23/4/20

Poroelasticity and Diffusion

in Elastic Solids

Shengqiang Cai

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

Weight, PWeight, P

Solvent

Pump,

l

M

Linear poroelasticity

Free energy

Material law

Force balance

Darcy’s law

Stress in a body induced by drying

B.C.

Stress in a thin film due to change of the humidity in the enviroment

0 5 10-1

-0.8

-0.6

-0.4

-0.2

0

(1-

v)/

E

A test of soil (Biot)

z

xz E

v

Ee

20 v

v

v

v

EZ

ue zz

1

1)

1

21(

20

)/

)2

12(exp()

2

12cos(

1

1)

1

21(

2

22

0

Mh

tn

h

Znb

v

vh

v

v

Eu

nnz

)12(

2

n

hab nn

tt

UU

Flory-Rehner free energy

Free energy of stretching

CWWCW ms FF,

FF detlog2321 iKiKs FFNkTW

Flory, Rehner, J. Chem. Phys., 11, 521 (1943)

vCχ

vCvC

vkT

CWm 11

1log

•Swelling decreases entropy by straightening polymers.

•Swelling increases entropy by mixing solvent and

polymers.

Free energy of mixing

Free-energy function

Nonlinear models for gels

322321321321

321

111 ])(

1)

11[log()(

vv

kTNkTs

212321321321

211

222 ])(

1)

11[log()(

vv

kTNkTs

212321321321

211

333 ])(

1)

11[log()(

vv

kTNkTs

Physically

Mathematically

Constitutive law

0

K

K

X

J

t

C

Kinetic process

1det FHHvkTD

M iLiKKL

KiK

i JF

Fj

det iK

iK

FxX

ii xkT

cDj

Diffusion in true quantities

LKLK X

MJ

Mass conservation

Force balance

Kinetics law

vv

kTNkTs

rrrrrr ]

)(

1)

11[log()(

21

r

rrrr vv

kTNkTs ]

)(

1)

11[log()(

21

r

rrrr vv

kTNkTs ]

)(

1)

11[log()(

21

Swelling process of a gel ball

Force balance

Geometrical relationship /r R dRdrr /

Constitutive law

])( )(][)1(2)1[(])([])(1[ 122

221112

2

2

R

rr

r

R

R

r

R

r

r

R

R

r

R

r

R

r

R

r

r

R

R

r

R

r

R

r

r

Nv

R

r

R

r

r

NvR

R

Force balance

Mass conservation( , )

2 0R RdJ JC R t

R dR R

Chemical potential driven flux r

cDj

kT r

2 2

r Rj r J R dR

r R dr

2( )

Cc

dr rdR R

2( )R

CDJ

dr RkTdR

Incompressibility assumption21 ( )r

dr rvC

dR R

Diffusion equation

12 22 2 2 2

2 2

1( ( ) )(1 ( ) ) ( ( ) 1) ( ) ( 2 ( )) 0

d dr r D dr dr r D dr dr d r

dR dR R kT R dR dR R kT dR R R dR dR R

My research: Swelling Packer in Oil Industry

“Swelling Elastomers” talk given by John Dewar in SPE conferenceA product of TAM Company

The most important problems engineers concerned are: How long does the swelling packer take to attain equilibrium?How long does the swelling packer take to attain equilibrium?How does the sealing force depend on the controllable parameters?How does the sealing force depend on the controllable parameters?

Stretches and Stress Distribution

B

A

b

a

Stretch distribution Stress distribution

B

A

b

a

Kinetics of Swelling Packer

b/B=1.2

Nv=0.01

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