Journal of Volcanology and Geothermal Research 202 (2011) 200–210

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Journal of Volcanology and Geothermal Research

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Application of fractal fragmentation theory to natural pyroclastic deposits: Insightsinto volcanic explosivity of the Valentano scoria cone (Italy)

Diego Perugini a,c,⁎, Alessandro Speziali a, Luca Caricchi b, Ulrich Kueppers c

a Department of Earth Sciences, University of Perugia, Piazza Università, 06100 Perugia, Italyb Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queen's Road, Bristol BS8 1RJ, United Kingdomc Department of Earth and Environmental Sciences, Ludwig-Maximilians Universität, Theresienstr. 41, 80333 Munich, Germany

⁎ Corresponding author at: Department of Earth ScPiazza Università, 06100 Perugia, Italy. Tel.: +39 075 585

E-mail address: diegop@unipg.it (D. Perugini).

0377-0273/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.jvolgeores.2011.02.008

a b s t r a c t

a r t i c l e i n f o

Article history:Received 26 September 2010Accepted 21 February 2011Available online 5 March 2011

Keywords:scoria conefragmentationfractalsgrain size distributioneruption dynamics

The extent of fragmentation triggered by basaltic volcanic eruptions has been studied by analyzing the grainsize distribution of natural pyroclastic deposits and applying the fractal fragmentation theory. For thispurpose, 75 samples have been collected from individual layers along horizontal section of the well-exposedValentano scoria cone (Italy). The cone is constituted by well-defined layers of fall deposits of up to 180 cm inthickness. An important feature of the stratigraphic sequence is the occurrence of several intercalated layers(called breccias) with clasts of different characteristics (angular shape, low porosity, and high crystallinity) inrespect to the “normal layers”. Results indicate that all achieved grain size distributions show fractal behaviorand can be clearly correlated with a single fragmentation event, namely the magmatic fragmentation duringthe eruption. The fractal dimension of fragmentation (D) increases as fragmentation efficiency increases.Values of D are found to vary significantly along the pyroclastic sequence defining two major trends: (1) ageneral, long-range increase of D during the course of the eruption and (2) a superimposed high-frequencyoscillating variation of fragmentation efficiency. Noteworthy is the fact that the highest values of D aremeasured for pyroclastic deposits following breccia deposits. Application of a conceptual model of fractalfragmentation, as claimed by natural data, indicates that larger values of D have to be associated with a higherprobability of fragmentation, thus suggesting that the D value can be used as a proxy for the “fragility” (i.e.ability to fragment) of samples. Fractal dimension of fragmentation of single layers is positively correlatedwith the vesicularity of clasts. This indicates that the bubble content in the ascending magma prior to and atfragmentation likely was the key factor modulating the variable fragmentation efficiency observed in thestudied pyroclastic sequence. We hypothesize that the long-range variation of D may be associated withmagma fragmentation triggered by expansion of gas bubbles progressively coalescing below solid plugsduring later stages of eruptive activity, whereas high-frequency variations of D are interpreted as due tomagma fragmentation during the “slug flow” regime characterizing the strombolian activity of the scoriacone.

iences, University of Perugia,2610; fax: +39 075 585 2603.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Since the introduction of fractal geometry techniques (Mandelbrot,1982) there has been a growing interest in the application of thesemethods to a variety of geological structures and processes. Fromstructural geology to petrology, geologists have applied fractal geom-etry methods and developed new techniques to quantify geologicalfeatures (e.g. Sammis et al., 1986; Turcotte, 1992; Korvin, 1992;Barton, 1995; Holtz et al., 2004; Perugini and Poli, 2005; Perugini et al.,2006, 2007). The major interest of geologists about fractal methodsis that they have been proven to quantify, often by a single parameter

(the fractal dimension,D), complex processes thatwould be otherwisedifficult to assess by classic geological techniques.

Among the research fields of Earth Sciences that benefited fromfractal techniques is the one studying the development of fragmen-tation processes of Earth materials. In particular, grain size distribu-tions generated by a number of geological processes, such as rockfragmentation (e.g. Matsushita, 1985; Turcotte, 1986; Sornette et al.,1990), fault gauge development (e.g. Sammis et al., 1986; Storti et al.,2003) and subsidence breccias (Barnett, 2004), have been successfullystudied by applying fractal statistics.

Recently, the application of fractal techniques has been proven tobe extremely efficient in the study of both morphology of basaltic ashparticles (Maria and Carey, 2002, 2007) and grain size distributionsresulting from volcanic eruptions both from experiments (Kuepperset al., 2006) and natural deposits (Taddeucci et al., 2004; Suzuki-Kamata et al., 2009). It is noteworthy that studies focused on the

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fractal analysis of grain size distributions mostly regarded felsic rockfragments generated by highly explosive volcanic eruptions, andstudies on basaltic pyroclastic natural deposits are still lacking.

In this contribution we present new data on fragmentationprocesses of basaltic explosive activity. We analyze in detail thepyroclastic succession of the Valentano scoria cone (RomanMagmaticProvince, Italy; Peccerillo, 2005) by applying the fractal fragmentationtheory. To the best of our knowledge this is the first attempt to applyfractal analysis to the grain size distribution of basaltic naturalpyroclastic deposits and thereby enhance our understanding ofbasaltic explosive eruptions. We use the variation of fractal dimensionof fragmentation (i.e. fragmentation efficiency) to reconstruct theeruptive history of this monogenic volcanic edifice.

