Back-analysis of a Rock Wedge Instability in the Ancient Parian Underground Marble Quarry

Abstract

The existence of multiple families of joints within the rock mass and the degradation of its properties create an additional risk of the collapse of peripheral rock wedges within a tunnel. Identifying and analyzing removable wedges is the first step in evaluating underground excavation safety. Then, the computation of the identified removable rock wedge’s safety factor (SF) follows. For this purpose, an analytical technique is applied for estimating safety factors in cases of falling or sliding wedges. The most dangerous wedge is identified by block theory, in which the geometric rules to locate removable wedges were implemented with the contribution of stereographic projections. The results, derived from the procedure above, are evaluated with Rocscience’s Unwedge software. SF calculation is made by using active/passive forces method for to define the strength properties of the famous marble of Parian Marble based on failed tetrahedral wedges in the ancient quarry in Nymphs. This paper aims to identify potential failure zones in this underground quarry using basic rock mechanics principles to estimate the necessary measures.

1 Introduction

Designing safe underground structures is challenging for mining engineers, especially with the complex works that a mining method demands. These works are either long term, such as access or ventilation shafts, inclines, or temporary, related to the development of each exploitation method. Numerical simulations are a common approach to analyzing the rock mass behavior in underground excavations. Simulations like finite elements [1, 2], distinct elements [3,4,5], or coupling different methods [6] are mainly applied. The complexity of the geometry, in addition to the parameters affecting the stability of the underground excavations, leads to a difficult-to-model problem. Empirical RMR rock mass classification ratings, Barton’s Q-system, and GSI are used in many analyses [7, 8] to quickly design the necessary support measures.

Apart from the degradation of the rock mass properties, joint systems increase the risk of creating removable wedges at the periphery or on the face of the opening, with the most dangerous being those near the roof. Other types of failures in underground openings are presented in Fig. 1 [9]. More frequent occurrences are shear failure, caving, or rapid failure of the rock mass, caused by a highly stressed brittle rock, local spalling of rock fragments, and flow of intensely fractured rocks or soils. These types of failures are characteristics in underground mining. For example, Majeed et al. [10] examine the strength of pillars in an underground salt mine in Pakistan.

Fig. 1

Failures in underground excavations, based on [9, 11]

The block theory [12] is crucial for identifying and analyzing removable wedges, used by many researchers, such as Yow and Goodman [13] for predicting the behavior of rock mass around a circular tunnel, Zhang [14] for stochastic wedge generation in an underground opening, Diederichs et al. [15] for hazard assessment of wedge instability, and Li et al. [16] for removable block identification based on laser point cloud images. The calculation of the wedge collapse/slip SF for the cases of a symmetric and non-symmetric prismatic wedge around a circular tunnel has been presented by Brady and Brown [17], using Kirsch’s analytical solution to calculate the restraint forces acting on the wedge. A similar technique was used by Sofianos [18] and Sofianos et al. [19], including the roughness of the surfaces of the discontinuities. The effect of soil–infilled joints on the stability of rock wedges appeared in a tunnel roof is examined analytically and numerically [20]. The three-dimensional calculation of the SF on a tetrahedral wedge is performed by estimating joint tractions and applying the Mohr–Coulomb (M-C) criterion [21]. Rocscience’s Unwedge program applies block theory to the geometric design of the most hazardous wedges and is used in various studies, such as (a) a hydroelectric project in India [22], (b) a pumped storage project located on the Rio Grande [23], and (c) the Kalatongke copper and nickel mine [24]. This methodology has also been generalized for verifying polyhedral wedges safety factor [25].

The primary objective of this paper is to propose a methodology to estimate dangerous failure zones based on observed wedge collapses in the underground excavation and in situ measurements. The ancient quarry of Nymphs, where the Parian marble (also known as Lychnitis) was extracted, is examined as a case study. Excavations in this quarry were initiated in the sixth century BC, and the produced Lychnitis or Parian Lithos was transported worldwide. Some of the most significant Greek sculptures are the Nike of Samothrace, the Hermes of Praxiteles, the Venus de Medici, and the Statue of Antinous was created on Parian marble. Regarding the status, there has been no quarrying activity since 1900 [26]. The block theory has been applied to the Nymph’s ancient marble quarry on Paros Island in Greece, using data available from the research by Marinos et al. [27]. Nymphs Quarry is a famous ancient excavation named after the bas-relief sculpture that was found in the cave of Agios Mua (Fig. 2); it could also be identified as a monument of the Greeks “Latomon” (quarrymen) and the special techniques they used to remove the marble blocks from the country rock. Underground mines like this are historical monuments to the evolution of underground mining and their restoration will increase visitor attendance that will contribute to the local economy. In the area operates a non-profit organization “Paros Ancient Marble Quarries Park” that promotes the protection of the ancient marble quarries in Marathi and the creation of an archeological and cultural park (https://parianmarble.com/). This paper is a first attempt to identify possible failure zones in Gallery I and propose steps and measurements needed for its restoration. Proposed steps and measurements could also be applied to several other ancient underground sites, making them safe and accessible. Subsequently, the complex potential technique [28,29,30,31] was used to estimate the stresses acting on the wedge surfaces for estimating safety factors by using an algorithm similar to Curran ‘s analysis [21].

Fig. 2

Hellenistic relief dedicated to nymphs (based on https://parianmarble.com, originates from [33])

In this paper, the basic principles of rock mechanics for fractured rock mass are used to quickly back-analyze the marble properties of this old quarry. Most existing studies use complex numerical algorithms that require time and effort to be properly adapted to the particular application. This article proposes using existing techniques for the rough assessment of risk areas, for immediate intervention, or to provide important information for a future more thorough investigation.

