Well water level changes associated with magmatic unrest can be interpreted as a result of pore pressure changes in the aquifer due to crustal deformation, and so could provide constraints on the subsurface processes causing this strain. We use finite element analysis to demonstrate the response of aquifers to volumetric strain induced by pressurized magma reservoirs. Two different aquifers are invoked – an unconsolidated pyroclastic deposit and a vesicular lava flow – and embedded in an impermeable crust, overlying a magma chamber. The time-dependent, fully coupled models simulate crustal deformation accompanying chamber pressurization and the resulting hydraulic head changes as well as flow through the porous aquifer, i.e. porous flow. The simulated strain leads to centimetres (pyroclastic aquifer) to metres (lava flow aquifer) of hydraulic head changes; both strain and hydraulic head change with time due to substantial porous flow in the hydrological system.

Well level changes are particularly sensitive to chamber volume, shape and pressurization strength, followed by aquifer permeability and the phase of the pore fluid. The depths of chamber and aquifer, as well as the aquifer's Young's modulus also have significant influence on the hydraulic head signal. While source characteristics, the distance between chamber and aquifer and the elastic stratigraphy determine the strain field and its partitioning, flow and coupling parameters define how the aquifer responds to this strain and how signals change with time.

We find that generic analytical models can fail to capture the complex pre-eruptive subsurface mechanics leading to strain-induced well level changes, due to aquifer pressure changes being sensitive to chamber shape and lithological heterogeneities. In addition, the presence of a pore fluid and its flow have a significant influence on the strain signal in the aquifer and are commonly neglected in analytical models. These findings highlight the need for numerical models for the interpretation of observed well level signals. However, simulated water table changes do indeed mirror volumetric strain, and wells are therefore a valuable addition to monitoring systems that could provide important insights into pre-eruptive dynamics.

Pre-, syn- and post-eruptive changes in water levels have been reported for
several volcanoes

Here, we focus on the very commonly suggested mechanism of strain-induced
water level changes. Examples include well level changes of more than
9

In volcanic environments, many processes can lead to substantial strain,
including pressure changes in magma reservoirs and intruding dykes.
Information about the local strain field is therefore highly valuable for
volcano monitoring and eruption forecasting, as it could allow derivation of
these subsurface magmatic processes

Previous studies have indeed utilized the poroelastic behaviour of aquifers
to infer magmatic processes from observed water level changes at volcanoes

However, oversimplification of the coupling between solid and fluid mechanics
may make these models inadequate. An example is the 2000 Usu eruption

Analytical solutions exist for only a few comparatively simple poroelastic
problems

We investigate the phenomenon of poroelastic responses to magmatic strain to better understand the hydrological signals one might observe in wells on a volcano before and during eruptions. We assess to what extent confined aquifers can serve as indicators of stress/strain partitioning in the shallow crust due to reservoir pressure changes and therefore if they could provide a tool to scrutinise pre-eruption processes.

Symbols.

Table

We present a set of generic models using finite element analysis to perform
parametric studies on several volcanic settings with an inflating magma
chamber affecting overlying rock layers and hydrology. The models solve
a series of constitutive equations that result from the full coupling of
continuum mechanics equations for stress–strain relations of a linear elastic
material with Darcy's law and mass conservation within the porous flow theory

Fluid flow is described by mass conservation

In both Eqs. (2) and (4), the terms including the Biot–Willis coefficient
describe the coupling between solid deformation and fluid flow, which
manifests in stress absorption by the fluid and pore pressure changes due to
the increase/decrease of pore space resulting from volumetric changes of the
porous medium. The coupling parameter

This set of equations is solved for solid deformation (

Input parameters: reference values and ranges for parametric studies (where performed).

