Stephen J. Martel

Slip distributions on faults: effects of stress gradients, inelastic deformation, heterogeneous host-rock stiffness, and fault interaction

Abstract Fault slip distributions are commonly assumed to be symmetrical about a central slip maximum, however, slip distributions in nature are often asymmetric. Although slip along an idealized fault is expected to follow an elliptical distribution after a single slip event in an elastic material, the slip distribution may be modified if the fault propagates or if additional slip events occur. Analytically and numerically computed fault-slip distributions in an elastic medium indicate that: (1) changes in the (frictional) strength along a fault; (2) spatial gradients in the stress field; (3) inelastic deformation near fault terminations; and (4) variations of the elastic modulus of the host rock can cause strong deviations from idealized symmetrical distributions along single-slip event faults. A relatively stiff body adjacent to or cut by a fault will tend to reduce fault slip in its vicinity and tends to flatten the slip profile where it is cut by the fault. Sharp slip gradients develop near the interface between relatively soft and stiff materials. The interaction of faults within about one fault radius of one another can strongly influence slip gradients. Inelastic processes, caused by stress perturbations in the stepover region of echelon faults, may link individual segments and thereby create a slip distribution resembling that of a single fault.

Formation of compound strike-slip fault zones, Mount Abbot quadrangle, California

Numerous strike-slip fault zones in granitic rocks of the Mount Abbot quadrangle, California, developed from steeply-dipping, subparallel joints. These joints generally were less than 50 m long and were spaced several centimeters to several meters apart. Some joints subsequently slipped and became small faults. Simple fault zones formed as oblique dilatant fractures (splay fractures) linked non-coplanar faults side-to-side and end-to-end. These simple fault zones are as much as 1 km long and accommodated displacements as great as 10 m. Compound fault zones formed as splay fractures linked small faults and simple fault zones. They are as much as several kilometers long and accommodated displacements as great as 100 m. These zones are distinctly different from ‘Riedel shear zones’ and the-mechanics of their formation are unlikely to be described well by Mohr—Coulomb mechanisms. Simple and compound fault zones are composed of non-coplanar segments that join at steps or bends; splay fracture length determines step widths. The longest splay fractures occur along the longest fault zones, allowing step widths to increase as the length and displacement across the zones increase. These findings are consistent with the structure of some active seismogenic faults, and they provide a mechanically consistent, field-based conceptual model for fault zones that grow in basement rocks from a preexisting set of joints.

Effects of volcano loading on dike propagation in an elastic half-space

We use laboratory experiments and numerical models to examine the effects of volcano loading on the propagation of buoyant dikes in a two-dimensional elastic half-space. In laboratory experiments we simulate the propagation of buoyant dikes in an isotropic regional stress field by injecting air into tanks of solidified gelatin. A weight resting on the surface of the gelatin represents a volcanic load. A numerical model is used to simulate these experiments. Both experiments and numerical simulations show that as a dike ascends, it begins to curve toward the load in response to the local stress field imposed by the load. The lateral distance over which dikes curve to the load increases with the ratio of average pressure at the base of the load to the dike driving pressure. For realistic volcano and dike dimensions this pressure ratio is going to be large, suggesting that dikes can converge to a volcano over lateral distances several times the load width. Numerical calculations involving an anisotropic regional stress field, however, predict that the lateral extent of dike attraction shrinks as the regional horizontal compressive stress decreases relative to the vertical compressive stress. Dike focusing will be substantial if the regional differential stresses are less than the average pressure at the base of the load. If this is the case, then our models predict a positive feedback between the size of volcanoes and the area of dike attraction. This feedback may promote the development of large discrete volcanoes and also predicts a positive correlation between the spacing and sizes of adjacent volcanoes. To test this prediction, we examine nearest-neighbor pairs of the 21 largest volcanoes in the Cascade Range. The 14 pairs examined show a large range in volcano spacing (6–115 km) and a statistically significant correlation between spacing and average volcano height. This result is consistent with our model results and suggests that the local compressive stress induced by these volcanoes may be an important factor in controlling magma transport in the lithosphere.

