Tunnels, drifts, drives, and other types of underground excavation are very common in mining as well as in the construction of roads, railways, dams, and other civil engineering projects. Planning is essential to the success of tunnel excavation, and construction time is one of the most important factors to be taken into account. This paper proposes a simulation algorithm based on a stochastic numerical method, the Markov chain Monte Carlo method, that can provide the best estimate of the opening excavation times for the classic method of drilling and blasting. Taking account of technical considerations that affect the tunnel excavation cycle, the simulation is developed through a computational algorithm. Using the Markov chain Monte Carlo method, the unit operations involved in the underground excavation cycle are identified and assigned probability distributions that, with random number input, make it possible to simulate the total excavation time. The results obtained with this method are compared with a real case of tunneling excavation. By incorporating variability in the planning, it is possible to determine with greater certainty the ranges over which the execution times of the unit operations fluctuate. In addition, the financial risks associated with planning errors can be reduced and the exploitation of resources maximized.
Underground mining represents a fundamental pillar of ore production in Chile. It is assumed that in the coming years the proportion of underground mining compared with opencast mining will increase as mineral resources accessible to surface exploitation become progressively exhausted.
One of the main activities involved in underground mining is tunnel construction, or, more generally, horizontal works, because this produces the infrastructure that provides access to the ore for extraction. Here, a “tunnel” should be understood as any underground excavation whose purpose is to join two points.
Given the importance of tunnels for mining, it is evident that there is a need to have a methodology that allows accurate planning of their excavation. To achieve this goal, the Markov chain Monte Carlo (MCMC) method is appropriate, since the construction of a tunnel is a cycle of activities consisting of unit operations, each of which exhibits a variability that can be represented in terms of a probability distribution function (PDF). Furthermore, the success or failure of the construction cycle is related to its actual duration compared with what was planned, which also depends on the time at which the cycle begins within the day’s work shift. Thus, the construction cycle of a tunnel is dependent on the success or failure of the immediately preceding cycle, and therefore the event’s probability of success is related to its predecessor, constituting an MCMC relation [
There are several methods of tunnel excavation. This paper will focus on the construction of tunnels by drilling and blasting [
The drilling and blasting cycles in tunneling.
The aim of the excavation cycle is to break up the rock with explosives, giving the required crosssectional shape while the tunnel advances proportionally to the length of the drilling in the tunnel face. In this way, with successive cycles, the infrastructure is built gradually until the tunnel has been completed. However, even with knowledge of the number of unit operations in each cycle, and the time generally taken to perform each one, it is very difficult to determine exactly the total time required to complete the construction of the tunnel, mainly because all of the operations are subject to variations that depend on unforeseen events (although they can be associated with a probability of occurrence).
On the other hand, the work cycles, and therefore each of the unit operations, are executed within welldefined time periods (work shifts), which are framed within a 24hour period. Usually, mining operates in continuous time periods without stoppages in production, and therefore tunnel construction proceeds in the same continuous manner. This is particularly important because the relation between the duration of the construction cycle and the period defined for the work shift will affect the efficiency of the cycle. Some of these inefficiencies result from changes in work shift that interfere with the working cycle. That is why Chilean law [
Configuration of evaluated shifts.
Shift ID  Shifts per day  Hour per shift  Effective hours per day 

