dynamics of a confined lava flow on kilauea volcano, hawaii · heslop et al.: dynamics of confined...

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Bull Volcanol (1989) 51:415-432 V01(/i 010 © Springer-Verlag 1989 Dynamics of a confined lava flow on Kilauea volcano, Hawaii Sally E Heslop 1, Lionel Wilson 1'2'3, Harry Pinkerton 1, and James W Head III 3 1 Institute of Environmental and Biological Sciences, University of Lancaster, Lancaster LA1 4YQ, United Kingdom 2 Planetary Geosciences Division, Hawaii Institute of Geophysics, Honolulu, HI 96822, USA 3 Department of Geological Sciences, Brown University, Providence, RI 02912, USA Abstract. This paper presents a new method of analysing lava flow deposits which allows the vel- ocity, discharge rate and rheological properties of channelled moving lavas to be calculated. The theory is applied to a lava flow which was erupted on Kilauea in July 1974. This flow came from a line of fissures on the edge of the caldera and was confined to a pre-existing gully within 50 m of leaving the vent. The lava drained onto the floor of the caldera when the activity stopped, but left wall and floor deposits which showed that the lava "banked up" as it flowed around each of the bends, Field surveys established the radius of cur- vature of each bend and the associated lava levels, and these data, together with related field and la- boratory measurements, are used to study the rheology of the lava. The results show the flow to have been fast moving but still laminar, with a mean velocity of just over 8 m s-l; the lava had a low or negligible yield strength and viscosities in the range 85-140 Pa s. An extension of the basic method is considered, and the possibility of su- percritical flow discussed. these effusive eruptions the study of the lava de- posits can be an important tool, especially when the flows were not carefully observed while ac- tive. There is, therefore, a need to develop reliable methods for obtaining information about the flIow dynamics from the deposits, based on a theoreti- cal understanding of the flow processes. An opportunuity to develop such a method was provided by a study of the remains of one particular flow near the summit of Kilauea vol- cano in Hawaii. This flow, erupted in July 1974 from fissures on the caldera rim, was channelled by local topography into a fairly steep, winding gully which led down into the caldera. When the eruption ended and most of the lava drained out, a distinctive pattern of deposits on the gully wall and floor showed that the lava level had been considerably higher on the outside wall of each bend than on the inside. Analysis of the geometry of these deposits allows calculation of the mean velocity, mass flux, friction factor, Reynolds num- ber, apparent viscosity and strain rate for the lava flow at each of several bends. Hence, a detailed rheological model for the lava may be built up. Introduction Much of our knowledge of the eruptive history of volcanoes and of the physical or chemical proper- ties of the erupted products comes from studying the resulting deposits. On volcanoes such as Ki- lauea, the characteristic style of activity is quiet effusion of lava, and it is possible to sample the flows and to make measurements on them while they are active. This is also true on Mount Etna and on many Icelandic volcanoes, but even for Offprint requests to: SE Heslop Description of the 1974 gully flow at Kilauea The July 1974 eruption The July 1974 eruption at Kilauea lasted for three days with most of the vents active only on 19 and 20 July. One line of fissures, running approxi- mately east-west, opened up across the floor of Kilauea caldera near the southern wall. This ]Line extended onto the southern rim and continued for a few hundred metres between the caldera and Keanakakoi crater (see Fig. 1). Other fissures ly- ing to the south and east of Keanakakoi crater fed

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Page 1: Dynamics of a confined lava flow on Kilauea volcano, Hawaii · Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii 417 there are two long lines (F1, F2) near

Bull Volcanol (1989) 51:415-432 V01(/i 010 © Springer-Verlag 1989

Dynamics of a confined lava flow on Kilauea volcano, Hawaii

Sally E Heslop 1, Lionel Wilson 1'2'3, Harry Pinkerton 1, and James W Head III 3

1 Institute of Environmental and Biological Sciences, University of Lancaster, Lancaster LA1 4YQ, United Kingdom 2 Planetary Geosciences Division, Hawaii Institute of Geophysics, Honolulu, HI 96822, USA 3 Department of Geological Sciences, Brown University, Providence, RI 02912, USA

Abstract. This paper presents a new method of analysing lava flow deposits which allows the vel- ocity, discharge rate and rheological properties of channelled moving lavas to be calculated. The theory is applied to a lava flow which was erupted on Kilauea in July 1974. This flow came from a line of fissures on the edge of the caldera and was confined to a pre-existing gully within 50 m of leaving the vent. The lava drained onto the floor of the caldera when the activity stopped, but left wall and floor deposits which showed that the lava "banked up" as it flowed around each of the bends, Field surveys established the radius of cur- vature of each bend and the associated lava levels, and these data, together with related field and la- boratory measurements, are used to study the rheology of the lava. The results show the flow to have been fast moving but still laminar, with a mean velocity of just over 8 m s - l ; the lava had a low or negligible yield strength and viscosities in the range 85-140 Pa s. An extension of the basic method is considered, and the possibility of su- percritical flow discussed.

these effusive eruptions the study of the lava de- posits can be an important tool, especially when the flows were not carefully observed while ac- tive. There is, therefore, a need to develop reliable methods for obtaining information about the flIow dynamics from the deposits, based on a theoreti- cal understanding of the flow processes.

An opportunuity to develop such a method was provided by a study of the remains of one particular flow near the summit of Kilauea vol- cano in Hawaii. This flow, erupted in July 1974 from fissures on the caldera rim, was channelled by local topography into a fairly steep, winding gully which led down into the caldera. When the eruption ended and most of the lava drained out, a distinctive pattern of deposits on the gully wall and floor showed that the lava level had been considerably higher on the outside wall of each bend than on the inside. Analysis of the geometry of these deposits allows calculation of the mean velocity, mass flux, friction factor, Reynolds num- ber, apparent viscosity and strain rate for the lava flow at each of several bends. Hence, a detailed rheological model for the lava may be built up.

Introduction

Much of our knowledge of the eruptive history of volcanoes and of the physical or chemical proper- ties of the erupted products comes from studying the resulting deposits. On volcanoes such as Ki- lauea, the characteristic style of activity is quiet effusion of lava, and it is possible to sample the f lows and to make measurements on them while they are active. This is also true on Mount Etna and on many Icelandic volcanoes, but even for

Offprint requests to: SE Heslop

Description of the 1974 gully flow at Kilauea

The July 1974 eruption

The July 1974 eruption at Kilauea lasted for three days with most of the vents active only on 19 and 20 July. One line of fissures, running approxi- mately east-west, opened up across the floor of Kilauea caldera near the southern wall. This ]Line extended onto the southern rim and continued for a few hundred metres between the caldera and Keanakakoi crater (see Fig. 1). Other fissures ly- ing to the south and east of Keanakakoi crater fed

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416 Heslop et al. : Dynamics of confined lava flow on Kilauea volcano, Hawaii

lkm I I

N

.~ ~ ~ Crater

Key:

J Jury 197~ fissures

Fig. 1. Map of Kilauea caldera, Hawaii, showing the line of the July 1974 fissures on the southern rim. The gully lies just south of the fissures at the point where there is a sharp inden- tation in the caldera rim; HVO Hawaiian Volcano Observa- tory

flows which can be seen from the present Chain of Craters road.

Field observations (R. I. Tilling, personal communication, 1986) indicate that the vents on

the caldera rim, which fed the gully flow, were ac- tive for a few hours on 19 July. A detailed de- scription of the eruption is not available, nor is there much film coverage of this particular activi- ty. Movie film taken during the eruption by scien- tists from the Hawaiian Volcano Observatory shows only the upper (eastern) end of the line of fissures -- the area of most interest was not filmed. Hence, there are no estimates of flow pa- rameters made at the time to compare with the de- duced values.

