Introduction

Lava flows are a major hazard associated with basaltic eruptions. Although the risk of human loss of life to lava inundation is typically low, particularly for well-monitored volcanoes, lava flows can cause major damage to buildings and infrastructure. This is exemplified by the 2018 eruption of Kīlauea on its lower East Rift Zone (LERZ), during which 716 structures were damaged, and millions of dollars were spent to clear highways buried by lava (Neal et al. 2019; Houghton et al. 2021b). Therefore, it is important to understand how lava flows propagate, in terms of path, velocity, and length. The path may be forecast, given topographic data of suitable resolution and accuracy as lava flows behave as gravity currents and follow the route of steepest descent (Favalli et al., 2005). However, the ability to predict flow advance velocity and final length with useful accuracy for a broad range of conditions remains difficult (Harris et al. 2016; Dietterich et al. 2017).

Lava flows are complex systems that are modulated by many factors, both internal (e.g., viscosity and yield strength) and external (e.g., substrate slope and roughness). Studies of actively flowing natural lava provide holistic views of emplacement processes, and field data are regarded as the gold standard for evaluating the efficacy of simple physical descriptions of flow dynamics. However, variables controlling lava flow advance tend to be correlated, making it difficult to isolate their effects. Understanding the influence and relative importance of individual variables is critical for hazard forecasting (e.g., Walker et al. 1973; Rowland and Walker 1990; Griffiths 2000). Thus, one branch of research has utilized analytical viscous flow theory (e.g., Huppert 1982a; 1982b; Lister 1992; Takagi and Huppert 2010) and lava analogs flowing in controlled experimental environments. For example, studies have successfully applied viscous flow equations to describe the balance of forces controlling advance velocity and ultimately, flow length of polyethylene glycol (PEG) wax, and PEG wax-solid particle slurries “erupted” in both subaerial and submarine settings (e.g., Fink and Griffiths 1992; Cashman et al. 2006; Kerr et al. 2006; Lyman and Kerr 2006; Rumpf et al. 2018; Lev et al. 2019). These quantitative descriptions of flow dynamics are suitably tuned to describe the propagation of PEG and PEG-solid particle slurry flows. However, their utility in forecasting the advance rate and lengths of lava flows has not been thoroughly evaluated.

Two sets of equations that describe one-dimensional flow propagation, or how flow length evolves through time, were developed by Lyman and Kerr (2006), “LK,” and Castruccio et al. (2013), “CEA,” using the same physical description of lava. The equation sets differ primarily regarding the time-dependency of input parameters. The diverse and high-frequency observations of the early LERZ events permit quantitative comparison of these viscous flow models. We take a hierarchal approach to testing whether a priori knowledge of input parameter time-dependency is needed to recover the observed time-length data, and secondly, to determine the level of specificity that is needed to characterize each parameter in the equation sets. Our objectives are to (1) discover the extent to which these simple equations can be used to accurately describe lava flow propagation and (2) infer the rheologic parameters that dominated the propagation of select lava flows during the LERZ eruption.

We start by quantifying key aspects of three lava flows from the early part of the LERZ eruption and use these measurements to test the equation sets’ ability to match the observed flow length as a function of time (“time-length data”). We chose the volume-limited (Guest et al. 1987) flows emanating from early fissure 8, fissure 17, and the fissure 20/22 complex because they span the full range in rheology exhibited in the 2018 sequence (Neal et al. 2019; Soldati et al. 2021), and they also most closely resemble the wax experiments (i.e., flows that have one main propagating front and relatively simple channel network morphology). We approach the fundamental efficacy question by presuming the validity of the equation sets and back-solving for the variable effusion rate relationships that most closely conform to the time-length data. We then consider the plausibility of the resulting flux curves in the context of other observations and a theoretical understanding of effusive eruption dynamics.

Kīlauea LERZ eruption summary

Drawing from the eruption chronology detailed in Neal et al. (2019), the 2018 eruption commenced on May 3rd, after a period of precursory activity that involved overflowing lava lakes, the collapse of the Puʻuʻōʻō cone, propagation of earthquakes downrift, and opening of ground cracks in the vicinity of the impending eruption. This activity was caused by pressurization of the magmatic plumbing system and subsequent failure of a blockage beneath Puʻuʻōʻō that allowed magma to travel farther downrift (Patrick et al. 2020; Roman et al. 2021). During the first 2 weeks of the eruption, the lava that erupted was viscous and rarely traveled more than tens of meters from the source vents. The high viscosity, cooler temperatures, and evolved bulk composition, relative to the lava that had been erupting from Puʻuʻōʻō, suggested that the lava that erupted during the first 2 weeks was sourced from leftover stored magma from previous LERZ eruptions (Gansecki et al., 2019). Among the compositions erupted during this phase was the first observed instance of andesite from Kīlauea. Fissure eruption durations during this early phase were generally short.

On May 18th, a shift in the temperature and composition of the magma supplied to the vents changed the behavior of the lava flows and the longevity of individual fissures. Fresher, hotter magma began to erupt at greater effusion rates (Gansecki et al., 2019; Dietterich et al. 2021), which produced fast-moving lava flows that rapidly reached the coast. This phase of the eruption was generally dominated by fissure reactivation, though a few new fissures opened. One of the early vents, fissure 8 (the cone formed during this phase was subsequently named Ahu’ailā’au), reactivated on May 27th and was the main center of eruptive activity by May 28th. This long-lived fissure fed a large, channelized flow that ultimately reached the ocean (Patrick et al. 2019). Most eruptive activity ceased on August 4th. Although not directly addressed in this paper, the LERZ eruption was accompanied by summit collapse events that also ceased when lava effusion ceased (Anderson et al. 2019).

Lava flow models

In this study, we examine two sets of equations that describe the same analytical theory of three rheologic regimes as restraining forces (so named because they “restrain” the flow from traveling farther at a given time interval) for lava flows. These two equation sets differ primarily in their ability to vary an input parameter through time. The three rheologic regimes we examine are viscosity limited, yield strength limited, and crust limited.

