ltiple interacting phases of embedded solid and gas inclusions. Multiphasepercolation theory provides a means of modeling assemblies of these different classes of magmatic inclusionsin a simple, yet powerful way. Like its single phase counterpart, multiphase percolation theory describes theconnectivity of discrete inclusion assemblies as a function of phase topology. In addition, multiphasepercolation employs basic laws to distinguish separate classes of objects and is characterized by its dependencyon the order in which the different phases appear. This paper examines two applications of multiphasepercolation theory: the first considers how the presence of bubble inclusions influences yield stress onset andgrowth in amagma's crystal network; the second examines the effect of bi-modal bubble-size distributions onmagma permeability. We find that the presence of bubbles induces crystal clustering, thereby 1) reducing thepercolation threshold, or critical crystal volume fraction, ϕc, at which the crystals form a space-spanningnetwork providing a minimum yield stress, and 2) resulting in a larger yield stress for a given crystal volumefraction above ϕc. This increase in the yield stress of the crystal network may also occur when crystal clustersare formed due toprocesses other than bubble formation, such as heterogeneous crystallization, synneusis, andheterogeneity due to deformation or flow. Further, we find that bimodal bubble size distributions cansignificantly affect the permeability of the system beyond the percolation threshold. This study thusdemonstrates that larger-scale structures and topologies, as well as the order inwhich different phases appear,can have significant effects on macroscopic properties in multiphase materials.
Near the Earth's surface, magma often exists as a multiphasemate-rial, typically comprisingbothgas bubbles and solid crystals embeddedin a liquid substrate (Fig. 1). Quantifying magma permeability andrelated gas flow rates through bubble networks, as well as the propa-gation of stresses through crystal networks, are key to understanding awide range of magmatic properties and processes. These includebubble and crystal content and relatedmagma rheology (Cashman andBlundy, 2000; Saar et al., 2001; Rust and Manga, 2002; Melnik et al.,2005), emissions of volcanic gases (Edmonds et al., 2003), coupled heatand volcanic gas transfer and related hydrothermal activity (Matsush-ima, 2003), focusing ofmagmatic-hydrothermal fluids (Candela,1991),oxidation and cooling of pumice (Tait et al., 1998), welding (Sparkset al., 1999), pressurization and destruction of conduit plugs andvolcano flanks (Melnik and Sparks, 2002), magma fragmentation involcanic conduits (Klug and Cashman, 1996) and subsequent expan-sion (Kaminski and Jaupart, 1997), and transitions in eruption dyna-mics (e.g., Eichelberger et al., 1986) or a lack thereof (Gonnermann andManga, 2003).
l rights reserved.
The macroscopic rheological properties of magma are influencedby interactions between embedded inclusions (e.g., Jerram et al.,2003; Noguchi et al., 2006). This is particularly evident when inclu-sions form clusters that span the substrate dimensions. The formationof space-spanning bubble networks, for example, coincides with theonset of magma permeability (Saar and Manga, 1999; Blower, 2001a).Such bubble networks may form open channels to the surface of themagma, preventing further bubble expansion (Gardner et al., 1996),causing the start of quenching (Rowland and Walker, 1990), and in-fluencing eruption dynamics (Hammer et al., 1999; Clarke et al., 2007).At depth, the formation of a connected bubble-network may allowdegassing into the surrounding rock, altering the melt rheology byinducing volatile exsolution and subsequent crystallization (Cashmanand Blundy, 2000; Hammer and Rutherford, 2002). Crystals also havea significant effect on magma rheology. At low volume fractions,crystals affect the magma viscosity as described by the Einstein–Roscoe equations (Einstein, 1906; Roscoe, 1953). The generation ofspace-spanning crystal networks marks a transition in magma rheol-ogy fromNewtonian to Bingham-like fluid flow behavior (Lejeune andRichet, 1995; Philpotts et al., 1998; Saar et al., 2001).
While the complex thermodynamic, rheological, and stress fieldconditions during magma transport are often elusive, it is possible toexamine the topological (i.e., geometric) conditions of object (e.g.,particle, bubble) assemblies required for certain threshold materialproperties to occur, independent of the underlying processes causing
Fig. 1. a) Thin section image of a scoria sample from a cinder cone in the CascadesVolcanic Range, Oregon, showing several connected bubbles (sub-spherical black), solidfeldspar crystals (ragged black), and glassy rock matrix, i.e., liquid before rapid chilling(white). The dashed line indicates a sample-spanning bubble network. Crystals likely donot form a network in this particular sample. Modified from Saar (1998). b) Example ofthe multiphase bubble and crystal simulations discussed in Section 3.
