## Abstract

In transtensional and transpressional deformation zones, bulk 3D strains are often kinematically partitioned into regions of wrench- and extension- or shortening-dominated faulting. Most strain models assume ideal incompressible materials with a Poisson's ratio (ν) of 0.5. It is well known from experimental and geophysical data, however, that natural rocks have values of ν <0.5 and that significant variations in the values of ν occur for different lithologies. We demonstrate that for non-coaxial, 3D transtension and transpression, this should lead to an expansion of the wrench-dominated strain field. The effect is especially marked in lithologies with very low Poisson's ratios (ν ≤0.15), where wrench-dominated deformation can occur even where the regional direction of divergence or convergence is only modestly oblique (e.g. 52°). The Carboniferous basin-bounding 90-Fathom Fault, NE England, was reactivated as a dextral transtensional structure during NE–SW regional stretching in post-Carboniferous times. Preferential dip-slip reactivation of pre-existing east–west structures in the underlying Carboniferous basement led to kinematic partitioning of the transtensional bulk strain. In addition, the geometric, spatial and kinematic patterns of minor faulting in Permian rocks located in the fault hanging wall are markedly influenced by the host lithology. Quartz-rich sandstones (ν=0.12) preserve complex faulting patterns consistent with a wrench-dominated transtension whereas immediately overlying dolostones (ν=0.29) preserve simpler patterns of Andersonian conjugate faults consistent with a more extension-dominated regime. We propose that the markedly different strain response during the same deformation reflects pronounced lithologically controlled variations in the value of Poisson's ratio in the adjacent rock units. Our findings illustrate that micro- to meso-scale faulting patterns are likely to be substantially influenced by lithology in all regions of oblique divergence or convergence.

In the last 30 years, geologists have increasingly realized that the three classic tectonic regimes predicted by the Andersonian model of faulting (extension, shortening, strike-slip; Anderson 1951) do not fully describe the 3D strain patterns that characterize crustal deformation zones where convergence or divergence is significantly oblique. There are two main reasons why oblique displacements are commonplace. (1) Oblique convergence or divergence is inevitable during motion of plates on a sphere (Dewey 1975; Dewey *et al*. 1998). Woodcock (1986) has shown, for example, that the relative plate motions at well over 50% of modern plate boundaries are significantly oblique. A similar percentage is likely in ancient settings. (2) Most crustal deformation zones contain pre-existing structures, such as layering, foliation, faults, fractures and shear zones, which may undergo reactivation when subjected to renewed stress (Holdsworth *et al*. 1997). Many of these pre-existing structures are likely to lie significantly oblique to the new regional transport direction. The Andersonian tectonic regimes have therefore been extended to include transtension (and transpression), which can be defined as strike-slip deformations that deviate from a simple shear as a result of a component of extension (or shortening) orthogonal to the deformation zone boundary (Fig. 1a; Dewey *et al*. 1998).

Sanderson & Marchini (1984) provided a basic vertically oriented, constant volume, homogeneous strain model that can be used for analysing regions of transpression and transtension. Many workers have further investigated and modelled 3D transtensional and transpressional strains, often by changing the boundary conditions of the original model. These include: using a strain matrix for simultaneous simple shear, pure shear and volume change (e.g. Fossen & Tikoff 1993); investigating the effects of strain partitioning (e.g. Tikoff & Teyssier 1994; Jones & Tanner 1995); modelling heterogeneous strain (e.g. Robin & Cruden 1994); including basal and lateral extrusion (Jones *et al*. 1997; Fossen & Tikoff 1998); investigating the effects of oblique simple shear (Jones & Holdsworth 1998; Lin *et al*. 1998) and incorporating inclined deformation zone boundaries (Jones *et al*. 2004). In transtensional settings, there have also been a number of field- and laboratory-based studies of deformation styles (e.g. Withjack & Jamison 1986; Dewey 2002; Ramani & Tikoff 2002), although these are less extensive compared with the equivalent literature for transpression zones (e.g. see Dewey *et al*. (1998) and references therein).

