1 Introduction

Tunneling projects around the world repeatedly encounter time and cost overruns, which frequently result in substantial losses or the mismanagement of public or private resources. The field of research that looks at time and cost overrun is not limited to tunneling projects, but covers all infrastructure and construction projects. A review of the literature on the factors affecting the magnitude of overrun reveals that project type, for example, rail, road or fixed link (bridge and tunnel), is an important determinant (Cantarelli et al. 2012a, 2012b; Flyvbjerg 2006; Flyvbjerg et al. 2002, 2003, 2004; Morris 1990; Odeck 2004; Skamris and Flyvbjerg 1997). Many researchers have attempted to identify the most important factors affecting overrun either by sending out questionnaires to experts or by conducting statistical analysis on a large number of completed projects (Gao and Touran 2020; Huo et al. 2018; Johnson and Babu 2020; Larsen et al. 2016; Memon et al. 2011; Zewdu and Aregaw 2015). After examining a large sample of projects, Flyvbjerg et al. (2002) concluded that technical explanations, such as estimation methods, are not important; instead, they contended that the most likely causes of time and cost overruns are either political factors (i.e., deliberate underestimation) or economic motives, whereby certain parties benefit from the approval of particular projects. The main reason Flyvbjerg et al. (2002) considered technical aspects to be unimportant was that forecast accuracy has not improved over the long term, and the magnitude of overrun in the aforementioned sample of projects was not distributed evenly around zero, that is, the distribution of overrun was biased.

However, according to a number of recent studies, forecast accuracy has in fact improved over time (Gao and Touran 2020; Sarmento and Renneboog 2017), and cost overruns for U. S. rail transit projects have become smaller (Dantata et al. 2006). As for the biased distribution of overrun, Flyvbjerg et al. (2002) acknowledge the necessity and viability of accounting for different risk factors (such as geological, environmental and safety problems) in the time and cost estimation process and point out that proponents of technical explanations for overruns should explain why the forecasters ignored these risks over the long term. However, the fact that forecasters have not taken risks into account over the long term is not a reason to ignore technical explanations. It can be an important point in explaining the reasons behind the occurrence of overruns from a technical point of view (Mohammadi, 2021). In addition, a gradual increase in project scope, called “scope creep,” results in increased construction costs and variance, and new technologies often increase the complexity of projects, resulting in cost escalation (Moret 2011). Moreover, project size has increased over time, and according to Flyvbjerg et al. (2004), large projects tend to have larger overrun, which may explain why no improvements in accuracy are evident over the long term. Therefore, we believe that technical explanations should not be disregarded in favor of political or economic reasons.

Tunneling projects are often characterized by a lack of knowledge of the geological conditions, known as epistemic uncertainty, and the time and costs of a certain construction class are in themselves variable, which is known as aleatory uncertainty (Efron and Read 2012; Kim and Bruland 2009; Membah and Asa 2015; Min et al. 2003; Paraskevopoulou and Benardos 2013; Zare and Bruland 2007). Thus, there are many risks associated with both sources of uncertainty, which can impede project goals, implying that the parties involved in underground projects should apply proper risk management strategies to achieve the goals of the project, namely, on-time completion while remaining within the budget and meeting the required standard of quality (Eskesen et al. 2004; Spross et al. 2018, 2020).

Probabilistic estimation is one possible approach that can be applied to deal with uncertainty in the time and cost estimation of tunneling projects. In recent years, a number of researchers have conducted such probabilistic time and cost estimations for tunneling projects (Einstein 2004; Einstein et al. 1999; Guan et al. 2014; Isaksson and Stille 2005; Mahmoodzadeh 2019; Naghadehi et al. 2016; Špačková et al. 2013). The outcome in all cases is a probability distribution that describes the total time and cost of the tunneling project. Such estimations can be very useful in aiding decision makers in client and contractor organizations regarding budget allocation, tender pricing and construction methods. However, all the published work in the literature can be divided into only three types of models:

  • Decision Aids for Tunneling (DAT) developed by Einstein et al. (1999), and some other published work (such as Mahmoodzadeh et al., 2019) that follow a similar process as the DAT. The process is to construct a large number of ground class profiles that represent the probabilities of occurrence of different ground conditions along the tunnel route. The ground class profiles are constructed using the Markov Chain Monte Carlo (MCMC) method and they are used to simulate the construction process to obtain time–cost scattergrams that represent the uncertainty.

  • The model developed by Špačková et al. (2013) that uses similar ground class profiles, but Dynamic Bayesian Networks (DBN) to facilitate the calculations.

  • The model developed by Isaksson and Stille (2005), which is based on analytical models and Monte Carlo simulation. In this model, the geological uncertainty is modeled according to the user’s description of the geological condition at site (see Sect. 4.2.1), and the proportions of different ground classes with respect to the total tunnel length are used for calculations instead of probabilities of ground classes.

Many applications of the DAT are discussed in the literature, including resource allocation, handling the excavated material, and other aspects that affect the total uncertainty in tunneling time and cost (Ritter et al., 2013; Min and Einstein, 2016). However, only one example of application exists for each of the models developed by Isaksson and Stille (2005) and Špačková et al. (2013). So, further development or application examples with real case studies of these two models developed by Isaksson and Stille (2005) and Špačková et al. (2013) contribute to the diversity of the existing literature, leading to a more holistic understanding of how to manage risk in time and cost estimations.

