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Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13448-2
10.1016/j.heliyon.2024.e37417
e37417
Research Article
Role of micropores within minerals in retardation of mass transfer by matrix diffusion and sorption in granitic rock
Yuguchi Takashi takashi_yuguchi@kumamoto-u.ac.jp
a⁎
Sasao Eiji b
Hibara Ryoko c1
Murakami Hiroaki d
Ozaki Yusuke d
a Faculty of Science, Kumamoto University, 2-39-1 Kurokami, Chuo-ku, Kumamoto, 860-8555, Japan
b Japan Atomic Energy Agency, 959-31, Jorinji, Izumi-cho, Toki, Gifu, 509-5102, Japan
c Faculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata, 990-8560, Japan
d Japan Atomic Energy Agency, 432-2 Hokushin, Horonobe, Hokkaido, 098-3224, Japan
⁎ Corresponding author. takashi_yuguchi@kumamoto-u.ac.jp
1 Present address: Earthquake Research Institute, University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan.

05 9 2024
15 9 2024
05 9 2024
10 17 e3741712 10 2023
3 9 2024
3 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Understanding the mass transfer characteristics of matrix diffusion and sorption is important in the safety assessment of geological disposal of high-level radioactive waste in crystalline rock (granite) by contributing to radionuclide retardation through mass transfer within the rock body. We present a comparative discussion of the effective diffusion coefficient (De), porosity, and petrological data for rock samples collected from the Toki Granite in central Japan, to evaluate the role of micropores within minerals in retardation by matrix diffusion and sorption in granitic rocks. De was derived from the through-diffusion experiments using uranine, barium, strontium, and chloride ions as tracers. Petrological data consist of the fracture frequency, the extent of hydrothermal alteration in the minerals, the micropore volume in the minerals, and the three-dimensional modal mineralogy (mineral assemblage and ratio) for the target rock samples. The relationship between the De, porosity, and petrological data has the following implications: 1) Micropores in minerals related to the alteration act as ‘storage pores’ that contribute to retardation due to matrix diffusion and sorption; 2) Once the uranine, cations (Ba2+ and Rb+), and anion (Cl−) penetrate the micropores in the minerals through matrix diffusion, the cations are sorbed on the micropore surfaces, whereas the uranine and conservative chloride anion is trapped at the end of the micropore network, resulting in retardation; 3) Regions with a high fracture frequency are associated with not only active advection–dispersion through fractures, but also retardation due to matrix diffusion and sorption; 4) The grain-boundary pores between colorless minerals act as ‘transport pores’ owing to matrix diffusion, and the retardation within grain-boundary pores is less than that within micropores in minerals.

Keywords

Matrix diffusion and sorption
Micropore
Hydrothermal alteration
Through-diffusion experiment
Toki granite
==== Body
pmc1 Introduction

The feasibility of the geological disposal of high-level radioactive waste in crystalline rock (granite) was studied in various countries (e.g., Sweden, Finland, and Japan). The safety assessment for geological disposal requires understanding the properties of granitic host rocks, particularly their potential to retard radionuclide migration through rock-matrix diffusion. This understanding must be incorporated into mass-transfer modeling of groundwater transport of radionuclides from the disposal facility to the biosphere [1].

Fractures in granitic rocks act as the major pathway for groundwater flow and mass transfer, and advection and dispersion through fractures were reported in many previous studies [e.g., [[1], [2], [3], [4], [5], [6], [7], [8], [9]]. Advection and dispersion through fracture networks have been identified as the major mechanisms of mass transfer in fractured granitic rocks [4,5,7]. Since the permeability and heterogeneity of fractured granite in the island arc of Japan are higher than those in the stable continent of Europe [4,5,8], the characterization of mass transfer through fractures has been the focus in Japanese granitic rocks. For example, Onoe et al. (2021) [8] presents the modeling methodology for hydrogeological heterogeneity within the deep fractured granite in Japan. Petrological and thermochronological methods for evaluating the fracture distribution within a granitic body were developed based on the relationship between the fracture distribution and the difference in cooling behavior within the pluton [10,11]. Contrastingly, the important phenomena that retard mass transfer within granite are matrix diffusion and sorption [e.g., [[12], [13], [14]]. Generally, rock masses surrounding fractures consist of minerals with negatively charged surfaces. Cationic radionuclide species (e.g., uranium and cesium) diffuse towards negatively charged mineral surfaces and are sorbed onto the surfaces (cation excess effect: [15]). Earlier experimental and modeling studies were conducted on granitic rock samples to obtain data that contribute to the evaluation of matrix diffusion and sorption [e.g., [[16], [17], [18]]. Through-diffusion experiments for rock samples yielded effective tracer diffusion coefficients (m2/s), and mercury intrusion porosimetry provided the porosity (%) [e.g., [19,20].

Radionuclide retardation due to matrix diffusion and sorption is enhanced in granitic rocks that have undergone significant hydrothermal alteration [21,22]. Tracer experiments using uranine for samples from the Toki Granite indicated that matrix diffusion occurs through the altered parts within the plagioclase as a mass transfer pathway [23]. However, the diffusion of uranium from fracture surfaces to the rock matrix has also been reported in granitic rocks, showing no significant alteration [[24], [25], [26]]. Furthermore, nuclide migration along biotite cleavage has also been reported in weakly altered Grimsel granodiorites [19,27]. Thus, matrix diffusion and sorption have been reported for granitic rock samples with varying degrees of alteration. Consequently, unresolved issues exist regarding the following: 1) the relationship between alteration and matrix diffusion and sorption and 2) the understanding of the pathways that contribute to radionuclide retardation due to matrix diffusion and sorption.

