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Buoyancy-driven fracture ascent: Experiments in layered gelatine
Journal of Volcanology and Geothermal Research 144 (2005) 273 – 285 www.elsevier.com/locate/jvolgeores
Buoyancy-driven fracture ascent: Experiments in layered gelatine E. Rivaltaa,T, M. Bfttingerb, T. Dahma a
Institut fu¨r Geophysik, Universita¨t Hamburg, Bundesstr. 55, 20146 Hamburg, Germany b Deutsches Klimarechenzentrum (DKRZ), Bundesstr. 55, 20146 Hamburg, Germany Received and accepted 16 November 2004
Abstract Laboratory experiments on air-filled fracture propagation in solidified homogeneous and layered gelatine have been carried out, providing an analogue model for magma-filled dikes ascending in the crust. The effects of layering on fracture velocity and shape have been analyzed in detail. The free surface is found to accelerate approaching fractures. Layering accelerates or decelerates fractures approaching discontinuities of the elastic parameters, depending on the value of the rigidity contrast. The shape of fractures are strongly influenced as they pass from one layer to another. The observed cross-sectional shape when crossing a layer interface and the acceleration with decreasing rigidity can be explained with theoretical models. Our experiments also reproduce the arrest of fractures in proximity of joints and the formation of sills in the layer below the interface. These findings could help in the interpretation of accelerated seismicity and deformation rates observed in volcanic areas. D 2005 Elsevier B.V. All rights reserved. Keywords: analogue experiments; layered media; fluid-filled fractures; dike propagation; sill
1. Introduction High contrasts in the elastic parameters are common at shallow depth in volcanic areas or at the crust–mantle transition. Several volcanic edifices lie on very stiff basaltic basements, whereas the edifices themselves are commonly much more compliant, as demonstrated by several tomographic studies (e.g. Patane` et al., 2002; Di Stefano and Chiarabba, 2002).
T Corresponding author. Now at Department of Physics, University of Bologna, V. le Berti Pichat 8, 40126 Bologna, Italy. E-mail address: [email protected] (E. Rivalta). 0377-0273/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2004.11.030
Vertical profiles of elastic parameters are therefore characterized by high rigidity contrasts. In some areas cold brittle rocks lie on hot viscoelastic materials. This situation is found for example in hot-spot areas, where upwelling plumes influence the rheology of the surrounding materials. Often these situations are manifested through lowvelocity layers for shear waves. Although these and other temperature-controlled interfaces may be smooth, a sharp interface approximation may be useful when dikes are considerably longer than the transition zone width. Numerical and analytical models (Tinti and Armigliato, 1998; Bonafede and Rivalta, 1999a,b; Gud-
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mundsson and Marinoni, 1999) show that such rigidity contrasts are responsible for stress concentrations and changes in fracture shapes. As a result of the interaction between magma-filled fractures and discontinuity surfaces, the enhanced deformation can trigger seismic events. Gelatine has been used several times in experimental works on fluid-filled fracture propagation (Johnson and Pollard, 1973; Pollard and Johnson, 1973; Takada, 1990, 1994a,b; Heimpel and Olson, 1994; Dahm, 2000b; Menand and Tait, 2002; Ito and Martel, 2002). Laboratory experiments with fluid injections into gelatine are analogous to dike ascent in the crust and provide a 3D analogue model with several advantages for observations: gelatine is transparent, brittle at room temperatures, and the typical length of propagating fractures is of the order of centimeters. Furthermore, the elastic parameters can be controlled by varying the concentration of gelatine in water. The injection phase, when the fracture is typically penny-shaped, and the propagation stage, when the fracture is arched-door-shaped and moves with constant velocity toward the surface, have already been documented in previous studies. The process is well described by the Weertman model (Weertman, 1971), that provides the fundamental equations governing buoyancy-driven fluid-filled fractures in solid materials and is able to predict the crosssectional shape of ascending fractures with high accuracy. In a detailed work Johnson and Pollard (1973) and Pollard and Johnson (1973) studied the growth of Laccolithic intrusions and performed laboratory experiments in layered gelatine. They injected oil or grease of nearly neutral buoyancy directly into the layer interface in order to study the lateral growth and the deformation of the overburden. In contrast to these experiments, our study concentrates on the buoyancydriven ascent of intrusions and the impact of layer interfaces on ascent velocity and emplacement. The aim of this work is thus to investigate experimentally how fluid-filled fractures propagate in layered media. We analyze both possible situations: (1) experiments H2L: the fracture propagates from a high rigidity towards a low-rigidity medium, (2) experiments L2H: the opposite layering.
We discuss the main influence of the discontinuity with respect to the homogeneous case and the relevance of the rigidity and fracture toughness contrasts. The acceleration or the stopping of fractures can also be observed in the experiments, as well as the formation of horizontal, sill-like intrusions just below the layer interface.
2. Experimental technique 2.1. Laboratory experiments Gelatine is transparent and at 4 8C brittle for the typical loading times of our experiments (b30 min). Density and bulk modulus are similar to that of water, i.e. qc1000 kg/m3 and K=2.2 GPa. The shear modulus is very small, in the range between 50 and 10 000 Pa, depending on the concentration of the gelatine. We used concentrations between 2% and 5% leading to 75 PaVG V2000 Pa (Table 1). Due to the large Poisson number of mi0.5 the relative deformation and the opening-length ratio of fractures is larger than for fractures in rock. The development of fractures crucially depends on the involved parameters: type and volume of injected viscous fluids, elastic constants and fracture toughness K C of the two layers. After Weertman (1971), the critical half-length necessary for propagation can be expressed by 23 KC aC ¼ pffiffiffi ð1Þ pDqg where Dq is the density difference between the solid (in our case gelatine) and fluid (in our case air) and g the gravity acceleration. We approximate Dq with q water. The fracture toughness K Cpis approximately pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi given by KC ¼ 4cs Gð1 þ mÞ ¼ cs 6G, where cs 1 J m2 is the surface energy of gelatine (see Menand and Tait, 2002). Typical critical fracture lengths in our experiments were between 2 and 10 cm and the opening-length ratio was between 0.01 and 0.07. For rock and magma-filled dikes the critical dike-lengths are in the order of kilometer having an openinglength ratio of about 0.001. For the experiments we have used two types of acrylic glass containers: a rectangular one (length=40 cm, width=30 cm, height=50 cm) and a cylindric one
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Table 1 Experimental parameters: subscripts 1 and 2 declare parameter of the lower and upper layer, respectively No.
G 1 (Pa)
G 2 (Pa)
F15% 5 6 7 9 10 12
920 920 610 130 130 310
2a 1 (mm)
2a 2 (mm)
F15%
F3%
F3%
75 290 130 610 1200 2100
86 83 48 29 40 –
52 64 39 – (104) –
v 1 (mm/s)
v 2 (mm/s)
l 1 (mm)
l 2 (mm) F3%
F3%
F3%
F3%
0.45 0.32 0.18 0.20 0.7 –
26.7 1.8 4.1 – (1.9) –
72 57 41 21 32 –
56 55 32 – (82)
G is the shear modulus, a the half length, v the fracture velocity and l the lateral fracture length. The crack opening has not been measured for technical reasons. The injected volume was between about 3 and 9 ml. The shear modulus has been measured by the manufacturer. We verified the estimates for selected samples by using a Physica MCR Rheometer (Anton Paar Company) and derived error estimates from that. In Experiment No. 10, values in parenthesis are referred to a further reinjection of air.