As a starting point to investigate hydrological responses to magma chamber
inflation, we build a 2-D-axisymmetric model geometry in COMSOL Multiphysics
following

2-D axisymmetric model set-up: a boundary load

Results of the reference simulation, shown as the initial (i.e.
10

The linear elastic material surrounding the magma chamber, from here on
called “host rock”, has elastic properties of a general granitic crust.
Depth- and temperature-dependent changes of Young's modulus of the crust are
ignored for simplicity. We chose a Young's modulus of 30

Temperature dependent water properties for a pressure of 4.5

In the parametric studies we investigated the effects of magmatic source
properties as well as poroelastic and geometric properties of the aquifer
(Table

When sweeping over one parameter, all others are kept constant. This entails
that in all geometric sweeps, the distance between magma chamber top and
aquifer was fixed, except for the sweep over magma chamber depth because
this distance is such an important parameter it would have otherwise
overwhelmed the pure effects of, for example, aquifer thickness. When
investigating the effects of magma chamber shape, we changed the vertical
semi-axis

To investigate the importance of parameters on hydraulic head change, we
performed a sensitivity analysis. The influence of lateral distance

Upper graphs: porous flow pattern shown for the reference simulation
at

The described model was run for each aquifer type, using reference values of
parameters given in Table

Figure

Exemplary plots used for the sensitivity analysis, showing the
influence of changing a parameter (whilst keeping all others constant) on the
central, initial hydraulic head change.

Parameter groups definition for the ranking resulting from sensitivity analysis.

Influence of Young's modulus and permeability of the aquifer on the
central hydraulic head change and its evolution with time.

Figure

priority 1 – parameters that belong to group A in

priority 2 – parameters that belong to group A in

priority 3 – parameters that belong to groups B or C in

priority 4 – parameters that belong to group C in

priority 1 – pressurization value, volume and aspect ratio of the chamber;

priority 2 – temperature of the pore fluid, permeability and Biot–Willis coefficient of the aquifer;

priority 3 – chamber depth, aquifer depth and Young's modulus of the aquifer;

priority 4 – Poisson's ratio and thickness of the aquifer.

Influence of the elastic stratigraphy on the central, initial
hydraulic head change, shown using the ratios of Young's Moduli
ER

This ranking is, however, only a relative one – even those parameters of the
last priority group have a non-negligible influence on the resulting
hydraulic head change. Furthermore, the ranking of a parameter depends partly
on the range of values tested for that parameter. This is particularly
important in interpreting the sensitivity to the Biot–Willis coefficient
(

The parametric sweeps provided a number of interesting insights; we are focusing here on describing the most important ones.

Of the aquifer's elastic properties, namely the Poisson's ratio

The non-monotonous influence of the Young's modulus stems from the fact that
not only its absolute, but also its value relative to the surrounding
lithology is important. We therefore also performed parametric sweeps over
the Young's Moduli of the host and cap rock,

Central hydraulic head change and its evolution with time for
different pore fluid temperatures.

By sweeping ER

The elastic stratigraphy determines the strain distribution in the domain,
visible in Fig.

Influence of the geometry, i.e. chamber radius, chamber depth

Influence of changing the aspect ratio of a spheroidal chamber (with
constant

Influence of lateral distance

Figure

Influence of lateral distance

Comparison of the development of central hydraulic change with time
due to an instantaneous chamber pressurization and a pressurization over
100

It is common that aquifers are heated in volcanic settings.
Figure

To demonstrate combined geometric effects we plot the central initial
hydraulic head change vs. the distance between magma chamber and aquifer for
different chamber radii and absolute chamber depths in Fig.

This relation is somewhat more complicated for the pyroclastic aquifers,
where hydraulic head depends on the distance between chamber and aquifer but
also on the chamber depth. The central hydraulic head in the pyroclastic
aquifers (Fig.

Normally, the maximum hydraulic head fall is directly above the chamber.
However, when considering a hydraulic head profile through the pyroclastic
aquifer for a shallow magma chamber without a sign-flipped strain (e.g.

We also evaluated the influence of the shape of the magma chamber by
incorporating tests for a prolate and oblate spheroid. Although chamber
volumes are constant, the shape can change the hydraulic head signal by 1
order of magnitude. Figure

Instead of having an “infinite” aquifer covering the whole volcano, we also
varied the lateral distance

For larger values of

Figure

The initial hydraulic head response linearly depends on pressurization
strength of the source. We assumed instantaneous pressurization for
simplicity; however, real magma chambers will more likely inflate over
longer time periods. In Fig.

