A new method for determining the pore-water pressure around twin shallow circular tunnels

A new method for determining the pore-water pressure around twin shallow circular tunnels | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article A new method for determining the pore-water pressure around twin shallow circular tunnels Tao Zhan, Shengbiao Shan, Peiyuan He This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4824150/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 26 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted 15 You are reading this latest preprint version Abstract In order to obtain the distribution rules of pore-water pressure for twin shallow circular tunnels, the determination approach in the pore-water pressure for the twin shallow circular tunnels buried in a semi-infinite space was proposed by using the bipolar coordinate system method and the Schwarz alternating method for the first time. The solution of pore-water pressure was also obtained by multiple iterations method. The proposed approach was validated by the results of numerical simulation. The maximum error between the results of the analytical solutions from the proposed method and the numerical simulation was only 2.15% when the accuracy was set to 1.0×10-3MPa. At last, influences of the number of holes and the tunnel center spacing on the pore-water pressure were presented. It is shown that the number of holes and the tunnel center spacing were the main factors for the distribution of pore-water pressure, and the pore-water pressure of twin shallow circular tunnels was only greater than the pore-water pressure of single shallow circular tunnel in the upper area of tunnels. And the maximum result for the pore-water pressure of twin shallow circular tunnels was obtained in the midpoint of the tunnel center spacing. Twin shallow circular tunnels Pore-water pressure Bipolar coordinate system Schwarz alternating method Steady seepage Semi-infinite space Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 1. Introduction In order to cope with the transportation problem in overcrowd urban, lots of metro tunnels are planning and constructing in many cities in China. The influence of groundwater is one of the most important concerns in tunnel engineering. During the water-rich tunnel excavation, due to the seepage effect, it is easy for groundwater to flow into the tunnel and generate the pore-water pressure in surrounding rock. And it will be negative to effective stresses in surrounding rock of tunnels and make the structures of tunnels failure. Therefore, for security, the seepage effect should be considered to keep the tunnel structures stable in the groundwater-rich areas. Up to now, some studies have been done on tunnel seepage. On the one hand, Shin et al. [ 1 ] studied the potential effect of pore water pressure on tunnel lining using the finite element method and proposed a method for estimating the design curve of pore water pressure for secondary lining. Pan et al. [ 2 ] analyzed the effect of pore water pressure on the stability of the palm face by using fluid mechanics and numerical simulation. Park et al. [ 3 ] derived the analytical solution of the seepage force around the circular tunnel by using the complex variable method. Lee et al. [ 4 ] proposed the calculation approach for groundwater seepage in surrounding rock by using the limit analysis upper limit method. Yang et al. [ 5 ] proposed the stability solutions taking into account of water seepage of shallow tunnels based on strength reduction theory. On the other hand, for twin tunnels, Zhang et al. [ 6 ] derived the analytical solutions of non-Darcy seepage field in shallow single-hole and double-hole circular tunnels by using the principle of mirror method. Zhu et al. [ 7 ] derive the stable seepage field of twin parallel tunnels by using the complex variable method. Zhang [ 8 ] derived the analytical solution for the stable seepage field of twin parallel tunnels in the semi-infinite plane by using the principle of mirror method. According to present results of studies, it can be seen that the boundary conditions of study methods are generally difficult to determine. In addition, the complex variable method is more complex, and it is also more difficult to apply in engineering. Therefore, it is necessary to make the studies on the calculation approach in the pore-water pressure of shallow twin circular tunnels to fill the gaps in current studies. The main objective of the current study is to obtain the accurate numerical solutions of pore-water pressure of twin shallow circular tunnels. On the basis of the bipolar coordinate system, taking the example of a shallow circular tunnel in a fully saturated, homogeneous and isotropous semi-infinite space, the calculation approach in the pore-water pressure of the shallow circular tunnel is proposed. Based on this, the calculation approach in the pore-water pressure of twin shallow circular tunnels can be proposed by Schwarz alternating methods. Through the comparison with the results of numerical simulation, the rationality of the calculation approach in this study is verified, and the influencing factors of the pore-water pressure is analysed. It is significant to propose the theoretical basis of the seepage of shallow tunnels for guiding the construction of tunnels securely. 2. Problem description and assumptions As shown in Fig. 1 , the radius, burial depth, tunnel spacing and surface water level of the twin shallow circular tunnels are r , d , L and h w , respectively. And the ratio of the tunnel burial depth to radius is not more than seven. Supposing that the length of the tunnel is long enough, the three-dimensional space problem can be simplified to the two-dimensional plane problem. Surface water level is higher than the ground, so the soil is fully saturated. And the permeability coefficient of soil too small to keep surface water level stable. Therefore, the shallow twin circular tunnels are buried in a fully saturated, homogeneous and isotropous semi-infinite space. Based on this, the x -axis is represented as the ground. And the y 1 -axis and y 2 -axis crossing the center of the twin circular tunnels are the symmetrical axis of the tunnels. As shown in Fig. 1 , the twin shallow circular tunnels are the object of study in a semi-infinite space. Because the tunnel spacing is smaller, the duplex tunnels should not be considered. 3. Analytic solutions in the pore-water pressure of the shallow circular tunnel Based on the study results of the literature [ 9 ], with the constant water head seepage mode, the approach in the pore-water pressure of the shallow circular tunnel is derived in the bipolar coordinate system to provide a theoretical basis for the solution of the twin shallow circular tunnel. 3.1. Stable seepage equations Because of the complexity of soil pores, the direction of water seepage is often random. Therefore, the seepage is usually analyzed macroscopically in the engineering. The main direction of seepage can be only considered in the analysis, and the influence of soil can be ignored. Before the continuous medium theory proposed, Bernoulli's equation was commonly used in hydraulics and expressed as follows: $$p+\frac{1}{2}\rho {v^2}+\rho gh=C$$ 1 where p is the fluid pressure, v is the fluid flow rate, ρ is the fluid density, g is the acceleration of gravity, h is the head height, and C is a constant. Because Eq. ( 1 ) is derived from the law of conservation of energy, it is only applicable to ideal fluids which are non-viscous or less viscous and non-compressible. And the pressure water head can be expressed the total water head of the fluids which is shown in Eq. ( 2 ). $$h=\frac{u}{{{\gamma _w}}}+z+\frac{{{{\overline {v} }^2}}}{{2g}}$$ 2 where u is the fluid pressure, \(\overline {v}\) is the permeation rate of water, and γ w is the density of water. z is the elevation head, which indicates the distance in the positive direction of the reference plane. If it lies below the x -axis, the values take the negative. It can be seen from Fig. 2 that water will flow from the high head to the low head through the connecting pore channel when there is a total head difference in the soil. And the total head difference (△ h ) can be expressed as follows: $$\Delta h{\text{=}}{h_A} - {h_B}$$ 3 with $${h_A}=\frac{{{u_A}}}{{{\gamma _w}}}+{z_A}+\frac{{{{\overline {{{v_A}}} }^2}}}{{2g}}$$ 4 $${h_B}=\frac{{{u_B}}}{{{\gamma _w}}}+{z_B}+\frac{{{{\overline {{{v_B}}} }^2}}}{{2g}}$$ 5 The head loss of the unit process is the hydraulic gradient which is shown in Eq. ( 6 ). $$J{\text{=}}\frac{{\Delta h}}{{\Delta L}}$$ 6 where J is the hydraulic gradient. Since the seepage which belongs to laminar flow in the soil tends to flow slowly with small hydraulic gradients, the momentum changes can be ignored and the flow rate can be considered as zero. For the seepage of laminar flow, the water head at any position in the whole seepage space can be expressed as follows: $$h=\frac{u}{{{\gamma _w}}}+z$$ 7 Darcy's law reflects the linear relationship between the seepage rate of water in the soil and the head loss. When water flows in the soil in a laminar state, the seepage of water follows Darcy's law and can be expressed as follows: $${\text{Q}}=kAJ$$ 8 $$v=kJ$$ 9 It can be seen from Darcy's law that the rate of passage is proportional to the magnitude of the hydraulic gradient and the permeability of the porous medium when water passes through the porous medium. However, the seepage rate of the fluid remains as a nonlinear relationship with the hydraulic gradient when the flow rate of the fluid is too fast. Therefore, Darcy's law is not applicable to the turbulence. Based on this, the fluid of study belongs to the laminar flow, with a constant flow rate per unit time. In the Cartesian coordinate system, water discharge velocity in the x -axis direction is u , and water discharge velocity in the y -axis direction is v . The water flow continuity equation can be obtained by $$\frac{{\partial u}}{{\partial x}}+\frac{{\partial v}}{{\partial y}}=0$$ 10 And the velocity potential energy equation can be expressed as follows: $$\phi (x,y)= - k\left( {\frac{p}{{{\gamma _\omega }}}+z} \right)= - kh$$ 11 According to Eq. ( 11 ), water discharge velocity in the x-axis direction ( u ) and water discharge velocity in the y -axis direction ( v ) can be obtained by $$u=\frac{{\partial \varphi }}{{\partial x}}= - k\frac{{\partial h}}{{\partial x}}$$ 12 $$v=\frac{{\partial \varphi }}{{\partial y}}= - k\frac{{\partial h}}{{\partial y}}$$ 13 And the Laplace equation for the two-dimensional stable seepage [ 11 ] can be obtained by combining Eqs. ( 12 ) and ( 13 ) with Eq. ( 10 ), which is shown in Eq. ( 14 ). $${\nabla ^2}\varphi =\frac{{{\partial ^2}\varphi }}{{\partial {x^2}}}+\frac{{{\partial ^2}\varphi }}{{\partial {y^2}}}=0$$ 14 In the bipolar coordinate system, Eqs. ( 11 ) and ( 14 ) can be expressed by the following expressions using the coordinate transformation technique: $$\phi (\alpha ,\beta )= - kh= - k(J\sinh \alpha +p/\gamma )$$ 15 $${\nabla ^2}\varphi =\frac{1}{{{J^2}}}\left[ {\frac{\partial }{{\partial \alpha }}(\frac{{\partial \varphi }}{{\partial \alpha }})+\frac{\partial }{{\partial \beta }}(\frac{{\partial \varphi }}{{\partial \beta }})} \right]$$ 16 3.2. Pore-water pressure solutions for the shallow circular tunnel Supposing that the permeability coefficient on the outside of lining structures is higher, the head at the tunnel boundary is constant, and the water head at infinity is only related to the tunnel depth. In the semi-infinite space, the Laplace equation for the water head can be expressed by the following expression using the coordinate transformation technique: $${\nabla ^2}h=\frac{1}{{{J^2}}}(\frac{{{\partial ^2}h}}{{\partial {\alpha ^2}}}+\frac{{{\partial ^2}h}}{{\partial {\beta ^2}}})=0$$ 17 According to the boundary conditions of stable seepage, the head of pore water in the surrounding rock can be obtained by $$h={h_0}+\frac{{{h_{{a_i}}} - {h_0}}}{{{\alpha _i}}}\alpha$$ 18 where h a is the tunnel boundary head, h 0 is the head of water at semi-infinite space infinity. Substituting Eq. ( 18 ) into Eq. ( 11 ), the pore-water pressure in the surrounding rock under constant water head conditions can be obtained by $$p={h_0}{\gamma _w}+\frac{{{h_{{a_i}}} - {h_0}}}{{{\alpha _i}}}{\gamma _{\text{w}}}\alpha - J\sinh \alpha {\gamma _w}$$ 19 4. The Calculation approach in the pore-water pressure of twin shallow circular tunnels 4.1. Calculation model As shown in Fig. 3 , the left tunnel is tunnel I and the right tunnel is tunnel II. Both tunnels I and II are located below the surface water level. The head of surface water on the tunnel is h w , and the water head in the surrounding rock is h 1 . In the bipolar coordinate system, the coordinates of any point of the tunnel are ( α , β ). The coordinate of the tunnel wall is α i . The coordinate of the ground is zero. In a coordinate system with the y -axis oriented vertically downward as the positive direction, α is constantly greater than zero and β ranges from -π to π. Supposing that the lining structures don’t have infiltration capacity, the boundary conditions belong to constant water head seepage mode. Because the twin tunnels belong to the multi-connected domain problem [ 13 – 15 ], the calculation approach in pore-water pressure of twin shallow circular tunnels can be obtained by using Schwarz alternating method to make the solutions of pore-water pressure of shallow circular tunnel performed for multiple iterations. 4.2. Schwarz alternating method The Schwarz alternating method was proposed for solving the Dirichlet problem and later proved its convergence in two and three dimensions by Mikhlin [ 10 ] and Soboldff [ 12 ] successively. Therefore, for the analysis of the elastic zone of the surrounding rock in a multi-connected domain, the sequence of loading and geometry changes have no effect on the stresses in the surrounding rock. Therefore, the Schwarz alternating method can transform the multi-connected domain problems into the single-connected domain problems [ 16 – 18 ] to simplify the solutions of the pore-water pressure of twin shallow circular tunnels in a semi-infinite space. And the solution of the pore-water pressure of shallow circular tunnel in the bipolar coordinate system is extended to the twin tunnels by the Schwarz alternating method. (1) Supposing that tunnel II does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, the pore-water pressure of tunnel I in the bipolar coordinate system can only be obtained and expressed as follows: $${p_{w11}}={h_0}{\gamma _w}+\frac{{{h_{{a_i}}} - {h_0}}}{{{\alpha _i}}}{\gamma _{\text{w}}}\alpha - J\sinh \alpha {\gamma _w}$$ 18 where p w 11 is the pore-water pressure of tunnel I. (2) Supposing that tunnel I does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, a water head ( h w 21 ) is applied around tunnel II to counteract the pore-water pressure around tunnel II which is generated by tunnel I. And the pore-water pressure around tunnel I which is generated by tunnel II can be expressed as follows: $${p_{w21}}={h_0}{\gamma _w}+\frac{{{h_{{a_i}}}+{h_{w21}} - {h_0}}}{{{\alpha _i}}}{\gamma _{\text{w}}}\alpha - J\sinh \alpha {\gamma _w}$$ 21 where p w 21 is the pore-water pressure around tunnel I which is generated by tunnel II. (3) Similarly, supposing that tunnel II does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, a water head ( h w 12 ) is applied around tunnel I to counteract the pore-water pressure around tunnel I which is generated by tunnel II. And the pore-water pressure around tunnel II which is generated by tunnel I can be expressed as follows: $${p_{w12}}={h_0}{\gamma _w}+\frac{{{h_{{a_i}}}+{h_{w12}} - {h_0}}}{{{\alpha _i}}}{\gamma _{\text{w}}}\alpha - J\sinh \alpha {\gamma _w}$$ 22 (4) Until the value of p w 12 small enough or equal to zero, and making p w 12 <[ p ] stand, the interaction can be stopped. Otherwise, the interaction will continue until p w achieving the standards. And the numerical solution of the pore-water pressure around the twin shallow circular tunnels can be obtained by superimposing the results of each iteration 4.3. Pore-water pressure solutions In the bipolar coordinate system, when the pore-water pressure is solved, the parameters of the twin shallow circular tunnels in the semi-infinite space are shown in Table 1 . Table 1 Parameters of the twin shallow circular tunnels. Parameters Value surface water level h w (m) 30 water head in the surrounding rock h 1 (m) 0 tunnel burial depth h (m) 20 tunnel radius r (m) 5 tunnel center spacing L (m) 20 water heaviness g (kN/m 3 ) 9.8 Since the surface water and groundwater are used twice during the calculation of the the Schwarz alternating method, h w 1 =0.5 h w = 15m is set. The effect of groundwater should be ignored in the second iteration, and the interaction between tunnels is only considered. When the interaction between tunnels is small enough, the numerical solution of the pore-water pressure can be obtained accurately. Therefore, the threshold value ([ p ]) is 10 − 4 MPa, which controls the iteration stopping. As shown in Fig. 4 , the value range of x -axis is set to [-100,100] and the value range of y -axis is set to [0,100], from which the two cross sections at the positions of y = 10m and y = 30m in the semi-infinite space are taken as the study area to observe the pore-water pressure distribution pattern at the same depth in the surrounding rock of the twin shallow circular tunnels. And the sections 1–2, 3–4, 5–6, and 7–8 are selected as the monitoring points of the pore-water pressure around tunnel I. 4.3.1 First iteration of the loop In the bipolar coordinate system, the pore-water pressure around tunnel I is solved when tunnel II is supposed not to be in the system. And the pore-water pressure in the Cartesian coordinate system can be expressed by using the coordinate transformation technique. The pore-water pressure around tunnel I belongs to the shallow circular tunnel. And the calculation results of the pore-water pressure around tunnel I in the above sections are shown in Fig. 5. When the influence of the pore-water pressure around tunnel I on tunnel II is analysed, the pore-water pressure acting around tunnel II should be the average value, and a water head ( h w 21 ) is applied around tunnel II to counteract the pore-water pressure around tunnel II. In the same way, the pore-water pressure around tunnel II is solved when tunnel I is supposed not to be in the bipolar coordinate system. And the results of the pore-water pressure around tunnel II should be superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 6. 4.3.2 Second iteration of the loop In the first iteration of the loop, the pore-water pressure around tunnel II has a larger impact on tunnel I. The pore-water pressure acting around tunnel I should be the average value, and a water head ( h w 12 ) is applied around tunnel I to counteract the pore-water pressure around tunnel I. In the Cartesian coordinate system, the pore-water pressure can be expressed by using the coordinate transformation technique. When the pore-water pressure around tunnel II is solved, the iteration should be stopped until the pore-water pressure less than [ p ] or equal to zero. Because the average pore-water pressure is 1.703e4Pa, which does not achieve the standard, a water head ( h ' w 21 ) should be applied around tunnel II to counteract the pore-water pressure. Then continue solving, the pore-water pressure around tunnel II can be obtained and superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 7. 4.3.3 Third iteration of the loop In the second iteration of the loop, the average value of the pore-water pressure acting around tunnel Ⅰ is 6.518e3Pa, which is greater than [ p ], then continuing to iteration. And a water head ( h ' w 12 ) should be applied around tunnel I to counteract the pore-water pressure, and the pore-water pressure around tunnel I can be solved. The average value of the pore-water pressure around tunnel II is 2.494e3Pa, which does not achieve the standard, and then continuing to iteration. And the pore-water pressure around tunnel II can be obtained and superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 8. 