Chapter 3 Origin of volcano-tectonic earthquakes

Chapter 3 Origin of volcano-tectonic earthquakes

Chapter 3: Origin of Volcano-tectonic Earthquakes 31 Chapter 3 Origin of Volcano-tectonic Earthquakes Volcanic e a r t h q u a k e s o c c u r w i t...

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Chapter 3: Origin of Volcano-tectonic Earthquakes

31

Chapter 3 Origin of Volcano-tectonic Earthquakes Volcanic e a r t h q u a k e s o c c u r w i t h i n a n d a r o u n d t h e v o l c a n i c edifice a n d reflect t h e i n t e r a c t i o n of two g e n e r a l geological p r o c e s s e s : t h e m a g m a m i g r a t i o n to t h e E a r t h ' s s u r f a c e a n d t h e c r u s t a l t e c t o n i c activity. Fig. 3.1 gives a g e n e r a l s c h e m e of m a g m a m e l t i n g in t h e m a n t l e a n d its r i s i n g t h r o u g h t h e r u p t u r e d c r u s t to t h e s u r f a c e a t z o n e s of o c e a n spreading and subduction.

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MANTLE convecVtion CORE Fig. 3. I. An illustration of the process of magma melting in the mantle and it's rising at the midocean spreading centers and subduction zones to the surface. Volcanoes overlie the zones where slowly moving oceanic lithosphere is pushed beneath the thicker lithosphere of the continent. The engines that drive this movement are rising heat currents in the mantle. The molten magma rises buoyantly towards the Earth's surface. Based on Fisher et al. (1998).

Chapter 3: Origin of Volcano-tectonic Earthquakes

32

3. I. Magma and its physical and chemical properties Magma is generated by a partial melting of rocks at depth. Migration of the melt then occurs in the Earth's interior by porous flow along grain-size channels, and by flow through larger conduits (Daines, 2000). Magma is formed in the mantle in two principal ways: the first way involves simple decompression of hot, solid mantle material; whereas the second involves lowering the melting temperature of the mantle material by the addition of volatiles (Perfit and Davidson, 2000). Plate tectonics show (Fig. 3.1) that magma is generated mainly below divergent plate boundaries (oceanic spreading ridges) and below convergent plate boundaries (subduction zones). Intraplate volcanism is supported by magma that is generated within so called hot spots producing mant/e plumes. The location of the magma generation constrains its physical properties and constitution. Partial melting of the mantle generates basaltic magmas; the silicic magmas are generated by partial melting of crustal rocks (Rogers and Hawkesworth, 2000). Common magmas vary nearly continuously in composition from rhyolitic (75% SiOg) through to basaltic (50% SiOg). Generally three types of magma are distinguished: acid (daciticandesitic) containing more than 66% of SiO2, intermediate (andesitobasaltic) containing between 52% and 66% of SiOg, and basic basaltic, containing less than 52% of SiO2. 10 I~

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Chapter 3: Origin of Volcano-tectonic Earthquakes

