To simulate the process of rock fracture under an externally applied loading, we are using the discrete element method . In this approach, a brittle–elastic material is modeled as a collection of spherical particles interacting with their nearest neighbors either by frictional–elastic interactions or by breakable elastic “bonded” interactions. Based on the force-displacement laws implemented in these interactions the forces and moments acting on each particle can be calculated. The resulting translational and rotational accelerations of the particles are then used to calculate particle movements from Newton's equations. For the breakable bonded interactions, a failure criterion is evaluated, and if the failure threshold has been exceeded, the affected bonded interactions are removed and, if the particles involved are still in contact, replaced by a frictional–elastic interactions.

A range of different implementations exist for each of the interaction types, differing mainly in the details of the force-displacement law and, in the case of the bonded interactions, the failure criterion. In this work, we are using a linear force-displacement law for the normal component of the frictional–elastic interactions and a Coulomb friction law for the tangential component as described by . For the bonded interaction, we are using the bond model by which takes normal, shear, bending and torsional deformation into account. The stiffness and strength of the bonds are parameterized using the approach of which calculates normal, shear, bending and torsional stiffness from the elastic parameters of an assumed bond material, specifically from Young's modulus Eb and Poisson's ratio νb, considering cylindrical bonds with a length and diameter controlled by the radii of the particles they are connecting. A Mohr–Coulomb failure criterion is used for the bonds based on the strength parameters of the bond material, i.e., cohesion Cb and friction angle Φb.

Because the size of the individual models is important in this work to obtain high-resolution roughness data from the simulated fracture surfaces, we are using the parallel DEM software ESyS-Particle , which enables the simulation of sufficiently large models.

The extraction of surface data from the numerical models requires two steps: (1) the identification of the individual fragments of the sample after fracturing (Fig. b) and (2) the calculation which groups of the particles contained in each fragment form an individual fracture surface. The fragments of the broken sample are extracted by constructing an undirected graph from the structure of the DEM model such that the particles form the nodes of the graph and the remaining unbroken bonds form the edges in the graph. The fragments can then be extracted by calculating the connected components of that graph . For each fragment larger than ≈10 % of the original model (Fig. c), a ray-casting method is used to determine which of the particles are forming the surface of the fragment. In this approach, a set of parallel lines or “rays” with their origin outside the fragment and a specific direction is defined. The first intersection between each line and one of the particles is calculated using the algorithm proposed by . The positions of the calculated intersection points then form the point cloud describing the fragment surface (Fig. ).

To get a complete coverage of the fragment surface (Fig. e), i.e., to avoid shadowing effects by “overhanging” parts of the fragment surface, the calculations are performed for multiple view directions of the rays. The directions from the mass centers of all neighboring fragments to the mass center of the fragment and directions deviating from those by 30∘ are used. To identify individual fracture surfaces two additional post-processing steps are performed. The initial outside surfaces of the intact model are removed by identifying each particle which was part of the surface of the model in the initial particle packing and removing the respective intersection points from the point cloud. In the final step, a calculation is performed for each particle contributing an intersection point to the surface point cloud to determine which other fragment is closest to this particle. This information is then used to split the point cloud into individual chunks, each representing an individual fracture surface (Fig. f). By performing this step for all fragments in the model, corresponding pairs of surfaces belonging to the same fracture can be identified.

The 3-D point clouds generated using this method are collections of (x,y,z) coordinates. However, for most further analysis steps, a representation of the surface as height field relative to a plane, i.e., as z′(x′,y′) is needed. To obtain such a representation, a “best-fit” plane for the point cloud is calculated. The location of such a plane is found by calculating the barycenter b=(x0,y0,z0) of the point cloud, i.e., 1(x0,y0,z0)=1n∑i=0n(xi,yi,zi), and its orientation is determined by the two major eigenvectors e1 and e2 of the covariance matrix C of the point cloud. The third eigenvector of the covariance matrix then determines the normal nfp=e3 of the plane. Using this, the in-plane coordinates (x′,y′) of each point p=(x,y,z) and its perpendicular distance z′ from the plane can be calculated as 2x′=(p-b)⋅e13y′=(p-b)⋅e24z′=(p-b)⋅e3. It should be noted that a surface can only be represented correctly as a height field in this way if there are no parts of the surface which are “overhanging” with respect to the normal of the fitted plane, i.e., if there are no points on the surface with identical (x′,y′) but different z′. However, this is generally the case for the fracture surfaces generated in the numerical models.

