Vibrational modes of hydraulic fractures: Inference of fracture geometry from resonant frequencies and attenuation

Abstract

Oscillatory seismic signals arising from resonant vibrations of hydraulic fractures are observed in many geologic systems, including volcanoes, glaciers and ice sheets, and hydrocarbon and geothermal reservoirs. To better quantify the physical dimensions of fluid‐filled cracks and properties of the fluids within them, we study wave motion along a thin hydraulic fracture waveguide. We present a linearized analysis, valid at wavelengths greater than the fracture aperture, that accounts for quasi‐static elastic deformation of the fracture walls, as well as fluid viscosity, inertia, and compressibility. In the long‐wavelength limit, anomalously dispersed guided waves known as crack or Krauklis waves propagate with restoring force from fracture wall elasticity. At shorter wavelengths, the waves become sound waves within the fluid channel. Wave attenuation in our model is due to fluid viscosity, rather than seismic radiation from crack tips or fracture wall roughness. We characterize viscous damping at both low frequencies, where the flow is always fully developed, and at high frequencies, where the flow has a nearly constant velocity profile away from viscous boundary layers near the fracture walls. Most observable seismic signals from resonating fractures likely arise in the boundary layer crack wave limit, where fluid‐solid coupling is pronounced and attenuation is minimal. We present a method to estimate the aperture and length of a resonating hydraulic fracture using both the seismically observed quality factor and characteristic frequency. Finally, we develop scaling relations between seismic moment and characteristic frequency that might be useful when interpreting the statistics of hydraulic fracture events.

Origin of Shallow Volcanic Tremor: The Dynamics of Gas Pockets Trapped Beneath Thin Permeable Media

“…Previous studies aimed to explain shallow volcanic tremor through frictional faulting (Dmitrieva et al, ; Hotovec et al, ) and fluid‐elastic processes (e.g., nonlinear oscillations of volcanic conduits and cracks during magma ascent (Balmforth et al, ; Julian, ; Lipovsky & Dunham, ), forced coalescence of bubbles (Ripepe & Gordeev, ), gas‐liquid two‐phase flow (Fujita et al, ; Lane et al, ), acoustic resonance of fluid‐filled crack/cavities (Chouet, ; Chouet, ), acoustic resonance of magmatic foams attached to the conduit walls (Bercovici et al, ; Jellinek & Bercovici, ), and the formation of turbulent eddies behind obstacles (Hellweg, )). Although we do not reject the possibility that different mechanisms may act simultaneously, our results suggest that the accumulation of gases beneath permeable caps generates shallow volcanic tremor; this is consistent with a localized seismic source and supports the idea of cap‐controlled tremor invoked by Hellweg (), Johnson and Lees (), and Valade et al ().…”

“…Several mechanisms have been proposed to explain volcanic tremor, including fluid‐elastic resonance (Balmforth et al, ; Bercovici et al, ; Chouet, ; Chouet, ; Fujita et al, ; Hellweg, ; Jellinek & Bercovici, ; Johnson & Lees, ; Julian, ; Lane et al, ; Lipovsky & Dunham, ; Ripepe & Gordeev, ) and frictional processes (Dmitrieva et al, ; Hotovec et al, ). These mechanisms are able to explain some tremor features.…”

Dynamics of the Askja caldera July 2014 landslide, Iceland, from seismic signal analysis: precursor, motion and aftermath

“…Reflecting these diverse settings, an equally diverse collection of physical processes may explain the source process responsible for creating seismic tremor. Possible sources of seismic tremor include (i) fluid-flow-induced oscillations of conduit or fracture walls (Julian, 1994;Hellweg, 2000;Rust et al, 2008;Matoza et al, 2010;Corona-Romero et al, 2012;Dunham and Ogden, 2012;Unglert and Jellinek, 2015); (ii) resonance of fluid-filled cracks or pipes with open or closed ends (Chouet, 1985(Chouet, , 1986(Chouet, , 1988Benoit and McNutt, 1997;Jousset et al, 2003;Neuberg, 2000;Jellinek and Bercovici, 2011;Röösli et al, 2014;Sturton and Neuberg, 2006;Lipovsky and Dunham, 2015); (iii) bubble growth or collapse due to hydrothermal boiling of groundwater (Leet, 1988;Kedar et al, 1998;Cannata et al, 2010); and (iv) continuously repeating processes such as stick-slip motion (Neuberg, 2000;Powell and Neuberg, 2003;Dmitrieva et al, 2013;Hotovec et al, 2013;Lipovsky and Dunham, 2016; see also reviews by Mc-Nutt, 1992 andKonstantinou andSchlindwein, 2003). We note that the first three of these processes are hydraulic in origin.…”

“…As the Askja landslide was not associated with any volcanic activity that would support this mechanical model of tremor generation through hydraulic processes, we conclude that a hydraulic source is unlikely to explain the phenomena observed at Askja. Furthermore, Lipovsky and Dunham (2015) analysed seismic tremor due to hydraulic resonance and found that the resonant frequencies of a hydraulic fracture are expected to be unevenly spaced following f n /f 1 = n 3/2 . Complementary, Lipovsky and Dunham (2016) showed that a simple application of the Fourier transform to a repeating sequence of slip pulses results in a frequency pattern of f n /f 1 = n. For a fundamental tone of f 1 = 2.3 Hz as observed for the Askja landslide tremor, we would expect its first harmonic at 6.5 Hz for a resonating fracture or at 4.6 Hz for a stick-slip source.…”

Dynamics of the Askja caldera July 2014 landslide, Iceland, from seismic signal analysis: precursor, motion and aftermath

“…Tremor has been observed in a variety of settings including volcanically active and glaciated 10 regions. Possible source mechanisms for tremor that have been put forward include: (i) fluid-flow-induced excitation and oscillations of volcanic conduit walls (Julian, 1994;Hellweg, 2000;Rust et al, 2008;Matoza et al, 2010;Corona-Romero et al, 2012;Unglert and Jellinek 2015); (ii) excitation and resonance of fluid-filled cracks or pipes with open or closed ends (Chouet, 1985(Chouet, , 1986(Chouet, , 1988Benoit and McNutt, 1997;Jousset et al, 2003;Neuberg, 2006;Jellinek and Bercovici, 2011;Röösli et al, 2014;Sturton and Walter et al, 2015);, (iii) bubble growth or collapse due to hydrothermal boiling of 15 groundwater (Leet, 1988;Kedar et al, 1998;Cannata et al, 2010); and (iv) continuously repeating processes such as rock and ice deformation and frictional faulting (Neuberg, 2000;Powell and Neuberg, 2003;Dmitrieva et al, 2013;Hotovec et al, 2013;Lipovsky and Dunham, 2015; see also reviews by McNutt, 1992 andKonstantinou andSchlindwein, 2003). Brace and Byerlee (1966) proposed that tremor can occur when stick-slip motion on a fault generates frequent small earthquakes.…”