Computers and Geotechnics 6 (1988) 95-129

COUPLED CONSOLIDATION ANALYSIS OF THE CONSTRUCTION AND SUBSEQUENT PERFORMANCE OF MONASAVU DAM

D. J. Naylor Department of Civil Engineering

University College of Swansea Singleton Park

Swansea SA2 8PP U.K.

D. J. Knight Sir Alexander Gibb & Partners

Farley House 427 London Road

Farley Reading RG6 1BL

U.K.

D. Ding Wuhan Institute of Hydraulic

and Electrical Engineering Wuhan,

People's Republic of China

ABSTRACT

A two--dimensional coupled consolidation finite element beck-analysis of the construction and first impounding of the dam is described. Interest centres on the relatively soft clay core. This was first modelled elastically, after which the analyses were repeated using an elasto-plastic critical state model for the clay. Comparisons are made of the measured and computed distributions of settlement, pore pressure, and vertical and horizontal stresses. Generally good agreement with measured values was obtained. The results show some arching across the core and the variations in stress (total and effective) across it. The elasto-plastic modelling supports the design expectation that the soft clay will yield plastically rather than develop tension. Conclusions are drawn that will assist designers in evaluating numerical models of embankment dams.

INTRODUCTION

Monasavu dam [1] is an 85 m high embankment with rockfill shells supporting a

very wet, and thus soft, halloysitic, residual clay core, situated at Monasavu Falls in

Viti Levu, the main island of Fiji. It was constructed for the Fiji Electricity Authority

95

Computers and Geotechnics 0266-352X/88/$03.50 © 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain

96

between 1979 and 1982. The dam site is underlain by a sequence of sedimentary rocks

into which is intruded a thick monzonite sill, constituting the river section foundations.

Immediately downstream is the top of a substantial cliff (Figure 1) where Nanuku Creek

once formed the spectacular 126 m high Monasavu Falls. Horizontally bedded

sandstones form the dam abutments. The sand grains consist of reworked tuffaceous

material and rock fragments, which the tropical climatic conditions have caused to

weather to a clay material.

The dam, its foundations and the downstream rock massif were extensively

instrumented to measure seepage, deformations, pore pressures, three-dimensional

stresses in the core and seismic behaviour during construction, impounding and

operation. Settlement cells, hydraulic piezometers and total earth pressure cells were

installed in the embankment. This instrumentation is located on three cross sections

(Figure 2). These are on the right abutment, valley centre and left abutment at the

respective Chainages 125, 175 and 210 m. The instruments were installed at three

levels: E1 705, E1 730 and E1 735, at each cross section, with a fourth at E1 680 at

Ch 175.

The results presented in this paper are obtained from coupled consolidation finite

element analyses. These are in two parts. The first part uses the Biot consolidation

theory with the soil idealised as isotropic and linear elastic [2-4]. In the second part

the soil skeleton is modelled by an elasto-plastic critical state constitutive law. The

description 'coupled' relates to the interaction between the soil skeleton stiffness and

Darcy's flow law due to the change of voids with time. With 'uncoupled' flow there

is no such interaction and the governing equations are the same as used to describe,

for example, heat flow.

The uncoupled or Terzaghi-Rendulic consolidation theory may be used to obtain

approximate solutions to two or three dimensional problems [5,6]. In view, however,

of the potential of the coupled theories to take into account the effect of soil skeleton

strains on the pore pressure field it was decided to use a coupled formulation in the

present work.

The first implementation of the Biot theory by the finite element method was

made by Sandhu and Wilson [7]. They based their method on previous work in fluid

mechanics and used Gurtin's variational principle [8]to derive a set of integral

equations. These formulations accounted for the coupling between deformation and

pore water pressure. Finite elements were used for spatial discretisation together with a

finite difference time-stepping scheme. Later, several different versions were published;

8001 700 4

600

97

L_ 500 m 5C)0 460 3(~3 2()0 I00 0 m

Section at Eh. 175

Ch.175

CIQy bor row __..---~ ~'~~~"gL'~'~,o~-~%~' <~ ~ < > ( Diversion

~7~o ' / / . ~ l l [ I / / / /J~'/JJ Morcsovu Foils Oh.175

Figure 1 Monasavu dam, plan and cross-section

98

in particular Christian et al. [9], Hwang et al. [10], Yokoo et al. [11,12], Booker [13],

Ghabonssi et al. [14] and Ding [15]. During this time nonlinear models for the soil

skeleton were being incorporated, for example, Lewis et at [16], Small et al [17,18].

