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1 Available online at Earth and Planetary Science Letters 262 (2007) Transient multiphase processes during the explosive eruption of basalt through a geothermal borehole (Námafjall, Iceland, 1977) and implications for natural volcanic flows S. Dartevelle, G.A. Valentine 1 Los Alamos National Laboratory, EES Division, NM 87545, USA Received 22 February 2007; received in revised form 10 July 2007; accepted 13 July 2007 Available online 7 August 2007 Editor: C.P. Jaupart Abstract Experimental and numerical studies have shown that vertical flows of gas particle mixtures are characterized by transient behavior, with development of waves of high particle concentration separated by regions of relatively clean gas. In contrast, most models of explosive flow in volcanic conduits either treat the multiphase mixture as a single fluid (pseudo-fluid approximation) and/or assume steady flow, thereby eliminating the potential for time-dependent effects related to multiphase dynamics. The 8 September 1977 explosive eruption of basaltic tephra through a geothermal borehole (Námafjall, Iceland) provides a unique test case for multiphase volcanic processes, given that its vertical extent ( 1 km) is similar to that of natural volcanic conduits and its geometry is exactly known. We model this eruption by solving separate, time-dependent governing equations for conservation of mass, momentum, and energy of the gas and particle phases, allowing for drag and heat transfer between the phases. Model results are consistent with the development of transient waves of high particle concentration that propagate up the borehole, resulting in complex compressible flow phenomena along with ejection of particles in pulses in a manner that is consistent with observations at Námafjall. These transient processes occur even though the influx of gas and particles at the base of the borehole is treated as constant. Our results indicate that transient multiphase behavior is likely to be common in volcanic conduit flows, and that a key topic of future research is quantifying the types of time-dependent behaviors and their impacts on eruption column dynamics. Published by Elsevier B.V. Keywords: multiphase; magma; turbulent; volcanism; shock 1. Introduction Explosive volcanic flows involve non-linear, transient, multidimensional, multiphase dynamics that evolve over a large range of spatial and temporal scales. Experiments Corresponding author. Tel.: ; fax: addresses: sdart@lanl.gov (S. Dartevelle), gav@lanl.gov (G.A. Valentine). 1 Tel.: ; fax: aimed at elucidating the flow of multiphase mixtures up volcanic conduits demonstrate the importance of such transient multiphase dynamics. For example, Anilkumar et al. (1993) showed that passage of a rarefaction wave into a bed of particles resulted in acceleration of particles in waves of relatively high concentration separated by regions of relatively low particle concentration. Alidibirov and Panov (1998) showed that fragmentation and acceleration of viscous magma under rapid decompression occurred as accelerating slabs of material rather than X/$ - see front matter. Published by Elsevier B.V. doi: /j.epsl

2 364 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) by propagation of a smooth fragmentation front. These observations suggest that fragmentation of magma and acceleration of a gas particle mixture up a conduit might result in highly transient behavior. A large body of engineering and fluid mechanics literature supports a view that vertical gas particle flows generally involve non-linear, chaotic, unsteady multiphase dynamics. It is well known in chemical engineering that gas particle or liquid-particle flows in vertical risers are inherently unstable (Homsy, 1998). Such fluidized systems manifest fluctuations over a wide range of length and time scales with a tendency for strong phase segregation, which leads to particle-free regions often referred to as bubbles (Homsy, 1998) and particle-rich regions referred to as clusters of solids (Breault et al., 2005). In narrow fluidized beds and in liquid fluidized systems, these successions of voidage fronts with discontinuous jumps in particle concentration form traveling disturbance bands (El-Kaissy and Homsy, 1976; Didwania and Homsy, 1981). Researchers have sought to capture these fluctuations through numerical simulation of continuum equations for gas particle flows (e.g. Ding and Gidaspow, 1990; Nieuwland et al., 1995; Benyahia et al., 2000; Goldschmidt et al., 2001; Zhang and VanderHeyden, 2001). An even larger experimental literature demonstrates that fluidized bed systems very commonly show the formation of bubbles and clusters which fall, get broken up, and lifted to result in repeating cycles of cluster formation/destruction as the solids progress towards the exit. (e.g. Anderson and Jackson, 1969; El-Kaissy and Homsy, 1976; Didwania and Homsy, 1981; Breault et al., 2005). Several investigators (e.g. Pigford and Baron, 1965; Anderson and Jackson, 1968; El-Kaissy and Homsy, 1976; Didwania and Homsy, 1981) have shown that a small disturbance imposed on a uniformly fluidized bed can grow with time to eventually form particle concentration heterogeneities. Based on linear instability analysis, some of these investigators (e.g. Pigford and Baron, 1965; Anderson and Jackson, 1968) have shown that the formation of voidage heterogeneities arises spontaneously from instabilities in the initial state of uniform fluidization. Anderson and Jackson (1968) hypothesized that these instabilities can be related to compression wave propagating upward through the fluidized bed and amplifying with time, also resulting in particle-free regions. Pigford and Baron (1965) were the first to suggest that the development of these heterogeneities could be interpreted as being analogous to shock-wave structures. This shock-wave analogy was confirmed by Fanucci et al. (1979) who use the method of characteristics applied to a full set of nonlinear two-phase flow equations. They show that a small disturbance changes with time and distance and can, eventually, produce a flow discontinuity similar to a shock wave in gases, i.e., the shock front and the bubble front are mathematically analogous. Most models of explosive volcanic conduit flows assume steady flow and/or treat the multiphase mixture as a single fluid (pseudo-fluid, where gas and dispersed particles are assumed to be perfectly coupled in terms of velocity and heat transfer and the mixture can be treated as a gas but with thermodynamic properties that account for the particles). For example, the well-known models of Wilson et al. (1980), Wilson and Head (1981), and Mastin and Ghiorso (2000) assume steady flow and treat the melt gas mixture as a pseudo-fluid. Macedonio et al. (1994) and Papale (1998) treat melt and gas as separate phases using a multifield approach (a set of mass, momentum, and energy conservation equations for each phase, accounting for the local volume fraction of each phase), but assume steady flow. Melnik and Sparks (2006) relax the assumption of steady flow, but treat the multiphase mixture as a pseudo-fluid. Each of these models are important for elucidating aspects of volcanic conduit flow, but their underlying assumptions eliminate the potential for time-dependent phenomena that are specifically related to multiphase processes. A notable exception is Dufek and Bergantz (2005), who used a twodimensional, time-dependent multiphase approach similar to the one presented in this paper to study the transition to steady flow for explosive eruption of rhyolitic magma. In this paper, we analyze the 1977 eruption of basaltic tephra from a borehole that was intercepted at depth by a laterally propagating dike at the Námafjall geothermal field (Iceland Larsen et al., 1979). This event can be viewed as a unique natural experiment in that the 1km depth of the borehole is similar in scale to the height of a volcanic conduit above a typical magma fragmentation surface, but, unlike the conduits of true volcanic eruptions, the shape of the borehole is known exactly (albeit much narrower than a natural volcanic conduit). Although quantitative measurements of parameters such as pressure, velocity, and particle volume fraction are not available for the event, qualitative observations provide clues to the dynamics of the eruption. We show that the observations of time-dependent, shock-like phenomena can be explained by instabilities in a rising gas particle mixture, even if the source flux of that mixture at the base of the borehole is steady. We begin by summarizing the 1977 borehole eruption, followed by a description of our multiphase modeling approach and results. Our results suggest that steady conduit flow in explosive eruptions might be atypical.

