Spring-slider System Research Articles

Toward quantitative characterization of simulated earthquake-cycle complexities

Earthquake cycle simulations based on the rate-and-state friction formulation are evolutions of nonlinear dynamical systems (NDS). The term “cycle” implies an overall stable structure that is a phase-space attractor naturally traced out by trajectories of NDS as it evolves. Quantitatively characterizing these attractors should be a basis for measuring complexities of the simulated earthquake cycles, i.e. to determine if and how regular or chaotic they are. I first revisit the textbook-standard quasi-dynamic spring-slider system from an NDS perspective, explicitly showing the attractors, their relationship with the parameters of the NDS, and how they can be characterized taken advantage of their low-dimensionality while aiming to extend the analysis to high-dimensionality. I evaluate two approaches, computing the Lyapunov exponents (LEs) and measuring correlation dimensions, with the simple spring-slider and earthquake-cycle examples whose phase-space attractors can be visually verified. I conclude LEs are too inconvenient and computationally expensive to use whereas measuring correlation dimensions is an easy and effective approach even with highly non-uniform time sampling present in all simulations. For earthquake-cycle simulations, an attractor reconstruction is performed based on Taken’s theorem to corroborate my correlation-dimension results. The current method is limited in its ability to detect chaos in a dichotomous manner, which illuminates the direction for future study. An improving ability to quantitatively characterize earthquake-cycle simulations as an overall stable structure offers new opportunities to understand exotic seismic observations such as slow-slip events and enables more informative comparison with real data, particularly from paleoseismology, which could have far-reaching implications in earthquake forecasting.

Analysis of Fluid-Injection-Induced Seismicity Using a Dynamic Sliding Model Incorporating the Rate- and State-Dependent Friction Law

Summary Earthquakes can be triggered by fluid injection into underground formations. Fluid injection can cause large changes in the underground volume that exert stresses on nearby preexisting faults, leading to seismic activity. Assuming an increase in underground development activities in the future, our understanding of the mechanism underlying induced seismicity must be improved, and methods must be developed to properly assess the risk of seismic events. The objective of this study is to develop a seismicity prediction model that calculates the magnitude and timing of triggered earthquakes or seismic events occurring during various subsurface fluid injection activities. We developed an injection-induced seismicity analysis model that predicts the dynamic earthquake nucleation caused by changes in stress and pore pressure that occur during various subsurface activities. The governing equations consisting of the dynamic motion of the poroelastic spring-slider system, rate and state friction laws and pore pressure diffusion equation were solved using the embedded semi-implicit Runge-Kutta (SIRK) method. The dynamic sliding model was also incorporated into the finite element method (FEM) model, considering the variations in the stresses and pore pressures in the formation. A field case study was also conducted to compare the model results with typical microseismicity responses observed from hydraulic fracturing treatments in shale fields. Contrary to the popular understanding derived from Amonton’s law, the dynamic friction model revealed that a large normal stress on the fault leads to rapid sliding. A larger normal stress accumulates a large amount of elastic energy until it slips owing to fluid injection, nucleating large seismic waves. The poroelastic spring-slider model estimated reasonable microseismic magnitudes for hydraulic fracturing treatment but overestimated the time required to trigger a microseismic event under field conditions. To improve the analysis results, the poroelastic spring-slider model was coupled with a linear elastic FEM that considered the complex interplay of stress changes from hydraulic fracturing and the associated pore pressure variation in the formation. Compared with the field data, the coupled simulation model estimated a reasonable timing for the induced microseismic events when the increasing pore pressure during hydraulic fracturing penetrated deep into the formation. These findings suggest the existence of permeable natural fractures in the formation, which intensify early frictional sliding during treatment. The seismicity prediction model presented in this study simulates the magnitude and timing of seismic nucleation, helping to manage and mitigate the environmental impacts of induced seismicity during various subsurface development activities, such as oil and gas extraction, hydraulic fracturing, geothermal, and carbon dioxide sequestration. Moreover, the case study results imply that the time series of seismic events predicted by the model can be used to understand the possible fracture geometry and extent of fluid invasion for field applications. Key Terms and Phrases induced seismicity, subsurface fluid injection, rate- and state-dependent friction law, embedded semi-implicit Runge-Kutta method, finite element analysis

Assessing Numerical Simulation Methods for Reinforcement–Soil/Block Interactions in Geosynthetic-Reinforced Soil Structures