In this paper we begin by investigating how changes in Poisson's ratio (ν) related to host-rock lithology may influence the development of brittle structures during transtensional deformation. We then document this lithological influence using an example where reactivation and strain partitioning are also important controls of deformation patterns.

## Strain modelling and rock parameters

In this paper we use the basic Sanderson and Marchini model for transtension (Fig. 1b) where the bulk strain can easily be factorized into pure shear and simple shear components. In all transtension (and transpression) zones, the relative displacement direction across the deformation zone, infinitesimal strain (or stress) and finite strain axes are all oblique to one another. However, predictable geometric relationships exist between the orientations of the deformation zone boundaries, the axes of infinitesimal strain (stress) and the relative displacement direction across the deformation zone. In many cases, therefore, it is useful to apply an analysis using infinitesimal strain, which is equivalent to the more conventional, widely used stress-inversion techniques (e.g. Angelier 1979, 1984; Michael 1984). It should be noted, however, that this approach is reasonable only in regions where bulk finite strain (or more correctly, finite non-coaxial strain) is reasonably low, so that the misorientation between finite and infinitesimal strain axes is limited.

Transtensional infinitesimal strain will occur when the bulk displacement is at an oblique angle α to the deformation zone boundary faults (i.e. 0° < α < 90°) (Figs 1a and 2a). When the divergence angle α is perpendicular (α=90°) or parallel (α=0°) to the boundary fault, we have pure shear coaxial extension (Fig. 2b) and non-coaxial wrench simple shear (Fig. 2c), respectively. These represent end-member strain states for transtension and both are considered in the present analysis to lead to plane strain (2D) deformation (Fig. 2b and c). When the divergence vector is at an oblique angle, non-coaxial 3D strain is always developed (Fig. 2a). In this case, the infinitesimal strain ellipsoid lies in the constrictional field (1 < *k* ≤ ∞) and the (horizontal) maximum principal extension axis (*e*_{H}*x*_{}) always lies in the horizontal plane during progressive deformation. The other horizontal infinitesimal principal extension axis can be either *e*_{H}*z*_{} (minimum principal extension axis) or *e*_{H}*y*_{} (intermediate principal extension axis), depending on the value of α. The switch of the minimum principal extension axis from a horizontal to a vertical orientation marks the transition between wrench- and extension-dominated transtension, respectively. The threshold angle α between wrench- and extension-dominated transtension has been termed the critical angle of displacement α_{crit} (Smith & Durney 1992). It has a value of 20° if one assumes no volume change during deformation and following the other assumptions of the Sanderson & Marchini model (McCoss 1986). The isovolumetric assumption implies that an ideal incompressible material (i.e. a liquid phase) is involved in the deformation, with a Poisson's ratio ν=0.5. Following Withjack & Jamison (1986), we propose a more general transtensional model in which the constant volume condition of the Sanderson & Marchini model (equation (1)) has been relaxed to allow volume change at fault initiation (equation (2)) as a result of the Poisson's effect (Fig. 3). The change in volume is expressed by $$mathtex$$\[{\Delta}V=e_{a}+e_{b}+e_{c}=e_{a}+0+e_{c}=0\]$$mathtex$$(1)

for constant volume transtension with no lateral extrusion (*e _{b}*=0), where

*e*

_{i}=(

*l*−

_{i}*l*

_{0})/

*l*

_{0}are the infinitesimal extension axes for

*i*=

*a*,

*b*,

*c*(see Fig. 3). If the effect of Poisson's ratio is included, equation (1) becomes $$mathtex$$\[{\Delta}V=e_{a}+0+Ce_{c}{\neq}0\]$$mathtex$$(2)

where the parameter *C*=[ν/(1−ν)]≠1 for ν≠0.5 (see Fig. 3). The incorporation of Poisson's effect will in general lead to positive volume change (Δ*V* > 0) at fault initiation during transtensional deformation (Jaeger 1964), because most rocks have values of ν < 0.5 (Table 1).

