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Home , Pressure solution, Vein (geology)

SHAPES OF ECHELON VEINS WITH COMPLEMENTARY SEAMS PROVIDE CLUES ABOUT THE STIFFNESS OF 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 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 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 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 , Comb 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 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 for their formation is propagation of veins with smaller spacing. Smaller vein based on simple kinematics (Bons et al., 2012;

Stanford 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 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 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 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 studies on these structures Sanderson, 1995; Srivastava, 2000). Ambiguity arises and emphasize the aptness of physics for describing regarding fracture positions and modes, the number of rock deformation. This is followed by descriptions and fractures, and their lengths and spacing (Figure 2b) measurements of echelon vein and pressure solution since no choice is made of the material properties and seam arrays observed in nature. We then demonstrate its inherent flaws, and it is impossible to know how the how to construct a mechanical model to best represent boundary conditions affect the distribution of stress. the geology, such as the field-measured geometries of Orientations of the principal remote stresses and veins and pressure solution seams, the mechanical their relative magnitudes are often inferred from the properties of limestone, the remote and crack surface interpreted shear strains and from the orientations of stress boundary conditions, and the displacements conjugate array sets using various adaptations of the caused by pressure solution seams. The resulting model Mohr failure criterion (Becker and Gross, 1999; Bons et crack surface displacements and crack tip stresses are al., 2012; Rickard and Rixon, 1983; Roering, 1968). shown as a function of the variables. The two modeling However, this criterion is based upon a stress state at a objectives are to compare the displacements on the given point or in a homogeneous field rather than a model crack boundaries to vein apertures, and to state of stress that varies spatially due to the presence of evaluate the near tip stress field of cracks to explain veins and pressure solution seams (Figure 2d). The straight vein traces. simple shear kinematic concept and the Mohr failure criterion for echelon vein formation are the Kinematics of Echelon Vein Arrays explanations given in current structural geology textbooks (Allmendinger et al., 2012 pp. 144-145, 216; Past studies of the formation of echelon veins in Davis et al., 2011 pp. 543-544, 573-574; Dennis, 1987 limestone aim to explain the initiation mode (tensile or p. 238; Fossen, 2010 pp. 289, 305-306; Mandl, 2005 shear fracture) of vein segments (Beach, 1975; Rickard pp. 185-203; Park, 1997 pp. 117-119, 126; Passchier and Rixon, 1983; Shainin, 1950), the relative timing and Trouw, 2005 pp. 172-173, 252; Price and between initiation and echelon fracture Cosgrove, 1990 pp. 44-45; Twiss and Moores, 2007 pp. formation (Lajtai, 1969; Roering, 1968), the 252, 446-447; Van der Pluijm and Marshak, 2004 p. distribution of strain within a shear zone (Hancock, 160). 1972; Ramsay, 1980), and the development of Kinematic quantities must be incorporated into a sigmoidal veins (Lajtai, 1969; Ramsay and Graham, complete mechanical model before any inferences are 1970; Shainin, 1950). These studies are limited by made about the relationship between vein inferences drawn from kinematic models consisting of a displacements and tectonic stresses (Fletcher and prescribed, homogeneous simple shear strain field Pollard, 1999). In order to test hypotheses about (Ramsay, 1967), as shown in Figure 2a, to explain vein echelon vein and pressure solution seam formation such orientations (Figure 2b). as fracture initiation, fracture opening or shearing, Simple shear models are the most widely used fracture rotation, fracture propagation, or even strains models in structural geology to describe echelon vein within and away from an array of echelon fractures, we formation (Belayneh and Cosgrove, 2010; Kelly et al., need to go beyond the limitations of kinematics and 1998; Mazzoli et al., 2004; Srivastava, 2000; and many utilize established physical laws and our observations others). The final orientation of a vein is known from a of rock behavior to explain the cause and effects of field measurement, and the initial orientation is inferred limestone deformation. from the postulated simple shear, where the vein is Similar to kinematic and mechanical models of presumed to be a passive marker. The angle of shear echelon veins, we also consider deformation in a two- and the shear strain are estimated from the change in dimensional plane that bisects and is orthogonal to vein orientation of echelon veins, ignoring perturbations of height. This plane displays the geometries of veins and veins on the local strain field (Figure 2d). Vein opening pressure solution seams in an array as typically is inferred to be parallel to the axis of greatest extension measured in the field. at 45° within the simple shear zone (Ramsay, 1967 pp.

83-88). Subsequent shearing rotates the supposed veins and each additional increment of hypothetical vein

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Figure 2. a) Simple shear kinematics. b) Hypothetical placement of cracks in a region of simple shear. Crack positions, lengths, and spacing are arbitrary. c) The maximum in-plane principal strain field within a material subjected to shear stresses on its longer boundaries. The contours map the magnitudes of maximum principal strain throughout the material, and the line segments represent the orientations of the maximum principal strain (extensional strain) at various positions. d) Including echelon cracks into a material perturbs the local strain field.

