1 INTRODUCTION

Personal computers have reached today a state of performance, which allows three-dimensional anal-yses to be performed systematically. User-friendly programs, like Z_Soil (Z_Soil 1985-2000) are avail-able, which allow parametric studies to be per-formed in a matter of hours, while taking into ac-count initial state, construction steps, aging, temperature, transient free surface flow, to mention just a few aspects. While more refined analyses are being undertaken new difficulties appear, which must be taken care of. We dicuss, in this paper, nu-merical aspects of the problem of two-phase consol-idation illustrated by a simulation of an earth-dam construction. Two additional case studies illustrate other aspects of numerical simulations performed with Z_Soil. Recent work performed at LSC-EPFL (Truty et al. 1997) shows that stabilized approaches, already extensively used in fluid mechanics, can be appropriate to overcome locking phenomena in nu-merical simulation of incompressible or dilatant media. At the same time, numerical pressure insta-bilities which are known to occur in simulation of consolidation problems when small time steps are necessary can be overcome similarly (Truty 1999).

Finite element solutions of consolidation problems can exhibit oscillating pore pressures, which tend to increase when the time steps are reduced. With clas-sical formulations, a lower limit for the time step has to be enforced (Vermeer et al. 1981), and this can be restrictive depending on the type of analysis which is being carried out. One aim of stabilized approach-es described in this report is therefore to eliminate spurious oscillations associated with small time steps in consolidation problems.

2 GOVERNING EQUATIONS

2.1 Saturated two-phase medium

The following set of equations applies, • overall equilibrium in terms of the total stress:

0, itot

jij b (1)

where totij is a component of total stresses and ib is

a component of the body loads vector. • effective stress concept after Terzaghi (tensile stresses are positive):

pijijtot

ij (2)

where ij is a component of the effective stress ten-

sor, p is the fluid pressure, ij the Kronecker delta;

and kkI 1 , ijij ssJ2

12 , with ij

kkijijs

3

(with summation over repeated indices), are associ-

ated effective stress invariants. • continuity:

FF

F

kk QpK

nvdiv (3)

Numerical simulations for dam constructions

Thomas Zimmermann, Stéphane Commend Laboratory of Structural and Continuum Mechanics (LSC), DGC-EPFL, CH-1015 Lausanne EPFL

Andrzej Truty Environmental Engineering Dept., Cracow University of Technology, Poland

Jean-Luc Sarf Stucky ingénieurs-conseils SA, Rue du Lac 33, CH-1020 Renens

ABSTRACT: In this paper we discuss practical aspects of 3D numerical analyses in relation with dam con-struction. In addition we propose a novel approach to overcome numerical instabilities in numerical simula-tion of consolidation problems in saturated two-phase media. Three applications illustrate, the first one a case of consolidation during an earth-dam construction, the second one a safety analysis during construction steps of an arch-dam, and the third one a deformation analysis under thermal loads.

where kk denotes the volumetric strain rate of the

solid skeleton, Fvdiv the divergence of the fluid velocity, n the porosity and FK the fluid bulk modu-

lus. • an elastoplastic constitutive relation for solid phase expressed in rate form:

pklkleijklij D with ijjiij uu ,,2

1 (4)

• Darcy's law :

j

Fij

F

i zp

kv,

(5)

• boundary conditions:

tractions on t: tn jij

displacements on u: uu , and = t + u

fluxes on q: qnv iF

i

pore pressures on p: pp , and = q + p

• initial conditions:

00 )( ii uttu

00 )( pttp

Depending on the type of analysis, only equilibrium or equilibrium coupled or uncoupled with continuity will be solved for. Thermal analysis, when applica-ble, is considered uncoupled and is performed as a preprocessing.

2.2 Galerkin least-squares (GLS) formulation for two-phase media

A discretized weak formulation is obtained using a now traditional derivation.

