# Fluid-induced seismicity: comparison of rate- and state- and critical pressure theory

## Abstract

Induced seismicity as generated by the injection of fluids in a homogeneous, permeable medium with faults with variable proximity to rupture conditions is simulated using the rate- and state-dependent frictional fault theory (RST) of Dieterich (J Geophys Res 99(B2):2601–2618, 1994) and the critical pressure theory (CPT) developed by Shapiro (Fluid-induced seismicity, Cambridge University Press, Cambridge, 2015). In CPT, the induced local seismicity density is proportional to the pressure rate, limited by the Kaiser Effect, and apparently un-related to the tectonic background seismicity. There is no time delay between a change in pressure rate and seismicity density. As a more complex theory, RST includes a time delay between a pressure change and induced seismicity and it is proportional to the natural tectonic background seismicity. Comparing both modelling approaches at a fixed location, this delay can be significant, dependent on a ‘free’ parameter that represents the lower threshold for pressure below which seismicity is not triggered. This parameter can be tuned so that the results of CPT and RST become similar. Approximations of the RST allow a new interpretation of the parameter ‘tectonic potential’ that controls the level of induced seismicity in CPT.

### Keywords

Fluid-induced seismicity Pressure diffusion Rate- and state-dependent frictional fault theory Critical pressure theory## Background

Gaucher et al. (2015) provide an overview on published methods employed for modelling induced seismicity in geothermal reservoirs. Here I focus exclusively on the rate- and state-dependent theory of Dieterich (1994), further referred to as RST for modelling fluid injection induced seismicity. For simplicity, I consider the case of fluid injection at the origin in a medium with isotropic pressure diffusion, characterized by constant diffusivity. Pressure depends on time and distance to the injection point only. I compare the induced seismicity obtained by RST with results of the critical pressure theory (CPT) developed by Shapiro et al. (2005) and subsequent papers and summarized in Shapiro (2015). RST and CPT differ mainly in the following aspects: (1) CPT postulates a density of seismogenic faults to which induced seismicity is proportional. RST assumes a tectonic background seismicity to which induced seismicity is proportional. (2) A change of stress to an individual fault has different consequences in RST and CPT. CPT considers only changes in normal stress, which is modified by the pore pressure of the injected fluid. If this pressure exceeds the criticality that is attributed to the fault it will rupture immediately. RST includes normal and shear stress changes and models the fault response with the rate- and state-dependent friction theory which assumes the friction on the fault to change with time and eventually lead to instability (=rapid rupture) but not instantaneously with the change of stress. In rate- and state-dependent friction theory faults are never at rest, they always slip with time but change from geologic slow slip rates to catastrophic slip rates, which are called earthquakes. (3) For an ensemble of seismogenic faults, the rate- and state-depended friction behaviour demands a non-uniform distribution of initial slip rate conditions. Exposed to a constant tectonic shear stress rate, this initial condition generates the constant background seismicity. Exposed to a specific time-dependent stress history, a time-dependent seismicity evolves. Dieterich (1994) provides the theory and the respective formulae.

In RST, faults are always slipping with a slip rate (=speed of differential motion across the fault). The friction on the fault (relating normal and shear stress) depends on this slip rate and one or more state parameters. The temporal evolution of the state parameter depends on the slip history. However, there is a steady state, where friction does not change, if the slip rate is constant and normal stress does not change. In a spring-slider model, a shear stress change is the product of the spring stiffness and the slip, or conversely an applied shear stress history results in slip history on the fault. There is a critical slip size beyond which the fault ‘runs away’, e.g. the slip rate become very high and the system unstable. The notion of instability in RST is not a transition from no slip to slip but the transition from slow to fast slip. Roy and Marone (1996) show that in response to a step-wise increase of shear stress, a fault experiences first a phase of quasi-static motion when the inertial force is negligible and later an inertia-dominated phase where friction can be neglected, the slip rate becomes very high. In terms of fault friction, the sudden increase in shear stress causes an immediate increase in friction, followed by the quasi-static decrease of friction with slip rates above geologic slip rates but below the inertia-driven fast rate. Only the latter one constitutes the earthquake. Rate- and state-dependent friction laws conform to laboratory observations (Linker and Dieterich 1992) and are required to model aftershocks, which was the initial application of RST. If shear stress is changed instantaneously by a main shock, the seismicity does not appear immediately in the area around the main fault where the stress change is large enough but evolves with time resulting in an Omori-type temporal distribution of seismicity (Omori 1894; Utsu 1961). This behaviour cannot be understood with a criticality model, as in this case all aftershocks would occur immediately after the main shock and not days, months, or years later.

