GEOPHYSICS, VOL. 74, NO. 6 共NOVEMBER-DECEMBER 2009兲; P. N41–N48, 5 FIGS., 3 TABLES.
10.1190/1.3207863
Poroelastic analysis of amplitude-versus-frequency variations
Haitao Ren1, Gennady Goloshubin2, and Fred J. Hilterman3
ry measurements, high attenuation and large velocity dispersion are
observed in the seismic frequency range 共Batzle et al., 2001兲.
Since Biot presented his work on poroelastic wave theory in the
early 1960s, significant research on poroelastic wave propagation
has been conducted.According to Biot’s theory, seismic waves propagating in a homogeneous poroelastic medium are not attenuated
significantly in the seismic frequency range of 0 – 200 Hz 共Biot,
1956a, 1956b, 1962; Dutta and Ode, 1979a, 1979b; Dutta and Seriff,
1979兲. Therefore, geophysicists have been examining other theories
to explain the high attenuation observed in field data and laboratory
experiments. Several theoretical models have been proposed since
the 1970s; examples are White’s gas-bubble model 共White, 1975;
White et al., 1975兲, the local fluid-flow or squirt-flow model
共O’Connell and Budiansky, 1974; Dvorkin et al., 1995兲, and the dual-porosity model 共Pride and Berryman, 2003a, 2003b兲.
Recently, the mesoscopic mechanism, which is the fluid-flow
mechanism caused by the P-wave-induced fluid-pressure difference
at different mesoscopic-scale inhomogeneities 共larger than the pore
size but smaller than the wavelength兲, has been studied extensively
共White, 1975; White et al., 1975; Dutta and Ode, 1979a, 1979b;
Dutta and Seriff, 1979; Norris, 1993; Gelinsky and Shapiro, 1997;
Gurevich et al., 1997; Shapiro and Muller, 1999; Muller and Gurevich, 2004; Pride et al., 2004; Carcione and Picotti, 2006; Muller et
al., 2007兲. White 共1975兲 and White et al. 共1975兲 first studied this mesoscopic mechanism. They introduced the patchy-saturation model,
numerous gas bubbles in water-saturated porous media, or thinly
layered porous media in which water-saturated layers alternate with
gas-saturated layers, to explain high attenuation in the seismic frequency band. This model is known commonly as White’s model.
Dutta and Ode 共1979a, 1979b兲 and Dutta and Seriff 共1979兲 numerically calculated the accuracy of White’s patchy-saturation model.
They find that the gas-bubble model and the thinly layered model are
comparable to some extent. Gurevich et al. 共1997兲 consider Biot’s
loss, scattering, and mesoscopic fluid-flow loss for 1D thinly layered
poroelastic media and study P-wave attenuation at normal incidence. They find that the attenuation peak for the mesoscopic loss
ABSTRACT
Although significant advancement has occurred in the
interpretation of seismic amplitude-variation-with-offset
共AVO兲 anomalies, a theory is lacking to guide the interpretation of frequency-dependent seismic anomalies. Using analytic equations and numerical modeling, we have investigated characteristics of the normal-incident reflection coefficient 共NI兲 as a function of frequency at an interface between a
nondispersive medium and a patchy-saturated dispersive medium. Because of velocity dispersion, the variation of NI
magnitude is divided into three general classes. These classes
are 共1兲 low-frequency dim-out reservoirs, in which NI magnitude decreases toward lower frequencies; 共2兲 phase-shift
reservoirs, in which NI is a small negative value at low frequencies but becomes positive at higher frequencies; and 共3兲
low-frequency bright-spot reservoirs, in which NI magnitude
increases toward lower frequencies. This classification could
provide insight for frequency-dependent seismic interpretation.
INTRODUCTION
For years, geophysicists have noticed low-frequency seismic
anomalies associated with hydrocarbon reservoirs 共Taner et al.,
1979兲, and this topic still maintains the interest of many 共van der
Kolk et al., 2001; Castagna et al., 2003; Korneev et al., 2004; Chapman et al., 2006兲. Although the mechanisms that cause frequencydependent anomalies have not been defined clearly, many authors
report that hydrocarbon-saturated reservoir zones often show anomalously high values of attenuation 共Castagna et al., 1993; Klimentos,
1995; Dasgupta and Clark, 1998; Dasios et al., 2001; Rapoport et al.,
2004兲.Associated with attenuation is velocity dispersion, and significant velocity dispersion is expected in hydrocarbon-saturated reservoirs that have high attenuation 共Chapman et al., 2006兲. In laborato-
Manuscript received by the Editor 1 July 2008; revised manuscript received 23 April 2009; published online 14 October 2009.