Mechanics of landslide initiation as a shear fracture phenomenon

Abstract A 3-D model of shear fracture in an elastic half-space provides insight into the initiation of sliding along weak pre-existing surfaces in rock or consolidated sediments. An elastic model is justified physically if regions of non-elastic deformation associated with sliding are small relative to the size of the shear fracture. A subsurface elliptical shear fracture parallel to the surface simulates sliding at depth along a pre-existing weakness (e.g. a bedding plane). Based on the stress concentration at the shear fracture perimeter, the model predicts landslide scars will tend to have elliptical shapes in map view and width-to-length values of 0.5–1, consistent with many observations. As a shear fracture spreads, the stress concentration at its perimeter promotes its propagation up towards the surface. The model predicts that sliding at depth causes and precedes fracturing at the surface. For a shear fracture less then twice as long as it is wide, surficial fracturing should start in the head and from there ‘unzip’ down along the slide flanks. Depending on the ambient stress state and the shear strength loss at the slide base, a shear fracture might need to become several or more times wider and longer than its depth to develop a sufficiently intense stress concentration to propagate out of plane to the surface. This accounts for the large length-to-thickness ratios of many natural slides. The model also accounts for the following generic landslide characteristics: a steep, arcuate, concave-downhill head scarp; an echelon pattern of opening-mode fractures along the flanks and subparallel to the head scarp trace; subsidence and normal faulting near the head of a slide; and uplift with thrust faulting near the slide toe.

Geophysical imaging reveals topographic stress control of bedrock weathering.

Bedrock weathering runs to the hills Fractures in bedrock drive the breakdown of rock into soil. Soil makes observations of bedrock processes challenging. St. Clair et al. combined a three-dimensional stress model with geophysical measurements to show that bedrock erosion rates mirror changes in topography (see the Perspective by Anderson). Seismic reflection and electromagnetic profiles allowed mapping of the bedrock fracture density. The profiles mirror changes in surface elevation and thus provide a way to study the critical zone between rock and soil. Science, this issue p. 534; see also p. 506 Geophysical survey data and stress modeling connect surface topography to Earth’s critical zone. [Also see Perspective by Anderson] Bedrock fracture systems facilitate weathering, allowing fresh mineral surfaces to interact with corrosive waters and biota from Earth’s surface, while simultaneously promoting drainage of chemically equilibrated fluids. We show that topographic perturbations to regional stress fields explain bedrock fracture distributions, as revealed by seismic velocity and electrical resistivity surveys from three landscapes. The base of the fracture-rich zone mirrors surface topography where the ratio of horizontal compressive tectonic stresses to near-surface gravitational stresses is relatively large, and it parallels the surface topography where the ratio is relatively small. Three-dimensional stress calculations predict these results, suggesting that tectonic stresses interact with topography to influence bedrock disaggregation, groundwater flow, chemical weathering, and the depth of the “critical zone” in which many biogeochemical processes occur.

Geometry and mechanics of secondary fracturing around small three‐dimensional faults in granitic rock

The orientations, locations, sizes, and relative abundances of secondary fractures observed along small natural faults can be accounted for by a three-dimensional elastic model. Secondary fractures along small subvertical left-lateral strike-slip faults in massive granitic rock of the Sierra Nevada of California (1) consistently strike 25°±10° counterclockwise from their host faults and dip at angles greater than 80°; (2) generally are absent along the central portions of the fault traces; (3) are numerous near the ends of some fault traces but absent along others; and (4) in rare cases form echelon arrays either centered along a fault trace or just past the fault trace ends. These observations are consistent with secondary fractures that nucleated near the perimeter of an elliptical fault along a “cohesive rim” of high slip resistance and propagated in three dimensions normal to the local most tensile stress. The fracture orientations relative to the faults reflect small stress drops during slip on the faults. The observations and model together have direct implications for how faults grow and conduct fluids. Secondary fractures are likely to be larger at the ends of small strike-slip faults rather than at their tops and bottoms. As a result, if strike-slip faults grow in an unrestricted manner, they are more likely to be linked end-to-end rather than top-to-bottom, especially where slip is small. Hydraulic conductivity is likely to be enhanced at the linkages between faults, so highly conductive regions along linked strike-slip faults are more likely to be vertical rather than horizontal.

