135
APPIJCATIONS OF FINITE DIFFERENCE
SYNTHETIC ACOUSTIC LOGS
by
R.A. Stephen· and F. Pardo-Casas
Earth Resources Laboratory
Department of Earth. Atmospheric and Planetary Sciences
Massachusetts Institute of Technology
Cambridge,lIA 02139
ABSTRACT
Finite difference synthetic acoustic logs are suitable for studying wave
propagation in vertically varying boreholes and in boreholes with continuously
varying properties. Snapshots for the traditional smooth bore in a
homogeneous rock show the standard phases in the borehole (compressional
and shear head waves, pseudo-Rayleigh waves, and Stoneley waves), and also
display the complex wave interaction which occurs in the rock. If a simple
gradient in elastic parameters and density replaces the sharp interface the
shear head wave and pseudo-Rayleigh wave are strongly attenuated. Also, offcentered receivers, washouts and horizontal fissures can have significant
effects on amplitudes. A thorough understanding of these effects by forward
modelling is essential in order to avoid pitfalls in interpretation and in order to
design robust schemes for obtaining elastic properties and attenuation from
acoustic logs.
INTRODUCTION
The finite difference method for synthetic acoustic logs was presented in
Stephen et al. (1983) and the stability and accuracy of the technique was
demonstrated. This paper discusses applications of the method by considering
some canonical examples.
The first set of examples consists of vertically homogeneous models with
step, gradient, and step-gradient velocity-radius functions. The effects of
smooth rather than sharp velocity contrasts are demonstrated. An example of
the effect of an off-centered receiver is included and the amplitude of multiples
is discussed.
The second set of examples demonstrates the effect of a washout at
different positions relative to the tool. The washout model is an example of a
medium which varies in two dimensions.
"Perma:neut address: P.O. 80" 587, 11'_ Falmouth, IIA 02574
136
Stephen and Pardo-Casas
The third set of examples demonstrates the effect of horizontal fractures
intersecting the well bore at the tool.
(
MODEL PARAllETERS
Figure 1 shows the geometry used for the calculations. Ail models
considered in this paper are 0.6 m wide by 3.0 m long (60 grid points by 300
grid points at 0.01 m per grid point) and time series are generated out to 2.5
msecs (1250 time steps at 0.002 msec per step). Unless otherwise noted, a
compressional point source is located at the center of a 0.1 m radius borehole
and receivers are placed at the center of the borehole at increments of 0.2 m
between 1.6 and 2.8 m below the source. (Pardo-Casas et at., this volume,
discuss the effects of a rigid tool in the borehole.) The source waveforms and
spectra are shown in Figure 2. The center frequency in pressure is 15 kHz and
the upper half power frequency is 20 kHz.
(
VERTICAUX HOMOGENEOUS MODELS
The first example is a sharp interface between a fiuld-filled borehole and a
homogeneous formation. This model represents a clean, mud-filled bore in a
rock with velocities similar to those of a well consolidated sandstone. Figure 3
shows (a) the velocity and density functions, (b) the time series plots and (c)
the snapshots for this model. The compressional and shear head waves and the
Stoneley and pseudo-Rayleigh waves can be identified on the time series plots.
This example was used for a comparison between finite difference and discrete
wavenumber methods by Stephen et al. (1983) and good agreement was
obtained.
The snapshots show how the various phases evolve as they propagate down
the well bore. The transmitted compressional wave can be readily identified as
the lowermost disturbance in the rock which reaches the bottom of the grid at
0.8 msec. This wavefront intersects the borehole at right angles. To the left of
this arrival, in the well bore, is the conical wave or compressional head wave.
Note also a wavefront at approximately thirty degrees from the interface, which
intersects the interface at the same point as the compressional transmitted
and head waves (at time steps 0.4, 0.6 and 0.8 msec). This is a shear wave
converted at the forward edge of the transmitted wave at the same time as the
compressional head wave. (It is described by Brekhovskikh, 1960, as P1P2S2).
If this wave front is followed back into the solid one sees that it is tangential to
a higher amplitude short wavelength arrival which is the "transmitted" shear
wave. This shear wave was converted at the borehole wall from the incident
compressional wave. Between the transmitted compressional and shear arrivals
one can also see the progressive deveiopment of compressional wave multiples
or "leaking" PL modes. The lowermost large amplitude arrival in the well bore,
which is coincident with the "transmitted" shear wave in the rock, is the onset
of the pseudo-Rayleigh wave packet. This packet becomes broader with time
demonstrating its dispersive nature. Towards the middle of the packet (for
example at 1.2 mse c) the wavelength shortens indicating the Stoneley wave.
