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Chapter 11

Magma–water interactions Ken Wohletz, Bernd Zimanowski and Ralf Büttner

energetics predicted by equilibrium and non- Overview equilibrium thermodynamics. Taken together, these approaches elucidate the relationships –water interaction is an unavoidable con- among aqueous environment, interaction phys- sequence of the hydrous nature of the ’s ics, and eruptive phenomena and landforms. , and may take place in environments ran- ging from submarine to desert regions, producing volcanic features ranging from passively effused to highly explosive events. Hydrovolcanism 11.1 Introduction: magma and is the term that describes this interaction at or the hydrosphere near the Earth’s surface, and it encompasses the physical and chemical dynamics that deter- The vast majority of volcanic eruptions take mine the resulting intrusive or extrusive behav- place under water because most ior, and the character of eruptive products and concentrates at mid-oceanic ridges where new deposits. The development of physical theory is produced. By definition, every describing the energetics and the hydrodynam- kind of extrusive subaqueous volcanism on ics (dynamics of fluids and solids at high strain Earth is hydrovolcanic since some degree of rates) of magma–water interaction relies on an water interaction must take place. The hydro- understanding of the physics of water behav- sphere also exists in continental areas, as the ior in conditions of rapid heating, the physics consequence not only of lakes and rivers, but of magma as a material of complex rheology, also of groundwater and hydrous fluids that cir- and the physics of the interaction between the culate in joints and faults in the upper crust and two, as well as detailed field observations and fill pore space in sedimentary rocks. Such loca- interpretation of laboratory experiments. Of pri- tions are typically referred to as geohydrologi- mary importance to address are the nature of cal environments. As a consequence, subaerial heat exchange between the magma and water volcanism is commonly influenced by magma– during interaction, the resulting fragmentation water interaction. Chapter 12 describes deep- of the magma, and the constraints on system eruptions in greater detail, whereas this chapter

Modeling Volcanic Processes: The Physics and Mathematics of Volcanism, eds. Sarah A. Fagents, Tracy K. P. Gregg, and Rosaly M. C. Lopes. Published by Cambridge University Press. © Cambridge University Press 2012.

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focuses on magma–water interaction in surface formalized by Abraham Werner). Initial ideas and near-surface environments. about the role of ground and surface water in vol- The explosive intensity of volcanic eruptions canism developed during the last century. These depends on the extrusion rate of magma and on perceptions resulted largely from observations the coupling of its thermal energy (i.e., heat flux) of unusually explosive periods of Hawaiian vol- to the surroundings. Because the thermal con- canism, during which groundwater entered ductivity of magma is very low, hydrovolcanic along which normal lava fountaining had previ- heat flux is mainly governed by the size of the ously occurred (Jaggar, 1949), as well as through interfacial area between magma and water and examination of fragmental found where its growth rate during interaction. Thus, the key lava had entered water (Fuller, 1931). Three well- process that determines the thermal energy flux documented eruptions during the late 1950s and from the magma to its surroundings is magma early 1960s brought an increased awareness of fragmentation. explosive hydrovolcanism: Capelinhos, Azores Water is the predominant thermodynamic (Tazieff, 1958; Servicos Geologicos de Portugal, working fluid on Earth, and practically every 1959), Surtsey, Iceland (Thorarinsson, 1964), power plant in the world uses water for con- and Taal, Philippines (Moore et al., 1966). Fisher verting thermal energy into mechanical energy and Waters (1970), Waters and Fisher (1971), and finally into electricity. Where rising magma and Heiken (1971) expanded the characteriza- contacts water (in contrast to rocks or atmos- tion of phreatomagmatic eruptions to include pheric ) the major effect is an increase in steam-rich eruption columns, base surges, and thermal energy flux and, by analogy to commer- typical landforms such as , rings, and cial power production, in the efficacy of heat tuff cones. As a result of this work, numerous and power generation. For this reason there is twentieth century phreatomagmatic eruptions a rich technical literature in science and engin- are now recognized, many of which formed eering that can be applied to understanding -like craters (e.g., Self et al., 1980). After magma–water interactions; this is the intent of cinder cones, phreatomagmatic constructs (tuff this chapter. rings, tuff cones, and maars) are perhaps the most abundant terrestrial volcanic landform. However, magma–water interaction is certainly not limited to explosive phreatomagmatic 11.2 Hydrovolcanism: from pillow eruptions – the hydrologic environment plays lava to maar a major role in determining the kind of inter- action that can occur. 11.2.1 History of hydrovolcanism Magma–water interactions occur deep within the 11.2.2 Hydrovolcanic environments Earth (hydromagmatism) as well as at or near its The wide variety of hydrovolcanic phenom- surface (hydrovolcanism). These terms are roughly ena suggests that interaction between water synonymous because in many cases the realm and magma or magmatic heat may occur in where interaction initiates and later manifests any volcanic setting and geohydrological envir- may span from deep within the Earth’s crust onment, and is not restricted to a particular to the surface. For this reason the term phreato- magma composition. From the formation of magmatic is used to designate interaction within and lineated, folded, and jumbled the phreatic realm of the Earth’s surface, which sheet flows in deep water, to the intrusion of includes the zone of saturation where ground- in dikes and sills deep in the crust, to water and surface water exist. the eruption of plumes of fine ash in desert, The topic of magma–water interaction may tropical, and shallow water environments, be considered to have its roots in the eighteenth- magma–water interaction includes both pas- ­Century Neptunists’ theory about the origin sive and dynamic phenomena. During ascent of basaltic rocks in oceans (which was later to the surface, magma commonly encounters

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Figure 11.1 Schematic illustration of common environments of hydrovolcanism (adapted from Wohletz, 1993).

0.1–0.5 0.3–1.5 km km 0–0.2 PLUME 0.1–2.0 km 20–50 HEIGHT km 10–40 0.5–3.0 km km km

subaqueous hawaiian and strombolian plinian columns phreatoplinian surtseyan surtseyan lava flows ERUPTIVE ballistic tephra fall column/collapse ash fall cypressoid jets (pillow, lineated, PHENOMENA tephra fall PDCs ashfall, PDCs lateral surges falls and surges folded, jumbed, sheet)

mm- to m-size fragments m- to mm-size fragments DOMINANT cm-size fragments µ condensing steam superheated steam PRODUCTS water

ROLE OF EXTERNAL Little/none Optimum Excessive WATER Increasing water abundance

Figure 11.2 Relationship of typical eruptive phenomena and products to water abundance (adapted from Sheridan magma fragments is called phreatic (Ollier, 1974) and Wohletz, 1983). Deposit features indicative of magma– or hydrothermal (Muffler et al., 1971; Nairn and water interaction may include planar to duneform bedding, Solia, 1980). Subglacial volcanism (Noe-Nygaard, impact sags or slumps and, particularly for greater water 1940; see Chapter 13) is best known from its abundances, accretionary lapilli, soft sediment deformation structures, and cohesive, altered, or vesiculated tuff. products, including massive floods jökulhlaups( ), table (tuyas or stapi), and ridges (tin- dars or mobergs). In all these environments, a major factor determining the expression of groundwater, connate (entrapped depositional) hydrovolcanism is the abundance of water water, marine, fluvial, or lacustrine water, ice, available to interact with the magma. Not only or rain water (Fig. 11.1). From this diversity are the eruptive phenomena affected by water of hydrovolcanic environments comes a wide abundance, but so are the characteristics of the range of terminology. For example, the subaque- interaction products, their dispersal, and the ous environment includes all activity beneath resulting landforms (Fig. 11.2). a standing body of water (Kokelaar, 1986); products of this activity have been called sub- 11.2.3 Hydrovolcanic eruption styles aquatic (Sigvaldason, 1968), aquagene (Carlisle, A wide variety of eruption styles result from 1963), hyaloclastite (for deep marine; Bonatti, magma–water interactions, ranging from pas- 1976), hyalotuff (for shallow marine; Honnorez sive lava effusion to highly energetic thermo- and Kirst, 1975), and littoral (Wentworth, 1938). hydraulic , depending on ambient Volcanism that heats groundwater to produce conditions and the proportions of water and steam explosions that do not eject juvenile magma interacting (Fig. 11.2). At the highest