Effects of cohesive zones on small faults and implications for secondary fracturing and fault trace geometry

Abstract Fracture mechanics theory and field observations together indicate that the shear stress on many faults is non-uniform when they slip. If the shear stress were uniform, then: (a) a physically implausible singular stress concentration theoretically would develop at a fault end; and (b) a single curved ‘tail fracture’ should open up at the end of every fault trace, intersecting the fault at approximately 70 °. Tail fractures along many small faults instead range in number, commonly form behind fault trace ends, have nearly straight traces and intersect a fault at angles less than 50 °. A ‘cohesive zone’, in which the shear stress is elevated near the fault end, can eliminate the stress singularity and can account for the observed orientation, shape, and distribution of tail fractures. Cohesive zones also should cause a fault to bend. If the cohesive zone shear stress were uniform, then the distance from the fault end to the bend gives the cohesive zone length. The nearly straight traces of the tail fractures and the small bends observed near some fault ends implies that the faults slipped with low stress drops, less than 10% of the ambient fault-parallel shear stress.

FORMATION OF JOINTS IN COOLING PLUTONS

Abstract The geometry, age, and mineralogy of steeply dipping joints in the Lake Edison Granodiorite of California indicate that thermal stresses played a key role in the formation of the joints. Joint traces curve to approach the pluton boundary at high angles at both large and small scales, and they generally terminate at or near the contact with an older pluton. Radiometric dates and the epidote and chlorite fillings of the joints tie jointing to the initial cooling of the pluton. A thermo-mechanical stress analysis assuming two-dimensional conductive cooling predicts thermal stresses of several tens of MPa, the same order of magnitude as plausible fluid pressures, regional stresses, and lateral normal stresses associated with the overburden. The orientation of the joints can be accounted for rather well by the stress field formed by superposing a uniform regional stress field on the predicted thermal stresses. The large-scale pattern of the early joints in many plutons should be predictable based on a plutons geometry, its age relative to the adjacent rock, and knowledge of the regional stress at the time of initial cooling. These findings bear on issues pertinent to mining, petroleum recovery, nuclear waste repository siting, and ground water flow.

Late Cretaceous age of fractures in the Sierra Nevada batholith, California

Regional sets of steeply dipping joints and faults are common throughout the Sierra Nevada batholith, yet relatively little is known about how or when they formed. Within some east-northeast-striking, left-lateral fault zones in the Mount Abbot quadrangle of the central Sierra Nevada, the host granodiorite is hydrothermally altered to a lower greenschist assemblage that contains muscovite. The muscovite yields a mean K-Ar and {sup 40}Ar/{sup 39}Ar age of 79 Ma, which provides a minimum age for the faulting. Field relations show that these faults developed from earlier formed, mineralized joints, so these ages also provide a minimum age for the jointing. Published ages of biotite, hornblende, and zircon from the host granodiorite of Lake Edison are 80 Ma (K-Ar), 85 Ma (K-Ar), and 90 Ma (U-Pb), respectively. The geochronology, field relations, and hydrothermal mineral assemblages together suggest that the mineralized joints and faults all formed between 85 and 79 Ma, soon after the host pluton was emplaced.

An inverse technique for developing models for fluid flow in fracture systems using simulated annealing

One of the characteristics of flow and transport in fractured rock is that the flow may be largely confined to a poorly connected network of fractures. In order to represent this condition, Lawrence Berkeley Laboratory has been developing a new type of fracture hydrology model called an “equivalent discontinuum” model. In this model we represent the discontinuous nature of the problem through flow on a partially filled lattice. This is done through a statistical inverse technique called “simulated annealing.” The fracture network model is “annealed” by continually modifying a base model, or “template,” so that with each modification, the model behaves more and more like the observed system. This template is constructed using geological and geophysical data to identify the regions that possibly conduct fluid and the probable orientations of channels that conduct fluid. In order to see how the simulated annealing algorithm works, we have developed a synthetic case. In this case, the geometry of the fracture network is completely known, so that the results of annealing to steady state data can be evaluated absolutely. We also analyze field data from the Migration Experiment at the Grimsel Rock Laboratory in Switzerland.