NP7.1 | Non-linear Waves and Fracturing and Scaling and Multi-Fractals in Hydrology and Meteorology
Non-linear Waves and Fracturing and Scaling and Multi-Fractals in Hydrology and Meteorology
Co-organized by TS2, co-sponsored by AGU and AOGS
Convener: Arcady Dyskin | Co-conveners: Elena Pasternak, Sergey Turuntaev, Arun Ramanathan, Jisun LeeECSECS, Alin Andrei Carsteanu, Vijay Prasad Dimri
| Wed, 26 Apr, 08:30–10:15 (CEST)
Room 0.16
Posters on site
| Attendance Mon, 24 Apr, 14:00–15:45 (CEST)
Hall X4
Posters virtual
| Attendance Mon, 24 Apr, 14:00–15:45 (CEST)
Orals |
Wed, 08:30
Mon, 14:00
Mon, 14:00
Waves in the Earth’s crust are often generated by fractures in the process of their sliding or propagation. Conversely, the waves can trigger fracture sliding or even propagation. The presence of multiple fractures makes geomaterials non-linear. Therefore the analysis of wave propagation and interaction with pre-existing or emerging fractures is central to geophysics. Recently new observations and theoretical concepts were introduced that point out to the limitations of the traditional concept. These are:
• Multiscale nature of wave fields and fractures in geomaterials
• Rotational mechanisms of wave and fracture propagation
• Strong rock and rock mass non-linearity (such as bilinear stress-strain curve with high modulus in compression and low in tension) and its effect on wave propagation
• Apparent negative stiffness associated with either rotation of non-spherical constituents or fracture propagation and its effect on wave propagation
• Triggering effects and instability in geomaterials
• Active nature of geomaterials (e.g., seismic emission induced by stress and pressure wave propagation)
• Non-linear mechanics of hydraulic fracturing
• Synchronization in fracture processes including earhtquakes and volcanic activity

Complex waves are now a key problem of the physical oceanography and atmosphere physics. They are called rogue or freak waves. It may be expected that similar waves are also present in non-linear solids (e.g., granular materials), which suggests the existence of new types of seismic waves.

It is anticipated that studying these and related phenomena can lead to breakthroughs in understanding of the stress transfer and multiscale failure processes in the Earth's crust, ocean and atmosphere and facilitate developing better prediction and monitoring methods.

The first part of the session is designed as a forum for discussing these and relevant topics.

The second part of the session aims to nurture the development of fractals, multifractals and related nonlinear methodologies applicable to a wide range of hydrological, meteorological systems and their multiscale interactions. Theories considering scalar and vector fields, applications in the area of hydrometeorology (e.g. rainfall extremes, urban flood control, water management etc.), analysis of in-situ, remotely sensed data and simulation techniques are of interest.

Orals: Wed, 26 Apr | Room 0.16

Chairpersons: Arcady Dyskin, Elena Pasternak, Arun Ramanathan
NP7.1. Non-linear Waves and Fracturing, 60 min
On-site presentation
Ioannis Stefanou, Georgios Tzortzopoulos, and Diego Gutierrez-Oribio

We propose a theory for preventing instabilities in frictionally unstable systems such as earthquakes are. We exploit the dependence of friction on fluid pressure and use it as a backdoor for provoking controlled, slow-slip over a single mature seismic fault. We use the mathematical Theory of Control and notions from passivity in order to (a) stabilize and restricting chaos, (b) impose slow frictional dissipation and (c) tune the system toward desirable global asymptotic equilibria of lower energy. Our control approach is robust and does not require exact knowledge of the frictional behavior of the system and its fluid diffusion properties (e.g. permeability, viscosity, compressibility) or of other parameters related to complex physical processes that are hard to determine in practice. We expect our methodology to inspire earthquake mitigation strategies regarding anthropogenic and/or natural seismicity.


[1] Stefanou, I. (2019). Controlling Anthropogenic and Natural Seismicity: Insights From Active Stabilization of the Spring‐Slider Model. Journal of Geophysical Research: Solid Earth, 124(8), 8786–8802.
[2] Tzortzopoulos G., Braun P., Stefanou I. (2021), Absorbent Porous Paper Reveals How Earthquakes Could be Mitigated, Geophysical Research Letters 48.
[3] Stefanou, I., Tzortzopoulos, G. (2022). Preventing instabilities and inducing controlled, slow-slip in frictionally unstable systems. Journal of Geophysical Research: Solid Earth.
[4] Gutiérrez-Oribio D., Tzortzopoulos G., Stefanou I., Plestan F. (2022). Earthquake Control: An Emerging Application for Robust Control. Theory and Experimental Tests.
[5] Papachristos, E., Stefanou, I. (2022), Controlling earthquake-like instabilities using artificial intelligence.
[6] Gutiérrez-Oribio D., Stefanou I., Plestan F. (2022). Passivity-based Control of a Frictional Underactuated Mechanical System: Application to Earthquake Prevention.

How to cite: Stefanou, I., Tzortzopoulos, G., and Gutierrez-Oribio, D.: Preventing earthquake instabilities and inducing controlled, slow-slip by active fluid pressure control in the vicinity of a single seismic fault, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-14939,, 2023.

