Volcanology and Geothermal Energy "d0e848"

Explosive Eruptions and Quantitative Models

For reasons that will be discussed later in Chapter 2, explosive volcanic eruptions are significant in the development of geothermal systems. Over the past two decades, our general knowledge of explosive eruption mechanisms has evolved from the application of theoretical models to quantitative field data. For example, a tripartite field classification scheme shown in Table 1.2 is based upon the assumption that products of explosive eruptions are emplaced as pyroclastic deposits by fallout, flow, and surge.

Fig. 1.8
Plot of rare-earth element (REE) partition
coefficients for clinopyroxene/glass in various
magma compositions. SiO₂ content greatly
affects these values; similar trends towards
high partition coefficients with increasing SiO₂ content are evident for other phases, including
Fe-Ti oxides, fayalite, and feldspars.
(Adapted from Hildreth, 1981.)

Walker (1973) showed how grain-size characteristics and dispersal area of a pyroclastic deposit can be used to deduce the type of volcanic eruption from which it was produced (Fig. 1.11).

A stylized explosive eruption system is depicted in Fig. 1.12. Although relatively little is known about subsurface processes in the volcanic conduit, the behavior of eruption columns has been deduced from observations; this information allowed Wilson (1976) and Sparks and Wilson (1976) to formulate physical conditions in explosive eruption columns (see also

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Fig. 1.9
Illustration of processes affecting magma chamber differentiation; idealized thermogravitational
column is at left. Early stage crustal heating by intermediate to basaltic volcanism triggers
crustal melting and buoyant rise of magmas (diapirism), followed by segregation of liquid
phases in a silicic magma chamber. Within the silicic chamber, convection enhances diffusion
processes such as that of Soret (between cold walls and hot center), volatile mass transport (dots),
and wallrock exchange. Stippled pattern depicts enriched zones in the magma chamber roof
and at both ends of the thermogravitational column. Such differentiation processes probably
last longer than the eruptive history of the associated volcanic field,
typically 10⁶ to 10⁷ years in large systems.
(Adapted from Hildreth, 1981.)

Wilson et al . 1980). The basic equations for the eruption are

which express conservation of mass and momentum, respectively, for one-dimensional flow along the subsurface volcanic conduit. The h and r_c = vertical distance and conduit radius, respectively; g = gravitational acceleration, u = the magma's velocity, p = r_g RT (perfect gas law pressure), and r = bulk density. f_h is the factor expressing frictional loses along the conduit walls. The relationship among bulk (r_b ), solid (r_p ), and gas (r_g ) densities is expressed as in Eq. (1-4).

Equations (1-5) and (1-6) earlier in this chapter are solutions for the conservation relationships of Eqs. (1-11) and (1-12). This quantitative approach to understanding volcanic phenomena is well summarized by Head and Wilson (1986) for a variety of eruption types, including effusive processes, Strombolian (scoria cone), Hawaiian (lava fountain), Plinian

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Fig. 1.10
Evolution of silicic magma chambers as a function of tectonic environment. These idealized
diagrams illustrate the profound effect of crustal stress on the size and geometry of evolving
magma bodies. The top diagrams depict basalt-rhyolite magmatism in regions of crustal
extension for (a) early and (b) advanced stages. The lower diagrams show two possible
stages of dominantly intermediate volcanism in convergent tectonic regions:
(c) early stage and (d) intermediate stage; the late stage shown in Fig. 1.9.
(Adapted from Hildreth, 1981.)

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Table 1.2. Tripartite Classification of Pyroclastic Materials^a
	*Emplacement Mechanism*	Areal Dispersal	Deposit Textures	*Grain-Size Characteristics*	*Eruption Mechanism*
*Class*
Fall
	Ballistic, aerodynamic drag modified; suspension	Symmetrical along wind vectors; relatively wide-spread	Mantles topography; normally and reversely (Plinian) graded beds	Well sorted by terminal velocity; coarse near vent, fines at distance	All
*Flow*
	Steady, lateral movement over substrate by grain flow, saltation, suspension	Directed, radial from vent, following drainages up to tens of kilometers	Massive; confined to topographic lows; fine base with reverse pumice grading; some bedded intervals	Poorly sorted fine to coarse ash with near vent breccia	Plinian, Vulcanian, Peléean, Merapian
*Surge*
	Unsteady, lateral blast over substrate by pulsating saltation, suspension, and grain flow locally accelerated by shocks	Directed, partially confined by drainages (some mantling), up to several vent radii from source	Thinly bedded, showing variety of bedforms: dunes, plane beds, massive beds, wet sediment deformations	Poor to moderate sorting of fine to coarse ash; zones of fine ash depletion	Vulcanian, Surseyan, Plinian, phreatic, hydrothermal
^a See Glossary (Appendix G) for definition of terms.

