ranbirsandhu_feanal_landandsubsid

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Compressibilities

f

clayb and som e means of predicting andpreoeniing subsidence

RIEKE,

.

.,

II

(1969): Compaction of Argillaceous Sediments.Ph. . Dissertation, Petrol. Eng.

Dept., Univ. South. Calif., Los Angeles, Calif.

RIEKE .H.,II, CHILINGAR, .V. nd ADAMSON,

L.G.

1966): Notes

on

applicationof electro-

kinetic phenomena in soil stabilization, roc. Internat.

Ctay

Conf., Jerusalem, Israel, pp. 381-

389.

RIEKE, .H.,

III,

CHILINGAR,

.V

.and

ROBERTSON

.O.,

r.

(1964): High-pressure (up to

500,000

psi) compaction studies on various clays,

Internat. Geol. Congress,

22nd Session,

New Delhi, India, in print.

RIEKE

.

. II, GHOSE, .

K.,

AHHAD, S.

.

nd

CHILINGAR,.

. (1969): Some dataon compres

sibility of various clays,Internat. Clay Conf., Tokyo, Japan.

SKEMPTON, A.W. (1953): Soil mechanics in relation to geology,

Proc. Yorkshire Geol. Soc.,

TERZAGHI,K. 1925): Erdbaumechanik auf bodenphysikalischer Grundlage. Deuticke, Leipzig.

THOMAS,

.W. (1966): Some effects of overburden pressure on soil shale during underground

TIMM, .C. and MARICELLI,.J. (1953): Formation waters in southwest Louisiana,

Bull. Amer.

TITKOV,.I., PETROV, .P. and NERETINA, .Y. 1964): Formation

of

Minerals

and

Structures

VAN

DER

KNAAP,. 1959): Nonlinear behavior of elastic porous media,

I M E rans.,ZI:

79-186.

VAN

DER KNAAP,.

nd

VAN DER

VLIS,

.

. (1967):

On

the cause of subsidence in oil-producing

areas. Seventh World Petroleum Congress,Mexico City,

Elsevier

Publ. Co., 3: 85-95.

WALLACE,

.

. (1962): Water production from abnormally pressured gas reservoirs

in

south

Louisiana,

Trans. Gulf Coast Assoc. Geol. Soc.,

XII:187-193.

WALLACE,.E. 1965): Application of

electric

log measured pressure to drilling problems and

a new simplified chart for

wellsite

pressure computation, The Log Analyst,6: 4-10.

WHITE,.E. (1965): Saline waters of sedimentary rocks.In:A. Young and J.E.alley(Editors),

Fluids in Subsurface Environments,

Am . ssoc. Petrol. Geologists, Memoir

No.

, Tulsa, Okla.,

WILLIAMS,.G.,

BROWN,

.O. and WOOD, .J. (1965): Cutting drilling costs in high-pressure

29:

33-62.

retorting, Jour. Soc. Petrol. Eng., 6(1): 1-8.

Assoc. Petrol. Geol.,

37(2): 394-409.

During Electrochemical induration of Unstable Rocks. Izdat. Nauka, Moscow.

pp. 342-366.

areas,

Oil

and Gas Jour.,63(41): 145-152.

FINITE ELEMENT ANALYSIS

OF

LAND SUBSIDENCE

Ranbir

S.

SANRHIJ and EdwardL.

WiJXON

ABSTRACT

Application of the finite element method to the problem of land subsidence

is

presented. The settlement

of

a land mass is viewed as an immediate or time dependent

surface deformation caused by direct application of surface loads or by the loss of support

associated with mining or withdrawal of pore fluids. Practically all cases involving static

or quasi-static subsidence can be treated. The method permits consideration

of

complex

geometrical configurations and arbitrary boundary conditions. Non-homogeneity, nisotropy,

viscoelasticity and creep, temperature effects, residual stresses, plastic behavior can be

allowed for. The method is applicable to two or three dimensional deformation and thus

takes into account horizontal as well as vertical movements.

1.