Thus, in this paper, we have improved upon Isaksson and Stille’s (2005) probabilistic time and cost estimation model for tunnels, where we now use a work breakdown structure (WBS), letting the unit activities be input parameters when assigning times (or costs). This simplifies considerably both the assessment of the input parameters and the simulation procedure. Furthermore, we describe the model’s length of geotechnical zones and exceptional time as stochastic variables, which were assumed to be deterministic values in the original model. We applied the improved model to estimate tunneling time in the headrace tunnel of the Uri hydropower project (constructed in India between 1991 and 1997) and compared the results with the actual tunneling time in a discussion from a risk management perspective.

2 Principles of KTH’s Probabilistic Time Estimation Model

In the time estimation model for tunnels developed by Isaksson and Stille (2005) at KTH Royal Institute of Technology, the total tunneling time (\(T\)) is split into normal time (\(T_{{\text{N}}}\)) and exceptional time (\(T_{{\text{E}}}\)). To estimate T, only the activities on the critical path need to be considered.

Normal time (\(T_{{\text{N}}}\)) is the time it takes to construct a tunnel without any disruption in the construction process due to the occurrence of disruptive events. The model facilitates the calculation of \(T_{{\text{N}}}\) by introducing the parameter production effort, \(Q\) [h/m], which is the time spent on the complete construction of a unit length (l) of the tunnel (not to be confused with the Q-system of rock mass classification). The fundamental assumption is that \(Q\) in tunnel section l is greatly affected by the more or less unknown geological and geotechnical conditions in the section. These conditions are described in the model by vector \({\mathbf{x}}\), the elements of which are a chosen set of geotechnical characteristics, for example, rock quality and water conditions. Conceptually, this gives the function \(Q = g\left[ {{\mathbf{x}}\left( l \right)} \right]\). This key function represents how the site conditions are accounted for in the assessment of the construction time. Thus, when preparing the time plan, its precision will be strongly connected to the level of knowledge about the geological characteristics in x. Since the nature of function \(g\left[ {{\mathbf{x}}\left( l \right)} \right]\) is uncertain, the model assigns \(Q\) to be a stochastic variable, which is estimated by the planning team (as detailed in Sect. 4).

Assuming that the same characteristics are relevant for a certain length of tunnel L, the \(T_{{\text{N}}}\) can be obtained by summing the production efforts of all tunnel sections, \({\mathbf{Q}} = \left[ {Q_{1} ,Q_{l} , \ldots ,Q_{L} } \right]\):

$$T_{{\text{N}}} = \mathop \int \limits_{L}^{{}} g\left( {\mathbf{x}} \right)dl \approx \mathop \sum \limits_{l = 1}^{L} Q_{l} .$$
(1)

As each \(Q_{l}\) is an identical stochastic variable with mean \(\mu_{Q}\) and standard deviation \(\sigma_{Q}\), the \(T_{{\text{N}}}\) will also be stochastic.

When \(T_{{\text{N}}}\) is the sum of a large number of production efforts, \(T_{{\text{N}}}\) tends toward a normal distribution; according to the central limit theorem:

$$T_{{\text{N}}} \to {\mathcal{N}}\left( {\mu_{{T,{\text{N}}}} ,\sigma_{{T,{\text{N}}}} } \right),$$
(2)

With mean \(\mu_{{T,{\text{N}}}} = L\mu_{Q}\) and standard deviation \(\sigma_{{T,{\text{N}}}} = \sqrt L \sigma_{Q}\). For the central limit theorem to hold, all samples in \({\mathbf{Q}}\) (one sample for each section) must be drawn from identical and independent distributions along L. However, the geological setting of a tunnel implies some spatial correlation (autocorrelation) along L, which can be determined using the scale of fluctuation (\(\delta\)) (see, e.g., Vanmarcke 1977). But according to Benjamin and Cornell (1970), the central limit theorem still holds if there is some degree of correlation that affects only a small number of the samples, and, in fact, the sample sum (\(T_{{\text{N}}}\)) will be close to normal unless the samples are highly dependent. As there can be clusters of such correlated samples in a rock mass due to the autocorrelation, \(\delta\) must, in practice, be considerably shorter than L for the assumption of normality to hold. For cases when this is not satisfied, the effect of autocorrelation can be explicitly accounted for using the variance reduction factor \({\Gamma }\), which can be approximated as (Fenton & Griffiths 2008):

$$\Gamma \approx \sqrt {\frac{\delta }{\delta + L}} .$$
(3)

The standard deviation of the average production effort for a unit length can then be written as \(\sigma_{{\overline{Q}}} = {\Gamma }\sigma_{Q}\), giving:

$$\sigma_{{T,{\text{N}}}} \left( \delta \right) = L{\Gamma }\sigma_{Q} = L\sqrt {\frac{\delta }{\delta + L}} \sigma_{Q} .$$
(4)

Evidently, as \(\delta\) approaches the unit length l = 1, we tend toward the uncorrelated case, \(\sigma_{{T,{\text{N}}}} \left( {\delta = 1} \right) \approx \sqrt L \sigma_{Q}\). The process of averaging over a certain length is commonly accounted for in the reliability-based design of geotechnical engineering structures.

