USING STRIATION DATA TO UNDERSTAND THE MECHANICS OF FAULTING IN HETEROGENEOUS STRESS FIELDS J. Ole Kaven, Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected]

For simplicity, he investigated only the 'strength Abstract anisotropism', or direction of maximum shear stress. Wallace concluded that the orientation of the remote Methods for stress inversion from striation data on stress in simple cases in nature can be determined from single or multiple faults are based on two fundamental the orientation of fault planes and the direction of slip assumptions: 1) the remote stress tensor is spatially on those fault planes. Techniques of paleo-stress uniform for the rock mass containing the faults and inversions have since evolved (Bott, 1959; Angelier, temporally constant over the history of faulting in that 1989; Lisle, 1992) but remain based on two region; and 2) the slip on each fault surface has the assumptions: 1) the remote stress tensor remains same direction and sense as the maximum shear stress uniform across the entire rock mass containing the fault resolved on each surface from the remote stress tensor. or faults and remains constant throughout the history of More than ten years ago it was demonstrated, using an faulting in the region, and 2) the direction of maximum analytical solution to the linear elastic boundary value shear stress resolved on the fault plane coincides with problem, that the second assumption is faulty: slip and the direction of slip (e.g. Shan et al., 2004). maximum shear stress directions differ because of Over ten years ago, analytic solutions to the linear anisotropy in fault compliance (caused by tipline elastic boundary value problem of a rectangular, blade- geometry), anisotropy in fault friction (caused by like fault were used to show that the second assumption surface corrugations), heterogeneity in host rock is wrong (Pollard et al., 1993). In the problem, both, the stiffness (caused by Earth’s surface, sedimentary aspect ratio of fault length to height and the layering, etc.), and perturbation of the local stress field compressibility of the host rock, render the assumption (caused by the mechanical interaction of adjacent of coincidence of direction of slip and direction of shear faults). It remains an open question, however, whether stress on the fault plane false. Nevertheless, as Wallace the errors introduced by ignoring these natural (1959) suggested, considering the regularity of some heterogeneities of the Earth’s crust lead to significant fault systems, the assumption may yield useful results. errors in the stress inversion for particular data sets. To test under what circumstances the results of paleo- Steady progress in the development of numerical stress inversions are useful, a numerical method which methods has supplied the structural geologists with the solves the linear elastic boundary value problem (BVP) tools to investigate the influence of natural is employed here. The numerical code solves the BVP heterogeneities on stress inversion results. These for geometrically complex faults or fault systems and methods are employed here, as we investigate the allows one to investigate faulting phenomena, including validity of inversions for commonly occurring paleo-stress inversions, with great rigor. This code is heterogeneities. Systematic forward models help to based on physical rather than empirical underlying better define criteria that can be used to choose assumptions. In addition to aspect ratios and appropriate analysis tools for field data. Two field compressibility of the host rock, the anisotropy of fault examples from Chimney Rock, Utah, and the Wytch compliance due to tipline geometry, frictional Farm Oil field, Southern England, are used to examine heterogeneity due to non-planar fault surfaces, and the the effect of fault interaction and non-planar fault heterogeneity due to the Earth’s free surface are tested surfaces on natural faults. in this study. To investigate the effects of the aforementioned variations on natural fault behavior, the Introduction angle between the resolved shear stress on the fault Structural geologists have tried to understand and plane and the orientation of slip on the fault plane is explain the origins and the evolution of specific evaluated (Fig.1). This angular difference is referred to structures and tectonic regions by employing paleo- as the discrepancy angle (γ). Methods such as those stress inversion techniques. These techniques were used here, may lead to a better understanding of the introduced in the middle of the last century (Wallace, faulting process, thereby allowing for the consideration 1951; Bott, 1959; Angelier, 1989). The relationship of a greater number of the 'innumerable variables' between fault plane orientation, maximum shear stress, mentioned by Wallace. and fault slip was first investigated by (Wallace, 1951). He noted that the correlation of shear stress and shear rupture along faults in nature is subject to 'innumerable variables' and that these variations appear 'bewildering'.

Stanford Rock Fracture Project Vol. 16, 2005 E-1 results to within a few percent (Crider and Pollard, 1998).

