ALBERT

Description: In the last decades, the development of the surface and satellite geodetic and geophysical observations brought a new insights into the seismic cycle, documenting new features of inter-, co-, and post-seismic processes. In particular since 2002 satellite mission GRACE provides monthly models of the global gravity field with unprecedented accuracy showing temporal variations of the Earth's gravity field, including those caused by mass redistribution associated with earthquake processes. When combined with GPS measurements, these new data have allowed to assess the relative importance of afterslip and viscoelastic relaxation after the Sumatra 26.12.2004 earthquake. Indeed the observed post-seismic crustal displacements were fitted well by a viscoelastic relaxation model assuming Burgers body rheology for the asthenosphere (60–220 km deep) with a transient viscosity as low as 4  x 10 17 Pas and constant ~10 19 Pas steady state viscosity in the 60–660-km depth range. However, even the low-viscosity asthenosphere provides the amplitude of strain which gravity effect does not exceed 50 per cent of the GRACE gravity variations, thus additional localized slip of about 1 m was suggested at downdip extension of the coseismic rupture. Post-seismic slip at coseismic rupture or its downdip extension has been suggested by several authors but the mechanism of the post-seismic fault propagation has never been investigated numerically. Depth and size of localized slip area as well as rate and time decay during the post-seismic stage were either assigned a priory or estimated by fitting real geodesy or gravity data. In this paper we investigate post-seismic rupture propagation by modelling two consequent stages. First, we run a long-term, geodynamic simulation to self-consistently produce the initial stress and temperature distribution. At the second stage, we simulate a seismic cycle using results of the first step as initial conditions. The second short-term simulation involves three substeps, including additional stress accumulation after part of the subduction channel was locked; spontaneous coseismic slip; formation and development of damage zones producing afterslip. During the last substep post-seismic stress leads to gradual ~1 m slip localized at three faults around ~100-km downdip extension of the coseismic rupture. We used the displacement field caused by the slip to calculate pressure and density variations and to simulate gravity field variations. Wavelength of calculated gravity anomaly fits well to that of the real data and its amplitude provides about 60 per cent of the observed GRACE anomaly. Importantly, the surface displacements caused by the estimated afterslip are much smaller than those registered by GPS networks. As a result cumulative effect of Burgers rheology viscoelastic relaxation (which explains measured GPS displacements and about a half of gravity variations) plus post-seismic slip predicted by damage rheology model (which causes much smaller surface displacements but provides another half of the GRACE gravity variations) fits well to both sets of the real data. Hence, the presented numerical modelling based on damage rheology supports the process of post-seismic downdip rupture propagation previously hypothesized from the GRACE gravity data.

Description: SUMMARY A new mathematical and numerical model is presented for the propagation of a pressure- and buoyancy-driven dyke filled with volatile-saturated magma and a gas cap at its upper part. The model accounts for coupling between conduit flow of a bubbly magma, gas filtration through the magma, gas accumulation in a gas cap and elastic deformation and fracturing of the host rock. All these processes allow studying different regimes of dyke propagation. The rate of propagation of dykes is controlled by the rate of the fracturing at the tip and by the flow rate of magma inside the dyke. When high energy is needed to fracture the host rock and magma viscosity is low, the rate of propagation is controlled by the rate of fracturing (fracture-controlled regime). When the energy to fracture the host rock is low, propagation is controlled by the magma flow rate (magma-controlled regime). We study the transition between these regimes for the case of a constant magma vesicularity and constant mass of gas in the cap. Under these conditions, the propagation of the dyke is self-similar. In the fracture-controlled regime the propagation rate only weakly depends on the amount of the gas in the gas cap, whereas at the magma-controlled regime it is significantly enhanced with increase the mass of gas at the cap. The gas pressure in the cap opens the dyke in front of the magma and allows magma flow rates that are significantly higher than predicted by models that ignore the gas cap. The maximum propagation rate is obtained at the transition between the fracture- and magma-controlled regimes. If the gas mass in the gas cap is high enough, a gas pocket can separate from the magma as a distinct unconnected pocket and propagate as a gas-filled crack at a constant velocity. Pressure decreases during ascent leads to higher vesicularity and faster gas filtration through the magma and into a gas cap. Gradual increase of the mass of gas in the cap is important in accelerating the propagation rate of dykes.