2. Fractal fragmentation theory

As reported by Mandelbrot (1982), Korcak (1940) performedempirical studies on the size distribution of the areas of islands anddeveloped the empirical relationship:

N≈ a−c; ð1Þ

where N is the total number of islands having sizes greater thana given comparative size, a, and c is a constant (Korvin, 1992).Mandelbrot (1982) found that c varied between island regions withc always being greater than 1/2 (cN1/2). In the light of fractal theory,he further realized that the size distribution of a population of islandswas a consequence of fractal fragmentation and that the empiricalconstant c correlated with the fragmentation fractal dimension. Hetherefore suggested that fractal fragmentation could be quantified bymeasuring the fractal dimension from cumulative distributions offragments through the equation

N≈ r−D; ð2Þ

where D is the fragmentation fractal dimension and N is the totalnumber of particles with linear dimension greater than a givencomparative size, r. It is noteworthy that D derived from Eq. (2) is nota measure of irregularity, but a measure of the size–number rela-tionship of the particle population or, in other terms, the fragmen-tation of the population.

Taking the logarithm of both sides of Eq. (2) yields a linearrelationship between N and r with D being the slope coefficient:

log Nð Þ≈−D log rð Þ: ð3Þ

The higher the value of D, the higher is the fragmentation efficiency.It is interesting to relate Eq. (2) to a power–law relationship,

which is extensively used as an empirical description for frequency–size distribution in the study of fragmentation processes (e.g. Turcotte,1992):

N≈m−b; ð4Þ

where N is the number of fragments with mass greater than m. Theconstant b is chosen to fit the observed distribution. It can be showedthat the constant b is equivalent to the fractal dimension D: sincefragments can occur in a variety of shapes, it is appropriate to define alinear dimension r as the cube root of the volume

r≈ V1=3 ð5Þ

and assuming constant density it follows that m≈r3. ComparingEq. (4) with the fractal distribution of Eq. (2) gives

D = −3b: ð6Þ

This implies that the power–law distribution of Eq. (4) isequivalent to the fractal distribution of Eq. (2):

N≈m−b ≈m−D=3 ð7aÞ

N≈ r−D: ð7bÞ

Many size distributions in nature follow this empirical law. Asan example, it was shown that fragmentation of rock material is aconsequence of the scale invariance of the fragmentation mechanism,in that the zones of weakness along which fragmentation occurs canbe found at all levels of scrutiny (Turcotte, 1992).

3. The Valentano scoria cone

The scoria cone of Valentano belongs to the magmatic activity ofthe Latera stratovolcano (Vulsini Volcanic Complex, RomanMagmaticProvince, Italy; e.g. Conticelli et al., 1991; Peccerillo, 2005) that wasactive during a time-span from 0.43 Ma to 0.15 Ma ago (Sparks, 1975;Metzeltin and Vezzoli, 1983; Conticelli et al., 1986). After a series ofhighly explosive eruptions, manifested by the presence of severaldeposits of pyroclastic density currents (Conticelli et al., 1987), thepresent-day Latera caldera formed. Thereafter, several eruptionsoccurred from vents located along the caldera ring fault as well asinside and outside the caldera in a short time-span around 0.15 Ma(Metzeltin and Vezzoli, 1983). The scoria cone of Valentano (Figs. 1and 2) is part of this final activity of the Latera volcano.

The Valentano deposits are mostly constituted by layers of ash andlapilli size scoria clasts, generated by strombolian eruptive activity. Theyare exceptionally well exposed because of recent quarrying activity(Fig. 1a) and represent an eligible outcrop to carry out a detailedinvestigation of an almost complete volcanic sequence. They areconstituted by loose ash and lapilli deposits, which can be readilysampled (Figs. 1b and 2a). Lapilli grains generally display cuspateand sharp margins and elongated shapes; rounded grains are rarelyobserved. A total of 75 samples has been collected along horizontaltransect (ca. 80 m), coveringmost of the accessible deposits forming thecone (Fig. 1a). Noteworthy is the occurrence of three layers of angularbreccia deposits (Fig. 2b; average diameter of fragments is 4–8 cm)occurring in the upper half of the stratigraphic sequence. Sampledpyroclastic layers display variable thickness, from ca. 15 to 100 cm andcan show massive, normal, normal to reverse, or reverse grading. Asummary of characteristics of the sampled pyroclastic deposits is shownin Table 1. Fromthefield record, each individual layerwas interpreted asreflecting a single explosive episode. The hiatus between individuallayers must have been fairly short, as we did not find any indication ofpaleosoil formation. Accordingly we hypothesize, for analogy withhistorical scoria cone eruptions (e.g. Schmincke, 2004), that theValentano scoria cone is a monogenetic cone, built during eruptiveactivity that may have lasted for a few days to a few weeks or months.

