Articles | Volume 26, issue 6
https://doi.org/10.5194/hess-26-1615-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/hess-26-1615-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
25 Mar 2022
Research article |
| 25 Mar 2022
| 25 Mar 2022
Stepping beyond perfectly mixed conditions in soil hydrological modelling using a Lagrangian approach
Alexander Sternagel
CORRESPONDING AUTHOR
Institute of Water Resources and River Basin Management – Hydrology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Ralf Loritz
Institute of Water Resources and River Basin Management – Hydrology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Brian Berkowitz
Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, Israel
Erwin Zehe
Institute of Water Resources and River Basin Management – Hydrology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Related authors
Alexander Sternagel, Ralf Loritz, Julian Klaus, Brian Berkowitz, and Erwin Zehe
Hydrol. Earth Syst. Sci., 25, 1483–1508, https://doi.org/10.5194/hess-25-1483-2021, https://doi.org/10.5194/hess-25-1483-2021, 2021
Short summary
Short summary
The key innovation of the study is a method to simulate reactive solute transport in the vadose zone within a Lagrangian framework. We extend the LAST-Model with a method to account for non-linear sorption and first-order degradation processes during unsaturated transport of reactive substances in the matrix and macropores. Model evaluations using bromide and pesticide data from irrigation experiments under different flow conditions on various timescales show the feasibility of the method.
Dan Elhanati, Erwin Zehe, Ishai Dror, and Brian Berkowitz
EGUsphere, https://doi.org/10.5194/egusphere-2025-3365, https://doi.org/10.5194/egusphere-2025-3365, 2025
This preprint is open for discussion and under review for Hydrology and Earth System Sciences (HESS).
Short summary
Short summary
Measurements of water isotopes are often used to estimate water transit time distributions and aquifer storage thickness in catchments. However, laboratory-scale measurements show that water isotopes exhibit transport behavior identical to that of inert chemical tracers rather than of pure water. The measured mean tracer and apparent mean water velocities are not necessarily equal; recognition of this inequality is critical when estimating catchment properties such as aquifer storage thickness.
Karl Nicolaus van Zweel, Laurent Gourdol, Jean François Iffly, Loïc Léonard, François Barnich, Laurent Pfister, Erwin Zehe, and Christophe Hissler
Earth Syst. Sci. Data, 17, 2217–2229, https://doi.org/10.5194/essd-17-2217-2025, https://doi.org/10.5194/essd-17-2217-2025, 2025
Short summary
Short summary
Our study monitored groundwater in a Luxembourg forest over a year to understand water and chemical changes. We found seasonal variations in water chemistry, influenced by rainfall and soil interactions. These data help predict environmental responses and manage water resources better. By measuring key parameters like pH and dissolved oxygen, our research provides valuable insights into groundwater behaviour and serves as a resource for future environmental studies.
Judith Nijzink, Ralf Loritz, Laurent Gourdol, Davide Zoccatelli, Jean François Iffly, and Laurent Pfister
Earth Syst. Sci. Data Discuss., https://doi.org/10.5194/essd-2024-482, https://doi.org/10.5194/essd-2024-482, 2025
Preprint under review for ESSD
Short summary
Short summary
The CAMELS-LUX dataset (Catchment Attributes and MEteorology for Large-sample Studies – LUXembourg) contains hydrologic, meteorologic and thunderstorm formation relevant atmospheric time series of 56 Luxembourgish catchments (2004–2021). These catchments are characterized by a large physiographic variety on a relatively small scale in a homogeneous climate. The dataset can be applied for (regional) hydrological analyses.
Svenja Hoffmeister, Sibylle Kathrin Hassler, Friederike Lang, Rebekka Maier, Betserai Isaac Nyoka, and Erwin Zehe
EGUsphere, https://doi.org/10.5194/egusphere-2025-1719, https://doi.org/10.5194/egusphere-2025-1719, 2025
Short summary
Short summary
Combining trees and crops in agroforestry systems can potentially be a sustainable option for agriculture facing climate change impacts. We used methods from soil science and hydrology to assess the effect of adding gliricidia trees to maize fields, on carbon content, soil properties and water availability. Our results show a clear increase in carbon contents and effects on physical soil characteristics and water uptake and retention as a consequence of the agroforestry treatment.