1
Formerly University of Houston, Department of Geosciences, Houston, Texas, U.S.A.; presently ExxonMobil Exploration Company, Houston, Texas, U.S.A.
E-mail: haitao.ren@exxonmobil.com.
2
University of Houston, Department of Geosciences, Houston, Texas, U.S.A. E-mail: ggoloshubin@uh.edu.
3
Geokinetics, Inc., Houston, Texas, U.S.A. E-mail: fred.hilterman@geokinetics.com.
© 2009 Society of Exploration Geophysicists. All rights reserved.
N41
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N42
Ren et al.
mechanism occurs at lower frequencies than that for scattering attenuation.
Shapiro and Muller 共1999兲 report effects of the fluid-mobility
property on P-wave attenuation for thinly layered, partially saturated
media and conclude that the frequency-dependent P-wave attenuation is sensitive to fluid-mobility properties and can be observed in
the seismic frequency range. Using numerical experiments, Muller
and Gurevich 共2004兲 show that dispersion and attenuation curves
vary more gradually with frequency for media with random fluid distribution than for media with regular fluid distribution. Carcione and
Picotti 共2006兲 study velocity dispersion and attenuation for thinly
layered, partially saturated media and find that the attenuation peak
for patchy saturation can occur in the seismic frequency range.
Although most recent studies are focused on velocity dispersion
and attenuation, a few authors consider plane-wave reflection coefficients in attenuative media 共Cooper, 1967; Krebes, 1984;
Nechtschein and Hron, 1997; Ursin and Stovas, 2002; Ruud, 2006;
Sidler and Carcione, 2007兲. For example, Cooper 共1967兲 obtains reflection and transmission coefficients at an interface between two
linearly viscoelastic materials. Krebes 共1984兲 and Nechtschein and
Hron 共1997兲 numerically compute reflection and transmission coefficients for viscoelastic waves. Ursin and Stovas 共2002兲 obtain reflection and transmission coefficients in a thinly layered viscoelastic
isotropic medium. Sidler and Carcione 共2007兲 present numerical
analyses of reflection and transmission coefficients for attenuative
vertical transverse isotropic media.
In general, reflection-coefficient equations for dispersive media
are very complicated. The problem, however, is simplified for normal incidence. Trapeznikova 共1985兲 provides simple expressions
for normal-incident reflection and transmission coefficients for an
interface between two attenuative media.
As mentioned above, attenuation and velocity dispersion might
have a substantial influence on seismic data in hydrocarbon-saturated zones. This study characterizes variations of the magnitude and
phase angle of the normal-incident reflection coefficient 共NI兲 as a
function of frequency at an interface between a nondispersive medium and a dispersive medium 共thinly layered, partially saturated poroelastic media兲. The examples presented span the range of velocitydispersion effects associated with patchy-saturated reservoirs in the
seismic frequency range. The geologic models associated with the
variation of amplitude-versus-frequency 共AVF兲 responses are very
similar to the geologic models associated with amplitude-variationwith-offset 共AVO兲 responses. Similar to the AVO classification 共Rutherford and Williams, 1989兲, the AVF responses are divided generally into three classes:
1兲
2兲
3兲
low-frequency dim-out reservoirs
small amplitude reservoirs that possibly can exhibit phase
shifts
low-frequency bright-spot reservoirs
In the first section of this study, we review wave-induced fluid
flow in porous media and introduce White’s patchy-saturation model 共White, 1975; White et al., 1975; Carcione and Picotti, 2006兲. In
the second section, we use three reservoir models to demonstrate
dispersion effects on the magnitude and phase angle of normal-incident reflections. The phase velocities of the dispersive reservoirs are
predicted by White’s patchy-saturation model. Based on the characteristics of AVF and phase angle versus frequency curves, we loosely
associate the AVF responses into three classes. In the third section,
wave-propagation numerical modeling based on Biot’s theory is performed on the three models, and the results confirm the characteristics of the three AVF classes.