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Figure 3. Thin-section images of straight, calcite-filled veins. a) A set of veins truncate into or traverse across a pressure solution seam. A vein has intersected a fossil that is suggestive of pure vein opening. b) Thin and thick veins are parallel to sub-parallel to one another. c) Thin veins are straight despite their overlap geometry. d) The vein tip comes to a point and has intersected a fossil near its tip suggestive of pure opening of the vein.

greatest compressive stress (Fletcher and Pollard, General Geologic Attributes 1981). Pressure solution seams form by dissolution of driven by the resultant normal stress at grain Two distinct sets of echelon vein and pressure boundaries in the presence of fluids with specific solution seam arrays are exposed in marine limestone solubility properties, followed by transportation of the strata near Mexican Hat, Utah across the Raplee solution to areas away from the seam with lower Anticline and Comb Monocline folds. The sets are pressures (Bathurst, 1975 p. 462; Nenna and Aydin, distinguishable from each other by the echelon step 2011; Weyl, 1959; Zhou and Aydin, 2010). geometries of veins and complementary pressure The veins have generally straight traces and are solution seams; right-stepping veins versus left- filled with calcite (Figure 1). Veins that are only visible stepping veins (Figure 1). Pressure solution seams at the thin-section scale also maintain straight traces in coexist with veins in all of the arrays observed in the the general orientation of centimeter-length veins, and field, and the veins and seams in a given array are are seemingly unhindered by neighboring veins (Figure nearly orthogonal to one another in bedding plane view. 3b,c). Pressure solution seams have curved traces From vein-fossil intersections in thin-section, veins (Figure 1) that can have up to 90° curves. formed predominantly as opening fractures (Figures 3a, The intersection of veins and pressure solution d), with few examples having a small component of seams display mutual crosscutting and abutting relative shear. In contrast, pressure solution seams are relationships (Figure 1). We infer that veins and closing mode structures that form perpendicular to the

Stanford Rock Fracture Project Vol. 25, 2014 D-5 pressure solution seams developed simultaneously. the components of displacement and resultant Mutually intersecting veins and pressure solution seams displacement at each position in the material are can be reconciled as structures responding to an calculated analytically as follows for a uniform remote identical stress state with the greatest compressive stress field (Pollard and Segall, 1987). stress parallel (and sub-parallel) to the vein and perpendicular (and sub-perpendicular) to the pressure 1 + � � = 1 − � �! − ��! � + ��! solution seam (Fletcher and Pollard, 1981; Katsman, ! � !! !! !" 2010). (1) In the third dimension, veins maintain their 1 + � � = 1 − � �! − ��! � + ��! orientation to their tip lines perpendicular to bedding ! � !! !! !" for millimeters to centimeters, but pressure solution (2) seams may have greatly curved planes over similar � = �! + �! distances beneath the limestone top surface.. Generally, ! ! additional veins and pressure solution seams exist (3) beneath those exposed at the limestone surface to form ! the array plane. Thus, the arrays extend perpendicular In this study, �!" is zero and the normal ! ! to bedding and are generally shorter in that direction components of remote stress (�!! and �!!) are principal than their bed-parallel length; limited by bedding planes ! ! stresses (�! and �! ). Compression is positive. or bed-parallel pressure solution seams. We have made For 2-D problems in elasticity the stress no observations of the continuation of echelon veins or components are not functions of the elastic material pressure solution seams into adjacent and properties (Timoshenko and Goodier, 1970 p. 19). For siltstone strata. Also, there is no systematic evidence of example, the analytical solution to the circumferential joints or faults in adjacent strata that are in the same stress field near the tip of a single, straight crack plane as array traces. Therefore, we negate mechanisms subjected to a uniform uniaxial perpendicular to of jointing or propagation into limestone strata to the crack trace (Anderson, 2005 pp. 81, 95; Jaeger et form the echelon vein and pressure solution seam al., 2007 p. 240; Lawn and Wilshaw, 1975 p. 53) is arrays. � � � = �! ∙ cos! Methods !! ! 2� 2 We build a 2-D mechanical model to describe the (4) formation of echelon vein and pressure solution seam arrays in terms of physical attributes representative of �!! is the tangential cylindrical component of stress at a the geology. This section is divided into six subsections radial distance, �, and angle, �, from the crack tip. The that explain the construction of this mechanical model. position around the crack tip where �!! is the minimum The six subsections are dimensional analysis, stress (tension is negative) is where the next increment of boundary conditions, limestone elasticity, pressure crack propagation is predicted. solution seam displacement, significant variables, and The angle of the crack to the remote stresses, �, and the model design. fluid pressure, �, are additional physical variables when considering fractures with various orientations in a rock Dimensional Analysis and the probability that there is fluid in the subsurface. For multiple, parallel, equal-length cracks in a left- A dimensional analysis is used to efficiently design stepping echelon array, number of cracks, �, spacing, the numerical experiments by narrowing the number �, the angle, �, of the crack traces to the array trend, and sequence of experiments to conduct based on the and the angle, �, of the array trend to �! are included sensitivity of the independent variables. The ! Buckingham Theorem of dimensional analysis relates (Figure 4). The crack and array angles, �, �, and � are the number of physical variables, �, and the number of related as � = � − �, so only two of these angles are independent dimensions occurring in those variables, �, necessary to evaluate. Since � and � are directly to give the dimensionless quantities or scale factors, �, associated with field measurements, � is omitted. The final physical variable that we consider is the for the physical process; such that � = � − � mechanical effect of the displacement discontinuity of (Anderson, 2005 pp. 18-21; Buckingham, 1915). For 2-D homogeneous elastic materials subject to seams, �!, that bisect cracks. uniform remote stresses, the displacements are a linear So, 13 physical variables (�) influence function of remote stresses �! and �! , the elastic displacements, �!, throughout the material. !! !! moduli (� and �), and the material coordinates (�, �). For example, in a quasi-static boundary value problem,

