Probabilistic Assessment of Void Risk and Grouting Volume for Tunneling Applications

Abstract

Tunneling in karstic geology confronts numerous challenges due to unpredictable occurrence of voids. The current approach of karstic void risk assessment is qualitative or semi-quantitative and lacks consideration of the spatial variability and distribution of voids. This often influences the pricing strategies, and design and construction activities on tunnel projects. This paper presents a geostatistical modeling-based methodology to develop a quantitative assessment of karstic void risk for a tunnel project in a karstic geological setting. The methodology is applied on an actual mixed-ground tunnel project situated in a karstic geological environment in Malaysia. The geology at the tunnel project site consists of sedimentary rock formations with limestone as the predominant rock type overlain by weak sedimentary residual soils. Pluri-Gaussian simulation (PGS) technique, a stochastic geostatistical-modeling algorithm, is applied to characterize the spatial distribution of voids in 3D along tunnel alignment. Simulations from PGS take into consideration the anisotropic distribution of voids on the tunnel project site. PGS utilizes void data from borehole investigations to model different void sizes (Vs) as categorical variables. The variability in multiple realizations from PGS technique is used to quantify the uncertainty in occurrence probabilities, number, and frequency of karstic voids. The proposed methodology demonstrates the ability to develop probabilistic estimates of occurrence frequency of different void sizes. Probabilistic assessments indicating 95% confidence interval (CI) on number of voids and respective occurrence probabilities are presented. The probabilistic assessment results are applied to estimate the grout quantity required for void treatment, while considering uncertainty in void occurrence. A minimum, mean, and maximum cumulative grout volume of about 2000 m3, 4000 m3, and 8000 m3 (for 95% CI), respectively, is estimated along the alignment.

Full text

Vol.:(0123456789)
1 3
Rock Mechanics and Rock Engineering
https://doi.org/10.1007/s00603-021-02528-6
ORIGINAL PAPER
Probabilistic Assessment ofVoid Risk andGrouting Volume
forTunneling Applications
RajatM.Gangrade1 · JacobG.Grasmick2· MichaelA.Mooney3
Received: 23 January 2021 / Accepted: 27 May 2021
© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021
Abstract
Tunneling in karstic geology confronts numerous challenges due to unpredictable occurrence of voids. The current approach
of karstic void risk assessment is qualitative or semi-quantitative and lacks consideration of the spatial variability and distri-
bution of voids. This often influences the pricing strategies, and design and construction activities on tunnel projects. This
paper presents a geostatistical modeling-based methodology to develop a quantitative assessment of karstic void risk for a
tunnel project in a karstic geological setting. The methodology is applied on an actual mixed-ground tunnel project situated
in a karstic geological environment in Malaysia. The geology at the tunnel project site consists of sedimentary rock forma-
tions with limestone as the predominant rock type overlain by weak sedimentary residual soils. Pluri-Gaussian simulation
(PGS) technique, a stochastic geostatistical-modeling algorithm, is applied to characterize the spatial distribution of voids in
3D along tunnel alignment. Simulations from PGS take into consideration the anisotropic distribution of voids on the tunnel
project site. PGS utilizes void data from borehole investigations to model different void sizes (Vs) as categorical variables.
The variability in multiple realizations from PGS technique is used to quantify the uncertainty in occurrence probabilities,
number, and frequency of karstic voids. The proposed methodology demonstrates the ability to develop probabilistic esti-
mates of occurrence frequency of different void sizes. Probabilistic assessments indicating 95% confidence interval (CI) on
number of voids and respective occurrence probabilities are presented. The probabilistic assessment results are applied to
estimate the grout quantity required for void treatment, while considering uncertainty in void occurrence. A minimum, mean,
and maximum cumulative grout volume of about 2000 m3, 4000 m3, and 8000 m3 (for 95% CI), respectively, is estimated
along the alignment.
Keywords Karstic features· Spatial variability· Void grouting· Tunnel risk assessment· Geostatistics
1 Introduction
1.1 Context andMotivation
For any tunnel project, the largest technical and financial risk
element usually lies in the ground, predominantly due to the
spatial variability in ground conditions and specific ground
features that affect tunneling. The spatial variability and the
associated uncertainty, especially of critical ground features,
need to be carefully assessed and considered in the planning,
design, and construction phases of tunnel projects. One such
ground feature affecting tunneling is the solution cavities
and voids formed due to the dissolution of limestone or gyp-
sum in karstic geological environments. For tunnel projects
planned in geological formations exhibiting karstic features,
the detection of voids in the driving path of the tunnel bor-
ing machine (TBM) has always been a critical issue. Tunnel
excavation in a karstic geological setting can induce severe
geotechnical hazards leading to high construction costs and
schedule delays (Yau etal. 2020). Ren etal. (2016), Zarei
etal. (2010), Shahriar etal. (2008), Day (2004), and Xei-
dakis etal. (2004) presented a summary of the karst-related
geotechnical hazards from tunneling in a karstic geological
setting for multiple tunnels in Iran and the adopted mitiga-
tion measures. Zabidi and De Freitas (2013), Boon etal.
* Rajat M. Gangrade
rajatgangrade@mines.edu
1 Center forUnderground Construction andTunneling,
Colorado School ofMines, Golden, CO, USA
2 Maxwell GeoSystems USA Ltd., Golden, CO, USA
3 Department ofCivil andEnvironmental Engineering,
Colorado School ofMines, Golden, CO, USA