ELm

800-

750-

700-

650-

600

Spillway 4 Embankment b L. .J

Instrumented sections Core borrowl I- area-

" " ~ . . ' - - ~oGtL . o _ _ ~ f - - - j ~- - ~--.~- • \.~ • ,,~/j" --~ ~.~-~ - ~ Sandstone

~ 1 --"; "" Core trench ~~r'--"~, . " ~"~.,~*~'~-~ ~ ....... 7-- - e×covat~on " ~ o o p & / - q , .~ , ~ 7 - 5 . ~ - - - - - - c o ~ e t ~ ,nf,Hs

.... ,Sandstone roc~: ~ o Conglomerate ~ . ; o ÷ . . . . o . . . . . . . . . .

~ ~ , ' ~ + Monzonite

I ; LEFT Diversion funnel ~o ~s ~5 RIGHT I I I I I I

+LO0 +300 +200 +100 0 -100

^'L.nalnage (m) ,Main instrument locations

Figure 2 Longitudinal profile along core trench

The linear-elastic stiffness parameters used for the analyses were determined from

the earlier studies [I]. The critical state model parameters (which applied to the clay

core only) were based on laboratory tests on samples taken from the dam during

construction, as detailed later in the paper. Permeability values, however, were

obtained by trial to reproduce measured pore pressure : time relations. The results

thus obtained provide both information on realistic values for the permeability

coefficients and a more comprehensive picture of the inferred distribution of stress and

pore pressure than that directly indicated by the instrumentation. It must be noted, of

course, that the instrumentation is the only means by which it is possible to measure

the dam's actual state.

99

THEORY

In this section the theory leading to the finite element formulation used for the

two types of analysis is outlined. The linear (Biot) formulation is first given and then

an incremental form is derived from it. This is suitable for both linear and non-linear

soil modeb. It applies to both sets of analyses.

The theory is based on (1) the equations of equilibrium for the total s t r e $ ~ (2)

the effective stress--strain relations for the soil skeleton and (3) the condition of

continuity for the movement of the solid and fluid components in a soil element.

a soil element b acted on by a set of ~rces and ~ in equilibrium then the

equations of equilibrium ~lating the components of the total stress and body force

intensity acting on the e~ment can be written as:

~ i j , j + FI " 0 (1)

where ~ i j - components o f the t e n s o r o f the t o t a l s t r e s s ,

compress ion c o n s i d e r e d as p o s i t i v e , and

F i - components o f the v e c t o r o f body f o r c e pe r

u n i t vo lume, u s u a l l y g r a v i t y .

The ~poated subscripts in the equation are summed from 1 to 3 and the comma

deno~s diffe~mlation with respect to the fol~wing component.

If the soil skeleton is assumed as linear elastic the effective stress--strain relation

can be written:

where

~ ' i j

61j

P

D i j k l

~kl

Uk

- 6 i j P - O i j k l ekl (2)

- components o f the e f f e c t i v e s t r e s s t e n s o r

- the K ronecke r d e l t a

- the pore w a t e r p r e s s u r e

- e l a s t i c c o e f f i c i e n t s i n the g e n e r a l i s e d Hooke 's

law f o r the s o i l s k e l e t o n

- - ( U k , 1 + U l , k ) / 2 , d i m e n s i o n l e s s components o f the

s t r a i n t e n s o r , and

- components o f the d i sp lacemen t v e c t o r

100

The relative f low of pore water is assumed to obey Darcy's law which can be

expressed generally as:

r s ~t r v i - v I - 7w ( j - p , j )

where

(3)

f v i

s v i

k i j

7w

Fj ( I f this is gravity and direction 2 is vertically upwards, F 2 = -Tw)

- c o m p o n e n t s o f t h e s u p e r f i c i a l v e l o c i t y v e c t o r ( i . e . f l ow p e r u n i t a r e a ) o f t h e f l u i d p h a s e

- c o m p o n e n t s o f t h e v e l o c i t y v e c t o r o f t h e s o l i d p h a s e

- c o e f f i c i e n t s o f p e r m e a b i l i t y i n t h e g e n e r a l D a r c y ' s law

- u n i t w e i g h t o f w a t e r

- c o m p o n e n t s o f body f o r c e on t h e w a t e r p e r u n i t vo lume

Note that velocities are measured relative to axes fixed in space, i.e. the

formulation is Lagranglan rather than Eulerian.

For engineering applications the simplification is usually made that the soil is

saturated and that the compressibility of the water and soil particles is negligible. The

divergence of the fluid velocity vector must therefore be zero, i.e.

f v i , 1 - 0

Adding the compressive strain rate of the soil skeleton,

~ v - -v~ i to b o t h s i d e s g i v e s J

f s s (v i - v t ) t - - v i , i (4)

This allows the left hand side of equation (3) to be replaced by iv-

Equations (1) to (4) are the governing equations on which the finite element

formulations are based.