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Figure 4. This is a conceptual model illustrating the 13 physical variables needed to describe the formation of echelon veins with complementary echelon pressure solution seams in a limestone from the a) initial stage to the b) final stage.

! ! �! = �! �! , �! , �, �, �, �, �, �, �, �, �, �, �! Normalized stress components are functions of 9 � = �, � independent dimensionless variables. (5) ! �!" �! � � � �! For an array of echelon cracks with complementary ! = �! ! , , �, ! , �, , �, �, �! �! � �! � � seams, the geometric variables, fluid pressure, and �, � = �, � displacement discontinuity of the seam also affect (8) stresses at the crack tips. The components of stress are linearly related to the To further improve model efficiency, we independent variables and are collectively represented demonstrate in the following subsections why some of by �!". the variables in Equations (7) and (8) can be held constant based on interpretations of the geology and the ! ! �!" = �! �! , �! , �, �, �, �, �, �, �, �, �! relative contribution of each variable to crack surface �, � = �, � displacements and crack tip stresses. (6) Stress Boundary Conditions We choose � and �! to normalize variables with ! Remote stress boundary conditions in the dimensions of length and stress, respectively. Each mechanical model are intended to reflect the burial and scales linearly with displacements and stresses and their tectonic conditions when arrays of echelon veins and effect can be understood in terms of complementary pressure solution seams developed. We infer that the parameters like �, �!, and �. Thus, the displacement is ! echelon vein and pressure solution seam arrays are a function of 11 dimensionless quantities. associated with the folding event that formed Raplee

Anticline and Comb Monocline during the Laramide � �! � � � � � � ! ! ! orogenic event beginning 70 million years ago; a 30 = �! ! , ! , �, , , ! , �, , �, �, � �! �! � � �! � � million year maximum period of regional compression � = �, � on the Colorado Plateau tangential to the Earth’s crust (7) oriented approximately east-west (Gregory and Moore, 1931; Kelley, 1955; Tweto, 1975). � is used to represent dimensionless functions; as Nuccio and Condon (1996) provide a depth and opposed to �. temperature history of the stratigraphy near the Monument Upwarp (Figure 5). The stratigraphic units

Stanford Rock Fracture Project Vol. 25, 2014 D-7 ! containing echelon vein and pressure solution seam within the range of 0° to 35° from the �! direction to arrays are shaded in Figure 5. During the Laramide open. This range is representative of vein angles orogenic event, the burial temperatures of these calculated from field measurements of � and �. A crack stratigraphic units were between 95°C and 114°C (as oriented at 35° (� = 50° and � = 15°) will be the most low as 40°C at the onset of rapid burial 95 Ma), and a resistant to opening in a biaxial remote compressive range of 3 km and 4 km depth (as shallow as 1.5 km, 95 environment, so we chose this angle to test the Ma). The confining pressures due to overburden combination of stress boundary conditions that will loading (assumed to be 25 MPa per kilometer) during allow all cracks that we model to open (Figure 6). For a the Laramide were between 75 MPa and 100 uniform fluid pressure, �, of 30 MPa, cracks oriented ! MPa (as low as 37.5 MPa, 95 Ma). Fluid pressures due between -35° and 35° from �! will open for ! ! to an assumed hydrostatic load (10 MPa per kilometer) �! = 65 ��� and for �! = 10 ���. Varying these were between 30 MPa and 40 MPa (as low as 15 MPa, three variables produces a fairly predictable, linear 95 Ma). relationship to the relative displacement distributions of Depending on the combination of stress boundary crack surfaces and to the stress fields at crack tips; as conditions, model cracks will either open or close. We can be inferred from Equations (1), (2), and (4). need to determine the conditions that will allow cracks

Figure 5. Burial history profile of the strata in Monument Valley (Nuccio and Condon, 1996). The inset is a magnified view of the rapid burial and uplift of strata after the Jurassic.

