Permeability evaluation for artificial single rock fracture according to geometric aperture variation using electrical resistivity

Hngok Lee, Jong-Won Lee, Te-Min Oh,*

a Deep Subsurface Research Center, Korea Institute of Geoscience and Mineral Resources, Daejeon, 34132, Republic of Korea

b Department of Civil and Environmental Engineering, Pusan National University, Busan, 46241, Republic of Korea

Keywords:Permeability Rock fracture Electrical resistivity Fracture geometry Long-term monitoring

ABSTRACT A convenient approach was proposed by which to evaluate and monitor the permeability of a rock fracture by verifying the quantitative correlation between the electrical resistivity and permeability at laboratory scale. For this purpose, an electrical resistivity measurement system was applied to the laboratory experiments using artificial cells with the shape of a single rock fracture.Sixty experiments were conducted using rock fractures according to the geometry, aperture sizes, wavelengths, and roughness amplitudes. The overall negative relationship between the normalized electrical resistivity values and the aperture sizes directly linked with the permeability,was well fitted by the power-law function with a large determination coefficient (≈0.86). The effects of wavelength and roughness amplitude of the rock fracture on the electrical resistivity were also analyzed.Results showed that the electrical resistivity was slightly increased with decreasing wavelength and increasing roughness amplitude. An empirical model for evaluating the permeability of a rock fracture was proposed based on the experimental data. In the field,if the electrical resistivity of pore groundwater could be measured in advance,this empirical model could be applied effectively for simple, quick monitoring of the fracture permeability. Although uncertainty may be associated with the permeability estimation due to the limited control parameters considered in this research, this electrical resistivity approach could be helpful to monitor the rock permeability in deep underground facilities such as those used for radioactive waste repositories or forms of energy storage.

1. Introduction

Understanding the flow behavior of deep subsurface groundwater is crucial for evaluating the safety of underground facilities such as those used for radioactive waste repositories, energy storage(i.e.hydrocarbon fluids),geological isolation of carbon dioxide,or an enhanced geothermal system (Rutqvist et al., 2002; Tsang,2012). Deep subsurface groundwater flow behavior is mainly dependent on the hydraulic properties of individual rock fractures(i.e. permeability and aperture) and the spatial distribution of the fracture network.This is because the rock mass is an impermeable medium with very low porosity and the fluid mainly moves through open spaces in fractures or joints.

1.1. Reviews of permeability of rock fractures

It is well known that the permeability of fractured rocks can be changed by various processes (thermal-hydraulic-mechanicalchemical, T-H-M-C) under deep geological conditions (Danko and Bahrami, 2012; Ghassemi, 2012). Several researchers have investigated the permeability variation induced by hydromechanical processes: deformation, growth of micro-sized fractures, and formation of excavation damage zones (EDZs) (Mahyari and Selvadurai, 1998; Rutqvist and Stephansson, 2003; Selvadurai,2004; Hudson et al., 2009; Liu et al., 2013). Moreover, the effect of thermal expansion and stress on change in the permeability has been considered in many investigations (Souley et al., 2001;Nguyen and Jing, 2007; Rutqvist et al., 2008; Chen et al., 2014;Najari and Selvadurai, 2014). They indicated that the formation or expansion of microcracks due to decay heat generated by radioactive waste can lead to increase of the fracture permeability,especially in fractured rock near radioactive waste repositories.Therefore,accurate estimation and monitoring of variations in the permeability of rock fractures is very important for securing the hydraulic stability and performance of deep subsurface facilities,such as the leakage/migration of radionuclides from a radioactive waste repository or the production efficiency of geothermal storage.

Generally,the permeability of rock fractures has been estimated using various hydrogeological techniques including pumping, slug tests, tracer tests, laboratory experiments using core specimens,and numerical simulation (Chandra et al., 2008; Khalil and Monterio Santos, 2009; Lu et al., 2019). However, these conventional methods usually require significant amounts of time and effort or large,complex equipment.Moreover,they can disturb the flow of groundwater systems due to artificial actions such as water injection or extraction.