4.3.4 Pore-water pressure solutions of twin shallow circular tunnels After the sixth iteration of the loop, the average value of pore-water pressure around tunnels is 20.463Pa, which satisfies the iterative stopping condition. And the pore-water pressure of twin shallow circular tunnels can be obtained by superimposing the results of first six iterations, which is shown in Fig. 9 . In the first three iterations, the results of pore water pressure superposition on the same horizontal plane tended to be symmetrically distributed. After the accuracy achieving the standard, the distribution of the pore-water pressure also confirmed the view. In the surrounding rock between tunnel I and tunnel II, the pore-water pressure within the plane from the ground to the center of the twin tunnels fluctuates due to the position of the twin tunnels, while the surrounding rock area below the center of the twin tunnels is consistent with the pore-water pressure distribution around tunnel I in Fig. 5(a). Therefore, the distribution of pore-water pressure of twin shallow circular tunnels is similar to the shallow circular tunnel when the depth of tunnel is large enough. It can be seen from Fig. 10 (a) to (c) that the pore-water pressure around tunnel I after the sixth iteration of the loop is similar to the pore-water pressure in the first iteration of the loop. And it can be seen from Fig. 10 (d) that the pore-water pressure is symmetrically distributed on the center line of twin shallow circular tunnels. 5. Validation In order to verify the rationality of the calculation approach in the pore-water pressure of twin shallow circular tunnels, the numerical model of the seepage in twin shallow circular tunnels has been established by FLAC 3D [ 19 – 21 ], which is shown in Fig. 11 . And in the condition of equal water head, the results of numerical simulation should be compared with the results of theoretical calculation. In the numerical model, the width is 100m, and the depth is 80m. The surface water level, depth of tunnel and radius are 30m, 20m and 5m, respectively. The rest of parameters are the same as the above case. And the permeability coefficient, porosity, fluid modulus and fluid density are 2×10 − 10 m/s, 0.5, 2×10 9 Pa and 9.81kg/m 3 , respectively. The calculation steps were set to 150,000. Considering the sufficient groundwater, the pore water pressure at the model boundary location is constant and uniformly distributed along the depth. Therefore, the results of numerical simulation in the pore-water pressure of twin shallow circular tunnels can be obtained and shown in Fig. 12 . Based on these, the results of numerical simulation in the pore-water pressure of the four sections were compared with the results of theoretical calculation, which is shown in Fig. 13. It can be seen from Fig. 13 (a) that the results of the numerical simulation are more consistent with the theoretical calculation when the pore-water pressure is near the tunnel and the ground, but there is still a certain deviation between the tunnel and the ground. And the maximum deviation occurs at the depth of 11.5m. The result of the theoretical calculation is 145.51kPa, but the result of the numerical simulation is 147.65kPa, whcih shows that the maximum error is about 1.45%. However, following the increase of the depth, the error is also reducing. It can be seen from Fig. 13(b) that the results of the theoretical calculation and the results of the numerical simulation are very close in the area where the depth is less than 65m, and the maximum error is only 0.398%. However, when the depth is more than 65m, the error between the theoretical calculation and the numerical simulation is gradually increasing. And the error is 2.15% when the depth is 80m, which will continue increasing with the depth. It is also found in Fig. 13(c) that the error between the theoretical calculation and numerical simulation is smaller within 25m from the left end of the tunnel I. But the error will increase as far away from the tunnel I. When x =-50m, the error between the theoretical calculation and numerical simulation is 0.365%. In addition, it is found in Fig. 13(d) that the results of the theoretical calculation and the numerical simulation are very close in the area where is close to the twin shallow circular tunnels or the midpoint of the center line. When x =-4m or x = 4m, the error is only 0.795%, which is the maximum error. Because of the finite boundary in FLAC 3D modeling, the calculation of the numerical simulation reaches the steady state faster than the theoretical calculation. And during the theoretical derivation, the average value of pore-water pressure around the tunnel was used in the iterations by the Schwarz alternating method, which can also cause the error in the results of the theoretical calculation. But on the whole, the error between the theoretical calculation and numerical simulation is very small. Therefore, the calculation approach in the pore-water pressure of twin shallow circular tunnels is verified to be reasonable. 6. Analysis and discussions of influencing factors of the pore-water pressure Twin tunnels are affected by inter-tunnel interactions, and their distribution of the pore-water pressure is different from the single shallow circular tunnel. In order to analyze the influence law of the number of holes and tunnel center spacing on the pore-water pressure of twin shallow circular tunnels, the control variable method is used to calculate the pore-water pressure in the different conditions of the number of holes and tunnel center spacing, respectively. The arrangement of monitoring positions is shown in Fig. 14 . 6.1. Influence of the number of holes The pore-water pressure around twin shallow circular tunnels and single shallow circular tunnel was calculated, and the values of parameters were shown in Table 2 . According to Fig. 14 , the distribution of the pore-water pressure in the four monitoring positions can be shown in Fig. 15. Table 2 Parameters of the number of holes. Parameters Value surface water level h w (m) 30 water head in the surrounding rock h 1 (m) 0 tunnel burial depth h (m) 20 tunnel radius r (m) 5 tunnel center spacing L (m) 20 water heaviness g (kN/m 3 ) 9.8 It is found in Fig. 15 that the pore-water pressure of the twin tunnels is slightly greater than the single tunnel in the upper area of tunnels. Besides, the pore-water pressure of the twin tunnels is lower than the pore-water pressure of the single tunnel at other monitoring locations, especially in Fig. 15(c). In the area beneath the tunnel, the pore-water pressure of the twin tunnels and the single tunnel are gradually converging with the depth increasing. 6.2. Influence of tunnel center spacing The tunnel center spacing of four conditions is 15m, 20m, 25m and 35m, respectively. And the values of parameters can be obtained by Table 2 . According to Fig. 14 , the distribution of the pore-water pressures at monitoring positions 1 and 2 can be shown in Fig. 16. And it is found from the comparison of four kinds of different tunnel center spacing that the pore-water pressure was gradually increasing with the tunnel center spacing increasing at the monitoring position 1. The maximum value of the pore-water pressure was at the midpoint of the center line of twin tunnels. At the monitoring position 2, the pore-water pressure was gradually reducing with the tunnel center spacing increasing. And the pore-water pressure around the twin tunnels in the conditions of different tunnel center spacing was gradually converging with the tunnel center spacing increasing. 7. Conclusions In this work, the bipolar coordinate system method and the Schwarz alternating method are adopted to calculate the pore-water pressure of the twin shallow circular tunnels in the semi-infinite space. The influence of the number of holes and the tunnel center spacing are analysed and discussed in this work. The main conclusions are as follows: The calculation approach in pore-water pressure of the twin shallow circular tunnels in the semi-infinite space is proposed using the Schwarz alternating method for the first time in the bipolar coordinate system. By multiple iterative cycles to counteract the interaction between the twin tunnels, the pore-water pressure of the twin shallow circular tunnels is obtained, which provides a new idea for solving the pore-water pressure under steady seepage. Compared with the existing methods, the efficiency of the proposed approach is higher, and the error is lower. When the accuracy is set to 1.0×10 − 3 MPa, the maximum error between the theoretical calculation and the numerical simulation is only 2.15%, which confirmed the rationality of the calculation approach in pore-water pressure of the twin shallow circular tunnels. And the proposed approach can provide theoretical guidance for the design and construction of tunnels under similar engineering geological conditions. The number of holes and the tunnel center spacing are major influence factors for pore-water pressure around tunnels. The influence of the number of holes mainly plays a role in the upper area of tunnels, which has an impact on the magnitude of the pore-water pressure. And in the area beneath tunnels, the pore-water pressure mainly varies with depth. The tunnel center spacing mainly determines the results of the pore-water pressure around the twin shallow circular tunnels, and the pore-water pressure around the twin shallow circular tunnels in the condition of different tunnel center spacing tends to be the same with the tunnel center spacing increasing. Declarations Data availability Data is provided within the manuscript or supplementary information files. 