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