3 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Description of the 1977 borehole eruption During the night of September 8, 1977, a basaltic dike, associated with rifting and a minor volcanic episode at Krafla caldera, intersected a borehole at a depth between 635 and 1038 m (most likely 1000 m) in the Námafjall geothermal field, Iceland. According to Larsen et al. (1979), the eruptive event at the wellhead began with an audible explosion followed by an incandescent column m high (note that due to the night time occurrence, only incandescent flows and objects were observed, and no observations of the lower temperature part of the eruptive column are available). As the column grew in width over a period of about 1 min, sparks and cinders were ejected and a constant roaring sound was heard. This was followed by a period of min during which apparently there was little to no activity at the eruption site ; toward the end of this second phase, red flashes were observed. The final phase consisted of a series of rapid explosions or shots of glowing scoria, wherein explosions were focused in groups over a total of about 1 min. Flow velocities of m/s were estimated by Larsen et al. (1979) based upon observed ejecta heights. The eruption produced a thin scoria fallout sheet that extended 150 m from the wellhead; Larsen et al. (1979) estimated that a total of 1.2 m 3 (dense rock equivalent) of basaltic magma was ejected. Median grain size of the deposits at 20 m from the wellhead was 18.4 mm. Clasts were reported as a highly vesicular glass froth with various types of quenched surface crusts. Although not quantified, it seems likely that geothermal waters mixed to some degree with the basaltic melt, generating abundant steam bubbles and quenching some clast surfaces as the mixture fragmented. The borehole itself suffered little damage and resumed steam production at a similar rate as before the eruption, suggesting that its geometry was not changed. 3. Modeling approach The 1977 borehole eruption was a multiphase flow of gas containing dispersed particles. We model this flow using a multifield approach that is similar in its foundations to other time-dependent, multiphase eruption models that have been developed during the past two decades (see reviews in Papale, 1998; Valentine, 1998; Neri et al., 2003; Dartevelle, 2004, 2005). Below, we present the governing equations, while the turbulence closure and constitutive models are provided in Appendices A and B, respectively. Appendices C and D further define operators, deviators, tensors, symbols, constants, and acronyms used in this manuscript. The numerical code that we use to solve the equations is referred to as GMFIX (Geophysical Multiphase Flow with Interphase exchange, v. 1.62), which is described in Section 3.2, followed by description of the initial and boundary conditions used in the calculations (Section 3.3) Multifield governing equations The multifield approach treats each phase (in our case, water vapor and dispersed particles) as a fluid field; the two fields interpenetrate, occupying the same control volume as volume fractions. Each instantaneous local point variable (e.g. mass, velocity, temperature, pressure) must be treated in a manner consistent with the volume fraction of the phase for which the variable is defined, by a smoothing process (Dartevelle, 2005). The approach we use to derive the multifield equations is laid out in detail in Dartevelle (2005), and is particularly useful because it allows flexibility in whether turbulence is modeled by an ensemble averaging approach (so-called Reynolds Average Navier Stokes or RANS framework, which is most appropriate for internal shear flows such as volcanic conduits) or a large eddy simulation approach (LES, which is most appropriate for unbounded flows with a large range of turbulence length scales, such as eruption columns and plumes). The core idea of deriving a set of universal multifield Navier Stokes Partial Differential Equations (PDE) formally compatible with different approaches to turbulence (RANS vs. LES) is based upon the function of presence of a given phase at any point in space and time (Dartevelle, 2005). The function of presence acts as a unique mathematical identifier of the presence of any phase in time and space, while the gradient of the function of presence acts as a unique identifier of the interface between phases, which allows to define mass and heat fluxes between phases at their respective interfaces. Dartevelle (2005) demonstrates that if one carefully defines an ensemble-average process (RANS) or a filtering process (LES) abiding by the mathematical properties of conservation of constant, linearity, and of commutativity with respect to the derivations, then the resulting multiphase RANS and LES Navier Stokes PDEs would be strictly identical for an eventual implementation in any multiphase computer codes. Only the turbulence closures would differ depending on whether the modeler works within the LES or RANS framework of turbulence.

