confined aquifer, the underlying mathematical framework is general and not limited to this specific case. The methodology can be extended to three-dimensional systems and applied to various types of aquifers, including unconfined and semi-confined aquifers.
- Adaptability to Different Settings: The stochastic modeling approach we propose is adaptable to different spatial dimensions and hydrogeological conditions. By adjusting the governing equations and numerical implementation, the method can accommodate the complexities of higher-dimensional models and more intricate geological formations.
- Generality and Applicability of the Method:
(b) The assumption stated following equation 6 is not defended (except for the case study). There is a suggestion to test it using Monte Carlo simulations, which defeats the entire purpose of the study.-
- Clarification on the Assumption and Use of Monte Carlo Simulations:
- Justification of the Assumption: The assumption following Equation 6 pertains to the stochastic structure of the transmissivity field and its treatment within our model. In the absence of comprehensive field data to validate this assumption, we relied on numerical experiments to assess its validity.
- Role of Monte Carlo Simulations: We employed Monte Carlo simulations as a benchmark to validate our proposed method under the same conditions. The use of Monte Carlo simulations in this context serves as a means of verification rather than comparison in terms of superiority. By comparing the results from our method with those obtained from Monte Carlo simulations, we demonstrate that our approach produces consistent and reliable estimates of the ensemble mean and variance of the hydraulic head.
- Purpose of Validation: The validation using Monte Carlo simulations helps to confirm the viability of the assumption in a controlled setting. It does not defeat the purpose of the study but rather strengthens it by providing evidence that our method can replicate results obtained from established stochastic modeling techniques.
- Revisions to the Manuscript: In the revised manuscript, we have provided a more detailed defense of the assumption following Equation 6, including theoretical justifications and references to related work. We have clarified the rationale behind using Monte Carlo simulations for validation purposes.
- Clarification on the Assumption and Use of Monte Carlo Simulations:
(c) The authors should list the models of spatial variability that can be accommodated: (1) does the method requires stationarity of some sort (must be, because of the assumed equivalence between upscaling and ensemble averaging, but it is not stated)? (2) what types of spatial covariances can be accommodated? (3) there is no mention of the type of spatial covariance used for the demo.
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- (1) Requirement of Stationarity: Our method does not require assumptions of stationarity or ergodicity. The probability density function (pdf) of the hydraulic head, obtained from the solution of the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE), varies in time and space. This allows us to model non-stationary processes where statistical properties change over time and spatial locations.
- (2) Types of Spatial Covariances: The method can accommodate various spatial covariance structures for the transmissivity field. Since the stochastic properties are directly incorporated into the LEFPE, we can model different types of covariance functions, including exponential, Gaussian, or other valid covariance models, provided they satisfy the requirement of finite covariance time (as discussed in Kavvas, 2003).
- (3) Spatial Covariance Used in the Demonstration: We apologize for the oversight in not explicitly mentioning the type of spatial covariance used in our demonstration. In the revised manuscript, we have included a detailed description of the spatial covariance model employed in our numerical example.
Specifically, we modeled the transmissivity random field as independent log-normal distributions with an exponentially decaying autocorrelation function. This approach is based on the stochastic differential equations (SDEs) model proposed by Zárate-Miñano and Milano (2016), which allows us to generate a log-normal transmissivity field with specified mean, variance, and correlation length, by numerically integrating these SDEs using the implicit Milstein scheme (Mil’shtejn, 1975). Thus, we generated transmissivity fields that exhibit the desired spatial variability and correlation structure.
(d) How to condition the simulations on data (multiple types, multiple scales, varying data quality)? The authors should take a deep look at this topic and show how it can be done. Such an example could start with a reasonable scenario of a real-life aquifer, proceed through model calibration and testing against data (validation)
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- In the light of this comment, the title of section 4 of the paper was changed to “Numerical application and discussion of its results” since in this section the authors provide the details of their model application, describing a) the model for the simulation of the 2D stochastic transmissivity field (the main parameter of confined aquifer flow in 2D) along with its assumed distribution, b) the physical setting of the confined aquifer, as summarized in Table 1 with respect to the aquifer dimensions, the assumed variability of the hydraulic conductivity, and its specific storage which were related to the literature (Todd, 1980). A size of 2km X 2km is a realistic setting for a regional confined aquifer.
There is no model calibration involved in the study since there is no field dataset to which the model is being fitted. The testing of the model performance is described in detail in Section 4 by means of the comparison of the time-space varying ensemble mean and ensemble standard deviation of the confined aquifer flow hydraulic head, determined from the developed model, against the corresponding ensemble mean and ensemble standard deviation of the hydraulic head, obtained from Monte Carlo simulations that used the same conditions as described in Table 1,
- In the light of this comment, the title of section 4 of the paper was changed to “Numerical application and discussion of its results” since in this section the authors provide the details of their model application, describing a) the model for the simulation of the 2D stochastic transmissivity field (the main parameter of confined aquifer flow in 2D) along with its assumed distribution, b) the physical setting of the confined aquifer, as summarized in Table 1 with respect to the aquifer dimensions, the assumed variability of the hydraulic conductivity, and its specific storage which were related to the literature (Todd, 1980). A size of 2km X 2km is a realistic setting for a regional confined aquifer.
(e) Top of p.26: “Therefore, this governing equation needs to be upscaled to the corresponding field scale to predict its behavior correctly”: again, what does that mean? And how does this upscaling affect the possibility to condition the simulations on local data at its locations.
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- Definition and Context: In our context, “upscaling” refers to the process of developing a stochastic model that represents the behavior of the hydrological process at the field or regional scale, starting from governing equations derived at the point scale (differential control volume). This involves transitioning from a deterministic point-scale description to a stochastic field-scale representation that accounts for heterogeneity and uncertainty.
