Analysis of Longyangxia dam deformation based on seepage and creep coupling method.

Haiqing, Guo ; Changcun, Gu ; Weiya, Xu 等

Dam failures are mainly caused by cracks or failures of their foundation rocks, which are directly related to water seepage in the rock. This kind of fluid-rock interaction has an important influence on deformation and stress characters of the dam-rock system. By use of visco-elastic constitutive models and finite element solution method, the stress and seepage fields of foundation rocks are studied as a coupled system in this paper. Using this coupled models, the deformation doubts of the continuous displacement of the 13th dam section of the Longyangxia dam are analyzed and explained reasonably.

INTRODUCTION

The Longyangxia hydropower gravity arch is the main part of the major hydropower generation and water resources utilization systems built along the Yellow River in China. However, continuous displacement towards the left bank at the 13th dam section puzzled both the dam safety administration and engineers. Using the seepage and creep coupling theory and finite element method, this paper represents the results of a research effort devoted to investigate its causes.

The theoretical foundations of the theory and the FEM formulations presented in this paper are based on works in Oda (1986), Ohnishi and Kobayashi (1993), Shen et al. (2000) and Wu et al. (2001).

BASIC EQUATIONS

The balance differential equations expressed by displacements and general water heads can be given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where, G is the shear module, E is the Young's modulus, v is the Poisson's ratio, is the density of water, [r.sub.c] is the saturated density

of the concrete or foundation rock, h is the water head. In addition, [[nabla].sup.2] is the Laplace operator, [[epsilon].sub.v] is the volume strain, [[delta].sub.i](I = x,y,z) are displacement components and [X.sub.0], [Y,sub.0], [Z.sub.0] are equivalent body force components caused by initial strain ([[epsilon].sub.0]}.

To get equation q, we have four hypotheses: a) the dam and its rock base are isotropic continuum media in different areas. b) The seepage follows the Darcy law. c) The grain skeletons' deformations of dam concrete and foundation rock are ignored. d) The deformations of dam concrete and its foundation rock are mainly caused by deformations of void spaces and cracks between grain skeletons mentioned above.

According to the law of mass conservation, the continuity equation of the water is derived as given in equation (2). For simplification, the equation is expressed along the main seepage directions denoted as coordinate axes x , and y z.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [k] is the permeability tensor, whose main components are [k.sub.x], [k.sub.y], and [k.sub.z]; n is the void ratio, and [beta] is the compression parameter of the water.

The basic equations for the coupling analysis of stress and seepage fields are composed of equations (1) and (2), whose boundary conditions include (a) displacement boundary condition {[dela]} = {[[delta].sub.0]} stress boundary condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where k, l = 1,2,3), (c) water head boundary condition h = [h.sub.0], (d) seepage boundary condition -[k.sub.n] [partial derivative]h/[partial derivative]n = [q.sub.0], which should be specified according to the site conditions.

For studying the viscous deformation caused by the creep of rock foundation upon time-dependent loading, different visco-elastic constitutive models are developed to identify the most suitable models and parameters for more accurate simulation of the time-dependent deformation of the dam-foundation system.

The time-dependent deformation of the foundation rock, caused by loadings, is described by a Burgers model, which is composed of a Kelvin model and Maxwell model in series. The partial strain expression of Burgers model is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Specially, when [S.sub.ij] = [S.sub.ij0] is a constant value, and [[??].sub.ij] = 0, the equation (3) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The visco-elastic strain of the Maxwell component will change into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at time [t.sub.0]. If t = [t.sub.0] + [DELTA]t and the stress remain the same during the increment of [DELTA]t, the visco-elastic strain increment of Burgers model can be derived from equation (4) as given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [E.sub.K], [[eta].sub.K], [[eta].sub.M] is the stretch (compression or shear) modulus and viscous parameters and [C] is the Poisson's ratio matrix, respectively.

