## 1 Introduction

## 2 Geothermal Power Plants Information

No. | GPP zone | Geothermal site | Country | Starting year | Longitude window (°) | Latitude window (°) | Maximum injection well depth (km) |
---|---|---|---|---|---|---|---|

1 | Geysers | Geysers | USA | 1960 | − 123.2 to − 122.39 | 38.6 to 38.9 | 3.7 |

2 | Imperial | Salton Sea | USA | 1982 | − 115.78 to − 115.41 | 32.8 to 33.4 | 3.2 |

Brawley | USA | 1982 | |||||

3 | Rhineland | Insheim | Germany | 2012 | 7.42 to 8.46 | 48.76 to 49.26 | 5.0 |

Landau | Germany | 2007 | |||||

Soultz-sous-Forêts | France | 2008 | |||||

Rittershoffen | France | 2016 | |||||

4 | Kawerau | Kawerau | New Zealand | 1993 | 176.65 to 176.85 | − 37.95 to − 35.25 | 3.0 |

## 3 Earthquake Catalogs

^{1}For the selected GPP zones, different online credible sources of earthquake event catalogs are used. For the Geysers and Imperial zones, the United States Geological Survey (2020) is used. In the case of the Rhineland zone, the data published by the International Seismological Center catalog (2020) is adopted. For the Kawerau GPP zone, the catalog presented by the Geological Hazard Information for New Zealand (2020) is employed to collect the microseismic data. Only some criteria are taken into account to select the seismic data: (1) Events related to the time period of GPP operation are considered for this study; (2) The events that occurred within the shallow focal depth (around 1−10 km) are collected based on the depth of GPP wells varying from 500 m to 5 km (see Table 1); and (3) A spatial window with a radius of approximately 25 km around the power plants is considered as a surrounding possible area of earthquake occurrence.

_{W}), local magnitude (M

_{L}), and duration magnitude (M

_{D}). To homogenize all data, scales of M

_{D}and M

_{L}are converted to M

_{W}, which is more familiar to the earthquake and structural engineers and scientists who are the target community of the proposed model. Different equations (Table 2) are applied to convert the magnitudes for each of the GPP regions.

Geothermal power plant (GPP) zone | Reported magnitude in earthquake catalog | Conversion equation(s) | References |
---|---|---|---|

Imperial | M _{W}, M_{L}, M_{D} | \({M}_{D}=\left(1.061\pm 0.02\right){M}_{W}+0.11\) \(\frac{{M}_{L}}{{M}_{W}}=\left\{\begin{array}{c}\begin{array}{cc}1.04\pm 0.030& {M}_{W}\ge 2.5\end{array}\\ \begin{array}{cc}1.46\pm 0.022& {M}_{W}<2.2\end{array}\end{array}\right.\) | Staudenmaier et al. (2018) |

Geysers | M _{W}, M_{D} | \({M}_{W}=0.9{M}_{D}+0.47\pm 0.08\) | Edwards and Douglas (2014) |

Kawerau | M _{W}, M_{L} | \({M}_{L}=(0.88\pm 0.03){M}_{W}+(0.73\pm 0.2)\) | Ristau (2009) |

Rhineland | M _{W}, M_{L}, M_{D} | \({M}_{W}=0.0376{M}_{L}^{2}\)+0.646 \({M}_{L}+0.53\) \({M}_{W}=1.472{M}_{D}-1.49\) | Grünthal et al. (2009) |

_{C}) of each catalog is calculated for the 10 year time-windows using the entire range magnitude method (Ogata and Katsura 1993), which is more comprehensive and stable than many other existing methods in the literature (Woessner and Wiemer 2005). The uncertainties of the completeness magnitude are also considered by obtaining their standard deviation using the Monte Carlo bootstrapping method (Efron and Tibshirani 1993). The results for each of the GPP zones during different time windows are shown in Table 3 along with the maximum moment magnitudes obtained by Kijko’s (2004) methodology. With the passing of time, the M

_{C}decreases in all catalogs, which implies the improvement of seismic networks responsible for recording the events during the decades since a GPP began operation. Fortunately, after eliminating the events with magnitudes lower than M

_{C}, an acceptable number of events remain for each of the GPP zones for further analysis (Table 3). The spatial distribution of the remaining data, showing their magnitudes, is depicted in Fig. 3. Clearly, earthquakes with higher magnitudes occur closer to the GPP stations’ wells.

