A hydrogeophysical simulator for fluid and mechanical processes in volcanic areas
 Armando Coco^{1, 2}Email author,
 Gilda Currenti^{3},
 Joachim Gottsmann^{2},
 Giovanni Russo^{4} and
 Ciro Del Negro^{3}
https://doi.org/10.1186/s133620160020x
© Coco et al. 2016
Received: 7 January 2016
Accepted: 14 April 2016
Published: 8 June 2016
Abstract
Efficient and accurate hydrothermal and mechanical mathematical models in porous media constitute a fundamental tool for improving the understanding of the subsurface dynamics in volcanic areas. We propose a finitedifference ghostpoint method for the numerical solution of thermoporoelastic and gravity change equations. The main aim of this work is to study how the thermoporoelastic solutions vary in a realistic description of a specific volcanic region, focusing on the topography and the heterogeneous structure of Campi Flegrei (CF) caldera (Italy). Our numerical approach provides the opportunity to explore different model configurations that cannot be taken into account using standard analytical models. Since the physics of the investigated hydrothermal system is similar to any saturated reservoir, such as oil fields or CO_{2} reservoirs produced by sequestration, the model is generally applicable to the monitoring and interpretation of both deformation and gravity changes induced by other geophysical hazards that pose a risk to human activity.
Keywords
poroelasticity finite difference fluid flow1 Introduction
Volcano geophysics focuses on integrating data from different monitoring techniques and developing numerical and physical models to explain the observations. The geophysical observations collected in volcanic areas are the surface expressions of processes that occur deeply within the volcanic edifice. Integration of different geophysical observations and mathematical models enables to identify renewed volcanic activities, forecast eruptions, and assess related hazards [1]. Unrest periods are defined as variations in the geophysical and geochemical state of the volcanic system with respect to the background behaviour. Usually, geophysical changes observed during unrest periods are modelled in terms of volume and pressure changes in a magma chamber embedded within an elastic medium [2–4]. Indeed, this approach often appears at odds with the complex processes accompanying volcanic unrest. Particularly, the interplay between magma chambers and hydrothermal systems may result in the heating and pressurization of hydrothermal fluids, which in turn induce ground deformation and variations in the rock and fluid properties [5–14]. A quantitative evaluation of this interaction is fundamental for a correct hazard assessment in volcanic areas.
A thermoporoelastic numerical model is here proposed to jointly evaluate ground deformation and gravity changes caused by hydrothermal fluid circulation in complex media with surface topography and mechanical heterogeneities in a 2D axis symmetric formulation.
Although the model is applicable to a generic volcanic system, in this paper we set up the model parametrisation on the Campi Flegrei caldera (CFc), a volcanic area situated to the west of Naples. After a long period of quiescence, two main uplift episodes (196972 and 198284) highlighted a reawakening phase of the CFc without culminating in an eruption [15]. The total amount of uplift of 3.5 m induced the evacuation of about \(40\mbox{,}000\) people from the town of Pozzuoli and surrounding areas. A slow subsidence followed the second uplift episode, periodically interrupted by miniuplifts. Since 2006 CFc started uplifting again, with a particularly increased rate from 2011 [16, 17].
To reduce the uncertainty related to hydrothermal activities, significant complexities must be accounted when a mathematical model is adopted, such as real topography and mechanical heterogeneities, which are indeed neglected by most of existing models [12, 14, 18, 19]. In addition, highly efficient and accurate numerical solvers are required when a large number of simulation runs must be executed for a single simulation scenario, such as for optimization purposes related to geophysical data inversion or when the thermoporoelastic response must be calculated at each time step in a coupled hydrothermal/mechanical model [20].
The thermoporoelastic numerical method proposed in this paper is second order accurate and based on a finitedifference ghostpoint discretization for complex geometries successfully adopted in [21] to solve elliptic equations and in [22] for elastostatic 2D problems with planestrain assumptions. Here, the methods proposed in [21, 22] are extended to account for thermal expansion and pore pressure effects (caused by the perturbation of the hydrothermal system) in a 2D axis symmetric framework. Although a 3D model would be suitable to represent a more accurate scenario, the axis symmetric assumption is a reasonable approximation for representing caldera systems and volcano edifices in general, which are usually characterized by radial structures.
