2
Physical Characteristics of Fractures and Fracture Patterns

The purpose of this chapter is to provide a geological and geomechanical understanding of fracture formation, characteristics of various fracture types, network patterns, and internal structure. The geometry of fractures, their internal architecture, and present-day state of stress control fluid flow in fractured rocks. A geomechanical understanding of these properties provides an intellectual platform for making sensible inferences and predictions about the nature and location of fractures in the subsurface and underlies the interpretation of data collected through other indirect characterization techniques.

The theme that underlies rock fracture mechanics is the notion of stress heterogeneities over a broad range of scales. This includes stress concentration around material flaws and other physical discontinuities as well as broad variations in the stress field. It is this heterogeneity of stress that controls the initiation and propagation of individual fractures and the localization and clustering of the fracture systems.

Rock properties also play an important role in the formation of fractures and in the structure of fracture zones. There are marked differences in the internal structures of fractures and fracture zones in different lithologic units.

This chapter will describe some of the most prominent modes of fracturing and fracture structures for common rock lithologies. The first section deals with the definition and classification of fractures to provide a common language for a multidisciplinary readership. A following section addresses mechanisms of fracture initiation and propagation based on the concept of stress concentration. Examples of fracture formation are given for a few common rock types. The geometric characteristics of fracture networks and fracture zones and their varia-



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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications 2 Physical Characteristics of Fractures and Fracture Patterns The purpose of this chapter is to provide a geological and geomechanical understanding of fracture formation, characteristics of various fracture types, network patterns, and internal structure. The geometry of fractures, their internal architecture, and present-day state of stress control fluid flow in fractured rocks. A geomechanical understanding of these properties provides an intellectual platform for making sensible inferences and predictions about the nature and location of fractures in the subsurface and underlies the interpretation of data collected through other indirect characterization techniques. The theme that underlies rock fracture mechanics is the notion of stress heterogeneities over a broad range of scales. This includes stress concentration around material flaws and other physical discontinuities as well as broad variations in the stress field. It is this heterogeneity of stress that controls the initiation and propagation of individual fractures and the localization and clustering of the fracture systems. Rock properties also play an important role in the formation of fractures and in the structure of fracture zones. There are marked differences in the internal structures of fractures and fracture zones in different lithologic units. This chapter will describe some of the most prominent modes of fracturing and fracture structures for common rock lithologies. The first section deals with the definition and classification of fractures to provide a common language for a multidisciplinary readership. A following section addresses mechanisms of fracture initiation and propagation based on the concept of stress concentration. Examples of fracture formation are given for a few common rock types. The geometric characteristics of fracture networks and fracture zones and their varia-

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications tions in terms of geological regimes and rock lithology are briefly discussed. These fracture network patterns are compared with those commonly used in reservoir simulations. DEFINITION AND CLASSIFICATION Fractures are mechanical breaks in rocks involving discontinuities in displacement across surfaces or narrow zones. Fracture is a term used for all types of generic discontinuities. This usage is common among scientists inside and outside the earth sciences and is used in other chapters of this report. However, different kinds of fractures exist, with different geometries, mechanical effects, and flow properties. Based on the nature of the displacement discontinuity, commonly encountered fractures can be classified into three geologically based major groups: (1) dilating fractures/joints, (2) shearing fractures/faults, and (3) closing fractures/pressure solution surfaces. (Pressure solution surfaces are fractures in sedimentary rock that are welded together by solution that occurs at the contact surfaces of grains [Bates and Jackson, 1980].) This chapter is concerned with the first two of these groups, joints and faults, illustrated schematically in Figure 2.1. Dilating fractures, which are also referred to as joints, can be idealized as two rough surfaces with normal displacement discontinuity; that is, the surfaces have moved away from each other in a direction perpendicular to the surfaces (Figure 2.1b). (They are also called mode I fractures in engineering fracture mechanics [Lawn and Wilshaw, 1975].) Shear fractures, which are also referred to as faults, are shear displacement discontinuities where the fracture surfaces move predominantly parallel to each other. This relative movement is either perpendicular (mode II) or parallel (mode III) to the fracture front (Figure 2.1b). Pressure solution surfaces, also referred to as stylolites, are known as anticracks in which the sense of the displacement discontinuity is opposite that of dilating fractures or mode I fractures (Fletcher and Pollard, 1981). For a review of pressure solution surfaces and their hydraulic properties, the reader is referred to Nelson (1985). Fractures with a combination of these modes (mixed-mode fractures) also are possible. Rock masses with complex deformational histories have fractures produced by two or more of these modes in a sequential manner (Figure 2.2 a–d). This yields fractures with overprinted displacement discontinuities (Barton, 1983; Dyer, 1983; Segall and Pollard, 1983a,b; Zhao and Johnson, 1992). All combinations of displacement discontinuities may occur in nature, but the most common ones are faulted joints and jointed faults. Although these two types of fractures are kinematically similar to faults and joints, which were defined previously, their total displacement discontinuities, geometries, and internal structures may be very different because of the overprinting of two different modes of deformation.

