---
title: "Tunnel alignment from three boreholes"
subject: "Engineering Geology II"
chapter: 6
key: "semester_4/engineering_geology_ii:tunnel-alignment-three-boreholes"
source: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/tunnel-alignment-three-boreholes
site: BCT Engineering
---

# Tunnel alignment from three boreholes

Engineering Geology II · Chapter 6: Measurement, analysis and interpretation of structural geological data

- Asked in: 2075 Baisakh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/tunnel-alignment-three-boreholes
- Whole chapter: https://www.bctengineer.com/md/semester_4/engineering_geology_ii/chapter-6

## Question

For a hydropower project, a tunnel alignment has to be selected. The overburden depth is confirmed from drilling of three boreholes, where the top of the bedrock is encountered as follows.
From BH#1 to BH#2, at distance of 1000 m along N32°E; from BH#1 to BH#3, at distance of 800 m along S73°E.
Depths of bedrock: BH#1 = -200 m, BH#2 = -300 m, BH#3 = -500 m.
Select a suitable alignment of the tunnel with respect to the attitude of bedrock.

## Short answer

Using three-point method, bedrock surface strikes N16.5°E and dips 20.6° towards ESE (107°). A tunnel aligned along the strike (N16.5°E) maintains constant rock cover and uniform conditions, making it the most suitable.

## Answer

A bed is a plane, so its level changes uniformly in plan: $z = z_B + a x + b y$, where $a$ and $b$ are the rates of change of level per metre east and north. This is the analytical form of the three-point (strike-line) method.

#### Coordinates (BH#1 at origin)
$$
\begin{aligned}
\text{BH\#2}&: x = 1000\sin 32^\circ = 529.92\ \text{m},\ y = 1000\cos 32^\circ = 848.05\ \text{m} \\
\text{BH\#3}&: x = 800\sin 107^\circ = 765.04\ \text{m},\ y = 800\cos 107^\circ = -233.90\ \text{m}
\end{aligned}
$$
Levels of top of bedrock: BH#1 = -200 m, BH#2 = -300 m, BH#3 = -500 m.

#### Plane of the top of bedrock
$z = -200 + a x + b y$:
$$
\begin{aligned}
-300 &= -200 + 529.92a + 848.05b \\
-500 &= -200 + 765.04a - 233.90b
\end{aligned}
$$
Solving: $a = -0.3595$, $b = 0.1067$.

#### Dip, dip direction and strike
$$
\tan\delta = \sqrt{a^2+b^2} = 0.3750 \Rightarrow \delta = 20.56^\circ
$$
The bed descends toward the direction $\arctan(-a/-b)$ = **106.5°** (ESE), so the strike is $\ 106.5^\circ - 90^\circ = 16.5^\circ$, i.e. **N16.5°E**.

So the top of the bedrock dips **20.6° towards 107° (ESE)**; strike N16.5°E.

#### Selecting the alignment
- Along the **strike** (about N16.5°E - S16.5°W) the surface of the bedrock is horizontal, so the depth of overburden and the rock cover above the tunnel stay constant, the tunnel stays in the same stratum, and the beds meet the tunnel roof uniformly. This gives the most uniform conditions and easiest support design.
- Along the dip (towards 107°) the bedrock drops $0.375$ m for each metre (about 20.6° slope), so the tunnel would pass from deep rock to shallow cover or soil, and would cross different beds.

**Answer: the bedrock surface strikes N16.5°E and dips 20.6° towards ESE (107°). A tunnel driven along the strike direction (N16.5°E) at a constant level below the bedrock surface is most suitable.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the three-borehole method, why can the bedrock level be written as z = z0 + a x + b y?
   A. Because every bed is exactly horizontal
   B. Because the boreholes lie on a straight line
   C. Because a bed is a plane, so its level changes uniformly with position
   D. Because the level depends only on the distance from BH#1
   Answer: C. A bed is treated as a plane, so level varies linearly with east and north coordinates; the three borehole levels fix a and b. (Key point: Bed as a plane)

2. What attitude of the top of bedrock did the solution find?
   A. Dip about 20.6° towards ESE (107°), strike N16.5°E
   B. Dip about 20.6° towards N16.5°E, strike ESE
   C. Dip about 0.375° towards 107°, strike N16.5°E
   D. Horizontal bedrock with no dip
   Answer: A. tan(dip) = sqrt(a^2+b^2) = 0.375 gives 20.6°; the dip direction is 107° and strike is perpendicular, N16.5°E. (Key point: Dip and strike result)

3. Which tunnel direction is most suitable?
   A. Along the dip direction, because it follows the steepest slope
   B. Perpendicular to the strike, to cross many beds quickly
   C. Along the strike, N16.5°E, because the bedrock surface is level in that direction
   D. Any direction, since the rock cover is the same
   Answer: C. Along strike the bedrock level is constant, so rock cover and strata stay uniform. (Key point: Best alignment)

4. Why is a tunnel along the dip direction unsuitable?
   A. The bedrock drops 0.375 m per metre, so cover changes from deep rock to shallow cover or soil and different beds are crossed
   B. The tunnel would be exactly parallel to the beds and collapse
   C. The overburden becomes constant, which weakens the roof
   D. It would be longer than the strike alignment by 20.6%
   Answer: A. A sloping bedrock surface along the tunnel gives varying cover and varying strata, making support design harder. (Key point: Problem with dip direction)

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