Biotite and plagioclase are common component minerals of granitic rocks. Micropores developed within chloritized biotite and altered plagioclase were previously described from hydrothermally-altered rock samples in the Toki Granite [[28], [29], [30]]. These micropores have apertures of a few micrometers or less, and the permeability is so low that the pore water is essentially stagnant. Therefore, the mass transfer mechanism that carries radionuclides through the micropore network involves matrix diffusion. The working hypothesis of this study is that matrix diffusion and sorption through micropores in minerals associated with alteration retard radionuclides most effectively. To evaluate this working hypothesis, we present a comparative discussion of the effective diffusion coefficients (De) of rock samples from the Toki Granite, central Japan (Fig. 1), based on through-diffusion experiments and petrological data. The petrological data include 1) the fracture frequency (fracture number per unit interval), 2) the extent of hydrothermal alteration in the minerals, 3) the volume of the micropores in the minerals, and 4) the modal mineralogy (volume ratio of constituent minerals in the rock sample) for the target rock samples.Fig. 1 Map of Southwest Japan showing the location of Mizunami Underground Research Laboratory (MURL) (a). Schematic figure of the MURL shafts and borehole 06MI03 (b) based on Yuguchi et al. (2015) [31].

Fig. 1

2 Toki Granite and MURL

The Toki Granite in central Japan is a stock of approximately 14 × 12 km2 [32] and is one of the Late Cretaceous plutonic bodies of the Sanyo Belt in Southwest Japan (Fig. 1a; [33]). The Toki Granite is a zoned pluton comprising three lithofacies: muscovite-biotite granite, hornblende-biotite granite, and biotite granite [32,34]. A detailed description of its petrography and geochronology is provided by Yuguchi et al. (2011a, 2011b, 2019, 2020) [11,[35], [36], [37]]. In the Mizunami Underground Research Laboratory (MURL), two 500-m long vertical shafts (main and ventilation shafts) were excavated in the Toki Granite, central Japan (Fig. 1). The MURL is located on the sedimentary Mizunami Group (Fig. 1b), which unconformably overlies the Toki Granite (Fig. 1b; [38,39]). Two geostructural domains have been identified in the Toki Granite based on macroscopic fracture frequency [40]: an upper highly fractured domain (UHFD) and a lower sparsely fractured domain (LSFD). The boundary between the UHFD and LSFD is located at a depth of approximately 265 m. Borehole 06MI03 was drilled to a depth of 191 m in the ventilation shaft before the shaft was excavated below 191 m (Fig. 1), which is 336 m long with a diameter of 123 mm.

3 Materials and methods

3.1 Sample descriptions

Thirteen rock samples were collected from the borehole 06MI03 of the MURL in the Toki Granite, central Japan (Fig. 1), for porosity measurements and through-diffusion experiments. The samples have a mineral assemblage consisting of quartz, plagioclase, K-feldspar, biotite, hornblende, and muscovite, with accessory minerals including zircon, apatite, ilmenite, and magnetite, and secondary minerals such as chlorite, titanite, epidote, allanite, illite, and calcite. Subhedral to euhedral plagioclase grains are 1–20 mm across and exhibit varying degrees of alteration. Subhedral to anhedral biotite grains are 0.1–15 mm across and have variable degrees of chloritization. Micropores are mostly observed in the altered areas of plagioclase and biotite. Subhedral–equigranular quartz with crystals 0.5–25 mm across and subhedral K-feldspar 1–15 mm across are observed in the rock samples.

Based on borehole television (BTV) investigations, borehole 06MI03 has 733 macroscopic fractures (microscopic fractures are not included) with fracture frequencies ranging from 0 to 48 fractures per 5-m interval [41,42]. Fracture frequencies at the sample locations were extracted from the BTV data (Table 1). Samples A1–A3, A5–A7, and A11–A12 were collected from the UHFD, ranging from 3 fractures per 5 m (No. A12) to 48 fractures per 5 m (No. A1). Samples No. 3 and 6–9 were collected from the LSFD, ranging from 0 fractures per 5 m (No. 7) to 16 fractures per 5 m (No. 9) (Table 1).Table 1 Sample information of the 06MI03 borehole in the Toki granite for using the through diffusion experiment, with the alteration indicators and areal fractions of microvoids for biotite and for plagioclase, respectively, in the corresponding samples.

Table 1Sample No.	Distance from the borehole mouth (mabh)a	Elevation (masl)b	Macroscopic fracture frequency	Bt chloritization Alteration indicatorsc	Bt chloritization Microvoidsc	Plagioclase Alteration indicatorsh	Altered plagioclase Microvoidsh	Fracture domaini	Litho-faciesj	
(N/5m)	Range (mabh)a	Nd	meane	Stdf	N	meang	Std	N	meanf	Std	N	mean	Std	
No.A1	21.02	−11.12	48	17.67	–	22.67	22	0.27	0.08	22	0.07	0.02	17	0.07	0.12	17	0.04	0.01	UHFD	HBG	
No.A2	44.16	−34.26	31	42.67	–	47.67	20	0.67	0.17	20	0.10	0.03	15	0.10	0.23	15	0.04	0.02	UHFD	HBG	
No.A3	49.22	−39.32	40	47.67	–	52.67	20	0.51	0.28	20	0.07	0.03	15	0.07	0.19	15	0.04	0.01	UHFD	HBG	
No.A5	99.31	−89.41	35	97.67	–	102.67	20	0.74	0.16	20	0.11	0.02	16	0.11	0.22	16	0.05	0.02	UHFD	HBG	
No.A6	121.17	−111.27	28	117.67	–	122.67	20	0.66	0.17	20	0.10	0.03	15	0.10	0.20	15	0.05	0.01	UHFD	HBG	
No.A7	132.96	−123.06	11	132.67	–	137.67	22	0.22	0.11	22	0.05	0.02	15	0.05	0.07	15	0.03	0.01	UHFD	HBG	
No.A11	222.16	−212.26	18	217.67	–	222.67	20	0.54	0.28	20	0.08	0.03	15	0.08	0.27	15	0.05	0.02	UHFD	HBG	
No.A12	253.14	−243.24	3	252.67	–	257.67	20	0.19	0.08	20	0.03	0.01	15	0.03	0.15	15	0.03	0.02	UHFD	HBG	
No.3	289.00	−279.10	2	287.67	–	292.67	20	0.16	0.12	20	0.05	0.02	15	0.05	0.09	15	0.03	0.02	LSFD	HBG	
No.6	304.00	−294.10	2	302.67	–	307.67	21	0.18	0.05	21	0.04	0.01	15	0.04	0.07	15	0.03	0.01	LSFD	HBG	
No.7	309.00	−299.10	0	307.67	–	312.67	20	0.22	0.08	20	0.06	0.02	15	0.06	0.08	15	0.03	0.01	LSFD	HBG	
No.8	314.00	−304.10	8	312.67	–	317.67	21	0.13	0.06	21	0.03	0.01	15	0.03	0.20	15	0.03	0.02	LSFD	HBG	
No.9	319.00	−309.10	16	317.67	–	322.67	22	0.27	0.13	22	0.06	0.02	15	0.06	0.13	15	0.04	0.01	LSFD	HBG	
a mabh: meters along borehole.