The gully flow

The area between Kilauea caldera and Keanaka- koi crater is shown in Fig. 2. This photograph looks east and was taken from a height of 200- 300 m in July 1985. In the bottom left-hand corner is the edge of the caldera and running across the upper half of the picture is Crater Rim Road. Beyond the road is Keanakakoi crater; the cars in the parking place just north of the crater give an idea of scale. Between the caldera and the road is a gully some 5-10 m wide and about 230 m long, which makes a sharp turn and runs down towards the caldera floor. This gully carried lava flows which originated from the lines of fissures which can be seen on the left side of the photograph:

Fig. 2. General view of the 1974 gully (G) and the surrounding area on the southern edge of Kilauea caldera (C). The photograph was taken looking approximately east and the other features marked are the rim fissures (F1, F2), the isolated vent (F3), the subsidiary flow into the gully (S), Keanakakoi crater (K) and Crater Rim Road (R)

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there are two long lines (F1, F2) near the edge of the caldera and a smaller, isolated vent (F3) near the parking place. F1 opened during the 1971 eruption in the area, while F2 and F3 opened in July 1974.

The mode of formation of the original gully is not known for certain, but an aerial photograph from 1960 suggests that the lower part of the gully may have been fissure controlled. This (low-reso- lution) photograph of the area shows several fis- sures running from Kilauea caldera to Keanaka- koi crater through the location of the present-day gully and parallel to its lower half. The trend of these cracks is east-west, the same as that of the fissures from the later 1971 and 1974 flows. H.V.O notes (J. Lockwood, personal communication) in- dicate that a flow from the 1971 eruption went down the gully, although the amount erupted is thought to have been comparatively small, partly because the rim vents were active for only 2.5 hours in 1971 and partly because much of the lava cascaded directly over the caldera rim. (The 1971 vents were fairly close to the edge of the caldera.) The gully may therefore have been shaped or en- larged by the 1971 flow, whose deposits were then completely covered by the 1974 flow. The channel geometry would have been more or less fixed dur- ing the 1974 flow due to a coating of solidified lava from previous activity, although how far up the walls the 1971 deposits extended is not cer- tain. A photograph of the 1971 deposits (Greeley 1974, p. 206) shows the area only around bend 4. In the section of gully shown the 1971 flow ap- pears to have reached levels comparable to those of the later flow, but there are fewer deposits on the inside of bend 4 and the floor seems to be slightly lower than after the 1974 event.

During the 1974 eruption the gully was fed from the top by lava erupting from the line of fis- sures on the rim (F2), which is a straight continua- tion of the main line on the caldera floor. These vents were active from 1225 to 1640 hours on 19 July. A vent slightly off the main line (F3) was ac- tive only from 1325 to 1600 hours on the same day and had a lower output than F2, so the F2 vents are thought to be the main source of lava for the gully flow. A tributary from F3 entered the gully about 25 m downstream from its top. This tribu- tary was initially unconfined and meandered for 100-200 m before entering the gully almost at right angles to the main flow.

The gully has a number of bends, with the sharpest about halfway down. The plan view in Fig. 3 shows all these bends, with the six studied in detail here numbered sequentially from the

N ~.....

10 m flow

6 5

Key:

Bends _~..--~- Level secfion

.,,_.,jArea of ponded Lava

1 - ~ - - ~ Edge of overriding Vz I'-'~,/'" . flow from

caldera boffom

Fig. 3. Plan of the July 1974 gully showing the bends used in the analysis and other features of interest. The "high points" (,) marked are the local highest lava levels on the outside of the bends

bottom of the gully. On each bend, the flow has tilted as it rounded the curve, so the deposits are considerably higher on the outside of the bend than on the inside: it is this feature which is used to examine the dynamics of the flow. Figure 4 is a ground view of the gully, looking downstream to- wards bend 3; the superelevation of the lava is particularly obvious at this bend, where the maxi- mum lava level is about 6.5 m above the floor of the gully.

Bend 6 corresponds to the place where the subsidiary flow enters the gully, and the high point here is on the same side as for bend 5, with no apparent high point on the opposite side be- tween these two bends. It is therefore possible that the entry of the second flow modified the pattern of the main flow. The high lava level at bend 5 is almost certainly due entirely to the cur- vature of the gully, but that at bend 6 may be ,due in part to overshoot of the subsidiary flow as it joined the main flow. However, the data fi,om bend 6 seem reasonably consistent with that from the other bends, so they are retained. Bends 5 and 6 may perhaps be considered as two parts of the same large bend.

On every bend, the highest lava level was downstream of the position of maximum curva- ture -- this was not obvious whilst in the gully

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Fig. 4. View of the gully from ground level showing bend 3

and only became evident later when the data were analysed. The high points lie on, or close to, the continuation of a straight line drawn along the centre of the preceding straight section. The im- plications of this are discussed later in the pa- per.

The striking pattern of the lava deposits at bend 3 illustrates another feature of the gully, i.e. the series of different lava levels recorded by small ledges in the wall deposits. At least two and sometimes three or more distinct levels are visi- ble; in between the ledges are columns where the lava seems to have stretched out as the level drop- ped, then cooled in that shape. These different levels may record decreases in eruption rate to- wards the end of the activity.

The slope of the floor varies from 4 ° to 7.5 ° with the steepest sections in the lower half of the gully, just below bend 3 and above bend 1. Fink and Fletcher (1978) drew attention to the develop- ment of ropy surface structures at the breaks in slope, and analysed the fo ld patterns in a small area round bend 3. There is an area of ponded lava just upstream of bend 3 on the outside of the curve. This is not likely to have had any effect on the flow once thedepress ion filled.

It was not possible to examine the flow where it had spread onto the caldera floor. Lava coming from the fissures on the caldera floor later in the eruption had overrun the gully deposits and ad-

vanced to the point where the gully widens and bends for the last time. This later flow shows up as a dark patch in the bottom right-hand corner of Fig. 2.

The fissure system

The main fissure system (marked F1 on Fig. 2), which fed the gully flow, consists of a series of wide and narrow cracks with a total length of about 190 m. Spatter ramparts 1-2 m high run pa- rallel to the fissures at the western end of the sys- tem, mainly on the northern side since the lava flowed out to the south. There is a short length of rampart on the southern side right at the western end where the lava did not flow downhill. The distance between fissure and spatter mound is 4- 5 m on the north side but up to 10 m on the south side. Since the prevailing winds at Kilauea are from the northeast this observfftion suggests that the material in the fire fountain was carried downwind.

At intervals along the vent, parts of a parallel- sided crack can be seen, defined clearly by chill margins. These features are a metre or two below present ground level with the vent widening rap- idly above them. The width of the parallel-sided parts varies between 250 and 300 mm, measure- ments being taken between the outer edges. These

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Table 1. Basic data for gully

419

Bend no 1 2 3 4 5 6

Radius of curvature, r / m 34.2 41.2 22.6 25.0 31.0 26.2 Cross-sectional area, A / m 2 11.7 24.7 34.5 14.0 17.3 18.4 Mean depth, h / m 1.48 2.6 3.35 1.7 2.48 2.8 Mean width, w / m 6.4 10.75 8.0 5.6 5.3 4.7 Wetted perimeter, P / m 12.0 13.8 19.2 13.0 11.2 11.6 Floor slope, a~ ° 7.0 7.0 7.5 5.5 5.0 4.5 Sin a 0.122 0.122 0.131 0.096 0.087 0.079 Surface tilt, t8/° 11.9 12.2 19.6 13.0 9.6 16.0

sections may be part of the original fissure and hence may define the width of the dike which fed the flow.

Basic data

Much of the field work for this study was carried out on two visits to the Big Island in July and Au- gust 1985. A more detailed survey of the whole gully in January and February 1987 gave a com- plete data set for the channel width at intervals of 2.5 m. This later survey also provided more accu- rate estimates of the floor slope at different sec- tions and allowed better estimates of the radii of curvature to be made.

