Introduction

Most model approaches for groundwater flow and transport have been developed for porous systems. Flow in fractured aquifers is considerably more difficult to describe due to the uneven distribution and discrete nature of its pathways. Such aquifers are, however, common in many regions worldwide and can play an important role for water supply (or hydrocarbon production) but they can also be affected by contamination (Lee and Farmer 1993; Berkowitz 2002; Krásny and Sharp 2007; Singhal and Gupta 2010; Adler et al. 2013; Sharp 2014). In both cases it is important to study the influence of the faults on the flow rates and velocities within the aquifer.

Several authors have tried to derive basic laws describing flow through fractures (e.g. Lomidze 1951; see also the reviews by e.g. Zimmerman and Bodvarsson 1996; Zimmerman and Yeo 2000; Berkowitz 2002; He et al. 2021). The most commonly used approach to describe laminar flow in a fractured aquifer is the cubic law in the form proposed by Snow (1969).

$$Q = \frac{1}{{12}} \cdot N \cdot \left[ {\frac{{g \cdot \rho }}{\mu }} \right] \cdot {b^3} \cdot w \cdot I$$
(1)

with

Q = flow rate [L³/T], μ = dynamic viscosity [M/(L·T)], g = acceleration of gravity [L/T²], ρ = fluid density [M/L³], b = fracture aperture (opening width) [L], w = length of fracture (normal to direction of flow) [L], N = number of equal fractures, and I = hydraulic gradient (see Fig. 1a for details).

The law takes its name from the cubic dependency of flow from the aperture, with the fracture planes being parallel. Even a slight change in aperture thus strongly influences flow and permeability of the system. The basic underlying assumption is that the fault planes are parallel and smooth, conditions rarely met in nature. Experiments by Witherspoon et al. (1980) showed the need for a correction term f to address deviations from the idealized concept. From their experiments with granite, basalt and marble, they arrived at values ranging from f = 1.04 to 1.65.

$$Q = \frac{1}{{12}} \cdot N \cdot \left[ {\frac{{g \cdot \rho }}{\mu }} \right] \cdot \left[ {\frac{1}{f} \cdot {b^3}} \right] \cdot w \cdot I$$
(2)

with

f = fracture surface characteristic factor [-].

An important contributor to this correction factor is the surface roughness (the term asperity is sometimes also used). Such protrusions from a basal surface can affect the single-most important parameter, aperture width. A distinction is thus often made between the mechanical aperture and the conducting (or hydraulically effective) aperture (Brown 1987; Renshaw 1995). Due to roughness, the conducting aperture is smaller than the mechanical aperture (Fig. 1b).

Despite or even because of its relatively simple form, the cubic law suffers from several problems. The first is that it addresses both macroscopic and microscopic scales at the same time. The number of fractures and their length can only be assessed on a macroscopic scale (e.g. an outcrop or a tunnel) but can be scale-dependent, raising the issue of the representative elementary volume (REV) of sampling (e.g. Piña et al. 2019). On the microscopic scale, parameters of the (individual) fracture are considered, i.e. its aperture and, indirectly through the factor f, its roughness. A parameter not explicitly included in the cubic law is the connectivity of fractures, i.e. the influence of a fracture network. Due to the aforementioned limits, many authors have tried to develop improved versions of the cubic law (e.g. Gee and Gracie (2022), see also review by He et al. 2021). For the sake of simplicity, effects of double porosity, i.e. the interaction of faults and intermittent porous blocks, e.g. in a sandstone, will also be ignored here.