4. Textural features of natural samples

Petrographically, scoria fragments have a porphiric texture inwhich phenocrysts (average length ca. 200–500 μm) of clinopyroxene,olivine, plagioclase, and phlogopite occur in a glassy groundmass(Fig. 3a–f). Phenocryst content does not vary greatly among studiedsamples being in the range of 3–5 vol.% (including vesicles). Microlitesare mostly constituted by plagioclase and clinopyroxene crystalswith acicular habit. As for phenocrysts, microlite contents are con-stant throughout the analyzed pyroclastic sequence with averagevalues of 2–5 vol.% (including vesicles). Among phenocrysts, glomer-oporphyric textures are typically observed and are mainly constitutedby plagioclase and clinopyroxene aggregates. Some plagioclase andclinopyroxene crystals display resorbed cores on which an oscillatory

Fig. 1. Study area. a) Arial view of the Valentano scoria cone (dissected dome-likemorphology at the center of the picture) showing outcrop exposure. The analyzedstratigraphic sequence runs along the dashed white arrow (approximate length ca.80 m) reported in the picture following the arrow direction (i.e. from left to right).The white circle in the middle of the cone represents the approximate location of theinferred volcanic conduit; (x) and (y) arrows indicate the approximate position ofthe pictures reported in Fig. 2a and b, respectively; geographic coordinates refer to thecorners of the picture. Inset in (a) shows the localization of the outcrop (white spot)within the Italian peninsula. b) Detailed view of portion of the stratigraphic sequencedisplaying consecutive overlapping of pyroclastic layers (from left to right) constitutingthe scoria cone.

Fig. 2. a) Detail of a typical pyroclastic layer sampled and analyzed in this work.b) Example of angular breccia deposit occurring in the external portions of the scoriacone. The length of ruler in the pictures is 30 cm.

202 D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

zoning pattern is often superimposed. Most plagioclase phenocrystsshow variable degrees of internal resorption patterns evidencedby boxy-cellular and sieve textures. As for olivine, phenocrysts aretypically idiomorphic although some skeletal crystals can be observed.Olivine phenocrysts often display hiddingsitic rims. Flogopite gener-ally occurs as elongated idiomorphic crystals, although in some casessub-idiomorphic phenocrysts showing opacitic rims can be observed.Clasts from the breccia deposits display drastically different petro-graphic features (Fig. 3g–h). The total crystal content (30% in average,including phenocrysts and microlites) and the average crystal size(500–900 μm) are much larger.

Selected samples have been analyzed for their vesicle content. Inparticular, several fragments from each sample have been sliced andthe sample surface covered with liquid white color epoxy in order tofill sample vesicles. After hardening of the epoxy, samples have beencarefully polished and subsequently scanned with a high-resolutionoptical scanner (1 pixel=0.075 mm; 1 mm=13.35 pixels). Obtaineddigital images have been analyzed by image analysis using thesoftware ImageJ (Abramoff et al., 2004) to obtain segmented binary(black and white) images in which vesicles have been rendered inblack color. The vesicle content (porosity, in %) has been measured bycounting the sample area covered by vesicles and dividing by the

total area of each image. Errors in the measurement of vesicle con-tents have been estimated by repeating the image analysis procedureon different fragments of the same sample and applying differentthreshold values during image segmentation. Results show thatvesicle contents can be estimated with an error better than 10%.Three representative samples of scoria deposits occurring at differentdistance along the studied pyroclastic sequence and characterizedby different vesicle contents are shown in Fig. 4. Sample vesicularityis highly variable among scoria deposits with values ranging fromca. 17% to 47%. Noteworthy is the fact that breccia deposits havemuch lower vesicle contents (in the range of 2–5%) with respectto scoria samples, further highlighting their textural differences, asalready indicated by petrographic analysis.

5. Grain size analysis

Samples (250 g) of the entire accessible outcrop were collectedfrom each pyroclastic layer along a horizontal transect from the centerof the scoria cone (with the inferred conduit) to the most distantlocations (dashed white arrow in Fig. 1). We assume that depositsare getting progressively younger with distance from the conduit.Therefore, collected samples are ordered according to their relativeeruption time. We sampled individual pyroclastic layers, which arewell marked in the field because of the presence, of gradations, mas-sive structure, or breccias (Table 1). Each analyzed sample belongsto a single pyroclastic layer. Grain size distributions were obtainedby dry sieving performed at half-Φ steps (Φ=−Log2(d), withd=particle diameter in millimeter; e.g. Cas and Wright, 1987, andreferences therein; Kueppers et al., 2006). In this study, particles inthe range of 0.250–5.660 mm have been used to allow for a

Table 1Summary of the main characteristics of samples from Valentano used to determine thegrain size distributions and fractal dimension of fragmentation. Thick. (cm)= thicknessof the pyroclastic layer; Por.%= porosity in vol.%; Crist.%= cristallinity in vol.% refers tothe total amount of phenocrysts plus microlites. Breccia deposits are indicated in italics.

