Three-dimensional models of reservoir sediment and effects on the seismic response of arch dams

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2004; 33:1103–1123 (DOI: 10.1002/eqe.392)

Three-dimensional models of reservoir sediment and e�ectson the seismic response of arch dams

O. Maeso1, J. J. Azn�arez1 and J. Dom��nguez2;∗;†

1Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num�ericas en Ingenier��a (IUSIANI);Universidad de Las Palmas de Gran Canaria; Campus Universitario de Ta�ra;

35017 Las Palmas de Gran Canaria; Spain2Escuela Superior de Ingenieros; Universidad de Sevilla; Camino de los Descubrimientos s/n;

41092 Sevilla; Spain

SUMMARY

The important e�ects of bottom sediments on the seismic response of arch dams are studied in thispaper. To do so, a three-dimensional boundary element model is used. It includes the water reservoiras a compressible �uid, the dam and unbounded foundation rock as viscoelastic solids, and the bottomsediment as a two-phase poroelastic domain with dynamic behaviour described by Biot’s equations.Dynamic interaction among all those regions, local topography and travelling wave e�ects are taken intoaccount. The results obtained show the important in�uence of sediment compressibility and permeabilityon the seismic response. The former is associated with a general change of the system response whereasthe permeability has a signi�cant in�uence on damping at resonance peaks. The analysis is carried outin the frequency domain considering time harmonic excitation due to P and S plane waves. The time-domain results obtained by using the Fourier transform for a given earthquake accelerogram are alsoshown. The possibility of using simpli�ed models to represent the bottom sediment e�ects is discussedin the paper. Two alternative models for porous sediment are tested. Simpli�ed models are shown to beable to reproduce the e�ects of porous sediments except for very high permeability values. Copyright? 2004 John Wiley & Sons, Ltd.

KEY WORDS: arch dams; seismic response; soil–structure interaction; bottom sediments; wavepropagation; boundary element method

INTRODUCTION

Among the di�erent factors having in�uence on the seismic response of concrete dams, thoserelated to hydrodynamic pressure on the dam upstream face are particularly important. Watercompressibility, reservoir geometry, foundation rock properties and bottom sediments play a

∗Correspondence to: J. Dom��nguez, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de losDescubrimientos s/n, 41092 Sevilla, Spain.

†E-mail: [email protected]

Contract=grant sponsor: Ministerio de Ciencia y Tecnolog��a, Spain; contract=grant number: DPI2001-2377-CO2-01=02

Received 12 September 2003Revised 10 February 2004

Copyright ? 2004 John Wiley & Sons, Ltd. Accepted 12 March 2004

1104 O. MAESO, J. J. AZN �AREZ AND J. DOM�INGUEZ

key role on the seismic behaviour. The importance of these factors has been analysed indi�erent papers by Hall and Chopra [1, 2]; Fenves and Chopra [3]; Fok and Chopra [4]; Tanand Chopra [5]; Dom��nguez and Maeso [6] and Dom��nguez et al. [7].The in�uence of bottom sediments is twofold: on the one hand, they change the actual

water reservoir geometry; on the other, they absorb the energy of the hydrodynamic wavesand therefore increase damping in the dam–reservoir–foundation system. Sediment e�ects onhydrodynamic pressure were analysed �rst by Cheng [8] who showed the in�uence of theircompressibility, highly dependent on the presence of undissolvable gases. Sediment e�ectson dam response are also taken into account in an indirect way by Fenves and Chopra [3]and Fok and Chopra [9] for 2D gravity dams and 3D arch dams, respectively. They use are�ection coe�cient �, based on the one-dimensional wave propagation theory, which is eval-uated from the bottom properties. Zhang et al. [10] have recently developed a 1D modelto simulate the re�ection behaviour of the sediment for � coe�cient evaluation. Bougachaand Tassoulas [11] and Dom��nguez et al. [7] studied the e�ects of sediments on gravitydam response using coupled 2D models where the sediment is a Biot’s poroelastic materialand water–sediment and foundation–sediment interaction are rigorously considered using 2Dequilibrium and compatibility conditions. Those authors concluded that bottom sediments canchange the dynamic behaviour of the system to a signi�cant extent, in particular when theyare partially saturated. No previous studies of this type exist for 3D arch dams.In the present paper a three-dimensional model for the analysis of sediment e�ects on the

seismic response of arch dams is presented. It is based on previous 3D Boundary Elementmodels developed by the authors for the seismic study of arch dams [12, 13], and on the 2Dmodel presented in Reference [7] for the study of sediment e�ects on gravity dams. All theregions of the dam–reservoir–sediment–foundation system are represented by boundary integralequations discretized into boundary elements. Dam and foundation rock are viscoelastic solids,water is a compressible �uid and the bottom sediment is a two-phase �uid-saturated porousmaterial whose behaviour is represented by Biot’s theory [14]. The integral equations fordynamic behaviour of this material were presented by Dom��nguez [15] and Cheng et al. [16],in slightly di�erent form. Interaction between di�erent materials is accounted for rigorouslyby setting equilibrium and compatibility conditions on interfaces. The model includes otherimportant e�ects such as foundation rock �exibility and space distribution of the excitation.The present technique can accommodate general geometries of the sediment region, irregularsurface and bottom topography, and di�erent foundation rock zones.The introduction of a two-phase poroelastic region in the model has a high computational