Fig. 1
figure 1

(a) schematic sketch of groundwater (blue) flow in a parallel plane fracture, matrix rock in grey (b) with non-parallel fracture walls, (c) role of roughness (green). Mechanical (b1) and hydraulically active aperture (b2), potential transition from laminar to non-linear flow. Blue arrows indicate flow direction and velocity profiles

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The number of fractures, their length and to some degree their aperture can be measured in the field, which is often done along a profile section. Drill cores and camera inspections of open boreholes are also used. However, effects of weathering and decompression in surface outcrops and cores may cause deviations from values representative for the subsurface, especially for the aperture (Singhal and Gupta 2010). Measuring the roughness is much more challenging, as it needs to be done in the laboratory and special sophisticated equipment is needed. Different standardized measuring methods developed mostly for material science applications are available (see methods section below, also ISO 25178-2:2021 (2021). Measurement techniques comprise both contact and non-contact methods, the former mostly mechanical feeler techniques, the latter mostly optical. The optical methods rely on different measurement principles and have differing spatial resolutions and measurable sample sizes. It is thus interesting to compare these methods and to see whether they yield comparable results.

Locally, roughness can cause more tortuous flow paths and eddies, especially at higher (non-Dracy) flow velocities, which can also affect transport processes (Brown 1987; Qian et al. 2005; Zou et al. 2015; Zhou et al. 2016; Briggs et al. 2017; Ni et al. 2018; Dou et al. 2019; Stoll et al. 2019; Li et al. 2020; Ma et al. 2023). Especially sharp discontinuities in surface profiles can result in complex flow patterns (Dijk et al. 1999; Dijk and Berkowitz 1999). Ignoring fracture roughness can introduce errors into model calculations based on the cubic law (Plouraboué et al. 2000; Méheust and Schmittbuhl 2003). According to Lee and Farmer (1993), the cubic proportionality between flow rate and aperture decreases to a square relationship for rough fractures. For non-linear laminar flow (non-Darcy or Forchheimer flow) the exponent decreases even further to 1.2 < n < 2.

Mechanical forces acting in the subsurface can also affect the aperture, e.g. load compression and shear forces (e.g. Witherspoon et al. 1980; Yeo et al. 1998; Lee and Cho 2002). Roughness can play a role here too, since the protrusions can prevent the full closure of a fracture under compressive stress due to the mismatch of the surfaces, the latter e.g. enhanced by shear displacement (Witherspoon et al. 1980; Cardona et al. 2001; Zhou et al. 2016; Al-Fahimi et al. 2018).

The influence of fracture roughness on groundwater flow is a widely discussed topic in hydrogeological literature, a SCOPUS search (February 20, 2024) for the search terms “fracture flow” and “roughness” yielded 222 studies, of which the majority have been published in the last ten years. Many studies, however rely on numerical modeling and/or artificially created roughness fields (e.g. Brown 1987; Brush and Thomson 2003; Li et al. 2020), including 3D-printed models (e.g. Zhang et al. 2022). If natural samples are used, the fractures are often freshly created by splitting (e.g. Rost et al. 2018).

Actual measurements of natural fracture roughness are comparably rare and usually rely on a single method. Many studies investigate freshly split samples, thus the bulk rock. Here, on the other hand, we study a natural, weathered surface. The aim of the present study is, therefore two-fold, (a) to compare different methods for measuring surface roughness operating on different scales, and (b) the effects of a natural, weathered joint surface (Bunter Sandstone from Southern Germany).

Methods

Sample material

The sample material was collected in surface outcrops (quarries) of the Triassic Lower to Middle Bunter Sandstone (Badischer Bausandstein) in the northern Black Forest, near the city of Rastatt, SW Germany. For the analysis of the fracture surfaces, several samples were chipped off the outcrop using hammer and chisel.

Optical analysis

The fracture surface was inspected by Scanning Electron Microscopy (SEM) with energy-dispersive X-ray spectroscopy (EDX) using a FEI Sirion D1625, operating in low-vacuum mode (0.6 mbar). It is equipped with an EDX (Energy Dispersive X-ray Spectroscopy)-system of the type Genesis 4000 by EDAX for chemical analysis. Additionally, petrographic thin sections were studied by a Zeiss Axioplan light microscope.

Chemical analysis

The chemical composition of powdered samples was determined by wavelength dispersive x-ray fluorescence spectrometry (WD-XRF), using a PANalytical Axios and a PW2400 spectrometer.

