Key factors influencing analytical solutions for predicting groundwater inflows in rock tunnels | Water Supply

Abstract

Predicting groundwater inflow into tunnels is essential to ensure the safe accessibility and stability of underground excavations and to attenuate any associated risks. Such predictions have attracted much attention due to their tremendous importance and the challenge of determining them accurately. Over recent decades, based on diverse methods, researchers have developed many relevant analytical solutions. Considering these research efforts, this article identifies and describes the most critical key factors that strongly influence the accuracy of groundwater inflow predictions in rock tunnels. In addition, it presents a synthesis of the latest advances in analytical solutions developed for this purpose. These key factors are mainly time dependency of groundwater inflows, water-bearing structures, aquifer thickness, hydraulic head and groundwater drawdown, rock permeability and hydraulic conductivity, fracture aperture, and rainfall data. For instance, groundwater inflows into tunnels comprise two stages. However, the transition between the stages is not always rapid and, for tunnels located in faulted karst terrains and water-rich areas, groundwater inflows can exceed 1,000 L/min/m. Under high stress, rock permeability can increase up to three times near the inevitable excavation-damaged zones, and groundwater inflows into tunnels can be significantly affected. Despite the enormous amount of research already conducted, improvements in the accuracy of predicting groundwater inflows into rock tunnels are still needed and strongly suggested.

HIGHLIGHTS

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The main factors influencing the accuracy of prediction of groundwater inflows into tunnels are comprehensively highlighted.
Accurate prediction of groundwater inflows into tunnels is not yet fully resolved.
The great need to continually improve the accuracy of these predictions is discussed.
The latest advances in analytical solutions for forecasting these inflows are presented.
A process for selecting an appropriate analytical solution to assess groundwater inflows in a given rock tunnel is proposed.

Graphical Abstract

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accurate prediction of groundwater inflows, groundwater inflows, key factors, long-term stability of tunnels, rock behavior

INTRODUCTION

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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.

TRIGGERING OF GROUNDWATER INFLOWS INTO ROCK TUNNELS

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Different conditions are generally propitious for triggering groundwater inflows into rock tunnels. Generally speaking, such triggering can be favored by three potential factors such as particular geological structures, presence of water-rich regions and perturbations due to tunneling (Peng et al. 2020). Excavation perturbations should be counted as predominant. In fact, when digging tunnels, existing stress field are redistributed. Subsequently, there is formation of the excavation-damaged zone (EDZ) and the excavation-disturbed zone (EdZ). The EdZ can also be called the excavation influence zone (EIZ); the EDZ is the plastic zone formed around the tunnel as shown in Figure 1.

Figure 1

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Stress–strain curve illustrating EDZ and EdZ (EIZ), after Wang et al. (2015), with permission from Elsevier.

In the EDZ, there are perennial deformations and significant depreciations of the physical, mechanical, hydraulic and geochemical properties of rocks (Cai & Kaiser 2005; Wang et al. 2015; Tian et al. 2019; Frenelus et al. 2021a). Depending on the geological and tunneling conditions, rockbursts can occur (Sun et al. 2007; Xiao et al. 2016), as the release of elastic deformation energy from the rocks was possible (Li et al. 2007). It should be noted that there is disturbance in the existing groundwater reserves, particularly when tunneling under the groundwater table. Groundwater inflows into tunnels are eased by the EDZ, which can create kinds of pathways (Armand et al. 2014; Gao et al. 2020). Since natural rocks are typically anisotropic (Barton & Quadros 2015), and made of discontinuities (Coli & Pinzani 2014), the paths created by the EDZ can be increased and groundwater inflows can be triggered into tunnels. The permeability of the surrounding rocks can be augmented thanks to the pathways generated by the EDZ. Note that depending on the rocky environment, existing faults zones can be reactivated by the redistribution of stresses, which also allows the flow of groundwater by increasing the permeability of rocks (Lianchong et al. 2011). The combination of the existing active fault zones and the EDZ can also generate groundwater flow paths in tunnels (Yu et al. 2020). Similarly, excavation fractures associated with existing ones can provoke groundwater flow paths in tunnels, and rock permeability can be sharply increased (Ni et al. 2016). As a result, various stages can lead to the triggering of groundwater influxes into tunnels. They are illustrated in Figure 2, from rock excavations by the tunnel boring machine (TBM) or drill-and-blast (DB) methods. In fact, referring to Sui et al. (2011), to effectively predict the groundwater into tunnels, the sources of water and flow-paths must be indicated.