Finite El©m~nt Formulat ion - Linear Case

The Blot theory for two phase linear elastic materials can be expressed as

follows:

101

Assume a finite element to have n nodes, and that i is the number of

dimensions in physical space, i.e. two or three. In addition to the displacement

variables there will be a pore pressure variable. This need not be assigned to every

node. Assume it is assigned to m nodes, m < n. The basic equations of a single

element for consolidation can then be expressed as:

l ( r + C r - - u - - p

c T r - - H r - - u

where K C

r_u

~p a_u

a_f

_-R ] --U

- - p gf

= stiffness matrix of the solid skeleton

= coupling matrix

= flow matrix

(5)

= vector of nodal displacement components (length i.n)

= rate of nodal displacement vector

= vector of nodal pore pressure (length m)

= vector of prescribed nodal forces

= vector of prescribed nodal flow

The first of the pair of matrix equations (5) expresses the equilibrium condition

and incorporates equations (1) and (2). The second represents continuity. It

incorporates equation (4), with Darcy's law of equation (3) included in the flow matrix

H .

Established procedures are used to assemble these element equations to form the

global equations to be solved for the unknown nodal displacements and pore pressures.

In the present work 8-noded quadrilaterals with some 6-noded triangles v0ere

used. These allowed quadratic displacement variations hut a pore pressure variation

that was linear for the triangles and bi-linear for the quadrilaterals. Thus there were

two displacement and a pore pressure variable at each corner node, with displacements

only assigned to midside nodes.

Incremental Formulat ion

This is derived from the above as follows. The first of equations (5) is replaced

by:

102

I( zlr u + C zl_rp - ~/-u (6)

K is now an incremental or tangential stiffness matrix. It will be the same as in

equation (5) for linear-elastic applications. In the context of elasto-plastic analysis it

will be denoted -~ep. A indicates the presumed small change in values in a time step

At. Note that, in general, loading will be applied over a period of time although load

stepping with At = 0 may be used to model ' instantaneous' loading.

[p in equation (5) can be expressed in terms of its value at tunes t and t+,%t as

follows:

+ r - ~ r + ( 1 - a ) r ; (7) - p - p

+ where -Pr - _rp(t + At )

r = r p ( t ) - - p - -

0 ~ < t ~ < l

A mid-difference scheme with c~ = ~ is used. This implies that for small time steps

r ~ £ p ( t + ~At) - p

- ; S i n c e AEp r + _p - £

r - r + c ~ r - p - p - p

e q u a t i o n (7) can be w r i t t e n

(8 )

Substituting equation (8) into the second of (5), writing £u = Ar-u/At, multiplying

through by At and rearranging, gives:

C T Ar u - ~ At H A r p - At(I t r p + _Rf) (9)

Combining equations (6) and (9) gives the incremental governing equations:

Note that the matrices K., C_, H and Rf should all be ~ for the time step

(strictly, Q = c ~ + + ( l - cOQ- where Q represents the matrix). In the present

applications C_C_ and H are constant. Mid-increment values are used for the other two.

(These are at an estimated mid- increment point in the case of _K).

Equations (10) are assembled for all the elements in the usual way.

103

FINITE ELEMENT IDEALISATION

~mlg.£ame, auMma

A plane strain idealization of the valley centre section (at chainage 175 which is

the highest part of the dam) has been made. The mesh is shown in Figure 3.

= El. in metres 752

~77&5 v73o

- - 705

. . - _ - i - i -

] I I I I L 0 50 100 150 200 250

680 ~7

x

w

300 metres

Figure 3 Finite element mesh

C o l t m ~ d o n aad l~el~rvoir Fillin¢

As large embankment dams are normally constructed in substantially horizontal

layers, this has to be incorporated in the modelling. This involves a sequence of finite

element analyses with the fill surface being raised to different element boundaries in

subsequent layers. Effective stresses, pore water pressures and displacements are

accumulated automatically from one analysis to the next. All the layers may be run in

a single batch job. Displacements are accumulated in such a way as to include

displacements of the previously placed material only. The procedure is explained by

Naylor and Mattar [19]. Eight layers were used in the mesh of Figure 3.

The continuous raising of the fill with time was approximated by assuming

instantaneous loading at the time when the fill was half way up the layer being added.

Thus an eight step loading sequence was used.

1 0 4

A similar treatment was used for the impounding analysis with the reservoir being

raised in eight stages.

It was found that a suitable length of time step was about 2% of the time for

the completion of primary consolidation. On this basis time steps of about 20 days

were used. This resulted in the use of a total of 27 time steps for the construction

analysis and 33 for the impounding and reservoir full stages.

