PROCEEDINGS, 45th Workshop on Geothermal Reservoir Engineering

Stanford University, Stanford, California, February 10-12, 2020

SGP-TR-216

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A Physical Basis for Gutenberg-Richter Fractal Scaling

P Leary & P Malin Advanced Seismic Instruments & Research, Dallas, USA

pcl@asirseismic.com; pem@asirseismic.com

Tero Saarno st1 Deep Heat, Helsinki, Finland

Tero.Saarno@st1.fi

Abstract

The Gutenberg-Richter (G-R) empirical law is a fractal scaling relation between earthquake frequency and earthquake size, but as such

provides no physical basis for the empiric. A physical basis for the G-R relation emerges from ambient crustal earthquake catalogues if

we note that (i) ambient crustal earthquake moments M are lognormally distributed, and (ii) the distribution of earthquake moments

corresponds closely to the lognormal distribution of ambient crust poroperm relation κ ∝ exp(αφ), where crustal permeability κ(x,y,z) is

a function of normally distributed porosity spatial distribution φ(x,y,z) about mean value φ, and parameter α obeys the empirical

condition αφ ~ 3-4 widely attested by crustal well-core. Computationally, if H (M) is a valid histogram representation of earthquake

catalogue moment M distribution, then H (exp(αφ)) provides a good least-squares fit to H (M) for parameter α fixed by the empirical

condition αφ ~ 3-4 for a normal distribution of random numbers φ. As the random number distribution φ can be taken as valid

representation of spatially-correlated porosity spatial distribution in a crustal volume, the G-R relation can be attributed a generic physical condition that ambient crustal earthquake moments are statistically congruent with crustal permeability.

Useful aspects of this physical interpretation of the G-R relation are:

The ambient crust inherently contains physically-valid distributions of low seismic moments that hitherto have gone

unrecognised in discussions of the G-R relation.

If the observed number of low seismic moments in an event catalog is fewer than predicted by the above-noted lognormal

distribution, the deficit provides a valid estimate of the observational ‘’incompleteness’’ of the event catalog.

Observational seismicity catalogs with an excess number of high moment events can be logically interpreted as indicating

events that occur on active tectonic faults within the ambient crustal volume surveyed by the catalog.

The correspondence of the actual lognormal G-R distribution (i.e., not its fractal interpretation) to the distribution of ambient

crust permeability structures implies that ambient crustal seismicity tends to occur on pre-existing poroperm-connectivity

structures generated by ambient rock-fluid interactions over long-range/long-duration scales of ambient crustal tectonics.

The multi-decadal fractal presentation of the G-R relation is less a physical statement than an artefact of suturing earthquake

catalogs in a log-log format across an artificially large range of seismic moments; what physical basis exists for the multi-

decadal G-R relation is an expression of scale-independent crustal fluid-rock interactions attested by well-log power-law

spectral scaling of ambient crust physical property spatial fluctuations.

Lognormal distributions of seismic moments offer a simple reason ‘’why earthquakes stop?’’: seismic slip tends to occur on

pre-existing populations of finite-length poroperm structures when and where fluid pressure imbalances occur within an

ambient poroperm-connectivity system.

While microseismicity spatial distributions reflect pre-existing poroperm-connectivity structures, they do not necessarily

reflect long-reach flow-connectivity pathways by which crustal fluids migrate through km-scale crustal volumes.

Long-reach flow-connectivity pathways by which crustal fluids actively percolate through km-scale crustal volumes

when/where stimulated by wellbore fluid injection can be detected by multichannel surface seismic sensor arrays.

In convective geothermal flow systems, crustal zones of high fluid flow clustering can logically be mapped by means of multichannel surface seismic sensor arrays, allowing production-well drilling to be much more efficient than at present.

Keywords: Gutenberg-Richter relation, microseismicity, fractals, lognormality

1. INTRODUCTION

As interesting as the long-standing Gutenberg-Richter (G-R) fractal scaling property of crustal seismicity is, it has failed to be

established in terms of plausible rock-physical processes. While physical origin of fractal scaling can be elusive – fractals describe

population numbers, e.g., the G-R relation N(m) ~ 10-b x m, rather than spatial organisation of physical elements within the system -- the

fact remains that crustal seismicity is a physical process constrained by internal energy conservation and application of the entropy

principle to internal spatial order. We consider here evidence that ambient physical processes in the crust generate lognormal

permeability distributions that closely fit crustal microseismicity magnitude distributions across the complete range of observed

magnitudes. This evidence indicates that a physics-based lognormal G-R relation arises from the spatial correlation empirics of crustal rock-fluid interactions.