Note that the velocity of this arrival is lower than the velocity of the onset of
the pseudo-Rayleigh wave. Above the Stoneley wave is the still shorter
wavelength Airy phase of the pseudo-Rayleigh wave. The amplitude of the high
6-2
(
FInite DitIerence Applications
137
energy packet dies away qUickiy in the rock leaving a complex pattern of body
waves which are the remanents of compressional and shear wave multiples
(leaking PL and SL modes).
The effect of an off-centered receiver for the sharp interface model is
shown in Figure 4. Receiver distance from the center of the borehole is
increased in increments of 0.02 m from 0.0 to 0.08 m. The amplitudes of all
phases decrease rapidly with increasing radius (approximately 12 db across the
borehole). Also the relative amplitudes of the multiples become more uniform
and the multiples themselves become less distinct. This example can be
compared with Willis et at. (1983). It is clear that if amplitudes are to be used
as a discriminant for particular phenomena correct centering of the tool is
essentiaL
The second example (Figure 5) is a linear velocity gradient between the
ftuid-filled well and the formation. This model represents a borehole in an
altered formation. The shear wave coupling is reduced by the gradient and the
shear head waves and pseudo-Rayleigh waves are not observed. The
compressional and Stoneley wave amplitUdes are greater than for the sharp
interface case (Figure 3) because less energy is lost to shear. The P head wave
velocity (3.9 lcm! s, Figure 5b) corresponds closely to the formation velocity
(4.0 lcm! s). However the Stoneley wave arrivaL which could be mistaken for a
shear or pseudo-Rayleigh wave arrival if attention was not paid to waveform, is
travelling at the borehole mud velocity (1.8 lcm! s). Care should be taken when
interpreting logs obtained in formations where this model may apply. Good
estimates of the formation shear wave velocity may be impossible to obtain
·from full waveform data. Also note the progressively higher amplitudes of the
compressional wave multiples. An event detecting deVice set to pick the
Stone ley wave arrival on an amplitude basis could easily pick the third multiple,
P". The higher amplitude multiples will be discussed further below.
The snapshots for the gradient model (Figure 5c) are simpler than for the
sharp interface case because of the absence of shear waves and pseudoRayleigh waves. The transmitted compressional wave, compressional head wave
and compressional wave multiples (leaky PL modes) can readily be identified.
The nature of the Stoneley wave without the interference of the pseudoRayleigh wave can also be observed. This wave shows considerably less
dispersion than the pseudo-Rayleigh wave in Figure 3c.
Similar phenomena are observed in the sharp-interface /linear-gradient
combination (Figure 6). This model is more realistic than the simple gradient
model because it includes a discontinuity in properties at the borehole wall.
The P head wave velocity (Figure 6b) corresponds to the formation (3.9lcm! s).
The next large amplitude pick lacks coherence but has a velocity of 1.8 lcm! s .
This corresponds to a shear or pseudo-Rayleigh wave travelling at the near hole
(or damaged zone) shear wave velocity (1.6-1.8 lcm! s). Poisson's ratio
estimates based on logs where this sort of model is valid will be biased towards
higher values. The pseudo-Rayleigh wave is about twice as large (6 db) for this
model as for the sharp interface model(Figure 3). Coupling to shear wave
effects is improved because the effective shear wave velocity in the solid is
closer to the compressional wave velocity in the fiuid (compare Figures 6a and
3a).
6-3
138
(
Stephen and Pardo-Casas
Compressional wave multiples are evident in the gradient model '(Figure
5b). The amplitude of the first P head wave decreases rapidly with depth and
the second and third P arrivals are progressively higher in amplitude. Why, for
example, is the second P arrival (the first mUltiple) larger than the first P
arrival (the P head wave)? One answer is that the P head wave spends more of
its time as a head wave than a body wave. For a Cartesian geometry head waves
decay much more rapidly with range (as 1/r 2 ) than body waves (as
..J 1/ r.dp / dr). (For example the compressional head wave in Figure 5 decays
to about a quarter (-12 db) between 1.6 and 2.8 m depth because the arrival at
the deeper receiver has travelled twice as far as a head wave.) To see this more
clearly consider the ray diagram (Figure 7) which corresponds to the velocityradius profile of the gradient model. The P head wave and the first multiple ray
paths are shown. The solid ray paths correspond to body waves and the dashed
ray paths to head waves. The paths are the same for the head wave and first
multiple except in the region marked "A." In this region the first multiple
travels 10.2 units as a body wave over the same vertical distance that 'the
primary wave travels as a head wave (7.5 units). Using the Cartesian
assumption, the head: wave has decayed by (1/7.5)2 or 1/56 over this distance.
