Simulation of the hydrothermal circulation is performed by the well-known TOUGH2 software, a fluid flow and heat transport simulator of multiphase multicomponent fluids in porous media accounting for phase changes, relative permeability of each phase and capillarity pressure . TOUGH2 solves a system of mass and energy balance equations that can be summarised as follows (for a general case of a fluid with k components): ∂Qα∂t+∇⋅Fα-qα=0,α=M1,…,Mk,E, where Q is the accumulation term, F the flux and q the source (or sink) term, while the subscript α=Mi or E refers to the mass balance equation for the ith component or the energy balance equation, respectively. The accumulation terms and fluid fluxes (based on the extended Darcy law) for mass balance equations are QMi=ϕ∑βρβSβχβi,FMi=∑βχβiFβ, with Fβ=-KKrβρβμβ-1(∇Pβ-ρβg^), where the subscript β=l or g refers to the liquid or gas phase respectively, ϕ is the porosity, ρβ the density, Sβ the saturation, χβi the mass fraction of the ith component in the β phase, K and Krβ are the absolute and relative permeability, respectively, μβ the viscosity, Pβ the fluid pressure and g^ the gravitational acceleration. For the energy balance equation, the accumulation term (QE) and the heat flux (FE) are QE=ϕ∑β(ρβeβSβ)+(1-ϕ)ρrCrT,FE=-λ∇T+∑βhβFβ, where eβ and hβ are the specific internal energy and enthalpy of the phase β, T is the temperature, and ρr, Cr and λ are the density, specific heat and the thermal conductivity of the rock respectively.

In this paper we simulate fluids of magmatic origin entering the domain as a mixture of two components (k=2): hot water and carbon dioxide. This mixture is simulated by the EOS2 module of TOUGH2. The depth of the domain for the hydrological model is 1.5 km, since the focus is the shallow hydrothermal activity, maintaining temperature and pore pressure of the entire simulation within the range considered by TOUGH2-EOS2 equation of state modules (which does not extend to super-critical fluids).

Atmospheric boundary conditions (P=0.101325 MPa and T=20 ∘C) are prescribed on the top of the domain z=0; lateral boundaries are assumed to be impervious and adiabatic. A heat flux of 0.195 W m-2 is assigned at the impervious bottom boundary during the entire simulation, specified in order to sustain a temperature gradient comparable to that estimated for CF – ∼130 ∘C km-1 .

Cell centres of the finite-volume mesh used in TOUGH2 for the 1.5 km depth domain are shown in Fig. a. Hydrological parameters (permeability, density and porosity) are obtained from averaging drilling data for AGIP's report , while the thermal properties of the rocks (thermal conductivity and specific heat) are derived from and (see Table ).

Although all parameter values are specified according to measured data at CF, the rock permeability may vary over several orders of magnitude, and this variation may substantially influence the fluid flow and heat transport in all the simulations. explore the sensitivity of the hydrological system to matrix (caldera fill) and fracture hydrological properties. However, exploration of a wide range of possible hydrological values goes behind the scope of this paper.

Fault zones are assigned the hydrothermal properties of the surrounding rock, except for the permeability, which is represented by an anisotropic tensor K of Eq. (): K=kr00kz, where kr and kz are the radial and vertical permeabilities, respectively. While kr equals the isotropic permeability of the surrounding rock (set at 5×10-15 and 10-15 m2 for layers A and B), a higher value of kz is chosen for those cells of the TOUGH2 finite-volume mesh whose centre falls into the core (kz=10-12 m2) and damage (kz=10-13 m-2) zones of the faults.

In order to simulate the fumarole activities at the centre of the domain, a central conduit with a higher permeability is placed at the centre of the domain and represented by a vertical cylinder with a radius of 200 m. A transition zone is specified between this conduit and the bulk of the caldera fill which has intermediate hydrothermal properties, as in previous simulations of and (Table ).