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.1 (a) Block diagram showing a fracture and its propagation front. (b) Three fundamental modes of fractures corresponding to joints (mode I) and faults (mode II or mode III). From Pollard and Aydin (1988). Joints, faults, and pressure solution surfaces filled by minerals are known as veins, seams, and filled pull-aparts. The mineral fillings have important consequences for fluid flow because they may alter the flow properties of the fractured rock. The mineral fillings may have different permeabilities than the host rock, and vein bridges may keep fractures open. The mineral fillings also provide information about the nature of the fluids flowing in the fractures, the original apertures of the fractures, and the physical and chemical conditions during precipitation. Joints and faults can be identified by distinctive surface features; joint surfaces are ornamented by the so-called plumose texture (Figure 2.3a), whereas through-

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.2 Development of faulted joints. Parts a and c are original joint patterns, and b and d are faulted joints with dilantant fractures in the overlap regions (b) and in between faulted joints (d). From Pollard and Aydin (1988). going fault surfaces are polished and are marked by linear features known as grooves and striations or slickensides (Figure 2.3b). Plumose patterns record fracture propagation directions and may be radial or axisymmetric or may have a more complex geometry (Kulander et al., 1979; Pollard and Aydin, 1988; Bahat, 1988). Grooves and striations on fault surfaces record slip directions (e.g., Patterson, 1958; Suppe, 1985). Subsequent motions different from the original movements along faults may cause mismatches between opposite faces of the fracture. This mismatch may produce open channels that are potential pathways for fluid flow. Joints and faults are fundamentally different in terms of their associated stress fields (Pollard and Segall, 1987). These differences provide a basis for

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.3 Left, a joint surface ornament known as plumose structure. From Pollard and Aydin (1988). Right, a normal fault surface with striations. From Aydin and Johnson (1978). understanding the initiation, propagation, interaction, and termination of joints and faults and the nature and distributions of the associated structures around them. GENESIS OF FRACTURES In general, fractures initiate and propagate when the stresses become equal to the strength of rock (a more precise propagation criterion is discussed later in this chapter, under ''Fracture Propagation and Internal Structures"). Several possible sources or mechanisms that are capable of producing high stresses in the earth's crust can be identified. Among these are (1) lithostatic (changes in the weight of overburden either by burial or removal caused by uplift and erosion); (2) fluid pressure; (3) tectonic forces associated with the movement of lithospheric plates; (4) thermal (cooling of intrusive and extrusive rocks, and cooling caused by uplift and erosion of the crust); (5) impact by extraterrestrial objects; and (6) other geological processes such as folding, volcanic activity, and salt intrusion. The relationship between fracture formation and a particular causal mechanism is obvious in some cases. For example, fractures that form a polygonal geometry (polygonal fractures) in volcanic rocks can be linked directly to thermal stresses produced during cooling of the rocks. In this case the patterns and systematic development of the fracture system can be understood in terms of the thermal history of the magma and rock fracture mechanics (Ryan and Sammis, 1978; DeGraff and Aydin, 1993). Igneous rocks that crystallize at depth may also fracture during cooling. For example, Segall et al. (1990) dated mineralized fracture fillings in a granitic body from the Sierra Nevada in California. This analysis showed that the fractures formed as the granite was emplaced or shortly thereafter, suggesting that they formed as the granite cooled. The Sierran fractures are nearly parallel to one another, quite unlike the polygonal fracture patterns that develop as a lava flow