(diameter=26 cm, height=50 cm). We injected air from the bottom through holes that could be closed during initial preparation with water proof screws. The container length and width is more than 5 times the fracture length and about 100 times the fracture opening, respectively. Boundary effects are therefore assumed to be small. The containers show different advantages and disadvantages with respect to our application: considering that the injected fracture orients itself in an unpredictable direction, the cylindric container can be rotated, which allows greater freedom to control the observation direction relative to the fracture, parallel or perpendicular to it. This permits characterisation of the 3D-shape of the crack. A disadvantage of the cylindrical shape is that optical distortion increases towards the sides of the container. This could have been minimised by placing the cylinder in a water bath (e.g. Cruden, 1988). Assuming that the fracture develops in a predictable way—parallel to the walls of the rectangular container—the optical properties of the gelatine body are much better suited to acquire clean undistorted images. It is possible to pre-determinate the alignment of the crack by smooth lateral movements of the injection needle (e.g. Ito and Martel, 2002), but this adds one more random element to the experiments. We filled the reservoirs with fluid gelatine (with concentration c and Bloom number BN) and put them into a refrigerator at 4 8C for at least 36–48 h, obtaining a stiff gelatine mass. After 36 h at constant temperature the rigidity of the gelatine is almost constant (Markus Lichtenfeld, Company Gelita, personal communication). The Bloom number is a technical term used by producers to
measure the strength of gelatine solid mass, representing the force necessary to deform a gel under standard conditions. To produce a layering, we carefully superposed warm and fluid gelatine on the already cold and stiff mass in a half-filled container. After further cooling for about 36 h the stiff gelatine layers are in welded contact. In the following, subscripts 1 and 2 correspond to the lower and upper layer, respectively. In our experiments the ascent process required from 2 to 30 min. The value of the fracture propagation velocity depends on the fracture length 2a and the medium parameters G, m, K C, on the fluid viscosity g and on Dqg. The relative change of fracture opening and ascent velocity between the two layers depend on G 2/G 1 and K C2/K C1 only. Thus, relative changes might directly be compared to situations in rock at small as well as large scales whenever the rigidity contrasts are similar to what we have modeled. We used rigidity contrast G 2/G 1 between 0.2 and 7. In order to accurately measure the velocity of the rising crack, we filmed the experiments with video cameras. The image resolution of the PAL system is 720576 (digital video). Photo cameras would allow for a much higher resolution of the resulting images, but we did not dispose of any other equipment able to continuously capture photos at regular intervals (e.g. one every second) for a 30 min long time interval. In order to capture the spatial crack geometry, the two cameras were set up perpendicular to each other (see Fig. 1). To minimize the geometrical distortion of the view and thus systematic errors it is necessary to keep the camera as far away as possible from the reservoir. We have used distances between 2.0 and 2.5 m.
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Fig. 1. Picture of the experimental setting: the container is lit by two spot lights. Air is then injected through a hole in the bottom with a syringe. Two video cameras record the experiments from perpendicular perspectives.
We have tried different setups for the position and direction of the lights (two halogen flood lights, 350 W and 650 W) relative to the gelatine reservoir. We achieved the best results (the highest contrast) with indirect back lights: white paper tables behind the reservoir lit by a flood light, with the room lights switched off. To gain a higher spatial resolution for the numerical analysis of the resulting images, we also did some experiments with both cameras pointing perpendicular to the crack, imaging the upper and the lower part of the gelatine body. 2.2. Data processing All video sequences were digitized on an Avid Xpress non linear editing system. The resulting pairs of video sequences were synchronized and composited into a side by side display. A time code was keyed into the resulting sequences before we exported single images at regular intervals. A cross-correlation algorithm (Matlab script) was used to locate upper tip of the fracture in every frame, in
order to determine the path of motion characteristic of each experiment. Since the finite resolution of the records produces discrete results, a least squares spline interpolation routine (Lawson and Hanson, 1974) was used to provide a continuous path of motion. This is necessary to define a velocity curve through differentiation. The following components have been considered for the estimation of the experimental errors: the discretisation of the images, the optical distortion caused by the fixed recording system and the light refraction, the numerical elaboration. The derivation of the interpolated curve is responsible for the biggest part of the whole final error, since it produces oscillations at the margins of the curve.
3. Experimental results 3.1. Case H2L: propagation from a stiff to a compliant medium Let us first analyze the dyke ascent from a highrigidity to a low-rigidity space. Such a layering can
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for instance be expected at the crust–mantle or at a basement–sediment boundary. A typical image sequence for this case is shown in Fig. 2 (Experiment No. 6, see Table 1). As predicted by numerical and analytical models (e.g. Bonafede and Rivalta, 1999b), we observe that as soon as the fracture reaches the discontinuity, it changes shape. Furthermore, we can measure a change in the propagation velocity. As long as the fracture is far from the discontinuity interface, the propagation is characterized by conservation of shape and velocity. The crack can be described by the model of a Weertman crack (e.g. Weertman, 1971; Takada, 1990; Heimpel and Olson, 1994; Dahm, 2000b). When the upper tip
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approaches the discontinuity, small modifications in shape and velocity begin to occur. While the fracture passes through the interface between the two layers, these changes become striking. Due to the lower rigidity of the upper layer, the fracture starts to enlarge its head, taking a more pronounced crosssectional tadpole and a frontal mushroom shape. The shape changes continuously, and re-assumes a teardrop cross-section after the completion of the transit. This shape is then conserved until the end of the ascent process, with small variations in the vicinity of the free surface. The opening in the middle of the crack has a typical shape during the passing of the interface,
Fig. 2. Experiment H2L (No. 6 in Table 1): In the picture, labels (L) indicate cross-sectional and (R) frontal perspective. (a) The injected fracture propagates in the first layer. (b, c, d) As soon as it reaches the interface, the shape changes considerably: the cross-section head of the fracture enlarges and frontally assumes a mushroom shape. (e, f) After the transit, the fracture moves on faster, reassuming the typical shape assumed in homogeneous media, with a different aspect ratio.
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where the crack head opens larger than its tail. In Fig. 3A and B an example for a rigidity contrast of r G =G 2/G 1=0.3 is shown (Experiment No. 6). The opening shape can be predicted by theoretical (e.g. Bonafede and Rivalta, 1999b) or numerical methods (e.g. Dahm, 1996). Fig. 3C shows the normalized theoretical opening for different ratios r G . Assuming a linear pressure gradient in the crack, the normalized opening of Weertman-cracks depends only on the position of the crack with respect to the interface and on the ratio G 2/G 1. It is independent from the magnitude of the gradient, and thus independent from the ascent velocity. Thus, the cross-sectional shape during the transition of the boundary can be used to estimate the ratio in shear moduli of the two gelatine layers (Fig. 3D).
Let us now analyze the fracture velocity. In this type of experiment two main phases can be identified, both characterized by an approximately constant velocity and shape far away from the layer interface. The fluid volume enclosed in the fracture is conserved during transition of the layer boundary. In Experiment No. 7 the ratio of the shear modulus is m=G 2/G 1=0.2. We find velocities of v 1=0.18 mm/s and v 2=4.1 mm/s far away from the layer interface (ratio v 2/v 1=23, Fig. 4 top panel). Values for other H2L-experiments are given in Table 1. To highlight the deviation from a constant velocity motion we removed the linear trend in the middle panel of Fig. 4. In proximity of the layer interface the curve deviates from linear trend. We find thus that the fracture accelerates even before it reaches the interface: the
Fig. 3. Panel A) Experiment H2L (No. 6 in Table 1): Snap shot of the cross-sectional (a) and frontal (b) shape of a gelatine crack during transit from a high to low-rigidity layer ( G 2/G 1=0.2). Panel B) Cross-sectional (left) and frontal (right) shape for Experiment No. 6 is shown before (bottom) and after (top) the transit through the interface. Fracture dimensions (width w, vertical length 2a and horizontal length l) are indicated in order to show the differences caused by the change of medium. The aspect ratio (w/2a) is 0.0035 and 0.06 for mediums 1 and 2 respectively. Panel C) Theoretical cross-sectional shape as predicted by theory and formulas given in Bonafede and Rivalta (1999b) and using different values of rigidity ratio of r G =G 2/G 1. Panel D) Theoretical prediction of the ratio between maximum fracture opening and opening at interface level for rigidity ratios of 0.1VG 2/G 1V1. The measured ratio for Experiment No. 6 is shown as a circle with error bars.