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Figure 6. The remote stress boundary conditions for crack opening for a set of crack surface pressure values. The star indicates the combination of traction boundary conditions selected for the models in this study.

These uniform boundary stresses are kept constant strain before inelastic deformation occurs between 370 for the model runs in this study. With constant values MPa and 490 MPa differential stresses. The elastic for the boundary stresses, the stress field, �!!, can be material stiffness (�) ranges between 49.1 GPa and normalized by the driving stress on the crack surfaces, 78.9 GPa (Birch, 1966; Deere and Miller, 1966 pp. 72, ∆�, for various crack angles where ∆� = � − 73, 78; Ide, 1936; Peselnick, 1962; Renner and ! ! �! sin � − � − �! cos (� − �). Normalizing by the Rummel, 1996). Poisson’s ratio (�) ranges between driving stress allows for a direct comparison of crack 0.12 and 0.36, but is kept constant in the following tip stress positions and magnitudes between echelon mechanical models since its effect on crack shape is cracks with different orientations. negligible over that range. Elastic deformation of the McKim Limestone Limestone Elasticity during the Laramide Orogeny was similar to the elastic response of the Solnhofen Limestone under Experimental data for limestone is experimental conditions. The temperatures in the rock abundant (Birch, 1966; Hatheway and Kiersch, 1989 mechanics experiments bracket the range of burial pp. 700-703; Ide, 1936; Kulhawy, 1975; Paterson and temperatures. Increasing temperature reduces the yield Wong, 2005 pp. 28, 32, 70, 125, 153, 214, 215; point of limestone and encourages strain-softening Robertson, 1955; Vajdova et al., 2004). The triaxial (Figure 7b), but has little effect on the elastic stiffness compressive test results from the Solnhofen Limestone below 500°C (Heard, 1960). The inferred geologic (Fischer and Paterson, 1989; Fredrich et al., 1990; confining pressures and fluid pressures on of Heard, 1960; Renner and Rummel, 1996) are used in the eastern Monument Upwarp during the Laramide this study to guide our choices of material properties Orogeny (Figure 5) were less than the confining and constitutive relationships to represent the pressures and fluid pressures used for tests on the limestones at Raplee and Comb folds that contain Solnhofen Limestone (Heard, 1960). Since variations in echelon vein and pressure solution seam arrays. The confining pressures and fluid pressures at 25°C and Solnhofen Limestone is made of micritic calcite grains 150°C show little effect on the elastic portion of with diameters from 1-6 to 1.5-5 m (Barnhoorn et al., deformation of the Solnhofen Limestone (Figure 7), we 2005; Green and Perkins, 1968; Ide, 1936; Robertson, infer the elastic response of limestones in the field to 1955). The of Solnhofen Limestone range have been similar during the formation of echelon from 1.7% to 5.9% (Barnhoorn et al., 2005; Baud et al., veins. In the field, elastic deformation is inferred from 2000; Green and Perkins, 1968). Similarly, the observations of numerous veins within arrays that have of the micritic McKim Limestone at Raplee Anticline, a small aperture-to-length ratios (Figure 3c,d). limestone unit containing most of the measured arrays of echelon veins and pressure solution seams, is between 2% and 7% calcite-filled and vacant pore Pressure Solution Seam Displacement space. Pressure solution seams cause displacements that Triaxial compression deformation experiments are predominantly parallel to veins, where the seam using the Solnhofen Limestone in wet conditions at “surfaces” move toward each other. This process will 25°C and 150°C are shown in Figure 7 (Heard, 1960). have an impact on the displacement and stress fields Deformation is approximately elastic up to 1% to 1.5% near the veins.

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Figure 7. Triaxial compression rock mechanics results of the Solnhofen limestone with interstitial fluids at a) 25°C and b) 150°C (from Heard, 1960 fig. 11). The Solnhofen limestone results are used as an analog for the physical properties of the McKim limestone based on their similar petrographic characteristics.