4 366 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Conservation of mass for the gas and solid particle phases (subscripts g and s, respectively) reads: 8 A ˆq >< g þ jd ˆq At g ũ g ¼ 0 >: A ˆq s þ jd ˆq At s ũ s ¼ 0 where ũ is the Favre mass-weighted averaged (RANS) or filtered (LES) velocity field as defined by Dartevelle (2005); ρˆ is the macroscopic bulk density of a given phase. We assume there is no exchange of mass between these two phases. The following relations hold in the two-phase system: ð1þ e s þ e g ¼ 1 a: ˆq s ¼ e s q s b: ˆq g ¼ e g q g c: ð2þ where ρ s and ρ g are respectively the ensemble-averaged or filtered microscopic density of the particle field and of the carrier phase; and ε s and ε g are respectively the volumetric concentration of the tephra material and of the carrier phase. Momentum conservation reads (Dartevelle, 2005): 8 A ˆq g ũ >< g þ jd ˆq g ũ g ũ g ¼ e g j P g jd e g ð s g þ tur=sg T g ÞþM drag g þ ˆq g g >: At A ˆq s ũ s At þ jd ˆq s ũ s ũ s ¼ e s j P g jd e s ð f T s þ tur=sg T s ÞþM drag s þ ˆq s g where P g is the ensemble-averaged or filtered thermodynamic pressure of the gas phase (assumed to be ideal gas, see Dartevelle, 2004, 2005), M drag is the interfacial momentum transfer rate between phases; g represents the body force (e.g. gravity); s g is a molecular viscous shear stress tensor; f T s is a frictional stress tensor defined from visco-plastic theory (Dartevelle, 2004; Dartevelle et al., 2004; Dartevelle, 2005); tur/sg T is the turbulence (RANS) or sub-grid (LES) stress tensors. At the gas particle interface: M drag s þ M drag g ¼ 0 ð4þ The key phenomenology to capture is the extra-dissipation either induced by the statistical turbulent variant motions of gas and grains around their respective ensemble-averaged mean value (provided by tur T within the RANS framework) or, within the LES framework, induced by the unresolved motions within the sub-grid (provided by SG T). Within a specific turbulence framework (RANS vs. LES), different constitutive equations must be specified for the turbulence/sub-grid stress tensor of the gas phase ( tur/sg T g ), the stress tensor of the solid phase ( tur/sg T s ), and the drag vector for all the phases (M i drag ). The closures for these functions are explained in detail by Dartevelle (2004, 2005, 2006a,b). Let T be the Favre mass-weighted averaged (RANS) or filtered (LES) temperature of a given phase, then conservation of specific internal energy is: f Cv;g C v;s " # A ˆq g T g þ jd ˆq At g T gũg ¼ e g w g e g P g jd ũ g jd e g ð q g þ tur=sg qþþ T H g A ˆq s T s þ jd ˆq At s T sũs ¼ jde s ð q s þ tur=sg q s Þþ T H s where C v is the specific heat at constant volume; q is the heat flux; w g is a mean or filtered viscous dissipation of the gas phase; and T H is the mean rate of interfacial heat transfer between phases. At the interface between phases: ð3þ ð5þ T H s þ T H g ¼ 0: ð6þ The heat fluxes are particularly important because they define a molecular heat flux (q k T, where k is the thermal conduction coefficient) and additional heat fluxes from either turbulence within RANS ( tur q) or from the sub-