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A)
0
1 t
50
d (mm)
279
v = 4.14 mm/s
2 →
100
↓ v = 0.18 mm/ s
150 200 250 0
B)
50
100
150
200
250
t (s) 4
d - vm*t (mm)
3 2
↓ d – 0.18*t
1 0
d – 4.14*t →
-1 -2 0
50
100
150
200
250
150
200
250
t (s)
v (mm/s)
C)
8 7 6 5 4 3 2 1 0 0
50
100
t (s) Fig. 4. Panel A) Path of motion of the fracture upper tip (Experiment No. 7, Table 1). Panel B) The segments of the motion path in mediums 1 and 2 have been detrended by subtracting the respective average velocities. The pixel discretization causes the scattering of the points. A spline interpolation with relative uncertainties is shown. The curves demonstrate that the path of motion diverges from linearity. Panel C) The velocity of the fracture upper tip is calculated as the derivative of the best interpolating curve for the path of motion. Notice the two time intervals, where the velocities are approximately constant. During the transit (t), and to a lesser degree in proximity of the free surface, the acceleration is particularly relevant.
fracture dfeelsT the compliance of the medium above, that opposes less resistance to the fracturing process. In Fig. 4 (bottom panel) the upper tip velocity is shown, as calculated by the differentiation of the interpolation curve for the path (top panel). The two phases of constant velocity are evident. The acceleration in proximity of the interface is hidden by the much greater acceleration relative to the transit phase. We can then expect that an ascending dike approaching a layer with lower rigidity would show the following characteristics: (1) increasing opening deformation),
(causing
increased
(2) increasing velocity (causing increasing seismicity rate). 3.2. Case H2L: propagation towards the free surface Since the last part of the curves in Fig. 4 is relative to a homogeneous gelatine mass, we can use it to investigate the effects of the free surface, which can be treated as the interface to an extremely compliant half-space with a rigidity G=0. The detrended curve for medium 2 shows an even bigger acceleration in proximity of the free surface. The effect is small, but very important, since it pertains to the last phases of propagation. Because of the
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relevance of this finding, we will discuss free surface effects in a separate paper. 3.3. Case L2H: propagation from a compliant to a stiffer medium Let us now discuss the opposite case, where a stiffer half-space lies on a more compliant one. For instance, a more compact solidified magmatic intrusion or sill lying on more compliant volcanic deposits or sediments. In contrast to the results of the case H2L, we can expect that the fracture decelerates in proximity of the interface. In our experiments we stated this deceleration, and observed further interesting aspects. If a is the half-length of the fracture and a Ck the critical half-length in layer k, we have to distinguish between two cases for model L2H: (a) a C1baba C2: the injected volume is only sufficient for a propagation in medium 1, (b) a C1ba C2ba: the injected volume is sufficient for propagation in both media.
For case a), the propagation will therefore stop at the interface, and an additional fluid injection is necessary to obtain aNa C2. We reproduced this in the Experiment No. 10 shown in Fig. 5 (rigidity contrast G 2/G 1=7.5). The fracture first decelerates, and then stops at the interface. After the injection of additional fluid, we observed a lateral spreading of the fissure, that develops an anvil shape. A further injection through the same channel allows the fracture to increase its vertical length and thus its overpressure at its tip, until it breaks through the interface and the stiff top layer. It moves to the surface re-assuming the typical shape observed in homogeneous media, in this case with a slight inclination. The path of motion, the detrended path and the velocity for this case are shown in Fig. 6. Case H2L was theoretically simpler, since if a given volume of fluid is sufficient for propagation in medium 1, it is necessarily also sufficient for medium 2 (aNa C1Na C2). The deceleration, and particularly the arrest of the propagation caused by the rigidity contrast is a very important result, because it represents a possible explanation for dikes that do not reach
Fig. 5. Case L2H ( G 2/G 1=5.5), Experiment No. 10. Labels (L) indicate cross-sectional and (R) frontal perspective. (a) A small volume of air is enough for the propagation to occur. Once it arrived at the interface, the fracture does not contain a sufficient volume for the propagation in the second medium and comes to a halt. (b, c, d) A further air injection enlarges the fracture that assumes an danvilT or dambosT shape. (e, f, g, h) After the injection of a sufficient air volume, the fracture propagates into the upper layer.
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281
A) 0
d (mm)
20 40 60
← vm = 0.73 mm/s
80 100 0
d - vm*t (mm)
B)
20
40
60
80
100
120
80
100
120
80
100
120
t (s) 4 3 2 1
↑ d – 0.73*t
0 -1
0
20
40
60
t (s)
C) 1.2 v (mm/s)
1 0.8 0.6 0.4 0.2
0
20
40
60
t (s) Fig. 6. Panel A: path of motion of the fracture upper tip (Experiment No. 10, shown in Fig. 5. Depth is measured from the interface). The average velocity is indicated in figure. Panel B: detrended path of motion (calculated in the same way as for Fig. 4). The interpolation curve shows a deceleration. Panel C: velocity of the fracture upper tip. In proximity of the interface with a very stiff medium, the fracture decelerates.
the surface. If several of such dikes accumulate next to each other, a magma reservoir can develop. Such reservoirs can then become instable, if additional magma is supplied.
crack abruptly cut the gelatine along a horizontal plane a short distance below the interface and rapidly developed a semicircular sill-structure. In this way we have been able to observe the formation of a sill, as illustrated in Fig. 8.
3.4. Case L2H: a very high shear modulus contrast When the critical lengths characterizing the two layers are very different (Experiment No. 12 in Table 1), the high rigidity contrast prevents propagation through the interface. The crack tip stops at the layer interface and thus a very high stress concentration can be built at the discontinuity. Further fracture development is illustrated in Fig. 7. The fracture first propagated laterally along the interface, analogous to case L2H above. After a repeated injection of air the
4. Discussion 4.1. The acceleration of fractures Both accelerated deformation and intrusioninduced seismicity have been identified, beside others, as precursor signals for volcanic eruption (e.g. Kilburn, 2003). Ascending and accelerated dikes are one candidate to cause such signals.
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Fig. 7. Case L2H ( G 2/G 1=7.5), Experiment No. 10, formation of a sill. Labels (B) cross-sectional and (T) frontal perspective. (aT and aB) A small fracture ascends towards the interface. (b, c) The fracture stops moving, and additional air is injected. The fracture enlarges and assumes an elongated shape. (d, e, f) Instead of breaking into the upper medium, it propagates horizontally slightly below the interface between the two layers. The formation of the sill is best observed on the cross-sectional views.
Fig. 8. Snap-shot of the sill from Fig. 7. Panel A: perspective view, panel B: top view.
Fig. 4 documents the acceleration of a buoyancydriven fracture in gelatine when passing the interface from a high to a low-rigidity layer. The shear modulus decreased by a factor of nearly 5, while the ascent velocity increased more than a factor of about 20. The crack tip velocity during the transition of the layer boundary is a non-stationary, complex function. Away from the boundary, however, the ascent velocity is constant. The cross-sectional shape of the cracks can there be described by the Weertman model. Dahm (2000b) studied the shape and velocity of Weertman cracks in a full space. He derived a theoretical model to predict the ascent velocity v of those fractures, depending on their length, the elastic parameter of the medium, the fracture toughness, and the viscosity and density of the dyke fluid. Fig. 9 shows the predictions of this model when adapting the parameter used in Dahm (2000b) to the gelatine of Experiment Nos. 5 and 6. The volume of the injected air was about Vc9 ml in both experiments. The measured velocities and crack lengths in the lower and
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Fig. 9. Theoretical predictions of ascent velocity (solid line, theory from Dahm, 2000b) compared to observations (Experiments No. 5 and No. 6, open circles). The dashed line shows the estimated length of the crack using an approximate formula for the volume of a 3D Weertman pffiffiffiffiffiffi crack (Dahm, 2000b). Crosses are the measured crack lengths. The volume of injected air was c9 ml in both experiments, and KC ¼ 6GPa m1=2 has been used.