To overcome the abovementioned drawbacks, much geophysical research has been performed on the correlation between hydrogeological parameters (i.e. permeability, hydraulic conductivity, and transmissivity) and electrical resistivity (Samouelian et al., 2005; Singh, 2005; Soupios et al., 2007; Doussan and Ruy,2009; Sikandar and Christen, 2012; Perdomo et al., 2014, 2018; Lu et al., 2019). Electrical resistivity technology was developed as an alternative field hydrogeological technique and has mainly been applied for particulate materials or alluvial aquifers. To monitor efficiently the variation of the permeability of rock aquifers around deep underground facilities, more effort (including systematic experiments and correlation analysis) is needed using evaluation of the quantitative connection between electrical resistivity and the permeability of rock fractures.

1.2. Basic concept of electrical resistivity

The basic mechanism of electrical resistivity monitoring used in this study is as follows.The electrical resistivity refers to a physical quantity of resistance that determines the amount of electrical current flow when a constant voltage is applied to a target object.In this concept of resistance,the water flow within a rock fracture can be considered as the same mechanism as for the flow of electrical current.Hence,the flow of electrical current in the rock fracture is not smooth if the water flow in the rock fracture is not smooth due to an obstacle.The deep underground is mainly composed of hard rock, and the fluid flow there is determined by the geometric characteristics (size, shape, and orientation) of the rock fractures.Fig. 1 shows conceptually the decrease in hydraulic conductivity and electrical conductivity (i.e. increase in electrical resistivity)with decreasing aperture size. At great depth, the electrical resistivity has a very high correlation with hydraulic properties (i.e.permeability, porosity, and diffusive tortuosity), and the electrical conductivity/resistivity of pore fluids with saturation levels(Campanella and Weemees, 1990; Gorman and Kelly, 1990;Boudreau,1996;Tiab and Donaldson,2004;Shen and Chen,2007).Kirkby et al. (2016) investigated the close relationship between electrical resistivity and permeability of rock fractures through random resistor network modeling.They found that small changes in aperture are associated with large changes in both the resistivity and permeability of the rock fault. Lu et al. (2019) suggested a simple method to estimate soil hydraulic conductivity by building the quantitative relationship between Hydraulic conductivity and electrical conductivity. Gernez et al. (2019) performed the comparison between hydraulic conductivity anisotropy and electrical resistivity anisotropy from tomography inverse modeling.

Compared to other forms of geophysical survey, electrical resistivity monitoring systems have considerable advantages. The electrode sensors installed in boreholes can monitor variation in the permeability to groundwater over the long term. In addition,this approach is economical because it requires small, simple equipment with low-cost electrode sensors.

Fig.1. Basic mechanism of electrical resistivity monitoring.

1.3. Research scope

In this study,a convenient method to evaluate and monitor the permeability of rock fractures was proposed by building a quantitative relationship between the electrical resistivity and permeability of rock fractures of various geometric qualities (aperture size,wavelength,and amplitude)at laboratory scale.To achieve this purpose,an electrical resistivity measurement system was applied for laboratory experiments using artificial cells with the shape of a single rock fracture. First, the electrical resistivity system was verified by analyzing changes in the resistance according to frequency. Then, a calibration process was performed to obtain the intrinsic electrical resistivity values. The electrical resistivity was measured according to three influencing factors related to the fracture geometry. Given these experimental results, empirical models were proposed for estimating the permeability of rock fractures based on correlation between the normalized electrical resistivity and fracture aperture. Finally, we discussed the feasibility of using the electrical resistivity system for long-term monitoring of hydraulic stability based on change in the permeability in a deep radioactive waste repository.

2. Experimental method

2.1. Artificial rock specimens

Artificial specimens,each with a single rock fracture with one of various shapes,were prepared using a plastic material and a threedimensional(3D)printer.Artificial specimens are composed of ABS(acrylonitrile-poly-butadiene-styrene)resin.This plastic material has non-conductive electrical properties, which have no effect on the electrical resistivity measurements. Pairs of the artificial rock specimens were designed for mounting inside a test cell. The fracture shape of an artificial specimen was determined by aperture size,joint wavelength,and joint roughness amplitude.Fig.2 shows the fracture shape of specimens according to the joint wavelength and roughness amplitude. Aperture sizes were adjusted by inserting rubber plates of a certain thickness between the specimen and the cell wall. Since this rubber plate is made of reinforced rubber rather than a soft material, it maintained its shape relatively well without being greatly deformed during the experiment.Moreover,the maintenance of a constant aperture was confirmed through measurement of separation distance between the fracture surface at the beginning and end of the experiment.The four aperture sizes selected ranged from 1 mm to 20 mm.It is common that permeable fracture distribution with large apertures may be rare in deep subsurface of the initial natural state. However, high permeable fractures with aperture sizes beyond the millimeter scale can exist due to the occurrence of EDZ by blasting or excavation during construction and operation of underground facilities such as a radioactive waste repository. Moreover, the increase in permeability, which is a factor that seriously threatens the stability of underground facilities,is caused by a large increase in the aperture of surrounding rock fracture due to thermal expansion or tectonic movement.