In order to avoid artifacts introduced by finite truncation of the domain, a coordinate transformation method is adopted in order to prescribe vanishing solutions at infinite distances maintaining the second order accuracy [22]. An extension of the multigrid solver described in [21, 22] is adopted as well.
Hydrothermal activity is simulated by TOUGH2, a well known multiphase multicomponent software for fluid flow and heat transfer in porous media [23]. The hydrothermal system is perturbed by injecting fluids of magmatic origin at the base of a central conduit for a prolonged period (until steadystate conditions), simulating the fumarolic activity at La Solfatara [24]. Following unrest periods are modelled by an increased injection rate. Variations in relevant geophysical signals (pore pressure, temperature and density) between the initial condition (steadystate) and unrest period are computed at each time step and fed into the thermoporoelastic model (oneway coupling) to compute associated ground deformations and gravity changes. A similar oneway approach has been implemented at CF in previous works (e.g., [12, 14, 19]).
The model proposed in this paper is widely applicable to a large number of relevant scientific and engineering problems for addressing subsurface flow and transport problems, such as geological carbon sequestration, nuclear waste disposal, energy production from geothermal, oil and gas reservoirs as well as methane hydrate deposits, environmental remediation, vadose zone hydrology. The coupling with the suite of TOUGH simulators makes the model suitable for simulating and interpreting geophysical changes induced by a wide range of alterations in the subsurface flows, such as CO_{2} sequestration and geothermal exploitation. However, the main scope of this paper is related to the methodology rather than the application to case studies and comparison with real data, and then the focus is restricted to the simulation of ground deformation and gravity changes in a generic volcanic area whose model parametrisation is set up on the Campi Flegrei caldera.
2 Hydrogeophysical model
The mathematical model is based on the governing equations of the thermoporoelasticity theory (Mechanical model, Section 2.1.1), which describes the elastic response of a porous medium to the propagation of hot fluid through pores (Hydrothermal model, Section 2.4) by the effects of variations in pore pressure and temperature. Alterations in the gravity field associated to fluid density variations are evaluated by the model described in Section 2.1.2.
2.1 Thermoporoelasticity model
2.1.1 Mechanical model
A suitable set of boundary conditions is posed to close the mathematical problem, namely zero displacements at infinity and stressfree boundary conditions \(\boldsymbol {\sigma} \cdot \boldsymbol {n}_{s}=0\) on the ground surface \(\Gamma_{M} = \{ z=f_{\mathrm{TOP}}(x,y) \}\).
Analyzing the stress tensor (second equation of (1)), we observe that two terms are added to the elastic stress tensor of the general Hooke’s law: the ΔP porepressure contribution from poroelasticity theory through the \(\alpha_{\mathrm{BW}}\) BiotWillis coefficient and the ΔT temperature contribution from thermoelasticity theory through the linear thermal expansion coefficient α. Both ΔP and ΔT refer to perturbations with respect to a previous equilibrium state.
2.1.2 Gravity model
2.2 Axis symmetric formulation
2.3 Numerical method
Pressure, temperature and density changes induced by fluid circulation are computed using the TOUGH2 numerical code [23]. Starting from these quantities, ground deformation and gravity changes are solved through Equations (8) and (12) using a finitedifference ghostpoint method developed in the context of elliptic problems [21] and recently extended to solve elastostatic equations in 2D plane strain configuration for unbounded domains [22]. Here, we extend the methods developed in [21, 22] to solve the axis symmetric formulations of Section 2.2.
The method consists of three stages: firstly, the unbounded domain problem is transformed in a bounded one by the coordinate transformation method. Secondly, the new set of equations and boundary conditions in the bounded domain is discretized by a finitedifference ghostpoint approach. Finally, the discrete equations are solved by a proper geometric multigrid technique.
2.3.1 Coordinate transformation method
By requirements (i) and (ii) the numerical method will achieve second order accuracy in a natural way, unlike the case when artificial truncation of the domain is used to assign zero displacement and zero potential boundary conditions.
2.3.2 Finitedifference ghostpoint method

First we compute the projection B of G to the boundary by the levelset function. In details, we compute the outward unit normal vector \(\boldsymbol {n}_{G}\) by discretizing Eq. (16) on G using a standard finitedifference scheme; then we apply the bisection method to solve the 1D nonlinear equation \(\phi(B)=0\) on the segment between points G and \(G_{1} = G\sqrt{2}\, h \, \boldsymbol {n}_{G}\), with h being the spatial step.