upper layer are indicated as circles and crosses, respectively. The increase in ascent velocity is accompanied by a decrease in crack length and can be approximated by the model of Dahm (2000b). The model can be used to predict the increase or decrease of the ascent velocity of a dike ascending in the earth and retaining its fluid volume. While shear modulus, density and Poisson’s ratio are relatively well known for the Earth’s upper mantle and crust, the fracture toughness is difficult to estimate and nearly unknown. For example, laboratory experiments of rock samples indicate a low-pressure tensile fracture toughness of c1 MPa m1/2 (c Sc10 to 30 J/m2) for limestone or c3 MPa m1/2 for basalt. Apparent K C values inferred from basaltic dikes are more in the range from 100 MPa m1/2 to 1 GPa m1/2 (see e.g. Parfitt, 1991; Heimpel and Olson, 1994, and references given in their Table 3), corresponding to surface energy in the order of 105 to 106 J/m2. Fialko and Rubin (1997) find indications that the surface energy increases with increasing pressure. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assuming KC c 4Gcs ð1 þ mÞ and a dike volume V, the fracture length and the ascent velocity can be predicted as a function of G and c S. G is about 70 GPa in the depth of about 200 km in the low-velocity zone and about 25 GPa in the upper crust. Assuming a volume of V=1d 106 m3 (see e.g. Battaglia and Bacherlery, 2003, for dike-volume estimates at Piton de la Fournaise) a dike would increase its ascent velocity by about a factor of 3 when sampling the scale from the LVZ to the upper crust, independently
from a constant c S. The dike length would decrease about 10% during such a hypothetic ascent. However, the effect of a pressure-dependent, decreasing apparent surface energy c S would be more dramatic and showing an opposite trend. Assuming that c S decreases about a factor of 100 from a depth of 200 km to 5 km in the earth, the ascent velocity would decrease a factor of about 20, while the dike-length would grow. Again, long and tall dikes are predicted to be slow while the short and fat ones are fast. Anyway, we think that surface energy variations for rocks in the crust are relatively small so that the knowledge of G is sufficient for a first order modeling of dike ascent there. 4.2. The formation of sills and magmatic underplating Figs. 5–8 document that buoyancy-driven fractures may decelerate or even stop when approaching a lowto-high rigidity contrast (or alternatively a low-to-high contrast in c S or K C). The formation of sill can be understood in terms of strong stress concentration at the crack tip. A small asymmetry of the stopped crack, in combination with an increasing horizontal compression, (e.g., produced by a small inclination of the stopped crack with growing opening), is able to enhance horizontal fracturing. Dahm (2000a) has shown by 2D numerical simulation that horizontal compressive stress acting on fluid-filled fractures with a weak buoyancy force promotes horizontal growth of the fractures and thus
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the formation of sills. Interestingly here, and in contrast to the simulations in Dahm (2000a) is that the sill forms in the bottom layer just below the interface. It does not form directly at the layer interface. The stress intensity factor at the fracture tip is too small to overcome the fracture toughness of the upper layer. Therefore, the dyke is arrested before cutting the layer boundary. This is a different process than the arrest of a vertical dike intrusion because of the decreasing of its apparent buoyancy at a short distance above the level of neutral buoyancy (LNB, e.g. Dahm, 2000a). Such a process has often been postulated for the Moho as a magma-trapping interface, explaining that high density, mantle derived magma intrudes into the lower crust and solidifies there. The rigidity barrier effect discussed in this study may additionally explain that smaller upper mantle intrusions are trapped below or at the Moho as long as the surface energy c S would increase in the crust. Other potential dike-stopping interfaces may exist at low-velocity zones in the upper mantle or within the crust or in the uppermost crust when compact sill layers overly soft sediments or highly fractured layers. The brittle–ductile transition is typically not associated with a shear-wave low-velocity zone and may thus not act as a dikestopping interface or transition. However, whenever the upper crust is loaded by tectonic compression, the ductile lower crust will not sustain large shear stresses and a compressional stress barrier may develop. Such a stress barrier is able to trap magma in a similar way as a LNB interface. The experiment of Fig. 8 shows that a low-to-high rigidity interface is able to generate a sill-like fluid reservoir below the layer boundary although the intruded fluid is lighter than the surrounding rock. It is difficult to give an estimation for a definite value of rigidity contrast promoting formation of sills. We did not systematically study the formation in function of the elastic parameters, and we cannot exclude that other factors may be even more important. After a buoyant sill has been formed, every additional fracture entering this sill would increase the volume of the reservoir but not cut the layer boundary (e.g. Weertman, 1980). For example, an extensive sill-like reservoir has been evidenced below
Mt. Vesuvius at about 10 km depth and associated with a low velocity layer (Auger et al., 2001). Buoyant sill-like reservoirs below the LNB bear a high risk for triggering volcanic eruption. The fluid trapped in the reservoir is less dense than the surrounding rock. After becoming instable, e.g. by the generation of an edge crack, a large volume of magma or fluid can be binjectedQ in a dike that will rise with high velocities and possibly break through the Earth’s surface. The formation of potentially instable reservoirs is, however, not only of interest for shallow levels in the upper crust. An unsolved problem is how magma-reservoirs may form in the mantle. Due to the small density variations in the mantle it is hard to generate a reservoir at a level of neutral buoyancy. A compressional stress barrier may be an alternative, but is not very likely in the mantle since high deviatoric stress would be released by viscous flow. Our experiments may give a third possibility to be discussed. Reservoirs in the mantle may be formed at binterfacesQ, where K C or G increases with decreasing depth, e.g. at the upper boundary of the low-velocity zone.
5. Conclusions The possibilities offered by the investigation of layered gelatine are numerous. In this paper we have only explored the simplest cases. Our analogue experiments for buoyancy-driven fractures in layered gelatine focus on the impact of layering on ascent velocity and dike emplacement. The effect of contrasts in rigidity and fracture toughness is studied. We observed accelerated and decelerated motion, arrest, accumulation of fluid and the formation of sill-like intrusions. The 3D-shape of the fractures changes in a characteristic manner during the different intrusion styles and has been documented by photographs and videos. As far as a 2D theory is applicable, it is in good agreement with our observations. The observation of propagation mechanisms in gelatine is helpful to answer questions relative to dike dynamics in a layered earth. The magmatic dike dynamics in the earth may be influenced by several additional mechanisms that cannot be introduced in
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our analogue model, e.g. temperature difference with the host rock, density difference between the several layers, tectonic stress field, topography. Nevertheless, we believe that our results are quite general and give fundamental information to understand magma ascent processes beneath volcanoes. Acknowledgements The review of Akira Takada and Alexander Cruden improved the work and is gratefully acknowledged, as well as useful comments by Agust Gudmundsson and Valerio Acocella. The authors would like to thank Michael Schnese, Martin Thorwart, Rainer Knut, Thomas Braun, Ali Dehgani, Christel Mynarik, Birgit Stffen-Vosberg for the help in the experiments. Marina Mu¨ller, Giorgio dalla Via and Massimiliano Pittore helped in the image analysis. Eberhard Dick and Markus Lichtenfeld from the company DGF Stoess AG provided the gelatine and measured the elastic parameters. The work has been financed by the EUproject VOLCALERT. References Auger, E., Gasparini, P., Virieux, J., Zollo, A., 2001. Seismic evidence of an extended magmatic sill under Mt. Vesuvius. Science 294, 1510 – 1512. Battaglia, J., Bacherlery, P., 2003. Dynamik dyke propagation deduced from tilt variations preceding the March 9, 1998, eruption of the Piton de la Fournaise volcano. J. Volcanol. Geotherm. Res. 120, 289 – 310. Bonafede, M., Rivalta, E., 1999a. The tensile dislocation problem in a layered elastic medium. Geophys. J. Int. 136, 341 – 356. Bonafede, M., Rivalta, E., 1999b. On tensile cracks close to and across the interface between two elastic half spaces. Geophys. J. Int. 138, 410 – 434. Cruden, A.R., 1988. Deformation around a rising diapir modelled by creeping flow past a sphere. Tectonics 7, 1091 – 1101. Dahm, T., 1996. Elastostatic simulation of dislocation sources in heterogeneous stress fields and multilayered media having irregular interfaces. Phys. Chem. Earth 21, 241 – 246. Dahm, T., 2000a. Numerical simulations of the propagation path and the arrest of fluid-filled fractures in the earth. Geophys. J. Int. 141, 623 – 638. Dahm, T., 2000b. On the shape and velocity of fluid-filled fractures in the earth. Geophys. J. Int. 142, 181 – 192. Di Stefano, R., Chiarabba, C., 2002. Active source tomography at Mt. Vesuvius: constraints for the magmatic system. J. Geophys. Res. 107, 2278 – 2292.
