Factors controlling the lengths of channel-fed lava flows

Bull Volcanol (1994) 56:108-120

Voli ology �9 Springer-Verlag 1994

Factors controlling the lengths of channel-fed lava flows H a r r y P i n k e r t o n , L i o n e l W i l s o n

Environmental Science Division, Institute of Environmental & Biological Sciences, University of Lancaster, Lancaster LA1 4YQ, U.K.

Received: May 18, 1993/Accepted: December 18, 1993

A b s t r a c t . Factors which control lava flow length are still not fully understood. The assumption that flow length as mainly influenced by viscosity was contested by Walker (1973) who proposed that the length of a lava flow was dependent on the mean effusion rate, and by Malin (1980) who concluded that flow length was dependent on erupted volume. Our reanalysis of Malin's data shows that, if short duration and tube-fed flows are eliminated, Malin's Hawaiian flow data are consistent with Walker's assertion. However, the length of a flow can vary, for a given effusion rate, by a factor of 7, and by up to 10 for a given volume. Factors other than effusion rate and volume are therefore clearly important in controlling the lengths of lava flows. We establish the relative importance of the other factors by performing a multivariate analysis of data for recent Hawaiian lava flows. In addition to generat- ing empirical equations relating flow length to other variables, we have developed a non-isothermal Bingh- am flow model. This computes the channel and levee width of a flow and hence permits the advance rates of flows and their maximum cooling-limited lengths for different gradients and effusion rates to be calculated. Changing rheological properties are taken into account using the ratio of yield strength to viscosity; available field measurements show that this varies systematically from the vent to the front of a lava flow. The model gives reasonable agreement with data from the 1983- 1986 Pu'u 'O'o eruptions and the 1984 eruption of Mauna Loa. The method has also been applied to andesitic and rhyolitic lava flows. It predicts that, while the more silicic lava flows advance at generally slower rates than basaltic flows, their maximum flow lengths, for a given effusion rate, will be greater than for basaltic lava flows.

K e y w o r d s : H a w a i i - l a v a - v i s c o s i t y - r h e o l o g y - flow

Correspondence to: H. Pinker ton

I n t r o d u c t i o n

During the past 20 years, a number of attempts have been made to determine the factors which control the lengths of lava flows. The initial assumption that flow length was mainly influenced by viscosity (e.g. Mac- donald 1972) was contested by Walker (1973) who proposed that the single most important factor which controlled the length of a lava flow was its average effusion rate. Malin (1980), using data from 84 historic lava flows on Mauna Loa and Kilauea in Hawaii, ques- tioned the validity of this relationship; he found a very poor correlation between length and average effusion rate for Hawaiian lava flows (Fig. 1), and concluded that flow length was more closely related to erupted volume. In this paper, we explore the relations between the various parameters which previous workers considered to be important in influencing the length of flows; we explain why Walker and Malin came to ap- parently conflicting conclusions; and we present a general framework within which lava flow length can be calculated.

100

E I0

t--

e-

1.0 _go IJ.

0.1

Malin's Hawai ian f lows

T u b e - f e d 2 0 0 h " / 100 h ~ j f flows / i - - ~ ~="

~, / / ~mmr. 7. m. ,1.~.. . / -

/ t ~ / / l N l u I �9 ~1~1 �9 , f ~ . . , , ~

I

1 10

I l

1 O0 1000 3 -1

Effusion ra te /m s

10000

Fig. 1. Malin's (1980) data for 84 Hawaiian lava flows, showing the relationship between duration, flow length and mean effusion rate for channel-fed lava flows. Tube-fed lava flows are grouped together at the top of the diagram, confirming that they should be considered separately

109

Preliminary analysis

When Malin's (1980) Hawaiian data are analysed, it can be seen that those lava flows which are longer than predicted using Walker's (1973) relationship were fed by mature tubes for more than 20 days, whereas those which are shorter than predicted were fed for less than two days by channels (Fig. 1). The latter observation is of fundamental importance in explaining the different conclusions reached by Malin and Walker. Walker ar- gued that high effusion rate lava flows cool proportion- ately less, as a function of distance, than those erupted more slowly. Thus, high effusion rate flows have the potential to flow further than low effusion rate flows before the cooling flow front prevents further advance. He also recognised that this difference would not be apparent in most short duration flows. Consequently he omitted from his data set all flows lasting less than about 30 h. If we apply the same restriction to Malin's data set, and if, in addition, we omit the mature tube- fed flows because they cool less rapidly and hence have the potential to flow further than channel-fed flows (Swanson 1973), then all of Malin's data fit within Walker's limits (Fig. 2). Thus, the apparent conflict between the analyses of Walker and Malin appears to be resolved; however, an uncritical acceptance of either a simple effusion rate-length relationship or a volume- length relationship is not justified, firstly because Walker's relationship cannot be applied to low duration or tube-fed flows, and secondly because the length of a lava flow can vary by up to a factor of 7 for any given effusion rate (Fig. 2) and by up to a factor of 10 for any given volume (Fig. 3 in Malin 1980). An additional limitation of Malin's relationship is that it is of limited usefulness in hazard analysis since it can only be used retrospectively. In addition, the imperfect cor- relations between length, effusion rate and volume support the views of Walker, Malin and other workers that other factors are important in controlling the lengths of lava flows.

100

E \ 10 c -

r "

1.0 I 0

ii

Malin's high duration channel-fed Hawaiian flows

0.1 [ 1 I

1 10 1 O0 1000 10000 3 -1

Effusion rate/m s

Fig. 2. Malin's (1980) data for high duration (> 45 h) channel-fed lava flows on Hawaii. The upper and lower lines derived by Walker (1973), shown as inclined lines in this figure, contain all of Malin's high duration, channel-fed flow data

Data collected during recent eruptions of the Ha- waiian volcanoes Kilauea (1983-1986) and Mauna Loa (1984) permit further investigations of the factors which affect the lengths of lava flows. By using these data, we avoid some of the uncertainties associated with the duration and hence effusion rates of older flows; we also minimise the compositional (and hence theological) variations.

A simple statistical analysis

For the first 20 of the 48 episodic eruptions between 1983 and 1986 from the Pu'u 'O'o vent at Kilauea, Wolfe et al. (1988) have compiled a comprehensive set of data which permits the flow front velocities and final lengths of flows to be determined. In the present analysis, their data have been supplemented by data from sean Bulletins for Episodes 21-48. We have calculated mean effusion rates for individual flows using the method used by Malin (1980) and we have estimated mean flow gradients from topographic maps.

