Probability of inducing given-magnitude earthquakes by perturbing finite volumes of rocks

Authors


Corresponding author: S. A. Shapiro, Fachrichtung Geophysik, Fachbereich Geowissenschaften, Freie Universität Berlin, Malteserstr. 74-100, 12249, Berlin, Germany. (shapiro@geophysik.fu-berlin.de)

Abstract

[1] Fluid-induced seismicity results from an activation of finite rock volumes. The finiteness of perturbed volumes influences frequency-magnitude statistics. Previously we observed that induced large-magnitude events at geothermal and hydrocarbon reservoirs are frequently underrepresented in comparison with the Gutenberg-Richter law. This is an indication that the events are more probable on rupture surfaces contained within the stimulated volume. Here we theoretically and numerically analyze this effect. We consider different possible scenarios of event triggering: rupture surfaces located completely within or intersecting only the stimulated volume. We approximate the stimulated volume by an ellipsoid or cuboid and derive the statistics of induced events from the statistics of random thin flat discs modeling rupture surfaces. We derive lower and upper bounds of the probability to induce a given-magnitude event. The bounds depend strongly on the minimum principal axis of the stimulated volume. We compare the bounds with data on seismicity induced by fluid injections in boreholes. Fitting the bounds to the frequency-magnitude distribution provides estimates of a largest expected induced magnitude and a characteristic stress drop, in addition to improved estimates of the Gutenberg-Richter a and b parameters. The observed frequency-magnitude curves seem to follow mainly the lower bound. However, in some case studies there are individual large-magnitude events clearly deviating from this statistic. We propose that such events can be interpreted as triggered ones, in contrast to the absolute majority of the induced events following the lower bound.

1 Introduction

[2] Fluid-induced microearthquakes in hydrocarbon or geothermal reservoirs, aftershocks of tectonic earthquakes or seismic emission in rock samples, are examples of seismicity resulting from a seismogenic activation of finite volumes of rocks. Fluid-induced earthquakes of magnitudes 3 to 4 occurred at several Enhanced Geothermal Systems (EGS) like those of Basel, Cooper Basin, The Geysers field, and Soultz [Giardini, 2009; Majer et al., 2007; Häring et al., 2008; Dyer et al., 2008; Baisch et al., 2009]. It seems that smaller but still perceptible events can be also observed by hydraulic fracturing of hydrocarbon reservoirs (see http://www.cuadrillaresources.com/news/). Induced seismic hazard becomes a topic of significance in the shale-gas industry (see http://www.energy.senate.gov/public/index.cfm/hearings-and-business-meetings?ID=2c908340-a9bb-40b4-bf7f-8308b272893d). Its understanding is of a considerable importance for mining of deep geothermic energy [Giardini, 2009; Majer et al., 2007; Cornet et al., 2007; Häring et al., 2008]. It is of significance for CO2 underground storage [see Zoback and Gorelick, 2012] and possibly also for other types of geo-technological activity [see Avouac, 2012].

[3] Similarly to the tectonic seismicity, statistics of the induced seismicity can be rather well described by the Gutenberg-Richter frequency-magnitude distribution [Shapiro et al., 2007, 2010, 2011; Shapiro and Dinske, 2009b; Dinske and Shapiro, 2013]. However, large-magnitude events deviate from these statistics [Shapiro et al., 2011]. In this paper we theoretically and numerically analyze the influence of the finiteness of a perturbed volume on the frequency-magnitude statistics of induced events. In contrast to exact specific mechanical models of rupturing a given pressurized fault (see the recent detailed analysis by Garagash and Germanovich [2012]), our analysis is a phenomenological one. It considers many faults and attempts to relate the geometry of the stimulated volume to observable statistical features of the induced seismicity using rather general heuristic assumptions. It is possibly applicable to different types of the seismicity triggering physics like a triggering by pore-pressure perturbations or stress perturbations or a triggering by rate-and-state processes modifying the friction. On the other hand, we describe different statistical scenarios of the triggering process. A tendency of real statistics of the induced seismicity to follow one or another scenario may be indicative for the physics of event nucleation.

[4] We start our analysis with a brief overview of a statistical model of induced seismicity, which ignores the fact that rupture surfaces and stimulated volumes are finite. This model adequately describes a general dependence of the seismicity on the volume of the injected fluid. It introduces also some other seismicity-controlling parameters like the seismogenic index. Then we shortly review data-based indications that the large-magnitude events have different statistical features than numerous small events. We observe that large-magnitude events are underrepresented compared with the predictions of the model mentioned above. Thus, we consider different scenarios of geometric relationships between the stimulated volume and potential rupture surfaces. On this basis, we propose lower and upper bounds of the occurrence probability of given-magnitude induced events. These bounds take into account the finiteness of the rupture surfaces and of the stimulated volumes. The bounds are computed from the statistics of arbitrary-size arbitrary-oriented penny-shaped inclusions (representing potential rupture planes) intersecting a finite ellipsoidal stimulated rock volume. A cuboidal stimulated volume is also considered for preferentially oriented rupture surfaces. We show how the finiteness of rupture surfaces and of the stimulated volume influences the frequency-magnitude relation of the induced seismicity. We also show how an estimate of an averaged stress drop can be obtained from the statistics of seismicity. Further we discuss applications of our results to some borehole-injection-based case histories. The observed frequency-magnitude curves seem to follow mainly the lower bound of the occurrence probability of given-magnitude induced events. However, in some case studies there are individual large-magnitude events clearly deviating from the lower-bound statistic. We propose that such events can be interpreted as triggered ones, in contrast to the absolute majority of other events which we call the induced ones. Then we discuss this interpretation.

2 Theoretical Introduction: Point-Like Preexisting Cracks

[5] In the following we review a model of magnitude distributions of fluid-induced earthquakes based on an assumption that their rupture surfaces are very small in respect to the size of the stimulated volume [Shapiro and Dinske, 2009b; Shapiro et al., 2011; Dinske and Shapiro, 2013]. In spite of its simplicity and schematic character, this model is able to well explain large-number statistics of induced events. In the next sections our theoretical analysis will not be any more based on the fluid-induced mechanism of event triggering. However, we will need this model as a reference point for understanding our observations.

[6] We consider a point-like pressure source switched on at the time τ=0 at the origin of an infinite medium. Preexisting cracks (defects) are assumed to be randomly distributed with a bulk concentration N. Each of these cracks is characterized by an individual critical value Cp of the pore pressure necessary in accordance with the Coulomb failure criterion for occurrence of an earthquake along such a defect. If Cp is high, the corresponding fracture will be stable. If Cp is low, the fracture will be close to failing. We assume that Cp(r) is a statistically homogeneous random field. If at a given point r of the medium pore pressure p(t,r) increases with time and at time t0 it becomes equal to Cp(r), then this point will become a hypocenter of an earthquake occurring at this location at time t0. We assume that no earthquake will be possible at this point again. This is equivalent to an assumption that the stress corrosion, tectonic load, tectonic deformation, and other relaxation phenomena (also related to the rate- and state-dependent friction, see, e.g., Dieterich [1994] and Segall et al. [2006]) leading to recharging critical cracks are much slower than the pore-pressure relaxation. Then, the probability of an earthquake occurring in the time interval 0<τ<t at a given point r will be equal to a probability of a critical pressure to be smaller than the maximum pore pressure at the point r achieved in this time interval: Wev(Cp(r)≤Maximum{p(t,r)}). If the pore-pressure perturbation caused by the fluid injection is a nondecreasing function, then

display math(1)

where f(Cp) is the probability density function (PDF in the following) of the critical pressure. Data analysis of Rothert and Shapiro [2007] indicates that Cp is usually of the order of 103–106 Pa and f(Cp) can be roughly approximated by a boxcar function f(Cp)=1/(CmaxCmin) if CminCpCmax and f(Cp)=0 elsewhere. Usually Cmax is of the order of several MPa. It is orders of magnitude larger than Cmin. Thus, f(Cp)≈1/Cmax (this is used in the last right-hand part of equation (1)). We also assume that Cmax is larger than the pressure perturbation (excluding maybe a small vicinity of the source). Therefore, the event probability is approximately proportional to the pore-pressure perturbation. In turn, a spatial density of events is proportional to the event probability. Thus, the spatial event density must be also proportional to the pore-pressure perturbation [see, e.g., Shapiro et al., 2005]. Thus, the total event number is given by a spatial integration of Wev:

display math(2)

where δ is the dimension of the space where the fluid flow takes place and Aδ is a corresponding proportionality coefficient of the volume differential (e.g., A3=4π). Finally, R denotes a radius of an effective injection cavity.