A roughness measure commonly used in the study of the mechanical behavior of rock surfaces is the JRC defined initially as a parameter relating the shape of a rock joint to its peak shear strength; see Eq. (9) in or Eq. (2) in . Its relation to the geometry of the joint surfaces was only qualitatively defined by assigning JRC values to a set of standard profiles Fig. 8. To estimate the JRC of an arbitrary profile from measured geometrical data, a wide range of empirical formulas have been developed in the literature Table 2. To calculate the approximate JRC of the fracture surfaces generated in the numerical models and the laboratory experiments for three of the 47 equations presented there have been chosen. The subscript of the JRC in the equations below shows the respective number of the equation in Table 2. 5JRC1=32.2+32.47log⁡(Z2)6JRC31=558.68Rp-557.137JRC34=92.97δ-5.25, where Rp is the “roughness profile index”, δ=Rp-1 the “profile elongation index” and Z2 the “root mean square of the first deviation of the profile”, all as defined in Table 1. Rp is therefore calculated as 8Rp=∑i=1N-1(xi+1-xi)2+(yi+1-yi)2∑i=1N-1xi+1-xi and 9Z2=1N∑i=1N-1(yi+1-yi)2xi+1-xi, where xi is the abscissa of profile point i, yi its height above a mean value, and N is the number of sample points. Given that those parameters are calculated along profiles, the irregular point clouds generated using the method described in Sect.  first need to be mapped to a regular grid.

Self-affine rough surfaces are characterized by the fact that they are statistically invariant under an affine transformation 10(x,y,z)→(ax,ay,aHz), where x,y are the “in plane” coordinates of the surface and z is the “height” of the surface above a given mean plane (Fig. a). The exponent H is the Hurst exponent or “roughness index” . A range of different method for the calculation of the Hurst exponent have been described in the literature , most of them either based on the evaluation of the power spectrum of the surface or correlation functions between the heights of points on the surface depending on their mutual distance. Because the point clouds describing the surfaces generated by the approach described in Sect.  do not form a regular grid, spectral methods would require an additional interpolation step. Aside from the additional computational effort required, this might also introduce some difficult to quantify errors in the calculation of the Hurst exponent .

In this work we therefore use the “height–height correlation function” method as described by . However, in contrast to the description in , the function is not calculated from 1-D profile data but directly from the 2-D surface. The radially averaged height–height correlation function is calculated as the root mean square (rms) averaged height difference of all point pairs within a given distance range, i.e., 11c(Δr)=1n∑Δr-w/2Δr+w/2Δz2, where Δr=Δx2+Δy2 is the “in-plane” distance between the points in the pair, Δz is the height difference between the points (Fig. b), w is the size of the distance bins over which the height differences are averaged, and n is the number of particle pairs in the respective distance bin. For the calculation of the angular dependence of the height–height correlation function, the direction between the two points of the pair is calculated as ϕ=arctan⁡(ΔyΔx) and the summation of the height differences is adjusted from 1-D distance bins in Eq. () to 2-D (distance, direction) bins. 12c(Δr,ϕ)=1n∑Δr-wr/2ϕ-wϕ/2Δr+wr/2ϕ+wϕ/2Δz2, where wr is the bin size with respect to the in-plane distance of the points and wϕ is the bin size with respect to the direction from one point of the pair to the other. Due to the large number of points contained in the surface point clouds, which is as large np≈500000 in some cases, and the resulting computational cost if all of the np2/2 particle pairs would be taken into account, only a random sample of 10 000 points (i.e., ≈5×107 point pairs) is used for each surface. Tests have shown that the reduction in the number of particle pairs evaluated does not impact the results.

Given that for a self-affine surface, the height–height correlation function follows a power law, i.e., c(Δr)∝ΔrH , the Hurst exponent H can be calculated by fitting a linear function to the straight part of the log–log plot of c(Δr) vs. Δr. The slope of the linear function is the Hurst exponent. The Hurst exponent H and the fractal dimension D of an object are related as D=2-H for a 1-D profile or, more generally, D=n+1-H, where n is the dimension of the object , i.e., n=1 for a profile and n=2 for a surface.

