Key factors influencing analytical solutions for predicting groundwater inflows in rock tunnels Open Access

Accurate prediction of groundwater inflows into tunnels is an urgent global need. Therefore, it has become a hotspot research topic. One of the causes of severe damage in underground space is the appearance of unforeseen groundwater influxes during and after the excavation of tunnels (Hwang & Lu 2007; Font-Capó et al. 2011). For instance, the influxes of water that have repeatedly eventuated in the Wulong tunnel of China have resulted in many casualties and economic losses (Li et al. 2016a). Likewise, several deaths and financial losses were recorded when a devastating influx of groundwater occurred on May 18, 2006 at the Xinjiang Coal Mine in Dating, Shanxi Province of China (Cui et al. 2020). From 2000 to 2018, referring to Cui et al. (2020), a total of 1,184 groundwater inflow accidents in Chinese coal mines caused considerable losses. In addition to China, in various other countries such as the United Kingdom, the United States of America, Canada, Germany, Russia, Poland, and Australia, the victims generated by the inflows of groundwater in coal mines, in particular, are a six-digit number (Ma et al. 2016). As substantial geological hazards (Jiang et al. 2010), groundwater ingress into tunnels is almost unavoidable during and after tunneling in most types of rock mass. They become growingly severe especially at great depth (Yu et al. 2020). Thereby, it remains a primary concern to accurately predict groundwater inflows into tunnels for optimal and sustainable management. More specifically, accurate predictions of these inflows are necessary to ensure safety, size adequate drainage systems and manage anticipated peak flows (Miladinović et al. 2015).

Indeed, given the plurality of their roles, underground tunnels are essential. Despite the associated risks, tunnel construction projects are on the rise worldwide (Sousa & Einstein 2021). Tunnels are usually required for water supply, reservoir emptying, hydropower stations, sanitary drainage, transport systems, mining, and so on (Lin et al. 2019). Whatever the function of a tunnel, groundwater inflows are of decisive importance in judging the degree of safety and stability. On the one hand, such inflows can generate instability and diminish the pertinent properties of the rocks surrounding the tunnels (Stille & Palmström 2008). On the other hand, they can engender exaggerated environmental impacts (Liu et al. 2018). Additionally, it should be noted that groundwater inflows increase the risks and challenges associated with tunnel construction (Li et al. 2020).

So far, despite many research efforts, the accurate prediction of groundwater inflows into rock tunnels remains a difficult task. This can be explained by the complexity linked to the behavior of the rocks located especially at great depth. The existing analytical solutions generally consider simplifying assumptions that do not always reflect actual rock characteristics and conditions. However, analytic solutions are very important for rapid estimates of groundwater inflows into tunnels. Importantly, they provide parameters governing groundwater inflows in tunnels. Technically, rapid assessment of groundwater inflows in tunnels is generally necessary (Gattinoni & Scesi 2010; Sedghi & Zhan 2021). In view of the significance of such solutions, it is important to examine the key factors that govern them.

The purpose of this article is to provide relevant information on the key factors that profoundly influence the accuracy of groundwater inflow predictions in tunnels. Furthermore, the paper summarizes important details about existing analytical solutions that predict such influxes into rock tunnels. This can make it possible to take into account the issues related to the prediction of said hazard, and the huge need to improve the accuracy of its predictions.

During dynamic events such as rockbursts that can be triggered near any latent water reservoir formation, groundwater inflows can spontaneously occur in rock tunnels (Odintsev & Miletenko 2015). This is very possible when the surrounding rocks of tunnels have been considerably altered. In karstic rocks, groundwater inflows can appear after the formation of cracks and water channels (Li et al. 2020). The more the rocks are weathered, the more easily they can be fractured. Groundwater inflow paths are favored by a large amount of fractures and faults in the rocks (Lipponen et al. 2005; Hornero et al. 2021; Rahimi et al. 2021; Su et al. 2022). Specifically, most groundwater inflows are primarily associated with geological features such as open faults and fractures (Zarei et al. 2011; Hornero et al. 2021). In such situations, even the rock support systems of tunnels can be affected. Indeed, without proper treatments, groundwater ingress can occur even in bolted tunnels in fractured rocky environments (Frenelus et al. 2022a). As stated by Zhang et al. (2018), the interface of different layers, deep weathered hollows and tectonically fractured zones are responsible for groundwater inflows into tunnels located in insoluble rocks. Importantly, the proportion of fragile tectonic overprint is a major factor controlling the rate of groundwater inflows into tunnels located in crystalline rocks (Masset & Loew 2010). The coupling of unfavorable geological conditions and excavation disturbances is also an important factor triggering groundwater inflows into tunnels (Zhang et al. 2017). In addition to fractures activated by excavation perturbations, newly opened rock cracks caused by changes in confining pressures and additional large shear stresses can be major precursors of groundwater ingress into tunnels (Guo et al. 2019). The distance that exists between the fracture zones and water-structures also affects groundwater inflows into the tunnels (Keqiang et al. 2011), regardless of the amount of water available. By excavation disturbances, the smaller this distance, the greater the possibility of triggering groundwater inflows in the tunnels. To sum up, groundwater inflows into rock tunnels located in non-water-rich areas are likely to trigger if:

• – The extent of rock damage during and after tunneling is considerable enough.

• – There is a wide distribution of fractures and faults around the tunnel route.

• – There is a favorable distance between the excavated areas and water available at the existing water-bearing structures.

• – The permeability of the rocks surrounding the tunnel is highly increased, and the water pressure is sufficient.

• – The safety thickness of the host rocks is ultimately no longer able to resist groundwater inflows.

It is important to mention that in non-water-rich water areas, high groundwater inflows are not commonly observed on a large scale. For these areas, there are not yet fixed minimum safety thickness values for host rocks to resist groundwater inflows. Further studies will be needed to find out more. However, for urban tunnels built in karst terrain, Table 1 presents the relevant expressions to estimate the safety thickness of the surrounding rocks to resist groundwater inflows.

Table 1

Safety thickness of rocks surrounding the tunnels built in karst terrains

Study .	Methods .	Safety thickness and parameters .	Comments .	Equation number .
Yang & Xiao (2016)	Clamped beam theory and engineering experiences	⁠: critical safety thickness of rocks; q: rock loads; E: elastic modulus of rocks; L: beam length.	The presence or effect of water is not considered. The caves are located in the tunnel floor.	(4)
Yang & Xiao (2016)	Clamped beam theory	⁠: softening factor of the rocks; ⁠: confined water downforce; other parameters: same as above.	The presence and effect of water are considered. The caves are located in the tunnel floor.	(5)
Li et al. (2020)	Numerical simulation and multiple regression analysis	S: reserved safety thickness in m; R: cutting head diameter adjustment (mm); P: water pressure en MPa; D: cave water dimeter (m); ⁠: factor associated to different types of cutter head.	Karst caves can be filled with water.	(6)

Study .	Methods .	Safety thickness and parameters .	Comments .	Equation number .
Yang & Xiao (2016)	Clamped beam theory and engineering experiences	⁠: critical safety thickness of rocks; q: rock loads; E: elastic modulus of rocks; L: beam length.	The presence or effect of water is not considered. The caves are located in the tunnel floor.	(4)
Yang & Xiao (2016)	Clamped beam theory	⁠: softening factor of the rocks; ⁠: confined water downforce; other parameters: same as above.	The presence and effect of water are considered. The caves are located in the tunnel floor.	(5)
Li et al. (2020)	Numerical simulation and multiple regression analysis	S: reserved safety thickness in m; R: cutting head diameter adjustment (mm); P: water pressure en MPa; D: cave water dimeter (m); ⁠: factor associated to different types of cutter head.	Karst caves can be filled with water.	(6)

