Title:

A Physical Basis for the Gutenberg-Richter Fractal Scaling

Authors:

P LEARY, P MALIN and Tero SAARNO

Key Words:

EGS, MEQs, wellbore stimulation, Gutenberg-Richter, fractals

Conference:

Stanford Geothermal Workshop

Year:

2020

Session:

Enhanced Geothermal Systems

Language:

English

Paper Number:

Leary1

File Size:

1679 KB

View File:

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 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-M 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-M 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.


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