To generate a set of model fracture surfaces, a large number of deformation experiments have been simulated. The set of simulations consists of unconfined compression (σ1>0, σ2=σ3=0), unconfined tension (σ3<0, σ1=σ2=0), standard triaxial compression (σ1>0, σ2=σ3>0) and true triaxial compression (σ1>σ2>σ3>0) experiments. In all compressive models, σ1 is parallel to the y axis, σ2 is parallel to the x axis, and σ3 is parallel to the z axis, whereas in the unconfined tensile models σ3, i.e., the extension direction, is parallel to the y axis, σ1 is parallel to the x axis, and σ2 is parallel to the z axis.

All models are using box-shaped samples with an aspect ratio of 1:2:1 contained between two servo-controlled plates in the case of the unconfined compression and tension experiments or six servo-controlled plates for the standard triaxial and true triaxial experiments. While deformation experiments in the laboratory usually use cylindrical samples, we decided in favor of box-shaped samples because they make it much easier to apply the two different confining stresses in the true triaxial tests. In the tension experiments, the plates are connected to the boundary particles of the sample by unbreakable bonds which only induce a force parallel to the normal of the plate but not perpendicular to it. This means the particle are free to move parallel to the loading plate, avoiding heterogeneous deformation (“necking”). In the compressive experiments, both the axial loading plates and, in the confined models, the plates along the x and z surfaces of the sample interact with the boundary particles by frictionless elastic interactions.

In the unconfined experiments (σ2=σ3=0), a simple loading procedure is used, applying a prescribed displacement rate to the plates at the y ends of the model to produce axial shortening or extension. During an initial phase, the plate speed is ramped up smoothly according to a cosine function until the chosen speed is reached and then it is held constant for the main phase of the experiment. In the confined experiments (σ2≥σ3>0), this loading procedure is preceded by a ramp-up of the stresses applied to the plates at the x- and z-sides of the sample until σxx=σ2 and σzz=σ3. For the smooth ramp-up of the applied stresses, the same cosine function is used as for the ramp-up of the axial deformation rate in order to minimize unnecessary vibrations in the model. During this phase, a stress is also applied to the loading plates at the y ends of the sample such that σ1=σ2. After a subsequent “rest” phase where the stress on all plates is held constant for given time to allow the particle movement introduced by the initial loading to dissipate, the same axial shortening as in the unconfined compression experiments is applied. A range of confining stresses from σ2=σ3=0 to σ2=σ3=15 MPa was used for the numerical models in this work. In order to avoid the effect of abrasion modifying the roughness of the fracture surfaces after their initial formation, the state of the model immediately after one or more through-going fractures have formed was used for the extraction of fracture surfaces described in Sect. . That is, there is no, or at least very little, post-failure slip on those surfaces.

A model size of 55×110×55 model units was chosen with a particle sizes ranging from Rmin=0.2 to Rmax=1.0, resulting in ≈950000 particles for those models (Fig. a). This model size was found in initial tests to provide a good balance between model resolution and computational cost. For the construction of the initial particle arrangement for the models, the insertion-based packing algorithm by was used. This algorithm generates dense particle packings having a power-law particle size distribution with an exponent of approximately -3; i.e., the number of particles with given radius r is roughly proportional to r-3.

In all deformation experiments, the final loading plate speed was set to ≈17cms-1. This is significantly higher than in real experiments, but using real lab values (µms-1…mms-1) would lead to impractically long computing times because the time step of the calculations is restricted to values of Δt≤3×10-8 s due to numerical stability constraints. Tests have shown that the increased velocities do not significantly influence the model results.

The mechanical properties of the DEM material have been calibrated to values similar to those of a typical sedimentary rock. The target values, Young's modulus E=30 GPa and unconfined compressive strength UCS=80 MPa, are within the range of sandstone or medium to high porosity limestone . The failure strength was found to vary by less than 1 % among samples. These parameters do not provide a direct match to the mechanical properties of the rocks used in the laboratory tests (Sect. ), but the important ratio between failure strength of the material and the confining stress applied in the laboratory experiments lies well within the range covered by the numerical models (Fig. b). Because the details of the fracture behaviors of individual samples in DEM models show a well-known dependence on the initial random particle arrangement , at least five simulations with different realization of the particle packing have been performed for each parameter set in order to quantify this variability.

