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Ground motion near an expanding preexisting crack
Journal of Volcanology and Geothermal Research, 19 (1983) 367--379
367
Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands
G R O U N D MOTION NEAR AN EXPANDING PREEXISTING CRACK
BERNARD CHOUET
Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received July 2, 1982; revised and accepted March 21, 1983)
ABSTRACT Chouet, B., 1983. Ground motion near an expanding preexisting crack. J, Volcanol. Geotherm. Res., 19: 367--379. We present a study of the motion of the ground in the near field of a preexisting vertical crack driven by excess tensile stress. Using the discrete wave number method, we make a complete representation of the three components of ground displacement result-ing from a small incremental extension of the b o t t o m tip of the crack and analyze the effects of the medium structure on the ground response. The results show the strong impulsive character of the dynamic motion near the source and demonstrate that the first motion is directed everywhere outward. Marked oscillations are observed in the ground response associated with the ringing of the crack faces triggered by the rupture. The displacement is dominantly vertical near the epicenter but becomes predominantly horizontal beyond the immediate source region. The presence of layers has a strong effect on the complexity and duration of the ground response. The static vertical and longitudinal displacements are respectively downward and inward in the epicentral area.
INTRODUCTION
Cooling and contraction in thick lava flows often produce tensile fractures which may grow vertically throughout the entire thickness of the flow dividing it into small-scale structures known as columnar joints. An example of the field appearance of such structures is given in Fig. 1 which illustrates a section of the vertical fracture surface of jointing within the Boiling Pots flow unit in the Wailuku valley near Hilo, Hawaii. Close examination of the surface morphology of the columns reveals that the fracture surface is covered by striations oriented perpendicular to the direction of crack advance. These striae are a well-known feature of fracture surfaces formed by incre mental crack growth and correspond to the inferred sequential stopping positions of the advancing fracture front (Ryan and Sammis, 1981). Another example of well-preserved fracture surface is illustrated in Fig. 2 which depicts the details of the vertical surface of jointing in the First Watchung flows near Orange, New Jersey. Notice that the surface is covered with four striations averaging 8 to 9 cm in height. These striae represent the increments of crack growth in this particular flow unit. 0377-0273/83/$03.00
© 1983 Elsevier Science Publishers B.V.
368
Fig. 1. An e x a m p l e o f c o l u m n a r jointing in basalt at the Boiling Pots flow unit in the Wailuku valley near Hilo, Hawaii. The 4-m-long stadia rod at the right gives the scale of the f o r m a t i o n . Horizontal striations, visible on individual columns, represent discrete increments o f crack growth in the thermal fracturing process. The flow base is at the bottom. ( R e p r o d u c e d with permission of Michael P. Ryan, U.S. Geological Survey. )
369
Fig. 2. Fracture surface of the basal colonnade of the First Watchung basalt near Orange, N e w Jersey. The length of the graduated portion of the scale is 20 cm. Four individual striations are visible on the fracture surface, each averaging about 8--9 c m in height. Each striation represents a successive stopping position of the incrementally advancing crack tip and therefore gives the increment of crack growth in this particular flow unit. The direction of fracture growth is from the bottom to the top. (Reproduced with permission of Michael P. Ryan, U.S. Geological Survey.)
The Hawaiian lava lakes formed by the eruptions of the Kilauea volcano are other well-documented examples of deep, ponded, thermally cracking basalt units. Recent detailed studies of the thermal fracturing process in the solidifying basalt of three of these ponds, namely Kilauea Iki, Makaopuhi, and Alae (Peck and Minakami, 1968; Chouet, 1979; Chouet and Aki, 1981; Ryan and Sammis, 1978, 1981) suggest that further attention should be given to the elastic radiation produced by this type of seismic source. The
370 aim of the present study is to provide some insight into this problem. While such a study is of direct interest to the detection of a cooling magma body buried in the earth and to the monitoring of the physical environment associated with it, it also offers a basis from which a better understanding of the mechanics of hydrofracturation and dike propagation may be gained through seismic observations made at close range. Using a two-dimensional model of an expanding preexisting crack developed by Aki et al. (1977) we shall apply the discrete wave number m e t h o d (Bouchon, 1979; Chouet, 1981, 1982) to obtain a complete representation of the surface displacements near a buried vertical crack that suddenly expands in length by a small increment. Our object will be to assess the major characteristics of the ground motion resulting from an incremental extension of the b o t t o m tip of a crack embedded in a layered structure similar to that found in a typical Hawaiian lava lake. GROUND MOTION RESULTING FROM THE EXTENSION