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Cite Share Download PDF Status: Published Journal Publication published 26 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 12 Sep, 2024 Reviews received at journal 10 Sep, 2024 Reviews received at journal 03 Sep, 2024 Reviews received at journal 30 Aug, 2024 Reviews received at journal 27 Aug, 2024 Reviewers agreed at journal 19 Aug, 2024 Reviewers agreed at journal 19 Aug, 2024 Reviewers agreed at journal 18 Aug, 2024 Reviewers agreed at journal 17 Aug, 2024 Reviewers agreed at journal 17 Aug, 2024 Reviewers invited by journal 17 Aug, 2024 Editor assigned by journal 17 Aug, 2024 Editor invited by journal 14 Aug, 2024 Submission checks completed at journal 12 Aug, 2024 First submitted to journal 29 Jul, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4824150","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":352258718,"identity":"7899a696-5a65-431c-8c7b-2c74f25c6e87","order_by":0,"name":"Tao Zhan","email":"","orcid":"","institution":"Metro Project Management Branch of Nanchang Rail Transit Group Limited Corporation","correspondingAuthor":false,"prefix":"","firstName":"Tao","middleName":"","lastName":"Zhan","suffix":""},{"id":352258719,"identity":"a12f0536-94fa-4942-93b4-73d473384377","order_by":1,"name":"Shengbiao Shan","email":"","orcid":"","institution":"Metro Project Management Branch of Nanchang Rail Transit Group Limited Corporation","correspondingAuthor":false,"prefix":"","firstName":"Shengbiao","middleName":"","lastName":"Shan","suffix":""},{"id":352258720,"identity":"138ce606-332b-44a8-a0ec-17db7aab10bd","order_by":2,"name":"Peiyuan He","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVRIiWNgGAWjYBACfvb2gw8+VNTU299/fIA4LZI9Z5INZ5w5lsBwIC2BOC0GNxLMpHnbmIFacgyIdNmBhATJGWfY8hgbzny88YbBTk63gYAOxoaDBww+VMgUMzP2bracw5BsbHaAgBZmxoaERKAtjG3MvNukeRgOJG4jpIWNmcHgMNAvjD1sPM+I08LDxmDYDNSSOIOHh404LRI8PMmMwEA2NpBgM7acY0CEX+zvPz/+AxiVcgYSzA9vvKmwkyOoBc1KYqMGSQupOkbBKBgFo2BEAADfKkSImnYy3QAAAABJRU5ErkJggg==","orcid":"","institution":"Central South University","correspondingAuthor":true,"prefix":"","firstName":"Peiyuan","middleName":"","lastName":"He","suffix":""}],"badges":[],"createdAt":"2024-07-29 19:30:59","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4824150/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4824150/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-03702-4","type":"published","date":"2025-12-26T15:57:07+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":64329003,"identity":"cb7ef2c8-8839-4a27-aac6-600025208225","added_by":"auto","created_at":"2024-09-11 17:45:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":30167,"visible":true,"origin":"","legend":"\u003cp\u003eTwin shallow circular tunnels in the semi-infinite space.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/f783caae5bceb3ec563f890b.png"},{"id":64329004,"identity":"820d9ecd-477a-4c9b-9237-3b288eb93e14","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":42791,"visible":true,"origin":"","legend":"\u003cp\u003eBernoulli's principle.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/2da120b89599106979a518ba.png"},{"id":64329005,"identity":"4a9d770d-99f3-4986-aa2f-08d3f2c8e044","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":29038,"visible":true,"origin":"","legend":"\u003cp\u003eCalculation model of pore-water pressure of twin shallow circular tunnels.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/3b474d01b640e96416d0a2ac.png"},{"id":64329300,"identity":"d3fde9aa-715b-4986-bded-4ec6572b5816","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":40401,"visible":true,"origin":"","legend":"\u003cp\u003eMonitoring points of pore-water pressure around tunnels.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/ce1070479c8edcefad80e7d3.png"},{"id":64329301,"identity":"488091d6-f4df-46ba-b346-bc607d3cecfc","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":497214,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure around tunnel I in the first iteration of the loop. (a) Sections I and section II; (b) Section 1-2; (c) Section 3-4; (d) Section 5-6; (e) Section 7-8.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/863241cc624b4bff69b36818.png"},{"id":64329302,"identity":"54f2fd22-3d7d-4f13-8025-a8fcc8fa4bd5","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":508335,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure of the twin shallow circular tunnels in the first iteration of the loop. (a) Sections I and section II; (b) Section 1-2; (c) Section 3-4; (d) Section 5-6; (e) Section 7-8.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/706ea56e875f5a76b3e4c1b7.png"},{"id":64329497,"identity":"7c1a975e-5f27-4cf5-b381-d2326f8c7141","added_by":"auto","created_at":"2024-09-11 18:01:54","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":515675,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure of the twin shallow circular tunnels in the second iteration of the loop. (a) Sections I and II; (b) Section 1-2; (c) Section 3-4; (d) Section 5-6; (e) Section 7-8.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/a33e1a94e92810a8a64ea71b.png"},{"id":64329007,"identity":"eb447bb1-3b1d-4899-b74e-33a4f9a09d9d","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":523219,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure of the twin shallow circular tunnels in the third iteration of the loop. (a) Sections I and section II; (b) Section 1-2; (c) Section 3-4; (d) Section 5-6; (e) Section 7-8.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/c140edf1bd3dbcb34e760920.png"},{"id":64329008,"identity":"12cb2405-b8ce-4e9c-b592-aa4ce0586b6d","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":241591,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure of twin shallow circular tunnels in sections I and II.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/f68fcb3b38e48302af36ef1a.png"},{"id":64329303,"identity":"38b2d3a0-2290-42e2-86e5-282489d296df","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":664698,"visible":true,"origin":"","legend":"\u003cp\u003ePore-water pressure of the twin shallow circular tunnels after the sixth iteration of the loop. (a) Section 1-2; (b) Section 3-4; (c) Section 5-6; (d) Section 7-8.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/bce16817a26968a33cd9badb.png"},{"id":64329016,"identity":"5c2d149e-ac16-4b93-acab-9c6a99680f26","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":568578,"visible":true,"origin":"","legend":"\u003cp\u003eNumerical model of the seepage of twin shallow circular tunnels.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/d8c202991c41b41c38d84b64.png"},{"id":64329015,"identity":"36aff330-c991-44b6-8108-f17927187d82","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":1231576,"visible":true,"origin":"","legend":"\u003cp\u003eResults of numerical simulation in the pore-water pressure of twin shallow circular tunnels.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/9e1163ff7d6f443e5778106a.png"},{"id":64329306,"identity":"a9411f8f-d528-47a7-9271-e3a9435841e3","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":968343,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the results in the pore-water pressure of twin shallow circular tunnels between the numerical simulation and the theoretical calculation. (a) Section 1-2; (b) Section 3-4; (c) Section 5-6; (d) Section 7-8.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/3daef0969add9c821fcea9b2.png"},{"id":64329013,"identity":"245621e6-7e4f-4b38-811b-dd3022f69083","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":331493,"visible":true,"origin":"","legend":"\u003cp\u003eThe arrangement of monitoring positions of the pore-water pressure of twin tunnels.\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/27e2af306403d791c58d8ac2.png"},{"id":64329018,"identity":"d0b4a2b3-8eb4-4afb-a919-1c42dd68365c","added_by":"auto","created_at":"2024-09-11 17:45:54","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":1135211,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the pore-water pressure of the single tunnel and the twin tunnels. (a) Pore-water pressure at monitoring location 1; (b) Pore-water pressure at monitoring location 2; (c) Pore-water pressure at monitoring location 3; (d) Pore-water pressure at monitoring location 4.\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/b603428f1836505b6cb6a6c8.png"},{"id":64329305,"identity":"010f984a-c9d7-4c5a-9676-87a3fa805911","added_by":"auto","created_at":"2024-09-11 17:53:54","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":545718,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the pore-water pressure of the influence of tunnel center spacing. (a) Pore-water pressure at monitoring location 1; (b) Pore-water pressure at monitoring location 2.\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/49e13a3b8fe1ff6a7a76e73a.png"},{"id":99172413,"identity":"329028a8-a3cd-43d8-923b-dab4bb80a4a4","added_by":"auto","created_at":"2025-12-29 16:09:04","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":10453927,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4824150/v1/61d1a751-5532-4f22-98ad-78ce614c71e8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A new method for determining the pore-water pressure around twin shallow circular tunnels","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn order to cope with the transportation problem in overcrowd urban, lots of metro tunnels are planning and constructing in many cities in China. The influence of groundwater is one of the most important concerns in tunnel engineering. During the water-rich tunnel excavation, due to the seepage effect, it is easy for groundwater to flow into the tunnel and generate the pore-water pressure in surrounding rock. And it will be negative to effective stresses in surrounding rock of tunnels and make the structures of tunnels failure. Therefore, for security, the seepage effect should be considered to keep the tunnel structures stable in the groundwater-rich areas. Up to now, some studies have been done on tunnel seepage. On the one hand, Shin et al. [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] studied the potential effect of pore water pressure on tunnel lining using the finite element method and proposed a method for estimating the design curve of pore water pressure for secondary lining. Pan et al. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] analyzed the effect of pore water pressure on the stability of the palm face by using fluid mechanics and numerical simulation. Park et al. [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] derived the analytical solution of the seepage force around the circular tunnel by using the complex variable method. Lee et al. [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] proposed the calculation approach for groundwater seepage in surrounding rock by using the limit analysis upper limit method. Yang et al. [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] proposed the stability solutions taking into account of water seepage of shallow tunnels based on strength reduction theory. On the other hand, for twin tunnels, Zhang et al. [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] derived the analytical solutions of non-Darcy seepage field in shallow single-hole and double-hole circular tunnels by using the principle of mirror method. Zhu et al. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] derive the stable seepage field of twin parallel tunnels by using the complex variable method. Zhang [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] derived the analytical solution for the stable seepage field of twin parallel tunnels in the semi-infinite plane by using the principle of mirror method. According to present results of studies, it can be seen that the boundary conditions of study methods are generally difficult to determine. In addition, the complex variable method is more complex, and it is also more difficult to apply in engineering. Therefore, it is necessary to make the studies on the calculation approach in the pore-water pressure of shallow twin circular tunnels to fill the gaps in current studies.\u003c/p\u003e \u003cp\u003eThe main objective of the current study is to obtain the accurate numerical solutions of pore-water pressure of twin shallow circular tunnels. On the basis of the bipolar coordinate system, taking the example of a shallow circular tunnel in a fully saturated, homogeneous and isotropous semi-infinite space, the calculation approach in the pore-water pressure of the shallow circular tunnel is proposed. Based on this, the calculation approach in the pore-water pressure of twin shallow circular tunnels can be proposed by Schwarz alternating methods. Through the comparison with the results of numerical simulation, the rationality of the calculation approach in this study is verified, and the influencing factors of the pore-water pressure is analysed. It is significant to propose the theoretical basis of the seepage of shallow tunnels for guiding the construction of tunnels securely.\u003c/p\u003e"},{"header":"2. Problem description and assumptions","content":"\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the radius, burial depth, tunnel spacing and surface water level of the twin shallow circular tunnels are \u003cem\u003er\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e, \u003cem\u003eL\u003c/em\u003e and \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e, respectively. And the ratio of the tunnel burial depth to radius is not more than seven. Supposing that the length of the tunnel is long enough, the three-dimensional space problem can be simplified to the two-dimensional plane problem. Surface water level is higher than the ground, so the soil is fully saturated. And the permeability coefficient of soil too small to keep surface water level stable. Therefore, the shallow twin circular tunnels are buried in a fully saturated, homogeneous and isotropous semi-infinite space. Based on this, the \u003cem\u003ex\u003c/em\u003e-axis is represented as the ground. And the \u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e-axis and \u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e-axis crossing the center of the twin circular tunnels are the symmetrical axis of the tunnels.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the twin shallow circular tunnels are the object of study in a semi-infinite space. Because the tunnel spacing is smaller, the duplex tunnels should not be considered.\u003c/p\u003e"},{"header":"3. Analytic solutions in the pore-water pressure of the shallow circular tunnel","content":"\u003cp\u003eBased on the study results of the literature [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], with the constant water head seepage mode, the approach in the pore-water pressure of the shallow circular tunnel is derived in the bipolar coordinate system to provide a theoretical basis for the solution of the twin shallow circular tunnel.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Stable seepage equations\u003c/h2\u003e \u003cp\u003eBecause of the complexity of soil pores, the direction of water seepage is often random. Therefore, the seepage is usually analyzed macroscopically in the engineering. The main direction of seepage can be only considered in the analysis, and the influence of soil can be ignored. Before the continuous medium theory proposed, Bernoulli's equation was commonly used in hydraulics and expressed as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$p+\\frac{1}{2}\\rho {v^2}+\\rho gh=C$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ep\u003c/em\u003e is the fluid pressure, \u003cem\u003ev\u003c/em\u003e is the fluid flow rate, \u003cem\u003eρ\u003c/em\u003e is the fluid density, \u003cem\u003eg\u003c/em\u003e is the acceleration of gravity, \u003cem\u003eh\u003c/em\u003e is the head height, and \u003cem\u003eC\u003c/em\u003e is a constant. Because Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) is derived from the law of conservation of energy, it is only applicable to ideal fluids which are non-viscous or less viscous and non-compressible. And the pressure water head can be expressed the total water head of the fluids which is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$h=\\frac{u}{{{\\gamma _w}}}+z+\\frac{{{{\\overline {v} }^2}}}{{2g}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e is the fluid pressure, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\overline {v}\\)\u003c/span\u003e\u003c/span\u003e is the permeation rate of water, and \u003cem\u003eγ\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e is the density of water. \u003cem\u003ez\u003c/em\u003e is the elevation head, which indicates the distance in the positive direction of the reference plane. If it lies below the \u003cem\u003ex\u003c/em\u003e-axis, the values take the negative.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt can be seen from Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e that water will flow from the high head to the low head through the connecting pore channel when there is a total head difference in the soil. And the total head difference (△\u003cem\u003eh\u003c/em\u003e) can be expressed as follows:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\Delta h{\\text{=}}{h_A} - {h_B}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewith\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${h_A}=\\frac{{{u_A}}}{{{\\gamma _w}}}+{z_A}+\\frac{{{{\\overline {{{v_A}}} }^2}}}{{2g}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${h_B}=\\frac{{{u_B}}}{{{\\gamma _w}}}+{z_B}+\\frac{{{{\\overline {{{v_B}}} }^2}}}{{2g}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe head loss of the unit process is the hydraulic gradient which is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$J{\\text{=}}\\frac{{\\Delta h}}{{\\Delta L}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eJ\u003c/em\u003e is the hydraulic gradient.\u003c/p\u003e \u003cp\u003eSince the seepage which belongs to laminar flow in the soil tends to flow slowly with small hydraulic gradients, the momentum changes can be ignored and the flow rate can be considered as zero. For the seepage of laminar flow, the water head at any position in the whole seepage space can be expressed as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$h=\\frac{u}{{{\\gamma _w}}}+z$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eDarcy's law reflects the linear relationship between the seepage rate of water in the soil and the head loss. When water flows in the soil in a laminar state, the seepage of water follows Darcy's law and can be expressed as follows:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${\\text{Q}}=kAJ$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$v=kJ$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIt can be seen from Darcy's law that the rate of passage is proportional to the magnitude of the hydraulic gradient and the permeability of the porous medium when water passes through the porous medium. However, the seepage rate of the fluid remains as a nonlinear relationship with the hydraulic gradient when the flow rate of the fluid is too fast. Therefore, Darcy's law is not applicable to the turbulence. Based on this, the fluid of study belongs to the laminar flow, with a constant flow rate per unit time.\u003c/p\u003e \u003cp\u003eIn the Cartesian coordinate system, water discharge velocity in the \u003cem\u003ex\u003c/em\u003e-axis direction is \u003cem\u003eu\u003c/em\u003e, and water discharge velocity in the \u003cem\u003ey\u003c/em\u003e-axis direction is \u003cem\u003ev\u003c/em\u003e. The water flow continuity equation can be obtained by\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\frac{{\\partial u}}{{\\partial x}}+\\frac{{\\partial v}}{{\\partial y}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAnd the velocity potential energy equation can be expressed as follows:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\phi (x,y)= - k\\left( {\\frac{p}{{{\\gamma _\\omega }}}+z} \\right)= - kh$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e), water discharge velocity in the x-axis direction (\u003cem\u003eu\u003c/em\u003e) and water discharge velocity in the \u003cem\u003ey\u003c/em\u003e-axis direction (\u003cem\u003ev\u003c/em\u003e) can be obtained by\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$u=\\frac{{\\partial \\varphi }}{{\\partial x}}= - k\\frac{{\\partial h}}{{\\partial x}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$v=\\frac{{\\partial \\varphi }}{{\\partial y}}= - k\\frac{{\\partial h}}{{\\partial y}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAnd the Laplace equation for the two-dimensional stable seepage [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] can be obtained by combining Eqs.