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water abundances, explosive activity is gener- or sheet lava, pillow breccia, and hyaloclastite. ally suppressed, producing passive quenching Posteruptive hydrothermal interaction with and thermal granulation of lava flows (but see fragmental materials may produce palagonitic Chapter 12 for further discussion of deep-sea and zeolitic tuff, silica sinter, and travertine. explosive activity). At somewhat more favorable Fragmental products are termed hydroclasts by water/magma mass ratios, surtseyan activity is a Fisher and Schmincke (1984), instead of pyro- common expression of explosive hydrovolcan- clasts, a term that refers solely to the fragmental ism in, for example, shallow marine environ- products of explosive eruptions driven by mag- ments. Dense, tephra-laden, cypressoid (“cock’s matic (juvenile) . Explosive hydrovol- tail”) jets and steam clouds contain a significant canic products commonly contain significant component of liquid water, both as a result of abundances of lithic fragments derived from vapor condensation and direct ejection from a explosive fragmentation of country in vari- shallow-water environment. Tephra is dispersed ous hydrovolcanic environments. as fall or surges, which may be expressed as Petrographic studies of hydroclastic prod- massive or planar-to-duneform bedded deposits. ucts involve the determination of particle size, Phreatoplinian eruptions are the result of more shape, componentry (magmatic vs. lithic) and intense interactions that produce a greater textural characteristics, and the chemical signa- vapor-phase component at water–magma mass tures caused by both rapid and slow alteration. ratios approaching the optimum for energy con- These data are indicators of the degree and type version efficiency. Groundwater or shallow sur- of water interaction. For example, the grain size face water is typically involved. Greater energy of hydroclasts is a function of the mass ratio of release allows for explosive vapor expansion, water and magma involved dynamically in the thorough magma fragmentation and the forma- interaction; grain textures are indicative of the tion of convecting columns, and significantly type of interaction – passive, explosive, brittle, more widespread tephra dispersal through fall or ductile. Field characterization of hydroclastic and pyroclastic density currents. Duneform bed- products focuses on analysis of deposit charac- ding becomes more common for energetic surge teristics, including bedding, grading, sorting, deposits. Transient explosive activity, i.e., vulca- lithification, and deposit thickness vs. distance nian eruptions, may also be driven in part by from the vent. Variations of these characteristics pressurized vapor derived from meteoric water within or among deposits can elucidate variabil- (Chapter 7). The deposits of phreatoplinian and ity in eruptive intensity and style (e.g., fall vs. vulcanian eruptions exhibit fewer indications of density current deposit), and degree of magma– a liquid water phase in comparison to surtseyan water interaction. deposits; the former are sometimes termed “dry” Experimental and field research revealed hydrovolcanic deposits, whereas the latter are a correlation between the median grain diam- termed “wet”. Subglacial eruptions, discussed in eters of hydroclasts and the interacting water/ detail in Chapter 13, commonly exhibit a range magma mass ratio (Fig. 11.3). In general, hydro- of hydrovolcanic styles (from pillow effusion to volcanic tephra are distinguishable from mag- surtseyan and phreatoplinian activity) during matic tephra by their much finer grain size. their eruption sequence, and understanding of Furthermore, for hydrovolcanic tephra there is magma–ice interactions benefitted greatly from an optimum water/mass ratio that produces the recognition of tephra and deposit characteris- most thorough fragmentation; ratios less than tics first mapped in phreatomagmatic tuff cones or greater than the optimum value result in less and tuff rings. finely fragmented tephra. Microscopic examination of grain shapes 11.2.4 Hydrovolcanic products and textures also reveals features indicative of Hydrovolcanic solid products include tephra, hydrovolcanic origins; whereas the products of blocks and bombs, breccia, pillow purely magmatic fragmentation are dominated

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Figure 11.3 Correlation of grain size and thermal-to-mechanical energy partitioning to water/melt ratio (adapted from Frazzetta et al., 1983, and Sheridan and Wohletz, 1983).

by highly vesicular clasts, hydroclast populations include poorly to non-vesicular clasts with a var- iety of textures (Fig. 11.4). Hydrovolcanic tephra Figure 11.4 Sketches of particle textures found in may show features of both magmatic and hydro- hydrovolcanic deposits. (a) A characteristic blocky and equant volcanic origin; in such cases, detailed statistical glass shard, (b) a vesicular grain with cleaved vesicle surfaces, study of tephra samples can help determine the (c) a platy shard, (d) a drop-like or fused shard, (e) a blocky relative proportions of the two end member with conchoidal fracture surfaces, and (f) a perfect crystal with layer of vesicular glass. In contrast, magmatically fragmentation processes (Büttner et al., 1999; fragmented deposits are dominated by highly vesicular clasts, Dellino et al., 2001). Some hydroclastic grain tex- but may contain minor proportions of particle types shown tures are also indicative of the type of magma– here. (Adapted from Wohletz and Heiken, 1992.) water interaction (e.g., whether a superheated vapor or liquid phase was involved; Wohletz, 1983). For example, quench cracks on grain sur- faces indicate quenching in contact with excess result in broader, lower landform (e.g., maars, liquid water, and chemical pitting is indicative tuff rings) due to the greater mobility of the of contact with corrosive fluids (Wohletz, 1987; products (in falls or surges), with limited alter- Dellino et al., 2001). ation of the hydroclasts. An individual eruption Field characteristics of deposits are also can evolve towards wetter or dryer condi- indicative of the proportion of liquid vs. vapor tions, which would be reflected in the deposit phases in phreatomagmatic eruptions. Broadly characteristics. speaking, liquid water will produce more cohe- Explosive hydrovolcanic eruptions may occur sive deposits that may contain accretionary as single, monogenetic events or at polygenetic lapilli, soft-sediment deformation structures, centers (volcanoes constructed of multiple erup- vesiculated tuffs (due to post-depositional tions). The former commonly produce maars, vaporization of liquid water), and may produce tuff rings or tuff cones, depending on the envir- steeper landforms (e.g., tuff cones) with some onment of interaction and availability of water. degree of hydrothermal alteration (e.g., palago- The latter may simply introduce a phreatomag- nitization). In contrast, more energetic erup- matic deposit into the record of activity at that tions producing a superheated vapor phase may particular .

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Experimental studies have made it possible to 11.3 Magma–water interaction quantify some controlling parameters by using physics field and laboratory measurements of hydro- volcanic products. The curves in Figure 11.3 11.3.1 Fuel–coolant interactions were derived from early experiments (Wohletz, Formulating an understanding of the physics 1983; Wohletz and McQueen, 1984) that dem- of magma–water interactions requires consid- onstrated that the explosive efficiency of the eration of the physical properties of magma, system (measured as the ratio of eruptive kin- the thermodynamic behavior of water, and the etic energy to magma thermal energy) is related physics of processes at the interface between to the mass ratio of interacting water and melt the two fluids. Much of the physical under- (thermite – a magma analog) and the confining standing of hydrovolcanism has developed from pressure. These experiments displayed a var- the combined insights derived from studies of iety of explosive and non-explosive behaviors molten fuel–coolant interaction (MFCI or FCI) and that are analogous to natural volcanic activity, laboratory experiments designed to replicate including classical strombolian, surtseyan, vul- hydrovolcanic phenomena (e.g., Wohletz et al., canian, and plinian phenomena. 1995a; Zimanowski et al., 1997a). The field of Experimental studies demonstrate that the FCI science arises from industrial applications, thermodynamics of heat transfer (see Section and concerns the interaction of two fluids for 11.4.1) is a significant aspect of hydrovolcanic which the temperature of one (fuel) is above systems and their physical and chemical effects. the vaporization temperature of the other (cool- The magmatic thermal energy transformed by ant) (Buchanan and Dullforce, 1973; Buchanan, interaction with external water is partitioned 1974). Application of FCI theory to hydrovolcanic into many possible forms, including mechanical eruptions is described in Wohletz et al. (1995a), fragmentation of the magma and , Zimanowski et al. (1991), and Zimanowski (1998). excavation of a crater, acceleration and disper- Figure 11.5 depicts a hypothetical geologic sys- sal of tephra, seismic and acoustic perturba- tem in which magma (fuel) explosively interacts tions, and chemical processes such as solution with water-saturated sediments (coolant). The and precipitation, and mass diffusion. Section stages of a FCI are as follows: 11.4.4 addresses energy partitioning further.

(1) Initial contact and coarse mixing of fuel and 11.3.2 Physical properties of magma coolant, growth of vapor film; Magma generally is treated as a three-phase sys- (2) Quasi-coherent collapse of all vapor films in tem of melt, solids, and gases (see Chapter 4). the premix caused by a triggering pressure The physical properties of magma are strongly pulse (seismically induced or by local over- controlled by chemical composition, the propor- expansion), leading to direct contact of fuel tions and types of solid content (, and coolant; xenocrysts, and ), presence of exsolved (3) Cycles of enhanced fuel–coolant heat trans- and dissolved gases, flow speed, and tempera- fer, rapid (<1 ms) coolant expansion, fine ture. Natural melts have temperatures fragmentation of fuel, producing super- that exceed the solidus by ~100–200 K, such heated and pressurized water, and explosive that the heat content of magma comprises a energy release; significant proportion of latent heat. However, (4) Volumetric expansion of fuel–coolant mix- during rapid cooling processes relevant to many ture as superheated water transforms to magma–water interactions, the release of latent superheated steam. heat is negligible, because quenching (and the formation of glass) takes place, rather than The process does not necessarily evolve through . all these stages and may be arrested, for instance, Magma is the result of interaction before mixing or explosion. between internal friction of the melt and

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Figure 11.5 Hypothetical setting of subsurface hydrovolcanic activity, showing (a) initial contact of magma with water-saturated sediments, (b) vapor film growth, (c) mixing of magma with the sediments, and (d) expansion of the high-pressure steam in an explosion (adapted from Wohletz and Heiken, 1992).