The occurrence of any incident, i.e., a disruptive event, imposes an exceptional time delay on the project, which is the time it takes to completely handle the consequences of the incident. For the \(i\)th disruptive event type, the total delay time \(T_{{{\text{E}},i}}\) can be expressed as:

$$T_{{{\text{E}},i}} = \mathop \sum \limits_{\upsilon = 0}^{{n_{{i,{\text{max}}}} }} p_{\upsilon } \left( {\upsilon K_{\upsilon } } \right),$$
(5)

where \(p_{\upsilon }\) is the probability of \(\upsilon\) number of occurrences of the \(i\) th disruptive event type, \(K_{\upsilon }\) is a stochastic variable representing the assessed total time delay for handling one occurrence of the event, and \(n_{{i,{\text{max}}}}\) is the largest possible number of events of the \(i\) th type. The total exceptional time (\(T_{{\text{E}}}\)) to handle all types of disruptive events is obtained by:

$$T_{{\text{E}}} = \mathop \sum \limits_{i = 1}^{n} T_{{{\text{E}},i}} ,$$
(6)

where \(n\) is the number of different disruptive event types that can occur during the construction of the tunnel.

Thus, the total time (\(T\)) for a given tunnel length (\(L\)) and specified construction method can be assessed as:

$$T = T_{{\text{N}}} + T_{{\text{E}}} \approx \mathop \sum \limits_{l = 1}^{L} Q_{l} + \mathop \sum \limits_{i = 1}^{n} T_{{{\text{E}},i}} .$$
(7)

Applying the model, the tunnel can be discretized into several geotechnical zones with similar geological condition. By applying the model for each geotechnical zone, \(T\) can be obtained for each and summed to yield the construction time for the entire tunnel length. The model can also straightforwardly be reformulated to address cost estimations, as detailed by Isaksson and Stille (2005).

3 The Uri Headrace Tunnel

The Uri hydroelectric power project was constructed in the state of Jammu and Kashmir in northwestern India. The project consisted of a headrace tunnel, a tailrace tunnel, concrete culverts, a barrage, siphons, an intake, desilting arrangements and open canal, vertical steel lined penstocks and an underground power station. The subject of this study is the shaded part of the headrace tunnel (Fig. 1), which is horseshoe shaped with an inner after-lining diameter of 8.4 m and a total length of 10.7 km (Brantmark and Stille 1996; Heiner et al. 1993). Excavation of the headrace tunnel was conducted using four adits.

Fig. 1
figure 1

a Schematic representation of excavation plan for headrace tunnel and its geological conditions. The shaded part is the subject of this study. Full-face excavation with the drill and blast method was used b Cross section of the Uri headrace tunnel

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Geological information about the bedrock formations was obtained through two holes along the headrace tunnel and the excavation of adits 2, 3 and 4. According to the investigations prior to start of construction work, the geological conditions along the tunnel were expected to be very homogenous. However, occasional faults containing heavily sheared rock material were expected to occur, and there was a possibility of encountering high-pressure ground water in some parts. The lithology in the section under study consisted of Lower Paleozoic to Pre-Cambrian arenaceous-argillaceous metamorphic rocks, comprising quartzitic schists, chloritic quartz schist, quartzites, gneissose, phyllites and grits. Lenses of graphitic schist were found occasionally. This formation is known as the Tanawal series.

For each prevailing rock condition, a rock mass rating (RMR) system was planned to be used to facilitate the selection of suitable support system. However, the RMR system was adjusted in terms of primary support (construction classes) to meet the specific requirements of the Uri project (see Table 1). Note that the main difference between rock classes IIA and IIB is the orientation of foliation with regard to tunnel axis.

Table 1 Rock classes and required primary support in each class for the Uri project
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The rock overburden varied between 100 and 1000 m. As a result, stress-related problems, such as rock burst or buckling, were expected to occur. The tunnel crossed under many Himalayan Mountain brooks, locally known as nallahs. Artesian water was observed in some nallahs during pre-investigation drillings. There was a potential to encounter water pressures equal to several hundred meters of water column. Some nallahs followed zones of fractured rock, meaning that heavy leakage of water into the tunnel could occur under those nallahs. Wherever necessary, pre-excavation grouting was planned to manage water leakage. In cases where the prevention of water leakage was not achieved fully by pre-excavation grouting, post-excavation grouting was planned. Guidelines for handling water leakage in the tunnel are presented in Table 2.

Table 2 Ground water conditions and planned control measures
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4 Improvement and Application of KTH’s Model: Normal Time Estimation

4.1 Improvements of the Model

The practical application of the original version of the KTH model used a concept called production classes, which meant that for each construction class, three states of production efforts were defined: the poor, fair, and good production class. Being in the good production class, it would take shorter time to construct a unit length of the tunnel, than in the fair and poor classes. For each production class in each construction class, experts would assess a triangular distribution in terms of the minimum, most likely, and maximum values of production effort, Q [h/m], based on their experience and knowledge about the influence of geotechnical characteristics such as rock quality and groundwater volume. Then, the probabilities of each production class occurring within a construction class were calculated. Accordingly, the normal time (TN) could be calculated. The following describes our improvements of the original model:

  • The construction classes are first described based on the required production activities, and these production activities are then broken down into their unit activities. Thereby, experts can now assess the input distributions in terms of unit activity time, instead of having to assess the aggregated time of the production classes, which was a more abstract concept.

  • The length of the geotechnical zones is usually unknown and we show how this can be modelled as a stochastic variable, using a Poisson distribution. Similarly, the delay time caused by the occurrence of disruptive events (i.e., \({K}_{\upsilon }\)) is here modeled as a stochastic variable in the assessment of TE. Both were deterministic in the original version.

  • Numerical simulation is used to sum the stochastic variables of unit activity times in the improved application, while the original version of the model uses a cumbersome and yet more approximate analytical approach.