R.M.Gangrade et al.
1 3
(2020), Cui etal. (2015), and Wang etal. (2020) discussed
the design and construction problems due to karstic features
in Kuala Lumpur and China. These case studies from Kuala
Lumpur and China indicate that unpreparedness in estimat-
ing karstic features has been an influencing factor in geo-
technical hazards due to karstic feature presence.
Eskesen etal. (2004) determined that geotechnical haz-
ards due to tunneling predominantly result from an inad-
equate level of site investigations or failure to comprehend
the information from site investigations. Typically, with
limited and sparse geotechnical-site investigations on tun-
nel projects, there exists significant uncertainty regarding
the location and occurrence frequency estimates of karstic
voids. Irregular distribution of the karstic voids and their
various shapes and sizes compounds the problem of evalu-
ating the void fraction within the tunnel envelope. Like any
other tunnel risk, karstic risk assessments are included in the
geotechnical baseline report (GBR). Interestingly, although
GBRs are used increasingly to demarcate risk allocation
and resolve any financial disputes, the karstic risk evalua-
tion incorporated in GBRs is qualitative or semi-quantitative
(Yau etal. 2020). More importantly, the GBR assessments
lack consideration of the spatial variability and distribution
of the karstic voids. The qualitative or semi-quantitative
risk assessments from the GBR may result in overbidding
or underbidding on the project, depending on the tunnel
contractor’s perspective. Tunnel design and construction
decisions based on GBR assessments can influence the pro-
ject pricing and construction means and methods and can
potentially result in an increased number of differing site
condition (DSC) claims.
1.2 Background
Karstic features are formed by the dissolution of sedimen-
tary formations of soluble carbonate rock, such as limestone
or dolomite. These features can also develop in other rock
types, such as gypsums/ salt-rich formations, or less com-
monly in quartizes formations (Piccini and Mecchia 2009).
Interaction of the mildly acidic water with weakly soluble
carbonate rocks results into development of karstic features.
Duringer etal. (2012) discussed a two-stage process—(i)
rain water interacting with carbon dioxide to form carbonic
acid, and (ii) percolation of acidic water through soil to dis-
solve the carbonate rocks. The rate of dissolution depends
on the strength of the acidic water and the presence of pre-
existing fractures, joints, bedding planes, and discontinuities
in carbonate rocks.
Over time, the openings in rock increase in size, and
intersect the groundwater regime, resulting into a significant
construction risk of water inflow in underground construc-
tion and tunneling projects. In tunnel construction, typically
watertight segmental concrete liners are installed around
the tunnel opening to arrest the water ingress. However, the
performance of the segmental liners can be affected due to
unevenly applied ground stress in void-filled ground. A lack
of knowledge of the karstic feature distribution can lead to
unforeseen changes in hydraulic pressures, resulting in face
stability issues, tunneling-induced settlement, subsidence,
and sinkhole formations during shield tunneling.
A number of studies have attempted to examine the spa-
tial variability of karstic features. Paraskevopoulou and
Benardos (2013) determined that the spatial variability and
the associated risk of uncertainty need to be carefully con-
sidered and assessed prior to tunnel design and construc-
tion as encountering unforeseen conditions could lead to
catastrophic delays and cost overruns. Ford and Williams
(2013) highlighted the uncertainty arising from the spatial
variability of karstic voids. Zabidi and De Freitas (2013)
developed a geospatial analysis-based methodology to gen-
erate deterministic estimates of karst void fraction in the
ground from volumetric analysis of the boreholes drilled
for a tunnel in Kuala Lumpur. Li etal. (2015) quantified the
risk of water inrush in tunnels situated in karstic geological
setting using fuzzy mathematics and analytical hierarchy
process (AHP). Cheng etal. (2017) proposed a fractal analy-
sis model describing the relationship between drilled bore-
holes and karst caverns’ diameters to predict the grouting
volume for treating karst caverns along a tunnel alignment.
Yau etal. (2020) used a random distribution of karstic voids
and developed a numerical model to quantify the impact of
karstic voids of water pressure expected and the structural
capacity of the tunnel liners. Kovačević etal. (2020) pre-
sented a customized neural network (NN) model, trained
to learn the complex non-linear relationship between rock
mass parameters in karstic environment and tunnel design
parameters. However, the study is limited to using empirical
correlations to determine reduced rock stiffness in presence
of karstic features.
1.3 Scope ofWork
It is evident from the abovementioned geotechnical haz-
ards that a quantified knowledge of karstic void size,
occurrence frequency, and locations of occurrence within
tunnel envelope would be beneficial for active risk man-
agement and decision-making on tunnel projects. Proba-
bilistic approaches capable of characterizing the spatial
variability and the associated uncertainty of subsurface
features could be implemented for a comprehensive quan-
titative assessment. Application and inclusion of proba-
bilistic assessments into the GBR can provide insights
into the expected tunnel excavation environment and
facilitate improved risk estimation and allocation. In tun-
neling applications, Grasmick (2019) and Gangrade etal.
(2020) employed geostatistical techniques to quantify

