1 Introduction

Segregation of geological fluids is an essential elementary process that controls the chemical differentiation and physical properties of the Earth’s interior. Melt segregation from a partially molten region controls the chemistry of magmas and the solid matrix in a two-phase flow (e.g., McKenzie and Bickle 1988; Iwamori 1993; Faul 2001). The excess pore fluid pressure produced by the dehydration of a subducting slab is a key factor controlling seismic activity within the slab (Kita et al. 2006; Peacock et al. 2011; Morishige and van Keken 2018). In turn, the pore fluids left in the slab contribute to the total amount of water downdragged into the Earth’s deep interior. Melt segregation is also important for the core formation and concomitant chemical evolution of the Earth and other planetary bodies (Stevenson 1990; Terasaki et al. 2008; Bagdassarov et al. 2009; Watson and Roberts 2011; Solferino et al. 2020).

In deep-seated rocks below the brittle–ductile transition, where rocks do not sustain fractures, the rock microstructure develops to reduce the interfacial energy (Watson and Brenan 1987) through a process of “textural equilibration”. The local textural equilibrium has been generally premised by the establishment of the dihedral angle (θ), the most elemental parameter that characterizes pore fluid geometry. In an ideal simplified fluid–rock system, with a uniform grain size and isotropic interfacial energy, θ is established at every solid–liquid interface with a constant mean curvature of the fluid–solid interfaces (Beere 1975; Bulau et al. 1979; Waff 1980; Park and Yoon 1985; Ghanbarzadeh et al. 2015; Rudge 2018). When θ < 60°, all the interstitial fluids are connected, and gravity drives fluid segregation and compaction of ductile solid matrix on geological timescales (McKenzie and Bickle 1988). When θ > 60°, to the contrary, fluids with less than the volume fraction at a percolation threshold can be isolated and retained in the grain boundaries, thereby escaping gravitational segregation. The latter case has been explicitly studied to examine the possibility of water being downdragged into the Earth’s deep interior along with the subducting slab (e.g., Watson and Lupulescu 1993; Ono et al. 2002; Mibe et al. 2003; Liu et al. 2018), because the dihedral angles between aqueous fluids and eclogite-constituent minerals in the slab dehydration condition are large. Here, unaddressed issues are mechanisms for the non-wetting, hard-to-connect fluids segregation, and the resultant proportion of retained fluids in the rocks.

Furthermore, the geometries of melts and aqueous fluids in the grain and interphase boundaries of real polycrystalline rocks often deviate from ideal values. Interfacial energy anisotropy results in the statistical diversity in the dihedral angles (Wark and Watson 1998) and faceting of flat solid–fluid interfaces, causing disruption of fluid networks and isolation of pore fluids (Waff and Faul 1992; Cmíral et al. 1998; Wark et al. 2003; Yoshino et al. 2006; Shangshang and Faul 2016; Price et al. 2006; Huang et al. 2023). In addition, the formation of fluid pools surrounded by more than three neighboring grains has been reported in olivine–basalt melt, quartz–albite melt, olivine-FeS melt, and quartz–H2O systems (Jurewicz and Watson 1984; Waff and Faul 1992; Cmíral et al. 1998; Shangshang and Faul 2016). In these experiments, large fluid pools were formed adjacent to small and rounded grains, suggesting the merging of neighboring pore fluids via the consumption of small grains. Similar fluid pool formation was also observed in situ for the analogous fluid–rock system of ethanol and norcamphor (Walte et al. 2003). These experimental results indicate that the microstructure of realistic fluid–rock systems develops to minimize the total interfacial energy of the system, allowing deviation in the local fluid geometry from ideal models with a single dihedral angle and a uniform liquid–solid interfacial energy. Thus, how and to what extent geological fluids are retained in deep-seated rocks is not fully constrained.

The permeabilities of geological bodies have been previously estimated using hydrological models based on geothermal and metamorphic data (e.g., Manning and Ingebritsen 1999; Ingebritsen and Manning 2010), laboratory measurements (e.g., Wark and Watson 1998; Katayama et al. 2012; Ganzhorn et al. 2019), and numerical simulations (e.g., von Bargen and Waff 1986; Zhu and Hirth 2003; Miller et al. 2014; Eberhard et al. 2022). Several theoretical models have been proposed for the permeability–porosity (\(k\)–\(\varphi\)) relationship by assuming a single dihedral angle and uniform liquid–solid interfacial energy in a system (hereafter, the “reference model”) to establish a mechanically and thermodynamically stable topology of fluids in a texturally equilibrated monomineralic rock (Beere 1975; Bulau et al. 1979; Waff 1980; Park and Yoon 1985; Ghanbarzadeh et al. 2015; Rudge 2018). Here, porosity is often replaced by the volume fraction of fluid or melt depending on the context. This assumption requires uniform space-filling tesselation of grains, such as a tetrakaidecahedral unit cell. The permeability is primarily controlled by the minimum cross-sectional area of grain-edge tubules (aperture size) Atb, which is a function of the grain diameter \(d\), \(\varphi\), and θ (Watson 1982; von Bargen and Waff 1986; Rudge 2018). An increase in the fluid fraction increases Atb, and thus, permeability. At a given \(\varphi\), Atb is larger for smaller dihedral angles (von Bargen and Waff 1986). Therefore, the absolute value of permeability (\(k_{{{\text{abs}}}} )\) depends on the grain size of the rocks, even if the rocks are geometrically similar. The relationship between \(k_{{{\text{abs}}}}\) and \(\varphi\) in textually equilibrated rocks can be expressed as:

$$k_{{{\text{abs}}}} = \frac{{d^{2} \varphi^{n} }}{C},$$
(1)

where \(C\) is a geometrical factor and \(n\) is an exponent, typically 2–3. The power-law exponent \(n\) depends on porosity: \(n = 2\) when porosity is low and fluid resides at triple-junction tubules along grain edges, and \(n = 3\) for higher porosity conditions with fluid-wetting grain faces (Rudge 2018). A theoretical model that considers variations in the dihedral angle and grain-scale heterogeneity of the fluid distribution also requires \(n = 3\) (Zhu and Hirth 2003). Permeability decreases with increasing dihedral angle at the same porosity because a larger dihedral angle decreased Atb by distributing more fluids at the grain corners than grain-edge tubules and grain faces, which are more effective for fluid interconnection. Notably, the dependence of permeability on the dihedral angle is prominent at low porosities, indicating that permeability is more sensitive to fluid topology at low porosities than at high porosities.

