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Evaluation of rock-mass permeability tensor and prediction of tunnel inows bymeans of geostructural surveys and nite element seepage analysis
N. Coli a,, G. Pranzini b, A. Al c, V. Boerio d
a Department of Chemical, Mining and Environmental Engineering (DICMA), University of Bologna, 40136, Bologna, Italyb Department of Earth Sciences, University of Florence, 50100, Florence, Italyc SPEA Engineering Consulting S.p.A., Hydraulic Ofce, 20170, Milano, Italyd SPEA Engineering Consulting S.p.A., Geoengineering Department, 20170, Milano, Italy
A B S T R A C TA R T I C L E I N F O
Article history:
Received 10 January 2008
Received in revised form 28 April 2008
Accepted 22 May 2008
Available online 29 May 2008
Keywords:
Tunnel inow
Permeability
Tensor
Finite element
Seepage analysis
In this paper, a new practical technical approach for the evaluation of hydraulic conductivity and tunnel
water inow in complex fractured rock masses is presented. This study was performed in order to evaluate
water ow into tunnels planned along the new highway project from Firenze Nord gate to Barberino di
Mugello (Tuscany). The results are based on detailed and comprehensive geostructural characterization of
rock masses, by means ofeld surveys, geological and hydrogeological studies.
Starting from discontinuities properties surveyed in the eld, the permeabilityKtensor was calculated using
the Kiraly equation, integrated with the introduction of the effective hydraulic opening of ssures (e).
Principal directions of tensor Kwere calculated for each geostructural survey station using an automated
software script especially developed for this purpose.
Available K data from Lugeon tests were also collected and analyzed with results that do not fully represent
the rock mass due to its structural (hydraulic) variability.
In order to evaluate water ow into tunnels planned for excavation under the water table (for a total number
of eight), a nite elements seepage analysis was performed on 38 representative geological sections
transverse to tunnel paths (-planes). Each section referred to the nearest and geologically most compatible
geostructural station.Principal directions ofKtensors were projected on the -planesby means of trigonometric transformations.
Unitary water inows were then evaluated for long-term steady-state, as well as for initial state immediately
after tunnel excavation. Inow values calculated for each unitary section were extended to geologically
homogeneous lengths of the tunnel, according to the variability of the water head above excavation, and then
summed up for the whole length of each tunnel. Inow values obtained with FE seepage analysis were also
compared to other inow evaluation methods.
2008 Elsevier B.V. All rights reserved.
1. Introduction
TheA1 Highway in Italy is the most important highway connecting
north to south. It is part of the recent Italian highway network
modernization program, which aims to improve the road structures in
order to guarantee a better service. In this context the doubling of the
A1 highway from Firenze Nord gate to Barberino di Mugello (Tuscany)
is planned.
The project, carried out by SPEA S.p.A., plans the construction of a
new highway beside and uphill to the present one on the left side of
the Marina river. The excavation of twelve tunnels is planned.
The new road path will cross the shaly Sillano Formation and the
calcareous Monte Morello Formation, which are part of the Super-
gruppo della Calvanastratigraphic unit (Cicali and Pranzini,1987), as
well as quaternary alluvial sediments and debris deposits.
Parallel to the mechanical and stability behaviour of the rock mass,
a very important engineering and environmental aspect which must
be taken into account in the project is the prediction of the waterow
into the tunnels. This kind of prediction is quite difcult due to the
inhomogeneous hydrogeological properties of fractured rock masses.
The hydraulic conductivity of a rock mass is a direct consequence of
the interlocking of discontinuities (bed planes, joints, fractures) and
therefore highly anisotropic.
Standard Lugeon tests give average and isotropic Kpermeability
values which strictly refer to the rock volume surrounding the length
of borehole where the test is performed. In fractured rock mass where
hydraulic conductivity is controlled by discontinuities, Lugeon K
values are not representative of the real permeability of rock mass
andnot extensibleto large volumes of rock. Therefore, we carried outa
new methodology for evaluating the permeability tensorK, based ona
Engineering Geology 101 (2008) 174184
Corresponding author. Tel.: +39 339 8029552; fax: +39 055 2479741.