Our improved version of the KTH model has three advantages for the user: (1) the production efforts of production activities are assessed from probability distributions of individual unit activities. This allows contractors to optimize the construction work by giving more insight about the individual effects of production activities on normal time. (2) Assessing unit activity time is more tangible for the experts, because unlike in the original model version, the experts do not have to consider the unit activities of more than one production activity at a time in their assessment. (3) The calculation process of the exceptional time (TE) is easier to understand and yields more information about TE, such as its distribution.

4.2 Practical Application of the Model

4.2.1 Geological Setting of the Uri Headrace Tunnel

For the application of the improved KTH model, it is important to clearly understand the geological setting of the project. According to the pre-investigations in the Uri headrace tunnel, the part of tunnel under study (Fig. 1) was expected to fall under rock classes I, IIA and IIB, referred to as base rock, represented by subscript b. However, rock classes III and IV could be encountered occasionally where the tunnel passes through faults and other weakness zones (for simplicity in the following, jointly referred to as faults and represented by subscript f). The lengths of these zones (Lf) were expected to vary between 2 and 12 m.

We identified rock mass quality, groundwater flow, and overburden to be the main geotechnical characteristics (components of x) that affect production effort in the Uri headrace tunnel. As it was believed that the conditions along the tunnel were very homogenous, we assumed a scale of fluctuation \(\delta\) = 500 m for all components of x.

4.2.2 Construction Classes in the Uri Headrace Tunnel

To simplify time estimation, construction classes are defined in terms of production activities. For each production activity, one or more components of x can be assumed to affect the construction time. The production activities are broken down further into their unit activities, to which experts can assign the minimum, most likely, and maximum production efforts.

For the Uri headrace tunnel the production activities include pre-excavation grouting, the excavation sequence and post-excavation grouting (if necessary). Groundwater flow is the component of x that affects the production effort of pre- and post-excavation grouting, while rock quality and overburden affect the excavation sequence. Pre- and post-excavation grouting can be broken down into the unit activities drilling and grout injection. But post-excavation grouting was planned to be executed in parallel with the excavation sequence behind the face. Thus, in this particular case post-excavation grouting is not on the critical path and does not affect production effort.

The excavation sequence can be broken down into the unit activities drilling, charging and blasting, ventilation, scaling, mucking, surveying, and installation of primary support. On some occasions, where a multiple drift method of excavation was to be used in faults (see Table 1), the excavation sequence would start with spiling, followed by a three-step excavation plan, where the top section is excavated in two steps, followed by excavation of the bench section. All three steps have the same unit activities as the usual excavation sequence.

The excavation sequence is affected by two components of x: rock quality and overburden. The rock quality affects the excavation sequence mainly due to the different requirements for installation of primary support in different rock classes (see Table 1). Overburden is associated with stress-related behavior, that is, rock burst in rock class I and buckling behavior in rock class IIB. Rock burst and buckling events were expected to be of different severity and were categorized as non-violent and violent cases. The non-violent cases were assumed to cause only minor delays in the normal production process and were to be dealt with by installing additional rock bolts. The violent cases were expected to cause major disruptions to the normal course of production. These cases were therefore treated as disruptive events, adding to \({T}_{\mathrm{E}}\).

4.3 Production Effort in Base Rock

To calculate the normal time TN, it is required to first calculate the production effort in base rock \(Q_{{\text{b}}}\), which is given by:

$$Q_{{\text{b}}} = Q_{{{\text{Wb}}}} + Q_{{{\text{Rb}}}} ,$$
(8)

where \(Q_{{{\text{Wb}}}}\) and \(Q_{{{\text{Rb}}}}\) are the production efforts of pre-excavation grouting and excavation sequence in base rock, which depend on their respective unit activities, as described in the following.

4.3.1 Production Effort of Pre-Excavation Grouting in Base Rock

The production effort of pre-excavation grouting in the \(k\)th water class (see Table 2) in base rock (\(Q_{{{\text{WCb}},k}}\)) is obtained by summing the times required for performing all \(j\) number of unit activities (\(q_{{{\text{WCb}},j}}\)) for a unit length:

$$Q_{{{\text{WCb}},k}} = \mathop \sum \limits_{j = 1}^{{{n_{q}{\left\{ {{\text{WCb}}} \right\}}} }} q_{{{\text{WCb}},j}} ,$$
(9)

where \(n_{q}^{{\left\{ {{\text{WCb}}} \right\}}}\) is the number of unit activities required to perform pre-excavation grouting in base rock. In the model, the values of \(q_{{{\text{WCb}},j}}\) should be assigned based on data from similar past projects and, if such data are not available, subjective assessments can be used based on expert experience. As there is uncertainty associated with these values, they can be modeled as stochastic variables using, for example, lognormal or triangular distributions. (Isaksson 2002; Moret and Einstein 2016). Usually, the engineers can easily assign the minimum, most likely and maximum times it takes to perform a unit activity per unit length of tunnel by using their knowledge gained from past projects in similar conditions. In our case, we assigned the minimum, most likely and maximum production efforts of the unit activities of pre-excavation grouting based on our experience as practicing engineers, considering the planned grouting works as defined in the detailed design and the expected characteristics of the machinery to be used (Table 3). Figure 2 demonstrates the histogram of triangular distributions that we assigned for modeling the uncertainty in production effort related to the unit activities of pre-excavation grouting in water class 2 in base rock.