1498 K but also strongly depends on the shear rate (Fig. 1473 K 11.6). At high confining pressures (i.e., at depth within the crust or beneath water), magma vis- a s) 102 cosity depends primarily on the behavior of

, η (P dissolved volatiles, and will generally decrease with increasing pressure. Although this pres-

Viscosity sure-dependence is not yet well characterized, it is the viscosity contrast between magma 101 and water that primarily governs interaction 10–2 10–1 100 101 dynamics, so little effect is expected on the

· –1 Shear rate, γ (s ) hydrodynamics of the magma–water system at increasing pressure. Figure 11.6 Shear stress dependent viscosity of a basaltic The intensity of magma–water interac- melt (<5 % crystals, <2 % bubbles), at two temperatures. tions reflects the efficiency of heat transfer The lines are viscosity calculations from a power-law model at the magma–water interface, which in turn (modified from Sonderet al., 2006). reflects the degree of magma fragmentation. Fragmentation can generally be viewed as tak- mechanically coupled compressible bubbles and ing place in one of two regimes, depending upon incompressible (see Chapter 4, Section whether the characteristic deformation time is 4.5). Crystals are not only characterized by vari- greater or less than the mechanical relaxation ous morphologies, but also by various interfacial time of the melt: (1) hydrodynamic fragmentation couplings (e.g., wetting angles). It is therefore is restricted to deformation of two-dimensional­ not surprising that the rheology of magma needs interfacial areas (boundaries between the melt to be addressed by a non-­Newtonian model. and ) and is most efficient in systems sub- Viscosity is not only temperature dependent, jected to high strain rates (rapid flow) at low

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, low interfacial tension, and high A crucial parameter for the behavior of water density contrast between the accelerating fluid at a high temperature interface is the phase and surrounding media; (2) brittle fragmenta- transition temperature of water into steam. In tion is the result of three-dimensional crack steady-state, equilibrium thermodynamic condi- growth caused by strain that exceeds the elas- tions, the phase transition occurs at the boiling tic properties of a medium (e.g., bulk modulus; point, which depends on the ambient pres- cf. the stress and strain-rate criteria discussed sure. However, dynamic (short duration) events in Chapter 4, Section 4.6). For hydrodynamic require application of quasi-steady-state thermo- fragmentation to produce fine ash (< 63 μm), dynamic models. Instead of the phase change the required accelerations are extremely high. occurring at the boiling point and spreading out- However, brittle fragmentation can readily ward from discrete nucleation sites, the water is produce fine ash particles through a steady heated so rapidly that it greatly overshoots its increase in strain. The features of natural parti- boiling temperature, reaching the homogeneous cles, experimental results, and theoretical mod- nucleation temperature (HNT), where all of the els clearly show ash generation is dominated by water spontaneously changes state. The boiling brittle fragmentation (Zimanowski et al., 2003). regime of water under atmospheric conditions The material parameter that controls the brittle- starts at 373 K and the HNT is reached at ~583 ness of magma (i.e., the mechanical deformation K. Little is known about the pressure depend- energy needed to produce new crack surfaces) ence of the HNT. During extremely rapid heat is the critical shear stress. The range of critical transfer, as occurs in explosive MFCI, the crit- shear stress of magma is as large as the vari- ical point of water (22 MPa and 647 K) may be ability of viscosity, but in contrast to viscosity, exceeded, so that water exists as a supercritical the critical shear stress decreases with increas- fluid, with no phase boundary separating vapor ing silica content. Therefore, basaltic magma and liquid. Experimental observations indicate can be more than three orders of magnitude that water can remain at liquid state densities stronger than rhyolitic magma. Bubbles and even at magmatic temperatures during such crystals both weaken the structural strength interactions (Zimanowski et al., 1997b). and therefore reduce the critical shear stress. Given that > 60% of the Earth’s surface is Further information on experimental and the- covered by oceans, magma–water interactions oretical fragmentation studies can be found in predominantly involve seawater. Seawater is Hermann and Roux (1990), Zimanowski et al. a solution dominated by the presence of salts (1997a, 1997b, 2003), and Büttner et al. (1999, (mostly NaCl), and its thermodynamic behav- 2002, 2006). ior can be approximated by the two-component system of pure water and NaCl. Figure 11.7(a) 11.3.3 Water physics illustrates a p–T phase diagram for the system

The thermodynamic behavior of water is well NaCl–H2O that shows phase boundaries of the known from the steam-locomotion era, and pure components and projections of the phase many volcanological studies have approached boundaries for intermediate compositions. The the problem of magma–water interaction rather salinity of seawater results in critical conditions simplistically, by direct application of the first occurring not at a single point but along a curve law of thermodynamics. While this approach that connects the critical points of the two pure does provide some limiting conditions, it ignores end members. These phase relationships (Fig. complex issues concerning the multi-phase 11.7(a)) show that at any temperature two fluid (steam, water, magmatic particles, melt) system. phases can coexist and a single critical point Consideration of multi-phase fluid mechanics does not exist if solid NaCl is present. Critical places further constraints on the application of behavior occurs at pressures and temperatures thermodynamics to this problem, as described elevated from those of pure water to values by Delaney (1982) and Wohletz (1986), and approaching ~30 MPa and ~680 K for seawater embodied in studies of MFCI (Section 11.3.1). with a salinity of 3.2 wt % NaCl (Bischoff and

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is heated, solid NaCl is precipitated, which may greatly affect vapor nucleation (cf. White, 1996). The two-phase boundary of seawater (Bischoff and Rosenbauer, 1984) is similar to that of pure water for subcritical conditions in pressure– temperature space (Fig. 11.7(b)), but unlike pure water, the critical point projects nearly linearly to higher pressures and temperatures (~680 K at 30 MPa to ~750 K at 50 MPa). Below the critical point the two-phase region of seawater consists of liquid and low-salinity vapor, and above the critical point it consists of brine and high-salinity vapor. These phase relationships indicate that a vapor phase can exist even at pressures above critical for seawater. The dissolved solids in seawater also reduce the heat capacity of seawater by several per- cent relative to pure water. However, this effect is relatively small compared to the factor of ~2 variation in the heat capacity of pure water over the range of pressures and temperatures rele- vant to hydrovolcanism. For quantitative calcu- lations, therefore, it is convenient to substitute Figure 11.7 (a) Pressure–temperature diagram for the pure water as a workable proxy for seawater system NaCl-H2O adapted from Krauskopf (1967) from because the heat capacity and phase transition experimental data of Sourirajan and Kennedy (1962). This of seawater appear to have offsetting effects in diagram shows both pure H2O and NaCl end members situations of rapid heating; however, one must (dashed lines). The solid lines (bold are experimental data) bear in mind that details of heat transfer dis- schematically represent the phase boundaries connecting the cussed in Section 11.3.4 will likely be greatly pure end members. TPw and CPw denote the triple point and affected by the vapor nucleation and solid pre- critical points of H2O, respectively; TPs and CPs those points for NaCl. (b) The two-phase curve for standard seawater cipitation behavior of seawater. (3.2% NaCl) as a function of pressure and temperature, based The hydrodynamic properties of water, on data from Bischoff and Rosenbauer (1984). Note that such as viscosity, heat content, expansion coef- in this plot, pressure increases downward. The solid curve ficients, do not change drastically with increas- designates the boundary where pure water and seawater ing pressure, but they do with increasing boundary are nearly coincidental. The boundary for pure temperature. Little is known about the effect water terminates at its critical point, whereas the boundary of increasing pressure on interfacial tension. for seawater extends (dashed curve) to its respective critical point, along which it separates the stability regions of liquid However, it can be expected that this influence and a mixture of low-salinity vapor and liquid. The phase is small over the range of pressures reflecting boundary extends (dotted curve) from seawater’s critical submarine volcanic scenarios. The viscosity of point to higher temperature and pressures, separating the water increases slightly with increasing pres- liquid region from that of a mixture of high-salinity vapor and sure but strongly decreases with increasing brine. temperature. A salt content of 3–5 % should have no significant influence on viscosity or interfacial tension. Rosenbauer, 1988). Critical pressure is expected Because magma–water interaction can result at a depth of ~3 km in the submarine environ- in a relatively high-pressure and high-temper- ment. Figure 11.7(a) also shows that as seawater ature water phase, it is important to consider