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications cools. This difference in fracture geometry may reflect the fact that the granites cooled at depth when they were subjected to other lithostatic or tectonic forces, whereas the lavas cooled under approximately homogeneous stresses at the surface. Mechanical analyses can help relate fracture patterns to the causative geological processes when direct information on the timing of fracturing is absent. For example, Delaney et al. (1986) investigated dikes (tabular igneous intrusions) and adjacent clusters of dike-parallel joints. By representing a dike as a pressurized crack and modeling the stress field around it, these investigators showed that dike-parallel joints could open ahead of a dike. Such joints would be juxtaposed along the dike if the dike front advanced. The task of relating a fracture system to a specific process is more difficult, however, for rocks and regions that have undergone multiple deformational events (Spencer, 1959; Wise, 1964). To establish a relationship between a particular fracture system and the mechanism responsible for its formation, it is necessary to establish the temporal and spatial relationship between the observed fracture system and the proposed process. When geological ages are not available or are uncertain, additional supporting arguments can be used. These are usually in the form of plausibility arguments based on geological and mechanical concepts or models. The sketch in Figure 2.4 illustrates a system of joints in which the mechanism of fracture formation can be discerned through such an analysis. The joints are perpendicular to the layer and are confined primarily in the convex part of the layer. This pattern is consistent with the distribution and orientation of the axial stresses for a buckled plate. Thus, a strong case can be made that buckling or bending is the origin of these joints. In a buckled layer, dilating fractures occur initially above and perpendicular to the neutral plane, which is consistent with the geometry and distribution of the joints in the layer (Figure 2.4). In laboratory compression tests, conjugate shear fractures and opening fractures both formed, with the opening fractures forming perpendicular to the least compressive stress and bisecting the acute angle between the shear fractures (e.g., Peng and Johnson, FIGURE 2.4 Sketch of a distorted layer and associated fractures. From Van Hise (1896).

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications 1972). Stearns and Friedman (1972) used these relationships to infer the stresses responsible for natural fracture patterns in folded rock layers. In some cases, regional distributions of joint and fault patterns mimic the trends of mountain belts, suggesting a causative relationship between the formation of fracture systems and a particular tectonic event responsible for the mountain belt. Nickelsen and Hough (1967) and Engelder and Geiser (1980) have made a case for such a correspondence between the Appalachian Mountains belt and the joints that are parallel and perpendicular to the strike of the belt. Additional examples for the genesis of fractures will be given after introduction of the basic concepts of rock fracture. For the time being it is sufficient to note that some regional or local fracture systems can be linked to one of the mechanisms listed above but that others cannot. There is a need to establish relationships between the common stress production mechanisms in the earth's crust and the resulting fracture patterns. This can be done most efficiently by case studies at the surface of the earth that take advantage of the availability of abundant direct information. Once the nature of the relationship between a process and a fracture system is established, it can be used for subsurface predictions and inferences. FLAWS, STRESS CONCENTRATION, AND FRACTURE INITIATION The concepts of stress concentration, amplification, and energy balance, which go back to Inglis (1913) and Griffith (1921), are essential to understanding rock fracture initiation as well as fracture propagation and distribution. Griffith showed experimentally that glass samples fractured at an applied stress level much lower than their theoretical strengths. He attributed this to the amplification of stresses around flaws in the glass, which is known as stress concentration. Most flaws in natural materials are slitlike. However, the stress distribution around a circular hole in a plate subjected to uniaxial tension (Figure 2.5) demonstrates the concept of stress concentration quite well. Because of symmetry, it is sufficient to consider the stresses on one quadrant of the hole (e.g., segments AB in Figure 2.5a). The part of the curve from A to B in Figure 2.5b shows the variation of normalized tangential stress from A to B along the perimeter of the circular hole. At a point on the circle diameter perpendicular to the direction of the applied remote stress (point A in Figure 2.5a), the tensile stress has increased threefold. The stress decreases gradually along the circular hole from point A to point B, changes its sign near point B, and finally becomes equal in magnitude but opposite in sign to the applied stress at point B. The stress concentration falls off rapidly away from the hole, as shown by curve segment BC in Figure 2.5b. Tensile stresses can be induced even under compressive remote loading systems (Pollard and Aydin, 1988; Einstein and Dershowitz, 1990). Figure 2.6b illustrates the tangential stresses on points from A to B at the perimeter of a circular hole in a plate subjected to uniaxial compression, as shown in Figure 2.6a. A tensile stress equal to the remote (measured at some distance from

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.5 (a) Circular hole of radius a in a plate subjected to uniaxial tension (0). (b) Stress distribution (tangential stress normalized by the remote stress around the hole). Because of the symmetry, only a quadrant of the hole (from A to B) is considered. Courtesy of Y. Du. the hole) uniaxial compressive stress (0) in magnitude is induced at point B. Furthermore, tensile stresses can be produced even under hydrostatically compressive loading systems (Figure 2.7a, b) if the internal fluid pressure, P, is greater than twice the applied hydrostatic compression (P > 20).

