Porosimetry of vesicular volcanic products by a water-expulsion method and the relationship of pore characteristics to permeability

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008jb005758, 2009 Porosimetry of vesicular volcanic products by a water-expulsion method and the relationship of pore characteristics to permeability Tadashi Yokoyama 1 and Shingo Takeuchi 2 Received 21 April 2008; revised 5 November 2008; accepted 17 November 2008; published 4 February [1] Porosimetry of vesicular volcanic products was conducted by a water-expulsion method to investigate the relationship between pore characteristics and gas permeability. The pores in a rock were categorized as total, connected, transport, dead-end, and isolated pores. Gas flow takes place only through the transport pores but the pores that do not provide a flow path are included in both total and connected porosities. Permeability correlated weakly with total porosity, connected porosity, and transport porosity. The characteristic maximum pore-throat radius r ch (m) was also determined. Both r ch and transport porosity correlated well with permeability k 1 (m 2 ) according to the following form of the Kozeny-Carman relation: k 1 = f tra r 2 ch. Permeability is thus strongly dependent on pore size, and estimation of pore size is essential for understanding the permeabilities of vesicular volcanic products. Citation: Yokoyama, T., and S. Takeuchi (2009), Porosimetry of vesicular volcanic products by a water-expulsion method and the relationship of pore characteristics to permeability, J. Geophys. Res., 114,, doi: /2008jb Introduction [2] Understanding the flow of gas and liquids through geologic media is important in diverse fields of earth science and engineering, such as petroleum engineering, groundwater hydrology, agricultural engineering, soil science, sedimentology, economic geology, and volcanology. The volume of published data on the permeability of volcanic products has increased since the concept of permeable degassing was first proposed by Eichelberger et al. [1986]. The permeability of magma controls the degassing process in volcanic conduits and is directly associated with the explosivity of volcanic eruptions [Eichelberger et al., 1986; Melnik et al., 2005]. [3] The permeability of volcanic products has been discussed often from the perspective of the permeabilityporosity relationship [Eichelberger et al., 1986; Klug and Cashman, 1996; Melnik and Sparks, 1999; Saar and Manga, 1999; Jouniaux et al., 2000; Blower, 2001; Klug et al., 2002; Rust and Cashman, 2004; Mueller et al., 2005; Takeuchi et al., 2005; Wright et al., 2006a, 2006b; Bernard et al., 2007]. It is generally recognized that permeability increases with increasing porosity, and empirical relationships between porosity and permeability have been proposed based on such as the Kozeny-Carman model and percolation theory [Klug and Cashman, 1996; Rust and Cashman, 2004; Mueller et al., 2005]. However, the validity of the empirical relationships is often unclear because permeabilities range 1 Department of Earth and Space Science, Osaka University, Osaka Japan. 2 Geosphere Sciences, Civil Engineering Research Laboratory, Central Research Institute of Electric Power Industry, Chiba, Japan. Copyright 2009 by the American Geophysical Union /09/2008JB005758$09.00 over two to four orders of magnitude for a given porosity (e.g., Figure 6f). Various factors such as the shape, size, distribution, and connectedness of pores can affect the permeability-porosity relationship; the reason for the considerable variation in the relationship is a topic of debate. [4] Permeability is usually determined by measuring the rate of fluid flow through a rock sample that is open at the top and bottom and peripherally sealed to prevent lateral flow. Figure 1 is a schematic cross section showing pores in a sample that has been prepared for analysis. The pores can be categorized as follows: [5] 1. Total pores. [6] 2. Connected or open pores: those pores that are open to the upper or lower end of the sample, or to both. [7] 3. Isolated or closed pores. [8] 4. Transport pores are connected pores that are open to both the upper and lower ends of the sample; fluid can be transported through the sample by these pores. Pores that branch off transport pores but come to a dead end are not included in our definition of transport porosity. These pores are also referred to as backbone pores. [9] 5. Dead-end pores are those pores that are connected to transport pores or directly to the upper or lower end of the sample but come to a dead end. [10] According to the above definitions, fluid flows only through the transport pores. However, the pores that are unavailable for flow contribute to both total porosity and connected porosity. Definitions of the categories of porosity by some researchers differ; therefore, care must be taken when comparing the results of different studies. In previous studies of the permeability of volcanic products, total porosity and connected porosity have often been considered, but transport porosity as defined above has not been reported. [11] To characterize different types of pores, we applied a water-expulsion method that analyzes the way in which 1of10

