Vector Analysis of Borehole and Joint Intersection Geometry for Pre-Excavation Grouting

Abstract Pre-excavation grouting (PEG) in jointed rock mass depends partly on whether boreholes intersect hydraulically significant joint sets at favourable angles. This geometric relationship can be described by hit rate, where a higher value indicates greater geometric compatibility between the borehole direction and joint plane orientation. Previous assessments of hit rate have relied on three-dimensional CAD models and manual measurement of borehole–joint geometry, making repeated evaluation of many boreholes and joint sets time-consuming. This study presents a closed-form vector-based method for calculating hit rate directly from borehole directions and joint plane orientations. The method is applied to the Haga Station section of the West Link railway tunnel. For Joint Set 1 (SG-1), the vector-based results show close agreement with the published CAD-based results, with a mean absolute difference of 0.99 percentage points and a Pearson correlation coefficient of r = 0.996. This agreement is interpreted within the limitations of the available geometric data and underlying assumptions. The method is then applied to Joint Set 2 (SG-2) and Joint Set 3 (SG-3) to examine how joint set orientation affects hit rate across the borehole fan. A bounded sensitivity analysis is also used to evaluate the reported variation in joint orientation. Borehole length, hit rate, and representative true joint spacing are combined to estimate an idealized expected number of borehole–joint intersections. The method is intended as a geometric screening and design-support tool and does not directly represent intersection probability, hydraulic connectivity, groutability, or grouting effectiveness.