OF A PREEXISTING CRACK To calculate the ground motion resulting from the extension of the crack we shall use parameters which are compatible with the seismic structure of Kilauea Iki determined by Aki et al. (1978). The depth of this unit is 111 m, of which the upper 45 m are known to be solidified as evidenced by drilling performed since 1976. Residual melt is still observed below this depth al though the precise extent of this zone remains a subject of debate (e.g. Zablocki, 1976; Flanigan and Zablocki, 1977; Smith et al., 1977; Aki et al., 1978; Colp, 1979; Chouet and Aki, 1981). We shall first analyze the ground response obtained for a crack buried in a homogeneous half-space. Then we shall compare these results with those derived for a crack embedded in a layered half-space. The general configuration of the problem is depicted in Fig. 3. The source is the tensile opening associated with the incremental extension by Al of a crack initially of length L. The crack is set on a rectangular vertical plane with the width of 1.5 m and a length L = 30 m extending between the depths of 10 and 40 m. Our choice of width is based on reconstructions for the exposed cliff section of the prehistoric Makaopuhi lava lake and is probably adequate to model the fractures in Kilauea Iki as well since both lakes have depths which are broad ly comparable (Ryan and Sammis, 1981). The depth to the b o t t o m of the source represents the maximum depth of penetration that we anticipate for the active cracks in the upper crust of Kilauea Iki. Although the tops of the thermal fractures in the upper crust of Hawaiian lava lakes are identically coincident with the surface, in our model the top of the source is assumed to remain buried below the free surface for the purpose of minimizing the computer time required in the calculation of the wave field. The representation of the displacements is carried out at 861 locations spaced at regular intervals of 4.5 m in the directions parallel and transverse
371 / /
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. . . . -L LI Fig. 3. C o n f i g u r a t i o n o f t h e source, m e d i u m a n d receivers used in t h e c o m p u t a t i o n o f t h e g r o u n d m o t i o n p r o d u c e d b y t h e i n c r e m e n t a l e x t e n s i o n ~ l o f a crack initially o f l e n g t h L. T h e s o u r c e is r e p r e s e n t e d b y t h e vertical r e c t a n g u l a r p l a n e w i t h t h e d i m e n s i o n s o f 30 x 1.5 m. T h e r e p r e s e n t a t i o n o f t h e g r o u n d d i s p l a c e m e n t s is carried o u t at 861 l o c a t i o n s over a n area o f 180 X 90 m. Each of t h e s e l o c a t i o n s is d e p i c t e d by o n e grid p o i n t in t h e figure. T h e s e p a r a t i o n b e t w e e n s t a t i o n s is 4.5 m in b o t h transverse a n d l o n g i t u d i n a l directions. T h e t w o planes at t h e d e p t h s o f 4 5 a n d 50 m r e p r e s e n t t h e i n t e r f a c e s of t h e t w o surficial layers used in o n e o f o u r models.
to the trace o f the crack plane. Each grid point represents a station in the figure. Because of the s y m m e t r y o f m o t i o n the wave field is depicted on one side o f the source only. Thus, the receiver array covers a rectangular area of 180 X 90 m, the f r ont edge of which coincides with the crack trace, the source epicenter being located midway along this segment. The u and v components are the horizontal displacements parallel and normal to the crack, respectively, and w is the vertical c o m p o n e n t o f motion. There are two axes of s y m m e t r y for horizontal m o t i o n with this source-receiver configuration i.e., u is symmetric, and v is antisymmetric a b o u t the crack trace, while u is antisymmetric and v is symmetric with respect to the orthogonal axis passing through the epicenter. The vertical c o m p o n e n t , w, itself has a symm e t r y with respect to b o t h axes. -
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Fig. 4. N o r m a l c o m p o n e n t o f the particle v e l o c i t y and d i s p l a c e m e n t p l o t t e d as a f u n c t i o n o f time and p o s i t i o n o n the surface o f a crack w h i c h s u d d e n l y e x t e n d s at o n e end in res p o n s e to the application o f an e x c e s s tensile stress. T h e crack is represented b y 1 0 grids equally spaced over its length. Position 10 c o r r e s p o n d s t o t h e tip being e x t e n d e d and p o s i t i o n 1 is at the o p p o s i t e end o f the crack. V e l o c i t y , displacement, and t i m e are given as dimensionless f u n c t i o n s o f the crack and m e d i u m parameters (see t e x t for details).
373
In our analysis the crack opening function is a variable of depth only and is taken as constant over the width of the crack plane. The source spacetime function, depicted in Fig. 4, is the two-dimensional finite-difference solution obtained by Aki et al. (1977) for a crack suddenly extending at one end in response to an excess tensile stress. These results show the normal c o m p o n e n t of the particle velocities, 5, and displacements, v, calculated at ten positions on the crack surface and given as dimensionless functions of the excess stress ~ P (tensile stress minus overburden pressure), lengths L and Al of the crack and of its extension, and the compressional velocity ~ and rigidity t~ of the solid. The grid labelled 10 is where the extension takes )lace. In this solution the initial static elastic field which exists prior to the U
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Fig. 5. Snapshots of the surface displacements produced by the jerky extension of the bottom tip of a vertical crack buried in a homogeneous half-space. The three components of displacement, u, v, w, shown at the top of the figure are parallel,transverse, and vertical, respectively. The snapshots are taken every 10 ms, starting 16 m s after the event origin time and ending 126 m s later. The static displacements resulting from this event are shown at the bottom of the figure.
