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Mass Transport Processes at Volcanoes
At volcanoes, processes of mass transport include porous flow in partially molten and deformable rocks, flow through fractures in elastic/brittle rocks, and diapiric ascent. Among these, transport in narrow fractures, or dykes, is the most eﬃcient mean of moving magma through cold lithosphere [for a review on magma transport see Rubin, 1995].
Measurements of seismic velocities at volcanoes indicate that the melt initially collects in reservoirs within the crust, where density discontinuities create favorable conditions for magma accumulation, i.e. where the density of the melt equals that of the surrounding rocks (named the Level of Neutral Buoyancy) [e.g. Ryan, 1987; Lister , 1990a,b; Hill et al., 2002]. Magma processes within the reservoir, such as buoyancy, refilling from depth, exolution of volatile components, crystallization, or bubble rising, induce pressure growth within the chamber. When the reservoir pressure exceeds a critical value, a fissure is created in the walls of the reservoir and propagates upwards [Lister , 1990a]. Such a fissure may reach the surface directly, leading to an eruption, or feed a shallower magma chamber, as it is frequently the case at basaltic volcanoes [e.g. Tilling and Dvorak , 1993; Hill et al., 2002]. From this latter, only a portion of the magma is subsequently erupted through secondary dykes rising to the surface (figure 1.2).
Dykes are tabular sheets through which magma rises across the solid matrix, their growth involve parting the host rock along pre-existing or magma-created fractures [Rubin, 1993b]. The surrounding solid matrix, subject to the ambient stress and usually considered to be elastic, is pushed apart with relatively little internal deformation, resulting in typical dyke thickness/length ratios of about 1/1000 [Rubin, 1993b]. Individual dykes in homoge-neous media grow thus as self-similar cracks normal to the direction of the least principal stress, acting mostly against the confining pressure rather than the intrinsic strength of the rock [Rubin et al., 1998]. The direction of crack growth is modified, however, by local stress concentrations [Pollard , 1973; Hill , 1977]. The propagation velocity is mostly controlled by the magma viscosity and can vary in the range 0.01 − 10 m/s at basaltic volcanoes [e.g. Klein et al., 1987; Rubin, 1995; Peltier et al., 2005, 2007], which allows dykes to propagate great distances before freezing [Rubin, 1993b, and references therein]. The diﬃculty of making direct observations of the plumbing system at volcanoes has limited our knowledge about the parameters and physical balances governing the magma movement at volcanoes [Lister and Kerr , 1991]. The principal forces at stake in controlling the crack propagation are the following [e.g. Lister , 1990b; Lister and Kerr , 1991; Rubin, 1995]:
• the pressure required to open the crack against elastic forces;
• the hydrostatic pressure due to the density diﬀerence between the magma and the host rock, i.e. the buoyancy force;
• the viscous pressure drop caused by flow in the crack;
• the magma driving overpressure;
• the tensile stress required for fracture extension against the strength of the host rock;
• the regional pre-existing stress field.
Several authors have applied analytical models of fluid-filled fracture to dyke em-placement and propagation to the eruption site. These models leans on diﬀerent initial conditions, which have been tested over the years.
Pioneer works [e.g. Weertman, 1971a; Rubin and Pollard , 1987; Pollard , 1988] neglect dynamical eﬀects such as the viscous pressure drop in the fluid, and focus on computing the stress field around a static fluid-filled crack and on evaluating the conditions under which the crack extends. Crack growth occurs when the stress intensity factor at the tip of the crack exceeds the critical value for the material. The vertical extent of the fluid-filled fracture cannot exceed a certain value without causing the upper tip of the crack to propagate or the lower tip to close [Weertman, 1971a]. For geological settings, this value is of order a hundred meters [Lister , 1990a]. This is not realistic for common tabular dykes, which are able to feed eruptions whose volume of lava emitted is often larger than the volume of the feeding dyke. The solutions proposed in these studies can be considered as equilibrium shapes of stationary fluid-filled cracks. They are therefore valid only once magma has come to rest, but is still molten [Lister and Kerr , 1991].
Initial attempts to include dynamic eﬀects have been carried out neglecting buoyancy forces [Spence and Sharp, 1985; Emerman et al., 1986; Spence and Turcotte, 1985]. The problem of density diﬀerence between melt and rock mass, therefore, is not considered. This makes them relevant only when considering propagation through sill (i.e. lateral dyke) [Lister and Kerr , 1991].
On the other hand, through an experimental approach in which various liquidus are injected into gelatin, Takada  concludes that buoyancy is the quantity that governs dyke propagation. In his experiments, Takada  tests two boundary conditions for crack growth: constant injection rate of fluid into the gelatin, and a constant volume for the fluid-filled crack.
