Computers and Geotechnics 4 (1987) 21-42

ANALYSIS OF PUMPING A COMPRESSIBLE PORE FLUID FROM A S A T U R A T E D ELASTIC H A LF SPACE

J.P. Carter and J.R. Booker School of Civil and Mining Engineering

University of Sydney N.S.W. 2006 AUSTRALIA

ABSTRACT

A solution method is presented for the consolidation of a sa turated, porous elastic ha l f space due to the pumping of a pore f lu id at a constant rate f rom a point s ink embedded beneath the surface. It is assumed that the sa tura ted medium is homogeneous and isotropic with respect to both its elastic and flow properties. The soil skeleton is modelled as an isotropic, l inear elastic solid obeying Hooke's Law while the pore f lu id is a s s u m e d to be compressible with its flow governed by Darcy 's Law. The solution has been evaluated for a soil with a value of Poisson's ratio of 0.25 and for a number of d i f f e ren t cases of pore f lu id compressibility. It is demonst ra ted that this compressibil i ty can have a s ign i f ican t inf luence on the rate of consolidation a round the sink. The solutions presented may have applicat ion in practical problems such as the extract ion of groundwater and other f lu ids f rom compressible geological media.

1. INTRODUCTION

To remove pore f lu id f rom the ground it is necessary to use a pump to

reduce the pressure in the pore f lu id in its vicinity. This establishes hydraul ic

gradients and causes flow of the pore f lu id towards the pump. One consequence

of the reduct ion of f lu id pressure in the sa turated ground is an increase in the

compressive e f fec t ive stress state, and this will cause consolidation of the

su r round ing mater ia l and may lead to large scale subsidence. In very many cases

the decrease in pore pressure will not occur immediate ly a f te r pumping, but will

decrease gradual ly below its ini t ial in situ value until eventual ly a steady state

d is t r ibut ion is reached. Hence the resul tant consolidation and surface subsidence

will be time dependent .

21

Computers and Geotechnics 0266-352X/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

22

Surface [Traction free. zero pore pressure)

/ / ,. f r - \ x y / X N V / /

i I

I i

Point Sink _ i -! Half Space Properties: Elastic - Shear Modulus G

- Poissons Ratio v - Lain=; Modutus k - Bulk Modulus K

Flow - Isotropic Permeability k - Unit Weight of Pore Fluid ~F - Bulk Modulus of Pore Fluid M

(adjusted for porosity) - Coefficient of Consolidation

c = (X*2G)(k/Yr)

F i gu re 1

Prob lem D e f i n i t i o n

Probably the best known examples of this phenomenon occur in Bangkok,

Venice and Mexico City where widespread subsidence has bccn caused by

withdrawal of water from aquifers for industrial and domestic purposes (e.g. Scott,

1978). However, the problem is not caused exclusively by the extraction of

groundwater; the withdrawal of petroleum, air and gas can also induce surface

subsidence (e.g. Bear and Pindcr, 1978). In the latter cases the presence of

trapped and dissolved gases in the pore fluid may render invalid the usual

assumption of an incompressible pore fluid in the theory of consolidation.

The purpose of th is paper is to p rov ide the comple t e so lu t ion for the

t r a n s i e n t e f f e c t s o f p u m p i n g f l u i d f r o m a po in t s ink in a s a t u r a t e d , isotropic

e las t ic h a l f space. In pa r t i cu l a r the e f f e c t of pore f l u i d compres s ib i l i t y is

i n c l u d e d in the ana lys i s in order to d e t e r m i n e its i n f l u e n c e on the conso l ida t ion

process, The p rob l em of an incompres s ib l e pore f l u id was dea l t w i t h i n a p rev ious

paper (Booker and Car te r , 1986). T he p rob lem solved he re is d e f i n e d in F igu re 1.

For the sake o f s impl i c i ty the p u m p is t r ea ted as a po in t s ink a n d it is a s sumed

tha t the h a l f space r e m a i n s s a t u r a t e d w i t h the wa te r table a lways at the su r face .

T h u s it is a s s u m e d tha t the ha l f space is c o n t i n u a l l y be ing r echa rged wi th pore

f l u i d to m a k e up for the pore f l u i d ex t r ac t ed f r o m the s ink. In o b t a i n i n g the

23

solution proper account has been taken of the coupling of the pore fluid flow

with the deformation of the solid skeleton.

