For this study we use a selection of samples that were archived from collaborative material property studies (e.g. ) in our laboratory and newly acquired samples from 14 laboratories worldwide that sent materials that are currently used for their analogue models. Some of these materials were characterised in terms of time-independent frictional properties in the original GeoMod benchmark study by , while others have been characterised in the framework of EPOS transnational-access activities (Table , ). Most of them have been subjected to standard ring shear tests to determine their time-independent friction parameters (friction coefficient and cohesion) and rate-dependent (velocity strengthening or weakening) friction parameters. To have consistent evaluation parameters, we re-picked and re-analysed the existing datasets with the newest software RST-Evaluation , which is the current standard for ring shear tests in our laboratory. New materials that were not in our test database were tested with the standard ring shear test outlined in Sect. . For this study, we tested all samples with the slide–hold–slide procedure developed in , which is outlined in Sect. .

We use a RST-01.pc ring shear tester to measure the frictional properties of the dry granular materials. The method is well established for analogue materials and follows international standards for powder and bulk material testing . The machine consists of a rotating, ring-shaped shear cell onto which normal stress is applied using a stationary lid. The shear stress required to hold the lid in place is measured using two tie rods that are each attached in series to force transducers. To improve the contact of the cell and lid with the material, the surface of the lid and the bottom of the shear cell are structured with slats and grooves. Normal stress is applied through a cantilever system with a moving mass and is therefore instantly adjusted by gravity, in contrast to other mechanical tests where the stresses are adjusted with a servo-hydraulic or electronic system. During each measurement, the shear velocity, shear stress, normal stress and lid position are monitored. Further information on the setup and device is available in , and .

The testing procedure follows standardised procedures for sample preparation , testing procedure and data analysis . All samples are first oven dried to remove excess humidity and then stored in the air-conditioned laboratory for several days to equilibrate with ambient laboratory conditions of T=25 ∘C and ≈50 % humidity. The samples are sieved using the SM sieve into the cell from a height of ca. 30 cm and above, ensuring a similar package density for each test . We do not use the same sieve as , called a “GeoMod” sieve, because for our measurements the samples are pre-sheared and the package density after sieving is not a primary concern. Excess material is scraped off, and the weight of the material is determined. After inserting the cell into the tester, the normal stress is applied and the shear procedure is started. Each property has a specific shear procedure, which is outlined in the respective subsections (Sect.  for friction μ and cohesion C and Sect.  for the healing rate b).

Most tectonic analogue models use dry granular materials as analogues for crustal rocks in the brittle regime , with only a few exceptions using wet clay or other non-granular material . The greatest advantage of granular materials is that they obey the empirical Mohr–Coulomb criterion (Eq. ): 1τ=μσ+C.

Granular analogue materials show friction coefficients μ and cohesion values C that are comparable (in case of C when scaled) to typical crustal rocks: specifically, sands with μ=0.6 to 0.7 and C in the order of tens of pascals (scaling to few MPa) or glass beads with μ=0.4 to 0.5 and C in the order of few pascals. Moreover, the granular materials show stress–strain relationships similar to crustal rocks involving strain weakening and static healing . This gives rise to three different coefficients of friction μ and cohesion C attributed to different stages of fault evolution . The highest strength μpeak is reached during initial shearing of undisturbed granular material and therefore is analogous to the strength of undeformed rock (static friction). With continued shearing the materials loses strength (strain weakening) and reaches a lower strength during stable sliding μstable corresponding to the sliding resistance of a fault zone (sliding, dynamic or kinetic friction). If a material has been sheared, the granular fault zone is persistent and leaves a heterogeneity with lower density in the bulk material. If re-sheared, a new peak strength μreactivation occurs that reflects the strength of a pre-existing fault zone. Usually, it is higher than the stable sliding strength but lower than the initial peak strength. Consequently, models with granular materials localise deformation into narrow shear bands because of their lower sliding resistance and reactivate these structures under favourable circumstances due to the lower reactivation strength. From now on, we will refer to these three friction coefficients by their shortened versions, i.e. μp, μs and μr, throughout this study. Note that corresponding cohesion values exist, i.e. Cp, Cs and Cr.

