Geomechanical analysis of the stability conditions of shallow cavities for Compressed Air Energy Storage (CAES) applications

Abstract

Compressed Air Energy Storage (CAES) systems compress air into underground cavities when there is an excess of energy production (e.g., in the electrical grid or in an electrical plant) and generate electrical energy using a turbine when the electricity demand exceeds the production. Underground air storage requires construction of new underground cavities or reconditioning of existing underground openings. This paper presents a study of geomechanical stability of shallow circular cavities carried out as part of a multidisciplinary project that investigated the feasibility of using existing underground mining works (drifts and shafts) from iron mining works dating back from the first half of the twentieth century in northern Minnesota (USA). The paper addresses the fundamental problem of establishing the stability conditions of shallow cylindrical or spherical openings excavated in cohesive ground, and subjected to either decreasing or increasing internal pressure, associated with the process of contraction or expansion of the cavities during operation of a CAES system. A statically admissible analytical model for a shallow circular opening in cohesive ground derived from the limit analysis lower bound theorem is presented, and key dimensionless groups of variables controlling the stability of the cavity, defined in terms of a scalar factor of safety, are identified. The analytical model allows several observations of practical interest to be made with regard to the stability of shallow cavities. Numerical finite-difference models are used to validate the various observations and to quantify the underestimation of factors of safety obtained with the proposed lower bound solution. The paper also presents a critical evaluation of limit equilibrium (Terzaghi’s type) models that are traditionally used to design cavities for gas and air storage. Comparisons of results obtained with existing limit equilibrium models, with the proposed analytical model and with numerical models, show that limit equilibrium models can lead to both over conservative (i.e., too safe or uneconomical) and to nonconservative (i.e., unsafe) cavity designs depending on the ranges of values considered for the dimensionless groups of variables governing the problem. Finally, the effect of various parameters such as water in the ground, frictional strength and others, on the stability of the cavity are discussed.

Appendices

Appendix 1: Derivation of the analytical solution for contracting and expanding cavities in dry cohesive ground

This appendix presents the derivation of the analytical solution for contracting cavities (Eqs. 12 through 15 and for expanding cavities (Eqs. 12 and 13; and Eqs. 17 and 18) in dry cohesive ground, using the scaling rules introduced by Eqs. (5) through (9).

Figure 29, which is equivalent to Fig. 6 discussed in the main text, shows the problem and variables involved in the derivation presented in this appendix.

Considering a polar (or spherical) coordinate system, $(\,\rho , \theta )$, the force equilibrium equations inside the integration circle in Fig. 29  that account for the self-weight of the material are partial differential equations involving the (scaled) radial and tangential (or hoop) stresses, $\tilde{\sigma }_{r}$ and $\tilde{\sigma }_{\theta }$, respectively (see, for example, Jaeger et al. 2007). For the radial direction, the equilibrium equation is

$$\begin{aligned} \frac{\partial \tilde{\sigma }_{r}}{\partial \rho } + k \, \frac{\tilde{\sigma } _{r} - \tilde{\sigma } _{\theta }}{\rho } + \sin \theta = 0 \end{aligned}$$

(43)

and for the tangential direction, it is

$$\begin{aligned} \frac{1}{\rho } \, \frac{\partial \tilde{\sigma } _{\theta }}{\partial \theta } + \cos \theta = 0 \end{aligned}$$

(44)

In Eq. (43), the parameter k is such that $k=1$ implies that the cavity is cylindrical, while $k=2$ implies that the cavity is spherical.

In the derivation that follows, the cavity in Fig. 29 is assumed to be at the verge of equilibrium; in particular, the cohesion, c, of the ground is assumed to be the critical cohesion that leads to collapse of the cavity. The Tresca shear failure criterion (see Eqs. 3 and 4) is expressed in terms of scaled principal stresses, $\tilde{\sigma }_1$ and $\tilde{\sigma }_3$, as follows

$$\begin{aligned} \tilde{\sigma }_1 = \tilde{\sigma }_3 + 2\, \tilde{c} \end{aligned}$$

(45)

where $\tilde{c}$ is the scaled cohesion defined by Eq. (7).

Note that in this appendix, in contrast with the notation used in the equations in Sect. 3.2, the critical scaled cohesion associated with the cavity in a critical state of equilibrium is denoted simply as $\tilde{c}$, rather than as $\tilde{c}_{cr}$.

