Sills and dikes are traditionally treated as extension fractures with a dilation vector normal to the fracture wall; i.e. they are extension fractures (Mode I; e.g. Anderson, 1951). This assumption has important implications for the use of sheet intrusions in constraining tectonic stress states because the minimum compressive stress (σ3) is perpendicular to the extension fracture walls; local deflections of the intrusion attitude are commonly inferred to represent local rotations of the stress axes. This is most commonly attributed to mechanical layering and the presence of pre-existing structures (e.g. Rubin, 1995; Gudmundsson, 2002; 2011a; Magee et al., 2016). This model implies that intrusions can locally propagate out of the regional σ1–σ2 plane, via Mode I failure of intact rock, or through Mode I dilation of pre-existing structures, producing intrusions that display variable dilation vectors along a single intrusion. Notably, many field examples of sills and dikes exhibit near-parallel dilation vectors, regardless of the intrusion attitude (e.g. Hoek, 1991; Walker, 1993; Airoldi et al., 2011; Martinez-Poza et al., 2014; Muirhead et al., 2014; Eide et al., 2016; Walker et al., 2017). Intrusions that demonstrate shear offset of markers across their margins indicate that during emplacement the dilation vector was inclined from plane normal (Muirhead et al., 2014; Stephens et al., 2017; Walker et al., 2017); this obliquity of opening can be characterised by the opening angle (Fig. 1).

Dilated structures have been studied in detail by Delaney et al. (1986), Baer et al. (1994), and Jolly and Sanderson (1997), who applied mechanical methods to estimate paleostress states using sheet intrusion attitudes. These mechanical models have been adopted for statistical constraints on paleostress axes, and the paleostress state, from vein or dike data (e.g. Sato et al., 2013; Yamaji, 2016). Although several methods exist to determine paleostress axes and a state of paleostress via fault, fracture, or dike data, subhorizontal sheet intrusions (sills) are typically omitted from such analyses. This is because sill emplacement is commonly associated with a local rotation of the principal stress axes, as a result of bedding or other layer heterogeneities, during regional extension or a low-deviatoric stress state (i.e. where σ1≈σ2≈σ3; e.g. Gudmundsson, 2011a; Magee et al., 2016). Several studies have demonstrated that sills may also form during tectonic horizontal shortening, in which the minimum principal stress is vertical (Walker, 2016; Walker et al., 2017; Stephens et al., 2017); hence sills can be used as paleostress indicators, provided they are accompanied by detailed kinematic characterisation.

Here, we present mechanical models, based on those of Jolly and Sanderson (1997), to determine paleostress state using the attitude of dilated fracture sets; we verify this using the measured opening angle of the intrusions. The method is applicable to any dilated fracture; here we focus on the stress state associated with sill emplacement in the San Rafael subvolcanic field (SRSVF), Utah, and compare these results to fault data in the same area to demonstrate the particular importance of subhorizontal igneous intrusions as records of paleostress.

Dilation of a cohesionless pre-existing planar structure occurs when the fluid pressure (Pf) exceeds the normal stress on the plane (σn; Fig. 1) (Delaney et al., 1986; Jolly and Sanderson, 1997). Normal stress is related to the plane attitude (θ) and the maximum (SH) and minimum (Sh) principal stresses acting on the plane (Jolly and Sanderson, 1997): σn=SH+Sh2+SH-Sh2cos⁡2θ, with the range of possible dilatant fracture attitudes controlled by the stress ratio (φ), ϕ=σ2-σ3σ1-σ3, and the driving pressure ratio (R′) (Baer et al., 1994): R=Pf-σ3σ1-σ3. The stress ratio, ϕ, is a non-directional value that describes the relative magnitudes of the principal stresses: σ3≤σ2≤σ1 (here, compressive stresses are positive). The driving pressure ratio R′ describes the relative magnitudes of the fluid pressure and the remote stress state: when R′<0 (i.e. Pf≤σ3) there is no dilation, and when R′≥1 (i.e. Pf≥σ1) all fracture attitudes could dilate. This assumes that an intrusion is emplaced during a deviatoric stress state (i.e. where σ1≠σ2≠σ3); if Pf>σ3 during a non-deviatoric stress state (i.e. where σ1=σ2=σ3), all pre-existing fracture attitudes could dilate.

