# Fluid processes highlighted by temporal variations of *b*-value during swarms and aftershocks sequences in the Ubaye Region (Western Alps, France)

Marion Baques, Clara Duverger, Louis de Barros, Hervé Jomard, Maxime Godano

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## ► To cite this version:

Marion Baques, Clara Duverger, Louis de Barros, Hervé Jomard, Maxime Godano. Fluid processes highlighted by temporal variations of *b*-value during swarms and aftershocks sequences in the Ubaye Region (Western Alps, France). Earth and Space Science, 2025, 12 (11), pp.e2025EA004250. 10.1029/2025EA004250 . hal-05391309

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Distributed under a Creative Commons Attribution 4.0 International License
# Earth and Space Science

## RESEARCH ARTICLE

10.1029/2025EA004250

**Key Points:**

* We study *b*-value evolution in the Ubaye Region (France), an area with a prolific and complex seismicity involving fluid-driven processes

* Seismic swarms show a similar increase of *b*-value

* *b*-value evolution during mainshock-aftershocks sequences is consistent with an interplay of fluid migration and stress transfer

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

M. Baques, marion.baques@unipd.it

**Citation:**

Baques, M., Duverger, C., De Barros, L., Jomard, H., & Godano, M. (2025). Fluid processes highlighted by temporal variations of *b*-value during swarms and aftershocks sequences in the Ubaye Region (Western Alps, France). *Earth and Space Science*, 12, e2025EA004250. https://doi.org/10.1029/2025EA004250

Received 4 FEB 2025
Accepted 7 NOV 2025

**Author Contributions:**

**Conceptualization:** Marion Baques, Clara Duverger, Louis De Barros, Hervé Jomard, Maxime Godano

**Formal analysis:** Louis De Barros

**Funding acquisition:** Clara Duverger, Louis De Barros, Hervé Jomard

**Methodology:** Marion Baques, Louis De Barros, Maxime Godano

**Project administration:** Louis De Barros

**Resources:** Clara Duverger

**Supervision:** Clara Duverger, Louis De Barros, Hervé Jomard, Maxime Godano

**Validation:** Louis De Barros

**Writing – original draft:** Marion Baques

**Writing – review & editing:**

Marion Baques, Clara Duverger, Louis De Barros, Hervé Jomard, Maxime Godano

© 2025. The Author(s).

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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# Fluid Processes Highlighted by Temporal Variations of *b*-Value During Swarms and Aftershocks Sequences in the Ubaye Region (Western Alps, France)

Marion Baques¹,² ID, Clara Duverger³,⁴ ID, Louis De Barros¹ ID, Hervé Jomard⁵, and Maxime Godano¹ ID

¹Université Côte d'Azur, CNRS, Observatoire de la Côte d'Azur, IRD, Géoazur, Valbonne, France, ²Now at Dipartimento di Geoscienze, Università degli Studi di Padova, Padova, Italy, ³CEA, DAM, DIF, Arpajon, France, ⁴Now at Lithium de France, Haguenau, France, ⁵Autorité de sûreté nucléaire et de radioprotection (ASNR), PSE-ENV/SCAN/BERSSIN, Fontenay-aux-Roses, France

**Abstract** The *b*-value from the Gutenberg-Richter law is a crucial parameter in the assessment of seismic hazard. Its temporal variations may also bring useful insights on the processes driving seismicity at depth, even if not yet fully understood. In this paper, we focus on the temporal evolution of the *b*-value in the Ubaye Region (French Western Alps) which was hit by seismic swarms (2003–2004) and complex sequences with several mainshocks (2012–2015). The swarm-like sequences show a common temporal behavior of *b*-value characterized by an increase and then a return to the initial level. The temporal *b*-value pattern for the mainshock-aftershock-like sequences is quite different. After a drop in the *b*-value that may follow the mainshock, the *b*-value increases above the background level before going back to it. Moreover, no precursory pattern can be identified before the mainshock. Fluid processes are recognized to play a crucial role in the driving mechanisms of these seismic sequences. Drawing parallel between swarms and aftershock sequences suggests that the *b*-value depicts fluid-processes in the Ubaye Region seismicity. We propose that *b*-value shows a complex behavior, with variations due to Coulomb stress-transfer from the mainshock and fluid-pressure processes. Therefore, even with a catalog made at the French national scale, the *b*-value variations may help to monitor the on-going processes at depth.

## 1. Introduction

One way of describing the evolution of seismicity over time is to use the *b*-value. This parameter comes from the Gutenberg-Richter law (Gutenberg & Richter, 1954), which describes the scaling relationship between the magnitude and the number of earthquakes, as expressed by:

$$ \log(N(M)) = a - bM \quad (1) $$

with $N(M)$, the number of earthquakes with magnitude larger or equal to $M$; $a$, a constant indicating the productivity of seismicity; $b$ (often called *b*-value), a scaling parameter giving the ratio between large and small magnitude events; and $M$, the magnitude. For a given region, based on past observations, extrapolation of this law toward higher magnitudes leads to an estimation of the rate of occurrence of a large magnitude earthquake occurrence. The *b*-value is a crucial parameter to evaluate the seismic hazard since it adjusts the number of large-magnitude earthquakes given the smaller magnitudes observed.