In order to investigate poroelastic aquifer responses to crustal deformation,
we made some simplifying assumptions. For one, the presented models only
consider single-phase, single-component flow under constant temperature
conditions. However, our parametric studies have shown that the pore fluid
properties significantly influence the resulting head changes. Hydrothermal
systems can contain steam, water and a number of solutes, and temperatures
can change substantially. This can also affect the solid matrix, as its
mechanical behaviour may deviate from elastic when it is sufficiently heated.
Additionally, the injection of hydrothermal fluids into the aquifer can lead
to a pore pressure increase, heating and further deformation (see e.g.

Secondly, the aquifer was fully saturated and confined. To keep this study feasible, we did not investigate unconfined aquifers as this would imply a non-saturated permeable zone, and the coupling of linear elastic behaviour with non-saturated porous flow is associated with a high computational effort and often the solvers fail to converge due to the high nonlinearity of the problem. The model also ignores any hydrological sources and sinks, such as meteoric recharge, which can significantly influence well level observations in reality.

The discussed models are most applicable to confined aquifers that do not
undergo extensive heating during the observation period (e.g. aquifers at
some distance of the volcanic centre). They present a good opportunity to
better understand poroelastic aquifer responses that have been used for
monitoring. Their advantage over previous models is the full two-way coupling
of flow and linear elastic behaviour and that we are able to simulate various
geometries. The comparatively short computation time (on the order of 10 to
15

Our simulations show that neither injections of fluids nor flow within a fracture
– hereafter termed “fracture flow” – is needed to induce hydraulic head
changes of several metres in an aquifer. Volumetric strain induced by a
quasi-instantaneous magma chamber pressurization causes immediate hydraulic
head changes in local aquifers. Dilation above the chamber, due to ground
uplift, leads to a fall in pore pressure, while the accompanying compression
at more than 5

Both the strain-induced pressure gradient in the aquifers and the topographic
gradient due to the ground uplift induce porous flow; groundwater flows from
larger to smaller pressure/hydraulic head and from higher to lower elevation
(Eq. 5). The chamber pressurization in our reference simulation leads to a
central ground uplift of about 4

Fluid flow leads to the changes of strain and hydraulic head signals with
time. Hydraulic head continues to fall in the pyroclastic aquifer as water
flows away from the centre, while the opposite flow direction in the lava
flow aquifer leads to a decrease of the initial hydraulic head fall with time
(i.e. head increases). In both aquifers the equilibrium hydraulic head is
approximately balancing the change in elevation (about 4

Strain changes simultaneously with hydraulic head due to the poroelastic
nature of the aquifers. As water flows away, the pyroclastic aquifer responds
to the removal of pore fluid with compaction – explaining the change of
strain from dilation to compaction. The volumetric strain increase in the
lava flow aquifer stems from the initial stress absorption by the pore fluid
(final term in Eq. 2), which manifests as the pore pressure change. With
equilibration of the pressure in the aquifer this stress absorption effect
vanishes and strain approximates an equilibrium value that represents the
strain value in an elastically equivalent, but dry material. Here, stress
absorption of the fluid leads to an initial strain reduction by about
15

As pointed out by, for example,

The above findings highlight the necessity of a full coupling of fluid and solid mechanics. Both the effect of ground deformation on the pore fluid, as well as the influence of a pore fluid on strain in the solid matrix need to be considered to fully understand well level and/or strain signals.

Parametric studies have shown that poroelastic aquifer responses are complex
processes that are strongly influenced by source, geometrical and aquifer
parameters as well as the elastic stratigraphy. Chamber radius and
pressurization determine the strength of the deformation source and the
subsurface strain it causes. Strain partitioning in the crust is regulated by
the elastic properties of the different layers; both the absolute and
relative elastic properties of the aquifer and its surrounding lithology have
a complex influence on the strain and head signals. A special case occurs
when the cap rock is sufficiently stronger than the aquifer. A stiff cap rock
prevents the dilation of the aquifer and turns the strain into compression,
hence causing sign-flipped signals. In the reference set-up, the cap rock
needs to be 2 orders of magnitude stiffer than the aquifer; this could be
fulfilled if an unconsolidated, permeable pyroclastic layer is overlain by a
lava flow. But other, perhaps more common, geological settings exist in which
a sign-flipped response can be expected, as the geometry plays an important
role as well: the thicker the cap rock, the smaller is the necessary ratio of
cap rock to aquifer stiffness to change the sign of strain. For example, a
cap rock that is only 3 times stiffer than the aquifer can already lead
to a sign-flip if the aquifer is about 1