The point sink problem treated here is of course an extreme idealisation of

most real situations. Nevertheless, the solution does allow an uncluttered look at

the basic physical processes in operation in this type of problem and provides an

assessment of the likely severity of various effects, e.g. pore f luid compressibility.

An addit ional benefi t of the solution presented here is its potential use as a

Green's funct ion in numerical analysis (e.g., using the boundary element method) of

problems with more complicated boundary and initial conditions.

2. GOVERNING EQUATIONS

The equations governing the consolidation of a poroelastic medium were first

developed by Biot (1941a, 1941b). A variety of numerical schemes have since been

proposed for the solution of these equations, and some even allow the incorporation

of a compressible pore fluid, e.g. Ghaboussi and Wilson (1973). Our aim here is

to provide a solution for the point sink problem in closed form, with the

evaluation of the solution requiring only some simple numerical integration.

When expressed in terms of a cartesian coordinate system Blot's equations take the

following forms.

2.1 Equil ibrium

In the absence of any increase in body forces the equations of equilibrium

can be writ ten as

~ = 0 (1 )

w h e r e b = 0 ~I~Y 0 ~ / ~ x ~ / ~ z

0 ~ / ~ z 0 ~ / ~ y o / ~ x J

T (7 = ( a x x , e y y , a z z , a x y , e y z , a z x )

is the vector of total stress components with tensile normal stress regarded as

positive (these quantit ies represent the increase over the initial state of stress).

2.2 Stra in-Displacement Relations

24

The strains are related to the displacement as follows

E = sTu ( 2 )

w h e r e T = ( e x x ' eYY' e z z , 7 x y , 7 y z , T z x )

is the vector of s t rain components of the soil skeleton, and

u T = (Ux, Uy, Uz) is the vector of cartesian displacement components

of the solid skeleton.

2.3 Ef fec t ive Stress Principle

It is assumed for the saturated soil that the ef fec t ive stress principle is valid,

i.e.

where

o = o" - pa ( 3 )

n a " = ( a x x " , a y y , a z z , a x y , a y z , a z x ) T

is the vector of e f fec t ive stress increments, (these quanti t ies

represent the increase over the initial state of e f fec t ive stress)

a T = ( l , l , l , 0 , 0 , 0 , )

and p is the excess pore f luid pressure.

2.4 Hooke's Law

The const i tu t ive behaviour of the solid phase (the skeleton) of the saturated

medium is governed by Hooke's Law, which is

w h e r e D

o" = D e

X+2G X

X+2G

s y m m e t r i c

X 0 0

X 0 0

X+2G 0 0

G 0

G

( 4 )

0

0

0

0

0

G

2 5

with X and G the Lame" modulus and shear modulus, of the isotropic soil skeleton,

respectively.

The moduli X, G can be expressed in terms of the more familiar Young's

modulus, E and Poisson's ratio v of the skeleton, by the relations:

), =

G -

Ev ( I - 2 u ) ( l + v )

E 2 ( I + v)

2.5 Darcy's Law

It will be assumed that the flow of pore water is governed by Darcy's law,

which for an isotropic soil takes the form:

k v = - - V p ( 5 )

'YF

where k is the coeff ic ient of permeability, TF is the unit weight of pore fluid

and the z coordinate direction is aligned vertically and v is the superficial

velocity vector of the pore fluid relative to the solid skeleton.

2.6 Displacement Equations

If Hooke's Law (4) and the equations of equilibrium (1) are combined it is

found that

G V2u + (X + G) Ve v = Vp (6)

w h e r e e v = exx + eyy + e z z

is the volume strain. This equation can be condensed to give the useful relation

(X + 2G) V 2 E v = V 2 p (7)

2.7 The Volume Constraint Equation

If the skeletal material is incompressible but the pore fluid is compressible

then the volume change of any element of soil must balance the di f ference

26

between the volume of f lu id leaving and enter ing the element by flow across its

boundaries plus the volume of f luid extracted f rom the element by some internal

sink mechanism and any change in the volume of pore fluid. Symbolically this

cont inui ty condit ion may be expressed as the volume const ra int equation, i.e.

t t

I VTv d t + ~v + P = I q d t M ( 8 )

where q is the volume of f lu id extracted per uni t volume per uni t t ime f rom the

porous mater ial by the sink mechanism and M is the bulk modulus (adjusted for

porosity) of the pore fluid.