The closing displacements caused by the pressure The seam-normal relative displacement, 2�!, is solution process are calculated from thin-section �! − �. Assuming mass conservation of the insoluble photographs of pressure solution seams. The calculated grains, �!" = �!!. Multiplying both sides of this values are used in the mechanical model of echelon equivalence by the areas in Equations (9) and (10) gives crack and seam arrays to simulate 1-D displacement of �! in terms of the measurable variables as follows. model limestone perpendicular to model seams. We then evaluate its effect on complementary crack surface �!" �!! displacements and crack tip stresses. � ∙ = � ∙ � + � ! � + � A kinematic description of the displacement !" !" !! !! (11) calculation method is illustrated in Figure 8. Grains of and alkali feldspars that are distributed The terms in parentheses can be represented by � throughout the micritic, calcite matrix accumulate at the ! pressure solution seam as apparently insoluble on the left side of the equivalence and by �! on the constituents (Figure 9). The concentration of insoluble right side. grains inside pressure solution seams are related to the �! concentration of insoluble grains in the neighboring �! = � ∙ calcite matrix as a basis for making the calculation of �! displacement; assuming mass conservation. (12)

The total areas of the seam (�!) and of the host rock Therefore, 2� = � ∙ � � − �. � is the (�!) are represented by the areas of insoluble and ! ! ! ! soluble material in each. displacement of limestone at one of the pressure solution seam “surfaces” normal to the pressure � = � ∙ � = � + � solution seam trace, and is used as a model boundary ! !" !" condition. (9)

�! = �! ∙ � = �!! + �!! � �! (10) �! = ∙ − 1 2 �! � is the seam thickness; � is a representative width (13) parallel to the seam trace; � is an unknown length ! � � is greater than one because � = � and perpendicular to the seam; � is the area of insoluble ! ! !" !! !" � < � . Note that the area of material dissolved is grains in the seam; � is the area of soluble matrix in !" !! !" � − � . the seam; � is the area of insoluble grains in the host !! !" !! Thin-section measurements were made from arrays rock; � is the area of soluble matrix in the host rock. !! with a range of vein geometries, in different limestone

Stanford Rock Fracture Project Vol. 25, 2014 D-10 strata, and from various geographic locations. The infinitely-long crack in plane strain conditions (Elliott, calculated values for �! range from 0.25 mm to 2.25 1947; Tada et al., 2000 pp. 125 & 342); given the same mm. boundary conditions and material properties. The model limestone is considered homogeneous since the Significant Variables limestone petrographic composition and texture vary little over decimeter lengths of an array containing tens We have demonstrated that �, �, �, �!, �!, and � ! ! to hundreds of veins and pressure solution seams, and can be held as constant parameters based on the there are few planes of discontinuities, such as joints, analysis of geologic data, rock mechanics data, and the that disrupt the array traces. Material isotropy is linear elastic solutions for deformation of a material. assumed because there are no apparent mineral The remaining values, �, �, � and � , are varied. ! lineations or planes in the limestone strata that Variations in � over the 15° to 25° range of measured would imply bedding-parallel directional variation in values provides little insight beyond the response of rock stiffness. crack opening and shearing to a crack inclined to the The exterior boundaries of the model are placed at remote stresses. This can be understood from the least 40 times the half-length of an array from the array overlap geometry of echelon cracks being only a center so the calculations of stress, strain, and function of crack length, crack spacing, and crack-array displacement within a distance 3 times the half-length angle as 2� − � tan � . This is visually represented in of the array from the center are within 5% error (Figure Figure 10. Furthermore, orienting a Cartesian 11). This is bench-marked using a comparison of the coordinate to be coincident with the crack length allows numerical solution (using Abaqus/CAE) for stresses at for evaluation of crack surface displacements using a the tip of a single pressurized crack with half-length, �, single coordinate position; x coordinates in this case. in a finite body to the analytical solution of a single Polar coordinate � is held constant when evaluating pressurized crack in an infinite body (Pollard and stresses at the near tip field of a crack; the placement of Segall, 1987). We determined that the stresses which is discussed in the following subsection. calculated within a distance 3� from the crack in the Therefore, the displacement and stress functions, numerical solution are within 5% error of the analytical Equations 7 and 8, can be simplified. solution for a boundary distance of 40�. We include an initial arrangement of echelon cracks �! � � � �! = �! , , �, , based on the hypothesis of a pre-existing, self- � � � ∆� � organization of fractures from discrete discontinuities � = �, � (flaws) in limestone forming the echelon geometry. An (14) array of cracks is used instead of a narrow zone of �!! � �! = � �, , �, weaker material (Ramsay and Lisle, 2000 pp. 766-768), ∆� ! � � (15) since there is no evidence of linear zones of weaker material in the field. Furthermore, an array of fractures There are just 5 independent quantities of the 13 localizes deformation as shown in Figure 2d, and as identified physical variables needed to quantify echelon demonstrated by interacting cracks from numerical vein shapes, and only 4 to explain echelon vein methods (Olson and Pollard, 1991; Pollard et al., 1982; propagation paths. Swain et al., 1974; Thomas and Pollard, 1993) and experimental methods (Brace and Bombolakis, 1963; Model Design Hoek and Bieniawski, 1965; Lange, 1968; Vajdova et al., 2010). We use a commercial finite element software, A single array of left-stepping echelon cracks and Abaqus FEA (v. 6.11-1 © Dassault Systèmes, 2011), complementary right-stepping echelon seams are for modeling static, 2-D plane strain deformation of modeled in each run (Figure 10). Due to the bilateral echelon cracks in an isotropic, homogeneous model symmetry of the problem, all results and insight gained limestone. The cracks are discontinuities in the material from left-stepping crack geometries can be concluded continuum and are considered to be long in the third about similar right-stepping echelon cracks. dimension, as are model seams. We choose to ignore The starting model geometries of echelon cracks three-dimensional effects because they would introduce (Figure 10) are � = 6 mm, � = 8 mm, � = 25°, and � = modest quantitative differences in the results. For 25°. They are within the range of measurements made example, a crack that is three-dimensional, such as the in the field and in thin-section. Cracks are prescribed penny-shaped crack (Sack, 1946; Sneddon, 1946; two, equal-length, coincident surfaces that are Sneddon, 1965), the stress intensity and the crack connected at their ends to create sharp tips and zero opening profile only vary by a factor of 2 � from the apertures. The crack surfaces are assigned uniform, similar analytical solutions for a two-dimensional, normal tractions to represent �.