Ambient crustal rock-fluid interactions link normally-distributed porosity spatial fluctuations φ(x,y,z) that are spatially correlated on all

scales to lognormally-distributed permeability spatial fluctuations, κ(x,y,z) ~ exp(αφ(x,y,z)) [1]. The value of the empirical parameter α

within a crustal volume is fixed by the condition αφ ~ 3-4, where φ is the volume mean porosity [2]. The αφ ~ 3-4 condition, observed

across porosity range .001 < φ < 0.3 spanning tight basement rock to reservoir rock, forces crustal permeability histograms H (κ ~

exp(αφ)) to be lognormal [3]. Field-scale crustal well-flow production statistics are observed to be lognormal for all typical flow

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systems: conventional and unconventional hydrocarbon reservoirs, groundwater aquifers, geothermal convection cells, and fossil flow

system mineral deposits [4]. We will see below that histograms of ambient crustal microseismicity magnitude m distributions, H(m), accord well with flow-related lognormal distributions, H(κ ~ exp(αφ)).

The original G-R relation identified a size-number relation, N(m) ~ 10-b x m expressed in the format of seismic event magnitudes m given

by the logarithm of the seismic wave amplitude that a given seismograph would register at a given distance from the event [5]. The

original magnitude scale was devised for California regional seismic events observed at distances up to 600km. The initial

magnitude/offset ranges were quickly expanded to include global-scale earthquake recordings spanning magnitudes across five

characteristic intervals, 7.8-8.5 7-7.7 6-7 5.3-6 <5.3 [6]. Shortly thereafter the magnitude distribution of California regional

earthquakes appeared in familiar form [7]:

log N ~ -2 + 0.88 (8 - m)

m 6.5 6 5.5 5 4.5 4 3.5

N 2 5 13 32 108 311 578+

Earthquakes in early G-R studies involved strain-release events on large active faults for a number of tectonic zones across the globe.

The hazard and cost associated with these faults and their occasional ruptures mandated a scientific quest to understand and, if possible,

anticipate large earthquake occurrence. The drive to achieve the quest created an anomalous situation in which interpretation of

earthquake catalogues became biased by the G-R power-law scaling phenomenology. If catalog numbers of small earthquake

magnitudes failed to obey the G-R relation established by large magnitude events, it was regarded as a problem with observing the small

events rather than absence of small events due to natural physical processes.

Figure 1 (left) reproduces an early instance of this interpretational bias. In the 1960s, an 60km distant seismic network recorded some

1000 crustal earthquakes induced by water flooding of a 20km-scale Colorado oil field [8-9]. In Figure 1 (left), solid dots plot

log(N(m)), the logarithm of the number of recorded events with magnitudes between m and m+.5, against event magnitude m. The

stated purpose of the plot was to establish that the recorded microseisms are ‘complete’ for events of magnitude m > 1: The plot of

magnitude versus frequency…shows that most earthquakes greater than magnitude 1 were [recorded]. The implication was that a large

number of earthquakes with m < 1 had occurred but were not adequately registered by the seismic network. The recorded G-R data

trend in Figure 1 (left) was recast in the form of the cumulative event count given by triangles. Cumulative counts plot the number of

events larger than a given magnitude m, a format that has no physical basis but which essentially masks the lack of small events.

Figure 1: (Left) Frequency-magnitude data for microseismicity induced during 1960s Rangely, Colorado, oil field water flood

[8-9]. Fall-off for the dotted event count curve below magnitude 1 is ascribed to failure of seismic network to record

small magnitude events. The triangle event count plots the number of events above a given magnitude; this data format

is used to estimate the data sample G-R scaling exponent indicated by the plot legend. ‘Missing’ events at magnitudes m

< 1 are ignored as being below detection threshold. (Right) Log-linear and log-log displays of Rangely oil field induced

seismicity magnitude distribution fit to a lognormal curve. A lognormal distribution allows the small event count fall-off

to be ascribed to source crustal processes rather than path and sensor effects.