The elastic response of a porous medium to pore pressure and temperature changes associated with the circulation of hot fluids is modelled by linear thermo-poroelasticity theory. The thermo-poroelastic effects are taken into account by including the pore pressure and temperature terms in Hooke's law : ϵ=12μσ-ν1+νtr(σ)I+131Kd-1KsΔP+βΔTI, where ϵ and σ are the strain and stress tensors, respectively, μ is the rigidity modulus, ν the Poisson's ratio, tr(σ)=σxx+σyy+σzz the trace of σ, I the identity tensor, ΔP and ΔT are pore pressure and temperature changes, respectively, Kd is the bulk modulus in drained conditions, Ks is the bulk modulus of the solid constituent , and β is the volumetric thermal expansion coefficient of the solid matrix. Since we assume that deformations occur slowly, the governing equations are represented by the equations of equilibrium ∇×σ=0 with σ obtained by the inversion of Eq. (), leading to the following set of Cauchy–Navier equations : ∇⋅σ=0,σ=2μν1-2νtr(ϵ)I+2μϵ-αΔPI-KdβΔTI,ϵ=12∇u+(∇u)T, where α=1-Kd/Ks is the Biot–Willis coefficient and u is the deformation vector, and where we have used the relation Kd=2μ(1+ν)3(1-2ν). The third Eq. () represents the linear approximation of the strain–deformation relation for small deformations.

Free-stress boundary conditions σ⋅n=0 are prescribed on the surface, where n is the outward unit vector orthogonal to the surface. Unlike the domain for the hydrothermal model (Fig. ), the computational domain of the problem defined by Eq. () is unbounded in the radial r and vertical z downward directions, and a vanishing displacement is assigned at infinite distance: lim⁡r→∞u=lim⁡z→-∞u=0. Since we assume that the problem is axi-symmetric, we solve the 2-D axi-symmetric version of Eq. () in the unknown u=(u,v), where u and v are the radial and vertical deformation, respectively.

The unbounded domain is discretised by a quasi-uniform grid (Fig. b), whose resolution is finest close to the axis of symmetry and smoothly decreases toward infinity . In this way artefacts introduced by artificial truncation of the domain are avoided. Equation () is discretised and solved by extending the finite-difference numerical method proposed by  for Cauchy–Navier equations to thermo-poroelasticity equations.

Heterogeneities in mechanical properties (μ and ν) are taken into account. In particular, the rigidity modulus μ for each layer of Fig.  is derived from seismic p wave velocity vp data by the application of the formula of : μ=Vp2ρ(1-2ν)2(1-ν). Density values of the porous medium ρ are derived from Vp by the Brocher equation : ρ=1.6612Vp-0.4721Vp2+0.067Vp3-0.0043Vp4+0.000106Vp5.

An appropriate value of the Poisson ratio for volcanic regions of ν=0.25 is specified for the whole domain, except in the damage and core zones of the fault areas, where higher values (0.30 and 0.40, respectively) are specified on the basis of the nature of the rock . Rigidity values are reduced in the fault zones (μ=0.385 and 0.0357 GPa for the fault core and damage zone, respectively, which correspond to E=109 and 108, where E is the Young modulus). The volumetric thermal expansion coefficient is β=10-5K-1 after and . All values are reported in Table .

In order to separate the contribution of pore pressure to the total ground deformation from thermal effects, we solve two different sets of differential equations for the mechanical simulation: ∇⋅σT=0σT=σ̃+3βΔTσT⋅n=0 on Γ,∇⋅σP=0σP=σ̃+αΔPσP⋅n=0 on Γ, where σ̃=λtr(ϵ)I+2μϵ is the elastic stress tensor (i.e. without taking into account pore pressure and temperature contributions). Let uT and uP be the solutions of the two problems Eq. (), respectively. As a result of the linearity of the stress–strain relationship σ̃(u) and the divergence operator, the total ground deformation u can be expressed as the sum of the solutions to the two problems (namely u=uT+uP). In practice, it is sufficient to solve only one of the problems Eq. () and obtain the other solution by difference.

Gravity changes Δg are computed by solving the following boundary value problem for the gravitational potential ϕg : ∇2ϕg=-4πGΔρ,ϕg=0 at infinity,Δg=-∂ϕg∂z where G is the gravitational constant and Δρ is the density distribution change. The finite-difference method presented by is applied to solve the problem Eq. () on an infinite domain, using the coordinate transformation method .