E-mail address:niccolo.coli@mail.ing.unibo.it(N. Coli).
0013-7952/$ see front matter 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2008.05.002
Contents lists available at ScienceDirect
Engineering Geology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o
mailto:niccolo.coli@mail.ing.unibo.ithttp://dx.doi.org/10.1016/j.enggeo.2008.05.002http://www.sciencedirect.com/science/journal/00137952http://www.sciencedirect.com/science/journal/00137952http://dx.doi.org/10.1016/j.enggeo.2008.05.002mailto:niccolo.coli@mail.ing.unibo.it8/12/2019 Coli Et Al 2008_rock Mass Permeability Tensor
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survey of the geostructural properties of the rock mass. This method-
ology takes into account the geological settingof the rock mass and its
anisotropy.
The permeability tensorKhas been calculated using the criterion
proposed byKiraly (1969, 1978, 2002)by means of geostructural data
collected with accurate eld surveys.
Once the principal Kcomponents had been calculated, prediction
of tunnel water inow was done by means ofnite element seepage
analysis, which was carried out for long-term steady-state inow and
also for inow immediately after tunnel excavation.
A detailed methodology of the study is explained in the following
chapters.
Fig.1.Geostructural frameworkof the CalvanaMonte Morelloreliefs. The thick black linerepresentsthe approximate path of thenew highway. Modied fromCicali andPranzini (1987).
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2. Geological framework
The geological reference map used in this work is the Geological
Map, scale 1:5.000, drawn by SPEA S.p.A. for the executive project.
The geological framework of the area is dominated by the Sillano
and Monte Morello Formations outcropping in the two contiguous
Monte Morello and Calvana reliefs (Fig. 1). These formations belong to
the Supergruppo della Calvana, an alloctonous stratigraphic unit
which slided onto the Tuscan Nappe for tens of kilometres (Boccalettiet al., 1980; Coli and Fazzuoli, 1983; Abbate, 1992).
These formations are intensely tectonized, and were subjected to
many deformational phases during their geological history (Coli and
Fazzuoli, 1983; Abbate, 1992). Some large recumbled folds are present,
trending NS with east vergence, minor folds are present in the limbs.
Fold structures are interlocked and dislocated by many faults. There
are relatively undisturbed rock masses as well as highly tectonized
ones with many tight and isoclinal folds of uncertain polarity (due to
the difculty in recognizing strata polarity).
The Firenze-Prato-Pistoia step-faults truncate the Supergruppo
della Calvana unit on the south-west side of the Marina river valley,
while on the north-east side the Mugello Basin develops ( Coli and
Fazzuoli, 1983; Briganti et al., 2003).
The stratigraphic sequence of the area is the following, from
bottom to top:
Sillano Formation (SIL) (Upper Cretaceous): scaly shales, often
chaotic and highly tectonized and sheared, including limestone
and arenaceous stratas and/or broken sequences (Bortolotti, 1962).
Monte Morello Formation (MML) (Paleocenemiddle Eocene): marly-
limestone and calcarenitic turbidites, fading upward to more pelitic
layers. The formation is characterized by marly-limestone beds(23 m
thick) alternating with marly beds (25 m thick) and thin pelitic
Fig. 2. Stereographic plotsof: A) Dip/Dip-Directionof mapped discontinuities (polesof planes, concentration contours,ciclographs of principal sets)and B) 3D frequencyof ubiquitary
ssures. Plots refer to survey station
S11
.
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interbeds. Calcarenites are also present, with bed-parallel laminations.
The turbidite Bouma sequence alternates thin Tb-e Tc-e pelitic
sequences with thick marly-limestone Tb stratas (Sestini and Curcio,
1965; Bortolotti, 1962; Coli and Fazzuoli, 1983).
Quaternary alluvial and uvial deposits, locally terraced; etero-
metric and eterogeneous debris in clayey matrix, most of which are
landslide and alluvial fan deposits.
In order to collect the structural rock-mass properties for calculat-
ingKtensor, a eld survey was developedlocating32 structural surveystations homogeneously distributed according to the tunnel paths and
geological framework.