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e) and (\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e13\u003c/span\u003e) with Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e), which is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e).\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$${\\nabla ^2}\\varphi =\\frac{{{\\partial ^2}\\varphi }}{{\\partial {x^2}}}+\\frac{{{\\partial ^2}\\varphi }}{{\\partial {y^2}}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the bipolar coordinate system, Eqs.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e) and (\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e) can be expressed by the following expressions using the coordinate transformation technique:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\phi (\\alpha ,\\beta )= - kh= - k(J\\sinh \\alpha +p/\\gamma )$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$${\\nabla ^2}\\varphi =\\frac{1}{{{J^2}}}\\left[ {\\frac{\\partial }{{\\partial \\alpha }}(\\frac{{\\partial \\varphi }}{{\\partial \\alpha }})+\\frac{\\partial }{{\\partial \\beta }}(\\frac{{\\partial \\varphi }}{{\\partial \\beta }})} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Pore-water pressure solutions for the shallow circular tunnel\u003c/h2\u003e \u003cp\u003eSupposing that the permeability coefficient on the outside of lining structures is higher, the head at the tunnel boundary is constant, and the water head at infinity is only related to the tunnel depth. In the semi-infinite space, the Laplace equation for the water head can be expressed by the following expression using the coordinate transformation technique:\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$${\\nabla ^2}h=\\frac{1}{{{J^2}}}(\\frac{{{\\partial ^2}h}}{{\\partial {\\alpha ^2}}}+\\frac{{{\\partial ^2}h}}{{\\partial {\\beta ^2}}})=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the boundary conditions of stable seepage, the head of pore water in the surrounding rock can be obtained by\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$$h={h_0}+\\frac{{{h_{{a_i}}} - {h_0}}}{{{\\alpha _i}}}\\alpha$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e is the tunnel boundary head, \u003cem\u003eh\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the head of water at semi-infinite space infinity.\u003c/p\u003e \u003cp\u003eSubstituting Eq.\u0026nbsp;(\u003cspan refid=\"Equ20\" class=\"InternalRef\"\u003e18\u003c/span\u003e) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e), the pore-water pressure in the surrounding rock under constant water head conditions can be obtained by\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$$p={h_0}{\\gamma _w}+\\frac{{{h_{{a_i}}} - {h_0}}}{{{\\alpha _i}}}{\\gamma _{\\text{w}}}\\alpha - J\\sinh \\alpha {\\gamma _w}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e"},{"header":"4. The Calculation approach in the pore-water pressure of twin shallow circular tunnels","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1. Calculation model\u003c/h2\u003e\n \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, the left tunnel is tunnel I and the right tunnel is tunnel II. Both tunnels I and II are located below the surface water level. The head of surface water on the tunnel is \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e, and the water head in the surrounding rock is \u003cem\u003eh\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e. In the bipolar coordinate system, the coordinates of any point of the tunnel are (\u003cem\u003e\u0026alpha;\u003c/em\u003e,\u003cem\u003e\u0026beta;\u003c/em\u003e). The coordinate of the tunnel wall is \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e. The coordinate of the ground is zero. In a coordinate system with the \u003cem\u003ey\u003c/em\u003e-axis oriented vertically downward as the positive direction, \u003cem\u003e\u0026alpha;\u003c/em\u003e is constantly greater than zero and \u003cem\u003e\u0026beta;\u003c/em\u003e ranges from -\u0026pi; to \u0026pi;. Supposing that the lining structures don\u0026rsquo;t have infiltration capacity, the boundary conditions belong to constant water head seepage mode. Because the twin tunnels belong to the multi-connected domain problem [\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e], the calculation approach in pore-water pressure of twin shallow circular tunnels can be obtained by using Schwarz alternating method to make the solutions of pore-water pressure of shallow circular tunnel performed for multiple iterations.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2. Schwarz alternating method\u003c/h2\u003e\n \u003cp\u003eThe Schwarz alternating method was proposed for solving the Dirichlet problem and later proved its convergence in two and three dimensions by Mikhlin [\u003cspan class=\"CitationRef\"\u003e10\u003c/span\u003e] and Soboldff [\u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e] successively. Therefore, for the analysis of the elastic zone of the surrounding rock in a multi-connected domain, the sequence of loading and geometry changes have no effect on the stresses in the surrounding rock. Therefore, the Schwarz alternating method can transform the multi-connected domain problems into the single-connected domain problems [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e] to simplify the solutions of the pore-water pressure of twin shallow circular tunnels in a semi-infinite space.\u003c/p\u003e\n \u003cp\u003eAnd the solution of the pore-water pressure of shallow circular tunnel in the bipolar coordinate system is extended to the twin tunnels by the Schwarz alternating method.\u003c/p\u003e\n \u003cp\u003e(1) Supposing that tunnel II does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, the pore-water pressure of tunnel I in the bipolar coordinate system can only be obtained and expressed as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e$${p_{w11}}={h_0}{\\gamma _w}+\\frac{{{h_{{a_i}}} - {h_0}}}{{{\\alpha _i}}}{\\gamma _{\\text{w}}}\\alpha - J\\sinh \\alpha {\\gamma _w}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e11\u003c/sub\u003e is the pore-water pressure of tunnel I.\u003c/p\u003e\n \u003cp\u003e(2) Supposing that tunnel I does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, a water head (\u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e21\u003c/sub\u003e) is applied around tunnel II to counteract the pore-water pressure around tunnel II which is generated by tunnel I. And the pore-water pressure around tunnel I which is generated by tunnel II can be expressed as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e$${p_{w21}}={h_0}{\\gamma _w}+\\frac{{{h_{{a_i}}}+{h_{w21}} - {h_0}}}{{{\\alpha _i}}}{\\gamma _{\\text{w}}}\\alpha - J\\sinh \\alpha {\\gamma _w}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e21\u003c/sub\u003e is the pore-water pressure around tunnel I which is generated by tunnel II.\u003c/p\u003e\n \u003cp\u003e(3) Similarly, supposing that tunnel II does not exist in the fully saturated, homogeneous and isotropous semi-infinite space, a water head (\u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e12\u003c/sub\u003e) is applied around tunnel I to counteract the pore-water pressure around tunnel I which is generated by tunnel II. And the pore-water pressure around tunnel II which is generated by tunnel I can be expressed as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e$${p_{w12}}={h_0}{\\gamma _w}+\\frac{{{h_{{a_i}}}+{h_{w12}} - {h_0}}}{{{\\alpha _i}}}{\\gamma _{\\text{w}}}\\alpha - J\\sinh \\alpha {\\gamma _w}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e(4) Until the value of \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e12\u003c/sub\u003e small enough or equal to zero, and making \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e12\u003c/sub\u003e\u0026lt;[\u003cem\u003ep\u003c/em\u003e] stand, the interaction can be stopped. Otherwise, the interaction will continue until \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e achieving the standards. And the numerical solution of the pore-water pressure around the twin shallow circular tunnels can be obtained by superimposing the results of each iteration\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003e4.3. Pore-water pressure solutions\u003c/h2\u003e\n \u003cp\u003eIn the bipolar coordinate system, when the pore-water pressure is solved, the parameters of the twin shallow circular tunnels in the semi-infinite space are shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eParameters of the twin shallow circular tunnels.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eValue\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003esurface water level \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ewater head in the surrounding rock \u003cem\u003eh\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel burial depth \u003cem\u003eh\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel radius \u003cem\u003er\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel center spacing \u003cem\u003eL\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ewater heaviness \u003cem\u003eg\u003c/em\u003e (kN/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eSince the surface water and groundwater are used twice during the calculation of the the Schwarz alternating method, \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e1\u003c/sub\u003e=0.5\u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;15m is set. The effect of groundwater should be ignored in the second iteration, and the interaction between tunnels is only considered. When the interaction between tunnels is small enough, the numerical solution of the pore-water pressure can be obtained accurately. Therefore, the threshold value ([\u003cem\u003ep\u003c/em\u003e]) is 10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003eMPa, which controls the iteration stopping. As shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e, the value range of \u003cem\u003ex\u003c/em\u003e-axis is set to [-100,100] and the value range of \u003cem\u003ey\u003c/em\u003e-axis is set to [0,100], from which the two cross sections at the positions of \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;10m and \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;30m in the semi-infinite space are taken as the study area to observe the pore-water pressure distribution pattern at the same depth in the surrounding rock of the twin shallow circular tunnels. And the sections 1\u0026ndash;2, 3\u0026ndash;4, 5\u0026ndash;6, and 7\u0026ndash;8 are selected as the monitoring points of the pore-water pressure around tunnel I.