Time window | Imperial | Geysers | Kawerau | Rhineland | ||||
---|---|---|---|---|---|---|---|---|

M _{C} | M _{max} | M _{C} | M _{max} | M _{C} | M _{max} | M _{C} | M _{max} | |

2011−2020 | 1.00±0.04 | 1.98 | 1.40±0.03 | 3.97 | 1.50±0.05 | 3.20 | 1.20±0.08 | 2.21 |

2001−2010 | 1.20±0.08 | 5.04 | 1.70±0.10 | 3.99 | 1.50±0.08 | 4.18 | 1.90±0.12 | 3.34 |

1991−2000 | 1.20±0.15 | 5.08 | 1.80±0.17 | 3.98 | 1.90±0.15 | 2.55 | – | |

1981−1990 | 1.70±0.36 | 3.29 | 1.80±0.31 | 3.91 | – | – | ||

1971−1980 | 1.70±0.42 | 3.50 | 1.90±0.35 | 3.80 | – | – | ||

Total number of events after applying M _{C} filtering | 5,660 | 33,175 | 943 | 204 |

## 4 Statistical Analysis of Induced Seismicity

_{W}), focal depth (D), and the distance of the epicenter of seismic events to the GPP site (R). All of these parameters are important and effective in the imposed seismic risk to the built environments in the vicinity of GPPs. M

_{W}is implicitly a measure of released energy in these earthquakes, while focal depth can show the relationship between the depth of injection wells and the hypocenter of earthquakes. In addition, the distance of the event epicenter to the GPP site shows the radius around the GPP zone, in which the earthquake occurrence is probable.

_{C}.

IR ^{*} (l/s) | Geysers | Imperial | Kawerau | Rhineland | ||||
---|---|---|---|---|---|---|---|---|

a | b | a | b | a | b | a | b | |

0−500 | 1.05±0.06 | 3.50±0.23 | 1.26±0.04 | 3.11±0.25 | 0.73±0.04 | 3.18±0.35 | 0.82±0.06 | 3.07±0.29 |

500−1,000 | 1.33±0.07 | 5.54±0.41 | 1.28±0.06 | 4.09±0.34 | 1.07±0.06 | 4.43±0.42 | ||

1,000−1,500 | 1.28±0.07 | 5.94±0.46 | 1.24±0.05 | 4.15±0.39 | ||||

1,500−2,000 | 1.26±0.09 | 5.85±0.39 | 1.06±0.04 | 4.37±0.33 |

Study | GPP location | Flow rate (l/s) | Duration of date recording (years) | Distance range (km) | Focal depth range (km) | Magnitude range |
---|---|---|---|---|---|---|

Megies and Wassermann (2014) | Bavaria, Germany | < 120 | 4 | < 4 | 4.2−5.1 | < 2.4 (M _{L}) |

Majer et al. (2007) | Geysers, USA | < 4,350 | 40 | – | – | < 4.6 (M _{L}) |

Cooper Basin, Australia | < 48 | 4 | – | – | < 3.7 (M _{L}) | |

Kwiatek et al. (2015) | Geysers (North western area), USA | < 115 | 7 | < 2.5 | 1.0−3.0 | < 3.8 (M _{D}) |

Cardiff et al. (2018) | Brady, USA | < 580 | 6 | < 2.0 | 1.0−3.5 | < 2.2 (M _{W}) |

Cheng and Chen (2018) | Salton Sea, USA | – | 6 | < 7.0 | 0.1−7.0 | < 4.0 |

Ellsworth (2019) | Pohang, South Korea | < 12 | 2.5 | – | 2.0−8.0 | < 5.4 (M _{L}) |

This study | Imperial | < 2,500 | 53 | < 24.0 | 1.0−10.0 | < 3.0 (M _{W}) |

Geysers | < 4,300 | 36 | < 7.0 | 1.0−4.0 | < 4.0 (M _{W}) | |

Kawerau | < 1,500 | 13 | < 20.0 | 1.0−10.0 | < 4.0 (M _{W}) | |

Rheinland | < 200 | 13 | < 25.0 | 1.0−9.5 | < 3.5 (M _{W}) |

## 5 Probabilistic Model for Simulating Random Seismic Events

### 5.1 Model Development

_{W}), focal depths (D), as well as the distance of the occurred earthquakes from the GPP zone (R) by considering their correlations. All of these parameters may be affected by the injection rate of GPPs. Thus, the proposed model considers this parameter as a physical characteristic of GPP activities that control the induced seismicity in the surrounding area. In such a model, the availability of data plays a key role. By considering the number of available data in previously introduced and studied databases, the Imperial, Geysers, and Kawerau GPPs are selected for the next steps. Additionally, 80% of the available data are selected for the model development and the remaining data are used as a part of the validation process.