Figure 3 (sketched from the detailed survey plan) shows a vertical view of the gully. Detailed measurements were made at the six marked bends and measurements of width and depth at five level sections in between the bends from below bend 1 to between bends 4 and 5. The bends were identified on the ground by looking for the point where the difference in lava level from one side of the channel to the other was a maximum. (As noted earlier, these positions turned out to be slightly downstream of the centre of the bend when viewed vertically.) At each bend, the chan- nel perimeter was measured and photographs taken with a person as scale and markers on the points of maximum and minimum lava level on that section.

The photographs were used later to check the perimeter measurements and to measure the other quantities not easily obtained in the field, namely the tilt of the lava surface and the cross-sectional area at each bend. Large drawings of each bend were made from projected slides and a typical section is shown in Fig. 5 with the measured pa- rameters marked. The cross-sectional area was measured using a planimeter, and the value used for each bend is the average of three measure-

ments. The mean depth was found from the draw- ing by dropping a vertical from the mid-point of the line PS which joins the highest and lowest lava levels. The wetted perimeter value is the aver- age of the field and photograph measurements -- the sets of measurements did not differ by more than 5% on any bend. The radii of curvature were estimated in several different ways but the values used here, considered to be the most accurate and corresponding to the centre of the channel, are those measured from the plan of the gully drawn from the detailed survey. The mean width at each bend is taken to be the average of about six meas- urements (made from the plan) at different points on the bend. The basic data for each bend are shown in Table 1.

The four lava samples collected gave density values of 650, 1035, 1077 and 1160 kg m - 3 , r e -

s p e c t i v e l y . All the samples were very vesicular: the first (lowest-density) specimen was colleclted from the centre of the channel, while the other three came from near the walls. These values of density are quite low, but measurements on the 1984 Mauna Loa flows gave very similar values (Lipman and Banks 1987). Quenched spatter and frothy lava from the Mauna Loa flows had densi- ties as low as 350 kg m - 3 ; o v e r the entire 1984 eruption most of the samples collected from near the vent showed densities in the range of 500-

P area °] i

Fig. 5. Sketch of a typical bend cross-section showing the mea- sured parameters; fl, tilt of surface; h, mean depth; PQ + QR + RS, wetted perimeter; w, width of base

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1200 k g m -3. The Mauna Loa samples also showed that vesicularity decreased downstream and the density measurements confirmed an in- crease in density along the flows.

No temperature measurements were made during the 1974 eruption and samples collected from the gully were found to be of no use for tem- perature determination because they were not quenched. Data from other Hawaiian flows give a range of values depending on the type of flow and the distance from the vent. Shaw et al. (1968) measured temperatures of 1130°-1135°C in the Makaopuhi lava lake. Thermocouple measure- ments gave temperatures of 1110 °-1148 ° C for the 1983 and 1984 Pu'u O'o flows at Kilauea (Wolfe et al. 1987) and 1120°-1144°C for the 1984 Mauna Loa flows (Lockwood et al. 1987). The highest temperatures in the Mauna Loa flows were recorded in lava tubes that issued from the base of spatter ramparts, although similar values were obtained several kilometres downstream in the main lava channels. Lava temperatures within a few hundred metres of the vents remained within the narrow range of 1140 ° _ 3 ° C through- out the eruption (Lipman and Banks 1987), while infrared pyrometer temperatures for the fountains were a few degrees hotter than the values mea- sured in near-vent lava. Since the gully flow was very close to its vents, we have tentatively as- signed it a temperature of 1145°C based on the above recent results.

ceeding to the rheological model and the results. If pursuing this course, the equations to note are Eqs. (4), (5), (14) and (20).

Calculation o f velocity and volume f lux

The superelevation in a curved channel may be related to the velocity by applying Newton's sec- ond law of motion to the fluid moving on a circu- lar path (e. g. Chow 1959). If the fluid is moving along the arc of a circle, then the force required to maintain such motion is mu2/r, where m is the mass of the fluid, u its velocity and r the radius of curvature of the bend: this force acts towards the centre of the circle. As the fluid rounds the bend, it tilts inwards so that the component of gravity now acting down the surface slope helps to pro- vide this centripetal force. Consider a bend where the angle of tilt is flo and the floor slope is a °. Then resolving along the surface,

(cosfl) (u2/r) = g (cos a) (sinfl) (1)

o r

u 2 = rg(tanfl) (cos a) (2)

since the component of gravity in the bend sec- tion is g(cosa). For small a, cosa-~ 1 and Eq. (2) becomes

Analysis

Aims o f the analysis

The data collected are used to obtain estimates of velocity, volume flux and viscosity at each bend. Under certain assumptions about the velocity dis- tribution and channel geometry, mean strain rates for the flow are calculated and a theological model for the lava obtained. The gully flow is as- sumed initially to be described by a rheological model without a yield strength and the possible effects of a different rheology are considered lat- er.

The section on determination of flow regime reviews some of the dynamics of pipe and open channel flow, while illustrating how these princi- ples may be used to interpret the deposits of the Kilauean flow. Those readers who are not parti- cularly interested in the theory may wish to read only the next few paragraphs (dealing with calcu- lation of velocity and volume flux) before pro-

u = [rg(tanfl)] 1/2. (3)

This simple formula is derived under the as- sumption that the fluid is inviscid and is more usually applied to water. Equation (3) was used by Fink et al. (1981) to calculate the velocity of the 1980 Mount St. Helen mudflows, by Lowe et al. (1986) to find the velocity of the Nevado del Ruiz mudflows and by Freundt and Schmincke (1986) in their study of the pyroclastic flows at Laacher See. It was also used by Komar (1969) to calculate the velocity of underwater turbidity cur- rents caused by the slumping of sediments on the continental slope. (Komar included a Coriolis force term in the equation, but since this is of the order [(1.5 x 10-4)U] m s -1, it can be ignored in most cases except when u is very large.) In apply- ing the equation to flows of viscous substances we need to check that the effects of viscosity do not render it invalid. This question has been consid- ered in some detail (Heslop 1987) and an expres- sion derived to test for the effects of viscosity when only the channel dimensions and degree of

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Table 2. Parameters calculated from basic data

421

Bend no 1 2 3 4 5 6

Velocity, u / m s - I 8.4 9.3 8.9 7.5 7.2 8.6 Volume flux, V / m 3 s - 1 98.3 230.8 306.4 105.3 124.0 157.9 Equivalent diameter, D / m 3.9 7.16 7.19 4.31 6.18 6.34 D / h 2.6 2.8 2.2 2.5 2.5 2.3 Friction factor, f 0.033 0.049 0.058 0.036 0.051 0.033

(Fanning)

superelevation are known (see Appendix A). The inequality obtained is likely to be satisfied for a channel of small equivalent radius (area/wetted perimeter) and /o r large radius of curvature and large width; if it is not satisfied, then a viscous term must be included in the velocity determina- tion. Preliminary calculations for the Kilauean gully flow indicate that neglect of the viscous term underestimates the mean velocity by 16% so the resulting rheological model is slightly modif- ied. The results presented in this paper, which are based on the approximation given by Eq. (3), may therefore be regarded as the first step in an itera- tive solution.

In an inviscid fluid the velocity is constant across the channel while in any real fluid it must be zero at the sides of the channel and increase to a maximum in the centre. The exact shape of the profile depends on the material, the type of flow and the channel size. In applying Eq. (3) to the gully flow (or any other flow for that matter) only an average value of the velocity can therefore be obtained, i.e. a bulk velocity. Detailed informa- tion on the velocity profile would require measur- ements on an active flow.

Equation (3) also assumes that the radius of curvature is constant across the channel. For this to be approximately true, the mean radius of cur- vature r must be much larger than the mean width w of the channel (i. e. r/w >> 1), so that the differ- ence in radius between the outside and inside of the bend is small. The measurements of r for the 1974 gully are all centre-line values, and the mean width (of the floor) for each bend is the arithmetic mean of six or seven measurements at intervals around the curve. The values of (r/w) for the gully are 5.3, 3.8, 2.8, 4.5, 5.9 and 5.6 (bends 1 to 6 in order) so the above inequality is not well satisfied for this flow. Again, this will tend to underesti- mate the velocity.

