Background and motivation
Although the model is applicable to any caldera system, the model
parametrisation in this paper is based on data available from the Campi
Flegrei (CF) caldera in Italy. The CF, situated to the west of Naples, formed
as a result of two structural collapses associated with the eruptions of the
Campanian Ignimbrite (39 ka) and the Neapolitan Yellow Tuff (14 ka)
. The CF has received growing
attention from the scientific community due to its reawakening in the last 50
years after a period of quiescence since the last eruption in 1538 with
background slow subsidence at a rate of ∼ 1.5 cm yr-1
. Renewed unrest was associated with two periods of
bradyseism (1969–1972 and 1982–1984), with a total vertical deformation of
about 3.5 m . To date these uplifts have not culminated
in an eruption. After 1984 a period of more than 20 years of general
subsidence followed, interrupted sporadically by a series of minor uplift
events. Since 2006 the caldera started uplifting again with an increased rate
from 2011 . Maximum ground deformation is recorded near
the town of Pozzuoli, while the main fumarolic activities occur
∼ 800 m away at La Solfatara.
Significant gravity changes associated with unrests are usually observed in
caldera systems , either at the centre of maximum deformation or at the
structural boundaries of the caldera complex, which are likely associated
with caldera ring faults e.g..
Ring faults significantly alter strain partitioning and fluid propagation and
hence must be considered for the interpretation of geophysical signals
. In this paper we explore the impact of vertical
and lateral mechanical heterogeneities in the shallow crust beneath the CF,
including ring faults, on monitoring signals at the surface (ground
deformation and gravity changes) as a consequence of unrest caused by a
perturbation of the shallow hydrothermal system. Unrest is modelled by the
injection of a mixture of hot water and carbon dioxide at the centre of the
caldera system, which is associated with the main fumarolic activity at La
Solfatara, and at the base of the ring faults, which simulates fluid release
from a deeper pressurised reservoir . We investigate the
separated contribution of pore pressure and thermal effects to total ground
deformation through a series of generic test cases which compare the single
(central) injection model with the simulation of multiple injection points.
We then show that different injection rates alter the timescales and
amplitudes of deformation and gravity changes during periods of unrest. A
sensitivity analysis of fault mechanical properties is also provided.
It is important to note that, while models are informed by data on the solid
and fluid mechanics of the CF, we do not attempt to replicate or fit
observations made during the ongoing unrest at CF.
Model parametrisation
In order to account for the complex mechanical structure of the shallow crust
and the caldera infill at a restless caldera (such as CF caldera), the
modelling domain is subdivided into several regions with different
hydrological and mechanical properties. The model is 2-D axi-symmetric and
defined by the coordinates (r,z), with r the radial distance and z the
vertical position. The hydrological model is 1.5 km deep and is closed to
heat and fluid flow in the radial direction and to fluid flow across much of
the basal boundary (Fig. , detailed in Sect. 3.1), whereas
the mechanical model is unbounded in the radial and downward vertical
direction (Fig. , detailed in Sect. 3.2). Both models are
based on information available for the CF and designed such that a central
fumarolic field is situated on its rotational axis.
Two high-angle faults (Faults A and B) are implemented with parameters
informed by data on the ring faults of the Neapolitan Yellow Tuff (14 ka)
and Campanian Ignimbrite (39 ka) eruptions, respectively
. The fault geometry
is represented in Fig. . Following the approach of
, the upper point P is placed at (r=3 km, z=-200 m) for Fault A and (r=6.5 km, z=0 m) for Fault B. Both faults
are steeply inclined (with dip angles of θ=80 and
θ=75∘, respectively) and penetrate the system up to a depth of
3 km (d=2.8 km for Fault A and d=3 km for Fault B). While Fault B
extends to the ground surface, Fault A tips out at a depth of z=-200 m
. The fault zone is divided into two
sub-zones with different hydrological and mechanical characteristics: a
central narrow (25 m wide) core zone is bordered on both sides by a
wider (100 m wide) damage zone, the latter having properties
intermediate between those of the core and the rock surrounding the fault
zone (Tables and ).
Geometry of a fault. Fault extends from a shallow point P, over a
vertical distance d and forms a dip-angle θ with the horizontal
axis. The fault structure comprises two units: a central narrow core zone
surrounded by a wider damage zone. Both units have different hydrological and
mechanical parameters to the surrounding rock.