To improve the coverage of the chosen range of stress conditions, data from a related study were integrated into the analysis (Fig. a). This study was using an identical model setup, except for slightly smaller models with dimensions of 40×80×40 model units (≈360000 particles) and 50×100×50 model units (≈710000 particles) compared to the roughly 950 000 particles used in most models in this work.

To compare the roughness of fractures created in the DEM models with the roughness of real fractures, we conducted a number of laboratory uniaxial and triaxial deformation experiments. For our study, we used a suite of fine grained, low porosity Upper Jurassic carbonate rock samples and additionally one Lower Triassic sandstone sample, both from Franconia, Germany. Sample size for the experiments were 55×110 mm cylinders. The main goal of the experiments was to produce fractures for given stress conditions which could then be used for roughness analyses.

The sandstone uniaxial compressive strength (UCS) experiment lead to an typical hourglass fracture pattern, splitting the sample into a small number of larger fragments, which could be used for further analyses (Fig. c). Unfortunately, for the UCS and most of the triaxial experiments of the carbonate rocks, the samples disintegrated into a very large number of very small fragments, leaving no suitable fracture surfaces to analyze. See Fig. S1 in the Supplement for a typical example. This applied in particular to the samples loaded with small confining pressures. Only in one experiment with a confining pressure of 30 MPa post-deformation fragments were large enough for our planned fracture surface analyses (Fig. b). From the suitable fragments, we constructed a digital three-dimensional surface model using photogrammetric methods. The models were built from more than 100 single pictures of the samples from different perspectives using a 12-megapixel SLR camera and a 40 mm macro lens. The photos were taken from a distance of 5 to 10 cm between front lens and the object, which is close to the minimum focus distance of the lens used. The models were then clipped to the fracture plane of interest. The remaining surface geometry was exported as 3-D point cloud data with approximately 2.2 million data points in total, resulting in a point density of approximately 28 000 pointscm-2 and an average point distance of 60 µm.

The generated point clouds were then used for roughness analyses of the fracture surfaces following the approach described in Sect. . Besides the creation of fracture surfaces the deformation experiments were also used to derive typical geomechanical properties of the carbonate and sandstone samples which were used for comparison with the DEM models. For the carbonate rocks, a UCS of ≈285 MPa was obtained and ≈85 MPa for the sandstone sample. For the limestone, a friction coefficient μ=0.7 was derived from experiments with confining pressures ranging between 0 to 30 MPa. Young's modulus was measured at E=48GPa for the limestone and E=12.5GPa for the sandstone.

Based on the data produced by a total of 131 numerical simulations, the geometrical properties of 388 fracture surfaces have been analyzed. The fracture orientations were as expected under the stress conditions. The dip angle was typically within 25–35∘ of σ1, i.e., 55–65∘ assuming σ1 to be vertical. The strike direction of the majority of the fractures was within ≈10∘ of σ2 in the true triaxial models (σ2≠σ3) and more or less randomly distributed under transverse isotropic stress conditions (σ2=σ3).

In an initial step, the joint roughness coefficients for a small set of surfaces were approximated using Eqs. ()–(). The results did show that the resulting JRC values were consistently above the range defined by , i.e., larger than 20, and therefore also outside the range of validity of the approximation equations in . Similarly, the geometric parameters Rp (Eq. ) and Z2 (Eq. ) from which the estimated JRC values were calculated, were outside the applicable ranges given there. While the roughness produced by the numerical models is therefore outside the range for which the fitting equations collected by were originally intended, Fig. 1a in their work suggests that Eq. () would be the best option to extend the range of approximate JRCs to the surface geometries observed here because it provides a particularly good fit at large values (i.e., Z2≈0.35–0.4, JRC≈20). Therefore, Eq. () was used to estimate the average JRC for each of 261 surfaces based on a total of ≈24 400 profiles. The remaining 127 of the 388 surfaces studied were found to be too small in at least one of the dimensions to allow the extraction of sufficiently long profiles. For each surface, profiles were generated in two orthogonal directions to check for a possible anisotropy of the surface roughness. The results did show that the mean estimated JRCs for the profiles differs by less than 10 % between the two direction, which is generally less than the standard deviation between the profiles within one direction. Plotting the estimated JRC for the analyzed surfaces against the mean confining stress of the models (Fig. a) shows that there is no clear trend of JRC vs. confinement, but that models with transversely isotropic confinement (σ2=σ3) generally have higher JRC values than models fractured under true triaxial conditions, i.e., σ2≠σ3. The directly calculated geometric roughness measures, i.e., Rp and Z2, show a very similar pattern (Fig. b and c).