33

After m a g m a generation, it c o m m e n c e s its j o u r n e y to the surface. Magma is not a pure liquid b u t h a s rheological properties a n d its ability to flow depends strongly on its viscosity. Fig. 3.2 (Spera, 2000) shows t h a t the viscosity of different melted rocks at high t e m p e r a t u r e (about 1000 oC) can change by 10 orders from komatite (less t h a n 1 Pa s) to rhyolite (10 ~o Pa s). This difference in viscosity determines the ability of different m a g m a types to migrate t h r o u g h the cold rocks of the interior of the Earth. At mantle depths, the m a g m a rises very slowly within convection c u r r e n t s without the fracturing of s u r r o u n d i n g material. Within the shallow c r u s t there is storage of the m a g m a and differentiation begins to occur. Seismic studies allow the location of the zones of m a g m a storage since they r e p r e s e n t a zone of low velocity. Fig. 3.3 and 3.4 show the locations of low velocity zones b e n e a t h the volcanoes Pinatubo and Mount St. Helens. A large zone representing a relative decrease of Pwave velocity (5 to 10%) b e n e a t h Pinatubo volcano (Fig. 3.3) was located between d e p t h s of 6 and 11 k m and is estimated to have a volume of 40 to 90 cubic k m (Mori et al., 1996a). For Mount St. Helens (Fig. 3.4), a Pwave low-velocity region began at 1.5 k m depth, directly below the crater. This localized low-velocity zone extends to a depth of 3.5 k m where it t h e n a p p e a r s to b r o a d e n (4-5 k m diameter) a n d c o n t i n u e s to a depth of 7.5 k m (Lees, 1992). Therefore, the regions of m a g m a storage are situated at very shallow depths. Two types of igneous rocks are distinguished: intrusive igneous rocks and extrusive igneous rocks. Intrusive igneous rocks crystallize w h e n molten rock i n t r u d e s into u n m e l t e d rock m a s s e s deep in the Earth's c r u s t while extrusive igneous rocks are formed w h e n m a g m a e r u p t s at the surface. The intrusive and extrusive rocks can have the s a m e content b u t different textures. The m a i n types of i n t r u s i o n s are sills and dykes. Sills are concordant, t a b u l a r bodies t h a t are emplaced essentially parallel to the bedding of the country rock and occur in relatively unfolded regions at shallow crustal level. A low viscosity is required to produce this sheetlike form. Dykes are thin, tabular, discordant intrusive bodies t h a t cut across the foliation or bedding of the country rock. They range in t h i c k n e s s from less t h a n a meter to several h u n d r e d meters, a n d a few have been traced along strike for tens of kilometers. Typically emplaced into already existing fracture systems, they m a y occur singly or in swarms. In some areas, dikes occur as radiating s w a r m s centered on an intrusion or on the flanks of a volcano. In rare cases, vertical- or outward-dipping ring dikes or inward-dipping cone sheets are distributed in oval or circular p a t t e r n s a r o u n d the i n t r u s i o n (Blatt a n d Tracy, 1996). Extrusive rocks form different types of lavas. The term lava is u s e d both for m a g m a t h a t h a s been erupted onto the E a r t h ' s surface a n d for

34

Chapter 3: Origin of Volcano-tectonic Earthquakes

Fig. 3.3. Cross-section of the P-wave velocity perturbations from the average velocity at a given depth beneath Pinatubo volcano, Philippines. The dots show the hypocenters of post-eruption seismicity from June 29 to August 16, 1991. From Mori et al. {1996a). the rocks t h a t h a s solidified from it (Blatt and Tracy, 1996). This i n t r u s i v e - e ~ s i v e activity of m a g m a indicates the seismic potential of m a g m a t i s m within the u p p e r c r u s t with well-developed tectonics. When m a g m a rises from its storage zone, it begins to move along the system of tectonic fractures and the volcanic conduit to the surface as a dike or sill. The difference in m a g m a viscosity becomes decisive in the choice of magmatic route at shallow depths. Low-viscosity basaltic m a g m a prefers to go along the stratified s t r u c t u r e s (or faults) in accordance with their strike and does not cut them. The intermediateviscosity andesite-basaltic m a g m a m a y choose both ways. High-viscosity dacitic m a g m a slowly cut the stratified s t r u c t u r e s a n d forms obelisks. The average a s c e n t rate varies from 0.001 m / s to 0.015 m / s for extrusive m a g m a s . During explosive eruptions, the a s s e s s m e n t of m a g m a a s c e n t rate is characterized by m u c h greater variations and leaves the top of conduit at essentially sonic velocities for truly explosive eruptions (Rutherford and Gardner, 2000). So the m a i n seismoproductive m a g m a at shallow depths is the andesite-basaltic m a g m a of

Chapter 3: Origin of Volcano-tectonic Earthquakes

35

i n t e r m e d i a t e viscosity t h a t is t r a n s p o r t e d in sills a n d dikes from the deep reservoir to the surface. The t h r e e m a i n types of volcanic e r u p t i o n s , Pelean, V u l c a n i a n a n d Hawaiian, are c h a r a c t e r i z e d the difference in c o n t e n t of e r u p t e d m a g m a t h a t c a u s e s the different s u r f a c e m a n i f e s t a t i o n s of t h e s e e r u p t i o n s (Table 3. I).

3.2. Volcanism and tectonics Volcanoes n o r m a l l y o c c u r within rift {or graben) type s t r u c t u r e s along long-lived faults. A g r a b e n is often formed above i n t r u d i n g dikes in volcanic rift zones b e c a u s e a dike c r e a t e s two zones of m a x i m u m t e n s i o n a l s t r a i n at the surface, parallel to t h e dike axis (Perfit a n d Davidson, 2000). The regional tectonic faults w h e r e the volcano is s i t u a t e d are a c c o m p a n i e d c o m m o n l y by n u m e r o u s local faults.

Fig. 3.4. Cross-section of P-wave velocity perturbations beneath Mount St. Helens (MSH) volcano, Cascades. Contours of the absolute P-wave velocities (background plus perturbation) are provided for reference. From Lees (1992).