Figure 2

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Steps leading to the triggering of groundwater inflows into rock tunnels (after Frenelus et al. 2021b).

It is important to emphasize that a safety thickness of the surrounding rocks can also be considered to understand the triggering of groundwater inflows into tunnels (Figure 3). With a safety thickness, fractures cannot easily disseminate (Tu et al. 2021), and thus the occurrence of groundwater inflows can be delayed into the tunnels. The safety thickness is generally the existing grouting length between the grouting zone without excavation and the zone without grouting (Liu et al. 2020). More precisely, it is the minimum necessary thickness that allows the hydraulic barrier to withstand the influx of water in tunnels (Xue et al. 2021). However, its effectiveness depends mainly on rock mass quality, the tunneling conditions and the groundwater pressure. Groundwater inflows can arise in tunnels particularly located in water-rich areas if the safety thickness of the surrounding rocks is not more than 4 m or 5 m (Liu et al. 2020). In other words, there is an influx of groundwater into tunnels when the minimum safety thickness is reached by the excavation effects (Sun & Liu 2011; Tu et al. 2021).

Figure 3

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Safety thickness of the rocks surrounding the tunnels, modified from Liu et al. (2020).

In water-rich areas, the minimum safety thickness of the surrounding rocks can be estimated as follows (Jiang et al. 2017):

(1)

where is the thickness of fracture zone and can be estimated by geophysical testing. It is generally in the range of 1–3 m.
is the protection zone thickness that depends on several factors and can be calculated as follows (Jiang et al. 2017):
(2)
where: R is the tunnel radius; is the water pressure in the water-rich region; is the critical splitting rupture water pressure and can calculated follows:
(3)
where ⁠: is the lateral pressure coefficient; a is half the length of the crack; is mode II fracture toughness of the rock type; is internal friction angle; is the angle between the major axis of the crack and the maximum principal stress ⁠.
In non-water-rich areas, after rock excavations, the possibility of groundwater inflows triggering in rock tunnels may also exist. Generally speaking, the inflows of groundwater into rock tunnels depend on several factors related to rock characteristics and conditions, tunnel conditions, topography, precipitation, and so on. However the safety thickness of rocks surrounding the tunnels located in non-water-rich areas will be different from those situated in water-rich areas. In fact, in non-water-rich zones, the water pressure can be low. As such, the triggering of groundwater inflows into tunnels can be retarded compared to the situation described in water-rich areas. However, as shown in Figure 2, rock permeability plays a fundamental role in triggering groundwater inflows into tunnels. Depending on how rock permeability has been enhanced, groundwater can trigger into tunnels. For example, as reported by Hong et al. (2021), initially impermeable areas can become permeable by the soft plastic Loess effect. It was reported by Kaya et al. (2017) that an influx of groundwater occurred in a portal tunnel in Turkey due to the change in weathered andesitic tuffs from impermeable to low permeability during rainy periods. In fact, when addressing the inflows of groundwater into tunnels, the availability of groundwater cannot be overlooked. Indeed, water is considered to be an important factor for triggering the inflows into tunnels (Zhang et al. 2017). Even when there is small availability of groundwater in the vicinity of rock surrounding the tunnels, a safety thickness is required to prevent impetuous groundwater inflows into tunnels. It should be noted that when a small amount of groundwater persists for a long time near the tunnels, the rocks may lose their strength and groundwater inflows may occur, even on a small scale. Referring to Cesano et al. (2000), the drainage of the rock mass is responsible for minor inflows of groundwater into tunnels located in hard rocks. It is possible that minor groundwater inflows can be found in tunnels excavated in non-water-rich water areas.

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)

TYPES AND MECHANISMS OF GROUNDWATER INFLOWS IN ROCK TUNNELS

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Based on the magnitude of the flows, groundwater inflows in tunnels can be categorized into six types. They are mainly the following for tunnels of 6 m diameter (Sharifzadeh et al. 2012): dripping (very low), leakage (low), inflow (medium), high inflow (high), inrush (very high), and water burst (extreme high). Figure 4 shows descriptions of groundwater flow ranges that can reflect their mechanisms. Normally, the flow of groundwater entering to tunnels can vary from dripping to water burst. From an engineering point of view, the name ‘groundwater inflow’ can refer to one of the aforementioned types. Although medium to very high flows levels are the most rapidly destabilizing and destructive, all have negative impacts on the long-term safety and stability of tunnels. The pertinent properties of the rocky environments and their conditions of exposure are the main factors that govern the mechanism of groundwater influx into tunnels.