[7] A consideration of the mass conservation of the injected fluid (an integration of the continuity equation, see Shapiro and Dinske[2009b]) yields

display math(3)

Here mc(t) and Qc(t) denote the cumulative mass and volume of the fluid injected till the time t, respectively. Further, ρ0 is a fluid density and Sp is a poroelastic compliancy of the fluid-saturated rock (the uniaxial storage coefficient). In the last two approximate terms we have neglected any pressure dependency of the fluid density and introduced the volumetric rate of the fluid injection, Qi(t). From equations (3) and (2) we further obtain

display math(4)

[8] To compute an amount of events with magnitude larger than a given one, we must introduce some assumptions about magnitude statistics. We assume further that the induced seismicity obeys the Gutenberg-Richter statistics [see Shearer, 1999, p. 288]. This means that the probability WM of events with the magnitude larger than M is given by log10WM=abM, where a and b are regional seismicity constants. Then, the number of events with magnitudes larger than M is equal to the product WMNev(t):

display math(5)

where Σ=a− log10(CmaxSp)+ log10N. This constant is completely defined by seismotectonic features of a given location. We call it a “seismogenic index.” The larger this index is, the larger is the probability of events of significant magnitudes. It is difficult to theoretically calculate Σ because of some unknown parameters (e.g., N). However, Σ can be estimated using equation (5) and parameters of seismicity induced by an injection experiment at a given location [see also Shapiro et al., 2007, 2010, 2011; Shapiro and Dinske, 2009b; Dinske and Shapiro, 2013].

[9] Equation (5) can be further rewritten in a more conventional form of the Gutenberg-Richter law but with the a value being time dependent:

display math(6)

with

display math(7)

Thus, for fluid injections the a value becomes a function of the time elapsed after the injection beginning. It it is given by the sum of the seismogenic index and of the time-dependent cumulative volume of the injected fluid.

[10] Finally, one more useful reformulation of equation (5) can be proposed. Let us introduce a specific magnitude MΣ defined as follows:

display math(8)

Note that similarly to the seismogenic index the specific magnitude is also completely defined by seismotectonic features of the injection site. The larger the specific magnitude, the larger is probability of significant induced events. Using the specific magnitude, the Gutenberg-Richter law modified for fluid injections (5) can be written in the following form:

display math(9)

Therefore, the number of events with MMΣ is given just by the cumulative volume of the injected fluid. Recently, Dinske and Shapiro [2013] observed this type of behavior in real data. Note that both the Σ and MΣ depend on the choice of the metric system for the injected volume. We work in the System International and measure Qc in m3. We usually observe Σ in the range from −8 to 1 and MΣ in the range from −4 to 1.

[11] The statistical model of induced seismicity we summarized above can be equally well applied for nonmonotonic injections. In such a case the probability (1) is given by a minimum monotonic majorant of the pore pressure [see Parotidis et al., 2004], and numerical computations are then required to estimate log10NM(t) for each particular situation.

3 Observations

[12] The derivation in the previous section and our practical experience show that during an active fluid injection with nondecreasing injection pressure equation (5) can be applied to approximately describe a large number of induced earthquakes. However, we observe systematic deviations from equation (5) for large-magnitude events. Their number is significantly smaller than expected, especially for short injection times [Shapiro et al., 2011].

[13] Figure 1 shows examples of distributions NM as a function of the injected fluid volume at different sites. In all examples, an approximate agreement of equation (5) with numbers for small-magnitude events can be demonstrated [see Shapiro et al., 2007, 2010, 2011]. Indeed, according to equation (5) in a range of not too large magnitudes, lines of log10NM are nearly mutually parallel and they are nearly regularly spaced. However, the number of large events is systematically smaller than a regular spacing of the lines log10NM would imply.

Figure 1.

Examples of NM as functions of the injected fluid volume at different sites. All curves correspond to the injection periods only.

[14] Large-magnitude events correspond to large-scale ruptures. Such events are less common than small-magnitude events. The statistics of potential rupture surfaces in rocks must correspond to a classical Gutenberg-Richter distribution of earthquakes produced on them. For a rupture surface a probability to intersect a stimulated volume of rocks will be the higher the larger the scale of this surface is. We will accept the following evident (but still heuristic) assumption: the probability of an earthquake on the corresponding rupture surface depends on the geometric relation between this surface and the stimulated volume. We will consider then two possible scenarios.

[15] In the first scenario we assume that to induce an event with a given rupture surface, it is enough to stimulate a small spot of this surface. In the induced seismicity then, the portion of large-magnitude events in respect to the portion of small-magnitude events should be higher than the proportion of potential large-scale rupture surfaces in relation to small ones in rocks. Therefore, large-magnitude events should be overrepresented in comparison to the expectations based on the Gutenberg-Richter statistics. This scenario seems to contradict with the observations. However, to prove this we must quantify this scenario.

[16] In the second scenario we assume that to induce an earthquake on a fault patch, a significant part of this patch must be stimulated. This is in agreement with the following formulation of the Coulomb failure criterion: to enable an earthquake along a given interface, an interface-integrated tangential stress must overcome a total friction force. As soon as a largest part of a potential rupture surface remains unperturbed, a probability of an earthquake remains low. Therefore, to enable an earthquake, a significant part of the corresponding rupture surface should belong to a stimulated volume. This scenario seems to be more adequate than the previous one. To prove this statement also, this scenario must be quantified and compared with the observations.

[17] To quantify both scenarios above and compare them with the frequency-magnitude statistics of induced events, we must modify the probability WM from the previous section. The modification of this probability must take into account the effect of the finiteness of the stimulated volume and of rupture surfaces. Note that both scenarios are not directly related to the physics of the stimulation of potential rupture surfaces. They just describe two different possible statistical patterns of the phenomenon. Therefore, they may be applicable to the seismicity induced by elastic stress or pore-pressure perturbations as well as to the seismicity induced by other processes like rate- and state-dependent friction alterations. On the other hand, a clear preference of the seismogenic process to follow one of these scenarios can provide us with a useful seismotectonic information.

4 Theory: Statistics of Potential Rupture Surfaces and of Seismic Events

[18] Let us assume that a finite volume of rocks V (the stimulated volume) has been somehow impacted so that seismogenic conditions in it have been changed sufficiently to produce seismicity. In practice we assume that the stimulated volume is approximately defined by an outer envelope of a cloud of hypocenters of induced seismicity. We consider the following simplified abstract model. A stimulated volume is an ellipsoid or a cuboid which can grow with time (e.g., due to a fluid injection). Rupture surfaces are randomly (or preferentially) oriented planar circular discs (penny-shaped inclusions with vanishing thickness). The spatial distribution of centers of the discs is random and statistically homogeneous with the bulk concentration N.

[19] We introduce firstly the probability Wf(X)=fX(X)dX, which is a probability of a given potential rupture surface in the unlimited medium to have a diameter X. fX(X) is a PDF of a rupture surface of the size X. For example, later, we will assume a power-law fX(X). Further, let Wc(X) denote a probability that the center of the rupture surface (of diameter X) belongs to the stimulated volume under the condition that this surface intersects the volume. Let us also for the moment assume that all ruptures have the size X. It is clear then that the product of Wc(X) with the number of all rupture surfaces intersecting the stimulated volume gives the number of rupture surfaces having their centers in the volume. On the other hand, this number is equal to the product NV. Thus, the ratio NV/Wc(X) is the number of all the rupture surfaces intersecting the volume. Recalling now that the rupture surfaces are statistically distributed over their size, we conclude that the ratio Wf(X)/Wc(X) gives the probability of a rupture surface intersecting the stimulated volume to have the diameter X.