(pumice and ash columns), Vulcanian (cannon-like explosions), and Peléean (lava dome destruction).

Pyroclastic Fallout

Pyroclastic fall deposits (Fisher and Schmincke, 1984) are characterized by their relatively well sorted size characteristics, topography-blanketing dispersal, and graded bedding, but lack of other internal bedforms. The emplacement characteristics of these deposits are controlled by the terminal fall velocities of individual pyroclasts (Walker et al ., 1971; Wilson, 1972). One important component for this modeling is the assumption that eruption columns behave as thermal plumes in which the height of the plume (h_t ) is proportional to the quarter root of the mass flux (d m/d t):

The constant of proportionality (k_h ) is ~43.7 for steady columns and 7.22 for discrete explosions when d m/d t is expressed in kilograms per second (1 kg/s @ 1.1 kW) and h_t in meters. For a convecting eruption column, a second important assumption is that vertical velocities (u_v ) fit a gaussian function of distance from the plume axis (Carey and Sparks, 1986):

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where u_c = the centerline velocity at height h as determined from solutions of Eqs. (1-5) and (1-6) (Wilson, 1980); x = the radial distance from the plume axis, and b_e = the e-folding distance of u_c ; 2b_e is the approximate distance from the plume axis to the visible edge of the plume (Sparks and Wilson, 1982). Superimposed upon u_v is u_r , the radial velocity of lateral plume spread, which is defined as

where r_p = the plume radius, r_a = the mean air density between h_t (the plume height) and h_b (the height at which the plume is neutrally buoyant and begins appreciable lateral movement). Figures 1.13 and 1.14 illustrate the features of this fallout model.

Pyroclastic Flows

Pyroclastic flows (ignimbrites) comprise some of the most voluminous explosive products in the geologic record, and one possible emplacement model is that for the gravitational collapse of an eruptive column (Sparks and Wilson, 1976; Wright, 1979). Based upon Prandtl's (1949) theory of turbulent fluid jets, in which ambient air is incorporated into the jet—thus changing its bulk density, the equation of motion for an eruptive column (Wilson, 1976) is written:

where q = a ratio of the average column velocity to its centerline velocity, r_b = the bulk density of the column, r_v = the vent radius, and r_a = the density of the ambient air. Numerical solutions to this equation, summarized by Sparks et al . (1978), relate column height to gas velocity, vent radius, and water content (Fig. 1.15). Column collapse is predicted for columns that do not continue their upward motion because buoyancy forces can no longer offset drag forces on the margins of the column.

Fig. 1.11
Classification (Walker, 1973) of eruptive mechanism by grain size and dispersal characteristics of
fallout deposits. F_t is the weight percent of tephra finer than 1 mm found along the dispersal axis
where the deposit thickness is 10% of its maximum. A_d is the area of the deposit where its
thickness is at least 1% of its maximum.
(Adapted from Wright et al ., 1981.)

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Figure 1.16 depicts the onset of gravitational collapse predicted by solutions to Eq. (1-16). Plinian eruptive column collapse can be precipitated by increases in vent radius or decreases in the water content of erupting materials; either condition decreases the initial velocities of the column and leads to its collapse.

Sheridan (1979) and Malin and Sheridan (1982) modeled the runout of pyroclastic flows and surges by employing an "energy line" concept (Fig. 1.17) derived by analogy to rock-fall debris streams (Hsu, 1975), which are dominantly gravity-driven flows. The maximum distance of runout is computed as the loci of points at which the potential energy surface of the flow intersects the topographic surface. The velocity of the flow at any increment (i) along its flow path [v(i)] is simply modeled as its gravitational potential velocity path: v(i) = [2gD h(i)]^1/2 , where D h(i) = height of the energy surface above the local topography; in general, this value is initially determined by height above the vent from which the pyroclastic flow collapses. For directed blasts (for example, Hoblitt et al ., 1981), the initial velocity [v(0)] can be taken as a calculated gas-dynamic velocity such as the blast's sound speed. The flow accelerates with incremental runout distance:

for which q (i) = the local slope and µ_h = the tangent of the energy surface slope (q_e ), called the Heim coefficient (Heim, 1932). This number can vary from 0.06, for highly mobile, large pyroclastic flows, to 0.74, for small pyroclastic flows with low mobility (Sheridan, 1979). The flow accelerates and decelerates depending upon the local slope, in such a way that it flows over a total runout distance (L_f ) to where its velocity v(i) = 0; v(i) = [v_o + 2a(i)L_e (i)]^1/2 , where L_e (i) is measured from topographic maps and t(i) = 2L_e (i)/v(i)].

