JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 95, NO. B4, PAGES 5141-5151, APRIL 10, 1990
Mechanical Stability of Shallow Magma Chambers G. SARTORIS,1,2J.P. PozzI, 3 C. PHILIPPE,4 AND J. L. LE MOOELs The mechanicalstability of the wall of a magma chamber dependson the value of the stresseslikely to be developedin the immediate vicinity of this boundary in relation to the conditionsfor failure to occur. Various structural and physical factors contributing to these stresses are systematically analyzed. Models of volcanic systemshaving an axial symmetry around a vertical axis with an isolated reservoir, filled with magma under pressure, in a homogeneouselastic medium are investigated. Spheroidal magma chambersof different aspect ratios (0.7-2) and approximately the same volume
(13.5-14km3),withtopsat 3.5-4.5kmdepth,areconsidered. Stressdistributions including theeffect of gravity, free surfaceof the Earth, and remote stressesincreasingwith depth are calculated by using a numerical finite element method. The developmentof tensile tangential stressesgreater than 10 MPa in the elements adjacent to the wall of the chamber is assumed to be a sufficient condition for its instability. The standardstate of stressassumedfor the crust (i.e., the boundary conditionsimposed at large horizontal distancefrom the chamber) is varied in a continuousrange having as lower limit a state of uniaxial strain in the vertical direction and as upper limit a state of hydrostatic stress. Calculationsare first performedby makingthe assumptionthat the magmapressureP acting at the top of a chamberequalsthe lithostaticpressureof the overlyingrocks. Alternatively, the upper limit of the valuesof P (critical pressurePc) for which the initial shapeof a chambermay be stable, accordingto the assumednecessaryconditionsfor stability, is determined.For boundary conditionscorresponding to uniaxial strain and Poisson'sratio v = 0.25, Pc turns out to be approximatelyhalf of the lithostatic pressureof the overlying rocks. A simple criterion is proposedto estimate if a chamber may evolve toward a new stable shapewhen P > Pc. This is highly improbableif P equalsthe lithostatic pressure of the overburden. Larger values of Poisson's ratio (0.30-0.35) favor stability, yielding critical pressuresexceedingthe lithostatic pressure. Stability is extremely sensitiveto the standard state of stressassumedfor the crust. Boundary conditionsprogressivelyapproachinga hydrostatic state of remote stressesyield increasingly higher critical pressures. It turns out that the density contrast between magma and host rocks is not crucial for stability (for Ap/p < 11%). For the rather large spheroidalchambersinvestigated,the shapeis not a critical factor either. The effect of topographical details (e.g., a volcanic edifice, 1.5 km high and 6 km in radius) is practically irrelevant.
INTRODUCTION
therefore neglecting the effects of the free surface of the Failure of the wall of a magmareservoir and development Earth, and only plane cross sections through magma bodies of fractures have been the object of a number of theoretical of simple geometric forms are considered. Less idealized studies. In these works, the stressfield around pressurized numerical models have only recently been proposed. Ryan [1988] performed mechanical analyses of dyke emplacement cavities is calculated within the framework of linear elasticwithin the rift zones of Kilauea, Hawaii, by using a finite ity and is used to explain the formation of secondary element method. However, in his calculations he was conintrusive systemsfed from the reservoir. Most of the models proposed to date [Anderson, 1936, cerned only with the changes of stress associated with dyke 1938; Od•, 1957; Robson and Barr, 1964; Roberts, 1970; intrusion. Chevallier and Verwoerd [1988] followed the same general approach as Anderson [1936, 1938] and developed a Pollard, 1973a, b; Phillips, 1974; Koide and Battacharji, behavior of an 1975; Gudmundsson, 1986; Sammis and Julian, 1987] are numerical simulation of the mechanical exactly solvable models. The advantage of exact solutions axisymmetric volcano, but their model suffers from the with straightforward relationshipsbetween variables is ob- assumption of uniform far-field stresses. M. Como and M. vious. On the other hand, exactly solvable models are Lembo (unpublished work, 1989) developed a similar finite extremely idealized and not necessarily applicable to real element simulation for Phlegraean Fields, Italy. Despite the problems.In particular, they sufferfrom two drastic approx- fact that this historical review is brief and undoubtedly imations. First, gravity is neglected. Second, the far-field incomplete, it should be clear that much work remains to be stresses are assumed to be uniform, whereas the standard done in this field of physical volcanology. state of stress within the crust is dependent on depth. In The mechanical evolution of a magma reservoir is govaddition, sometimes the medium is assumed to be infinite, erned by several structural and physical factors, such as the geometry of the volcanic system, the properties of the 1Institutde Physique du Globede Paris,Observatoires Volca- magmaand of the host medium, the magmaticpressure, and nologiques,Paris, France.