Virtual presentation
Kuan Jiang and Cheng-zhi Qi

Rock mass has complex block-hierarchical structure involving various scale levels, which should be considered both in dynamic and static conditions. Because the interlayer has weak mechanical properties compared with rock blocks, the deformation of rock mass mainly concentrates at the interlayers both in dynamic and static conditions, which provides the possibility of translation and rotation for rock blocks. The basic carriers of pendulum-type wave in rock mass are geoblocks with translational and rotational degrees of freedom involving various hierarchical levels. The major part of the energy of a blast is spent to fragmentation of rocks and is transferred to rock blocks of the stressed geomedium in the form of kinetic energy (including translational kinetic energy and rotational kinetic energy). The in-situ experimental data has shown that the block-rock mass has significant angular deformation under dynamic impact, and the rotation of blocks can deeply affect the wave propagation and dynamic behavior of rock mass. Previous research on 1D dynamic model of block-rock mass cannot reflect the rotation effect of blocks, and the new 2D dynamic model should take into account the rotation of blocks and energy transfer. Consequently, aiming at the investigation of rotation of blocks, the 2D dynamic model of block-rock mass is established based on the accurate consideration of rotation effect. The research based on this model reveals the mechanism of the rotation of blocks, and determines the characteristics of energy transfer and the influence of the rotation of blocks on the inhomogeneous deformation of interlayers. Research shows that the rotation of blocks is not directly related to whether the structure of rock mass is symmetrical, or whether the interlayer is deformed or not, or the form of external loads, but is caused by the non-equilibrium shear between interfaces in the absence of the external torque. The rotation of blocks results in the inhomogeneous deformation of interlayers, and has a significant influence on the shear deformation of interlayers. At some local positions, in addition to the deformation of the interlayer caused by translation, the block-rock mass also produces additional tension and compression deformation caused by the rotation of blocks, which may lead to the phenomenon of rock crushing. This study theoretically solves the problems of wave propagation in block medium under arbitrary loads and torque, and is helpful for the research of seismic wave propagation in block medium with inhomogeneous complex structures.

How to cite: Jiang, K. and Qi, C.: Study on the rotation of blocks in two-dimensional block-rock mass, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-4271,, 2023.

Virtual presentation
Tao Ming, Luo Hao, Zhao Rui, and Xiang Gongliang

Underground chambers or tunnels often contain inclusions, the interface between the inclusion and the surrounding rock is not always perfect, which influences stress wave propagation. In this study, the spring model and Ricker wavelet were adopted to represent the imperfect interface and transient seismic wave. Based on the wave function expansion method and Fourier transform, an analytical formula for the dynamic stress concentration factor (DSCF) for an elliptical inclusion with imperfect interfaces in infinite space subjected to a plane SH-wave was determined. The theoretical solution was verified via numerical simulations using the LS-DYNA software, and the results were analyzed. The effects of the wave number (k), radial coordinate (ξ), stiffness parameter (β), and differences in material properties on the dynamic response were evaluated. The numerical results revealed that the maximum DSCF always occurred at both ends of the elliptical minor axis, and the transient DSCF was generally a factor of 2-3 greater than the steady-state DSCF. Changes in k and ξ led to variations in the DSCF value and spatial distribution, changes in β resulted only in variations in the DSCF value, and lower values of ωp and β led to a greater DSCF under the same parameter conditions. In addition, the differences in material properties between the medium and inclusion significantly affected the variation characteristics of the DSCF with k and ξ.

How to cite: Ming, T., Hao, L., Rui, Z., and Gongliang, X.: Dynamic response of an elliptic cylinder inclusion with imperfect interfaces subjected to plane SH wave, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-7726,, 2023.

Virtual presentation
Yuxin Ban, Jun Duan, Qiang Xie, Xiang Fu, and Weichen Sun

The key to increasing shale gas production is to construct fracture networks in shale reservoir to provide channels for shale gas. Understanding the cracking characteristics of shale is necessary for oil and gas exploitation engineering. Given this, uniaxial compression tests were conducted on Longmaxi shale in China to study the mechanical properties and cracking behaviors affected by bedding layers and pre-existing slot. Sandstone specimens with different pre-existing slot angles were also tested as a comparison. A mechanical-optical-acoustical comprehensive data acquisition system consisting of a rigid hydraulic machine, high-speed industrial camera and acoustic emission acquisition instrument was established to monitor the cracking behaviors in real time. The results show that the cracking behaviors of shale specimens are quite different from sandstone specimens in the uniaxial compression tests. Crack initiation is predominantly controlled by the pre-existing slot and is also affected by bedding layers. Crack propagation is mainly controlled by bedding layers and stress field distribution. When the bedding layers are vertical, the cracks are most likely to propagate along the direction of the bedding and tensile cracks are observed. When the bedding is 30°, the shale specimens are most likely to be controlled by the bedding layers, resulting in shear slip failure along the bedding layers. The experimental results contribute to the understanding of cracking properties in layered anisotropic materials.

How to cite: Ban, Y., Duan, J., Xie, Q., Fu, X., and Sun, W.: Cracking properties of shale influenced by bedding layers and a pre-existing slot, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6464,, 2023.

