A new interpretation for formation of orthogonal joints in quartz sandstone

Le Li, Shaocheng Ji

Département des Génies Civil, Géologique et des Mines, école Polytechnique de Montréal, Montréal, Québec, H3C 3A7, Canada

Keywords:Rock fractures Orthogonal joints Sandstone Auxetic effects Potsdam group

ABSTRACT Two vertical and orthogonal systematic joint sets are generally arrayed in a grid pattern on the bedding surface,which are the significant features of flat-lying sandstone terrains.Although extensive researches are reported on this topic, many fundamental problems have still not been solved.Such mutually perpendicular opening-mode fractures are an obvious manifestation of effective tensile stresses in two orthogonal directions in the horizontal bedding plane.A good understanding of these orthogonal joint systems is a key to structural analysis, landscape interpretation, and guidance of resolving a number of very practical problems in engineering, mining and hydrologic projects.Based on an anatomic investigation on the orthogonal joints in the Potsdam sandstone of Cambrian age at Ausable Chasm (New York State, USA) and Beauharnois (Quebec, Canada), we proposed that the orthogonal joints may result from the auxetic effects of quartz-rich sandstone rather than local or regional rotation of the maximum tensile stress (σ3) direction by about 90°.The sandstone beds with negative Poisson’s ratios are so fascinating that,when placed under vertical burial compression and layer-parallel extension in one direction(σ3),it becomes stretched in the transverse direction (σ2), producing two orthogonal sets of mutual abutting and intersecting joints (J1 and J2 normal to σ3 and σ2, respectively),and both are normal to the bedding surface.Joint set J1 is more closely-spaced than J2 by a factor of ～3.3,which is correlated with an average Poisson’s ratio of －0.3 for the Potsdam sandstone at the time of joint formation.

1.Introduction

Orthogonal joints, which consist of two sets of nearly vertical,opening-mode, brittle fractures aligned at almost right angles, are very common in flat-lying sandstones on platforms (e.g.Hancock,1985; Gauthier and Angelier, 1986; Hancock et al., 1987; Bahat,1989; Dunne and North, 1990; Rives et al., 1994; Caputo, 1995).Orthogonal joints with a variety of abutting and cross-cutting relationships result in a characteristic landform named as‘tessellated pavement’,because the sandstone bedding surface is fractured into more or less regular rectangular blocks that resemble tiles of a mosaic floor or tessellations (e.g.Branagan,1983).Famous examples of tessellated pavement (Fig.1a-c) are observed at the Eaglehawk Neck (Tasmania, Australia) and St.Mary’s Chapel(Caithness,Scotland,UK),which are geological wonders attracting tourists.Orthogonal joints have been produced in brittle analogue materials such as varnish by orthogonal loadings (Fig.1d, Rives et al., 1994).The origin of orthogonal joints is worth studying(e.g.Hancock et al.,1987;Dunne and North,1990;Rives et al.,1994;Caputo,1995), not only because of their important role in forming fabulous landscapes,but also because they tremendously influence the permeability of fluid (e.g.water, gas, oil, and pollutants(Thunvik and Braester,1990;Bear et al.,1993)),rock mass strength and mechanical stability(e.g.Einstein et al.,1983;Shang et al.,2016,2018), and erosion rate of the uppermost crustal rocks (e.g.Bahat,1989; Ji and Saruwatari, 1998; Ji et al., 1998; Rawnsley et al.,1998; Tang et al., 2008).Thus, the knowledge of joints is important for rock engineering and hydrologic applications (excavations of underground openings and chambers, open pits of mines,quarrying operations, slope stability, and groundwater and toxic waste leakage and flow)and fundamental geological research(e.g.determination of paleostress field, and structural control on landforms and drainage patterns).