[20] Now we can formulate a probability WE(X) of an induced seismic event to have a rupture surface of diameter X:

display math(10)

Ws denotes a probability of the corresponding rupture surface to be sufficiently stimulated to produce the event. This probability is a conditional one. It implies that the rupture surface has something to do with the stimulated volume, i.e., at least intersects it (intersecting includes also touching). The presence of Wc(X) in equation (10) shows that there are much more rupture surfaces intersecting the stimulated volume than just the product NV.

[21] The quantity Wg(X)=Ws(X)/Wc(X) describes the influence of the geometry of the stimulated volume. Let us now concentrate on its part, the probability Ws(X). Shapiro et al. [2011] assumed that seismic events will be possible only if their rupture surfaces are located completely within the stimulated volume. Consequently, they investigated the probability Wvol(X) that a disc of a diameter X is completely contained within a given stimulated volume under the condition that a center of this disc is located within the stimulated volume. They have neglected a possibility of seismic events with rupture surfaces intersecting the stimulated volume partially only. Here we will further develop their approach. Wg will also take into account rupture surfaces intersecting the stimulated volume.

[22] Let us consider all potential rupture surfaces intersecting or located within the stimulated volume. We recall that Wc(X) is a probability of such a surface of diameter X to have its center within the stimulated volume. Let us also define a probability of a seismic event along this whole rupture surface, We1(X), under the condition that the rupture center is located within the stimulated volume but the rupture surface is not entirely contained in the volume. Respectively, We2(X) will denote the event probability for such a rupture surface under the condition that its center is outside of the stimulated volume. We compute further the probability Ws(X) of a seismic event on a rupture of diameter X. Such an event occurs on a rupture located completely within the stimulated volume. Its probability is given by the product Wvol(X)Wc(X). Such an event can also occur on a rupture with the center inside the volume but rather intersecting the volume only. Corresponding probability is given by (1−Wvol(X))Wc(X)We1(X). Finally, such an event can occur on a rupture intersecting the stimulated volume and having its center outside of the volume. Corresponding probability is given by (1−Wc(X))We2(X). Therefore, the probability of a seismic event along a rupture of diameter X is given by the sum

display math(11)

Note that under assumptions of Shapiro et al. [2011], Wc(X)=1 and We1(X)=We2(X)=0, and we obtain Ws(X)=Wvol(X). This corresponds to the lower bound of Ws under the condition that all potential ruptures have their centers inside the stimulated volume. In a general case of accounting for any potential rupture surfaces intersecting the stimulated volume (i.e., no any limitation for locations of rupture centers), the lower bound of Wsl(X) will be given by an arbitrary Wc(X) along with We1(X)=We2(X)=0:

display math(12)

For the upper bound Wsu(X) several alternatives can be considered. The first and simplest one is We1(X)=We2(X)=1, and thus,

display math(13)

This corresponds to the situation discussed in the previous section, where stimulation of an arbitrary small spot of a potential rupture surface is enough for a corresponding seismic event. This would mean an overrepresentation of the large-magnitude events in respect to the standard Gutenberg-Richter distribution. It can be seen in equation (10), where Wc(X) becomes especially small for large X.

[23] The next simple assumption would be We1(X)=1 and We2(X)=0. This assumption means that for triggering an event, the centrum of its potential rupture surface must be within the stimulated volume. Such a restriction is a reasonable formalization of the intuitive requirement that a “significant part” or a “nucleation spot” of the rupture surface must be within the stimulated volume (note also a topological equivalence between a disc centrum and any other “nucleation centrum” placed inside the rupture). It leads to the following estimate

display math(14)

Corresponding to equation (10), this would mean that the statistics of induced events should be given by Wf(X), i.e., given by a standard Gutenberg-Richter distribution.

[24] Other estimates involving more assumptions on We1 and We2 are possible. However, we will restrict our consideration to the three bounds (12)(14) as more natural ones. These three bounds represent three different scenarios of the development of induced seismicity. Their comparison with real data will clearly show which of the scenarios is more preferable for the induced seismogenesis.

[25] The relation between the different estimates of Ws is

display math(15)

Correspondingly with the bounds of the probability Ws, we obtain the bounds for the quantity Wg=Ws/Wc:

display math(16)

where

display math(17)
display math(18)

and, finally,

display math(19)

[26] In the following we estimate probabilities Wc(X) and Wvol(X).

4.1 Probability of a Given-Size Rupture Surface in a Stimulated Volume

[27] Shapiro et al. [2011] investigated the probability Wvol(X) that a disc of a diameter X is completely contained within a given stimulated volume under the condition that its center belongs to the volume. They found an exact expression of the function Wvol(X) for a spherical stimulated volume of the diameter L:

display math(20)

This function (the subscript “sp” indicates a spherical stimulated volume) is quickly decreasing with increasing inline image. Shapiro et al. [2011] have numerically computed Wvol(X) for ellipsoidal volumes with principal axes Lmin<Lint<Lmax. They found that

display math(21)

often provides a good estimate of a characteristic scale Y such that Wvol(Y) becomes nearly vanishing (<<0.1). If Lmin is sufficiently small, then it will provide a dominant contribution to γ. The function Wsp(X/γ) will give a good approximation if the axes of the stimulated ellipsoid are close to each other. Frequently (especially in the case of a hydraulic fracturing), one of the axes is extremely small: Lmin<<Lint<Lmax. In this case further corrections are required to approximate Wvol(X). In the Appendix, equation (A4) defines a function Wel(X) providing such an approximation (see also Figure 2).

Figure 2.

A comparison of numerically computed (crosses) and theoretically estimated (lines) probabilities Wvol as functions of disc diameters normalized to the minimum principal axes of the volume. (left) For an ellipsoidal geothermal-type stimulated volume. The dashed line represents the approximation Wsp(X/γ) (see the text below equation (21). The solid line shows the result of equation (A4). (right) The same, but for a hydraulic-fracture-like ellipsoid. The parameters of the ellipsoids and resulting size γ are given on the plots.

[28] Until now we have considered chaotically oriented potential rupture surfaces. Let us assume that such surfaces tend to be inclined by an angle ±φ(defined by the friction coefficient) to a plane of the maximum and intermediate tectonic stresses. Usually this angle is close to ±30°. Thus, the rupture surfaces have a larger angle to the minimum stress axis. To simplify the consideration, we assume that all the potential rupture surfaces have these inclinations. Further, we will approximate the stimulated volume by a cuboid with sides Lmin, Lint, and Lmax, rather than by an ellipsoid. Then, it is simple to show that the sought-after probability is given by

display math(22)

For example, if φ=0, all potential rupture surfaces will belong to the same plane. Such a geometry seems to be less relevant for seismicity induced by fluid stimulations of rocks. However, it is more adequate for aftershocks of earthquakes in subduction zones.

4.2 Probability of a Given-Size Rupture Surface to Have its Center Within a Stimulated Volume

[29] Let us firstly consider chaotically oriented rupture surfaces and a spherical stimulated volume of the diameter L. The sought-after probability Wc(X) (Figure 3) is given by the ratio of the total number of rupture surfaces with the centers within the stimulated volume to the total number of all rupture surfaces having any intersections with (or completely located within) this volume (see section A2):

display math(23)
Figure 3.

Probability Wc(a) of a rupture surface of the diameter a to have its center within a spherical stimulated volume of the diameter L: analytical and numerical results.

[30] We have numerically investigated this probability for ellipsoidal volumes with principal axes Lmin<Lint<Lmax. Numerical results show that substituting into equation (23) instead of L the following quantity

display math(24)

often provides a good estimate of Wc. Note again that if Lmin is sufficiently small, then it will provide a dominant contribution to γc.

[31] For rupture surfaces inclined by an angle ±φ to a plane of the maximum and intermediate tectonic stresses we will approximate the stimulated volume by a cuboid with sides Lmin, Lint, and Lmax rather than by an ellipsoid. Then, one can show that the sought-after probability is given by

display math(25)

Also here, if φ=0, then all potential rupture surfaces belong to the same plane. Equation (25) provides then probability Wc for a stimulated area of a rectangular form. Such a geometry may be relevant for aftershocks of tectonic earthquakes.

