Positive Feedback During Progressive Rock-Fracture Growth and the Emergence of Power-Law Size Distributions

Session: Faults, Fractures, and Geomechanics for the Energy Transition

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Abstract:

Natural fracture size distributions commonly, though not exclusively, follow power-law or “fractal” size distributions. In such populations, over some broad range of sizes, the logarithm of fracture abundance is inversely proportional to the logarithm of fracture size. Fractal patterns emerge from non-fractal or “characteristic” size populations via positive feedback during progressive fracturing, whereby large fractures are favored to grow at the expense of small fractures. Such feedback can occur via fragmentation of rock masses, in which large fractures physically block the lengthening of small fractures, creating abutting intersections. Sets of parallel—thus non-intersecting—extension fractures can also be power-law distributed. Fracture lengthening amid constant remote stress involves inherent positive feedback, whereby the crack-tip stress intensity factor increases as the fracture lengthens. In this presentation I argue that creep processes, particularly solution-precipitation creep, moderate this inherent positive feedback, and therefore largely control the type of size distribution that results.

Specifically, three size-scaling domains are possible under increasing rates of creep, relative to fracture strain rate. At very slow creep rates, crack-tip stress intensities produce excessive positive feedback between fracture length and growth propensity. Runaway growth occurs: fractures that begin to lengthen quickly become critically stressed and outpace the growth of the rest. A characteristic size distribution results as these few large fractures are limited in size by layer thickness or other geological boundaries. At very fast creep rates, stress concentrations are rapidly homogenized, neutralizing the positive feedback between fracture size and growth potential. Fractures grow subcritically and a characteristic size distribution also emerges as small differences in fracture size are not amplified. It is at intermediate creep rates that parallel fractures grow into power-law distributed sets. Here a stable level of positive feedback promotes the growth of already-large fractures without leading to runaway growth. Lengthening remains subcritical, but size-dependent stress concentrations remain, and a power-law size distribution emerges. This phenomenon is not limited to opening-mode fractures; numerical modeling of subduction zone earthquake sizes switches from characteristic to power-law when interfacial asperities grow by mineral precipitation. Thus progressive geologic fracture patterns are sensitively controlled by geochemical processes.