MECHANISM OF ATTENUATION OF COIPRESSIONAL AND SHEAR WAVES
FOR DRY, WATER AND BENZENE SATURATED ROCKS.
by
Christophe Gonguet
Ingdnieur de L'Ecole Nationale Sup6rieure de Gbologie
Nancy,France
(1983)
SUBMIVITTED TO THE DEPARTMENT OF
EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in
Earth and Planetary Science
at the
( MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May , 1985
Signature of Author......... . ..
............................
.........................................
Departmrnt of Earth, Atmospheric, and Planetary Sciences
May ,1 985
-^
Certified by ................................ ................
..... ................................... .......
M. Nafi Toksoz
Thesis Advisor
S/
..................
Accepted by ............- ................ ,.........
................................
Theodore R. Madden
Chairman
Departmental Committee on Graduate Students
MT
Li.
r
Iindgren
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Table of Contents
3
ABSTRACT..............................................................................
4
..................................................
ACKNOW LEDGEMENTS .....................................................
Chapter 1.
55
....................................................
INTRODUCTION .......................
Chapter 2.
LABORATORY MEASUREMENTS AND SAMPLE DESCRIPTION
A
Experimental technique...........................
B
Samples studied ....................................................................................
Chapter 3.
A
8
10
DETERMINATION OF ATTENUATION COEFFICIENT
Spectral ratio technique
13
A.1 Analytical presentation .....................................................
15
............................
A.2 Practical problems ........................................................
17
A.3 Comparison with other techniques ............................................
B
Rise time technique
17
B.1 Analytical presentation ....................................................
B.2 Practical problems ....................................................
19
20
B.3 Comparison with other techniques ............................................
C
Sources of error
C.1 Spherical spreading ...................................................... 22
27
C.2 Sidewall reflections and geometrical artifacts ...................................
C.3 Scattering by grains ...................................................... 27
-1-
27
C.4 Effects of noise .......................................
Chapter 4.
RESULTS AND INTERPRETATION
29
A
Spectral ratio attenuation- Results...........................................................
B
Effects of fluid saturants ................................................................................. 34
C
Comparison with other ultrasonic determinations ..................................... 40
Chapter 5.
CONCLUSIONS ..........
............................................................................................ 44
FIGURE CAPTIONS ...............................
...............
FIGURES ........................................................
-2-
....................45
...........
REFERENCES ...................................
....................
...................... 48
51
MECHANISM OF ATTENUATION OF COIVPRESSIONAL AND SHEAR WAVES
FOR DRY, WATER AND BENZENE SATURATED ROCKS.
by
Christophe Gonguet
Submitted to the department of Earth, Atmospheric and Planetary
Sciences on May, 1985 in partial fulfillement of the
requirements for the degree of Master of Science in Geophysics
Abstract
The spectral ratio method is used to calculate the quality factor (Q) in porous rock
samples at ultrasonic frequencies (0.3 - 1.5 MHz). The data were collected using the
pulse transmission' technique with aluminum used as a high Q standard. The data set
consists of dry, water and benzene saturated rocks at differential pressures from zero
to one kilobar. Four sandstones, Berea, Kayenta, Navajo and Weber, in addition to
Bedford limestone, Webatuck dolomite and Westerly granite are studied. Water and
benzene were chosen as pore fluid saturants to contrast the effects of two different
pore fluids (density, compressibility, viscosity, dielectric constant, and wetting
properties) at ultrasonic frequencies. The main features observed are: 1) The quality
factor Q increases with increasing confining pressure; at low pressures the rate of
increase is larger. 2) Q for saturated samples is generally lower than for dry samples.
This effect could be counterbalanced by a "crack-healing" effect for rocks which have
a high crack-porosity and low pore-porosity. 3) The introduction of a fluid saturant into
a dry rock increases S-wave attenuation more than P-wave attenuation. (4) We
observe that benzene saturated attenuations are higher than water-saturated values at
lower pressures, and conversly slightly lower at higher pressures.This observation is
well correlated with a similar behavior of the benzene- and water-saturated results for
non-linear stress-strain and velocity curves. The difference between benzene- and
water-saturated attenuations is generally not important, it is larger for rocks with a
higher clay content. (5) Bulk and extensional attenuations are useful to discriminate
between compressional and shear attenuation and therefore enhance the effects of
fluid saturants. (6) The rise time technique and the spectral ratio technique are in
relatively good agreement. The rise time technique requires a precise determination of
the constant c in order to adjust well to the spectral ratio technique. However, rise time
attenuations are easier to calculate for very high attenuations.
Thesis Supervisor : M.N. Toksiz
Title : Professor of Geophysics
Acknowledgements
The completion of this work was made possible because of some self determination
and a great deal of stimulation, support and fundamental insight by my advisor, Prof. Nafi
Toks6z, and by Dr. Karl Coyner and because of a generous financial aid from the oil
company TOTAL.
I am also grateful for all the good friends I made in the departement and at ERL in
particular, who made this working environment something exciting and enjoyable.
Finally, I must thank my parents, uncle Louis and aunt Juliet who encouraged and
supported me all the way, giving to me this unique opportunity of discoveries and
enrichment.
-4-
CHAPTER 1 :
INTRODUCTION
The contribution of pore fluid to attenuation in rocks has been a subject of interest
and intensive study (Bulau et al., 1984; Murphy, 1982; Spencer, 1981; Winkler and Nur,
1979; Mavko.and Nur, 1979; Johnston et al., 1979; Johnston and Toksiz, 1980;
O'Connell and Budiansky, 1977; Biot, 1956a,b; Wyllie et al,., 1962).
Pore fluid can contribute to attenuation in many ways: inertial decoupling, pressure
induced flow; viscoelastic relaxation; local flow between adjacent pores ("squish
flow"); and wetting of grain surfaces that can contribute to chemical and physical
attenuation mechanisms. Although these mechanisms have been studied extensively,
neither the experimental data nor the theoretical studies have been able to determine
quantitatively the relative significance of different mechanisms (see summary in Toks6z
and Johnston, 1981). Attenuation in fully or partially saturated rocks is greater than
1
attenuation in dry rocks. For dry rocks, attenuation (Q- ) varies by orders of magnitude
whetter the rock is "air dry" (containing at least a monolayer of water) or totally
outgassed by high-vacuum and thermal pulsing (Pandit and King, 1979; Clark et al.,
1980). These data confirm the role of water on the scale of monolayer thicknesses.
Going from air-dry to fully fluid saturated case, attenuation increases (Johnston et
al., 1980; Toksbz et al., 1979; Winkler and Nur, 1979). The largest effect of saturation
occurs in Qs.
In the air-dry case Qp ; Qs and, in fully saturated case
Qs
<
Qp.
Partial
saturation may reduce Qp relative to full saturation, but this effect is generally small
(Winkler and Nur, 1979; Johnston and Toksiz, 1980; Frisillo and Stewart, 1980;
-5-
Murphy, 1984).
There is little experimental data on attenuation with fluids other than water and
brine. Wyllie et al., (1962) measured resonant bar attenuation in alundum rods (25%
porosity) saturated with soltrol and water and found that attenuation was lower with
soltrol saturation. Nur and Simmons (1969) measured the relative change in ultrasonic
signal amplitudes transmitted through glycerin- and water-saturated Barre granite as a
function of temperature and concluded that a resonant peak existed as a function of
fluid viscosity. Spencer (1981) measured the low-frequency phase-lag between stress
and strain in Navajo sandstone with a sequence of different fluids and correlated
increasing attdhuation 'with larger reductions in surface ftee energy of silica when
immersed in the various fluids, with water being the highest.
Bulau et al., 1984,
water and iso-octand' and noted that
measured resonant bar attenuation with
attenuation with Water was higher.
To evaluate whether water versus another type of liquid pore fluid affects
ultrasonic attenuation significantly, we made a systematic study of the dynamic bulk
and shear moduli, (K,.A), and the quality factors, Qp and Qs,
and benzene-saturated rock samples.
K and QE, in dry, water
Measurements are made over a hydrostatic
confining pressure range up to 1 kbar.
Rock samples include sandstones (Berea,
Kayenta, Navajo and Weber), a limestone (Bedford), a dolomite (Webatuck) and a fine
grained granite (Westerly).
The measurements are made at ultrasonic frequencies
(300-1500 kHz).
The effects of water versus benzene on P- and S-wave velocities and dynamic and
shear moduli have been discussed by Coyner and Cheng (1985).
In short, the
differences between velocity measurements with benzene and water are not easily
interpreted strictly on the basis of density, compressibility, or viscosity differences. At
-6-
low pressures water- saturated P- and S-wave velocities are higher than benzenesaturated values.
At high pressures (greater than about 300 to 500 bars) in the
sandstones the water-saturated velocities tend to fall below the benzene-saturated
values. Conversion of velocities into dynamic shear and bulk moduli as a function of
pressure indicates that the effect is concentrated in the shear modulus.
Water-
saturated bulk moduli are consistently higher than benzene-saturated values while
water-saturated shear moduli, initially slightly higher at low pressures, fall below
benzene-saturated values and even the dry shear moduli at higher pressures.
In this study a portion, of the signal waveforms collected by Coyner (1984) were
analyzed for P- and S-Wgave ultrasoniq attenuation with the hope that a similar effect
would be measured. Contrary to what was expected, benzene- saturated attenuation is
higher than with water, particularly at low pressures for the P-wave,when most cracks
have not been closed and the different bulk properties of water and benzene have a
predominant effect. At high pressures water-saturated attenuation tends to be slightly
higher although the amount is on the order of experimental error.
-7-
CHAPTER 2 :
LABORATORY MEASUREMENTS AND SAMPLE DESCRIPTIONS.
A. Experimental technique
In this study we determine attenuation of P- and S-waves using ultrasonic. pulse
transmission.
The experimental arrangements shown schematically in Figure 1, representing a
sample loaded in the high pressure apparatus. In this method, a jacketed sample is
placed in the pressure vessel with source and receiver transducer attached. The
source and receiver piezoelectric crystals are contained within a titanium alloy housing.
The main characteristics of this experimental combination are:
(i) The samples used are cylinders 2 inches in length and 3 inches in width
(ii) Independant hydrostatic confining pressure and pore pressure or vacuum are
applied. The differential pressure was chosen to vary from 25 to 1000 bars, by
increments of 25 bars up to 250 bars, then by increments of 50 bars up to 500 bars
and finally by increments of 100 bars up to the maximum pressure. Pore pressure for
the fluid saturated samples was maintained at 100 bars.
(iii)
The ceramic transducers are isolated from pressure, thus avoiding effects on the
source spectrum caused by the change of pressure.
(4i) At any given time or pressure, P or either S-wave can be recorded.
(5i) The experiment is computer monitored.
The design of this apparatus and all the measurements were completed by Karl Coyner
(Phd thesis, 1984). A more detailed description is available in his PhD thesis. In addition
-8-
to these specifications, it may be useful to remark that two perpendicular S-wave
recordings were made at each pressure. In this study only one of those has been used,
but the processing of both recordings reveal some anisotropic attenuation if for
instance the rock microfractures are preferably oriented in some direction (Lo, Coyner,
Toksiz, 1985).