the remote
stresses. Information
on these factors
is inevita-
2Nowat Dipartimento di Scienze Fisiche,Universith di Napoli, bly incomplete, and a number of assumptions have to be Naples,Italy.
made in order to construct a model. That these assumptions may be interrelated is generally disregarded. For instance, in 4EcoleNationale Sup•rieure d'Artset M•tiers,Paris,France. modeling mechanical stability (prolonged existence) of res5Institutde Physique du Globede Paris,Observatoires Volca- ervoirs, any assumption on the physical properties of the nologiques,Paris, France. host medium and the far-field stressestacitly implies restricCopyright 1990 by the American GeophysicalUnion. tions on the pressure that the wall of the chamber can bear without failing. This aspect of the problem appears to have Paper number 89JB03293. 0148-0227/90/89JB-03293 $05.00 been overlooked by Chevallier and Verwoerd [1988]. 'Ecole Normale Sup6rieure, D6partement de G•ologie, Paris,
France.
5141
5142
SARTORIS ET AL.: MECHANICAL
STABILITY
Rather low magma pressuresare expected to provide for mechanical stability and long-term evolution of shallow reservoirs. The reason is that once the yield strengthof the overlying rocks is exceeded, the volcanofails and pressureis released. On the other hand, there is some evidence that pressures within exploding magma chambers may be great enough to form shocked minerals [Carter et al., 1986; Rice, 1987, also unpublished work, 1988]. This would imply that
just before explosion, pressuresmay increaseso rapidly as to prevent a progressivedynamic responseof the host rocks and a premature failure of the volcano [Rice, 1987, 1988; Loper and McCartney, 1988]. In this paper, we shall ignore such extremely rapid and poorly known processes. We consider models of volcanic systemscharacterized by axial symmetry around a vertical axis with an isolated magma reservoir in a homogeneous elastic medium. Stress distributions, including the effect of gravity, free surface of
OF MAGMA CHAMBERS
0
c•8
I0
the Earth, and remote stressesincreasingwith depth, are calculated by using a numerical finite element method. Since the main purpose of this work is to analyze the interplay of the various factors controlling the stability of the wall of the reservoir, only necessary conditions for stability are considered, and no attempt is made to use the computedstressfield to explain the development of secondary intrusions (cone sheets, ring dykes, etc.). Spheroidalmagma chamberswith different aspectratios are considered.Calculationsare performed for a range of boundary conditions(magmapressure and remote stresses) and for various values of magma density and Poisson's ratio of the country rocks. Our interest in this problem was initially related to the interpretation of the considerable ground deformation recently observed at Phlegraean Fields and to the risk of eruption [Bianchi et al., 1987]. Most of the models considered in this work were constructed having in mind this early interest. Therefore, despite the fact that they are still too idealized to be directly applied to PhlegraeanFields, their analysisis also a preliminary steptoward the developmentof a more realistic model of this particular volcanic system. THE MODELS
We consider isolated magma chambers of spheroidal shape with different aspect ratios (0.7-2). The chambers
Fig. 1. Vertical cross section through the symmetry axis of a finite element mesh. The example refers to a sphericalchamber, 1.5 km in radius with center at 5.4 km depth. The meshesadoptedfor the various
models considered
in this work
are rather similar.