An analysis of the channel-fed flows erupted during the 48 episodes confirms that, while there is no statistically significant relationship between flow length and average volumetric effusion rate for all Pu'u 'O'o flows, the 11 flows lasting more than two days lie with, in Walker's (1973) limits. There is also a statistically significant relationship between the measured length (L 0 and total volume (V~) of all 74 flows given by:

L~ = 0.60 V ~ (correlation coeficient, Cc = 0.75).

Thus the recent Kilauea data confirm the validity of both the Walker and the Malin relationships. It should be noted that the above equation, together with the other equations derived in this section, are not dimensionless and that SI units are used in all such equations.

A less simplistic investigations of the factors influencing flow length can be achieved by performing a multiple regression analysis on all of the Pu'u 'O'o data. Since multiple regression analysis requires that the data used approximate to normal distributions, Kolmogorov-Smirnov normality tests were applied to the variables that influence the advance rates of lava flows. While the raw data fail the normality tests, the transformed data form lognormal distributions; hence the analysis was performed on the logarithms of each variable. Another requirement of this type of analysis is that the data do not vary significantly during events. This is a potentially serious limitation in lava flow analysis, because, as outlined above, one of the main factors that influences flow length is effusion rate, and this can change by orders of magnitude during some eruptions (Wadge 1981). However, this was not a major problem in the present analysis; apart from Episodes 1-3, the effusion rates of the Pu'u 'O'o lava flows did not change significantly during individual eruptive events (Wolfe et al. 1988).

If we consider the relationship between flow length, average volumetric effusion rate (E) and duration of

110

the eruption of the fow (t0, the best fit equation is:

Lf = 0.66 E ~176 t ~176 (Cc = 0.83).

This is significantly better than the relationship between length and volume noted earlier.

Since the advance rate of any flow will be dependent on the mean gradient of the underlying ground (a), the effects of this variable should also be included in the analysis. The best fit equation now becomes:

Lf = 2.75 E 0"43 tf T M Ol 0"53 (Cc: 0.88).

Another factor which will influence flow advance rates is the rheological properties of the lava. The absence of any rheological data for the Pu'u 'O'o lava flows prevents us from incorporating rheology directly into this analysis, though we show later that differences in eruptive chemistry appear to have affected the initial rheological properties and hence the mean thickness of different batches of lava during Episodes 1-48, as shown previously for Episodes 2-5 (Fink and Zim- belman 1990). Rheological properties of lavas are also dependent on the eruptive temperature and thermal history of the flow. Since conductive heat loss rates from a lava flow are a function of the mean thickness of the flow, both the initial rheology and subsequent cooling-induced rheological properties of a lava flow are related to flow thickness. For many of the Pu'u 'O'o lava flows, mean flow thickness (d) was estimated; we will therefore use this as an indirect indicator of rheology and cooling. We accept that, unlike the other variables used until now in the multivariate analysis, mean flow thickness is not an independent variable. Consequently, the improvement in the equations will be small. This is confirmed by our analysis which gives the following relationship between all of the parameters:

L f : 1.32 E ~ t 0'71 0/051 d--0.34 (Cc = 0.89).

Calculated lengths based on the final regression equation are compared with measured flow lengths in Fig. 3.

Using this simple statistical analysis, we have developed equations which are considerably better predic- tors of flow length for the channel-fed Pu'u 'O'o flows than any peviously published equations. These equations also permit the relative importance of those factors which affect the lengths of lava flows to be determined. For example, the final equation suggests that the length of a flow at any given time will be greater, by 50%, if the mean effusion rate is increased by a factor of 2.4, or if the lava flows over slopes with a gradient which is greater by a factor of 2.2.

If the equations are transformed to permit advance rates to be calculated, the results are in agreement with the observed decrease in flow advance rates with time during an eruption. For example, the final equation suggests that advance rates are reduced by 50% for each order of magnitude increase in flow duration. However, the equations derived above must be used with caution. Since they are not dimensionless, the equations can be used only for flows from Kilauea with

10

"- - . 8 -

6 -

"7.

�9 ~ 2 1 ~ m [ ]

I 1 I i

2 4 6 8 Measured length / km

gl

Fig. 3. Comparison between measured lengths of Episodes 1-48 lava flows from Pu'u 'O'o and those obtained using a multivariate analysis

gradients, eruption rates and durations similar to those erupted during Episodes 1-48. In addition, it is clear from Fig. 3 that some flows have lengths which cannot be predicted accurately from our regression equations. Additional processes are therefore involved in determining the lengths of lava flows, and we can make further progress only by understanding these processes. We will show later that the final equation resulting from this multivariate analysis is very similar to that based on a theoretical non-isothermal Bingham flow model developed later in this paper.

Effects of topography, cooling and other factors on flow length

It is clear that many lava flows are volume-limited (Guest et al. 1987), that is they advance until the supply of fresh lava from the vent is terminated, after which further advance of the flow front may take place at a diminishing rate as the lava from the relatively uncooled central parts of the channels drains towards the flow front (Borgia et al. 1983). The resulting drained lava channels are easily recognised in the field, and they are characteristic of many of the Pu'u 'O'o flows. From our statistical analysis, it is clear that the majority of Pu'u 'O'o flows are volume-limited.

By contrast, if lava continues to be supplied from a vent for a sufficient time for a significant amount of cooling to take place, the flow front thickens, advance rates decrease, and the front eventually stops, even though lava is still being added to the active channel. This can result in overflows, reactivation of levees, and ultimately in the breakout of a new flow from the sides or possibly the front of the now stationary original flow. This process has been described for small flows on Mount Etna (Pinkerton and Sparks 1976) and for other volcanoes, including an excellent account of breakouts at a stationary flow front in Paricutin (Krauskopf 1948). These flows, termed cooling-limited (Guest et al. 1987) are important from the point of haz-

111

ard analysis. If we perform a multivariate analysis on the seven Pu'u 'O'o lava flows which, using criteria established later, are considered to be cooling-limited, the relationship between effusion rate and length im- proves and is given by Lo = 1230 E ~ (Cc = 0.76).

