Magma flow between summit and Pu‘u ‘Ō‘ō at K̄lauea Volcano, Hawai‘i


  • The images in Figures A2 and A3 were transposed when this article was published online on 29 July 2013. In addition, the captions for several figures were truncated in the issue PDF version of this article available on 3 September 2013. Both of these errors were corrected on 23 October 2013.


[1] Volcanic eruptions are often accompanied by spatiotemporal migration of ground deformation, a consequence of pressure changes within magma reservoirs and pathways. We modeled the propagation of pressure variations through the east rift zone (ERZ) of K̄lauea Volcano, Hawai‘i, caused by magma withdrawal during the early eruptive episodes (1983–1985) of the ongoing Pu‘u ‘Ō‘ō-Kupaianaha eruption. Eruptive activity at the Pu‘u ‘Ō‘ō vent was typically accompanied by abrupt deflation that lasted for several hours and was followed by a sudden onset of gradual inflation once the eruptive episode had ended. Similar patterns of deflation and inflation were recorded at K̄lauea's summit, approximately 15 km to the northwest, albeit with time delays of hours. These delay times can be reproduced by modeling the spatiotemporal changes in magma pressure and flow rate within an elastic-walled dike that traverses K̄lauea's ERZ. Key parameters that affect the behavior of the magma-dike system are the dike dimensions, the elasticity of the wall rock, the magma viscosity, and to a lesser degree the magnitude and duration of the pressure variations themselves. Combinations of these parameters define a transport efficiency and a pressure diffusivity, which vary somewhat from episode to episode, resulting in variations in delay times. The observed variations in transport efficiency are most easily explained by small, localized changes to the geometry of the magma pathway.

1. Introduction

[2] Magma movements in reservoirs and pathways beneath active volcanoes cause changes in the pressure of those systems. On eruptive time scales, the surrounding rock deforms elastically in response to such pressure variations, which are thus transmitted to the Earth's surface, where they can be recorded as ground deformation and seismic signals [Dzurisin, 2000, 2003; Segall, 2010].

[3] The interaction of the ambient stress field with dike propagation has been extensively studied, and much effort has been directed toward understanding dike ascent and propagation mechanisms [e.g., Weertman, 1971; Spence and Turcotte, 1985; Rubin, 1993; McLeod and Tait, 1999; Segall et al., 2001; Jellinek and DePaolo, 2003; Rivalta and Segall, 2008; Karlstrom et al., 2009]. Models of dike propagation, which couple the flow dynamics to the elastic properties of the crust, indicate that the surrounding wall rock exerts a normal stress on the dike. This stress is proportional to the dike opening, provided that the dike itself is elongated in one direction compared to the other two [Lister and Kerr, 1991; Pinel and Jaupart, 2000; Bokhove et al., 2005; Woods et al., 2006]. Attention has also been directed toward the study of subhorizontal, long-lived dikes [Wilson and Head, 1988; Mériaux and Jaupart, 1995; Pinel and Jaupart, 2004; Costa et al., 2007], which may be part of the magmatic systems of basaltic volcanoes with long rift zones such as K̄lauea [Parfitt and Wilson, 1994; Segall et al., 2001; Montgomery-Brown et al., 2010], Krafla [Einarsson and Brandsdottir, 1980; Tryggvason, 1986; Buck et al., 2006], or Afar [Grandin et al., 2012].

[4] Building upon the work of Pinel and Jaupart [2000] and Bokhove et al. [2005], we study how variations in magma pressure and flow rate, caused by changes in magma withdrawal or supply, are transmitted in space and time through a horizontal dike surrounded by linearly elastic wall rock. We evaluate how the propagation rate and attenuation of such changes in pressure and flow rate depend on the physical characteristics of the system (rock elasticity, magma viscosity, dike dimensions), as well as on the pressure changes themselves (magnitude, duration).

[5] We apply this elastic dike model to the early episodes (1983–1985) of the Pu‘u ‘Ō‘ō-Kupaianaha eruption at K̄lauea Volcano, Hawai‘i. Although the model is a geometrically idealized representation of the complex magmatic pathway that traverses K̄lauea's east rift zone (ERZ), it reproduces the fundamental patterns of deflation and inflation at Pu‘u ‘Ō‘ō and at K̄lauea's summit. Tilt measurements show that the onset of eruptive episodes at the Pu‘u ‘Ō‘ō vent [Heliker et al., 2003] was typically accompanied by abrupt deflation and followed by sudden onset of gradual inflation during repose periods [Wolfe et al., 1987]. Approximately 15 km to the northwest, K̄lauea's summit underwent similar patterns of deflation and inflation, albeit with a time delay of several hours [Dvorak and Okamura, 1985; Wolfe et al., 1987].