Laboratory measurements of permeability using synthesized, texturally equilibrated quartzite and calcite have shown that permeability is reduced in moderately faceted systems by segregating fluids into grain-scale pools, resulting in Atb decrease (Yoshino et al. 2006; Price et al. 2006). For example, in a moderately faceted quartz–water (brine) system, the values of n and C were 3 and 270, respectively (Wark and Watson 1998; Liang et al. 2001). Thus, the \(k\)–\(\varphi\) relationship showed that the permeability decreased from those of the reference models by a factor ranging 5–10 at the same porosity. These previous studies indirectly interpreted the measured permeability by observing 2D cross sections. Direct computation of permeability using the 3D tomographic images, a digital rock physics approach, would bridge the gap and contribute to a better understanding of how pore microstructures control the permeability and isolated fluid fraction of deep-seated rocks. In the present study, we used quartzite, a polycrystalline quartz aggregate. This well-studied system works as a semi-analog of silicate rocks because quartz’s mineralogical and microstructural properties at elevated pressures and temperatures are well established. Although quartzite remains simpler than actual rocks, it exhibits nearly “textually equilibrated” (interfacial energy-minimized) microstructures through normal grain growth in the piston-cylinder syntheses (e.g., Watson and Brenan 1987; Wark and Watson 1998; Nakamura and Watson 2001), thus allowing to investigate the microstructural evolution of natural rocks.

2 Methods

2.1 Quartzite synthesis

We utilized some of the quartzite samples, which were synthesized to examine the capsule-scale chemical redistribution under high pressure and temperature in the companion paper (Fujita et al. 2022). Other details of the piston-cylinder synthesis were reported by Fujita et al. (2022). The starting material was a powder of fine-grained (< 45 μm) quartz and amorphous silica mechanically mixed with a weight ratio of 3:1. The amorphous silica, prepared from the sol–gel method, was adequate for rapid achievement of steady grain growth (Nakamura and Watson 2001; Ohuchi and Nakamura 2007a, b). We added the fluid source to produce three different CO2 fractions (XCO2 in mol): XCO2 = 0 (pure water), 0.27–0.29, and 0.44–0.47. Brucite powder was used as the water source for the pure water experiments. Although enstatite will be formed at the bottom of the capsule after dehydrated brucite reacts with silica, it would not affect the fluid distribution in quartzite (Fujita et al. 2022) or other microstructures of interest in this study. For the CO2-bearing runs, oxalic acid dihydrate powder was used as a fluid source. Two geometries were tested for mixing with SiO2 powder (i.e., initial mixing by grinding in a mortar or placing at the capsule bottom). However, no systematic difference effect of the fluid distribution was found, as reported by Fujita et al. (2022). The added volumetric fluid fraction \(\varphi_{{{\text{ad}}}}\) was controlled between 0.019 and 0.101 at each experimental condition by changing the amount of fluid sources. The fluid fraction calculation included the absorbed H2O in the amorphous silica (~ 0.6 wt%). Hereafter, we refer to the CO2-free and CO2-bearing systems as wetting and non-wetting systems, respectively. All the synthetic experiments were conducted at 900 °C and 1 GPa. From the series of experiments with run durations of 24–382 h reported by the companion paper (Fujita et al. 2022), we selected and prepared 11 representative experimental products with run durations of 192 h, added fluid fractions of 0.025–0.101 and XCO2 = 0, 0.28, and 0.39–0.44, for non-destructive imaging of the interstitial pores with synchrotron radiation X-ray computed microtomography (SRμCT—e.g., Lindquist et al. 2000; Zhu et al. 2011; Miller et al. 2014; Skemer et al. 2017; Thomson et al. 2018) (see Table 1).

Table 1 Summary of experimental conditions and analytical results for subvolumes (stacks of SRμCT slices)
Full size table

2.2 Synchrotron X-ray microtomography

The three-dimensional (3D) pore structure was obtained by using SRμCT. Details of sample preparation for imaging are presented in Fujita et al. (2022). The SRμCT was conducted at the BL20XU experimental hutch2 of SPring-8 (Uesugi et al. 2012) with the 510 nm edge length of the cubic voxels in the obtained images. The details of the beamline setting were described by Huang et al. (2021).

2.3 Microstructure characterization

Cross sections of quenched samples that differed from those used for SRμCT were observed using backscattered electron (BSE) imaging using a field-emission type scanning electron microscope (FE–SEM; JSM-7100F, JEOL Ltd., Akishima, Tokyo, Japan) equipped with an energy dispersive X-ray spectrometer (EDS, JEOL JED-2300F). The details of the sample preparation procedures are reported by Fujita et al. (2022). For the 3D characterization of the pore microstructure and numerical simulation of fluid flow, cubic subvolumes with a dimension 512 × 512 × 512 voxels (i.e., 261 × 261 × 261 µm3) were cut out from the reconstructed images. For run QW13-13, whose grain size was coarser than that of the others, a larger subvolume of 600 × 600 × 600 voxels (i.e., 306 × 306 × 306 µm3) was cut out. The edge lengths of each subvolume were 4.4–6.7 times larger than their mean grain size, which are too small to represent the capsule-scale fluid distribution (Fujita et al. 2022) but can represent multi-grain-scale fluid distributions. To check representativeness, we examined two to six subvolumes from different positions in each sample. Only one subvolume was cut from QW13-13 because the CT-scanned area was much smaller than those of the other samples. A smoothing filter was applied using the ImageJ software (National Institutes of Health, USA; https://imagej.net/), and the subvolumes were converted into a binary format. Dilation cracks perpendicular to the pressurization direction, formed upon quenching and pressure release in piston-cylinder experiments, were found only in QW15-13. For QW15-13, subvolumes were chosen from areas without dilation cracks. The open grain boundaries were broadly recognized especially along relatively large grains, and hence, they were eliminated by dilating and subsequently eroding the solid phase by 1-pixel layer using the software package “Slice” (Nakano et al. 2006).

We measured the connected porosity (\(\varphi_{{{\text{con}}}}\)) in addition to the total porosity (\(\varphi_{{{\text{mes}}}}\)) from the corrected 3D images using “Slice”. Here, the \(\varphi_{{{\text{con}}}}\) is a volume fraction of the pores connected from one side to the opposite side of the cubic subvolume and thus should directly affect permeability. The relationship between \(\varphi_{{{\text{mes}}}}\) and \(\varphi_{{{\text{con}}}}\) highlighted the microstructural difference between the wetting and non-wetting systems.