If equat ion (8) is combined with Darcy's Law, equat ion (5), and Laplace

t ransforms are taken of the result ing equation, we f ind that

o r

m

v ' ~ = S(Tv + ~) + ~ (9) 7F

c ~ 2 ~ = (x + 2 G ) [ s ( ~ v + ~ ) + ~ ] ( 1 0 )

w h e r e c = k(X + 2 G ) / ~ F

is the coeff ic ient of consolidation of the saturated porous elastic medium.

The superior bar is used here to indicate a Laplace t ransform, i.e.

~ ( s ) = i f ( t ) c - s t d t ( l l

3. SOLUTION METHOD

In proceeding to the solution of the equations of consolidation for the case

of a point sink embedded in a saturated elastic half space, we introduce triple

Fourier t ransforms of the type

P * ( ~ , f l , V ) = ( 1 / 2 ~ r ) 3 ~ ~ ~ e - i ( c e x + flY + 3 ' Z ) p ( x , y , z ) d x d y d z

- ~ -oo -co ( 1 2 a )

27

The corresponding inversion formula is

p(x,y,z) - ~ y ~ e'i(~x+'Y+TZ) P*(c~,B,7)d~dBd7 -CO -cO -co

Use will also be made of double Fourier t ransforms of the type

P ( o ~ , B , z ) ffi (I/21r) 2 T T ¢ - i ( ( x x + / ~ y ) p ( x , y , z ) d x d y - c O - ¢ 0

and the corresponding invers ion formula

(12b)

( 1 3 a )

p(x,y,z) ffi T ~ ci(c~x+BY) P(c~,B,z)dotdB -CO -00

If we compare equations (12, 13) we see that

P*(",8,7) ffi (I/2.) ~ e'i7 z P(~,B,z)dz -co

and conversely

P ( ~ , B , z ) ffi ~ e+i7 z P * ( ~ , B , 7 ) d 7

- c o

(13b)

(14a)

( 1 4 b )

Sometimes it will be convenient to introduce the coordinates (p, e) where

Ot == p COS

O " p sin e ( 1 5 )

in which case equations (13b) become, for polar coordinates (r, 0,z),

i l p ( r , O , z ) = e i p r c o s ( O - e) p pdpd~ ( 1 6 )

Quite of ten the t r ans form P will be able to be represented in the form

P ffi c o s n ( 0 - c) F ( o , z ) ( 1 7 )

and thus

= 2T i n ~ p F ( p , z ) P v

where Jn represents the Bossol func t ion of order n.

J n ( p r ) d p ( 1 8 )

28

In the analysis which follows solutions for the equations of consolidation are

found in terms of the Laplace t ransforms of the triple Fourier Trans fo rms of the

field quanti t ies. Partial inversion of the triple Fourier t ransforms is then carried

out in closed form using equation (14) or (18) and the inversion is completed

using a single numerical integration. This leaves us with the Laplace t ransforms

of the field quant i t ies which in turn are inverted numerical ly using the technique

developed by Talbot (1979), giving the t ime-dependent field quanti t ies.

The complete solution for a point source embedded in a half space is built

up by f i rs t considering the case of a point sink in an inf in i te medium and then

the case of a ha l f space with no sink. The solutions for these problems are given

in the following sections.

4. SOLUTION FOR A POINT SINK

Let us consider a sink of s t rength F k located at the point (Xk, Yk, Zk)

wi thin an inf in i te medium, so that

q = FkS(X " Xk) 6(Y " Yk) 5 ( z - Zk) ( 1 8 )

where 5 indicates the Dirac delta funct ion. We introduce triple t ransforms having

the form of equat ion (12a) and thus we see, for example, that the t ransform of q

is

Fk "i(~xk + 6Yk + VZk) Q* - ( 2 , ) 3 e

It will be convenient for our purposes to write this in the form

Q* = ~ e ' i ~ z k

Fk w h e r e Q = ( 2 x ) 2 e ' i ( ~ x k + ~Yk)

4.1 Displacement Equations

( 1 9 )

( 2 0 )

In terms of triple t ransforms the displacement equations (6) become

G D2U * + (X + G) ic~E* = ic~P* x v

29

0 D2U; + (X + G) i 3 E : ," i 3 P *

- G D 2 U : + (X + G) i 3 ' r : " i31P"

( 2 1 )

i~o" + i ~ u ; + i ~ o : - E;

w h e r e D 2 = Oe 2 4- i~ 2 + , y2 a n d

(~-, ~y o: ~" ~;) .

