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Simple fluid dynamic models of volcanic rift zones
EPSL ELSEVIER
Earth and Planetary Science Letters 136 (1995) 223-240
Simple fluid dynamic models of volcanic rift zones Catherine MCriaux, Claude Jaupart Instirut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris Cedex 05. France Received 30 January
1995; revised 17
July 1995; accepted5 August 1995
Abstract The dynamics of magma flow through a reservoir connected to a fissure in an elastic medium is investigated. Magma is supplied to the reservoir at a rate which may vary with time. The distribution of fluid pressure, the elastic deformation of the reservoir walls and the characteristics of dyke intrusion are calculated for a range of reservoir sizes and supply rates. Magma is stored in the reservoir whilst magma is intruded in the fissure. At high supply rates, the reservoir pressure increases by large amounts and only small amounts of magma flow into the fissure. Conversely, at low supply rates, most of the magma supplied to the system is injected into the fissure. For a given supply rate, the amount stored depends weakly on the reservoir size, and is largest for small reservoirs. The delay between the onset of reservoir inflation (i.e., the start of magma input) and the opening of the fissure decreases with increasing reservoir size. Rapid deflation of the reservoir occurs if the supply rate decreases with time. All else being equal, the largest reservoir pressures are reached for high supply rates, which suggests that these conditions lead to summit eruptions. Rift-zone intrusions are most extensive at low supply rates.
1. Introduction In large basaltic volcanoes such as Kilauea (Hawaii) and Krafla (Iceland) magma erupts either in the summit caldera or at large distances from the summit in a rift zone. In both cases, magma is fed first into a shallow reservoir below the summit [l-3]. Magma motion along the rift zone has been explained by a buildup of a critical magma pressure in this reservoir, which then empties into a horizontally propagating dyke [4-71. This model undoubtedly contains a large element of truth, but leaves open several questions. These may be discussed using Kilauea’s eruption behaviour, which has been monitored in detail over the last thirty years. As the measurement techniques become more sophisticated and as the data accumulate, it is becoming clear that magma flow between the summit and the rift zones 0012-821X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0012-821X(95)00170-0
proceeds in a number of different ways (e.g. [8]). One important fact is that summit inflation sometimes leads to an eruption at the summit instead of through the rift zones. Another fact is that dyke injection does not always imply summit deflation. There are numerous examples of this, but we shall only cite two. From January 1966 to October 1967, precise levelling detected inflation in both the summit area and near Makaopuhi crater, some 12 km away [9]. The late 1982 evolution which led to the 1983 Puu 00 eruption was such that the summit was swelling whilst earthquake activity was migrating dowmift and, some time later, whilst the middle east rift zone was inflating [8]. In fact, a series of geophysical, petrological and structural observations suggests that there are no blockage zones in the shallow conduit system which connects Kilauea’s summit reservoir to its rift zones [8,10- 141. A recent
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analysis of deformation data shows that many inflation events involve not just a single reservoir, but a conduit system which includes a summit reservoir and a dyke, and sometimes two dykes [ 151. In a dynamic situation, pressure variations are coupled to the flow of magma. The behaviour of a reservoir which feeds an actively propagating dyke depends on the rates at which it is being tilled and emptied, and on its volume. It also depends on magmatic buoyancy, on the cooling of magma and on the ambient stress field, which may not be lithostatic. In this paper, we focus on the effects of magma motion. We study the behaviour of a deformable plumbing system, which depends on the dimensions of the system and on the rate at which it is supplied with liquid. The model introduces two variables, the supply rate and the volume of the reservoir. This is already quite complicated, and all the more so as there is no reason to assume that the supply rate remains constant over time. We shall show that, indeed, a range of behaviour is predicted, depending on how the supply rate varies with time.
2. Model description
as a propagating dyke. In principle, a dyke is supposed to form because of rock fracturing. However, in the present framework there is no difference between the two situations. Volcanic rift zones develop around the level of neutral buoyancy where magma and country rock have the same density [20,21]. In keeping with this situation, the initial state is assumed to be one of mechanical equilibrium with a lithostatic stress field. The magma chamber lying beneath Kilauea’s summit caldera is probably elongated vertically [1,22]. The model reservoir has an elliptic cross section in a vertical plane ( y,z> with a minor axis b, along the y direction and a major axis a, along Oz, and extends to x = +L (Fig. 1). Axes a,, and b,, are allowed to change as a function of x. With this, we are able to consider a wide range of shapes, from a circle to an ellipsoid with a large aspect ratio. The reservoir is connected to a fissure which is initially closed. This corresponds to the limit case of an ellipse with a vanishing minor axis. This simple model provides a continuous description of the reservoir-fissure system. The geometry may be specified by two functions of X, one for each ellipse axis. The closed fissure system is necessarily such that b,, is zero.
2.1. Basic assumptions
a, = hf( ~1 b,,=dg(x)
We consider a reservoir connected to a fissure in an infinite medium. The fissure is closed when no overpressure is applied to its edges. The model entails two simplifications. In reality, both the reservoir and the crack are close to the Earth’s surface. The consequences for the stress and deformation fields in the vicinity of the reservoir and dyke system are examined in Appendix A. For the parameter range relevant to, for example, Kilauea volcano, differences with calculations in an infinite medium are shown to be small. The other simplification is that the strength of the country rock is ignored in the stress balance for magma motion (the validity of this simplification has been discussed in [16-191). In volcanic rift zones, the effective fracture resistance is likely to be very small because magma is repeatedly injected in the same region. In this case, magma propagation proceeds by the reopening of old pathways. Throughout the following, for the sake of simplicity we refer to the gradually opening fissure
(la> forOlxllL
b,, = 0 for x > L
(lb) (14
The shape functions fl x> and g(x) are non-dimensional and take the value of 1 at x = 0. They may be specified arbitrarily to investigate the influence of the shape of the plumbing system on the dynamics of the flow. The magma is taken to be an incompressible Newtonian liquid with viscosity CL,as in previous studies [6,17,18]. In Kilauea’s case, this is reasonable because basaltic magma contains small amounts of volatiles, and hence carries only a very small proportion of gas bubbles, if any, at depth. 2.2. Governing equations In the conditions of interest here, flow is laminar throughout the plumbing system. For an ellipse of semi-axes a and b and volume flux Q, the typical
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and Planetary Science Letters 136 (1995) 223-240
velocity is Q/ah and one may write the Reynolds number as: Re= p(Q/ab)b
=;- PQ
(2)
I-L pa where a is the major axis of the ellipse. Taking the example of Kilauea volcano, we take a density of 2700 kg rne3, a magma viscosity of 10’ Pa-s, and a major semi-axis of 500 m [1,5,23]. For supply rates in the range 0.3-6 m3 s- ’ [24-261, the Reynolds number is less than 1. Thus, inertial effects can be ignored in the momentum balance. Rift zones extend over more than 20 km, which is much larger than the value of either a or b. Flow is therefore essentially in the x direction with velocity U, and the velocity components in the other directions may be safely disregarded. Thus, the fluid pressure, which represents a pressure excess with respect to the ambiant pressure in the absence of motion, depends only on x and time t. In these conditions the Navier-Stokes equations reduce to:
aP o= -ax+pv214
(3)
The system is elongated in the x-direction (Fig. l>, and hence we follow Lister [17,18] in assuming that, at each abscissa X, elastic deformation may be described in the plane strain approximation. We use the solution for an ellipse subjected to a constant internal overpressure ([27], p. 345). The elliptical shape is maintained and semi-axes with initial values a, and b, become, respectively: a=a,+
&(-(l-2+0+2(l-u)b0} e (4a)
b=b,+
+-{2(1-“)a~-(I-2v)b,,}
(4b)
225
Lastly, we write the continuity equation: +rb) ,7r-=--
84
(6)
ax
at
Eiq. (4-6) provide relationships between the variables a, b, p and q, which are functions of x and t. We obtain an equation for pressure:
(C+Dp)$
-e$[;G)
(7a)
where C and D are coefficients which depend on the rigidity and on the dimensions a,, and b,: C= ~((a.+bO)2+(3-4~)(aO-bO)2J e D= 5{-8(‘-u)(l e
(7b)
-2v)(ai+b,Z)
+4(5 - 12uf 8~‘) a,b,}
(74
3. Analysis 3.1. Discussion E@. (7) describes flow through both reservoir and fissure, but obscure the fact that the dynamics are different in the two situations. To illustrate this, we assume that minor axis b,,(x) is everywhere much smaller than major axis a,(x), and write explicitly that b,(x) is zero in the fissure. We further assume that elastic deformation contributes little change to the reservoir dimensions, or to the fissure height a,(x). Both assumptions are not unrealistic, as shown by Eq (4). Thus, Eq. (7a) reduces to the following equations in the reservoir and in the fissure respectively:Reservoir:
e where u denotes Poisson’s ratio and CL,the rigidity. As b,, goes to zero, this solution tends to that for a closed crack of length a,. This provides a description of both the magma reservoir and its fissure system. Through an elliptical conduit of semi-axes a and b, the laminar flux of magma is ([28], p. 123): r ‘=
ap
-4/L%a2+b2
a3b3 (5)
(1 - u) af ap z --Pe
at
@a)
Fissure:
Comparison between these two equations shows the strong difference between the pressure effect in the
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a
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and Planetary Science Letters 136 (1995) 223-240
view of the system in 3 dimensions
2000
1000 ._ :
0
.k P
x
-1000
1500 -2000 1500
x direction z direction
b Dike
Reservoir
‘t
9 x
L
X>L
a
>
Y
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227
Table I Parameters and physical properties used in the calculations (8)
(9)
loo0
556
556
400
500
500
25
100
250
250
1.4 106
4.1 107
1.2 10’
10s
Refcrcnccd figure
(2)
(3)
(4)
(5N
(5b)
(6)
(7)
Geometrical parameters Dcmi length of the chamber (m.)