Pre-existing fractures, with poles parallel to the σ3 axis, will show plane-normal dilation (i.e. dilation parallel to the normal stress vector). Fractures inclined to the plane of σ3 resolve a shear stress on their surface (Delaney et al., 1986). During dilation, shear stress is reduced to zero through a plane-oblique dilation vector (extensional shear). The angle between the dilation vector and normal stress is defined as the opening angle (μ; Fig. 1a–c) and represents the ratio of shear to dilation (Delaney et al., 1986). Extensional shear therefore acts to reduce the amount of dilation. For an intrusion comprising inclined and subhorizontal sections, the inclined sections will therefore be thinner than subhorizontal sections oriented perpendicular to σ3 (e.g. Fig. 1c, d) for a given fluid pressure, providing the stress state is deviatoric and 0<R′<1 (England, 1988). The dilation vector can be measured using traditional compass techniques in the field based on offset piercing points or unique contact geometries, such as recognisable corners in the intrusion walls (e.g. Figs. 1c, 2a, b). True offsets result from extensional-shear opening (Fig. 2c, d); however apparent shear offset can be produced by marker units that are oriented at an oblique angle to the intrusion plane (Fig. 2e) or via dilation of pre-existing faults, which produce a larger-than-expected opening angle (Fig. 2f). To determine whether offset and the opening angle are true or apparent, multiple manual measurements of opening angle and intrusion thickness must be made along strike of an intrusion (Figs. 1c, 2g). The opening angle (μ) of a dilated fracture can be calculated as the inverse cosine of the true thickness (t, normal to the plane) and the vertical thickness (tv, parallel to the dilation vector; Fig. 1c): μ=cos⁡-1(t/tv). Alternatively, μ can be measured as a rake of the obtuse angle between the intrusion contact and the dilation vector, minus 90∘. Importantly, this measurement only accounts for the opening angle in two dimensions; movement in or out of the face will not be recorded; therefore dip-parallel sections should ideally be used.

The opening angle is related to the shear stress (τ), normal stress (σn), and fluid pressure (Pf) acting on that plane at the time of intrusion (Delaney et al., 1986; Jolly and Sanderson, 1997): μ=tan-1τPf-σn. Equation (5) shows that if the fluid overpressure (Pf–σn) is equal to the shear stress, the opening angle is 45∘, and the shear-to-dilation ratio is unity. If the overpressure is greater than the shear stress, the opening angle is less than 45∘, and the fracture will show a greater component of dilation to shear. When μ is negative, the fracture will remain closed as the fluid pressure did not exceed the normal stress. An intrusive segment, however, may inflate against a closed fracture (where Pf<σn), causing a local contractional shear and a blunt intrusion tip (e.g. Fig. 2g).

Equation (5) can be represented in three-dimensional (3-D) space, for any given stress state, stress ratio, and fluid pressure: Fig. 3 shows the opening angle (μ) of all possible fracture planes in 3-D space, plotted as colour-contoured pole-to-plane values of μ, on equal-area lower-hemisphere stereographic projections and on 3-D Mohr diagrams. Fluid pressure was calculated at five equal intervals, which are expressed as R′ values (Fig. 3a), relative to the ambient stress state. Figure 3 shows plots of opening angles for three tectonic regimes in which ϕ is 0.5: (1) a thrust-fault regime (σv=σ3) (Fig. 3b), (2) a strike-slip regime (σv=σ2) (Fig. 3c), and (3) a normal-fault regime (σv=σ1) (Fig. 3d). The principal stress attitudes are constant in all models, with the azimuth for the maximum horizontal stress (σH) trending E–W. All planes are modelled as cohesionless surfaces.