Model geometry and boundary conditions The assignment of the remote stress field and the resolution of tractions on boundary elements make use of several coordinate axes systems (CAS). The g notation, xi refers to the global reference (Fig.2). A e local reference frame, xi , is located at the centers of the e e polygonal elements, has x1 pointing down-dip, x2 e pointing along the strike of the element, and x3 pointing normal to the element plane (Fig. 2).

Figure 1. Illustration of discrepancy angle γ on a slip patch. The slip vector is defined by ∆u in the fault plane at an angle β from the strike. The shear traction vector tsh is oriented at an angle α from the strike.

Method In this study fracture mechanics principles are employed that were first introduced by Griffith (1921) and have been used to explain a variety of rock fracture phenomena (e.g. Pollard and Aydin, 1987). The analyses are carried out by utilizing Poly3D, a three- dimensional boundary element method (BEM) numerical code (Thomas, 1993). In this method, the fault or fracture surface is divided into contiguous polygonal elements, each of which accommodates a constant magnitude of relative displacement (Fig. 2). The displacement discontinuities across all elements are found by solving a system of linear equations that describes the influence of the elements on one another and that simultaneously satisfies the given boundary conditions. These so-called boundary element solutions satisfy the governing partial differential equations for linear elasticity in a half-space. The domain requires no discretization except for the fault surface and the number of linear equations solved is smaller than for Figure 2. Discretized elliptical fault with (a) global other numerical techniques for solving partial g coordinate system, x1 , and (b) local element differential equations such as finite element methods e e coordinate system, x1 . x1 points down dip of the (Crouch and Starfield, 1983). Poly3D allows two types e element, x2 points along the strike of the element, of boundary conditions to be specified at the center of e x3 is normal to the element. each element: Burger's vector components or tractions vector components. The code calculates the In this study, a complete shear stress drop is displacement vectors and the stress tensor (tension is assumed for each element making up a fault. This positive) at points on a defined observation grid. A maximizes slip and yields the maximum stress limitation is that Poly3D calculates the stress and perturbation in the surrounding rock. All fault elements displacement field for a single slip event, but cannot are kept from opening or interpenetrating by include the effects of previous slip events on the e prescribing a Burger's vector component, b3 = 0, at surrounding material if there has been any stress each element center. Most faults in this study are relaxation due to viscoelastic or plastic deformation. subjected to a fault strike normal horizontal extension. Tests on simple fault shapes, for which analytical In the model setup of figure 2 the only non-zero solutions exist, show that Poly3d reproduces analytical assigned remote stress would be σ22, where the

Stanford Rock Fracture Project Vol. 16, 2005 E-2 subscripts refer to the global coordinate system, effect of the free surface is thus only felt for shallow g coordinate system, xi . faults and magnitudes of discrepancy angle are small. The values converged on in the case of the circular fault represent error associated with the discretization Model results of the fault surface. This error is due to odd triangular shapes of particular elements that comprise the fault The isolated effects of heterogeneity in host rock surface. The fact that the magnitudes of γ are twice as stiffness, aspect ratios of faults, and the compressibility high for the rectangular fault suggests that irregular of the host rock are evaluated for sets of rectangular and tipline geometry has a more significant effect on the elliptical faults. Both geometries are tested to constrain discrepancy angle than do discretization errors in the the effects of a smoothly varying tipline shape in the case of the circular fault. case of an elliptical fault and of irregular fault tipline shapes in the case of rectangular faults. To investigate the effects of non-planar fault surfaces, we evaluate the idealized model of a regularly corrugated fault.

Figure 4. Statistics of discrepancy angle with varied depth, (a) for a square fault, and (b) for a circular fault. Maximum values are plotted with lines and diamonds, standard deviations, σγ, are plotted with lines and triangles, mean values are plotted with lines and squares.