-co -co -co

Equa t ions (21) have the solut ion

U* = ¢ i c ~ • x " , b - ~ , Ev

U ; . i B . - * " " { D 2 ) ~ v

U* . i_.~. _ * z = " (D2) l~v

(22)

P* = (X + 2 G ) E * v

4.2 Volume Constraint Equation

In te rms o f the t r ans fo rms equa t ion (10) becomes

--* --* .X + 2G.- , - * c D 2 E v = sE v [1 + t ~ ) J + Q

I f we now in t roduce the var iab les

.X + 2G.-~ #2 = p2 + s [ l + ( ~ ) j / c

p2 = or2 + ~3

we see tha t

v c (~ ,2 + # 2 )

( 2 3 )

( 2 4 )

4.3 Stress C o m p o n e n t s

30

The stress components may bc obtained directly from Hookc's Law, equations

(4), and so

S * 012 x x = 2G(~-~ - l)E*v

s" : 1 ) : YY

2

S* = 2G D2 E~ x y

s* = E; y z

S* = 2G ~--~ zx D2 E~

(25)

where Sjk = (l/2x) a ~ ~ ~ e'i(ax + flY + "rz)Ojkdxdydz -co -co - o o

and j, k denote any of the indices x, y, z.

4.4 Pa r t i a l Inversion

All t he field quantities determined in this section can be expressed in terms of

the three functions H*, K*, L* where

~* 1 e" iTzk = (~~-~) 3,2 + ~2 (26a)

e" iTzk K* = (~-~) (7,/2 + p2)D2

. e'iTzk e-iVZk = 21r (#2 . p 2 ) ( 2 6 b )

= %-;) i , y e - i ' t z k (,y2 + # 2 ) D 2

31

l 2 x ( F 2 p 2 )

- i ~ z k . i ~ e ~2 + #2

i 'g¢ " l"YZk] + , ~ 2 + p 2 3

( 2 6 c )

Now fo r the doub le F ou r i e r t r a n s f o r m s

(H, K, L) = ~ eivz (~*, ~*, ~*) dy

a n d it c an be s h o w n (see A p p e n d i x ) tha t

1 e " # Z _

2

- # Z 1 f e " p Z e "_]

E s g n ( z k - z ) e "OZ e E = 2(# 2 - P 2) P

( 2 7 a )

( 2 7 b )

( 2 7 c )

w h e r e Z = IZ - Z k l

T h u s on c o m b i n i n g e q u a t i o n s (13a, 22, 25, 27) we have

m

iU x = ~ K Q / c

i ~ y = e

U z = L Q/¢

= - (X + 2G) H Q / c

S x x = - 2 0 ( ~ 2 K - H) Q / ¢

~yy = 2o(fl2E- H) Q/c

S x y = - 2G (~flK Q / c

i S y z = - 2G B L Q / c

i S z x = - 2G a L Q / c

( 2 8 )

32

5. S O L U T I O N F O R A H A L F SPACE WITH NO S I N K

To ana ly se th i s p rob lem we i n t r o d u c e double Fou r i e r t r a n s f o r m s l e ad ing to

r e p r e s e n t a t i o n s of the f o r m g iven by e q u a t i o n (13b). It will also be u se fu l to

i n t r o d u c e a u x i l i a r y quan t i t i e s :

w h e r e

U( = c o s e U x + s i n e Uy

U~ = s i n e U x + c o s e Uy

S ~ z = c o s e S x z + s i n e Sy z

S~z = - s i n e S x z + c o s e Sy z

C O S E = - -

P

s i n e = P

( 2 9 )

5.1 D i s p l a c e m e n t E q u a t i o n s

In t e rms of these doub le t r a n s f o r m s e q u a t i o n s (6) become

w h e r e

2U~ G(~--~-- p u U ~ ) + (X + G) i o E v = i p P

~2U~ o ( a - 7 ~ - o ~ u ~ ) = o

a 2 U z a g v ~ G(~-~-~- - p 2 U z ) + (x + 2G) bz - bz

- ~U z E v - ~ + io U~

( 3 0 a )

( 3 0 b )

( 3 0 c

(30d)

(In the p rob lem cons ide red here it is f o u n d t ha t U~7 = 0).