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
spatial uncertainties in geotechnical conditions within
the tunneling envelope. Grasmick etal. (2020b) used
geostatistical simulations to quantify uncertainties and
confidence intervals for geotechnical properties relevant
to tunneling risks. Gangrade and Mooney (2020) inter-
preted the geostatistical modeling results to quantify the
uncertainty in soil transition locations in the longitudi-
nal and transverse directions to tunneling. Grasmick etal.
(2020a) and Gangrade and Mooney (2020) discussed and
proposed the incorporation of probabilistic assessments
into the GBRs. The study by van der Pouw Kraan (2014)
discussed the risk-sharing mechanism on tunnel projects
and presented a strong case for incorporating probabil-
istic assessments and geotechnical uncertainty in tunnel
project GBRs. The existing studies on karstic voids are
limited in characterizing a qualitative and/or semi-quan-
titative distribution only.
This paper presents a quantitative assessment of karstic
void risk within the tunnel envelope using a geostatistical
modeling-based probabilistic approach. The probabilistic
assessment applies the pluri-Gaussian simulation (PGS)
technique, a stochastic geostatistical algorithm, on bore-
hole data from a tunnel project situated in a karstic geo-
logical environment. Multiple realizations from stochastic
modeling are post-processed to quantify the locations, the
sizes, the number, and the occurrence frequency of karstic
voids within the tunnel excavation envelope. The paper’s
motivation is to implement a probabilistic approach to
characterize karstic void spatial variability and develop
quantitative evaluations for GBR baseline statements such
as, “There is a 40% probability of encountering at least
three voids, with a maximum size of approximately 0.5m
between tunnel length X and Y.” The study attempts to
contribute to the applicability of probabilistic methods
towards an elaborate risk assessment of encountering
karstic voids within a tunnel envelope, which is currently
not taken into account in tunnel projects. The quantita-
tive assessment of karstic voids is utilized to estimate the
grout volume required as a part of pre-excavation grout-
ing for treating karstic voids. The improved probabilistic
knowledge of voids would allow the tunnel contractors to
better target the required pre-excavation grouting quanti-
ties and respective costs.
The next section describes the methodology of the
geostatistical modeling-based probabilistic approach to
generate quantitative assessments of karstic voids within
the tunnel excavation envelope. We then present an appli-
cation of the proposed methodology on geotechnical-site
investigations from a tunnel project situated in karstic
geology. The probability of occurrence, the expected
number, and the occurrence frequency of karstic voids of
multiple sizes are quantified and presented.
2 Overview ofProject Dataset
This study considers a 3km tunnel section of a 5.8 m
diameter tunnel project set in a karstic geological envi-
ronment and excavated using a shield machine. The exact
location and name of the tunnel project cannot be dis-
closed as authors are bound by a non-disclosure agreement
(NDA) with the project owner and involved stakeholders.
The geology at the tunnel project site consists of sedi-
mentary rock formations with limestone as the predomi-
nant rock type. Based on the borehole investigations, the
bedrock consists of limestone, with highly folded and
faulted sandstone and shale overlying the limestone. The
sedimentary rocks have been regionally metamorphosed to
form meta-sediments. The degree of metamorphism var-
ies across the region. Under low grade metamorphism,
quartzites (from sandstone) and phyllite (from shales)
have evolved. On the other hand, schists (from shales)
have evolved under high grade of metamorphism. Weak
alluvium material, deposited by the major rivers in the
region, is found at the surface. The material comprises
ofloose, unconsolidated soil or sediments. Due to natu-
ral sedimentation process, alluvium is typically found to
consist of a mixture of clay, silt, sand, and gravels. The
bedrock is overlain by sedimentary residual soils that cov-
ers most parts of the land in Malaysia. The residual soils
in the area are composite soils of sand, silt and clay in
varied proportions depending on the geological setting of
the soil. For the purpose of this study, weak alluvium and
the residual soils are grouped into soil. Various rock types
encountered within the tunnel project site are grouped and
denoted by rock in the borehole profile. The tunnel align-
ment is expected to traverse predominantly through soil/
limestone interface with intrusions of granite, quartzite,
phyllite and schist. Interaction of soil water and rainwa-
ter with limestone, especially at the soil–rock interface,
causes the formation and enlargement of voids. Waltham
and Fookes (2003) discussed that the upper part of the
rock mass is more fissured and are progressively isolated
from the neighboring rock mass. This leads to formation
of isolated, undercut pinnacles of limestone surrounded
by soil. Between the remnant pinnacles of limestone, fis-
sures are enlarged and are filled with soil or remain open.
Narrow vertical, soil-filled sections are particularly com-
mon in porous limestone. At the project site, the karstic
voids are typically found to occur within isolated lime-
stone blocks that are left as dissolution remnants within
the soil. For simplicity, this study does not account for
the void fillings of cohesive or non-cohesive material, and
groundwater. Geohazards associated with the karstic lime-
stone on this project include the pinnacle zone (i.e., highly
irregular rock head profile, pinnacles alternating with deep

R.M.Gangrade et al.
1 3
troughs) and collapsed weak soil zones or slumped mate-
rial near the soil–rock contact zone. The high variability of
the ground conditions within several meters of tunneling
is a major issue of tunneling in karstic limestone on this
project. No fault zones exist in the tunnel section discussed
in this study.
Historical and recent geotechnical investigations on the
tunnel section consisted of 115 continuous-core boreholes
drilled vertically at least 20m to 30m below the tunnel
springline. However, a few boreholes were terminated just
about at or above the tunnel springline. Figure1 presents
a plan layout of the boreholes vis-à-vis tunnel alignment.
Three to four clusters of closely drilled boreholes are
observed along the tunnel alignment. The rock quality des-
ignation (RQD) of cores ranged between 11% and 100%,
with a mean, median, and standard deviation of 55%, 60%,
and 22%, respectively. The horizontal spacing between the
drilled boreholes ranges from 8mto 150m with a mean and
standard deviation of 35m and 28m, respectively. The mean
vertical sampling interval within the boreholes for the soil
samples is about 1.3m, whereas the rock cores were con-
tinuously logged. This study considers the tunnel envelope to
be about 0.2 times the tunnel diameter above and below the
tunnel crown and invert, to account for karstic voids present
just outside the excavation envelope. For the 5.8m diam-
eter tunnel, the tunnel envelope is 8.1m. Figure2 shows a
distribution of the horizontal spacing between the nearest-
neighbor boreholes and vertical sampling, as observed from
investigations. Figure3 shows the boreholes’ longitudinal
profile and the groundwater level along the tunnel alignment.
The groundwater level is consistently found to be near the
ground surface.
The karstic voids are predominantly found at the con-
tact zone of soil and rock formation and within the rock
formation. However, a few karstic voids are observed
to occur within the soil formation. Tunneling-induced
Fig. 1 Plan view of the tunnel
alignment with borehole loca-
tions
Fig. 2 Distribution of nearest-
neighbor horizontal spacing
between the boreholes and verti-
cal sampling within boreholes
Fig. 3 Longitudinal profile
of the boreholes along tunnel
alignment in terms of samples
observed during investigations