Orthogonal joints viewed on a bedding-parallel surface can be geometrically classified into two types:(i)Ladder-pattern networkin which cross joints extend across intervals between long parallel joints but do not cut across these joints(e.g.Gross,1993;Rives et al.,1994; Rawnsley et al.,1998); and (ii) Fracture grid-lock system in which both orthogonal joints systematically abut and crosscut each other (e.g.Hancock et al., 1987; Dunne and Hancock, 1994; Rives et al., 1994).According to linear elastic fracture mechanics (e.g.Jaeger and Cook,1979),an extension or mode I fracture form,which is perpendicular to the minimum principal stress(σ3)trajectory and parallel to the principal stress plane containing the maximum principal stress σ1and intermediate principal stress σ2directions(e.g.Hancock,1985; Gross,1993; Ji and Saruwatari,1998; Ji et al.,1998; Jain et al., 2007; Tang et al., 2008).As each of the orthogonal joint sets is extension fractures, presence of the orthogonal joints has been argued due to rotation of the local or regional σ3direction by about 90°.The local stress rotation is presumably caused by stress release on newly formed systematic fractures(e.g.Simon et al., 1988; Dunne and North, 1990; Rives et al., 1994;Caputo,1995;Tang et al.,2006)or reduction in joint spacing to bed thickness ratio(e.g.Bai et al.,2002;Boersma et al.,2018).Although the existent models may explain some features of ladder-pattern cross joints that extend across intervals between systematic joints but do not cut across the systematic joints, it is difficult for the workers to convincingly interpret the formation of grid-lock-type orthogonal joints that mutually abut and intersect (Fig.1a-c).Is there an alternative interpretation for formation of fracture gridlock system? Does the fracture grid-lock system arise as a consequence of roughly simultaneous extension along two orthogonal directions along the layer? In flat-lying sandstone terrains, for example,whether does the presence of orthogonal joints indicate a stress field where effective (compressive) stress σ1* is vertical whereas both σ2* and σ3* are horizontal and tensile (σ3* ≤σ2*＜0).

Fig.1.Traces of orthogonal systematic joints defining a grid-lock pattern.(a-c)Views on a bedding surface of quartz sandstone from Eaglehawk Neck,Tasmania,Australia(a,b)and St.Mary’s Chapel, Caithness, Scotland, UK (c).(d) Orthogonal fractures produced in brittle varnish by orthogonal loadings (Rives et al., 1994).Courtesy Carolyn Bell, Stephanie Sykora, and Mike Norton for pictures (a), (b) and (c), respectively.

Based on an anatomic investigation on the brittle deformation of quartz sandstone from the Potsdam Group at Ausable Chasm(New York State, USA) and Beauharnois (Quebec, Canada) (Fig.2), we proposed a new alternative interpretation that formation of orthogonal joints may result from auxetic effects of quartz sandstone rather than the rotation of the maximum tensile stress direction by about 90°.The flat-lying sandstone layers had presumably negative values of Poisson’s ratio at the time of joint formation so that,when deformed by compression of the overburden,transverse contraction induced horizontal tensile stresses in the bedding plane, producing two roughly orthogonal sets of vertical opening-mode fractures that initiate at various flaws.

2.Geological setting

The orthogonal joints were observed and measured from two sites in Potsdam sandstone(Fig.2):Ausable Chasm(New York State,USA;44.60569°N,73.46330°W)and Beauharnois(Quebec,Canada;45.31613°N, 73.91696°W).The Potsdam sandstone, which is bedded in moderately thick, uniform layers, was deposited along the edge of Ottawa Embayment and Quebec Basin on the Laurentian margin during the Cambrian age(～520-500 Ma)(Landing et al., 2009; Sanford and Arnott, 2009).Rivers located on the northwestern continental shoreline flowed into the shallow sea and deposited deltas of sand.Fisher (1968) reported a cumulative stratigraphic thickness of 450 m for the Potsdam sandstone in the northern Champlain Lowland just between sites 1 and 2 in Fig.2.In the northeastern Adirondack margin, however, the Potsdam sandstone has a thickness of nearly 300 m(Chiarenzelli and Selleck,2016).The sandstone exposed at the sites consists of clean washed quartz sand cemented by silica,and some beds contain a little clay and other impurities (e.g.Globensky, 1987; Sanford and Arnott,2009).The quartz-rich sandstone of the Potsdam Group, which has a porosity between 3% and 3.5%, displays abundant ripple marks caused by tide on sandy beach (Fig.3) and cross beds and dewatering structures formed by migration of the subaqueous dunes during sand deposition.The tide-dominant shallow marine quartz arenite of the Potsdam Group overlies non-conformably on the Precambrian metamorphic basement consisting of gneisses and intrusive rocks such as anorthosite, granite and tonalite of 1.35-1 Ga (Chiarenzelli and Selleck, 2016).The Potsdam Group is conformably but abruptly overlain by dolostone, dolomitic limestone, sandstone and shale of lower Ordovician age (theBeekmantown Group,about 305 m thick),which were deposited in a shallow sea environment (Globensky,1987; Selleck,1993; Salad Hersi et al., 2002, 2003; Dix et al., 2004; Lavoie,2008).

