Introduction

Water inrush is a sudden and often overwhelming flow of water into underground workings (the Mindat Glossary). Water inrush (usually with mud gushing) is one of the geological disasters that can be faced in underground engineering works which might impose huge impacts on construction timetable, financial plan, and on project safety as well (Hou et al. 2016). This event is particularly emphasized in karst terrains where, because of high heterogeneity and anisotropy of karstified rock masses (Bakalowicz 2005; Ford and Williams 2007), the occurrence of unexpected events is prevalent (Parise et al. 2015a). Serious water inrushes during the tunnel construction can result in human casualties and financial losses (Hou et al. 2016). Table 1 represents a history of water inrush events in some tunnels with causes, casualties, and consequences.

Table 1 Documented water inrush events in some tunnels with their causes and consequences
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Therefore, the resumption of excavation depends on the relative discharge of water inrush to the tunnel face. Because in tunneling projects, downtime of the tunnel boring machine (TBM) causes high financial loss, knowing the discharge time after the water inrush is of high importance. Water discharge time estimation will help project managers to instantly make appropriate decisions and be able to maintain the excavation process as scheduled or, at least, mitigate the delays.

The occurrence of water inrush during tunnel excavation could be prevented where there is accurate and detailed knowledge on the local geology, from both the hydrogeological and engineering geological point of view. However, in complex geological settings, there might always be a certain amount of unpredictability, due to insufficient data, spatial and temporal variability of hydrogeological properties, complexity in the groundwater flow paths, the position of main conduits, and speleogenesis (Milanović and Vasić 2016; Milanović 2016), or due to a combination of the above, especially in karst terrains. Based on the statistics data, up to the end of 2008 in China, nearly half of tunnels in construction and operation had encountered large-scale water inrushes in karst areas, because of complexities in these natural setting, and difficulty in reaching a full understanding of the hydrogeological properties (Li et al. 2016).

Water inrush in tunnels can be particularly difficult to manage in the karstified rock mass and weakened fault zones, rich in groundwater and with high permeability (Zhao et al. 2013; Li et al. 2016; Wang et al. 2017). In detail, karst environments are highly prone to these situations, due to the great heterogeneity of karst aquifers, and unpredictability of the groundwater paths in karst aquifers, even with the possibility to reactivate the previously dry passages and conduits, in consequence of the hydrogeological changes caused by the tunnel construction (Waltham and Smart 1988; Milanović 2000, 2002; De Waele et al. 2011; Parise et al. 2015a, b, 2018). In addition to imposing delays in the construction of engineering works, related damages, and casualties, these events might also progressively move upward, eventually producing consequences at the ground, and originating sinkholes (Wang et al. 2008; Vigna et al. 2010a; Gutierrez et al. 2014; Parise 2019).

Similar to the flow rate recession of karst springs (Fiorillo 2014; Stevanović 2015), there is a sudden drop in inrush rate in a short period, which is due to draining of that part of the aquifer in the immediate surroundings of the tunnel. Over time, water inrush to the tunnel decreases slowly and reaches a constant rate, when the impacted aquifer reaches a new condition of equilibrium with the tunnel. Due to the need for a proper method that can be used to estimate the groundwater discharge time after the inrush, a new method considering the nature of the recession curve is introduced in this study.