cost when the region is large and is in contact with viscoelastic regions (foundation rock anddam). The number of elements is large due to the existence of a very short length secondP-wave (Biot’s wave) in the porous material. To improve the computational e�ciency, twosimpli�ed models for the porous domain will also be tested within the 3D coupled system.In the �rst one, the sediment is represented as a viscoelastic solid with frequency-dependentcomplex-valued P- and S-wave velocity equal to those of the largest P-wave and S-wavevelocity, respectively, of the porous medium. In the second, the sediment is a compressiblescalar domain with a frequency-dependent complex-valued wave velocity equal to that of thelargest wave of the porous material.A dam–reservoir–sediment–foundation rock system with a realistic geometry is studied in

the present paper. The Morrow Point dam has been chosen for the analysis since it is wellknown from previous studies. Its geometry is taken from Hall and Chopra [2]. Fok and

Copyright ? 2004 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2004; 33:1103–1123

EFFECTS OF SEDIMENTS ON SEISMIC RESPONSE OF ARCH DAMS 1105

Chopra [9] carried out a FE analysis of this dam including bottom absorption by means of aone-dimensional wave absorption coe�cient. The frequency response of this dam to impingingwaves was also studied by the authors using 3D BE models [6, 12, 13]. Bottom sediment e�ectswere not considered in those studies. To the best of our knowledge an analysis of bottomsediment e�ects, including a proper analytical or numerical model of this region as a Biot’sporoelastic material, has never been done before for a 3D arch dam. The present study iscarried out in the frequency domain and the system response to a time excitation obtained bya Fourier transform.The main objectives of this study are: �rst, present a coupled 3D boundary element model

able to represent all the regions of the problem and the important dynamic interaction phe-nomena existing between them; second, analyse the e�ects of bottom sediments on the seismicresponse of 3D arch dams for di�erent reservoir geometries and sediment properties; and third,study the possibility of using simpli�ed models for the sediment region in this context.

BOUNDARY ELEMENT MODEL

Morrow Point dam is a quasi-symmetric 142 m high arch dam that is assumed symmetricfor the present study. The canyon and surface topography are also symmetric. Thus, onlyone half of the problem is discretized into boundary elements. Data related to the dam andreservoir geometry can be found in Reference [2]. A porous sediment bottom layer with adepth equal to 20% of the maximum dam height H , and extending in the upstream directionup to a distance of 172 m from the dam is assumed. The reservoir is full of water and theresponse compared to that of the case where there is no bottom sediment. The boundaryelement discretization of the dam–water–sediment–foundation system is shown in Figure 1,where the di�erent regions of the system, interfaces and external boundaries are also shown.All the elements in the model are nine-node quadratic quadrilateral or six-node quadratictriangular elements.The dam concrete is assumed to be a linear isotropic viscoelastic material with the following

properties: density �d = 2481:5kg=m3, Poisson’s ratio �d = 0:2, shear modulus Gd = 11500MPaand internal damping ratio �d = 0:05. The foundation rock is also assumed to be linearviscoelastic with the same shear modulus, Poisson’s ratio and damping ratio as the damconcrete and density �f = 2641:65 kg=m3. Water is assumed to be an inviscid compressible�uid under small amplitude motion with density �w =1000 kg=m3 and pressure wave veloc-ity cw =1438 m=s. In the �rst part of the study, the reservoir is assumed to extend fromthe end of the BE model to in�nity as a water channel with uniform cross-section [6]. Thebottom sediment is a two-phase porous material with the following properties (taken fromReference [11]): porosity �=0:6, shear modulus of the solid skeleton Gs = 7:7037× 106N=m2,Poisson’s ratio �s = 0:35, internal damping ratio of the solid skeleton �s = 0:05, solid mate-rial density �s = 2640 kg=m3, pore water density �w =1000 kg=m3, added density �a = 0, anddissipation constant b=3:5316× 106 Ns=m4 (corresponding to a permeability k=10−3 m=s).Two di�erent saturation conditions are assumed: one is fully saturated sediment (Biot’s con-stants Q=8:2944× 108 N=m2 and R=1:24416× 109 N=m2, obtained from a �uid compress-ibility Kf = 2:0736× 109 N=m2); the other is a partially saturated sediment with a satura-tion degree s=0:995, and Biot’s constants Q=8:9328× 107N=m2 and R=1:3399× 108N=m2;



Figure 1. Boundary element model for coupled dam–water–sediment–foundation rock system.



corresponding to an equivalent �uid compressibility K ′f obtained according to [17]:

1K ′f=1Kf+1− spo

(1)

where po is the hydrostatic pressure of the mean depth of the sediment layer (127:8 m).Boundary integral equations for time harmonic behaviour are written for the four di�erent

regions shown in Figure 1: foundation rock, dam, water and porous sediment. The seismicexcitation is assumed to be time-harmonic plane waves impinging the model from in�nity.The wave �eld in the foundation rock (F), displacement uFT and tractions p

