the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Feature scale and identifiability: How much information do point hydraulic measurements provide about heterogeneous head and conductivity fields?
Abstract. We systematically investigate how the spacing and type of point measurements impacts the scale of subsurface features that can be identified by groundwater flow model calibration. To this end, we consider the optimal inference of spatially heterogeneous hydraulic conductivity and head fields based on three kinds of point measurements that may be available at monitoring wells: of head, permeability, and groundwater speed. We develop a general, zonation-free technique for Monte Carlo (MC) study of field recovery problems, based on Karhunen-Loève (K-L) expansions of the unknown fields whose coefficients are recovered by an analytical, continuous adjoint-state technique. This allows unbiased sampling from the space of all possible fields with a given correlation structure and efficient, automated gradient-descent calibration. The K-L basis functions have a straightforward notion of period, revealing the relationship between feature scale and reconstruction fidelity, and they have an a priori known spectrum, allowing for a non-subjective regularization term to be defined. We perform automated MC calibration on over 1100 conductivity-head field pairs, employing a variety of point measurement geometries and evaluating the mean-squared field reconstruction accuracy, both globally and as a function of feature scale. We present heuristics for feature scale identification, examine global reconstruction error, and explore the value added by both the groundwater speed measurements and by two different types of regularization. We find that significant feature identification becomes possible as feature scale exceeds four times measurement spacing and identification reliability subsequently improves in a power law fashion with increasing feature scale.
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Status: open (until 18 Jul 2024)
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CC1: 'Comment on egusphere-2024-337', Giacomo Medici, 13 Mar 2024
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General comments
Good theoretical research with implication on groundwater flow modelling and the engineering of the reservoirs where the geological flow heterogeneities are of paramount importance. Please, follow my guidance to improve the manuscript.
Specific comments
Line 6. I suggest “This technique allows unbiased”. Add the word “technique”.
Lines 20-22. Mini-permeameter, slug, packer and pumping tests can be also used to identify flow heterogeneities and determine the hydraulic conductivity. Specify this point.
Line 22. “Point-to-point tracer tests” to detect flow heterogeneities. Please, add recent literature on the topic:
- Deleu, R., Frazao, S. S., Poulain, A., Rochez, G., & Hallet, V. (2021). Tracer Dispersion through Karst Conduit: Assessment of Small-Scale Heterogeneity by Multi-Point Tracer Test and CFD Modeling. Hydrology, ((4)
- Lorenzi, V., Banzato, F., Barberio, M. D., Goeppert, N., Goldscheider, N., Gori, F., Lacchini A., Manetta M., Medici G., Rusi S., Petitta, M. (2024). Tracking flowpaths in a complex karst system through tracer test and hydrogeochemical monitoring: Implications for groundwater protection (Gran Sasso, Italy). Heliyon, 10(2)
- Poulain, A., Rochez, G., Van Roy, J.P., Dewaide, L., Hallet, V. and De Sadelaer, G., 2017. A compact field fluorometer and its application to dye tracing in karst environments. Hydrogeology Journal, 25
Lines 48-88. The literature on the topic is much broader.
Line 84. Disclose the 3 to 4 specific objectives of your research by using numbers (e.g., i, ii and iii).
Line 90. “Feature scale”. This expression is difficult to understand. Do you mean “observation scale”?
Line 199. “theoretical observations”. Can you re-call the key equations instead?
Line 501. “Other geophysical fields”. (i) remind to the reader that the principal implications are in the calibration of groundwater flow models, (ii) which other implications/applications in geophysics? You can look at my general comments.
Line 514. Please, add relevant literature on the topic.
Figures and tables
Figure 1. Very nice figure, but it needs some changes. (i) Make the rectangles closer, (ii) make words and numbers larger, and (iii) thicker the black nodes.
Figures 2-4. Make words and figures larger on x and y axes.