Samplename

Thick.(cm)

Texture and grading Clast size Por.(%)

Crist.(%)

Val 01 31 Normal graded Lapilli dominatedVal 02 59 Reverse graded Lapilli dominated 19 7Val 03 89 Reverse to normal graded Fine ash–lapilli–fine ashVal 04 31 Massive Lapilli dominated 6Val 05 28 Massive Lapilli dominated 6Val 06 15 Normal graded Lapilli di coarse ashVal 07 33 Reverse graded Fine ash to lapilliVal 08 31 Massive Lapilli dominated 9Val 09 64 Massive Lapilli dominatedVal 10 61 Reverse graded Fine ash to lapilliVal 11 59 Normal graded Lapilli di coarse ashVal 12 72 Normal graded Lapilli to fine ash 21 8Val 13 72 Massive Lapilli dominatedVal 14 27 Normal graded Lapilli to fine ashVal 15 85 Massive Lapilli dominated 9Val 16 42 Normal graded Lapilli di coarse ashVal 17 48 Reverse to normal graded Fine ash–lapilli–fine ashVal 18 68 Reverse to normal graded Fine ash–lapilli–fine ashVal 19 76 Massive Lapilli dominated 23 10Val 20 24 Massive Lapilli dominated 9Val 21 68 Normal graded Lapilli di coarse ashVal 22 16 Reverse graded Coarse ash to lapilliVal 23 16 Reverse to normal graded Fine ash–lapilli–fine ashVal 24 89 Massive Lapilli dominatedVal 25 85 Normal graded Lapilli to fine ash 17 8Val 26 68 Normal graded Lapilli di coarse ashVal 27 59 Reverse graded Fine ash to lapilliVal 28 78 Massive Lapilli dominated 7Val 29 54 Massive Lapilli dominatedVal 30 55 Normal graded Lapilli di coarse ashVal 31 33 Massive Lapilli dominatedVal 32 89 Reverse to normal graded Fine ash–lapilli–fine ash 29 7Val B01 150 Massive Breccia deposit 5 28Val 33 68 Normal graded Lapilli to fine ashVal 34 96 Massive Lapilli dominatedVal 35 72 Massive Lapilli dominated 5Val 36 68 Massive Lapilli dominatedVal 37 55 Reverse graded Coarse ash to lapilli 8Val 38 46 Reverse to normal graded Fine ash–lapilli–fine ashVal 39 55 Reverse graded Fine ash to lapilliVal 40 48 Massive Lapilli dominated 7Val 41 72 Reverse to normal graded Fine ash–lapilli–fine ashVal 42 23 Normal graded Lapilli to fine ash 32 6Val 43 32 Normal graded Lapilli to fine ashVal 44 14 Reverse graded Coarse ash to lapilliVal 45 16 Reverse to normal graded Fine ash–lapilli–fine ash 7Val 46 98 Normal graded Lapilli di coarse ashVal 47 44 Massive Lapilli dominatedVal 48 67 Normal graded Lapilli di coarse ash 36 9Val B02 180 Massive Breccia deposit 4 31Val 49 64 Reverse graded Fine ash to lapilliVal 50 55 Massive Lapilli dominatedVal 51 66 Massive Lapilli dominated 10Val 52 72 Normal graded Lapilli di coarse ashVal 53 46 Reverse graded Coarse ash to lapilli 34 8Val 54 32 Reverse to normal graded Fine ash–lapilli–fine ashVal 55 53 Normal graded Lapilli to fine ashVal 56 23 Reverse graded Fine ash to lapilli 8Val 57 53 Massive Lapilli dominatedVal 58 16 Reverse graded Fine ash to lapilliVal B03 210 Massive Breccia deposit 3 26Val 59 57 Normal graded Coarse ash to lapilli 46 7Val 60 18 Massive Lapilli dominatedVal 61 59 Reverse graded Fine ash to lapilliVal 62 32 Normal graded Lapilli to fine ash 9Val 63 27 Reverse to normal graded Fine ash–lapilli–fine ashVal 64 51 Reverse graded Coarse ash to lapilli 8Val 65 46 Massive Lapilli dominatedVal 66 25 Massive Lapilli dominatedVal 67 48 Normal graded Lapilli to fine ash 28 10Val 68 79 Reverse graded Coarse ash to lapilli

(continued on next page)

Table 1 (continued)

Samplename

Thick.(cm)

Texture and grading Clast size Por.(%)

Crist.(%)

Val 69 51 Reverse to normal graded Fine ash–lapilli–fine ash 6Val 70 48 Reverse graded Fine ash to lapilliVal 71 64 Massive Lapilli dominatedVal 72 44 Massive Lapilli dominated 33 8Val 73 20 Massive Lapilli dominated 7Val 74 16 Normal graded Lapilli di coarse ashVal 75 74 Reverse to normal graded Fine ash–lapilli–fine ash 8

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comparison with the analysis of experimentally produced pyroclastsKueppers et al. (2006).

In order to apply the number-based size relationship Eqs. (2)and (3) to the analysis of the scoria cone deposits, assumptionsmust be made regarding the unit weight of individual grains, sincegrain size distributions are determined by mass comparison, i.e. massretained on successive sieve sizes. The number-based size relationshipcan be achieved by modeling individual grains as uniform shapes,thereby developing uniform standard densities. By adjusting the sizeof the grain to coincide with sieve aperture, the number of grainsbounded by each sieve can be determined by dividing the total weightof material retained on each sieve by the assumed density. Once thenumber of grains for each granulometric class has been calculated,Eqs. (2) and (3) can be applied to determine the value of fractaldimension of fragmentation. It is noteworthy that a constant densityof particles irrespective of the particle size may be generally notexpected, but needs to be tested for each case study (e.g. Dellinoet al., 2005). In our case, a strong evidence for the validity of thisassumption is that data points are disposed along a straight line,defining an excellent fractal relationship (Fig. 6). If this assumptionwas not valid, a deviation of data points from the straight line shouldbe observed. The absolute value of density of pyroclasts does notaffect the value of fractal dimension of fragmentation (D). The sameprocedure was utilized by Kueppers et al. (2006) in their fractalanalysis of experimentally generated pyroclasts.