FT, consists of two

parts: the known incident �eld (uFI ; pFI ), and the unknown scattered �eld (u

FS; p

FS). Thus, the

total �eld in the foundation rock can be written as: uFT = uFI +u

FS, p

FT = p

FI +p

FS; whereas the total

�eld for all other regions coincides with the scattered �eld: uDT = uDS , p

DT = p

DS for the dam

(D),pWT = p

WS , U

WnT =U

WnS for the water

(W) and uST = uSS, p

ST = p

SS for the porous sediment

(S). The BEintegral equations for the foundation rock are written for the scattered �eld. Thus, radiationconditions are satis�ed and the BE discretization can be truncated at a rather small distancefrom the zone of interest. In the present case, the foundation rock free surface discretizationextends to a distance equal to 2.5 times the dam height. This distance has been revealedas very adequate in previous arch dam studies and other dynamic soil–structure interactionproblems [6, 12].The equations for the Boundary Element Method solution of the problem written in matrix

notation are:

Foundation rock HFuFT −GFpFT =HFuFI −GFpFIArch dam HDuDT −GDpDT = 0Water reservoir HWpWT −GWUWnT = 0Bottom sediments HSuST −GSpST = 0

(2)

where uFT, pFT, u

DT , p

DT are vectors containing displacement and traction nodal values of the

foundation rock and the dam, respectively; pWT , UWnT contain water hydrodynamic pressure and

normal displacement nodal values, respectively; uST is a vector containing nodal values of thethree components of the solid displacement plus the �uid equivalent stresses, and pST is avector containing nodal values of the three traction components at the solid plus the normal�uid displacement. The elements of the system matrices HF, GF, HD, GD, HW, GW, HS, GS

are obtained by integration over the BE of the time harmonic fundamental solution for eachkind of material, times the corresponding shape function. Details of the BE formulation fortime harmonic �uid, viscoeastic and poroelastic regions can be found in Reference [18].There are six interfaces in the boundary element discretization shown in Figure 1: dam–

water, dam–foundation rock, dam–sediment, foundation rock–sediment, water–sediment andwater–foundation rock. Boundary element equations are written at interface nodes twice perdegree of freedom; i.e. once for each region shearing the node. Coupling conditions at theseinterfaces are imposed by enforcing equilibrium of tractions (pressure) and displacementcompatibility in the following way.



Poroelastic material–viscoelastic solid interface

The equilibrium and compatibility conditions between a two-phase poroelastic region (denotedby superscript p) and an impervious viscoelastic solid (denoted by superscript s) are thefollowing:

Equilibrium:t sn + (t

pn + �)=0

tst + tpt = 0

(3)

where the subscripts n and t stand for one normal and two tangential components, respectively.

Compatibility:u sn = u

pn =U

pn

ust = upt

(4)

Poroelastic material–water interface

Equilibrium:

pw =pp

t pn + (1− �p)pw =0tpt = 0

(5)

The second one can also be written as:

t pn + (1− �p)pp = t′pn = 0 (6)

where t′pn is the normal component of Terzaghi’s e�ective stress.

Compatibility: Uwn = (1− �)upn + �U p

n (7)

Viscoelastic solid–water interface

Equilibrium:t sn + p

w =0

tst = 0(8)

Compatibility: u sn =Uwn (9)

Viscoelastic solid–viscoelastic solid interface

Equilibrium:ts1n + t

s2n = 0

ts1t + ts2t = 0

(10)

Compatibility:us1n = u

s2n

us1t = us2t

(11)



The closed-form half-space time harmonic fundamental solution is used for the �uid domain.Thus, the water-free surface zero-pressure condition is automatically satis�ed and only theinterface boundaries have to be discretized: dam–water, foundation rock–water and sediment–water. The geometry shown in Figure 1 corresponds to the case of a very long water reservoir.In such a case, the boundary of a zone close to the dam, which can be rather extensiveand irregular, is discretized into elements and the rest of the reservoir is assumed to be auniform section in�nite channel. A closing boundary taking into account the hydrodynamicwave radiation is located at that point [6]. In the following, the term ‘open reservoir’ willbe used for this type of unbounded water domain. However, when the reservoir geometryis not very long in the upstream direction or the depth becomes small at a not very largedistance, the whole �nite water domain including all signi�cant reservoir boundaries will bediscretized. Reservoirs of this type will be called ‘closed reservoir’.The size of the elements used in the discretization is determined by the wavelength in each

material; i.e., by its mechanical properties and the highest frequency analysed. In the presentmodel the smallest elements are at the boundary of the poroelastic sediment (10 m long),whereas elements in boundaries which belong only to viscoelastic and=or water regions are,at least, four times larger (Figure 1). The use of non-conforming elements simpli�es the meshde�nition. A frequency range between zero and four times the dam’s �rst natural frequency,has been analysed. In cases where simpli�ed models are used for the porous sediment, the sizeof the smallest elements can be increased to reduce the total number of degrees of freedomof the model.For the sake of shortness only excitations consisting of vertical P- or S-waves will be

considered. They produce vertical (P-wave), upstream or cross-stream (S-wave) free-�eldmotion. Other wave propagation directions and other types of excitations such as Rayleighwaves, can be studied without additional di�culties as was done in Reference [13].