ATTENUATION AND VELOCITY DISPERSION
IN THINLY LAYERED POROUS MEDIA
When a P-wave travels through a porous medium, some parts of
the porous rock are under compression and some experience dilatation. If the pores in the rock are fluid saturated, the pore pressure at
the compressed parts is higher than that where it is relaxed. When the
pores are connected, the fluid will flow from the higher pore-pressure regions to the lower. The relative fluid flow will induce energy
losses if the fluid is viscous. In other words, the wave attenuates; in
addition, velocity dispersion occurs.
Biot’s theoretical model predicts the attenuation and dispersion in
a poroelastic medium. In his theory, the displacement vectors of the
rock frame and pore fluid are denoted by u and U, respectively. The
displacement of the pore fluid relative to the rock frame is denoted
by w共w ⳱ ␾ 共U ⳮ u兲兲, which is measured in terms of volume per
unit area of the bulk medium 共Biot, 1962兲. For a saturated rock having a bulk density ␳ b, porosity ␾ , and permeability ␬ , the equations
of motion for the porous media are written in the form 共Biot, 1956b;
Stern et al., 1985; Schmidt, 2004兲
␮ⵜ2u Ⳮ 共H ⳮ ␮兲 ⵜ e ⳮ C ⵜ ␰ ⳱ ␳ b
C ⵜ e ⳮ M ⵜ ␰ ⳱ ␳f
⳵ 2u
⳵ 2w
Ⳮ
␳
,
f
⳵ t2
⳵ t2
⳵ 2u c m␳ f ⳵ 2w F ␩ ⳵ w
,
Ⳮ
Ⳮ
⳵ t2
␾ ⳵ t2
␬ ⳵t
共1兲
共2兲
where e ⳱ ⵜ · u and ␰ ⳱ ⳮⵜ · w; ␳ b ⳱ ␳ g共1 ⳮ␾ 兲 Ⳮ ␳ f␾ . Here ␳ g is
the density of the mineral grain; ␳ f is the pore-fluid density; ␩ is the
pore-fluid viscosity; cm is the tortuosity parameter, which is a constant depending on the pore structure 共Dutta and Ode, 1979a兲; and ␮
is the shear modulus of the saturated rock. Because we assume that
the pore fluid does not affect the shear modulus of the rock, the shear
modulus of the saturated rock is the same as the shear modulus of the
drained rock frame 共 ␮dry兲.
Following Stern and his coauthors 共1985兲, M, C, and H are elastic
moduli calculated from the following expressions:
M⳱
Kg
,
Kg
Kdry
1ⳮ
Ⳮ␾
ⳮ1
Kg
Kf
冉
冊
4
H ⳱ Kdry Ⳮ ␮dry Ⳮ BC, and C ⳱ BM,
3
共3兲
共4兲
where Kg, Kf, and Kdry are the bulk moduli of the mineral grain, the
pore fluid, and the drained rock frame, respectively; and B ⳱ 1
ⳮ Kdry / Kg, which often is called Biot’s coefficient. Finally, F is the
frequency-dependent coefficient which characterizes the drag force
between the solid and fluid as the frequency increases 共Biot, 1956b;
Stern et al., 1985兲. It is 共Stern et al., 1985兲
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Amplitude-versus-frequency variations
␨ T共␨ 兲
冋
F⳱
4 1 Ⳮ 2i
T共␨ 兲
␨
册
V⳱
共5兲
,
ber⬘共␨ 兲 Ⳮ ibei⬘共␨ 兲
,
ber共␨ 兲 Ⳮ ibei共␨ 兲
␨ ⳱a
冑
␻␳ f
␩
共6兲
冋
E0 ⳱
冉
p1
p2
Ⳮ
H1 H2
冊
册
ⳮ1
共8兲
,
ⳮ1
共9兲
,
where pS ⳱ ds / 共d1 Ⳮ d2兲 with s ⳱ 1,2 representing media 1 and 2.
Omitting the subscript s, we have for each medium
r⳱
I⳱
BM
,
H
共10兲
冉 冊
␩
kd
coth
,
k␬
2
共11兲
where k is the complex wavenumber of the slow P-wave velocity and
is given by
k⳱
冑
i␻ ␩ H
.
4
␬ M Kdry Ⳮ ␮dry
3
冉
冊
共12兲
Table 1. Rock-frame and pore-fluid properties of models 1, 2, and 3.