2 Figure 1. Schematic illustration of pore structures in a rock and the formulas used for calculation of the different categories of porosity. pore water is extruded from a water-saturated sample [e.g., Gelinas and Angers, 1986; Innocentini and Pandolfelli, 2001]. The water-expulsion method allows estimation of transport porosity and the characteristic radius of pore throats, as well as total porosity and connected porosity. Our results show that permeability is only weakly correlated with total, connected, and transport porosities, whereas the characteristic maximum pore-throat radius is strongly correlated with permeability. Our study shows that the relationship among permeability, porosity, and pore-throat radius can be expressed by the Kozeny-Carman relation. 2. Methods 2.1. Permeability Measurements [12] Figure 2a is a schematic illustration of the apparatus we used for permeability measurements. Compressed air was used as the working fluid. For laminar flow in a rock sample, the relationship between the specific flow rate n (volumetric gas flow rate per cross-sectional area of the sample, m s 1 ) and the inlet and outlet pressures of the sample is described by Darcy s law for a compressible fluid: P 2 2 P2 1 2P 0 L ¼ m gas k a n; ð1þ where P 1 and P 2 (Pa) are the gas pressures at the outlet and the inlet, respectively, of the sample, P 0 (Pa) is the gas pressure at which flow rate is measured (Pa), L (m) is the length of the sample, m gas (Pas) is the viscosity of gas, and k a (m 2 ) is the apparent permeability. In this study, P 0 and P 1 were atmospheric pressure at the time of measurement. The left side of equation (1) accounts for the compressibility of Figure 2. (a) Schematic illustration of the apparatus used for gas permeability measurement. (b) Setup for water-expulsion method of measuring transport porosity. (c) Setup for measuring characteristic maximum pore-throat radius. (d) The process used to saturate pores with water. Note that closed pores do not contain water. 2of10

3 Table 1. Summary of Experimental Results and Description of Samples a Sample Abbreviation Sk para Koma 7.8 Koma 9.7 Nj My Sk perp AsFg1S Sk MTX Usu17.1 Iki 2 Kz Kb Kz Tj AsFg1L Viscous permeability (k1: m 2 ) 8.6E E E E E E E E E E E E 15 Inertial permeability (k2: m) 3.2E E E E E E E E E E E E 13 Bulk density (10 3 kg m 3 ) Solid density (10 3 kg m 3 ) Sample length (10 3 m) Sample area (10 6 m 2 ) Sample volume (10 6 m 3 ) Total porosity (%) Connected porosity (%) Transport porosity (%) Dead-end porosity (%) Isolated porosity (%) Scale of typical pore radius 100 mm to 1mm 10 mm to 10 mm to 100 mm to 1mm Tens to Tens to Several mm to tens 10 mm to a few hundred mm Tens to 10 mm to a few hundred mm 10 mm to Tens mm to Breakthrough pressure (Pa) Characteristic maximum pore-throat radius (10 6 m) a Sample descriptions: Sk para and Sk perp, juvenile tube pumice blocks from Shikotsu pumice flow (parallel and perpendicular to pore elongation) [Katsui, 1959]; Sk MTX, weakly welded matrix from Shikotsu pumice flow; Koma 7.8 and Koma 9.7, phenocryst-rich, 1929 plinian-fall pumice from Komagatake volcano [Takeuchi and Nakamura, 2001]; As Fg1L and As Fg1S, andesitic low vesicularity lapilli from 2004 vulcanian explosion of Asama volcano [Shimano et al., 2005]; Usu 17.1, phenocryst-poor, 1977 plinian-fall pumice of Usu volcano [Tomiya and Takahashi, 2005]; Iki 2, air-fall scoria from Kilauea Iki, Hawaii; Nj My, pumicious lava from 886 AD eruption of Mukaiyama volcano, Nijima [Isshiki, 1987]; Kz Kb and Kz Tj, pumicious lavas from 1.8 ka eruption of Kobeyama volcano and 838 AD eruption of Tenjyosan volcano, Kozushima [Yokoyama and Banfield, 2002]. 3of10