Breakouts at the flow margins can also develop for reasons other than cooling. Observations of active flows (Guest et al. 1987; Frazzetta and Romano 1984; Lipman and Banks 1987) have shown that active channels can become blocked by portions of the solidified channel wall or vent deposits which become dislodged and subsequently dam the main channel. Lava then spills over the levees and, in cases where the blockage becomes total and permanent, a new flow is formed. These flows are termed here accidentally breached flows.

While many flows spread out on eruption from the vent and attain a width which is dependent on the non- newtonian behaviour of the lava (Hulme 1974, 1982), others encounter a depression (which is commonly defined by pre-existing flow margins). These depressions can effectively channel the flow for part, or possibly all, of its length (Wilson et al. 1987). This was very common during the Episode 1-48 eruptions of Pu'u 'O'o and is considered to be one of the reasons for the relatively poor relationship between length and volume. Flows which are restricted to pre-existing topographic depressions for more than 25% of their length are termed here captured flows.

Thus accidentally breached flows will have lengths which are shorter than otherwise identical flows where this process does not take place, whereas captured flows will generally be longer than similar flows which are permitted to attain their calculated widths. Careful observation of lava flows will determine whether one or both processes have affected a flow. Finally, as recognised earlier, we acknowledge that the lengths of tube-fed flows can be considerably greater than their channel-fed counterparts because of the reduced cooling rates within tubes. In essence, the tubes become part of the sub-surface plumbing system. While accept- ing that accidentally breached flows are difficult to model, we shall see that it is possible to calculate the lengths of the other types of lava flow by modelling them as non-isothermal Bingham materials.

Lava flow models

During the past 20 years considerable advances have been made in modelling the behaviour of lava flows. Earlier models were based on the assumption that lava flows could be approximated as newtonian fluids (Shaw and Swanson 1970; Danes 1972; Harrison and Rooth 1976). Evidence in support of the newtonian behaviour of basaltic lava is provided by Rowland and Walker (1988) and Walker (1989) who use crystal and vesicle distribution studies to show that proximal pahoehoe and aa flows in Hawaii have properties which are close to newtonian. However, a newtonian model is clearly not valid for medial and distal basaltic and an-

desitic lava flows where field measurements have shown that there are pronounced thermal and rheological gradients in the margins of active lava flows and lakes (Shaw et al. 1968; Pinkerton and Sparks 1976, 1978; Borgia et al. 1983).

Huppert et al. (1982) argue that, even though lavas are non-newtonian, their yield strengths will be considerably lower than the large shear stresses applied at the base of large silicic domes. Consequently, they argue that a newtonian model is valid for the growth of large domes, and they model the growth of the Sou- friere of St Vincent using a newtonian model. Howev- er, the inferred viscosities on Soufriere were several orders of magnitude higher than typical measured values; they attributed this difference to the presence of a cool, high viscosity skin on the outer surface of the dome. This conclusion is supported by Stasiuk et al. (1993a) who used cooled glucose syrup to examine the effects of effusion and cooling rates on the advance rates and morphology of radial flows.

Iverson (1990) developed a membrane model which assumes that a dome can be modelled as a newtonian fluid surrounded by a continuous brittle skin. Howev- er, Delinger (1990) recognised that the cool, outer shell will develop fractures, and he modified the model to show how fracturing of this skin can result in episodic dome growth. Kilburn and Lopes (1991) have shown how basaltic lava flow fields can develop by approximating the behaviour of basaltic lavas as newtonian fluids surrounded by a brittle skin. Fink and Griffiths (1990, 1992) and Griffith and Fink (1992a, b) investi- gated the effects of effusion and cooling rates on the behaviour and morphology of cooled polyethylene glycol flows. A skin forms rapidly on the surface of their model flows, and the resulting flow morphologies are similar to many seen in subaerial and subaqueous lava flows.

An alternative approach to modelling lava flows is based on the observation that levees develop on the margins of isothermal Bingham fluids which flow down an inclined plane (Hulme 1974) and on the margins of proximal lava flows (Sparks et al. 1976). The development of isothermal Bingham flow models (Hulme 1974; Moore and Schaber 1975; Blake 1990; Dragoni et al. 1992) has been used to predict the factors which control the dimensions of lava flows and domes. We argue later that an isothermal Bingham flow model is inappropriate for modelling most lava flows, though we will describe, in this contribution, how an isothermal Bingham flow model can be modified to incorpo- rate cooling.

We have succesfully reproduced, in the laboratory, most features encountered during the eruption of lava flows using Bingham fluids (viscous kaolin-water sus~ pensions). In our experiments, water is evaporated from the surface of the flows by forced convection, si- mulating crustal cooling. The yield strength and apparent viscosity of the flow margins change systematically during flow, and we are able to reproduce thickening and widening of the flows and breakouts when flows attain a maximum length which is controlled by the

112

amount of simulated cooling experienced by the flow front. The results of these experiments will be described elsewhere. They are sufficiently encouraging to justify the development, in this paper, of a model based on the flow, down an inclined plane, of a cooling Bingham fluid.

If we begin by approximating lavas as isothermal Bingham fluids which are erupted at a constant rate onto a slope with a uniform gradient, the widths of their initial levees (Sparks et al. 1976) can be calculated (Hu!me 1974) using:

W b = r/2 g p ol 2 (1)

and the widths of channels can be calculated (Wilson and Head 1983) from:

wc = {(24 E q)4gp/(~ o26)}1/11 (wdZWb > 1) (2)

o r

wo = {(24 E ~7/(rol2)} 1/3 (0 < Wo/2Wb < 1) (3)

where ~1 is the Bingham viscosity of the lava, p is its density and r is its yield strength. Hulme (1974, 1982) also argues that the centre-line flow thickness can be adequately approximated by:

dr = ~-/g poz (4)

Using these equations, flow advance rates of isothermal Bingham flows can readily be calculated. While these equations may be appropriate for unconfined, high effusion rate, short duration flows, for most flows, including those considered here, cooling and the consequent change in the theological properties of the flow margins (Pinkerton and Sparks 1978) will cause an increase in the channel and levee dimensions and hence a reduction in the advance rate of the flow front (Pinkerton 1987; Fink and Griffiths 1990, 1992). Evi- dence in support of this can readily be obtained by ob- serving active basaltic flows, some of which experience breakouts of relatively uncooled lava from the flow fronts (Pinkerton and Sparks 1976). Similar features have been reproduced in the laboratory using polyethylene glycol waxes (Fink and Griffiths 1990, 1992). The relatively rapid advance rate of the new front, together with the concomitant reduction in channel height behind the old front, confirms that the lava in channels behind established flow fronts has a height which is greater than would be achieved by an isothermal, unconfined flow (Pinkerton and Sparks 1976). The amount of cooling controls the rheological properties of the front, and this in turn is one of the main factors controlling the thickness and advance rates of the rest of the flow and the cooling-limited length of a lava flow. Thus the advance rates and maximum cooling-limited lengths of lava flows can be calculated accurately only if cooling, and its effect on theology and hence flow dimensions, is taken into account.