E q u a t i o n s (30) can be c o m b i n e d to give

(X + 2 G ) [ ~ - - ~ - 0 2 E v ] = L ' ~ - "

33

5.2 V o l u m e C o n s t r a i n t E q u a t i o n

E q u a t i o n (10) becomes

c t ~ - - ~ - p 2 ~ ) . (X + 2G) s ( E v + ~ ) ( 3 2 )

5.3 So lu t ion

T h e so lu t ions o f e q u a t i o n s (31, 32) w h i c h r e m a i n b o u n d e d as z -~ -co are:

w h e r e

2G E v = A e # Z + ( ~ - ~ - ~ ) 6BeP z

= (X + 2G) A e # z + 2GBePZ

.k + G. 6 = - ~ - - - - - f f - - j

I f we s u b s t i t u t e e q u a t i o n s (33) in to e q u a t i o n (30e) we f i n d

( 3 3 )

~2 U b z uz p 2 U z = # A e ~ z + 2 B p ( I - 6 ) c O z

a n d t hus

PUz = (/z2 . 0 2 ) Ae # z + 0 z B ( I

F u r t h e r m o r e , it is no t d i f f i c u l t to show tha t

6 ) e p z + Ce p z ( 3 4 )

_p2 ipU~ = ( # 2 . p 2 ) Ae/ZZ + B [ ( ) 6 - (1 + p z ) (1

~ZZ 2G - (# 2

6 ) ] e p z - Ce~Z

(35)

a 2 ) A e # z + B [ ( ) 6 - 1 + (1 - 8 ) ( 1 + p z ) ] e O Z + CePZ

i ' ~ z = ( - - - . - : . P - 8 ~ 2G 2 0 /z2 _ p 2 ) Ae/XZ + B [ ( ~ - ~ - - ~ + G) 6 - (1

( 3 6 )

6) (1 + p z ) ] e P Z - C e P Z

( 3 7 )

34

6. SO L U T IO N FOR A SINK IN A H A L F SPACE

The solut ion to this problem can be synthesised by super impos ing the solutions

found in the previous sections. To do this it is convenient to in t roduce the

fo l lowing change of notat ion,

N = S z z / 2 G

T = iS~ z / 2 G ( 3 8 )

U = iU~

W = U z

The completc solution for the Laplace t r ans fo rms of the double Fourier

t r a n s f o r m s can then be wr i t ten in the fo rm

No T T O

7 : - ( ~ / c ) ~o

,w Wo

71 4- P1

N2 N3

72 73 P2 P3 u2 G3 w2 w3

B

F2

F~ ( 3 9 )

where the func t ions No, To, ..., W 3 are specif ied in Table 1. The coeff ic ients

F~, F2, F 3 may be obta ined f rom the boundary condit ions, i.e. zero t ract ions and

pore pressure at the sur face of the ha l f space, z = 0. Thus we have

72

73

= + (Qlc)

No 7o Po

(40 )

where all of the coeff ic ients in the above equat ion are evaluated at z = 0. Once

the u n k n o w n coeff ic ien ts F1, F2, F 3 have been found as the solution to equat ion

(40), any of the t r ans fo rms of the field quant i t ies may be evaluated f rom

equat ions (39). These solut ions should be precisely the same independent of which

al ternat ive*, specif ied in Table 1, is used.

Al te rna t ive 1 corresponds to a single sink at z = -h in an unbounded medium while a l te rna t ive 2 corresponds to a single sink and an image source placed at z = +h in an unbounded medium.