Using Eq. (3), estimates of velocity at each bend are obtained and then used with the mea- sured areas to calculate a volume flux V for each bend. The values of u and V are shown in Table 2.

One might expect the volume fluxes to be nearly equal corresponding to the maximum flux condi- tion, and the range of values gives an indication of the accumulated errors in the quantities from which they are derived. A brief consideration of these errors is therefore in order.

The assumptions made in deriving the expres- sion for velocity and the condition imposed on the ratio of radius of curvature to width have al- ready been discussed. The other parameter which appears in Eq. (3) is the surface tilt angle ft. This should increase with decreasing r since the force required to cause motion in a circular path iin- creases with the curvature. A glance at Table 1 shows that this is only partially reflected in the data. In order of increasing fl the bends rank 5, 1, 2, 4, 6, 3, but in order of decreasing r they rank 2, 1, 5, 6, 4, 3. Thus, bend 3 has the largest tilt angle and smallest radius of curvature, but of the others only bend 1 retains the same place in both series. Bends 4 and 6 exchange places, as do bends 5 and 2.

The estimates offl involved joining the highest and lowest levels of the lava, and it is possible that at any given bend this does not represent an instantaneous flow surface. The eye-witness re- port states that the flow was relatively steady, but that a few short-lived surges did occur, probably related to temporary increases in effusion rate. The other reported cause of surges in Hawaiian flows are temporary obstructions formed by col- lapsed wall material (Lipman and Banks 1987). Therefore, the highest high level would represent increased (maximum) flow during the occasional surge. If the velocity stayed the same during these events, the low level would rise by the same amount. However, to accomodate a likely in- crease in velocity, the tilt angle must increase to provide the greater force. This would mean the low level on the inside of the bend would drop below its usual line. Similarly, any decrease in the velocity would lead to a decrease in the tilt angle. Thus, the tilt angle may be over- or underesti- mated and this will affect the estimates of u. For

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Table 3. Values of area, mean depth, wetted perimeter, hydraulic diameter and friction factor re-calculated for the average volume flux of 170 m 3 s -1. The velocities are unchanged from Table 2 but are included here for ease of reference

Bend no 1 2 3 4 5 6

Velocity, u / m s-1 8.4 9.3 8.9 7.2 7.2 8.6 Area, a / m 2 20.23 18.19 19.14 22.60 23.72 19.81 Mean depth, h / m 2.56 1.92 1.86 2.74 3.4 3.01 Wetted perimeter, P /m 16.75 12.35 13.84 18.23 13.42 12.17 Equivalent diameter, D / m 4.83 5.89 5.53 4.96 7.07 6.51 D / h 1.9 3.1 3.0 1.8 2.1 2.2 Friction factor, f 0.041 0.040 0.045 0.041 0.059 0.034

example, if the tilt angle at bend 3 is 2 ° too large, the velocity would drop from 8.9 m s -1 to 8.4 m s - 1. However, there is no sure way of knowing where the flow surface was at any given moment and estimates can only be based on what is seen in the deposits. Similarly, the values for area could be too high due to surges or too low due to the presence of cooled lava of unknown thickness on the floor and walls.

The velocity estimates are in the range 7.2-9.3 m s-1, showing a decrease around bends 5 and 4. The results are reasonably consistent and the flow may well have exhibited velocity variations of this sort. The volume flux has a range 98-306 m 3 s - 1 ,

whereas it should have been more or less constant (except during the surges). The best estimate of the actual flux is, therefore, taken to be the arith- metic mean of the six values. This is 170 m 3 s - 1 ,

with one standard deviation of error of +82 m 3 s -1 or +48%. The implied mass flux is 1.1 x 105+48% kgs -1 to 1.87x 105+48% kgs -1 over the density range 650-1100 kg m -3 .

Using the mean volume flux and estimated velocities, the cross-sectional area at each bend is recalculated in order to give consistent results and the depths and wetted perimeters are then fac- tored according to the change in area. These mod- ified values are shown in Table 3 and all are sub- ject to variation of + 48% of the given value since all are linear functions of the mean volume flux.

Determination of flow regime

In order to proceed further some way of relating the velocity to the viscosity of the flow is needed. Both quantities appear in the dimenionless Rey- nolds number defined by

Re =puL/#, (4)

where L is a characteristic length, and u, p and/.t are the fluid velocity, density and viscosity, re-

spectively. Hence, the problem becomes one of finding Re, which then determines/z for given u and L.

The characteristic length appearing in Eq. (4) is taken to be the equivalent diameter D of the conduit. This quantity is defined as

D=4(A/P), (5)

where A is the cross-sectional area and P is the wetted perimeter (the length of surface in contact with the fluid in the cross-section). A related quantity, the hydraulic radius, is given by

R =A/P=D/4. (6)

Thus, in a circular pipe the hydraulic radius is half the pipe radius and the equivalent diameter is equal to the pipe diameter, which is why D is so called. The value of D for each bend is calculated using Eq. (5) and these equivalent diameters are shown in Table 2 for the original measured areas and in Table 3 for the recalculated areas and per- imeters based on the average volume flux. The values of D in Table 3 are subject to variation of not more than _+ 25% due to the likely error in mean volume flux. (The quantity D is used here in preference to the depth in order to facilitate com- parison between pipe flow and open channel flow.)

The Reynolds number is a measure of the balance between inertial and viscous forces and therefore its value indicates the flow regime, namely laminar (low Re) or turbulent (high Re). Numerous experiments have shown that for New- tonian fluids the flow in a pipe changes from one type to the other at a Reynolds number of around 2000 or higher depending on the flow conditions. Thus, the critical value of Re is taken to be 2000, and values below this indicate laminar flow.

The laminar, turbulent and transitional states of flow can be expressed by a diagram that shows

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Heslop et al." Dynamics of confined lava flow on Kilauea volcano, Hawaii 423

a relationship between the Reynolds number and another dimensionless quantity called the friction factor f Such a plot is called a Stanton diagram and, like the concept off, was developed for flow in pipes. However, the laws of flow resistance are essentially the same in open channels as in closed pipes running full; in particular, a resistance equation may be derived by balancing the retard- ing shear force at the boundary against a propul- sive force acting in the direction of flow (Hender- son 1966). In pipe flow this propulsion is supplied by a pressure gradient, while in open channel flow it is provided by the weight of the fluid re- solved down the slope.

The concept of the friction factor was sug- gested by many studies on friction losses. The ba- sis of the definition is the fact that in turbulent flow friction losses are proportional to the kinetic energy per unit volume of the fluid pu2/2 and the area A~ of the solid surface in contact with the fluid. The force of resistance is expressed as

F= f (.puZ/2)Aw, (7)

where f is a proportionality factor. Rearranging,

f(pu2/2)=F/Aw=Tw, (8)

where rw is the shear stress at the wall. Thus, a general expression for the Fanning friction factor (Knudsen and Katz 1958) is

For the gully, if it is assumed as a first approxima- tion that the energy gradient is equal to the floor gradient (which we take as being locally constant over a bend), then

f=gD sinot/2u 2. (13)

This is equivalent to saying that the friction gra- dient equals the driving pressure gradient and so implies no acceleration down the flow. (The max- imum rate of change of velocity down the gully is about 0.1 m s -2 between bends 6 and 5, giving a convective acceleration term u(du/dx) of 0.8 s - 1

This is very much less than pg(sina) and so is ne- glected.) The values of f for each bend found from Eq. (13) are shown in Tables 2 and 3. The values in Table 3 are subject to variation of not more than +25% due to the likely error in mean volume flux.