b masl: meters above sea level.

c Alteration indicators and areal fraction of microvoids of chloritized biotite are extracted from Yuguchi et al. [29].

d Number of measured minerals.

e Mean value of alteration indicator.

f Standard deviation.

g Mean areal fraction of microvoids in the alteration minerals.

h Alteration indicators and areal fraction of microvoids of altered plagioclase are extracted from Yuguchi et al. [30].

i Two geostructural domains, i.e., an upper highly fractured domain (UHFD) and a lower sparsely fractured domain (LSFD), have been identified in the Toki granite based on macroscopic fracture frequency [40]. In borehole 06MI03, the boundary between the UHFD and LSFD is at a depth of approximately 265 m.

j Hornblende biotite granite (HBG).

Thirteen samples were collected from borehole 06MI03 for this study (Table 1), and the extent of alteration and volume of micropores in the altered minerals for the same samples was analyzed by Yuguchi et al. (2021; 2022) [29,30]. Yuguchi et al. (2021, 2022) [29,30] presented the petrographic alteration indicators of chloritized biotite and altered plagioclase to evaluate the extent of alteration in granite quantitively. The alteration indicators were obtained as the ratio between the alteration product domain and the original mineral domain via backscattered electron (BSE) image analysis. The alteration indicators in the range of 0–1 represent the extent of hydrothermal alteration, where a relatively weak alteration is indicated by values closer to zero and relatively strong alteration by values close to 1. The volume of the micropores in the altered minerals was estimated using the areal fraction of micropores in the grains [29,30]. The areal fractions of micropores were obtained as the ratio between the micropore domain and the original mineral domain via BSE image analysis of the altered minerals accompanied by an alteration indicator. The areal fraction of micropores in the mineral is represented in the range of 0–1, where relatively small and large volumes are indicated by values closer to 0 and 1, respectively. Twenty chloritized biotite grains and fifteen altered plagioclase grains were employed fundamentally in each sample, and the alteration indicator and the areal fraction of micropores for the target minerals are represented by the mean value and standard deviation. The detailed methods were described in Yuguchi et al. (2021; 2022) [29,30]. For the 13 samples, data for the alteration indicators and proportions of micropores in the chloritized biotite and the altered plagioclase were taken from Yuguchi et al. (2021; 2022) [29,30].

3.2 Measurement of porosity

The porosity of the rock samples was measured using the water re-saturation method described by Yamaguchi et al. (1997) [43] and Vilks and Miller (2007) [44]. The rock samples used for the porosity measurement and through-diffusion experiment were cylindrical, with a 25 mm diameter and 5 mm thickness. The rock samples were placed in beakers filled with pure water for saturation. The beakers were placed in a vacuum desiccator and degassed for approximately 48 h using a vacuum pump. The weights of the water-saturated samples (water-saturated submerged weight; Wu) were measured in a bucket filled with water (cf. Fig. 7 of Vilks and Miller, 2007 [44]). After weight measurement, excess water was gently wiped from the sample with damp, lint-free tissue, and the sample was placed on an electronic balance. The weights of the samples were recorded every 30 s until no further weight loss could be determined. The water-saturated surface-dry weight (Ws), which represents the condition at where the surface is dry and the sample remains completely saturated, is determined by the intersection of two approximately straight lines on the drying curve; this indicates that drying only occurred on the surface and pore spaces (cf. Fig. 8 of Vilks and Miller (2007) [44]). The dry weight (Wd) of the samples was determined after drying at 105 °C for approximately 71 h and leaving to rest at room temperature (20 °C) for 1 h. The sample porosity (ε) was calculated by following equation (1):(1) ε=(Ws−Wd)/(Ws−Wu)

The measurement error of the porosity (0.05 %) was referred from Yamaguchi et al. (1997) [43].