1012 S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
these geometric conditions. Such geometric properties of embedded,space-spanning clusters of discrete inclusions are open to investigation
Fig. 2. Exclusive assemblies of 100 circular and 100 rectangular inclusions: a) circular inclusiofirst. d–f) Voronoi diagrams for the figures above illustrating the distribution of particle centVoronoi cells for the circular particle centers. Periodic boundary conditions are enforced.
bycontinuumpercolation theory (Meester,1996). Percolation theory isconcerned with the connectedness of assemblies of objects. It isemployed in a wide range of fields, including astrophysics, quantummechanics, epidemiology, fire management, and traffic flow studies(Shante and Kirkpatrick, 1971; Meester, 1996). While percolationtheory has also been applied to the study of magma rheology andpermeability, much of this work has focused on single-phasepercolation (e.g., Saar and Manga, 1999; Blower, 2001a; Saar et al.,2001; Rust and Cashman, 2004; Walsh and Saar, 2008). Single-phasepercolation deals only with objects of a particular type, for exampleeither bubbles or crystals alone — ignoring codependent effectsbetween separate embedded phases.
Here, we consider multiphase (or polychromatic, Zallen, 1977)continuum–percolation theory, which accounts for the separate naturesof individual phases. As further discussed in Section 2, multiphasepercolation differs from single-phase percolation in that it is pathdependent: the structure of the inclusion clusters are a function of theorder inwhich embedded objects appearwithin thematerial. This paperexamines how path dependency influences magmatic yield stress andpermeability, by considering two applications ofmultiphase percolationtheory. The first, given in Section 3, examines the effects of bubble–crystal codependencies on the development of magmatic yield stress.The simulationsdemonstrate that, in addition to any direct contribution,bubbles also increase the yield stress of the magma by reducing thecrystal volume fraction required for crystal network formation and byrearranging the resultant crystal network structure. The second exam-ple, presented in Section 4, demonstrates how multiphase percolationtheory can be applied to separate inclusion types of the same phase byexamining the permeability of a bubble network with bimodally distri-buted radii. Although bimodal bubble populations alter the volumefraction at the onset of permeability only slightly, the distribution ofbubble sizes strongly influences permeability development beyond thepercolation threshold.
2. Multiphase percolation
Continuum percolation is concerned with the connectedness ofassemblies of objects that are embedded within a continuous spatialdomain (in contrast to discrete percolation which deals with the
ns first; b) alternating circular and rectangular inclusions; and c) rectangular inclusionsers: lighter regions are Voronoi cells for rectangular particle centers, darker regions are
Fig. 3. Two-dimensional illustration of a) the excluded volume, vexab, of twoobjectsa andb asthe region about object awithin which the center of the second object, b, cannot be placedwithout causing overlap of the objects. b) The excluded volume of object awith respect toobject b equals the excluded volume of object bwith respect to object a, i.e., vexab=vexba.
1013S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
connectedness of lattices of discrete nodes). Of particular interest isthe behavior of these assemblies at, and close to, critical thresholds.One such threshold is the percolation threshold — at which objects inan assembly connect to form a space-spanning cluster. This thresholdis typically expressed in terms of a critical object volume fraction, ϕc.
Most continuum percolation studies examine assemblies construc-ted from a single population of inclusions. However, several materials –including many magmas – contain multiple interacting embeddedphases. Multiphase percolation offers a simple way to model separatephases through the introduction of basic interaction laws. Like single-phase percolation theory, multiphase percolation theory examines thebehavior of particle clusters, but does so accounting for the differingrelationships between the individual particle types.
A distinguishing feature of multiphase percolation is path depen-dency. With single-phase percolation, the assembly microstructure isunaffected by the order in which particles are added. In contrast, theorder inwhich inclusions are added significantly influences themicro-structure of assemblies withmore than one phase. This is illustrated inFig. 2, which shows three two-dimensional assemblies constructedfrom separate populations of 100 circular and 100 rectangular inclu-sions that are not allowed to overlap. If inclusions are added by alter-nating between the two phases, the particle centers of both phases aredistributed equally (Fig. 2b). Conversely, if inclusions of one phase areadded before the inclusions from the other, thenmost of the regionwillbe occupied by the first phase, while the second phase is confinedwithin the remaining space (Fig. 2a and c), as is illustrated by theVoronoi diagrams given in Fig. 2d–f.