The red curves in Figure 1 centre and right interpret the observed event magnitude frequency decline for magnitudes m < 1 as a

lognormal distribution. The close centre log-linear and right log-log fits suggest a physical process underlies the distribution of seismic

slip event magnitudes in line with the distribution of crustal fluid permeability κ(x,y,z) ~ exp(αφ(x,y,z)) in the waterflooded ambient

crustal volume. In this putative relationship, beginning at scale lengths associated with seismic moments of magnitude m ~ 1 events, the

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G-R power-law scaling seismic slip distribution can be viewed as affected by physical crustal processes rather than by observational

deficiency. Prospectively, the decline in event count occurs because in the ambient crust fluid connectivity flow structures below the

scale corresponding to m ~ 1 slip events no longer pass sufficient waterflood fluid pressures and fluid volumes to activate dislocation

slip events that emit seismic energy.

The Figure 1 lognormal distribution of earthquake magnitudes is not isolated. Figure 2 shows published examples of reporting actual

event counts, accompanied by explicitly interpretations of the observed event roll-off as due to events being ‘’incompletely’’ detected

by the observation network [10-11]. Figure 3 further exhibits event magnitude distributions from a southern California regional

network covering zones of active faulting [12]. Lognormal distribution fits to the catalogue magnitude distribution give evidence for (i)

natural roll-off (red, blue, gold traces), (ii) observational roll-off in excess of natural roll-off (blue dots linked by magenta traces), and

(iii) large magnitude events occurring on active faults in excess of expectations for ambient crustal seismicity (individual blue dots at m > 4).

Figure 2: Microseismicity event numbers log10(N) as function event magnitudes m observed at Parkfield on the San Andreas

fault (left, [10]) and an Indonesian geothermal field (right, [11]). Pluses/triangles denote recorded event numbers for

each magnitude interval; diamonds/squares denote cumulative recorded event numbers N>m, the number of events

having magnitudes ≥ m. For Parkfield, the ‘?’ and arrow at left, and for Indonesia ‘’MC’’, indicate the magnitudes at

which the seismic network is perceived to lose event count due to observational deficiencies. For Parkfield, the ‘?’ and

arrow at right indicate magnitudes at which the G-R relation predicts too few events.

Figure 3: Log-log plots of lognormal fits to an ambient crust seismic event magnitude catalogue for southern California [12].

Data given by blue dots; 3 lognormal curves given by red, blue, and gold traces. Large circles indicate, left, event

number roll-off augmentation due to observational incompleteness, and, right, large magnitude events on active tectonic

faults recorded by the observational network.

We argue here that Figs 1-3 earthquake magnitude distributions are the result of ambient crustal properties tied to rock-fluid

interactions. We first show that the lognormal magnitude distribution occurs for microseismicity generated by controlled injection of

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fluid at depth in crystalline basement for which there is clear evidence that the event magnitude catalogue is complete. We then show

that standard fractal interpretation of the G-R relation can be derived from the ambient permeability properties of crustal rock; in

particular, the standard G-R relation b-value can be derived from the poroperm properties of ambient crustal rock given by the empirical

parameter α. We conclude that the power-law-scaling or fractal interpretation of the G-R relation is an artefact that systematically

ignores direct data evidence and obscures recognition of the role that crustal fluids play in the physics of crustal seismic slip. Failure to

recognise the role fluids in crustal microseismicity has strong commercial consequences for reservoir operations.

2. LOGNORMAL G-R DISTRIBUTION EVENT-COMPLETENESS OF AMBIENT CRUST SEISMICITY

We now consider microseismicity induced in isolated ambient crust by slow controlled injection of fluid over successive 200m wellbore

intervals at 6km depth in crystalline basement near Helsinki, Finland [13-14]. Figure 4 shows the distribution of some 4200 magnitude

-1.5 < m < 1.5 seismic events. The events are embedded in a (4km)3 cube of near-uniform (~6.25km/s) seismic velocity with low

intrinsic and scattering attenuation (2kHz waveforms for 3km source-to-downhole-sensor travel paths). Event waveforms were

recorded at 2kHz for a 2.5km-deep wellbore sensor array and at 500Hz for a dozen 300m-deep near-surface sensors with low back

ground noise. The spatially-correlated fluctuation rock properties at the injection site are consistent with well-log porosity fluctuation

power that scales inversely with spatial frequency k, Sφ(k) ~ 1/k, and with well-core spatial fluctuation poroperm properties that link

permeability to porosity, κ(x,y,z) ~ exp(αφ(x,y,z)) [1-3]. Well-core and field-scale well-productivity distributions fix the poroperm

connectivity parameter α by the ambient crust condition αφ ~ 3-4 for site volume mean porosity φ ~ 0.01[1-3].