Fig. 2. Geometric fracture shapes of the artificial rock specimens.

The artificial fractures used in the experiment consisted of one smooth specimen(Set A)and fourteen specimens(Sets B-D)with differing wavelength and amplitude (Table 1). As the joint wavelength becomes smaller and the amplitude becomes larger, the fracture roughness (as an indicator that fluid flow has been interrupted)increases.Compared to realistic rock fracture geometry,the extremes of wavelength and amplitude in this study were used to evaluate clearly the effect of roughness on electrical resistivity.

Table 1 Fracture types of the artificial rock specimens.

2.2. Test setup

The multichannel electrical resistivity monitoring system consisted of three parts: (1) LCR (inductance (L), capacitance (C) and resistance (R)) meter devices capable of measuring electrical resistance up to a few third digits; (2) a DAQ (data acquisition)device for continuously collecting multichannel data; and (3) a controller for the LCR meter device and for storing the acquired data. This system was adjustable from a frequency of 0 Hz (direct current, DC) to a maximum of 2 MHz, and the input voltage range was adjustable from 0.1 V to 20 V.For multiple measurements,the maximum number (20) of channels was prepared to enable electrical resistivity monitoring.

The electrical resistivity system was applied to the test cell where the artificial rock specimens were inserted(Fig.3). The test cell,150 mm long,100 mm wide, and 100 mm high, was made of transparent polyvinyl chloride(PVC)material.The inside of the test cell was observable visually from all directions to ensure that the specimen was positioned correctly and accurately in the cell.Stainless steel electrodes with diameter of 5 mm were installed at intervals of 12.5 mm on both sides of the test cell. The electrode array was composed of 13 channels in a cross shape, and was connected to the electrical resistivity system.

2.3. Experimental procedures

Fig. 4 shows a conceptual diagram of the procedure of the electrical resistivity tests used to evaluate permeability. Electrical resistances were measured using the LCR meter by applying an alternating current (AC) voltage to the test section and by sensing the current flow at that time.In this experiment,the resistance was measured under the conditions of 1 V at 1 kHz. Multichannel data were collected by the DAQ device and stored in the controller.Prior to the main experiment,the shape factor was first estimated using a calibration process and then was used to calculate the inherent value of electrical resistivity. For this, the test cell was filled with different concentration levels of NaCl solution, and electrical resistivity and electrical resistance of the fluid were measured simultaneously.An increase in the electrolyte concentration in the pore fluid was achieved by stepwise increasing the amount of NaCl dissolved in distilled water.

Fig. 3. Complete experimental setup.

Fig. 4. Diagram of the electrical resistivity monitoring for rock specimen.

Fig.5. Correlation of pore fluid resistance with resistivity for the test cell(at 1 kHz and 1 V).

After that, the bulk electrical resistivities were measured according to various geometric qualities of the rock fractures including aperture size, wavelength, and amplitude. A total of 60 experimental trials were performed with the selected parameters,which included 4 different aperture sizes (1.5 mm ± 0.5 mm,3.2 mm ± 0.8 mm, 8 mm ± 1.6 mm, and 17.9 mm ± 0.8 mm), 4different wavelengths(10 mm,30 mm,75 mm,and 150 mm),and 5 different roughness amplitudes(5 mm,10 mm,15 mm,20 mm,and 30 mm). In this study, all other geometric values of fracture specimen (length, height and width), as well as the aperture-related geometry (aperture size, wavelength and amplitude), were measured. The fracture aperture(the void space between a pair of specimens) was completely saturated with an aqueous solution of NaCl mixed with water (36.3 Ω m of averaged resistivity with 8.5 Ω m of standard deviation).The time interval for data collection was 1 s, and the laboratory room temperature was maintained at(18 ± 1)°C to minimize the temperature effects on the resistivity.