Methodology

Hydrodynamic of conduit flow

Water inrush in tunnels is often due to an encounter of the tunnel face with a karst conduit or conduit fault (McCallum et al. 2016; Milanović 2016). In conduit flow systems, where predominantly turbulent flow conditions prevail, especially at the local, applying Darcy’s law (and equations developed on it) is not correct due to the high proportion of turbulent flow, which contradicts the limitations of the laminar flow assumptions (Fanchi 2002; Giese et al. 2018). Conduit flow especially prevails in the first minutes or hours of the inrush, but due to emergency, usually, no monitoring is possible at that moment. After that first period, a higher proportion of groundwater inrush generates from the rock mass (consisting of numerous mutually connected discontinuities and karstic voids), and the inrush becomes more predictable; in some approximate way, the flow can be subjected to mathematical approach study. The scale issue becomes more and more important (Sauter 1992; White 2002), also due to the difficulty in identification of the hydrologic catchment boundaries in karst. Results of investigations showed that the hydrodynamics of flow in conduit-flow prevailing media can be described or simulated based on the hydrodynamics of pipes or in open channels. For instance, McCormack et al. (2017) developed a two conduit network model based on the hydrodynamics of open channels that simulated the main karst conduit flow system and discharge from a conduit fault in the Bell Harbour watershed in western Ireland, using a complex network of pipes (representing karst conduits), drainage channels (representing faults) and tanks (representing lakes). Also, 3D and mathematical modeling of karst conduit spatial position, as well as model of flows, was obtained and serve as a foundation for the simulation of technological processes in defining groundwater seepage pathways in the Višegrad dam rehabilitation process (Milanović and Vasić 2016). Besides, Perriquet (2014), using recession analysis, observed conduit hydrodynamic behavior in some boreholes drilled in the Bell Harbour watershed. On the other hand, it is generally accepted that the flow under pressure karst conduits can be simulated by the Darcy-Weisbach equation (Harmon and Wicks 2006).

Determining water inrush discharge time during the tunnel construction

By experience, it can be stated that the drawdown of tunnel inrush water level is exponential within in time (Li et al. 2018). Recession curve analyses are common in karst hydrogeology, especially exercised for estimation of karst spring regime. There are several equations, used for over a century (Boussinesq 1904; Maillet 1905; Mangin 1975; Dewandel et al. 2003). Still, using them as they are for the water inrush calculations during the tunnel construction works does not seem plausible, because the monitoring time is too short and does not comprise all the components of the recession curve, and especially not the master regression curve. Nevertheless, the relation of the inrush recession in a tunnel over time, similarly to the discharge recession relation of a karst spring (Maillet 1905), can be described by an exponential equation as follows:

$$ \mathrm{Q}\left(\mathrm{t}\right)={\mathrm{Q}}_0{\mathrm{e}}^{-\propto \mathrm{t}} $$
(1)

where Q(t) and Q0 are the inrush rate [L3T−1] at time t and the initial inrush rate at an earlier time, respectively and α is the recession coefficient [T−1].

Using this equation, it is possible to assess the time required for reaching the inrush to a level that makes possible recovery of the tunnel operation. The equation can be achieved by the following steps after a short period of daily groundwater level (GWL) monitoring in boreholes and water inrush rate into the tunnel.

Step 1: Determining the relation of groundwater level drawdown in time

During the recession of water inrush, the GWL hydrograph component in nearby boreholes had a recession that can be approximated by a simple exponential equation as below (Degallier 1966):

$$ \mathrm{H}={\mathrm{H}}_0{\mathrm{e}}^{-\propto \mathrm{t}} $$
(2)

where H0 and H are the measured GWL [L] in the borehole at the beginning of the measurement period and after a certain time (t), respectively. The linear form of the above equation can be determined as follows:

$$ \ln \left(\mathrm{H}\right)=\ln\ \left({\mathrm{H}}_0{\mathrm{e}}^{-\propto \mathrm{t}}\right)\to \ln \left(\mathrm{H}\right)=-\propto \mathrm{t}+\ln \left({\mathrm{H}}_0\right)\to \ln \left(\mathrm{H}\right)=-\propto \mathrm{t}+\mathrm{D} $$
(3)

where D is a constant equal to ln(H0) To obtain this equation, one should draw a natural logarithm of the observed GWL against time in a graph and determine its linear regression.