2.4 Numerical computation of flow velocity and permeability

The velocity and flow patterns of a fluid can be described by the Reynolds number \(Re\), which is the ratio of inertial forces to viscous forces and is defined as:

$$Re = \frac{{\rho vd_{{{\text{ch}}}} }}{\eta },$$
(2)

where \(\rho\) is the density, \(v\) is the velocity, \(\eta\) is the viscosity of the fluid, and \(d_{{{\text{ch}}}}\) is the characteristic internal diameter of the fluid channel. For all simulations, we employed a fluid viscosity of 1 Pa s as the computed permeability does not depend on the viscosity value. Reynolds numbers typically range from 10–9 to 10–10 for magmas (Glazner 2014) and from 10–8 to 10–5 for groundwater flow. Therefore, owing to the narrow fluid channels of our samples, \(Re\) is expected to be small (< 1); thus, the fluid flow should be laminar (Bear 1988). Under these conditions, the incompressible Navier–Stokes equations can be simplified to the Stokes equations (ignoring gravity).

$$\frac{{\partial v_{i} }}{{\partial x_{i} }} = 0;$$
(3)
$$\frac{\partial }{{\partial x_{j} }}\left[ {\eta \left( {\frac{{\partial v_{i} }}{{\partial x_{j} }} + \frac{{\partial v_{j} }}{{\partial x_{i} }}} \right)} \right] - \frac{\partial P}{{\partial x_{i} }} = 0,$$
(4)

where \(x_{{\text{i}}}\) and \(x_{{\text{j}}}\) are the spatial coordinates corresponding to the Cartesian axes.

Permeability was computed using a finite-difference Lithospheric and Mantle Evolution Model (LaMEM) (Kaus et al. 2016; Eichheimer et al. 2019). LaMEM employs a staggered finite-difference discretization that defines the pressure P at the cell centers and the velocities \(v\) at the cell faces. The computational grid cell corresponded to an image voxel. A free-slip boundary condition was employed on the sides of the subvolume. The Eqs. (3) and (4) were non-dimensionalized, discretized, and then solved for a given pressure difference using the multigrid solvers in the PETSc library (Balay et al. 2019) and the V-cycle geometric multiplicative multigrid solver (Fedorenko 1964; Wesseling 1995). The applied non-dimensional fluid pressure was 1 on a subvolume face at the inlet side perpendicular to the flow direction, and 0 on a subvolume face at the outlet side. A single-grid solver was used in the case of QW13-13, which had a larger subvolume. The relative and absolute convergence tolerances were specified to be 10−9 and 10−10, respectively, sufficiently low for calculation convergence (Eichheimer et al. 2019). From the computed velocity field in the flow direction z, the volume-averaged velocity component \(v_{{\text{m}}}\) was calculated as follows (e.g., Osorno et al. 2015):

$$v_{{\text{m}}} = \frac{1}{{V_{{\text{f}}} }}\int_{{V_{{\text{f}}} }} \left| {v_{{\text{z}}} } \right|{\text{d}}v,$$
(5)

where \({\varvec{V}}_{{\mathbf{f}}}\) denotes the volume of the fluid phase. Permeability values were then calculated using Darcy’s law:

$$k_{{{\text{abs}}}} = \frac{{\eta v_{{\text{m}}} L}}{\Delta P},$$
(6)

where \(L\) is the height of the subvolume and \(\Delta P\) is the pressure difference. A more detailed explanation of the computational method is provided by Eichheimer et al. (2019).

Because the absolute permeability is length-scale dependent, it is normalized by grain size to extract the geometric features of the system. In this study, the computed permeabilities were normalized to a mean grain diameter of 1 mm so the results can be compared with those with those of Wark and Watson (1998) and Liang et al. (2001):

$$k = k_{{{\text{output}}}} \times \frac{{1000^{2} }}{{{\textit{d}}_{{{\text{mean}}}}^{2} }} ,$$
(7)

where k is the normalized permeability in m2, koutput is the computed absolute permeability for the subvolume with mean grain diameter of dmean in µm. The error bars of \(\varphi_{{{\text{mes}}}}\) were calculated as half the difference between coating and scraping the solid (quartz) surface by a one-pixel layer. To investigate the anisotropy of the permeability, a pressure difference was applied to the three orthogonal directions (\(k_{{\text{x}}}\), \(k_{{\text{y}}}\), and \(k_{{\text{z}}}\)) for all subvolumes. Here, z was the vertical direction (the pressurized direction of the piston-cylinder experiments), and x and y were the horizontal directions perpendicular to each other. The mean values were then obtained to represent the sample.

The flow velocity distributions of the four representative subvolumes (QW15-22#1, QW13-13#1, QW13-11#1, and QW15-13#2) were investigated to understand the mechanism which governs permeability. The streamlines of the four subvolumes were visualized using the open-source visualization application, ParaView (https://www.paraview.org/).

3 Results

3.1 Shapes and occurrences of pore fluids

The mean grain size, grain size distribution, and dihedral angles measured in the polished sections of the synthesized quartzite are described in Fujita et al. (2022). The grain size distribution indicated that near-steady-state grain growth was achieved in both the CO2-free (wetting) and CO2-bearing (non-wetting) systems. The dihedral angles between the fluids and quartz (θ), given by the median values excluding facetted pores, were 52°, 61°, and 71° for XCO2 = 0, 0.28, and 0.44, respectively. The fluid fractions that minimize the total interfacial energy of the fluid-bearing polycrystalline system (MEFF) are ~ 0.035 and 0.015 for the dihedral angles of 61°, and 71°, respectively (Park and Yoon 1985; Watson 2015). Based on the BSE images of the polished sections of the run products (Fig. 1a, b) and the two-dimensional SRμCT slices (Fig. 1c, d), we can classify the pore fluids in the quartz aggregate into three types based on the number of surrounding grains: (1) fluid lenses between two-grain boundaries, (2) grain-edge tubules surrounded by three grains, and (3) fluid pools surrounded by more than four grains. The fluid lenses were occasionally distributed as lenticular islands (Fig. 1a–d) and sometimes exhibited necklace microstructures (German 1985). Among run products with similar fractions of fluid remaining in quartzite, fluid lenses occurred more commonly in samples with high XCO2 than in the other two types of samples. Grain-edge tubules were commonly observed in both wetting and non-wetting systems. However, dry triple junctions existed even in the wetting system, in which the dihedral angle was < 60° (Fig. 1a, b), as reported by previous studies (Yoshino et al. 2006; Price et al. 2006). The fluid pools were often surrounded by several to multiple grains (Fig. 1c, d), forming irregular shapes. The size of the fluid pools is generally larger than those of the fluid lenses and cross section of the grain-edge tubules, sometimes comparable to or larger than the mean grain diameter, and occasionally as large as 50–100 µm. The plucking-free SRμCT slices demonstrated that the grain-scale pools were often adjacent to small and rounded quartz grains (Fig. 1c, d). The grain-scale pools appeared to form within quartzites with relatively large added fluid fraction \(\varphi_{{{\text{ad}}}}\). In all runs, faceted grain-edge tubules were commonly found together with an equilibrium shape surrounded by three grains with a constant interface curvature, as assumed in the reference models. Faceting was more common in the CO2-bearing quartz aggregates.

