⚡ electrical engineering

A vector-based analytical method for evaluating borehole–joint hit rate in pre-excavation grouting of jointed rock mass

This study presents a transparent, reproducible vector-based analytical method for efficiently calculating borehole–joint hit rates in pre-excavation grouting, offering a practical alternative to manual CAD assessments for optimizing borehole orientations and evaluating intersection probabilities in jointed rock masses.

Original authors: Abishek Shrestha, Kristin H. Holmøy, Johan Funehag

Published 2026-07-22

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Abishek Shrestha, Kristin H. Holmøy, Johan Funehag

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to catch a specific type of fish in a massive, dark ocean. You have a net (your drill holes), but the fish (water-filled cracks in the rock) are swimming in very specific directions. If you throw your net straight down, you might miss the fish swimming sideways. If you throw it sideways, you might miss the ones swimming up. In the world of building tunnels deep underground, engineers face this exact problem. They need to pump a special glue-like substance called "grout" into the ground to seal up cracks before they dig, stopping dangerous water from flooding the tunnel. But to do this, the holes they drill must hit the cracks at the perfect angle. If the hole runs parallel to the crack, it's like trying to slice a loaf of bread by sliding the knife along the crust; you won't cut through. If the hole hits the crack straight on, it's like a perfect slice. The challenge is that underground rock is messy, and the cracks can be anywhere, pointing in any direction.

For a long time, figuring out if a drill hole would hit a crack "perfectly" was like trying to solve a 3D puzzle by hand. Engineers had to build giant, complicated computer models (CAD) and measure every single angle with a digital ruler. It was slow, tedious, and hard to change if you wanted to try a different drill pattern. This paper introduces a much faster, smarter way to solve this puzzle using a "vector-based" method. Think of it as switching from measuring every slice of bread with a ruler to using a magic calculator that instantly tells you the perfect angle just by knowing which way the bread and the knife are pointing. The authors tested this new math on a real railway tunnel project in Sweden and found it works just as well as the old, slow way, but it's much easier to use and can quickly tell engineers if they need to tweak their drill holes to catch more "fish."

The Problem: The "Hit Rate" Puzzle

When engineers build tunnels in hard rock, they often have to stop and inject grout to seal up water leaks before they dig further. They do this by drilling a fan of holes ahead of the tunnel face, like a porcupine's quills sticking out. The goal is to hit the water-filled cracks (joints) in the rock. But here's the catch: hitting a crack isn't just about where you drill; it's about how you hit it.

If you drill a hole straight into a crack (perpendicular), you get a nice, round circle of contact. This is the "gold standard" because it gives the grout the best chance to spread out and seal the leak. We call this a 100% hit rate.

However, if you drill at a slant, the hole slices through the crack at an angle, creating a long, stretched-out oval (an ellipse). The more slanted the angle, the longer and skinnier the oval gets. This is a "bad hit." The grout has a harder time getting into the crack, and you might miss the water entirely. If you drill parallel to the crack, you barely touch it at all, and the hit rate drops to near zero.

The big question for engineers is: "How good is our current drill plan at hitting these cracks?" To answer this, they need to calculate the hit rate for every single hole against every single set of cracks.

The Old Way vs. The New "Magic" Way

Previously, to figure this out, engineers had to build a detailed 3D computer model of the tunnel, the drill holes, and the cracks. Then, they had to manually measure the angles and areas for every single hole. It was like trying to count every grain of sand on a beach by picking them up one by one. It worked, but it was slow and frustrating, especially if they wanted to test a new drill pattern or if they discovered a new type of crack.

This paper proposes a new, vector-based analytical method. Instead of building a 3D model and measuring it, the authors use a clever mathematical shortcut. They treat the direction of the drill hole and the direction of the crack as simple arrows (vectors). By doing a quick math operation called a "dot product" on these arrows, they can instantly calculate the hit rate.

Think of it like this: The old way was like trying to measure the shadow of a tree by drawing the tree and the sun on paper and measuring the shadow with a ruler. The new way is like having a formula that says, "If the sun is at this angle and the tree is this tall, the shadow is this long," and you just punch the numbers into a calculator.