3.3 Through-diffusion experiment

The through-diffusion experiments allow the direct measurement of actual tracer diffusion; that is, effective diffusivities [45,46]. Through-diffusion experiments were performed according to previous experimental procedures [19] to obtain a De for understanding matrix diffusion. Uranine (fluorescein sodium salt), rubidium ion (Rb+), barium ion (Ba2+), and chloride ion (Cl−) were used as tracers. Rock samples were fixed in acrylic sample holders using epoxy resin. The samples were sandwiched between two reservoirs (inlet and outlet reservoirs) (Fig. 2). The tracer solutions (RbCl, BaCl2, and uranine) of approximately 100 mL were added to the high-concentration reservoir. The initial tracer concentrations in the inlet reservoirs of RbCl, BaCl2, and uranine were 1 mmol/L, 1 mmol/L, and 500 mg/L, respectively. The outlet reservoirs were filled with approximately 100 mL of ultrapure water to balance the water levels in both reservoirs. The reservoirs were then stored in a box at room temperature (20–25 °C) to prevent uranine degradation. The solution (5 mL) was collected from outlet reservoirs at 12 times (on days 1, 3, 10, 18, 27, 39, 52, 66, 83, 101, 123, and 151) for chemical analysis of uranine, rubidium, barium, and chloride ions. These solutions were analyzed by inductively coupled plasma mass spectrometry (ICP-MS; Agilent7700x, Agilent Technologies Japan, Ltd., Tokyo, Japan), ion chromatography (ICS-1500, Thermo Fisher Scientific K.K., Tokyo, Japan), and fluorometry (Trilogy Laboratory Fluorometer, Turner Designs, California, USA), at the Tono Geoscience Center, Japan Atomic Energy Agency. Ultrapure water (5 mL) was added to the low-concentration reservoirs to maintain the balance of the water levels of both reservoirs immediately after solution sampling. The dilution of tracer concentrations caused by sampling of the solution and the addition of ultrapure water was corrected. The results of the through-diffusion experiments were plotted on a diagram of the time versus tracer concentration change in the outlet reservoirs.Fig. 2 Schematic figure of the through-diffusion experiment. The samples were sandwiched between inlet and outlet reservoirs.

Fig. 2

3.4 Calculation of effective diffusion coefficient

Assuming a linear isotherm, the governing equation for one-dimensional diffusion in a reversibly sorbable porous material is described by Equation (2):(2) ∂C∂t=Deε+ρKd∂2C∂x2=Deα∂2C∂x2

where, C is the ionic concentration in the pore fluid, t is the time [s], De denotes the effective diffusion coefficient [m2/s], ε stands for the effective porosity [−], ρ is the dry density [kg/m3], Kd denotes the solid-liquid distribution coefficient [m3/kg], x stands for the distance [m], and α is the capacity factor [47]. The solid–liquid distribution coefficient (Kd) implies solid-phase concentration against liquid-phase concentration in a solid–liquid two-phase mixture. A high Kd value implies a high concentration from the solid phase to the liquid phase. The capacity factor, α, is the contribution to the retardation from the porosity and the sorption described by Equation (3) [48]:(3) α=ε+ρKd

The total mass conservation and mass conservation at the outlet reservoir were used to calculate the ionic concentrations at the inlet and outlet, respectively, as shown in Equations (4) and (5):(4) Cin,0Vin=CinVin+∫0LSαCdx+CoutVout

(5) Vout∂Cout∂t=−SDe∂C∂x,atx=L

where, Cin,0 denotes the initial ionic concentration in the inlet reservoir, Vin is the volume of the inlet reservoir, Cin stands for the ionic concentration in the inlet reservoir at a certain time, L denotes the length of the sample, S is the cross-sectional area of the sample, Cout stands for the ionic concentration in the outlet reservoir at a certain time, and Vout denotes the volume of the outlet reservoir. The solute transport in the rock sample described by Equation (2) was solved using the finite-difference method. The ionic concentration in the experiment was set to the cell located at the inlet reservoir side (x = 0), and zero concentration was assigned to the other cells as the initial condition of the simulation. The boundary condition of Equation (5) was assigned to calculate the ionic concentration in the outlet reservoir. The total mass conservation law of Equation (4) was solved to calculate the ionic concentration in the inlet reservoir. The repeated water sampling process dilutes the ionic concentration in the outlet reservoir. The effect was considered by modifying the ionic concentration in the outlet reservoir at the time steps after the water sampling as follows:(6) Cf=Ci(1−VsVout)

where, Ci and Cf are the concentration before and after the water sampling, respectively, and Vs is the volume of water sampling [49].

The least-squares method was used to estimate the De and capacity factor. Powell's method [50] was applied to minimize the least squares error between the simulated and measured data. The minimum values of the capacity factor were constrained to be larger than the measured porosity, i.e., Kd > 0, for the appropriate estimation of parameters because the Kd value is theoretically not negative as indicated by Equation (3).

3.5 Three-dimensional modal mineralogy for the rock samples

Modal mineralogy (volume % of the constituent mineral in the rock sample) is required for comparison with the De and porosity of the rock sample. Generally, the modal mineralogy of the rock sample is determined using the point-counting method with thin sections produced from the sample. The resulting percentage areal proportions of the minerals in the thin section is defined as the volume ratio of the minerals in the rock sample. However, it is challenging to evaluate whether the two-dimensional (2D) data obtained from thin sections are consistent with three-dimensional (3D) information from rock samples. This study presents the use of a three-dimensional (3D) modal mineralogy based on X-ray computed tomography (CT) modal analysis as a new method for quantitatively evaluating the modal mineralogy of target samples employed in the through-diffusion experiments.

X-ray CT is a powerful technique for investigating inner structures by producing 2D and 3D images of material regions. During the penetration of irradiated X-rays through a rock sample, the attenuation of the X-rays depends on the density difference of the minerals. High-density minerals cause a larger attenuation in X-rays and vice versa [51]. CT images with visualized cross-sectional slices are represented by grayscale values based on the differences in X-ray attenuation. CT images were obtained using the non-destructive CT scanner system (RF Naomi-CT 3D-M, Nagano, Japan) housed at Yamagata University (tube voltage of 60 kV; tube current of 4 mA; pixel number of 1216 × 1232 pixels; resolution capability of 100 μm) (Video S1). The 3D modal mineralogy of the CT images was determined by binarization and volumetric measurements using the image processing software Fiji/ImageJ. The target minerals for the modal determination were plagioclase, K-feldspar, quartz, and biotite. The ranges of the grayscale values were measured for recognizable plagioclase, K-feldspar, quartz, and biotite located on the disc surface and were defined as the internal standards of the target minerals (Video S1), that is, the grey levels of the ‘3D voxels’ corresponding to the X-ray density of target minerals within the rock sample were determined based on the greyscale of the target minerals that can be recognized wherever exposed on the surface of the sample. The pixel numbers of the target minerals, which differed in the ranges of the grayscale values, were counted, and the volume ratio of the target minerals in the rock sample was determined based on the difference in pixel counts (Video S2).