5 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) grid within the LES framework ( SG q). Closures for k, H, and q functions are given in Dartevelle (2004, 2005, 2006a,b) and are not repeated here. In Eq. (3), the viscous contribution to the gas momentum, ε g (s g+ tur/sg T g ), is modeled as: jd e g ð s g þ tur=sg TÞc jd 2e g eff l g D ¼ g ð7þ while, for the solid phase, the viscous contribution to the momentum, ε s ( f T s+ tur/sg T s ), is modeled as: jd e s ð f T s þ tur=sg T s Þc jð f P s þ tur P s Þ jd ð 2 eff l s D ¼ s eff l b s jd ũ s IÞ ð8þ In these equations, eff μ and eff μ b are respectively an effective shear and bulk viscosity modeled from the RANS framework of turbulence; D ¼ ¼ 1 2 ½jũ þ jũt Šþ 1 jd ũi is the deviator of the rate-of-strain tensor where the superscript T 3 denotes the operation of transpose of matrix and I is the unit tensor (see Appendices C and D). The reasons of the approximatively equal sign,, in Eqs. (7) and (8) can be found in Appendix 5 of Dartevelle (2005). As explained before, these partial differential equations have been derived in a manner that is universally compatible for both LES and RANS frameworks of multiphase turbulence (Dartevelle, 2005). For the analysis of the borehole eruption, with its confined geometry, the RANS framework is the appropriate choice in this paper. The closures of RANS turbulence are built upon a set of partial differential equations for the production and dissipation of turbulence in the gas phase (k 1 ε 1 ) and solid phase (k 2 ) with a supplementary covariance equation (k 12 ) to provide a non-linear coupling between turbulence in the solid and gas phase (Simonin, 1996; Ferschneider and Mege, 2002; Benyahia et al., 2005; Dartevelle, 2006a,b). For the gas phase, eff μ g is the sum of the molecular ( mol μ g ) and turbulent or sub-grid ( tur/sg μ g ) contributions: eff l g ¼ mol l g þ tur=sg l g ð9þ with, "!# mol l g ¼ 1: T g T g þ 110 ð10þ and a RANS gas phase turbulence model being coupled with the granular temperature model for the solid phase (Appendix A). From the k 1 ε 1 model for the gas phase: tur l g ¼ C l q g k 2 1 e 1 ð11þ where k 1 and ε 1 are respectively the turbulent kinetic energy and the dissipation of the turbulent kinetic energy of the carrier phase; and C μ is a constant (see Appendices A, C, and D, and (Simonin, 1996; Benyahia et al., 2000; Ferschneider and Mege, 2002; Dartevelle, 2005, 2006a,b). For the solid phase, within the RANS framework, a granular kinetic-collisional model alongside a plastic model for high-concentration granular flows (Dartevelle, 2005) provides all the necessary closures: P i ¼ f P i þ k=c P a: ð12þ eff l i ¼ f l i þ k=c l i b: eff l b i ¼ f l b i þ k=c l b i c: which are defined in Dartevelle (2004, 2006a,b). Within the RANS framework, tur μ s = k/c μ s ; in other words, the kineticcollisional model provides the RANS turbulence closures for the dispersed (solid or tephra) phase (Dartevelle, 2005) Numerical implementation GMFIX (version 1.62) is a set of multiphase computational fluid dynamics codes evolved from the MFIX family of codes (Symlal, 1998; see also in order to model geophysical multiphase fluid dynamic problems. The

6 368 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Table 1 Cylindrical geometrical setup Radial length X (m), including the casing wall 0.12 Radial resolution ΔX (m) 0.04 Number of grid-points in the X-direction 3 Vertical length Y (m) a 1001 Vertical resolution ΔY (m) 1.0 Number of grid-points in the Y-direction 1001 Borehole radius (m) b 0.08 Inlet radius (m) 0.08 Outlet radius (m) 0.08 a Inferred from (Larsen et al., 1979): The superstructure of the borehole suffered minor damaged due to the eruption and the borehole continued to produce steam at a similar rate as before. Hence, one had to conclude that the dyke did not intersect any other (upper) parts of the borehole but only its lower end. b Given by (Larsen et al., 1979). first applications of GMFIX to volcanological problems focused on full scale Plinian eruption columns in the atmosphere and on the flow and deposition of pyroclastic density currents (Dartevelle et al., 2004), and more recently to explore the dynamics of a rising fragmented basalt gas mixture interacting with underground openings at the proposed Yucca Mountain radioactive waste repository (Dartevelle and Valentine, 2005). Dartevelle (2006a,b) provides a detailed description of the physical models and numerical solution techniques in GMFIX 1.62, as well as a detailed user manual. Table 2 Initial and boundary physical properties for all simulations Simulations A (coarse particles) B (fine particles) Borehole Pressure (Pa) Temperature (K) a Calculated gas density (kg/m 3 ) The mass fraction of water in vapor phase b 1 1 Volumetric solid concentration (vol.%) b 0 0 Kappa (gas phase turbulent production) (m 2 /s 2 ) Epsilon (gas phase turbulent dissipation) (m 2 /s 3 ) Inlet Mixture temperature (K) Gas pressure (Pa) c Calculated gas density (kg/m 3 ) The mass fraction of water in vapor phase d Grain diameter (m) a Grain/magma density (kg/m 3 ) a Magmatic mass fraction of water (wt.%) b 3 3 Inferred volumetric solid concentration (vol.%) Mass flux (kg/s) a Inferred vertical speed (m/s) Calculated mixture density (kg/m 3 ) Calculated mixture speed of sounds (m/s) Calculated mixture static pressure (Pa) Calculated mixture specific heat ratio Theta, granular temperature (solid phase turbulent production) (m 2 /s 2 ) Solid inelastic collisional dissipation coefficient Kappa (gas phase turbulent production) (m 2 /s 2 ) Epsilon (gas phase turbulent dissipation) (m 2 /s 3 ) Outlet Gas pressure (Pa) Gas temperature (K) a Calculated gas density (kg/m 3 ) a Given by (Larsen et al., 1979). b Although unspecified by (Larsen et al., 1979), it is much likely that before the magma ascended into the borehole, it must have been filled by liquid water at the bottom and water vapor. c This corresponds to lithostastic plus magmatic water vapor pressure. d This corresponds to magmatic gas made of pure water vapor.