4.3 Magnitude-Length Scaling and Frequency-Magnitude Distributions

[32] A spatial scale of the rupture surface controls a magnitude of the corresponding earthquake. A relationship between a rupture size X and an earthquake magnitude M can be found using a standard relationship for seismic moment magnitudes [Shearer, 1999; Lay and Wallace, 1995; Kanamori and Brodsky, 2004]. A seismic moment is given by a product of the shear modulus, G, of a slip displacement, D, and of an involved rupture surface area, S. The moment magnitude is given by [Lay and Wallace, 1995; Shearer, 1999; Kanamori and Brodsky, 2004]

display math(26)

for seismic moments measured in Nm. In the last part of the equation we conventionally assume that the slip displacement D scales as a characteristic length X of the slipping surface (this is a result of the linear elastic theory of the fracture mechanics; see also equation 9.26 and Table 9.1 from Lay and Wallace[1995]). The quantity Δσ is usually defined as a static stress drop, and C is a geometric constant of the order of 1. We will use a shorter form of equation (26):

display math(27)

where we introduced a convenient notation Cσ for the cubic root of the reciprocal stress drop: Cσ=1084C1/3σ1/3≈103Δσ−1/3.

[33] In the following we are interested in the statistic of magnitudes. Thus, we have to consider M, X, and Cσ as random variables. Equation (27) defines the magnitude M as a function of two random variables, X and Cσ. It can be also written in the following form:

display math(28)

This equation defines the rupture length X as a function of two random variables, M and Cσ.

[34] There are two transformation equations relating the pair of random variables (M;Cσ) to another pair, (X;Cσ). The first relation, X(M,Cσ), is given by equation (28). The second relation is given by the trivial statement Cσ=Cσ. These two relations define a coordinate transformation from the system (M;Cσ) to the system (X;Cσ). The Jacobian of this transformation is equal to X(M,Cσ)/M. This Jacobian and the transformation equation (28) yield the PDF of magnitudes, fM:

display math(29)

where we accepted ln101/2≈1.151 and assumed that the random variables X and Cσ are statistically independent. Further, we introduced the following notations: fX(X) is a PDF of the rupture length, and fC(Cσ) is a PDF of Cσ. Thus, a probability WM of events with the magnitude larger than an arbitrary M is equal to

display math(30)

[35] Let us firstly assume that the following factorization is possible:

display math(31)

where f1 and f2 are two independent functions. This will be the case if fX(X) is a power-law function. Then

display math(32)

with the proportionality coefficient

display math(33)

Thus, under the factorizing assumption for fX, the randomness of the stress drop influences the distribution of magnitudes by modifying its proportionality factor (33) only.

[36] Let us assume further a power-law PDF of a size of potential rupture surfaces in an unlimited medium: fX(X)≈AXXq (here q>0 and AX is a proportionality constant). Note that such a PDF cannot be exactly valid because of an integration singularity at X=0. We assume that a power-law function is a good approximation of a real PDF of potential rupture surfaces above a certain very small size (which corresponds to a magnitude significantly smaller than M). Thus, a PDF fX(X) is strongly decreasing with X. Power-law size distributions are typical for natural fractal-like sets [Scholz, 1990; Shapiro and Fayzullin, 1992; Shapiro, 1992]. This type of self-similarity has been already related to the Gutenberg-Richter frequency-magnitude distribution of earthquakes [Shearer, 1999; Turcotte et al.2007; Kanamori and Brodsky, 2004].

[37] Indeed, a power-law PDF fX(X) with q=2b+1 along with equations (27) and (31) gives the following form of the functions f1 and f2:

display math(34)

and

display math(35)

Then equation (32) provides the Gutenberg-Richter law:

display math(36)

where

display math(37)

and

display math(38)

[38] Therefore, a power-law size distribution of rupture surfaces leads to the Gutenberg-Richter magnitude distribution in a rather general case of an arbitrary statistically distributed stress drop. This power-law distribution also takes into account a possibility that the same earthquake would be possible on any larger fault including exactly the same sufficiently perturbed rupture area. In the following we will include in our model a possibility of events of different size by assuming that the Gutenberg-Richter magnitude distribution is due to a power-law size distribution of all rupture surfaces spanned by earthquake events.

[39] In order to account for a finiteness of the stimulated volume, we must include into the consideration the influence of the geometry. Consequently, we must include the quantity Wg (see equation (10) and the consideration below this equation) as a factor into the PDF of the magnitude (30) and, therefore, under the integral in equation (30). Taking also into account equation (31) and results (34)(38), we obtain

display math(39)

Note that the quantity Wg is usually a function of a ratio of a potential-rupture scale X and a characteristic scale of the stimulated volume Y (e.g., Y=L, γ, Lmin, etc.; see equations (20)(25)). Thus, the explicit dependence of Wg on Cσ can be eliminated by introducing a characteristic magnitude MY so that inline image. Using this, the quantity Wg can be expressed (at least, approximately) as a function Wgm(MMY), which is directly obtained from Wg by corresponding substitution of the argument. For example, the most important for our discussion of the case studies is the lower bound of Wg(X). This bound is given by Wgl(X)=Wvol(X)≈Wsp(X/γ). In turn, inline image, where Y=γ. Therefore, substituting in equation (39) function Wg(Cσ10M/2) by the function Wgm(MMY), we can write

display math(40)

with the proportionality coefficient inline image.

[40] Generally, the magnitude MY is an unknown quantity effectively representing the range of induced magnitudes. It is defined by the condition of equivalence of equations (39) and (40). If inline image tends to a narrow δ-function like distribution around a representative value Cσ, then in accordance with equation (28), MY will be directly given by inline image. In reality Cσ is restricted to a limited range between approximately 1 and 1000. A fitting of equation (40) to a real frequency-magnitude distribution of an induced seismicity yields estimates not only of the b value but also of the magnitude MY. Using its relation (equation (28)) to the scale Y, one can estimate the representative value of Cσ and compute corresponding estimate of the stress drop. Note that in the case of a lower-bound probability Wgl, due to the vanishing probability Wvol for X>Y, the magnitude MY is a limiting value for a largest possible magnitude of an induced earthquake:

display math(41)

[41] Equation (40) can be also represented in the following form:

display math(42)

where

display math(43)

is a function correcting the magnitude distribution for the finiteness of the stimulated volume (and m is an integration variable). This function can be very roughly estimated in the following way. The exponential function under the integral is a quicker decreasing function than Wgm(m+MMY). Thus, by the integration we can very roughly assume that the last function is a constant equal to Wgm(MMY) and we obtain Δ(MMY)≈ log10Wgm(MMY). It shows clearly that if Mis significantly smaller than MY(so that MMY<<−1), the magnitude distribution will be indistinguishable from Gutenberg-Richter's classical one (because Wgm→1). By MMY the magnitude distribution will quickly drop down in the case Wgm→0 (this is the case for Wg=Wgl, see equation (17)).

[42] We use equation (42) to further modify equation (5):

display math(44)

[43] Figure 4 shows theoretical cumulative frequency-magnitude curves (i.e., the quantities log10NM as functions of M) for a given time elapsed since the injection start. In the Figure, the elapsed time has been involved implicitly only. It defines corresponding geometrical sizes (Lmin, Lint, Lmax) reached by the growing cloud of the seismicity. It defines also a value of the Gutenberg-Richter quantity a(t)= log10Qc(t)+Σ. For the particular examples shown in Figure 4 we assigned to these quantities such values that a=4.5 (e.g., Σ=−0.5 and Q=105 m3). We also assume b=1.5. Then we numerically computed different functions Δ(MMY) using equation (43). The functions Wgm(m+MMY) were obtained using the substitution (28) into the three functions Wg(X) given by equations (17)(19), respectively. To compute the lower bound of the quantity log10NM for the case of an ellipsoidal stimulated volume, the function Wg(X) was substituted by the approximating function Wel(X) defined by equation (A4). To compute the lower bound of the quantity log10NM for the case of a cuboidal stimulated volume, the function Wg(X) was substituted by the function Wcub(X) defined by equation (22). To compute the uppermost bound of log10NM for the case of a cuboidal stimulated volume, the function Wg(X) was substituted by the 1/Wcc(X), which is reciprocal to the one given by equation (25). Finally, to compute the uppermost bound of log10NM for the case of an ellipsoidal stimulated volume, the function Wg(X) was substituted by the quantity 1/Wc(X), which is reciprocal to the approximative function given by equation (23) along with the quantity γc from equation (24). Two situations are represented: geothermal- and hydraulic-fracturing types of stimulated volumes. Both parts of the figure contain the five following curves (from the lowest to the uppermost ones): a lower bound for an ellipsoidal stimulated volume, a lower bound for a cuboidal stimulated volume, a Gutenberg-Richter straight line, an upper bound for a cuboidal stimulated volume, and finally, an upper bound for an ellipsoidal stimulated volume.