Step 2: Determining the relation of water inrush recession and groundwater level drawdown

Since conduit flow hydrodynamics in karstic conduits and conduit faults can be modeled in a way similar to the hydrodynamics of pipes or open channels, after extensive evaluations the developed simple power equation for Parshall Flume (Parshall 1928) and the Darcy-Weisbach equation was used to express the relationship of the inrush rate to the tunnel and its corresponding hydraulic head measured in adjacent boreholes (Fig. 1):

$$ \mathrm{Q}=\mathrm{C}{\mathrm{H}}^{\mathrm{n}} $$
(4)
$$ \mathrm{r}\ {\mathrm{Q}}^2=\mathrm{H} $$
(5)
$$ \mathrm{r}=\raisebox{1ex}{$8\mathrm{fL}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{D}}^5{\uppi}^2\mathrm{g}$}\right. $$
(5a)
Fig. 1
figure 1

Head-discharge relationship for a a 1.2-m Parshall Flume device and b the Nosud Tunnel

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where C is the free-flow coefficient for Parshall Flume, n is Parshall Flume power constant, f is Darcy–Weisbach friction factor, L is pipe length [L], D is hydraulic diameter [L], and g is the gravitational acceleration [LT-2]. If inrush is fully turbulent, r can be considered a constant. The linear form of the above equations can be determined as follows:

$$ \ln \left(\mathrm{Q}\right)=\ln\ \left(\mathrm{C}{\mathrm{H}}^{\mathrm{n}}\right)\to \ln \left(\mathrm{Q}\right)=\mathrm{n}\ \ln \left(\mathrm{H}\right)+\ln \left(\mathrm{C}\right)\to \ln \left(\mathrm{Q}\right)=\mathrm{n}\ \ln \left(\mathrm{H}\right)+\mathrm{E} $$
(6)
$$ \ln \left(\mathrm{r}\ {\mathrm{Q}}^2\right)=\ln\ \left(\mathrm{H}\right)\to 2\ln \left(\mathrm{Q}\right)+\ln \left(\mathrm{r}\right)=\ln \left(\mathrm{H}\right)\to \ln \left(\mathrm{Q}\right)=0.5\ \ln \left(\mathrm{H}\right)+\mathrm{A} $$
(7)

where E and A are constants equal to ln (C) and -0.5 ln (r), respectively. As seen, Eqs. 6 and 7 are similar; however, the logarithmic form of the Parshall Flume equation is the general form, which is considered in this study. Therefore, one should draw a natural logarithm of the observed inrush rates and a natural logarithm of the observed GWL in the graph, and determine its linear regression equation.

Step 3: Determining the relation of water inrush recession in time

By integrating Eqs. 3 and 6, the exponential equation of the water recession can be determined.

$$ {\displaystyle \begin{array}{c}\ln \left(\mathrm{Q}\right)=\mathrm{n}\ \left(-\propto \mathrm{t}+\mathrm{D}\right)+\mathrm{E}\to \ln \left(\mathrm{Q}\right)=-\left(\propto \mathrm{n}\right)\mathrm{t}+\left(\mathrm{nD}+\mathrm{E}\right)\to \ln \left(\mathrm{Q}\right)=-\mathrm{F}\ \mathrm{t}+G\\ {}\to \mathrm{Q}={\mathrm{e}}^{-\mathrm{F}\ \mathrm{t}+\mathrm{G}}\to \mathrm{Q}={e}^G\times {\mathrm{e}}^{-\mathrm{F}\ \mathrm{t}}\to \mathrm{Q}=\mathrm{I}{\mathrm{e}}^{-\mathrm{F}\ \mathrm{t}}\end{array}} $$
(8)

where F, G, and I are constants equal, respectively, to αn, nD+E, and eG.

For a clearer understanding, the method of determining water discharge time is described in the three following case studies.

Case studies

The described method was first developed for the Nosud tunnel (Iran) and later for further validation applied to the Headrace tunnel (Sri Lanka) and the Glas tunnel (Iran).