6. Results of grain size analysis

Fragmentation efficiency as displayed in the grain size distributiondata was variable during the Valentano eruption as can be clearlyseen in some representative granulometry plots presented in Fig. 5.The plots clearly show shifts in the relative percentage of differentgranulometric classes. This feature is much more evident in thecumulative distributions that plots of particles displayed in Fig. 6used to quantify the degree of fragmentation by fractal analysis. Fromthese graphs it is clear that fragments generated by the eruptiveactivity follow a fractal distribution. The degree of fragmentation ofeach pyroclastic layer can be, hence, quantified by the value of fractaldimension of fragmentation (D): the higher the D value, the finer theaverage grain size indicating efficient fragmentation. To evaluateuncertainty associated with fractal dimension (D), repeated analysesand D estimates for several samples from the same pyroclastic layerhave been carried out. Results indicate that D values can be measuredwith an error better than 3%. Before moving to further investigationof data it may be worth discussing whether the studied grain sizedistributions are fully representative of the original volcanic frag-mentation event or additional processes that may have acted todetermine the size–number relationships presented in Fig. 6 and,hence, the value of fractal dimension of fragmentation. At least twomain processes need to be considered: (1) the wind dispersal ofparticles after the fragmentation process and (2) the influence oftransport of particles during sedimentation. Both processes canpotentially affect the original grain-size distribution. Regardingwind dispersal, this process is expected to operate much moreefficiently on the ash-sized fraction of particles. In this work, we

Fig. 3. Thin section micrographs showing crystallinity and vesicularity of some representative pyroclastic samples (a–f) and breccia deposit (g–h). Top and bottom panels in eachcolumn are respectively plane and cross-polarized light pictures of the same sample. Sample labels as in Fig. 7. The sample labelled as ‘breccia’ corresponds to the breccia depositlocated at ca. 38 m along the pyroclastic sequence in Fig. 7.

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Fig. 4. Digital images of sections of three representative samples at different distances along the studied pyroclastic sequence, showing the % value of porosity. Sample labels as inFig. 7.

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analyzed fragments down to 0.21 mm in size. Fig. 6 shows that thepower–law (fractal) relationship holds very well for the entire rangeof analyzed fragment sizes (i.e. data points are perfectly disposed

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Fig. 6. (a–c–e) variation of N against r for three representative samples collected at progressively increasing distance from the center of the scoria cone; (b–d–f) Log(N) vs. Log(r)graphs showing the fractal nature of grain size distributions of studied pyroclastic deposits. D and r2 (correlation coefficient) values resulting from linear fitting are also reported inthe graphs. Plots presented in this paper were prepared by using the software Petrograph (Petrelli et al., 2005).

206 D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

law relationship. In other words, a kink in the straight line (Fig. 6)should have been observed. This feature is not observed in the studiedsamples, indicating that, at least for the range of fragment sizesstudied in our work, the effect of dispersal of ash-sized particles bywind did not influence results. Regarding transport (and/or remobi-lization), the sampled quarry was the most proximal location possibleand the influence of secondary transport processes on the grain sizedistribution should be negligible. In addition, as shown in Fig. 6,

the distribution of data points defines a single straight line; thisfeature is consistent only with a single fragmentation process. In fact,if a second fragmentation process (i.e. due to transport) would haveoverlapped to the original ‘volcanic’ fragmentation process, we shouldhave observed a kink in the line for particles having the largestsize, since they tend to fragment more easily than smaller particles(e.g. Turcotte, 1992). This kink is not observed in the grain sizedistributions of Valentano, arguing against the hypothesis that

Fig. 8. Frequency histogram of fractal dimension of fragmentation values (D) measuredfor the 75 studied pyroclastic deposits. The number of classes to divide D values toconstruct the histogram has been calculated using the relationship derived by Scott(1979).

207D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

further fragmentation due to transport was an essential process todetermine the observed grain size distributions. In addition, second-ary grain transport would have caused abrasion and led to roundedgrains. As can be seen from Fig. 2a, the morphology of grains is notrounded; on the contrary they show cuspate margins and elongatedshapes corroborating the idea that grain transport did not play afundamental role in producing the observed grain size distributions.Therefore, the fractal relationship observed for the grain sizedistributions of Valentano deposits can be considered inheritedfrom the volcanic fragmentation of the magma during the eruptiveactivity.

The graph of Fig. 7 reports the variation of fractal dimension (D)along the analyzed stratigraphic sequence; to better appreciate thevariability of D values, these are also reported in the histogram ofFig. 8. In Fig. 7 the pyroclastic deposits are progressively youngermoving from the left to the right side of the graph. D values rangefrom 1.512 to 2.614 and they are found to change extensivelythroughout the sequence defining two major variation patterns: (1) ashort-range, high frequency, variation in which D values oscillatedefining a zig-zag pattern which can be recognized along the entirelength of the pyroclastic sequence and (2) a long-range variation inwhich D values tend to progressively increase passing from the olderto younger deposits. This latter variation is marked by spikes of Dreaching values of up to approximately 2.6. It is important to note thatthese spikes of D always belong to the layer directly above a brecciadeposit (indicated by the gray lines in the plot of Fig. 7). This indicatesthat fragmentation efficiency was not constant during explosiveactivity that progressively built the scoria cone, nor fragmentationoscillated steadily between limiting values (i.e. D increases towardsmost recently erupted deposits). The main question arises as to whatprocesses may have acted to produce the observed variation of D inthe pyroclastic sequence.