NUMERICAL STUDIES

In order to analyse the bottom sediment e�ects as independently as possible from other fac-tors, a dam on rigid foundation rock will be assumed �rst. Thus, dam–foundation interactionand travelling wave e�ects do not exist and the in�uence of sediment layers with di�erentproperties can be analysed. The use of simpli�ed models will be tested in this case. A morerealistic dam–reservoir system on compliant foundation rock will be studied in another sectionof this paper where unbounded and �nite reservoir geometries will be considered. To end thenumerical analyses and in order to determine the in�uence of sediments on the system timeresponse, the response to a given earthquake for full reservoir conditions, with and withoutbottom sediment, will be studied.

Response quantities represented

The amplitudes of the complex-valued frequency-response functions are represented. They areacceleration components of two points located at the dam crest on the upstream face of thedam due to harmonic P- or S-waves coming vertically from in�nity and producing a unitground acceleration at points of the horizontal free surface far from the canyon. The case ofthe vertical P-wave is denoted as vertical excitation in the �gures, whereas the vertical S-wave



cases are denoted as upstream or cross-stream excitation depending on the direction of thefree-�eld motion. The upstream acceleration of the point located on the plane of symmetryis shown for the vertical and upstream excitations. For the cross-stream excitation case, theupstream acceleration of the point located at an angle 13:25◦ from the plane of symmetry isshown.In each case the amplitude of the acceleration is plotted versus the dimensionless frequency

for a signi�cant range. The dimensionless frequency for vertical and upstream excitation isde�ned as !=!s1, where ! is the excitation frequency and !s1 is the fundamental resonantfrequency of the dam-on-rigid-foundation and empty-reservoir conditions for a symmetricmode. For the cross-stream excitation cases, the dimensionless frequency is de�ned as !=!a1,where !a1 is the fundamental resonant frequency of the dam-on-rigid-foundation and empty-reservoir conditions for an antisymmetric mode.

DAM ON RIGID FOUNDATION

The BE discretization for this case is as the one shown in Figure 1 except for the soilfree-surface and the downstream part of the canyon which do not have to be discretized.The excitation is an in-phase motion of the rigid foundation rock in vertical, upstream orcross-stream direction.

In�uence of sediment saturation degree

The variation of the pore-�uid compressibility produces changes of di�erent signi�cance onthe three wave propagation velocities of the porous medium. Figure 2 shows these changesfor the three types of waves, saturation degree from 100% to 99%, and an angular frequencycorresponding to the �rst natural frequency of the dam. The saturation degree clearly modi�esthe modulus of the P1 wave velocity but not that of the other two waves (P2 and S).Figure 3 shows the dam crest upstream ampli�cation for a dimensionless frequency range

going from zero to four. Three di�erent situations are represented: (1) full reservoir and no bot-tom sediment; (2) full reservoir and fully saturated bottom sediment layer; and(3) full reservoir and partially saturated (99.5%) bottom sediment layer. The sediment layer

Figure 2. Wave propagation velocity amplitudes in the sediment versus degree of saturation.



Figure 3. In�uence of sediment saturation degree. Upstream response at dam crest to: (a) upstream;(b) vertical; and (c) cross-stream excitation. Rigid foundation rock.

has a thickness of 0:2H =28:4 m. It can be seen from the �gure that the existence of a fullysaturated sediment has very little in�uence on the dam response. Only a very small increasein the system �rst resonant frequency and small changes in the response in the upper part ofthe frequency range are observed. The e�ect of having a fully saturated sediment instead ofwater at the bottom of the reservoir is very small in this case. More signi�cant changes occurin the system response when a partially saturated sediment exists at the bottom. The response



Figure 4. Wave propagation velocity amplitudes in the sediment versus permeability.

is clearly changed by the existence of a sediment layer of this type as shown in Figure 3.The amplitude and position of the resonance peaks are modi�ed. It should be concluded thatsediment compressibility must be carefully evaluated. The signi�cant in�uence of partiallysaturated sediments was also observed in 2D models for concrete gravity dams by Bougachaand Tassoulas [11] and Dom��nguez et al. [7]. In the following a 99.5% saturation degree isassumed for all partially saturated sediments.

In�uence of sediment permeability

A brief analysis of sediment permeability e�ects on the characteristics of the waves in thesediment is done �rst. Figure 4 shows P1-, P2- and S-wave propagation velocity versuspermeability k. Changes of the order of 20% exist in P1- and S-wave velocities for the shownpermeability range. Wave velocity amplitude variation for two di�erent saturation degrees isshown. The short wave velocity (P2) presents the most important variation with permeability.It can be seen from the �gure that this velocity grows very fast for permeabilities between5× 10−3 m=s and 5× 10−1 m=s; the P2-wave velocity being larger than the S-wave velocityfor values greater than 5× 10−2 m=s. The wave velocity variations shown in Figure 4 havebeen obtained for a frequency equal to the �rst natural frequency of the dam. Results forother frequencies are similar. Little in�uence of the sediment permeability can be expectedfor values of k below 10−3.Next, the in�uence of permeability of partially saturated sediments on the system response

is evaluated. It can be expected that the changes observed in sediment P2-wave velocity in-�uence the system response. Two limit values of permeability k=10−3 m=s and k=10 m=sare assumed. Dam crest ampli�cation for vertical excitation is shown in Figure 5. It can beobserved that an increase in permeability reduces the sediment dissipation e�ect and con-sequently the damping of the system. This in�uence is particularly important for the �rstresonance peak. Similar conclusions can be drawn for upstream and cross-stream excitation.Permeability e�ects on the system response for fully saturated sediments are less importantthan for partially saturated sediments, since the relative importance of P2-waves is smaller inthat case.