The first equation set, utilized by Lyman and Kerr (2006; “LK”), assumes a constant value for all input parameters. The three equations that govern length (L) in m vs. time (t) in s for viscosity (Lvisc) limited, lava yield strength (Lys) limited, and crust strength (Lcrust) limited flows are given by, respectively:

$$\begin{array}{c}{L}_{visc}={C}_{1}{\left(\frac{g\Delta \rho sin\beta {q}^{2}t}{\eta }\right)}^{1/3}\end{array}$$
(1)
$$\begin{array}{c}{L}_{ys}={C}_{2}\frac{qg\Delta \rho sin\beta }{{\sigma }_{0}}\end{array}$$
(2)
$$\begin{array}{c}{L}_{crust}={{C}_{3}\left(\frac{{q}^{2}\rho g\mathrm{sin}\beta }{{\sigma }_{c}\sqrt{\kappa t}}\right)}^{1/2}\end{array}$$
(3)

where g is the acceleration due to gravity in m s−2, Δρ is the density difference between the lava and air in kg m−3, taken generally to be ρ, the density of the lava, β is ground slope, q is volumetric flow rate normalized to flow width in m3 m−1 s−1, η is bulk viscosity in Pa s, σ0 is bulk yield strength in Pa, σc is crust yield strength in Pa, κ is thermal diffusivity in m2 s−1, and C1, C2, and C3 are constants. A summary of all symbols and their units can be found in Table 1. Time is not included in the yield strength equation (Eq. 2) as overcoming the yield strength requires a certain amount of force imparted by the mass of lava; without that force, the flow will not move, regardless of the time that passes. Kerr and Lyman (2007) utilized Eqs. 1 and 3 to infer material properties of the 1988–1989 Lonquimay, Chile, andesite lava flow, and concluded that flow propagation was predominantly controlled by the crust.

Table 1 Summary of variables
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In contrast, the second equation set, utilized by Castruccio et al. (2013; “CEA”) to describe historic lava flows, including some Hawaiian flows, is parameterized to allow input parameters to vary at each calculation interval. These equations use the same governing physical laws as the LK calculations. However, Castruccio et al. (2013) used the equations to derive lava material properties, given knowledge of the other parameters, rather than solving for flow length given the knowledge of material properties. The length of the flow is computed as a summation of the contribution of each calculation interval. The equations that govern the three rheologic regimes are as follows:

$${L}_{visc}={C}_1\sum_{i=1}^n{\left(\frac{Q_i^2\rho g\sin {\beta}_i{t}_i}{\eta {W}_i^2}\right)}^{1/3}$$
(4)
$${L}_{ys}={C}_2\sum_{i=1}^n\left(\frac{Q_i\rho g\sin {\beta}_i}{\sigma_{0i}{W}_i}\right)$$
(5)
$${L}_{crust}={C}_3\sum_{i=1}^n{\left(\frac{Q_i^2\rho g\sin {\beta}_i}{W_i^2{\sigma}_{ci}\sqrt{\kappa t}}\right)}^{1/2}$$
(6)

Here, instead of a single q term, there are instead Q (effusion rate in m3 s−1) and W (flow width in m). This approach was also utilized by Magnall et al. (2017) to infer rheological properties of the 2001 Etna, Sicily, and 2011–2012 Cordón Caulle, Chile, lava flows.

Lava flows selected for study

We selected three lava flows to span a range of composition, particle content, and distance traveled (Fig. 1; Table 2). Here, we use the term “particle” to indicate both bubbles and crystals. The early fissure 8 (EF8) flow, from the first sluggish phase of the eruption (phase 1; ~ 4–5 wt. % MgO), is a crystal-rich lava with a relatively evolved liquid (Gansecki et al., 2019). We note that we use the qualifying term “early” on this flow to differentiate it from the voluminous flow produced when the fissure reactivated (phase 3; > 6 wt. % MgO). The fissure 17 (F17) flow is the most viscous due to its evolved composition (< 4 wt. % MgO) and high crystallinity (Gansecki et al., 2019). The final flow is from the flow field produced by fissures 20 and 22 (F20/22) during a transitional period of increasing effusion rate and magma temperature (phase 2; 4.5–6 wt. % MgO). We treat these fissures as one system because there was significant overlap between the flows produced by these fissures, and it is difficult to determine the contributions of each fissure to the flow. This flow, sourced from the fresh magma supplied ~ 2 weeks after the eruption started, is crystal-poor and lowest in viscosity of the three flows (Gansecki et al., 2019). In contrast to the other two flows, we conclude our calculations of F20/22 propagation once the flow reaches the ocean, rather than when propagation ceases.

Fig. 1
figure 1

Map of the three studied flows and associated samples used for textural analysis. The black and white area, also outlined by a dashed line, is the extent of the thermal map (see the “Data sources” section). Temperature in the thermal image is displayed as gray-scale values, with the brightest pixels indicating the hottest areas. The base is a copyrighted color satellite image (used with permission) provided by DigitalGlobe. Flow outlines represent the final extent considered for each flow, and red dots are the collection locations of each sample. Our use of the shorter length for F17 is described in the “Key flow attributes” section. The base thermal image is of the lava flow field as of May 21, 2018. The outline for EF8 is from May 7, the outline for F17 is from May 15, and the outline for F20/22 is May 21. Red box on the inset map of the Island of Hawai’i shows the extent of the map

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Methods

Data sources

Imagery of the flows comes from three sources: (1) still-shots captured from a helicopter; (2) composite ortho-rectified images, “orthomosaics,” assembled from unoccupied aircraft systems (UAS) overflight imagery; and (3) thermal orthomosaics from helicopter overflight imagery. Both data sources (1) and (2) are in the visible range. The UAS imagery was captured and processed following Turner et al. (2017). The thermal imagery was captured and processed following Patrick et al. (2017).

Samples were collected both syn- and post-eruption (Fig. 1; Table 2). All samples for EF8 were collected during emplacement, and additional samples are not available; the flow was buried entirely by the reactivated fissure 8 flow. Syn-eruptive quenched samples were collected by the U.S. Geological Survey (USGS). Post-eruptive samples were collected during a helicopter field campaign, and we selected the glassiest possible samples. Glassier samples are more representative of the crystallinity of the flow during emplacement and are less influenced by groundmass crystallization once the flow halts and cools.

