3.1 Magmatic System
We refer as an archetypal case to the Phlegraean Fields magmatic system, where seismic imaging and attenuation tomographies have identified a huge (probably around 10 km wide) magma reservoir at a depth of around 8 km (Zollo et al. 2008; De Siena et al. 2010), while a variety of geophysical and geochemical evidence suggests that smaller (probably less than 1 km3), shallower batches of magma have been forming throughout the caldera history at virtually any depth smaller than 9 km (Arienzo et al. 2010; Di Renzo et al. 2011). These shallow magma bodies have been identified as actively involved in past eruptions, which at least in some cases shortly followed the arrival of volatile-rich, less differentiated magmas from the deep feeding system (Arienzo et al. 2009; Fourmentraux et al. 2012). Chemical compositions of erupted magmas range from shoshonitic to trachytic to phonolitic; geochemical analyses on melt inclusions suggest a variety of processes contributing to this variability, such as recharge from depth, intra-chamber mixing, syn-eruptive mingling (Arienzo et al. 2010; Fourmentraux et al. 2012). The same analyses show that deep magmas are typically rich in gas, especially CO2 (Mangiacapra et al. 2008), while shallow magmas are unusually crystal-poor, down to less than 3 wt% (Arienzo et al. 2009). To study the magmatic dynamics occurring as a consequence of a recharge event, we simplify the magmatic system retaining its most peculiar features. We model the injection of CO2-rich shoshonitic magma coming from a deep reservoir into a shallower, much smaller chamber, containing more evolved and partially degassed phonolitic magma (see Table 1 for compositions). The two chambers are connected by a dyke. This idealized layout captures several first-order characteristics of prototype magmatic systems, including a composite structure, vertical extension, and heterogeneous composition, and it approximates systems composed by long-lived, interconnected multiple reservoirs believed to exist at many active volcanoes (Elders et al. 2011).
Table 1 Composition of the phonolite and shoshonite magma types employed in the simulations
Figure 1 shows the system domain for the numerical simulations. We assume one of the horizontal dimensions of the magmatic system to be much larger than the other, so that our domain is two-dimensional. The deep chamber is elliptical, 1 km thick and 8 km wide; its top is at 8 km depth. The geometry of the shallow chamber has been varied as shown in Fig. 1, keeping its surface area fixed. In the elliptical cases, the semi-axes measure 400 and 800 m, respectively, while the circular chamber has a radius of 283 m.
The initial conditions of the system are also shown in Fig. 1. The shallow chamber hosts a differentiated, volatile-poor phonolitic magma. Its volatile content has been varied from 0.3 wt% CO2 and 2.5 wt% H2O to 0.1 wt% CO2 and 1 wt% H2O. In the feeding dyke and deep reservoir is a less evolved, basaltic shoshonite, containing 1 wt% CO2 and 2 wt% H2O. Such a low water content in the phonolitic end-member derives from melt inclusion data from Phlegrean Fields (Arienzo et al. 2010). Typically, more evolved magmas are expected to have a relatively larger water content (Signorelli et al. 2001; Cannatelli et al. 2007; Pappalardo et al. 2007; Mollo et al. 2015), resulting possibly in smaller density contrasts at the interface among the two magmas thus less efficient mixing dynamics.
Volatiles partitioning between gaseous and liquid phases is computed following Papale et al. (2006) as a function of composition and pressure. Pressure at time 0 consists of a depth-dependent magmastatic contribution superimposed to the host rock confining pressure. The interface between the two magmas, at the inlet of the shallow chamber, is gravitationally unstable, the lower magma being less dense due to its higher gas content. The dynamics is solely driven by buoyancy, without any external forcing.
The density contrast at the interface varies for each simulated scenario, as it depends on both volatiles content and their partitioning between liquid and gaseous phases; volatiles exsolution in turn depends on pressure, thus on the depth at which the interface is placed, which is different for each geometry of the shallow chamber (Fig. 1). Temperature differences between interacting magmas are often negligible (Sparks et al. 1977), particularly at Phlegraean Fields (Mangiacapra et al. 2008; Arienzo et al. 2010), thus the system is assumed isothermal. As a result, there is no need to speculate on the thermal status of the surrounding rock, thus reducing model uncertainties. Moreover, heat transfer effects are expected to play a minor role on the short simulated time scales (hours; Di Renzo et al. 2011).
3.2 Magma Dynamics
Interaction among the two magmas develops as a consequence of the initial gravitational instability at the interface. We solve numerically the two-dimensional space-time evolution of the system, consisting of a mixture of two different magmatic components, each of them including a liquid (silicate melt and dissolved volatiles) and a gaseous (exsolved volatiles) fractions. The equations of motion for the mixture express conservation of mass for each component k = 1, 2, and momentum for the whole mixture (Longo et al. 2012a):
In the above, t is time; \( \rho \) is mixture density,
is mass fraction of component k, u is fluid velocity, Dk is the k-th coefficient of mass diffusion, p is pressure, μ is viscosity and g is gravity acceleration.