Isotropic elasticity was assumed for the first set of analyses. Isotropy was also

assumed for the permeability. The parameters involved are therefore the Young's

modulus, E' , Poisson ratio ~,' and permeability k.

The continuous plasticity critical state model (c.p.c.s.m.) described by Naylor [20]

was used for the second set of analyses. This requires the specification of six

parameters. Four of these are material properties and comprise the two elastic

parameters E ' and ~', the critical state angle of shearing resistance, ~cs, and a

parameter, ~, which is a measure of the irreversible compressibility of the soil skeleton.

A fifth parameter, t~co, represents the preconsolidation stress state. It is more

conveniently classified as an 'initial stress' and will be considered under that heading

below. The sixth, n, is a 'tuning' parameter for the c.p.c.s.m. It controls the

amount of plastic flow generated at below yield stress states. The value 2, suggested

for n in reference [20] for lack of evidence for a different value, has been used in the

present work.

The parameter ~ replaces X, ~ and the initial voids ratio, e, (or specific volume

v) used in standard texts on the critical state model [21,22,23]. It is defined as

C C X - K c - e

X - 1 + e " 2 . 3 ( 1 + e )

where

X or C c

or C c

= voids ratio (either the initial or, preferably, an

average value for the analysis)

= the slope of the virgin consolidation line on respectively

e : Lno-' or e : Log 10 a ' plots

= corresponding slopes of the assumed reversible unloading-

reloading e : Lna ' or e : Log 1 o t~' lines.

105

The rockfill was modelled as linear--elastic for both sets of analyses. This is

justified on the basis that it is very much stiffer than the clay so that displacements in

it are relatively small and are of little significance. The values of E ' and v' are the

same as used in ref.[l ] and are given in Table I.

The clay core E ' and v' for the elastic analyses were essentially those used in

Ref, 1. E ' was made to vary automatically with the vertical effective stress according

to Figure 4. These analyses were not therefore strictly linear. Hence they are

described simply as 'elastic'.

The clay core parameters for the c .p.c .s .m, were determined by the trial and

error matching of the laboratory stress-strain curves from tests on samples taken from

the clay core during construction. These parameters are given in Table I .

E I

HPa

5 - -

3 -

2

Ado

I I I I 100 200 300 ~00

a ~y kPa

o = E' caicutated from oedometer tes ts for v'=0.~

I 500

Figure 4 Variation of core Youn~'s modulus with vertical effective stress

106

TABLE 1

Parameter values

Ana ly se s P a r a m e t e r Core R o c k f i l l Comments

E'(MPa) 2-5 60 See F ig . 4 L i n e a r v' 0 .4 0 .3 e l a s t i c a ' o ( k P a ) 20 0 See t e x t

k ( n m / s ) * 3-9 ~ See t e x t

R o c k f i l l l i n e a r e l a s t i c

c . p . c . s . m .

E'(MPa) I ' f

~ ' c s ( d e g )

~co(kPa) a ' o ( k P a ) k ( n m / s ) *

24 0 .3 35 0 .02 40 60

0 . 3 - 1 2

60 0 .3

0 c o

See t e x t See t e x t

* 1 nm/s = 10 -9 m/s

Initial Stresses

For the rockfill it is reasonable to assume that the stress state is fully due to the

application of gravity. Consequently it was not assigned any initial stress.

For the clay core, h~ ever, it was necessary to assign an initial isotropic

compressive effective stress (a'xo = ~'yo = t~'o), and an equal initial pore suction (-Uo)

to model the effect of compaction. For the finite element analyses this initial suction

needs to be higher than the actual since the clay is idealized as saturated and no gain

in shear strength can occur under undrained loading. Dissipation of pore pressure is

required before an increase in shear strength can be modelled. In reality a core

material, even as wet as was used in Monasavu, will not be fully saturated when placed

and there will be a gain in shear strength during the placing of the first few metres

even if there is insufficient time for significant consolidation. Such a gain was

demonstrated on one occasion during construction when removal of fill revealed

underlying stronger material.

The initial stress ~'o for the elastic analyses was set at 20 kPa. This was

chosen fairly arbitrarily as the deformations and total stresses computed would be

unaffected by it. The selection of o ' o was considered more carefully for the c.p.c.s.m.

work, and 60 kPa was chosen. In cot:junction with this the pre-consolidation

parameter Crco was assigned the value 40 kPa. The selection of these values will now

be justified.