556
556
556
556
556
556
Semi-axis of the ellipse (tn.)
500
500
MO
500
500
100
loo
10
l&l00
0.5
5
2.9@
2.916
1.4 I#
Volume of the.reservoir (m3)
2.9 106
1.4 10.5
Elastic properties Poisson’s ratio
Shear modulus (Pa.)
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
5 10s
5 10s
5 10s
5 16
510s
5 16
5 108
5 10s
5 10s
1
1
1
1
0.1-20
1
5
5
Dynamical Vnriab1P.s Viscosity (Pas.)
(I&‘)
Q
1’
SeslS Dike Xmc Scale (s.)
ItI
5 lot
5 lop
5 104
5 lo”
5 101
5.6 1@ _ ld
8.5 104
1.5 Ior
1.5 1CP
Prcssurc scale (MPa.)
[PI
0.18
0.18
0.18
0.18
0.18
0.5 - 1.9
0.26
0.27
0.27
Fissure Width Scale (m.)
lb1
0.18
0.18
0.18
0.18
0.18
0.10.0.38
0.21
0.27
0.27
Flux
two parts of the plumbing system. This is the critical feature of the problem. To gain insight into the governing equations, we scale x by the reservoir length L and major axis a( n,t) by its initial value at x = 0, h (see Eq. la). The choice of L for length scale in the x-direction emphasizes that what matters is the balance between the amount of magma fed into the reservoir and into the dyke. If the dyke opens over only a small length, it contains a small quantity of magma and hence does not significantly affect the reservoir pressure. Using a typical flux value Q, we obtain the following scales for time, pressure and fissure width:
L (i.e., over a length equal to that of the reservoir). Pressure scale [ p] represents the pressure needed to open the fissure in order to let a flux Q through it, and length scale [b] is the corresponding value of the fissure width. With these scales, the governing equations may be made dimensionless, which introduces two dimensionless numbers, ratios h/[ b] and d/[ b]. When these numbers are small, the solution is affected only weakly by the presence of the reservoir, and hence is basically that of a propagating dyke. When both of these numbers are large, the reservoir is significantly larger than the fissure it feeds and the solution is changed. If the supply rate varies over some time scale T, a third dimensionless number is ratio T/[ t]. When this ratio is small, the supply of magma is not able to sustain flow in a long fissure, and pressures are small everywhere at all times. 3.2. Geometrical shape functions and boundary conditions
Time scale [t] is for opening the fissure over length
Various shape functions f(x) and g(x) are used, and have a small effect on the numerical results
Fig. 1. (a) General view of the reservoir and dyke system investigated in this paper. The reservoir has an elliptical cross section in the vertical ( y,z) plane and extends over distance L in the horizontal direction X. The units shown are metres. (b) The dimensions of the reservoir and conduit in two different planes. The width of the reservoir is given by function Mx) over distance L. The height of the reservoir and fissure is given by function a(x).
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(Appendix B). In the following, we discuss results for two specific choices. Ax) is constant and equal to 1 everywhere (i.e., the reservoir and fissure have the same height everywhere). The width shape function g(x) is taken to be: g(x)
= 1- 5)2 (
forOlx41
At time t = 0, pressure p is zero everywhere and hence the fissure is considered to be initially closed. A flux function is specified at x = 0:
q(W) =
x=0
=
Q@(r)
(11)
where G(t) is a dimensionless function. Various forms of Q(t) may be specified. The characteristics of deformation and flow depend on both the input function Q(t) and on the geometrical shape functions. For a given set of these functions, the parameters of the problem are the reservoir dimensions h, d, L, and the material parameters viscosity p and rigidity pe, and finally the flux scale Q. As shown by the dimensional analysis, changing one of these parameters acts to change the
characteristic scales of the problem. For example, increasing the magma viscosity implies an increase in all three scales, because viscous stresses are larger. In contrast, increasing the supply rate leads to an increase in pressure and width, but to a decrease in the characteristic time. The geometrical factors enter dimensionless numbers and define different behaviours. We present our results in dimensionless form, with one exception. For ease of comparison with a natural situation, Table 1 gives a set of numerical values for the parameters of each calculation discussed in the text, indexed with the relevant figure number. Other combinations of parameter values give the same dimensionless numbers, and hence the same dimensionless solutions. Poisson’s ratio has a value of 0.3, which is on the high side, but this has little effect on the quantitative results, as shown by the governing equations. 3.3. Numerical method Eq. (7) was solved numerically with a predictorcorrector iteration coupled to an implicit finite difference scheme. The dimensionless space step had a
\
40T
I
Dike Terminw
4
5
6
I
8
9
10
Dimens lionless Distance pressure for a constant magma supply at dimensionless times of 0.4T, 4T and 4OT, where T is time scale from Eq, (9a). The parameter values for this case are given in Table 1, and correspond to a large reservoir.
Fig. 2. Horizontal variation of dimensionless the dynamic
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typical value of 10e4. Convergence was verified by comparing runs with different time steps and space increments. At each step, the total volume of magma is calculated, in order to check for the overall magma budget. Width is calculated at each nodal point and hence the front of the propagating dyke is tracked directly. 3.4. A typical solution We set the flux to a constant value and select a large reservoir with L = 556 m, h = 500 m and d = 100 m, corresponding to a dimensionless height and width of 4.2 X lo3 and 8.3 X 10’ respectively. At time t = 0, magma is fed into the reservoir, which swells as pressure builds up. The fissure opens up after some finite length of time. The pressure gradient has the same sign everywhere, and pressure decreases continuously from a maximum value at the entry point x = 0 to zero at the dyke tip (Fig. 2). The reservoir sees its pressure increase, even though it is losing magma to the fissure. The flux of magma into the fissure increases with time and tends towards a
229
dimensionless value of 1 (i.e., it tends to balance the supply rate into the reservoir, Fig. 3). However, this situation is only achieved for dimensionless times larger than 1, which, in practice, implies a rather long transient. Thus, in the general case it is improper to assume a constant input rate for a dyke which is fed from a reservoir. This first example shows two simple features which are of importance in a volcanological context. One is that magma may become stored in the reservoir even while dyke propagation occurs. The other feature is that the greatest pressure is achieved in the reservoir.
4. Discussion The behaviour of the reservoir-fissure system depends on many different parameters and variables, but there are two main effects. One is the competition between storage in the magma reservoir and dyke injection, which is a function of two variables, the initial volume of the reservoir and the supply rate of magma. The two variables do not act indepen-
0.2
1
0,.
I
Dimensionless
1.5
Time
Fig. 3. Dimensionless flux into the dyke as a function of time with a slightly smaller reservoir than in Fig. 2 (Table 1). The flux is zero for a small time before the dyke opens, and then increases towards 1. At large values of time, therefore, the flux of magma into the dyke balances the rate of magma supply into the reservoir.
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dently of each other, as shown by the dimensional analysis. To study this first effect, we consider that the supply rate of magma is constant. The second effect is due to variations of supply rate with time, and this is treated in a second set of calculations. 4.1. Size of the reservoir In the present model, ‘reservoir’ only means the widest part of the conduit system. If the reservoir is very small, for all practical purposes the system behaves as a single fissure. Fig. 4 shows how the flux into the fissure varies with time for different values of b,, and for a given flux Q. At large values of b,,, the volume of magma which is fed into the system can be stored with a small pressure increase in the reservoir. Flow through the reservoir is rapid and pressure increases almost uniformly inside the reservoir. The fissure opens up at an early time. With a smaller value of b,, the reservoir is narrow and viscous friction is increased. Thus, to accommodate the same volume flux, pressure must be higher, which leads to greater deforma-
tion of the reservoir walls and hence to a larger volume of stored magma. The fissure opens up at a later time and receives a smaller flux of magma than in the case of a larger reservoir (Fig. 4). The difference between the supply rate and the flux into the dyke gives directly the amount of magma accumulating in the reservoir. The paradoxical result is, therefore, that at early times a small reservoir is able to store more magma than a large one. This is due to the different pressure regimes and is only valid if the supply rate is fixed. At large reservoir width, pressure is almost uniform in the reservoir and only starts decreasing noticeably in the fissure (Fig. 2). This behaviour remains at all times. In this case, the transition between the reservoir and the fissure is marked by a sharp change in pressure gradient. With decreasing reservoir width, this feature becomes less apparent and eventually disappears completely (Fig. 5). In this case, pressure decreases at a similar rate everywhere, and the spatial distribution of pressure does not indicate the presence of a reservoir. As might have been expected intuitively, the transition between the
0.8 -
0.7 -
0.6 2 lz 2 0.5 s g ‘Z c 0.4 ._B P
0.3 /’
/’
0.1 -
,’ I’
0
0
’
0.05
0.1
0.15
0.2
( !5
Dimensionless Time Fig. 4. Dimensionless flux of magma into the dyke as a function of time for two different reservoir widths (see Table 1). The dashed curve corresponds to b, = 555. The solid line is for b,, = 5.55.
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b -I
3
4 5 Dimensionless
6 Distance
I
8
9
10
Fig. 5. (a) Horizontal distribution of pressure at dimensionless times of T and IOT, where T is 35, for b, = 2.8. In this case, the reservoir is narrow and pressure decreases smoothly with increasing distance through both reservoir and dyke. (b) Same as (a), bit for b, = 28.0. For this wider reservoir there is a sharp break in the pressure distribution at the edge of the reservoir. The reservoir pressure is a decreasing function of reservoir width.
