the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 03 Feb 2026
A unified physically based recession model reveals contrasting functioning of monsoon and Mediterranean karst aquifers
Abstract. Karst aquifers supply drinking water to hundreds of millions of people but remain among the least understood freshwater systems because of their strongly heterogeneous conduit–matrix structure. Spring recession analysis is widely used to infer karst storage dynamics, yet most existing models either treat different parts of the hydrograph separately or rely on empirical formulations with limited physical interpretability. Here we derive a unified analytical model for karst spring recession by combining turbulent conduit flow, represented by a Forchheimer-type relationship, with linear drainage from a porous matrix. The resulting governing equation, dQ/dt = -αQn + γe-λt, simultaneously describes nonlinear conduit depletion and delayed conduit–matrix exchange. The parameters α, n, γ and λ can be related to aquifer properties, reflecting conduit drainage efficiency, flow nonlinearity, and the magnitude and timescale of matrix drainage.We apply the model to hourly discharge records (2013–2023) from climatically and geologically contrasting karst systems in monsoon-influenced southwest China and Mediterranean central Italy. Event-based calibration shows that the unified recession model reproduces complete recession limbs and consistently outperforms a classical dual-reservoir benchmark. Beyond goodness of fit, the inferred parameter patterns reveal systematic regional contrasts: Chinese spring exhibits higher nonlinear exponents (n = 2) and larger α, indicating strongly turbulent, conduit-dominated drainage with short memory, whereas Italian spring is characterised by n = 1 and negligible γ, consistent with large, slowly draining matrix storage. Using regional-average parameter sets without further calibration, the unified recession model also reproduces independent multi-week drought recessions in both regions, demonstrating that these parameter contrasts are robust and transferable within each hydrogeological setting. Mapping events in the (n, S~) parameter space delineates distinct functional regimes of karst aquifers that can be related to differences in drought resilience and baseflow support. The unified recession model thus provides a diagnostic tool to infer karst aquifer functioning from discharge data alone, supporting water-resource assessment and climate-impact studies in data-scarce regions.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2025-6044', Anonymous Referee #1, 23 Feb 2026
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AC1: 'Reply on RC1', Liangjie Zhao, 27 Feb 2026
We thank the reviewer for the constructive comments and the positive assessment of our preprint. We respond point-by-point to clarify our current implementation and interpretation; the requested clarifications, sensitivity checks, and unit improvements will be incorporated in the revised manuscript submitted after the Discussion.
For Comment 1: We confirm that we enforced physically motivated bounds during calibration: α≥ 0, n ≥ 1 (n<1 is not physically meaningful), and γ ≥ 0 (and λas a positive timescale parameter). Under these constraints, the Italian events converging to n=1 and γ≈0 emerge naturally as the linear-reservoir limit of the model rather than being imposed. We agree that when γ→0, the exchange term becomes negligible and λ is practically unidentifiable. In the revision, we will explicitly state the imposed parameter bounds in Sect. 3.2 and add a note to Table 2 explaining that is practically unidentifiable when .
For Comment 2: We agree and will add a brief sensitivity check (Table S1 in the Supplement) by repeating event selection with modest changes in the initial-discharge threshold (±30%) and dry-spell length (7–14 days). Preliminary tests indicate that while the number of selected events changes, the key regional contrasts and the main conclusions remain unchanged.
For Comment 3: In our framework, n = 1 corresponds to the linear-reservoir limit, whereas many strongly curved early-time recessions are well described by n close to 2 (quadratic-type nonlinearity). We therefore used n = 1.5 as a simple operational midpoint to separate “near-linear” from “clearly nonlinear” recession behavior in the regime diagram. Similarly, in the revision we will add a one-sentence physical justification in Sect. 4.3: “We interpret as a transition because compares the time-integrated exchange contribution to the event discharge scale , so indicates that exchange and drainage are of comparable magnitude.”