Case study 1: The Nosud Tunnel

The Nosud project is one of the most important water supply projects in Iran, located in the northwest of Kermanshah province, near the Iran-Iraq border. According to the plan, it is expected to transfer the Sirwan River water after having it stored at the Daryan Dam, using the Nosud gravity water conveyance tunnel for a length greater than 49 km, with a slope of about 0.082%. The tunnel consists of three parts. Excavation of the second stretch of the tunnel is approximately 25.7 km long, with a boring diameter of 6.73 m and a final diameter of 6 m. The Zimakan Valley study area is located approximately at the chainage 19 to 23 km, of the second stretch of the tunnel, as shown in Fig. 2.

Fig. 2
figure 2

Location map and satellite image of the second stretch of the Nosud tunnel

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Geological characteristics of the study area

The rock outcrops along the tunnel way in the Zimakan Valley include the Surmeh (Jsm), the Nairiz (Jnz), and the Khan-e-Kat (Trkh) Formations. (Fig. 3). The Surmeh Formation consists of marly limestone and shale with an overall thickness of about 400 m. Below this unit, marls of the Nairiz Formation are presented with an overall thickness of about 50 m (Sadeghi et al. 2020). The oldest lithotypes in the area are the Triassic dolomites belonging to the Khan-e-Kat Formation. In the Zimakan Valley, most of the tunneling problems have occurred in the Khan-e-Kat Formation, affected by faulting and karst.

Fig. 3
figure 3

Geological map and profile of the Nosud Tunnel in the Zimakan Valley (chainage 19 to 23 km)

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Event of groundwater inrush into the tunnel

In late May 2015, excavation operation of the Nosud tunnel at chainage 19.632 km in the Zimakan Valley stopped due to the sudden water inrush of about 4.1 m3/s to the TBM face leading to serious damages to construction equipment and machinery and imposing a serious delay of about four months to the project (Fig. 4). At this point, TBM encountered a karst conduit in the Khan-e-Kat formation (Figs. 5 and 6). In crisis management meetings held after the event, all emergency plans and decisions depended upon the time needed for water inrush discharge; therefore, estimation of the time became a priority for the committee.

Fig. 4
figure 4

Water inrush to a TBM Back up utilities and b the tunnel cavern area

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Fig. 5
figure 5

a Schematic proposal and b 3-D block diagram of the Zimakan Valley, showing the tunnel route and stopping location of the inrush site in the Zimakan Valley

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Fig. 6
figure 6

A view of the karst conduit causing the water inrush to the tunnel at chainage 19.632 km

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Determining the water inrush discharge time

According to GWL monitoring in boreholes along the tunnel route in Zimakan Valley (Fig. 7), the BH-S4 and BH-D boreholes had a similar GWL decrease rate which was different from the BH-S5 borehole (location of boreholes is presented in Fig. 5). Figure 7 shows hydrographs of GWL in BH-S5, BH-D, and BH-S4 boreholes. According to the BH-S4 and BH-D level changes, a GWL loss of 11.5 m occurred from May 28 to June 21, 2015, over 25 days. The GWL decrease in the BH-S5 borehole, on the other hand, was about 0.2 m. The reason for this difference was unknown, but it can assume that BH-S5 is out of the main karst preferential flow paths. It is also notable that the GWL decrease was close to linear, pointing to the fact that the outflow was mostly dictated by the hydraulic resistance of the karst features and the hosting rock mass. Figure 8 shows the hydrograph of the measured inrush rate at different locations in the tunnel. An amount of 7.1 million m3 of water, almost equal to the water consumption of Tehran metropolitan in three days, discharged from the tunnel in these 25 days. Water inrush measurement was carried out using the water current meter.

Fig. 7
figure 7

GWL recession in the BH-S5, BH-S4, and BH-D boreholes from May 28 to June 21, 2015

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Fig. 8
figure 8

Inrush rate measured at different locations in the tunnel from May 28 to June 21, 2015

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Twenty-seven days after the inrush event, based on daily monitoring data, the required time for tunnel inrush recession in the tunnel was calculated, using the method here presented. Therefore, the linear equation of the natural logarithm of GWL against time was derived for the BH-D borehole (Eq. 9; Fig. 9). Then, changes in the natural logarithm of the inrush rates were fit to the changes in the natural logarithm of GWL (measured in the BH-D), and the linear equation correlating these parameters was determined (Eq. 10; Fig. 10). Finally, the exponential recession equation of water inrush against time was derived by substituting two equations, and through related simplification (Eq. 11).