_{N}is the residuals of the equation modeled as a random variable following zero-mean normal probability distribution with a standard deviation of σ

_{N}.

^{2}) of different models range between 0.72 and 0.88, which confirm the acceptable accuracy of the fitting process. In addition, the histograms of residuals of equation (2) prove that the consideration of zero-mean normal distribution for their behavior is an acceptable presumption. Finally, for the case of considering all GPPs together, the mean and standard deviation of obtained t-student distribution for coefficient c are − 1.14 and 0.085, while these values for the coefficient d are 0.99 and 0.26, respectively. In addition, the standard deviations of residuals (σ

_{N}) are calculated as equal to 0.164, 0.243, 0.261, and 0.237 for the cases of Geysers, Imperial, Kawerau, and all GPPs together.

_{W}, D, and R, follows the lognormal distribution (Pasari 2019) as a well-known function for simulating natural phenomena with the mean and standard deviations of \(({\mu }_{Mw}.{\sigma }_{Mw})\), \(({\mu }_{D}.{\sigma }_{D})\), and \(({\mu }_{R}.{\sigma }_{R})\), respectively. To consider the dependency of output variables of the model to the injection rate, these parameters, for all of the variables (M

_{W}, D, and R), are considered as a function of injection rate and modeled using a linear Bayesian regression method.

_{Mw}, σ

_{Mw}, μ

_{D}, σ

_{D}, μ

_{R}, and σ

_{R}, respectively. W is the regression coefficients’ probability distribution functions (PDFs) following the t-student distribution, and ϕ is the regressors of the model. ε

_{i}is the ith parameter regression error, which is a random variable with zero mean and variance of σ

^{2}.

^{2}coefficients of regression correlation are in the suitable range, which implies the acceptable accuracy of the modeling approach. It is depicted that the simulating model follows the general trend of observed data. A very slight increase is seen in the mean value of moment magnitudes with the increase of injection rate for different GPPs. This implies that the event magnitude is not significantly dependent on the injection rate. The mean values of distance between the event epicenters and GPP locations increase with the increase of injection rate, which means that the higher injection rate leads to a farther propagation of high-pressure water in the rock layers and fault activates in the far distance. In the case of focal depth, both increase and decrease are observable with the higher injection rates, which could be inferred as a sign that shows that focal depth is not a function of injection rate directly, and perhaps injection well depth and fault geometry are more important parameters. The obtained model parameters, that is, the mean and standard deviation of the t-student PDF of all regression coefficients, are reported in Table 6. The correlation coefficient and covariance matrices of model output variables are also shown in Table 6. The arrays of correlation matrix with values higher than 0.3 are bolded in this table. The mean value of focal depth and distance have a slight dependency on the moment magnitude with the correlation coefficient varying from 0.34 to 0.36, while the focal depth and distance show a higher-level dependency on each other with a coefficient of around 0.5.

GPP zone | Parameter | Output variables | ||||||
---|---|---|---|---|---|---|---|---|

μ _{Mw} | σ _{Mw} | μ _{D} | σ _{D} | μ _{R} | σ _{R} | |||

Imperial | W _{0} | Mean ^{a} | 0.842 | − 0.553 | 0.650 | − 3.728 | − 8.154 | − 8.812 |