371 crack extension is not considered. Consequently the static opening depicted in the figure only represents the increment AV in crack volume resulting from the extension of the crack. Note also that the static opening of grid 1 is n o t zero because this position is offset from the crack edge by half a grid owing to the discretization. This solution was derived assuming a Poisson ratio of 0.25 in the rock matrix and according to Aki et al. (1977) it is reliable down to the wavelength ~ = L / 2 or wavenumber k = 2~/~ = 4 ~ / L . Various tests o f the validity, stability, and dispersion of the finite-difference scheme used in this calculation can be found in Madariaga {1976) and Fehler and Aki (1978). Snapshots o f the ground motion pr oduced by the sudden extension of the b o t t o m tip of the crack are shown in Fig. 5. The medium considered is a homogeneous half space with the compressional and shear velocities a = V ~ km/s, ~ = 1 km/s, and a density p = 2.5 g/cm 3, The snapshots are taken every 10 ms, starting 16 ms after the onset of the rupture and ending 126 ms later. The static displacements resulting from the opening of the crack and calculated using Maruyama's (1964) formulation are shown at the b o t t o m of the figure for comparison. By convention the displacements outward from the source are positive. The sequence gives an overall picture of the dynamic ground response to the opening crack, clearly showing the onset of the disturbance and the propagation of the various wave fronts over the area. Of particular interest is the st[ong overshoot of the vertical and transverse horizontal displacements in relation to their static values. It is worth noting also that the static vertical displacement is downward and the static horizontal displacement parallel to the crack is inward directly above the source. For a more detailed interpretation, the seismograms of the three components of displacement obtained at 49 stations set along 7 profiles transverse to the crack are shown in Fig. 6. The profiles, listed at the top of the figure, are spaced 15 m apart and from left to right appear at increasing distance from the trace of the plane of s y m m e t r y transverse to the source, the first profile lying in this plane and the last one being located at a distance of 90 m. Similarly, the stations are set at distances of 0, 15, 30, 45, 60, 75, and 90 m from the crack trace, shown f r om t o p to b o t t o m for each individual c o m p o n e n t of motion. The records start with the arrival of the P wave and last 160 ms from this arrival, covering most of the duration of the ground response at each location. All the displacements have been normalized by the p ro d u ct APAI of excess stress applied on the crack times the length of the crack extension. The main characteristics of these records are the first motion which is outward everywhere, the marked oscillations in the signal associated with the ringing of the crack surfaces, the rapid damping of the vertical c o m p o n e n t of m o t i o n with range, the presence of a clear SH wave in the seismograms obtained at the most distant stations, and the large overshoot of the ground response near the source. The peak displacement occurs at the epicenter on the vertical c o m p o n e n t and within 15 m of the epicenter on the two horizontal components. The directions of profile 1 and of the
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Fig. 6. S y n t h e t i c seismograms o f t h e three c o m p o n e n t s o f ground d i s p l a c e m e n t resulting f r o m the jerky e x t e n s i o n o f the b o t t o m tip o f a vertical crack buried in a h o m o g e n e o u s half space. T h e records are o b t a i n e d at 4 9 stations spaced at intervals o f 15 m in the directions parallel and normal t o the source. For each c o m p o n e n t marked at the left the t o p r o w o f stations c o i n c i d e s w i t h t h e crack trace and the b o t t o m r o w is 9 0 m away. Similarly the first c o l u m n o f stations d e p i c t e d at the left lies o n the trace o f the transverse vertical plane bisecting the crack and the last o n e at the right is at a distance o f 9 0 m. T h e records start w i t h the arrival o f the P wave f r o m the h y p o c e n t e r and last 1 6 0 ms. N o t e that the d i s p l a c e m e n t a m p l i t u d e s have b e e n n o r m a l i z e d b y the p r o d u c t APAl o f e x c e s s stress t i m e s the i n c r e m e n t o f crack length.
crack trace are nodes for SH motion because of the antisymmetry of the problem for horizontal displacements about these two lines. Further details on the P-SV and P-SH motions of the ground axe given in
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Fig. 7. Polar plots of the particle trajectories in the three orthogonal planes defined by u, v, w. The data shown are taken from Fig. 6 for four stations located 90 m away from the crack trace and set from left to right at the distances of 0, 30, 60, and 90 m from the trace of the transverse vertical plane bisecting the crack. The scale of the displacements indicated at the bottom is given for the particular value ~PAl = 1 bar m. Fig. 7, w h i c h s h o w s p o l a r p l o t s o f t h e p a r t i c l e t r a j e c t o r i e s in t h e t h r e e o r t h o g o n a l planes d e f i n e d b y u, v, a n d w. F o u r s t a t i o n s are d e p i c t e d w i t h parallel and t r a n s v e r s e c o o r d i n a t e s as i n d i c a t e d at t h e t o p o f t h e figure; t h e s e s t a t i o n s are at a d i s t a n c e o f 90 m f r o m t h e c r a c k t r a c e a n d f r o m left t o right c o r r e s p o n d t o profiles 1, 3, 5, a n d 7. T h e r e t r o g r a d e elliptical m o t i o n appearing at t h e tail e n d o f t h e t r a j e c t o r i e s s h o w n in t h e b o t t o m p l o t s r e p r e s e n t s t h e c o n t r i b u t i o n o f t h e R a y l e i g h wave. T h e m a x i m u m a m p l i t u d e o f the R a y l e i g h m o t i o n is f o u n d a l o n g t h e first p r o f i l e w h i c h lies o n t h e axis o f t h e m a i n l o b e o f r a d i a t i o n f o r this wave. T h e c h a r a c t e r o f t h e g r o u n d r e s p o n s e evolves f r o m d i s p l a c e m e n t s w h i c h are p u r e l y P-SV t o d i s p l a c e m e n t s w h i c h are p r e d o m i n a n t l y P-SH as we m o v e t o w a r d t h e a z i m u t h w h i c h bisects t h e t w o planes o f s y m m e t r y o f t h e source. This a z i m u t h is t h e d i r e c t i o n o f m a x i m u m r a d i a t i o n f o r t h e far field SH wave a n d this wave is t h e d o m i n a n t c o n t r i b u t i o n t o the m o t i o n o b s e r v e d at t h e m o s t d i s t a n t s t a t i o n d e p i c t e d on p r o f i l e 7.
