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\DOI{10.5802/crgeos.214}
\datereceived{2022-08-19}
\daterevised{2022-12-04}
\dateaccepted{2023-04-27}
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%\copyeditor{RK}

\dateposted{2023-06-12}
\begin{document}

\begin{noXML}

%\TopicEF{Magma degassing and its impact on the Earth's atmosphere: from
%magma oceans to lava lakes}{Impact atmosph\'erique  du d\'egazage
%magmatique : des oc\'eans de magma aux lacs de lave}

\title{Bubble connectivity in experimentally-sheared crystal-bearing
silicic melts}

\author{\firstname{Camille} \lastname{Daffos}}
\address{Institut des Sciences de la Terre d'Orl\'{e}ans (ISTO), Univ. Orl\'{e}ans, CNRS, BRGM, UMR~7327, Orl\'{e}ans, France}
\email[C. Daffos]{camille.daffos@univ-orleans.fr}

\author{\firstname{Caroline} \lastname{Martel}\CDRorcid{0000-0002-7230-8413}\IsCorresp}
\addressSameAs{1}{Institut des Sciences de la Terre d'Orl\'{e}ans (ISTO), Univ. Orl\'{e}ans, CNRS, BRGM, UMR 7327, Orl\'{e}ans, France}
\email[C. Martel]{caroline.martel@cnrs-orleans.fr}

\author{\firstname{Laurent} \lastname{Arbaret}\CDRorcid{0000-0003-4904-9035}}
\addressSameAs{1}{Institut des Sciences de la Terre d'Orl\'{e}ans (ISTO), Univ. Orl\'{e}ans, CNRS, BRGM, UMR 7327, Orl\'{e}ans, France}
\email[L. Arbaret]{laurent.arbaret@univ-orleans.fr}

\author{\firstname{R\'{e}mi} \lastname{Champallier}\CDRorcid{0000-0002-7822-7155}}
\addressSameAs{1}{Institut des Sciences de la Terre d'Orl\'{e}ans (ISTO), Univ. Orl\'{e}ans, CNRS, BRGM, UMR 7327, Orl\'{e}ans, France}
\email[R. Champallier]{remi.champallier@cnrs-orleans.fr}

\shortrunauthors 

\keywords{\kwd{Degassing}
\kwd{Mush}
\kwd{Rhyolite}
\kwd{Shear}
\kwd{Experiments}}

\begin{abstract}
The explosivity of an eruption is mainly controlled by the ability of
gases to escape the magma column. Indeed, magmas able to evacuate gases
mostly erupt effusively whereas magmas that retain pressurised gases are
likely to trigger explosive events. In order to evaluate the explosive
potential of magmas residing at shallow level, we investigated the
influence of crystal content and shear on the development of bubble
connectivity in bubble- and crystal-bearing silicic melts. The
pre-deformed samples contain 0 to 50~vol\% of plagioclase crystals
(40--90~$\rmmu$m size) in a hydrated haplogranitic melt with 20--30
vol\% vesicularity mainly consisting of decompression-induced
H\tsub{2}O bubbles (${\sim}$20--250~$\rmmu$m in diameter). The samples
were deformed in torsion at a temperature of 650~\textdegree C
(crystal-free) or 750~\textdegree C (crystal-bearing), confining
pressure of 50~MPa, constant moderate shear rate of $2 \times
10^{-4}~\mathrm{s}^{-1}$, and low strains ($\gamma < 2$). The sample
microtextures and three-dimensional pore network show that bubbles are
mostly isolated in crystal-poor (0--10~vol\%) samples, whereas bubble
connection reaches more than 70\% in crystal-rich (30--50~vol\%)
samples, whether deformed or not. With increasing strain from $\gamma =
0$ to 2, bubbles re-organise in shear zones by forming channels.
Therefore, moderately-porous (20--30 bulk~vol\%) crystal-rich magmas
emplacing at shallow depths, such as in upper conduits or lava domes,
may be highly permeable via a process of gas channelling effective at
very low strains ($\gamma < 2$). This implies that violent explosions
of lava domes producing devastating surges require additional
mechanisms of gas pressurisation in moderately-porous crystal-rich
magmas.
\end{abstract}

\maketitle

\vspace*{-10pt}

\twocolumngrid

\end{noXML}

\section{Introduction}

The explosivity of an eruption is mainly controlled by the ability of
gases to escape the magma column, with overpressured gases trapped in melt
likely to lead to an explosive eruption, whereas efficient outgassing
may lead to effusive activity 
\citep{Eichelbergeretal1986,WoodsKoyaguchi1994}. The conditions under
which gas escapes the magma is thus a prerequisite to predict the
eruptive dynamics, and depend on an interplay of many parameters
involving magma characteristics [e.g., composition, temperature,
volatile and crystal contents; 
\citealp{Takeuchietal2021,Popaetal2021}] and emplacement conditions
[e.g. flow and conduit properties, rates of decompression and shear;
\citealp{Cassidyetal2018}].

An ascending volatile-saturated magma exsolves gases that are
accommodated through bubble nucleation or growth of pre-existing
bubbles in case of an initial volatile phase in excess. In highly
viscous magmas (viscosity ${>}10^{6}~\mathrm{Pa}{\cdot}\mathrm{s}$),
bubbles are usually trapped in the melt and cannot easily migrate
independently of the liquid to outgas via separated flow. If physical
and time conditions permit, bubbles connect by coalescing and can
develop gas channels that ultimately make the magma permeable to gases,
thus allowing outgassing through fractured dome rocks or conduit walls.
Rates of gas escape in silica-rich melts are difficult to assess
because they depend on intrinsic parameters, such as the bubble
characteristics [e.g., vesicularity, \citealp{KlugCashman1996}; pore
aperture size, \citealp{Wrightetal2009,Degruyteretal2010}; bubble
number density; \citealp{Bainetal2019}; bubble network, 
\citealp{Burgisseretal2017}], and external parameters related to magma
emplacement kinetics. Among these external parameters, decompression
rate has been demonstrated to affect bubble connectivity and
permeability. The studies dealing with decompression-induced bubbles
developing isotropically highlighted permeability increase at
vesicularities mostly ${>}$50~vol\% 
\citep{BurgisserGardner2005,Takeuchietal2005,Okumuraetal2012,MartelIaconoMarziano2015}.
Thereafter, shear has been demonstrated to drastically affect bubble
connectivity and permeability, by promoting bubble coalescence 
\citep{Okumuraetal2008} and outgassing 
\citep{Okumuraetal2009,Shieldsetal2016}, but also by alternately
favouring and preventing connectivity where deformation resulted in
magma compaction \citep{Gonnermannetal2017}. If magma viscosity and/or
shear rate are high enough, permeability development may proceed
through magma fracturing 
\citep{Stasiuketal1996,Tuffenetal2003,Castroetal2012,Gauntetal2014,Kushniretal2017}. 



Many studies have been dedicated to crystal-free or -poor melts, but
magmas can be highly crystallized, which affects the development of
bubble connectivity. Estimates of the percolation threshold commonly
range from 30~vol\% \citep{SaarManga1999,Blower2001} to 80~vol\%
vesicularity 
\citep{WestrichEichelberger1994,Takeuchietal2008,Wrightetal2009}, with
the highest values characterizing rapidly-decompressed crystal-free
magmas. \citet{Okumuraetal2012} experimentally decompressed rhyolitic
melts containing 30 and 50~vol\% corundum crystals, and concluded that
gas permeability remained low until vesicularity reached ${>}$68~vol\%.
\citet{Lindooetal2017} and \citet{deGraffenriedetal2019} decompressed
mafic and rhyolitic magmas, respectively, and reported a decrease of
the percolation threshold from ${>}$60 to ${<}$50~vol\% in the presence
of more than about 20~vol\% crystals. \citet{Parmigianietal2017}
numerically modelled the outgassing efficiency of a magmatic volatile
phase in crystal-rich (mush-like) magmas, highlighting gas permeability
from very low porosities (${\sim}$10~vol\%) via a mechanism of
channelling by which crystals build sustainable channels for gas
percolation. Such a permeability development and outgassing at low
vesicularity and high crystal content could explain the low porosity
values inferred to occur in the volcanic conduit prior to Vulcanian
eruptions~\citep{Collombetetal2021}. 