Table 2 Microfeatures of the three flows
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Macro-scale feature characterization

The still-shots were collected at oblique angles to the ground surface, so the images were corrected for look angle by georeferencing using QGIS (version 3.4.5). Reported error in flow front location is the maximum error in the final georeferenced image location (Online Resource 1).

A primary vent location for flow length measurements was selected for each flow on the basis of continuous effusion and centralized location to flow propagation (Fig. 2). All subsequent measurements of flow length were evaluated relative to that point, following the flow route as discerned by flow features, such as channels, within images, rather than straight-line distances to the flow front. Flow width location was chosen to be as close to the active flow front as possible, but where the width was not changing rapidly. The ground slope between images was also calculated along the flow path, using the USGS National Elevation Dataset 1983 10 m DEM.

Fig. 2
figure 2

Example of flow outline and its evolution during the early fissure 8 flow. Outlines indicate the shape of the flow at the labeled time (HST), and the blue lines (labeled “Flow Width”) indicate the flow width at the time next to the line. The line labeled “Flow Length” shows the measured path of flow length over the duration of the flow, following the observed flow paths in the studied imagery. Light blue arrows point to rapidly advancing lobes that are explained in the “Discussion” section. The base thermal image is from May 7

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Total flow volumes were also calculated from the thermal imagery. Surface area of the flow was measured in QGIS then multiplied by average flow thickness (Tables 3–5). Average flow thickness was determined from field measurements and a digital surface model of the F17 flow constructed from UAS imagery (Fig. S3). Volume was corrected for vesicularity by approximating 25 vol. % vesicularity across all three flows, as an intermediate value between all flows (Table 2), such that all volumes are dense rock equivalent (DRE) volumes. Mean output rate (Harris et al. 2007) was calculated by dividing the DRE volume by the time at which the measured length vs. time curve flattened, indicating no additional flow propagation (Fig. 3). Without more robust estimates of effusion rate variations in time for the flows of interest, we use this estimation of average effusion rate for our modeling as it requires the fewest assumptions. Other lava flow models commonly use only one effusion rate value (e.g., FLOWGO; Harris and Rowland 2015) and effusion rate measurements can be sparse during an eruption (e.g., the early stages of the Kīlauea 2018 eruption—Dietterich et al. 2021; 2011–2012 Cordón Caulle eruption – Magnall et al. 2017), so it is important to evaluate when this simplification is valid. An important note is that our effusion rate estimate is based on a posteriori knowledge of the final flow volume and total propagation time. Implementation of our approach during an eruption crisis would be to estimate the flow volume at a given point in time from its outline, as was provided regularly by the HVO staff during the 2018 Kīlauea eruption (e.g., https://www.usgs.gov/volcanoes/kilauea/maps), and thickness.

Table 3 Physical measurements for early fissure 8 flow
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Fig. 3
figure 3

Plot of measured flow length vs. time data for each flow. Error bars for flow front location are smaller than the symbols (Online Resource 1). The lines extend from the final point for EF8 and F17 to indicate that the flow may have continued expanding slightly for a time, but the main propagation was finished at the time of the last plotted point. In contrast, the F20/22 line is truncated at the last point to indicate the ocean entry and cessation of flow propagation over land

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Micro-scale feature characterization

A total of five samples were imaged for microtexture analysis—two each from EF8 and F17, and one from F20/22 (Table 2). Back-scattered electron (BSE) images were collected from thin sections of each sample using the JEOL Hyperprobe JXA-8500F electron microprobe located at the University of Hawai’i at Mānoa (Fig. 4). Image collection at variable magnifications followed the methodology in Shea et al. (2010). Glasses were analyzed by wavelength-dispersive spectroscopy on the same instrument, with an accelerating voltage of 15 keV, a current of 10 nA, and a spot size of 10 μm. Time-dependent intensity corrections were applied for Na when necessary. Two basaltic glass standards, VG2 and A99, were measured as unknowns during analysis to check for accuracy and evaluate drift.

Fig. 4
figure 4

Representative back-scattered electron images from the five samples used to measure flow microtexture. The scale bar in all of the images is 100 μm. Distances (proximal, medial, and distal) refer to relative distances from the vent. Although the absolute grayscale values are different between the images, the relative grayscale of the different phases is the same. Ol olivine, Pg plagioclase, Px pyroxene (orthopyroxene and clinopyroxene were not differentiated), V vesicle, and G glass

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BSE images were subsequently processed in Adobe Photoshop and imported to NIH ImageJ to measure the area and aspect ratio of particles. Only particles > 50 pixels in surface area were included in measurements. These measurements were conducted separately; the surface area measurement included all particles in the image, whereas the aspect ratio measurement excluded crystals touching the edge of the image (Hammer et al. 1999; Muir et al. 2012). Vesicles were all approximated as spheres, so their aspect ratios were not measured.

Modeling

Our modeling used a parallel approach, with each round of calculations corresponding to the equations from either the LK or CEA equation set (Fig. 5). As a subset within each calculation round, particles were treated with differing levels of complexity, both in terms of assumed shape and which particle types were incorporated into the calculations (Fig. 5). We use the term “scenario” to refer to a specific calculation branch, whereas “model” refers to the process of calculating the length vs. time curves. Measurements from the BSE images informed our characterization of particle content and crystal shape.

Fig. 5
figure 5

Flow chart illustrating the calculation approach in this study. The schematic BSE images in the “Simplification to Particle Inclusion” column shows the treatment of particles for each calculation type, and it is the same between both the LK and CEA calculations. Abbreviations next to each terminus indicate the scenario, and are used later in the text. Dashed line and stars indicate the paths used for calculations associated with variable effusion rate in the “Discussion” section

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Glass composition was used to calculate the initial liquid viscosity (Giordano et al. 2008), liquid density (Iacovino and Till, 2019), and eruption temperature (Beattie 1993) (Table 2). Water content for viscosity and density was assumed to be 0.2 wt. %, though this is likely a slight overestimation based on typical Hawaiian basalt lava flow water contents of ~ 0.1 wt. % (Seaman et al. 2004). However, volatile contents from erupted products have been as high as 0.3 wt. % (Ferguson et al. 2016); thus, our melt viscosity values are likely an underestimation, but the difference in melt viscosity between 0.1 and 0.2 wt. % water is less than a factor of two. Relative viscosity with the incorporation of solid particles was calculated using the model of Costa et al. (2009), using the fit parameters from Cimarelli et al. (2011) that most closely matched the measured crystal aspect ratios as well as the spherical particles. Calculated capillary numbers for our samples are < 0.1, so we incorporate bubbles into our calculations, where appropriate, as solid spherical particles (Mader et al., 2013; Online Resource 1). However, we note that some of the larger vesicles are deformed (Fig. 4), so locally, capillary number may be greater than 1.