Figure 4: Spatial distribution of 4200 microseismic (-1.5 < m < 1.5) events relative to a data centroid centred on 1km-long

wellbore fluid injection interval at 6km depth in Fennoscandian basement rock [13-14]. Magenta tint denotes 2000

events selected for Figure 6 spatial/temporal block display of G-R distribution.

Figure 5 indicates that standard processing the Figure 4 site magnitude distribution for some 40,000 events conforms to the standard G-

R relation for the magnitude range -1.2 < m < 1.5 [13]. The data are displayed in the cumulative data format and the b-value is given

with 2% precision. The cumulative data format has effectively truncated the observations in the traditional manner, with no attention to

event magnitudes below the cut-off mc ~ -1.21.

Figure 4 event count completeness over the full 3-magnitude unit range -1.5 < m < 1.5 is, however, validated by event counts from the

deep seismic sensor array operating in uniform, high-velocity, high-Q, high-sensor-signal/noise environment. In light of several tens of

thousands of events detected by the downhole sensor count but too small to locate with the surface net, we may safely ascribe G-R event

count declines to inherent physical source processes rather than by ancillary path and sensor processes. Further evidence for event

completeness is provided by a power-law-scaling two-point spatial correlation function, Γ(r) ~ 1/r1/2, observed for event-pair offsets r

across the range 30m < r < 500m in the crustal volume computed for event marked by magenta dots [14]. The observed n ~ ½ power-

law-scaling correlation function exponent can be derived from the power-law scaling spatial correlation poroperm properties of crustal rock observed to exist in the site basement rock.

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Figure 5: Cumulative format G-R magnitude-frequency distribution for 40,000 stimulation seismicity events across magnitude

range -1.5 < m < 1.5 at the Figure 4 basement site [13]. The G-R b-value cut-off magnitude is m ~ -1.2. Figure 6 looks at

the G-R details of the Figure 4 magenta subset of these events for the full magnitude range -1.5 < m < 1.5.

Given the magnitude-completeness of the Figure 4 catalogue, we can explore the full range of the observed magnitude distributions in

terms of the lognormal function expressed by the crustal permeability population κ(x,y,z) ~ exp(αφ(x,y,z)), αφ ~ 3-4, at the injection

site. Histograms of the empirical function, H (κ ~ exp(αφ)), can be fit to the standard lognormal number distribution N(κ) ~

1/√(2π)/(σκ) exp(−(log(κ) −μ)2/2σ2) for a given permeability range κmin < κ < κmax [15]. Expressing this relation as log(N(κ)) ~ A

(log(κ))2 + B log(κ) + C, with A ≡ -1/2σ2, B ≡ (μ/σ2 – 1), C ≡ − log(√(2π)σ) − μ2/2σ2, shows that crustal rock has an inherent population

number scaling relation for the variable log(κ). If the coefficients A and B for the histogram population distribution obey the condition

A x log(κ) << B for a range of permeability κ, we obtain matching population statements between permeability distributions and the

standard G-R seismic magnitude distributions.

A different picture from standard Figure 5 G-R fractal distributions emerges in the Figure 6 display for the 2000 magenta events of

Figure 4. The Figure 6 magnitude-frequency data are not ganged together as in Figure 5, but are separated into spatial and temporal

batches. Each data batch is fit to a generic lognormal distribution rather than being truncated to conform to a power-law distribution

plus dangling low-magnitude extras obscured by the cumulative data format. The Figure 6 plots show that the Figure 4 seismic events -

- which were slowly and carefully stimulated in homogeneous ambient deep basement rock with no path or sensor effects to confuse the

physical magnitude-frequency relation -- do not conform to the standard G-R fractal interpretation.