Supplementary data related to this article can be found online at https://doi.org/10.1016/j.heliyon.2024.e37417

The following are the Supplementary data related to this article:Video 1

X-ray CT images of sample A5. CT images with visualized cross-sectional slices are represented by grayscale values based on the differences in X-ray attenuation. The difference in grayscale range reflects the difference in constituent minerals in rock samples. The target minerals for the modal determination were plagioclase, K-feldspar, quartz, and biotite. The ranges of the grayscale values were measured for recognizable plagioclase, K-feldspar, quartz, and biotite located on the disc surface and were defined as the internal standards of the target minerals. That is, the grey levels of the 3D voxels corresponding to the X-ray density of target minerals within the rock sample were determined based on the greyscale of the target minerals that can be recognized wherever exposed on the surface of the sample. In the video, the establishment of a designated area to measure the gray values of plagioclase which is located on the sample surface.

Video 1

Video 2

Processed X-ray CT images of Sample A5, displaying plagioclase (Pl: yellow color), K-feldspar (Kfs: purple color), quartz (Qtz: red color), and biotite (Bt: green color). The CT image processing was conducted using the image processing software Fiji/ImageJ. The pixel numbers of the target minerals (plagioclase, K-feldspar, quartz, and biotite) and their volume ratio in the rock sample were determined based on the difference in pixel counts.

Video 2

4 Results

4.1 Porosity measurement

Porosities of the studied samples range from 0.20 to 1.06 % with mean values of 0.57 ± 0.06 % (Table 2). In the Toki Granite, Kuwabara et al. (2015) [52] reported porosity data ranging from 0.40 to 4.95 % with a mean value of 1.21 ± 0.07 % (N = 175) of rock samples in the MURL. The porosity data of rock samples in the borehole MIZ-1 and DH-2, which are adjacent to the MURL, were reported to be in the range of 0.64–3.41 % with mean value of 1.12 ± 0.40 % (N = 180; [52]) and the range of 0.67–10.20 % with mean value of 0.83 % (N = 60; [53]), respectively. Thus, the porosities of the studied samples are lower than those reported earlier. This discrepancy may be caused by different measurement methods; this study used the water immersion method, while earlier studies used the water saturation method, which utilizes dry and water-saturated weights.Table 2 Effective diffusion coefficient, porosity and mode for the samples collected from the 06MI03 borehole in the Toki granite.

Table 2Sample No.	Porosity (%)	Effective diffusion coefficient	Modal mineralogy (vol%)	
Uranine	Ba2+	Rb+	Cl-	Pld	Kfse	Qtzf	Btg	
Dea	Stdb	αc	De	Std	α	De	Std	α	De	Std	α	
No.A1	0.32	2.4E-13	1.7E-15	1.7E-02	3.3E-13	1.0E-15	7.7E-01	9.1E-13	2.6E-15	5.9E-01	7.8E-13	1.5E-14	3.2E-03	31.8	20.7	38.5	9.0	
No.A2	0.69	1.2E-13	4.8E-16	1.4E-02	1.2E-13	5.3E-16	2.9E-01	6.0E-13	4.1E-15	6.8E-01	5.6E-13	1.5E-14	6.9E-03	25.1	33.3	34.8	6.9	
No.A3	0.56	1.7E-13	3.1E-15	5.6E-03	1.5E-13	3.1E-15	4.4E-01	5.8E-13	1.8E-15	8.2E-01	5.8E-13	8.8E-15	5.6E-03	31.7	34.9	28.0	5.4	
No.A5	0.66	6.4E-13	2.1E-14	6.6E-03	2.2E-13	5.1E-15	6.6E-03	1.4E-12	2.6E-14	8.7E-01	1.4E-12	4.0E-14	6.6E-03	12.2	39.2	42.4	6.4	
No.A6	0.63	4.1E-14	4.4E-16	6.3E-03	2.9E-13	1.2E-15	1.0E+00	1.1E-12	1.4E-15	1.5E+00	3.0E-13	9.1E-15	6.3E-03	21.0	20.4	50.5	8.2	
No.A7	0.20	5.4E-14	1.0E-15	2.0E-03	1.1E-13	1.2E-15	2.2E-01	5.1E-13	1.2E-15	5.6E-01	4.1E-13	9.0E-15	2.0E-03	26.1	26.6	41.2	6.2	
No.A11	1.06	3.8E-13	9.7E-15	1.1E-02	1.2E-13	2.7E-15	1.2E-02	2.0E-12	1.3E-14	9.9E-01	1.2E-12	1.9E-14	1.1E-02	40.0	21.8	33.6	4.6	
No.A12	0.66	5.9E-13	1.9E-14	6.6E-03	4.0E-13	1.3E-14	6.6E-03	1.7E-12	2.8E-14	6.6E-03	1.2E-12	3.1E-14	6.6E-03	39.8	18.2	34.3	7.8	
No.3	0.29	1.3E-12	2.1E-14	8.7E-03	1.2E-12	3.4E-15	7.1E-01	2.4E-12	1.3E-14	7.5E-01	2.2E-12	1.9E-14	4.9E-03	22.7	33.1	39.9	4.5	
No.6	0.45	1.5E-13	8.4E-16	2.3E-02	5.2E-13	4.6E-15	3.6E-01	2.4E-12	5.5E-15	7.4E-01	1.1E-12	1.2E-14	4.5E-03	35.2	28.1	30.8	6.0	
No.7	0.64	6.0E-13	2.3E-14	6.4E-03	6.2E-13	1.4E-14	6.4E-03	2.0E-12	2.3E-14	6.4E-03	1.8E-12	4.3E-14	6.4E-03	23.0	34.9	36.6	5.6	
No.8	0.61	1.1E-13	2.4E-15	6.1E-03	5.9E-13	9.9E-16	8.2E-01	1.2E-12	1.1E-14	7.3E-01	5.1E-13	8.8E-15	6.1E-03	28.2	44.0	20.9	7.0	
No.9	0.62	1.8E-13	4.8E-15	6.2E-03	1.6E-13	9.0E-16	7.9E-02	2.8E-12	1.7E-14	3.8E-01	9.0E-13	2.3E-14	6.2E-03	24.9	29.9	40.9	4.4	
a Effective diffusion coefficient (m2/s).

