David D. Pollard

Formation and interpretation of dilatant echelon cracks

The relative displacements of the walls of many veins, joints, and dikes demonstrate that these structures are dilatant cracks. We infer that dilatant cracks propagate in a principal stress plane, normal to the maximum tensile or least compressive stress. Arrays of echelon crack segments appear to emerge from the peripheries of some dilatant cracks. Breakdown of a parent crack into an echelon array may be initiated by a spatial or temporal rotation of the remote principal stresses about an axis parallel to the crack propagation direction. Near the parent-crack tip, a rotation of the local principal stresses is induced in the same sense, but not necessarily through the same angle. Incipient echelon cracks form at the parent-crack tip normal to the local maximum tensile stress. Further longitudinal growth along surfaces that twist about axes parallel to the propagation direction realigns each echelon crack into a remote principal stress plane. The walls of these twisted cracks may be idealized as helicoidal surfaces. An array of helicoidal cracks sweeps out less surface area than one parent crack twisting through the same angle. Thus, many echelon cracks grow from a single parent because the work done in creating the array, as measured by its surface area, decreases as the number of cracks increases. In cross sections perpendicular to the propagation direction, echelon cracks grow laterally, each crack overlapping its neighbors, until the mechanical interaction of adjacent cracks limits this growth. Dilation of each crack pinches the tips of adjacent cracks into an asymmetrical form and introduces local stresses that can cause lateral growth along a curving, sigmoidal path. Sigmoidal echelon cracks may link at tip-to-plane intersections, leaving a step in the through-going crack wall. The geometry of dilatant echelon cracks may be used to infer spatial or temporal changes in the orientation of principal stresses in the Earth.

Surface deformation in volcanic rift zones

Abstract The principal conduits for magma transport within rift zones of basaltic volcanoes are steeply dipping dikes, some of which feed fissure eruptions. Elastic displacements accompanying a single dike emplacement elevate the flanks of the rift relative to a central depression. Concomitant normal faulting may transform the depression into a graben thus accentuating the topographic features of the rift. If eruption occurs the characteristic ridge-trough-ridge displacement profile changes to a single ridge, centered at the fissure, and the erupted lava alters the local topography. A well-developed rift zone owes its structure and topography to the integrated effects of many magmatic rifting events. To investigate this process we compute the elastic displacements and stresses in a homogeneous, two-dimensional half-space driven by a pressurized crack that may breach the surface. A derivative graphical method permits one to estimate the three geometric parameters of the dike (height, inclination, and depth-to-center) and the mechanical parameter (driving pressure/rock stiffness) from a smoothly varying displacement profile. Direct comparison of measured and theoretical profiles may be used to estimate these parameters even if inelastic deformation, notably normal faulting, creates discontinuities in the profile. Geological structures (open cracks, normal faults, buckles, and thrust faults) form because of stresses induced by dike emplacement and fissure eruption. Theoretical stress states associated with dilation of a pressurized crack are used to interpret the distribution and orientation of these structures and their role in rift formation.

Joint formation in granitic rock of the Sierra Nevada

A single steeply dipping joint set in the Mount Givens Granodiorite, central Sierra Nevada, was studied to clarify the mechanics of fracture and joint formation in granitic rocks. The joints were filled with fluid during, or immediately following, formation; these fluids deposited epidote and chlorite within the joints. Examination of lithologic markers in outcrop and thin section demonstrates that relative displacements are normal to the joint surfaces. These observations rule out a shear origin for these joints. The measured extensional strain acommodated by joint dilation is on the order of 1 × 10 −4 to 5 × 10 −4 . A few joints in the area exhibit small strike-slip offsets. In these joints, the mineral fillings are sheared, indicating the strike-slip motion postdated the jointing. Individual joints consist of numerous subparallel, planar segments. The lengths of joints range from metres to nearly 100 metres. Shorter joints are more abundant than longer joints. The observed distribution of joint lengths is thought to result from the elastic interaction of adjacent joints. Shorter joints are prevented from further propagation by their long neighbors. Between mapped joints, small cracks that have lengths of several centimetres are found parallel to the longer joints. These cracks represent a growth stage between grain-scale microcracks and macroscopic joints. A method is developed for estimating the tensile stress responsible for initiating joint growth. The method requires knowledge of the extensional strain accommodated by joint dilation and the spatial density of joints, both of which can be determined by field observations. Calculations based on observations of joints in the Florence Lake area indicate relative tensile stresses (average remote stress plus internal fluid pressure) of approximately 1 MPa to 40 MPa. These values of stress and estimates of initial crack length are used to estimate the quasi-static fracture toughness of the granodiorite. The calculated fracture toughnesses range from 0.04 Mpa·m 1/2 to 4 Mpa·m 1/2 . The stress and fracture toughness estimates are compatible with existing data from laboratory fracture experiments.