The subsurface stress and strain fields are also substantially dependent on
the shape of the chamber. For oblate chambers, the aquifer area that is
exposed to vertical stress is larger than for prolate chambers and it is
therefore subject to stronger strain. Additionally, the centre of the oblate
chambers is shallower than the centre of the prolate chambers (as the depth
of the chamber top is fixed in the simulations). The distance between aquifer
and magma chamber is another factor contributing to the strength of the
strain field affecting the aquifer. Generally, the closer the aquifer to the
source, the stronger the strain and hence its pressure response. However,
if elastic properties are close to values causing a sign-flipped signal,
i.e. if the aquifer is rather soft, a sufficiently close aquifer-source
distance can lead to a sign-flipped strain (because ER

The elastic properties of the solid matrix as well as the pore fluid together with the Biot–Willis coefficient of the aquifer determine the initial pressure response of the aquifer to the strain. Permeability then determines the velocity of pressure equilibration and gravitational flow and therefore the development of head and strain signals with time. Of particular interest is the influence of pore fluid temperature. It can change the hydraulic head response by 1 order of magnitude as well as influence the flow behaviour in the aquifer. This is especially important in volcanic environments, where heat flow is high and therefore temperature changes are likely. Changing the temperature means changing compressibility, density and viscosity of the water. We attempted to distinguish their individual influence with simulations in which only one of the three parameters was changed to a value corresponding to steam, while the others were kept at values corresponding to water. The viscosity does not influence the initial head fall, but affects the speed of equilibration, which is slower for a liquid water viscosity than for a lower steam viscosity (compare Eq. 5). Changing the phase of the pore fluid does not have this straightforward effect however, as flow velocities are also determined by the initial pressure gradients. These are influenced by fluid density and compressibility: decreasing the density of the pore fluid increases the initial hydraulic head change, while increasing the compressibility decreases it. At higher temperatures high enough for a phase change from liquid to steam, fluid density is reduced, while its compressibility rises. We therefore see a complex combination of these two effects. In the pyroclastic aquifer, the density effect dominates (hence hydraulic head change is larger), while in the lava flow aquifer the compressibility effect is more important (hence hydraulic head change is smaller). The stronger initial hydraulic head fall in the steam saturated pyroclastic aquifer is then large enough to overcome the topographic gradient, such that flow is towards the volcano. Therefore, as opposed to the reference simulation, the initial fall in hydraulic head diminishes with time in the soft aquifer just as it does in the stiffer aquifers. Hydraulic head falls in the lava flow and pyroclastic aquifers are of the same order of magnitude when saturated with steam, suggesting that the elastic properties of the solid matrix are less important and the processes are now governed by the fluid properties. While we investigated the temperature effect, other processes could also change pore fluid properties, such as dissolved minerals, and thereby play a role in determining the hydraulic head change.

Porous flow in the lava flow aquifer and therefore evolution of signals with
time is also significantly influenced by the lateral distance between the
magma chamber and the aquifer, even though initial hydraulic head values at
respective locations are the same. For

Horizontal flow directions in the softer pyroclastic aquifers are not
affected by the changed initial strain and head gradient for different

We only briefly studied the effect of long-term inflation, but results show that the time scale of pressurization is non-negligible as flow processes act simultaneously with the response to increased pressurization and can significantly change the signals. While hydraulic head responses in soft aquifers, where flow mostly follows the topographic changes, are comparable to instantaneous pressurization, the hydraulic head signals in stiffer, strain-dominated aquifers are reduced as the flow quickly equilibrates strain-induced pressure changes. Flow works against the increased pressurization and the rate of change of inflation determines which effect dominates, i.e. whether hydraulic head continues to fall (pressurization dominates) or starts to reach its equilibrium value (flow dominates).

