Geophysical Research Letters

Influence of pore-pressure on the event-size distribution of induced earthquakes


Corresponding Author: C. E. Bachmann, Swiss Seismological Service, ETH Zurich, Sonneggstrasse 5, CH-8092 Zurich, Switzerland. (


[1] During an Enhanced Geothermal System (EGS) experiment, fluid is injected at high pressure into crystalline rock, to enhance its permeability and thus create a reservoir from which geothermal heat can be extracted. The fracturing of the basement caused by these high pore-pressures is associated with microseismicity. However, the relationship between the magnitudes of these induced seismic events and the applied fluid injection rates, and thus pore-pressure, is unknown. Here we show how pore-pressure can be linked to the seismic frequency–magnitude distribution, described by its slope, theb-value. We evaluate the dataset of an EGS in Basel, Switzerland and compare the observed event-size distribution with the outcome of a minimalistic model of pore-pressure evolution that relates event-sizes to the differential stressσD. We observe that the decrease of b-values with increasing distance of the injection point is likely caused by a decrease in pore-pressure. This leads to an increase of the probability of a large magnitude event with distance and time.

1. Introduction

[2] Enhanced Geothermal Systems (EGS) represent a promising alternative for clean energy. For such systems, two boreholes are drilled to depths below 3 km into the basement, between which a fluid, typically water, is circulated to extract the heat at temperatures well above 100°C. The fluid is pumped at high pressure into the first borehole, to increase the permeability of the rock in a process called reservoir stimulation. This process can be accompanied by micro-earthquakes. Their hypocenters reveal critical information about the ongoing evolution of the reservoir. Few events eventually happen to be large enough to be felt at the surface, creating nuisance for the population or even damage to the building stock. The seismic hazard and risk associated with the induced seismicity is in some cases, such as for the EGS project beneath the city of Basel (Switzerland), estimated as too high to be acceptable for society [Baisch et al., 2009]. Induced seismicity is currently the largest obstacle to the widespread installation of EGS systems near urban centers. Enhancing our understanding of the physical processes that control the generation of induced seismicity is therefore an urgent scientific challenge.

[3] The basic principles of pore-pressure induced seismicity are well understood: the increase in pore-pressure decreases the normal stress on the rock volume, resulting in a lower effective stress [Pearson, 1981]. The failure releases pre-existing tectonic stresses with the consequence of nucleating sudden slip – in other words, micro-earthquakes. However, the current statistical, physical and numerical models of induced seismicity cannot sufficiently explain the critical parameters that ultimately control seismic hazard and risk, i.e., the scaling of earthquake sizes, the temporal occurrence of events, the amount of induced seismicity and the maximum possible magnitude. To investigate the physical processes controlling these parameters, we analyse the seismicity related to the EGS experiment conducted in Basel in December 2006.

[4] The seismicity induced during the experiment in Basel was closely monitored by a six-sensor borehole array (seeFigure 1a), which recorded over 11,000 events of which over 3,500 could be located [Häring et al., 2008]. The located events range from moment magnitudes Mw 0.1 to 3.2, with three events above Mw3. Today, seismicity at the site is still slightly above the assumed long-term background, its decay being indistinguishable from a typical aftershock sequence [Bachmann et al., 2011].

Figure 1.

Overview. (a) Overview of the experiment showing the depth (a.s.l.) of the 3560 located events (circles) and the location of the seismic stations (triangles). The red plane marks the top of the crystalline basement, within which all events occurred. The colorscale indicates the recording completeness ranging from Mc0.7 to 1. (b) Close-up of the events with the overallb-value distribution based on all events. While values range from 0.8 to 3.5, the colorbar is limited from 1 to 2 for a clearer visibility.