The applied method used here was compared with the results of the Unwedge program [32]. Back-analysis of Parian’s marble strength on the bedding plane was based on in situ shear strength properties measurements of the joints creating the two failed wedges of Gallery I, as discussed in Marinos et al. [27]. The proposed analysis includes the following:

• Block theory and stereographic projection are used not only to identify dangerous wedges but also to dimension the maximum removable wedges.

• Tetrahedral wedge surfaces are discretized with triangular elements.

• Analytical stress solution (complex potential method) for longitude tunnel is applied.

• Normal and shear forces acting on the joint’s surface are computed.

• The Safety Factor (SF) of the falling/sliding wedge is estimated.

• Back-analysis is applied in the ancient quarry for marble strength definition from critical SF of failed wedges.

• Estimation of zones of possible collapses based on back-analysis of wedges and involving finite element method (FEM).

2 Geology: Tectonics

Paros Island belongs to Cyclades group in the central Aegean Sea. The geology of Paros is dominated by marble, characterized by its fine-grained texture and its uniform white color, which is the result of the absence of impurities such as iron oxides and other minerals. The marble was quarried on the island from ancient times and was used to create a variety of sculptures, buildings, and other structures, including the famous Venus de Milo. A detailed mapping of the ancient quarry in Paros (Fig. 3) was created by Korre [26].

Fig. 3

Plan view of Nymphs quarry. The upper opening is Gallery I and the lower opening is Gallery II based on [26]

Paros Island is geologically structured by the gneiss and schists formations, while the Marmara (uppermost) nappe is characterized by metamorphic carbonate formations [34].

Regarding the geodynamic setting, Paros Island is in the back volcanic arc region [35], where extensional tectonics occur [36, 37], accompanied by onshore and offshore dip-slip normal faulting. However, active tectonics is limited within this area, which is confirmed by the low upper-crust deformation [38] and the lack of strong seismic events [39, 40]. Furthermore, the offshore fault analysis confirms the limited tectonic motions within the surrounding Paros Island region [41].

3 Block Theory

Block theory uses simple geometric rules to identify dangerous wedges in underground openings. The two fundamental theorems of block removability [12] are presented in Fig. 4:

Fig. 4

Block theory application in underground excavations: a five wedges formed around tunnel section, b finiteness for block pyramid—removability theorem for joint pyramids

The case study concerns the best known of the Marathi quarries, namely the one of the Nymphs, which takes its name from the votive relief carved into the natural stone (fourth century BC). The current status of the quarry of the Nymphs remains the same after the activities of the Belgian Company and the Greek Marble Company of Paros, which attempted to re-exploit the quarry in the nineteenth century [42]. Data are available from [27] for North Gallery I, with a length of 68 m and an average inclination of 20°. More specifically, data include the geometry of the tunnel cross-section with plane 90°/105° and the identification of three joint family systems and their bedding plane (Table 1).

Table 1 Joint orientation parameters

The two theorems of block theory are applied in this case study by implementing stereographic projections, where each possible Joint Pyramid (JP) is named by use four indexes XXXX representing the four joint planes. Each index takes the value 1 for describing the lower semi-half space of the respective joint and 0 for the upper. There are 2⁴ = 16 combinations of possible wedges around the tunnel for the four joints. If the JP, determined by the intersection of the circles of the joints, does not appear on the projection, then this wedge is not removable regardless of the opening alignment. If an excavation circle is added in JP, then the finite and infinite blocks around the tunnel could be defined. Based on these, the stereographic projection of the three joint planes, plus the bedding plane of the surrounding rock, can be created as shown in Fig. 5:

Fig. 5

The use of equal angle stereographic projections (upper pole projection) in Gallery I of Nymphs quarry

• Wedges 1111 (falling), 1011 (slip on J2), and wedge 0111 (slip on J1) can appear on the roof of the tunnel; these wedges are inside the reference circle (thick circle), which represents the space. In other words, the Block Pyramid is empty, and the wedges are finite. The wedge 1011 is the one that appeared in Fig. 6a. The mirror wedges 0000, 1000, and 0100 can form removable blocks on the floor. These wedges are safe, as their own weight helps to support them.

• Wedge 1100 can appear on the SW tunnel section, such as containing the tunnel axis. In this case, excavation made in the NE direction is not applicable. The mirror wedge 0011 could appear on the NE front.

• Wedges 1110, 1010, and 1000 can form removable blocks on the NW side, such as appearing in the projection entirely on the left side of the tunnel axis. The mirror wedges 0001, 0101, and 0111 can form removable blocks on the SE side. Wedge 0101 has slid into the upper right side of the tunnel (Fig. 6). Plane 3 does not appear on the design wedge (cut the wedge corner with plane 3).

• Blocks 1101 and 0010 are forming infinite wedges in all tunnel surfaces.

• Wedges 1001 and 0110 do not form a removable block on any tunnel surface (not appear on projection).

Fig. 6

Compare maximum wedges with actual. a Wedge failures along Gallery I (modified from [27]), b The geometry of the maximum wedges on the roof and the right side of the tunnel

It is necessary to eliminate joint 3 to obtain the tetrahedral wedges of Fig. 6a, which creates the larger wedge in the roof (1111). Due to the extensive contact surfaces and friction, it is safer than 1011 and 0101 (Fig. 5), while a wedge with an undefined joint plane exists in the image in the circle in Fig. 6a. This plane seems to be sub-vertical and cuts the tunnel’s axis vertically.