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Chapter 3: Origin of Volcano-tectonic Earthquakes

Table 3.1 Characteristic features of three t y p e s of volcanic e r u p t i o n s T y p e of eruption ' Pelean

Characteristic feature Central eruption, pyroclastic flows, lava dome

Type of lava Dacitic and andesitic

Vulcanian (or Strombolian)

Central and flank eruptions, strong explosions without pyroclastlc flows, b u t with a great cauliflower-shaped dark eruption cloud (white for Strombolian type), lava flow, s o m m a structure

Andesito-basaltic

Hawaiian

Central, fissure and flank eruptions, lava flow and fountains, lava lake

Basaltic

These s t r u c t u r e s could be mid-oceanic ridges or rift valleys (depressions) along the convergent b o u n d a r y (subduction zone) or above a hot spot (See Fig. 3.1). They could be the basis for a long volcanic c h a i n s or a small volcanic group. Basaltic volcanoes have a well-defined framework of local faults where the flank cones are formed. Fig. 3.5 shows the s t r u c t u r a l m a p of Mt. E t n a volcano, Sicily (Azarro, 1999). This volcano h a s grown at the intersection of two regional faults: the Malta E s c a r p m e n t a n d the Messina-Fiumefreddo, striking respectively NNW a n d NE, a n d at the front of the A p e n n i n e - M a g h r e b i a n t h r u s t belt, the crustal scale fault. The tectonic setting of E t n a ' s flanks results from the interaction of regional a n d local tectonics. The Timple s y s t e m at the volcano's e a s t e r n flank r e p r e s e n t s the n o r t h e r n m o s t prolongation of the Malta E s c a r p m e n t a n d forms a NNW-SSE-trending s y s t e m of parallel east-facing step-faults of considerable length (8-10 km). To the n o r t h this s y s t e m is i n t e r r u p t e d by the E-W P e m i c a n a fault, t h a t cuts a large part of the volcanic edifice a n d by the NE-SW Ripe della Naca system. Two isolated s t r u c t u r e s , the Trecastagni a n d Tremestrieri faults are s i t u a t e d at the s o u t h e r n flank of the volcano. The g r o u n d b r e a k a g e t h a t was observed in the volcanic region generally follows pre-existing active fault traces. Azarro (1999) shows also evidence of the m o v e m e n t along the h i d d e n a n d s e m i - h i d d e n local faults. The flank volcanic eruptions occur normally along the local tectonic faults, mainly on the e a s t e r n flank of the volcano (Gresta et al., 1990). Among t h e m was the 1991-1993 eruption t h a t occurred along the

Chapter 3: Origin of Volcano-tectonic Earthquakes

37

N N W - S S E T i m p l e s y s t e m of f a u l t s ( F e r r u c c i a n d P a t a n d , 1993). T h e n u m e r o u s c i n d e r c o n e s i l l u s t r a t e t h e p r o c e s s of m a g m a i n t r u s i o n a l o n g t h e local faults. lyrrhenian Sea P

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Fig. 3.5. Simplified structural map of Mt. Etna. 1, faults with downthrown side; 2, eruptive fissures; 3, cinder cones; 4, coseismic surface faulting zones; 5, strike-slip components; 6, caldera rims; 7, limit of Etna volcanics; C.C., central craters. Box indicates area of the Timple fault system. Insert map (a) shows the regional geological setting: AMF, front of the Apennine-Maghrebian t h r u s t belt; IF, Iblean Foreland; ME, Malta Escarpment; MFL, Messina-Fiumefreddo line. From Azarro (1999).

Chapter 3: Origin of Volcano-tectonic Earthquakes

38

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Fig. 3.6. Kilauea volcano, Hawaii, and its rifts and fault systems. From Parfitt and Peacock (200 I). Another basaltic volcano, Kflauea, Hawaii (Fig. 3.6) is situated within a complex tectonic framework formed by the Ka'Oki, Koa'E, a n d Hilina fault s y s t e m s and the S o u t h w e s t and East Rift zones. The n o r m a l fault s y s t e m s at the s o u t h flank of Kilauea do not participate directly in m a g m a t i c activity b u t reflect longer-term deformation of the volcanic edifice. The fault s y s t e m s consist of e n e c h e l o n fault s e g m e n t s t h a t have significant intersection and linkage (Parfitt and Peacock, 2001). The rift zones, which intersect the flanks of the volcano, are filled by dikes t h a t supply m a g m a and form cinder cones. The dikes propagate downrfft at narrow bladelike intrusions from shallow m a g m a reservoirs b e n e a t h the s u m m i t caldera. Rift eruptions and m a n y s u m m i t eruptions begin w h e n the m a g m a issues from long extension cracks t h a t fracture the surface immediately prior to the eruptions. These eruptive fissures are oriented parallel to the regional trend of the rift zones and m a r k the intersection of propagating dikes with the Earth's surface (Dieterich, 1988).