Fig.2.Simplified geological map showing the distribution of the Potsdam sandstone and the sites of study.Modified from Lowe et al.(2015).Stratigraphic column of the study region can be found from Globensky (1987) and Salad Hersi et al.(2003).

Fig.3.Tide-formed ripple marks observed on the upper bedding surface of quartz sandstone at Ausable Chasm (New York State, USA).The mechanical pencil is 14.5 cm long.

Site 1 is located at Ausable Chasm which is a gorge developed in the Potsdam sandstone.Through this gorge,the Ausable River runs for several kilometers from the area just northeast of Keeseville(New York State, USA) toward Lake Champlain.The gorge is 90-150 m deep in places and only 6-15 m wide for narrow parts.The occurrence of such high vertical side walls indicates that the quartz arenites of the Potsdam Group are both mechanically and chemically stable;otherwise,the walls should be broken down and form a valley with the normal V-shaped profile.The rock slope stability results principally from nearly horizontal orientation of the sandstone layers.Since it was first seen by non-native Americans in 1765, Ausable Chasm has been regarded as among the natural wonders of eastern North America and presents an important tourist attraction for geological heritage (Resser,1942).Like many canyons and gorges in the world(e.g.the Grand Canyon carved by the Colorado River in Arizona, USA), Ausable Chasm results from erosion of running water of a swift river along well-developed joints in flat-lying sandstones.Site 2 is a large continuous outcrop located near the Dam of Beauharnois Power Plant of Hydroquebec,Canada (Fig.2).

3.Observations and measurements

The bedding in the Potsdam quartz arenite gently dips northwestward with an attitude(314°,5°)at Ausable Chasm(Figs.4 and 5).Two sets of planar vertical joints(one set aligned at ～140°and the other set at ～50°, which will be referred to as sets J1 and J2,respectively; see Fig.6) at almost right angles to each other, cut systematically the sandstone so that almost everywhere the rock can be taken out in more or less rectangular cuboidal blocks(Fig.7).The blocks bounded by orthogonal joints and bedding surface more readily fall down and then are carried away by running water along the stream.The channel has formed by removal of block after block since the end of the Pleistocene Epoch ice age (which ended up there around 15,000 years ago) (Resser,1942).

Field observations display that NW-SE trending joint set J1 is more persistent than NE-SW trending joint set J2, although both sets are interpreted to be systematic joints.These orthogonal joints are organized essentially in grid-lock pattern (see Figs.4 and 5;Gauthier and Angelier, 1986; Hancock et al., 1987; Caputo, 1995),rather than ladder-pattern (Gross, 1993; Bai et al., 2002).Mutual abutting/cutting relationships (e.g.both sets of joints abut each other) indicate that the orthogonal joints are geologically coeval extension fractures(Hancock et al.,1987;Caputo,1995).Each set of orthogonal joints displays a uniform strike, and is straight and continuous over several meters.As shown in Fig.8, the joint spacing for the NW-SE-oriented set(J1)is statistically about 1/3 of that for the NE-SW-oriented set(J2).Furthermore,these joints are opening-mode fractures and display no detectable shear displacement along them (Figs.4 and 5).No joints contain detectable mineral fill(Figs.4 and 5),indicating that brittle fracturing occurred at shallow depths where temperature was so low that the solubility of quartz or calcite was too low to form veins along the joints(e.g.Dunne and North,1990; van Noten and Sintubin, 2010).Rawnsleyet al.(1998) took this phenomenon as an indicator that fluid pressure was relatively low during joint formation.

Fig.4.Characteristics of orthogonal joints(sets J1 and J2)observed in quartz sandstone at Ausable Chasm(New York State,USA).Joint set J1 is generally more persistent than joint set J2.Note that the observation surface in(d)is parallel to joint plane of joint set 1.In central part of(d),a throughgoing open fracture developed late by coalescence and linkage of joint set J2 across bed boundaries.The pen is 13.5 cm long.

Fig.5.Geometrical relationships between two sets of orthogonal systematic joints(sets J1 and J2)observed on bedding-parallel surface of quartz sandstone at Ausable Chasm(New York State, USA).Mutual intersecting and abutting suggest that the orthogonal joints are geologically coeval.NSJ: Non-systematic joint.