Eduardo Acuña Espinoza, Frederik Kratzert, Daniel Klotz, Martin Gauch, Manuel Álvarez Chaves, Ralf Loritz, and Uwe Ehret
Hydrol. Earth Syst. Sci., 29, 1749–1758, https://doi.org/10.5194/hess-29-1749-2025, https://doi.org/10.5194/hess-29-1749-2025, 2025
Short summary
Short summary
Long short-term memory (LSTM) networks have demonstrated state-of-the-art performance for rainfall-runoff hydrological modelling. However, most studies focus on predictions at a daily scale, limiting the benefits of sub-daily (e.g. hourly) predictions in applications like flood forecasting. In this study, we introduce a new architecture, multi-frequency LSTM (MF-LSTM), designed to use inputs of various temporal frequencies to produce sub-daily (e.g. hourly) predictions at a moderate computational cost.
Evgeny Shavelzon, Erwin Zehe, and Yaniv Edery
EGUsphere, https://doi.org/10.22541/essoar.173687429.91307309/v1, https://doi.org/10.22541/essoar.173687429.91307309/v1, 2025
Short summary
Short summary
We analyze how chemical reactions and fluid movement interact in porous materials, focusing on how water paths form in underground environments. Using a thermodynamic approach, we track energy dissipation and entropy changes to understand this process. Over time, water channels become more defined, reducing chemical mixing and energy loss. Eventually, the system stabilizes, with flow concentrated in efficient pathways, minimizing further reactions and energy use.
Maria Staudinger, Anna Herzog, Ralf Loritz, Tobias Houska, Sandra Pool, Diana Spieler, Paul D. Wagner, Juliane Mai, Jens Kiesel, Stephan Thober, Björn Guse, and Uwe Ehret
EGUsphere, https://doi.org/10.5194/egusphere-2025-1076, https://doi.org/10.5194/egusphere-2025-1076, 2025
Short summary
Short summary
Four process-based and four data-driven hydrological models are compared using different training data. We found process-based models to perform better with small data sets but stop learning soon, while data-driven models learn longer. The study highlights the importance of memory in data and the impact of different data sampling methods on model performance. The direct comparison of these models is novel and provides a clear understanding of their performance under various data conditions.
Eduardo Acuña Espinoza, Ralf Loritz, Frederik Kratzert, Daniel Klotz, Martin Gauch, Manuel Álvarez Chaves, and Uwe Ehret
Hydrol. Earth Syst. Sci., 29, 1277–1294, https://doi.org/10.5194/hess-29-1277-2025, https://doi.org/10.5194/hess-29-1277-2025, 2025
Short summary
Short summary
Data-driven techniques have shown the potential to outperform process-based models in rainfall–runoff simulations. Hybrid models, combining both approaches, aim to enhance accuracy and maintain interpretability. Expanding the set of test cases to evaluate hybrid models under different conditions, we test their generalization capabilities for extreme hydrological events.
Sanika Baste, Daniel Klotz, Eduardo Acuña Espinoza, Andras Bardossy, and Ralf Loritz
EGUsphere, https://doi.org/10.5194/egusphere-2025-425, https://doi.org/10.5194/egusphere-2025-425, 2025
Short summary
Short summary
This study evaluates the extrapolation performance of Long Short-Term Memory (LSTM) networks in rainfall-runoff modeling, specifically under extreme conditions. The findings reveal that the LSTM cannot predict discharge values beyond a theoretical limit, which is well below the extremity of its training data. This behavior results from the LSTM's gating structures rather than saturation of cell states alone.
Ashish Manoj J, Ralf Loritz, Hoshin Gupta, and Erwin Zehe
Hydrol. Earth Syst. Sci. Discuss., https://doi.org/10.5194/hess-2024-375, https://doi.org/10.5194/hess-2024-375, 2024
Revised manuscript under review for HESS
Short summary
Short summary
Traditional hydrological models typically operate in a forward mode, simulating streamflow and other catchment fluxes based on precipitation input. In this study, we explored the possibility of reversing this process—inferring precipitation from streamflow data—to improve flood event modelling. We then used the generated precipitation series to run hydrological models, resulting in more accurate estimates of streamflow and soil moisture.