4 Stress Estimation

The adit (access drift) is considered horizontal despite its overall slope of 20° [26]. Moreover, the overburden height is 20 m at the entrance and 120 m at the back of the adit [26]. The studied wedge has been assumed in the center of the adit. The properties of the marble of Paros used are unit weight $\gamma =27\text{kN}/{\text{m}}^{3}$, modulus of elasticity E = 20 GPa, v = 1/3. At the same time, the analytical solution was derived by using lateral coefficient perpendicular to the tunnel’s axis ${k}_{x}=\frac{v}{1-v}=0.5$ and parallel to the tunnel’s axis ${k}_{y}=0$. Based on the above parameters, the average stress field around the tunnel is (for depth 73 m):

$$\begin{array}{l}{p}_{xx}=0.98 \text{MPa}\\ {p}_{zz}=1.96 \text{MPa}\\ {p}_{yy}=0 \text{MPa}\end{array}$$

(1)

For the calculation of stresses around a tunnel of any cross-section, the method of complex potential functions is employed. This starts with the use of the conformal transformation of the cross-section into an equivalent unitary inner circle (disk), where any point of the medium $w$ corresponds to the point of the plane $\zeta$ based on the following transformation:

$$w=\omega \left(\zeta \right)=\frac{{a}_{0}}{\zeta }+\sum_{k=1}^{n}{a}_{k}\bullet {\zeta }^{k}$$

(2)

where:

$w=x+i\bullet z$:

actual coordinates of plane vertical into tunnel axis,

$\zeta =\rho {e}^{-i\theta }, 0<\rho \le 1$:

conformal plane polar coordinates ($\rho$, $\theta$),

$i=\sqrt{-1}$:

imaginary unit,

${a}_{0}$:

real constant and ${a}_{1}, {a}_{2},\dots , {a}_{n}$ complex constants of conformal mapping.

Based on the theory of Muskhelisvili [31], the unit disk is used by using the complex potential functions:

$$\begin{array}{c}\phi \left(\zeta \right)=\Gamma \omega \left(\zeta \right)+{\phi }_{0}\left(\zeta \right)\\ \psi \left(\zeta \right)={\Gamma }^{\prime}\omega \left(\zeta \right)+{\psi }_{0}\left(\zeta \right)\end{array}$$

(3)

where $\Gamma =\frac{1}{4}\left({p}_{xx}+{p}_{zz}\right), {\Gamma }^{\prime}=-\frac{1}{4}\left({p}_{xx}-{p}_{zz}\right)$

The complex potential functions ${\phi }_{0}\left(\zeta \right)$ and ${\psi }_{0}\left(\zeta \right)$ are defined by solving Cauchy integrals based on the boundary conditions ${f}_{0}\left(\sigma \right)$ for every point $\sigma$ belong to the boundary of the disk $\gamma$.

$$\begin{array}{c}{\phi }_{0}\left(\zeta \right)+\frac{1}{2\pi i}{\int }_{\gamma }\frac{\omega \left(\sigma \right)}{\overline{\omega^{\prime}\left(\sigma \right)}}\frac{\overline{{\phi^{\prime}}_{0}\left(\sigma \right)}}{\sigma -\zeta }d\sigma =\frac{1}{2\pi i}{\int }_{\gamma }\frac{{f}_{0}\left(\sigma \right)}{\sigma -\zeta }d\sigma \\ {\psi }_{0}\left(\zeta \right)=\frac{1}{2\pi i}{\int }_{\gamma }\frac{\overline{{f }_{0}\left(\sigma \right)}}{\sigma -\zeta }d\sigma -\frac{1}{2\pi i}{\int }_{\gamma }\frac{\overline{\omega \left(\sigma \right)}}{\omega^{\prime}\left(\sigma \right)}\frac{{\phi^{\prime}}_{0}\left(\sigma \right)}{\sigma -\zeta }d\sigma - \overline{{\phi }_{0}\left(0\right)}\end{array}$$

(4)

where ${f}_{0}\left(\sigma \right)=-2\Gamma \omega \left(\sigma \right)-\overline{\Gamma ^{\prime}\omega \left(\sigma \right)}$

If the above integrals are solved, the final solution of the elastic stresses at any point inside the unitary disk $(\rho , \theta )$ and global cartesian systems $(x, z)$, are defined as follows:

$$\begin{array}{c}{\sigma }_{\rho }+{\sigma }_{\theta }={\sigma }_{xx}+{\sigma }_{zz}=4\mathfrak{R}\left(\frac{{\phi }^{\prime}{\left(\zeta \right)}}{{\omega }^{\prime}{\left(\zeta \right)}}\right)\\ {\sigma }_{\rho }-i{\tau }_{\rho \theta }=2\mathfrak{R}\left(\frac{{\phi}^{\prime}{\left(\zeta \right)}}{{\omega }^{\prime}{\left(\zeta \right)}}\right)-\frac{{\zeta }^{2}}{{\rho }^{2}\overline{{\omega }^{\prime}{\left(\zeta \right)}}}\left(\overline{\omega \left(\zeta \right)}\frac{{\phi }^{{\prime}{\prime}}\left(\zeta \right){\omega }^{\prime}\left(\zeta \right)-{\phi }^{\prime}{\left(\zeta \right)}{\omega }^{{\prime}{\prime}}{\left(\zeta \right)}}{{{\omega }^{{\prime}{2}}}\left(\zeta \right)}+{\psi }^{\prime}\left(\zeta \right)\right)\\ {\sigma }_{zz}-i{\tau }_{xz}=2\mathfrak{R}\left(\frac{{\phi }^{\prime}{\left(\zeta \right)}}{{\omega }^{\prime}{\left(\zeta \right)}}\right)-\frac{1}{{\omega}^{\prime}{\left(\zeta \right)}}\left(\overline{\omega \left(\zeta \right)}\frac{{\phi }^{{\prime}{\prime}}\left(\zeta \right){\omega }^{\prime}\left(\zeta \right)-{\phi }^{\prime}{\left(\zeta \right)}{\omega }^{{\prime}{\prime}}{\left(\zeta \right)}}{{{\omega }^{{\prime}{2}}}\left(\zeta \right)}+{\psi }^{\prime}\left(\zeta \right)\right)\end{array}$$

