COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Implementation of a 3D Multilaminated Hydromechanical Model
for Analysis of an Unlined High Pressure Tunnel
N. Schclar Leitão, L. N. Lamas *
LNEC – Laboratório Nacional de Engenharia Civil, Av. do Brasil 101, 1700-066 Lisboa, Portugal
Email: [email protected], [email protected]
Abstract: The purpose of this paper is to report the first stage of the implementation within FLAC3D of a
hydromechanical model using an iterative procedure between two independent hydraulic and mechanical
sub-models. The fracture network of the rock mass is considered using the multilaminated medium concept.
The implemented model is illustrated through its application to the hydraulic circuit of the Venda Nova II
hydroelectric power scheme, in Portugal.
Key words: hydromechanical, permeability, pressure tunnel
INTRODUCTION
In fractured rock masses in the presence of water, the excavation of an underground opening corresponds to
a disturbance of the fracture medium, which introduces a change in the stress field, a change in the water
flow boundary conditions as well as a local damage in the rock mass structure. The changes in the water flow
conditions cause variations in the seepage forces, which are mechanical loadings that introduce changes in
the stress field, thus inducing deformations in the fractured rock mass. In turn, these deformations are
responsible for changes in the rock mass permeability, which will again introduce changes in the water flow
conditions. The existence of this interaction between the hydraulic and the mechanical problems makes it
important to study this problem as an integrated process.
A review of hydromechanical couplings in fractured rock, with special emphasis on hydromechanical
interactions as a result of human activities such as underground injection and underground construction can
be seen in Rutqvist et al. [1].
The purpose of this paper is to report the first stage of the implementation within FLAC3D [2] of a
hydromechanical model using an iterative procedure between two independent hydraulic and mechanical
sub-models. The implemented model was applied to the high pressure hydraulic circuit of the Venda Nova II
hydroelectric power scheme, which has an unlined, high pressure tunnel at great depth.
HYDROMECHANICAL MODEL
The hydromechanical behaviour of fractured rock masses is strongly influenced by the presence of joints. In
fact, the joints are responsible, to a large extent, for the seepage that occurs through the rock mass and they
are also the most sensitive elements of the rock mass with respect to deformation under stress changes. In
order to adequately simulate the behaviour of the joints, it is advantageous to consider them individually, in
an explicit manner.
However, this is only conceptually possible when a limited number of major discontinuities, such as faults or
shear zones, are to be considered. If the rock mass has several joint sets and the critical dimension of the
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structure under analysis is large compared to the spacing of the joints, then it is more appropriate to consider
a constitutive model that accounts for the fabric of the joints. This can be done using the multilaminated
medium concept, which was introduced by Zienkiewicz & Pande [3] for the mechanical behaviour, and
extending it also to the hydraulic behaviour. In this way, the permeability tensor of the rock mass can be
considered as the sum of two terms: one relative to the contribution of the rock material (a strictly continuous
medium), K ijc , and the other relative to the contribution of the total number n of joints sets, K ijs [4]:
K ijec
=
n
s
∑ K ij
s =1
+
K ijc
with
K ijs
g I s (e s ) 3
=
12 ν ω
(1)
where g is the acceleration of gravity, I s is the mean intensity (equal to the inverse of the mean spacing of
the set along the normal to the planes), e s is the mean hydraulic aperture of the set, and ν ω is the kinematic
viscosity of water.
The law proposed by Wei [5] was adopted for representation of the influence of the mechanical behaviour on
the fluid flow in the strictly continuous medium. This law considers that the permeability in each of its
principal directions, Ki, is a function of its initial value along that direction, Koi, and of the sum of the strain
increments in the orthogonal plane, Δεj and Δεk:
K i = K oi e
β i ( Δε j + Δε k )
(i = 1,2,3;
j = 2,3,1; k = 3,1,2)
(2)
where βi (i = 1,2,3) are empirical parameters that, according to Wei, depend on the deformability and on the
shape of the rock pores.