- Revisions Made: to avoid ambiguity, we have removed the term “upscaling” from the manuscript and provided a clearer explanation of how we transition from point-scale equations to a field-scale stochastic model using the LEFPE.
- Impact on Conditioning Simulations on Local Data: The methodology allows for the inclusion of local data at field scale. This means that conditioning the simulations on local observations is feasible and can enhance the model’s accuracy at those locations.
- We appreciate your thorough review and valuable suggestions, which have helped us improve the clarity and comprehensiveness of our manuscript. We have made revisions to address each of your points and believe that these changes enhance the presentation and applicability of our proposed method.
Citation: https://doi.org/10.5194/egusphere-2024-1411-AC2 - Thank you for your thoughtful comment. We appreciate your perspective on the mathematical derivations presented in our paper. We acknowledge that the derivations are extensive and may be challenging to follow. However, we believe that this mathematical rigor is essential and forms a significant part of our contribution of this study.
Status: closed
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RC1: 'Comment on egusphere-2024-1411', Anonymous Referee #1, 29 Aug 2024
The authors present an interesing approach for the modeling of the PDF of hydraulic
for 2-dimensional groundwater flow under spatially heterogeneous transmissivity. Formulating
the groundwater flow equation as an advection-diffusion equation, the latter rewritten in the
characteristic system. The characteristic equations represent stochastic differential
equation. Then a non-local Lagrangian-Eulerian equation for state variables is presented.
The authors then present a numerical scheme to solve this equation. The model is then
applied to a numerical experiment of two-dimensional groundwater flow. The authors state
that they propose a novel stochastic model. However, from the presentation of the model
in Sections 2 and 3 it is not entirely clear which model developments refer to the present
paper, and which refer to previous work for the second author. This needs to be clarified in
a revised version of the manuscript.Comments:
* Eq. 8-9: No partial derivative on x and y.
* Eq. 10: A different notation should be used for the coordinates in the
characteristic system. The expression for the diffusion terms in the characteristic
system should be given.* Eq. 14: Do the authors mean x_t as the x-coordinate at time t or the vector x at time t? Please
clarify.* Eqs. 19 and 21: These equations a linear and local in P. What is meant by non-locality?
* Eq. 40 and following: x refers to the x-ccordinate or vector x? Please clarify.
Citation: https://doi.org/10.5194/egusphere-2024-1411-RC1 -
AC1: 'Reply on RC1', Joaquin Meza, 23 Sep 2024
Comment on egusphere-2024-1411: Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head’s probability density function
Joaquín Meza (1) and Levent Kavvas (2)
(1) Departamento de Obras Civiles, Universidad Técnica Federico Santa María, Valparaiso, Chile
(2) Department of Civil and Environmental Engineering, University of California, Davis, CA, 95616, USA
Dear Editor and Reviewers,
We express our gratitude for your correspondence and the comprehensive feedback provided by the esteemed reviewers regarding the manuscript entitled "Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function.''
The constructive insights and comments offered by the reviewers have proven to be invaluable, significantly contributing to the enhancement of the manuscript. We have diligently reviewed each comment and incorporated necessary revisions to align the manuscript with the requisite standards. The amended sections have been distinctly highlighted in blue within the revised manuscript.
The ensuing sections delineate the principal corrections made in the paper, accompanied by our responses to the respective reviewer's comments:
Authors’ Response to Referee #1's Comments:
The authors present an interesing approach for the modeling of the PDF of hydraulic for 2-dimensional groundwater flow under spatially heterogeneous transmissivity. Formulating the groundwater flow equation as an advection-diffusion equation, the latter rewritten in the characteristic system. The characteristic equations represent stochastic differential equation. Then a non-local Lagrangian-Eulerian equation for state variables is presented. The authors then present a numerical scheme to solve this equation. The model is then applied to a numerical experiment of two-dimensional groundwater flow. The authors state that they propose a novel stochastic model. However, from the presentation of the model in Sections 2 and 3 it is not entirely clear which model developments refer to the present paper, and which refer to previous work for the second author. This needs to be clarified in a revised version of the manuscript.
- Thank you for your valuable feedback. In order to clarify which model developments pertain to the present work and which refer to the previous contributions of the second author, we have provided a detailed breakdown below:
Novel Developments in the Present Paper: The Lagrangian-Eulerian Extension to the Fokker Planck Equation (LEFPE), that was developed in Kavvas (2003), was then applied to the modeling of stochastic one-dimensional (1D) groundwater flow in Cayar and Kavvas ( 2009a), and then to two-dimensional (2D) stochastic groundwater flow in Cayar and Kavvas (2009b). For modeling a real-life aquifer a 2D setting is necessary. However, in order to model the 2D stochastic groundwater flow by the symmetry method Cayar and Kavvas (2009b) had to restrict the 2D groundwater flow to a symmetric model domain with symmetric boundary conditions, and had to restrict the hydraulic conductivity, the main parameter, to a random constant in space (a random variable) while the hydraulic conductivity or its vertically-integrated form, transmissivity, are random functions of spatial location.In the current manuscript under review, co-authored by Joaquín Meza and M. Levent Kavvas, a new stochastic method of characteristics is applied to the 2D stochastic groundwater flow governing equation in order to render the resulting model applicable to the realistic setting of 2D groundwater flow with generally asymmetric boundary conditions and with the hydraulic conductivity or transmissivity as random functions of the spatial location over the model domain.
Secondly, instead of utilizing the conventional regular perturbation methods which usually have closure problems, for the first time in stochastic groundwater modeling, this study utilizes a one-to-one correspondence between the LEFPE and the stochastic ordinary differential equations, resulting from the application of the method of characteristics to the 2D stochastic groundwater flow governing equation to convert the stochastic modeling problem to the solution of a deterministic linear evolution equation for the time-space evolution of the probability density function (pdf) of the state variable, the hydraulic head. Once this pdf is solved from the LEFPE under specified initial and boundary conditions, then it is possible to determine the time-space varying ensemble mean and ensemble variance of the hydraulic head over the whole model domain.