When a second set of inclusions is added after an initial populationof inclusions is present, the distribution of the second set of inclusionsis described by a non-homogeneous Poisson process (Meester, 1996),such that the probability that the number of inclusions, N, within avolume, V, or area in two dimensions, equals some number, m, is
P N Vð Þ ¼ mð Þ ¼ exp −n′ ∫Ω∫VΛΦ dxdΘ
� �n′∫Ω∫VΛΦdxdΘð Þm
m!; ð1Þ
where n′ is defined as the local number density, i.e., the number ofparticles per unit volume available to the second phase, and Φ=Φ(Θ)is a probability density distribution function describing the probabili-ty of finding a particle with a given orientation, Θ. The functionΛ=Λ (x,Θ) is defined such that Λ=1 if a particle from the second phasewith an orientation Θ can be placed at position x without overlappingthe first phase, and Λ=0 otherwise. For large V,
∫Ω∫VΛΦdxdΘ ¼ V 1−�exab
� �; ð2Þ
where ϕexabis the excluded volume fraction occupied by the excluded
volume, vexab, of the region about object a inwhich the center of objectb cannot be placed without causing overlap of the objects (Fig. 3).
An average excluded volume fraction, ⟨ϕexab⟩, is then given by theaverage excluded volume, ⟨vexab⟩, obtained by averaging vexab over allpossible relative orientations between objects a and b. However, if atleast one of the two objects is a circle (in two-dimensional systems)or a sphere (in three-dimensional systems), then ⟨vexab⟩=vexab
andhence ⟨ϕexab⟩=ϕexab, because spheres are orientation-independent.Thus for systems in which neither phase a nor phase b are comprisedof spheres, all vexab and ϕexab
discussed in this paper would be replacedby ⟨vexab
⟩ and ⟨ϕexab⟩, respectively.
Assuming the first phase is itself constructed from a Poissonprocess, ϕexab
is expressed in terms of vexabas
�exab ¼ 1−exp −navexab� �
; ð3Þ
where na is the particle number density of the first phase. Note thatalthough the excluded volume of one particlewith respect to the otheris dependent only on the choice of particles, the same is not neces-sarily true for the excluded volume fractions of the phases, i.e.,
vexab ¼ vexba ; but ð4Þ
�exab≠ �exba : ð5Þ
An interesting consequence of Eqs. (3) and (4) is that if twoassemblies have phases with identical number densities, but the shapesof their first and second phases are reversed, then the distributions ofthe particle centers of both phases will be statistically identical. Thisexplains why the centers of the particles in the first and second phasesare similarly distributed (as demonstrated by the Voronoi diagrams inFig. 2d and f ), although the original assemblies appear very different.
The macroscopic particle density, n, of any phase is defined as
n ¼ limVY∞
∑mmP N Vð Þ ¼ mð Þ=V ; ð6Þ
giving the following relationship between the local and macroscopicparticle number densities for phase a:
n′a ¼ na= 1−�exab
� �: ð7Þ
Two examples of multiphase percolation are considered in the fol-lowing sections. The first example investigates the effect of interactingbubble and particle inclusions on the onset and growth of yield stress. Inthe second example, two separate populations of spherical particles areused to model fluid flow through a network of bubbles with bimodalsize distributions. Both sections consider multiphase systems similar toFig. 2a and c, in which the second set of inclusions are added after thepositions of the inclusions from the first phase are fixed.
3. Bubble and crystal networks
Nucleation and growth of bubbles inmagma causes crystallization inresponse to the exsolution of volatiles from the melt (Cashman andBlundy, 2000). The crystals in magma in turn influence bubble deve-lopment: by lowering the critical supersaturation pressure required forbubble nucleation (Gonnermann andManga, 2007); and by altering theeffective melt viscosity (Pinkerton and Stevenson, 1992).
In the absence of bubbles, the onset of a finite yield stress marks atransition in magma rheology from Newtonian to Bingham flow asso-ciated with the formation of a space-spanning crystal network (Lejeuneand Richet, 1995; Philpotts et al., 1998). Elsewhere, this transition hasbeen modeled as the single-phase percolation of assemblies ofinterpenetrating cuboids (Saar et al., 2001; Baker et al., 2002; Saar andManga, 2002; Walsh and Saar, 2008). The particles in these models areallowed to interpenetrate fully, a so-called “soft core”model of contact,
Fig. 4. a) Crystal percolation threshold in the presence of bubbles, ϕcba, versus bubblevolume fraction, ϕa. Bubble radii are one tenth the domain width, the particles arerandomly oriented cuboids with dimensions (0.02, 0.004, 0.004) relative to the domainwidth and periodic boundary conditions are enforced. Simulated results are given bythe circles, and the approximate solution, given in Eq. (8), is represented by the solidcurve. The dashed line indicates magmas with a fixed ratio of crystal volume fractions,ϕb, to interstitial melt volume, 1−ϕa, due to volatile exsolution as described by Eq. (12),while the arrows indicate the potential effects of increased crystallization, volatileexsolution, or a combination thereof, as further discussed in the main text. b) Minimumbubble separation that must be traversed by a percolating cluster, wmin, as a function ofbubble volume fraction, ϕa, for the bubble and crystal dimensions used in a.