The phase velocity VP and quality factor Q of
the periodic stratification are given by
冋 冉 冊册
VP ⳱ Re
1
V
Re共V2兲
Q⳱
,
Im共V2兲
ⳮ1
,
共13兲
共14兲
where Re共V兲 and Im共V兲 represent the real and
imaginary parts of the complex velocity V, which
is expressed as
共15兲
To illustrate dispersion effects on the magnitude and phase angle
of the NI, three reservoir models have been selected so that they represent three types of reservoirs commonly encountered in oil exploration. Model 1 represents a deep consolidated sand reservoir; model
2, a middepth sand reservoir; and model 3, a shallow unconsolidated
sand reservoir. The model 1 reservoir has a larger acoustic impedance 共AI兲 than the overlying shale, model 2 has a slightly lower impedance than shale, and model 3 has a considerably smaller impedance. For each model, the reservoir consists of 1-m-thick layers with
the same rock frame, but brine-saturated layers alternate with gassaturated layers. Although the stratified model might not be realistic
physically, it does represent the attenuation associated with White’s
patchy-saturation model 共Dutta and Seriff, 1979兲, and the stratified
layering simpifies the numerical modeling.
The bulk modulus and density of the sand grain are 38 GPa and
2.65 g / cm3, respectively. The rock-frame and pore-fluid properties
of the gas- and brine-saturated sands are shown in Table 1. In the
rock-frame properties column, Kdry and ␮dry are the bulk and shear
moduli 共in GPa兲 of the drained rock frame, respectively; ␾ is porosity; cm is the tortuosity parameter; and a is the pore-size factor 共in m兲.
In the fluid-properties column, K, ␳ , and ␩ are the bulk modulus 共in
GPa兲, density 共in g / cm3兲, and viscosity 共in cP兲 of the pore fluid, respectively; where the subscripts g and w represent gas and brine saturation 共wet兲, respectively. The P-wave 共VP兲, S-wave 共VS兲 velocities
and densities 共 ␳ 兲 of the overburden nondispersive shale are listed in
Table 2. The P-wave velocities for the reservoir rocks were computed with equation 13 and the rock-frame and fluid properties in Table
1. The acoustic impedance of the reservoir rocks at very low frequency 共0.01 Hz兲, along with that of the overburden shale, is listed
in Table 3.
Because of velocity dispersion and attenuation, reflection amplitudes from the interface between the nondispersive overburden and
the dispersive reservoir rock are a function of angular frequency 共 ␻ 兲.
In addition, because it is assumed that the overburden medium is
nondispersive, the attenuation of the overburden medium is neglect-
共7兲
,
2共r2 ⳮ r1兲2
1
Ⳮ
E0 i␻ 共d1 Ⳮ d2兲共I1 Ⳮ I2兲
E
,
p1␳ b1 Ⳮ p2␳ b2
AMPLITUDE VERSUS FREQUENCY
where a is the pore-size factor, ␻ is angular frequency, and i ⳱ 冑ⳮ1.
The functions ber共 兲 and bei共 兲 are Bessel functions of the first kind,
and ber⬘共 兲 and bei⬘共 兲 are the first derivatives of ber共 兲 and bei共 兲
functions 共Abramowitz and Stegun, 1972, p. 379-381兲.
For simplicity, we consider two porous media 共1 and 2兲 with thicknesses d1 and d2. Here, d1 and d2 are much smaller than the seismic
wavelength. These two media are layered periodically with the period of 共d1 Ⳮ d2兲. Based on White’s model, Carcione and Picotti
共2006兲 present the formula for the complex modulus for a P-wave
traveling perpendicular to the thinly layered stratification. It is given
by
E⳱
冑
where ␳ b1 and ␳ b2 are the bulk densities of media 1 and 2.
The equations above are used to calculate the phase velocities and
quality factors of the porous media presented below.
where T共 ␨ 兲 is the Kelvin function 共Stern et al., 1985兲
T共␨ 兲 ⳱
N43
Rock-frame properties
Pore-fluid properties
Properties Model 1 Model 2 Model 3
Kdry 共Gpa兲
␮dry 共Gpa兲
16.90
13.96
7.49
5.67
1.56
1.10
␾
␬ 共darcy兲
cm
0.20
0.20
1.62
0.26
1.60
1.72
0.33
2.00
1.82
ⳮ6
ⳮ6
ⳮ6
a 共m兲
10
10
10
Properties
Kg 共GPa兲
␳ g 共g / cm3兲
␩ g 共cP兲
Kw 共GPa兲
␳ w 共g / cm3兲
␩ w 共cP兲
Model 1 Model 2 Model 3
0.29
0.34
0.13
0.28
0.03
0.15
0.01
2.80
1.02
0.01
2.58
1.01
0.01
2.42
1.00
1.60
1.30
1.00
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Ren et al.