Furthermore, we only used the transmitted events for attenuation
measurements, while P and S-waveforms have been recorded for all all first order
reflected events, refecting from the titanium plates - sample core interfaces. A study of
attenuation would therefore be possible by comparatively measuring amplitudes for
events with and without reflection or by using the phase velocity determination and the
method described by Winkler and Plona, 1982. A specific manipulation was conducted in
order to estimate the travel time through the titanium plates at all pressures, taking the
coupling effects into account, by measuring the delay time, with visual threshold
detection, from source to receiver, for aluminum samples of decreasing thickness, finally
interpolating the results to a zero-length aluminum core. The results agree well with the
approximate
-calculation
of
At
using
Al =3.6cm
and
vp=6.07 km/ sec, vs=3 .1 3 km/ sec.
Ultrasonic P- and S-wave signals are propagated through cylindrical cores or rock.
An aluminum cylinder, same size as the sample, is used as a high-Q standard for
calculation of the spectral ratio. Center frequencies of the signals are approximately
400 to 850 KHz depending on pressure and sample.
Temperatures are laboratory
ambient. Confining pressures are systematically cycled up to levels of 1000 bars and
pore pressures are either vacuum (20 u m Hg) or 100 bars for the benzene and water
saturations. The difference between confining and pore pressures is taken to be the
"effective" pressure in determining attenuation.
Each sample was first measured
vacuum dry over a cycle of confining pressure, then saturated with benzene reagent
-9-
(100 bars) and again measured over a cycle of confining pressure. Samples were then
removed from the sample apparatus, soaked and washed with acetone, and dried for at
least 24 hours in a vacuum at 800C. Next the sample was rejacketed, measured dry
again as a check for repeatability, and then measured over a cycle of confining
pressure after saturating with water at 100 bars pore pressure.
Specially developed transducers provide broadband P and S-waves. Examples of
waveforms for dry, benzene and water saturated Kayenta sandstone at 100 and 500
bars, for Westerly granite at 100 bars
are shown in Figures 2, 3 and 4.
The experimental procedures used for both the sample and the aluminum reference are
similar. Therefore, we have relative measurement of attenuation and such we minimize
the effects of the transducer, the coupling between transducer, titanium holders and
the sample. The Q of aluminum is about 1 50,000 (Zemanek and Rudnick, 1961) , roughly
3 to 4 orders of magnitude higher than those of rocks in this study, and is constant with
frequency (Savage and Hasegawa, 1967).
B. Samples studied
Table 1: Rock Samples in this Study
The standard characteristics of the rocks studied are given in next table:
-10-
Table 1
Density
Grain Size
Porosity
4
(g/cc)
(mm)
(%)
(Mbar)
Westerly Granite (blue)
2.641
0.75
0.8
0.56
Webatuck Dolomite
2.846
0.45
0.5
0.97
R Bedford Limestone
2.360
0.75
11.9
0,65
Weber Sandstone
2.392
0.05
9.5
0.385
Navajo Sandstone
2.316
0.15
11.8
0.36
Berea Sandstone
2.197
0.1
17.8
0.39
Kayenta Sandstone
2.017
0.15
22.2
0.33
Rock
In addition to the information contained in the above table, it is useful to point out some
extra characteristics of these rocks':
Westerly granite: this is a fine grained equigranular, biotite granite of very uniform
texture and grain size.
Webatuck dolomite: this metamorphic foliated dolomite has a very low pore porosity and
a relatively abundant amount of cracks (confer to the low Poisson's ratio for dry
sample). The matrix structure is susceptible of collapsing at very high pressure, after
all the pores have been closed.
Bedford
limestone:
this
is
a
coarse-grained,
poorly-sorted,
well
cemented
calcarenite,brecciated limestone with fossil debris. The rock contains almost no cracks
and is very stiff. Accordingly, Poisson's ratio is nearly constant along pressure
-11-
variations. The pores have a high aspect ratio.
Weber sandstone: this is an extremely fine grained sandstone and probably should be
called a siltstone; It contains a substancial amount of cracks, more near grain boundary
cracks than all the other sandstones studied here.
It is finely layered and the
cylindrical core sample has been cut with its axis perpendicular to the bedding. This
anisotropy has been the source for underestimating compressional velocity (measured
perpendicular fo'the bedding), and overestimating shear velocity (the shear stresses
and dispacements are measured parralel to the bedding). Henceforth we obtained
negative Poisson's ratio for the drysample at pressures below 250 bars. This non
physical result =was corrected by increasing slightly the values 'of the compressional
velocities in order °tobetter inhterpret the results of attenuations for bulk and Young's
moduli, where Poisson's ratio is introduced in the calculation of these variables.
Navajo sandstone. this is a very clean, well-silicified and isotropic sandstone. The
compacity of the rock is reflected in the Poisson's ratio which is nearly unaffected by
pressure increase.
Berea sandstone: this is a well sorted, fine-grained, submature protoquartzite containing
less than 10% of fine-grained clays. This is the only rock where the core sample has
been cut with its longitudinal axis parralel to-the bedding orientation.
Kayenta sandstone: this is a very friable rock, coarser grained than the Berea, higher in
porosity and not as well cemented. Poisson's ratio varies a lot below 250 bars, in
particular for the dry sample. Therefore, we conclude that the rock may contain a fair
amount of cracks.
-12-
CHAPTER 3 : DETERMNATION OF ATTENUATION COEFFICIENT
A Spectral ratio technique
A.1 Analytical presentation
Attenuation values were first measured using the spectral ratio technique (T,oksoz
et al., 1979).
For plane waves, wave amplitudes can be expressed as:
(1)
ei(2rft -k)
)
Al(f) = G,(f ,) ea(f
and
-a
A 2 (f)
where
A = amplitude, f
= G2 (f,)
2
i(21ft-k
(f)z
2
X)
(2)
e
e
= frequency, z = distance of propagation, k=2rrf/v
wavenumber, v = velocity, G(f,z) is a geometrical factor which includes beam
spreading. Subscripts 1 and 2 refer to the reference and the rock sample respectively.
ac(f) is the attenuation factor and related to Q by
Tr
(3)
Qv
Taking the ratio of amplitudes given by equations (1) and (2) and the natural logarithm
of this ratio gives
In(
A1
A2
)-In(
G
=(
G2
-13-
2
1)X,
(4)
If we assume that a is a linear function of the frequency, i. e. constant Q, then
(5)
a(f )=-f ,
where y is a constant, and
(6)
Q= 7v
We can rewrite (4) as
A,
In( -n(
A2
The term In(A
1/
C1
-)=(72
-Y1)ZX.
(7)
G2
A 2 ) is the ratio of the spectral amplitudes of the reference to the
sample. The term In(G,/G
2)
, may be frequency dependent because of beam
spreading.
F(f )=(Y2--71)Zf,
where F(f) = In(
A1
A2
G1
)-In(
(8)
). Since Q of the standard is very high, /1 can be taken
G2
equal to 0.
For
plane waves the
geometric
factors
G1 and G2
have similar frequency
dependence, In(G1 / G2 ) is independent of frequency.
In practice, a window is selected over the signals, aluminum and sample, and then
Fourier transformed.
The natural log of the amplitude ratios are computed versus
frequency. Attenuation constant, 7, is determined over the frequency interval where
both the sample and reference spectra have adequate power (i.e. when signal to noise
ratio is high). An example of spectra, the ratio, and frequency window over which a least
squares linear fit is made are shown in Figure 2. The data quality is good for all samples.
-14-
A.2 Practical problems
Some of the problems occuring are directly related to the field of digital signal
processing. One of the objectives is to obtain the best resolution in frequency domain,
where the amplitude ratio is to be calculated at each frequency. In order to increase the
sampling density in of the frequency spectra without loosing information we zeropadded the time domain series of point and took a longer FFT (Fast Fourier Transform
algorithm), up to 1024 points where in most cases the minimum was 256 points. In
addition we undersampled the time domain signal by a factor of 10, without approaching
the Nyquist frequency too closely: we operate on signals recorded over 4 to 5
microseconds, we select a rectangular window containing the first 3 half-cycles of the
pulse, we sinus-tape the edges of this window; we now have about than 400 points
over 3 to 4 microseconds, the initial digitization frequency is thus around 100MHz; the
signal does not contain any information above 1.5 MHz, then the Nyquist frequency is
below 3MHz and we see that a 10 to 1 compression of the signal cannot cause any
aliasing problem.
The rectangular window has been chosen rather than a Blackman or a Hamming window
since the signal is relatively noise free and the zero-crossings of the initial waveform
are non ambiguous.
The next problem is to determine over which portion of the amplitude spectra the ratio is
to be calculated and a line to be adjusted to the data in the least-square sense. Ideally
we would like yo take the longest window, where the signal to noise ratio is reasonable
and at the same time get the best fit. The windowing process should thus respect the
folowing conditions
(i) The spectral window should be centered around the common area limited by the two
amplitude peaks (aluminum and rock).
(ii) We should get the best numerical adjustment in order to satisfy to the predicted
-15-
linear theoty.
(iii) Between different pressures and between different saturation conditions for the
same sample of rock, the window determination should be consistent so that the results
are comparable.
One choice was to select this window visually, while respecting the above criteria. This
the time consuming solution. The other solution was to automatize the computer program
completely. The interpolated slope of the fitted line is then taken to be a weighted
average of the slope values from three overlapping windows defined as such:The first
one is between the 50% levels of the peak amplitudes on both sides of the peaks. (This
level values are taken on the average from the two peaks: aluminum and rock sample).
The next two windows are taken respectively between the [20% - 50%] and [50% 20%] level amplitudes astride the common area of the peaks. The weigting factor is
taken to be the correlation factor for the fit on each window. The averaged slope is
normalized by dividing the sum of the weighted slopes by the sum of the weighting
factors.
It is difficult to provide an estimation of the accuracy of the spectral
ratio
measurements. This accuracy depends mainly on the quality of the least square fit
between the spectral amplitude ratio and a straight line. As adressed above, the choice
of a spectral window is essential in this sense. In the absolute, the relative error in
calculating the slope is greater for smaller slope and therefore very low attenuation
values, like those encountered for Webatuck dolomite are to be interpreted with
caution.
Finally, in order to obtain interpretable results, we must must insist on the importance of
the standardization of the method: the experimental conditions should be kept constant
(temperature, source spectrum, sample length, initial signal filtering and recording) as
well as the processing techniques, particularly for the spectral window determination.
-16-
A.3 Comparison with other techniques
Little comparative information is available in the literature about the comparison of
the different methods of attenuation measurements in the ultrasonic field, in terms of
accuracy, reliability and limits of the methods. However, it appears that the spectral
ratio is well suited for laboratory measurements because the signals are relatively free
of noise and the single pulse events easy to identify. Nevertheless, the accuracy
becomes uncertain for high quality factors, since they correspond to very small -slopes
where there becomes to be numerical problems in estimating that slope. Within the limits
of this study, we did not find any Q below 7, but we suspect that for very low Q's, for
instance below 5 the broadening of the pulse.
B Rise time technique
B.1 Analytical presentation
This method is based on the quantification of the change in the shape of the
transient waveform propagating through the rock, modification which is due to anelastic
absorption.
Gladwin and Stacey (1974), have proposed a linear relationship between Q, the quality
factor, and 7, the rise time: -7increases linearly with the distance from the source and
the slope of this linear relationship is inversly proportional to Q. This is valid only if we
assume a constant-Q or nearly constant-Q over the frequency range concerned.