Figure 4

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Types of groundwater inflows into tunnels for 6 m diameter, including their associated geological features and hydrology conditions, data from Sharifzadeh et al. (2012).

As shown in Figure 4, for tunnels with 6 m in diameter, groundwater inflows can occur in different forms including the water burst, which is the most devastating. According to Sousa & Einstein (2021), when the influx of groundwater entering the tunnels is under pressure, it is thus the water burst. Such a form of groundwater inflow leading to flooding of tunnels has already been recorded in China (Sousa & Einstein 2021). A thorough understanding of the features linked to groundwater inflows into rock tunnels is necessary for better predictions (Figure 5).

Figure 5

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Features related to groundwater inflows into rock tunnels, adapted from Sharifzadeh et al. (2012).

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.

KEY FACTORS INFLUENCING THE ACCURACY OF GROUNDWATER INFLOWS INTO ROCK TUNNELS

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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.

Time dependency of groundwater inflows

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Groundwater inflows into tunnels are time dependent (Hwang & Lu 2007; El Tani 2010; Liu et al. 2018; Xia et al. 2018; Golian et al. 2021; Han et al. 2022). According to Aston & Lafosse (1985), they are generally composed of temporary inflows and persistent inflows particularly at depth. In fact, below the safety thickness of the rocks near the tunnels, the final phase of groundwater inflows into tunnels is the steady state (Liu et al. 2020). However, only that phase is mainly considered in the establishment of many analytical solutions predicting groundwater inflows in tunnels. It is therefore difficult for these many analytical solutions to accurately predict such inflows. Indeed, the time-dependent character of groundwater inflows into rock tunnels cannot be neglected. Since it is itself a tributary of other relevant factors such as groundwater drawdown and hydraulic conductivity, it should be considered a key factor that strongly influences the accuracy of the mentioned predictions. It is important to consider that the passage between the transitory state and the steady state of groundwater inflows in tunnels is not always rapid. This is illustrated in Figure 6.

Figure 6

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Illustration of time-dependent groundwater inflow in Nosud tunnel in Iran from May 26 to August 15, 2015 (data from Golian et al. (2021), with permission from Springer Nature).

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)

Depending on the characteristics of the rock mass and the excavation conditions, groundwater inflows into tunnels can be stabilized after a considerable time. For example, during the digging of the Yungchuen tunnel in Taiwan, groundwater inflows varied greatly and took nearly six months to progressively stabilize (Wang et al. 2011). In some situations, groundwater inflows in tunnels remain substantial for a notable time. Therefore, tunnel projects are abandoned due to the uncontrollable effects of high groundwater pressures and flows. This was the case for the Alfalfal tunnel in Chile in which excessive damage generated by extreme groundwater inflows forced the abandonment of the project (Holmøy & Nilsen 2014). In order to better plan the tunneling project and prevent huge damage and financial loss, a prior assessment of the stabilization time of groundwater inflows into the tunnels is required. By examining the rate of inflows in three tunnels in the level of groundwater in neighboring boreholes, Golian et al. (2021) proposed an analytical approach to estimate the necessary time of the downturn of groundwater inflows in tunnels. This proposition is established using exponential laws as follows (Golian et al. 2021):

(7)

where: Q is the groundwater inflow rate in tunnels; t is the recession time of groundwater inflows; I and F are constants parameters which can be estimated as

⁠;

⁠; G is also a constant estimated as

⁠; D correspond to the hydraulic diameter; E is also a constant that can be determined in specific situations;

is the coefficient of recession; n is a power coefficient.

Water-bearing structures

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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.

