Oda’s permeability tensor theory
A quartz vein is assumed to be a fracture that once served as a fluid pathway, and its shape is approximated as a disk with two parallel plates in this study. The fracture geometry shown in Fig. 1 was used to calculate the permeability tensor using Oda's permeability tensor (Oda 1985; Oda et al. 2002), where \(r\) is the radius of the fracture and \({\varvec{n}}\) is the normal unit vector of a fracture, which is given by \(n_{1} = \cos \alpha \sin \beta\), \({n}_{2}= \sin \alpha \sin \beta\), and \({n}_{3}= \cos \beta\), where \(\alpha\) is the dip direction and \(\beta\) is the dip angle.
The host rock is assumed to be impermeable when considering the permeability of a fractured rock mass. The water flow along the fracture is assumed such that water movement is idealized to be laminar flow between two parallel plates with an aperture \(t\), and its mean velocity is given by the cubic law. The apparent flow velocity \({\overline{v} }_{i}\) (Darcy’s velocity) under hydraulic gradient \({J}_{j}\), which is approximated by uniform distribution over a flow domain, is given as (Oda et al. 2002):
$$\begin{array}{c}{\bar{v} }_{i}={\frac{\lambda g}{12\nu }}\left\{\pi \rho \int\limits_{0}^{{t}_{m}} \int\limits_{0}^{{r}_{m}}\int\limits_{\Omega /2}^{ }{r}^{2}{t}^{3}\left({\delta }_{ij}{n}_{i}{n}_{j}\right)\times 2E\left({\varvec{n}},r,t\right){{\text{d}}}{\Omega} {{\text{d}}}r{{\text{d}}}t\right\}{J}_{j},\end{array}$$
(1)
where \(\lambda\) is the connectivity index (note that \(\lambda\) = 1 corresponds to the cubic law (e.g., Snow 1969), and the connectivity index satisfies the inequality 0 ≤ \(\lambda\) ≤ 1; \(g\) is the gravitational acceleration; \(\nu\) is the kinematic viscosity of water; \(\rho\) is the number of fractures per unit volume; \({\delta }_{ij}\) is the Kronecker delta; \({n}_{i}\) is the projected length of a unit vector \({\varvec{n}}\) on the reference axes \({x}_{i} (i=1, 2, 3)\); \(\Omega\) is the entire solid angle; and \(E\left({\varvec{n}},r,t\right)\) is a density function that describes the statistical distribution of \({\varvec{n}}\), \(r\), and \(t\). Here we compare Eq. (1) with Darcy's law, assuming that there is no head loss at the intersection of the fractures, and that \(\lambda\) is determined independently of \({\varvec{n}}\) and \(r\), such that the permeability tensor \({k}_{ij}\) can be expressed with the summation convention as follows:
$$\begin{array}{c}{k}_{ij}=\frac{\lambda }{12}\left({P}_{kk}{\delta }_{ij}{P}_{ij}\right),\end{array}$$
(2)
where \({P}_{ij}\) is a secondrank symmetric tensor that depends only on the geometric aspect of the fractures (Oda 1985) and is given by:
$$\begin{array}{c}{P}_{ij}=\pi \rho \int\limits_{0}^{{t}_{m}}\int\limits_{0}^{{r}_{m}}\int\limits_{\Omega /2}^{ }{r}^{2}{t}^{3}{n}_{i}{n}_{j}\times 2E\left({\varvec{n}},r,t\right){\rm d}\Omega {\rm d}r{\rm d}t.\end{array}$$
(3)
Determination of \({{\varvec{P}}}_{{\varvec{i}}{\varvec{j}}}\) from the observable geometric information
The parameters in \({P}_{ij}\) are related to the fracture (or vein) geometry. However, \({P}_{ij}\) is not easy to determine because \(\rho\), \(r\), and \({\varvec{n}}\) are defined in threedimensional space. The aperture \(t\) of the fracture (or vein) can be used to determine \({P}_{ij}\) in this study, as observed in the twodimensional section.