Therefore, they derive the equation, valid for an impulsive source:
-17-
7=
0
+
CT
(9)
T being the travel time from the source.
media.
To is the rise time at the source or before propagation in the studied attenuative
To and C are two source dependant parameters.
In our case, the distance of propagation in the rock medium is is close to 2 inches, the
measured travel.times vary between 8 and 20 microseconds.
Kjartanson's work
(1979),
yields a value for C of .298 for measurements of
acceleration pulse rise times and for Q factors higher than approximately 10 as is
generally, the case in this study. The ceramic transduzers are sensitive to pressure
variations and hence to accelerations in the particle movements.
Note that the value of 70 must be strongly dependent upon the high frequency
component of the initial source impulse since at T=0, there must be a non zero rise time.
In our experimental setting, a titanium plate is interposed between the source and the
rock, thus To is different from 0, as would be the case if the source was perfectly
impulsive and the source was directly in contact with the rock or the titanium had no
relevant attenuation. Furthermore we are interested over obtaining a Q-value relative to
the aluminum standard, for which the same linear relationship is also valid:
T(aluminum)=To(aluminum)+--- (aluminum)
Q
(1 0)
Q is very large, around 150,000, then the the second term in this equation is negligible
in front of the other two terms. We deduce that rT(aluminum)=T(alumninum) at each
working pressure.
T(aluminum) has been measured for the entire range of differential pressures used in
this work, with the same source and apparatus as for the rock samples, therefore we
will assume Tr(Rcck)=7(aluminum) at the same pressLre.
To is also corrected for the small discrepancy existing in terms of sample length
-18-
between aluminum reference and rock core. This correction corresponds at most to 8%
of the initial value of 0The rise time parameter is calculated using two different techniques:
(i) using the definition from Gladwin and Stacey (1974), with -7 being the ratio of the
maximum ordinate of the first peak to the maximum slope in the first quarter cycle of the
recorded pulse.
u max
(du/ dt)max
(11)
(ii) using the definition from Blair and Spathis (1982), with T being the time difference
between 10% and the 90% amplitude levels of the pulse onset.
tau -_
At
(12)
The use of the two definitions yield values of
-that agree within 10% of error for all
'data and within 5% variation in most cases.
Therefore, we shall limit all further
discussion on Q-values to the results obtained by applying Gladwin and Stacey's
definition. As a remark we could state that the common 0Cvalue of .298, that was used
in both cases has been calculated analitically by Kjartanson (1979) for the restricted
case of Gladwin and Stacey's definition for 7-, thus some divergence between our two
series of results is not surprising.
B.2 Practical problems
In order to gain precision on the measurement of the rise time, we adjusted a fifth
degree polynomial to the first period of the waveform. This accounts for the fact that
the discrete nature of the signal ( sampled every 1.0 to 2.0 nanoseconds according to
the rock ) and the short duration of each period ( less than 2 microseconds ) , makes it
dificult to find the exact point of steepest tangent, or the 100%, 90% and 10% peak
-19-
amplitude levels. The point of steepest tangent and the point of maximum amplitude are
then easily determined by finding the roots of the second and first derivatives of the
polynomial.
This method was relatively simpler to implement than the previous one. For the spectral
ratio technique, the sensitive part was the selection of a spectral window, where to
observe a linear adjustment. For the rise time technique, the sensitive part is to
determine the point of maximum slope, under the assumption that the shape of the pulse
is not distorted by the recording system. The signal shape is more faithfully reproduced
on a broad-band'system, where the energy is not delayed by the phase response of the
instrument. But again this is a relative measurement method: The value of 7- considered
is the differential with the aluminum standard.
B.3 Comparison with other methods
We compared the results provided by the spectral ratio and rise time techniques.
The agreement is good provided that we chose the appropriate value of th
constant C
for the latest method. In the next table we give the C-value that provides attenuation
coefficients Q's with an identical mass on the range of pressures considered, for both
rise-time and spectral-ratio techniques. Together with this estimation of C which
practically corresponds to the best coincidence between the two group of values, we
give the deviation for the those two
groups, considering rise-time attenuation
coefficients calculated with the corrected C constant. We observe that rise time
measurements provide attenuations that are higher in the case of S-waves for
Webatuck dolomite. The reliability of this result is uncertain but the measurement is
probably more accurate than for the spectral ratio technique since in this iast case the
incertitude on the measurements of very small slopes (corresponding to very high Q's) is
very large.
-20-
Table 2
Q variation( in %)
C-estimate
S-wave
P-wave
P-wave
S-wave
Weber sandstone
dry
water
benzene
.61
.25
.24
.24
.15
.18
23
6
6
24
12
15
Navajo sandstone
dry
benzene
.61
.33
.41
.21
10
5
11
17
,Kayenta sandstone
dry
water
benzene
.42
.. 31
.26
.25
.19
.19
4
6
9
12
4
10
Berea sandstone
dry
water
benzene
.70
.33
.32
.3.
.34
.19
.20
10
11
21
28
12
15
Bedford limestone
dry
water
benzene
.41
.34
.41
.20 ..
.18
.20
2
6
2
8
7
8
Webatuck dolomite
dry
benzene
.75
.64
70
33
184
242
Westerly granite
dry
water
benzene
.50
.45
.39
8
3
5
4
9
22
1.78
1.56
i
.27
.23
.23
We will discard the results from Webatuck dolomite for the calculation of an average
value of C, because it is difficult to normalize the rise time Q-values to the very high Qvalues obtained by spectral ratio analysis.In this respect, the rise time technique seems
to give more plausible results than the spectral ratio technique for high quality factors
(above 200).
-21-
From this values, we conclude that the average C-value is .40 for P-waves, and .24 for
S-waves. Those two values are in a 5 to 3 ratio with each other. This result is purely
informative and has little physical significance. It is more interesting to point out that
the estimated C-values are higher for P than for S-waves, and higher for dry than for
saturated rocks. From Kjartanson (1979), we observe that the value we considered for
C is based on a Nearly-Constant-Q assumption (NCQ), and accordingly it appears that C
calculated is independant of Q for any Q greater than 20. In conclusion, normalizing
rise-time to spectral ratio values is not necessary legitimate. However, if we take the
spectral ratio calculations as a reference, it appears that the divergence of C from the
theoretical value of .298 could be explained by the non rigorous validity of the NCQ
hypothesis.
It is interesting to note that the beam spreading effect due to a non strictly plane
wave, as well as the scattering effects should not affect the rise time measurements
since they do not affect the pulse shape or its time duration. Finally, the rise-time
method has the clear advantage over spectral methods that the measurement may be
done on a clearly defined phase of the waveform (Gladwin and Stacey,1 974).
C Sources of error
C.1 Spherical spreading
In this experiment the transducer diameter is 2.5 cm (1 inch), and the sample
diameter is 7.62 cm. As a result the energy beam spreads and this spreading is
dependent on wavelength. Seki et al., (1956) and Papadakis (1975) studied the beam
spreading problem for a cylindrical source. They found that amplitude decrease due to
-22-
beam spreading is about 1 db per
a 2 /X,
where a = source diameter and X =
wavelength.
Calulation of the diffraction correction: Due to the short wavelengths,it is required that
the opposite faces of the sample be flat and parrallel. Truell et al., 1969, have shown
that the limit to the deviation from parralel faces depends on Q-
1
for the studied
material and on the frequency. We assume that both source and receiver transducers
are coaxial, have the same circular shape and same dimensions, that the source
behaves pistonlike, uniformly and harmonically, that the electrical response of the
receiver is proportional to the average pressure over the surface of the transducer at a
given time. Diffraction occurs when the product ka gets large, i.e., the wavelength is
much shorter than the radius of the source (the source therefore has a radiation
pattern, the displacements are dependant on the angle of emergence).
2rr
If k is the wavenumber, a is the source diameter; in our case k = 2=5 mm for vp=5.0
km/sec and f=1 Mhz, a=25 mm, thus ka=1 0 ir=31.4, the transducer diameter is about 3
to 4 times the wavelength.
The diffraction correction could be made for each data point in the term In(G, / G2), but
since this correction is nonmonotonic in nature, it is contrary to our linearity assumption.
Therefore this correction is to be made after Q is calculated.
We will use the following table given by Seki et al.
-23-
Table 3
I
S
d (dB)
S
d (dB)
0.2
0.868
1.8
2.600
0.3
1.070
2.0
2.478
0.4
1.198
2.2
2.386
0.5
1.342
2.4
2.358
0.6
1.554
2.6
2.404
0.7
1.548
2.8
2.536
0.8
1.776
3.0
2.708
0.9
1.910
3.2
2.904
1.0
1.848
3.4
3.118
1.1
1.862
3.6
3.364
1.2
2.014
3.8
3.607
1.4
2.458
j
4.0
3.862
2.664
_i _
1.6
___
_
In this table, S is the distance between source and receiver measured in units of (
22
),
a is the diameter of the transducers (2.5 cms in our case), d is the drop in amplitude
measured in dB.
This loss, as can be readily seen from the values is not linear. Indeed the curve, 'd as a
function of S, although overall approximately monotonic (tends to be exponential),
presents local minima, at S= 0.73, 1.05, 2.4.
at a given frequency:
d, being the drop in amplitude in dB for the rock sample,
-24-
d 2 being the drop in amplitude in dB for the aluminum reference. Therefore d, and d 2
are negative quantities.
d2 -d
=20log(
d 2 -d
1
A2
A,
)
-)-201og(
A2
A
=20log(
A 2 A,
A
A 1.
(14)
)
A2
A 2 A2
then 10
, A,Al
In(-
(13)
(13)
20
(15)
2
A
A 2 . ..
)+ln(10
)=n(
A
d2-d,1
20
)
..
. .(16)
(16)
At a given frequency, we get the correction to apply to the natural logarithm of the
spectral ratio. We can calculate this term at both extremities of the spectral window
where the linear interpolation is made. This correction comes in terms of adding or
substracting a normalized value to ordinate for the interpolate line at both ends oi the
window, an addition occuring when the correction due to diffraction is larger for the
aluminum and on the other hand, a substraction is made when diffraction effect is
prominent for the rock sample. Finally, the slope is recalculated from:
Slopecarrected=slopeinzt
Az max -Az min
+ A
(1 7)
(17)
with
A2
Azmin=Aln( A,
a t fmin
(1 8)
and
Azmax =AIn(
A2
)
(19)
at max
nsthe
frequency
span
correspondng to the spectral wndow. Then we
Af =f max -f min is the frequency span corresponding to the spectral window. Then we
-25-
can recalculate Q from:
Q=
Tz
v slope
(20)
We deduce:
Qcorrected
r
Tz
+v slope
(21)
We chose to make our corrections over the standard frequency range, on a window
centered around .8 MHz and with 1 MHz width. It is noted that the titanium plate h'o!ders
introduce by themselves a drop in amplitude of 2 dB (20,6%) and 1.55 dB (16.3%) for
P-waves respectively at 0.3 and 1.3 Mhz, and .9 dB (9,8%) and .8 dB (8.8%) for Swaves at the same frequencies; the velocities for this metal being 6.07 km/sec fo
compressicnal waves and 3.12 km/sec for shear waves, the total interposed thickness
being 3.6 cms. However, we do not correct for this contribution to the total spreading
since it does not introduce a differential effect between the aluminum reference and
the rock sample.