Figure 4.

Theoretical frequency-magnitude curves: the lower bound for the case of an ellipsoidal stimulated volume, the lower bound for the case of a cuboidal stimulated volume, the Gutenberg-Richter distribution, the uppermost bound for the case of a cuboidal stimulated volume, and the uppermost bound for the case of an ellipsoidal stimulated volume. (left) A geothermal type of a stimulated volume. (right) A hydraulic-fracturing type of a stimulated volume.

[44] Note that the curves for ellipsoidal stimulated volumes are approximations only. In contrast, the curves for cuboidal volumes are exact. However, the equations for cuboidal stimulated volumes assume rupture surfaces inclined under an angle φto the plane of the maximum and intermediate axes. For Figure 4 we accepted φ=30°.

[45] To demonstrate the influence of the angle, we show such curves in Figure 5 for different values of φ. Finally, Figure 6 shows an example of how a sophisticated geometric form of the stimulated volume can influence the frequency-magnitude distribution (a lower bound). Here a situation corresponding to two intersecting ellipsoids has been numerically modeled.

Figure 5.

The same as Figure 4 but cuboidal stimulated volumes and different angles φ.

Figure 6.

Theoretical frequency-magnitude curves (lower bounds) for a stimulated volume in a form of two intersecting ellipsoids.

[46] A consideration of Figures 4-6 along with Figure 2 shows that if the seismicity statistics tends to the lower bound, then a fitting of the Gutenberg-Richter straight line will produce a systematically overestimated b value. The smaller the size of the stimulated volume, the stronger this effect will be. Especially important is the Lmin scale. Thus, the effect will be especially strong for the hydraulic-fracturing type of the geometry. This effect will be also strong for small time periods elapsed from an injection start. For small injection times, stimulated volumes are small. Thus, the effect will result in a decrease of b value estimates with injection times. This effect can be easily understood from equation (42). For the lower bound the quantity Δ(MMY) is negative. The smaller the size of the stimulated volume, the smaller the MY is, and therefore, the larger is the absolute value of Δ(MMY). By fitting the Gutenberg-Richter straight line, this quantity will directly contribute to the values of the parameters b and a. It will decrease a and increase b. This effect will act in an opposite direction if the event statistics follows the uppermost bound. It would increase a and decrease bvalues.

[47] In the following section we compare several observed frequency-magnitude distributions to the theoretical bounds. The differences between the theoretical curves for ellipsoidal and cuboidal volumes are not significant. Moreover, the angle φ is not known. In spite of the fact that φ=30° seems to be a reasonable approximation, in reality the angle can be broadly distributed. Thus, we attempted to fit real data by the theoretical approximations for ellipsoidal volumes. We assume that in practice the stimulated volume can be satisfactory represented by an approximate outer ellipsoidal envelope of the cloud of hypocenters of induced seismicity.

5 Discussion of Case Studies

[48] We have compared equation (44) to frequency-magnitude distributions in several case studies. We considered two geothermal locations in crystalline rocks, Basel [Häring et al., 2008] and Soultz 1993 [Baria et al., 1999]. Further, we have included a Paradox Valley data set obtained by an injection of saline water into deep carbonate rocks [Ake et al., 2005]. Finally, we have also included three hydrocarbon locations: a hydraulic-fracturing stage (A) in gas shales (Canada), a hydraulic fracturing stage (B) of a tight-gas reservoir at the Cotton Valley [Rutledge and Phillips, 2003], and a stage of an atypical hydraulic fracturing from the Barnett Shale [Maxwell et al., 2009].

[49] Corresponding fitting results are shown in Figures 7-12. For our analysis we take microseismic clouds at the injection-termination time t0 (later in this section we will discuss a possibility of using for the analysis intermediate elapsed times). In the first step we attempted to fit the real data by a standard Gutenberg-Richter cumulative distribution. For this, several techniques can be applied. We started with finding a longest linear segment providing an approximation minimizing deviation area per unit length in the (log10NM,M) space. On this way we improve estimates of the completeness magnitude Mc and obtain initial approximations of the a and b values of the Gutenberg-Richter distribution. Further, for a comparison, we apply the maximum likelihood technique originally proposed by K. Aki in 1965 [see Utsu, 2002]. Both methods provide very close results. However, in some situations (e.g., the Paradox Valley case study) the last technique provides visibly worse results. A clear reason for this is the fact that the real magnitude statistics do not satisfy the Gutenberg-Richter distribution, for which the approach of Aki is designed. Thus, we show in all the corresponding figures the results of the first technique only. We call then the resulting straight line and its parameter as apparent parameters of the Gutenberg-Richter distribution.

Figure 7.

Fitting the frequency-magnitude distribution of the seismicity induced by the Soultz 1993 injection. Results of a fitting of the Gutenberg-Richter distribution (dashed line) and of the lower bound Wgl (solid line) are shown on the plot below the acronyms GR and LB, respectively, and discussed in the text. The dotted line shows the Gutenberg-Richter distribution with parameters a and b estimated from the lower-bound curve. Mc, MY, and Δσ are estimated values of the completeness magnitude, the maximum induced magnitude, and the stress drop, respectively.

Figure 8.

Fitting the frequency-magnitude distribution of the seismicity induced by a hydraulic fracturing stage (A) in a gas-shale deposit in Canada. Notations are explained in Figure 7.

Figure 9.

Fitting the frequency-magnitude distribution of the seismicity induced by hydraulic fracturing at one location in Barnett Shale. Notations are explained in Figure 7.

Figure 10.

Fitting the frequency-magnitude distribution of the seismicity induced by the Paradox Valley injection. Notations are explained in Figure 7.

Figure 11.

Fitting the frequency-magnitude distribution of the seismicity induced by the Cotton Valley Stage B injection. Notations are explained in Figure 7.

Figure 12.

Fitting the frequency-magnitude distribution of the seismicity induced by the Basel injection. Notations are explained in Figure 7.

[50] Then we try to fit the data by the theoretical curve of the lower bound. For this we try to find a curve simultaneously satisfying least-square deviations and restricting the (log10NM) from below. Using the a and b values from this fit, we attempt to reconstruct “real” Gutenberg-Richter cumulative distribution. All the six data sets allowed to reestimate the Gutenberg-Richter quantities a and b. The reestimated b values are systematically lower than the parameters obtained by the apparent Gutenberg-Richter fit. We observe that the real frequency-magnitude distributions are usually well restricted between the lower bound and the fitting straight line corresponding to the apparent Gutenberg-Richter distribution. Note that this line is located lower than the bound Wgu0 (see equation (18)) corresponding to the reconstructed Gutenberg-Richter distribution. Moreover, nearly all data sets show a tendency of the seismicity to be better represented just by the lower bound.

[51] The fitting of the lower bound yields also estimates of maximum expected induced magnitudes and of the corresponding stress drop. The stress drop is computed from equation (41) by substituting the estimates of characteristic length γ and of the maximum induced magnitude MY. In reality the values of stress drop of induced events can be distributed in a broad range. For example, Goertz-Allmann et al. [2011] estimated stress drop of selected 1000 events from the above mentioned Basel injection experiment. Their Figure 2 shows values in the range of 0.1–102 MPa. Jost et al. [1998] estimated stress drops of events induced by the 1994 KTB fluid injection experiment. Their Table 1 gives values distributed in the range 5×10−3 to 6MPa. On the other hand, our lower-bound-based stress-drop estimate represents an average value in the sense of equation (33). It depends on a real distribution of the stress drop and will be dominated by the most probable stress-drop values. This can enhance a contribution of numerous small-magnitude events (which can have small stress drop and which are frequently not analyzed because of low signal-noise relations). It can lead to even smaller estimates of stress drops than those measured at the KTB site. This tendency can be further enhanced by the fact that the quantity γ is usually overestimated due to event location errors. Still, because of its independence of any rupture model and of any estimates of the corner frequency, our approach to estimate the stress drop can yield a reasonable additional constraint of this insufficiently well understood quantity.