Description: SUMMARY A new mathematical and numerical model is presented for the propagation of a pressure- and buoyancy-driven dyke filled with volatile-saturated magma and a gas cap at its upper part. The model accounts for coupling between conduit flow of a bubbly magma, gas filtration through the magma, gas accumulation in a gas cap and elastic deformation and fracturing of the host rock. All these processes allow studying different regimes of dyke propagation. The rate of propagation of dykes is controlled by the rate of the fracturing at the tip and by the flow rate of magma inside the dyke. When high energy is needed to fracture the host rock and magma viscosity is low, the rate of propagation is controlled by the rate of fracturing (fracture-controlled regime). When the energy to fracture the host rock is low, propagation is controlled by the magma flow rate (magma-controlled regime). We study the transition between these regimes for the case of a constant magma vesicularity and constant mass of gas in the cap. Under these conditions, the propagation of the dyke is self-similar. In the fracture-controlled regime the propagation rate only weakly depends on the amount of the gas in the gas cap, whereas at the magma-controlled regime it is significantly enhanced with increase the mass of gas at the cap. The gas pressure in the cap opens the dyke in front of the magma and allows magma flow rates that are significantly higher than predicted by models that ignore the gas cap. The maximum propagation rate is obtained at the transition between the fracture- and magma-controlled regimes. If the gas mass in the gas cap is high enough, a gas pocket can separate from the magma as a distinct unconnected pocket and propagate as a gas-filled crack at a constant velocity. Pressure decreases during ascent leads to higher vesicularity and faster gas filtration through the magma and into a gas cap. Gradual increase of the mass of gas in the cap is important in accelerating the propagation rate of dykes.

Description: SUMMARY We study the interaction of acoustic pressure waves with an expanding bubbly magma. The expansion of magma is the result of bubble growth during or following magma decompression and leads to two competing processes that affect pressure waves. On the one hand, growth in vesicularity leads to increased damping and decreased wave amplitudes, and on the other hand, a decrease in the effective bulk modulus of the bubbly mixture reduces wave velocity, which in turn, reduces damping and may lead to wave amplification. The additional acoustic energy originates from the chemical energy released during bubble growth. We examine this phenomenon analytically to identify conditions under which amplification of pressure waves is possible. These conditions are further examined numerically to shed light on the frequency and phase dependencies in relation to the interaction of waves and growing bubbles. Amplification is possible at low frequencies and when the growth rate of bubbles reaches an optimum value for which the wave velocity decreases sufficiently to overcome the increased damping of the vesicular material. We examine two amplification phase-dependent effects: (1) a tensile-phase effect in which the inserted wave adds to the process of bubble growth, utilizing the energy associated with the gas overpressure in the bubble and therefore converting a large proportion of this energy into additional acoustic energy, and (2) a compressive-phase effect in which the pressure wave works against the growing bubbles and a large amount of its acoustic energy is dissipated during the first cycle, but later enough energy is gained to amplify the second cycle. These two effects provide additional new possible mechanisms for the amplification phase seen in Long-Period (LP) and Very-Long-Period (VLP) seismic signals originating in magma-filled cracks.