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Fialko, Y.A., Rubin, A.M., 1997. Numerical simulation of highpressure rock tensile fracture experiments: evidence of an increase in fracture energy with pressure? J. Geophys. Res. 102, 5231 – 5242. Gudmundsson, A., Marinoni, L.B., 1999. Geometry, emplacement and arrest of dykes. Ann. Tecton., 71 – 92. Heimpel, M., Olson, P., 1994. Buoyancy-driven fracture and magma transport through the litosphere: models and experiments. In: Ryan, P. (Ed.), Magmatic Systems. Academic Press, pp. 223 – 240. Ito, G., Martel, S., 2002. Focusing of magma in the upper mantle through dike interaction. J. Geophys. Res. 107, B10, ECV 16-1–ECV 16-17. Johnson, A., Pollard, D.D., 1973. Mechanics of growth of some laccolithic intrusions in the Henry Mountains, Utah: I. Field observations, Gilbert’s model, physical properties and flow of the magma. Tectonophysics, vol. 18, 261 – 309. Kilburn, C.R.J., 2003. Multiscale fracturing as a key to forecast volcanic eruptions. J. Volcanol. Geotherm. Res. 2623, 1 – 19. Lawson, C.L., Hanson, R.J., 1974. Solving Least Square Problems. Prentice-Hall, Englewood Cliffs, New Jork, p. 224. Menand, T., Tait, S.R., 2002. The propagation of a buoyant liquid-filled fissure from a source under constant pressure: an experimental approach. J. Geophys. Res. 107, B11, ECV 16-1–ECV 16-14. Parfitt, E.A., 1991. The role of rift zone storage in controlling the site and timing of eruptions and intrusions of Kilauea volcano, Hawaii. J. Geophys. Res. 96, 10101 – 10112. Patane`, D., Chiarabba, C., Cocina, O., De Gori, P., Moretti, M., Boschi, E., 2002. Tomographic images and 3D earthquake locations of the seismic swarm preceding the 2001 Mt. Etna eruption: evidence for a dyke intrusion. Geophys. Res. Lett. 29 (10), 135 – 138. Pollard, D.D., Johnson, A.M., 1973. Mechanics of growth of some laccolithic intrusions in the Henry Mountains, Utah: II. Bending and failure of overburden layers and sill formation. Tectonophysics 18, 311 – 354. Takada, A., 1990. Experimental study on propgation of liquid-filled crack in gelatine: shape and velocity in hydrostatic stress conditions. J. Geophys. Res. 95, 8471 – 8481. Takada, A., 1994. Accumulation of magma in space and time by crack interaction. In: Ryan, P. (Ed.), Magmatic Systems. Academic Press, pp. 241 – 257. Takada, A., 1994. Development of a subvolcanic structure by the interaction of liquid-filled cracks. J. Volcanol. Geotherm. Res. 62, 207 – 224. Tinti, S., Armigliato, A., 1998. Displacement and stress induced by a point source across a plane interface separating two elastic semi-infinite spaces: an analytical solution. J. Geophys. Res. 103 (B7), 15109 – 15125. Weertman, J., 1971. Theory of water-filled crevasses in glaciers applied to vertical magma transport beneath oceanic ridges. J. Geophys. Res. 76, 1171 – 1183. Weertman, J., 1980. The stopping of a rising liquid-filled crack in the Earth’s crust by a freely slipping horizontal joint. J. Geophys. Res. 85, 929 – 932.
Buoyancy-driven fracture ascent: Experiments in layered gelatine E. Rivaltaa,T, M. Bfttingerb, T. Dahma a
Institut fu¨r Geophysik, Universita¨t Hamburg, Bundesstr. 55, 20146 Hamburg, Germany b Deutsches Klimarechenzentrum (DKRZ), Bundesstr. 55, 20146 Hamburg, Germany Received and accepted 16 November 2004
Abstract Laboratory experiments on air-filled fracture propagation in solidified homogeneous and layered gelatine have been carried out, providing an analogue model for magma-filled dikes ascending in the crust. The effects of layering on fracture velocity and shape have been analyzed in detail. The free surface is found to accelerate approaching fractures. Layering accelerates or decelerates fractures approaching discontinuities of the elastic parameters, depending on the value of the rigidity contrast. The shape of fractures are strongly influenced as they pass from one layer to another. The observed cross-sectional shape when crossing a layer interface and the acceleration with decreasing rigidity can be explained with theoretical models. Our experiments also reproduce the arrest of fractures in proximity of joints and the formation of sills in the layer below the interface. These findings could help in the interpretation of accelerated seismicity and deformation rates observed in volcanic areas. D 2005 Elsevier B.V. All rights reserved. Keywords: analogue experiments; layered media; fluid-filled fractures; dike propagation; sill
1. Introduction High contrasts in the elastic parameters are common at shallow depth in volcanic areas or at the crust–mantle transition. Several volcanic edifices lie on very stiff basaltic basements, whereas the edifices themselves are commonly much more compliant, as demonstrated by several tomographic studies (e.g. Patane` et al., 2002; Di Stefano and Chiarabba, 2002).
T Corresponding author. Now at Department of Physics, University of Bologna, V. le Berti Pichat 8, 40126 Bologna, Italy. E-mail address: [email protected] (E. Rivalta). 0377-0273/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2004.11.030
Vertical profiles of elastic parameters are therefore characterized by high rigidity contrasts. In some areas cold brittle rocks lie on hot viscoelastic materials. This situation is found for example in hot-spot areas, where upwelling plumes influence the rheology of the surrounding materials. Often these situations are manifested through lowvelocity layers for shear waves. Although these and other temperature-controlled interfaces may be smooth, a sharp interface approximation may be useful when dikes are considerably longer than the transition zone width. Numerical and analytical models (Tinti and Armigliato, 1998; Bonafede and Rivalta, 1999a,b; Gud-
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mundsson and Marinoni, 1999) show that such rigidity contrasts are responsible for stress concentrations and changes in fracture shapes. As a result of the interaction between magma-filled fractures and discontinuity surfaces, the enhanced deformation can trigger seismic events. Gelatine has been used several times in experimental works on fluid-filled fracture propagation (Johnson and Pollard, 1973; Pollard and Johnson, 1973; Takada, 1990, 1994a,b; Heimpel and Olson, 1994; Dahm, 2000b; Menand and Tait, 2002; Ito and Martel, 2002). Laboratory experiments with fluid injections into gelatine are analogous to dike ascent in the crust and provide a 3D analogue model with several advantages for observations: gelatine is transparent, brittle at room temperatures, and the typical length of propagating fractures is of the order of centimeters. Furthermore, the elastic parameters can be controlled by varying the concentration of gelatine in water. The injection phase, when the fracture is typically penny-shaped, and the propagation stage, when the fracture is arched-door-shaped and moves with constant velocity toward the surface, have already been documented in previous studies. The process is well described by the Weertman model (Weertman, 1971), that provides the fundamental equations governing buoyancy-driven fluid-filled fractures in solid materials and is able to predict the crosssectional shape of ascending fractures with high accuracy. In a detailed work Johnson and Pollard (1973) and Pollard and Johnson (1973) studied the growth of Laccolithic intrusions and performed laboratory experiments in layered gelatine. They injected oil or grease of nearly neutral buoyancy directly into the layer interface in order to study the lateral growth and the deformation of the overburden. In contrast to these experiments, our study concentrates on the buoyancydriven ascent of intrusions and the impact of layer interfaces on ascent velocity and emplacement. The aim of this work is thus to investigate experimentally how fluid-filled fractures propagate in layered media. We analyze both possible situations: (1) experiments H2L: the fracture propagates from a high rigidity towards a low-rigidity medium, (2) experiments L2H: the opposite layering.
We discuss the main influence of the discontinuity with respect to the homogeneous case and the relevance of the rigidity and fracture toughness contrasts. The acceleration or the stopping of fractures can also be observed in the experiments, as well as the formation of horizontal, sill-like intrusions just below the layer interface.
2. Experimental technique 2.1. Laboratory experiments Gelatine is transparent and at 4 8C brittle for the typical loading times of our experiments (b30 min). Density and bulk modulus are similar to that of water, i.e. qc1000 kg/m3 and K=2.2 GPa. The shear modulus is very small, in the range between 50 and 10 000 Pa, depending on the concentration of the gelatine. We used concentrations between 2% and 5% leading to 75 PaVG V2000 Pa (Table 1). Due to the large Poisson number of mi0.5 the relative deformation and the opening-length ratio of fractures is larger than for fractures in rock. The development of fractures crucially depends on the involved parameters: type and volume of injected viscous fluids, elastic constants and fracture toughness K C of the two layers. After Weertman (1971), the critical half-length necessary for propagation can be expressed by 23 KC aC ¼ pffiffiffi ð1Þ pDqg where Dq is the density difference between the solid (in our case gelatine) and fluid (in our case air) and g the gravity acceleration. We approximate Dq with q water. The fracture toughness K Cpis approximately pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi given by KC ¼ 4cs Gð1 þ mÞ ¼ cs 6G, where cs 1 J m2 is the surface energy of gelatine (see Menand and Tait, 2002). Typical critical fracture lengths in our experiments were between 2 and 10 cm and the opening-length ratio was between 0.01 and 0.07. For rock and magma-filled dikes the critical dike-lengths are in the order of kilometer having an openinglength ratio of about 0.001. For the experiments we have used two types of acrylic glass containers: a rectangular one (length=40 cm, width=30 cm, height=50 cm) and a cylindric one
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Table 1 Experimental parameters: subscripts 1 and 2 declare parameter of the lower and upper layer, respectively No.
G 1 (Pa)
G 2 (Pa)
F15% 5 6 7 9 10 12
920 920 610 130 130 310
2a 1 (mm)
2a 2 (mm)
F15%
F3%
F3%
75 290 130 610 1200 2100
86 83 48 29 40 –
52 64 39 – (104) –
v 1 (mm/s)
v 2 (mm/s)
l 1 (mm)
l 2 (mm) F3%
F3%
F3%
F3%
0.45 0.32 0.18 0.20 0.7 –
26.7 1.8 4.1 – (1.9) –
72 57 41 21 32 –
56 55 32 – (82)
G is the shear modulus, a the half length, v the fracture velocity and l the lateral fracture length. The crack opening has not been measured for technical reasons. The injected volume was between about 3 and 9 ml. The shear modulus has been measured by the manufacturer. We verified the estimates for selected samples by using a Physica MCR Rheometer (Anton Paar Company) and derived error estimates from that. In Experiment No. 10, values in parenthesis are referred to a further reinjection of air.