It is interesting to note that the Ausable River at site 1 is not straight and makes a right-angle turn from the northwest direction to the northeast direction near the highway bridge(Fig.9a)and the Whirlpool basin (Fig.9b).The NW- and NE-oriented stream segments are roughly parallel to the vertical joints sets J1 and J2,respectively.The chasm walls stand vertical, straight and smooth because the walls are parallel to one set of the orthogonal joints.Thus, Ausable Chasm provides a good example of the control of systematic vertical joints on orientation of gorge.

At site 2,the bedding in the Potsdam quartz arenite gently dips southwestward with an attitude(236°,3°)and vertical orthogonal joints can be clearly seen on the Google map(Fig.10a)with one set striking about 301°and the other set striking about 212°(Fig.10b).

4.Discussion

As orthogonal joints provide pathways for underground flow of water and hydrocarbons (e.g.Thunvik and Braester, 1990; Bear et al., 1993), and the joint spacing controls sizes of rock blocks and in turn affects rock mass stability and permeability (e.g.Einstein et al., 1983; Shang et al., 2016, 2018), the origin of the orthogonal joints has received considerable attention(e.g.Hancock et al.,1987;Rives et al.,1994;Caputo,1995;Bai et al.,2002).These two sets of joints cannot be interpreted as a conjugate shear fracture system formed by a stress field because the dihedral angle between the two joint sets is approximately 90°, which yields a unrealistic frictional coefficient μ ≈ 0 for the quartz arenite.Two sets of coeval shear joints in a system commonly form an acute angle (θ) described by the following equation:Taking μ ＝ 0.6 (Byerlee, 1978) for example, we have θ ＝ 59°.Furthermore, these orthogonal joints show no evidence of shear displacement.

Fig.6.Preferential orientations of orthogonal joints in quartz sandstone at Ausable Chasm (New York State, USA).N is the number of measurements.

4.1.Existing models

It is generally accepted that each of orthogonal opening-mode joint sets forms perpendicular to the minimum principal stress(σ3) when the effective tensile stress （σ*3＝ σ3－Pfwhere Pfis the pore fluid pressure)exceeds the tensile fracture strength of the rock(T) at the time of failure (e.g.Jaeger and Cook, 1979; Dunne and Hancock, 1994; Tang et al., 2006).Opening-mode fractures develop when σ1－ σ3≤ 4T (Price and Cosgrove,1990), otherwise only shear fractures form.In the Potsdam sandstone studied, the two orthogonal joint sets are roughly perpendicular to the bedding plane that is approximately horizontal.Thus,the stress field at the time of joint formation was characterized by vertical orientation of σ1and layer-parallel alignment of σ2and σ3.Previous researchers proposed that the vertical orthogonal joints could be formed by temporary rotation of the σ3direction by around 90°in the horizontal plane but the maximum principal stress σ1remained vertical(Hancock et al., 1987; Dunne and North, 1990; Stewart and Hancock,1990; Rives et al.,1994; Caputo,1995).Such swapping of σ2and σ3repeatedly occurs to form mutual abutting of joint sets,which were roughly coeval in a geological time scale (Caputo,1995).Accordingly, geometrical relationships display that some numbers of one joint set are older than those in the other set,and vice versa.

The 90°rotation of σ3direction can be caused by various processes:

(1) A switch between σ2and σ3in the horizontal plane normal to the vertical stress σ1is caused by some kind of stress release on newly formed systematic fractures (Simon et al., 1988;Dunne and North, 1990; Stewart and Hancock, 1990; Rives et al.,1994; Caputo,1995).Joints formed in this manner are most probably cross joints that extend across the intervals between existing systematic sets.In other words,cross joints initiate at an existing systematic joint, then propagate laterally,and finally terminate at an adjacent systematic joint(Fig.11a).Cross joints generally do not cut across the systematic joints, forming so-called T and H shapes of joint traces (Hancock, 1985).Thus, their maximum lengths are limited by the separation distance between the two adjacent initial joints.The above process may be periodically repeated.