Another key advancement in our work is the explicit incorporation of the spatial correlation in the hydraulic conductivity field. This development is crucial for capturing the realistic variability and interdependence of hydrological properties across the study area, which traditional models often oversimplify. Unlike previous works, which assumed that spatial correlation was non-existent (uncorrelated hydraulic conductivity field) in order to simplify the calculation of covariance from variance, our approach acknowledges and integrates these correlations directly.
Furthermore, our paper introduces an appropriate numerical method for solving the Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE), providing a practical tool for modeling confined groundwater systems under uncertainty. This capability is a significant extension beyond the studies of Kavvas (2003) and his co-workers (Cayar and Kavvas, 2009a,b), allowing for more accurate and robust predictions in hydrological modeling - We hope this clarifies the contributions and delineates the extensions made in the current manuscript relative to the previous studies of the second author. Additional details on specific equations and methodologies developed distinctly in this manuscript are outlined in Sections 2 and 3, which explicitly describe the integration of nonlocal stochastic elements into the modeling framework for confined aquifers.
Comments:
- Eq. 8-9: No partial derivative on x and y
- Thank you for your careful review and comment regarding the partial derivatives in equations 8 and 9. After revisiting the formulations and the intended mathematical representation, we realized that including partial derivatives was indeed a mistake in the manuscript. We have corrected this error in the revised manuscript by removing any incorrect indications of partial derivatives, ensuring the equations now accurately reflect the intended differentiated model.
- Eq. 10: A different notation should be used for the coordinates in the characteristic system. The expression for the diffusion terms in the characteristic system should be given.
- Thank you for your insightful comment regarding the notation used for the coordinates in the characteristic system. We appreciate your concern about the potential for confusion with the current notation. In the initial manuscript, we indeed used x, y, and t to denote the coordinates in the characteristic system. To address your concerns and to eliminate any ambiguity, we propose to introduce a clearer notation where λ explicitly denotes the parameter along the characteristic curves, which allows us to describe the dynamics more transparently.
From the derivation in the file attached, it follows that, aligning the characteristic system with the representation used in the manuscript. This approach ensures that each variable clearly corresponds to its role in the characteristic framework, separating them distinctly from their usage in the general field equations.We have revised the notation in the manuscript accordingly to reflect this clarification, ensuring that the representation is consistent and clear across all sections. We appreciate your guidance on this matter and hope that the revised notation and the inclusion of the diffusion terms will meet the rigorous standards of clarity and precision required for our study.
- Thank you for your insightful comment regarding the notation used for the coordinates in the characteristic system. We appreciate your concern about the potential for confusion with the current notation. In the initial manuscript, we indeed used x, y, and t to denote the coordinates in the characteristic system. To address your concerns and to eliminate any ambiguity, we propose to introduce a clearer notation where λ explicitly denotes the parameter along the characteristic curves, which allows us to describe the dynamics more transparently.
- Eq. 14: Do the authors mean xt as the x-coordinate at time t or the vector x at time t? Please clarify.
- We appreciate the valuable feedback from the reviewer. In response, we have significantly improved the notation along the document to clearly differentiate the coordinate “x” and the "vector x". In the case of equation 14 it effectively refers to "vector x” at time “t”, which was already modified in the whole document to ensure clarity in the use of the variables.
- Eqs. 19 and 21: These equations a linear and local in P. What is meant by non-locality
- We value the reviewer’s suggestion and have addressed it by extending the explanation about "non-locality". This concept refers to the influence of conditions or parameters at a specific point that depend on information from distant points in space or earlier moments in time. In contrast to purely local equations, where the evolution of a variable in time or space only depends on the conditions at the same location, non-local processes involve the dependence of variables on their interaction with conditions at other locations or past times.
This study involves covariances over space and time, indicating that the evolution of variables at a given location is influenced by their interaction with other variables at different time-space locations. For example, the covariance terms in equation (15) described in the document imply that to understand the evolution of a system at a particular location, one must account for the system’s past states or its state at different spatial positions.Thus, the concept of non-locality emphasizes that the system's behavior is not just a function of local conditions but is interconnected with the system's behavior across space and time. In summary, non-locality refers to the fact that the behavior of a system at a specific point location is not independent but is influenced by the broader spatial and temporal context, as reflected in the integral terms and covariances in the equation (21).
We have expanded the discussion around these equations and explained the concept of non-locality as it applies to our model. The revisions are intended to clarify this concept, emphasizing how the probability density function "P" is influenced across a range of spatial interactions.
- We value the reviewer’s suggestion and have addressed it by extending the explanation about "non-locality". This concept refers to the influence of conditions or parameters at a specific point that depend on information from distant points in space or earlier moments in time. In contrast to purely local equations, where the evolution of a variable in time or space only depends on the conditions at the same location, non-local processes involve the dependence of variables on their interaction with conditions at other locations or past times.
- Eq. 40 and following: x refers to the x-ccordinate or vector x? Please clarify.
- Thank you for your observation concerning the clarification needed in Equation 40 and the subsequent equations regarding whether "x" refers to the spatial coordinate or a vector.
In the initial manuscript, the notation "x" was indeed ambiguous, and we appreciate your pointing this out. To clarify, in Equation 40 and the following equations, "x" and "y" refer specifically to the spatial coordinates in the x-direction and y-direction, respectively, not to vectors. These equations model the stochastic dynamics along each spatial dimension independently but within the context of the two-dimensional field.To eliminate any potential confusion in the revised manuscript, we have adjusted the notation as follows:
- We use xi to denote the spatial coordinates, where i = {1, 2} corresponds to the x- and y-directions, respectively.