Cooling of lava flows

The rate at which heat is lost from a lava flow is largely dependent on the rate at which it can be conducted

from the interior of the flow. Pinkerton and Sparks (1976), Hulme and Fielder (1977) and Wilson and Head (1983) conclude that conductive: heat loss can best be Characterised using the dimensionless Grfitz number (Knudsen and Katz 1958 ). The Gr~itz number permits an assessment of the temperature and hence rheological profile of any flowing material to be derived and is defined as:

(5) where u is the mean flow velocity; K is the thermal diffusivity of lava (taken here tO be 7.10-7.m2s-1); L is the length of the lava flow; and de is the equivalent diameter of the flow, which is equal to four times the cross sectional area divided by the wetted perimeter (Knud, sen and Katz 1958). For lava flowing in tubes, de is the same as the mean flow thickness, d, However, for all other flows, de is defined as de =2wcd/(wo+d). In all cases d<d~<2d. For example; for a channel width/ thickness ratio of 3, ddd=l:5; and for :ratios greater than 40, ddd.~ 2.

It is convenient, when analysing heat loss in la~?a flows, to reexpress Eq. 5 in terms of time or effusion rate as follows:

Gz =(nd) 2/K t = n 2 Ed/(Kwo L) (6)

where w~ is the mean channel width, t is the time from initiation of the flow, E is the:mean effusion rate, and n = ddd.

Pinkerton and Sparks (1976) have shown that the cooling-limited flows erupted on Etna :in 1975 stopped advancing when their Gr/itz numbers had fallen to a critical value (Gzcrit) of 300, at which point their fl0w fronts had attained a strength which was sufficient to inhibit any further advance of the flow front. Before we analyse the Hawaiian flows, it is essential that we determine the appropriate critical Gr~itz number for Hawaiian lava f lows.

Determination of the critical Griitz numbers for Hawaiian lava flows

Examination of length-time data for a number of Pu'u 'O'o lava flows reveals that they developed either single or multiple breakouts a short distance from their flow fronts. Using published mean flow depths we calculate Gr~itz numbers of these cooling-limited lava flows in the range 280-370 (see Figs. 4a, b). As discussed earlier, breakouts can occur for reasons other than marginal cooling. The flows which we have analysed here, however, are not considered to be breached flows, firstly because no accidental blockage of the ear- ly Pu'u 'O'o flows has been recorded and secondly because no instance of continued advance of the main flow has been recorded once a breakout has taken place.

An additional method of estimating the critical Gr~itz number of Hawaiian lava flows relies on studying data from volume-limited flows. In common with other volcanoes, some of the flows erupted from

113

Episode 7 NE flow

E .-~ 6 \

r--

4. (D

O 2' U.-

It,

Gz = 367

I I I 1 I 10 20 30 40 50

Time from start of the erupt ion/h

Episode 2 NE 1123 vent

60

Gz = 285

E ~ ~ _ . . . . + v 3 \

t -

r" 2 '

1.i_

0 I I I I 1 I I 1 40 60 80 100 120 140 160 180 200 220

b Time from the start of the erupt ion/h

Fig. 4a, b. Two examples illustrating the flow length-durat ion relationships for flows erupted f rom Pu 'u ' O ' o during 1983-1984, based on data given in Wolfe et al. (1988). Solid lines indicate the movemen t of the main flows, while dashed lines denote the lengthening of flows which break out f rom the margins of the main channel. The main flows are considered to be cooling-limited, and their Gr~tz numbers are given in the diagrams. Figure 4a shows the advance rates of the NE flow from Episode 7 and Fig. 4b shows the same information for the NE flow from the 1123 vent of Episode 2

Pu 'u 'O 'o were observed to advance after supply to the vent had ceased. The lava supplied to these flow fronts came from the draining of the isothermal interior (see Pinkerton and Sparks 1976). Channel drainage has been described in detail for small flows erupted on Ar- enal, Costa Rica by Borgia et al. (1983) who developed a model to predict how rapidly flows would advance once supply to the main channel ceased. In their model, Borgia et al. (1983) had to introduce a correction factor, q, to explain the difference between the theoretical and actual flow distances achieved by beheaded lava flows. This factor q will, of course, be a function of the amount of cooling that a flow has undergone when the supply is terminated, and it will be related to the amount of drainage required to reduce drained channel depths to ~-/pgd (see Eq. 4). There should, therefore, be a relationship between the increase in length caused by drainage and the pre-drainage Gr~itz numbers of the lava flows. The limited data available for Pu 'u 'O 'o flows suggest that there is a linear relation-

700 . ~ . t i . - , ~ 7

I

600 / /

E t I

r- 500 1 1 " ~ 1 8 �9 4 . /~ "

(_~ 400 t , ~ . i ~"

300 I I t I I I 1 I I

0 100 200 300 400 500 600 700 800 900 1000

Drainage distance/m

Fig. 5. When eruption ceases at the vent, volume-limited lava flows can continue to advance. The relationship between the resulting drainage distance and the Grfitz number of those Episode 1-20 Pu'u 'O'o lava flows for which drainage distances are available (Wolfe et al. 1988) is shown above. Extrapolating the trend back to zero drainage, which is when we consider that we are dealing with a cooling-limited lava flow, a critical Grfitz number of N 300 is obtained. This figure is in agreement with the value obtained from Figs. 4a, b and from previous studies of Etnean lava flows

ship between Grgtz number and drainage distance, and extrapolation to zero drainage results in a Gr~itz number of approximately 320 (Fig. 5).

Thus, on the basis of both sets observations of cooling-limited and volume-limited lava flows, we are con- fident that flows from Hawaii cease to advance on account of marginal cooling when they attain a critical Gr~itz number of -300 .