1014 S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
that simulates particle intergrowth in a quasi-static environment, assu-med for simplicity and first-order insight (Saar et al., 2001). Morecomplex contact laws (e.g., so-called cherry-pit models) as well asphysical or chemical processes may be applied as well, but for now, wefocus on purely topological (geometrical) phase relationships, indepen-dentof the processes that caused them, in order to evaluate the effects ofthese relationships on magma rheology and permeability.
The current model simulates the microstructure of bubbles andcrystals inmagmawith two populations of soft-core particles: sphericalparticles that represent bubbles; and cuboid particles that represent anetwork of intergrowing crystals as discussed in Saar et al. (2001). Bothsets of particles are placed at random locations and orientations insidethe simulated domain, approximating a zero-shear environment asmaybe expected in the center of plug flow. The spherical bubbles are placedfirst, followed by the cuboid crystals (Fig.1b), i.e., the three-dimensionalequivalent of Fig. 2a, meant to resemble, for example, a scoria samplewith large subspherical bubbles and small crystals as shown in Fig.1a. In
contrast to that particular physical sample, crystal volume fractions insimulations canbe increased to, andbeyond, their percolation threshold.Similarly, bubble volume fractions can be varied numerically. If a crystalis found to overlap one or more of the bubbles then its position andorientation are reselected randomly until a non-overlapping configura-tion is found.
In a single-phase assembly at the percolation threshold, the space-spanning cluster of connected particles propagates densely through-out the assembly domain. Thus, a good approximation for the perco-lation threshold in the multiphase assembly is obtained by assumingthat percolation occurs when the local volume fraction, 1−ϕexab
,reaches the percolation threshold, i.e., when
�cba≈�cb 1−�exab
� �; ð8Þ
where ϕcb is the single-phase percolation threshold for the crystalassembly. Note that the local volume fraction available for crystalnetwork formation is not given by simply subtracting the bubble volumefraction, ϕa, from the system volume fraction. Rather, the local volumefraction is derived by subtracting the excluded volume fraction forcrystal placements around bubbles, ϕexab, from the system volume frac-tion. The value of ϕexab is found from Eq. (3), where vexab is given by
inwhich li are the dimensions of the cuboid crystal particles and R is thebubble radius. The theoretical (solid line in Fig. 4a) and simulated per-colation thresholds (circles in Fig. 4a) are in good agreement for lowbubble volume fractions, however, the two diverge at higher bubblevolume fractions. Eq. (8) accounts for the volumeoccupied by thebubblephase, but not the effect of the bubbles on the crystals' ability to form apercolating cluster. At small bubble volume fractions the crystals' abilityto form a connected network is relatively unimpeded. However, athigher bubble volume fractions, the diminished space between bubbleshas a greater impact on the connected crystal network. The sketchwithin Fig. 4b illustrates that for a given bubble number density, n, thereis a critical bubble radius, Rc, at which the interstitial melt surroundingthe bubbles will itself cease to form a connected network from one sideof the assembly to the other. This critical bubble radius is given by
Rc ¼ −3
4πnln 1−�cm
� � �1=3; ð10Þ
where ϕcm is the critical percolation threshold for the melt phase withrespect to interpenetrating spherical bubbles. The critical threshold forthe melt phase is given as ϕcm≈0.97 by Kertesz (1981) and Elam et al.(1984). This implies that the crystal cluster must traverse a gap betweentwo bubbles separated by a minimumwidth
wmin ¼ 2 Rc−Rð Þ ¼ 2Rln 1−�cm
� ln 1−�að Þ
� �1=3−1
!; ð11Þ
for Rc>R, to form a percolating network. Plotting wmin as a function ofthe bubble volume fraction for the given bubble and crystal dimensions(Fig. 4b), we see that the simulated and estimated values of theexcluded volume begin to diverge as wmin falls below 10 times themaximum crystal dimension (when ϕa≈0.35, see dotted lines inFig. 4ab), approximately the same point at which finite size effects areobserved in single-phase percolation simulations (Saar and Manga,2002).
Fig. 4a is somewhat analogous to a phase diagram: the percolationthreshold delineates suspensions that behave as Newtonian fluids,from those that behave as Bingham liquids. Paths within this plotrepresent families of assemblies that share particular characteristics.For example, assemblies with a fixed ratio, λ, of crystal volume
Fig. 5. Log–log plot of the crystal assembly yield stress (normalized by the maximumsimulated yield stress) versus ϕb−ϕcb (1−ϕexab) for different bubble volume fractions,ϕa, above the percolation threshold. As shown in the inset, the resultant power-lawexponent, ν, from Eq. (14) is not constant, but instead varies between 3.6 and 4.6 for theinvestigated values of bubble volume fraction, ϕa.