(2) A switch between σ2and σ3is due to changes in local or regional stress field(Hancock et al.,1987;Dunne and North,1990).Mechanisms to cause the local stress switch comprise folding,faulting and erosion(e.g.,Isachsen,1975;Rives et al.,1994).The orthogonal joints formed by the regional stress rotation are systematic joints over a broad region (e.g.in orogenic forelands).If the regional stress rotation takes a few million years as inferred from previous results (e.g.Mercier et al., 1987; Tapponnier et al., 1990), one set of joints may be readily distinguishable to have formed prior to the other based on their intersecting and abutting relationships.Joint spacing of each set, after the fracture saturation has been reached,depends mainly on the magnitudes of σ*3during the brittle deformation before and after stress rotation.The higher the magnitude of σ*3in one direction is,the closer the joints perpendicular to this direction are.According to Dunne and Hancock (1994), the square-shaped fracture grid-lock system (e.g.Rawnsley et al., 1992) may result from 90°switching of σ*2and σ*3axes of approximately equal magnitude.However, independent field evidence to support the rotation of regional stress field is often absent.

Bai et al.(2002)proposed a critical joint spacing to bed thickness ratio (s/t)cfor the local tensile stress (σ3) switches from being perpendicular to parallel to the strike of the systematic joints in a brittle layer.When s/t ＞ (s/t)c,the local stress(σ3)is perpendicular to the systematic joints,and sequential infilling of systematic joints occurs.When s/t ＜ (s/t)c, the local tensile stress(σ3) becomes parallel to the systematic joints,orthogonal cross joints form across the intervals between two adjacent systematic joints (Fig.11b).As a result, the systematic joints formed prior to the cross joints.Numerical computations of Bai et al.(2002) suggested that (s/t)cranges from 1 to 2 considering the stress field within the sedimentary rocks.However, their model cannot explain our field observations.As shown in Fig.12, no orthogonal cross joints have formed between the systematic joints in greywacke layers consisting of quartz,feldspar and clay,even though the joint spacing to bed thickness ratio is as low as 0.1-0.2.Bai et al.(2000)proposed s/t ＝1 as the critical joint spacing to bed thickness ratio for joint saturation after which no new tensile fractures can form between two existing fractures because the stress becomes compressive rather than tensile.Wu and Pollard (1995) performed a series of extension experiments using a brittle coating material(methylene chloride) on a polymethyl methacrylate substrate and found that the critical joint spacing to bed thickness ratio for joint saturation ranges from 7.4 to 11.7.These values are much higher than that proposed by Bai et al.(2000) from numerical modeling.The numerical modeling of Bai et al.(2000) assumed that an isotropic,homogeneous,elastic whole space with equally spaced fractures is elastically deformed under horizontal principal tensile stresses(i.e.both σ2and σ3are tensile).Whether their input rock mechanical parameters(e.g.tensile strength,Young’s modulus E ＝ 30 GPa,and Poisson’s ratio ν ＝ 0.25) are representative of in situ sedimentary rocks during the formation of the orthogonal joints remainsuncertain(e.g.Freund,1992;Ji et al.,2002;2018;van der Pluijm and Marshak, 2003; Bandyopadhyay and Abdullah, 2013; Mahmoud et al., 2019).More critically, Bai et al.(2000) assumed that the opening-mode fractures have a zero tensile strength, disregarding the effects of ‘rock bridges’ in natural rocks (Zheng et al., 2016;Shang et al., 2018).With these points in mind, it is not surprising that the numerical results from oversimplified models are not necessarily applicable to real rocks in nature.

Fig.7.Rectangular cuboidal sandstone blocks bounded by vertical orthogonal joints and horizontal bedding surface at Ausable Chasm (New York State, USA).

Fig.8.Plot of joint spacing for set J1 (sJ1) versus joint spacing for joint set J2 (sJ2) for quartz sandstone at Ausable Chasm (New York State, USA).Spacings sJ1 and sJ2 are respectively the length and width of each rectangle bounded by joint sets J1 and J2,measured on the bedding-parallel surface.

4.2.A new interpretation

Previous researchers observed that orthogonal tensile joints formed by a strain field with λ1≥ λ2＞ 1 and λ3≤ 1(e.g.Simon et al.,1988), where λ1, λ2and λ3are the major, intermediate and minor quadratic elongations,respectively.Analogue experiments of Rives et al.(1994) using analogue brittle materials such as varnish demonstrated that two orthogonal extensile loadings could induce an orthogonal fracture pattern with abutting and intersecting relationships (Fig.1d).How do the tensile stresses appear simultaneously along two orthogonal direction in a sandstone layer within the Earth’s upper crust?