Effusion rate for the CEA equations was maintained as a volume per unit time parameter. However, mass flux was treated differently for the LK equations. Instead of separate volume and width terms, the LK equations use the volumetric flow rate normalized to a constant width (i.e., the flume width in the experiments; “q” of Lyman and Kerr 2006). The constant width could either be the fissure length or the average width of the flow. We have chosen to use the average flow width as it is most comparable with the CEA calculations, and the portions of a fissure that actively contribute to the flux generally decrease with time (e.g., vent focusing; Jones and Llewellin 2021).

Yield strength of natural silicate melts remains poorly constrained, with measurements ranging from 101 (Robert et al. 2014) to 104 Pa (Fink and Zimbelman 1990) for basalts; we follow the approximation that incorporates both temperature and crystallinity employed by an existing lava flow propagation model, FLOWGO (Harris and Rowland 2015). The yield strength of the basaltic liquid is approximated as a function of eruption temperature relative to the liquidus, and the crystals can be incorporated by adding a term that accounts for their effect of increasing yield strength at greater fraction (Dragoni, 1989; Pinkerton and Stevenson 1992):

$$\begin{array}{c}{\sigma }_{0}=B\left[{e}^{C\left({T}_{0}-{T}_{e}\right)}-1\right]+\left[6500{\varphi }^{2.85}\right]\end{array}$$
(7)

Here, B and C are empirical constants of 0.01 and 0.08, respectively, T0 is liquidus temperature, Te is eruption temperature, and φ is crystal fraction. The empirical constants were derived from numerical modeling and are valid for a temperature range that includes the temperatures of our chosen flows (Dragoni, 1989). This equation assumes that all crystals are 1 mm long on their longest axis and have tabular shapes with axis length ratios of 10:5:1. This approximation likely overestimates the impact of crystals on yield strength for our samples as most crystals were < 1 mm. However, a more accurate 3D determination of crystal size distribution and aspect ratio would be a time-consuming exercise in the context of an emergency response effort. In our case, the overestimation of yield strength by approximating the crystals as 1 mm could be by as much as an order of magnitude (Pinkerton and Stevenson 1992), but as seen in our modeling results below, even the maximum possible yield strength does not impact flow propagation in most cases.

The effective strength of the crust can be estimated, to an order of magnitude, from the flow width using empirical equation of Kerr et al. (2006):

$$\begin{array}{c}{\sigma }_{c}\sim \frac{1}{W}{\left(\frac{{\left(g\Delta \rho \right)}^{5}{\left(\mathrm{cos}\beta \right)}^{5}{Q}^{7}{\eta }^{3}}{{\kappa }^{4}}\right)}^{1/8}\end{array}$$
(8)

The mass flux (Q) used for this calculation is the mean output rate for each flow.

Results

Key flow attributes

The initial opening of EF8 occurred on May 5, and lava began erupting from the fissure at ~ 21:00 HST (Houghton et al. 2021a). The flow ceased moving by 11:30 HST on May 7 after achieving a final length of 1230 m (Table 3).

F17 began erupting on May 13 at 4:30 HST (HVO Staff 2018). Although the fissure kept erupting for over a week, we consider the flow to have finished its initial advance by 6:45 HST on May 15. After this time, the fissure was weakly fountaining, and any additional lengthening of the flow occurred as localized “ooze-outs” around the edge of the flow. On May 18, when fresh magma reached the LERZ system (Gansecki et al., 2019), there was a noticeable increase in fountain height and thus volumetric output of lava, and lava extruded from the edges of the flow at a higher rate. However, the analytical theory we are applying is relevant to initial propagation, rather than expansion by break-outs, so we do not model this behavior. At the end of the initial period of advance, the flow achieved a length of 2260 m; subsequent expansion increased the flow length to 2340 m before all activity at the fissure ceased (Table 4).

Table 4 Physical measurements for fissure 17 flow
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F20/22 opened at around 18:30 HST on May 17 (C. Parcheta, pers. comm.), and the first approximately 24 h of the fissure’s eruption life was characterized by lava ponding near the fissure. A small flow propagated along the northern rampart of the fissure during the day on May 18, but the main, ocean-bound flows started during the late afternoon/early evening and continued into the night. A complex flow field was built of several branching flows that started and stalled throughout the night. The continuous branch that we model, outlined in Fig. 1, began around 00:00 HST on May 19, based on UAS footage, thermal satellite observations from the Visible Infrared Imaging Radiometer Suite (VIIRS), and estimates of flow front velocity (Online Resource 1). The UAS footage showing the flow in motion was captured at ~ 00:30 HST, and the visible flow length indicates that the flow was likely in motion for longer than 30 min. However, without further constraints on the exact start time of the flow branch, we use the start time of 00:00 HST estimated from the flow velocity. Approximately 22 h later, the eastern branch of the flow reached the ocean after traveling 5400 m (Table 5).

Table 5 Physical measurements for fissure 20/22 complex flow
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The three selected lava flows differ with respect to SiO2 wt. %, temperature, crystal content, vesicularity, and average effusion rate (Tables 2–5). The F20/22 flow represents the most fluid and voluminous of the flows, with low SiO2 (52.1 wt. %) and crystallinity (6.4 vol. %), and high vesicularity (53 vol. %), temperature (1121 °C), and average effusion rate (34.8 m3 s−1). The F17 flow is the most viscous of the flows, with high SiO2 (56.9 wt. %) and crystallinity (23.2–45.0 vol. %), and low vesicularity (1.1–11 vol. %) and temperature (1071 °C). The F17 flow was longer in length than the EF8 flow due to the higher average effusion rate (13.1 m3 s−1) and longer duration. The EF8 flow is less viscous than the F17 flow, due to the lower SiO2 (51.6 wt. %) and higher temperature (1117 °C), though the crystal content (25.3–71.1 vol. % for EF8) and vesicularity (2.8–3.8 vol. % for EF8) are similar. However, the low average effusion rate (3.7 m3 s−1) and short duration ensured that the flow was relatively short.

