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[SSS07-P09] A method for estimating slip distribution and friction constitutive parameters from shear stress distribution along a fault
Keywords:Fault slip, Boundary integral equatuon method, Fault friction constitutive law
In recent years, artificially triggered earthquakes by underground water injection associated with shale oil / shale gas extraction and geothermal power generation become serious problems. To reduce the risk of these triggered earthquakes, we need to investigate the fault rupture process. Especially, constitutive relation of fault friction plays an important role. In the laboratory experiments, it is possible to measure the shear strain distribution close to the 2D fault plane, but it is difficult to measure the slip distribution directly on the two-dimensional fault plane (Fukuyama et al., 2018). Here, we conducted numerical experiments to examine whether these information can be extracted from the near-fault shear strain data such as those obtained by Fukuyama et al. (2018). In this study, we prepare the spatio-temporal distribution of shear stress from the spatio-temporal distribution of shear strain, and investigate whether the spatio-temporal distribution of fault slip and the constitutive law of fault friction can be estimated or not.
We used the boundary integral equation method (Hok and Fukuyama, 2010) to calculate the dynamic rupture propagation on a planar fault. In this method, the fault plane is divided into triangular elements, and the slip rate and shear stress of the entire fault plane are calculated. This method can handle two kinds of problems: 1) to calculate the spatio-temporal distribution of shear stress from the spatio-temporal distribution of fault slip, and 2) to calculate the spatio-temporal distribution of shear stress and slip from the initial shear stress and friction constitutive law. For the friction constitutive law, we use the sliding-weakening friction constitutive law (Ida, 1972; Andrews, 1976). This constitutive law is defined by the static friction, the dynamic friction, and the slip weakening distance.
First, we construct the assumed slip function using the regularized Yoffe function (Tinti et al., 2005). This becomes a true slip function in the experiment. A uniform slip of 0.1 mm is set over the entire fault surface. The slip initiated at the center of the fault plane, and propagated radially from the initiation position with a rupture velocity of 70% of the S-wave velocity. The fault plane is 1.5 m long and 0.5 m wide, which is the same as the experimental sample of Fukuyama et al. (2018). From the spatio-temporal distribution of the slip, the spatio-temporal distribution of the shear stress was calculated using the boundary integral equation of Hok and Fukuyama (2010). Since the calculated shear stress distribution is relative, we need to add the offset to the shear stress value to make the stress distribution at the end of the sliding uniform. This was used as the pseudo-observed shear stress data.
Next, from the pseudo-observed shear stress data, the location and the size of the initial crack were set and the initial stress distribution was extracted. The static friction was fixed to be slightly smaller than the initial stress within the initial crack. The values of static friction, dynamic friction, and slip weakening distance outside the initial crack were searched by a Monte Carlo method, and the slip distribution was calculated by the dynamic rupture propagation program of Hok and Fukuyama (2010). The friction parameters were determined by trial and error by finding the calculated shear stress distribution fits best to that of the pseudo-observed shear stress data. Finally, the slip distributions obtained by using the best friction parameters were compared with the true slip distributions to evaluate the accuracy of the method.
Although the amount of slip and shear stress distribution could be obtained accurately close to the slip initiation point, the error became larger as we moved away from the rupture initiation point. This is due to the assumption that the frictional constitutive law parameters are uniform on the fault. From the regularized Yoffe function propagating at a constant speed, the constitutive parameters are expected to be spatially variable, and it is necessary to incorporate this point into the method. In future, we would like to apply this method to actual experimental data to estimate directly the slip distribution on a two-dimensional fault plane.
References
Andrews, J. D. (1974) https://doi.org/ 10.1029/JB081i032p05679
Fukuyama, E. et al. (2018) https://doi.org/10.1016/j.tecto.2017.12.023
Hok, S. and E. Fukuyama (2010) https://doi.org/10.1111/j.1365-246X.2010.04835.x
Ida, Y, (1972) https://doi.org/10.1029/JB077i020p03796
Tinti, E. et al. (2005) https://doi.org/10.1785/0120040177
We used the boundary integral equation method (Hok and Fukuyama, 2010) to calculate the dynamic rupture propagation on a planar fault. In this method, the fault plane is divided into triangular elements, and the slip rate and shear stress of the entire fault plane are calculated. This method can handle two kinds of problems: 1) to calculate the spatio-temporal distribution of shear stress from the spatio-temporal distribution of fault slip, and 2) to calculate the spatio-temporal distribution of shear stress and slip from the initial shear stress and friction constitutive law. For the friction constitutive law, we use the sliding-weakening friction constitutive law (Ida, 1972; Andrews, 1976). This constitutive law is defined by the static friction, the dynamic friction, and the slip weakening distance.
First, we construct the assumed slip function using the regularized Yoffe function (Tinti et al., 2005). This becomes a true slip function in the experiment. A uniform slip of 0.1 mm is set over the entire fault surface. The slip initiated at the center of the fault plane, and propagated radially from the initiation position with a rupture velocity of 70% of the S-wave velocity. The fault plane is 1.5 m long and 0.5 m wide, which is the same as the experimental sample of Fukuyama et al. (2018). From the spatio-temporal distribution of the slip, the spatio-temporal distribution of the shear stress was calculated using the boundary integral equation of Hok and Fukuyama (2010). Since the calculated shear stress distribution is relative, we need to add the offset to the shear stress value to make the stress distribution at the end of the sliding uniform. This was used as the pseudo-observed shear stress data.
Next, from the pseudo-observed shear stress data, the location and the size of the initial crack were set and the initial stress distribution was extracted. The static friction was fixed to be slightly smaller than the initial stress within the initial crack. The values of static friction, dynamic friction, and slip weakening distance outside the initial crack were searched by a Monte Carlo method, and the slip distribution was calculated by the dynamic rupture propagation program of Hok and Fukuyama (2010). The friction parameters were determined by trial and error by finding the calculated shear stress distribution fits best to that of the pseudo-observed shear stress data. Finally, the slip distributions obtained by using the best friction parameters were compared with the true slip distributions to evaluate the accuracy of the method.
Although the amount of slip and shear stress distribution could be obtained accurately close to the slip initiation point, the error became larger as we moved away from the rupture initiation point. This is due to the assumption that the frictional constitutive law parameters are uniform on the fault. From the regularized Yoffe function propagating at a constant speed, the constitutive parameters are expected to be spatially variable, and it is necessary to incorporate this point into the method. In future, we would like to apply this method to actual experimental data to estimate directly the slip distribution on a two-dimensional fault plane.
References
Andrews, J. D. (1974) https://doi.org/ 10.1029/JB081i032p05679
Fukuyama, E. et al. (2018) https://doi.org/10.1016/j.tecto.2017.12.023
Hok, S. and E. Fukuyama (2010) https://doi.org/10.1111/j.1365-246X.2010.04835.x
Ida, Y, (1972) https://doi.org/10.1029/JB077i020p03796
Tinti, E. et al. (2005) https://doi.org/10.1785/0120040177