Virtual presentation
Xiang Fu, Yuxin Ban, Qiang Xie, and Weichen Sun

After the impoundment of a high dam reservoir, the water pressure environment of the rock masses in dam base and reservoir bank changes, which may easily induce engineering problems such as bank slope instability and dam collapse. In order to investigate the differences and mechanisms of different constant water pressures on the rock mass of dam base, triaxial loading tests were conducted on sandstone with initial damage under different high constant porewater pressures, and the multidirectional fracture mechanism was analyzed by combining CT and electron scans. The test results show that:(1) Under the confining pressure of 80 MPa, the greater the pore water pressure, the more brittle the sandstone is, the lower the peak strength, the smaller the volume expansion stress, the pore water pressure increases from 10 MPa to 50 MPa, and the peak strength decreases by 33%.  (2) For different pore water pressure, there are significant differences in sandstone internal deterioration range and deterioration effect  as the fracture surfaces of sandstone specimens have various forms and directions. Due to CT scaning results, with the pore water pressure increases, the deterioration effect spreads from specimen middle to both ends. When the water pressure-confining pressure ratio is less than 25.0%, the deterioration of pore water pressure is mainly concentrated in the middle 1/3 of the specimen. When the water pressure-confining pressure ratio is bigger than 62.5%, the pore water pressure has obvious deterioration effect on the whole specimen. (3) Electron microscopy scanning reveals that with the increase of pore water pressure: the microgranular structure of sandstone changes from shear slip failure to shear fracture failure, and the microcrystalline structure of sandstone changes from cauliflower to rice granular. The macroscopic failure mode changes from plastic failure to brittle failure, and multidirectional fracture plane is formed, which is related to the migration of fine particles and the fracture of large particles in the meso-particle structure under pore water pressure. The formation of the multidirectional fracture plane is directly related to the shear strength of the microscopic crystal structure.

How to cite: Fu, X., Ban, Y., Xie, Q., and Sun, W.: Triaxial compression mechanical properties and multidirectional fracture mechanism of sandstone under different pore pressure, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6852,, 2023.

Virtual presentation
Denis Taurenis and Victor Nachev

Over the past years in the development of oil and gas fields, there has been a trend towards an increase in the development of unconventional low-permeability reservoirs. In this regard, it is becoming increasingly important to study the problems associated with the use of hydraulic fracturing technology (HF) in rocks with a complex internal structure. To achieve the maximum oil and gas production rate and increase the drainage zone in the near-wellbore space, it is necessary to carry out hydraulic fracturing with the most extensive system of fractures.

In this work the authors investigate the propagation of a hydraulically driven fracture in a fully saturated, permeable, and porous medium at the pore scale. To achieve a goal, at the first stage, we set a system of determining ratios and a crack propagation criterion. At the next stage, a three-dimensional numerical poroelastic model of a rock sample is prepared based on a three-dimensional image of the pore space of rock samples. Then numerical poroelastic modeling of the processes of one- and two-phase filtration and rock destruction using the extended finite element method is performed. For a more accurate description of filtration processes, the authors have prepared a physico-mathematical model that takes into account the flow rate and leakage of fluid into the rock during fracture growth at the pore scale. The obtained numerical results are compared with the previously conducted results of laboratory studies.

As a result of the numerical simulation, the authors prepared a digital rock model (DRM) based on microCT data, performed numerical simulation of the filtration process in the DRM and numerical simulation of fracture propagation in a fully saturated, permeable, and porous medium at the pore scale. Then, the dependences of filtration, initiation and fracture propagation were investigated depending on various conditions of HF fluid injection.

How to cite: Taurenis, D. and Nachev, V.: Three-dimensional numerical simulation of multiphase filtration and fracture propagation at the pore scale, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9483,, 2023.

(Former NP3.3). Multi-Fractals in Hydrology and Meteorology. 40 min
On-site presentation
Ángel García Gago, Auguste Gires, Paul Veers, Ioulia Tchiguirinskaia, and Daniel Schertzer

Wind fields are extremely variable in space and time over a wide range of scales. This extreme variability is transferred to the wind turbine torque and ultimately to wind energy production. The Universal Multifractal (UM) framework is a powerful tool that allows to characterise and simulate the extreme variability of geophysical fields across scales with the help of only three parameters (α, C1 and H) with physical interpretation; while the 4th, the power a of a conservative flux, is absorbed by the empirical estimation of the mean singularity over a non-conservative field.

The main challenge is to simulate over 2D space plus time vector fields which realistically reproduce observed spatial and temporal variability of wind fields. The outer scale of the simulated fields should basically correspond to the size of the wind turbine in space and ten minutes in time. To achieve that, we combine two broad classes of stochastic processes: stable Levy processes and Clifford algebra. We use as input characteristic parameters obtained from the multifractal analysis of the data collected by two high-resolution 3D anemometers with approx. 33 m vertical distance on a meteorological mast. The data is collected as part of the RW-Turb measurement campaign (, supported by the French National Research Agency (ANR-19-CE05-0022). The expected behaviour of the simulated field is confirmed by multifractal analysis. 