Here we proposed alternatively that the two sets of mutual abutting and crosscutting orthogonal joints formed due to auxetic behavior of in situ quartz arenite layers that possessed negative values of Poisson’s ratio during horizontal contraction caused by overburden pressure (Fig.11c).Such an auxetic material is so fascinating that, when placed under layer-normal compression, it becomes shortened in the layer-parallel direction (Lakes, 1987;Alderson and Alderson,2007;Greaves et al.,2011;Ren et al.,2018).In the case of flat-lying Potsdam sandstone,the effective stress σ*1is vertical and given by

where ρ is the average density of the overlying rocks from the surface to depth z (e.g.ρ ＝ 2500 kg/m3for sandstone), g is the gravitational acceleration,and λ is the pore fluid factor(i.e.ratio of pore fluid pressure to overburden pressure, λ ＜ 1, and λ ＝ 1/2.5 ＝0.4 for hydrostatic conditions).Assuming that the elastic strain in the direction along the horizontal layer is fully constrained, the effective minimum principal stress σ*3can be estimated from the following equation (Jaeger and Cook,1979):

Fig.9.Control of orthogonal joints on gorge orientation at Ausable Chasm (New York State, USA).The Ausable River makes a right-angle turn from the northwest direction(joint set J1)to a northeast direction(joint set J2)near the highway bridge(a)and the Whirlpool basin (b).

Fig.10.Orthogonal joints (a) and their preferential orientations (b) in sandstone at Beauharnois, Quebec, Canada.

where ν is the Poisson’s ratio.

As shown in Fig.13, σ*3becomes tensile (negative) only when the Poisson’s ratio is negative regardless the λ value (λ ＝ 0 and λ ＝ 0.4 for lithostatic and hydrostatic conditions, respectively).Furthermore,the absolute magnitude of σ*3decreases with raising pore fluid pressure.Superposing a horizontal tectonic stress (Δσ)over the gravity-induced horizontal stress will slightly change the calculated σ*3value, depending on whether the tectonic stress is compressive or tensile within the shallow crust(≤5 km depth).For the same reason,an auxetic(negative Poisson’s ratio)quartz arenite layer becomes expanded in the direction of σ*2when stretching in the direction of σ*3,where both σ*2and σ*3are parallel to the layer.The layer-parallel tectonic stress (Δσ) can result from tectonic uplift, flexure due to folding, or updoming of sandstone body by underlying magmatic intrusion or salt diapirism (e.g.Isachsen,1975;Rives et al.,1994).

A conventional material or rock with a positive Poisson’s ratio becomes thinner in directions orthogonal to the applied extension.However,an auxetic material with a negative Poisson’s ratio shows a ‘counterintuitive’ behavior: it becomes stretched in directions orthogonal to the applied extension(Lakes,1987;Ren et al.,2018).The transverse stretching results in transverse deformations with tensile stresses in the directions both parallel and perpendicular to σ3due to its unusual mechanical response caused by the auxeticeffects (Evans and Alderson, 2000; Alderson and Alderson, 2007;Greaves et al., 2011).Natural rock layers such as the Potsdam sandstone were abundant in flaws(e.g.microcracks,microcavities,defects, inhomogeneity, and irregularity on bedding or fracture plane).Each extension fracture nucleated preferentially at such a flaw where the material had locally a low tensile strength or stress concentration occurred.Not all flaws could initiate tensile fractures.Only those whose long axes were aligned perpendicular or high angles to tensile stress σ3or σ2were preferentially chosen and successfully propagated to become continuous joints.In addition,the materials at different flaws of varying size and shape have different tensile strengths.A younger joint could crosscut either a rock bridge or a healed fracture.Repeated failure events, which were related to cyclic accumulation-relaxation of stresses, eventually generated grid-lock pattern orthogonal joints in each auxetic sandstone layer subjected to vertical compression and horizontal extension under elastic-brittle conditions.