ber. The finite element meshes adopted for the various models are similar (Figure 1). Surrounding the chamber is a region (r < 6 km, z < 11 km) of fine-scale quadrangular elements. In the far field, a coarser mesh turns out to be adequate (we shall come back to this point in a subsequent section). Stress boundary conditions corresponding to a uniform pressure P in addition to the hydrostatic pressure due to the weight of the magma are imposed on the chamber wall. Displacements are unconstrained on the top of the cylinder, correspondingto the Earth's surface, while only traction-free radial displacements are allowed on the bottom (plane at z = 13.5 km). On the vertical side of the cylinder we impose stress boundary conditions correspondingto a confining radial pressureproportional to the lithostatic load, i.e.,
o'r -- -Apgz and O'rz-- 0 (cylindricalcoordinates;minussign
haveapproximately equalvolume(13.5-14km3) andsame for compression)and we investigate the range (v/(1 - v) _
T, where T is the uniaxial tensile strength (positive). The Griffith-Murrell criterion applied by Roberts [1970] implies instead tr0 or % > trn + T, with trn (negative)given by (3). The tensile strength of rocks is commonly of the order of 10 MPa [Touloukian e! al., 1981]. However, the values obtained in the laboratory on small soundsamplesshouldbe consideredas upper limits for the intensively fractured rocks of a volcanic system. Moreover, as mentioned before, our numerical results underestimate the values of tr0 and o't on the surfaceof the chamber. Therefore, without performing a detailed analysis of failure, we assume that a net horizontal
tension
of the order of 10
MPa (100 bars) in the elements near the polar zones is sufficient
to make
the chamber
unstable.
Since the rocks
surroundingthe chamber are expectedto be highly fractured as a result of previous limited failures or eruptive and tectonic events, we also assume that horizontal tensions
exceeding 10 MPa are sufficient for pieces of rocks to become detached. On these grounds, stability and evolution of the chamber shape are investigated according to the following procedure: (1) the stress distribution around the chamber is calculated for each model; (2) the elements at the margins of the chamber for which the normal stress on a
Fig. 2.
Empirical relationship between frictional shear strength and normal stress [Barton, 1976].
that the P equals the lithostatic load pgZ. Alternatively, the upper limit of the values of P for which the initial shape of a chamber may be stable (according to the assumed necessary conditions) was determined. As mentioned before, in areas where volcanic activity has been intense with formation of large calderas, as at Phle-
graeanFields, the rocks above the chamber are presumably highly fractured. In this case, frictional failure may play a relevant role in the distribution of microseismicity and in the stability of the whole volcanic system. A detailed investigation of this problem is beyond the scopeof the present work. On the other hand, adopting the empirical law of friction [Barton, 1976] shown in Figure 2, we shall evaluate for each model where frictional failure might occur, that is the zones
of potential weaknessof the volcanic system. RESULTS
The results presentedin this section refer--unless otherwise stated--to models characterized by (1) host rock pa-
rameters p -- 2700kg/m3, v = 0.25;(2) magma densityP2u-p; (3) flat Earth's surface; (4) external boundary conditions with A = v/(1 - v) (i.e., no horizontal displacementin the far field). In the first three subsectionswe present the results obtainedfor various spheroidalchambersunder the assumption that the magma pressure P acting at the top of a long-lived chamber equals the lithostatic pressure of the overlying rocks. Results corresponding to different pressures, topography, mechanical parameters, and external boundary conditions are presented successively.
