Geophysical Characterization of the Effects of
Fractures and Stress on Subsurface Reservoirs
by
Xinding Fang
B.S. Geophysics, University of Science and Technology of China, 2008
M.S. Geophysics, Massachusetts Institute of Technology, 2010
Submitted to the Department of Earth, Atmospheric, and Planetary Sciences
in partial fulfillment of the requirements for the degree of
t
IMA SSACHUSETTh
Doctor of Philosophy
INSTITUTE
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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September 2013
© Massachusetts Institute of Technology, 2013. All rights reserved.
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Signature of Author:
Department of Earth, Atmospheric, and Planetary Sciences
July 19, 2013
Certified by:
I
12
Michael Fehler
Senior Research Scientist
Thesis Supervisor
Accepted by:
Robert D. van der Hilst
Schlumberger Professor of Earth Sciences
Head, Department of Earth, Atmospheric and Planetary Sciences
2
Geophysical Characterization of the Effects of Fractures and Stress on
Subsurface Reservoirs
by
Xinding Fang
Submitted to the Department of Earth, Atmospheric, and Planetary Sciences
on 19 July 2013, in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy in
Geophysics
Abstract
We study the effect of fractures on reservoir characterization and subsurface rock
property measurements using seismic data. Based on the scale of a fracture relative to
seismic wavelength, we divide the dissertation into two parts: larger scale fractures
and microcracks.
In the first part, we study the sensitivity of seismic waves and their time-lapse
changes in hydraulic fracturing to the geometrical and mechanical properties of fractures that have dimensions comparable to the seismic wavelength. Through our analysis, we give the general seismic reponse of a fracture with a linear slip boundary and
introduce the fracture sensitivity wave equation for optimal time-lapse survey design.
Based on the characteristics of scattering from fractures, we develop an approach to
determine the fracture properties using scattered seismic waves. The applicability and
accuracy of our method is validated through both numerical simulations and laboratory experiments. Application of our approach to the Emilio Field shows that two orthogonal fracture systems exist and the field data results are consistent with well data.
In the second part, we study the effects of microcracks and in situ stress on the
formation properties measured from borehole sonic logging. Formation property measurements in a borehole could be biased by the borehole stress concentration, which
alters the near wellbore formation properties from their original state. To study this
problem, we first develop an iterative approach, which combines a rock physics model and a finite-element method, to calculate the stress-dependent elastic properties of
the rock around a borehole when it is subjected to an anisotropic stress loading. The
validility of this approach is demonstrated through a laboratory experiment on a Berea
sandstone sample. We then use the model obtained from the first step and a finitedifference method to simulate the acoustic response in a borehole. We compare our
numerical results with published laboratory acoustic wave measurements of the azimuthal velocity variations along a borehole under uniaxial loading and find very good
agreement. Our results show that the variation of P-wave velocity versus azimuth is
different from the presumed cosine behavior due to the preference of the wavefield to
propagate through a higher velocity region.
Thesis Supervisor: Michael Fehler
Title: Senior Research Scientist
3
4
Biography
Education
"
B.S. Geophysics, University of Science and Technology of China. June 2008.
*
M.S. Geophysics, Massachusetts Institute of Technology. June 2010.
Employment
*
Research Assistant, Department of Earth, Atmospheric, and Planetary
Sciences, Massachusetts Institute of Technology. August 2008 - June 2013.
"
Summer Intern, ExxonMobil Upstream Research Company, Houston, Texas.
June 2012 - August 2012.
*
Summer Intern, Halliburton, Houston, Texas. June 2011 - September 2011.
Journal publications
*
Fang, X.D., M. Fehler, and A. Cheng, 2013. Effect of borehole stress concentration on sonic logging measurements: submitted to Geophysics.
*
Fang, X.D., X.F. Shang, and M. Fehler, 2013. Sensitivity of time lapse seismic
data to fracture compliance in hydraulic fracturing: submitted to GJI.
*
Fang, X.D., M. Fehler, Z.Y. Zhu, Y.C. Zheng, and D.R. Burns, 2013. Reservoir
fracture characterization from seismic scattered waves: submitted to GJI.
*
Zheng, Y.C, X.D. Fang, M. Fehler, and D.R. Burns, 2013. Seismic characterization of fractured reservoirs by focusing Gaussian beams: Geophysics, 78(4),
A23-A28.
*
Fang, X.D., M. Fehler, Z.Y. Zhu, T.R. Chen, S. Brown, A. Cheng, and M.N.
Toksoz, 2013. An approach for predicting stress-induced anisotropy around a
borehole: Geophysics, 78(3), D143-D150.
*
Fang, X.D., M. Fehler, T.R. Chen, D.R. Bums, and Z.Y. Zhu, 2013. Sensitivity
analysis of fracture scattering, Geophysics, 78, Tl-T1O.
*
Willis, M., D. Pei, A. Cheng, X.D. Fang, and X.F. Shang, 2013. Advancing the
use of rapid time-lapse shear-wave VSPs for capturing diffractions from Hydraulic fractures to estimate fracture dimensions: Reservoir Innovations, 2, 1821.
5
*
Chen, Tianrun, M. Fehler, X.D. Fang, X.F. Shang and D.R. Bums, 2012. SH
Wave Scattering from 2D-fractures using boundary element method with linear
slip boundary condition: Geophysical JournalInternational, 188, 371-380.
*
Yu, C.Q., W.P. Chen, J.Y. Ning, K. Tao, T.L. Tseng, X.D. Fang, Y.S. Chen and
R. Hilst, 2012. Thick crust beneath the Ordos Plateau: Implications for instability of the North China Craton: Earth and Planetary Science Letters, 357358, 366-375.
Conference publications
*
Fang, X.D., X.F. Shang, and M. Fehler, 2013. Sensitivity of time lapse seismic
data to the compliance of hydraulic fractures: 8 3rd Annual InternationalMeeting, SEG ExpandedAbstract (accepted).
*
Fang, X.D., M. Fehler, and A. Cheng, 2013. Effect of borehole stress concentration on compressional wave velocity measurements: 8 3 rd Annual International Meeting, SEG ExpandedAbstract (accepted).
*
Zheng, Y.C., X.D. Fang, and M. Fehler, 2013. Seismic characterization of reservoirs with variable fracture spacing by double focusing Gaussian beams:
83rd Annual InternationalMeeting, SEG ExpandedAbstract (accepted).
*
Kang, P.K., Y.C. Zheng, X.D. Fang, R. Wojcik, D. McLaughlin, S. Brown, M.
Fehler, D. Bums, and R. Juanes, 2013. Joint flow -seismic inversion for characterizing fractured reservoirs: theoretical approach and numerical modeling:
83'd Annual InternationalMeeting, SEG ExpandedAbstract (accepted).
*
Zhan, X., X.D. Fang, R. Daneshvar, E. Liu, C.E. Harris, 2013. Full elastic finite-difference modeling and interpretation of karst system in a sub-salit carbonate reservoir: 83'd Annual InternationalMeeting, SEG Expanded Abstract
(accepted).
*
Liu, Y., M. Fehler, and X.D. Fang, 2013. Estimating the fracture density of
small-scale vertical fractures when large-scale vertical fractures are present:
83rd Annual InternationalMeeting, SEG Expanded Abstract (accepted).
*
Fang, X.D., M. Fehler, Z.Y. Zhu, T.R. Chen, S. Brown, A. Cheng, and M.N.
Toks~z, 2012. Predicting stress-induced anisotropy around a borehole: 8 2nd
Annual InternationalMeeting, SEG ExpandedAbstract.
*
Fang, X.D., M. Fehler, Z.Y. Zhu,Y.C. Zheng, and D. Bums, 2012. Reservoir
fracture characterization from seismic scattered waves: 8 2 nd Annual International Meeting, SEG ExpandedAbstract.
*
Zheng, Y.C., X.D. Fang, M. Fehler, and D. Burns, 2012. Seismic characterization of fractured reservoir using 3D double beams: 8 2nd Annual International
Meeting, SEG ExpandedAbstract.
*
Brown, S., and X.D. Fang, 2012. Fluid flow property estimation from seismic
scattering data: 82 Annual InternationalMeeting, SEG ExpandedAbstract.
6
*
Willis, Mark, X.D. Fang, D.H. Pei, X.F. Shang, and A. Cheng, 2012. Advancing the use of rapid time lapse shear wave VSPs for capturing diffractions
from hydraulic fractures to estimate fracture dimensions: 74th EAGE Conference & Exhibition,EAGE, ExpandedAbstract.
*
Zheng, Y.C., X.D. Fang, M. Fehler, and D.R. Burns, 2012. Efficient doublebeam characterization for fractured reservoir: 74th EAGE Conference & Exhibition, EAGE, ExpandedAbstract.
*
Wang, Hua, X.F. Shang, X.D. Fang, and G. Tao, 2011. On perfectly matched
layer schemes in finite difference simulations of acoustic LoggingWhileDrilling: 81' Annual InternationalMeeting, SEG Expanded Abstract.
*
Chen, Tianrun, M. Fehler, X.D. Fang, X.F. Shang and D.R. Burns, 2011. SH
wave scattering from fractures using boundary element method with linear slip
boundary condition: 8 1st Annual International Meeting, SEG Expanded Abstract.
.
Zheng, Y.C., X.D. Fang, M. Fehler and D.R. Burns, 2011. Double-beam stacking to infer seismic properties of fractured reservoirs: 8 1' Annual International Meeting, SEG ExpandedAbstract.
*
Fang, X.D., M. Fehler, T.R. Chen and D.R. Burns, 2010. Sensitivity analysis
of fracture scattering: 8 0 th Annual InternationalMeeting, SEG, Expanded Abstracts.
*
Chen, T., M. Fehler, S. Brown, Y. Zhang, X.D. Fang, D. Burns, and P. Wang,
2010. Modeling of acoustic wave scattering from a two-dimensional fracture:
8 0 th Annual InternationalMeeting, SEG ExpandedAbstract.
7
8
Acknowledgements
I would like to thank many people for their help towards the completion of this thesis
during my five years of study in the Earth Resource Laboratory at MIT.
First of all, I would like to thank my research advisor, Dr. Michael Fehler for his
support and guidance throughout my study at MIT. I always feel lucky to meet and work
with such a reputable scientist as Dr. Fehler who offered me the chance to enter MIT as
well as the field of geophysics. Whenever I had problem regarding my research, he was
always there for giving me help and support.
I also would like to thank Mark Willis, Daniel Bums, Zhenya Zhu, Tianrun Chen,
Stephen Brown, Yingcai Zheng, Oleg Poliannikov, Sudish Bakku, Andrey Shabelansky
and Ahmad Zamanian for many helpful comments and fruitful suggestions on my fracture research.
I would like to thank my thesis committee members: Prof. M. Nafi Toks6z, Prof.
Robert D. van der Hilst, Prof. Alison Malcolm and Dr. Arthur Cheng for their valuable
suggestions and make my defense possible in such a short time. Specially, I want to thank
Dr. Arthur Cheng for supporting my research and offering me the opportunity to work
with the intelligent scientists in Halliburton and also giving me sincere advice on my future research career.
9
I would like to thank Eni and Halliburton for funding my research and ERL
Founding Member Consortium for supporting me during these years.
I would like to thank all my colleagues and friends at MIT for their help and encouragement in my life and study.
In the end, I would like to give my especial thanks to my wife Jing Jiang, who is
always there to support and back me up. Her comfort and encouragement helps me to go
through many ups and downs during my research at MIT.
10
Contents
1
2
31
1.1
Objective and approach.........................................................................
1.2
Outline of thesis.....................................................................................36
Sensitivity Analysis of Single Fracture Scattering......................................
39
2.1
Introduction ...........................................................................................
40
2.2
M ethodology.........................................................................................
43
2.3
Numerical results and discussions........................................................
46
2.4
3
31
Introduction ...................................................................................................
2.3.1
Fracture scattering pattern.......................................................
46
2.3.2
Scattering strength ..................................................................
50
2.3.3
Effect of fracture height on fracture scattering ........................
51
2.3.4
Single fracture scattering in three-dimensions.........................
52
2.3.5
Born scattering analysis ..........................................................
54
56
Summary ...............................................................................................
Sensitivity of Time-lapse Seismic Data to Fracture Compliance in Hydraulic
Fracturing...................................................................................................
...... 73
3.1
Introduction ...........................................................................................
74
3.2
M ethodology.........................................................................................
76
3.2.1
Fracture sensitivity wave equation ..........................................
76
3.2.2
Relation between time lapse seismic data and fracture
compliaace... .......................................................................................
11
80
3.2.3
3.3
3.4
4
Relative strength of normal and tangential compliance
sensitivities..........................................................................................
83
3.2.4
84
Separation of P- and S-energies ..............................................
Numerical results...................................................................................
86
3.3.1
Sensitivity patterns of different rocks......................................
87
3.3.2
Distribution of sensitivity energy in different acquisition
configuration ........................................................................................
90
Summary ...............................................................................................
94
Reservoir Fracture Characterization Using Fracture Transfer Function .... 115
4.1
Introduction ............................................................................................
116
4.2
M ethodology ...........................................................................................
118
4.2.1
Analysis of the SI method.........................................................
118
4.2.2
M odification of the SI method ..................................................
121
4.3
Laboratory experiment............................................................................124
4.4
Numerical modeling................................................................................
128
4.4.1
Parallel vertical fractures ..........................................................
129
4.4.2
Orthogonal vertical fractures.....................................................
133
4.4.3
Orthogonal sub-vertical fractures with randomly varying fracture
spacing and compliance ..........................................................................
4.4.4
4.5
4.6
5
134
Presence of random heterogeneity in a background model ........ 136
Emilio Field application..........................................................................
137
4.5.1
Background of the Emilio Field ................................................
137
4.5.2
Acquisition geometry................................................................
138
4.5.3
Data processing ........................................................................
138
Summary ................................................................................................
141
Calculation of Stress-induced Anisotropy around a Borehole.......................181
5.1
Introduction ............................................................................................
182
5.2
W orkflow for the numerical modeling.....................................................
184
5.3
Laboratory experiment............................................................................
188
12
5.3.1
Measurement of P- and S-wave velocities under hydrostatic
compression............................................................................................189
5.3.2
Strain measurement of intack rock under uniaxial stress loading190
5.3.3
Strain measurement of the rock with a borehole under uniaxial
loading....................................................................................................191
5.4
6
7
Summ ary ................................................................................................
193
Effect of Borehole Stress Concentration on Sonic Logging Measurements.. 205
6.1
Introduction ............................................................................................
206
6.2
M odeling building ..................................................................................
207
6.3
Comparison with laboratory m easurements .............................................
211
6.4
Num erical results for a monopole source ................................................
216
6.5
Num erical results for a dipole source ......................................................
220
6.6
Sum mary ................................................................................................
221
Conclusions .....................................................................................................
247
7.1
Conclusions ............................................................................................
247
7.2
Contributions..........................................................................................250
7.3
Future work ............................................................................................
253
255
Appendix A
Linear Slip Fracture M odel..............................................................
Appendix B
Mavko's Method for Calculating Stress-induced Anisotropy....... 257
References....................................................................................................................
13
259
14
List of Figures
Figure 2-1: (a) is the fracture model and (b) is the reference model; these two models are
exactly the same except for the presence of a fracture in (a) indicated by the vertical bar.
Solid circles are receivers and they are equidistant (500 m) from the fracture center, stars
indicate sources at different angles of incidence to the fracture. The distance between receivers and fracture center is 2.5 times of the fracture height which is 200 m. Incident angles are measured from the normal of the fracture (e.g. a source directly above the fracture is considered to have a 900 incident angle).................................................58
Figure 2-2: Fpp(O,co) for different ZN/ZT at different incident angles. Incident angles,
which are shown on top of the figure, are 00, 300, 600 and 900 for each column. The value
of ZN/ZT for each row is shown at the left side of each row. Fpp(O,co) is plotted in polar
coordinates, the radial and angular coordinates are frequency(o/(2n)) and 0 respectively.
The range of frequency in each panel is from 0 Hz to 50 Hz. The magnitude of each fracture response function is normalized to 1 in plotting. Fppmax denotes the P-to-P maximum scattering strength of each panel. ZT is fixed at 1010 m/Pa, ZN varies. ............ 59
Figure 2-3: Same as Figure 2-2 except that Fps(O,co) is plotted. Fpsmax is the P-to-S max60
imum scattering strength of each panel. ....................................................
Figure 2-4: Comparison of Fpp(6,wo) (1st row) and Fps(O,wO) (2nd row) for fractures with
different compliance values at 300 incidence. ZN is equal to ZT for these models. Fpp(O,cO)
and Fps(O,w) are normalized by the corresponding ZT value in plotting. Fppmax and
Fpsmax are, respectively, the P-to-P and P-to-S maximum scattering strength of each
61
pan el. ................................................................................................
Figure 2-5: Cartoon showing the scattering from two fracture tips. i and 12 represent the
paths from the two fracture tips to the receiver, I indicates the distance from receiver to
fracture center, d is fracture height, 0 is the radiation angle with respect to the fracture
norm al. .............................................................................................
62
Figure 2-6: Fracture tip P-to-S scattering interference at 00 incidence. Radial and angular
coordinates are frequency and azimuth. Red star indicates the source at 00 incidence. Blue
lines show the constructive interference frequencies, of which the corresponding wave
15
lengths satisfy A =
(n=1,2,...), of the P-to-S scattered waves from the two fracture
tips at all directions. ...............................................................................
63
Figure 2-7: Cartoon showing how incident P-waves are scattered by a fracture. Scattering
energy mainly includes three parts: (i) fracture tip scattering; (ii) P-to-P forward scattering; (iii) P-to-S back scattering. ..............................................................
64
Figure 2-8: (a) is a homogeneous isotropic model with 21 non-parallel fractures, red lines
indicate fractures and asterisk is the source. Parameters for the background medium are
shown in (a) and ZN and ZT are 5x 1010 m/Pa and 10-9 m/Pa, fracture height is 200 m, the
source wavelet is a Ricker wavelet with a 40 Hz central frequency; (b) and (c) show
snapshots of the divergence and curl of the scattered displacement field at 0.52 s.........65
Figure 2-9: P-to-P scattering strength for different ZT and different ZN/ZT. Horizontal and
vertical axes are incident angle and frequency. The scattering strength for each panel is
normalized to 1 for plotting, the number above each panel is the scaling factor (maximum
scattering strength). ............................................................................
66
Figure 2-10: Same as Figure 2-9 but for P-to-S scattering strength. ....................
67
Figure 2-11: Comparison of P-to-P (solid) and P-to-S (dashed) scattering at 20 Hz for different ZN/ZT at 20 (black) and 600 (blue) incidences. ZT is fixed at 10~9 m/Pa. The corresponding P-wave wavelength for 20 Hz is 200 m, which is the same as the fracture height.
. ..................................................................................................
. . 68
Figure 2-12: Comparison of the Fpp(6,o) of two fracture models with different fracture
heights at four different incident angles. The first and second rows, respectively, show the
Fpp(O,(o) of fractures with 100 m and 200 m lengths at the incident angles of 00, 300, 600
and 900. In each panel, Fpp(O,a) is normalized by its maximum scattering strength, which
is shown by the number below each panel. ZN and ZT of these two models are 1010 m/Pa,
other model parameters are the same as the model shown in Figure 1. .................. 69
Figure 2-13: Same as Figure 2-12 but for Fps(6,w). .......................................
70
Figure 2-14: Comparison of 2D and 3D fracture response functions at 300 incidence. ZT
and ZN are 10-9 m/Pa and 5x1040 m/Pa respectively. L=200m, 100m and 50m indicate
the fracture length in the fracture strike direction for three different models. Note that a
point source in 2D is equivalent to a line source in 3D, so the scattering strength of 2D
results is larger. ..................................................................................
71
Figure 2-15: P-to-P and P-to-S scattering strength as a function of frequency. The scattering strength is normalized by the maximum strength of all cases. Solid and dash curves
correspond to P-to-P and P-to-S scattering strength respectively. The value of ZN/ZT is
shown on top of each panel. ....................................................................
72
16
Figure 3-1: (a), (b), (c) and (d) Variations of AN (black solid curve) and AT (black dashed
curve) for sandstone, carbonate, shale and granite, respectively, with fracture compliance
Z4 ( -N,T) varying from 10-4 m/Pa to 10-9 m/Pa. In each panel, blue and red dashed lines
indicate the values of r/N and r/T, respectively. Properties of the four types of rock are
listed in Table 3.1. Both horizontal and vertical axes are in log scale. ................... 97
Figure 3-2: (a) (b) and (c) Variations of AN/AT (equation 3.19) with ZN for ZN/ZT =0.5, 0.1
and 0.05, respectively. Black, blue, red and magenta curves show the variations for sandstone, carbonate, shale and granite, respectively. The properties of rocks are listed in Table 3.1. Horizontal axes are in log scale. Vertical axes are in linear scale... ............... 98
Figure 3-3: Schematic showing the layout of sources and receivers surrounding a single
fracture (black bar) with 100 m length. Source (red star) is 550 m away from the fracture
center. Receivers (blue circles) covering 3600 are 500 m away from the fracture center. 0
is the source incident angle with respect to the fracture normal.....................
... 99
Figure 3-4: Absolute values of S (equation 3.29) and Sf (equation 3.30) for the two
fracture models in Table 3.2 at 00, 300, 600 and 900 incident angles. Background matrix is
sandstone (Table 3.1). In these polar coordinate plots, the radial and angular coordinates
are frequency and radiation angle, respectively. The range of frequency in each panel is
from 0 Hz at the center to 60 Hz at the edge. The dashed white circle indicates the freav
aV
quency of 40 Hz. Red star indicates source position.
and - indicate the sensitivity
"ZN
aZT
fields for fracture normal and tangential compliances, respectively. Fracture length is 100
m............................................................
..
....................... 10 0
Figure 3-5: Absolute values of
(equation 3.29) and S (equation 3.30) for fracture
model 1 (Table 3.2) in four rock samples (Table 3.1) at 30 and 600 incident angles. Polar
plots are as described in Figure 3-4. Fracture length is 100 m.................. ............ 101
Figure 3-6: Red and green lines represent 100 m long vertical and horizontal fractures
centered at X=0 m and Z=1050 m, respectively. Red stars represent sources at Z=Om.
Black and blue lines represent a vertical well at X=-200m and a horizontal well at Z=1 100
m , respectively. ....................................................................................
102
Figure 3-7: Energy distribution for a vertical fracture. al, a2 and a3 show the recorded Pwave energy (equation 3.20) at the surface, in a horizontal well and in a vertical well, respectively, for 41 shots with source horizontal position varying from -1 km to 1 km. bl,
b2 and b3 show the corresponding S-wave energy (equation 3.21). Model geometry is
shown in Figure 3-6. .......................
........................................
103
17
Figure 3-8: P- and S-wave sensitivity energies (equations 3.25 and 3.26) at the surface for
a vertical fracture are shown in a and b, for normal compliance, and c and d, for tangential compliance. EP and Es are multiplied by 4 for plotting. Model geometry is shown in
F igure 3-6. ..........................................................................................
104
Figure 3-9: Same as Figure 3-8 but for the sensitivity energy in the horizontal well. EP
and ES are multiplied by 4 for plotting. Model geometry is shown in Figure 3-6........105
Figure 3-10: Same as Figure 3-8 but for the sensitivity energy in the vertical well. EP and
Es are multiplied by 4 for plotting. Model geometry is shown in Figure 3-6.............106
Figure 3-11: aO and bO show the X and Z displacement components recorded in the vertical well (VSP) of the model containing a vertical fracture for a shot at X=-1000 m. al,
a2 and a3 show the changes of the X displacement when the fracture compliances (ZN and
ZT) increase by 10%, 20% and 30%, respectively. bl, b2 and b3 show the corresponding
changes of Z displacement. Scales are the same for all plots. Model geometry is shown in
F igure 3-6. ..........................................................................................
107
Figure 3-12: Black, red and blue curves show the percentage changes of the seismic scattered waves recorded in the vertical well when a vertical fracture is present and the fracture compliance increases by 10%, 20% and 30%, respectively. ........................... 108
Figure 3-13: Energy distribution for a horizontal fracture. al, a2 and a3 show the recorded P-wave energy (equation 3.20) at the surface, in a horizontal well and in a vertical
well, respectively, for 41 shots with source horizontal position varying from -1 km to 1
km. bi, b2 and b3 show the corresponding S-wave energy (equation 3.21). Model geometry is show n in Figure 3-6. .......................................................................
109
Figure 3-14: P- and S-wave sensitivity energies (equations 3.25 and 3.26) at the surface
for a horizontal fracture are shown in a and b, for normal compliance, and c and d, for
tangential compliance. EP and ES are multiplied by 8 for plotting. Model geometry is
show n in Figure 3-6. ..............................................................................
110
Figure 3-15: Same as Figure 3-14 but for the sensitivity energy in the horizontal well. EP
and ES are multiplied by 8 for plotting. Model geometry is shown in Figure 3-6........111
Figure 3-16: Same as Figure 3-14 but for the sensitivity energy in the vertical well. EP
and ES are multiplied by 8 for plotting. Model geometry is shown in Figure 3-6........112
18
Figure 3-17: aO and bO show the X and Z displacement components recorded in the vertical well (VSP) of the model containing a horizontal fracture for a shot at X=0 m. al, a2
and a3 show the changes of the X displacement when the fracture compliances (ZN and
ZT) increase by 10%, 20% and 30%, respectively. b I, b2 and b3 show the corresponding
changes of Z displacement. Scales are the same for all plots. Model geometry is shown in
113
F igure 3-6 . ..........................................................................................
Figure 3-18: Black, red and blue curves show the percentage changes of the seismic scattered waves recorded in the vertical well when a horizontal fracture is present and the
fracture compliance increases by 10%, 20% and 30%, respectively. ..................... 114
Figure 4-1: Cartoon showing how incident waves I are scattered by a fractured layer,
which is denoted by the gray horizontal zone with nearly vertical black lines. O and 02
are, respectively, waves reflected by layers above and below the fractured layer, whose
reflectivities are r 1 and r 2, respectively. T is the transmitted waves. ........................ 144
Figure 4-2: Photo of the Lucite model. The cuts representing fractures can be seen about
half way along the block in the vertical direction..............................................145
Figure 4-3: (a) Schematic showing the geometry of the Lucite model. (b) Expanded view
of the fracture zone. The fracture zone was built by cutting parallel notches with 0.635
cm (± 0.05 cm) spacing at the bottom of the upper Lucite block. The properties of the upper and lower Lucite blocks are identical. P- and S-wave velocities of Lucite are 2700
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
rn/s and 1300 m/s, density is 1180 kg/m . .................... . . . . .
Figure 4-4: (a) Source wavelet excited from the transducer. (b) Amplitude spectrum of
the source wavelet. The amplitude spectrum is expressed in decibels by evaluating ten
times the base- 10 logarithm of the ratio of the amplitude to the maximum amplitude... 147
Figure 4-5: Schematic showing the layout of CMP measurements at the surface of the
Lucite fracture model. CMP measurement was conducted at 10 different azimuths, the
acquisition lines of 00 and 900 correspond to the orientations parallel to and normal to the
148
fracture strike, respectively. .....................................................................
Figure 4-6: (a) and (b) Data acquired at the surface of the Lucite model in the direction
normal and parallel to the fracture strike, respectively. For each azimuth, 8 traces at different offsets are plotted. (c) Data measured in a region without fractures. In (b), labels
identify (1) direct P arrivals, (2) surface waves, (3) P-to-S converted waves from the interface below the fracture zone and (4) P-wave reflections from the model bottom......149
19
Figure 4-7: (a) CMP stacks at 10 different azimuths. (b) Expanded view of the signals in
the red window of (a). The acquisition angles are denoted above each trace, 'Parallel'/'Normal' indicates the acquisition is parallel/normal to the fracture strike. 'Average'
and 'Reference' represent the average of the ten black traces and the stack of traces collected in a region without fractures, respectively. ............................................
150
Figure 4-8: (a) and (b) FTF obtained using the average of CMP stacks and the CMP
stack at normal direction to represent the background reflection, respectively. 00 and 900
represent the directions parallel and normal to the fracture strike, respectively. Color bar
is in linear scale. ...................................................................................
15 1
Figure 4-9: Solid and dashed black curves are the average of FTF shown in Figures 4-8a
and 4-8b, respectively. ...............................
...........................................
152
Figure 4-10: Five layer model. Red star and black triangles are source and receivers respectively. Properties of each layer are listed in Table 4-1. (Modified from Willis et al.
(2006)). .............................................................................................
153
Figure 4-11: Layout of parallel fractures (black lines) in the third layer of the model
shown in Figure 4-10. Fractures with equal spacing are vertical and parallel to the X-axis.
........................................................................................................
15 4
Figure 4-12: (a) and (b) Shot records (pressure) acquired parallel (along X-axis) and
normal (along Y-axis) to the fracture strike, respectively. (c) Shot record obtained from a
reference model without fractures. Fracture spacing is 40 m. ............................... 155
Figure 4-13: CDP stacks of the model with 40 m fracture spacing. The numbers of traces
stacked at each azimuth are 5, 10 and 20 for (a), (b) and (c), respectively. The trace spacings are 80, 40 and 20 m for (a), (b) and (c), respectively. 00 and 900 are the directions
along X and Y axes, respectively. ..........................................................
156
Figure 4-14: CDP stacks of six models with different fracture spacings, which are shown
above each panel. 00 and 900 are respectively the directions along X and Y axes, respectively. In each panel, the blue trace is the average of the ten azimuthal stacks. The red
trace in (a) is the stack for the reference model without fractures. The dashed red window
is the selected window for computing FTF. The waves inside the red window arriving at
about 0.4 s are reflections from the bottom of the fractured layer. ......................... 157
20
Figure 4-15: FTF (equation 4.13) of six models with different fracture spacings. The
number above each panel denotes fracture spacing. 00 and 900 are respectively the direc158
tions along X and Y axes......................... ................................................
Figure 4-16: FTF of the six models with different fracture spacings. The number above
each panel denotes the fracture spacing. In each polar plot, the blue curve shows the azimuthal variation of FTF. Radial and angular coordinates are the value of FTF and azimuth, respectively. The radial coordinate is 0 at the center and 0.6 at the solid circle. 00
and 900 are the X and Y directions respectively. In plot (a), FTF is multiplied by 10 for
159
plottin g . .............................................................................................
Figure 4-17: Plot of the maximum of FTF for six models with different fracture spacings
when the fracture compliance varies from 10-10 m/Pa to 10-9 m/Pa........................160
Figure 4-18: Layout of orthogonal fractures (black lines) in the third layer of the model
shown in Figure 4-10. Two sets of vertical fractures intersect with each other in this model. They are parallel to the X and Y axes respectively. Each fracture set has equal fracture
161
spacing and com pliance. .........................................................................
Figure 4-19: CDP stacks of six orthogonal fracture models with different fracture spacings and compliances. Dx and Dy indicate the fracture spacings for fracture sets X and Y,
respectively. 00 and 900 are respectively the directions along X and Y axes..............162
Figure 4-20: FTF of the six orthogonal fracture models. Dx and Dy indicate the fracture
spacings for fracture sets X and Y, respectively...............................................163
Figure 4-21: FTF of the six orthogonal fracture models. The radial coordinate is 0 at the
164
center and 0.5 at the solid circle............. ....................................................
Figure 4-22: Layout of orthogonal sub-vertical fractures (black lines) in the third layer of
the model shown in Figure 4-10. Fracture spacing, compliance and dip angle randomly
vary w ithin the specified ranges.................................................................165
Figure 4-23: FTF of 25 orthogonal fracture models with randomly varying fracture spacing, compliance and dip angle. The number above each polar plot denotes the model
number. The horizontal and vertical dashed lines in each plot indicate the X and Y direc166
tions, respectively. ................................................................................
21
Figure 4-24: (a), (b) and (c) P-wave velocities of the layer model shown in Figure 4-10
after adding random heterogeneity with 25 m, 50 m and 100 m correlation lengths, respectively. ..........................................................................................
167
Figure 4-25: FTF of 15 heterogeneous models containing orthogonal fractures with randomly varying fracture spacing, compliance and dip angle. The correlation lengths of the
heterogeneity are 25 m, 50 m and 100 m for results shown in the first, second and third
rows, respectively. The number above each polar plot denotes the model number......168
Figure 4-26: Em ilio field location. ..............................................................
169
Figure 4-27: Emilio ID P-wave velocity model...............................................170
Figure 4-28: Direction of the maximum horizontal stress derived from earthquake focal
mechanism solutions. (Modified from Gerner et al. (1999)) ................................ 171
Figure 4-29: Emilio field acquisition configuration. Red and blue lines are the source and
receiver lines, respectively. The green region is the target region where wells have been
drilled . ..............................................................................................
172
Figure 4-30: Seismic profile (vertical component) for a shot at Inline 1376 and Xline
2796. The most prominent event is the reflection from the Gessoso-Solfifera at about 2
seconds. The data are strongly contaminated by the ocean bottom Scholte waves at small
offsets and water multiples at large offsets. ...................................................
173
Figure 4-31: Gessoso horizon picked from Kirchhoff pre-stack depth migration.........174
Figure 4-32: Number of traces at each CMP bin whose size is 50 x.50 m2 . ......
. . . . . . ..
175
Figure 4-33: Average number of traces stacked at each azimuth at each CMP bin......176
Figure 4-34: FTF at four locations. Radial and angular coordinates are frequency and
azimuth, respectively. 'N', 'E' and 'W' indicate the directions of north, east and west,
respectively. Color bar is in linear scale. ......................................................