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The mechanisms of groundwater inflows in rock tunnels are mostly controlled by the features presented in Figure 5. As reported by Sharifzadeh et al. (2012), geological features, rock mass conditions and groundwater are closely interrelated. The local hydrogeology surrounding the tunnels are important for groundwater inflow mechanisms. In fact, to properly predict groundwater inflows into tunnels, the hydrogeological regime is required (Nwankwor et al. 1988). Around the tunnels, the hydrogeological regime varies with the local hydrogeology. Primarily, as presented in Figure 5, the local hydrogeology that surrounds the tunnels can consist of geological features, regional hydrology, rock mass conditions and groundwater. Therefore, the mechanism of groundwater inflows into tunnels is governed by the local hydrogeology and tunnel features (Sharifzadeh et al. 2012). Tunnel features include not only geometric characteristics such as radius, depth or shape, but also the excavation methods used. Each excavation method has adverse effects on the characteristics of the rock mass and therefore on the groundwater inflow into the tunnels. For example, compared to TBM tunneling, DB excavation generally has more adverse effects on rocks surrounding tunnels (Frenelus et al. 2021a). Accordingly, under the coupling effect of local hydrogeology and tunnel features, the extent of groundwater inflows in tunnels can be due to one of the forms presented in Figure 4. Indeed, when the groundwater inflow mechanism is well predicted before tunneling, adequate measures can be taken in order to avoid unwanted damage that could be related to huge flow rates.

Groundwater inflows into rock tunnels depend on several key factors. The actual conditions of these factors are instrumental in accurately predicting these inflows. It is very interesting to identify and describe such factors. Their proper consideration in the development of solutions can provide accurate predictions of groundwater inflows. The key factors that should be given great attention are: time dependency of groundwater inflows, water-rich structures, aquifer thickness, hydraulic head and groundwater drawdown, rock permeability and hydraulic conductivity, fractures aperture, and rainfall data. All of these key factors vary depending on the relevant geological and hydrogeological conditions.

The greatest damage and economic loss are usually caused by the sudden influx of groundwater into tunnels. Table 2 presents a summary of potential damage caused by large unexpected inflows of groundwater into rock tunnels. It is very important to highlight the potential damage generated by unforeseen groundwater inflows in rock tunnels. Indeed, the greater the damage, the greater the impact and the more complicated management. Not only can excessive damage lead to excessive financial losses, but it could also delay the stabilization phase of groundwater inflows and thus influence the time-dependency of inflows. That is, depending on the scope of potential damage from groundwater inflows into tunnels, persistent inflows may occur more or less quickly or be delayed further. It should be noted that the potential damage caused by groundwater inflows mainly depends on geological, hydrogeological and tunneling conditions of rocks surrounding the tunnels. For instance, geological features such as fractures and faults are among the factors related to unexpected groundwater inflows into tunnels (Bilgin & Ates 2016). The time taken for these inflows to change from temporary to persistent inflows depends not only on the aforementioned conditions, but also on the extent of the damage generated. Besides that, temporary groundwater inflows are in a state of non-equilibrium, their transformation into a stable state is progressive (Li et al. 2021). In some situations, when the duration of temporary groundwater inflows with excessive damage is so long, the construction of the tunnels is abandoned. Such a situation has occurred in the case of Alfalfal tunnel in Chile (Holmøy & Nilsen 2014). Therefore, when studying the time dependency of groundwater inflows into tunnels, it is very important to consider the potential damage generated by the unforeseen inflows.

Table 2

A summary of potential damage caused by unexpected inflows of groundwater during tunneling

Tunnel name .	Location .	Predominant lithology .	Maximum inflow (⁠⁠) .	Potential damage .	Source (year) .
Qiyueshan tunnel	China	Limestone, shale, and coal	87,000.00	Flooding, fractures on lining.	Lin et al. (2019)
Junchang tunnel	China	Granite		Mud inrush and ground collapses, tunnel construction delayed.	Yuan et al. (2019)
East–West tunnel	USA	Sandstone, claystone and shale		Failure by buckling of the supporting conduit, which caused 20 working days of repair.	Marsters & Dornfest (2018)
Yichang-Wanzhou tunnels	China	Carbonate		Mud inrush, failure of karstic zones in many sections of tunnels, casualties, equipment damaged, and project completion delayed.	Fan et al. (2018)
Zagros tunnel	Iran	Limestone, marlstone		Collapses and delayed tunnel construction.	Bayati & Hamidi (2017)
Wulong tunnel	China	Limestone		Considerable casualties and economic losses.	Li et al. (2016a)  Tu et al. (2021)
Aica-Mules tunnel	Italy	Granite		The construction of the tunnel was delayed as the TBM was blocked for six months, economic losses.	Perello et al. (2014)
Arrowhead Tunnels	USA	Granite and gneiss		Complicated excavations, project delayed for 2 years, financial losses.	Holmøy & Nilsen (2014)
Tanum and Skaugum tunnels	Norway	Limestone and shale		Corrosion and failure of accessories, ground settlements, economic losses.	Dammyr et al. (2014)
Alfalfal tunnel	Chile	Granite		Excessive damage, tunnel abandoned and eventually used to alleviate high groundwater pressure.	Holmøy & Nilsen (2014)
Burnley tunnel	Australia	Mudstone		Observation of cracks and uplift up to 200 mm in slab panels.	Mothersille & Littlejohn (2012)
Barcelona subway L9	Spain	Weathered granite		Instability at the tunnel face, TBM blocked, project delayed, tunnel route modified, groundwater pumping performed.	Font-Capó et al. (2011)

Tunnel name .	Location .	Predominant lithology .	Maximum inflow (⁠⁠) .	Potential damage .	Source (year) .
Qiyueshan tunnel	China	Limestone, shale, and coal	87,000.00	Flooding, fractures on lining.	Lin et al. (2019)
Junchang tunnel	China	Granite		Mud inrush and ground collapses, tunnel construction delayed.	Yuan et al. (2019)
East–West tunnel	USA	Sandstone, claystone and shale		Failure by buckling of the supporting conduit, which caused 20 working days of repair.	Marsters & Dornfest (2018)
Yichang-Wanzhou tunnels	China	Carbonate		Mud inrush, failure of karstic zones in many sections of tunnels, casualties, equipment damaged, and project completion delayed.	Fan et al. (2018)
Zagros tunnel	Iran	Limestone, marlstone		Collapses and delayed tunnel construction.	Bayati & Hamidi (2017)
Wulong tunnel	China	Limestone		Considerable casualties and economic losses.	Li et al. (2016a)  Tu et al. (2021)
Aica-Mules tunnel	Italy	Granite		The construction of the tunnel was delayed as the TBM was blocked for six months, economic losses.	Perello et al. (2014)
Arrowhead Tunnels	USA	Granite and gneiss		Complicated excavations, project delayed for 2 years, financial losses.	Holmøy & Nilsen (2014)
Tanum and Skaugum tunnels	Norway	Limestone and shale		Corrosion and failure of accessories, ground settlements, economic losses.	Dammyr et al. (2014)
Alfalfal tunnel	Chile	Granite		Excessive damage, tunnel abandoned and eventually used to alleviate high groundwater pressure.	Holmøy & Nilsen (2014)
Burnley tunnel	Australia	Mudstone		Observation of cracks and uplift up to 200 mm in slab panels.	Mothersille & Littlejohn (2012)
Barcelona subway L9	Spain	Weathered granite		Instability at the tunnel face, TBM blocked, project delayed, tunnel route modified, groundwater pumping performed.	Font-Capó et al. (2011)