(5)

The cross-section of the adit (Fig. 6b) was approximated using a horseshoe shape with dimensions: width 3.4 m, height 4.7 m, and the upper part simulated with a semicircular arc with a radius of 1.7 m. The optimal conformal transformation is approximated using the algorithm presented in [28]. Based on this analysis, the above constants $a$ (Table 2) are estimated by using $n=7$ terms in Eq. (2):

Table 2 Conformal mapping parameters

The terms used to approximate the opening in conformal mapping play a vital role in the distribution of stresses around the opening. Intense stress concentration is expected in areas with abrupt geometry change, such as right angles at the floor of the cross-section where the radius of curvature is theoretically zero. In the conformal transformation, the radius of curvature ${r}_{0}$ around the tunnel cross-section can be calculated according to Eq. (6) [43]:

$${r}_{0}=\frac{{\left({x^{\prime}}^{2}\left(\theta \right)+{z^{\prime}}^{2}\left(\theta \right)\right)}^{3/2}}{x"\left(\theta \right)\bullet z^{\prime}\left(\theta \right)-x^{\prime}\left(\theta \right)\bullet z"\left(\theta \right)}$$

(6)

The Cartesian coordinates at the periphery of the tunnel (ρ = 1) are obtained from Eq. (2) by substituting $\zeta =cos\theta -i\bullet sin\theta$.

The tunnel cross-section derived with 7 terms ($n=7$) in Fig. 7a is transformed to a unit disk in Fig. 7b, with the smallest radius of curvature occurring at the right corner of the tunnel’s floor with r₀ = 0. 159 m (Fig. 7c). At this point, there is also the maximum concentration of tangential stresses (Fig. 7d) with the value ${\sigma }_{\theta max}=10.3 MPa$.

Fig. 7

Conformal mapping by using $\mathit{n}=7$ terms a in w plane, b in $\upzeta$ plane, c minimum curvature position, and d tangential stress distribution around tunnel surface

Forty-five points (black dots in Fig. 7c) of the circumference at equal distances were used to approximate the tunnel cross-section. Based on the conformal mapping, it was observed that the arch cross-section is approximated by using more than three terms. In this analysis, up to seven (7) terms were considered, as more terms would require denser discretization of the aperture to upgrade the approximation significantly, also:

• 1. The cross-section geometry approximates the actual cross-section.

• 2. Using too many terms would result in a curvature radius close to zero resulting in very high stresses leading to fracture of this region (resulting in a larger curvature radius).

• 3. Using too many terms ($n>15$) can create numerical instabilities away from the tunnel’s surface ($\zeta \ll 1\Rightarrow {\zeta }^{-n}\to \infty$).

5 Wedge Stability

5.1 Safety Factor Estimation

A commonly used technique applied for the estimation of the Safety Factor of rock wedges is Rocscience’s Unwedge program [32], the initial stage requires the calculation of the normal and shear stresses in each of the triangular leaf elements created through the process of discretization of the three joints of the tetrahedral wedge using Cauchy’s relationship:

$${{\varvec{t}}}_{i}={{\varvec{\sigma}}}_{ij}\bullet {{\varvec{n}}}_{j}\to \left\{\begin{array}{l}{\sigma }_{n}={{\varvec{t}}}_{j}{\bullet {\varvec{n}}}_{j}\\ {\sigma }_{sA}={{\varvec{t}}}_{j}{\bullet {\varvec{m}}}_{j}\\ {\sigma }_{sB}={{\varvec{t}}}_{j}\bullet {{\varvec{k}}}_{j}\end{array}\right.$$

(7)

where ${{\varvec{t}}}_{j}$, stands for traction vector, ${n}_{i}$ the outward unitary normal vector, ${m}_{i}$ is the unitary vector in the dip direction and ${{\varvec{k}}}_{l}={{\varvec{n}}}_{i}\times {{\varvec{m}}}_{j}={{\varvec{\varepsilon}}}_{lij}\bullet {{\varvec{n}}}_{i}\bullet {{\varvec{m}}}_{j}$, where ${{\varvec{\varepsilon}}}_{lij}$ stands for the Levi-Chivita permutation symbol.

The initial forces acting on the surface (Fig. 8) of the wedge will be calculated using the complex potential method. The forces acting on the wedge will be calculated by applying Eq. (7) to the centers of gravity and the area of the triangles, assuming constant stress on the triangle element.