4 Figure 3. Cross-sectional images of selected samples. CCD scanner images perpendicular to the experimental flow direction for (a) Sk para and (b) Sk perp. Backscattered electron images (BEI) of polished surfaces of samples from (c) Komagatake volcano and (d) Usu volcano. Note that images (c) and (d) are not of the samples used for permeability measurement, but are from the same sources as Koma 7.8 and Koma 9.7, and Usu Pores (black) in images (c), (d) and (e) were impregnated with resin. (e) BEI of the polished surface of Kz Tj. (f) Scanning electron microscope topographic image of Kz Kb. gas flowing through the sample (modified pressure gradient). For an incompressible fluid, the left side is simplified to (P 2 P 1 )/L. In some cases, however, k a decreases with increasing pressure gradient because of energy loss due to inertial effects. The dependency of k a on pressure gradient was reported previously for volcanic products by Rust and Cashman [2004] and Takeuchi et al. [2008]. To account for inertial effects, Forchheimer s equation [Forchheimer, 1901] has been used to determine the permeability of volcanic products [Rust and Cashman, 2004]: P 2 2 P2 1 2P 0 L ¼ m gas n þ r gas n 2 ; ð2þ k 1 k 2 where k 1 (m 2 ) is the viscous (Darcian) permeability, k 2 (m) is the inertial (non-darcian) permeability, and r gas (kg m 3 ) is the density of the gas used for the measurement. [13] Twelve vesicular volcanic products with total porosities ranging from 11% to 82% were used in our experiments. Sample descriptions and localities are provided in the footnotes of Table 1. Images of some of the samples are provided in Figures 3a 3f. Variations of pore size within our samples are generally large. The pores were formed by exsolution of bubbles from magma at the time of eruption and some became elongated or collapsed during eruptive processes. In determination of permeability, differential pressure DP (= P 2 P 1 ) (Figure 2a) was increased at intervals and n was measured at each pressure. For DP 5000 Pa, the differential pressure was adjusted by a pressure regulator. Delicate pressure adjustment by the pressure regulator was difficult for DP < 5000 Pa; for this range a pressure stabilizer (Figure 2a) was used to adjust the differential pressure. The pressure stabilizer was composed of acrylic inner and outer pipes. The inner pipe was connected to the main gas-flow line and water was inserted via the outer pipe. When gas pressure in the inner pipe exceeded the water head pressure, bubbles spilled from the bottom of the inner pipe so that the gas pressure was stabilized at the water head pressure. The differential pressure was thus adjusted within an error margin of about ±<5%. The measurements were made at C. At this temperature m gas was approximately Pas and r gas was 1.2 kg m 3 ; for both values, there were possible variations of up to a few percent. Further details were described by Takeuchi et al. [2008]. Figure 4 shows the results of several of our permeability measurements. The observed parabolic increase of the modified pressure gradient with increasing n indicates that there were inertial effects. k 1 and k 2 were determined by fitting to equation (2). If the inertial effect is not considered (i.e., equation (1) is used instead of equation (2)), k a differs from k 1 by a factor of for sample nos. 4, 5, 6, 7, 8, 10, 11, and 12; by a factor of for sample nos. 1, 3, and 9, and by a factor of 1 22 for sample no Porosimetry Porosity Measurements [14] Porosimetry employed in our study requires that pores are well saturated with water. In saturating rock pores with water, the following procedure is commonly used: a sample is immersed in a vessel of water, the vessel is inserted in a vacuum chamber, the chamber is degassed, and air is then drawn out from the pores. This process is simple, but there is a risk of incomplete saturation because 4of10

5 Figure 4. Plot of modified pressure gradient vs. gas flux for selected permeability measurements. The numbers correspond to the sample numbers listed in Table 1. The curves are fitted based on Forchheimer s equation for determination of viscous permeability k 1 and inertial permeability k 2. some air might remain in the pores. Thus, in our study, each sample was first degassed in a vacuum chamber before water was inserted into the chamber (Figure 2d). Each of the porosities defined in Figure 1 were determined as follows: [15] (1) Total porosity (f tot ) = 1 (bulk density/solid density), where bulk density is sample dry weight/bulk volume, and solid density is measured by water pycnometry (see below); (2) Connected porosity (f con ) = (water-saturated weight dry weight)/bulk volume; (3) Isolated porosity (f iso ) = total porosity connected porosity; (4) Transport porosity (f tra ) was determined by the water-expulsion method (see below); (5) Dead-end porosity (f end ) = connected porosity transport porosity. For water pycnometry measurements, a powdered rock sample of known weight W sam (kg) and