Figure 6: G-R distributions for 6 blocks of seismic events from the Figure 4 magenta display. Data denoted by plus sign; curves

denote lognormal distribution fits to the data. Blue tints denote 1st 3 weeks of stimulation injection; red tints the 2nd 3

weeks of stimulation. Successive panels denote radial sectioning of the Figure 4 events; section radii given as y-axis

labels. There is a slight trend towards a higher population of low magnitude events for earlier compared with later

stimulation periods; the trend may indicate the earlier disturbance of the crust involves a higher number of small slip

reactivations that trend downward as the crustal disturbance continues.

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The Figure 6 blue data lognormal curve fits denoted by + are for the first three weeks of stimulation, and the red data lognormal curve

fits denoted by + for the second three weeks of stimulation; [13] details the stimulation time and space sequence. The plot panels show

data for three concentric radial domains about the median event location located near the stimulation wellbore; the radial domains are

given as y-axis labels. There is a spatiotemporal trend in which the earlier data (blue) are more populated at low-magnitudes than are

the later data (red), with the effect being greatest near the wellbore (top panel). It is possible, but by no means certain, that this trend

reflects a crustal disturbance progression in which earlier (blue) events result from stimulating long-undisturbed rock adjacent to the

wellbore, while later stimulation events (red) and events further from the wellbore (lower panels) occur when and/or in rock that has

been disturbed.

Key Figure 6 results for the G-R distribution of ambient crust EGS project induced seismicity are:

The data require the full lognormality parameterisation log(N(M)) ~ A (log(M))2 + B log(M) + C rather than the truncated

version log(N(M)) ~ B log(M) + C as given in Figure 5 and routinely accepted for seismicity processing elsewhere;

The data correlate well with the lognormality of ambient crustal permeability distributions determined by power-law scaling

poroperm properties of crustal rock manifested in the rock-fluid interaction empirics.

The G-R relation N(m) ~ 10-bm can be extracted from ambient crustal poroperm properties if the seismic slip moment M of

magnitude m ∝ log(M) at a location (x,y,z) is proportional to the permeability at that location, M(x,y,z) ∝ κ(x,y,z).

The next section extends the Figure 6 theme by using ambient microseismicity magnitude distributions to estimate the empirical

parameter α via the condition 4 < αφ < 5; then explaining the observed range of G-R power-law-scaling distribution b-values, 0.5 < b <

3.5, in terms of the observed value-range of poroperm empirical parameter α centred on 4 < αφ < 5.

3. AMBIENT CRUSTAL POROPERM PARAMETER CONDITION ΑΦ ~ 3-4 DETERMINES STANDARD G-R B-VALUES

An essential aspect of crustal rock-fluid interaction is that the well-core derived poroperm relation κ(x,y,z) ~ exp(αφ(x,y,z)) represents a

lognormal distribution. While in principle the poroperm distribution can be normal for small α, exp(αφ(x,y,z)) → 1 + αφ(x,y,z) as α →

0, the observed crustal values αφ ~ 3-4 guarantee that crustal permeability κ(x,y,z) ~ exp(αφ(x,y,z)) is lognormal [2]. Figs 7-8 derive

this lognormal fluid-rock interaction empiric in terms of ambient crustal G-R magnitude distributions. Figs 9-11 then take this

connection a step further to give physical meaning to the standard G-R b-values. Heretofore b-values have been evaluated extensively

in attempts to represent crustal microseismicity but rarely if ever leading to secure or testable physical interpretations.

In Figure 7, Indonesia geothermal field microseismicity magnitude-frequency distribution (red circles) are matched to a sequence of five

crustal permeability distributions (blue traces). The crustal permeability frequency distribution curves are given by the permeability

function κ ~ exp(αφ) evaluated for empirical parameter values α = 19:1:23 for the fixed normal porosity φ distribution shown in the

lower right plot. The best fit distribution curve parameter, α ~ 21 with φ ~ 0.2, corresponds to the empirical condition αφ ~ 4.2. The

observed decline of microseismicity event numbers for low magnitudes now can be seen as occurring for reasons of crustal physics.

Figure 7: Geothermal field microseismicity magnitude-frequency distribution (red circles) matched to a sequence of five crustal

permeability distributions (blue traces). Crustal permeability distribution curves H (κ(x,y,z)) ~ H (exp(αφ(x,y,z))) are

given for empirical parameter values α = 19:1:23 for normal porosity φ distribution at lower right. The best fit gives αφ

~ 4.2. Data from Figure 2 (right) [11].