Spence et al.  and Lister [1990a] give then analytic solutions governing the steady upward propagation of a two-dimensional buoyancy-driven dyke from a prescribed constant flow rate source. Lister [1990a] shows that the width and rate of propagation of the crack are determined by the geometry of the source feeding the crack and the magma supply rate. Lister [1990a,b] show that the pressure associated with elastic deformation and the strength of the country rock only aﬀect the vicinity of the dyke tip. This allows for simple solutions of dyke shape far from its tip [Spence and Turcotte, 1990]. Lister [1990b] derives then expressions for the lateral extent and the lateral cross-section of a dyke rising from a localized source, as well as for the rate of a fluid-filled crack laterally propagating in a stratified solid at the Level of Neutral Buoyancy (LNB).
Lister and Kerr  study of the eﬀect of fluid viscosity, elasticity and buoyancy on the fluid-filled crack propagation, concluding that none of these eﬀects can be neglected. They demonstrate that magma ascent is mainly driven by buoyancy. Indeed, after the vertical extent of the dyke exceeds a value of order a hundred of meters, the buoyancy forces are much greater than fracture resistance of rocks. From then on, the dominant resistance to further crack growth is provided by the viscous pressure drop in the melt as it flows towards the dyke tip. Near the dyke tip, on the other hand, the balance between viscous and elastic pressures controls crack growth. However, the authors show that is fluid dynamics who governs dyke propagation, and the ascent ceases near the LNB of the magma. Here dykes can propagate laterally driven by the buoyancy forces arising from the density diﬀerence between the magma and the underlying and overlying rocks.
Judging the boundary conditions of constant magma supply from the reservoir into the dyke [Lister , 1990a,b; Lister and Kerr , 1991] geologically inappropriate, Meriaux and Jaupart  propose a dyke propagation driven by buoyancy from a constant over-pressure reservoir through an elastic plate of finite thickness. In this configuration both, the dyke width and the magma injection rate (which depends on the conduit width) increases as dyke ascends to the surface. They identify two diﬀerent fracturing mechanism during crack growth, (i) the fracture initiation (sub-critical crack growth) and (ii) the subsequent propagation by tensile hydrocracking. Consequently, elastic stresses in the dyke conduit cannot be neglected as the dyke extends.
Ida  point out however that a very large volume reservoir would be required for the magma overpressure in the reservoir to remain constant as the dyke propagates.
M´eriaux et al.  consider dyke propagation within host rock with distributed damage. They conclude that the rate of magma-driven propagation is indeed determined by fluid dynamics [as proposed by Lister and Kerr , 1991]. This is because the host-rock response is linear except in the damaged tip region. A part from this small zone, therefore, the rock resistance can be neglected, and the resistance to dyke propagation is given by the viscous head loss.
The same conclusion is attained by Menand and Tait , who go back to the problem of dyke growth from a constant overpressure chamber, but from an experimental point of view. They conclude that, initially, the crack growth is controlled by a balance between the chamber over-pressure and the fracture toughness of the host rock. Once the buoyancy pressure overcomes the source pressure, a steady state is achieved, in which the fissure develops a bulbous head fed by a thinner tail. This steady state depends on the source overpressure.
Recently Roper and Lister  considered analytically the case of crack propagation under the influence of buoyancy and overpressure in an infinite impermeable solid. They find solutions which depend on the length of the crack relative to the buoyancy length, which measures the relative importance of the elastic pressure gradient and buoyancy. In both cases, short and large cracks, the overpressure at the source acts to make the width of the crack grow in proportion to its length. This leads to an increase in the flux, whose driving force is dominated either, by the elastic pressure gradient for short cracks, or by buoyancy for large cracks. In agreement with the experimental results obtained by Menand and Tait , Roper and Lister  find that, for large cracks, the solution develops a head-and-tail structure: in the tail the elastic pressure gradient is negligible and the flow is buoyancy-driven. In the head the elastic pressure gradient becomes comparable to buoyancy.
In conclusion, the studies that have been carried out since the early 70’s have led to fundamental understandings of the processes driving and accompanying dyke ascent from the reservoir to the surface. In particular they have established that the influence of of the toughness of rocks on dyke propagation only aﬀects fracture growth at an initial stage.
Subsequently the resistance of the country rock can be neglected and the fluid dynamics determine the propagation rate of the dyke. It means that the dyke ascent is driven by buoyancy, which allows for pushing the host rock aparts against elastic stresses, while resistance to dyke growth is given by the viscous head loss. One point is still debated, i.e. whether the feeding of the dyke from the magma reservoir can occur at constant magma injection rate, or whether the best initial condition is given by a constant overpressure at the dyke inlet.
Origin, Mechanics and Characteristics of Earth-quake Occurrence
The aim of this section is to describe the state of the art about theory of earthquake occurrence, in order to provide the reader with a background about all the tools we use throughout the following chapters. Most of the tools we describe here come from the ”classic” tectonic seismicity, and have been adapted here to suit volcano seismology. We briefly describe useful notions of fracture mechanics, the processes related to earthquake generation and the statistical features of seismic occurrence.