unknown volume V sam (m 3 ) was inserted in a pycnometer with a volume V pic of m 3. To completely wet the powder, the pycnometer was half-filled with water and degassed in a vacuum chamber. Care was taken during degassing to avoid leakage or boiling over of the pycnometer. After degassing, water was added to fill the pycnometer and the total weight W tot (sample + water, kg) was measured. Then, V sam was determined as follows: V sam = V pic (W tot W sam )/r water, where r water is the density of water (kg m 3 ). Finally, solid density (kg m 3 ) was calculated as W sam /V sam. [16] Total porosity was determined from rock samples with all surfaces unsealed. For these, the length of the sample was measured with a slide gauge and the sample volume was calculated from its cross-sectional area and length. The cross-sectional areas of cylindrical samples (sample nos. 2, 3, 4, 7, 8, 10, and 11) were calculated from their radii and those of rectangular columnar samples (sample nos. 1, 5, 6, 9, and 12) were determined by scanned-image analysis. The error in calculated total volume was estimated to be 4%. [17] After total porosity measurement, the rock samples were mounted with Technovit 4004 resin to provide lateral seal. Their top and bottom surfaces were polished; connected and transport porosities, permeability, and porethroat radii were then determined. Some pores that were open to the sides of the samples, and may have been connected pores in their natural state, were closed by the sealing resin. Therefore, the calculated porosities were affected by sealing of the sides of the samples. Because permeability was measured from the sealed samples, connected and transport porosities were also determined from those samples Water-Expulsion Method [18] We determined transport porosity by using a waterexpulsion method [e.g., Innocentini and Pandolfelli, 2001]. When gas pressure is applied to one end of a water-saturated sample, water in transport pores is expelled from the other end and water in dead-end pores remains in the sample. Thus, the total volume of the transport pores is equal to the volume of expelled water. [19] To determine the transport porosity of our samples, they were degassed, saturated with water (Figure 2d), and weighed. The sample was then placed in a sample holder (Figure 2b) and air was pumped into the lower end of the sample; the water expelled at the top of the sample was wiped off with a wet tissue. When water expulsion ceased, the sample was weighed. The transport porosity was calculated as (water-saturated sample weight sample weight after gas penetration)/sample volume. The final pressure applied to the samples was 0.4 MPa, except for sample no. 1, for which it was 0.1 MPa. To avoid the possibility of destruction of pore structures by the application of high pressure, the maximum pressure applied was 0.4 MPa; consequently, some residual water might have remained in the samples Determination of Pore-Throat Radius [20] The water-expulsion method can be used to estimate pore-throat radius. When gas pressure is applied at one end of a water-saturated sample and the pressure exceeds the resistance due to the surface tension of water, water is expelled from the other end of the sample. The relationship between differential pressure and pore-throat radius is expressed by the following equation of capillarity [e.g., Gelinas and Angers, 1986; Innocentini and Pandolfelli, 2001] (Figure 5a): DP ¼ 2g cos q=r; where DP (Pa) is the pressure difference at the gas water interface, g (N m 1 ) is the interfacial tension, q (degrees) is the contact angle between the solid and liquid, and r (m) is the pore-throat radius for the ideal cylindrical pore. The value of r represents the theoretical characteristic radius of the pores with respect to surface tension and does not necessarily represent exactly the radii of the pores. According to equation (3), water is expelled first from the largest pores, then from successively smaller pores as DP increases. Figure 5b shows, in incremental steps, how water ð3þ 5of10