Estimating the viscosity and yield strength of Hawaiian lava flows

Few reliable field measurements have been made of the rheological propert ies of lava. On Hawaii, the pioneering work by Shaw and colleagues (1968) still provides the most useful measurements of the viscosity and yield strength of lava close to the vent. Work by Fink and Zimbelman (1986) permits the rheological properties along the lengths of four lava flows to be estimated, and a detailed study of the 1984 Mauna Loa eruption by Moore (1987) shows how the viscosity and yield strength of these flows changed during their advance. These, and complementary studies on Mount Etna by Walker (1967, 1971), Robson (1967), Sparks et al. (1976) and Pinkerton and Sparks (1978) have confirmed that, although the bulk behaviour of lava close to its vent is more accurately depicted by a viscoplastic model (Pinkerton 1987), a Bingham model is a useful approximation to the behaviour of lava flows. Field data for Hawaiian lavas (Shaw et al. 1968; Fink and Zimbelman 1986; Moore 1987) confirm that their viscosities and yield strengths change by several orders of magnitude during their emplacement. The yield strength, for example, varies f rom a few tens of Pa at the vent to in excess of 30 kPa at the flow front. Rather surprisingly, in spite of these large variations in yield strength and viscosity, the yield strength/viscosity ratio

114

0.6"

0.5"

O'h 0.4"

0.3"

0.2'

0.1 I ........ I '1 0 2 4 6 8 10 12 14 16 18

Flow length/k in

Fig. 6. Relationship between the rheological properties of Ha- waiian lava flows and the distance from the vent at which the measurements were made, expressed in the form (W~/)a/~. It can be seen that, even though the viscosity and yield strength of lava flows increase by orders of magnitude, this ratio undergoes a relatively small change. Data are from Shaw et al. (1968), Fink and Zimbelman (1986), and Moore (1987)

changes by a significantly smaller amount. In Fig. 6 we show that (~./~/)u3, a factor which we will use in the next section, varies between 0.57 and 0.17s -v3. While there are, at present, insufficient data to permit a rigorous analysis of the relationship between cooling and rheology, a regression analysis using existing data for Hawaiian flows indicates that the following empirical relationship is a useful approximation:

(,-1-/7) 1/3 ~_. A a 0.o9 (7)

where A has the value 0.167s 1/3.

A non-isothermal Bingham flow model

For flows in which 0 < wd(2Wb)< 1 we can see, from Eq. 3, that channel width is already expressed in terms of (~-h?) u3. However, for flows with Wc/2Wb > 1, Eqs. 2 and 4 need to be combined to produce a relationship which includes (r/~) as follows:

We = {244E4/[0{ 7 d (~'/'r)4]} in1 (wd2 Wb > !) (8)

Similarly, Eq. 1 can be re-expressed as:

W b = d/2 ~ (9)

For any flow, the appropriate value of (r ~/3 can be calculated using Eq. 7. This can then be used in Eqs. 8 (or 3) and 9 to calculate variations in channel, levee and hence total width at any distance from the vent for appropriate combinations of effusion rate, slope and flow thickness. Flow thickness has been used in these equations instead of yield strength because localised topographic channelling results in measured depths which are considerably greater than calculated depths, at least for proximal flows. Flow lengths can now be calculated using:

L = Et/[(wc + Wb)d] (10)

if we approximate levee cross sections as triangular. It is convenient to re-express this as:

L = Et/{[wr (1 + 0.5/(wr Wb))]d } (11)

Application to the recent f lows on Pu'u '0 'o

For Wc/(2 Wb)~---2, a typical value for many of the Pu'u 'O'o lava flows, we can obtain the following dimensionless relationship by combining Eqs. 8 and 11:

L = 0.8{E 7 t ll G7/[244 d 1~ (~7/~-) 4]} lnl (12)

This can be re-expressed as:

L = 0.252 E 0"64 t 0{ 0.64 d - 0.91 [(7"/7) 1/311.09 (13)

where [(Th?)l/3] 1~ has values in the range 0.063 (for distal f lows) to 0.136 (for proximal flows).

Equation 13 has a similar from to the statistical relationship Lf = 1.32 E ~ tf ~ 0{o.sa d -o.3~ derived earlier. Differences between the statistical and theoretical relationships are due to the inclusion, in the data set used to derive the statistical relationship, of all the Pu'u 'O'o flows, including the captured, accidentally breached, and co�9 flows. The implied constant advances rates in Eq. 13 are offset by the incorpo- ration of the changing theology of the flow using Eq. 7. The cooling limited lengths of the Kilauea lava flows are readily obtained by calculating the channel dimensions of the flow using the distal version of Eq. 8 and then inserting this into Eq. 6 using a critical Grfitz number of 300.

Calculated flow lengths using Eq. 13 (modified, where appropriate, for calculated values of wd(2 Wb) are plotted against measured flow lengths in Fig. 7. The only flows omitted from this analysis are those which were described as having occupied a pre-existing channel (i.e. captured flows). While these results are very encouraging, we recognise that one of the reasons for the scatter in Fig. 7 is that three types of lava flow were erupted during Episodes 1-20 from Pu'u 'O'o. Those fed by remobilised spatter had a lower eruptive temperature than other flows and were consequently considerable thicker and hence shorter than the other lava flows. In addition, there is strong evidence for a

10

7.5-

2.5"

" ' O < � 9

. ' / �9 ' 0 ' 0 / " / O �9

�9 0 / - �9

~ J O O # � 9 �9

0Oo 0o

I I I 2.5 5 7.5 10

Measured length / km

Fig. 7. Comparison between measured and calculated lengths of Episodes 1-48 lava flows from Pu'u 'O'o

progressive increase in the crystallinity, a decrease in the eruptive temperature and hence a progressive change in the rheological properties of the lavas erupted during Episodes 1-20 (Wolfe et al. 1988). In the absence of theological data for these eruptions, it is not possible to model these differences accurately, though the effects of rheology are apparent when a multivariate analysis is per formed on the different theological groups erupted during Episodes 1-48 from Pu 'u 'O'o. The correlation improve from 0.90 when the flows are considered as a single rheological group to 0.94 for the 14 flows erupted during Episodes 1-10; 0.96 for the 10 flows erupted during Episodes 11-18; and 0.92 for the 46 flows erupted during Episodes 1-10 when each group are analysed separately. We will in- vestigate the effects of rheological differences further by looking at the dimensions and advance rates of non- basaltic lava flows.