b Standard deviation.

c Capacity factor.

d Plagioclase.

e K-feldspar.

f Quartz.

g Biotite.

4.2 Effective diffusion coefficient

Fig. 3 shows the normalized concentration of uranine, barium, strontium, and chloride ions in the outlet reservoir as a function of time (hours) in the through-diffusion experiment for the 13 samples (Fig. 3a–m) to determine the De. The De of each tracer was determined as follows (Table 2): uranine ion range from 4.2E-14 ± 4.4E-16 to 1.3E-12 ± 2.1E-14 m2/s, barium ion range from 1.1E-13 ± 1.2E-15 to 1.2E-12 ± 3.4E-15 m2/s, rubidium ion range from 5.2E-13 ± 1.2E-15 to 2.8E-12 ± 1.7E-14 m2/s, and chloride ion range from 3.0E-13 ± 9.2E-15 to 2.2E-12 ± 1.9E-14 m2/s. The earlier reported De were compiled in the JAEA Diffusion Database (DDB) [54]. The DDB stored De values of the tracers as follows: uranine (order of 10−14 m2/s), barium ion (order of 10−13 m2/s), rubidium ion (5.8E-12 and 1.7E-11 m2/s), and chloride ion (range of 1E-13–2E-10 m2/s). The measured De values obtained in this study are approximately in accordance with the database values. The capacity factor (α) of each tracer is as follows (Table 2): uranine (range from 2.0E-3 ± 1.1E-5 to 2.3E-2 ± 2.5E-4), barium ion (range from 6.4E-3 ± 3.1E-5 to 1.0E+00 ± 1.4E-3), rubidium ion (range from 6.4E-3 ± 1.9E-5 to 1.5E+00 ± 1.6E-3), and chloride ion (range from 2.0E-3 ± 1.4E-5 to 1.1E-2 ± 5.1E-10). In the experimental results, the De of uranine is relatively low, whereas that of the rubidium ion (cation) is high. The capacity factors of barium and rubidium ions (cations) are larger than those of uranine and chloride ion (anion). This result is consistent with cation excess and anion exclusion effects [cf. 19].Fig. 3 Normalized tracer concentration (Cout/Cin,0) of uranine, barium, strontium, and chloride ion in outlet reservoir as a function of time (hours) in the through-diffusion experiment to determine the effective diffusion coefficient (De): samples A1 (a), A2 (b), A3 (c), A5 (d), A6 (e), A7 (f), A11 (g), A12 (h), 3 (i), 6 (j), 7 (k), 8 (l), and 9 (m). Cin,0 denotes the initial ionic concentration in the inlet reservoir and Cout stands for the ionic concentration in the outlet reservoir at certain time.

Fig. 3

Fig. 4a–d shows the relationship between the De (uranine, barium, rubidium, and chloride ions) and the porosity of the samples. The De of barium ion has a poor positive relationship with porosity (R2 = 0.13), and the De of uranine, rubidium, and chloride ions show no clear relationship with porosity.Fig. 4 Relationship between the effective diffusion coefficient (De) (uranine (a), barium (b), rubidium (c), and chloride ions (d)) and porosity in 13 rock samples collected from the 06MI03 borehole. The measurement error of the porosity (0.05 %) was referred from Yamaguchi et al. (1997) [43].

Fig. 4

4.3 Three-dimensional modal mineralogy for the rock samples

Table 2 displays the 3D modal mineralogy of the 13 samples that were employed in the porosity measurements and through-diffusion experiments. Plagioclase ranges from 12.2 to 40.0 vol%. The ranges of K-feldspar are 18.2–39.2 vol %, and that of quartz are 20.9–50.5 vol %. Biotite ranges from 4.4 to 9.0 vol %.

5 Discussion

Fig. 5 shows the relationship between the De (uranine, barium, rubidium, and chloride ions) and the fracture frequency in the samples, indicating that smaller fracture frequencies result in larger De of uranine, barium ion (cation), rubidium ion (cation), and chloride ion (anion) in the rock samples. The De of uranine and chloride ion have a low correlation with fracture frequency relative to barium and rubidium ions (Fig. 5). Fig. 6 shows the relationship between the De and alteration indicators in chloritized biotite (A) and altered plagioclase (B), showing that a smaller alteration indicator results in larger De for uranine, barium, rubidium, and chloride ions. The De of uranine and chloride and rubidium ions have low correlations with the alteration indicator relative to barium ion (Fig. 6). Fig. 7 shows the relationship between the De and the areal fraction (volume) of micropores in the chloritized biotite (a-1 to a-4) and altered plagioclase (b-1 to b-4), generally demonstrating that a smaller volume of micropores in the mineral results in larger De for uranine, barium, rubidium, and chloride ions. The De of uranine and chloride and rubidium ions have low correlations with the volume of micropores relative to barium ion (Fig. 7).Fig. 5 Relationship between the effective diffusion coefficient (De) (uranine (a), barium (b), rubidium (c), and chloride ions (d)) and fracture frequency (N/5 m) in the sample location of the 06MI03 borehole.

