## Abstract

An example of sheet-like intrusion emplacement at very shallow crustal levels on Elba Island, Italy, is described. The Eastern Elba Dyke Complex (EEDC) consists of decimetre- to metre-thick sheeted aplites emplaced within intensely folded low-grade metamorphic rocks. Field data indicate that sill and dyke emplacement was controlled by mechanical discontinuities, represented by fractures in the host rocks, and was strongly favoured by magma overpressure. The occurrence of angular fragments of host rocks in the dyke border zones and the branching of sills testify to hydraulic fracturing. Analysis of the spatial distribution and geometry of EEDC sills and dykes provides clues on fluid pressure conditions and the stress state at the time of magma emplacement, as well as on the depth of emplacement. The calculated stress ratio and driving pressure ratio were used to estimate a magma overpressure of 6–54 MPa at the time of emplacement of the EEDC at a depth of about 2 km.

Dykes and veins (i.e. liquid-filled cracks) typically form perpendicular to the least compressive stress (σ_{3}), and are commonly used to map patterns and variations in regional stress fields (e.g. Zoback & Zoback 1980). However, in some instances dykes are not simply related to the regional stress field. Magma cannot only exploit the existing fracture network, but it may also generate new fractures (e.g. Delaney *et al.* 1986). Shallow-level sheet-like intrusions (i.e. dykes and sills) represent a record of magma transport in fractures, and their development results from the competition between magma pressure, host-rock elastic properties and the stress field at depth (Delaney *et al.* 1986; Rubin & Pollard 1988; Jolly & Sanderson 1997; Babiker & Gudmundsson 2004).

We report on the structural analysis of an intrusive complex consisting of a network of sills and dykes intruding shallow crustal rocks. This complex, the Eastern Elba Dyke Complex, is hosted in pelitic–psammitic hornfels rocks and is exposed along the eastern coast of Elba Island (central Italy). Sills and dykes mainly exploit pre-existing fractures and show paraconcordant–discordant contacts with the foliation in the host rock. Geometric relationships between the host rock and dykes, and the geometry and structure of dykes and sills were used to constrain the local fluid pressure and stress state at the time of magma emplacement, as well as the depth of emplacement.

## Magmatic overpressure, dyke injection and geometry, and stress state

The condition for initiation of a dyke may be described by the equation (Jaeger & Cook 1979; Gudmundsson 2002):
1
where *P*_{t} is the total magmatic pressure, and σ_{3} and *T*_{0} are the minimum principal stress and the *in situ* host-rock tensile strength, respectively. The difference between the total magma pressure (*P*_{t}) at the time of dyke formation and lithostatic stress (*p*_{l}) yields the excess magma pressure (*p*_{e}). Equation (1) may therefore be rewritten as:
2

When conditions (1) and (2) are met at any point on the roof (or walls) of a magma reservoir, and depending on the local stress field (as indicated by stress trajectories) around the source magma reservoir (e.g. Gudmundsson 2002), a dyke (or an inclined sheet) is initiated. In the case of shallow-dipping sheet-like intrusions and sills, two requirements must be met for their injection. First, the driving pressure must exceed either the strength of the rock or the tensile strength of a subhorizontal pre-existing plane of weakness, such as a bedding plane or joint; it must also exceed vertical stress at the level of sill injection. The tensile strength of pre-existing weaknesses (e.g. bedding planes or joints) tends to be lower than that of the intact rock. Second, sill emplacement requires that the least compressive stress (σ_{3}) must have a subvertical orientation.

It is assumed that sills and dykes are extension fractures (modelled as a mode I crack), an assumption supported by field observations (see below). For an extension fracture, the magmatic overpressure, *P*_{o}, is also referred to as net pressure or driving pressure. *P*_{o}, is a measure of how much the magmatic pressure exceeds the minimum principal compressive stress, σ_{3}, which is normal to the dyke in an extension fracture, and is the pressure responsible for the aperture of the dyke fracture at a particular point.

