Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper is interesting, since it uses EnKF for inverse modeling in geothermal applications, with good results. Few comments below, to be addressed before the paper can be reconsidered for publication. 

• Equation (2). I assume that the term “W” is a forcing term describing a source or a sink, right?
• Authors honestly state that “physical processes in geothermal systems are complex, which makes it difficult to achieve convergence in inversion simulations”. Moreover, authors admit that “the estimated ensemble mean of the tracer concentration curve is far from the observations” (lines 220-221). It seems that a significant model error occurs in this problem. To this purpose, authors could take advantage of the approach proposed in Berardi et al CPC 2016, https://doi.org/10.1016/j.cpc.2016.07.025 which has been exactly designed for these cases, where a significant model error arises, due to the uncertainty of the model. A different method, with same objective, has been proposed in Jamal and Linker VZJ 2020 https://doi.org/10.1002/vzj2.20000 : these two papers would improve the discussion of this topic.
• Authors claim that SGSIM is used to generate the number of realizations: isn’t this number fixed in advance? If not, according to which rule is this number set?
• In figures 1 and 3, label for unit of measurement of K seems wrong, if K is a permeability,  therefore it should not be .
• Page 7: I would set separately boundary conditions for flow and heat equations.
• Authors refer to pumping and injection wells. Section 4.2.1. should be linked to the 3.2 section, in my opinion.

 

Author Response

Comment 1: Equation (2). I assume that the term “W” is a forcing term describing a source or a sink, right?

Response 1: We thank the reviewer for the remark. Yes, the interpretation is correct: in Eq. (2), the W denotes the volumetric forcing term that represents internal sources or sinks. We have clarified this explicitly in the manuscript (please see line 143).

Comment 2: Authors honestly state that “physical processes in geothermal systems are complex, which makes it difficult to achieve convergence in inversion simulations”. Moreover, authors admit that “the estimated ensemble mean of the tracer concentration curve is far from the observations” (lines 220-221). It seems that a significant model error occurs in this problem. To this purpose, authors could take advantage of the approach proposed in Berardi et al CPC 2016, https://doi.org/10.1016/j.cpc.2016.07.025 which has been exactly designed for these cases, where a significant model error arises, due to the uncertainty of the model. A different method, with same objective, has been proposed in Jamal and Linker VZJ 2020 https://doi.org/10.1002/vzj2.20000: these two papers would improve the discussion of this topic.

Response 2: We appreciate the reviewer highlighting related studies in soil hydrology. These works indeed provide valuable methodological advances. Their studies focused on the unsaturated flow governed by Richards’ equation, while our work addresses hydrothermal parameter estimation in saturated porous media using Darcy’s law and the Oberbeck–Boussinesq approximation. Because the two systems involve different physical processes, initial and boundary conditions, and hydro-thermal properties, direct comparison might not be straightforward. In the revised manuscript, we have addressed the future applications of our studies and added a discussion of cases of unsaturated flow and rock formations in the introduction (lines 89-91).

References

BE Berardi, M.; Andrisani, A.; Lopez, L.; Vurro, M. A new data assimilation technique based on ensemble Kalman filter and Brownian bridges: An application to Richards’ equation. Comput. Phys. Commun. 2016, 208: 43-53. https://doi.org/10.1016/j.cpc.2016.07.025

Jamal, A.; Linker, R. Inflation method based on confidence intervals for data assimilation in soil hydrology using the ensemble Kalman filter. Vadose Zone J. 2020, 19.1: e20000. https://doi.org/10.1002/vzj2.20000:

Comment 3: Authors claim that SGSIM is used to generate the number of realizations: isn’t this number fixed in advance? If not, according to which rule is this number set?