Chapter 3: Origin of Volcano-tectonic Earthquakes

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Fig. 3.7. Clustering of volcano-tectonic earthquakes (crosses) along the eruptive fissure (thick line) during the 1983 dike intrusion in East rift zone of Kilauea. A, epicenters of earthquakes; B, cross section along I-II line of Fig. 3.7A. The cinder cones are shown by ovals. From Rubin et al. (1998). The precise location of small (magnitude 1 to 2 ) v o l c a n o - t e c t o n i c e a r t h q u a k e s t h a t were observed during the 1983 dike i n t r u s i o n fissure eruption (Rubin et al., 1998) shows a p r o n o u n c e d tightening of seismicity along the dike trend (Fig. 3.7A). The foci of e a r t h q u a k e s are located a b o u t 0.5 k m s o u t h of the eruptive fissure, b u t a dike t h a t dipped only 7 ~ from vertical would p a s s t h r o u g h them. The h y p o c e n t e r s of the e a r t h q u a k e s lie between 3 a n d 4 k m d e p t h (Fig. 3.7B) t h a t indicate the origin of seismicity from a very n a r r o w d e p t h interval, despite the fact t h a t the dike b r e a c h e d the surface. The andesitic stratovolcano Volc~.n de Colima is located within the NS-trending Colima Rift Zone, in the w e s t e r n p a r t of the Mexican Volcanic Belt, a n d together with the Pleistocene volcano Nevado de Colima forms the Colima Volcanic Complex (Fig. 3.8). The Colima Volcanic Complex is s i t u a t e d above the s u b d u c t i o n zone where the Rivera plate goes b e n e a t h the North American plate. At the s a m e time, the Colima Rift r e p r e s e n t s the e a s t e r n b o u n d a r y of the Jalisco block t h a t is moving towards ocean from the continent. The Colima volcano m a r k s also the intersection of

40

Chapter 3: Origin of Volcano-tectonic Earthquakes

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Fig. 3.8. A s t r u c t u r a l position of Volcmn de Colima in s o u t h w e s t e r n Mdxico. Three major rift zones are shown: CRZ, Colima Rift Zone; ChRZ, C h a p a l a Rift Zone; a n d TZRZ, Tepic-Zacoalco Rift Zone. The dotted line r u n n i n g t h r o u g h Volc~_n de Colima is the proposed T a m a z u l a Fault zone (Gardufio-Monroy et al., 1998). O t h e r major volcanic centers from the w e s t e r n Mexican Volcanic Belt are s h o w n as triangles: 1, Nevado de Colima; 2, Volcmn C~_ntaro; 3, Sierra la Primavera; 4, Volc~n Tequila; 5, Volc~_n Ceboruco; 6, Volc~_n S a n Pedro; 7, Volc~.n Tepetiltic; 8, Volc~_n Sangangfiey; 9, Volc~_n las Navajas; 10, Volc~_n S a n J u a n ; 11, S a n S e b a s t i a n Volcanic Field; 12, Mascota Volcanic Field; 13, Los Volcanes Volcanic Field. From Zobin et al. (2002c).

Fig. 3.9. Epicenters of e a r t h q u a k e s recorded at Volc~_n de Colima, M6xico prior to the November 1998 block-lava extrusion in J u n e - J u l y (circles) a n d OctoberNovember, 1998 (stars). VC a n d NC are Volcmn de Colima a n d Nevado de Colima, respectively. The c o n t o u r s of 3 0 0 0 m a n d 3500 m are shown. From Zobin et al. {2002b).

