Effect of pore geometry on VP/VS: From equilibrium geometry to crack

Authors


Abstract

[1] Seismic wave velocities of melt or aqueous fluid containing systems are studied over a wide range of pore shapes, including oblate spheroids, tubes, cracks, and an equilibrium geometry controlled by a dihedral angle. The relative role of liquid compressibility and pore geometry on the “VP/VS” velocity ratio is clarified. The result clearly indicates that P and S velocity structures determined by seismic tomography can be used to verify whether interfacial energy-controlled melt or fluid geometry (equilibrium geometry) is achieved. Relationships between the diverse models are clearly established by relating each model to the oblate spheroid model in terms of the equivalent aspect ratio. As a function of the aspect ratio, a significant effect of pore geometry on “d ln VS/d ln VP”, the ratio of the fractional changes in VS and VP, is shown. Equilibrium geometry of the partially molten rocks, characterized by a dihedral angle of 20°–40°, corresponds to an aspect ratio of 0.1–0.15. The value of d ln VS/d ln VP expected for the texturally equilibrated partially molten rocks is shown to be 1–1.5, which is much smaller than that expected for cracks and dikes with an aspect ratio of <10−2–10−3. In the upper mantle low-velocity regions the seismologically obtained value of d ln VS/d ln VP is within this range beneath the Bolivian Andes (1.1–1.4) but is as high as 2 beneath Iceland (1.7–2.3) and beneath northeastern Japan (2.0). The former region can be regarded as a region where equilibrium geometry is achieved, and the latter regions can be regarded as regions where dikes and veins typical of a system far from the textural equilibrium dominate.

1. Introduction

[2] Recent seismic tomographic results reveal the VP and VS structures of subduction zones [e.g., Zhao et al., 1992; Myers et al., 1998; Nakajima et al., 2001], active volcanic regions [e.g., Benz et al., 1996; Miller and Smith, 1999; Dawson et al., 1999], and earthquake source regions [e.g., Zhao et al., 1996]. The existence of melt or aqueous fluid phase in these regions is indicated not only from low-velocity features but also through changes in the VP/VS velocity ratio. In some zones the spatial resolution of the tomographic images is good enough to obtain quantitative information about the VP/VS structure [e.g., Myers et al., 1998; Nakajima et al., 2001]. What information about the liquid phase can be derived from these data?

[3] The liquid phase affects VP and VS differently, which explains the changes of VP/VS ratio. If the intrinsic elasticities of the solid phase are given, VS is determined by the structural factors (liquid volume fraction and pore geometry), while VP additionally depends on liquid compressibility [e.g., Takei, 2000]. While the importance of liquid compressibility in changing VP/VS has been widely recognized, the effect of pore geometry on VP/VS has not been commonly recognized. The significance of pore geometry can be demonstrated by an apparent contradiction between the studies of O'Connell and Budiansky [1974] and Watanabe [1993]. In the former, with a crack configuration, VP/VS is increased by water-saturated pores, whereas it is decreased by dry pores. In the latter, with a tube geometry proposed by Mavko [1980], VP/VS is decreased by water-saturated pores, whereas it is increased by pores saturated by silicate melt, which has much smaller compressibility than water. Even for the same liquid we cannot predict the sign of VP/VS change without specifying the pore geometry. The relative importance of the liquid property and the pore geometry was dealt with by Budiansky and O'Connell [1976] within the framework of the crack model. The effects of pore geometry should be clarified, not only for each model individually but also in a unified form applicable to all possible models.

[4] In addition to the crack and tube models an oblate spheroid model with a variable aspect ratio has been used [Wu, 1966; Kuster and Toksöz, 1974; Berryman, 1980; Schmeling, 1985a]. In petrology, equilibrium microstructures of melt or aqueous fluid containing systems, at which interfacial energy is at a minimum, have been studied intensively [e.g., Bulau et al., 1979; von Bargen and Waff, 1986; Watson and Brenan, 1987; Holness, 1993, 1997]. Variation of this geometry with temperature, pressure, and/or chemical composition is described in terms of dihedral angle. A granular model with variable grain boundary contiguity has been developed for dealing with the equilibrium geometry as a function of dihedral angle [Takei, 1998]. The agreement of this model with experimental data on texturally equilibrated partially molten media has been confirmed [Takei, 2000].

[5] In this study, relationships between the diverse models are clearly established by relating each model to the oblate spheroid model in terms of the “equivalent aspect ratio” (section 2). Then, as a function of the aspect ratio, I present a diagram clarifying the relative role of liquid compressibility and pore geometry in changing VP/VS (section 3). A general approach to the VP/VS ratio, irrespective of each individual model, is presented; this clarifies the underlying physics determining the VP/VS change. I demonstrate the importance of interpreting the observed change of the VP/VS ratio in the two-dimensional parameter space of liquid compressibility and pore geometry. The tube model and the contiguity model results show that compared to the crack-like geometry, the interfacial energy-controlled equilibrium geometry has much less effect on the VP/VS ratio. Hence VP/VS data can be used to verify whether textural equilibrium is achieved. These results are applied to the VP and VS structures of the upper mantle low-velocity regions (section 4).

2. Relationships Between Diverse Models in Terms of Equivalent Aspect Ratio

[6] The change of shear and longitudinal wave velocities caused by the existence of a liquid phase is given by

equation image
equation image

where

equation image

and γ = μ/k = {3(1 − 2ν)}/{2(1 + ν)}. equation image and equation image represent the wave velocities of the solid phase, where k, μ, ν, and ρ are bulk modulus, shear modulus, Poisson's ratio, and density, respectively, of the solid phase. For ν = 0.25, γ = 0.6. Bulk modulus and density of the liquid phase are expressed as kf and ρf, respectively. These quantities represent the intrinsic properties of the solid or liquid phase. The ϕ represents liquid volume fraction, and equation image represents total density, (1 − ϕ)ρ + ϕρf. Kb and N are the bulk and shear moduli, respectively, of the solid skeleton, which are evaluated by replacing the regions actually containing liquid with a vacuum. (The term “skeleton” or “skeletal” is used to denote the solid framework with vacuum pores.) Keff is the bulk modulus effective for the solid-liquid aggregate.

[7] Equations (1)(3) are derived from a continuum mechanical approach for two-phase media or from poroelastic theory [e.g., Biot, 1956; Johnson, 1986; Johnson and Plona, 1982; Murphy, 1984]. In this approach, the heterogeneity of the liquid pressure in the pore size scale is not taken into account; this is why the effect of the microstructure appears only in the form of Kb and N and hence why it can be evaluated irrespective of the liquid behavior. This situation, often referred to as the relaxed state, occurs when pores are not isolated and when the frequency of the wave is low compared to the relaxation frequencies of the fluid flow within each pore and between the neighboring pores (i.e., squirt flow) [Mavko and Nur, 1975]. Even when pores are isolated, if the shape is almost spherical, the heterogeneity in the liquid pressure caused by wave field is negligible. The present study assumes the relaxed state.