Heterogeneous hydrological domain. Two transition zones are placed
between the central conduit and Layers A and B, with intermediate
hydrological parameters (Table ). Atmospheric boundary
conditions are fixed on the top (which is open to fluid and heat flow),
lateral boundaries are assumed to be impervious and adiabatic, while a heat
flux is assigned at the bottom impervious boundary at a rate of
0.195 W m-2 to ensure a temperature gradient comparable to that
estimated for CF (∼ 130 ∘C km-1,
).
Hydrological parameters for the domain of Fig. :
rock density ρr (kg m-3), porosity ϕ,
permeability K (m2), thermal conductivity λ
(W (m × K)-1), specific heat capacity Cr
(J (kg × K)-1). Matrix permeability is isotropic, but enhanced in the
vertical direction kz by almost 2 orders of magnitude in the fault
damage zone and by 3 orders of magnitude in the core of the faults. In other respects the fault zones have the same
hydrological characteristics as the matrix (star symbol ⋆).
Rock
Porosity
Permeability
Thermal
Specific heat
density
conductivity
capacity
ρr (kg m-3)
ϕ
K (m2)
λ (W (m × K)-1)
Cr (J (kg × K)-1)
Central conduit
1800
0.10
10-14
1.15
900
Layer 1 – Pyroclastic material
1700
0.35
5×10-15
1.15
900
Layer 2 – Tuffs and marine deposits
2300
0.15
10-15
1.50
1000
Transition 1
1700
0.15
8×10-15
1.15
900
Transition 2
1700
0.10
5×10-15
1.50
1000
Faults – damage zone
⋆
⋆
kz=10-13
⋆
⋆
Faults – core zone
⋆
⋆
kz=10-12
⋆
⋆
Heterogeneous mechanical domain. Mechanical parameters are reported
in Table . Inclination and radial placement of faults
are not in scale. The domain extends toward infinity in the radial and
vertical (downward) directions. Free-stress boundary conditions are ascribed
at the top boundary, while vanishing displacements are assigned at infinite
distances.
Mechanical parameters for the domain of Fig. :
seismic p wave velocity vp (km s-1), rock density
ρr (kg m-3), rigidity modulus μ (GPa),
Poisson ratio ν.
Seismic p wave
Rock density
Rigidity modulus
Poisson ratio
velocity
vp (km s-1)
ρr (kg m-3)
µ (GPa)
ν
Layer 1 – Pyroclastic material
1.60
1700
1.24
0.25
Layer 2 – Tuffs and marine deposits
3.44
2300
7.79
0.25
Layer 3 – Thermo-metamorphic rocks
4.78
2490
16.3
0.25
Layer 4 – Crystalline basement
5.76
2650
25.1
0.25
Layer 5 – Melt zone
2.80
2180
4.87
0.25
Layer 6 – Mantle
6.50
2810
33.9
0.25
Faults – damage zone
⋆
⋆
0.385
0.30
Faults – core zone
⋆
⋆
0.0357
0.40
Hydrothermal model
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 .
(a) Cell centres of the quasi-uniform mesh used in TOUGH2
to solve the equations of the hydrological model of Sect. 3.1. It is composed of 5848 cells, spaced on a basis of a composed
exponential distribution in such a way the radial spacing is finer adjacent
to the central conduit and faults and the vertical spacing is finer around
injection points and towards the surface. (b) Exponential
distribution for the quasi-uniform mesh used for the unbounded domains of the
geomechanical model (Eq. ), composed of 66 049 grid points.
The two ring faults are shown in red. Yellow box represents the hydrological
domain. The same mesh, but extended toward infinity also in the upward
direction, is used for the gravity model (Eq. ).
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 ).
Geomechanical and gravity models
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: limr→∞u=limz→-∞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 .
Numerical simulation scenarios and results
The background hydrothermal fluid circulation is driven by the injection of a
mixture of hot water and carbon dioxide at a temperature of about
350 ∘C from the base of the central high-permeable conduit,
simulating the input of fluids of magmatic origin. A heat flux is assigned at
the bottom impervious boundary at a rate of 0.195 W m-2. The steady-state solution, obtained after a long-lasting injection period (c. 4000
years), is used as the initial condition for the unrest simulations (run-up
to a final time of 100 years), which are divided into the three scenarios
described below.