107

For undrained loading the path AC in Figure 5 would be followed. The choice

of a aco value slightly less then a ' o represents the soil as being lightly

over-consolidated. This will be realistic due to the compaction of the clay in thin

layers. The model therefore incorporates some negative dilatancy (or contractancy) so

that the effective stress path is deflected to the left under undrained loading as shown

in Figure 5. Yield will eventually occur at the critical state C. At this stage the

deviator stress, ~d, is 2~cSin ~cs" Neglecting the small increase of ~r c above its initial

value of 40 kPa due to strain hardening, and taking ~cs as 35", ~r d at failure is

approximately 46 kPa. c u is therefore 23 kPa. This may be compared with the

measured average of 17 kPa deduced from unconfined compression tests on samples

from the core [1]. A higher value for the model is appropriate since, as has been

mentioned, the real situation of a partially saturated soil at low stress levels will result

in the strength being increased above the measured value when a few metres of fill are

placed, quite apart from the increase in strength associated with consolidation.

O'd= 0"1-0" 3

/+0

20

//" Crii'ica[ sfafe line

/4/J 2 Sin ~cs C /

' Yield s u r f a c e

20 ~0 60 80 ~'~=c~+c~; Sfresses in kPa

Figure 5 Effective stress path for undrained shearing of point %n clay core using c.p.c.s, mode]

108

Seleetim of Permeal~Uitv O~

The assumption of isotropy for the permeability was based on preliminary tests

which showed that the results were not sensitive to variations in the horizontal

permeability, kx, if the vertical, ky, was kept constant. Consequently, k = k x = ky

was assumed for the main analyses.

For the elastic analyses it was decided to base k on the assumption of a constant

coefficient of consolidation, Cv, where

kE' ( 1 - v ' ) (11) Cv = 7 w ( l + v ' ) ( 1 - 2 v ' )

Since v' has been assumed constant this implies that k varies inversely with E' .

The program was modified to impose this requirement so that k was varied

automatically as E ' varied. Best overall agreement was obtained with c v = 3.9 mm2/s.

This gave k values reducing from 9 nm/s for the lowest E ' of 2 MPa. See Table 1

and Figure 4. The same values were used in the construction and reservoir filling

stages. This is considerably higher than laboratory-measured values which were in the

region of 0.1 nm/s.

For the c.p.c.s.m, analyses different permeability values were used for the

construction and reservoir filling stages. For the former k varied over the range 12

nm/s to 1.5 nm/s. For the latter the range was 1.5 to 0.3 nm/s.

RESULTS

The computed displacements, pore pressures and stresses are compared with

measurements, and their variation with time and in space is plotted in Figures 6 to 23

inclusive. The data are arranged as follows:

Figures 6 to 8

Figure 9

Figures 10 and 11

Figures 12 to 23

Settlement results

Pore pressure v. time on core centre line

Effective stress v. time on core centre line

Spatial distribution of stress and pore pressures (as the

difference between total and effective stress) at the end of

construction (April 1982) and the end of 1983 (impounding

complete in March 1983).

Computed

750

7gO Measured

730 3

720 >

i 710 700

690

680

670 I I 0 1 2

SettLement (m)

109

I - Mar. 1981 2 - Mar. 1982 3 - Mar. 1982 (end of construction)

• Mar. 1981 A Sep. 1981 • Mar. 1982 (end of construction)

Elastic - - - - c.p.c.s.m.

Figure 6 Settlement v. elevation on core centre line

110

In some of the figures the elastic and c.p.c.s.m, results are both shown, using a full

line for the former and broken for the latter. Where this would make the plots too

congested they are separated into parts (a) and (b). This applies to Figures 14 to 17

and 20 to 23.

0

- o . s - ~E

- 1 , 0 - E OJ

• -" - I . 5 -

-2,0 --

Core I_ L I _

-r I I I

A

o ~ (a) Etasf ic

Downst ream h v

200m [ I I J

A A

"6 -0.5 -

-1.0 OJ

'~ -1.5

-2.0 --

150

I I I

2

(b) c.p.c.s.m.

200m I I I I

A _ o o

Computed 1 - Sep. 1981 2 - Mar. 1982

3 - Mar. 1983

Measured A Sep. 1981

• Mar. 1982 0 Mar. 1983

Figure 7 Settlement profiles at el. 705

111

1981 1982 1983 =~- --i= --

0 IJIFIMIAIMIJIJIAIslOINIOIJIFIMIAIMIJIJIAIslO]NIOIJIFIMI,41M[JIJIAIsloINIDI

o . . . . J e ' - _ . - - - -

~_-1.5 . . . . . . $'-2.0 ] a A A /% /%

Construction .]_ Impounding ..[_ Operation

Computed a - EL 730 Measured • Et. 730 -- Etastic b - El. 705 /% El. 705 - - - - c.p.c,s.m. c- Et. 680 O Et. 680

Figure 8 Settlement v. time at three points on core centre line

750

Construction Impounding Operation

-F" -I- =

7~0

i 730 720

710 E_ 700

69~; Q.