In crystal-bearing silicic magmas, the bubble network mostly rearranges
under decompression in the central part of the volcanic conduit and
simple-shear deformation at conduit edges or in lava domes where the
rearrangement of the stressed crystal framework is facilitated by
strain localization. To understand the processes governing the
development of degassing pathways in crystal-rich silicic magmas, few
studies have been dedicated to deform three-phase (melt, bubbles,
crystals) magmas in simple shear using high-temperature and
high-pressure deformation rigs of Paterson type. 
\citet{Laumonieretal2011} deformed a haplotonalitic melt containing 50
vol\% plagioclase crystals and 11~vol\% porosity, at temperature ($T$)
of 600~\textdegree C, confining pressure ($P$) of 200~MPa, bulk finite
shear strains ($\gamma$) of 1.3 and shear rates ($\gamma_{r}$) from $3
\times 10^{-5}$ to $1\times 10^{-3}~\mathrm{s}^{-1}$. They highlighted
gas accumulation in local microstructures caused by shear-induced
crystal fabric (local $\gamma$ up to ${\sim}$9). 
\citet{Shieldsetal2014} sheared haplogranitic melts spanning crystal
contents from 0 to 42~vol\% and CO\tsub{2}-bubble contents from 12 to
36~vol\%, at $T < 600~\text{\textdegree}\mathrm{C}$, $P$ of
150--200~MPa, $\gamma$ up to 10, and $\gamma_{r}$ from $1 \times
10^{-4}$ to $5 \times 10^{-4}~\mathrm{s}^{-1}$. They did not observe
bubble coalescence and outgassing took place via sample-wide
fracturing. These results of sample permeability reached by fracturing
agreed with those obtained by \citet{Kushniretal2017} using
crystal-free haplogranitic melts with ${\sim}$15~vol\% argon bubbles
deformed at magmatic $T$ of $880~\text{\textdegree}\mathrm{C}$, $P$ of
60~MPa, $\gamma$ up to 7, and $\gamma_{r}$ of $1 \times 10^{-4}$ to $8
\times 10^{-4}~\mathrm{s}^{-1}$. \citet{Pistoneetal2012} sheared
haplogranitic melts with 24 to 65~vol\% quartz crystals and 9--12~vol\%
CO\tsub{2} \mbox{bubbles,} at $T$ of 450--750~\textdegree C, $P$ of 200~MPa,
$\gamma$ up to 8, and $\gamma_{r}$ from $5\times 10^{-6}$ to $4\times
10^{-3}~\mathrm{s}^{-1}$. In the crystal-poor (24--44~vol\%) samples,
the authors reported bubble stretching with increasing strain,
facilitating coalescence (for $\gamma >2$ and for increasing
$\gamma_{r}$) and gas channel formation, eventually leading to
outgassing along the walls of the sample container. In the crystal-rich
(55--65~vol\%) samples, however, bubble stretching was localized in
shear bands where the crystal framework (including crystal breakage)
prevented gas\unskip\break  loss. 


These experimental studies provided information on the deformation of
three-phase magmas at $P$ of ${\sim}$200~MPa, $T$ mostly close to the
glass transition, for bubble contents mostly ${<}$15~vol\%, and for
$\gamma$ up to ${\sim}$10. We aim at extending these studies to the
investigation of the effect of a crystal network on bubble connectivity
and gas percolation in bubbly crystal-bearing magmas emplacing at
shallow depth, such as in upper volcanic conduits or lava domes, from
which explosive eruptions may be triggered. We performed simple shear
experiments in a Paterson rig, using three-phase magmas consisting of 0
to 50 bulk~vol\% plagioclase crystals and 15--30 bulk~vol\% H\tsub{2}O
bubbles (20--45~vol\% recalculated on the melt phase) in a
haplogranitic melt, under magmatic $T$ of 650--750~\textdegree C,
confining $P$ of 50~MPa, $\gamma$ up to ${\sim}$2, and $\gamma_{r}$ of
$2\times 10^{-4}~\mathrm{s}^{-1}$. We characterized the sample
microstructures and bubble connectivity as a function of crystal
content and shear strain, and have discussed implications for the natural
magmas and eruptive dynamics. 


\section{Methods}
\subsection{Experimental methods}


The experiments consisted of three main phases: (i) synthesis of
crystal-bearing hydrated glasses, (ii)~decompression-induced
H\tsub{2}O-bubble formation, and (iii) deformation in simple shear of
the three-phase magmas. 


\subsubsection{Synthesis of crystal-bearing hydrated glasses} 

The method of preparation is detailed in Supplementary Section~1.1.1
and a summary is given below. The anhydrous starting glass is a
haplogranite (HPG8; composition in wt\%: 78.6 SiO\tsub{2}, 12.5
Al\tsub{2}O\tsub{3}, 4.6 Na\tsub{2}O, 4.2 K\tsub{2}O). The HPG8 glass
was chosen because (i) it is rheologically well-characterized
\citep{HessDingwell1996}, (ii) its eutectic  composition allows
experiments at relatively  low $T$ without crystallization, and (iii) it
simulates late-stage crystallization of evolved liquids such as
rhyolitic melts. The HPG8 hydrations were performed in
internally-heated pressure vessels (at the Institut des Sciences de la
Terre d'Orl\'{e}ans, ISTO) in order to obtain glasses with 5.0 and 9.6
wt\% H\tsub{2}O, to which different amounts of plagioclase crystals
(size of 50--90~$\rmmu$m) were manually added. Plagioclase crystals
were chosen because they represent the main crystalline phase of
silicic magmas, as phenocrysts or microcrysts. The crystal contents
recalculated on the melt are 0, 21, 50, and 70~vol\%, leading to bulk
crystal volume ($\Phi_{\mathrm{c}\_\mathrm{bulk}\_3\mathrm{D}}$,
recalculated on a bubble-bearing basis) of about 0, 10, 30, and 50~vol\%. 



\subsubsection{Synthesis of a bubble-bearing magmas}
\vspace*{-1pt}

The powder mixtures of plagioclase crystals and hydrated HPG8 glass
were decompressed from 300 to 50~MPa at 850~\textdegree C in the
Paterson gas-medium apparatus [\citealp{PatersonOlgaard2000}; Australian
Scientific Instruments Pty Ltd, at ISTO; Figure~\ref{fig1}a], in order to
trigger a bubble nucleation event of supercritical H\tsub{2}O fluid
(hereafter referred as to gas bubbles). The strategy was to obtain bulk
amounts of bubbles around 20--40~vol\%, ideally not connected to each
other, in order to check whether further deformation will promote gas
connectivity. The experimental details are given in Supplementary
Section~1.1.2.



\begin{figure*}
\includegraphics[scale=0.95]{fig01}
\caption{Experimental device and strategy. (a) Scheme of the
Paterson gas-medium press, (b) Photography of the sample assembly, and
(c) Pressure--temperature--time path showing the isothermal step
(850~\textdegree C) of decompression-induced (300 to 50~MPa) bubble
nucleation, followed by an isobaric (50~MPa) and isothermal (750 or
650~\textdegree C) step of sample torsion ($\gamma < 2$) at a rate of
$2 \times 10^{-4}~\mathrm{s}^{-1}$.\label{fig1}}
\end{figure*}


\subsubsection{Torsion experiments}



Keeping the experimental setup as it was after decompression,
deformation in right-lateral simple shear was carried out at $P$ of
50~MPa, $T$ of 650 or 750~\textdegree C, $\gamma_{r}$ of $2 \times
10^{-4}~\mathrm{s}^{-1}$, and $\gamma$ from 0 to 2. Such $\gamma$ are
rather small compared to those observed at natural conduit margins or
dome bases, but the aim was to determine the gas percolation thresholds
occurring during the initiation of deformation. The experimental
details and calculations confirming limited relaxation of the bubbles
during quenching are given in Supplementary Section~1.1.3.




\subsection{Analytical methods}
\vspace{-5pt}
After experiment, the sample was cut following a section parallel to
the shear plane and exposing the longitudinal tangential surfaces
(maximum shear strain) for microstructural analyses using a scanning
electron microscope (SEM; Merlin Compact Zeiss at ISTO). The whole  
section was imaged at high\unskip\break resolution using the SEM, and the images 
were segmented using the SPO2003 image-processing software 
\citep{LauneauRobin1996,LauneauCruden1998}, in order to determine the
\mbox{two-dimensional} (2D) contents, sizes, and orientations of the bubbles
and the crystals, as detailed in Supplementary Section~1.2.1. A core sample bored
perpendicularly to the shear plane was used for \mbox{three-dimensional} (3D)
analyses using X-ray computed tomography (XCT; Phoenix NanoTOM at
ISTO). The 3D bulk porosity ($\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$)
and the amount of bubbles connected to the sample borders
($\Phi_{\mathrm{b}\_\mathrm{connect}\_3\mathrm{D}}$) were determined by
segmenting the XCT images using commercial software, such as VGStudio
Max and Blob3D, as detailed in Supplementary Section~1.2.2. The plagioclase crystals
were hardly discernible due to their low contrast of density with the
glass and were not subjected to 3D analyses.

The glass H\tsub{2}O contents of the bubble-bearing samples were
checked following the ``by-difference'' method using an electron
microprobe (EMP; Cameca SX Five at ISTO). As some of the residual
glasses were unexpectedly rich in H\tsub{2}O, they were double-checked
by micro-Raman spectroscopy (ISTO), as detailed in Supplementary Section~1.2.3.


\vspace{-5pt}
\subsection{Calculations}

The calculations mainly concerned the gas fraction and the bulk
viscosity of the samples. The expected gas fraction was determined
using the equation of \citet{JaupartAllegre1991}, 
which calculates the
gas volume ($\alpha_{\mathrm{melt}}$) resulting from a closed-system
degassing (in that gas bubbles remain in contact with the melt), as
described in Supplementary Section~1.3.1. 


The samples were not viscous enough at 750~\textdegree C to allow
viscosity measurements in the Paterson press, so we estimated the
bulk viscosity ($\eta_{\mathrm{bulk}}$) from \citet{HessDingwell1996}
for the hydrated melt and using the equation of \citet{Trubyetal2015}
for the three-phase suspensions, as detailed in Supplementary Section~1.3.2.


