Coli Et Al 2008_rock Mass Permeability Tensor

8/12/2019 Coli Et Al 2008_rock Mass Permeability Tensor

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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:[email protected](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:[email protected]://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:[email protected]


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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).

175N. Coli et al. / Engineering Geology 101 (2008) 174184


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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

.

176 N. Coli et al. / Engineering Geology 101 (2008) 174184


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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.



11/11

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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