177
Figure 4-35: Amplitude spectrum of the field data. ..........................................
22
178
Figure 4-36: Map of fracture orientations derived from the FTF. Dark and light background colors indicate high and low confidences of the result, respectively. White bars
represent the fracture orientation. Two intersecting white bars indicate the orientations of
179
two potential crossing fracture sets. ............................................................
Figure 4-37: Comparison of the fracture directions derived from well data (upper row)
................................. 180
and FTF (lower row) in three wells.................
Figure 5-1: Workflow for computation of stress-induced anisotropy around a borehole.
'FEM' and 'M' represent finite-element method and the method of Mavko et al. (1995),
195
respectively. See text for explanation. ..........................................................
Figure 5-2: Schematic showing two ways to predict the velocity under tensile stress. Solid curves represent the data measured in a compression (stress>0) experiment, dashed
lines indicate the extrapolation of data to tensile stress (stress<0) region. (a) velocity decreases with the decreasing of stress and (b) velocity is constant when stress<0.........196
Figure 5-3: Seismograms recorded for measuring the compressional and shear velocities
of the Berea sandstone sample. Acoustic measurements were conducted in three orthogonal directions. Pi (i=X,Y,Z) indicates the measurement of P-wave along i direction,
and Si; (ij=X,Y,Z) indicates the measurement of S-wave along i direction with polariza197
tion in j direction. .................................................................................
Figure 5-4: Measurements of P-wave (squares) and S-wave (triangles and circles) velocities of the Berea sandstone core sample under hydrostatic compression. All measurements were conducted along the core axis direction. Shear wave velocities S, and S2 were
measured along the same propagation direction but with orthogonal polarization directions. Blue and red curves are the fitting curves (equation 5.5) to the P- and S-wave velocities (average of S1 and S2), respectively. The root-mean-square misfits are, respectively, 38 m/s and 18 m/s for the fits to P- and S-wave velocities................................198
Figure 5-5: Photo of experiment setup. The Berea sandstone sample has dimensions
101.4(X)xl00.6(Y)x102.3(Z) mm3 . The size of the strain gage is about 2 mm. The aluminum foil between the press and the rock is used to make the loading pressure distribu199
tion m ore uniform on the rock surface.......... ................................................
Figure 5-6: Average normal strains of the intact rock sample under uniaxial loading. Solid and open squares are the measured strain in the directions parallel and normal to the
loading stress respectively. Error bars represent estimates of errors from uncertainty in
the measurement of loading stress (~5%) and the error of the gage factor (-%). Solid
curves and dashed curves are the predicted values obtained from the anisotropic model
200
and isotropic model, respectively. ..............................................................
23
Figure 5-7: Schematic showing uniaxial stress loading on a rock sample with a borehole.
The borehole axis, which is along the X-axis, is normal to the loading stress direction.
Strain measurements are conducted at locations A, B, C and D. B and C are 20 mm away
from the borehole center, A and D are 30 mm away from the borehole center. Borehole
radius is 14.2 m m. .................................................................................
20 1
Figure 5-8: Convergence of the iteration scheme under different loading stress strength.
Convergence, which is defined by equation 5.1, describes the percentage change of the
model stiffness after each iteration..............................................................202
Figure 5-9: (a) and (b) show the distribution of oay and q, which are the normal stresses
in Y and Z directions, under 10.56 MPa uniaxial stress loading in the Z direction (see
Figure 5-7). (c) is the sum of ay, and az.. ......................................................
203
Figure 5-10: Comparison of laboratory measured strains and numerical results at locations A, B, C and D, which are shown in Figure 5-7. Solid and open squares are the
measured strain in the directions parallel and normal to the loading stress respectively.
Error bars represent estimates of errors from uncertainty in the measurement of loading
stress (-5%) and the error of the gage factor (-%). Solid curves and dashed curves are
the predicted values obtained from the anisotropic model and isotropic model, respectively . ...................................................................................................
20 4
Figure 6-1: Borehole model geometry. A piston source (red circle) is located at the borehole center. Receivers (blue circles) are 0.7 cm away from the borehole center along the
piston source direction, which is indicated by the red arrow. A uniaxial stress is applied
in the X direction. Borehole diameter is 2.86 cm. ............................................
223
Figure 6-2: Schematic showing the 1/4 inch diameter piston source used in the simulation. Arrows indicate source excitation direction. Dashed curve represents a Hanning
window, which is used to taper the source amplitude from the center to the edges. Source
direction i is measured from the positive X direction. .......................................
224
Figure 6-3: Snapshot showing the pressure field in the borehole excited by a piston
source pointing at 300. Red and blue colors indicate peak and trough, respectively......225
Figure 6-4: Color hue of each box indicates the average value of the corresponding component (Cij) in the stiffness tensor of the formation around a borehole under 10 MPa
stress loading. The components inside the dashed green lines are about two orders of
m agnitude larger than the others. ................................................................
226
24
Figure 6-5: Variations of the nine dominant components of the stiffness tensor around a
borehole on the X-Y plane when a 10 MPa uniaxial stress is applied in the X direc2 27
tio n ...................................................................................................
Figure 6-6: Solid and dashed curves show the numerical error for P- and S-wave velocities, respectively, on the X-Y plane at four frequencies. V and V,"d represent the true velocity (i.e. formation velocity) and the grid velocity (i.e. velocity in numerical simula228
tion), respectively. .................................................................................
Figure 6-7: (a) Comparison of the seismograms simulated from two models respectively
containing 21 (dashed red) and 9 dominant (solid black) elastic constants for a piston
source at 300 and 10 MPa stress loading; (b) Difference (multiplied by 105) between the
229
seism ogram s show n in (a). .......................................................................
Figure 6-8: Pressure profiles recorded in the borehole for sources at ten different directions when a 10 MPa uniaxial stress is applied in the X direction (i.e. 00). The number
above each panel indicates the source direction. 00 and 900 are along the X and Y direc230
tions, respectively. ................................................................................
Figure 6-9: Schematic showing the wave paths of direct (dashed line) and refracted (solid
lines) P-waves. Z is the vertical distance between source and receiver. O is the critical
angle for refraction to occur at the wellbore. r is borehole radius. Red star and blue circle
231
represent source and receiver, respectively. ...................................................
Figure 6-10: (a) and (b) Seismograms recorded at Z=7 cm and 10 cm, respectively, when
the model is subjected to 10 MPa stress loading. Source is at Z= 0 cm. Red circles indicate the arrival times of the refracted P-wave. 00 and 900 are along the X and Y axes di232
rections, respectively. .............................................................................
Figure 6-11: Azimuthal variation of the normalized P-wave velocity (normalized by the
velocity measured at zero stress state) in a borehole under 10 MPa uniaxial stress loading.
Squares and circles are the P-wave velocities obtained from the numerical simulation and
the experiment of Winkler (1996), respectively. P-wave velocities are determined by
measuring the delay time between receivers located 7 and 10 cm along the Z axis from
233
the source. Applied stress is along 00 and 1800. ..............................................
Figure 6-12: Azimuthal variation of the normalized P-wave velocity (normalized by the
velocity measured at zero stress state) around a borehole at three different loading
stresses. (a) shows the results obtained from the finite-difference simulations. (b) shows
the best fits to the laboratory measured data (modified from Winkler (1996)). .......... 234
25
Figure 6-13: (a), (b) and (c) Advance of the first break arrival time between sources at 00
and other directions for 5, 10 and 15 MPa uniaxial stresses, respectively. Horizontal and
vertical axes are source direction and receiver depth, respectively. ........................ 235
Figure 6-14: Seismograms recorded at ten azimuths and at different offsets for 10 kHz
monopole source and 10 MPa loading stress. Source is at Z=0 m. Azimuth indicates the
receiver angle measured from the loading stress direction, which is along 00. The refracted P-waves are marked by the two dashed red lines in each panel. .................. 236
Figure 6-15: (a), (b), (c) and (d) Time semblances of data recorded at 00, 300, 600 and 900,
respectively, for 10 kHz monopole source and 10 MPa stress loading. 'P' and 'S' indicate
the refracted P- and S-waves, respectively. ...................................................
237
Figure 6-16: Expanded view of the refracted P-wave semblances for data recorded at four
different azimuths, which are denoted above each column. The first, second, and third
rows show the results for 10, 15, and 20 kHz monopole sources, respectively. Red and
blue colors respectively indicate high and low amplitudes. All plots are in the same color
scale. ................................................................................................
2 38
Figure 6-17: Same as Figure 6-16 except that the refracted S-wave semblances are plotted. ...................................................................................................
23 9
Figure 6-18: (a), (b) and (c) Dispersion of the Stoneley wave recorded at ten different
azimuths for 5, 10 and 15 MPa loading stresses, respectively. Loading stress is along 00.
........................................................................................................
24 0
Figure 6-19: (a), (b), (c) and (d) Variations of the refracted P-wave amplitude versus azimuth for data recorded at Z=2, 3, 4 and 5 m, respectively. The 10 kHz monopole source
is located at Z=0 m. Black, red and blue curves show the results for 5, 10 and 15 MPa
loading stresses, respectively. The amplitudes of all data are normalized by the same
valu e. ................................................................................................
24 1
Figure 6-20: Same as Figure 6-19 except that results for the 15 kHz monopole source are
plotted. ........................................................................
242
Figure 6-21: Same as Figure 6-19 except that results for the 20 kHz monopole source are
p lotted . .............................................................................................
24 3
26
Figure 6-22: (a) and (b) Simulated seismograms for dipole sources at 00 and 900, respectively. Receivers are positioned at the corresponding dipole source direction. ........... 244
Figure 6-23: Dipole flexural wave dispersion for three different loading stresses. Solid
and open circles represent the dispersion of the flexural wave at 00 (stress direction) and
245
900, respectively. ..................................................................................
27
28
List of Tables
Table 3.1: Properties of four types of rock...................................................96
Table 3.2: Compliance values for two fracture models........................................96
Table 4.1: Parameters of the model shown in Figure 4-10...................................143
Table 4.2: Parameters of six orthogonal fracture models.....................................143
Table 5-1: Summary of parameters of the Berea sandstone sample in an unstressed state
.........................................................................................................
29
19 4
30
Chapter 1
Introduction
1.1 Objective and approach
Characterization of fractures in an oil or gas reservoir is very important for field
development and exploration because natural fracture systems can dominate the fluid
drainage pattern in a reservoir. A fracture refers to a local discontinuity surface or
defect that is caused by stress exceeding the rock strength and causing the rock to lose
cohesion along a plane that meets the sliding condition. Propagation of a fracture
usually starts from micro-cracks, which are the weakest part of a rock, when the stress
concentration at the crack tips exceeds the rock strength (Jaeger et al., 2007). The
region confined by the two irregular surfaces of a fracture can be modeled as a
localized zone containing a large number of micro-cracks, each with size that is
orders of magnitude smaller than the seismic wavelength (Hudson and Heritage, 1981;
Oda, 1982; Wu, 1982; Wu, 1989; Brown, 1995; Benites et al., 1997; Yomogida et al.,
31
1997; Liu and Zhang, 2001; Yomogida and Benites, 2002). Thus micro-cracks are the
origin and also micro-scale components of fractures and they can be treated as microfractures with similar boundary conditions of a large scale fracture (Mavko et al.,
1995; Schoenberg and Sayers, 1995; Sayers, 1999; Sayers, 2002). However, the
effects of fractures and micro-cracks on seismic wave propagation are very different
due to the difference of their dimensions relative to the seismic wavelength. Fractures
having characteristic lengths on the order of the seismic wavelength are seismic
scatterers
that generate scattered seismic waves (Fang et al.,
2013),
whose
characteristics can be used to characterize the fracture's geometrical and mechanical
properties (Willis et al., 2006; Bums et al., 2007; Grandi et al., 2007; Zheng et al.,
2011; Fang et al., 2012; Zheng et al., 2012). Micro-cracks, which are the compliant
part of a rock, cause stress-dependence of the formation elastic properties (Mavko et
al., 2003). The stress dependence becomes more prominent when the local in situ
stress becomes complicated, such as in a borehole environment (Sinha and Kostek,
1996; Winkler, 1996; Tang et al., 1999; Brown and Cheng, 2007), and can bias the
formation velocity measurements (Mavko et al., 1995; Sayers, 1999; Sayers, 2002).
We study different aspects of the effects of fractures and micro-cracks on
geophysical exploration in the thesis. For fractures, we conduct research on their
scattering characteristics and develop methods to monitor their changes during
hydraulic fracturing and to determine their properties from seismic scattered waves.
For micro-cracks, we study their effect on formation property measurements from
borehole sonic logging, which is conducted in a complicated environment with
spatially varying stress field induced by drilling.
32
Traditional seismic methods for fracture characterization, such as AVOA (Riger
1998; Shen et al. 2002; Hall & Kendall 2003; Liu et al. 2010; Lynn et al. 2010) and shear
wave birefringence (Gaiser & Van Dok 2001; Van Dok et al. 2001; Angerer et al. 2002;
Crampin & Chastin 2003; Vetri et al. 2003), are based on the equivalent medium theory
with the assumption that fracture dimensions and spacing are small relative to the seismic
wave length, so a fractured unit is equivalent to a homogeneous anisotropic medium.
However, fractures on the order of the seismic wave length are also very important for
enhanced oil recovery, and they are one of the important sources of scattered seismic
waves. In the thesis, we conduct research on the characteristics of fractures on this scale.
We first use a finite-difference method to study the sensitivity of scattered seismic waves
to the geometrical and mechanical properties of a fracture with characteristic length on
the order of seismic wavelength. We also introduce the fracture sensitivity wave equation
to study the sensitivity of time-lapse seismic data to changes in fracture compliance
accompanying hydraulic fracturing. Based on the characteristics of fracture scattering, we
develop an approach to determine the fracture orientation through calculating the fracture
transfer function using surface recorded seismic coda waves. The applicability and
accuracy of our method is validated through both numerical simulations and laboratory
experiments. Our results show that fracture direction can be robustly determined by using
our approach even
for heterogeneous models
containing complex non-periodic
orthogonal fractures with varying fracture spacing and compliance. The application of
our method to the Emilio Field shows that the fracture orientations derived from the
fracture transfer function are consistent with well data.
33
Micro-cracks play an essential role in controlling the stress-dependence of formation
elastic properties (Jaeger et al., 2007). The crack-related stress-dependent effect becomes
more prominent in borehole environments as formation elastic properties near a borehole
could be altered from their original state due to the stress concentration around the
borehole. Accurate measurements of formation velocity from sonic logs are essential for
field production because they provide direct measures of rock properties that are taken as
true values in well-ties to constrain the reservoir velocity model constructed from other
data. However, the concentration of stress around a borehole caused by drilling can lead
to a biased estimation of formation elastic properties measured from sonic logging data.
A thorough analysis of the sonic logs needs to consider the constitutive relationship
between an anisotropic applied stress field and the stiffness tensor of a rock with microcracks embedded in the matrix (Brown and Cheng, 2007), which is not considered in most
previous studies (Sinha and Kostek, 1996; Winkler et al., 1998; Tang et al., 1999). To
study the effect of stress concentration around a borehole on sonic logging, our first step
is to develop an iterative approach, which combines a rock physics model and a finiteelement method, to calculate the stress-dependent elastic properties of the rock around a
borehole when it is subjected to an anisotropic stress loading. The validity of this
approach is demonstrated through laboratory experiments on a Berea sandstone sample.
Then we use the anisotropic elastic model obtained from the first step and a finitedifference method to simulate the acoustic response in a borehole. We first compare our
numerical results with published laboratory measurements of the azimuthal velocity
variations caused by borehole stress concentration and find very good agreement. Both
numerical and experimental results show that the variation of P-wave velocity versus
34
azimuth has broad maxima and cusped minima, which is different from the presumed
cosine behavior that is expected from the azimuthal dependence of elastic properties
around the borehole. This is caused by the preference of the wavefield to propagate
through a higher velocity region. We then study the borehole azimuthal acoustic
responses for monopole and dipole sources. The monopole simulation results show that
mode velocity is sensitive to the azimuthal variation of formation properties only when
frequency is above about 15 kHz, but the refracted P-wave amplitude has large azimuthal
variation at all source frequencies studied in the simulation and is much more sensitive to
the azimuthal variation of formation properties than the mode velocity. The dipole
simulations show that the overall velocity of the flexural wave increases with increasing
loading stress but the frequency of the flexural wave dispersion crossover is not sensitive
to the loading stress strength.
35
1.2 Outline of thesis
The thesis is divided into 7 chapters. In Chapters 2, 3 and 4, we discuss fracture
scattering analysis and characterization of fractures whose dimensions are on the order of
the seismic wavelength. In Chapters 5 and 6, we study the effect of micro-cracks and
stress on borehole sonic logging measurements.
In Chapter 2, we study the sensitivity of seismic waves to the properties, such as
fracture orientation, fracture compliance and fracture
size, of a fracture with
characteristic length on the order of seismic wave length. A finite-difference method is
used to numerically calculate the seismic response of a single finite fracture with a linearslip boundary in a homogeneous elastic medium.
In Chapter 3, we study the sensitivity of seismic waves to changes in fracture
compliances by analyzing and numerically solving the fracture sensitivity wave equation.
We analyze the relationship between the changes of fracture compliance and seismic
time-lapse data in hydraulic fracturing for different types of reservoir rock and also
discuss the effects of different fracture and acquisition geometries on seismic time-lapse
monitoring.
In Chapter 4, we present an approach for detecting fracture orientation through
computing the fracture transfer function using surface recorded seismic scattered waves.
The validity and accuracy of this approach is demonstrated using both synthetic and
experimental data. We also show results for the application of this approach to the Emilio
Field in the Adriatic Sea of Italy.
36
In Chapter 5, we develop an iterative numerical approach to calculate the stressinduced anisotropy around a borehole by combing a rock physics model and a finiteelement method. The accuracy of the approach is validated through comparing the
numerical results with laboratory measurements on a Berea sandstone sample, which
contains a borehole and is subjected to a uniaxial stress loading.
In Chapter 6, we use the approach developed in Chapter 5 to construct an
anisotropic elastic borehole model that is subjected to an anisotropic stress loading, and
then use a finite-difference method to simulate the acoustic response in a borehole. The
numerical results are compared with published laboratory measurements of the azimuthal
velocity variations caused by borehole stress concentration. We then discuss the influence
of stress-induced anisotropy on monopole and dipole sonic logging.
In Chapter 7, we first draw conclusions for the work included in this thesis, then
discuss the contributions made in the thesis and the future work on this topic.
37
38
Chapter 2
Sensitivity Analysis of Fracture
Scattering'
We use finite difference modeling to numerically calculate the seismic response of a
single finite fracture with a linear-slip boundary in a homogeneous elastic medium. We
use a point explosive source and ignore the free surface effect, so the fracture scattered
wave field contains two parts: P-to-P scattering and P-to-S scattering. The elastic
response of the fracture is described by the fracture compliance. We vary the incident
angle and fracture compliance within a range considered appropriate for field
observations and investigate the P-to-P and P-to-S scattering patterns of a single fracture.
We show that P-to-P and P-to-S fracture scattering patterns are sensitive to the ratio of
normal to tangential fracture compliance and incident angle, while the scattering
amplitude is proportional to the compliance, which agrees with the Born scattering
'(the bulk of this Chapter has been) published as: Fang, X.D, M. Fehler, T.R. Chen, D.R. Bums, and Z.Y.
Zhu, 2013. Sensitivity analysis of fracture scattering, Geophysics, 78, TI-T10.
39
analysis. We find that, for a vertical fracture, if the source is located at the surface, most
of the energy scattered by the fracture propagates downwards. We also study the effect of
fracture height on the scattering pattern and scattering amplitude.
2.1
Introduction
The understanding of seismic scattering from fractures is very important in reservoir
fracture characterization. Analytical solutions for scattering from realistic fractures are
not available. Scattering from a system of fractures involves not only scattering from
individual fractures but also interaction of the scattered wavefields with other fractures in
the system. Examination of the wave field scattered by fractures has been carried out by
Willis et al. (2006) and Grandi-Karam (2008) who showed how to process the complex
scattered wavefield to determine some characteristics of a fractured layer. They were
only interested in the spatial-temporal pattern of the wavefield and did not study how
scattering is influenced by the properties of individual fractures. Scattering of seismic
waves from an object is influenced by the ratio of the characteristic scale of the object to
the incident wavelength, the geometry of the object, and the mechanical properties of the
object. We seek to better understand the scattering from a single fracture, which will
provide important insights for interpreting the observed scattered waves. The first step is
to understand scattering from individual fractures. Thus, in this paper, we present a
numerical study of the seismic response of a finite fracture.
A fracture usually refers to a localized zone containing lots of micro-cracks or
inclusions, each with size that is orders of magnitude smaller than the seismic
40
wavelength. The boundary element method (Benites et al., 1997; Yomogida et al., 1997;
Liu and Zhang, 2001; Yomogida and Benites, 2002), which solves for the accurate
solution by satisfying the boundary conditions, and the Born approximation method
(Hudson and Heritage, 1981; Wu, 1982; Wu 1989), which considers only the single
scattered field and ignores the multiply scattered field, have been used for the study of the
seismic response of cracks. The seismic response of a fracture can be approximated as the
superposition of the responses of individual cracks on the fracture plane. But it is not
practical to represent a fracture as a system of micro-cracks in a reservoir scale
simulation due to the tremendous computational cost. In numerical simulation, a fracture
is usually represented as an imperfectly bounded interface (Schoenberg, 1980) between
two elastic media.
The way fractures affect seismic waves depends on fracture mechanical
parameters, such as compliance and saturating fluid, and on their geometric properties,
such as dimensions and spacing. When fractures are small relative to the seismic
wavelength, waves are weakly affected by individual fractures. Effective medium theory
can be used to model the fractured layer as a zone comprised of many small fractures,
which is equivalent to a homogeneous anisotropic zone without fractures (Hudson, 1991;
Coates and Schoenberg, 1995; Schoenberg and Sayers, 1995; Grechka and Kachanov,
2006; Grechka, 2007; Sayers, 2009). When fractures are much larger than the seismic
wavelength, then we can take fracture interfaces as infinite planes and apply plane wave
theory to calculate their reflection and transmission coefficients and interface wave
characteristics (Schoenberg, 1980; Fehler, 1982; Pyrak-Nolte and Cook, 1987; Gu et al.,
1996). In field reservoirs, fractures having characteristic lengths on the order of the
41
seismic wavelength can be the scattering sources that generate seismic codas. SdnchezSesma and Iturrarain-Viveros (2001) derived an approximate analytical solution for
scattering and diffraction of SH waves by a finite fracture, and Chen et al. (2010) derived
an analytical solution for scattering from a 2D elliptical crack in an isotropic acoustic
medium. However, there are no analytical elastic solutions for scattering from a finite
fracture with a linear-slip boundary and characteristic length on the order of the seismic
wavelength. Although fractures are usually present as fracture networks in reservoirs, the
interaction between fracture networks and seismic waves is very complicated, scattering
from a single fracture can be considered as the first order effect on the scattered wave
field. Therefore, study of the general elastic response of a single finite fracture is
important for reservoir fracture characterization, and we will do it numerically in this
study.
Here, we adopt Schoenberg (1980)'s linear-slip fracture model (see Appendix A)
and use the effective medium method (Coates and Schoenberg, 1995) for finite-difference
modeling of fractures. In this model, a fracture is considered as an interface across which
the traction is taken to be continuous, yet displacement is discontinuous. The
displacement discontinuity vector and the traction vector are linearly related by the
fracture compliance matrix Z;. For a rotationally symmetric fracture, the fracture
compliance matrix only has two independent components: the normal compliance ZN and
the tangential compliance ZT. The accuracy of our numerical simulation software has
been validated by comparison with the boundary element method (Chen et al., 2012). In
our study, density and velocity are assumed to be constant over the whole model, and
fracture scattered waves are induced by the change of elastic property on the fracture.
42
2.2 Methodology
We assume the complete wave field recorded at receivers in the fracture model shown in
Figure 2-la is u(',t, 6in,), and the corresponding wave field recorded in the reference
model, Figure 2-lb, without a fracture, is
UO (-, t, Oinc ), then
i(r, t, inc = i
,t, Oinc) + -(?, t, Oinc)
(2.1)
where S(?, t, inc) is the scattered wave field, Oinc is the incident angle.
In the frequency domain, equation 2.1 can be written as
, Oin)
0)(,
= UO (jZ*,J,
inc) + S(w, inc)
(2.2)
where U, UO and S are the Fourier transformations of u, UO and S, respectively, and o is
angular frequency.
Thus, the scattered wave field can be expressed as
(),
U)* , O,inc)(23
O,inc ) = U(, (t, Oinc
We assume the source is a pressure point source and we ignore the Earth's free surface, so
the scattered wave field $ (
, inc ) includes two parts: P-to-P scattered wave field
Pp (V, U, Oinc ) and P-to-S scattered wave field ps (V,(,
inc).
In a homogeneous isotropic medium, the total displacement field in the frequency
domain can be expressed as
j(j, o)
= - (
V[V7.
U(?,
(
where Vp and Vs are P- and S-wave velocities.
43
)7 + (V)2 V x
[V x U(, a)]
(2.4)
In a homogeneous isotropic medium, we can separate the P- and S-wave energy by
simply calculating the divergence and curl of the whole displacement field, thus, equation
2.4 can be written as
Uiz,w,tGine)
=
UptiObinc) +
s(0,Oino
)
(2.5)
with
Up ( &, in)
V[V - U(r, O)]
= -
(2.6a)
is the P-wave displacement, and
Us(r, wOine) = (
Vx[VxU( r,w)]
(2.6b)
is the S-wave displacement.
Therefore,
$(,w
Bn
SpF
Sn
S,
,
(2.7)
nc
with
,Up)=
E,Wtine)
Sps(r, w, Oinc)
Note that
SP (
UO(VW, Oin
,
=
Us(, ,
-
0o iWOinc)
(2.8)
(2.9)
6Onc)
) is the reference wave field, and it has no S-wave component.
and g (,
, Ofi) ) are frequency dependent, and we wish to obtain the
far field fracture response function which is independent of the source pulse used in
simulation.
Thus, we write
|Spp (V,w,
n )i
ISpS(rw,)inc)
=
aFpp(0,
, Oinc)I(W, Oinc)
(2.10)
=
aFPs(OP,,6min)I((, 1 nc)
(2.11)
with
44
V =
1/r,
where Fpp(0,w, 6Oi)
and Fps(0,,
(2.12)
for 2D,
for 3D.
inc) are P-to-P and P-to-S fracture response
functions, respectively, a is the geometrical spreading factor which is a function of the
distance (r) from the receiver to the fracture center, and I(o, 0,c) is the incident wave
field recorded at the center of the fracture, 0 and Oie are radiation angle and incident
angle respectively.
We will fix 0i, when we calculate Fpp and Fps by evaluating the wavefield at a
fixed r from the fracture for all angle 0. For convenience, hereafter Fpp and Fps will only
be expressed as functions of 0 and o, but they depend on the incident angle.
The fracture response function Fpp(0,co) and Fps(,co) can be expressed as
Fpp (8, a)) =
'''C"
'p
aIt(w,Oine)I
Fps (0, o) =
'S(r,'W'&L"CI
(2.13)
(2.14)
Here, we want to emphasize that fracture response functions 13 and 14 are frequency
dependent but are source-wavelet independent, we can get the same solution for a given
frequency even though we use different source wavelets to numerically calculate 13 and
14. Fpp(0,wo) and Fps(0,wo) could be functions of frequency, radiation angle, incident
angle, fracture compliance and wavelength to fracture-length ratio. We visualize the
scattering pattern by plotting them in polar coordinates.
45
2.3
Numerical results and discussions
We first conduct 2D simulations using Vp= 4 km/s, Vs= 2.4 km/s and p = 2.3 g/cm 3 , and
we investigate scattering for frequencies ranging from 0 to 50 Hz. The source wavelet is a
Ricker wavelet with 40 Hz center frequency in our simulation. In all the 2D simulations,
sources and receivers are 550 m and 500 m away from the fracture center, respectively,
so the incident wavefield magnitude and geometrical spreading factor a are identical for
all simulations. Results for 3D simulations are discussed later.
2.3.1 Fracture scattering pattern
We first study the scattering from a single 200 m long fracture in a homogeneous 2D
medium. In this paper, we study fractures with ZT varying from 10-12 m/Pa to 10-'
m/Pa
(Worthington, 2007) and ZN/ZT varying from 0.1 to 1 (Lubbe et al., 2008; Gurevich et al.,
2009). We first investigate the influence of ZN/ZT on the scattering pattern by fixing the
tangential compliance ZT at 10-10 m/Pa and varying the normal compliance ZN from 1011
m/Pa to 1010 m/Pa. Figure 2-2 shows the P-to-P fracture response functions for five
different ZN/ZT at four different incident angles 6ic. For 300 and 600 incidences, P-to-P
back scattering changes significantly for different ZN/ZT. For 00 and 900 incidences, the Pto-P fracture scattering patterns are similar for different ZN/ZT. For all angles of incidence,
the P-to-P scattering magnitude increases with the increasing of ZN/ZT. When ZN/ZT
changes from 0.1 to 1, the P-to-P scattering magnitude increases by about an order for 00
and 900 incidences. For 300 and 600 incidences, the P-to-P scattering magnitude,
respectively, increases by about 5 times and 3 times when ZN/ZT varies from 0.1 to 1.
46
Figure 2-3 shows the corresponding P-to-S fracture scattering patterns. As opposite to Pto-P scattering, P-to-S forward scattering patterns change with ZN/ZT while P-to-S back
scattering patterns are similar for different ZN/ZT when the incident angle is larger than 00.
For 300 and 600 incidences, P-to-S back scattering is much stronger than P-to-S forward
scattering and the P-to-S scattering magnitude increases by about 2.5 times and 3 times,
respectively, when ZN/ZT varies from 0.1 to 1. For both P-to-P and P-to-S scattering, the
scattering strength increases with increasing compliance magnitude.
Figure 2-4 shows the comparison of Fpp(,w) and Fps(O,wO) for different
compliance values with ZN/ZT=l. By using ZT to normalize Fpp(6,wo) and Fps(O,O), the
scattering patterns for fractures with different compliance values are almost identical, but
the normalized amplitude of the case of 10-9 m/Pa compliance is smaller than that of the
others. In our numerical study, we compare Fpp(,o) and Fps(O,w)) for fractures with ZT
varying from 10-12 m/Pa to 10-9 m/Pa and ZN/ZT varying from 0.1 to 1. We find that, for a
given incident angle, if we only consider the influence of fracture compliance on the
fracture response functions (keeping other conditions, such as background medium,
fracture height, etc., unchanged), then the fracture scattering pattern is controlled by the
ratio of normal-to-tangential compliance (ZN/ZT). In other words, Fpp(O,aO) and Fps(O,wO)
of different fractures have similar scattering patterns if their ZN/ZT has the same value, but
the scattering strength depends on the magnitude of ZN and ZT. In Figure 2-4, the
magnitude of the fracture response function is linearly proportional to the fracture
compliance when the compliance is less than 10-9 m/Pa. A fracture of 10~9 m/Pa
compliance represents a strong scatterer which makes its scattering strength depart from
the linear variation trend. We will discuss this in a later section.
47
The belt-shaped pattern shown in Figure 2-3 at 00 incidence is caused by the
interference of the scattered waves from the two fracture tips. Figure 2-5 is a cartoon
showing the ray paths of the scattered waves from the fracture tips to a receiver for the
case of 00 incidence. Constructive interference appears at the receiver if the distance
difference S
(51
=ll
- 121 is integer multiple of wave length A. 61 can be expressed as
|l1 - 121 = V12 + 1 . d - sinO + d2/4 _ V12 - 1 - d - sinO + d2/4
The scattered waves from the fracture tips with wave length
=
(2.15)
1 (n=1,2,...) have
constructive interference at the receiver. We calculate the constructive interference
frequencies in all directions and plot them in polar coordinates in Figure 2-6. In Figure 26, radial and angular coordinates respectively are frequency and azimuth. The blue lines,
which represent the constructive interference frequencies, agree with the P-to-S scattering
pattern at 00 incidence. The fracture tip P-to-P scattering should have a similar
interference pattern, but the fracture tip P-to-P scattering at 00 incidence is relatively
weak compared to the entire scattered wave field, so we cannot see it in Figure 2-2. When
the incident angle departs from 00, the fracture tip scattering becomes weak compared to
the entire scattered wave field. Gibson (1991) modeled Born scattering from fractured
media using an approach similar to that described here.
From Figures 2-3 and 2-4, we can find that, when the incident angle is between 00
and 900, P-to-P forward scattering is much stronger than back scattering, however, for Pto-S scattering, back scattering is much stronger than forward scattering. In this paper,
forward scattering and back scattering refer to the scattering energy propagating to the
right side and left side (source on the left side) of the fracture, respectively. In the field,
most fractures are close to be vertical (Bredehoeft et al., 1976; Rives et al., 1992) and the
48
source is at the surface. In this case, as illustrated in Figure 2-7, the observed scattered
seismic waves at the surface can be separated into three parts based on the complexity of
their wave path: (1) the upward propagating scattered waves (mainly P-to-P scattering)
from fracture tips; (2) the downward propagating P-to-P scattered waves; (3) the
downward propagating P-to-S converted scattered waves. The downward propagating Pto-P and P-to-S scattered waves are reflected back to surface by reflectors below the
fracture zone. The P-to-P and P-to-S scattering energy, respectively, propagates in the
forward and backward directions with respect to the source, as shown in Figure 2-7. Both
fracture tip scattering and downward propagating scattered waves have been observed in
the laboratory experiments of Zhu et al. (2011).