0

)(),,(1

dd

dpR

TeuTeu

epTTTehhhf

twNfwN

pN1BσBwwu

(6)

where B defines the strain-displacement relation in

the usual form:

eh Buuε (7)

The discretized weak form of the fluid flow continu-ity equation can be written:

0

),,(2

dQdq

dK

n

d

dqpR

FT

epT

ep

Tep

F

Tep

FT

ep

hTT

ephhhf

q

qNqN

pNqN

vqN

uε1qNu

(8)

The above equations can be written in a combined

form as:

0

,

,

2

1

)q,p(Rt

),p(R),q,p,(R

hhhf

hhhfhhhhf

u

wuwu

(9)

where fR2 has been premultiplied by a factor t ,

where is an algorithmic integration coefficient

and t the time increment.

2.3 Stabilization technique

The discretized variational formulation modified by a least-squares term now takes the form:

0

,

,

,

2

1

),q,p,(R

)q,p(Rt

),p(R),q,p,(R

hhhhGLSf

hhhf

hhhfhhhhf

wu

u

wuwu

(10)

and it has to be satisfied at each time step. The fol-lowing least-squares term based only on the residu-um of the fluid flow continuity equation is intro-duced:

eFhhhF

Th

kk

N

e

T

hhT

F

h

kk

hhhhGLSf

dQpczp

qcq

),q,p,(R

el

e

kku

kw

wu

1

1

1

*

,

(11)

In the weighting part the term hzk is omitted since

the elevation hz for geometrically linear problems

remains unchanged. The scalar factor * has mean-

ing of a stabilization factor. A more detailed devel-

opment of this stabilization technique can be found

in (Truty 1999). The matrix equations resulting from previous der-

ivations result in a mixed formulation which in-volves displacements and pore pressures as main variables. These field variables are approximated here using quadrilateral elements with linear interpo-lation functions.

3 APPLICATIONS

3.1 Consolidation due to the construction of an earth dam

A 15 ft depth of fill is placed over a large lateral area having the profile shown in Figure 1. This example is based on an actual field case in Lagunillas, Vene-zuela, reported in (Lambe et al. 1969).

Figure 1. Lagunillas field case geometry (Lambe et al. 1969)

The type of analysis is the following: first, an initial

state analysis is performed. Then, the load represent-

ing the fill is applied and the consolidation process is

initiated (tinit. = 0.01, tfinal = 10000). Soil is consid-

ered linear in this analysis. Material data are given in

Table 1.

E [kN/m2] [-] K [m/day] e0 [-]

Silt 6900 0 5 10-1

1.00

Clay 690 0 5 10-5

1.92

d [kN/m3] b [kN/m

3] cv [m

2/day]

Silt 16.34 11.30 2.59

Clay 18.22 14.80 3.46 10-3

Table 1. Material properties

where E is Young’s Modulus, Poisson’s ratio, e0 the initial void ratio, k the permeability, d the dry body weight, b the buoyant body weight and cv the coefficient of consolidation.

Remarks:

1) Because the overlying and the underlying soils are much more permeable than clay, there is double drainage (p = 0 at both the top and the bottom of the mesh) and the considered domain for the analysis will be retricted to the clay layer.

2) Poisson ratio is taken here equal to zero as the

problem is essentially one-dimensional.

3) The whole domain is considered fully saturated, and the water is considered incompressible.

The consolidation process which takes place almost

exclusively in the clay layer has the following ana-

lytical solution for the evolution of the pore pressure

p (Lambe et al. 1969, Z_Soil 1985-2000):

TMm

w eZMMq

tZp 2sin

2),(

0

(12)

with 122

mM

(13)

H

zZ (14)

and 2H

tcT v

(15)

The critical time step after (Vermeer et al. 1981) is:

voed

w

critc

h

kE

ht

22

4

1

4

1 (16)

211

1

EEoed (17)

h is the element size taken here in vertical direction.

is an algorithmic coefficient (here = 1).

In our case, 4.13 critt days.