Fig. 9
figure 9

Correlation diagram of the natural logarithm of GWL in the BH-D borehole versus time

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Fig. 10
figure 10

Correlation diagram of the natural logarithm of GWL in BH-D borehole and the natural logarithm of inrush rate to the tunnel face

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Step 1: determining the linear form of simple exponential equation of observed groundwater level drawdown in time (refer to Eq. 3):

$$ \ln \left(\mathrm{H}\right)=-0.0007\ \mathrm{t}+6.5188 $$
(9)

Step 2: determining the linear form of simple power equation between observed water inrush and observed groundwater level drawdown (refer to Eq. 6):

$$ \ln \left(\mathrm{Q}\right)=30.151\ \ln \left(\mathrm{H}\right)-195.08 $$
(10)

Step 3: determining the simple exponential equation of water inrush recession in time by substituting two above equations (refer to Eq. 8):

$$ {\displaystyle \begin{array}{c}\ln \left(\mathrm{Q}\right)=30.151\ \left(-0.0007\ \mathrm{t}+6.5188\right)-195.08\to \ln \left(\mathrm{Q}\right)=-0.0211\ \mathrm{t}+1.47\\ {}\to \mathrm{Q}=4.34{\mathrm{e}}^{-0.0211\ \mathrm{t}}\end{array}} $$
(11)

The noble design of water drainage equipment in double shield TBM of Nosud tunnel made it possible to pump out the water at a maximum rate of 0.8 m3/s. Therefore, the required time to decrease the inrush rate to 0.8 m3/s was calculated using the exponential equation in at least 80 days, as shown in Table 2 (equal to August 18, 2015). In Fig. 11, the calculated rescission of water inrush from May 28 to August 18, 2005, was compared with the observed one. It can be noted that from August 11, 2015, the inrush rate reached a constant value of about 1 m3/s.

Table 2 Calculated inrush rate to the Nosud tunnel face over time
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Fig. 11
figure 11

Calculated versus observed water inrush rate over time (from May 26 to August 18, 2015)

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Following the tunnel excavation process, TBM encountered several caves and karst conduits, which were dry or with a small amount of water.

Case study 2: The Headrace Tunnel

The Headrace tunnel (HRT), with a length of about 15.4 km, is located in the southern part of Badulla administrative district in the province of Uva, Sri-Lanka (Fig. 12). HRT is part of the hydro-mechanical Uma Oya Multipurpose Development Project (UOMDP), operating as a pressure tunnel to convey water from the Dyraaba dam to a pressure shaft followed by a 120MW underground powerhouse. This project aims to improve the irrigation of 20.2 km2 of agricultural land, transfer 145 million m3 water, and generate 290 GW/h of power in a year (Golian et al. 2021b). Excavation of HRT was finished by two Double Shield TBMs (called M1684 and M1685) with an excavation diameter of 4.3 m from two sides (inlet and outlet). The main hazards concerning HRT were unpredictable geological conditions and the possibility of water inrush.

Fig. 12
figure 12

Geographical location and satellite image of the HRT

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Geological characteristics of the study area

The HRT is located in the Highland Complex (HC) lithological unit of central Sri-Lanka, characterized by the presence of gneisses, ranging from charnockitic gneiss to quartzofeldspathic gneiss and garnet-sillimanite-biotite-graphite gneiss (the Khondalite unit), with some parts containing undifferentiated gneiss. Generally, these rocks are locally interlayered with marble and quartzite. From the structural-geological standpoint, the study area shows a poly-phase ductile deformation history, characterized by large scale folding and thrusting. Rock bands of the area display their main foliation (S) which could be related to the main deformation phase. The geological map and profile of the tunnel from chainage 8 km to the outlet are shown in Fig. 13.