Figure 8 repeats the Figure 7 evaluation of the αφ condition using the lin-log format for microseismicity magnitude distributions from

Figure 4, Figure 3, and Figure 7 catalogues, respectively. The microseismicity magnitude distribution fits provide an essentially

independent assessment of the αφ ~ 3-4 condition derived from well-core poroperm sequences from hydrocarbon reservoir rock and

crystalline basement rock [1-3].

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Figure 8: Lognormal curve fits to normalised lin-log curves for ambient crustal microseismicity magnitude distributions for

Figure 4 basement crust (left, [13]), Figure 3 southern California tectonic zone (centre, [12]), and Figure 2 Indonesian

geothermal field (right, [11]). The fits to lin-log microseismicity magnitude distributions give values of αφ ~ 5 in

comparison with αφ ~ 3-4 derived from well-core sequence poroperm data [2].

To establish an explicit relation between standard G-R b-values and the crustal poroperm α parameter, we consider a sequence of

numerical G-R distributions from a synthetic 3D crustal porosity field φ(x,y,z). The porosity field is normally distributed in population

between 0.1 < φ < 0.3 and spatially correlated so that numerical wellbore sequences through the porosity volume have power-law-

scaling spectra Sφ(k) ~ 1/kβ, β ~ 1, for spatial frequencies 1/1000 < k < 1 over 3 decades of scale length. The resulting permeability field

κ(x,y,z) ~ exp(αφ(x,y,z)) for a given value of α parameter is then partitioned into spatial domains for which the median permeabilities

are determined. The resulting population histogram H for each value of α generates a permeability distribution across the computed

range κmin < κ < κmax. The distributions following removal of the smallest 5% permeability values are then fit to a b-value exponent.

Figure 9 shows the results of this computation for 7 < α < 40. Values of α ~ 7-13 and α ~ 35-40 are values respectively smaller and

larger than have been observed in oil/gas field reservoir data. Values 15 < α < 35 are attested by reservoir and basement rock [2].

Figure 9: Evaluation of standard G-R b-values for synthetic permeability distributions, H (κ(x,y,z)) ~ H(exp(αφ(x,y,z))) for

sequence of α parameter values 7 < α < 40. Red/blue curves are respectively in the number/cumulative-number

distribution format (see Figure 1 for distinction). Black lines denote standard G-R fractal fits to the distributions. The

b-value corresponding to each α-value is given above plots.

Three properties for b-values emerge from Figure 9 ambient crust poroperm distribution synthetics:

Observed range of b-values derives naturally from observed range of α values;

b-values are large for small values of α and small for large values of α;

b-values have an asymptotic limit for high values of α.

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Figure 9 b-value systematics derived from physically attested values of α agree field data (e.g., standard attested α ~ 25 gives standard

attested b-value ~ -1.15), thus connecting b-values to crustal physics for the first time. The physical nature of the α parameter provides

a simple interpretation for the Figure 9 systematics. Small α corresponds to fluid-rock interaction systems of low connectivity

throughout the spatially-correlated porosity field; for low connectivity systems, a unit change in connectivity has a large effect on

permeability throughout the crustal volume. Large α corresponds to fluid-rock interaction systems of high connectivity for a few flow

structures in the spatially-correlated porosity field; a unit change in connectivity parameter systems affects the connectivity of the few

high-flow structures with very limited effect in the crustal volume at large. This contrast in the effect of changes in connectivity for

low/high values of α is shown graphically in Figure 10 and illustrated in Figure 11.

Figure 10: Dependence of b-value on ambient crust poroperm connectivity parameter α displayed in Figure 9. At low values of

α small changes have large impact on b-value; conversely at high values of α changes have small effect on b-value. As

illustrated in Figure 8, the low poroperm connectivity at small α means that increases in connectivity affect the entire

poroperm flow structure, while at high poroperm connectivity at high α means that increases in connectivity affect only

the few conductivity pathways.

Figure 11: G-R distributions of ambient crust permeability κ against frequency of κ for low and high values of connectivity

parameter α. Increasing α by two units has big effect on permeability for low α (i.e., high b-values in Figs 9-10) and

small effect on permeability for high α (i.e., low b-values in Figs 9-10).