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It is well known that water-rich structures such as aquifers play an important role in triggering groundwater inflows into tunnels (Liu et al. 2017). Convinced of that, Shi et al. (2017) pointed out that it is crucial to precisely control and predict such structures. The characteristics and the positioning of these structures in relation to the tunnel route are of tremendous importance. When they are detected and characterized appropriately, groundwater inflows can be predicted with good accuracy. In fact, huge groundwater inflows are unavoidable when tunnels pass through such structures (Liu et al. 2017). Groundwater ingress can also increase through the connection of aquifers and faults (Li et al. 2017; Ma et al. 2020). In the worst cases, serious overflows in excavated areas can be catastrophic when the large permeable zones are tied to major groundwater sources (Zabidi et al. 2019). Such situations are significant and cannot be overlooked in the assessment of groundwater inflows into tunnels. The lack of such information can provoke failure of the pre-grouting techniques. The latter are usually applied to the surrounding rocks in order to limit groundwater inflows into the openings (Forth 2004; Li et al. 2016b; Hognestad & Kieffer 2019; Han et al. 2022; Sousa & Einstein 2021; Zheng et al. 2022). They also reinforce the joints of the rocks by sealing the existing interstices (Deng & Chen 2021). Thus, the quality and control of grouting must be considered to ensure that groundwater inflows can be effectively minimized. In fact, unsuitable grouting cannot reduce the flows of groundwater entering tunnels (Daw & Pollard 1986; Liu et al. 2017). In karst terrains, particularly, advanced grouting should be employed for the best results (Ren et al. 2016). The general characteristics of the surrounding rocks, the quantity and quality of groundwater must be taken into account for an appropriate selection of the grouting (Li & Zhou 2006). Attention should be paid as the grouting holes are sources for groundwater inflows (Wu et al. 2014). It is therefore more than necessary to estimate the characteristics and conditions of underground water storage structures. In addition, different expansions can be envisaged in the appearance of groundwater inflows in tunnels when water channels such as fractures and faults develop near the aquifer structures (Xue et al. 2021). The accurate prediction of groundwater inflows into tunnels requires accurate information on the types and conditions of water-bearing structures and the availability of groundwater stored in them near the openings.

Equations (10) and (11) clearly show that the hydraulic head is one of the key factors which influence the extent of groundwater inflow into tunnels. Indeed, the groundwater inflows will vary with the variation of the hydraulic head at the tunnel perimeter. However, due to the difficulty of correctly capturing the variation in hydraulic head and also for simplification, the hydraulic head is generally considered to be constant in most assessments of groundwater inflow into tunnels.

It is important to note that, as excavation progresses and groundwater flows into a given tunnel, the groundwater level decreases. There is drawdown of the existing groundwater level which could occur even after the use of pre-grouting techniques into the surrounding rocks (Yoo et al. 2012). In deep tunnels, especially where the rock conditions are usually complex, the influx of groundwater into the openings can take place even if the most appropriate grouting techniques have been applied (Sousa & Einstein 2021). Groundwater drawdown is therefore unavoidable when groundwater inflows occur during or after excavations (Frenelus et al. 2021b). As an example, in Switzerland, significant drawdowns of 230 m and 300 m were observed respectively in the Rawyl Exploratory Adit and Campo Valle Maggia (Preisig et al. 2014). According to Wang et al. (2021), controlling the maximum water supply of a given tunnel is possible by estimating the groundwater drawdown. The constant level ordinarily considered for the groundwater table cannot be maintained in real situations. It is therefore important to consider the groundwater level as a variable, not a constant parameter (Fernandez & Moon 2010). Thereby, the accuracy of the analytical solutions for groundwater inflows into tunnels can be enhanced. Note that one of the serious effects of groundwater drawdown is that the deformed areas of the surrounding rocks that are connected to the upstream section may receive increased groundwater inflows (Bockgård et al. 2014).

The increase in rock permeability also depends on the lithology and solubility (Zarei et al. 2013). When exposed to water, carbonate and soluble rocks in particular can be karstified, and groundwater inflows can occur more easily in the tunnels. It should be noted that the karstification of rocks can be seen at different levels which are most governed by the flood of water mixed to detritus (Zini et al. 2015; Fan et al. 2018). Precisely, three jointed stages govern the level of rock karstification namely: rock water exposure time, kind and amount of rainfall, and the concerned hydraulic gradient (Zini et al. 2015). Tunnels built in karst areas are risky. They generally cause excessive groundwater inflows, structure instability, and more expense in tunnel construction (Kaufmann & Romanov 2020). In tunneling, nearly 50% of groundwater inflows are generated by karst terrains (Xue et al. 2021).

It is interesting to consider the variation of such a parameter in the evaluation of groundwater inflows into tunnels. Otherwise, such assessments could not be realistic (Song et al. 2013; Cheng et al. 2019), nor accurate. Table 4 shows some propositions for estimating the hydraulic conductivity of rocks.

Table 4

Expressions of hydraulic conductivity and their applicability

Hydraulic conductivity .	Equation number .	Applicability and comments .	Source (year) .
⁠: Hydraulic conductivity tensor; ⁠: average frequency for ith set of discontinuities; ⁠: average hydraulic aperture of the ith set of discontinuities; ⁠: related conversion matrix; ⁠: water density; ⁠: gravitational acceleration; ⁠: dynamic viscosity of water; ⁠: total number of sets of discontinuities.	(15)	Circular tunnels in discontinuous rocky environments. The hydraulic conductivity is defined by its tensor and takes account of its anisotropic character.	Shahbazi et al. (2021); Coli et al. (2008)
⁠: postpeak hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: coefficient of unexpected leap of hydraulic conductivity after giving in; ⁠: coefficient which takes into account the reduction in the rate of hydraulic conductivity; ⁠: volume strain.	(16)	Coal mines in fractured rocks affected by roof water. Hydraulic conductivity varies mainly with the change in the stress-strain process due to mining disruption.	Lu et al. (2020)
⁠: Stress-dependent hydraulic conductivity (m/s); ⁠: initial hydraulic conductivity; Effective stress (Pa); ⁠: effective fracture closure stress (Pa); ⁠: Coefficient of statistical distribution of the length of asperities.	(17)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies mainly with stress.	Preisig et al. (2014)
⁠: Pressure head-dependent hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: factor related to the elastic compressive strength of rocks; ⁠: initial pressure head; h: final pressure head.	(18)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies with pressure.	Preisig et al. (2014)
⁠: Stress-dependent hydraulic conductivity (m/s); ⁠: initial hydraulic conductivity; Effective stress (Pa); ⁠: effective fracture closure stress (Pa); ⁠: Coefficient of statistical distribution of the length of asperities.	(19)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies mainly with stress.	Preisig et al. (2014)
⁠: Pressure head-dependent hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: factor related to the elastic compressive strength of rocks; ⁠: initial pressure head; h: final pressure head.	(20)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies with pressure.	Preisig et al. (2014)