Scenario I:

central injection at the base of the conduit (radius of 200 m) at the same temperature but at an increased rate with respect to that used for the steady-state quiescent solution (see Table );

At each time step of the unrest simulations we evaluate changes in pressure (ΔP=P-P0), temperature (ΔT=T-T0) and density (Δρ=ρ-ρ0), relative to initial conditions (subscript 0) and use these to compute ground deformation and gravity changes at the surface by Eqs. () and (). Density change is in practice computed as Δρ=ϕ∑β(ρβSβ-ρβ0Sβ0), where subscript β=l or g refers to the liquid or gas phase, respectively. We observe that Δρ is mainly driven by the gas saturation change, since densities of liquid and gas do not significantly change during the simulation. For this reason we plot the gas saturation change ΔSg=Sg-Sg0 rather than Δρ (Figs.  and ).

Analysing Scenario I (Fig. ), we observe that after 6 months of simulated unrest the zone of perturbed pore pressure has already approached the surface (z=-500 m) at the central conduit, with a maximum ΔP of about 4 MPa observed at the injection point. Temperature and gas saturation changes remain small and confined to the areas surrounding the injection point. The maximum ΔP of the entire simulation (about 5 Mpa) is observed at 3 years. At the same time, gas saturation changes reach the shallower layer (z=-400 m), while no changes in temperature are apparent. After 3 years ΔP decreases; at 10 years hot fluid (warmed by up to ΔT∼100 ∘C) rises up to about z=-1000 m and the gas region extends up to the surface. At 100 years, which is the end of the simulation, ΔP continues to decrease towards a new steady state, while ΔT keeps increasing (with a maximum ΔT∼130 ∘C), extending the central plume laterally by up to 250 m. Gas saturation changes approach the steady-state solution, and a single-phase gas region is forming close to the surface. We do not observe any significant variation in pore pressure, temperature or density close to the faults, where the values remain the same as the initial condition.

The location of regions where significant changes in pore pressure, temperature and density are observed depends on the background simulation. During the steady-state simulation, fluids are injected only at the centre of the model, and thus a two-phase plume develops only in the central conduit, with complete liquid saturation within the fault zones. During the unrest the increased rate of injection at the conduit leads to an increase in pore pressure most markedly at depth within the conduit, but increases in temperature and gas saturation occur at the border of the expanding two-phase plume.

If we vary injection rates in Scenario I (Fig. ), the amplitudes of ΔP, ΔT and Δρ are strongly (nonlinearly) affected. Regardless of the injection rate, ΔT continues to increase for the entire simulation (100 years), while ΔP peaks at ∼ 3 years. Therefore, the timescale for pore pressure changes to reach the maximum value does not significantly depend on the injection rate. In particular, the maximum ΔP is 2.15 for Unrest ×0.5, 9.85 for Unrest ×2 and 14.1 MPa for Unrest ×3 (all at t=3 years). The maximum ΔT is observed at the final simulation time (t=100 years) and is 92.1 for Unrest ×0.5, 171 for Unrest ×2 and 181 ∘C for Unrest ×3. The extent of the central plume increases for the entire simulation: after t=100 years the plume has extended laterally by up to 200 m for Unrest ×0.5, 450 for Unrest ×2 and 550 m for Unrest ×3.

In contrast to Scenario I, in Scenarios II and III injection at the base of the faults induces a perturbation in pore pressure, temperature and density at the fault zones (mainly located on the hanging wall), while the behaviour at the central conduit is similar in all three scenarios (Fig. ). Due to the higher injection rates at the base of the faults, Scenario III shows more pronounced perturbations than Scenario II. Both faults behave similarly in Scenario III: the region with significant pore pressure change approaches the surface after 6 months (with a maximum ΔP of about 2.5 MPa), while temperature and gas saturation changes remain confined around injection points for up to 10 years. Similar to the central conduit, ΔP near the faults starts decreasing after 3 years towards a new steady state condition. For Scenario III, at t=100 years ΔT has reached 170 ∘C and extends up to 200 m from the faults, while a single-phase gas region has formed near the surface. Important differences however exist between Scenarios II and III. In Scenario II, maximum ΔP is about 1 for Fault A and 0.4 MPa for Fault B, and ΔT is about 100 for Fault A and 60 ∘C for Fault B, while gas saturation does not exceed 0.4 for either fault. Hence not only are there differences between the magnitudes of perturbations near Fault A and Fault B but also the time needed to observe the perturbations at the surface is greater than 100 years.