The first step to derive the solution of the problem in Fig. 29 is to solve for the stress quantities along the vertical segment defined by points A and B. The superscript ‘AB’ is used to denote stresses along this segment (see Fig. 29). For the case of contracting cavities, the hoop stresses are considered to be major principal stresses, while the radial stresses are considered to be minor principal stresses—i.e., $\tilde{\sigma }_\theta ^{\,AB}=\tilde{\sigma }_1$ and $\tilde{\sigma }_r^{\,AB}=\tilde{\sigma }_3$, respectively. For the case of expanding cavities, the opposite assumption is made—i.e., $\tilde{\sigma }_\theta ^{\,AB}=\tilde{\sigma }_3$ and $\tilde{\sigma }_r^{\,AB}=\tilde{\sigma }_1$, respectively. Therefore, because of Eq. (45), the relationship between the (scaled) hoop stresses, $\tilde{\sigma }_\theta ^{\,AB}$, and the (scaled) radial stresses, $\tilde{\sigma }_r^{\,AB}$, is written as¹

$$\begin{aligned} \tilde{\sigma }_\theta ^{\,AB}= \tilde{\sigma }_r^{\,AB}\pm 2\, \tilde{c} \end{aligned}$$

(46)

Along the segment AB in Fig. 29, the equilibrium equation for the radial direction is obtained from Eq. (43), making $\theta =\pi /2$, to yield the following total differential equation of the variable $\rho$,

$$\begin{aligned} \frac{{\mathrm {d}}\tilde{\sigma }_r^{\,AB}}{{\mathrm {d}}\rho } + k \, \frac{\tilde{\sigma }_r^{\,AB}- \tilde{\sigma }_\theta ^{\,AB}}{\rho } + 1 = 0 \end{aligned}$$

(47)

Replacing Eq. (46) in Eq. (47), a total differential equation of the unknown variable $\tilde{\sigma }_r^{\,AB}$ only is obtained. This equation can be solved considering the following boundary condition

$$\begin{aligned} \tilde{\sigma }_r^{\,AB}=\tilde{q}_s \quad \text {at}\quad \rho =\xi \end{aligned}$$

(48)

In this way, the solution for $\tilde{\sigma }_r^{\,AB}(\,\rho )$ results (see footnote associated with Eq. 46)

$$\begin{aligned} \tilde{\sigma }_r^{\,AB}(\,\rho )= \tilde{q}_{s} + \xi - \rho \pm 2 \, k \, \tilde{c}\, \ln \frac{\rho }{\xi } \end{aligned}$$

(49)

Also, because of Eq. (46), the solution for $\tilde{\sigma }_\theta ^{\,AB}(\,\rho )$ results (see footnote associated with Eq. 46)

$$\begin{aligned} \tilde{\sigma }_\theta ^{\,AB}(\,\rho )= \tilde{\sigma }_r^{\,AB}(\,\rho )\pm 2 \, \tilde{c} \end{aligned}$$

(50)

Note that Eqs. (49) and (50), with positive last terms, are the same Eqs. (14) and (15) in the main text for the case of contracting cavities, while Eqs. (49) and (50), with negative last terms, are the same Eqs. (17) and (18) in the main text for the case of expanding cavities. As explained earlier on, the scaled cohesion is denoted as $\tilde{c}_{cr}$ in the equations in the main text, while the same scaled cohesion is denoted as $\tilde{c}$ in the equations in this appendix.

The scaled internal pressure at the crown of the cavity, $\tilde{p}_s^{\,A}$, (see Fig. 29) can be found by applying the following boundary condition to Eq. (49)

$$\begin{aligned} \tilde{\sigma }_r^{\,AB}=\tilde{p}^{\,A}_s \quad \text {at} \quad \rho =1 \end{aligned}$$

(51)

This yields²

$$\begin{aligned} \tilde{p}_s^{\,A} = \tilde{q}_s + \xi - 1 \mp 2 \, k \, \tilde{c}\, \ln \xi \end{aligned}$$

(52)

Note that Eq. (52) with a negative last term is the same Eq. (16) presented in the main text for the case of contracting cavities, while Eq. (52) with a positive last term is the same Eq. (19) presented in the main text for the case of expanding cavities.