The corresponding decay for the first multiple in this section is only
..J1/r.dp/dr or 1/14.3. (It is assumed here that the first multiple does not lose
energy at its turning point. This is consistent with WKBJ theory, e.g. Chapman,
1978, where turning rays are only subject to a phase shift of ninety degrees.)
Thus the primary path is attenuated four times as much as the first multiple
path. This ratio is confirmed by inspection of Figure 5. The amplitude of the
first multiple at 2.8 m depth is very similar to the amplitude of the primary at
1.6 m depth.
(
WASHour MODElS
The next series of models investigates the effects of washouts in the
borehole wall on acoustic logs (Figure 8). The continuous gradient model
(Figure 4) is perturbed by introducing a depth: dependent borehole radius. The
washout is placed at four locations relative to the source of the tool: directly
opposite the source at depth 0.0 m, at 0.8 m, at 1.6 m, and at 2.4 m below the
source. Strangely enough the washout directly opposite the source has little
effect on the compressional wave signals (Figure 8a). The primary
compressional wave is twice as large (6 db) as when the washout was not
present but the first and second multiples are considerably less affected.
Presumably the steeper ray paths which contribute to the multiples are less
affected by the washout. The Stoneley wave on the other hand is dramatically
affected, being less in amplitude by about 3 or 4 db and at the 1.6 m receiver is
about half the pulse-width. Energy seems to be lost by reverberation in the
cavity, which is evident on the snapshots (Figure 8b).
The washout at 0.8 m below the source (Figure 8c) has essentially no effect
on the primary compressional wave but the uppermost 0.4 m of the multiples P'
and P" are larger by 3-4 db than without the washout. The washout has an
apparent tendency to focus the compressional wave energy to certain depths.
As with the 0.0 m washout the Stone ley wave is much lower in amplitude
(approximately -6db) and is less dispersed. The Stoneley waves tend to
oscillate being first smaller (at 1.6 m), then larger (at 1.8 m), then smaller (at
2.0 m),then larger (at 2.2 m), then smaller again (at 2.4 m). The presence of
6-4
(
Finite Difference Applications
139
the washout apparently causes a "beating" of these phases as they propagate
down the well bore. Stoneley wave retlections from the washout can be seen in
the snapshots (Figure 8d).
The washout at 1.6 m (Figure 8e) is directly opposite the top receiver.
Note that the trace at the top receiver for this model is identical to the top
receiver trace for the washout at 0.0 m. This is a demonstration of sourcereceiver reciprocity. I=ediateiy below the washout (at 1.8 m) the
compressional wave phases are higher in amplitude by 3-9 db. Below this the
primary compressional wave returns to normal. the tlrst P-wave multiple is
anomalously low (at 2.2 m), and the second P wave multiple and Stoneley wave
are dramatically higher. Beating in these latter phases is not evident.
The final washout model (Figure 8g) has the washout at 2.4 m depth,
opposite the middle of the array. Again the tlrst and second P arrivals are
higher in amplitude just below the washout (2.6 m) but decay quickly below
that. There Is no evidence of retlections of these phases back up the borehole.
These examples demonstrate some of the effects which washouts can cause
on full waveform logs. Amplitudes of compressional waves can increase and
decrease by up to 6 db depending on the location of the receiver relative to the
washout. Washouts tend to decrease the dispersion and amplitude of the
Stoneley waves but beating effects are apparently stimulated. The
compressional body waves tend to follow the washout deformation and, except
for some focussing. are little affected. The Stoneley waves, on the other hand.
retlect from the change in thickness of the waveguide.
HORIZONTAL FISSURES
The effect of horizontal fissures on full waveform logs was studied using two
models: a 10.0 em thick tlssure was placed at 0.8 and 2.2 m below the source.
