stress analysis of multiply fractured porous rocks
DESCRIPTION
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Technical University Crete Jan. 2002. STRESS ANALYSIS OF MULTIPLY FRACTURED POROUS ROCKS. Exadaktylos, G. & Liolios P. TUC). Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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)
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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.