Aquifer thickness

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Aquifer thickness is another key factor influencing the magnitude of the groundwater inflows into tunnels. In fact, the thicker the aquifer, the greater the inflows of groundwater into the tunnels can be. Ying et al. (2018) demonstrated that for tunnels with shallow or medium burial depth, the thickness of the aquifer has clear effects on the extent of groundwater inflow. Nevertheless, no matter how deep the tunnel is buried, the effects of aquifer thickness on groundwater inflows must be considered. Figure 7 shows the increase in groundwater inflow with increasing aquifer thickness. More broadly, the variation in the thickness of confined aquifers undoubtedly generates a variation in groundwater inflows into tunnels (Liu et al. 2018). Determining the thickness of aquifer, as well as their capacity and conditions are of utmost importance to achieve fairer prediction of groundwater inflows into tunnels. Normally, an accurate prediction of groundwater inflows should include the thickness of the aquifers involved as one of the key parameters. Indeed, in real rock conditions, the thickness of aquifers can vary throughout the rocks surrounding the tunnels. Besides, it varies as the burial depth of the tunnels increases. Therefore, such situations need to be further investigated to provide analytical solutions of groundwater inflows into tunnels which could properly reflect them. Thereby, the prediction of groundwater inflows into tunnels can be accurately improved.

Figure 7

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Influence of aquifer thickness on the magnitude of groundwater inflow into tunnels. For the symbols, is the aquifer thickness; is the hydraulic conductivity; is the drawdown at the heading of the tunnel; is the tunnel radius; is the specific storage capacity of the aquifer. Reprinted from Hwang & Lu (2007), with permission from Elsevier.

Hydraulic head and groundwater drawdown

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Groundwater inflows into tunnels also depend on the hydraulic head and groundwater drawdown. In fact, the hydraulic head generally varies during groundwater influxes entering the tunnels. Its careful estimation is required to correctly evaluate the groundwater inflow into tunnels. The geological and hydrogeological conditions of the excavations are of great importance for controlling the hydraulic head. The hydraulic head is usually increased by the presence of significant permeable zones in the vicinity of the tunneling areas (Moon & Jeong 2011). Specifically, the hydraulic head varies according to the geological and hydrogeological conditions existing around the tunnels. It should be noted that the sum of pressure head (P) and elevation head (⁠

⁠) represents the total hydraulic head (H) (Su et al. 2017; Tang et al. 2018; Li et al. 2021), and can be written as follows (Tang et al. 2018):

(8)

By considering a semi-infinite aquifer (Figure 8), where the medium is completely saturated, homogeneous and isotropic, Ming et al. (2010) established a relevant equation to assess the hydraulic head distribution and pore pressure boundary condition at the perimeter of a circular tunnel, based on conformal mapping methods and complex variables. Under such conditions, according to Ming et al. (2010), the hydraulic head is given as follows:

(9)

where,

⁠: hydraulic head;

: water pressure at the perimeter of tunnel;

⁠: water specific weight; h : tunnel depth;

tunnel radius;

⁠: water depth above the tunnel;

: variable of the Laplace function.

Figure 8

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Illustration of a circular tunnel excavated in a semi-infinite aquifer (adapted from Ming et al. 2010; Li et al. 2021).

Equation () proposed by Ming et al. (2010) can evaluate a constant hydraulic head which depends on several parameters. It illustrates the complexity associated with hydraulic head when assessing groundwater inflows in tunnels. In reality, the prediction of groundwater inflows in a given tunnel is an approximation by considering the hydraulic head as constant. Indeed, the hydraulic head generally decreases linearly with the increase in the burial depth of the tunnels (Guo et al. 2021; Li et al. 2021). Such a situation should be taken into account in the calculation of the actual groundwater inflows aimed at designing reliable and accurate drainage systems for deep tunnels or for those especially built in mountainous areas. When these types of tunnels are built on water-rich areas, they may experience high hydraulic heads which can be influenced by existing faults (Zhang et al. 2021). In most of these tunnels, the burial depth usually varies from entrance to exit or along the routes. Computing groundwater inflows for such tunnels with consideration of constant hydraulic head, could not meet the needs for reliable drainage systems in different sections. In a semi-infinite saturated medium, the influence of hydraulic head on the magnitude of groundwater inflows into circular tunnels can be expressed as follows (Guo et al. 2021):

(10)

where Q: total groundwater ingress in the tunnel,

⁠: thickness of the aquifer,

⁠: rock permeability, and

⁠,

⁠: respectively polar radius and polar angle related to the polar coordinate system considered in physical plane.

Considering each position of a tunnel located in water-rich fault zones, Zhang et al. (2021) proposed the following relationship between groundwater inflows (Q) and hydraulic head (H):

(11)

where

⁠: seepage cross-section area;

⁠: the radial seepage radius, and

⁠: rock permeability coefficient

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).