As the adopted tinit. (= 0.01) is smaller than critt ,

we expect the Standard Galerkin solution for the

pore pressure to oscillate around the exact solution

at an early stage of consolidation, and the GLS

solution to overcome this problem and give the

exact solution (which will be close to q = -96

kN/m2). This is verified in Figure 2.

0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

4.500

-200.000-150.000-100.000-50.0000.000

Pore overpressure [kN/m2]

Z [

m]

t = 0.2 days (GLS) t = 0.2 days (SG)

Figure 2. Pressure for t = 0.2 days, comparison between Stand-

ard Galerkin and Galerkin Least-Squares (GLS) schemes

3.2 Shahid Rajaee dam – Iran, case study 1 (by Stucky Consulting Engineers Ltd)

The Shahid Rajaee dam is a concrete arch structure impounding a reservoir with a capacity of 190 mil-lion m

3. The dam foundation is heterogeneous.

Whereas the dam toe is located on marly sandstone with low compressive strength (c ~12.5 MPa), the flanks are founded on sound sandstone (c ~7.0 MPa) and the upper arches are abuted against lime-stone (c ~85 MPa).

The pronounced heterogeneity of the foundation strength and also its deformability necessitated the simulation of the induced displacements in order to compare them with in situ measures. 3D numerical modelling of the dam and the rock foundation was performed using the Z_SOIL 3D (Z_Soil 1985-2000) finite element program to simulate the con-struction stages (block by block with more than 150 stages). The purpose was also to estimate the safety of the structure against a general shear failure in the marly sandstone.

Figure 3 shows the general layout of the 3D model and the material properties used for numerical simu-lation.

E 2 ( rock) = 7’000 MPa

E 4 ( rock) = 2’500 MPa

E1 (concrete) = 20’000 MPa

E 5 ( rock) = 2’500 MPa

E 3 ( rock) = 5’000 MPa

LEFT BANK

RIGHT BANK

Figure 3. General layout

Figures 4 and 5 represent respectively construction stage after 210 and 485 days.

Figure 4. Construction stage after 210 days

Time = 1162.00 s.

Properties

Figure 5. Construction stage after 210 days

Figure 6 shows the comparison of in situ measured displacements with calculated values for the numeri-cal model.

Shahid Rajaee Dam - Vertical displacements

Block 11 (level ~375 masl)

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

[mm

]

3 m onths

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

Measured

Calculated

Figure 6. Calculated vs. in situ values

Time = 668.00 s.

Properties

3.3 Shahid Rajaee dam – Iran, case study 2 (by Stucky Consulting Engineers Ltd)

In order to perform a rapid control of the behavior of the dam, the measured deflections are compared to the calculated values. These deflections are defined for 2 load types :

1. Water pressure on the upstream face 2. Variation of the concrete temperature with re-

spect to the reference state.

A complete 3D analysis was performed using finite element program Z_SOIL 3D. The first load case studied is the effect of the lake level. The lowest level is chosen at elevation 405. The reservoir level is then raised step by step up to the maximum oper-ating level at 493.00 masl. These steps are shown for the central block 13 in Figure 7.

85 mm

16.5 mm3.7 mm

Water Level : 420 Water Level : 493Water Level : 450

Figure 7. Displacement vectors

Displacements obtained for full lake conditions (lev-

el 493) are shown in Figure 8.

Dir. X : Right bank - Left bankDir. Z : upstream - downstream

8.53e-02

7.58e-02

6.63e-02

5.68e-02

4.73e-02

3.78e-02

2.84e-02

1.89e-02

9.40e-03

-8.7e-05

Time = 8.00 s.

Displacement (Z)

LEFT BANK RIGHT BANK

Displacements Z

+ 85.3 mm

-0.087 mm

LEFT BANK RIGHT BANK

1.07e-02

7.95e-03

5.22e-03

2.50e-03

-2.2e-04

-2.9e-03

-5.7e-03

-8.4e-03

-1.1e-02

-1.4e-02

Time = 8.00 s.