Ralf Loritz, Alexander Dolich, Eduardo Acuña Espinoza, Pia Ebeling, Björn Guse, Jonas Götte, Sibylle K. Hassler, Corina Hauffe, Ingo Heidbüchel, Jens Kiesel, Mirko Mälicke, Hannes Müller-Thomy, Michael Stölzle, and Larisa Tarasova
Earth Syst. Sci. Data, 16, 5625–5642, https://doi.org/10.5194/essd-16-5625-2024, https://doi.org/10.5194/essd-16-5625-2024, 2024
Short summary
Short summary
The CAMELS-DE dataset features data from 1582 streamflow gauges across Germany, with records spanning from 1951 to 2020. This comprehensive dataset, which includes time series of up to 70 years (median 46 years), enables advanced research on water flow and environmental trends and supports the development of hydrological models.
Dan Elhanati, Nadine Goeppert, and Brian Berkowitz
Hydrol. Earth Syst. Sci., 28, 4239–4249, https://doi.org/10.5194/hess-28-4239-2024, https://doi.org/10.5194/hess-28-4239-2024, 2024
Short summary
Short summary
A continuous time random walk framework was developed to allow modeling of a karst aquifer discharge response to measured rainfall. The application of the numerical model yielded robust fits between modeled and measured discharge values, especially for the distinctive long tails found during recession times. The findings shed light on the interplay of slow and fast flow in the karst system and establish the application of the model for simulating flow and transport in such systems.
Svenja Hoffmeister, Rafael Bohn Reckziegel, Ben du Toit, Sibylle K. Hassler, Florian Kestel, Rebekka Maier, Jonathan P. Sheppard, and Erwin Zehe
Hydrol. Earth Syst. Sci., 28, 3963–3982, https://doi.org/10.5194/hess-28-3963-2024, https://doi.org/10.5194/hess-28-3963-2024, 2024
Short summary
Short summary
We studied a tree–crop ecosystem consisting of a blackberry field and an alder windbreak. In the water-scarce region, irrigation provides sufficient water for plant growth. The windbreak lowers the irrigation amount by reducing wind speed and therefore water transport into the atmosphere. These ecosystems could provide sustainable use of water-scarce landscapes, and we studied the complex interactions by observing several aspects (e.g. soil, nutrients, carbon assimilation, water).
Eduardo Acuña Espinoza, Ralf Loritz, Manuel Álvarez Chaves, Nicole Bäuerle, and Uwe Ehret
Hydrol. Earth Syst. Sci., 28, 2705–2719, https://doi.org/10.5194/hess-28-2705-2024, https://doi.org/10.5194/hess-28-2705-2024, 2024
Short summary
Short summary
Hydrological hybrid models promise to merge the performance of deep learning methods with the interpretability of process-based models. One hybrid approach is the dynamic parameterization of conceptual models using long short-term memory (LSTM) networks. We explored this method to evaluate the effect of the flexibility given by LSTMs on the process-based part.
Samuel Schroers, Ulrike Scherer, and Erwin Zehe
Hydrol. Earth Syst. Sci., 27, 2535–2557, https://doi.org/10.5194/hess-27-2535-2023, https://doi.org/10.5194/hess-27-2535-2023, 2023
Short summary
Short summary
The hydrological cycle shapes our landscape. With an accelerating change of the world's climate and hydrological dynamics, concepts of evolution of natural systems become more important. In this study, we elaborated a thermodynamic framework for runoff and sediment transport and show from model results as well as from measurements during extreme events that the developed concept is useful for understanding the evolution of the system's mass, energy, and entropy fluxes.
Judith Meyer, Malte Neuper, Luca Mathias, Erwin Zehe, and Laurent Pfister
Hydrol. Earth Syst. Sci., 26, 6163–6183, https://doi.org/10.5194/hess-26-6163-2022, https://doi.org/10.5194/hess-26-6163-2022, 2022
Short summary
Short summary
We identified and analysed the major atmospheric components of rain-intense thunderstorms that can eventually lead to flash floods: high atmospheric moisture, sufficient latent instability, and weak thunderstorm cell motion. Between 1981 and 2020, atmospheric conditions became likelier to support strong thunderstorms. However, the occurrence of extreme rainfall events as well as their rainfall intensity remained mostly unchanged.
Ralf Loritz, Maoya Bassiouni, Anke Hildebrandt, Sibylle K. Hassler, and Erwin Zehe
Hydrol. Earth Syst. Sci., 26, 4757–4771, https://doi.org/10.5194/hess-26-4757-2022, https://doi.org/10.5194/hess-26-4757-2022, 2022
Short summary
Short summary
In this study, we combine a deep-learning approach that predicts sap flow with a hydrological model to improve soil moisture and transpiration estimates at the catchment scale. Our results highlight that hybrid-model approaches, combining machine learning with physically based models, are a promising way to improve our ability to make hydrological predictions.