If \({\varvec{n}}\), \(r\), and \(t\) in \(E\left({\varvec{n}},r,t\right)\) are statistically independent, Eq. (3) can be rewritten as follows (note that the method for confirming this assumption is described in detail in the next section):
$$\begin{array}{c}{P}_{ij}=\pi \rho \langle {r}^{2}\rangle \langle {t}^{3}\rangle {N}_{ij},\end{array}$$
(4)
where:
$$\begin{array}{c}{N}_{ij}=\int\limits_{\Omega /2}^{ }{n}_{i}{n}_{j}2E\left({\varvec{n}}\right)\mathrm{d\Omega },\end{array}$$
(4a)
$$\begin{array}{c}\langle {r}^{2}\rangle =\int\limits_{0}^{{r}_{m}}{r}^{2}f\left(r\right)\mathrm{d}r,\end{array}$$
(4b)
$$\begin{array}{c}\langle {t}^{3}\rangle =\int\limits_{0}^{{t}_{m}}{t}^{3}g\left(t\right)\mathrm{d}t,\end{array}$$
(4c)
where \({N}_{ij}\) is the fabric tensor (Oda 1982, 1984), which is defined using the dip direction \(\alpha\) and dip angle \(\beta\) of the mineral veins. Therefore, we can determine \({N}_{ij}\) from field observations using clinometer measurements. Thus, the fabric tensor represents the stereonet of the strike and dip of a fracture. However, \(\rho\) and \(r\) are parameters that cannot be determined from the observations. A solution to the problem of determining \(\rho\) and \(r\) was proposed based on the stereological method (Oda 1985; Oda et al. 2002). Equation (4) can be rewritten as follows:
$$\begin{array}{c}{P}_{ij}={S}_{0}\langle {t}^{3}\rangle {N}_{ij},\end{array}$$
(5)
where:
$$\begin{array}{c}{S}_{0}=\pi \rho \langle {r}^{2}\rangle =\frac{m\left({\varvec{q}}\right)}{\langle \left{\varvec{n}}\cdot {\varvec{q}}\right\rangle },\end{array}$$
(5a)
$$\begin{array}{c}\langle \left{\varvec{n}}\cdot {\varvec{q}}\right\rangle ={\int }_{\Omega }^{ }\left{\varvec{n}}\cdot {\varvec{q}}\rightE\left({\varvec{n}}\right)\mathrm{d\Omega },\end{array}.$$
(5b)
where \({S}_{0}\) is the fracture density (m^{2}/m^{3}) defined by the surface area per unit volume of the analyzed fracture (or vein); \({\varvec{q}}\) is the unit vector of the scanning line; \(m\left({\varvec{q}}\right)\) is the intersection number per unit length; and \({\varvec{n}}\cdot {\varvec{q}}\) is the inner product between unit vectors \({\varvec{n}}\) and \({\varvec{q}}\).
In summary, the threedimensional fracture structure \({P}_{ij}\) can be determined by measuring the aperture \(t\), dip direction \(\alpha\), dip angle \(\beta\), and intersection number per unit length \(m\left({\varvec{q}}\right)\), from the outcrop. When estimating the paleopermeability, the following assumptions are made in this study with reference to those of Ioannou and Spooner (2007).

1.
The host rock is considered impermeable.

2.
The measured fracture apertures reflect the original fracture aperture before infilling and the effect of mineral precipitation and dissolution on permeability is not considered (Ioannou and Spooner 2007).

3.
All fractures are interconnected (Cox et al. 2001) as all the veins in this study are traces of fluid flow.

4.
The cubic law of fluid flow is assumed to hold true. It is known that if the fracture surfaces are in contact or have roughness, the cubic law should be modified using a coefficient, which is determined by fracture surface characteristic factors and/or the fractional contact area and smoothness (e.g., Witherspoon et al. 1980; Zimmerman et al. 1992). However, since the veins in this study are tensile cracks, with a smooth roughness (e.g., Fujii et al. 2007), no contact area, and form an extension stress field with high water pressure, the fracture aperture is considered to approximate that of a parallel plate.
Considering these assumptions, we interpreted the calculated paleopermeability to have the first order of magnitude at its maximum value. In addition, Oda et al. (2002) compared experimentally determined and theoretically predicted permeabilities of granite samples with microcracks. Although the permeability tensor in this study is obtained for fractures in mudstone, the results are still valid, as the permeability tensor \({k}_{ij}\) depends only on the geometry of cracks in the domain. Assuming that pores in the host rock are not the major pathways, the results obtained from this approach are therefore reliable.
Statistical analysis
\(E\left({\varvec{n}},r,t\right)\) can be written as \(E\left({\varvec{n}}\right)\text{, }f\left(r\right)\), and \(g(t)\) when \({\varvec{n}}\), \(r\), and \(t\) are statistically independent. A Chisquare test was conducted for each combination to confirm the independence of the measurements. The dip direction (measured using a clinometer) and the aperture and length of the quartz veins (measured in the field) were used in the Chisquare test. The Chisquare test is a onetailed test calculated at a significance level of 5%.
The fracture density \({S}_{0}\) is calculated in three dimensions using \(m\left({\varvec{q}}\right)\), which can be measured from a twodimensional cross section using stereology. The scan lines used to calculate \({S}_{0}\) and the number of quartz veins crossing the scan lines \(m\left({\varvec{q}}\right)\) must both be statistically homogeneous and represent the elementary volume of a given grid. Therefore, we confirmed the convergence of \({S}_{0}\) by setting the length of the scanning line, which was necessary to determine \({S}_{0}\), as a variable. Furthermore, the Oda’s permeability tensor determined by \({S}_{0}\) is the equivalent permeability of a grid that satisfies the representative elementary volume.