The *b*-value is observed to vary temporally and spatially (e.g., El-Isa & Eaton, 2014; Gulia & Wiemer, 2019; Wiemer & Wyss, 1997), which may help to anticipate how the seismicity may evolve. Many studies of mainshock-aftershocks seismic sequences highlight *b*-value variations before and after the occurrence of the mainshock (e.g., Gulia & Wiemer, 2019). However, these variations differ from one sequence to another, since *b*-value has been observed to either decrease or increase before and after the mainshock (De Santis et al., 2011; Görgün, 2013; Gulia & Wiemer, 2019; Hui et al., 2021). For instance, Görgün (2013) observed a decrease of *b*-value before the Mw 7.2 Van mainshock (23/10/2013, Turkey) and an increase following it. Hui et al. (2021) observed a similar behavior for the 2008 Ms 8.0 mainshock in Wenchuan area and for the 2013 Ms 7.0 mainshock in Lushuan area (China). Particularly, a decrease of *b*-value before mainshocks has been proposed to be a potential precursor (Gulia & Wiemer, 2019), even if this idea is still debated (van der Elst, 2021). On the contrary, other
studies observed an increase in *b*-value before the mainshock and a decrease afterward. This pattern was described by Ma et al. (2018) in Creighton Mine and Kidd Mine (Canada) for magnitude between −2 and ~4 and by Enescu et al. (2011) for the 1995 Kobe earthquake (Mw 7.2). As *b*-value variation does not show a generic behavior before a mainshock, its use as a precursor is delicate (e.g., Gulia & Wiemer, 2019; van der Elst, 2021).

Seismic swarms are sequences of spatially and temporally clustered earthquakes. They are characterized by an absence of mainshock, with several events having a magnitude close to the largest one. The spatial distribution of the seismic cloud is also much larger than the rupture area of the highest magnitude earthquake (Vidale & Shearer, 2006). This type of seismic sequence may last from a few days (e.g., Hatch et al., 2020) to several years (e.g., Zheng et al., 2017). To maintain such prolonged activity requires external forcing mechanisms. Specifically, seismic swarms are often interpreted as driven by fluid processes. By analogy between natural swarms and induced seismicity (Danré et al., 2022; Zang et al., 2014), this model is validated through direct fluids observations: West-Bohemia swarm (Fischer et al., 2017; Hainzl et al., 2016) and for Strasbourg induced seismicity (Lengliné et al., 2023; Schmittbuhl et al., 2021) and through indirect evidences: seismicity migration (Danré et al., 2024; Duverger et al., 2015; Ross et al., 2020) or misoriented mechanism (Baques et al., 2023). There is a dominant behavior observed in the temporal variation of *b*-value during the swarm sequences. The *b*-value usually shows an increase during the beginning of the swarm followed by a decrease toward its end (Aswini et al., 2021; Herrmann et al., 2019; Ibáñez et al., 2012; Jacobs et al., 2010; Peng et al., 2021). For example, Ibáñez et al. (2012) observed such a behavior for several swarms during the 2011–2012 submarine eruption off the coast of El Hierro (Canary Island, Spain). Yet, for some swarms, no clear time variations in *b*-values were observed, as during the 2008 West Bohemia swarm (Fiedler et al., 2018) or a monotonous decrease of *b*-value with time for the 2000 West Bohemia swarm (Hainzl & Fischer, 2002).

Different mechanisms have been proposed to explain the temporal variations of *b*-value. Globally speaking, the *b*-value seems correlated to the effective normal stress (difference between total stress and pore pressure). An increase of *b*-value may be associated with an increase of fluid-pressure (e.g., Bachmann et al., 2012; Herrmann et al., 2019; Maxwell et al., 2009; Smirnov et al., 2022). The temporal variation of *b*-value may also be associated with differential stress variation (e.g., Colledge et al., 2023; Scholz, 2015; Tan et al., 2019; Yoshida et al., 2017), which corresponds to the difference between the largest ($\sigma1$) and the smallest ($\sigma3$) principal stress components. Scholz (1968) demonstrated that the *b*-value decreases linearly with increasing differential stress, and therefore with depth (e.g., Minetto et al., 2022; Sharma & Biswas, 2022; Spada et al., 2013) and with the faulting mechanism (normal, reverse or strike-slip fault, e.g. Ishibe et al., 2008; Kulhanek, 2005; Schorlemmer et al., 2005). Mukuhira et al. (2024) studied the Basel EGS sequence in Switzerland and compared the temporal variation of *b*-value against several geomechanical parameters. They found that the *b*-value seems to be inversely correlated to the shear stress, even if they also observed some loose correlation with pressure. The *b*-value may be seen as a stress-meter due to its dependency on stress, which may explain its decreasing trend before mainshocks (Gulia & Wiemer, 2019). Additionally, temperature changes (e.g., Lin et al., 2007; Murru et al., 2007) and on fault/off fault seismicity (Collettini et al., 2022; King et al., 1983) may also lead to variation of *b*-values. Laboratory tests have demonstrated that strain localization (e.g., Kwiatek et al., 2014), roughness (e.g., Goebel et al., 2017; Sammonds & Ohnaka, 1998) and geometrical complexity (e.g., Goebel et al., 2015) of the faults also strongly contribute to changes in frequency-magnitude distributions. The large variety of behaviors in the temporal variations of the *b*-value, and of the processes that may explain them, make their interpretation difficult and ambiguous.