The correction should be greater for samples with slower velocities such as porous
sandstones, since the difference in velocities with aluminum is greater. Indeed, we
found a correction smaller than 2% in most cases, in the range of pressures considered,
except fci Berea, Navajo and Weber sandstones, in laboratory dry state where the the
relative contribution of beam dispersion to P-wave attenuation reaches 10 to 17%
decrease of the Q-value at the higher pressures. For the fastest rocks considered,
Westerly granite and Webatuck dolomite, the correction corresponds to a Q-increase
due to velocities relatively higher than for the aluminum reference (for the aluminum,
Vp=6.42km/sec, vS=3.04km/sec ) in the case of the dolomite and due to velocities
comparable with aluminum for the granite with in addition a sligthly longer trave! path
(increased by 2.1%).
-26-
C.2 Sidewall reflections and other geometric artifacts
The geometry of the samples (2" long by 3" large), was designed to avoid sidewall
refections, even if we take the beam spreading effect into account. The samples are
also considered as homogeneous and isotropic and thereby no inside reflection occuring
on occur on a bedding interface, should take place. Nor any major crack was detected
in those samples. Furthermore the time domain signals as well as their frequency spectra
do not show any apparent superimposed event.
C.3 Scattering by grains
In this study, rocks with larger grain size are the Westerly granite and the Bedford
limestone. Both contain grains with an average size of 0.75 millimeters. When we
compare the wavelength of the pulses (dominent wavelength, X = 6mm for P-waves in
granite) with the size of the grains, we find that the rayleigh condiion: ka <0.1 ( k being
the real wavenumber and a the size of the grains ), is not met in the strict sense. Thus
there may be some contribution due to scattering. Another rule of the thumb is that
Xa <3, X being the wavelength. This latter condition is generally satisfied. Based on
these, we can say that scattering may be present but its effect is probably small. We
consider this effect to be negligible.
C.4 Effects of noise
The recorded signals are represented in Figures 2 to 4 for Kayenta sandstone and
Westerly granite. It can be readily seen that they are at least visually free of noise.
The same conclusion can be drawn when looking at the shape of the spectra. Some
-27-
additional information seems to come only at frequencies below 100 or 200 kilohertz or
above 2 megahertz. However, we observe some variation in the results according to
how many arches ( half-cycles ), we process into the Fourier transform for the spectral
ratio method, or on how many of these arches we try to fit a fifth degree polynomial in
the case of the rise time method. We cannot assert wether the difficulty to find a
perfectly linear portion on the spectral ratio, on the larger possible section of the
spectrum peak, is due to numerical noise or to noise attributable to the recording
instruments or is inherent to the technique. To give an idea of the quality of the
spectral ratio fit, we get on the average a correlation coefficient of .99 for the fit in the
window corresponding to the 50% -50% levels of the peak amplitude, which corresponds
to about 50 points scattered on a range covering 0.8 megahertz.
-28-
CHAPTER 4 : RESULTS AND INTERPRETATION
A. Spectral ratio attenuation - results
A brief lithological description of the samples studied is given in chapter 2-C. The
properties of the saturating fluids are given in Table 4, chapter 4.
In order to separate the P-wave attenuation into bulk and shear attenuation
(Spencer, 1979) we calculated QE and QK using observed Qp, QS, and Poisson's ratio v
from the velocity data. K and Q refer respectively to bulk and Young's moduli.
(1 +V)
(1-v)(1-2v) 2(2-v)
=
Qp
K
3(1 -v)
(22)
QS
(1+v)
(23)
2(1 -2 v)
QP
QS
with
V=
2(24)
2(v_2-v)
At first
1000
QP
1000
QS
we present
1000
QK 0
1000
Q
all the main
versus
features observed on the
differential
pressure
displays
(difference
of
between
confining and pore pressure). We consider the effects of pressure, fluid saturation
versus dry , benzene versus water saturation , shear versus compressional contribution
towards absorption.
-29-
Weber sandstone
As for all sandstones studied, pressure dependence is greatest for benzene saturated
attenuations and smallest for dry attenuation. Here, dry S-attenuation is almost
independant of pressure, while dry K-attenuation is strongly dependant on pressure
variations.
At highest pressures, saturation affects more shear than bulk or compressional
attenuation: at 1000 bars, saturated'shear and Young's attenuations are twice the dry
attenuations, while compressional and bulk attenuation are less affected.
Benzene attenuation is greater than water attenuation for all Q variables. The
divergence between benzene and water decreases to almost zero towards the highest
pressures (over 800 bars). At 800 bars, we observe a cross-over for S-attenuation,
above this pressure benzene S-attenuation becomes sligthly smaller than water Sattenuation.
This
last
observation
is
made
almost
within
error limits
for
our
measurements. If we compare bulk attenuation to P- and S-attenuation, the difference
between water and benzene is enhanced. In addition water attenuation unexpectedly
falls lower than dry attenuation. Except for the water-dry reversal for bulk attenuation,
all other attenuation coefficients are greater for benzene and lower for dry and the
difference is quite important at the low pressures.
P and S-wave attenuations are similar for benzene saturation, S-attenuation is greater
than P-attenuation for water saturation (the difference becomes negligible at high
pressures) and P-attenuation is greater than S-attenuation for the dry case. The same
observations can be made between K- and E-attenuations.
.Finally, we notice that in the dry case, there is the greatest difference between K- and
S-attenuation: at all pressures, dry K-attenuation is more than twice dry S-attenuation.
Navalo sandstone
-30-
A single fluid saturation was measured, using benzene as fluid saturant. This rock has
the lowest attenuation of all our sanastones. For all type of attenuations, 1000 Q-!
values for benzene are well above dry values. At 1000 bars, benzene S-attenuation is
three times
dry S-attenuation and benzene P-attenuation is two times
dry P-
attenuation. Pressure dependence, at least for the lower pressures (P < 250 bars), is
greater for dry P- and K-attenuation than for dry S-attenuation. For dry S-attenuation,
the pressure dependence is almost linear.
All P-attenuations are greater than S-attenuations. (The difference becomes negligible
for benzene saturation when pressure is close to the lowest values). The same results
are also valid when we compare K and E-attenuations.
Berea sandstone
The first observation we can make is that for all measurements of attenuation, dry
values are significantly lower than saturated values (even at the higher pressures).
Pressure dependence is greater for saturated values. The greaterdDry attenuation
pressure dependence can be observed for S-attenuation, the least for K-attenuation
(almost constant values at all pressures in this last case) and intermediate values are
found for P-attenuation.
Benzene and water attenuations are comparable. The deviation is nearly within error
limits but, nevertheless we are able to observe a cross-over feature occuring at 250
bars for QsandQE and 450 bars for QpandQK: at lower pressures, benzene saturation is
greater and at higher pressures, water saturation is greater. This cross-over is less
obvious for P and even more for K-attenuations than for S- and E-attenuations.
S-attenuation is greater than P-attenuation for saturated data. The same is true for dry
data below 500 bars. Above this threshold, dry P and S values are identical. The same
comparison can be made for K- and E-attenuations respectively.
-31-
Kayenta sandstone
The pressure dependence is remarquably similar for all different Q's calculated. The
data curves are parrallel to each other. As for other sandstones, benzene saturation
has the greatest pressure dependence and dry the lowest one. As for Berea sandstone,
saturated attenuations are well above dry attenuations at all pressures. The increase in
attenuation due to fluid saturation is proportionally larger for S-waves than fcr Pwaves, especially at high pressures.
Benzene saturated attenuations are greater than water saturated attenuations. The
difference is -more noticeable for P and K-attenuations rather than for S and Eattenuations. In these last cases, the benzene-water deviation disappears above 300
bars, but we do not observe a real cross-over as for other sandstones.
In the case of K-attenuation, water saturated and dry attenuations are nearly identical,
dry attenuatiQn.tends to be greater than water saturated attenuation at low pressures.
This observation is to be associated with the inversion in K-attenuations between dry
and water saturated case for Weber sandstone.
Dry P-attenuation is slightly higher than dry S-attenuation. On the opposite, water
saturated P-attenuation is lower than water saturated
S-attenuation.
Benzene
saturated attenuations are identical for P and S-waves. The same observations can be
made between K and E-attenuations.
,-
Bedford limestone
First, we can observe a very similar and almost linear pressure dependence for all
saturations and for all attenuation parameters., Then we can point out the small
difference between water and benzene saturations.
Saturated attenuations are always greater than dry attenuation. The presence of fluids
increases mostly shear attenuation . Benzene saturated values are greater than water
-32-
saturated values for P and K-attenuations. As for Weber sandstone and to a lesser
extent for Kayenta sandstone, dry K-attenuation is greater than water K-attenuation.
For S and E-attenuations, a cross-over can be observed: Benzene is more attenuative
than water below 200 bars and conversly at pressures above 200 bars.
Attenuation is significantly greater for P than for S-waves at all saturations. Identically,
K-attenuation is greater than E-attenuation.
Webatuck dolomite
The only fluid saturant used for measurements was benzene. The first observation we
can make is that dry P and K-attenuations are much greater than corresponding benzene
saturated attenuations, particularly at pressures below 200 bars. Above 300 bars,
benzene P and K-attenuations become slightly higher than dry P and K-attenuations. For
S and E-attenuations, the result is opposite below 250 bars. Above 400 bars the
values are so low (Q is of the order of 1000),that we cannot compare dry and
saturated values accurately.
The effect of pressure variations is negligible above 400 bars.
For the dry- measurements we observe that the pressure dependence is greater for Pattenuation
than. for
S-attenuation and
that P-attenuation
is greater than
S-
attenuation; identically, K-attenuation is greater than E-attenuation. For the benzene
saturated measurements, the pressure dependence has an opposite behavior: it affects
more S and E-attenuations than P and K-attenuations. Furthermore, S-attenuation is
greater than P-attenuation below 200 bars, and conversly above 200 bars. The same
result is observable for E and K-attenuation.
Westerly granite
The pressure dependence is greater for S,E than P,K-attenuations for saturated
measurements,
greater for P,K than for S,E-attenuations for dry measurements.
Saturated attenuations are higher than dry attenuations for S and E measurements,
lower than dry attenuation for P and K measurements below 175 and 400 bars
respectively, -greater than dry
P and K-attenuations
above 200 and 400 bars
respectively.
A surprising behavior can be observed for dry K-attenuations where the attenuation is
increasing with pressure.
Similarly to Weber sandstone and to a certain extent to Kayenta sandstone and Bedford
limestone, we observe a dry K-attenuation greater than water and benzene saturated
K-attenuation, at for pressures lower than 400 bars. Above 400bars, dry K-attenuation
becomes slightly lower than water and benzene attenuations.
1000/ Qp never gets below 20 for dry measurements when it is well under 5 for
Webatuck dolomite. P-attenuations present no difference between water and benzene
saturations.
S an E-attenuations exhibit a difference between water and benzene
saturations that decreases with increasing pressure, this deviation becomes negligible
above 500 bars. For dry data water saturated measurements, P and K-attenuations are
higher than S and E-attenuations respectively. For benzene saturated measurements,
the same result is true above 350 bars .