Fig. 13
figure 13

Geological map and profile of the HRT

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Event of groundwater inrush to the tunnel

On December 24, 2014, a water inrush of about 0.4 m3/s occurred at a chainage of 11.160 km (Fig. 14). This event caused serious delays in the excavation process. At the site, TBM encountered a high permeable fault zone in gneiss. After 80 days, the measured water outflow by Parshall Flume at the tunnel outlet reduced to about 0.216 m3/s. Subsequently, GWL in DT-08 borehole decreased to approximately 2 m. GWL drawdowns due to the tunnel excavation caused ground subsidence, and domestic water wells drying up near the tunnel route (Golian et al. 2021b; Fig. 15).

Fig. 14
figure 14

Water inrush to the outlet TBM face at chainage 11.160 km of the tunnel

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Fig. 15
figure 15

Some environmental and geotechnical consequences of the HRT excavation; a drying up of domestic wells near the tunnel route (Rathnayake 2016), and b ground subsidence and damage to residential buildings (Ismail 2017)

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Recession time estimation of the water inrush

When water inrush occurred in the tunnel, changes in its rate and the GWL drawdown in DT-08 borehole were monitored daily (Fig. 16). Using the gathered data from January 3 to 12, 2015 (10 days), the presented method was applied with the calculations presented in Eqs. 12 to 14, while the hydrographs, and diagrams, and the various comparisons are shown in Figs. 17 and 18.

$$ \ln \left(\mathrm{H}\right)=-0.00003\ \mathrm{t}+6.9173 $$
(12)
$$ \ln \left(\mathrm{Q}\right)=483.51\ \ln \left(\mathrm{H}\right)-3345.6 $$
(13)
$$ {\displaystyle \begin{array}{c}\ln \left(\mathrm{Q}\right)=483.51\ \left(-0.00003\ \mathrm{t}+6.9173\right)-3345.6\to \ln \left(\mathrm{Q}\right)=-0.0145\ \mathrm{t}-1.02\\ {}\to \mathrm{Q}=0.36\ {\mathrm{e}}^{-0.0145\ \mathrm{t}}\end{array}} $$
(14)
Fig. 16
figure 16

Hydrographs of measured GWL in DT-08 borehole and observed water inrush rate to the tunnel face

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Fig. 17
figure 17

Correlation diagram of the natural logarithm of GWL in DT-08 borehole versus time

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Fig. 18
figure 18

Correlation diagram of the natural logarithm of GWL in the DT-08 borehole and natural logarithm of inrush rate to the tunnel face

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Employing this exponential equation, water inrush recession from January 13 to March 23, 2015, was calculated and then compared with that observed (Table 3 and Fig. 19). It can be noted that, since February 7, 2015, the decreasing trend of groundwater inrush into the tunnel was slowed down, reaching a constant value, so that after 45 days, the inrush rate into the tunnel has decreased by only about 0.02 m3/s. This may be due to a change in the flow system from the conduit to the diffuse, or it could arise from the tunnel being at equilibrium with the groundwater system. Such circumstance points out to validity of this method only in the case when the rate of decreasing inrush in tunnels is an exponential function of time (conduit dominant flow); when the rate of decreasing inrush is near-constant, the method is no longer valid.

Table 3 Calculated inrush rate to the Headrace tunnel face over time
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Fig. 19
figure 19

Comparison diagram of observed water inrush rate to the tunnel face over time versus calculated ones

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Case study 3: The Glas Tunnel

The Glas water conveyance tunnel is a 35.661 km tunnel located in the West Azerbaijan Province, Iran (Fig. 20). The tunnel is under construction to transfer about 650 million m3 of water annually from the reservoir of Kanisib Dam on Lavin River to the Urmia Lake, with the main purpose of reviving the lake (Golian et al. 2021a). Excavation of this tunnel from two directions (inlet and outlet), with Double Shield and mixed EPB (Earth Pressure Balance)/Open Mode TBMs, with a drilling diameter of 6.42 m and a final diameter of 5.5 m on a slope of 0.008% is currently being carried out. Excavation of the inlet section is from 0 to 15 km, and the outlet is from 15 to 35.661 km.