For each station a detailed mapping of outcropping discontinuities
was carried out, in particular:
Evaluation of principal discontinuity sets
Dip/Dip-Direction of each discontinuity
Aperture, spacing, persistence and roughness [by means ofJRCindex
(Barton et al., 1974)] of discontinuities
A considerable amount of data was collected for each discontinuity
set in order to obtain reliable statistical elaborations. Dip/Dip-
Direction data were plotted in stereographic projections using
software Dips ( Rocscience Inc.) (Fig. 2A). Average aperture and JRC
were also calculated. Three dimensional ssures frequency calculationwas also performed using a self-developed software tool (Fig. 2B),
which calculates 3D frequency of ubiquitary ssures according toLa
Pointe and Hudson (1985), andHudson and Harrison (1997).
3. Hydrogeological framework
Water ow into MML and SILformations, andin general, inssure-
permeable rock masses, is directly related to the frequency and
physical properties of discontinuities.
However water ow is also affected by the relativeposition of inow
areas and drainage areas. Water tends to ow through discontinuities
whose direction is more favourable on a hill slope. Fractures almost
parallel toow gradient in karsticable rocks tend to widen due to the
dissolution of carbonate. This mechanismcan alsolead to thedepositionof carbonates and thus the closure ofssures which are not favorably
oriented towards the water ow.
Waterow in MML is also perturbed by lithological variability and
tectonic asset: the wide interblocking of the formation caused by
faults causes permeable limestone blocks to come in contact with
impermeable pelitic barriers. In this context discontinuities persis-
tence never exceeds a few hundred meters (Pranzini, 2002).
Surveyed geological boreholes also pointed out the presence of
highly fractured zones of MML, called MMLc, with a higher mean of
hydraulic conductivity.
A MMLpfr member is also differentiated, which refers to palaeo-
landslide accumulations of MML material. This unit does not show
signicant difference from MML in the valueof hydraulic conductivity,
because the high level of fracturing is balanced by the presence of a
silty-clay weathering matrix.
Almost all of the tunnels are planned to be excavated from a few
metersto around40 m deep into a slope which is generally steep. Only
eight outof twelve tunnels have thegroundwater table level above the
invert and the piezometric head never exceeds 30 m. In some cases
water table is lowered by the drainage of the present A1 highway
tunnels.
4. Lugeon tests
Lugeon permeability tests were done in MML and SIL and the result
values from these tests were collected and analyzed allowing for some
observations.Figs. 3 and 4 show the distribution of the Km classes in
MMLand SIL, whereKm canbe referred to as theequivalentpermeability
of the rock mass.
Through Lugeon tests, in fact, it is possible to evaluate a mean
permeability coefcient (Km) for the length of the boreholes where
tests were performed. This coefcient is susceptible to the presence of
structural features like wide opening ssuresand it has no orientation
in the space.Kmclasses show a high range of variability in MML with the modal
class of 107 m/s, while in SIL the variability is lower with a modal
class of 106 m/s. The geometricalmean (more reliablethan thesimple
mean fora wide-range specimen) givesthe values of 3.20 106 m/sfor
MML and 1.30106 m/s for SIL. Even ifKmpermeability for MML and
SIL appears to be very similar, this is in contrast with hundreds of past
work experiences, wellow rates andeld observations, which show
that MML canrepresent a good acquifer while SILis always a poor one.
This similarity in the mean Km values for MML and SIL can be partially
explained taking into account the different average depth where
Lugeon tests were done.
Moreover Lugeon tests are much less reliable in shaly rocks (i.e.
SIL) than in limestone formations (i.e. MML). In fact, the presence of
imbibition phenomena in the shaly rock surrounding the length of theborehole where the test is performed lead to the softening of the
material which in turn causes a water ow between the packers and
the borehole walls and therefore an overestimation of the measured
Km values. These kind of phenomena are well known in other
appenninic shaly formations like the Argille a Palombini (Palombini
shales) and Unit Argilloso-Calcarea (Shale-limestone complex).
