A quasi-static axisymmetric solution is obtained of the fully coupled diffusion-deformation system of equations for a poroelastic half-space possessing anisotropic permeability and compressible solid and fluid constituents. This solution is used to study analytically the deformation of a half-space by surface loads. The problem of normal disc loading is discussed in detail. The effects of the compressibility of the solid and fluid constituents and the anisotropy in permeability are studied numerically

Kumar, R.

Rani, S.

Singh, S.J.

English

Publications of the IAS Fellows

 The 12 th International Conference of 
International Association for Computer Methods and Advances in Geomechanics (IACMAG) 
1-6 October, 2008 
Goa, India   
 
Quasi-static Axisymmetric Deformation of a Poroelastic Half-space 
with Anisotropic Permeability and Compressible Constituents by 
Surface Loads 
 
S.J. Singh 
Dept. of Mathematics, University of Delhi, South Campus, New Delhi, India 
 
S. Rani, R. Kumar 
Dept. of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar, India  
Keywords: anisotropic, compressible, poroelastic 
ABSTRACT: A quasi-static axisymmetric solution is obtained of the fully coupled diffusion-deformation system of 
equations for a poroelastic half-space possessing anisotropic permeability and compressible solid and fluid 
constituents. This solution is used to study analytically the deformation of a half-space by surface loads. The problem 
of normal disc loading is discussed in detail. The effects of the compressibility of the solid and fluid constituents and 
the anisotropy in permeability are studied numerically. 
1 Introduction 
Deformation of a poroelastic half-space by surface loads has been studied extensively (see, e.g., Wang, 2000). 
However, in most of the investigations, the permeability is assumed to be isotropic. On account of the sedimentation 
process producing horizontal stratification planes, the permeability in the horizontal and vertical directions may differ. 
Therefore, it is useful to investigate the effect of the anisotropy in permeability on the quasi-static deformation of a 
half-space by surface loads. Recent studies on the deformation of a poroelastic half-space with anisotropic 
permeability include Chen (2004, 2005) and Singh et al. (2007). While Chen (2004) assumed the solid constituents of 
the poroelastic medium to be incompressible, Chen (2005) assumed both the fluid and solid constituents as 
incompressible. Singh et al. (2007) discussed the consolidation of a poroelastic half-space with anisotropic 
permeability and compressible constituents by two-dimensional surface loads. 
The purpose of the present paper is to study the quasi-static deformation of a poroelastic half-space by axisymmetric 
surface loading. The permeability in the vertical direction may be different from the permeability in the horizontal 
direction. The fluid and solid constituents are assumed to be compressible. The problem of normal disc loading is 
discussed in detail. An explicit analytical sol ution in the Laplace-Hankel transform domain is obtained. Schapery’s 
formula (Schapery, 1962) is used for the Laplace transform inversion and the extended Simpson’s rule for the Hankel 
transform inversion. Detailed numerical computations are performed to study the effects of the anisotropy in 
permeability and the compressibility of the fluid and solid constituents. 
2 Governing equations 
Let (r, q, z) denote the cylindrical polar coordinates and (u
r
, u
q
, u
z
) the corresponding displacement components. For 
axial symmetry, ¶ / ¶ q º 0. A poroelastic material with compressible fluid and solid constituents can be characterized 
by four constitutive constants. Let these constants be: the shear modulus (G), the drained Poisson’s ratio (n), the 
undrained Poisson’s ratio (n
u
) and the Biot-Willis coefficient (a). For axial symmetry we have the following governing 
equations in which s denotes the total stress tensor and p the pore pressure. 
1248
 2.1 Equilibrium equations 
   0
21
11
2
2
22
2
=
¶
¶
-
¶
Î¶
-
+
¶
¶
+-
¶
¶
+
¶
¶
r
p
Grz
u
r
u
r
u
rr
u rrrr a
n
 (1) 
   0
21
11
2
2
2
2
=
¶
¶
-
¶
Î¶
-
+
¶
¶
+
¶
¶
+
¶
¶
z
p
Gzz
u
r
u
rr
u zzz a
n
 (2) 
   hÑ2p = G Ñ2Î (3) 
where  
   Î = div u, 
( )an
n
h
-
-
=
12
21 . (4) 
2.2 Constitutive equations 
   p
r
u
G rrr an
n
s -÷÷
ø
ö
çç
è
æ
Î
-
+
¶
¶
=
21
2  
   p
r
u
G r a
n
n
s qq -÷÷
ø
ö
çç
è
æ
Î
-
+=
21
2  
   p
z
u
G zzz an
n
s -÷÷
ø
ö
çç
è
æ
Î
-
+
¶
¶
=
21
2  
   0==÷
ø
öç
è
æ
¶
¶
+
¶
¶
= qq sss zrzrrz ,r
u
z
u
G . (5) 
2.3 Darcy’s law 
   q
r
 = -c
r
 ¶ p / ¶ r, q
q
 = 0, q
z
 = -c
z
¶ p / ¶ z (6) 
where q is the fluid flux and (c
r
, c
z
) is the Darcy conductivity in the (r, z) direction. 
2.4 Fluid diffusion equation 
   