T1  3  8  24 
T2  2  10  20 
T3  2  12  24 
Currently, to plan the construction of a tunnel, whatever its purpose, fixed values of the relevant parameters are used, giving consistent results. This is reflected in the following equation, which gives the construction speed of the tunnel in days related to the drilling length unit (
As already mentioned, tunnel construction involves excavation cycles consisting of unit operations that can be represented by PDFs, and it is clear that this process can be simulated using a Monte Carlo method [
Assuming that the number of excavation cycles required to construct the total length of the tunnel is known, since the advance achieved in each cycle is determined by the drilling length, which remains unchanged throughout the construction, the theoretical time taken for construction (
However, the duration of tunnel construction cannot be estimated using the Monte Carlo method alone, because this method does not take account of an aspect that is extremely important, namely, the fact that the probability of success of a cycle depends on the preceding cycle.
As already mentioned, the use of Markov chain theory is appropriate in this context, considering the characteristics of tunnel construction. Construction in mining takes place continuously 24 hours per day, and in general the working day is broken up into two or three periods (shifts), depending on the chosen workday (see Table
In underground mining, owing to the specific characteristics of the work, excavation cycles are generally kept as multiples of working shifts, for a duration of less than 8 hours, for example, with preference being given to a T1 type of shift over T2 or T3 (Table
It is possible that, owing to particular aspects of operational interference or inefficiency, an excavation cycle will not fit the established workday, increasing the duration of the cycle and affecting the next cycle. On the other hand, if success in the execution of an excavation cycle is represented by its completion within the established work shift or group of work shifts (with the cycle otherwise being considered a failure), then this condition in turn reduces the possibility of success of the following shift, because it takes time away from the latter and furthermore adds unproductive time due to activities interrupted as a result of the change of work. These situations can be considered as processes that can be modeled using Markov chain theory [
The algorithm presented in (
To predict tunnel construction time, the Monte Carlo method appears to be an appropriate tool to use together with the Markov chain principle, given that it is a stochastic simulation that allows analysis of complex systems with several degrees of freedom. The Monte Carlo method [
To use the Monte Carlo method, the unit operations are identified and each is assigned a PDF that depends on its nature and on the results of field sampling.
If the inverse functions of the PDFs of each unit operation of the excavation cycle are fed with random numbers, they will give as a result the duration of each operation. If the times thus obtained are added, this gives the total duration of the excavation cycle.
Once we know the duration of the excavation cycle, we must also consider another very important variable, namely, the distance advanced, or the real advance, after blasting (
Thus, if the durations of all the excavation cycles and the corresponding distances advanced are known, it is possible to determine the time taken for tunnel construction. Simulating this as many times as possible, a large number of scenarios are produced, generating a PDF of the time taken for tunnel construction.
The algorithm (see Algorithm
BEGIN
WHILE (
END WHILE
END
Variables:
nsim: number of simulation
ltunnel: lenght of tunnel (meters)
DI.op.m: inverse distribution of operation unit “m”
DI.rec: inverse distribution of efficiency of the blast
Le: lenght of the drilling
dt: during of the work shift (minutes)
rand#: random number.
The proposed scheme consists of three inclusive loops dependent on each other. The operating form is that the first loop, which contains the other two, controls the number of required simulations, on the basis that each simulation is the construction of a tunnel with specified length.
The second loop imposes the condition that the construction does not exceed the defined tunnel length, with every advance being estimated by the PDF of the performance of the blast multiplied by the drilling length. The drilling length is a fixed value, and is added consecutively until the required tunnel length is achieved.
Finally, the third loop has the function of adding consecutively the times for the unit operations in each cycle and comparing this total time with the established work shift, a fundamental aspect of this work.
This last loop is the key to the simulation, because it constructs the cycle within the shift. This procedure is carried out using the Monte Carlo method, where the inverses of the PDFs associated with the execution times of the unit operations are applied. The times obtained from this simulation are added, and it is determined whether the cycle can finish during the operating shift.