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the variability of water properties near critical conditions. Figure 11.8 illustrates the variation of heat capacity, viscosity, and expansion coeffi- cients at 30 and 60 MPa, analogous to deep sub- marine environments at about 3000 and 6000 m depth, respectively. Note that heat capacity, viscosity, and the isobaric expansion coefficient vary rapidly in the range 600–800 K, near the critical-point temperature (647 K). This means that small changes in temperature produce large changes in properties that determine the thermodynamic and hydrodynamic behav- ior of water. The large thermal gradients near the magma–water contact will produce large Figure 11.8 Variation of physical properties of water at (a) 30 MPa (~3000 m depth) and (b) 60 MPa (~6000 m depth) pressure and velocity gradients, and perturba- as a function of temperature. The properties are shown as tions in water movement may result in hydro- curves designated by symbols and their corresponding SI dynamic instability. For example, a supercritical units and scale factors are: α, isobaric expansion coefficient 3 −1 –2 −1 fluid subjected to small pressure perturbations (×10 K ); α/β, pressure coefficient ((dp/dt)v; ×10 MPa K ) will tend to oscillate in density (Greer and where β is the isothermal expansion coefficient;ν , kinematic 7−7 2 −1 Moldover, 1981) between liquid and vapor states viscosity (×10 m s ); and cp, the constant pressure heat (e.g., growth and collapse of vapor bubbles). capacity (4.184 kJ kg−1 K−1). Note the sharp inflections The speed at which water flows from higher and discontinuities apparent near the critical temperature. to lower pressure regimes depends not only on (Adapted from Wohletz and Heiken, 1992.) the magnitude of the pressure gradient but also on the viscosity; rapid viscosity fluctuation may enhance or dampen convective currents. These magma thermal conductivity (~1 W m−1 K−1; pronounced fluctuations in properties are crit- Büttner et al., 1998, 2000, Ebert et al., 2003), ical to address with respect to magma–water and limited convection due to its high viscosity. interaction. This thermal-hydraulic system has Consequently, the temperature at the hot side received considerable attention for application of the interface drops rapidly, leading to solidifi- to coolant flow stability in nuclear reactors (e.g., cation of the magma. Because the thermal con- Ruggles et al., 1989, 1997). ductivity of quenched melt is more than twice that of its liquid state, a cooling front migrates 11.3.4 Physics of the magma–water into the magma. Once a certain thickness has interface solidified, the strain caused by the 1–3% decrease Magma provides a high temperature heat in volume at the glass transition temperature source, such that the magma–water system is leads to the formation of cooling cracks (ther- somewhat analogous to the thermal coupling mal granulation) and thus to fragmentation of and heat transfer to a liquid coolant (water or the magma into relatively coarse particles. This aqueous ) used in power plant systems process has the potential to greatly increase (e.g., Baierlein, 1999, Bejan and Eden, 1999). the interfacial area between magma and water. The major difference between magma–water The balance between quenching and surface interactions and power plant heat flow prob- area growth rates determines whether the heat lems is that the interface conditions in a boiler transfer is relatively steady, or whether the heat are of a fixed geometry, whereas the magma– transfer rate escalates. Section 11.4.1 discusses water interface is always dynamic. The heat flux the thermodynamic regimes within which the from the magma–water interface to the water heat transfer takes place. It is not only ther- greatly exceeds the heat flux from the magma mally induced fragmentation that modifies the to the interface because of the relatively low heat flux, but also external hydro-mechanical

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forcing, such as differential movement between (2) Metastable-film boiling: At a temperature con- magma and water, and seismic impulses (earth- trast below the HNT, a macroscopic vapor quakes), which can affect the stability of the film does not form; however, the thermal magma–water interface. Experimental stud- coupling is modified by local formation of ies suggest that external triggering is a major steam at the interface. At a temperature con- mechanism leading to the escalation of heat trast well above the HNT, metastable-film transfer into explosive conditions. For more boiling can take place in two cases. In Case 1, details on magma–water mingling/mixing and high ambient pressure causes a reduction of fragmentation physics see Morrissey et al. (2000) the thickness of the superheated steam layer and Zimanowski and Büttner (2002, 2003). of the film so that it reaches the same thick- In contrast to the hydrodynamics discussed ness range as the thermal boundary layer. in Section 11.3.3, the thermodynamic behav- In Case 2, hydrodynamic disturbances cause ior of the magma–water interface changes interfacial instability waves with amplitudes dramatically with increasing temperature and of the same scale as the total film thickness. hydrostatic pressure, and the effects of min- (3) Direct contact: At a temperature contrast eralization (e.g., seawater) are generally small below the boiling point of water and/or at by comparison. The thermal coupling between ambient pressures exceeding the critical magma and water is largely dependent upon point of water, the thermal coupling can be the interface temperature and development described using equilibrium thermodynam- of a vapor film between the magma and the ics, and it depends on the thermal proper- water. The vapor film consists of two regions: ties of the liquids and the hydrodynamics (1) a thicker hot layer of superheated steam at at the interface. The thermal properties of the same temperature as and directly in con- magma are not strongly affected by increas- tact with the magma, and (2) a thinner cool ing ambient pressure, however, the prop- layer (the thermal boundary layer) between the erties of water change markedly (Fig. 11.8). superheated steam layer and water, within For magma and water in direct contact, the which condensation and vaporization balance. thermal coupling is one to two orders of The temperature of the thermal boundary layer magnitude higher than that of stable and can be only microscopically defined because its metastable film boiling. thickness may be much less than a few mil- limeters. Because the interface temperature is Considering the effect of increasing hydrostatic a hypothetical value, dependent upon thermal pressure on the thermal coupling of magma and diffusivities of magma and vapor, it is useful water, the stable-film boiling regime is restricted to characterize the vapor-film by its tempera- to ambient pressures below about 1 MPa where ture contrast with the magma (i.e., the magma the vapor film thickness at the magmatic tem- surface temperature minus water tempera- peratures of the hot layer becomes critical. ture). In so doing, the transfer of heat energy Above ambient pressures of 10 MPa, the moder- between substances, known as thermal coupling, ating effect of metastable-film boiling can prac- can be classified in three regimes, with coup- tically be excluded. At thermal power plants, ling increasing from (1) to (3): where an optimum heat flux (i.e., direct con- tact) is required, the water pressure of the heat (1) Stable-film boiling: At a temperature contrast exchange system usually is set to 15–20 MPa for well above the water HNT and an ambient temperature contrasts comparable to magmatic pressure well below the critical pressure of conditions. The thermal coupling at a constant water, a macroscopic vapor film forms at the magma–water interface at water depths exceed- interface between magma and water and ing ~1 km (equivalent to 10 MPa) can therefore restricts the thermal coupling. The thickness be described well by the direct contact regime. of the vapor film is greater for high tempera- Magma fragmentation is an important ture contrasts and low ambient pressures. aspect of the interface physics. Wohletz (1983)

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discusses a number of hydrovolcanic fragmen- material relative to the shock front, and ur must tation mechanisms that create a complex inter- equal the sonic velocity of the shocked mater- face geometry, but the actual fragmentation ial (Courant and Friedrichs, 1948; and Zel’dovich mechanisms involved are generally categorized and Raizer, 1966). as hydrodynamic (ductile) and brittle (Section The C–J condition can be evaluated on a pres- 11.3.2). With increasing fragmentation, the sure–volume diagram (Fig. 11.9) that shows the interfacial surface area increases exponentially. shock adiabat (termed the Hugoniot and defined This increase in surface area is often referred to as the locus of points representing pressure– as mixing. The resulting mixture evolves from volume states achievable by shocking a mater- a pre-expanded combination of magma frag- ial from an initial state) and the release adiabat ments and high-pressure water and vapor to a (called the Rankine–Hugoniot curve or detonation post-expanded mixture of quenched fragments adiabatic). These adiabats are concave upward and steam, which typically forms an eruption and the detonation adiabatic exists at higher column. Although Kokelaar (1986) distinguishes volume states than the Hugoniot. The deton- contact-surface interaction (dynamics along an inter- ation adiabatic is defined by classical Rankine– face between a free body of water and magma) Hugoniot jump conditions for conserving mass, from bulk interaction (dynamics of a volume of momentum, and energy across a shock wave magma that confines water or water-rich - clas (Landau and Lifshitz, 1959; Zel’dovich and tic materials), both cases involve heat transfer Kompaneets, 1960; and Zel’dovich and Raizer, along the interface between magma and water. 1966). These conditions can be stated as: Experimental evidence indicates that most of ρρuu= ;(mass) (11.1) the fragmentation occurs during development 11 22

of interface dynamics and prior to expansion 2 2 pu+=ρρpu+ ;(momentum) (11.2) and eruption. Nevertheless, Mastin (2007) devel- 111 222 ops a strong argument for the occurrence of tur- 1 (11.3) EE21−=(pp12+ )(VV12− );(energy) bulent shedding of glassy rinds from fragments 2 in erupted jets of steam and ash. The physics of the dynamic interface are complex and poorly understood; however, experimental studies of shock waves associated with interface dynam- ics suggest the physics may involve detonation, Von Neuman Rankine–Hugoniot Curve i.e., the formation of an exothermic shock front Point (detonation adiabatic) whose propagation creates an explosion. Wohletz (2003) discusses how film boiling Ra

can be intimately linked to fragmentation and yleigh Line C–J Point

interface surface area increase. The interface Pressure

becomes not only thermodynamically unstable Hugoniot with metastable film boiling, it is also hydrody- (shock adiabatic) namically unstable because of large pressure, density, sound speed, and conductivity gra- (p , V ) dients produced by the film. Such instability 1 1 is prone to a kind of detonation, termed ther- mal detonation, especially if perturbed by some Volume external pressure wave, such as that produced by volcanic seismicity. In a thermal detonation, Figure 11.9 Pressure–volume diagram showing the the acceleration of the particle mixture by the relationships among the initial state of the magma–water

shock wave must produce a relative velocity, ur, mixture (p1, V1), the shock Hugoniot, the Rayleigh line, the high enough to satisfy the Chapman–Jouguet Von Neumann and Chapman–Jouguet (C-J) points, and the detonation adiabatic. (C–J) condition: ur is the speed of the shocked