7. Discussion

A recent experimental work (Kueppers et al., 2006) has shownthat fractal dimension of fragmentation can be used to quantify theexplosive energy of eruptions. They analyzed the grain size distribu-tion of pyroclasts, experimentally generated from dacitic volcanic

Fig. 7. Variation of fractal dimension of fragmentation (D) along the studied pyroclasticsequence. The first analyzed sample at position zero was collected at ca. 15 m from thecenter of the scoria cone to minimize potential sampling problems related to transienteruption stages associated with the beginning of volcanic activity. The dashed linerepresents the linear fitting of data and indicates an increase of fragmentation efficiencymoving towards the external layers of the volcanic edifice. Capital letters in the graphindicate samples selected for porosity measurements (Fig. 4; see text). Note the highestspikes of D values marked by E, G, and I systematically following breccia deposits(vertical gray lines).

rocks of Mt. Unzen volcano (Japan) by rapid decompression of gas-pressurized samples. We applied the same method for the analysis ofnatural deposits of explosive, basaltic volcanism. Their resultsevidenced that grain size distributions followed a fractal distributionwith measured values of fractal dimension of fragmentation (D)directly related to the applied overpressure and sample porosity.Experimental data (Kueppers et al., 2006) and data derived fromnatural deposits share an important basic feature: at magmaticconditions, the expanding gas exerts a stress on the melt. If this stressis too high or too rapidly applied to be compensated by viscousdeformation, the magma experiences a transition from viscous tobrittle behavior and fragmentation takes place.

7.1. Application of the Renormalization Group approach tofragmentation

Fragmentation is a catastrophic phenomenon. If the stress on agiven volume of material (rock, magma, etc.) is increased, a criticalstress will be reached at which fragmentation will occur. The fractalbehavior of particle size distributions studied in this work indicatesthat the fragmentation process is scale invariant over a range of scales.A possible conceptual model for fragmentation can be derived byusing the Renormalization Group approach (RG). RG techniques havebeen successfully applied to scale invariant processes that exhibitsudden or catastrophic change, including rock fragmentation (Allegreet al., 1982; Turcotte, 1986). In order to model fragmentation, thematerial is hypothesized to develop microfracturing under appliedstress. In our case, the applied stress is given by the expansion of gasin the magma just before and during fragmentation and eruption. Asa hypothetical model, in the following we will consider a cube witha linear dimension h, as illustrated in Fig. 9. This cube is referred to as azero-order cell; there are N0 of these cells. As an example, each zero-order cell may be divided into eight zero-order cubic elements eachwith a dimension of h/2. The volume V1 of each of these elements isgiven by

V1 =18V0; ð8Þ

where V0 is the volume of the zero-order cells. The probability that azero order cell will fragment to generate eight zero-order elements is

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2.8

3.0

Fig. 10. Variation of probability of fragmentation (p) as a function of fractal dimensionof fragmentation (D) for the samples studied in this work.

h

0

1

2

h/4

h/8

Fig. 9. Illustration showing the application of the Renormalization Group approach tofragmentation. A zero order cubic cell with dimension h is divided into eight cubicelements with dimension h/2. The critical probability that a zero-order cell will befragmented into eight elements is p. The fragments with dimension h/2 become firstorder cells; each of these first-order cells is divided into eight first-order elements withdimension h/4. The critical probability that a first-order cell will be fragmented intoeight elements is also p. The process is repeated to higher orders.

208 D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

p. Thus, the number of zero-order elements produced by the frag-mentation is

N1 = 8p N0: ð9Þ

After fragmentation the number of zero-order cells that have notbeen fragmented (N0a) is given by

N0a = 1−pð ÞN0: ð10Þ

Each first-order cell can be fragmented into eight first-order cubicelements each with a dimension of h/4 (Fig. 9). The fragmentationprocess is repeated for these smaller cubes. The problem is thenrenormalized (see Turcotte, 1986 for details) and the cubes withdimension h/2 are treated in exactly the same way as the cubes withlinear dimension hwere treated above (Fig. 9). Each of the fragmentedcubic elements with linear dimension h/2 is taken as a first-ordercell; each of these cells is divided into eight first-order cubic elementswith linear dimension h/4 (Fig. 9). The volume of each first-orderelement is

V2 =18V1 =

182 V0: ð11Þ

The probability that a first-order cell will fragment is again takento be p in order to preserve scale invariance. The number offragmented first-order elements is

N2 = 8pN1 = 8pð Þ2N0: ð12Þ

After fragmentation the number of first order cells that have notbeen fragmented is

N1a = 8p 1−pð ÞN0: ð13Þ

This process is repeated at successively higher orders. The volumeof the nth-order cell Vn is given by

Vn =18n V0: ð14Þ

After fragmentation the number of nth-order cells is

Nna = 8pð Þn 1−pð Þ N0 = 8pð ÞnN0a: ð15Þ

Taking the logarithm of both sides it is possible to write Eqs. (14)and (15) as

lnVnV0

= −n ln 8; ð16Þ

lnNna

N0a= n ln 8p: ð17Þ

Elimination of n from Eqs. (16) and (17) gives

Nna

N0a=

Vn

V0

� �−ln8p= ln8:ð18Þ

Comparison of Eq. (18) with Eqs. (7a) and (7b) indicates that thisis a fractal distribution with