Fig. 20
figure 20

Geographical location and satellite image of the Glas tunnel

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Geological characteristics of the study area

Based on the geological studies of the tunnel route, the excavation from 0 to 3 km and from 12.6 to 14.8 km will cross alluvial deposits, whilst the remaining will be in rock units. These latter are very diverse and can be divided into three classes of sedimentary (conglomerate and limestone), metamorphic (schist, slate, and marble) and igneous (granite, granodioritic and andesitic intrusions) rocks (Fig. 21).

Fig. 21
figure 21

Geological map and profile of the outlet stretch of the Glas tunnel

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Event of groundwater inrush to the tunnel

On March 1, 2017, the tunnel encountered a karst conduit (1 m in width, 1.5 m high) in the marble unit at chainage 29.231 km (Fig. 22). The initial volume of water entering the tunnel front was approximately 0.25 m3/s (Fig. 23). Along with water inrush, a considerable amount of silt and clay sediments entered the tunnel, too, imposing a delay in the tunnel excavation. The water inrush also caused subsidence at the ground surface of agricultural lands in the vicinity (Fig. 24).

Fig. 22
figure 22

A view of the karst conduit causing the water inrush to the tunnel at chainage 29.231 km

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Fig. 23
figure 23

Water inrush to a TBM Back up utilities, and b the tunnel outlet

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Fig. 24
figure 24

Ground subsidence near the tunnel route due to the water inrush

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Estimation of recession time of the water inrush

When water inrush into the tunnel took place, changes in the inrush rate and GWL drawdown in KST-08 borehole were being monitored on a daily basis (Fig. 25). Using the observed data from March 1 to 10, 2017 (10 days), according to the presented approach, the related calculations are shown in Eqs. 15 to 17, and the outcomes in Figs. 26 and 27.

$$ \ln \left(\mathrm{H}\right)=-0.006\ \mathrm{t}+4.5252 $$
(15)
$$ \ln \left(\mathrm{Q}\right)=3.942\ \ln \left(\mathrm{H}\right)-19.22 $$
(16)
$$ {\displaystyle \begin{array}{c}\ln \left(\mathrm{Q}\right)=3.942\ \left(-0.006\ \mathrm{t}+4.5252\right)-19.22\to \ln \left(\mathrm{Q}\right)=-0.0237\ \mathrm{t}-1.38\\ {}\to \mathrm{Q}=0.251\ {\mathrm{e}}^{-0.0237\ \mathrm{t}}\end{array}} $$
(17)
Fig. 25
figure 25

Hydrographs of measured GWL in KST-08 borehole and observed water inrush rate to the tunnel face

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Fig. 26
figure 26

Correlation diagram of the natural logarithm of GWL in the KST-08 borehole versus time

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Fig. 27
figure 27

Correlation diagram of the natural logarithm of GWL in the KST-08 borehole and the natural logarithm of water inrush rate to the tunnel face

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As presented in Fig. 28 and Table 4, 12 days after the water inrush event, the inrush into the tunnel reached a constant value of 0.19 m3/s.

Fig. 28
figure 28

Comparison diagram of observed water inrush rate to the tunnel face over time versus calculated ones

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Table 4 Calculated inrush rate to the Glas tunnel face over time
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Discussion and results

The statistics data of the observed (Qob) and calculated (Qcal) groundwater inrushes into tunnels are summarized in Table 5.