For the basic equations of coupled stress-flow analysis mentioned above, the finite element method is used to solve the coupled partial differential equations in this paper. The FEM solution scheme for the coupled equations of displacement and seepage fields is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [theta] is the integration parameter, [[bar.X]]] is the general stiffness matrix, [K'] is the general coupling matrix, [S] is the general compression matrix, [[??]] is the general seepage matrix, {[F.sub.0]} is the equivalent nodal load vector by the initial strain, is the equivalent nodal load vector, is the equivalent nodal discharge vector, and [{h}.sub.m+1] is the general water head vector at time [t.sub.m+1], respectively.

If displacement increment {[DELTA][delta]} and super-static water pressure {[DELTA]p} are taken as the unknown quantities, and the full Hermit differences are adopted with [theta] = 1, the equation (6) can be re-written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where {[R.sub.t]} refers to the part of loads balanced by the stresses related to the displacement happened before the time [t.sub.m], and are calculated by the following equation

{[R.sub.t]} = [[t.sub.m].summation over (l = 0)] [[[[bar.X].sub.t]].sub.l][{[DELTA][delta]}.sub.l] (8)

where l is the number of calculating periods of time before the time [t.sub.m]. According to the incremental initial-strain method used in equation (7), the initial stress field and seepage field of dam-foundation system need to be created by iterations starting from the beginning of the time-marching process. The element stress remains unchanged during the subsequent time step [DELTA]t, so that the seepage coefficients, which are decided by the state of stress at the beginning of the time step, remain the same value. They change step by step with loads increasing gradually at each time step. The water heads at the end of every time step are the initial ones of the next time step.

ANALYSIS OF LONGYANGXIA DAM'S DEFORMATION

The main part of Longyangxia dam is a gravity arch dam of 396 meters in length, 29.2/80 meters in width (at the top/base), 2610 meters for DCL and 178 meters in height. The dimension of the FEM model is 540 m in both length and width and 360 m in height. When dividing the FEM mesh, the geological structures in the foundation rock and their engineering treatment measures were taken into account. While laying out the element nodes, the locations of the in situ measuring points of displacement, temperature and stresses are considered. The FEM mesh, which is consisted of 21,189 eight-node-hexahedron elements with 24,873 nodes, is illustrated in Figure 1. The material parameters are listed in Table 1.

[FIGURE 1 OMITTED]

From April 16, 1990 to May 1992, the water level of Longyangxia reservoir dropped from 2575.04 m to 2533.15 m. From May 1992 to December 1994, the water level rose from 2533.15 m to 2577.58 m. From January 1995 to July 1998, the water level dropped from 2577.58 m to 2533.54 m again. After July 1998, the water level rose from 2533.54 m to 2581.08 m. These water level variations were used for deriving the water head loading conditions.

The tangential displacements of Longyangxia dam's typical dam section on April 16, 1989, April 16, 1990, and April 16, 1996, and December 31, 1999 were calculated and compared with measured data. The results are listed in Table 2 and illustrated in Figure 2. It can be seen that: the calculated values are close to the measured ones, and he calculated tangential displacements above the 2500m level on April 16, 1996 are as follows:--3.04mm, -6.91mm, -6.33mm, -5.77mm, and -4.68mm, respectively. These data show that the 13th dam section moved towards the left bank. This was caused by the continuing high water level, about 100m ~ 110m higher than adjacent uplift pressure, and the creep displacements of dam body and rock foundation.

[FIGURE 2 OMITTED]

The calculated results, which agreed well with the measured ones, clearly indicated that the main reason for the continuous displacement of Longyangxia dam towards the left bank after July 1989, was caused by the influence of the seepage-stress combined operations on its rock foundation.

REFERENCES

Oda, M. 1986. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock massed. Water Resource Research 22(13): 1845-1856

Ohnishi, Y. & Kabayashi, A. 1993. Thermal-hydraulic-mechanical coupling analysis of rock mass. In Hudson J. A. (ed), Comprehensive Rock Engineering, Pergamon Press: 191-208

Shen Zhenzhong. Xu Zhiying. & Luo Cui. 2000. Coupled analysis of viscoelasticity stress field and seepage field for the Three Gorges dam's foundation. Engineering Mechanics. 17(1): 105-113.