For the equivalent continuum used in this model, the mechanical and hydraulic behaviour for the joint sets
have also to be considered. With the values of the normal stress acting on each of the joint sets, the normal
displacement caused by the normal stress, δ n , is computed and added to the initial aperture of the joint of the
set, Eos , in order to determine the final mechanical aperture:
E s = Eos + δ n
(3)
To relate the hydraulic aperture and the mechanical aperture the law proposed by Elliot et al. [6] is used, but
the term that refers to the residual hydraulic aperture of the model of Witherspoon et al. [7] is also
considered. Thus the law used here can be written as:
s
e s = f E ( E s + eres
)
(4)
s
where f E is Elliot’s hydromechanical coupling parameter, E s is the mechanical aperture and eres
is the
residual aperture.
IMPLEMENTATION OF THE MODEL USING FLAC3D CODE
The equivalent continuous model described above allows the use of a continuum-based modelling package.
For this analysis the FLAC3D code was adopted. FLAC3D is a widely used commercial code that is designed
for rock and soil mechanics and can also handle hydraulic and fully coupled hydromechanical processes.
For the proposed hydromechanical model, based on an iteratively coupled approach, the FLAC3D code is
executed sequentially to model the mechanical and the hydraulic behaviour. Although the FISH
programming language embedded within FLAC3D enables the computation of the strain-induced changes in
permeability, it was decided to do it externally using a FORTRAN program. The use of a more robust
programming language such as FORTRAN will facilitate further developments in the proposed model.
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The information between the FLAC3D and the FORTRAN codes is passed using ASCII files, which can be
read or written by FISH subroutines during FLAC3D execution. The set of commands which control the
running of FLAC3D is given through the “FLAC3D.INI” initialization program which is automatically
accessed upon starting FLAC3D with a run-time FORTRAN function.
The FLAC3D grid is configured for anisotropic fluid flow and the computed permeability tensor, Eq. (1),
is given in terms of its principal values. According to FLAC3D definitions, the hydraulic properties are
given in terms of the mobility coefficient, k, expressed in its principal directions, k1, k2, k3. The mobility
coefficient k (m2/(Pa sec)) is related to the hydraulic permeability K (m/s), given by Eq. (1), by the
following expression, where ρ f is the water density:
k=
K
gρ f
(5)
APLICATION OF THE MODEL TO A HYDRAULIC CIRCUIT
1. Venda Nova II scheme
Venda Nova II scheme is a new hydroelectric upgrading project recently built in the north-western region of
Portugal. Venda Nova II that takes advantage of the 420 m difference in level between two existing
reservoirs established at the beginning of the 1950s and separated by a distance of only 4,500 m. It was built
almost exclusively underground inside the north face of the Cabreira Mountain’s granite rock mass. Its
construction involved several tunnels, covering a total length of about 7.5 km, several vertical and inclined
shafts of about 700 m in total length and two caverns incorporating the power-house complex (Fig. 1).
Figure 1: Venda Nova II overview, where: A = water intake, B = pressure tunnel, C = upper surge chamber,
D = powerhouse caverns, E = tailrace tunnel, F = access adit, G = ventilation tunnel, H = water outlet
The main powerhouse cavern is located in an intermediate position in the hydraulic circuit, at a depth of
about 350 m. The unlined headrace pressure tunnel is 2.8 km long and has a slope of 15%; its section is a
modified circumference of 6.3 m in diameter [8]. The maximum internal water pressure is 4.5 MPa.
The purpose of this application is to simulate the behaviour of the rock mass for the service internal pressure
installed in the pressure tunnel and to compare it with the behaviour observed during the first infilling of the
hydraulic circuit, which took place between November and December 2004.
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2. Finite difference grid
To study the hydromechanical behaviour of the Venda Nova II scheme, a segment of the pressure tunnel,
three access tunnels and the main powerhouse cavern were represented embedded into a 391×438×220 m3
rock block (Fig. 2).