Fig. 5

Fig. 6 Relationship between the effective diffusion coefficient (De) (uranine (a-1), barium (a-2), rubidium (a-3), and chloride ions (a-4)) and alteration indicator in the chloritized biotite (Bt), and that between the De (uranine (b-1), barium (b-2), rubidium (b-3), and chloride ions (b-4)) and alteration indicator in the altered plagioclase (Pl).

Fig. 6

Fig. 7 Relationship between the effective diffusion coefficient (De) (uranine (a-1), barium (a-2), rubidium (a-3), and chloride ions (a-4)) and areal fraction of micropores in the chloritized biotite (Bt), and that between the De (uranine (b-1), barium (b-2), rubidium (b-3), and chloride ions (b-4)) and areal fraction of micropores in the altered plagioclase (Pl).

Fig. 7

Fig. 8 Relationship between the effective diffusion coefficient (De) (uranine (a-1), barium (a-2), rubidium (a-3), and chloride ions (a-4)) and modal mineralogy of biotite (Bt), and that between the De (uranine (b-1), barium (b-2), rubidium (b-3), and chloride ions (b-4)) and modal mineralogy of colorless minerals (plagioclase (Pl), K-feldspar (Kfs), and quartz (Qtz)).

Fig. 8

Yuguchi et al. (2021; 2022) [29,30] described that positive correlations are present 1) between the alteration indicator and fracture frequency, 2) between the alteration indicator and areal fraction of micropores, and 3) between the areal fraction of micropores and fracture frequency, that is, samples with high fracture frequencies correspond to a high number of volume fractions of micropores and large alteration indicators. Micropores occurred at temperatures between 350 and 780 °C, and resulted in alteration progress [29,30]. Subsequent faulting and unloading developed micropores into macroscopic fractures [29,30]. Fig. 5, Fig. 6, Fig. 7 indicate the relationship among the De of the rock sample, fracture frequency, degree of alteration in minerals, and volume of micropores in the mineral. Therefore, the micropores in minerals related to biotite chloritization and plagioclase alteration act as pathways that contribute to retardation owing to matrix diffusion and sorption. Kita et al. (1989) [31] described two kinds of pores in rock: the ‘transport pore,’ which contributes to diffusion, and the ‘storage pore,’ which traps diffusing nuclides. Thus, the micropores in minerals act as storage pores.

Barium and rubidium act as cations (Ba2+ and Rb+) in the fluid and are easily adsorbed onto negatively charged micropore surfaces in minerals. Contrastingly, chloride ions (Cl−) are characterized by poor sorption on micropore surfaces in minerals owing to the anion exclusion effect. Because mineral surfaces are negatively charged, anions are excluded from some parts of the micropores [15]. Uranine has a larger particle size than the other tracers and is not positively charged. However, uranine and chloride ion have mass transfer characteristics similar to those of barium and rubidium ions (Fig. 5, Fig. 6, Fig. 7); micropores in minerals contribute to the retardation of not only barium and rubidium ions but also uranine and chloride ion in mass transfer. Once the uranine, cations and anions penetrate the micropores in the minerals through matrix diffusion, the cations are sorbed onto the micropore surfaces, and the uranine and conservative chloride anion are trapped at the end of the micropore network, resulting in the radionuclide retardation. Regions with large fracture frequencies are accompanied by not only active advection-dispersion through fractures [7] but also retardation due to matrix diffusion and sorption in mass transfer.

Fig. 8 shows the relationship between the De and the modal content of biotite, indicating that a smaller volume of biotite results in larger De for uranine (Figs. 8a–1), barium (Figs. 8a–2), rubidium (Figs. 8a–3), and chloride ions (Figs. 8a–4) in the rock samples. A smaller biotite volume indicates a smaller volume of micropores in the biotite. Thus, the relationship between the De and biotite volume (Figs. 8a–1 to 8a-4) is consistent with the argument that the micropores in the minerals act as storage pores, thereby contributing to retardation via matrix diffusion and sorption. Contrastingly, no relationship is observed between the plagioclase volume and De, while a relationship between the volume (areal fraction) of micropores in plagioclase and the De was confirmed (Figs. 7b–1 to 7b-4). This is attributed to the difference between the formation mechanism of micropores and the alterations in plagioclase and biotite. The hydrothermal alteration of biotite is predominantly constrained by dissolution–precipitation processes during the penetration of hydrothermal fluids through micropores [21,29,55]. Biotite cleavage corresponds to micropores in biotite [29] and occurs during the biotite crystallization stage. Therefore, the biotite micropore volume is associated with the biotite volume. The altered plagioclase contains ‘incipient micropores’ and ‘alteration micropores’ [30]. The incipient micropores, which occurred before plagioclase alteration, acted as a pathway for hydrothermal fluid within the plagioclase, resulting in the alteration progress, and hydrothermal alteration resulted in the production of new alteration micropores [28,30]. The formation of altered micropores leads to a poor relationship between the plagioclase volume and micropores (sum of incipient and altered micropores).