Fig. 1
figure 1

Representative BSE, synchrotron X-ray microtomography, and CT images. Representative BSE images of polished cross sections (a, b; Table 1 of Fujita et al 2022) and slice images of synchrotron X-ray microtomography images (c, d) of the run products. a QW5-13: XCO2 = 0, \(\varphi_{{{\text{ad}}}}\) = 0.102, 382 h; b QW9-12: XCO2 = 0.44, \(\varphi_{{{\text{ad}}}}\) = 0.094, 192 h; c QW13-13: XCO2 = 0, \(\varphi_{{{\text{ad}}}}\) = 0.101, 192 h; and d QW13-11: XCO2 = 0.44, \(\varphi_{{{\text{ad}}}}\) = 0.094, 192 h. White arrows with “L”, fluid lenses; “T”, triple-junction tubules; “P”, fluid pools; white dotted arrows, dry triple junctions. Polished sections inevitably suffered from grain plucking (black arrows). Many fluid lenses were observed as dimples on the quartz grain surfaces as in (a). The voxel size of CT images was 0.513 µm3; fluid lens and triple-junction tubules that were smaller than 0.51 µm are not resolved in (c, d). Note that the CT images are free from grain plucking. The fluid pools were often adjacent to small and rounded quartz grains as indicated by the white arrows marked “R.” Scale bars represent 40 µm for all images

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3.2 Pore connectivity

The connected porosity (\(\varphi_{{{\text{con}}}}\)) roughly coincided with the measured total porosity \(\varphi_{{{\text{mes}}}}\) at larger porosities, while \(\varphi_{{{\text{con}}}}\) became lower than \(\varphi_{{{\text{mes}}}}\) at lower porosities (\(\varphi_{{{\text{mes}}}}\) \({ \lesssim }\) 0.07) and reached 0 (Fig. 2), being consistent with the prediction from the presence of pinch-off threshold for the measured permeability and observations of 2D cross sections (Wark and Watson 1998). In the wetting system, the pore interconnection was sharply cut off at \(\varphi_{{{\text{mes}}}}\) between 0.031 and 0.037. At \(\varphi_{{{\text{mes}}}}\) ≥ 0.037, most subvolumes except QW15-21#1 and QW15-22#7 had \(\varphi_{{{\text{con}}}} /\varphi_{{{\text{mes}}}}\) > 0.9. In contrast, in the non-wetting system, \(\varphi_{{{\text{con}}}}\) extensively varied from 0.008 to 0.050 against a small range of \(\varphi_{{{\text{mes}}}}\) (= 0.048–0.059), which corresponded to a rapid increase in \(\varphi_{{{\text{con}}}} /\varphi_{{{\text{mes}}}}\) from 0.13 to 0.91. In the subvolumes with \(\varphi_{{{\text{mes}}}}\) above this range (\(\varphi_{{{\text{mes}}}}\) ≥ 0.079), \(\varphi_{{{\text{con}}}} /\varphi_{{{\text{mes}}}}\) was above 0.98, whereas at \(\varphi_{{{\text{mes}}}}\) ≤ 0.044, \(\varphi_{{{\text{con}}}}\) was 0. Notably, QW15-11#2 had the lowest \(\varphi_{{{\text{con}}}} /\varphi_{{{\text{mes}}}}\) (0.14), but still maintained pore interconnection.

Fig. 2
figure 2

Relationship between the measured total porosity (\(\varphi_{{{\text{mes}}}}\)) and connected porosity (\(\varphi_{{{\text{con}}}}\)). Error bars for both \(\varphi_{{{\text{mes}}}}\) and \(\varphi_{{{\text{con}}}}\) were calculated by shrinking and expanding the solid phase by half a pixel layer. Blue and orange symbols represent subvolumes of the wetting (XCO2 = 0) and non-wetting (XCO2 = 0.28–0.44) systems, respectively. Filled symbols represent subvolumes having pore interconnection between the two facing sides. Open symbols show subvolumes that have isolated pores without a flow pathway

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In the present study, the pinch-off thresholds were 0.037 and 0.048 in the wetting and non-wetting systems, respectively, which were significantly higher than the values in the reference model (i.e., 0 for θ < 60° and ~ 0.03 for θ = 70°; Rudge 2018). The strict threshold value depends on the resolution of 3D images. However, the fine voxel size of SRμCT for the present study (510 nm) ensures that the permeability significantly deceases at this fluid fraction range.

The pore interconnection behavior, along with the geometry of each pore fluid, were visualized through volume renderings. The largest interconnected pores of the subvolumes are depicted in the upper row of Fig. 3, and the remaining pores are shown in the lower row. The largest connected pore was one that connected from one side of the subvolume to the opposite side (i.e., that penetrated the subvolume all the way through).

Fig. 3
figure 3

Three-dimensional fluid distribution of representative subvolumes. The upper row (a–f) and the lower row (g–l) show the maximum volume pore and other pores of each subvolume, respectively. Three subvolumes of the wetting system—QW14-21#1 (a, g), QW13-13#1 (b, h), and QW15-23#5 (c, i)—and non-wetting system—QW13-11#1 (d, j), QW15-11#1 (e, k), and QW15-14#2 (f, l)—are shown

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As observed in the 2D images (Fig. 1), the pores in the non-wetting systems (Fig. 3d–f, j–l) were more rounded and the fluid tubules along the grain edges were thicker than those in the wetting system as expected from the larger dihedral angle (Fig. 3a–c, g–i). In both systems, fluid networks were established by interconnecting the fluids at the grain corners and relatively large fluid pools that often had more than four faces, whereas small fluids were left unconnected. In the wetting system, the fluid networks became thinner and sharper with decreasing total fluid fraction; however, the connectivity remained high as long as the interconnection was maintained (Fig. 3a, b, g, h). In contrast, in the non-wetting system, a portion of the pores [0.45 = 0.28/(0.28 + 0.34)] in Fig. 3e maintained the pore connection from one side to the opposite side. The isolated fluid fractions after the connectivity reached 0 were larger in the non-wetting system (Fig. 3c, i) than in the wetting system (Fig. 3f, l).

