A Study on Dynamics of Cracks in Three Dimensions for Estimating Reservoir Cracks in EGS Kazuo Hayas

1
A Study on Dynamics of Cracks in Three Dimensions
for Estimating Reservoir Cracks in EGS Kazuo
Hayashi and Shinya OnoderaInstitute of Fluid
Science, Tohoku University, Sendai 980-8577,
Japan
2
Characterization of Cracks Active Acoustic Method
Model
Figure 1 Example of set up for active acoustic
measurements. (Schematic view of EGS model field,
Tohoku University)
Frequency domain
Model prediction
Field observation
Size, aperture.
3
10cm
400m
Figure 3. A planer crack created by hydro- frac.
Figure 2. Subsurface system at HDR test site of
Tohoku Univ..
4
Objective
Full 3D Approach to Crack Oscillation
So far, crack shape and deformation have been
assumed to be circular and axisymmetric .
node
crack periphery
1st mode
In fully 3D cases, anti-symmetric oscillation is
expected.
node
crack periphery
An approach based on axisymmetric deformation may
lead us to serious misunderstanding.
5
Full 3D Approach to Crack Oscillation
Assumptions
- Elliptic crack with and without fluid inside
the crack
- Opening mode deformation

• No fluid leakage from the crack periphery

Elliptic Crack
(Major radius a, minor radius b)
Approach
The problem is reduced to solving an eigen value
problem.
BIEM in Laplace image space with the aid of
collocation method
6
Fundamental frequencies and modes
- The eigen values are obtained explicitly as the
zero points of a known function of the Laplace
parameter.
- Physical meanings of eigen values and eigen
vectors
Imaginary part of eigen values
fundamental frequencies Eigenvectors
fundamental modes Real part of eigen values
attenuation
7
Fundamental mode of oscillation(penny shaped
crack)
2nd mode
1st mode
3rd mode
4th mode
5th mode
6th mode
8
Fundamental mode of oscillation(elliptic dry
crack)
1st mode
3rd mode
2nd mode
node
2D crack
5th mode
4th mode
9
Fundamental mode of oscillation(elliptic dry
crack, b/a0.8)
1st mode
2nd mode
3rd mode
4th mode
5th mode
6th mode
10
Fundamental mode of oscillation(elliptic dry
crack, b/a0.8)
7th mode
8th mode
9th mode
11th mode
10th mode
12th mode
11
Response to an impulsive load (dry crack)
receiver
source
1st mode
5th mode
1st
Displacement gap at the points (hollow circles)
in response to an impulsive load applied at A.
5th
Two peaks corresponding to 1st and 5th modes are
predominant.
12
Wet Crack
The following supplemental condition is imposed,
instead of solving fluid flow problem
average of fluid pressure,
bulk modulus of fluid,
initial aperture.
shear modulus of rock,
The above condition describes the mass
conservation of the fluid.

Since
the following values are used in
numerical calculation

30 GPa,
3.0 GPa,
13
Response to an impulsive load (wet crack)
1
1
Displacement gap at the points (hollow circles)
in response to an impulsive load applied at A.
There is a peak at O5.
14
Penny shaped wet crack
Displacement gap at O5.0
Similar to axisymmetric mode
15
Conclusions
1.  So far only eigen frequency were determined.
The corresponding fundamental mode was left
undetermined. In the present paper, a new method
was developed to determine the fundamental mode
of oscillation explicitly by reducing the problem
to an eigen-value problem. 2.  The axisymmetric
oscillation modes are predominant when there is
not fluid in the crack. When there is fluid in
the crack, this tendency becomes much sharper. 
3. The oscillation modes assumed in the 2D mode
hardly appears in elliptic cracks even when the
aspect ratio is fairly small, i.e. the crack is
fairly slender.  
16
Zero points of
Im(ap/CT)
Fundamental angular freq.
Re(ap/CT)
Intensity of attenuation
The fundamental frequency and intensity of
attenuation increase with decreasing aspect
ratio (b/a)
17
Fundamental frequencies and modes
Set the non-dimensional displacement gap as in
the form
where
BIE is reduced to
,
where

,
,
.
18
The displacement gap is given in the time domain
by
.
.
Eigen vectors fundamental modes
Real part of eigen values attenuation
19
(No Transcript)