Response 3: According to the SHEMAT-Suite, the Stochastic Geostatistical Simulation (SGSIM) is used to generate initial parameter estimates from geostatistical data. The SGSIM often produces these initial values across a range of realizations, from 1 to the final realization. Consequently, the Monte Carlo Simulation (MCS) uses these initial values to conduct forward modeling. Afterward, evaluation metrics such as Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Squared Error (MSE) are calculated from the simulation results. According to section 4, our study explored a wide range of realizations, from 100 to 2000 (see lines 350-351). As a result, the key message is that a reasonable number of realizations is 1000, based on the stabilization of the evaluation metrics (lines 357-359). Specifically, the number of realizations should be gradually increased until the evaluation metrics stabilize and no longer change. Therefore, the number of realizations is set at 1000 (lines 357-359). This study has emphasized the principle of identifying the appropriate number of realizations as a focal point.

Comment 4: In figures 1 and 3, label for unit of measurement of K seems wrong, if K is a permeability, therefore it should not be…

Response 4: We appreciate the reviewer’s attention to detail. We confirm that the unit shown for permeability is correct as presented. Permeability is expressed in square meters (m²), and the range of values used in our study (5E–13 to 1.5E–12 m²) is consistent with those commonly adopted for high-permeability sediments and fractured rocks in comparable thermo-hydraulic modeling studies. For example, Bauer et al. (2019) investigate permeabilities ranging from 1E-15 to 1E-11 m², and several reservoir and hydrothermal flow simulations likewise consider permeabilities on the order of 1E-13 – 1E-12 m² (Vogt et al., 2010; Zhou et al., 2021) (see lines 264-266)

References

Bauer, J. F.; Krumbholz, M.; Luijendijk, E.; Tanner, D. C. A numerical sensitivity study of how permeability, porosity, geological structure, and hydraulic gradient control the lifetime of a geothermal reservoir. Solid earth 2019, 10(6), 2115-2135. https://doi.org/10.5194/se-10-2115-2019.

Vogt, C.; Mottaghy D.; Wolf, A.; Rath, V.; Pechnig, R.; Clauser, C. Reducing temperature uncertainties by stochastic geothermal reservoir modelling. Geophys. J. Int. 2010, 181(1), 321-333, https://doi.org/10.1111/j.1365-246X.2009.04498.x.

 

Zhou, D.; Tatomir, A.; Sauter, M. Thermo-hydro-mechanical modelling study of heat extraction and flow processes in enhanced geothermal systems. Advances in geosciences 2021, 54, 229-240, https://doi.org/10.5194/adgeo-54-229-2021, 2021.

 

Comment 5: I would set separately boundary conditions for flow and heat equations.

Response 5: In our study, the flow and heat transport processes are fully coupled through Darcy flow and the energy conservation equation, and therefore, they share consistent physical boundary conditions. This problem is solved successfully by SHEMAT-Suite (Keller et al. 2020). The physical process of coupling requires integrating the boundary conditions for the flow and heat equations. Hence, we use a unified boundary-condition framework that simultaneously governs both flow and heat transport, and satisfactorily integrates them in solving the coupled governing equations.

References

Keller, J.; Rath, V.; Bruckmann, J.; Mottaghy, D.; Clauser, C.; Wolf, A., ... Klitzsch, N. SHEMAT-Suite: An open-source code for simulating flow, heat and species transport in porous media. SoftwareX 2020, 12, 100533. https://doi.org/10.1016/j.softx.2020.100533

 

Comment 6: Authors refer to pumping and injection wells. Section 4.2.1 should be linked to Section 3.2, in my opinion.