[6] In section 'Elastic Dike Model', we introduce the governing equation for the dike model and perform a parametric analysis, wherein we illustrate the parameter dependence of the rate at which pressure perturbations travel along the length of the dike. In section 'Early Episodes of the Pu‘u ‘Ō‘ō-Kupaianaha Eruption', selected episodes from the early stage of the Pu‘u ‘Ō‘ō-Kupaianaha eruption are presented, and in section 'Delay Times', delay times are introduced. In sections 'Modeling Episode 18' and 'Model Results', the model is applied to episode 18 of the Pu‘u ‘Ō‘ō-Kupaianaha eruption, and in section 'Sensitivity to Model Parameters', we show how model parameters affect the observed variability in delay times.

2. Elastic Dike Model

2.1. Governing Equations

[7] We investigate horizontal magma flow through a dike of variable width, b. The dike is assumed to have an elliptical cross section and b is much smaller than the constant horizontal length H, as well as the vertically oriented major axis, a. The problem can be treated as one-dimensional in the horizontal direction x. Consequently, all quantities vary only along x, including the width-averaged horizontal velocity of magma flow, u [Pinel and Jaupart, 2000; Bokhove et al., 2005; Woods et al., 2006].

[8] In addition, we assume that the magma viscosity, math formula, is constant and Newtonian, and that the flow is isothermal [Pinel and Jaupart, 2000; Bokhove et al., 2005; Woods et al., 2006; Costa et al., 2007]. We also neglect magma compressibility (Appendix Compressibility) and thus assume a constant magma density, math formula.

[9] The mass and momentum conservation equations that describe this idealized system are [Bokhove et al., 2005]

display math(1)
display math(2)

where p is magma pressure (see Table 1 for definitions of variables and parameters). The last term of equation (2) represents viscous friction, evaluated for an incompressible, steady flow. The coefficient math formula is appropriate in the approximation of flow between two parallel plates, and its derivation is provided in Appendix Viscous Friction Term. For smaller aspect ratios, a/b, other values of math formula apply [Pinel and Jaupart, 2000].

Table 1. Definition of Symbols (hat indicates a dimensionless quantity).
adike height
B0characteristic dike width
bdike width
b0initial dike width
Gshear modulus of host rock
Hdike length
L0characteristic dike length
Pperturbation amplitude
pfluid pressure
p0initial fluid pressure
math formulapressure in the Halema‘uma‘u reservoir
math formulaeruptive flow rate
math formulamagma flow rate from the dike into the Pu‘u ‘Ō‘ō reservoir
qmagma flow rate in the dike
T0characteristic time scale
math formulaeruption duration
math formulaperturbation duration
udike width averaged flow velocity
math formulavolume of the Halema‘uma‘u reservoir
xhorizontal coordinate
math formulaequation parameter
math formulacompressibility of the elastic dike
math formulamagma compressibility
math formulafriction coefficient
math formuladike width variation
math formulaamplitude of magma pressure variation
math formulavolume variation
math formuladelay time
math formuladelay time for the onset of deflation
math formuladelay time for the onset of reinflation
math formulaamplitude of tilt variation
math formulapressure diffusivity
math formulacompressibility ratio
math formulaelasticity of the host rock
math formulamagma viscosity
math formulaPoisson's ratio for host rock
math formulamagma density
math formulastart of fountaining episode
math formulaend of fountaining episode
math formulatransport efficiency
math formulatilt
math formulasmoothed tilt
math formulawavelength of pressure pertubation

[11] The system of equations (1) and (2) is closed by relating the dike width to the magma pressure at the dike walls. For an elliptical dike cross section and a linearly elastic surrounding material [Muskhelishvili, 1963; Pinel and Jaupart, 2000]

display math(3)

where math formula is the Poisson's ratio for the host rock and G is the shear modulus. If the aspect ratio a/b is large, this relation reduces to the dike width b being proportional to p, that is

display math(4)

[12] Here, math formula represents the elasticity of the host medium. Equation (3) holds strictly only for pressure variations with wavelengths that are greater than the smaller between dike height and length [Dunham and Ogden, 2012], and far from the tip of the dike. Otherwise, the coupling between p and b becomes nonlinear and nonlocal [Lister, 1990]. Local coupling as assumed here represents a first-order approximation, valid only for long-wavelength perturbations.

[13] The magma flow rate through K̄lauea's ERZ is small enough for the inertial term in equation (2) to be negligible (Appendix Inertial Forces). The velocity is thus given by [Bokhove et al., 2005]

display math(5)

[14] Upon substitution into equation (1), this yields

display math(6)


display math(7)

[15] Equation (6) is a nonlinear diffusion equation that governs the spatiotemporal change in pressure within the elastic-walled dike.

2.2. Pressure Diffusivity and Transport Efficiency

[16] The key parameter that determines the rate of propagation for pressure changes, as they travel along the dike, is pressure diffusivity, which can be defined as

display math(8)

[17] A second parameter of importance for magma transport is the proportionality constant between magma flow rate and pressure gradient. It represents a transport efficiency and can be defined as

display math(9)

[18] Both math formula and math formula are strongly dependent on variations of the dike width b.

2.3. Dimensionless Form

[19] Equation (6) can be nondimensionalized by means of a characteristic dike width, B0, dike length, L0, and magma travel time, T0, resulting in

display math(10)

[20] Here, math formula, and math formula.