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Figure 8. This conceptual model illustrates the aggregation of insoluble grains in the limestone matrix to the localized area of pressure dissolution from an a) undeformed state to c) the final, deformed state. The length and area components used to calculate the kinematic displacements are labeled.

Figure 9. Thin-section images of two pressure solution seams within an array. a) and c) were made using plane polarized light where quartz, feldspars, pore spaces, and some large calcite grains are light in color and the micritic calcite matrix is dark. b) and d) were made using cross-polarized light using a , wave-retardation plate for greater contrast between quartz and feldspar grains. Quartz and feldspar grains have a higher concentration at the pressure solution seams than in the matrix.

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Figure 10. These are ranges of echelon crack spacing, crack-array angles, and array angles tested in the mechanical models. The ranges of values are representative of field-measured values.

The model seam geometry is constructed as a of �!. � is not applied on the portion of the crack rhombus of homogeneous, isotropic material with the surface that is coincident with the seam. Including or long axis bisecting the crack length (Figure 12) and an excluding � at the seam portion of the crack has a initial half-length, �!, of 2.5�. The displacements across negligible effect on crack surface displacements away the model seam boundaries are continuous, and the from the seam, but does affect the magnitude of �! and boundary displacement condition, �!, is applied does affect the crack tip stress field. indirectly. Stiffness of the seam, �!, is assigned to be In this finite element analysis, the model materials 0.1 GPa. This value is much less than the surrounding (main body and seams) are discretized (meshed) using model limestone stiffness to facilitate displacement of quadrilateral elements everywhere except within the seam boundaries at the seam-crack point-of- ~0.00025c and 0.005c mm radial distance, �, from a tip intersection in a closing mode by amounts calculated of the middle crack where triangular elements are for �! 2. The � values used for the model limestone assigned (Figure 13). The crack surface normal and range from 1 GPa to 60 GPa. Lower, less-realistic � crack surface tangent components of displacement are values are used for admitting larger seam displacements collected at 120 nodes of the two middle crack surfaces and for testing its sensitivity. Poisson’s ratio for the for each model run. The spacing along the crack length seam, �!, is 0 to restrict transverse strain when the seam between each node decreases toward the crack tips and is under compression. The initial half-width of a seam, toward the model seam (Figure 13c). A node of one max �!, is measured at the crack-seam intersection crack surface shares the initial coordinates of a node (Figure 12a) and is 0.125� to admit as much as 1.5 mm

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Figure 11. An initial configuration of the mechanical model with an echelon crack geometry example.

belonging to the other complementary crack surface. which is within the near tip region of the crack (Pollard Subtracting the components of displacement from each and Segall, 1987). The crack propagation hypothesis node pair gives the relative opening and relative shear used in this study follows Erdogan and Sih (1963). The value at points along the crack length (Figure 12b). position, �, of the maximum circumferential tensile An orthogonal cylindrical mesh is placed a distance stress (min. �!!) a distance � from the crack tip predicts � = 0.005c to 0.01c from the tip of the middle crack the initial propagation direction. Since we observe (Figure 13d) to evaluate the circumferential stress straight veins in the field (Figures 1 and 3), we seek to component using a polar coordinate system with the determine the range of model parameters that would origin at the crack tip (Figure 12c). Stresses at the tip of predict straight crack propagation. the middle crack in each model run are collected from 360 evenly-spaced nodes at � = 0.005� (Figure 13d);

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Figure 12. a) Model attributes of a crack and seam. b) Method for plotting crack surface displacements collected at element nodes in the model. c) Method for plotting crack tip stresses.