Thus the calculated angle α_{crit} is now additionally controlled by the parameter *C*, which is related to the Poisson's ratio value as follows: $$mathtex$$\[0.5(w{/}l_{0})(\ sin\ {\alpha}_{crit}{-}1)={-}C(w{/}l_{0})\ sin\ a_{crit}.\]$$mathtex$$(3)

This represents the situation where *e*_{H}*z*_{}=*e*_{V}*y*_{} (i.e. the transition strain-state when α=α_{crit}, equations (2) and (4) in appendix 1 of Withjack & Jamison (1986)). *w* is the infinitesimal displacement in a direction at an angle α to the fault boundary and *e*_{H}*z*_{} and *e*_{V}*y*_{} are the horizontal and vertical infinitesimal minimum and intermediate principal extension axes, respectively.

The likely importance of Poisson's ratio in determining deformation patterns is illustrated by the α_{crit} values obtained during experimental analogue modelling studies investigating structures formed during oblique divergence. These values are significantly greater than the 20° angle predicted by the McCoss (1986) model, e.g. 30° (Withjack & Jamison 1986; Ramani & Tikoff 2002) and 45° (Smith & Durney 1992). A mean ν value of *c*. 0.3 represents a typical mean value for most rocks in nature (see Table 1). Introducing this value into equation (3), we obtain α_{crit}=33°, which is consistent with the values observed in the analogue experiments.

We can extend this analysis to investigate the potential control that variations in lithology (e.g. variations in Poisson's ratio) might exert upon faults forming in a compositionally heterogeneous bedded rock sequence. Following Christensen (1996) we calculated dynamic Poisson's ratio values (see Table 1) using compressional-wave (*V*_{p}) and shear-wave (*V*_{s}) velocities for a range of typical igneous, metamorphic and sedimentary rocks (Christensen 1996, for igneous and metamorphic rocks; Johnston & Christensen 1992, for sedimentary rocks). This is achieved using the equation $$mathtex$$\[{\nu}=0.5[1{-}\frac{1}{(V_{p}{/}V_{s})^{2}{-}1}].\]$$mathtex$$(4)

The solutions of equation (3) have been plotted on a ν v. α_{crit} graph (Fig. 4a) to show the influence of various lithologies on the strain regime formed during transtensional (and transpressional) deformations. A further equation (equivalent to equation (3) in appendix 1 of Withjack & Jamison (1986)) relates α to β_{x}, the angle between the infinitesimal horizontal maximum extension strain axis and the *b*-axis of the Cartesian co-ordinate frame, which corresponds to the deformation zone boundary (Fig. 2a): $$mathtex$$\[{\beta}_{x}=90{^\circ}{-}0.5\ tan\ ^{{-}1}(\ cot\ {\alpha}).\]$$mathtex$$(5)

The solutions to equations (3) and (5) are plotted on an α v. β* _{x}* diagram (Fig. 4b) for four representative Poisson's ratio values (equivalent to lithologies) listed in Table 1 and also shown in Figure 4a (incompressible material, dolostones, the lowest static value measured for real rocks and quartz-rich sandstones where quartz content is

*c*. 90%). In all cases, compared with the ideal incompressible material, the reduced value of the Poisson's ratio leads to an expansion of the wrench-dominated field at the point of fault initiation (Fig. 4a and b). This suggests that in quartz-rich sandstones, for example, faulting could initiate in a wrench-dominated transtensional regime even where displacements are only modestly oblique (e.g. α values up to 52°; Fig. 4b). Furthermore, in any basin containing markedly differing lithological units (and therefore Poisson's ratios) it is possible that adjacent rock units experience very different infinitesimal strain fields at fault initiation for the same regional value of oblique divergence (e.g. angle α). The geometry and kinematics of minor brittle structures, at least during the early stages of deformation, might then be controlled very significantly by lithology. We now present a possible example of this effect from NE England that additionally illustrates the importance of pre-existing mechanical anisotropies in bringing about strain partitioning on different scales.