Fig. 1.12
Schematic of an idealized volcanic eruptive system.
Although analytical solutions for subsurface
flow of magma and volatiles can be made,
the exact physical conditions of this flow are
unknown, and this lack of information limits
the calculation of mass and energy transport
within the erupted jet and plume.
(Adapted from Wilson et al ., 1980.)

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Fig. 1.13
Clast trajectories from the umbrella region of a Plinian eruption column; clast sizes
are given in centimeters. Note that the dispersal is greater for the 35-km-high
column than for the 21-km-high column.
(Adapted from Carey and Sparks, 1986.)

Pyroclastic Surge

Relatively thin bedding (generally less than a decimeter), and a multiplicity of bedforms distinguish the deposits of pyroclastic surges (Fisher and Waters, 1970; Wohletz and Sheridan, 1979). These textural features are thought to indicate unsteady flow and rapid variations in particle-to-gas volume ratios—flow conditions that are especially prevalent during eruptive blasts such as those that may occur during the initial moments of Plinian eruption (Kieffer, 1981; Wohletz et al ., 1984) and explosive hydrovolcanic activity (Waters and Fisher, 1971).

Kieffer (1984b) showed that some volcanic blasts have a jet structure when they emanate from the vent orifice. The conditions of the jet can be initially supersonic and will vary with decompression of the magma reservoir. As Kieffer (1977) showed, the sound speed of multiphased fluids (c_s ), such as steam loaded with solid particles found in volcanic columns, can be substantially less than that of the constituent phases (Marble, 1970). The sound speed may be several tens to several hundreds of meters per second for steam and tephra mixtures. Because observed velocities of volcanic ejecta are in the range of 100 to 500 m/s, their flow is internally supersonic and the effects of gas compressibility are important. The Bernoulli Eq. (1-3) can be written to show the effect of Mach number (M = u/c_s ):

in which p_o = the stagnation pressure (the pressure of the erupting mixture at zero velocity; for example, the chamber overpressure), p_s = the static pressure, and g , the isentropic exponent (ratio of heat capacities at constant pressure and constant volume), expresses the degree to which the erupting mixture approaches isothermal expansion (g = 1.0). In contrast to the incompressible Bernoulli Eq. (1-3), in which the pressure is a function of velocity only, the compressible form shows that pressure is also a function of thermodynamic parameters. For eruption columns modeled by incompressible equations, the pressure along the axis of the column is nearly atmospheric, but for columns erupted as supersonic jets, the effects of compressibility cause pressure and density to vary by large factors along the column's axis.

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To understand flow conditions for surge-producing blasts, it is necessary to solve non-linear forms of the equations of motion. In simplified form (Kieffer, 1984b), these equations express

where r = density,

= the velocity vector, and

= the nabla operator that signifies spatial differentiation. Unlike previous models of eruption columns (for example, Wilson et al ., 1980; Wilson and Walker, 1986; Woods, 1988), these equations cannot be solved analytically, which is the main reason previous researchers used incompressible approximations. However, using the classical method of characteristics, Kieffer (1984b) obtained solutions for the continuous ranges of the equations to show their profound effect upon the flow of tephra and gas during blast eruptions (Fig. 1.18). A more complete formulation of this problem (Fig. 1.19) involves the complete set of multiphase, Navier-Stokes equations and employs a high-speed computer (Valentine and Wohletz, 1989). However, the emplacement of pyroclastic surges, a topic of great importance in volcanic hazard analysis, has not been so completely analyzed that quantitative models can predict field relationships.

The above discussion of important quantitative models includes those that have had wide applications in recent years and are frequently cited. With improved modeling approaches and close development of theory in conjunction with field observation, it will be possible to use field measurements to constrain eruptive mechanisms and subsurface conditions that are needed to understand the thermal regime and hydrothermal

Fig. 1.14
Plots of maximum clast isopleths show the
effect of crosswind velocities (v) of 30, 20,
and 10 m/s on a 28-km-high eruption column.
The isopleth contours are for clast diameters (in
centimeters) and clast densities of 2500 kg/m³ .
(Adapted from Carey and Sparks, 1986.)

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systems associated with volcanoes. Progress towards these latter goals has been greatly aided by the development of a hydrovolcanism theory that links quantitative models of explosive eruption with the hydrological character of the volcano. Through this theory, both the heat resource and water necessary for a geothermal system can be simply assessed by characterization of explosive eruption products.