Changing the aspect ratios of both deeply buried rectangular and elliptical faults reveals a small effect Figure 3. Distribution of discrepancy angle γ, on due to aspect ratios (Fig. 5). Values of maximum γ shallow (d/a = 0.05) faults. The top figure, a, depicts quickly approach 5° and 2.5° for rectangular and the distribution on a square fault, in b the elliptical faults, respectively. However, as Pollard et al. distribution on a circular fault is depicted. (1993) showed, long, rectangular faults are more strongly affected by aspect ratio as the compressibility The effect of the free surface is evaluated by of the host rock increases. In figure 5 the Poisson’s varying the depth, d, of equi-dimensional faults of ratio of the host rock was set to υ=1/4. length, a, dipping at 70° under fault-normal extension. For shallow faults, the distribution of discrepancy angle γ, reveals an asymmetry over a horizontal axis and symmetry about a dip-parallel axis (Fig. 3). The largest magnitudes of γ are observed along the upper, and outermost parts of both the square (Fig. 3a) and the circular fault (Fig. 3b). The asymmetry is attributed to the free surface effect and is insignificant for depths of about twice the width or down-dip length, d/a >2, of the fault (Fig. 4). The maximum values and the standard deviation of the discrepancy angle converge on values of 5.2° and 1.9° in the case of the rectangular fault, and 2.2° and 0.7° in the case of the elliptical fault. The

Stanford Rock Fracture Project Vol. 16, 2005 E-3 magnitudes of γ vary from 0° at locations where the fault surface is parallel to the mean strike and dip of the fault to 90° on the corrugation related hinges of the fault surface (Fig. 7a). The large angle between slip direction and resolved shear stress is due to slip being directed towards the troughs on the limbs of the corrugated fault, while the resolved shear stress there is mostly approximately parallel to the local strike.

Figure 5. Statistics of discrepancy angle with varied aspect ratio, (a) for rectangular faults, and (b) for elliptical faults.

Varying the compressibility (Poisson’s ratio) of the host rock while the fault aspect ratio is fixed at length/width = 2/1 reveals that the maximum values and standard deviation of the discrepancy angle on the fault planes increases significantly with decreased compressibility (Fig. 6). Maximum discrepancy angles and standard deviations for rectangular faults are about 3 times greater than for elliptical faults.

Figure 7. Elliptical fault with sinusoidal corrugations down dip: (a) shows the geometry and non-planar surface (shading accentuates surface irregularities), and (b) shows the distribution of discrepancy angle γ on the fault surface.

Figure 6. Statistics of discrepancy angle with varied The investigation of several model parameters Poisson's ratio, (a) for rectangular faults, and (b) for reveals that depth and aspect ratio only have a minor elliptical faults. effect on the discrepancy angle. Irregular tipline geometry and compressibility of the host rock have a To investigate the effect of frictional heterogeneity greater effect but not exceeding about ±15° except very on a fault plane, an elliptical fault that is sinusoidally near sharp corners in the tipline. The discrepancy angle corrugated down-dip is considered (Fig. 7a). A natural may be strongly affected by the geometry of the fault example of corrugations in the slip direction of a fault surface, tested here by regular corrugations. The results surface is the Northridge thrust, California (Carena and of these idealized models suggest that one of the basic Suppe, 2002). The elliptical geometry is chosen to assumptions of paleo-stress inversions is significantly eliminate the effects of irregular tipline geometry. The

Stanford Rock Fracture Project Vol. 16, 2005 E-4 violated. Irregular tipline shapes, including corners with faults: a northwest striking set, a southwest striking set, perpendicular limbs (Kattenhorn and Pollard, 2001; a southeast striking set and a northeast striking set. The Walsh and Watterson, 1998), non-planar fault surfaces larger faults have fault traces between 1 and 6 km in (Carena and Suppe, 2002), and the interaction of length and displace the Jurassic Navajo Sandstone and adjacent faults (Maerten et al. 2001) are commonly the overlying Carmel Formation, comprised of observed in nature and are investigated in subsequent limestone, siltstone, and mudstone (Maerten et al., sections. 2001). Several faults intersect the Blueberry fault striking Field examples west-southwest, including the Little fault, and the La Sal fault, both striking west-northwest. The west- Two field examples are used to illustrate the northwest striking faults terminate at the Blueberry distribution of discrepancy angle in natural settings. fault. Along all three faults data on the dip-slip The first example from the Chimney Rock fault system magnitudes and striation rake or direction of slip (Maerten et al., 2001), central Utah, aims to illustrate manifested by slickenlines, were collected (Maerten, the effect of adjacent and interacting faults on the 2000). Along the Blueberry fault, both the dip-slip discrepancy angle. The second example from the Wytch distribution and the direction of slip are highly variable Farm oil field (Kattenhorn and Pollard, 2001), southern near intersections (Fig. 9a). The rake systematically England, illustrates the effect of non-planar fault varies from purely down-dip away from the surfaces and interaction within a complicated fault intersections to 70° when approaching the intersections system. from the ends of the Blueberry fault, to -70° to -80° between the two intersections (Fig. 9a). Numerical models were employed by Maerten (2000) to simulate the interaction between the three faults and to understand the systematic changes in slip direction. Little is known about the three-dimensional geometry of the faults, but, for simplicity, elongate fault surfaces were used to simulate the three faults. Slip magnitudes and directions along a horizontal line along the model fault surface are compared to field data (Fig.9b). Although the maximum deviation of the slip direction in the models is about half of that observed in nature, the same systematic change in slip direction is reproduced.