B. Effects of fluid saturants
Many authors agree with the fact that frictional grain sliding alone does not
account significantly in the mechanism for stress wave attenuation in saturated porous
rocks. A major argument for this assertion is that frictional attenuation being non linear
in strain, we will not observe a Q dependence on strain for strains lower than 10
-34-
Therefore pore fluids control attenuation even in laboratory-dry rocks. The question
remains whereas what are the particular interactions of fluids pore or crack surfaces
which contribute to the stress wave energy dissipation.
Tittmann et al. (1980),
suggest that at the molecular level the energy loss can be ascribed to stress-induced
diffusion of absorbed polar fluid, the energy being dissipated in the breakage of
hydrogen bonds between the surface hydroxyls and the fluid molecules, i.e. the silanol
and the water molecules in the case of silicates. The removal of water molecules from
the surface of grains increases surface free energy and thereby stiffens the mineral at
the grain contacts. Murphy (1982) regroups under the qualification of micro capillary
hysteresis the phenomenons of bond breaking at the grains surfaces or the viscous
dissipation as a fluid fllm moves over surfaces asperities in the case of incomplete
saturation. These mechanisms take place mostly for relatively low partial pressures of
fluid as is probably the case of our laboratory-dry rocks although the experimental
zero-pore, pressure is as low as 20u of Hg and the dry samples have been thoroughly
dried during 24 hours in a vacuum at 800 C.
At significant saturations (we insured total saturation as much as possible, by applying
a hydrostatic pore-pressure of 100 bars), Murphy (1982) tells us that frequency
dependant flow mechanisms are probably predominant. Biot (1956)
suggested an
oscillation of the pore walls in their plane, inducing fluid, diffuse vorticity and thereby
energy dissipation, but this mechanism which predicts a strong frequency dependance
centered around 104 -105 Hz and QP<Qs is certainly quantitatively minor compared to
the fluid squirt or squish mecanism: the dissipative fluid flow is generated by a local
pressure field from the compression of grain contacts and fine capillaries. Since cracks
have different orientations to the passing wave or different aspect ratios, the squirt
ocurs when those cracks undergo a differential compression. This mechanism shculd
cause more shear than compressional attenuation (Winkler, 1979). Accordingly, Winkler
-35-
finds fully saturated bulk attenuations greater than dry bulk attenuations. At full
saturation, the bulk compression losses are smaller than the shearing losses, because
shear stress provides stronger gradients between adjacent contacts or capillaries with
different orientations and aspect ratios while longitudinally the compressibility of pores
and thus the pore pressure gradient is small. Hence this mechanism explains QS<Qp.
Bulau et al. (1984), confirm the existence of a viscous dissipation mecanism with a
characteristic relaxation time proportional to pore fluid viscosity. The utilization of a
non-polar saturating liquid, iso-octane yields significantly lower attenuation, showing
that near-surface processes as surface-adsorbed molecular effects are non negligible.
They measured attenuation over a wide range of frequencies and fluid viscosities, for
Coconino sandstone which is similar to Navajo sandstone. A graphic representation is
given of log[1000/ Q] as a function of log[f I].The attenuation corresponding to our
frequency range (centered around 0.8 Mhz) and to the viscosities for the pore fluids we
consider, water (7=1.0021 0- 2 Poises) and benzene (r7=.60210-2Poises) are found to
be respectively 23 and 28, at an effective pressure of 1000 psi (about 70 bars). Our
results yield a 1000/Q value of 48 for benzene saturated Navajo sandstone at 75 bars.
Therefore attenuation for benzene saturated Coconino sandstone appears to be greater
than for the same sandstone water saturated.
We do not have water saturated
measurements for Navajo sandstone, but the results for the other sandstones also show
a greater attenution for benzene fluid than for water saturating fluid at the same
effective pressure. A water-glycerol mixture was used as pore fluid saturant on the
Coconino sandstone. A viscous dissipation mechanism with a characteristic relaxation
time was used with a good agreement to describe the results. Nevertheless, with a
non-polar fluid (iso-octane), the attenuation was found to be significantly lower than
values predicted from simple frequency-viscosity scaling, as is the case for waterglycerol system. This was explained by the presence of near surface processes
-36-
attenuation mechanism which also depends on partial-saturation (Mavko and Nur, 1979)
may be present. This may explain the higher P-wave attenuation but not the higher Swave attenuation, which has not been observed in partially-saturated rock at ultrasonic
frequencies (Spencer, 1979; Johnston, 1978). It is clear that no single mechanism alone
could explain our results, we may have to consider various contributions to attenuation,
including in the case of benzene, partial saturation and presence of water in thin cracks
and pores and possibly as a monolayer coating on silicates (particularly clay minerals).
The interpretation of our data raises a number of questions that need to be addressed.
(1)
Coyner,
1984, pointed out that the shear-modulus in the water saturated
sandstones fell below the benzene-saturated values at the higher pressures. The
water-saturated bulk modulus
is always larger than the benzene-saturated bulk
modulus. This was explained as a water "softening" of the water-saturated shear
modulus which was not apparent in the water-saturated bulk-modulus. This "crossover" feature is also observed for our attenuation data. It is prominent for Berea
sandstone and Bedford limestone. The observation diverges with Coyner's remark in the
sense that the cross-over is also slightly visible for P- and K-attenuations, although it is
less visible than for S- and E-attenuations and it occurs at a higher effective pressure
for P- and K- than for S- and E-attenuations. The same cross-over appears for the
velocity data: water saturated velocities are higher than benzene saturated velocities
at lower pressures, and this difference is reversed above a threshold pressure which is
about the same
as the cross-over pressure for shear-moduli
and
attenuation
measurements. As for attenuation, a cross-over is also abserved for P-wave velocity
measurements
although the phenomenon has
a smaller
amplitude and at higher
pressures.
(2)
The comparison of the attenuations for all sandstones shows that benzene
saturated attenuation is higher than water-saturated attenuation in the case of
-38-
Kayenta and Weber sandstones, while for Berea'sandstone as well as for Bedford
limestone water- and benzene-saturated attenuation measurements yield approximately
the same values.
One explanation could be that both Weber and Kayenta sandstones contain a fair
amount of clays (about 15%) compared to Berea (about 5%) or Navajo sandstones (less
than 1%) and Bedford limestone has no clay content. The effect of clay content on
attenuation may be that clay-water interaction could result in a stiffening of the rock
structure (swelling of hydratated clays) while benzene non polar molecules do not wet
the clay surfaces. Another explanation is that Weber and Kayenta sandstones have
much more cracks than Berea sandstone or bedford limestone: this can be seen by
looking at the volume of microstrain on static stess-strain curves.
The microstrain
volume is not an exact measurement of an hypothetical cracks volume, it reflects more
the reduction and flattening of asperities at grain contacts while pressure is increasing.
-6
Nonetheless, the volume of microstrain for the sandstones is approximately 6000 10
for Weber, 5200 10-6for Kayenta, 3500 10-6for Berea, 700 1 O-for Navajo and less
than 100 10-6for Bedford limestone. Benzene, due to its lower viscosity induces more
attenuation than water particularly at low pressures, when the porosity due to cracks is
important as is seen for Weber and Kayenta sandstones (although Weber and Kayenta
sandstones have very different average grain size: 0.15 mm for Kayenta and 0.05 mm
for Weber sandstone, which means that the difference in terms of permeability and
hydraulic radius must be greater than just in terms of porosity, a for Kayenta sandstone
is 22.2%, 4 for Weber sandstone is 9.5%).
(3) We also observe that fluid saturated P- and K-attenuations are less than dry P- and
K-attenuations for Webatuck dolomite and Westerly granite, at lower pressures. This
effect may be due to the low porosities of these rocks, respectively 0.5 and 0.8%, and
consecutively their permeability must be small.
-39-
Henceforth, the effect of fluid
saturation could be a crack healing effect. The elastic coupling between grains across
cracks may be improved as well as the wave scattering effect may be reduced by the
presence of fluid.
As a matter of fact, this affects only compressional and bulk
attenuartions, for which the propagated wave is most likely to close cracks.
(4) We finally address the possibility of an eventual partial saturation in the case of
clay rich sandstones, as an explanation
for the higher attenuations observed for
benzene saturation than for water-saturation as is seen for Kayenta and Weber
sandstones. It is unlikely that partial saturation is responsible for this observation since
S-wave attenuation is affected when it should not be as much (Frisillo and Stewart,
1980). In adition, the 100
bars pore pressure maintained along the saturated
measurements should insure a complete saturation.
C. Comparison with other ultrasonic determinations.
If we compare our results for Berea sandstone with other authors, we observe: Jones
and Nur, 1983, found at 220 C, fluid saturant 7=.95cpoises , 1000/ Q varies from 45 at
25 bars to 18 at 200 bars, for 100 and 80 at the same pressures in our measurements.
Winkler, 1983, measured attenuation on a Berea sample with 20.3% porosity (our sample
had 17.8% porosity), 2.65 density (2.20 for us), 150-200 A average grain size (100 p.
for us), and found brine saturated 1000/ Q values varying from 85 at 25 bars to 10 at
400 bars, compared to 95 and 40 at the same pressures for our data. Dry attenuations
are found to be much lower (less than 20), but no numerical value is given in the
frequency range where we performed our measurements.
Tittman et al. , 1981, measured 1000/ QE at a central frequency of 7 Khz for Berea
sandstone and found, in dry conditions, 12 at 25 bars and 5 at 700 bars (against 35
and 13 for us) and under saturation 110 at 25 bars and 30 at 700 bars (against 45
-40-
and 30 for us). They conclude that attenuation caused by fluid flow is strongly
frequency dependent and increases with frequency. Thus the discrepancy with our
results could be explained by the much higher frequency we have been using. We can
point out that, aside from the shift in absolute value between their measurements and
ours, the shape and slopes of the.curves (attenuation versus pressure) are very similar.
At last, we compare our results with values from Spencer, 1979. Spencer computed
1000/ QKandl 000/ QE from the P- and S-attenuations obtained by Johnston, 1979,
measured on Berea sandstone. The results are as follows: dry P-attenuation is sligthly
lower than in our case, brine saturated P-attenuation is higher. S-attenuation is identical
to our results for both dry and brine saturated measurements.
The results differ
significantly when it comes to K-attenuation. Adversely from our results, Spencer finds
saturated K-attenuations greater than dry K-attenuations. Spencer proposes the
following explanations for the low (even negative) dry K-attenuations he obtains: he
suggests that the drying process might have increased the defects in the adhesion
between grains or that the fluid saturated core is not perfectly elastic: fluid flow
increases S-attenuation but not P-attenuation as much.
We can also compare our results for Navajo with Spencer, 1981. E-attenuation
(1000/ QE) is found to be equal to 150 for dry and 130 for water saturated sample.
Their conclusion is that a small amount of water can reduce the stiffness as much as full
saturation: water molecules by bonding to the surface, reduce the surface free energy
and thus generate frequency dependent softening in rocks.
We notice that the
reduction of water-saturated shear modulus so apparent in the velocity data for the
sandstones is not strongly reflected in the shear attenuation. The high-pressure low Pand K-attenuations in benzene saturated Webatuck dolomite may reflect unrelaxed
stresses in the fluid at ultrasonic frequencies (O'Connell and Budiansky, 1977; Murphy,
1982). Crack geometry and surface interactions between water and matrix minerals
-41-
(silica, feldspar and clay versus carbonates) apparently play an important role.
