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Shapes of Echelon Veins with Complementary Pressure Solution Seams Provide Clues About the Stiffness of Limestone and the Remote Stresses

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Shapes of Echelon Veins with Complementary Pressure Solution Seams Provide Clues About the Stiffness of Limestone and the Remote Stresses SHAPES OF ECHELON VEINS WITH COMPLEMENTARY PRESSURE SOLUTION SEAMS PROVIDE CLUES ABOUT THE STIFFNESS OF LIMESTONE AND THE REMOTE STRESSES Solomon Seyum and David D. Pollard Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected] spacing requires less pressure solution seam Abstract displacement for straight vein propagation than larger vein spacing. With a vein spacing of 8 mm, the A 2-D mechanical model shows the effect that limestone stiffness needs to be 3 GPa and admit 0.3 mm geometries, limestone material properties, boundary of pressure solution seam displacement. In contrast, a conditions, and pressure solution seam displacements 19 GPa limestone stiffness (u = 0.1 mm) produces have on echelon fracture propagation and vein shape. s straight vein propagation when vein spacing is 5 mm. We present a range and combination of geologically We observe most vein spacing to be less than the crack substantiated values for these physical parameters to geometries that we model, therefore suggesting that the reproduce the geometries of echelon veins observed in limestone could have been stiffer than 20 GPa. the field. Particularly, triangular vein shapes and straight vein traces angled to the remote maximum principal compressive stress direction. Keywords: A complete description of echelon vein and pressure Echelon veins, pressure solution seams, fracture solution seam formation reveals that limestone stiffness, mechanics, limestone stiffness, stress, finite element pressure solution seam displacement, and vein model, elasticity, Raplee Anticline, Comb Monocline interaction, in terms of vein length, vein spacing, and vein-array angle, are significant parameters. For veins Introduction in a left-stepping geometry oriented clockwise from the Arrays of echelon veins with complementary �! direction, straight vein propagation requires a ! echelon pressure solution seams are common features specific amount of seam displacement. Displacement of in limestone rocks (Figure 1). Here, the term “array” the pressure solution seam is a function of the limestone refers to the linear arrangement of echelon veins and stiffness. We find that for cracks at 0° (in line with �!), ! pressure solution seams; connecting vein or pressure E must be relatively soft with a value of 1.5 GPa. solution seam midpoints approximates a straight line Cracks that are at 10° (� = 35°) to the �! direction, E ! when viewed at the decimeter to meter scale. Their can be much stiffer (19 GPa). systematic distribution in the stratigraphy encourages Echelon veins angled to the remote maximum structural geologists to relate the formation of these principal compression direction are more likely to structures to a tectonic stress state (e.g. Jackson, 1991; propagate in their own plane than veins oriented Roering, 1968; Wiltschko et al., 2009); similar to the parallel to the maximum compression direction when stress state inferences made for the formation of joint they are coupled with pressure solution seams. This sets (Mynatt et al., 2009). For joints, the relationships implies that for the formation of veins in echelon between driving stresses and joint opening and joint arrays, such as those identified at the eastern propagation are well known (Pollard and Segall, 1987), Monument Upwarp, pressure solution seam and are based on the analytical solutions of linear displacements can cause veins to be straight and angled elastic fracture mechanics (Sneddon and Lowengrub, to the remote maximum principal compressive 1969). However, the physical mechanism for the direction. These results explain the common opening and propagation of echelon veins in an array interpretation of �! bisecting the acute angle between ! with pressure solution seams is largely unknown. With conjugate array sets to cause their synchronous the exception of a few studies known to the authors that formation using the method that explicitly relates consider the physical causes of echelon vein formation deformation (displacements and strain) to the causative (Chau and Wang, 2001; Fleck, 1991; Mandal, 1995; forces as functions of the material properties. Olson and Pollard, 1991; Rogers and Bird, 1987; Zhang Small vein spacing provides clues about limestone and Sanderson, 2002 pp. 171-174), the widely accepted stiffness. Limestone stiffness can be greater for straight hypothesis in structural geology for their formation is propagation of veins with smaller spacing. Smaller vein based on simple shear kinematics (Bons et al., 2012; Stanford Rock Fracture Project Vol. 25, 2014 D-1 Lisle, 2013; and many