STDV ^{a} | 0.482 | 0.379 | 1.790 | 7.564 | 14.460 | 16.15 | ||

W _{1} | Mean | 0.062 | 0.227 | 1.184 | 2.006 | 4.100 | 5.924 | |

STDV | 0.136 | 0.107 | 1.923 | 2.141 | 6.078 | 8.200 | ||

σ | 0.065 | 0.051 | 0.911 | 1.010 | 2.886 | 3.380 | ||

Geysers | W _{0} | Mean | 1.684 | 0.279 | 3.594 | 2.265 | 0.277 | − 0.408 |

STDV | 0.078 | 0.048 | 0.249 | 0.215 | 0.411 | 0.703 | ||

W _{1} | Mean | 0.012 | 0.010 | − 0.052 | − 0.439 | 0.344 | 0.525 | |

STDV | 0.025 | 0.015 | 0.080 | 0.069 | 0.132 | 0.227 | ||

σ | 0.028 | 0.017 | 0.088 | 0.072 | 0.145 | 0.251 | ||

Kawerau | W _{0} | Mean | − 1.557 | 1.126 | 4.095 | − 0.194 | 2.406 | 6.222 |

STDV | 2.474 | 1.028 | 7.377 | 7.455 | 28.800 | 3.957 | ||

W _{1} | Mean | 1.117 | − 0.241 | 0.751 | 0.585 | 1.199 | − 0.348 | |

STDV | 0.843 | 0.351 | 0.604 | 2.541 | 0.815 | 1.385 | ||

σ | 0.491 | 0.207 | 3.895 | 1.502 | 1.256 | 0.636 |

Covariance coefficient matrix | ||||||
---|---|---|---|---|---|---|

μ _{Mw} | σ _{Mw} | μ _{D} | σ _{D} | μ _{R} | σ _{R} | |

μ _{Mw} | 0.059 | − 0.013 | 0.173 | − 0.048 | 0.042 | − 0.128 |

σ _{Mw} | − 0.013 | 0.007 | − 0.050 | 0.006 | − 0.008 | 0.000 |

μ _{D} | 0.173 | − 0.050 | 1.697 | − 0.339 | 1.368 | − 0.130 |

σ _{D} | − 0.048 | 0.006 | − 0.339 | 0.555 | 0.074 | 0.825 |

μ _{R} | 0.042 | − 0.008 | 1.368 | 0.074 | 1.700 | 0.500 |

σ _{R} | − 0.128 | 0.000 | − 0.130 | 0.825 | 0.500 | 1.700 |

Correlation coefficient matrix | ||||||
---|---|---|---|---|---|---|

μ _{Mw} | σ _{Mw} | μ _{D} | σ _{D} | μ _{R} | σ _{R} | |

μ _{Mw} | 1.000 | 0.255 | 0.351 | − 0.291 | 0.357 | − 0.237 |

σ _{Mw} | 0.255 | 1.000 | − 0.345 | 0.171 | − 0.356 | 0.019 |

μ _{D} | 0.351 | − 0.345 | 1.000 | − 0.015 | 0.550 | 0.011 |

σ _{D} | − 0.291 | 0.171 | − 0.015 | 1.000 | − 0.296 | 0.508 |

μ _{R} | 0.357 | − 0.356 | 0.550 | − 0.296 | 1.000 | 0.178 |

σ _{R} | − 0.237 | 0.019 | 0.011 | 0.508 | 0.178 | 1.000 |

_{θ}is the mean value of parameters of lognormal distributions obtained via regression equation (3) without considering ε. L

_{θθ}is a lower triangular matrix obtained by Cholesky decomposition of the covariance matrix of model parameters (θ) proposed in Table 6, and y is a realization of uncorrelated standard normal random variables vector.

### 5.2 Model Validation

^{2}) and, consequently, highest accuracy level, respectively.

_{W}, D, and R) in these months are calculated. Then by applying the model and the Monte Carlo sampling method concurrently, and considering different numbers of samples in which 7, 25, 50, 100, 150, 200, 300, and 500 events are regenerated, the mean and standard deviation values from all samples are calculated. In Fig. 10, the mean and standard deviation values of output variables of the model (M

_{W}, D, and R) are compared with the target values of real observed data for different numbers of generated samples in the selected months. It is seen that among different variables, M

_{W}, R, and D converge faster, respectively. Generally, by generating more than 100 realizations, the Monte Carlo simulation results converge for all variables. In addition, the average error of simulated events for estimating the mean of moment magnitude, focal depth, and distance for all GPPs are 6.2%, 10.2%, and 13.4%, respectively, while these values for the standard deviation of variables are 27.2%, 18.5%, and 19.2%, respectively.

_{W}, D, and R for all events in a month) via the procedure that was described in the previous paragraph. The results, shown in Fig. 11, emphasize the acceptable accuracy of modeled events. Almost all of the estimated parameters are located in the range of mean ± standard deviation of the observed data, except a few data of focal depth and distance in Kawerau and Imperial GPPs. It is noteworthy that the proposed model is developed for the purpose of seismic risk assessment of buildings and structures via sampling methods such as Monte Carlo. In this case of seismic loss evaluation, the error of 0.1 to 0.3 in estimating the magnitude, or 1−2 km in estimating the distance of epicenter-to-GPP will not have a significant effect on seismic risk assessment results, which is shown in previous studies (Khansefid 2018, 2021). That is, the accuracy of the presented model is within the acceptable range.