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: nschclar@lnec.pt, llamas@lnec.pt 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), c

ijK , and the other relative to the contribution of the total number n of joints sets, sijK [4]:

cij

n

s

sij

ecij KKK +∑=

=1 with

ων12)( 3ss

sij

eIgK = (1)

where g is the acceleration of gravity, sI is the mean intensity (equal to the inverse of the mean spacing of the set along the normal to the planes), se is the mean hydraulic aperture of the set, and ων 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:

)3,1,2;2,3,1;1,2,3()( ==== + kjieKK kjioii

εΔεΔβ (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, nδ , is computed and added to the initial aperture of the joint of the

set, soE , in order to determine the final mechanical aperture:

nso

s EE δ+= (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:

)( sres

sE

s eEfe += (4)

where Ef is Elliot’s hydromechanical coupling parameter, sE is the mechanical aperture and srese 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 fρ is the water density:

fgKkρ

= (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).

A

B

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

sI = 0.5 m-1, the initial aperture sE = 200 μm, the normal stiffness kn = 100 MPa, the residual aperture srese = 5μm and Elliot’s hydromechanical coupling parameter Ef = 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.

80º80º

75º

66º

58º

72º70º

80º80º

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.

(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.

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

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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 Flow rate (litres/sec) Case G4 G5 Main access tunnel

1 0,20 - - 2 0.16 - - 3 9.69 0.07 0.25

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(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.

0

2

4

6

8

01-Nov 16-Nov 01-Dez 16-Dez 31-Dez 15-Jan 30-Jan

Flow

rate

s (li

ters

/sec

)

200

250

300

350

400

450

500

550

600

650

700

Wat

er L

evel

in t

he p

ress

ure

tunn

el (

m)

Infiltrations tunnel G4Infiltrations tunnel G5Water level

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. REFERENCES

1. Rutqvist J., Stephansson O. The role of the hydromechanical coupling in fractured rock engineering. Hydrogeology Journal, 2003; 11: 7-40.

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4. Lamas L.N. Contribution to Understanding the Hydromechanical Behaviour of Pressure Tunnels. PhD thesis, University of London, U.K., 1993.

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6. Elliot G.M., Brown E.T., Boodt P.I., Hudson J.A. Hydromechanical behaviour of joints in the Carnmenellis granite, S.W. England. Proc. Int. Symp. on Fundamental of Rock Joints, Björkliden, 1985; 249-258.

7. Witherspoon P.A., Wang J.S.Y., Iwai K., Gale J.E. Vality of cubic law for fluid flow in a deformable rock fracture. Water Resources Res., 1980; 16(6): 1016-1024.

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