35

T A B L E I

T f l t n l f o r m 0 A I t . I

0 A i r . 2

" P = ( K b - Ka) p = c F Z

# 3 . p~ c P Z [ ( ~ x ÷ G) 6 - I + (1 - a ) ( l + p z ) ] © P z

- . _ ~ c . Z [ ( x G P~b p t ' L b " La) F= " PZ " cPZ * G) & " (1 - 6 ) ( I + p z ) ] c P z

(x ÷ 2 G ) H b (X + 2G) (Hb " I ] a ) (X + 2 G ) ¢ # z 0 2OcPZ

P r b p t K ~ - r a ) - ~ c~" epZ i r 2o p ~ L ( x + G ) 6 - ( ] . 5 ) ( I • p z ) ] c p z

F : " P : - ( L b - L a )

cPZ P "~b z ( l - a ) c P z

where c " # Z a ,,-~zb .~ , -½ ;

Hb 2

~ , , , ( . , . ,~ [ , . , ,~. o.,z~] ~, , , , ~ , . ,~ [o.,~,. , - , , , ] " 2(tJ~ - # = ) ' " 2(/~= . p Z )

Z b - I z + h i , Z a = I z - h i

7. CALCULATION OF FIELD QUANTITIES

Expressions for N, T, P, U, W were developed in the previous section. It

will be observed for a point sink that these are all functions of p. Thus we see

from equation (18) that

(Fzz, P, Uz) = 2r ~ p(N, P, W) J0(Pr)dp

Now we can easily establish that U~ = 0 and thus that

Ux = c o s , U ~

Uy = s i n , ~

(4 ] )

36

Thus the expressions for the Laplace transforms of displacement can be written as

and hence

~X = ~ ~ ei(C~X + 6Y)cos~(P) dc~d~ -,3o - o o

Uy = e i ( ~x + ~ Y ) s i n ~ U ~ (p) dc~d/3

- oo -ore

o~ 2~r

-Ur = i i e i p r c ° s ( O " E cos (O - c)U$ ( p ) p d ~ d p (42)

: 2~r i p J l ( p r ) U ( p dp

It is not d i f f icul t to show that u 0 = 0. Similarly we may show for the stresses

that

O r z

aOz

= 2~r i p J, (o r ) T(p

= 0

do (43)

The single inf ini te integrals contained in equations (41-43) have been evaluated

numerically, using Gaussian quadrature.

Evaluation of the field quantities is finally achieved by inversion of the

appropriate Laplace transforms. As mentioned earlier, this is also done

numerically, using the eff icient algorithm developed by Talbot (1979).

8. RESULTS

The solutions have been evaluated for the particular case where the solid

skeleton has a Poisson's ratio v = 0.25 and the results have been summarised in

Figures 2, 3 and 4.

37

4nkh ' l

-10 - ! -6 - t . -2 i I I I I I j

. . . . ;., / i Ct . . -~/ / i - o - ~

= 0.1

J / # - -0 .75

• 125

Figure 2

Isochrones of Excess Pore Pressure on the Vert ical Axis for Times c t / h 2 = 0.1 and

1

-0.1 ~

-0.2 a' /

" 3 ' /

~l" " i . / , , I.II/ -0.~~

- 0 . 5

2 3 4 5 r/h

v=0.25

N/K

1

. . . . 0.1

Figure 3

Isochrones of Surface Sett lement

3 8

-OA 01 hi3 -01 "0i5 -OiZ' -013 -0i2 I I I 0i2 _Oi 5 .OlZ, / .0i2 0 0ii OlZ

/ --0.25 -025

/ / j o, z / - -

Sink Sink ~ -

v:0.25 v:025 .1254\ ! / Curve H/K Curve K

- - T " I/!/ ' 01 -1.5

.0 - 20 "1 ' t

-0.5 -Or,

' / 'I/\, Sink . ~ X

v=O.Z5

Curve H/K

1 . . . . 0.1

(c) el" 10 '~2 =

[~]o. r ~ l o , t l~O. l

.o i] .oiz .0ii o11 o12 4.s -oiz, -o i] -oiz -oi~

i~,Ts - z /h

t~.0_

/

- - 025

- - 0 , 5

- - 0 . 75

- -1 ,0

v=0.25

Sink

-1.i

-1. z/~

Figure 4

Vertical Displacements along the Vertical Axis Containing the Sink

--025

--0.5

--075

--1.0

5-

5-

5-

.0-

01 02 I t

39

In discussing the ef fec ts of pore f lu id compress ib i l i ty it is convenient to

def ine the re la t ive compress ib i l i ty as M/K where M is the bulk modulus (adjusted

for porosi ty) of the pore f lu id and K is the bulk modulus of the elastic solid

skeleton, given by

E K -

3 ( 1 2 v )