Associate Professor, Departement of Civil Engineering, The Ohio Srate University, Co-

lumbus, Ohio, U.S.A.

2. Associate Professor, Department of Civil Engineering, University of California, Berkeley,

California. U.S.A.

393


2/8

Ranbir S. andhu and Edward

L.

Wilson

RESUME

Les auteurs prsentent une

appl i cati on de

la mt hode des l ment s f i ni s au

probl me des aff ai ssements.

Le t assement d une masse

de

sol

est consi dr

comme

une

dformati on i mmdi at e ou dpendant des

t emps causs par

l appl i cati on di recte

de

charges en surf ace

ou

par

la

perte de support provoque par l exploi tati on mni re

ou

l extracti on

du

f l ui de des

pores.

Prati quement

tous

l es

probl mes

d af fai ssements

stati ques

ou

quasi

statiques peuvent tre trai ts.

La

mt hode permet la

pri se

en consi d-

ration

de conf i gurati ons gomtr i ques

compl exes et

de

condi ti ons aux l i mtes arbi trai res.

On peut tenr compt e

de la

non- homogni t,

de

l ani sotropi e,de

la

visco-lasti ci tet du

f l uage,

des effets de temprature, des contrai ntes rsi duel l es, du comportement pl asti que.

La mt hode est appl i cabl e la dformati on deux ou

trois

di mensi ons

et tient

compt e

par consquent tant des mouvement s hori zontaux

que

verti caux.

I NTRODUCTI ON

Land subsi dence, natural or artifi ca ly i nduced,

i s

of consi derabl e i mportance in engi -

neeri ng. However , anal yti cal sol uti ons are avai l abl e onl y f or the cases of si mpl es l oadi ngs,

i deal i zed homogeneous l i nearl y easti c materi al s and si mpl e geometry. Probl ems i nvol vi ng

non- homogeneous materi al s and compl ex geometry are i ntractabl e by the classical

approach. For these numeri cal procedures have to be used to generate approxi mate

sol uti ons.

Settl ement of a l and mass may resul t f r oma natural process or may be caused by man.

Creati on of l arge reservoi rs, l oadi ng of soft so l to consol i date t faster

to

i mprove its

supporti ng qual i ties, constructi on of structures that appl y l arge l oads to the surface are

some exampl es of external l oads on the l and mass. Removal

of

materi al by mni ng or deep

excavati on in a stressed medi um

or

a change in the pressure- f l ow egi me of the pore fluids

resul ts in l and-mass def ormat i ons due to , loss of support . The actual def ormati ons

evi dentl y wll depend upon the manner of l oadi ng and al so upon the mechani cal proper-

ties of the consti tuent materi al s. Presence of i nternal or external boundari es and the con-

di ti on of saturati on prof oundl y i nf l uence the magni tude of subsi dence.

For a sati sfactory anal ysi s of l and subsi dence, therefore, we need a met hod that

wll

al l owconsi derati on of external l oads, resi dual stresses, i ncremental excavati onor mni ng,

arbi trary geometri cal conf i gurati on, presence and i nf l uence of pore fl uids, l ack of homo-

genei ty, and actual mechani cal behavi or of the materi al s i nvol ved. Appr oxi mat e sol uti on

techni ques usi ng the finte el ement i deal i zati on hol d the best promse. The theory

of

the

finte el ement met hod is wel l document ed in l i terature

[3, 17,

201. t has been used to sol ve

boundary val ue probl ems intwoand three-di mensi onal asti ci ty

[3,5, 17,201

and seepage

[7,

3,

16, 191.

Creep and vi scoelasti ci ty [8,151, i ncremental constructi on [4, , 151, non

l i near e asti ci ty [5,

181,

effect of oi nts

[6]

ave been anal yzed by ths met hod. The met hod

has been appl i ed to predi ct hori zontal as wel l as verti cal settl ement

[9],

and to determne

the t i me dependent def ormati ons caused by excavati ng a l arge cavi ty i n a materi al havi ng

l arge in tia stresses and nonl i near stress-strai n-ti me aw

[lo].