The contents of sulphur and total carbon were measured with a LECO CS-444-Analysator. The organic carbon content was determined with the same machinery after dissolving the carbonates by repeated hydrochloric acid treatment at 80 °C.

The specific surface area was determined by nitrogen adsorption using a five point BET (Brunauer-Emmett-Teller surface area) method. Measurements were performed with a Micromeritics Gemini III 2375 on about 300 mg of the ground material.

Measures of roughness

In geological media, the roughness of a fracture or fault surface can be defined as small-scale vertical deviations from an ideally smooth surface. It can be caused by the undulations of the mineral grains of the broken rock matrix at the surface but also by secondary precipitates.

The usual measure of roughness is the vertical distance (deviation) of a measuring point from a baseline (see ISO 25178-2:2021 (2021) for details on terminology). It is therefore usually measured as a unit of length. Roughness parameters along a measurement profile (cross section) are indicated by the letter R, e.g. Ra for the average roughness, while the letter S (= surface) is used for values obtained from areal measurements. R and S values from the same sample need not be identical, due to the natural anisotropy of surfaces and the effects of the user-dependent selection of the position of the profile on the surface. Areal measurements are therefore preferred, although they take longer and are computationally more demanding. Following ISO 25178-2:2021, commonly used parameters are the arithmetical mean deviation Ra or Sa (Eq. 3) and the root mean squared deviation Rq or Sq (Eq. 4). The maximum height above the base(line) is also often given (Rz or Sz).

$$Ra = \frac{1}{n} \cdot \sum\nolimits_{i = 1}^n {\left| {{y_i}} \right|}$$
(3)
$$Rq = \sqrt {\frac{1}{n} \cdot \sum\nolimits_{i = 1}^n {y_i^2} }$$
(4)

with

n = number of equally spaced measuring points along a profile.

yi = vertical distance of i-th measuring point from the base line [L].

There is a multitude of roughness indices, which cannot all be described here (e.g. Li and Zhang 2015; Magsipoc et al. 2020). The oldest but still most common index is the dimensionless Joint Roughness Coefficient (JRC) introduced by Barton (1973). An increasing JRC indicates more roughness and thus more obstruction to groundwater flow over a fracture plane. The JRC is obtained visually by comparing measured roughness profiles to a set of standard profiles by e.g. Barton and Choubey (1977). This has the disadvantage that the outcome is prone to the subjective influences of the user (Beer et al. 2002; Stigsson and Mas Ivars 2019). Another popular index is the roughness profile index of a fracture, Rp (Eq. 5a), with Rp ≥ 1 (El Soudani 1978). The profile elongation index δ is a variation (Eq. 5b).

$${R_p} = {L_t}/L$$
(5a)
$$\delta = ({L_t} - L{\rm{)/L}}$$
(5b)

with

Lt = actual length of rough profile [L].

L = projected length of profile in fracture plane [L].

Several studies have shown that JRC and Rp are highly correlated and several empirical equations have been proposed (see review by Li and Zhang 2015). Photogrammetric studies by Maerz et al. (1990) yielded Eq. 6, which will be used here

$$JRC = 411({R_p} - 1)$$
(6)

Several studies indicate that the correlation of JRC with Ra and Rq is significantly weaker (Li and Zhang 2015; Mo and Li 2019; Luo et al. 2022). Only a few empirical equations have been proposed, e.g. the ones by Tse and Cruden (1979), where Ra and Rq are given in centimeters.

$$JRC = 2.76 + 78.87 \cdot {R_a}$$
(7a)
$$JRC = 2.37 + 70.97 \cdot {R_q}$$
(7b)

Ni et al. (2018) established an empirical relation between the JRC and the Forchheimer coefficient β, an important, but difficult to obtain parameter for non-Darcy flow.

$$\beta = \frac{{0.042 \cdot JRC}}{b}$$
(8)

with

β = Forchheimer coefficient [L− 1].