[8] Kb and N are calculated with specific assumptions about the pore geometry; Table 1 shows a summary of the models. The oblate spheroid model with variable aspect ratio α [Wu, 1966; Kuster and Toksöz, 1974; Berryman, 1980; Schmeling, 1985a] gives Kband N as functions of ϕ and α. The tube model with a variable cross-sectional shape [Mavko, 1980] gives Kb and N as functions of ϕ and tube geometry ϵ. The contiguity model applied to the equilibrium geometries [Takei, 1998] gives Kb and N as functions of ϕ and dihedral angle θ. In the crack model [e.g., O'Connell and Budiansky, 1974], Kb and N are given by only one parameter: crack density parameter κ. This model presents the asymptotic solution of the oblate spheroid model at the limit of the small aspect ratio (α → 0); at this limit the effects of ϕ and α appear only in the form of ϕ/α, which is proportional to the crack density parameter (3ϕ/(4πα) = κ) [Walsh, 1969; Watt et al., 1976]. If α < 10−2, Kb and N calculated from the oblate spheroid model and plotted as functions of 3ϕ/(4πα) agree with the crack model results obtained from equations (5), (7), and (8) of O'Connell and Budiansky [1974], differing a few percent or less. In this sense, the crack model can be included in the oblate spheroid model.

Table 1. Theoretical Models for Solid-Liquid Composites
ModelStructural ParametersReferences
  • a

    Aspect ratio α is the minor radius/major radius, dimensionless.

  • b

    Typographical errors in the former studies were corrected.

  • c

    As a dimensionless parameter ϵ varies from 0 to ∞, the cross section of the tube changes from a triangular shape with three cusps to a circle.

  • d

    Contiguity φ is the ratio of the area of each grain being in contact with the neighboring grains relative to the total surface area; dimensionless.

  • e

    Crack density parameter κ = na3, where n is the number of cracks per unit volume and a is the radius of circular crack; dimensionless.

Oblate spheroidliquid volume fraction ϕ, aspect ratio αafor example, Berryman [1980]b
Tubeliquid volume fraction ϕ, tube geometry ϵcMavko [1980]
Granularcontiguityd φdTakei [1998]
(Equilibrium geometry) (liquid volume fraction ϕ, dihedral angle θ)  
Crackcrack density parametere κeO'Connell and Budiansky [1974]

[9] Among these models the tube model and the contiguity model describe the equilibrium textures at which interfacial energy is at a minimum. The dihedral angle θ is a material parameter characterizing the equilibrium geometry (Figure 1); at 0° < θ < 60° an interconnecting network of grain edge channels develops. Most of the partially molten rocks have θ between 20° and 40° and most of the rock + aqueous fluid systems have θ between 40° and 100° [Holness, 1997]. In the tube model, various tube geometries (given by 0 ≤ ϵ ≤ ∞) were chosen for mathematical convenience, and ϵ cannot be directly related to θ. Contiguity φ, determining the skeleton properties of the granular aggregates, reflects the actual geometry and can be determined as a function of liquid volume fraction and dihedral angle; the explicit form of φ(ϕ, θ) is available for a range of 20° ≤ θ ≤ 80° and ϕ ≤ 0.05 by using the theoretical results of von Bargen and Waff [1986]. In this paper, the contiguity model applied to the equilibrium textures is referred to as an equilibrium geometry model, and Mavko's [1980] model is referred to as a tube model. Further discussion on the equilibrium geometry model used in this study is presented in Appendix A.

Figure 1.

Equilibrium liquid geometry at which interfacial energy is at a minimum. It varies with dihedral angle θ. After Watson and Brenan [1987] (modified).

[10] Relationships between the oblate spheroid model, the equilibrium geometry model, and the tube model can be established by the following procedures. Let x be a geometrical parameter representing aspect ratio α, dihedral angle θ, or tube geometry ϵ. Under a fixed x, Kb and N decrease with ϕ almost linearly, and Kb(ϕ, x) and N(ϕ, x) can be approximated by

equation image

for x = α, θ, or ϵ. The slopes denoted by equation image and ΛN, which are the functions of x, take positive values larger than unity; Kb and N given by equation image= ΛN = 1 are equal to the upper bound of the skeleton moduli estimated by the Voigt model [e.g., Watt et al., 1976]. Figure 2 shows Kb(ϕ, α)/k and N(ϕ, α)/μ for the oblate spheroid model calculated under a self-consistent scheme. Also shown are Kb(ϕ, θ)/k and N(ϕ, θ)/μ for the equilibrium geometry model at various θ (bold lines); the limitation with respect to ϕ comes from that of von Bargen and Waff's [1986] result. Figure 3 shows equation image and ΛN obtained from these data. For the equilibrium geometry model, equation image and ΛN were calculated by (1 − Kb/k)/ϕ and (1 − N/μ)/ϕ, respectively, at ϕ = 0.03 (solid symbols in Figure 3b); since the curves shown in Figure 2are slightly concave upward, equation image and ΛN calculated at ϕ = 0.05 are also shown (open symbols). In this calculation, von Bargen and Waff's [1986] formulae derived at θ ≥ 20° are used down to θ = 10°; the validity of this extrapolation is shown in Appendix A. For the tube model with a self-consistent scheme we obtain equation image= 7.2 and ΛN = 4.5 at ϵ = 0 and equation image = 2.6 and ΛN = 2.2 at ϵ = ∞ (circular tube), which are plotted on the right side of Figure 3b. By comparing equation image(x) and ΛN(x) (x = θ or ϵ) to equation image(α) and ΛN(α) the equivalent aspect ratio α(x) is determined to satisfy

equation image

for x = θ or ϵ. As shown in Figure 3, the ratio between equation image and ΛN is not sensitive to the change of models, and a value of α satisfying both equations of (5) can be determined. Pore shapes affect VP and VS only through the effects on Kb and N. Therefore as far as these velocities are concerned, the oblate spheroid model having an aspect ratio satisfying (5) is equivalent to the original model. In this sense, the relations between the parameters determined in this way give the relations between the models. In the procedures described above, Kb and N calculated for the solid Poisson's ratio ν of 0.25 are used. If the calculations are made for a different value of ν, the slopes take different values. However, the effects of ν appear almost equally for all models, and the equivalent aspect ratio can be determined almost regardless of ν.

Figure 2.

(a) Normalized bulk modulus Kb/k and (b) normalized shear modulus N/μ of skeleton versus liquid volume fraction ϕ. Poisson's ratio of the solid phase, ν, is taken to be 0.25. The solid lines show the oblate spheroid model with aspect ratio α calculated under a self-consistent scheme. The bold solid lines show the equilibrium geometry model with dihedral angle θ.