Table 5 Data statistics in the case studies
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To validate the estimated groundwater inrushes by the proposed method, first, the correlation of the observed and calculated values was determined using regression analysis (Golian et al. 2019; Fig. 29 and Table 6). As can be seen, there is a very high correlation in the Nosud tunnel and a high correlation in the Headrace and Glas tunnels between Qob and Qcal data (Shayib 2013). Besides, line 1:1 (the identity line) is used as a reference in the correlation plot to compare two datasets under ideal conditions. When the corresponding data points from the two datasets are equal to each other, the corresponding scatters fall exactly on the identity line. As shown in Fig. 29, according to the identity line, the best dataset is for the Nosud Tunnel, where the flow was dominantly conduit.

Fig. 29
figure 29

Correlation curve of the observed and calculated groundwater inrush rates to the front of a the Nosud tunnel, b the Headrace tunnel, and c the Glas tunnel faces

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Table 6 Observed (Qob) and calculated (Qcal) groundwater inrush rates, with the calculation error
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The application of regressions techniques for model evaluation has been questioned by some authors (Mitchell 1997; Kobayashi and Salam 2000). It has been suggested that the evaluation of the model alone using the regression model may not be accurate and appropriate so that utilization of the error analysis is strongly recommended (van Tongeren 1995; Knightes and Cyterski 2005). In this study, the error analysis was investigated by different methods such as ME, MSE, MAE, RMSE, and the Percent Decline (or the Percentage Change), the results of which are listed in Table 6. Results of the error analysis indicate that there is a good relationship between Qob and Qcal data.

The important point to consider in this study is that for the determination of discharge duration in tunnels, the simple exponential regression equation can also be used. In this case, one should fit the measured groundwater inrush against the time to direct and determine its correspondence regression equation. However, it should be noted that due to the very high turbulence of the water inrush, the water inrush measurement may include high errors. For this reason, the exponential equation obtained from the inrush-time regression provides estimations with lower accuracy compared to the method presented in this study (Table 7 and Fig. 29). In the presented method, to increase the accuracy of the results, the effect of the groundwater level, which is measured with very high accuracy, is also considered. The discharge duration in the tunnels is also calculated by power, logarithmic, and linear regressions and the results are presented in Fig. 30. Linear regression has not been effective in any of the case studies and represents a significant computational error. However, power and logarithmic regressions represent the best fit with the observation curves of Glas and Headrace tunnels with the lowest error, despite the Nosud tunnel with significant errors. In conclusion, considering that the proposed method exhibited acceptable results in all three case studies, the application of this method is recommended instead of the regression models.

Table 7 Tunnel water inrush calculation error comparison for the proposed method and simple exponential, linear, power, and logarithmic regression models
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Fig. 30
figure 30

Comparison of observations with tunnel water inrush calculated with the proposed method and simple exponential, linear, power, and logarithmic regression models

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Conclusions

The event of water inrush to tunnels in karst terrains and some cases of tunneling in fault zones is a subtle hazard, especially due to inappropriate site investigation, deriving from a combination of human activities (engineering works, in this case, the tunnel excavation) and the natural asset. This latter is extremely difficult to be assessed in karst environments, where the anisotropy and heterogeneity of the rock mass (Andriani and Parise 2015, 2017), and its hydrogeological characteristics (Stevanović et al. 20072010; Zini et al. 2015), make carbonate aquifers one of the most problematic situations to be encountered in engineering works (Milanović 2000).

Water inrush in tunnels may cause severe time delays, and financial and life casualties as well. It is of crucial importance to engineers and geologists to calculate the downtime of TBM, as a consequence of water inrush (Palma et al. 2012). This lasts until the flow into the tunnel decreases to the level in which excavation is possible again. To determine water inrush recession time in tunnels, an analytical approach, based on observed inrush rate in tunnels and GWL in nearby boreholes, was introduced and applied to three case studies. The good fit between the observed and estimated curves indicates the efficiency of the applied analytical method. In principle, our conclusions should be validated by a larger number of cases than the considered ones; the authors will be grateful to colleagues who, after trying the method, will want to inform them of the results.

Furthermore, the accuracy of using the Parshall Flume equation within hydrodynamic equations of inrush to tunnel shows that the relationship between discharge and hydraulic head for conduit karstic media and fault zones is a simple power relation.