Recent l y the met hod has

been extended to consol i dati on,natural or anti fial , of an intia ly stressed saturated l and

mass

[13, 41.

El asto-pl asti cmateri al bahavi or has al so been consi dered

[i].

BASI C EQUATI ONS OF DEF ORMAT I ON

AND

FLOW

We shal l consi der the stresses and deformati on of a f l ui d-saturatedmedi um The case of

fluid free l and mass wll be i ncl uded as a parti cul ar appl i cati on of the general f ormul at i on.

Negl ecti ng chemcal reacti on and inertia affects, the equi l i bri umequati ons are

[13]:

394


3/8

Finite element analysis

of

land subsidence

where:

oki re the components of the symmetric stress tensor for the solid phase;

r~

p

Fi

hik the Kronecker delta.

Subscripts after a co mm a denote spatial differentiation in the indicia1 notation and

repeated indices indicate summation. n cases where there

is

no pore fluid or w e have a free

draining soil mass, 1 =O and the equilibrium equations reduce to the usual form

the hydrostatic stress in the pore fluid;

the total mass density of the saturated material;

are the components

of

the body force and

~kj,k pFi=O

The equation of flow

is

a generalization of Darcy's law.

vi

=

Kij(?T,j+PZFj)

where:

v i are the components of the relative velocity vector;

ICijare the components of the permeability tensor and

p 2 the mass density of the fluid, assumed to be incompressible.

Taking divergence of equation

3,

we note that

vi,i=

- u .

=

-e..

1.1

where:

ui

eiiare the components of the strain tensor, for the solid phase,

and a superposed dot denotes differentiation with respect to time. Hence

are the components of the displacement vector;

(3)

(4)

ci,i+CKij ~,j+~zFj)li

=0 (5)

Equation

5 is

the condition of saturation. In integral form

t

can be written, assuming

an initially undeformed system:

ui,i

+ CKij(n,j+p~Pj)lidt

=

0 (6)

tinns

1 can he

written in terms of the three components of displacement. Then these equations along

with equation

6

are sufficient for determination of deformations and fluid pressures.

Evidently for a free-draining soil mass or no pore fluid, equations 2 written in terms

of

displacements are sufficient to govern the problem.

so

Assuming

a

strain disr>lacement a n h t r w

For the case of infinitesimal deformations, the strain-displacement aw is:

eij=&(ui,j+uj,i) (7)

Oij =Cijkl

(8)

(9)

For linear elasticity, the general stress strain law is:

where Cijkl are the components of the anisotropic elasticity tensor such that

C Jkl =Ckaj =Cjikl =CiJlk

It

is

to be noted that

Kij

nd Cijklre completely unrelated i.e. hydraulic anisotropy

is

independent of mechanical anisotropy.

395


4/8

Ranbir S. Sandha and Edward L. Wilson

THE

FINITE

ELEMENT METHOD

The finite element method involves the replacement of the actual continuum by a finite

number of discrete subregions called elements. The geometry of the elements is completely

defined by a

set

of points in space called the nodal points of the system. he values of the

unknown field variables within each element are expressed in terms of the nodal point

values by means of suitable interpolation functions. hus the entire solution can be expres-

sed in terms of the nodal point values which are evaluated through solution of governing

equations written in the matrix form. ariational methods provide a convenient procedure

for setting up the matrix equations and have been widely used in finite element analysis

[7,

1, 13,

14,

181. However, in what follows, w e shall adopt the direct approach [16,20].

Interpolation scheme for displacements within an element m can be expressed as

where:

u? ( i ,

) is the ith component of the displacement vector at

(x,

) in the element;

{

@

is

the set of displacement interpolation functions and

{ui t)} is

the

set

of ith components of nodal point displacements fo the entire system.

The symbolsx and in the parentheses indicate the space and time dependence of the

quantities.u? are functions of space and time and ui re functions of time only.

