the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
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: open (until 26 Sep 2024)
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RC1: 'Comment on egusphere-2024-1411', Anonymous Referee #1, 29 Aug 2024
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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 -
RC2: 'Comment on egusphere-2024-1411', Anonymous Referee #2, 06 Sep 2024
reply
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
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