Whereas RST requires a specific slip rate distribution to model a constant tectonic background seismicity that changes in response to a stress field superimposed on the tectonic shear loading, CPT requires assumptions on the spatial distribution of criticality only and includes no tectonic background seismicity. Whereas CPT is designed for changes in normal stress by changing pore pressure, Dieterich (1994) derives a general equation for seismicity changes, as compared to the background seismicity if shear and/or normal stresses are changed. The RST has been used in the simulation of aftershocks and earthquake swarms (Catalli et al. 2008; Daniel et al. 2011; Dieterich et al. 2000; Kilb et al. 2002; Toda et al. 2002, 2003) and recently also utilized for seismicity modelling in geothermal reservoirs (Hakimhashemi et al. 2014). In this model, the change in Coulomb failure stress (King 2007) is combined with the RST theory. Segall and Lu (2015) reformulated the RST seismicity evolution equation and included the Coulomb failure stress to model the seismicity as consequence of dyke intrusion in volcanic systems. Dieterich et al. (2015) addressed the seismicity caused by fluid injection, however, without employing their seismicity formula. The CPT has emerged from the study of induced seismicity in relation to permeability for instance in (Shapiro et al. 1999, 2002; Rothert and Shapiro 2003; Shapiro and Dinske 2009).

The next section represents a summary of the essentials of the critical pressure theory (CPT) including the Kaiser Effect and the rate- and state-dependent theory (RST), which was originally set up for sudden change of shear stress (earthquakes) and has been modified for the case of pressure changes by Wenzel (2015). The evolution of seismicity can be written as a Ricatti Equation (Ince 1956, pp. 23–35), which is used for approximations as well as for numerical implementation. The approximations allow (1) to understand the similarity of results in modelling seismicity with CPT and RST, and (2) to establish a scaling relation between RST and CPT that leads to a new interpretation of the ‘tectonic potential’ that controls the level of induced seismicity in CPT.

It is important to understand the differences between the stress changes in modelling aftershocks and pressure-induced seismicity. Pressure affects normal stress on faults. Released shear stress, after an earthquake, mostly affects shear stress on faults in its vicinity. The main difference in terms of induced seismicity is that the latter is a sudden change in stress whereas the former is a smooth change provided by pressure diffusion. The implications of this, on RST seismicity is demonstrated in the section on ‘Approximate behaviour of rate and state-dependent Ricatti solutions’. Induced seismicity after pressure shut-in is relevant as seismicity does not stop with shut-in; moreover, in many cases, the maximum magnitudes are observed. The post-shut-in seismicity is controlled by the diffusion constant of the pressure and the lower cut-off pressure below which no seismicity can be triggered. This applies to both theories and does not constitute a major difference. The implications for the seismicity evolution are studied numerically.

## Comparison of critical pressure theory and rate- and state-dependent theory

I briefly review the key features of the critical pressure theory (CPT) for induced seismicity. A permeable medium contains faults with density (faults per unit volume) \(n_{f} \left( {\vec{x}} \right)\) with location vector \(\vec{x}\). The fault density is understood as the density of seismogenic faults. There may be faults that slip in an aseismic way but they are not subject of induced seismicity. Faults have different sizes and it is often assumed that the size distribution follows an exponential law so that the magnitude distribution in each source volume is compatible with a Gutenberg/Richter (1956) distribution. As seismicity is defined as the number of earthquakes within a source volume above a certain magnitude threshold, we set this magnitude threshold for simplicity of the mathematics to zero. If the actual threshold for lower magnitude, usually constrained by the magnitude level where seismic monitoring provides complete observations, is different, the fault density can easily be scaled from magnitude zero with the Gutenberg/Richter distribution. The injection of fluid into the medium is modelled as a diffusion process. The modelling of fault ruptures includes two aspects: (1) An individual fault can be triggered by pressure changes associated with fluid injection. The shear stress on these faults provided by the tectonic environment is characterized by a criticality parameter *C* > 0; this value must be exceeded by the pore pressure to trigger an earthquake. According to Shapiro (2015, p. 202) the simplest assumption on the spatial distribution of seismogenic faults and criticality is that the faults are randomly distributed in space, statistically homogeneous so that the fault density fully characterizes their statistics. Each of these faults is associated with a criticality value. If it is high, the fault is rather stable and requires a high pore pressure during fluid injection to trigger it. If criticality is low, the fault is close to rupture condition and requires only a small pressure for triggering. The size of criticality is modelled by a distribution function and assumed to be independent on location. The cumulative distribution \(F_{C} \left( C \right)\) is the probability that the criticality parameter is less than *C*. It is further assumed that there is a lower trigger threshold \(C_{\text{CPT}}\) (typically 1–100 kPa) below which no triggering occurs. It is also reasonable to assume an upper threshold \(C_{\text{CPT}} + \Delta C\) beyond which the fault ruptured even if it not pre-stressed. \(\Delta C\) should be on the order of the stress drop typically released by earthquakes: 3–10 MPa.