3.3 Permeability

The anisotropy of computed permeability, i.e., the difference between the maximum and minimum directions, was lowest (9% of the mean) for QW15-21#2 and highest (128%) for QW15-23#3 (Table 1). Here, the samples for which permeability was obtained in only one or two directions were excluded. No systematic tendency was found in the vertical and horizontal directions of the subvolumes, indicating that the fluid distribution was not affected by the pressurization direction during high-pressure syntheses. The anisotropy showed a weak negative correlation with \(\varphi_{{{\text{con}}}}\) and was relatively high at \(\varphi_{{{\text{con}}}} < 0.040\). Ten out of eleven samples had \(\varphi_{{{\text{con}}}} > 0.040\) and showed an anisotropy < 65%. The small difference in log scale among \(k_{{\text{x}}}\), \(k_{{\text{y}}}\), and \(k_{{\text{z}}}\) implied that the subvolume sizes (261 × 261 × 261 µm and 306 × 306 × 306 µm) were reasonable and could represent the grain-scale fluid distribution.

Figure 4 shows the computed permeability k against the measured total porosity (\(\varphi_{{{\text{mes}}}}\)) and connected porosity (\(\varphi_{{{\text{con}}}}\)) for all subvolumes with pore connection across the subvolumes. The computed permeabilities generally increased with increasing \(\varphi_{{{\text{mes}}}}\) and \(\varphi_{{{\text{con}}}}\) in both the wetting and non-wetting systems, except for subvolumes with relatively low \(\varphi_{{{\text{mes}}}}\) in the non-wetting system. The computed permeabilities were smaller by approximately half an order of magnitude than those of the reference \(\varphi - k\) model calculated for \(\theta\) = 50°, 60°, and 70° (Rudge 2018). However, most subvolumes showed an excellent agreement with the \(\varphi_{{{\text{mes}}}}\)–\(k\) relationship obtained by direct laboratory measurements of the quartz + H2O (i.e., wetting) system (Wark and Watson 1998), except for five subvolumes (QW15-13#3, QW15-11#1, QW15-21#1, QW15-13#2, and QW15-11#2) whose permeability agreed with or were lower than the laboratory measurements for the calcite + H2O (i.e., non-wetting) system (Wark and Watson 1998). These subvolumes exhibited low pore connectivity except for QW15-13#3 (Fig. 2). Thus, the variation in \(k\) in the non-wetting system stemmed from the variation in pore connectivity. Notably, QW15-13#2 and QW14-11#2 had the same \(\varphi_{{{\text{mes}}}}\) (0.053) but different \(k\), ranging from 1.35 × 10–13 to 5.66 × 10–13 m2, owing to the differences in \(\varphi_{{{\text{con}}}}\) of 0.015 and 0.048, respectively. QW15-13#3 lacked connected pores in x directions, suggesting the microstructural heterogeneity was large for its relatively high \(\varphi_{{{\text{con}}}}\) (0.050). Among the subvolumes of the wetting system, the most noticeable gap in \(k\) at similar \(\varphi_{{{\text{mes}}}}\) was between QW15-21#1 and QW15-23#1 (2.35 × 10–13 m2 at \(\varphi_{{{\text{mes}}}}\) = 0.056 and 7.73 × 10−13m2 at \(\varphi_{{{\text{mes}}}}\) = 0.048, respectively).

Fig. 4
figure 4

Computed permeabilities. Computed permeabilities in log units as a function of a the measured porosity \(\varphi_{{{\text{mes}}}}\), and b connected porosity \(\varphi_{{{\text{con}}}}\). The permeabilities are mean values of kx, ky, and kz obtained from the flow simulations in three orthogonal directions (x, y, and z) and normalized to a mean grain diameter of 1 mm. Error bars for both \(\varphi_{{{\text{mes}}}}\) and \(\varphi_{{{\text{con}}}}\) were calculated by shrinking and expanding the solid phase by half a pixel layer. Blue and orange symbols represent subvolumes of the wetting (XCO2 = 0) and non-wetting (XCO2 = 0.28–0.44) systems, respectively. Gray dotted lines are from the reference model by Rudge (2018) for dihedral angles of 50°, 60°, and 70° from left to right. Gray dashed lines are the results of the direct permeability measurements of synthetic quartzite (left, wetting) and calcite (right, non-wetting), respectively (Wark and Watson 1998; Liang et al. 2001)

Full size image

In Fig. 4b, the variation in \(k\) at relatively low \(\varphi_{{{\text{mes}}}}\) (Fig. 4a) disappears against \(\varphi_{{{\text{con}}}}\). The results mostly fall onto the \(\varphi_{{{\text{con}}}} - k\) relationship of quartz analog (Wark and Watson 1998) or are slightly higher at \(\varphi_{{{\text{con}}}}\) > 0.03. The two subvolumes with the lowest \(\varphi_{{{\text{con}}}}\) (QW15-13#2, \(\varphi_{{{\text{con}}}}\) = 0.015 and QW15-11#2, \(\varphi_{{{\text{con}}}}\) = 0.008) were off to the higher side of the \(\varphi_{{{\text{con}}}} - k\) relationship of the quartz + H2O system.

