Data-driven estimations of ground deformations induced by tunneling: a Bayesian perspective

Abstract

Estimating tunneling-induced ground deformations is a key issue in tunnel engineering. Many analytical approaches, including empirical models and physical models, have been developed to predict tunneling-induced ground vertical and lateral displacements. However, the most suitable model complexity level and their associated predictive ability have not been fully plumbed. This paper aims to perform a statistically rigorous model comparison of several representative predicting models in the framework of Bayesian model selection, and a probabilistic assessment of the information gain of different types of monitoring data by assessing the Kullback–Leibler divergence. The results of the calculated model evidences show that the Loganathan–Poulos model is the most suitable one when predicting tunneling-induced ground deformations in the illustrative example even though it has the least model parameters. The analyses of the estimated Kullback–Leibler divergences indicate that the measured ground vertical deformations are more informative than the measured ground horizontal deformations. The finding of this study is a first step to clarifying the role of model complexity in tunneling-induced ground deformation analysis and is helpful to provide guidance for ground deformation monitoring in future tunneling engineering.

Appendix

1.1 Peck model

The classical method for estimating the tunneling-induced ground movements is the Peck empirical equation proposed by Peck [45]. According to a large amount of in situ measurement data, Peck found that the profile of ground surface settlements induced by tunneling resembles a Gaussian distribution curve, as shown in Fig. 

Fig. 3

Tunneling-induced settlement trough curve

3. The ground surface settlement trough profile is expressed as

$$s_{z} = S_{\max } \exp \left( { - \frac{{x^{2} }}{{2i^{2} }}} \right)$$

(7)

where $S_{\max }$ is the maximum settlement occurring at the tunnel centerline; $x$ is the distance to the tunnel centerline; $i$ is the settlement trough width, which is the distance between the inflection point to the tunnel centerline. The settlement trough width is related to the tunnel diameter D and the tunnel cover depth C as follows,

$$i = \frac{D}{2}\left( {\frac{C}{D} + 0.5} \right)^{n}$$

(8)

where $n$ is a fitting parameter to field measurements.

In the subsequent physical models, the settlement trough width is defined as the distance between the tunnel centerline and the point at which the settlement magnitude is equal to $s_{i} = S_{\max } \exp \left( { - \frac{1}{2}} \right)$.

The maximum settlement $S_{\max }$ is linked to the Volume loss ratio, $V_{{\text{L}}}$, which is defined as the ratio of the volume of settlement trough (per unit length) to the tunnel excavation volume. The volume loss can be expressed as

$$V_{{\text{L}}} = \frac{{4\sqrt {2\pi } iS_{\max } }}{{\pi D^{2} }}$$

(9)

Thus, the settlement trough profile can be rewritten as

$$s_{z} \left( {z = 0} \right) = \frac{{\pi D^{2} V_{L} }}{{4\sqrt {2\pi } i}}\exp \left( { - \frac{{x^{2} }}{{2i^{2} }}} \right)$$

(10)

1.2 Verruijt–Booker model

By simultaneously considering a uniform radial displacement and an ovalization of the tunnel lining structure, Verruijt and Booker [62] extended the method of Sagaseta [53] from incompressible soils to compressible soils. According to the Verruijt–Booker model, the horizontal displacement component $s_{x}$ and vertical displacement component $s_{z}$ of the ground are expressed as

$$s_{x} = - 2\varepsilon R^{2} x\left[ {\frac{1 - 2v}{{r_{2}^{2} }} - \frac{{2zz_{2} }}{{r_{2}^{4} }}} \right] - \frac{{2\delta R^{2} xH}}{{\left( {1 - v} \right)}}\left[ {\frac{{z_{2} \left( {1 - 2v} \right)}}{{r_{2}^{4} }} + \frac{{z\left( {x^{2} - 3z_{2}^{2} } \right)}}{{r_{2}^{6} }}} \right]$$

(11)

$$s_{z} = 2\varepsilon R^{2} \left[ {\frac{{2\left( {1 - v} \right)z_{2} }}{{r_{2}^{2} }} - \frac{{z\left( {x^{2} - z_{2}^{2} } \right)}}{{r_{2}^{4} }}} \right] - 2\delta R^{2} H\left[ {\frac{{x^{2} - z_{2}^{2} }}{{r_{2}^{4} }} + \frac{{zz_{2} \left( {3x^{2} - z_{2}^{2} } \right)}}{{\left( {1 - v} \right)r_{2}^{6} }}} \right]$$