Modeling results

Following Lyman and Kerr (2006), the “dominant” restraining force regulating the flow advance is that which predicts the shortest length for a given time point. The length of a given natural flow should correspond to the length of the dominant restraining force. Changes to the flow properties through cooling and crust growth produces transitions in dominant restraining force through time and space. Two criteria are used to assess whether a set of calculations describes a flow well: (1) whether the calculation matches the final flow length and (2) whether the calculation produces the observed time-length curve. Final length in the context of our modeling refers to the predicted flow length at the time we consider as the “end” of a given flow. The predicted flow length for each equation set and restraining force is given in Table 6. The best-fitting set of calculations, as evaluated relative to the above two criteria, for each flow and each equation set are presented in Fig. 6. Although only the rheologic regime that predicts the shortest flow length at any given time is considered the dominant regime, we plot all regimes for the best-fitting scenario to show how predicted flow length evolves with respect to each restraining force.

Table 6 Modeling results
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Fig. 6
figure 6

Best-fitting results for the calculations for each flow, labeled with the particle treatment. Thicker lines represent calculations that incorporated vesicles (BE bubbles excluded, BI bubbles included) into the total particle fraction. The final lengths of all flows are underpredicted, based on the shortest predicted flow length, even for these best-fitting scenarios. The black arrow indicates the time at which a change in dominant regime is predicted. Labels that coincide with the vertical axis denote a regime that increases very rapidly in length

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Early fissure 8

For the LK calculations, the dominant regime is always crust-limited flow (Fig. 6a; Fig. S4), regardless of crystal incorporation. The predicted final length is 450 m, which is a significant underestimation of the measured final length of 1230 m. However, for the scenario that treats crystals as parallelepiped particles, the viscosity-limited flow calculation approximates flow propagation reasonably well until approximately 20 h after the flow initiated (Fig. 6a). The viscosity-limited calculation does not replicate the deceleration of the flow after 20 h that is evident in the measured data. None of the calculations generates a curve that replicates the time-length data.

The CEA calculation results are similarly independent of particle treatment (Fig. 6d; Fig. S5). For all calculations, crust is the dominant regime and predicts a final length of 550 m. Although this is still an underprediction of the final flow length, it is less of an underprediction than the LK model. Both viscosity- and yield strength-dominated models overpredict the final flow length, even with the incorporation of particles. None of the models adequately tracks the flow propagation at any point in time (Table 6).

Fissure 17

The general shape of the time-length curve is sigmoidal, and the LK models do not replicate its shape (Fig. 6b; Fig. S4). However, the first 10 h of the flow are reasonably well approximated by the crust-limited calculation (Fig. 6b). Crust-limited flow is predicted to restrain the flow for the entirety of its propagation, regardless of crystal incorporation. The final flow length predicted by the crust-limited flow calculation is 540 m, which is a significant underestimation of the measured final flow length of 2260 m. Although the bubble-bearing viscosity calculation is the closest calculation to replicating some of the flow propagation (hours 12–25 after flow initiation), it still overpredicts the measured data for all times. Thus, none of the calculated flow regimes matches the actual flow propagation.

Similarly with the CEA model, the dominant regime across all times and crystal shapes is the crust regime with a final flow length of 890 m (Fig. 6e; Fig. S5). Like the LK calculations, the initial propagation of the flow is modeled reasonably well, though the rapid increase in length that starts ~ 10 h after the flow initiates is not replicated. Regardless of particle shape, both yield strength and viscosity regimes greatly overpredict flow length. However, the calculations that incorporate all particles as spheres perform the best overall (Table 6).

Fissure 20/22

Because the physical properties (e.g., flow width, ground slope) of the fissure 20/22 flow did not vary greatly through time or space, the LK and CEA results are very similar (Fig. 6c and 6f; Figs. S4–S5). For both equation sets, crust strength is predicted to be the dominant regime for the whole flow in all scenarios, except for calculations that incorporate bubbles as spherical particles. The final length for the LK crust calculation is 2270 m, and the final length for the CEA crust calculation is 3250 m; both are underpredictions of the measured flow length of 5400 m (Table 6).

Including bubbles into the spherical particle calculations for the CEA equations produces curves that predict the initial ~ 5 h of the flow is dominated by viscous flow then switches over to a crust-dominated flow regime (Fig. 6f). For both the LK and CEA bubble-bearing calculations, the viscosity-limited model curves relatively well approximate flow propagation, except for the slowing of the flow near the end of modeling. The final predicted flow lengths for the bubble-bearing, viscosity-limited models are 6190 for the LK calculation and 6500 for the CEA calculation.

Discussion

We first consider the uncertainty in the final flow length predicted by the different rheologic regimes, as a function of the uncertainty associated with our estimation of the flow material properties. We also compare with other lava flow propagation studies to determine the reasonableness of our results. The poor fit of the length vs. time trends for the EF8 and F17 flows likely stems from the fact that effusion rate was not constant during these flows, yet we treated it as such. Previous work highlights the significant impact of effusion rate on lava flow propagation (e.g., Walker et al. 1973; Harris et al. 2007; Tarquini and de’ Michieli Vitturi 2014; Harris and Rowland 2015). Therefore, we explore the impact of allowing effusion rate to vary through time in additional, targeted calculations. The results of these additional calculations, coupled with the initial constant effusion rate calculations, offer insights into three-phase flow. However, the tested equation sets are by no means comprehensive in accounting for conditions present in nature; many other natural complexities have a demonstrated influence on lava flow propagation, explored below. Innovations in technology, such as UAS, can aid measurements of important lava flow propagation parameters, such as effusion rate (Patrick et al. 2019; Dietterich et al. 2021).