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two behaviours occurs at a dimensionless width of about 1. The difference between narrow and wide reservoirs is small at large times. Fig. 4 shows that the flux into the fissure is basically the same at dimensionless times larger than about 0.2. At dimensionless times of 1 and more, the values of pressure are similar for all cases. At a given time, the reservoir pressure, at x = 0 say, decreases weakly with increasing reservoir width (Fig. 5). To summarize, a reservoir of large width is characterized by a weak variation of pressure and by a
8
6
0.1
10
1
100
Flux (m %-’ )
comparatively small delay between the onset of inflation and dyke initiation. A small width leads to a larger horizontal variation in pressure, and hence in surface displacement and tilt, and a larger delay between the onset of inflation and dyke initiation. 4.2. Storage of magma The behaviour of the system depends on the magma input rate as well as on the reservoir dimensions. Increasing the input rate at fixed reservoir volume leads to a change in behaviour equivalent to decreasing the reservoir volume at fixed input rate. A large input rate leads to a large overpressure in the reservoir and hence to a large volume stored there, implying in turn that the fissure takes a correspondingly small magma volume. This is illustrated in Fig. 6, where a given volume of magma (set equal to lo6 m3, or 70% of the initial volume) has been injected in the same system at various values of supply rate. The results are shown with their true dimensions because it is the supply rate which is being varied, and this acts on all scales. The state of the reservoir-fissure system is shown at the time when injection stops, which is inversely proportional to the supply rate. At low supply rate, the fissure propagates to a very large distance (Fig. 6, top) and contains a large volume of magma, whereas the reservoir has a low pressure (Fig. 6, bottom) and hence a small volume change. With a larger input rate, the fissure length which has been opened is markedly smaller and the reservoir pressure larger. Transient storage of magma in the summit reservoir is therefore enhanced by large supply rates. 4.3. Eruption behaviour
0.1
1
10
100
Flux (m’s_‘) Fig. 6. (Top)Distance reached by the dyke tip for a fixed volume of magma injected into a given reservoir and dyke system, as a function of supply rate. The huger the supply rate, the shorter the distance, which shows that magma storage in the reservoir is enhanced. (Bottom) Reservoir pressure for a fixed volume of magma injected into the same system, as a function of supply rate. The huger the flux, the larger the reservoir pressure.
At constant supply rate, the reservoir pressure increases steadily. As explained earlier, this results from the fact that the pressure gradient has the same sign everywhere as magma flows away from the injection point. The consequence is that, if all parts of the system have the same strength, failure should always occur above the injection point. In geological terms, such a situation leads to a summit eruption. If the input of magma is not sustained indefinitely, and is sufficiently low, a summit eruption will not occur and the conduit system may be able to store all the
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Dimensionless Time
0
2
4
6
8
10
12
14
16
18
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250
loo
50
Dimensionless Time
0.6 ?z z 0.5 2 & I 2B 0.4 .s g .g 0.3 n 0.2
0.1
n
“0
2
4
6
8
10
12
14
16
8
Dimensionless Distance Fig. 7. (a) Three functions for a time-dependent supply rate, numbered from (11 to (3). Time T takes a dimensionless value of 35. Function (1) is a step function over time T, function (2) decreases exponentially with a decrement of T/4, and function (3) increases exponentially towards 1. (b) Horizontal variation of pressure at times T and 10T for the step function. At these large time values, no magma is supplied to the reservoir and the dyke propagates at the expense of magma stored in the reservoir. (cl Reservoir pressure (at x = 01 as a function of time for the step function (upper curve) and for the exponentially decreasing function (lower curve). In all cases, the reservoir pressure reaches a maximum and then decreases as magma drains into the dyke. In the case of an exponentially decreasing supply rate, the time of reservoir deflation does not coincide with the cessation of magma input. (d) Horizontal variation of pressure at times of 10T and 30T for the exponentially decreasing supply rate. (e) Horizontal variation of pressure for the case of an increasing supply rate.
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6
12 8 10 Dimensionless Distance
14
16
18
Fig. 7 (continued).
new magma. Flank eruptions require that roof failure occurs somewhere along the magma path, at a location which depends on the stress field and on the local nature of the country rock, but also on depth [5]. This, however, is outside the scope of the present study. The important result here is that, for a given geometry, the same system may or may not produce a summit eruption depending on the supply rate. 4.4. Time dependence
of the input rate
So far, the input rate function G(t) has been kept at a constant value of 1. In a true system this cannot be sustained over an infinite interval of time, because it would imply that the reservoir pressure increases indefinitely. We now illustrate how the time dependence of the input rate affects the dynamics of magma flow and the deformation pattern. We consider a step function, an exponential decrease towards zero and an exponential increase towards 1 (Fig. 7a). Time dependence occurs over some characteristic time T, whose dimensionless value is 35 for the calculations discussed below (Table 1). This rather large dimensionless value implies that the supply rate varies over a time which is long com-
pared to the time it takes to inject a dyke. In other words, the dyke is actively propagating while the supply rate is changing. This differs from a case in which the dimensionless time T would be of order 1 or less. In such a case, the supply rate is cut off before significant dyke motion occurs, and hence the evolution is dominated by a phase of pressure relaxation in both reservoir and dyke, with little dyke motion. For the step function, the input rate is constant for time T (Table 1) and is then instantaneously dropped to zero. At that time, the reservoir pressure drops as magma drains into the fissure (Fig. 7b). However, pressure is at all times highest in the reservoir as the sign of the pressure gradient follows the direction of magma motion. In this example, the onset of reservoir deflation coincides with the cessation of magma input. We consider next that the input rate decreases in an exponential manner with a time decrement equal to T/4. This particular choice leads to an evolution of reservoir pressure which is almost identical to the one for the step function (Fig. 7~). In this second example, the reservoir deflates even when it is still supplied with magma. Reservoir deflation begins
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and Planetaty
when the flux into the fissure becomes larger than the supply rate. Note that the onset of deflation is very rapid. Although the reservoir deflates, its pressure remains higher than in the fissure and magma follows the same outward-directed motion (Fig. 7d). Comparing the step function and exponential decrease cases at the same time of 10T (Fig. 7b and 7d), we can see that the latter leads to lower reservoir pressures and shorter dyke lengths than the former. This is due to the fact that the overall volume fed into the reservoir is smaller (Fig. 7a). From these two calculations, we draw two general conclusions. The first is that, in the present framework, reservoir deflation requires a decrease in supply rate and, vice-versa, that a decrease in supply rate leads to reservoir deflation. The second conclusion is that it may be difficult to deduce how the supply rate changes with time using only deformation data from the vicinity of the reservoir. The case of an increasing supply rate reveals no real surprises: the reservoir inflates continuously (Fig. 7e). At any given time, the dyke length is much greater than in the two other cases, because the initial phase of small supply rate has led to efficient transport towards the dyke.
5. Volcanological 5.1. Kilauea’s
applications
eruption record
Kilauea has been erupting very frequently over the last 120 years [3]. The supply rate has been rather variable, and the eruptive regime has been changing, with episodes of summit eruptions, intrusions without eruption, and rift zone eruptions. The internal structure of the volcano is certainly more complicated than our hypothetical structure [29], but the same basic physical principles should apply. We review below a few relevant observations. It has been suggested that the alternation between summit and flank eruptions may be due to variations in the regional stress field. However, it is instructive to review the occurrences of summit eruptions over a long time interval. Dzurisin et al. [25] find that magma supply was high during 1959- 1961, in midto late 1971, and in 1974-1975. There were four summit eruptions between 1959 and 1961. The next
Science Letters 136 (1995) 223-240
one started in late 1967 and was associated with a small peak in supply. The following summit eruption occurred in mid-1971 and was again associated with a local peak in supply rate. Finally, two summit eruptions occurred in July and September 1974, and one in late 1975. Thus, summit eruptions are associated with periods of high supply rate. In contrast, there were numerous flank eruptions from 1962 to 1965 when the supply rate was low. The period spanning the years from 1977 to 1980 had the lowest rate of magma supply and many small flank eruptions and intrusions, and saw no summit eruption. This is consistent with our physical expectation that summit eruptions correlate with pulses of magma supply f As illustrated by the 1977-1982 record [30], the same relationship between supply rate and defonnation regime appears on a shorter time scale. Starting in 1977, the east-west summit tilt exhibited a steady increase for three years, with minor flank eruptions having no significant effect. After a major intrusion episode in August 1981, levelling and tilt data indicated that there was a rapid increase in supply rate, and this led to a small summit eruption in August 1982. After this eruption, the supply rate picked up again, but remained smaller than during the preceding period, and showed signs of slowing down. This did not lead to a summit eruption, but to a large southwest-rift intrusion. Following this event, the supply rate increased again, and there was a summit eruption in the southernmost part of the Kilauea caldera. Subsequently, the supply rate seemed slightly lower, and this led to inflation in the east-rift zone, and eventually to the Puu 00 flank eruption. 5.2. Long-term
changes in eruption behaviour
In a large volcanic edifice, magma storage is most efficient during periods of low supply rate, because it is achieved through the whole conduit system. In such conditions, the reservoir increases its volume both by deformation and by its physical extension as the fissure system is widened. Over a long period of time, therefore, the reservoir gradually enlarges. If the same supply rate is maintained, it is increasingly unlikely that summit eruptions will occur. If, on the other hand, the supply of magma is cut off for an extended length of time, the reservoir will gradually
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237
0 400E ._ n 300200 Symmetry CenterLine
0.2
0.4 0.6 0.8 Dimensionless Distance
1.2
1
Fig. 8. Two different shapes for the magma reservoir.
“0
2
4
6
8 10 12 Dimensionless Distance
14
16
Fig. 9. Horizontal variation of pressure for dimensionless times of 13.3 and 133 and for a constant supply rate (see Table 1) and for the two reservoir shapes illustrated in Fig. 8. There is no significant difference between the two cases.