What They Found: Testing the New Method

The authors tested their new math method on a real-life project: the Haga Station section of the West Link railway tunnel in Gothenburg, Sweden. They compared their new "magic calculator" results against the old, slow 3D computer model results that had already been published.

The Results:

It Works: The new method produced almost the exact same pattern of results as the old 3D model. It correctly identified which drill holes were hitting the cracks well (high hit rate) and which ones were missing them (low hit rate).
Small Differences: There were tiny differences in the exact numbers (a few percentage points), but the overall picture was the same. The new method is fast, transparent, and easy to repeat.
The "Reorientation" Test: The authors asked, "What if we just tweak the direction of a few bad holes?" They simulated moving the tips of some drill holes slightly to aim better at the cracks. They found that by changing the direction of just a few holes, they could boost the hit rate by up to 12 percentage points in the worst areas. For example, one hole that was barely hitting the crack (2% hit rate) was improved to a decent 14% just by turning it slightly.
The "Spacing" Factor: They also added in the distance between cracks (joint spacing). They found that even if cracks are very close together (every 0.5 meters), if your drill hole is pointing the wrong way, you still won't hit them often. But if you aim well, you might hit 16 or 17 cracks in a single hole!

Why This Matters

This paper doesn't invent a new type of glue or a new way to drill. Instead, it gives engineers a better map.

Before, if an engineer wanted to check if their drill plan was good, they had to spend hours building a 3D model. Now, they can use this new vector method to quickly screen their plans. They can instantly see, "Hey, holes 3 through 5 are pointing the wrong way for the cracks we found; let's turn them a bit."

The authors are careful to say that this is just a geometric screening tool. It tells you if the hole is pointing in the right direction, but it doesn't guarantee the grout will work (that depends on how wide the crack is, how much water is flowing, and the type of glue used). However, by quickly identifying the "bad angles," engineers can save time, money, and potentially prevent water disasters in tunnels.

In short, this paper turns a slow, manual 3D puzzle into a fast, smart calculation, helping engineers aim their "nets" more accurately at the underground "fish" before they start digging.

Technical Summary: A Vector-Based Analytical Method for Evaluating Borehole–Joint Hit Rate in Pre-Excavation Grouting

Problem Statement
Pre-excavation grouting (PEG) in jointed rock masses relies on the geometric compatibility between grouting boreholes and hydraulically significant discontinuities. Effective grouting requires boreholes to intersect target joint sets at favorable angles, ideally perpendicular, to maximize the intersection trace area and facilitate grout penetration. Existing assessment methods, such as the three-dimensional Computer-Aided Design (CAD) models used by Funehag and Sjöli (2023), require manual construction and measurement of borehole–joint geometry. While accurate for individual cases, these CAD-based approaches are computationally inefficient and impractical for the repeated evaluation of multiple boreholes, varying joint sets, or alternative fan layouts. There is a need for a transparent, reproducible, and computationally efficient analytical framework to quantify the "hit rate"—the ratio of the borehole cross-sectional area to the elliptical intersection area on a joint plane—without relying on manual 3D modeling.

Methodology
The study proposes a closed-form, vector-based analytical workflow to calculate hit rates in a common three-dimensional coordinate system. The method integrates three primary input groups:

Tunnel Alignment: Defined by tunnel azimuth ( $AZ_t$ ) and plunge ( $PZ_t$ ), establishing a tunnel axis unit vector ( $\mathbf{t}$ ) and an orthogonal tunnel-face coordinate system ( $\mathbf{t}_1, \mathbf{t}_2$ ).
Joint Set Orientation: Defined by strike ( $\theta$ ) and dip ( $\delta$ ), from which a unit normal vector ( $\mathbf{n}$ ) for each joint plane is derived.
Borehole Fan Geometry: Defined for each borehole $i$ by its inclination angle ( $\beta_i$ ) relative to the tunnel axis and its circumferential angle ( $\psi_i$ ) within the tunnel-face plane, yielding a borehole direction unit vector ( $\mathbf{l}_i$ ).