7 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) GMFIX 1.62 has been extensively tested to ensure that the results of simulations are accurate representations of the physical systems modeled. The model implemented into the GMFIX codes has been verified against analytical solutions such as the Sod and shock tube problems (Sod, 1978; Toro, 1999), and has been validated against analog experimental data of overpressured pure gas and gas particle jet dynamics and of particle-laden turbulent plumes. Details of verification and validation tests are provided in Dartevelle (2006c). The documented ability of GMFIX 1.62 to accurately model compressible, turbulent, gas particle flows provides confidence that the modeling results described in Section 4 are good representations of the 1977 borehole eruption within the context of uncertainties in the initial and boundary conditions Initial and boundary conditions Based on the description of Larsen et al. (1979) of the borehole and the ejected scoria, we have reconstructed the initial and boundary conditions as shown in Tables 1 and 2, and Fig. 1. Two simulations were performed with different particles sizes, 18.4 mm and 18.4 μm. The former particle size case is exactly the median grain-size reported by Larsen Fig. 1. Boundary and initial conditions; and geometrical and spatial discretization setups. Drawing not to scale. As indicated, at the inlet, the multiphase flow has a gas pressure of atm, a temperature of 1428 K, a solid volumetric concentration of 30.8 vol.%, a phasic weighted vertical speed of 1.41 m/s, and a phasic mass flux of 25 kg/s.

8 370 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Table 3 Numerical properties for all simulations Geometry Spatial discretization Time discretization Linear-equation solver Inlet boundary Outlet boundary Wall 1 Cylindrical MUSCL (2nd order accurate) 1st order accurate BI-CIGSTAB Parallelization Scheme SMP / OPEN-MP Constant mass flux inflow (MI) Constant pressure and temperature outflow (PO) Free-slip wall (all phases) et al. (1979) at a distance of 20 m from the wellhead (Section 2), while the latter size was modeled in order to provide comparison with a particle field that is much better coupled to the gas phase. In Table 2, we have used a 3 wt.% water mass fraction as an initial condition. We were unable to find petrologic data on the basaltic products of the activity associated with the borehole eruption that would allow us to constrain the magmatic water content, but infer that it might have been 1 wt.%. The larger water content value that we use allows for the likelihood of some interaction between the magma and externally derived geothermal waters (see Section 2). The conservation equations are solved in the vertical dimension only, which is a reasonable approximation since the length of the borehole conduit (1000 m) is much larger than its diameter (0.16 m). In addition, within such a small diameter conduit, all turbulent fluctuations are strongest in the vertical direction. The choice of the 1000 m depth for the intersection of the borehole and the dyke is supported by the fact that the structure of the borehole suffered little to no damage indicating that the dyke crossed the borehole near its bottom; we assume that intersection at higher levels in the borehole would have resulted in some plugging due to down flow and chilling of magma. Larsen et al. (1979) estimate an average mass flux for the event of 25 kg/s, which we apply as a boundary condition at the base of the computational domain. We assume that this source flux was constant, given the lack of any information on potential source transience. Borehole walls are treated as free slip boundaries. Indeed, numerical experiments suggest that there is negligible difference between results obtained with no-slip and free-slip boundary conditions for the gas phase. In addition, visual observations of flows through vertical pipes indicate that particles do slip near the wall, supporting a partial or free-slip boundary condition for the particle phase. It is generally believed that in high-velocity flows of densely loaded gas particle mixtures through large risers the vertical pressure gradient is largely due to the particle holdup and that wall shear is only weakly relevant (Chang and Louge, 1992). Numerical simulation snapshots are not much different from those obtained with no-slip boundary conditions, suggesting that the gross features of the fluctuating flow pattern in a statistical steady state are not driven by specific choice of wall boundary conditions (Sundaresan, 2004). Table 3 provides the numerical properties for other parameters used in the simulations. All simulations described herein have been tested over different grid resolutions to ensure that these numerical solutions are strictly grid-size independent. 4. Results Simulation of the borehole flow with a very small particle size ( 18.4 μm, or about three orders of magnitude smaller than the mean particle size of the actual eruption) should approximate a pseudo-fluid model and provides a reference case for comparison with flows with more realistic particle size. After a simulated time of 180 s, particle volume fraction decreases relatively gradually from the inlet (base of borehole) value of 0.31 to , with small fluctuations of about a factor of 2 around that vertical decrease (Fig. 2A). Density and phase weighted velocity (Fig. 2B,C) vary smoothly with height. This simulated flow field is relatively stable after an initial transient phase. The same initial and boundary conditions, but with a particle size of 18.4 mm intended to better mimic grain size produced by the actual event (Section 3.3), result in more complex flow phenomena. After a simulated time of 40 s the flow field is one of smooth variations with height in the borehole (Fig. 3). Particle volume concentration (Fig. 3A) smoothly decreases upward, mainly reflecting smooth expansion of the water vapor field with nearly isothermal decompression (Fig. 3B,C). As the vapor phase expands it also accelerates (Fig. 3D) in order to conserve mass in the constant diameter borehole. Particles also accelerate because of drag forces, but due to their inertia the particles lag behind the vapor (Fig. 3D); the velocity of vapor exiting the top of the borehole is 200 m/s, while particles exit at 58 m/s. The flow field illustrated in Fig. 3 represents a temporary steady state that is reached after an initial s period which is the time needed to transport particles throughout the entire length of the borehole. By