3.4 Power-law exponent n

The power-law exponent \(n\) in this study was derived for both the \(\varphi_{{{\text{mes}}}}\)–\(k\) (Fig. 5a) and \(\varphi_{{{\text{con}}}}\)–\(k\) (Fig. 5b) relationships. For the wetting system, \(n\) was calculated by applying the least-squares method to all subvolumes where the permeability was obtained. As \(\varphi_{{{\text{mes}}}}\) values mostly coincide with \(\varphi_{{{\text{con}}}}\) in the wetting system, the derived \(n\) values were almost the same for \(\varphi_{{{\text{mes}}}}\) – \(k\) and \(\varphi_{{{\text{con}}}}\)–\(k\), at 3.0 and 3.1, respectively. This similarity is consistent with the previous \(\varphi\) – \(k\) relation of the quartz (wetting) and calcite (non-wetting) analogs (Wark and Watson 1998). For the synthetic calcite, \(\varphi\) was replaced with \(\varphi - 0.015\), where 0.015 is the pinch-off threshold below which no fluid interconnection is allowed. One subvolume of the non-wetting system (QW15-11#2) with significantly low connectivity (\(\varphi_{{{\text{con}}}} { }/\varphi_{{{\text{mes}}}}\) < 0.15) was excluded from the fitting. The obtained \(n\) value was 3.4 for \(\varphi_{{{\text{mes}}}}\)–\(k\). For \(\varphi_{{{\text{con}}}}\)–\(k\) in the non-wetting system, the data did not fit well into the single power-law relation of Eq. (1). Instead, it could be fit by using two lines switching at around \(\varphi_{{{\text{con}}}}\) = 0.04; \(n\) was determined to be 0.76 and 3.0 for the low \(\varphi_{{{\text{con}}}}\)(= 0.0075–0.028) and high \(\varphi_{{{\text{con}}}}\)(= 0.042–0.09), respectively. The obtained power-law exponent \(n\) for the low \(\varphi_{{{\text{con}}}}\) range was extremely low, indicating that permeability was insensitive to \(\varphi_{{{\text{con}}}}\), but rather controlled by other geometrical parameters such as aperture size.

Fig. 5
figure 5

Plot for the power-law exponent \(n\) of Eq. (1) for the a \(\varphi_{{{\text{mes}}}}\)–\(k\) and b \(\varphi_{{{\text{con}}}}\)–\(k\) relationships.The non-wetting system in (b) was fitted by using two lines with \(n\) = 0.76 and 3.0 for the low \(\varphi_{{{\text{con}}}}\)(= 0.0075–0.028) and high \(\varphi_{{{\text{con}}}}\)(= 0.042–0.090), respectively

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3.5 Fluid flow visualization

The results of flow velocity visualization showed that the velocity distribution was not homogeneous and was concentrated in some channels (Fig. 6). On average, these high-velocity channels were thicker than the other channels, suggesting that the fluid selectively flowed through wider pore spaces. Bottlenecked segments have a high velocity along the channels owing to the flow concentration. The volumetric flow rate \(q\) in a channel of radius \(R\) is given by integration of the velocity of the fluid \(v\), expressed as,

$$q = \mathop \smallint \limits_{0}^{R} 2\pi rvdr.$$
(8)

Therefore, fluid flow occurs effectively in a channel with a large \(R\), and \(v\) becomes highest at a point with a small \(r\) in the channel. A remarkable difference in the compared flow pattern was found in QW15-13#2 (Fig. 6d), where the flow lines consisted of significantly fewer channels, which is consistent with its low pore connectivity (Fig. 2, \(\varphi_{{{\text{con}}}}\) = 0.015).

Fig. 6
figure 6

Streamlines of the fluid velocity using open-source application ParaView for four representative subvolumes. a QW13-13#1, b QW14-21#2, c QW14-11#1, and d QW15-13#2. The high and low velocity channels are shown in red and blue, respectively. High velocity commonly occurs in thick channels, suggesting that fluids selectively flow in wider pore spaces. Subvolume QW15-13#2 in (d) shows a remarkable difference in flow patterns compared with those of the other three subvolumes, as it consists of significantly fewer channels

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The volumetric flow rate along the flow direction was examined for the selected subvolumes where the permeability was computed. First, computational grid cells were sliced into layers perpendicular to the applied pressure gradient. The computed local velocity of the fluid was stored at the cell faces (see Fig. 1 of Eichheimer et al. 2019). In the solid phase, the velocity was 0. Second, all velocities stored at the cell faces located in the ith cross section were summed to calculate the volumetric flow rate in the ith cross section of the channel. The summed-up volumetric flow rate in all cross section is equal to the total flow rate \(Q\):

$$\mathop \sum \limits_{i} q_{i} = Q.$$
(9)

To ignore computational artifacts in the flow calculation, cross-sectional channels that accounted for < 1% of \(Q\) were excluded. The values of \(q_{{\text{i}}} /Q\) were obtained from every 10th slice of grid cells perpendicular to the pressure gradient (i.e., the distance between slices is 5.1 µm). In Fig. 7, the values of \(q_{{\text{i}}} /Q\) are shown along the flow direction for the four subvolumes, whose streamlines were also visualized. The values of \(q_{{\text{i}}}\) were calculated for three orthogonal directions (x, y, and z), except for the z-direction of QW13-13#1, in which the connected pores were absent. In QW15-22#1, QW13-13#1, and QW13-11#1, each channel occupied < 30% of the total flow rate (i.e., \(q_{i} /Q\) < 0.3) from the inlet to the outlet. The values of \(q_{{\text{i}}} /Q\) along the flow direction were the lowest (~ 0.1) and most homogeneous in QW13-11#1. In contrast, \(q_{{\text{i}}} /Q\) fluctuated significantly along the flow direction in QW15-13#2, indicating flow focusing. The maximum \(q_{{\text{i}}} /Q\) reached 1 near the outlet of the flow in the x-, y-, and z-directions, showing that the flows were concentrated in one channel at these distances.

Fig. 7
figure 7

Ratio of volumetric flow rate to the total flow rate. Ratio of volumetric flow rate in the i-th cross section of a channel \(q_{{\text{i}}}\) to the total flow rate \(Q\) along the flow direction showing the flow velocity distribution. a QW15-22#1, b QW13-13#1, c QW13-11#1, and d QW15-13#2

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3.6 Number of channels

The permeability computation (Figs. 4 and 5), fluid flow visualization (Fig. 6), and flow rate investigation (Fig. 7) demonstrated that the heterogeneity and anisotropy resulting from non-uniform grain size, interfacial energy anisotropy of minerals affected fluid connectivity and permeability especially when the fluid fraction is low. To examine these effects in the present experiments, the number of channels in a slice of a subvolume \(N_{{{\text{ch}}}}\) was obtained from every 10 computational grid cells along the flow directions for the subvolumes with \(\varphi_{{{\text{con}}}}\) > 0. To ignore the computational artifacts in the flow calculation, only cross-sectional channels that accounted for > 1% of the total volumetric flow rate \(Q\) were counted as \(N_{{{\text{ch}}}}\). The mean counted \(N_{{{\text{ch}}}}\) of the each subvolume was normalized to the mean grain diameter of all the subvolumes (55 μm). Assuming a packing of hexagonal grains with a side length of 27.5 μm, the number of triple junctions in a slice with a side of 261 μm is approximately 70. Figure 8 shows that the normalized \(N_{{{\text{ch}}}}\) increased with increasing \(\varphi_{{{\text{con}}}}\) both in the wetting and non-wetting systems (from 4 to 28 and from 4 to 38, respectively).

