Analysis of faulting in three-dimensional strain field

Multiple faults, composed of three, four or more sets of faults, have been observed at a wide range of scales, from clay experiments to rift valleys. Multiple faults usually are explained by multiple phases of deformation. However, in several cases the multiple faults develop simultaneously; therefore, they cannot be explained by the common theories of faulting. Furthermore, these theories were derived for plane strain, whereas, multiple faults are associated with three-dimensional strain.An elementary analysis of faulting in three-dimensional strain field is presented here. The analysis considers the deformation of an idealized model due to slip along sets of faults; the model is subjected to strain boundary conditions. The analysis shows that (1) three or four sets of faults are necessary to accommodate three-dimensional strain, (2) there is a combination of four fault sets which minimize the dissipation of the deformation; the orientation of these faults depend on the strain state, and (3) if the resistance to slip along these four sets of faults is cohesive, then the stresses which cause slippage along them are equal or larger than the yielding stresses of a Tresca rigid-plastic with the same cohesion.The analysis presented here is too elementary to be directly applied to field observations; however, it indicates that multiple faults and rhomboid patterns of faults probably form when a body is strained three-dimensionally.     

Earthquake Rupture at Focal Depth,Part II: Mechanics of the 2004 M2.2 Earthquake Along the Pretorius Fault,TauTona Mine,South Africa

We analyze here the rupture mechanics of the 2004, M2.2 earthquake based on our observations and measurements at focal depth (Part I). This event ruptured the Archean Pretorius fault that has been inactive for at least 2 Ga, and was reactivated due to mining operations down to a depth of 3.6 km depth. Thus, it was expected that the Pretorius fault zone will fail similarly to an intact rock body independently of its ancient healed structure. Our analysis reveals a few puzzling features of the M2.2 rupture-zone: (1) the earthquake ruptured four, non-parallel, cataclasite bearing segments of the ancient Pretorius fault-zone; (2) slip occurred almost exclusively along the cataclasite-host rock contacts of the slipping segments; (3) the local in-situ stress field is not favorable to slip along any of these four segments; and (4) the Archean cataclasite is pervasively sintered and cemented to become brittle and strong. To resolve these observations, we conducted rock mechanics experiments on the fault-rocks and host-rocks and found a strong mechanical contrast between the quartzitic cataclasite zones, with elastic-brittle rheology, and the host quartzites, with damage, elastic–plastic rheology. The finite-element modeling of a heterogeneous fault-zone with the measured mechanical contrast indicates that the slip is likely to reactivate the ancient cataclasite-bearing segments, as observed, due to the strong mechanical contrast between the cataclasite and the host quartzitic rock.     

Non-linear elastic behaviour of damaged rocks

The pervasive damage of rocks by microcracks and voids strongly affects their macroscopic elastic properties. To evaluate the damage effects, we derive here the macroscopic stress-strain relations for a 3-D elastic solid with non-interacting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky & O'Connell (1976). The solution describes the stress-strain relations in terms of Λ^d and μ^d, which are the modified Lame constants, and an additional parameter Λ. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ɛ < 0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension.
We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.     

Slip Sequences in Laboratory Experiments Resulting from Inhomogeneous Shear as Analogs of Earthquakes Associated with a Fault Edge

Faults are intrinsically heterogeneous with common occurrences of jogs, edges and steps. We therefore explore experimentally and theoretically how fault edges may affect earthquake and slip dynamics. In the presented experiments and accompanying theoretical model, shear loads are applied to the edge of one of two flat blocks in frictional contact that form a fault analog. We show that slip occurs via a sequence of rapid rupture events that initiate from the loading edge and are arrested after propagating a finite distance. Each successive event extends the slip size, transfers the applied shear across the block, and causes progressively larger changes of the contact area along the contact surface. Resulting from this sequence of events, a hard asperity is dynamically formed near the loaded edge. The contact area beyond this asperity is largely reduced. These sequences of rapid events culminate in slow slip events that precede a major, unarrested slip event along the entire contact surface. We suggest that the 1998 M5.0 Sendai and 1995 off-Etorofu earthquake sequences may correspond to this scenario. Our work demonstrates, qualitatively, how the simplest deviation from uniform shear loading may significantly affect both earthquake nucleation processes and how fault complexity develops.     