Technically, the drawdown of the groundwater level can be seen as the level difference between the initial water level and the current level. Nevertheless, as it is correlated with time (Liu et al. 2018), its exact determination remains a difficult task (Table 3). Figure 9 illustrates the drawdown of the groundwater level after occurrence of groundwater inflows in a confined aquifer.

Table 3

Relevant analytical solutions developed to estimate the groundwater drawdown

Drawdown S and parameters	Equation number	Remarks	Study (year)
⁠: Groundwater inflow into tunnel given by Equation (32); rock hydraulic conductivity; ⁠: Reservoir level above the tunnel; ⁠: Reservoir level above an impermeable layer; ⁠: distance between tunnel and reservoir; ⁠: Tunnel radius or seal extrados; horizontal coordinate.	(12)	Equations derived from particular analytic developments. Equations adapted to circular tunnels in faulted rocky environments.	El Tani et al. (2019)
⁠: Groundwater inflow per linear meter in the tunnel (⁠⁠); tunnel length; ⁠: time from the start of water leakage (d); ⁠: pressure conductivity coefficient; ⁠: specific yield; ⁠: half-length of tunnel in x-axis; half-width of tunnel; ⁠: minor axis; ⁠: a defined function based mainly on Gaussian error function.	(13)	Equations based on area-well theory. The first equation is the transverse drawdown and the second one is the longitudinal drawdown at the center of the tunnel. They are suitable for cylindrical tunnels in supposedly homogeneous rocks.	Cheng et al. (2019)
groundwater inflow from the initial infiltration surface (⁠⁠); ⁠: time (d); thickness of confined aquifer (m); ⁠: specific storage (⁠⁠); ⁠, ⁠: respectively, horizontal hydraulic conductivity along the x, y, z axes (⁠⁠).	(14)	The equations are derived from the Taylor expansion and allow the estimation of groundwater drawdown for circular tunnels in rocky media taking into account the variation in hydraulic conductivity.	Liu et al. (2018)

Drawdown S and parameters	Equation number	Remarks	Study (year)
⁠: Groundwater inflow into tunnel given by Equation (32); rock hydraulic conductivity; ⁠: Reservoir level above the tunnel; ⁠: Reservoir level above an impermeable layer; ⁠: distance between tunnel and reservoir; ⁠: Tunnel radius or seal extrados; horizontal coordinate.	(12)	Equations derived from particular analytic developments. Equations adapted to circular tunnels in faulted rocky environments.	El Tani et al. (2019)
⁠: Groundwater inflow per linear meter in the tunnel (⁠⁠); tunnel length; ⁠: time from the start of water leakage (d); ⁠: pressure conductivity coefficient; ⁠: specific yield; ⁠: half-length of tunnel in x-axis; half-width of tunnel; ⁠: minor axis; ⁠: a defined function based mainly on Gaussian error function.	(13)	Equations based on area-well theory. The first equation is the transverse drawdown and the second one is the longitudinal drawdown at the center of the tunnel. They are suitable for cylindrical tunnels in supposedly homogeneous rocks.	Cheng et al. (2019)
groundwater inflow from the initial infiltration surface (⁠⁠); ⁠: time (d); thickness of confined aquifer (m); ⁠: specific storage (⁠⁠); ⁠, ⁠: respectively, horizontal hydraulic conductivity along the x, y, z axes (⁠⁠).	(14)	The equations are derived from the Taylor expansion and allow the estimation of groundwater drawdown for circular tunnels in rocky media taking into account the variation in hydraulic conductivity.	Liu et al. (2018)

Figure 9

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Drawdown (⁠⁠) of groundwater in a section of a given tunnel located in a confined aquifer, data from Liu et al. (2018), with permission from John Wiley & Sons.

Rock permeability and hydraulic conductivity

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Groundwater inflows into tunnels are strongly influenced by rock permeability. As shown through Figure 2, when the permeability of surrounding rocks are augmented and become considerable, groundwater inflows can be triggered into tunnels. Mainly, rock lithology and pore pressure are factors on which permeability depends. When such factors are favorable, rock permeability can quickly increase around the surrounding rocks of tunnels. Moreover, owing to the weakened zones created by the tunneling, the relevant characteristics of rocks are modified and the permeability can also increase more rapidly. In fact, the permeability of rocks considerably varies (Coli & Pinzani 2014; Barton & Quadros 2015), due to their common anisotropy (Barton & Quadros 2015). It can be even tripled from its initial value in the vicinity of the EDZ and under high stresses (Chen et al. 2015). Instead of a material property and due of its variation with exposure conditions, rock permeability can even be thought of as a process (Heiland 2013). Note that the anisotropy of rock permeability is due to the complex apportionment that exists in rock massifs (Ma et al. 2020). The variation of rock permeability can cause non-Darcian groundwater flows in tunnels (Ni et al. 2016). This is a major consequence of the variability in the hydraulic conductivity of rocks. Taking into account the real nature of the groundwater flow regime is necessary for suitable predictions. Figure 10 shows an illustration of the influence of the variability of rock permeability on the rate of groundwater inflow into tunnels.