D = 3ln 8pln 8

: ð19Þ

Although the model is much idealized and the division of the unitcell is obviously not unique, it is possible to show that the lineardimension of elements does not influence results that, hence, areindependent of the renormalization configuration chosen (Turcotte,1986). For instance, the division of the unit cell into two or 16fragments gives the same result (Turcotte, 1992). Thus, whatever thechosen configuration is, Eq. (19) highlights a very important feature:it shows that the value of fractal dimension of fragmentation Dstrongly depends on the probability for fragmentation p. Consideringthe data shown in Fig. 7 for the natural case study and the relationshipbetween D and p, hence, it can be said that fragmentation probabilityduring the course of the eruptive activity changed in time followingthe two major trends already identified in the variation of D values(i.e. short- and long-range variation). Fig. 10 shows the fragmentationprobability of studied samples calculated by Eq. (19) as a function offractal dimension of fragmentation (D) measured on the studiedpyroclastic deposits, according to Eq. (19).

This discussion highlights an important additional significanceof the fractal dimension parameter: if D is correlated with the frag-mentation probability of the starting material, it can be interpretedas a proxy for its “fragility” (i.e. ability to fragment into smaller

209D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

fragments), with higher p values associated with more fragilesamples. Therefore, the observed variation of fractal dimension(Fig. 7) along the pyroclastic sequence also reflects a variability ofthe fragility of material that is fragmented during the progression intime of the scoria cone eruption.

7.2. Magma “fragility” and vesicle content of erupted products

A possible source for different fragility, and hence, different Dvalues of the fragmented material may be the amount of vesicles(porosity) in the different samples. Porosity of the samples is, indeed,directly related to the amount of gas expanding in the magma andtriggering the fragmentation process (e.g. Kueppers et al., 2006). Inparticular, the larger the amount of vesicles in the magma, the higherits fragility, since the average bubble wall thickness is decreasing.Bubble walls are those sample portions on which gas overpressureacts and, thus, they can be considered as “weakness” zones enhancingthe production of smaller magma fragments (i.e. increasing fragmen-tation efficiency). This also means that probability p is related to theenergy available for fragmentation.

The graph of Fig. 11 shows the variation of sample porosity (%)against fractal dimension of fragmentation (D). It can be seen that Dis positively correlated with sample porosity corroborating the ideathat higher vesicle contents imply higher fragmentation efficiency(Kueppers et al., 2006).

7.3. Correlation between D and explosive activity

In an attempt to give a volcanological explanation to the abovediscussed change in fragmentation efficiency during the formationof the scoria cone, it is thus necessary to consider the processesresponsible for the variation of porosity in the magma as the eruptionof the studied pyroclastic deposits progressed in time. Strombolianeruptive activity is driven by the rise of gas in a magma-filled conduit.Two main regimes can be identified depending on the dimension ofbubbles moving in the magma column: (1) rise of many individualbubbles and (2) rise of gas slugs. The first process may evolve intothe second one upon coalescence of many small bubbles eventuallyforming large slugs (e.g. Wilson, 1980; Jaupart, 2000; Parfitt, 2004).The bubble/magma ratio and the relative speed and flow directiondefine the eruptive dynamics. As reported in Section 3 we can clearlydistinguish separated eruptive pulses. Based on the clear separation(short hiatus) and the large volumes for each single explosion(inferred from the fairly large thickness of pyroclastic deposits;

1.6 1.8 2.0 2.2 2.4 2.615

20

25

30

35

40

45

50

A

BC

D

EF

G

H

I

J

K

Fig. 11. Variation of sample porosity (%) as a function of fractal dimension offragmentation (D) for the samples studied in this work. Sample labels as in Fig. 7.

Table 1), it is likely that large slugs were themain driving force behindthe explosions. Therefore, it appears reasonable to suppose that, afterthe initial transient stages of volcanic activity characterized by theopening of the conduit, eruption initially mainly developed in the so-called ‘slug-flow’ regime. Cooling from the top of the magma columnmay induce the formation of an interface with a finite strength. Theinterface strength depends on how much cooling occurs before thearrival of the next gas pocket. If the interval between gas pocketarrivals is short enough, each slug will deform and possibly rupturethe thin interface. If the interval between the arrivals of gas pockets islonger, the interface will cool and thicken, eventually forming a plug-like structure. This way, rising bubbles and slugs accumulate and aretrapped before sufficient pressure is built up to fragment the plug onthe top of the conduit. Repeated cycles of cooling and bubbleaccumulation followed by bubble bursting can lead to the series oftransient explosions which characterize scoria cone eruptive activity.The amount of slugs and bubbles, hence, can be variable in timeleading to oscillating fragmentation probability and, thus, to oscillat-ing values of fractal dimension of fragmentation (D). This process mayexplain the high frequency variation of D values along the pyroclasticsequence (Fig. 7).