vertical plane exceeds 10 Mpa are consideredto be detached and are suppressed;i.e., they are considered part of the The Sphere chamber; (3) steps 1-2 are iterated for the new chamber; and Consider the spherical chamber of Figure 1 (without (4) iteration is stopped when there are no elements to be suppressedor when the volume of the elements suppressed volcanic edifice). The magma pressureacting on its roof, at in the whole procedure representsan appreciablefraction of depth Z = 3.9 km, is assumedto be P = p gZ = 103 MPa. the chamber volume. The corresponding distributions of stress are shown in In the iteration procedure, the magmapressureP actingon Figure 3. It can be seenthat at relatively short distancefrom the roof of the chamber is kept constant. This assumption the chamber (r --• 7 km, z "• 10 km) the normal state of stress has a justification in the fact that, while the volume of the of the crust is practically unperturbed.This confirmsthat the chamber increases during the evolution, the volume occu- dimensions of the cylinder and the mesh used in the finite pied by the magma may be considered roughly constant. element calculation are adequate to the problem. At the top Calculationswere first performed by making the assumption and at the bottom of the chamber, rrr and rr0 are tensile
5144
SARTORIS ET AL.' MECHANICAL
STABILITY OF MAGMA CHAMBERS
K I LOMETERS 8
I0
!
CONTOUR INTERVAL 35
O'rz CONTOUR INTERVAL 7 Mill
Fig. 3.
Distributions of stress(cylindrical coordinates;minus sign for compression)in a vertical plane trough the
symmetry axisof thesystem in Figure1 (without volcanic edifice)' p = PM= 2700kg/m3'v = 0.25'A = 1/3;P = pgZ = 103 MPa.
(25-30 MPa), the zones where •r0 > 0 being more extended toward the equatorial plane than the zones where •rr > 0. The chamber is mechanically unstable. The first two stepsof its evolution are shown in Figure 4. Step 1 is the initial state: the hatched zones correspondto elementsin which either •rr or •r0 exceeds 10 MPa; the heavily dotted zones correspond to elementsin which the valuesof either •rr or •r0 are in the range 0-10 MPa; the lightly dotted zones correspond to elements in which frictional failure might occur. According to the procedure described in the previous section, the results of step 2 were obtained by suppressingthe hatched elements of step 1 and repeating the calculation. The extent of the unstable zones turns out to be still rather large and further
iterations
do not lead to a stable situation.
The Prolate Spheroid
Consider a chamber having the shapeof a prolate spheroid with
horizontal
minor
semiaxis
a =
1.3 km and vertical
major semiaxis b = 1.9 km. As in the case of the sphere, the center is assumed to be at 5.4 km depth. The magma pressure at the top of the chamber (depth Z = 3.5 km) is taken to be P = rr gZ = 90 MPa. The associateddistributions of stress are shown in Figure 5. Figures 3 and 5 are very similar and the commentsmade for the sphereapply as well
to the prolate spheroid. In particular, while the flanks of the chamber experience compressive stresses, two zones of tensile stress(30-40 MPa) immediately adjacent to the major axis terminationsmake the systemunstable. Its evolution is shown in Figure 6. Although step 4 correspondsto a still unstableshape,the iteration procedurecannot be reasonably pushed beyond this limit. In fact, the assumption that P remainsconstantduringthe evolution processis most probably untenable when failure affects a considerablezone and the number of suppressedelements becomes as large as in Figure 6, step 4. The Oblate Spheroid
Let us now consider an oblate chamber having a horizontal major semiaxisa = 1.9 km, a vertical minor semiaxisb = 0.95 km and center at 5.4 km depth. As in the previous cases, a magmapressureP = p gZ = 118 MPa is assumedto act on the roof of the chamber. The correspondingdistributionsof stressare rather similar to the distributions shown in Figures
3 and 5. On the top of the chamber,rrr and rr0 are lesstensile (20-25 MPa) than for the casespreviously considered, but the chamber is still unstable (Figure 7, step 1) and does not evolve toward a stable shape. As for the prolate spheroid,
SARTORISET AL..' MECHANICAL STABILITY OF MAGMA CHAMBERS
5145
o
7
•-
4
L•
Step 1
Step 2