Table 3 Production effort for unit activities of pre-excavation grouting in base rock and faults (same for both \({\text{q}}_{{{\text{WCb}},{\textit{j}}}}\) and \({\text{q}}_{{{\text{WCf}},{\textit{j}}}}\)), and time delay for unfavorable water status. The unit activities of water class 2 are exemplified in Fig. 2
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Fig. 2
figure 2

a Assigned triangular distribution of production effort for drilling in water class II in base rock. b Assigned triangular distribution of production effort for grout injection in water class II in base rock

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As the actual water class in a specific tunnel section is unknown, the production effort for pre-excavation grouting (\(Q_{{{\text{Wb}}}}\)) needs to be obtained as a mixture distribution of the underlying distributions of production efforts of the classes (\(Q_{{{\text{WCb}},k}}\)), weighed with respect to their estimated proportions along the tunnel length, \(p_{{{\text{WCb}},k}}\) (Table 4):

$$Q_{{{\text{Wb}}}} = \mathop \sum \limits_{k = 1}^{{n_{{{\text{WCb}}}} }} p_{{{\text{WCb}},k}} Q_{{{\text{WCb}},k}} ,$$
(10)
Table 4 Proportion of water and rock classes in base rock and faults and proportions of multiple drift in rock class III and IV (ideally, these proportions should be stochastic variables, as they are uncertain
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where \(n_{{{\text{WCb}}}}\) is the number of water classes.

4.3.2 Production Effort for Excavation Sequence in Base Rock

The production effort of the excavation sequence in the kth rock class (see Table 1) in base rock (\(Q_{{{\text{RCb}},k}}\)) is also obtained by summing the times required for performing all \(j\) number of unit activities (\(q_{{{\text{RCb}},j}}\)) for a unit length:

$$Q_{{{\text{RCb}},k}} = \mathop \sum \limits_{j = 1}^{{{n_{q}{\left\{ {{\text{RCb}}} \right\}}} }} q_{{{\text{RCb}},j}} ,$$
(11)

where \(n_{q}^{{\left\{ {{\text{RCb}}} \right\}}}\) is the number of unit activities of the excavation sequence in base rock. The \(q_{{{\text{RCb}},j}}\) are assigned as detailed in Table 5. The corresponding mixture distribution of production effort related to excavation sequence (\(Q_{{{\text{Rb}}}}\)) can then be obtained from:

$$Q_{{\text{b}}} = \mathop \sum \limits_{k = 1}^{{n_{{{\text{RCb}}}} }} p_{{{\text{RCb}},k}} Q_{{{\text{RCb}},k}}$$
(12)

where \(n_{{{\text{RCb}}}}\) is the number of rock classes in base rock, and \(p_{{{\text{RCb}},k}}\) is the estimated proportion of rock class \(k\) along the tunnel route (Table 4).

Table 5 Production effort for unit activities of normal excavation sequence in base rock and faults (\({q}_{\mathrm{RCb},j}\) corresponds to classes I, IIA and IIB, while \({q}_{\mathrm{RCf},j}\) corresponds to III and IV)
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4.4 Production Effort in Faults

4.4.1 Production Effort of Pre-Excavation Grouting in Faults

The production effort of pre-excavation grouting in the \(k\)th water class (see Table 2) in faults (\(Q_{{{\text{WCf}},k}}\)) is obtained by summing the time required to perform all \(j\) number of unit activities (\(q_{{{\text{WCf}},j}}\)) for a unit length:

$$Q_{{{\text{WCf}},k}} = \mathop \sum \limits_{j = 1}^{{{n_{q}{\left\{ {{\text{Wf}}} \right\}}} }} q_{{{\text{WCf}},j}} ,$$
(13)

where \(n_{q}^{{\left\{ {{\text{Wf}}} \right\}}}\) is the number of unit activities required to perform pre-excavation grouting in faults. The \(q_{{{\text{WCf}},j}}\) are assigned as detailed in Table 3.

For faults, we included the additive time delay (\({\rm K}_{{{\text{Wunf}}}}\)) related to unfavorable water status in nallahs, which, however, is a disruptive event by definition. \({\rm K}_{{{\text{Wunf}}}}\) is the time required to handle the consequences of the occurrence of unfavorable water status (see Table 2). As this time delay is connected to water class 3, it is necessary to model it as part of TN and not as a separate event in TE. The uncertainty of \({\rm K}_{{{\text{Wunf}}}}\) can be modelled using triangular distribution (Table 3). Given that water class 3 is present, \({\rm K}_{{{\text{Wunf}}}}\) occurs with the assessed proportion \(p_{{{\text{Wunf}}}}\). The production effort of pre-excavation grouting in faults (\(Q_{{{\text{Wf}}}}\)) can then be obtained as a mixture distribution from:

$$Q_{{{\text{Wf}}}} = p_{{{\text{WCf}},1}} Q_{{{\text{WCf}},1}} + p_{{{\text{WCf}},2}} Q_{{{\text{WCf}},2}} + p_{{{\text{WCf}},3}} \left[ {p_{{{\text{Wunf}}}} \left( {Q_{{{\text{WCf}},3}} + {\rm K}_{{{\text{Wunf}}}} } \right) + \left( {1 - p_{{{\text{Wunf}}}} } \right)Q_{{{\text{WCf}},3}} } \right]$$
(14)

where \(p_{{{\text{WCf}},1}}\), \(p_{{{\text{WCf}},2}}\), and \(p_{{{\text{WCf}},3}}\) are the respective proportions of water classes 1, 2 and 3 in faults (Table 4), and \(Q_{{{\text{WCf}},1}}\), \(Q_{{{\text{WCf}},2}}\) and \(Q_{{{\text{WCf}},3}}\) are the production efforts of pre-excavation grouting in the respective water class.