Notions of Fracture Mechanics
Understanding about rock strength properties dates back to ancient times, but the pioneer work of Griﬃth [1921, 1924] poses this wonder at a more fundamental level, in the form of an energy balance for crack propagation [for a detailed description see Scholz , 2002; Janssen et al., 2003]. According to the Griﬃth’s theory, then modified by Irwin  and Rice , in order the crack growth to occur, the potential energy G released by the extension of the crack is suﬃcient to provide the energy necessary for fracture Gc (i.e. the instantaneous elastic stress field surrounding the crack tip defined on the basis of a global energy change [Rice, 1968]). Owing to practical diﬃculties of this energy approach, later in the 1950’s, Irwin develops the stress intensity approach, according to which the crack extension occurs when the crack-tip stress intensity factor K reaches a critical value Kc. The factor K gives the magnitude of the elastic stress field, and depends on the crack size and loading configuration [e.g. Janssen et al., 2003; Rubin, 1993a, 1995], while Kc is the rock fracture toughness. The macroscopic strength of a material is thus related to the intrinsic strength of the material through the relationship between the applied stresses and the crack-tip stresses [Scholz , 2002]. According to the displacement field generated by the crack extension, the fracture can be divided into three basic types, or modes (figure 1.3). Mode I is tensile, or opening, Mode II is in-plane shear, and Mode III is anti-plane shear.
Faults correspond to mode II fractures, in which the displacements are in the plane of the discontinuity. Dykes, on the other hand, correspond to mode I fractures, in which the displacements are normal to the discontinuity walls. Earthquakes recorded on Earth surface are thus the expression of natural shear and opening mode cracks.
Mechanics of Earthquakes
According to the Elastic Rebound theory [Reid , 1911], the tectonic plate motion induces stress accumulations on faults, due to the fact that friction on the fault plane ”locks” it and prevents the sides from slipping. Eventually the strain accumulated in the rock is more than the rocks on the fault can withstand, and the fault slips, resulting in an earthquake that relaxes all the available energy. However, strain released by earthquakes can occur in a variety of forms resulting from e.g. the nature of crust materials and the local stress field [e.g. Kanamori , 1973].
According to more recent views, earthquakes are indeed triggered as a consequence of stress perturbations. Such stress perturbations are, however, due to both, external forcing (i.e. either tectonic plate motion, or volcano processes in tectonic and volcanic environments, respectively) and earthquake interactions. From the simple view point of an isolated homogeneous fault loaded at constant stress rate, characteristic earthquakes occur periodically by rupturing the whole fault, with a period equal to ratio of the stress drop divided by the rate of stress loading [e.g. Scholz , 2002, and references therein]. These earthquakes are the signatures of the tectonic loading [e.g. Helmstetter , 2002]. However, statistical and geological studies show that faults are complex structures organized into complex and interacting networks [for a review see Bonnet et al., 2001]. There are in-deed abundant evidences of fault (and thus earthquake) interacting through their static stress field, which results in earthquakes triggered by the static stress change induced by a previous event. However, the physics driving these interactions is not fully understood and various diﬀerent mechanisms have been proposed [e.g. Helmstetter et al., 2005, and references therein]. The manner in which fracture system properties at diﬀerent scales relate to each other, has thus recently received increased attention motivated by the promise of statis-tical prediction that scaling laws oﬀer [e.g. Bonnet et al., 2001]. Recent studies on tec-tonic earthquake occurrence through a stochastic model of seismicity (see section 1.3.4), demonstrate that earthquakes have a key role in the triggering of other earthquakes [e.g. Helmstetter and Sornette, 2002a; Helmstetter , 2003; Helmstetter and Sornette, 2003].
Assuming the validity of the stress triggering mechanism for earthquakes, the Coulomb stress change Δσf defined below, will enhance or retard the potential for rupture to nucleate on a given fault. Δσf = S = Δτ − µ(Δσn − ΔP ). (1.1) where Δτ is the shear stress change on a fault (positive in the direction of fault slip), Δσn and ΔP are the changes in normal stress and pore pressure on the fault (positive for compression), and µ is the friction coeﬃcient. Failure is encouraged if Δσf is positive, and inhibited if Δσf is negative. Both, increased shear and unclamping of faults promote failure [Stein, 1999]. Coulomb stress changes refer to static stress changes that occur instantaneously and permanently [Steacy et al., 2005].
Computations of Coulomb stress change indeed show that enhancement of seismicity rather occurs in areas of stress increase, while seismic quiescence is observed in stress drop shadow areas [e.g. King et al., 1994; Harris, 1998; Stein, 1999; King and Cocco, 2000]. Steacy et al. , commenting on Toda et al.  results, aﬃrm that triggering primarily represent clock-advanced failure, rather than creation of new fractures.