1015S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
fraction, ϕb, to interstitial melt volume, 1−ϕa, are given by curves ofthe form:
�b ¼ λ 1−�að Þ: ð12Þ
Such curves can be used to gain insight into how these systemsrespond toparticular physical processes. For example, taking apoint onthe curve given by Eq. (12) and moving in the direction of increasingbubble-volume fraction, represents a depressurizing, and thus expan-ding, magma, where changes in liquid mass and volume are negligiblewhile volatiles exsolve, forming expanding gas bubbles. There exists arange of λ values for which the line of assemblies described by Eq. (12)will cross the critical percolation threshold — an example of one suchset of assemblies is given by the dashed line in Fig. 4a. As the bubblevolume fraction is increased in this set of assemblies, the rheologychanges from a Newtonian fluid, to developing an initial yield stresswhich then grows as the percolation threshold is crossed and theBingham liquid field is entered. This change in rheology is due to theincreased clustering of crystals within the suspension and is in addi-tion to any direct contribution the bubbles may make to the onset andgrowth of yield stress (e.g., Ryerson et al., 1988; Gardiner et al., 1998),which is not accounted for by this model.
Any additional crystallization with or without further exsolution ofvolatiles from themelt, would shift the assembly such thatϕb>λ(1−ϕa),thereby moving the point upwards to higher crystal volume fractions,ϕb, and facilitating the transition from Newtonian fluid to Binghamliquid. Therefore, the exact onset of yield stress will depend on thecrystallization and volatile exsolution path and on the location of thepercolation threshold curve, where the latter is itself dependenton crystal and bubble volume fractions, but also on crystal and bubbleshapes, orientations, and size distributions. The effect of the latter isfurther discussed in Section 4.
The formation of a macroscopic, but fragile, crystal network at thepercolation threshold, discussed so far, marks the onset of a minimumyield stress, σyield→0 Pa (Saar et al., 2001). Walsh and Saar (2008)investigate, how σyield may grow above the percolation threshold forvarious crystal topologies in the absence of bubbles. In the following, wediscuss the effect of bubble-induced crystal rearrangement on yieldstress growth above the percolation threshold. Yield stress growth isinvestigated with a crystal network model that simulates the mechan-ical properties of the assembly. In this model, described in greater detailin Walsh and Saar (2008), the strength of particle bonds is determinedby the amount of overlap betweenparticles. Each bond is also assigned afailure criterion that controls the strain at which the bond is fractured.
Under single-phase percolation, the simulated yield stress, σyield,due to phase b, experiences power-law growth for volume fractions,ϕb, above phase b's percolation threshold, ϕcb (Walsh and Saar, 2008),i.e.,
σyield~ �b−�cb
� �m; ð13Þ
for ϕb>ϕcb. During multiphase percolation, however, a preexistingphase, a (here bubbles), reduces phase b's multiphase percolationthreshold, ϕcba, compared to the single phase (b) percolation threshold,ϕcb. This can be seen by replacingϕcb withϕcba in Eq. (13), and combiningit with the earlier expression for the multiphase percolation threshold,given in Eq. (8), yielding
σyield~ �b−�cb 1−�exab
� �h im; ð14Þ
for ϕb>ϕcb(1−ϕexab). As required, Eq. (14) reduces to Eq. (13) when thebubble phase, a, is not present (ϕexab=0).
The value of the power-law exponent, ν, is roughly constant insingle phase assemblies (∼3.5), with little variation (∼±0.2) in res-ponse to changes to particle alignment and particle shape aspect ratiofor both oblate and prolate particle assemblies (Walsh and Saar, 2008).
However, this is not true for the multiphase assemblies studied here.As the bubble volume fraction is increased from ϕa=0.0 to ϕa=0.5, themeasured value of the exponent increases from 3.6 to 4.6 (Fig. 5).
The earlier onset of the percolating crystal network and the increasein the power-law exponent observed in the multiphase assembly aredue to the heterogeneous spatial distribution of the crystals caused bythe bubbles' presence. Both effects increase the crystal network yieldstress for a given crystal volume fraction. This is at least as importantas other microstructural characteristics such as particle shape andorientation, particularly in determining the growth of yield stress withincreasing crystal volume fractions. Heterogeneity in the crystalnetwork is also a factor in determining the yield stress of single phasesystems. Crystals tend to cluster in systemswithoutbubbles as a resultofheterogeneous crystallization (Hoover et al., 2001; Jerram et al., 2003)and synneusis (clustering due to hydrodynamic interactions) as dis-cussed by Schwindinger and Anderson (1989). Multiphase percolationmay also be employed in simulating such systems containing distinctpopulations of inclusions belonging to the same phase, as illustrated inthe following section.