Chapter 3: Origin of Volcano-tectonic Earthquakes

41

two large local tectonic s t r u c t u r e s , the N-S-trending Colima Rift a n d NESW-trending T a m a z u l a fault (Gardufio et al., 1998). The November 1998 block-lava extrusion at Volc~n de Colima was p r e c e d e d by a 1 2 - m o n t h period of seismic activity t h a t included five e a r t h q u a k e s w a r m s (Zobin et al., 2002b). The epicenters of the e a r t h q u a k e s t h a t were recorded d u r i n g J u n e - J u l y 1998 s w a r m clustered across the N-S-trending Colirna Rift, while the epicenters of the OctoberNovember 1998 e a r t h q u a k e s e q u e n c e were located along the rift s t r u c t u r e (Fig. 3.9). This e a r t h q u a k e clustering m a y reflect the m a g m a m o v e m e n t along the two local faults t h a t were active at the different stages of m a g m a uplift to the surface. The dacitic volcano Usu, Hokkaido, is s i t u a t e d on the s o u t h e r n rim of the nearly circular Toya caldera t h a t is of a b o u t 10 k m in diameter, s u r r o u n d e d by a steep wall a n d partially filled now by Lake Toya (Katsui et al., 1981). U s u volcano h a s a characteristic distribution of its central a n d flank cones (Fig. 3.10) t h a t outlines a s y s t e m of circular faults in the volcanic edifice. The craterlets of the 1977 e r u p t i o n s were formed within the s u m m i t caldera; w h e r e a s a c h a i n of dacitic cones w a s formed in 1910 a n d 1 9 4 3 - 1 9 4 4 along the 2 0 0 - m c o n t o u r line of volcanic edifice at

jm.

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200

1, 2 & 3

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F Fig. 3. I0. A system of cones that formed at Usu volcano, Hokkaido, during the 1910, 1943-1944 and 1977 eruptions. Contours with dots show approximate outlines of cryptodomes. MS is the 1910 upheaval; K and E have no dates. Thick contours show lava domes. SS indicates the 1943 lava dome; Gn and NM are the 1977 Gin-numa craterlet and one of the 1977 peaks, respectively. The stars indicate the wells. Double broken lines show the lateral extent of volcanic activity. From Yokoyama and Seino (2000).

Chapter 3: Origin of Volcano-tectonic Earthquakes

42

/-

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Fig. 3.11. The deep slices of earthquake foci located during the 1977-1978 eruption of Usu volcano for depths from 0 km (sea level) to 2.0 km. Numbers in the left comer show the number of located events for each depth slice. From Okada et al. (1981). a d i s t a n c e of a b o u t 2-3 k m from the s u m m i t (Yokoyama a n d Seino, 2000). T h e s e s u r f a c e m a n i f e s t a t i o n s of volcanic activity reflect the m o v e m e n t of dacitic m a g m a along the circular faults. The e a r t h q u a k e s located d u r i n g the 1 9 7 7 - 1 9 7 8 s u m m i t e r u p t i o n of U s u (Fig. 3.11) s h o w the circular d i s t r i b u t i o n of e a r t h q u a k e c l u s t e r s a r o u n d its c a l d e r a (Okada et al., 1981). The e a r t h q u a k e s s u r r o u n d a n event-free zone b e n e a t h the c e n t r a l p a r t of the s u m m i t a n d indicate the deep c i r c u l a r fault distribution. E x a m i n i n g the slice at the 0.5 k m depth, the e p i c e n t e r s for a n arc in the SW p a r t of volcano are seen, while with the 1.0 k m a n d 1.5 k m d e p t h slices, the NE arc is outlined.

3. 3. Models of volcano-tectonic earthquake sequences It h a s b e e n s h o w n t h a t volcano-tectonic e a r t h q u a k e s o c c u r d u e to m a g m a t i c activity a n d tectonic fracturing. The m o v e m e n t of m a g m a to the s u r f a c e o c c u r s w i t h i n a dike-type body. R u b i n a n d Gillard (1998) p r o p o s e d a model of d i k e - i n d u c e d seismicity b a s e d on dike i n t r u s i o n o b s e r v a t i o n s at Kilauea Volcano, Hawaii (Rubin a n d Gillard, 1998; R u b i n et al., 1998). They s t u d y the inelastic o u t c r o p - s c a l e d e f o r m a t i o n p r o d u c e d within several m e t e r s of the dike tip w h e n the i n t e r n a l

Chapter 3: Origin of Volcano-tectonic Earthquakes

43

p r e s s u r e variations due to viscous flow of the m a g m a m u s t be considered. The s c h e m a t i c d i a g r a m of blade-like dike is s h o w n in Fig. 3.12. In this model, the dike moves within the h o s t rocks t h a t are s u b j e c t to the action of a m b i e n t stress. The region n e a r the tip of dike (tip cavity) where the m a g m a c a n n o t p e n e t r a t e m a y be occupied by m a g m a t i c volatiles or h o s t rock pore fluids. The region influenced by the increase in s t r e s s is c o m p a r a b l e in size to the length of the tip cavity. The possibility of the m a g n i t u d e 1 to 2 e a r t h q u a k e s with the source d i m e n s i o n of a b o u t 100 m in the vicinity of the tip cavity is d i s c u s s e d for three c a s e s (Fig. 3.12): 1. Fault slip away from the tip cavity. 2. F a u l t slip n e a r the tip cavity. 3. S h e a r failure of intact rock.