Observations of aftershocks occurring in stress shadow areas seem to contradict the stress triggering mechanism [Hardebeck et al., 1998; Catalli et al., 2008]. However, Helm-stetter and Shaw ; Marsan  demonstrate that small-scale slip variability, which might not be directly measured, may explain the absence of quiescent regions in the first period of aftershock activity.
Based on this model and on the observation that stress and earthquake rate changes are not linearly correlated, Deterich  proposes a constitutive law for the rate of earthquake production, leaning on experimentally derived rate- and state-dependence of fault friction. He models the seismicity as a sequence of nucleation events whose occurrence depends on the distribution of initial slip conditions on the fault and on the stress history to which the fault is subjected. An evolving variable representative of the state of the fault over time allows to quantify the rate of earthquake production resulting from an applied stress history.
According to this formulation, the earthquake rate R in a specified magnitude range is given by R = r , (1.2) γ Sr where dγ = 1 [dt − γdS], (1.3)
γ is a state variable, t is time, the constant r is the steady-state earthquake rate at the reference stressing rate Sr , A is a dimensionless fault constitutive parameter, and S is the Coulomb stress function of equation (1.1). See appendix B for details on equations about the eﬀect of stressing history on Earthquake rate).
On the bases of this formulation, Dieterich et al.  uses seismicity rate changes to compute stress changes prior and contemporary to the 1983 flank eruption at Kilauea volcano. The results they obtain well agree with the deformation model obtained for the same episode by Cayol et al. . It evidences that accompany this eruption are promoted by Coulomb stress changes induced by the expansion of a dyke-like magma system within the Kilauea rift zones, coupled with aseismic creep. Toda et al.  show a linear relationship between the stressing rate change induced by the 2000 dyke opening at Izu Islands (Japan), and the increase in seismicity rate. Feuillet et al.  show that earthquakes recorded at Alban Hills volcano (Italy) are promoted by elastic stress changes induced by a magmatic intrusion.
All these argue for the fact that just as earthquake occurrence perturb the stress state in surrounding areas, so does mass transport and volcano processes at volcanic en-vironments. Here, stress perturbations induced by magmatic processes may thus promote faulting and earthquake activity. Indeed, major volcanic events are generally associated with dramatic increases of seismic activity. Conversely, a volcanic system may be per-turbed by stress changes induced by neighboring earthquakes [e.g. Dieterich et al., 2003; Hill et al., 2002; Lemarchand and Grasso, 2007].
Table of contents :
1 Theoretical background
1.1 Volcano seismology
1.2 Mass Transport Processes at Volcanoes
1.3 Origin, Mechanics and Characteristics of Earthquake Occurrence
I Seismic Signature of Dyke Propagation
2 Brittle creep damage as the seismic signature of dyke propagations within basaltic volcanoes
2.3 Seismicity patterns during dyke intrusions
2.4 A generic model for dyke propagation in basaltic volcanoes as mapped from VT seismicity patterns
2.5 Concluding remarks
2.6 Data and Resources
3 A Constant Influx Model for Dyke Propagation. Implications for Magma Reservoir Dynamics
3.2 Models of dyke propagation
3.3 Case study: The August 22 2003, Piton de la Fournaise eruption
4 Space and Time Seismic Response to a 60-Day-Long Magma Forcing. The 2000 Izu dyke Intrusion Case
4.3 From the earthquake rate to the stress history
4.4 The forcing rate as a tool to estimate the background seismicity in a point process
4.5 The effect of a long duration forcing on a system
4.6 Discussion and conclusions
II Seismic Signature of Simple Volcano Processes
5 Seismic Signature of Magma Reservoir Dynamics at Basaltic Volcanoes, lesson from the Piton de la Fournaise Volcano
5.2 Brittle damage models for PdlF reservoirs dynamics, the State of the Art
5.3 Testing models for PdlF reservoirs dynamics
5.4 Concluding remarks
6 How is Volcano Seismicity Different from Tectonic Seismicity?
6.3 Analysis of interevent time distributions
6.4 The ETAS model
6.6 Concluding remarks
6.7 Data and Resources
7 Short-Term Forecasting of Explosions at Ubinas Volcano
7.2 Ubinas volcano
7.4 Long Period seismicity patterns before explosions
7.5 Predictability of explosions from LP earthquake rate on Ubinas volcano
7.6 Discussion and conclusions
A Line Creep in Paper Peeling
A.3 Creep velocity
A.4 Statistical distributions
A.5 Measures of correlated dynamics
B Rate-and-State friction model
B.1 Earthquake nucleation
B.2 Effect of Stressing History on Earthquake Rate
C Change Point Analysis