A sheet-like intrusion is thus modelled as a two-dimensional through crack. The geometry of shallow-level sheet-like intrusions (sills, dykes and laccoliths) is defined in terms of their width (thickness or aperture) and length. The width is generally the smallest dimension of the intrusion. For dykes the plan length, considered smaller that the height, is the controlling dimension, whereas sills are imaged as flat ellipsoidal or circular sheet-like intrusions.

The maximum thickness (aperture) *W* of the sheet-like intrusion and its controlling dimension (*L*) are related to magmatic overpressure through the equation (Gudmundsson 1999; 2000; Babiker & Gudmundsson 2004; Valentine & Krogh 2006):
3
where *P*_{o} is the magmatic overpressure in the dyke at the time of emplacement, *E* is Young's modulus and ν is Poisson's ratio for the host rock.

The aspect ratio of sheet-like intrusions (*W*/*L*) is thus an important geometric feature and can be used to derive the static fluid overpressure (driving pressure) during dyke formation. This parameter can vary from 10^{−2} to 10^{−5} (e.g. Delaney *et al.* 1986; Maaloe 1998). Dykes along the Red Sea, in the East African Rift, have aspect ratios of 4.3×10^{−4}–4×10^{−3}, with an average of 1.1×10^{−3} (Babiker & Gudmundsson 2004). Dykes in the Colorado Plateau Province have aspect ratios of 3×10^{−5}–6×10^{−3} (Delaney *et al.* 1986).

Although dykes propagate perpendicular to the least compressive stress, they can also invade pre-existing fractures misaligned with respect to the principal stresses (e.g. Ziv *et al.* 2000). In this case, the stress ratio (Φ=(σ_{2}−σ_{3})/(σ_{1}−σ_{3})) and the driving stress ratio (*R*′=(*P*_{f}−σ_{3})/(σ_{1}−σ_{3})), which are related to both the effective stress field and fluid pressure (*P*_{f}), can be derived from the analysis of the geometric features of sills and dykes, such as attitude and distribution (e.g. Baer *et al*. 1994; Jolly & Sanderson 1997). The stress ratio Φ ranges from 0 to 1, and describes the Mohr circle configuration (Angelier 1984; Baer *et al.* 1994; Orife & Lisle 2003). The driving stress ratio *R*′ (Baer *et al.* 1994) varies from −1 (no opening of fractures) to 1 (re-opening of pre-existing fractures), and describes the equilibrium between *P*_{f} and the minimum (σ_{3}) and maximum (σ_{1}) stresses.

The plots of poles to sills and dykes may thus show two quite peculiar distributions described by three angles: θ_{1}, θ_{2} and θ_{3}. θ_{1} is the angle between the σ_{2} stress axis and the border of the sill and dyke distribution in the σ_{2}−σ_{3} plane; θ_{2} is the angle between the σ_{1} stress axis and the border of the sill and dyke distribution in the σ_{1}−σ_{3} plane; and θ_{3} is the angle between the σ_{1} stress axis and the border of the sill and dyke distribution in the σ_{1}−σ_{2} plane (Jolly & Sanderson 1997; André *et al.* 2006). The orientation of the principal stress axes is derived from the sill and dyke distribution by calculating the Bingham distribution (e.g. Press *et al.* 1986) of poles to sills and dykes, and assigning each eigenvector to the appropriate principal stress. The attitude distribution of sills and dykes may thus be used to infer the stress ratio (Φ) and the driving stress ratio (*R*′), which are defined in terms of the principal stresses (σ_{1}, σ_{2}, σ_{3}), the distribution of poles to sills and dykes, and the fluid pressure (*P*_{f}) (Baer *et al.* 1994; Jolly & Sanderson 1997):
4
and
5
the driving pressure ratio is defined as:
6