The second step to derive the solution of the problem in Fig. 29 is to integrate the equilibrium equation for the tangential direction (Eq. 44) throughout the integration circle. Integrating Eq. (44) with respect to the variable $\theta$, the general solution for the scaled hoop stress, $\tilde{\sigma }_\theta (\,\rho ,\theta )$, results

$$\begin{aligned} \tilde{\sigma }_\theta (\,\rho ,\theta )=-\rho \, \sin \theta + \tilde{C}_1(\,\rho ) \end{aligned}$$

(53)

where $\tilde{C}_1(\,\rho )$ is an integration function of the variable $\rho$ (i.e., of the variable disregarded in the integration). At $\theta =\pi /2$ (on the segment AB) the solution for $\tilde{\sigma }_\theta (\,\rho ,\theta )$ given by Eq. (53) must be equal to the solution for $\tilde{\sigma }_\theta ^{\,AB}(\,\rho )$ given by Eq. (50). In this way, the integration function, $\tilde{C}_1(\,\rho )$, is found to be

$$\begin{aligned} \tilde{C}_1(\,\rho )= \rho + \tilde{\sigma }_\theta ^{\,AB}(\,\rho ) \end{aligned}$$

(54)

Replacing Eq. (54) into Eq. (53), the final solution for the scaled hoop stress, $\tilde{\sigma }_\theta (\,\rho ,\theta )$, becomes

$$\begin{aligned} \tilde{\sigma }_\theta (\,\rho ,\theta )= \tilde{\sigma }_\theta ^{\,AB}(\,\rho )+ \rho \, (1-\sin \theta ) \end{aligned}$$

(55)

Note that Eq. (55) is the same Eq. (13) in the main text that applies to both contracting and expanding cavities.

The third step to derive the solution of the problem in Fig. 29 is to integrate the equilibrium equation for the radial direction (Eq. 43) throughout the integration circle. Replacing Eq. (55) into Eq. (43) and integrating with respect to the variable $\rho$, the general solution for the scaled radial stress, $\tilde{\sigma }_r(\,\rho ,\theta )$, results (see footnote associated with Eq. 46)

$$\begin{aligned} \tilde{\sigma }_r(\,\rho ,\theta )=\tilde{q}_s + \xi \pm 2 \, k \, \tilde{c} \, \ln \frac{\rho }{\xi } - \rho \, \sin \theta + \rho ^{-k} \, \tilde{C}_2(\theta ) \end{aligned}$$

(56)

where $\tilde{C}_2(\theta )$ is an integration function of the variable $\theta$ (i.e., of the variable disregarded in the integration). At $\theta = \pi /2$ (on the segment AB) the solution for $\tilde{\sigma }_r(\,\rho ,\theta )$ given by Eq. (56) must be equal to the solution for $\tilde{\sigma }_r^{\,AB}(\,\rho )$ given by Eq. (49). In this way the integration function, $\tilde{C}_2(\theta )$, is found to be

$$\begin{aligned} \tilde{C}_2(\theta )= 0 \end{aligned}$$

(57)

Replacing Eq. (57) into Eq. (56), the final solution for the scaled radial stress, $\tilde{\sigma }_r(\,\rho ,\theta )$, becomes

$$\begin{aligned} \tilde{\sigma }_r(\,\rho ,\theta )= \tilde{\sigma }_r^{\,AB}(\,\rho )+ \rho \, (1-\sin \theta ) \end{aligned}$$

(58)

Note that Eq. (58) is the same Eq. (12) in the main text that applies to both contracting and expanding cavities.

In the remainder of this appendix some observations about particular features of the solution derived above are highlighted.

The first observation to be made is about the distribution of internal pressure for the cavity predicted by the proposed solution. The scaled internal pressure function, $\tilde{p}_s(\theta )$, is obtained by making $\rho =1$ in Eq. (58). This yields

$$\begin{aligned} \tilde{p}_s(\theta ) = \tilde{p}^{\,A}_s + 1 - \sin \theta \end{aligned}$$

(59)

As expected, Eq. (59) predicts that at the crown of the cavity (i.e., $\theta =\pi /2$, see point A in Fig. 29) the scaled internal pressure becomes equal to $\tilde{p}^{\,A}_s$. Also, Eq. (59) predicts that at the spring line of the cavity (i.e., at $\theta =0$, see point C in Fig. 29) and at the invert of the cavity (i.e., at $\theta =-\pi /2$, see point D in Fig. 29) the scaled internal pressures $\tilde{p}^{\,C}_s$ and $\tilde{p}^{\,D}_s$ are, respectively

$$\begin{aligned} \tilde{p}^{\,C}_s = \tilde{p}^{\,A}_s + 1; \quad \quad \tilde{p}^{\,D}_s = \tilde{p}^{\,A}_s + 2 \end{aligned}$$