## Case study: the 90-Fathom Fault, NE England

### Regional geological setting

The early Carboniferous Northumberland Basin, NE England, is one of the northernmost basins that developed in the foreland of the Variscan orogenic belt (Fig. 5a). The basin has an asymmetric shape and can be described as a half-graben. It is bounded to the south by the Stublick–90-Fathom normal fault system, which dips to the north and trends ENE–WSW to east–west (Fig. 5b and c; Collier 1989; Kimbell *et al*. 1989; Leeder *et al*. 1989). The fault system appears to be segmented, with the main movement transferred south and east along-strike from the western Stublick Fault to the easternmost 90-Fathom Fault (Fig. 5c). Thickness changes in the lower Carboniferous strata (Dinantian) are recorded across the fault system, with more than 4.2 km of Dinantian sedimentary rocks in the hanging wall, compared with a few hundred metres overlying the Alston block, the structural high in the footwall (Kimbell *et al*. 1989). This is taken as evidence of syndepositional fault activity (Kimbell *et al*. 1989). Other intrabasinal faults are recognized based on geophysical evidence, variations in stratigraphical thickness, concentration of channel bodies and dewatering structures (Leeder *et al*. 1989). Collectively, these structures suggest that the early Carboniferous rifting involved north–south-oriented extension that is believed to have ended during the Namurian. This was followed by a thermal subsidence sag phase during the Westphalian, with thickening of the basin fill towards the basin centre (Kimbell *et al*. 1989). Both the Stublick and 90-Fathom faults offset Upper Carboniferous (Coal Measures) and overlying Permian strata, suggesting that renewed extensional or transtensional faulting occurred in Permian to Mesozoic times (hereafter referred to as ‘post-Carboniferous’), probably associated with the early stages of rifting in the North Sea basin (Collier 1989). A lack of exposure and high-quality subsurface data means that the overall geometry of the major faults at depth is uncertain, especially in those segments reactivated during post-Carboniferous times (e.g. Collier 1989; Kimbell *et al*. 1989; Leeder *et al*. 1989).

### Deformation patterns

At Cullercoats the 90-Fathom Fault is an east–west-striking normal fault, dipping to the north and juxtaposing a hanging-wall sequence of Permian sandstones and dolostones against a footwall of Carboniferous Coal Measures (Fig. 5b). The fault plane is well exposed in the cliffs and foreshore and preserves dip-slip slickenlines (Fig. 5d and e). According to Collier (1989), Coal Measures strata are offset by 260 m, whereas the base of the Permian is estimated to exhibit 90 m of dip-slip normal displacement on the basis of cross-sections constructed from British Coal mine plans and borehole data.

Kinematically the overall pattern of deformation associated with the 90-Fathom Fault at Cullercoats has previously been interpreted to be consistent with a dextral transtensional deformation caused by post-Carboniferous reactivation of a pre-existing east–west-trending Dinantian normal fault at depth (Collier 1989). However, in the aeolian sandstones of the immediate hanging-wall region, ascribed to the mid-Permian (Collier 1989), the faulting patterns appear much more complex compared with the immediately overlying dolostones (Fig. 6a and b). The sandstones contain: (1) significantly higher numbers of closely spaced faults (i.e. higher fault densities); (2) geometrically different patterns of conjugate fault sets, e.g. quadrimodal (Fig. 6a) v. bimodal (Fig. 6b). Two distinct sets of mutually cross-cutting, and therefore broadly contemporaneous, cataclastic faults are recognized in the sandstones (Figs 5, 6a and 7): east–west-trending normal faults; and ESE–WNW dextral strike-slip faults. The cataclastic, deformation-band style of faulting in the sandstones is generally consistent with strain hardening behaviour and intense grain-size reduction along localized brittle faults (Collier 1989; Knott *et al*. 1996). At the concordant sedimentary contact with the overlying dolostone units, in a zone where sandstone and dolostone are interbedded, many of the smaller faults observed in the sandstones appear to die out as they are traced into the dolostones (Fig. 6c–e). In the latter lithologies a much more straightforward, low-density pattern of conjugate ESE–WNW-trending extensional–oblique displacement faults occurs (Fig. 6b, d and e). The observed difference in complexity and density of faulting clearly suggests that lithology has exerted a significant control on the faulting pattern. Previous workers have attributed these differences to changes in rheology, i.e. brittle faulting in sandstones v. more ductile folding and faulting in dolostones (Collier 1989).