In the second step, we investigate the effect of small-scale wind variability on the wind turbine torque computation by imputing the simulated vector fields to three modelling chains with increasing complexity. The first one only considers the temporal variability, averaging the wind field and considering it at hub height. The second one is based on the angular moment definition and allows us to consider both spatial and temporal variability by computing the torque at each blade point and integrating it along the radius for each time step. Finally, the third one uses the realistic software OpenFAST developed by the US National Renewable Energy Laboratory (NREL). To analyse and physically interpret wind variability's effect, we compared the torque obtained by the three modelling chains focused on the small scales. As we expected, we found pronounced differences on small scales with stronger fluctuations exhibited in the second modelling chain, followed by OpenFAST and the first one. 

How to cite: García Gago, Á., Gires, A., Veers, P., Tchiguirinskaia, I., and Schertzer, D.: Small Scales Space-Time Variability of Wind Fields: Simulations with Vector Fields and Transfer to Turbine Torque Computation, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-11767,, 2023.

On-site presentation
Auguste Gires, Ioulia Tchiguirinskaia, and Daniel Schertzer

Rainfall fields exhibit extreme variability over wide range of space-time scales which make them complex to characterize, model and even measure. Furthermore, rainfall, as most geophysical fields, is strongly anisotropic. Fortunately, scaling anisotropy has been developed for a few decades to generalise scaling in an anisotropic framework, e.g., in the simplest case iso-surfaces become self-affines ellipsoids instead of self-similar spheres. This is particularly straightforward for continuous in scale cascades. For them, as well as for discrete in scale cascades, Universal Multifractals (UM) have been widely used to analyse and simulate such geophysical fields with the help of a very limited number of physically meaningful parameters. Recently blunt cascades have been introduced. They enable to remain in the simple framework of discrete cascades while partly overcoming their well known strong limitations such as non-stationnarity. It basically consists in geometrically interpolating over moving windows the multiplicative increments at each cascade steps.

Here we suggest to incorporate observed features in blunt 2D and 3D (space-time) blunt discrete cascade simulations. The data analysis corresponds to a 1D analysis along various directions ,considering each lof them as a different “sample” of the process. Analysing how the UM parameters change with the angle of the chosen direction enables to unveil underlying rainfall anisotropy features. Impacts, and notably potential biases, of these features on standard spatial analysis in 2D are also explored and discussed. For this purpose high resolution space-time rainfall data collected with help of a dual polarisation X-band radar operated by HM&Co-ENPC is used .

To simulate anisotropy features with the help of blunt extension of discrete UM cascades, we tentatively suggest to use moving window shaped as ellipses instead of squares. Tuning the eccentricity and orientation of the ellipses enables to introduce various levels of anisotropy within the simulated fields. First, multifractal expected behaviour is theoretically established and then it is numerically confirmed with the help of ensembles of stochastic simulations and the previously developed analysis approach.

Authors acknowledge the RW-Turb project (supported by the French National Research Agency - ANR-19-CE05-0022), for partial financial support.

How to cite: Gires, A., Tchiguirinskaia, I., and Schertzer, D.: Characterizing and simulating with blunt extension of discrete cascades rainfall anisotropy in a Universal Multifractals framework, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-9020,, 2023.

On-site presentation
Arun Ramanathan S, Pierre-Antoine Versini, Daniel Schertzer, Ioulia Tchiguirinskaia, Remi Perrin, and Lionel Sindt


Several equations and their simplified versions already exist for estimating evapotranspiration. Still, the practical difficulty in using them is that they contain too many variables and empirical parameters including some non-atmospheric vegetation-based ones which may not be appropriate for all plant types. Therefore, a simple empirical equation is suggested here to approximately estimate evapotranspiration loss in a deterministic manner as a function of the green roof’s water content, ambient air temperature, wind speed, relative humidity, and total net radiation flux. For nonlinear processes such as evapotranspiration clearly, such deterministic estimates are not representative of the extreme values observed in evapotranspiration losses. Therefore, a universal multifractal-based simulation procedure is proposed here to improve such deterministic estimates, so that the simulated evapotranspiration loss has realistic intermittency and temporal scaling behaviour, while preserving its diurnal variability.



Multifractals, Non-linear geophysical systems, Cascade dynamics, Scaling, Hydrology, Meteorology.


How to cite: Ramanathan S, A., Versini, P.-A., Schertzer, D., Tchiguirinskaia, I., Perrin, R., and Sindt, L.: Simulating Evapotranspiration in Green roofs using a Multifractal approach, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-7767,, 2023.

On-site presentation
Jerry Jose, Auguste Gires, Ernani Schnorenberger, Ioulia Tchiguirinskaia, and Daniel Schertzer

Wind power production plays an important role in achieving UN’s (United nations) Sustainable development goal (SDG) 7 - affordable and clean energy for all; and in the increasing global transition towards renewable and carbon neutral energy, understanding the uncertainties associated with wind and turbulence is extremely important. Characterization of wind is not straightforward due to its intrinsic intermittency: activity of the field becomes increasingly concentrated at smaller and smaller supports as the scale decreases. When it comes to power production by wind turbines, another complexity arises from the influence of rainfall, which only a limited number of studies have addressed so far suggesting short term as well as long term effects. To understand this, the project RW-Turb (; supported by the French National Research Agency, ANR-19-CE05-0022) employs multiple 3D sonic anemometers (manufactured by Thies), mini meteorological stations (manufactured by Thies), and disdrometers (Parsivel2, manufactured by OTT) on a meteorological mast in the wind farm of Pays d’Othe (110 km south-east of Paris, France; operated by Boralex). With this simultaneously measured data, it is possible to study wind power and associated atmospheric fields under various rain conditions.