Figure 13. a) Finite element mesh of the entire model with mostly quadrilateral elements. b) Mesh around echelon crack and seam array. c) Mesh around a single crack (c = 6mm) and seam. d) Mesh near a crack tip. e) Mesh at a crack tip.

crack opening by an order of magnitude. The Results contribution � has on crack opening is as predicted from the analytical solutions of the 2-D displacement For echelon cracks without model seams, crack opening field described by Equations 1 and 2. increases with decreasing �, increasing �, and Of the significant variables, crack shearing is a decreasing � (Figures 14a, b, e). Crack opening function of �, �, and � . Therefore, they influence the distributions and magnitudes vary little with changes to ! crack tip stress field. Varying angle � will cause a �. The crack shape becomes less elliptical and more change in the sense of shear that is related to the angle triangular when � is reduced from 0.83� to 0.50� at between the crack and the direction of �! (Figure 14d). � = 25°; less crack opening near the crack tips and ! The position of min. � is toward the adjacent crack more opening in the middle for a crack spacing of !! for larger � and away from the adjacent crack for 0.54�. Decreasing � by an order of magnitude increases smaller � (Figure 15b). Decreasing spacing increases

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Figure 15. Distributions of ��� around the crack tip. The circumferential position, θ, of the minimum ��� for the various models are labeled. a) Varying crack spacing. b) Varying crack-array angle. c) Increasing seam displacement.

crack shearing, and its distribution is not elliptical (Figure 14c). For a left-stepping echelon crack geometry, decreased spacing induces left-lateral relative shear. Thus, decreasing � causes the position of min. �!! to be more toward the adjacent crack in the echelon array (Figure 15a). �! increases crack opening and crack shearing by an order of magnitude (Figures 14f, h). In combination

Figure 14. Opening distribution along the middle with a small � value that causes a triangular crack crack in a five-crack echelon array without opening profile, increasing �! results in greater opening seams for a) various crack spacing and b) and straighter crack surfaces. Furthermore, increasing various crack-array angles. Crack shearing �! will encourage min. �!! at the crack tip to be distribution for c) various crack spacing and d) positioned away from the adjacent crack, countering the various crack-array angles. Opening magnitudes effect of mechanical interaction between adjacent for e) decreasing model limestone stiffness and cracks (Figure 15c). Increased displacement at the f) increasing displacement at seams. Crack seams rotates the position of min. � toward the shearing is g) unchanged with varying model !! limestone stiffness and h) enhanced with adjacent crack again. This suggests that a balance exists displacement at seams. between echelon vein spacing, vein-array angle, and displacements induced by pressure solution seams to encourage straight vein propagation. Figure 16 is an example of the change in position of min �!! for various �! values applied to the echelon crack geometry � = 35° and � = 5 ��. Figure 17

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Figure 16. a) ��� distribution at the tip of a crack in a left-stepping echelon crack array with geometries � = ��° and � = � ��. No seams are in this model. b) Model seams are introduced and the concentration of negative ��� values at the crack tip has shifted to the right side of the crack tip. c) Increasing the seam displacement shifts the stress concentration counterclockwise. d) Minimum ��� is directly ahead of the crack. e) With greater seam displacement, minimum ��� is to the left of the crack tip again.

Figure 17. The amount of model seam displacement and limestone stiffness needed to encourage straight crack propagation for crack-array angles 25° and 35°, over a range of crack spacing values. The upper half of the graph colored in gray encompasses an area where the model limestone stiffness has to be very low and might be unrealistic. The shaded areas at the bottom left of the graph cover an area related to the crack-array angle and crack spacing geometry combinations that are unallowable using vein and seam geometries used in the models (see Figure 10).

