the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Jun 2026
Mechanistic modeling of the impact of rainfall pumping on soil solute remobilization into runoff
Abstract. The remobilization of solutes from the soil to surface runoff is a critical process for surface water contamination, yet it remains challenging to model mechanistically. This study explores a novel mechanistic modeling approach that explicitly accounts for local advective transport driven by pressure fluctuations induced by raindrop impacts (“rainfall pumping”) on the soil surface. We coupled the HYDRUS-1D model with a time variable, surface pressure head boundary condition, which combines runoff depth and the semi empirical, sinusoidal rainfall pumping wave of Higashino and Stefan (2014) whose amplitude and frequency are estimated from runoff depth and rainfall intensity. This boundary condition was tested for the first time against experimental data on saturated soils using two benchmark studies: one for model development and another for independent verification. Two additional datasets were used to verify the physical consistency of the pressure head values estimated by the pumping formulation. The results were compared with those from a conventional “no-pumping”, fixed runoff depth boundary condition. The rainfall-pumping boundary condition improved estimates of the final soil bromide concentration profile, solute extraction depth, and total remobilized mass, with the pumping model achieving the best predictions across all soils (final concentration profile NSE = 0.997, 0.977, 0.620, 0.881 for the sandy loam, loam and clay soils in the development phase and the loamy soil the verification phase, respectively). The results support the importance of accounting for the physical rainfall pumping process to explain enhanced solute extraction from soils during storms and to avoid unrealistic transport parametrization.
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Status: open (until 11 Sep 2026)
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RC1: 'Comment on egusphere-2026-3253', Anonymous Referee #1, 08 Jul 2026
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AC1: 'Reply on RC1Response to Reviewer Comments We thank the reviewer for the insightful comments. Below we address the conceptual and physical concerns raised, detailing how we will enhance the manuscript to clarify the physical basis of the model. Major Comme', Lucie Guertault, 23 Jul 2026
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Response to Reviewer Comments
We thank the reviewer for the insightful comments. Below we address the conceptual and physical concerns raised, detailing how we will enhance the manuscript to clarify the physical basis of the model.
Major Comments
- The physical mechanism linking oscillatory pressure to irreversible solute transport is not demonstrated.
Response:
The mechanism that causes irreversible solute transport at the upper boundary is the mass removal by surface runoff. In our model, we adopt the common assumption that surface runoff instantaneously removes any mass extracted from the soil matrix to the surface interface. This is implemented via the top boundary condition where solute concentration is maintained at zero (C=0), representing runoff continuously washing off the chemical at the interface.
As discussed around Line 194 in our manuscript, this assumption is physically consistent with the experimental setup of Ahuja and Lehman (1983), where the chemical breakthrough curve measured at the outlet of the system shows that the chemicals released at the surface are transported by the lateral surface runoff. Furthermore, as noted in the Conclusions (Line 520), we acknowledge the importance of the process of mass removal by runoff. We also recommend that future research from this work aims to explicitly couple the soil domain with overland flow hydrodynamic equations to refine this boundary interaction.
To address this comment, in the manuscript revision we will expand the Methodology and Discussion sections to explicitly articulate how the coupling of the oscillatory pressure head with surface runoff results in net solute transport from the soil surface because of instantaneous lateral transport with overland flow.
- The manuscript inherits assumptions from Higashino and Stefan (2014) without critically evaluating their validity.
Response:
We agree that the underlying assumptions of the Higashino and Stefan (2014) formulation must be critically evaluated. In fact, rather than accepting the assumptions we address and test these limitations in our manuscript. Specifically:
- Line 80: We explicitly state that this formulation has never been tested against experimental data before.
- Line 495: We identify and discuss that independent experimental datasets from Palmer (1965) and Yu and Hopkins (2018) question the inverse relationship between the sine wave amplitude (h0) and runoff depth (d) proposed by Higashino and Stefan (2014) and recommend that this assumption needs further study and refinement.