- We have ensured that all instances of the coordinate "x" and vector "\underline{x}" in the manuscript are now clearly defined, with an explicit definition provided in Section 2.2.
The corrected equations are presented with clear differentiation between the handling of the x- and y-directions, ensuring that the interpretation aligns with the physical processes being modeled. This notation not only enhances readability but also aligns with the mathematical precision required for stochastic modeling.
- Thank you for your observation concerning the clarification needed in Equation 40 and the subsequent equations regarding whether "x" refers to the spatial coordinate or a vector.
- We hope that this adjustment resolves the ambiguities and thank you for helping us improve the clarity and accuracy of our manuscript. Your meticulous attention to our manuscript is sincerely appreciated,
- Thank you for your valuable feedback. In order to clarify which model developments pertain to the present work and which refer to the previous contributions of the second author, we have provided a detailed breakdown below:
-
AC1: 'Reply on RC1', Joaquin Meza, 23 Sep 2024
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RC2: 'Comment on egusphere-2024-1411', Anonymous Referee #2, 06 Sep 2024
I have several suggestions for the authors to consider.
- The paper is presenting (what seems to me) a tedious math derivation that may not be easy for many readers to follow. I am not against tedious derivations in principle, but the authors should convince the reader that the effort is worthwhile.
- The open science questions this paper attempts to address should be stated clearly. And then answered. As it is now, it seems that the paper is trying to show computational speed under (what sems to me) a very limited set of conditions and limiting assumptions.
- The title of the paper states “probability density function”. However, the text deals with mean and standard deviations. There is equivalence here only if the probability density function is Gaussian. But there is no proof provided for this assumed/implied equivalence.
- Page 4 (last paragraph) states: “upscales the governing stochastic differential equations from a point-scale (at which they are valid) to a field scale.” There are also no explanation of what upscaling means or how it is carried out.
- The authors need to define “point-scale” and “field local-scale”.
- The follow-up paragraph (page 5) suggests (through references to previous works) that upscaling and ensemble averaging are equivalent. This can be true under strict conditions which need to be stated. These conditions, once met, pose severe limitations on the applicability of the proposed approach.
- The paper attempts to show the superiority of the proposed method compared to Monte-Carlo simulations. This could be true under a very limited set of conditions which casts doubts on the value and generality of this approach:
- The method is demonstrated for a 2D, confined aquifer.
- The assumption stated following equation 6 is not defended (except for the case study). There is a suggestion to test it using Monte Carlo simulations, which defeats the entire purpose of the study.
- The authors should list the models of spatial variability that can be accommodated: (1) does the method requires stationarity of some sort (must be, because of the assumed equivalence between upscaling and ensemble averaging, but it is not stated)? (2) what types of spatial covariances can be accommodated? (3) there is no mention of the type of spatial covariance used for the demo.
- How to condition the simulations on data (multiple types, multiple scales, varying data quality)? The authors should take a deep look at this topic and show how it can be done. Such an example could start with a reasonable scenario of a real-life aquifer, proceed through model calibration and testing against data (validation)
- Top of p.26: “Therefore, this governing equation needs to be upscaled to the corresponding field scale to predict its behavior correctly”: again, what does that mean? And how does this upscaling affect the possibility to condition the simulations on local data at its locations.
Citation: https://doi.org/10.5194/egusphere-2024-1411-RC2 -
AC2: 'Reply on RC2', Joaquin Meza, 27 Sep 2024
Comment on egusphere-2024-1411: Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head’s probability density function
Joaquín Meza (1) and Levent Kavvas (2)
(1) Departamento de Obras Civiles, Universidad Técnica Federico Santa María, Valparaiso, Chile
(2) Department of Civil and Environmental Engineering, University of California, Davis, CA, 95616, USA
Dear Editor and Reviewers,
We express our gratitude for your correspondence and the comprehensive feedback provided by the esteemed reviewers regarding the manuscript entitled "Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function.''
The constructive insights and comments offered by the reviewers have proven to be invaluable, significantly contributing to the enhancement of the manuscript. We have diligently reviewed each comment and incorporated necessary revisions to align the manuscript with the requisite standards. The amended sections have been distinctly highlighted in blue within the revised manuscript.
The ensuing sections delineate the principal corrections made in the paper, accompanied by our responses to the respective reviewer's comments:
Authors’ Response to Referee #2's Comments:
1) The paper is presenting (what seems to me) a tedious math derivation that may not be easy for many readers to follow. I am not against tedious derivations in principle, but the authors should convince the reader that the effort is worthwhile.
- Thank you for your thoughtful comment. We appreciate your perspective on the mathematical derivations presented in our paper. We acknowledge that the derivations are extensive and may be challenging to follow. However, we believe that this mathematical rigor is essential and forms a significant part of our contribution of this study.
The governing equation of a hydrologic process is derived by conserving mass and momentum/energy over a differential control volume. This derivation over a differential control volume represents the process at “point scale” since it is at the scale of a differential control volume. When the governing equation for a specific process is applied over a field or a region of several square kilometers, this “point-scale” equation becomes uncertain at any location over the field or regional model domain due to the heterogeneity of the material of the model domain, rendering the equation’s parameters uncertain, due to the uncertainty in the generally time-space varying sources and sinks into and from the model domain, and due to the spatially varying boundary conditions over the field or regional model domain.