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cool. At the beginning of a new cycle of activity, a summit eruption may occur even for a relatively small supply rate if the liquid reservoir has been reduced to a small size. 5.3. Discussion The model is valid for a reservoir and fissure system where hydraulic connection is maintained between the different parts. It serves to emphasize that the behaviour of a reservoir and fissure system depends on how the supply rate of magma changes with time. The supply rate function may be quite complicated, and may in fact fluctuate with time. This is difficult to establish reliably because one must isolate the effects of a variable supply rate from all the other effects which may operate in a complex system such as Kilauea. Clear-cut interpretations come from events which do not involve an eruption, because no mass is lost by the conduit system. Consider the following example. From September 21, 1977 to May 3, 1978 Kilauea was deflating with no detectable new intrusion and without eruption [15]. Magma was draining from the reservoir into shallow fissures, which implies that the supply rate was smaller than the rate of drainage. After this episode, Kilauea started to inflate over a broad region, which included the summit and both the south and upper east rift zones. It is difficult to explain this change other than by an increase in supply rate. We may also note that, in this case, summit inflation was accompanied by swelling in two rift zones. Clearly, assumptions made regarding the supply rate function must be made less arbitrary, which entails the study of a larger system encompassing the deep source of magma. The supply rate may in fact depend on the reservoir pressure. This would be the case, for example, if magma is being supplied from a large and deep source where pressure stays more or less constant. The flux of magma from the source to the summit reservoir would then be a function of the pressure difference between source and reservoir, which decreases as the reservoir swells.
pressure regimes depending on the size of the reservoir and on how the supply rate changes with time. A few simple and robust results emerge from our analysis. All else being equal, summit eruptions are most likely during periods of high supply rate. The time evolution of dyke intrusion depends on the dimensions of the reservoir, which may or may not operate as a buffer zone. Dyke injection may not always lead to summit deflation. One cause of reservoir deflation is a decrease in supply rate as dyke propagation occurs. [FA]
Appendix A. Effect of a free surface
The presence of the free surface acts to modify the stress distribution in the vicinity of the conduit system, and hence the displacements of its walls. This effect of a free surface may be evaluated by using the solution of Jeffery for a cylindrical hole [31]. We maximize the effect by encasing the reservoir or the fissure in a cylinder of radius r, where we apply an overpressure p. Let H be the depth of the cylinder axis. By definition, the normal stress is equal to the fluid pressure, and hence is the same as that in an infinite medium. The hoop stress at the edge of the hole, o, varies between the two following bounds ([27], pp. 265-293): 1 (H/r)2-
1
(‘41)
In an infinite medium, this stress would be equal to p everywhere on the cylinder surface. This equation
gives an upper bound on the fractional change of stress due to the free surface, and, in this linear elasticity problem, an upper bound for the fractional change of displacement. This change is less than 10% if the following condition is met: 1 2 (H/r)’
_ 1 “*’
(AT)
which may be recast as follows: 6. Conclusion Simple fluid dynamic models of a reservoir connected to a closed fissure show a variety of flow
H 2 4.5 r
(A3)
For r = 500 m, corresponding to a total vertical extent of 1 km, the depth of the reservoir axis should
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be more than 2.3 km. At Kilauea, according to the distribution of fracturing events [l] and the recent tomographic images of Rowan and Clayton [151, magma presence is detected mostly in the upper 4 km of the volcanic edifice, with a centreline lying approximately at a depth of 3 km. Thus, we conclude that the effect of the free surface is small.
Appendix B. Effect of reservoir
shape
We compare two shape functions for the reservoir, which describe dimensionless width b(x) for x between 0 and L (Fig. 8):
&w=(l-;)z g,(x)=-cus
1
1
X
7r-
wa>
+-
@lb)
2 ( Li These functions were selected so that they represent different width gradients near the end of the reservoir (i.e., near the fissure entry). The calculated pressure profiles for the same supply rate are weakly sensitive to the shape function (Fig. 9). 2
References [I] M.P. Ryan, R.Y. Koyanagi and R.S. Fiske, Modeling the three-dimensional structure of macroscopic magma transport systems: Application to Kilauea Volcano, Hawaii, J. Geophys. Res. 86, 7111-7129, 1981. [2] A. Bjomsson, G. Johnsen, S. Sigurdsson, G. Thorbersson and E. Tryggvason, Rifting of the plate boundary in north Iceland, J. Geophys. Res. 84, 3029-3038, 1979. [3] R.I. Tilling and J.J. Dvorak, Anatomy of a basaltic volcano, Nature 363, 125-132, 1993. [4] D.A. Swanson, W.A. Duffield and R.S. Fiske, Displacement of the south flank of Kilauea volcano: The result of forceful intrusion into the rift zones, U.S. Geol. Surv. Prof. Pap. 963, 1-39, 1976. [5] A.M. Rubin and D.D. Pollard, Origins of blade-like dikes in volcanic rift zones, U.S. Geol. Surv. Prof. Pap. 1350, 14491470, 1987. [6] L. Wilson and J.W. Head, Nature of local magma storage zones and geometry of conduit systems below basaltic eruption sites: Pu’u ‘0’0, Kilauea East Rift, Hawaii example, J. Geophys. Res. 93, 14785-14792, 1988. [7] E.A. Partitt and L. Wilson, The 1983-86 Pu’u’O’o eruption of Kilauea Volcano, Hawaii: A study of dike geometry and eruption mechanisms for a long-lived eruption, J. Volcanol. Geotherm. Res. 59, 179-205, 1994.
239
(81 A.T. Okamura, J.J. Dvorak, R.Y. Koyanagi and W.R. Tanigawa, Surface deformation during dike propagation, in: The Puu 00 Eruption of Kilauea volcano, Hawaii: Episodes 1 through 20, January 3, 1983, through June 8, 1984, U.S. Geol. Surv. Prof. Pap. 1463, 165-181, 1988. [9] R.S. Fiske and W.T. Kinoshita, Inflation of Kilauea volcano prior to its 1967-1968 eruption, Science 165, 341-349, 1969. [lo] D.B. Jackson, D.A. Swanson, R.Y. Koyanagi and T.L. Wright, The August and October 1968 East Rift Eruption of Kilauea Volcano, Hawaii, U.S. Geol. Surv. Prof. Pap. 890, I-32, 1975. [II] D.A. Swanson, D.B. Jackson, R.Y. Koyanagi and T.L. Wright, The February 1969 East Rift eruption of Kilauea Volcano. Hawaii, U.S. Geol. Surv. Prof. Pap. 891, l-29, 1976. [12] F.W. Klein, R.Y. Koyanagi, J.S. Nakata and W.R. Tanigawa. The Seismicity of Kilauea’s magma system, U.S. Geol. Surv. Prof. Pap. 1350, 1019-l 185, 1987. [13] G.P.L. Walker, Three Hawaiian calderas: an origin through loading by shallow intrusions?, J. Geophys. Res. 93, 1477314784, 1988. [14] L.R. Rowan and R.W. Clayton, The three-dimensional structure of Kilauea Volcano, Hawaii, from travel time tomography, J. Geophys. Res. 98, 4355-4375, 1993. [15] X. Yang, P.M. Davis, P.T. Delaney and A.T. Okamura, Geodetic analysis of dike intrusion and motion of the magma reservoir beneath the summit of Kilauea volcano, Hawaii: 1970-1985, J. Geophys. Res. 97, 3305-3324, 1992. [16] D.A. Spence and D.L. Turcotte, Magma-driven propagation of cracks, J. Geophys. Res. 90, 575-580, 1985. [17] J.R. Lister, Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors, J. Fluid Mech. 210, 263-280, 1990. [I81 J.R. Lister, Buoyancy-driven fluid fracture: Similarity solutions for the horizontal and vertical propagation of fluid-filled cracks, J. Fluid Mech. 217, 213-239, 1990. [191 J.R. Lister and R.C. Kerr, Fluid-mechanical models of crack propagation and their application to magma transport in dykes, J. Geophys. Res. 96, 10049-10077, 1991. 1201 M.P. Ryan, Elasticity and contractancy of Hawaiian olivine tholeiite and its role in the stability and structural evolution of subcaldera magma reservoirs and rift systems, U.S. Geol. Surv. Prof. Pap. 1350, 1395-1447, 1987. [21] M.P. Ryan, Neutral buoyancy and the mechanical evolution of magmatic systems, in: Magmatic Processes: Physicochemical Principles, B.O. Mysen, ed., Geochem. Sot. Spec. Publ. 1, 259-287, 1987. [22] P.M. Davis, Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half-space, with reference to Kilauea, Hawaii, J. Geophys. Res. 91,7429-7438, 1986. [23] M.P. Ryan and J.Y.K. Blevins, The viscosity of synthetic and natural silicate melts and their glasses at high temperatures and one bar pressure and at higher pressures, U.S. Geol. Surv. Bull. 1764, 565 pp., 1987. [24] D.A. Swanson, Magma supply rate at Kilauea Volcano, 1952-1971, Science 175, 169-170, 1972.
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and Planetary Science Letters 136 (1995) 223-240
D. Dzurisin, R.Y. Koyanagi and T.T. English, Magma supply and storage at Kilauea volcano, Hawaii, 1956-1983, J. Volcanol. Geothenn. Res. 21, 177-206, 1984. [26] J.J. Dvorak and D. Dzurisin, Variations in magma supply rate at Kilauea Volcano, Hawaii, J. Geophys. Res. 98, 22255-22268, 1993. (271 N.I. Muskhelishvili, Some Basic Problems in the Mathematical Theory of Elasticity, pp. 345-350, Noordhof, Leiden, The Netherlands, 1963. [28] F.M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974. [25]
[29] M.P. Ryan, The mechanics and three-dimensional internal structure of active magmatic systems: Kilauea Volcano, Hawaii, J. Geophys. Res. 93, 4213-4248, 1988. [30] E.W. Wolfe, M.O. Garcia, D.B. Jackson, R.Y. Koyanagi, CA. Neal and A.T. Okamura, The Puu 00 eruption of Kilauea Volcano, episodes l-20, January 3, 1983 to June 8, 1984, U.S. Geol. Surv. Prof. Pap. 1350, 471-508, 1987. [31] G.B. Jeffery, Plane stress and plane strain in bipolar coordinates, Trans. R. Sot. London 221, 265-293, 1920.