Modelling scenarios
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 );
Scenario II – constant mass rate:
Scenario I plus injection at the bases of each fault core zone of a
total mass flow rate equal to that of the central injection (see
Table );
Scenario III – constant flux rate:
Scenario I plus injection at the bases of each fault core zone at a
specific (per square metre) mass flow rate equal to that of the central
injection (see Table ).
Injection at the base of the faults (core zone of Fig. ,
25 m wide) for Scenarios II and III simulates the possible release of gas
from a deeper reservoir ascending along preferential pathways of the fault
zone during unrest periods.
Different injection values (mass and flux rate) for the central
conduit and faults, normalised to the injection of a mass of 1 kg of fluids.
Base area of the central conduit is π×2002=125 664 m2, of
the Fault A core zone is 2π×3000×25=471 239 m2,
of the Fault B core zone is 2π×6500×25=1021 018 m2.
Scenario I
Scenario II
Scenario III
central conduit – mass (kg s-1)
1
1
1
central conduit – flux rate (kg m-2 s-1)
7.96×10-6
7.96×10-6
7.96×10-6
Fault A – mass (kg s-1)
0
1
3.75
Fault A – flux rate (kg m-2 s-1)
0
2.12×10-6
7.96×10-6
Fault B – mass (kg s-1)
0
1
8.13
Fault B – flux rate (kg m-2 s-1)
0
9.79×10-7
7.96×10-6
Injection rates
Once the rates of the central injection are established, the corresponding
injection rates at the base of the faults are determined by
Table . Rates of hot water and carbon dioxide central
injection for both the steady-state and unrest simulations are selected in
order to match observed data at CF, following other models present in the
literature for simulating the unrest activity associated with the perturbation
of the hydrothermal system e.g.. In particular, the injection rates for the
steady-state simulation are chosen so that the total flux
(3400 tons day-1) and the molar ratio CO2/H2O
of 0.17 (equating to 1000 tons day-1 of CO2 and
2400 tons day-1 of H2O) are based on average
degassing measured prior to the 1982–1984 bradyseismic crisis, while an
increased molar ratio of 0.40 is used for the unrest simulation. Regarding
the magnitude of the injection rates, several values have been adopted in the
literature in different contexts, albeit the rates used in
and (6000 tons day-1 of
CO2 and 6100 tons day-1 of H2O) provide a
good match to observed data. Recently, a constraint on the magnitude of the
injection rates has been discussed by . Although there
are many other parameters that can influence the mechanical response
(including depth of injection and temperature of the injected fluid), in this
paper we focus on the influence of the injection rates on the timescale and
amplitude of the deformation (Table ). Where not
specified, the injection rates of the unrest × 1 column of
Table are used.
Injection rates (tons day-1) for different unrest
simulations. Molar ratio is 0.17 for the Steady-State simulation and 0.40
for all the Unrest simulations.
Steady-State
Unrest ×1
Unrest ×0.5
Unrest ×2
Unrest ×3
H2O (tons day-1)
2400
6100
3050
12 200
18 300
CO2 (tons day-1)
1000
6000
3000
12 000
18 000
Molar ratio
0.17
0.40
0.40
0.40
0.40
Initial conditions
Initial conditions for the unrest simulation are the same for all scenarios
and represented in Fig. . Due to the injection of hot
fluids, the central conduit shows a higher temperature with respect to the
rest of the domain, while the pressure approaches hydrostatic, indicative of
a steady-state condition. A slight temperature variation is observed at the
fault zones, where the locally increased permeability focuses convective
fluid flow, with downward flow of cold water via the
fault . A two-phase plume forms close to the central
conduit, according to the results of previous fluid flow simulations
.
Changes in pore pressure, temperature and gas saturation relative to
the steady-state initial condition at different times after the initiation of
unrest. Initial conditions are obtained as the steady-state solutions of
central injection of 2400 of H2O and 1000 tons day-1 of
CO2, through a cylindrical conduit with radius 200 m. Unrest is
simulated by injecting 6100 of H2O and 6000 tons day-1 of
CO2 through the central conduit (Unrest ×1 column of
Table ) and additionally for Scenarios II and III
injecting at the base of the core zone of the two faults according to
Table . Note that the colour scale of initial conditions is
different from the respective colour scale of unrest plots.