## Geological outline of Elba Island

Elba Island lies in the north Tyrrhenian Sea and is part of the Neogene northern Apennine chain. The geological framework, resulting from the Late Oligocene–Mid-Miocene main collisional phase of Apennine orogenesis, is characterized by a stack of five eastward-facing tectonic units derived from both oceanic (Ligurian) and continental (Tuscan) domains. From top to bottom they are as follows (Pertusati *et al.* 1993):

Upper Ligurian Unit: Paleocene–Eocene and upper Cretaceous flysch, the latter intruded by Late Miocene porphyritic dykes;

Lower Ligurian Unit: ophiolitic rocks with a Mesozoic sedimentary cover (limestone and slate);

Tuscan Nappe: very-low-grade metamorphic rocks (slates and limestones) derived from a sedimentary sequence dating from the Late Carboniferous–Dogger; and

Tuscan Metamorphic Unit: low-grade metamorphic rocks (slate, metasandstones, metavolcanites, limestones and calc-schists) derived from a sedimentary sequence dating from the Late Carboniferous–Dogger; and

Calamita Schist: a unit belonging to the Tuscan domain and consisting of pelitic–psammitic hornfels derived from low-grade metamorphic rocks (metasandstones and phyllites).

The architecture of the tectonic stack (Fig. 1a) is characterized by westward-dipping thrust faults with top-to-the-east displacement that led to the superposition of the Ligurian units on the Tuscan units and to the imbrication of Tuscan units, as observed in other portions of the northern Apennines. During the Late Miocene two major plutonic bodies, namely the Mt Capanne pluton (8–6.8 Ma) and the Porto Azzurro pluton (6.0 Ma), along with their related dyke systems, were emplaced in the tectonic units, with the consequent development of contact aureoles (Dini *et al.* 2002 and references therein). Granite emplacement was coeval with, or just prior to, the extensional tectonics associated with the opening of the northern Tyrrhenian Sea (Keller & Coward 1996). The main extensional structure in eastern Elba Island is the Zuccale Fault, a low-angle normal fault that strikes north–south and dips gently (5°–15°) to the east, cross-cutting the tectonic stack (Keller & Coward 1996; Pertusati *et al.* 1993). The fault is characterized by a meter-thick zone in which breccias containing hornfels as tectonized clasts within foliated cataclasites and clay-rich gouge suggest that displacement along the Zuccale Fault post-dates contact metamorphism and granite emplacement.

### The Calamita Schist and Eastern Elba Dyke Complex

The Calamita Schist, cropping out in the Calamita Peninsula and Porto Azzurro area (southeastern Elba Island; Fig. 1b), consists of hornfels rocks resulting from low pressure–high temperature (LP–HT) contact metamorphism that affected a sequence of metasandstones and phyllites which experienced alpine deformation under low-grade metamorphic conditions. Contact metamorphism related to the emplacement of the Porto Azzurro pluton (Fig. 1b) is characterized by biotite + andalusite+K-feldspar+cordierite-bearing mineral assemblages. According to Pattison & Tracy (1991), the cordierite+andalusite + K-feldspar mineral assemblage in the pelitic hornfels indicates that contact metamorphism developed under LP–HT conditions (*P*_{max}<0.20 GPa, 550 °C <*T*_{max}<600 °C). The Calamita Schist constitutes a NE-trending monoclinal structure that dips gently to moderately to the NW. A NE- to NNE-trending schistosity testifies to intense deformation prior to contact metamorphism (Fig. 2a), and is the axial-plane foliation of mesoscopic tight to isoclinal east-verging folds.

The Eastern Elba Dyke Complex (EEDC) emplaced within the Calamita Schist consists of centimetre- to metre-thick, inequigranular, medium- to fine-grained leucocratic sills and dykes with a mineral assemblage consisting of quartz, K-feldspar, plagioclase and tourmaline, with rare biotite and muscovite. The EEDC mainly crops out in the eastern portion of the Calamita Schist, along the coastline, where there are good exposures of dykes and host rocks. The following describes the structural features of the EEDC cropping out in the northeastern portion of the Calamita Peninsula; the complex was investigated at five selected sites (Fig. 1b) where structural features of sills and dykes, as well as relations with host rocks, are well exposed.