\vspace{-5pt}
\section{Results}

Four series with different bulk crystal contents
($\Phi_{\mathrm{c}\_\mathrm{bulk}\_3\mathrm{D}}$) of 0, 10, 30, 50~vol\%,
hereafter referred as to serie0, serie10, serie30, and serie50,
respectively, were decompressed in the Paterson press for bubble
formation and subsequent deformation in simple shear ($P = 50$~MPa, $T
= 650\ndash 750~\text{\textdegree}\mathrm{C}$, $\gamma_{r} = 2 \times
10^{-4}~\mathrm{s}^{-1}$) to finite deformations $\gamma = 0$ (no
deformation), $\gamma = 0.6$, and $\gamma = 1.3\ndash 1.8$ (no run
for serie10). The experimental conditions of the deformed bubble- and
crystal-bearing magmas are given in Table~\ref{tab1}. The analytical
results are reported in Table~\ref{tab2} and detailed below.


\begin{table*}[t!]
\caption{Experimental conditions\label{tab1}}
\fontsize{9.8}{12}\selectfont
\begin{tabular}{ccccccccccc}
\thead
\xmorerows{2}{Run \#}  & \xmorerows{2}{Series \#}  &
\raisebox{.6pc}{\morerows{2}{\parbox[t]{1.4cm}{\centering \smath{\Phi_{\mathrm{c}\_\mathrm{melt}\_3\mathrm{D}}}$^{\mathrm{a}}$ (vol\%)}}}  &
\raisebox{.6pc}{\morerows{2}{\parbox[t]{1cm}{\centering H\tsub{2}O\tsub{i}$^{\mathrm{b}}$ (wt\%)}}}  &
\multicolumn{5}{c}{Pre-deformation bubble forming} & 
\multicolumn{2}{c}{\morerows{1}{\parbox[t]{2.5cm}{\centering Shear at $P$--$T$ ($\gamma_{r} = 2 \times 10^{-4}~\mathrm{s}^{-1}$)}}} \\
\cline{5-9}
& & & & \multicolumn{3}{c}{\parbox[t]{4cm}{\centering Decompression (from 300~MPa and 850~\textdegree C)$^{\mathrm{c}}$}} &
\multicolumn{2}{c}{Cooling at $P$} & \vspace*{2pt}\\
\cline{5-7}\cline{8-9}\cline{10-11}
& & & & \parbox[t]{1cm}{\centering $P$ (MPa)} &
\parbox[t]{1.5cm}{\centering $\tau_{\mathrm{DP}}$\smath{^{\mathrm{d}}} (min)} &
\parbox[t]{1cm}{\centering Dwell1 (min)} & 
\parbox[t]{.5cm}{\centering $T$ (\textdegree C)} &
\parbox[t]{1cm}{\centering Dwell2 (min)} &
\parbox[t]{1cm}{\centering $\tau_{\mathrm{def}}$\smath{^{\mathrm{d}}} (min)} & 
\parbox[t]{3pt}{\centering $\gamma$} \vspace*{2pt}\\
\endthead
Pp17 &  serie0 &  \00 &  5.0 &  50 &  19 &  45 &  650 &  \015 &  - &  0 \\ 
Pp16 & serie0 &  \00 &  5.0 &  50 &  18 &  45 &  650 &  \015 &  \060 &  0.6 \\ 
Pp15 &  serie0 &  \00 &  5.0 &  50 &  18 &  45 &  650 &  \015 &  120 &  1.3 \\ 
Pp5 &  serie10 &  21 &  9.6 &  50 &  16 &  45 &  750 &  240 &  - &  0 \\ 
Pp7 &  serie10 &  21 &  9.6 &  50 &  18 &  45 &  750 &  \015 &  \060 &  0.6 \\ 
Pp6 &  serie30 &  50 &  9.6 &  50 &  \09 &  45 &  750 &  \015 &  - &  0 \\ 
Pp8 & serie30 &  50 &  9.6 &  50 &  19 &  45 &  750 &  \015 &  \060 &  0.6 \\ 
Pp9 &  serie30 &  50 &  9.6 &  50 &  16 &  45 &  750 &  \015 &  164 & 1.8 \\ 
Pp12 &  serie50 &  70 &  9.6 &  50 &  18 &  45 &  750 &  \015 &  - &  0 \\ 
Pp11 &  serie50 &  70 &  9.6 &  50 &  20 &  45 &  750 &  \015 &  \062 & 0.6 \\ 
Pp10 &  serie50 &  70 &  9.6 &  50 &  17 &  45 &  750 &  \015 &  145 & 1.5 
\botline
\end{tabular}
\tabnote{$^{\mathrm{a}}\Phi_{\mathrm{c}\_\mathrm{melt}\_3\mathrm{D}}$ is the
content of plagioclase crystals manually added to the hydrated
haplogranitic (HPG8) glass powder.}
\tabnote{$^{\mathrm{b}}$H\tsub{2}O\tsub{i} is the H\tsub{2}O content of
the hydrated glasses that were used for the Paterson experiments,
calculated after \citet{NewmanLowenstern2002} at 950~\textdegree C and
170~MPa (for the crystal-free series) or 400~MPa (for the
crystal-bearing series).}
\tabnote{$^{\mathrm{c}}$Decompression follows a 15-min dwell at 300~MPa
and 850~\textdegree C.} 
\tabnote{$^{\mathrm{d}}\tau_{\mathrm{DP}}$\smath{^{\mathrm{c}}} and
$\tau_{\mathrm{def}}$\smath{^{\mathrm{c}}} are the duration of the decompressions and
deformations, respectively.}
\end{table*}




\vspace{-5pt}
\subsection{Crystal-free series (serie0)}


The SEM pictures of the samples from serie0 show relatively homogeneous  
spatial distributions of the bubbles (Figure~\ref{fig2}a--c), in
agreement with the absence of strain localization expected from the
Newtonian behaviour of pure melts. Melt porosity,
$\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}}$, is $31 \pm 3$, to $24 \pm 2$
and $30 \pm 4$ area\%, at  $\gamma = 0$, 0.6, and 1.3, respectively
(Figure~\ref{fig3}a), in relatively good agreement with the porosities
measured from the XCT images, $\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$,
which are $30 \pm 3$, $21 \pm 3$, and $15 \pm 3$~vol\%, respectively.
The content of bubbles connected to the sample outside measured from
the XCT images, $\Phi_{\mathrm{b}\_\mathrm{connect}\_3\mathrm{D}}$, represents
about half of the $\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$ (i.e., $49
\pm 3$, $52 \pm 3$, and $50 \pm 3$~vol\%, at $\gamma = 0$, 0.6, and
1.3, respectively). The 2D bubble number densities measured from the
SEM images, BND\tsub{\_2\mathrm{D}}, are of $10^{5.6\ndash 5.7\pm
0.1}~\mathrm{m}^{-2}$ (Figure~\ref{fig4}a) and those determined using the
XCT, BND\tsub{\_3\mathrm{D}}, are $10^{10.5\ndash 10.6}~\mathrm{m}^{-3}$.
In the undeformed sample, 95\% of the bubbles have equivalent diameters
of 20--40~$\rmmu$m, representing about half of the total vesicularity.
The other half volume is made of bubbles with up to 300~$\rmmu$m
diameter (Supplementary Figure~3a). Bubble eccentricity, $R_{b}$, increases from
$1.3 \pm 0.1$ to $1.9 \pm 0.2$ and $2.1 \pm 0.3$ with increasing
$\gamma$, with a slight preferential orientation of the bubbles,
$\theta_{b}$ of ${\sim}22 \pm 6\text{\textdegree}$ (Figure~\ref{fig4}b). Glass
H\tsub{2}O contents measured following the EMP by-difference method are
$3.6 \pm 0.7$~wt\% at $\gamma = 0$ and $4.1 \pm 0.7$~wt\% at $\gamma
= 1.3$, giving bulk viscosities of $10^{7.3}$ and
$10^{7.0}~\mathrm{Pa}{\cdot}\mathrm{s}$, respectively.


\begin{figure*}
\vspace*{-2pt}
\includegraphics{fig02}
\vspace*{-2pt}
\caption{SEM pictures of the samples cut following the sketch shown in
Supplementary Section~1.2.1. Increasing shear strain from $\gamma = 0$ to 2 (from
left to right) and crystal content from 0 to 50~vol\% (from top to
down) promotes the development of non-spherical bubbles and
heterogeneities in bubble spatial distribution. All main pictures have
a yellow scale bar of 4~mm and subpanels are enlargements (with
different coloured scale bars). The red arrows in (i) and (k) point to
crystal fragmentations.\label{fig2}}
\vspace*{-2pt}
\end{figure*}