4. Bimodal bubble networks
Multiphase percolation analysis is not confined to assemblies ofdifferent physical phases. It is also applicable to problems involvingdistinct populations of inclusions from the same phase. Here, this isillustrated by considering gas flow through bubble assemblies withbimodal size distributions.
Magmas display both exponential (Mangan and Cashman,1996) andpower-law bubble-size distributions (Klug et al., 2002; Namiki et al.,2003). Exponential bubble-size distributions are attributed to bubbledevelopment under steady-state conditions (Marsh, 1988), whilepower-law distributions may be due to bubble coalescence (Gaonac'het al., 1996) or successive waves of nucleation (Blower et al., 2002). Thelatter scenario is considered in this section, where we simulate thesimplest case in which two separate waves of bubble nucleations occur(Fig. 6). However, while details may vary, our general conclusions,regarding the effects of differing bubble sizes onpermeability, should beapplicable to any systemwith non-uniform bubble sizes.
The two rounds of nucleation are modeled by two populations ofspherical bubbles. The centers of the first population are chosen atrandom, the centers of the second are also selected randomly with thecondition that they do not lie within a given radial distance from thecenters of the first population. The excluded region about the centers of
Fig. 6. Schematic two-dimensional illustration of two rounds of bubble nucleation andgrowth: (a) the first bubbles nucleate and (b) begin to grow; (c) a later round ofnucleation is triggered with new bubbles that appear between the old; (d) followed by asecond round of bubble expansion. Actual simulations contain from 1000 to 72,000three-dimensional bubbles and employ periodic boundary conditions.
Fig. 7. Plot of the percolation threshold (i.e., critical bubble volume fraction, ϕc=ϕs+ϕL)for assemblies of bimodally sized spheres, generated as described in Fig. 6, versus thelarge bubble volume fraction, ϕL. The small bubble volume fraction at percolation, ϕs, isalso shown. Results are based on 20 simulations for each value of ϕc.
1016 S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
the first population of bubbles represents the size of the first set ofparticles at themomentwhen the secondset nucleates.Once the centersof the second set of bubbles have been selected, both sets of bubbles areincreased to their final sizes and allowed to interpenetrate. As beforewith soft-core crystals, interpenetratingbubbles are a simplification thatdoes not account for physical or chemical processes causing, forexample, bubble resistance or tendency to coalesce or drain inter-bubble films. However, particularly for high viscosity rhyolitic melts, aswell as for rapidly chilled scoria formedduring fountain eruptions of lowviscosity basalts (Fig. 1a), this topological approach allows somefundamental insights into geometric requirements, necessary for theformation of bubble networks and related finite system permeabilities.
Fig. 7 shows the volume fractions of the two bubble phases at thepercolation threshold. There is relatively little difference in the overallbubble-volume fraction atwhichpercolation occurs, despite thebimodalsize distribution. Similar results have also been observed elsewhere (e.g.,Lorenz et al.,1993; Saar andManga, 2002), although thismay not hold inbimodal assemblies with vastly different bubble radii, RL≫Rs (Phani andDhar, 1984), or non-spherical particles of significantly varying sizes.
Although the percolation threshold is not altered dramatically, thebimodal distribution affects the assembly's macroscopic permeability.We calculate the permeability of the simulated assemblies using anetwork model based on that given by Blower (2001b). In this model,the permeability of the system is calculated by assigning a resistanceto flow to each of the apertures between the bubbles according to anapproximate expression given by Feng et al. (1987).
Fluid is transmitted through the percolating backbone of theassembly, i.e., the space-spanning bubble network, excluding dead-ends (Fig. 8). The ratio of large bubbles to small bubbles in thepercolating backbone differs from the ratio of bubbles in the assemblyas a whole: the ratio of large to small bubbles in the backbone isapproximately 1 to 3, whereas the ratio over the entire assembly isapproximately 1 to 26. From this itmight be deduced thatflow throughthe percolating backbone is controlled by the larger bubbles. However,the permeability of the pore network is complicated by the presence ofthe smaller bubbles, which create bottlenecks that restrict flow.
Fig. 9 shows simulated permeabilities, k, of bimodal bubble assem-blies with different number densities of small spherical bubbles ofradius Rs=0.01L against the volume fraction of the large-bubble po-pulation of radius RL=0.04L, where L is the side length of thesimulation bounding box. The second round of nucleation is assumedto occur when the large bubble radii are RL=0.035L. The number ofsmall bubbles in the simulations range between 0 and 70 000 inincrements of 3500; the number of large bubbles range between 1000and 2000 in increments of 100. Permeabilities for each combination ofsmall and large bubble number densities are calculated based on themedian value obtained from twenty simulations. Median, rather thanmean, values are used as they show less variation in the calculatedpermeabilities, particularly at volume fractions close to the percola-tion threshold of the system.