(diameter=26 cm, height=50 cm). We injected air from the bottom through holes that could be closed during initial preparation with water proof screws. The container length and width is more than 5 times the fracture length and about 100 times the fracture opening, respectively. Boundary effects are therefore assumed to be small. The containers show different advantages and disadvantages with respect to our application: considering that the injected fracture orients itself in an unpredictable direction, the cylindric container can be rotated, which allows greater freedom to control the observation direction relative to the fracture, parallel or perpendicular to it. This permits characterisation of the 3D-shape of the crack. A disadvantage of the cylindrical shape is that optical distortion increases towards the sides of the container. This could have been minimised by placing the cylinder in a water bath (e.g. Cruden, 1988). Assuming that the fracture develops in a predictable way—parallel to the walls of the rectangular container—the optical properties of the gelatine body are much better suited to acquire clean undistorted images. It is possible to pre-determinate the alignment of the crack by smooth lateral movements of the injection needle (e.g. Ito and Martel, 2002), but this adds one more random element to the experiments. We filled the reservoirs with fluid gelatine (with concentration c and Bloom number BN) and put them into a refrigerator at 4 8C for at least 36–48 h, obtaining a stiff gelatine mass. After 36 h at constant temperature the rigidity of the gelatine is almost constant (Markus Lichtenfeld, Company Gelita, personal communication). The Bloom number is a technical term used by producers to
measure the strength of gelatine solid mass, representing the force necessary to deform a gel under standard conditions. To produce a layering, we carefully superposed warm and fluid gelatine on the already cold and stiff mass in a half-filled container. After further cooling for about 36 h the stiff gelatine layers are in welded contact. In the following, subscripts 1 and 2 correspond to the lower and upper layer, respectively. In our experiments the ascent process required from 2 to 30 min. The value of the fracture propagation velocity depends on the fracture length 2a and the medium parameters G, m, K C, on the fluid viscosity g and on Dqg. The relative change of fracture opening and ascent velocity between the two layers depend on G 2/G 1 and K C2/K C1 only. Thus, relative changes might directly be compared to situations in rock at small as well as large scales whenever the rigidity contrasts are similar to what we have modeled. We used rigidity contrast G 2/G 1 between 0.2 and 7. In order to accurately measure the velocity of the rising crack, we filmed the experiments with video cameras. The image resolution of the PAL system is 720576 (digital video). Photo cameras would allow for a much higher resolution of the resulting images, but we did not dispose of any other equipment able to continuously capture photos at regular intervals (e.g. one every second) for a 30 min long time interval. In order to capture the spatial crack geometry, the two cameras were set up perpendicular to each other (see Fig. 1). To minimize the geometrical distortion of the view and thus systematic errors it is necessary to keep the camera as far away as possible from the reservoir. We have used distances between 2.0 and 2.5 m.
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Fig. 1. Picture of the experimental setting: the container is lit by two spot lights. Air is then injected through a hole in the bottom with a syringe. Two video cameras record the experiments from perpendicular perspectives.
We have tried different setups for the position and direction of the lights (two halogen flood lights, 350 W and 650 W) relative to the gelatine reservoir. We achieved the best results (the highest contrast) with indirect back lights: white paper tables behind the reservoir lit by a flood light, with the room lights switched off. To gain a higher spatial resolution for the numerical analysis of the resulting images, we also did some experiments with both cameras pointing perpendicular to the crack, imaging the upper and the lower part of the gelatine body. 2.2. Data processing All video sequences were digitized on an Avid Xpress non linear editing system. The resulting pairs of video sequences were synchronized and composited into a side by side display. A time code was keyed into the resulting sequences before we exported single images at regular intervals. A cross-correlation algorithm (Matlab script) was used to locate upper tip of the fracture in every frame, in
order to determine the path of motion characteristic of each experiment. Since the finite resolution of the records produces discrete results, a least squares spline interpolation routine (Lawson and Hanson, 1974) was used to provide a continuous path of motion. This is necessary to define a velocity curve through differentiation. The following components have been considered for the estimation of the experimental errors: the discretisation of the images, the optical distortion caused by the fixed recording system and the light refraction, the numerical elaboration. The derivation of the interpolated curve is responsible for the biggest part of the whole final error, since it produces oscillations at the margins of the curve.
3. Experimental results 3.1. Case H2L: propagation from a stiff to a compliant medium Let us first analyze the dyke ascent from a highrigidity to a low-rigidity space. Such a layering can
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for instance be expected at the crust–mantle or at a basement–sediment boundary. A typical image sequence for this case is shown in Fig. 2 (Experiment No. 6, see Table 1). As predicted by numerical and analytical models (e.g. Bonafede and Rivalta, 1999b), we observe that as soon as the fracture reaches the discontinuity, it changes shape. Furthermore, we can measure a change in the propagation velocity. As long as the fracture is far from the discontinuity interface, the propagation is characterized by conservation of shape and velocity. The crack can be described by the model of a Weertman crack (e.g. Weertman, 1971; Takada, 1990; Heimpel and Olson, 1994; Dahm, 2000b). When the upper tip
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approaches the discontinuity, small modifications in shape and velocity begin to occur. While the fracture passes through the interface between the two layers, these changes become striking. Due to the lower rigidity of the upper layer, the fracture starts to enlarge its head, taking a more pronounced crosssectional tadpole and a frontal mushroom shape. The shape changes continuously, and re-assumes a teardrop cross-section after the completion of the transit. This shape is then conserved until the end of the ascent process, with small variations in the vicinity of the free surface. The opening in the middle of the crack has a typical shape during the passing of the interface,
Fig. 2. Experiment H2L (No. 6 in Table 1): In the picture, labels (L) indicate cross-sectional and (R) frontal perspective. (a) The injected fracture propagates in the first layer. (b, c, d) As soon as it reaches the interface, the shape changes considerably: the cross-section head of the fracture enlarges and frontally assumes a mushroom shape. (e, f) After the transit, the fracture moves on faster, reassuming the typical shape assumed in homogeneous media, with a different aspect ratio.
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where the crack head opens larger than its tail. In Fig. 3A and B an example for a rigidity contrast of r G =G 2/G 1=0.3 is shown (Experiment No. 6). The opening shape can be predicted by theoretical (e.g. Bonafede and Rivalta, 1999b) or numerical methods (e.g. Dahm, 1996). Fig. 3C shows the normalized theoretical opening for different ratios r G . Assuming a linear pressure gradient in the crack, the normalized opening of Weertman-cracks depends only on the position of the crack with respect to the interface and on the ratio G 2/G 1. It is independent from the magnitude of the gradient, and thus independent from the ascent velocity. Thus, the cross-sectional shape during the transition of the boundary can be used to estimate the ratio in shear moduli of the two gelatine layers (Fig. 3D).
Let us now analyze the fracture velocity. In this type of experiment two main phases can be identified, both characterized by an approximately constant velocity and shape far away from the layer interface. The fluid volume enclosed in the fracture is conserved during transition of the layer boundary. In Experiment No. 7 the ratio of the shear modulus is m=G 2/G 1=0.2. We find velocities of v 1=0.18 mm/s and v 2=4.1 mm/s far away from the layer interface (ratio v 2/v 1=23, Fig. 4 top panel). Values for other H2L-experiments are given in Table 1. To highlight the deviation from a constant velocity motion we removed the linear trend in the middle panel of Fig. 4. In proximity of the layer interface the curve deviates from linear trend. We find thus that the fracture accelerates even before it reaches the interface: the
Fig. 3. Panel A) Experiment H2L (No. 6 in Table 1): Snap shot of the cross-sectional (a) and frontal (b) shape of a gelatine crack during transit from a high to low-rigidity layer ( G 2/G 1=0.2). Panel B) Cross-sectional (left) and frontal (right) shape for Experiment No. 6 is shown before (bottom) and after (top) the transit through the interface. Fracture dimensions (width w, vertical length 2a and horizontal length l) are indicated in order to show the differences caused by the change of medium. The aspect ratio (w/2a) is 0.0035 and 0.06 for mediums 1 and 2 respectively. Panel C) Theoretical cross-sectional shape as predicted by theory and formulas given in Bonafede and Rivalta (1999b) and using different values of rigidity ratio of r G =G 2/G 1. Panel D) Theoretical prediction of the ratio between maximum fracture opening and opening at interface level for rigidity ratios of 0.1VG 2/G 1V1. The measured ratio for Experiment No. 6 is shown as a circle with error bars.