Hydraulic conductivity .	Equation number .	Applicability and comments .	Source (year) .
⁠: Hydraulic conductivity tensor; ⁠: average frequency for ith set of discontinuities; ⁠: average hydraulic aperture of the ith set of discontinuities; ⁠: related conversion matrix; ⁠: water density; ⁠: gravitational acceleration; ⁠: dynamic viscosity of water; ⁠: total number of sets of discontinuities.	(15)	Circular tunnels in discontinuous rocky environments. The hydraulic conductivity is defined by its tensor and takes account of its anisotropic character.	Shahbazi et al. (2021); Coli et al. (2008)
⁠: postpeak hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: coefficient of unexpected leap of hydraulic conductivity after giving in; ⁠: coefficient which takes into account the reduction in the rate of hydraulic conductivity; ⁠: volume strain.	(16)	Coal mines in fractured rocks affected by roof water. Hydraulic conductivity varies mainly with the change in the stress-strain process due to mining disruption.	Lu et al. (2020)
⁠: Stress-dependent hydraulic conductivity (m/s); ⁠: initial hydraulic conductivity; Effective stress (Pa); ⁠: effective fracture closure stress (Pa); ⁠: Coefficient of statistical distribution of the length of asperities.	(17)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies mainly with stress.	Preisig et al. (2014)
⁠: Pressure head-dependent hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: factor related to the elastic compressive strength of rocks; ⁠: initial pressure head; h: final pressure head.	(18)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies with pressure.	Preisig et al. (2014)
⁠: Stress-dependent hydraulic conductivity (m/s); ⁠: initial hydraulic conductivity; Effective stress (Pa); ⁠: effective fracture closure stress (Pa); ⁠: Coefficient of statistical distribution of the length of asperities.	(19)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies mainly with stress.	Preisig et al. (2014)
⁠: Pressure head-dependent hydraulic conductivity; ⁠: initial hydraulic conductivity; ⁠: factor related to the elastic compressive strength of rocks; ⁠: initial pressure head; h: final pressure head.	(20)	Circular tunnels in discontinuous rocky environments. Hydraulic conductivity varies with pressure.	Preisig et al. (2014)

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As shown in Figure 10, groundwater inflows into tunnels vary with variation in rock permeability. Not only does the groundwater inflows into the tunnels decrease over time under the influence of the rock permeability but, more importantly, it drastically decreases under the effect of a considerable reduction in permeability. This can illustrate that, in predicting groundwater inflows, the variability of rock permeability plays a crucial role. As expressed by Hwang & Lu (2007), groundwater inflow values can be skewed by any error in estimating the permeability of rocks.

Figure 11 clearly shows that groundwater inflows into tunnels vary with variation in rock hydraulic conductivity. When the hydraulic conductivity (⁠⁠) increases, groundwater inflows considerably increase. Similarly, if the hydraulic conductivity decreases, groundwater inflows also considerably decreases. Excavation disturbances are unavoidable in the construction of rock tunnels and play an important role in varying the hydraulic conductivity of rocks. As already mentioned, due to rock anisotropy, the hydraulic conductivity has a random variation and can be more pronounced at depth. It remains a difficult task to assess the exact values of hydraulic conductivity of rocks for a given rock tunnel. However, for tunnels with varying burial depth, it is necessary to be more careful, because the variation of the hydraulic conductivity of the rocks can be much more marked.

In particular in fractured rock environments, the aperture of fractures has a predominant role in the variation of the hydraulic conductivity of rocks (Bai & Elsworth 1994). According to Cesano et al. (2003), the extent of the groundwater flow velocity increases due to the increase in the variation of the fracture aperture, mainly for the rocky environments mentioned above. Therefore, fracture aperture of rocks should be taken into account in the evaluation of groundwater inflows into tunnels particularly constructed in fractured aquifers. In fact, natural rocks generally contain discontinuities (Huang et al. 2019). In addition to pre-existing fractures that exist in such rocks, new fractures are generally created due to stress disturbances caused by the effects of excavations. Accordingly, it is inevitable that groundwater enters the interior of the tunnels through fractures or discontinuities. The opening of rock fractures can improve the accuracy of predictions of groundwater inflows in tunnels. As simulated by Huang et al. (2013), fracture apertures considerably affect groundwater inflows into tunnels. More precisely, referring to Shahbazi et al. (2021), groundwater inflows into tunnels increase as fracture apertures increase. Proper consideration of such a factor can significantly improve the accuracy of groundwater inflow predictions in tunnels built in particularly discontinuous rocky environments.

In general, the occurrence of precipitation influences the extent of groundwater inflows in tunnels. For tunnels located in particular in unfavorable geological environments, the influence of rainfall on groundwater inflows is stronger (Zhao et al. 2013). For instance, owing to considerable rainfalls, groundwater inflows were more serious in a diversion tunnel of the Jinping II hydropower station during 2012–2013 (Hou et al. 2016). Similarly, following heavy rains, terrible groundwater inflows have occurred in the Daba tunnel located in western Hunan in China (Li et al. 2018b). In the Yichang–Wanzhou tunnels (China), there was mainly the appearance of groundwater inflows during periods of rainfalls (Fan et al. 2018). Such situations can illustrate the effects of rainfalls in the extent of groundwater inflows into tunnels. In fact, owing to heavy rainfalls, the increase in groundwater inflow is serious (Chiu & Chia 2012; Yap & Ngien 2017). The drainage systems of the openings can receive much more water flow during heavy rains (Polak et al. 2016), and can be weakened and put the tunnel lining under high pressure (Lan et al. 2021). Hence, the maximum inflow of groundwater is required to design optimal and reliable drainage systems.

Although groundwater inflows in tunnels are time dependent, there are few analytical solutions that take into account such behavior of groundwater inflows. Table 5 shows the relevant analytical solutions taking into account this behavior to predict groundwater inflows into tunnels. Salient details regarding the applicability and conditions of these solutions are also included in Table 5.

Table 5

Summary of the salient analytical solutions taking into account the time dependency of groundwater inflows

