STRESS ANALYSIS OF MULTIPLY FRACTURED

POROUS ROCKS Exadaktylos, G. & Liolios P. TUC)

ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Technical University Crete Jan. 2002

Introduction

A class of important geomechanical problems involves the quantification of critical conditions for crack initiation in the vicinity of singularities and stress concentrators in the geomaterial. This class of problems may be extended if the diffusion of pore pressure and/or heat is further considered.

Governing Equations of plane strain linear Elasticity

yyxx2

2

2

2

w tp

yx211

EK

0pyx1

1

yx 2

2

2

2

yyxx2

2

2

2

p

p

ijijij

ijijij

'

'

Simplification of the governing equations in short term and steady state states Steady state

This problem requires the solution for the total stresses (essentially the bi-harmonic eqn) and pore pressure (harmonic) which are uncoupled

0pyx 2

2

2

2

0yx

yyxx2

2

2

2

0/ t

Formulation of the two problems with the complex variable technique (isotropy)

zzzi

zzz

xyxy

yx

00

000

22

Re42

zpy

izpx

Kzq

zFzp Re2

zz

zz

0

0

ieNN

NN

221

21

2

14

1

Formulation of the two problems with the complex variable technique (anisotropy)

202101

2010

202210

21

Re2

Re2

Re2

zz

zz

zz

xy

y

x

zz

zz

0

0

ieNN

NN

221

21

2

14

1

3Re2 zFzp

yxz

yxz

yxz

33

22

11

Boundary Conditions (i.e. 2nd kind for the elasticity problem and Dirichlet for pore pressure)

000000

00000tn ttt

dt

dttti '

00Re2 tptF

Cauchy integral representation of the solution of stresses and pore pressure

zz

z

1

zz

z

1

L

dtzt

t

iz

21

L

dttt

t

itt

ttt

000

000

1

Formulation of the system of S.I.E’s

020

0

00

0

00

21

111

tdttt

ttt

i

dttt

t

idt

dtdt

tt

t

idt

tt

t

i

L

LLL

0

00 '222

dt

dtit ttnn

L tt

dttftp

000

1

Numerical Integration of the system of singular integral equations of the 1st kind Separation of density into singular and

regular part

L

dttt

t

tit

020

ˆ

1

1

2

1

L

dttt

tfttF

0

20

ˆ1

2

1

Gauss-Chebyshev integration scheme for the stresses

1,,2,1,

ˆ

1

ˆ

1 0

1

10

2

1

10

nr

tt

t

ndt

ttt

tdt

tt

tI

n

j rj

nj

n

jt j ,,2,1,

2

12cos

1,,2,1,cos0 nrn

rt r

Gauss-Chebyshev integration scheme for the pore pressure

1,,2,1,

ˆˆ1

1 0

1

10

21

10

nr

tt

tftwdt

tt

tftdt

tt

tfI

n

j rj

j

1

sin1

2

n

j

ntw j

njn

jt j ,,2,1,

1cos

1,,2,1,

12

12cos0

nrn

rt r

Transformation of the integration and collocation points to cracks of arbitrary shapeLinear transformation from the [-1,1] space to the

length s of the curvilinear crack

Liolios, P. and Exadaktylos, G. (2005), A solution of steady-state fluid flow in multiply fractured isotropic porous media, Int. J. Solids Structures, in print.

Convergence of the integration scheme for the pore pressure

Comparison of the integration scheme with the analytic solution for the stress (30 int. points)

221Re

z

zny

Example 1: Parallel cracks under uniform pore pressure

Example 2: Multiple (40) cracks ahead of a horizontal crack under uniform pore pressure (steady state)

Simplification of the governing equations in short term and steady state states Short term (Undrained Conditions)

Hence, the solution of only the compatibility equation for the total stresses gives the full short term solution of the problem

0E

121yyxxvol

xxyyyyxx 0

yyxxyyxx 2

1

yyxxyyxx 2

1pp2

Example 3 – Inclined open crack in uniaxial compression and undrained conditions

Conclusions - Recommendations A fast solver has been developed for multiply

fractured media that takes into account pore pressure, stresses and temperature in steady state.

The consideration of time and crack propagation will be performed in a next step.

Interaction of holes, thin inclusions, cracks in anisotropic media will be also included in the numerical code.