Variations of wind velocity, power available at the wind farm, power produced by wind turbines and air density are examined here during rain and dry conditions using the framework of Universal Multifractals (UM). UM is a widely used, physically based, scale invariant framework for characterizing and simulating geophysical fields over wide range of scales which accounts for the intermittency in the field. While statistically analysing the power produced by turbine, rated power acts like an upper threshold resulting in biased estimators. This is identified and quantified here using the theoretical framework of UM along with the actual sampling resolution of instruments under study. Further, from event based analysis, differences in UM parameters were observed between rain and dry conditions for the fields illustrating the influence of rain. This is further explored using joint multifractal analysis and an increase in correlation exponent was observed between various fields with increase in rain rate.

How to cite: Jose, J., Gires, A., Schnorenberger, E., Tchiguirinskaia, I., and Schertzer, D.: Joint multifratcal analysis of available wind power and rain intensity from an operational wind farm, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-7495,, 2023.

Posters on site: Mon, 24 Apr, 14:00–15:45 | Hall X4

Chairperson: Arun Ramanathan
Multi-Fractals in Hydrology and Meteorology
Shengjie Hu, Zhenlei Yang, Sergio Torres, Zipeng Wang, and Ling Li

Power law size distribution, associated with important system behaviors including scale-invariance, critical tipping and self-organization, has been observed in many complex systems. Such distribution also emerges from natural lakes, with potentially important links to the dynamics of lake systems. But the driving mechanism that generates and shapes this feature in lake systems remains unclear. Moreover, the power law itself was found inadequate for fully describing the size distribution of lakes, due to deviations at the two ends of size range. Based on observed and simulated lakes in China’s 11 hydro-climatic zones, we established a conceptual model for lake systems, which covers the whole size range of lake size distribution and reveals the underlying driving mechanism. The full lake size distribution is composed of three components featured by exponential, stretched-exponential and power law distribution. These three distributions are referred to as three phases which represent system (size) states with successively increasing degrees of heterogeneity and orderliness, and more importantly, indicate the dominance of exogenic and endogenic forces in lake systems, respectively. As the dominant driving force changes from endogenic to exogenic, a phase transition occurs with lake size distribution shifted from power law to stretched-exponential and further to exponential distribution. Apart from compressing the power law phase, exogenic force also increases its scaling exponent, driving the corresponding lake size power spectrum into the regime of “blue noise” with reduced system resilience. Besides, the change may also lead to a rising proportion of small lakes in the whole size distribution, which would increase the overall greenhouse gas emissions from natural lakes.

How to cite: Hu, S., Yang, Z., Torres, S., Wang, Z., and Li, L.: Size distributions reveal regime transition of dominant driving force in lake systems, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-4009,, 2023.

Abrar Habib, Athanasios Paschalis, Adrian P. Butler, Christian Onof, John P. Bloomfield, and James P. R. Sorensen

Using a physically based recharge-groundwater flow model, a multiplicative random cascade rainfall model and robust detrended fluctuation analysis (r-DFA), the effect of the fractal behaviour of rainfall on recharge and groundwater levels is investigated. The study site selected for this work is in Wallingford, United Kingdom, where groundwater levels in a shallow riparian aquifer and meteorological data of high temporal resolution are monitored.

The rainfall model is calibrated to the observed rainfall and used to simulate 40 synthetic rainfall series exhibiting different scaling behaviour (with r-DFA scaling exponents between 0.6 and 1.05). The scaling behaviour of the rainfall series are then objectively quantified using r-DFA. The synthetic rainfall is used as forcing to run the recharge-groundwater flow model which is calibrated to the observed groundwater levels.

It is found that small changes in the fractal behaviour of rainfall has a significant effect on the fractal behaviour of recharge and this in turn results in a small change in the fractal behaviour of groundwater levels. The significant effect on the fractal behaviour of drainage is attributed to the extended recharge periods which correspond to more frequent rain events in rainfall with higher scaling exponents. This effect is more subdued in groundwater level fluctuations due to attenuation of the recharge signal as it percolates through the unsaturated zone.

How to cite: Habib, A., Paschalis, A., Butler, A. P., Onof, C., Bloomfield, J. P., and Sorensen, J. P. R.: Effect of Rainfall Fractal Behaviour on that of Recharge and Groundwater Levels, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-8426,, 2023.