As Fig. 9a demonstrates, at low volume fractions of large bubbles,the permeability of the assembly is mostly determined by the numberdensity of the small bubbles. Indeed, for some of the assemblies withhigh number densities of small bubbles, there is a slight dip in thepermeablility with the increase in the large bubble volume fraction,ϕL. This may be a result of the variation in the percolation threshold,ϕc, seen in Fig. 7, but is more likely a numerical artifact as the variationis within the measured error for each run. Conversely, at large-spherevolume fractions of ϕL≳0.3, the number density of the small bubbleshas a negligible effect on the overall permeability. The shift in beha-vior is most easily explained by critical path analysis (e.g., Ambegaokaret al., 1971; Pollak, 1972; Hunt, 2005). When the large-bubble volumefraction is insufficient to form a percolating network on its own, anyfluid flowmust necessarily pass through the small bubble population.Thus, the flow runs in series through the large and small bubblepopulations, where permeability is dominated by the smaller aper-tures of the smaller bubbles. Once the volume fraction of larger bub-bles reaches the percolation threshold for assemblies of spheres,ϕL=0.2895 (Rintoul and Torquato, 1997), there is an increase in themacroscopic permeability. At this point the flow runs in parallelthrough the large and small bubble networks and is thus dominatedby the larger apertures between the large bubbles.
For a given total bubble volume fraction, ϕ, the dimensionless per-meability increases as the small bubble number density, ns, increases forassemblies below the large-bubble critical volume fraction, but de-creases with increasing ns for assemblies above the large-bubble criticalvolume fraction. This is illustrated in Fig. 9b by the approximatelyhorizontal lines (indicating assemblies where ϕLb0.2895) whosenormalized permeabilty increases with increasing ns, while the
Fig. 9. Calculated dimensionless permeability, k/RL2, of bimodal bubble assemblies fordifferent populations of large (RL=0.04L) and small (Rs=0.01L) bubbles in a boundingbox of side length, L, as a function of a) large bubble volume fraction, ϕL, and b) totalvolume fraction, ϕ. Each line represents assemblies with the same number density of
1017S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
normalized permeabilites decrease with increasing ns for assemblieswhere ϕL>0.2895.
Comparisons with real pumice and vesicular basalt samples underdifferent conditions show that single phase percolation can both over-and underestimate the onset and development of percolation inmagma (Saar, 1998; Saar and Manga, 1999; Rust and Cashman, 2004).Higher permeability estimates have been attributed to the collapse ofthe vesicular network, resulting in more elongated pores and prefe-rential flow paths at a reduced pore fraction (Saar, 1998; Saar andManga, 1999). In simulations of fully interpenetrating polydispersebubble assemblies, Blower (2001a) also found that a distribution ofbubble sizes resulted in an increase in the permeability. However, aspredicted by Blower et al. (2001), it is theoretically possible to developarbitrarily low-permeability/high-porosity bubble assemblies, if the newbubble positions are dependent on existing bubble locations. The resultsof our multiphase simulation support that prediction. In the assembliessimulatedwith large volume fractions of large bubbles, the small-bubblepopulation adds to the void space without contributing significantly tothe overall permeability. Thus, the permeability of the multiphaseassembly is less than that of a single-phase assembly of large bubbleswith the same volume fraction (Fig. 9b).
We also note in passing that in this paper we have chosen tonormalize the permeability, k, by the square of the larger bubbleradius, whereas the earlier papers by Blower (Blower, 2001a,b; Bloweret al., 2001) use the average bubble radius. While the choice of radiusdoes change the value given for the dimensionless permeability, itdoes not overly affect the qualitative relationships discussed in thispaper. This is because in all but a few of the assemblies examined thenumber of small bubbles is much greater than the number of largebubbles, thus for most cases Raυ≈Rs=RL/4 and hence k/RL2≈k/16Raυ2 .