6 Figure 5. (a) Relationship between surface tension of water and DP. The equation for capillarity is derived from DPpr 2 =2prg cosq. (b) Schematic illustration of the expulsion of water with increasing DP. Surface tension occurs at the water air solid interface. At the appearance of the first bubble, the upper and lower ends of the sample are connected by air flow. Characteristic maximum pore-throat radius r ch is determined from DP break. is expelled as DP is increased. The breakthrough of gas occurs when the differential pressure overcomes the surface tension arising at the characteristic maximum pore throat (the position of smallest radius in the widest transport pore). Thus, by monitoring the differential pressure at which the first bubble appears (breakthrough pressure, DP break ), a characteristic maximum pore-throat radius r ch can be determined. Strictly speaking, the weight of the water above the gas water interface should be considered in equation (3), but this effect was not considered here. This might have caused DP errors of up to a few hundred pascals in our study. [21] For our determination of r ch, a water-saturated sample was placed in a sample holder and water was poured onto the upper end of the sample to create a water layer of about 2 mm above the sample (Figure 2c). This water layer helped in the detection of bubbles. Then, gas pressure was gradually increased until the first appearance of bubbles. This provided a first estimate of the pressure DP break at which bubbles first appear. The sample was then saturated again with water and a pressure of about 90% of the pre-estimated DP break was applied. If no bubbles were observed after 3 min or more, the pressure was incrementally increased until bubbles were detected. The measurements were taken at C; for this temperature range g = to N m 1. In common with the work of Gelinas and Angers [1986] and Innocentini and Pandolfelli [2001], q was assumed to be 0 for our study. [22] A method similar to the water-expulsion method applied here has been used in several previous studies for a variety of sample types; for example, sintered spherical glass [Gelinas and Angers, 1986], natural and synthetic clays [Tanai et al., 1997; Horseman et al., 1999; Gallé, 2000], refractory castables [Innocentini and Pandolfelli, 2001], and sedimentary rocks [Hildenbrand et al., 2002]. Some of these studies call the method the gas breakthrough method. 3. Results [23] Our experimental results are summarized in Table 1. Sample numbers were assigned according to permeability. The characteristic maximum pore-throat radius r ch of sample SK para (no. 1) was calculated to be 291 mm. The shape of pores in this sample can be deduced from Figures 3a and 3b, where Figure 3a is a cross section perpendicular to the experimental flow direction and Figure 3b is analogous to a cross section parallel to the experimental flow direction (note that Figure 3b is a cross section of sample Sk perp). Quantitative evaluation of r ch is difficult from these images, but the visually observed pore shapes appear to be compatible with the measured r ch. Similarly, the images of the other samples shown in Figure 3 do not provide quantitative information about r ch, but do suggest that most of the r ch values we calculated are smaller than the largest pore radii observed in the two-dimensional images. [24] Figures 6a and 6b show f tot, f con, and f tra for each sample. The variation of f tra (8 35%) is smaller than those of f con (11 74%) and f tot (11 82%). Many f con values lie near the line defined by f con /f tot = 1 (Figure 6b), which indicates that a substantial number of pores are interconnected and open to the upper or lower end of the sample. A similar relationship between f con and f tot was reported by Saar and Manga [1999], Rust and Cashman [2004], and Mueller et al. [2005]. In contrast, no meaningful relationship is apparent between f tra and f tot. [25] Plots of viscous permeability versus f tot, f con, and f tra (Figures 6c 6e) show that viscous permeability generally increases with increasing porosity for all types of porosity, although at a given porosity there is commonly dispersion of the permeability values over about three orders of magnitude. The proportion of the volume of pores available as flow paths increases according to the order f tot f con f tra, but no comparable improvement of the correlation between viscous permeability and porosity is apparent. Figure 6f shows a comparison of our plot of viscous permeability versus f tot with those of other published analyses of volcanic products. Only the data that account for inertial effects are plotted in Figure 6f, because permeability is likely underestimated without considering the effect. Our data are generally within the scatter of the previously reported values. A previous interpretation of the correlation between k and f by Klug and Cashman [1996], using a power-law fitting curve, is also shown in Figure 6f (solid 6of10