Figure 3.

(a) Slopes, equation image = (1 − Kb/k)/ϕ and ΛN = (1 − N/μ)/ϕ, versus aspect ratio α for the oblate spheroid model. (b) equation image and ΛN versus dihedral angle θ for the equilibrium geometry model. The solid and open symbols show the values obtained at ϕ = 0.03 and 0.05, respectively. On the right side of Figure 3b, equation image and ΛN of the tube model are shown. The arrows from Figure 3b to Figure 3a illustrate the concept of equivalent aspect ratio.

[11] The obtained relationships between the models are summarized in Figure 4a. The aspect ratio corresponding to the texturally equilibrated partially molten rocks (θ ≃ 20°–40°) is ∼0.1–0.15. The equilibrium geometries of the rock + aqueous fluid systems (θ ≃ 40°–100°) correspond to the aspect ratio of ∼0.15–0.5. The equilibrium geometries with θ ∼ 30°–80° are almost covered by the tubes with ϵ = 0 − ∞, and ϵ = 0 corresponds to θ ≃ 30°. Figure 4a also shows the range corresponding to the crack model; the detailed procedure used to obtain this range is stated in the latter part of section 3.

Figure 4.

(a) Relationships between various models shown by the correspondence of each geometrical parameter to the aspect ratio α of the oblate spheroid model. Solid and open symbols show the correspondence determined from equation image and ΛN, respectively. (b) Ratio of the fractional changes in VS and VP, d ln VS/d ln VP, versus aspect ratio α for various β. The β is the ratio of solid to liquid bulk moduli, k/kf; β ≃ 5–10 corresponds to rock + melt system at ∼50–0 km depth, β ≃ 10–40 corresponds to rock + water system at ∼30–0 km depth, and β ≃ 50–105 corresponds to rock + gas system at ∼30–0 km depth; ν = 0.25.

3. Relative Role of Liquid Compressibility and Pore Geometry in d ln VS/d ln VP

[12] The different effects the liquid phase has on VP and VS are evaluated by the ratio of the fractional changes in VS and VP: d ln VS/d ln VP. This ratio is obtained almost directly from the seismic tomography. The relation between d ln VS/d ln VP and the fractional change in VP/VS is given by d(VP/VS)/(VP0/VS0) = (d ln VS/d ln VP − 1)(−dVP/VP0). Since dVP/VP0 < 0 (low velocity), d ln VS/d ln VP >1 and < 1 correspond to the increase and decrease, respectively, of the VP/VS ratio. By using d ln VS/d ln VP rather than VP/VS we can reduce the dimension of the problem by one at the expense of losing the explicit dependence on liquid volume fraction ϕ. Several recent studies have pointed the importance of d ln VS/d ln VP as a diagnostic of the physical cause of a seismic anomaly [e.g., Masters et al., 2000].

[13] By substituting (4)into (1)(3) and by considering the velocity change to be small enough to be approximated by a linear function of ϕ, we obtain

equation image

where

equation image

Equation (6) shows that d ln VS/d ln VP is independent of ϕ. Generally, liquid bulk modulus kf is smaller than solid bulk modulus k. While variation of k is not so large, kf takes a wide range of value corresponding to various types of liquid phase (melt, aqueous fluid, and gas) at various depths. Hence β, which takes a wide range of value larger than 1, can be referred to as (normalized) liquid compressibility. equation image(≥1) and ΛN(≥1) also vary over a wide range according to the variation of pore shape. In this section, equation image and ΛN are considered to be the functions of aspect ratio α; the results can be immediately applied to the other models by the equivalence shown in Figure 4a. Equation (6) clearly shows that the essential factors causing the variation of d ln VS/d ln VP are aspect ratio α, affecting equation image and ΛN, and liquid compressibility β. Although ρf/ρ takes a wide range of value usually smaller than 1, the effect of this factor appearing in both numerator and denominator is not significant in a practical sense, and in the following discussion this factor is set to unity; the validity of this is confirmed in section 4.1. The variation of γ can be considered small; this effect is discussed briefly in the latter part of this section.

[14] Figure 4b shows d ln VS/d ln VP as a function of α and β. This diagram is calculated from (6) with γ = 0.6 and ρf/ρ = 1. Also shown are the liquid types corresponding to each value of β; kf used to evaluate β is summarized in Table 2. The relative role of α and β in d ln VS/d ln VP is clearly shown in Figure 4b. Under a given α, d ln VS/d ln VP takes the maximum value (=(1 + 4γ/3)/(4γ/3) = 2.25) at β = 1. This value of d ln VS/d ln VP corresponds to Keff/k = 1 and hence corresponds to the situation where both VP and VS are reduced by the reduction of skeleton shear modulus N. As β increases up to ∼∞, d ln VS/d ln VP decreases to a value of the dry case (=(1 + 4γ/3)/(equation imageN + 4γ/3)); this value is <1, since equation imageN > 1. As α decreases from 1 to 5 × 10−4 or less, the dependence of d ln VS/d ln VP on β significantly changes: at a nearly spherical pore shape (α ∼ 1) d ln VS/d ln VP is nearly 1 regardless of β unless β takes a very small value around unity; at a very thin pore shape (α ≪ 1), d ln VS/d ln VP approaches the maximum value regardless of β unless β takes a very large value comparable to α−1. The liquid phase may be called soft if d ln VS/d ln VP is less than or nearly 1 and hard if d ln VS/d ln VP is close to the maximum value. Almost all liquids are soft when contained in spherical pores and are hard when contained in sufficiently thin pores. At α ranging from ∼5 × 10−3 to ∼0.1, d ln VS/d ln VP is sensitive to the change of the liquid compressibility corresponding to the change of the liquid types.

Table 2. Liquid Bulk Modulus kf
Depth, kmP, GPaT, °Ckf, GPa
   GasaWaterbMeltc
  • a

    Adiabatic bulk modulus estimated by 1.3 P.

  • b

    Isothermal bulk modulus estimated at each (P, T) condition. Data from Schäfer[1980].

  • c

    Data from Stolper et al. [1981]. Data at P = 2 GPa are estimated from ∂kf/∂P = 6–7.

  • d

    Numerals in the parentheses show β = k/kf evaluated for k ≃ 40–120 GPa.