377
Our next model, shown in Fig. 8, studies the effect of a more complex structure on the ground response. The disposition of the source and stations is the same as in Fig. 6 but the medium structure n o w consists of t w o layers overlying a h o m o g e n e o u s half-space (see Fig. 3). The source is located in the t o p m o s t layer which has a thickness of 45 m and whose elastic parameters are identical to those of the half-space considered in our first model. The second layer, which is 5 m thick, has the compressional velocity, shear velocity, and density of 0.3 km/s, 0.2 km/s, and 2.3 g/cm 3, respectively. Finally, 2
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;37~ the density in the half-space is 2.4 g/cm 3 and the wave velocities are 0.9 km/s for P waves and 0.5 km/s f or S waves. The pat t ern o f radiation produced by the b o t t o m extension of the crack is analogous to that obtained in a homogeneous half-space but the c o m p l e x i t y o f the signals is greatly increased by the interference of multiply reflected P, SV, and SH arrivals. Close to the source the vertical displacement pulse associated with the first arrivals is followed by a lower-frequency c o m p o n e n t propagating with a low phase velocity. At increasing range in the direction transverse to the crack the vertical m o t i o n evolves into a wave train which consists primarily of inversely dispersed Rayleigh waves. As not e d earlier the horizontal displacements are larger than the vertical displacements outside the immediate source region and the Love waves and SH waves become the dom i nant contributions to the m o t i o n at the most distant station. The displacements are amplified by a factor of 3/2 with respect to those obtained in Fig. 6. The lateral e x t e n t and amplitude o f the downward static displacement is also more p r o n o u n c e d owing to the softness of the deep-seated structure. CONCLUSION The prime object o f our study was to provide a picture o f the m o t i o n of the ground induced by the small jerky extension of a buried preexisting crack. Our calculations disclosed the impulsive character and marked oscillation o f the ground response associated with the rupture. Among the major features f o u n d in the ground displacements are the strong overshoot of the dynamic m o t i o n in relation to the static d e f o r m a t i o n directly above the crack, the dominance of the vertical c o m p o n e n t at the epicenter, the predominance o f horizontal m o t i o n at some distance from the source, and the direction of the first m o t i o n which is everywhere outward. The static vertical displacement is dow nw a r d and the static longitudinal displacement is inward in the epicentral area. ACKNOWLEDGEMENTS I am grateful to Michael P. Ryan for helpful c o m m e n t s and for the permission to r ep r o duce two o f his photographs. This work was supported by the D e p a r t m e n t of Energy under Contract DE-AC02-76-ER02534.
REFERENCES
Aki, K., Fehler, M. and Das, S., 1977. Source mechanism of volcanic tremor: fluid-driven crack models and their application to the 1963 Kilauea eruption. J. Volcanol. Geotherm. Res., 2: 259--287. Aki, K., Chouet, B., Fehler, M., Zandt, G., Koyanagi, R., Colp, J. and Hay, R.G., 1978. Seismic properties of a shallow magma reservoir in Kilauea Iki by active and passive experiments. J. Geophys. Res., 83: 2273--2282.
379 Bouchon, M., 1979. Discrete wave number representation of elastic wave fields in threespace dimensions. J. Geophys. Res., 84: 3609---3614. Chouet, B., 1979. Sources of seismic events in the cooling Lava Lake of Kilauea Iki, Hawaii. J. Geophys. Res., 84: 2315--2330. Chouet, B., 1981. Ground motion in the near field of a fluid-driven crack and its interpretation in the study of shallow volcanic tremor. J. Geophys. Res., 86: 5985---6016. Chouet, B., 1982. Free surface displacements in the near field of a tensile crack expanding in three dimensions. J. Geophys. Res., 87 : 3868--3872. Chouet, B. and Aki, K., 1981. Seismic structure and seismicity of the cooling lava lake of Kilauea Iki, Hawaii. J. Volcanol. Geotherm. Res., 9: 41--56. Colp, J.L., 1979. Magma energy research project: FY 1979 annual progress report. Rep. SAND 79-2205, Sandia Lab., Albuquerque, NM, 96 pp. Fehler, M. and Aki, K., 1978. Numerical study of diffraction of plane elastic waves by a finite crack with application to location of a magma lens. Bull. Seismol. Soc. Am., 68 : 573--598. Flanigan, V.J. and Zablocki, C.J., 1977. Mapping the lateral boundaries of a cooling basaltic lake, Kilauea Iki, Hawaii, U.S. Geol. Surv., Open File Rep. 77-94, 21 pp. Madariaga, R., 1976. Dynamics of an expanding circular fault. Bull. Seismol. Soc. Am., 66: 639--660. Maruyama, T., 1964. Statical elastic dislocations in an infinite and semi-infinite medium. Bull. Earthquake Res. Inst. Univ. Tokyo, 42: 289--368. Peck, D.L. and Minakami, T., 1968. The formation of columnar joints in the upper part of Kilauean Lava Lakes, Hawaii. Geol. Soc. Am. Bull., 79: 1151--1166. Ryan, M.P. and Sammis, C.G., 1978. Cyclic fracture mechanisms in cooling basalt. Geol. Soc. Am. Bull., 89: 1259--1308. Ryan, M.P. and Sammis, C.G., 1981. The glass transition in basalt. J. Geophys. Res., 86: 9519---9535. Smith, B.D., Zablocki, C.J., Frischknecht, F.C. and Flanigan, V.J., 1977. Summary of results from electromagnetic and galvanic soundings on Kilauea Iki lava lake. U.S. Geol. Surv., Open File Rep. 77-59, 27 pp. Zablocki, C.J., 1976. Some electrical and magnetic studies on Kilauea Iki lava lake, Hawaii. U.S. Geol. Surv., Open File Rep. 76-304, 19 pp.
367
Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands
G R O U N D MOTION NEAR AN EXPANDING PREEXISTING CRACK
BERNARD CHOUET
Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received July 2, 1982; revised and accepted March 21, 1983)
ABSTRACT Chouet, B., 1983. Ground motion near an expanding preexisting crack. J, Volcanol. Geotherm. Res., 19: 367--379. We present a study of the motion of the ground in the near field of a preexisting vertical crack driven by excess tensile stress. Using the discrete wave number method, we make a complete representation of the three components of ground displacement result-ing from a small incremental extension of the b o t t o m tip of the crack and analyze the effects of the medium structure on the ground response. The results show the strong impulsive character of the dynamic motion near the source and demonstrate that the first motion is directed everywhere outward. Marked oscillations are observed in the ground response associated with the ringing of the crack faces triggered by the rupture. The displacement is dominantly vertical near the epicenter but becomes predominantly horizontal beyond the immediate source region. The presence of layers has a strong effect on the complexity and duration of the ground response. The static vertical and longitudinal displacements are respectively downward and inward in the epicentral area.