Fig. 8

Discretized wedge with triangular elements to estimate initial forces acting on the wedge (based on [21])

Based on this analysis, the safety factor of the sliding wedge can be found from the passive and active forces method by dividing strength in the direction of wedge motion with forces acting in this direction. More specifically, the strength characterizing each element is defined as the inner product of (a) the shear strength developed along the ${{\varvec{m}}}_{i}$ direction and (b) the tensile strength developed along the normal direction ${{\varvec{n}}}_{i}$, with the vector $\widehat{{\varvec{s}}}$ coincide with ${{\varvec{m}}}_{i}$ with the lowest strength for the case of sliding and with the vertical axis $\widehat{{\varvec{z}}}$ for the case of falling. The forces acting on the wedge are estimated based on the tractions acting on each element (Eq. (7)) and the wedge self-weight as follows:

$$SF=\frac{\sum_{i=1}^{3}\sum_{j=1}^{li}{a}_{ij}\bullet \left\{\left({c}_{ij}+{\sigma }_{nij}\bullet tan{\phi }_{ij}\right)\bullet {{\varvec{m}}}_{i}-{\sigma }_{tij}\bullet {{\varvec{n}}}_{i}\right\}\bullet \widehat{{\varvec{s}}}}{\left\{W\bullet \widehat{{\varvec{z}}}+\sum_{i=1}^{3}\sum_{j=1}^{li}{a}_{ij}\bullet \left({{\sigma }_{sAij}\bullet {\varvec{m}}}_{i}+{{\sigma }_{sBij}\bullet {\varvec{k}}}_{i}-{\sigma }_{nij}\bullet {{\varvec{n}}}_{i}\right)\right\}\bullet \widehat{{\varvec{s}}}}$$

(8)

$i=\text{1,2},3$:

wedge joint number index,

$j=\text{1,2}\cdots ,{l}_{i}$:

element index for joint iwith l_i triangles,

${c}_{ij}$, ${\phi }_{ij}$:

shear strength parameters,

${\sigma }_{tij}$:

tensile strength,

${{\varvec{m}}}_{i}$, ${{\varvec{k}}}_{i}$, ${{\varvec{n}}}_{i}$:

are sliding and normal unitary vectors of joint i

$\widehat{{\varvec{s}}}$:

sliding/falling direction vector,

$\widehat{{\varvec{z}}}=[\text{0,0},-1]$:

vertical direction vector,

${a}_{ij}$:

triangle j (of joint i) area and

${\sigma }_{nij}$:

normal stress

${\sigma }_{sAij}, {\sigma }_{sBij}$:

shear stresses derived from traction vector (Eq. (7)).

Additional constraints must be applied for each element to check that either shear or tensile strength is exceeded, that means that the specific element lose the cohesion forces [44]:

$$\begin{array}{c}{\sigma }_{nij}<0{\to {\phi }_{ij}=0, -\sigma }_{nij}\ge {\sigma }_{tij}\to {\sigma }_{tij}={\sigma }_{nij}=0, {c}_{ij}=0\\ \Vert {\sigma }_{sAij}\Vert ,\Vert {\sigma }_{sBij}\Vert \ge {c}_{ij}+{\sigma }_{nij}\bullet tan{\phi }_{ij}\to {\sigma }_{tij}=0, {c}_{ij}=0\end{array}$$

(9)

Finally, for the case of the sliding wedge for the joint of sliding in shear strength component, the weight component (denoted by $W)$ must be added to the normal stress vector:

$${\sigma }_{nij}={\sigma }_{nij}+\frac{W}{\sum_{j=1}^{ki}{a}_{nij}}\bullet \widehat{{\varvec{z}}}\bullet {{\varvec{n}}}_{i}$$

(10)

5.2 Parian Marble Weak Plane Strength Estimation

According to the field research conducted by Marinos et al. [27], and the removable wedges identified by the use of block theory (Figs. 5 and 6), the two wedges that failed are:

• a) the wedge on the tunnel roof, which is identified as 1011 and is characterized by a potential slide along joint plane 2 and

• b) the wedge on the SE side of the tunnel, which is identified as 0101 and is characterized by a potential slide along joint plane 1.

Correspondingly, tetrahedral wedges are formed by excluding joint plane 3, which intersects them, forming smaller volume wedges. The block theory was used to size the wedges, using the maximum possible dimensions forming removable blocks. The joint shear strength properties are developed based on [27] and include the Joint Compression Strength JCS = 80–100 MPa based on Schmidt hammer measurements, the residual friction angle assumed typical values range for marbles $\phi ={20-40}^{o}$, and the roughness of the joints measured as JRC = 10–12. Based on these properties and the nonlinear Barton-Bandis model [45,46,47], the friction along the sides of each triangle by assuming cohesionless the joints 1 and 2 (except the bedding plane) is:

$${\phi }_{ij}=\phi +JRC\bullet {log}_{10}\left(\frac{JCS}{{\sigma }_{nij}}\right)$$

(11)

The two failed wedges were defined by block theory and designed in AutoCAD (Fig. 9a). The side wedge is referred to as (011) and the roof wedge is referred to as (101) in the Unwedge program (Fig. 9b). The tangential stresses distribution around the tunnel section is presented in Fig. 9c, with normal forces acting on the two wedges like that estimated Fig. 9b.

Fig. 9

Forming maximum wedges 101 and 011 a in AutoCAD Civil, b in Unwedge, and c normal forces acting on wedge

The steps for the analytical calculations conducted for the three sides of each wedge are presented below:

• Form potentially unstable tetrahedral wedges by using block theory and discretize with triangular elements.

• The roof wedge 101 presented with blue in Fig. 9a, has a volume ${V}_{1}=0.268 {\text{m}}^{3}$, and the surface of three joints are ${A}_{1}=1.47 {\text{m}}^{2}$, ${A}_{2}=0.36 {\text{m}}^{2}$ and ${A}_{3}=1.09 {\text{m}}^{2}$, respectively.

• The side wedge 011 presented with green in Fig. 9a, has a volume ${V}_{2}=0.483 {\text{m}}^{3}$, and the surface of three joints is ${B}_{1}=2.31 {\text{m}}^{2}$, ${B}_{2}=1.11 {\text{m}}^{2}$ and ${B}_{3}=0.89 {\text{m}}^{2}$, respectively.