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A)
0
1 t
50
d (mm)
279
v = 4.14 mm/s
2 →
100
↓ v = 0.18 mm/ s
150 200 250 0
B)
50
100
150
200
250
t (s) 4
d - vm*t (mm)
3 2
↓ d – 0.18*t
1 0
d – 4.14*t →
-1 -2 0
50
100
150
200
250
150
200
250
t (s)
v (mm/s)
C)
8 7 6 5 4 3 2 1 0 0
50
100
t (s) Fig. 4. Panel A) Path of motion of the fracture upper tip (Experiment No. 7, Table 1). Panel B) The segments of the motion path in mediums 1 and 2 have been detrended by subtracting the respective average velocities. The pixel discretization causes the scattering of the points. A spline interpolation with relative uncertainties is shown. The curves demonstrate that the path of motion diverges from linearity. Panel C) The velocity of the fracture upper tip is calculated as the derivative of the best interpolating curve for the path of motion. Notice the two time intervals, where the velocities are approximately constant. During the transit (t), and to a lesser degree in proximity of the free surface, the acceleration is particularly relevant.
fracture dfeelsT the compliance of the medium above, that opposes less resistance to the fracturing process. In Fig. 4 (bottom panel) the upper tip velocity is shown, as calculated by the differentiation of the interpolation curve for the path (top panel). The two phases of constant velocity are evident. The acceleration in proximity of the interface is hidden by the much greater acceleration relative to the transit phase. We can then expect that an ascending dike approaching a layer with lower rigidity would show the following characteristics: (1) increasing opening deformation),
(causing
increased
(2) increasing velocity (causing increasing seismicity rate). 3.2. Case H2L: propagation towards the free surface Since the last part of the curves in Fig. 4 is relative to a homogeneous gelatine mass, we can use it to investigate the effects of the free surface, which can be treated as the interface to an extremely compliant half-space with a rigidity G=0. The detrended curve for medium 2 shows an even bigger acceleration in proximity of the free surface. The effect is small, but very important, since it pertains to the last phases of propagation. Because of the
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relevance of this finding, we will discuss free surface effects in a separate paper. 3.3. Case L2H: propagation from a compliant to a stiffer medium Let us now discuss the opposite case, where a stiffer half-space lies on a more compliant one. For instance, a more compact solidified magmatic intrusion or sill lying on more compliant volcanic deposits or sediments. In contrast to the results of the case H2L, we can expect that the fracture decelerates in proximity of the interface. In our experiments we stated this deceleration, and observed further interesting aspects. If a is the half-length of the fracture and a Ck the critical half-length in layer k, we have to distinguish between two cases for model L2H: (a) a C1baba C2: the injected volume is only sufficient for a propagation in medium 1, (b) a C1ba C2ba: the injected volume is sufficient for propagation in both media.
For case a), the propagation will therefore stop at the interface, and an additional fluid injection is necessary to obtain aNa C2. We reproduced this in the Experiment No. 10 shown in Fig. 5 (rigidity contrast G 2/G 1=7.5). The fracture first decelerates, and then stops at the interface. After the injection of additional fluid, we observed a lateral spreading of the fissure, that develops an anvil shape. A further injection through the same channel allows the fracture to increase its vertical length and thus its overpressure at its tip, until it breaks through the interface and the stiff top layer. It moves to the surface re-assuming the typical shape observed in homogeneous media, in this case with a slight inclination. The path of motion, the detrended path and the velocity for this case are shown in Fig. 6. Case H2L was theoretically simpler, since if a given volume of fluid is sufficient for propagation in medium 1, it is necessarily also sufficient for medium 2 (aNa C1Na C2). The deceleration, and particularly the arrest of the propagation caused by the rigidity contrast is a very important result, because it represents a possible explanation for dikes that do not reach
Fig. 5. Case L2H ( G 2/G 1=5.5), Experiment No. 10. Labels (L) indicate cross-sectional and (R) frontal perspective. (a) A small volume of air is enough for the propagation to occur. Once it arrived at the interface, the fracture does not contain a sufficient volume for the propagation in the second medium and comes to a halt. (b, c, d) A further air injection enlarges the fracture that assumes an danvilT or dambosT shape. (e, f, g, h) After the injection of a sufficient air volume, the fracture propagates into the upper layer.
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281
A) 0
d (mm)
20 40 60
← vm = 0.73 mm/s
80 100 0
d - vm*t (mm)
B)
20
40
60
80
100
120
80
100
120
80
100
120
t (s) 4 3 2 1
↑ d – 0.73*t
0 -1
0
20
40
60
t (s)
C) 1.2 v (mm/s)
1 0.8 0.6 0.4 0.2
0
20
40
60
t (s) Fig. 6. Panel A: path of motion of the fracture upper tip (Experiment No. 10, shown in Fig. 5. Depth is measured from the interface). The average velocity is indicated in figure. Panel B: detrended path of motion (calculated in the same way as for Fig. 4). The interpolation curve shows a deceleration. Panel C: velocity of the fracture upper tip. In proximity of the interface with a very stiff medium, the fracture decelerates.
the surface. If several of such dikes accumulate next to each other, a magma reservoir can develop. Such reservoirs can then become instable, if additional magma is supplied.
crack abruptly cut the gelatine along a horizontal plane a short distance below the interface and rapidly developed a semicircular sill-structure. In this way we have been able to observe the formation of a sill, as illustrated in Fig. 8.
3.4. Case L2H: a very high shear modulus contrast When the critical lengths characterizing the two layers are very different (Experiment No. 12 in Table 1), the high rigidity contrast prevents propagation through the interface. The crack tip stops at the layer interface and thus a very high stress concentration can be built at the discontinuity. Further fracture development is illustrated in Fig. 7. The fracture first propagated laterally along the interface, analogous to case L2H above. After a repeated injection of air the
4. Discussion 4.1. The acceleration of fractures Both accelerated deformation and intrusioninduced seismicity have been identified, beside others, as precursor signals for volcanic eruption (e.g. Kilburn, 2003). Ascending and accelerated dikes are one candidate to cause such signals.
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Fig. 7. Case L2H ( G 2/G 1=7.5), Experiment No. 10, formation of a sill. Labels (B) cross-sectional and (T) frontal perspective. (aT and aB) A small fracture ascends towards the interface. (b, c) The fracture stops moving, and additional air is injected. The fracture enlarges and assumes an elongated shape. (d, e, f) Instead of breaking into the upper medium, it propagates horizontally slightly below the interface between the two layers. The formation of the sill is best observed on the cross-sectional views.
Fig. 8. Snap-shot of the sill from Fig. 7. Panel A: perspective view, panel B: top view.
Fig. 4 documents the acceleration of a buoyancydriven fracture in gelatine when passing the interface from a high to a low-rigidity layer. The shear modulus decreased by a factor of nearly 5, while the ascent velocity increased more than a factor of about 20. The crack tip velocity during the transition of the layer boundary is a non-stationary, complex function. Away from the boundary, however, the ascent velocity is constant. The cross-sectional shape of the cracks can there be described by the Weertman model. Dahm (2000b) studied the shape and velocity of Weertman cracks in a full space. He derived a theoretical model to predict the ascent velocity v of those fractures, depending on their length, the elastic parameter of the medium, the fracture toughness, and the viscosity and density of the dyke fluid. Fig. 9 shows the predictions of this model when adapting the parameter used in Dahm (2000b) to the gelatine of Experiment Nos. 5 and 6. The volume of the injected air was about Vc9 ml in both experiments. The measured velocities and crack lengths in the lower and
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Fig. 9. Theoretical predictions of ascent velocity (solid line, theory from Dahm, 2000b) compared to observations (Experiments No. 5 and No. 6, open circles). The dashed line shows the estimated length of the crack using an approximate formula for the volume of a 3D Weertman pffiffiffiffiffiffi crack (Dahm, 2000b). Crosses are the measured crack lengths. The volume of injected air was c9 ml in both experiments, and KC ¼ 6GPa m1=2 has been used.