## Structural features of the Eastern Elba Dyke Complex

In the northeastern portion of the Calamita Peninsula the structural grain of the Calamita Schist is a NE-trending metamorphic foliation that dips gently to moderately to the NW, with an average strike of nearly N60E (Figs 2a and 3a). The foliation is cross-cut by two fracture systems identified in all the examined sites. They correspond to: (i) homogeneously distributed decimetre-spaced east- to NE-trending fractures (system a) that dip gently to the east and west; and (ii) NNW- to NE-trending, centimetre-spaced steeply dipping fractures (system b) generally filled by tourmaline (Fig. 3b) and distributed within metre- to decametre-wide fracture zones (Fig. 2b).

The EEDC consists of NE-SW-trending sills gently dipping towards the NW (Figs 2c and 3c). Vertical or subvertical dykes only occur at sites 2 and 5, where they represent feeders between sills (Fig. 3d and f). Distribution patterns at all examined sites vary from evenly spaced (every 0.6–0.8 m) sills (sites 1, 3, 4 and 5) to a dense network of sills and dykes (site 2). Sills and dykes cross-cut the metamorphic foliation. The contact between sills and the host-rock foliation ranges from discordant, with an angular unconformity of up to 40° (Fig. 3e), to paraconcordant (10°; Fig. 3f). Most contacts are discordant at the metre-length-scale. Paraconcordant contacts are less diffuse and occur at the centimetre- to decimetre-length-scale. No fully concordant contacts were observed between the host-rock foliation and sills.

Noteworthy features of sills are the single tapering terminations (Fig. 3c), which are clearly identified when entire lengths of sills are exposed in the outcrop, as well as the angular contacts between sills and the host-rock foliation, and the decimetre- to metre-scale host-rock septa and off-shoots (Figs 3g, h).

As for the relationship with fracture systems, sills and host rocks are cross-cut by steeply dipping fractures (system b; Fig. 3b), while sill orientations partially overlap with those of gently dipping fractures (system a; Fig. 2d), which are in several cases invaded by centimetre- to decimetre-wide sills.

Sill width (*W*) measured in the field (93) ranges from 0.06 to 2.1 m, with an average of 0.24 m. Centimetre- to metre-wide sills are variably distributed in all sites. Although sills tend to maintain a nearly constant width throughout the exposure (Fig. 3h), in a few cases they also show large variations in width over short distances (Fig. 3c). The exposed sill lengths vary from 0.62 to 9.0 m, with an average of 2.86 m. We assume that these lengths considerably underestimate the true sill lengths, as terminations are rarely exposed.

## Physical condition of EEDC emplacement

The geometry of EEDC sills and the displacement of the wall rock indicate that they correspond to mode I crack fractures (e.g. Beach 1980), with the opening direction normal to the fracture/crack plane. Sills therefore intrude tensile fractures whose orientations partially correspond to gently dipping fractures (system a), and the occurrence of feeder dykes between sills indicates hydraulic connection at the time of their emplacement (Fig. 3d). On this basis, the different patterns and geometries observed for the EEDC can be used to constrain the stress-field and fluid-pressure conditions at the time of magma emplacement.

### EEDC aspect ratio

The EEDC sills and dykes are mainly exposed along the coast, where only the width of sills is well constrained. As a consequence, the aspect ratio (*W*/*L*) computed using the measured length of sills is poorly constrained in the range 0.005–0.187. This measurement, which is strongly biased because of the inability to observe sill terminations, represents an overestimation of the true aspect ratio of sills. In order to overcome this problem, as in the case of plutons and laccoliths, we tentatively assumed a power-law relationship between the width (thickness) and length of dykes and sills (e.g. McCaffrey & Petford 1997; Cruden & McCaffrey 2001):
7
where *c* is a normalization constant, *W* and *L* are the laccolith width and length, and *n* is the power-law exponent. Equation (7) can also be expressed as:
8