In addition to the dependence on effective normal stress σ, sliding and reactivation friction coefficients μs and μr show a measurable dependence on slip rate δ˙ and hold time th, respectively. This is highly non-linear and described using the rate-and-state framework . This formulation is widely accepted as a good heuristic approximation of laboratory shear tests and natural phenomena where a time and strain rate dependence is observed. In general, there are two additional contributions to shear resistance: the rate effect aln⁡δ˙δ*˙ and the state effect bln⁡θθ*. Both are defined by a ratio with respect to reference constants (denoted by asterisks) and added to the reference friction μ0: 2τ=σμ0+aln⁡δ˙δ*˙+bln⁡θθ*.

The direct effect a, healing rate b and μ0 are derived empirically from experimental measurements. The evolution of state θ can take several forms as a function of time (ageing law, ), slip (slip law, ), time-dependent healing (Kato law, ) or stressing rate (Nagata law, ), which has to be assessed from experiments.

The change in strength over time in granular materials is known as time consolidation and can be very large depending on the material . For this study we assume purely time-dependent healing (ageing law) and use the healing rate b to calculate the time consolidation for each sample. We use slide–hold–slide tests to measure the healing rate following the procedure outlined in , and references therein. After sample preparation (Sect. ) and loading with σN=1 kPa the sample is sheared by 10 mm at a loading velocity vL=0.5 mm s-1, leading to a fully developed shear zone. The sample is then subjected to several slide–hold–slide intervals (Fig. a). The hold intervals are increased exponentially from th=101 to 104 s at increments of half a 10-fold increase in time and repeated three times per interval. An additional single hold interval of th=36000 s = 10 h ≈104.55 s gives a long term data point to each series. The full test duration is approximately 23 h.

We pick the shear stress needed to reactivate the shear zone after each hold time th and normalise it to the mean stress during stable sliding: Δμ=μr-μs‾ (Fig. b). This results in an effective stress measure assuming no cohesion. The healing rate is then the change in Δμ in comparison to the natural logarithm of hold time th and is obtained from the slope of Δμ versus ln⁡th (Fig. c): 3b=δΔμδln⁡th.

We calculate the compaction rate in the same manner by using the difference in lid position between the start and end of the hold phase. The same power law relation is found with strong compaction for short hold phases. Values for the compaction rate are negative because higher compaction results in smaller sample heights. All data are automatically picked and evaluated using a dedicated Python code published as part of the open-source software “RST-Stick-Slipy” .

Ultimately, the frictional response of a bulk material is the result of granular interactions and therefore depends primarily on the geometric characteristics of the grains. The shear zones forming in the ring shear tester usually span 11 to 16 times the mean grain size . The grains react to stress by creating force chains that frequently change their orientation . As a result, frictional resistance between individual grains and material elasticity has major implications on the bulk behaviour.

However, it is technically very challenging to measure friction between the individual grains, and therefore we assess the tendency of each material to create locked states by using a qualitative index for several key features. We categorise the materials with three different parameters: sphericity, roundness and surface roughness. The definition of the shape parameters follows that of clastic sediments. Sphericity is defined as the closeness to a perfect sphere. Roundness is given by the shape of the edges of a grain. Surface roughness is a measure of the small-scale surface properties (not of the grain shape itself). Each parameter is assigned a score from 1 to 4 that expresses the materials proneness to locking with 4 being “low impact” and 1 being “high impact”. Some materials have grains with different characteristics: for example, sands that are crushed are usually mixtures of rounded and angular grains. Therefore, depending on the degree of mixing, they are given an intermediate quality score. Grain size distribution was not taken into account because comparable measurements do not exist for all materials. A heterogeneous grain size distribution changes the bulk density of the material, which can influence the frictional characteristics . However, the effect is minor . Table  shows an overview of the parameters, criteria and associated literature.

The quality score is a qualitative measure that only gives a general tendency of material behaviour and follows an arbitrary scale. We use this scale to turn a qualitative observation into a quantity. Here, the purely subjective observation of the grains, without capturing a truly measurable quantity, represents a major source of error. The quality score is therefore only a theoretical support to find possible correlations between the parameters. In no way should the score be seen as an objective quantity. It could serve as a preliminary stage for future measurable variables and give a rough direction.