2.4. Calibration process

The calibration process was needed to convert the electrical resistance to a resistivity value. The electrical resistance value can vary depending on the geometric shape of the test cell and of the electrode. Therefore, a calibration process provides the inherent electrical resistivity indicated by the pore water characteristics.

To observe the resistivity changes with fracture geometry characteristics(e.g.aperture size),the reference resistivity has to be determined based on the resistivity of the cell without artificial fracture.In the experiment,the fracture geometry plays an obstacle role in flowing of the electrical current of the calibrated cell.

After filling the test cell with NaCl solution, the electrical resistivity and resistance were simultaneously measured to obtain the calibration (shape) factor used in the calibration. The shape factor of the test cell(150 mm×100 mm×100 mm)and electrodes(5 mm in diameter) with the 150 mm distance between two electrodes was determined by the following electrical resistanceresistivity correlation equation:

where ρ is the electrical resistivity,α is the shape factor,and R is the electrical resistance.

Because the increase in electrical resistivity and resistance was linear, the relationship was measured by gradually increasing the amount of NaCl dissolved in distilled water. Fig. 5 shows the correlation between the electrical resistance and resistivity obtained from the calibration experiments. The shape factor of the fracture test cell was 0.0049 m under the measurement conditions of 1 V at 1 kHz in series mode.

2.5. Channel selection for reliable analysis of the experimental data

In these experiments, the electrical resistivity values were measured using 13 channels, but not all values were used to evaluate the permeability relationship. Even under the same conditions, the measured electrical resistivity values can be different according to the position of the electrodes because of boundary conditions. Therefore, a channel data value appropriate to reflect the permeability characteristics of the rock fractures had to be selected.

Fig. 6a shows the bulk resistivity values along the horizontal lines for the specimen of Set A (wavelength of 150 mm and amplitude of 0 mm).With the exception of Ch4,the other channels located on the horizontal lines had very high resistivity values,and this tendency was the same even though the fracture aperture size was changed. As shown in Fig. 4, because all horizontal channels(Ch8-Ch13) except Ch4 were located in front of the artificial rock specimen,they could not accurately reflect the electrical resistivity characteristics of the water pathway of the rock fractures.Thus,the data from the horizontal line channels were excluded when analyzing the results of this study.

Fig.6b shows the bulk resistivity values along the vertical lines for the specimen of Set A. Data from both ends of Ch1 and Ch7,which were affected by the boundary effect were excluded from this study. For Ch1, the resistivity was affected by air boundary effects, and for Ch7, the resistivity was affected by bottom (solid)boundary effects. Given the preliminary experiments, some resistivity data from Ch5 and Ch6 were also disturbed by bottom boundary effects; the resistivities show unstable results and somewhat higher values.Therefore,the resistivity values obtained from Ch2-Ch4 were determined and averaged to provide reliable analysis of the resistivity data in this study.

3. Results of bulk electrical resistivity changing with fracture geometry

In the field, the groundwater that saturates rock fractures may differ slightly in electrical conductivity initially depending on the local characteristics of the environment.The electrical conductivity(electrical resistivity) of the solution used for saturation has a significant effect on the total electrical resistivity measurement (Oh et al., 2014). Therefore, electrical resistivity values should be expressed as normalized electrical resistivity for accurate comparison between different cases.This concept of normalized resistivity,also called the formation factor, has been extensively studied to quantify the relationship between the hydraulic parameters(permeability,porosity,hydraulic conductivity,and transmissivity)and the electrical resistivity signals by several researchers (Tong and Tao, 2008; Bernabé et al., 2011; Choo et al., 2016; Shedid,2018; Kassab and Weller, 2019). The normalized electrical resistivity was obtained by dividing the measured bulk resistivity values at each experimental case by the inherent electrical resistivity value of initial pore fluid saturated with NaCl solution.

3.1. Effect of aperture size

Fig.7 shows the change of normalized electrical resistivity(bulk resistivity/pore fluid resistivity) according to the aperture sizes of the artificial rock specimens with smooth and rough surfaces.Fig. 7a is the result of the normalized electrical resistivity test for the smooth specimen(A1),whereas Fig.7b shows the result of the normalized electrical resistivity test for a specimen with constant roughness(wavelength of 10 mm and amplitude of 10 mm)in the fracture (D3).