2
2
2
2 1
z
p
r
p
rr
p
zr ¶
¶+÷÷
ø
ö
çç
è
æ
¶
¶+
¶
¶ cc  = ÷
ø
öç
è
æ +Î
¶
¶
p
Mt
1
a  (7) 
where 
   M = 
( )
( ) ( )u
uG
nna
nn
2121
2
2 --
- . (8) 
3 Solution 
A solution of the governing equations can be obtained by following a modified version of the procedure used by 
Rajapakse and Senjuntichai (1993) and elaborated by Wang (2000, Sect. 9.6). We define the Laplace 
transform ( )s,z,rf~ of a function ( )t,z,rf by the relation 
   ( ) ( )ò dtet,z,rfs,z,rf~ st
¥
-=
0
. (9) 
Omitting the details, the Laplace transform of the solution of the governing equations which is bounded for zà¥ can 
be expressed in the form  
1249
    ( ) ( ) ( )ò
¥
-- +=
0
021 kdkkrJeAeAs,z,rp
~ kzmz  (10) 
    ( ) ( ) ( )ôõ
ó
ú
û
ù
ê
ë
é
++
-
-=
¥
---
0
146122
kdkkrJeAzeAeA
kmG
k
s,z,ru~ kzkzmzr
h  (11) 
   ( ) ( ) ( )ôõ
ó
ú
û
ù
ê
ë
é
++
-
-=
¥
---
0
056122
kdkkrJeAzeAeA
kmG
m
s,z,ru~ kzkzmzz
h  (12) 
   ( ) ( )ôõ
ó
ê
ë
é
-
-
=
¥
--
0
6122
2 kzmzrz kzeAeAkmG
mk
Gs,z,r~
hs ( ) ( )kdkkrJekAA kz 143 22
1
úû
ù-+ -  (13) 
   ( ) ( ) ( )ôõ
ó
ê
ë
é
+-
-
=
¥
--
0
6122
2
12 kzmzzz ekzAeAkmG
k
Gs,z,r~
hs ( ) ( )kdkkrJekAA kz 043 22
1
ú
û
ù
-+ -  (14) 
where  
   ( ) ( )( ) 23
1
211 A
s
s
G
A a
u
u
ú
û
ù
ê
ë
é
-
-
--=
nn
n
nh , A
6
 =
( )
( ) 2
1
1
2
A
s
s
G u
au
ú
û
ù
ê
ë
é
-
-
+
nn
nh  
   ( ) ( )( ) 245
1
431
2
A
s
s
Gk
AA
u
au
ú
û
ù
ê
ë
é
-
-
-+-+=
nn
n
n
h
 