7 Figure 6. (a) Comparison of f tot, f con, and f tra for the 12 samples. (b) Plot of f con and f tra vs. f tot. (c) to (e) Plots of viscous permeability k 1 vs. f tot, f con, and f tra, respectively. (f) Previously published data on viscous permeability k 1 vs. f tot. See text for explanation of the fitted curve. curve), with the upper and lower limits of their data (dashed curves), although the inertial effect was not considered in their data. There might be a weak correlation between k and f tot, as expressed by the curve of Figure 6f, but there is considerable dispersion. Thus, the gas permeability is clearly not controlled by porosity alone. [26] Figure 7a shows the relationship between viscous permeability k 1 and characteristic maximum pore-throat radius r ch. Viscous permeability is strongly correlated with r ch. Sk para (sample no. 1) and Sk perp (sample no. 5) have similar porosities, but their r ch values differ by a factor of 21; accordingly, their k 1 differ by three orders of magnitude (Table 1). The empirically determined equation k 1 = r ch may be useful to approximate the relationship between r ch and k 1. Figure 7b shows the relationship between inertial permeability k 2 and r ch k 2 is also well correlated with r ch. These results clearly indicate that r ch is a critical factor that influences k 1 and k 2 of volcanic products. A plot of k 1 versus k 2 (Figure 7c) shows that k 1 is strongly correlated with k 2. Rust and Cashman [2004] also reported the good correlation between k 1 and k 2. For our samples, the relationship between k 1 and k 2 can be approximated by the equation k 1 = k Discussion [27] Our results showed that the permeability does not depend on porosity alone. Although this concept is not new, the permeability of volcanic products has been often correlated with porosity alone and quantitative evaluation of the effect of the size of flow path remains insufficient. Through our characterization of pores, we can quantitatively discuss the relationship among permeability, porosity, and porethroat radius. [28] The Kozeny-Carman relation has been widely used to analyze the permeability of porous materials [e.g., Carman, 1956; Paterson, 1983; Walsh and Brace, 1984]. For the Kozeny-Carman relation, the fluid flow path in a rock is approximated by a network of tubes with smooth internal walls. The following equation is a form of the Kozeny- Carman relation: f3 k ¼ fr2 h bt 2 ¼ bt 2 S 2 ; where R h (m) is the hydraulic radius (explained below), b is a shape factor (generally 2 to 3; unitless), t (unitless) is the tortuosity of the pore tubes, and S (m 1 ) is the surface area per unit volume of sample. b equals 2 for circular tubes and 3 for flat cracks. t is the average length of actual flow path divided by the length of the sample. The hydraulic radius has been defined as either the ratio of the cross-sectional area of normal to flow to the wetted perimeter, or the ratio of the volume filled with fluid to the wetted surface area. For a circular tube of radius r, R h is equal to 1/2r [Carman, 1956]. If the surface of the pore wall is not smooth, S may not represent the physical surface area that would be measured by methods such as nitrogen ð4þ 7of10

8 Figure 7. (a) Viscous permeability k 1 vs. characteristic maximum pore-throat radius r ch. (b) Inertial permeability k 2 vs. r ch. (c) Viscous permeability k 1 vs. inertial permeability k 2. adsorption [Berryman and Blair, 1987]. We evaluated the applicability of equation (4) to our results by plotting k against f tot r ch and f tra r ch (Figures 8a and 8b). The 2 correlation between k 1 and fr ch is reasonably good. The scatter of k 1 values is somewhat smaller for f tra than for f tot. If it is assumed that the flow path is a circular tube (i.e., by inserting R h = 1/2 r ch into equation (4)), t is calculated from the regression lines of Figures 8a and 8b to be about 7 to 12, although the meaning of these values in the physical sense is unclear. The good correlation between k and fr 2 implies that the approximations that are made in the derivation of the Kozeny-Carman relation successfully explain the pore structure of our samples. In other words, the flow paths can be approximated by smooth-walled tubes of radius r ch, even though the actual pore geometry is very complex. [29] Several studies have discussed the k f relationship for vesicular volcanic products in terms of the Kozeny- Carman relation. Klug and Cashman [1996] studied the permeability of pumice samples with f tot in the range 30 92% and reported that k = f 3.5 predicted well the trend of their data (Figure 6f). Mueller et al. [2005] showed that the measured permeabilities of samples with f tot in the range 3 30% from several dome lavas lie between the bounds defined by k = f 3 tot and k = f 3.8 tot, based on the Kozeny-Carman relation. The permeability of other samples with