In both cases, it was confirmed that the electrical resistivity rapidly increased when the aperture size decreased. As the aperture size decreases, the movement of fluid through a rock fracture becomes relatively difficult due to the narrow channel. Moreover,the relationship between electrical resistivity and aperture size shows a nonlinear tendency. This nonlinear correlation indicates that the electrical resistivity value of a low fracture size(e.g.<5 mm)may be more sensitive than that of a greater fracture size(e.g.>10 mm).Compared to the large fracture size,the ratio of the normalized resistivity value with change of the aperture size was observed to be as high as 9.5 times that at the smaller fracture size.

3.2. Effect of fracture wavelength

Fig. 8 shows the change in the normalized electrical resistivity according to the fracture wavelength at the aperture size of 19 mm and amplitude of 10 mm. The specimen with a wavelength of 150 mm was set as an experimental control with no fracture roughness. From the experimental results, it was confirmed that the fracture roughness and the normalized electrical resistivity value increase simultaneously as the fracture wavelength decreases.The normalized electrical resistivity value of the specimen with a wavelength of 10 mm increases by about 43% compared to that of the smooth specimen (amplitude of 0 mm).As the fracture wavelength decreases,the fracture roughness also increases,which means that the degree of fluid flow obstruction increased. Therefore, there was an overall negative correlation between the normalized electrical resistivity value and the fracture wavelength.

Fig.9 shows the electrical resistivity sensitivity due to change in the fracture wavelength according to the aperture size at amplitude of 10 mm.The degree of influence on the electrical resistivity by the fracture wavelength under the condition of low aperture size was higher than that of a large aperture size. This result indicated that the wavelength effect on the electrical resistivity may be relatively small for large apertures(i.e.20 mm).On the other hand,under the small aperture condition (i.e. 2 mm), the electrical resistivity increased very sensitively with decreasing fracture wavelength.This is because the effect of the rough surface on the fluid flow resistance for the entire flow through the channel depends on the aperture size of a rock fracture. As the size of flow channel (aperture) decreases, the portion of flow resistance due to roughness formed by the wavelength becomes larger,which leads to increase of the electrical resistivity in a similar context.

3.3. Effect of fracture amplitude

As the fracture amplitude increases at a certain wavelength value, the roughness of the rock fracture increases and the degree of disturbance to the fluid flow also increases.To evaluate the exact extent of the impact,the influence of fracture amplitude should be compared at the same wavelength and aperture size.Therefore,the experimental data satisfying the same conditions of fracture wavelength and aperture size were extracted to analyze the influence of amplitude on the electrical resistivity value.

Fig.10 shows the relationship between the normalized electrical resistivity and fracture amplitude according to wavelength at similar aperture size (3.2 mm ± 0.4 mm). There was an overall positive correlation between the normalized electrical resistivity value and fracture amplitude,although there was some variation of the normalized resistivity depending on the wavelength and aperture size of the artificial rock specimens. However, the degree of influence(sensitivity)of the fracture amplitude on the electrical resistivity became larger as the wavelength became shorter. In other words, both the decrease of wavelength and the increase of amplitude increase the fracture roughness,which also increases the resistance of the fluid flow and results in an increase of the electrical resistivity value.

Fig. 6. Bulk resistivity values according to the channel location for the specimens of Set A: (a) Horizontal and (b) vertical lines.

In the general steady-state where the groundwater flow velocity is slow,the effect of aperture size of a rock fracture on the electrical resistivity value is the greatest, while the effect of the roughness due to the variation of wavelength and amplitude on the resistivity value is relatively small.However,in extreme environments where the groundwater flow velocity is fast at great depth,the nonlinear or turbulent flow caused by roughness restricts the fluid flow.

4. Analysis of relationship between bulk electrical resistivity and fracture permeability

4.1. Relationship between the bulk electrical resistivity and aperture size

4.1.1. Quantitative correlation between the normalized electrical resistivity and aperture size

Fig. 9 clearly shows that the bulk resistivity value was most affected by the aperture size rather than the shape of the rock fracture.However,the electrical resistivity is also strongly affected by the aperture shape, especially for the smaller aperture size. In other words, the degree of influence of aperture shape such as wavelength on the electrical resistivity value varies depending on the aperture size of rock fractures. This study indicates that the most important factor affecting the permeability of rock fractures is the aperture size.