Having found a value of f, it now remains only to determine the corresponding Reynolds num- ber. The relationship between f and Re depends on the flow regime and on the roughness of the conduit, although this second factor does not matter in laminar pipe flow, and the f: Re diagram for pipe flow is shown in Fig. 6. The behaviour of f in circular pipes has been very thoroughly ex- plored experimentally and the following f: Re re- lations are thought to hold with a good degree of accuracy (Knudsen and Katz 1958).

f= 2Tw/pU 2, (9)

or in terms of the pressure gradient due to friction dpf/dx

f= (D /2pu 2) dpf/ dx, (lo)

where D is the equivalent diameter defined by Eq. (4). This may also be written as

f= gDs/2u 2, (11)

where s is the energy gradient or friction gradient such that pgs = dpf/dx.

The Fanning friction factor is related to the Darcy friction factor fD used in some texts (Chow 1959; Henderson 1966) by

fD = 4f. (12)

The equation for f may also be applied to open channels, though it only applies strictly to uniform or nearly uniform flows (Chow 1959).

f = gOs 2u z

0.1" 0.06 •

0•03-

0.01-

0.001

~ f = / * 5 Furry t-urbutenf (rough)

- o ooo Laminar : ....~.--..-- ~ ~ K . . . . .

Transition Blasius~ ~

10 2 1~03 2x103 10 ~ 105 1'06

Re : p uO la

Fig. 6. The friction factor: Reynolds number diagram for flow in pipes showing the laminar, transitional and fully turbulent flow regimes. The lower line in the laminar region ( f= 16/Re) represents pipe flow data and the upper line ( f=45/Re) is an average for rough open channel data. The Blasius and Prandtl- yon Karman curves fit data from turbulent flow in smooth pipes; the series of different (D/e) lines fit data from rough pipes with the roughness inversely proportional to (D/e). The dotted line divides transitional from fully turbulent flow

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424 Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii

The relation in the laminar region is given by

f= 16/Re, (14)

which reflects the influence of viscosity in laminar flow and holds for Re < 2000.

For turbulent flow the friction depends in gen- eral on the Reynolds number and on the relative roughness e/D, where D is the equivalent diam- eter and e is the average height of the roughness projections on the wall. For smooth tubes, this quantity is small enough to be neglected, since it is of the order (10-4), and the Prandtl-von Kar- man equation can be used as follows:

f - 1/2 = 4 log(Re fl/2)__ 0.4. (15)

This expression was modified from von Karman's theoretical equation to agree more closely with data obtained by Nikuradse (in Knudsen and Katz 1958) and holds for Re>3000 up to the highest Re investigated, 3.24 x 10 6. A simpler ex- pression, the Blasius equation

f= 0.079/Re 1/4, (16)

is accurate for 3000 < Re <100 000 but deviates from the experimental data (and Eq. (15)) for Re>100000. The curves corresponding to Eqs. (15) and (16) are shown in Fig. 6.

In rough tubes the relative roughness becomes important and an individual f:Re curve is ob- tained for each value of e/D. In fully turbulent flow well beyond the transition region, f becomes independent of Re and reaches a limiting value given by

f - 1/2 = 4 log(D/e) + 2.28, (17)

which holds for (D/e)/(Refl/2)<O.O1. The curves are plotted as functions of D/e, the reciprocal of the relative roughness, so the smallest values of D/e correspond to the roughest pipes.

The transition from laminar to turbulent flow is unaffected by the roughness of the pipe and oc- curs at the same point as in smooth tubes. For val- ues of D/e down to a few hundred, the rough pipe curves follow the smooth pipe equation for a short range of Re, then go through a transition re- gion to fully turbulent flow. For D/e less than a few hundred, the rough curves lie completely above the smooth curve. The best empirical rela- tionship for this transition region is given by

f - 1/2 = 4 log(D/e) + 2.28 - 4 log [1 + 4.67(D/e) / (Re fl/2)], (18)

which holds for (D/e)/(Refl/2)>O.O1. As Re in- creases, Eq. (18) tends to Eq. (17). The rough pipe curves are shown in Fig. 6 for values of D/e of 1000, 100 and 10, showing the progressive depar- ture from Eq. (15) as the roughness increases.

Corresponding equations have been devel- oped for flow in open channels and appear to be similar to those given above. However, due to the free surface and the interdependence of the hy- draulic radius, discharge and slope, the f: Re rela- tion does not follow exactly the simple concepts that hold for pipe flow. Chow (1959, p. 10) con- tains diagrams showing the f:Re relation based on experimental data (a few hundred points) for smooth and rough open channels.

The following points may be noted for smooth channels: 1) The discontinuity of the plot and the spread of data characterise the transitional region, as for pipe flow, although this region is not so well de- fined due to a greater spread of data. The lower critical Reynolds number appears to be slightly greater than 2000, in general, but varies with channel shape. 2) The data in the laminar region can be repre- sented by a general equation

f= K/Re, (19)

where K is a number which depends on the chan- nel shape. K= 14 for the triangular section chan- nel tested, and K-~24 for the rectangular sec- tions. 3) The data in the turbulent region correspond closely to Eq; (15) so the law for turbulent flow in smooth pipes may be approximately representa- tive of flow in all smooth channels. The shape of the channel does not influence the friction factor in turbulent flow.

For rough channels, the following observa- tions may be made: 1) In the laminar region the data lie between f=33/Re and f=60/Re so the value of K is higher than for pipe flow or smooth channel flow. Thus, the channel roughness influences the fric- tion factor. 2) In the turbulent region the shape of the channel has a large effect on the friction factor. The spread of data is roughly similar to that of pipe flow.

For the gully the values of f calculated all lie in the range of 0.03-0.06, which implies that the flow is laminar whichever f-Re curve is used. If the gully were smooth, then f--16/Re given by the pipe flow Eq. (14) is a reasonable approxima-

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Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii 425

tion to the data for flow in smooth channels, given that the channel is roughly trapezoidal in shape. However, if the gully is considered as rough, then rough channel data suggests that the factor K in Eq. (19) should have a higher value than 16. The 1974 gully deposits are quite rough with an average size of perhaps 10 cm for the pro- jections. If this reflects the surface of the channel when the flow travelled down it (bearing in mind that the walls were probably covered by lava de- posits from the 1971 flow), then a suitable value for e is 0.1 m. With the range of values of equival- ent diameter found for the bends this implies that D/e lies in the range of 40-70. The relation

1 I W

Fig. 7. Semi-circular and wide rectangular channel cross-sec- tions, showing strips of fluid undergoing a given strain

f = 45 /Re (20)

is therefore used to model a rough gully, this be- ing an average taken from the boundary lines through the rough channel data (Chow 1959).

Determination of rheologieal model

To determine the rheological model, values of strain rate are required. The velocity profile for the laminar flow of a Newtonian fluid in a chan- nel is given by

U = Uo[l - (y/d)2], (21)

where y is the distance measured from the centre of the surface of the flow, d is the distance to the boundary and Uo is the maximum velocity on the centre line. The local strain rate is given by

e L = ( - d U / d y ) = ( 2 U o / d 2 ) y (22)

and thus varies linearly from zero at the centre to a maximum value 2Uo/d at the boundary. Then the mean strain rate e is found by averaging eL over the moving fluid, allowing for the amount of fluid undergoing a given strain. For a flow whose boundary is an arc of a circle which subtends an angle 0, we consider a strip of fluid of thickness 6y and area Oy~Sy; for a rectangular section much wider than it is deep, the corresponding area is w6y, where w is the channel width. Figure 7 illus- trates two common channel geometries (a semi- circular and a wide rectangular section) and shows the strip of fluid undergoing a given strain in each case. Hence, for a semi-circular channel (0=Jr)

e = (4/3) Uo/d (23)

and for a wide rectangular channel

e = Uo/d. (24)

The maximum velocity is related to the mean vel- ocity u by u = Uo/2 for a semi-circular geometry and u- -2Uo/3 for a wide rectangle. The expres- sions for e can then be written in terms of the mean velocities; for the semi-circle

e = 8u/3d (25)

and for the wide rectangle

e = 3u/2d. (26)

In turbulent flow the velocity profile is more complex and e is no longer a linear function of position across the flow.