Wu Zhongru. Gu Chongshi. & Wu Xianghao. 2001. Theory and its applications of safety monitoring of roller concrete dam. Science Press.

GUO HAIQING

Geotechnical Institute, Hohai University, 1, Xikang Road Nanjing, Jiangsu, 210098, China

GU CHANGCUN, XU WEIYA

Geotechnical Institute, Hohai University, 1, Xikang Road Nanjing, Jiangsu, 210098, China

Table 1. Parameters of Longyangxia dam and foundation materials

 Elastic constants

 Density Poisson's
 kg/ Modulus ratio
 [m.sup.3] GPa [micro]

Dam concrete 2400 20 0.18

Rock 2580 m 2400 8 0.25
foundation ~ 2560 m

 2560 m 2650 12 0.23
 ~ 2540 m

 2540 m 2700 16 0.22
 ~ 2500 m

 Below 2750 22 0.22
 2400 m

 [G.sub.4] * 2755 3 0.25

 [F.sub.18] 2600 4.5 0.25

 [F.sub.71],
 [F.sub.73],
 [F.sub.32],
 [F.sub.67] 2600 3.2 0.25

 [A.sub/2] +
 [F.sub.120] 2600 5.4 0.25

 Viscous constants

 [E.sub.K] [[eta].sub.K] [E.sub.M]
 GPa GPa x S GPa

Dam concrete 250 2.3 x 66.7
 [10.sup.5]
Rock 2580 m 50 4.0 x 15.0
foundation ~ 2560 m [10.sup.4]

 2560 m 20 3.6 x 20.0
 ~ 2540 m [10.sup.4]

 2540 m 200 2.8 x 40.0
 ~ 2500 m [10.sup.4]

 Below 300 2.0 x 51.0
 2400 m [10.sup.5]

 [G.sub.4] * 30 5.4 x 10.0
 [10.sup.4]
 [F.sub.18] 40 4.0 x 15.0
 [10.sup.4]
 [F.sub.71],
 [F.sub.73],
 [F.sub.32],
 [F.sub.67] 30 5.4 x 10.0
 [10.sup.4]

 [A.sub/2] +
 [F.sub.120] 40 4.0 x 15.0
 [10.sup.4]

 Seepage
 parameters

 [[eta].sub.M]
 GPa x S cm/s

Dam concrete 1.5 x Dam 1.0 x
 [10.sup.9] [10.sup.-7]
Rock 2580 m 8.5 x Rock 1.0 x
foundation ~ 2560 m [10.sup.9] [10.sup.-9]

 2560 m 5.4 x Rock
 ~ 2540 m [10.sup.9]

 2540 m 5.0 x Crack 1.0 x
 ~ 2500 m [10.sup.9] [10.sup.-4]

 Below 1.8 x Crack
 2400 m [10.sup.9]

 [G.sub.4] * 9.5 x Curtain 1.0 x
 [10.sup.9] [10.sup.-8]
 [F.sub.18] 9.0 x Curtain
 [10.sup.9]
 [F.sub.71],
 [F.sub.73],
 [F.sub.32],
 [F.sub.67] 9.5 x Drainage 1.0 x
 [10.sup.9] [10.sup.-2]

 [A.sub/2] +
 [F.sub.120] 9.0 x Drainage
 [10.sup.9]

Table 2. Calculated values of time-dependent viscous displacements (mm)

Date/
Elevation 1989-4-16 1190-4-16 1996-4-16 1999-12-31

2463.3 m 0.12 -0.28 -1.01 -1.75
2497 m 1.05 -0.44 -2.85 -3.82
2530 m 1.24 -0.79 -4.44 -5.97
2560 m 0.89 -0.61 -3.69 -5.48
2585 m 0.57 -0.36 -1.84 -2.26
2600 m 0.06 -0.31 -4.14 -5.49