F igure 2 shows isoehrones of excess pore pressure on the vert ical axis th rough

the point s ink for non-d imens iona l t imes c t / h 2 = 0.1 and ~x The symbol t is

used here to represent the elapsed t ime since the commencement of pumping. In

all cases the changes in pore pressure due to pumping are actual ly suct ions and

this is indicated by the negat ive values of p. When the excess pore pressures are

normal i sed as indicated in Figure 2 the s teady state response (at c t / h 2 = co) is

i ndependen t of the degree of compress ib i l i ty of the pore f lu id (i.e. M/K). Indeed

it is possible to f ind a closed f o r m expression for the excess pore pressure

d i s t r ibu t ion at large t ime and this has been shown by the au thors (Booker and

Carter , 1986) to be

° , F [ l i ] P = - (~-~-~) (41 ' )

I r 2 + ( z + h ) 2 I r 2 + (z - h ) 2

Along the axis r = 0, this of course reduces to

QVF [ 1 1 ] P = "(T¢-k ) iz + hi )z - hl (42)

At in te rmedia te times the normal ised excess pore pressures along the axis are

a f unc t i on of the relat ive compress ib i l i ty of the pore f lu id M/K, as i l lus t ra ted in

F igure 2 fo r the time cor responding to c t / h 2 = 0.1. The results show that the

more compress ible the pore f luid, i.e. the smaller the value of M/K, then the

s lower is the deve lopment of the excess pore suct ions and hence the slower will be

the consol idat ion of the soil a round the sink.

Typical results for sur face displacement are indicated on F igure 3 where

isoehrones or vert ical d isplacement have been plotted agains t radial dis tance f rom

the vert ical axis. The set t lement values have been plotted in non-d imens iona l fo rm

and selected cases cor responding to M/K = 0.1, 1 and co have been shown. It is

obvious that the relat ive compress ibi l i ty of the pore f lu id has a marked inf luence

on the t ime dependent sur face sett lements. General ly , the more compressible the

pore f lu id (i.e. smaller M/K) , the s lower is the movement . The rate of set t lement

4O

in a poroe las t ic ha l f space for w h i c h M / K = 0.i is more t h a n 10 t imes s lower

t h a n in a ha l f space h a v i n g an incompres s ib l e pore f l u id ( M / K = ~o).

The long t e rm s u r f a c e s e t t l emen t s are i n d e p e n d e n t of the compres s ib i l i t y of

pore f l u i d and it is pe rhaps wor th no t ing the i r closed f o r m express ions . For the

genera l case the a u t h o r s r epor t ed (Booker and Car te r , 1986) tha t the ver t ica l

d i s p l a c e m e n t o f a po in t on the s u r f a c e at large t ime is g iven by

I Q'YF ] 1

u z ( r ' ° ) = L4~rk(X + G) l r 2 + h 2 ( 4 3 )

Solu t ions for the v a r i a t i o n of ver t ica l d i s p l a c e m e n t a long the ver t i ca l axis

c o n t a i n i n g the po in t s ink are shown in F igu re 4, c o r r e s p o n d i n g to n o n - d i m e n s i o n a l

t imes, c t / h 2 = 0.1, 1, 10 and o~ R esu l t s have been p lo t ted in n o n - d i m e n s i o n a l

f o r m for the fo l l owing cases: M / K = 0.1, 1 ~ Aga in it is c lear f r o m th is

f i gu re tha t the re la t ive compres s ib i l i t y of the pore f l u i d has a s i g n i f i c a n t

i n f l u e n c e on the ra te of ver t ica l d i sp lacement . Also no t i ceab le on F igu re 4 is the

d i s c o n t i n u i t y in ve r t i ca l d i s p l a c e m e n t tha t occurs at all t imes at the loca t ion of

the s ink (z /a = -1). Th i s is a m a t h e m a t i c a l s i n g u l a r i t y tha t ar ises f r o m the

a s s u m p t i o n of a poin t s ink; in rea l i ty the p u m p wou ld have a f i n i t e size and for

th is case no s i n g u l a r i t y would arise. It is also no tab le t ha t in the so lu t ion for a

po in t s ink the ma t e r i a l benea t h the s ink moves u p w a r d s at ea r ly t ime, whereas at

large t imes and at the s t eady s ta te cond i t i on ( c t / a 2 = oo) the en t i r e mass moves

ve r t i ca l ly d o w n w a r d s due to the pumpi ng .