The purpose of this study is to assess effects of two different simulation methods (i.e., interfaces with a single spring-slider system and interfaces with double spring-slider systems) for interactions between reinforcement and the surrounding medium on the performances of geosynthetic-reinforced soil (GRS) structures when conducting numerical analyses. The fundamental difference between these two methods is the number of the spring-slider systems used to connect the nodes of structural elements simulating the geosynthetic reinforcement and the points of solid grids simulating the surrounding medium. Numerical simulation results of pull-out tests show that both methods reasonably predicted the pullout failure mode of the reinforcement embedded in the surrounding medium. However, the method using the interfaces with a single spring-slider system could not correctly predict the interface shear failure mode between the geosynthetics and surrounding medium. Further research shows that these two methods resulted in different predictions of the performance of GRS piers as compared with results of a laboratory load test. Numerical analyses show that a combination of interfaces with double spring-slider systems for reinforcement between facing blocks and interfaces with a single spring-slider system for reinforcement in soil resulted in the best performance prediction of the GRS structures as compared with the test results. This study also proposes and verifies an equivalent method for determining/converting the interface stiffness and strength parameters for these two methods.

Precursory predictors of the onset of stick-slip frictional instability

Defining precursory phenomena that forewarn the onset of stick slip events remains an elusive but central goal in earthquake and other failure forecasting. We present a method to define the timing of instability by separately decoupling velocity and inertial effects in a spring-slider system. System state is defined in phase-space by velocity and inertia that in turn control critical stiffness and define the onset of instability while sliding velocity V = VLk/(k + dF/du) remains finite. A general energy analysis defines velocity and inertial effects (coefficients) that act against each other to promote/suppress instability by respectively increasing/decreasing critical stiffness. Repetitive stick-slip experiments define features of stick-slip cycles and identify a precursory trend in accelerating slip that precedes the onset of unstable sliding. We represent this precursory acceleration (dV/dt) immediately preceding instability by a general power-law relation dV/dt = AVα that reduces to V = B(ci + ti-t)-β. This represents the “true” timing of the onset of stick slip, ti relative to that “projected” from the linearization of the precursory data, tf, with tf=ci + ti. ci represents the small offset between the “predicted” and “true” timing of the onset of stick slip resulting from velocity and inertial effects and may be considered as the error in prediction. This error is one-order of magnitude smaller (∼3 ms) than the lead-time that forewarns of the event (∼40 ms) in our experiments and thus is useful as a predictor. Three independent methods using power-law, linear and criticality relationships confirm the fidelity of the timing of the slip instability transition evident in the stick slip data. This theoretical treatment suggests that the underlying physical meaning of the parameter c in the modified Omori law R∼(c + tM-t)-q is the time of the main shock in advance of the timing of the singularity predicted for an ideal response.

Obstacle contained system (OCS) design method and its application in valve core orifice design of pilot-control globe valve

With the development of the society and the times, traditional parametric design methods are witnessing a severe challenge due to the more and more complex physical systems. Thus, developing novel parametric analysis methods is very important for dealing with complex physical systems, refining useful parameters from numerous data, and proposing accurate prediction formulas. A spring slider system, a direct-current circuit system, a pipeline pressure drop system and a steady heat transfer model of flat plate system were described from the point of systemic parametric analysis method. Then, the key physical parameters in above four systems were summarized. Based on the comparative results, a novel systemic parametric design method, obstacle contained system (OCS) design method, was proposed. The OCS was made up of three elements:an obstacle element, a pass body element and a D-value element. With an abundant accurate data pole, the OCS design method could build the direct relationship of the obstacle element and the D-value element, which meant the simplification of the physical models and much easier to get relatively accurate results. Meanwhile, the design of pilot-control globe valve orifice was checked with both the OCS design method and the numerical simulation. The diameter of orifice on the valve core could influence the pressure difference and the maximum vapor rate inside pilot-control globe valves. Achieved by two different methods, the OCS design method and the numerical simulation, the results showed that the effects of orifice diameters on the pressure difference and the maximum vapor rate under different inlet velocities, were within 2% errors, which was reasonable and acceptable for the engineering application. In other words, the OCS design method was credible for parametric analysis. In future, the OCS design method has a broad application prospect to analyze various types of physical models especially in the era of big data.

Effect of varying normal stress on stability and dynamic motion of a spring-slider system with rate- and state-dependent friction