Geogas transport in fractured hard rock – Correlations with mining seismicity at 3.54 km depth, TauTona gold mine, South Africa

An on-site gas monitoring study has been conducted in the framework of an earthquake laboratory (The International NELSAM–DAFGAS projects) at the TauTona gold mine, South Africa. Five boreholes up to 60 m long were drilled at 3.54 km depth into the highly fractured Pretorius Fault Zone and instruments for chemical and seismic monitoring installed therein. Over the span of 4 years sensitive gas monitoring devices were continuously improved to enable the direct observation of geogas concentration variations in the DAFGAS borehole. The major gas concentrations are constant and air-like with about 78% N₂, 21% O₂, 1% Ar. The geogas components CO₂, CH₄, He and H₂ show the most interesting trends and variations on the minute-by-minute basis and significantly correlate with seismic data, while the ²²²Rn activity remains constant. Time series and cross correlation analysis allow the identification of different gas components (geogas and tunnel air) and the identification of two processes influencing the borehole gas composition: (1) pumping-induced tunnel air breakthrough through networks of initially water-saturated fault fractures; and (2) seismicity induced permeability enhancement of fault fractures to above ∼5 × 10^-10 m². The current set-up of the gas monitoring system is sensitive enough to quantify the resulting geogas transport during periods of intense blasting activities (including recorded blasts with seismic moment ?1 × 10⁹ Nm, located within 1000 m of the cubby) and, it is suggested, also during induced earthquakes, a final goal of the project.     

Holocene seismic and tectonic activity in the Dead Sea area

The Dead Sea is a large, active graben within the Dead Sea rift, which is bounded by two major strike-slip faults, the Jericho and the Arava faults. We investigated the young tectonic activity along the Jericho fault by excavating trenches, up to 3.5 m deep, across its trace. The trenches penetrate through Late Pleistocene and Holocene sediments. We found that a zone, up to 15 m wide, of disturbed sediments exists along the fault. These disturbed sediments provide evidence for two periods of intensive activity or more likely, for two major earthquakes, that occurred during the last 2000 years. The earthquakes are evident in small faults, vertical throw of a few layers, cracks, unconformities and wide fissures. We further documented evidence for recent sinistral shear along the Jericho fault in deformed sediments and damage to an 8th Century palace on a subsidiary fault. We suggest that the two earthquakes may be correlated with the 31 B.C. earthquake and the 748 A.D. earthquake, reported by the ancients.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: VI. Asymmetric folding and monoclinal kinking

One of the rules of thumb of structural geology is that drag folds, or minor asymmetric folds, reflect the sense of layer-parallel shear during folding of an area. According to this rule, right-lateral, layer-parallel shear is accompanied by clockwise rotation of marker surfaces and left-lateral by counterclockwise rotation. By using this rule of thumb, one is supposed to be able to examine small asymmetric folds in an outcrop and to infer the direction of axes of major folds relative to the position of the outcrop. Such inferences, however, can be misleading. Theoretical and experimental analyses of elastic multilayers show that symmetric sinusoidal folds first develop in the multilayers, if the rheological and dimensional properties favor the development of sinusoidal folds rather than kink folds, and that the folded layers will then behave much as passive markers during layerparallel shear and thus will follow the rule of thumb of drag folding. The analyses indicate, however, that multilayers whose properties favor the development of kink folds can produce monoclinal kink folds with a sense of asymmetry opposite to that predicted by the rule of thumb. Therefore, the asymmetry of folds can be an ambiguous indicator of the sense of shear.The reason for the ambiguity is that asymmetry is a result of two processes that can produce diametrically opposed results. The deformation of foliation surfaces and axial planes in a passive manner is the pure or end-member form of one process. The result of the passive deformation of fold forms is the drag fold in which the steepness of limbs and the tilt of axial planes relative to nonfolded layering are in accord with the rule of thumb.The end-member form of a second process, however, produces the opposite geometric relationships. This process involves yielding and buckling instabilities of layers with contact strength and can result in monoclinal kink bands. Right-lateral, layer-parallel shear stress produces left-lateral monoclinal kink bands and left-lateral shear stress produces right-lateral monoclinal kink bands. Actual folds do not behave as either of these ideal end members, and it is for this reason that the interpretation of the sense of layer-parallel shear stress relative to the asymmetry of folds can be ambiguous.Kink folding of a multilayer with contact strength theoretically is a result of both buckling and yielding instabilities. The theory indicates that inclination of the direction of maximum compression to layering favors either left-lateral or right-lateral kinking, and that one can predict conditions under which monoclinal kink bands will develop in elastic or elastic—plastic layers. Further, the first criterion of kink and sinusoidal folding developed in Part IV remains valid if we replace the contact shear strength with the difference between the shear strength and the initial layer-parallel shear stress.Kink folds theoretically can initiate only in layers inclined at angles less than to the direction of maximum compression. Here φ is the angle of internal friction of contacts. For higher angles of layering, slippage is stable so that the result is layer-parallel slippage rather than kink folding.The theory also provides estimates of locking angles of kink bands relative to the direction of maximum compression. The maximum locking angle between layering in a nondilating kink band and the direction of maximum compression is . The theory indicates that the inclination of the boundaries of kink bands is determined by many factors, including the contact strength between layers, the ratio of principal stresses, the thickening or thinning of layers, that is, the dilitation, within the kink band, and the orientation of the principal stresses relative to layering. If there is no dilitation within the kink band, the minimum inclination of the boundaries of the band is to the direction of maximum compression, or to the direction of nonfolded layers. Here α is the angle between the direction of maximum compression and the nonfolded layers. It is positive if clockwise.Analysis of processes in terminal regions of propagating kink bands in multilayers with frictional contact strength indicates that an essential process is dilitation, which decreases the normal stress, thereby allowing slippage and buckling even though slopes of layers are low there.     