INTRODUCTION
Cooling and contraction in thick lava flows often produce tensile fractures which may grow vertically throughout the entire thickness of the flow dividing it into small-scale structures known as columnar joints. An example of the field appearance of such structures is given in Fig. 1 which illustrates a section of the vertical fracture surface of jointing within the Boiling Pots flow unit in the Wailuku valley near Hilo, Hawaii. Close examination of the surface morphology of the columns reveals that the fracture surface is covered by striations oriented perpendicular to the direction of crack advance. These striae are a well-known feature of fracture surfaces formed by incre mental crack growth and correspond to the inferred sequential stopping positions of the advancing fracture front (Ryan and Sammis, 1981). Another example of well-preserved fracture surface is illustrated in Fig. 2 which depicts the details of the vertical surface of jointing in the First Watchung flows near Orange, New Jersey. Notice that the surface is covered with four striations averaging 8 to 9 cm in height. These striae represent the increments of crack growth in this particular flow unit. 0377-0273/83/$03.00
© 1983 Elsevier Science Publishers B.V.
368
Fig. 1. An e x a m p l e o f c o l u m n a r jointing in basalt at the Boiling Pots flow unit in the Wailuku valley near Hilo, Hawaii. The 4-m-long stadia rod at the right gives the scale of the f o r m a t i o n . Horizontal striations, visible on individual columns, represent discrete increments o f crack growth in the thermal fracturing process. The flow base is at the bottom. ( R e p r o d u c e d with permission of Michael P. Ryan, U.S. Geological Survey. )
369
Fig. 2. Fracture surface of the basal colonnade of the First Watchung basalt near Orange, N e w Jersey. The length of the graduated portion of the scale is 20 cm. Four individual striations are visible on the fracture surface, each averaging about 8--9 c m in height. Each striation represents a successive stopping position of the incrementally advancing crack tip and therefore gives the increment of crack growth in this particular flow unit. The direction of fracture growth is from the bottom to the top. (Reproduced with permission of Michael P. Ryan, U.S. Geological Survey.)
The Hawaiian lava lakes formed by the eruptions of the Kilauea volcano are other well-documented examples of deep, ponded, thermally cracking basalt units. Recent detailed studies of the thermal fracturing process in the solidifying basalt of three of these ponds, namely Kilauea Iki, Makaopuhi, and Alae (Peck and Minakami, 1968; Chouet, 1979; Chouet and Aki, 1981; Ryan and Sammis, 1978, 1981) suggest that further attention should be given to the elastic radiation produced by this type of seismic source. The
370 aim of the present study is to provide some insight into this problem. While such a study is of direct interest to the detection of a cooling magma body buried in the earth and to the monitoring of the physical environment associated with it, it also offers a basis from which a better understanding of the mechanics of hydrofracturation and dike propagation may be gained through seismic observations made at close range. Using a two-dimensional model of an expanding preexisting crack developed by Aki et al. (1977) we shall apply the discrete wave number m e t h o d (Bouchon, 1979; Chouet, 1981, 1982) to obtain a complete representation of the surface displacements near a buried vertical crack that suddenly expands in length by a small increment. Our object will be to assess the major characteristics of the ground motion resulting from an incremental extension of the b o t t o m tip of a crack embedded in a layered structure similar to that found in a typical Hawaiian lava lake. GROUND MOTION RESULTING FROM THE EXTENSION
OF A PREEXISTING CRACK To calculate the ground motion resulting from the extension of the crack we shall use parameters which are compatible with the seismic structure of Kilauea Iki determined by Aki et al. (1978). The depth of this unit is 111 m, of which the upper 45 m are known to be solidified as evidenced by drilling performed since 1976. Residual melt is still observed below this depth al though the precise extent of this zone remains a subject of debate (e.g. Zablocki, 1976; Flanigan and Zablocki, 1977; Smith et al., 1977; Aki et al., 1978; Colp, 1979; Chouet and Aki, 1981). We shall first analyze the ground response obtained for a crack buried in a homogeneous half-space. Then we shall compare these results with those derived for a crack embedded in a layered half-space. The general configuration of the problem is depicted in Fig. 3. The source is the tensile opening associated with the incremental extension by Al of a crack initially of length L. The crack is set on a rectangular vertical plane with the width of 1.5 m and a length L = 30 m extending between the depths of 10 and 40 m. Our choice of width is based on reconstructions for the exposed cliff section of the prehistoric Makaopuhi lava lake and is probably adequate to model the fractures in Kilauea Iki as well since both lakes have depths which are broad ly comparable (Ryan and Sammis, 1981). The depth to the b o t t o m of the source represents the maximum depth of penetration that we anticipate for the active cracks in the upper crust of Kilauea Iki. Although the tops of the thermal fractures in the upper crust of Hawaiian lava lakes are identically coincident with the surface, in our model the top of the source is assumed to remain buried below the free surface for the purpose of minimizing the computer time required in the calculation of the wave field. The representation of the displacements is carried out at 861 locations spaced at regular intervals of 4.5 m in the directions parallel and transverse
371 / /
/
/
/
. . . . -L LI Fig. 3. C o n f i g u r a t i o n o f t h e source, m e d i u m a n d receivers used in t h e c o m p u t a t i o n o f t h e g r o u n d m o t i o n p r o d u c e d b y t h e i n c r e m e n t a l e x t e n s i o n ~ l o f a crack initially o f l e n g t h L. T h e s o u r c e is r e p r e s e n t e d b y t h e vertical r e c t a n g u l a r p l a n e w i t h t h e d i m e n s i o n s o f 30 x 1.5 m. T h e r e p r e s e n t a t i o n o f t h e g r o u n d d i s p l a c e m e n t s is carried o u t at 861 l o c a t i o n s over a n area o f 180 X 90 m. Each of t h e s e l o c a t i o n s is d e p i c t e d by o n e grid p o i n t in t h e figure. T h e s e p a r a t i o n b e t w e e n s t a t i o n s is 4.5 m in b o t h transverse a n d l o n g i t u d i n a l directions. T h e t w o planes at t h e d e p t h s o f 4 5 a n d 50 m r e p r e s e n t t h e i n t e r f a c e s of t h e t w o surficial layers used in o n e o f o u r models.