\vspace{-5pt}
\subsection{Series with 10\% crystals (serie10)} 


Serie10 consists of only a $\gamma = 0$ (undeformed) sample and a
$\gamma = 0.6$ sample, since higher shear strains inevitably conducted
to capsule tearing. The deformed sample shows incipient shear zones
marked by discrete local reorganisation of the bubbles and crystals
(Figure~\ref{fig2}e). $\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}}$ slightly
decreases from $31 \pm 3$ area\% at $\gamma  = 0$ to $27 \pm 3$ area\%
at $\gamma = 0.6$ (Figure~\ref{fig3}a). Recalculating bulk porosities
(i.e. including crystals) give $\Phi_{\mathrm{b}\_\mathrm{bulk}\_2\mathrm{D}}$
of $29 \pm 3$ area\% at $\gamma = 0$ and $24 \pm 3$ area\% at $\gamma =
0.6$, in good agreement with the XCT
$\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$ of $28 \pm 3$~vol\% at $\gamma =
0$ and $18 \pm 3$~vol\% at $\gamma = 0.6$. BND\tsub{\_2\mathrm{D}} is $10^{5.6
\pm 0.2}~\mathrm{m}^{-2}$ at $\gamma = 0$ and $10^{5.2 \pm
0.2}~\mathrm{m}^{-2}$ at $\gamma = 0.6$ (Figure~\ref{fig4}a), and
BND\tsub{\_3\mathrm{D}} is $10^{9.2\ndash 9.3}~\mathrm{m}^{-3}$. In the
undeformed sample, about 80\% of the bubbles have diameters of
20--40~$\rmmu$m, but their volumes only represent less than 10\% of
total vesicularity. More than one third of the vesicularity is occupied
by bubbles with equivalent diameters of 220--240~$\rmmu$m 
(Supplementary Figure~3b). The bubbles have $R_{b}$ of $1.0 \pm
0.1$ at $\gamma = 0$ and $1.3 \pm 0.1$ at $\gamma = 0.6$,   
showing no preferential orientation ($\theta_{b}$ of $20 \pm
43\text{\textdegree}$) in the unsheared sample (Figure~\ref{fig4}b).
The glass H\tsub{2}O contents are of $6.1 \pm 0.7$~wt\% at $\gamma = 0$
and $5.3 \pm 0.7$~wt\% at $\gamma = 0.6$. The crystal contents, 
$\Phi_{\mathrm{c}\_\mathrm{bulk}\_2\mathrm{D}}$, are of $9 \pm 1$
area\% at $\gamma = 0$ and $12 \pm 2$ area\% at $\gamma= 0.6$. The
crystal number densities, CND, are  $10^{4.8 \pm 0.2}~\mathrm{m}^{-2}$
at $\gamma = 0$ and $10^{5.1 \pm 0.3}~\mathrm{m}^{-2}$ at $\gamma =
0.6$ (Figure~\ref{fig4}a). The crystals have eccentricities, $R_{c}$,
that are not varying from $1.1 \pm 0.1$ with $\gamma$, and their main
orientations, $\theta_{c}$, is $15 \pm 33\text{\textdegree}$ at $\gamma
= 0$ and $18 \pm 3\text{\textdegree}$ at $\gamma = 0.6$
(Figure~\ref{fig4}b).%\pagebreak



\begin{figure}
\includegraphics{fig03}
\caption{Bubble formation. (a) Measured porosity versus
porosity calculated for a H\tsub{2}O degassing from the initial content
(5.0~wt\% for the crystal-free samples and 9.6~wt\% for the others) to
the final content measured in the residual glasses, after the equation
of \citet{JaupartAllegre1991}; the two arrows on the $X$-axis give
the gas content calculated for an equilibrium H\tsub{2}O exsolution
(i.e., final content is H\tsub{2}O solubility at 50~MPa); the vertical
error bars are not statistical uncertainties but span the whole range
of the measured values; (b) Difference between the H\tsub{2}O
contents measured in the residual glasses (circles and stars using
electron microprobe and Raman spectrometry, respectively; technical
details in Supplementary Section~1) and H\tsub{2}O solubility at final pressure
(dashed box), showing incomplete H\tsub{2}O exsolution in some
samples.\label{fig3}}
\end{figure}


\begin{figure}
\includegraphics{fig04}
\caption{Bubble and crystal characteristics. (a)~Number
density and (b) Preferential orientations ($0\text{\textdegree}$ is
parallel to shear direction), with the dashed red line giving the
theoretical angles between particle major axis and shear direction
calculated for solid particles with eccentricity of 1.1, using the
formulation of \citet{Fernandezetal1983}; the $X$-axis values have been
slightly shifted for clarity; the vertical bars are not statistical
uncertainties but span the whole range of the measured
values.\label{fig4}}
\end{figure}



\subsection{Series with 30\% crystals (serie30)}

Serie30 shows shear zones of reorganised bubbles and crystals,
evidencing an intensification of strain localisation from $\gamma =
0.6$ to 1.8 (Figure~\ref{fig2}g,h).
$\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}}$ are $20 \pm 3$, $26 \pm 3$, $26
\pm 12$ area\% at $\gamma = 0$, 0.6, and 1.8, respectively
(Figure~\ref{fig3}a). Recalculating bulk porosities gives
$\Phi_{\mathrm{b}\_\mathrm{bulk}\_2\mathrm{D}}$ of $15 \pm 3$, $20 \pm 3$, $23
\pm 12$ area\% at $\gamma = 0$, 0.6, and 1.8, respectively. The XCT
$\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$ gives $22 \pm 3$, $25 \pm 3$ and
$15 \pm 3$~vol\% at $\gamma = 0$, 0.6, and 1.8, respectively.  
BND\tsub{\_2\mathrm{D}} is $10^{5.6\ndash 5.8\pm 0.1}~\mathrm{m}^{-2}$
(Figure~\ref{fig4}a) and BND\tsub{\_3\mathrm{D}} is $10^{9.3}~\mathrm{m}^{-3}$ at
$\gamma = 0$ and 0.6, and $10^{9.9}~\mathrm{m}^{-3}$ at $\gamma = 1.8$.
The undeformed sample shows more than 90\% of bubbles with
20-$\rmmu$m diameters, the volume of which represents one third
of total vesicularity. The rest of the vesicularity is mostly occupied
by bubbles with equivalent diameters from 40 to 140~$\rmmu$m, with some
larger bubbles up to 240~$\rmmu$m diameter (Supplementary Figure~3f). The bubbles
have $R_{b}$ of $1.0 \pm 0.1$ at any $\gamma$, showing no preferential
orientations ($\theta_{b}$ of $33\ndash 38\text{\textdegree}$;
Figure~\ref{fig4}b). Glass H\tsub{2}O content was only analysed at
$\gamma = 1.8$, giving $3.0 \pm 0.7$~wt\%.
$\Phi_{\mathrm{c}\_\mathrm{bulk}\_2\mathrm{D}}$ is $28 \pm 1$ area\% at $\gamma
= 0$, $29 \pm 4$ area\% at $\gamma = 0.6$, and $33 \pm 4$ area\% at
$\gamma = 1.8$. The CND is $10^{5.5\ndash 5.6\pm
0.2}~\mathrm{m}^{-2}$ at any $\gamma$ (Figure~\ref{fig4}a). $R_{c}$ are
not varying from $1.1 \pm 0.1$ with $\gamma$. $\theta_{c}$ are $44 \pm
31\text{\textdegree}$, $58 \pm 38\text{\textdegree}$, and $26 \pm 13\text{\textdegree}$ 
in the samples at $\gamma = 0$, 0.6, and 1.8, respectively  (Figure~\ref{fig4}b).






\subsection{Series with 50\% crystals (serie50)}

Serie50 shows strong features of strain localization in the deformed
samples, with alternation of strongly bubble-depleted zones and
channelized bubble zones (Figure~\ref{fig2}j,k).
$\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}}$ increases from $21 \pm 2$, $23
\pm 10$, $44 \pm 17$ area\% with $\gamma$ increasing from 0, to 0.6 and
1.5 (Figure~\ref{fig3}a). Bulk $\Phi_{\mathrm{b}\_\mathrm{bulk}\_2\mathrm{D}}$ 
gives $14 \pm 2$, $16 \pm 10$, and $23 \pm 17$ area\% at $\gamma = 0$,
0.6, and 1.5, respectively, in good agreement with the XCT
$\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$ of $19 \pm 3$, $16 \pm 3$ and
$25 \pm 3$~vol\%. BND\tsub{\_2\mathrm{D}} is $10^{6.1\pm 0.1}~\mathrm{m}^{-2}$
at $\gamma = 0$ and 0.6, and $10^{5.9\pm 0.3}~\mathrm{m}^{-2}$ at
$\gamma = 1.5$ (Figure~\ref{fig4}a). BND\tsub{\_3\mathrm{D}} is
$10^{10.0\ndash 10.2}~\mathrm{m}^{-3}$ at $\gamma = 0$ and 0.6, and
$10^{9.5}~\mathrm{m}^{-3}$ at $\gamma = 1.5$. In the undeformed sample,
95\% of the bubbles have diameters of 20~$\rmmu$m, representing
about 25\% of the total vesicularity. Another third of the volume is
occupied by bubbles with  equivalent diameters of 240--260~$\rmmu$m
(Supplementary Figure~3i). The bubbles have $R_{b}$ of $1.0 \pm 0.1$ at any
$\gamma$, not showing significant preferential orientations
($\theta_{b}$ from $-$13 to ${+}33\text{\textdegree}$, with standard deviations
${>}35\text{\textdegree}$; Figure~\ref{fig4}b). The glass H\tsub{2}O contents are
$4.9 \pm 0.7$~wt\% at $\gamma = 0$ and $2.7 \pm 0.7$~wt\% at $\gamma =
1.5$. $\Phi_{\mathrm{c}\_\mathrm{bulk}\_2\mathrm{D}}$ is of 39--48 area\%
whatever $\gamma$. The CND increases from $10^{5.5\pm
0.3}~\mathrm{m}^{-2}$ at $\gamma = 0$ to $10^{5.8\pm
0.1}~\mathrm{m}^{-2}$ at $\gamma = 0.6$ and $10^{6.1\pm
0.2}~\mathrm{m}^{-2}$ at $\gamma = 1.5$ (Figure~\ref{fig4}a). $R_{c}$ are
not varying from $1.1 \pm 0.1$ with $\gamma$. $\theta_{c}$ are $40 \pm
27\text{\textdegree}$, $27 \pm 8\text{\textdegree}$, and $28 \pm 2\text{\textdegree}$ 
in the samples at $\gamma = 0$, 0.6, and 1.5, respectively (Figure~\ref{fig4}b).