A general modeling framework in terms of the probability density function (pdf) of the state variable of the process, which, in general, may vary with time and spatial locations, would be a useful approach for deriving the ensemble mean and ensemble variance of the process over the model domain since the process mean and variance are the most often addressed process moments in hydrologic practice that attempts to address process uncertainty in a physically-based framework. One fundamental advantage of this approach is that the resulting pdf of the process is obtained directly from the one-to-one correspondence between a Lagrangian-Eulerian extension to the conventional Fokker-Planck Equation (LEFPE) of statistical physics and the subject governing equation of a particular hydrologic process. Although the LEFPE has been developed about two decades ago (Kavvas, 2003) and could be useful in determining the generally time-space varying pdf of the state variable of any particular hydrologic process under specified initial and boundary conditions, it has been largely unknown to the hydrologic community. Accordingly, in this article we tried to provide as much information as permitted by the paper length limit imposed by the publisher, on the LEFPE in terms of the confined aquifer flow process, and its application toward the derivation of the time-space varying ensemble mean and ensemble variance of the process under spatially varying process parameter (transmissivity) and boundary conditions around the model domain. An important aspect of this methodology is that each hydrologic process will have a one-to-one correspondence to a LEFPE. As such, this approach provides a general framework for determining the generally time-space varying pdf of a hydrologic process under specified initial and boundary conditions without needing any assumption about the underlying distribution of the state variable’s pdf. As explained in detail in Kavvas (2003), the fundamental approximation of this methodology is that it is valid to the order of the covariance time of a specified hydrologic process (to the order of the second cumulant of the process). - Justification of the Mathematical Derivations
- Necessity for Addressing Fundamental Challenges: The heterogeneity of geological formations and the uncertainties in hydrological parameters present significant challenges in accurately modeling groundwater flow. Our detailed mathematical derivations are necessary to develop a robust framework that can address these challenges and provide accurate simulations.
- Contribution to Hydrological Modeling: Our paper introduces a novel methodology that establishes a one-to-one correspondence between the stochastic governing equations of confined groundwater flow and a Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE). This approach allows us to derive the time-space evolution of the probability density function (pdf) of the hydraulic head without making assumptions about its underlying distribution, which is a significant advancement in stochastic hydrology.
- Mathematical Rigor as a Core Contribution: The rigorous mathematical development is integral to our contribution. It ensures the validity and applicability of our methodology. By thoroughly presenting these derivations, we enable other researchers and practitioners to understand the foundations of our approach and potentially extend it to other hydrological processes.
- We believe that the mathematical rigor presented in our paper is justified and worthwhile, as it addresses fundamental uncertainties in hydrological modeling and offers a substantial contribution to the field. Our work fits well within the scope of HESS by providing innovative research that enhances the understanding of hydrological systems and processes.
We appreciate your feedback, which has prompted us to performed a major revision of our paper to clarify the whole methodology, which hopefully will describe its value to the readerAnd to better communicate the importance and applicability of our work.
2) “The open science questions this paper attempts to address should be stated clearly. And then answered. As it is now, it seems that the paper is trying to show computational speed under (what sems to me) a very limited set of conditions and limiting assumptions.”
- We appreciate the reviewer’s insightful comment and the opportunity to clarify the open scientific questions our paper aims to address. We recognize the importance of explicitly stating these questions and demonstrating how our work contributes to advancing hydrological science.
- Open Scientific Questions Addressed:
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- How can we determine the ensemble mean and variance of a hydrological process without making assumptions about the underlying probability distribution of the state variable or assuming stationarity/ergodicity? Traditional stochastic modeling approaches often rely on specific assumptions about the probability distributions and stationarity of hydrological processes. However, these assumptions may not hold in heterogeneous media with spatially varying parameters and boundary conditions. Our study seeks to develop a methodology that overcomes these limitations by directly deriving the time-space evolution of the probability density function (pdf) of the hydraulic head from the governing equations.
- How does the heterogeneity of aquifer materials and uncertainty in source/sink conditions affect the time-space varying probabilistic structure of confined groundwater flow, and how can we model this behavior in a physically-based framework? Understanding the influence of spatial variability and uncertainty on groundwater flow is crucial for accurate modeling and prediction. We aim to explore how these factors impact the ensemble behavior of the hydraulic head and to develop a modeling framework that captures these effects without oversimplifying assumptions.
- How Our Paper Addresses These Questions:
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- We introduce a novel methodology that establishes a one-to-one correspondence between the stochastic governing equations of confined groundwater flow and a Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE). This approach allows us to determine the time-space evolution of the pdf of the hydraulic head under specified initial and boundary conditions, without assuming any specific underlying distribution or stationarity.
- By applying this methodology, we derive the ensemble mean and variance of the hydraulic head directly from the governing equations, accounting for the heterogeneity of the aquifer medium and the uncertainty in source/sink terms.
- Our numerical application demonstrates how the ensemble mean and variance evolve over time and space in a two-dimensional, asymmetric flow domain with spatially varying transmissivity and boundary conditions.
- Addressing Concerns About Limited Conditions and Assumptions: While our numerical example focuses on a specific scenario to illustrate the methodology, the approach itself is general and not limited to particular conditions. The methodology does not rely on restrictive assumptions about the system’s properties, such as homogeneity or stationary processes. Instead, it provides a framework that can handle time-space varying parameters and conditions, making it applicable to a wide range of hydrological processes and settings.
Our study aims to move beyond traditional methods that may equate ensemble means with spatial averages or require ergodicity assumptions. By not making such assumptions, we address the complexities inherent in natural systems, where parameters and conditions can vary significantly across space and time. - Revisions Made to the Manuscript: In response to your valuable suggestion, we have substantially revised the manuscript to clearly state the open scientific questions our paper addresses. Specifically:
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- Abstract and Introduction: We have explicitly outlined the fundamental scientific questions at the beginning of the manuscript, providing a clear context for our study.
- Summary and conclusions: We have expanded these sections to directly answer the stated questions, highlighting the significance of our findings and their implications for hydrological modeling.