In order to obtain the fractal dimension of dimensionally variable data, the well-established Hurst exponent H is used (Hurst 1951). Here, a simplified approach, modified from Malinverno (1990), is applied, which relates the roughness Rx to the Hurst exponent H via

$$Rx = A \cdot {L^H}$$
(9)

with

Rx = Ra or Rq = roughness [L].

A = proportionality coefficient.

L = length of observation window (profile) [L].

H = Hurst exponent.

Laboratory roughness measurements

It should be noted that all measurements of this study were performed on independent sub-samples of one sample of a fracture surface. Using the same sub-sample for all roughness measurements is not feasible, since the individual methods vary in image size, resolution, and sample preparation. Finding the same profile locations with different methods is practically impossible. Results can thus be potentially affected by the differences (heterogeneity) amongst the sub-samples.

For a first rough measure of the surface texture, a mechanical feeler gauge (Barton´s profile comb) was used (e.g. Al-Fahimi et al. 2018). It consists of a row of 150 steel pins of 1 mm width each, which can reveal the 2D height profile of a sample in the chosen direction. When removed from the object, the pins remain in the position of the measured profile and can be copied to a sheet of paper for later evaluation against templates by Lee and Farmer (1993).

Interferometric Microscopy with True Color Imaging (CSI-TC) combines Coherence Scanning Interferometry with capture of true color information (details see Beverage et al. 2014; de Groot 2014). Capturing of true color information together with a virtual 3D topography allows distinguishing variations in material properties and the visualization of photorealistic textures. Scans were performed using a nexview™ surface profiler provided by ZygoLOT GmbH (now Ametek GmbH), Germany. The lateral optical resolution was 0.34 μm (100X objective) and the spatial sampling distance 0.04 μm (100X objective, 2X zoom).

Optical 3D Micro Coordinate Measurements (3D-MCM) were performed with an InfinteFocusSL provided by Alicona Imaging GmbH, Austria. The method is based on focus-variation, which means that the change of sharpness is determined while the distance between the objective and the sample is varied. The high measurement point density is also applicable for form and roughness measurements over relatively large sample areas. Uncertainty information is automatically included for every value. The lateral and vertical resolution are 4 μm and 250 nm for both.

For Laser Scanning Microscopy (LSM) a Keyence VK-9700 (Keyence International, Japan) confocal LSM of the Institute for Soil Science of the Leibniz University of Hannover, Germany, was used. The system allows a magnification of up to 24,000x and delivers true color images.

Vertical Scanning Interferometry (VSI) with a white light source provides very high vertical spatial resolution (∼ 1 nm). Like all other light optical profilometers (confocal microscopy, coherence microscopy), the lateral spatial resolution is limited due to diffraction. For all these methods, this leads to the fact that relatively steep flanks of the topography cannot be detected. “Steep” in this context means that the undetected height difference occurs within one pixel width and is thus not detectable. Here, a ZeMapper Vertical Scanning Interferometer (VSI) from Zemetrics Inc. of Tuscon, Arizona, was used for surface analysis of the fractures (installed at MARUM, University of Bremen, Germany). It is equipped with four Mirau objectives for 10x, 20x, 50x, and 100x magnification and a 5x Michelson objective. With this setup, a field of view of up to 3 mm x 3 mm can be achieved. The square pixels have a virtual length of 147 nm when using the 50x objective. More details on this instrument can be found in Fischer et al. (2012).

Results

Optical and chemical inspection of fracture surfaces

Under the electron microscope, the weathered fracture surface it is coated by a dense material, obscuring the grains of the underlying sandstone (Fig. 2, upper row). While the fracture surface looks rather smooth under low magnifications (and less rough than the sandstone matrix underneath), detailed images show a variety of very fine grained materials in the coating, including iron oxides (Fig. 2, lower left), clay minerals (Fig. 2, lower right) and finely dispersed gypsum crystals (not shown). The mentioned minerals are in good accordance with the chemical EDX analyses, which show that the material is locally enriched in iron and sulphur as well as, additionally, in (organic) carbon. These phases are more reactive in terms of sorption and redox reactions compared to the abundant quartz and may thus influence the quality of groundwater flowing along it.