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
ground settlement or sinkhole development is highly likely
to occur, given the weak alluvium material overlying the
limestone with karstic voids. As illustrated in Fig.3, there
exists significant variability in the number of voids within
the tunnel excavation envelope. A relatively higher den-
sity of voids is observed from Chainage 800m to 1000m,
Chainage 1300m to 1500m, and Chainage 1800m to
2000m. Figure4 presents the proportion of investigation
samples categorized as rock, soil, and voids, and the dis-
tribution of void sizes encountered during investigations.
The investigations revealed the karstic void size (Vs) to
range between 0.1m and 1.5m. It is to be noted that void
sizes reported and recorded from the drilled boreholes
are based on only vertical dimensions. For this study, the
voids have been subdivided into three categories based on
the void length intercepted by the vertical boreholes: (1)
category V1 voids with a maximum size between 0.10m
and 0.25m inclusive (0.10m ≤ Vs ≤ 0.25 m), (2) category
V2 voids with a maximum size between 0.25 mand 0.75m
(0.25m < Vs < 0.75m), and (3) category V3 voids with a
maximum size equal to or larger than 0.75m (Vs ≥ 0.75m).
About 48% of the voids identified belong to category V2.
Category V1 and V3 account for 30% and 22%, respectively,
of the voids identified during investigations.
The voids are assumed to have a spherical shape, and the
void diameter is considered equal to the length of the void
intercepted by the borehole. We note that the void’s recorded
size represents only a certain length of the void intercepted
by the vertical borehole. This size is rarely expected to rep-
resent the largest dimension of the void, i.e., the diameter in
the case of spherical voids. Hence, the measured sizes are
likely underestimates of void sizes. Prior work on probabil-
istic assessment of karstic voids includes applying a ran-
domly generated distribution of spherical-shaped karstic
cavities for liner stability assessment and water inflow (Yau
etal. 2020). Medley (2002), and Tang etal. (1986) previ-
ously investigated the estimation of boulder size in 3D from
the 1D measurement of intercepted borehole lengths. The
statistically rigorous techniques identified boulder shapes,
boulder orientation, and volumetric proportions as the most
important factors in accounting for statistical uncertainty.
Felletti and Beretta (2009) assumed spherical boulders and
considered the length intercepted by the vertical borehole as
its maximum size. The methodology was applied to charac-
terize the boulder occurrence frequency for a tunnel project
in glacial till. Figure5 shows a schematic representation
considering the intercepted void length as the spherical void
diameter. In this study, a void size equal to the void diam-
eter has been used for setting up the 3D grid for geostatis-
tical simulations and probabilistic assessment of the void
occurrence frequency. Though the probabilistic modeling
approach presented herein is limited to the assumption of
a spherical shape for the karstic voids, the approach is not
limited to a specific void shape and can be applied to any
void geometry.
Figure6 presents the voids’ longitudinal profile, subdi-
vided into the three abovementioned categories, vis-à-vis
tunnel alignment. Category V1 and V2 voids are observed
in the initial 100m of the tunnel envelope (tunneling left to
right). Category V3 voids are frequently observed for the
next 800m of the tunnel envelope. A higher density of all
three void categories is observed from Chainage 800m to
1500m and from Chainage 1800m to 2200m. All three void
categories are observed in the borehole cluster from Chain-
age 800m to 1000m and from Chainage 1800m to 2200m.
The borehole cluster between Chainage 1200m and 1500m
exhibits category V2 and V3 voids in dominant proportions.
From Chainage 2200m onwards, the number of observed
voids decreases up to about Chainage 2600m. This could be
due to the lower density of boreholes within the section. A
cluster of voids is observed after Chainage 2600m. The high
density of logged boreholes within the project site provides
an excellent opportunity to assess the karstic void occur-
rence quantitatively. However, as observed in Figs.3 and 6,
Fig. 4 Distribution of a samples
observed during investigations
and b void sizes encountered
within boreholes

R.M.Gangrade et al.
1 3
there are gaps in the void occurrence data due to the absence
of dense boreholes from Chainage 400m to 800m, Chain-
age 1000m to 1300m, and Chainage 1500m to 1800m.
3 Methodology
The methodology begins with exploratory data analysis of
the geotechnical investigations on the project site. The data
analysis aims to identify the range of karstic void sizes to
set up a 3D simulation grid for geostatistical modeling. The
decision on the resolution of the 3D grid in the longitudinal
(x–y plane), transverse, and vertical (y–z plane) directions
is based on the karstic void sizes. Karstic voids encountered
within boreholes are subdivided into categories based on
the void sizes encountered during investigations. The aim
of categorizing the voids is to comprehend the occurrence
frequency of different sizes of karstic voids. The methodol-
ogy assumes a void shape and size based on the intercepted
length of void within a borehole. Conversion of the inter-
cepted borehole length to void size is discussed in Sect.2.
Next, the borehole data are discretized into the 3D
simulation grid cells, also known as voxels. The karstic
voids are coded as geological–geotechnical units or cat-
egorical variables for stochastic geostatistical modeling.
The methodology employs the PGStechnique, a stochas-
tic geostatistical modeling method to simulate the spatial
variability of the geological–geotechnical units. Chiles
and Delfiner (2009) discussed the pixel-based methods—
truncated or PGS technique (Armstrong etal. 2011), mul-
tipoint geostatistics, and Markov chain methods such as
T-PROGS (Carle and Fogg 1997) to simulate the spatial
variability stratigraphic units, or so-called categorical
variables. Among the techniques, the PGS technique is a
complete approach as geological constraints can be applied
on the simulated domains. The technique overcomes most
categorical simulation method limitations of not captur-
ing the spatial changes in stratigraphic unit proportions,
stratigraphic contact relationships and geological realism
within realizations (Madani and Emery 2015; Madani
etal. 2019). Due to advantages of better characterizing
spatial transitions and contact relationships between strati-
graphic units, PGS is used within this study. With a sto-
chastic modeling framework, the PGS technique attempts
to preserve and reproduce the heterogeneity of parameter
of interest (Pyrcz and Deutsch 2014; Ma 2019). Appli-
cation of stochastic geostatistical modeling techniques
allows for generating multiple equiprobable realizations of
ground conditions. Each realization from the geostatistical
modeling captures the degree of heterogeneity and spatial
variability of the stratigraphic units. The PGS technique,
capable of simulating multiple stratigraphic units, has
been used in modeling petroleum reservoirs and environ-
mental science problems (Emery 2007; Armstrong etal.
2011). Madani etal. (2019) presented tools to validate
the reproduction of stratigraphy in individual realizations
Fig. 5 Void geometry con-
sidered for the probabilistic
estimation of void number and
occurrence frequency
Fig. 6 Longitudinal profile of
the voids along the tunnel align-
ment. Voids are categorized
based on the sizes observed
during investigations