Impact of rheologic parameter uncertainty on calculated flow length

No matter the approximation chosen for each rheological parameter (e.g., viscosity, melt yield strength, crust strength), there is some inherent error in the value we use. To assess the impact of these uncertainties in our values of final flow length, we recalculate final flow length for each flow while varying each parameter to the maximum and minimum possible value within our chosen approximation. For these calculations, we use the LK formulations for each restraining force (Eqs. 1–3) and treat crystals as spherical particles. Bubbles are not included in the particle concentration for these calculations. The potential variation in melt viscosity, based primarily on our choice of dissolved water concentration, is a factor of two. The estimates for yield strength and crust strength are accurate to an order of magnitude. Thus, viscosity was multiplied by a factor of 2 and 0.5 to evaluate the reasonable extremes, and yield strength and crust strength were multiplied by a factor of 5 and 0.2 (Table S2). These factors produce error envelopes that are slightly larger than the inherent error in a given approximation. However, there is also some uncertainty between the different approximation models for each parameter, so these larger multiplication factors help account for this uncertainty.

As both the viscosity- and yield strength-limited calculations overpredicted flow length, we discuss these estimates with respect to the maximum possible value. An increase in melt viscosity by a factor of two resulted in a predicted flow length that was shorter by approximately 20% (3750 m length with the initial value vs. 2980 m with the 2 × increased value; Table S2). A 20% decrease in flow length predicted by the viscosity-limited calculations improves the estimate of the final flow length for the LK EF8 calculation with parallelepiped crystals, and both the LK and CEA F20/22 calculations with bubble-bearing spherical particles. A 5 × increase in yield strength corresponds to a 5 × reduction in predicted flow length (Table S2). This severely impacts the estimates of yield strength-limited final flow length for the EF8 flow in all cases that incorporate crystals. Although the maximum increase by 5 × does produce an underestimate of the final flow length for the EF8 case (yield strength-limit predicted length of 820 m compared to the measured value of 1230 m; Table S2), that indicates that there is some yield strength value, within error, that can accurately reproduce the measured flow length.

Conversely, the crust-limited calculations all produced underestimates of the final flow length, so we discuss these estimates with respect to the minimum possible value. The approximation we use is accurate within an order of magnitude, so we consider a 0.2 × reduction in crust yield strength. This produces an increase in predicted flow length by a factor of two (Table S2). The general error between measured and predicted final flow lengths can almost be entirely accounted for with this decrease in crust strength for the CEA EF8 and the LK F20/22 calculations.

With these considerations, the relative impacts of the different rheologic parameters are less clear for the EF8 flow, as much of the general error in predicted final length can be accounted for by the uncertainties in the input parameters. Although the F20/22 flow is relatively well approximated by the values chosen already, the fits can be improved by moderate variation in the values for melt viscosity and crust strength. However, parameter uncertainty cannot account for the generally poor approximation of final flow length for F17. Additionally, no variation in rheologic parameter value produces the sigmoidal shape of the measured EF8 and F17 length–time data.

Comparison with other lava flows

Although our modeling of flow propagation is not sufficient to describe the lava flows from EF8 and F17, general trends in dominant regimes based on composition and crystallinity are comparable to previous studies (Kerr and Lyman 2007; Castruccio et al. 2013). For all of the flows considered and both equation sets, the crust is a dominant restraining force during most or all of the propagation, assuming that our approximation of crust strength is valid (Fig. 6). With the uncertainty in our viscosity and yield strength values, EF8 could also be significantly controlled by these parameters. For the F20/22 flow, the CEA equation set predicts that the early part of flow propagation is dominated by the bubble-bearing viscosity and then transitions to crust-limited. Uncertainty in our crust strength and viscosity values could shift the transition between these regimes to different times in the flow for the CEA calculation, and introduce a period of time when the bubble-bearing viscosity is the dominant restraining force in the LK calculation.

The predicted dominance of crust in determining flow propagation for the more evolved lavas erupted from EF8 and F17 is consistent with results from Kerr and Lyman (2007) and Castruccio et al. (2013). Both Kerr and Lyman (2007) and Castruccio et al. (2013) find that the propagation of the andesitic lava flow from Lonquimay in 1988–1989 is best fit by crust strength as the dominant restraining force. Additionally, Castruccio et al. (2013) finds that the more crystal-rich basaltic flows from Etna, Italy, in 2001 and 2006 are best fit by an initial phase in which bulk viscosity restrains the flow that later transitions to a phase in which crust strength restrains the flow. Both of these cases (Lonquimay and Etna) are most analogous to the EF8 and F17 flows, and we note similar restraining forces inferred for these flows, although the flow propagation itself was not well-fit with our modeling conditions.

The best modeling results were achieved with the flow from F20/22 (Fig. 6). Although the interplay of forces predicts that crust strength determines the final flow length, the data were reasonably well-fit by the viscosity-limited regime that includes bubbles for both LK and CEA calculations. This is consistent with results from Castruccio et al. (2013). They find that basaltic flows with high effusion rate and low crystal content from Puʻuʻōʻō (episodes 11 and 17) are well-fit by viscosity-limited flow.

Influence of time-variable effusion rate

Misfit to final flow length can largely be accounted for with uncertainty in rheologic parameter estimation (described above), but we may also consider time evolution by assessing the misfit between the calculated and the measured flow propagation curves. We hypothesize that the misfit can be attributed to treating effusion rate as a constant value, even though it was not constant during flow emplacement. Both temperature and density are also held constant in the initial calculations, but this simplification is less likely to account for the general misfit in both the LK and CEA calculations. Cooling primarily impacts the material properties of lava flows by promoting crystallization, and this is accounted for by the steadily increasing crystal content in the CEA calculations. Although temperature impacts both viscosity and melt yield strength, the contribution of crystals is more pronounced (e.g., Gonnermann and Manga 2007). Flow density is most impacted by changes to the vesicle content, but that property either effectively does not change or even decreases downflow in our measured samples (Table 2). Thus, the most likely property that accounts for the model misfit is the effusion rate.