upper layer are indicated as circles and crosses, respectively. The increase in ascent velocity is accompanied by a decrease in crack length and can be approximated by the model of Dahm (2000b). The model can be used to predict the increase or decrease of the ascent velocity of a dike ascending in the earth and retaining its fluid volume. While shear modulus, density and Poisson’s ratio are relatively well known for the Earth’s upper mantle and crust, the fracture toughness is difficult to estimate and nearly unknown. For example, laboratory experiments of rock samples indicate a low-pressure tensile fracture toughness of c1 MPa m1/2 (c Sc10 to 30 J/m2) for limestone or c3 MPa m1/2 for basalt. Apparent K C values inferred from basaltic dikes are more in the range from 100 MPa m1/2 to 1 GPa m1/2 (see e.g. Parfitt, 1991; Heimpel and Olson, 1994, and references given in their Table 3), corresponding to surface energy in the order of 105 to 106 J/m2. Fialko and Rubin (1997) find indications that the surface energy increases with increasing pressure. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assuming KC c 4Gcs ð1 þ mÞ and a dike volume V, the fracture length and the ascent velocity can be predicted as a function of G and c S. G is about 70 GPa in the depth of about 200 km in the low-velocity zone and about 25 GPa in the upper crust. Assuming a volume of V=1d 106 m3 (see e.g. Battaglia and Bacherlery, 2003, for dike-volume estimates at Piton de la Fournaise) a dike would increase its ascent velocity by about a factor of 3 when sampling the scale from the LVZ to the upper crust, independently
from a constant c S. The dike length would decrease about 10% during such a hypothetic ascent. However, the effect of a pressure-dependent, decreasing apparent surface energy c S would be more dramatic and showing an opposite trend. Assuming that c S decreases about a factor of 100 from a depth of 200 km to 5 km in the earth, the ascent velocity would decrease a factor of about 20, while the dike-length would grow. Again, long and tall dikes are predicted to be slow while the short and fat ones are fast. Anyway, we think that surface energy variations for rocks in the crust are relatively small so that the knowledge of G is sufficient for a first order modeling of dike ascent there. 4.2. The formation of sills and magmatic underplating Figs. 5–8 document that buoyancy-driven fractures may decelerate or even stop when approaching a lowto-high rigidity contrast (or alternatively a low-to-high contrast in c S or K C). The formation of sill can be understood in terms of strong stress concentration at the crack tip. A small asymmetry of the stopped crack, in combination with an increasing horizontal compression, (e.g., produced by a small inclination of the stopped crack with growing opening), is able to enhance horizontal fracturing. Dahm (2000a) has shown by 2D numerical simulation that horizontal compressive stress acting on fluid-filled fractures with a weak buoyancy force promotes horizontal growth of the fractures and thus
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the formation of sills. Interestingly here, and in contrast to the simulations in Dahm (2000a) is that the sill forms in the bottom layer just below the interface. It does not form directly at the layer interface. The stress intensity factor at the fracture tip is too small to overcome the fracture toughness of the upper layer. Therefore, the dyke is arrested before cutting the layer boundary. This is a different process than the arrest of a vertical dike intrusion because of the decreasing of its apparent buoyancy at a short distance above the level of neutral buoyancy (LNB, e.g. Dahm, 2000a). Such a process has often been postulated for the Moho as a magma-trapping interface, explaining that high density, mantle derived magma intrudes into the lower crust and solidifies there. The rigidity barrier effect discussed in this study may additionally explain that smaller upper mantle intrusions are trapped below or at the Moho as long as the surface energy c S would increase in the crust. Other potential dike-stopping interfaces may exist at low-velocity zones in the upper mantle or within the crust or in the uppermost crust when compact sill layers overly soft sediments or highly fractured layers. The brittle–ductile transition is typically not associated with a shear-wave low-velocity zone and may thus not act as a dikestopping interface or transition. However, whenever the upper crust is loaded by tectonic compression, the ductile lower crust will not sustain large shear stresses and a compressional stress barrier may develop. Such a stress barrier is able to trap magma in a similar way as a LNB interface. The experiment of Fig. 8 shows that a low-to-high rigidity interface is able to generate a sill-like fluid reservoir below the layer boundary although the intruded fluid is lighter than the surrounding rock. It is difficult to give an estimation for a definite value of rigidity contrast promoting formation of sills. We did not systematically study the formation in function of the elastic parameters, and we cannot exclude that other factors may be even more important. After a buoyant sill has been formed, every additional fracture entering this sill would increase the volume of the reservoir but not cut the layer boundary (e.g. Weertman, 1980). For example, an extensive sill-like reservoir has been evidenced below
Mt. Vesuvius at about 10 km depth and associated with a low velocity layer (Auger et al., 2001). Buoyant sill-like reservoirs below the LNB bear a high risk for triggering volcanic eruption. The fluid trapped in the reservoir is less dense than the surrounding rock. After becoming instable, e.g. by the generation of an edge crack, a large volume of magma or fluid can be binjectedQ in a dike that will rise with high velocities and possibly break through the Earth’s surface. The formation of potentially instable reservoirs is, however, not only of interest for shallow levels in the upper crust. An unsolved problem is how magma-reservoirs may form in the mantle. Due to the small density variations in the mantle it is hard to generate a reservoir at a level of neutral buoyancy. A compressional stress barrier may be an alternative, but is not very likely in the mantle since high deviatoric stress would be released by viscous flow. Our experiments may give a third possibility to be discussed. Reservoirs in the mantle may be formed at binterfacesQ, where K C or G increases with decreasing depth, e.g. at the upper boundary of the low-velocity zone.
5. Conclusions The possibilities offered by the investigation of layered gelatine are numerous. In this paper we have only explored the simplest cases. Our analogue experiments for buoyancy-driven fractures in layered gelatine focus on the impact of layering on ascent velocity and dike emplacement. The effect of contrasts in rigidity and fracture toughness is studied. We observed accelerated and decelerated motion, arrest, accumulation of fluid and the formation of sill-like intrusions. The 3D-shape of the fractures changes in a characteristic manner during the different intrusion styles and has been documented by photographs and videos. As far as a 2D theory is applicable, it is in good agreement with our observations. The observation of propagation mechanisms in gelatine is helpful to answer questions relative to dike dynamics in a layered earth. The magmatic dike dynamics in the earth may be influenced by several additional mechanisms that cannot be introduced in
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our analogue model, e.g. temperature difference with the host rock, density difference between the several layers, tectonic stress field, topography. Nevertheless, we believe that our results are quite general and give fundamental information to understand magma ascent processes beneath volcanoes. Acknowledgements The review of Akira Takada and Alexander Cruden improved the work and is gratefully acknowledged, as well as useful comments by Agust Gudmundsson and Valerio Acocella. The authors would like to thank Michael Schnese, Martin Thorwart, Rainer Knut, Thomas Braun, Ali Dehgani, Christel Mynarik, Birgit Stffen-Vosberg for the help in the experiments. Marina Mu¨ller, Giorgio dalla Via and Massimiliano Pittore helped in the image analysis. Eberhard Dick and Markus Lichtenfeld from the company DGF Stoess AG provided the gelatine and measured the elastic parameters. The work has been financed by the EUproject VOLCALERT. References Auger, E., Gasparini, P., Virieux, J., Zollo, A., 2001. Seismic evidence of an extended magmatic sill under Mt. Vesuvius. Science 294, 1510 – 1512. Battaglia, J., Bacherlery, P., 2003. Dynamik dyke propagation deduced from tilt variations preceding the March 9, 1998, eruption of the Piton de la Fournaise volcano. J. Volcanol. Geotherm. Res. 120, 289 – 310. Bonafede, M., Rivalta, E., 1999a. The tensile dislocation problem in a layered elastic medium. Geophys. J. Int. 136, 341 – 356. Bonafede, M., Rivalta, E., 1999b. On tensile cracks close to and across the interface between two elastic half spaces. Geophys. J. Int. 138, 410 – 434. Cruden, A.R., 1988. Deformation around a rising diapir modelled by creeping flow past a sphere. Tectonics 7, 1091 – 1101. Dahm, T., 1996. Elastostatic simulation of dislocation sources in heterogeneous stress fields and multilayered media having irregular interfaces. Phys. Chem. Earth 21, 241 – 246. Dahm, T., 2000a. Numerical simulations of the propagation path and the arrest of fluid-filled fractures in the earth. Geophys. J. Int. 141, 623 – 638. Dahm, T., 2000b. On the shape and velocity of fluid-filled fractures in the earth. Geophys. J. Int. 142, 181 – 192. Di Stefano, R., Chiarabba, C., 2002. Active source tomography at Mt. Vesuvius: constraints for the magmatic system. J. Geophys. Res. 107, 2278 – 2292.
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