Figure 8. Structural map of the Chimney Rock fault system, San Rafael Swell, central Utah, from Maerten et al. 2001.

Chimney Rock, Utah The Chimney Rock fault system is located in central Utah in the northern part of the San Rafael Figure 9. Measured dip-slip (black squares) and Swell. Most faults near Chimney Rock (Fig. 8) are measures striation rake (grey diamonds) in a. exposed as linear fault scarps and crop out throughout Computed dip-slip distribution (black squares) and 2 computed striation rake (grey diamonds) in b, from an area of approximately 25km . The fault system is Maerten, 2000. comprised of four sets of north- and south dipping

Stanford Rock Fracture Project Vol. 16, 2005 E-5 Near the ends of the Blueberry Fault, slip The Arne fault, one of the larger normal faults in directions converge on almost purely dip slip. This the Wytch Farm fault system, is used to illustrate the suggests that without mechanical interaction the distribution of discrepancy angle γ. Figure 11a depicts Blueberry fault is likely subjected to a roughly fault the interpreted geometry of the fault. The Arne fault normal, horizontal extension. Along the non-interacting extends to a depth of more than 1750m, below which portions of the fault, slip direction and the direction of the exact geometry cannot be well constrained due to the maximum shear stress resolved on the fault plane poor data. The changes from lighter to darker areas are approximately coincident. Field data show that the show the irregularities of the fault surface. The model mechanical interaction of intersecting faults fault is subjected to a horizontal extension systematically affects the discrepancy angles (Fig. 9a) perpendicular to the average fault strike with boundary The same systematic influence can be reproduced by conditions identical to those for previous models. mechanical models (Fig. 9b). We conclude that the Figure 11b depicts the distribution of γ. Magnitudes of γ presence of nearby and intersecting faults may render are as great as 50° may be attributed to the fault one of the basic assumptions of paleo-stress inversion geometry; still greater values probably are due to odd inadequate. geometries of triangular elements. Even under these simple loading conditions and without interaction with other faults the discrepancy reaches significant values. Wytch Farm, England The Wytch Farm oil field is located in the Wessex basin, an extensional basin of Paleozoic to Mesozoic age in southern England. The oil field itself is defined by a 2.5km wide horst block that is dissected by east- west striking normal faults. It is bounded to the south by the south-dipping Wytch Farm Fault, and to the north by the north-dipping Northern Bounding fault (Kattenhorn and Pollard, 2001).

Figure 10. Geological setting of the Wytch Farm oil field, southern England. The gridlines in the boxed area represent seismic survey lines in the area.

The fault system was characterized by a three- dimensional seismic data set of an area of 2 Figure 11. Arne Fault, Wytch Farm fault system: a approximately 66km . Horizon interpretations were shows location and non-planar surface (Lighter based on strong reflector horizons and faults were areas are portions of the fault which are orientated interpreted every 62.5m, on survey lines that trend toward a light source to the left of the figure.), and b roughly normal to the strikes of most faults (Kattenhorn distribution of discrepancy angle, γ, on the fault and Pollard, 2001). The high resolution of the data set surface. and the careful interpretations of faults make the Wytch Farm data a great resource for investigating the mechanical behavior due to irregular tipline geometry and non-planar surfaces of naturally occurring faults.