Earth and Planetary Science Letters 136 (1995) 223-240
Simple fluid dynamic models of volcanic rift zones Catherine MCriaux, Claude Jaupart Instirut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris Cedex 05. France Received 30 January
1995; revised 17
July 1995; accepted5 August 1995
Abstract The dynamics of magma flow through a reservoir connected to a fissure in an elastic medium is investigated. Magma is supplied to the reservoir at a rate which may vary with time. The distribution of fluid pressure, the elastic deformation of the reservoir walls and the characteristics of dyke intrusion are calculated for a range of reservoir sizes and supply rates. Magma is stored in the reservoir whilst magma is intruded in the fissure. At high supply rates, the reservoir pressure increases by large amounts and only small amounts of magma flow into the fissure. Conversely, at low supply rates, most of the magma supplied to the system is injected into the fissure. For a given supply rate, the amount stored depends weakly on the reservoir size, and is largest for small reservoirs. The delay between the onset of reservoir inflation (i.e., the start of magma input) and the opening of the fissure decreases with increasing reservoir size. Rapid deflation of the reservoir occurs if the supply rate decreases with time. All else being equal, the largest reservoir pressures are reached for high supply rates, which suggests that these conditions lead to summit eruptions. Rift-zone intrusions are most extensive at low supply rates.
1. Introduction In large basaltic volcanoes such as Kilauea (Hawaii) and Krafla (Iceland) magma erupts either in the summit caldera or at large distances from the summit in a rift zone. In both cases, magma is fed first into a shallow reservoir below the summit [l-3]. Magma motion along the rift zone has been explained by a buildup of a critical magma pressure in this reservoir, which then empties into a horizontally propagating dyke [4-71. This model undoubtedly contains a large element of truth, but leaves open several questions. These may be discussed using Kilauea’s eruption behaviour, which has been monitored in detail over the last thirty years. As the measurement techniques become more sophisticated and as the data accumulate, it is becoming clear that magma flow between the summit and the rift zones 0012-821X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0012-821X(95)00170-0
proceeds in a number of different ways (e.g. [8]). One important fact is that summit inflation sometimes leads to an eruption at the summit instead of through the rift zones. Another fact is that dyke injection does not always imply summit deflation. There are numerous examples of this, but we shall only cite two. From January 1966 to October 1967, precise levelling detected inflation in both the summit area and near Makaopuhi crater, some 12 km away [9]. The late 1982 evolution which led to the 1983 Puu 00 eruption was such that the summit was swelling whilst earthquake activity was migrating dowmift and, some time later, whilst the middle east rift zone was inflating [8]. In fact, a series of geophysical, petrological and structural observations suggests that there are no blockage zones in the shallow conduit system which connects Kilauea’s summit reservoir to its rift zones [8,10- 141. A recent
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analysis of deformation data shows that many inflation events involve not just a single reservoir, but a conduit system which includes a summit reservoir and a dyke, and sometimes two dykes [ 151. In a dynamic situation, pressure variations are coupled to the flow of magma. The behaviour of a reservoir which feeds an actively propagating dyke depends on the rates at which it is being tilled and emptied, and on its volume. It also depends on magmatic buoyancy, on the cooling of magma and on the ambient stress field, which may not be lithostatic. In this paper, we focus on the effects of magma motion. We study the behaviour of a deformable plumbing system, which depends on the dimensions of the system and on the rate at which it is supplied with liquid. The model introduces two variables, the supply rate and the volume of the reservoir. This is already quite complicated, and all the more so as there is no reason to assume that the supply rate remains constant over time. We shall show that, indeed, a range of behaviour is predicted, depending on how the supply rate varies with time.
2. Model description
as a propagating dyke. In principle, a dyke is supposed to form because of rock fracturing. However, in the present framework there is no difference between the two situations. Volcanic rift zones develop around the level of neutral buoyancy where magma and country rock have the same density [20,21]. In keeping with this situation, the initial state is assumed to be one of mechanical equilibrium with a lithostatic stress field. The magma chamber lying beneath Kilauea’s summit caldera is probably elongated vertically [1,22]. The model reservoir has an elliptic cross section in a vertical plane ( y,z> with a minor axis b, along the y direction and a major axis a, along Oz, and extends to x = +L (Fig. 1). Axes a,, and b,, are allowed to change as a function of x. With this, we are able to consider a wide range of shapes, from a circle to an ellipsoid with a large aspect ratio. The reservoir is connected to a fissure which is initially closed. This corresponds to the limit case of an ellipse with a vanishing minor axis. This simple model provides a continuous description of the reservoir-fissure system. The geometry may be specified by two functions of X, one for each ellipse axis. The closed fissure system is necessarily such that b,, is zero.
2.1. Basic assumptions
a, = hf( ~1 b,,=dg(x)
We consider a reservoir connected to a fissure in an infinite medium. The fissure is closed when no overpressure is applied to its edges. The model entails two simplifications. In reality, both the reservoir and the crack are close to the Earth’s surface. The consequences for the stress and deformation fields in the vicinity of the reservoir and dyke system are examined in Appendix A. For the parameter range relevant to, for example, Kilauea volcano, differences with calculations in an infinite medium are shown to be small. The other simplification is that the strength of the country rock is ignored in the stress balance for magma motion (the validity of this simplification has been discussed in [16-191). In volcanic rift zones, the effective fracture resistance is likely to be very small because magma is repeatedly injected in the same region. In this case, magma propagation proceeds by the reopening of old pathways. Throughout the following, for the sake of simplicity we refer to the gradually opening fissure
(la> forOlxllL
b,, = 0 for x > L
(lb) (14
The shape functions fl x> and g(x) are non-dimensional and take the value of 1 at x = 0. They may be specified arbitrarily to investigate the influence of the shape of the plumbing system on the dynamics of the flow. The magma is taken to be an incompressible Newtonian liquid with viscosity CL,as in previous studies [6,17,18]. In Kilauea’s case, this is reasonable because basaltic magma contains small amounts of volatiles, and hence carries only a very small proportion of gas bubbles, if any, at depth. 2.2. Governing equations In the conditions of interest here, flow is laminar throughout the plumbing system. For an ellipse of semi-axes a and b and volume flux Q, the typical
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and Planetary Science Letters 136 (1995) 223-240
velocity is Q/ah and one may write the Reynolds number as: Re= p(Q/ab)b
=;- PQ
(2)
I-L pa where a is the major axis of the ellipse. Taking the example of Kilauea volcano, we take a density of 2700 kg rne3, a magma viscosity of 10’ Pa-s, and a major semi-axis of 500 m [1,5,23]. For supply rates in the range 0.3-6 m3 s- ’ [24-261, the Reynolds number is less than 1. Thus, inertial effects can be ignored in the momentum balance. Rift zones extend over more than 20 km, which is much larger than the value of either a or b. Flow is therefore essentially in the x direction with velocity U, and the velocity components in the other directions may be safely disregarded. Thus, the fluid pressure, which represents a pressure excess with respect to the ambiant pressure in the absence of motion, depends only on x and time t. In these conditions the Navier-Stokes equations reduce to:
aP o= -ax+pv214
(3)
The system is elongated in the x-direction (Fig. l>, and hence we follow Lister [17,18] in assuming that, at each abscissa X, elastic deformation may be described in the plane strain approximation. We use the solution for an ellipse subjected to a constant internal overpressure ([27], p. 345). The elliptical shape is maintained and semi-axes with initial values a, and b, become, respectively: a=a,+
&(-(l-2+0+2(l-u)b0} e (4a)
b=b,+
+-{2(1-“)a~-(I-2v)b,,}
(4b)
225
Lastly, we write the continuity equation: +rb) ,7r-=--
84
(6)
ax
at
Eiq. (4-6) provide relationships between the variables a, b, p and q, which are functions of x and t. We obtain an equation for pressure:
(C+Dp)$
-e$[;G)
(7a)
where C and D are coefficients which depend on the rigidity and on the dimensions a,, and b,: C= ~((a.+bO)2+(3-4~)(aO-bO)2J e D= 5{-8(‘-u)(l e
(7b)
-2v)(ai+b,Z)
+4(5 - 12uf 8~‘) a,b,}
(74
3. Analysis 3.1. Discussion E@. (7) describes flow through both reservoir and fissure, but obscure the fact that the dynamics are different in the two situations. To illustrate this, we assume that minor axis b,,(x) is everywhere much smaller than major axis a,(x), and write explicitly that b,(x) is zero in the fissure. We further assume that elastic deformation contributes little change to the reservoir dimensions, or to the fissure height a,(x). Both assumptions are not unrealistic, as shown by Eq (4). Thus, Eq. (7a) reduces to the following equations in the reservoir and in the fissure respectively:Reservoir:
e where u denotes Poisson’s ratio and CL,the rigidity. As b,, goes to zero, this solution tends to that for a closed crack of length a,. This provides a description of both the magma reservoir and its fissure system. Through an elliptical conduit of semi-axes a and b, the laminar flux of magma is ([28], p. 123): r ‘=
ap
-4/L%a2+b2
a3b3 (5)
(1 - u) af ap z --Pe
at
@a)
Fissure:
Comparison between these two equations shows the strong difference between the pressure effect in the
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a
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view of the system in 3 dimensions
2000
1000 ._ :
0
.k P
x
-1000
1500 -2000 1500
x direction z direction
b Dike
Reservoir
‘t
9 x
L
X>L
a
>
Y
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and Planetary Science Letters 136 (1995) 223-240
227
Table I Parameters and physical properties used in the calculations (8)
(9)
loo0
556
556
400
500
500
25
100
250
250
1.4 106
4.1 107
1.2 10’
10s
Refcrcnccd figure
(2)
(3)
(4)
(5N
(5b)
(6)
(7)
Geometrical parameters Dcmi length of the chamber (m.)
556
556
556
556
556
556
Semi-axis of the ellipse (tn.)