Then, we identify a ninepoint stencil made by internal and ghost points, containing G and whose convex hull (the smallest convex set containing the stencil) contains B. In Figure 3 the ninepoint stencil is represented by red circles. Although there are different stencils satisfying these conditions, an appropriate choice will often avoid numerical instabilities in the solution (see [22] for more details and examples).

Stressfree boundary conditions (9) are then imposed on B, providing then the following two equations of the linear system for \(u_{r}^{G}\) and \(u_{z}^{G}\):$$ \textstyle\begin{cases} ( \tilde{\sigma}_{rr} \frac{\partial\tilde{\phi }}{\partial r} +\tilde{\sigma}_{rz} \frac{\partial\tilde{\phi }}{\partial z} )\vert _{B} = 0 ,\\ ( \tilde{\sigma}_{rz} \frac{\partial\tilde{\phi }}{\partial r} +\tilde{\sigma}_{zz} \frac{\partial\tilde{\phi }}{\partial z} )\vert _{B} = 0, \end{cases} $$(17)
2.3.3 Multigrid solver
The discrete linear systems involving the unknowns \(\{ u_{r}^{P}, u_{z}^{P} \}\), with P varying within the sets of internal and ghost points, is solved by an efficient geometric multigrid solver, which is an extension of the multigrid approach proposed in [22] (to which the reader is referred for more details).
2.4 Hydrothermal model
Notation  Description  Status 

ϕ(x⃗)  Porosity  Assigned 
\(\rho_{r}(\vec{x})\)  Density of the rock  Assigned 
\(C_{r}(\vec{x})\)  Specific heat of the rock  Assigned 
λ(x⃗)  Thermal conductivity of the rock  Assigned 
K(x⃗)  Permeability of the rock  Assigned 
ĝ  Gravity acceleration  Assigned 
\(\rho_{\beta}(\vec{x},t)\)  Density of the phase β  Unknown 
\(S_{\beta}(\vec{x},t)\)  Saturation of the phase β  Unknown 
\(\chi^{i}_{\beta}(\vec{x},t)\)  Mass fraction of the ith component in the phase β  Unknown 
\(e_{\beta}(\vec{x},t)\)  Internal energy of the phase β  Unknown 
\(h_{\beta}(\vec{x},t)\)  Enthalpy of the phase β  Unknown 
\(k_{\beta}(\vec{x},t)\)  Relative permeability of the phase β  Unknown 
\(\eta_{\beta}(\vec{x},t)\)  Viscosity of the phase β  Unknown 
\(P_{\beta}(\vec{x},t)\)  Pressure of the phase β  Unknown 
T(x⃗,t)  Temperature  Unknown 
We call \(\Omega_{H}\subset \mathbb {R}^{3}\) the spatial domain in which we solve Eq. (18) (the domain of the hydrothermal model). Due to the axissymmetric configuration of the problem, the domain can be expressed in terms of the radial distance \(r=\sqrt{x^{2}+y^{2}}\) and the vertical variable z, as \(\Omega_{H} = \{ 0 \leq r \leq R, z_{\mathrm{min}} \leq z < f_{\mathrm{TOP}}(r) \}\), where \(f_{\mathrm{TOP}}(r)\) represents the topography function (see Figure 1). Since TOUGH2 does not account for infinite domains, \(\Omega_{H}\) differs from \(\Omega_{M}\) and is truncated at a radial distance of \(r=R\) and a depth of \(z=z_{\mathrm{min}}\). On the portion of the ground surface within a radial distance \(r \leq R\), \(\Gamma_{H}\), atmospheric (Dirichlet) boundary conditions for pressure and temperature are prescribed.
3 Simulation of unrest at Campi Flegrei
In the thermoporoelastic and gravity models the domain extends toward infinity (Figure 1), warranting zero displacement and potential. The computational grid is vertexcentered and has a maximum resolution of \(\Delta r = \Delta z = 38.76\) m at \(r=z=0\).
Firstly, TOUGH2 is run to evaluate temperature, pressure and density variations with respect to their initial distributions, which, then, are fed into the thermoporoelastic solver to compute the deformation and gravity changes. Due to the different grids adopted, these quantities are interpolated from the TOUGH2 grid to the thermoporoelastic solver nodes.