Fig. 2
figure 2

(a) broken edge showing transition from bulk sandstone (bottom, granular) to fracture surface (top, smooth), (b): perpendicular view of fracture surface, (c): detail view of fracture surface (example): spherical aggregates, probably goethite (d): detail view of fracture surface (example): finely foliated clay minerals in a depression surrounded by quartz grains

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Thin sections show that the surface material has invaded up to 1 mm into the sandstone matrix (Fig. 3). The dark colors could be an indication of the organic carbon mentioned above. Light-optical inspection of thin sections showed that the intergranular space between the quartz grains located near the fracture is filled by a dark phase, which could not be observed farther than 1 mm away from the fracture surface. Based on the above presented electron-optical investigations this phase may be either organic material or iron oxyhydroxide or even a mixture of both.

Fig. 3
figure 3

Thin-section microscopy (unpolarized light): (a) fracture surface with black opaque material between quartz grains, (b): pore-filling clay minerals and sand grains (right)

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Table 1 Comparison of selected chemical and physical parameters of bulk sandstone material and fracture coating
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For the chemical and physical analyses, samples of the bulk sandstone matrix and the fracture coating material were studied separately (Table 1). They show that the latter is enriched in iron, manganese and organic carbon and slightly enriched in sulphur and carbonate. This is in good accordance with the findings from the electron microscopy discussed above. Some trace metal contents are also elevated for the coating material, including barium, copper, nickel, and lead (Table 1). They could be contained in carbonate or sulphate phases (barium) or in the observed iron oxides (heavy metals). The higher specific surface area of the fracture surface reflects the somewhat higher content of fine.grained and microporous phases such as ironoxyhydroxide, carbonate, and clay minerals (Fig. 2; Table 1). This indicates that its roughness is higher than that of the bulk sandstone, despite its overall more smooth appearance.

Mechanical roughness analysis

Figure 4 shows roughness profiles measured with a feeler gauge on 22 surfaces from the same fracture surface. Through visual comparison against templates by Lee and Farmer (1993), an average JRC after Barton (1973) of about 8 was obtained. This is in the typical range of JRC = 5 to 14 for sandstones (e.g. Bandis et al. 1983; Vattai and Rozgonyi-Boissinot 2018). The roughness profile index of a fracture Rp, would be 1.019, following Eq. 6. With an aperture of 1 mm, the Forchheimer coefficient derived from Eq. 8 would assume a value of β = 336 m− 1.

Fig. 4
figure 4

Fracture plane roughness profiles measured with feeler gauge (digitized)

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Optical roughness analysis

The measurements with Coherence Scanning Interferometry with True Color Imaging (CSI-TC) by Zygolot GmbH were performed for two different sample sizes and several replicates each, with three samples of a size of 0.336 mm x 0.336 mm (0.113 mm²) and two samples of 0.841 mm x 0.841 mm (0.707 mm²). The observed data are summarized in Annex 1. Figure 5 shows example images of the measurement. The results show a general trend of increasing roughness with sample size. However, the increase of the sample size by a factor of 6.26 only leads to an increase of roughness by a maximum factor of around two. The measurements at the smaller sample size do show pronounced variations, e.g. for Sa from 11 to 17 μm, while the admittedly only two larger-scaled samples are closer together.

Fig. 5
figure 5

(courtesy of ZygoLOT GmbH)

Example of Coherence Scanning Interferometry with True Color Imaging of fracture surface: (a): sample size 0.336 mm x 0.336 mm, (b) right sample size 0.841 mm x 0.841 mm.

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The roughness measurements using Optical 3D Micro Coordinate Measurements (3D-MCM) by Alicona Imaging GmbH were done both for an area (5.7 mm x 7.5 mm) and a profile (12 mm length). Figure 6 and Annex 2 show example images and results for the former, Fig. 7 and Annex 3 for the latter. No replicate measurements were carried out. The areal 3D-MCM measurement is the one with the largest area (42.75 mm²) of all methods considered. Although this area is larger by factor of 60 compared to the larger samples (0.707 mm²) measured with CSI-TC, the roughness parameters only increase by a factor of around two. The results of the profile measurement based on 3D-MCM are Ra = 35.9 and Rq = 45.4 μm, respectively. They are somewhat smaller than the values obtained from the areal measurements with Sa = 43.2 and Sq = 54.8 μm, respectively.