Fig.11 shows the relationship between the normalized electrical resistivity and aperture size with data gathered from all the experimental cases. There was an overall negative correlation between the normalized electrical resistivity and aperture size,although there were some scattered values of normalized resistivity due to the wavelength and amplitude effects. Although there were some discrepancies and uncertainties at small apertures, they did not deviate significantly from the overall trend of correlation.In future investigations,we will enhance the reliability of the correlation by expanding the application to small-scale experiments using the micro-sized fractures. As the aperture size decreases, the movement of fluid through rock fractures becomes relatively difficult due to the narrower channels, which increases the electrical resistivity.

The relationship fit well with a nonlinear curve of power-law function. A correlation (R2≈0.83) was shown in the given relationship (Fig. 11). The relationship tends to be more dispersed at small aperture size(i.e.<5 mm)compared to that at large aperture size (i.e. >15 mm).

Fig. 7. Normalized resistivity according to aperture size: (a) Smooth and (b) rough fractures.

Fig. 8. Relationship between the normalized resistivity and the wavelength at the same aperture size of 19 mm ± 0.5 mm and amplitude of 10 mm.

4.1.2. Comparative verification using actual granite fractures

In the previous experiment,the artificial specimens made using 3D printers were used to easily and precisely realize various geometries, instead of using actual rock fracture specimens.Although both granite mass and artificial specimens made of plastic material have almost no effect on the electrical resistivity measurements due to their non-conductive properties, a comparative verification experiment using actual granite specimens was needed to improve the reliability of the results of our study using artificial specimens, and to increase the applicability of this application in actual field work in a fractured-rock environment.

Fig.12 shows the bulk resistivity measurement using the actual granite fracture specimens. A steel ball with diameter of 20 mm connected with a cable was placed in the middle of the test cell,and was used as an electrode for the measurement of electrical resistivity.Aperture sizes were determined by controlling the open space between two specimens by inserting a non-conductive rubber plate withacertain thicknessbetween the rock specimens and the cellwall.The aperture sizes selected were of four different sizes ranging from 10 mm to 50 mm. However, because the roughness of the granite specimens could not easily be controlled,the degree of roughness in this case was fixed to a moderate level (joint roughness coefficient JRC=9.1±0.9)suchasisoften foundinrealinsitu rockenvironments.

Fig.13 shows the normalized electrical resistivity according to the aperture sizeincluding the resistivityof the actualgranite fractures.In the mixed data, although there were some differences in surface roughness between the artificial and granite specimens, the normalized electrical resistivity values according to aperture size were all placed in almost the same power-law relationship as follows:

Fig. 10. Relationship between the normalized resistivity and fracture amplitude according to wavelength at the same aperture size of 3.2 mm ± 0.4 mm.

Fig. 11. Relationship between the normalized resistivity and aperture size under different wavelength and amplitude conditions.

where ERnis the normalized electrical resistivity; bgis the geometrical aperture size(mm)of single rock fractures;and γ and β are the parameters to be fitted given the experimental data.Given all the data from Figs.11 and 13,the values of γ and β were shown to be 8.1-8.4 and approximately 0.54, respectively.

Consequently, the influence of the material properties of both the artificial and granite specimens on the electrical resistivity measurement was not significant. Thus we can infer that this empirical equation showing the quantitative relationship between the normalized electrical resistivity and aperture size obtained using artificial and granite specimens may be valid for actual rock environments. However, the generalization through many experimental cases using small-sized natural fractures is necessary in order to increase the reliability and feasibility of application to in situ rock.

4.2. Evaluation of approximate permeability according to geometrical aperture using bulk resistivity

This section discusses the functional relationship between the permeability of a rock fracture and the normalized electrical resistivity.As the depth of a rock aquifer becomes deeper,fresh rock develops and the number of rock fractures is reduced. In addition,the aperture size of individual rock fractures becomes smaller due to the increase of underground pressure. In this environment, the fluid flow is mostly through open rock fractures, and the permeability is also determined by the geometry of the rock fractures.The rock permeability can be evaluated through the approximation of the smooth plate model suggested by Zimmerman and Bodvarsson(1996) as follows:

where bhis the hydraulic aperture size(mm),F is the factor related to the roughness/nonlinear flow turbulence effect, and k is the permeability (mm2) through single rock fractures. At very small fractures on the micron scale, the size of hydraulic aperture is generally less than that of the geometrical aperture due to the F factors (Zimmerman and Bodvarsson,1996; Crawford et al., 2017).However,in this study,the approximation of permeability obtained using only the concept of geometrical aperture size with overwhelming weight compared to the roughness/turbulence factors was performed on a scale of millimeters or larger. Since the permeability flow test was not conducted directly in this study,the concept of the hydraulic aperture was not used in calculating the permeability.In other words,there is a limitation in evaluating the approximate permeability based only on the size of the geometrical aperture while excluding the effect of roughness and turbulence in large-scale rock fractures in this study.