Dnyanesh Borse and Basudev Biswal

Power-law distributions occur in a diverse range of phenomena. Natural drainage networks also exhibit distinctive fractal properties and certain power-law scaling relationships irrespective of the underlined controls, such as geology, topography, and climate. Here we study the distribution of basin areas of continents as well as some islands. We used area-fraction vs. rank distribution, where the area fraction represents the area of a basin with respect to the total landscape area. To obtain the basin area distribution, we used HydroRivers data for the nine continent regions and performed DEM analysis for 12 islands. The results show that basin area distribution follows a power law in the case of all continents with scaling exponent ranging from -1.15 to -1.4. In the case of islands, the majority of them followed power law scaling with exponent ranging from -1.2 to around -2.5; however, distributions of some islands deviated from the power laws.

We also looked at the basin area distribution with the optimal channel network model with all boundary pixels modelled as outlets. We got the scaling exponent around -1.8. Our recently proposed probabilistic model for drainage network evolution (Borse & Biswal, 2023) shows the capability to produce networks with different distributions. This model can capture the varying range of exponents with its flexible parameters. Further studies would be needed to understand the significance of this basin area distribution scaling exponent and whether it could be used as a metric to characterize landscapes.

How to cite: Borse, D. and Biswal, B.: Power law Scaling in Drainage Basin Areas of Independent landscapes, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-11210,, 2023.

Posters virtual: Mon, 24 Apr, 14:00–15:45 | vHall ESSI/GI/NP

Chairpersons: Sergey Turuntaev, Elena Pasternak, Arcady Dyskin
1-6: Non-linear Waves and Fracturing
7: Multi-Fractals in Hydrology and Meteorology
Sergey Turuntaev and Vasily Riga

The conditions for the transition from slow slip to seismic generation motion along a tectonic fault as a result of fluid injection through a well located near the fault are studied.

Movements along the fault caused by fluid injection can occur in the form of slow slips or lead to earthquakes. The implementation of a particular type of movement is dependent on the injection parameters and the fault friction and stress conditions. Numerical calculations were performed in which the consequences of fluid injection lasting from 1.5 months to 6 years were modeled. The calculations varied the total volume of the injected fluid, the flow rate during injection, the rate-state friction law properties of the fault, tangential stresses on the fault. It was found that under certain combinations of fault parameters and fluid flow, seismic generations occur. The transition to such a mode within the framework of the considered model occurs abruptly, a further increase in the injection rate does not lead to an increase in the rate of seismic movement, reaching values of 0.1-1 m/sec, depending on tectonic tangential stresses.

With fixed parameters of the rate-state friction law, the magnitude of the maximum displacement velocity depends on the rate of the pressure perturbation on the fault. Until the sliding velocity reaches a value of the order of 10-6 m/sec, the dependence of the logarithm of the sliding velocity on the rate of the pressure perturbation is linear or close to it, then there is a significant more dramatic increase in sliding velocity depending on the rate of the perturbation growth. The influence of the rate-state friction law parameters on the movements along the fault is not so unambiguous. However, it can be said that the sliding is determined by a combination of the following parameters: the critical length at which the stiffness of the fault section reaches the value of critical stiffness, and the characteristic response time determined by the parameters of the friction law.

How to cite: Turuntaev, S. and Riga, V.: Study of tectonic fault transition from aseismic to seismic slip due to fluid injection, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-16812,, 2023.

Elena Pasternak, Arcady Dyskin, Roman Pevzner, and Boris Gurevich

Field observations show that power spectra of the response to high amplitude harmonic excitation contain peaks at frequencies multiple to the driving frequency, e.g. [1]. This phenomenon is conventionally attributed to the effect of mechanical non-linearity of the Earth’s crust. Given that there exist various types of non-linearity it is important to identify the types of non-linearities that can produce multiple harmonics in response to high power excitation and thus ensure the correct interpretation of the monitoring data.

One type of non-linearity capable of producing multiple resonances is bilinearity of stiffness, the simplest representation of which is a bilinear oscillator – the oscillator with different stiffnesses for compression and tension. In the Earth’s crust the role of bilinear oscillators can be played by pre-existing fractures initially closed by the in-situ compression but capable of being opened by the tensile phase of the applied high amplitude harmonic excitation.

Bilinear oscillators possess multiple resonances, e.g. [2], however these are multiples of the natural frequency. We note that extreme damping effected by the presence of fluids in fractures and porous rocks can quickly eliminate the effect of the natural frequency leaving only the stationary oscillations with the driving frequency in each linear (tensile or compressive) stage of oscillations. The transition from one stage to another is characterised by short transients, which gives rise to multiple spectral peaks. This mechanism is investigated in asymptotics of high damping ratio. It is shown the existence of the following spectral peaks: if f0 is the driving frequency, the peaks will be observed at 2f0, 3f0, 5f0 and further at all odd multiples of f0.

The theory developed is essential for identifying the prevailing mechanisms of non-linearity in the Earth’s crust and determining their parameters.

1. Yurikov, A., B. Gurevich, K. Tertyshnikov, M. Lebedev, R. Isaenkov, E. Sidenko, S. Yavuz, S. Glubokovskikh, V. Shulakova, B. Freifeld, J. Correa, T.J. Wood, I.A. Beresnev and R. Pevzner, 2022. Evidence of nonlinear seismic effects in the earth from downhole distributed acoustic sensors. Sensors 2022, 22, 9382.