[21] To evaluate how a well established and persistently open dike responds to relatively small variations of magma pressure, we rewrite equation (4) by separating the initial dike width b0 and pressure p0 from their variations math formula and math formula, resulting in

display math(11)

[22] The importance of the dike width, b0, prior to an eruptive pressure change can thus be separated from the effects caused by local variations in dike width caused by the propagation of pressure changes, math formula. We will solve equation (10) for different combinations of b0 and math formula in order to highlight their different contributions.

2.4. General Solutions and Parameter Dependencies

2.4.1. Boundary Conditions

[23] To examine the response of the dike model to the aforementioned changes in pressure, we use an idealized model configuration. We impose a sinusoidal pressure change of amplitude P and period math formula as a time-dependent boundary condition at math formula

display math(12)

[24] At the opposite end of the dike, math formula, we use a Neumann boundary condition

display math(13)

which allows p to vary with time, in response to the propagation of the imposed pressure change within the dike. This model configuration isolates the response of the dike to an imposed pressure change at one end.

2.4.2. Delay Time

[25] A key observation is the time delay in deflationary or inflationary tilt changes between Pu‘u ‘Ō‘ō and K̄lauea's summit. For the purpose of this general analysis, we define the delay time as the time it takes for the minimum of the pressure variation to reach the other side of the dike. In the subsequent sections where we apply this model to tilt data from individual eruptive episodes, the definition of delay time will be somewhat different.

2.4.3. Parameter Dependencies

[26] Equation (10) is solved numerically using the pdepe partial differential equation solver of MATLAB®. The solver used is ode15s, which is an implicit solver, suitable for stiff problems where the time step and the integration formula adapt dynamically, to deal with rapid changes in the solution [Shampine and Reichelt, 1997]. The accuracy in space is second order [Skeel and Berzins, 1990].

[27] Two illustrative examples for the spatiotemporal propagation of a pressure perturbation are shown in Figure 1. All else being the same, the initial sinusoidal pressure change becomes more attenuated in amplitude and propagates at a lower speed with decreasing initial dike width math formula. This is a consequence of both math formula and math formula (equations (8) and (9)) being smaller.

Figure 1.

Propagation of an idealized sinusoidal change in pressure using equation (10). Boundary condition at math formula is math formula. Here, math formula is dimensionless pressure, math formula dimensionless time, and math formula dimensionless distance along the length of the dike. In both cases, the dimensionless duration of the pressure change at the boundary math formula has a value of 1 and dimensionless magnitude of pressure change math formula. (a) The dimensionless unperturbed dike width is math formula and (b) math formula. Note the slower propagation speed and larger attenuation in the case of smaller math formula.

[28] We show in Figure 2 how the dimensionless delay time math formula changes with respect to math formula, and the dimensionless duration of the pressure perturbation, math formula. Because math formula scales as b3 and math formula as b2, the value of math formula most strongly affects the rate at which the pressure variation travels through the dike, with a decrease (increase) in math formula resulting in longer (shorter) delay times. math formula is less affected by the amplitude of the perturbation, but does increase with increasing math formula. For values of math formula, the delay time also increases with decreasing math formula.

Figure 2.

Parameter dependence of dimensionless delay time, math formula, the time for the minimum of the pressure variation to reach the other side of the dike. (a) Delay time versus dimensionless initial dike width, math formula. (b) Delay time versus dimensionless change in pressure math formula, where math formula denotes the end of the dike where the perturbation is imposed as boundary condition. (c) Delay time versus dimensionless duration of the change in pressure, math formula.

3. Application of the Elastic Dike Model to K̄lauea's ERZ

3.1. Early Episodes of the Pu‘u ‘Ō‘ō-Kupaianaha Eruption

[29] Although the elastic dike model is an idealization of a more complex east-rift zone plumbing system, we apply it to the dike through which magma is thought to have flowed from K̄lauea's summit to the Pu‘u ‘Ō‘ō eruptive vent during the early high fountaining episodes (1983–1985) of the Pu‘u ‘Ō‘ō-Kupaianaha eruption (Figure 3) [Dvorak and Okamura, 1987; Wolfe et al., 1987; Okamura et al., 1988; Parfitt and Wilson, 1994; Heliker et al., 2003].

Figure 3.

Schematic diagram of the ERZ magma pathway connecting K̄lauea's summit and the Pu‘u ‘Ō‘ō vent.

[30] Before 1983, magma had been accumulating in the ERZ with successive dike intrusions since 1956 [Jackson, 1988; Wilson and Head, 1988; Cayol et al., 2000]. The intrusion on 3 January 1983 [Okamura et al., 1988; Heliker and Mattox, 2003] initiated the ongoing ERZ eruption. Dimensional estimates of the dike, which hydraulically connected K̄lauea's summit with a shallow magma body beneath the Pu‘u ‘Ō‘ō vent [Dvorak and Okamura, 1987], fall in the range 2–3.5 m for width and 2.4–4.4 km for hight [Wolfe et al., 1987; Okamura et al., 1988; Parfitt and Wilson, 1994; Heliker et al., 2003]. Seismicity indicated that this dike had propagated more or less horizontally [Wolfe et al., 1987], thus justifying the idealized geometry of a horizontal dike for our model.