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where ρ is mixture density, V is mixture spe- their MFCI experiments. In order to assess the cific volume = 1/ρ, p is pressure, u is mixture role of ambient pressure in hydrovolcanism velocity, E = E(p,V) is mixture internal energy, using detonation theory, the method of Board and subscripts 1 and 2 indicate the unshocked et al. (1975) can be used to calculate (using and shocked states. All notation is given in the Rankine–Hugoniot jump conditions) the Section 11.6. For an inert material, the shocked speed of a propagating shock wave through a thermodynamic states given by Eq. (11.3) define magma–water mixture. For mixture condi- the Hugoniot, which is typically determined tions of R ~ 0.5 (where R is the mass ratio of experimentally to derive the equation of state water to magma), a shock speed is ~300 m s−1. of a material. Two points on the Hugoniot, one This velocity defines a Rayleigh line slope, and

at the initial pressure (p1,V1 in Fig. 11.9) and the then the Rankine-Hugoniot conditions must be other at the pressure of the shock front (the specified in order to predict a detonation adia- von Neumann point or spike, since there is a tran- batic that touches the Rayleigh line at a sin-

sient pressure peak at this location), define the gle point of tangency, the C–J condition of (pcj,

Rayleigh line. For a reactive material in which Vcj). To induce mechanisms for melt breakup the shock induces vapor production in its wake (Fig. 11.10), the C-J condition requires that the

(either chemically for common explosives, or slip velocity us between the shocked melt frag-

physically for a magma–water mixture where ments and water is at least as large as ur. Using nearly instantaneous heat transfer produces the classical detonation theory described by vapor), the energy equation (Eq. 11.3) gives Eqs. (11.1)–(11.3), the relative velocity of the all possible states of the detonation adiabatic shocked mixture leaving the front is a function (Fig. 11.9). of the mixture pressure and specific volume at By combining Eqs. (11.1) and (11.2), one initial and C–J conditions:

obtains the value of (p2 – p1)/(V1 – V2), which is

a constant defined by the slope of the Rayleigh uprc=−()jcpV11()− V j . (11.4) line. The square root of this slope is propor- tional to the velocity of the detonation wave. For an idealized thermal detonation in which

Zel’dovich and Kompaneets (1960) show that the pcj ≈ 100 MPa (Board et al., 1975), Eq. (11.4) sets ur only possible steady state (in which a detonation at ~100 m s−1. For volcanic MFCIs, the approach

wave is sustained) is where the Rayleigh line is of Corradini (1981) yields a minimum ur of 60 a tangent to the detonation adiabat at the C–J m s−1 (Wohletz, 1986). Drumheller (1979) com- point. The points behind a propagating shock at bined the requirements for relative velocity and which the C–J condition exists define a surface melt breakup time into a critical Bond number (the known as the C–J plane or the detonation front (not Bond number is ratio of body and surface ten-

to be confused with the shock front). sion forces). By assuming a constant pcj, Wohletz Figure 11.10 shows a generalized concep- (1986) evaluated the critical Bond number with tual view of thermal detonation that includes respect to MFCI experimental data (Wohletz observed phenomena such as thermohydrau- and McQueen, 1984) to predict the effects of R lic fracturing and brittle reaction, described and ambient pressure on the development of by Zimanowski et al. (1997b) and Büttner and relative velocities and magma particle sizes. Zimanowski (1998), which lead to enhanced Optimal conditions for thermal detonation heat transfer and catastrophic vapor expan- exist at 0.5 < R < 2.0, for ambient pressures sion. Zimanowski et al. (1997a) document ≤ 40 MPa (Wohletz, 2003). With increasing ambi- experiments that show development of intense ent pressure, predicted relative velocities fall to shock waves in less than a millisecond under < 60 m s−1 (which is considered the lower limit extreme rates of cooling (> 106 K s−1) and stress for sustaining a detonation; Wohletz, 1986), and (> 3 GPa m−2). Yuen and Theofanous (1994) particle fragmentation decreases, meaning less demonstrate that application of detonation thermal energy is released in the wake of the theory successfully predicts results of many of shock wave.

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THERMAL DETONATION MODEL (1D) Figure 11.10 Schematic illustration Shock wave of thermal detonation (adapted from propagation Vapor explosion Wohletz, 2003; Board et al., 1975), showing Coarse mixture of propagation of a shock wave through magma and water a coarse mixture of magma fragments and water (vapor and liquid). The shock wave moves at velocity u and differentially accelerates the water and magma to

velocities of uw and um, respectively,

resulting in a slip velocity us, which decays behind the shock. The slip velocity must be of sufficient magnitude to cause fine fragmentation of the magma fragments by Pressure mechanisms that shear away the boundary Distance Magma breakup layer and cause interpenetration of the uw u ucj • Boundary layer water and magma (Taylor instability) before stripping the arrival of the C–J plane. At the C–J um • Taylor instability plane, the average mixture velocity is just (Film collapse) sonic with respect to the shock wave. Fine fragmentation causes an exponential rise

us = uw– um C–J Plane: ucj = u – c in heat transfer from the magma fragments to the water (characteristic conductive heat transfer times are ~ tens of µs for fine fragmentation) and catastrophic vapor expansion. 11.4 Modeling magma–water interaction 11.4.1 Experimental insights into magma–water interaction Experimental research on magma–water inter- action has been strongly influenced by FCI stud- ies in the context of nuclear safety research (e.g., Board et al., 1975; Corradini, 1981; Henry and Fauske, 1981; Theofanous, 1995). Since the early 1980s, laboratory experiments have been conducted on magma–water interaction (e.g., Wohletz et al., 1995a; Zimanowski et al., 1997a; Büttner and Zimanowski, 1998), in which high- temperature melt (predominantly remelted vol- canic rocks at ~1270 K) is brought into contact with water (Fig. 11.11). The philosophy of the experiments was to achieve a mesoscale dimen- sion, minimizing danger, costs, and experimen- tal effort, while fully representing the relevant Figure 11.11 Explosive interaction after injection of water physics. As discussed in Section 11.2.4, import- into molten basaltic rock (Physikalisch-Vulkanologisches ant information on natural explosive magma– Labor, Universität Würzburg). See color plates section. water interaction is found in the characteristics of the products (tephra). In the experiments, the physical processes can be observed and

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Figure 11.12 Products of thermal granulation experiments using remelted (Physikalisch-Vulkanologisches Labor, Universität Würzburg).

measured directly. To verify the relevance to direct contact producing about two orders of natural volcanic processes, the experiments magnitude greater heat flux per unit area than were designed to produce artificial tephra in stable film boiling (Section 11.3.4). Three heat statistically relevant quantities. Furthermore, flux regimes in the magma–water system can be the products were nearly identical to the nat- defined to allow classification of magma–water ural analogs in terms of grain-size, morphology, interaction and facilitate theoretical descrip- and chemical composition (Fig. 11.12). tion of the thermodynamics and hydrodynam- Experimental results demonstrate that the ics of the system (Wohletz, 1995; Büttner and intensity of magma–water interaction, and thus Zimanowski, 1998; Zimanowski et al., 2003; the danger potential of hydrovolcanic erup- Büttner et al., 2005): tions, depends on the efficacy of the heat trans- (1) Non-explosive interaction regime. The heat flux fer from magma to water, i.e., the amount of in the system does not create a water over- heat transferred per unit volume and time. As pressure (subcritical state of coolant). Steady discussed in Section 11.3.4, this heat transfer is thermodynamic sink conditions are main- directly correlated to the interfacial area (i.e., tained (i.e., water can always be described the size of the contact area between magma and as a passive heat sink and steady-state water per unit volume) and the interfacial coup- thermodynamic models are applicable). ling conditions. Heat transport in the magma to Fragmentation of magma is governed by its the interface is controlled by the temperature- rheology in an aqueous environment (e.g., dependent magma thermal conductivity; con- pillow formation) and/or by thermal con- vection can be neglected because of the high traction of melt (thermal granulation), lead- viscosity of magma and the short timescale. ing to passive fragmentation (Schmid et al., Heat transport in the water (having a viscos- 2010; Sonder et al., 2011). ity at least three orders of magnitude less than (2) Subsonic explosive interaction regime. The heat magma) away from the interface is controlled by flux in the system creates a water overpres- both conduction and convection, and therefore sure (critical state of coolant). Superheated is in principle much more effective. The ther- water is generated and complex phase tran- mal coupling at the interface can be described sitions occur. Steady-state thermodynamic in the three regimes: of stable film boiling, models need significant modifications, but metastable film boiling, and direct contact, with still are generally applicable. Fragmentation