f tot in the range 30 80% were described by Mueller et al. [2005] in terms of a percolation model (discussed below). It appears that these studies attempted to describe k as a simple power-law function of f as follows: k = cf n, where c is proportionality constant. This is a simplified form of the Kozeny-Carman relation. However, because permeability is strongly dependent on pore size (Figure 7a), use of the Kozeny-Carman relation should take into account not only f but also pore-throat radius. [30] The k f relationship for volcanic products has been discussed also in terms of percolation theory [Saar and Manga, 1999; Blower, 2001; Rust and Cashman, 2004; Mueller et al., 2005; Bernard et al., 2007; Walsh and Saar, 2008]. Percolation theory deals with the connectedness of pores in media of given shapes and spatial arrangements of pores. The permeability can be expressed in the following general form: k ¼ Cðf f cr Þ e ; ð5þ Figure 8. Viscous permeability k 1 plotted against (a) f tot r 2 ch and (b) f tra r 2 ch. Here, f tot and f tra are expressed as fraction. 8of10

9 pores. Such requirement might be satisfied in a sequence of volcanic products obtained from a single volcanic event. Figure 9. Normalized permeability (k 1 /r ch 2 ) vs. f tot. Dotted and solid curves are examples of computations of equation (5); they provide a guide to the trend expected according to percolation theory. where C (m 2 ) is a magnitude-defining constant, f cr is the critical porosity or percolation threshold (k = 0 for f < f cr and k > 0 for f f cr ), and e (unitless) is a critical exponent [e.g., Sahimi, 1995; Saar and Manga, 1999]. From equation (5), k at a given f is determined by C, f cr, and e, where each of these factors reflects different physical parameters. C is related to the cross-sectional area of a pore (thus, to the square of pore radius) and also to the tortuosity of the flow path [Niimura, 2006]. f cr and e are essential parameters of percolation theory and their values depend on the model employed. For a medium in which spherical void spaces (e.g., bubbles) are randomly placed and permeation takes place through interconnected spheres (the fully penetrable sphere model), values of e = 2 and f cr 30% have been reported [Rintoul and Torquato, 1997; Blower, 2001]. As the shape of void spaces deviates from spherical (e.g., spheres are elongated or flattened to form ellipsoids), f cr decreases markedly [Saar and Manga, 2002]. We have already seen that permeability is sensitive to pore-throat radius. Hence, to properly evaluate e and f cr in equation (5), the effect of the magnitude-defining constant must be removed; this is achieved by dividing k by the crosssectional area of a pore [Saar and Manga, 1999] or by the square of the characteristic pore radius [Blower, 2001]. The 2 relationship between permeability normalized by r ch and total porosity (Figure 9) for our samples does not show the systematic trend predicted by equation (5), indicating that neither f cr nor e can be properly evaluated for our samples. Various factors, such as the size of the sample used, pore size, pore-size distribution, pore shape, barriers to pore coalescence, the spatial arrangement of pores, and the presence of crystals, can affect the values of f cr and e [Blower, 2001; Walsh and Saar, 2008]. Thus, it is not unexpected that the data do not plot along a single trend because our samples have a variety of origins and their pore structures are different. For the data to lie along one trend, the samples must have different porosities, but must also have similar pore shapes and spatial arrangements of 5. Conclusion [31] Porosimetry of vesicular volcanic products was conducted by using a water-expulsion method and the relationship of the pore characteristics to gas permeability was investigated. Permeability was only roughly correlated with total porosity, connected porosity, and transport porosity, but was well correlated with the characteristic maximum pore-throat radius as follows: k 1 = f tra r 2 ch. This correlation represents a form of the Kozeny-Carman relation. The permeability of volcanic products is thus very sensitive to pore-throat radius, and estimation of pore size is essential when considering the degassing processes associated with volcanic activity. [37] Acknowledgments. This manuscript was greatly improved by the comments of M. O. Saar and two anonymous reviewers. References Bernard, M. L., M. Zamora, Y. Géraud, and G. Boudon (2007), Transport properties of pyroclastic rocks from Montagne Pelée volcano (Martinique Lesser Antilles), J. Geophys. 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