67O

- • • 4

f ~-~--~ ~ --~-~- ~--

) . ' ~ For tegend see Figure 8

J IFIMIA IMIjIjIAISIOLNfDIJIFIMIA IMIJ FJIAIsIoINIOEJ IpIMIA[MIJIJ~ A Is101N[O

1981 ~ 1982 . ]_ 1983 d

Figure 9 Piezometric el. v. time at three points on core centre line

W ~00 o _

350

300

250

200

150

100

50

112

Consfrucfion j _ Impounding ~t_ Operation

- i--- 5- 0--.2_0 o o < ~" O m .

~ . . . . ~ = - - . . . . _ _ . " _ _ ' _ .

> 0 j IJ1FIMIAIMbblAISlOINIOIJIFIMIAIMIjblAIslolNIDtJIFIMIAIMIjIj1AIsloINIo L 1981 ..L. 1982 ~ 1983 _I

For legend see Figure 8

Figure 10 Vertical effective stress v. time at

three points on core centre line

250

200

150

~ 1 0 0

50

~ o ~

I -

Consfrucfion --L.. Impounding _L. Operation

.- ~^m--O~ O -- ----Ore--O-- --~

- c o.- "~o ~

- . . o - - > I . . . . . - - / - - - - = L = = - -

:7"r'FIMIA rMTTr~A [S~OIMDIJ IFIMIA IMIJ Ia IA Islo IN IDlE IF IMIA IMI J Ij IA Is IOINID I

1981 j _ 1982 j _ 1983

For legend see Figure 8

Figure ii Horizontal effective stress v. time at three points on core centre line

750

7~0

730

-~ 720

.~ 710

700 LtJ

690

680

670

113

Computed ~ Elastic . . . . c.p.c.sm.

Measured ~

o" = Total stress ~ \ ~ o"= Effective stress

100 300 500 700

Vertical stress (kPa)

750

740

730

720 E

c 710 0

700 OJ

LU

69O

680

670

For legend see Figure 12

i 100 300 500 700

Horizontal stress (kPa)

Figure 12 Vert. stress v. el. on core centre line at end of construction

Figure 13 Hot. stress v. el. on core centre line at end of construction

2000

7 % 1500

* z

5 1000

I 2

k ’ 500

0

2000

Upstream H 4

-

-

/

- ./ ./

0’ -

114

Core

7--== I

‘1

_.-. ,/’ ‘\.

A? -_e 0 _’ Pore pressure

I 100 150

(a1 Elastic

200 m

Computed ------Effective stress -Total stress -.-.- Overburden (gh)

Measured o Effective stress A Total stress

01

100 150 200 m

(b) c.p.c.s.m

Figure 14 Vertical stress profile at el. 580 at end of construction

115

2000

t For legend see Figure 14

G 1000

L

>” 500

0

(a) Elastic

7% 1500 % VI

z 5 1000

0

150

(bl c.p.c.s.m.

200 m

Figure 15 Vertical stress profile at el. 705 at end of construction

116

r

For Legend see Figure 14 2000

0

2000

3 a r :

1500

2 zi -iG 1000 + s .N

P 500

0

Upstream Core Downstream u

_I_

w

f'\ /' '1. ._.-.-'

./ \

.I /

100 150 200m

(a) Elastic

r /“. ,

.-.-. 0’

1’ ‘\.,

0.

I 1 I

Figure 16

100 150

(b) c.p.c.s.m.

200m

Horizontal stress profile at el. 680 at end of construction

117

2000 r

2000

0

For legend see Figure 14

100 150

(a) Elastic

200m

.A.

A’-- .J \ ‘\

A I

\ -0e

/

100 150

(b) c.p.c.s.m.

200m

Figure 17 Horizontal stress profile at el. 705 at end of construction

750

7&O

730

E 720

.~ 710

700

690

680

670

Corn Juted - - Elastic . . . . cp.c.s,m. MeasuredA ~

100 300 500 700

Vertical stress (kPa)

118

750

7~0

730

720

g 710

700

690

680

670

For legend see m I Figure 1B

100 300 500 700

Horizontal stress (kPa)

Figure 18 Figure 19 Hor. stress v. el. on

core centre line at the

end of 1983

Vert. stress v. el. on

core centre line at the

end of 1983

O.

,.¢,_

,.¢--

>

2000

1500

1000

500

119

Upstream C o r e Downstream

For tegend see

Figure 1/+

/ ' ~ . /14"" --'~"~~'~~ - - !

I CL_~,"~- Pore pressure I /

I / I /

I v I I 100 150 200m

(a) Etastic

~ L

n

~u ¢ . .