Figure 2-8 shows a numerical simulation of wave propagation in a uniform medium
containing 21 non-parallel fractures, Figure 2-8a shows the geometry of the model,
Figure 2-8b and Figure 2-8c show snapshots of the divergence of the scattered
displacement wave field and curl of the scattered wave field at 0.52 s (the scattered wave
field is obtained by subtracting the whole wave field from the reference wave field of the
same model without fractures). We can see that most of the P-to-P scattering energy
propagates down and forward and most of the P-to-S scattering energy propagates down
and backward. In the field, we could have reflectors below the fracture zone to reflect the
down going scattered waves back to the surface. Therefore, in the field, most scattered
signals observed at the surface come from fracture tips and scattering energy reflected off
events below the fracture zone. If we want to use both P-to-P and P-to-S scattered waves
to study fractures, we should search for P-to-P and P-to-S scattered waves at different
offsets based on their scattering direction.
49
2.3.2 Scattering strength
We define scattering strength for a given frequency as the average of the fracture
response function over all radiation angles. Figure 2-9 shows the scattering strength of Pto-P scattering for different
ZT
and
ZN/ZT.
We find that P-to-P scattering is usually
stronger at small angles of incidence except for the case of a small
ZN/ZT
(0. 1). Figure 2-
10 shows the corresponding P-to-S scattering strength, where regardless of the variation
of ZN/ZT, P-to-S scattering is always strongest when the incident angle is about 400. The
strength of P-to-P scattered waves at small incident angles is mainly determined by
but the influence of
ZN
ZN,
on P-to-P scattered waves decreases with increasing incident
angle, and at intermediate incident angles,
ZT
has stronger impact on the scattered
wavefield, because P-to-S conversion at the fracture surface is more efficient at
intermediate incident angles (Gu et al., 1996), so stronger P-to-S scattered waves are
generated at intermediate incident angles. When
ZN«ZT
(e.g. ZN/Zr0.1), the scattered
wavefield is mostly influenced by ZT, ZN has relatively small impact on the incident
wavefield, therefore, the P-to-P scattering is strong only at intermediate incident angles
when ZN/ZT =0.1.
By comparing P-to-P and P-to-S scattering strength in Figures 2-9 and 2-10, we
find that P-to-S scattering is stronger than P-to-P scattering when
ZN/ZT
is smaller than
0.5. For both P-to-P and P-to-S scattering, the scattering strength increases about one
order of magnitude when the compliance increases
I order, the scattering strength and
compliance have a very good linear relationship when the compliance is smaller than 10-9
m/Pa, while the scattering strength deviates from this linear relationship a little when the
compliance is 10-9 m/Pa. We will discuss this discrepancy in the Born scattering analysis.
50
ZN/ZT is strongly influenced by the way the fracture surfaces interact, so this ratio
may be of use for fluid identification. Both numerical simulations (Sayers et al., 2009;
Gurevich et al., 2009) and laboratory measurements (Lubbe et al., 2008; Gurevich et al.,
2009) suggest that the compliance ratio ZN/ZT of reservoir fractures should be less than 1.
Based on laboratory experimental data, Lubbe et al. (2008) suggested that a ZN/ZT ratio of
0.5 is appropriate for simulation of gas filled fractures, and ZN/ZT can be less than 0.1 for
fluid saturated fractures. Figure 2-11 shows the comparison of P-to-P and P-to-S
scattering at 20 Hz, of which the corresponding P-wave wavelength is the same as the
fracture height, for different ZN/ZT at 200 and 600 incidences. At 200 incidence, P-to-S
scattering is stronger than P-to-P scattering when ZN/ZT is small, while this reverses with
the increasing of ZN/ZT. At 600 incidence, P-to-S scattering is always stronger than P-to-P
scattering regardless the change of ZN/ZT. From our simulations, we find that, when ZN/ZT
is <0.5, generally, P-to-S scattering is stronger than P-to-P scattering when the incident
angle is larger than 200. This implies that we could detect strong P-to-S scattered waves
at the surface although S waves generally attenuate more rapidly than P waves.
2.3.3 Effect of fracture height on fracture scattering
Figures 2-12 and 2-13 show comparisons of the fracture response functions of models
with two different fracture heights at different incident angles. Fracture heights are 100 m
and 200 m, which correspond to the wavelength of waves at frequencies of 40 and 20 Hz,
respectively. In these two models, except for the fracture height, all other model
parameters are the same. From these two figures, we can see that the dominant scattering
directions of two fractures with different fracture heights are very similar, but both the P-
51
to-P and P-to-S scattering patterns for the 100 m long fracture have a broader distribution
of high amplitude regions. So the scattering of a fracture will approach that of the
scattering from a point scatterer when the length of the fracture decreases, so the
scattering pattern of a shorter fracture at a given frequency is comparable to that of a
longer fracture at lower frequency. We also find that the scattering strength increases
with the increasing fracture height, such a pattern is evident in Figures 2-9 and 2-10,
which show that higher frequencies (shorter wavelengths) are more strongly scattered by
a fracture. Although we only show the comparison of two models, the same phenomena
can be observed from all other models.
2.3.4 Single fracture scattering in three-dimensions
We began our study on single fracture scattering using the 2D case because it is easier to
calculate and visualize the fracture response function in 2D. We now discuss scattering in
3D. In our 3D simulation, a fracture is represented as a rectangular plane with finite area,
all model parameters are the same as those used for the 2D simulation, sources are 550 m
away from the fracture center and receivers cover a spherical surface centered at the
fracture center with a 500 m radius. Because it is very time consuming to compute the 3D
fracture response function, we compute a suite of models with different incident angles,
then the fracture response function at any incident angle is obtained through
interpolation. Because both the incident and radiation directions are unit 3D vectors, it is
difficult to visualize the fracture response functions. From our numerical study, we find
that the P-to-P and P-to-S fracture scattered waves have characteristics that are generally
similar to those shown in Figure 2-7. A point source and a finite fracture in a 2D model
52
are equivalent to a line source and an infinite long fracture in 3D. In order to study the
influence of the finite fracture length on the scattered wavefield, in the 3D model, we
vary the fracture length in the fracture strike direction (the direction perpendicular to the
2D plane shown in Figure 2-1) from 50 m to 200 m while keeping the fracture height
(200 m) equal to that of the 2D model. Figure 2-14 shows the comparison of 2D and 3D
fracture response functions for a fracture with Z 7=10~9 m/Pa and Z=5x1O'0 m/Pa. We
only show a profile of the 3D fracture response function, which contains the source and is
perpendicular to the fracture plane and has the same geometry as the 2D model. From
Figure 2-14, we can see that, compared to the 2D case, the 3D fracture response functions
have a broader distribution of high amplitude regions, the 3D effect is similar to
shortening the fracture height in 2D. When the fracture length is changed from 200 m to
50 m, the scattering pattern does not change while the scattering strength decreases.
In order to quantitatively study the scattering strength, we average the fracture
response function of the 200 m x 200 m fracture over different incident angles and
radiation angles and take the mean value as the average scattering strength. Figure 2-15
shows the average scattering strength as a function of frequency and fracture compliance.
For each panel, ZN/ZT IS fixed, black, blue and red curves show the scattering strength for
different ZT. For a given ZT, both the P-to-P and P-to-S scattering strength increase with
increasing ZN, and the curves of P-to-P and P-to-S scattering strength merge when the
frequency increases. The P-to-S scattering is stronger than the P-to-P scattering for all
ZN/ZT. For both P-to-P and P-to-S scattering, the scattering strength is almost linearly
dependent on the compliance.
53
2.3.5 Born scattering analysis
We have numerically investigated the characteristics of single fracture scattering. In this
section, we use Born scattering approximation to derive the wave equation for fracture
scattered waves, which can help us to gain insights on the observations from the
numerical simulations.
The average strain of a rock containing fractures can be expressed as (Schoenberg
and Sayers, 1995):
(2.16)
(Sb + Sf)
where a and c are stress and strain, Sb and Sf are the background medium compliance and
the fracture induced extra compliance. Sf is a linear function of ZN and ZT, its expression
can be found in the paper of Schoenberg and Sayers (1995).
After simple algebraic manipulations, we can rewrite equation 2.16 as
a = [Cb - (I + S - Cb
-E
(2.17)
where C = SL-1 is the background stiffness and I is identity matrix.
By assuming Sf
Cb<1 and expanding the term (I + Sf - Cb)
in a Taylor series, we
obtain the first order approximation of equation 2.17 as
a ~ (C + Cf) - e
where Cf
=
-Gb
(2.18)
S - Cb is the extra stiffness caused by the presence of fractures.
Negative Cf means the medium becomes more compliant due to fractures.
Thus, the wave equation for the displacement wavefield
pnii, t) - ;[
54
Ui(i,t) is
+ Cl)
0
(2.19)
.
where cbjk and c i are the 4th rank tensor notations of Cb and Cf, and dj represents
We can solve equation 2.19 by using the Born approximation (Sato and Fehler, 2009).
The total wavefild
U'is written as a sum of an incident wave UG and a scattered wave Ui:
(2.20)
U +U1
U
The incident wave satisfies the homogeneous wave equation
pU
(t)
where Ekl is the strain associated with
a1(cdk1
-
(2.21)
0k
-E
Ul0.
Substituting equation 2.20 in equation 2.19 and using equation 2.21 and assuming
U Ill
I U 0 1,we get the wave equation for the scattered wave:
pi1 (-, t)
where the cross term cql -I
-
aj (C
kI
is neglected, Sfi
body force for the scattered wave field and
4)
(,
=
t) =
Sf (2,t
(]kIC
(2.22)
- Eki) is the equivalent
4kl is the strain associated with U'.
Because Sf is a linear function of ZN and ZT, and if ZN/ZT is fixed, then Sfi (, t) has the
same form for different compliance values except for being scaled by a constant factor,
which explains why the scattering pattern is independent of the compliance value when
ZNIZT is fixed.
From equation 2.22, we can infer that
U1 c Sf oc Cf = -Cb
- Sf - Cb OC Sf
(2.23)
Therefore, the amplitude of scattered wave is proportional to the fracture compliance.
However, the Born scattering approximation is based on weak scattering
assumption. If the extra fracture stiffness is comparable to or larger than the background
stiffness, a fracture is a strong scatterer. Born scattering approximation does not hold for
55
this case, and extra nonlinear terms have to be considered in the scattering wave equation,
which breaks the linear relationship between scattered waves and fracture compliance in
equation 2.22. This explains the discrepancies, which are observed in the numerical
study, between the relationship of the fracture response functions and compliance values
of very compliant fractures (>10-10 m/Pa) and less compliant fractures (<1010 m/Pa). For
a fracture system with large compliance, waves are trapped among fractures, so in a
sense, multiple scattering becomes important. In the next chapter, we will further discuss
the validity of weak scattering assumption for fracture scattering.
2.4
Summary
We studied scattering from a single fracture using numerical modeling and found that the
fracture scattering pattern is controlled by ZN/ZT and the fracture scattering strength varies
linearly with fracture compliance for fractures with compliance
-1-0
m/Pa. This is well
explained by the Born approximation. For compliance greater than about 10-10 m/Pa,
scattering strength does not scale with compliance. For small value of ZN/ZT (<0.5), P-toS scattering dominates P-to-P scattering when the incident angle is larger than 200; for
large value of ZN/ZT (>0.5), P-to-P scattering dominates. Due to the gravity of overburden
and the regional stress field, fractures tend to be vertical in a fractured reservoir. Since
seismic sources are normally located at the surface, waves scattered from vertical
fractures propagate downwards, specifically, the P-to-P scattering energy propagates
down and forward while the P-to-S scattering energy propagates down and backward.
56
Therefore, for a vertical fracture system, most of the fracture scattered waves observed at
the surface are, first scattered by fractures, and then reflected back to the surface by
reflectors located below the fracture zone, so the fracture scattered waves have complex
wave paths and are influenced by the reflectivity of the reflectors.
57
(a)
(b)
'4
U)***-.
S*0
*..0-'00--...0
%
0
**0000000
0000
0e,..
*........
km/s
=2.3g/c0
.. e*
*e..
*e.....e0
V =4 km/s Vs =2.4
Figure 2-1: (a) is the fracture model and (b) is the reference model; these two models are
exactly the same except for the presence of a fracture in (a) indicated by the vertical bar.
Solid circles are receivers and they are equidistant (500 m) from the fracture center, stars
indicate sources at different angles of incidence to the fracture. The distance between
receivers and fracture center is 2.5 times of the fracture height which is 200 m. Incident
angles are measured from the normal of the fracture (e.g. a source directly above the
fracture is considered to have a 900 incident angle).
58
0
UI,,,=W"
;,,e=30
Fppmax=0.283
Fppnax=0.452
Fppnitax-=0.378
Fppmax=0.0327
Fppnax=0.819
Fppnax=0.829
Frppax=0.523
Fppnax=0.0982
Fppmax=1.35
Fppmax=1.21
Fppmar-0.667
Fppnax=0.163
6;,,,=900
0.1
0.3
0.5
AMEMA0.7
Fppinax=1.88
Fppnax=1.59
Fppnax=0.812
Fp1 nmar-2.67
Fppynax=-2.15
Fppinar1.03
Fppnax=0.228
I
0
0.2
Figure 2-2: Fpp(O,eo) for different
0.6
0.4
Normalized strength
ZN/ZT
0.8
Fppnax=0.325
1
at different incident angles. Incident angles,
which are shown on top of the figure, are 00, 300, 600 and 900 for each column. The value
of ZN/ZT for each row is shown at the left side of each row. Fpp(6,>) is plotted in polar
coordinates, the radial and angular coordinates are frequency(o/(2n)) and 0 respectively.
The range of frequency in each panel is from 0 Hz to 50 Hz. The magnitude of each
fracture response function is normalized to 1 in plotting. Fppmax denotes the P-to-P
maximum scattering strength of each panel. ZT is fixed at 10-10 m/Pa, ZN varies.
59
Z-#/-T
'
'' ^"
'
^ ^^"
K AH
6,&~60u
0.1
FPFax=o.174 I
Fnar=0.912
Fpjrmax=0.649
Frna-=0.0986
Fppnai=01253
Frgna-1.22
FPPrnax=0.926
FPona=0.296
Fppnax=0.372
Fpvnwx=1.53
Fppnar=1.21
FPfnax=0.493
Fppnax-0.491
Fppnax=1.84
FPPuax=l.5
Fonx=O.689
0.3
0.5
0.7
I
F.w
Fppaax=-0.668
0
Fppinax=2.31
0.2
Fppoiax-1 93
0.6
0.4
Normalized strength
0.8
FpPinax=0.982
1
Figure 2-3: Same as Figure 2-2 except that Fps(6,w) is plotted. Fpsmax is the P-to-S
maximum scattering strength of each panel.
60
Z7 '=0~"
Z7=10-12 M/Pa
Zr10in/P
n/Pa
z7=10-11 M/Pa
Zy=10
Z=109 M/Pa
oM/Pa
Fppinax=0.0218
Fppmax=0.218
Fpjwar215
Fppinax=13 2
FPsnax=0.0234
Fjssnax=0.234
Fynaxa2 31
Fpgnax 14.8
i
W
0
5
10
15
Scattering strength
20
23.4
Figure 2-4: Comparison of Fpp(O,co) (1st row) and Fps(O,co) (2nd row) for fractures with
different compliance values at 300 incidence. ZN is equal to ZT for these models. Fpp(O,a)
and Fps(6,co) are normalized by the corresponding ZT value in plotting. Fppmax and
Fpsmax are, respectively, the P-to-P and P-to-S maximum scattering strength of each
panel.
61
receiver
fracture
11
OP
source
Figure 2-5: Cartoon showing the scattering from two fracture tips. 14 and
paths from the two fracture tips to the receiver,
fracture center, d is fracture height,
12
represent the
1 indicates the distance from receiver to
6 is the radiation angle with respect to the fracture
normal.
62
2
so
S
goS
ITO
Figure 2-6: Fracture tip P-to-S scattering interference at 00 incidence. Radial and angular
coordinates are frequency and azimuth. Red star indicates the source at 00 incidence. Blue
lines show the constructive interference frequencies, of which the corresponding wave
lengths satisfy A
=
11-12
n
(n=,2,...), of the P-to-S scattered waves from the two fracture
tips at all directions.
63
source
incident P-wave
scattering at fracture tips
fracture
P
S
layer interface
Figure 2-7: Cartoon showing how incident P-waves are scattered by a fracture. Scattering
energy mainly includes three parts: (i) fracture tip scattering; (ii) P-to-P forward
scattering; (iii) P-to-S back scattering.
64
(a)
(c)
divergence
(b)
model geometry
0
0
CUl
0
I IVIlI ROMiV\ 11
E
SV
1
S11
=4 km/s
km/s
V=2.4
S
3
p=2.3 g/cm
-1
0
offset (km)
1
0
offset (km)
-1
1
-1
0
offset (km)
1
Figure 2-8: (a) is a homogeneous isotropic model with 21 non-parallel fractures, red lines
indicate fractures and asterisk is the source. Parameters for the background medium are
shown in (a) and ZN and ZT are 5x10-
0
m/Pa and 10-9 m/Pa, fracture height is 200 m, the
source wavelet is a Ricker wavelet with a 40 Hz central frequency; (b) and (c) show
snapshots of the divergence and curl of the scattered displacement field at 0.52 s.
65
ZT=10-12 M/Pa
0.00109
1
-10
ZT=10- m/Pa
I
m/Pa
ZT=10~
0.0109
ZT=10~9 m/Pa
0.107
0.888
V-
1
0
30
60
60
0.0126
0
90
30
0.00126
90
0
60 90
30
113
Cv,
~30
120
10
10
0
30
60
90
0 w0 0 %00
0.00209
0.0208
30
60 90
0
30
6
0.207
1.65
0.407
233
3C
1C
0
30
60
901
0
30
0.00416
60
90
0.0416
V30
NZ
120
20
10
20
110
0 30 60 90
1
_
0.0
90
(deg)
0.2
ti (aegj
0.4
0.6
0.8
1L
0
30 60
e (deg)
90
0
30
60
90
0(deg)
1
Figure 2-9: P-to-P scattering strength for different ZT and different ZN/ZT. Horizontal and
vertical axes are incident angle and frequency. The scattering strength for each panel is
normalized to 1 for plotting. The number above each panel is the scaling factor
(maximum scattering strength).
66
Z=10~1 m/Pa
-12 M/Pa
ZT
0.963
0.122
0.0124
0.00125
50
Z =10~9 M/Pa
ZT=10~10 m/Pa
50
50
50
40
40
40
40
30
30
130
30
20
AA20
2020
N
'10
10
10
0
30
60
0
90
30
60
90
1o
0
60
0
90
so
50
50
150
40
40
40
40
30
30
30';
30,
20
20
20
20
10
10
10
110
N
0
30
90
60
0
30
60
0
90
60
90
0.17
0.0172
0.00173
30
0
50
50
50
50
40
40
140
140
10
30
10
10
0
30
0
90
60
0.00298
50
30
60
90
0.0297
50
0
5050
30
60
0
90
40
40
30
30
30
30
20
10
20
20
N
10
0o
10
0
30 60
0 (deg)
0.0
0.2
0.4
0
90
0.6
30 60
0 (deg)
0.8
90
0
30 60
0 (deg)
30
60
30
60
90
0
30 60
0 (deg)
1
Figure 2-10: Same as Figure 2-9 but for P-to-S scattering strength.
67
90
90
90
1.93
0.291
40
S
60
1.38
40
20
30
1.13
0.139
0.0141
0.00142
30
90
1.6
20 0
1 .4
..-- -
-- --- - -.
-----
60
S12 ----- -
0.1
0.2
0.3
-.
---
- ----
----
0.4
0.5
0.6
0.7
0.8
0.9
Figure 2-11: Comparison of P-to-P (solid) and P-to-S (dashed) scattering at 20 Hz for
different ZN/ZT at 200 (black) and 600 (blue) incidences. ZT is fixed at 1 0~ rn/Pa. The
corresponding P-wave wavelength for 20 Hz is 200 m, which is the same as the fracture
height.
68
0,=30C
Oiw=o
F.=603
F
0*.=9a0
0
0
0
1.19
Frpoalx=1. 5
Fppnax
Fppnax-2.67
Fppnax=2.15
Fprnax=0.585
Fppnav=0.219
Fppnax=-1.03
Fppnax=0.325
E
0
0
*0
0
0.2
0.6
0.4
0.8
1
Normalized strength
Figure 2-12: Comparison of Fpp(O,)
of two fracture models with different fracture
heights at four different incident angles. The first and second rows, respectively, show the
Fpp(O,wo) of fractures with 100 m and 200 m lengths at incident angles of 0', 300, 600 and
900. In each panel, Fpp(O,wo) is normalized by its maximum scattering strength, which is
shown by the number below each panel. ZN and ZT of these two models are 10-10 m/Pa,
other model parameters are the same as the model shown in Figure 2-1.
69
I
I
Om,=3OP
E
Fppnax=0.567
Fpprnarl.41
Fpponax 1.04
Fppoax=0.522
Fppynax0.668
Fppnax=2.31
Fppnax 1 93
Fpprnax=0.982
E
CD
0
0.2
0.4
0.6
0.8
1
Normalized strength
Figure 2-13: Same as Figure 2-12 but for Fps(O,wo).
70
|3D (L=-200 m)
2D
Fppmax=-9.8
Fpsax=12.2
0
|Fpiax=-3.24
Fpsnax=-5.4
0.2
3D (L=100 m)
3D (L=50 m)
Fppmnax=1.87
Fppnax=0.983
Fsnax=3.35
Fprynar=1.75
0.6
0.4
Normalized strength
0.8
1
Figure 2-14: Comparison of 2D and 3D fracture response functions at 300 incidence. ZT
and ZN are 10~9 m/Pa and 5x 1010 m/Pa respectively. L=200m, I00m and 50m indicate
the fracture length in the fracture strike direction for three different models. Note that a
point source in 2D is equivalent to a line source in 3D, so the scattering strength of 2D
results is larger.
71
(a)
(b)
lop
ZN/ZT = 0-1
............
:........................ ..............
.................
....
10-1
..
10-2
10 0
10-1
............ ......
.............
...............
........
......
.............
10-3
10-3
...........
..............
..................
........................................................ ......
........................
20
(C)
10
.G
.............. ......
60
10 -4
80
10 0
..........
1
10-2
10-3
10-4
.........................
...................
................ I........ .....................
............
...............
; ...........
40
60
freq (Hz)
Z, = 10-9 m/Pa
................................................... .......
...........................................................
....................... .........................
20
40
60
80
....................
..................
.............
..........
.............. .........
..........................
10-2
..........
..............................
. .... ...................
.... .
..................
..............
10 -3
..........
..........
......
..........
.................
.................. .............. ......
...............I ........................... ......
80
20
- - - - ZT = 10-'0 m/Pa
P-to-P scaftering
ZN/ZT = 1
7. ..............
... .............. .............
...............
........
.............. ................
............. ........
......................
......................
01
10-1
............
..................
..............
..........I.......
I............
................ ---------- ---.......... .............. ............... ......
..... .
...........
............ ............
I.... ..
..............
............. ..............
... ........... ................. :..........
r ............... ............
........
................. : ................... .
..................
.............. ............
................. ........
........ . .........
....................................... ... .........
...........
20
........
..00,
(d)
z N/ZT = 0-5
0
10-
..............
40
...........
...........
......... ......
-0 ..........
............................... ......
........... ..............
..........
.................
10 -2 ........
..............
, - ... ...
............
....
:--. ---...
......... --- --------........
..........
...........
..............
..................... : .............. ......
............;;:
..........
...... ;...
..........
.. .. . . ....................
...........
....
.......... :..
................... ...
.. . ..........
......
--------.................
---.................
...............
.
* ..........
0.3
...............
............ ..................
................
........ ......
.........
................. ......
.. .........
10-4
ZW/ZT
.............
................
- - - -
40
60
freq (Hz)
80
ZT = 10-1' m/Pa
- - - - - - P-to-S scaftering
Figure 2-15: P-to-P and P-to-S scattering strength as a function of frequency. The
scattering strength is normalized by the maximum strength of all cases. Solid and dashed
curves correspond to P-to-P and P-to-S scattering strength, respectively. The value of
ZNIZTis
shown on top of each panel.
72
Chapter 3
Sensitivity of Time-lapse Seismic
data to Fracture Compliance in
Hydraulic Fracturing2
In Chapter 2, we studied the characteristics of scattered waves from a single finite
fracture with a linear slip boundary condition. In this chapter, we study the sensitivity of
seismic waves to changes in fracture normal and tangential compliances accompanying
hydraulic fracturing by analyzing and numerically solving the fracture sensitivity wave
equation, which is derived by differentiating the elastic wave equation with respect to the
fracture compliance. The sources for the sensitivity wavefield are the sensitivity moments,
which are functions of fracture compliance, background elastic properties and the stress
2(the bulk of this Chapter has been) submitted as: Fang, X.D., X.F. Shang, and M. Fehler, 2013. Sensitivity
of time lapse seismic data to fracture compliance in hydraulic fracturing: GJI.
73
acting on the fracture surface. Based on the analysis of the fracture sensitivity wave
equation, we give the condition for the weak scattering approximation to be valid for
fracture scattering. We study the sensitivity of P- and S-waves to fracture normal and
tangential compliances respectively by separating the seismic wave field and the
sensitivity field into P- and S-components. In our numerical simulations, we study the
effect of fracture compliances, source incident angle and background elastic properties on
the sensitivity field. We also discuss the sensitivity of seismic data to the compliances of
vertical and horizontal fractures, respectively, for surface and borehole acquisitions.
Under the weak scattering approximation, we find that the percentage change of fracture
compliance in hydraulic fracturing is equal to the percentage change of the amplitude of
the recorded time-lapse seismic data. This could provide a means for designing and
interpreting experiments for monitoring the opening/closing of fractures in hydraulic
fracturing through time-lapse seismic surveys.
3.1
Introduction
For low permeability reservoirs such as tight shale gas, hydraulic fracturing is frequently
conducted to develop more connected fracture networks to enhance oil and gas recovery
(King, 2010). Currently, microseismic monitoring of hydraulic fracturing is the primary
method for characterizing the fracturing process (Fisher et al., 2004; Song and Toks6z
2011). However, this method can only reveal the locations where rocks break or where
existing faults are reactivated (Willis et al., 2012), and microseismicity may not have a
direct relation to changes in reservoir properties during fluid injection, it only reflects the
74
changes of stress around a fracture. Recent studies have shown that scattered (diffracted)
waves from fractures in a reservoir can be detected and characterized from either surface
seismic data (Willis et al., 2006; Fang et al., 2012; Zheng et al., 2012) or vertical seismic
profile (VSP) data (Willis et al., 2007; Willis et al., 2008; Willis et al., 2012). Field
experiments have also demonstrated that the strength of fracture scattered waves can
change notably during hydraulic fracturing, and such changes are attributed to the
opening or closing of fluid induced fractures (Dubos-Sallde and Rasolofosaon, 2008;
Willis et al., 2012).
Time-lapse seismic surveys play an important role in the evaluation of elastic
parameter changes during hydrocarbon production and in monitoring fluid migration after
carbon sequestration (e.g. Arts et al., 2004; Calvert, 2005; Shang and Huang, 2012).
Time-lapse surveys consist of the collection of two or more seismic acquisitions recorded
by the same source-receiver configuration but at different times. The changes in elastic
properties between surveys can then be determined using an inversion method. Denli and
Huang (2010) introduced the elastic-wave sensitivity equation for time-lapse monitoring
study, which quantifies the seismic sensitivity with respect to the change of some
physical parameter for a given monitoring target. Shabelansky et al. (2013) modified the
method of Denli and Huang (2010) and proposed a data-driven sensitivity estimation
approach.
The elastic properties of fractures can be described by a compliance matrix
(Schoenberg, 1980). Previous studies have mainly focused on characteristics of scattering
from fractures as a means of fracture characterization (e.g. Chen et al., 2012; Fang et al.,
2013). In this paper, we propose a quantitative analysis of the sensitivity of time-lapse
75
seismic data to the fracture compliance. The effects of fracture orientation, compliance,
and source and receiver locations are investigated in detail.
3.2
Methodology
3.2.1 Fracture sensitivity wave equation
The propagation of seismic wave through an elastic medium is governed by the elastic
wave equation and the constitutive relation (Aik and Richard, 1980)
Pax
(3.1)
o-i; = Cik all
where C,ij is the stiffhess tensor, p is density, ui is the i-th component of displacement, U,-i
is the stress tensor, xi is the i-th Cartesian coordinate component, and t is time.
Denli and Huang (2010) developed an elastic wave sensitivity analysis approach
for designing optimal seismic monitoring surveys. They studied the sensitivity of the
seismic wave field to the reservoir properties by numerically solving a sensitivity wave
equation, which is obtained by differentiating the elastic wave equation (i.e. equation 3.1)
with respect to geophysical parameters. We follow their idea to study the sensitivity of
seismic waves to the fracture compliance. The sensitivity wave equation for fracture
compliance is obtained by differentiating equation 3.1 with respect to the fracture
compliance (ZN, ZT),
Ia
2
(aLL
Pat2 az )
aua
=
a
ax1
(Oeiq)
az'
)6
76
(3.2)
with
M
=
(3.3)
azf aUk
axI
-CIk
-u' represents the sensitivity of displacement to fracture compliance Z (4 =N,T),
where azf
My is defined as the sensitivity moment which is a function of fracture compliance and
strain at the fracture surface.
source for the wave
Equation 3.2 is analogous to a wave equation for au.
az The
equation is the term My, which has non-zero values only on the fracture plane and must
be determined by solving first for ax, in equation 3.1. Hereafter, seismic wave field and
sensitivity field refer to ui and
, respectively.
From equation 2.3, we can obtain az ' as (in Voigt's contracted index notation)
ac
aZN
1
Yy
Y Y2 Y2
0
0
0-
0
Y2
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0-
Y2
-A2Y
0
0
0
(3.4)
and
-0
0
0
0
0
0-
0 0 0 0 0 0
ac
azr
A2 0
AT 0
0
-0
0
0
0
0
0
0
0
0
0
0
0
0
0 1
0 0
0
1-
77
0
0
(3.5)
with
A = z1
(3.6)
and
1/( + 2y), for =N
for T
1/a,
(3.7)
is the matrix compliance.
In equation 3.6, Zf = Zf has the unit of Pa-.
1 m2
is the effective compliance averaged in a I m
3
cube and
1 is not shown in the equation for the sake of
The number
convenience. Please see Coates and Schoenberg (1995) for details.
Substitute equations 3.4 and 3.5 into equation 3.3, the sensitivity moments for ZN and ZT
are obtained as
0
MN
01
0y
A2
=A
0
(3-8)
0
for ZN, and
2E12
MT = Af 2212
2E13
for ZT,
78
0
0
2E1 3
01
0
(3.9)
where
Z = Ell + YE22 + yE33, Eij is the ij component of the strain tensor, the superscripts
'N' and 'T' indicate the sensitivity moments for normal and tangential compliance
sensitivity, respectively.
The strain tensor Eij at a fracture can be represented as (Schoenberg and Sayers, 1995)
EL
(3.10)
= Sijkl OkI
where Sijkl is the inverse of the effective stiffness tensor in equation 2.3 and akI is the
stress tensor at the fracture. The expression for SijkI can be found in Schoenberg and
Sayers (1995).
Substituting equation 3.10 into equations 3.8 and 3.9 after some algebraic manipulation,
we have
MN =AN
01
0
[all
0
0
Yah
0
0
y01
(3.11)
and
0
MT =UA
U1
U13
0
0
0
0
12
U13
(3.12)
For fracture normal compliance, the strength of the sensitivity field is proportional to AN
and the normal stress resolved on the fracture plane, while its pattern is controlled by the
parameter y which is a function of background medium elastic moduli. For fracture
tangential compliance, the strength of the sensitivity field is proportional to AT and the
79
shear stresses acting on the fracture plane, and its pattern is controlled by the ratio of the
two shear stresses.
3.2.2 Relation between time lapse seismic data and fracture compliance
If we assume
YUz,
12 and cr3 have similar values, the strength of the sensitivity field
(equation 3.2) is proportional to A . For weak scattering that satisfies Z < r7,
we
expand A in a Taylor series,
A
~-(3.13)
Equation 3.13 indicates that the amplitude of the sensitivity field is independent of
fracture compliance for weak scattering given that the presence of fracture has negligible
effect on the stress field. Figure 3-1 shows the variations of AN (black solid curve) and AT
(black dashed curve) with fracture compliances varying from 10-14 m/Pa to 10-9 m/Pa for
sandstone, carbonate, shale and granite, respectively, which are typical reservoir rocks for
oil, gas or geothermal fields. The properties of these four types of rock are listed in Table
3.1. Both AN and AT are independent of fracture compliance when compliance is smaller
than about
10-12
m/Pa for the four types of rock, while AN and AT decrease significantly
when normal and tangential compliances exceed qN and t7T, respectively, which are
illustrated using dashed blue and red lines ,respectively, in each panel.
For a flat planar fracture, the relation between fracture aperture, which is assumed
to be much smaller than the seismic wave length, and fracture compliance is given as
(Schoenberg 1980)
80
(3.14)
with
' 1/( + 2y'), for , =N
for (=T
t/p',
(3.15)
where r; is the compliance of fracture infill, A' and y' are the Lame moduli of the
fracture infilling material, h is fracture aperture.