Application to non-basaltic lava flows

The main differences between basaltic and more silicic flows that we need to consider here are that silicic flows tend to be thicker; they have lower channel width to thickness ratios; and they have higher yield strengths and viscosities than basaltic flows. In addition, they are generally erupted at lower effusion rates, and the regional slopes of andesitic volcanoes are generally greater than for other compositions. These differences are summarized in Table 1.

Recent work on Arenal volcano, Costa Rica (Wadge 1983; Cigolini et al. 1984; Borgia et al. 1983) permits some of the methods described in this paper to be applied to andesitic volcanoes. Measurements of

115

bulk viscosity and yield strength by Cigolini et al. (1984) confirm the usefulness of a Bingham model and show that, in the distal regions, the yield strength was 79 kPa and the Bingham viscosity was 18 MPa s. Thus, ('T/7) 1/3 is 0.17 s --1/3, i.e. identical to the value that we found for distal basaltic lava flows. Measurements of channel dimensions and velocity profiles permit an es- t imated effusion rate of 0.94 m3s -1 to be calculated for the flow studied by Cicolini et al. (1984). Using a (T/ ~)1/3 value of 0.17 s-1/3 and a mean measured gradient of 0.5, a theoretical channel width of 26 m is calculated. This compares favourably with the value of 29 m measured by Cigolini et al. (1984).

The maximum and minimum cooling-limited lengths of the Arenal lava flows erupted during 1974- 1980 (Wadge 1983) are readily calculated using the values shown in Table 2. The calculated values of 1.14- 2.42 km compare favourably with the measured range of 1.0-2.5 km.

Using the data in Table 1, it is possible to calculate the maximum cooling-limited lengths of lava flows of different compositions. The cooling-limited lengths of lava flows increase with increasing mean channel width to thickness ratio, flow thickness, gradient and effusion rate. Although mean effusion rates and channel width to thickness ratios generally decrease with increasing silica content, this is generally offset by the greater mean depths and, for andesitic flows, higher gradients. Consequently, for a given effusion rate, it can be shown that, in general, the higher the silica content of a lava flow the greater will be the potential flow length. Using optimum values for mean thickness and gradient which satisfy the maximum yield strengths given in Ta- ble 1, the maximum flow lengths of basalts, andesites and rhyolites can be calculated. These are shown on

Table 1. Summary of the important differences between the physical properties and dimensions of basaltic, andesitic and rhyolitic lava flows, based on data given in Moore and Schaber (1975) and others. Ranges in effusion rates are based partly on field measurements and partly on estimates based on flow morphology (see Moore and Schaber 1975). No field measurements have been

made of the viscosity and yield strength of rhyolite lavas and the only reliable measurements on andesites are by Cigolini et al. (1984) on Arenal and Naranjo et al. (1992) for Lonquimay, By contrast, a number of estimates exist for the yield strengths of stationary lava flows (Moore et al. 1978; Hulme and Fielder 1977, etc)

Basaltic lava Andesitic lava Rhyolitic lava

Effusion rate/m 3 s - I 0 . 1 - 1 0 3 0.3--80 1-10 Mean thickness/m 1-10 8-800 200-550 Width/thickness of channel 1.3-15 1-3 - 2 Gradient/radians 0.007-0.03 0.031-0.60 0.067-0.17 Yield strength/Pa 50-2.5.105 0.5" 105-3.6 �9 105 1.2" 105-3.6 . 105 Apparent viscosity/Pa s 200-2.3.105 105-109 108-1012

Table 2. Dimensions of the flows erupted on Arenal, Costa Rica, between 1974 and 1980, based on data in Wadge (:1983), Borgia et al. (1983) and Cigolini et al. (1984)

Channel width/ Thickness/ Gradient/ Effusion rate/ (z/~q) 1/3/ Calculated Measured thickness m radians m3s -1 s -z /3 length/kin length/km

Thickest flow 2 15 0.6 0.33 0.17 2.42 2.5 Thinnest flow 2 8 0.5 0.33 0.17 1.14 1.0

116

IO0 , ~ ~ ~ / M ? ~ ..... E ~ .... d " ( ~ . ~

~ " ' ~ " - - ~ i ....... ~: 10- ~ ~ �9 - ~7--_ ,, ..........

. . . . . . . �9 x

�9 " . " " \ t

B

~o 1.0

LI_

0.1 1 I I A

0.1 1.0 10 100 1000

Effusion rate/m 3s-1

Fig. 8. Selected data from Walker (1973) showing the maximum flow lengths predicted for basaltic, andesitic and rhyolitic lava flows. These are based on Gr~itz numbers of 300 and use the data in Table 1. AII channel-fed basaltic flows lie below the basalt line, the three basaltic flows lying closest to the line being represented by squares. The presence of two rhyolitic flows (shown as filled circles) on or above the basaltic line supports the argument that silicic flows have the ability to flow further than basaltic flows with comparable mean effusion rates. Also shown on this diagram are the different flow lengths which can be achieved by two flows, both of which have the same initial effusion rate and the same duration. Line ABC represents a constant effusion rate flow, whereas ABD represents the more common case of a flow with a decreasing effusion rate. The dotted lines represent Walk- er's upper and lower limits. An andesite from Arenal is represented by a cross

Fig. 8 where it can be seen that they can explain the observation that rhyolites and andesites with effusion rates less than 1 m3s -1 have lengths which are considerably greater than channel-fed basaltic flows with comparable effusion rates (Walker 1973).

The reason why all long-duration flow data fall within the limited region shown in Fig. 1 can now be understood. If the effusion rate of a lava flow remains constant throughout an eruption, and if the flow can be considered to behave essentially as an isothermal Bingham fluid, then its length will increase with time at a constant rate. For the majority of lava flows, a correction is required to take into account the changing rheological propert ies of the lava as it flows and cools. Thus the lengths of constant effusion rate lava increase with time along vertical lines on Fig. 8, whereas for flows whose effusion rate decreases with time, a curved t rend similar to the example shown on Fig. 8 is fol- lowed. As the length increases, the Grfitz number decreases until the flow attains the critical Gratz number of - 300 at which point it stops advancing and becomes breached upstream. Flows which attain this length are cooling-limited, whereas those which stop before they attain this length are volume-limited. The lower limit in Fig. 8 is simply due to Walker 's rejection of short- duration flows. The right boundary reflects the highest recorded effusion rate. The upper boundary is determined by the maximum erupted volume of volume- limited lava flows.