We may further observe that individual ambient crust G-R data sets are generally limited to 3 decades of magnitude, e.g., Figs 1-3, 6;

[16-17]. While the standard G-R fractal distribution affords no good reason for a limited magnitude range, it is heuristically

understandable in terms of ambient crust permeability distributions. In a given time-frame, stimulation or activation of ambient crust

microseismicity is closely tied to fluid effects controlled by crustal permeability. Higher fluid mobility at larger scales rapidly activates

larger events. Mid-range fluid mobility at mid-range scales activates numerous mid-range events over a longer period of time.

Eventually small-scale low-permeability flow ceases to activate small-scale permeability flow structures, hence a systematic magnitude

count decrease. In contrast, standard G-R fractal relations focus only on larger-scale events at each of a number of tectonic zones.

The large range of magnitudes in standard G-R plots arises from suturing many independent crustal volumes that collectively have no

common fluid connectivity structure that can effectively limit the range of fluid-activated seismic slip event magnitudes. The well-

known spectral scaling of earthquake radiation indicates that some form of ambient crust fluid-permeability-control continues to

influence earthquake ruptures even at the most spatially extensive and disrupted fault structures hosting the highest magnitude events

[18].

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4. DISCUSSION/CONCLUSIONS – THE SPATIALLY-CORRELATED FLUID-ROCK INTERACTIVE AMBIENT CRUST

Shortly after Mandelbrot [19] popularised fractal distributions, Aki [20] and King [21] noted that the G-R relation codifying the power-

law scaling distribution of seismic event magnitudes was fractal-like, with the G-R scaling exponent ‘b-value’ putatively related to a

fractal dimension. Many others quickly followed this line of thought [e.g., 22-27].

Because the G-R relation appeared to hold for a large range of earthquake magnitudes occurring in diverse geological and tectonic

settings [28], it seemed that some form of fundamental self-similar scaling for earthquakes was indubitable. The fact that as in Figure 1

the number of low-magnitude event numbers regularly fell below the G-R power-law trend was essentially reflexively attributed to

failure of seismic networks to detect the full complement of small events. That fractals are essentially mathematical descriptors without

a clear physical meaning was not widely discussed.

In reviewing the practice of seismicity b-value estimation and interpretation, Utsu [22] simply skirts the issue of low magnitude event

incompleteness:

[When] no complete data are available below a certain magnitude….., the distribution must be truncated at this level.

and

Under the assumption that the magnitudes are distributed in accordance with the G-R formula, the b value is only the

parameter which characterizes the distribution….

While adjustments to the standard G-R relation based on a cut-off magnitude antedated the advent of fractals [8-9], the influence of

fractal scaling of earthquake magnitudes over many decades of scale length appears to have effectively relegated discussion of low-

magnitude seismic event data incompleteness to a shadow zone. In the spirit of ignoring low-amplitude event numbers, Utsu [22]

explicitly considers modifications to the G-R fractal relation only for higher magnitude events (e.g., the magnitude m > 4 events of

Figure 3).

Figure 12: G-R relation for southern California earthquakes surrounding a wellbore instrumented with seismometers at 1.5 and

2.5 km depths. The event count for wellbore seismometers followed the G-R trend with b-value b ~ 1 for 1.5 magnitude

units below the low-magnitude cut-off for the surface sensors [28]. For these data, the surface seismometer count for

low-magnitude events was affected by scattering and intrinsic attenuation of high frequency seismic waves and by

elevated background seismic noise at the surface. As in Figs 1-3, the cumulative format of the G-R data serves to

effectively mask the down-trending event count for low magnitudes events.

When the question of seismic catalogue incompleteness was addressed, it was assumed that only field-scale complexities such as

seismic wave attenuation and sensor background noise were at work. Figure 12 compares the G-R relation for seismicity in a crustal

volume near the San Andreas fault in southern California [28]. The ‘cumulative’ frequency-magnitude data format (the number of all

events above a given magnitude (e.g., Figs 1-3,5) is used to plot number distributions for earthquakes recorded by surface sensors and

for earthquakes recorded by wellbore sensors at 1.5 and 2.5 km depths. The Figure 12 data clearly show that the surface sensor count is

deficient compared with the borehole sensor count for reasons of pathway signal attenuation and sensor background noise levels. The

2.5km deep sensor records a standard b-value = 1 event count trend over a full 1.5 magnitude units lower magnitude range than do the

surface sensors. There is thus no question that in some cases attenuation and background noise acts to limit sensor count for a given set

of seismic activity. The question nonetheless remains whether attenuation and background noise are the only processes at work in

limiting the observed sensor counts. Our above discussion shows quite clearly that it was premature to eliminate natural physical crustal

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processes from consideration of magnitude distributions. In consequence, it is the imposed fractal interpretation of the G-R relation

based on uncritical treatment of low magnitude event counts, not observational means, that is deficient.