\u003c/p\u003e\n \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e\n \u003ch2\u003e4.3.1 \u003cem\u003eFirst iteration of the loop\u003c/em\u003e\u003c/h2\u003e\n \u003cp\u003eIn the bipolar coordinate system, the pore-water pressure around tunnel I is solved when tunnel II is supposed not to be in the system. And the pore-water pressure in the Cartesian coordinate system can be expressed by using the coordinate transformation technique. The pore-water pressure around tunnel I belongs to the shallow circular tunnel. And the calculation results of the pore-water pressure around tunnel I in the above sections are shown in Fig.\u0026nbsp;5.\u003c/p\u003e\n \u003cdiv\u003eWhen the influence of the pore-water pressure around tunnel I on tunnel II is analysed, the pore-water pressure acting around tunnel II should be the average value, and a water head (\u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e21\u003c/sub\u003e) is applied around tunnel II to counteract the pore-water pressure around tunnel II. In the same way, the pore-water pressure around tunnel II is solved when tunnel I is supposed not to be in the bipolar coordinate system. And the results of the pore-water pressure around tunnel II should be superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 6.\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e\n \u003ch2\u003e4.3.2 \u003cem\u003eSecond iteration of the loop\u003c/em\u003e\u003c/h2\u003e\n \u003cp\u003eIn the first iteration of the loop, the pore-water pressure around tunnel II has a larger impact on tunnel I. The pore-water pressure acting around tunnel I should be the average value, and a water head (\u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e12\u003c/sub\u003e) is applied around tunnel I to counteract the pore-water pressure around tunnel I. In the Cartesian coordinate system, the pore-water pressure can be expressed by using the coordinate transformation technique.\u003c/p\u003e\n \u003cp\u003eWhen the pore-water pressure around tunnel II is solved, the iteration should be stopped until the pore-water pressure less than [\u003cem\u003ep\u003c/em\u003e] or equal to zero. Because the average pore-water pressure is 1.703e4Pa, which does not achieve the standard, a water head (\u003cem\u003eh\u003c/em\u003e\u003csup\u003e\u0026apos;\u003c/sup\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e21\u003c/sub\u003e) should be applied around tunnel II to counteract the pore-water pressure. Then continue solving, the pore-water pressure around tunnel II can be obtained and superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 7.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e\n \u003ch2\u003e4.3.3 \u003cem\u003eThird iteration of the loop\u003c/em\u003e\u003c/h2\u003e\n \u003cp\u003eIn the second iteration of the loop, the average value of the pore-water pressure acting around tunnel Ⅰ is 6.518e3Pa, which is greater than [\u003cem\u003ep\u003c/em\u003e], then continuing to iteration. And a water head (\u003cem\u003eh\u003c/em\u003e\u003csup\u003e\u0026apos;\u003c/sup\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e12\u003c/sub\u003e) should be applied around tunnel I to counteract the pore-water pressure, and the pore-water pressure around tunnel I can be solved.\u003c/p\u003e\n \u003cp\u003eThe average value of the pore-water pressure around tunnel II is 2.494e3Pa, which does not achieve the standard, and then continuing to iteration. And the pore-water pressure around tunnel II can be obtained and superimposed with the pore-water pressure around tunnel I, which are shown in Fig. 8.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e\n \u003ch2\u003e4.3.4 \u003cem\u003ePore-water pressure solutions of twin shallow circular tunnels\u003c/em\u003e\u003c/h2\u003e\n \u003cp\u003eAfter the sixth iteration of the loop, the average value of pore-water pressure around tunnels is 20.463Pa, which satisfies the iterative stopping condition. And the pore-water pressure of twin shallow circular tunnels can be obtained by superimposing the results of first six iterations, which is shown in Fig. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e. In the first three iterations, the results of pore water pressure superposition on the same horizontal plane tended to be symmetrically distributed. After the accuracy achieving the standard, the distribution of the pore-water pressure also confirmed the view. In the surrounding rock between tunnel I and tunnel II, the pore-water pressure within the plane from the ground to the center of the twin tunnels fluctuates due to the position of the twin tunnels, while the surrounding rock area below the center of the twin tunnels is consistent with the pore-water pressure distribution around tunnel I in Fig.\u0026nbsp;5(a). Therefore, the distribution of pore-water pressure of twin shallow circular tunnels is similar to the shallow circular tunnel when the depth of tunnel is large enough.\u003c/p\u003e\n \u003cdiv\u003eIt can be seen from Fig. 10 (a) to (c) that the pore-water pressure around tunnel I after the sixth iteration of the loop is similar to the pore-water pressure in the first iteration of the loop. And it can be seen from Fig. 10 (d) that the pore-water pressure is symmetrically distributed on the center line of twin shallow circular tunnels.\u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e"},{"header":"5. Validation","content":"\u003cp\u003eIn order to verify the rationality of the calculation approach in the pore-water pressure of twin shallow circular tunnels, the numerical model of the seepage in twin shallow circular tunnels has been established by FLAC\u003csup\u003e3D\u003c/sup\u003e [\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e], which is shown in Fig. \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e. And in the condition of equal water head, the results of numerical simulation should be compared with the results of theoretical calculation.\u003c/p\u003e\n\u003cp\u003eIn the numerical model, the width is 100m, and the depth is 80m. The surface water level, depth of tunnel and radius are 30m, 20m and 5m, respectively. The rest of parameters are the same as the above case. And the permeability coefficient, porosity, fluid modulus and fluid density are 2\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;10\u003c/sup\u003em/s, 0.5, 2\u0026times;10\u003csup\u003e9\u003c/sup\u003ePa and 9.81kg/m\u003csup\u003e3\u003c/sup\u003e, respectively. The calculation steps were set to 150,000. Considering the sufficient groundwater, the pore water pressure at the model boundary location is constant and uniformly distributed along the depth. Therefore, the results of numerical simulation in the pore-water pressure of twin shallow circular tunnels can be obtained and shown in Fig. \u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e. Based on these, the results of numerical simulation in the pore-water pressure of the four sections were compared with the results of theoretical calculation, which is shown in Fig. 13.\u003c/p\u003e\n\u003cp\u003eIt can be seen from Fig. 13 (a) that the results of the numerical simulation are more consistent with the theoretical calculation when the pore-water pressure is near the tunnel and the ground, but there is still a certain deviation between the tunnel and the ground. And the maximum deviation occurs at the depth of 11.5m. The result of the theoretical calculation is 145.51kPa, but the result of the numerical simulation is 147.65kPa, whcih shows that the maximum error is about 1.45%. However, following the increase of the depth, the error is also reducing. It can be seen from Fig. 13(b) that the results of the theoretical calculation and the results of the numerical simulation are very close in the area where the depth is less than 65m, and the maximum error is only 0.398%. However, when the depth is more than 65m, the error between the theoretical calculation and the numerical simulation is gradually increasing. And the error is 2.15% when the depth is 80m, which will continue increasing with the depth. It is also found in Fig. 13(c) that the error between the theoretical calculation and numerical simulation is smaller within 25m from the left end of the tunnel I. But the error will increase as far away from the tunnel I. When \u003cem\u003ex\u003c/em\u003e=-50m, the error between the theoretical calculation and numerical simulation is 0.365%. In addition, it is found in Fig. 13(d) that the results of the theoretical calculation and the numerical simulation are very close in the area where is close to the twin shallow circular tunnels or the midpoint of the center line. When \u003cem\u003ex\u003c/em\u003e=-4m or \u003cem\u003ex\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4m, the error is only 0.795%, which is the maximum error. Because of the finite boundary in FLAC\u003csup\u003e3D\u003c/sup\u003e modeling, the calculation of the numerical simulation reaches the steady state faster than the theoretical calculation. And during the theoretical derivation, the average value of pore-water pressure around the tunnel was used in the iterations by the Schwarz alternating method, which can also cause the error in the results of the theoretical calculation. But on the whole, the error between the theoretical calculation and numerical simulation is very small. Therefore, the calculation approach in the pore-water pressure of twin shallow circular tunnels is verified to be reasonable.