2. Earthquake-Size Distribution

[5] The cumulative number of earthquakes, N, in a given volume generally follows a power law distribution and can be expressed as logN = a − bM [Gutenberg and Richter, 1942], where a and b are constants that describe the productivity and the relative size distribution, respectively. Higher b-values indicate more small events relative to larger events and vice versa. Equation (1) and slight modifications thereof are used in essentially all seismic hazard studies [e.g.,Giardini et al., 2004] as it allows extrapolation from the observed smaller events to the infrequent larger ones. Studies of micro-earthquakes on faults [e.g.,Schorlemmer and Wiemer, 2005] have shown that the b-value, when mapped with high quality data at high resolution, varies in the Earth's crust over distances of a few kilometers or less. These studies, combined with the analysis of regional and global focal mechanism data [Gulia and Wiemer, 2010; Schorlemmer et al., 2005] as well as laboratory work [Amitrano, 2003] indicate that the b-value is inversely proportional to the differential stressσD and thus may qualitatively be used as a stress meter at depth in the Earth's crust, where generally no direct measurements are possible. In subduction zones [van Stiphout et al., 2009] and volcanic systems [Wiemer and McNutt, 1997], it has been argued that high b-values are also related to the presence of fluids. In this study we apply for the first time high-resolutionb-value mapping to induced seismicity in order to obtain information on the stress regime and possibly the pore-pressure evolution.

[6] To identify the most prominent gradients in the earthquake-size distribution, we need to investigate the dataset with respect to both, time and space. Throughout this study, we first describe the observations and then introduce a geomechanical model to explain the spatio-temporalb-value variability comparing the match for observed and simulated seismicity (Figures 2 and 3). The migration of fluids away from the injection point can be illustrated as the evolution of the observed seismicity as a function of time and distance (Figure 2a). We find that the b-value substantially decreases from the co-injection period (bco = 1.57 ± 0.06) to the post-injection period (bpost = 1.14 ± 0.06) [Bachmann et al., 2011] (inset to Figure 2a). This indicates an increase of the seismic hazard for the post-injection phase relative to the co-injection phase. Further systematic behavior of the seismicity is documented by analysis in one dimension, the distance to the casing shoe, separately for the co- and post-injection period (Figure 2c). While a relation between distance and b-value is not obvious for the shortest distances during the injection, we find a systematic decay ofb-values in both periods beyond a distance of 200 meters that also holds when considering the uncertainty in the data.

Figure 2.

Evolution of seismicity in one dimension. (a) Radial distance from the injection point versus time. The inset marks the frequency-magnitude distribution of the co- and post-injection period separately. The values arebco = 1.57 ± 0.06 (brown) and bpost = 1.14 ± 0.06 (green) [Bachmann et al., 2011]. (b) Equivalent figure for the simulated events, we find bmco = 1.21 ± 0.06 (brown) and bmpost = 1.12 ± 0.07 (green). The contours represent the pressure based on a linear diffusion model and linearly increasing wellhead pressure [MPa] [Dinske et al., 2010]. b-values against radial distance from the injection point for (c) observed events and (d) simulated events. Theb-value decays with distance from the injection point starting around 200 m for the data and the simulated events. Events are separated in co- and post-injection periods in all four panels (brown and green, respectively).

Figure 3.

The b-value distribution. Cross section along the N-S axis, withb-value distribution based on (a) co-injection and (b) post-injection events, stars in Figure 3b indicate events withMw ≥ 2.5. (c) Examples of frequency–magnitude distributions for two sampled volumes with a high b-value of 2.93 ± 0.40 (red) and a lowb-value of 0.83 ± 0.1 (blue). The potential for larger events such as Mw 3.5, estimated solely by extrapolating the FMDs, is up to 10,000 times higher in the blue volume. (d–f) Equivalent plots to Figures 3a–3c with the simulated events. The sample volumes in Figure 3f show b-values of 2.31 ± 0.30 (red) and 1.00 ± 0.10 (blue). The colorbar of theb-values is limited from 1 to 2 in all subplots. All distances are relative to the injection point at [7.59; 47.59; 4.37] for the observed events and at [0; 0; 0] for the simulated events.