The depth below surface level is a major factor that inuences
hydraulic conductivity in fractured rock masses because the progres-
sive increasing of lithostatic load cause the progressive closure ofssures.
The average testing depth for MML is 28 m under surface level
while for SIL is 18 m under surface level, thus the meanKmvalue for
SIL is expected to be higher than the value obtained for MML at a
greater average testing depth.Fig. 3.Kmclasses distribution in SIL.
Fig. 4.Kmclasses distribution in MML.
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The variation ofKmwith depth for MML is shown in Fig. 5.
Analyzing Fig. 5 it can be seen that shallower Km values are
widespread because of the high variability ofssure opening caused
by the unloading of rock mass. It is clearly visible that with increasing
depthKmvalues have the tendency to be reduced.
Therefore Km
values obtained by means of Lugeon tests conrm
the uncertainty of this kind of permeability test in order to
characterize the whole rock mass. Taking into account geostructural
complexity and discontinuity properties, there is the need to perform
more accurate investigations.
5. Determination of the permeability tensorK
In order to evaluate the hydraulic conductivity of an anisotropic
fracturedrock mass, the approach proposed by Kiraly (1969, 1978) was
adopted in this study. Thepermeability in a fractured rock mass canbe
represented by a second order tensorKreferred to either a global or a
local coordinate system:
K
kxx kxy kxz
kyx kyy kyzkzx kzy kzz
24 35
Thus, as long as the tensor is symmetric, eigenvalues and
eigenvectors can be calculated in order to nd the directions and
magnitudes of principal permeability:
K
k1k2
k3
24
35
The magnitude of tensor components are strictly related to the
distribution and physical properties of discontinuities of the rockmass.
Tensor K can be estimated from physical properties of the
discontinuities, but some assumptions must be made (Snow, 1968,
Kiraly, 1969): The global permeability of the intact rock must be much less, almost
null, compared to the permeability of the discontinuities. Discontinuities must be persistent in a representative element of
volume of the rock mass.
The permeability must be isotropic on the plane of discontinuities. The mean velocity vector V
mof the ow must have a linear variation,
with theprojection of thegradient vectorJ
on the discontinuity planes.
Once these assumptions are made, we can express theKtensor for
Nsets of parallel discontinuities with the following equation (Kiraly,
1969):
K
g
12m
dN
i1
fidd3id I
!ni
!nih i 1
where:
g gravity acceleration (9.81 m/s2)
v kinematic viscosity of water (3.20 106 m2/s)N Total number of discontinuities sets
f average frequency of thei-set of discontinuities (m1)
d average aperture of thei-set of discontinuities (m)
I identity matrix!n n1; n2; n3 dimensionless unitary vector normal to the average
plane of the discontinuity set
The term [In
n
] can be expressed in the matrix form as:
1 0 00 1 00 0 1
24
35 n
21 n1n2 n1n3
n2n1 n22 n2n3
n3n1 n3n2 n23
24
35
TensorKcanbe referredto a Cartesian coordinate systemas well as
direction cosines or azimuthal (Dip/Dip-Direction) spherical coordi-
nate system. The parameter d used in Eq. (1) refers to ssures
characterized by smooth and parallel surfaces.
Taking into consideration a set of perfectly parallel and smoothssures in a non-permeable matrix, the permeability of the ssures
can be easily expressed by:
k gdd3df
12m2
where
g gravity acceleration (9.81 m/s2)v kinematic viscosity of water (3.20 106 m2/s)
d average aperture of planarssures (m)
f average frequency ofssures (m1)
The average physical aperture Eofssures measured in the eld
cannot be considered as a representative value for the d parameter
due to the fact that natural discontinuities are characterized by wavy-
Fig. 5.Kmvariability with depth in MML.
Fig. 6.Graphical explanation of the effective hydraulic opening (fromBarton, 2004a).
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rough surfaces which lead to a great lowering of the magnitude ofk
(Barton, 2004a,b).The effect of roughness can be taken into account by means of an
effective hydraulic opening (e)(Fig. 6) dened as follows (Barton,
2004a,b):
e E2
JRC2:503
where:
E average physical aperture ofssures (mm)
JRC0 7Jr3 (whereJr=Barton Q-System roughness index [Barton
et al., 1974]) in this study JRC0was assumed equal to Barton
JRC index (Barton and Choubey, 1977).