Gradients similar to those used in Figure 5 were introduced above and below
the tlssure for its full length.
The effects of the fissure at 0.8 m are essentially indistinguishable from
the washout at the same depth (cf. Figure 9a with Figure 8c). The first P wave
amplitude is the same as the unmodified borehole amplitude. The second P
wave arrival has very large amplitude at the 1.6 m receiver but decays qUickly
with depth. The same beating phenomena is present in the Stoneley wave
arrival that was mentioned above. Stoneley waves refiect from the fissure but
remarkably little energy is actually trapped in the fissure.
When the fissure is placed opposite the middle of the receiver section the
effect of transmissions and retlections on the compressional waves can be
observed (Figure 9c). Directly opposite the fissure (at 2.2 m) the P wave
arrivals are in a shadow zone, but below the fissure the amplitude is unchanged.
8-5
(
140
Stephen and Pardo-Casas
CONCLUSIONS
Synthetic full wave form acoustic logs are a useful instructive tool for
studying wave propagation in boreholes. The snapshot formats show clearly the
evolution and conversion of the various phases. Even for depth independent
media the snapshots provide interesting insights into the propagation.
The finite difference technique is well suited to depth dependent problems
and some examples were presented above. Body waves seem to be little affected
by washouts and horizontal fissures. Refiections and "beating" phenomena
associated with the waShouts and fissures has been demonstrated for the
Stone ley waves.
(
Frequency-wavenumber analysis in a similar fashion to Kelly (1983) would
give further insight into the effects on the guided wave phenomena. Dispersion
relations for vertically inhomogeneous models couid be compared with the
analytical results of Biot (1952) for vertically homogeneous models. Future
work should also include attenuation and applications to real data. The true
value of synthetic acoustic logs cannot be assessed until some comparison with
real data is made.
(
ACKNOWLEDGEMENT
This research was supported by the Full Waveform Acoustic Logging
Consortium at M.LT.
(
REFERENCES
Biot, M.A., 1952, Propagation of elastic waves in a cylindrical bore containing a
fiuid: J. appl. Phys., 23, 997-1005.
Brekhovskikh, L.M., 1960, Waves in layered media. Academic Press, New York.
Chapman, C.H. 1978, A new method for computing synthetic seismograms:
Geophys. J.R Astr. Soc., 54,481-518.
Kelly, K.R, 1983, Numerical study of Love wave propagation: Geophysics, 48,
833-853.
(
Stephen, RA., Pardo-Casas, F., Cheng, C.H., 1983, Finite difference synthetic
acoustic logs: M.LT. Full Waveform Acoustic Logging Consortium Annual
Report, Paper 4.
Willis, M.E., Toksoz, M.N., and Cheng,C.H., 1983, Approximate et!ects of otl'-center
acoustic sondes and elliptic boreholes on full waveform logs: M.LT. Full
Waveform Acoustic Logging Consortium Annual Report, Paper 5.
6-6
l
141
FInite Difference Applications
Point Compressional Source
Axis of Symmetry
"
Homogeneous
Liquid
A
X
I
o
F
S
Y
M
M
E
T
R
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Homogeneous
T
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t
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- radius (r)
Solid
A
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.
0
R
u
e
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9
i
0
0
A
R
y
n
Absorbing
Boundary
;,
depth (z)
Figure 1: Outline of the geometry used for finite difference synthetic acoustic
logs. A compressional point source is located at the intersection of vertical and horizontal axes of symmetry. A vertical transition region
separates the fiuid in the well bore from the solid rock. The transition region has arbitrary thickness and can represent sharp vertical interfaces
or continuous two dimensional variations of compressional and shear wave
velocity and density.
6-7
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142
Stephen and Pardo-Casas
Waveforms
Potential at Origin
Ampli tude Spectra
(
(
(
Far Field Displacement
(
Pressure
(
..
..
o
0.2 msec
45
Frequency (kHz)
Figure 2: Source waveforms and amplitude spectra for the examples shown in
this paper. The peak frequency in pressure is 15 kHz and the upper halfpower frequency is 20 kHz. The pressure pulse duration is 0.2 msec.
(
6-8
l
143
Finite Di1rerence Applications
DEPTH (M)
0.2
0.4
0.0
o
0.6
,..-r--T---.----,---..,..---,
I
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..... 1.0
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::E
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(!)