[7] We introduce a new focus-centered mapping technique to determine the three-dimensional distribution of theb-values. The local magnitude of completeness Mc is determined using the closest 150 events in space and is calculated with the maximum curvature method [Woessner and Wiemer, 2005]. Mc defines the catalogue threshold for determination of the b-value, and ranges from Mc0.7 - closest to the deepest borehole sensor, to Mc 1 furthest away from the sensor (Figure 1a). To determine the b-value with the maximum likelihood method [Utsu, 1999], we require at least 25 events with M ≥ Mc in each sample. The resulting b-values range from 0.8 to 3.5 (Figure 1b) with a mean standard error of σ(b) = 0.2. We observe a very systematic behavior: highest b-values are observed close to the casing shoe, forming a toroidal (doughnut-shaped) area, and lowest values are located at the outermost edges of the seismicity cloud. However, the variability is high within the event cloud. We also observe lower values close to the borehole and larger values further out, but the general behavior is constant. The temporal component of this variability is shown by analyzing the co- and post-injection period separately (Figures 3a and 3b, respectively); high b-values are observed near the injection point and occur during the injection, while areas with lowerb-values are found further away from the borehole and develop after shut-in. Large magnitude events (LME; defined in our context as events felt at the surface, i.e., events withMw ≥ 2.5) are located in regions with below average b-values (b ≤ 1.3). We evaluate this by calculating the causal b-value for each event, where we determineb-values based on the closest 150 events in space,preceding the event. This method implies that we miss the earliest LME, if they are within the first 150 events. By comparing the distribution against a random correlation of b-value and magnitude, we find that it is significant at the 99.95% level to observe five LME in regions with below average b-value.

[8] Based on all these observations, the first conclusion we draw is that b-values are spatially highly variable in the induced seismicity sequence of the EGS in Basel, decreasing systematically with time and distance from the casing shoe. The current state of knowledge onb-values of induced sequences is that they are generally higher than normal tectonic events [Wyss, 1972; Shapiro et al., 2011]. Our analysis unravels a much more complex yet systematic pattern; values range from typical tectonic b-values at the edges of the seismicity cloud to extremely high values near the injection point. Our detailed mapping of the spatial distribution of theb-value can be used to estimate the probability of occurrence of felt events more accurately than possible with a bulk analysis.

3. Geomechanical Model

[9] To develop a geomechanical understanding of the observed systematic behavior of the b-values with space and time, we simulate the induced seismicity cloud in space and time under the hypothesis that highb-values near the injection point are a response to the pore-pressure perturbation. A similar model was previously introduced byRothert and Shapiro [2003], however here we include the earthquake sizes in the model to be able to analyse b-value distributions.

[10] We randomly distribute potential failure points (we call them seed faults), representing pre-stressed faults, in a three-dimensional space centered around the injection point. Each seed fault is assigned a minimum and maximum principal stressσ3 and σ1, based on a background stress regime according to Häring et al. [2008]and a Gaussian perturbation of 10% (K. F. Evans, personal communication, 2011). Using a constant cohesion (7 MPa) and coefficient of friction (0.85 for intact rock), we can analytically calculate the Mohr-Coulomb diagrams and thus a failure criterion for each seed fault. In case of an instable seed fault, we randomly reassign the principal stresses until we acquire the desired number of stable seed faults. Here we use 300,000 seed faults in a volume of 1 km3. We introduce a time-dependent point pressure source at the injection point. The pressure-time function is linearly increasing over 6 days, a simplified version of the actual injection pressure time series at Basel [Dinske et al., 2010]. We propagate the pore-pressure through the model space based on linear diffusion in a hydraulically isotropic medium with an effective diffusivity of 0.05 m2/s. Dinske et al. [2010] introduced an analytical solution to the diffusion equation [Wang, 2000] for the case of a linearly increasing source time function. We use their analytical solution assuming an effective source radius of 70 m, in order to reach realistic values of the pore-pressure perturbation. The naturally occurring pore-pressure variation in the reservoir due to tides is assumed to be on the order of 2000 Pa, which has been measured at 1 km depth in the middle of the Furka tunnel, Switzerland (Evans, personal communication, 2011). Therefore we assume that the triggering front of the induced seismicity should closely follow this isobar.