:,v-

2000

1500

1000

500

-- A

--I"""~ '~ II I

I - - i

I 100

b A l /~. a_Li" Pore pressure

[ I 150 200m

(bl c.p.c.sm.

Figure 20 Vertical stress profile at el. 680 at end of 1983

120

t

For legend see Figure 14 2000

G 1500 a Y

: ; 1000 VI 6 2

5 500

0

2000

0

50 '100 150m

t

(a1 Elastic

150

(b) c.p.c.s.m.

200m

Figure 21 Vertical stress profile at el. 705 at end of 1983

121

For legend see Figure 14

0

2000

2 r 2 1500

: VI

E 1000 z 'C 2

500

Downstream

150

(a) Elastic

200m

0

100 150 200m

(b) c.p.c.s.m.

Figure 22 Horizontal stress profile at el. 680 at end of 1983

122

2000

x 1500 a r

:

2 1000

0

2000

0

For legend see Figure 14

Upstream Core Downstream W4 d--L-

/

lOO_----'

k-1 I

150 200m

(a) Elastic

r .p. \

.’ .-‘1.

\./ ‘1

Pore pressure

0 __--

100 150 200m

(b) c.p.c.s.m

Figure 23 Horizontal stress profile at el. 705 at end of 1983

123

DISCUSSION

Oeaeral

Knight et al [1] compared measured settlements and stresses against computed

values. These were obtained by an effective stress analysis which used the measured

pore pressures as part of the input data. A key feature of the consolidation analysis

described here is that the pore pressures are calculated. They can therefore be

compared with the measurements , thus adding the time dimension to the calibration

process.

A further feature of the present work is that it provides information on the

spatial distribution of pore pressures, stresses and displacements over the full section of

the dam. Consequently, if reasonable agreement with the measurements is obtained at

the instrumentation points, credibility is given to the computed values elsewhere.

Caution, however, is necessary as the analysis cannot take into account local

inhomogeneities in, for example, permeability which could cause local pore pressure

variations.

Stress and Pore Pressure

The variation with time of pore pressure (expressed as piezometric elevation), and

vertical and horizontal effective stresses on the core centre line are shown in Figures 9,

I0 and II respectively. Agreement with measurement is generally good. The main

differences are in the pore pressures and effective stresses towards the end of, and

following, reservoir filling. During these stages the pore pressures are less than

measured at El. 680 and 765 for both the elastic and c.p.c .s .m, analyses (Figure 9),

the vertical effective stresses are also less than measured at El. 705 for both models

(Figure I0), and the horizontal effective stresses at El. 680 are less than measured for

the elastic modelling.

A possible explanation of the lower calculated pore pressures during reservoir

filling is a reduction in compressibility associated with the dissolving of any remaining

air bobbles in the core. This could not be reproduced in the analyses. Also it is

likely that the core is negatively dilatant in the lower parts of the dam. This means

that shearing will reduce the voids under drained conditions or induce excess pore

pressures if undrained. This phenomenon is not incorporated in the elastic modelling.

Pore pressures greater than calculated would therefore result from the shear stresses in

the core caused by the rising reservoir. Negative dilatancy is, however, incorporated in

124

the c.p.c.s.m, and it is significant that this has reduced the discrepancy (Figure 9).

The undercalculation of the vertical effective stresses (and also the total stresses)

at the core centre is probably due to the actual arching being less than calculated (see

below). This may be due tO undercalculation of shear strains within the initially soft

core which would allow it to settle relative to the shoulders thus reducing hang up in

the upper regions and allowing the core to be supported within itself rather than by

shear and arching to the shoulders. There may be rheological effects which are not

incorporated in the numerical models. This is reassuring as it implies a greater

resistance to hydraulic fracturing after impounding than indicated by the analyses.

Regarding the horizontal effective stresses, the discrepancy in the elastic analyses at the

bottom level disappears with the c.p.c.s.m. This is the most marked difference in the

results from the two sets of analyses and suggests that the c.p.c.s.m, work is the more

realistic.

Figures 12 and 13 show the variation of total and effective vertical and horizontal

stresses, and the pore pressures as the differences between them, with elevation on the

core centre at the end of construction. The same information is provided on

horizontal sections at the lower two instrumentation levels (El. 680 and 705) in Figures

14 to 17 inclusive. These data extend the information available from the

instrumentation to other parts of the dam as has been mentioned above. There are no

very great differences between the analyses and the measurements at this stage.