This article aims to improve our understanding of how *b*-value temporal variation reflects processes driving mainshock-aftershocks and swarm sequences. For that, we apply the recent *b*-positive method (van der Elst, 2021) to compute the temporal evolution of the *b*-value in the Ubaye area. This area, in the French Alps, shows a prolific seismicity that involves fluid processes in its driving mechanism (e.g., Baques et al., 2021). The *b*-value variations we observe in swarms and mainshock-aftershocks sequences supports the role of fluids.

## 2. The Ubaye Region

The Ubaye Region, located in the south-western Alps (France), is one of the most seismically active areas in France, with the strongest recorded earthquake (ML 5.3) occurring in 1959 (Nicolas et al., 1998) causing some damage to the surrounding villages. Globally, the seismicity in this area is complex, as it shows alternating swarms and mainshock-aftershocks sequences (e.g., Baques et al., 2021).
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Map (a) and temporal distribution (b) of the seismicity in the Ubaye Region. The seismicity catalog comes from the CEA/LDG for the period 2000–2019 (Baques et al., 2021; Duverger et al., 2021). In panel (a), the colors correspond to the time the events occurred. The red line shows the Penninic front which delimits the Extern and Intern parts. The colored ellipses highlight the location of the 2003–2004 swarm (orange) and the 2012–2015 sequence (blue). In (b), the dots correspond to the events with local magnitude larger or equal to 4.</figcaption></figure>

This area is at the transition of two zones, with different geological units separated by the Penninic thrust front. These two areas show different seismicity patterns (Figure 1a). The Intern zone, mainly composed of the Briançonnais geological unit, shows a diffuse and regular seismicity with some small mainshock-aftershocks
sequences and swarms. The Extern zone is composed of the Dauphinois units covered by sedimentary nappes (Embrunais-Ubaye nappes) with the seismicity developing mainly as large and long swarms as well as complex sequences (e.g., Baques et al., 2021).

For instance, in the Extern zone, a long swarm occurred in 2003–2004 with more than 16,000 recorded events and a maximal local magnitude of 2.7 (Figure 1a, orange ellipse; Jenatton et al., 2007). The seismicity showed a migration from northwest toward southeast (Daniel et al., 2011; Jenatton et al., 2007; Leclère et al., 2012, 2013). The seismic front of the spatio-temporal distribution shows an apparent diffusivity of 0.005 m²/s (Baques et al., 2021; Jenatton et al., 2007), which is compatible with fluid-driving processes (e.g., Duverger et al., 2015; Shapiro et al., 2002). The fluid-overpressure needed to explain the migration was calculated by Daniel et al. (2011), Leclère et al. (2012, 2013), and Baques et al. (2023). They found that an overpressure changes from 8 MPa to about 60 MPa is required to drive this seismicity. Danré et al. (2023) found that a fluid volume circulation of about 1e7 m³ may explain this long-lasting sequence. In view of these independent observations, the 2003–2004 swarm is interpreted to be primarily driven by fluid-pressure diffusion.

Between 2012 and 2015, the Extern zone was also affected by a long and complex seismic sequence, mixing mainshock-aftershocks and swarm sequences (Figure 1a, blue ellipse; Figure 1b). It is composed of three mainshocks (ML 4.8 in 2012, ML 5.2 in 2014, and ML 4.3 in 2015) followed by long-lasting aftershock sequences (Baques et al., 2021; De Barros, Baques, et al., 2019; Thouvenot et al., 2016). The 2012 aftershock sequence displayed a migration pattern, which was fitted by a diffusion law with a diffusivity of 0.005 m²/s (Thouvenot et al., 2016). For the 2014 aftershock sequence, De Barros, Baques, et al. (2019) found that part of the aftershock clusters behave like swarms. Their migration can be explained by a diffusivity law, with diffusivity between 0.002 and 0.06 m²/s. Yet, other clusters behave like aftershock sequences, with Omori's law-type decay.

Baques et al. (2023) calculated the fluid-overpressure needed to trigger this sequence. They found that all events, including the mainshocks, require fluid-overpressure. They used a fault-valve model (Sibson, 1990) to explain this seismic sequence. They proposed that fluids are trapped inside the main fault plane by hydraulic barriers. Therefore, pressure within the faults tends to be lithostatic, which triggers the mainshock rupture. Around the rupture area, aftershocks are generated by Coulomb static stress (Stein, 1999). As the mainshock also broke the hydraulic barriers, fluid-pressure diffused upwards inducing seismicity within swarm sequences (De Barros, Baques, et al., 2019).

In the Ubaye Region, fluid processes seem to play a crucial role in both swarms and mainshock-aftershocks sequences. Therefore, given this knowledge of the driving processes, we aim to explore how the *b*-value behaves in this context. Indeed, *b*-value temporal variations were only explored by Jenatton et al. (2007) and are limited to the 2003–2004 swarm.