In hydraulic fracturing, fracture aperture is on the order of 1 mm (Perkins and
1 mm aperture, its normal compliance
Kern, 1961). For a water saturated fracture with
can be estimated as 4.4x 10-13 m/Pa using equation 3.14. The normal compliance
increases with fracture aperture and reaches 4.4x
10-12
m/Pa for a 10 mm wide fracture.
The tangential compliance estimated from equation 3.14 goes to infinity because p' of
water is zero. This is unrealistic. The tangential compliance of a realistic fracture has a
finite value due to the existence of asperities on the fracture surfaces, which are not
considered in the flat planar fracture model. The tangential compliance of a realistic
fracture is found to be larger than its normal compliance and it can be one order of
magnitude larger than the normal compliance for a fluid saturated fracture (Lubbe et al.,
2008). If we consider a fracture with aperture no larger than 10 mm, then both its normal
and tangential compliances fall in the weak scattering regime for typical reservoir rocks,
as shown in Figure 3-1. Therefore, the weak scattering assumption is valid for the study
of scattering from fractures in hydraulic fracturing.
81
Assuming that the presence of a fracture has negligible effect on the stress field,
then the strength of the fracture scattered waves is mainly affected by A in equations
3.11 and 3.12. By integrating A with respect to Z , we have
U C InL1 + -)
For weak scattering that has Z
(3.16)
<ia, equation 3.16 can be simplified as
U c Zf
(3.17)
Equation 3.17 indicates that the strength of fracture scattered waves is linearly
proportional to the fracture compliance for weak scattering. Fang et al. (2013) found the
same linear relationship between fracture scattering strength and fracture compliance
when compliance is less than 1040 m/Pa, and they argued that the departure of this linear
relationship at large compliance is due to the breakdown of the Born approximation. We
here demonstrate that Zf
«<j7
is the necessary condition for the Born approximation (or
weak scattering) to be valid for scattering from fractures. If the weak scattering condition
is not satisfied, the relation between fracture scattering strength and fracture compliance
deviates from the linear relationship (i.e. equation 3.17) by following a logarithmic
variation (equation 3.16).
From equation 3.17, we can obtain the relation between the change of time lapse
wave field and the change of fracture compliance in hydraulic fracturing as
Al
Ubase
-
lapse
~Ubase
Ubase
82
AZ
.ae Zs
bs
as
(3.18)
where Ubase
and
UTapse represent the baseline and time lapse fracture scattered
wavefields, respectively, AU' is the change between the two time lapse wavefields, Zbase
is the fracture compliance before fracturing and AZf represents the change of fracture
compliance in hydraulic fracturing.
Equation 3.18 indicates that the percentage change of time lapse data is equal to
the percentage change of fracture compliance in hydraulic fracturing for weak scattering.
Based on equation 3.18 and an appropriate rock physics model, we may be able to obtain
information about fracture opening in hydraulic fracturing based on the percentage
change of time lapse seismic data since fracture compliance is a function of fracture
aperture.
As compliance becomes larger, the amplitude of the scattered wavefield increases.
However, when scattering becomes strong, the breakdown of the Born approximation
means that the sensitivity of the scattered wavefiled, i.e. the proportional change of the
scattered wavefield with change in compliance, decreases.
3.2.3 Relative strength of normal and tangential compliance sensitivities
Assuming ol,
072
and U13 in equations 3.11 and 3.12 have similar values, the relative
strength of AMv and M is approximately equal to
AN
AT
-
1+2pu+p(A+2pu)ZT
p+
(3.19)
(A+ 2 p)ZN
The relative magnitudes of AN and AT depend on the values of ZN and ZT. The ZN/ZT ratio
is strongly influenced by the way the fracture surfaces interact and it can be taken as an
83
indicator representing the fracture saturation condition (Dubos-Sallee and Rasolofosaon,
2008; Fang et al., 2013). Both numerical simulations (Sayers et al., 2009; Gurevich et al.,
2009) and laboratory measurements (Lubbe et al., 2008; Gurevich et al., 2009) suggest
that ZN is generally smaller than ZT for reservoir fractures. Based on laboratory
experimental data, Lubbe et al. (2008) pointed that the ZN/ZT ratio is close to 0.5 for gasfilled fractures, and ZN/ZT can be less than 0.1 for fluid-saturated fractures. Figures 3-2a,
b and c, respectively, show the variations of ANAT with ZN for ZN/74=0.5, which
represents a gas-filled fracture, and ZN/ZT=O.1 and 0.05, which represent fluid-saturated
fractures.
AN/AT
is larger than
1 regardless of the compliance value and fracture saturation
condition and its value increases with decreasing ZN/ZT ratio. Also, the normal stress
a,,
is generally larger than the shear stresses J2 and UJ3 since P-wave source is commonly
used in exploration. Therefore, the sensitivity of seismic surveys to fracture normal
compliance is always larger than that to the tangential compliance.
3.2.4 Separation of P- and S-energies
In time lapse monitoring of hydraulic fracturing, we are interested in the sensitivity of
seismic data to fracture compliance, which contains information about fracture opening
and fluid contents. Following Denli and Huang (2010), we evaluate the criteria for
seismic monitoring by separating the seismic wave field and sensitivity field into P- and
S-components. The P- and S-wave energies recorded over a period of time t are given as
(Fang et al., 2013)
EP= f 0t p (iP)2 dt
84
(3.20)
Es _
tp
(-s) 2 dt
(3.21)
with
F-
--P =
+s _
V=
V)
(3.22)
2 Vx (V x-)1
(3.23)
)2 V(V
[Q'S)
U = V +
(3.24)
s
where V is the Fourier transform of particle velocity v, v" and 's are the P- and S-wave
-1 [*] indicates inverse Fourier transform, p, Vp and Vs
particle velocities, respectively,
are the density and P- and S-wave velocities at the receiver location.
The corresponding P- and S-wave sensitivity energies are defined as
(3.25)
p (t ) 2 dt
E
(3.26)
with
Sf azf
-+S
S==
avS
=
SP
-
*
[=F
p=
-
-
(-71
85
(3.28)
a]
~P(V)2
$f =
(3.27)
S+
(3.29)
~s(
where
sfP and
s
)
V x (V x
(3.30)
are the P- and S-components of the sensitivity field, respectively, _"P
and $f are the Flourier transforms of Sg and s , =N,T represent normal and tangential
compliance, respectively.
and S represents the sensitivity of P and S scattered waves to normal
compliance, for
(=N,
and tangential compliance, for 4,=T, respectively. EP and Es
indicate the P- and S-wave energies recorded at an observation location during the
recording time t, respectively. While EP and ES represent the sensitivity of the seismic
data recorded over the same time period to fracture compliance Zf (4=NT).
3.3
Numerical results
We now investigate the sensitivity of seismic data to fracture compliance by using the
finite-difference method of Coates and Schoenberg (1995) to numerically solve the
elastic wave equation (i.e. equation 3.1) and the sensitivity wave equation (i.e. equation
3.2) to obtain the seismic wave field il and the sensitivity field a-, respectively. These
two equations have to be solved simultaneously, because the sensitivity moment in
equation 3.2 is a function of the strain from the incident wave, which is obtained by
solving equation 3.1. Denli and Huang (2010) gave a detailed description of the
procedures for solving a sensitivity wave equation.
86
We only consider fractures that satisfy the weak scattering condition Zf
r<77
because this is the case appropriate for hydraulic fracturing as we have discussed
previously. We use 0.05 for ZN/ZT ratio to represent a fluid saturated fracture. Table 3.2
gives the compliance values of two fracture models that will be used in the following
study. The corresponding apertures of a flat planar fracture (equation 3.14) are 1 mm and
10 mm, respectively, for fracture models
1 and 2.
First, we will investigate the effect of incident angle, fracture compliance and
background elastic properties on the sensitivity field. Second, we will discuss the
distribution of sensitivity energy in different acquisition configurations, such as surface
and borehole acquisitions. In our simulation, a point explosive source with a 40 Hz center
frequency Ricker wavelet is used. Perfectly match layer (PML) is used at all model
boundaries to avoid boundary reflection.
3.3.1 Sensitivity patterns of different rocks
The sensitivity field is excited by the sensitivity moments (equations 3.11 and 3.12),
which are functions of ZN, ZT, background elastic properties (A, p) and the incident stress
au;. It also depends on the incident stress field direction. We first study the effect of ZN, ZT,
and incident angle and then discuss the difference of the sensitivity field in different
rocks. Figure 3-3 shows the model used to study the sensitivity field of a single fracture.
In this model, we have receivers (blue circles) located at equal distance (500 m) from the
fracture center and at angles of 00 to 3600 measured from the fracture normal to record
the sensitivity field. We vary the source incident angle from 00 (fracture normal direction)
87
to 900 (fracture strike direction) to study the effect of incident angle on the sensitivity
field.
Figure 3-4 shows SP and Ss, which represent the absolute values of Sf (equation
3.29) and SS (equation 3.30), for the two fractures (Table 3.2) in sandstone (Table 3.1) at
300 and 600 incident angles. S[ and SS are plotted in polar coordinates. The radial and
angular axes are frequency and radiation angle, respectively. SP and Ss of the two
fractures are identical at all incident angles, which demonstrates that the sensitivity field
is independent of fracture compliance for weak scattering. The patterns of SF and SS vary
significantly with incident angle due to the change of the incident stress field. We can
also see that the sensitivity to normal compliance (aV/OZN) is always larger than that to
tangential compliance (aV/aZT). Comparing SF and Ss at different incident angles, we
can see that SN is strongest when the incident wave field is normal to the fracture plane
(i.e. 00) and its magnitude decreases with increasing incident angle, while both SN and SS
have larger values at 300 and 600 incidences.
The magnitude of SjN is comparable to SN at 300 incident angle and becomes
larger than SN at 600 and 900 incident angles. SN and SS are stronger at intermediate
angles of incidence because P-to-S scattered waves are stronger at these incident angles
(Fang et al. 2013). In our study, we vary both the incident angle and background elastic
properties and find that, generally, SN is larger than SN when the incident angle is
smaller than 300, while SjN is larger than SN when the incident angle is larger than 400.
This suggests that S-waves are generally more sensitive to the normal compliance than Pwaves for a vertical fracture while this reverses for a horizontal fracture if a source is
88
excited at the surface near the fracture location. Sj is always larger than SP regardless of
the source incident angle. This indicates that P-waves are not sensitive to the fracture
tangential compliance. Since the sensitivity moment (equation 3.3) in the sensitivity wave
equation is defined at the fracture plane, the sensitivity patterns of S4 and S are always
symmetric with respect to the fracture plane. This indicates that back (right side of the
fracture plane in Figure 2-3) and forward (left side of the fracture plane in Figure 3-3)
scattered wave fields have the same sensitivity to fracture compliance. Although we only
show the case of ZN/ZI=0.05 here, the sensitivity patterns are independent of the ZN/ZT
ratio as long as ZN and ZT satisfy the weak scattering condition. S and SS are only
affected by background rock properties and incident angle.
Figure 3-5 shows the SP and Ss of fracture model 1 (Table 3.2) in four different
rock samples (Table 3.1). We can see that the magnitudes of Sip and S4 increase with the
decrease of the rock velocity, because the sensitivity field has larger amplitude in a
slower medium. Although the sensitivity field varies with rock type due to the
dependence of the sensitivity moment on y, which is a function of the Poisson's ratio, we
find that the radiation direction of the dominant sensitivity energy (high amplitude of the
sensitivity field) is almost the same for rocks with different properties and different
Poisson's ratios. This suggests that, for a given incident angle, the spatial distribution of
the sensitivity energy for different rocks should be similar, so the design of time lapse
monitoring surveys based on a given reservoir rock is applicable to other fields with
different reservoir rocks.
89
3.3.2 Distribution of
sensitivity
energy
in
different
acquisition
configurations
In 3.3.1, we know that the spatial variation of the sensitivity field is mainly determined
by the incident angle, so an optimal acquisition strategy for a time lapse survey is
essential for maximizing the sensitivity of time lapse data to fracture compliance. Seismic
monitoring of hydraulic fracturing is usually conducted either at the surface or in a
borehole (e.g. VSP). In order to study the general characteristic of the sensitivity field, we
use a homogeneous model, shown in Figure 3-6, to study the sensitivity of seismic waves
to fracture compliance. Since the sensitivity patterns of a fracture in different rocks are
similar and they are not sensitive to the fracture compliance, we will only discuss the case
of fracture model
1 (Table 3.2) embedded in a homogeneous sandstone (Table 3.1)
background. In the following, we will discuss two representative scenarios: (1) a vertical
fracture and (2) a horizontal fracture. Fractures generally tend to be vertical and subparallel to the horizontal maximum stress direction (Crampin & Chastin 2000). They can
also be horizontal, particularly at shallow depths (Baisch et al. 2009). As shown in Figure
3-6, we excite explosive sources at different positions at the surface and record the
synthetic seismic data at the surface, in a vertical well (black line), and in a horizontal
well (blue line). The red and green lines, respectively, represent 100 m long vertical and
horizontal fractures centered at X= 0 m and Z = 1050 m. The vertical well is 200 m away
from the fracture center and the horizontal well is 50 m below the fracture center.
90
Vertical fracture
We first discuss the case of a vertical fracture. We simulate 41 shots with source
horizontal position varying from -1000 m to 1000 m, as shown in Figure 3-6. For each
shot, we record the synthetic seismic and sensitivity data at the surface and in the
horizontal and the vertical wells and then calculate the seismic energy and sensitivity
energy. Figure 3-7 shows the P- and S-wave scattered energy (equations 3.20 and 3.21)
for the vertical fracture. The direct P-wave is muted from the synthetic data before
calculation of the energy. We can see that S-wave energy is generally stronger than Pwave energy for a vertical fracture, because the tangential compliance is much larger than
the normal compliance for a fluid saturated fracture. Both Ep and Es have larger values
when sources are at -1000 m and 1000 m, because the fracture scattered waves are
stronger at these angles of incidence (Fang et al. 2013). Comparing Ep and ES at the
surface and in the wells, we can see that the energy in the horizontal and the vertical
wells are much stronger than that at the surface, because most of the scattered energy
propagates downward for a vertical fracture (Fang et al., 2013).
Figures 3-8, 3-9 and 3-10 show the sensitivity energies at the surface, in the
horizontal well and in the vertical well, respectively. ETP and Es are multiplied by 4 for
plotting. Because the sensitivity field is always symmetric with respect to the fracture
plane, the distribution of sensitivity energy is always symmetric with respect to the
horizontal position (X= 0 m) of the vertical fracture. The sensitivity energy in the wells
(Figures 3-9 and 3-10) is about three orders of magnitude larger than that at the surface
(Figure 3-8). From Figures 3-8, 3-9 and 3-10, we can see that, for a vertical fracture,
shots at the surface with larger horizontal distance to the fracture have higher sensitivity
91
to the fracture compliance for all kinds of acquisition. The sensitivity energy for normal
compliance is always larger than that for tangential compliance regardless of source and
receiver positions. For normal compliance sensitivity, ENS is larger than EN except for
sources at about ±1000 m and receivers near ±2000 m at the surface. For tangential
compliance sensitivity, Ejs is always larger than EP.
To verify equation (3.18), we increase the normal and tangential compliances of
the vertical fracture by 10%, 20% and 30%, respectively, to mimic the hydraulic
fracturing process and study the corresponding change in seismic data. Figures 3-11 a0
and 3-1 ibO show the X and Z displacements recorded in the vertical well of the model
containing the vertical fracture for a shot at X=- 1000 m. Figures 3-11 al, 3-11 a2 and 311 a3 show the changes of X displacement comparing to the baseline case (Figure 3-11 aO)
when ZN and ZT increase by 10%, 20% and 30%, respectively. Figures 3-11 bI, 3-11b2
and 3-1 1b3 show the corresponding changes of Z displacement. Comparing these three
time lapse cases with the baseline case in Figure 3-11, we can see that the displacement
difference increases with increasing fracture compliance. To investigate further, we
calculate at each receiver the average absolute amplitude of the time lapse displacement
difference and the baseline displacement, in which the direct P arrivals are muted, and
then use the average amplitudes to compute the percentage change of the time lapse data.
In Figure 3-12, the black, red and blue curves show the average percentage changes of
the seismic data recorded in the vertical well when the fracture compliances increase by
10%, 20% and 30%, respectively. From this figure, we can see that the percentage change
of the time lapse data is equal to the percentage change of fracture compliance. Although
we only show the time lapse change of the data in the vertical well, the data recorded at
92
the surface and in the horizontal well have the same relationship with the fracture
compliance.
Horizontal fracture
The horizontal fracture is shown as the green line in Figure 3-6. Figure 3-13 shows the
distribution of P- and S-wave energies at the surface (13al and 13bl), in the horizontal
well (13a2 and l3b2) and in the vertical well (13a3 and 13b3). Similar to the case of a
vertical fracture, S-wave energy is stronger than P-wave energy. S-wave energy is
strongest when sources are at about ±600 m, the corresponding incident angles at the
fracture plane are about 300. P-to-S scattered waves are strongest at this angle of
incidence. As shown in Figures 3-13b2 and 3-13b3, the S-wave energy recorded in the
horizontal well only shows high amplitude at receivers right below the fracture at X=0 m,
while the high amplitude region spreads out over a broader area in the vertical well.
Because most of the P-to-S scattered waves propagate backward (Fang et al. 2013) and
recorded in the vertical well.
Figures 3-14, 3-15 and 3-16 show the sensitivity energies at the surface, in the
horizontal well and in the vertical well, respectively. ET2 and ES are multiplied by 8 for
plotting in these three figures. Similar to the vertical fracture case, the sensitivity energy
for normal compliance is always larger than that for the tangential compliance regardless
of source and receiver positions. For normal compliance sensitivity, ENP is larger than Ej
except for sources near ±1000 m. For tangential compliance sensitivity, ES is always
larger than ET.
93
Same as for the vertical fracture case, we increase the normal and tangential
compliances of the horizontal fracture by 10%, 20% and 30%, respectively, to study the
corresponding change in the seismic waves. Figures 3-17a0 and 3-17b0 show the X and Z
displacements, respectively, of the baseline model for a shot at X=0 m and the other six
panels in Figure 3-17 show the displacement changes for three time lapse cases. We can
see that the displacement changes increase with increasing fracture compliance. The
average percentage changes, whose calculation is similar to Figure 3-12, of the seismic
data in the vertical well for these three time lapse cases are plotted in Figure 3-18. Similar
to Figure 3-12, the percentage change of the time lapse data for the horizontal fracture is
equal to the percentage change of fracture compliance.
3.4 Summary
Under the weak scattering approximation, which we have shown is valid for a range of
fracture compliance values of relevance to subsurface conditions, we have demonstrated
that the percentage change of time lapse seismic data is equal to the percentage change of
fracture compliance in hydraulic fracturing. This could provide a means for determining
fracture opening using time lapse data since fracture compliance is a function of fracture
aperture.
We have found that the sensitivity field is mainly affected by the source incident
angle and background medium elastic properties but not sensitive to the fracture
compliance. This suggests that the spatial variation of the sensitivity field is not sensitive
to the fluid content of a fracture. The sensitivity field is always symmetric with respect to
94
the fracture plane which is determined by the sensitivity wave equation. Also, we
demonstrate that the sensitivity of seismic waves to fracture normal compliance is always
larger than that to tangential compliance regardless of source and receiver positions. For
normal compliance sensitivity, P-waves are more sensitive than S-waves if the source
incident direction is close to normal to the fracture plane, while this reverses when the
incident direction is close to parallel to the fracture strike. For tangential compliance, Swave sensitivity is always larger than P-wave sensitivity.
We also discuss the sensitivity of seismic data to the compliances of vertical and
horizontal fractures, respectively, for surface and borehole (vertical and horizontal wells)
acquisitions. For both vertical and horizontal fractures, sensitivity to normal compliance
is always larger than sensitivity to tangential compliance. For surface explosion sources,
S-wave sensitivity to normal compliance is generally larger than that for P-wave for a
vertical fracture, while it reverses for a horizontal fracture. For tangential compliance, Swave sensitivity is always larger than P-wave sensitivity. Because the P-to-S scattered
energy is much larger than the P scattered energy for both vertical and horizontal
fractures, as shown in Figures 3-7 and 3-13, S scattered waves are easier to be detected.
This may suggest that S waves can be more reliable than P waves for retrieving the
knowledge of both fracture normal and tangential compliances from seismic data.
Although we only show the results for a homogenous model, the conclusions made from
this study are valid for heterogeneous models, because the sensitivity field is governed by
the sensitivity wave equation. The complexity of a background model only alters the
wave path and introduces converted waves into the data, but it does not change the main
feathers of the sensitivity field.
95
Table 3.1: Properties of four types of rock.
Rock type
VP (m/s)
Vs (m/s)
p (kg/m 3 )
(Mavkandsto.2003)
Carbonate
(Mavko et al., 2003)
Muderong shale
(Dewhurst and Siggins, 2006)
Chelmsford granite
(Lo et al., 1986)
4090
2410
2370
59
5390
90
2970
29
2590
3090
1660
2200
5580
3430
2610
Table 3.2: Compliance values for two fracture models.
ZN
Fracture model 1
Fracture model 2
(m/Pa)
4.4x10'
4.4x10-12
96
ZT (m/Pa)
ZNIZT
8.8x108.8x1011
0.05
0.05
(a)
(b)
Sandstone
Carbonate
1 11
11
10-
--
1010
-----
...
--
--
. %.....
7 N----- ---
-,,,*:
10
108
108
1410131012
10
(c)
11 I.:110
10
i0
10
(d)
11
10 t
Shale
11
-----. . ---.-.-.-........
---..
1010
-
.
10-13
-10 10-9
101
102
Granite
-J
1010
....
....
...
....
10
1
IU
a I
10
-14
8
13
10~ 101 10~ 10
Z
(m/Pa)
-10
-9
10~
10
-14
0-10
101-13 100-12 100-11 10
Z(rn/Pa)
10~
Figure 3-1: (a), (b), (c) and (d) Variations of AN (black solid curve) and AT (black dashed
curve) for sandstone, carbonate, shale and granite, respectively, with fracture compliance
Z4 ( =N,T) varying from 10-14 m/Pa to 10-9 m/Pa. In each panel, blue and red dashed lines
indicate the values of
17N
and qT, respectively. Properties of the four types of rock are
listed in Table 3.1. Both horizontal and vertical axes are in log scale.
97
ZZZT.05
ZWZ
=O.1
(b)
(a)a
;ZZ=0.05
(C)
20
r
15
15
15
10
10
10
4
<-
1 4, 1
5
5
5
13
-12
-11
-1 10
-.0
202
14 io13
1
P
12
R-11
101
F
2-14
10
-
10
-13
-12
-1
(;!a)
10 -1 0 10 -
Figure 3-2: (a) (b) and (c) Variations of AN/AT (equation 3.19) with ZN for ZN/ZT =0.5, 0.1
and 0.05, respectively. Black, blue, red and magenta curves show the variations for
sandstone, carbonate, shale and granite, respectively. The properties of rocks are listed in
Table 3.1. Horizontal axes are in log scale. Vertical axes are in linear scale.
98
900
*
-' - -
.
,f00
fractu re
00
Figure 3-3: Schematic showing the layout of sources and receivers surrounding a single
fracture (black bar) with 100 mn length. Source (red star) is 550 mn away from the fracture
center. Receivers (blue circles) covering 3600 are 500 mn away from the fracture center. 0
is the source incident angle with respect to the fracture normal.
99
aV/aZN
sN
SSN
SPT
SsT
00
300
0
E
LL
60'
90
00
|1 M
('0
0
30
Z
)
Ca
U-
60'
go,
0
0.5
E ' l -.-. J
1
1.5
x10
2
Figure 3-4: Absolute values of SP (equation 3.29) and Sf (equation 3.30) for the two
fracture models in Table 3.2 at 00, 300, 600 and 900 incident angles. Background matrix is
sandstone (Table 3.1). In these polar coordinate plots, the radial and angular coordinates
are frequency and radiation angle, respectively. The range of frequency in each panel is
from 0 Hz at the center to 60 Hz at the edge. The dashed white circle indicates the
frequency of 40 Hz. Red star indicates source position.
-V
OIZN
and Ov indicate the
OZT
sensitivity fields for fracture normal and tangential compliances, respectively. Fracture
length is 100 m.
100
aV/aZN
SpN
SPT
SSN
SST
C
0
300
C
0
W
Qu
0
60 '
x
0
0.5
1
1.5
2
Figure 3-5: Absolute values of Sf (equation 3.29) and
07
2.5
54
(equation 3.30) for fracture
model 1 (Table 3.2) in four rock samples (Table 3.1) at 300 and 600 incident angles. Polar
plots are as described in Figure 3-4. Fracture length is 100 m.
101
A.
Cr
A.
&
&
-
-
N
1000
1100
.-.-.- .--
--.--.--.--.--................
..-.-.----..-.-..
. . . . . ..-.
. . . . . . +-
-
u~I
.-.
-.-.-.-.
-
I
-
4 C mfl
'tmoo
-.-.-.--
-1000
-200 0
X (M)
1000
2000
Figure 3-6: Red and green lines represent 100 m long vertical and horizontal fractures
centered at X=0 m and Z=1050 m, respectively. Red stars represent sources at Z=Om.
Black and blue lines represent a vertical well at X=-200m and a horizontal well at Z= I100
m, respectively.
102
(al)
E, (surface)
x 19
E
1
.
(bi)
Es (surface)
x 10-9
8
8
6
I
6
o0
4
C)
2
W
0
-2
-1
u
0
1
0
Receiver position (km)
-1
U
E. (horizontal well)
(b2)
X 10-6
4
0
C',
0
C.
ci)
0
C.)
0
C')
1
2
Receiver position (km)
E, (vertical well)
(a3)
-1
0
0
(b3)
1-_X
4
I
2
0.
2
X 10-6
I
E
C
0
2
1
Receiver position (km)
Ep (horizontal well)
(a2)
2
4
2
0
0
1
2
Receiver position (kin)
ES (vertical well)
1
4
0
x 10-6
4
E
3
0a.
0S0.
a.
2
C)
2
1
0
CO)
-1
0
0.5 1
1.5
Receiver depth (km)
0
0 0.5
1
1.5
Receiver depth (kin)
0
Figure 3-7: Energy distribution for a vertical fracture. al, a2 and a3 show the recorded Pwave energy (equation 3.20) at the surface, in a horizontal well and in a vertical well,
respectively, for 41 shots with source horizontal position varying from -1 km to 1 km. b 1,
b2 and b3 show the corresponding S-wave energy (equation 3.21). Model geometry is
shown in Figure 3-6.
103
(a)
EP
(b)
E1
1
N1n
Nx
10"
7
0
1-"Rom
5
_
CL
(c)
ET (x4)
(d)
Es x4
5
.2
0
-1
-2
Receiver position (km)
0
Receiver position (ki)
Figure 3-8: P- and S-wave sensitivity energies (equations 3.25 and 3.26) at the surface for
a vertical fracture are shown in a and b, for normal compliance, and c and d, for
tangential compliance. Ei and Ej are multiplied by 4 for plotting. Model geometry is
shown in Figure 3-6.
104
(a)
EP
(b)
NN
1
(n 0
0~
a..
Es
X
1
5
04
4
16
0
-1--3
0
1
2
ETp (x4)
(c)
0
1
2
ETs (x4)
(d)
12
E
0
01
S0
a. 0
CL)
--1
0
1
Receiver position (km)
2
-1
0
1
Receiver position (kin)
2
0
Figure 3-9: Same as Figure 3-8 but for the sensitivity energy in the horizontal well. ET
and Es are multiplied by 4 for plotting. Model geometry is shown in Figure 3-6.
105
(a)
EP
(b)
N
0
0
0.5
0.5
ESN
x10
3
(D
(D
1.5
-10
(c)
0
1
E (x4)
P
1.5
-1
0
(d)
ET(4
0
T1
0.5
0.5
a)
1.5
-1
0
Source position (kin)
1
1.5
-1
0
Source position (kin)
1
0
Figure 3-10: Same as Figure 3-8 but for the sensitivity energy in the vertical well. EP and
ES are multiplied by 4 for plotting. Model geometry is shown in Figure 3-6.
106
.
(aO)
(al)
0
ux (baseline)
0
#c
0.5
(a2)
0
0.5,
Scattered
1waves
O 0.5
(bO)
0
Aux (10%)
1
1.5
uz (baseline)
,-Direct P
0.5U
.
......
.....
(a3)
0
Aux (20%)
0.5
0.5
15
1
Aux (30%)
1
10
(bi)
0
0.5
1
Auz (10%)
1.5
50
(b2)
0
0.5
0.5
1
1.5
Auz (20%)
50
(b3)
0
0.5
0.5
1
1.5
Auz (30%)
0
0.5
Scattered
N
waves
1
1.51
0
0.5
1
Time (s)
1
1
1.5
0
0.5
1
Time (s)
1.5
1.5
0
1
0.5
Time (s)
1.5
1.50
-1
0.5
1
Time (s)
1.5
Figure 3-11: aO and bO show the X and Z displacement components recorded in the
vertical well (VSP) of the model containing a vertical fracture for a shot at X=-1000 m.
al, a2 and a3 show the changes of the X displacement when the fracture compliances (ZN
and ZT) increase by 10%, 20% and 30%, respectively. bl, b2 and b3 show the
corresponding changes of Z displacement. Scales are the same for all plots. Model
geometry is shown in Figure 3-6.
107
.........
0.4
0.........
...........--I...
00om
~z-
-
-
- 02
0)
I
ance nreases 30%
Compliarce increases 10%
-~
.
0.
.
increases1%
.Comparice
.
0
0
0.5
1
Z (kin)
1.5
Figure 3-12: Black, red and blue curves show the percentage changes of the seismic
scattered waves recorded in the vertical well when a vertical fracture is present and the
fracture compliance increases by 10%, 20% and 30%, respectively.
108
(al)
EP (surface)
x
E
C
0
0
- -2
-2
CI
EP (horizontal well)
(a2)
ES (surface)
Lbl)
x 10
10
0
10
5
U) 0
0
5
2
1
0
-1
Receiver position (kin)
107
x 10-6
:0-2
-1
0
2
1
Receiver position (km)
ES (horizontal well)
(b2)
x 10~-
E
4
0
4
0
0
0
0
2
U,
0
1
2
C)
0
2
Receiver position (kin)
E, (vertical well)
(a3)
0
2
1
0
Receiver position (km)
(b3)
-6
Es (vertical well)
1
2
2
E
0
0
0
1.5
0
0
4)
0.
U)
0.5
-1
1
1.5
0
0.5
Receiver depth (km)
0.5
0
(n2
-1
0
0
0.5
1
1.5
Receiver depth (km)
0
Figure 3-13: Energy distribution for a horizontal fracture. al, a2 and a3 show the
recorded P-wave energy (equation 3.20) at the surface, in a horizontal well and in a
vertical well, respectively, for 41 shots with source horizontal position varying from -1
km to 1 km. bl, b2 and b3 show the corresponding S-wave energy (equation 3.21).
Model geometry is shown in Figure 3-6.
109
(a)
(b)
EP
ES
1
1
00
0
x 1016
E
CL
2
0
-2
0
-1
(c)
1
2
E (x8)
-1
(d)
0 0
0
1
2 21.5
E (x8)
0
O.
-1
-2
-2
-1
0
1
Receiver position (kin)
2
-10
-2
0.5
-1
0
1
Receiver position (km)
2
Figure 3-14: P- and S-wave sensitivity energies (equations 3.25 and 3.26) at the surface
for a horizontal fracture are shown in a and b, for normal compliance, and c and d, for
tangential compliance. ET and ETS are multiplied by 8 for plotting. Model geometry is
shown in Figure 3-6.
110
(a)
EP
(b)
ES
N
N
1
X 10
1
E
2
0 0
0
a.
0
CI)
-1
-
12
0
EP (x8)
(C)
0
(d)
1
12
Ey (x8)
1
E
_0
00
0
0
0
00
-1
0
1
Receiver position (km)
2
-
0
1
Receiver position (kin)
2
Figure 3-15: Same as Figure 3-14 but for the sensitivity energy in the horizontal well. ET
and EjT are multiplied by 8 for plotting. Model geometry is shown in Figure 3-6.
111
(a)
EP
N
(b)
0
0
0.5
0.5
1.5
1.5
ES
N
x1&
E
a)
3
Z
(D
(c)
70
EP (x8)
0
(d)
0
4= 0.5
0.5
EyS (x8)
a)
1.5
-1
01
Source position (kin)
1.5
1
0
Source position (kin)
0
Figure 3-16: Same as Figure 3-14 but for the sensitivity energy in the vertical well.
and ETS are multiplied by 8 for plotting. Model geometry is shown in Figure 3-6.
112
E
. .....
..............
......
................
(aO)
-
(al)
ux (baseline)
(a2)
0
0.5
0.5.
0Direct P
-0.5
1
(bO)
0
0.5
1
1.5
0
(a3)
0
Aux (30%)
0.5
uz (baseline)
1.50
0.5
(b2)
0
Auz (20%)
0
0.5.
0.5
1
15
0.5
(bi)
Auz (10%)
Direct P
.5
1
Time (s)
1.5
1
1.50
waves
1.5
Aux (20%)
Scattered
waves
15LX~
0
Aux (10%)
0
1
0.5
1
Time (s)
1.5
1.5
0
1
1.5
.0
(b3)
0
0.5
1
1.5
Auz (30%)
0
0.5
0.5
1
Time (s)
1.5
0
0.5
1
Time (s)
1.5
Figure 3-17: aO and bO show the X and Z displacement components recorded in the
vertical well (VSP) of the model containing a horizontal fracture for a shot at X=0 m. al,
a2 and a3 show the changes of the X displacement when the fracture compliances (ZN and
ZT)
increase by 10%, 20% and 30%, respectively. b 1, b2 and b3 show the corresponding
changes of Z displacement. Scales are the same for all plots. Model geometry is shown in
Figure 3-6.