Stanford Rock Fracture Project Vol. 25, 2014 D-17 shows the amount of �! needed to cause the position of crack geometries that we model, therefore suggesting min �!! to be directly ahead of the crack tip (pure crack that the limestone could have been stiffer than 20 GPa. opening) for values of � and �. So, straight crack We also determined how triangular vein shapes propagation is predicted for � = 25° ��� 35° along the observed in the field (Figure 1 and 3a) are formed. two dashed lines in Figure 17. For � less than 25°, Small crack spacing causes the crack surfaces to be increased seam displacement will enhance right-lateral straighter rather than elliptical (Figure 14a). When seam shear on cracks and will never result in pure crack displacement is included and increased, the crack opening and straight crack propagation Figure 16. surfaces become increasingly straight. A mechanical description of echelon vein and Discussion pressure solution seam formation provides several additional insights. An example is that veins of an array Echelon veins angled to the remote maximum that parallel the complementary conjugate array principal compression direction are more likely to orientation likely formed in limestone that was stiffer propagate in their own plane than veins oriented than limestone that formed veins parallel to one another parallel to the maximum compression direction when across conjugate arrays. they are coupled with pressure solution seams. Left- Regarding sigmoidal vein shapes, for any vein stepping echelon veins oriented counterclockwise from ! geometry with some pressure solution seam the �! direction cannot propagate in their own plane. displacement, if there is not enough displacement at the This suggests that for a vein to propagate in its own seam (or the limestone is too stiff) the veins would tend plane, there is a vein orientation limit counterclockwise ! to propagate in a sigmoidal shape opposite of what has to the �! direction at 0° for a left-stepping array of been commonly described in the literature (Beach, veins. By rules of symmetry, there is a clockwise limit 1975; Bons et al., 2012; Lisle, 2013; Shainin, 1950; for right-stepping echelon veins, with pressure solution Smith, 1996, 1999). With a softer rock (or greater seams, to the σ1∞ direction at 0° for the potential of pressure solution seam displacement), the more straight propagation of a vein. commonly documented sigmoidal shapes may be For veins in a left-stepping geometry oriented ! formed. clockwise from the �! direction, straight vein A finite element method is used in this study since it propagation requires a specific amount of seam is capable of solving nonlinear constitutive equations displacement (Figure 17). Displacement of the pressure that compute inelastic deformation and admit larger solution seam is a function of the limestone stiffness. ! strains; specifically at the crack tips. The stresses at the We find that for cracks at 0° (in line with �! ), E must tips of echelon cracks under the stress and displacement be relatively soft with a value of 1.5 GPa. Cracks that boundary conditions presented in this study (Figure 16) ! are at 10° (� = 35°) to the �! direction, E can be much greatly exceed the fracture toughness of all tested stiffer (19 GPa). This implies that for the formation of limestones. Inelasticity is a material behavior that can veins in echelon arrays, such as those identified at the further explain large vein aperture to vein length ratios eastern Monument Upwarp shown in Figure 1, pressure that we observe (Figure 1 and 3), and vein rotation and solution seam displacements can cause veins to be sigmoidal shapes inferred from images in other studies angled to the remote maximum principal compressive (Beach, 1975; Bons et al., 2012; Lisle, 2013; Shainin, direction. These results explain the common 1950; Smith, 1996, 1999). This is the subject of future ! interpretation of �! bisecting the acute angle between work on this problem. conjugate array sets to cause their synchronous formation using the method that explicitly relates Acknowledgments deformation (displacements and strain) to the causative forces as functions of the material properties. Funding for this research was provided by a Small vein spacing provides clues about limestone National Science Foundation grant (award stiffness. Limestone stiffness can be greater for straight number EAR 1250447), a National Science Foundation propagation of veins with smaller spacing. Smaller vein Collaborations in Mathematical Geosciences grant spacing requires less pressure solution seam (award number EAR 0417521), the Stanford University displacement for straight vein propagation than larger Rock Fracture Project, and a Stanford University, vein spacing (Figure 17). With a vein spacing of 8 mm, School of Earth Sciences McGee Research grant. the limestone stiffness needs to be 3 GPa and admit 0.3 Akhtar Zaman and Brad Nesemeier of the Navajo mm of pressure solution seam displacement. In Division of Natural Resources Minerals department contrast, a 19 GPa limestone stiffness (us = 0.1 mm) provided permission in 2009 and 2013 to collect rock produces straight vein propagation when vein spacing is samples on Navajo Nation land for this research. Shang 5 mm. We observe most vein spacing to be less than the Yip contributed to the preliminary measurements and calculations of pressure solution seam displacements

Stanford Rock Fracture Project Vol. 25, 2014 D-18 from thin-sections. Christopher Zahasky, Nathaniel Elliott, H. A., 1947, An analysis of the conditions for rupture Levine, Meredith Townsend, and Emily Pope assisted due to griffith cracks: Proceedings of the Physical us with fieldwork. The 2011 Stanford Rock Fracture Society, v. 59, no. 2, p. 208. Project members contributed insight to this research Erdogan, F., and Sih, G. C., 1963, On the Crack Extension in Plates Under Plane Loading and Transverse Shear: through discussions while in the field. Discussion with Journal of Basic Engineering, v. 85, no. 4, p. 519-525. Johanna Nevitt about finite element modeling helped to Fischer, G. J., and Paterson, M. S., 1989, Dilatancy during improve this study. rock deformation at high temperatures and pressures: Journal of Geophysical Research: Solid Earth, v. 94, no. References B12, p. 17607-17617. Fleck, N. A., 1991, Brittle Fracture Due to an Array of Allmendinger, R. 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