\onecolumngrid

\begin{sidewaystable}[p!]
\caption{Analytical results\label{tab2}}
\tabcolsep=2.5pt\fontsize{5.9}{6.3}\selectfont
\begin{tabular}{cccccccccccccccccccccc}
\thead
\xmorerows{2}{Run \#} &
\xmorerows{2}{Series \#} & \xmorerows{2}{$\gamma$}  & 
\multicolumn{3}{c}{Glass} & \multicolumn{4}{c}{Crystals} &
\multicolumn{10}{c}{Bubbles} & \multicolumn{2}{c}{Viscosity$^{\mathrm{e}}$} \\
\cline{4-6}\cline{7-10}\cline{11-20}\cline{21-22}
& & & \morerows{1}{\parbox[t]{1.2cm}{\centering $\Phi_{\mathrm{g}\_\mathrm{bulk}\_2\mathrm{D}~(\mathrm{calc})}$ (vol\% $\pm$3)}}  &
\morerows{1}{\parbox[t]{.8cm}{\centering H\tsub{2}O\tsub{\mathrm{calc}}$^{\mathrm{a}}$ (wt\%)}}  &
\morerows{1}{\parbox[t]{.8cm}{\centering H\tsub{2}O\tsub{\mathrm{meas}}$^{\mathrm{a}}$ (wt\%)}} &
\morerows{1}{\parbox[t]{.7cm}{\centering $\Phi_{\mathrm{c}\_\mathrm{bulk}\_2\mathrm{D}}$ (area\%)}} &
\morerows{1}{\parbox[t]{.9cm}{\centering Log CND\tsub{\_2\mathrm{D}} ($\mathrm{m}^{-2}$)}}  &
\xmorerows{1}{$R_{c}$\smath{^{\mathrm{b}}}}  & 
\xmorerows{1}{$\theta_{c}$\smath{^{\mathrm{b}}} (\textdegree)}  &
\morerows{1}{\parbox[t]{1cm}{\centering \smath{\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}}\smath{^{\mathrm{c}}} (vol\% $\pm$3)}}  &
\morerows{1}{\parbox[t]{1.1cm}{\centering \smath{\Phi_{\mathrm{b}\_\mathrm{connect}\_3\mathrm{D}}}\smath{^{\mathrm{c}}} (\%, $\pm$3)}}  & 
\morerows{1}{\parbox[t]{.9cm}{\centering $\Phi_{\mathrm{b}\_\mathrm{bulk}\_2\mathrm{D}}$ (area\%)}} &
\morerows{1}{\parbox[t]{1.2cm}{\centering $\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}~(\mathrm{calc})}$ (area\%)}} &
\morerows{1}{\parbox[t]{.6cm}{\centering Log BND\tsub{\_2\mathrm{D}} ($\mathrm{m}^{-2}$)}} & 
\morerows{1}{\parbox[t]{.6cm}{\centering Log BND\tsub{\_3\mathrm{D}} ($\mathrm{m}^{-3}$)}} &
\xmorerows{1}{$R_b$\smath{^{\mathrm{b}}}}  &
\xmorerows{1}{$\theta_{b}$\smath{^{\mathrm{b}}} (\textdegree)}  & 
\multicolumn{2}{c}{$\alpha_{\mathrm{melt}}$\smath{^{\mathrm{d}}} (vol\%)} & 
\morerows{1}{\parbox[t]{.5cm}{\centering Log $\eta_{\mathrm{melt}}$ (Pa${\cdot}$s)}}  &
\morerows{1}{\parbox[t]{.5cm}{\centering Log $\eta_{\mathrm{bulk}}$ (Pa${\cdot}$s)}}  \\
\cline{19-20}
& & & & & & & & & & & & & & & & & &
\parbox[t]{.5cm}{\centering  with H\tsub{2}O\tsub{\mathrm{calc}} (wt\%)} &
\parbox[t]{1cm}{\centering  with H\tsub{2}O\tsub{\mathrm{meas}} (wt\%)}\vspace*{2pt} & &  \\ 
\endthead
Pp17 & serie0 & 0 & 69 & 3.2 & 3.6 & 0 & \textit{n.a.} & \textit{n.a.} & \textit{n.a.} & 30(28) & 49 
& 31 $\pm$ 3 & 31 $\pm$ 3 & 5.6 $\pm$ 0.1 & 10.6 & 1.3 $\pm$ 0.1 & $-$10 $\pm$ 21 & 27 & 22 & 7.1 & 7.3 \\ 
Pp16 &  serie0 &  0.6 &  76 &  3.2 &  \textit{n.a.} &  0 & \textit{n.a.} &  \textit{n.a.} &  \textit{n.a.} &  21(20) 
&  52 &  24 $\pm$ 2 &  24 $\pm$ 2 &  5.7 $\pm$ 0.1 &  10.6 & 1.9 $\pm$ 0.2 &  $+$22 $\pm$ 6 &  27 &  \textit{n.a.} &  \textit{n.a.} & \textit{n.a.} \\ 
Pp15 &  serie0 &  1.3 &  79 &  3.2 &  4.1 &  0 &  \textit{n.a.} & \textit{n.a.} &  \textit{n.a.} &  15(21) 
& 50 & 30 $\pm$ 4 & 30 $\pm$ 4 &  5.7 $\pm$ 0.1 &  10.5 &  2.0 $\pm$ 0.3 &  $+$23 $\pm$ 6 &  27 &  15 &  6.8 &  7.0 \\ 
Pp5 & serie10 & 0 & 63 & 2.8 & 6.1(5.1) & 9 $\pm$ 1 & 4.8 $\pm$ 0.2 & 1.1 $\pm$ 0.1 & $+$15 $\pm$ 33 & 28(29) 
& 59 & 29 $\pm$ 3 &  31 $\pm$ 3 &  5.6 $\pm$ 0.2 &  9.3 &  1.0 $\pm$ 0.1 &  $+$20 $\pm$ 43 &  60 &  41 &  4.9 &  5.2 \\ 
Pp7 &  serie10 &  0.6 &  65 &  2.8 &  5.3(5.9) &  12 $\pm$ 2 &  5.1 $\pm$ 0.3 &  1.1 $\pm$ 0.1 &  $+$18 $\pm$ 3 
&  18(23) &  64 &  24 $\pm$ 3 &  27 $\pm$ 3 &  5.2 $\pm$ 0.2 &  9.2 & 1.3 $\pm$ 0.1 &  $+$35 $\pm$ 4 &  60 &  46 &  5.2 &  5.4 \\ 
Pp6 &  serie30 &  0 &  63 &  2.8 &  \textit{n.a.}(4.3) &  28 $\pm$ 1 & 5.5 $\pm$ 0.2 &  1.1 $\pm$ 0.1 &  $+$44 $\pm$ 31 &  22(22) &  37 
& 15 $\pm$ 3 &  20 $\pm$ 3 &  5.8 $\pm$ 0.1 &  9.3 &  1.0 $\pm$ 0.1 & $-$8 $\pm$ 74 &  60 &  \textit{n.a.} &  \textit{n.a.} &  \textit{n.a.} \\ 
Pp8 &  serie30 &  0.6 &  57 &  2.8 &  \textit{n.a.} &  29 $\pm$ 4 & 5.5 $\pm$ 0.1 &  1.1 $\pm$ 0.1 &  $+$58 $\pm$ 38 &  25(27) 
&  83 & 20 $\pm$ 3 &  26 $\pm$ 3 &  5.6 $\pm$ 0.1 &  9.3 &  1.0 $\pm$ 0.1 &  $+$33 $\pm$ 17 &  60 &  \textit{n.a.} &  \textit{n.a.} & \textit{n.a.} \\ 
Pp9 &  serie30 &  1.8 &  62 &  2.8 &  3.0 &  33 $\pm$ 4 &  5.6 $\pm$ 0.2 &  1.1 $\pm$ 0.1 &  $+$26 $\pm$ 13 &  15(10)
&  80 &  23 $\pm$ 12 & 26 $\pm$ 12 &  5.7 $\pm$ 0.1 &  9.9 &  1.0 $\pm$ 0.1 &  $+$38 $\pm$ 22 &  60 &  57 &  6.2 &  6.9 \\ 
Pp12 &  serie50 &  0 &  53 &  2.8 &  4.9 &  48 $\pm$ 3 &  5.5 $\pm$ 0.3 &  1.1 $\pm$ 0.1 &  $+$40 $\pm$ 27 &  19(19) &  87 
& 14 $\pm$ 2 &  21 $\pm$ 2 &  6.1 $\pm$ 0.1 &  10.2 &  1.0 $\pm$ 0.1 & $-$13 $\pm$ 42 &  60 &  49 &  5.3 &  6.5 \\ 
Pp11 &  serie50 &  0.6 &  54 &  2.8 &  \textit{n.a.} &  43 $\pm$ 4 & 5.8 $\pm$ 0.1 &  1.1 $\pm$ 0.1 &  $+$27 $\pm$ 8 &  16(15) 
&  76 & 16 $\pm$ 10 &  23 $\pm$ 10 &  6.1 $\pm$ 0.1 &  10.0 &  1.0 $\pm$ 0.1 &  $+$33 $\pm$ 36 &  60 &  \textit{n.a.} &  \textit{n.a.} & \textit{n.a.} \\ 
Pp10 &  serie50 &  1.5 &  49 &  2.8 &  2.7 &  39 $\pm$ 7 &  6.1 $\pm$ 0.2 &  1.1 $\pm$ 0.1 &  $+$28 $\pm$ 2 &  25(27) &  99 
& 39 $\pm$ 17 & 44 $\pm$ 17 &  5.9 $\pm$ 0.3 &  9.5 &  1.0 $\pm$ 0.1 & $-$1 $\pm$ 36 &  60 &  58 &  6.4 &  7.4 
\botline 
\end{tabular}
\tabnote{\textit{n.a.} indicates not analyzed;}
\tabnote{$^{\mathrm{a}}$H\tsub{2}O\tsub{\mathrm{calc}} is the H\tsub{2}O
solubility at 650 or 750~\textdegree C and 50~MPa calculated after
\citet{NewmanLowenstern2002}; H\tsub{2}O\tsub{\mathrm{meas}} is the glass
H\tsub{2}O content measured by EMP (analytical uncertainty of 
$\pm$0.7~wt\%) and by Raman spectroscopy in bracket (average of 2 to 3
analyses with statistical error from $\pm$0.2 to  $\pm$0.7~wt\%).}
\tabnote{$^{\mathrm{b}}\theta_{c}$ and $\theta_{b}$ are the
orientations to the horizontal of the bubbles and crystals,
respectively; $R$ gives the bubble mean eccentricity as the ratio of
the long axes to the short axes.}
\tabnote{$^{\mathrm{c}}\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$
gives the bulk porosity measured on cubic samples of 1400-$\rmmu$m side
volume and at smaller scale (600-$\rmmu$m side) in brackets;
$\Phi_{\mathrm{b}\_\mathrm{connect}\_3\mathrm{D}}$ gives the percentage
of bubbles connected to the sample outside, as defined in Supplementary Section~1.}
\tabnote{$^{\mathrm{d}}\alpha_{\mathrm{melt}}$ is calculated after
\citet{JaupartAllegre1991} for a degassing from the H\tsub{2}O content
of the pre-hydrated glass (H\tsub{2}O\tsub{i} in Table~\ref{tab1}) to
either H\tsub{2}O\tsub{\mathrm{calc}} or H\tsub{2}O\tsub{\mathrm{meas}}.}
\tabnote{$^{\mathrm{e}}\eta_{\mathrm{melt}}$ is the viscosity of the
hydrated melt calculated at 650 or 750~\textdegree C after 
\citet{HessDingwell1996} using H\tsub{2}O\tsub{\mathrm{meas}};
$\eta_{\mathrm{bulk}}$ is the viscosity of the three-phase (crystal,
bubble, melt) suspension calculated after \citet{Trubyetal2015} using
$\eta_{\mathrm{melt}}$,
$\Phi_{\mathrm{c}\_\mathrm{bulk}\_2\mathrm{D}}$, and
$\Phi_{\mathrm{b}\_\mathrm{melt}\_2\mathrm{D}}$ (see text and 
Supplementary Section~1 for details).}
\end{sidewaystable}