For a channel which is semi-circular the equi- valent diameter is related to the depth by D = 2h, while in a wide flow this becomes D - 4h. For the gully D = 2.5h (see Tables 2 and 3) so the gully is regarded as semi-circular, and Eq. (25) is used to calculate the mean strain rate for each bend, tak- ing d to be equal to h. The resulting values of e lie in the range 7.1-15.1 s - 1 using individual volume fluxes, and in the range 5.6-13.0 s -1 for the aver- age volume flux.

Values of Re are found using Eqs. (14) and (20), and the apparent viscosity at each bend is then calculated from

I-t,, =puD/Re (27)

using two values of the density, 650 kg m - 3 and 1100 kg m - 3 , to cover the range suggested by the gully samples. The variation in Re will be not more than ___ 25%, and the variation in ]-/a will be

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426 Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii

+55%-60% due to the likely error in mean vol- ume flux.

The values of apparent viscosity/,t~ and mean strain rate e found from the model may be used to give a relation between these two quantities. The implied stress:strain rate curve can then be found by substituting for ~,/a (as a function of 6) in the equation

r =/.t.e, (28)

where r is the shear stress.

Results and discussion

Results of basic model

Using the individual volume fluxes for a smooth gully ( f= 16/Re) #~ lies in the range 44-151 Pa s with a mean value of 90 Pa s (p=650 kg m -3 ) o r 74-255 P a s with a mean value of 153 P a s (/9= 1100 kg m - 3 ) . For a rough gully ( f = 4 5 / R e ) #a drops to 16-54 Pa s with a mean value of 32 Pa s for the lower density or 26-91 Pa s with a mean value of 54 Pa s for the higher density. The strain rates lie in the range 7.1-15.1 s-1. Using the average volume flux the smooth gully model gives iza in the range 62-120 Pa s with a mean value of 85 Pa s (p=650 kg m -3) o r 106-205 Pa s with a mean value of 143 Pa s (p= 1100 kg m-3). For a rough gully these values drop to 22-43 Pa s with a mean value of 30 Pa s (/9=650 kg m -3 ) o r 38-73 Pas with a mean value of 51 Pas (/9=1100 kgm-3) . The strain rates are between 5.6 and 13 s-1. Note that in using the average volume flux all values of #~ calculated have errors of _ 56% due to the likely error in mean V. These results show that the lava had a fairly low viscosity and also that the use of the f: Re curves implied by the rough channel data reduces the viscosities to about one-third of the values given by the pipe f low/smooth channel curve.

The results from the different volume fluxes do not give any meaningful stress:strain rate curve, and the data points are very scattered, as might be expected. However, the best-fit line (us- ing the method of least squares) through the re- suits from the average volume flux model (see Fig. 8) gives

o r

log/.t = - 0.09 l o g s + l og C

]2 -~- C 6 -0"09,

(29)

(30)

500

400

300

200

100

p./Pa s

50

40

30

20

5

~5 4 .6 oi

,:x p = 650, f=16/Re

• p = 1100, f = 16/Re [] p = 650, f = z,5/Re

o p = 1100, f : z , S / R e

32 ¸ ** P : 170E-°'°9 1

L ~I ~t: 100~ -0.09 f

5 26 1 o /+ zx 32

_ _ oO W = 60E -°'°9 "~

o t, °6 1 I o 32 , , I f

.... o°la = 35.5E -0.

t~ °6 []

f : 16/Re

= ~5/Re

6 7 8 9 10 20 E/s -I

Fig. 8. V i scos i ty : s t r a in rate g r a p h for the gul ly f low for the m e a n v o l u m e f lux (170 m 3 s - l ) . Values o f dens i ty are in kg m -3

where C takes different values depending on the density and f: Re curve adopted. Thus, for

f=45/Re, p= 6 5 0 k g m -3,

f = 4 5 / R e , p= 1100 kg m -3,

f = 1 6 / R e , p = 6 5 0 k g m - 3 ,

f = 16/Re, p = 1100 kg m -3,

C=35.5, (31a)

C-- 60, (31b)

C = I O 0 and (31c)

C = 170. (31d)

From Eq. (28), curves are

r = 35.5 6 0"91,

"t'= 60 e °'91,

T = 1 0 0 e 0"91

and v = 170 6 °'91.

the corresponding rheological

(32a)

(32b) (32c) (32d)

These are power law relationships indicating a pseudoplastic material (without a yield stress). The factor of 0.91 indicates that the rheology is not too far from Newtonian. These results should, however, be treated with some caution due to the small data set and the wide spread of volume flux values which gave rise to the large error bars on the mean volume flux.

Extension of model to include a yield strength

For fluids without a yield strength the Reynolds number is the only dimensionless parameter r e -

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Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii 427

quired to specify the viscous effects. If instead one considers a Bingham fluid model, where the relation between stress and strain rate is

v = ry +/-the, ( 33 )

then two parameters appear in the rheological model: the yield strength ry, and the plastic vis- cosity #b. This implies that the flow regime can be described adequately only by two dimensionless groups (Wilkinson 1960; Holland 1973). One of these is the Reynolds number, now defined expli- citly in terms of #b,

Reb =puD/#b (34)

and the other is the Hedstrom number given by

He=D2pfy/#b 2. (35)

The Hedstrom number may be thought of as the ratio of shearing forces to viscous forces. Since the viscosity is a measure of the resistance to shear once the fluid is moving, the Hedstrom number will define the onset of shearing in a fluid with a yield strength in much the same way as the Reynolds number defines the onset of turbu- lence.

Under laminar flow conditions f is related to Reb and He by (Wilkinson 1960; Holland 1973)

f = a/Reb + b(He/Reb 2) - c(He/Reb2) ". (36)

fore used here, because it gives a better feel for the explicit effect of a yield strength.

For a given value of f Eq. (36) may be solved for Reb for different values of He or vice versa. The process of solving for Reb is simplified by no- ting that from Eqs. (34) and (35)

He /Re 2 = ry/pU 2, (38)

which leaves only a linear term in Reb and relates it directly to f and ry. This leaves a simple equa- tion to solve for Reb for different values of ry.

For the gully flow a further simplification is possible. Measurements on Hawaiian lavas show that ry is commonly quite small, about 200 Pa or less, particularly near a vent (Shaw et al. 1968; Moore 1987). The gully flow was close to its vent and R. I. Tilling (personal commication, 1986) was "impressed by the great fluidity of the flow", which is not a characteristic associated with a high-yield-strength flow. Considering the terms in Eq. (36) the first term is of order 10 -2, and the second term of order lO-3(ry/U2). For a yield strength of 200 Pa, the second term is of order 10-3; the third term is then of order 10-1%. In the laminar region f is greater than 0.01 (see Fig. 6), so c has a maximum value of (16/3)x 106 which makes the third term around 5 x 10 -6. This is at least two orders of magnitude less than the other terms, so in the case of the gully flow may be neglected.

The simplified version of Eq. (36) may now be written as

For flow in a circular tube, a=16, b = 1 6 / 6 , c = 16/3f 3 and n =4. For flow in a wide shallow section, a=24 , b = 3 , c = 4 / f 2 and n--3. In line with the analysis so far, the first set of coefficients is used. As the Hedstrom number increases, tran- sition to turbulence takes place at higher values of the Reynolds number; a plot of friction factor as a function of Reynolds number and Hedstrom number may be found in Hulme (1982).

Moore and Schaber (1975) used a modified Reynolds number which combined the viscosity and yield strength in one dimensionless parame- ter giving

Rem = 2/[(2llb/pUO) -F ('ry/2pu2)]. (37)

When ry = 0, Rem reduces to Reb. This approach may simplify the problem in that it reduces the number of parameters, but it could obscure the fundamental relations involved in the different rheologies. The two-parameter approach is there-

Reb = 16/[f-- 16Ty/6pu2], (39)

and the f values at the bends used to find the cor- responding Reb, and hence plastic viscosities, for yield strengths up to 200 Pa.