# 3. Data and Methods

## 3.1. Data

The catalog used in this study is the CEA/LDG catalog (http://www.dase.cea.fr; Baques et al., 2023; Duverger et al., 2021) for the period from 01/01/2000 to 31/12/2019 (Figure 1). It is composed of 7,065 events that have an absolute location, local magnitude ranging from 0.3 to 5.2, and an average magnitude uncertainty of 0.25. In the framework of this study, this catalog was preferred to other richer catalogs (based on temporary station records) as it has been computed using the same detection method with almost identical stations maintained for this region throughout the whole period studied by the CEA (Duverger et al., 2021).

## 3.2. *b*-Positive Method

Several methods exist to calculate the *b*-value. Most of them first need to estimate the magnitude of completeness, as for the MLE (Maximum-likelihood estimation, Aki (1965)) and the OK93 (Ogata & Katsura, 1993). These methods require a large number of events and a stable magnitude of completeness. Given the relatively small study area, the magnitude range in the catalog (Figure S1a in Supporting Information S1), and the low rate of seismic activity during some quiet periods, it is difficult to estimate the evolution of the completeness magnitude over time in the Ubaye Region. We therefore chose to use the *b*-positive method (*b*⁺-value), recently proposed by van der Elst (2021). This method relies on the use of the positive magnitude difference between two consecutive events, the first one having a smaller magnitude than the second one (Figure S1b in Supporting Information S1;
Lippiello & Petrillo, 2024; van der Elst, 2021). This method proves particularly efficient following large mainshocks, as a detection bias due to nearly simultaneous events leads to highly changeable magnitude of completeness. The spread of differences in magnitude leads to a Laplace distribution. Its slope is equal to the *b*-value and is measured by the MLE method (Aki, 1965; van der Elst, 2021). We then calculate the *b*⁺-value for the window containing 70 positive magnitude differences where ΔMpos is greater than or equal to the minimum positive magnitude difference where ΔMmin = 0.25 (average magnitude uncertainty). The windows are then moved by 5ΔMpos, allowing for an overlap of 90% (65 ΔMpos in common). On average, each resulting window contains around 230 events.

### 3.3. Validation of *b*-Positive Method on the Ubaye Region Data Set

In order to check the *b*-positive method for our study area and catalog, we first compare its results with the *b*-values computed with the OK93 method (Ogata & Katsura, 1993) and the MLE method (Aki, 1965) for the 2002–2006, 2012–2015, and 2010–2019 periods (Figures S2a–S2c in Supporting Information S1). The magnitude of completeness (with 66% of detection rate) is obtained following Ogata and Katsura (1993). For each period, the *b*-positive method leads to a *b*⁺-value between the OK93 and the MLE methods. While the OK93 method describes correctly the first part of the decrease of the frequency-magnitude distribution, the MLE and *b*-positive methods seem to better describe the general decrease of the frequency-magnitude distribution (Figures S2a–S2c in Supporting Information S1). Therefore, the *b*-positive approach used here seems to lead to results consistent with other methods. Moreover, as shown by statistical tests, the *b*-positive approach proves to be robust (Lippiello & Petrillo, 2024; van der Elst, 2021), even for incomplete catalogs such as the one used here.

Before looking at the evolution of the *b*⁺-value, we first qualitatively check if the distributions of the event magnitudes within the 70-ΔMpos windows can be fitted by a linear model, hence if they can be described by a Gutenberg-Richter law. Qualitatively, Figures S2d–S2f in Supporting Information S1 show a few examples where the magnitude distribution can be correctly fitted by a Gutenberg-Richter law. To quantify it, we compute the misfit between the observations and the reconstructed fitted line through the mean squared error (MSE). Text S1 in Supporting Information S1 describes how we calculate the MSE for each *b*⁺-value, and Figure S3 in Supporting Information S1 shows the temporal variation of the MSE together with the *b*⁺-value. We obtain an average MSE of ~0.7% for our data, showing that the data can be correctly fitted by a linear model, with the *b*-value slope calculated by the *b*-positive method.

The *b*⁺-value uncertainty is calculated for each temporal window of 70 positive differences, using a bootstrap method (Efron, 1992). From the 70 positive magnitude differences in the temporal window, we randomly select 70 positive magnitude differences, allowing for duplication, creating a new bin of 70 positive magnitude differences. We generate 500 catalogs for each window. The *b*⁺-value is then calculated for each of these 500 synthetic catalogs, which allows us to compute the mean and standard deviation on these 500 new *b*⁺-values. In Figures 2–4, we represent the two standard deviations (±2σ) with uncertainty shown as vertical gray bars. In Figure 2, the significant variations in the *b*⁺-value exceed the calculated uncertainties, indicating that the observed variation in *b*⁺-values cannot be attributed solely to statistical error.