We have shown that wells can reflect the deformation at volcanoes, suggesting
that their implementation in volcano monitoring systems could provide
insights into subsurface processes causing the strain. However, prior to the
interpretation of well signals, one needs to carefully consider that water
levels can also be changed by several other processes, e.g. meteorological
influences (rainfall), hydrothermal fluid injection, heat transferred
conductively through the crust or changes in flow conditions due to the
opening or closure of fractures. These processes can also act simultaneously
and overcome hydraulic head changes caused by a poroelastic response. Under
certain circumstances the different processes can be distinguished. First,
the general hydrological behaviour – i.e. the meteorological responses – and
up-to-date meteorological information should be tracked and therefore be
reasonably well known if wells are to be included in a monitoring system.
Then, a water level response to strain will be a transient signal on top of
the background behaviour. Well level monitoring can form an important
component for volcano monitoring in conjunction with geophysical or
geochemical observations to track magma reservoir evolution. For example,
ground deformation data will be useful for identifying hydrothermal
injections. When hydrothermal fluids from a magma reservoir are injected into
surrounding rocks, hydraulic head in the hydrothermal systems will rise and
the ground will be uplifted

Our parametric studies show how poroelastic aquifer responses are influenced by a variety of source, geometrical and aquifer parameters, which each have the potential to significantly alter the signal amplitude and development with time and space making the poroelastic processes highly complex. Consequently, a change in any of these parameters could lead to a change in an observed hydraulic head. In addition, the porous flow alters the initial hydraulic head signal with time. Therefore, not all observed aquifer pressure transients are necessarily related to a change in the magmatic system, which needs to be carefully considered when interpreting observed water level changes.

Yet another limitation lies in the fact that chamber inflation generally is not instantaneous. As the focus of this study was to identify the different influences of model parameters on hydraulic head signals, we assumed instantaneous pressurization for simplicity. However, it is clear that the analysis of signals during long-term inflation is even more complicated as one needs to decipher flow and inflation effects – again emphasizing that well data should ideally only be used in conjunction with surface deformation data.

Strain sensitivity in the aquifers, determined by dividing simulated
hydraulic head change by the volumetric strain, along a profile through the
aquifers.

Strain sensitivity in the aquifers, determined at a point centrally
above the chamber for different simulation times.

If the level changes are thought to be caused by strain, our models suggest
that volumetric strain in the aquifer can be directly inferred from measured
water level changes, as the simulated initial hydraulic head change perfectly
mirrors the strain. This requires a known strain sensitivity, the change of
hydraulic head in the aquifer in metres per unit applied strain, which can be
assessed by tracking water level changes as a result of predictable
excitations. Figure

However, the influence of flow on strain sensitivity is problematic;
Fig.

The better the local hydrology is known, the more value lies in well monitoring. Our simulations show how different soft and stiff aquifers behave in a strain field. Hence, knowledge of the lithology and/or determination of the strain sensitivity is important to discriminate between aquifer types. A low sensitivity value would indicate an aquifer similar to the presented pyroclastic example, which entails crucial information: the topographic gradient due to ground deformation can easily dominate over a strain-induced pressure gradient, and if this is the case, dilatational strain will quickly change to compression. Additionally, softer aquifers are more prone to the sign-flip effect.

Central vertical surface deformation and hydraulic head change with time for the pyroclastic aquifer overlain by different cap rocks, showing the effect of a sign-flipped strain in comparison to the reference case.

Information about flow in the aquifer is important and the acquisition of permeability data, e.g. via pumping tests, should be part of hydrological monitoring efforts as it can help decipher flow processes. While these tests usually provide only a local value, high-resolution time series of head data can in fact provide information on the effective permeability of the aquifers. If a poroelastic head response has been identified, one can observe the equilibration of the pressure change with time – giving information on groundwater flow velocities. In any case, observing the flow behaviour in local aquifers, by installing several observation wells, is a valuable addition to existing monitoring efforts as they can reveal flow patterns caused by head changes, be they strain-induced or caused by other (volcanic) processes. Finally, it is important to know aquifer geometry as the models show that the flow pattern can strongly depend on the lateral distance from the aquifer to the source.

Even if strain sensitivity has been accurately used to infer volumetric
strain, we still face the problem of interpretation of this signal. To invert
for the source of volumetric strain, analytical volcano deformation models
can be applied. However, these models commonly assume a source in
a homogeneous half space and some only consider spherical or point-like
chambers

The stress absorption of a pore fluid leads to a reduction of initial strain in the aquifer when compared to an elastically equivalent dry layer. If the initial strain is used to infer the magmatic source based on a model for dry deformation, its strength can therefore be underestimated. “Dry” strain is reached in the lava flow aquifer after porous flow has equilibrated the strain-induced pressure gradient. So, this problem could be solved when sufficiently dense time series of hydraulic head data are available: strain sensitivity can be combined with the evolution of signals with time to infer initial as well as equilibrium “dry” strain. In aquifers that are dominated by the topographic gradient, the presence of the pore fluid even leads to a reversal of dilatational to compressional strain and the application of dry deformation models is not possible.