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(of magma is dominated by brittle proc- (e.g., water vaporization and magma crystalliza- esses caused by the hydraulic forcing of the tion). During explosive interaction the energy coolant, but governed by the mechanical exchange happens rapidly, resulting in a min- properties of the magma (i.e., mechanical imum of energy going towards phase transitions deformation does not exceed critical shear and most going towards raising the temperature conditions and propagation of cracks is sub- of the water and cooling the magma. For this sonic with respect to the shear-wave vel- simple situation, the equilibrium temperature,

ocity). This regime leads to subsonic active Te, is between the temperature of the magma,

fragmentationAustin-Erickson et al., 2008). Tm, and that of the water, Tw. Therefore, simple (3) Supersonic explosive interaction regime. The heat energy conservation can be expressed as: flux in the system increases rapidly, due to a thermohydraulic feedback mechanism mcww()TTew− =−mcmm()TTme, (11.5) and high overpressure is generated in the water (supercritical state of coolant). Non- where m is the mass and c is the specific heat equilibrium thermodynamic models are capacity of the water (subscript w) and the recommended. Fragmentation of magma magma (subscript m), respectively. Equation (11.5) shows that T varies with the mass ratio is driven by hydraulic forcing of the cool- e of water to magma, m /m , denoted as R. In add- ant exceeding the mechanical properties w m and propagation of cracks is supersonic. ition, the specific heat ratio of water to magma Consequently a significant proportion (up is typically in the range of 3.0 to 4.0, depending to 80%) of the mechanical energy is released upon magma composition and the range of typ- as shock waves. This regime leads to super- ical water states at the surface of the Earth. For most cases, it is assumed that the ratio = c /c sonic active fragmentation. ξ w m ~ 3.5. Accordingly, we can rearrange Eq. (11.5)

The consequences for the eruptive behavior of to predict Te: the magma–water system depend on the inter- ξRTwm+ T Te = . (11.6) acting magma/coolant volume ratio, the rheo- 1 + ξR logical properties of the magma, the thermal and rheological properties of the coolant, the The value of Te can be thought of as an idealized hydrodynamic mingling energy, the ambient initial thermal equilibrium prior to explosive pressure, and the geometry of the mingling vaporization of the water, which allows thermo- space. In the following section, we discuss dynamic prediction of the resulting mechan- modeling techniques for steady state regimes ical energy of the interaction. Figure 11.13(a) 1 and 2 and the non-equilibrium conditions of shows some predictions of Te as a function of regime 3. R. However, this thermal equilibrium is just an idealization and many factors can cause initial 11.4.2 Multiphase equilibrium interaction temperatures to be higher or lower, thermodynamic models one of which is the composition of the water. Simple conservation of energy provides for a Water at the surface of the Earth may con- first-order assessment of magma–water interac- tain dissolved or suspended constituents (e.g., tions. For most cases, the magma temperature muddy water), or it may be contained within exceeds the water vaporization temperature, pores of rock or sediments. The effect on the which varies somewhat with composition thermal equilibrium can be evaluated as follows. and ambient pressure. During interaction, the For interactions between magma and saturated internal energy of the water increases by an rocks or impure water (White, 1996), rock and amount equal to that lost by the magma. While impurities act as heat sinks, and the mass ratio, most of that internal energy is involved in heat- Rr, is given as ing the water and quenching the magma, some mr Rr = , (11.7) of it may also be involved in phase transitions mm

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S φρ x = ww, w ρ (11.9) r

where Sw is the rock saturation (the volume frac- tion of pores filled by water), and ϕ is the rock porosity. For impure water with suspended rock

particles one can consider Sw = 1 and ϕ some- what less than unity (depending on the volume fractions of water and particles). From Eqs. (11.8) and (11.9), R may be approximated as:  ρ   V  RS w r , ≈ wφ     (11.10)  ρm   Vm  such that R can be constrained by measuring the volume ratio of lithic constituents and magma in samples of tephra or consolidated sediments such as peperite (a sedimentary rock contain- ing igneous fragments formed during contact of magma and wet sediments). Besides the effect on the value of R, one must

Figure 11.13 Thermal equilibrium (Te) is the idealized also consider the heat capacity of water impur- temperature that water (in saturated rock) can reach during ities or the porous rock containing the water. In

interaction with magma, and it is shown as a function of Rr Eq. (11.6), the heat capacity ratio ξ is replaced (mass ratio of wet-rock/magma). These plots show cases by the ratio involving wet rock instead of pure where the initial water temperature water is 298 K and the water, cr/cm. Here cr is the effective heat capacity −1 magma heat capacity cm is ~ 1 kJ kg . The range in critical of the impure water or water–rock mixture, temperature reflects the effect of dissolved solids. (a)T is e which can be approximated as a function of x shown for different initial temperatures of magma interacting w by: with water (100% volume fraction, where Rr = R). (b) For basalt (solid curve; 1473 K) and (dashed curve; 1173 cxrw=+cxww()1,− cs (11.11) K) interacting with wet rocks, Te is shown as a function of rock water volume fraction, for which 10 and 70% bound a where cs denotes the heat capacity of the solid

range centered on 40% (xw ≈ 0.2; Eq. (11.9)), representing a constituents, and cs ≈ cw / 4. porous, fully saturated sandstone. For a typical magma in contact with

water-saturated rock at 298 K, Te decreases with

Rr, as shown in Figure 11.13(b). It is evident that,

where mr is the mass of rock and pore water (or for magma interacting with wet rocks, Te can water plus impurities). In these situations, the exceed critical temperature (647 K (pure) to 720

water/magma mass ratio is K (5 wt.% dissolved solids)) where Rr < 1.0 (bas-

alt) and Rr < 0.5 (rhyolite). Where critical tem- ρrrV (11.8) Rx==wrRxw , perature is exceeded during interaction prior ρ V mm to explosive expansion, supercritical pressures

where xw is the water mass fraction in a rock will be created.

(or impure water) volume Vr having bulk dens- Thermal equilibrium is probably never

ity ρr, interacting with a magma volume Vm hav- reached during the time span of interaction

ing density ρm. For magma interacting with pure because of the insulating property of the vapor

water alone, xw = 1, ρr = ρw, and Vr = Vw. In order to film that forms at the magma–wet-rock inter- cast this relationship into terms of rock porosity face. Because of its relatively low thermal con- and saturation, which are commonly measured ductivity, a vapor film can greatly decrease the quantities, we write the following equation: rate of heat transfer from the magma to the wet

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rock, allowing gradual quenching. With grad- In addition, the rapid heat loss from the magma ual quenching, the film slowly heats water in by this continued vapor-film instability leads to the rocks near the magma at a rate balanced magma quenching and possible granulation. by the heat transfer away from it by the con- With these interface phenomena, the magma vective movement of pore water, provided that becomes increasingly fragmented, leading to pores are interconnected. However, such passive larger surface areas for heat transfer and larger quenching is not always the case. Consider the volumes of superheated water and vapor. hypothetical instantaneous interface tempera- Further thermodynamic predictions about

ture, TI, attained by the initial contact of magma magma–water interactions can be derived from with water that can be estimated (Cronenburg, analysis of the thermodynamic work done by 1980) by: the expansion of water from its initial thermal equilibrium. We have assumed that nothing Tk Tk mm()κκm + ww()w is known about the mechanism of the contact TI = , (11.12) kk between magma and water, but that it results in ()m κκm + ()w w production of high-temperature and high-pres- where k and κ are thermal conductivity and sure water that may explosively decompress. diffusivity, respectively, and subscripts m and Thermodynamic work is manifested in the frac- w refer to magma and water, respectively. For ture and excavation of country rock to form a the contact of a typical basalt magma with pure crater, fragmentation of the magma into fine-

water, TI approaches 1000 K. Because of the rap- grained debris, and ejection of these fragments idity of heat exchange, water may exist in the in an expanding jet of steam. For a hydrovol- metastable state of superheating in which it is a canic eruption, it is necessary to find the work liquid well above its vaporization temperature. potential for expansion of the steam to atmos- A consequence of this superheated state is that pheric pressure. Calculation of the true poten- it continues to absorb heat at a high rate, reach- tial requires determination of a complex set of ing temperatures well in excess of its spontan- boundary conditions unique to each eruption, eous nucleation temperature (i.e., HNT ~583 but for simplicity and generality, one can make K). As temperatures approach the critical tem- some standard assumptions that allow analyt- perature (647 K), instantaneous vaporization by ical calculation of maximum potential. For the homogeneous nucleation produces a vapor film system consisting of a mixture of magmatic par- that expands rapidly and is highly unstable – ticles and water and the surroundings being the that is, it can expand well beyond the thermo- volcano vent structure and the atmosphere, we dynamic equilibrium thickness. In so doing, the assume that: vapor becomes supercooled, leading to spon- (1) All heat lost by the magma during the erup- taneous condensation. The condensation then tion is transferred to external water; leads to a rapid collapse such that liquid water (2) Liquid water and magma are incompressible, impacts the magma surface with a finite amount so that for each the specific heats remain con- of kinetic energy, leading to a second spontan- stant with changing pressure and volume; eous vaporization event. This cyclic vapor film (3) The specific volume (reciprocal of density) of growth and collapse is repeated continuously, liquid water is small compared to that of its typically with a frequency of up to 1 kHz (analo- vapor; gous to the Leidenfrost phenomenon of a drop of (4) behaves as a perfect gas and the water vibrating on a hot metal surface). Vapor magma volume does not change during the film instability can generate enough kinetic eruption (i.e., it does not vesiculate or con- energy to distort the interface between the tract during cooling). magma and wet rock, as well as cause failure of the host rock. In some cases, film collapse can Some heat is lost from the magma to both the lead to jets of water-saturated rock fragments country rock and the atmosphere during erup- penetrating the magma surface (White, 1996). tion, and magma does exhibit a finite volume