Volumetric flow rate and parameters .	Equation number .	Tunnel structure and media type .	Flow regime and state of groundwater level .	Remarks .	Study (year) .
⁠: discharge rate (⁠⁠); ⁠: coefficient reflecting the magnitude of initial discharge rate; ⁠: a decreasing exponent; ⁠: time (⁠⁠).	(21)	Circular tunnel. Heterogeneous media.	Darcy flow. Groundwater level drawdown.	Equations based on dynamic modeling, and validated by numerical analysis.	Xia et al. (2018)
Identical parameters are similar to those below.	(22)	Circular tunnel. Isotropic confined aquifer.	Non-Darcian flow. Drawdown of groundwater level.	Same remarks as below.	Liu et al. (2018)
⁠: total groundwater inflows into tunnel ⁠; ⁠; ⁠; ⁠: hydraulic conductivities along the axis x, y, z (⁠⁠); ⁠: groundwater drawdown; ⁠: time; ⁠: confined aquifer thickness (m); ⁠: specific storativity (⁠⁠); ⁠: distance from a point source along the y-axis (m); ⁠: drawdown at distance (m); ⁠: distance between initial water head and tunnel centre (m).	(23)	Circular tunnels. Anisotropic confined aquifer.	Non-Darcian flow. Drawdown of groundwater level.	Equations based on the horizontal well theory with assumptions such as infinite lateral boundaries and there is no flow boundaries in the top and bottom of aquifer.	Liu et al. (2018)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: drilling speed; ⁠: time; ⁠: Heaviside step-function (⁠⁠, if ⁠; ⁠, if ⁠); hydraulic conductivity; ⁠: porosity; ⁠: coordinate along the tunnel axis; thickness of the saturated aquifer.	(24)	Circular tunnels located above an impermeable layer. Homogeneous media.	Darcy flow. Transient flow.	Hydrostatic initial conditions, constant drilling speed, low piezometric disturbances ahead of the drilling front of excavations.	Maréchal et al. (2014)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: discharge at an infinite length; ; ⁠: time of sector ⁠; ⁠: time; ⁠: Heaviside step-function.	(25)	Circular tunnels located above an impermeable layer Heterogeneous media.	Non-Darcy flow. Transient flow.	These equations are derived from superposition principles.	Maréchal et al. (2014)
⁠: Groundwater inflow into tunnel (L/s); T: rock transmissivity; S: elastic storage; ⁠: tunnel radius; ⁠: time.	(26)	Circular tunnel. Granitic and similar rocks.	Darcy flow. Non-steady flow.	Equations derived from retrospective analysis of tunnel excavation flow systems.	Perello et al. (2014)
⁠: Laplace transform of the volumetric flow rate; ⁠: tunnel depth; r: tunnel radius; ⁠; ⁠: storage capacity, ⁠: hydraulic conductivity, ⁠: Laplace dual time variable; ⁠: Fourrier coefficients of the single layer potential; In, : respectively modified Bessel functions of the first and second kind of order n; ⁠, ⁠: respectively modified Bessel function for the 1st kind of order 0 and 1.	(27)	Circular tunnel Isotropic and homogeneous aquifer.	Transient flow due to the excavation of a tunnel generating rock mass consolidation.	Exact solution in the Laplace dual time	El Tani (2009, 2010)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: tunnel radius; ⁠: Heaviside step-function (⁠⁠, if ⁠; ⁠, if ⁠); ⁠: drilling speed; ⁠: drilled speed at sector i; ⁠: time; ⁠: time of sector i; ⁠: coordinate along the tunnel axis; ⁠: hydraulic conductivity; ⁠: Specific storage coefficient; ⁠: Thickness of saturated zone; ⁠: length at a sector i.	(28)	Circular tunnel. Heterogeneous media.	Non-Darcy flow. Transient flow.	These equations are derived from convolution and superposition principles. Consecutive sectors are considered.	Perrochet & Dematteis (2007)
⁠, ⁠: respectively specific drawdown and storage coefficient; tunnel radius. Other parameters: same as above.	(29)	Circular tunnels. Homogeneous media.	Darcy flow. Transient flow.	Consideration on progressive drilling excavation. Convolution integral employed.	Perrochet (2005)

Volumetric flow rate and parameters .	Equation number .	Tunnel structure and media type .	Flow regime and state of groundwater level .	Remarks .	Study (year) .
⁠: discharge rate (⁠⁠); ⁠: coefficient reflecting the magnitude of initial discharge rate; ⁠: a decreasing exponent; ⁠: time (⁠⁠).	(21)	Circular tunnel. Heterogeneous media.	Darcy flow. Groundwater level drawdown.	Equations based on dynamic modeling, and validated by numerical analysis.	Xia et al. (2018)
Identical parameters are similar to those below.	(22)	Circular tunnel. Isotropic confined aquifer.	Non-Darcian flow. Drawdown of groundwater level.	Same remarks as below.	Liu et al. (2018)
⁠: total groundwater inflows into tunnel ⁠; ⁠; ⁠; ⁠: hydraulic conductivities along the axis x, y, z (⁠⁠); ⁠: groundwater drawdown; ⁠: time; ⁠: confined aquifer thickness (m); ⁠: specific storativity (⁠⁠); ⁠: distance from a point source along the y-axis (m); ⁠: drawdown at distance (m); ⁠: distance between initial water head and tunnel centre (m).	(23)	Circular tunnels. Anisotropic confined aquifer.	Non-Darcian flow. Drawdown of groundwater level.	Equations based on the horizontal well theory with assumptions such as infinite lateral boundaries and there is no flow boundaries in the top and bottom of aquifer.	Liu et al. (2018)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: drilling speed; ⁠: time; ⁠: Heaviside step-function (⁠⁠, if ⁠; ⁠, if ⁠); hydraulic conductivity; ⁠: porosity; ⁠: coordinate along the tunnel axis; thickness of the saturated aquifer.	(24)	Circular tunnels located above an impermeable layer. Homogeneous media.	Darcy flow. Transient flow.	Hydrostatic initial conditions, constant drilling speed, low piezometric disturbances ahead of the drilling front of excavations.	Maréchal et al. (2014)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: discharge at an infinite length; ; ⁠: time of sector ⁠; ⁠: time; ⁠: Heaviside step-function.	(25)	Circular tunnels located above an impermeable layer Heterogeneous media.	Non-Darcy flow. Transient flow.	These equations are derived from superposition principles.	Maréchal et al. (2014)
⁠: Groundwater inflow into tunnel (L/s); T: rock transmissivity; S: elastic storage; ⁠: tunnel radius; ⁠: time.	(26)	Circular tunnel. Granitic and similar rocks.	Darcy flow. Non-steady flow.	Equations derived from retrospective analysis of tunnel excavation flow systems.	Perello et al. (2014)
⁠: Laplace transform of the volumetric flow rate; ⁠: tunnel depth; r: tunnel radius; ⁠; ⁠: storage capacity, ⁠: hydraulic conductivity, ⁠: Laplace dual time variable; ⁠: Fourrier coefficients of the single layer potential; In, : respectively modified Bessel functions of the first and second kind of order n; ⁠, ⁠: respectively modified Bessel function for the 1st kind of order 0 and 1.	(27)	Circular tunnel Isotropic and homogeneous aquifer.	Transient flow due to the excavation of a tunnel generating rock mass consolidation.	Exact solution in the Laplace dual time	El Tani (2009, 2010)
⁠: Total groundwater inflow into tunnel (L/s); ⁠: tunnel radius; ⁠: Heaviside step-function (⁠⁠, if ⁠; ⁠, if ⁠); ⁠: drilling speed; ⁠: drilled speed at sector i; ⁠: time; ⁠: time of sector i; ⁠: coordinate along the tunnel axis; ⁠: hydraulic conductivity; ⁠: Specific storage coefficient; ⁠: Thickness of saturated zone; ⁠: length at a sector i.	(28)	Circular tunnel. Heterogeneous media.	Non-Darcy flow. Transient flow.	These equations are derived from convolution and superposition principles. Consecutive sectors are considered.	Perrochet & Dematteis (2007)
⁠, ⁠: respectively specific drawdown and storage coefficient; tunnel radius. Other parameters: same as above.	(29)	Circular tunnels. Homogeneous media.	Darcy flow. Transient flow.	Consideration on progressive drilling excavation. Convolution integral employed.	Perrochet (2005)

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Many existing analytical and semi-analytical solutions for predicting groundwater inflows into tunnels do not take into account the temporal dependence of groundwater inputs. As a result, they generally consider the steady stage of groundwater inflows. They therefore propose solutions for the phase stable of groundwater inflows into tunnels. It should be noted that such solutions must also consider the real properties of the rock masses concerned in order to be precise. Regardless of the type of analytical or semi-analytical solutions, the accuracy of the predictions remains a challenge. The reported solutions can constitute a source of motivation for the continuous search for new solutions whose accuracy can be increasingly improved. Table 5 presents a summary of the most relevant existing analytical and semi-analytical solutions that uniquely predict the stable stage of groundwater inflows in rock tunnels. Salient details regarding the applicability and conditions of these solutions are also included in Table 6.