9. C O N C L U S I O N S

A so lu t ion has been f o u n d for the conso l ida t ion o f a s a t u r a t e d e las t ic ha l f

space b r o u g h t abou t by the c o m m e n c e m e n t of p u m p i n g of the pore f l u i d f r o m a

s ink e m b e d d e d w i t h i n the h a l f space. The m e d i u m was a s s u m e d to be

h o m o g e n e o u s and isot ropic wi th a compress ib le pore f lu id . The g o v e r n i n g equa t ions

of the p rob l em have been solved in Laplace t r a n s f o r m space r e q u i r i n g the use of

double and t r ip le F ou r i e r t r a n s f o r m s . Inve r s ion of some of these t r a n s f o r m s has

been ca r r i ed out u s ing n u m e r i c a l in tegra t ion .

Some p a r t i c u l a r so lu t ions have been e v a l u a t e d for an e las t ic m e d i u m h a v i n g a

Poisson ' s ra t io v = 0.25. These ind ica te t ha t the compres s ib i l i t y of the pore f lu id

can have a s i g n i f i c a n t i n f l u e n c e on the ra te of conso l ida t i on of the soil

41

surrounding the point sink and thus on the settlement of the surface of the half

space. Generally the more compressible the pore fluid then the slower will be the

development of excess pore suctions and hence the slower the rate of surface

settlement.

The solutions presented may have application in practical problems such as

dewatering operations in compressible soils and in the extraction of f luid and gas

from petroleum bearing deposits. In such cases the time to reach steady state and

the final profile of surface settlement could be of great interest.

10. R E F E R E N C E S

3.

4.

5.

6.

7.

8.

BEAR, J. and PINDER, G.F. (1978). Porous Medium Deformation in Multiphase Flow, J. Eng. Mech. Divn, ASCE, Vol. 104, EM4, pp. 881-894.

BIOT, /VLA. (1941a). General Theory of Three Dimensional Consolidation, J. Appl. Physics, Vol. 12, No. 2, pp. 155-164.

BIOT, M.A. (1941b). Consolidation Settlement Under Rectangular Load Distribution, J. Appl. Physics, Vol. 12, No. 5, pp. 426-430.

BOOKER, J.R. and CARTER, J.P. (1986). Long Term Subsidence Due to Fluid Extraction from a Saturated, Anisotropic, Elastic Soil Mass. Q.J. Mechanics Applied Mathematics, Vol. 39, pp. 85-97.

BOOKER, J.R. and CARTER, J.P. (1986). Consolidation of a Saturated Elastic Half Space due to Fluid Extraction from a Point Sink, Proc. Int. Conf. on Compressibility Phenomena in Subsidence, Ed. S.K. Saxena, Henniker, New Hampshire, USA, July 29 - August 3, 1984.

GHABOUSSI, J. and WILSON, E.L. (1973). Flow of Compressible Fluid in a Porous Elastic Media, Int. J. Numerical Methods Engineering, Vol. 5, pp. 419-442.

SCOTT, R.G. (1978). Subsidence - A Review. Evaluation and Prediction of Subsidence, Ed. S.K. Saxena, ASCE, New York, pp. 1-25.

TALBOT, A. (1979). The Accurate Numerical Inversion of Laplace Transforms, J. Instit. Math. Applic., Vol. 23, pp. 97-120.

42

APPENDIX

The aim of this appendix is to verify the expressions for H, K, L contained in

equations (27). We proceed as follows.

Le t ~ = T e ' i 7 z e - P l Z l 2p dz

where p has a positive real part. Then

I e " p l z l 1 = e o s T z P d z p2 + 72

Thus using the Fourier Inversion Theorem

- P l Z l 1 ~ e i 7 z 2p - 2r 72 + p2 d7

Also, let

= ~ e . i 7 z s g n ( z ) -Pl 2 e Z l d z

i i s i n 7 z e ' P l Z l d z

i7 p2 + 72

Thus from the Fourier Inversion Theorem

s g n ( z ) - p l z l 1 f i 7 2 e = 2-~- p2 + 72

The results of equations (27) then follow.

Received 23 March 1987; revised version received and accepted 1 May 1 987