In addition, we want to emphasize that the primary objective of this manuscript is to provide a mechanistic proof of concept showing that treating rainfall-induced pumping as a dynamic boundary condition significantly improves soil transport predictions over the classical diffusion-only or empirical mixing-layer models. In the revision we will modify the introduction and Discussion to further emphasize that the Higashino and Stefan formulation was used as a starting point to test the modeling framework and further discuss the limitations of the formulation.
- Rainfall impacts are localized, stochastic, and spatially heterogeneous, while the model assumes a spatially uniform boundary condition.
Response:
Our goal is to propose an effective macroscopic, averaged boundary condition that parsimoniously represents the physical impact of rainfall pumping. For this study, we intentionally selected a 1D model to test if a parsimonious modeling approach can eventually be integrated into larger decision-support and watershed management tools. Capturing the precise 3D spatial and temporal stochasticity of individual raindrop impacts would require high-resolution multi-dimensional simulations, with limited transferability or predictive capability for practical, real-world applications.
To address this reviewer’s comment, in the revision we will add a section to the Discussion highlighting the implications of this 1D uniform simplification and defining the conditions under which it successfully approximates macroscopic reality.
- Is Richards equation appropriate for this frequency?
Response:
We thank the reviewer for bringing up this interesting point. Darcy’s law is valid when viscous forces dominate over inertial forces. Pressure pulsations can introduce an accelerating fluid force and we must verify that viscous forces still dominate.
Darcy’s law holds if the frequency of the pressure pulsations is less than the characteristic frequency (fc) of the porous medium, which corresponds to the threshold where fluid inertia can no longer be neglected (Johnson et al., J. Fluid Mech., 1987). The characteristic frequency is defined as:
fc=Φν/2πk
where Φ is the porosity, ν is the kinematic viscosity of water, and k is the intrinsic soil permeability, defined as k=Ksatν /g
In our application, Φ≈0.5, ν≈ 10-6 m2/s and Ksat ≈ 10-4 m/s, so k ≈ 10-11 m2.
We find that fc ≈ 8000 s-1 (Hz)The rainfall pumping forcing angular frequency calculated using Higashino and Stephan (2014) formula is ωh=6.48 rad/s (Table 5), which corresponds to a frequency of approximately 1.03 Hz. This forcing frequency is 3 orders of magnitude below the threshold value (fc ), which confirms that Darcy’s law is valid.
To address this comment, we will incorporate this response in the revised manuscript text.Minor Comments
Other rainfall-related mechanisms can also contribute to removal
Response:
We agree that alternative processes that contribute to the removal should be discussed. In the manuscript, we explicitly noted that surface physical modifications, such as the swelling of clay observed in Ahuja and Lehman experiments can alter Ksat near the interface (Line 492).
A preliminary assessment of other advective removal processes was done as part of this study. We identified surface flushing (caused by shear stress) and microtopographic pumping as other potential mechanisms for contaminant removal. Based on typical conditions observed in practical applications and in Ahuja and Lehman experiments (runoff depths from 0.1 - 1 cm, rainfall intensity from 5-100 mm/h, Ksat from 10-4 to 10-2 cm/s, runoff velocity from 1 - 10 cm/s , and a microtopographic height-to-depth ratio of 0.25 to 0.75, with a 5 cm wavelength), we estimated the order-of-magnitude of fluxes caused by each advective processes to be :
- Rainfall pumping fluxes: ∼ 10-4 to 1 cm/s, (Higashino & Stefan, Water. Resourc. Res, 2014 corrected formulation)
- Surface shear-driven fluxes: ∼ 10-6 to 10-2 cm/s,(Higashino et al., Water. Resourc. Res, 2009)
- Microtopographic pumping fluxes: ∼ 10-7 to 10-4 cm/s (Elliott & Brooks, Water. Resourc. Res, 1997)
Because the fluxes induced by rainfall pumping are dominant under these specific experimental boundary conditions, we focused exclusively on this mechanism in this study. We will revise the manuscript introduction to present these mechanisms and include the fluxes order of magnitudes to justify the focus of this study on the rainfall pumping.