Table 6

Summary of the pertinent steady-state analytical solutions of groundwater inflows

Volumetric flow rate and parameters .	Equation number .	Tunnel structure and media type .	Flow regime and state of groundwater level .	Remarks .	Study (year) .
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: water head outside the concerned area of the drainage system (m); ⁠: drainage coefficient; ⁠, ⁠, ⁠: respectively permeability coefficient of rocks, primary lining and the drainage system (m/s); t: thickness of geotextile (m); ⁠, ⁠: respectively outer radius of primary and secondary lining (m); ⁠: radius of the drainage concerned zone (m); ⁠: distance between two vicinal drainage pipes (m).	(30)	Lined circular deep tunnel. Continuous porous media.	Darcy flow. Constant water table.	The effects of waterproofing and drainage systems on groundwater inflows are taken into account.	Liu & Li (2021)
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: respectively permeability of the shotcrete lining, the grouting and the rocks (m/s); ⁠: inner water head of the ground surface (m); ⁠: inner water head of the shotcrete lining (m); ⁠, ⁠: respectively distance between center and the border for image tunnel (A) and original tunnel (B); ⁠, ⁠: respectively distance between center and the border of lining for image tunnel and original tunnel (m); ⁠, ⁠: respectively distance between center and the border of grouting for image tunnel and original tunnel (m).	(31)	Circular tunnel with shotcrete and concrete lining. Homogeneous and isotropic rocks.	Darcy flow. Constant water table.	Equations derived from mirror method, and validated by numerical simulation.	Qin et al. (2020)
⁠: groundwater inflow into tunnels (⁠⁠); rock hydraulic conductivity; ⁠: reservoir level above the tunnel; ⁠: reservoir level above an impermeable layer; d: distance between tunnel and reservoir; ⁠: tunnel radius or extracted seal; medium pressure on the tunnel edge that depends on the lining and grouted layer if not is zero.	(32)	Circular tunnels in active seismic zones.	Darcy flow. Water table drawdown.	Equations derived from particular analytic developments. The tunnel is recharged by a reservoir.	El Tani et al. (2019)
⁠: groundwater inflow into tunnels (⁠⁠); ⁠: boundary head of the lining; ⁠: Tunnel diameter; ⁠: lining diameter; ⁠: outer diameter of the grouting ring; ⁠, ⁠, ⁠: respectively equivalent hydraulic conductivity of lining; hydraulic conductivity of the grouting area; hydraulic conductivity of surrounding rock.	(33)	Deep circular tunnels with lining and grouting. Homogeneous media.	Darcy flow. Constant water table.	Different grouting ring thickness are considered.	Xu et al. (2019)
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: permeability of the aquifer (m/s); parameter which can be determined from suitable polar coordinates.	(34)	Lined circular tunnels. Homogeneous and isotropic media.	Darcy flow. Constant water table.	Semi-analytical equations based on conformal mapping techniques and numerically verified.	Ying et al. (2018)
⁠: Effective groundwater inflow into tunnel (⁠⁠); ⁠: joints aperture (mm); ⁠: joint aperture surface (⁠⁠); ⁠: Shape perimeter from joint strike intersection and tunnel axis (m); ⁠: joints spacing (m); ⁠: water head (m); ⁠,…, ⁠, ⁠: joints sets. ⁠: hydraulic conductivity (m/s); ⁠: effective discharge length (m).	(35)	Deep circular tunnel. Jointed rocks media.	Darcy flow. Constant water table.	These equations are derived from groundwater seepage rating (SGR). Geological and hydraulic parameters, as well as tunnel properties are required.	Maleki (2018)
Q; groundwater inflow into tunnels; ⁠: Total hydraulic head; ⁠, ⁠: respectively internal and external radius of the grouted zones; ⁠: tunnel depth; ⁠: constant water pressure; ⁠, ⁠: respectively permeability of aquifer and grouted zone; ⁠: unit weight of water.	(36)	Circular subsea grouted tunnels.	Darcy flow. Constant water table.	Complex variable, mirror image and axisymmetric modeling methods are considered.	Li et al. (2018a)
⁠: groundwater inflow into tunnel (L/min/m); ⁠: hydraulic conductivity (m/s); ⁠: initial piezometric head above the tunnel center (m); ⁠: radius of tunnel (m).	(37)	Circular tunnel. Homogeneous media.	Darcy flow. Water table drawdown.	Semi-analytical equations obtained by numerical simulations.	Su et al. (2017)
⁠: groundwater inflow into tunnel; ⁠: tunnel radius; ⁠: initial piezometric head above the tunnel center; ⁠: parameters.	(38)	Circular tunnel. Homogeneous media.	Non-Darcy flow. Constant water table.	Atkinson equations are used to determine the experimental constants a, b.	Joo & Shin (2014)
⁠: groundwater flow into tunnel; ⁠: coefficient linked to the shape and depth of tunnel; ⁠: another coefficient linked to the shape and depth; ⁠: hydraulic conductivity; ⁠: water head at the upper limit.	(39)	Deep horseshoe tunnels and caverns. Homogeneous rocky media.	Darcy flow. Constant water table.	Semi-analytical equations.	Xu et al. (2013)
⁠: groundwater inflow into tunnel, ⁠: hydraulic conductivity; ⁠: constant parameter; ⁠: piezometric head above the tunnel centre; ⁠: tunnel radius; ⁠: specific water height; ⁠: modified Bessel function for the 2nd kind of order zero; ⁠: modified Bessel function for the 1st kind of order zero.	(40)	Circular tunnels. Heterogeneous media with different behaviors.	Darcy flow. Constant water table.	Equations derived from integral solution. Transient consolidation is considered.	El Tani (2010)
⁠: groundwater inflow per m of tunnel length; ⁠: isotropic permeability coefficient; ⁠: tunnel canter depth to the water table; ⁠: energy head of the tunnel drained perimeter; ⁠: head of water table; ⁠: tunnel radius. Water inflow does not converge and becomes infinite when the tunnel depth equals its radius that is or equivalently the water table is tangent to the tunnel.	(41)	Circular tunnel in a semi-infinite aquifer with a constant potential at the tunnel edge.	Darcy flow. Homogeneous media. Constant water table.	Conformal mapping technique is the basis of this equation.	Rat (1973); Lei (1999); Kolymbas & Wagner (2007)
⁠: groundwater inflow per m of tunnel length; ⁠: isotropic hydraulic conductivity; with h depth of the tunnel center to the water table and r its radius. Water inflow converges to when the tunnel depth is equal to the radius, that is or equivalently when the water table is tangent to the tunnel.	(42)	Circular tunnel in a semi-infinite aquifer with a zero pressure at the tunnel edge	Darcy flow.	Exact solution derived considering the integral formulation in combination with the Mobius transformation.	El Tani (2003)