>r-
en 2.0
Z
r-
W
Cl
Figure 3a: The velocity and density profiles, typical of a mud filled hole in sandstone, are shown. This is a sharp interface model for a vertically homogeneous case.
6-9
144
(
Stephen and Pardo-Casas
(
!
Time
1.6
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.\ \/ !'"
I
\ ,I i
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Sharp Interface
Sharp Interface
(x 1)
(x 10)
Figure 3b: The time series format for the sharp interface modei shows the traditional waves expected for a point source in a borehole. The compressional
p, shear S,and Stoneley St arrivals are indicated. The velocities are 4.0,
2.1,1.8 km/s respectively. The receivers are located on the axis of symmetry at the depths indicated.
6-10
Finite Difference Applications
145
Figure 3c: The complete picture of wave interaction around the borehole for the
sharp interface model is shown in the snapshot format. A description of
this figure is given in the text. The first six frames show the amplitude
distribution of the the vertical displacement field in radius-depth space.
Each frame is 0.6 m wide by 3,0 m deep. Time progresses from 0,2 msec
to 1.4 msec. The eighth frame is a representation of the velocity field.
6-11
146
(
Stephen and Pardo-Casas
RECEIVER
RADIUS
(0 m)
fTl1:-'I.'1,"l ~
".\ 1 ,:!.';
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(x 1 0)
SI:iARP INTERFACE (2.2 M DEPTH)
(x 1)
Figure 4: The variation of response across the borehoie is shown here at two
gains (xl. xlO) for the sharp interface model. The amplitude of all phases
decreases rapidly with increasing radius, Note that the Airy phase is
essentially non existent at the 8.0 em location,
6-12
147
Finite Di1I'erence ApplicatioIlB
DEPTH (M)
0.0
-
°
'\
0.4
0.6
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z 2.0
w
Cl
Figure 5a: The veiocity density functions are given for a simple gradient model
which represents a severely altered borehole.
6-13
Stephen and Pardo-Casas
148
Time (msec)
1.6
(
1.8
2.0
2.4
2.6
2.8
Gradient
Gradient
(x 1)
(x 10)
Figure 5b: The time
head wave P,
wave St. The
Stoneley wave
series plots for the gradient model show the compressional
the compressional wave multiples p', p", and the Stoneley
compressional waves have a velocity of 3.9 kml s and the
has a velocity of 1.45 kml s.
(
6-14
Finite Difference Applications
149
Figure 50: The snapshots for the simple gradient model are shown. The dimensions are the same as for Figure 30.
6-15
Stephen and Pardo-Casas
150
DEPTH (M)
0.0
0.2
0.6
0.4
O.----,..--,..--.,..--"T"""--.---,
I
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" 2.0
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(
Cl
Figure Sa: The velocity and density functions for the combination of a sharp interface With a gradient zone behind it. This model is more representative
of an altered borehole than the simple gradient model (Figure 5).
(
6-16
Finite Di1ference Applications
151
Time (msec)
0.8
1 6
____- L
,
1.6
~
,"
I/\f~~~---
vv
\
i
1.8
_
,
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\
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al ....
e
p
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2.4
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V\ ~
2.6
--+----./'-Ar--J\"-J'vV'v\JV~WN
2.8
Sharp Interface / Gradient
Sharp Interface / Gradient
(x 1)
(x 10)
Figure Sb: The time series lormat lor the sharp interlace/ gradient modeL The
velocities 01 the compressional P and shear S arrivals in the time series
plot are 3.9 and 1.8 kml s respectively. The P wave is sensitive to the lormation but the shear wave is "Ieeling" the altered zone.
6-17
J:P2
Stephen and Pardo-Casas
(
(
Figure 6c: Snapshots for the sharp interface/gradient model.
6-18
(
FInite Di1Ierence Applications
153
Velocity
LRadiUS
Source
:
1
rRadiUS
Depth
1
•
1:- primary
I:
I
1
first
multiple
I:
-T
I:
1
A
1
1
I
1
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
I
I
I
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1
I
Receiver
Figure 7: For the simpie gradient model, multiples are larger in amplitude than
the primary because they travel more of their path as body waves and less
as head waves. In the section marked "A" where the primary and first
multiple differ. the first multiple loses less energy as a body wave (solid
line) even though it travels farther than the primary which travels as a
head wave (dashed line).