Fig. 3.12. Schematic diagram of blade-like dike (vertical section) and the three types of deformations that could be related to the moving dike: (1) slip on existing faults away from the tip cavity; (2) slip on existing faults adjacent to the tip cavity; and (3) shear failure of intact rock adjacent to the tip cavity. The least principal stress 03 is perpendicular to the dike. The tip cavity of dike is shown by empty space. From Rubin and Gfllard (1998).

44

Chapter 3: Origin of Volcano-tectonic Earthquakes

The e x a m i n a t i o n of elastic s t r e s s fields s u r r o u n d i n g p r o p a g a t i n g fluid-filled c r a c k s s h o w e d t h a t the s t r e s s s t a t e is favorable to failure o c c u r r i n g n e a r the tip cavity w h e n the cavity p r e s s u r e is m a i n t a i n e d by the influx of h o s t rock pore fluids, r a t h e r t h a n by exsolution of m a g m a t i c volatiles. Even in this case, however, s h e a r f r a c t u r e of previously i n t a c t rock s e e m s unlikely. According to t h e s e results, m o s t d i k e - i n d u c e d seismicity s h o u l d r e s u l t from slip s u i t a b l y aligned along existing f r a c t u r e s (Rubin a n d Gillard, 1998). Therefore, R u b i n a n d Gillard s u g g e s t t h a t the d i s t r i b u t i o n of e a r t h q u a k e foci of dike seismicity reflects the d i s t r i b u t i o n of a m b i e n t s t r e s s e s t h a t are n e a r to failure, a n d does n o t n e c e s s a r i l y reflect the extent of the dike. They reject f r a c t u r i n g a s the trajectory of the dike m o v e m e n t . At the s a m e time, the dike c a n trigger the f r a c t u r i n g in the vicinity of its tip along existing fractures. We could c o n s i d e r t h e s e f r a c t u r e s as m a g m a t i c fractures. R u b i n a n d Gillard (1998) c o n s i d e r t h a t slips could o c c u r at the d i s t a n c e s over 7 0 - 1 0 0 m from the dike if a favorably oriented fault c o n n e c t e d two en echelon dike s e g m e n t s .

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Fig. 3.13. Schematic representation of dikes and conjugate fault planes with respect to greatest and least principal stresses oi and 03. A and B illustrate typical patterns of crack interactions near adjacent dike tips in homogeneous media. From Hill (1977).

Chapter 3: Origin of Volcano-tectonic Earthquakes

45

The model t h a t describes this situation was proposed by Hill (1977). He suggested t h a t the clusters of magma-ffiled dikes exist within brittle volumes of the c r u s t in regions of active m a g m a t i c intrusions, or volcanism, and the dikes within a cluster are systematically oriented in a regional deviatoric stress field. Their long dimension is parallel to the greatest principal stress ~i. Hill (1977) considered s h e a r failures occurring along oblique fault planes connecting adjacent tips of offset dikes w h e n a critical combination of the fluid p r e s s u r e P in the dikes and the regional stress difference oi - o3 is attained. Fig. 3.13 shows the schematic representation of the model. The dikes are forced open by a gradual increase in the tectonic stress difference, m a g m a is passively drawn into the dikes. A s w a r m develops w h e n the stress field within the volume reaches the critical state, and a series of s h e a r failures occur along cracks between the dikes. Hill (1977) suggests t h a t the distribution of e a r t h q u a k e m a g n i t u d e s will be related to the size and spacing of dikes in the s w a r m volume in addition to s u c h factors as the "roughness" in the variation of frictional properties along individual fractures. These two models allow a description of the relationship between magmatic and tectonic fracturing. For basaltic volcanoes, where the liquid low-viscosity m a g m a prefers to migrate along the stratified s t r u c t u r e s (or faults) in accordance with their strike and does not cut them, the Rubin-Gillard model works. For andesitic a n d dacitic volcanoes, the intermediate- a n d high viscosity m a g m a s m a y choose both ways; therefore, the application of both Rubin-Gillard's and Hill's models could be more realistic.