- Clarification of Methodology: We have enhanced the description of our methodology to emphasize its generality and applicability beyond the specific example presented.
- We believe that these revisions improve the clarity of our manuscript and better communicate the importance and scope of our work. We are grateful for your thoughtful feedback, which has helped us strengthen our manuscript. We hope that the revisions address your concerns.
3) “The title of the paper states “probability density function”. However, the text deals with mean and standard deviations. There is equivalence here only if the probability density function is Gaussian. But there is no proof provided for this assumed/implied equivalence.”
- Clarification on the Use of the Probability Density Function: Our paper focuses on determining the time-space evolution of the probability density function (pdf) of the hydraulic head in a confined groundwater flow system under uncertainty. The pdf is obtained directly from the solution of the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE) under specified initial and boundary conditions. This approach does not assume any specific form for the underlying probability distribution of the hydraulic head; in particular, it does not assume that the distribution is Gaussian.
Relationship Between the PDF, Mean, and Standard Deviation: While the ensemble mean and standard deviation are statistical moments commonly used to describe a distribution, they do not uniquely define the distribution unless additional information or assumptions are made (such as normality). In our study, we derive the pdf of the hydraulic head without assuming it follows a Gaussian distribution. From this derived pdf, we calculate the ensemble mean and variance (and hence the standard deviation) of the hydraulic head using the standard definitions:
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- Ensemble Mean: \langle h(x, y, t) \rangle = \int h(x, y, t) P(h, x, y; t) \, dh
- Ensemble Variance: \text{Var}[h(x, y, t)] = \int h^2(x, y, t) P(h, x, y; t) \, dh - \langle h(x, y, t) \rangle^2
These expressions allow us to compute the mean and variance directly from the pdf obtained through the LEFPE, regardless of the distribution’s shape.
- No Assumption of Gaussianity: Our methodology does not assume or imply that the hydraulic head follows a Gaussian distribution. The LEFPE provides a framework to determine the pdf of the hydraulic head based on the stochastic properties of the transmissivity field and source/sink terms, without specifying the form of the distribution a priori.
- Revisions Made to the Manuscript: To address this concern and enhance clarity, we have made the following revisions:
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- Emphasized the Role of the PDF: We have updated the manuscript to highlight that the pdf of the hydraulic head is a primary outcome of our methodology, and the mean and standard deviation are calculated from this pdf without assuming any specific distribution.
- Clarified the Absence of Gaussian Assumption: We have explicitly stated in the text that no assumption is made regarding the Gaussianity of the hydraulic head distribution. This clarification is included in the methodology section and reinforced in the discussion of results.
- Included Discussion on the Shape of the PDF: Where appropriate, we have included discussions on the potential shape and properties of the pdf obtained from the LEFPE, acknowledging that it may not be Gaussian and that higher-order moments could be calculated if needed.
- We hope that this explanation clarifies that our study does not assume an equivalence between the pdf and its mean and standard deviation based on Gaussianity. Instead, we derive the pdf directly from the LEFPE and compute the statistical moments from this pdf. This approach allows us to capture the full probabilistic behavior of the hydraulic head under uncertainty, without restrictive assumptions about the form of its distribution.
4) “Page 4 (last paragraph) states: “upscales the governing stochastic differential equations from a point-scale (at which they are valid) to a field scale.” There are also no explanation of what upscaling means or how it is carried out.”
- Clarification on the Use of “Upscaling”: In the original manuscript, we used the term “upscaling” to describe the process of developing a stochastic model that determines the probabilistic behavior of a hydrological process at the field or regional scale, starting from its governing equations derived at the point scale (i.e., over a differential control volume). Specifically:
- How Upscaling Was Addressed: In our initial manuscript, we intended to convey that our methodology transitions from the deterministic point-scale equations to a stochastic representation suitable for field-scale applications. However, we recognize that the term “upscaling” was not adequately explained and may have caused confusion.
- Revisions Made to Address This Issue: In response to your valuable feedback, we have made the following revisions to the manuscript:
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- Removed the Term “Upscaling”: To avoid ambiguity and enhance clarity, we have removed the term “upscaling” from the manuscript. We acknowledge that without proper explanation, the term can be misleading or confusing.
- Provided Detailed Explanation of the Methodology: We have revised the text to provide a detailed explanation of how we develop the stochastic model for the confined groundwater flow process, starting from the point-scale governing equations. This includes:
•Derivation of the LEFPE from Point-Scale Equations: We explain how the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE) is derived directly from the point-scale governing equations of confined groundwater flow. This derivation is presented step-by-step to ensure that readers understand the transition from deterministic to stochastic modeling.
•Accounting for Field-Scale Heterogeneity and Uncertainty: We elaborate on how heterogeneity in the transmissivity field, uncertainties in source/sink terms, and spatially varying boundary conditions are incorporated into the stochastic framework. This clarifies how the model represents the probabilistic behavior of the hydraulic head at the field scale. - Clarified Terminology and Concepts: Throughout the manuscript, we have ensured that all terms and concepts are clearly defined and explained. We avoid using technical jargon without proper context and provide explanations where necessary.
- By removing the term “upscaling” and providing a more detailed explanation of our methodology, we aim to enhance the clarity and accessibility of our manuscript. Our goal is to ensure that readers can follow the development of our stochastic modeling approach from the point-scale governing equations to the field-scale representation without confusion.
5) The authors need to define “point-scale” and “field local-scale”.
- The “point-scale” is the scale of the differential control volume within which the governing equation of the particular process is derived. What was meant by field or regional scale is the area size of the modeling domain which, in the case of the reported confined aquifer flow modeling study, corresponds to several square kilometers.
6) The follow-up paragraph (page 5) suggests (through references to previous works) that upscaling and ensemble averaging are equivalent. This can be true under strict conditions which need to be stated. These conditions, once met, pose severe limitations on the applicability of the proposed approach.