While this study of non-basaltic flows has permit ted us to understand the trend established by Walker (1973), and to explain additional aspects of his rela-

tionship, it has also highlighted the importance of de- veloping a model which permits the incorporat ion of changes, during flow, of the rheological properties, effusion rates and other relevant variables.

Cumulative flow model

In all of the calculations so far we have assumed that the effusion rates, flow depths and gradients can be considered to be constant. While effusion rates during most of the Episode 1-48 eruptions from Pu 'u 'O 'o did not vary significantly, the assumption of constant values for the other parameters will result in differences between measured and modelled flow dimensions, as discussed above. Thus, a more rigorous approach is required which permits advance rates to be calculated using the changing dimensions of flows as the cool. The model will also allow any changes in effusion rate or gradient to be taken into account.

The refined model determines the conductive heat loss f rom the flow by calculating cumulative Grfitz numbers (see Guest et al. 1987) f rom which the changing channel and levee widths can be calculated itera- tively using Eqs. 3 (or 8) and 11. These can then be used to calculate the cumulative flow length from:

Lc~m = 2{E At/[ (wc + Wb) d]} (14)

where E, wc, wu and d are now the average values during each of a series of time intervals At. In order to test the validity of this approach, we have applied the method to the 1984 eruption of Mauna Loa.

Determination of the advance rates and maximum length o f Flow 1A, Mauna Loa, 1984

The 1984 eruption of Mauna Loa has been comprehen- sively documented by Lipman and Banks (1987) who argue that Flows 1 and 1A (the longest flows formed during the 1984 eruption) were breached as a consequence of the permanent blockage of the main feeder channel by detached portions of the channel walls. As confirmatory evidence that Flow 1 was an accidentally breached lava flow, we use Moore 's (1987) observation that it continued to flow for i km after the breach had taken place. For Flow 1A, however, there is no evidence of continued advance after breaching. In fact, the marked reduction in advance rate of the flow front before breaching developed supports our contention that Flow 1A was cooling-limited. Using the channel dimensions and velocities published by Lipman and Banks (1987) and Moore (1987), the cumulative Gr~itz number of lava in the channel is readily calculated. The data used are given in Table 3, and the cumulative Gr~itz number construction in Fig. 9 demonstrates that the lava would attain a length of 27 km when the Gr~itz number decreases to 300, i.e. the same as the observed length, supporting the claim that this flow was indeed cooling-limited. It is therefore an appropriate flow on which to test Eq. 4. F rom the comprehensive accounts

117

Table 3. Dimensions of the channels which formed during the 1984 eruption of Mauna Loa Flow 1A on 3 and 4 April, based on data in Lipman and Banks (1987) and Moore (1987). These values are used to calculate the cumulative Gr~itz number of Flow 1A in Fig. 10

Station Distance Mean Equivalent from vent/ velocity/ thickness/ km ms-~ m

11 0.2 15.0 5.3 8 3.8 5.2 6.7 7 6.2 4.2 7.5 4 11.0 0.94 10.1 2 14.8 0.33 9.1 1 16.8 0.31 9.1

Distal flow 1727 0.016 13.5

Mauna Loa Flow 1A 107.

Proximal channel 6 . "--~-~-"~._ .~rend Cumulative Gratz

10 . "'"-'-~.~''[~-'~----..~ "~ " ~ - ' < - - number trend 5 'Trend s?-- - .2>- /

104

="~ D i s t a l ~ - " ' " ~ - . . " " ' - Z e 1~ 10 1 I

0.1 1.0 10 27 00 Flow length/k in

Fig. 9. The Gr~tz number of Flow 1A which was erupted on Mau- na Loa in 1984 as a function of distance from the vent. The values, which are calculated using the data in Table 3, show that the heat loss rate from the proximal channel was relatively small compared with that from the distal channel. Thus, if lava was permitted to flow indefinitely in a channel having the dimensions of the proximal Mauna Loa channel and at the same velocity as lava in the proximal channel, then it would be able to flow for several hundred kilometres before cooling prevented further movement. Lava flowing throughout its life at the same velocity and in a channel having the dimensions of a distal channel, on the other hand, would be able to flow only 11 km before it had cooled to a Gr~itz number of 290. The solid line shows the cumulative Gr~itz number of Flow 1A at increasing distance from the vent. Using the construction described in Guest et al. (1987), the maximum length corresponding to a Gr~itz number of 300 and a mean value of (~./~)u3 of 0.28 is 27 km, which is in agreement with the observed length

E

ID ._1

o u.

Calculated leng

2010 ~ r e d length

0 i J 0 100 200

Time/h 300

Fig. 10. Calculated and measured of Flow 1A, Mauna Loa, 1984

1201 J 100

. 0

60 -I ~ 222

56 20

14 0

0 2 0 4 0 6 0 8 0

T i m e / h

Fig. 11. Calculated flow front positions of basaltic lava flows erupted at different effusion rate (shown in m3s-1) down a slope of 5.5 ~

It should be stressed that the m e t h o d o u t l i n e d above can be applied no t only to old flows for which the re levant data are known, bu t also to flows which are current ly active where m e a s u r e d effusion rates, depths and gradients can be used to mode l fu ture advance rates and m a x i m u m cooling-l imited flow lengths. This has clear implicat ions for haza rd assessment.

Discuss ion

in L i p m a n and Banks (198)) and M o o r e (1987), the thickness, effusion ra te and gradient data in Table 3 were de te rmined , and these permi t the advance rates at different distances f r o m the vent to be calculated. The calculat ions are readi ly p e r f o r m e d on a spread- sheet with i terat ion facilities. Calcula ted and m e a s u r e d advance rates of M a u n a L o a F low 1A c o m p a r e very favourab ly (Fig. 10). Us ing this mode l we can calculate the dimensions, advance rates and m a x i m u m flow di- mens ions of typical Hawai i an lava flows (Fig. 11).