to the trace o f the crack plane. Each grid point represents a station in the figure. Because of the s y m m e t r y o f m o t i o n the wave field is depicted on one side o f the source only. Thus, the receiver array covers a rectangular area of 180 X 90 m, the f r ont edge of which coincides with the crack trace, the source epicenter being located midway along this segment. The u and v components are the horizontal displacements parallel and normal to the crack, respectively, and w is the vertical c o m p o n e n t o f motion. There are two axes of s y m m e t r y for horizontal m o t i o n with this source-receiver configuration i.e., u is symmetric, and v is antisymmetric a b o u t the crack trace, while u is antisymmetric and v is symmetric with respect to the orthogonal axis passing through the epicenter. The vertical c o m p o n e n t , w, itself has a symm e t r y with respect to b o t h axes. -
-
37 2 3
2 <~
2
J
~L >-
0 _J LU O~ O0 w _J Z 0 Z W
O'
5
i
i
4 DIMENSIONLESS
6 TIME
8
Io
Ott/L
12
<~ <:]
0,.. _
:::L I--z
_
to
.8
W L~ <~ _J O_ uD
09 o9 W
.4
Z 0
~45
Z W
o
L4
,
2
4
DIMENSIONLESS
,
,
6
8
TIME
I0
aH/L
Fig. 4. N o r m a l c o m p o n e n t o f the particle v e l o c i t y and d i s p l a c e m e n t p l o t t e d as a f u n c t i o n o f time and p o s i t i o n o n the surface o f a crack w h i c h s u d d e n l y e x t e n d s at o n e end in res p o n s e to the application o f an e x c e s s tensile stress. T h e crack is represented b y 1 0 grids equally spaced over its length. Position 10 c o r r e s p o n d s t o t h e tip being e x t e n d e d and p o s i t i o n 1 is at the o p p o s i t e end o f the crack. V e l o c i t y , displacement, and t i m e are given as dimensionless f u n c t i o n s o f the crack and m e d i u m parameters (see t e x t for details).
373
In our analysis the crack opening function is a variable of depth only and is taken as constant over the width of the crack plane. The source spacetime function, depicted in Fig. 4, is the two-dimensional finite-difference solution obtained by Aki et al. (1977) for a crack suddenly extending at one end in response to an excess tensile stress. These results show the normal c o m p o n e n t of the particle velocities, 5, and displacements, v, calculated at ten positions on the crack surface and given as dimensionless functions of the excess stress ~ P (tensile stress minus overburden pressure), lengths L and Al of the crack and of its extension, and the compressional velocity ~ and rigidity t~ of the solid. The grid labelled 10 is where the extension takes )lace. In this solution the initial static elastic field which exists prior to the U
V
W
U
V
W
Fig. 5. Snapshots of the surface displacements produced by the jerky extension of the bottom tip of a vertical crack buried in a homogeneous half-space. The three components of displacement, u, v, w, shown at the top of the figure are parallel,transverse, and vertical, respectively. The snapshots are taken every 10 ms, starting 16 m s after the event origin time and ending 126 m s later. The static displacements resulting from this event are shown at the bottom of the figure.