Samuel Schroers, Olivier Eiff, Axel Kleidon, Ulrike Scherer, Jan Wienhöfer, and Erwin Zehe
Hydrol. Earth Syst. Sci., 26, 3125–3150, https://doi.org/10.5194/hess-26-3125-2022, https://doi.org/10.5194/hess-26-3125-2022, 2022
Short summary
Short summary
In hydrology the formation of landform patterns is of special interest as changing forcings of the natural systems, such as climate or land use, will change these structures. In our study we developed a thermodynamic framework for surface runoff on hillslopes and highlight the differences of energy conversion patterns on two related spatial and temporal scales. The results indicate that surface runoff on hillslopes approaches a maximum power state.
Brian Berkowitz
Hydrol. Earth Syst. Sci., 26, 2161–2180, https://doi.org/10.5194/hess-26-2161-2022, https://doi.org/10.5194/hess-26-2161-2022, 2022
Short summary
Short summary
Extensive efforts have focused on quantifying conservative chemical transport in geological formations. We assert that an explicit accounting of temporal information, under uncertainty, in addition to spatial information, is fundamental to an effective modeling formulation. We further assert that efforts to apply chemical transport equations at large length scales, based on measurements and model parameter values relevant to significantly smaller length scales, are an unattainable
holy grail.
Erwin Zehe, Ralf Loritz, Yaniv Edery, and Brian Berkowitz
Hydrol. Earth Syst. Sci., 25, 5337–5353, https://doi.org/10.5194/hess-25-5337-2021, https://doi.org/10.5194/hess-25-5337-2021, 2021
Short summary
Short summary
This study uses the concepts of entropy and work to quantify and explain the emergence of preferential flow and transport in heterogeneous saturated porous media. We found that the downstream concentration of solutes in preferential pathways implies a downstream declining entropy in the transverse distribution of solute transport pathways. Preferential flow patterns with lower entropies emerged within media of higher heterogeneity – a stronger self-organization despite a higher randomness.
Jan Bondy, Jan Wienhöfer, Laurent Pfister, and Erwin Zehe
Hydrol. Earth Syst. Sci. Discuss., https://doi.org/10.5194/hess-2021-174, https://doi.org/10.5194/hess-2021-174, 2021
Manuscript not accepted for further review
Short summary
Short summary
The Budyko curve is a widely-used and simple framework to predict the mean water balance of river catchments. While many catchments are in close accordance with the Budyko curve, others show more or less significant deviations. Our study aims at better understanding the role of soil storage characteristics in the mean water balance and offsets from the Budyko curve. Soil storage proved to be a very sensitive property and potentially explains significant deviations from the curve.
Alexander Sternagel, Ralf Loritz, Julian Klaus, Brian Berkowitz, and Erwin Zehe
Hydrol. Earth Syst. Sci., 25, 1483–1508, https://doi.org/10.5194/hess-25-1483-2021, https://doi.org/10.5194/hess-25-1483-2021, 2021
Short summary
Short summary
The key innovation of the study is a method to simulate reactive solute transport in the vadose zone within a Lagrangian framework. We extend the LAST-Model with a method to account for non-linear sorption and first-order degradation processes during unsaturated transport of reactive substances in the matrix and macropores. Model evaluations using bromide and pesticide data from irrigation experiments under different flow conditions on various timescales show the feasibility of the method.
Samuel Schroers, Olivier Eiff, Axel Kleidon, Jan Wienhöfer, and Erwin Zehe
Hydrol. Earth Syst. Sci. Discuss., https://doi.org/10.5194/hess-2021-79, https://doi.org/10.5194/hess-2021-79, 2021
Manuscript not accepted for further review
Short summary
Short summary
In this study we ask the basic question why surface runoff forms drainage networks and confluences at all and how structural macro form and micro topography is a result of thermodynamic laws. We find that on a macro level hillslopes should tend from negative exponential towards exponential forms and that on a micro level the formation of rills goes hand in hand with drainage network formation of river basins. We hypothesize that we can learn more about erosion processes if we extend this theory.