Citation: https://doi.org/10.5194/egusphere-2024-337-CC1 -
RC1: 'Comment on egusphere-2024-337', Anonymous Referee #1, 19 Apr 2024
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The authors present an approach for using Karhunen-Loeve explanations to estimate spatial variability of properties and system states on observations of head, permeability, and velocity magnitude, both with and without regularization. They apply the adjoint approach for efficiently determining gradients of the objective function, allowing for more efficiency and, presumably, high-order approximations. The methods are explained thoroughly, including the assumptions and simplifications. I just have a few minor comments.
1. A substantial amount (about 20%) of the paper was devoted to developing the adjoint equation and the adjoint-based form of the derivatives of the objective function. Since that was just a tool to be used in the analysis, the detailed development seemed to detract from the main focus of the paper. Could Section 2 be moved to an appendix?
2. Figure 1 is very helpful as an example of the recovery of spatial distributions of head and permeability through the approach presented in the paper. All other figures show "error" between measures and fits, so Figure 1 is very useful as means of showing the reader the intermediate step. However, Figure 1 is out of place - it appears on p. 11, but it isn't mentioned in the text until page 20. Also more explanation can be provided to make the link between what appears in Figure 1 and how that is related to the data point plotted in the other figures. I wonder also why the two subplots of Figure 1 use different true ln K fields. If subplot a has regularization and subplot b does not, it would be more informative to see that results from the same true ln K field so that the reader can see the benefit of regularization. It would also be helpful to see the numerical value of "error" (the quantities that are plotted in the other figures) to get a sense for where these fits fall in those plots, compared to all other realizations that appear in the plots in Figures 2 and above.
3. I would like to see some explanation of the practicality of this method. What measurement density is needed? Is it different for measurements of different quantities?
Citation: https://doi.org/10.5194/egusphere-2024-337-RC1 -
RC2: 'Comment on egusphere-2024-337', Anonymous Referee #2, 17 Jun 2024
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This is a well written paper (until the discussion section) on the study of the worth of different types of data for inverse modeling in hydrogeology. Its aim is to try to identify the necessary sampling density in order to identify the underlying hydraulic conductivity structure. While I enjoyed reading the paper and the detailed derivations of the different equations, I had a hard time trying to find the novelty of the work. This question has been studied as early as in the late 1990s, and the application proposed, while very elegant, is limited to a very narrow and little interesting case of steady groundwater flow in an aquifer with prescribed head at the boundary and with an underlying hydraulic conductivity field drawn from a multi Gaussian random function with an exponential covariance. For the paper to be worth to be considered for publication it should have addressed a tougher problem: transient, non-Gaussian field, larger variance, anisotropic, generic boundary conditions, addressing the issue of measurements taken at different supports. As is, the a paper is a very nice mathematical exercise with little practical interest.
As a minor point, there was an earlier benchmarking of inverse models not mentioned in the opening, which addressed a tougher problem than the one discussed here published in Water Resources Research by Zimmerman et al. In 1988.
Citation: https://doi.org/10.5194/egusphere-2024-337-RC2 -
CC2: 'Reply on RC2', James Hambleton, 17 Jun 2024
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Thanks heartily for the feedback. Can the Reviewer be more specific about what work in the early 1990's endeavoured (and perhaps succeeded) in addressing the aim of the paper, so succinctly identified in the Reviewer's appraisal?
Citation: https://doi.org/10.5194/egusphere-2024-337-CC2 -
CC3: 'Reply on CC2', James Hambleton, 17 Jun 2024
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Apologies for the substantive error in my previous reply with respect to late 1990s versus early 1990s. Perhaps more importantly, apologies for the typographical error of writing "1990's" instead of "1990s," an error now immortalised on this platform.
Citation: https://doi.org/10.5194/egusphere-2024-337-CC3
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CC3: 'Reply on CC2', James Hambleton, 17 Jun 2024
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CC2: 'Reply on RC2', James Hambleton, 17 Jun 2024
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