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change on quenching. The combined effect of these caveats may account for energy discrepan- cies of several percent. The other assumptions concern the exact thermodynamic properties of water. At supercritical states a deviation from ideality of ≥ 10% is expected, but since most volume change occurs at subcritical states for adiabatic expansion, the assumptions intro- duce very little error into calculations, espe- cially when extended or extrapolated steam table data are considered, such as by using a modified Redlich–Kwong equation of state (Burnham et al., 1969; Holloway, 1977; Kieffer and Delany, 1979). Online Supplement 11A (see end of chapter) provides a derivation of the thermodynamic work of the interaction. From this analysis, the following predictions can be made for magma interacting with water or water-saturated rock. The conversion ratio of a magma–water interaction is the fraction of magmatic heat converted to thermodynamic work, and is a measure of the mechanical energy released by the interaction. Figure 11.14(a) shows conver- Figure 11.14 (a) Conversion ratio and (b) condensed

sion ratios as a function of the wet-rock/magma water fraction vs. wet-rock/magma mass ratio, Rr. Solid

mass ratio Rr for the interaction of a basaltic curves are isentropic values and dashed curves are magma with saturated rocks (with 40% porosity pseudo-isothermal values (labeled isothermal). Conversion ratio is the percentage of the magma’s heat energy that is and xw = 0.2). Also shown are curves depicting converted to thermodynamic work during interaction with pure water–magma interaction (i.e., xw = 1). Both water. This plot shows results for a basaltic magma at 1473 isentropic expansion (in which steam separates K interacting with water at 298 K (solid curves, 100% water from fragmented magma) and pseudo-isother- by volume; xw = 1) compared with those for water-saturated mal expansion (in which steam and fragments sediments (dashed curves; 40% porosity, 100% saturated; remain at the same temperature, denoted in xw ≈ 0.2; Eq. (11.9). The condensed water fraction figures as isothermal) are calculated. Note that, represents the fraction of interacting water that condenses compared to pure water–magma interactions, to a liquid state after expansion. For water-saturated wet-rock–magma interactions have lower con- sediment interactions having Rr > 1.3 (isentropic) to 3 version ratios (are less energetic) with optimum (pseudo-isothermal), a dominant fraction (> 0.6) of vaporized water will condense to liquid during expansion to ambient conversion ratios near Rr = 1.0. Figure 11.14(b) pressures, and wet sediments have the ability to convectively shows the fraction of vaporized water that con- carry heat from the magma, behaving as fluid substances denses back to liquid during expansion. For rather than explosive vapor-rich ones. Using this criterion, interactions at low Rr values, most of the water an arbitrary region, separating explosive from non-explosive

remains in the vapor state after expansion, behavior, may be drawn over the range 1.3 ≤ Rr ≤ 3.0. leading to the likelihood of explosive behavior.

In contrast, for Rr > 3.0 (wet rock), a dominant portion of the steam condenses during expan- Wet-rock–magma interaction may also occur sion, producing a liquid–particle system. This at depths below the Earth’s surface in hydro- latter behavior allows convective heat transfer thermal systems of elevated temperature and that promotes passive cooling, which is less pressure. Figure 11.15(a) shows that when pore likely to be explosive. water is at elevated temperatures (e.g., 358 K), a

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the pseudo-isothermal expansion of the vapor– fragment mixture follows paths intermediate to those of isentropic and pure isothermal expan- sions (Kieffer and Delany, 1979). The phase diagrams also illustrate the three regimes of heat flux described in Section 11.4.1. The non- explosive interaction regime generally corres-

ponds to interaction ratios Rr > 10.0, for which isentropic expansion is simply a vertical path in Figure 11.16(b) and water expansion is always subcritical. The subsonic explosive interaction regime approximately corresponds to interac-

tions where 1.0 < Rr < 10.0. For adiabatic and pseudo-isothermal expansions in this regime, water is converted to steam at high pressure but then some of it condenses during expansion.

The supersonic explosive interaction regime (Rr < 1.0) involves significant expansion in super- critical states, especially for pseudo-isothermal conditions. The supercritical state is consid- ered further in Section 11.4.3. It is important to note that the correspondence of these heat- flux regimes to the range of water/magma mass Figure 11.15 (a) Conversion ratio and (b) condensed ratios and modeled thermodynamic states is

water fraction vs. wet-rock/magma mass ratio Rr for very approximate because other factors contrib- hydrothermal conditions of elevated pore-water temperature ute to determining the actual heat-flux regime, (358 K) and elevated hydrostatic pressure (30 MPa). These namely the rheology of the magma, and con- curves represent a saturated sediment containing 40% by ditions of contact and subsequent mixing and volume water (x 0.2); solid curves are for isentropic w ≈ fragmentation. expansion and dashed curves are for isothermal expansion. Figure 11.16 also illustrates an important point that experimental studies confirm: the critical point of water is not necessarily a limit- slightly greater fraction of the magma’s thermal ing factor in vapor explosions. It has been com- energy is converted to thermodynamic work and monly assumed that interactions occurring at optimum peaks occur at slightly higher values confining pressures above the critical point

of Rr when compared to values shown in Figure of water (22 MPa) cannot result in explosions, 11.14(a). However, elevated hydrostatic pres- because water exists as a supercritical fluid for sure (e.g., 30 MPa) does not have much effect. which there is no liquid–vapor phase boundary. Still, conversion ratios at hydrostatic pressures Considering the thermodynamic paths illus- exceeding critical pressure are sufficiently large trated in Figure 11.16(c), where interactions that such interactions could be explosive. are nearly an order of magnitude in excess To further illustrate these calculations, of critical pressure, release of the interaction Figure 11.16 depicts water phase diagrams with pressure involves a large volume increase, pressure-temperature-volume-entropy (p-T-V-S) especially for pseudo-isothermal expansion, relationships for theoretical initial equilibrium at rates determined by local sound speeds and final states. Both isentropic (Fig. 11.16(a)) (Kieffer and Delany, 1979). Although the inter- and pseudo-isothermal (Figs. 11.16(b,c)) paths actions extend to pressures exceeding critical, are shown. Isentropic expansion follows paths it is not yet clear if explosive expansion can of constant entropy (Fig. 11.16(a)), whereas, initiate at these high pressures (equivalent to

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~2.2 km ocean depth). However, the presence of other pressure perturbations, such as those caused by seismicity, host media failure, and the vapor-film dynamics, add to the likelihood that expansion will lead to thermohydraulic explosion.

11.4.3 A non-equilibrium thermodynamic model In the supersonic explosive interaction regime (regime 3; Section 11.4.1), water reacts as a super- critical fluid that undergoes complex phase tran- sitions that are difficult to model. Consequently, the steady-state models described in Section 11.4.2 do not fully capture some important physics. Experimental observations reveal that magma quenches with little or no phase change and can therefore be described as a supercooled liquid (i.e., glass; Section 11.3.2), the physical properties of which are well known and/or can easily be measured. It is therefore much simpler to model the non-equilibrium behavior of the magma rather than the complex phase reac- tions of the water. The basis of the model is the heat conduc- tion equation of standard, non-equilibrium ther- modynamics applied to the individual magma fragments:

∂T(,xy,,zt) 2 (11.13) =∇kT , ∂t mx(,yz,,t)

where T(x,y,z,t) is the time-dependent temperature-

field inside the fragment and km is the thermal conductivity (approximately constant for super- cooled liquids). The solution of this partial dif- Figure 11.16 Phase diagrams illustrating calculations of ferential equation can be found in Büttner et al. wet-rock–magma interaction, using the method of Wohletz (2005). Inserting the result into the heat flux (1986). Labeled points (diamonds) are the theoretical initial equation yields: equilibrium condition for interactions of various R (from 0.01 r ∆Q ∆T to 50.0; x 0.2). The critical point (CP), liquid (L), vapor (V), ()t (,xy,,zt) (11.14) w ≈ = mcm , and liquid plus vapor two-phase (L+V) regions are shown. (a) A ∆t ∆t p–V diagram shows expansion volumes and release isentropes where ΔQ /Δt represents the time-dependent (kJ kg−1 K−1) followed during isentropic expansion of vapor. (t) For all interactions, water expands into the two-phase region. heat flux,m is the mass of the fragment, cm is the (b) A T–S diagram for pseudo-isothermal expansion shows specific heat capacity, and ΔT(x,y,z,t)/Δt is the vari- the increase in entropy as water stays in thermal equilibrium ation of the temperature field in the fragment

with magma fragments. For Rr < 1.3 water expands into the with time. Analysis of experimentally produced vapor field. (c) Ap–T diagram shows the variation in water particles yields the size, shape and number of temperature during pseudo-isothermal expansion. For more particles produced for a given starting mass. detail on these diagrams see Kieffer and Delany (1979). Summing the heat transferred from all particles