The perpendicular distance or “height” of the points of the fracture surfaces above a fitted fit plane is calculated according to Eq. (). Analysis shows that the heights are normally distributed (Fig. ) as expected for fracture surfaces , allowing a “rms roughness” hrms=1n∑n(z′2) to be calculated.

Plotting the rms roughness hrms of all models against the mean confining stress (Fig. ) shows that there is no clear dependence between the two parameters, except for a difference between transverse isotropic (σ2=σ3) and true triaxial (σ2≠σ3) stress conditions. In the case of the transverse isotropic confinement, the observed rms roughness hrms=2.35±0.78 model units is higher than in the case of true triaxial conditions where hrms=1.63±0.48 model units. It can also be observed that the rms roughness of the models subjected to unconfined extension (blue marker in Fig. ) is smaller at hrms=1.51±0.44 model units than that of the models subjected to unconfined compression with hrms=2.76±0.88 model units. This difference, however, is possibly at least in part an artifact of the different size of the fracture surfaces between the two model groups. In the tensile case, the fractures tend to be roughly normal to the extension direction, i.e., the long axis of the model and their size is therefore restricted to the small cross section of the model. In contrast, the fracture surfaces in the compressive case tend to be oriented such that their normal is at an angle of ≈55…60∘ to the compression direction and can therefore grow as large as a plane diagonally across the model, i.e., more than twice the size compared to the tensile case. Plotting the height–height correlation function (Eq. ) of the surfaces in a log–log plot (Fig. ) shows a clear linear section which, for most surfaces analyzed, ranges from Δrmin≈1.5–2 model units, i.e., somewhat more than the maximum particle size, to about half of the smaller dimension of the surface, which in most cases means Δrmax≈20–30 model units. This linear section in the log–log plot, representing a power-law dependence c(Δr)∝Δrh shows that the surface is indeed self-affine, at least for the range of scales covered by the linear section.

In order to verify that the observed self-affine structure of the fracture surfaces generated in the numerical model is indeed a result of the fracture process and not an artifact caused by the intrinsic roughness of surfaces in the particle model, a number of “quasi-planar” surfaces were generated in the particle model and their roughness was analyzed. For this purpose, one of the blocks of packed particles used in the DEM simulations of the triaxial tests (Fig. a) was cut with an arbitrarily oriented plane; i.e., the particles on one side of the plane were removed. The remaining fragment of the block then underwent the same surface extraction and roughness analysis procedures as the fracture surfaces produced in the deformation experiments. The result (Fig. ) shows that the height–height correlation function of the cut surface is essentially flat from the particle scale up to the model size. Performing this analysis on multiple cut surfaces did show that this is independent of the orientation of the cut plane and the details of the particle packing. Only the absolute value of the roughness of the cut surfaces depends somewhat on the size range of the particles. Calculating the joint roughness coefficients for the cut surfaces according to Eq. () did, as expected, produce non-zero values of the JRC. However, the JRC values for the cut surfaces are in the range of 11.5–12, which is much smaller than the values observed in the fracture surfaces generated in the numerical models (JRC≈23–32, Fig. a). It can therefore be assumed that, while there is some contribution of the intrinsic particle-scale roughness to the total roughness of the fracture surface, the self-affine structure of the fracture surfaces as well as the major part of their total roughness is due to the fracture process and not the particle structure of the model as such.