Volumetric flow rate and parameters .	Equation number .	Tunnel structure and media type .	Flow regime and state of groundwater level .	Remarks .	Study (year) .
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: water head outside the concerned area of the drainage system (m); ⁠: drainage coefficient; ⁠, ⁠, ⁠: respectively permeability coefficient of rocks, primary lining and the drainage system (m/s); t: thickness of geotextile (m); ⁠, ⁠: respectively outer radius of primary and secondary lining (m); ⁠: radius of the drainage concerned zone (m); ⁠: distance between two vicinal drainage pipes (m).	(30)	Lined circular deep tunnel. Continuous porous media.	Darcy flow. Constant water table.	The effects of waterproofing and drainage systems on groundwater inflows are taken into account.	Liu & Li (2021)
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: respectively permeability of the shotcrete lining, the grouting and the rocks (m/s); ⁠: inner water head of the ground surface (m); ⁠: inner water head of the shotcrete lining (m); ⁠, ⁠: respectively distance between center and the border for image tunnel (A) and original tunnel (B); ⁠, ⁠: respectively distance between center and the border of lining for image tunnel and original tunnel (m); ⁠, ⁠: respectively distance between center and the border of grouting for image tunnel and original tunnel (m).	(31)	Circular tunnel with shotcrete and concrete lining. Homogeneous and isotropic rocks.	Darcy flow. Constant water table.	Equations derived from mirror method, and validated by numerical simulation.	Qin et al. (2020)
⁠: groundwater inflow into tunnels (⁠⁠); rock hydraulic conductivity; ⁠: reservoir level above the tunnel; ⁠: reservoir level above an impermeable layer; d: distance between tunnel and reservoir; ⁠: tunnel radius or extracted seal; medium pressure on the tunnel edge that depends on the lining and grouted layer if not is zero.	(32)	Circular tunnels in active seismic zones.	Darcy flow. Water table drawdown.	Equations derived from particular analytic developments. The tunnel is recharged by a reservoir.	El Tani et al. (2019)
⁠: groundwater inflow into tunnels (⁠⁠); ⁠: boundary head of the lining; ⁠: Tunnel diameter; ⁠: lining diameter; ⁠: outer diameter of the grouting ring; ⁠, ⁠, ⁠: respectively equivalent hydraulic conductivity of lining; hydraulic conductivity of the grouting area; hydraulic conductivity of surrounding rock.	(33)	Deep circular tunnels with lining and grouting. Homogeneous media.	Darcy flow. Constant water table.	Different grouting ring thickness are considered.	Xu et al. (2019)
⁠: groundwater inflow into tunnel (⁠⁠); ⁠: permeability of the aquifer (m/s); parameter which can be determined from suitable polar coordinates.	(34)	Lined circular tunnels. Homogeneous and isotropic media.	Darcy flow. Constant water table.	Semi-analytical equations based on conformal mapping techniques and numerically verified.	Ying et al. (2018)
⁠: Effective groundwater inflow into tunnel (⁠⁠); ⁠: joints aperture (mm); ⁠: joint aperture surface (⁠⁠); ⁠: Shape perimeter from joint strike intersection and tunnel axis (m); ⁠: joints spacing (m); ⁠: water head (m); ⁠,…, ⁠, ⁠: joints sets. ⁠: hydraulic conductivity (m/s); ⁠: effective discharge length (m).	(35)	Deep circular tunnel. Jointed rocks media.	Darcy flow. Constant water table.	These equations are derived from groundwater seepage rating (SGR). Geological and hydraulic parameters, as well as tunnel properties are required.	Maleki (2018)
Q; groundwater inflow into tunnels; ⁠: Total hydraulic head; ⁠, ⁠: respectively internal and external radius of the grouted zones; ⁠: tunnel depth; ⁠: constant water pressure; ⁠, ⁠: respectively permeability of aquifer and grouted zone; ⁠: unit weight of water.	(36)	Circular subsea grouted tunnels.	Darcy flow. Constant water table.	Complex variable, mirror image and axisymmetric modeling methods are considered.	Li et al. (2018a)
⁠: groundwater inflow into tunnel (L/min/m); ⁠: hydraulic conductivity (m/s); ⁠: initial piezometric head above the tunnel center (m); ⁠: radius of tunnel (m).	(37)	Circular tunnel. Homogeneous media.	Darcy flow. Water table drawdown.	Semi-analytical equations obtained by numerical simulations.	Su et al. (2017)
⁠: groundwater inflow into tunnel; ⁠: tunnel radius; ⁠: initial piezometric head above the tunnel center; ⁠: parameters.	(38)	Circular tunnel. Homogeneous media.	Non-Darcy flow. Constant water table.	Atkinson equations are used to determine the experimental constants a, b.	Joo & Shin (2014)
⁠: groundwater flow into tunnel; ⁠: coefficient linked to the shape and depth of tunnel; ⁠: another coefficient linked to the shape and depth; ⁠: hydraulic conductivity; ⁠: water head at the upper limit.	(39)	Deep horseshoe tunnels and caverns. Homogeneous rocky media.	Darcy flow. Constant water table.	Semi-analytical equations.	Xu et al. (2013)
⁠: groundwater inflow into tunnel, ⁠: hydraulic conductivity; ⁠: constant parameter; ⁠: piezometric head above the tunnel centre; ⁠: tunnel radius; ⁠: specific water height; ⁠: modified Bessel function for the 2nd kind of order zero; ⁠: modified Bessel function for the 1st kind of order zero.	(40)	Circular tunnels. Heterogeneous media with different behaviors.	Darcy flow. Constant water table.	Equations derived from integral solution. Transient consolidation is considered.	El Tani (2010)
⁠: groundwater inflow per m of tunnel length; ⁠: isotropic permeability coefficient; ⁠: tunnel canter depth to the water table; ⁠: energy head of the tunnel drained perimeter; ⁠: head of water table; ⁠: tunnel radius. Water inflow does not converge and becomes infinite when the tunnel depth equals its radius that is or equivalently the water table is tangent to the tunnel.	(41)	Circular tunnel in a semi-infinite aquifer with a constant potential at the tunnel edge.	Darcy flow. Homogeneous media. Constant water table.	Conformal mapping technique is the basis of this equation.	Rat (1973); Lei (1999); Kolymbas & Wagner (2007)
⁠: groundwater inflow per m of tunnel length; ⁠: isotropic hydraulic conductivity; with h depth of the tunnel center to the water table and r its radius. Water inflow converges to when the tunnel depth is equal to the radius, that is or equivalently when the water table is tangent to the tunnel.	(42)	Circular tunnel in a semi-infinite aquifer with a zero pressure at the tunnel edge	Darcy flow.	Exact solution derived considering the integral formulation in combination with the Mobius transformation.	El Tani (2003)

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For ease of understanding, an example is shown in Table 7.

After selecting Equation (37), the calculation of groundwater inflows in rock tunnels can be performed by considering the associated relevant information. The tunnel radius is approximatively 5 m, the hydraulic conductivity for the surrounding rocks is about ⁠. The total length of the tunnel is 662 m, the burial depth below the initial level of groundwater is 80 m. Owing to the excavation-drawdown, different piezometric heads above the tunnel are taken into account. The simple results are presented in Table 8.

Table 8

Results of groundwater inflows in Weilai tunnel based on Equation (37)

Piezometric heads (m) .	Groundwater inflows .
80
77	93.55
72	87.50
70	85.13
63	76.69

Piezometric heads (m) .	Groundwater inflows .
80
77	93.55
72	87.50
70	85.13
63	76.69

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This example illustrates how to select an appropriate analytical solution to estimate the groundwater inflows into a given rock tunnel, and at the same time it shows the influence of the groundwater drawdown in the variability of these inflows.