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.6 (a) A circular hole of radius a in a plate subjected to uniaxial compression (0). (b) Stress distribution around the hole. The amplification of stresses around holes is related to the curvature of the aspect ratio (a/b) of the opening. For an elliptical hole in a plate subjected to a uniaxial stress, the amplification of the stress at the tips is (2a/b) + 1, where a and b are the longest and shortest axes, respectively. Consequently, the magnitude of the stress at one of the tips of a long and thin discontinuity (Figure 2.8) can

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.7 (a) A circular hole of a radius of a in a plate subjected to hydrostatic compression (0) and an internal fluid pressure (P). (b) Stress distribution around the hole. Note that the stresses at points A and B are equal and will be negative if P is greater than twice the hydrostatic stress.

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications FIGURE 2.8 Regions of stress around a fracture tip. Two regions are defined: one region immediately surrounding the tip (r1 < < l) and one region extending to a distance roughly equal to the half-length (r2 ~ l). be many times that of the applied remote stress. For this case the expression for stress in a region immediately surrounding the tip based on the assumption of linear elasticity is given by (Lawn and Wilshaw, 1975): fracture half-length, l, and m = I, II, and III. Here, Km is the stress intensity factor, that is, a measure of the intensity of the stress concentration, which depends on the applied load and fracture geometry, and is known as KI, KII, and KIII, for modes I, II, and III, respectively. The symbols r and l are defined in Figure 2.8. For the case of a uniformly loaded fracture of half-length l, the stress intensity factor is The term m is the stress driving the relative displacement of the fracture walls for a particular mode of fracture (Lawn and Wilshaw, 1975). This equation, together with the previous one, shows that stresses increase near a fracture tip as the fracture lengthens. The remaining terms depend on the geometric factors about the fracture tip; r is the radial and is the angular component (Figure 2.8). Some of the characteristic properties of the fracture tip stress field can be seen from this expression, for example, the dependence on r-1/2, and the singularity as r goes to zero. For larger r this relationship breaks down. For example, for r > l, the distance dependence of the stress is r-2 on average. Recall that

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications APPENDIX 2.CROLE OF PORE FLUIDS IN THE SAN ANDREAS FAULT Fundamental questions exist about the mechanics of deformation and slip in the San Andreas fault (SAF), such as: Why is the fault segmented at all scales into creeping and locked sections? What factors control the localization of deformation and the nucleation and recurrence of earthquakes? Why is the SAF weak both absolutely and in a relative sense compared with the adjacent plate interiors, as evidenced by heat flow measurements (Brune et al., 1969; Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1973, 1980, 1992) and by apparent principal stress orientations in the adjacent plates (Mount and Suppe, 1987; Zoback et al., 1987; Oppenheimer et al., 1988; Wong, 1990)? The effective large-scale permeability of the fault zone and its adjacent country rock are key issues in the debate on the seismic behavior of the SAF. Discovering the causes of the weakness of the SAF zone will perhaps be key to understanding the fundamental behavior of the system. Existing data (e.g., Healy and Zoback, 1988; Zoback and Healy, 1992) indicate that the frictional strength of the crust is generally consistent with the Mohr-Coulomb faulting theory, with friction coefficients in the range of 0.6 to 0.9 (from Byerlee's law). In contrast, the apparent low strength of the SAF would require a coefficient of friction of 0.1, yet the friction coefficient of even the weakest material envisioned to exist in the fault zone, montmorillonite gouge, is found to be no less than about 0.4 (Wang and Mao, 1979; Morrow et al., 1982, 1992). Furthermore, temperatures at seismogenic depths may too high to preserve this mineral. One favored alternative cause of the apparent low strength of the SAF is a very low effective stress in the fault zone caused by high fluid pore pressure. Byerlee (1990) and Rice (1992) proposed models in which permanently high fluid pore pressure in the SAF zone can coexist with near-hydrostatic pressure in the adjacent country rock without causing hydrofracturing of the fault wall rocks. The models are identical mechanically but rely on different mechanisms to maintain the high pressure differential between the fault zone and the country rock. In Rice's model flow is allowed from the fault zone to the country rock, but the in-plane permeability of the fault zone is much higher than fault-perpendicular permeability, and the high pore pressure in the fault zone is maintained by fluid recharge from the lower crust at the root of the fault. In Byerlee's model the fault zone is sealed (i.e., there is zero effective cross-fault permeability). Fault zone fluids originally derived from the country rock are maintained at high pore pressure by compaction of fault zone materials during shearing. Fournier (1990), Blanpied et al. (1992), Sibson (1992), Sleep and Blanpied (1992), Byerlee (1993), and Chester et al. (1993) have proposed mechanisms whereby intermittent high pore pressures are developed during the earthquake cycle by the formation of impermeable barriers through pressure-induced sealing and healing processes. Dynamic weakening mechanisms that act only during