[52] We start with the data sets permitting rather simple interpretation. They are shown in Figures 79. We consider fitting the frequency-magnitude distribution of the seismicity induced by the Soultz 1993 injection (Figure 7). The axes of the ellipsoid are 440m, 1400m, and 1740m. The effective sphere scale γwas approximately 630m. A conventional Gutenberg-Richter fitting yields a=2.5,b=1.4. The completeness magnitude is approximately Mc=−0.7. The bound Wgl yields a=2.9,b=1.1. The estimated maximum induced magnitude is MY≈1.3. Substituting this estimate of the magnitude MY along with the value of γand the assumption that C=1 into equation (41) yields an estimate of the characteristic stress drop Δσ≈400 Pa. The complete data set is well described by the lower-bound approximation. Thus, indeed the inducing of events seems to require pore-pressure perturbation involving a nearly total rupture plain.

[53] A similar tendency can be seen in a data set from hydraulic fracturing of gas shales in Canada (Figure 8). The axes of the ellipsoid are 10m, 100m, and 650m. The effective sphere scale γwas approximately 15m. A conventional Gutenberg-Richter fitting yields a=0.9,b=1.3. The bound Wgl yields a=1.9, b=0.9, MY=0.15, and Δσ≈0.6 MPa.

[54] Fitting the frequency-magnitude distribution of the seismicity induced by hydraulic fracturing at one location in Barnett Shale (Figure 9) seems to be also similar. The axes of the ellipsoid are 70m, 340m, and 1000m. The effective sphere scale γwas approximately 100m. A conventional Gutenberg-Richter fitting yields a=−5.0,b=2.7. The bound Wgl yields a=−0.5, b=1.3, MY=−1.8, and Δσ=1 Pa. This case study is distinguished by an especially low estimate of the stress drop. In addition to the reasons already mentioned above, one more reason for this can be the following. The minimum principal size of the stimulated volume in this particular case study represents the total thickness of the shale reservoir. In reality the induced seismicity is concentrated in several layers of the thickness of 10m each [see Shapiro and Dinske, 2009a, Figure 3]. Thus, effectively the quantity γ is possibly 10 times less than the one used for the estimate of the stress drop. This would yield the stress drop of the order of 2500Pa.

[55] In all the three examples above the lower bound seems to be well suited. We explain the fact that the lower-bound curve somewhat underestimates the number of events in the intermediate- to high-magnitude range by a too rough analytical approximation of the real rupture statistics. Also, an influence of the geometry which is more complex than just an ellipsoid (see Figure 6) or a rather restricted angular spectrum of the rupture orientations (see Figure 5) could also contribute to this effect.

[56] Somewhat more sophisticated interpretation seems to be required by the data sets shown in Figures 1012. Fitting the frequency-magnitude distribution of the seismicity induced by the Paradox Valley injection (Figure 10) yields the following Gutenberg-Richter parameters: a=3.6, b=0.83. The axes of the ellipsoid are 3000m, 4000m, and 7000m. The effective sphere scale γ was approximately 3800m. Fitting of equation (44) corresponding to the lower bound provides nearly unchanged Gutenberg-Richter parameters: a=3.6,b=0.75. In addition, we estimate a characteristic stress drop, Δσ=0.01 MPa and the maximum magnitude defining the lower bound: MY=3.8.

[57] Again, we observe that a dominant majority of events follows well the lower-bound approximation. However, two data points corresponding to high-magnitude events return backward to the classical Gutenberg-Richter distribution. Thus, apparently, the corresponding three large-magnitude events were triggered by just a pore-pressure-related perturbation of nucleation spots on their rupture surfaces.

[58] A very similar situation can be observed on the data set corresponding to a hydraulic fracturing of the tight sand gas deposit of the Cotton Valley, Stage B (Figure 11). The axes of the ellipsoid approximating the stimulated volume are 10m, 40m, and 480m. The effective sphere scale γ is approximately 15m. Fitting the Gutenberg-Richter frequency-magnitude distribution yields a=−0.9, b=1.9 Fitting equation (44) corresponding to the lower bound provides the following update of the Gutenberg-Richter parameters: a=1.1,b=1.2. In addition, we estimate a characteristic stress drop, Δσ=0.016 MPa and the maximum magnitude defining the lower bound: MY=−0.9. It seems that the rupture surfaces of three large events were not completely included into the stimulated volume. For their triggering, an excitation of rather large nucleation domains was sufficient (we conclude this from the fact that they are still below the Gutenberg-Richter distribution, indicating that their rupture surface is possibly larger than just a nucleation spot expected by this distribution).

[59] One more example of a similar situation is given by the Basel data set (Figure 12). A conventional Gutenberg-Richter fitting yields a=4.3,b=1.4. For the fitting of the lower-bound equation (44), we took the axes of the stimulated ellipsoid being equal to 100m, 760m, and 920m. The effective sphere scale γ is approximately 150m. The bound Wgl yields close results: a=4.3,b=1.3. Additionally, Δσ=12.5 MPa, and the maximum magnitude defining the lower bound MY=3.05. It seems that for the triggering of a majority of events in Basel, stimulation of their nucleation spots was sufficient.

[60] A comparison of these two groups of case studies indicates a possibility to distinguish between triggered and induced events. McGarr et al. [2002] defined induced and triggered seismicity in respect to the stress impact of a stimulation. If this impact is of the order of the ambient shear stress, they speak about induced seismicity. If this impact is significantly smaller than the ambient shear stress, they speak about triggered seismicity. In this terminology nearly the total fluid-induced seismicity from our case studies could be considered as triggered one. We use here a definition which is related to the one of McGarr et al. [2002]. However, our definition is more specific in respect to the geometry of the stimulated volume. Our definition is closer to the one used by Dahm et al. [2013]. We define as “induced” ones events resulting from perturbing their nearly complete rupture surfaces. Then their statistic should follow the lower bound. We define as “triggered” ones events resulting from perturbing nucleation spots of their rupture surfaces only. Their statistic should follow the reconstructed Gutenberg-Richter distribution. It seems that some geothermal-reservoir case studies include triggered events (e.g., the Basel case study). This is seldom but also possible for hydraulic fracturing of hydrocarbon reservoirs. Note also that for an event triggering, a perturbation of an arbitrary element of its rupture surface is not sufficient. This would correspond to the uppermost bound (19). This bound strictly contradicts with our observations. The data show that triggering requires a perturbation of a significant part of the rupture surface necessarily including the nucleation domain (which we model by the rupture center). It seems also that the induced events are much more common than triggered ones. Summarizing these observations, we propose to distinguish between induced and triggered events in the following way. Usually the absolute majority of the events is induced, and their statistic follows the lower bound. We identify as triggered ones those individual events whose high magnitudes deviate clearly from this statistic. For example, Figure 10 indicates that all events with magnitude larger than 3 are triggered. All other events are induced. On the other hand, the size of a stimulated part of the rupture surface of a triggered event could be possibly interpreted as a nucleation size necessary to activate fault weakening mechanisms necessary for a rupture propagation outside of the stimulated volume (see the recent study of Garagash and Germanovich [2012]). This could be a subject of interesting future investigations.

[61] As already commented, we used for our analysis all microseismic events that occurred until the final time moment of the fluid injection (termination time t0). The main reason for this is that such data sets have larger number of events than catalogs collected till any intermediate elapsed times. One more argument in favor of this time moment is the fact that the completeness magnitude Mc is a function of the time elapsed after the injection. One of reasons for this is changing position of the seismicity cloud in respect to the observation system. Time moment t0 provides usually the best statistic in respect to Mc. On the other hand, equation (44) shows that the frequency-magnitude statistic of induced earthquakes is time dependent. This is so not only due to the term containing Qc(t). Another time-dependent factor is the characteristic magnitude MY. This magnitude is a function of a geometrical scale of the stimulated volume. This scale is in turn a function of time (or, equivalently, of the injected volume). This means, for example, that both the quantities a and b of a conventionally fitted Gutenberg-Richter frequency-magnitude distribution will be time dependent. We should expect a systematic increase of the a values and decrease of the b values with increasing time elapsed since the injection start. A careful analysis of such intermediate-time data will be required to account for the issues with the Mc mentioned above. This is outside of the frame of this paper and will be a subject of another publication. However, statistics of large-magnitude events at intermediate-time moments are more robust. For example, such data can be included into a consideration of the maximum observed magnitudes as a function of the scale of stimulated volumes.