4.4.2 Production Effort for Excavation Sequence in Faults

Similar to \(Q_{{{\text{RCb}},k}}\), production effort for the excavation sequence in each rock class (\(Q_{{{\text{RCf}},k}}\)) in faults (i.e., rock classes III and IV) is the sum of the time it takes to perform each unit activity (\(q_{{{\text{RCf}},j}}\)) per meter of tunnel length:

$$Q_{{{\text{RCf}},k}} = \mathop \sum \limits_{j = 1}^{{{n_{q}{\left\{ {{\text{Rf}}} \right\}}} }} q_{{{\text{RCf}},j}} ,$$
(15)

where \(n_{q}^{{\left\{ {{\text{Rf}}} \right\}}}\) is the number of unit activities for the excavation sequence in faults. The \(q_{{{\text{RCf}},j}}\) are assigned as detailed in Table 5.

Multiple drift excavation was expected to be required, with the proportions \(p_{{{\text{md}},{\text{III}}}}\) and \(p_{{{\text{md}},{\text{IV}}}}\) in rock classes III and IV, respectively (Table 4). The production effort related to multiple drift excavation of faults (i.e., \(Q_{{{\text{md}}}}\)) is obtained by summing the time it takes to perform all relevant unit activities (Table 6), similar to Eq. 15. Production effort of the excavation sequence in faults (\(Q_{{{\text{Rf}}}}\)) is therefore obtained as a mixture distribution by:

$$Q_{{{\text{Rf}}}} = p_{{{\text{RCf}},{\text{III}}}} \left[ {p_{{{\text{md}},{\text{III}}}} Q_{{{\text{md}}}} + \left( {1 - p_{{{\text{md}},{\text{III}}}} } \right)Q_{{{\text{RCf}},{\text{III}}}} } \right] + p_{{{\text{RCf}},{\text{IV}}}} \left[ {p_{{{\text{md}},{\text{IV}}}} Q_{{{\text{md}}}} + \left( {1 - p_{{{\text{md}},{\text{IV}}}} } \right)Q_{{{\text{RCf}},{\text{IV}}}} } \right],$$
(16)
Table 6 Production effort for unit activities of multiple drift excavation sequence in faults (\(q_{{{\text{md}},j}}\))
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where \(p_{{{\text{RCf}},{\text{III}}}}\) and \(p_{{{\text{RCf}},{\text{IV}}}}\) are the respective proportion of rock classes III and IV in faults (Table 4)\(, {\text{and}} Q_{{{\text{RCf}},{\text{III}}}}\) and \(Q_{{{\text{RCf,IV}}}}\) are the production efforts of excavation sequence in rock classes III and IV, respectively.

Similar to base rock, the distribution of production effort in faults (\(Q_{{\text{f}}}\)) can be obtained as:

$$Q_{{\text{f}}} = Q_{{{\text{Wf}}}} + Q_{{{\text{Rf}}}} .$$
(17)

4.5 Calculation of Normal Time

4.5.1 Total Length of Faults

Considering the lack of knowledge of geological conditions, that is, epistemic uncertainty, the number of faults Nf can be assumed to follow a Poisson process, as the events are discrete, and the arrival time of each event can be assumed to be independent of the previous event:

$$P\left( {N_{{\text{f}}} = n} \right) = \frac{{{\text{exp}}\left( { - \lambda_{{\text{f}}} L} \right)\left( {\lambda_{{\text{f}}} L} \right)^{n} }}{n!}, n = 0, 1, 2, \ldots$$
(18)

where n is the possible number of faults occurring with the respective probabilities described by the Poisson distribution, and \(\lambda_{{\text{f}}} L\) is the average number of faults in the tunnel. We assigned \(\lambda_{{\text{f}}} = 0.0145\) faults per meter of tunnel. Since \(N_{{\text{f}}}\) is a stochastic variable, the total length of faults (\(L_{{\text{f}}}\)) also becomes a stochastic variable:

$$L_{{\text{f}}} = N_{{\text{f}}} l_{{\text{f}}} ,$$
(19)

where \(l_{{\text{f}}}\) is the average length of one fault, here assigned as 7 m. Thus, the length of base rock is also uncertain:

$$L_{{\text{b}}} = L - L_{{\text{f}}}$$
(20)

4.5.2 Calculation of Normal Time in Base Rock

For all possible outcomes of Lb, represented by the vector \({\mathbf{L}}_{{\mathbf{b}}} = \left[ {L_{{{\text{b}},{\text{min}}}} , \ldots ,L_{{{\text{b}},n}} , \ldots ,L_{{{\text{b}},{\text{max}}}} } \right]\), a corresponding normal time \(T_{{{\text{Nb}},n}}\) is calculated, approximated by normal distributions with standard deviations affected by Γ, using Eqs. 2–4:

$$T_{{{\text{Nb}},n}} \sim {\mathcal{N}}\left( {\mu_{{T,{\text{Nb}},n}} ,\sigma_{{T,{\text{Nb}},n}} } \right)$$
(21)

with \(\mu_{{T,{\text{Nb}},n}} = L_{{{\text{b}},n}} \mu_{{Q,{\text{b}}}}\) and \(\sigma_{{T,{\text{Nb}},n}} = L_{{{\text{b}},n}} \sqrt {\delta /\left( {\delta + L} \right)} \sigma_{{Q,{\text{b}}}}\), where \(\mu_{{Q,{\text{b}}}}\) and \(\sigma_{{Q,{\text{b}}}}\) are the mean and the standard deviation of \(Q_{{\text{b}}}\) (Eq. 12). The theoretical boundaries of \({\mathbf{L}}_{{\mathbf{b}}}\) are \(L_{{{\text{b}},{\text{min}}}} = 0\), which represents the extreme case of no base rock and only fault-like rock along the whole tunnel, and \(L_{{{\text{b}},{\text{max}}}} = L\), which represents the case of encountering no faults. Note that the number of faults does not affect the assumption of normality, as the spatial correlation within the base rock is not affected by the intrusion of faults.