371 crack extension is not considered. Consequently the static opening depicted in the figure only represents the increment AV in crack volume resulting from the extension of the crack. Note also that the static opening of grid 1 is n o t zero because this position is offset from the crack edge by half a grid owing to the discretization. This solution was derived assuming a Poisson ratio of 0.25 in the rock matrix and according to Aki et al. (1977) it is reliable down to the wavelength ~ = L / 2 or wavenumber k = 2~/~ = 4 ~ / L . Various tests o f the validity, stability, and dispersion of the finite-difference scheme used in this calculation can be found in Madariaga {1976) and Fehler and Aki (1978). Snapshots o f the ground motion pr oduced by the sudden extension of the b o t t o m tip of the crack are shown in Fig. 5. The medium considered is a homogeneous half space with the compressional and shear velocities a = V ~ km/s, ~ = 1 km/s, and a density p = 2.5 g/cm 3, The snapshots are taken every 10 ms, starting 16 ms after the onset of the rupture and ending 126 ms later. The static displacements resulting from the opening of the crack and calculated using Maruyama's (1964) formulation are shown at the b o t t o m of the figure for comparison. By convention the displacements outward from the source are positive. The sequence gives an overall picture of the dynamic ground response to the opening crack, clearly showing the onset of the disturbance and the propagation of the various wave fronts over the area. Of particular interest is the st[ong overshoot of the vertical and transverse horizontal displacements in relation to their static values. It is worth noting also that the static vertical displacement is downward and the static horizontal displacement parallel to the crack is inward directly above the source. For a more detailed interpretation, the seismograms of the three components of displacement obtained at 49 stations set along 7 profiles transverse to the crack are shown in Fig. 6. The profiles, listed at the top of the figure, are spaced 15 m apart and from left to right appear at increasing distance from the trace of the plane of s y m m e t r y transverse to the source, the first profile lying in this plane and the last one being located at a distance of 90 m. Similarly, the stations are set at distances of 0, 15, 30, 45, 60, 75, and 90 m from the crack trace, shown f r om t o p to b o t t o m for each individual c o m p o n e n t of motion. The records start with the arrival of the P wave and last 160 ms from this arrival, covering most of the duration of the ground response at each location. All the displacements have been normalized by the p ro d u ct APAI of excess stress applied on the crack times the length of the crack extension. The main characteristics of these records are the first motion which is outward everywhere, the marked oscillations in the signal associated with the ringing of the crack surfaces, the rapid damping of the vertical c o m p o n e n t of m o t i o n with range, the presence of a clear SH wave in the seismograms obtained at the most distant stations, and the large overshoot of the ground response near the source. The peak displacement occurs at the epicenter on the vertical c o m p o n e n t and within 15 m of the epicenter on the two horizontal components. The directions of profile 1 and of the
375 I
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3 4 5 DISTANCE ALONG CRACK TRACE ( M ) 30 45 60
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7
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~,~
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TqME (MSEC)
Fig. 6. S y n t h e t i c seismograms o f t h e three c o m p o n e n t s o f ground d i s p l a c e m e n t resulting f r o m the jerky e x t e n s i o n o f the b o t t o m tip o f a vertical crack buried in a h o m o g e n e o u s half space. T h e records are o b t a i n e d at 4 9 stations spaced at intervals o f 15 m in the directions parallel and normal t o the source. For each c o m p o n e n t marked at the left the t o p r o w o f stations c o i n c i d e s w i t h t h e crack trace and the b o t t o m r o w is 9 0 m away. Similarly the first c o l u m n o f stations d e p i c t e d at the left lies o n the trace o f the transverse vertical plane bisecting the crack and the last o n e at the right is at a distance o f 9 0 m. T h e records start w i t h the arrival o f the P wave f r o m the h y p o c e n t e r and last 1 6 0 ms. N o t e that the d i s p l a c e m e n t a m p l i t u d e s have b e e n n o r m a l i z e d b y the p r o d u c t APAl o f e x c e s s stress t i m e s the i n c r e m e n t o f crack length.
crack trace are nodes for SH motion because of the antisymmetry of the problem for horizontal displacements about these two lines. Further details on the P-SV and P-SH motions of the ground axe given in
376
(0,90)
(30,90)
(60,90)
(90,90}
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::~
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y ~u
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&
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Fig. 7. Polar plots of the particle trajectories in the three orthogonal planes defined by u, v, w. The data shown are taken from Fig. 6 for four stations located 90 m away from the crack trace and set from left to right at the distances of 0, 30, 60, and 90 m from the trace of the transverse vertical plane bisecting the crack. The scale of the displacements indicated at the bottom is given for the particular value ~PAl = 1 bar m. Fig. 7, w h i c h s h o w s p o l a r p l o t s o f t h e p a r t i c l e t r a j e c t o r i e s in t h e t h r e e o r t h o g o n a l planes d e f i n e d b y u, v, a n d w. F o u r s t a t i o n s are d e p i c t e d w i t h parallel and t r a n s v e r s e c o o r d i n a t e s as i n d i c a t e d at t h e t o p o f t h e figure; t h e s e s t a t i o n s are at a d i s t a n c e o f 90 m f r o m t h e c r a c k t r a c e a n d f r o m left t o right c o r r e s p o n d t o profiles 1, 3, 5, a n d 7. T h e r e t r o g r a d e elliptical m o t i o n appearing at t h e tail e n d o f t h e t r a j e c t o r i e s s h o w n in t h e b o t t o m p l o t s r e p r e s e n t s t h e c o n t r i b u t i o n o f t h e R a y l e i g h wave. T h e m a x i m u m a m p l i t u d e o f the R a y l e i g h m o t i o n is f o u n d a l o n g t h e first p r o f i l e w h i c h lies o n t h e axis o f t h e m a i n l o b e o f r a d i a t i o n f o r this wave. T h e c h a r a c t e r o f t h e g r o u n d r e s p o n s e evolves f r o m d i s p l a c e m e n t s w h i c h are p u r e l y P-SV t o d i s p l a c e m e n t s w h i c h are p r e d o m i n a n t l y P-SH as we m o v e t o w a r d t h e a z i m u t h w h i c h bisects t h e t w o planes o f s y m m e t r y o f t h e source. This a z i m u t h is t h e d i r e c t i o n o f m a x i m u m r a d i a t i o n f o r t h e far field SH wave a n d this wave is t h e d o m i n a n t c o n t r i b u t i o n t o the m o t i o n o b s e r v e d at t h e m o s t d i s t a n t s t a t i o n d e p i c t e d on p r o f i l e 7.
377
Our next model, shown in Fig. 8, studies the effect of a more complex structure on the ground response. The disposition of the source and stations is the same as in Fig. 6 but the medium structure n o w consists of t w o layers overlying a h o m o g e n e o u s half-space (see Fig. 3). The source is located in the t o p m o s t layer which has a thickness of 45 m and whose elastic parameters are identical to those of the half-space considered in our first model. The second layer, which is 5 m thick, has the compressional velocity, shear velocity, and density of 0.3 km/s, 0.2 km/s, and 2.3 g/cm 3, respectively. Finally, 2
6
3 4 5 DISTANCE ALONG CRACK TRACE [M)
~5
30
45
75
60 "
fZ
7
~
90 -
' ~ , . ~
- o
0 ~,
W'T
,q
°°i
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0 Z
o
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~
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~
15 50 45
/~v~,-,,~,,~
' ~ " ~ -
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~
~
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(~ 5'0 ~oo TIME (MSEC)
Fig. 8. T h e s a m e as Fig. 6 for a crack e x t e n d i n g at the b o t t o m in a structure w h i c h consists o f t w o layers over a half space. The source is in t h e t o p layer.