The interpretation of the increase of fragmentation efficiencytowards the latest stages of eruptive activity may appear moreproblematic. In fact, a typical conceptual model invoked for volcaniceruptions is that they begin with violent explosion phases in whichthe fragmentation process is very efficient and fade out progressivelyin time leading to progressively lower fragmentation efficiencies (e.g.Jaupart and Allegre, 1991). However, recent studies indicate thatbasaltic eruptions can follow more complex patterns in whichfragmentation efficiency does not necessary decrease with time (e.g.Taddeucci et al. 2002; Valentine et al., 2005; Cimarelli et al., 2010). Inour case we observe that fragmentation efficiency was increasingduring the course of the eruption. In the attempt of providing anexplanation for this feature, we need to recall that the highest valuesof fractal dimension of fragmentation (D; i.e. fragmentation efficien-cy) characterize systematically those pyroclastic layers occurringstratigraphically above breccia deposits (Fig. 7). As reported above,clasts from these breccia deposits have different characteristics thanthe “normal” eruption products; they exhibit an angular shape, have alower porosity and a higher crystal content; Figs. 2b, 3h–g) indicatingthat these rocks had considerably lower temperatures. Consideringthat passing from the left to the right side of the plot of Fig. 7corresponds to moving to the latest stages of volcanic activity, it isevident that the breccia deposits characterize mostly the last stagesof the scoria cone eruption. A hypothesis that may explain this featureis the formation of solid plugs (i.e. solid thick interfaces) in theconduit, considering a decrease of eruption rate in time, as it shouldbe expected from the time evolution of eruption dynamics of amonogenic volcanic edifice. The decrease of eruption ratewould allowthe top of themagma in the conduit to solidify forming plugs of a solidbasaltic rock. Evidence of long residence time of the solidifyingmagma in the conduit is provided by the texture of breccia depositsshowing a very low vesicle content and higher crystallinity relative tolapilli-size deposits. However, gas slugs and bubbles accumulateprogressively and possibly coalesce below the solid plug. When gasoverpressure overcomes the strength of the overlying plug, anexplosive eruption is initiated ejecting first fragments of the plug(breccia layer) followed by juvenile material (layer above with highD values). This idea is in agreement with previous studies indicatingthat the formation of plugs during basaltic explosive eruptive activitycan cause changes in the degree of fragmentation (Taddeucci et al.,2004; Andronico et al., 2009).Therefore, the general increase offractal dimension during the eruptive history of this monogeneticvolcano could be related to the “aging” of the volcanic system. Thedecrease of eruption rate with time causes the magma to spend moretime in the volcanic conduit. During this stall, volatiles released from

210 D. Perugini et al. / Journal of Volcanology and Geothermal Research 202 (2011) 200–210

magma further down in the conduit will rise and accumulate. Theresulting foamy magma will be fairly easily fragmented, causing areduction in the mean diameter of pyroclasts (i.e. high fragmentationefficiency).

8. Summary and conclusions

In this contribution we measured the grain size distribution ofnatural pyroclastic deposits within a basaltic scoria cone. Grain sizedistributions have been studied by the fractal fragmentation theoryand the main results can be summarized as follows:

(1) The grain size distribution of pyroclasts is consistent withfractal fragmentation theory in that size distributions are fractals andthey can be univocally quantified bymeasuring a single parameter, i.e.the fractal dimension of fragmentation (D). This result is recognized inthis study for grain size distribution generated by basaltic eruptionsand, together with previous works based on application of fractalstatistics to more silicic pyroclastic deposits (Kueppers et al., 2006;Suzuki-Kamata et al., 2009), points towards potentially wideapplicability of fractal methods to quantify fragmentation processesduring explosive volcanic eruptions.

(2) Fractal dimension of fragmentation (i.e. fragmentationefficiency) varies extensively along the studied stratigraphic sequenceshowing the superimposition of two major patterns: a high frequencyoscillation and an overall increase towards the last erupted pyroclasticlayers. The highest values of fractal dimension of fragmentation havebeen measured on pyroclastic layers following breccia deposits, thelatter likely representing fragments of solid thick plugs forming in thevolcanic conduit during decreasing eruption intensity. The highfrequency variation is interpreted as a consequence of the slug-flowregime that typically characterizes strombolian eruptive activity. Theincrease of fragmentation efficiency towards the latest stages ofvolcanic activity is interpreted as related to a decrease of eruption rateand the formation of solid plugs on the top of the conduit. This way,the residence time of magma increases and allows the nucleation of ahigher amount of bubbles, whose expansion leads to the removal ofsolid plugs and more efficient fragmentation of magma.

In conclusion, application of fractal fragmentation theory tonatural pyroclastic deposits may give new insights into the processesassociated with volcanic explosions as it provides clues for theinterpretation of possible transitions of eruptive style with time andmay furnish additional methods to evaluate potential explosivity ofvolcanic deposits, a basic issue in volcanic hazard assessment.

We proposed a fractal fragmentation model to explain the variableexplosivity during the growth of theValentano scoria cone (Italy). Basedon this, more sophisticated models may be developed in the future andaccount more in detail for additional features, such as crystal content,crystal size distribution and spatial distribution of vesicles.

Acknowledgments

This work was funded by MIUR (Ministero Italiano dell'Universitàe della Ricerca), University of Perugia grants, and Humboldt Foun-dation (Germany) that awarded DP a Humboldt Fellowship to carryout part of this research activity. We thank an anonymous reviewerand Daniele Andronico for their thorough comments that helped toimprove the manuscript.

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