W e have shown that the empir ical relat ionships proposed by Walke r (1973) and Malin (1980) p rov ide useful insights into the pa ramete r s which affect the lengths of lava flows. Howeve r , they are of l imited usefulness because Walke r ' s t rend takes no account of factors o ther than effusion rate, and Malin 's t rend is applica- ble only to flows of comparab le rheological proper t ies to Hawai ian flows and which f low down slopes of similar gradient . Mal in ' s (1980) conclus ion that f low length was p ropor t iona l to e rup ted vo lume for the Hawai ian

118

lava flows will be true for volume-l imited lava flows. H o w e v e r his equat ion cannot be applied to other volcanic regions [compare, for example, Fig. 4 in Lopes and Gues t (1982) with Fig. 3 in Malin (1989)1, since the constant and exponent in his equat ion will depend on factors such as the rheological proper t ies of the flow, the rate of effusion, flow thickness and erupt ion duration. A relationship which takes into account all of these pa ramete r s is considerably more useful than a set of empirical length-volume relationships for those few volcanoes for which accurate flow data are available.

We have developed non- isothermal Bingham flow equations to de termine advance rates of lava flows. These take into account the effects of marginal cooling. Appl icat ion of these equations to recent Hawai ian lava flows is encouraging. The m a x i m u m lengths of andesitic flows on Arenal , Costa Rica have also been calculated, and they are found to be close to those attained in the field. Manley (1992) has recently developed a cooling model for rhyolite flows, and he has shown that silicic lava flows are capable of flowing distances in excess of 20 km. Similarly, we argue that that the more siliceous lava flows, in spite of their higher apparent viscosities, will, for a given effusion rate, be capable of flowing further than a basaltic lava flow erupted at the same effusion rate before marginal cooling prevents fur ther advance. For any lava flow, once the channel dimensions and regional slope are known, the positions of the critical Gr~itz num ber can be plot ted on Fig. 8 and the m a x i m u m flow length can be estimated. As emphasised in this paper , errors can occur as a consequence of topographic chanelling, regional variations of slope and accidental breaching. In addition, it is possible for some cooling-limited flows to become breached at the flow front (Pinker ton and Sparks 1976), thereby causing them to extend fur ther than would be expected using the methods outlined here. For active lava flows, these problems can be minimised by adopt ing the more rigorous cumulat ive Gr~itz number approach used for the 1983 E tna lavas (Gues t et al. 1987) and for Flow 1A of the 1984 Mauna Loa lava flows. This involves calculating the changing dimensions of a lava flow by comput ing the cumulative cooling of the advancing lava flow front. This model also permits variat ions in effusion rate and topography to be included and it can be used to compute the advance rates and cooling-limited lengths of lava flows (Fig. 10 and 11) with greater confidence than simpler models based on mean parameters . These models can be used to assist in the decision-making process during future eruptions involving the flow of lava towards inhabited areas.

Al though the methods outlined in this pape r permi t the lengths of lava flows to be calculated more accurately than existing empirical models, we accept that, while a considerable amount of progress has been made in lava flow modell ing during the past decade (e.g. H u p p e r t et al. 1982; Pieri and Baloga 1986; Crisp and Baloga 1990; Iverson 1990; Kilburn and Lopes 1991; Fink and Griffiths 1990, 1992), the definitive flow model has yet to be developed. This will need to take

into account the factors causing changes in effusion rates during eruptions (Stasiuk et al. 1993b) and the changing theological propert ies caused by crystallisa- t ion and gas loss f rom flows (Sparks and Pinker ton 1978; L ipman et al. 1985; Pinker ton and Stevenson 1992). It will also model heat loss f rom the surface and interior of lava flows using appropr ia te finite difference models and it will take into account the different heat loss f rom different types of crust which has frac- tured by differing amounts (Crisp and Baloga 1990). It will use t empera tu re -dependen t thermal propert ies of lavas and it will use more accurate field rheological data for lava flows of various compositions. This will result in improved relationship be tween the rheological propert ies, gas loss and cooling than the empirical relationships which we have used. These data can then be used in appropr ia te finite e lement models to simu- late lava flows more accurately than is possible at the present time.

Acknowledgements. The authors wish to thank EW Wolfe and his co-authors for sending a pre-print of their papers on the Pu'u 'O'o eruption Episodes 1-20; this was the stimulus for many of the ideas contained in this paper. We also thank G Ulrich and colleagues for allowing us to use their data from Episodes 21M8, and D Clague, Scientist in Charge at HVO for his permission to use their very useful data set from the Pu'u 'O'o eruptions. Dis- cussions of preliminary ideas with HJ Moore, CA Neal, HR Shaw, GPL Walker, D Pieri, S Baloga and AJ Bond, and com- ments on earlier drafts by S Blake, SK Rowland, P Delaney, TH Druitt, JE Guest, A Freundt, JH Fink and an anonymous review- er were useful in refining some of the ideas presented here. LW acknowledges support from the Royal Society through a Lever- hulme Trust Senior Fellowship. This study was completed with the assistance of a grant for work on Climatology and Natural Hazards under Contract No. EV5V CT92-0190 of the EC Envi- ronment Programme 1991-1994.

Appendix

List of variables used in the paper

Symbol d do de

E g Gz

Gzcrit

L Lf Lo /,/

t tf

Meaning the estimated mean thickness of a lava flow centre-line flow thickness the equivalent diameter of the flow, which is equal to four times the cross sectional area divided by the wetted perimeter (Knudsen and Katz 1958) mean volumetric effusion rate acceleration due to gravity the Grfitz number, which is defined in Eqs. 5 and 6. The Grfitz number, which reflects the relative amount of uncooled material within a cooling flow, decreases from c~ at the vent to - 300 when a lava flow attains its cooling-limited length the critical Gr~itz number at which a cooling-limited flow stops. This has a numerical value of -300 for flows on Kilauea and on Mount Etna the length of the lava flow the measured final length of a stationary lava flow the final cooling-limited length of a lava flow defined as did the time from initiation of the flow the total time over which lava is being supplied to a flow from the vent the mean advance rate of the front of a lava flow

119

V~ Wc

Wb

Ol

K

the estimated final volume of a stationary lava flow the mean channel width the mean width of levees the mean gradient of the ground beneath a lava flow the bulk Bingham viscosity of the lava flow the mean thermal diffusivity of lava (taken here to be 7.10 -7mZs -1) the bulk density of a lava flow the bulk yield strength of a lava flow

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Factors controlling the lengths of channel-fed lava flows

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