Fig. 7shows the effect of varying JRC on the value ofk using theeffective hydraulic opening in Eq. (2).
Since roughness has a big impact on permeability it must be taken
into account in the mainKformulation.
Therefore Eq. (1) is replaced with:
K g
12md
N
i1
fide3id I
!ni
!ni
h i 4
Wheree (m) is the effective hydraulic opening.
Eq. (4) permits the calculation ofKtensor for a given geostructural
survey station once discontinuity properties have been mapped.
Eq.(4) hasbeen implementedin an automatedscript developed for
the software Mathematica ( Wolfram). For each geostructural survey
station the script inputs are in this form:
Dip/Dip-Direction () of each set of discontinuities.
Average frequencyf(m1) for each of thei-set of discontinuities.
Average effective hydraulic opening e (m) for each of the i-set of
discontinuities.
The script performs the calculation of tensorKand the determina-
tion of the principal K= [k1,k2,k3] directions by evaluating eigenvec-
tors and eigenvalues of the matrix. This process involves two
coordinate transformations.
Fig. 8illustrates the script workow.
The model described above has been validated by means of simple
tests. Discontinuity properties assumed in tests are described in Table 1.
Therst test (Test A) was performed using a single discontinuity set
(Set1) inorder tocompare k numericalvalues with those given by Eq.(2).
In the second test (Test B) two sets of perpendicular discontinuities
were assumed with different physical properties in order to achieve
very different permeability values. It is clear that the maximum
hydraulic conductivity will be in the plane ofSet1which has a greater
effective hydraulic opening and the minimum hydraulic conductivity
will be in the plane ofSet2. Therefore we expect that simulation will
produce compatible results.
Fig. 9. shows stereonets of the average Set1 and Set1 planes and
computed principalKmax and min directions.
It can be seen that computed principalKdirections perfectly agree
with expected results in bothTests A and B. Moreover k values given
byTest Aagree with values given by Eq. (2) (Table 2).
Other similar tests were carried out and all gave expected results,
thus the model can be considered valid and functional.
Therefore for each of the 32 geostructural survey stations tensor
K= [k1,k2,k3] was calculated (Table 3) and plotted in stereographic
stereonets (Fig. 10).
Some stations arenot included because they were discarded after a
geological consistency verication. These stations in fact refer to very
local geological structures or accidents and are not representative of
the dominant rock-mass structure.
6. Finite element 2D seepage analysis
2D nite element seepage analysis (Phase2, Rocscience Inc.) was
used in order to evaluate water ow into tunnels, once principal K
directions had been calculated for each geostructural survey station. The
model provides tunnel inow through unitary sections of rock mass (1 m).
Tunnelswere rstsubdivided intogeologically homogeneouslengths,
as many as needed foran exhaustivecoverage of the geological variability
of each tunnel. Then geological sections of each homogeneous length,
transverse to tunnel paths (-plane), were developed for a total number
of 38 sections.
Initial phreatic levels utilized in the nite-element models, referred
to the water table prole developed by SPEA in the Geotechnical
Longitudinal Prole of the project. Where more than one phreatic level
were present, only the highest one was taken into account in order to
simulate the worst inow scenario.Geological sections (-plane) were then associated to the nearest
and most representative gestructuralstationsand thusto the relativeK
tensors.
Then principalKvalues projected to the -planewere calculated.
In fact, Phase2 ( Rocscience Inc.) seepage module requires only two
components of hydraulic conductivity as inputs (k1,k2), mutually
Fig. 8.Exemplicative scheme of the script workow.
Table 1
Discontinuities properties assumed for validation tests
Set 1 Set 2
Dip-Direction/Dip () 190/89 100/89
f(m1) 3.30 4.00
e(m) 0.05 0.00000028
Fig. 7. Effect of JRC on the nal value ofk for a perfectly parallel set ofssures with
f=3.30 andE=10 mm.kEis calculated using the physical aperture ofssures (E) while
keis calculated using the effective hydraulic opening (e).