For time-variable fluid injection rates, the seismicity has to be calculated by convolving the input pressure signal with the Green’s Function for the diffusion in a homogeneous 3D medium and the spatial integration must be done numerically. However, (2) is very useful for scaling CPT and RST.

*D*to demonstrate main features of CPT. Figure 1 shows the volume flow rate time history at the injection point (with 20 l/s as maximum) and the pressure at 200, 300, and 400 m calculated with the analytic solution of the Green’s Function convolved with the source volume flow time history. With growing distance from the injection point (

*r*), the pressure amplitude becomes smaller. In addition, the diffusion process with \(D = 0.1\;{\text{m}}^{ 2} /{\text{s}}\) modifies the cyclic behaviour significantly at distances in excess of 100 m. This follows from the relation of characteristic distance and time in diffusion \(r^{2} = D \cdot \tau\) with the period of the pressure cycle as characteristic time. The pressure peaks at 200 m are shifted by 30 h, at 300 m by 45 h. The second maximum is larger than the first one, and at 400 m there is no distinct first peak and the second is shifted by 70 h.

\(C_{\text{RST}}\) controls the lower pressure \(p\left( t \right)\) below which no seismicity will be triggered. Its role in the evolution of seismicity is similar to the lower pressure threshold \(C_{\text{CPT}}\) in CPT. This is not obvious in Eq. (3) but will become evident later in the paper. The assumptions in deriving (3) are: The tectonic shear stress rate is much smaller than the pressure rates within the medium and pressure \(p\left( {r,t} \right)\) is much smaller than the normal stress \(\sigma_{n}^{0}\).

*t*= 0 \(R\left( 0 \right) = 1\) and after a constant pressure level is achieved at say \(p_{\infty },\) where \(\dot{p} = 0\) the induced seismicity density change should vanish \(\dot{R}\left( {t \to \infty } \right) = 1\). Wenzel (2015) shows that expression (8) can be approximated by

It is thus evident that \(\Delta C\) represents the static stress drop of earthquakes, typically in the range of 3–10 MPa for tectonic events (Kanamori and Brodsky 2004). Shapiro et al. (2007) call the ratio of fault density to stress drop the ‘tectonic potential’ as it characterizes the tectonics of the region within which fluid injection is done. As I show it is identical to the ratio of tectonic shear stress rate to tectonic background seismicity density. Thus the ‘tectonic potential’ can be better understood and also measured by geophysical ad geodetic means.

^{−3}, storativity in Pa

^{−1}, and \(\Delta C\) in Pa. Once a value for the storativity (\(S = 10^{ - 11} \,{\text{Pa}}^{ - 1}\) in the calculations) is assumed, a given SI requires a specific value of the background seismicity. For instance, \(\varSigma = - 1\) results in \(\dot{\nu }_{\text{tec}} = 0.125\,{\text{km}}^{ - 3} {\text{year}}^{ - 1}\). Figure 3 shows the comparison between the CPT and RST results for total seismicity of the two-cycle injection with

*D*= 0.1 m

^{2}/s (see Fig. 1). The lower trigger threshold for CPT is 1 kPa; and also \(C_{\text{RST}} = 1\;{\text{kPa}}\). There are no differences visible. The comparison is surprisingly good given the circumstance that the methods as reflected in (2) and (4) are rather different.