We also look at the influence of the size of the sliding bin (Figure S4 in Supporting Information S1), by computing the temporal variations using bin size ranging from 50 to 90 positive magnitude differences. We observe similar patterns for the different bin sizes, even if both uncertainty and resolution decrease with increasing size bin. A bin size of 70 positive magnitude differences was finally retained. van der Elst (2021) applied the *b*-positive method to the 2019 Ridgecrest sequence, for which a much more exhaustive seismicity catalog is available in comparison to the one used in this study. We therefore checked whether the higher magnitude of completeness and the presence of large events in our study area could bias the results or not. For that, we compare the temporal variations of the *b*⁺-values calculated with the full catalog, a catalog without the largest events (*M* ≤ 3.5), and a catalog without the smallest events (*M* ≥ 1.75) by considering only events of magnitude larger or equal to the completeness magnitude. The magnitude of completeness is estimated at 1.75 by the method from Ogata and Katsura (1993) during the 2000–2021 period (Figure S5 in Supporting Information S1) As we observe the same trend for the different catalogs (Figure S5 in Supporting Information S1), the method does not seem to be strongly influenced by the completeness of the catalog or the largest events, confirming the observations by van der Elst (2021).
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Temporal evolution of the $b^+$-value for the Intern zone (a) and the Extern zone (b). The blue bars show the monthly number of events. The red squares correspond to the calculated $b^+$-value positioned at the end of the bins containing 70 positive magnitude differences, together with their uncertainties (two standard deviations). The pink horizontal dashed line shows the mean $b^+$-value for each zone (Intern: 1.17; Extern: 1.15). In (b), the red vertical lines indicate events with a local magnitude equal to or greater than 4 on which Figure 4 is focused. The colored time range zones correspond to periods containing swarms (light red) or mainshocks (light gray), on which Figures 3 and 4 focus, respectively.</figcaption></figure>

# 4. Results

The result of the temporal evolution of $b^+$-value may be affected by the size of the study area in which the $b$-value is calculated. In our case, the areas are selected based on geological features and seismic behaviors. In the Intern part the seismicity is mainly diffuse, but the reverse is true for the Extern part, which is dominated by spatially focused activity around the 2003–2004 and 2012–2015 sequences.

Figures 2a and 2b represent the temporal evolution of $b^+$-value in the Intern and Extern parts, respectively, alongside the temporal distribution of the seismicity. Background seismicity in the Extern part is characterized by a relatively constant $b^+$-value of $b \sim 1.15$, only disrupted by the occurrence of seismic sequences in 2003–2004 and 2012–2015. On the contrary, the $b^+$-value in the Intern zone is always fluctuating, as it seems to be constantly perturbed by continuous seismic activity. This highlights the difference in seismic behavior between these areas, with the activity being more constant in the Intern part. In both zones, the main swarms (red frames in Figure 3) and the mainshock-aftershocks sequences (gray frames in Figure 3) show strong temporal variations of $b^+$-value.

Figure 3 focuses on the temporal evolution of $b^+$-value for three swarms that occurred in the Intern and Extern zones. For the first swarm (Figure 3a), occurring in 2001–2003 in the Intern zone, the $b^+$-value increases from 1.3 to 1.5 during the initial months. The $b^+$-value then stays at a high level, but with some fluctuations, for over a year. It finally slowly decreases, going back to $b \sim 1.3$ during the second year of the swarm. The two other swarms (Figures 3b and 3c) show a sharp increase of 0.3 in $b^+$-value correlated with the increased activity of the swarm, as
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Temporal evolution of the *b*⁺-value during three swarms. The blue bars show the monthly number of events. The red squares show the calculated *b*⁺-values located at the end of the bins containing 70 positive magnitude differences. Vertical bars show the uncertainties (two standard deviations).</figcaption></figure>

the *b*⁺-value goes from 1.1 to 1.4 for the 2003–2004 swarm (Figure 3b) and from 1.0 to 1.3 for the 2008–2010 swarm (Figure 3c). The 2003–2004 swarm shows a dramatic decrease of *b*⁺-value going from 1.45 to 1.15 in less than 2 months while the seismic activity still remains higher than the background level. For the 2008–2010 swarm (Figure 3c), the *b*⁺-value stays high for a longer period (~6 months) but then falls from 1.3 to 1.0 over a 2–3-month period. Such variations appear significant, as their amplitudes are larger than the calculated uncertainty. Despite differences in the rate of the variation of *b*⁺-value, the swarms show a similar behavior, with an increase in *b*⁺-value during the beginning of the swarm and returning to a lower value, close to that before the swarm, after a certain period of time.
<figure><img src="image_7.png" /><figcaption>Figure 4.</figcaption></figure>
We then focused on the mainshock-aftershocks sequences in the Extern zone, and particularly on the largest mainshocks within the complex 2012–2015 seismic sequence. Figures 4a, 4c, and 4e show the $b^+$-value variations around the 26 February 2012 (ML 4.8), 7 April 2014 (ML 5.2), and 6 November 2015 (ML 4.3) earthquakes, respectively. However, as the bins are defined independently of the mainshock position (as in Figure 2), some bins contain events both before and after the mainshock. To be sure that these results are not biased by the mixing of these events, we redefined the bins position according to the time of the mainshock (Figures 4b, 4d, and 4f). These bins thus contain only seismicity before and after the mainshock. For the 2012 earthquake, an apparent $b^+$-value decrease is observed before the mainshock (Figure 4a). This decrease disappears when seismicity before and after the mainshock are separated (Figure 4b). No precursory signals can be identified either before the 2014 or the 2015 events. This confirms that mixing pre- and post-mainshock seismicity within the same bins may lead to spurious observations. In the aftershock sequences, a clear drop of $b^+$-value is observed just after the 2012 earthquake (Figure 4b), going from 1.1 to 0.7. After the 2014 and the 2015 earthquakes, a slight $b^+$-value decrease seems to occur (Figures 4c and 4e), but briefly and maybe insignificantly given the uncertainty. For all three earthquakes, the $b^+$-value then increases to get slightly (2012, Figure 4b) or significantly above its initial level ($b$ = 1.3 in 2014, Figure 4d; $b$ = 1.5 in 2015, Figure 4f). This increasing phase lasts less than 1 month in 2014, several months after the 2015 mainshock, to more than 1 year in 2012. A recovery phase is then observed after the 2015 earthquake with the $b^+$-value decreasing back to its initial value. Such a decrease of the $b^+$-value phase is also observed for the 2012 and 2014 sequences (Figure 2b), even if the large earthquake following them (2014 and 2015, respectively) prevents the observation of the full recovery. In summary, a similar pattern may be observed for all three mainshocks. First, there is no significant variation of $b^+$-value preceding the mainshock. Hence, the $b^+$-value does not show a predictive power. Second, the aftershocks may be associated with a decrease of $b^+$-value of various amplitudes and durations, followed in all cases by an increase above the initial $b^+$-value level, and a recovery phase.