Substituting all the experimental data from Fig. 13 in Eq. (3),permeability values of single rock fracture specimens were evaluated using the normalized electrical resistivity (Fig. 14). Permeability was nonlinearly decreased according to the normalized electrical resistivity value,which can be described by a power-law function with high correlation (R2≈0.86).

Higher values of the normalized electrical resistivity indicate smaller aperture size and more severe roughness of a rock fracture.Under actual field conditions, it is difficult to measure the permeability of natural fractures with a complicated structure having varying roughness, and the deviation of permeability values obtained under such conditions may be quite large according to the measuring techniques, operational proficiency, and field conditions. Although there was some deviation of permeability according to the normalized electrical resistivity,the relationship between the rock permeability and the normalized electrical resistivity was similar to the relationship between the aperture size and the normalized electrical resistivity(Fig.13), from the point of view of almost identical power-law function and determination coefficient(R2≈0.86).In other words,because the electrical resistivity value is much more sensitive to the change of aperture size than of the wavelength or amplitude,the relationship between the normalized electrical resistivity value and the fracture permeability predominantly determined by aperture size is reasonable and reliable.

Fig. 14. Relationship between the normalized resistivity and approximate permeability evaluated by geometrical aperture.

In conclusion, this quantitative relationship can be applied for the evaluation of permeability for rocks with a single fracture. In further study, we will generalize the correlation between the permeability and the bulk resistivity value by collecting more signal data under small-scale conditions, and to perform comparative verification through actual flow experiments.

Fig.12. The bulk resistivity measurement using the actual granite specimens: (a) Top and (b) front views of the test cell.

Fig.13. Relationship between the normalized resistivity and aperture size in the actual granite fractures, compared to the case of artificial specimens.

4.3. Field applicability of the electrical resistivity monitoring system

The hydraulic characteristics of a fractured-rock aquifer can vary due to the T-H-M-C reactions under deep subsurface conditions.The proposed mechanism of permeability variation in rock fractures was previously mentioned in Section 1.1.Due to the thermalhydraulic-mechanical (T-H-M) interaction, the change in permeability of fractured-rock environments can affect the hydraulic stability(i.e.leakage/migration of radionuclides from a radioactive waste repository)and the performance(i.e.production efficiency of geothermal storage) of deep subsurface facilities. Therefore, variation of the permeability of rock fractures near underground facilities need to be accurately monitored for a long time using simple and cost-effective methods.

Fig.15 shows the concept of field applicability of the electrical resistivity system to the long-term monitoring of hydraulic stability in a deep subsurface facility such as a radioactive waste repository.In the field, several monitoring and test boreholes were installed around the radioactive waste repository,which are penetrated and connected by a few unexpected permeable rock fractures.Although grouting is performed for hydraulic stability in the waste repository site, unexpected permeable fractures may exist. The electrodes were installed as close as possible to the rock fracture distributed within the boreholes.This is because the signal sensitivity becomes lower and consequently an accurate electrical resistivity value cannot be obtained because the location of electrodes in the borehole becomes farther away from the permeable fracture.Temperature change is also one of the important factors that can affect the signal sensitivity of electrical resistivity. Therefore, it is necessary to perform complementary monitoring and calibration of temperature together, in order to distinguish between the geometrical change of the rock fracture and the influence of other factors such as temperature change.When electrical current is sent through a pair of electrodes (a designated electrode and the other electrode), the electrical resistivity value can be measured.

If there is no signal(e.g. very high resistivity) indication of electrical resistivity between the two electrodes(i.e.A-B,A-E,and D-E),it means that the boreholes are not connected by the fracture.Fractured rock aquifers have very complex structure and high predictive uncertainties,thus the hydraulic connectivity is often not found,even though the boreholes are adjacent and the locations of the internal fractures are similar. If the electrical resistivity is measured at the electrodes of B-C or C-D and applied to the quantitative relationship between the permeability and the normalized electrical resistivity obtained in the experiments with temperature, the actual permeability of the connected rock fracture could be evaluated.