500
500
MO
500
500
100
loo
10
l&l00
0.5
5
2.9@
2.916
1.4 I#
Volume of the.reservoir (m3)
2.9 106
1.4 10.5
Elastic properties Poisson’s ratio
Shear modulus (Pa.)
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
5 10s
5 10s
5 10s
5 16
510s
5 16
5 108
5 10s
5 10s
1
1
1
1
0.1-20
1
5
5
Dynamical Vnriab1P.s Viscosity (Pas.)
(I&‘)
Q
1’
SeslS Dike Xmc Scale (s.)
ItI
5 lot
5 lop
5 104
5 lo”
5 101
5.6 1@ _ ld
8.5 104
1.5 Ior
1.5 1CP
Prcssurc scale (MPa.)
[PI
0.18
0.18
0.18
0.18
0.18
0.5 - 1.9
0.26
0.27
0.27
Fissure Width Scale (m.)
lb1
0.18
0.18
0.18
0.18
0.18
0.10.0.38
0.21
0.27
0.27
Flux
two parts of the plumbing system. This is the critical feature of the problem. To gain insight into the governing equations, we scale x by the reservoir length L and major axis a( n,t) by its initial value at x = 0, h (see Eq. la). The choice of L for length scale in the x-direction emphasizes that what matters is the balance between the amount of magma fed into the reservoir and into the dyke. If the dyke opens over only a small length, it contains a small quantity of magma and hence does not significantly affect the reservoir pressure. Using a typical flux value Q, we obtain the following scales for time, pressure and fissure width:
L (i.e., over a length equal to that of the reservoir). Pressure scale [ p] represents the pressure needed to open the fissure in order to let a flux Q through it, and length scale [b] is the corresponding value of the fissure width. With these scales, the governing equations may be made dimensionless, which introduces two dimensionless numbers, ratios h/[ b] and d/[ b]. When these numbers are small, the solution is affected only weakly by the presence of the reservoir, and hence is basically that of a propagating dyke. When both of these numbers are large, the reservoir is significantly larger than the fissure it feeds and the solution is changed. If the supply rate varies over some time scale T, a third dimensionless number is ratio T/[ t]. When this ratio is small, the supply of magma is not able to sustain flow in a long fissure, and pressures are small everywhere at all times. 3.2. Geometrical shape functions and boundary conditions
Time scale [t] is for opening the fissure over length
Various shape functions f(x) and g(x) are used, and have a small effect on the numerical results
Fig. 1. (a) General view of the reservoir and dyke system investigated in this paper. The reservoir has an elliptical cross section in the vertical ( y,z) plane and extends over distance L in the horizontal direction X. The units shown are metres. (b) The dimensions of the reservoir and conduit in two different planes. The width of the reservoir is given by function Mx) over distance L. The height of the reservoir and fissure is given by function a(x).
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C. Jaupart / Earth and Planetary Science Letters 136 (1995) 223-240
(Appendix B). In the following, we discuss results for two specific choices. Ax) is constant and equal to 1 everywhere (i.e., the reservoir and fissure have the same height everywhere). The width shape function g(x) is taken to be: g(x)
= 1- 5)2 (
forOlx41
At time t = 0, pressure p is zero everywhere and hence the fissure is considered to be initially closed. A flux function is specified at x = 0:
q(W) =
x=0
=
Q@(r)
(11)
where G(t) is a dimensionless function. Various forms of Q(t) may be specified. The characteristics of deformation and flow depend on both the input function Q(t) and on the geometrical shape functions. For a given set of these functions, the parameters of the problem are the reservoir dimensions h, d, L, and the material parameters viscosity p and rigidity pe, and finally the flux scale Q. As shown by the dimensional analysis, changing one of these parameters acts to change the
characteristic scales of the problem. For example, increasing the magma viscosity implies an increase in all three scales, because viscous stresses are larger. In contrast, increasing the supply rate leads to an increase in pressure and width, but to a decrease in the characteristic time. The geometrical factors enter dimensionless numbers and define different behaviours. We present our results in dimensionless form, with one exception. For ease of comparison with a natural situation, Table 1 gives a set of numerical values for the parameters of each calculation discussed in the text, indexed with the relevant figure number. Other combinations of parameter values give the same dimensionless numbers, and hence the same dimensionless solutions. Poisson’s ratio has a value of 0.3, which is on the high side, but this has little effect on the quantitative results, as shown by the governing equations. 3.3. Numerical method Eq. (7) was solved numerically with a predictorcorrector iteration coupled to an implicit finite difference scheme. The dimensionless space step had a
\
40T
I
Dike Terminw
4
5
6
I
8
9
10
Dimens lionless Distance pressure for a constant magma supply at dimensionless times of 0.4T, 4T and 4OT, where T is time scale from Eq, (9a). The parameter values for this case are given in Table 1, and correspond to a large reservoir.
Fig. 2. Horizontal variation of dimensionless the dynamic
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typical value of 10e4. Convergence was verified by comparing runs with different time steps and space increments. At each step, the total volume of magma is calculated, in order to check for the overall magma budget. Width is calculated at each nodal point and hence the front of the propagating dyke is tracked directly. 3.4. A typical solution We set the flux to a constant value and select a large reservoir with L = 556 m, h = 500 m and d = 100 m, corresponding to a dimensionless height and width of 4.2 X lo3 and 8.3 X 10’ respectively. At time t = 0, magma is fed into the reservoir, which swells as pressure builds up. The fissure opens up after some finite length of time. The pressure gradient has the same sign everywhere, and pressure decreases continuously from a maximum value at the entry point x = 0 to zero at the dyke tip (Fig. 2). The reservoir sees its pressure increase, even though it is losing magma to the fissure. The flux of magma into the fissure increases with time and tends towards a
229
dimensionless value of 1 (i.e., it tends to balance the supply rate into the reservoir, Fig. 3). However, this situation is only achieved for dimensionless times larger than 1, which, in practice, implies a rather long transient. Thus, in the general case it is improper to assume a constant input rate for a dyke which is fed from a reservoir. This first example shows two simple features which are of importance in a volcanological context. One is that magma may become stored in the reservoir even while dyke propagation occurs. The other feature is that the greatest pressure is achieved in the reservoir.
4. Discussion The behaviour of the reservoir-fissure system depends on many different parameters and variables, but there are two main effects. One is the competition between storage in the magma reservoir and dyke injection, which is a function of two variables, the initial volume of the reservoir and the supply rate of magma. The two variables do not act indepen-
0.2
1
0,.
I
Dimensionless
1.5
Time
Fig. 3. Dimensionless flux into the dyke as a function of time with a slightly smaller reservoir than in Fig. 2 (Table 1). The flux is zero for a small time before the dyke opens, and then increases towards 1. At large values of time, therefore, the flux of magma into the dyke balances the rate of magma supply into the reservoir.
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dently of each other, as shown by the dimensional analysis. To study this first effect, we consider that the supply rate of magma is constant. The second effect is due to variations of supply rate with time, and this is treated in a second set of calculations. 4.1. Size of the reservoir In the present model, ‘reservoir’ only means the widest part of the conduit system. If the reservoir is very small, for all practical purposes the system behaves as a single fissure. Fig. 4 shows how the flux into the fissure varies with time for different values of b,, and for a given flux Q. At large values of b,,, the volume of magma which is fed into the system can be stored with a small pressure increase in the reservoir. Flow through the reservoir is rapid and pressure increases almost uniformly inside the reservoir. The fissure opens up at an early time. With a smaller value of b,, the reservoir is narrow and viscous friction is increased. Thus, to accommodate the same volume flux, pressure must be higher, which leads to greater deforma-
tion of the reservoir walls and hence to a larger volume of stored magma. The fissure opens up at a later time and receives a smaller flux of magma than in the case of a larger reservoir (Fig. 4). The difference between the supply rate and the flux into the dyke gives directly the amount of magma accumulating in the reservoir. The paradoxical result is, therefore, that at early times a small reservoir is able to store more magma than a large one. This is due to the different pressure regimes and is only valid if the supply rate is fixed. At large reservoir width, pressure is almost uniform in the reservoir and only starts decreasing noticeably in the fissure (Fig. 2). This behaviour remains at all times. In this case, the transition between the reservoir and the fissure is marked by a sharp change in pressure gradient. With decreasing reservoir width, this feature becomes less apparent and eventually disappears completely (Fig. 5). In this case, pressure decreases at a similar rate everywhere, and the spatial distribution of pressure does not indicate the presence of a reservoir. As might have been expected intuitively, the transition between the
0.8 -
0.7 -
0.6 2 lz 2 0.5 s g ‘Z c 0.4 ._B P
0.3 /’
/’
0.1 -
,’ I’
0
0
’
0.05
0.1
0.15
0.2
( !5
Dimensionless Time Fig. 4. Dimensionless flux of magma into the dyke as a function of time for two different reservoir widths (see Table 1). The dashed curve corresponds to b, = 555. The solid line is for b,, = 5.55.
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b -I
3
4 5 Dimensionless
6 Distance
I
8
9
10
Fig. 5. (a) Horizontal distribution of pressure at dimensionless times of T and IOT, where T is 35, for b, = 2.8. In this case, the reservoir is narrow and pressure decreases smoothly with increasing distance through both reservoir and dyke. (b) Same as (a), bit for b, = 28.0. For this wider reservoir there is a sharp break in the pressure distribution at the edge of the reservoir. The reservoir pressure is a decreasing function of reservoir width.