The method used previously to calculate the strain rates is only strictly correct for a Newtonian fluid; for a lava with an appreciable yield strength a much more complex treatment is needed which takes account of the presence of an unsheared "plug" of fluid in the upper part of the flow. Johnson (1970, 1984) gives some examples of methods of treating these problems in channels, and Wilkinson (1960) and Skelland (1967) give the basic relationships for simple geometries. However, it may be shown that at low Hedstrom numbers (He < Reb) the factors multiplying u/D to give the mean strain rate differ from those given previously by less than 2%. The error rises to 4% at He = 2Reb and 5.3% at He = 3Reb (L. Wil- son, unpublished results).

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For the range of yield strengths modelled here the Hedstrom numbers implied by the gully flow parameters are not more than about three times the corresponding Reynolds numbers. For a yield strength of 200 Pa a density of 650 kg m - 3 gives Hedstrom numbers in the range 828-2045 and Reynolds numbers in the range 372-700. The ratio (He/Reb) therefore lies between 1.8 and 3.3. A density of 1100 kg m - 3 gives ranges of 369-842 (Hedstrom number) and 323-584 (Reynolds num- ber) so that (He/Reb) lies between 0.95 and 1.6. For lower yield strengths the Hedstrom number drops considerably and becomes approximately equal to the Reynolds number, even at the lower density. As a result, the error in strain rate caused by using Eq. (25) is not going to be more than 6% so the strain rates were calculated in the same way as before.

Results for yield strengths of 30-200 Pa were calculated using the average volume flux model for the gully. The shear stresses were found using Eq. (33) for each combination of yield strength and strain rate. For a density of 650 kg m -3 the plastic viscosities lie in the range 59-116 Pa s for ry=30 Pa, 53-106 Pas for ry=90 Pa and 40-89 Pa s for ry=200 Pa. For a density of 1100 kg m - 3

the corresponding viscosity ranges are 102-200 Pa s, 96-190 Pa s and 83-173 Pa s with error bars of about +55% due to the likely error in mean V.

In order to find a yield strength for the gully lava, the shear stress was plotted against the strain rate for yield strengths of 30, 50, 70, 90 and 200 Pa and both density values. The data points and the best-fit lines for ry = 30 Pa (solid lines) are shown in Fig. 9: the results for all the yield strengths be- tween 30 and 90 Pa plot so close together that only one set of data is shown here for clarity. The lines of best fit for a model yield strength of 200 Pa are also shown (dotted) in Fig. 9. The lines of best fit through the data points intersect the stress axis below 30 Pa and well below 200 Pa; the data are thus plotting below the implied rheological curves for each yield strength. The trend of these best fit lines therefore suggests that the yield strength of the lava that flowed down the gully was fairly low, probably less than 50 Pa. At 50 Pa, the mean values of plastic viscosity give the Bingham rheology models

r = 50 + 796 (40)

for a density of 650 kg m - 3 and

T= 50 + 1376 (41)

2000 - J i /

1800 - • ry = 30 Pa, p = 650 kg m -3 o'ry = 30Pa, p= 1100kgrn -3

1600 -

1400. ///'/ o / ?

"r /Pa / , ooo

800- / /~o o / /

600 ,// /~ •

,?# ~ - 0 0 •

200

0 2 /* 6 8 10 12 1/,

£/$-1

Fig. 9. Stress:strain rate graph for the gully flow data using a Bingham rheology model. Data points are for a model yield strength of 30 Pa and solid lines are lines of best fit through data points ; dot ted lines are for a model yield strength of 200 Pa

for a density of 1100 k g m -3. At strain rates of 7-13 s -1 these expressions give apparent viscosi- ties of 83-86 Pa s or 141-144 Pa s, depending on the density. These must be compared with average apparent viscosities of 85 Pa s and 143 Pa s from the model without a yield strength. The rheology of the lava that flowed down the 1974 gully may therefore be described adequately by a pseudo- plastic model without a yield strength.

If it is suspected that the yield strength of a lava is significantly higher than 200 Pa, the above analysis should be repeated without any of the simplifying assumptions.

Comparison of results with other Kilauea flows

Shaw et al. (1968) made rheological measure- ments on the lava lake at Makaopuhi, which was formed by another flow from Kilauea. Over the measured range of strain rates (0.1-1 s-1), the two sets of data obtained can be approximated by Bingham models having yield strengths of 120 Pa and 70 Pa and plastic viscosities of 650 Pa s and

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Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii 429

750 Pa s, respectively. Alternatively, the data may be fitted by a power law model. A re-analysis of the data (H. Pinkerton, unpublished results) con- firms that a power law model provides a better fit than a Bingham model, and the relevant equa- tions are

r = 974 ~,0.75 (42)

and

r = 716e °54. (43)

If these models are valid for strain rates greater than 1 s-1, the apparent viscosities at 10 s-1 (an average value for the gully) will be 548 Pa s and 279 Pa s, respectively. These are somewhat larger than the values implied for the gully, but the con- ditions in the lake would have been different. The Makaopuhi data refers to lava which had flowed some distance before forming the lake and which had cooled somewhat: Shaw et al. (1968) gave a temperature of 1130-1135 ° C. The gully lava was very close to the vent and so was probably hotter than the lake lava -- we suggested a temperature of 1145°C for the gully flow based on recent measurements on active Hawaiian flows. The gully lava was certainly gas-rich and therefore less dense than the lava in the lake: Shaw et al. (1968) assumed a value of 2000 kg m -3 for the lake-lava density, about twice as large as the values for the gully. Both these factors would tend to make the viscosity in the lake higher than in the gully.

Estimates of plastic viscosities using flow de- posits are given for two other Kilauean flows: the July 1974 flow south of Keanakakoi crater (Moore and Kachadoorian 1980) and the 1977 Pu'u Kia'i flow (H. J. Moore et al. 1980). In both cases, the thickness of the flow was used to calcu- late a yield strength on the assumption that the lava behaved as a Bingham fluid. The plastic vis- cosity was then obtained using an assumed model relating yield strengths and plastic viscosities to the volume fractions of solids in basalts (Moore and Schaber 1975). For the July 1974 flow, Moore and Kachadoorian (1980) estimated a yield strength of 620 Pa and a plastic viscosity of about 2000 Pa s. This would give an apparent viscosity of 2620 Pa s at a strain rate of 1 s - 1 and 2062 Pa s at a strain rate of 10 s -1. For the 1977 flow H. J. Moore et al. (1980) gave yield strengths which in- creased from 400 Pa near the source to 15000 Pa near the tip. The corresponding plastic viscosities were 2000-13000 Pa s. The apparent viscosities would be 2400-28000 Pas at a strain rate of

1 s -1, and 2040-14500 Pas at a strain rate of 10 s -1. Since neither strain rate estimates nor the dimensions of the channel were given, it is :not clear which of these apparent viscosities are the most appropriate, but even the smallest value is an order of magnitude higher than the values found for the gully. The yield strengths and visco- sities are also much higher than those found for the Makaopuhi lava (Shaw et al. 1968). The 1974 flow south of Keanakakoi and the 1977 flow must therefore have been considerably cooler and much denser than either the gully flow or the lava lake. A density of 2200 kg m-3 was given for the 1974 flow (Moore and Kachadoorian 1980), and H. J. Moore et al. (1980) commented that their re- sults were consistent with the observations that the 1977 lava was one of the most differentiated Kilauean lavas ever produced and that it was rela- tively cold and degassed (R. B. Moore et al. 1980).

Preliminary estimates of rheological paramet- ers of the 1984 Mauna Loa flows indicate that ap- parent viscosity of the lava on a given day in- creased along the length of the flow and with time at some locations (Moore 1987). Apparent viscos- ity at the vents increased from 100 Pa s on April 2 (eight days after the eruption started) to 2000 Pa s on April 13. The early values are comparable to those found for the gully flow, which came from vents which were active only for a few hours.