   ( ) 222 kccss,
c
sk
c
c
m zra
zz
r -+=+÷÷
ø
ö
çç
è
æ
=  
   ( ) ( )( )
( ) ( )
( )zr
u
u
zr ,
Gc,c cc
nna
nnn
--
--=
121
12
22
. (15) 
Three arbitrary constants, A1,  A2 and A4, appear in the solution. These constants can be determined from the 
boundary conditions. 
4 Normal disc loading 
Consider a poroelastic half-space z ³ 0 with z-axis vertically downwards. Suppose a total normal  
force Q
0 
is uniformly applied over a circular surface area (z = 0, r £ a). If the surface is permeable and the load is 
applied in the positive z-direction, the boundary conditions yield 
1250
 0.65
0.70
0.75
0.80
0.85
0.90
-5 -4 -3 -2 -1 0 1 2 3
g = 0.1
g = 1
g = 5
g = 10
g = 20
RUHR SANDSTONE
Log
10
T
W
 
Figure 1. Effect of the permeability anisotropy on the time-settlement. 
 
   p = 0, s
rz
 = 0, 
( )
î
í
ì
>
£-
=
ar,
ar,aQ
zz 0
2
0 ps   (16) 
at z = 0. Using the boundary conditions, we obtain 
   ( )( ) 021 N
s
kmAA u Wh
nn +-=-=  
   ( )( ) ( )( )( )[ ]22224 2112 kmsmkmksA uau ---++---= nnnn ( )WkmGk
N
-2
0  (17) 
where 
   ( )úû
ù
êë
é-=
ak
akJQ
N 100 p
 
   ( )( ) ( )( )mksmks uau +----= nnnW 1 . 
Explicit expressions for the pore pressure and the vertical displacement are: 
   ( ) ( ) ( ) ( ) ( ) ( )ôõ
ó +--=
¥
--
0
10
0 dkkmkaJkrJee
a
sQs,z,rp~ kzmzu
Whp
nn  (18) 
1251
 0.725
0.730
0.735
0.740
0.745
0.750
-5 -4 -3 -2 -1 0 1 2 3 4
a = 1.0
a = 0.8
a = 0.5
a = 0.3
a = 0.1
n = 0.25, n
u
 = 0.27
g = 1
Log
10
T
W
 
Figure 2. Effect of the compressibility of the solid constituents on the time-settlement. 
   ( ) ( ) ( )ôõ
ó
êë
é -
-
-=
¥
--
0
0 kzmzu
z eemk
ms
aG
Qs,z,ru~ nn
p
( ) ( ){ }( ) kzuau zekmss -+-+-- nnn 12
1  
          ( ) ( ) ú
û
ù
÷
ø
ö
ç
è
æ +--- -kzau ek
kmsnn 11  ( ) ( ) dkkaJkrJ
W
1
10 . (19) 
5 Numerical results 
The solution obtained is in the Laplace-Hankel transforms domain. We have used Schapery’s approximate formula 
(Schapery, 1962) for the Laplace transform inversion. The Hankel transform inversion has been performed 
numerically by using the extended Simpson’s rule. 
We have computed the surface settlement u
z
 at the centre of the disc load (r = z = 0) and pore pressure at various 
points on the z-axis (central line). We define the following dimensionless quantities  
  p
Q
a
P ÷÷
ø
ö
çç
è
æ
=
0
2p
,        zuQ
aG
W ÷÷
ø
ö
çç
è
æ
=
0
p
,       
a
z
Z = ,       t
a
G
T z ÷
ø
ö
ç
è
æ
=
2
c ,       
z
r
c
c
g =2 . (20) 
Figure 1 shows the time-settlement W for five values of the permeability anisotropy parameter g for Ruhr sandstone (n 
= 0.12, n
u
 = 0.31, a = 0.65). When g = 1, the vertical permeability is equal to the horizontal permeability. We notice 
that the permeability anisotropy has no effect on the initial settlement or the final settlement. However, if the 
horizontal permeability c
r
 is greater than the vertical permeability c
z
, the permeability anisotropy accelerates the 
consolidation process. Figure 2 depicts the effect of the value of the Biot-Willis coefficient a on the time-settlement W 
for g = 1, n = 0.25, n
u
 = 0.27. For a poroelastic material with incompressible solid constituents, a = 1. As a decreases, 
the compressibility of the solid constituents increases. Figure 2 shows that the compressibility of the solid 
constituents accelerates the consolidation process. The influence of the value of the undrained Poisson’s ratio n
u
 on 
the time-settlement W is displayed in Figure 3 for g =1, n = 0.12, a = 0.65. It is known that n £ n
u
 £ 0.5. The upper limit 
n
u
 = 0.5 corresponds to a poroelastic material with incompressible fluid constituents. As expected, the final settlement 
is independent of the value of n
u
. However, the compressibility of the fluid constituents of the poroelastic medium has 
a strong influence on the consolidation process. The initial settlement for a compressible fluid cons tituents model is 
greater than the initial settlement for the corresponding incompressible fluid constituents model. 
1252
 0.5
0.6
0.7
0.8
0.9
-5 -4 -3 -2 -1 0 1 2
n
u
 = 0.5
n
u
 = 0.4
n
u
 = 0.3
n
u
 = 0.2
n = 0.12, a = 0.65
g = 1
Log10T
W
 