\twocolumngrid

\clearpage

\section{Discussion}
\subsection{Bubble nucleation and growth}

Exsolving H\tsub{2}O from 9.6 to 2.8~wt\% [solubility at 50~MPa and
750~\textdegree C; \citealp{NewmanLowenstern2002}] corresponds to a
calculated vesicularity of 58~vol\% [\citealp{JaupartAllegre1991};
Supplementary Section~1 (Equation~(1))]. Yet the average porosities measured in the
decompressed samples are mostly below 35~vol\% (Figure~\ref{fig3}a). We
propose two explanations to account for this deviation; incomplete
H\tsub{2}O exsolution and partial loss of bubbles.

\subsubsection{Incomplete H\tsub{2}O exsolution}

Checking the H\tsub{2}O contents of the residual glasses in the
undeformed samples using both electron microprobe and Raman
spectroscopy, revealed contents in the crystal-bearing samples higher
by 1.5 to 3.5~wt\% than expected for an equilibrium H\tsub{2}O
exsolution (Figure~\ref{fig3}b). This suggests incomplete H\tsub{2}O
exsolution during the ${\sim}$1.5~h ($\tau_{\mathrm{DP}}$~$+$ Dwell2
$+$ Dwell3 in Table~\ref{tab1}) allocated to decompression-induced
bubble nucleation. Yet, all series show equilibrium H\tsub{2}O
exsolution (${\sim}$3.0~wt\% H\tsub{2}O) at $\gamma$ of 1.5--1.8, that
is, for another couple of hours spent at 50~MPa, thus confirming that
H\tsub{2}O exsolution under the chosen experimental conditions requires
more than 1~h to reach equilibrium. This strongly disagrees with the
conclusions of \citet{MartelSchmidt2003} suggesting H\tsub{2}O
exsolution timescales of less than 1~min in microlite-bearing rhyolitic
melts at 860~\textdegree C, but agrees with complete bubble expansion
requiring several hours. Therefore, we infer that in our experiments,
the timescale of bubble expansion controlled H\tsub{2}O exsolution from
the melt. In contrast, the undeformed crystal-free samples degassed to
equilibrium, which we rather attribute to their lower initial
H\tsub{2}O content (5.0 instead of 9.6~wt\%) rather than the absence of
crystals, since serie10 which contains 10~vol\% crystals, shows the highest
H\tsub{2}O exsolution disequilibrium. Further investigation would be
required to validate the positive correlation between the initial
H\tsub{2}O content of the melt and the timescale of bubble expansion,
which in turn controls the H\tsub{2}O content of the residual melt.


\subsubsection{Partial loss of bubbles}

Knowing that H\tsub{2}O exsolution was not completed in the
H\tsub{2}O-rich samples, the gas fractions were\unskip\break calculated using the
final H\tsub{2}O contents measured in the residual glasses instead of
using H\tsub{2}O solubility at 50~MPa. The results show that the
measured porosities are lower by 10 to 30~vol\% than
expected for an equilibrium degassing (Figure~\ref{fig3}a), suggesting
that these samples lost a part of their bubble cargo. One explanation
could be that the melt partially outgassed by bubble connection at some
point during bubble growth, and partially collapsed via compaction. Yet
the released gas pockets cannot escape the sealed capsule and were
considered in the measurement of total vesicularity. Therefore, we
infer that these gas pockets were heterogeneously distributed in the
capsule corners and were not systematically visible on our SEM pictures
(for instance, a large gas pocket is visible below the capsule roof in
Figure~\ref{fig2}c), so that they may miss from some porosity budget. This
hypothesis is supported by the lack of clear relationships between the
bubble deficiency and parameters such as H\tsub{2}O content, crystal
content, or finite deformation.

\subsubsection{Pre-deformation porosity}

The undeformed samples show melt vesicularities of 19--30~vol\%
(Figure~\ref{fig3}a) and bubble number densities not varying by more than
0.5 log unit (Figure~\ref{fig4}a), which suggests a comparable and
reproducible process of bubble formation. Within this 0.5 log unit,
however, BND increases with crystal content, as experimentally observed
by \citet{Caceresetal2021}. The bubble number densities of
$10^{9.3\ndash 10.6}~\mathrm{m}^{-3}$ measured from the XCT images
are lower by 1 log unit compared to those predicted in crystal-free
magmas by the formulation of \citet{Toramaru2006} for decompression
rates of 0.1--1.0~MPa/s (i.e., BNDs of
$10^{10.5\ndash 11.5}~\mathrm{m}^{-3}$), highlighting a possible role
of the crystals in controlling nucleation sites for bubbles, as already
pointed out by \citet{Caceresetal2021}. In the undeformed samples from
all series, more than half (and up to 95\% in serie0 and serie50) of
the bubbles have diameters of about 20~$\rmmu$m. These small bubbles
occupy about one third of total porosity in all series but serie0. In
serie0, the volume of the small bubble population nearly disappeared at
the expense of larger (220--240~$\rmmu$m diameter) bubble volume
(Supplementary Figure~3a). In serie50, the population of 20-$\rmmu$m diameter
bubbles coexists with some 240--260~$\rmmu$m diameter bubbles
(Figure~\ref{fig2}i) representing one third of total porosity 
(Supplementary Figure~3i). The homogeneous spatial distribution of
the small 20--40~$\rmmu$m bubbles (Figure~\ref{fig2}a) confirms a
process of homogeneous nucleation \citep{MourtadaBonnefoiLaporte2002},
in agreement with the observations in natural and experimental products
that plagioclase is not a wetting phase for nucleating H\tsub{2}O
bubbles \citep{Navonetal1998,Caceresetal2021}. The population of
200--300~$\rmmu$m bubbles may have three possible origins: coalescence
of the 20--40~$\rmmu$m bubbles, a separate nucleation event, or
pre-decompression bubbles. A separate nucleation event or bubble
coalescence would lead to more or less similar bubble size
distributions in the different series, which is not the case 
(Supplementary Figure~3a,d,f,i). Therefore, we argue in favour of
bubbles that formed before decompression, as often observed in
experiments starting with powdered material that is able to trap
H\tsub{2}O and air in the intergranular voids before melting [i.e., the
``hydration bubbles'' in \citealp{Gardneretal1999}]. The number and size
of these bubbles would then be affected by the crystal content in the
glass powder. The undeformed samples of serie30 and serie50 show that
bubbles started channelling along the crystals (Figure~\ref{fig2}f,i).
According to the model of \citet{Burgisseretal2017}, crystal-free melts
containing 20--30~vol\% bubbles, number density of $10^{9\ndash
11}~\mathrm{m}^{-3}$, and size of 200--300~$\rmmu$m, would be
impermeable, even with 50--70\% connected porosity. In contrast, gas
permeability measurements in natural crystal-bearing samples with
connected porosities from 50 to 80\% give permeabilities from
${\sim}10^{-14~\mathrm{to}~-10}~\mathrm{m}^{2}$ \citep{Bainetal2019}.
Therefore, in absence of permeability measurements in our samples, it
is not clear whether the gas percolation thresholds were met before
deformation. 