To test whether variable effusion rate could improve the fit of the initial calculations, we iteratively back-solved for instantaneous discharge rate curves (Harris et al. 2007) that adequately fit the time-length data. The plausibility of the derived effusion rate curves was evaluated by comparison with qualitative observations of the imagery and similarity to previously observed patterns of effusion rate through time (Wadge, 1981; Bonny and Wright 2017). Changes in effusion rate with time often reflect changes to magma supply at depth and pressure changes within the magma reservoir (Wadge, 1981; Anderson and Poland 2016; Bonny and Wright 2017). The “Wadge-type” effusion rate curve (i.e., rapid initial increase to a maximum, then subsequent exponential decrease) was used for the EF8 and F17 flows as it reflects the proposed mechanism during phase 1 of the eruption of ejecting magma from pressurized pockets (Gansecki et al., 2019), and pressure within the magma pocket decreasing as the supply is erupted (Wadge, 1981; Harris et al. 2000). Many eruptions exhibit more complex patterns in effusion rate (Bonny and Wright 2017), but this idealized pattern requires the fewest assumptions while still fitting the data. Potentially, the F20/22 overall effusion pattern follows a similar trend, but for the window of time we consider, the best fit to the data is achieved through an essentially constant effusion rate. The effusion rate trends are generally within previously determined values for the different eruption phases, with phase 1 effusion rates ~ 3–6 m3/s increasing to ~ 65 m3/s at phase 2 (Plank et al. 2021; Dietterich et al. 2021). It is important to note that these studies collected total rates for all active fissures, and the temporal resolution is lower than what we are trying to resolve.

For this test, the overall best-fitting scenarios from the initial calculations were used to calculate length vs. time. Insufficient data are available to create robust effusion rate curves with time. However, there are qualitative differences in effusion rate, both variance through time and total output, apparent from the visible and thermal images used to measure flow length (Online Resource 1). Incorporating these curves (Fig. 7) into the model calculations produced qualitatively good fits to the time-length data.

Fig. 7
figure 7

Variable effusion rate curves constructed for each flow. The criteria used to construct these curves are based on qualitative observations of the flows and described in detail in Online Resource 1. These relationships were subsequently used to calculate length versus time relationships for each flow

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The Wadge-type effusion rate curve was able to produce the sigmoidal shape of the time-length curves for the EF8 and F17 flows that the constant effusion rate was unable to replicate (Fig. 8). However, only the crust-limited regime is capable of fitting the data. With the additional flux at earlier times in the flows’ propagations, the viscosity and yield strength regimes greatly overpredict flow length. The calculations for F20/22 deviated slightly from the calculations for EF8 and F17 due to the relative goodness of fit produced by incorporating bubbles into the viscous flow regime in the initial calculations. We find that the first ~ 10 h of the F20/22 flow are predicted to be dominated by viscous flow, and the rest of the flow propagation is dominated by the crust. The slight decrease in effusion rate is needed to replicate the inflection point in the slope of the data at ~ 17 h.

Fig. 8
figure 8

Calculated length vs. time curves for each flow using the variable effusion rate relationships displayed in Fig. 7. Abbreviations next to each curve indicate the scenario in Fig. 5 used to calculate the line. Note that the conditions used for F20/22 are slightly different than those used for EF8 and F17. Variable effusion rate is able to reproduce the sigmoidal shape of the measured data. The black arrow indicates the time at which a change in dominant regime is predicted

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Although we find that incorporating a variable effusion rate improves the modeling results, this interpretation requires the inherent assumption that the equations themselves were adequate to describe the flows, an assumption we were unable to test without better estimates of effusion rate variations. Most importantly for hazard management, the constant effusion rate simplification tended to produce an underestimate of flow length. Thus, measurements of effusion rate during an eruption are needed to approach a credible prediction of flow propagation.

Implications for controls on three-phase flow propagation

Our calculations demonstrate that incorporation of bubbles into viscosity calculations can improve model fits in certain cases. The F20/22 flow was best fit by the viscosity-limited equations, both LK and CEA, that incorporate vesicles as solid spheres (Fig. 6). Considering that the vesicles in these lavas are relatively spherical (Fig. 4), this observation is consistent with previous studies of vesicle impacts on lava viscosity (e.g., Manga et al. 1998; Rust and Manga 2002; Soldati et al. 2020).

The relative importance of crystal incorporation in the viscosity-limited regime is unclear because of the impact of rheologic parameter uncertainty. However, the constant increase in crystal content for the CEA calculations did not produce as good of a fit for the measured data as the average crystal content in the LK calculations (Fig. 6a and 6d), so three phase flow may be better approximated by the average crystal content over the course of the total flow duration.

Simple vs. complex models: conditions for usage

The complexity of calculation needed to accurately reproduce the measured time-length curves varies based on the state of the fissure itself during the window of time modeled. The timing of both flows from EF8 and F17 corresponds to the waxing and waning of lava effusion. Conversely, the flow emanating from F20/22 was active during a period of quasi-stability. The flow did not begin until after the fissure had been active for > 24 h, and the window of time that we are considering in our modeling does not include the effusion waning phase. We conclude that simplified models and average input values work well for periods of steady-state effusion, but tend to underpredict flow advance rate in periods in which effusion rate varies.

For most of our models, there is no systematic improvement to the fit to the data from incorporating the time-dependency of input parameters, aside from effusion rate (Fig. 6). Potentially, this can be attributed to the uncertainty associated with the effusion rate variation for the crystal-rich flows from EF8 and F17 that would be most impacted by down-flow crystallization. However, our results indicate that average values work just as well as time-dependent values, for the wide range of material properties and ground slopes that were present for the three tested flows.

If we consider the F17 flow, for which our results are not sensitive to uncertainty in rheologic parameter estimation, there is little systematic benefit to modeling crystals as parallelepipeds, rather than spheres. Thus, for a more dilute crystal content (25–45 vol. % crystals), the spherical approximation for crystals does not introduce significant inaccuracy. Although the EF8 results are obfuscated by rheological parameter uncertainty, there is an improvement to model fit by treating the crystals as their true parallelepiped shape, rather than simplifying them to spheres. Thus, for more crystal-rich melts (> 45 vol. %), simplifying parallelepiped crystals to spheres could introduce another source of error in flow propagation models.