Fig. 6
figure 6

Results of optical 3D micro coordinate measurement, lateral resolution 4 μm, vertical resolution 250 nm: (a) color-coded height representation of areal measurement of the fracture surface. Sample size 5.7 × 7.5 mm, 43 million measuring points. (b) image window for profile measurement (black line), window size 16.8 × 2.0 mm, 34.3 million measuring points, (c) result of profile measurement (12 mm). Images by courtesy of Alicona Imaging GmbH

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The single roughness measurement with a ZeMapper white light interferometer (VSI) was evaluated over a field of view of 30 μm x 196 μm (0.0059 mm²). The field of view size was chosen small enough to exclude sections with many missing data points, which can be caused by e.g. locally lower reflectivity of the surface. Compared to the previously discussed CSI-TC measurements with an area 0.113 mm², this area is smaller by a factor of around 19. However, the Sq of 14.7 μm obtained by VSI is not much smaller, but rather in the same range as the values obtained by the CSI-TC method (13.5–21.7 μm), although on the lower side.

The confocal laser scanning microscope (LSM) was used for 15 areal measurements (Fig. 7a, Annex 4) and 14 profiles (Fig. 8, Annex 5) of the same sample. They show a rather large scatter but a weak trend of increasing roughness with increasing imaging area.

Discussion

The areal roughness measurements are shown together in Fig. 7. It is evident that both roughness parameters, Sa and Sq, increase with increasing sample (or image) size. The difference between Sa and Sq is relatively small. Sample reproducibility is generally good, although only available for CSI-TC. It is remarkable, that the different methods do no show a notable dependence of roughness on image size, a testament to the usefulness of all applied techniques. The profile measurements (Fig. 8) show a steady increase of roughness parameters Ra and Rq with increasing profile length. As expected, values for Rq are consistently higher than those for Ra.

A comparison of our roughness values to published data is somewhat hampered by the fact that the majority of studies prefer JRC measurements or parameters that correlate well with JRC. Applying Eq. 7a and 7b for our observed JRC = 8, we obtain Ra = 66 μm and Rq = 79 μm. Since Tse and Cruden (1979) apparently worked on a centimeter scale, these values have to be compared to the higher values on the x axis of Fig. 8. The calculated values thus coincide well with the range shown there.

Fig. 7
figure 7

Areal roughness measurements from different methods: (a) laser scanning microscopy, (b) other methods. The text labels indicate the methods used (acronyms see methods section). Note the logarithmic scales and the different scale ranges

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Fig. 8
figure 8

Roughness profile measurements from different methods. The text labels indicate the methods used (acronyms see methods section). The transparent symbols (red = Rq, black = Ra) indicate literature values plotted for comparison: (1) = Haeri et al. (2020), (2) = Rost et al. (2018), values see text. The literature data were not considered for the regression analysis. Note the logarithmic scales

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Measurements of amplitude-based roughness parameters on samples similar to ours are somewhat scarce. Rost et al. (2018) gave values of Ra = 32.4 μm and Rq = 50.1 μm for the Gildehaus sandstone (Germany) and Ra = 18.3 μm and Rq = 35.1 μm for the quartz-rich Fontainebleau sandstone (France), each for a 1.7 × 2.3 mm sample window (in Fig. 8, a mean length of 2 mm was assumed). It should be noted, however, that Rost et al. (2018) studied the surface of freshly split samples, not weathered fracture surfaces like ours, which have undergone secondary geochemical processes (Figs. 2 and 3). Haeri et al. (2020) measured the Rq of six sandstones, namely the Navajo, Nugget, Bentheimer, Bandera Brown, Berea and Mt. Simon sandstones and obtained values of 23.3, 17.8, 34.4, 35.5, 30.4 and 123.8 μm, respectively (sample size 0.95 × 1.3 mm, plotted as 1.1 mm in Fig. 8). These literature values are somewhat higher than the range for our data shown in Fig. 8, maybe an effect of weathering which our samples experienced.