Dike-induced faulting in rift zones of Iceland and Afar

Geodetic data and field observations demonstrate that the emplacement of dikes in volcanic rift zones frequently generates normal faulting and graben subsidence at the Earth9s surface. Elastic modeling of the vertical ground-surface displacements above dikes and faults indicates that the extent of graben subsidence can be achieved only if fault slip extends virtually to or beyond the dike plane at depth. A mechanical model that includes dikes and frictional faults shows that dike opening tends to compress and lock faults located to either side of the dike. Therefore, slip extending into or beyond the dike cavity must occur either (1) on faults that intersect the dike near its top, above the zone of dike-induced compression, or (2) on faults that slip ahead of the dike as it propagates laterally. Data from Iceland indicate that slip occurred on deep faults that presumably slipped in advance of the laterally propagating dike.

Mechanics of growth of some laccolithic intrusions in the Henry mountains, Utah, I: Field observations, Gilbert's model, physical properties and flow of the magma

The shapes of sills and laccolithic intrusions and associated host rock deformation were studied at several locations on the flanks of the Henry Mountains. Diorite sills range from 0.5 to 10 m in thickness, are less than 1 km2 in areal extent, and have blunt terminations. The laccolithic intrusions range from 10 to 200 m in thickness, and from 1 to 3 km2 in areal extent. The host rock, principally sandstone and shale, is deformed along closely spaced cataclastic shear planes. This deformation is concentrated at contacts, especially near sill terminations and over laccolith peripheries. The diorite contains plagioclase phenocrysts which are usually sheared in a thin zone adjacent to each contact. Field observations suggest that sills are the forerunners of laccolithic intrusions which form only after magma has spread far enough laterally (greater than about 1 km2 in the Henry Mountains) to gain leverage to bend the overburden upward. Further injection of magma results in laccolithic peripheries or terminations with one of three distinct cross-sectional forms: (1) blunt termination of the diorite accompanied by bending and minor faulting of the host rock; (2) termination at a peripheral diorite dike cutting upward across the host rock; or (3) abrupt termination of the diorite against a nearly vertical fault zone. In order to study some of the processes of sill and laccolith intrusion, mechanical models for the driving pressure, physical properties, and flow behavior of the diorite magma are derived and discussed. A static driving pressure (equal to the difference between total magma pressure and lithostatic pressure) of up to 700 bar is estimated. The rheological behavior of the magma in the Henry Mountains is unknown. However, flow behavior is calculated assuming three of the more common models for fluids: Newtonian viscous, pseudoplastic, and Bingham. Suspended crystals probably contributed to the finite strength of the magma (estimated to be at least 103 dyn/cm2 for the Henry Mountains magma) which enables it to support dense zenoliths and also fixes maximum limits on the lengths of sills or dikes. Pressure in magma flowing along tabular intrusions of uniform thickness drops linearly in the flow direction for all three rheological materials. Thickening of tabular intrusions tends to make the pressure drop less rapidly, but pressure drops more rapidly in the tapered region near a termination. Pressure distributions under these and other conditions are derived in order to use them in the models of host rock deformation presented in Part II.

Fracture spacing in layered rocks: a new explanation based on the stress transition

Opening-mode fractures (joints and veins) in layered sedimentary rocks often are periodically distributed with spacings linearly related to the thickness of the fractured layer. To better understand this linear relation, we have investigated the stress distribution between two adjacent opening-mode fractures as a function of the fracture spacing to layer thickness ratio using a three-layer elastic model with a fractured central layer. The results show that when the fracture spacing to layer thickness ratio changes from greater than to less than a critical value (approximately 1.0) the normal stress acting perpendicular to the fractures changes from tensile to compressive. This stress state transition precludes further infilling of fractures unless there are existing flaws and/or the fractures are driven by an internal fluid pressure or other mechanisms. Hence, for fractures driven by tectonic extension, the critical fracture spacing to layer thickness ratio defines a lower limit, which also defines the condition of fracture saturation. The critical value of the fracture spacing to layer thickness ratio is independent of the average strain of the fractured layer, and it increases with increasing ratio of Youngs modulus of the fractured layer to that of the neighboring layers. The critical value increases with increasing Poissons ratio of the fractured layer, and with increasing overburden stress (depth), but it decreases with increasing Poissons ratio of the neighboring layers. For representative variation of the elastic constants of the fractured layer and the neighboring layers, and overburden stress, the critical fracture spacing to layer thickness ratio varies between 0.8 and 1.2. This range encompasses the often cited spacing to layer thickness ratios in the literature for well-developed fractures sets.