010−4201.3 × 10−42.27–25
   (β=105−106)d(18–50)(4–10)
50.15750.23.1 
   (200–600)(13–40) 
100.31500.41.8 
   (100–300)(22–66) 
3515001.34.5 
   (30–100)(9–25) 
702   20–40
     (3–6)

[15] To examine the underlying mechanism causing these variations of d ln VS/d ln VP, we write Keff as

equation image

which simplifies d ln VS/d ln VP to be (1 + 4γ/3)/(equation imageN + 4γ/3). By substituting (4) into (3) we obtain

equation image

When equation image ≫ β − 1, equation image = β − 1. In this case, equation imageN = (β − 1) /ΛN ≪ 1, and hence d ln VS/d ln VP takes the maximum value corresponding to β = 1. When equation image ≪ β − 1, equation image = equation image, and hence d ln VS/d ln VP reaches the lower limit corresponding to the dry case (β = ∞). These are summarized as

equation image
equation image

The critical factor governing the transition of d ln VS/d ln VP between these limits is given by (β − 1)/equation image; the liquid intrinsic property β is to be evaluated by the pore geometry dependent scale equation image. In order to clarify the physical meaning of this factor we introduce Skempton's coefficient B, which is given by B = (k/Kb − 1)/{(k/Kb − 1) + ϕ(β − 1)} [e.g., Mavko et al., 1998]. This coefficient represents the ratio of the induced pore pressure relative to the applied confining pressure under the undrained condition and is a measure of solid-liquid coupling in the poroelastic media. By using (4) and by assuming ϕ to be small we obtain B−1 − 1 = (β − 1)/equation image. Hence (β − 1)/equation image ≫ 1 corresponds to B ≃ 0, where the confining pressure applied by the wave field is mostly supported by the skeletal frame (this situation is hereinafter referred to as “uncoupled support”). Also, (β − 1)/equation image ≪ 1 corresponds to B ≃ 1, where the applied pressure is supported uniformly by the two phases (“coupled support”). Under the uncoupled support the medium response is similar to the dry case, and d ln VS/d ln VP takes a small value near the lower limit. Under the coupled support, d ln VS/d ln VP takes a large value near the upper limit. The transitions of d ln VS/d ln VP with decreasing α, with decreasing β, and with both are all understood by the transition of the pressure support mechanism from uncoupled support to coupled support. At the transient stage ((β − 1)/equation image ∼ 1), d ln VS/d ln VP varies sensitive to α and/or β. It may be worth noting that (over the range of ϕ considered here) the pressure support mechanism, as well as d ln VS/d ln VP, is independent of ϕ. This can be understood intuitively by considering the following two effects of ϕ: under a given change of confining pressure the change of pore volume increases with increasing ϕ because the skeleton stiffness decreases with ϕ; under a given change of pore volume the change of pore pressure decreases with increasing ϕ. These effects compensate each other and B is independent of ϕ.

[16] At α ≪ 1, since equation image ∝ α−1 (Figure 3a), critical factor (β − 1)/equation image approaches ∼(β − 1)α ∼ αβ. This is equal to the reverse of the hardness parameter ω introduced by O'Connell and Budiansky [1974] and Budiansky and O'Connell [1976] in the crack model. The present study has derived in a much simpler way a general form of this factor applicable to any geometry. In the crack model, Keff/k = 1 is used under the saturated condition [O'Connell and Budiansky, 1974]. This requires αβ to be sufficiently small for the coupled support to be realized. For water (β ≃ 25) and for melt (β ≃ 10) at 1 atm, d ln VS/d ln VP exceeds 95% of the maximum value at α ≃ 0.0007 and 0.0017, respectively. Therefore, in Figure 4a, α ≤ 0.0007 or α ≤ 0.0017 is labeled “crack.”

[17] If Figure 4b is written for a different value of ν (that is, for a different value of γ), it slightly expands or shrinks vertically without changing the intermediate values. As (10) shows, the upper limit of d ln VS/d ln VP increases with increasing ν (decreasing γ). The lower limit of d ln VS/d ln VP decreases with increasing ν because equation image increases and ΛN decreases with ν [e.g., Takei, 1998, Figure 4]. As a result of these two opposite effects, the intermediate values of d ln VS/d ln VP (∼1–1.7) are relatively independent of ν.

[18] The present linearized analysis, approximating the effects of liquid as linear functions of ϕ, has successfully simplified the problems: The characterization of each model in terms of equation image and ΛN has enabled us to establish the relationships between the diverse models; the independence of d ln VS/d ln VP from ϕ has enabled us to obtain the perspective shown in Figure 4b; using the linearized form of B, the transition of d ln VS/d ln VP controlled by (β − 1)/equation image has been related to the transition of the pressure support mechanism. The present analysis is valid as long as the velocity changes are not so large. The actual variation of d ln VS/d ln VP with ϕ, which may be caused by the higher-order terms of ϕ (ϕn with n ≥ 2), was checked by calculating (1 − VS/VS0)/(1 − VP/VP0) directly from (1)(3) without using (4). This variation is small in the equilibrium geometry model because the higher-order effect of ϕ neglected in (4)(that is, the concavity in Figure 2) cancels the higher-order effect neglected in taking the square roots in (1)(2). The variation of d ln VS/d ln VP with ϕ is <0.1 up to ϕ ∼ 0.15, which corresponds to the VS variation of ≤35%. In discussing the equilibrium geometries, Figures 4a and 4b hold true over this range. (In this calculation the approximation formulae of von Bargen and Waff [1986], derived at ϕ ≤ 0.05, were used up to ϕ ∼ 0.2; the validity of this extrapolation is discussed in Appendix A.) The applicable range of Figures 4a and 4b is not so wide (≲20% in VS) for the oblate spheroid, tube, and crack models. These models, however, well satisfy the linearity assumed in (4). In this case, a small modification given in Appendix B allows us to apply Figure 4 to the large velocity perturbations or large ϕ data. The correspondence between d ln VS/d ln VP and the pressure support mechanism obtained from the linearized analysis cannot be extended to the whole range of ϕ. Even when (β − 1)/equation image ≫ 1, if ϕ is increased up to equation image, Kb vanishes and the coupled support is realized (B = 1). This change with ϕ does not increase d ln VS/d ln VP.