Nicolas Björn Rodriguez, Laurent Pfister, Erwin Zehe, and Julian Klaus
Hydrol. Earth Syst. Sci., 25, 401–428, https://doi.org/10.5194/hess-25-401-2021, https://doi.org/10.5194/hess-25-401-2021, 2021
Short summary
Short summary
Different parts of water have often been used as tracers to determine the age of water in streams. The stable tracers, such as deuterium, are thought to be unable to reveal old water compared to the radioactive tracer called tritium. We used both tracers, measured in precipitation and in a stream in Luxembourg, to show that this is not necessarily true. It is, in fact, advantageous to use the two tracers together, and we recommend systematically using tritium in future studies.
Ralf Loritz, Markus Hrachowitz, Malte Neuper, and Erwin Zehe
Hydrol. Earth Syst. Sci., 25, 147–167, https://doi.org/10.5194/hess-25-147-2021, https://doi.org/10.5194/hess-25-147-2021, 2021
Short summary
Short summary
This study investigates the role and value of distributed rainfall in the runoff generation of a mesoscale catchment. We compare the performance of different hydrological models at different periods and show that a distributed model driven by distributed rainfall yields improved performances only during certain periods. We then step beyond this finding and develop a spatially adaptive model that is capable of dynamically adjusting its spatial model structure in time.
Conrad Jackisch, Samuel Knoblauch, Theresa Blume, Erwin Zehe, and Sibylle K. Hassler
Biogeosciences, 17, 5787–5808, https://doi.org/10.5194/bg-17-5787-2020, https://doi.org/10.5194/bg-17-5787-2020, 2020
Short summary
Short summary
We developed software to calculate the root water uptake (RWU) of beech tree roots from soil moisture dynamics. We present our approach and compare RWU to measured sap flow in the tree stem. The study relates to two sites that are similar in topography and weather but with contrasting soils. While sap flow is very similar between the two sites, the RWU is different. This suggests that soil characteristics have substantial influence. Our easy-to-implement RWU estimate may help further studies.
Stephanie Thiesen, Diego M. Vieira, Mirko Mälicke, Ralf Loritz, J. Florian Wellmann, and Uwe Ehret
Hydrol. Earth Syst. Sci., 24, 4523–4540, https://doi.org/10.5194/hess-24-4523-2020, https://doi.org/10.5194/hess-24-4523-2020, 2020
Short summary
Short summary
A spatial interpolator has been proposed for exploring the information content of the data in the light of geostatistics and information theory. It showed comparable results to traditional interpolators, with the advantage of presenting generalization properties. We discussed three different ways of combining distributions and their implications for the probabilistic results. By its construction, the method provides a suitable and flexible framework for uncertainty analysis and decision-making.
Uwe Ehret, Rik van Pruijssen, Marina Bortoli, Ralf Loritz, Elnaz Azmi, and Erwin Zehe
Hydrol. Earth Syst. Sci., 24, 4389–4411, https://doi.org/10.5194/hess-24-4389-2020, https://doi.org/10.5194/hess-24-4389-2020, 2020
Short summary
Short summary
In this paper we propose adaptive clustering as a new method for reducing the computational efforts of distributed modelling. It consists of identifying similar-acting model elements during the runtime, clustering them, running the model for just a few representatives per cluster, and mapping their results to the remaining model elements in the cluster. With the example of a hydrological model, we show that this saves considerable computation time, while largely maintaining the output quality.
Cited articles
Adams, R. E., Hyodo, A., SantaMaria, T., Wright, C. L., Boutton, T. W., and
West, J. B.: Bound and mobile soil water isotope ratios are affected by soil
texture and mineralogy, whereas extraction method influences their measurement, Hydrol. Process., 34, 991–1003, 2020.
Allen, S. T. and Kirchner, J. W.: Potential effects of cryogenic extraction biases on inferences drawn from xylem water deuterium isotope ratios: case studies using stable isotopes to infer plant water sources, Hydrol. Earth Syst. Sci. Discuss. [preprint], https://doi.org/10.5194/hess-2020-683, 2021.
Benettin, P., Nehemy, M. F., Asadollahi, M., Pratt, D., Bensimon, M.,
McDonnell, J. J., and Rinaldo, A.: Tracing and closing the water balance in
a vegetated lysimeter, Water Resour. Res., 57, e2020WR029049, https://doi.org/10.1029/2020WR029049, 2021.
Berkowitz, B., Cortis, A., Dentz, M., and Scher, H.: Modeling non-Fickian
transport in geological formations as a continuous time random walk, Rev.