[11] Figure 4adepicts the model schematically. The seed faults are represented by the points closest to failure and are color-coded according to their respective differential stress, defined asσD = σ1σ3. The pore-pressure evolution with time reduces the normal stresses shifting the Mohr circles towards the failure envelope. An event is induced once the Mohr circle touches the failure envelope. For simplicity, we here assume that all seed faults are optimally oriented and each seed fault can only be triggered once. To assign a magnitude to an event, we exploit the observation from laboratory and natural earthquake analysis thatb-values are inversely related toσD [Amitrano, 2003; Schorlemmer et al., 2005]. As scaling law, we use a simple linear scaling between the observed ranges of b-values (0.8 to 3.5) andσDvalues of our model (20 to 150 MPa). The magnitudes are then randomly drawn from a power-law distribution with the assignedb-value for each failure occurrence. Using a linear diffusion model with homogeneous and isotropic hydraulic diffusivity is a strong simplification for two reasons: (i) the opening of fractures potentially modifies the diffusivity, creating non-linearity [Hummel and Müller, 2009] and (ii) actual pore-pressure propagation can be highly anisotropic, occurring preferentially along zones of weakness [Evans et al., 2005]. However, the model serves as a first-order approximation and can explain some features of the observed seismicity well [Goertz-Allmann et al., 2011].

Figure 4.

Model. (a) Mohr-Coulomb plot for the 3D model. Circles denote the points closest to failure for all 300,000 seed faults. Increasing the pore pressure (ΔP) shifts the potential seed to the left; an event is triggered if a Mohr-Coulomb circle touches the failure envelope. This is indicated with the half circle (dashed black line) representing the mean stress state. The arrow indicates the minimum pore pressure perturbation needed to trigger this seed fault. The ranges of the minimum and maximum principal stressσ3 and σ1 are indicated with a solid and dashed green line, respectively. (b) Result of one simulation of the model. A total of 791 seed faults were triggered. The b-values for events triggered with high ΔP (red rectangle) and low ΔP (green rectangle) are significantly different, bhighP = 1.45 ± 0.105 (red) and blowP = 1.11 ± 0.0 (green) (indicated by the FMD inset).

[12] We run a total of 100 simulations, whereas results of only one simulation are usually presented. The evolution of the seismicity as a function of time and distance of the simulated seismicity (Figure 2b) resembles the one we find from the data (Figure 2a). The same holds for the evolution of the b-value with distance;b-values of the simulation decay with distance (Figure 2d), however we do not reproduce the complexity of the b-values for the nearest 200 m to the injection point. Likewise, the spatial distribution of theb-values (Figures 3d and 3e) resembles the one we find from the data. While we reproduce the general patterns like the highest values close to the injection point and the lower values during the post-injection period, we do not replicate the finer details.

[13] To understand the geomechanical reasons for the good match between model and observation, we show the state-of-stress of all seed faults that were triggered in one simulation of the model (Figure 4a). Out of the 300,000 seed faults, 791 ruptured. We divide these into events triggered with pore-pressure changes ΔP above (red square) and below (green square) the mean value of 2.74 MPa and determine the corresponding FMDs of the two subsets (inset of Figure 4b). The corresponding b-values arebhighP = 1.45 ± 0.10 and blowP= 1.11 ± 0.05. This can be understood when viewing the available seeds: The subset of faults triggered at high pore-pressures contains fewer large differential stresses compared to the one at lower pore-pressuresand thus less large magnitude events are induced.

4. Discussion

[14] Our results suggest the following conceptual model for the scaling of induced seismicity. Near the borehole (0–200 m), the pore-pressure rapidly rises once the stimulation starts, and numerous events are induced. These failures sample a wide range of differential stresses, including very smallσD that were originally far away from the failure criterion but still ruptured due to the high pressure perturbation. It is these small σD values that cause a systematic bias towards higher b-values, such that events close to the injection point exhibit a higherb-value, if we assume thatb-values are inversely related toσD. As the pore-pressure front progresses, seed faults experience only a moderate increase in pore-pressure. The differential stresses of these faults follow more closely a typical tectonic earthquake–size distribution. Once the pumping is stopped, pore-pressures near the well decrease rapidly, and this area is then essentially devoid of seismicity [Parotidis et al., 2004]. At distances exceeding 200 m, pore-pressures continue to increase gradually, although absolute values remain low. We suggest that static and dynamic stress changes caused by the numerous earthquakes themselves will contribute to inducing (or in some cases inhibiting) events in these regions. These stress changes also contribute in the vicinity of the injection, but there they are negligible when compared to the pore-pressure. The observed exponential decay of seismicity [Bachmann et al., 2011] with typical aftershock parameters is then a combination of the continued gradual expansion of the pore-pressure front and the gradual decay of aftershock activity caused by static and dynamic stress changes.