9 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Fig. 2. Vertical profiles of flow properties in borehole after 3 min of simulation time, using a very fine particle size (18.4 μm) that is well coupled to the gas flow field. A. Particle volume concentration (dimensionless) vs. height in the borehole (m). B. Gas phase density (kg/m 3 ) vs. height. C. Phase weighted vertical speed (m/s) vs. height. Both gas density and phase weighted speed change smoothly with height because of the pressure gradient. The particle volumetric concentration smoothly decreases with heights with only minor local heterogeneities. this time (40 s) the flow is similar to a steady fluidized bed, albeit with a lower particle concentration than is normally considered for such beds. The difference in particle and vapor accelerations sets up a situation that cannot be maintained and soon instability will set in; viz., the particulate phase tend to

10 Fig. 3. Results for the reported mean particle size (at a distance of 20 m from the wellhead) of 18.4 mm at a simulated time of 40 s. A. Profiles of the particle volumetric concentration (dimensionless); B., of the gas density (kg/m 3 ); C., of the phase weighted temperature (K); and, D., of the vertical speeds (m/s) of the tephra phase (red) and of the gas phase (blue) vs. height in the borehole. All these profiles are smooth and characterized by a reduction of particle concentrations and of the gas density with height, a constant acceleration of all phase towards the top, and a quasi-constant temperature in the borehole system. 372 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007)

11 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) lag behind, hence to drag and slow down the mixture, while at the same time the gas phase expands and tends to accelerate the mixture towards the surface. The onset of unstable flow is best demonstrated by focusing on the lower 200 m of the borehole and comparing variables at 40 s, just before the beginning of unstable flow, and at 45 s, just after the onset of instability (Fig. 4). Compared to the smooth decrease in particle concentration at 40 s, by 45 s the concentration profile is developing waves with wavelengths of 20 m (Fig. 4A). Likewise, both particle and gas velocity profiles indicate the onset of waves between 40 and 45 s (Fig. 4B). The instabilities begin near the bottom of the borehole and propagate upward (Fig. 5; see details of basal part of flow field at 60 s in Fig. 6). The small, early variations in particle concentration shown in Fig. 4A grow into waves with variations of more than two orders of magnitude (compare with the results of the 18.4 μm simulation, Fig. 2, where the volume fraction fluctuations are limited to a factor of 2 and do not amplify within the simulated time). The waves are essentially high concentration, narrow slugs of particles, separated by relatively clean gas. The slugs reach sufficiently high particle concentration to begin to enter into a plastic-like rheological behavior (Symlal, 1998; Dartevelle et al., 2004; Dartevelle, 2005), a state in which the resistance to upward movement increases substantially. Phase weighted velocity shows one effect of the segregation of particles into slugs; as the gas expands and accelerates upward, it encounters slugs of particles that exert large drag forces and rapidly decelerate the gas, which then re-accelerates once it passes through a slug. The coupling between gas and particle momentum creates a nonlinear feedback; increasing particle concentration slows the gas, which reduces drag on the particles and further slows a slug and increases its concentration. Compressible flow effects are Fig. 4. A. Profiles of particle volumetric concentration (dimensionless) with height between 0 and 200 m, sampled in the borehole at 40 s (dashed curve) and 45 s (plain). Within 5 s, this system switched from a homogenous fluidized system to a chaotic and heterogeneous one characterized by propagating upwards concentration waves. B. Vertical speeds (m/s) of particle (red) and gas phase (blue) within the first 200 in the borehole, sampled at 40 s (dashed curve) and 45 s (plain). Notice that the gas phase speed is the most sensitive of the solid volumetric concentration variations owing its lower inertia.

12 374 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Fig. 5. Time sequence (from 50 s to 180 s) of, A., particle volumetric concentration (dimensionless); B., phase weighted vertical speed (m/s); and, C., multiphase Mach number (dimensionless) profiles vs. the borehole height (m). Notice, with time, the vertical propagation and amplification of particle waves. There is a one-on-one relation between higher concentration wave and higher Mach number (the speed of sound is smaller in more concentrated system).

13 Fig. 6. Detail of Fig. 5 taken at 60 s and between 0 and 400 m of: the particle volumetric concentration, the vertical speed of solid (red) and gas (blue) phase, of the phase weighted multiphase Mach number, and of the gas density. S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007)