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
generated from PGS technique. Madani and Emery (2015)
presented a split-sample validation of the individual reali-
zations using calibration plots, thus showing the deviation
of simulated proportion of stratigraphic units from actual
proportions.
Next, the tunnel envelope is digitized to a cylindrical vol-
ume to extract the tunnel envelope voxels from the 3D grid.
The voxels’ information from the realizations is post-pro-
cessed to evaluate the occurrence probability of the geologi-
cal–geotechnical units, the most probable ground conditions,
and the uncertainty in the units’ occurrence. The number of
voxels belonging to each geological–geotechnical unit along
the tunnel alignment’s longitudinal length is identified and
segregated for every realization generated from the geosta-
tistical modeling. The number of voids within the tunnel
envelope are calculated from the number of voxels belong-
ing to the respective void categories and the void category
area. The number of occurrences of each void category is
recorded from the individual realizations to estimate uncer-
tainty in the occurrence probability of the void category.
The total volume of karstic voids is calculated to estimate
the required pre-excavation grout volume to treat the karstic
voids. Factors, such as the presence of loose infill material
within voids, grouting methodologies, and specifics of the
grouting material, are beyond this study’s scope. The steps
to estimate the void number for each void category and the
volume of pre-excavation grouting required within the tunnel
excavation envelope are outlined below.
Step 1: Observe the minimum size of the karstic voids.
Set up the 3D grid resolution to be the same as the mini-
mum karstic void size in the longitudinal, transverse, and
vertical directions.
Step 2: Model the karstic voids as a geological–geotech-
nical unit, or so-called categorical data, and generate mul-
tiple realizations of ground conditions using a stochastic
geostatistical modeling algorithm.
Step 3: Extract the voxels and the associated simulations
within the tunnel envelope volume.
Step 4: Post-process the realizations from the geostatisti-
cal modeling to estimate the occurrence probabilities of
the voids.
Step 5: Find the number of voids for each void category
as follows:
where V denotes the volume, i is the void category, c
denotes the longitudinal ID length of tunnel alignment
(chainage, in this case), and Δx, Δy, and Δz represent the
resolution of the 3D grid in the longitudinal, transverse,
and vertical directions, respectively.
(1)
V
ic
=(Number of simulation voxels with i)c
×Δx×Δy×Δz
where r is the mean radius, and A is the area of the void
category. The methodology assumes a spherical void
shape.
Step 6: Find the number of voids expected along the tun-
nel alignment and within the tunnel envelope.
where Ni denotes the number of voids for void category i.
Step 7: Calculate the volume of pre-excavation grouting
(Vg) required from the volume of karstic voids along the
tunnel alignment.
4 Geostatistical Modeling
In this study, the 3D reconstruction of geological–geotech-
nical conditions and karstic void distribution was simulated
using the PGS technique. The PGS requires the definition
of (i) geological–geotechnical units for simulations, (ii) the
simulation grid, (iii) the relative proportions of units and the
transition probability matrix, (iv) the spatial variability char-
acterization, and (v) the number of simulations. In this tech-
nique, the correlation of the geological–geotechnical units
is quantified using indicator variograms generated for the
vertical and horizontal directions. Variograms generated in
multiple directions capture the anisotropy within the avail-
able data. The variogram range, the distance beyond which
no correlation exists between the data in each direction and
sill, is the maximum value of the variogram and defines it.
The range of a stratigraphic unit indicator variogram is a
function of heterogeneity and stratigraphic unit occurrence
frequency. Dubrule (2017) provided an overview of indicator
variograms for geostatistical modeling in three dimensions.
Maleki etal. (2017) discussed that indicator variograms
convey valuable information on proportions of geological-
geotechnical units, spatial correlation structure and contact
relationships between the units. Indicator variograms can
also be used to validate the consistency of the interpreted
geostatistical model (Madani etal. 2019).
As per the central limit theorem, each lag distance bin
in a variogram must contain a minimum of 30 point pairs
(Cressie 1985). In addition, the binned lags should be of
equal spacing and the number of pairs should be similar
across all binned lag distances. The lag distances at which
the experimental variograms can be determined with an
acceptable number of data pairs indicates the scale at which
(2)
Ai
=𝜋×
r2.
(3)
N
i=
V
i
/
A
i
(4)
V
gc
=
c
∑
c=1
Vic
=
c
∑
c=1
Nic
×A
i

R.M.Gangrade et al.
1 3
the spatial variation can be modelled. For horizontal vari-
ograms, a decision on the angular tolerance and the slicing
height is made to achieve the balance between the number
of data points available and number of pairs found in the
variogram search. In this study, an angular tolerance of 15°
corresponding to a 30° span for the data points is considered
to develop the horizontal variograms. To compensate for
the lack of close spacing of the boreholes, a relaxation of
the angular tolerance width is required, without affecting
the directionality and anisotropy of the variograms. Based
on the number of data points available, a decision on the
angular tolerance width and slicing height can be made to
increase the precision of the horizontal variograms. Web-
ster and Oliver (1992) discussed that practitioners should
aim to sample 150 to 200 points within a region to estimate
the variograms. Due to higher density of data points in the
vertical direction, vertical variograms are relatively easier
to comprehend. Spherical and exponential models are found
suitable to model Gaussian random functions as these mod-
els generally satisfy the positive definiteness for covariance
and conditional positive definiteness for variograms (Chiles
and Delfiner 2009). Accordingly, spherical model is used to
fit the experimental variograms.
Following the indicator variogram model, the geologi-
cal–geotechnical units are transformed into Gaussian values
in appropriate intervals using the Gibbs sampler algorithm.
A search neighborhood is defined, and the available Gauss-
ian values within the neighborhood, called conditioning
data, are used to simulate Gaussian values within the neigh-
borhood. The simulated Gaussian values are transformed
back into geological–geotechnical units using the factors
mentioned above in the PGS technique. The realizations
generated from the PGS technique are constrained to the
available data, therefore exhibiting realistic representations
of ground configurations. The geostatistical analyses in this
study was performed through Isatis, a geostatistical toolkit
by Geovariances software (https:// www. geova rianc es. com/
en/ softw are/ isatis- geost atist ics- softw are/).
The three void categories—V1, V2, and V3—and soil
and rock were considered asthe geological–geotechnical
units in the PGS. Since the study’s interest was to charac-
terize the distribution of karstic voids larger than 0.10m, a
3D simulation grid of resolution 0.10m in the longitudinal,
transverse, and vertical directions was generated. The 3D
simulation grid extended to about 50m on either side of
the tunnel alignment. About 150 million voxels were gener-
ated in the 3D simulation grid enclosing the complete tunnel
alignment. The boreholes were discretized and digitized into
voxels, such that voxels along the borehole length were iden-
tified with a geological–geotechnical unit. Due to computa-
tional capacity constraints, the tunnel alignment was divided
into seven subsections, and a 3D grid was generated for each
subsection. Within PGS modeling, units are modeled by
random fields that incorporate the geological–geotechnical
knowledge from information on the probability of finding
a specific unit in a specific proportion at any given loca-
tion. This information is converted into conditioning data to
ensure that each unit’s spatial variation within the modeling
domain is reproduced.
Figure7a presents the global vertical proportion curve
(VPC) of the geological–geotechnical units from all the
boreholes within the tunnel project site. The tunnel enve-
lope’s minimum and maximum elevations are presented to
quantify each unit’s proportion within the tunnel envelope.
VPCs are a simple tool for quantifying the proportions of
geological–geotechnical units present as a function of eleva-
tion or depth (Huber etal. 2015). The VPCs are computed
along the lines vertical to a chosen reference level. The
tunnel alignment is subdivided into multiple sections, and
within each section, multiple VPCs are generated to capture
the lateral variation in the unit’s proportions for geostatisti-
cal modeling. Figure7b represents an interpretation of the
available borehole data and conceptual geological–geotech-
nical model. The rule derived from the available boreholes
controls the permissible and forbidden contacts between
pairs of units and reproduces the units’ ordering. Relative
proportions presented in Fig.7c indicate the proportions
of borehole samples belonging to each unit within the tun-
nel envelope. Utilizing PGS technique,200 equally prob-
able realizations of the ground conditions were generated
in terms of the coded geological–geotechnical units within
the seven 3D grids. Each realization differs from the others
within a specific range of variation. The quantification of the
variation indicates the uncertainty of the realization results.
For this study, at about 200 realizations, the uncertainty dif-
ference of the realization results was within 1%, indicating
that 200 realizations were sufficient for geostatistical mod-
eling. A similar methodology for identifying the number
of realizations has been applied by Gangrade and Mooney
(2020) to characterize the location uncertainty in soil transi-
tion boundaries within the tunnel excavation envelope.
The simulation of geological–geotechnical units proceeds
as follows:
(1) Gaussian fields are iteratively co-simulated at points
within the modeling domain that contain sampled
borehole data. Within each iteration, a random point
among the borehole samples is selected, and the Gauss-
ian fields are updated based on the values taken by the
Gaussian fields at other sampled locations. Gibbs sam-
pler, an optimization algorithm, is used to perform the
Gaussian fields’ iteration and modeling.
(2) Next, the Gaussian fields are simulated at the unsam-
pled locations. Since the Gaussian fields are independ-
ent, each field is conditioned to its previously simulated