2 2
␻ 2共VP2␳ 2 ⳮ VP1␳ 1兲2 Ⳮ VP1
VP2共␳ 1␣ 2兲2
,
2 2
2
2
␻ 共VP2␳ 2 Ⳮ VP1␳ 1兲 Ⳮ VP1VP2共␳ 1␣ 2兲2
共16兲
␺ 共␻ 兲 ⳱ tanⳮ1
2
2␳ 2VP1␳ 1␻ ␣ 2VP2
2 2 2
2 2 2
␳ 1共VP2␣ 22 Ⳮ ␻ 2兲
VP2
␳ 2␻ ⳮ VP1
,
共17兲
where VP and ␳ are the vertical P-wave phase velocity and density,
respectively, and the subscripts 1 and 2 represent the upper and lower
media. Attenuation ␣ of the reservoir rock is related to the quality
factor Q by 共Carcione, 2001, p. 55-59兲
␣ ⳱ 共冑Q2 Ⳮ 1 ⳮ Q兲
␻
.
VP2
共18兲
The vertical phase velocity 共VP兲 and the reciprocal of the quality
factor 共1 / Q兲 for the stratified porous medium were computed based
on equations 13 and 14, respectively. Then the phase velocity VP
from equation 13 was inserted as VP2 in equations 16–18. The expression VP1 is the P-wave velocity of the overburden shale. The attenuation ␣ 2 of the stratified porous medium is given by equation 18.
The VP and 1 / Q for the porous medium are plotted against frequency in Figure 1. It shows that velocity dispersion and attenuation
are larger for model 3, the unconsolidated sand, than for models 2 or
1, the consolidated sand.
The magnitude and phase angle of the normal-incident reflection
coefficient are plotted as a function of frequency in Figure 2 for models 1, 2, and 3, respectively. For model 1, the reservoir is consolidated and the porosity and permeability are small. The attenuation and
dispersion of the reservoir rock also are very small compared to the
moderately consolidated 共model 2兲 and unconsolidated sands 共model 3兲. For model 1, the reservoir impedance is higher than that of the
overburden rock. Because of dispersion, the reservoir velocity increases toward higher frequencies. Correspondingly, the impedance
contrast becomes larger toward higher frequencies. Therefore, the
NI magnitude increases toward higher frequencies. The center curve
Table 2. Overburden shale properties for reservoir models 1,
2, and 3.
Models
VP 共m/s兲
VS 共m/s兲
␳ 共g / cm3兲
Model 1
Model 2
Model 3
3100
2650
2190
1520
1160
820
2.40
2.27
2.16
Table 3. Impedances of reservoir models 1, 2, and 3.
a)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
b) 0.16
Model 1
0.12
Model 2
1/Q
冑
0
20
40
Overburden shale
Reservoir sand
Acoustic impedance 共共m / s兲 · 共g / cm2兲兲
Model 1
7404
9060
Model 2
6016
5780
Model 3
4730
2500
Model 2
Model 1
60
0.00
80
0
Frequency (Hz)
20
40
60
80
Frequency (Hz)
Figure 1. 共a兲 Vertical phase velocity and 共b兲 1 / Q versus frequency
for reservoir models 1 共circle兲, 2 共square兲, and 3 共triangle兲. The velocity and 1 / Q are given by equations 13 and 14.
a) 0.40
b)
0.30
Model 3
0.20
Model 1
0.10
Model 2
0.00
0
20
40
60
Frequency (Hz)
Rock type
Model 3
0.08
0.04
Model 3
Phase angle (radian)
兩NI共␻ 兲兩 ⳱
in Figure 2a for model 1 shows that the NI magnitude increases
about 0.007 from 1 Hz to 60 Hz. The phase angle of the reflection
varies only slightly as a function of frequency because of the small
dispersion. For model 1, phase angles are positive and relatively
small at all frequencies.
In model 2, the reservoir is moderately consolidated rock with
good porosity and permeability. The attenuation and velocity dispersion of the reservoir rock is larger than that of model 1, as shown in
Figure 1. From 1 Hz to 60 Hz, the velocity increases about 0.13 km/s.