# 5. Discussion

## 5.1. *b*-Value During Swarms and Mainshock-Aftershocks Sequences

We show in this study that a common pattern can be observed in the $b^+$-value evolution during the swarms in the Ubaye Region. The $b^+$-value increases during the beginning of the sequence, then decreases to its initial level toward the end of the swarm. Jenatton et al. (2007) also calculated the *b*-value evolution for the 2003–2004 swarm and observed the same behavior as described here. A similar temporal variation of *b*-value is also observed for other swarms worldwide. For example, Aswini et al. (2021) observed such behavior for three swarms of off Nicobar Region. They proposed different mechanisms to explain it, with one of them being related to the microfracturing of highly porous rocks, allowing fluids to generate seismicity. The strong clues for fluid-driven swarms in the Ubaye area (Section 2), make it likely that the $b^+$-value increase is related to fluid-pressure. This hypothesis is supported by analysis of the induced seismicity in Basel (Switzerland) that occurred during December 2006 due to geothermal stimulation (Bachmann et al., 2012). The authors looked at the behavior of the *b*-value close to and further away from the injection point. They found that the *b*-value is extremely high close to the injection point and similar to the tectonic *b*-value at the boundary of the seismic cloud, which demonstrates that fluid-pressure tends to increase *b*-value.

For the mainshock-aftershocks sequence in the Ubaye Region, there is no clear signature in the $b^+$-value that can be seen as a precursor of a mainshock, as there is no short-term decrease or increase of the $b^+$-value before the mainshock. It can be noted that $b^+$-value decreases below its background level for a few months after the recovery phase following the 2012 and the 2014 mainshocks (Figure 2b), hence preceding the 2014 and 2015 mainshocks, respectively. However, the amplitude of these decreases stays limited and follows a period with much larger variations. Therefore, if precursory signals exist in the *b*-value variations, as stated in Gulia and Wiemer (2019), it may be masked by the large number of aftershocks following the 2012 and the 2014 mainshocks, preventing a

**Figure 4.** Temporal evolution of the $b^+$-value around three mainshocks. In (a), (c), and (e), the blue bars show the monthly distribution of events. The red squares show the calculated $b^+$-values, located at the end of the bins containing 70 positive magnitude differences. Vertical gray bars show the uncertainties (two standard deviations). In (b), (d) and (f), same as in (a), (c), and (e), but by defining the bins from the mainshock time, in order to separate the seismicity to get bins containing events only before (cyan square) and after (purple square) the mainshock. The gray squares correspond to bins with mainshock included in the calculation of $b^+$-value while the mainshock is discarded for the cyan and purple squares. A same number of bins (20) are shown before and after the mainshock to allow for comparison of the $b^+$-value. In all panels, the red line shows the time of the mainshock. In (a), (c), (e), the time windows are globally centered on the mainshock time, while same number of bins (20) are shown before and after the mainshock (b, d, f).
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Schematic model linking temporal behavior of *b*⁺-value and depth processes during a mainshock-aftershocks sequence. In (a) to (e), evolution of *b*-value as function of time for each step of the conceptual models (f to j). In (f) to (j), conceptual cross-sections showing the processes generating the seismicity in the Ubaye Region. The gray areas represent the hydraulic barriers while the blue surfaces represent pressurized fluid distribution, the red lines show the rupture area, the red dots correspond to earthquakes. In all panels, the purple star represents the mainshock hypocenter.</figcaption></figure>

clear and unambiguous signature that may announce the subsequent mainshock. Moreover, the magnitude of the mainshocks here is about 5, smaller than earthquakes for which precursory signatures were observed (Gulia & Wiemer, 2019).