Despite a reasonably good understanding of the general geological setting of the 90-Fathom Fault, three key issues remain concerning the post-Carboniferous deformation. (1) The relationship between the dip-slip displacements along the major fault plane and the multiple kinematics of the mesoscale pattern of deformation exhibited in its hanging wall is uncertain. (2) The significance of the contemporaneous, but kinematically very different fault sets in the sandstones requires explanation. (3) Does lithology account for the very different patterns of faulting in the Permian sandstones and dolostones? A detailed 3D strain analysis has been carried out in the attempt to address these issues and give some new insights into the development of complex fault patterns in transtensional settings. Our findings illustrate the importance of strain partitioning acting simultaneously and on different scales during transtensional deformation.

### Strain analysis: strain partitioning, reactivation and lithological control

In both sandstones and dolostones, individual fault displacements are small, rarely exceeding a few tens of centimetres, suggesting low finite strain intensities. This observation, together with a lack of evidence for significant bulk shear-induced rotations (i.e. no sigmoidal vein arrays observed), means that an approximate coincidence between the stress and infinitesimal strain axes may reasonably be assumed, i.e. σ_{1}=*e _{z}*; σ

_{2}=

*e*; σ

_{y}_{3}=

*e*.

_{x}Stress inversion has been applied to the fault-slickenline dataset from all faults in the sandstones (Fig. 7). The software package used (Daisy 2 of Salvini 2001) automatically separated the faults into two groups, each associated with very different stress fields: respectively, extensional (vertical σ_{1}, horizontal north–south-trending σ_{3}) (Fig. 7a–c) and dextral strike-slip (horizontal NW–SE-trending σ_{1} and NE–SW-trending σ_{3}) (Fig. 7d–f). These two groups correspond exactly to the two-fold subdivision of faults recognized in the field on kinematic grounds; i.e. the stress inversion results and the field observations suggesting strain partitioning are consistent.

The quartz content of sandstones at Cullercoats has been estimated at about 80% (T. Needham, pers. comm.) and lies close to the quartz content of the sandstone shown in Table 1. Assuming that the Poisson's ratio of this sandstone (ν=0.12) is representative of those at Cullercoats, we use an appropriate version of the α v. β_{x} diagram in the following strain analysis (Fig. 8a).

The preferential accommodation of north–south extension by east–west-trending dip-slip normal faults could be related to reactivation of the suitably oriented pre-existing structures associated with the 90-Fathom Fault in the Carboniferous rocks immediately below the Permian strata (Fig. 8b). This is consistent with the dip-slip slickenlines preserved on the present-day exposed fault plane (Fig. 5e). This condition is fixed by the relation α_{1}=β_{x}=90°, which expresses the partitioned extensional component of displacement, α_{1} (Fig. 8a and b).

We suggest that the dextral ESE–WNW faults in the hanging-wall region are the product of a residual wrench-dominated strain (with a displacement component α_{2}) left over after the extensional component (α_{1}) of the bulk regional strain was taken up by fault reactivation (Fig. 8a–c). Repeated cycles of reactivation and strain partitioning led to the observed mutually crosscutting relationships exhibited by the kinematically different sets of structures in the sandstones. The stress inversion applied to the ESE–WNW dextral strike-slip faults in the sandstones yields an angle β_{x}=70° between the horizontal infinitesimal principal extension axis direction (taken here as equivalent to σ_{3}) and the main strike-slip fault trend (Figs 7f–8c). When plotted on the α v. β_{x} diagram calculated for quartz-rich sandstones (Figs 4b and 8a), with an appropriate value of Poisson's ratio, this suggests a divergence angle α_{2} of *c*. 50° between the local extension direction and the ESE–WNW-trending strike-slip faults (Fig. 8a–c). The local extension direction expressed as the angle α_{2} *c*. 50° matches the condition α < α_{crit} *c*. 52° required to develop a wrench-dominated transtensional strain in the sandstone units (Fig. 8a–c). The local wrench component α_{2}=50° is measured relative to the trend of strike-slip faults (ESE-dextral) because they represent the active boundary structures accommodating the local wrench component of strain (Fig. 8c).

The bulk regional transport direction can be estimated using the calculated partitioned components, α_{1} (extensional) and α_{2} (wrench), respectively; it must have trended approximately NE–SW, somewhere between the two partitioned components of the deformation (Fig. 8d).