\u003c/p\u003e"},{"header":"6. Analysis and discussions of influencing factors of the pore-water pressure","content":"\u003cp\u003eTwin tunnels are affected by inter-tunnel interactions, and their distribution of the pore-water pressure is different from the single shallow circular tunnel. In order to analyze the influence law of the number of holes and tunnel center spacing on the pore-water pressure of twin shallow circular tunnels, the control variable method is used to calculate the pore-water pressure in the different conditions of the number of holes and tunnel center spacing, respectively. The arrangement of monitoring positions is shown in Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\n \u003ch2\u003e6.1. Influence of the number of holes\u003c/h2\u003e\n \u003cp\u003eThe pore-water pressure around twin shallow circular tunnels and single shallow circular tunnel was calculated, and the values of parameters were shown in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. According to Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e, the distribution of the pore-water pressure in the four monitoring positions can be shown in Fig. 15.\u003c/p\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eParameters of the number of holes.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eValue\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003esurface water level \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ewater head in the surrounding rock \u003cem\u003eh\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel burial depth \u003cem\u003eh\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel radius \u003cem\u003er\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003etunnel center spacing \u003cem\u003eL\u003c/em\u003e (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ewater heaviness \u003cem\u003eg\u003c/em\u003e (kN/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003eIt is found in Fig.\u0026nbsp;15 that the pore-water pressure of the twin tunnels is slightly greater than the single tunnel in the upper area of tunnels. Besides, the pore-water pressure of the twin tunnels is lower than the pore-water pressure of the single tunnel at other monitoring locations, especially in Fig.\u0026nbsp;15(c). In the area beneath the tunnel, the pore-water pressure of the twin tunnels and the single tunnel are gradually converging with the depth increasing.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\n \u003ch2\u003e6.2. Influence of tunnel center spacing\u003c/h2\u003e\n \u003cp\u003eThe tunnel center spacing of four conditions is 15m, 20m, 25m and 35m, respectively. And the values of parameters can be obtained by Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. According to Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e, the distribution of the pore-water pressures at monitoring positions 1 and 2 can be shown in Fig. 16. And it is found from the comparison of four kinds of different tunnel center spacing that the pore-water pressure was gradually increasing with the tunnel center spacing increasing at the monitoring position 1. The maximum value of the pore-water pressure was at the midpoint of the center line of twin tunnels. At the monitoring position 2, the pore-water pressure was gradually reducing with the tunnel center spacing increasing. And the pore-water pressure around the twin tunnels in the conditions of different tunnel center spacing was gradually converging with the tunnel center spacing increasing.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"7. Conclusions","content":"\u003cp\u003eIn this work, the bipolar coordinate system method and the Schwarz alternating method are adopted to calculate the pore-water pressure of the twin shallow circular tunnels in the semi-infinite space. The influence of the number of holes and the tunnel center spacing are analysed and discussed in this work. The main conclusions are as follows:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe calculation approach in pore-water pressure of the twin shallow circular tunnels in the semi-infinite space is proposed using the Schwarz alternating method for the first time in the bipolar coordinate system. By multiple iterative cycles to counteract the interaction between the twin tunnels, the pore-water pressure of the twin shallow circular tunnels is obtained, which provides a new idea for solving the pore-water pressure under steady seepage. Compared with the existing methods, the efficiency of the proposed approach is higher, and the error is lower. When the accuracy is set to 1.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eMPa, the maximum error between the theoretical calculation and the numerical simulation is only 2.15%, which confirmed the rationality of the calculation approach in pore-water pressure of the twin shallow circular tunnels. And the proposed approach can provide theoretical guidance for the design and construction of tunnels under similar engineering geological conditions.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe number of holes and the tunnel center spacing are major influence factors for pore-water pressure around tunnels. The influence of the number of holes mainly plays a role in the upper area of tunnels, which has an impact on the magnitude of the pore-water pressure. And in the area beneath tunnels, the pore-water pressure mainly varies with depth. The tunnel center spacing mainly determines the results of the pore-water pressure around the twin shallow circular tunnels, and the pore-water pressure around the twin shallow circular tunnels in the condition of different tunnel center spacing tends to be the same with the tunnel center spacing increasing.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData is provided within the manuscript or supplementary information files.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors gratefully acknowledge that the financial support from the Nanchang Rail Transit Group 2020 Annual Research Program (2020HGKYB002).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eShin, J.H.; Potts, D.M.; Zdravkovic, L. 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Analysis of the limit support pressure of a shallow shield tunnel in sandy soil considering the influence of seepage. \u003cem\u003eSymmetry-Basel\u003c/em\u003e. \u003cstrong\u003e2020\u003c/strong\u003e, 12, 1023.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Twin shallow circular tunnels, Pore-water pressure, Bipolar coordinate system, Schwarz alternating method, Steady seepage, Semi-infinite space","lastPublishedDoi":"10.21203/rs.3.rs-4824150/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4824150/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn order to obtain the distribution rules of pore-water pressure for twin shallow circular tunnels, the determination approach in the pore-water pressure for the twin shallow circular tunnels buried in a semi-infinite space was proposed by using the bipolar coordinate system method and the Schwarz alternating method for the first time. The solution of pore-water pressure was also obtained by multiple iterations method. The proposed approach was validated by the results of numerical simulation. The maximum error between the results of the analytical solutions from the proposed method and the numerical simulation was only 2.15% when the accuracy was set to 1.0\u0026times;10-3MPa. At last, influences of the number of holes and the tunnel center spacing on the pore-water pressure were presented. It is shown that the number of holes and the tunnel center spacing were the main factors for the distribution of pore-water pressure, and the pore-water pressure of twin shallow circular tunnels was only greater than the pore-water pressure of single shallow circular tunnel in the upper area of tunnels. And the maximum result for the pore-water pressure of twin shallow circular tunnels was obtained in the midpoint of the tunnel center spacing.\u003c/p\u003e","manuscriptTitle":"A new method for determining the pore-water pressure around twin shallow circular tunnels","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-11 17:45:48","doi":"10.21203/rs.3.rs-4824150/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-09-12T09:05:45+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-11T03:46:16+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-03T09:05:13+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-08-30T05:15:34+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-08-27T14:22:52+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"88916663057191861885647937417315541566","date":"2024-08-19T23:57:36+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"332736501993419884148440128948011180900","date":"2024-08-19T13:17:24+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"302824712334031199516464389949614218903","date":"2024-08-18T05:39:58+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"173398464301748603542783465461641557129","date":"2024-08-18T00:16:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"271388816634923729990823925976668026839","date":"2024-08-17T13:18:57+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-08-17T13:12:02+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-08-17T13:00:41+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-08-14T16:03:26+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-08-12T10:24:47+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-07-29T19:29:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"663ed923-dae9-43c1-b3e3-d533624d1254","owner":[],"postedDate":"September 11th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-12-29T16:03:55+00:00","versionOfRecord":{"articleIdentity":"rs-4824150","link":"https://doi.org/10.1038/s41598-025-03702-4","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-12-26 15:57:07","publishedOnDateReadable":"December 26th, 2025"},"versionCreatedAt":"2024-09-11 17:45:48","video":"","vorDoi":"10.1038/s41598-025-03702-4","vorDoiUrl":"https://doi.org/10.1038/s41598-025-03702-4","workflowStages":[]},"version":"v1","identity":"rs-4824150","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4824150","identity":"rs-4824150","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}