\subsubsection{Bulk viscosity}

The bulk viscosities calculated using the formulation of 
\citet{Trubyetal2015} range from
$10^{5.2\ndash 5.4}~\mathrm{Pa}{\cdot}\mathrm{s}$ for the series
containing 10~vol\% crystals at 750~\textdegree C,
$10^{6.5\ndash 7.4}~\mathrm{Pa}{\cdot}\mathrm{s}$ for the series
containing 30 and 50~vol\% crystals at 750~\textdegree C, to
$10^{7.1\ndash 7.3}~\mathrm{Pa}{\cdot}\mathrm{s}$ for the
crystal-free series at 650~\textdegree C (Table~\ref{tab2}). The regime
of low capillary number ($Ca < 1$) calculated for our sample
deformation conditions (see Supplementary Section~1.3.2 for details) results in
considering the 20--30 bulk~vol\% bubbles as rigid particles that
increase the hydrated melt viscosity by about 0.2 log unit (e.g. serie0
in Table~\ref{tab2}). Adding crystals further increases bulk viscosity,
from ${\sim}$0.3 log unit for 10~vol\% added crystals to ${\sim}$0.7
log unit for 30~vol\% crystals, and 1.0--1.2 log unit for 40--50~vol\%
crystals, which agrees with most of the common models of particle
suspension rheology \citep[see][]{Maderetal2013}. 

\subsection{Effect of the crystal network on porosity development under small shear 
$(\gamma < 2)$}

\subsubsection{Crystalline fabric}

The crystals were nearly isotropic in shape ($R_{c} = 1.1 \pm 0.1$;
Table~\ref{tab2}), suggesting that the cutting of the labradorite block
using the Selfrag apparatus did not permit to reproduce the elongated
plagioclase crystals commonly observed in the natural magmas. This
small anisotropy likely limited the development of strong crystalline
fabrics in the sheared samples. Nevertheless, the crystals show main
orientations that agree with those calculated as a function of $\gamma$
by \citet{Fernandezetal1983} for solid particles of $R_{c} = 1.1$
(Figure~\ref{fig4}b). Yet, the high standard deviations associated with 
these angles, especially at $\gamma = 0$ and 0.6, suggest that the
crystalline fabric is very weak, with the exception of serie50 at
$\gamma = 0.6$ and 1.5, for which the crystalline fabrics are well
marked (standard deviations ${<} 10\text{\textdegree}$). The serie30 and serie50
samples show crystal breakage features, evidenced by the appearance of
a population of small crystals (Figure~\ref{fig2}i,k) that are also
visible in the 2D crystal size distributions (Supplementary Figure~4). Crystals
breakage was also observed in crystal-rich samples deformed under low
strains by \citet{Forienetal2011}, likely resulting from intense
stress localization at some grain contact.


\subsubsection{Bubble spatial distribution}

The melt porosity of 20--30 area\% is maintained in whatever crystal
content and $\gamma$, with the exception of one sample
(Figure~\ref{fig5}a), which agrees with the results of 
\citet{Pistoneetal2012} obtained on their samples sheared at $T$ high
enough to prevent bubble loss through shear fracturing. Yet in details,
increasing both bulk crystal content ${\geq}$30~vol\% and finite
strain, increases heterogeneities in bubble spatial distribution, as
reflected by porosity variations up to ${\pm}$20\% between bubble-poor
and bubble-rich zones, with the variation being positively correlated
to $\gamma$ (Figure~\ref{fig5}a). For these samples, the gas accumulated
in local microstructures caused by the shear-induced crystal fabric
(Figure~\ref{fig2}h,k), in agreement with the observations of
\citet{Laumonieretal2011}. In samples with bulk crystal contents
${<}$30~vol\%, however, our results suggest that the spatial
distribution of the bubbles remain homogeneous (variations below
${\pm}$5~vol\%; Figure~\ref{fig5}a).


\begin{figure}
\includegraphics{fig05}
{\vspace*{-6pt}}
\caption{Effects of the crystals on (a)
melt porosity and (b) connected porosity; the solid line indicates 
the linear fit of the data but one point ($\gamma=0$) from serie30 with
an anomalously low connected porosity; the dashed lines contour the
evolution of the connected porosity as a function of bulk crystallinity
(0--50~vol\%) for bulk vesicularities of 20--30~vol\% and shear strain
of $0 < \gamma < 2$.\label{fig5}}
{\vspace*{-2pt}}
\end{figure}



\subsubsection{Connectivity development}




At $\gamma = 0$, bubble connectivity increases with crystal content
(Figure~\ref{fig6}), reaching ${\sim}$87\% connected porosity in the
sample with bulk crystal contents of 50~vol\% (Figure~\ref{fig5}b; one
exception is the undeformed sample of serie30 with only 37\% connected
porosity, for which we have no explanation). These results agree with
those of \citet{Caceresetal2021} who observed that phenocryst-bearing
magma noticeably showed a higher degree of bubble coalescence, but
disagree with those of \citet{Okumuraetal2012} who reported no clear
effect of crystallinity (30--50~vol\% recalculated with respect to the
melt volume) on the degree of bubble coalescence and connectivity until
melt vesicularities reached ${\sim}$68~vol\%. This disagreement may be
explained by differences in bubble texture. Indeed, their crystal-rich
samples with vesicularity ${\sim}$20--30~vol\% (i.e. those decompressed
to 20~MPa; number densities of ${\sim}10^{12}~\mathrm{m}^{-3}$) do not
show bubble channelling features, such as those observed in the
undeformed samples of our serie50 (Figure~\ref{fig2}). This could also
result from different bulk viscosities, different bubble to crystal
size ratios, or unimodal versus polymodal bubble size distributions
(i.e. the ${\sim}$200--300~$\rmmu$m bubble population in our samples
may have played a key role in early connectivity development).


\begin{figure*}
\includegraphics{fig06}
\vspace*{-2pt}
\caption{XCT image processing of the core samples (3~mm diameters) for
increasing shear strain from $\gamma = 0$ to 2 (from left to right) and
crystal content from 0 to 50~vol\% (from top to down). The left blocks
were processed using VGStudioMax, with the melt and crystals in yellow,
the isolated bubbles in red and the bubbles connected to the sample
outside in grey. $\Phi_{\mathrm{b}\_\mathrm{bulk}\_3\mathrm{D}}$ and
$\Phi_{\mathrm{b}\_\mathrm{connect}\_3\mathrm{D}}$  give the 3D percentage of
bulk and connected bubbles, as defined in Supplementary Section~1. The upper right
insets show details of the bubbles processed using Blob3D, with another
colour code that show one colour per isolated bubble or identified
network of connected bubbles. The black line in (e) highlights an
alignment of bubbles connected to the sample outside, which denotes a
post-experimental fracture.\label{fig6}}
\vspace*{-2pt}
\end{figure*}

In our samples, increasing bubble connectivity with increasing crystal
content before shearing ($\gamma = 0$) highlights the role of the
crystal network on the decompression-induced formation of bubbles.
Firstly, the crystals possibly influence bubble nucleation, by
increasing bubble number density by up to 0.5 log unit with increasing
crystal content (see log BND at $\gamma = 0$ in Figure~\ref{fig4}a). We
speculate that physical processes at the origin of more bubble nuclei
in crystal-richer magmas could involve modified gas-melt interfacial
tensions or excess free energy. After nucleation, bubble growth must
have been partly dictated by the crystal network, forcing bubble
coalescence and channelling, eventually approaching full bubble
connection in serie50 (i.e. ${>}$80\% connectivity; Figures~\ref{fig5}b
and \ref{fig6}i). These observations confirm the results of the
numerical modelling of \citet{Parmigianietal2017} predicting maximum
outgassing efficiency via 40--50\% connected porosity in crystal-rich
(40--70~vol\%) magmas. 