The Role of Adsorbed Water on the Friction of a Layer of Submicron Particles

Anomalously low values of friction observed in layers of submicron particles deformed in simple shear at high slip velocities are explained as the consequence of a one nanometer thick layer of water adsorbed on the particles. The observed transition from normal friction with an apparent coefficient near ???=?0.6 at low slip speeds to a coefficient near ???=?0.3 at higher slip speeds is attributed to competition between the time required to extrude the water layer from between neighboring particles in a force chain and the average lifetime of the chain. At low slip speeds the time required for extrusion is less than the average lifetime of a chain so the particles make contact and lock. As slip speed increases, the average lifetime of a chain decreases until it is less than the extrusion time and the particles in a force chain never come into direct contact. If the adsorbed water layer enables the otherwise rough particles to rotate, the coefficient of friction will drop to ???=?0.3, appropriate for rotating spheres. At the highest slip speeds particle temperatures rise above 100°C, the water layer vaporizes, the particles contact and lock, and the coefficient of friction rises to ???=?0.6. The observed onset of weakening at slip speeds near 0.001?m/s is consistent with the measured viscosity of a 1?nm thick layer of adsorbed water, with a minimum particle radius of approximately 20?nm, and with reasonable assumptions about the distribution of force chains guided by experimental observation. The reduction of friction and the range of velocities over which it occurs decrease with increasing normal stress, as predicted by the model. Moreover, the analysis predicts that this high-speed weakening mechanism should operate only for particles with radii smaller than approximately 1???m. For larger particles the slip speed required for weakening is so large that frictional heating will evaporate the adsorbed water and weakening will not occur.     

Evolution of Wear and Friction Along Experimental Faults

We investigate the evolution of wear and friction along experimental faults composed of solid rock blocks. This evolution is analyzed through shear experiments along five rock types, and the experiments were conducted in a rotary apparatus at slip velocities of 0.002–0.97 m/s, slip distances from a few millimeters to tens of meters, and normal stress of 0.25–6.9 MPa. The wear and friction measurements and fault surface observations revealed three evolution phases: A) An initial stage (slip distances <50 mm) of wear by failure of isolated asperities associated with roughening of the fault surface; B) a running-in stage of slip distances of 1–3 m with intense wear-rate, failure of many asperities, and simultaneous reduction of the friction coefficient and wear-rate; and C) a steady-state stage that initiates when the fault surface is covered by a gouge layer, and during which both wear-rate and friction coefficient maintain quasi-constant, low levels. While these evolution stages are clearly recognizable for experimental faults made from bare rock blocks, our analysis suggests that natural faults “bypass” the first two stages and slip at gouge-controlled steady-state conditions.