Case studies on thrust fault
The permeability tensor was estimated from the geometric information of quartz veins on the outcrop around the Nobeoka Thrust. Here, the estimated permeability tensor refers to the paleopermeability when the fracture acted as the pathway for fluid migration before the quartz veins formed. The Nobeoka Thrust is a ~ 90kmlong outofsequence thrust in Kyushu, Japan. Surface exposures of the Nobeoka Thrust are found in east–westoriented coastal outcrops in the northeastern part of Nobeoka City, Miyazaki Prefecture, southwest Japan. Murata (1996) reported that the strike and dip of the fault plane is approximately 240°/10°. Recent outcrop surveys and borehole investigations in this coastal region have provided a wealth of geological and geophysical data (e.g., Hamahashi et al. 2013; Kimura et al. 2013; Kitajima et al. 2017). In the study area (Fig. 2), the Eocene Kitagawa Group, which is composed of phyllitedominated terrigenous sediments, including sandstone layers, is exposed in the hanging wall, and the Eocene Hyuga Group, which comprises a shale matrix and a mélange of sandstone and basaltic blocks, is exposed in the footwall (Kondo et al. 2005). Kondo et al. (2005) calculated that the total displacement of the fault plane is 8.6–14.4 km by measuring the vitrinite reflectance (hanging wall, 320 °C; footwall, 250 °C). Mukoyoshi et al. (2009) calculated that the displacement is 6.7–11.6 km by measuring the illite crystallinity (hanging wall, 300 °C; footwall, 250 °C). These were determined considering the assumption that the geothermal gradient is 28–47 °C/km and the dip of the thrust is 10° (Kondo et al. 2005). The stress inversion of the tensile crackfilling vein shows that these fractures were opened by the slip of the Nobeoka Thrust during the earthquake (Otsubo et al. 2016). In addition to these tensile crackfilling veins, the veins observed in the outcrop of the damage zone are classified as faultfilling veins or postmélange veins (Hamahashi et al. 2015; Kondo et al. 2005; Otsubo et al. 2016; Yamaguchi et al. 2011).
In situ measurements were conducted for the following geometric information of the veins. Of geometric information such as dip direction \(\alpha\), dip angle \(\beta\), aperture \(t\), and length \(l\) were acquired for the quartz veins. Here, \(t\) was measured around the middle of the vein observed in the outcrop by placing the crack width ruler, with a lower limit of 0.05 mm (Fig. 3a). Since the veins observed in cross section were protruded by differential erosion, the true aperture could be directly measured for each vein (Fig. 3b). In addition, \(l\) was taken as the length of the straight vein that can be observed at the outcrop surface; quartz veins longer than 30 mm, for which strike and dip could be obtained accurately, were measured. For bent veins, they were counted by dividing them into two. Here, the tensile crackfilling vein is considered to have been generated during a single seismic event (Saishu et al. 2017); therefore, the veins with slicken lines on the surface that preserve the shear movement were excluded. Seven grids were then defined to obtain geometric information on quartz veins. Quartz veins that crossed an entire grid were not included, although quartz veins longer than 10 m were observed in coastal outcrops. The outcropscale quartz veins were treated as reference values, and thin sections of quartz veins were evaluated along with the type of veins, i.e., whether they were multiple or single, using microscopic observations (Fig. 4).
Seven grids were set up in an east–west direction across the fault [A–G: footwall; Fig. 2c]. The fabric tensor in a given grid was considered homogeneous if one edge was more than 10 times the average fracture length observed in cross sections within the region (Takemura and Oda 2004, 2005). Therefore, the length of one side of the grid was set as 2.5 m to fully satisfy this condition. Here, the maximum lengths of the veins were measured in advance for all grids, and the roughly estimated average length was used to determine the length of one side of the grid. Furthermore, when setting up the grids, we selected areas without uneven ground to prevent interference with the observation of the veins, and to ensure that these would not be submerged at high tide.
We measured the number of intersections of the scanning line \(m\left({\varvec{q}}\right)\) with quartz veins on the trace map to obtain \({S}_{0}\). Here, the quartz vein trace map was drawn using the images of each grid as a base image to measure \(m\left({\varvec{q}}\right)\), with both north–south and east–west scanning lines. Furthermore, \(m\left({\varvec{q}}\right)\) and \(\langle \left{\varvec{n}}\cdot {\varvec{q}}\right\rangle\) were calculated, and the coordinates of the endpoints of the quartz veins on the base trace map were determined using a digitizer. Here, \({S}_{0}\) was estimated using the orientations of both scanning lines. The error due to the placement of the grid area may be larger than that due to the scanning line because the angles between the scanning line and quartz veins are adjusted by \(\langle \left{\varvec{n}}\cdot {\varvec{q}}\right\rangle\) (Oda 1985). The results of the calculations in Takemura and Oda (2004) indicated that the error in the estimated fracture density is within 0.2 if the length of one side of the grid area is at least 10 times the average fracture length.