## Approximate behaviour of rate- and state Ricatti solutions

*t*limited to days. Expression (11) is similar to the seismicity density evolution derived in Dieterich (1994) for a step-increase of shear stress \(\tau \left( t \right) = \tau_{0} \cdot H\left( t \right)\) if \(g_{0} \cdot p_{0}\) is replaced by \(\tau_{0} \cdot\). The solutions show an Omori-type of behaviour with Omori exponent being 1. The seismicity density jumps to its highest level \(\frac{{g_{0} \cdot p_{0} }}{{C_{\text{RST}} }}\) immediately after time 0. Then it decays to 0 with a time constant \(t_{\text{ST}}\) that can be defined as the time within which the initial seismicity level reduces to 50%.

*t*= 0 and approaches the CPT solution with a time constant \(t_{\text{G}}\) that can be quantified as

^{−3}s

^{−1}. The lower threshold of CPT delays the onset of seismicity by \(t = {{C_{\text{CPT}} } \mathord{\left/ {\vphantom {{C_{\text{CPT}} } q}} \right. \kern-0pt} q}\). 1 kPa as lower threshold causes only 40 s delay. Figure 4 shows the behaviour of RST for different values of the lower threshold. If I used the value of 1 kPa for the threshold that has been used in Fig. 3, there would be no visible difference to the CPT response: an immediate increase as soon as the pressure starts to grow, and an immediate reduction to zero, once the pressure remains constant. For a value of 6 kPa, the seismicity density reaches the CPT level of 0.025 km

^{−3}s

^{−1}in a few hours, then shows a slight decay to 0.024 km

^{−3}s

^{−1}at 111 h and after the end of the pressure increase a rapid decay within a few hours. As expected from (12), the delay for seismicity to rise grows with growing lower threshold (50, 100, 170 kPa) and the maximum seismicity densities become smaller. The time constants for the decay of seismicity, after 111 h, are also proportional to the lower threshold. If I use the formulae for the time constants for growth and decay of seismicity, a threshold value of \(C_{\text{RST}} = 100\,{\text{kPa}}\) provides 43 h until RST seismicity density grows to half the CPT level and 4 h to drop after 111 h. These estimates match the full numerical solution of the Ricatti Equation in Fig. 4 well.

The key insight from the previous discussion is that if \(C_{\text{RST}}\) is in the range of 10 kPa or even smaller the time constants involved in RST are quite small and do not influence the overall seismicity pattern strongly.

## Post-shut-in behaviour

*D*of the medium: \(r_{F} \left( t \right) = \sqrt {6 \cdot D \cdot t}\) After shut-in, there is still a pressure front; however, its shape cannot be expressed analytically and in addition to

*D*, it depends on the time of shut-in. The location where the pressure rate turns from positive values so that seismicity can be generated to negative values where no seismicity ceases is called the back front (Parotidis et al. 2004) and has the following form:

Seismicity at a specific time after shut-in can only occur within a spherical shell with the outer radius controlled by the seismicity front, and the inner radius by the back front. The actual size of the shell is controlled by the diffusivity and the lower threshold of the pressure \(C_{\text{CPT}}\) that must be exceeded before seismicity can be triggered. It causes a maximum distance beyond which not triggering will occur even if injection is maintained and thus has a significant influence on the outer radius but little influence on the back front (inner radius).

As formula (2) indicates, the level of seismicity before shut-in is independent on diffusivity, but would change with a lower threshold \(C_{\text{CPT}}\) required for the pressure before seismicity can be induced. The seismicity after shut-in, however, depends on both the injection rate and the diffusivity. In addition, \(C_{\text{CPT}}\)has a significant influence. Langenbruch and Shapiro (2010) have studied the post-shut-in seismicity in the CPT context in detail and hypothesized an Omori-type decay of the seismicity after the end of fluid injection. A higher diffusivity causes a more rapid decay of seismicity. The diffusion front propagates faster for high diffusivity and thus reaches the lower threshold earlier in time. A higher threshold leads to an earlier stop and thus shortens the duration of seismicity after shut-in.