Extrinsic effects on flow propagation

The implementation of the equation sets that we have studied represents a simplified treatment of all the natural complexities that are possible. Additionally, both of the flows from EF8 and F17 were essentially single lobes, though the flow from F20/22 had significant branching (Fig. 1). Surface roughness (e.g., Rumpf et al. 2018), interactions with trees (e.g., Chevrel et al. 2019; Biren et al. 2020), and natural terrain that diverts lava flows (e.g., Hamilton et al. 2013; Dietterich and Cashman 2014; Dietterich et al. 2015) are all potential complexities that can influence lava flow propagation and branching.

The impact of all of these factors combined can qualitatively be seen in the outlines of the EF8 propagation (Fig. 2). While the flow was propagating in an area that had paved roads, offshoot lobes from the main flow traveled down the roads. Although the bulk of the flow propagated along the path of steepest descent through trees and homes, there was clearly a secondary preferential downslope path along the paved road. At least two of these lobes are readily apparent in our aerial observations (Fig. 2), and they correspond to when the flow encountered a road, even when the road is at an oblique angle to the flow direction. Similar propagation patterns were observed during the 1973 eruption on Heimaey, Iceland (Williams and Moore, 1983). Smoother surfaces allow for easier flow spreading (Rumpf et al., 2018; Richardson and Karlstrom 2019), and the confinement provided by the dense vegetation on the sides of the road serves to promote channelization (e.g., Williams and Moore, 1983), so these roads represent areas of faster flow propagation. This phenomenon presents a somewhat unique hazard to populated areas; these rapidly advancing lobes at oblique angles to the main flow advance could prove a risk to people attempting to evacuate a lava inundation area.

The other two flows, emanating from F17 and the F20/22 complex, propagated primarily through sparsely inhabited, densely forested land. As demonstrated by Chevrel et al. (2019) and Biren et al. (2020), lava-tree interactions can enhance cooling and, if the trees are densely packed, lava trees can become obstacles. Their results pertain to pāhoehoe flows, however, as ‘a‘ā flows, like the ones studied here, do not typically leave behind lava trees to study. Bulldozing and burial of the dense forest during propagation of these flows could potentially have increased the flow substrate surface roughness, imposed by the bulldozed trees. Biren et al. (2020) demonstrate lava in contact with trees experience enhanced cooling rates, which promotes crystallization and increases viscosity. This is coupled with the results of Rumpf et al. (2018), which indicate that rough surfaces (such as that produced by bulldozed trees) also enhance cooling and subsequent crystallization by increasing the surface area available for conductive cooling. Rumpf et al. (2018) find a complex relationship between surface roughness and apparent lava viscosity, with a maximum increase in apparent viscosity by a factor of four. Thus, the viscosities used in our calculations may be an underestimation as we do not account for any surface roughness.

Looking to the future of lava flow monitoring

Lava flow monitoring has improved greatly in the past few years with the use of aerial imaging to track lava flow evolution (e.g., Patrick et al. 2017; Turner et al. 2017; De Beni et al. 2019). UAS have gained increasing use in volcanological monitoring due to their utility for monitoring hazardous phenomena with exceptional spatial resolution (up to two orders of magnitude greater) as compared to space-based observations (James et al. 2020), and without the risks and higher costs associated with crewed aviation. Additionally, UAS are not impacted by cloud cover, which can make satellite observations challenging to use. UAS were utilized to monitor the 2018 eruption with great success, accomplishing many tasks such as aiding ground crews, monitoring flow propagation direction, and conducting gas emission measurements (Zoeller et al., 2018; DeSmither et al. 2021). Although UAS were primarily utilized for qualitative observations in the early, chaotic phases of the eruption, the reactivated fissure 8, and its corresponding lava flow were observed extensively with UAS to quantify changes in flow velocity and channel dynamics (Patrick et al. 2019; Dietterich et al. 2021). These parameters were used to calculate effusion rate at variable time intervals to constrain the effusion rate variations through time (Dietterich et al., 2021). Many of the early qualitative UAS observations are also used now for more quantitative analysis, such as in this study. Incorporating UAS monitoring earlier in an eruption crisis can help provide better constraints on effusion rate variations. As noted in our study and many others (e.g., Walker et al. 1973; Harris et al. 2007; Tarquini and de’ Michieli Vitturi 2014; Harris and Rowland 2015), constraints on effusion rate are critical for accurate lava flow propagation forecasting.

Best-fitting lava flow regimes can be difficult to determine, but modeled flow thickness through time can be used as a discriminating factor (e.g., Castruccio et al. 2013) as some regimes produce unreasonable thicknesses. Although not implemented in this study due to a paucity of corresponding field measurements, these measurements could be conducted with UAS as well (e.g., Fig. S3). Thus, UAS are flexible tools that can collect multiple data streams that aid with lava flow monitoring and modeling, and are critical to the future of understanding lava flow evolution.

Conclusion

We present analyses of three lava flows from the 2018 eruption of Kīlauea and model the length vs. time evolution of the flows. We use two equation sets that utilize simple physics of three potential flow-restraining forces, viscosity, yield strength, and crust. Our strategy pursues two goals: (1) to test the versatility of simple equations to provide a useful analysis for future eruptive crises and (2) to infer the dominant rheological parameters for the studied flows. Based on the equations of Lyman and Kerr (2006) and Castruccio et al. (2013), we find that the simpler calculation of Lyman and Kerr (2006) is reasonably accurate for steady-state flows, such as the flow from F20/22. For the flows that are impacted by both the waxing and waning phase of the fissure, like the flows from EF8 and F17, neither set of equations can reasonably reproduce the measured length vs. time relationship without information about the temporal evolution of effusion rate. The interplay of forces that control the propagation of each flow is complicated by uncertainty in our estimates of each rheological parameter, but crust strength is likely important at some stage of propagation for each flow. Additionally, the propagation of the F20/22 flow is likely influenced by the bubble-bearing viscosity. Modeling fits are greatly improved when reasonable variations in effusion rate trends are incorporated. The sensitivity of flow propagation on effusion rate emphasizes the need for such data in accurate lava flow propagation forecasting. If a single effusion rate is used in a crisis to model propagation of a flow that has variable effusion rate patterns, as in LERZ EF8 and F17, the flow may be much longer than predicted. Future effusion rate measurements can potentially be conducted with greater temporal resolution with the utilization of UAS, as has been demonstrated by recent studies.