O'r > 10MPa •
O'randO' 0 > 10MPaI
O' 8 > 10MPaI•
0 v/(1 - v). By assuming A = 0.6, i.e., by applyinga horizontalcompression 0.6pgzat r = 13.5km, the distributions of stressshownin Figure12wereobtained.A comparison with the corresponding distributionsof Figure3 showsthat the tensilezonesnear the polesof the spherehave disappeared.In fact, for this case the critical pressureturns out to be P - 220 MPa, approximately twicethe lithostaticload.Highervaluesof A, in particulara hydrostaticstandardstate of stress,imply
the chamber are only negligibly affected by the density contrast.This result may also be regardedas a justification larger values of Pc. for neglectingany density gradient in the magma. We found that for P = 103 MPa, v = 0.25 and A = v/(1 v) the sphericalchamberwas unstable.The previouscalcuDISCUSSION AND CONCLUSIONS lationswere then repeatedfor differentvaluesof Poisson's Ground deformation and other surface observations are ratio. For v = 0.15, the systemis extremely unstable:trp is greaterthan 10 MPa in a thick shellaroundthe chamber,trr often interpretedin terms of pressurechangesin a magma and trpexceeding 60 MPa nearthe poles.For v = 0.35, the chamber without making any assumptionon the total presnecessaryconditionsfor stabilityare satisfied.This resultis sureactingon its wall. In the frameworkof linear elasticity, easily explained.In fact, the assumptionA = v/(1 - v), thisapproachisjustifiedif the totalpressuredoesnotexceed correspondingto a conditionof no horizontal displacement the instabilitythreshold.More generally, whenever failure on the vertical boundary at r = 13.5 km, implies a remote mayplaya relevantrole, a preliminaryanalysisof stabilityis horizontalcompressiontrh = -vpgz/(1 - v). Therefore,by required and total pressures, rather than overpressures,
5150
SARTORIS ET AL.' MECHANICAL
STABILITY
OF MAGMA
CHAMBERS
Km 2
4
2L I
6
Kin 8
IO
12
4
6
8
10
12
!
i
i
i
i
- 21
-585
-60----------1335
I•1
2085
- 140
10
246 •'"•
____
12
•
•
_too•
28• ••
CONTOUR INTERVAL 20MPa
--
0"z
__.___••.•CONTOUR INTERVAL 37.6 MPa
Krn
Krn 2
4
6
8
IO
12
2
4
i
i
6
8
i
i
IO i
12 i
2
4
8
o
2
CONTOUR INTERVAL 20MPa
CONTOUR INTERVAL 4 MPa
Fig. 12. Distributions of stressfor A = 0.6; other parametersas for Figure 3.
have to be considered. This is the case of any study concerning the formation of secondary intrusions or the factors which may trigger eruptions. The mechanicalstability of a magma chamberdependson the value of the stressesacting on its wall in relation to the conditions for failure to occur. The main purpose of this paper was not a detailed analysis of failure and formation of secondary intrusions but rather the assessmentof the relative importance of various structural and physical factors controlling stability. To this end, we focused our attention on the stresseslikely to be developed on the surface of a chamber in relation to only necessary conditions for its stability, namely, tensiletangentialstressesnot exceeding10 MPa. Our analyses were mainly performed by assuminga
chambersinvestigated. In the vicinity of the chambers (z > 3 km), the effect of a volcanic edifice, 1.5 km high and 6 km in radius, is practically not felt. 3. Stability is strongly affected by gravitational body forces. On the other hand, in spite of the appreciablevertical dimensionsof the chambers considered, a density contrast betweenmagmaand country rocks Ap/p -< 11% (of the order of the density contrast between liquid and solid magma) has negligibleeffects on stability. This suggeststhat, in developing more realistic models, layered chambers would be an unnecessarycomplication. 4. For external boundary conditionscorrespondingto no horizontal displacementin the far field and for v = 0.25, the necessaryconditionsfor stability are satisfiedonly for rather standard state of stress of the crust rather unfavourable for low critical pressuresPc (half the lithostaticpressurepgZ of stability,i.e., rr• - rr2 = •rr • 3 = -«pgz,corresponding to the overlying rocks). For pressuresslightlygreater than Pc, uniaxial strain in the z direction if the usual value v = 0.25 is a chamber may evolve toward a new stable shape. For P -adoptedfor Poisson'sratio. The more importantconclusions pgZ--• 2Pc, the volcanic system turns out to