During the three sequences of mainshock-aftershocks studied here, the *b*⁺-value variations show some similarities, even if the overall pattern differs in amplitude and duration. First, there is no significant variation before the mainshocks that could be used as precursory signals. Second, a decrease of *b*⁺-value right after the mainshock is clearly visible for the 2012 mainshock, but within uncertainties for the 2014 and 2015 events. Similarly, Dascher-Cousineau et al. (2020) observed a sharp *b*-value decrease after the mainshock before a slow recovery for an earthquake in Puerto Rico. Third, an increase of the *b*⁺-value above the pre-mainshock value is observed for all sequences (Figure 4). An increase of *b*-value following the mainshock is also observed in different studies, like those for the Lushan Ms 7.0 (Hui et al., 2021) or for the Van Ercis Mw 7.2 (Görgün, 2013) earthquakes.

These successive phases in the *b*⁺-value variations agree with the role of fluids highlighted in 2012 and 2014 studies (De Barros, Baques, et al., 2019; Thouvenot et al., 2016), and with the model proposed by Baques et al. (2023) to explain how aftershocks are generated in the Ubaye Region. Figure 5 relates the temporal *b*⁺-value variations to the processes at play, following De Barros, Baques, et al. (2019) and Baques et al. (2023) and based on a fault-valve behavior (Sibson, 1990). Baques et al. (2023) showed that the rupture of the mainshocks requires fluid-pressure to be trapped within the fault (Figure 5f). Once pressure triggered the mainshock, the rupture directly generates aftershocks by Coulomb stress-transfer (King et al., 1994). Therefore, the fast decrease of the *b*⁺-value immediately after the mainshock (Figures 5b and 5g) might be related to an increase in differential stress, associated with the Coulomb stress-transfer, which directly generates aftershock-like seismicity (Figure 5g). As the mainshock also breaks the hydraulic barriers that allow the fluid to be pressurized within the main fault, the pressurized fluid is released and may diffuse in the medium. This drives aftershocks (De Barros, Cappa, et al., 2019; Hainzl et al., 2016; Miller, 2020; Ross et al., 2020), that behave as small swarm sequences. Due to the diffusion time of the fluid-pressure, fluid-driven seismicity is delayed from aftershocks that are directly triggered by the stress perturbations from the mainshock (De Barros, Baques, et al., 2019). Therefore, fluid-
pressure slowly becomes the predominant trigger of the seismicity, leading to the gradual increase of the $b^+$-value (Figures 5c and 5h). By analogy with the $b^+$-value variations during the swarms, this increase of $b^+$-value, followed by a recovery period, may highlight fluid processes (Figures 5c–5e). As fluid-pressure eventually dissipates to return to an equilibrium state, the $b^+$-value decreases to pre-mainshock $b^+$-values (Figures 5d, 5e, 5i, and 5j). This is consistent with the decrease of fluid-overpressure with time during a mainshock-aftershock sequence computed by Baques et al. (2023).

As stress-transfer and fluid-overpressure are both triggering aftershocks, the $b^+$-value variations show a trade-off response between these processes according to the number of events associated with each process. The differences in the $b^+$-value variations among the three aftershocks sequences (amplitude of the decrease following the mainshock and in the slow level of increase after) may be explained by this competition. Following the 2012 mainshock, aftershocks are dominantly triggered by stress-transfer, while fluid-pressure diffusion only slowly becomes predominant. On the contrary, fluid-pressure is the most efficient driver of seismicity following the 2014 and the 2015 mainshocks. Therefore, the $b^+$-value drop is more pronounced following the 2012 aftershocks, while the level of increase is higher in 2014 and 2015.

This trade-off between fluid processes and stress-transfer can be confirmed by the decay rate of the aftershocks. Indeed, Miller (2020) linked the *p*-value of the modified Omori's law (Utsu, 1961) to the presence of fluid-pressure, being a mechanism triggering the aftershocks. In this instance, we calculated the *p*-value for the three mainshock-aftershock sequences using seismicity data for a period of 1 year after the mainshocks. We find *p*-values of 0.74, 0.62, and 0.66 for the 2012, 2014, and 2015 sequences, respectively. These values, much smaller than 1, imply that stress-transfer alone cannot explain the aftershock sequences. Following Miller (2020), such rich and long-lasting sequences reflect fluid processes, that are even more evident after the 2014 and 2015 than in the 2012 aftershocks.

Following Gulia and Wiemer, this decrease in *b*-value may suggest that a larger shock may occur. However, here it is observed to be more transient and followed by an increase of *b*-value. Therefore, on average, the *b*-value after the mainshock tends to be similar or higher than the background one. Finally, the pattern we described in Figure 5 comes from sequences occurring in a context where fluid processes play an important role while tectonic rate deformation is very limited. Therefore, an extrapolation of this behavior to larger magnitude events occurring in high deformation rate settings may prove difficult as a universal behavior may not exist.