others), as introduced by Ramsay (1967), that describes the motions of cracks as passive markers; or, imaginary lines that rotate. To appropriately relate observed, systematic, deformation structures in rock to the regional tectonic stresses, a physical mechanism of formation for a single, representative structure should be understood. The deformation criterion should include a geologically appropriate constitutive relationship between applied stresses and the resulting strain that, with other geologic constraints, would reproduce the observed geometries. The constitutive relationship satisfies Newton’s laws of motion, such as static equilibrium of forces and conservation of momentum. These physical laws apply to deformation of rocks in the Earth’s crust, and are applied here to describe the formation of echelon veins with complementary pressure solution seams in limestone. Studies of echelon veins that use mechanics include Rogers and Bird (1987), Fleck (1991), Olson and Pollard (1991), Mandal (1995), Ramsay and Lisle (2000 pp. 766-768), and Chau and Wang (2001). All of these studies use 2-D models. Ramsay and Lisle (2000 pp. 766-768) use a finite element model to show how strain is deflected in a narrow zone of softer, elastic material relative to the stiffer surrounding material as a way to infer the orientation of potential initial echelon fractures. Rogers and Bird (1987) illustrate the complexity of stress distributions at crack tips for a Figure 1. Conjugate arrays of left-stepping and geometrically irregular set of echelon cracks using a right-stepping echelon veins, with boundary element model (isotropic, linear elastic complementary pressure solution seams, on the material) to reproduce echelon dike geometries. Chau top surface of the McKim Limestone at the and Wang (2001) describe interactions between echelon northern end of Raplee Anticline. cracks and the limits on straight crack growth patterns using an analytical solution to solve for the critical mechanical responses of echelon cracks to applied ratios of crack size and crack spacing for a variety of forces. The model input values are varied within the boundary configurations. Mandal (1995) provides an range of geologically appropriate values in order to analytical solution for the stress field in an elastic produce crack shapes (crack surface displacements) that material containing cracks to show how echelon crack most closely approximate vein shapes measured in the spacing, or mechanical interaction of neighboring crack field. Since vein surface displacements and the remote tips, affects the direction of infinitesimal crack stresses are explicitly related an interpretation of the propagation. Fleck (1991), using dislocation theory, and regional tectonic setting at the time echelon veins and Olson and Pollard (1991), using a boundary element pressure solution seams formed is defensible. model, show that the crack surface displacements can Some of the terminology used in this paper is be calculated given the stiffness of the material, and explained here. “Fracture” is used when referring to the similar to Chau and Wang (2001) and Mandal (1995), separation of rock to form two surfaces without any show that crack propagation is a function of the near-tip specification to relative motion of those surfaces stress field and the near-tip stress field is perturbed by beyond initial opening. The term “joint” refers to purely neighboring cracks. opening mode fractures identified in the field. “Vein” is We use a commercial finite element software to a term used for a fracture that has been filled with calculate displacements on crack surfaces and stresses mineral precipitates. The term “crack” is used when near crack tips to show how the resulting crack shape is referring to a discrete, material discontinuity in the a function of the initial crack orientation, the crack mechanical models and consists of two surfaces and length and spacing, the boundary conditions, and the two tips. Cracks in the models are compared to veins amount of displacement at model seams. Using observed in the field. “Pressure solution seam” refers to elasticity, we record the values to illustrate the simplest the field-identification of a two-dimensional trace in Stanford Rock Fracture Project Vol. 25, 2014 D-2 limestone along which we infer rock has been dissolved propagation is perpendicular to the direction of greatest and transported in solution. A “seam” refers to the extensional strain; forming sigmoidal vein shapes. mechanical model representation of a pressure solution More recent kinematic models have included normal seam. components of homogeneous strain across shear zones This is a mechanical study of echelon vein and (transtension and transpression) to explain smaller vein- pressure solution seam arrays in its simplest form. First, array angles (Kelly et al., 1998; Peacock and we introduce past kinematic
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