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the surface energy consumed by fragmentation, and S is the entropy generated by creation of new surfaces during fragmentation. In order to apply this general function, one must assume: (1) Magma breaks up due to internal stress (i.e., in a brittle regime); (2) Thermodynamic equilibrium is not achieved during crack growth within the magma until its fragmentation into a number, n, of spher- ical particles; (3) Isothermal conditions exist during fragmentation; (4) Surface energy is not recoverable, and the Figure 11.17 Results of heat flow modeling of a MFCI entropy of the system increases; and experiment, using the non-equilibrium thermodynamic approach. The different curves represent end members of (5) During increments of time the stress is fragment morphology (e.g., cubes, spheres), which show treated as constant. the limited influence of shape on the results. The dashed Following the methods of Grady (1982, 1985), line represents the total measured heat transfer during the each term in Eq. (11.15) can be evaluated as fol- experiment, which ended in explosion after 700 μs. (After Büttner et al., 2005.) lows. The value of U is taken as the strain energy in the magma just prior to failure, and it is a function of n, the number of particles formed by fragmentation:

allows calculation of the total energy release and 223 µε    6 −23 53 (11.16) therefore the explosivity of this regime. Figure U =     nV, 11.17 shows the results of modeling the heat 8  c   π transfer during an MFCI experiment. The model where μ is the magma bulk modulus (μ = ρ c2; curves predict well the total heat flow measured m ρm is density and c is sound speed), ε̇ is the during the experiment (dashed line), confirm- strain rate, and V is the total volume. Yew and ing that this technique is robust with respect Taylor (1994) give a simple geometric argument to the magma properties and fragment size and to constrain B, the total energy dissipated by morphology. It is possible to extend this treat- fragmentation: ment to natural eruptions to evaluate the total 13 2 energy release during explosive events by using  π  3K  13 23 (11.17) B = c nV , grain size and shape analysis of natural tephra    2   6  ρmc  samples, together with appropriate magma properties and an assessment of the total mass where Kc is the stress intensity factor. The of the deposit. entropy S of new surfaces can be evaluated by the configuration entropy ofn cracks that could 11.4.4 A brittle fragmentation model occupy N + n sites in a volume V (Varotsos and The above non-equilibrium behavior is intim- Alexopoulos, 1986): ately tied to magma fragmentation by appli-  Nn+ ! Skln ( ) , (11.18) cation of the Gibbs thermodynamic function = B    Nn!! (Yew and Taylor, 1994), which is expressed­

as: where kB is the Boltzmann constant and N is the total number of normalized new surfaces. GUBTS, =+− (11.15) In order to apply Eq. (11.15) to the state where G is the Gibbs free energy, U is the pre- where fragmentation has occurred, thermo- fragmentation internal energy of the magma, B is dynamic equilibrium is assumed at a constant

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temperature and pressure where the Gibbs function is at a minimum: 11.5 Summary  ∂G = 0. The following points summarize a basic under-   (11.19) ∂n Tp, standing of magma–water interactions during volcanism (hydrovolcanism). The numerous By taking the derivative of Eq. (11.15), setting references cited in this chapter will permit the the average fragment size (s; Grady, 1982, 1985; reader to gain a deeper appreciation of the phys- Yew and Taylor, 1994) to: ics of magma–water interaction. 13 23  6V   3 K  s = ≈ 2 c , (11.20) • Magma–water interaction is not a rare pro-      πn  2 ρεm c  cess, and is relevant for nearly every volcanic system. Earth is a water planet and most of the strain rate associated with fragmentation its volcanic activity is found underwater, that produces particulates of an average size s especially at the mid-oceanic ridges (see is expressed by: Chapter 12). • A well-developed catalog exists of diagnostics K 12  c (11.21) that are useful to characterize magma–water ε = 3 . ρcs interaction from analysis of hydrovolcanic products, including vent/construct morph- The value of s can be constrained by inspection ology, deposit dispersal, tephra bedding of tephra sample subpopulation modes, dis- sequences, and ash analyses. However, a quan- tinguished either by shapes and textures (e.g., titative analysis in terms of energy balances Heiken and Wohletz, 1985, 1991) or by mode and hazard mitigation needs experimental dispersion (e.g., Wohletz et al., 1995b). Using data and theoretical concepts. an estimated stress intensity value Kc, fragmen- • Whereas understanding of water physics is tation strain rates can be determined from Eq. highly evolved because of the rich tradition of (11.21). The strain rates are then related to Ef, power plant technology, magma physics are the fragmentation energy, by: less well understood, and volcanologists are just beginning to develop the complex phys- 1 6 23    −23 53 (11.22) ics needed to handle this multiphase, multi- Enfm≈   ρε V . 120  π component, subliquidus system. Further data are required for basic physical parameters This treatment can be applied to natural such as magma heat conductivity and viscos- hydrovolcanic deposits if the particle density ity, without which the dynamic heat transfer and whole deposit volume and grain size dis- at the magma–water interface cannot be reli- tribution are known (see Büttner et al., 2006, ably quantified. for details). Summation of the specific frag- • The key process for understanding the energy mentation energy for each grain size fraction balance of hydrovolcanism is magma frag- and for the total deposit volume allows calcu- mentation. To make the thermal energy avail- lation of the total fragmentation energy for able for conversion to mechanical energy via the eruption. Using an assessment of the par- the working fluid water, heat must be trans- titioning of total eruption energy among frag- ferred. Because heat transfer is dominated by mentation (50–75%), kinetic (15–30%), seismic conduction, the energy transfer rate increases (10–20%), and acoustic (<5%) energies, Büttner with the increased surface area produced by et al. (2006) calculated total eruption energies fragmentation. Energy release on a short time- on the order 1014 J (~ 25 Mt of high explosive scale (explosion) during hydrovolcanic proc- equivalent) for two eruptions at Campi Flegrei, esses is likely a result of a positive feedback Italy. mechanism in which heat exchange drives

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fragmentation, which in turn drives escalat- N number of new surfaces produced by ing heat transfer rates. fragmentation event • Three heat-flux regimes can be defined to p pressure (Pa)

cover the range of magma–water interaction pcj pressure at Chapman–Jouguet point (Pa) dynamics from non-explosive passive cooling R mass ratio of water to magma

to supersonic thermal detonation. Modeling Rr mass ratio of saturated rock (or impure techniques comprise steady-state thermody- water) to magma namics for the non-explosive to explosive inter- Q heat (J) action regimes, based on thermal power plant s average fragment size (m) physics. Modeling of the supersonic explosive S entropy generated by fragmentation (J)

regime, however, benefits from application Sw rock saturation (pore volume fraction of non-equilibrium thermodynamics, which filled with water) develops a strong link between explosion t time (s) energy and fragment size distributions. T temperature (K)

Te equilibrium temperature (K)

TI instantaneous interface temperature (K)

11.6 Notation Tm magma temperature (K)

Tw water temperature (K) B surface energy consumed by u mixture velocity (m s−1) fragmentation (J) ur velocity of shocked material relative to c sound speed (m s−1) shock wave (m s−1) cm specific heat capacity of magma U internal energy of pre-fragmentation (J kg−1 K−1) magma (J) cp specific heat capacity at constant V mixture volume (m3) or specific volume pressure (J kg−1 K−1) (m3 kg−1) c specific heat capacity of saturated rock r Vcj volume at Chapman–Jouguet (J kg−1 K−1) point (m3) c specific heat capacity of solids dissolved 3 s Vm volume of magma (m ) in water (J kg−1 K−1) Vr volume of porous rock or impure cw specific heat capacity of pure water water (m3) (J kg−1 K−1) 3 Vw volume of water (m ) E mixture internal energy (J) W thermodynamic work (J) E fragmentation energy (J) f xw mass fraction of pure water G Gibbs free energy (J) α isobaric expansion coefficient (K−1) –23 kB Boltzmann constant (1.3806488 × 10 J β isothermal expansion coefficient (Pa−1) −1 kg ) ε̇ strain rate (s−1) km thermal conductivity of magma ϕ porosity −1 −1 (W m K ) γ̇ shear rate (s−1) kw thermal conductivity of water η dynamic viscosity (Pa s) (W m−1 K−1) 2 −1 κm thermal diffusivity of magma (m s ) K stress intensity factor (Pa m1/2) 2 −1 c κw thermal diffusivity of water (m s ) m mass of particle (kg) μ magma bulk modulus (Pa) mm mass of magma (kg) ν dynamic viscosity (m2 s−1) mr mass of porous rock (or impure water) ρ mixture density (kg m−3) (kg) −3 ρm magma density (kg m ) m mass of water (kg) −3 w ρr rock density (kg m ) n number of particles produced by −3 ρw water density (kg m ) fragmentation event ξ ratio of water and magma specific heats

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