113
- - --- -----.
.........
......
0.4
. . ......
p... . ..
-0.
Compianceincrass 230%
.....
0.
0
0
0.5
1
1.5
Z (kin)
Figure 3-18: Black, red and blue curves show the percentage changes of the seismic
scattered waves recorded in the vertical well when a horizontal fracture is present and the
fracture compliance increases by 10%, 20% and 30%, respectively.
114
Chapter 4
Reservoir Fracture
Characterization from Seismic
Scattered Waves 3
In Chapters 2 and 3, we studied the characteristics of fracture scattering and the
time-lapse change of fracture compliance in hydraulic fracturing. In this chapter, we
will discuss the characterization of fractures using the fracture scattered waves.
Traditional seismic methods for fracture characterization, such as AVOA and shear
wave birefringence, are based on the equivalent medium theory with the assumption
that fracture dimensions and spacing are small relative to the seismic wave length, so
a fractured unit is equivalent to a homogeneous anisotropic medium. However,
fractures on the order of the seismic wave length are also very important for enhanced
oil recovery, and they are one of the important subsurface scattering sources that
generate scattered seismic waves. In this study, we present an approach for detecting
fracture direction through computing the fracture transfer function using surface
3
(the bulk of this Chapter has been) submitted as: Fang, X.D., M. Fehler, Z.Y Zhu, Y.C. Zheng, and
DR. Burns, 2013. Reservoir fracture characterization from seismic scattered waves: GJI.
115
recorded seismic scattered waves. The applicability and accuracy of this approach is
validated through both laboratory experiment and numerical simulation. Our results
show that fracture direction can be robustly determined by using our approach even
for heterogeneous models containing complex non-periodic orthogonal fractures with
varying fracture spacing and compliance. We also show results for the application of
our approach to the Emilio Field.
4.1
Introduction
With the advance in seismic data processing technology, studying subsurface
anisotropy, especially fracture-induced anisotropy, has become more and more
important in drilling and enhanced oil recovery. Natural fracture systems in an oil or
gas reservoir can significantly affect the fluid drainage pattern and matrix
permeability. For low permeability reservoirs, hydraulic fracturing is frequently
conducted to enhance the fracture systems. Helbig and Thomsen (2005) gave a very
good review of the study of anisotropy in exploration geophysics.
Due to regional stress, fractures tend to be vertical and sub-parallel to the horizontal
maximum stress direction (Crampin and Chastin, 2000). Seismic waves propagating
through a formation containing aligned fractures are significantly affected by both the
mechanical and geometric properties of the fractures. Traditional seismic methods for
fracture characterization include shear wave birefringence (Gaiser and Van Dok, 2001;
116
Van Dok et al., 2001; Angerer et al., 2002; Crampin and Chastin, 2003; Vetri et al.,
2003) and amplitude variations with offset and azimuth (AVOA) (Riiger 1998; Shen
et al., 2002; Hall and Kendall, 2003; Liu et al., 2010; Lynn et al., 2010). These
methods are based on the equivalent medium theory with the assumption that fracture
dimensions and spacing are small relative to the seismic wave length, so a fracture
zone behaves like an equivalent anisotropic medium. This assumption breaks down
when the fracture size or spacing are comparable to the seismic wave length (Chen et
al., 2012; Fang et al., 2013). Fractures on the order of seismic wave length are one of
the important subsurface scattering sources that generate scattered seismic waves,
which can play an important role in reservoir fracture characterization and enhanced
oil recovery.
In field data, it is very difficult to clearly observe the fracture scattered waves due
to the nearly continuously changing nature of subsurface reflectivity and to the
potentially low amplitudes of the scattered energy (Willis et al., 2006). Willis et al.
(2006) developed the Scattering Index (SI) method to extract the fracture scattering
characteristics by calculating the transfer function of a fracture zone. The signals that
have propagated through a fracture zone can be expressed as a convolution of the
signals incident on the fracture zone and a transfer function. Willis et al. (2006) obtain
the transfer function through deconvolution and then define the scattering index to
quantify the fracture scattering strength.
Zheng et al. (2011, 2012) proposed the double-beam method for determining the
fracture orientation and spacing by extracting the fracture scattered energy from
117
source and receiver beams, which are formed through dynamic ray tracing, from a
selected target region. Their approach is based on the theory of Bragg scattering by
periodic structures. Further study is needed to demonstrate the applicability of this
approach to field reservoir applications in which complex and non-periodic fracture
systems may present.
In this chapter, we will extend the SI method. The physical meaning of the
transfer function in the SI method is not clear and this leads to some drawbacks of this
method, which will be discussed later. We will first try to understand the physical
meaning of the transfer function and then will propose a modification of the SI
method that can overcome its disadvantages and lead to a more robust fracture
characterization.
4.2
Methodology
4.2.1 Analysis of the SI method
We use a simplified conceptual model, as shown in Figure 4-1, to study the
relationship between the transfer function of a fractured layer and fracture scattering.
We assume that fractures are only present in a single layer, which is represented as the
gray layer with nearly vertical fractures (black lines) in Figure 4-1, and structural
reflectors exist both above and below the fractured layer. For simplicity, we start our
analysis from a 2D acquisition geometry; the azimuthal effect will be discussed later.
In Figure 4-1, the wave field reflected from layers above the fractured layer can be
118
represented as
0i(o) = r (0)
(4.1)
-1(w)
where r1 (o) is the reflectivity of the layers above the fractured layer and I(w) is
the incident wave field, o is angular frequency.
When the incident wave field propagates into the fractured layer, it generates
fracture scattered waves, if fractures are vertical or sub-vertical, most of the scattered
waves propagate downward (Fang et al., 2013), so the downgoing transmitted waves
can be expressed as
T(O) = I(o) + S(o)
(4.2)
where S(w) denotes the downgoing scattered waves.
Based on the Born approximation, we can represent the downgoing scattered waves as
S(o) = FTF(w) - I(o)
(4.3)
where FTF(w) is the fracture transfer function that represents the capability of a
fractured layer for generating scattered waves.
By substituting equation 4.3 into equation 4.2, we get
T(o) = I(o) - [1 + FTF(w)]
(4.4)
The downgoing wave field T(w) is reflected back to the fractured layer by reflectors
below it, and the upgoing transmitted waves generate scattered waves again, thus we
can express the upgoing wave field 02 as
02 ( )
=
=
r 2 (o) - T(a)
[1 + FTF(a)]
r 2 (o) - I(o) - [1 + FTF(0)]
2
where r 2 (w) is the reflectivity of the layers below the fractured layer.
119
(4.5)
In equation 4.5, we assume the fracture transfer function FTF(co) is identical for
downgoing and upgoing waves. The physics behind this assumption is that fractures
are sub-vertical and have no preferential inclination direction.
In the frequency domain, the transfer function defined in the SI method (Willis et al.
2006) can be expressed as
TF(w) = 72(W)
(4.6)
where 0(w) and 02(co) are the reflected waves from above and below the fractured
layer, respectively.
Substitute equations 4.1 and 4.5 into equation 4.6, we have
TF(o) = TF 0 (o)- [1+ FTF(6)]
2
(4.7)
with
TF 0 (w)
= r2(6)
ri (w)
(4.8)
represents the transfer function of background layers. In equation 4.7, TF(O)
includes the effects of background layer reflectivity (i.e. TF0 (co)) and fracture
scattering (i.e. FTF(co)). Fractures have two contributions to the entire wave field: (1)
generating fracture scattered waves; (2) changing background reflectivity. The
presence of fractures makes a fractured layer softer than it would be in an unfractured
state. The reduced stiffness of the fractured layer will alter the value of r2 (0j) since
the reflectivity includes the specular reflections at the bottom of the fractured layer,
and it can also affect the value of r1 (w)
if the selected time window for 0(co)
includes the reflections from the top of the fractured layer. Thus, both FTF((O) and
TF0 (co) in equation 4.8 are affected by fractures. From equation 4.7, we can see that
120
TF(w) is affected by background reflectivities because r1 (a) is generally unlikely to
be equal to r2 (o) and thus the transfer function TF(co), which was used by Willis et
al. (2006), is not equivalent to the fracture transfer function FTF(o), which represents
the true response of a fracture system.
4.2.2 Modification of the SI method
From the previous analysis, we know that the ratio 0 2 (o)/0j(co) does not give the
fracture transfer function FTF(o) unless r1 (&)=r2 (&). To derive the formula for
calculating FTF(o), we start from equation 4.5 and rewrite it as
O(w) = 00 (o) - [1 + FTF(w)] 2
where 0(a)
indicates
convenience and 00 (&)
02(6)
(4.9)
where the subscript is removed for the sake of
= 1(&) - r2 (a)
represents the background reflected waves
excluding the effect of fracture scattered waves.
Equation 4.9 is obtained through the analysis of a 2D geometry (Figure 4-1) in
which acquisition is conducted along one azimuth. We need to further consider the
azimuthal response of a fracture system. Willis et al. (2006) pointed out that the
fracture scattered wave field acquired along the fracture strike direction is more
coherent than that acquired normal to the strike direction. After azimuthal stacking of
the data with normal moveout correction applied, the scattered waves are enhanced in
the fracture strike direction, while they are eliminated significantly in the direction
normal to the fracture strike. Based on this observation, fracture direction can be
determined by identifying the acquisition direction with shot records containing
121
coherent scattered waves (Willis et al., 2006). With multi-azimuth data, we stack the
data into multi-azimuthal stacks, as in Willis et al. (2006). We further assume that the
azimuthal variation of the azimuthal stacks is mainly caused by fracture scattering but
not specular reflection. The azimuthal dependence of r 2 (w) is weak compared to
scattering from a preferentially-aligned fracture system with fracture
spacing
comparable to the seismic wavelength. Then at azimuth 6, equation 4.9 can be
represented as
O(o, 6) = O"(o) - [1 + FTF(w, 6)]2
(4.10)
where O(w,6) represents the data stacked along azimuth 6 and 0 0(w) represents
background reflected waves which are assumed to be azimuthally independent.
Willis et al. (2006) showed that O(w,9) in the direction normal to the fracture
strike should be very close to 0 0(w) because fracture scattered waves are eliminated
after stacking along this direction. However, fracture direction is unknown in our
problem. We will show below that the average of O(w,6) over all azimuths can also
give us a good approximation of 0 0(w) provided that a significant number of traces is
stacked for each azimuth, because the fracture scattered waves are reduced after
stacking at most azimuths except for the fracture direction. We will show some
examples in the numerical simulation section.
If we assume 0 0(w) in equation 4.10 can be approximated by azimuthal
averaging of O(co,9), then FTF(w) can be directly obtained from equation 4.10 by
doing a spectral division. However, the direct calculation of FTF(w,6) using equation
4.10 is unstable due to the high sensitivity of the phase spectrum to the selected
122
analysis time window. We avoid this instability by ignoring the phase part and taking
the absolute value of equation 4.10 to get
1 + FTF(a, 0)- =
16(o111
(4.11)
where 0(w) represents the azimuthal averaging of O(w, 6), IO(o, 0)1 and |0(w)|
are respectively the amplitude spectra of 0 (w, 6) and 0(w).
We then use the multi-taper spectral method (Park et al., 1987), which eliminates the
windowing effect, to compute the amplitude spectra of 0(o, 6) and 0(o).
For weak scattering, it is reasonable to assume that IFTF(co,0) <1. We further
assume that FTF(o,9) is real although FTF(o,O) can be complex. Then, we can
rewrite equation 4.11 as
FPTJ(w, 6) =O(o8)11 1 - 1
The tilde
(4.12)
above FTF(w,O) in equation 4.12 means that TTF(w, 6) is
an
approximation of FTF(wo,0). By adding a small number (water level) to the
denominator for stabilizing the spectral division, we have
FTF(o, 6) =
'(6,O)
1/2_I
(0)1/2
11/2 +W1
16(to)
(4.13)
where w/ is water level.
FiTN(w, 6) describes the azimuthal response of a fracture system on the
azimuthally stacked data. Calculation of FIfT (w, 6) is more stable than the direct
calculation of FTF(co,0) from equation 4.10, because only the calculation of the
amplitude spectra of data is needed in equation 4.13. In the following experimental
and numerical simulation sections, we will show that the response of a fracture system
can be obtained correctly using equation 4.13.
123
To evaluate the azimuthal variation of FTPF(o,0), we define the average of
FTF(co,6) as
=
NT(6 =F_(&n
(4.14)
where [Wj, CON] is the selected analysis frequency window, N is the total number of
frequencies in the selected window. The fracture strike direction is given by the
maximum of FTF(6), which indicates the direction that has the strongest scattering
energy. The frequency window [col,
WNr]
should be chosen as the one that contains the
fracture scattered energy.
We want to emphasize that our analysis is based on two important assumptions:
(1)
Azimuthal variation of the scattered energy is induced by subsurface fracture
systems;
(2) Fracture scattered waves are preserved and enhanced if stacking is conducted
along the fracture strike direction.
In the following sections, we use both laboratory and numerical simulation data to
explore PFTI(o, 0) and FTF(0).
4.3
Laboratory experiment
We built a parallel fracture network model by cutting parallel notches with 0.635 cm
(± 0.05 cm) spacing and 0.635 cm depth in a Lucite block. We then put this Lucite
block on top of another intact Lucite block to form a two-layer model. Figure 4-2 is a
photo of the Lucite model and Figure 4-3 is a schematic showing the geometry of the
124
model. These two Lucite blocks were coupled by a very thin water layer with
thickness less than 0.05 cm, but the fractures were air-filled.
We used a P-wave source with 500 kHz central frequency (P-wave wavelength
=
0.54 cm) and a vertical component receiver to generate and record seismic waves at
the top surface of the model. Figure 4-4a shows the recorded source wavelet excited
from the transducer and Figure 4-4b shows the source spectrum. Data were collected
at 10 different azimuths, as shown in Figure 4-5. For each azimuth, 8 traces with
common midpoint were collected at the offsets of 4 cm, 6 cm, 8 cm, 10 cm, 12 cm, 14
cm, 16 cm and 18 cm. Figures 4-6a and 4-6b show the recorded seismograms in the
direction normal to fracture strike and fracture strike direction, respectively. Figure
4-6c shows reference traces measured in a region without fractures. In Figure 4-6b,
the blue dashed lines mark (1)
direct P arrivals, (2) surface waves, (3) P-to-S
converted waves from the interface below the fracture zone and (4) P reflections from
the model bottom. Signals between 0.08 ms and 0.12 ms include the scattered waves
from the fracture zone along with P-to-P reflections from the interface below the
fractures. Because the fracture scattered waves of the traces collected at offsets larger
than 10 cm are significantly affected by surface waves, as shown in Figures 4-6a and
4-6b, they are not used in the following processing. At each azimuth, the remaining 4
traces are stacked into a common midpoint (CMP) stack after normal moveout.
Figure 4-7 shows the 10 CMP stacks corresponding to acquisitions at 10 different
azimuths. 'Parallel' and 'Normal' indicate the directions parallel and normal to the
fracture direction, respectively. The blue trace labeled 'Average' is the average of the
125
ten black CMP stacks. For comparison, the CMP stack of 4 traces from a region
without fractures (Figure 4-6c) is shown as the red trace labeled 'Reference'. In Figure
4-7, direct arrivals and surface waves have been muted. The strong signals that arrive
at about 0.18 ms are the reflection from the bottom of the lower Lucite block. The
signals inside the red window contain the waves scattered from the fracture zone. The
coherent signals arriving right after 0.08 ms, which are not shown in the reference
stack (red trace), are the specular reflections from the fracture zone top. Their
frequency is about 250 kHz, the corresponding wave length is 1 cm, which is twice of
the fracture height and spacing. They arrive earlier than other waves, because, for low
frequency components, the fracture zone behaves like an equivalent anisotropic
transition zone, which makes the actual reflection interface shallower. Compared to
the reference stack, the reflection from the interface below the fracture zone is
difficult to identify from the ten CMP stacks since it is disrupted by the fracture
scattered waves. Also, we can see that fractures generate a long coda in the data.
We apply equation 4.13 and use the data inside the red window of Figure 4-7 to
calculate FTF. The length of the analysis time window is 0.05 ms and the time
sampling is 0.4x 10-ims, so the frequency resolution is 20 kHz. A higher frequency
resolution can be obtained by increasing the length of the time window, however, the
window length is restricted by our Lucite model, because the results will be
influenced by the P-to-S converted waves from the interface and the surface waves if
a longer window is chosen. Figures 4-8a and 4-8b show the FTE obtained by using
the blue averaging trace and the stack in normal direction as 0(co) in the calculation,
126
respectively. In Figure 4-7b, we can see that the blue trace is close to the stack at
normal direction but there is some difference. Figure 4-8a is different from Figure
4-8b but they have similar patterns. The value of FTF below 100 kHz and above 900
kHz may not be reliable, because the source has little energy in these frequency
ranges, as shown in Figure 4-4b, and spurious high amplitude could be caused by the
spectral division.
From both Figures 4-8a and 4-8b, we can see that FTF has high amplitude at 00,
which indicates the coherent nature of fracture scattered waves along the fracture
strike direction. The amplitude of FTP has a little increase at around 900, as shown
in Figure 4-8a, which conflicts with our expectation that fracture scattered waves
should be eliminated after stacking along 900. This may be caused by the low fold in
stacking. The spacing of the measured traces, which is 2 cm, is larger than the
wavelength at the peak frequency, which is 0.54 cm. So the stacks are aliased spatially.
Along 900, fracture back scattering is strong (Grandi et al. 2007), so we need
sufficient number of traces collected at a smaller spacing to stack out the scattered
waves. However, the stacking fold and trace spacing are limited by the size of the
model and transducer in our experiment. In the numerical simulation section, we will
show that the average of the azimuthal stacks is very close to the stack at normal
direction and a very good result can be obtained if the trace spacing is smaller than the
seismic wavelength and twenty or more traces are stacked for each azimuth.
In Figure 4-9, the solid and dashed curves are FTF(8) (equation 4.14)
calculated from the FTF(>,0) shown in Figures 4-8a and 4-8b, respectively. For
127
both (a) and (b) in Figure 4-9, the fracture direction can be clearly determined from
the maximum of FTF(6). We choose 100 kHz ~ 900 kHz as the analysis window for
computing FTF(9) based on the source spectrum (Figure 4-4b), whose amplitude
decays by 10 dB (i.e. drops to 10% of the peak amplitude) at 100 kHz and 900 kHz.
The fluctuation of FTF(6) at around 900 is caused by the low fold in stacking.
4.4
Numerical modeling
In our numerical modeling study, we use the finite difference method of Coates and
Schoenberg (1995) to simulate the propagation of seismic waves through an elastic
medium containing discrete fractures. The five layer model, which is shown in Figure
4-10, studied by Willis et al. (2006) is used in our numerical study. The properties of
each layer are summarized in Table 4-1. Fractures are in the third layer. An explosive
source, shown as a red star in Figure 4-10, is used in the simulation. The source time
function is a Ricker wavelet with a 40 Hz central frequency. Receivers (black
triangles in Figure 4-10) spread along the surface in a rectangular area of 1000 m
x
1000 m with 20 m spacing. Only the recorded pressure is used in our study. A
perfectly match layer (PML) absorbing boundary condition is added to all boundaries
of the model.
We consider fractures with compliance varying from 10-1im/Pa (stiff fracture) to
10-9 m/Pa (compliant fracture) (Daley et al., 2002). In order to reduce the complexity
in the following analysis, we take normal compliance to be equal to tangential
128
compliance, which may represent gas-filled fractures (Sayers et al., 2009). Here and
after, fracture compliance means both fracture normal and tangential compliances. We
will show the results for four different scenarios: (1) parallel vertical fractures; (2)
orthogonal vertical fractures; (3) orthogonal sub-vertical fractures with randomly
varying facture spacing and compliance; (4) presence of random heterogeneity in the
background model of scenario (3).
4.4.1 Parallel vertical fractures
Figure 4-11 shows the layout of parallel vertical fractures in the third layer of the
model shown in Figure 4-10. In this model, fractures with equal spacing are parallel to
the X axis. Figures 4-12a and 4-12b show examples of the shot records acquired along
the X and Y axes, respectively, for a model with 40 m fracture spacing and 5x1010
m/Pa fracture compliance. Figure 4-12c shows a shot record for a model without
fractures. Comparing these three panels, we can see that significant amount of
scattered waves are generated by the fractures and the scattered waves are less
coherent in the shot record normal to the fracture direction (Figure 4-12b) than for the
record parallel to the fracture strike (Figure 4-12a). Following the same azimuthal
stacking approach used in the laboratory experiment, we stack the shot data acquired
at ten different azimuths after normal moveout. Figures 4-13a, 4-13b, and 4-13c show
the CDP stacks at ten different azimuths obtained by stacking 5, 10 and 20 traces,
respectively, for each azimuth for the model with 40 m fracture spacing. 00 and 900
are the X and Y axis directions, respectively. The intervals of stacked traces (receiver
129
spacing) are 80 m (- one wavelength), 40 m (- half wavelength) and 20 m (- quarter
wavelength), respectively, for the CDP stacks shown in Figures 4-13a, 4-13b and
4-13c. The maximum offset of the stacked traces is 400 m. From Figure 4-13a, we can
see that the fracture scattered waves in the CMP stacks are still very strong near the
fracture normal direction (i.e. 900) if only five traces are stacked for each azimuth.
Similar to the case in our laboratory experiment, it is due to the spatial aliasing in
stacking. However, the scattered waves are reduced significantly at most azimuths
except for the fracture strike direction (i.e. 00) when the number of stacked trace
increases to ten, as shown in Figure 4-13b. We find that generally stacking of twenty
traces with spacing of about a quarter of wavelength is sufficient to eliminate the
scattered waves in the fracture normal direction.
Figure 4-14 shows the CDP stacks for six models with different fracture spacings.
Fracture compliance is 5x10
10
m/Pa for all six models. In each panel of Figure 4-14,
the ten black traces are the CDP stacks at ten different azimuths. For each azimuth,
twenty near offset (from 20 m to 400 m) traces are stacked after normal moveout. The
blue trace, which is labeled 'Ave', is the average of the ten azimuthal stacks. It is used
for estimating the background reflectivity in our calculation. The red trace in Figure
4-14a, which is shown for comparison, is the stack for the reference model without
fractures. From Figure 4-14a, we can see that, when the fracture spacing (20 m) is
much smaller than the dominant seismic wavelength (100 m), the fractured layer is
equivalent to a homogeneous anisotropic layer and very weak scattered waves are
generated. When the fracture spacing is comparable to the dominant wavelength, the
130
fractured layer generates strong scattered waves. The dashed red window, having
length of 0.2 s, is the analysis window chosen for computing FTF (equation 4.13). It
contains reflections from the third and fourth interfaces and the fracture scattered
waves. Figure 4-15 shows the FTF of the six models. FTF has higher amplitude,
which indicates strong fracture scattering energy in the CDP stack, along the fracture
direction (i.e. 00) for all models except for the case of 20 m fracture spacing. For the
model with 20 m fracture spacing, FTF is close to zero at all azimuths. This
indicates that very weak fracture scattered waves are contained in the data, which is
consistent with the CDP stacks in Figure 4-14a.
We use equation 4.14 to calculate FTF and use it for determining the fracture
direction. In the calculation of FTF, we choose 8 Hz ~ 88 Hz as the averaging
frequency window based on the amplitude spectrum of the source wavelet (Ricker),
which drops by a factor of 10 at 8 and 88 Hz from that at the peak frequency of 40 Hz.
Thus the frequency window 8 Hz ~ 88 Hz contains most of the source energy and is a
reasonable window for our analysis. Figure 4-16 shows the polar plots of FTF. FTF
is multiplied by 10 for plotting in Figure 4-16a. Our simulations do not cover a
complete 3600 acquisition, so we replicate the first quadrant (from 00 to 900) to the
other three quadrants in the polar plots. These results show that there is a clear
maximum in the fracture direction (i.e. 00) even for the case of 20 m fracture spacing,
which does not show obvious azimuthal variation of scattered waves in the CDP
stacks.
We have only shown the results for models containing fractures of 5x10-10 m/Pa
131
fracture compliance here. We found that the azimuthal variation trend of both PT
and FTF does not change much when the fracture compliance varies between 1010
m/Pa to 10~9 m/Pa, but their amplitudes change with fracture compliance. Figure 4-17
shows the maximum of FTF (i.e. FTF(00)) for the six models with different fracture
spacings when the fracture compliance changes from 10-10 m/Pa to 10~9 m/Pa. We can
see that FTF is very small for the case of 20 m fracture spacing for all compliance
values. For the models of 32 m, 40 m, 60 m, 80 m and 100 m fracture spacing, the
maximum of FTF increases nearly linearly with fracture compliance. This indicates
that fracture scattering strength and fracture compliance have a nearly linearly
variation trend (Fang et al., 2013). But for the case of 100 m fracture spacing, the
value of
FTF maximum is much larger than that of the other models. This is
because the fracture spacing is equal to the dominant wavelength in the fractured
layer in this case, so the fractured layer generates the strongest scattered waves. From
this comparison, we can see that the fracture scattering strength depends on both the
fracture compliance and spacing. The relationship between fracture scattering strength
and fracture spacing is complicated. For traditional AVOA analysis, if the compliance
of individual fractures is known, then the relationship between reflection amplitude
and fracture spacing can be determined (Lynn et al., 1996; Hunt et al., 2010), because
the fracture induced formation excess compliance is linearly proportional to the
fracture density (Schoenberg and Sayers, 1995; Dubos-Sallde and Rasolofosaon, 2008)
under the effective medium assumption. However, the linear relationship breaks down
when the fracture spacing is comparable to the seismic wavelength. The effective
132
medium assumption is not valid in this regime and scattering theory is needed for
explaining the behavior of seismic waves.
4.4.2 Orthogonal vertical fractures
In 4.4.1, we have shown that the direction of a parallel fracture system can be
determined through computing FTF of a fractured layer. Multiple fracture sets with
different orientations are common in nature (Pollard and Aydin, 1988) and may exist
in a reservoir (Bakulin et al., 2000; Bai et al., 2002; Grechka and Kachanov, 2006), so
it is also important to investigate the applicability of our approach to a fractured layer
containing fractures with more than one direction. We study a model with two
orthogonal fracture sets as shown in Figure 4-18. We will call the fracture set striking
parallel to the X axis fracture set X and the other fracture set Y. We vary the fracture
spacing and compliance of both fracture sets and study the effect of the orthogonal
fracture system on the seismic waves. Figure 4-19 shows the CDP stacks of six
different orthogonal fracture models. Table 4-2 lists the parameters of these six
models. In models (al), (a2) and (a3), all parameters are the same except for the
fracture spacing of fracture set Y, which are 40, 60 and 80 m, respectively. Compared
to models (al), (a2) and (a3), the fracture compliance of fracture set X increases by a
factor of two in models (bl), (b2) and (b3). In Figure 4-19, we can see that the
fracture scattered waves are stronger at 00 and 900 while weaker at intermediate
azimuths for all six models. The dashed red window is the selected analysis time
window for computing FTF which is shown in Figure 4-20. All parameters used in
133
the calculation of FTF are the same as those used in 4.1. High amplitudes appear at
both 00 and 900 for all six models in Figure 4-20. This indicates that the scattered
waves from both fracture sets X and Y are preserved in the azimuthal stacks, which is
also observed by Burns et al. (2007).
Similar to Figure 4-16, we plot the azimuthal variation of FTF in polar
coordinates in Figure 4-21. These results show that there are clear two maxima in the
X (00) and Y (900) directions, which indicate the orientations of the two orthogonal
fracture sets. For (bl) and (b3), the maximum in the X direction is larger than that in
the Y direction, while the reverse is true for (b2). This indicates that the value of
maxima is affected by both fracture compliance and spacing. As we have discussed in
4.1, there appears to be no simple linear relationship between the amplitude of FTF
and fracture compliance. Thus the relative compliances of these two fracture sets
cannot be determined from the relative ratio of the two maxima shown in FTF.
Moreover, we should point out that the fracture scattered waves are affected by both
fracture sets. That means the fracture scattered wave field from an orthogonal fracture
system is not equivalent to the superposition of the scattered wave fields from the two
parallel fracture systems that form the orthogonal fracture system.
4.4.3 Orthogonal sub-vertical fractures with randomly varying
fracture spacing and compliance
In 4.4.1 and 4.4.2, we have shown that the fracture strike of an equally-spaced
periodic fracture system can be determined using our approach. The two models in
134
4.1 and 4.2 are simplified prototypes for the field case of aligned fractures that are
likely to be neither equally-spaced nor exactly vertical. To study the robustness of our
approach to a more realistic problem, we vary the fracture spacing, compliance and
dip angle to construct a non-periodic orthogonal fracture model. Figure 4-22 shows
the layout of the fractured layer containing orthogonal sub-vertical fractures with
varying fracture spacing. The fracture spacing of fracture set X, which is measured in
the middle of the fractured layer at Z = 500 m, randomly varies from 48 m to 72 m
and the fracture spacing of fracture set Y also has a random variation within the range
of 32 m to 48 m. The dip angle of all fractures randomly varies from -150 to 150. The
fracture compliances of both fracture sets also have a random distribution. The
9
compliance variation ranges for fracture sets X and Y are 0.9x10~ m/Pa ~ 1.1x10
9
m/Pa and 4.5x10- 0 m/Pa ~ 5.5x 1040 m/Pa, respectively. We take the compliance to be
constant over the fracture plane for an individual fracture. The distributions of fracture
spacing, compliance and dip angle are uniformly random.
We randomly vary the fracture spacing, compliance and dip angle within the
specified ranges and generate 25 different fracture models. For each model, we
simulate the synthetic seismic wave field and then calculate FTF and FTF of the
fractured layer as we did in 4.1 and 4.2. Figure 4-23 shows the polar plots of FTF
for these 25 models. For all models, FTF has two maxima in the X and Y directions,
which indicate the directions of the two fracture sets. The maximum in the X direction
is larger than that in the Y direction for most cases, since fracture set X is more
compliant than fracture set Y. These two maxima have similar values for some models,
135
such as models 1, 2, 9 and 20. This again suggests that the relative amplitude of FTF
in the two fracture directions is affected by both fracture compliance and spacing.
From this study, we can see that the fracture direction can be robustly determined
using our approach even for a non-periodic model containing inclined fractures with
varying fracture spacing and compliance.
4.4.4 Presence of random heterogeneity in a background model
To study the effect of background heterogeneity on our method, we add random
heterogeneity into the background layer model shown in Figure 4-10 using the von
Kdrmin autocorrelation function (Sato and Fehler, 1998). We apply the same
perturbation function to both P- and S-wave velocities while keeping the density
unperturbed. The maximum velocity perturbation is limited at 10%. Heterogeneities
with different correlation lengths are added to the model studied in 4.3, which
contains orthogonal fractures with varying fracture spacing, compliance and dip angle.
Figures 4-24a, 4-24b and 4-24c show the P-wave velocities of three heterogeneous
models with 25 m, 50 m and 100 m correlation lengths, respectively. The spatial
variation of S-wave velocity is similar to that of P-wave. Figure 4-25 shows the polar
plots of FTF for 15 models, whose heterogeneity and fracture system are generated
randomly and independently. The heterogeneity correlation lengths are 25 m, 50 m
and 100 m for the results shown in the first, second and third rows, respectively.
Comparing to Figure 4-23, we can see that although the existence of heterogeneity
may increase the uncertainty of fracture orientation detection due to broadening the
136
peaks with maximum value of FTF, the two orthogonal fracture sets are still able to
be identified clearly.
4.5
Emilio Field application
4.5.1 Background of the Emilio Field
The Emilio field, which is operated by ENI S.P.A., is an offshore oil field that is
located in the central part of the Adriatic Sea, near the eastern coast of Italy. It is about
25 km offshore and the water depth is about 80 m. Its location is shown in Figure 4-26.
PS-wave anisotropy study (Gaiser et al., 2002; Vetri et al., 2003) and PP scattering
analysis (Willis et al., 2006) indicate that two orthogonal fracture sets oriented
east-northeast and north-northwest exist in this field, which is confirmed by the well
data. Well data also indicate that most fractures show near-vertical (600-900) dips.
Figure 4-27 shows the Emilio ID P-wave velocity model, which was provided by ENI.
The most prominent reflector that can be identified from seismic data is the
Gessoso-Solfifera, which is a high-veloicty evaporite layer at the depth of about 2 km
as shown in Figure 4-27. The Gessoso-Solfifera layer was built up in the Miocene and
then a thick sequence of overburden sediments, mainly composed by sand-clay, were
deposited during the Pliocene. The reservoir unit, which is a fractured carbonate layer,
is at the depth of about 2850 m with the thickness of tens of meters. The current
tectonic stress, as shown in Figure 4-28, determined from earthquake focal
mechanism solutions (Gerner et al., 1999) indicates that the maximum horizontal
137
stress direction changes from nearly east-west in the north part of Emilio to
northeast-southwest in the southern part. The abrupt change of stress direction in
Emilio may result in intersecting fracture systems, which have been observed in wells.