Changes in pore pressure, temperature and gas saturation relative to
the steady-state initial condition at different times after the initiation of
unrest for Scenario I. Initial conditions are obtained as the steady-state
solutions of central injection of 2400 of H2O and
1000 tons day-1 of CO2, through a cylindrical conduit with
radius 200 m. Unrest is simulated by injecting a mixture of H2O
and CO2 through the central conduit. Injection rates for the unrest
simulation are listed in Table . Note that the colour scale of
initial conditions is different from either the respective colour scale of
unrest plots or the colour scale of Fig. .
Pore pressure, temperature and density changes during unrest
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.
Ground deformation
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 .
Ground deformation computed for Scenario I at the surface after
t=0.5, 3, 10 and 100 years of unrest: total vertical deformation (a), total horizontal deformation (b), vertical deformation
due to pore pressure (c), and vertical deformation due to thermal
effects (d). Vertical lines refer to the boundary of the central
conduit and to the injection and shallowest points of faults.
Computed vertical deformation at the centre of the model (r=0,z=0)
over 100 years of unrest for Scenario I with different injection rates (see
Table ). The solid line is the total vertical
displacement v=vP+vT, while the dashed and dotted lines are the
vertical displacement due to pore pressure vP and thermal effects vT,
respectively.
Ground deformation computed for Scenario II at the surface after
t=0.5, 3, 10 and 100 years of unrest: total vertical deformation (a), total horizontal deformation (b), vertical deformation
due to pore pressure (c), and vertical deformation due to thermal
effects (d). Vertical lines refer to the boundary of the central
conduit and to the injection and shallowest points of faults.
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.
Ground deformation computed for Scenario III at the surface after
t=0.5, 3, 10 and 100 years of unrest: total vertical deformation
(a), total horizontal deformation (b), vertical deformation
due to pore pressure (c), and vertical deformation due to thermal
effects (d). Vertical lines refer to the boundary of the central
conduit and to the injection and shallowest points of faults.
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.
Sensitivity analysis on fault rigidity modulus
In this section we analyse the influence of rigidity of fault core and damage
zone on ground deformation. For simplicity we restrict our analysis to the
vertical component of deformation. In detail, μc, μd and
μ¯ are the rigidity values of the core zone, damage zone and the
surrounding rock, respectively. We reduce the rigidity on the fault core (and
damage) zone with respect to the surrounding rock by s1 (and s1/2)
orders of magnitude, i.e.
μc=μ¯10s1,μd=μ¯10s1/2.
In the baseline simulation the rigidity of the faults is the same as the
surrounding area (i.e. s1=0). For each value of s1 in the range 0≤s1≤3 we obtain a variation in the ground deformation of s2 orders
of magnitude, i.e.
v=v0×10s2,
where v0 is the uplift observed for the baseline simulation (i.e.
s1=0). Figure shows the values of s1 and s2
computed at the centre of the model and at faults for simulation times t=3
and t=100 years. Reducing the rigidity values (i.e. increasing s1), the
deformation increases for the simulations at t=3 years and decreases for
t=100 years. At t=3 years the deformation is mainly driven by pore
pressure changes (Figs. and )
close to injection points (therefore at a depth of ∼ 1.5 km), while at
t=100 years deformation is mainly driven by temperature changes, which
constitute a shallow source of deformation (thermal effects reach the surface
at t=100 years, see Fig. ). In the latter case, the
region where the rigidity is reduced (fault core and damage zones) is below
the source of deformation, causing less uplift than that observed for the
baseline simulation. After t=3 years sensitivity of deformation to fault
rigidity is greater for Fault B than for Fault A, whilst the reverse is true
at t=100 years. Changes in deformation at the centre of the domain are
minimal throughout all simulations, showing the limited lateral influence of
the mechanical properties at the faults.
Gravity changes
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
.
Uplift variations against variations in rigidity at faults for
Scenario III. Decreasing the rigidity by s1 orders of magnitude (i.e.
dividing the rigidity by 10s1), the corresponding uplift changes by
s2 orders of magnitude (i.e. by a factor of 10s2). Blue lines refer
to the simulation at t=3 years, while red lines refer to t=100 years.