[31] The model objective is to constrain the characteristic transport properties of the ERZ magmatic system by using equation (6) to model the observed changes in tilt, at both Pu‘u ‘Ō‘ō and K̄lauea's summit, in response to individual eruptive episodes. In particular, we focus on the difference in time delays between the onset of deflation at Pu‘u ‘Ō‘ō and K̄lauea's summit as well as the subsequent onset of reinflation.

[32] The eruptive episodes and associated deflation at Pu‘u ‘Ō‘ō typically lasted less than 24 h [Heliker and Mattox, 2003]. They were characterized by average eruption rates of about 106 m3 h math formula and by lava fountaining with heights of up to 500 m. During repose periods, averaging approximately 24 days, tilt measurements at Pu‘u ‘Ō‘ō recorded a gradual reinflation beginning at the end of the eruption.

[33] Deflation and inflation at Pu‘u ‘Ō‘ō were a consequence of magma withdrawal and accumulation, respectively, within a magma body that may have been a remnant of the first intruded dike [Wilson and Head, 1988; Hoffmann et al., 1990; Garcia et al., 1992; Mittelstaedt and Garcia, 2007]. The precise geometry of this magma body remains uncertain in detail [Heliker et al., 2003].

[34] The pattern of deflation and inflation at Pu‘u ‘Ō‘ō was typically repeated at K̄lauea's summit, albeit with time delays of few hours to approximately half a day [Wolfe et al., 1987], as shown in Figure 4. At K̄lauea's summit, changes in the volume of stored magma produced pressure changes within a shallow reservoir of math formula in volume [Johnson, 1992; Ohminato et al., 1998; Poland et al., 2009, 2012; Gonnermann et al., 2012] and located beneath the southeastern edge of Halemau‘ma‘u crater [Almendros et al., 2002; Cervelli and Miklius, 2003]. It is referred to as the Halemau‘ma‘u reservoir. These pressure decreases resulted in deflationary summit tilt changes and were presumably a consequence of increased magma flow rates into the ERZ, caused by the eruptive magma withdrawal at Pu‘u ‘Ō‘ō. Subsequent refilling of the coupled K̄lauea-ERZ-Pu‘u ‘Ō‘ō magmatic system during repose periods resulted in a gradual increase in pressure and in inflation.

Figure 4.

(a) Tilt data at K̄lauea's summit (blue) and at Pu‘u ‘Ō‘ō (red) for selected episodes, normalized to maximum tilt amplitude. Gray bars indicate eruptive periods. (b) Eruption rate and duration. Bar heights represent erupted volumes and widths represent eruption durations, also indicated by numbers. (c) Reinflation delay time, math formula, measured from the end of eruption to the onset of reinflation at the summit. (d) Modeled initial dike width, b0, required to match the observed math formula.

3.2. Delay Times

[35] We hypothesize that the differences in time for the onset of deflation and reinflation between Pu‘u ‘Ō‘ō and K̄lauea's summit were a consequence of the time required for changes in magma pressure to be transmitted through the elastically deforming dike, which hydraulically connected the shallow magma body beneath Pu‘u ‘Ō‘ō with K̄lauea's summit [Thurber, 1984; Dvorak and Okamura, 1987; Johnson, 1992; Barker et al., 2003; Poland et al., 2013, Figure 3]. Figure 4 shows those episodes for which tilt data at both Pu‘u ‘Ō‘ō and K̄lauea's summit have a distinct deflation-inflation pattern. Delay times for the onset of reinflation, math formula, were typically easier to quantify than deflation delay times, math formula, and were therefore chosen as the key observational constraint for our analysis.

3.3. Modeling Episode 18

[36] To illustrate the utility of the elastic dike model for characterizing the transport properties of the ERZ, we present detailed model results for episode 18 of the Pu‘u ‘Ō‘ō-Kupaianaha eruption. Because the deflationary patterns are similar among all the episodes shown in Figure 4, we show in section 'Sensitivity to Model Parameters' how the range in observed delay times from different episodes can be modeled by varying model parameters.