(60)

Considering the scaling rule for internal pressure (see left-side Eq. 8), Eq. (60) predicts the following (unscaled) internal pressures at the spring line and invert of the cavity, respectively,

$$\begin{aligned} p^{\,C}_s = p^{\,A}_s + \gamma \,a; \quad \quad p^{\,D}_s = p^{\,A}_s + 2\, \gamma \,a \end{aligned}$$

(61)

Equation (61) indicates that the internal pressure increases with depth inside the cavity in a lithostatic manner, as if the internal pressure is provided by an imaginary pressurized gas which has the same unit weight as the ground surrounding the cavity.

The second observation to be made is about the stress state of the ground inside the integration circle in Fig. 29. When deriving the solution of the problem stated in this figure, the assumption was made that the radial and hoop stresses on the vertical segment AB were compatible with a plastic state (see Eq. 46); nevertheless no assumption was made about the relationship between radial and hoop stresses elsewhere throughout the integration circle (e.g., whether these stresses are compatible with a plastic or with an elastic state). Subtracting the solution for scaled hoop and radial stresses given by Eqs. (55) and (58), respectively, gives (see footnote associated with Eq. 46)

$$\begin{aligned} \tilde{\sigma }_\theta (\,\rho ,\theta ) - \tilde{\sigma }_r(\,\rho ,\theta ) = \tilde{\sigma }_\theta ^{\,AB}- \tilde{\sigma }_r^{\,AB}= \pm 2 \, \tilde{c} \end{aligned}$$

(62)

Equation (62) indicates that in the proposed analytical solution the ground is in plastic state throughout the entire integration circle and not just on the segment AB.

The third and last observation to be made is about the continuity of the stress field across the integration circle boundary (see Fig. 29). For the proposed analytical solution to be a statically admissible solution, continuity of radial and shear stresses must exist across the integration circle boundary; the hoop stress, though, may be discontinuous (see, for example, Davis and Selvadurai 2002; Pietruszczak 2010).

The scaled radial stress on the integration circle boundary, inside the integration circle, is obtained from Eq. (58) making $\rho =\xi$. This gives

$$\begin{aligned} \tilde{\sigma }_r(\xi ,\theta ) = \tilde{q}_s + \xi \, (1-\sin \theta ) \end{aligned}$$

(63)

The scaled hoop stress on the integration circle boundary, inside the integration circle, is similarly obtained from Eq. (55) (see also Eq. 62) and results (see footnote associated with Eq. 46)

$$\begin{aligned} \tilde{\sigma }_\theta (\xi ,\theta ) = \tilde{\sigma }_r(\xi ,\theta ) \pm 2\,\tilde{c} \end{aligned}$$

(64)

The scaled vertical stress on the integration circle boundary, outside the integration circle, is obtained from the left-side Eq. (9) considering $y/a = \xi \sin \theta$ (see Fig. 29). This gives

$$\begin{aligned} \tilde{\sigma }^{\,o}_y(\xi ,\theta ) = \tilde{q}_s + \xi \, (1-\sin \theta ) \end{aligned}$$

(65)

The scaled horizontal stress on the integration circle boundary, outside the integration circle, is similarly obtained from the right-side Eq. (9) and from Eq. (65) and results

$$\begin{aligned} \tilde{\sigma }^{\,o}_x(\xi ,\theta ) = K_o\, \tilde{q}_s + K_o\, \xi \, (1-\sin \theta ) \end{aligned}$$

(66)

The scaled radial and shear stresses, $\tilde{\sigma }^{\,o}_n$ and $\tilde{\tau }^{\,o}_s$, respectively, on the integration circle boundary, outside the integration circle, can be obtained with the following classical stress transformation formulae (see, for example, Jaeger et al. 2007)

$$\begin{aligned} \tilde{\sigma }^{\,o}_n= \tilde{\sigma }^{\,o}_x\, \cos ^2\theta + \tilde{\sigma }^{\,o}_y\, \sin ^2\theta \end{aligned}$$

(67)

and

$$\begin{aligned} \tilde{\tau }^{\,o}_s= \frac{ \tilde{\sigma }^{\,o}_y- \tilde{\sigma }^{\,o}_x}{2} \, \sin 2\theta \end{aligned}$$