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two behaviours occurs at a dimensionless width of about 1. The difference between narrow and wide reservoirs is small at large times. Fig. 4 shows that the flux into the fissure is basically the same at dimensionless times larger than about 0.2. At dimensionless times of 1 and more, the values of pressure are similar for all cases. At a given time, the reservoir pressure, at x = 0 say, decreases weakly with increasing reservoir width (Fig. 5). To summarize, a reservoir of large width is characterized by a weak variation of pressure and by a
8
6
0.1
10
1
100
Flux (m %-’ )
comparatively small delay between the onset of inflation and dyke initiation. A small width leads to a larger horizontal variation in pressure, and hence in surface displacement and tilt, and a larger delay between the onset of inflation and dyke initiation. 4.2. Storage of magma The behaviour of the system depends on the magma input rate as well as on the reservoir dimensions. Increasing the input rate at fixed reservoir volume leads to a change in behaviour equivalent to decreasing the reservoir volume at fixed input rate. A large input rate leads to a large overpressure in the reservoir and hence to a large volume stored there, implying in turn that the fissure takes a correspondingly small magma volume. This is illustrated in Fig. 6, where a given volume of magma (set equal to lo6 m3, or 70% of the initial volume) has been injected in the same system at various values of supply rate. The results are shown with their true dimensions because it is the supply rate which is being varied, and this acts on all scales. The state of the reservoir-fissure system is shown at the time when injection stops, which is inversely proportional to the supply rate. At low supply rate, the fissure propagates to a very large distance (Fig. 6, top) and contains a large volume of magma, whereas the reservoir has a low pressure (Fig. 6, bottom) and hence a small volume change. With a larger input rate, the fissure length which has been opened is markedly smaller and the reservoir pressure larger. Transient storage of magma in the summit reservoir is therefore enhanced by large supply rates. 4.3. Eruption behaviour
0.1
1
10
100
Flux (m’s_‘) Fig. 6. (Top)Distance reached by the dyke tip for a fixed volume of magma injected into a given reservoir and dyke system, as a function of supply rate. The huger the supply rate, the shorter the distance, which shows that magma storage in the reservoir is enhanced. (Bottom) Reservoir pressure for a fixed volume of magma injected into the same system, as a function of supply rate. The huger the flux, the larger the reservoir pressure.
At constant supply rate, the reservoir pressure increases steadily. As explained earlier, this results from the fact that the pressure gradient has the same sign everywhere as magma flows away from the injection point. The consequence is that, if all parts of the system have the same strength, failure should always occur above the injection point. In geological terms, such a situation leads to a summit eruption. If the input of magma is not sustained indefinitely, and is sufficiently low, a summit eruption will not occur and the conduit system may be able to store all the
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Dimensionless Time
0
2
4
6
8
10
12
14
16
18
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250
loo
50
Dimensionless Time
0.6 ?z z 0.5 2 & I 2B 0.4 .s g .g 0.3 n 0.2
0.1
n
“0
2
4
6
8
10
12
14
16
8
Dimensionless Distance Fig. 7. (a) Three functions for a time-dependent supply rate, numbered from (11 to (3). Time T takes a dimensionless value of 35. Function (1) is a step function over time T, function (2) decreases exponentially with a decrement of T/4, and function (3) increases exponentially towards 1. (b) Horizontal variation of pressure at times T and 10T for the step function. At these large time values, no magma is supplied to the reservoir and the dyke propagates at the expense of magma stored in the reservoir. (cl Reservoir pressure (at x = 01 as a function of time for the step function (upper curve) and for the exponentially decreasing function (lower curve). In all cases, the reservoir pressure reaches a maximum and then decreases as magma drains into the dyke. In the case of an exponentially decreasing supply rate, the time of reservoir deflation does not coincide with the cessation of magma input. (d) Horizontal variation of pressure at times of 10T and 30T for the exponentially decreasing supply rate. (e) Horizontal variation of pressure for the case of an increasing supply rate.
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6
12 8 10 Dimensionless Distance
14
16
18
Fig. 7 (continued).
new magma. Flank eruptions require that roof failure occurs somewhere along the magma path, at a location which depends on the stress field and on the local nature of the country rock, but also on depth [5]. This, however, is outside the scope of the present study. The important result here is that, for a given geometry, the same system may or may not produce a summit eruption depending on the supply rate. 4.4. Time dependence
of the input rate
So far, the input rate function G(t) has been kept at a constant value of 1. In a true system this cannot be sustained over an infinite interval of time, because it would imply that the reservoir pressure increases indefinitely. We now illustrate how the time dependence of the input rate affects the dynamics of magma flow and the deformation pattern. We consider a step function, an exponential decrease towards zero and an exponential increase towards 1 (Fig. 7a). Time dependence occurs over some characteristic time T, whose dimensionless value is 35 for the calculations discussed below (Table 1). This rather large dimensionless value implies that the supply rate varies over a time which is long com-
pared to the time it takes to inject a dyke. In other words, the dyke is actively propagating while the supply rate is changing. This differs from a case in which the dimensionless time T would be of order 1 or less. In such a case, the supply rate is cut off before significant dyke motion occurs, and hence the evolution is dominated by a phase of pressure relaxation in both reservoir and dyke, with little dyke motion. For the step function, the input rate is constant for time T (Table 1) and is then instantaneously dropped to zero. At that time, the reservoir pressure drops as magma drains into the fissure (Fig. 7b). However, pressure is at all times highest in the reservoir as the sign of the pressure gradient follows the direction of magma motion. In this example, the onset of reservoir deflation coincides with the cessation of magma input. We consider next that the input rate decreases in an exponential manner with a time decrement equal to T/4. This particular choice leads to an evolution of reservoir pressure which is almost identical to the one for the step function (Fig. 7~). In this second example, the reservoir deflates even when it is still supplied with magma. Reservoir deflation begins
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when the flux into the fissure becomes larger than the supply rate. Note that the onset of deflation is very rapid. Although the reservoir deflates, its pressure remains higher than in the fissure and magma follows the same outward-directed motion (Fig. 7d). Comparing the step function and exponential decrease cases at the same time of 10T (Fig. 7b and 7d), we can see that the latter leads to lower reservoir pressures and shorter dyke lengths than the former. This is due to the fact that the overall volume fed into the reservoir is smaller (Fig. 7a). From these two calculations, we draw two general conclusions. The first is that, in the present framework, reservoir deflation requires a decrease in supply rate and, vice-versa, that a decrease in supply rate leads to reservoir deflation. The second conclusion is that it may be difficult to deduce how the supply rate changes with time using only deformation data from the vicinity of the reservoir. The case of an increasing supply rate reveals no real surprises: the reservoir inflates continuously (Fig. 7e). At any given time, the dyke length is much greater than in the two other cases, because the initial phase of small supply rate has led to efficient transport towards the dyke.
5. Volcanological 5.1. Kilauea’s
applications
eruption record
Kilauea has been erupting very frequently over the last 120 years [3]. The supply rate has been rather variable, and the eruptive regime has been changing, with episodes of summit eruptions, intrusions without eruption, and rift zone eruptions. The internal structure of the volcano is certainly more complicated than our hypothetical structure [29], but the same basic physical principles should apply. We review below a few relevant observations. It has been suggested that the alternation between summit and flank eruptions may be due to variations in the regional stress field. However, it is instructive to review the occurrences of summit eruptions over a long time interval. Dzurisin et al. [25] find that magma supply was high during 1959- 1961, in midto late 1971, and in 1974-1975. There were four summit eruptions between 1959 and 1961. The next
Science Letters 136 (1995) 223-240
one started in late 1967 and was associated with a small peak in supply. The following summit eruption occurred in mid-1971 and was again associated with a local peak in supply rate. Finally, two summit eruptions occurred in July and September 1974, and one in late 1975. Thus, summit eruptions are associated with periods of high supply rate. In contrast, there were numerous flank eruptions from 1962 to 1965 when the supply rate was low. The period spanning the years from 1977 to 1980 had the lowest rate of magma supply and many small flank eruptions and intrusions, and saw no summit eruption. This is consistent with our physical expectation that summit eruptions correlate with pulses of magma supply f As illustrated by the 1977-1982 record [30], the same relationship between supply rate and defonnation regime appears on a shorter time scale. Starting in 1977, the east-west summit tilt exhibited a steady increase for three years, with minor flank eruptions having no significant effect. After a major intrusion episode in August 1981, levelling and tilt data indicated that there was a rapid increase in supply rate, and this led to a small summit eruption in August 1982. After this eruption, the supply rate picked up again, but remained smaller than during the preceding period, and showed signs of slowing down. This did not lead to a summit eruption, but to a large southwest-rift intrusion. Following this event, the supply rate increased again, and there was a summit eruption in the southernmost part of the Kilauea caldera. Subsequently, the supply rate seemed slightly lower, and this led to inflation in the east-rift zone, and eventually to the Puu 00 flank eruption. 5.2. Long-term
changes in eruption behaviour
In a large volcanic edifice, magma storage is most efficient during periods of low supply rate, because it is achieved through the whole conduit system. In such conditions, the reservoir increases its volume both by deformation and by its physical extension as the fissure system is widened. Over a long period of time, therefore, the reservoir gradually enlarges. If the same supply rate is maintained, it is increasingly unlikely that summit eruptions will occur. If, on the other hand, the supply of magma is cut off for an extended length of time, the reservoir will gradually
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237
0 400E ._ n 300200 Symmetry CenterLine
0.2
0.4 0.6 0.8 Dimensionless Distance
1.2
1
Fig. 8. Two different shapes for the magma reservoir.
“0
2
4
6
8 10 12 Dimensionless Distance
14
16
Fig. 9. Horizontal variation of pressure for dimensionless times of 13.3 and 133 and for a constant supply rate (see Table 1) and for the two reservoir shapes illustrated in Fig. 8. There is no significant difference between the two cases.