Mechanics of growth of some laccolithic intrusions in the Henry mountains, Utah, II: Bending and failure of overburden layers and sill formation

Deformation of host rocks during growth of a laccolithic intrusion is analyzed using the theory of bending a stack of thin elastic plates. The theoretical model suggests that magma spreading laterally in the form of a sill will eventually gain sufficient leverage on the overlying strata to deflect them upward and form a laccolith. The amount of bending increases as the fourth power of the distance the magma spreads, whereas the overburden resists bending as the third power of its effective thickness. Effective thickness is the thickness of a single layer which has the same resistance to bending as a multilayer of similar length and elastic modulus. The effective thickness of overburden in the Henry Mountains is estimated as between 17 and 23 of the actual thickness. The form of bending is similar for Newtonian, pseudoplastic, and Bingham magmas. The magnitude of the bending depends upon the total upward force and its distribution and is not simply related to magma viscosity as has been suggested by several previous investigators. After elastic bending strata should fail over the periphery of an intrusion, the site of maximum bending strain and differential stress predicted by the theory. Field observations described in Part I correlate well with these predictions. Because bending strains are proportional to layer thickness, strata of comparable strength but different thicknesses fail at different stages of laccolith development. This leads to the different cross-sectional forms of laccoliths observed in the field. The effect of host rocks on sill form and growth is analyzed using the elastic solution for an elliptical hole under uniform pressure. The theory suggests that sill thickness increases in proportion to length. The concentration of high stresses near the sill termination should induce permanent deformation and account for the blunt terminations described in Part I. This blunting is most likely to occur in relatively ductile rocks whereas sills simply split brittle rocks and maintain sharp terminations. The driving pressure in sills can be calculated from measurements of length and termination radius of curvature, if the yield strength of the host rocks can be estimated. This driving pressure must be greater than the overburden pressure, but sills apparently do not form or propagate by lifting their overburdens. Instead they propagate by locally deforming the host rock. After spreading over a distance about three times the effective overburden thickness, the overlying layers begin to bend upward significantly. This stage marks the transition from a sill to a laccolithic intrusion.

Microstructure of deformation bands in porous sandstones at Arches National Park, Utah

Abstract At Arches National Park it is possible to distinguish three kinds of deformation bands on the basis of their distinctive microstructure: (1) deformation bands with little or no cataclasis; (2) deformation bands with cataclasis; and (3) deformation bands with clay smearing. The micromechanics of deformation band development consist of initial dilatancy followed by grain crushing and compaction. This process may be developed to different stages according to the interplay of porosity, confining pressure, clay content and amount of strain. Low porosities and low confining pressures promote the formation of dilatant bands with no cataclasis. High porosities and high confining pressures promote compaction and cataclasis. Two generations of deformation bands were documented. The older generation has little or no cataclasis and formed in relatively undisturbed sandstone probably under conditions of low confining pressure. The younger generation exhibits cataclasis, appears to be localized in proximity to major faults and seems to have developed under conditions of high confining pressure. The temporal sequence of deformation band development can be related to the regional geology of the area; where the first generation probably formed during growth of the salt anticline, and the second generation during its collapse.

An experimental study of the relationship between joint spacing and layer thickness

Two methods for measuring joint spacing are described and compared. The area method is a constant for a given outcrop area and is not affected by joint distribution within that area; in contrast, the line method depends on the location of the linear traverse. Two kinds of joint sets are distinguished on bedding surfaces: (1) a poorly-developed set represents the early stages of development when typical joint lengths are less than typical spacing; (2) a well-developed set represents later stages when lengths are much greater than spacing. The area method is applicable to both poorly- and well-developed sets, whereas the line method produces inconsistent results for poorly-developed joint sets. Surface textures on many joints indicate point fracture origins and propagation parallel to bedding, whereas most numerical model studies of spacing assume linear origins and propagation perpendicular to bedding. The laboratory experiments described in this paper do not suffer from these restrictions. An important concept, confirmed during these experiments, is fracture saturation. When the applied strain reaches a certain value, fracture spacing stops evolving and remains nearly constant: these fracture sets are well-developed. Spacing at saturation is a function of layer thickness but is independent of strain, whereas spacing before saturation varies strongly with applied strain. Thus, plotting spacing vs thickness and comparing the slopes of lines fit to such data for poorly-developed joint sets in different layers is unlikely to be a diagnostic test for differences in material properties. On the other hand, spacing may be a sensitive indicator of strain for layers with poorly-developed joint sets. Assessing fracture saturation is a lirst-order consideration when gathering spacing data.

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.