Compared with the sandstones, the dolostones preserve a much simpler pattern of deformation (Fig. 6a and b) with main faults having a similar orientation to the main dextral strike-slip faults in the sandstones. Unfortunately, no slickenlines are preserved on these fault planes, but the bimodal conjugate style of faulting and the extensional stratigraphic offsets seem to suggest an extension with a small component of dextral shear. Significantly, this is consistent with what is predicted if we plot the previously calculated value α_{2} for the partitioned wrench direction on an α v. β_{x} diagram for a material with a Poisson's ratio appropriate for a dolostone (Fig. 9a–c). In this case, an angle α_{2} *c*. 50° matches the condition α_{2} > α_{crit} *c*. 34° required to develop an extension-dominated transtension (Fig. 9a). Thus, for the same value of α_{2}, the dolostone will experience a markedly less non-coaxial deformation compared with that experienced by the adjacent sandstones undergoing wrench-dominated transtension (Fig. 9d–f). Inclusion of the Poisson's effect mainly results in a change in the simple shear–pure shear ratio, i.e. the kinematic vorticity. For a fixed regional displacement direction, this should lead to changes in the orientation and shape of the infinitesimal strain ellipsoid in different lithologies. Unfortunately, we cannot quantitatively analyse this further in the present case study because of the lack of exposed kinematic indicators in the limited outcrops of dolostones.

Independent validation of the NE–SW direction of bulk regional extension derived from our analysis (Fig. 8d) is provided by the patterns of offshore normal faulting in the immediately adjacent Mesozoic rocks of the southern North Sea (Petroleum Exploration Society of Great Britain 2000).

In summary, we have demonstrated that the mesoscale pattern of deformation in the Permian hanging wall of the 90-Fathom Fault can be interpreted as resulting from partitioning of a bulk transtensional strain, induced by the obliquity between the pre-existing east–west-striking, basin-bounding fault and the NE–SW regional direction of post-Carboniferous extension. This transtension has been partitioned into an extensional strain with a displacement component α_{1}, manifested by the dip-slip normal reactivation of the 90-Fathom Fault and associated structures, together with a more highly oblique displacement component α_{2}, which led to wrench- and extension-dominated transtension in immediately adjacent sandstone and dolostone units, respectively. Lithology has exerted a clear influence on the style, density, geometric and kinematic complexity of faulting in adjacent units, as the same displacement component (α_{2}) resulted in very different patterns of faulting. Lithologically controlled changes are here explained mainly by differences in the value of the Poisson's ratio, which are seen as being particularly significant as they lead to changes in the threshold angle α_{crit} between wrench- and extension-dominated transtension. In our interpretation, the 90-Fathom Fault represents a system where mechanical decoupling occurred during oblique regional extension. A perfect partitioning and mechanically decoupling of the different fault sets seems unlikely but seems to represent a reasonable approximation.

## Discussion

Our interpretation of the complex faulting patterns associated with the 90-Fathom Fault is relevant to the continuing debate concerning whether it is stress or the imposed displacement (strain) that controls the faulting process (e.g. Tikoff & Wojtal 1999). The 90-Fathom Fault illustrates that the only parameter not affected by the orientation of structures accommodating local deformation is the regional imposed displacement, i.e. the NE–SW opening direction during post-Carboniferous times. The stress (equivalent to infinitesimal strain) distribution in the hanging wall of the 90-Fathom Fault appears to be highly partitioned (Fig. 7a–d) and depends on local controls such as lithology and reactivation of pre-existing zones of mechanical weakness in the subjacent basement. In an ancient structure of this kind we have no independent evidence to constrain the orientation of the regional stress. Traditionally, the observed heterogeneities in apparent stress or strain patterns encountered here might be interpreted as being due to polyphase deformation. This is at odds with the field observations.

More generally, our findings illustrate that kinematic partitioning is particularly likely to occur during the deformation of heterogeneous anisotropic crust where pre-existing structures are often significantly oblique to regional tectonic transport directions (e.g. Dewey *et al*. 1998; Jones *et al*. 2004). It is important to note that the direction of the infinitesimal principal extension axes (β_{x}) does not correspond to the bulk extension direction accommodated by the overall fault system; this is a consequence of strain partitioning and non-coaxial strain component present during 3D transtensional strain (Dewey *et al*. 1998).