Description: Travelling seismic waves and Earth tides are known to cause oscillations in well water levels due to the volumetric strain characteristics of the ground motion. Although the response of well water levels to S and Love waves has been reported, it has not yet been quantified. In this paper we describe and explain the behaviour of a closed artesian water well (Gomè 1) in response to teleseismic earthquakes. This well is located within a major fault zone and screened at a highly damaged (cracked) sandstone layer. We adopt the original Skempton approach where both volumetric and deviatoric stresses (and strains) affect pore pressure. Skempton's coefficients 〈 tex – mathid = " IM 0001" 〉 B and 〈 tex – mathid = " IM 0002" 〉 A couple the volumetric and deviatoric stresses respectively with pore pressure and 〈 tex – mathid = " IM 0003" 〉 BK u and 〈 tex – mathid = " IM 0004" 〉 N are the equivalent coupling terms to volumetric and deviatoric strains. The water level in this well responds dramatically to volumetric strain ( P and Rayleigh waves) as well as to deviatoric strain ( S and Love waves). This response is explained by the nonlinear elastic behaviour of the highly damaged rocks. The water level response to deviatoric strain depends on the damage in the rock; deviatoric strain loading on damaged rock results in high water level amplitudes, and no response in undamaged rock. We find high values of 〈 tex – mathid = " IM 0005" 〉 N = 8.5 GPa that corresponds to –0.5 〈 A 〈 –0.25 expected at highly damaged rocks. We propose that the Gomè 1 well is located within fractured rocks, and therefore, dilatency is high, and the response of water pressure to deviatoric deformation is high. This analysis is supported by the agreement between the estimated compressibility of the aquifer, independently calculated from Earth tides, seismic response of the water pressure and other published data.

Description: We present a thermodynamically based formulation for modelling dynamic rupture processes in the brittle crust using a continuum damage-breakage rheology. The model combines aspects of a continuum viscoelastic damage framework for brittle solids with a continuum breakage mechanics for granular flow within dynamically generated slip zones. The formulation accounts for the density of distributed cracking and other internal flaws in damaged rocks with a scalar damage parameter, and addresses the grain size distribution of a granular phase in the slip zone with a breakage parameter. A dynamic brittle instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid strain energy, localization and transition to a granular phase associated with lower energy level. The continuum damage-breakage rheology model treats the localization to a slip zone at the onset of dynamic rupture and post-failure recovery process as phase transitions between solid and granular states. The model generates sub- and supershear rupture velocities and pulse-type ruptures seen also in frictional models, and additional important features such as strong dynamic changes of volumetric strain near the rupture front and diversity of nucleation mechanisms. The propagation of rupture front and slip accumulation at a point are correlated with sharp dynamic dilation followed by a gradual decay to a level associated with the final volumetric change associated with the granular phase transition in the slipping zone. The local brittle failure process associated with the solid–granular transition is expected to produce isotropic radiation in addition to the deviatoric terms. The framework significantly extends the ability to model brittle processes in complex geometrical structures and allows analysing the roles of gouge thickness and other parameters on nucleation, rupture and radiation characteristics.

Description: During hydraulic stimulations, a complex interaction is observed between the injected flux and pressure, number and magnitude of induced seismic events, and changes in seismic velocities. In this paper, we model formation and propagation of damage zones and seismicity patterns induced by wellbore fluid injection. The model includes the coupling of poroelastic deformation and groundwater flow with damage evolution (weakening and healing) and its effect on the elastic and hydrologic parameters of crystalline rocks. Results show that three subsequent interactions occur during stimulation. (1) Injected flux–pressure interaction: typically, after a flux increase, the wellbore pressure also rises to satisfy the flux conditions. Thereafter, the elevated pore pressure triggers damage accumulation and seismic activity, that is, accompanied by permeability increase. As a result, wellbore pressure decreases retaining the target injected flux. (2) Wellbore pressure–seismicity interaction: damage processes create an elongated damage zone in the direction close to the main principal stress. The rocks within the damage zone go through partial healing and remain in a medium damage state. Damage that originates around the injection well propagates within the damage zone away from the well, raising the damage state of the already damaged rocks, and is followed by compaction and fast partial healing back to a medium damage state. This ‘damage wave’ behaviour is associated with the injected flux changes only in early stages while fracture's height ( h ) is larger than its length ( l ). The ratio h / l controls the deformation process that is responsible for several key features of the damage zone. (3) Stress- and damage-induced variations of the seismic P -wave velocities ( V p ). V p gradually decreases as damage is accumulated and increases after rock failure as the shear stress is released and healing and compaction are dominant. Typically, V p decreases within the damage zone and increases in most regions outside the damage zone. After a ‘damage wave’ that is originated at the well, V p rises back and may exceeds its initial values. Similar transient variations of the elastic parameters and the effects of h / l are observed at the Soultz-sous-Forêts Enhanced Geothermal System (EGS) records of induced seismicity during hydraulic injection.