Stanford Rock Fracture Project Vol. 16, 2005 E-6 Summary and Conclusions Maerten, L, Pollard, D.D., and Maerten, F., 2001, Digital Mapping of three-dimensional structures of the Chimney Several faulting parameters, such as depth, aspect Rock fault system, central Utah. J. Struct. Geology, 23, ratios of faults, and compressibility of the host rock 585-592 have small effects on the discrepancy angle. In the Maerten, L., 2000, Variation in slip on intersecting faults: absence of irregular tipline geometry, non-planar fault Implications for paleostress inversion, Journal of surface, interacting faults, high values of Poisson’s Geophysical Research, 105, 25,553-25,565 ratio, and combinations of these conditions, paleo-stress Lisle, R.J., 1992, New method for estimating regional stress inversions based on a zero discrepancy angle are likely orientations: applications to focal mechanism data of to be a reliable tool, if the stress field was homogeneous recent British earthquakes, Geophysical Journal International, 110, 276-282 throughout the faulting history of a certain tectonic Pollard, D.D., Saltzer, S.D., and Rubin, A.M., 1993, Stress region. However, field data such as those presented inversion methods: are they based on faulty assumptions. here call this second assumption into question. J. Struct. Geology, 15, 1045-1054 Half a century after Wallace (1951) stated that Pollard, D.D. and Aydin, A., 1987, Progress in understanding paleo-stress inversions can be investigated by relating jointing over the past century, Geological Society of fault slip to the ambient stress state, the understanding America Bulletin, 100, 1181-1204 of the mechanics of faulting has improved as have the Shan, Y., Li, Z, and, Lin, G., 2004, A stress inversion tools available to structural geologists. The structural procedure for automated recognition of polyphase geologist can now undertake more complex fault/slip data sets. J. Struct. Geology, 26, 919-925 Thomas, A.L., 1993, Poly3D: A three-dimensional, polygonal investigations of the relation between faulting and the element, displacement discontinuity boundary element stress states using a complete mechanical analysis. computer program with applications for fractures, faults, Using tools such as those employed in this study will and cavities in the Earth's crust, M.S. Thesis, Stanford yield more insight into the phenomenon of faulting, of University, Stanford, California which paleo-stress inversion is only one. We urge Wallace, R.E., 1951, Geometry of shearing stress and relation workers interested in paleo-stress inversions to utilize to faulting, Journal of Geology, 59, 118-130 these tools that are readily available. Walsh, J.J. and Waterson, J., 1988, Analysis of the relationsship between displacement and dimensions on faults. J. Struct. Geology, 10} 239-247

Acknowledgements Financial support for this project was provided by the Stanford Rock Fracture Project. Special thanks to Dave Pollard for suggesting and reviewing this project. Thanks also to Patricia Fiore for reviewing the manuscript.

References Angelier, J., 1989, From orientation to magnitudes in paleostress determination using fault slip data. J. Struct. Geol. 11, 37-50. Bott, M.H.P., 1959, The mechanics of oblique slip faulting. Geological Magazine 96, 109-117. Carena, S. and Suppe, J., 2002, 3D imaging of active structures using earthquake aftershocks: the Northridge thrust. J. Struct. Geol., 24, 887-904. Crider, J.G. and Pollard D.D., 1998, Fault linkage: Three- dimensional mechanical interaction between echelon normal faults, J. Geophysical Res., 103, 24373-24391. Crouch. S.L. and Starfield, A.M., 1983, Boundary Element Methods in Solid Mechanics, George Allen and Unwin, London Griffith, A.A., 1921, The phenomena of rupture and flow in solids. Phil. Transactions of the Royal Society, A221, 163-198 Kattenhorn, S.A. and Pollard, D.D., 2001, Integrating 3-D seismic data, field analogs, and mechanical models in the analysis of segmented faults in the Wytch Farm oil field, southern England, United Kingdom, American Association of Petroleum Geologist Bulletin, 85, 1183- 1210

Stanford Rock Fracture Project Vol. 16, 2005 E-7