14 376 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) also complex. Sound speed in the mixture is very sensitive to particle concentration, and as a result the particle slugs result in narrow zones where the flow is supersonic. These zones (Fig. 5) are shock waves that travel upward with the slugs of high concentration. Comparison of profiles at successive times (Figs. 5 and 7) shows that each wave or slug tends to grow in amplitude as it propagates upward until it finally exits the top of the borehole. The multiphase coupling processes are illustrated in more detail in snapshots focusing on the lower 400 m of the borehole at a time of 60 s (Fig. 6). For example between 353 and 380 m, particles accumulate to produce a drastic dynamic change of the particulate flow behavior from collisional (low viscosity flow) to plastic (high viscosity flow) (Dartevelle et al., 2004). At 380 m the particle concentration is 56%, a true granular plastic flow. As a consequence, the vertical speed of particles decreases from 40 m/s (at 377 m) to 5.7 m/s (at 383 m), which is also accompanied by a drop of gas pressure and density in this confined environment shared between the gas and solid phase (between 372 m and 383 m, the gas density drops by 55%). This sharp decrease, within 10 m, of P g and ρ g first accelerates the gas phase from 51 m/s (at 370 m) to 76 m/s (at 380 m), but above that height the drag from the particle slug forces the gas phase to decelerate to 61.6 m/s (at 383 m). Immediately above the partially clogged area (higher than 383 m, not shown in Fig. 6), the concentration, gas pressure and density profiles show a smoother decreasing trend while both phases smoothly re-accelerate towards the surface (see 60 s snapshot in Fig. 5). Sharp variations of particle concentration are also accompanied by sharp variations of the multiphase Mach number (defined in Appendix B, Eq. (B.8)); within a few meters the flow changes from subsonic (diluted) to supersonic (concentrated) and viceversa (Fig. 6). Outlet conditions as a function of time at the top of the simulated borehole (Fig. 8) offer some comparison with observed phenomena at the wellhead. Particle volume concentration (Fig. 8A) indicates that the first particles exit the wellhead 25 s after flow is introduced at the base of the domain in the simulation. This is preceded by a blast of pure steam, as indicated by the phase weighted exit velocity (Fig. 8B). After an early peak in particle concentration exiting the wellhead there is a period (35 70 s) during which exiting concentration is relatively constant (Fig. 8A). This period reflects the early period of steady flow and smooth vertical profiles (Fig. 3) plus the time for the first slug to reach the well head after the onset of instability Fig. 7. A fully developed and amplified system of particle volume concentrations sampled between 112 and 116 s. Notice how a given solid volume concentration train-wave progressively rises upwards towards the outlet, it takes 4 s (between 112 and 116 s) to travel a distance of 250 m (or with an averaged solid wave speed of 63 m/s).

15 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) Fig. 8. A. Time sequence profiles of the particle volume concentration (dimensionless) and, B., of the phase weighted vertical speed (m/s) sampled at the top outlet of the borehole every second. Note at 4 s, an initial water vapor explosion (no tephra) with an ejection vertical speed of 215 m/s. The tephra will be ejected from the borehole at 31 s with a phase weighted speed of 31 m/s. Also, it is worth noting that a quiet phase up to 75 s, then this eruptive system becomes pulsating with an explosion or tephra shot every 6 s or so. low in the domain. After 75 s the exiting flow is highly unsteady, with major pulses of ejected particles (concentrations up to 0.35) tending to occur in clusters of two or more pulses within 15 s. Each one of these major pulses represents the ejection of one of the slugs seen in the vertical profiles (e.g., Fig. 7), and observed at night, would have appeared as pulses or shots of incandescent particles. Smaller pulses of

16 378 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) particle ejection occur at a rate of about one every 3 s. Overall (Fig. 8A), at the wellhead, the particle volume concentration varies between values up to 0.35 (maximum) and reflecting dynamic flow behavior switching back and forth from collisional to kinetic regimes (Dartevelle, 2004; Dartevelle et al., 2004). Phase weighted exit velocity varies between 50 and 100 m/s, with the peak velocities corresponding to jets of relatively clean gas (between particle slugs). Interestingly, the average vertical speed computed over the length of the borehole is 31 m/s, which compares well with Larsen et al.'s (1979) estimate of m/s based upon qualitative estimates of ejecta heights. 5. Interpretation and implications Our simulations produce wellhead phenomena similar to those documented at the Námafjall event (Fig. 8A and B), namely during the final phase which Larsen et al. (1979) describe as follows: The first phase [ ] started with an explosion and a thin incandescent column was seen. [ ]. This phase, accompanied by a continuous roar, lasted no longer than 1 minute. From Fig. 8B, one can see an initial blast of pure steam exiting the wellhead at speeds 220 m/s, which is followed 25 s later by the ejection of the tephra (Fig. 8A). The second phase, lasting minutes, there was little or no activity. Occasional red flashes may have occurred during the latter part of this phase. From Fig. 8A and B, one can see that, after the initial blast explosion, between 35 and 70 s, there is a period of relative quietness with decreasing speed during which the borehole mimics a steady state period during which instabilities gradually set in at the bottom of the borehole and finally rise towards the wellhead. The modeled period of relative steady state quietness is much shorter than the 10 min observed by Larsen et al. (1979). This difference in time might be caused by some supplementary unsteadiness in the borehole system such as temporary disruption of magma supply from the dyke, or from water entering the borehole and temporarily stopping upward magma flow (i.e., potentially representing a phase of pre-mixing of melt and water). This quiet period precedes the last stage described in (Larsen et al., 1979). The final phase consisted of a series of very rapid explosions or shots of glowing scoria. A few groups of explosions were observed each consisting of several individual shots. This final phase of the eruption, which qualitatively compares well with the last stage in Fig. 8, corresponds to the arrival of the magmatic slugs followed by voidage (bubble) formed initially at the bottom of the borehole and amplifying as this succession of magmatic pulses rise towards the surface. The numerical results that we have presented in this paper are consistent with this large body of experimental, numerical, and theoretical analyses of gas particle systems summarized in Section 1. We have shown that even with a constant flux of material at the bottom, a vertical flow of gas and pyroclasts is likely to eventually result in pulsatory flow at the vent (in this case, the wellhead), even in a flow configuration such as the Námafjall borehole eruption where the conduit geometry is well constrained and was constant throughout the eruption. Volcanic conduit flow is likely to be significantly more complex due to factors such as the huge range in particle sizes, potential unsteadiness in magma supply, bubble, volatile content, the fragmentation zone, the mechanical failure of wall rocks and the avalanching of vent debris into a vent. These factors may provide further time-dependent sources of particulates as well as transient variations in conduit geometry. Such processes would be expected to occur to varying degrees in all types of pyroclastic eruptions from small scoria coneforming events to caldera eruptions. This all suggests that while the conduit flow models summarized in Section 1 are useful to some degree, they should also be used with a perspective on the likely role of unsteadiness in the multiphase flows. We note that the work of Dufek and Bergantz (2005) also used a time-dependent multiphase approach, but focused on the transition from an initial shock propagation phase to a steady state phase in conduit flow. It seems plausible that their calculations would have resulted in a sort of post-steady flow transient behavior, similar to that obtained in our calculations, if their simulations had been carried to longer times, on a much finer grid (their grid seem to be rather coarse, 4 m 4 m, with respect to the size of their particles, 0.2, 2, 20 mm, hence preventing their simulation from covering the whole range of granular behaviors), and/or with a fully advective multiphase turbulent model as presented here. As we have seen, the frequency of particle bursts is strongly dependent on the particle size, i.e., the larger the particle sizes, the more decoupled the phases, hence the more amplified the unsteadiness and explosive bursts. It is also worth noting the complex feedback mechanism caused by the large variations of the mixture speed of sound on the overall unsteady dynamic of this multiphase system and on the particle burst frequency (e.g., wave amplification). Indeed, the initial variations of the sonic regime in the conduit (subsonic to supersonic and vice-versa) is the initial result of the inherent decoupling between phases (for instance caused by the