In all the TOUGH2 simulations, atmospheric boundary conditions \(P=0.1\) MPa and \(T=20^{\circ}\)C are prescribed on the surface \(\Gamma_{H} = \{ z=f_{\mathrm{TOP}}(r) \}\), while adiabatic and impervious conditions are set on the remaining boundaries \(\{r=R \}\) and \(\{ z=z_{\mathrm{min}} \}\), except at the inlet near the symmetry axis, where inflow boundary conditions are prescribed.
Rock properties used in calculations of the hydrothermal model
Density  2000 kg/m^{3} 
Permeability  10^{−14} m^{2} 
Conductivity  2.8 W m^{−1} K^{−1} 
Porosity  0.2 
Specific heat  1000 J kg^{−1} K^{−1} 
Elastic properties of the medium
Region  Rigidity [GPa]  Poisson ratio 

L1  4  0.25 
C1  2.3  0.33 
L2  5  0.25 
C2  3.6  0.33 
L3  6.5  0.25 
C3  5  0.33 
L4  20  0.25 
Although hydrothermal fluids are not pure substances, but generally mixtures of several mass components or chemical species, the dominant fluid component is usually water, and it is often reasonable to ignore other components. However, in volcanic regions the aqueous phase generally contains also dissolved incondensable gases, such as CO_{2}. To investigate the effect of a CO_{2} component in water, two different scenarios are investigated, simulating: (1) a pure water injection using the TOUGH2/EOS1 module, which models a water system in its liquid, vapor, and twophase states; (2) a mixture of water and carbon dioxide injection using the TOUGH2/EOS2 module, which accounts for the nonideal behavior of gaseous CO_{2} and dissolution of CO_{2} in the aqueous phase.
We observe that the depth of \(\Omega_{H}\) is chosen in such a way the focus of the simulations is on the shallower hydrothermal activity, in order to maintain temperature and pore pressure within the range considered by TOUGH2, which does not take into account supercritical fluids. Therefore, in all the simulations of this paper we investigate only subcritical fluids.
Observe that we do not aim to simulate the subsidence period following the main unrest phase, and therefore the simulation is confined to the first years of unrest.
3.1 Injection rates
In the following sections two scenarios will be studies: pure water injection and water and carbon dioxide injection. The injection rates adopted in the scenario of water and carbon dioxide injection are obtained from previous works according to geochemical data collected at the CFc and providing a good matching to observed data [12, 18, 19]. The injection rates adopted for the scenario of pure water injection are chosen in such a way the total mass of fluid injected is the same for both scenarios.
3.2 Pure water injection
In this first scenario the fluid is composed of pure water, whose properties and phase transitions are calculated based on the thermodynamic conditions, according to the steam table equation [32]. Initial conditions are obtained reaching a steadystate solution by simulating a 5 thousand year long phase with a deep injection of hot water with a constant flux rate of 39.4 kg/s at temperature of 350°C. The fluid is injected in a 150 m wide inlet located at the bottom of the domain around the symmetry axis. This extent has been estimated from the area affected by fumaroles activity [18].
The deformations are accompained by significant gravity changes induced by the fluid density variations. The gravity changes computed at the axis of symmetry on the ground surface (Figure 6) increase almost linearly through time and reache about 135 μGal after 3 years of constant injection. The ratio \(\Delta g / \Delta h\) increases monotonously reaching a value of about 27 μGal/cm. These results are in fully agreement with those obtained by Coco et al. [33] using the HYDROTHERM flow code [23].
3.3 Water and carbone dioxide injection
In this second scenario the fluid is a mixture of water and carbon dioxide. The initial conditions are achived by injecting for 5 thousand years a fluid at temperature of 350°C composed of a water component with a flux rate of 27.8 kg/s and a CO_{2} component with a flux rate of 11.6 kg/s. Then, the unrest is simulated increasing the flux rates to 70.6 kg/s for the water and to 69.4 kg/s for the CO_{2}. The distributions of saturation and pressure, temperature, and density variations after 3 years from the unrest are displayed in Figure 10.
The distributions are quite different with respect to the first scenario, when only water is injected (Figure 6). The pressure changes affect a wider area, whereas the temperature changes are almost similar to the previous scenario. The saturation shows two areas of gassaturated zones, one at the inlet and the other one below the ground surface. The fluid density variations are almost negative at the edge of the rising plume, where liquid phase is substituted by gas phase. The pressure and temperature changes greatly affect the ground deformation as shown in Figure 11. These behaviors are in agreement with the results obtained in [12].