Our findings further illustrate that the use of 2D strain ellipse models to describe faulting patterns (Wilcox *et al*. 1973; Harding 1974) is inappropriate when dealing with complex deformation patterns arising from 3D strain. Such 2D methods are still used widely in the interpretation of faulting patterns in sedimentary basins, including the 90-Fathom Fault (e.g. Collier 1989). Given the marked differences between faulting patterns that arise during 2D and 3D finite strain, this practice is unwise in any areas where there is evidence for obliquely divergent plate motions or rifting oblique to reactivated basement faults.

Our findings also have significant implications for structural models of fracture interconnectivity and fluid flow in hydrocarbon reservoirs. For example, the association of contemporaneous normal and strike-slip faults produces a potential mixture of preferential structural permeability patterns. In 2D (plane strain) extensional or wrench regimes, the horizontal or vertical σ_{2} directions are generally thought to form preferential flow paths for migrating fluids (e.g. Sibson 2000). Fracture interconnectivity models, based on simple Andersonian faults, are not sufficient to predict the structural permeability pattern in the hanging wall of the 90-Fathom Fault. A 3D model of fault pattern that incorporates the intersections between the various conjugate fault sets will reproduce a more reliable structural permeability model, with multiple flow directions likely to develop. If the faults act as effective pressure seals, as seems likely, this will result in a highly compartmentalized reservoir with a low chance of being economically productive, as each fault-bounded sector will not communicate with the others. A similar case has been recently presented by Olsson *et al*. (2004) for progressive deformation in sandstone units showing different, almost simultaneously developed, fault patterns.

Thus in cases where there is evidence for transtensional (or transpressional) strain, 3D strain modelling should be used as a tool to investigate and predict faulting and structural permeability patterns. It is very likely that many offshore basins have experienced transtensional deformation and have been influenced by reactivation of pre-existing structures oblique to direction of extension.

## Conclusions

Traditional transtensional strain modelling assumes a Poisson's ratio ν=0.5 (ideal incompressible material) and predicts that the threshold angle α_{crit} between faults and transport direction is 20° for wrench-dominated (α < 20°) and extension-dominated (α < 20°) transtension (Fig. 4). α_{crit} values recalculated for dynamic ν values from a range of real rocks show a general expansion of the wrench-dominated field for most rock types (α_{crit} *c*. 30°), except in quartz-rich rocks, which display a substantial increase (α_{crit} up to 52°).

The 90-Fathom Fault, NE England, has been used as a field example to test this model and to discuss the significance of geometrically and kinematically complex fault patterns as well as the influence of lithology on faulting. Regional post-Carboniferous NE–SW extensional strain has been partitioned into an extensional component accommodated along the reactivated 90-Fathom Fault, with a residual oblique component accommodated in the hanging wall of the fault. Three-dimensional strain analysis suggests that the calculated divergence angle α_{2}=50° for the residual oblique component of extension has led to wrench-dominated transtension for quartz-rich (*c*. 90%) sandstones (ν *c*. 0.12) where α_{crit} is *c*. 52° (Figs 8 and 9). In immediately adjacent dolostone units (ν *c*. 0.29), however, the same analysis suggests that extension-dominated transtension has occurred, where α_{crit} is *c*. 34° (Fig. 9). Field data match the assumptions of the strain modelling and seem to explain well both the kinematically complicated patterns of deformation and the lithological control on style of faulting.

## Acknowledgements

The authors would like to thank the following for help during fieldwork and many discussions: P. Clegg, J. Imber, R. Wilson and R. Jones. N.D.P. gratefully acknowledges the financial and scientific support provided by M. Barchi and the University of Perugia (Italy). The RRG acknowledges the continuing support of NERC, Statoil and BP. Finally, the authors very much appreciate the detailed reviews provided by the journal referees S. Giorgis and S. Wojtal, and the editor A. Maltman.

- © 2005 The Geological Society of London