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.