Figure 5. In�uence of sediment permeability. Upstream response at dam crest to: (a) upstream;and (b) vertical excitation. Rigid foundation rock.

Simpli�ed sediment models

The use of Biot’s poroelastic model within a boundary element context is very adequate forthe study of bottom sediment e�ects on the seismic response of concrete dams. There are,however, computational e�ciency reasons to explore the possible use of simpler models torepresent the sediment behaviour and its e�ects on dam response. The main drawback oflarge sediment regions discretization is that a small element size, as compared to the wateror viscoelastic region elements, is required for the analysis. In addition to that, the vectorand tensor components are four per node instead of three or one as in viscoelastic or scalarregions, respectively. Small elements are needed because wavelengths are small in the porousdomain; in particular the P2-wave is very short. In order to reduce the number of elementsand unknowns of the problem two alternative models are tested in the following. The use ofthese models will simplify the task of running realistic 3D dam–reservoir–sediment modelson a desktop computer.A simple idea is representing the sediment as a scalar medium able to transmit pres-

sure waves in which velocity is complex valued and frequency-dependent, equal to the P1-wave velocity of the two-phase porous material (this model will be called ‘Scalar-1’). Thesediment is a dissipative compressible �uid where wave velocity is obtained for each fre-quency from porous material properties. Preliminary results obtained with this model forsimpler dam–reservoir geometries showed that the resonant frequencies were accurately



Figure 6. Simpli�ed models for porous sediment. Scalar models. Response at dam crest to upstreamexcitation. Rigid foundation rock.

represented but not the dissipation at those frequencies. The system response is under-damped.A more useful scalar model can be obtained by increasing material damping. To do so,

the imaginary part of the pressure-wave velocity is increased. The equivalent scalar medium(‘Scalar-2’ model) has a frequency-dependent pressure-wave velocity with a real part equalto that of the P1-wave velocity of the porous material, and an imaginary part which has beenchosen as equal to 5% of the real part. The increase in the imaginary part is intended toreproduce the energy dissipation e�ect associated with the porous material. The 5% valuewas set by comparison of some preliminary results with those obtained with the two-phaseporous material representation.Figure 6 shows the dam crest ampli�cation for upstream excitation and three sediment mod-

els: two-phase poroelastic, Scalar-1 and Scalar-2. Sediment permeability is set to k=10−3m=s.Other properties are as indicated above. It can be observed in Figure 6(a) that the threemodels give very similar results for fully saturated sediments; the response being very sim-ilar to that of the non-sediment case (Figure 3). In the partially saturated sediment case(Figure 6(b)), the e�ect of sediment is important and it is very well represented not only bythe two-phase poroelastic model but also by the Scalar-2 model. The simpler Scalar-1 modelyields an under-damped solution with spurious oscillations. Very similar results are obtainedfor vertical and cross-stream excitations.



It can be concluded that porous sediment can be represented, in this type of problems andfrequency range, by a scalar model. However, this model does not represent adequately thebottom sediment absorption e�ect unless the imaginary part of its frequency-dependent wavevelocity is increased to take into account the actual dissipation in the porous material. The useof a scalar model for the sediment layer improves the numerical e�ciency of the BE analysissince each node contains only one degree of freedom instead of four as in the poroelasticmodel. The minimum element size is also signi�cantly bigger for the scalar model than forthe poroelastic model.An equivalent frequency-dependent viscoelastic solid can also be assumed to represent the

poroelastic material behaviour. This model is based on the simple idea of considering thesediment as a viscoelastic solid with frequency-dependent properties. The dynamic proper-ties of the equivalent solid are de�ned by the porous sediment density and the two wavepropagation velocities corresponding to the P1- and S-wave velocities of the porous mate-rial. The two wave velocities are complex valued and frequency dependent. Their values arecomputed for the porous material at each frequency of the analysis. Thus the model accu-rately represents the e�ects of the P1- and S-waves and ignores the P2-waves. Bardet [19]presented a Kelvin–Voigt viscoelastic model with linear frequency dependency of dissipation.Instead of the frequency dependency, the present model is based on the actual values of realand imaginary (damping) parts of the wave velocities, computed at each frequency for theporous material. Results have been obtained for fully and partially saturated sediments, withpermeability values of 10−3 m=s and 10 m=s.In the following, this model will be tested for the dam–reservoir–sediment system. Figure 7