Variation in uplift is computed at the centre of the model r=z=0
(diamonds), Fault A (circles) and Fault B (stars). Linear best fits
(constrained through the origin s1=s2=0) are represented by solid, dashed and dotted lines for the centre of the model, Fault A and Fault B, respectively. The slopes of the best fit lines for simulations at t=3 years
are about 0.0849 for Fault A, 0.149 for Fault B and 0.00756 at the centre of
the model, while for simulations at t=100 years are about -0.0663 for
Fault A, -0.0108 for Fault B and -0.00656 at the centre of the model.
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).
Computed (residual) gravity changes at the centre of the model
(r=0,z=0) during 100 years on unrest (a) for Scenario I with
different injection rates (see Table ): unrest ×0.5, unrest ×1, unrest ×2, unrest ×3. (b) Close-up of the first 2.5 years of unrest (boxed on a).
(a, b, c) (residual) gravity gradient Δg/v on the surface after t=0.5, 1, 3, 5, 10 and 20 years of unrest for
Scenarios I, II and III. The respective (residual) gravity changes Δg
(solid lines) and vertical deformation v (dashed lines) are reported in
double y axis plots for t=0.5, 1 and 3 (d, e, f) and t=5,
10 and 20 (g, h, i) years of unrest. Vertical lines refer to the
boundary of the central conduit and to the injection and shallowest points of
faults. In Scenario I we observe that the gravity gradient starts to
oscillate at r∼2500 m. This behaviour is a purely numerical artefact,
since for r>2500 m the uplift approaches to zero and the gravity gradient
becomes singular. For this reason the plot is limited to 0<r<3500 m.
For the interpretation of the legend the reader is referred to the colour
version of the paper.
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).
Discussion
Heterogeneities in hydrological and mechanical properties as well as the
presence of faults within caldera forming volcanoes in the model
substantially affect the hydrothermal circulation of hot fluids and the
consequent variation in geophysical signals.
Models of the CF caldera suggest that the higher permeability of a central
conduit at La Solfatara favours the uprising of hot fluids from the deep
portion of the reservoir to the surface. Steady-state simulations show
formation of a two-phase plume
, with radius and gas
composition that depend on the permeability structure of the caldera
fill . According to our simulations the two-phase plume
occupies the entire central conduit and part of the transition zone, leading
to a 300 m radius plume at 1.5 km depth. The radius of the plume reaches
500 m in a shallow region close to the surface. Two gas regions form at the
edges of the plume: one surrounding the injection point and a shallower
region which extends to the surface, simulating the gas discharging observed
during the fumarolic activities at La Solfatara. The transition zone of
intermediate hydrological properties favours pressurisation of the system
during the first 3 years of the unrest and then allows depressurisation
as injected fluids ascend and discharge at the surface
(Fig. ).
This behaviour is reflected by the fast initial vertical deformation at the
centre of the domain, which is followed first by a rapid and then by a slower
subsidence period (Fig. ). This pattern would not be
observed if the permeability contrast between the central conduit and the
rest of the domain was stronger. The close relationship between deformation
and fluid flow is highlighted in this simulation. If we lower the
permeability of the caldera fill, subsidence after uplift does not occur. In
fact lower-permeability caldera fill would inhibit the recharge of cold water
to the base of the domain and the plume would be confined to a considerably
narrower area, resulting in a hotter gas-saturated region, as shown
in . Pressure release after the initial uplifting would
not be present and the following period of subsidence would not be observed.
Although the deformation profile observed in Scenario I reflects the solution
of a Mogi-type source in the first years of the unrest,
over time it develops into a more complex pattern that cannot be explained by
a simple deformation source (Fig. ). In the long timescale the ground deformation is therefore mainly driven by the
thermo-poroelastic response of the hydrothermal system.
Usually deformation observed at the centre of the model and associated with a
central source located at the axis of symmetry is amplified by the mechanical
heterogeneities of lateral fault zones . This behaviour is
not observed in the simulations of this paper, due to the small ratio between
the central source depth (injection depth of 1.5 km) and radial distance of
the closest fault (∼3 km), although in Scenario III the higher
injection rate at the base of the faults gives a small amplification of
deformation.