• Define a conformal mapping variable $\zeta$ based on triangle centroid coordinates $w$ by finding the root of the polynomial with $\rho =\Vert \zeta \Vert \le 1$:

  $${a}_{0}-w\bullet \zeta +{a}_{1}\bullet {\zeta }^{2}+{a}_{2}\bullet {\zeta }^{3}+\cdots +{a}_{k}\bullet {\zeta }^{k+1}=0$$

  (12)

• Compute the stress tensor using Eq. (1) and estimating tunnel axis normal stress by applying general plain strain modification:

  $${\sigma }_{yy}={p}_{yy}+v\bullet \left({{\sigma }_{xx}+\sigma }_{zz}-{p}_{xx}-{p}_{zz}\right)$$

  (13)

• Define three vectors ${{\varvec{n}}}_{i}$, ${{\varvec{m}}}_{{\varvec{i}}}$, and ${{\varvec{k}}}_{{\varvec{i}}}$ based on dip direction, dip angle, and tunnel axis (see Appendix).

• Define the normal and shear components of traction vector for each triangle. For the wedge 101, the total normal and shear forces are:

  $$\begin{array}{c}{N}_{1}=1211 \text{kN},{N}_{2}=86\text{ kN}, {N}_{3}=1865 \text{kN} \\ {S}_{1}=1281 \text{kN},{S}_{2}=171\text{ kN}, {S}_{3}=567 \text{kN}\end{array}$$

  (14)

For the side wedge (011):

$$\begin{array}{c}{N}_{1}=672 \text{kN},{N}_{2}=83 \text{kN}, {N}_{3}=1012 \text{kN} \\ {S}_{1}=1543\text{ kN},{S}_{2}=-157 \text{kN}, {S}_{3}=1210 \text{kN}\end{array}$$

(15)

• Apply the restrictions of Eq. (9) for each triangle by defining the strength parameters of each joint: $\phi =20-{40}^{0}$ for all joints, $c=0$, ${\sigma }_{t}=0$, $JRC=10-12$, and $JCS=80-100$ MPa for joint 1 and 2. The cohesion for the bedding plane is estimated from back analysis of unstable wedges, while tensile strength is defined from Mohr-Coulomb criterion as:

  $${\sigma }_{t3}=\frac{2\bullet {c}_{3}\bullet cos\phi }{1+sin\phi }$$

  (16)

• Change ${c}_{3}$ for the roof wedge for limit $SF=1$ for two roughness parameters cases, $JRC=10$, $JCS=80$ MPa and $JRC=12$, $JCS=100$ MPa.

The above analysis shows that the side wedge has a lower factor of safety than the roof wedge, so the critical strength parameters are defined based on the roof wedge (101) and presented in Fig. 10a, b.

Fig. 10

Bedding plane strength parameters as a function of friction angle a cohesion, b tensile strength

The Unwedge software uses the shear strength (circles in Fig. 11) to estimate the safety factor [44]. This solution is compared to the respective method (red line) by using the stresses defined by the complex potential method and the solution of Eq. 6 (blue line) in Fig. 11. The parameters used for the estimation of the safety factor are:

• $c=0$, ${\sigma }_{t}=0$ for all planes,

• ${\phi }_{ij}$ friction angle defined by use Eq. (11) for the 1 and 2 joints,

• $\phi$ angle is used for joint 3.

Fig. 11

Comparison of the safety factor calculated with the modified equation and Unwedge code

Safety factor estimations based on the stress solution of the complex potential method and by using the shear strength approach, coincide with Unwedge estimations. On the other hand, safety factors estimations when using forces acting in the direction of motion are much lower than the ones obtained using the shear strength method.

6 Stability Analysis

6.1 Analytical Rough Estimation

Based on the above analysis, the properties of the weak plane of the marble of Paros on the exposed surface of the ancient excavation were estimated. The next step is to estimate failure regions around the cross-section based on the stress distribution around the opening in Fig. 9c. Based on the strength of the intact rock 80–100 MPa and the stress field around the tunnel, it is evident that the intact rock does not fail. It is then necessary to consider the failure at the weak plane, where the strength is assumed based on the Mohr–Coulomb criterion with a limit cycle shown in Fig. 12b. Figure 12 a shows the manner of the calculation of the angle $\beta$ of the plane of weakness with the plane of maximum principal stress using the pole P in Mohr’s circle.

Fig. 12

Weak plane strength estimation a shear stress acting on weak plane, b critical Mohr circles with angle $\upbeta$

Based on the properties of the plane of weakness and considering the friction angle ${\varphi }_{w}={40}^{o}$, which is a typical friction value of marbles, the cohesion ranges from ${c}_{w}=0-260$ kPa so based on the analysis of Fig. 12b, the above relation for $\beta \in \left({\varphi }_{w},90\right)$ [17]:

$${\sigma }_{1max,intact}\ge {\sigma }_{1max}=\frac{2\bullet \left({c}_{w}+{\sigma }_{3}\bullet tan{\varphi }_{w}\right)}{\left(1-\frac{tan{ \varphi }_{w}}{tan\beta }\right)\bullet sin\left(2\bullet \beta \right)}\ge 0$$

(17)

6.2 Numerical Simulation

The use of finite element methods is a widespread technique for calculating instabilities in mines [48,49,50,51,52,53]. In the present study, the stability of the underground opening was investigated through the numerical simulation technique, utilizing the PLAXIS 2D Version 2016 software and the Jointed rock model. The model is an anisotropic elastic perfectly plastic model, especially designed to simulate the behavior of stratified and jointed rock layers. The intact rock is considered to behave as a transversely anisotropic elastic material, and in the major joint directions, it is assumed that shear stresses are limited according to Coulomb’s criterion. When the maximum shear stress is reached, plastic sliding occurs [54]. The joint sets are parallel, not filled with fault gouge, and their spacing is small compared to the characteristic dimension of the underground structure [54].