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perpendicular, lying on the 2D model plane (-plane) and allowed to
rotate by an angle from the+ xaxes of the model datum.Transformation from the absolute principal Kcomponents to the
-planeprincipalKcomponents involves these steps:
each of the three principal Kcomponents (k1,k2,k3) were projected
to the corresponding -plane(k1,k2,k3) through trigonometric
transformations. Once k1, k2,k3were calculated the component with minimum
magnitude (usuallyk3) was discarded.
k1, k2 values were then scaled accordingly to tunnel depth,
comparing them to the nearest Lugeon data; scaled k1,k2values
have a mean value of (k1, + k2)/2, which is equal to the closest
Lugeon permeability value, and (k1/k2) = (k1/k2).
FE seepage simulations were then performed: geological sections
were reproduced into nite element software and specic materials
were assigned to different lithologies according to computedk1,k2
vectors.
Phase2 seepage module can only evaluate steady-state ow
conditions and it cannot directly simulate tunnel inow immediately
after excavation. Initial inow was then calculated by means of a
specic conguration of the model: two vertical xed piezometriclevels at the same level as the undisturbed ones were placed close to
tunnel boundaries. With this simulation the FE model reaches the
Fig. 9. Stereonets of Test A and Test B. Straight lines represent ciclographs of average
Set1 and Set2 planes, circles and triangles are the traces of the three principal k1,k2,k3
directions. Calculated k directions perfectly agree with expected results.
Table 2
Values ofk calculated by the model (Test A) and using Eq. (2)
Simulation Eq.(2)
k(m/s) 2.30 102 1.50102 Fig.10. Example stereonet for station S11.PrincipalKdirectionsare plotted withrelative
magnitudes.
Table 3
Values of principal k directions calculated for all geostructural survey stations
Station k1 k2 k3
m/s DD/Dip m/s DD/Dip m/s DD/Dip
S3 1.00 104 35 /13 1.00 104 300/21 1.00106 333/64
S4 3.30 104 5 5/0 2 3.30 104 324/20 4.401013 331/70
S5 1.20 106 331/47 8.40107 134/71 3.80107 57/17
S6 1.61 106 228/78 1.22106 346/6 6.60 107 257/10
S7 1.78 105 77/07 1.17105 339/50 1.80108 353/38
S8 2.04 106 250/73 2.01106 287/14 2.99108 194/10S11 5.30 104 261/8 5. 30 104 169/10 1.00109 29/77
S12 1.16 104 181/71 1.16104 306/11 3.60108 219/15
S13 5.84 108 207/90 5.79108 24/31 6.151010 291/5
S14 8.20 108 123/40 8.12108 65 /38 8.50 1010 357/26
S15 1.16 104 350/15 1.16104 252/27 2.38108 287/58
S19 4.00 106 327/10 3.80106 240/10 2.00107 207/71
S23 1.74 107 109/75 1.47107 83 /13 2.83 108 355/6
S25 6.56 105 293/6 6. 56 105 212/55 4.70107 199/34
S26 1.16 104 352/15 1.15104 254/28 4.61107 287/58
S27 7.24 106 277/25 7.24106 120/63 6.28106 192/09
S28 2.04 106 250/73 2.01106 287/14 2.99108 194/10
S29 1.40 108 325/32 1.30108 321/57 4.801010 234/02
S30 5.10 104 2/78 5. 00 104 335/09 8.13108 245/05
S31 4.25 107 5 9/6 4 4.25 107 276/20 5.221012 181/14
S32 3.76 107 4/ 35 3.75 107 74/26 1.34107 317/43
Values below 109 must be considered as non-permeable.
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steady-stateow condition with a piezometric level over tunnel head
very close to the undisturbed one. The two xed vertical piezometric
levels were placed symmetrically from the boundary of the tunnel
section at a distance equal to the tunnel radius, considering that
during excavation the most relaxed part of rock mass, where water
ow is more relevant, is about one diameter all around tunnel
boundaries. Obviously during excavation the piezometric level above
the tunnel head will not be equal to the undisturbed phreatic level of
the water table, due to the lowering of the phreatic level caused by
tunnel overall drainage. Thus initial inow evaluated in the above
method is referable to maximum theoretical inow at thebeginning of
excavation.