Rock expansion due to temperature changes is slower than that due to changes
in pore pressure. Temperature changes are confined to the areas surrounding
the injection points during the first 10 years of the unrest and take more
than 50 years to reach the surface. Thermal contribution to the total ground
deformation is therefore almost negligible within the first 10 years but
becomes dominant after some decades of unrest (Fig. ).
suggest that the relative contribution of temperature
and pore pressure is directly proportional to the injection depth.
modelled the effect of a short unrest period (20 months)
of high injection rate, and showed that the pore pressure declines
immediately after cessation of fluid injection, while the temperature
continues to increase until hot fluids discharge at the surface. Most
recently, examined the accelerating rate of ground
deformation affecting CF between 2005 and 2014, and suggested that the
observed deformation pattern requires both an extended period of heating of
the rock and short pulses of injection of magmatic fluids into the
hydrothermal system.
In our simulations, maximum temperature change is located close to the edge
of the plume (Fig. ). Consequently, the maximum uplift
observed at t=100 years is slightly displaced from the centre. The shape of
this temperature change is elongated in the vertical direction, resembling a
prolate source, and causes the rapid decay of the vertical deformation. The
same behaviour is observed for the gravity changes at the centre of the
domain. Density changes are localised at the boundary of the plume, where
replacement of water by gas over an increasingly large area occurs and
gravity changes present a local extreme. Gas saturation changes are small
during the first years of unrest and restricted to an area close to the
injection point (Fig. ). As a consequence, gravity changes
take about 2 years to exceed 50 µGal in absolute value (for the
Unrest ×1 case). Indeed, the initial period of the unrest is
characterised by an increase in density, since a substantial amount of water
is rapidly introduced to regions with positive gas saturation, following the
increase in injection relative to the background rate. This perturbation is
amplified for Unrest ×2 and ×3 models since a larger mass of
water is injected, as inferred by the positive sign of gravity changes at the
beginning of the unrest in Fig. . After a transient
period this pattern is inverted, since the higher molar ratio of
CO2/H2O of the fluid injected during unrest pushes
the system toward a steady-state solution in which a substantial amount of
gas will replace fluid-saturated regions, causing a negative change in
density and consequently in gravity changes. In contrast, gravity changes
over the fault zones are negative for the whole simulation time, since the
base of the faults are liquid saturated at the beginning of the unrest (no
background injection is performed at the base of the faults).
The inclusion of faults in the model fundamentally alters the dynamics of
fluid flow and heat transfer in the surrounding of fault areas.
show that the permeability contrast between the fault zone
and surrounding rock affects local temperature gradients, causing faults to
act as preferential pathways for either recharge or discharge of groundwater,
depending both on fault/matrix permeability ratio and on the vertical
extension of the fault. Temperature changes are more pronounced around the
faults than at the central conduit, since the background hydrothermal
circulation in the fault zones is not driven by any fluid injection, locally
enhancing the contrast between the steady-state and unrest simulations.
Gravity change and deformation associated with thermal effects are thus
larger on the faults than close to the axis of symmetry.
Fault mechanical properties strongly influence the deformation profile in the
vicinity of faults. In particular, a lower rigidity for the fault core and
damage zones is associated with increased uplift on the fault where the
source of deformation is deep (as in the case of pore pressure change during
the first years of unrest, mainly localised around injection points) but with
reduced uplift where the source of deformation is adjacent to the surface (as
in the case of temperature changes after a long period of unrest). There is
only minor perturbation of uplift observed at the centre of the domain,
showing that mechanical properties of faults have a limited lateral
influence. Such influence would be amplified if a deeper domain was
considered .
Fault geometry (inclination, vertical extension, penetration depth, radial
distance, etc.) also influences the amplitude and pattern of deformation and
gravity changes. Profiles of vertical deformation vary smoothly on Fault A,
while a sharper contrast is present at the Fault B, likely because Fault B
extends up to the surface z=0. This sharp behaviour is mainly associated
with the mechanical heterogeneities of fault core and damage zones rather
than with hydrological causes (it would not be observed if the mechanical
heterogeneities do not reach the surface).
Although the simulations performed in this paper provide a qualitative
assessment of the contribution of hydrothermal fluid circulation at restless
calderas, a more quantitative study and comparison with observed data from a
particular caldera (such as the CF) is beyond the scope of this study.
It is important to consider limitations of the approach adopted in this
paper. First, the shallow fluid injection (only 1.5 km deep) is constrained
by the range allowed by TOUGH2 (which does not account for supercritical
fluids), while several studies at CF have speculated that there is a deeper
source, between 2.7 and 5 km . recently investigated
deep supercritical regions of the hydrothermal system at CF using MUFITS, a
multiphase multicomponent fluid flow in porous media simulator that accounts
for high-temperature processes , more
realistic for representing restless calderas.