B
A
C
Figure 2: Schematics of the pressure tunnel (A); main powerhouse cavern (B); and access tunnels (C)
The mesh was built using only brick primitives1. At first, the excavation was created using FISH functions to
move the locations of the gridpoints for each primitive to fit the tunnel shapes and the main cavern. Sixteen
primitives were used to fit the tunnels shapes and 59 primitives were used to fit the main cavern shape. Then,
the rock mass surrounding the excavation was represented. This was done first with the bricks at the same
level of the tunnels, and then with the bricks below and above the tunnels level. The creation of the mesh was
thought as a “cut-and-fold process” in order to obtain a mesh as smooth and regular as possible (Fig. 3). In
total 641 brick primitives, 94,352 grid points and 82,254 zones were used. A better explanation of the mesh
generation process can be seen in Leitão et al. [9], although some modification had to be introduced in order
to ensure the stability of the hydraulic analysis.
Figure 3: Grid at tunnels level and main powerhouse cavern
1
Primitive: grid shapes of specific connectivity available in FLAC3D which can be connected and conformed to create complex
three-dimensional geometries.
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3. Mechanical analysis
For the mechanical problem, roller boundary conditions were applied on all sides of the domain except the
top one, where the weight of the overburden was applied. The rock mass was considered as an elastic
material, with elastic modulus E = 30 GPa, Poisson’s ratio ν = 0.15 and unit weight γ = 0.027 MN/m3.
Although it is also possible to use a multilaminated model for the mechanical simulation, its influence was
not considered relevant for this stage of the study and will be implemented in future works.
The analysis was divided into two stages. In the first stage, the model was brought to a pre-excavation stress
state with a vertical stress σzz equal to the self weigh, a horizontal stress parallel to the pressure tunnel
σxx = 2σzz and a horizontal stress normal to the pressure tunnel σyy = σzz. In the second stage, the tunnels and
the cavern were excavated.
The mechanical properties considered in this model resulted from a comprehensive test programme carried
out in the zone of the powerhouse cavern during the design stage.
4. Hydraulic analyses
Three different hydraulic analyses were carried out. In cases 1 and 2 the rock mass was modelled as a
continuous medium, but case 1 considers the hydromechanical coupling parameter β = 0, i.e. a strain
independent permeability, whereas case 2 adopted β = 5,000. An initial permeability coefficient
K = 10-8 m/s was adopted and this is equivalent to an initial mobility coefficient
ko = 1.02 × 10-6 m2/(MPa sec). This corresponds to permeability values typical for sound granitic rock
masses with nearly closed joints.
In case 3, a joint set was considered in order to simulate the main discontinuity surfaces found during
construction (Fig. 4). A dip direction and a dip angle of 75º measured in the global x y z system used in the
FLAC3D model was adopted. The joint set parameters used in the computation were the intensity
I s = 0.5 m-1, the initial aperture E s = 200 μm, the normal stiffness kn = 100 MPa, the residual aperture
s
eres
= 5μm and Elliot’s hydromechanical coupling parameter f E = 1. Additionally, a strain independent
(β = 0) permeability coefficient K = 10-9 m/s (10 times smaller then in case 1) was assumed in order to
account for the rock mass behaviour excluding this particular joint set.
80º
80º
72º
70º
58º
75º
66º
80º
80º
Figure 4: Main discontinuity surfaces
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To determine the boundary conditions it was considered that seepage was established from the pressure
tunnel into the rock mass during the pressurization of the tunnel, thus establishing unconfined flow
conditions. A water pressure of 4.5 MPa was imposed in the pressure tunnel and zero water pressures were
imposed in all other underground openings. Impervious boundaries were adopted for all sides of the domain
except at the bottom, where a pore pressure corresponding to an initial position of the water table at an
elevation of 1 m was considered.
5. Modelling results
In the hydraulic analyses corresponding to cases 2 and 3 the permeability tensor was first calculated from the
results obtained in the mechanical analyses. Figs. 5(a) and 5(b) represent contours of the global component of
the permeability tensor Kzz around the pressure tunnel, obtained in cases 2 and 3. The purpose of these
figures is to illustrate the permeability changes in the domain in the several situations considered. Contour
colours are graded from blue to red (blue is the minimum value and red is the maximum value). In case 2 the
application of Eq. (2) makes the vertical permeability values change from the original 10-8 m/s to
values that are up to 3 times higher and lower. For case 3, the consideration of the joint set increases the
vertical permeability by several orders of magnitude and the effect of the joint orientation is evident.