The third assumption of a homogeneous half space is precarious as volcanoes
are strongly heterogeneous – several previous studies have already shown
that mechanical heterogeneities in the subsurface affect the ground
deformation at volcanoes

The above considerations hint that the apparent inconsistency of observed
well data and model predictions in the 2000 Usu case

In summary, while water level data can be a valuable addition to monitoring systems and give indications on subsurface strain, one needs to be careful when interpreting the head as well as strain data. We need to take into account that many parameters influence water level changes and that most of the commonly used analytical dry deformation models might fail to explain them.

In this study we presented fully coupled numerical models to investigate the interaction between solid mechanics and fluid flow in porous media. We have shown that strain due to the inflation of a magma chamber leads to significant hydraulic head changes and porous flow in the local hydrology. The flexibility of the finite element analysis method allowed us to perform extensive parametric studies providing detailed insights in these poroelastic processes. Parameters controlling aquifer behaviour are in order of importance (i) the shape, volume and pressurization strength of the magma chamber (ii) the phase of the pore fluid and the permeability of the aquifer (iii) chamber and aquifer depths and the aquifer's Young's modulus. Magmatic source properties and the distance between chamber and aquifer determine the strain field; strain partitioning is defined by the elastic stratigraphy of the crust. Elastic and flow parameters of the aquifer define its response to this strain and how head and strain signals change with time due to porous flow.

One aim of this study was to investigate the accuracy of the method to combine strain sensitivities with deformation models to interpret observed hydraulic head changes. Our models show that volumetric strain in the aquifer can indeed be inferred from hydraulic head changes using strain sensitivities, under certain conditions. Firstly, other causes for hydraulic head change have to be excluded, ideally by consulting other monitoring systems. Dense time series of well level data need to be acquired in order to account for flow processes and to measure the initial hydraulic head change. Additionally, we need to ensure that strain sensitivities have been accurately determined and have not changed with time due to changes in the hydrology.

However, using common analytical deformation models for the interpretation of
this strain information is problematic, as several assumptions of these
models can lead to substantial misinterpretation. They are only applicable
for a comparatively homogeneous crust (i.e.

The hydraulic head signal is very sensitive to source volume, shape and pressurization value. This suggests that if we have a detailed knowledge on the hydrology, some information about the source can be gained from hydraulic head changes – although solutions will always be non-unique. Our analysis has shown the necessity of numerical models to account for the large number of parameters that significantly influence the results. Nevertheless, well water levels and groundwater flow reflect subsurface strain and therefore are a valuable complement to other monitoring systems.

The influence of the Biot–Willis coefficient is quite complex, as it defines
the coupling terms in the constitutive equations and is involved in the
definition of specific storage of the aquifer as well.
Figure

Dependence of central, initial hydraulic head change on the
Biot–Willis coefficient.

The different dependence of

For a shallow magma chamber, in a situation with no sign-flipped strain, the
maximum head change in the pyroclastic aquifer is no longer central, but
laterally offset by up to 1

Hydraulic head change profile in the pyroclastic aquifer for

When the central portion of the aquifer is replaced with an area of zero
permeability, the change of head and strain with time due to porous flow in
the outer aquifer leads to a mechanical boundary at the lateral aquifer
onset. Especially in the pyroclastic aquifer, where strain undergoes
significant flow-induced changes, this leads to a discontinuity in strain
(Fig.

Volumetric strain after 1000

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007–2013) under the project NEMOH, REA agreement no. 289976. Additional funding was provided by the MED-SUV project, under grant agreement no. 308665, and the VUELCO project under grant agreement no. 282759, both part of the European Union's Seventh Framework Programme. We thank Micol Todesco, Maurizio Battaglia and Maurizio Bonafede for their constructive reviews that helped to significantly improve the manuscript. Edited by: M. Heap