The results of our multiphase simulations of bimodal bubble-sizedistributions show that reproducing the bubble-size distribution isnot sufficient to correctly simulate magma permeability; it is alsoimportant to account for the influence of larger-scale heterogeneousstructures in the bubble network. As with crystal assemblies discussedin Section 3, these heterogeneities may also arise as a result of pro-cesses other than the successive rounds of nucleation discussed here.Bubble deformation as well as preferred bubble orientations andspatial distributions, for example, occur in response to shearing (Rustand Manga, 2002; Tucker and Moldenaers, 2002; Quilliet et al., 2005)and bubble coalescence (Gardner, 2007). Therefore, correctly charac-terizing these larger-scale structures in rock samples or deriving such
small bubbles, from ns=0 to ns=70,000 as the lines vary from light gray to black. Alongeach line the number density of large bubbles varies from nL=1000 to nL=2000 inincrements of 100. Some permeabilities are not given below the percolation thresholdfor the large-bubbles, ϕLb0.2895. In these cases, k=0, as the assemblies fail to form apercolating cluster.
Fig. 8. Normalized fluid pressure change through the percolating backbone, ΔP,calculated based on the network model of Blower (2001b). The bubble volume fractionis 0.33, approximately 0.01 higher than the critical volume fraction for an assembly withthe same volume fraction of large bubbles. The ratio of large to small bubbles in thebackbone is approximately 1 to 3, whereas the ratio over the entire assembly is ap-proximately 1 to 26. Arrows indicate bottleneck locations caused by smaller bubbles.
structures in simulations of magma transport, is vital in determiningthe overall permeability of magmatic systems.
5. Conclusions
Multiphase percolation theory offers a simple way to simulate theeffects of inclusion interactions on the macroscopic properties ofmagma.Here this is demonstratedbyconsidering two cases of interest:1) the effect of bubbles on the onset and growth of yield stress in crystalmelt suspensions, and 2) the role of bimodal bubble size distributionson the permeability of bubble networks.
The results from the simulations of multiphase bubble and crystalassemblies suggest that the presence of bubbles in a magma acts toincrease its yield stress by forcing greater clustering in the crystalnetwork. This contribution is in addition to any increase in the overallyield stress of the magma due to the presence of the bubble networkitself. In these simulations, the excluded volume of the bubblescontrols the onset of crystal percolation, rather than the bubble
1018 S.D.C. Walsh, M.O. Saar / Journal of Volcanology and Geothermal Research 177 (2008) 1011–1019
volume, as might be expected. Thus, increasing the bubble volumefraction lowers the crystal percolation threshold for a given ratio ofcrystal to melt volume. As in studies of single-phase percolation, theyield stress is shown to have a power-law growth for crystal volumefractions above the crystal percolation threshold. However, as thebubble phase is increased the exponent of the power-law is larger thanthat measured in single-phase simulations, and increases withincreasing bubble volume fractions. Therefore, clustering or clumpingof crystals, bywhatevermechanism,maybe a significant contributor tomagmatic yield stress both by lowering the percolation threshold andby increasing subsequent yield stress growth.
While different ratios of large to small bubbles in the simulatedbimodal bubble assemblies do not dramatically alter the percolationthreshold, the effect on the permeability of the assembly is significant.Although larger bubbles are overrepresented in the percolating cluster, itis the smaller bubbles that determine the overall permeability close tothe percolation threshold. Before the larger bubble population hasreached its percolation threshold, any percolating pathway must neces-sarily pass through the set of smaller bubbles, resulting in bottlenecksthat control the permeability. There is an increase in the permeability inassemblies with larger number densities of large bubbles due to theformation of a percolating cluster in the larger bubbles. When thisoccurs,flow through the assembly bypasses the smaller-bubble network.Through thismechanism, assemblies of polydisperse bubble sizes exhibitlower permeabilities than predicted by single-phase percolation theory.
In both sets of simulations, the multiple interdependent phasesconsidered introduce larger-scale heterogeneities absent in single-phase assemblies. The discussed nucleation and growth paths areexamples of mechanisms whereby such larger-scale structures mightarise; other processes include synneusis, heterogeneous crystalliza-tion and bubble nucleation, as well as heterogeneities arising fromdeformation and flow. Independent of the underlying cause, this studydemonstrates that such larger scale structures can have a significantrole in determining the macroscopic properties of magmas and othermultiphase materials. As a result, it may be critical to consider multi-ple coexisting phases and their topologies (e.g., shape and size distri-bution, clustering, and connectivity), i.e., not merely the mass orvolume fractions of phases, when studyingmagma flow and degassingas well as other multiphase processes.
Acknowledgments
MOS thanks the George and Orpha Gibson endowment for itsgenerous support of the Hydrogeology and Geofluids research group.In addition, this material is based upon support by the NationalScience Foundation (NSF) under Grant No. EAR-0510723 and Grant No.DMS-0724560. Any opinions, findings, and conclusions or recom-mendations expressed in this material are those of the authors and donot necessarily reflect the views of the NSF. We also gratefullyacknowledge the use of resources from the University of MinnesotaSupercomputing Institute (MSI). We thank Jim Gardner and ananonymous reviewer for providing excellent reviews of this paper.
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