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
values. In this study, the turning bands algorithm is
used to simulate the Gaussian fields.
(3) The regionalized VPCs and the geological–geotechni-
cal unit rule (shown in Fig.7) are applied to the simu-
lated Gaussian fields to convert Gaussian variables to
the actual geological–geotechnical unit.
(4) The process is repeated multiple types leading to the
generation of multiple realizations of the ground condi-
tions.
It is to be noted that the realizations are only equally
probable mathematically for a given modeling method and
the given inputs, and not equally probable physically (Ma
2019).
5 Results andDiscussion
5.1 Quantitative Assessments
The set of 200 realizations from PGS and a tunnel excavatio-
nand a tunnel excavation occurrence probability of the units
and identify the most frequently occurring unit at each simu-
lation grid’s voxel. The simulations of the geological–geo-
technical units within the 3D tunnel envelope are presented
using 2D profiles developed through the center of the tunnel
alignment. The longitudinal 2D profile of an example reali-
zation and the most frequently occurring unit (i.e., the most
probable unit within each simulation voxel) is presented in
Fig.8. The elevations within the tunnel envelope are nor-
malized with respect to the tunnel invert level. As shown
in Fig.8a and b, category V1 voids are expected to occur
Fig. 7 a Global proportions of
the geological–geotechnical
units considering all the bore-
holes along the tunnel align-
ment, b geological–geotechnical
unit modeling rule generated
based on the probability of
transition and juxtaposition
tendency between units from
boreholes, and c relative propor-
tions of the units as observed
from boreholes
Fig. 8 Longitudinal profile of
most frequently occurring unit
within the tunnel envelope

R.M.Gangrade et al.
1 3
at the start of the tunnel alignment below, at, and above the
tunnel springline level. The model predicts the presence of
category V1 voids at about Chainage 2050m at the spring-
line level. Category V2 voids are expected to occur at about
Chainage 2050m just above and at the tunnel springline
level. Individual realization shows that category V2 voids
predominantly occur from Chainage 1500m to 1600m and
from Chainage 1900m to 2100m. Category V3 voids are
expected to occur between Chainage 500m and 600m, just
about at the tunnel springline level. The individual realiza-
tion shows the presence of category V3 voids from about
Chainage 900m to 1000m.
Figure9 presents, for each void size category, the occur-
rence probabilities and the number of voids expected spa-
tially within the tunnel excavation envelope. The occurrence
probabilities are calculated as the ratio of the number of
occurrences of the void category, at each simulation voxel,
to the total number of geostatistical realizations. It is noted
that the occurrence probabilities shown in Fig.9 are along
the tunnel centerline only. The number of voids is calculated
using Eqs.(1)–(3), where the volume of the simulation grid
is normalized with the mean void area of the respective void
category. The results in Fig.9 are for the most probable
ground conditions within the 3D tunnel envelope. Contour
plots are used to represent the expected number of voids
within the 3D tunnel envelope. The contour plots provide
a 2D profile view in which tunnel sections with the same
number of voids are connected to produce contour lines.
The gaps between each pair of neighboring contour lines
are filled with color to make it easier to identify the num-
ber of voids. As shown in Fig.9a, the occurrence probabil-
ity of category V1 voids, P (V1), ranges between 35 and
80% from Chainage 0m to 50m. Two to five category V1
voids are expected within the 50m interval. Two to foud
category V1 voids are expected with 40% < P (V1) < 80%,
between Chainage 2000m and 2200m. P (V1) is relatively
low between Chainage 900m and 1800m. From Fig.9b P
(V2) > 40% at the start of the tunnel alignment and between
Chainage 900m and 1800m. At Chainage 2050m, cat-
egory V2 are expected with 50% < P (V2) < 80%. Two to
eight category V2 voids are expected at or below the tunnel
springline between Chainage 600m and 1300m. Results
reveal that a relatively high number of category V2 voids are
expected to be encountered between tunnel springline and
invert. Figure9c shows that 40% < P (V3) < 90%, between
Chainage 500m and 700m. For the rest of tunnel alignment,
P(V3) is relatively low. Between one and four category V3
voids are expected within the tunnel envelope, with maxi-
mum expected between Chainage 600m and 900m.
Individual realizations are utilized to capture the uncer-
tainty in the occurrence probability of void categories from
equally probable ground conditions. This study’s probabilis-
tic approach highlights significant variations in the inferred
probability of encountering voids during tunneling. Confi-
dence interval (CI) bands are a robust measurement of the
uncertainty estimates. Figure10 presents a distribution of
occurrence probabilities within the 3D tunnel envelope from
200 realizations through CI band sets. The black line indi-
cates the median occurrence probability of the void catego-
ries as extracted from 200 realizations. The most extensive
band reflects the 95% CI i.e., a 95% probability that the void
category’s actual occurrence probability within the tunnel
envelope is captured within the band. As an example, in
Fig.10a at Chainage 50m (95% CI), the occurrence prob-
ability of category V1 voids within the tunnel envelope lies
between 0 and 80% with a median value of about 20%.
Similar significant variations in the occurrence probabili-
ties of category V2 and V3 voids are observed in a few sec-
tions along the tunnel alignment. A wider CI band reflects
relatively higher uncertainty in the occurrence probability
of the void categories. The visualization of the occurrence
probabilities and the CI bands help interpret the uncertainty
in the void occurrence within the tunnel envelope. As illus-
trated, all three void categories show higher uncertainty
at chainage locations with a low density of boreholes (see
Figs.3 and 6), resulting in fewer available data points to
condition the geostatistical realizations. The results can
be interpreted quantitatively as, “At Chainage 1000m, the
occurrence probability for category V2 voids (for 95% CI)
within the tunnel envelope ranges between 0 and 60%.” The
observations indicate a quantifiable occurrence probabil-
ity for the three void categories within the tunnel envelope
rather than the qualitative or semi-quantitative assessments
indicated in the GBR.
Figure11 presents the variability in the number of voids
expected within the tunnel envelope in terms of a 95% prob-
ability interval (PI) band. From the set of 200 geostatisti-
cal realizations, the minimum, the mean, and the maximum
number of voids expected for every 100m of longitudinal
tunnel excavation is calculated from Eq.(3). The results
quantify the variability in the number of voids for about
5300 m3 of tunnel excavation (every 100m of longitudinal
length and full-face excavation of 53 m2). The abscissa in
Fig.11 indicates the length of the longitudinal tunnel exca-
vation. Discussion on the number of voids for each category
is provided below.
Category V1 voids Geostatistical modeling predicts that
up to about 40 category V1 voids
could be expected in the first 100m
of tunneling. About the same number
of category V1 voids is expected at
the end of the tunnel alignment. The
maximum concentration of category
V1 voids is expected to be between
Chainage 1000m and 1600m, where