## 5.2. Other Possible Processes

We here interpret the $b^+$-value variations as primarily due to a competition between the increase of stress by Coulomb stress-transfer and fluid processes, on the basis of accumulating evidence showing the role of fluids in this area (Baques et al., 2023; Daniel et al., 2011; De Barros, Baques, et al., 2019; Jenatton et al., 2007; Leclère et al., 2013; Thouvenot et al., 2016). However, fluid-pressure can directly decrease the effective stress, but it may also trigger aseismic deformation (Guglielmi et al., 2015) by modifying frictional properties (Cappa et al., 2019) and seismic nucleation length (De Barros et al., 2024). Therefore, it indirectly modifies the shear stress on the faults (Duboeuf et al., 2021). The link between fluid processes and *b*-value variations may thus be more complex and ambiguous than just being directly linked to effective stress changes (Mukuhira et al., 2024). Alternatively, the *b*-value has been shown to depend on the differential stress, and on depth of the seismicity (e.g., Minetto et al., 2022). As shown by De Barros, Baques, et al. (2019), the 2014 aftershocks directly triggered by stress-transfer are globally deep, while fluid diffusion induces seismicity at a shallower depth. Therefore, the increase of $b^+$-value after the mainshocks can ambiguously be related to the shallower depth of the seismicity or to fluid processes. However, an increase of *b*-value is also observed during swarm sequences, in which the mean depth stays constant as seen during the 2003–2004 swarm (Jenatton et al., 2007). This suggests that fluid-pressure is likely to be the dominant factor in the increase of the *b*-value observed during both swarms and aftershock sequences. Finally, *b*-value changes were also interpreted as being due to changes in the location of the seismicity, which when occurring on large faults leads to a smaller *b*-value than off-fault seismicity (Collettini et al., 2022; King et al., 1983) and because *b*-value also depends on the roughness and structural heterogeneities of the faults (Goebel et al., 2017). However, the 2014 aftershocks are all relocated in the damaged volume around the main fault, without any events directly on the main fault (De Barros, Baques, et al., 2019). Additionally, no clear changes in the aftershock locations are observed in the 2003–2004 swarm, there is instead a radial migration (Daniel et al., 2011; Jenatton et al., 2007). Therefore, seismicity being on-fault or off-fault, or changes of fault
properties, seems unlikely to alone explain variations in *b*-value. In conclusion, even if several effects may explain changes in *b*-value, effects of fluid pressure, together with stress changes, seem to be the dominant factors to explain the fluctuations observed in this case. This statement applies to the Ubaye region, but it might be generalized to other areas where fluid processes are present. Given the variability in observed *b*-values and the complexity of the *b*-value interpretation, a generic explanation, valid in all contexts, seems unlikely.

## 6. Conclusion

The Ubaye Region seismicity is peculiar due to its alternating swarm and mainshock-aftershocks sequences. This seismicity in a low deformation rate environment reflects the important role of fluid processes at depth. In this paper, we explore how the temporal variation of *b*-value, computed through the *b*-positive method, may be related to the processes at play that are inferred from several previous observations. First, we show that the different swarms recorded in the Ubaye Region all show an increase in the *b*-value before returning to the initial level. We associate this behavior with a fluid-driven process before a return to the background level once pressure eventually dissipates, which can take months or years. Second, in the aftershock sequences there seems to be a competition between Coulomb stress-transfer that induces an immediate sharp decrease of *b*-value (clearly seen in the 2012 sequence) and fluid processes that lead to a delayed increase of *b*-value in all sequences. Therefore, *b*-value variations may help to discriminate the primary cause of the aftershocks, even with a catalog done at the national French scale.

We here only focus on the temporal variation of the *b*-value. To further complete this study of the *b*-value in the Ubaye Region, it could be of great interest to also have the spatial variations. It might bring some new insights on the location of the fluids in the medium. However, a complete catalog with a sufficient number of events to enable statistical analysis and, above all, precise localization is needed.

## Conflict of Interest

The authors declare no conflicts of interest relevant to this study.

## Data Availability Statement

Some of the figures were made using GMT (Wessel et al., 2019), MATLAB (MathWorks, 2023), Inkscape (Harrington et al., 2004–2005) and the color scales used in the different figures come from the Scientific Colour Map of Crameri (2018). All the codes created for this study are available here: https://github.com/BaquesM/b_value.git (Baques, 2025). Data sets for this research are included in this paper (and its supplementary information files): Baques et al. (2023). Supplemental material for this article includes distribution and difference of magnitude for consecutive events for the CEA/LDG catalog for the 2000–2019 period, temporal evolution of *b*-value from *b*-positive method for several size of bin, temporal evolution of the *b*-values for three specific catalogs, and MSE calculated for each temporal variation of *b*-value and the supplementary material S1 text, showing how the MSE is calculated.

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We thank A. Helmstetter for the discussion about the statistical view of *b*-value. Most of this work was done thanks to the funding grant of M. Baques PhD thesis from the IRSN and CEA. This work was also supported by the ANR INSeis (PI L. De Barros) under contract ANR-22-CE49-0018. We also thank the two anonymous reviewers and the editor Helen Glaves for their time and their thoughtful comments that have helped to improve the manuscript. Open access publishing facilitated by Universita degli Studi di Padova, as part of the Wiley - CRUI-CARE agreement.
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