The proposed simulation algorithm is detailed in the following section.
As already mentioned, the function of the first loop is to control the number of simulations required, taking into account that each simulation estimates the time required to build a tunnel with an already defined length. This loop, which contains the other two, starts with the variable “
Some variables are then defined to start the execution of the algorithm. The variable “
In the first loop, there is the variable “
On the other hand, the variable “
Finally, the variable “
The second loop imposes the condition that the simulated construction advance does not exceed the proposed length, and to that end the construction takes place in the third loop, whose purpose is to control the duration of the cycle in terms of the duration of the work shift “
The verification expression is true as long as the sum of the times of the cycle’s operations “
The cycle’s duration in the work shift is built successively in a selection routine “CASE” that adds the time for each operation until the end of the cycle; then, the following cycle can start again within the same shift or stop the execution to retake it on the following shift, and this depends on the operational tolerance “
The “CASE” routine in the third loop has the function of arranging the operations so they take place one after the other and also of evaluating the time taken under the loop’s condition, in order to see if this is still within the duration of the chosen shift.
The times for each operation belong to the probability distributions used to represent the process. In our case, the variable “
By means of the variable “
At the end of the third loop, there is a conditioning routine depending on whether the cycle ends together with the shift or is interrupted. This routine evaluates whether “
Also, at the end of the second loop, a work shift is added in the variable “
In this way, the successive simulations are constructed, delivering the time taken for each simulated tunnel, and accounted for in work shifts.
Taking in account the possibility of iterating as many times as necessary, we will have a representative sample of the population from which we can infer the most probable duration of the tunnel’s construction.
Minera San Pedro Limitada (MSP) has several copper ore deposits in the Lohpan Alto district, located in the Coastal Range of Central Chile. One of these deposits is Mina Romero, where the ore will be removed by underground mining [
To gain access and prepare the mineralized body for its exploitation, MSP has planned the construction of a 560 m access tunnel with no slope and in a straight line, with a cross section of approximately 3.5 m × 3.0 m.
The equipment used for drilling is an electrohydraulic drill of 45 mm diameter and the explosives are ammonium nitratefuel oil (ANFO) and Tronex (a derivative of dynamite), initiated by a nonelectrical shock tube detonator (NONEL). The mucking equipment is a load haul dump (LHD) of 6 yd^{3}.
To estimate the duration of construction of this tunnel, MSP, based on its experience in similar projects, has considered that, for every three work shifts with a duration of 9 actual hours each, it is able to carry out four cycles with a drilling length of 1.8 m, giving a rate of advance of 2.4 m per shift, so, with two work shifts per day, an advance of 4.8 m/day would be achieved.
Taking the above figures into account, MSP has estimated that the project should take 117 days, although previous experience in mines close to Mina Romero has shown that this estimate is not precise, because there are often delays that have not been considered at the time of planning.
To apply the proposed methodology, data from areas with similar geological and operational characteristics to those that will be faced in Mina Romero have been used. For that purpose, exploratory tunnels have been made in the upper part of the deposit, with the same cross section of the one that will be built and in rocks with similar characteristics to those of the Mina Romero access tunnel.
The cycle has been divided into five activities: drilling, loading and blasting, ventilation, scaling, and mucking. Support is not considered, because of the good quality of the rock. The rock mass rating (RMR) geomechanical classification of the rock mass [
After measurements had been made of the operational times of the cycle in the exploratory activities with characteristics similar to those that will be simulated, under the guidance of the Engineering Department of MSP, each of the activities was characterized in a statistical analysis that allowed determination of the probability distribution that best fitted the performance of each of the unit operations. Table
Summary of statistical adjustment for unit operation.
Unit operation  Parameters  Probability Distribution  