shows the dam crest ampli�cation for upstream excitation of the system. It can be seen fromFigure 7(a) that results obtained with this equivalent viscoelastic sediment for a permeabilityvalue k=10−3 m=s are almost identical to those obtained with the two-phase poroelastic sedi-ment model. A similar behaviour can be observed for vertical and cross-stream excitation. Aslong as the permeability is not very high (k6 5× 10−2 m=s) a frequency-dependent viscoelas-tic model is a good alternative for porous sediment representation. It has the advantage overthe Scalar-2 model in that material properties are directly obtained from the porous materialproperties (no damping assessment is required), but it has the drawback that the number ofunknowns is not reduced as much as in the scalar case.It is worth remembering that the use of simpli�ed models (scalar or viscoelastic) for the

sediment is adequate for low and moderate permeability values. Sediments with very highpermeability must be represented by a porous material model since P2-waves play an impor-tant role in those cases and must be taken into account. Figure 7(b) shows some discrepanciesbetween results obtained with a two-phase poroelastic material and the approximate viscoelas-tic model for a sediment with extremely high permeability (k=10m=s). A similar conclusionwas reached by Bardet [19]. Results presented in the following correspond to a permeabilityvalue of k=10−3 m=s.

DAM ON COMPLIANT FOUNDATION ROCK

Foundation rock compliance introduces two basic di�erences on the dam–reservoir behaviour[6, 12]: (1) the existence of dam–foundation rock, water–foundation rock and sediment–foundation rock interactions, and (2) the space distribution of the excitation. As a consequence



Figure 7. Simpli�ed models for porous sediment. Viscoelastic model. Response at dam crest to upstreamexcitation. Rigid foundation rock.

the resonant frequencies of the system decrease because of the �exibility increase, the peaksamplitude decreases because of radiation damping and the dam response at some of the uppermodes changes because of the travelling excitation. In the following, the e�ects of bottomsediments for the compliant foundation rock situation are analysed.

Sediment e�ects

The BE discretization for the coupled dam–water–foundation rock–sediment system is asshown in Figure 1. The system is under the e�ects of vertically incident P- or S-wavesproducing vertical, upstream or cross-stream excitation. Figure 8 shows the dam crest ampli�-cation for these three excitations and three di�erent situations: full reservoir without sediment,full reservoir with a fully saturated bottom sediment layer, and full reservoir with a partiallysaturated bottom sediment layer. The sediment properties are the same as in the previoussection. It can be seen from Figure 8 that the existence of a fully saturated sediment layerhas a very small in�uence on the system response. However, the e�ects of a partially satu-rated sediment layer are important. It reduces the �rst symmetric natural frequency, reducesthe peak amplitude at that frequency, changes the position of other natural frequencies andreduces the system ampli�cation at higher resonant frequencies except for some parts of theupstream excitation case. This e�ect (second and third peaks in Figure 8(a)) can be due to the



Figure 8. In�uence of sediment saturation degree. Upstream response at dam crest toground motion. Compliant foundation rock: (a) upstream excitation; (b) vertical excitation;

and (c) cross-stream excitation.

ampli�cation of the input motion at some frequencies produced by the foundation rock–sediment layer system.The existence of a partially saturated sediment layer changes very signi�cantly the hydro-

dynamic pressure on the dam and consequently its response. Therefore, the seismic analysisof a 3D arch dam requires the identi�cation of bottom sediments, the adequate evaluation oftheir properties (in particular saturation degree and consequently compressibility), and the useof a numerical model which includes proper representation of these bottom sediments.



Figure 9. Simpli�ed models for porous sediment. Scalar models. Response at dam crest to cross-streamground motion. Compliant foundation rock.

Simpli�ed sediment models

The same bottom sediment simpli�ed models as in the rigid foundation case are testedfor the dam–reservoir system on the compliant foundation. The dam crest ampli�cation forfully saturated sediment and partially saturated sediment when the system is under a cross-stream excitation, are presented in Figures 9(a) and (b), respectively. It can be seen fromFigure 9(a), that both scalar models give accurate results for the fully saturated sediment,but this only means that they do not change the system response as the sediment e�ect isnegligible. In the partially saturated sediment case (Figure 9(b)), the Scalar-1 model oncemore gives an under-damped response whereas the Scalar-2 model reproduces very accuratelythe poroelastic sediment results. This behaviour of the simpli�ed models is also observed forthe other two excitations considered in the analysis (not shown) and is in agreement with thatobserved in the rigid foundation rock case. Results obtained with the frequency-dependentcomplex velocity viscoelastic model reproduce very accurately those obtained with Biot’sporoelastic sediment model. For the sake of brevity, these results are not shown.

Closed reservoir

Some reservoir geometries are far from being a uniform channel extending to in�nity. They arecloser to a �nite dimensions geometry which can be fully represented by a boundary element



Figure 10. Boundary element discretization. Closed reservoir.

model. A situation of this type is shown in Figure 10 where the reservoir geometry is thesame as in the previous examples up to a distance of 212m from the dam. From that pointthe canyon is assumed to decrease in depth and width as a spherical surface. This geometrywill be referred to as ‘closed reservoir’ in the following. Dom��nguez and Maeso [6] showedthe important di�erences taking place in the dam response when the reservoir is partially orcompletely closed at a certain distance from the dam. These di�erences are more signi�cantfor vertical and upstream excitation than for cross-stream excitation. Bottom sediment e�ectscan be expected to be more important when some of the other energy dissipation mechanismsdo not exist in the system. Thus, in the closed reservoir situation sediment e�ects should bemore important as there is no wave radiation towards in�nity along the water channel. Allthe energy must be dissipated through internal damping, foundation rock radiation and bottomsediments.Dam crest ampli�cation for upstream and vertical excitation is shown versus frequency in