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
the maximum expected number ranges
between 20 and 40. The mean and the
maximum cumulative count of cat-
egory V1 voids expected within the
tunnel envelope is about 200 and 500,
respectively.
Category V2 voids Results from geostatistical modeling
indicate a maximum concentration of
category V2 voids between Chainage
1000m and 1500m and at the end
of the tunnel alignment. In contrast,
the expected number of category V2
Fig. 9 Longitudinal profile of occurrence probability and contours indicating void number for each void category within the tunnel envelope.
Occurrence probability and number of voids from 200 geostatistical realizations within the 3D tunnel envelope are presented in 2D profiles

R.M.Gangrade et al.
1 3
voids ranges between zero and four
for the rest of the tunnel alignment. A
cumulative maximum of 140 voids of
category V2 is expected to be encoun-
tered within the tunnel envelope for
the complete tunnel excavation.
Category V3 voids Compared to the number of category
V1 and V2 voids, fewer category V3
voids are expected. A cumulative
maximum of about 55 category V3
voids are predicted to be encountered
within the tunnel envelope. Results
indicate that two to three category
V3 voids are expected within every
100 m of tunneling from the start
of tunneling up to about Chainage
1700m. Less than two voids of this
category are expected between Chain-
age 1700m and 2500m, with three to
four voids expected for the last 200m
of tunneling.
An integrated review of the probabilistic assessment
results provides quantitative insights to understanding the
void occurrence and the tunnel excavation environment,
which cannot be obtained from the ground profiles and
assessments from the GBR. For example, the results from
the probabilistic approach in Figs.10 and 11 can be inter-
preted as, “Between Chainage 2000m and 2100m, about
six category V2 voids are expected with an occurrence prob-
ability between 0 and 80% (for 95% CI). A total of about 40
category V2 voids are expected in 2100m of longitudinal
tunneling”. The probabilistic assessment results help inter-
pret the spatial locations where voids are expected to be
encountered, the occurrence probability of void categories,
the occurrence frequency, and the cumulative number of
voids within the 3D tunnel envelope.
5.2 Application ofProbabilistic Assessment
Figure12 presents the volumetric fraction of voids within
the tunnel envelope for every 50m of tunnel excavation. The
volumetric void fraction is evaluated from the total volume
of each void category and volume of excavation for 50m
longitudinal distance. For most of the tunnel alignment, the
average void fraction within the tunnel envelope remains at
or below 6% with an exception between Chainage 1400m
and 1600m and around Chainage 2600m. The 95% prob-
ability interval of expected void fraction shows a maximum
value of about 12% at Chainage 1500m. The expected void
fraction from Chainage 2300m to 2600m remains about 3%
and increases to about 9% at the end of the tunnel section.
The results from probabilistic assessment aid in quantify-
ing the occurrence probability, void count, and void fraction
within the tunnel envelope.
In a karstic geological setting, pre-excavation grout-
ing can be utilized as an effective measure to mitigate the
geohazards of water ingress and face collapse during tun-
neling. Within pre-excavation grouting, the grout is typi-
cally injected in front of the shield machine to improve water
tightness and stiffness at the excavation face. This is to avoid
any geohazards and maintain an optimal rate of tunneling
without lengthy stoppages. Cui etal. (2015) and Yang etal.
(2020) discussed the pre-excavation grouting methodologies
Fig. 10 Confidence interval (CI)
profile of occurrence probability
of void categories within the
3D tunnel envelope using 200
geostatistical realizations

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
Fig. 11 Number of voids expected within tunnel envelope for every 100m of tunnel excavation. These values have been obtained using the 200
individual geostatistical realizations and a tunnel excavation area of 53 m2 (for an 8.1m diameter tunnel)
Fig. 12 Longitudinal profile of
the void fraction for every 50m
of tunnel excavation with tunnel
excavation area of 53 m2 (for an
8.1m diameter tunnel)