Kzz min = 0.33 x 10-8 m/sec
Kzz max = 2.96 x 10-8 m/sec
Kzz min = 0.71 x 10-8 m/sec
Kzz max = 1.63 x 10-6 m/sec
(a)
(b)
Figure 5: Global component of the permeability tensor Kzz contours, (a) case 2, (b) case 3
Figs. 6(a), 6(b) and 6(c) illustrate the pore pressure distribution obtained in cases 1, 2 and 3 for the total
domain. The free surface is represented by the line indicated in the scale with a zero pore pressure. Zones of
the domain with an indicated negative pore pressure are outside the hydraulic domain. In case 1 the free
surface has a usual bell shape. In case 2, owing to the variability of the permeability, the calculated free
surface is not so smooth and the hydraulic domain is smaller and has nearly vertical lateral boundaries. There
is a much faster decrease in pore pressure around the pressure tunnel, while in the rest of the domain the pore
pressure variation is very small. In case 3 all the equal-pressure lines (including the free surface) have a
shape that is much influenced by the orientation of the joints. In this case it can be observed that higher
pressures reach further downstream and further elevations then in case 1.
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(a)
(b)
(c)
Figure 6: Pore pressure contours, (a) case 1, (b) case 2, (c) case 3
The pore pressure distributions around the pressure tunnel for the three cases are shown in detail in Fig. 7.
In the strain dependent permeability analysis, case 2, the pore pressure decrease in the radial direction is
much faster then in case 1. This results from the decreased radial permeabilities considered in case 2, which
are a consequence of the increase in the hoop compressive strains caused by the tunnel excavation. For case
3, the large difference between the permeabilities along the direction of the joints (subvertical) and normal to
the joints is clearly observed.
Table 1 indicates, for the three cases that were studied, the calculated flow rates into tunnel G4, tunnel G5
and the main access tunnel. For cases 1 and 2, the values of flow rates into tunnel G5 and the main access
tunnel are negligible and therefore all the water that flows from the pressure tunnel is infiltrated into tunnel
G4. This is the closest tunnel to the pressure tunnel and, as could be expected, functions as a large drain in
the rock mass and collects nearly all the water. The slight decrease in the flow rates from case 1 to case 2 is
justified by the effect of the smaller radial permeability in the strain dependent calculation.
Table 1: Flow rates into the access tunnels
Case
1
2
3
G4
0,20
0.16
9.69
Flow rate (litres/sec)
G5
0.07
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Main access tunnel
0.25
(a)
(b)
(c)
Figure 7: Pore pressure contours, (a) case 1, (b) case 2, (c) case 3
In case 3 the flow rate into the tunnel G4 is approximately 50 times higher than in cases 1 and 2, as a result of
the dominant contribution of the discontinuity set for the overall hydraulic behaviour of the rock mass. The
flow rates into tunnel G5 and into the main access tunnel are no longer negligible, but are several orders of
magnitude smaller.
6. Comparison of the calculation results with the observed behaviour
The first infilling of the hydraulic circuit of the Venda Nova II hydroelectric scheme was done in several
stages. First, the water was allowed into the hydraulic circuit from the tailrace tunnel until it reached the
level of the downstream reservoir. Then, the infilling continued from upstream, in several steps. A large
number of measurements was taken during this process in order to control the safety of the underground
structures and the operating conditions of the hydraulic circuit. Of relevance for the present application of
the numerical model are the total values of the water infiltrations in the underground openings in the vicinity
of the pressure tunnel. Fig. 8 shows the water level in the pressure tunnel and the total measured
infiltrations in tunnels G4 and G5 during the whole process of the first infilling, until they reached stability.
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The measured values of infiltrations in other tunnels and in the powerhouse were negligible when compared
with these two tunnels.