- Thank you for your insightful comment. We appreciate the opportunity to clarify the relationship between upscaling and ensemble averaging in our methodology. Due to the time-space variation of the pdf and all of the moments of the confined aquifer flow process (in terms of the hydraulic head’s pdf), the time-space varying ensemble mean (ensemble average) of the process is not equivalent to the areal average of the process. In the reported methodology there is no stationarity or ergodicity assumption, or any assumption about the underlying distribution of the process state variable, the hydraulic head. The fundamental assumption of the reported modeling methodology is the approximation of the process to the covariance time of the process (second-order cumulant expansion). This approximation is stated in the revised version of the paper. The derivation of the LEFPE that corresponds to any particular governing equation of a process is given in Kavvas (2003).
- We appreciate your feedback, which has prompted us to clarify the assumptions and limitations of our approach. By explicitly stating the conditions under which our methodology operates and ensuring that our explanations are clear and accurate, we aim to provide a robust and transparent framework for modeling stochastic groundwater flow.
7) The paper attempts to show the superiority of the proposed method compared to Monte-Carlo simulations. This could be true under a very limited set of conditions which casts doubts on the value and generality of this approach:
- Thank you for your insightful comment. We appreciate the opportunity to clarify the value and generality of our proposed method in comparison to Monte Carlo simulations.
- Clarification on the Purpose of Our Method: Our paper does not aim to demonstrate the superiority of our proposed method over Monte Carlo simulations. Instead, we present an alternative approach for modeling stochastic groundwater flow that offers certain characteristics making it a valuable tool for evaluating temporal and spatial uncertainties in parameters and their influence on the system’s response.
- Value and Generality of Our Approach:
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- Alternative Framework for Uncertainty Analysis: Our methodology provides a different perspective on handling uncertainties in hydrological modeling. By directly deriving the time-space evolution of the probability density function (pdf) of the hydraulic head from the governing equations, we offer a framework that can complement existing methods like MC simulations. This approach can be particularly useful in scenarios where assessing the influence of parameter uncertainties on the system’s behavior is essential.
- Direct Evaluation of Temporal and Spatial Uncertainties: The method allows for explicit incorporation of temporal and spatial uncertainties in model parameters, such as transmissivity and source/sink terms. By solving the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE), we can directly observe how these uncertainties propagate through the system and affect the hydraulic head’s response over time and space.
- Applicability to Realistic Conditions: The proposed approach does not rely on restrictive assumptions like stationarity, ergodicity, or specific probability distributions of the state variables. This generality enhances its applicability to a wide range of hydrological systems, including those with heterogeneous aquifer properties and complex boundary conditions.
- Context of Monte Carlo Simulations in Our Study: Due to the lack of comprehensive publicly available datasets for confined aquifers with sufficient spatial and temporal resolution, we employed Monte Carlo simulations to validate our developed stochastic model. The MC simulations served as a benchmark to compare the results obtained from our method under realistic initial and asymmetric boundary conditions in a two-dimensional confined aquifer flow over a 4 sq. km region.
- Addressing the Concern About Limited Conditions: While our numerical example focuses on a specific scenario, the methodology itself is general and not limited to particular conditions. Our intention is to provide an alternative tool that can be applied to various hydrological problems, offering insights into the effects of parameter uncertainties on groundwater flow without the need for extensive computational resources often associated with MC simulations.
- Our proposed method offers a complementary approach to existing techniques for modeling stochastic groundwater flow. By providing a framework that can effectively evaluate temporal and spatial uncertainties and their influence on the system’s response, we contribute to a more comprehensive understanding of groundwater flow under uncertainty.
We appreciate your feedback, which has helped us clarify the purpose and scope of our work. We have revised the manuscript to emphasize that our method is an alternative approach rather than a replacement for Monte Carlo simulations, highlighting its potential benefits and applications without overstating its superiority.
(a) The method is demonstrated for a 2D, confined aquifer.
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- Relevance of the 2D Confined Aquifer Scenario:
- Practical Significance: Modeling a 2D confined aquifer allows us to capture essential features of groundwater flow in a manner that is both computationally manageable and practically relevant. Many real-world groundwater problems involve large-scale aquifers where horizontal flow dominates, making a 2D approach appropriate for initial investigations.
- Illustrative Purposes: The 2D scenario provides a clear and focused context to demonstrate the capabilities of our proposed method. It enables us to effectively illustrate how the methodology handles spatial heterogeneity, uncertain parameters, and complex boundary conditions without the added computational complexity of a three-dimensional model.
- Relevance of the 2D Confined Aquifer Scenario:
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- Generality and Applicability of the Method:
- Method Generality: While we have demonstrated the method using a 2D confined aquifer, the underlying mathematical framework is general and not limited to this specific case. The methodology can be extended to three-dimensional systems and applied to various types of aquifers, including unconfined and semi-confined aquifers.
- Adaptability to Different Settings: The stochastic modeling approach we propose is adaptable to different spatial dimensions and hydrogeological conditions. By adjusting the governing equations and numerical implementation, the method can accommodate the complexities of higher-dimensional models and more intricate geological formations.
- Generality and Applicability of the Method:
(b) The assumption stated following equation 6 is not defended (except for the case study). There is a suggestion to test it using Monte Carlo simulations, which defeats the entire purpose of the study.-
- Clarification on the Assumption and Use of Monte Carlo Simulations:
- Justification of the Assumption: The assumption following Equation 6 pertains to the stochastic structure of the transmissivity field and its treatment within our model. In the absence of comprehensive field data to validate this assumption, we relied on numerical experiments to assess its validity.