Fig. 8 shows the relationship between the De and the colorless mineral mode (sum of plagioclase, K-feldspar, and quartz), indicating that a larger volume of plagioclase, K-feldspar, and quartz results in larger De for uranine (Figs. 8b–1), barium (Figs. 8b–2), rubidium (Figs. 8b–3), and chloride ions (Figs. 8b–4) in the rock samples. There is no relationship between the volume of individual minerals (plagioclase, K-feldspar, and quartz) and the De. This suggests that the grain-boundary pores between colorless minerals act as pathways for mass transfer due to matrix diffusion. Möri et al. (2003) [15] described the occurrence of grain-boundary pores in granitic rocks. The formation of grain-boundary pores is derived from the expansion and contraction of quartz and feldspar during the uplift process of the rock body [15]. Grain-boundary pores are the most abundant pores in the Grimsel granodiorite [15]. The large volume of colorless minerals in the equigranular granite sample with a holocrystalline texture indicates that the grain-boundary pores between the colorless minerals provide a short pathway without meandering. Therefore, Figs. 8b-1–8b-4 indicates that the grain-boundary pores between colorless minerals function as ‘transport pores’ [31] due to matrix diffusion. It also shows that radionuclide retardation through grain-boundary pores is less effective compared to micropores in minerals. In summary, in the mass transfer of uranine, barium, rubidium, and chloride ions in granitic rock, grain-boundary pores act as transport pores, while the micropores within minerals act as storage pores. There is a weak positive relationship between De of barium and porosity (Fig. 4b). The variations in De of barium ions with the petrological parameters (fracture frequency: Fig. 5b, alteration indicator: Figs. 6a–2 and b-2, areal fraction of micropores: Figs. 7a–2 and b-2) are larger than those of uranine (Fig. 5a, 6a-1, 6b-1, 7a-1, and 7b-1), rubidium (Fig. 5c, 6a-3, 6b-3, 7a-3, and 7b-3), and chloride ions (Fig. 5d, 6a-4, 6b-4, 7a-4, and 7b-4). Therefore, the micropores in the minerals effectively affect the retardation of barium ion (i.e., divalent cation).

Matrix diffusion and sorption of radionuclides are taken up at 1) micropores in the minerals, 2) grain-boundary pores, and 3) trans-granular microfractures in the rock body based on numerous natural analog studies [e.g., [15, [56], [57], [58]]. Trans-granular microfractures, which cross grain boundaries between adjacent mineral grains, can provide a more extensive connected microfracture network [58]. Our data show that the micropores within minerals effectively act as a radionuclide retardant. However, there are low correlation coefficients between De and petrological data (Fig. 5, Fig. 6, Fig. 7, Fig. 8). This suggests a complex interaction of matrix diffusion and sorption in various kinds of porosities (micropores in the minerals, grain-boundary pores, and trans-granular microfractures) within the rock sample.

6 Conclusions

In the safety assessment of the geological disposal of high-level radioactive waste in crystalline rock (granite), it is important to understand the mass transfer characteristics with matrix diffusion and sorption of granitic rock because matrix diffusion and sorption yield radionuclide retardation in mass transfer. Unresolved issues exist regarding 1) the relationship between alteration and matrix diffusion and sorption, and 2) pathways that contribute to radionuclide retardation due to matrix diffusion and sorption. Micropores in altered (chloritized) biotite and altered plagioclase occur in the Toki Granite in central Japan. This study established the working hypothesis that matrix diffusion and sorption through micropores in minerals associated with alteration provide the most effective radionuclide retardation. To evaluate this hypothesis, this study presents a comparative discussion of the De, porosities, and petrological data of granitic rock samples. De values were derived from through-diffusion experiments using uranine, barium, strontium, and chloride ions as tracers. The petrological data consist of 1) the fracture frequency, 2) the extent of hydrothermal alteration in the altered biotite and plagioclase, 3) the volume of micropores in the altered biotite and plagioclase, and 4) the modal mineralogy of the target rock samples.

The relationship between the De and volume (areal fraction) of micropores in the minerals demonstrates that a smaller volume of micropores in the mineral results in larger De. Micropores in minerals related to alteration act as ‘storage pores’ that contribute to retardation owing to matrix diffusion and sorption. Once the uranine, cations (barium and rubidium ions), and anion (chloride ion) penetrate the micropores within the minerals through matrix diffusion, the cations are sorbed on the micropore surfaces, and the uranine and conservative chloride anions become trapped at the end of the micropore network, resulting in radionuclide retardation. In particular, the micropores in the minerals effectively affect the retardation of barium ion (i.e., divalent cation). Regions with large fracture frequencies are accompanied by not only active advection–dispersion through fractures but also radionuclide retardation due to matrix diffusion and sorption.

The relationship between the De of uranine, barium, rubidium, and chloride ions and the colorless mineral mode (the sum of plagioclase, K-feldspar, and quartz) indicates that a larger volume of colorless minerals results in larger De. The large volume of colorless minerals in the equigranular granite sample with a holocrystalline texture indicates that the grain-boundary pores between the colorless minerals provide a short pathway without meandering. Therefore, the grain-boundary pores between colorless minerals act as ‘transport pores’ owing to matrix diffusion, and the radionuclide retardation through grain-boundary pores is less than that through micropores in minerals.

Data availability

Data associated with this study were not deposited in a public repository. All data relevant to this study were published in the article. If additional data are required beyond those presented, we are happy to provide them upon request.

CRediT authorship contribution statement

Takashi Yuguchi: Writing – original draft, Validation, Methodology, Investigation, Funding acquisition, Data curation. Eiji Sasao: Writing – review & editing, Validation, Investigation, Funding acquisition, Data curation. Ryoko Hibara: Validation, Investigation. Hiroaki Murakami: Validation, Investigation, Data curation. Yusuke Ozaki: Validation, Investigation, Data curation.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:Takashi Yuguchi reports financial support was provided by 10.13039/501100001691 Japan Society for the Promotion of Science . Eiji Sasao reports financial support was provided by 10.13039/501100001691 Japan Society for the Promotion of Science . If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We would like to acknowledge the constructive reviews by three anonymous reviewers and Dr. Daniela Ducci (Associated editor – Earth Science), which greatly improved the quality of the manuscript. We also would like to thank Editage (www.editage.jp) for their English language editing services. This work was supported by the 10.13039/501100001691 Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (B) [grant number 21H01865 ] to T.Y. and the 10.13039/501100001691 JSPS Grant-in-Aid for Scientific Research (C) [grant number 20K05410 ] to E.S.
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