17 S. Dartevelle, G.A. Valentine / Earth and Planetary Science Letters 262 (2007) larger particle size); yet, at a later stage, these mixture sound speed variations must be also one of the leading cause of the amplifications of the particle burst frequency and, hence, cause of further developments of unsteadiness. Other factors may also have an important control on the frequency burst, e.g., the ratio of particle size and conduit diameter and, perhaps, the vertical length of the conduit itself. Overall, this work has implications for eruption column models that use conduit models to provide boundary conditions. We suggest that major advances in volcanic conduit flow will derive from a focus on timedependent multiphase dynamics and characterization of the sources and character of unsteady behavior. Acknowledgments This work was funded by the U.S. Department of Energy's Yucca Mountain Project, which is managed by the Sandia National Laboratory. The authors thank Don Hickmott, Ed Gaffney, and Gordon Keating for their thorough review of a preliminary draft of this work. Two EPSL anonymous reviewers are also thanked for their comments and suggestions which have helped to improve this manuscript. Appendix A. RANS turbulence PDE The two-phase flow turbulence model can be understood as a competition between different time scales defined from key physical phenomenology between the gas and the dispersed phases (Simonin, 1996; Ferschneider and Mege, 2002; Benyahia et al., 2005): The gas-phase turbulence, which is characterized by gas phase turbulence time-scale (Lagrangian timescale or eddy-turn over time-scale), t 1 ; The coupling between the fluctuating motion of the gas and the agitation of the particles, characterized by the fluctuation time of the fluid as seen by the particles (eddy particle interaction time), t t 12; The entrainment of the solid particles by the motion of the gas, characterized by the particle relaxation time t x 12, direct function of the inertia of the particles and the drag. At the limit, if t x 12, particles are ballistic and if t x 12, the particles are passive tracers; The collisions between particles, characterized by the characteristic time between the collisions (interparticle collision time), t c 2; The last two mechanisms are in competition and can be characterized by the particle dissipation time, t 2, defined as the harmonic mean of the particle relaxation time and of the particle collision time, 1 f 1 t 2 t þ 1 12 x t2. c In order to quantify the importance of the different physical mechanisms involved in gas solid flow turbulence, their time-scales can be compared: t 1 b : t 2, the dominant mechanisms are inherent to the gas phase. The time between two collisions is large; therefore, the motion of the particles is considered to be statistically independent. This regime is called the dilute regime and is encountered only at low solid volume fractions ε s b0.1 vol.%. Depending on their relaxation times the particles behave differently from the gas turbulence. If t x 12, particles are not affected by the gas turbulence (whether collisions exist or not); but if t 1 bt x 12bt c 2, the main dissipative scale is the gas turbulence and the drag between the particle and the gas (particle collision is non-existent). t 1 N : t 2, this situation emerges at large solid volumetric fractions ε s N10 vol.%. This is the collisional regime. In this case, the internal momentum transport in the solid phase is dominated by particle particle collisions. The granular motion is only slightly perturbed by the presence of the gas. Indeed, if t x 12N and t c 2b, then the gas particle interactions are small; if t x 12 t 2 and t c 2, then the motions of the two-phases are coupled but collisions is the only dissipative mean. t 1 c : t 2, the gas particle and particle particle interactions are in competition. This intermediate regime is called the kinetic regime. The granular motion is highly perturbed by the presence of the gas. If t x 12 t 1 t c 2, the motion of the particles is controlled by the turbulence of the gas (barely by collisions). In the opposite case, t c 2 t 1 t x 12, each phase's behavior seem to be uncorrelated, but the gas must influence the transport properties of the particles in limiting the mean-free path. It can be easily seen that eddy-viscosities of the gas and dispersed phases can be written as functions of these time scales and turbulent kinetic energies. A.1. Granular temperature equation (or k 2 -equation) Let k 2 be the fluctuating kinetic energy of the dispersed phase (Dartevelle, 2005), then A ˆq s k 2 þ jd ˆq At s k 2 ũ s ¼ ˆq s ð Ts:jũ s Þ jd q H þ P H R H ða:1þ