be highly which can be drawn are as follows: unstable. Larger values of Poisson's ratio (0.30-0.35) yield 1. For the rather large spheroidal chambers considered critical pressuresexceeding the lithostatic pressure. (aspect ratios 0.7-2) the shapeis not crucial for stability. In 5. According to the assumptionsmade and the models particular, the values of the critical pressure,normalizedto investigated,the magma pressurethat the wall of a chamber the lithostatic load, are practically constant. can bear without failing depends critically on the external 2. Topography affects the stress field near the free sur- boundary conditions. For v- 0.25 and external boundary face of the Earth but is not critical for the stability of the conditions progressively approaching a hydrostatic state of
SARTORIS ET AL.: MECHANICAL
STABILITY
stress, the necessaryconditions for stability can be satisfied for increasinglyhigher magma pressures(up to a few kilobars). This result is consistent with both observations of lithostatic stresses [McGarr, 1988] and the fact that magma bodies do often appear to remain stable. However, these caseswould require a more complete analysis of stability. Our descriptionof the mechanicalevolution of a chamber is obviously schematic. In particular, the assumptionof a uniform mechanical response of the surroundingmedium is most probably a reasonableapproximationonly for the rocks immediately adjacent to the chamber. Farther away, nonhomogeneous rocks, fractures, faults, and fluids may make the response of the medium nonuniform. In the context of our finite element calculations, this would imply an early failure
of the weakest
elements
and a different
evolution
of
the chamber. The development of more realistic models to describe specific volcanic systemsrequires therefore a detailed knowledge of the subsurface structure. On the other hand, even in mechanically uniform rocks, a nonuniform responsemay arise from pressure and temperaturesgradients. Appreciable temperature gradients are to be expected in the vicinity of a chamber and a realistic model shouldtake their effect into account. This implies a knowledge of the mechanicalproperties of the country rocks under the physical conditions existing at depth. Even for our reference example of Phlegraean Fields, despite the large amount of data obtained from deep geothermal drillings, geophysical prospecting, and laboratory measurements, only limited information is available on this subject. In fact, while laboratory measurements concern small samples of usually soundrocks, a knowledgeof effectivelarge-scalemechanical properties is required to develop realistic geophysicalmodels. Fractures and fluids may make large-scale properties quite different from small-scaleproperties. Moreover, while detailed seismologicalanalysesmay provide some information on large-scale, nonuniform dynamic responses, the information required to model stability concerns essentially static properties.
Acknowledgments. We acknowledgethe financial supportof the Gruppo Nazionale di Vulcanologia (CNR, Italy) and the Programme Interdisciplinaire de Recherche pour la Pr6vision et la Surveillance des Eruptions Volcaniques (CNRS, France). IPGP contribution 1117.
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Geochemical Society, University Park, Pa., 1987. Ryan, M.P., The mechanics and three-dimensional internal structure of active magnetic systems: Kilauea volcano, Hawaii, J. Geophys. Res., 93, 4213-4248, 1988. Sammis, C. G., and B. R. Julian, Fracture instabilities accompanying dike intrusion, J. Geophys. Res., 92, 2597-2605, 1987. Touloukian, Y. S., W. R. Judd, and R. F. Roy, Physical Properties of Rocks and Minerals, Data $er. on Mater. Properties, vol. 1-2, McGraw-Hill, New York, 1981. J. L. Le Mofiel, Institut de Physique du Globe, 4, place Jussieu, 75252 Paris, France. C. Philippe, Ecole Nationale Sup6rieure d'Arts et M6tiers, 151 Blvd. de l'Hopital, 75013 Paris, France. J.P. Pozzi, Ecole Normale Sup6rieure, D6partement de G6ologie, 24 rue Lhomond, 72531 Paris, France. G. Sartoris, Dipartimento di Scienze Fisiche, Universith di Napoli, Mostra d'Oltremare pad. 19, 80125 Napoli, Italy. (Received June 5, 1989; revised October 16, 1989; accepted October 18, 1989.)