[62] Indeed, we have reported already on an important role of Lmin for the maximum magnitudes [Shapiro et al., 2011]. It follows also from equations (20)-(25) defining the bounds of Wg. Figure 13 shows the values of Mmax as a function of log10(Lmin)2for all data sets we have at our disposal and several case studies which detailed descriptions we found in the literature. The error bars show possible impact of errors in magnitudes and event locations. For magnitudes we assumed the error of the order of 0.5 (this roughly corresponds to possible differences between local and moment magnitudes taken from different literature sources; see, e.g., Grünthal and Wahlström [2003]). For the principal axes we assumed the error bars of the order of seismicity location errors: 10m for hydraulic fracturing sites Barnet Shale [Maxwell et al., 2009] and Cotton Valley [Rutledge and Phillips, 2003]; 50m for geothermic sites Basel [Häring et al., 2008], Soultz, Cooper Basin [Baisch et al., 2009], Fenton Hill [Phillips et al., 1997], and Berlin [Bommer et al., 2006]; and 100m for the Paradox Valley [Ake et al., 2005]. The red star corresponds to the largest event of the Basel injection. The blue crosses correspond to data we have evaluated ourselves. In addition (red crosses), we included information that we evaluated using literature sources [Baisch et al., 2009; Phillips et al., 1997; Bommer et al., 2006]. In addition to all these final-moment (i.e., t0) related data, we included data indicated by the black crosses. They show maximum induced magnitudes for the time moments t2/3 defined, so that the cumulative injected volume Qc(t2/3) numerically satisfies the following condition: log10Qc(t2/3)=(2/3) log10Qc(t0) (all volumes are measured in m3). We see that the intermediate-time data support well the general trend.

Figure 13.

Largest observed magnitudes of induced earthquakes as functions of the minimum principal axes of corresponding stimulated rock volumes for different case studies. The error bars show possible impact of errors in magnitudes and event locations. The red star corresponds to the largest event of the Basel injection. The blue crosses correspond to the data we have at our disposal and evaluated ourselves. The red crosses correspond to case studies that we were able to evaluate using literature sources. The black crosses correspond to seismicity clouds at time moments t2/3, which are defined so that the cumulative injected volume Qc(t2/3) at these time moments numerically satisfies the following condition: log10Qc(t2/3)=(2/3) log10Qc(t0), where t0 is the termination time of the stimulation and all volumes are measured in m3. Maximum magnitudes, minimum principal axes (at t0), and injection sites are given in Table 1 of Shapiro et al. [2011].

[63] We observe a good agreement of the data points with equation (41) for Y=Lmin (the dotted line in Figure 13). The corresponding values of Δσ are of the order of 0.0001–10MPa. Substituting a highest probable limit of stress drops of the order of 10MPa into equation (41), we obtain a rough estimate of the maximum probable magnitude limit (it would correspond to the upper envelope in Figure 13) of an induced earthquake for a given location:

display math(45)

[64] This result can be useful for estimating and constraining induced seismic hazard. For example, to restrict the hazard, one could attempt to keep the minimal principal axis of the stimulated volume restricted by terminating the injection if this size achieves a planned critical value. However, because this result addresses mainly induced events, its application requires a careful analysis of the seismotectonic and geologic situation in each particular case. By such an analysis a probability of triggered events should be carefully constrained. Factors like scales of faults intersecting the stimulated volume, tendencies of the faults to influence the shape and scales of the stimulated volume, and seismotectonic parameters (e.g., the seismogenic index, b values, stress states of significant faults, and modifications of these parameters during the stimulation) are of importance for this task.

6 Conclusions

[65] In this paper we analyze influence of the finiteness of rupture surfaces and of stimulated volumes on the statistics of magnitudes of induced earthquakes. A power-law size distribution of potential rupture surfaces leads to the Gutenberg-Richter magnitude distribution in a rather general case of an arbitrary statistically distributed stress drop.

[66] We consider both rupture surfaces located completely within or intersecting only the stimulated volume. We propose lower and upper bounds of frequency-magnitude distributions of induced events, taking different types of geometric relations between a stimulated volume and a potential rupture surface into account.

[67] Observations show that by borehole fluid injections at geothermal and hydrocarbon reservoirs, the frequency-magnitude statistics of induced events tends to be in agreement with the lower bound of the event probability. This indicates that a rupture of a fluid-induced earthquake seems to be mainly probable along a potential rupture surface located nearly completely inside a stimulated rock volume.

[68] Fitting the lower bound to the magnitude distributions can provide an estimate of a largest expected induced magnitude and a characteristic stress drop, in addition to improved estimates of the Gutenberg-Richter a and b parameters. Because the statistics of induced seismicity tends to the lower bound, a direct fitting of the Gutenberg-Richter straight line to frequency-magnitude data will produce a systematically overestimated b value. The smaller the size of the stimulated volume, the stronger this effect will be. An overestimating of the b value will be especially strong for the hydraulic-fracturing type of the geometry of stimulated volumes. This effect will be also strong for small time periods elapsed from the injection beginning.

[69] The main geometric factor limiting the probability to induce a large-magnitude event is the minimum principal axis of the stimulated rock volume. The controlling role of this geometric scale is supported by the observations.

[70] Our results indicate a possibility to separate between triggered and induced events. We identify the induced events as those for which perturbing of their nearly complete rupture surface is necessary. Their statistics follows the lower bound. Triggered ones are events which occur due to perturbing nucleation spots of their rupture surfaces only. These are individual large-magnitude events clearly deviating from the lower-bound statistic. Their statistics is given by the reconstructed Gutenberg-Richter distribution. It seems that geothermal reservoirs include some triggered events. This is seldom but also possible for hydraulic fracturing of hydrocarbon reservoirs. Note also that for an event triggering, a perturbation of an arbitrary element of its rupture surface is not sufficient. A triggering requires a perturbation of a significant part of the rupture surface necessarily including the nucleation domain. It seems also that the induced events are much more common than triggered ones.

[71] Our approach is not restricted to fluid-induced seismicity. We hypothesize that it is applicable for any type of seismicity induced in a restricted rock volume, e.g., aftershock series of tectonic events.

Appendix A: Probability of an Arbitrary-Oriented Disc Belonging to or Intersecting With a Stimulated Volume

A1 Probability of an Arbitrary-Oriented Disc Belonging to an Ellipsoidal Volume

[72] In contrast to the exact results (20) and (22)), here we propose an approximation of the sought-after probability Wel(X) (here X is the diameter of the rupture). We consider an ellipsoid with the principal axes Lmin<Lint<Lmax. If the axes are close to each other, then a good approximation of the Wel(X) will be given by Wsp(X/γ) where γis given by equation (21). Let us consider another quite realistic geometry of stimulated volumes: Lmin<<Lint<Lmax.