However, the relative importance of the grain parameters sphericity, roundness and surface roughness can be estimated by comparing studies that systematically varied them as part of either numerical or experimental studies (references in Table ). To account for the impact of each parameter, we take a weight average that uses the estimated effect on bulk friction from the references. We assume that this weighted average tends to reflect the amount of inter-particle locking during a hold phase and therefore also is a proxy to healing rate b. The weights are estimated by comparing the influence of each parameter on the value of friction in the respective studies. report an increase in sliding friction from μd=0.56 for spherical particles to μd=0.59, which is a 10 % increase in friction. Particle roundness has been shown by to increase friction from μ=0.45 for round particles to μd=0.6, which is an increase of 30 %. show an increase from μd=0.23 for slightly roughened particles to μd=0.37 for highly roughened particles, reflecting a 60 % increase. This is the strongest influence of all examined parameters. As a result, the relative weighting of the parameters is given as the ratio between the three percentages to each other summed up to three. Coincidentally, all three percentages sum up to one, and therefore the percentages given previously also reflect their relative proportion for the weighted average (10 %, 30 % and 60 % or 0.3, 0.9 and 1.8 given as weights, respectively).

The reactivation of pre-existing faults in nature and in analogue models is primarily governed by (i) fault geometry, (ii) the surrounding stress field and (iii)the fault's frictional properties . For natural fault systems, an additional mechanism is fluid overpressure, e.g. in basin sediments. However, fluids and fluid pressures cannot be accurately modelled in a scaled fashion and are thus rarely implemented in tectonic analogue models. We therefore focus on the aforementioned three factors (i) to (iii) to estimate the tendency to reactivate a pre-existing fault instead of forming a new fault. We do this in a typical basin inversion scenario and use a simplified Amonton wedge model to estimate the stresses needed to push a sidewall with a reactivated pre-existing fault and compare it to the stress needed to form a new fault .

The first step is to calculate the optimal fault dip angle θa with respect to the horizontal at which the fault forms during extension to approximate the preferred fault geometry in the first phase of the basin inversion scenario (Eq. , Fig. a, after ). This fault angle is defined by the friction μ of the material so that higher friction leads to steeper faults in the model. 4θa=90∘-12tan⁡-11μ

After the extensional phase, the basin switches to compression, and the optimal angle for a new fault θp with respect to the horizontal is as follows (Fig. b): 5θp=12tan⁡-11μ.

To calculate what is energetically more favourable, reactivation of an inherited fault versus formation of a new fault, we calculate the horizontal force FR required to move the wedge of material formed by the side wall, surface and fault angle θ (Fig. c). This force additionally incorporates the weight W=ρgh of the material . For the calculations we assume a normal stress of σN=1000 Pa, analogous to the slide–hold–slide tests corresponding to a height h=3.5 to 5.5 cm for materials with a bulk density of ρ=1800 to 3000 kg m-3, respectively: 6FR=ρgh2cot⁡θ2μ+tan⁡θ1-μtan⁡θ.

A fault is considered severely misoriented when its angle is twice as large as the optimal fault angle. In this case, the term (1-μtan⁡θ)→0 leading to extremely large values for FR. In the locking region, (1-μtan⁡θ)<0 and therefore FR<0, which is unrealistic.

To account for healing, the friction for a reactivated fault is time dependent (i.e. increases with hold time) with the healing rate b as the power law coefficient: 7μr(t)=μ0tb.

This methodology neglects possible edge effects, such as shear stresses along the sidewall, because we assume a continuous granular layer that is cross-cut by several normal faults, which are going to be inverted. Additionally, we do not incorporate changes in the stress field due to differences in elasticity of the material after healing and the formation of lower-density shear bands that could lead to stress concentrations. Another important constraint of this simple model is that it is not suitable for materials with higher cohesion because these tend to form surface cracks during extension, leading to a change in fault angle with depth.

All calculations incorporate full uncertainty propagation through the Python module “uncertainties” assuming normally distributed variables. For the frictional parameters μ, C and b the error given is 2 standard deviations calculated from the covariance of the fit and averaged per material (quartz sand, feldspar sand, glass beads, etc.). For density ρ the value is the arithmetic mean and error is 2 standard deviations of density for each material.

The materials are well sorted and very homogeneous because they are standardised industrial products for specific purposes. As a consequence, they contain few to no impurities, such as clay or pebbles, and are mostly monomineralic. Therefore, the properties presented here should apply to all batches from the same manufacturer. See Table A1 for a more detailed description of each sample and Fig.  for a comparison of the quality index with the frictional properties.