Fig.11.Schematic illustration of different models to interpret the development of ladder-pattern (cross joints, a and b) and grid-lock pattern (c) orthogonal joints in a flat-lying sandstone bed.(a)Cross joints develop perpendicular to the systematic joints due to regional stress rotation.(b)Cross joints form laterally between adjacent systematic joints due to stress rotation caused by local stress release or reduced joint spacing to bed thickness ratio(s/t).(c)Grid-lock joints formed by repeated extension along two orthogonal directions in sandstone due to the auxetic effects (negative Poisson’s ratio).The effective maximum principal stress σ1* (overburden) is compressive and normal to the sandstone layer,whereas both effective immediate and minimum principal stresses （σ*2 and σ*3）are tensile and perpendicular to widely-and narrowly-spaced joint sets,respectively,at the time of fracturing.Joint sets J1 and J2,which are normal to σ*2 and σ*3 ,respectively,display spacings of sJ1 and sJ2.Different joints nucleated at flaws with varying shape and size,where the materials had different tensile strengths and stress concentration factors.A new joint could crosscut “rock bridges” and healed or partially sealed fractures.

Fig.12.Examples of jointed calcareous greywacke beds with joint spacing to bed thickness ratio s/t ≤0.2 but without cross joints that developed as predicted by numerical models of Bai et al.(2002).Calcareous greywacke consisting of quartz,feldspar, calcite and clay has a positive Poisson’s ratio(Ji et al.,2002).Samples collected from the External Humber Zone of the Appalachian orogen at Petite Vallée,Quebec,Canada(GPS coordinates:49.22133°N,65.03569°W).(a,b)View from a plane perpendicular to the bedding surface and to the systematic joints.(c, f) Jointed sandstone resembling a sliced-up loaf of bread.(d, e) View from the bedding surface.

The proposed process explains reasonably the formation of the vertical orthogonal joints in flat-lying sandstone.The major and subordinate sets of systematic joints are roughly normal, respectively,to σ3and σ2;both are tensile(Fig.11c).Furthermore,the ratio of joint spacing of J1 to J2 is correlated with the value of negative Poisson’s ratio.The sandstone exposed at Ausable Chasm shows a typical sJ2/sJ1ratio of 3.3 (Fig.8), suggesting that a Poisson’s ratio(ν ＝ -ε2/ε3)of－0.3 corresponds to-sJ1/sJ2,assuming reasonably an average tensile strength of sandstone in the bedding plane.This value is consistent with the experimental data of sandstones measured at confining pressures of below ～100 MPa(Fig.14)or at depths of less than about 4.5 km.

Poisson’s ratios illustrated in Fig.14 are the dynamic values obtained by measuring velocities of P-and S-waves(VPand VS)propagating through the rock.The method is non-destructive, costeffective,and time-efficient(e.g.Ji et al.,2002;Yu et al.,2016).In a static test of Poisson’s ratio measurements, however, the strain is larger and an inelastic deformation of the sample(e.g.microcrackinduced transverse dilatation) occurs (e.g.Simmons and Brace,1965;Cheng and Johnston,1981;Li and Fj?r,2012).The static and dynamic Poisson’s ratios are roughly equal in magnitude if the effects of crack-induced dilatancy have been eliminated or corrected in the static deformation tests (e.g.Ciccotti et al., 2004; Xu et al.,2016; Ji et al., 2019).Thus, there is generally no significant dependence of the Poisson’s ratio on either frequency or strain amplitude because it depends on only the ratios of E/G (E is the Young’s modulus and G is the shear modulus)or VP/VSrather than their absolute values (Ciccotti and Mulargia, 2004).Recently, Tinni et al.(2019) conducted experimental measurements on acrylic blocks and reported that propagation velocities of hydraulic fractures are as high as 45-96 m/s.Thus, it would be reasonable to use dynamic rather than static properties when dealing with problems such as initiation and propagation of fractures in materials and rocks.

Isotropic aggregates of α-quartz with zero porosity has extremely low Poisson’s ratio (0.06, Heyliger et al., 2003).Laboratory measurements (Fig.14) show that quartz-rich sandstone and siltstone containing microcracks and micropores display negative

Poisson’s ratios (ranging between －0.4 and －0.1) at low confining pressures (＜150-200 MPa) (Freund,1992; Ji et al., 2018).For the same reason, the in situ sandstones, which presumably possessed negative Poisson’s ratios, acted as auxetic materials at the time of fracturing by vertical burial compression and horizontal NE-SW extension (Fig.9).The sandstone beds stretched in the direction not only parallel but also perpendicular to the maximum tensile stress(σ3)direction in the flat-lying layers,forming closely-spaced fracture set J1 and widely-spaced fracture set J2.