The normal time in base rock (\(T_{{{\text{Nb}}}}\)) is then a mixture distribution of all possible \(T_{{{\text{Nb}},n}}\) distributions, weighted with the respective probabilities of occurrence of faults:

$$T_{{{\text{Nb}}}} = \mathop \sum \limits_{n = 0}^{{N_{{\text{f}}} }} P\left( {N_{{\text{f}}} = n} \right)T_{{{\text{Nb}},n}}$$
(22)

4.5.3 Calculation of Normal Time in Faults

For the normal time in faults, the assumption of normality does not typically hold because unless the number of faults is very large, the total fault length would be too short for the central limit theorem to be applicable (implicitly assuming that Γ = 1 for faults). Therefore, for each possible outcome of \(L_{{\text{f}}}\), represented by the vector \({\mathbf{L}}_{{\mathbf{f}}} = \left[ {L_{{{\text{f}},{\text{min}}}} , \ldots ,L_{{{\text{f}},n}} , \ldots ,L_{{{\text{f}},{\text{max}}}} } \right]\), a normal time in faults (\(T_{{{\text{Nf}},n}}\)) can be calculated as:

$$T_{{{\text{Nf}},n}} = L_{{{\text{f}},n}} Q_{{\text{f}}} ,$$
(23)

which consequently does not become a normal distribution.

The normal time in faults is a mixture distribution of all \(T_{{{\text{Nf}},n}}\), again weighted with the respective probabilities for the occurrence of faults:

$$T_{{{\text{Nf}}}} = \mathop \sum \limits_{n = 0}^{{N_{{\text{f}}} }} P\left( {N_{{\text{f}}} = n} \right)T_{{{\text{Nf}},n}}$$
(24)

Finally, the normal time \(T_{{\text{N}}}\) can be obtained as:

$$T_{{\text{N}}} = T_{{{\text{Nb}}}} + T_{{{\text{Nf}}}}$$
(25)

5 Improvement and Application of the KTH Model: Exceptional Time

For the tunnel examined here, two types of disruptive events were expected to cause exceptional time, namely, rock burst and buckling in rock classes I and IIB, respectively. The occurrence of rock burst and buckling was assumed to follow Poisson processes according to the following format, yielding the probabilities \(p_{\upsilon }\) in Eq. 5:

$$P\left( {N_{{{\text{RB}}}} = m} \right) = \frac{{{\text{exp}}\left( {\lambda_{{{\text{RB}}}} L} \right)\left( {\lambda_{{{\text{RB}}}} L} \right)^{m} }}{m!},{ }m = 0,{ }1,{ }2,{ } \ldots$$
(26a)
$$P\left( {N_{{{\text{BK}}}} = m} \right) = \frac{{{\text{exp}}\left( {\lambda_{BK} L} \right)\left( {\lambda_{{{\text{BK}}}} L} \right)^{m} }}{m!},{ }m = 0,{ }1,{ }2,{ } \ldots$$
(26b)

where \(N_{{{\text{RB}}}}\) and \(N_{{\text{BK }}}\) are the number of occurrences of rock burst and buckling behavior, and \(\lambda_{{{\text{RB}}}} L\) and \(\lambda_{{{\text{BK}}}} L\) are the average number of occurrences of rock burst and buckling behavior in the total tunnel length L. For both \(\lambda_{{{\text{RB}}}}\) and \(\lambda_{{{\text{BK}}}}\), we assigned the value of 0.001. The exceptional time for rock burst (\(T_{{{\text{RB}}}}\)) and buckling (\(T_{{{\text{BK}}}}\)) are calculated using Eq. 5, with the time delay input (KRB and KBK) detailed in Table 7.

Table 7 Time delays for handling disruptive events
Full size table

Total exceptional time is then:

$$T_{E} = T_{{{\text{RB}}}} + T_{{{\text{BK}}}} .$$
(27)

Finally, Eq. 7 is used to obtain total time (\(T\)). The summary of the estimation procedure is presented in Fig. 3. To numerically calculate T, we used Monte Carlo simulation with a sampling size of 100,000, where each sample represents one potential outcome of T.

Fig. 3
figure 3

The process of probabilistic time estimation of tunneling projects using the improved KTH model. The cells on left side of the figure demonstrate the general process of estimation, and the cells on the right side present the corresponding application in the Uri headrace tunnel

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6 Application of the KTH Model: Estimation Results and Comparison with Actual Data

The Poisson distribution of number of faults (\(N_{{\text{f}}}\)), histograms of production effort in base rock (\(Q_{{\text{b}}}\)) and faults (\(Q_{{\text{f}}}\)), normal time in base rock (\(T_{{{\text{Nb}}}}\)), and faults (\(T_{{{\text{Nf}}}}\)) and exceptional time (\(T_{{\text{E}}}\)) are shown in Fig. 4. In Fig. 4f, the number of simulations in which no disruptive events occur is higher than the number of simulations with the occurrence of one or more disruptive events, as the probability of the occurrence of disruptive events is low. Consequently, in the histogram, the bar related to zero number of disruptive events is quite high. The histogram of total time (\(T\)) is shown in Fig. 5, where the actual tunneling time (12,800 h) is indicated.