The models complement the results of Jolly and Sanderson (1997) and demonstrate that within a given stress state, increasing the fluid pressure increases the range of pre-existing fracture attitudes that can dilate, provided the dilated fracture is linked to the magmatic source. The models also show that for any given fracture, increasing Pf decreases μ. Where σ3<Pf≤σ2 (Fig. 3b–d, i–ii), the dilation zone, delineating the poles to fractures that are predicted to dilate forms an ellipse about the σ3 axis and is elongate in the direction of the intermediate stress (σ2); only fractures with poles parallel to the σ3 axis show Mode I opening. If σ2<Pf<σ1 (Fig. 3biii–diii) the dilation zone forms a girdle parallel to the axis of σ2, with two defined zones of near-Mode I opening (0–10∘) surrounding the σ2 and σ3 poles. Fractures of all attitudes can dilate if Pf=σ1 (Fig. 3biv–div). The models suggest that Mode I opening of pre-existing fractures should only be common if the fluid pressure exceeds σ1 (e.g. Fig. 3bv–dv). For dilation of misoriented cohesionless fractures, with planes oriented at a high angle to σ3, a higher fluid pressure is required to overcome the normal stress acting on their surface (e.g. Gaffney et al., 2007). Importantly, if misoriented planes do not become linked to the fluid network, high fluid overpressures will be relaxed by dilation of preferentially oriented pre-existing structures, or via failure of intact rock (e.g. Sibson, 2012). Dilational fracture data should therefore plot stereographically with the greatest density of poles to planes about the σ3 axis (Jolly and Sanderson, 1997; Yamaji et al., 2010). Propagating intrusions may cut across or terminate against misoriented structures, which plot in the zone of no dilation (white areas on the Mohr circles and stereonets; Fig. 3), as the normal stress acting on these planes exceeds the fluid pressure, preventing dilation (e.g. Gaffney et al., 2007; Gudmundsson, 2011a). Terminated intrusions may inflate against closed fractures if magma supply continues; dilation will be accommodated by local shear along the closed fracture, producing a blunt intrusion tip (Baer, 1991; Stephens et al., 2017). Through measurement of the opening angles of veins or intrusions, and determining the distribution of fracture attitudes that are in contact with intrusions, but not dilated, it is therefore possible to constrain the fluid pressure relative to the ambient stress state.

In addition to changes in fluid pressure, fracture dilation is also sensitive to changes in remote stress, particularly at lower Pf. For instance, where σ3<Pf≤σ2 (Fig. 3b–d, i–ii) each stress regime is defined by a particular range of fracture attitudes, with a unique opening angle pattern. Where σ2<Pf≤σ1 (Fig. 3b–d, iii–iv), the thrust-fault regime (Fig. 3biii–iv) and strike-slip regime (Fig. 3ciii–iv) have similar opening angle patterns, whereas the distribution of attitudes and opening angles for the normal-fault regime (Fig. 3diii–iv) is unique. Applied carefully, using detailed kinematic field analyses of sills, and their associated deformation, the method has the potential to discriminate among intrusions related to different remote stress states. Here, we have applied this method to igneous sills in the SRSVF in Utah.

Dilated fractures, created during the same dilational event, will produce a unique distribution of opening angles when plotted stereographically as poles to planes (e.g. Jolly and Sanderson, 1997; Yamaji et al., 2010; Sato et al., 2013). Whether this is a tight cluster or a distributed set of poles, the pattern is usually interpreted to represent a single event, governed by one fluid pressure (e.g. Jolly and Sanderson, 1997); however, it is possible that the data represent multiple events caused by fluid pressure pulses of varying magnitude (Yamaji et al., 2010). As the opening angle represents the ratio of shear to dilation (Delaney et al., 1986), magmatic events of varying magnitude within the same governing stress state could produce cross-cutting intrusions with similar geometries but different opening angles (e.g. Fig. 3). The attitude and opening angle of dilated fractures, along with cross-cutting relationships, can therefore be used to identify whether a network of dilated fractures represents one, or multiple, fluid pressure pulses and the stress state during each pulse.