Fig.13.Horizontal effective stress （σ*3） within a flat-lying layer due to the vertical overburden （σ*1） at a depth of 4 km.Conditions with different pore fluid factors (λ) have been considered.λ ＝ 0 and λ ＝ 0.4 represent lithostatic and hydrostatic conditions,respectively.Tensile stress σ*3 occurs only when the material is auxetic with negative Poisson’s ratios or Pf ＞ σ3 (not shown).The density is taken to be 2.5 g/cm3.

Fig.14.Sandstones with zero or negative values of Poisson’s ratio.Poisson’s ratio as a function of density(a),porosity(b),and confining pressure(c,d).Experimental data in(a-c)from Freund(1992)and(d)from Ji et al.(2018).Sample numbers are also indicated in(c,d).Data of Ken-Betwa sandstone from Bandyopadhyay and Abdullah(2013)are indicated in(b).

The fracture pattern of the study region seems to indicate that quartz arenite layers are very possibly natural auxetic materials that demonstrate similar mechanical behavior to artificial auxetic materials fabricated in the laboratory(Lakes,1987;Alderson and Alderson,2007; Greaves et al., 2011; Ren et al., 2018).These rocks exhibit a natural ability to form geologically coeval, orthogonal joints upon vertical burial compression and horizontal extension.For this reason,the model proposed here for the Potsdam sandstone is believed to be applicable to the interpretation for the origin of similar fractures in other terrains of quartz arenite such as Eaglehawk Neck (Tasmania,Australia), St.Mary’s Chapel (Caithness, Scotland, UK) and Southwestern USA(e.g.Utah,Nevada and Arizona).

4.3.Implications

Orthogonal joints play an extremely important role in enhancing fracture interconnectivity and fluid flow in rocks.Weathering and erosion through ages have preferentially attacked the walls of the orthogonal joints in sandstones,giving rise to some of the most spectacular landforms on the Earth(e.g.Ji,2019):tower karst in northern Australia, Danxia-style landscapes in China,buttes and rocky pillars, pinnacles and needles in southwest USA(e.g.the Canyonlands National Park, Arches National Park).The visual impression of the tall,flat-topped,steep,often vertical sided tables and towers of sandstone catches people’s imagination, so these joint-bounded sandstone blocky structures have been dubbed with names like ruiniform reliefs in the western and central Sahara, and Southern Brazil, and rock cities in the Bohemian Cretaceous Basin in the Czech Republic,Germany and Poland(Melo and Coimbra,1996;Adamoviˇc and Coubal,2015;Migo′n et al.,2017).Furthermore, the effects of orthogonal joints should be taken into consideration when modeling underground migration of water,gas, oil, and pollutants in the sandstone terrains.In addition, the presence of mutual intersecting orthogonal joints in sandstones can substantially reduce the rock mass strength and slope stability,and hence increase the regional erosion rate.

5.Conclusions

Although a significant progress has been made, during the last hundred years, in understanding of brittle fractures or joints in terms of stress and strain, many important and interesting problems still require further investigations.For example,the presence of grid-lock pattern,opening-mode joints has perplexed geologists for decades because they cannot satisfactorily explain why two sets of tensile fractures developed perpendicularly to each other in the same stress field.Based on an anatomic investigation on the orthogonal joints in the Potsdam sandstone of Cambrian age at Ausable Chasm (New York State, USA) and Beauharnois (Quebec,Canada),we proposed that the vertical orthogonal joints may result from the auxetic effects of quartz-rich sandstone in the absence of local or regional stress rotation.In the terrains of flat-lying sandstone with widely-and narrowly-spaced vertical orthogonal joints,both σ*2and σ*3are tensile and perpendicular to widely- and narrowly-spaced sets, respectively, whereas the vertical stress(overburden pressure) is always compressive (σ1), at the time of fracturing.The aspect-ratio of sandstone blocks bounded by orthogonal joints is correlated with the value of negative Poisson’s ratio of the sandstone.Actually, no independent geological evidence from the field has been systematically documented to support the local or regional rotation of the maximum tensile stress(σ3)due to changes in the ratio of joint spacing to bed thickness of the systematic joints.Furthermore, this study highlights that mechanical properties of rocks other than layer thickness, applied stress, bulk strain and pore fluid pressure, play an extremely important role in affecting the joint arrays in natural rocks.

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

Shaocheng Ji thanks the Natural Sciences and Engineering Research Council of Canada for a discovery grant, and Dr.Terry Engelder for helpful discussion.