2. Dyskin, A.V., E. Pasternak and E. Pelinovsky, 2012. Periodic motions and resonances of impact oscillators. Journal of Sound and Vibration 331(12) 2856-2873.

Acknowledgement. EP, AVD and BG acknowledge support from the Australian Research Council through project DP190103260. RP and BG acknowledge financial support from the Australian Department of Industry, Science and Resources for the 2021 Global Innovation Linkage (GILIII000114) grant and the Sponsors of the Curtin Reservoir Geophysics Consortium.

How to cite: Pasternak, E., Dyskin, A., Pevzner, R., and Gurevich, B.: Harmonics multiple to the driving frequency of damped bilinear oscillators, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6515,, 2023.

Maoqian Zhang, Elena Pasternak, and Arcady Dyskin

Fragmentation of rocks, e.g. splitting into blocks, is a common occurrence at a range of scales from rock fragmentation caused by rockbursts or blasting to blocky rock mass produced by systems of fractures to rubble-pile asteroids. Common in these diverse objects is the ability of blocks (fragments) to assume relatively independent displacement and/or rotation.


Modelling deformation of blocky/fragmented rocks is complicated by the phenomenon of elbowing [1] whereby the rotating block pushes away the neighbouring blocks. The direction of the push can be independent of the direction of block rotation making the problem strongly non-linear (the “absolute value” type non-linearity). In order to investigate elbowing we constructed a simple 1D physical model of a chain of blocks with one translational and one rotational degrees of freedom. It is found that when one block (the initial block) is rotated, the neighbouring blocks may not rotate, only displace, depending on the magnitude of friction and the number of blocks in the chain. A discrete element (3DEC) model of the chain is developed. It shows the conditions of rotation of the blocks and the rotational wave propagation following a pulse rotation of the initial block.


  • Pasternak, E., Dyskin, A.V., Estrin, Y. (2006) Deformations in transform Faults with rotating crustal blocks. Pure Appl. Geophys. 163 2011–2030.


Acknowledgement. The authors are grateful to Dr I. Shufrin and School of Engineering workshop for help with designing and manufacturing of the physical model. EP and AVD acknowledge support from the Australian Research Council through project DP210102224.

How to cite: Zhang, M., Pasternak, E., and Dyskin, A.: Rocks with rotating blocks: 1D displacement, rotation and wave propagation, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6080,, 2023.

Qiang Xie, Weichen Sun, Kai Wu, Zhilin Cao, Xiang Fu, Alessio Fumagalli, and Yuxin Ban

This research aims to study the deformation and fracture behavior of shale by a novel anisotropic regular lattice spring model (ARLSM). The novel ARLSM applies the normal and tangential coupling spring to release the Poisson's ratio limitation in the traditional regular lattice spring model. Meanwhile, a nonlinear strength criterion is introduced into ARLSM to simulate the fracture failure of shale. Two benchmark problems are tested to implement the research. The study shows that ARLSM has larger range of Poisson's ratio and better effects comparing with the existing anisotropic lattice spring model. Moreover, ARLSM can accurately predict the deformation and fracture behavior of shale under different conditions.

How to cite: Xie, Q., Sun, W., Wu, K., Cao, Z., Fu, X., Fumagalli, A., and Ban, Y.: Study of deformation and fracture behavior of shale by a novel anisotropic regular lattice spring model, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6565,, 2023.

Yide Guo, Elena Pasternak, and Arcady Dyskin

Heating of rock surface (e.g., flame heating) induces compressive stresses in the surface layer and tensile stresses of lower magnitude in the layer beneath. If the heating temperature is large enough (around 900 deg for shales), the compressive stresses initiate spallation produced by pre-existing cracks that and extensively grow parallel to the surface under compression. The extensive cracks separate thin layers from different parts of the heated surface which eventually buckle opening a new surface which starts being subjected to flame heating. Then the spallation process repeats itself producing a cavity of approximately cylindrical shape growing into the rock normal to the surface.


The presentation reports the results of tests on flame heating of shales, which demonstrate that the spallation process is accompanied by emergence of a large tensile fracture normal to the surface. In order to check whether the fracture can be produced by tensile thermal stresses induced in the layer situated under the compressed layer we conducted a series of finite element simulations of thermal stresses for different spallation depths (depths of the cavity). The modelling shows that: (1) as the spallation cavity deepens the magnitudes of maximum compressive and tensile stresses remain approximately the same except of two peaks at the spallation depths of about 6% and 30% of the diameter of the heating flame; (2) the magnitude of the maximum tensile stresses is about half of the compressive stress. Given that the spallation strength is about half of the UCS (e.g., [1]) and that the tensile strength is often up to an order of magnitude lower than the UCS, the induced tensile thermal stresses can be considered as sufficient to produce the tensile fracture.


The experiment and computer modelling suggest that the production of tensile fractures is an intrinsic feature of the spallation process. These results can assist in understanding large scale spallation-like processes in the Earth’s crust and design rock cutting based on thermal spallation.


  • Wang, H., A.V. Dyskin, Pasternak, P. Dight and B. Jeffcoat-Sacco, 2021. Fracture mechanics of in-situ spallation. Engineering Fracture Mechanics, 260, 108186.