For Comment 4: We agree and will add a short plain-language interpretation paragraph in Sect. 2.1. “ In the unified recession model, the exponent controls the degree of nonlinearity in conduit drainage: reduces to the linear-reservoir limit, whereas produces a more strongly curved recession consistent with nonlinear conduit resistance. The coefficient sets the overall drainage efficiency of the conduit system. The exchange term represents a delayed matrix-to-conduit contribution whose initial magnitude is set by , while is the characteristic timescale over which this exchange decays, reflecting how quickly matrix exchange relaxes during recession.”
For Comment 5: We clarify that θ=0.2 is used primarily as an illustrative operational threshold rather than a universal ecological threshold. In the revision we will state explicitly that θ is user-defined and can be selected for local needs.
For Comment 6: We will ensure consistent notation between Sect. 2 and Sect. 4.3 and state units when variables are first introduced.
Citation: https://doi.org/10.5194/egusphere-2025-6044-AC1
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AC1: 'Reply on RC1', Liangjie Zhao, 27 Feb 2026
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RC2: 'Comment on egusphere-2025-6044', Stefan Hergarten, 11 Mar 2026
This manuscript presents a new model for the recession of spring hydrographs. It is mostly well written and brings some novelty. But to be honest, I expected a bit more from a "unified physically based recession" model and from the scientific statement "reveals contrasting functioning of monsoon and Mediterranean karst aquifers."
(A) Scientific results
If I read Sect. 4.1 correctly, the main result is that the Chinese springs are dominated by the power-law part with n = 2 initially, while the Italian springs are like linear reservoirs (decaying exponentially). So some springs show a strongly nonlinear recession characteristics and some do not. Is this really a new finding?
(B) The model itself
(1) The results of Sect 4.1 and the details given in Table 2 suggest that the model performance is worse than that of the old model of Mangin (1975) for the Chinese springs. For me, this means that the new model can only be justified by its better/clearer physical basis. As discussed below, I am only partly convinced that this is really the case.
(2) The values of lambda show a strong variation for the Chinese springs. This is a problem of most of the models in this field. The exponential tail differs from event to event, which is difficult to understand and somehow tells that the models do not capture the dynamics well. It seems not to be worse than in other models here, but also not better. Similar to the previous point, this means for me that the physical basis must be better/clearer.
(3) I am not completely happy with the derivation of the model equation in its full form (Appendix A). First, I was not able to follow the derivation completely. By combining Eqs. A1-A3, I arrived at
dV/dt = kappa*L^phi*(a*Q+b*Q^2)^(phi-1)*(a+2*b*Q)*dQ/dt = -Q + Q_ex.
But then, I only obtain the first term in Eq. A7 and was not able to reproduce where the second term in D(Q) (S_m/...) is coming from.
(4) In lines 317-322, an approximation for the long-term recession is made with the result that the exchange term decreases exponentially in the late recession phase. Immediately after this, however, this approximation is assumed to be valid during the entire recession. I would not claim that this is fundamentally wrong, but I am not fully convinced why this should be the case.
(5) Finally, the model is simplified in a quite sloppy way to the form of Eq. 1/A8, which is basically the model proposed by Wittemberg (1999) (according to the list in Table 1) extended by an exponential tail. The full derivation and the simplifications do not convince me fully that writing the rate of change in discharge dQ/dt as a sum in this way is really a "physically-based" approach.
To be clear, this is still a nice piece of work in my opinion. There is still a need for better and more physically-based recession models since this is essential for interpreting recession curves. In this sense, the proposed model is definitely not useless in my opinion. However, I feel that the paper neither keeps the promise of a "unified physically based recession model" and the scientific claim made in the title. So my recommendation would be to make the physical basis clearer and to be more realistic concerning the model and the scientific results.
