Poroelasticity and Diffusion
in Elastic Solids
Shengqiang Cai
2016/5/30
Examples of diffusion in elastic solids

Migration of water into sponges

Consolidation of soil

Swelling of gels

Tissues of animals and plants

Insertion of Li-ions in electrodes
Thermodynamic framework of poroelasticity
Pump, m
Solvent
M
l
Weight, P
Linear poroelasticity
Free energy
Material law
Force balance
Darcy’s law
Stress in a thin film due to change of the humidity in the enviroment
Stress in a body induced by drying
0
B.C.
(1-v)/ E
-0.2
-0.4
-0.6
-0.8
-1
0
5

10
A test of soil (Biot)
z
ez 
0
E

2v
 x  m
E
ez 
u z  0
2v 2
1 v

(1 
)
m
Z
E
1 v 1 v
U
0
2v 2
1 v
2n  1 Z
2n  1 2
t
uz 
(1 
)h 
  bn cos(
 ) exp( (
) 2
)
E
1 v
1 v n
2
h
2
h  /M
bn  an
t
2h
(2n  1)
Flory-Rehner free energy
•Swelling decreases entropy by straightening polymers.
•Swelling increases entropy by mixing solvent and polymers.
Free-energy function
W F, C   Ws F   Wm C 
Free energy of stretching
Ws F   21 NkTFiK FiK  3  2 logdet F 
Free energy of mixing
Wm C   
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
kT
v

1 
χ 


vC log 1 

 vC  1  vC 

Nonlinear models for gels
Physically
Mathematically
Constitutive law
s1  NkT (1  11 ) 
kT
1
1

m
23[log( 1 
)

]  23
2
v
123 123 (123 )
v
s2  NkT (2  21 ) 
kT
1
1

m
12 [log( 1 
)

]  12
2
v
123 123 (123 )
v
s3  NkT (3  31 ) 
kT
1
1

m
12 [log( 1 
)

]

12
v
123 123 (123 ) 2 v
Force balance
J K   M KL
Kinetics law
Kinetic process
Diffusion in true quantities
ji 
FiK
JK
det F
M KL 
Mass conservation
ji  
m
X L
cD m
kT xi
m
m

FiK
X K xi
D
H iK H iL det F  1
vkT
C J K

0
t X K
Swelling process of a gel ball
Geometrical relationship
    r / R
r  dr / dR
Force balance
Constitutive law
sr  NkT (r  r 1 ) 
kT
1
1

m
  [log( 1 
)

]

 
v
r   r   (r   ) 2 v
s  NkT (  1 ) 
kT
1
1

m
r  [log( 1 
)

]

r 
v
r   r   (r   ) 2 v
s  NkT (  1 ) 
kT
1
1

m
r  [log( 1 
)

]  r  
2
v
r   r   (r   )
v
Force balance
2
m NvR  2 r
r 2 Nv r
r 1 r R
r r
r r
R
r 1
1
1 R r  2  r

[
1

(
)
]

[

(
)


]

[(

1
)

2

(

1
)
][
(
)


(
r
) ]
R
r R 2
R
r R R
R r
R R
R R
r R R 2 r 2
R
Mass conservation
Chemical potential driven flux
J
C ( R, t ) dJ R

2 R 0
R
dR
R
jr  
cD m
kT r
JR  
Incompressibility assumption
jr r 2  J R R 2
m m dR

r R dr
c
C
dr r 2
( )
dR R
CD m
dr
kT ( ) 2 R
dR
1  vC  r   
dr r 2
( )
dR R
Diffusion equation
d dr r 2
D m dr 2
dr r 2
D dr 2  2 m
m dr
( ( ) )(1 
( ) )  ( ( )  1)
( ) ( 2 2
(
dR dR R
kT R dR
dR R
kT dR
R
R dR
1
d 2r 1
 ))  0
dR 2 R
My research: Swelling Packer in Oil Industry
A product of TAM Company
“Swelling Elastomers” talk given by John Dewar in SPE conference
The most important problems engineers concerned are:
How long does the swelling packer take to attain equilibrium?
How does the sealing force depend on the controllable parameters?
Stretches and Stress Distribution
b
B
A
Stretch distribution
a
Stress distribution
Kinetics of Swelling Packer
B
A
b
a
b/B=1.2
Nv=0.01
Summary

Thermodynamic framework for diffusion in
porous media has been established

Linear model for porouselasticity has been
analyzed

Non linear model for hydrogel has been setup