(12)

where R is the tunnel diameter; $v$ is the soil Poisson’ ratio; $H = C + D/2$ is the depth to the spring line of the tunnel; $\varepsilon$ is the radial strain, from which the volume loss can be obtained by $V_{L} = 4\left( {1 - v} \right)\varepsilon$; $\delta$ is the ovalization parameter and the relative ovalization is defined by $\rho = \delta /\varepsilon$; $z_{2} = z + H$,$r_{2}^{2} = x^{2} + z_{2}^{2}$. Setting z = 0 in Eq. (12) leads to the ground surface settlement,

$$s_{z} \left( {z = 0} \right) = 4\varepsilon R^{2} \left( {1 - v} \right)\frac{H}{{x^{2} + H^{2} }} - 2\delta R^{2} \frac{{H\left( {x^{2} - H^{2} } \right)}}{{x^{2} + H^{2} }}$$

(13)

1.3 Loganathan–Poulos model

Considering that the actual soil behavior is nonlinear and the radial ground deformation around the tunnel section is not uniform, Loganathan and Poulos [32] improved the solution of Verruijt and Booker [62] by introducing an oval-shaped ground deformation at the soil-tunnel interface. In the Loganathan–Poulos model, the horizontal component $s_{x}$ and the vertical component $s_{z}$ of ground deformations are written as

$$s_{x} = - x\left\{ { - \frac{1}{{x^{2} + \left( {z - H} \right)^{2} }} + \frac{{\left( {3 - 4v} \right)}}{{x^{2} + \left( {z + H} \right)^{2} }} - \frac{{4z\left( {z + H} \right)}}{{\left[ {x^{2} + \left( {z + H} \right)^{2} } \right]^{2} }}} \right\}\frac{{4gR + g^{2} }}{4}{\text{exp}}\left\{ { - \left[ {\frac{{1.38x^{2} }}{{\left( {H + R} \right)^{2} }} + \frac{{0.69z^{2} }}{{H^{2} }}} \right]} \right\}$$

(14)

$$s_{z} = \left\{ { - \frac{z - H}{{x^{2} + \left( {z - H} \right)^{2} }} + \left( {3 - 4v} \right)\frac{z + H}{{x^{2} + \left( {z + H} \right)^{2} }} - \frac{{2z\left[ {x^{2} - \left( {z + H} \right)^{2} } \right]}}{{\left[ {x^{2} + \left( {z + H} \right)^{2} } \right]^{2} }}} \right\}\frac{{4gR + g^{2} }}{4}{\text{exp}}\left\{ { - \left[ {\frac{{1.38x^{2} }}{{\left( {H + R} \right)^{2} }} + \frac{{0.69z^{2} }}{{H^{2} }}} \right]} \right\}$$

(15)

where $g$ is the gap parameter defined by Lee et al. [27, 28], $g = G_{p} + U_{{{\text{3D}}}} + \omega$, $G_{p}$ representing the physical clearance between the outer skin of the shield and the lining, $U_{{{\text{3D}}}}$ representing the equivalent 3D elastoplastic deformation at the tunnel face and $\omega$ reflecting the quality of workmanship. In Loganathan–Poulos model, the gap parameter can be linked to the Volume loss through an approximated relation of $g = 2R\left( {\sqrt {1 + V_{L} } - 1} \right)$ [32, 75]. This relation is employed in this study in the subsequent calculations.