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cool. At the beginning of a new cycle of activity, a summit eruption may occur even for a relatively small supply rate if the liquid reservoir has been reduced to a small size. 5.3. Discussion The model is valid for a reservoir and fissure system where hydraulic connection is maintained between the different parts. It serves to emphasize that the behaviour of a reservoir and fissure system depends on how the supply rate of magma changes with time. The supply rate function may be quite complicated, and may in fact fluctuate with time. This is difficult to establish reliably because one must isolate the effects of a variable supply rate from all the other effects which may operate in a complex system such as Kilauea. Clear-cut interpretations come from events which do not involve an eruption, because no mass is lost by the conduit system. Consider the following example. From September 21, 1977 to May 3, 1978 Kilauea was deflating with no detectable new intrusion and without eruption [15]. Magma was draining from the reservoir into shallow fissures, which implies that the supply rate was smaller than the rate of drainage. After this episode, Kilauea started to inflate over a broad region, which included the summit and both the south and upper east rift zones. It is difficult to explain this change other than by an increase in supply rate. We may also note that, in this case, summit inflation was accompanied by swelling in two rift zones. Clearly, assumptions made regarding the supply rate function must be made less arbitrary, which entails the study of a larger system encompassing the deep source of magma. The supply rate may in fact depend on the reservoir pressure. This would be the case, for example, if magma is being supplied from a large and deep source where pressure stays more or less constant. The flux of magma from the source to the summit reservoir would then be a function of the pressure difference between source and reservoir, which decreases as the reservoir swells.
pressure regimes depending on the size of the reservoir and on how the supply rate changes with time. A few simple and robust results emerge from our analysis. All else being equal, summit eruptions are most likely during periods of high supply rate. The time evolution of dyke intrusion depends on the dimensions of the reservoir, which may or may not operate as a buffer zone. Dyke injection may not always lead to summit deflation. One cause of reservoir deflation is a decrease in supply rate as dyke propagation occurs. [FA]
Appendix A. Effect of a free surface
The presence of the free surface acts to modify the stress distribution in the vicinity of the conduit system, and hence the displacements of its walls. This effect of a free surface may be evaluated by using the solution of Jeffery for a cylindrical hole [31]. We maximize the effect by encasing the reservoir or the fissure in a cylinder of radius r, where we apply an overpressure p. Let H be the depth of the cylinder axis. By definition, the normal stress is equal to the fluid pressure, and hence is the same as that in an infinite medium. The hoop stress at the edge of the hole, o, varies between the two following bounds ([27], pp. 265-293): 1 (H/r)2-
1
(‘41)
In an infinite medium, this stress would be equal to p everywhere on the cylinder surface. This equation
gives an upper bound on the fractional change of stress due to the free surface, and, in this linear elasticity problem, an upper bound for the fractional change of displacement. This change is less than 10% if the following condition is met: 1 2 (H/r)’
_ 1 “*’
(AT)
which may be recast as follows: 6. Conclusion Simple fluid dynamic models of a reservoir connected to a closed fissure show a variety of flow
H 2 4.5 r
(A3)
For r = 500 m, corresponding to a total vertical extent of 1 km, the depth of the reservoir axis should
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be more than 2.3 km. At Kilauea, according to the distribution of fracturing events [l] and the recent tomographic images of Rowan and Clayton [151, magma presence is detected mostly in the upper 4 km of the volcanic edifice, with a centreline lying approximately at a depth of 3 km. Thus, we conclude that the effect of the free surface is small.
Appendix B. Effect of reservoir
shape
We compare two shape functions for the reservoir, which describe dimensionless width b(x) for x between 0 and L (Fig. 8):
&w=(l-;)z g,(x)=-cus
1
1
X
7r-
wa>
+-
@lb)
2 ( Li These functions were selected so that they represent different width gradients near the end of the reservoir (i.e., near the fissure entry). The calculated pressure profiles for the same supply rate are weakly sensitive to the shape function (Fig. 9). 2
References [I] M.P. Ryan, R.Y. Koyanagi and R.S. Fiske, Modeling the three-dimensional structure of macroscopic magma transport systems: Application to Kilauea Volcano, Hawaii, J. Geophys. Res. 86, 7111-7129, 1981. [2] A. Bjomsson, G. Johnsen, S. Sigurdsson, G. Thorbersson and E. Tryggvason, Rifting of the plate boundary in north Iceland, J. Geophys. Res. 84, 3029-3038, 1979. [3] R.I. Tilling and J.J. Dvorak, Anatomy of a basaltic volcano, Nature 363, 125-132, 1993. [4] D.A. Swanson, W.A. Duffield and R.S. Fiske, Displacement of the south flank of Kilauea volcano: The result of forceful intrusion into the rift zones, U.S. Geol. Surv. Prof. Pap. 963, 1-39, 1976. [5] A.M. Rubin and D.D. Pollard, Origins of blade-like dikes in volcanic rift zones, U.S. Geol. Surv. Prof. Pap. 1350, 14491470, 1987. [6] L. Wilson and J.W. Head, Nature of local magma storage zones and geometry of conduit systems below basaltic eruption sites: Pu’u ‘0’0, Kilauea East Rift, Hawaii example, J. Geophys. Res. 93, 14785-14792, 1988. [7] E.A. Partitt and L. Wilson, The 1983-86 Pu’u’O’o eruption of Kilauea Volcano, Hawaii: A study of dike geometry and eruption mechanisms for a long-lived eruption, J. Volcanol. Geotherm. Res. 59, 179-205, 1994.
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(81 A.T. Okamura, J.J. Dvorak, R.Y. Koyanagi and W.R. Tanigawa, Surface deformation during dike propagation, in: The Puu 00 Eruption of Kilauea volcano, Hawaii: Episodes 1 through 20, January 3, 1983, through June 8, 1984, U.S. Geol. Surv. Prof. Pap. 1463, 165-181, 1988. [9] R.S. Fiske and W.T. Kinoshita, Inflation of Kilauea volcano prior to its 1967-1968 eruption, Science 165, 341-349, 1969. [lo] D.B. Jackson, D.A. Swanson, R.Y. Koyanagi and T.L. Wright, The August and October 1968 East Rift Eruption of Kilauea Volcano, Hawaii, U.S. Geol. Surv. Prof. Pap. 890, I-32, 1975. [II] D.A. Swanson, D.B. Jackson, R.Y. Koyanagi and T.L. Wright, The February 1969 East Rift eruption of Kilauea Volcano. Hawaii, U.S. Geol. Surv. Prof. Pap. 891, l-29, 1976. [12] F.W. Klein, R.Y. Koyanagi, J.S. Nakata and W.R. Tanigawa. The Seismicity of Kilauea’s magma system, U.S. Geol. Surv. Prof. Pap. 1350, 1019-l 185, 1987. [13] G.P.L. Walker, Three Hawaiian calderas: an origin through loading by shallow intrusions?, J. Geophys. Res. 93, 1477314784, 1988. [14] L.R. Rowan and R.W. Clayton, The three-dimensional structure of Kilauea Volcano, Hawaii, from travel time tomography, J. Geophys. Res. 98, 4355-4375, 1993. [15] X. Yang, P.M. Davis, P.T. Delaney and A.T. Okamura, Geodetic analysis of dike intrusion and motion of the magma reservoir beneath the summit of Kilauea volcano, Hawaii: 1970-1985, J. Geophys. Res. 97, 3305-3324, 1992. [16] D.A. Spence and D.L. Turcotte, Magma-driven propagation of cracks, J. Geophys. Res. 90, 575-580, 1985. [17] J.R. Lister, Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors, J. Fluid Mech. 210, 263-280, 1990. [I81 J.R. Lister, Buoyancy-driven fluid fracture: Similarity solutions for the horizontal and vertical propagation of fluid-filled cracks, J. Fluid Mech. 217, 213-239, 1990. [191 J.R. Lister and R.C. Kerr, Fluid-mechanical models of crack propagation and their application to magma transport in dykes, J. Geophys. Res. 96, 10049-10077, 1991. 1201 M.P. Ryan, Elasticity and contractancy of Hawaiian olivine tholeiite and its role in the stability and structural evolution of subcaldera magma reservoirs and rift systems, U.S. Geol. Surv. Prof. Pap. 1350, 1395-1447, 1987. [21] M.P. Ryan, Neutral buoyancy and the mechanical evolution of magmatic systems, in: Magmatic Processes: Physicochemical Principles, B.O. Mysen, ed., Geochem. Sot. Spec. Publ. 1, 259-287, 1987. [22] P.M. Davis, Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half-space, with reference to Kilauea, Hawaii, J. Geophys. Res. 91,7429-7438, 1986. [23] M.P. Ryan and J.Y.K. Blevins, The viscosity of synthetic and natural silicate melts and their glasses at high temperatures and one bar pressure and at higher pressures, U.S. Geol. Surv. Bull. 1764, 565 pp., 1987. [24] D.A. Swanson, Magma supply rate at Kilauea Volcano, 1952-1971, Science 175, 169-170, 1972.
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D. Dzurisin, R.Y. Koyanagi and T.T. English, Magma supply and storage at Kilauea volcano, Hawaii, 1956-1983, J. Volcanol. Geothenn. Res. 21, 177-206, 1984. [26] J.J. Dvorak and D. Dzurisin, Variations in magma supply rate at Kilauea Volcano, Hawaii, J. Geophys. Res. 98, 22255-22268, 1993. (271 N.I. Muskhelishvili, Some Basic Problems in the Mathematical Theory of Elasticity, pp. 345-350, Noordhof, Leiden, The Netherlands, 1963. [28] F.M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974. [25]
[29] M.P. Ryan, The mechanics and three-dimensional internal structure of active magmatic systems: Kilauea Volcano, Hawaii, J. Geophys. Res. 93, 4213-4248, 1988. [30] E.W. Wolfe, M.O. Garcia, D.B. Jackson, R.Y. Koyanagi, CA. Neal and A.T. Okamura, The Puu 00 eruption of Kilauea Volcano, episodes l-20, January 3, 1983 to June 8, 1984, U.S. Geol. Surv. Prof. Pap. 1350, 471-508, 1987. [31] G.B. Jeffery, Plane stress and plane strain in bipolar coordinates, Trans. R. Sot. London 221, 265-293, 1920.