At each time step of the unrest simulations, changes in pore pressure and temperature are interpolated from the finite-volume mesh of the hydrological model to the finite-difference mesh of the mechanical model (the two meshes are represented in Fig. ) and fed into Eq. (). This is known as one-way coupling between hydrological and mechanical models, as used previously by a number of studies . It is a simplified approach compared with a fully coupled model that also takes into account the influence of stress and strain on permeability and porosity during the simulation .

In Scenario I (Fig. ), for the first 10 years of unrest the uplift is maximum at the centre of the domain and decays radially. Vertical and horizontal displacements reflect the Mogi solution for a small spherical source . The profile obtained at t=100 years does not reflect a Mogi solution and presents a maximum total uplift of 21 cm at r=300 m, decaying rapidly as radial distance increases. Temporal evolution of the ground deformation at the centre of the domain throughout 100 years of unrest (Fig. ) indicates that the contribution of thermal effects (vT) to the total ground deformation is almost negligible with respect to the pore pressure contribution (vP) during the first years of the unrest, but increases in time and eventually becomes dominant. In particular, for lower injection rates (unrest ×0.5 of Table ) the vertical deformation due to thermal effects only exceeds the pore pressure contribution after more than 100 years, while for higher injection rates (unrest ×2 and ×3 of Table ) it takes less than 50 years. The amplitude of the deformation is nonlinearly dependent on the injection rate, while the timescale of the first local maximum is largely independent of injection rate, occurring after ∼ 3 years of unrest in all simulations. Vertical displacement due to pore pressure effects (vP) increases very rapidly with the onset of unrest. After this strong initial pressurisation (lasting about 3 years), vertical deformation starts decreasing towards a new steady-state value. Thermal effects strongly affect the long-term behaviour and their importance increases with increasing injection rates. Consequently, the timing of the local minimum, prior to the thermally induced later monotonic increase, occurs earlier for higher injection rates. Although we show only the temporal variation of the vertical deformation at the centre of the model for Scenario I, a similar pattern is observed localised around both faults for Scenarios II and III.

In Scenario II (Fig. ) the deformation profile reflects the injection of fluids at the fault zones. Maximum vertical deformation is observed at the centre of the model and two local maxima correspond to the faults (Fig. a). Magnitude of peak displacements both horizontal and vertical reduces from centre to Fault A and from Fault A to Fault B, reflecting the different injection rates.

After about t=3 years the vertical deformation at the centre of the model reaches a temporary maximum (see solid line in Fig.  for Scenario I), then decreases toward a lower value (at about t=10 years) while deformation on faults continues to increase. At t=100 years the vertical displacement at the centre of the model increases again toward a steady-state solution (solid line of Fig. ), while deformation on faults decreases toward a lower value. We observe in Fig. c, d that the vertical deformation profile at t=100 years is almost exclusively attributable to thermal effects, which are negligible in the first years of the unrest simulation. Horizontal deformation (Fig. 9b) shows a Mogi-like pattern close to the central conduit , while two peaks are observed close to the fault zones. For both peaks the deformation profile is steeper on the side towards the centre of the domain due to the fault inclination (dip-angle smaller than 90∘, Fig. ), since the steeper deformation profile is always observed in the hanging wall.

We finally observe for all the plots that the deformation profile is relatively smooth above Fault A, while there is a sharp kink above Fault B, because such fault reaches the surface (Fig. ). Vertical deformation at the centre of the domain throughout the entire simulation (100 years) is practically the same as for Scenario I (Fig. ), indicating that pore pressure and temperature changes along the faults do not significantly affect the mechanical behaviour of the fumarole.