This equation can be used to derive the sill length relative to each measured sill width. Results may then be used to calculate the aspect ratio for the EEDC sills. Considering *c*=0.026 and the fractal exponent *n*=1.36 proposed for laccoliths in central Elba Island (Rocchi *et al.* 2002), the computed EEDC *W*/*L* ratio ranges from 0.013 to 0.002, with an average of 0.006. The EEDC aspect ratio computed using the exponents of Rocchi *et al.* (2002) and equation (8) is very close to the known range of dyke aspect ratios (see above), and is here assumed to be representative of EEDC sills.

### EEDC magma overpressure

The magma overpressure can be calculated using equation (3) and the derived aspect ratio for sills. The result is strongly dependent on the elastic properties of the host rock.

In equation (3) the magma overpressure depends on the dyke aspect ratio, and on Young's modulus and Poisson's ratio for the host rock. In particular, magma overpressure estimates depend linearly on Young's modulus.

Young's modulus (*E*), Poisson's ratio (ν) and tensile strength (*T*_{0}) values for the Calamita Schists were derived from the elastic parameters of lithologies that best approximate the mechanical behaviour of this unit (i.e. quartzites, shales, sandstones and schists). The following ranges of elastic parameters were derived from the literature (Lama & Vutukuri 1978; Jaeger & Cook 1979; Turcotte & Schubert 2002; Nasseri *et al.* 2003; Gercek 2007): *E*=5–20 (GPa), ν=0.11–0.40 and *T*_{0}=5–10 (MPa). Fluid overpressures, *P*_{o}, of 6–54 MPa were obtained using the calculated *W*/*L* ratio of 0.006 and taking into account the variability of host-rock elastic parameters (see earlier). The wide range of variation mainly depends on the elastic parameters of host rocks. Considering that the host rock was fractured at the time of intrusion (e.g. fracture system a, see above), a Poisson's ratio of 0.25 and a Young's modulus of 10 GPa were assumed for the Calamita Schist, yielding a magma overpressure of 32 MPa.

### EEDC emplacement depth

As a general mode I aperture of sills was observed, the general equation for the maximum depth of formation can be applied (e.g. Gudmundsson 1999):
9
where *h* is the maximum depth of sill formation, Δρ is the difference between rock and magma densities, Δσ is the difference between principal stresses (σ_{1}−σ_{3}), *p*_{e} is the difference between total magma pressure (*P*_{t}) and lithostatic pressure (*p*_{l}), and *g* is the acceleration due to gravity.

For the Calamita Schist we assumed a bulk rock density of 2600–2700 kg m^{−3} and for the felsic EEDC sills a density of 2200–2400 kg m^{−3} (e.g. Lange 1994); the resulting Δρ is 300–500 kg m^{−3}. The term *p*_{e} is usually equal to the *in situ* tensile strength, *T*_{0}, and the differential stress, Δσ, for hydrofracturing is assumed to be 4*T*_{0} (e.g. Jaeger & Cook 1979; Gudmundsson 1999). Since the tensile strength measured *in situ* is usually much lower than that measured in the laboratory (see Gudmundsson 1999), the adopted *T*_{0} for the Calamita Schist was 5 MPa. A depth range of 1.8–3.1 km was thus obtained for the EEDC using the computed range of *P*_{o} (6–54 MPa) in equation (9). Considering 32 MPa as the most likely value for *P*_{o}, EEDC emplacement occurred at a depth of 1.4–2.4 km.

### Stress state during EEDC emplacement

The EEDC sills and dykes intruded fractured host rocks, as suggested by field observations, and the distribution and the attitude of sills and dykes may be used to infer the stress state at the time of EEDC emplacement (e.g. Jolly & Sanderson 1997).

As the poles to sills and dykes are dispersed and do not define a clear clustered or girdle distribution (Fig. 4), the 1% distribution contour was used to calculate the θ_{1} or θ_{3} and θ_{2} angles. Contours higher than 1% (e.g. 2%) produced inconsistent stress ratios (e.g. Φ>1).