Supercritical flow

The values of velocity obtained for the gully indi- cate that the flow was moving fairly fast, which, coupled with observations on the position of the highest lava levels, suggest that the flow may have been supercritical. A flow is defined to be super- critical (i. e. the fluid is moving faster than a wave propagating downstream) if the dimensionless Froude number

F = u / ( g h ) l / 2 (44)

is greater than 1. Using the velocity values and mean depth values from Table 1, values of F in the range 1.5-2.2 are obtained. As noted earlier, the gully cross-sections containing the maximum lava superelevation were all slightly downstream of the centre of the bends. In supercritical flow, the fluid surface begins to rise from its normal depth at the start of the bend and reaches a maxi- mum height at a point below the bend (Chow 1959). This is in contrast to subcritical flow where

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430 Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii

the maximum height rise occurs at the start of the bend.

Supercritical flows exhibit characteristic cross- wave disturbance patterns on the surface, which increases the superelevation effect. This might help to account for the big difference in lava lev- els between the ouside and inside of the bends, particularly at bend 3. There was no obvious pat- tern of maxima and minima on the gully walls but any change in flow rate (such as a surge) would tend to obscure this.

It is not known exactly what effect supercriti- cal flow would have on the flow resistance in the laminar range. In the turbulent range, the friction factor is likely to increase with increasing Froude number for F > 3 (Chow 1959). A detailed study of this aspect of the gully flow, or any other simi- lar flow, would need a great deal more data than it was possible to acquire from the deposits. Ob- servations on an active flow would be required and future studies might bear this in mind. It is mentioned here as an interesting feature of this particular flow and to provoke further thought on the subject.

Conclusions

A method of analysing lava flow dynamics has been applied to deposits in a channel at Kilauea. The results imply that this flow was relatively fast moving but still laminar with viscosities in the range 85-140 Pa s. There is some evidence that it had a pseudoplastic rheology with a low or negli- gible yield strength of no more than 50 Pa.

The method may be applied quite generally to lavas or other volcanic flows confined in pre-ex- isting channels. It has been developed mainly for laminar flow but may be extended to turbulent flows with the inclusion of suitable expressions for the estimate of strain rates under turbulent conditions. The lack of data on velocity distribu- tions precludes a really detailed analysis of the fluid dynamics of a lava flow. Siutable laboratory experiments could help to alleviate this problem, and model flows could also provide data on the superelevation of viscous flows in curved chan- nels.

Appendix A

Effect o f viscosity on the motion around a bend

In deriving an expression for velocity the effect of a non-negli- gible viscosity was neglected. To look at the effect of viscosity on the motion we need to formulate a general expression for

the motion of the fluid as it rounds the bend. Consider a small fluid element as it moves round the bend and the forces acting on it. Let x be the main flow direction, y be the cross-stream direction and z the perpendicular (positive upwards) direction. The resultant force towards the inside of the bend must be equal to the force required to constrain the element to move in a circle, so

[p(y) --p(y -}- 67) ] ~X~Z -- [~yy( Z) -- CYyy(Z "~ ~Z)] ~x~y = p f x ~ y 6 z ( u ~ ) / r . (45)

Here, p is the pressure acting on the fluid (assumed to be hy- drostatic), ux is the velocity in the x-direction, r is the (mean) radius of curvature of the path on which the element is mov- ing, &x, 6y, 6z are the dimensions of the element and ffyy i8 the shear stress acting in the x-y plane. Thus, t3ryy=fl(duy/dz), where uy is the transverse component of velocity and # is the apparent viscosity. There will be similar stresses, given by crzz =#(duz /dy ) , acting in the x-z plane due to vertical motion of the fluid. Simplifying Eq. (45) gives

- d p / d y + dcryr/dz = p u ] / r (46)

o r

pg( tan fl) + dff yy/ dz = pu~ l r. (47)

So to use Eq. (3) with accuracy, the shear gradient term must be much smaller than the other two terms in Eq. (47).

The velocities of the fluid in the y or z directions will, in general, be considerably less than the velocity in the x direc- tion. Some idea of the order of magnitude of these velocities and their gradients may be obtained by considering the dimen- sions of the channel and the height to which the flow has ris- en. Referring to Fig. 10, as the fluid rounds the bend, the max- imum distance moved across the flow by any particle is ap- proximately w / 2 and the maximum vertical distance is ap- proximately w(tanf l) /2 , where w is the mean base width of the channel. (The base width is used here, as previously, because

5h = w t a n

J

Fig. 10. Parameters used to estimate the effect of viscosity on the motion of a fluid around a bend

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Heslop et al.: Dynamics of confined lava flow on Kilauea volcano, Hawaii 431

the data set for base widths is complete and accurate, and the gully is fairly steep sided over most of its length. The actual horizontal distance moved may well be less than w/2, so the resultant velocity estimate may in any case be regarded as an upper limit.) The time taken for such motion is roughly the time rO/ux taken for the fluid to move half-way round the bend in the main flow direction. Here, ~b is half the bend angle in radians so that the distance travelled in the main direction of flow is r~. Hence, the maximum transverse velocities are given, to an order of magnitude, by

Uy = wux/ 2rd~ (48)

and

us = WUx(tanfl)/2rq). (49)

In order to find the shear gradients, the second (spatial) derivatives of Uy and uz are required. The size of these deriva- tives may be estimated by considering representative lengths in the z and y directions, respectively, namely w(tanfl)/2 and w~ 2. The result may be written as

dtTyy/dg =#[2ux/rOw(tan2 fl)] (50)

and

dcrzz/ dy =#[2ux(tanfl)/wrq)]. (51)

If these terms are much smaller than the pressure gradient term pg(tanfl), they may be ignored, otherwise they should be included. The conditions under which the viscous terms may be ignored are

It "~pgwr~(tan3 fl)/2Ux (52)

and

# ,~pgwr~/2Ux. (53)

Clearly, the more stringent condition is Eq. (52), fo r f l<45 °, so if the viscosity satisfies this inequality, then it is not an impor- tant factor in determining the velocity from the superelevation. Since # is not usually known in advance, rewrite Eq. (52) in terms of the channel geometry using Eq. (13) and f = 16/Re to give

D2(sin a) ~ 16rq~w(tan3fl). (54)

This inequality may be used to test for the effect of viscosity when only the channel dimensions and degree of supereleva- tion are known. It is likely to be satisfied for small equivalent diameter and/or large radius of curvature and large width. If it is not satisfied, then the viscous term should be included in the determination of velocity. The values of viscosity obtained from the simple model may be used as a starting point and Eqs. (50) and (47) used to iterate to a solution.

This treatment of the effects of viscosity is really a first attempt - - the magnitudes of transverse velocities in lava flows are not known and until such data become available, we can only make educated guesses. We hope that the above treatment will provide a basis for future work. The analysis carried out in this appendix may be applied to any kind of fairly viscous flow and the modified method of determining velocities and rheological properties may be used equally well for debris flows or pyroclastic flows.

Acknowledgements. S.E.H. was supported by a NERC Re- search Studentship; L.W. and J.W.H. acknowledge funding from NASA grants NAGW 437 and NGR-40-002-088 at the University of Hawaii and Brown University, respectively. L.W. was partly supported by a Research Grant from the Royal So- ciety. J.W.H. thanks the William F. Marlar foundation and S.E.H. the University of Hawaii for partial support for field work. Staff of the Hawaiian Volcano Observatory kindly pro- vided access to their records of the 1974 eruptions; we are grateful for discussions with T.L. Wright (Scientist-in-Charge) and J. P. Lockwood. We thank R. I. Tilling who resurrected his field notes and supplied information about the eruption. Thanks to G. Giberti and I. Williams for helping with field data collection; to G. P. L. Walker and P. J. Mouginis-Mark for discussions based on the first draft of this paper and to the last named for co-operation in organising field work. We also thank P. Delaney and J. H. Fink for detailed reviews.

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Received April 25, 1988/Accepted February 2, 1989