Numerical investigation of heating effect on the earthquake faulting based on the Chester-Higgs model
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摘要: 速率和状态相依赖的摩擦定律是本文采用的重要定律。结合Chester-Higgs摩擦模型和McKenzie-Brune摩擦生热模型，在一维弹簧-滑块-断层近似模型下，利用四阶变步长的Dormand-Prince算法，研究探讨了断层摩擦生热对断层演化的影响。结果表明：与忽略温度影响的情形相比，摩擦生热造成的温度上升可导致断层滑移时刻的略微提前，并伴随着摩擦系数和状态变量的下降，同时也使得断层的滑移量和应力降略有减小，而滑移速率有所增大；另外，在考虑温度影响时，有效正应力和临界滑移距离也会影响断层的演化过程，断层上的有效正应力越大，断层失稳时刻越提前，温度上升越明显；断层的临界滑移距离越大，断层失稳时刻则越迟，温度上升越显著，但当临界滑移距离超过5 cm时，具有不同临界滑移距离的断层，失稳时的温度则基本保持一致。
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关键词:
- Chester-Higgs摩擦模型 /
- McKenzie-Brune摩擦生热模型 /
- 一维弹簧-滑块-段层模型 /
- 摩擦生热 /
- 有效正应力 /
- 临界滑移距离
Abstract: Rate- and state-dependent friction （RSF） law is an empirical law derived from labo-ratory experiments related to rock friction. RSF law has been used to quantitatively describe complex fault friction processes. Currently, it has emerged as the theoretical basis for the study of seismogenesis and earthquake faulting. With a combination of the Chester-Higgs friction model and the McKenzie-Brune frictional heat generation model, in this study we have investi-gated the effect of frictional heating process on the fault temporal evolution based on a spring-slider-fault system subjected to a rate- and state-dependent friction law. The system equations are solved efficiently by Dormand-Prince method with adaptive steps. The results show that, compared with the case in which the temperature effect is neglected （unheated fault）, the rise of temperature caused by frictional heating can lead to a slight time advance of fault instability, accompanied by abrupt decreases of the friction coefficient and state variable, respectively. In the case when the temperature effect is taken into consideration （heated fault）, the slip and stress drop on the fault are slightly smaller than that on the unheated fault, while the slip rate becomes larger. In addition, the effective normal stress and critical slip distance can also affect the fault temporal evolution. The greater the effective normal stress on the heated fault is, the earlier the fault instability occurs, accompanied with higher temperature rising. The larger the critical slip distance of the heated fault is, the later the fault instability occurs with a significant temperature increase. However, when the critical slip distance is larger than 5 cm, the peak temperatures are almost the same when the fault is unstable. -
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图 1 断层力学模型
（a） 一维断层模型；（b） 一维弹簧-滑块-断层模型
Figure 1. Schematic illustration of the mechanical behavior of earthquake fault
（a） 1D fault model；（b） 1D spring-slider-fault model
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图 3 小时间尺度下失稳前后断层模型的演化
（a） 滑移速率
${\dot \delta}$ 演化图；（b） 摩擦系数μ演化图；（c） 状态变量Θ演化图；（d） 温度T演化图Figure 3. Simulated time histories of fault evolution from small time scale
The evolution of system around the onset of instability is shown. Figs. （a），（b），（c） and （d） show the evolutions of slip rate
${\dot \delta}$ ，frictional coefficient μ，state variable Θ and temperature change T，respectively![]()
图 4
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（a） 摩擦系数-滑动速率相图；（b） 摩擦系数-位移相图；（c） 位移-滑动速率相图
Figure 4.
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图 2 大时间尺度下断层模型的演化图
（a） 滑移速率
${\dot \delta}$ ；（b） 摩擦系数μ；（c） 状态变量Θ；（d） 温度TFigure 2. Simulated time histories of fault evolution from large time scale
（a），（b），（c） and （d） show the evolutions of slip rate
${\dot \delta}$ ，frictional coefficient μ，state variable Θ and temperature change T，respectively![]()
图 5 与温度改变量有关的相图
（a） 温度改变量-滑动速率相图；（b） 温度改变量-位移相图
Figure 5. Phase diagram related to temperature change
（a） Phase diagrams of slip rate versus temperature change; （b） Slip displacement versus temperature change
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图 6 不同有效正应力对应的断层模型的演化（小时间尺度）
（a） 滑移速率
${\dot \delta}$ 演化图；（b） 摩擦系数μ演化图；（c） 状态变量Θ演化图；（d） 温度T演化图Figure 6. Simulated time histories of fault evolution corresponding to different effective normal stresses （small time scale）
Figs. （a），（b），（c） and （d） show the evolutions of slip rate
${\dot \delta}$ ，frictional coefficient μ，state variable Θ and temperature change T，respectively![]()
图 7 有效正应力对应力降Δτ （a）、断层半径rc （b）、地震矩M0 （c）和矩震级MW （d）的影响
Figure 7. Influences of effective normal stress on variation of stress drop Δτ （a），fault radius rc （b），seismic moment M0 （c） and moment magnitude MW （d）
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图 8 不同临界滑移距离所对应的断层模型的演化（大时间尺度）
（a） 滑移速率
${\dot \delta}$ 演化图；（b） 摩擦系数μ演化图；（c） 状态变量Θ演化图；（d） 温度T演化图Figure 8. Simulated time histories of fault evolution corresponding to different critical slip distances （large time scale）
Figs. （a），（b），（c） and （d） show the evolutions of slip rate
${\dot \delta}$ ，frictional coefficient μ，state variable Θ and temperature change T，respectively![]()
图 9 临界滑移距离对地震参数的影响
（a）—（d）分别为应力降Δτ、断层半径rc、地震矩M0和矩震级MW随临界滑移距离Dc的变化
Figure 9. Influences of critical slip distance on seismic parameters
Figs.（a） to （d） are variation of stress drop Δτ，fault radius rc，seismic moment M0 and moment magnitude MW plotted against critical slip distance Dc
表 1 一维弹簧-滑块-断层模型参数的定义和取值
Table 1 Definitions and values of parameters for 1D spring-slider-fault model
参数 含义 取值 参数 含义 取值 a 直接影响系数 0.012 ${\dot \delta }$ u
失稳速率 1 m/s b 演化影响系数 0.017 T* 参考温度 550 K σ 有效正应力 100—1 000 MPa Tini 初始温度 550 K Dc 临界滑移距离 0.01—10 cm Qa 直接影响的表面激活能 105 J/mol k/kc 断层刚度与临界刚度的比值 ≈1 Qb 演化影响的表面激活能 105 J/mol μ0 T＝T*时以 ${\dot \delta} $ ＝
${\dot \delta }$ *稳定滑动的摩擦系数
0.6 R 气体常数 8.314 μini 初始摩擦系数 0.623 P 介质密度 2 600 kg/m3 ${\dot \delta} $ 0
远场加载速率 3.5 cm/a c 介质比热 1 000 J/（kg·K） ${\dot \delta }$ *
参考速率 3.5 cm/a κ 介质固体热扩散系数 10−6 m2/s ${\dot \delta }$ ini
初始速率 0.035 cm/a 
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