4. Discussion

4.1. The d ln VS/d ln VP of the Texturally Equilibrated Partially Molten Rocks

[19] The aspect ratio corresponding to the texturally equilibrated partially molten rocks has been obtained as 0.1–0.15. From Figure 4b, d ln VS/d ln VP is nearly 1 for β = 10 and 1.5 for β = 3. Judging from ∂kf/∂P = 6–7 [Stolper et al., 1981; Rigden et al., 1989], β < 3 seems to be hardly expected, even at ∼70 km depth. I conclude that the value of d ln VS/d ln VP expected for the texturally equilibrated partially molten rocks is 1–1.5. This is the case even for θ = 10°, which is smaller than the experimentally obtained values of θ [Holness, 1997]. The present conclusion is true for a wide range of velocity change (≤35% in VS) and is not changed even if the possible variations of solid Poisson's ratio ν and ρf/ρ are taken into account. Also, possible uncertainties of the equilibrium geometry model do not change the conclusion, as shown in Appendix A. Measurements of both VP and VS on the texturally equilibrated partially molten systems were performed by Takei [2000] using an organic binary system (borneol + diphenylamine) having moderate values of dihedral angle (∼35°–17°). In this experiment, d ln VS/d ln VP ≃ 4 was obtained. This large value of d ln VS/d ln VP is realized by the combination of small β(= 1.16) and large ν(= 0.37) of the borneol + melt system, both of which are peculiar to the organic materials. Such a large value of d ln VS/d ln VP can no more be expected for the combination of silicate minerals and melt having much larger β and much smaller ν. Although Hammond and Humphreys [2000] estimated d ln VS/d ln VP to be 2.3 for the tubes with ϵ = 0, an unrealistically small value of β(∼1) may be assumed in their calculation.

[20] In crystalline aggregates, interfacial energies are anisotropic due to crystal anisotropy, and this modifies the equilibrium textures; flat crystalline interfaces and completely wetted two-grain boundaries coexist with smoothly curved solid-melt boundaries and dry grain boundaries predicted by the isotropic theory [Waff and Faul, 1992]. The detailed effects of such modification on the contiguity and hence on the macroscopic properties have not yet been clarified. The effects of the crystal anisotropy were also observed in the equilibrium textures of the borneol + melt system [Takei, 2000]. Although their effects on the wave velocities were reported to be small, this cannot be immediately applied to the Earth system because in this experiment the textural change by the crystal anisotropy was not quantified nor was the possible effect of velocity dispersions due to fluid flow fully examined. Valuable data are given by Hirth and Kohlstedt [1995], who measured the ratio of the solid-solid interfacial area to the solid-liquid interfacial area, Ass/Asl, on the actual system (olivine + enstatite + melt) affected by the crystal anisotropy. Although the measurements were made on the deformed aggregates, they stated that the diagnostic features of the melt topology in the deformed aggregates are very similar to those in the undeformed aggregates, and hence I use their data to evaluate the effect of crystal anisotropy. Contiguity, φ = (2Ass/Asl)/(2Ass/Asl + 1), can be obtained as 0.41, 0.36, and 0.22 for ϕ = 0.072, 0.081, and 0.115, respectively. These values are not so different than those expected from the isotropic theory; they are almost equal to the results of von Bargen and Waff [1986] on θ ≃ 20°. Hence, even when the crystal anisotropy is taken into account, d ln VS/d ln VP is obtained as 1–1.5.

[21] Figures 4a and 4b show that the difference between θ = 20° and 40°, for example, is difficult to detect seismologically. However, it is also shown that a difference between the equilibrium geometries and the disequilibrium geometry of cracks can be detected seismologically. The ranges expected for texturally equilibrated rock + melt and rock + aqueous fluid systems are shown in Figure 5, which is drawn by almost the same procedure as Figure 4b but which additionally takes into account the effects of ρf/ρ. (The difference from Figure 4b is very small.) Also shown is −d ln VS/d ϕ versus α as a practical convenience in evaluating ϕ from the absolute reduction of VS/VS0. The value of d ln VS/d ln VP predicted for the texturally equilibrated partially molten rocks is much smaller than 2. If d ln VS/d ln VP is observed to be as large as 2, it means that the system is far from textural equilibrium. In the low-velocity region observed at a depth of 20–50 km beneath Iceland, VP was reduced compared to the normal mantle by ∼13% (from 8.39 to 7.3 km/s), and VP/VS was increased from ∼1.76 to 1.96 (or 2.2) [Gebrande et al., 1980]. This corresponds to the VS reduction of 22–30% and hence corresponds to d ln VS/d ln VP = 1.7–2.3. By considering these data on the basis of the oblate spheroid model, Schmeling [1985b] stated that α > 0.03 cannot explain both changes of VP and VP/VS consistently; this can be easily confirmed from Figure 5, where the range of 1.7–2.3 is shown on the right side. Beneath northeastern Japan, Nakajima et al. [2001] reported low-velocity regions which are mutually related to the arc volcanism. At the upper mantle depth the velocity reduction is 6% (from 7.90 to 7.42 km/s) for VP and 12% (from 4.57 to 4.02 km/s) for VS, which gives d ln VS/d ln VP = 2.0. They stated that this can be explained by the melt and/or H2O inclusions with small α (10−2–10−3), which can be easily confirmed from Figure 5. The important consequence of the present study is that the partially molten rocks possibly existing in these regions are far from textural equilibrium. In the upper mantle low-velocity region related to the arc volcanism in the Bolivian Andes, VP and VS are reduced by 6.0% (from 8.3 to 7.8 km/s) and 8.5–6.4% (from 4.7–4.6 to 4.3 km/s), respectively [Myers et al., 1998]. This shows d ln VS/d ln VP = 1.4–1.1, which can be explained by the texturally equilibrated partially molten rocks. (Another low-velocity spot with d ln VS/d ln VP ∼ 0.9 was also observed in this region.)

Figure 5.

(bottom) Range of texturally equilibrated rock + melt and rock + aqueous fluid systems shown in d ln VS/d ln VP versus α diagram (hatched regions); ρf/ρ = 1, 0.92, 0.33, and 0 are assumed for β = 1, 2–10, 25–400, and 105, respectively; ν = 0.25. The seismologically observed values of d ln VS/d ln VP are shown on the right side. (top) The −d ln VS/d ϕ versus α calculated by (ΛN − (1 − ρf/ρ))/2, where ρf/ρ = 0.92, 0.33, and 0 are assumed for melt, water, and gas (0 km), respectively.

[22] Further we cannot tell whether the disequilibrium structure is formed in the grain size scale or in a larger scale such as dikes and veinlets; the spatial scale of the inclusions, as long as it is sufficiently smaller than the wavelength, does not affect the wave velocities. Various disequilibrium textures have been reported, including the disequilibrium structures in the grain size scale [e.g., Jin et al., 1994; Daines and Kohlstedt, 1997; Zimmerman et al., 1999; Takei, 2001], larger structures described by a heterogeneous distribution of melt fraction [e.g., Sleep, 1988; Stevenson, 1989], and dikes and veins without pervasive flow [e.g., Rubin, 1995]. Once a liquid phase is segregated into the large-scale dikes and veins, it is considered to be difficult for the system to stay near textural equilibrium. Hence one possible interpretation may be that the difference in d ln VS/d ln VP observed in the upper mantle low-velocity regions reflects the difference between the region where the melt migrates by pervasive flow [e.g., McKenzie, 1984] and the region where the melt is segregated into dikes and veins. Although only randomly oriented inclusions are considered in this study, in discussing disequilibrium textures, data on seismic anisotropy become important since the deviation from the equilibrium texture usually involves some structural anisotropy.