For an artificially cemented sandstone, Konstantinou et al. (2021) found values for Ra and Rq, ranging between 500 and 2500 μm, but for a high scan length of 50 mm, which is far beyond our range. For a fresh, artificially induced joint in an unspecified Chinese sandstone, Fang et al. (2019) found values of Sa = 517 μm and Sq = 666 μm. These much higher values can also be explained by their much larger sample size (90 × 90 mm = 8100 mm²), which is almost two orders of magnitude higher than our maximum sample size. The observed dependence of roughness on the scale of imaging is a reflection of the fractal (self-affine) nature of roughness (e.g. Magsipoc et al. 2020). Our study is not the first to identify such a behavior (e.g. Brown 1987; Kumar and Bodvarsson 1990; Develi and Babadagli 1998; Beeler 2020). Already Schmittbuhl et al. (1993) clearly stated that “… roughness increases continuously with the size of the window over which it is estimated. No absolute roughness scale can be defined independently of the sample size.” Our study, which takes into account both profile and areal measurements, confirms these previous findings, even when using several independent methods.

Despite the large spread of data and the use of different methods (Figs. 7 and 8), we investigated the fractal dimension of our samples via the well-established Hurst exponent H (Hurst 1951).

A regression analysis of our data shown in Fig. 8 with Eq. 9, yields for Ra values of A = 0.124 (± 0.057) and H = 0.601 (± 0.053), with a correlation coefficient of R² = 0.90. For Rq, we obtain very similar values of A = 0.152 (± 0.073) and H = 0.605 (± 0.056), correlation coefficient R² = 0.89. It should be noted that we could only use four different profile lengths, fewer than recommended by Kulatilake and Um (1999). The Hurst exponent is related to the fractal dimension D via Eq. 10 (Magsipoc et al. 2020).

$${\rm{H}}\,{\rm{ = }}\,{\rm{E}}\,{\rm{-}}\,{\rm{D}}$$
(10)

with

D = fractal dimension.

E = number of spatial dimensions (E = 2 for profiles).

By definition, self-affine fractals have a spatial dimension of D = 1.5, while typical values for rock joints fall in the range D = 1.0-1.5 (Brown 1987; Magsipoc et al. 2020). With H = 0.6, we obtain D = 1.4, a value that fits well into this range. The results thus confirm the fractal, self-affine nature of our fracture surface. It should, however, be noted that our material was weathered, while most literature data are for freshly-split material.

Conclusions

Fracture roughness is an important but not fully understood factor influencing flow in fractured aquifers, since it can lead to deviations from the commonly used cubic law.

Unlike most other studies, we investigated a natural, weathered fracture surface. Despite its overall smooth appearance, the fracture coating on the Bunter Sandstone from Southern Germany exhibited a higher specific surface area than the bulk rock. This is due to the presence of many small and potentially microporous phases such as iron oxyhydroxides, organic matter, clay minerals, and fine grained gypsum, which all provide an increased geochemical reaction potential.

Mechanical feeler gauge measurements and the resulting JRC values compared favorably to the results from the optical amplitude methods with their much higher spatial resolution.

Despite the relatively small number of measurements, areal and profile measurements of fracture surface roughness both indicate that typical roughness parameters such as Ra/Sa and Rq/Sq increase with increasing sample (or image) size. This scale effect is a result of the fractal, self-affine nature of roughness. For the first time, this effect could be shown, although several different measuring methods were used, and image sizes differed by several orders of magnitude. Albeit at different scales, all methods are thus able to provide useful data, which confirms their applicability in this context, something which was not investigated before.

The observed scale-dependence of roughness should be considered when modeling fractured aquifers.