;37~ the density in the half-space is 2.4 g/cm 3 and the wave velocities are 0.9 km/s for P waves and 0.5 km/s f or S waves. The pat t ern o f radiation produced by the b o t t o m extension of the crack is analogous to that obtained in a homogeneous half-space but the c o m p l e x i t y o f the signals is greatly increased by the interference of multiply reflected P, SV, and SH arrivals. Close to the source the vertical displacement pulse associated with the first arrivals is followed by a lower-frequency c o m p o n e n t propagating with a low phase velocity. At increasing range in the direction transverse to the crack the vertical m o t i o n evolves into a wave train which consists primarily of inversely dispersed Rayleigh waves. As not e d earlier the horizontal displacements are larger than the vertical displacements outside the immediate source region and the Love waves and SH waves become the dom i nant contributions to the m o t i o n at the most distant station. The displacements are amplified by a factor of 3/2 with respect to those obtained in Fig. 6. The lateral e x t e n t and amplitude o f the downward static displacement is also more p r o n o u n c e d owing to the softness of the deep-seated structure. CONCLUSION The prime object o f our study was to provide a picture o f the m o t i o n of the ground induced by the small jerky extension of a buried preexisting crack. Our calculations disclosed the impulsive character and marked oscillation o f the ground response associated with the rupture. Among the major features f o u n d in the ground displacements are the strong overshoot of the dynamic m o t i o n in relation to the static d e f o r m a t i o n directly above the crack, the dominance of the vertical c o m p o n e n t at the epicenter, the predominance o f horizontal m o t i o n at some distance from the source, and the direction of the first m o t i o n which is everywhere outward. The static vertical displacement is dow nw a r d and the static longitudinal displacement is inward in the epicentral area. ACKNOWLEDGEMENTS I am grateful to Michael P. Ryan for helpful c o m m e n t s and for the permission to r ep r o duce two o f his photographs. This work was supported by the D e p a r t m e n t of Energy under Contract DE-AC02-76-ER02534.
REFERENCES
Aki, K., Fehler, M. and Das, S., 1977. Source mechanism of volcanic tremor: fluid-driven crack models and their application to the 1963 Kilauea eruption. J. Volcanol. Geotherm. Res., 2: 259--287. Aki, K., Chouet, B., Fehler, M., Zandt, G., Koyanagi, R., Colp, J. and Hay, R.G., 1978. Seismic properties of a shallow magma reservoir in Kilauea Iki by active and passive experiments. J. Geophys. Res., 83: 2273--2282.
379 Bouchon, M., 1979. Discrete wave number representation of elastic wave fields in threespace dimensions. J. Geophys. Res., 84: 3609---3614. Chouet, B., 1979. Sources of seismic events in the cooling Lava Lake of Kilauea Iki, Hawaii. J. Geophys. Res., 84: 2315--2330. Chouet, B., 1981. Ground motion in the near field of a fluid-driven crack and its interpretation in the study of shallow volcanic tremor. J. Geophys. Res., 86: 5985---6016. Chouet, B., 1982. Free surface displacements in the near field of a tensile crack expanding in three dimensions. J. Geophys. Res., 87 : 3868--3872. Chouet, B. and Aki, K., 1981. Seismic structure and seismicity of the cooling lava lake of Kilauea Iki, Hawaii. J. Volcanol. Geotherm. Res., 9: 41--56. Colp, J.L., 1979. Magma energy research project: FY 1979 annual progress report. Rep. SAND 79-2205, Sandia Lab., Albuquerque, NM, 96 pp. Fehler, M. and Aki, K., 1978. Numerical study of diffraction of plane elastic waves by a finite crack with application to location of a magma lens. Bull. Seismol. Soc. Am., 68 : 573--598. Flanigan, V.J. and Zablocki, C.J., 1977. Mapping the lateral boundaries of a cooling basaltic lake, Kilauea Iki, Hawaii, U.S. Geol. Surv., Open File Rep. 77-94, 21 pp. Madariaga, R., 1976. Dynamics of an expanding circular fault. Bull. Seismol. Soc. Am., 66: 639--660. Maruyama, T., 1964. Statical elastic dislocations in an infinite and semi-infinite medium. Bull. Earthquake Res. Inst. Univ. Tokyo, 42: 289--368. Peck, D.L. and Minakami, T., 1968. The formation of columnar joints in the upper part of Kilauean Lava Lakes, Hawaii. Geol. Soc. Am. Bull., 79: 1151--1166. Ryan, M.P. and Sammis, C.G., 1978. Cyclic fracture mechanisms in cooling basalt. Geol. Soc. Am. Bull., 89: 1259--1308. Ryan, M.P. and Sammis, C.G., 1981. The glass transition in basalt. J. Geophys. Res., 86: 9519---9535. Smith, B.D., Zablocki, C.J., Frischknecht, F.C. and Flanigan, V.J., 1977. Summary of results from electromagnetic and galvanic soundings on Kilauea Iki lava lake. U.S. Geol. Surv., Open File Rep. 77-59, 27 pp. Zablocki, C.J., 1976. Some electrical and magnetic studies on Kilauea Iki lava lake, Hawaii. U.S. Geol. Surv., Open File Rep. 76-304, 19 pp.