R.M.Gangrade et al.
1 3
for tunnels in karstic geological environments. The method-
ologies involve injecting the grouting slurry from the ground
surface into the karstic cavities and voids within a specific
transverse distance from the tunnel alignment. This study
only estimates the grouting volume required to fill the karstic
voids prior to tunnel construction. The quantity of grout vol-
ume required for voids within the 3D tunnel envelope is cal-
culated using Eq.(4). The void number estimates presented
in Fig.11 are used for grout volume calculations. Figure13a
presents the minimum, the mean, and the maximum grout
volume estimates for every 50m of longitudinal tunnel exca-
vation (about 2650 m3 of tunnel volume) and filling karstic
voids within the tunnel envelope. The increase in the esti-
mated grout volume between Chainage 1400m and Chain-
age 1500m is due to the possible presence of category V1
and V2 voids within the tunnel section. A higher estimated
concentration of category V2 and V3 voids from Chainage
2300m onwards leads to an increase in the estimated grout
volume required within the tunnel section. Figure13b pre-
sents an estimate of the cumulative volume of grout required
to fill the karstic voids within the tunnel envelope. The gra-
dient of the cumulative grout volume curve is indicative of
the tunnel sections with relatively higher void concentra-
tions. A consideration of spherical void shape and the mean
dimension of the void are the predominant limiting factors
for grout volume estimations.
No empirical studies or rule of thumb estimate the extra
time and cost of risk mitigation strategies, such as pre-exca-
vation grouting requirements, on a tunnel project. Since each
tunnel project is unique, very few statistical data is readily
available concerning the occurrence probability and conse-
quences of unexpected subsurface features, such as karstic
voids. A number of decision-support tools, such as Decision
Aids in Tunneling (DAT) (Einstein etal. 1987), Successive
Method (Lichtenberg 1990), and Isaksson (2002), developed
probabilistic estimates of tunnel construction time and costs
whilst planning and bidding for a tunnel project. The results
from the probabilistic approach discussed in the manuscript
can be fed into the available decision–support tools to get
improved estimates of tunnel construction cost. An improved
quantitative probabilistic assessment of voids would allow
tunnel contractors to better target the pre-excavation grout-
ing quantities, avoid over-use, and optimize costs towards
the risk mitigation strategy.
6 Conclusion
A geostatistical modeling-based probabilistic approach is
proposed and is applied to geotechnical-site investigations
from a 3km section of a tunnel project situated in karstic
geology. Karstic voids observed within the project site are
classified into three size categories (V1, V2, and V3). The
PGS technique, a stochastic geostatistical algorithm, is
applied to develop multiple realizations of the ground con-
ditions, including the karstic voids. The realizations are used
to characterize the occurrence probability, the number, and
the frequency of karstic voids within the 3D tunnel envelope.
Confidence intervals (CI) and contour plots characterize the
occurrence probability and void number within the tunnel
envelope. The study then uses the probabilistic assessment
results to develop estimates of the pre-excavation grout vol-
ume required to treat karstic voids.
This study’s results contribute to the applicability of
probabilistic methods towards a quantitative risk assess-
ment and compensation/mitigation of karstic features for
tunneling applications. The conclusions from this study are
as follows:
(1) The geostatistical modeling-based probabilistic assess-
ment allowed for developing assessments of occurrence
probabilities and the number of expected voids for each
void category. The results also aided in identifying the
spatial locations of the occurrence of karstic voids
within the tunnel envelope.
Fig. 13 Pre-excavation grouting
volume estimates of karstic
voids treatment within the
tunnel envelope. The grout
volume required for every 50m
of longitudinal excavation and
the cumulative grout volume are
estimated using 200 geostatisti-
cal realizations

Probabilistic Assessment ofVoid Risk andGrouting Volume forTunneling Applications
1 3
(2) Quantitative interpretations of the results were given in
the following format: “Between Chainage 2000m and
2100m, about six category V2 voids are expected with
an occurrence probability between 0 and 80% (for 95%
CI). A total of about 40 category V2 voids are expected
within 2100m of longitudinal tunneling.”
(3) Contour plots, providing a 2D profile view of the 3D
tunnel envelope, were generated to identify voids’
number and spatial locations with respect to the shield
machine cutterhead.
(4) Interpretations of the results of contour plots were given
in the following format: “Just below and at the tunnel
springline elevation, four to six category V2 voids are
expected between Chainage 600m and 1300m. Cat-
egory V2 voids ranging between two and six in number
are expected to occur along the rest of the tunnel align-
ment.”
(5) Probabilistic assessment results are used to evaluate
the expected volumetric fraction of voids within the
tunnel envelope. For every 50m of tunnel excavation,
an average volumetric void fraction between 2% and
6% is expected in the tunnel section. A 95% PI indi-
cates about 12% volumetric void fraction at Chain-
age 1500m and at end of the tunnel section. For the
remaining alignment, the maximum volumetric void
fraction remains at about 6%.
(5) A cumulative number of voids for each category were
used to generate estimates of pre-excavation grouting
volume for void treatment. Absolute grout volume
required for every 50m of longitudinal excavation
(about 2650 m3 of tunnel volume) was calculated.
Estimates of cumulative grout volume indicated that
an average of 4000 m3 and a maximum possible value
of 8000 m3 would be required for karstic void treatment
along the complete tunnel alignment.
Application of the proposed probabilistic approach can
help develop baseline statements in the GBR for karstic risk
assessment. Tunnel contractors can utilize the improved
knowledge of the tunnel excavation envelope to (1) plan
additional geotechnical investigations, (2) identify grouting
locations and techniques, (3) develop appropriate financial
and technical bids, and (4) allocate risk and resolve contrac-
tual disputes. The work presented herein was developed to
be applied by tunnel designers, contractors, and involved
stakeholders to evaluate karstic feature risk and improve
ground awareness prior to tunnel project procurement and
construction.
Author contributions RG: conceptualization, methodology, software,
exploratory data analysis, visualization, and writing—original draft.
JG: conceptualization, resources, writing-review, and editing. MAM:
resources, supervision, funding acquisition, writing—review, and
editing.
Funding The authors gratefully acknowledge the University Transpor-
tation Center for Underground Transportation Infrastructure (UTC-
UTI) at the Colorado School of Mines for funding this research under
Grant No. 69A3551747118 from the U.S. Department of Transpor-
tation (DOT). The opinions expressed in this paper are those of the
authors and not of the DOT.
Data Availability All data used for this study are available from the
corresponding author by reasonable request.
Code Availability All code generated for this study is available from
the corresponding author by reasonable request.
Disclosures
Conflict of interest The authors declare that they have no known com-
peting financial interests or personal relationships that could have in-
fluenced the work reported in this paper.
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