We find that most materials exhibit healing rates in the range of b≤0.03 (Fig. ), i.e. an increase in strength of less than 3 % per 10-fold increase in hold time (e.g. 10 s versus 100 s). Only two samples show healing rates that are not significantly different from zero, “Freiburg” and “Bern, CarloAG (100–300 µm)”, and these are therefore not considered to show healing over experimental timescales. Most quartz sands show lower than average healing rates of roughly 1 % per 10-fold increase in hold time. Foam glass shows healing rates generally lower than those of quartz sands. Glass beads exhibit higher healing rates and show considerable spread between b=0.015 and 0.025. Corundum sands are distributed over the full range of healing rates, and feldspar sands overlap with glass beads. However, the number of samples for materials other than quartz sands and glass beads is too small to generalise.

Compaction rates are more diverse and show a clearer separation between the materials (Fig. b). Glass beads show the strongest compaction rates of c=-0.055 to -0.04 at the reference normal stress of σN=1 kPa. This means that sample thickness decreases by 4 % to 5.5 % per 10-fold increase in hold time. The only exception are the very fine glass beads (Prague, 0–50 µm), which show a very low compaction in comparison to the other glass bead samples. Quartz sands show small compaction rates between c=-0.04 and -0.02, with the majority of quartz sands having c≤-0.03. Feldspar sands and foam glass compaction rates are again comparable with quartz sands. In general, corundum sands exhibit the lowest compaction rates.

There is a strong influence of normal stresses on compaction, which leads to much stronger compaction during hold intervals. However, the healing rate is not significantly correlated with higher compaction, and therefore the measurements with higher normal stresses form a distinct cluster when plotting compaction rate versus healing rate (Fig. ). Most samples do not show a clear distinction between sample material, compaction and healing rate. They all plot in a single cluster, with no significant correlation of compaction rate and healing rate. The glass beads are exceptional because they show high healing rates and high compaction rates and therefore form a separate cluster. Using the main cluster and glass bead cluster, a weak negative correlation of healing rate and compaction rate is visible. An increase in healing with stronger compaction might be recognisable, but this is not statistically significant. The most notable exception is again the very fine glass beads (Prague, 0–50 µm). These show a significantly stronger healing at small compaction rates.

For the reactivation properties we summarised the samples into seven groups by taking the average and the standard deviation 2σ of all properties (friction coefficients μ, healing rate b and density ρ). As material height we choose h=0.05 m, which is comparable to a normal stress of σN=1000 Pa for most materials and lies in the range of typical analogue model setups.

For the first phase of a typical basin inversion model, the material is subjected to extensional stresses (σ1∥Sv, Eq. ), while fault angles are defined by the materials peak friction of μp=0.50 to 0.75, which results in θa=58 to 64∘ (dashed coloured lines in Fig. a–g). During compression the stress field changes (σ1∥Shmax) and so does the angle of newly generated faults (Eq. ). The friction coefficient is the same because new faults mainly cross cut undisturbed material and angles are in the range of θp=26 to 32∘, which leads to flatter faults in the model (solid coloured lines in Fig. a–g).

The locking regions of all materials become increasingly larger as reactivation friction μr increases over time. We find that for all materials, with the exception of glass beads, the angles of the pre-existing faults fall within the locking region. As a result, none of the faults that were created during extension should reactivate because they are severely misoriented. The optimal angle for reactivation using the time-dependent μr is similar to the optimal angle of a new fault with μp. This means that the difference in friction due to healing is not large enough to facilitate sliding along the inherited normal faults.

We find the same for the force required to move a wedge of material along the pre-existing fault or the creation of a new fault (Fig. h–n). For new faults the shear force per unit area ranges between FR=50 and 150 N m-1, which means that for a triangular wedge of 5 cm height and 100 cm length in the direction of σ2 between 50 and 150 N is required to initiate a new fault. While the height of the wedge has an influence on the absolute values of forces, the ratio between the forces FnewFinherited is independent of height. For reactivated faults most materials show negative values, and therefore reactivation is not possible.

The only exception is glass beads, which are still in the field of possible reactivation up to th=104 s after shear. However, the stress required to reactivate these faults is roughly twice as high as for creating a new fault and reaches extremely high values already after th=103 s (Fig. l). A close inspection of the individual glass bead samples shows that glass beads with low grain sizes (e.g. GFZ, GB 70–110 µm) are fully outside the locking region. The stresses to reactivate these faults are only 1.5 to 2 times higher than the stress for new faults. The fault orientation is still more than 20∘ away from the optimal value, but under certain conditions a reactivation is possible for this sample. In general, reactivation is very unlikely despite the small difference between optimal and inherited angles.