Displacement (X)+ 10.7 mm

-14 mm

Water Level : 493

Displacements X

Figure 8. Displacements X and Y

The second load case is considered in order to calcu-

late the displacements due to the effect of the tem-

perature variations of the dam body. A unit tempera-

ture increase is applied to each horizontal arch of the

dam and its influence at different elevations is calcu-

lated. This is shown in Figure 9.

1.00e+00

8.89e-01

7.78e-01

6.67e-01

5.56e-01

4.44e-01

3.33e-01

2.22e-01

1.11e-01

0.00e+00

Time = 2.00 s.

Temperature

1°C

1°C

0°C

1°C

1°C

0°C

1.00e+00

8.89e-01

7.78e-01

6.67e-01

5.56e-01

4.44e-01

3.33e-01

2.22e-01

1.11e-01

0.00e+00

Time = 3.00 s.

Temperature

0°C

1°C

0°C

0°C

1°C

0°C1.00e+00

8.89e-01

7.78e-01

6.67e-01

5.56e-01

4.44e-01

3.33e-01

2.22e-01

1.11e-01

0.00e+00

Time = 4.00 s.

Temperature

0°C

1°C

0°C

0°C

1°C

0°C

ARCH 480 ARCH 465 ARCH 450

Figure 9. Temperature distribution

Unit temperature increase is applied to each of the 7 arch levels considered (480, 465, 450, 435, 420, 405 and 390). This temperature variation is applied to the whole length of the arch, say i. These 7 temperature loadings enable the radial deflections at the points where plumbline stations are located to be deter-mined. At each level of the different plumbline sec-tions corresponding displacements (a1 to a7) are ob-tained. In order to compute the actual effect of a temperature difference in the dam body, it is then sufficient to determine the mean temperature differ-ence Ti at levels given previously and to compute displacements at a given point j as:

7711 ..... TaTa jjj (18)

(j representing levels 493, 483, 465, 444, 423, 402 and 384)

Calculated displacements have been determined for the upstream / downstream (US / DS) direction.

Measured displacements are then compared to the calculated curves on graphs as functions of the lake level (fig. 10).

4 CONCLUSION

In this paper we illustrate the usefulness of three-dimensional nonlinear finite-element analysis for actu-al dam structures. In addition, the potential of Galerkin Least-Squares based finite element formulations to overcome numerical oscillations in numerical simula-tion of consolidation problems is demonstrated.

4.01

6.11

8.47

11.13

14.07

17.33

20.91

24.81

29.06

33.67

38.65

44.01

49.77

53.42

Reference reading : 03.08.1997 / lake : 416.12

Reserv

oir

Level

370

380

390

400

410

420

430

440

450

460

470

480

490

-40 -30 -20 -10 0 10 20 30 40 50 60 70 80

US (-) DS (+) Displacements in mm

12.04.1997

01.04.1998

Econcrete=20'000 N/mm2

concrete=0.77x10-5

Elevation 444

PLUMBLINE P13 (U/S-D/S)(Reference point at 300.00)

CENTER

Measured displacement : dm

dwater+dtemp

Calculated displ. : dwater

04.11.1998

Figure 10. Displacement as function of lake level (plumbline P9, elevation 483, left bank)

REFERENCES

Lambe T.W., Whitman R.V., 1969. Soil Mechanics. Wiley & Sons Inc.

Truty A., Zimmermann Th., 1997. A robust formulation for FE-analysis of elastoplastic media. Numerical Models in Geomechanics (NUMOG VI), ed. Pande & Pietruszczak, pp. 381-386

Truty A., 1999. A stabilized FE-formulation for fully satu-rated two-phase media. LSC Internal report 99/01.

Vermeer P.A., Verruijt A, 1981. An accuracy condition for consolidation by finite elements. International Journal For Numerical And Analytical Methods in Geomechanics, 5, pp. 1-14.

Z_Soil.PC User Manual, 1985-2000. Zace Services Ltd., Lau-sanne, Switzerland.