6-19
Stephen and Pardo-Casas
Time (msec)
o8
1.6
--~'-r-~--1.6
1.8
2.0
-,fM~~V/~
(
-J\J"\IW\i\~\r-1
1\ f'
--Jv--\A.,vV~~~A-,
j~ -_v~~j\J\l\jVLr-
,
'Ii"
--~~ji)V\j:::i1y
\11)
'\
2.4
--Jv'4!VvI/~vJ
- -----;.',
JV-.~,J.,~
j'N-'fl'
2.6
Washout at O.Om
Gradient Without Washout
(x 1)
(x 1)
Figure 8a: A comparison of time series and snapshots for gradient modeis with
washouts at four depths relative to the tool is shown in this sequence of
figures, The effects are discussed in the text. This figure is the time
series format for the washout at 0.0 m.
6-20
Finite Difference Applications
Figure Bb: Snapshots for the washout at 0.0 m.
6-21
155
(
156
Stephen and Pardo-Casas
Time (msec)
~-_.
__
0 18 ,
1.6
~\fV\!JIJ~-,
1\
1\11
1.8
~jVV\J
2.0
~A.Aj'VVVJ~
-
.t::
.-.
Q.E
(1) ....
Cl
Jij'v-(
~J~J~V\j~\JV~
f,
2.4
(
"
~WV\rJVI
(
- - - - - - . ' v..--".. .\ .
Washout at 0.8m
Gradient Without Washout
(x 1)
Figure 8e: Time series format for the washout at 0.8 m.
6-22
l
Yinite DI1Ierence Applications
Figure 8d: Snapshots lor the washout at 0.8 m.
6-23
157
158
Stephen and Pardo-Casas
Time (msec)
__°-lL1__~_ _11_6__
1.6
1.8
(
1 6
_--l[
_
--JvvV~~'\-
(
-JV_vV\t;~~~
2.0~~~\ll~
(
i~~~~I~
2.4
(
II-t' :\
~VV"\V\
2.8
Washout at 1.6m
Gradient
(x 1)
Without Washout
(x 1)
Figure Be: Time series format for the washout at 1.6 m.
6-24
Finite Ditrerence Applications
Figure 8f: Snapshots for the washout at 1.6 m.
6-25
159
Stephen and Pardo-Casas
160
Time (msec)
0.8
1.6
___I " .
1.6
1.8
2.0
- ....E
.c
Co
Ql
o
....
(
I
~~~\~~V~'----
-VV4Vv\j11~1ik­
~~VJWlr,
(
"
Ii
~J~IIJVV\\liJ
(
2.4
2.6
~JWv\
J1t--'\j\rvij\
(
--vf\,.../,..,"'j"/
2.8
(
Washout at
Gradient Without Washout
2.4m
(x 1)
(x 1)
Figure 8g: Time series format for the washout at 2.4 m.
8-26
(
Finite Ditlerence Applications
Figure 8h: Snapshots lor the washout at 2.4 m.
6-27
161
Stephen and Pardo-Casas
162
Time (msec)
__
1.6
~[
-----.JV-.--J\ 1'1
t _-
_-----'1
f\rv!\r~-
\ II
•
\
1.8
~J\~~\r~
2.0
~~"\fvVIr~
1~ ~JY,J~tlJ\V\jl
---~J\-NIJ\,;VV\i\!v
-
j
:'
J
t\
,"!\
~\i\.J\j\;\
,
\1 \
----~J'v--ov'jwv)
Horizontal Fissure at 0.8m
(x 1)
Gradient Without Fissure
(x 1)
Figure 9a: This sequence of figures shows a comparison of time series and
snapshots for gradient models with horizontal fissures at two depths relative to the tool. The effects are discussed in the text. This figure is the
time series format for the fissure at 0.8 m_
6-28
(
Finite Difference Applications
Figure 9b: Snapshots for the fissure at 0.8 m.
6-29
1'63
Stephen and Pardo-Casas
164
(
Time (msec)
(
1.6
1.8
2.0
.c
.........E
Q.
al
o
.....
(
2.4
(
2.6
Horizontal Fissure at 2.2m
(x 1)
Gradient Without Fissure
(x 1 )
Figure 9c: Time series format for the fissure at 2.2 m.
8-30
Yinite Di1Ierence Applications
Figure 9d: Snapshots for the fissure at 2.2 m.
6-31
165.
166
(
(
(
(
(
(