Flow rates (liters/sec)
6
650
Infiltrations tunnel G4
Infiltrations tunnel G5
Water level
600
550
500
450
4
400
350
2
300
250
0
01-Nov
Water Level in the pressure tunnel (m)
700
8
200
16-Nov
01-Dez
16-Dez
31-Dez
15-Jan
30-Jan
Figure 8: Evolution of the flow rates into tunnels G4 and G5 during the first infilling
In the first stages of the first infilling the values of the water infiltrations measured in tunnel G4 were high
and much larger then in tunnel G5. It was observed that the water flowed into the tunnel mainly through a
number of large conducting discontinuities with the orientation represented in Figure 4. In order to control
the water inflow, it was decided to grout some or these discontinuities around tunnel G4. This resulted in the
expected decrease in the water inflow into G4, but in an increase in the water inflow into G5. This means that
the effect of main drain, which in the beginning was being performed by G4, was partially transferred to G5
due to the barriers created to the inflow into G4 by the grouting. Having this in mind, comparisons between
the values of the measured infiltrations and the flow rates calculated with the numerical model can only be
done for the total water infiltrations in G4 + G5, since this transfer of flow from G4 to G5 was not considered
in the model.
The total infiltrations measured in tunnels G4 and G5 at the end of the first infilling was approximately
equal, with a value of around 4 litres/sec, which makes a total of 8 litres/sec in the two tunnels. The slight
flow rate decrease after completion of the first infilling was due to grouting works. In case 3, the total
calculated flow rate into these tunnels was 9.7 litres/sec, which is very close to the measured flow rate, with
a difference of around 20%. In cases 1 and 2, the calculated flow rates are 50 times lower that the measured
ones.
From the results obtained it is clear that consideration of the main conductive joint set, in case 3, was
essential for modelling the hydraulic behaviour of the rock mass. Consideration of this joint set in the model,
with the properties (orientation, intensity, aperture, stiffness) that were assumed, allowed to calculate flow
rates into tunnels G4 and G5, considered together, which match well the measured values.
On the other hand, when the rock mass was considered as an isotropic equivalent continuum, in case 1, with
permeability values typical for sound granitic rock masses, the calculated flow rates were 50 times lower. In
other words, in order to calculate flow rates similar to the measured ones, an isotropic permeability
coefficient K = 5 × 10-7 m/s would have to be considered for the rock mass, which is a clearly large value.
In this application case, the influence of a strain dependent permeability, considered in case 2, did not affect
much the calculated flow rates. However, it clearly changed the pore pressure distribution and the flow
pattern in the rock mass. In a coupled analysis, this would result in seepage forces different from case 1.
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CONCLUSIONS
The evaluation of the hydraulic behaviour of an unlined high pressure tunnel was addressed by numerical
modelling with FLAC3D. The implemented formulation relates permeability with strain for the continuum. It
also allows considering the influence of joint sets in the hydraulic behaviour by means of the multilaminated
medium concept. Relations between the closure of the joints under normal stress and its hydraulic properties
were also implemented.
A large mesh was generated to simulate the granitic rock mass where the Venda Nova II hydraulic circuit
was constructed, including a pressure tunnel, several other access tunnels and the powerhouse.
The 3 different hydraulic calculation cases that were simulated correspond to three different ways to
consider the rock mass permeability: isotropic and strain independent in case 1; strain dependent in case 2;
anisotropic using the multilaminate concept to consider one joint set, with an aperture dependent on the
normal stress, in case 3. Different pore pressure distributions in the domain and flow rates into the openings
were obtained in the three cases.
The results were compared with the measurements obtained during the infilling of the hydraulic circuit. The
formulation implemented in case 3 allowed a reasonable simulation of the flow rates, since it considered the
dominant role played by a hydraulically conductive joint set on the water infiltration into the tunnels.
The work presented in this paper corresponds to the early stages of implementation, within FLAC3D, of a
more complex model of the hydromechanical behaviour of rock masses, using an iterative procedure
between two independent hydraulic and mechanical sub-models and considering the fracture network of the
rock mass using the multilaminated medium concept.
Acknowledgements
The permission by EDP Produção EM to use the data relative to the Venda Nova II project is acknowledged.
This research is partly funded by the Portuguese Foundation for Science and Technology, under the project
POCI/ECM/57495/2004.
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