4.5.2 Acquisition geometry
In the Emilio field, a cross shooting acquisition with shot line perpendicular to
receiver lines was used in the OBC survey. In Figure 4-29, the red and blue lines show
the source and receiver positions, respectively. The target area where wells have been
drilled is highlighted in green. The Inline and Xline increments are 12.5 m and 25 m,
respectively. The average shot line spacing is 200 m and the receiver line spacing is
250 m. Shot and receiver intervals are 25 m and 50 m, respectively. Source and
receiver lines were oriented parallel to the principal axes of anisotropy determined
from borehole analysis.
4.5.3 Data processing
Figure 4-30 shows a shot profile at Inline 1376 and Xline 2796. The strong event that
can be directly identified
in the seismic data is the reflection
from the
Gessoso-Solfifera layer that arrives at about 2 s. The Gessoso-Solfifera reflection
signals become weak and are contaminated by the water multiples when the offset is
beyond 2000 m, as shown in Figure 4-30. At offsets less than 500 m, the signals are
strongly influenced by the 'slow noise', which is interpreted as Scholte waves that
138
propagate along the sea floor with a velocity smaller than the shear wave velocity of
sea floor sediments. We pick the Gessoso horizon, which is shown in Figure 4-31,
based on the Kirchhoff pre-stack depth migration conducted using the 1D P-wave
velocity model shown in Figure 4-27 and then shift it by 600 m downward to
approximate the reservoir top as inferred from well data. The Gessoso horizon shows
that the reservoir is associated with an anticline structure which was formed by
east-west compression (Gaiser et al., 2002). The reflection from the reservoir top is
very weak and its approximate position is interpreted from the borehole and seismic
data by ENI. We do not need to know the exact depth of the reservoir layer for
fracture characterization using the fracture transfer function method. We only need to
select a time window that contains the signals that have passed through the reservoir
region. In the processing, we choose a 50x50 m2 bin size. At each common-midpoint
(CMP) bin, we only select the data with offsets varying from 500 m to 2000 m for the
stacking due to the poor data quality at short and long offsets, as shown in Figure 4-30.
There are at least 180 traces included at each CMP bin. Figure 4-32 shows the
variation of the total number of traces within the offset range from 500 m to 2000 m
at each CMP location. At each CMP location, we then stack the CMP gather after
normal moveout into 18 azimuth bin gathers from 00 (East) to 1700 in steps of 100. In
order to make the azimuth coverage fold to be uniform, we stack the data within ±200
from the central direction of each azimuthal bin. By doing this, the azimuthal
coverage fold is fairly uniform and there are at least 40 traces stacked at each azimuth,
as shown in Figure 4-33. The average trace spacing in each azimuthal bin gather is
139
about 30 m, which is small enough to avoid spatial aliasing in stacking. We use the
data within a 400 ms window that starts from the reservoir top reflection to calculate
the fracture transfer function. Figure 4-34 shows a polar plot of the fracture transfer
function at four different locations. Radial and angular coordinates are frequency and
azimuth, respectively. From Figure 4-35, which shows the amplitude spectrum of the
data, we can see that the dominant frequency of the data is at about 10 Hz and the
signals decay by 10 dB at 70 Hz. However, the FTF shows very weak amplitude
below 30 Hz, because the low frequency components do not generate fracture
scattered waves. We pick the fracture orientation by analyzing the amplitude of the
FTF from 30 Hz to 70 Hz. The value of FTF is not reliable at frequencies greater than
70 Hz due to the lack of seismic energy for these frequencies. In Figures 4-34a and
4-34b, the fracture orientations can be easily determined from the strong peaks at
about 300 and 400, respectively. In Figures 4-34c and 4-34d, we can see that two peaks
(at
00
and 600 for 8c and 00 and 1200 for 8d) having similar amplitudes appear. This
may indicate the existence of two fracture sets oriented in different directions. If the
FTF shows more than one peak with similar amplitudes, we pick all of them and
interpret them as potential intersecting fractures. Figure 4-36 shows the fracture
orientations derived from the FTF. Dark and light background colors indicate high and
low confidence of the result, respectively. Confidence is defined as the standard
deviation of the FTF divided by its mean value. High confidence means the
azimuthal variation of the FTF is larger while low confidence indicates the FTF
shows weak directionality. White bars represent the fracture strike orientation. Two
140
intersecting white bars indicate the orientations of two potential crossing fracture sets.
The long and short bars indicate the dominant and secondary fracture sets,
respectively. From Figure 4-36, we can see that fractures tend to be oriented
east-northeast or north-northwest at most locations and sub-orthogonal fracture
systems are present in this field. Figure 4-37 shows the comparison of the fracture
directions derived from the well data (upper row) and the FTF (second row) in three
wells, whose locations are shown in Figure 4-36. We generally expect that recently
created tensile fractures should have azimuths that are parallel to SHmax and this is
what is most commonly seen in the well data. The fracture directions derived from the
FTF are generally consistent with the well data but with about 100 to 150 mismatch,
which is a reasonable uncertainty of the result obtained from FTF as the overlapping
angle between adjacent azimuthal bin gathers is 200.
4.6
Summary
We have shown that the response of a fracture system can be obtained through
computing the fracture transfer function using equation 4.13 and then the fracture
direction can be determined by identifying the maximum value of the average fracture
transfer function (equation 4.14). Both laboratory experiments and numerical
simulations show that fracture direction can be determined robustly using our
approach. In the numerical simulation section, we demonstrate that our approach can
determine the directions of both fracture sets of an orthogonal fracture system. Also,
141
the study of an orthogonal fracture model containing sub-vertical fractures with
randomly varying fracture spacing and compliance shows that our approach can be
applied to not only periodic fracture systems but also non-periodic fracture systems,
which represent the real case of aligned fractures in a reservoir. Moreover, our
approach is shown to be robust even in the presence of random heterogeneity in a
background model. The consistency between the fracture directions derived from the
fracture transfer function and the well data in the Emilio Field further demonstrates
the potential of our approach for field applications.
We have only discussed the application of our approach for determining fracture
orientation. The amplitude of FTF defined by equation 4.13 also gives information
about fracture compliance, which was discussed in 4.1 of the numerical simulation
section. The relationship between scattering strength and fracture compliance and
spacing is not straightforward when the problem falls into the scattering regime.
Further study is needed to understand their relationship. The interference pattern of
fracture scattered waves in the seismic data might provide us the knowledge about
fracture spacing. However, the interference pattern is altered by the stacking.
Therefore, fracture spacing cannot be determined from our approach. We need to use
the pre-stack data with a method like the double beam method of Zheng et al. (2011,
2012) to retrieve fracture spacing information.
142
Table 4.1: Parameters of the model shown in Figure 4-10.
Vp (m/s)
Model
Vs (m/s) p (kg/m3 ) Thickness (m)
Layer 1
3000
1765
2200
200
Layer 2
3500
2060
2250
200
Layer 3
4000
2353
2300
200
Layer 4
3500
2060
2250
200
Layer 5
4000
2353
2300
200
Table 4.2: Parameters of six orthogonal fracture models.
Fracture set Y
Fracture set X
Fracture set X Fracture set Y
compliance (m/Pa) compliance (m/Pa)
spacing (m)
spacing (in)
al
40
40
5x10-10
5x10-10
a2
40
60
5x10' 0
5x10~"4
a3
40
80
5x10~40
0
5x101'
bl
40
40
10~9
5x10-4
b2
40
60
10-9
5x10-10
b3
40
80
10-9
5x10
143
Figure 4-1: Cartoon showing how incident waves I are scattered by a fractured layer,
which is denoted by the gray horizontal zone with nearly vertical black lines. 0 and
02
are, respectively, waves reflected by layers above and below the fractured layer,
whose reflectivities are r, and r2 , respectively. T is the transmitted waves.
144
Figure 4-2: Photo of the Lucite model. The cuts representing fractures can be seen
about half way along the block in the vertical direction.
145
(a)
Luck*
Vp 2.7 kmls
Vs. 1.3 kmls
p -1.18 gem*
22
1 .3 cm
0.3
cm 50
6.3 mm
7.5
c
15.6
c
7.5
cm
I .2cm
30.6 cm
lb)
0..3
--5
-e--
mm
635m
Fracture zone
Figure 4-3: (a) Schematic showing the geometry of the Lucite model. (b) Expanded
view of the fracture zone. The fracture zone was built by cutting parallel notches with
0.635 cm (± 0.05 cm) spacing at the bottom of the upper Lucite block. The properties
of the upper and lower Lucite blocks are identical. P- and S-wave velocities of Lucite
are 2700 m/s and 1300 m/s, density is 1180 kg/M3 .
146
(a)
1
S0.5
--
--
0
..
......
.... ...
-.- --..-.--..-.-.
..-..- -............
E- 0.5
0
--------
z
0.05
0.04
0.03
0.02
Time (ms)
0.01
0
(b)
0
--..-- -.--.
----....
-.
-.
. ---..
- 2 .....-- - ----- ---. ...
......................
0
100
200
..
300
.
.
400 500 600
Frequency (kHz)
.
.-.-.
.--
......
.
700
800
900
1000
Figure 4-4: (a) Source wavelet excited from the transducer. (b) Amplitude spectrum of
the source wavelet. The amplitude spectrum is expressed in decibels by evaluating ten
times the base-10 logarithm of the ratio of the amplitude to the maximum amplitude.
147
fracture
zone
0
I
60
o0
74
01
,0
I
-
0M 1
I
Figure 4-5: Schematic showing the layout of CMP measurements at the surface of the
Lucite fracture model. CMP measurement was conducted at 10 different azimuths, the
acquisition lines of 00 and 900 correspond to the orientations parallel to and normal to
the fracture strike, respectively.
148
(a)
Normal
18
-A
6
E 14
CO,
0
8
6
4
0
0.05
(b)
E
0.1
0.15
0.2
0.15
0.2
Parallel
18
a)
0
16
14
)
o
18
12
0.05
0.1
A- A
Reference
--
-A
AA
16
14
0 129
10
Co
8
0
6
4
0
0.05
0.1
Time (ms)
0.15
0.2
Figure 4-6: (a) and (b) Data acquired at the surface of the Lucite model in the
direction normal and parallel to the fracture strike, respectively. For each azimuth, 8
traces at different offsets are plotted. (c) Data measured in a region without fractures.
In (b), labels identify (1) direct P arrivals, (2) surface waves, (3) P-to-S converted
waves from the interface below the fracture zone and (4) P-wave reflections from the
model bottom.
149
(a)
(b)
Parallel
Paralle!,,I
100
100
200
20LO
300
V
300
AJ
4&
400
0
700
70 0
800
80
Normal
Norma
Average
Aver
Reference
I
I
,
0
0.05
Referen_
I
I
I
0.1
,
0.15
0.2
0.07
Time (ms)
0.08
0.09
0.1
Time (ms)
0.11
0.12
Figure 4-7: (a) CMP stacks at 10 different azimuths. (b) Expanded view of the signals
in the red window of (a). The acquisition angles are denoted above each trace,
'Parallel'/'Normal' indicates the acquisition is parallel/normal to the fracture strike.
'Average' and 'Reference' represent the average of the ten black traces and the stack
of traces collected in a region without fractures, respectively.
150
(a)
1000r."
(b)
1000
.9 ...
900
800
800
700
700
600
600
500
500
400
400
300
300
01
200
100
100
00
0 10 20 30 40 50 60 70 80 90
Azimuth (degree)
0 10 20 30 40 50 60 70 80 90
Azimuth (degree)
-1
Figure 4-8: (a) and (b) FTF obtained using the average of CMP stacks and the CMP
stack at normal direction to represent the background reflection, respectively. 00 and
900 represent the directions parallel and normal to the fracture strike, respectively.
Color bar is in linear scale.
151
.
0. 1.. . ... .........
U
0-
-0.11
0
10
20
30 40 50 60
Azimuth (degree)
70
80
90
Figure 4-9: Solid and dashed black curves are the average of FTF shown in Figures
4-8a and 4-8b, respectively.
152
0
0.2
E0.4
N
m
0.8
11.5
Y (km)
1.5
0.5
0.5
1X (km)
Figure 4-10: Five layer model. Red star and black triangles are source and receivers
respectively. Properties of each layer are listed in Table 4-1. (Modified from Willis et
al. (2006)).
153
2
Y (km)
X (kin)
0 0
Figure 4-11: Layout of parallel fractures (black lines) in the third layer of the model
shown in Figure 4-10. Fractures with equal spacing are vertical and parallel to the
X-axis.
154
(a) 0
(b)0
(c) 0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
F 0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
E
0.70
0.5
Offset (km)
1
0.70
0.5
Offset (km)
1
0.70
0.5
Offset (km)
1
Figure 4-12: (a) and (b) Shot records (pressure) acquired parallel (along X-axis) and
normal (along Y-axis) to the fracture strike, respectively. (c) Shot record obtained
from a reference model without fractures. Fracture spacing is 40 m.
155
(a)
Stack of 5 traces
0
10
D 20
6> 30
R 40
50
E 60 --- '
70
80
90
0
(b)
Stack of 10 traces
0
0
-
0.4
0.2
Time (s)
0.6
Stack of 20 traces
(c)
10
10-
20
30
40
50
60
70
80
90 -A-
20
0
'V
0.2
0.4
Time (s)
0.6
30
40
50
60
70
80
90
0
0.2
0.4
Time (s)
0.6
Figure 4-13: CDP stacks of the model with 40 m fracture spacing. The numbers of
traces stacked at each azimuth are 5, 10 and 20 for (a), (b) and (c), respectively. The
trace spacings are 80, 40 and 20 m for (a), (b) and (c), respectively. 00 and 900 are the
directions along X and Y axes, respectively.
156
20 m
(a)
0
10
$ 20
030
S40
---
0.4
60
--
T
80 m
0
0
0.6
0.4
0.2
(e)
-1wv
10
10
S20
30
940
50
S60
70
80
90
70
80
A,-
90
0
0.6
60 m
(d)
0
90
0.2
40 m
(c)
10
20
30
40
50
50
60
70
80
x 60
70
80
90
32 m
(b)
0
10
20
30
40
0.2
0
20
30
40
50 60
70
80
90
0
10
20
30
40
50
60
70
80
90
IOAA
A
I
0
0.2
0.4
Time (s)
0.6
0
0.2
0.4
Time (s)
0.6
0.6
100 m
(f)
t
0.4
0
I
0.2
0.4
Time (s)
0.6
Figure 4-14: CDP stacks of six models with different fracture spacings, which are
shown above each panel. 00 and 900 are respectively the directions along X and Y
axes, respectively. In each panel, the blue trace is the average of the ten azimuthal
stacks. The red trace in (a) is the stack for the reference model without fractures. The
dashed red window is the selected window for computing FTF. The waves inside the
red window arriving at about 0.4 s are reflections from the bottom of the fractured
layer.
157
20 m
(a)
(b)
2Gm
40
m
4Gm
:AT
80
80
80
6
60
60
40
40
40
U-20,
20
20
0
0
0L
30
60
90
60m
(d)
0
80
601
60
C) 40-
60
90
80 m
(e)
80
30
0
30
60
90
100 m
()
80
40
40
LI~20.
20
01
30 60 90
Azimuth (degree)
0
0
30
60
90
Azimuth (degree)
-1
0
0
0
30 60 90
Azimuth (degree)
1
Figure 4-15: FTF (equation 4.13) of six models with different fracture spacings. The
number above each panel denotes fracture spacing. 00 and 900 are respectively the
directions along X and Y axes.
158
(a)
180
20 m (x10)
-
--
(b)
0
180
--
0
(e)
---
0
270
(f)
--
270
--
0
270
80 m
90
180 --
40 m
180 --
270
60 m
0
-
(C)
180 ---
270
(d)
32 m
0
100 M
0
180
0
270
Figure 4-16: FTF of the six models with different fracture spacings. The number
above each panel denotes the fracture spacing. In each polar plot, the blue curve
shows the azimuthal variation of FTF. Radial and angular coordinates are the value
of FTF and azimuth, respectively. The radial coordinate is 0 at the center and 0.6 at
the solid circle. 00 and 900 are the X and Y directions respectively. In plot (a), FTF is
multiplied by 10 for plotting.
159
0.9
.
-*-20m
-A-32m
0.8
-
---
-v-40m
0.7
-E-60m
UL 1 0 .6 --.
-8-80m
>5
0 .4
-O-100m
- -
-
... .............
-.-.-.-.-.-
--.
~
--
.
-
-
0.3- ---
0.2
0.1
- -
"
-----------
4'
.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fracture compliance (10~9 m/Pa)
Figure 4-17: Plot of the maximum of FTF for six models with different fracture
spacings when the fracture compliance varies from 10-10 m/Pa to 10
160
9
n/Pa.
N
2
2
X (kin)
0
0
Figure 4-18: Layout of orthogonal fractures (black lines) in the third layer of the
model shown in Figure 4-10. Two sets of vertical fractures intersect with each other in
this model. They are parallel to the X and Y axes respectively. Each fracture set has
equal fracture spacing and compliance.
161
(al) Dx=40m, Dy=40m
(a2) Dx=40m, Dy=60m
)
-Ci
10
20-_
30
2
3
4
5
6(
E 78
8
90
0
0
I
1
20
30
40
5060-
40 -
50
60
70
0.4
0.6
0
(bi) Dx=40m, Dy=40m
80
90
0A
9(
0.2
(a3) Dx=40m, Dy=80m
0.2
0.4
0.6
0
(b2) Dx=40m, Dy=60m
= 60
E 70
~480
< 90
0
--
"***
0.2 0.4 0.6
Time (s)
0.4
0.6
0
28
2
i- 50
0.2
illyr
(b3) Dx=40m, Dy=80m
1
L28
S30
-40
I
3011
30
4(
5(
6
7C
8(
9(
40
50
M,
A
-
60
78
8
90
0
0.2 0.4 0.6
Time (s)
0
0.2 0.4 0.6
Time (s)
Figure 4-19: CDP stacks of six orthogonal fracture models with different fracture
spacings and compliances. Dx and Dy indicate the fracture spacings for fracture sets
X and Y, respectively. 00 and 900 are respectively the directions along X and Y axes.
162
(al) Dx=40m, Dy=40m
Cr
0
a)
(a2) Dx=40m, Dy=60m
(a3) Dx=40m, Dy=80m
80-
80
80
60
60
40
40
40
20
20-
20
60-
e
U-
30
0
60
90
(bi) Dx=40m, Dy=40m
0
0
30
60
90
(b2) Dx=40m, Dy=60m
0L
0
:' j'
30
-L" -
60
90
(b3) Dx=40m, Dy=80m
80
80
80
60
30
60
a 40
0
40
40
2U
-U
C
Q)
U-
20
0L
0
30
60
90
Azimuth (degree)
-1
0
30
60
90
Azimuth (degree)
0
30
0
60
90
Azimuth (degree)
1
Figure 4-20: FTF of the six orthogonal fracture models. Dx and Dy indicate the
fracture spacings for fracture sets X and Y, respectively.
163
(al) Dx=40m, Dy=40m
90
(a2) Dx=40m, Dy=60m
90
0
0 180
0 180
180
(a3) Dx=-40m, Dy=80m
90
270
270
270
(b1) Dx=40m, Dy=40m
9
(b2) Dx=40m, Dy=60m
90
(b3) Dx=40m, Dy=80m
90
0 180
180
270
0- 180
-
270
0
270
Figure 4-21: FTF of the six orthogonal fracture models. The radial coordinate is 0 at
the center and 0.5 at the solid circle.
164
-0.4
_e0.6
N 2
2
1
Y (km)
X (km)
0 0
Figure 4-22: Layout of orthogonal sub-vertical fractures (black lines) in the third layer
of the model shown in Figure 4-10. Fracture spacing, compliance and dip angle
randomly vary within the specified ranges.
165
2
3
4
4
14
16
17
18
19
20
21
22
23
24
25
Figure 4-23: FTF of 25 orthogonal fracture models with randomly varying fracture
spacing, compliance and dip angle. The number above each polar plot denotes the
model number. The horizontal and vertical dashed lines in each plot indicate the X
and Y directions, respectively.
166
(a) 25m
(b) 50m
(c) 1OOm
V (km/s)
2.7
Figure 4-24: (a), (b) and (c) P-wave velocities of the layer model shown in Figure
4-10 after adding random heterogeneity with 25 m, 50 m and 100 m correlation
lengths, respectively.
167
2
3
4
5
7
8
9
10
13
14
15
25m I
6
50m
12
F OOm
Figure 4-25: FTF )f 15 heterogeneous models containing orthogonal fractures with
randomly varying fracture spacing, compliance and dip angle. The correlation lengths
of the heterogeneity are 25 m, 50 m and 100 m for results shown in the first, second
and third rows, respectively. The number above each polar plot denotes the model
number.
168
witzerlind
,;Lusanne
wian Brescia
nVerona
Tur
Pari
1-logna
Bono
Pat
Florec
Can
SItaly
A
Temi
Figure 4-26: Emilio field location.
169
0
Sand-clay
1
2
Evaporite
Carbonate
3
4
5L --------
1
2
3
4
Velocity (km/s)
5
6
Figure 4-27: Emilio ID P-wave velocity model.
170
45 0 N
Adriati Sea
A
42 0N I
12 0E
L
L
/
16 0E
Figure 4-28: Direction of the maximum horizontal stress derived from earthquake
focal mechanism solutions. (Modified from Gerner et al. (1999))
171
3000
I
2900
1
2800 -
2700
2600-
2500'
'
1100 1200 1300 1400 1500 1600 1700 1800
Inline
Figure 4-29: Emilio field acquisition configuration. Red and blue lines are the source
and receiver lines, respectively. The green region is the target region where wells have
been drilled.
172
W- 2
E
'3
5
-4
-3
-2
-1
0
Offset (km)
1
2
3
Figure 4-30: Seismic profile (vertical component) for a shot at Inline 1376 and Xline
2796. The most prominent event is the reflection from the Gessoso-Solfifera at about
2 seconds. The data are strongly contaminated by the ocean bottom Scholte waves at
small offsets and water multiples at large offsets.
173
4
2.4
2.3
1
2.2
1.5
2.1
-L 2
2
2.5
2800
1500
2750
1.9
27001450
XMine
2650
1400
Inline
Figure 4-31: Gessoso horizon picked from Kirchhoff pre-stack depth migration.
174
2800
U260
m..
250
2750
140
1
240
230
270021
200
190
2650
1400
1450
Inline
%
1500
1180
1540
Figure 4-32: Number of traces at each CMP bin whose size is 50 x.50 m2
175
fI
2800
58
56
54
2750
152
50
48
2700
46
44
42
2650
0
1400
1450
Inline
40
1500
15 40
Figure 4-33: Average number of traces stacked at each azimuth at each CMP bin.
176
(a)
N
1200'
-
30
W-E
30Hz
N
70Hz
W
(d)
60 0
1
30
150
Inline 1480 Xine 2672
1200,'
Inline 1424 Xline 2774
600
150
(c)
(b)
Inline 1416 Xline 2712
30Hz
70Hz
E
Inline 145%Xline 2726
120'
600
0
150
W--
300
-A
30Hz
70Hz-E
1500,1
W-
300
30Hz
70Hz
E
Figure 4-34: FTF at four locations. Radial and angular coordinates are frequency and
azimuth, respectively. 'N', 'E' and 'W' indicate the directions of north, east and west,
respectively. Color bar is in linear scale.
177
0
<c -2
4-0
-12
0
10
20
30 40 50
Frequency (Hz)
60
70
Figure 4-35: Amplitude spectrum of the field data.
178
80
2800
E
S
0
2750
ell 4
ell 7dir
ell 8dir
WA
Xline
2700
2650
1400
1500 1540
1450140
Inline
Figure 4-36: Map of fracture orientations derived from the FTF. Dark and light
background colors indicate high and low confidences of the result, respectively. White
bars represent the fracture orientation. Two intersecting white bars indicate the
orientations of two potential crossing fracture sets.
179
Well derived
fracture
directions
(from Vetri et
al., 2003)
Well 4
Breakout NSHm
Well 7dir
Well 8dir
(
FTFderived
fracture
directions
Figure 4-37: Comparison of the fracture directions derived from well data (upper row)
and FTF (lower row) in three wells.
180
Chapter 5
Calculation of Stress-induced
Anisotropy around a Borehole
4
Formation elastic properties near a borehole may be altered from their original state
due to the stress concentration around the borehole. This could result in a biased
estimation of formation properties but could provide a means to estimate in situ stress
from sonic logging data. In order to properly account for the formation property
alteration, we propose an iterative numerical approach to calculate the stress-induced
anisotropy around a borehole by combining the rock physics model of Mavko et al.
(1995) and a finite-element method. We show the validity and accuracy of our
approach by comparing numerical results to laboratory measurements of the
stress-strain relation of a sample of Berea sandstone, which contains a borehole and is
subjected to a uniaxial stress loading. Our iterative approach converges very fast and
can be applied to calculate the spatially varying stiffness tensor of the formation
around a borehole for any given stress state.
4
(the bulk of this Chapter has been) published as: Fang, X.D., M. Fehler, Z.Y Zhu, T.R. Chen, S.
Brown, A. Cheng, and M.N. Toksoz, 2013. An approach for predicting stress-induced anisotropy
around a borehole: Geophysics, 78(3), D143-D150.
181
5.1
Introduction
Borehole logging data provide an important way to interpret the rock anisotropy and
estimate the in situ stress state (Mao, 1987; Sinha and Kostek, 1995). Typically, the
anisotropy in intact rocks includes intrinsic and stress-induced components (Jaeger et
al., 2007). Intrinsic anisotropy can be caused by bedding, microstructure, or aligned
fractures, while stress-induced anisotropy is caused by the opening or closing of the
compliant and crack-like parts of the pore space due to tectonic stresses. Most
unfractured reservoir rocks, such as sands, sandstones and carbonates, have very little
intrinsic anisotropy in an unstressed state (Wang, 2002). Drilling a borehole in a
formation significantly alters the local stress distribution. When the in situ stresses are
anisotropic, drilling causes the closure or opening of cracks in rocks around the
borehole and leads to an additional stress-induced anisotropy. In order to properly
include this additional stress-induced anisotropy during inversion for formation
properties and the in situ stress estimation from logging data, a thorough analysis
needs to consider the constitutive relation between the complex stress field applied
around a borehole and the stiffness tensor of a rock with micro-cracks embedded in
the matrix (Brown and Cheng, 2007).
Three theoretical approaches have been proposed to calculate the stress-related
anisotropy around a borehole. The first approach (Sinha and Kostek, 1996; Winkler et
al., 1998) used the acoustoelastic model, which gives a non-linear stress-strain
relationship (Johnson and Rasolofosaon, 1996), to calculate the stress-induced
182
azimuthal velocity changes around a borehole. The velocity variation with applied
stresses is accounted for through the use of the third order elastic constants, which are
obtained through compression experiments on rock samples. The second approach
(Tang and Cheng, 2004; Tang et al., 1999) used an empirical stress-velocity coupling
relation to estimate the variation of shear elastic constants (C 44 and Css) as a function
of stress. In this approach, the square of the shear wave velocities propagating along a
borehole with different polarizations are assumed to be linearly proportional to the
stresses applied normal to the borehole axis. This approach is used for studying shear
wave splitting in a borehole and only gives the values of shear elastic constants (i.e,
C44 and C55) instead of the full elastic stiffness tensor. However, these two approaches
have no rock physics basis as they ignore the constitutive relationship between an
anisotropic applied stress field and the stiffness tensor for a rock (Brown and Cheng,
2007). They thus give approximate solutions. Also, they are based on an assumption
of plane strain, which considers formation properties to be invariant along the
borehole axis and the applied stresses are normal to the borehole axis. Brown and
Cheng (2007) proposed the third approach to calculate stress-induced anisotropy
around a borehole embedded in an anisotropic medium. In their model, the
stress-dependent stiffness tensor of anisotropic rocks is calculated using a general
fabric tensor model (Oda, 1986; Oda et al., 1986). The intrinsic relation between
stress and stiffness of a rock is accounted for through the use of a rock physics model,
which is not included in the methods of Sinha and Kostek (1996) and Tang (1999).
The approach of Brown and Cheng (2007) reflects the physics of stress-induced
183
anisotropy and thus is more accurate. However, the general fabric tensor model
requires the prior knowledge of crack geometries (i.e, crack shapes and aspect ratio
spectra) and distributions, which may not always be available in the field applications.
In this paper, we replace the general fabric tensor model with the model of Mavko et
al. (1995), which assumes the crack orientation distribution in the rock matrix in an
unstressed state to be uniform and isotropic. Under this assumption, rocks are still
isotropic when subjected to a hydrostatic stress and become anisotropic under an
anisotropic stress loading. Anisotropy is induced through closing of the compliant part
of the pore space, which includes micro-cracks and grain boundaries (Sayers, 1999;
Sayers, 2002). The detailed information about pores/cracks is not required in Mavko's
model and the effect of pore/crack closure is implicitly determined by the relation
between elastic wave velocities of the rock and the applied hydrostatic pressure
obtained from laboratory data. Another major assumption of Mavko's model is that
the anisotropy induced by pore/crack opening is negligible.
5.2
Workflow for the numerical modeling
Mavko et al. (1995)
proposed a simple and practical method to estimate the
generalized pore space compliance of rocks using experimental data of rock velocity
versus hydrostatic pressure (see Appendix B). But their method is only applied to
calculate the stress-induced anisotropy in homogeneous intact rocks. When a borehole
is drilled in a rock subjected to an anisotropic stress loading, the local stress field
184
around the borehole is changed and causes rock anisotropy. Similar to the procedure
proposed by Brown and Cheng (2007),
in this paper, we investigate this
stress-induced anisotropy around a borehole at a given stress state by combining the
method of Mavko et al. (1995) and a numerical approach illustrated in Figure 5-1.
We first begin with a homogeneous isotropic intact rock model C"' , on which
Mavko's model is based. The intact rock refers to the rock before the borehole is
drilled. After experimentally obtaining the P and S-wave velocity data as a function of
hydrostatic pressure, we apply equation B.3 to calculate the anisotropic stiffness
tensor Cant of the intact rock under stress a, which can be anisotropic. Next, we drill
a borehole in the model and use the calculated Cjk
containing a borehole. The currentC
as the input in our initial model
does not include the effect from stress
change due to the borehole. We apply a finite-element method (FEM) to calculate the
spatially varying stress field within the model including the borehole for a given stress
loading a and the initial anisotropic CIIj.
From the output of FEM, we can obtain the
local stress tensor a(x) and then calculate a new stiffness tensor Cai (x) as a
function of space applying equation B.3. The new Cajk(x) becomes heterogeneous
and includes the effect of the borehole. We keep iterating the above steps by calling
FEM and applying equation B.3 until Cant (X) converges. We use the following as a
convergence criterion
2
N
Convergence(m) = -()C
S
E ijk
l (xn)--C l
>iijkl [Cijkf(
k
)
(5.1)
)]
where -m indicates the mth iteration, N is the total number of spatial sampling points of
the model,
Zijkl
means the summation over 21 independent elastic constants.
185
Convergence(m) indicates the percentage change of the model stiffness after the m
iteration comparing to the model at the (m-I)th iteration. We define Cijki to have
converged when Convergence(m) < 1%. Convergence means that
Cijl
and the stress
field are consistent and Hooke's law is satisfied, the model is under static equilibrium.
Finally, we can obtain the spatial distribution of the anisotropic elastic constants
Cijk (x) around a borehole for the given stress state as the output of our numerical
simulation. In our approach, density is assumed to be independent of the applied stress,
because the change of density caused by stress loading is negligible (Coyner, 1984).
In our approach, we assume that stress induced anisotropy is caused by the
closure of cracks/pores due to the applied compressive stress on their surfaces and the
effect of tensile stress is negligible. This assumption brings out two issues: (1) how
important is the tensile stress in the earth? (2) how do we deal with the tensile stress in
our calculation? We will discuss these below.
For a homogeneous isotropic elastic rock, the circumferential stress UO and the
radial stress ar around a circular borehole subjected to minimum and maximum
principal stresses (Sh and SH) are given by (for example, Tang and Cheng (2004))
O1 =+ (S+ S) (1'+
Ur = $(SH + Sh
1
-
(5.2)
(S,- S,) (1+ 3 )cos2
)+(SH - Sh) (1 - 4
+
3 )cos20
(5.3)
where R is borehole radius, r is the distance from the center of the borehole, 0 is
azimuth measured from the direction of SH.
The compressive stress ao+u, around the borehole provides an indication of how
velocity around the borehole is affected by stress concentration.
186
U9+qr
has maximum
and minimum values at the wellbore, and arO=
at r-R, so the stress field at the
wellbore is dominated by co, which has the maximum value ao=3SH-Sh at 0 =±900 and
the minimum value ao= 3 Sh-SH at 0=04 and 1800. In situ, both SH and Sh are present,
and SH 3Sh in most case (Brace and Kohlstedt, 1980; Zoback et al., 1985), thus the
minimum ao=3Sh-SH 0 is compressive. In this sense, there is no tensile stress around
the borehole.
However, the condition SH<3Sh may not be satisfied in laboratory experiments.
Uniaxial compression experiments (i.e. Sh=O), which would induce significant tensile
stress around the borehole, have been conducted for the study of stress induced
velocity change around a borehole by many researchers (Tang and Cheng, 2004;
Winkler, 1996; Winkler et al., 1998). The change of rock elastic properties caused by
tensile stress is usually unknown. Traditional methods (Sinha and Kostek, 1996; Tang
et al., 1999) for calculating the stress dependent velocity around a borehole use the
data measured from compression experiments to estimate either the third order elastic
constants or empirical coefficients, which relate the rock velocity change to the
applied stresses. For the case of uniaxial stress, they based their equations on
compression experiment data to predict the velocity in the tensile stress regions. This
kind of extrapolation has no physical basis and could result in underestimation of the
velocity in the regions around 0=00 and 1800. A schematic explanation is shown in
Figure 5-2a. The solid curve represents the data measured in a compression
experiment, and the dashed curve indicates the extrapolation of the data to the tensile
stress (stress < 0) region, which may incorrectly predict low velocity in this region.