Geophys., 44, RG2003, https://doi.org/10.1029/2005RG000178, 2006.
Berkowitz, B., Dror, I., Hansen, S. K., and Scher, H.: Measurements and
models of reactive transport in geological media, Rev. Geophys., 54, 930–986, 2016.
Beven, K. and Germann, K.: Macropores and water flow in soils, Water Resour. Res., 18, 1311–1325, 1982.
Beven, K. and Germann, P.: Macropores and water flow in soils revisited, Water Resour. Res., 49, 3071–3092, https://doi.org/10.1002/wrcr.20156, 2013.
Bonell, M., Pearce, A. J., and Stewart, M. K.: The Identification Of Runoff-Production Mechanisms Using Environmental Isotopes In A Tussock Grassland Catchmant, Eastern Otago, New-Zealand, Hydrol. Process., 4, 15–34, 1990.
Boso, F., Bellin, A., and Dumbser, M.: Numerical simulations of solute transport in highly heterogeneous formations: A comparison of alternative
numerical schemes, Adv. Water Resour., 52, 178–189, 2013.
Bowers, W. H. and Mercer, J. J.: A combination of soil water extraction
methods quantifies the isotopic mixing of waters held at separate tensions
in soil, Open Science Framework [data files and R code], https://doi.org/10.17605/OSF.IO/ET3G5, 2020.
Bowers, W. H., Mercer, J. J., Pleasants, M. S., and Williams, D. G.: A combination of soil water extraction methods quantifies the isotopic mixing of waters held at separate tensions in soil, Hydrol. Earth Syst. Sci., 24, 4045–4060, https://doi.org/10.5194/hess-24-4045-2020, 2020.
Chou, H., Wu, L., Zeng, L., and Chang, A.: Evaluation of solute diffusion
tortuosity factor models for variously saturated soils, Water Resour. Res., 48, W10539, https://doi.org/10.1029/2011WR011653, 2012.
Currie, I. G.: Fundamental mechanics of fluids, CRC Press, ISBN 146651700X, 9781466517004, 2016.
Edery, Y., Guadagnini, A., Scher, H., and Berkowitz, B.: Origins of anomalous transport in heterogeneous media: Structural and dynamic controls, Water Resour. Res., 50, 1490–1505, 2014.
Engdahl, N. B., Benson, D. A., and Bolster, D.: Lagrangian simulation of
mixing and reactions in complex geochemical systems, Water Resour. Res., 53, 3513–3522, 2017.
Engdahl, N. B., Schmidt, M. J., and Benson, D. A.: Accelerating and Parallelizing Lagrangian Simulations of Mixing-Limited Reactive Transport,
Water Resour. Res., 55, 3556–3566, 2019.
Gouet-Kaplan, M. and Berkowitz, B.: Measurements of interactions between
resident and infiltrating water in a lattice micromodel, Vadose Zone J., 10, 624–633, 2011.
Green, C. T., LaBolle, E. M., Fogg, G. E., and Davis, C.: Random walk particle tracking for simulating reactive transport in heterogeneous aquifers: effects of concentraction averaging, in: Proceedings of the International Groundwater Symposium, International Association of Hydraulic Research and American Geophysical Union, 25–28 March 2002, Berkeley, California, USA, 446–450, 2002.
Harman, C. J.: Time-variable transit time distributions and transport: Theory and application to storage-dependent transport of chloride in a watershed, Water Resour. Res., 51, 1–30, https://doi.org/10.1002/2014wr015707, 2015.
Hasegawa, T., Nakata, K., and Gwynne, R.: Measurement on Diffusion Coefficients and Isotope Fractionation Factors by a Through-Diffusion
Experiment, Minerals, 11, 208, https://doi.org/10.3390/min11020208, 2021.
Hrachowitz, M., Savenije, H., Bogaard, T. A., Tetzlaff, D., and Soulsby, C.: What can flux tracking teach us about water age distribution patterns and their temporal dynamics?, Hydrol. Earth Syst. Sci., 17, 533–564, https://doi.org/10.5194/hess-17-533-2013, 2013.
Jackisch, C. and Zehe, E.: Ecohydrological particle model based on representative domains, Hydrol. Earth Syst. Sci., 22, 3639–3662,
https://doi.org/10.5194/hess-22-3639-2018, 2018.