Fig. 11.Finite elements seepage analysis performed on a single section. A) Initial inow immediately after tunnel excavation, B) Steady-state inow with impermeable invert.
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The following FE seepage simulations were performed:
Initial inow at excavation. Long-term steady-state inow with impermeable invert: real long-
term steady-state inow condition. Hypothetical long-term steady-state inow with fully-permeable
tunnel boundaries: this simulation was performed in order to have a
comparison with other standard analytical inow computational
methods.
Fig. 11shows an example of a FE seepage analysis.
The unitary inow values obtained for each section were then
extended for the length of the tunnel with the same geological setting
of the section. The total inow value for each tunnel was obtained by
the sum of inow values calculated for each geologically homo-
geneous lengths of the tunnel path.
This calculation was performed taking into account water table
variability within every geologically homogeneous length. When a
phreatic level within a geologically homogeneous length suffered
important variability, its inuence in the water inow calculation was
taken into account by varying initial phreatic levels in the correspond-
ing section during FE analysis.
Total inow values for tunnels were estimated, for both the initial
and the long-term steady-state conditions.
7. Discussion
Every mathematical model obtains results whose reliability is
directlylinked to input data reliability. For hydrogeological models the
most important parameter is the hydraulic conductivity K.
Kvalues evaluated with the procedure presented in this paper are
more reliable than those derived from Lugeon tests, because they are
calculated from discontinuities properties and they have a dened
spatial orientation, therefore these data were used for the nite
element seepage analysis models.
The tunnel water inow values derived from the nite element
seepage models have been compared to those derived from other
standard analytical methods: The Goodman formula (Goodman et al.,
1965) and the Heuer abacus (Heuer,1995). The Goodman formula is a
very simple mathematical relationship deriving from the Darcy's law:
Q2dKdL
ln
2L
r0
where:
Q unitary ow at steady-state ow (for tunnel meter)
K permeability of homogeneous and isotropic rock mass
L water head above tunnel
r0 tunnel radius
The value ofQrefers to a theoretical inow in an idealized circular
tubeintoa perfectlyhomogeneousand isotropicrockmass.This formula
is very simple and many authors warn about using it in fractured rock
masses because this kind of rocks are neither homogeneous nor
isotropic. In fact the formula usually provides higher Q-values than the
real ones, even by some greater order of magnitude.
Due to the poor reliability of the Goodman formula in fractured
rock masses the Heuer abacus was developed (Fig. 12). It is an
empirical relationship between water inow and equivalentKvalues
obtained fromeld Lugeon tests. It derives from a long experience in
tunnelling even if in a geological framework different from the object
of the present study.
Fig. 12.The Heuer abacus.
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The values of tunnel inows obtained for six sections (-plane)
with FE analysis were compared in Table 4with those calculated with
Goodman formula and Heuer abacus.
Of course these methods arevery differentin their formulation and
they shall not give the same values, anyway some interesting
considerations can be made.
Looking at Figs. 13 and 14 Q-values distributions have the same
trend for all of the three methods. This is an important result because,
excluding the relative magnitudes, it means that there is a global
coherence and therefore methods can be compared even if they are
methodologically very different.Inparticular thetwo simple methodsof GoodmanFormulaand Heuer
abacus have exactly the same trend (Fig.14) while the FE analysis, which
is much more complicated than the others involving a greater number of
variables, is only a bit different. This means that theworkow developed
to perform FE seepagesimulationswasmethodologicallycorrect and that
results are reliable and consistent with the overall behaviour given by
traditional and widely used inow evaluation methods.
As expected, values derived from the Goodman formula are always
higher than the others, even tentimes higher, while values derived from
the Heuer abacus are of the same magnitude as those in the FE analysis.
Values computed with FE analysis can be therefore considered reliable
and not outscaled.