This article highlights relevant keys factors influencing the accuracy of groundwater inflows into rock tunnels. Additionally, it presents the latest advances in the development of analytical solutions commonly used to predict groundwater in rock tunnels. One can observe that the equations reflecting the analytical solutions are numerous. The analysis helps us to understand that the accuracy of these predictions remains a high priority. In fact, given the vital importance of groundwater inflow into tunnels, there are many ongoing considerations that must be taken into account. Indeed, in view of the developed equations, due to enormous difficulties, the key factors that govern the reliability of groundwater inflow predictions are not properly taken into account. Therefore, many of the analytical solutions either overestimate or underestimate these predictions. However, precise prediction is highly necessary not only for safety, but also for the long-term stability of tunnels. Likewise, the design of effective dewatering systems that will serve as facilitators throughout tunnel excavation depends on the accuracy of predictions (Park et al. 2020).

Among the simplifying hypotheses, it can be noted that most of these equations consider that the groundwater inlets in the tunnels obey Darcy's law. However, such an assumption is not always fulfilled (Karay & Hajnal 2016; Xue et al. 2019). For example, in faulted areas where groundwater inflows can be extremely high, Darcy's law is unsuitable and cannot provide accurate predictions (Shi et al. 2018). In fact, in many situations, even the steady-state groundwater inflow prediction could achieve acceptable degrees of accuracy if the actual rock conditions were properly taken into account. However, as the behavior of rock masses remains complex, especially at great depth, their real characteristics are also complex to determine with precision. Their actual conditions are very difficult to model. Thereby, different interpretations are usually considered to approximate these characteristics and conditions in order to circumvent the difficulties. Hence many analytical equations are developed, but their accuracy is generally questionable with respect to their real applicability. Furthermore, it is important to specify that the steady state is part of the process which characterizes groundwater inflows in tunnels, it is not the whole process. Groundwater inflows into tunnels is firstly transitory (Aston & Lafosse 1985; Liu et al. 2020). When the flows are very high leading to water inrush, it is worth noting the phase of collection followed by a phase of instability in the coupled interactions between water and rocks (Liang et al. 2016).

The key factors govern the prediction of groundwater inflow and the long-term safety and stability of tunnels. For example, as related by Zingg & Anagnostou (2018), the variability in rock permeability is detrimental to stability. When the stability of tunnels is attacked, safety cannot be ensured. In fact, constant rock permeability considerations cannot guarantee accurate solutions for predicting groundwater inflows into tunnels. In the case of underestimated forecasts, the designed drainage or dewatering systems will be inefficient and unsuitable and groundwater may seep into the tunnels in the form of dripping or leakage at least. Note that one of the negative effects of any form of seepage flows is the instability generated at the tunnel faces (Perazzelli et al. 2014). Thereby, the overall resistance of the rocks surrounding the tunnels will be gradually reduced. Then, channels could also progressively evolve and serious groundwater inflows could occur into the tunnels. Indeed, disasters could still happen in such situations. Frequent economic losses are also caused by groundwater seepage inside underground structures (Guo et al. 2017). On the other hand, if overestimated, the drainage systems will also be unsuitable and uneconomical. In all situations, precise inflows of groundwater into tunnels are essential and required prior to tunneling. In fact, proper and effective drainage systems can be conceived if precise groundwater inflows into tunnels are determined in advance (Yang & Yeh 2007). All the key factors presented through this paper are required to accurately predict groundwater inflows into rock tunnels.

Tunnels are usually built to operate for decades. Their lifespan is sometimes projected for more than a hundred years. However, their durability depends mainly on their drainage systems. The efficiency of the latter stems from their adequate design which must take account of precise information on the maximum flows that they will have to transport. Thus, accurate prediction of groundwater inflows is a major requirement for decision-making leading to preservation of tunnel life. Knowing that the construction of such structures is associated with very substantial mobilizations and expenses, their longevity is extremely desirable.

Through this article, key factors that strongly influence the level of accuracy of the groundwater inflow predictions have been exposed. Also, an overview of the latest advances in analytical solutions concerning the prediction of groundwater inflows in rock tunnels has been presented. The main conclusions drawn are the following:

• – Despite the existence of various innovations in the prediction of groundwater inflows in tunnels, the ideal analytical and semi-analytical solutions are not yet fully achieved. The risks and damage associated with inaccurate predictions of groundwater inflows are still considerable. Above all, when the distance between the water table and the edges of the tunnels is not considerable, the approximate solutions are inefficient for engineering applications and for the overall management of the associated risks. The need to develop new analytical solutions with much more precision is more and more urgent. With the problem of climate change, rainfall varies randomly in many parts of the world and floods are very common. Analytical solutions including the key factors can be very helpful in sizing reliable dewatering systems facing huge groundwater inflows in tunnels.

• – The realistic characteristics of the rocky environments and the time-dependency behavior of groundwater inflows are like barriers to overcome in order to develop accurate predictions of groundwater inflows into tunnels. New solutions must effectively reflect the real tendency of groundwater inflows into tunnels. They must include a phase predicting the time-dependent flow and a steady-state phase predicting the groundwater stabilization flow. The temporal solution consists of dealing with the short- or medium-term action of groundwater inflows according to the geological and hydrogeological characteristics of the rocks near the tunnels. The steady-state solution must be a sustainable solution if the key factors influencing groundwater inflows in tunnels are effectively considered.

• – Since groundwater inflows into tunnels are frequent, especially at great depths, and rock masses are naturally discontinuous, non-Darcian flows must be prioritized in order to improve prediction accuracy. Likewise, in case of high inflow, inrush and water burst, attention must be paid to non-Darcian groundwater flows. Moreover, in all situations, accurate solutions of groundwater inflows must take into account the variability of relevant rock properties, even if it remains a challenge.

• – It is important to estimate the overall effects of the relevant aquifers on the magnitude of groundwater inflows into deep tunnels. A good prediction of the position of the water-rich areas relative to the projected edges of the tunnels is strongly required. In fact, the precise prediction of water inflows into the tunnels also depends on the safety thickness of the surrounding rocks. It is of extreme importance to determine this safety thickness with precision. Severe damage associated with groundwater inflows such as collapses, mud inrushes, human casualties, and financial losses can be minimized if the safety thickness is accurate and the groundwater inlets are predicted with sufficient accuracy.

• – Recently, researchers have combined numerical and analytical methods to obtain more reliable analytical solutions of groundwater inflow into tunnels. However, the need to improve accuracy is still considerable. It is observed that nowadays, tunnels are placed at greater and greater depths and the evaluation of the real properties of the rocks becomes more and more difficult. Laboratory testing does not always reflect actual rock exposure conditions, and monitoring time is usually too short to accurately estimate all relevant rock properties. The use of appropriate and verified remote sensing techniques associated with geographic information systems to monitor and control the behavior and relevant properties of rocks is strongly needed. The correct use of these techniques will detect the actual variation in rock properties as well as the potential risks of groundwater release as a result of excavation. Then, such techniques can be combined with analytical and numerical methods for the development of much more precise analytical solutions in the evaluation of groundwater inflows in tunnels.

• – The enrichment of real and reliable data relating to groundwater inflows into tunnels must be a priority. It is therefore necessary to categorize the different case studies and provide appropriate, secure and accessible databases. Data collection can be done efficiently with the help of groundwater inflow apps that need to be developed for this purpose. Thereby, analytical solutions will be designed with great success and their application will be very fruitful in different situations.

The authors would like to express their thanks to Dr Mohamed El Tani for early corrections, comments and suggestions of the article.

This research was supported by the Special Topics of National Key Research and Development Program of China (Grant Number 2018YFC1508801-4 and 2019QZKK020705).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.