4 Conclusion
A finitedifference ghostpoint method for solving the elastostatic and Poisson equations on an arbitrary unbounded domain has been presented. The proposed strategy, which adopts the coordinate transformation method, has been applied to thermoporoelastic models to evaluate deformation and gravity changes in volcanic areas. The method has been coupled with the TOUGH2 code to investigate the role of pressure, temperature and density changes on the geophysical observables. The results from two scenarios have been compared in order to explore the effect of the presence of carbon dioxide in the fluid mixture. Although the total flux rate is similar in both scenarios, the presence of CO_{2} strongly alter the solutions. Particularly, the injection of the CO_{2} engenders an enhancement in the deformation field and perturbs the temporal evolution of the gravity changes. This last feature could be a useful signature to constrain the relative ratio between water and carbon dioxide content. It turns out that the ratio between gravity and vertical deformation decreases for higher ratios of \({\mathrm{CO}}_{2}/{\mathrm{H}}_{2}{\mathrm{O}}\). The comparisons between gravity and deformation may help to discriminate between injection with or without carbon dioxide content and to provide inferences on the nature of the source. Since unrest periods are accompanied by increases of the \({\mathrm{CO}}_{2}/{\mathrm{H}}_{2}{\mathrm{O}}\) ratio, a decrease in the ratio between gravity and vertical deformation could help in detecting the onset of unrest phases. Moreover, our findings seem able to justify what observed in some volcanic regions, where major gravity changes appear without any significant deformation. In many volcanoes worldwide, there are evidences that the ratio between gravity and height changes is far beyond what could be predicted by simple models, in which volume and pressure changes in a magma chamber are considered. When significant gravity changes occur without any significant deformation, or vice versa, it is often difficult, if not impossible, to jointly explain the observations using the popular Mogi model. Our results may provide an alternative explanation to the observations and help in resolving the controversy between geodetic and gravity observations as a volcano moves from rest to unrest state.
However, there are some limitations of the model presented in this paper that must be considered. The shallow hydrothermal system is only 1.5 km deep, while a deeper model is more realistic to represent volcanic areas [7, 17]. The shallow injection depth considered in this paper is constrained by the range of applicability of TOUGH2, which does not account for supercritical fluids. Some recent models investigated the influence of supercritical fluids to the results, allowing to consider deeper domains [35]. The 2D axisymmetric representation may not be able to describe the complex 3D network that characterises caldera systems. However, to a first approximation calderas present a radial structure and then can be approximated by the axis symmetric assumption. The oneway coupling between the hydrothermal model and the thermoporoelastic model represents a reasonable simplification when a short period of unrest is simulated, but it is not adequate for simulations of longer processes, where stress and strain alterations may induce a significant variation in the relevant hydrological parameters (permeability, porosity), modifying the long term processes of fluid flows in the porous medium and then the associated geophysical signals [36, 37].
The effects of these limitations may be reduced by considering a more realistic fully coupled 3D model that accounts for supercritical fluids. The implementation of a more effective simulator is part of our future investigations, with the aim to extend the simulation to more realistic cases of multicomponent fluids.