[23] One may consider the present result of d ln VS/d ln VP = 1–1.5 to be small compared to that expected previously. In modeling the partially molten upper mantle, α much smaller than 0.1 has been frequently postulated, which predicts much larger values for d ln VS/d ln VP. Gebrande et al. [1980] stated that several researchers used α = 0.01, which leads to d ln VS/d ln VP = 1.8–2.1. A model presented by O'Connell and Budiansky [1977] assumes the distribution of α from α2 = 10−1 to α1= 10−4 uniform in ln α. This model is called “film model” and is frequently used. The effective aspect ratio of this model is 0.014, which can be calculated by equation image α v(α)dα with normalized density function v(α) = (α2 − α1)/(ln α2 − ln α1) [e.g., Schmeling, 1985a, equation (17)]. Owing to the small aspect ratio and owing to the mathematical simplification (Keff/k = 1) thus introduced, the reduction of VS/VS0 predicted by the film model is 2.25 times larger than that of VP/VP0. The present study has shown that such a small α means a disequilibrium texture. Small α has been preferred in explaining the high attenuation of the partially molten regions by the melt squirt flow mechanism [e.g., O'Connell and Budiansky, 1977; Schmeling, 1985a]. Therefore, although the attenuation is beyond the scope of this paper, I briefly discuss the present result of α = 0.1–0.15 from the view point of attenuation due to squirt flow. The effect of the squirt flow is significant if relaxation strength is large enough and also if relaxation frequency fC is not so far from the seismic band; fC is evaluated by kfα3/η with melt viscosity η (= 1–103 Pa s). If α has no distribution, this latter condition directly constrains α to ∼10−2–10−3 [Schmeling, 1985a]. This, however, is not the case when α has a distribution; α effective for fC can be much smaller than that effective for d ln VS/d ln VP. Intuitively, this is because fluid flow in a pore with variable thickness can be dominantly affected by the thinnest part. If the film model mentioned above is modified to α2 = 1 and α1= 10−4, the α effective for d ln VS/d ln VP becomes equation image αv(α)dα = 0.11. Although this α is much larger than that of the original model, the attenuation (and VS reduction) predicted by the original model with ϕ = 0.006, 0.018, and 0.03 can almost be predicted by the modified model as well with ϕ = 0.045, 0.135, and 0.225, respectively (ϕ is increased by 0.11/0.014 times). Therefore α effective for d ln VS/d ln VP cannot be a strict constraint on fC. This is considered to be the case for partially molten rocks whose equilibrium textures are complicated due to crystal anisotropy, and hence the relatively large value of α concluded for the equilibrium melt geometries does not necessarily mean the insignificance of the squirt flow mechanism.

[24] There remain several relevant problems to be clarified which are not discussed in this paper. These include the effects of temperature and/or chemical composition on the elastic properties of the solid phase. Clarifying these effects is important in interpreting the tomographic images, since a lateral heterogeneity by partial melting can be mutually coupled with the thermal and/or chemical heterogeneities. Estimation of these effects involves relatively large uncertainties, since the contribution of the anelastic (including viscoelastic) effects has not been clarified. At this stage the most reliable evaluation of the temperature effect (both by anharmonic and anelastic mechanisms) may be the theoretical one obtained by Karato [1993] as a function of Q. Although several experimental studies have measured the thermal reduction of the solid shear modulus at the seismic frequency range (10−3–1 Hz) [e.g., Gribb and Cooper, 1998, Gribb and Cooper, 2000; Jackson, 2000], an appropriate extrapolation of the experimental results obtained for fine-grained samples with a very small viscosity (∼1012–1014 Pa s) to the Earth's condition has not been made. Karato [1993] obtained δ ln VPT = 1.01 × 10−4 K−1 and δ ln VST = 1.54 × 10−4 K−1 at the upper mantle depth (QP ∼ 200 and QS ∼ 100). For δT = 200 K, VP and VS reductions are estimated to be 2% and 3%, respectively. When corrected for the thermal effect of 200 K [Gebrande et al., 1980], the data obtained beneath Iceland show that the velocity reduction by the poroelastic effect is 11% for VP and 19–27% for VS, which again indicate that the melt geometry is far from the textural equilibrium (d ln VS/d ln VP = 1.73–2.45). Since d ln VS/d ln VP estimated for the thermal effect is ∼1.5 and is near the upper bound of the texturally equilibrated partially molten rocks (Figure 5), the corrections for the thermal effect do not affect the qualitative results of the above discussion. The thermal and chemical effects still have large uncertainties and should be clarified in future studies.

4.2. Limitation of the Crack Model

[25] The crack model is shown to cover α ≲ 10−3. As discussed in section 3, α should be in this range to assume Keff/k = 1. In other words, this restriction on α is imposed for mathematical simplicity and may not reflect the usage of the term “crack” in geological observations; pores with α ∼ 10−2 might be called crack in field or laboratory observations. Figure 4b demonstrates that the difference between α = 10−3 and α = 10−2 is significant for d ln VS/d ln VP; the situation of α = 10−2 where Keff/k cannot be approximated by 1 but should be an explicit function of β does reflect the sensitivity of Keff on the liquid types, which enables us to derive valuable information on the liquid phase in the Earth. It is important to remember the limitation of the crack model and to use a more general formulation covering the actually possible values of α.

[26] Saturation rate [O'Connell and Budiansky, 1974] has been used to interpret the variation of VP/VS within the framework of the crack model [e.g., Zhao and Mizuno, 1999]. This factor (which is usually written as ξ) represents the ratio of the number of saturated cracks to the total number of the cracks, where the other cracks are considered to be dry or soft. In other words, the observed value of d ln VS/d ln VP is explained by changing the fraction of the hard pores relative to the soft pores. This is possible only in the shallow regions where β of a gas phase can be extremely large (∼105–106) and hence can be soft unless α ≪ 10−5–10−6. At a depth of more than several kilometers, even a gas phase has a moderate value of bulk modulus and no liquid exists which can remain soft at α < 10−3. (Since the thin pores are mechanically close to the series arrangement, the pore pressure should be close to the lithostatic pressure.) In this case, variation of d ln VS/d ln VP can no more be explained within the framework of the crack model, and it is important to consider the two-dimensional parameter space of α and β.

[27] Schmeling [1985a] discussed the limitation of the crack model from a different point of view; the effective aspect ratio can be significantly increased by the asperity contacts between the rough crack surfaces. This point should be also taken into account in considering the aspect ratios of cracks effective for d ln VS/d ln VP.