Figure 10

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Influence of rock permeability on the magnitude of groundwater inflow into tunnels, after Hwang & Lu (2007), with permission from Elsevier. For the symbols, is the aquifer thickness; is the hydraulic conductivity; is the drawdown at the heading of the tunnel; is the tunnel radius; is the specific storage capacity of the aquifer.

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).

The hydraulic conductivity of rocks is usually proportional to the permeability of rooks. Therefore, any variation in the permeability of the rocks affects the hydraulic conductivity. For the same reasons above evoked, the realistic hydraulic conductivity of the surrounding rocks is not constant, and according to Coli et al. (2008), it is strongly anisotropic. In porous media, it varies due to the variability in grain size (Chandel et al. 2022). Furthermore, the hydraulic conductivity varies randomly and shows a tendency to devaluation at depth (Bai & Elsworth 1994; Jiang et al. 2010). However, it increases under the effect of significant water pressure (Xue et al. 2021). Its distribution can even be employed as means of predicting groundwater inflows into tunnels (Jiang et al. 2010; Masset & Loew 2013). Additionally, it is important to note, as studied by Lu et al. (2020), under excavation disturbances where stress–strain processes generally evolve, variations in hydraulic conductivity are observed. Simultaneously, the variation in hydraulic conductivity caused by excavation disturbances inevitably leads to considerable variations in groundwater inflows into rock tunnels (Figure 11).

Figure 11

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Influence of hydraulic conductivity on the magnitude of groundwater inflow into tunnels, after Lu et al. (2020). Advanced distance refers to the consideration of step by step excavation.

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 i^th set of discontinuities; ⁠: average hydraulic aperture of the i^th 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 i^th set of discontinuities; ⁠: average hydraulic aperture of the i^th 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)

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.

Fracture aperture

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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.

Rainfall data

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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 rainfall data strongly influence the magnitude of groundwater inflow into tunnels, they rarely integrate the existing analytical solutions. It is important to have accurate rainfall data related to the location of underground tunnels. Such data can help to know the amount of rainfall that could recharge the aquifers or any water-bearing structures. In fact, the more the aquifers are recharged, the higher the risk of occurrence of groundwater inflows into tunnels during and after the excavations. Consequently, developing solutions without taking into account recent or historical rainfall data in the vicinity of the excavations risks being imprecise. Aquifer recharge is usually estimated using the water budget imposed by the water cycle by taking into account precipitation, evapotranspiration, runoff and variation in soil water content (Lo Russo et al. 2013). The groundwater budget is considered as the main origin of groundwater inflow (Sammarco 1986; Lo Russo et al. 2013). Underground excavations perturb water stored in aquifers. A response to these perturbations may be the influx of groundwater into excavated areas. According to Lin et al. (2019), the duration of precipitation is one of the important types of rainfall data that greatly influences the extent of groundwater entering tunnels. Particularly for karst areas, the longer the rainfall duration, the greater the magnitude of groundwater inflows into the tunnels (Lin et al. 2019). For different rainfalls, Figure 12 shows the influence of rainfall duration on the extent of groundwater inflows into tunnels constructed especially in karst zones with fissures.

Figure 12

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Influence of rainfall duration on groundwater inflow into tunnels for different rainfalls (Rf) especially in karst environments with fissures. Data from Lin et al. (2019), with permission from ASCE.

RELEVANT ANALYTICAL SOLUTIONS FOR GROUNDWATER INFLOWS INTO ROCK TUNNELS

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Analytical solutions considering the time-dependency of groundwater inflows

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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 1^st 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 1^st 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)

Steady-state analytical and semi-analytical solutions of groundwater inflows

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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 2^nd kind of order zero; ⁠: modified Bessel function for the 1^st 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 (⁠⁠);