3.3.1. Boundary Conditions

[37] Because of the aforementioned uncertainties in geometry, we refrain from modeling the Pu‘u ‘Ō‘ō magma storage reservoir explicitly. Instead, we use existing estimates of pressure variations during the early Pu‘u ‘Ō‘ō eruptive episodes [Hoffmann et al., 1990] as scaling factors to obtain a relationship between the observed tilt changes and the time-varying pressure changes at the Pu‘u ‘Ō‘ō end of the dike. Consistent with conventional linearly elastic kinematic deformation models [Mogi, 1958], we assume that the total pressure variation during the eruptive episode, math formula, is proportional to the total tilt variation, math formula, and to the total change in stored volume of magma, math formula. This proportionality relation is valid only if, as we assume, the magma within the system is incompressible. Changes in tilt, math formula, are caused by volume changes in the reservoir, so that math formula, where math formula is the time-dependent flow out of the dike and math formula is the eruptive flow rate. We use the dimensionless tilt, defined as math formula [Dvorak and Okamura, 1985], to obtain the change in pressure with respect to time at the Pu‘u ‘Ō‘ō end of the dike

display math(14)

[38] This normalization of math formula avoids some of the difficulties associated with obtaining reliable absolute values for ERZ tilt amplitudes for the time period between 1983 and 1985 (A. Miklius, pers. comm.).

[39] To model episode 18 we use a value of math formula MPa, which represents the independently estimated average value of deflationary pressure change during the early Pu‘u ‘Ō‘ō eruptive episodes [Hoffmann et al., 1990]. It should be noted that the estimates of math formula do not correlate with other parameters, such as the length of the eruptive episode [Hoffmann et al., 1990]. Furthermore, precise values of math formula do not significantly affect our results, because the delay time is relatively insensitive to the magnitude of pressure variations (section 'Sensitivity to Model Parameters').

[40] The pressure boundary condition at the Pu‘u ‘Ō‘ō end of the dike, that is x = 0, is thus given as

display math(15)

where math formula is a linear fit to the normalized tilt, math formula, at math formula. For the time period math formula we use a smooth cubic spline fit to the dimensionless tilt, constrained to match the linearly fitted tilt at math formula and math formula (Figure 5). At the onset of deflation at Pu‘u ‘Ō‘ō, that is math formula, we have math formula and at the onset of reinflation, that is math formula, the value is math formula. Different methods of processing the raw tilt data to remove noise and diurnal variations, such as sinusoidal corrections, do not provide any improvement over the fit employed here.

Figure 5.

Observed and predicted normalized tilt at Pu‘u ‘Ō‘ō and at K̄lauea's summit for episode 18 (shaded area indicates the eruptive period). Red filled circles are normalized tilt at Pu‘u ‘Ō‘ō and the red line is the smoothed fit for the model boundary condition at Pu‘u ‘Ō‘ō (equation (15)). The blue line is the normalized tilt at K̄lauea's summit, whereas the cyan line is the normalized tilt at the summit predicted by the model.

[41] At K̄lauea's summit, which corresponds to x = 15 km, we impose a Dirichlet boundary condition. The specified value of math formula is based on the pressure within the Halema‘uma‘u reservoir, pH, which is calculated from its change in volume due to magma flow into the dike. The relationship between this change in volume and the change in pressure is based on the assumption that deformation is the consequence of volume and pressure changes of incompressible magma within an uniformly pressurized spherical reservoir embedded in a homogeneous rock medium [Mogi, 1958]. The resultant relationship between pressure and volume change is

display math(16)

where math formula km3 is the assumed volume of the Halema‘uma‘u reservoir (section 'Early Episodes of the Pu‘u ‘Ō‘ō-Kupaianaha Eruption').

3.4. Model Results

[42] The model is calibrated to match the observed tilt at K̄lauea's summit by adjusting b0, a, and G. These variations correspond to specific changes in the transport properties of the ERZ, as discussed later. A best fit is obtained by grid search, minimizing the difference between predicted and observed onset of reinflation at K̄lauea's summit. The observed and modeled normalized tilts for episode 18 are shown in Figure 5. In addition to matching the overall shape of the observed tilt curve, the model reproduces both delay times math formula h and math formula h. Table 2 lists the model parameters used to obtain the best fit to the data for episode 18. The predicted time-delayed pressure change at K̄lauea's summit is of similar magnitude as that imposed at Pu‘u ‘Ō‘ō (Figure 6a) and would result in a few tens of math formula deflationary tilt change at Uwēkahuna tilt station, consistent with the magnitude of the east-west component of tilt variations recorded during the eruptive episode [Wolfe et al., 1987; Parfitt and Wilson, 1994].

Table 2. Values of Parameters for the Best Fit to Episode 18 Tilt Data
dike height a math formula
magma viscosity math formula math formula
friction coefficient math formula12
shear modulus of host rock G math formula
Poisson's ratio for host rock math formula0.25
elasticity of the host rock math formula math formula
initial dike width b0 math formula
amplitude of pressure variation math formula math formula
delay time for the onset of deflation math formula3.5 h
delay time for the onset of reinflation math formula11.5 h
eruption duration math formula2.5 days
Figure 6.

(a) Modeled change in magma pressure within the dike during episode 18 as a function of horizontal position between Pu‘u ‘Ō‘ō and at K̄lauea's summit and time. (b) Corresponding magma flow rate.

