APPLYING POWER-LAW RELATIONSHIPS IN GEOTECHNICAL CHARACTERIZATION

Many complex physical systems measured using nonlinear concepts, such as fractal geometry and self-ordered criticality, demonstrate straight-line trends of magnitude and frequency in log-log space. A system exhibits fractal behavior over regions in which the fractal dimension is less than the corresponding Cartesian dimension. Most physical systems exhibit more than one straight-line magnitude-frequency trend in log-log space. The Gutenberg-Richter relationship describing earthquake magnitude and frequency is a well-known example of one-dimensional data in which the fractal dimension (the ‘b’ value) is less than 1.0. At great earthquake magnitudes, the slope of the log-log plot exceeds 1.0, and the behavior is no longer fractal. A critical magnitude can be identified at the point separating the two straight-line trends, below which the system exhibits fractal behavior. At the critical magnitude, the system exhibits self-ordered criticality, which might include an abrupt change in the system behavior.

Power-law relationships have been applied to water pipeline vibrations, traffic vibrations, laboratory grain-size distributions, scan-line particle-size distributions, fracture spacing in rock core, and scan-line fracture spacing. The desired parameter (peak particle velocity, size, distance, weight) is measured over a selected period of time, distance, or interval, and the measurements are ordered according to rank. The log of the size or magnitude of the parameter is plotted on the abscissa, whereas the log of the cumulative number of measurements is plotted on the ordinate.

Behavior of the physical system can be inferred from the slope of the power-law trends and the critical size. Rock-mass rippability index can be calculated as the product of the unconfined compressive strength of the rock material and the square of the critical core length. Maximum anticipated particle size for excavations, such as trenches or tunnels, can be estimated by extrapolating from the power-law trend of the largest particles sampled.