Performing the calculations for all 388 fracture surfaces extracted from the numerical models produced Hurst exponents ranging from 0.2 to 0.6. To investigate possible dependencies on the stress conditions under which the fractures were created, the average Hurst exponents over all surfaces generated in each set of simulations with identical boundary conditions have been calculated. The mean value of H for the sets varies between 0.3 and 0.45, with the variation between top and bottom quartile within each set typically in the range of 0.05…0.1. Due to the relatively small number of data points within each set of models, i.e., between 8 and 28 surfaces, and the observed asymmetry of the error distribution in some instances, quartiles have been calculated and plotted (Figs.  and ) instead of standard deviations. Plotting the calculated Hurst exponents against the mean confining stress (σ2+σ3)/2 (Fig. ) shows a weak trend towards lower Hurst exponents with increasing confinement. No dependence of the Hurst exponent on the ratio between the confining stresses σ3/σ2 could be observed (Fig. ).

To characterize the roughness of the fracture surfaces produced in the laboratory deformation tests, we examined the photogrammetrically produced point clouds of the single sample fragments. For each of the limestone and sandstone samples, one fracture surface was chosen. The maximum sampling area for the roughness investigation was 14 cm2 for the sandstone and ≈25 cm2 for the limestone sample. The analyses of the heights distances of the single points of the point clouds above their fitted mean planes revealed a normal distribution of the heights. Thus, a calculation of the rms roughness is justified (Fig. ). The height–height correlation functions of these surfaces have a well-defined linear section in a log–log plot proving a self-affine geometry in a distance range between ≈0.1 and ≈1 cm, both for the limestone and sandstone sample (Figs. S2 and S3 in the Supplement). With distances larger ≈1 cm, a flattening of rms curve can be observed, marking the upper end of the power-law relationship between the point distance and the rms height difference. From the linear slope segments of the correlation functions, similar Hurst exponents could be deduced with H=0.66 for the sandstone and H=0.69 for the limestone when analyzing the maximum sampling area on the respective fracture.

To check whether the size of the investigation area on the fracture surfaces has an effect on calculated Hurst exponents, we analyzed the height–height correlation functions and Hurst exponents for a suite of different area sizes (Fig. ). For the limestone sample, the mean H value results in H=0.73 with a standard deviation of 0.08. The sandstone sample shows a clearly lower mean H value of H=0.6 and a standard deviation of 0.05. A stronger scatter of Hurst exponents can be observed for the smallest analyzed sample area size of ≈1 cm2, ranging between H=0.43 and 0.64 for the sandstone surface and between H=0.67 and 0.85 with two outliers of H≈0.5 for the limestone surfaces. For these outliers, a closer investigation of the corresponding rms–distance curves shows that two different linear sections could be derived, one with a higher Hurst exponent for smaller distances and one lower Hurst exponent for larger distances.

For both sample surfaces, the JRC was estimated using the same methods as for the numerical models. The results show that the estimated JRC is dependent on the sampling resolution, i.e., the number of sampling points on the profile, specifically that the calculated value of the JRC is increasing with smaller sampling intervals (Fig. ). This is a known effect which is caused by the dependence of the underlying geometric parameters Rp and Z2, from which the estimated JRC is calculated, on the sampling interval used . It is also to be expected based on the fact that the analyzed surfaces are self-affine. In that case, the dependence of Rp on the sampling interval is directly described by the “compass dimension” of the profile. For the fracture surfaces in the numerical models (Sect. ), a similar analysis of the resolution dependence of the JRC was not done because of the lower intrinsic resolution of the point clouds which limits profiles to less than 100 sample points in most cases.

The empirical equations used for the calculation of JRC from measured geometric parameters are usually derived based on sampling resolutions between 100 and 400 points per profile . Specifically, the equation used in this work to estimate JRC from Z2 (Eq. ) was derived by Tse and Cruden using a sample interval of 1.27 mm at a profile length of 25 cm, i.e., slightly less than 200 points. The surfaces analyzed here have dimensions of about 7cm×5cm for the sandstone and approximately 10cm×4.5cm for the limestone. Therefore, a sampling interval of between 0.25 and 0.5 mm will produce a similar number of sample points along the profiles. Therefore, the best estimates for the average JRC of the fracture surfaces produced in the laboratory experiments are for the sandstone JRC≈9–11 in the direction parallel to shortening direction in the deformation experiment and JRC≈11–13 perpendicular to it (Fig. ). For the limestone, the estimates are JRC≈6.5–7.5 in the parallel direction and JRC≈16–17 in the perpendicular direction. In both cases, the JRC shows a clear anisotropy between the two directions. However, this anisotropy is much larger in the limestone compared to the sandstone sample.