Citation: https://doi.org/10.5194/egusphere-2026-3253-AC1
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AC1: 'Reply on RC1Response to Reviewer Comments We thank the reviewer for the insightful comments. Below we address the conceptual and physical concerns raised, detailing how we will enhance the manuscript to clarify the physical basis of the model. Major Comme', Lucie Guertault, 23 Jul 2026
reply
Data sets
digitized datasets Lucie Guertault and Rafael Muñoz-Carpena https://github.com/carpena1/pumping_repository
model outputs Lucie Guertault and Rafael Muñoz-Carpena https://github.com/carpena1/pumping_repository
Model code and software
R scripts to run simulations Lucie Guertault and Rafael Muñoz-Carpena https://github.com/carpena1/pumping_repository
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This manuscript presents a numerical investigation of rainfall-induced enhancement of solute transport across the soil-water interface. The manuscript is generally well written, clearly organized, and the numerical implementation appears careful and comprehensive. The validation against independent experimental datasets is also a strength.
However, I have substantial concerns regarding the physical basis of the proposed mechanism. The manuscript builds upon the rainfall-pumping hypothesis originally proposed by Higashino and Stefan (2014), treating it as the governing physical process responsible for the observed enhancement of solute transport. While the numerical framework developed here is considerably more sophisticated than previous studies, the underlying physical assumptions remain largely unchanged and, in my opinion, have not been sufficiently justified.
My primary concern is therefore not with the numerical implementation, but with the physical assumptions on which the entire model is based.
Major Comments
The central assumption of the manuscript is that rainfall-induced pressure oscillations at the soil surface generate enhanced upward solute transport. While raindrop impacts undoubtedly generate transient pressure fluctuations, it does not automatically follow that periodic pressure forcing produces net solute exchange. In a porous medium, oscillatory Darcy flow is generally reversible, and purely periodic motion does not necessarily generate irreversible transport. The manuscript assumes that oscillatory flow directly translates into enhanced exchange but does not identify the physical mechanism responsible for breaking this reversibility. For example, no discussion is provided regarding the possible roles of pore-scale heterogeneity, hysteresis, nonlinear hydraulic conductivity, diffusion during oscillatory motion, or other processes that could produce net mass transfer. Since this assumption underlies the entire modeling framework, I believe it requires considerably stronger physical justification.
The physical model is largely based on Higashino and Stefan (2014), where rainfall impacts are represented as a spatially uniform sinusoidal pressure boundary condition acting on the soil surface. That earlier work presented the mechanism primarily as a conceptual hypothesis, supported by order-of-magnitude arguments. Several key assumptions (including the conversion of raindrop kinetic energy into a uniform pressure signal and the interpretation of oscillatory pore-water velocities as an effective exchange velocity) were introduced with relatively limited physical validation. The present manuscript adopts these assumptions as established without discussing their limitations or providing additional evidence that they are appropriate under the conditions considered. Because the conclusions depend directly on these assumptions, I believe their applicability should be discussed much more critically.
Rainfall impacts are localized, stochastic, and spatially heterogeneous. The model instead assumes a spatially uniform sinusoidal pressure boundary over the entire soil surface. Although such a simplification may be useful for analytical development, it is not obvious that it adequately represents the actual forcing generated by individual raindrop impacts. The manuscript would benefit from a discussion of the implications of this simplification and the conditions under which it may be expected to approximate reality.
Richards assumes quasi-static Darcy flow. The applicability of Richards' equation under (relatively) high-frequency pressure oscillations raises questions: local inertial effects, dynamic capillary pressure, non-equilibrium flow. The manuscript never discusses whether Darcy/Richards remains valid at these oscillation frequencies.
One small comment:
The discussion should acknowledge that other rainfall-related mechanisms (surface flushing, preferential flow activation, increased hydraulic connectivity…) may also contribute to the observed enhancement of solute transport.
Recommendation
The manuscript addresses an interesting problem and presents a careful numerical implementation. However, because the conclusions rely heavily on a physical mechanism that I do not believe has been adequately established, I find it difficult to assess the predictive capability of the model without a stronger justification of its underlying assumptions. I therefore recommend revising with emphasis on strengthening the discussion of the physical basis of the rainfall-pumping mechanism, clarifying its assumptions and limitations, and distinguishing between demonstrated physics and modeling hypotheses.