^{2}/s and for lower thresholds of 1 kPa (blue), 5 kPa (green), and 20 kPa (red). Injection increases linearly during 50 h to 20 l/s, remains constant for 20 h, and after this drops to zero. The decay of seismicity after 70 h follows a power law as claimed by Langenbruch and Shapiro (2010). For given diffusivity, a higher threshold leads to a lower level of seismicity between 50 h and 70 h and a more rapid decay. This also applies to the lower diffusivity of 0.1 m

^{2}/s. However, in this case, the role of the threshold becomes less significant. This is important for the assessment of post-shut-in seismicity. If a probabilistic estimate of the exceedance probability of a certain magnitude is demanded one needs to know the crustal volume that will be affected by pressure diffusion and the seismicity rate after shut-in. Both ingredients of hazard assessment depend on the lower threshold. The observation of the temporal and spatial evolution of seismicity during injection allows inferring the value of the diffusivity, but not the lower threshold value, which requires the observation of the decay of seismicity. Therefore, it appears difficult to assess the probability of post-shut-in earthquakes before the actual shut-in. However, for realistic diffusivities in rocks the dependency of post-shut-in earthquakes on the lower threshold becomes smaller and probabilistic assessments more feasible. As these aspects are not the topic of this paper, they are not discussed further.

^{2}/s. For 0.2 kPa (blue), the maximum seismicity is 6 events per hour and the tail after shut-in in the range of days. For 5 kPa (red), the maximum value of seismicity is 1.8 events per hour but the tail is only in the range of hours. In general, a lower value of \(C_{\text{RST}}\) results in a higher level of maximum seismicity and a longer tail after shut-in. Conversely, a higher value results in a lower level of maximum seismicity and a longer tail after shut-in. In this sense RST, with \(C_{\text{RST}}\) properly set, can produce very similar results as compared to CPT. Similar to what has been said earlier, it is possible to tune the lower threshold parameter for RST such that seismicity very similar to CPT evolves during the injection phase and after shut-in.

## Discussion and conclusion

The critical pressure theory (CPT) developed by Shapiro (2015) for a quantitative description of the evolution of seismicity during fluid injection under pressure in permeable rocks assumes a Mohr–Coulomb-type rupture model for earthquakes. If the excess pressure exceeds the criticality of the fault, rupture occurs. This leads to a model where the seismicity density at a site is proportional to the temporal pressure derivative at this site. In addition, the Kaiser Effect must be considered. A quite different approach is provided by the rate- and state-dependent frictional fault theory (RST) of Dieterich (1994), which has been developed originally for modelling aftershock activity of larger earthquakes in response to sudden shear stress release. As RST has been formulated for shear and normal stresses acting on a medium, it can be modified for the case of fluid injection (Wenzel 2015) and results in a Ricatti Equation for the enhancement of the natural tectonic background seismicity at a site by fluid injection. The non-linear equation includes the Kaiser Effect and contains the time histories of pressure and its temporal derivative.

The comparison of both methods leads to the following conclusions:

Mathematical approximations allow simplifying RST to CPT solutions so that seismicity density becomes proportional to the temporal pressure derivative. With this results CPT can also be interpreted as enhancement of the natural tectonic background seismicity so that both theories predict a higher level of induced seismicity in areas where the natural seismicity is higher and conversely a lower level of induced seismicity in areas where the natural seismicity is lower.

Both theories include a parameter that controls the lower pressure threshold below which no seismicity is induced. The numerical comparison—without approximations—shows that the respective threshold value of RST can always be specified in a way that both RST and CPT provide very similar seismicity time histories for given injection time functions. This fact is established for the periods of active injections as well as for the shut-in phase. One can thus conclude that the computationally more demanding RST is not necessary for modelling-induced seismicity. The rate- and state-dependent friction behaviour of faults that is highly relevant in modelling aftershocks can be ignored in induced seismicity. This, in turn, is related to the key difference in stress histories to which a seismogenic medium is exposed in case of fluid injection and aftershocks. In the latter case, the medium experiences a stress shock as the shear stress release from the main earthquake occurs within seconds. In contrast, pressure changes resulting from a diffusion process are always ‘smooth’; rate and state dependency is not relevant and occurs on short time scales and at minor rates of seismicity. Thus, CPT is a fine theory for modelling seismicity in an enhanced geothermal, i.e. petrothermal, reservoir.

## Notes

### Authors’ information

Friedemann Wenzel is Prof. of Geophysics at Karlsruhe Institute of Technology (KIT). His main research fields are seismology, earthquake hazard, and risk.

### Acknowledgements

I am indebted to two anonymous reviewers whose comments and suggestions helped significantly in improving the manuscript.

### Competing interests

There are no potential competing interests. The paper has not been submitted anywhere else.

### Availability of data and materials

Does not apply as no data are used; the study compares with synthetic examples.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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