Figure 11. Results correspond to the same three cases as before: no sediment, fully saturatedsediment and partially saturated sediment. The important di�erences in the system responsedue to the change of reservoir geometry can be appreciated by comparison of Figure 11(closed reservoir) and Figure 8 (open reservoir). Di�erent to the open reservoir situation, theexistence of a fully saturated sediment modi�es the system response to a signi�cant extent inthe closed reservoir case as can be seen in Figure 11; in particular, in the intermediate andhigher parts of the frequency range represented. Sediment e�ects are even more important inthe 99.5% partially saturated sediment case as shown also in Figure 11. The value of the �rstresonant frequency and the system response at that frequency are both reduced. The systemresponse at higher frequencies is damped by the sediment layer, in particular for verticalexcitation.The three bottom sediments simpli�ed models have also been tested for the close reservoir

geometry. They (Scalar-1, Scalar-2 and frequency-dependent viscoelastic) represent very wellthe fully saturated sediment e�ect as compared to Biot’s porous material model. In the partiallysaturated sediment case, the simpli�ed models have a similar behaviour to before. Scalar-1yields an underdamped response (shown in Figure 12 for cross-stream excitation) whereasScalar-2 produces a system response very much in agreement with that of the porous material



Figure 11. In�uence of sediment saturation degree. Upstream response at dam crest to ground motion.Compliant foundation rock. Closed reservoir: (a) upstream excitation; and (b) vertical excitation.

Figure 12. Simpli�ed models for porous sediment. Scalar models. Response at dam crest to cross-streamground motion. Compliant foundation rock. Closed reservoir.

model (Figure 12). The same good representation of the sediment e�ects is obtained withthe frequency-dependent viscoelastic model. This behaviour of the simpli�ed models is alsoobserved for the other two excitations (not shown).



TIME RESPONSE

The frequency-domain analyses carried out in previous sections have shown the importanceof bottom sediment e�ects, the in�uence of sediment properties and reservoir geometry onthose e�ects, and the possibility of using simpli�ed models to represent porous sediments. Itis important to analyse the in�uence of these factors on the system time response when thedam site is reached by an earthquake. To do so, the above model has been assumed to beunder an earthquake with the N–S motion component of the El Centro 1940 earthquake. Theaccelerogram has been taken from Reference [20]. A system of plane S-waves propagatingvertically and producing an upstream free-�eld motion as given by that accelerogram has beenassumed. The use of a FFT algorithm in combination with the frequency-response functionsshown in previous sections led to the system time response in several di�erent situations.Figure 13 shows the dam crest mid-point upstream displacement relative to the free-�eld

displacement for the �rst 15seconds. Displacements represented are dynamic and do not includethose due to hydrostatic pressure and dam self-weight. First, the e�ect of dam–foundationrock interaction is shown in Figure 13(a), where the dam responses for rigid and compliantfoundation rock conditions are compared. Second, the e�ect of dam–reservoir–foundation rockinteraction is shown in Figure 13(b) by comparison of the dam response for empty reservoirand full unbounded reservoir conditions. The in�uence of the hydrodynamic pressure is veryimportant as expected. Finally the e�ect of the partially saturated sediment layer is shown inFigure 13(c) for an unbounded reservoir. It is observed in this �gure how the existence of asediment layer modi�es the system response to a signi�cant extent; in particular, there is areduction of the peaks amplitude in the �rst half of the time range represented.

CONCLUSIONS

A three-dimensional boundary element technique for the seismic analysis of arch dams in-cluding dam–reservoir–bottom sediment–foundation rock interaction has been presented in thispaper. The technique is an enhanced version including porous sediment of a procedure previ-ously developed by the authors [6, 12, 13]. The porous sediment is represented as a two-phaseporoelastic region in which the dynamic behaviour is described by Biot’s equations.The present model has made possible the study of the e�ects that bottom sediments have

on the seismic response of arch dams, in a context where water compressibility, foundationcompliance, local topography and travelling wave excitation are taken into account. It hasbeen shown how sediment e�ects are important and depend on sediment geometry and me-chanical properties, such as saturation degree and permeability, and on reservoir geometry.Global response is clearly in�uenced by the sediment saturation degree which determines com-pressibility and consequently the length of the dominant volumetric waves in the sediment.Changes in permeability have little in�uence on the system natural frequencies (it mainlyoccurs for higher modes) but clearly modify the dissipation properties of the sediment andconsequently the damping at resonant frequencies.Results for an unbounded reservoir (open) and a bounded reservoir (closed) have been

compared. It has been shown how fully saturated sediments have a little in�uence on thedam response unless the reservoir geometry is closed and this energy absorption mechanismbecomes signi�cant. Partially saturated sediments have an important in�uence on the dam



Figure 13. Displacement response at dam crest centre point due to the N–S component of theEl Centro ground motion: (a) dam–foundation interaction; (b) dam–water–foundation interaction;

and (c) dam–water–sediment–foundation interaction.

response in any case. As a general rule they reduce the system response, in particular for thevertical excitation case.Permeability of partially saturated sediments has a signi�cant e�ect on the system response.