187
Different kinds of rock would respond to tensile stress differently due to varying
micro-crack structure and rock strength. For Berea sandstone, which is used in our
experiments, tensile stresses are relatively less efficient in opening micro-cracks
(Winkler, 1996). We assume the rock elastic constants under tensile stress remain the
same as in a zero stress state in our calculation, shown as the dashed line in Figure
5-2b, in other words, crack opening is neglected. Our results will show that good
results can be obtained with this assumption on Berea sandstone.
5.3
Laboratory experiment
In this section, we present results from static strain measurement on a Berea sandstone
under uniaxial loading to verify the validity and reliability of our numerical approach.
The dimensions of the Berea sandstone sample used in this experiment are
11.4x100.6x102.3 mm3 . P- and S-wave velocities of the unstressed rock sample
were measured in three directions. Figure 5-3 shows the measured P- and S-waves in
three orthogonal directions. We pick the first breaks from the seismograms and
calculate the P- and S-wave velocities and find that P- and S-wave anisotropy are only
0.7% and 1.8%, respectively. The measured parameters of the rock are summarized in
Table 5-1.
First, we measure P- and S-wave velocities under varying hydrostatic stress.
These data are used to estimate the normal and tangential crack compliances
(equations B. 1 and B.2) as functions of hydrostatic pressure, which are required by
188
the method of Mavko et al. (1995). Then, we perform strain-stress measurements of
the intact rock under uniaxial loading. This step is to benchmark our measurement
setup with the method of Mavko et al. (1995). Finally, we measure the strain-stress
behavior of the rock containing a borehole subjected to a gradually increasing uniaxial
stress and compare it with our numerical calculations.
5.3.1 Measurement of P- and S-wave velocities under hydrostatic
compression
In order to measure P- and S-wave velocities versus hydrostatic pressure, we cut a 2
inch long and 1 inch diameter cylindrical core from our rock sample that will also be
used for the subsequent experiments. We measured P- and S-wave velocities parallel
to the core axis. The S-wave velocity measurements were made using two orthogonal
polarization directions, as shown in Figure 5-4. The velocity and hydrostatic pressure
have the following empirical relation (Birch, 1961)
(5.4)
V = alog(p) + b
where V represents both compressional and shear velocities and p is hydrostatic
pressure, a and b are coefficients related to the porosity and mineralogy of the rock.
We find that equation 5.4 cannot fit the hydrostatic data very well for pressure < 1
MPa, so we modify equation 5.4 to equation 5.5 as
V =
a p + bl,
a 2110g(p) + b2 ,
p ; 1 MPa
p > 1 MPa
(5.5)
where al, bl, a2 and b2 are constants to be determined through least-squares method by
adding the constraint that the two fitting functions are equal at p=I MPa. The fits to
the P- and S-wave velocities (average of S and S2) are shown as the blue and red
189
curves, respectively, in Figure 5-4. Given equation 5.5, we can now analytically
calculate the P-and S-velocities at any given hydrostatic pressure.
5.3.2 Strain measurement of intact rock under uniaxial stress loading
In the intact rock experiment, four two-component strain gages were mounted at
different positions on the rock sample for measuring the normal strains in the
directions parallel and normal to the direction of loading stress, as shown in Figure
5-5 (a photo of our experiment setup). We use standard amplified Wheatstone bridge
circuits with an analog-to-digital converter to collect signals from all strain gages
simultaneously. Before performing the experiment, the rock was stress-cycled several
times in order to minimize hysteresis. During the experiment, the uniaxial loading
stress was gradually raised from 0 to 10.56 MPa in steps of 0.96 MPa. We limited the
maximum loading stress to 10.56 MPa to prevent permanent deformation in the rock.
The strains measured under uniaxial loading by the four strain gages were almost the
same. This suggests that the loading stress was evenly distributed on the rock surface.
Figure 5-6 shows the comparison between the strains (black solid curves)
calculated using the method of Mavko et al. (1995) and the measured strains (solid
and empty squares). In the direction parallel to the loading axis, the black solid curve
(calculated values) matches the solid squares (measured data) very well. In the
direction normal to the loading axis, the measured data (empty squares) seem larger
than the calculated values, especially at higher loading stress ranges. This could be
caused by neglecting the effects of the opening of new cracks aligned parallel with the
190
loading axis (Mavko et al., 1995; Sayers et al., 1990). The dashed curves, which are
shown for comparison, are the strain values calculated under the assumption that rock
properties remain isotropic during the experiment but Vp and Vs are given by equation
5.5 in different stress state, which is corresponding to hydrostatic compression and
will be referred to as the isotropic model hereafter. The absolute values of these
dashed curves are always smaller than those of the solid curves. For the isotropic
model, the normal stress causes the closure of all cracks independent of orientation,
while the anisotropic model assumes smaller closure of cracks oriented in directions
not perpendicular to the loading direction. Therefore, hydrostatic compression leads to
a stiffer rock compared to the rock under uniaxial stress compression.
5.3.3 Strain measurement of the rock with a borehole under uniaxial
loading
A borehole with 14.2 mm radius was drilled through the rock along the X-axis at the
center of the Y-Z plane, as shown in Figure 5-7. Uniaxial stress, which is applied
along the Z-axis, is perpendicular to the borehole axis. The stress is also raised in
steps of 0.96 MPa up to 10.56 MPa. Strain measurements were made in two
orthogonal directions at four locations represented by A, B, C and D, as shown in
Figure 5-7. We applied our work flow illustrated in Figure 5-1 and used a FEM
software to numerically calculate the stress-induced anisotropy around the borehole
subjected to a uniaxial stress.
191
Figure 5-8 shows the convergence (equation 5.1) of the iterations at eleven
loading stresses. We found that the convergence is very fast and the change of model
stiffness is less than 1% after the first two iterations. We will show the results
obtained after the fifth iteration. Figure 5-9 shows the simulated principal normal
stresses ay and ozz on the borehole model surface under 10.56 MPa stress loading in
the Z direction. Let 0-00 define the direction of the applied stress. As seen in Figure
5-9c, the circumferential stress is highly compressive at 0-900 while it is tensile at
0-00 and 1800. The stress around the borehole now is strongly spatially dependent. As
a result, the initially elastic isotropic rock in the unstressed state becomes anisotropic
at each point in space due to the varying local stress field.
In Figure 5-10, the strains measured at four different positions A, B, C and D are
compared to the numerical simulations, similar to Figure 5-6. We find a good match
between the measurements and numerical simulations. Strains measured at B and C,
roughly 6 mm away from the borehole edge, are strongly affected by the stress
alteration around the borehole. The strain ell at B in absolute value is much larger than
those at A, C and D, and it reaches a minimum value at C. This is because stress is
highly concentrated at B and released at C, as shown in Figure 5-9b. The strain elI at D
is smaller than that at A. This is again due to the alteration of stress concentration
around the borehole. The strain eL always seems to be underestimated in the
numerical calculations, perhaps due to the neglect of crack opening, similar to Figure
5-6. Our numerical results, however, are a very reasonable match with the
192
measurements. This suggests that the neglect of crack opening has a minor effect in
our approach.
5.4
Summary
An isotropic rock becomes anisotropic when subjected to an anisotropic applied stress,
which causes closure or opening of pores and cracks and induces elastic anisotropy.
The presence of a borehole alters the local stress field and leads to inhomogeneous
anisotropy distribution around it. In this chapter, we presented a numerical approach
to predict the stress-induced anisotropy around a borehole given a stress state by
applying the method of Mavko et al. (1995). Our method uses hydrostatic data (i.e. Vp
and Vs), which are easy to obtain, to calculate the distribution of this stress-induced
anisotropy around a borehole. The accuracy of our method is validated through
laboratory experiments on a Berea sandstone sample. Our approach can predict the
stress-strain relation around a borehole in Berea sandstone under uniaxial stress
reasonably well. Our method can be applied to calculate the spatially varying
anisotropic elastic constants which are required for the forward modeling of wave
propagation in a borehole under a given stress state. Also, our results can contribute
towards the development of a physical basis for using acoustic cross-dipole logging to
estimate the in situ stress state.
193
Table 5-1: Summary of parameters of the Berea sandstone
sample in an unstressed state.
Dimensions (mm)
VP
Vs
Density
Poission's ratio
Porosity
Permeability
101.4x100.6x102.3
2.83km/s
1.75km/s
2198kg/m3
0.19
17.7%
284 mD
194
~-
~- ---
~------~
- - - ---
~-----
-
Generate initial model for the iteration
I-M
FEM
FEM
me
iterate
Canl(p4
Outputr
het
eplace Ck
Figure 5-1: Workflow for computation of stress-induced anisotropy around a borehole.
TEM' and 'M' represent finite-element method and the method of Mavko et al. (1995),
respectively. See text for explanation.
195
(a)
(b)
A(a)
(b)
/0100
~1
I
I
Il
-i
U
Stress
U
Stress
Figure 5-2: Schematic showing two ways to predict the velocity under tensile stress.
Solid curves represent the data measured in a compression (stress>O) experiment,
dashed lines indicate the extrapolation of data to tensile stress (stress<O) region. (a)
velocity decreases with the decreasing of stress and (b) velocity is constant when
stress<O.
196
Y
Y
P
x
S
S
S
Y
YX
xZ
S
XY
0
0.02
0.04
0.06
0.08
Time (ms)
Figure 5-3: Seismograms recorded for measuring the compressional and shear
velocities of the Berea sandstone sample. Acoustic measurements were conducted in
three orthogonal directions. Pi (i=X,Y,Z) indicates the measurement of P-wave along i
direction, and SU (i-j=X,Y,Z) indicates the measurement of S-wave along i direction
with polarization inj direction.
197
4000 -.-.-.-.-.-.---
3600
AS1
3200
S2
0
0
10
20
30
40
50
Stress (MPa)
Figure 5-4: Measurements of P-wave (squares) and S-wave (triangles and circles)
velocities of the Berea sandstone core sample under hydrostatic compression. All
measurements were conducted along the core axis direction. Shear wave velocities Si
and S2 were measured along the same propagation direction but with orthogonal
polarization directions. Blue and red curves are the fitting curves (equation 5.5) to the
P- and S-wave velocities (average of S, and S 2), respectively. The root-mean-square
misfits are, respectively, 38 m/s and 18 m/s for the fits to P- and S-wave velocities.
198
Press
Berea sandstone
Strain gage
Aluminum foil
z
X
Figure 5-5: Photo of experiment setup. The Berea sandstone sample has dimensions
101.4(X)x 100.6(Y)x 102.3(Z) mm 3. The size of the strain gage is about 2 mm. The
aluminum foil between the press and the rock is used to make the loading pressure
distribution more uniform on the rock surface.
199
0.1
-0.1
Or
-0.2
-----
--
.. . . . . . . . .
-
-0.3
-0.4
-0.5 (
...
* e
-0-
2
--............
...... .....
6
4
Stress (MPa)
8
10
Figure 5-6: Average normal strains of the intact rock sample under uniaxial loading.
Solid and open squares are the measured strain in the directions parallel and normal to
the loading stress respectively. Error bars represent estimates of errors from
uncertainty in the measurement of loading stress (-5%) and the error of the gage
factor (-1%). Solid curves and dashed curves are the predicted values obtained from
the anisotropic model and isotropic model, respectively.
200
100.4 mm
+-
Uniaxial stress
101.5 mm
3
30mm
z
Boreo[o
A
Y
B
Ir3
Ox -
A -D---- -----
Figure 5-7: Schematic showing uniaxial stress loading on a rock sample with a
borehole. The borehole axis, which is along the X-axis, is normal to the loading stress
direction. Strain measurements are conducted at locations A, B, C and D. B and C are
20 mm away from the borehole center, A and D are 30 mm away from the borehole
center. Borehole radius is 14.2 mm.
201
0.96 MPa
_10
1
_2
10
-
....- -..-..-....
.......
...-
10
--
~3.85
1. -----.- ----
-
---. ---.....
...
2 .88 M Pa .
MPa
4.81 MPa
5 .77 M Pa
6.73 MPa
7 .69 M Pa -
----- -- --- ..
..--
..... ........
..
10
10
MPa
-1.92
---------- ----...---. ----.-.----.----.--- - . -
8.65 MPa
-
2M
10
1
2
3
4
5
6
Number of iteration
7
8
9.61 MPa
10.6 MPa
9
10
Figure 5-8: Convergence of the iteration scheme under different loading stress
strength. Convergence, which is defined by equation 5.1, describes the percentage
change of the model stiffness after each iteration.
202
(a)
(b)
zz
yy
10
10
C. 50
N
N
a +a
(C)
yy
zz
Stress (10MPa)
10
2
N
0
5
Y (cm)
10
Figure 5-9: (a) and (b) show the distribution of qy and a,,, which are the normal
stresses in Y and Z directions, under 10.56 MPa uniaxial stress loading in the Z
direction (see Figure 5-7). (c) is the sum of ay, and a,,.
203
(A)
6,
C
0.2
(B)
0.2
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
II
--- .....
- -- - ---- --..
..
-1
0
e
2
4
6
8
10
(IC)
2
0
0
-r
6
4
8
10
(D)
0.2
0
-0.8
-U-*
0.2
-0.2
---
--.-.-
-0.4
----
- -.......
-..
.
-0.6........
me I
-0.8
0
2
4
6
8
Stress (MPa)
.
. ... ...........
-02 . .. .
-0.2
- . .............. .......
- 0 .4
-0.6 -
. .-..
.
--..
-...-..
-...
.
- --
.e
.. -.. --
-nnA
10
-~0
2
4
6
8
Stress (MPa)
10
Figure 5-10: Comparison of laboratory measured strains and numerical results at
locations A, B, C and D, which are shown in Figure 5-7. Solid and open squares are
the measured strain in the directions parallel and normal to the loading stress
respectively. Error bars represent estimates of errors from uncertainty in the
measurement of loading stress (-5%) and the error of the gage factor (-1%). Solid
curves and dashed curves are the predicted values obtained from the anisotropic
model and isotropic model, respectively.
204
Chapter 6
Effect of Borehole Stress
Concentration on Sonic Logging
Measurements 5
In Chapter 5, we developed an iterative approach, which combines a rock physics model
and a finite-element method, to calculate the stress-dependent elastic properties of the
rock around a borehole when it is subjected to an anisotropic stress loading. To further
study the effect of stress concentration around a borehole on sonic logging, we use the
anisotropic elastic model obtained from the method described in Chapter 5 and a finitedifference method to simulate the acoustic response of the borehole. We first compare
our numerical results with published laboratory acoustic wave measurements of the
azimuthal velocity variations along a borehole under uniaxial loading and find very good
5
(the bulk of this Chapter has been) submitted as: Fang, X.D., M. Fehler, and A. Cheng, 2013. Effect of
borehole stress concentration on sonic logging measurements: Geophysics.
205
agreement. Both numerical and experimental results show that the variation of P-wave
velocity versus azimuth has broad maxima and cusped minima, which is different from
the presumed cosine behavior. This is caused by the preference of the wavefield to
propagate through a higher velocity region at high frequency. We then study the borehole
azimuthal acoustic responses for monopole and dipole sources. Our results indicate that
the velocities of refracted P-wave and Stoneley waves show obvious azimuthal variation
when frequency is above about 15 kHz, and the dipole dispersion crossover of flexural
waves appears at about 4.5 kHz, which is not affected by the strength of the loading stress.
6.1
Introduction
Borehole acoustic logging data provide an important way to interpret formation elasticity
(Mao, 1987; Sinha and Kostek, 1995). Monopole and cross-dipole measurements are
widely used for determining the formation P-wave velocity and S-wave anisotropy (Sinha
and Kostek, 1995; Sinha and Kostek, 1996; Tang et al., 1999; Tang et al., 2002; Winkler
et al., 1998). Most conventional unfractured reservoir rocks, such as sands, sandstones
and carbonates, show very little intrinsic anisotropy in an unstressed state (Wang, 2002).
However, the stress-induced anisotropy, which is caused by the opening or closing of the
compliant and crack-like parts of the pore space due to tectonic stresses, significantly
affects the elasticity of rocks. Drilling a borehole in a formation strongly alters the local
stress distribution. When the in situ stresses are anisotropic, drilling causes the closure or
opening of cracks in the formation around a borehole and leads to an additional stressinduced anisotropy. Winkler (1996) experimentally measured the azimuthal variation of
206
the P-wave velocity in a direction parallel to a borehole that was subjected to a uniaxial
stress loading and showed that the borehole stress concentration has a strong impact on
the velocity measurements.
To fully understand the effect of borehole stress concentration on borehole sonic
logging, a thorough analysis of the propagation of waves in a three dimensional borehole
embedded in a medium with stress-dependent elastic properties needs to be conducted.
The elasticity of the formation around a borehole is described by the stiffness tensor that
is governed by the constitutive relation between the stress field applied around the
borehole and the elasticity of the rock with micro-cracks embedded in the matrix. Several
approaches (Sinha and Kostek, 1996; Winkler et al., 1998; Tang et al., 1999; Brown and
Cheng, 2007; Fang et al., 2013) have been proposed to describe the stress-dependent
response of the elastic properties of the rock around a borehole when it is subjected to
anisotropic stress loading. In this chapter, we use the method developed in Chapter 5 to
calculate the stiffness tensor of the formation around a borehole. Then we use a finitedifference method to simulate the wave propagation in the borehole. We analyze the
results by first comparing with the laboratory measurements of Winkler (1996) and then
studying the effect of stress on mode velocity and amplitude.
6.2
Model building
We will first compare our numerical simulations with the laboratory experiments of
Winkler (1996), in which the P-wave velocity versus azimuth around a borehole in a
207
Berea sandstone sample with and without applied uniaxial stress was measured. In his
experiment, a block of Berea sandstone having dimensions of 1 5x 1 5x 13 cm and with a
2.86 cm (1.125 inch) diameter borehole parallel to the short dimension was placed in a
water tank for conducting acoustic measurements. The P-wave velocity at each azimuth
was measured along the borehole axis by using a directional transducer and two receivers.
P-wave velocity is calculated from the travel time delay of the refracted P-waves
recorded at the two receivers, which are 7 and 10 cm, respectively, away from the
transducer. The porosity of his Berea sandstone sample is 22%. The P-wave velocity
variation with azimuth is very small before applying the stress, and its average value is
about 2.54 km/s when no external stress was applied. When his model is scaled to a 20
cm (8 inch) borehole, the corresponding frequency of the received acoustic signals is 30
kHz.
In the approach described in Chapter 5, the stress-induced anisotropy around a
borehole is obtained through an iterative process that combines the method of Mavko et
al. (1995), which is used to calculate the crack compliance from the P- and S-wave
velocities versus hydrostatic pressure data, and a finite element method. P- and S-wave
velocities of the rock sample versus hydrostatic pressure, which are the necessary input
for our iterative approach, were not measured by Winkler (1996). We use the data
measured from a Berea sandstone sample, which has similar properties to the sample
used in the experiment of Winkler (1996), to construct a model for our wave propagation
simulation. Table 5-1 lists the properties of our rock sample. We build a borehole model
with the exact geometry of the experiment configuration of Winkler (1996), so that the
numerical results are comparable to the laboratory measurements. There are two steps to
208
calculate the elastic properties of the rock around a borehole. First, we take a core sample
from our Berea sandstone sample and measure the P- and S-wave velocities at different
hydrostatic pressures. These data are shown in Figure 5-4. Second, we use the data
measured in the first step and the iterative approach to calculate the stiffness tensor of the
rock around a borehole when it is subjected to a uniaxial stress applied normal to the
borehole axis as in the experiment of Winkler (1996). The output stiffness tensor is
anisotropic and inhomogeneous due to the spatially varying stress field around the
borehole.
Figure 6-1 shows the geometry of our borehole model. The formation is Berea
sandstone and the borehole is water saturated. A 2.86 cm (1.125 inch) borehole is at the
center of the model along the Z direction. A uniaxial stress is applied normal to the
borehole in the X direction. The direction of applied uniaxial stress is defined as 00. A
0.64 cm (1/4 inch) diameter piston source, which mimics the 1/4 inch diameter
directional transducer in the experiment of Winkler (1996), is used in the simulation. A
schematic of the piston source is shown in Figure 6-2. The source amplitude is tapered
from the center to the edges by using a Hanning window, which is shown as the dashed
curve in Figure 6-2. Source time function is a Ricker wavelet with a 213 kHz center
frequency, the corresponding frequency is 30 kHz in a 20 cm (8 inch) borehole. Figure 63 shows a snapshot of the pressure field in the borehole excited by a piston source
pointing at 300. We can see that the wave field excited by the piston source has good
directionality and is anti-symmetric with respect to the source plane. Receivers, which are
shown as the blue circles in Figure 6-1, are 0.7 cm away from the borehole axis along the
209
source direction. Perfectly match layer (PML) is used at all model boundaries to avoid
boundary reflection.
The elastic model obtained from our iterative approach contains 21 independent elastic
constants, which are functions of the applied stress and position. For 10 MPa stress
loading, we calculate the average value of the stiffness tensor of the formation around the
borehole and plot it in Figure 6-4. Color intensity of each box in Figure 6-4 represents the
average value of the corresponding component in the stiffness tensor (6
x
6 matrix
notation). The nine components inside the dashed green lines of Figure 6-4 are about two
orders of magnitude larger than the others. This indicates that only 9 elastic constants (i.e.
CII, C 12 , C 13 , C 22 , C 23 , C 33 , C 44 , C 55 and C 66 ) in the stiffness tensor are important. We call
these 9 elastic constants the dominant components. In the next section, we will show that
the 9 dominant components determine the characteristics of wave propagation in a
borehole and the remaining 12 components in the stiffness tensor have negligible effect.
Figure 6-5 shows the variations of the 9 dominant elastic constants around the borehole
on the X-Y plane for a 10 MPa uniaxial stress applied along the X direction. Properties of
the model are invariant in the Z direction because of model symmetry. As seen in Figure
6-5, the rock around the borehole becomes inhomogeneous and anisotropic under a stress
loading. The diagonal components in the stiffness tensor (i.e. Cii, i=1,2,...,6) are
significantly different from each other. Due to the stress concentration at ±900, the
stiffness of the rock increases from the stress loading direction (00) to the direction
normal to the loading stress (±900).
210
6.3
Comparison with laboratory measurements
In our simulation, we use a staggered grid finite-difference method with fourth-order
accuracy in space and second-order accuracy in time (Cheng et al., 1995). The grid
spacing is 0.0176 cm (1/162 borehole diameter) and time sampling is 0.0176 ps.
Numerical velocity error (Moczo et al., 2000) for P- and S-waves on the X-Y plane at
zero stress state is illustrated by the solid and dashed curves, respectively, in Figure 6-6.
Numerical error in the Z direction is of the same order of magnitude of that on the X-Y
plane, because grid spacing is uniform in all three directions. From Figure 6-6, we can
see that the difference between the true velocity (i.e. formation velocity) and the actual
velocity (i.e. grid velocity) of wave propagation in the simulation is less than 0.01% for
both P- and S-waves in all directions within the source frequency range. The azimuthal
variation of P-wave velocity in the borehole caused by 10 MPa uniaxial stress is about 10%
(Winkler, 1996), which is several orders of magnitude larger than that caused by
numerical error. Thus the numerical error has negligible effect on the results. Figure 6-7
shows a comparison of the seismograms simulated from two models respectively
containing 21 (dashed red) and 9 dominant (solid black) elastic constants for a piston
source at 300 and 10 MPa uniaxial stress loading. As shown in Figure 6-7b, the difference
(multiplied by 105) between the wavefields recorded in these two models is negligible.
This indicates that the wave propagation in the borehole is reliably determined using only
the 9 dominant elastic constants and the rest of the stiffness tensor can be neglected.
Therefore, we only use the 9 dominant components and assume the other components in
the stiffness tensor are equal to zero in the simulations below.
211
Figure 6-8 shows the data recorded in the borehole for sources at ten different
orientations when the model is subjected to 10 MPa uniaxial stress. 00 and 900 are along
the X and Y directions, respectively. The refracted P waves, which have nearly linear
moveouts, can be identified clearly in Figures 6-8a - 6-8e, but become very weak when
the source angle is larger than 400. The amplitude of the refracted P wave decreases with
the increase of source angle due to the increase of rock stiffness. Depending on the
distance of a receiver from the source, the recorded first arrival can be either the refracted
P wave propagating along the wellbore or the direct P-wave that travels from the source
to the receiver through the borehole fluid. The wave paths for direct and refracted Pwaves are schematically shown as the dashed and solid lines in Figure 6-9. From Snell's
law, we can obtain the direct P-wave travel time Td and the refracted P-wave travel time
Tr as
(6.1)
Td =Z-+(rl2)2
Z
Tr =F+
3
r
1
-
1
(6.2)
where V, and Vp are the P-wave velocity of the borehole fluid and the formation,
respectively, Z is the vertical distance between source and receiver and r is borehole
radius.
By making Td= Tr and then solving for Z, we have
z=
r(2vVp+3V,)
2 vj -VI
212
(6.3)
Equation 6.3 gives the receiver position where the direct and refracted P-waves
arrive at the same time. We can obtain Z=3.7 cm from equation 6.3 if we take the P-wave
velocity of the rock at zero stress state as Vp. This gives us an estimation of the beginning
of the refracted P-wave in the synthetic data. As shown in Figure 6-8, we can see that the
refracted P-waves start to appear at about Z=3-4 cm.
Figure 6-10 shows the simulated data at two receivers located at Z=7 and 10 cm
(source at Z=0 cm), which are the positions of the near and far receivers in the
experiment of Winkler (1996), for sources at ten different orientations. At each source
orientation, we divide the distance between the two receivers by the delay of the refracted
P-wave arrival times, which are indicated by the red circles in Figure 6-10, to get the Pwave velocity. Figure 6-11 shows the azimuthal variation of the normalized P-wave
velocity obtained from our numerical simulations together with the data measured by
Winkler (1996) for 10 MPa uniaxial stress. The numerical data (squares) and the
measured data (circles) are normalized separately by the corresponding P-wave velocity
of the rock sample at zero stress state. We simulate ten sources with orientation varying
from 00 to 900 in steps of 100. Using symmetry, we replicate the data from 00 to 900 to the
other three quadrants for plotting. As shown in Figure 6-11, our numerical results agree
well with the laboratory measurements of Winkler (1996). This indicates that the
constitutive relation between the formation stiffness around a borehole and the applied
stress field is accounted for correctly in the method described in Chapter 5. Figures 6-12a
and 6-12b show our numerical results and the laboratory measurements of Winkler
(1996), respectively, for 5, 10 and 15 MPa uniaxial stresses. For 5 and 15 MPa uniaxial
stresses, Winkler (1996) does not show the original measured data but only the best fits.
213
As shown in Figure 6-12b, Winkler (1996) finds that the P-wave measured velocities
have broad maxima and cusped minima and can be better fit by using an exponential
function instead of a cosine function, which is deduced from the cosine dependence of
stress near a borehole (Jaeger et al., 2007). Our numerical results shown in Figure 6-12a
have very similar azimuthal variation as the measured data shown in Figure 6-12b. The
overall variation of our numerical results with azimuth is a little bit smaller than that of
the measured data because the rock sample used in the experiment of Winkler (1996) is
more compliant than our rock sample, as the porosity of our sample is lower and the
velocity before stress applied is higher. Another difference between the numerical results
and the measured data occurs at 00 and 1800, where the measured velocities for 10 and 15
MPa uniaxial stresses are smaller than that for 5 MPa uniaxial stress. This may be caused
by the opening of micro cracks induced by tensile stresses, whose effect becomes
significant at large loading stress but is negligible at small loading stress. However, the
effect of tension on rock velocity is still not clear and needs to be further investigated.
Crack opening caused by tensile stress is neglected in our iterative approach, so the
normalized velocities in the numerical results increase with the increase of loading stress
at 00 and 1800.
If the propagation of the refracted P-wave follows a straight wave path along the
wellbore at the source excitation direction, then the P-wave velocity versus source
direction should show a cosine function variation, which has been predicted by the
theoretical calculations of Sinha and Kostek (1996) and Fang et al. (2013). The broad
maxima and cusped minima shown in both the numerical and measured data in Figure 612 suggest that the propagation of the refracted P-wave does not follow a straight wave
214
path along the wellbore. A wave tends to propagate through a higher velocity zone and
finds the fastest path to reach a receiver. Figure 6-13 shows the advance of the first break
arrival time at each azimuth compared to that at 00 for 5, 10 and 15 MPa uniaxial stresses.
At each azimuth, the arrival time advance at a given receiver depth is calculated by
subtracting the travel time of the first arriving P-wave at that azimuth from that at 00,
where the P-wave arrival time has maximum value as the velocity is minimum. The time
advance of the first arriving P-wave is zero when Z is small (<3.7 cm), because the first
recoded P-waves at near receivers are the direct P-wave, which propagates at water
velocity. As shown in Figure 6-13, the arrival time advance increases with Z until it
reaches a maximum, which appears at about Z=20, 14 and 11 cm for 5, 10 and 15 MPa
stress loading, respectively. At far receivers near Z=25 cm, the refracted P-waves for
sources at all azimuths arrive at almost the same time, because the refracted P-wave
propagating through the highest velocity zone at ±900 is faster than that traveling along
the wellbore following a straight wave path not at 1900. The maximum of the arrival time
advance is associated with the strength of the applied stress which determines the
magnitude of the velocity variation around the borehole. This indicates that the P-wave
velocity calculated by using the time delay between two receivers depends on the
selected positions of the two receivers.
215
6.4 Numerical results for a monopole source
In the previous section, we have shown that the numerical simulation results can match
the laboratory measurements of Winkler (1996) very well. This demonstrates the
applicability of the approach described in Chapter 5 for estimating the azimuthal
variation of anisotropic elastic properties around a borehole. However, the source
frequency (i.e. 30 kHz for a 20 cm borehole) used in the above simulations is much
higher than the conventional logging frequency used in the field. In this section, we
conduct the simulation by up scaling the borehole model constructed previously to a 20
cm borehole and replacing the piston source with a monopole source, which is commonly
used in borehole sonic logging (Tang and Cheng, 2004). The center frequency of the
monopole source used in wireline sonic logging is usually about 10 kHz. In order to
investigate the acoustic response in a borehole at a higher frequency, we will simulate
monopole sources of 10, 15 and 20 kHz center frequencies. A 20 kHz source can be
achieved in a logging-while-drilling (LWD) tool. The source wavelet is a Ricker wavelet.
In the following simulations, we have receivers with offsets from
1 m to 5 m positioned
at every 100 from 00 to 900 relative to the applied stress direction (00). In order to collect
dense enough data and avoid spatial aliasing in the following analysis, we choose 0.51
cm (0.2 inch) receiver spacing, which is much smaller than that of a real logging tool. All
receivers are 5 cm away from the borehole center. We will treat receivers at each azimuth
as an individual array and process the data separately for each azimuth.
Figure 6-14 shows the seismograms recorded at five different offsets and at ten
different azimuths for 10 kHz monopole source when the model is subjected to 10 MPa
loading stress. The refracted P-waves, which are marked by the two dashed red lines in
216
each panel, are dispersive and extend over a longer time window as source-to-receiver
distance increases. At a given source-to-receiver distance, the difference of the P-wave
arrival times at different azimuths is too small to be directly measured from the trace data
in Figure 6-14. We apply time semblance analysis (Tang and Cheng, 2004) to estimate
the velocities of different wave modes. Figure 6-15 shows the time semblances of the
data recorded at 00, 300, 600 and 900 for 10 MPa loading stress. At each azimuth, we only
use the data recorded from Z=2 m to Z=4 m to calculate the semblance. We can see that
the refracted P- and S-waves propagate at the slowness (inverse of velocity) of about 300
us/m and 500 us/m, respectively. The Stoneley wave slowness has small variation and is
close to the water slowness (667 us/m). The semblances at these four azimuths do not
show significant difference. Figures 6-16 and 6-17 respectively show the semblances of
the refracted P- and S-waves for monopole sources with different center frequencies
when the model is subjected to 10 MPa loading stress. The source center frequency is
shown at the bottom right corner of each panel. In Figure 6-16, the red high amplitude
regions, which indicate the refracted P-wave, do not show obvious azimuthal variation
for 10 and 15 kHz monopole sources. For the 20 kHz monopole source, the onsets of the
high amplitude regions at about 0.75 ms give similar slowness values at the four azimuths,
while the P-wave slowness picked from the high amplitude tails after 1 ms change from
about 335 us/m at 00 to about 325 us/m at 900, which is about 3% difference. In Figure 617, semblances of the refracted S-wave do not show clear azimuthal variation for all three
source frequencies, because they are influenced by the pseudo-Rayleigh waves, which
arrive at a time following the S-waves and have stronger amplitude. However, the Swave semblances show two arrivals with different slowness for the 15 and 20 kHz
217
sources. This may indicate the splitting of S-waves. For 5 and 15 MPa loading stresses,
the semblances, which are not shown, of both P- and S-waves are similar to these at 10
MPa loading stress except that the overall slowness of the high amplitude region has a
small shift.