The SRSVF is located in the north-western Colorado Plateau, central Utah (Fig. 4a), and comprises numerous sills, dikes, and volcanic breccia bodies (Fig. 4b, c). The close spatial association of these different intrusive bodies has been suggested to reflect emplacement during a low deviatoric stress state (Delaney and Gartner, 1997; Richardson et al., 2015). No crustal magma chambers have been identified as the source for the intrusive complex (Delaney and Gartner, 1997). The exposed intrusive system was emplaced at ∼ 1 km depth, within the Middle Jurassic sedimentary rocks of the San Rafael Group, which comprises four formations: the Carmel Formation (siltstone and shale), Entrada Sandstone (interbedded sandstone, siltstone, and claystone), Curtis Formation (interbedded glauconitic sandstone, and siltstone), and the Summerville Formation (siltstone, sandstone, and shale) (Gartner, 1986; Richardson et al., 2015). The sequence represents a paralic environment, with near-shore and shallow marine deposits (Gartner, 1986; Peterson, 1988). The sills are mainly emplaced within the Entrada Sandstone; however they also cut across formation boundaries to intrude the Carmel and Summerville formations (e.g. Fig. 4d; Walker et al., 2017). Richardson et al. (2015) highlighted an example of a dike-to-sill transition for the Cedar Mountain sills (Fig. 4b). Our study focuses on sills in the southern SRSVF (Fig. 4b), and we cannot confirm that relationship here. In our study area, sills and dikes display mutual cross-cutting relationships (e.g. Fig. 4e; Gartner, 1986; Walker et al., 2017), and we have observed no dike–sill transitions for the major sills (e.g. Fig. 4c). Delaney and Gartner (1997) determined ages of dike emplacement between 4.6 and 3.7 Ma; the field relationships therefore constrain sill emplacement to a ∼ 1 Myr interval.

Walker et al. (2017) interpret the sills in the SRSVF as low-angle conjugate intrusions, and concluded that the sills record a phase of horizontal tectonic shortening, rather than relating to local deflections due to material layering. The sills range from < 10 cm to ∼ 30 m thick and are discordant to bedding, with dips of 1–25∘ (Fig. 4d, f). Based on sill attitude measurements over kilometre-scale outcrops, Walker et al. (2017) showed that sill poles cluster about a near-vertical axis with two defined clusters: a NW- and SE-dipping set in the northern SRSVF and a NE- and SW-dipping set in the southern SRSVF. This study focuses on the NE- and SW-dipping sills in the southern SRSVF (Fig. 4b, c).

Across all scales, we find mutual cross-cutting relationships between sills with bimodal dips (e.g. Figs. 1c, 5a–c), and en-echelon sill segments which define low-angle (∼ 20∘ dip) conjugate sets (Fig. 5a). Deformation bands and gypsum veins have comparable attitudes to sills and thrust faults in the field area and define low-angle conjugate sets with NE and SW dips (Fig. 5d–i). Sills intruded and have been cut by low-angle fractures, thrust faults, and reverse faults (all referred to as thrust faults henceforth) and either cut or abut against sub-vertical fractures (Figs. 4d, 5f; Walker et al., 2017). A constant near-vertical opening direction is noted across all sill segment attitudes (Figs. 6, 7a). Local sill contact data display a range of dips from 4 to 87∘ with horizontal to subhorizontal sill segments (0–20∘ dip) showing plane-normal (Mode I) and extensional-shear dilation (μ = 0–20∘), respectively (Figs. 6a, 7a). Inclined segments (20–56∘ dip) are consistently thinner than adjoining horizontal segments and display reverse-sense shear offset of piercing points (20∘<μ≤75∘; Fig. 7a). Subvertical to vertical fractures (< 70∘ dip) were not intruded (Fig. 7a), suggesting that fluid pressure was less than the normal stress on those planes during intrusion, and the opening angle was < 0∘.

Since the range of dilated sill attitudes is known, we can determine the relative stress and fluid pressure using the parameters ϕ and R′. We calculate these parameters stereographically, using an adapted method from Jolly and Sanderson (1997). The local sill contact data are plotted as poles to planes and coloured by their μ value; the pole cluster showing Mode I opening (μ=0–10∘) is fitted with an ellipse (Fig. 7b). For data with a clustered distribution, this ellipse geometry provides a guide for the total range of dilated fractures, for which blunt tips define the limit of fracture attitudes able to dilate; beyond this is the zone of no opening (where μ<0∘; Fig. 7b). The minimum compressive stress (σ3) plots in the centre of the Mode I ellipse (which likely coincides with the centre of the data cluster); σ1 and σ2 are mutually orthogonal to this (Fig. 7c). If the data have a girdled distribution, the 0–10∘ ellipse should be used as an approximation for the geometry of the zone of no opening. In these cases, there will be two clusters of poles where μ=0–10∘ (see Fig. 3biii–diii); the larger ellipse signifies the location of σ3. A distributed set of dilated fractures may contain several zones of Mode I opening (μ=0–10∘; see Fig. 3): σ3 will plot in the larger of these ellipses and σ1 in the zone of highest opening angle, or the smallest Mode I ellipse when Pf exceeds σ1 (see Fig. 3bv–bv).