[15] Based on the simple geomechanical simulation, we can also address the question of how large future events will be and how likely they will be. Therefore, we evaluate the synthetic seismicity cloud in time and space for a and b-values within specific time- and distance bins. For each time- or distance bin, we can estimate the probabilityp of an event exceeding a certain magnitude M as p = 1 − e−1/x, where x = 1/(a − bM) [Wiemer, 2000]. The probability of an event exceeding M 4 is shown in Figure 5. We choose time bins of 105 s, moving at 104 s intervals, and distance bins of 100 m, moving across distance in 10 m increments. We compare the probability based on our synthetic event cloud with varying b-values (white) with the probability based on an event cloud synthesized from a constantb-value (1.2, gray). We find that the probability for LME increases with time during the injection and then decays after the shut-in. A varyingb-value causes a substantial increase of the mean probability for the time period right after shut-in relative to before shut-in (Figure 5a), and shifts the maximum of the probability to further distances from the injection point (Figure 5b). These calculations are consistent with observations of LME during hydraulic stimulations in geothermal systems not only in Basel [Deichmann and Giardini, 2009], but also elsewhere [Baisch et al., 2010]. Our observations and the stochastic model approach presented can form the basis for and improved probabilistic real time forecasting system of induced seismicity in future EGS experiments.

Figure 5.

Probability for large magnitude events. Occurrence probability of M 4 event, varying with (a) time and (b) radial distance from the injection point for a varying b-value (white) and a constantb-value (gray). The errorbar denotes the standard deviation based on 100 simulations of the geomechanical model. The dashed line in Figure 5a marks the shut-in time and in Figure 5b the location of the largest observed Basel event.

[16] The observed, highly systematic distance and time evolution of the b-values in Basel calls in our opinion for a fundamental mechanism as an explanation. The link between differential stress andb-value is a fundamental mechanism, established across a large range of scales, from the analysis of individual rock samples in the laboratory to global seismicity [Amitrano, 2003; Schorlemmer et al., 2005]. Adding this link as the only independent ingredient to our minimalistic model of induced earthquakes is sufficient to explain the observed b-values evolution in space and time. Furthermore, the ability to explain the size distribution of induced events is strong additional evidence that differential stress is one of the governing parameters that influencesb-values for a wide range of tectonic settings.

[17] We propose that our conceptual model, developed for the well-recorded induced seismicity in the natural laboratory of the Basel EGS, is universally able to explain a wide range of observations of the earthquake-size distribution. We speculate that the same mechanism of pore-pressure changes is responsible for other highb-values anomalies in the Earth's crust: in aftershock zones near the areas of highest slip [Wiemer and Katsumata, 1999]; in volcanic regions near magma chambers [Wiemer and McNutt, 1997] and in subduction zones [van Stiphout et al., 2009]. It is worthwhile noting that in the immediate vicinity of the rupture plane of larger earthquakes, static and dynamic stress changes are expected to be large enough to trigger events with a wider than usual range of differential stresses, thus leading to higher b-values with in the aftershock sequence close to the rupture fault plane.


[18] The authors would like to thank Geothermal Explorers Ltd., who provided the seismicity data and C. Langebruch and one anonymous reviewer for comments that greatly improved the paper. This study is part of the project GEOTHERM, funded by the Competence Center of Environment and Sustainability (CCES) of ETH Zurich and it was supported by the European Commission through funds provided within the FP7 project GEISER, grant agreement 241321-2.

[19] The Editor thanks Cornelius Langenbruch and an anonymous reviewer for assisting with the evaluation of this paper.