The magmatic mixture is considered ideal. Its density is evaluated as weighted sum of the components’ densities; for each component, density is calculated using a non-ideal equation of state for the liquid phase, real gas properties and ideal mixture laws for multiphase fluids. Mixture viscosity is computed through standard rules of mixing for one phase mixtures and with a semi-empirical relation in order to account for the effect of non-deformable gas bubbles. Liquid viscosity is modeled as in Giordano et al. (2008), and it depends on liquid composition and dissolved water content. The assumption of Newtonian rheology is justified by the very low strain rates and the crystal-free nature of the magmas. The generalized Fick’s law is used to describe mass diffusion. Volatile partitioning between gaseous and liquid phases is evaluated at every point in the space-time domain as function of mixture composition and pressure as in Papale et al. (2006). All the physical properties of the two magmas are evaluated at every point in the space-time domain depending on the local conditions of pressure, velocity and mass fractions, which are the unknowns in Eqs. (1) and (2). The equations are solved numerically using GALES, a finite element C++ code specifically designed for volcanic fluid dynamics (Longo et al. 2012a).
The evolution in space and time of the system is complex and presents a number of interesting features. Figure 2 summarizes the results regarding magma dynamics, showing the evolution of composition in time in the shallow chamber for the five different simulation scenarios.
The initial inverse density contrast at the contact interface between the two magmas gives rise to convective mass transfer from the deeper parts of the system to shallower depths and vice versa. The unstable density contrast is solely due to the different volatile content of the two mixtures: the shoshonitic melt has an higher density than the phonolitic. The role played by volatiles is crucial, and it is exsolved gases that ultimately determine the buoyant dynamics. A Rayleigh-Taylor instability develops, which acts to bring the system to gravitational equilibrium by overturning it. The instability develops starting from the perturbed interface, with a first plume of light material that rises into the chamber. Depending on the initial density contrast as well as on the geometry of the shallow chamber, the initial plume starts developing at different times. The dynamics is strongly enhanced by higher density contrasts; geometry also plays an important role when density contrasts are similar, with horizontally elongated, sill-like chambers favouring convection with respect to more dyke-like setups (see also Fig. 2).
Plumes of light magma coming from depth keep entering the shallow reservoir as discrete filaments, following irregular trajectories and showing typical convective patterns. The lighter material tends to rise into the chamber, thereby decreasing more and more its density as volatiles exsolve in lower-pressure environments; on the other hand, the denser magmatic mixture initially residing in the chamber sinks into the feeder dyke, increasing its density by the reverse process of volatile dissolution at higher pressures. The plumes thus progressively increase their buoyancy, enhancing their expansion and acceleration. During the rise, vortexes form at the head of the plumes and subsequent plumes interact among themselves, further favouring mixing. The dynamics creates complicated patterns that maximize the interaction among the two different magmatic mixtures (Petrelli et al. 2011). Mingling is evident for all simulated conditions both within the chamber itself and even more in the feeding dyke (Fig. 2), and it is strongly intensified by the chaotic patterns that form as a consequence of deep magma injection.
Independently from system geometry or density contrast at the interface, mingling is very efficient in the feeding dyke, more than inside the upper chamber. Figure 2 shows that since the very beginning of the simulations, the magma entering the chamber is already a mixture of the two initial end-members, and not the pure shoshonitic composition.
As the dynamics proceeds, faster for higher density contrasts and sill-like setups, the gas-rich mixture tends to accumulate at the top of the chamber, thereby originating a stable density stratification that has indeed been testified at various magmatic systems (Arienzo et al. 2009). The stratification is more prominent in vertically elongated, dyke-like reservoirs (Fig. 2). The density profile along the vertical direction, evaluated averaging along horizontal planes (Fig. 3), illustrates that a quasi-stable profile is reached after some hours of simulated time.
As time proceeds, convection slows down due to smaller buoyancy of the incoming already mixed component, and the instability proceeds in time asymptotically: the more the two end-members have mingled, the less intense is convection.
The evolution of pressure in the system is highly heterogeneous in space and time. Alternating phases dominated by buoyancy and sinking at chamber inlet result in pressure fluctuations with periods of hundreds of seconds and amplitudes decreasing with time (Fig. 4). Typically pressure variations are smaller than 1 MPa; under these conditions, it is unlikely that rejuvenation can trigger eruption, as the stresses needed to create a pathway to the surface in the host rock are typically larger than that (Gudmundsson 2006).
3.3 Ground Deformation
Determining the time–space-dependent ground displacement requires modeling the magma–rocks boundary conditions and the mechanical response of rocks, the latter depending on heterogeneous rock properties, presence and distribution of faults, interfaces, fluids, and volcano topography (e.g., O’Brien and Bean 2004). A first-order analysis performed here assumes magma–rock one-way coupling and adopts the Green’s functions formulation for a homogeneous, infinite medium (Aki and Richards 2002).
We consider as point sources the fluid dynamics computational grid nodes located at the reservoir walls. As source time functions, we use the respective temporal evolutions of magmatic forces computed from pressures and stresses provided at those nodes by the numerical simulations of magma convection and mixing dynamics. Ground displacement at a series of virtual receivers is finally obtained by integrating, over all sources, the Green’s functions associated with individual sources.
Continuity of pressure and stress is taken as the boundary condition along the non moving magma–rock interface. Physical properties of rocks are homogeneous averages that describe the volcanic edifices within the range of considered depths (<10 km, vP = 3000 m/s; vP/vS = 1/√3, ρ = 2500 kg/m3).
Propagation of pressure disturbances in the host rock medium reveals that the computed pressure oscillations, originated by the ingression of buoyant magma in the magma chamber, translate into Ultra Long Period ground displacement dynamics with amplitudes of millimeter to micrometer order (Fig. 5; Longo et al. 2012b). ULP ground movements like those predicted by the present modeling could not be detected by classical broadband seismometers (although more recent seismometers extend their working range up to 100–200 s periods), while they are visible in the records from other instruments, especially borehole dilatometers characterized by high signal-to-noise ratio (Sacks et al. 1971).