The strain-displacement elationship, equation 7, can then be written as

{e (&, t > > =

C4Y U W I

(11)

where {em

(x, is

the reduced strain tensor

{ern x,t>)=

ey

(12)

LY

and [

3'

is the transformation matrix derived from the displacement interpolation

functions by suitable differentiation and re-arrangement

of

terms. Application of equation

8

gives the reduced stress tensor for the element as:

{cm(X,

>

=CH"1 C4Tl {U(t>l

(13)

where IH' ]s the reduced elasticity tensor for the element.

If residual stresses are present in the soil mass, e.g. horizontal stresses caused by

geologic factors, gravity load stresses, etc., these can be allowed for by simply adding

these to the stresses associated with the deformations. hus equation 13

will

be modified to

{crn(X,

>

=CH"1 C4ZI {. +{cXX, > (14)

where

{er

x,

)}

are the residual stresses for element

m.

It can be shown

[17]

that the transformation matrix relating nodal point forces to

element stresses is transpose of the matrix relating element strains to nodal point dis-

placements. Thus,

{PY)

=Mel*

crn(X,> >

396


5/8

Finite element analysis of land subsidence

where:

{PT}

[A' ]

{My}

is the vector of nodal poi nt l oads;

i s

the el ement sti ffness and

i s

the vector of l oads correspondi ng

to

rel ease of resi dual stresses.

Si ml arl y,we use an i nterpol ati on schemefor pore-f l ui d pressures and carry out appro-

pri ate di fferenti ati onsto establ i sh a rel ati onshi p between nodal poi nt fl uxes and the f iuid

pressures. Summ ng up the rel ati ons for the enti re systemand i ncl udi ng effect of pore

flud pressure and body forces, we obtai n:

where:

[Al

[CI is the coupl i ng matr i x;

[BI is the lowmatri x;

-{MI}

{M}

{Pi}

Local l y appl i ed l oads can be i ncl uded in {Pi}.n the second equati on {M3}

s

the

vector of nodal poi nt f l ows due tobody forces general l y gravi ty) and

{P,}

s the vector of

speci fied boundary f l owand l ocal drai nages

(sinks

or sources).

Assumng l i near i nterpol ati on over a short t i me i nterval to

to

t,, and wri ti ng

At = t,

- t o ,

equati on

17

becomes:

is the sti ffness matri x

for

the sol i d phase;

is the l oad vector due to resi dual stresses;

i s the l oad vector due

to

the body forces, and

is

due

to

the boundary pressures.

At

At

Equat i ons 16 and 18 express the val ues of the unknown nodal di spl acements and fluid

pressures at any t i me in terms

of

the prescri bed data and the val ues at the previ ous t i me

step. It i s noted that the boundary data may

be

vari ed w th each t i me step. These equati ons

can be sol ved by standard procedures. Prescri bed boundary val ues for di spl acements

and pressures can be al l owed for usi ng techni ques expl ai ned by W l son [lq.

The above deri vati on appl i es to the l i near elasti c case. Other materi al l aws can be

i ncorporated in the anal ysi s as desi red fol l ow ng st andard procedures 1,

5,6,8,

10, 12,

15,20).

I n the case of free drai ni ng soi l or no pore fl ui ds,equati on

16

is the onl y one to sol ve.

The terms contai ni ng

7~

t) are dropped. Thus:

397


6/8

Ranbir S. Sandhu and Edward L. Wilson

o.Surfoce loading

b.

Collapse of underground

supports

pq

c.Mining excavotion

d.

Own

excovotion

F3oter level

e .Consolidotion

Saturated sail

f.

Induced compaction

I

*,,

g.

Induced compocth

with Ioterol droiroge

h. Chonge in pore fluid

Numbers indicote different

materiol types.Loyers m e

shown

in

the sketches bui

octuolly the system may

be

completely h e te r og e m

FIGURE.

Typical land subsidence

-Lood

D

Der unit oreo

oturoted soil

A

t l

FIGURE

. System analyzed or settlement

.5P

.4

.3P

.2

.IP

O

1.5

.5L I.0L

FIGURE.