In any case, I hope that my comments will help to improve the paper further.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere-2025-6044-RC2 -
AC2: 'Reply on RC2', Liangjie Zhao, 17 Mar 2026
We thank the referee for the careful, constructive, and encouraging review of our manuscript. We appreciate the positive assessment that the manuscript is generally well written and contains some novelty, and we are equally grateful for the reviewer’s thoughtful concerns regarding the strength of our physical claims, the interpretation of the scientific results, and the clarity of the derivation. These comments are very helpful and will help us improve the manuscript substantially. Below we respond point by point and indicate how we plan to address these issues in the revised manuscript.
Author response to Referee comment of (A):
We thank the referee for this important comment. We agree that the existence of strongly nonlinear versus near-linear recession behavior is not, by itself, entirely new. Our intention was not to claim novelty in the mere existence of this contrast, but rather to show that both behaviors can be represented and interpreted within a single unified framework using a common parameterization. In the revised manuscript, we will present the regional contrast more cautiously as a diagnostic interpretation supported by the fitted parameters, rather than as a major standalone discovery. And we will emphasize that the contribution of the study lies in the unified recession framework and in the interpretation of recession events within a common parameter space, rather than in claiming that nonlinear versus linear recession behavior has been discovered here for the first time.
Author response to Referee comment1 of (B):
We agree that the URM “consistently outperforms” the benchmark is too strong and will be revised. In the revised manuscript, we will state more carefully that the URM achieves competitive or broadly comparable performance, while offering a compact framework with clearer process interpretation. We agree with the referee that the contribution of the model should not be justified primarily by benchmark superiority.
Author response to Referee comment2 of (B):
We agree that this point deserves a more careful discussion. In the revised manuscript, we will clarify that should be interpreted as an event-scale effective parameter rather than as a fixed aquifer constant. We will also discuss that its variability may reflect changing antecedent storage conditions, varying conduit–matrix disequilibrium, and the limitations of the present single-timescale lumped approximation. We agree that this event-to-event variability is an important limitation of many recession models, including ours, and we will make this clearer in the discussion.
Author response to Referee comment3 of (B):
We agree that the derivation in Appendix A is currently too compact and that the origin of the second term in is not sufficiently transparent. In the revised manuscript, we will expand the intermediate algebraic steps and clarify more explicitly how the matrix-storage contribution enters the damping term . In particular, we will distinguish more clearly between the conduit-storage contribution and the matrix-storage contribution in the coupled derivation.
Author response to Referee comment4 of (B):
We agree that this transition is not sufficiently justified in the current text. In the revised manuscript, we will distinguish more clearly between the late-time asymptotic argument used to motivate the exponential exchange term and the subsequent event-scale approximation in which this form is adopted over the full recession. We will make explicit that this is a physically motivated closure assumption introduced for tractability, rather than an exact statement valid over all recession stages. In this way, the role of the approximation will be stated more clearly, and the limits of its validity will be more transparent.
Author response to Referee comment5 of (B):
We thank the referee for this thoughtful comment and agree that the phrase “physically based” is too strong in view of the simplifying assumptions used in the derivation. We also agree that the final form of the model is structurally close to earlier nonlinear recession formulations, including a Wittenberg-type power-law recession augmented by an exponential tail. In the revised manuscript, we will therefore adopt more cautious wording and describe the URM as a physically motivated or physics-informed lumped recession model. We will also clarify that the final additive form is the result of asymptotic reasoning and simplification, and should be understood as a tractable, process-interpretable approximation rather than a fully exact mechanistic derivation.
We hope that these planned revisions address the referee’s concerns and improve the manuscript accordingly.
Citation: https://doi.org/10.5194/egusphere-2025-6044-AC2
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AC2: 'Reply on RC2', Liangjie Zhao, 17 Mar 2026
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This preprint proposes a unified physically based recession model that combines nonlinear conduit depletion with an exponentially decaying matrix-to-conduit exchange term. The manuscript well writting and I think it maybe suitable for publishing after addressing a few number of clarifications and robustness checks that would further strengthen reproducibility and interpretation.