[43] Figure 6a shows predicted pressure variations within the dike for episode 18 as a function of time and position between Pu‘u ‘Ō‘ō and K̄lauea's summit, whereas Figure 6b shows the corresponding predicted magma flow rate math formula. Magma withdrawal during the eruptive episode causes a decrease in magma pressure at Pu‘u ‘Ō‘ō. The resultant increase in pressure gradient, math formula, causes an increase in magma flow rate, q. Mass balance requires that both the pressure change and “surge” in magma flow propagate along the ERZ from Pu‘u ‘Ō‘ō to K̄lauea's summit. The time required to reach the summit results in a delay in the onset of deflation at the summit, math formula, relative to Pu‘u ‘Ō‘ō. Given the estimated size of the Halema‘uma‘u reservoir, mass balance calculations indicate that during the eruptive episode, only a modest volume of magma, of the order of 105 m3, was withdrawn from the Halema‘uma‘u reservoir. The majority of the erupted volume was derived from within the deflating dike, combined with magma stored within the Pu‘u ‘Ō‘ō reservoir.

[44] Once there is no more magma withdrawal at Pu‘u ‘Ō‘ō, tilt changes reverse from rapid deflationary to slow inflationary, because of gradual magma flow from K̄lauea's summit into the dike and shallow Pu‘u ‘Ō‘ō reservoir. This is represented in the model by the gradual increase in specified pressure at the Pu‘u ‘Ō‘ō end of the dike. The change in math formula from deflation to inflation travels uprift from Pu‘u ‘Ō‘ō to K̄lauea's summit and arrives with a slightly longer delay, math formula, than the deflation delay. This increase in delay time is a consequence of the elastic decrease in dike width (a function of magma pressure), which results in more viscous resistance to magma flow and, hence, a decrease in the pressure diffusivity of equation (6).

3.5. Sensitivity to Model Parameters

[45] Although we show the modeled pressure and tilt changes for episode 18 only, the model was applied to other episodes as well. To illustrate the changes in model parameters that are required to match the range in observed delay times, we provide a sensitivity analysis of model predictions to parameters. This analysis not only shows the trade-offs between parameters, but it also demonstrates how relatively subtle variations thereof can result in changes in delay times that span the whole range of observed values.

[46] Relevant parameters are math formula (magnitude of deflation), math formula (duration of the eruption), math formula (elasticity of the dike), and b0 (initial dike width), whereas math formula (magma viscosity) is relatively well constrained and not expected to change significantly. b0 is of key importance, because it provides a quantitative measure of the pressure diffusivity, math formula, and the characteristic transport efficiency, math formula, of the ERZ magma pathway during the early eruptive episodes. Changes in delay times are most strongly affected by changes in math formula (Figure 7), which we speculate to be caused by changes in b0, perhaps due to thermal erosion of dike walls or to freezing of magma against them during repose periods [Costa et al., 2007]. Because it is unlikely that the dike-like ERZ magma pathway was of uniform geometry, as is our idealized model, the predicted change in math formula should be viewed as representative of either an average change throughout the ERZ or of a spatially localized change, perhaps due to opening or removal of a constriction or wall-rock collapse [Parfitt and Wilson, 1994].

Figure 7.

Parameter sensitivity of reinflation delay times, math formula. (a) math formula as a function of dike width, b0, and magnitude of pressure change, math formula. (b) math formula as a function of dike width, b0, and eruption duration, math formula. (c) math formula as a function of dike width, b0, and rock elasticity math formula. The observed values of math formula range between 1 and 12 h. All parameters not shown are kept constant at the values used for episode 18: math formula m, math formula MPa, math formula days, math formula km, math formula GPa, and math formula Pa s. Note the predominant model sensitivity to the transport capacity, math formula.

[47] Figure 7 also demonstrates that changes or uncertainty in either TE or math formula will not significantly affect predicted delay times, but that model results are sensitive to math formula. We used math formula m Pa math formula. Given an assumed dike height of 3 km [e.g., Wolfe et al., 1987; Koyanagi et al., 1988; Okamura et al., 1988; Parfitt and Wilson, 1994; Heliker et al., 2003] and math formula, this would correspond to a shear modulus of 3 GPa. Although the effective shear modulus of K̄lauea's summit or ERZ are difficult to constrain [Johnson, 1987], K̄lauea is predominantly built of lava flows with abundant void space [Ryan et al., 1983], and its effective shear modulus is expected to be lower than the intrinsic shear moduli of laboratory-scale basalt samples [Johnson, 1987], which range between a few and several tens of gigapascals, depending on porosity [Manghnani and Woollard, 1968]. Because it is unlikely that the effective value of math formula will change significantly between individual episodes, the observed changes in delay time are best explained by changes in b0.

[48] Observed delay times during the early Pu‘u ‘Ō‘ō eruptive episodes range between 3 and 12 h (Figure 4). Figure 7 indicates that these delay times can be modeled for dike widths between math formula and math formula, a variation of less than 20%. Delay time also increases somewhat with the duration of the eruption (Figure 7). Although there is no unequivocal correlation between episode duration and delay time for the episodes shown in Figure 4, it should be noted that episode 18 has the longest delay time as well as the longest duration.