How to cite: Guo, Y., Pasternak, E., and Dyskin, A.: Thermal spallation and fracturing of rocks produced by surface heating, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6358,, 2023.

Arcady Dyskin and Elena Pasternak

Spallation is a type of surface rock failure under uniaxial and biaxial compression manifested by successive production and ejection of spalls/fragments. This type of failure is observed in laboratory experiments on uniaxial/biaxial compression of rocks and mortar as well as in rock masses. In the latter case spallation is seen in slopes and in the walls of underground openings. In its unstable phase the spallation can lead to such a dangerous phenomenon as strain rockburst.

Spallation is caused by formation and extensive growth of wing cracks parallel to a free surface (e.g., excavation wall) under the applied compressive load. Their growth amplified by the strong interaction with the surface leads to separation of thin layers whose subsequent buckling produces the spalls and opens a new surface. This produces new wing cracks extensively growing parallel to the new surface, thus enabling the process that repeats itself, e.g. [1].

A critical role in this mechanism is played by the interaction of the wing crack with the free surface. The interaction is the stronger the closer the wing crack to the free surface. The closeness to the free surface is limited by the sizes of the largest pre-existing defects seeding the wing cracks. Therefore, the wing cracks inducing each step of spallation are approximately coplanar. Subsequently, the layer separated from the bulk of the rock can be considered as a plate connected to the main part of the rock by bridges formed by intact rock sections remaining between the wing cracks. In the first approximation the effect of bridges can be modelled by Winkler layer [2]. The cracks are assumed to be disc-like; the interaction with the free surface is computed using the beam asymptotics [3].

The velocity of flexural wave propagation depends upon the Winkler layer stiffness and the frequency of oscillations. There exists a minimum frequency, below which the wave does not propagate.  Both parameters depend upon the average crack radius and the number of wing cracks. If the monitoring of the wave velocities and the minimum frequency is complemented by monitoring of the average surface deformation (for instance using non-contact methods such as the digital image correlation) the parameters of the spallation process can be determined, and the approaching buckling phase identified. Results of this research will be instrumental in developing methods of monitoring and predicting strain rockbursts.

1. Wang H, A.V. Dyskin, E. Pasternak, P. Dight and B. Jeffcoat-Sacco, 2022. Fracture mechanics of spallation. Engineering Fracture Mechanics, 260:108186.

2. He, J., Pasternak, E. and A.V. Dyskin, 2020. Bridges outside fracture process zone: Their existence and effect. Engineering Fracture Mechanics, 225, 106453.

3. Dyskin, A.V., L.N. Germanovich and K.B. Ustinov, 2000. Asymptotic analysis of crack interaction with free boundary. J. Solids Structures, 37, 857-886.

4. Lloyd J.R. and Miklowitz, 1962. Wave Propagation in an Elastic Beam or Plate on an Elastic Foundation. J. Applied Mechanics, 459-464.

Acknowledgement. The authors acknowledge support from the Australian Research Council through project DP210102224.

How to cite: Dyskin, A. and Pasternak, E.: Monitoring of spallation processes in rocks by continuous measurements of surface deformation and wave parameters, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-6427,, 2023.

Juan Mauricio Bedoya-Soto and Heli Steven Ocampo-Zapata

The frequency and severity of extreme hydrometeorological events in the Colombian Andes have increased due to the combined effects of climate change and climate variability, with the El Niño-Southern Oscillation (ENSO) being the main contributor. To address this issue and improve hydrologic and hydraulic infrastructure designs, it is necessary to develop better tools for accurately predicting the impact of these events. Hydrological scaling and similarity, based on relatively simple mathematical laws, can synthetically translate the high heterogeneity of hydrological processes into equations that are particularly useful in ungauged catchments, a widespread problem in the Colombian Andes. This research proposes the use of specific hydrological scaling tools, including the Geomorphological Instantaneous Unit Hydrograph (GIUH) and the equivalent Horton-Strahler (H-S) ratios, to calculate peak flows. These ratios express the self-similarity of channels and basins, independent of the threshold area for channel initiation, which the classical bifurcation ratio (RB), length ratio (RL), and area ratio (RA) depend on. Using digital elevation model (DEM) data from NASA's ALOS-PALSAR mission, which provides terrain elevation at a resolution of 12.5m x 12.5m, we analyzed regional patterns of extreme event scaling on various slopes of the Andes Mountains (Colombia) using the GIUH/equivalent H-S theory. With this DEM data, we developed a methodology for automatically extracting the equivalent H-S (RBe, RLe, RAe) in several catchments of the Western, Central, and Eastern ranges that compose the Colombian Andes, while simultaneously validating the self-similarity assumption of their channel networks. Our results highlight the importance of the equivalent H-S ratios as self-similarity indices and regional indicators of the intrinsic relationship between geomorphology and hydrology in the Colombian Andes and their usefulness for hydrological design engineering purposes.

How to cite: Bedoya-Soto, J. M. and Ocampo-Zapata, H. S.: Using Equivalent Horton-Strahler Ratios to Predict Extreme Events in Colombian Andes Catchments, EGU General Assembly 2023, Vienna, Austria, 23–28 Apr 2023, EGU23-10989,, 2023.