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[1] Observations of natural fracture dimensions have sparked a continuing debate as to the nature of the fundamental relationship between fracture aperture (maximum opening) and length. On the basis of theoretical fracture mechanics, some have argued aperture-tolength scaling should be linear. This relationship implies that all fractures in a given population have the same driving stress regardless of fracture length, arguably a state that is difficult to reconcile with fracture propagation criteria. Also, some field observations indicate sublinear aperture-to-length scaling that is apparently inconsistent with the linear elastic fracture mechanics theory. In this work, a nonlinear aperture-to-length relationship is derived, still based on linear elastic fracture mechanics in a homogeneous body, but incorporating subcritical and critical (equilibrium law) fracture propagation criteria. The new hypothesis postulates that fractures of different lengths preserved in a body of rock are all in the same condition with respect to propagation (i.e., they all have the same stress intensity factor). This requires that fractures have driving stresses that vary inversely with the square root of fracture length, producing fracture apertures that scale with length to the 1/2 power. Under these conditions, fracture aspect ratio (aperture/length) decreases with increasing fracture length to the negative 1/2 power. Linear aperture-to-length scaling is still considered a possibility but is attributed to a relaxed, postpropagation mechanical state. Deviations in fracture aperture-to-length relationships from these idealized models can result from mechanical fracture interaction, fracture segmentation into en echelon arrays, and three dimensional effects in stratabound fractures.

Fracture networks are examined in the light of subcritical crack growth theory. Examples of equilibrium crack geometries are generated using a fracture mechanics model that explicitly tracks the propagation of multiple fractures. It is determined that propagation velocity as modeled using a subcritical fracture growth law exerts a controlling influence on fracture length distributions and spacing. Velocity is modeled as proportional to the n‐th power of the mode I stress intensity. Numerous, closely spaced, similar length fractures result for n=1, with many en echelon arrays forming due to fracture interaction. Increasing the value of n results in the growth of fewer fractures that are more widely spaced. Fractures tend to cluster in narrow zones, with limited fracture growth in the intervening areas. The spacing between zones is controlled by the stress shielding effects of longer fractures on shorter ones. The amount of time required for fracture pattern development is also influenced by the subcritical velocity exponent, n. At low n, patterns take seconds to minutes to develop, while patterns generated at higher n can require hundreds of years or more.

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In November 2013, a series of earthquakes began along a mapped ancient fault system near Azle, Texas. Here we assess whether it is plausible that human activity caused these earthquakes. Analysis of both lake and groundwater variations near Azle shows that no significant stress changes were associated with the shallow water table before or during the earthquake sequence. In contrast, pore-pressure models demonstrate that a combination of brine production and wastewater injection near the fault generated subsurface pressures sufficient to induce earthquakes on near-critically stressed faults. On the basis of modelling results and the absence of historical earthquakes near Azle, brine production combined with wastewater disposal represent the most likely cause of recent seismicity near Azle. For assessing the earthquake cause, our research underscores the necessity of monitoring subsurface wastewater formation pressures and monitoring earthquakes having magnitudes of ∼M2 and greater. Currently, monitoring at these levels is not standard across Texas or the United States.

Swarms or clusters represent an exception to the widely accepted idea that fracture spacing in sedimentary rock should be proportional to mechanical layer thickness. Experimental studies and static stress analysis do not provide adequate explanation for fracture swarm occurrence. The problem is re-examined numerically, accounting for the dynamics of pattern development for large populations of layer-confined fractures. Two crucial aspects of this model are: (1) the inclusion of three-dimensional effects in calculating mechanical interaction between simultaneously propagting fractures; and (2) the use of a subcritical crack-propagation rule, where propagation velocity during stable growth scales with the crack-tip stress intensity factor. Three regimes of fracture spacing are identified according to the magnitude of the subcritical index of the fracturing material. For low subcritical index material (n = 5) numerous fractures propagate simultaneously throughout a body resulting in irregular spacing that is, on average, much less than layer thickness. For intermediate subcritical index (n = 20) one fracture propagates at a time, fully developing its stress shadow and resulting in a pattern with regular spacing proportional to layer thickness. For high subcritical index cases (n = 80) fractures propagate in a fashion analogous to a process zone, leaving a fracture pattern consisting of widely spaced fracture clusters.

Long-standing racial differences in US life expectancy suggest that black Americans would be exposed to significantly more family member deaths than white Americans from childhood through adulthood, which, given the health risks posed by grief and bereavement, would add to the disadvantages that they face. We analyze nationally representative US data from the National Longitudinal Study of Youth (n = 7,617) and the Health and Retirement Study (n = 34,757) to estimate racial differences in exposure to the death of family members at different ages, beginning in childhood. Results indicate that blacks are significantly more likely than whites to have experienced the death of a mother, a father, and a sibling from childhood through midlife. From young adulthood through later life, blacks are also more likely than whites to have experienced the death of a child and of a spouse. These results reveal an underappreciated layer of racial inequality in the United States, one that could contribute to the intergenerational transmission of health disadvantage. By calling attention to this heightened vulnerability of black Americans, our findings underscore the need to address the potential impact of more frequent and earlier exposure to family member deaths in the process of cumulative disadvantage.race | life expectancy | bereavement | family | disparities