[49] The elastic dike model does not account for magma compressibility, which is expected to increase the compliance of the system [Rivalta and Segall, 2008]. Using an incompressible approximation, delay times become very short as math formula decreases (Figure 7). This is a consequence of the large increase in stiffness of the combined magma-wall rock system. For gas-poor magmas, the dominant effect on the overall compliance of the system is the elasticity of the wall rock (Appendix Compressibility).

[50] The volume fraction of exsolved magmatic gases within the ERZ is poorly constrained. Although the majority of exsolved magmatic volatiles, predominantly CO2, is thought to escape to the surface before entering the ERZ [e.g., Poland et al., 2012, and references therein], some volatile bubbles may be transported into the ERZ. Of these, only bubbles smaller than ∼1–10 mm in diameter will rise buoyantly at significantly smaller velocities than the average horizontal flow velocity of magma within the dike and, thus, get transported for significant distances within the ERZ. Changes in magma pressure within the dike are unlikely to result in sufficient volatile exsolution to significantly affect magma compressibility (Appendix A). Therefore, although the ERZ magma may contain some fraction of exsolved volatiles, it is unclear if there was a sufficient volume of exsolved volatiles to significantly affect our overall model results. However, explicit inclusion of magmatic volatiles and magma compressibility are desirable model enhancements for future work along these lines.

4. Conclusions

[51] We have shown that a one-dimensional model of magma flow in an elastic-walled dike can reproduce spatiotemporal patterns of tilt at K̄lauea Volcano during the early eruptive episodes of the ongoing Pu‘u ‘Ō‘ō-Kupaianaha eruption. Simplifying assumptions of the model are an idealized uniform dike geometry and an incompressible magma (Appendix Compressibility). Furthermore, equation (4) is strictly valid only for long-wavelength pressure pertubations and far from the dike tip [Dunham and Ogden, 2012]. Although future work along these lines should eliminate some of these approximations, our results, despite potential inaccuracies, demonstrate that spatiotemporal deformation patterns at active volcanoes bear important information about magma transport properties of the magmatic plumbing system. In particular, the time delays of deflation and inflation between the Pu‘u ‘Ō‘ō vent and K̄lauea's summit can be explained by the propagation of pressure transients within such an elastically deformable dike.

[52] The delay times depend on the dike dimensions, the elasticity of the wall rock, the magma viscosity, as well as the amplitude and duration of the deflationary phases themselves. The most important of these parameters is the dike width, which is a measure of the characteristic magma transport efficiency. Consequently, delay times and changes thereof can be used to provide constraints on the average magma transport properties of the ERZ.

[53] Since the eruption started in 1983, the ERZ has seen other dike intrusions [Segall et al., 2001; Montgomery-Brown et al., 2010] as well as changes in the eruptive style [Heliker and Mattox, 2003]. However, the pathway connecting the summit to the Pu‘u ‘Ō‘ō vent has remained an effective means for magma transport. Similar to the delay in deflation described herein, K̄lauea's summit is now experiencing episodes of deflation and inflation on the time scales of hours to days [Cervelli and Miklius, 2003], which are also being recorded along the ERZ, albeit with time delays relative to the summit. Although the dike that is thought to have existed during the early eruptive episodes may have evolved into a more pipe-like conduit [Cervelli and Miklius, 2003], the propagation of pressure changes in an elastically deforming conduit may also be a viable explanation for these more recent observations.

[54] Similar records of delayed transmission of pressure variations have been measured elsewhere, for example, during the 1960 Kapoho eruption at K̄lauea [Eaton, 1962], as well as during the 1984 eruption of Krafla Volcano in Iceland [Tryggvason, 1986]. The model presented here could also provide constraints on the transport properties of the magmatic pathways during those eruptions.

Appendix A: Compressibility

[55] Magma supply to K̄lauea's summit is thought to constitute an open system wherein bubbles of exsolved magmatic volatiles decouple from the melt phase and rise buoyantly to the surface [e.g., Poland et al., 2012, and references therein]. Consequently, it is not clear what fraction of the already exsolved volatiles will actually enter the ERZ [e.g., Gerlach, 1986]. Magma is thought to flow into the ERZ at a few kilometers depth [Cervelli and Miklius, 2003; Montgomery-Brown et al., 2010; Poland et al., 2013] where, assuming that the melt is equilibrated with CO2 rich bubbles derived from deeper within the system, the dissolved CO2 content of the magma will be a few hundred parts per million, and the H2O and S content will be in the thousands parts per million. Most of this H2O and S will be lost upon eruption at the Pu‘u ‘Ō‘ō vent and is thought to play a dominant role in the upper few hundred meters of eruptive ascent during sustained fountaining episodes [Wilson et al., 1980; Head and Wilson, 1987, 1989]. Here we provide a brief analysis of the effect of volatile exsolution due to expected pressure changes associated with the early eruptive episodes at Pu‘u ‘Ō‘ō as well as of the role of magma compressibility due to the presence of exsolved volatiles.