Incorporating rate and state friction laws, stability of linearly stable (i.e., with stiffness greater than the critical value) spring-slider systems subjected to triggering perturbations was analyzed under variable normal stress condition, and comparison was made between our results and that of fixed normal stress cases revealed in previous studies. For systems associated with the slip law, the critical magnitude of rate steps for triggering unstable slips are found to have a similar pattern to the fixed normal stress case, and the critical velocity steps scale with a/(b − a) when k = kcr for both cases. The rate-step boundaries for the variable normal stress cases are revealed to be lower than the fixed normal stress case by 7 %–16 % for a relatively large α = 0.56 with (b − a)/a ranging from 0.25 to 1, indicating easier triggering under the variable normal stress condition with rate steps. The difference between fixed and variable normal stress cases decreases when the α value is smaller. In the same slip-law-type systems, critical displacements to trigger instability are revealed to be little affected by the variable normal stress condition. When k ≥ kcr(V*), a spring-slider system with the slowness law is much more stable than with the slip law, suggesting that the slowness law fits experimental data better when a single state variable is adopted. In stick-slip motions, the variable normal stress case has larger stress drops than the constant normal stress case. The variable normal stress has little effect on the range of slip velocity in systems associated with the slowness law, whereas systems associated with the slip law have a slowest slip velocity immensely smaller than the fixed normal stress case, by ~10 orders of magnitude.

Stability analyses and numerical simulations of the single degree of freedom spring-slider system obeying the revised rate- and state-dependent friction law

The linear and nonlinear stabilities of the single degree of freedom spring-slider system which accords to the revised rate- and state-dependent friction law (RSF) (Nagata et al. J Geophys Res 117 (B2):B2314, 2012) are analyzed. The revised ageing law obtained by Nagata et al. (J Geophys Res 117 (B2):B2314, 2012) incorporates the effects of changes in shear stress. Numerical simulations on the cyclic stick–slip motions of the system are developed and compared with the results of the systems according to the original ageing law or the slip law. From the insight of the stability analyses and numerical simulations, it is found that the revised ageing law integrates the “healing effect” feature of the original ageing law and the dynamic slip features of the slip law. In the stick–slip cycles, the velocity decreases with non-constant states during the dynamic overshoot for the revised ageing law, which is different from both the original ageing law and the slip law. Although the revised ageing law concluded from the low velocity friction experiments cannot account for the earthquake-like high velocity friction experiments, it can be used in earthquake nucleation with low velocity. The stability analyses and the results of numerical simulations are helpful to understanding the implications of the revised ageing law.

Stability analysis of slip of a one-body spring-slider model in the presence of thermal pressurization

This study is focused on stability of slip in a one-body spring-slider system, with stiffness k of the slider, in the presence of thermal pressurizationfor two end-member models of thermal pressurization, i.e., the adiabatic-undrained- deformation (AUD) model and slip-on-a-plane (SOP) model. Analytic studies based on the functions of frictional stress, t<sub>f</sub>, versus slip, d, of the two models show that k<sub>cr</sub>=|dt<sub>f</sub>/dd| at d=0 is the critical stiffness of the system. k<sub>cr</sub> is a finite positive value for the AUD model and infinity for the SOP model. Slip is stable when k>k<sub>cr</sub> and unstable when k<k<sub>cr</sub>. Analytic stability conditions of slip from the equations of motions of a one-body spring-slider model with the t<sub>f</sub>–d function of the ADU model are also those from |dt<sub>f</sub>/dd| at d=0. Under ADU thermal pressurization, modeled final slip and maximum velocity for k=5 to 31 MPa/m, with k<sub>cr</sub>=30 MPa/m, both decrease with increasing k. The motions become very weak when k≥k<sub>cr</sub>, thus being unable to result in unstable motions. The phase portraits of v versus d for k<30 MPa/m show the existence of a non-zero unstable fixed point.

Effects of permeability and porosity evolution on simulated earthquakes

Numerical simulations are a fundamental tool to access the typical conditions attained during earthquake instabilities and to simulate the large number of dissipative processes taking places during faulting. In this study we consider a single-degree-of-freedom spring-slider system, a simplified fault model which can describe the whole seismic cycle and the dynamics of a fault with spatially homogeneous properties. We assume a rate- and state-dependent friction in which we incorporate the effects of pore fluid pressure, thermally-pressurized as a consequence of the frictional heat produced during sliding. We explore, in a single framework, the role of the time variations of the porosity, permeability or both, ultimately leading to changes in hydraulic diffusivity, which has been recognized as one of the key parameters in thermally-pressurized faults. Our synthetic ruptures show that the changes in the hydraulic diffusivity only due to porosity variations do not markedly affect the earthquake recurrence (cycle time), the traction evolution and the thermal history of the fault. On the contrary, when the evolutions of both the porosity and the permeability are accounted for, the cycle time is significantly reduced. This result has a clear implication in the context of the hazard assessment.