We compare the quality score of each material with all frictional parameters to recognise possible influences. To quantify correlations we calculate Pearson r and Spearman ρ correlation coefficients. The Pearson correlation coefficient assumes a linear correlation between two variables, but this is not always the case. Therefore, we also use the Spearman correlation coefficient that is independent of the underlying distribution. For both parameters the p value is determined, which gives an indication of the significance of correlation. Generally, if p<0.05 one assumes that the found correlation is not possible through random permutation of the original data and therefore statistically significant. It should be noted, however, that in this paper the quantification of the quality score is subjective, while the friction parameters (including their errors) are based on real measured values. Thus, the use of correlation coefficients is only an aid to identify trends more easily in the set of measurements and samples.

We find that the healing rate b does show a weak positive correlation with sphericity (r=0.39, p=0.05; ρ=0.40, p=0.04; Fig. a) and is otherwise not correlated with any other quality measure. This means that with higher sphericity the materials tend to have higher healing rates. This is probably related to the compaction rate, which shows a similar negative correlation with sphericity. Spherical grains compact more than aspherical beads, as indicated by the higher compaction rates of glass beads (Figs.  and e). This is consistent with a tendency of glass beads to compact more easily during cyclic axial loading tests reported by . Higher sphericity is associated with lower void ratios before and after shearing , and therefore the individual grains are compacted more during a hold phase. As a consequence, the stress required to mobilise these materials is higher because a larger amount of dilation is needed.

All three friction types (peak, reactivation and static; see Fig. b-d) show a moderate negative correlation with all quality scores (r≈ρ≈-0.6, p<0.05). As a result, there is a tendency to get lower frictional coefficients for materials that have high sphericity, higher roundness and smoother surfaces. The negative correlation of quality index with cohesion is very weak and has a much higher p value, leading to a higher uncertainty in the correlation (Fig. f–h). These results are in accordance with previous studies on granular characteristics (, and references in Table ). We find a high correlation of peak friction with average quality score (r=-0.70, ρ=-0.67, p=0.01) mostly influenced by the correlation of peak friction with roundness and surface quality.

The exceptionally high healing rate for very fine glass beads (Prague, 0–50 µm) is probably the result of a strong increase in cohesion. In contrast to the other glass bead samples, they only show a small compaction rate, i.e. only a small reduction in density. Because of the wide particle size distribution and large amount of fine materials, the void space is largely filled with finer glass powder. Additionally, very fine granular materials are susceptible to electrostatic effects that increase the attractive forces between particles.

Another outlier is the sand from Utrecht that shows stick–slip behaviour. Due this behaviour, we assume an increased healing rate, as in this case there can theoretically be several effects. The higher the healing rate b, the greater the difference a-b, with the direct effect a being constant. In theory, this makes the material more velocity weakening and thus more unstable . Additionally, with higher b the critical stiffness increases. Therefore, a system that was previously stable, such as the ring shear tester, now needs to have a higher stiffness to sustain stable sliding. As a result the system shows a higher tendency for stick–slip behaviour. However, due this behaviour it is not possible to measure the healing rates adequately. This would require a much higher apparatus stiffness, which was not possible to change for our ring shear tester.

Geodetic studies on post-seismic surface motion demonstrate that faults quickly relock after a large earthquake . A similarly high post-slip healing rate is required to explain the observations in analogue models of subduction zone kinematics using sticky rice as a fault zone analogue . We observe the same strong healing in the first few hold intervals caused by the power law relation of strength and hold time (Eq. ), which is also observed for laboratory experiments on synthetic and natural fault rocks . Healing rates for dry synthetic and natural fault gouges range between 0.001 and 0.01 at room temperature . Under wet conditions this healing rate can increase by an order of magnitude at temperatures >300 ∘C (, and references therein). The rates are even higher for gouges containing salt or salt–muscovite layers. A general correlation between the healing properties and the behaviour of fault zones thus seems plausible.

Quartz sands and foam glass show healing rates that are generally low and closer to dry gouges, while the glass beads, especially the very fine glass beads, have values that are closer to hydrothermal healing rates. However, the frictional strength is too low to use glass beads as well-scaled analogues of self-healing faults in a model. A possibility might be the addition of a small percentage of fine material to sands, which would thus achieve cohesion-driven healing while retaining the overall higher bulk strength of sand. Such material mixes were not part of this study but can easily be tested with the methodology outlined here.