- Role of Monte Carlo Simulations: We employed Monte Carlo simulations as a benchmark to validate our proposed method under the same conditions. The use of Monte Carlo simulations in this context serves as a means of verification rather than comparison in terms of superiority. By comparing the results from our method with those obtained from Monte Carlo simulations, we demonstrate that our approach produces consistent and reliable estimates of the ensemble mean and variance of the hydraulic head.
- Purpose of Validation: The validation using Monte Carlo simulations helps to confirm the viability of the assumption in a controlled setting. It does not defeat the purpose of the study but rather strengthens it by providing evidence that our method can replicate results obtained from established stochastic modeling techniques.
- Revisions to the Manuscript: In the revised manuscript, we have provided a more detailed defense of the assumption following Equation 6, including theoretical justifications and references to related work. We have clarified the rationale behind using Monte Carlo simulations for validation purposes.
- Clarification on the Assumption and Use of Monte Carlo Simulations:
(c) The authors should list the models of spatial variability that can be accommodated: (1) does the method requires stationarity of some sort (must be, because of the assumed equivalence between upscaling and ensemble averaging, but it is not stated)? (2) what types of spatial covariances can be accommodated? (3) there is no mention of the type of spatial covariance used for the demo.
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- (1) Requirement of Stationarity: Our method does not require assumptions of stationarity or ergodicity. The probability density function (pdf) of the hydraulic head, obtained from the solution of the Lagrangian-Eulerian extension of the Fokker-Planck Equation (LEFPE), varies in time and space. This allows us to model non-stationary processes where statistical properties change over time and spatial locations.
- (2) Types of Spatial Covariances: The method can accommodate various spatial covariance structures for the transmissivity field. Since the stochastic properties are directly incorporated into the LEFPE, we can model different types of covariance functions, including exponential, Gaussian, or other valid covariance models, provided they satisfy the requirement of finite covariance time (as discussed in Kavvas, 2003).
- (3) Spatial Covariance Used in the Demonstration: We apologize for the oversight in not explicitly mentioning the type of spatial covariance used in our demonstration. In the revised manuscript, we have included a detailed description of the spatial covariance model employed in our numerical example.
Specifically, we modeled the transmissivity random field as independent log-normal distributions with an exponentially decaying autocorrelation function. This approach is based on the stochastic differential equations (SDEs) model proposed by Zárate-Miñano and Milano (2016), which allows us to generate a log-normal transmissivity field with specified mean, variance, and correlation length, by numerically integrating these SDEs using the implicit Milstein scheme (Mil’shtejn, 1975). Thus, we generated transmissivity fields that exhibit the desired spatial variability and correlation structure.
(d) How to condition the simulations on data (multiple types, multiple scales, varying data quality)? The authors should take a deep look at this topic and show how it can be done. Such an example could start with a reasonable scenario of a real-life aquifer, proceed through model calibration and testing against data (validation)
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- In the light of this comment, the title of section 4 of the paper was changed to “Numerical application and discussion of its results” since in this section the authors provide the details of their model application, describing a) the model for the simulation of the 2D stochastic transmissivity field (the main parameter of confined aquifer flow in 2D) along with its assumed distribution, b) the physical setting of the confined aquifer, as summarized in Table 1 with respect to the aquifer dimensions, the assumed variability of the hydraulic conductivity, and its specific storage which were related to the literature (Todd, 1980). A size of 2km X 2km is a realistic setting for a regional confined aquifer.
There is no model calibration involved in the study since there is no field dataset to which the model is being fitted. The testing of the model performance is described in detail in Section 4 by means of the comparison of the time-space varying ensemble mean and ensemble standard deviation of the confined aquifer flow hydraulic head, determined from the developed model, against the corresponding ensemble mean and ensemble standard deviation of the hydraulic head, obtained from Monte Carlo simulations that used the same conditions as described in Table 1,
- In the light of this comment, the title of section 4 of the paper was changed to “Numerical application and discussion of its results” since in this section the authors provide the details of their model application, describing a) the model for the simulation of the 2D stochastic transmissivity field (the main parameter of confined aquifer flow in 2D) along with its assumed distribution, b) the physical setting of the confined aquifer, as summarized in Table 1 with respect to the aquifer dimensions, the assumed variability of the hydraulic conductivity, and its specific storage which were related to the literature (Todd, 1980). A size of 2km X 2km is a realistic setting for a regional confined aquifer.
(e) Top of p.26: “Therefore, this governing equation needs to be upscaled to the corresponding field scale to predict its behavior correctly”: again, what does that mean? And how does this upscaling affect the possibility to condition the simulations on local data at its locations.
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- Definition and Context: In our context, “upscaling” refers to the process of developing a stochastic model that represents the behavior of the hydrological process at the field or regional scale, starting from governing equations derived at the point scale (differential control volume). This involves transitioning from a deterministic point-scale description to a stochastic field-scale representation that accounts for heterogeneity and uncertainty.
- Revisions Made: to avoid ambiguity, we have removed the term “upscaling” from the manuscript and provided a clearer explanation of how we transition from point-scale equations to a field-scale stochastic model using the LEFPE.
- Impact on Conditioning Simulations on Local Data: The methodology allows for the inclusion of local data at field scale. This means that conditioning the simulations on local observations is feasible and can enhance the model’s accuracy at those locations.
- We appreciate your thorough review and valuable suggestions, which have helped us improve the clarity and comprehensiveness of our manuscript. We have made revisions to address each of your points and believe that these changes enhance the presentation and applicability of our proposed method.
Citation: https://doi.org/10.5194/egusphere-2024-1411-AC2 - Thank you for your thoughtful comment. We appreciate your perspective on the mathematical derivations presented in our paper. We acknowledge that the derivations are extensive and may be challenging to follow. However, we believe that this mathematical rigor is essential and forms a significant part of our contribution of this study.
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