Field data for sills in the SRSVF can be fitted to an elliptical region on the stereonet with a NW–SE long axis (Fig. 7b), giving horizontal NE–SW maximum compression where σ1 plunges 3∘ towards 068∘; σ3 is vertical, plunging 87∘ towards 265∘; and σ2 is horizontal, plunging 1∘ towards 158∘ (Fig. 7c). Using the stereonet data in Fig. 7c, we derive the angles θ1 and θ2, which are used to calculate the stress ratio (ϕ) and driving pressure ratio (R′) by Jolly and Sanderson (1997): ϕ=1+cos⁡(2θ2)1+cos⁡(2θ1), and R′=1+cos(2θ2)2, where θ1 is the angle between the σ2 axis and the perimeter of the dilational ellipse and θ2 is the angle between the σ1 axis and the dilational ellipse (Jolly and Sanderson, 1997). When an ellipse can only be fitted to the zone of no dilation, Jolly and Sanderson (1997) provide an alternative method for calculating ϕ. The R′ value is two-dimensional (2-D); it does not take into account the magnitude of σ2. The three-dimensional relationship between the fluid pressure and all principal stresses is illustrated by the ellipse geometry on the stereonet (Fig. 7c, d), and can also be visualised through construction of a 3-D Mohr circle (Fig. 7e).

The calculated ϕ (0.77) and R′ (0.68) values define the stress ratio and driving stress used to create the opening angle mechanical model (Fig. 7d). For the model, we assigned a minimum vertical stress of 25 MPa to simulate a ∼ 1 km emplacement depth (Gartner, 1986). Development of extensional-shear fractures requires a low differential stress (σD: σ1–σ3): 4T<σD<5.66T; compressional shear faults require that σD>5.66T (T is the host rock tensile strength; Sibson, 2003). Due to the bimodal (conjugate) dip distribution of the sills, their consistent near-vertical opening direction, and the mutual cross-cutting and intrusive relationship between thrust faults and sills, we estimated that σD=6T. We estimate the tensile strength of the host rock at 1 km depth to be 3 MPa, giving σD=18 MPa. It should be noted, however, that due to the nature of the model, providing all parameters are scaled relative to T, the value of T does not change the resulting opening angle pattern, only the magnitudes of the principal stresses and fluid pressure. ϕ (0.77) and R′ (0.68) derived in the model indicate a mild horizontal radial compression during emplacement, with fluid pressure less than σ2 (Fig. 7d, e).

The measured opening angles of the studied sills fit with the contouring of a single ellipse (Fig. 7d), suggesting that during the intrusion of sills in the study area, the fluid pressure and stress state remained relatively constant. However, we acknowledge that a broader study across the SRSVF may reveal spatio-temporal fluctuations in either the stress regime or magma pressure.

Our mechanical models build upon the work of Delaney et al. (1986) and Jolly and Sanderson (1997) to use fracture geometry and opening angles to derive the principal stress axes during sill emplacement and provide crucial new constraints on the stress state and fluid pressure, applicable to dikes, sills, and veins. The geometry of sills in the SRSVF records a continuous horizontal shortening that is otherwise accommodated by contractional faults in the area. Contoured regions on stereonets for opening angles suggest that in a high deviatoric stress state, it should be relatively rare for intrusions to be purely Mode I structures. In such settings, it should be commonplace for intrusions to accommodate a component of shear during their emplacement. Our opening angle model is particularly useful in determining paleostress states for regions where there is little brittle deformation (i.e. faulting), other than intrusions, and it may therefore present a useful and important tool in tectonic and magmatic studies.