(68)

Replacing Eqs. (65) and (66) into Eqs. (67) and (68), the scaled radial stress results

$$\begin{aligned} \tilde{\sigma }^{\,o}_n= \left( \sin ^2 \theta + K_o \cos ^2 \theta \right) \left[ \tilde{q}_s + \xi \, (1-\sin \theta ) \right] \end{aligned}$$

(69)

and the scaled shear stress results

$$\begin{aligned} \tilde{\tau }^{\,o}_s= -\frac{K_o-1}{2} \sin 2\theta \left[ \tilde{q}_s + \xi \, (1-\sin \theta ) \right] \end{aligned}$$

(70)

Considering the case $K_o = 1$ (i.e., the horizontal and vertical in situ stresses are the same) in Eqs. (69) and (70), the stresses $\tilde{\sigma }^{\,o}_n$ and $\tilde{\tau }^{\,o}_s$ result to be, respectively

$$\begin{aligned} \tilde{\sigma }^{\,o}_n= \tilde{q}_s + \xi \, (1-\sin \theta ) \end{aligned}$$

(71)

and

$$\begin{aligned} \tilde{\tau }^{\,o}_s= 0 \end{aligned}$$

(72)

Equation (71) is the same as Eq. (63), implying continuity of radial stresses across the integration circle boundary (for the assumed case $K_o = 1$). Also, Eq. (72) and the fact that the shear stresses are assumed null inside the integration circle (see Fig. 29), imply that there is also continuity of shear stresses across the integration circle boundary (again for the assumed case $K_o = 1$). With regard to the hoop stresses on the integration circle boundary, Eq. (64) indicates these stresses are discontinuous across the boundary; the difference in hoop stresses when passing from the external to the internal sides of the integration circle boundary are equal to $2\,\tilde{c}$ for contracting cavities and equal to $-2\,\tilde{c}$ for expanding cavities. The observations regarding the continuity of radial and shear stresses across the integration boundary show that the proposed solution is a statically admissible solution when the lateral earth pressure coefficient is considered to be equal to one.

Fig. 29

Problem of stability of cylindrical or spherical (contracting or expanding) shallow cavity. Note the representation of the stress states along the segment AB and on the integration circle boundary

Appendix 2: Derivation of limit equilibrium (Terzaghi) solution for contracting and expanding cavities in dry cohesive ground

This appendix presents the derivation of Eqs. (32) and (38) in the main text. The derivation follows a similar analysis (including the use of the same notation) as in Terzaghi (1943) when describing arching effect in soils.

Figure 30 shows the problem to be analyzed. A block (representing the ground above the opening) of height, D, and unit weight, $\gamma$, rests above the (assumed flat) roof of an opening of (in-plane) width $2\,B$. The flat roof of the opening in question is either the flat roof of long tunnel (of in-plane width $2\,B$ and unit length along the out-of-plane direction) or the flat roof of a square cavern (of width $2\,B$ in both in-plane and out-of-plane directions). The cases of long tunnel and square cavern are distinguished by introduction of the same parameter k used in the formulation in “Appendix 1”.

A support pressure, $p_s$, is assumed to act at the base of the rectangular prismatic block in Fig. 30 (i.e., on the roof of the tunnel/cavern). A surcharge load, $q_s$, is assumed to act at the top of the block (i.e., on the ground surface). The rectangular prismatic block in Fig. 30 is assumed to be at the limit state of equilibrium. Therefore, full shear resistance is assumed to develop on the sides of the prismatic block. Considering the case of purely cohesive ground, the shear resistance is provided by the cohesion, c, of the ground. For the case of contracting openings, the shear resistance per unit area of prism side is represented by vectors, c, pointing upward, as indicated in Fig. 30. For the case of expanding openings, the shear resistance (per unit area of prism side) is represented by the same vectors, c, but pointing downward (note that in Fig. 30, the symbol ± is used as a prefix of the label of the cohesion vector to indicate either of the cases mentioned above).

Note that in this appendix, in contrast with the notation used in the equations in Sect. 5, the critical cohesion associated with the cavity in a critical state of equilibrium is denoted simply as c, rather than as $c_{cr}$.