Description: Sandstones display non-linear and inelastic behaviour such as hysteresis when subjected to cyclic loading. We present three hydrostatic compaction experiments with multiple loading–unloading cycles on Berea and Darley Dale sandstones and explain their hysteretic behaviour using non-linear inelastic compaction and dilation. Each experiment included eight to nine loading–unloading cycles with increasing maximum pressure in each subsequent cycle. Different pressure–volumetric strain relations during loading and unloading were observed. During the first cycles, under relatively low pressures, not all of the volumetric strain is recovered at the end of each cycle whereas at the last cycles, under relatively high pressures, the strain is recovered and the pressure–volumetric strain hysteresis loops are closed. The observed pressure–volumetric strain relations are non-linear and the effective bulk modulus of the sandstones changes between cycles. Observations are modelled with two inelastic deformation processes: irreversible compaction caused by changes in grain packing and recoverable compaction associated with grain contact adhesion, frictional sliding on grains or frictional sliding on cracks. The irreversible compaction is suggested to reflect rearrangement of grains into a more compact mode as the maximum pressure increases. Our model describes the ‘inelastic compaction envelope’ in which sandstone sample will follow during hydrostatic loading. Irreversible compaction occurs when pressure is greater than a threshold value defined by the ‘inelastic compaction envelope’.

Description: 〈span〉〈div〉SUMMARY〈/div〉We discuss analytical results on seismic radiation during rapid episodes of inelastic brittle deformation that include changes of elastic moduli in the source volumes. The full source tensor rate is shown to be the sum of (i) a moment rate term associated with the product of the rate of transformational strain and current local tensor of elastic moduli and (ii) the rate of damage-related term given by the product of the time derivative of the elastic moduli and current local elastic strain. Order of magnitude estimates indicate that the damage source term can be larger than the moment term for small events and the region around rupture front. However, the moment term integrated over the entire rupture zone is likely to be considerably larger than the damage term for large crack-like ruptures. The formulation provides rigorous definitions that can be used to estimate different source terms and associated radiated seismic fields in numerical simulations, experiments and field studies that have information on changes of elastic moduli in brittle source regions. Using elastic moduli taken from reference earth models in the analysis of seismic moment can lead to overestimation of the moment and artificial spatial variations.〈/span〉

Description: 〈span〉〈div〉Summary〈/div〉We discuss analytical results on seismic radiation during rapid episodes of inelastic brittle deformation that include changes of elastic moduli in the source volumes. The full source tensor rate is shown to be the sum of (i) a moment rate term associated with the product of the rate of transformational strain and current local tensor of elastic moduli, and (ii) the rate of damage-related term given by the product of the time derivative of the elastic moduli and current local elastic strain. Order of magnitude estimates indicate that the damage source term can be larger than the moment term for small events and the region around rupture front. However, the moment term integrated over the entire rupture zone is likely considerably larger than the damage term for large crack-like ruptures. The formulation provides rigorous definitions that can be used to estimate different source terms and associated radiated seismic fields in numerical simulations, experiments and field studies that have information on changes of elastic moduli in brittle source regions. Using elastic moduli taken from reference earth models in analysis of seismic moment can lead to overestimation of the moment and artificial spatial variations.〈/span〉