Distribution of excess pore water pressures immediately after loading

398


7/8

Finite element analysis

of

land subsidence

This equation represents the equilibrium equation under the effect of residual stresses,

body forces and surface loads. In general, gravity is the only body force. If the vertical

residual stresses are also due to gravity alone, these effects will cancel the body force

effect. However, when mining

is

involved, the differencewill represent the negative load

due to removal of support. Similarly, mining

will

involve negative loads in the horizontal

direction over surface of the cavity to represent relief of stress, with associated displa-

cements.

DISCUSSION

The finite element method furnishes a powerful tool for the analysis of land subsidence

in a variety of situations and due to a wide range of causes. Some of these situations aie

illustrated in figure 1 The method permits complex geometry, arbitrary time varying

boundary conditions, non-homogeniety as well as anisotropy to be considered. Nonlinear

and time-dependent material behavior can be allowed for.

The procedure outlined

will

predict land subsidence and ground movement associated

with mining, surface loading, fluid withdrawal, and changes in fluid pressures, in the

presence of residual stresses. Any or all of these factors can be simultaneously allowed for.

Overconsolidated or consolidating soil masses can be considered. The method can be

extended to situations where the degree of saturation changes during application of the

cause of subsidence. The analysis presented treats the saturated soil mass as a mixture and

is

free from the semi-empirical assumptions often made in such analyses to allow for the

pore fluid pressures.

As

an illustrative example, the finite element method was used to investigate the

influence of elastic properties

on

the consolidation settlement of a thick clay layer under

a rectangular surface lead. Figure 2 shows the system analyzed.It was found that over

a

range of values of the elastic modulus and Poisson's ratio, the distribution of pore

f

lood

f lood

0.5L I.0L 1.5L

S

3.4

' o. Time: L2/64c

3R T S

2.4

2s

b. Time -4(I /64c)

--

3s

c . Time =9 /64c)

0.5L

I.0L

1.5L

S

2s

3s

-

s

d.Time=16( /64c)

62500 0.0.625 62500

56250 0.2.625 62500

29170 0.4,625 62500

4 62500 0.2.694 9440

S=Consolidotion settlement

in

time =

l /64c

if iood were(-m,a>)= o p L / 4 w

FIGURE. Influenceof elastic properties on consolidation settlement

399


8/8

Ranbir

S.

Sandhu and Edward

L.

Wilson

pressures immediately after application of load was as shown in figure 3.

A

history of

consolidation settlements

is

shown in figure

4.

It

is seen

that the settlements vary with

Poissons ratio even

for

the same value of the coefficient

of

consolidation.

In

the

example the soil is assumed to be linearly elastic, isotropic and homogeneous.

However, as explained earlier, the approach

is

quite general and complex material

laws can be considered.

REFERENCES

1.

2

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

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and SHIEH,W.

.

(1969): Application of Elasto-plasticity in

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Berkeley,

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Jour.

App. Phys.,

Vol. 12, May.

CLOUGH,

.W. 1965): The Finite Element Method in Structural Mechanics,

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f

Stress Analysis,

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Diu., SCE, oi.

93,

SM4.

FELIPPA, .A. (1966): Refined Finite Element Analysis of Linear and Non-linear Two-

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, SESM Report 66-22,

niv. of California, Berkeley.

GOODMAN,

.E.,TAYLOR,.L. and

BREKKE,

.L. (1968): A Model for the Mechanics of

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Soil Mech. and Found.Diu.,

SCE, ol. 94,

SM3.

JAVANDEL, I.

and WITHERSPOON,.A. (1968): Application of the Finite Element Method

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.P.

(1965): Finite Element Analysis of Two-Dimensional, ime-Dependent tress

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Sym.

Rock Mech.,

Austin,

Texas, May.

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TONG,.

1969): Basis of Finite Element Methods for Solid Continua,

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ethods

in Engg.,

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F. and DEERE,

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by the Finite Element Method,

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SANDHU, R.

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(1968): Fluid Flow in Saturated Porous Elastic Media,

Ph.D. Thesis,

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of California, Berkeley.

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C.B.

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Surface,

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Hyd.

Diu., SCE,

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400

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