Fig. 4
figure 4

a Histogram of number of faults encountered in tunnel length. b Histogram of production effort in base rock. c Histogram of production effort in faults. d Histogram of normal time in base rock. e Histogram of normal time in faults. f Histogram of exceptional time

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Fig. 5
figure 5

Histogram of estimated time (\(T\)). The dashed line indicates the actual time

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The mean value and standard deviation of estimated total time T are equal to 12,410 and 1460 h, which, according to the monthly working hours in the Uri project, correspond to 24 and 2.9 months, respectively. The 10th and 90th percentiles of estimated total time are 11,000 and 14,300 h (21 and 28 months). The actual excavation time (12,800 h or 25 months) is close to the mean value of the estimated time (µT). The project did not encounter differing site conditions with regard to geology. However, the percentage of rock classes III and IV, that is, faults and weakness zones, was more than expected. In reality, 18% of the tunnel length (488 m) consisted of faults, which is in the upper tail of the distribution of \(N_{{\text{f}}}\) (Fig. 4a). Within the base rock, we assessed the proportions of rock classes I, IIA and IIB to be 20%, 50% and 30%, respectively. However, the actual proportions of these rock classes were 0.7%, 2.7% and 96.6%, respectively; however, the difference in production effort among rock classes I, IIA and IIB is small, so the effect of this deviation on T was minor. The project did not report any cases of violent rock burst, but several cases of buckling behavior and unfavorable water status was reported, which overall led to delays of 5–10 weeks.

7 Discussion

7.1 General Modelling Considerations

Using the KTH model, every project would have its own specific conditions and characteristics. For instance, in the Uri case, the rock mass was divided into base rock and faults because the entire tunnel length was expected to mostly consist of base rock with only occasional cases of disruption with faults. This is certainly not the case for many other tunneling projects, and the user of the model would need to consider the specific conditions of the respective project. However, by understanding the geological setting of the projects and defining production activities and unit activities, the model can contribute to effective risk management in both the client’s and the contractor’s respective organizations, showing how the prevailing uncertainty affects the possibility to meet budget and time plan for different decision alternatives. Examples include the client’s selection of tunneling method and the contractor’s bid preparation. The importance of carefully interpreting the geotechnical context when planning a geotechnical engineering project has recently been discussed by Spross et al. (2022).

7.2 The Shape of the Histogram of Total Time

The shape of the histogram of total time (Fig. 5) is considerably affected by the estimated average number of faults per meter of tunnel (\({\lambda }_{\mathrm{f}}\)). Since there is a large difference in the primary support components between faults and base rock (see Table 1), the weakness zone is associated with a higher level of uncertainty. Therefore, an increasing value of \(\lambda_{{\text{f}}}\) increases the uncertainty of total time. In Fig. 6, the mean value, along with the 10th and 90th percentiles of T, are graphed versus \(\lambda_{{\text{f}}}\). Note that the increase rate in the 90th percentile is steeper than that of mean value and 10th percentile. This means that by increasing \(\lambda_{{\text{f}}}\), the uncertainty in total time increases sharply, which affects the final shape of the histogram of total time.

Fig. 6
figure 6

Effect of \({\uplambda }_{{\text{f}}}\) on estimated total time (\(T\))

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7.3 Multiple Excavation Faces

The focus of this paper was on the application of the model itself. Therefore, we applied the model for the estimation of T for only one excavation face (see Fig. 1). However, in practice, multiple excavation faces in a long tunnel must be considered for estimation purposes. In such cases, it is necessary to identify one or several potential critical paths that can be the major path for determining total time. Yet, the dependency of multiple faces in terms of machinery, equipment and working personnel needs to be considered. By delineating the production activities and relevant unit activities of multiple excavation faces with respect to each other in terms of shared machinery and equipment, the model can be applied to estimate time and cost in projects where multiple excavation faces exist. Further research can be beneficial to determine the usefulness of the model for quantifying the risk related to the shared use of machinery for multiple faces.

8 Conclusions

Epistemic uncertainty, that is, the lack of knowledge of geological conditions along the tunnel route, is one of the main sources of uncertainty in tunneling projects. One way of dealing with this uncertainty is to apply probabilistic time estimation at the early stages of tunneling projects for the sake of proper risk management. In this paper, we improved upon the KTH model and applied it to the probabilistic time estimation of the headrace tunnel in the Uri hydropower project. The outcome of the estimation is a mixture distribution for the total tunneling time.

The results of estimation were satisfactory, as the mean value of estimated total time, 24 months, was very close to the actual total time of 25 months. The geological knowledge of the area, the epistemic uncertainty, played an important role in the performance of the model. In this particular case, the length of faults and weakness zones had a considerable effect on the estimation, especially the standard deviation of the total time. This calculation example shows how important it is for planning teams to have an adequate understanding and sufficient knowledge of the geological setting and the prevailing uncertainty of the project at hand. The improved KTH model allows the user to analyze in depth the effect of this prevailing uncertainty on the estimated construction time and cost, making the KTH model a practical risk management tool for both clients and contractors. We believe that an increased use of such probabilistic time and cost estimation models can reduce the common problem of cost overrun and schedule delay in tunneling projects.