It must be considered that inows derived from F.E analysis are
probably a bit higher than the real inows that will occur in tunnelsbecause:
The FE model assumes a constant inow head in the upstream
boundary of the model. Even if the boundary was kept at the
maximum allowable distance according to the hydrogeological
framework, it is possible that the actual recharge of the acquifer
might not be enough to permanently provide the constant inow
estimated by the model.
In the choice of input data, the higher values were always assumed
in order to be as conservative as possible.
8. Conclusions
In this paper a new practical approach to the evaluation of rock-
mass permeability tensor and the prediction of tunnel inow was
presented.
For therealization of thenew highway from Firenze to Barberino di
Mugello (Tuscany), twelve tunnels are planned to be excavated in the
shaly Sillano Formation (SIL) and in the calcareous Monte Morello
Formation (MML), our target was to predict the water ow into those
tunnels.
The determination of hydraulic conductivity is very difcult for
those kinds of geological formations, because the water ow into the
rock mass is controlled by the discontinuity network, therefore
permeability is high anisotropic and it changes with the variation ofdiscontinuity properties and the geological structure of the formation.
Measured Kvalues resulting from Lugeon tests have shown to be
not fully representative of the mass permeability in fractured and
structurally complex rocks, because LugeonKmvalues can be referred
only to a very local volume and they do not agree with the real
hydrogeological behaviour of rocks. Moreover Km values measured
with Lugeontests arestrongly affected by the lithologyand thedegree
of fracturing of rock masses where tests are performed. Shaly (i.e. SIL)
or highly fractured rock masses (i.e. fault zones in appenninic ysh
formations) cause the water ow between boreholewalls and packers,
with the consequence of an overestimation of the measured Kmvalues.
To evaluate the hydraulic conductivity tensorKof the studied rock
masses, the approach proposed byKiraly (1969, 1978)was used andimproved with the introduction of the effective hydraulic opening (e)
(Barton, 2004a,b). Discontinuities properties for Ktensor calculation
were collected by means of a eld survey in 32 structural survey
stations located according to tunnel paths and geological framework.
Once structural data were collected, principal K directions were
calculated for each geostructural survey station.
Tunnels were then subdivided into geologically homogeneous
lengths, and geological sections of each homogeneous length,
transverse to tunnel paths (-plane) were developed for a total
number of 38 sections. Geological sections were associated to the
nearest and most representative gestructural stations and to the
relative Kvalues, which were then projected to the -plane.
In order to evaluate water ow into tunnels, 2D nite element
seepage analysis (Phase2
, Rocscience Inc.) were performed on the38
Table 4
Inow values obtained by means of FE analysis, Goodman formula and Heuer abacus
Q(l/min/m)
FE analysis Goodman formula Heuer abacus
Section 1 0.23 1.29 0.11
Section 2 0.86 12.09 1.12
Section 3 36.00 307.92 9.04
Section 4 1.60 2.86 0.55
Section 5 0.18 0.86 0.14
Section 6 0.42 1.35 0.17
Fig.13.Distribution ofQ-values fromTable 4. The left side small graph is a enlargement
for a better view of FE analysis and Heuer abacus trends.
Fig. 14. Distribution ofQ-values fromTable 4, logarithmic y-axis scale. Trends of the
three different methods are almost the same.
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geological sections (-plane), using theKvalues calculated for each-plane.
Three kinds of simulation were carried out: initial inow
immediately after the excavation, long-term steady-state inow
with impermeable invert, long-term steady-state inow with fully-
permeable tunnel boundaries.
Inow values given by the FE model were then compared and
validated with inow values given by two classical inow prediction
methods: the Goodman formula (Goodman et al.,1965) and the Heuerabacus (Heuer, 1995).
In conclusion, the workow presentedin this paper resultedto be a
valid approach in the determination of hydraulic conductivity values
in fractured rock masses and in evaluating water ow into tunnels,
because it takes into account the geological variability of the rock
mass, the properties of the discontinuities and the hydrogeological
context.
Acknowledgements
The study presented in this paper was supported by SPEA S.p.A..
The authors are thankful to Prof. F. Rosso for the support in the
mathematical aspects of the study, especially for the development of
the automated script of the software Mathematica ( Wolfram).
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