[56] Figure A1 shows the expected fractional change in volume of exsolved volatiles at a given pressure for a decrease in pressure of 1 MPa. The calculations account for the combined exsolution of CO2, H2O, and S and are based on the solubility models of Gerlach [1986] and Dixon [1997], together with ideal gas behavior. Local changes in pressure, as modeled within the dike, are math formula MPa and will result in approximately a 0.1% change in the volume of exsolved volatiles.

Figure A1.

Fractional change in volume of exsolved volatiles per 1 MPa change in pressure over a range of pressures corresponding to the approximate depth range of the ERZ dike that is approximately 1–4 km depth. The upper curve is for water-saturated conditions, assumed to be 7000 ppm, whereas the lower curve is for water concentrations in equilibrium with a CO2-rich volatile phase with a CO2 mole fraction of 0.99.

[57] Dunham and Ogden [2012] define the compressibility ratio as the ratio of the compressibility of the elastic dike, math formula, to the compressibility of the magma, math formula as

display math(A1)

[58] Here math formula and math formula is the wavelength of the pressure pertubation. For values of math formula, magma compressibility is negligible. Figure A21 indicates that this is the case for pertubation wavelengths greater than 1000 m. As shown in Figure A2, the pressure pertubation extends along the entire length of the dike (15 km), hence, the approximation of an incompressible magma is not unreasonable.

Figure A2.

math formula, the ratio of the magma compressibility, math formula, to the elastic compressibility of the dike [Dunham and Ogden, 2012], scaled by the wavelength of the pressure pertubation math formula, for different values of the volume fraction of exsolved volatiles, where math formula. Curves are for pressures of 25, 50, 75, and 100 MPa, roughly spanning the range in depths from the top to the bottom of the ERZ dike. Calculations are based on a CO2-rich gas phase and the equation of state of Holloway [1977] with modifications by Flowers [1979]. For wavelengths math formula m, the value of math formula, indicating that magma compressibility is negligible.

[59] In the long-wavelength approximation, Woods et al. [2006] provide a formulation for compressible magma flow within an elastic-walled dike, which indicates that magma compressibility cannot be neglected if math formula. Figure A31 shows the ratio of math formula to math formula as a function of the volume fraction of exsolved volatiles, assuming ideal gas behavior and math formula equal to the value used for the Pu‘u ‘Ō‘ō model (Table 2). It can be seen that for a wide range of conditions the effect of magma compressibility is small, because of the compliance of the large-aspect ratio dike.

Figure A3.

Ratio of math formula to math formula as a function of the volume fraction of exsolved volatiles. Curves are for pressures of 25, 50, 75, and 100 MPa, roughly spanning the range in depths from the top to the bottom of the ERZ dike. Calculations are based on a CO2-rich gas phase and the equation of state of Holloway [1977] with modifications by Flowers [1979]. For a wide range of exsolved gas content math formula and based on the formulation of Woods et al. [2006], the effect of magma compressibility should be negligible.

[60] Based on Figures A1-A3, it can be concluded that our assumption of negligible magma compressibility is first order justified.

Appendix B: Viscous Friction Term

[61] The friction term that appears in the momentum equation (equation (2)) can be derived from the unidirectional steady flow between two parallel plates that extend indefinitely along the x and z directions. The momentum equations for flow in the x direction are [White, 1991]

display math(B1)
display math(B2)

[62] Here vx is the velocity along the x direction, varying along the y direction, perpendicular to the plates, whereas math formula. From equation (B2) we find that math formula, so that equation (B1) can be integrated to give

display math(B3)

[63] The boundaries are at math formula and math formula, where math formula, which gives math formula and math formula. Thus, the velocity has a parabolic profile given by

display math(B4)

[64] The mean velocity is math formula, and the pressure drop due to viscous friction is

display math(B5)

Appendix C: Inertial Forces

[65] The scaling used to nondimensionalize equation (6) can be used to evaluate the relative importance of the individual terms appearing in the momentum conservation equation (equation (2)). The assumption of a viscously dominated flow is justified if viscous forces are much larger than inertial forces. The first term of the right-hand side of equation (2) is the fluid pressure gradient

display math(C1)

[66] The second right-hand side term in equation (2) is the viscous force, which can be written as

display math(C2)

for a characteristic flow velocity of math formula [Bokhove et al., 2005].

[67] Both A and B are of the same order. The left-hand side of equation (2), denoted as C, comprises fluid acceleration and inertial forces and is given by

display math(C3)

[68] The ratio between inertial and dissipative forces is

display math(C4)

[69] Reasonable values of parameters that are applicable to K̄lauea are math formula m2 s2 kg math formula m math formula s math formula km, and math formula m. The resultant force balance results in a ratio math formula, which justifies our assumption of neglecting the inertial terms in equation (2).


[70] The authors thank M. Poland and A. Miklius as well as two anonymous reviewers for valuable suggestions and comments. The authors also thank A. Miklius and the Hawaiian Volcano Observatory staff for providing the tilt data used in the analysis. H.M.G. and C.P.M. were in part supported by NSF grant EAR 1145187.