Increasing $\gamma$ from 0 to 2 does not significantly
influence the amount of connected porosity (Figure~\ref{fig5}b), thus
highlighting the large predominance of crystal content over shear
strain up to $\gamma = 2$ in favouring bubble connections. Fitting all
the data but one (the low-connected sample at $\gamma = 0$ from
serie30) shows a positive linear correlation between connected porosity
and bulk crystal content ($y = 0.84 x + 52$; Figure~\ref{fig5}b),
thus valid for silicic magmas with 20--30~vol\% bulk vesicularity and
sheared at low strain ($\gamma < 2$) with shear rates of
$10^{-4}~\mathrm{s}^{-1}$ under a low~confining pressure of 50~MPa. The
rather poor \mbox{coefficient} of determination, $R^{2}=0.84$, mostly reflects
a large variability of the connected porosity at high crystallinity
(${\pm}$15\% for crystal content ${\sim}$45~vol\%) with respect to
crystal-free samples (${\pm}$2\%; Figure~\ref{fig5}b).




\begin{figure*}
\vspace*{-2pt}
\includegraphics{fig07}
\vspace*{-2pt}
\caption{Schematic representation of bubble connectivity in magmas with
(a) 0--10~vol\% crystals or (b)~30--50~vol\% crystals, deformed under
shear at magmatic temperatures (650--750~\textdegree C), moderate
shear-strain rates ($2 \times 10^{-4}~\mathrm{s}^{-1}$) and low
pressures (50~MPa, that is at level of upper conduit to lava dome
interior). The gas bubbles in crystal-poor magmas with 20--30~vol\%
porosity hardly connect under strain below $\gamma = 2$, so that
outgassing requires fracture opening under high strains or strain
rates, likely only reachable at conduit margins 
\citep{Kushniretal2017}. In contrary, a dense crystalline fabric
favours bubble connectivity under strains as low as $\gamma = 2$ and
bulk porosity of 20--30~vol\%, so that outgassing may be an early
process upon magma emplacement. Crystal fragmentation commonly occurs
in crystal-rich magmas and heightens with increasing crystal content
and strain. If temperature decreases and/or strain increases as
expected at conduit margins, fractures may open and assist outgassing 
\citep{Pistoneetal2012,Shieldsetal2014}.\label{fig7}}
\vspace*{-4pt}
\end{figure*}



\subsection{Implications for natural volcanic systems}

The experimental samples are silica-rich melts containing 20--30~vol\%
bubbles and 0 to 50 bulk vol\% plagioclase crystals that cover a range
from natural moderately-porous obsidians to crystal-rich (mush-like)
rhyolitic magmas. The 50--90~$\rmmu$m crystal size typically scales
microlites to microphenocrysts 
\citep[e.g.][]{Hammeretal1999,MartelPoussineau2007}. The low confining
pressure of 50~MPa and the low shear strains ($\gamma < 2$) applied to
the samples simulate a shear initiation at shallow depth (about 2~km
deep, considering rock density of $2400~\mathrm{kg}/\mathrm{m}^{3}$),
such as in a volcanic upper conduit. Overall, these conditions may be
representative of Vulcanian events or dome-related eruptions, during
which a moderately-porous magma degasses and crystallizes microcrysts
at shallow depths, eventually emplacing effusively (as domes or plugs)
or erupting explosively (as surges), depending on outgassing
efficiency. In particular, vesicularities of 20--30~vol\% are
representative of those measured in lava domes and blast-generated
pyroclastic density currents, such as the ones generated in 1350 AD
(P1) and 1902 at Montagne Pel\'{e}e, Martinique \citep{Marteletal2000}
and in 1980 at Mount St Helens, USA \citep{HoblittHarmon1993}.\looseness=-1


The implications of our results to volcanic systems are schematized in
Figure~\ref{fig7} and detailed\unskip\break below. Our results suggest that H\tsub{2}O
bubbles nucleating in undeformed aphyric or crystal-poor (0--10~vol\%)
magmas by decompression at rates of 0.1--1.0~MPa/s (i.e. bubble number
densities around $10^{9 \ndash 11}~\mathrm{m}^{-3}$) can connect to 
50\% (Figure~\ref{fig6}a,d), but the magma could still be impermeable
mainly due to the moderate bulk porosity of 20--30~vol\% 
\citep{Burgisseretal2017}. Such moderately-porous crystal-poor melts
may reach permeability via a process of fracturing at high strains
($\gamma > 2$) along the conduit margins, as experimentally
demonstrated by \citet{Kushniretal2017}. Adding crystals drastically
increases bubble connectivity (${>}$70\%; Figure~\ref{fig5}b), by
favouring channelling even at low shear strains (Figure~\ref{fig2}h,k).
Channelling creates gas permeability of the order of
$10^{-10}~\mathrm{m}^{2}$, which is two to five orders of magnitude
higher than the other permeability processes at the equivalent bubble
content \citep{Collombetetal2021}. Therefore, magmas with matrix
crystallinity ${>}$30~vol\% and vesicularity of 20--30~vol\%, as
commonly observed in lava domes \citep{Boudonetal2015}, may show bubble
connectivity ${>}$70\% (Figure~\ref{fig5}b) and gas permeabilities
${>}10^{-12}~\mathrm{m}^{2}$~\citep{Bainetal2019}. Our data suggest
that bubble connectivity in low-porous (20--30~vol\%) silica-rich
magmas slightly sheared ($\gamma < 2$; strain rate of
$10^{-4}~\mathrm{s}^{-1}$) at low confining pressure (50~MPa; about
2~km deep) increases linearly with crystal content (following $y = 0.84
x + 52$, with $y$ being the connected porosity in vol\% and $x$ being
the bulk crystal content in area\%; Figure~\ref{fig5}b). In natural
samples, connectivity would be enhanced by unequal bubble sizes that
favour bubble coalescence \citep{LinLin2009}, microtextures made of
crystals with high aspect ratios \citep{Lindooetal2017}, and
polydisperse crystal size distributions involving phenocrysts and
microlites \citep{Caceresetal2021}. Therefore, in comparison to
crystal-poor magmas, crystal-rich melts are able to outgas much more
easily (at lower $\gamma$) for comparable temperature, pressure, and
strain rate, confirming the effusive mode of crystal-rich lava-dome
emplacement and the unnecessity to invoke dominant outgassing
mechanisms other than channelling \citep{Collombetetal2021}. From a
crystallinity point of view and regardless of melt H\tsub{2}O content
and magma storage pressure, the gas-permeable nature of the 30--50
vol\% crystal-bearing rhyolitic melts investigated in this study agrees
with the eruptions of low magnitude reported by
\citep{Takeuchietal2021} and the crystallinity-controlled effusive
eruptions reported by  \citet{Popaetal2021}. Yet, the present
experiments highlight the major control of an exsolved volatile phase
(20--30~vol\%) and low shear ($\gamma < 2$) on the permeability
development, with consequences on eruptive style that may deviate from
those expected from the sole consideration of crystallinity and melt
H\tsub{2}O content. Since low-sheared crystal-rich magmas at upper
conduit level is permeable to gases, this poses the question of the
mechanisms behind dome-related explosive eruptions, such as Vulcanian
or Pelean explosions. External factors can be invoked, such as
depressurization due to unloading by gravitational collapse of a part
of the dome \citep{Voightetal1981}. Among internal factors of
self-explosivity, one could invoke processes capable of building up
overpressure in gas bubbles from an initially permeable magma, such as
a second boiling event in response to late-stage extensive microlite
crystallization \citep{Sparks1997} and gas volume reduction by
silica-phase deposition in bubbles  \citep{Boudonetal2015}. Another
hypothesis for self-explosivity is permeability barriers and pressure
accumulation via vertical rheological and lithological gradients in the
magma column\unskip\break  \citep[e.g.,][]{Voightetal1999}.


\section{Conclusions}

By deforming silicic melts with 20--30~vol\% H\tsub{2}O bubbles and 0
to 50~vol\% crystals at magmatic temperatures (650--750~\textdegree C)
and low confining pressures (50~MPa), we highlighted that:
\begin{itemize}
\item Torsion experiments using three-phase (bubbles, crystals, and
melt) suspensions are valuable analogues to shear initiation in
moderately-porous aphyric to crystal-bearing silicic magmas at shallow
depth, such as in lava domes or upper conduits.
\item Bubble connectivity linearly increases with crystal content. 
\item Increasing strain from $\gamma = 0$ to $2$ in
crystal-bearing magmas drastically increases the heterogeneity in
bubble spatial distribution, by concentrating pores in shear zones. 
\item Crystal-rich magmas develop gas permeability via a process of
channelling in shear zones, even at moderate porosities (20--30~vol\%)
and low strains ($\gamma < 2$).
\item Natural crystal-rich magmas stalling in upper conduits or lava
domes are thus likely permeable, so that violent dome explosions into
devastating surges require additional mechanisms of gas pressurisation
in shallow crystal-rich magmas.
\end{itemize}


\section*{Conflicts of interest}
Authors have no conflict of interest to declare.


\section*{Acknowledgements}

The authors greatly thank I. di Carlo, S. Erdmann, and P. Benoist for
assistance with the SEM and~EMP; M.~Hatton for the analyses using the 
elemental \mbox{analyser;} A. Slodczyk and F. Faranda for the analyses \mbox{using}
Raman spectroscopy; P. Penhoud for the technical assistance with the 
X-ray microtomograph; and M. Beaulieu for the crystal block grinding
using the Selfrag device at BRGM. This work is part of CD PhD
thesis that was funded by the French MNRT grant. The study costs
were provided by the CNRS-INSU TelluS program, the Agence Nationale de
la Recherche (ANR-19-CE31-0007; LA), the EQUIPEX PLANEX project
(ANR-11-EQPX-0036; B. Scaillet), and the LABEX VOLTAIRE project
(ANR-10-LABX-100-01; B. Scaillet). The authors greatly thank B. J.
Andrews and two anonymous reviewers for their helpful comments on a
previous version of the manuscript, as well as R. Cioni and an
anonymous reviewer for their comments on the present manuscript. We
would also like to thank B. Scaillet for the editorial handling.

\CDRGrant[ANR]{ANR-19-CE31-0007}
\CDRGrant[ANR]{ANR-11-EQPX-0036}
\CDRGrant[ANR]{ANR-10-LABX-100-01}

\back{}

\section*{Supplementary data}
Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.  

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crgeos-214-suppl.pdf}}

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