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications rapid fault slip also have been proposed. These include transient high fluid pressure caused by shear heating during slip (Lachenbruch, 1980; Mase and Smith, 1987), reduction of normal stress accompanying the propagation of dilational waves along the fault (Brune, 1993), and acoustic fluidization of fault zone materials (Melosh, 1979). The latter two mechanisms do not require high fluid pressures. The critical factor in discriminating between Rice's fault-weakening model, Byerlee's (1990) model, and cyclic high pore pressure models is the origin of fault zone fluids and the hydrogeological connectivity between the fault zone and the country rock. Rice's model requires a deep crustal or upper-mantle source, whereas in the other models fluids are derived from the country rock. REFERENCES Antonellini, M. 1992. Geometry and distribution of deformation bands in porous sandstones, Delicate Arch area, Arches National Park, Utah. Pp. A1-A7 in the Rock Fracture Project, D. D. Pollard and A. Aydin, eds. Vol. 3. Stanford University, Stanford, Calif. Antonellini, M., and A. Aydin. 1994. Effect of faulting on fluid flow in porous sandstones. Petrophysical Properties, American Association of Petroleum Geologists Bulletin , 78:355–377. Antonellini, M. A., A. Aydin, and D. D. Pollard. 1994. Microstructure of deformation bands in porous sandstones at Arches National Park, Utah. Journal of Structural Geology, 16:941–959. Aydin, A. 1978. Small faults formed as deformation bands in sandstone. Pure and Applied Geophysics, 116:913–930. Aydin, A., and J. M. DeGraff. 1988. Evolution of polygonal fracture patterns in lava flows. Science, 239:471–476. Aydin, A., and A. M. Johnson. 1978. Development of faults as zones of deformation bands and as slip surfaces in sandstone. Pure and Applied Geophysics, 116:931–942. Aydin, A., and A. Nur. 1982. Evolution of pull-apart basins and their scale independence. Tectonics, 1:91–105. Aydin, A., and B. Page. 1984. Diverse Pliocene-Quaternary tectonics in a transform environment, San Francisco Bay Region, California. Geological Society of America Bulletin, 95(11):1303–1317. Aydin, A., and Z. Reches. 1982. Number and orientation of fault sets in the field and in experiments. Geology, 10:107–112. Aydin, A., and R. A. Schultz. 1990. Effect of mechanical interaction on the development of strike-slip faults with echelon patterns. Journal of Structural Geology, 12:123–129. Aydin, A., R. A. Schultz, and D. Campagna. 1990. Fault-normal dilation in pull-apart basins: implications for the relationship between strike-slip faults and volcanic activity. Annales Tectonicae, IV:45–52. Bahat, D. 1988. Fractographic determination of joint length distribution in chalk. Rock Mechanics and Rock Engineering, 21:79–94. Bahat, D., and T. Engelder. 1984. Surface morphology on cross-fold joints of the Appalachian Plateau, New York and Pennsylvania. Tectonophysics, 104:299–313. Barker, J. 1988. A generalized radial flow model for pumping tests in fractured rock. Water Resources Research, 24:1796–1804. Barton, C. C. 1983. Systematic jointing in the Cardium Sandstone along the Bow River, Alberta, Canada. Ph.D. thesis, Yale University, 301 pp. Barton, C. C., and P. A. Hsieh. 1989. Physical and Hydrologic-Flow Properties of Fractures. 28th International Geological Congress Field Trip Guidebook T385. P. 36. Washington, D.C.: American Geophysical Union.