-42-
Table 4: Comparison of water and-benzene properties
unit
benzene
water
density
g/cc
.87
.99
velocity
km/sec
1.295
1.497
viscosity(1)
cpoises
.602
1.002
compressibility(2)
Mbar-1
96.70
45.24
bulk modulus
Kbar
12.1
22.3
(1) measured at 100 bar, 20 0 C
(2) measured at 1 bar,25 0 C
(3) measured at 100 bars, 20 0 C
All other parameters are at conditions of temperature
(20 C) and pressure (1 bar).
-43-
CHAPTER 5 : CONCLUSIONS
Our measurements of ultrasonic attenuation for several rocks as a function of
pressure for vacuum dry, water and benzene saturated conditions confirm previous
observations while raising questions concerning water versus non-aqueous pore fluid
saturants. - Our observations show that for the Kayenta, Berea, Weber and Navajo
sandstones, benzene-saturated attenuation, in particular for P-wave, is larger than
water-saturated attenuation. For the S-wave the same relationship-holds although the
differences are less. In addition, as pressure increases the differences decrease.
Crack geometry and surface interactions between water and matrix minerals (quartz,
feldspar, clay, and .carbonates)
may play an important role in how fluids affect
attenuation. The fluid properties are also essential: viscosity is a determinant factor for
the fluid flow and fluid squirt mechanisms, molecular electrical properties may have a
great contribution to attenuation relatively to the wetting capacity, grain surface
softening and full or partial saturation of cracks and pores. In this respect, even under
petroleum saturation, we can suspect that some residual water occupying large aspect
ratio pores and cracks and adsorbed on minerals, still plays a major role towards
attenuation.
-44-
References
Biot, M.A., 1956a, Theory of elastic waves in a fluid-saturated porous solid. I., J. Acoust.
Soc. Am., 28, 168-178.
Biot, M.A., 1956b, Theory of elastic waves in a fluid-saturated porous solid. II. J. Acoust.
Soc. Am., 28, 179-191.
Blair, D.P., 1982, Measurement of rise times of seismic pulses in rock, Geophysics, 47,
1047-1058
Blair, D.P., Spathis, A.T., 1982, Attenuation of explosion-generated pulse in rock masses,
J. Geophys. Res., 87, 3885-3892
Blair, D.P., 1984, Rise time of attenuated seismic pulses detected in both empty arid
fluid-filled cylindrical boreholes, Geophysics, 49, 398-410
Blair, D.P., Spathis, A.T., 1984, Seismic source influence in pulse attenuation studie3, J.
Geophys. Res., 89, 9253-9258
Born, W.T., Owen, J.E., 1935, Effect of moisture upon velocity of elastic waves in
Amherst sandstone, Bul. Am. Assoc. Petroleum Geol., 19, 9-18
Budiansky, B., Sumner, E.E.Jr., O'Connell, R.J., Bulk thermoelastic attenuation
composite materials, J. Geophys. Res., 88, 10343-10348
of
Bulau, J.R., Tittmann, B.R., and Abdel-Gawad, M., 1984, Modulus and attenuation in
sandstone with hydrocarbon and aqueous pore fluids: Presented at the 54th Annua!
SEG Meeting, Atlanta.
Christensen, N.I., Wang, H.F., 1985, The influence of pore pressure and confining
pressure on dynamic elastic properties of Berea sandstone, Geophysics. 50, 207213
Clark, V.A., Spencer, T.W., Tittmann, B.R., 1981, The effect of thermal cycling on the
seismic quality factor Q of some sedimentary rocks: J. Geophys. Res., 86, 70877094
Clark, V.A., Spencer, T.W., Tittmann, B.R., Ahlberg, L.A., and Coombe, L.T., 1980, Effect of
volatiles on attenuation and velocity in sedimentary rocks: J. Geophys. Res., 85,
5190-5198.
Coyner, K.B., 1984, Effects of stress, pore pressure, and pore fluids on bulk strain,
velocity, and permeability in rocks: Ph.D. thesis, M.I.T.
Coyner, K.B., and Cheng, C.H., 1984, New laboratory measurements of seismic velocities
in porous rocks: Presented at the 54th Annual SEG Meeting, Atlanta.
Devaney, A.J., Levine, H., Piona, T., 1982, Attenuation due to scattering of ultrasonic
compressional waves in granular media: ch.8 in Elastic wave scattering and
propagation by Varadan V.K. and V.V., Ann Harbor Sc.
-45-
Elsley, R.K., Tittmann, B.R., Nadler, H.L., Ahlberg, L.A., 1977, Defect characterization by
ultrasonic signal processing techniques, Proc. Ultrasonics Symp., IEEE ,48-52, 1977
Frisillo, A.L., Stewart, T.J., 1980, Effect of partial gas/brine saturation on ultrasonic
absorption in sandstone, J. Geophys. Res., 85, 5209-5211
Gladwin, M.T., Stacey, F.D., 1974, Anelastic degradation of acoustic pulses in rock,
Physics of the Earth and Planetary Interiors, 8, 332-336
Johnston, D.H., and Toksiz, M.N., 1980, Ultrasonic P and S wave attenuation in dry and
saturated rocks under pressure: J. Geophys. Res., 85, 925-936.
Jones, T., and , Nur, A., 1983, Velocity and attenuation in sandstone at elevated
temperatures and pressures, Geophys. Res. Letters, 10, 140-1 43
Kjartanson E., 1979, Constant Q-wave propagation and attenuation, J. Geophys. Res.,
84, 473 7-474
Mason, W.P., Beshers, D.N., Kuo, J.T., 1970, Internal friction in westerly granite: relation
to dislocation theory, J. Applied Phys., 41, 5206-5209
Mavko, G.M., and Nur, A., 1979, Wave attenuation in partially saturated rocks:
Geophysics, 44, 161-174.
Murphy, W.F., III, 1982,. Effects of microstructure and pore fluids on the acoustic
properties of granular sedimentary materials: Ph.D. thesis, Stanford Univ.
Murphy, W.F., III, 1982, Effects of partial saturation on attenuation in Massillon
sandstone and Vycor porous glass, J. Acoust. Soc. Am., 71, 1458-1468
Murphy, W.F., III, 1984, Acoustic measures of partial gas saturation in tight sandstones,
J. Geophys. Res., 89, 11549-11559
O'Connell, R.J., and Budiansky, B., 1977, Viscoelastic properties of fluid- saturated
cracked solids: J. Geophys. Res.. 82, 5719-5735.
O'Doherty, R.F., Anstey, N.A., 1971, Reflections on amplitudes, Geoph. Pros., 19, 430458
Pandit, B.I., and King, M.S., 1979, The variation of elastic wave velocities and quality
factor of a sandstone with moisture content: Can. J. Earth Sci., 16, 2187-2195.
Papadakis, E.P., 1975, Ultrasonic diffraction from single apertures with application to
pulse measurements and crystal physics: in Physical Acoustics, vol. 11, Academic
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Rafavich, F., Kendall, C.H.St.C., Todd, T.P., 1984, The relationship between acoustic
properties and the petrographic character of carbonate rocks, Geophysics, 49,
1622-1636
Savage, J.C. and Hasegawa, H.S., 1967, Evidence for a linear attenuation mechanism:
Geophysics, 6, 1003-1 01 4
-46-
Sears, F.M., Bonner, B.P., 1981, Ultrasonic attenuation measurement by spectral ratios
utilizing signal processing techniques, IEEE Trans. Geoscience and Remote Sensing,
GE-19, No 2
Seki, H., Granato, A., Truell, R., 1956, Diffraction effects in the ultrasonic field of a
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Acoust. Soc. Am., 28-2, 230-238
Spencer, J.W., 1979, Bulk and shear attenuation in Berea sandstone: the effects of pore
fluids, J. Geophys. Res.; 84, 7521-7523.
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; 76, 793-805.
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reprint series, No. 2.
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sliding, Geophysics, 47, 1-15 rocks, Geophys. Res. Lett., 6, 1-4, 1979.
Winkler, K.W., and Nur, A., 1979, Pore fluids and seismic attenuation in rocks, Geophys.
Res. Lett., 6, 1-4, 1979.
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attenuation spectra in rocks under pressure: J. Geophys. Res., 87, 10776-10780.
Winkler, K.W., 1983, Frequency dependant ultrasonic properties of high-porosity
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cylindrical bar: J. Acoust. Soc. Am., 33, 1283-1288
-47-
FIGURE CAPTIONS
Figure 1: Experimental apparatus used to measure the velocities and record the
waveforms for the aluminum standard and the rock samples, with variable confining
pressure and pore pressure.
The next four figures represent examples of recorded waveforms.
Figure 2: P- and S-wave signals transmitted through Kayenta sandstone at 100 bars
net confining pressure for dry, benzene and water-saturations.
Figure 3: P- and S-wave signals transmitted through Kayenta sandstone at 500 bars
net confining pressure for dry, benzene and water-saturations.
Figure 4: P- and S-wave signals transmitted through Berea sandstone at 100 bars net
confining pressure for dry, benzene and water-saturations.
Figure 5: P- and S-wave signals transmitted through Westerly granite at 100 bars net
confining pressure for dry, benzene and water-saturations.
Figure 6: Spectral ratio (long-short dash) with amplitude spectra for aluminum (dashedline) and vacuum dry Kayenta sandstone (solid line) at 500 bars confining pressure.
The next 49 figures consist in a series of measurements for the 7 rocks studied.
For each rock the results are presented in this order:
- Poisson's ratio as a function of net confining pressure, for dry, water* and benzenesaturations.
- P-wave attenuation measurements versus confining pressure compared between
spectral ratio (sr) and rise time (rt) techniques, for dry, waters
saturations.
-48-
and benzene-
- S-wave attenuation measurements versus confining pressure compared between
spectral ratio (sr) and rise time (rt) techniques, for dry, water'
and benzene-
saturations.
- P-wave attenuation measurements versus confining pressure from spectral ratio
technique, for dry, water* and benzene saturations.
- P-wave attenuation measurements versus confining pressure from spectral ratio
technique, for dry, water" and benzene saturations.
- S-wave attenuation measurements versus confining pressure from spectral ratio
technique, for dry, water * and benzene saturations.
- K-attenuation measurements versus confining pressure from spectral ratio technique,
for dry, water" and benzene saturations.
- E-attenuation measurements versus confining pressure from spectral ratio technique,
for dry, water" and benzene saturations.
* Water saturation has not been measured for Navajo sandstone and for Webatuck
dolomite.
Figures 7 to 13: Previously indicated measurements for Weber sandstone.
Figures 14 to 20: Previously indicated measurements for Navajo sandstone.
Figures 21 to 27: Previously indicated measurements for Berea sandstone.
Figures 28 to 34: Previously indicated measurements for Kayenta sandstone.
Figures 35 to 41: Previously indicated measurements for Bedford limestone.
Figures 42 to 48: Previously indicated measurements for Webatuck dolomite.
Figures 49 to 55: Previously indicated measurements for Westerly granite.
-49-
In the next six figures the P- and S-attenuations measurements versus confining
pressure are compared between the four sandstones for the saturations available.
Figure 56: Dry P-wave attenuation for the four sandstones.
Figure 57: Dry S-wave attenuation for the four sandstones.
Figure 58: Water saturated P-wave attenuation for three sandstones.
Figure 59: Water saturated S-wave attenuation for three sandstones.
Figure 60: Benzene saturated P-wave attenuation for the four sandstones.