Fig. 8
figure 8

Number of cross-sectional channels (\(N_{{{\text{ch}}}}\)) in each subvolume against connected porosity (\(\varphi_{{{\text{con}}}}\)). a Wetting and b non-wetting systems. Bars and solid circles represent the range and mean of each subvolume, respectively

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3.7 Flow concentration

We evaluated the degree of flow concentration in a single channel by calculating \(q_{{\text{i}}}^{{{\text{max}}}} /Q\), where \(q_{{\text{i}}}^{{{\text{max}}}}\) is the maximum volumetric flow rate \(q_{{\text{i}}}\) in a slice of computational grid cells. In a similar manner to \(N_{{{\text{ch}}}}\) counting, \(q_{{\text{i}}}^{{{\text{max}}}}\) was obtained from every 10th slice of the computational grid cells along flow directions with connected pores from inlet to outlet sides of the subvolume (\(\varphi_{{{\text{con}}}}\) > 0). The mean \(q_{{\text{i}}}^{{{\text{max}}}}\) for each subvolume is plotted in Fig. 9. In the wetting system, \(q_{{\text{i}}}^{{{\text{max}}}} /Q\) showed no or weak negative correlation with \(\varphi_{{{\text{con}}}}\), while \(q_{{\text{i}}}^{{{\text{max}}}} /Q\) decreased clearly from 0.70 to 0.17 with increasing \(\varphi_{{{\text{con}}}}\) in the non-wetting system. These results are consistent with the tendency of increasing \(N_{ch}\) with increasing \(\varphi_{{{\text{con}}}}\), especially for the non-wetting system. The two subvolumes of the non-wetting system with the lowest \(\varphi_{{{\text{con}}}}\) and highest \(q_{{\text{i}}}^{{{\text{max}}}} /Q\), QW15-11#2 and QW15-13#2, consisted of fewer channels than those in the other subvolumes, as shown for QW15-13#2 in Figs. 6d and 7d. These subvolumes typically represent concentrated fluid flows rather than pervasive flows. The other subvolumes of the non-wetting system showed a gradual shift to pervasive flow with increasing \(\varphi_{{{\text{con}}}}\).

Fig. 9
figure 9

Flow rate concentration to a single channel \(q_{{\text{i}}}^{{{\text{max}}}} /Q\) against connected porosity (\(\varphi_{{{\text{con}}}}\)) of each subvolume. Bars and solid circles represent the range and mean of each subvolume, respectively

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4 Discussion

4.1 Reliability of permeability computation

The reliability of the permeability values computed using the 3D thermomechanical code LaMEM has been examined in previous studies. The permeability of pipe flow computed using LaMEM has been confirmed to converge with the analytically calculated permeabilities of the Hagen–Poiseuille flow with increasing computational resolution (Eichheimer et al. 2019). Then, experimentally measured permeabilities of sintered glass bead media with a wide range of porosities were compared with the permeabilities computed by LaMEM using subvolumes of micro-computed tomographic images of the glass bead media (Eichheimer et al. 2020). The computed values agreed to an order of magnitude with the measured values, demonstrating that the numerical model works well not only for a simple pipe flow but also for a porous flow. As the quartzite pores in this study were finer than those of the glass bead media, the computed fluid velocity error was expected to be larger than that for the glass bead media. Nevertheless, the computed values fit well with the empirical permeability–porosity relationship obtained from laboratory measurements (Wark and Watson 1998). This empirically indicates the applicability of permeability computation using the LaMEM for porous media featuring small pore spaces.

4.2 Fluid channel radius

Yokoyama and Takeuchi (2009) demonstrated that the characteristic maximum pore throat radius strongly correlates with the permeability of porous volcanic pyroclasts because the flux of a single flow channel is limited by its pore throat radius (i.e., the minimum aperture size) along the flow, whereas the total flux through the sample is strongly controlled by the maximum throat radius of the available channels in the sample. In the present study, we used a simplified method to estimate the maximum pore throat radius characteristic of the subvolumes with the SRμCT slices by gradually eroding the pore pixel layers until the pore interconnection broke off. The number of erosion layers at which the interconnections between the two facing sides of subvolumes are finally lost (i.e.,\(, \varphi_{{{\text{con}}}}\) = 0), defined as \(R_{{{\text{min}}}}\), approximates the characteristic maximum pore throat radius of the fluid tubules. Figure 10 shows the relationship between the number of erosion layers and the connected pore volume divided by the total pore volume (\(V_{{{\text{con}}}} /V_{{{\text{total}}}}\)). For example, if the pore interconnection is lost after two layers were eroded (\(R_{{{\text{min}}}}\) = 2 voxels = 1.02 µm), the maximum pore throat area corresponds to 5–16 voxels. The results shown in Fig. 10 indicate that \(R_{{{\text{min}}}}\) is not always higher for subvolumes with larger connected porosities. For example, in the wetting system, QW14-21#1 has no interconnection after three layers of erosion even featuring a relatively high \(\varphi_{{{\text{con}}}}\) (0.067), while QW15-21#2 with a lower \(\varphi_{{{\text{con}}}}\) (0.057) still maintained the interconnection even after erosion of four layers. This may explain why QW15-21#2 had higher permeability than QW14-21#1. It is clear that the wetting system tends to have a larger \(R_{{{\text{min}}}}\) than that of the non-wetting system. The maximum \(R_{{{\text{min}}}}\) for most of the wetting system was 1.02–2.04 μm, whereas that for the non-wetting system was 0.51–1.02 μm.