[73] We consider further a rupture in a form of a plane disc of diameter X of an arbitrary orientation with a center at a point P inside of such an ellipsoidal stimulated volume. We will concentrate firstly on large discs with X>Lmin. The centers of large discs completely belonging to the ellipsoid are approximately located inside the following ellipsoidal volume:

display math(A1)

[74] We consider a sphere S1 defined by normals of the length X/2 (the sphere's radius) at the point P for all possible orientations of the disc. If X>Lmin (i.e., large discs), then such a sphere will always intersect the surface of the ellipsoid. We estimate approximately a part of the surface of this sphere (in relation to the sphere's complete surface), where a normal can have its end point under the condition that a corresponding disc still belongs to the stimulated volume. For this we consider such a sphere intersecting with “side surfaces” of the volume. The side surfaces are two ellipsoid's surface halves spanned on the axes Lint and Lmax. Further, we approximate these surfaces just by planes (we call them side planes). Let us then consider a sphere S1 of radius X/2 with the center at P at a minimum distance y from a side plane. This sphere intersects this side plane along a circle. In order to belong to the volume, a disc must have a normal located inside of a cone with the symmetry axes coinciding with the normal from P to the side plane. The sphere S1 and this cone define a spherical segment of the height inline image and the surface πXh. A probability of a disc to have an orientation necessary for belonging to the stimulated volume is equal to the ratio of this surface to the surface of the half of the sphere S1, i.e., inline image. Note that the effect of a possible intersecting of the sphere S1 with the second side plane is taken automatically into account. Indeed, the largest intersection is of importance only because of its symmetric effect on permitted orientations of the discs. The probability of large discs inside of the stimulated volume can be estimated then by the following integral over y:

display math(A2)

The integration yields

display math(A3)

From the derivation it is clear that with increasing XLint the estimate Wlarge(X) will adequately decrease to zero. However, for small X the function Wlarge(X) becomes inadequate. In the point X=Lmin it must be reasonably combined with the function Wsp(X/γ). Thus, we propose the following approximation of Wel(X):

display math(A4)

A2 Probability of a Disc Intersecting a Sphere With the Center Inside the Sphere

[75] We consider a spherical stimulated volume of a radius R=d/2 with a center at a point O. To calculate a number Ns of discs of radius r=X/2 intersecting this volume, we must locate their centers (possible points P) inside or outside of the stimulated volume. The number Ns can be divided into two different parts. These are rupture surfaces with centers P belonging to the stimulated volume. Their number is given by 4πNR3/3, where N is a bulk concentration of potential rupture surfaces.

[76] The second part is discs with centers outside of the stimulated volume. Let us consider a sphere S1 of radius r with the center at point P. An orientation of the disc is given by the orientation of the normal to its plane at point P. A necessary condition for the disc to intersect (or to touch) the stimulated volume is that this sphere S1 completely includes the stimulated volume or intersects the surface of the stimulated volume along a circle (or just touch the volume). We introduce a variable s=|O1P|=|OP|−R(see Figure A1) and consider further a plane including the straight line OP. In this plane there exist two radii of S1 which (or continuations of which) are tangential to the spherical volume. We denote one of the touching points as A (see Figure A1a). There are two such points located symmetrically to the line OP. From the triangle OAP we have inline image. We consider then two possible situations: (1) τr (shown in Figure A1a) and (2) τ>r(shown in Figure A1b; note that here A is just an intersection point of S1 and the stimulated volume). In case (1) the limiting orientations of a disc intersecting the volume are given by the tangential positions of the radii of S1 (Figure A1a). Note that this yields the following restrictions for the variable s: inline image. A disc will intersect the stimulated volume if a disc's normal is located within the spherical sector defined by the rotation of the plane section A1PA2 around the line PA4. Therefore, such a normal can intersect the surface of the right-hand (upper) half sphere of S1 everywhere excluding the surface of the spherical segment covered by the rotation of the semi-arc A2A4 around the point A4. Taking into account that sinOPA= sinA1PA2= sinPA2A3=R/(R+s), we obtain |A3A4|=rs/(R+s). For the segment surface and the sector surface, this yields 2πr2s/(R+s) and 2πr2R/(R+s), respectively. The probability of a disc to have an intersection with the volume is given by a relation of the spherical sector surface to the surface of the half sphere. This is equal to R/(R+S). The number of rupture surfaces intersecting the stimulated volume is given then by the following integral:

display math(A5)
Figure A1.

Geometrical sketches for computing probability of an arbitrary-oriented disc intersecting with a spherical volume to have its center inside the volume.

A3 A List of Main Notations

Table 1. 
SymbolDefinition
f(ζ)probability density function of the variable ζ
Wevprobability of a point-like event
Nbulk concentration of preexisting defects
Nevtotal number of events
NM

number of earthquakes with a magnitude

larger M

WMprobability of an earthquake with a magnitude larger M
Cpcritical pore pressure
Qccumulative injected fluid volume
Sp

poroelastic compliancy/uniaxial storage

coefficient

Mmoment magnitude
Σseismogenic index
a

a parameter of the Gutenberg-Richter

distribution

b

a parameter of the Gutenberg-Richter

distribution

MΣspecific magnitude
Vstimulated volume
Xdiameter of a circular rupture surface
Ldiameter of a spherical stimulated volume
Lminminimum principal axis of an ellipsoidal/shortest dimension of a cuboidal stimulated volume
Lintintermediate principal axis of an ellipsoidal/intermediate dimension of a cuboidal volume
Lmaxmaximum principal axis of an ellipsoidal/+longest dimension of a cuboidal volume
WE(X)probability of an induced event to have rupture diameter X
Wf(X)probability of a potential rupture surface in unbounded medium to have a diameter X
Wc(X)probability of a rupture surface to have its center within a stimulated volume
Ws(X)probability of a rupture surface to be sufficiently stimulated to produce an event
Wg(X)the part of function WE(X) that is influenced by the geometry of the simulated volume: Wg=Ws/Wc
Wvol(X)probability of a rupture surface to be completely inside the stimulated volume
We1(X)probability of a seismic event along the entire potential rupture surface under the condition that
 the center of the rupture surface is inside the stimulated volume
We2(X)probability of a seismic event along the entire potential rupture surface under the condition that
 the rupture surface is intersecting stimulated volume but its center is outside
Wsl(X)lower bound of Ws(X)
Wsu(X)upper bound of Ws(X), arbitrary intersection of rupture surface and stimulated volume can
 trigger an event
Wsu0(X)upper bound of Ws(X), center of rupture surface has to be inside the stimulated volume to
 trigger an event
Wgl(X)lower bound of Wg(X)
Wgu(X)upper bound of Wg(X), arbitrary intersection of rupture surface and stimulated volume can
 trigger an event
Wgu0(X)upper bound of Wg, center of rupture surface has to be inside stimulated volume V to
 trigger an event
WspWvolfor a sphere
Welapproximation of Wvol for an ellipsoid
WcubWvol for a cuboid
WccWc for a cuboid
γdiameter of a spherical volume that approximates Wvol of an ellipsoidal volume
γcdiameter of a spherical volume that approximates Wc of an ellipsoidal volume
Gshear modulus
Dslip displacement
Srupture surface area
Δσstatic stress drop
Cgeometric constant
Cσa quantity proportional to the cubic root of the reciprocal stress drop
Ycharacteristic scale of the stimulated volume

[77] The additional number of the rupture surfaces intersecting the volume is given by case (2) τ>r. This situation corresponds to Figure A1b. Correspondingly, inline image. Note that PA is not anymore tangent to the stimulated volume. The height of the excluded spherical segment (defined by the rotation of the semi-arc A2A4) is given by r−|PA3|. The length h1=|PA3|=|AO2| can be found from the triangle OAP. The probability of a disc to have an intersection with the volume is given by inline image. The number of rupture surfaces intersecting the stimulated volume is given then by the following integral:

display math(A6)

[78] Finally, the probability Wc(X) is given by the following ratio taking into account all the contributions discussed above:

display math(A7)

Acknowledgments

[79] This work was funded in part by the PHASE project of the Freie Universität Berlin and in part by the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety in the frame of the project MAGS. The data from Cooper Basin are courtesy of H. Kaieda (CRIEPI). The data from Basel are courtesy of U. Schanz and M. Häring (Geothermal Explorers). The data from the Paradox Valley are courtesy of K. Mahrer (formerly, Bureaux of Reclamation). The data from Cotton Valley are courtesy of J. Rutledge (formerly, LANL). The data from Barnett Shale are courtesy of S. Maxwell (formerly, Pinnacle). The data from Soultz are courtesy of A. Gerard and R. Baria (formerly SOCOMINE) and of EEIG Heat Mining. A. Jupe (EGS Energy) also provided great assistance with accessing these data. The data from Canada are courtesy of T. Urbancic, A. Baig, and A Wuestefeld (ESG). We appreciate also corrections and suggestions of the Associate Editor and of two anonymous reviewers that improved the manuscript.