The creation of the geometry model representing the underground opening is based on the cross-section depicted in Fig. 6. The generated model guarantees that stresses and deformations around the opening are not significantly affected by boundary conditions by setting the boundaries 20 m and 26 m away from the center of the excavation in horizontal and vertical direction, respectively. After inputting the material specifications, a finite element mesh was generated using the Plaxis software. Plane strain 15-node triangular elements were used, and the created mesh is deemed to be capable of providing a reliable analysis (Fig. 13). The initial stage of calculation involves gravity loading, while the second stage involves the deactivation of the mesh clusters (removing material) to simulate the evolution of the opening. A vertical load equivalent to 48 m of overburden was applied to the upper boundary of the model. It should be mentioned that the weak bedding plane direction is marked with red lines in Fig. 13, where the spacing of these lines is not to scale.

Fig. 13

Numerical anisotropic model after the material extraction

6.3 Comparing the Numerical and the Analytical Solution

The analytical solution indicates that there are two zones of potential slip that on the upper right side and at the crown. These zones are represented with brown color in Fig. 14a, at the weakness plane. The presence of these zones could lead to rock detachments/collapses (top of Fig. 14a), while these zones are larger if it is assumed that the bedding plane has lost its cohesion around the tunnel due to long-term weathering (lower of Fig. 14a). Also, there is a zone of instability in the lower left section that may have led to the failure in Fig. 6a that has been noted within the circle.

Fig. 14

Instabilities zones a estimated from analytical calculations and b from numerical model with c_w = 260 kPa and for cohesionless bedding plane, respectively

In the case of a cohesionless bedding plane (lower of Fig. 14a), there is also a failure zone in the tunnel floor that does not involve the risk of detachment but can lead to large deformations of the tunnel. Finally, it should be noted that the above risks increase at deeper parts of the tunnel when the overburden depth increases, which means that also the stresses around the opening will be larger.

A comparison with the numerical solution yields similar outcomes for the dangerous failure zones. Figure 14b displays the calculated incremental shear strains, which clearly delineate the boundary between the area with no displacement (zero value) and the area with displacements (non-null values). The structural deformation localization pattern validates the findings obtained from the analytical solution. Specifically, a higher degree of displacement is observed at the lower left edge of the opening.

Regarding the case of geological formation with c_w = 260 kPa, φ_w = 40° and angle of inclination a = 30° the maximum relative shear stress locally approaches unity at the lower left edge of the model. This indicates shearing zones of a local character that probably cause small-scale differential movements. In contrast, in the case where the cohesion of a geological formation is zero, mass failure results in large-scale movements.

Besides the qualitative comparison of the results of Fig. 14 for the unstable zones observed in analytical and numerical approximations, there is also a satisfactory convergence of the results of the tangential stresses (Fig. 15) compared at the periphery of the opening. In detail, the anisotropy of the material in the numerical solution leads to higher stresses on the left side of the opening (180°) as in this region, it is more difficult to show slip in the bedding plane as clearly shown in the four solutions of Fig. 14. On the other hand, the solutions for cohesion c_w = 260 kPa (marked with “o” in Fig. 15) gives higher stresses around the opening compared to the case of zero cohesion (marked with “x” in Fig. 15), as expected as in the second case a failure was detected in the numerical model indicating that in the specific zones in the lower Fig. 14b slip in the plane of weakness is present. In the numerical model with cohesion, no failure was observed, indicating no slips in this case. It is noted that the presence or absence of slip zones affects the stress distribution as the material properties are locally altered (following the ideal plasticity based on the M-C criterion), which eventually leads to the relaxation of the stresses around the opening.

Fig. 15

Comparison of analytical and numerical solutions around the tunnel boundary

7 Discussion

Two-dimensional wedge analysis around underground excavations using the forces acting on the wedge surfaces is not an easy task. Estimating the forces in the case of tetrahedral wedges is even more challenging, as discontinuities modify the stress field around the excavation. The stress relaxation technique has been proposed by Brady & Brown [17] to include the reduction of wedge restraint forces during movement. However, these approaches are for prismatic wedges and are difficult to apply in three dimensions. Based on the above, a complete solution to the problem would require using a three-dimensional numerical code with the possibility of introducing surfaces using contact elements to determine the safety factor accurately. This paper uses rock mechanics fundamentals to quickly (roughly) estimate the strength parameters of the marble and identify areas of weakness in the old underground excavation. The findings of the simple analysis proposed here are verified using a 2D numerical model designed using the commercial finite element software Plaxis. It must be noted that Plaxis has the option of 3D modeling, but more extensive analysis and field measurements are needed to support a 3D model. Also, this option is out of the scope of this paper, and based on the available data, simplifying assumptions that already exist on a 2D model have to be made.

The present analysis helps highlight the risks of the collapse of the ancient cave beyond the already collapsed wedges so that measures can be implemented to support the historic mine in the future. Various techniques of using anchors exist in supporting underground openings, such as the method used in the USA [55], while other techniques include the use of wooden or metal supports, hydraulic support systems, use of shotcrete, etc. [56]. Choosing the most appropriate support measure can be difficult, but today, modern tools can help in the optimal choice, such as the method presented by Gligorić et al. [57] and the new support system that simultaneously can monitor the rock motion [58]. It should be noted that in historic caves and mines, the choice becomes even more difficult as there is a need not to disturb the esthetics. In the future, an archeological and cultural park will be created by the co-operations of the Ministry of Culture and the Municipality of Paros (https://parianmarble.com/). In addition, the surfaces are chronically exposed to natural weathering, which makes the excavation more unstable. In future works, a more detailed analysis could be designed to assess the probability of failure and propose support measures.