In Scenario III (Fig. ) vertical deformation on faults is greater than at the centre of the model (up to t=10 years). Although pore pressure change at the faults shows a lower value compared with that close to the injection point, it is more vertically extensive (Fig. ) due to the lower vertical permeability of the central conduit compared to the faults, causing a larger uplift. Vertical deformation at the axis of symmetry is also slightly amplified (by the mechanical influence of faults) with respect to the one observed in Scenarios I and II.

Except for faults, the mechanical heterogeneities described so far depend only on depth, resulting in a 1-D heterogeneity structure. A complex mechanical structure for CF could be used, taking into account the lateral variation in mechanical properties to reflect differences between the two caldera infills, as proposed in the models of , based on tomographic studies of . Some simulations (not shown) have been performed with different matrix properties around faults, maintaining the same mechanical properties for fault core and damage zones. No significant differences were obtained close to fault areas, highlighting that the amount of deformation is mainly driven by the values of μ and ν assigned to the fault core and damage zones, especially when these values are much smaller than those assigned to the surrounding area (Table ). A sensitivity analysis of the rigidity modulus on faults is provided below.

The solution of Eq. () is the gravity change Δg=g-g0, where g0 and g are the gravity distributions observed at the initial condition and at a fixed time of unrest, respectively. Evaluating Δg at a particular point of the surface (r,z=0) means that also g and g0 refer to the same geometric location (r,z=0). Gravity change measured in the field Δg=g̃-g0 is actually influenced by ground deformation, since g̃ is measured at the same material point of g0, but at a different geometric (translated) point (r,z=0)+u(r,z=0), which takes into account the absolute movement of the gravimetry associated with the ground displacement. The value Δg=g-g0 is often referred in literature as residual gravity , since it does not include the gravity change associated with the ground deformation .

Gravity changes computed at the centre of the model (r=0,z=0) for different injection rates (Table ) are reported in Fig. . After a transient increase (maximum 16.1 µGal for the Unrest ×1 model) over the first months of unrest (Fig. b), gravity changes become negative and decrease monotonically towards a steady-state value, although this is not reached within 100 years (Fig. a). The modulus of the gravity changes is more pronounced for higher injection rates, with a maximum increase after 0.5 years of 33.9 µGal for the Unrest ×3 model and a much larger negative value. The behaviour is, however, nonlinear at a fixed time with respect to injection rates, due to both the change of molar ratio from the steady state to the unrest phase and the nonlinearity of the hydrothermal model. The increase in injection rate causes only a minor increase in the time needed to change sign (from positive to negative, Fig. b).

Figure  compares the gravity changes computed at the surface for different simulation times and three injection scenarios (D–I) and the associated vertical gravity gradient (A–C), computed as Δg/v, where v is the vertical deformation computed in Sect. 4.3. Again, this is usually referred as the residual gravity gradient, since it does not take into account the free-air correction . Data are plotted for up to 20 years of unrest, since after a long period of unrest the gravity gradient becomes unstable in most of the domain due to very small vertical deformation far from the faults and central conduit. Maximum values in modulus are observed at a radial distance of ∼ 570 m at the boundary of the two-phase plume, and are almost equal for the three scenarios. However, local maxima of the modulus of the signals are present at the faults for Scenarios II and III. The absolute value is significantly higher for Scenario III, reflecting the higher mass flux.

The sign of the vertical gravity gradient is the same as that of the gravity changes, since the sign of ground deformation is almost always positive (i.e. uplift) in all the simulations. The pattern observed close to the axis of symmetry is similar to that for the gravity changes, presenting a local extreme at the border of the plume. In Scenarios II and III, the gravity gradient presents local extremes on the faults (most evident for t>10 years) because of local extremes in both gravity changes and vertical deformation (see Appendix). In Scenario II the local extreme on Fault A is a minimum, since the wavelength of the gravity change profile on Fault A is lower than that of the vertical displacement, after a proper normalisation (see Appendix for more details). Local extreme on Fault B is a maximum, since Δg has a greater wavelength than v (see, for instance, the Δg and v profiles at t=20 years in Fig. h). In Scenario III both extremes are minima, since the wavelength of the gravity change profile on the faults is lower than that of the vertical displacement, after a proper normalisation (see Appendix for more details). The value observed at the faults is much greater (due to greater gravity changes associated with greater injection rates).