Declarations
Acknowledgements
This work has been supported by the VUELCO and MEDSUV projects, which are funded by the EC FP7 under contracts #282759 and #308665.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Del Negro C, Currenti G, Solaro G, Greco F, Pepe A, Napoli R, Pepe S, Casu F, Sansosti E. Capturing the fingerprint of Etna volcano activity in gravity and satellite radar data. Sci Rep. 2013;3:3089. doi:10.1038/srep03089. Google Scholar
 Bonafede M, Dragoni M, Quareni F. Displacement and stress fields produced by a centre of dilatation and by a pressure source in a viscoelastic halfspace: application to the study of ground deformation and seismic activity at Campi Flegrei, Italy. Geophys J R Astron Soc. 1986;87:45585. View ArticleGoogle Scholar
 Bianchi R, Coradini A, Federico C, Giberti G, Lanciano P, Pozzi JP, Sartoris G, Scandone R. Modeling of surface deformation in volcanic areas: the 19701972 and 19821984 crises of Campi Flegrei, Italy. J Geophys Res. 1987;92(B13):1413950. View ArticleGoogle Scholar
 De Natale G, Pingue F, Allard P, Zollo A. Geophysical and geochemical modelling of the 19821984 unrest phenomena at Campi Flegrei caldera (southern Italy). J Volcanol Geotherm Res. 1991;48:199222. View ArticleGoogle Scholar
 Dzurisin D, Savage JC, Fournier RO. Recent crustal subsidence at Yellowstone caldera, Wyoming. Bull Volcanol. 1990;52(4):24770. View ArticleGoogle Scholar
 Watson IM, Oppenheimer C, Voight B, Francis PW, Clarke A, Stix J, Miller A, Pyle DM, Burton MR, Young SR, et al.. The relationship between degassing and ground deformation at Soufriere Hills Volcano, Montserrat. J Volcanol Geotherm Res. 2000;98(1):11726. View ArticleGoogle Scholar
 Gottsmann J, Camacho AG, Tiampo KF, Fernández J. Spatiotemporal variations in vertical gravity gradients at the Campi Flegrei caldera (Italy): a case for source multiplicity during unrest? Geophys J Int. 2006;167:108996. View ArticleGoogle Scholar
 Chiodini G, Vilardo G, Augusti V, Granieri D, Caliro S, Minopoli C, Terranova C. Thermal monitoring of hydrothermal activity by permanent infrared automatic stations: results obtained at Solfatara di Pozzuoli, Campi Flegrei (Italy). J Geophys Res. 2007;112:B12206. doi:10.1029/2007JB005140. View ArticleGoogle Scholar
 Gottsmann J, Camacho AG, Martí J, Wooller L, Fernández J, Garcia A, Rymer H. Shallow structure beneath the Central Volcanic Complex of Tenerife from new gravity data: implications for its evolution and recent reactivation. Phys Earth Planet Inter. 2008;168(3):21230. View ArticleGoogle Scholar
 Hutnak M, Hurwitz S, Ingebritsen SE, Hsieh PA. Numerical models of caldera deformation: effects of multiphase and multicomponent hydrothermal fluid flow. J Geophys Res. 2009;114:B04411. doi:10.1029/2008JB006151. View ArticleGoogle Scholar
 Ingebritsen SE, Geiger S, Hurwitz S, Driesner T. Numerical simulation of magmatic hydrothermal systems. Rev Geophys. 2010;48:RG1002. doi:10.1029/2009RG000287. View ArticleGoogle Scholar
 Rinaldi A, Todesco M, Bonafede M. Hydrothermal instability and ground displacement at the Campi Flegrei caldera. Phys Earth Planet Inter. 2010;178:15561. View ArticleGoogle Scholar
 Rinaldi A, Todesco M, Bonafede M, Vandemeulebrouck MJ, Revil A. Electrical conductivity, ground displacement, gravity changes, and gas flow at Solfatara crater (Campi Flegrei caldera, Italy): results from numerical modeling. J Volcanol Geotherm Res. 2011;207:93105. View ArticleGoogle Scholar
 Troiano A, Di Giuseppe MG, Petrillo Z, Troise C, De Natale G. Ground deformation at calderas driven by fluid injection: modelling unrest episodes at Campi Flegrei (Italy). Geophys J Int. 2011;187:83347. View ArticleGoogle Scholar
 Troise C, De Natale G, Pingue F, Tammaro U, De Martino P, Obrizzo F, Boschi E. A new uplift episode at Campi Flegrei caldera (southern Italy): implications for unrest interpretation and eruption hazard evaluation in Caldera Volcanism: analysis, modelling and response. Dev Volcanol. 2008;10:37592. View ArticleGoogle Scholar
 De Martino P, Tammaro U, Obrizzo F. GPS time series at Campi Flegrei caldera (20002013). Ann Geophys. 2014;57(2):S0213. doi:10.4401/ag6431. Google Scholar