The core calculation defines the hit rate ( $h_{ij}$ ) for a specific borehole $i$ and joint set $j$ as the absolute value of the dot product between the borehole direction vector and the joint plane normal vector:
$h_{ij} = |\mathbf{l}_i \cdot \mathbf{n}_j|$
This formulation yields a dimensionless value between 0 and 1, where 1 represents a perpendicular intersection (maximum hit rate) and 0 represents a parallel intersection (minimum hit rate). The method extends to multiple joint sets by generating a hit rate matrix. Furthermore, the study incorporates borehole length ( $L$ ) and representative joint spacing ( $s_j$ ) to estimate the expected number of intersections ( $N_{ij}$ ) using the relationship:
$N_{ij} = \frac{L \cdot h_{ij}}{s_j}$

Case Study and Validation
The workflow was applied to the Haga Station section of the West Link railway tunnel in Gothenburg, Sweden. The case involved a PEG fan intersecting three distinct joint sets (SG-1, SG-2, and SG-3).

Validation: The vector-based results for SG-1 were compared against published CAD-based geometric modeling results. The analytical method successfully reproduced the spatial distribution of hit rates, identifying the same low-hit rate sectors (e.g., boreholes 2–4 and 16) and high-hit rate sectors (e.g., boreholes 26–28). While moderate magnitude differences existed (the vector method yielded slightly higher values in favorable sectors), the spatial ranking was consistent.
Reorientation Analysis: A scenario was tested where selected boreholes in low-hit rate sectors were reoriented toward SG-1 while maintaining original collar positions. This adjustment increased hit rates by up to 12 percentage points for specific boreholes (e.g., from ~2% to ~14% for borehole 3), demonstrating that circumferential angle adjustments can significantly improve geometric compatibility.
Multi-Set Analysis: The method was applied to SG-2 and SG-3. Results showed that the same fan layout produced vastly different hit rate distributions depending on the joint set orientation. SG-2 yielded uniformly high hit rates (~89–100%) across the fan, whereas SG-3 showed high variability, and SG-1 contained distinct sectors with very low hit rates.

Key Contributions and Results

Analytical Formulation: The study provides a vector-based formulation that eliminates the need for manual 3D CAD construction, allowing for rapid, reproducible calculation of hit rates for any number of boreholes and joint sets.
Geometric Sensitivity: The analysis confirms that joint set orientation is the primary driver of hit rate distribution. A fan layout favorable for one joint set may be geometrically unfavorable for another, necessitating separate evaluations for each hydraulically significant set.
Intersection Estimation: By integrating joint spacing, the method moves beyond simple angular compatibility to estimate the expected frequency of borehole–joint intersections. The results indicate that closely spaced joints cannot compensate for highly unfavorable borehole orientations; low hit rates result in low intersection counts regardless of joint density.
Design Optimization: The reorientation scenario demonstrates that targeted adjustments to borehole direction (primarily circumferential angles) can significantly improve geometric access to specific joint sets without altering the general fan arrangement or collar locations.

Significance and Limitations
The paper positions this workflow as a geometric screening and design-support tool intended to complement, not replace, hydrogeological characterization, hydraulic testing, and engineering judgment. Its primary significance lies in providing a transparent framework for comparing alternative fan layouts, identifying sectors with poor geometric coverage, and assessing the impact of joint orientation on grouting feasibility.

The authors explicitly acknowledge limitations: the method idealizes joint sets using mean orientations and uniform spacing, ignoring spatial variability, persistence, and clustering. It is purely geometric and does not account for hydraulic properties (aperture, connectivity), grout rheology, or drilling deviations. Consequently, hit rate is presented as a relative metric for geometric access rather than a direct predictor of grouting success or water ingress control. The study concludes that while the method offers a robust basis for preliminary design and iterative assessment, field validation against actual borehole intersections and grouting performance data is required for future development.

The Problem: The "Hit Rate" Puzzle

The Old Way vs. The New "Magic" Way

What They Found: Testing the New Method

Why This Matters

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