Response 6: We thank the reviewer for this suggestion. The structure of the presentation in the manuscript has been reevaluated. In the current version, Section 4.2.1 is based on the injection- and pumping-well configuration already described in Section 3.2. Because the information in Section 3.2 provides the necessary background for Section 4.2.1, we considered the connection between the two sections sufficiently established in the current structure. Therefore, we prefer to keep the existing organization while ensuring the text reads clearly.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper analyzes two approaches to estimating key parameters characterizing flow in porous media, such as permeability, thermal conductivity, and porosity distribution. These parameters play a key role in studying the reservoir's capacitive properties and help develop an optimal operating strategy. The authors focus on the application of two of the most common methods—Monte Carlo Simulation (MCS) and Ensemble Kalman Filter (EnKF). They rely on synthetic data obtained through numerical simulations. Their numerical simulations rely on Darcy's law and the equation of continuity, which uses the Oberbeck-Boussinesq approximation. To describe the heat transfer process, this paper uses the approach developed in the publications (Clauser, C., (Ed.), Numerical Simulation of Reactive Flow in Hot Aquifers. SHEMAT and PROCESSING SHEMAT, Springer, New York, 2003) and (Beardsmore, G. R.; Cull, J. P., Crustal Heat Flow, Cambridge University Press, Cambridge, U.K., 2001).
Monte Carlo simulation was used to conduct numerical modeling to obtain synthetic data subsequently used to test the proposed approaches to parameter estimation. In this paper, the authors, as noted, utilized Sequential Gaussian Simulation (SGSIM) from the Geostatistical Software Library GSLIB to generate a number of realizations. The authors state that their objective is to assess the accuracy of the inverse models that could reproduce the target hydrothermal parameter fields (either true or reference fields). As I already noted, the parameter reconstruction was based on two main methods: Monte Carlo Simulation (MCS) and Ensemble Kalman Filter (EnKF). The analysis showed a slight advantage of the latter.
Unfortunately, the authors do not provide any results on parameter estimation for in-situ observations applicable to specific reservoirs. In my opinion, such results would significantly enhance the relevance of this publication.

Author Response

Comment 1: This paper analyzes two approaches to estimating key parameters characterizing flow in porous media, such as permeability, thermal conductivity, and porosity distribution. These parameters play a key role in studying the reservoir's capacitive properties and help develop an optimal operating strategy. The authors focus on the application of two of the most common methods - Monte Carlo Simulation (MCS) and Ensemble Kalman Filter (EnKF). They rely on synthetic data obtained through numerical simulations. Their numerical simulations rely on Darcy's law and the equation of continuity, which uses the Oberbeck-Boussinesq approximation. To describe the heat transfer process, this paper uses the approach developed in the publications (Clauser, C., (Ed.), Numerical Simulation of Reactive Flow in Hot Aquifers. SHEMAT and PROCESSING SHEMAT, Springer, New York, 2003) and (Beardsmore, G. R.; Cull, J. P., Crustal Heat Flow, Cambridge University Press, Cambridge, U.K., 2001).
Monte Carlo simulation was used to conduct numerical modeling to obtain synthetic data subsequently used to test the proposed approaches to parameter estimation. In this paper, the authors, as noted, utilized Sequential Gaussian Simulation (SGSIM) from the Geostatistical Software Library GSLIB to generate a number of realizations. The authors state that their objective is to assess the accuracy of the inverse models that could reproduce the target hydrothermal parameter fields (either true or reference fields). As I already noted, the parameter reconstruction was based on two main methods: Monte Carlo Simulation (MCS) and Ensemble Kalman Filter (EnKF). The analysis showed a slight advantage of the latter.
Unfortunately, the authors do not provide any results on parameter estimation for in-situ observations applicable to specific reservoirs. In my opinion, such results would significantly enhance the relevance of this publication.

Response 1: We thank the reviewer for the thoughtful summary and for highlighting the importance of parameter estimation using in-situ observations. We fully agree that applying the proposed approaches to real reservoir data would further enhance the practical relevance of the study. However, the purpose of this paper is to conduct a controlled assessment of MCS and EnKF under the assumption that the true parameter fields are known. Synthetic data allows us to rigorously quantify performance, model uncertainty, and evaluate reconstruction accuracy without the additional complexities and unknowns from field data.

The inclusion of in-situ case studies is an important direction for future work, and we are currently preparing a follow-up research that applies our inversion framework to real hydrothermal reservoir data. We believe this staged approach - first establishing method performance under controlled synthetic conditions, followed by real-world validation - provides a clearer foundation for understanding the strengths and limitations of the techniques.

We appreciate the reviewer’s valuable suggestion and emphasize this point in the discussion to clarify the intended scope of the present study (see lines 599-602).

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Authors replied wisely to my comments. 

The paper has been improved in its contents and I feel satisfied by their review. 

Just pay attention to some typos (for instance, authors name in reference 22 is wrong). A careful proofreading is needed