 Amoruso A, Crescentini L, Sabbetta I, De Martino P, Obrizzo F, Tammaro U. Clues to the cause of the 20112013 Campi Flegrei caldera unrest, Italy, from continuous GPS data. Geophys Res Lett. 2014;41:30818. doi:10.1002/2014GL059539. View ArticleGoogle Scholar
 Chiodini G, Todesco M, Caliro S, Gaudio CD, Macedonio G, Russo M. Magma degassing as a trigger of bradyseismic events: the case of Phlegrean fields (Italy). Geophys Res Lett. 2003;30:14347. View ArticleGoogle Scholar
 Todesco M, Rinaldi AP, Bonafede M. Modeling of unrest signals in heterogeneous hydrothermal systems. J Geophys Res. 2010;115:B09213. doi:10.1029/2010JB007474. View ArticleGoogle Scholar
 Rutqvist J. Status of the TOUGHFLAC simulator and recent applications related to coupled fluid flow and crustal deformations. Comput Geosci. 2011;37:73950. doi:10.1016/j.cageo.2010.08.006. View ArticleGoogle Scholar
 Coco A, Russo G. Finitedifference ghostpoint multigrid methods on Cartesian grids for elliptic problems in arbitrary domains. J Comput Phys. 2013;241:464501. View ArticleGoogle Scholar
 Coco A, Currenti G, Del Negro C, Russo G. A second order finitedifference ghostpoint method for elasticity problems on unbounded domains with applications to volcanology. Commun Comput Phys. 2014;16:9831009. MathSciNetGoogle Scholar
 Pruess K, Oldenburg C, Moridis G. TOUGH2 User’s Guide, Version 2.0. Paper LBNL43134, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. 1999. Google Scholar
 Chiodini G, Frondini F, Cardellini C, Granieri D, Marini L, Ventura G. CO2 degassing and energy release at Solfatara volcano, Campi Flegrei, Italy. J Geophys Res. 2001;106:1621321. View ArticleGoogle Scholar
 Jaeger JC, Cook NGW, Zimmerman RW. Fundamentals of rock mechanics. 4th ed. Oxford: Blackwell Publishing; 2007. Google Scholar
 Fung YC. Foundations of solid mechanics. Englewood Cliffs: PrenticeHall; 1965. Google Scholar
 Currenti G. Numerical evidences enabling to reconcile gravity and height changes in volcanic areas. Geophys J Int. 2014; 197(1):16473. doi:10.1093/gji/ggt507. View ArticleGoogle Scholar
 Osher S, Fedkiw R. Level set methods and dynamic implicit surfaces. New York: Springer; 2002. MATHGoogle Scholar
 Brooks A, Corey AT. Hydraulic properties of porous media. Colorado State 599 University Hydrology. Paper No. 3, Fort Collins, Colorado, USA. 1964. Google Scholar
 Zollo A, Judenherc S, Auger E, D’Auria L, Virieux J, Capuano P, Chiarabba C, De Franco R, Makris J, Michelini A, Musacchio G. Evidence for the buried rim of Campi Flegrei caldera from 3d active seismic imaging. Geophys Res Lett. 2003;30:2002. doi:10.1029/2003GL018173. View ArticleGoogle Scholar
 Trasatti E, Giunchi C, Bonafede M. Structural and rheological constraints on source depth and overpressure estimates at the Campi Flegrei caldera, Italy. J Volcanol Geotherm Res. 2005;144:10518. View ArticleGoogle Scholar
 Committee IF. A formulation of the thermodynamic properties of ordinary water substance. ICF Secretariat. 1967. Google Scholar
 Coco A, Currenti G, Gottsmann J, Del Negro C, Russo G. Geophysical changes in hydrothermalvolcanic areas: a finitedifference ghostcell method to solve thermoporoelastic equations. Progress in Industrial Mathematics at ECMI 2014. In print. Google Scholar
 Todesco M, Berrino G. Modeling hydrothermal fluid circulation and gravity signals at the Phlegraean fields caldera. Earth Planet Sci Lett. 2005;240:32838. doi:10.1016/j.epsl.2005.09.016. View ArticleGoogle Scholar
 Afanasyev A, Costa A, Chiodini G. Investigation of hydrothermal activity at Campi Flegrei caldera using 3D numerical simulations: extension to high temperature processes. J Volcanol Geotherm Res. 2015;299:6877. View ArticleGoogle Scholar
 Neuzil CE. Hydromechanical coupling in geologic processes. Hydrogeol J. 2003;11(1):4183. View ArticleGoogle Scholar
 Rutqvist J, Wu YS, Tsang CF, Bodvarsson G. A modeling approach for analysis of coupled multiphase fluid flow, heat transfer, and deformation in fractured porous rock. Int J Rock Mech Min Sci Geomech Abstr. 2002;39:42942. View ArticleGoogle Scholar