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Rock Fractures and Fluid Flow: Contemporary Understanding and Applications Bates, R. L., and J. A. Jackson, eds. 1980. Glossary of Geology, 2d ed. Falls Church, Va.: American Geological Institute. Bilham, R., and G. King. 1989. The morphology of strike-slip faults: examples from the San Andreas fault, California. Journal of Geophysical Research, 94:10,204–10,216. Birkeland, P. W., and E. E. Larson. 1989. Putman's Geology, 5th ed. New York: Oxford University Press. Black, J., O. Olsson, J. Gale, and D. Holmes. 1990. Site Characterization and Validation, Stage 4, Preliminary Assessment and Detail Predictions. Stripa Project Technical Report 91-08, Swedish Nuclear Fuel and Waste Management Company, Stockholm, p. 248. Blanpied, M. L., D. A. Lockner, and J. D. Byerlee. 1992. An earthquake mechanism based on rapid sealing of faults. Nature, 358:574–576. Borg, I. Y., and J. C. Maxwell. 1956. Interpretation of fabrics of experimentally deformed sands. American Journal of Science, 254:71–81. Boyer, S. E., and D. Elliott. 1982. Thrust systems. American Association of Petroleum Geologists Bulletin, 66:1196–1230. Brown, N. N., and R. H. Sibson. 1989. Structural geology of the Octillo Badlands antidilational fault jog, southern California. Pp. 94–109 in Fault Segmentation and Controls of Rupture Initiation and Termination. USGS Open-File Report 89-315, U.S. Geological Survey, Menlo Park, Calif. Brown, A., N. M. Soonawala, R. A. Everitt, and D. C. Kamineni. 1989. Geology and geophysics of the Underground Research Laboratory site, Lac du Bonnet Batholith, Manitoba. Canadian Journal of Earth Sciences, 26:404–425. Brune, J. N. 1993. Rupture mechanism and interface separation in foam rubber models of earthquakes: A possible solution to the heat flow paradox and the paradox of large overthrusts. Tectonophysics, 218(1–3):59–67. Brune, J. N., T. L. Henyey, and R. F. Roy. 1969. Heat flow, stress, and the rate of slip along the San Andreas fault, California. Journal of Geophysical Research, 74:3821–3827. Byerlee, J. 1990. Friction, overpressure and fault normal compression. Geophysics Research Letter, 17:2109-2112. Byerlee, J. 1993. Model for episodic flow of high pressure water in fault zones before earthquakes. Geology, 21:303–306. Caine, J. S., C. B. Forster, and J. P. Evans. 1993. A classification scheme for permeability structures in fault zones. Transactions of the American Geophysical Union, 74:677. Chester, F. M., J. P. Evans, and R. L. Beigel. 1993. Internal structure and weakening mechanisms of the San Andreas fault. Journal of Geophysical Research, 98:771–786. Choukrone, P., and D. Gapais. 1983. Strain pattern in the Aar granite (central Alps): orthogenesis developed by bulk flattening. Journal of Structural Geology, 5:411–418. Cloos, E. 1955. Experimental analysis of fracture patterns. Geological Society of America Bulletin, 66:241–256. Cox, B. L., and J. S. Y. Wang. 1993. Fractal surfaces: measurement and application in the earth sciences. Fractals, 1:87–115. Cowie, P. A., and C. H. Scholz. 1992. Displacement-length relationship for faults: data analysis and discussion. Journal of Structural Geology, 14:1149–1156. Cruikshank, K. M., and A. Aydin. 1994. Role of fracture localization in arch formation, Arches National Park, Utah. Geological Society of America Bulletin, 106:879–891. Cruikshank, K. M., and A. Aydin. 1995. Unweaving the joints in Entrada Sandstone, southwest limb of the Salt Valley anticline, Arches National Park, Utah. Journal of Structural Geology, 17:409–421. Cruikshank, K. M., G. Zhao, and A. M. Johnson. 1991. Duplex structures connecting fault segments in Entrada sandstone. Journal of Structural Geology, 13:1185–1196. DeGraff, J. M. 1987. Mechanics of columnar joint formation in igneous rock. Ph.D. thesis, Purdue University, West Lafayette, Ind., p. 221.

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