Question 9 of 12: Tunnel Breakthrough Error and its Prediction

Question 9 (Part B): Tunnel Breakthrough Error and its Prediction (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

What it is. When a tunnel is driven from two (or more) headings toward a common meeting face, the two headings will not meet exactly: the breakthrough error is the vector misclosure between the two surveyed centre-lines at the holing-through point. It has three components — longitudinal (along the tunnel, least critical, absorbed by chainage), vertical (governs the invert/grade match), and transverse (perpendicular to the axis, the most critical, because it decides whether the two bores align laterally). Design tolerances (often a few centimetres) are set on the transverse and vertical components.

Given. Independent survey error contributions to the transverse position of the breakthrough: surface control $\sigma_s$, orientation/coordinate transfer down the shafts $\sigma_o$, and the underground traverse to the face $\sigma_u$.

Find. The predicted (1σ and 95%) transverse breakthrough error.

Approach. Model each survey stage as an independent random error source and propagate them to the breakthrough point; independent contributions combine in quadrature, and the 95% figure is about twice the 1σ value.

1. Enumerate the error sources along each route. From the surface network at each portal/shaft, through the orientation transfer into the underground (the weakest link when access is by shaft), to the underground traverse driving the heading — each contributes a transverse uncertainty at the face. These are obtained by pre-analysis (propagating the design covariance $\mathbf{Q}_{xx}=(\mathbf{A}^{\mathsf T}\mathbf{P}\mathbf{A})^{-1}$ of the planned traverses and orientation).
2. Predict each underground-traverse contribution from its geometry. Take the $x$-axis along the tunnel and $y$ transverse. An angle error $\sigma_\beta$ at traverse station $i$ swings the rest of the traverse about that station, moving the breakthrough point sideways by $R_{x,i}\,\sigma_\beta/\rho$, where $R_{x,i}$ is the distance from station $i$ to the breakthrough point projected on the tunnel axis; a distance error moves it along each leg, of which only the transverse projection $d_{y,j}$ of leg $j$ matters. Hence, for one heading, $$\sigma_{y\beta}=\frac{\sigma_\beta}{\rho}\sqrt{\sum R_{x,i}^2},\qquad \sigma_{yl}=\frac{\sigma_l}{l}\sqrt{\sum d_{y,j}^2},$$ with $\rho=206\,265''$. For a straight tunnel $d_y\approx0$, so the angle errors govern the transverse breakthrough; the starting-azimuth (orientation) error $\sigma_{A}$ adds $R_x\,\sigma_A/\rho$ over the full heading length. The surface-network contribution comes from the covariance of the portal/shaft points and their orientation lines. Each heading is evaluated this way, and the two headings' contributions are added.
3. Combine in quadrature. Being independent, the transverse variances add: $$\sigma_y=\sqrt{\sigma_s^2+\sigma_o^2+\sigma_u^2}.$$ Example: $\sigma_s=8$ mm, $\sigma_o=15$ mm, $\sigma_u=20$ mm give $$\sigma_y=\sqrt{8^2+15^2+20^2}=\sqrt{689}=\boxed{26.2\ \text{mm}}.$$
4. Express at the required confidence. The 95% expected breakthrough error is about $2\sigma_y$: $$E_{95\%}\approx 2\sigma_y = \boxed{52\ \text{mm}},$$ which is compared against the project tolerance. If it exceeds the tolerance, the design is strengthened (better shaft orientation — e.g. gyro azimuth or a two-shaft base, tighter underground traverse, more sets), i.e. the design is iterated exactly as in Q1.