Figure 3. Effect of the compressibility of the fluid constituents on the time-settlement. 
0
0.02
0.04
0.06
0.08
0.10
0.12
-4 -3 -2 -1 0 1 2 3
g = 0.3
g = 0.7
g = 1.0
g = 1.5
g = 2.0
g = 3.0
Z = 2
RUHR SANDSTONE
Log
10
T
P
 
Figure 4. Effect of the permeability anisotropic on the diffusion of the pore pressure. 
Figure 4 shows the effect of the permeability anisotropy on the diffusion of the pore pressure at the point r = 0, z = 2a. 
The pore pressure vanishes in the drained state (T à ¥). Moreover, anisotropy has no effect in the undrained state 
(T à 0). From Figure 4 we notice that instead of decreasing monotonically with time, the pore pressure rises above 
the initial undrained value before it decays to zero as T à ¥. This is in accordance with the Mandal-Cryer Effect 
(Cryer, 1963). This effect is more pronounced at greater depths and for smaller values of the permeability anisotropy 
parameter g = ( ) 21zr / cc . Since, in general, cr > cz, the theoretical prediction of the Mandel-Cryer Effect may get 
diluted in materials with anisotropic permeability. 
6 Acknowledgements 
One of the authors (SJS) is thankful to the Indian National Science Academy for financial support under its Senior 
Scientist Scheme. 
7 References 
Chen G.J. 2004. Consolidation of multilayered half space with anisotropic permeability and compressible  
   constituents, Int. J. Solids Struct., 41, 4567-4586. 
 
Chen G.J. 2005. Steady -state solutions of multilayered and cross -anisotropic poroelastic half -space due to a point  
   sink, Int. J. Geomech., 5, 45-57. 
1253
 
Cryer C.W. 1963. A comparison of the three-dimensional consolidation theories of Biot and Terzaghi, Q. Jl. Mech. Appl. Math., 16,  
     401-412. 
 
Rajapakse R.K.N.D., Senjuntichai T. 1993. Fundamental solutions for a poroelastic half -space with compressible constituents, J.  
     Appl. Mech., 60, 844-856. 
 
Schapery R.A. 1962. Approximate methods of transform inversion for viscoelastic stress analysis. Proc. 4 th US Nat. Congr. Appl.  
     Mech., 2, 1075-1085, ASME Pub., Berkely (USA). 
 
Singh S.J., Rani S., Kumar R. 2007. Quasi-static deformation of a poroelastic half -space with anisotropic permeability by two- 
     dimensional surface loads, Geophys. J. Int., 170,1311-1327. 
 
Wang H.F. 2000. Theory of Linear Poroelasticity, Princeton University Press, Princeton, NJ (USA). 
1254


Quasi-static axisymmetric deformation of a poroelastic half-space with anisotropic permeability and compressible constituents by surface loads

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Quasi-static axisymmetric deformation of a poroelastic half-space with anisotropic permeability and compressible constituents by surface loads

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