Fig. 10
figure 10

Degree of pore interconnection. Degree of pore interconnection expressed as the connected pore volume divided by the total pore volume (\(V_{{{\text{con}}}} /V_{{{\text{total}}}}\)), for a wetting (XCO2 = 0) and b non-wetting (XCO2 = 0.28–0.44) systems, respectively. Only subvolumes having a pore connection from one side to the opposite side are shown. The horizontal axes are the number of erosion layers of voxels removed from the pores’ surface. Erosion was performed until the final pore lost interconnection. The minimum pore throat radius is given by the thickness corresponding to the maximum number of erosion layers (the right dead end of the symbols) plus 0–1 layer (0.51 µm)

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4.3 Small power-law exponent

The power-law exponent \(n\) obtained from the \(k\)–\(\varphi_{{{\text{con}}}}\) relationship of the non-wetting system was not determined to be a single value but could only be fit by two straight lines with slopes increasing from 0.76 to 3.0 at around \(\varphi_{{{\text{con}}}}\) = 0.04 (Fig. 5b). A small \(n\) value indicates that permeability is insensitive to changes in porosity. Generally, a power-law exponent \(n\) of < 2 reflects the failure of new interconnections with increasing porosity (Zhu and Hirth 2003), being consistent with the flow focusing in the small \(\varphi_{{{\text{con}}}}\) range. In a rock in which all channels are represented by N pipes with radius \(r\), the absolute permeability can be expressed as follows:

$$k_{{{\text{abs}}}} = N\frac{{\pi r^{4} }}{8A},$$
(10)

where \(A\) represents the cross-sectional area of the rock (Mavko et al. 2020), and the tortuosity is assumed to be 1 for simplicity. When the number of channels in the cross section is \(N\), it is related to \(\varphi\) as:

$$N\pi r^{2} = A\varphi .$$
(11)

Combining these equations and Eq. (1), \(N\) as a function of \(\varphi\) is solved as:

$$N\left( \varphi \right) = \frac{AC}{{8\pi d^{2} \varphi^{n - 2} }}.$$
(12)

An \(n <\) 2 is needed for \(N\) to increase with increasing porosity \(\varphi\). Therefore, \(n =\) 0.76 (< 2) for the non-wetting system is plausible. At \(\varphi\) ~ 0.05–0.06, where the variation in pore connectivity exists, permeability is controlled by pore connectivity rather than porosity.

4.4 Fluid isolation in the polycrystalline rocks and its implications for the fluid transport to deep earth

The results of this study depicted the processes in which interstitial pore fluids are left isolated in the experimental polycrystalline rocks. In the reference models of both wetting and non-wetting systems, all fluid tubules were uniformly narrowed with decreasing fluid fraction until the pinch-off threshold, where the pore interconnection was lost at every grain-edge tubule. Figures 2 and 3 demonstrate, however, that the grain-edge tubules were partly cut off at a fluid fraction near the threshold, while a limited number of fluid tubules across the subvolume were maintained. This feature in the experimental system suggests that the aperture size of the grain-edge tubules was not uniform. Relatively wider tubules maintained the pore interconnection, while the narrower tubules were pinched off. The normalized \(N_{{{\text{ch}}}}\) increased with increasing \(\varphi_{{{\text{con}}}}\) both in the wetting and non-wetting systems (Fig. 8) also supported a picture in which the fluids gradually wetted the new grain edges with increasing fluid fraction, even when the mean dihedral angle was < 60°, where all the grain edges were supposed to be wetted in the reference model. The maximum \(N_{{{\text{ch}}}}\), the number of channels in a slice of a subvolume, was smaller than that for homogeneous hexagonal packing (70) in all subvolumes even at the highest \(\varphi_{{{\text{con}}}}\). This supports the BSE image observation (Fig. 1) that a considerable proportion of triple junctions were dry or narrower than the resolution of SRμCT.

This study experimentally simulated grain growth and geological fluid segregation in deep-seated rocks, which include dehydrating slabs, rocks under prograde metamorphism, and partially molten upper mantle. The ranges of fluid fractions in these settings are often lower than a few percent (e.g., Ganzhorn et al. 2019). The results of our study confirm that the amount of unsegregated fluid left in practical rocks with heterogeneity and interfacial energy anisotropy is not determined solely by MEFF. Even in a wetting system where all interstitial fluids are expected to interconnect for gravitational segregation in an ideal isotropic case, a percolation threshold was observed for the total fluid fraction (0.037 in volume). This result suggests that fluids can be transported into the deep Earth’s interior in a wetting system, contrary to the prediction made with a homogeneous isotropic system. In the non-wetting system, the heterogeneous cutoff of the pore interconnection during fluid segregation resulted in an isolated fluid fraction of 0.048–0.062. This value indicates the amount of non-wetting fluid transported into the deep Earth’s interior in subduction zones, although the specific amount depends on many parameters such as lithology, fluid composition, and fluid separation processes.

5 Conclusions

To better understand the segregation mechanisms and retention fraction of aqueous fluids in deep-seated rocks, we applied a combined approach of 3D imaging using SRμCT and LaMEM computations of permeability and flow velocity distribution in the synthetic quartzite as a semi-analog of polycrystalline silicate rocks. In the wetting system (dihedral angle θ = 52°) with \(\varphi_{{{\text{mes}}}}\) > 0.037, we found that the calculated permeability–total porosity (\(\varphi_{{{\text{mes}}}}\)) relationship was mostly consistent with those obtained by laboratory measurements using quartzite (Wark and Watson 1998). Below this \(\varphi_{{{\text{mes}}}}\) (percolation threshold), fluid interconnection was lost, mainly owing to the interfacial energy anisotropy of quartzite, and the pore fluids were left isolated. In the non-wetting system (dihedral angle θ = 61–71°), grain-scale fluid pools were effectively connected to allow similar permeability to that in the wetting system at a fluid fraction > 0.062. Below this fluid fraction, the cutoff of heterogeneous fluid tubules formed the isolated fluid, and the percolation threshold was 0.048. Streamline visualization revealed that the fluid flow just above this threshold was focused into a few channels, establishing efficient channelized fluid pathways. These results demonstrated that the isolated fluids after the interconnection was lost in the wetting system can reach ~ 0.037, although the specific value depends on the anisotropy and heterogeneity of the system. In the non-wetting system, on the other hand, the fluid isolation process with decreasing fluid fraction was visualized for the first time. As a result, the fraction of retained fluid was presented to be 0.048, providing an estimate for the fraction of downdragged fluid in a subducting slab. This fraction was determined through grain growth of polycrystalline rocks, beyond the previous assessment solely based on the dihedral angle. The water circulation in the Earth is thus potentially influenced by factors such as lithology and fluid separation processes in subducting slabs.