The present discussion of a lognormal G-R relation between number and size of crustal earthquakes shows that crustal fluids have a

decisive role in both earthquake size distribution and in spacing distribution [14]. We can now see that a related aspect of earthquakes

phenomenology, understanding why crustal slip events stop slipping, may be at hand. The standard concept of earthquake slip focuses

on a stress singularity at the tip of the rupture dislocation [18, 32-33], and merely assumes that eventually some physical condition

prevents the rupture from continuing. Our present perspective gives a simple stopping mechanism in accord with both the size and

spacing distributions.

When subjected to fluid injection, crustal sites of small permeability either don’t slip, or if they do, they slip with reduced energy

emission so that they don’t register in isolated/distinct observed slip events. Because of an effective permeability floor to slip energy

emission, the distribution of fluid-induced slip event energy does not cascade to ever lower slip event levels as expected from, say,

stress-singularity mechanics. In terms of slip energy event population, injection fluid pressure tends to not activate sites where

permeability is low.

Continuing the line of thought involving slip dislocations in ambient crust permeability structures, we can then ask what happens when

dislocation front encounters a large-scale fracture-connectivity permeability pathway through which fluids can slowly traverse a crustal

volume. Such fracture connectivity structures need not slip as if they were well-established zones of weakness. In an encountering a

large-scale but disseminated flow-connectivity structure, participating low permeability structures along the pathway can be activated as

low-level slip events which are individually too small to be seen, but can generate spatially and temporally aggregates of seismic energy

emission.

Evidence for just such a temporally/spatially defined low-level seismic emission phenomenon is seen in Figure 13. As discussed in [34-

35] and references therein, a surface array of seismic sensors can passively record low-level seismic emissions associated with large-

scale flow-connectivity structures without the need for fluid injection. In the case of Figure 13, prior to stimulation of the horizontal

well A, passive seismic emission data recorded above a Marcellus gas shale formation were processed to map the skeleton fracture-

connectivity structure denoted by coloured formation voxels. Upon Well A stimulation at well A sites intersecting the skeleton fracture

connectivity structure, flow into Well B was observed.

Figure 13: Plan view of horizontal Marcellus shale formation transected by wells A and B. Well A has a long horizontal leg in

the shale formation (black line). Well B had no known connection to Well A. Colourised voxels denote shale formation

locations where passive seismic data indicate the presence of low-level seismic emissions. During systematic stimulation

of Well A, fluid injection at the points of intersection with the seismic-emission structure caused Well B to flow.

In light of the present discussion, the implications of Figure 13 for convective geothermal fields are great. Lognormal distribution of

well productivity for all types of fluid reservoir settings show that crustal fluids moving through spatially erratic pathways are subject to

high degrees of spatial clustering [1-4]. As lognormal well distributions mean that most production is from a few wells and many wells

are poor producers, problems arising from poorly flowing wells can be costly. The most pronounced cost problems occur for

geothermal power production, where only the highest flowing wells can drive turbines. The cost problem could be solved if the flow

clustering of convective geothermal systems could be mapped in advance of production well drilling. It is thus of considerable

importance exploit the coupling between crustal seismicity and crustal permeability discussed here with a view to using remote seismic

source detection and location as a guide to active fluid flow volumes. Key to this advance is that fluid flow structures of interest are

continuous in both time and space. In many crustal environments fluid flow structures are erratic and clustered, but if active fluid flow

is not spatially/temporally continuous, there is little interest in their detection. Convective flow systems are clear examples of

spatially/temporally continuous flow structures. Temporally/spatially continuous flow structures as detectable sources of low-level

seismic emission are worthy targets for crustal reservoir operations.

Leary et al.

11

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