Figure 61: Benzene saturated S-wave attenuation for the four sandstones.
-50-
METERING VALVES
POC
CONFINING
PRESSURE
PRESSURE
TRWISOCER
MACOR CERAMIC
INSULATORS
PORE PRESSURE
TUBING
PZT-SA ULTRASONIC
TRANSOUCERS
PRESSURE
VESSEL
CORETITANIUM ALLOY
CORE HOLDERS
POLYURETHANE JACKET
2"l x 3"d
PORE PRESSURE
VOLUMOMETER
ACCJULATOR
TRANSOUCER
,
A
HIGH PRESSURE
RESERVOIR
METERING PUMP
Figure 1.
-51-
KAYENTA SANDSTONE
(100 bars)
-0.1
water sat S-wave
I
*
0.8
t
I ,
benzene sat. P-wave
0.1
I. I . . . .
.
T.
0
1
a
i l l.
3
S
4
0
time in microsec.
Figure 2.
-52-
I .
8
. I
3
I4
*
$
$
KAYENTA SANDSTONE
()
C,
I
(500
bars)
I
water sat S-wave
0
C
.$
E-
E
-0.5
-,..
....IV. .II
W.. ...
..
benene
3
1.0
benzene sat. P-wave
.
4
benzene sat. S-wave
s
e.s-
-A.5--
-0.s-
I
.
.
.
I
I
.
I
.
"
I
.
I
.
.
3
.
1
4
.
.
.8
.
.,
.
5
Time in microsec.
Figure 3.
-53-
II
..
.
II.
.
I
I
3
4
s
BEREA SANDSTONE
0.
6
(100 bars)
U
dry P-wave
0
0. 0-
-0
6-1
0. 6-
0
water sat. S-wave
water sat. P-wave
0.
0 0
-0.
'I
CL
1
0.
0 6
-0. 6
benzene sat. S-wave
benzene sat. P-wave
o.
-0.
07
6
0
314
1
2
3
5
4
-o
5
0,
1.. 2, . 3,!.
..4 .. 5 . .6
0
1
Time in microsec.
Figure 4.
-54-
2
3
4
5
6
(100 bars)
WESTERLY GRANITE
it. P-wave
C')
O
0.
Eu-f.
-I.S
-
a
E
5
4
1.S -
1
t. P-wave
benzene sat. S-wave
0.5
1.0 -
-10.5--
4
Time in microsec.
Figure 5.
-55-
1
a
3
4
KAYENTA SANDSTONE
4
Z
I
I-C
m
,)
so
-I
5
O.s
1.e
FREQUENCY
Figure 6.
-56-
1.,$
(MHz)
I
t1
I
t
1
L
1
Weber
Sandstone
a Benzene
o Waker
0
a Dry
o0.3L
(n
C
0
00000
O
O
a
0.2
0.1
O
0
O O
D
(
-a
AAAA
4
AAAw---- , XZA,
i-
•
g
-
0
8
50
25
75
!
Pressure,
Figure 7.
-57-
MPa
100
150
I I 1 I1 I
1I
I
I
I
Il!lljlllllllll
I I
I
I
I
I
Jo
Weber
Sands tone
*a
C
oa B enz . sr&r
sr&r
sr&r
o* Water
Av Dry
1000
C
c
*0
V -O
0
a
"' X=
V
A
V
A
V
A
A
V
I 1-
0
50I
so
25
Pressure
Figure 8.
-58-
--
II
I
75
MPa
I
I
I
100
- "
"l
150-
"
l
13
l
"
"
"
Weber
Sandstone
C
sr&r
sr&rE
sr &rt-
OBenz.
C>
o*
\
Waker
av Dry
0 100
a
o
C
o
O
a
*0*0 O
.....,
a
0
*
+e
V
0
M
9
-VVV
V
a
V
V
V
0-T
0
50
25
Pressure,
Figure 9.
-59-
75
MPa
100
150
I,
1
I~
SI
t
Lt
I
I
I
I
I
I
I
I
I
I
-
I-
-
I
I
D
Weber
Sandstone
El
0
a Benzene
C
O Water
A Dry
100C
0
-0
0
SO
"0
o
-4<c
0
00000
AAAA
A
A
SI
0
I
1
50
25
Pressure,
Figure 10.
-60-
A
75
MPa
II
100
150
I
---I
II
I1
I
I I
1
1
I
I
I
I
I
I
II
Weber
Sands tone
o Benzene
C3
0
Dry
100-
CED
Water
0I,,
0
C
C
0
0
C
0
0
~
2
j
o0
0
00
A&A
Si
0
i
I
A
A A
_I
,
50
25
Pressur e
Figure 1 1.
-61-
I
75
MPa
i Il
II
l
I
k
100
150
*
1I
t
I
ci
-
3
F
J
1
I
1
f
I
I
I
I
I
I
01
Weber
Sandstone
a Benzene
O Water
100
A Dry
7
c31
EO
C
0
A
c
50--
00
[]
-4-)
0 co
cc
!
0
~o0
o
0 0 0 0
I
oa
0A
• I
I
1
25
50
25
so
Pressure,
Figure 12.
-62-
i 75
I
75
MPa
I-
100
10
150
i
I
I
SI
I
1
I
I
I
Weber
Sandstone
LU
a Benzene
O Water
A Dry
18800-
C
0
0
0
0
3
O
C
0
00
--
OO
0
-4-
O0
*
4
O
n~ AA
O
A
-0
AAJ
25
0
75I
50
25
Pressure,
Figure 13.
-63-
75
MPa
I
100
18
0.5
0
ec
0L.
0.2
0
75
25
Pressure,
Figure 14.
-64-
MPa
100
60
40
0L
C
0
0
20
<
0
50
25
r essur e
Figure 15.
-65-
75
MPa
100
60
I I I I 1 I I I I I 1 I
I t I
f 1 I I
Navaj o
Sandstone
0
(I)
** Benz. SR, RT
C0
SR, RT
&" Dry
0
0
0
*00
0
00
C
0)
13
20 0
4J:
vvvvrvooAV A
00
I i
ai
9 I
25
1
I
50
Pressure,
Figure 1 6.
-66-
4
I
I
-
rcl 1
X
x X
*
I
I
75
MPa
I
I
1
1,I
100
SI
I
IL
L
I
t,
l
I
r
I
I
r
Navajo
Sandstone
O
CL
0a
C
Benzene
0
SDry
40-
0
C
0
c3)
20+
-4-
A
AAA A AA~
A
0
25
58
Pressure,
Figure 17.
-67-
75
MPa
100
60
I
1
1
t
1
I
1
1
I 1
1
I
.
I
f
.
1
I
1
1
1
Navajo
Sandstone
a
0
o Benzene
[]a
40-
A
Dry
0
CD
Ea
28
-.
0
AAAA AA A A A
AA
0
A
A
0
n
n
I
I
I
I
I
0
25
50
Pressure,
Figure 1 8.
-68-
1
I
I i
I
I75
75
MPa
100
---
I
SI
I
I
I
I
I
I
I
L
I
I
I
I
I
..
I
I"
Navajo
Sandstone
0
o Benzene
A Dry
CO
O3C
40-
•
AO
00
-C
0
El
A
A
A
C
3
c) 20-
A
AA
A&
-4-J
<)
0
I
1I
eI
50
25
Pressure,
Figure 19.
-69-
75
MPa
1
100
60
I
--
I 1 1
L 1 I
II
I
-
I1
I
11I
r
Z
I
Navajo
Sandstone
L
o Benzene
ADry
0
40-
0
0
0
C
0
C
00
0
c
20--
h
'AS
0
I
S
25
0
25
I
L
I
-
5
0
50
Pressure,
Figure 20.
-70-
7
75
MPa
1
'
'0
100
I
0.5
_
.
i,,,
.
,1
i
I
II
Ber ea
Sandstone
4-
a Benzene
o Water
A Dry
0
S0.3-
L
0
c
0(n
0.2-
a0,
0000
00000
C
0
AAAAAA
CL
O
O
O0 0
0
a
0
a
A
A
A
A
"W&xm
0.1
8.0
I
I
aI
I_
I
U
a
*
I
50
25
Pressure,
Figure 21.
-71-
I
75
MPa
*
100
""
""
150--
Berea
Sandstone
a
* Benz. sr&rt
o* Water sr&rt
\a
CD
sr&rk -
v Dry
S100-O0
CD
o
C
0
1
0
S + +x x
0
50
25
Pressure,
Figure 22.
-72-
x
x
75
MPa
X
100
150
100
3
C
c
5so
50
25
Pressure,
Figure 23.
-73-
75
MPa
100
I
1SO
[
I
I
1
I I
!
I
I
I
I
I
I
L
I
I
I
I
I
I
i
Il
i
I
I
I
L
I.
I.
I'.I-
I
I.
Ber ea
Sandst one
CL
a Benzene
O Water
A Dry
100- 03
0C
O
0
-4J
O
C
C
I_
88
<
B
f~
hAAAAA
0-
0
SI
A
A
I
I
25
50
Pressure,
Figure 24.
-74-
I
I
A
-
I
75
MPa
100
158
I
I
I
I
!I
I
I_
I
I
)
L
i
I
I
I
I
.f
1
1
1
L
I
1
I
I
I
I
I
L
1I
Ber ea
Sandstone
O3
(n
o Benzene
O Water
A Dry
1000
O
1
0C
0
-a
80
o
o
C
0
C
E3
50-
29o
13
0
..- )
A AAAA
I
0
I
.1,
50
so
25
Pr essur e
Figure 25.
-75-
I
I
75
MPa
II
ii
I
100
L
I
I
I
S
I
1s50
I
*I l -
I
I
t
-
I
t
i
I
I
A
A
*1
I1
I
Ber ea
Sandstone
v
0
o Benzene
o Water
aDry
100C
C
0
0
0
0
C
0
50+ -
<[
coc
o
Do
E!
A
A
Y
1~~~~-
I
j
Pressure,
Figure 26.
-76-
16
5 1
50
25
A
W
75
MPa
W
w
v
100
150
I
....
1,,,1
!
I
I
'
i
-
II
-
Ber ea
Sandstone
LUL
O Water
O
100-
0
a Benzene
a
C3
0
A Dry
0
C
0
o
CGB
0
O
AA"AA
A
I
-1
4
..
1
1
I
1
50
Pressure,
Figure 27.
-77-
A
A
I
-
,-
I
75
MPa
I
I
100
100
0.5
.
.
.
.
-
1~1~~
I
-
1
-
9
1
,
*,
|
Kayent a
Sandstone
a Benzene
o Water
a Dry
0
0
a
-
00000
qkO
00
00000
C 0.2-
Caa
l
0
a
0
a
9A
0
A
AAA
0
CL
A
h:
A
A
A
0.0
- -I- -1-~.
-I
0
a
I
25
50
Pressure,
Figure 28.
-78-
75
MPa
I
I
I
100
200
CL
150
C 100
0
C
ct
0(D
D
-c
50
50
25
Pressure
Figure 29.
-79-
75
MPa
100
,
200
,I
,
,I
(
I
I
I
I
I
I
I
I
i
I
L
I
I
I
.I
I
I
I
I
I
I
Kayenta
Sandstone
(n
0C)
SBenz.
150-
sr cr t
sr &r
sr &r
o* Water
Av
0
Dry
a
0
C
.0
100
-
-)
8
C
*O~
V
cD
0
-- )
-4c
50 -
0
BtX
.