4.3. Observability of Pore Geometry by d ln VS/d ln VP

[28] When VP and VS are each considered separately, the increase of liquid content ϕ and the decrease of pore aspect ratio α can be the same effect. Hence neither ϕ nor α can be determined definitely from a VP (or VS) structure. However, the ratio d ln VS/d ln VP obtained by the combination of the two velocities is independent of ϕ and is sensitive to the possible change of the pore geometry. This has practical importance because d ln VS/d ln VP can be used for determining the actual pore geometry in the Earth's interior. (Then, ϕ can be determined by using Figure 5 (top).) Previous studies to determine the pore geometry were based on the combination of electrical resistivity data and seismological data; the electrical data give an additional constraint on ϕ [Schmeling, 1985b; Shankland et al., 1981]. The present study points out the potential ability to determine α only from velocity data. This point has not been widely recognized; there exist several studies in which the observed large and small values of d ln VS/d ln VP are immediately related to the melt and water, respectively, without taking into account a possible variation of pore geometry. By considering the increasing availability and accuracy of the P and S velocity structures we will soon be able to obtain more and more information about the pore geometry in the Earth's interior. Physical models which can predict actual pore geometry or larger-scale melt or aqueous fluid structure in the Earth with sufficient accuracy to distinguish α = 10−2 from 10−3 seem to be lacking. Establishing such models is important in interpreting seismologically observed pore geometries.

5. Conclusion

  1. The relationships between the oblate spheroid model, crack model, tube model, and equilibrium geometry model have been clearly established in terms of the equivalent aspect ratio. Texturally equilibrated melt and aqueous fluid geometries with a dihedral angle of 20°–40° and 40°–100°, respectively, correspond to an aspect ratio of 0.1–0.15 and 0.15–0.5, respectively.
  2. The significant contributions from liquid compressibility and pore geometry to d ln VS/d ln VP have been clarified as a function of aspect ratio α: for α between ∼5 × 10−3 and ∼0.1, d ln VS/d ln VP variations are sensitive to the compressibility of the liquid; outside this range almost all liquids become soft (at α ≳ 0.1) or hard (at α ≲ 5 × 10−3).
  3. The critical factor governing the variation of d ln VS/d ln VP is (β − 1)/equation image, which is mutually related to Skempton's coefficient and specifies the pressure support mechanism of the poroelastic media between the uncoupled support and the coupled support.
  4. The value of d ln VS/d ln VP expected for the texturally equilibrated partially molten rocks is 1–1.5, which is much smaller than the value expected for the dikes and veins with an aspect ratio of <10−2–10−3. Seismologically observed d ln VS/d ln VP can be used to verify the textural equilibrium.

Appendix A:: Equilibrium Geometry Model

[29] The equilibrium geometry model used in this study is presented with discussion on the points affecting the accuracy of this study. Detailed descriptions of this model are given by Takei [1998].

[30] For a granular aggregate, contiguity φ is the essential geometrical factor determining the macroscopic elasticities of the skeletal framework. The bulk and shear moduli of the skeleton used in the text, Kb and N, are given by

equation image
equation image

where ksk and μsk have been derived as functions of φ. The explicit forms of ksk(φ) and μsk(φ) used in this study are given by

equation image
equation image

where powers nk and nμ depend on φ as

equation image
equation image

Coefficients ai and bi(i = 1, 2, 3) are given by polynomial functions of the intrinsic Poisson's ratio of the solid phase, ν, as

equation image
equation image

where âij(i = 1–3, j = 0–3) and equation imagei(i = 1–3, j = 0–2) are given in Table A1. (Equations (A3)(A8) were obtained by fitting the numerical results calculated for 0.1 ≤ φ ≤ 1 and 0.05 ≤ ν ≤ 0.45.) The relationship of φ to liquid volume fraction ϕ and dihedral angle θ, φ(ϕ, θ), is obtained from the theoretical results of von Bargen and Waff [1986]. They calculated the areas of solid-solid and solid-liquid interfaces per unit volume, AssVand AslV, for a range of 20° ≤ θ ≤ 80° and ϕ ≤ 0.05 by assuming tetrakaidekahedral grains and isotropic interfacial energy; φ is obtained as φ = 2 AssV/(AslV + 2 AssV). By substituting φ(ϕ, θ) into ksk(φ) and μsk(φ), Kb and N are derived as functions of ϕ and θ.

Table A1. Fitting Parameters for ksk and μsk
  ksk  μsk 
jequation imageequation imageequation imageequation imageequation imageequation image
01.86254.5001−5.65121.61224.5869−7.5395
10.52594−6.15516.91590.135273.6086−4.8676
2−4.8397−4.363429.59500−4.3182
300−58.96

[31] The ksk(φ) and μsk(φ) given by (A3)(A8) are briefly discussed. When the dependence of φ on ϕ under fixed θ is written as (1 − φ) ∝ equation image, the exponent n1 is nearly 1/2 for θ < 60°. Since (1 − φ) and ϕ represent the interfacial area and the volume, respectively, of the liquid phase, this value of n1 reflects the two-dimensional or tube-like configuration of the liquid phase obtained at θ < 60°. The value of n1 obtained at θ > 60° (three-dimensional isolated pores) is nearly 2/3. Calculation of ksk and μsk requires contact function XC, which specifies in detail the grain-to-grain contact state; XC is defined on the grain surface and takes 1 on the contact faces and 0 on the wetted areas. Two types of XC with a packing coordination of 12 were considered by Takei [1998]; in one type, called standard type, the area of XC = 0 becomes several isolated islands at large φ (>0.804), whereas in the other type the area of XC = 0 forms a continuum network even at large φ. Therefore the pore geometry at φ → 1 is three-dimensional in the former model and two-dimensional in the latter model. The results on the former model are presented in Appendix F of Takei [1998], and those on the latter model are given by (A3)(A8). Although both models give almost the same results on ksk(φ) and μsk(φ), a systematic difference appears at 0.75 < φ < 1 corresponding to the different dimension assumed on the pore geometry. The behavior of Msk(M = k or μ) at φ → 1 can be written as (1 − Msk/M) ∝ equation image. The numerical results on Msk show n2 ≃ 1.3 for the three-dimensional case and n2 ≃ 1.8 for the two-dimensional case. This study uses the results with n2 ≃ 1.8 so as to match the two-dimensional pore geometry at θ < 60°. Since (1 − φ) ∝ equation image, (1 − Msk/M) ∝ equation imageat ϕ → 0. For (n1, n2) = (1/2, 1.8) and (2/3, 1.3) the exponent n1 · n2 is 0.9 and 0.87, respectively, which are slightly smaller than unity. In the inclusion models the effective moduli are reduced in proportion to ϕ1 near ϕ = 0. As yet, I cannot tell whether the difference in exponents comes from the essential difference between the models or from a numerical error.