Transient behavior and stability analyses of halite shear zones with an empirical rate-and-state friction to flow law

Generation of large earthquakes involves with behaviors of whole plate boundaries or faults from brittle to ductile regimes. This paper reports stability analyses of halite shear zones using a recently developed rate-and-state friction to flow law with an emphasis on the behaviors across the brittle–ductile transition. The law smoothly connects the friction law with pressure-insensitive flow law without any additional constitutive parameter. Behavior upon a velocity step is characterized by an instantaneous change in shear resistance followed by transient behavior toward a steady-state. These transient behaviors are in opposite directions between friction and flow regimes, resulting in variable transient behaviors across the brittle–ductile transition. Linear stability analyses of a spring-slider system around steady-state solutions predict pressure and temperature conditions for unstable fault motion that are consistent with experimental results. The condition for potential instability is not equal to, but includes that for rate-weakening. A nonlinear analysis at the stable-unstable boundary has revealed that a sub-critical Hopf bifurcation takes place and thus a permanently sustained oscillation around a destabilized steady-state solution does not exist although experimental results suggest it. This issue deserves further study including the investigation of the friction law and construction of a physical model for brittle–ductile transition.

Correction to “The role of porosity evolution in frictional instabilities: A parametric study using a spring-slider dynamic system”

Correction to “The role of porosity evolution in frictional instabilities: A parametric study using a spring-slider dynamic system”

One-dimensional dynamical modeling of slip pulses

Propagating slip pulses in earthquake ruptures (Heaton, 1990) are studied based on a one-dimensional N-degree-of-freedom dynamical spring-slider system with a constant ratio of static to dynamic frictional forces. Results show that for almost all cases in the study, the rise time of ruptures at a particular site is much smaller than the duration of ruptures along the whole fault. A propagating slip pulse is generated along the fault. Hence, simulation results are in agreement with Heaton's propagating slip-pulse model.

High-order synchronization of stick–slip process: experiments on spring–slider system

In the last years it has been shown that the synchronization and triggering of dynamic events by weak external forcing is ubiquitous and is observed in biological systems, lasers, electronic networks, etc. In the present paper, new experimental data on the phase synchronization in frictional system induced by a weak electromagnetic or mechanical periodic forcing are analyzed. For quantitative analysis of stick–slip time series, modern tools of nonlinear dynamics were used. Stick–slip events were identified by recording acoustic emissions, which accompany slip displacements. The spring–slider system in stick–slip regime is considered as a proxy of active tectonic fault, generating earthquakes.

Phase synchronization of slips by periodical (tangential and normal) mechanical forcing in the spring-slider model

In the present study, the character of slip regimes under weak external periodical (tangential and normal) mechanical forcing has been investigated in a laboratory spring-slider system. We report the experimental evidence of phase synchronization in a slip dynamics, induced by the external periodic mechanical impact. At certain conditions, we have a stick-slip effect in the spring-slider system. To describe this effect, we can use rate-and state-dependent friction law.

Electromagnetic Synchronization of Slip

Character of changes of slip regimes in experiments on laboratory spring-slider system under external weak periodical forcing have been investigated. Here we report the experimental evidence of phase synchronization in slip dynamics, induced by external periodic electromagnetic (EM) impact.

Phase synchronization of slip in laboratory slider system

Abstract. In the present study the character of slip regimes in laboratory spring-slider system under weak external periodical forcing has been investigated. We report the experimental evidence of phase synchronization (PS) in a slip dynamics, induced by the external periodic electromagnetic (EM) impact. The quality of synchronization depends on the intensity and frequency of the applied field; the corresponding Arnold's tongue region is constructed. Application of special techniques (measuring phase differences, phase diffusion coefficient, Shannon entropy, Recurrence Quantification Analysis) allows quantitative assessment of the strength of synchronization of microslips with EM impact. It is also shown that the character of power law relationship in acoustic emission amplitude (energy) distribution also undergoes significant changes at changing excitation intensity.

One-dimensional dynamic simulations of slip complexity of earthquake faults

Slip complexity of earthquake faults is studied based on an N-degree-of-freedom dynamical spring-slider system in the presence of slip-law-type, velocity- and state-dependent friction. Simulation results based on such a friction law show that slip complexity depends on the inhomogeneous distribution of the breaking strengths (including its pattern and degree) along the fault and nonlinear velocity- and state-dependent friction. However, for the given model parameters the former is more important than the latter in controlling slip complexity. Frictional effects obviously appear only when the distribution of the breaking strengths is inhomogeneous. In addition, the stiffness ratio, defined as the ratio of the coil spring strength, Kc, to the leaf spring strength, Kl, is also a factor in controlling slip complexity.

Effects of loading velocity, stiffness, and inertia on the dynamics of a single degree of freedom spring-slider system : Gu, Y; Wong, T F J Geophys ResV96, NB13, Dec 1991, P21677–21691

Effects of loading velocity, stiffness, and inertia on the dynamics of a single degree of freedom spring-slider system : Gu, Y; Wong, T F J Geophys ResV96, NB13, Dec 1991, P21677–21691