Used data  Mean  Median  Mode  Standard deviation  Coefficient of variability  Minimum  Maximum  
Drilling  137  195.45  193.89  190.23  38.08  0.19  70.98  392.79 



Load and blasting  133  56.09  55.76  54.98  11.64  0.20  14.77  113.51 



Ventilation  136  90.00  90.00  90.00  0  —  —  — 



Scaling  135  26.08  25.24  23.54  6.77  0.25  8.04  — 



Mucking  135  70.03  65.66  58.33  20.22  0.28  30.95  — 

Ventilation was kept constant in time, because MSP provides lunch for workers at the same time, and the duration of both lunch and ventilation is 90 min.
The distributions shown in Table
In general, the analysis presented in Table
The simulation is involved between 10^{5} and 10^{6} iterations. It was found that the variability between the first and last simulations from this interval was not significant, considering the mean and the mode of the results (see Table
Sensitivity analysis for iteration determination for the simulation.
Number of iterations  Tunnel length  Shift duration  Tolerance  Reset  Average  Mode  Standard deviation  Min.  Max. 

(m)  (min)  (min)  (min)  (shifts)  (shifts)  (shifts)  (shifts)  (shifts)  

560  540  60  30  266.89  267  2.31  257  277 

560  540  60  30  266.88  267  2.31  256  277 

560  540  60  30  266.89  267  2.31  256  278 

560  540  60  30  266.89  267  2.31  256  279 

560  540  60  30  266.89  267  2.31  256  277 

560  540  60  30  266.89  267  2.31  256  279 
As stated in the model, every simulated event involves the construction of a tunnel with the specified length, taking into account the duration of the shift and the tolerance (“Tolerance” in Table
Finally, it is also necessary to consider the restarting time (“Reset” in Table
Both the tolerance and the restarting time can be modeled as Markov chain processes [
For the simulation, a tunnel of 3.5 m × 3.0 m with a length of 560 m is considered, with two work shifts per day, each of 540 min duration, with a tolerance of 60 min, which represents the time between the end of a cycle and the end of the shift. If the tolerance time is less than 60 min, the activities are finished and service or other activities related to the operation, such as cleaning, are carried out.
A 30 min time is considered for resumption of activities, implying that if the cycle time is longer than the shift time, this value is added to the cycle time, because all the distributions presented in Table
According to the data presented in Table
Probability distribution for the simulation in Mina Romero.
In contrast to the conventional planning method, which determines one value for the required number of shifts, one of the advantages offered by the simulation method presented here is that it is possible to have both pessimistic and optimistic scenarios with respect to the number of shifts required for carrying out the work. These scenarios can be considered as the lower and upper limits of the confidence interval that describes the construction time for the tunnel under study in shifts.
The simulation produces a histogram with a mean of 266.89 shifts, a mode of 267 shifts, a median of 267 shifts (with a minimum of 257 and a maximum of 277), and a standard deviation of 2.31, with a distribution that is symmetric in form. This histogram is fitted into a gamma distribution (Figure
For the case in question, 267 shifts, which is the value of the mean as well as the median and the mode, were considered. It was decided to use this value because it is a good representation of the simulated case and is the most repeated value. Figure
Cumulative distribution function of simulated data.
The gamma distribution is not of symmetrical type, but in this case is the best fit to the simulation data using the AndersonDarling method—this is why the mean probability of success is 60% rather than 50%. Figure
Once the tunnel construction, which took 133 days, was finished, the means resulting from the simulation could be compared with the MSP plans and with the actual data (Table
Summary of real data, planned data, and simulation data.
Description  Cycles per shift  Advance rate [m/shift]  Daily advance rate [m/day]  Planned time [days]  Average shifts  % Error with real case 

Planning  1.33  2.4  4.8  117  233  12.28% 
Simulation  1.16  2.1  4.2  134  267  0.37% 
Real  1.18  2.1  4.2  133  266 
It should be mentioned that the plan made by the Engineering Department of MSP is not represented in the histogram shown here. A more careful analysis would show that there is no event similar to what was planned and indeed that it would be very difficult for the event planned by the mining company to occur.
As can be seen, this simulation methodology based on the Monte Carlo method can estimate the time required for the construction of a tunnel used in underground mining.
The data analysis performed at MSP shows that this kind of stochastic simulation is a very effective tool for planning the construction time of a tunnel.
Beyond the accuracy of the means, the range of minima and maxima obtained by the simulation is interesting, because it delivers a potentially useful parameter for establishing planning criteria.
Based on the minimum and maximum values obtained from the simulation, optimistic and/or pessimistic scenarios can be proposed that they can serve as background information for making mine planning decisions.
Because of the random nature of the execution times of the operations involved in mining construction, it has been determined that a planning methodology based on the Monte Carlo method provides a better fit to real conditions than a conventional approach, because, with its use of probability distributions, it incorporates the variability inherent in the planning process.
By incorporating variability into planning, it is possible to determine with greater certainty the ranges over which the execution times of the different operations fluctuate. It is thereby possible to reduce the financial risks due to planning errors while maximizing the exploitation of resources.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by DICYT/Universidad de Santiago de Chile, USACH (code 051515VN), and PPGE3M Universidade Federal do Rio Grande do Sul.