The most prominent signals in the synthetic data are the Stoneley waves, whose
amplitudes are much larger than those of the refracted P- and S-waves. Figure 6-18
shows the dispersion of the Stoneley waves at ten azimuths for three different loading
stresses. We only show the Stoneley dispersion above 10 kHz, because they do not show
any azimuthal variation below that frequency. The overall slowness of the Stoneley wave
decreases with increasing loading stress. For all three loading stresses, we can see that the
Stoneley wave slowness increases from 900 to 00 when the frequency is above 12 kHz.
The percentage change of the Stoneley wave slowness from 900 to 00 at 20 kHz increases
from about 4% for 5 MPa loading stress to about 5% for 15 MPa loading stress.
From the time semblance and dispersion analysis, we can see that both refracted
P-wave and Stoneley wave slowness shows measurable azimuthal variations when the
frequency reaches 20 kHz. This indicates that a high frequency (e.g. 20 kHz) source is
necessary to measure the azimuthal variation of formation velocity around a borehole.
Besides of the dependence of wave velocity on azimuth, from Figures 6-10 and 614, we also notice that the amplitude of P-wave shows significant azimuthal variation.
Figures 6-19, 6-20 and 6-21 show the variations of the refracted P-wave amplitude versus
azimuth for monopole sources of 10, 15 and 20 kHz center frequencies, respectively, at
four different source-to-receiver distances. The value of amplitude used in plotting is the
218
normalized mean absolute value of the amplitude of the refracted P-wave in the selected
time window, which is shown as the dashed red window in Figure 6-14. We only record
the data from 00 to 900 in the simulations.
In order to compare with the velocity
measurements in Figure 6-12, we replicate the data from 00 to 900 to the other three
quadrants in plotting based on the model symmetry. In Figures 6-19, 6-20 and 6-21, black,
red and blue curves show the variations of the P-wave amplitude for 5, 10 and 15 MPa
loading stresses, respectively. For each source frequency, all amplitudes are normalized
by the same value. In Figures 6-19, 6-20 and 6-21, we can see that the P-wave amplitude
decreases with increasing source-to-receiver distance and increases with increasing
loading stress. The P-wave amplitude shows maxima at 00 and 1800 while has minima at
900 and 2700, which is contrary to the P-wave velocity variation. For 10 and 15 kHz
monopole sources, the amplitude versus azimuth variations shown in Figures 6-19 and 620 are similar to a cosine function. However, the amplitude shows broad minima and
cusped maxima for the 20 kHz source, as shown in Figure 6-21. This is similar to the Pwave velocity variation shown in Figure 6-12 except that the roles of maxima and
minima reverse. Compared to the azimuthal variation of velocity, the amplitude is much
more sensitive to the change of formation properties around the borehole. It varies by
several times from the loading stress direction to the direction normal to the loading
stress even for the 10 and 15 kHz monopole sources, whose time semblances show very
little azimuthal sensitivity. This suggests that the P-wave amplitude can be used to study
the in situ stress state around a borehole.
219
6.5
Numerical results for a dipole source
We use the same model as in the monopole study to simulate cross-dipole logging, in
which dipole sources oriented in two orthogonal directions are excited separately. Data
are recorded in the source excitation direction. The center frequency of the dipole source
is 3 kHz. Figure 6-22 shows the recorded seismograms for dipole sources at 00 (stress
direction) and at 900 for 10 MPa uniaxial stress. Receivers are 5 cm away from the
borehole center in the source inline direction. The most prominent signals shown in the
recorded data are the flexural waves. Figure 6-23 shows the dispersion of the flexural
wave at 00 and 900 for three different loading stresses. The flexural wave at 00 is faster
than that at 900 at low frequency (<4 kHz) while this reverses at high frequency (>5 kHz).
This is because low frequency waves are sensitive to the formation away from the
borehole, which is less affected by the borehole stress concentration. The low frequency
shear wave with polarization in the loading stress direction propagates along the borehole
at a velocity faster than that polarized in the direction normal to the stress. But high
frequency waves are sensitive to the near borehole formation, whose properties are
altered from their original state. This results in a crossover, which appears at about 4.5
kHz, between the two dispersion curves. The dipole dispersion crossover was discovered
by Sinha et al. (1995) and Sinha and Kostek (1996) and later observed in field logging
data (Sinha et al., 2000). In Figure 6-23, we can see that the overall slowness of the
flexural wave decreases with increasing loading stress, but the crossover frequency does
not change. The crossover frequency is not sensitive to the stress strength because the
overall stress distribution is scaled by the strength of the loading stress. Both near and far
away from borehole stresses increase with increasing loading stress, this results in the
220
shifting of the dispersion curves over all frequencies and makes the crossover frequency
not sensitive to the change of the loading stress.
6.6
Summary
We studied the borehole azimuthal acoustic response caused by borehole stress
concentration using a finite-difference method and the approach developed in Chapter 5.
We compared our numerical results with the laboratory acoustic wave measurements of
Winkler (1996). The consistency of the azimuthal variation of the normalized P-wave
velocity between the numerical and the experimental data suggests that the constitutive
relation between an applied stress field and stiffness of the rock around a borehole can be
accounted for correctly using the approach described in Chapter 5. Due to the preference
of the wave field to propagate through a higher velocity region, the variation of P-wave
velocity versus azimuth shows broad maxima and cusped minima, which is observed in
both the numerical simulations and the laboratory experiments. This suggests that a
correct interpretation of the P-wave velocity measured from borehole sonic logging needs
to consider the effect of borehole stress concentration which results in azimuthally
varying stress-induced anisotropy in the formation around a borehole and deviation of the
wave path of the refracted P-wave from a straight wave path along the borehole axis
direction.
From the monopole simulations, we have found that refracted P-wave and
Stoneley wave velocities have little dependence on azimuth for frequency below about 15
221
kHz, while they show a few percent azimuthal changes when the frequency goes up to 20
kHz. The refracted S-wave velocity does not show clear azimuthal variation for all source
frequencies due to the influence of the pseudo-Rayleigh wave. However, the refracted Pwave amplitude strongly varies with azimuth regardless of the source frequency and has
maxima and minima in directions parallel and normal to the loading stress, respectively,
which is contrary to the variation of velocity versus azimuth. For 10 and 15 kHz
monopole sources, the variation of P-wave amplitude versus azimuth is similar to a
cosine function, while it shows broad minima and cusped maxima for 20 kHz monopole
source. The high sensitivity of P-wave amplitude to the azimuthal variation of formation
properties may suggest that the variation of P-wave amplitude versus azimuth, which is
easy to measure, can be used to study the in situ stress state around a borehole. The
dipole simulations show that the overall velocity of the flexural wave increases with
increasing loading stress but the frequency of the flexural wave dispersion crossover is
not sensitive to the loading stress strength.
222
Figure 6-1: Borehole model geometry. A piston source (red circle) is located at the
borehole center. Receivers (blue circles) are 0.7 cm away from the borehole center along
the piston source direction, which is indicated by the red arrow. A uniaxial stress is
applied in the X direction. Borehole diameter is 2.86 cm.
223
Y
NI
40
N~
A
Figure 6-2: Schematic showing the 1/4 inch diameter piston source used in the simulation.
Arrows indicate source excitation direction. Dashed curve represents a Hanning window,
which is used to taper the source amplitude from the center to the edges. Source direction
i is measured from the positive X direction.
224
Time = 0.011 ms
E
-2
-2
-1
0
X (cm)
1
2
Figure 6-3: Snapshot showing the pressure field in the borehole excited by a piston
source pointing at 300. Red and blue colors indicate peak and trough, respectively.
225
r -- - - - - - - - -1
C13| C14
C15 C16
C23 C24 C25 C26
:C34 C35 C36
Stiffness (GPa)
25
20
15
C45 C46
10
C56
--
--
5
0
Li
Figure 6-4: Color hue of each box indicates the average value of the corresponding
component (Cij) in the stiffness tensor of the formation around a borehole under 10 MPa
stress loading. The components inside the dashed green lines are about two orders of
magnitude larger than the others.
226
C11
32
GPa
30
25
GPa
30
Y
20
30
25
25
20
20
X
X
,GPa
GPa
5.5
C44
GPa
3
5.5
,F2
,23
GPa
5.5
5
5
5
4.5
4.5
4.5
',5
GPa2
'6
GP~a2
GPa2
10
10
10
8
8
8
X
X
Figure 6-5: Variations of the nine dominant components of the stiffness tensor around a
borehole on the X-Y plane when a 10 MPa uniaxial stress is applied in the X direction.
227
,Vg
S
S
/V-1 x1 05
35
- 3
Hz
25 H
-2
H
1
4
z
7
Figure 6-6: Solid and dashed curves show the numerical error for P- and S-wave
velocities, respectively, on the X-Y plane at four frequencies. V and V8fid represent the
true velocity (i.e. formation velocity) and the grid velocity (i.e. velocity in numerical
simulation), respectively.
228
Seismograms
(a)
5
E 10
N
15
-
9 Cij
-- -21
20
Cij
0
0.07
0.14
Difference (x10 5)
(b)
5
__0 _ _~
MI ii
I'
-OWN
I- SWN-
E 10
N
15
20
0
0.07
Time (ms)
0.14
Figure 6-7: (a) Comparison of the seismograms simulated from two models respectively
containing 21 (dashed red) and 9 dominant (solid black) elastic constants for a piston
source at 300 and 10 MPa stress loading; (b) Difference (multiplied by 105 ) between the
seismograms shown in (a).
229
(a)
00
100
(b)
0
0
20
20
0
(f)
0
0.07
0.14
500
0
(g)
0
(C)
200
(d)
0.14
600
0
(h)
(e)
400
O-3unmfl
20
20
0.07
300
0.07
0.14
700
0
(i)
20
0.07
0.14
0
vi
0.07
j)
800
0
0
i
.14
960
0
10
10
10
10
10
20
20
20
20
20
s
N
0
0.07 0.14
Time (ms)
0
0.07 0.14
Time (ms)
0
0.07 0.14
Time (ms)
0
0.07 0.14
Time (ms)
0
0.07
).14
Time (ms)
Figure 6-8: Pressure profiles recorded in the borehole for sources at ten different
directions when a 10 MPa uniaxial stress is applied in the X direction (i.e. 00). The
number above each panel indicates the source direction. 00 and 900 are along the X and Y
directions, respectively.
230
Figure 6-9: Schematic showing the wave paths of direct (dashed line) and refracted (solid
lines) P-waves. Z is the vertical distance between source and receiver. 0, is the critical
angle for refraction to occur at the wellbore. r is borehole radius. Red star and blue circle
represent source and receiver, respectively.
231
Z=7 cm
(a)
90
-
80
w,7 0
D0 6 0
-, 500
=i 4 0
330
A 20
<1 0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.06
0.07
0.08
Z=10 cm
(b)
90
-
70
60
T 50
(D,
3C0
20
10
01
0
0.01
0.02
0.03
0.04 0.05
Time (ms)
Figure 6-10: (a) and (b) Seismograms recorded at Z=7 cm and 10 cm, respectively, when
the model is subjected to 10 MPa stress loading. Source is at Z= 0 cm. Red circles
indicate the arrival times of the refracted P-wave. 00 and 900 are along the X and Y axes
directions, respectively.
232
0 FD
1.15
-
00-
0 Winkler
0
Q0
0E
008
0
an
0
oqu
eo a
000
0
0
0
0
1
0
:0
90
0
180
Azimuth (degree)
270
360
Figure 6-11: Azimuthal variation of the normalized P-wave velocity (normalized by the
velocity measured at zero stress state) in a borehole under
10 MPa uniaxial stress loading.
Squares and circles are the P-wave velocities obtained from the numerical simulation and
the experiment of Winkler (1996), respectively. P-wave velocities are determined by
measuring the delay time between receivers located 7 and 10 cm along the Z axis from
the source. Applied stress is along 00 and 1800.
233
Finite-difference
(a)
-.- 1.M..
1.15
-
.....
..
1 a
5 -MP a . -
1.1
...........
..
-...
CU
0
N
- ....
.. -.
1.05
1
0
90
...
180
270
. -..
-.
36 0
Laboratory
(b)
1.15
10
N
~0
Zi
Pa
a
1.1
...
.....
....
1.05
C
90
180
Azimuth (degree)
270
360
Figure 6-12: Azimuthal variation of the normalized P-wave velocity (normalized by the
velocity measured at zero stress state) around a borehole at three different loading
stresses. (a) shows the results obtained from the finite-difference simulations. (b) shows
the best fits to the laboratory measured data (modified from Winkler (1996)).
234
(a)
0
5 MPa
5
10 MPa
(b)
-...
5
(c)
...
15 MPa
Time (ps)
5
3
N
10-
101
10
15-
15-
15
20-.
20 -
20
25 -
25[
0
30
60
Azimuth (degree)
2
1
J
25
0
30
60
90
Azimuth (degree)
0
30
60
Azimuth (degree)
90
0
Figure 6-13: (a), (b) and (c) Advance of the first break arrival time between sources at 00
and other directions for 5, 10 and 15 MPa uniaxial stresses, respectively. Horizontal and
vertical axes are source direction and receiver depth, respectively.
235
Z=1m
Z=2m
90
I
I
I
I
I
I
Z=3m
I
I
I
I
I
I
I
I
I
I
I
-550
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
*
I
I
I
I
I
I
I
40
30
20
10
I
I
0
I
I
I
I
L~L
0.5
1
I
I
I
I
70
( 60
Z=5m
I
I
I
80
Z=4m
I
1.5
2
2
3
Time (ms)
Figure 6-14: Seismograms recorded at ten azimuths and at different offsets for 10 kHz
monopole source and 10 MPa loading stress. Source is at Z=O m. Azimuth indicates the
receiver angle measured from the loading stress direction, which is along 00. The
refracted P-waves are marked by the two dashed red lines in each panel.
236
(a)
00
(b)
300
800
800
0
(c)
(I)0.400
80080
200
0
600
(d)
900
400
1
2
3
4
5
Time (is)
200
OZ
1
2
3
4
5
Time (is)
Figure 6-15: (a), (b), (c) and (d) Time semblances of data recorded at 00, 300, 600 and 900,
respectively, for 10 kHz monopole source and 10 MIPa stress loading. 'P' and 'S' indicate
the refracted P- and S-waves, respectively.
237
900
600
300
00
350
325
300
E
S350
S325
S300
350
3
(b1)ime2(in)b4
325
300
0.75 1
1.25
0.75 1
1.25
0.75 1
1.25
0.75 1
1.25
Time (ms)
Figure 6-16: Expanded view of the refracted P-wave semblances for data recorded at four
different azimuths, which are denoted above each column. The first, second, and third
rows show the results for 10, 15, and 20 kHz monopole sources, respectively. Red and
blue colors respectively indicate high and low amplitudes. All plots are in the same color
scale.
238
600
300
90
550
525
500
550
525
Tme(--
500
550
525
500
1.25 1.5 1.75
1.25 1.5 1.75
1.25 1.5 1.75
1.25 1.5 1.75
Time (ms)
Figure 6-17: Same as Figure 6-16 except that the refracted S-wave semblances are plotted.
239
0
740 I-a)- SM Pa-OOO.OOO
720
.
700 F----
...
E'
)-0.
k
" o 0*
**
*
"
0000.
'
-- ..
740 1b)-1 0MPa 720OLQ A
740
o. 0 0
-
10
6
400
686----
&~~~~.
.......
0 . ..... .... . . .... ....9
---
I
-0
I
.00006**
10
11
12
13
50
600
......
680 ------
200
*300
.
* **
700
I
0
14 15 16 17
Frequency (kHz)
18
19
0
6-..0
20
Figure 6-18: (a), (b) and (c) Dispersion of the Stoneley wave recorded at ten different
azimuths for 5, 10 and 15 MPa loading stresses, respectively. Loading stress is along 00.
240
(a)
<0
(b)
Z=2m
1
15MPa
0.8
0.8
a
10
E 0.6
0
0.4
-.--.----.--....
-~.
5MPa
--
0.2
0
C
90
180
(C).Ns
270
0
Z
0.4
-0
90
180
270
360
Z=5m
-
1
-.-..
....
.-....---.-.
..
-..
.
--
-..
...
--
-..
.-
....
---.
.. ..
-..
-....
--...
-- .....
. . ... .
-..- ---- - .-....
0.8 --
---..
0.6
0.4-
.....
--.
0.2
A
. . --...
----. ....
-..-.-.-.-..... .
(d)
1
N
-. -- .-.-.
. -- -. --.-.-.
0
C
360
Z=4m
--... - . ..
-- . ..-
0.4
E 0.2
E 0.6
-. ..-...
0.6
z
0.8
Z=3m
1
-- --
-..
- -..
------...
..-..-
---......
-----...
-- .
- -...
0.2
90
180
270
Azimuth (degree)
360
A
0
90
180
270
Azimuth (degree)
360
Figure 6-19: (a), (b), (c) and (d) Variations of the refracted P-wave amplitude versus
azimuth for data recorded at Z=2, 3, 4 and 5 m, respectively. The 10 kHz monopole
source is located at Z=0 m. Black, red and blue curves show the results for 5, 10 and 15
MPa loading stresses, respectively. The amplitudes of all data are normalized by the
same value.
241
(a)
(b)
Z=2m
1
15MPa
0.8
0.8
0.6
-....
- 10
. ---...- 0.6
.r~0.4
0.4 . -- -
~~~ ...-----..
~~....
.. .......
.....
t 0.2
z
0
Z=3m
1
90
180
270
0C
360
. 0.8
0.8
0.6 -0
a 0.6
0
z
-----.
.
-.--
....
.... .- .
--.-- - -.
------
0.4
.---.
180
270
360
Z=5m
1
-.
.. .--. .
90
(d)
Z=4m
1
0.2
--- - ---- - - -
0.2
(c)
!A0.4
----.
90-1-....
0--.-.---
--
0- ----
---
0.2
A
'0
90
180
270
360
Azimuth (degree)
*0
90
180
270
360
Azimuth (degree)
Figure 6-20: Same as Figure 6-19 except that results for the 15 kHz monopole source are
plotted.
242
(a)
Z=2m
1
15MPa
0.8
1 OMPa
W0.6
5MPa
----..
6 0.2
0
90
180
(C)
-----.. ..
..
-- -----270
4)
N
0
0.2
0.6
-..
..
-..
---..-.--
-..
.
....
-
+ --.- .----.--
0.2
C6
90
(d)
Z=4m
0.4-
- .-- - -.. -.-.---.-.. -.-.---------..-
0
360
1
E 0.6CU
0.8
0.4
.N 0.4
-. 0.8
Z=3m
(b)
1
180
270
360
Z=5m
1
--..
..
.. -.
..--..
-- . -.--------.
..
..
---
---...
.
--- ...
-- ......
-
---.
. -..
-..
-...
. -..
.....
- ...- -.
-- - . -..-..
.-.
..
---.
...--....-...-...------ --- -...
.. -.
.
0.8
0.6
0.4
-----...
-.
---..
-.-- ....
-.-..
-
..
-.
..--- -....
- -- --.
-...
.-
.-.-
0.2
z
0
0
90
180
270
Azimuth (degree)
360
0
0
90
180
270
Azimuth (degree)
360
Figure 6-21: Same as Figure 6-19 except that results for the 20 kHz monopole source are
plotted.
243
(a)
3E
4
5,
0
1
2
3
4
5
6
4
5
6
90
(b)
2
3-
E
N
50
1
2
3
Time (ms)
Figure 6-22: (a) and (b) Simulated seismograms for dipole sources at 00 and 900,
respectively. Receivers are positioned at the corresponding dipole source direction.
244
:
650 a. 5MPa..
600
900:
.OY.................................
0~5 Q...... .h
I! 650 (b)IMPa
0600
---.--.-) ..... ............
550
550.......
.....
............. 0
500
-.....-.-.
W 6 00
650
.
900
904,
0-8888
.)...MP..
**--spectivel
respctiely
9
245
246
Chapter 7
Conclusions
7.1
Conclusions
In this thesis, we studied the effects of fractures on both a large scale (i.e. seismic
scatterer) and a micro-scale (i.e. micro-crack) on geophysical exploration. This work was
divided into two parts to cover the study on the two different scales. In the first part, we
began by studying the characteristics of fracture scattering and the associated time-lapse
change, and then we developed methods for time-lapse monitoring survey design and
fracture orientation determination. The applicability of the fracture transfer function
method is validated through numerical simulations, laboratory experiments and field data
application. In the second part, we developed an approach, which accounts for the
interaction between micro-cracks in a rock and in situ stress, to calculate the stressinduced anisotropy around a borehole when it is subjected to an anisotropic stress
loading, and then we study the effect of stress-induced anisotropy on borehole sonic
logging through numerical simulation. The accuracy of this approach is demonstrated
247
through laboratory experiments and comparison with published laboratory measurements
of the azimuthal velocity variations caused by borehole stress concentration.
From the work presented in this thesis, we can draw the following conclusions:
*
The condition for the weak scattering approximation to be valid for fracture
scattering is that the fracture compliance needs to be much smaller than the matrix
compliance. We showed in Chapter 3 that this condition is valid for a range of
fracture compliance values of relevance to subsurface conditions.
*
Fracture scattering pattern is controlled by the ratio of normal to tangential
compliance Z/ZT and the fracture scattering strength varies linearly with fracture
compliance for weak scattering. When sources are located at the surface, P-to-S
scattering generally dominates P-to-P scattering for vertical or sub-vertical
fractures assuming that ZNIZr<0.5, and most of the fracture scattered waves
propagate downward and then are reflected back to the surface by reflectors
below the fracture zone. This means that the surface recorded fracture scattered
waves are influenced by the reflectivity of the reflectors below the fracture zone.
* The percentage change of the amplitude of scattered waves in time-lapse seismic
data is equal to the percentage change of fracture compliance in hydraulic
fracturing. The fracture sensitivity field is mainly affected by the source incident
angle and background medium elastic properties but not sensitive to the fracture
compliance. The sensitivity of seismic waves to fracture normal compliance is
always larger than that to tangential compliance regardless of source and receiver
positions.
248
*
We showed that the fracture direction can be determined from the fracture transfer
function, which describes the seismic response of a fracture system, computed
from the surface recorded seismic data. Study of the orthogonal fracture models
containing heterogeneity and sub-vertical fractures with randomly varying
fracture spacing and compliance indicates that this approach can be applied to the
study of complicated realistic fracture systems. This is further demonstrated by
the field application.
*
Borehole stress concentration has a significant impact on sonic logging
measurements as it causes azimuthally varying stress-induced anisotropy in the
formation around a borehole and deviation of the wave path of the refracted Pwave from a straight wave path along the borehole axis. This makes the azimuthal
variation of P-wave velocity differ from the presumed cosine variation. Thus, a
correct interpretation of the formation velocity measured from borehole sonic logs
needs to consider the effect of in situ stress.
*
The borehole monopole simulation results showed that not only the mode velocity
but also the P-wave amplitude is sensitive to the azimuthal variation of formation
properties around a borehole. This may indicate that both velocity and amplitude
of P-wave can give us the knowledge of in situ stress. The dipole simulations
show that the overall velocity of the flexural wave increases with increasing
loading stress but the frequency of the flexural wave dispersion crossover is not
sensitive to the loading stress strength.
249
7.2
Contributions
In this thesis, we put a lot of effort to understand the effects of fracture and micro-crack
on rock elastic properties and seismic wave propagation through both numerical
simulations and laboratory experiments. We proposed new methods for quantitative
analysis and calculation of several important subjects. In summary, the contributions
made in this thesis are:
1. We studied the general seismic response of a single fracture with a linear slip
boundary condition, especially, we studied the P-to-P and P-to-S fracture
scattering separately. When sources are located at the surface, P-to-S scattering
generally dominates P-to-P scattering for vertical or sub-vertical fractures
assuming that ZN/Zr<0.5, which is satisfied for fluid-saturated fractures. We found
that fracture scattering pattern is controlled by the ratio of normal to tangential
compliance and the fracture scattering strength varies linearly with fracture
compliance for weak scattering.
2. We derived the fracture sensitivity wave equation and introduced an approach for
designing time-lapse surveys to monitor the changes of fracture compliance
accompanying hydraulic fracturing. We demonstrated that fracture compliance <<
matrix
compliance
is the necessary
condition for the
weak
scattering
approximation to be valid for fracture scattering. Based on the analysis of the
fracture sensitivity moments, we found that the fracture sensitivity field is mainly
affected by the source incident angle and background medium elastic properties
but not sensitive to the fracture compliance. The sensitivity of seismic waves to
fracture normal compliance is always larger than that to tangential compliance
250
regardless of source and receiver positions. We further demonstrated that the
percentage change of the amplitude of scattered waves in time-lapse seismic data
is equal to the percentage change of fracture compliance in hydraulic fracturing.
3. We developed the fracture transfer function method to determine fracture
orientation using surface recorded seismic scattered waves. Compared to the
Scattering Index method, our approach is more robust and easier to implement as
only one analysis time window is used and all processing is done in the frequency
domain. The applicability and accuracy of this approach is validated through
laboratory experiment, numerical simulation and field data application. Our
results show that fracture orientation can be robustly determined by using our
approach even for heterogeneous models containing complex non-periodic
orthogonal fractures with varying fracture spacing and compliance.
4. We developed an approach for calculating the stress-induced anisotropy around a
borehole. This method gives the stiffness tensor of the rock around a borehole for
any given stress state and provides the anisotropic elastic borehole model for
forward modeling study. The accuracy of this approach is validated through
laboratory experiments on a Berea sandstone sample. Compared to other methods
(Sinha and Kostek, 1996; Winkler et al., 1998; Tang et al., 1999; Brown and
Cheng, 2007) for studying the stress-dependent response of the elastic properties
of the rock around a borehole when it is subjected to anisotropic stress loading,
our approach is easier to implement, as the knowledge of crack geometry and
distribution is not needed in the calculation. Our method also accounts for the
physics of rocks, as the constitutive relation between the stress field applied
251
around the borehole and the elasticity of the rock with micro-cracks embedded in
the matrix is considered.
5. We are the first ones to do elastic full-wave simulation in a borehole environment
when the borehole is subjected to an anisotropic stress loading. We are also the
first ones to discover that the azimuthal variation of P-wave velocity in a borehole
does not follow the presumed cosine behavior at high frequency due to the
preference of the waves to propagate through a higher velocity region. It is
important to consider the propagation of sonic waves in a three-dimensional
borehole geometry. Our study provides new insight into the propagation of sonic
waves in a borehole and also the interpretation of sonic logging data.
6. Through the borehole monopole simulations, we pointed out that the velocities of
compressional head wave and Stoneley wave are sensitive to the azimuthal
variation of formation properties around a borehole when frequency is above
about 15 kHz. We also found that the compressional head wave amplitude is
much more sensitive to the azimuthal variation of formation properties than its
velocity. The high sensitivity of P-wave amplitude to the azimuthal variation of
formation properties may suggest that the variation of P-wave amplitude versus
azimuth, which is easy to measure, can be used to study the in situ stress state
around a borehole. The dipole simulation results show that the frequency of the
flexural wave dispersion crossover is not sensitive to the strength of the stress
applied on the borehole because the overall stress distribution is scaled by the
strength of the loading stress.
252
7.3
Future work
The work presented in this thesis can be further advanced in the following directions:
*
We need to seek for more sophisticated numerical methods, such as mesh free
finite-difference method, to simulate the propagation of seismic waves in
complicated fracture systems. In the method of Coates and Schoenberg (1995), if
fracture planes are dipping or not aligned with either of the coordinate axes, the
effective stiffness tensors of those grid cells intersected by fractures may have
more than nine non-zero components, which introduce not only additional
computational cost but also complexity in the finite-difference calculation.
*
We can only extract the knowledge of fracture orientation from post-stack data as
stacking may change the interference patterns of fracture scattered waves. In order
to obtain information about fracture spacing.and compliance, we have to study the
fracture scattered signals in pre-stack data. The double-beam method (Zheng et
al., 2012) is a promising technique to meet this purpose and needs to be further
developed.
*
Worthington and Lubbe (2007) found out that the value of fracture compliance
scales with the fracture size. That means the value of fracture compliance
increases with increasing fracture size. But the cause of this scaling effect is still
not clear. One possible explanation is that the value of fracture compliance is
frequency dependent. Seismic waves of different wave lengths sample cracks and
contacts of different sizes on the fracture surfaces. We can better understand this
through numerical simulation.
253
*
We only discussed the effect of borehole stress concentration on monopole and
dipole waves. It is also important to study its effect on quadrupole waves.
*
For the problem of borehole stress concentration, we assume that a rock is
isotropic at zero stress state. Although most reservoir rocks satisfy this
assumption, some rocks, such as shale, have significant intrinsic anisotropy. To
extend our approach to include the effect of intrinsic anisotropy, we need to adopt
other rock physics models (e.g. Oda et a. (1986)) rather than Mavko's model to
calculate the stress-induced anisotropy.
*
It will be interesting to study the case of a tilted borehole when it is subjected to
the compression of horizontal stresses. The complicated stress field may induce
some interesting features in the spatial variation of formation elastic properties.
*
It is important to study the effect of borehole stress concentration on sonic
logging in Logging-While-Drilling (LWD) environments. The presence of an
LWD tool may introduce significant complexity into the acoustic wave field.
254
Appendix A
Linear Slip Fracture Model
In the linear slip fracture model proposed by Schoenberg (1980), a fracture is represented
as an imperfectly bonded interface between two elastic media. Traction is continuous
across the fracture, while displacement is discontinuous. The displacement discontinuity
across the fracture is given by
Au, = Zij oj knk
(A.1)
where Au, is the i-th component of the displacement discontinuity, Yjk is the stress tensor,
Z; is the fracture compliance matrix, nk is the fracture normal.
For a rotationally invariant planar fracture with its symmetry axis parallel to
x,
direction, the fracture compliance matrix Z has the following simple form (Schoenberg,
1980; Schoenberg and Sayers, 1995),
ZN
Z
ZT
(A.2)
ZT-
where ZN and ZT are the fracture normal and tangential compliances, respectively.
For a single planar fracture in an isotropic background, the medium in the vicinity
of the fracture can be considered to be transversely isotropic with the symmetry axis
255
perpendicular to the fracture. In Voigt's contracted index notation, the effective stiffness
matrix of the medium with the fracture symmetry axis parallel to the x, direction can be
expressed as (Schoenberg and Sayers, 1995),
(A
+ 2/)(1 -.
A(
C
=
(1
-
SN)
-
'6N)
N)
A(-8N)
(A+2(1-Y
A(1
A('
2
6
(A + 21t)(1
Y N)
-
~(N)
A(, - YN)
N)
2
-y
&N)Y(A3
-~
&T)
P(1 -
&7)
with
SN
(A.3a)
ZN (A+2p)
1+ZN(A+2p)
=ZT
(A.3b)
(A.3c)
Y=
where A and p are the Lame moduli of background medium.
The corresponding compliance matrix, which is the inverse of the stiffness matrix A.3, is
A_+___+__
pA+pg
p(3A+2p)
S=
+
ZN
-A
-A
2p(3A+2p)
A +I
2p(3A+2p)
Mp(3A+2p)
-A
-A
2p(3A+2p)
2p (3+2g)
-A
2,u(3A+2p)
-A
2p(3A+2p)
A+P
(A.4)
u(3A+2p)
+
ZT
-
+ ZT-
If the fracture symmetry axis is not parallel to the x] direction, we can use the Bond
transformation matrix (Mavko et al., 2003) to transform the stiffness matrix to a given
coordinate system.
256
Appendix B
Mavko's Method for Calculating
Stress-induced Anisotropy
Mavko et al. (1995) proposed a simple and practical method to estimate the generalized
pore space compliance of rocks using experimental data of rock velocity versus
hydrostatic pressure. The approach proposed by Mavko et al. (1995) for calculating the
stiffness tensor with stress-induced anisotropy at a stress state u uses the following steps:
(1)
Calculate the pressure-dependent
isotropic elastic compliances S f (p) from
measurements of compression (P) and shear wave (S) velocities versus hydrostatic
pressure. The compliance S0 i at the largest measured pressure, under which most of the
compliant parts of the pore space are closed, is chosen as a reference point. The
additional compliance AS"
to be Siiski
(P) - S1
(p) due to the presence of pore space at pressure p is defined
. Note that at a pressure p less than the largest measured pressure,
there is more pore volume than at the highest pressure and the compliance is larger.
(2) Calculate the pressure-dependent crack normal compliance WN (p) and crack
tangential compliance WT (p) from ASjh (p) via
WN(P) =2
jjkk (p)
257
(B.1)
and
(P)-ASjjk
WT (p) = WN (p) . AS4jskk
(B.2)
(
where the repeated indices in ASj"k (p)and ASk"o (p) mean summation. The factor
comes from the average of crack compliance over all solid angles.
(3) Calculate the stress-induced compliance ASijkl ()
ir/2
ASijkI (a) =
through
21r
=of
WN(mTM) mm Mkmisin0d65dp
M
0=0 f =o
fS§fef 4
W TmiTam) [Vikm"mI + S
(5m
1
mk
+6 ;kmtmi + 6jI mimk - 4mimjmkmkM] sin0d6d#
(B.3)
where a is a 3x3 stress tensor, m = (sin6cos4, sin~sin#, cos6)T is the unit normal to
the crack surface, 0 and P are the polar and azimuthal angles in a spherical coordinate
system. Note that WN(p) and WT(p) in equations B.1 and B.2 have been replaced by
WN(mT am) and Wr(mT am) in equation B.3, assuming that the crack closure is
determined by the normal stress, mT am, acting on the crack surface. The stress tensor a
needs to be projected onto the normal directions of the crack surfaces.
(4) Obtain the stiffness tensor Cijkl (a) by inverting SO
258
+ AS
(a)
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