[32] Possible uncertainty of the equilibrium geometry model comes from the uncertainty of packing coordinations. The present study used ksk(φ) and μsk(φ) calculated for XC with a packing coordination of 12 (rhombic dodecahedral grain) and φ(ϕ, θ) calculated for a packing coordination of 14 (tetrakaidekahedral grain) because VP and VS calculated from this combination of ksk(φ), μsk(φ), and φ(ϕ, θ) show an excellent agreement with the experimental results [Takei, 2000]. (Although the theoretical predictions made by Takei [2000] were based on Appendix F of Takei [1998], ksk(φ) and μsk(φ) given by (A3)(A8) also well explain the experimental data, since the data points exist only at ϕ ≥ 0.03 and hence at φ ≲ 0.75.) If the packing coordinations are assumed differently, the equivalent aspect ratio for each θ becomes slightly larger. The ksk(φ) and μsk(φ) calculated for XC with a packing coordination of 14 are larger than those for 12 [Takei, 1998]. I calculated φ(ϕ, θ) for the rhombic dodecahedral grains by using the same procedures as von Bargen and Waff [1986] (Figure A1). (In the obtained solid-liquid interfacial geometries the fluctuation of the mean curvature was <0.5%; only for θ = 5°, fluctuations of 2% and 10% remained at ϕ = 0.047 and at ϕ = 0.105, respectively.) The φ(ϕ, θ) obtained (Figure A1e) is larger than that calculated for the tetrakaidekahedral grains. Figure A2 shows the equivalent aspect ratios obtained from the combination of ksk(φ), μsk(φ), and φ(ϕ, θ) calculated for the tetrakaidekahedral grains (denoted 14-14), and from the combination of these calculated for the rhombic dodecahedral grains (12-12). These are slightly larger than the aspect ratios obtained in the text (12-14). These changes do not affect the conclusion of this study.

Figure A1.

(a) A rhombic dodecahedral grain. The shaded areas show the grain-to-grain contact faces. O represents the center of the inscribed sphere of the rhombic dodecahedron. △ABC is 1/4 of the area of one rhombic face, where line segment equation image intersects the face at a right angle. (b) Element with 1/48 of the volume of the rhombic dodecahedral unit, where O, A, B, and C are shown in Figure A1a. The solid-liquid interface (bold line) is parameterized by a nonorthogonal (u, v, w) coordinate system. A constant u parameter plane (shaded plane) intersects the dihedral edge at a right angle; the intersection of this plane with △ABC forms the v axis, and the distance between the u axis (= equation image) and the dihedral edge is represented by b(u); cos γ = −equation imageb′(u). (c) One constant u parameter slice. The distance from the u-v plane to the solid-liquid interface is given by the function w(u, v); cosα = [4 − 2(b′(u))2]−1/2. (d) Equilibrium interfacial geometry shown by the dihedral edge curve on △ABC and the section of solid-liquid interface cut by u = 0.5 plane. The angle formed at the dihedral edge is equal to θ/2. The solid and dotted lines correspond to ϕ ≃ 0.1 (0.103 ± 0.002) and 0.05 (0.046 ± 0.001), respectively. (The interface was represented by grid points located on 70 u slices and 16 v slices at even intervals.) (e) Grain boundary contiguity φ versus liquid volume fraction ϕ at dihedral angle θ between 5° and 58°. The symbols show the numerical results.

Figure A2.

Equivalent aspect ratios of the equilibrium geometry model assuming various packing coordinations. For 14-14 (12-12), ksk(φ), μsk(φ), and φ(ϕ, θ) for the tetrakaidekahedral grains (rhombic dodecahedral grains) are used. For 12-14, ksk(φ) and μsk(φ) for the rhombic dodecahedral grains and φ(ϕ, θ) for the tetrakaidekahedral grains are used. Solid and open symbols show the correspondence determined from equation image and ΛN, respectively.

[33] The results of von Bargen and Waff [1986] are limited to θ ≥ 20° and do not give a behavior of φ at θ → 0. It was verified that the decrease of φ with decreasing θ is gradual, at least down to θ = 5°, for the rhombic dodecahedral grains (Figures A1d and A1e). In other words, although φ vanishes at θ = 0, the singular behavior of φ appears at θ < 5°. This supports the validity of applying the formula of von Bargen and Waff [1986]down to θ = 10° as a smooth extrapolation. Also, using φ(ϕ, θ) calculated for the rhombic dodecahedral grains, the concavity of the curves of Kb/k and N/μ versus ϕ and the independence of (1 − VS/VS0)/(1 − VP/VP0) from ϕ were confirmed up to ϕ = 0.2. These support the similar results obtained in section 3 by extrapolating von Bargen and Waff's [1986]formula up to ϕ = 0.2.

Appendix B: Application of Figures 4 and 5 to Large Velocity Perturbations

[34] In the last part of section 3 the applicable range of Figures 4a and 4b is shown to be the VS variation of ≤35% for the equilibrium geometry model and ≤20% for the oblate spheroid, tube, and crack models. In the oblate spheroid, tube, and crack models the linearity assumed in (4) is well satisfied. In this case, even when the velocity perturbations are larger than the above limit, small modifications enable us to accurately determine aspect ratio α (or tube geometry ϵ) and the liquid volume fraction ϕ by using Figures 4 and 5.

[35] It is easy to show that if (4) holds, [1 − (VS/VS0)2]/[1 − (VP/VP0)2] is strictly equal to the right-hand side of (6). Hence, when the velocity perturbations are large, by using [1 − (VS/VS0)2]/[1 − (VP/VP0)2], rather than (1 − VS/VS0)/(1 − VP/VP0), to calculate the vertical position of the observed data, α (or ϵ) can be accurately determined from Figures 4a and 4b. Note that in this situation the value of θ corresponding to α (or ϵ) is slightly smaller than that shown in Figure 4a. Also note that if a detailed behavior with ϕ in the vicinity of d ln VS/d ln VP = 1 is of importance, a small deviation from (4) is not negligible.

[36] If (4) holds, [1 − (VS/VS0)2]/[ΛN − (1 − ρf/ρ)(VS/VS0)2] is strictly equal to ϕ. Hence, when the velocity perturbations are large, it is accurate to estimate ϕ from this relation rather than to use the linearized form ϕ = [1 − (VS/VS0)]/[(ΛN − (1 − ρf/ρ))/2]. ΛN is given in Figure 3a; Figure 5 (top) shows (ΛN − (1 − ρf/ρ))/2.

Acknowledgments

[37] I gratefully acknowledge stimulating and helpful discussions with S. Karato, B. Chouet, Y. Fukao, R. N. Edwards, I. Shimizu, K. Mibe, and J. Nakajima. D. L. Kohlstedt and an anonymous reviewer have provided thoughtful reviews.

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