Optimizing the Terzaghi Estimator of the 3D Distribution of Rock Fracture Orientations

Orientation statistics are prone to bias when surveyed with the scanline mapping technique in which the observed probabilities differ, depending on the intersection angle between the fracture and the scanline. This bias leads to 1D frequency statistical data that are poorly representative of the 3D distribution. A widely accessible estimator named after Terzaghi was developed to estimate 3D frequencies from 1D biased observations, but the estimation accuracy is limited for fractures at narrow intersection angles to scanlines (termed the blind zone). Although numerous works have concentrated on accuracy with respect to the blind zone, accuracy outside the blind zone has rarely been studied. This work contributes to the limited investigations of accuracy outside the blind zone through a qualitative assessment that deploys a mathematical derivation of the Terzaghi equation in conjunction with a quantitative evaluation that uses fractures simulation and verification of natural fractures. The results show that the estimator does not provide a precise estimate of 3D distributions and that the estimation accuracy is correlated with the grid size adopted by the estimator. To explore the potential for improving accuracy, the particular grid size producing maximum accuracy is identified from 168 combinations of grid sizes and two other parameters. The results demonstrate that the 2° × 2° grid size provides maximum accuracy for the estimator in most cases when applied outside the blind zone. However, if the global sample density exceeds 0.5°⁻², then maximum accuracy occurs at a grid size of 1° × 1°.