In addition, whilst assuming that simple layering of rock properties is
appropriate in the absence of detailed subsurface data, in reality it is
probable that the stratigraphy of the caldera fill is more complex.
Representing the effects of such heterogeneity, and in particular the strong
local contrasts in the vicinity of the faults, is difficult using standard
gridding approaches . Small-scale geological
heterogeneities observed in nature, usually modelled by geostatistical
methods , cannot be correctly represented
by a coarse cell blocks and identifying appropriate upscaling methods is
challenging . On the other hand, using an
extremely fine grid would radically increase the computational cost, making
the model unusable for practical purposes where a number of simulation runs
is required, such as optimisation and uncertainty reduction
.
The 2-D axi-symmetric representation of ring faults is obviously not able to
describe the complex fault networks which characterise collapse calderas. For
example, circulation along fault planes is a purely 3-D
phenomenon that cannot be represented by a 2-D model. However, this study
provides a first approximation of the influence of fluid flow mechanics
around faults on relevant geophysical observations and indicates the
importance of this area for future research.
Last but not least, the one-way coupling adopted in this paper, although
provides a reasonable simplification for short period unrests, is not
appropriate for the simulation of prolonged processes, since a significant
variation in key hydrological parameters (permeability, porosity) can be
induced by a change in stress and strain ,
altering the long-term behaviour of fluid flow in the porous medium and the
consequent evaluation of geophysical signals. For example, since an increase
in the effective stress may cause a permeability and porosity reduction
, a drop in these hydrological parameters
is expected where higher deformation are observed, namely at the centre of
the domain and close to the fault zones. This may reduce the deformation and
gravity change profiles over time. In addition, since these changes in
permeability and porosity would be less pronounced where deformation is
lower, the permeability contrast between the central conduit and the
transition zones would be attenuated, modifying the dynamics of the rapid
uplift and subsequent deflation observed in Fig. .
However, a qualitatively analysis is difficult to perform at this stage for a
number of uncertainties, such as the sensitivity to parameters regulating the
relationship between effective stress and permability/porosity.
Conclusions
The model proposed in this paper is targeted at evaluating the variations in
geophysical parameters associated with the perturbation of the hydrothermal
system in a restless caldera. A correct evaluation is fundamental to
discriminate between magmatic and hydrothermal unrest. Although the model can
refer to a generic system, parameters have been chosen on the basis of the CF
caldera, to simulate a behaviour proposed to explain the periodic unrests at
the CF caldera since 1969. This periodic behaviour can be explained by a
series of brief injections of hot fluids into the hydrothermal system
or after a long thermal
process following an increase in rock heating, as highlighted by
. Similarly, show that periodic
behaviour of gas composition can be associated with sharp increase of the heat
flux, with periodicity comparable to the decennial cycle observed at CF.
Simulations performed in this paper evaluate the ground deformation and
gravity changes caused by a long period of unrest associated with a prolonged
injection of fluid of magmatic origin into the shallow hydrothermal system at
a higher rate compared to that of the background simulation. To represent the
inherent complexities at collapse calderas, we considered the effects of
heterogeneities in the vertical and lateral distribution of hydrological and
mechanical parameters and the effect of faults. Permeability contrasts
considerably affect the fluid flow pattern
and consequently ground deformation and gravity changes at the surface.
The presence of the ring faults formed as a consequence of the episodes of
collapse can significantly alter the behaviour of the system in the
surrounding of the fault zones. Higher permeability parallel to the plane of
the fault favours convection and upward discharge of hot fluids from depth,
perturbing the hydrothermal system by changing pore pressure, temperature and
fluid density, dependent on injection rate (compare Scenarios II and III).
These perturbations, together with weaker mechanical properties of fault core
and damage zones, substantially alter geophysical signals (ground
deformation, gravity changes) at the surface close to the faults;
furthermore, in Scenario III, a minor influence on the centre of the model is
observed.
Investigation of different scenarios shows that the qualitative and
quantitative perturbations of the fluid dynamics are sensitive to fluid
injection rates, whose correct evaluation is one of the key challenges to
improve the understanding of restless caldera systems.