The orientation of the principal stress axes was derived calculating the Bingham distribution of sill and dyke poles and considering that σ_{3} corresponds to the maximum concentration of poles. As shown in Figure 4, θ_{2} (the angle between σ_{1} and the 1% contour in the σ_{1}−σ_{3} plane) and θ_{3} (the angle between σ_{1} and the 1% contour in the σ_{1}−σ_{2} plane) can be measured. The stress ratio (Φ=0.62) and driving pressure ratio (*R*′=0.55) were obtained by applying equations (5) (for *P*_{f} lower than the intermediate stress) and (6).

## Discussion and conclusions

The EEDC was emplaced at shallow crustal levels (*P*<0.2 GPa) in the hornfels rocks of eastern Elba Island within the context of the late Miocene magmatism that affected the island as well as other portions of the inner northern Apennine chain. Sills in the examined area represent mode I fractures, as evidenced by: (i) paraconcordant–discordant contacts with the host-rock foliation; (ii) off-shoots; and (iii) host-rock septa. These features indicate that sill emplacement was primarily driven by magma overpressure and favoured by pre-existing, gently dipping fracture systems in the host rocks. This is highlighted by the close spatial and geometrical relationship between sills and fractures, the latter sometimes exploited by sills.

The physical conditions of sill emplacement and the stress state were derived by geometrical analysis. The reported example highlights the critical importance of determining the dimensions of the intrusion in the field. In the case of the EEDC, only the sill width was well constrained. The difficulty in directly observing the tip regions of the exposed sheet-like intrusions in several cases hampered accurate estimation of their length. The problem was overcome using the known power-law relationship between the width and length of tabular intrusions (e.g. McCaffrey & Petford 1997; Rocchi *et al.* 2002), which allowed us to reliably estimate the EEDC aspect ratio. Considering the derived sill aspect ratio of 0.006, a magma fluid overpressure of 6–54 MPa was estimated. A magma overpressure of 32 MPa was obtained assuming a Poisson's ratio of 0.25 and a Young's modulus of 10 GPa for the Calamita Schist. The calculated depth of approximately 2 km for sill emplacement derived from equation (9) indicates that the EEDC was emplaced at very shallow depths after peak LP–HT contact metamorphism, for which pressures of 0.1–0.2 GPa (constrained by mineral assemblages) suggest depths of 3.8–7.8 km. The obtained value for the EEDC emplacement depth is strongly dependent on the knowledge of host-rock elastic parameters. Future investigations will include also laboratory geotechnical analyses on host-rock samples.

The computed stress ratio, Φ (0.62), and the driving pressure ratio, *R*' (0.55), indicate *P*_{f}<σ_{2} during magma emplacement. Moreover, EEDC sill and dyke distributions testify to a subvertical attitude of the least principal stress (σ_{3}). Considering the computed magma overpressure, supra-hydrostatic pressure conditions (σ_{3}<*P*_{f}<σ_{2}) are proposed for the EEDC. On this basis, the above-reported magma overpressure of 32 MPa is consistent with sill emplacement under supra-hydrostatic pressure conditions at a depth of approximately 2 km.

In conclusion, in the examined area corresponding to upper crustal rocks affected by diffuse magmatism and LP–HT contact metamorphism, the EEDC represents an example of a shallow-level sill and dyke system emplaced after peak contact metamorphism. Further constraints on the elastic parameters of the host rock will provide better estimation of the emplacement depth of the EEDC.

Field data and calculated physical parameters suggest that EEDC emplacement was determined by both supra-hydrostatic magma overpressure and the pre-existing fracture system. The former overcame local stress conditions and rock strength, while the latter played a major role in controlling the site of magma emplacement.

## Acknowledgments

The authors gratefully acknowledge A. Bunger and an anonymous referee for their reviews, and K. Thomson for helpful editorial assistance.

- © The Geological Society of London 2008