As described in the previous section,because of the decay heat generated by radioactive waste for a long period of time, the aperture size of the surrounding rock fractures can be increased by thermal cracking/stress or decreased by geochemical reactions between the groundwater and rock minerals. This can induce variation of the permeability of rock fractures near the radioactive waste repository, which is difficult to be evaluated precisely and continuously because the permeability change happens very slowly over a long time. In this regard, the electrical resistivity measurement system could offer a proactive and simple method to monitor more effectively the change in permeability. For a more accurate evaluation of permeability, it will be necessary to consider the effects of unexpected factors on the electrical resistivity. By placing the temperature sensor together with the resistivity electrode,it is possible to exclude the influence of temperature when the electrical resistivity value changes.Moreover,it is also needed to obtain utilization data in advance through water-rock interaction tests using local in situ rocks and groundwater, because the concentration of the fluid within the aperture may increase or decrease due to dissolution or precipitation.

If the electrical resistivity is initiated at the electrodes of B-C or C-D, these changes may be reflected in the electrical resistivity when the permeability varies due to change of the fracture aperture size. We could continuously evaluate the change of permeability and surrounding groundwater flow system in real time through the timing and degree of change of resistivity values.If it is judged that there is a major problem(threat)to the hydraulic stability due to a large change of permeability, more detailed inspection could be performed at the target section. This role of warning alarm could enhance the economics and performance efficiency of securing the stability of deep underground structures such as a radioactive waste repository.

Fig.15. Concept of field applicability of electrical resistivity system to the long-term monitoring of hydraulic stability in a deep subsurface facility.

In summary,the electrical resistivity monitoring method could be applicableinavarietyof fieldsituationsinwhich itisdifficultto obtain initial information in advance.This is because this method does not use an absolute value,but instead uses the degree and timing of signal change compared to the initial normalized electrical resistivity value.Moreover, this technique has the advantage of requiring simple equipment and only a small amount of time and effort, unlike the classical hydraulic tests that disturb steady-state groundwater flow systems through artificial actions such as water injection or extraction. Therefore, this method could be useful for evaluating and monitoring hydraulic characteristics that are difficult to be measured directly,especially in deep underground environments where access is very difficult. In addition, because the installed electrodes can be used almost permanently, the electrical resistivity system could be very suitable as a long-term monitoring method for evaluating the hydraulic stability of a radioactive waste repository.

5. Conclusions

The aim of this study was to provide a convenient method for estimating and monitoring the permeability of rock fractures by establishing a quantitative relationship between the bulk resistivity and permeability at laboratory scale. To achieve this purpose, the compact electrical resistivity measurement system was applied to artificial cells with various fracture shapes. For the experiments,three factors were considered: aperture size, wavelength, and roughness amplitude.Given the experimental results,an empirical model for estimating the permeability of a rock fracture was proposed by analyzing the relationship between the normalized electrical resistivity and fracture aperture size. Ultimately, this study is expected to be used to develop monitoring technology for improving the hydraulic stability of underground facilities at great depth. The main conclusions are as follows:

(1) The nonlinear negative relationships between the bulk resistivity values and the aperture sizes directly linked with the permeability value were well fitted by the power-law function with a large correlation coefficient (R2≈0.83).

(2) The bulk resistivity values were slightly increased with decreasing wavelength and increasing amplitude because the increase of roughness due to the above two factors caused increase of the resistance to fluid flow.

(3) The bulk resistivity value was most sensitive to the change of aperture size and the highest correlation was between the resistivity value and aperture size rather than the wavelength or amplitude.

(4) The empirical model for evaluating the permeability of a rock fracture was proposed by building the relationship between the permeability and the normalized electrical resistivity values. This model was also expressed as a power-law function with a large correlation coefficient (R2≈0.86).

(5) The electrical resistivity method could be used as a longterm monitoring system for evaluating the hydraulic stability of deep underground structures such as a real radioactive waste repository.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was supported by the National Research Foundation of Korea(KRF) grant funded by the Korea government (MSIT) (No.NRF-2019R1G1A1100517)and the Basic Research and Development Project of the Korea Institute of Geoscience and Mineral Resources(KIGAM), which was funded by the Ministry of Science and ICT,Korea.