Setting z = 0 in Eq. (15) leads to the ground surface settlement as follows:

$$s_{z} \left( {z = 0} \right) = \left( {1 - v} \right)\left( {4gR + g^{2} } \right)\frac{H}{{x^{2} + H^{2} }}{\text{exp}}\left[ { - \frac{{1.38x^{2} }}{{\left( {H + R} \right)^{2} }}} \right]$$

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1.4 Gonzalez–Sagaseta model

Sagaseta [53] seems to be the first to derive a closed-form analytical solution to tunneling-induced ground movements using a virtual image technique in isotropic and homogeneous incompressible soils. This method was later improved by Gonzalez and Sagaseta [19] who assumed that the ground deformations can be considered as the superposition of three components, namely the ground loss caused by uniform radial contractions, the tunnel ovalization or distortion and the tunnel downward uniform movements. The corresponding horizontal component $s_{x}$ and vertical component $s_{z}$ of ground deformations in the Gonzalez–Sagaseta model are expressed by

$$\frac{{s_{x} }}{{2\varepsilon R\left( \frac{R}{H} \right)^{2\alpha - 1} }} = - \frac{{\overline{x}}}{{2\overline{{r_{1} }}^{2} \alpha }}\left( {1 - \rho \frac{{\overline{x}^{2} - \overline{{z_{1} }}^{2} }}{{\overline{{r_{1} }}^{2} }}} \right) - \frac{{\overline{x}}}{{2\overline{{r_{2} }}^{2} \alpha }}\left( {1 - \rho \frac{{\overline{x}^{2} - \overline{{z_{2} }}^{2} }}{{\overline{{r_{2} }}^{2} }}} \right) + \frac{{2\overline{x}\overline{z}}}{{\overline{{r_{2} }}^{2} \alpha }}\left( {\frac{{\overline{{z_{2} }} }}{{\overline{{r_{2} }}^{2} }} - \rho \frac{{\overline{x}^{2} - 3\overline{{z_{2} }}^{2} }}{{\overline{{r_{2} }}^{4} }}} \right)$$

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$$\frac{{s_{z} }}{{2\varepsilon R\left( \frac{R}{H} \right)^{2\alpha - 1} }} = - \frac{{\overline{{z_{1} }} }}{{2\overline{{r_{1} }}^{2} \alpha }}\left( {1 - \rho \frac{{\overline{x}^{2} - \overline{{z_{1} }}^{2} }}{{\overline{{r_{1} }}^{2} }}} \right) + \frac{{\overline{{z_{2} }} }}{{2\overline{{r_{2} }}^{2} \alpha }}\left( {1 + \rho \frac{{\overline{x}^{2} - \overline{{z_{2} }}^{2} }}{{\overline{{r_{2} }}^{2} }}} \right) - \frac{1}{{2\overline{{r_{2} }}^{2} \alpha }}\left[ {2\left( {\overline{z} + \rho } \right)\frac{{\overline{x}^{2} - \overline{{z_{2} }}^{2} }}{{\overline{{r_{2} }}^{2} }} + 4\rho \overline{z}\overline{{z_{2} }} \frac{{3\overline{x}^{2} - \overline{{z_{2} }}^{2} }}{{\overline{{r_{2} }}^{4} }}} \right]$$

(18)

where $\varepsilon$ is a radial contraction which can be calculated by $\varepsilon = \frac{{V_{{\text{L}}} }}{2}$, $\rho$ is the relative ovalization, $\alpha$ is a parameter accounting for the volumetric strains in the plastic phase, $\overline{x} = x/H$, $\overline{z} = z/H$, $\overline{{z_{1} }} = \overline{z} - 1$, $\overline{{z_{2} }} = \overline{z} + 1$, $\overline{{r_{1} }}^{2} = \overline{x}^{2} + \overline{{z_{1} }}^{2}$, $\overline{{r_{2} }}^{2} = \overline{x}^{2} + \overline{{z_{2} }}^{2}$. Please note that the value of $\alpha$ is set to 1.0 in clays (at least for undrained condition) and ranges between 1.0 and 2.0 for granular soils [19].

Setting z = 0 gives the ground surface settlement

$$s_{z} \left( {z = 0} \right) = 2\varepsilon R\left( \frac{R}{H} \right)^{2\alpha - 1} \frac{1}{{\left[ {1 + \left( {x/H} \right)^{2} } \right]^{\alpha } }}\left[ {1 + \rho \frac{{1 - \left( {x/H} \right)^{2} }}{{1 + \left( {x/H} \right)^{2} }}} \right]$$

(19)

1.5 Databases

See Tables 4, 5 and 6.

Table 4 The database of volume loss and settlement trough width

Table 5 The database of the radial strain $\varepsilon$ and the relative ovalization $\rho$

Table 6 The database of the gap parameter g