8 Conclusions

The Parian marble is famous around the world, and the examined ancient quarry is a touristic attraction in Paros Island. In this paper, a three-dimensional model of active and passive forces is used to calculate the SF of wedges that occur at the tunnel roof and periphery. The derived equation, based on forces acting in the direction of wedge motion, gives lower estimations of safety factor compared to the analysis based on the Unwedge program which considers the shear strength of the wedge. These two techniques are more appropriate than the relaxation technique for the two non-symmetric tetrahedral wedges and were implemented in the ancient underground quarry of Paros, where two failed wedges were identified. The bedding plane strength of marble is estimated based on available sources from the international literature, in situ measurements of roughness and strength parameters of the joints.

Based on the present stability analysis, protection measures are essential to protect this archeological monument from future collapse. The main objective of the present work is to develop a rough estimation of marble properties based on minimum available data. Based on simple rock mechanics principles combined with numerical solution approximations, potential zones of failures are defined in Gallery I adit. These failure zones are on the crown and in the left upper side of the horseshoe-shaped tunnel section.

Appendix

1.1 Vector Calculation

Normal vector outward the tetrahedral wedge surface estimated by subtracting from direction angles the tunnel axis azimuth angle of 15°, resulting for the roof wedge (101):

$$\begin{array}{c}{{\varvec{n}}}_{1}=\left[\begin{array}{ccc}sin80\bullet sin247& sin80\bullet cos247& cos80\end{array}\right]\\ {{\varvec{n}}}_{2}=-\left[\begin{array}{ccc}sin85\bullet sin165& sin85\bullet cos165& cos85\end{array}\right]\\ {{\varvec{n}}}_{3}=\left[\begin{array}{ccc}sin35\bullet sin70& sin35\bullet cos70& cos35\end{array}\right]\end{array}$$

(18)

While the vector in dip direction:

$$\widehat{{\varvec{s}}}=\begin{array}{c}{{\varvec{m}}}_{1}=\left[\begin{array}{ccc}cos80\bullet sin247& cos80\bullet cos247& -sin80\end{array}\right]\\ {{\varvec{m}}}_{2}=\left[\begin{array}{ccc}cos85\bullet sin165& cos85\bullet cos165& -sin85\end{array}\right]\\ {{\varvec{m}}}_{3}=\left[\begin{array}{ccc}cos35\bullet sin70& cos35\bullet cos70& -sin35\end{array}\right]\end{array}$$

(19)

The sliding direction defined by dip direction of second joint.

Third vector is defined by using external product of vectors ${{\varvec{n}}}_{{\varvec{i}}}$ and ${{\varvec{m}}}_{i}$:

$$\begin{array}{c}{{\varvec{k}}}_{1}={{\varvec{n}}}_{1}\times {{\varvec{m}}}_{1}=\left[\begin{array}{ccc}-cos247& sin247& 0\end{array}\right]\\ {{\varvec{k}}}_{2}={{\varvec{n}}}_{2}\times {{\varvec{m}}}_{2}=\left[\begin{array}{ccc}cos165& -sin165& 0\end{array}\right]\\ {{\varvec{k}}}_{3}={{\varvec{n}}}_{3}\times {{\varvec{m}}}_{3}=\left[\begin{array}{ccc}-cos70& sin70& 0\end{array}\right]\end{array}$$

(20)

For the side wedge (011), normal outward vectors are defined by:

$$\begin{array}{c}{{\varvec{n}}}_{1}=-\left[\begin{array}{ccc}sin80\bullet sin247& sin80\bullet cos247& cos80\end{array}\right]\\ {{\varvec{n}}}_{2}=\left[\begin{array}{ccc}sin85\bullet sin165& sin85\bullet cos165& cos85\end{array}\right]\\ {{\varvec{n}}}_{3}=\left[\begin{array}{ccc}sin35\bullet sin70& sin35\bullet cos70& cos35\end{array}\right]\end{array}$$

(21)

The sliding direction for the side wedge is the first joint, while the third joint plane points upward so the opposite vector from di direction is used:

$$\begin{array}{c}{\widehat{{\varvec{s}}}={\varvec{m}}}_{1}=\left[\begin{array}{ccc}cos80\bullet sin247& cos80\bullet cos247& -sin80\end{array}\right]\\ {{\varvec{m}}}_{2}=\left[\begin{array}{ccc}cos85\bullet sin165& cos85\bullet cos165& -sin85\end{array}\right]\\ {{\varvec{m}}}_{3}=-\left[\begin{array}{ccc}cos35\bullet sin70& cos35\bullet cos70& -sin35\end{array}\right]\end{array}$$

(22)

Third vector defined as previous:

$$\begin{array}{c}{{\varvec{k}}}_{1}={{\varvec{n}}}_{1}\times {{\varvec{m}}}_{1}=\left[\begin{array}{ccc}cos247& -sin247& 0\end{array}\right]\\ {{\varvec{k}}}_{2}={{\varvec{n}}}_{2}\times {{\varvec{m}}}_{2}=\left[\begin{array}{ccc}-cos165& sin165& 0\end{array}\right]\\ {{\varvec{k}}}_{3}={{\varvec{n}}}_{3}\times {{\varvec{m}}}_{3}=\left[\begin{array}{ccc}cos70& -sin70& 0\end{array}\right]\end{array}$$

(23)