.
1 .
-
.
50
25
Pr essure,
Figure 30.
-80-
_I
I
I
75 I
75
MPa
a
f
1
1
-
100
I
_.
.
I
I
.
-I
i
rrl
I ,
~
I.
.
I
I.
I
-
I
I
Kayen a
Sandstone
0C
a
a Benzene
o Water
ADry
150-
C11
am.-
0
O
0
ao
000 0
000 0
C
58 -
A&
- 0.S
8
0
&
•
1
A
50
25
Pressure,
Figure 31.
-81-
S
a
w
75
MPa
5_
a
5
I
S
a
100
0
*\
150-
o Benzene
0 Water
^ Dry
o
( o
0
0
S40
C
0
we
A
0
50
25
Pressure,
Figure 32.
-82-
75
MPa
100
0 13 1.50
C
.9
ieeO
(SJ
C
-4 J
75
25
Pressure,
Figure 33.
-83-
MPa
100
2800
....
I M ..
OWa
I-
-V.
..
I
I
.
A
.
,.
L -f
9
1
f
--.-
l
g
I
-1
t
I
--
i
I
I
I
-
Kayenta
Sandstone
bJ
a Benzene
O Water
SDr y
QMW4
01"
c
OM-0
Q
o0 a
0
03
a
Sa
88
CAA
50
A
A
ILIL
I
1
Sr
9
S
i
7
a
S
I
75
25
Pressure,
Figure 34.
-84-
MPa
i111
e0
100
Bedford
L i mestone
-
0.4
O
Benzene
o Water
a Dry
o
03
L
2
M
a
0
25
75
so
Pressure,
MPa
Figure 35.
-85-
100
150
-i -
I
I
I
I
.
..
I
I
I
.
I
I.
I
I
1
-- ~-
1
Bedford
L i mestone
0
a* Benz . sr&rE
o* Water sr &rt
sr&rk
Av Dry
C
C
C 100O
0a
4J
)<
V04
$ j
*
88
a
50
0
1
f
A~
ISI
I
50
S25
Pressure,
Figure 36.
-86-
51
75
MPa
100
100
I
I
I
I
I
I I ( 1 r
B
1
t
1 _.I LL *
-
Bedford
L i mestone
0V)
*° Benz . sr&rE
Water sr&rE
75-
sr&rk
Av Dry
0
0
0
(-
so-
v
a
^
VV
O
**
A,
80o
vv~i
*Xv
V
V
04
.
.
.
.
.
.
25
#
W
i
I_-
s0
Pressure,
Figure 37.
-87-
I-
I
I
I
75
MPa
I
i
i
100
10080
0Bedford
oa13
L i mesone
08
S75-
Co
^A08
c
50
o Benzene
'25
O
Water
A
Dry
0
0
50
25
Pressure,
Figure 38.
-88-
75
MPa
100
IIII
I~,lr
I
I
I
I
II
I
I
i
I
1
I
1
I
1
Bedford
L imestone
0 Benzene
Wa ter
75
Dr y
O
00
AO
-
[] NfO
0
ZsL
00
AA
_ r
i
.t
s
s
I
I
i
i
r
I
A
-7
I
I
50
r essur e
Figure 39.
-89-
Ij
.I
75
MPa
I
I.
I
t
I
158-
II
lt
t
I
I
1
-
I
,
,
t
I
~t
f
I
,
,
,
,
I
Bedford
L i mest one
C
a Benzene
0 Water
aDry
125-
C ,
.
CQ
0
-
10°:
=8
,*1
~Oo
03
J
O0 0A
75.-
0
0
50
25
Pressure,
Figure 40.
-90-
75
MPa
100
...
100
I1.1
•.
1.
.
.1
!
CLJ
1
1
Il
I
I
I
I-
Bedford
13
L i mestone
3
O
2
I
Benzene
o Water
a
75-
C3
A Dry
00
03
o o
0
0
c3 s500
O
0
C)
-
4J
25-
-
0
_l
-1
i
Ir
i.
.-
a
.
A
-
-r
•
25
Ir
9 9
.I
50
Pressure
Figure 41.
-91-
I
I
75
MPa
•
w
100
0.5
--
I
I
I ~
-- ,
IIIIIII
1
,
.
,
I
I
r
IL
I
1
IIIL_~
Webatuck
Dolomi te
0.4
o Benzene
ADry
0
00000
qil-
-113 [
-
0
A
L
AA
AAA
AA
C
c 0.20
0
--
CL
0.0
a-
a
i
5
25
Pressure,
Figure 42.
-92-
I
I
75
MPa
i
100
100
I
l
•L
I
I
I
I
I
I
I
I
I
I
I
I
I
I
L1I
I
1
1
1
_
'4ebatuck
Dol om ite
75
0o
Be nz . SR
Av Dr y
C
SR,
RTT
I
0
0
D
0
<c
13
SVv
25-
av
8 34 1 1
0- -
.
..
I
I
i
.
a
I-
.
I
I
I
25
50i
so
Pressure
Figure 43.
-93-
75
MPa
I
I
S00
100
I
100-
I
weba uck
Doiomi te
<n
C)
S75--
C* Benz.
SR RT
RT-L
Dry SR,
CP
01
c
0
5--
25
0
50
25
Pressure,
Figure 44.
-94-
75
MPa
100
I
1008
r
I
I
1
I
t
I
r
I
I
1
I
I
I
L
t
I
I
I
1
I
L
F
Weba t uck
Dolomi te
CL
A Benzene
75-A
A
Dry
sA
C
0
58--
0,
[
C
-c
25-
A
A
0
r'0
003
A
4
0
I I
0
25
I
I
25
50
58
22
75
Pressure, MPa
Figure 45.
-95-
100
101
I
I'I
''
Webaku
Dolomi
o
BenzE
75-C
C
& Dry
50--
O
00
0
C
40
-2-
--
a
022
0
50
25
7
Pressure, MPa
Figure 46.
-96-
SI
100
"
"I
"
" 1
!
I
A
I
,
I _
Webatuck
Dolomi te
C
o Benzene
A Dry
75-
(C
I)
A
C
0
5so-
AI
(3
C
-a
L30000
A
ea
-
2
0
I 75
I
50
25
Pressure,
Figure 47.
-97-
75
MPa
1
100
•
100
•
.
•
I
I
I
I
I
II
1
I
I
I
I
a
I
I
aI
I
Webatuck
Dolomite
LUJ
0
A
W
C
C
0O
Be nzene
Dr y
50 A
E0
C
0
4C
-J
C
-25 A
ODA
0A
0
I
. ..
an
.- t
a
a
so
25
Pressure
Figure 48.
-98-
aa
75
MPa
a
a
n
100
-66-
n6t
Jn6!i
JrlsseJd
.1~.
i
i
O8
SL
I
J
L0J1 a
1
B
&
J
I
-
.....
IL
800
_...
I"
0
1)
(I)
0
vvvvV
V
V
8
8
:3
VVV
VV
-
S'
0
A (M
Je l
m
euezueg D
-t-~' a
SI JeISM
S
i
S
I
II
I
I
I
I
S
I
it
i
S
S
I
S
So0
60
I
I
.
i
,
1
1
I
I
f
I
i
I
I
I
I
I
I
)
_I
I
I
I
I-
I
I
I
I
I
I
I
I
1.
I.
t-
Wester .ly
Gran i t
sr&rt
o* Waoer sr&rk
sr&rt
Av Dry
3enz .
a_
C
C3
40 -
C
V.
c0
3
C
VV
Vv
20-
II
II
II
II
1
2
I
I
I
I
S
1
I
I
9
25
-
58
Pres sure
Figure 50.
-100-
..
I
75
MPa
1
m
00
00
68
!.
I
a
I
lI
U
U
1
.
I
I
1
I
,
I
.
I
a
a"I
Wester I
Grani te
0
=* Benz .
a
.
40
sr&rk
sr&rk
sr&rk-
o* Water
Av Dry
a-
C
0
a
=
68 0
0
*
V~S
~hg~
13i~
't~V h h
*
88s
O
Ix
I
i
i
I
21
25
1
1
1
so
Pressure,
Figure 51.
-101-
75
MPa
111I
00
Wester l y
Gran i te
o00
Benzene
1A
(S
40--
C
C
0
Water
A Dry
--
A
20
I
,0-1I
0
50
25
Pressure,
Figure 52.
-102-
,I
75
MPa
100
60 -
I "I
Wester I
Grani tE
a
3
40
O
o Benze
0 Water
Dry
0~c
0
n
0
0A
.
C
20--
0
O8
AAA
AA
50
25
Pressure,
Figure 53.
-103-
75
MPa
,I
60
I
I
II I1I
S
II-
I1
1
i
i
i
I
I
I
I
I
I
Wester ly
Gran i
o Benzene
o Water
40-
A Dry
ALA&
A
0C
r"6
0o
C
- 40
J
0
1300[31
20-
-4-)
A
I
l
I
I
I
I
i
rUI
I
i
U
I
I
50
25
ressure,
Figure 54.
-104-
1
li1
75
MPa
I
100
68
I
I
I
I
1....1
I
f
I
I
I .
.
.
U
I
I
I
I
I
I
,
,
I I
I
i
I
.
I I
Wester I y
Gran i te
0
LU
0C
0
0
o Benzene
Wa ter
Dr y
40-
C
A
0
f3
N
A
A
C
O
c
A A
20 -
CI h
o4
0-
III
V
I
I
i
I
1
I
0
-I-
50
50
25
Pressure
Figure 55.
-105-
1
5
1
I
75
MPa
I.
I
I
j
100
200
DRY
S150CD
+
Weber
o
Navajo
Kayenta Berea
a
c 100-0
o
Q
0
O0
so
0 0
50
000
+ + + ++
ffh
0
50
25
75
Pressure, MPa
Figure 56.
-106-
188
DRY
weber
+
Navajo -Kayenta -
S150-
o
0
D
Berea
%A
-
e
0
.
-0
C00
Saaaaca:
a
0
I
50
25
Pressure,
Figure 57.
-107-
C
,
75
MPa
a
100
200
8
I
I
I
I
I
I
I
I
WATER SAT.
150vieber
Kayen
ai
Berea
0
0
OI
000000
so+ - +,
O
25
S8
Figuressr
8.Pa
Figure 58.
-108-
0
0
75
00
,
. ,
i
,
f
fI
I
t
t
i
WATER SAT.
(/
Weber
Kayen a
er
eo
O
j3
00
d2o
0
C
J
Ooo
c'
4-
So.-
A
-t-F
hh
+4
AA
A
+
+
0
v
50
25
ressure,
Figure 59.
-109-
75
10C
200
0
3a
"'
D
CD
c
158 -
0
+
+
a
0
Weber
Nava jo
Kayen ta
A
Berea
1oo--
oto
Q
+
o000
C
BENZENE SAT.
0
+
Pressure, MPa
+00
00
50
25
Pressure,
Figure 60.
-110-
75
MPa
0
00
2008
-I
I
I
I
I
I
BENZENE SAT.
Weber
avaj o
Kayenka
+
o
150 -
0o -4-)
3
o
Berea
A
c
00
a
00
0
+
aaJ
0
25
Pr essur e,
7
MPa
Figure 6 1.
Pressre,
-111-
~4
~l
100