Kapetas, L., Dror, I., and Berkowitz, B.: Evidence of preferential path formation and path memory effect during successive infiltration and drainage
cycles in uniform sand columns, J. Contam. Hydrol., 165, 1–10, 2014.
Klaus, J. and McDonnell, J. J.: Hydrograph separation using stable isotopes: Review and evaluation, J. Hydrol., 505, 47–64, https://doi.org/10.1016/j.jhydrol.2013.09.006, 2013.
Klaus, J., Chun, K. P., McGuire, K. J., and McDonnell, J. J.: Temporal dynamics of catchment transit times from stable isotope data, Water Resour. Res., 51, 4208–4223, https://doi.org/10.1002/2014wr016247, 2015.
Lin, Y. and Horita, J.: An experimental study on isotope fractionation in a
mesoporous silica-water system with implications for vadose-zone hydrology,
Geochim. Cosmochim. Ac., 184, 257–271, 2016.
Lin, Y., Horita, J., and Abe, O.: Adsorption isotope effects of water on
mesoporous silica and alumina with implications for the land-vegetation-atmosphere system, Geochim. Cosmochim. Ac., 223, 520–536, 2018.
McGlynn, B. and Seibert, J.: Distributed assessment of contributing area and riparian buffering along stream networks, Water Resour. Res., 39, WR001521, https://doi.org/10.1029/2002WR001521, 2003.
McGlynn, B., McDonnell, J., Stewart, M., and Seibert, J.: On the relationships between catchment scale and streamwater mean residence time,
Hydrol. Process., 17, 175–181, 2002.
Mennekes, D., Rinderer, M., Seeger, S., and Orlowski, N.: Ecohydrological
travel times derived from in situ stable water isotope measurements in trees
during a semi-controlled pot experiment, Hydrol. Earth Syst. Sci., 25,
4513–4530, https://doi.org/10.5194/hess-25-4513-2021, 2021.
Mills, R.: Self-diffusion in normal and heavy water in the range 1–45. deg,
J. Phys. Chem., 77, 685–688, 1973.
Mualem, Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, 513–522, 1976.
Oerter, E., Finstad, K., Schaefer, J., Goldsmith, G. R., Dawson, T., and
Amundson, R.: Oxygen isotope fractionation effects in soil water via interaction with cations (Mg, Ca, K, Na) adsorbed to phyllosilicate clay
minerals, J. Hydrol., 515, 1–9, 2014.
Ogden, F. L., Allen, M. B., Lai, W., Zhu, J., Seo, M., Douglas, C. C., and
Talbot, C. A.: The soil moisture velocity equation, J. Adv. Model. Earth Syst., 9, 1473–1487, 2017.
Orlowski, N. and Breuer, L.: Sampling soil water along the pF curve for
δ2H and δ18O analysis, Hydrol. Process., 34, 4959–4972, 2020.
Orlowski, N., Frede, H.-G., Brüggemann, N., and Breuer, L.: Validation and application of a cryogenic vacuum extraction system for soil and plant water extraction for isotope analysis, J. Sens. Sens. Syst., 2, 179–193, https://doi.org/10.5194/jsss-2-179-2013, 2013.
Penna, D., Geris, J., Hopp, L., and Scandellari, F.: Water sources for root
water uptake: Using stable isotopes of hydrogen and oxygen as a research tool in agricultural and agroforestry systems, Agr. Ecosyst. Environ., 291, 106790, https://doi.org/10.1016/j.agee.2019.106790, 2020.
Rodriguez, N. B. and Klaus, J.: Catchment travel times from composite StorAge Selection functions representing the superposition of streamflow generation processes, Water Resour. Res., 55, 9292–9314, 2019.
Rodriguez, N. B., Pfister, L., Zehe, E., and Klaus, J.: A comparison of catchment travel times and storage deduced from deuterium and tritium tracers using StorAge Selection functions, Hydrol. Earth Syst. Sci., 25, 401–428, https://doi.org/10.5194/hess-25-401-2021, 2021.
Schmidt, M. J., Pankavich, S. D., Navarre-Sitchler, A., and Benson, D. A.: A
Lagrangian method for reactive transport with solid/aqueous chemical phase
interaction, J. Comput. Phys. X, 2, 100021, https://doi.org/10.1016/j.jcpx.2019.100021, 2019.
Sklash, M. G., Beven, K. J., GILMAN, K., and DARLING, W. G.: Isotope Studies Of Pipeflow AT Plynlimon, Wales, UK, Hydrol. Process., 10, 921–944, 1996.