The stress distributions at the end of construction shown in Figures 14, 15, 16

and 17 are of particular interest for two reasons. Firstly the vertical stress distribution

(Figures 14 and 15) indicates some load transfer, or arching, from the core to the rock

fill shoulders. The difference between the total vertical stress (full lines) and the

theoretical overburden pressure, i.e. the self weight of a column of overlying material,

(chain dotted line) demonstrates this. (The overburden pressure distribution will differ

significantly from the triangular shape of the cross section; furthermore the density of

the core material is only about 75% of the density of the shoulders.) Secondly the

data reveal any tendency for tensions to develop in the core by the minimum

compressive effective stress developed. This is everywhere positive at the end of

construction. This is not quite the case at the end of impounding as will now be

considered.

The reservoir-full stress and pore pressure distributions presented in Figures 18 to

23 inclusive show limited regions near the upstream edge of the core where the

effective stresses are at or close to zero. They are marginally tensile in the elastic

125

modelling (Figures 2Oa, 21a, 22a and 23a), These stresses are higher for the

c.p.c .s .m, and are all compressive except for the horizontal effective stresses in Figures

22b and 23b which just touch zero at the core/upstream fill interface. In view of the

fact that these tensile regions are ~ r y localized, being sharp dips in the calculated

stress profiles which would disappear if smoothed lines were drawn, it may be

concluded that tension is not indicated by the modelling. Furthermore, as already

discussed, the effective stresses measured in the core are generally higher than those

calculated, thus providing reassurance on this point.

D e t o ~ a o ~

The computer deformations shown in Figures 6 and 7 give good agreement

between measurement and calculation at the end of construction. The elastic results

tally closely with those presented in Reference 1. This is a consequence of the good

agreement in the calculated pore pressures at this stage. (In Reference 1 measured

pore pressures were used as data). The c.p.c .s .m, deformations are little different

from the elastic at the core centre.

The measured settlements slightly exceed those calculated for the during and after

impounding stages (Figures ? and 8). This is attributed to the rheological effects not

incorporated in the modelling.

CONCLUSIONS

Two-dimensional coupled consolidation analyses, incorporating f'wst elastic and

then the continuous plasticity critical state elasto-plastic model for the clay core, have

been used to provide comparison with measured values of settlement, pore pressure and

stress during construction, reservoir filling, and for most of the subsequent year.

The generally good agreement between the computed and measured values

demonstrates the ability of the modelling techniques to reproduce realistic distributions

of settlement, pore pressure and stress in space and time. Since the analyses were

beck-analyses, with some of the parameters adjusted to provide a good overall fit with

the measurements, conclusions cannot be drawn about the predictive role of the

modelling. The tests on the clay core samples did, however, provide realistic stiffness

and strength parameters for the critical state model, but the permeability values

obtained from these tests were much lower (about one order of magnitude) than the

best overall fit values.

126

The results enhance the data from the instrumentation by allowing these to be

extrapolated across the dam section for various idealisations.

The calculated distributions of stresses across horizontal sections provide some

indication of hang-up at the edges of the core due to the relatively stiff shoulders.

These show low, but positive, effective stresses in the unusually low density core.

I.zcalised tensile values at the core/upstream shoulder interface calculated by the elastic

analyses are concluded to be unrealistic. Any tendency for tensile stresses to develop

will be prevented by plastic yielding. This is supported by the elasto-plastic

(c.p.c.s.m.) analyses, which show compressive effective stresses in this region.

The tendency for the calculated pore pressure to be less than measured towards

the end of impounding is tentatively attributed to an actual reduction in pore-fluid

compressibility as the core became fully saturated in the early stages of impounding.

This change in compressibility could not be modelled. A further reason in the case of

the elastic analyses was their inability to model negative dilatancy (or contractancy), the

existence of which would in reality cause an increase in pore pressure as the core

deformed in shear. The higher pore pressures from the c.p.c.s .m, are attributed to its

ability to model this.

The undercalculation of effective vertical stresses in the core at the later stages is

attributed to undercalculation of shear strains at yield stress states in the clay core, and

to the inability of the modelling to incorporate rheologicai (creep) effects which would

in practice reduce the load transfer from the core to the relatively rigid shells.

The undercalculation of core settlements at the later stages is also attributed to

these factors.

ACKNO~ ~nG[~.MENTS

The authors thank the Fiji Electricity Authority and their Consulting Engineers

Gibb Australia (Pry) Ltd for permission to publish this paper. They are also grateful

to Sir Alexander Gibb & Partners for their support in the joint collaboration involved.

Thanks are also due to the Science and Engineering Research Council, and to

Mr. S. L. Tong who was employed as a Research Assistant under S .E.R.C. ' s Grant

OR/D/23886. Parallel research on the Monasavu Dam was done under this grant which

made a significant contribution to the work presented in this paper.

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4.

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127

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