This paper demonstrates a stochastic optimizing process to determine a set of two horizontal in situ principal stresses by minimizing misfits between the probability model and the measured density data of borehole breakout width in a segment of a vertical well bore. The probability as a function of breakout angular width is obtained from a probability density function of rock strength at the wellbore wall that is given by far-field principal stresses. The probability of rock strength is approximated by the Gaussian model that can be obtained from sonic velocity logging data with their empirical relation to rock strength. The aggregation of all misfits between the probability model and measured density of breakout width is represented as an objective function of the magnitudes of two horizontal principal stresses. The minimum of the objective function yields the best set of the two horizontal principal stresses. This process provides not only the optimized values of both horizontal in situ stresses but also their statistical reliability with their uncertainty and sensitivity.
Most rock mechanics theories have been developed on a postulate of existence of representative elementary volume (REV), over which a single number is chosen as an effective property of the REV without consideration of small variation in rock properties and behaviors (e.g., Wang, 2000). If the vertical stress is a principal stress, the tangential principal stress at the wall of a vertical well bore is theoretically expressed as a function of the angular orientation θ measured from the direction of the maximum far-field horizontal principal stress (Jaeger et al., 2007): (Equation 1).