Changes in the permeability do not change to a signi�cant extent the resonance frequenciesof the system but modify the damping e�ect of the sediment by changing the amplitude ofthe peaks. An increase of permeability leads to an increase of the amplitude of the highermode peaks.In order to make the BE approach more e�cient, several simpli�ed models for bottom

sediments have been tested. They are: a scalar medium with a frequency-dependent complex-valued wave velocity equal that of the P1-wave of the porous material and some additionaldamping; and a frequency-dependent viscoelastic solid with P- and S-wave velocities equal tothe P1- and S-wave velocities of the porous material. The former leads to a smaller number



of degrees of freedom than the latter, but requires an assessment of the imaginary part of itswave velocity by comparison with some results obtained with the two-phase porous material.Both sediment models produce accurate results except for cases where the permeability of thesediment is extremely high (k¿10−3 m=s). In such cases, the two-phase poroelastic modelbased on Biot’s equations must be used.

ACKNOWLEDGEMENTS

This work was supported by the Ministerio de Ciencia y Tecnolog��a of Spain (DPI2001-2377-C02-01=02). The �nancial support is gratefully acknowledged.

REFERENCES

1. Hall JF, Chopra AK. Hydrodynamic e�ects in the dynamic response of concrete gravity dams. EarthquakeEngineering and Structural Dynamics 1982; 10:333–345.

2. Hall JF, Chopra AK. Dynamic analysis of arch dams including hydrodynamic e�ects. Journal of EngineeringMechanics (ASCE) 1983; 109:149–163.

3. Fenves G, Chopra AK. E�ects of reservoir bottom absorption and dam–water–foundation rock interaction onfrequency response functions for concrete gravity dams. Earthquake Engineering and Structural Dynamics1985; 13:13–31.

4. Fok K, Chopra AK. Water compressibility in earthquake response of arch dams. Journal of StructuralEngineering (ASCE) 1987; 113:958–975.

5. Tan H, Chopra AK. Earthquake analysis of arch dams including dam–water–foundation rock interaction.Earthquake Engineering and Structural Dynamics 1995; 24:1453–1474.

6. Dom��nguez J, Maeso O. Earthquake analysis of arch dams II: Dam–water–foundation interaction. Journal ofEngineering Mechanics (ASCE) 1993; 119:513–530.

7. Dom��nguez J, Gallego R, Jap�on BR. E�ects of porous sediments on seismic response of concrete gravity dams.Journal of Engineering Mechanics (ASCE) 1997; 123:302–311.

8. Cheng AHD. E�ects of sediments on earthquake induced reservoir hydrodynamic response. Journal ofEngineering Mechanics (ASCE) 1986; 112:645–665.

9. Fok KL, Chopra AK. Frequency response functions for arch dams: hydrodynamic and foundation �exibilitye�ects. Earthquake Engineering and Structural Dynamics 1986; 14:769–795.

10. Zhang C, Yan C, Wang G. Numerical simulation of reservoir sediment and e�ects on hydro-dynamic responseof arch dams. Earthquake Engineering and Structural Dynamics 2001; 30:1817–1837.

11. Bougacha S, Tassoulas JL. Seismic response of gravity dams II: E�ects of sediments. Journal of EngineeringMechanics (ASCE) 1991; 117:1839–1850.

12. Maeso O, Dom��nguez J. Earthquake analysis of arch dams I: Dam–foundation interaction. Journal of EngineeringMechanics (ASCE) 1993; 119:496–512.

13. Maeso O, Aznarez JJ, Dom��nguez J. E�ects of the space distribution of the excitation on the seismic responseof arch dams. Journal of Engineering Mechanics (ASCE) 2002; 128:759–768.

14. Biot MA. The theory of propagation of elastic waves in a �uid-saturated porous solid I. Low-frequency range.Journal of the Acoustical Society of America 1956; 28:168–178.

15. Dom��nguez J. An integral formulation for dynamic poroelasticity. Journal of Applied Mechanics (ASME) 1991;58:588–591.

16. Cheng AHD, Badmus T, Beskos DE. Integral equation for dynamic poroelasticity in frequency domain withBEM solution. Journal of Engineering Mechanics (ASCE) 1991; 117:1136–1157.

17. Verruijt A. Elastic Storage of Aquifers. Flow Through Porous Media. de Weist RJM (ed.) Academic Press:London, 1969.

18. Dom��nguez J. Boundary Elements in Dynamics. Computational Mechanics Publications: Southampton andElsevier Applied Science: New York, 1993.

19. Bardet JP. A viscoelastic model for the dynamic behaviour of saturated poroelastic soils. Journal of AppliedMechanics (ASME) 1992; 59:128–135.

20. Chopra AK. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice Hall:New Jersey, 2001.


Three-dimensional models of reservoir sediment and effects on the seismic response of arch dams

Documents

Transcript of Three-dimensional models of reservoir sediment and effects on the seismic response of arch dams