Figure 30 shows the free body diagram of a horizontal slice of block of differential height, ${\mathrm {d}}z$, at a depth, z, below the ground surface. On the top surface of this (differential) slice, the acting force is the vertical stress, $\sigma _v$, multiplied by the slice area, $(2\,B)^k$, where $k=1$ for the tunnel case, and $k=2$ for the square cavern case. On the bottom surface of the (differential) slice, the acting force is the vertical stress, $\sigma _v + {\mathrm {d}}\sigma _v$, multiplied by the same area. On the sides of the (differential) slice, the acting forces are $2\,c\,{\mathrm {d}}z$ for the tunnel case, and $4\,(2B)\,c\,{\mathrm {d}}z$ for the square cavern case. The weight of the (differential) slice is the unit weight, $\gamma$, of the slice multiplied by the volume of the slice, $(2\,B)^k\,{\mathrm {d}}z$, again, with $k=1$ for the tunnel case and $k=2$ for the square cavern case.

Considering equilibrium of the acting forces for the horizontal slice in Fig. 30, the following total differential equation of the unknown function $\sigma _v$ is obtained³

$$\begin{aligned} \frac{{\mathrm {d}}\sigma _v}{{\mathrm {d}}z} \pm k \frac{c}{B} - \gamma = 0 \end{aligned}$$

(73)

Equation (73) can be integrated to obtain the solution for the function, $\sigma _v$, using the following boundary condition

$$\begin{aligned} \sigma _v=q_s \quad \text {at} \quad z=0 \end{aligned}$$

(74)

The integration and application of the boundary condition gives⁴

$$\begin{aligned} \sigma _v(z) = q_s + z \left( \gamma \mp k \frac{c}{B} \right) \end{aligned}$$

(75)

The cavity roof pressure, $p_s$, required to maintain equilibrium of the prismatic block in Fig. 30 is found using the following condition

$$\begin{aligned} \sigma _v=p_s \quad \text {at} \quad z=D \end{aligned}$$

(76)

Application of Eq. (76) in Eq. (75) gives (see footnote associated with Eq. 75)

$$\begin{aligned} p_s = q_s + D \left( \gamma \mp k \frac{c}{B} \right) \end{aligned}$$

(77)

Equation (77) allows definition of the equations for the required support pressure presented in the main text for the cases of contracting and expanding openings as follows.

For the case of contracting cavity, the limit equilibrium model by Terzaghi specifies that the required pressure on the flat roof of the opening is to be computed considering a prismatic block of width $2\,B_1$ (see Fig. 4). This width is obtained by adding the width of the cavity, $2\,B_0$, and the top width of two active wedges developing on the sides of the opening (see Fig. 4). For the case of purely cohesive ground, the top width of the two active wedges is two times the height of the opening, H (see Fig. 4). Based on Terzaghi’s specifications, Fig. 25 shows how the limit equilibrium model by Terzaghi (Fig. 4) applies to the case of contracting circular or spherical openings of radius, a, and depth, h, as considered in Sect. 3. In such case, the width, B, of the prismatic block in Fig. 30 becomes (see Fig. 25)

$$\begin{aligned} B = B_1 = 3\, a \end{aligned}$$

(78)

while the height, D, of the prismatic block in Fig. 30 becomes

$$\begin{aligned} D = h - a \end{aligned}$$

(79)

Equation (32) in the main text is obtained by replacing Eqs. (78) and (79) into Eq. (77) and applying the scaling rules introduced by Eqs. (6) through (8); also, as explained earlier on, the critical cohesion in Eq. (77) is simply referred to as c rather than as $c_{cr}$.

Figure 27 shows how the limit equilibrium model proposed by Terzaghi (1943) (see Fig. 4) applies to the case of circular or spherical expanding openings of radius, a, and depth, h, as considered in Sect. 3. For expanding cavities, the most conservative estimation of the maximum pressure, $p_s$, is obtained considering the smallest possible width of the prismatic block above the opening in Fig. 5; this is because that the self-weight of the block contributes favorably to the stability, as it opposes the internal pressure. In such case, the width, B, of the prismatic block in Fig. 30 becomes

$$\begin{aligned} B = a \end{aligned}$$

(80)

Equation (38) in the main text is obtained by replacing Eqs. (79) and (80) into Eq. (77) and applying the scaling rules introduced by Eqs. (6) through (8); also, as explained earlier on, the critical cohesion in Eq. (77) is simply referred to as c rather than as $c_{cr}$.

Fig. 30

Limit equilibrium analysis of a rectangular prismatic block representing the ground above the flat roof of a tunnel or square cavern—adapted from Terzaghi (1943)