Sprenger, M., Leistert, H., Gimbel, K., and Weiler, M.: Illuminating hydrological processes at the soil-vegetation-atmosphere interface with
water stable isotopes, Rev. Geophys., 54, 674–704, 2016.
Sprenger, M., Tetzlaff, D., Buttle, J., Laudon, H., Leistert, H., Mitchell, C. P. J., Snelgrove, J., Weiler, M., and Soulsby, C.: Measuring and Modeling
Stable Isotopes of Mobile and Bulk Soil Water, Vadose Zone J., 17, 170149, https://doi.org/10.2136/vzj2017.08.0149, 2018.
Sprenger, M., Stumpp, C., Weiler, M., Aeschbach, W., Allen, S. T., Benettin,
P., Dubbert, M., Hartmann, A., Hrachowitz, M., and Kirchner, J. W.: The
demographics of water: A review of water ages in the critical zone, Rev. Geophys., 57, 800–834, 2019.
Sternagel, A.: KIT-HYD/last-model: v0.1.1 (v0.1.1), Zenodo [code], https://doi.org/10.5281/zenodo.6375769, 2022.
Sternagel, A., Loritz, R., Wilcke, W., and Zehe, E.: Simulating preferential
soil water flow and tracer transport using the Lagrangian Soil Water and
Solute Transport Model, Hydrol. Earth Syst. Sci., 23, 4249–4267,
https://doi.org/10.5194/hess-23-4249-2019, 2019.
Sternagel, A., Loritz, R., Klaus, J., Berkowitz, B., and Zehe, E.: Simulation of reactive solute transport in the critical zone: a Lagrangian model for transient flow and preferential transport, Hydrol. Earth Syst. Sci., 25, 1483–1508, https://doi.org/10.5194/hess-25-1483-2021, 2021.
Talbot, C. A., and Ogden, F. L.: A method for computing infiltration and
redistribution in a discretized moisture content domain, Water Resour. Res., 44, W08453, https://doi.org/10.1029/2008WR006815, 2008.
Uffink, G. J. M.: Analysis of dispersion by the random walk method, PhD Dissertation, Delft University of Technology, Delft, https://ci.nii.ac.jp/naid/10026170615/en/ (last access: 22 February 2022), 1990.
van der Velde, Y., Heidbuchel, I., Lyon, S. W., Nyberg, L., Rodhe, A., Bishop, K., and Troch, P. A.: Consequences of mixing assumptions for time-variable travel time distributions, Hydrol. Process., 29, 3460–3474, https://doi.org/10.1002/hyp.10372, 2015.
van Genuchten, M. T.: A closed-form equation for predicting the hydraulic
conductivity of unsaturated soil, Soil Sci. Soc. Am. J., 44, 892–898, 1980.
Weiler, M., McGlynn, B. L., McGuire, K. J., and McDonnell, J. J.: How does
rainfall become runoff? A combined tracer and runoff transfer function approach, Water Resour. Res., 39, 1315, https://doi.org/10.1029/2003wr002331, 2003.
Willmann, M., Carrera, J., and Sánchez-Vila, X.: Transport upscaling in
heterogeneous aquifers: What physical parameters control memory functions?,
Water Resour. Res., 44, W12437, https://doi.org/10.1029/2007WR006531., 2008.
Zehe, E. and Jackisch, C.: A Lagrangian model for soil water dynamics during rainfall-driven conditions, Hydrol. Earth Syst. Sci., 20, 3511–3526, https://doi.org/10.5194/hess-20-3511-2016, 2016.
Zinn, B. and Harvey, C. F.: When good statistical models of aquifer heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in
connected and multivariate Gaussian hydraulic conductivity fields, Water
Resour. Res., 39, 1051, https://doi.org/10.1029/2001WR001146, 2003.
Short summary
We present a (physically based) Lagrangian approach to simulate diffusive mixing processes on the pore scale beyond perfectly mixed conditions. Results show the feasibility of the approach for reproducing measured mixing times and concentrations of isotopes over pore sizes and that typical shapes of breakthrough curves (normally associated with non-uniform transport in heterogeneous soils) may also occur as a result of imperfect subscale mixing in a macroscopically homogeneous soil matrix.
We present a (physically based) Lagrangian approach to simulate diffusive mixing processes on...