The reservoir impedance at zero frequency is slightly smaller than
that of the overburden rock. When the phase velocity increases toward higher frequencies because of velocity dispersion, the impedance of the reservoir becomes larger than that of the overburden
rock. Therefore, a phase shift occurs from low frequencies to high
frequencies, as shown in Figure 2b for model 2.
In model 3, the reservoir impedance at low frequencies is much
lower than that of the overburden rock. Figure 1a illustrates that the
velocity dispersion is large for model 3. From 1 Hz to 60 Hz, the
velocity increases about 0.21 km/ s. When velocity becomes faster
at the higher frequencies, the impedance contrast becomes smaller
and consequently the NI magnitude decreases. As depicted in Figure
2, the NI magnitude decreases about 0.05 from 1 Hz to 60 Hz, and
the phase angles of the NI are all negative values and are fairly constant above 10 Hz.
Similar to the AVO classification by Rutherford and Williams
共1989兲, the amplitude-versus-frequency 共AVF兲 curves might be divided loosely into three classes: low-frequency dim-out reservoirs,
possible phase-shift reservoirs, and low-frequency bright-spot reservoirs. Here are the reflection characteristics of these three classes.
Phase velocity (km/s)
ed and the velocity is independent of frequency. The magnitude of
the normal-incident reflection coefficient 兩NI兩 and the phase angle ␺
are expressed by 共see Appendix A兲
NI magnitude
N44
80
π
—
2
π
—
4
Model 2
Model 1
0
π
⫺
—
4
π
⫺—
2 0
Model 3
20
40
60
80
Frequency (Hz)
Figure 2. 共a兲 Magnitude, and 共b兲 phase angle, of the normal-incident
reflection coefficient 共NI兲 versus frequency for reservoir models 1
共circle兲, 2 共square兲, and 3 共triangle兲. The magnitude and phase angle
are given by equations 16 and 17. 共Note that for model 2, a phase
shift occurs from low frequencies to high frequencies because of dispersion.兲
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Amplitude-versus-frequency variations
Phase-shift reservoirs
In our examples, a phase-shift reservoir 共model 2兲 has a slightly
lower impedance than that of the overburden material at zero frequency. As the frequency increases, the impedance of the reservoir
becomes larger than that of the overburden rock because of dispersion effects. Such moderately compacted and consolidated sands often have moderate porosities and permeabilities and are often associated with class 2 AVO responses. A characteristic of the phase-shift
reservoir is that from low to high frequencies, a negative to positive
shift in the phase spectra occurs, which could lead to an apparent
phase reversal. As illustrated in Figure 2, the magnitude of the reflection for this type of reservoir is small, and depending on the signalto-noise ratio, the phase shift in the frequency domain could be difficult to observe.
Low-frequency bright-spot reservoirs
A low-frequency bright-spot reservoir has a much lower impedance than the overburden medium at zero frequency. As frequency
increases, the impedance of the reservoir becomes larger. Consequently, the impedance contrast becomes smaller, but the impedance
of the reservoir still is lower than that of the overburden. Such reservoirs often have class 3 AVO responses. The reservoir rock usually is
undercompacted and unconsolidated with high porosity and high
permeability. The velocity dispersion and attenuation of such reservoirs can be very large. The curves with square symbols in Figure 2
represent the low-frequency bright-spot reservoir responses. Because of velocity dispersion, the NI magnitude increases toward low
frequencies, but it does not have a significant phase shift. For this
type of reservoir, high-amplitude anomalies often appear at low frequencies. This could explain the low-frequency seismic anomalies
observed from shallow unconsolidated sand reservoirs that are hydrocarbon saturated.
a)
Wavelet center frequency (Hz)
0.40
0.42
0.44
0.46
0.48
0.50
15
25 35 45 55
b)
MAX (magnitudes)
A low-frequency dim-out reservoir 共model 1兲 has a higher impedance than the overburden medium. The interface between the upper
nondispersive material and the lower dispersive reservoir has a positive reflection coefficient at zero frequency. Because of velocity dispersion in the reservoir, the impedance increases with frequency.
Consequently, impedance contrast becomes larger, which results in
the NI magnitude increasing with frequency, as shown in Figure 2.
The phase angle of the NI is positive.
Low-frequency dim-out reservoirs are associated with class 1
AVO responses. These reservoir rocks usually are stiffer and have
small porosity and permeability values. Therefore, velocity dispersion and attenuation are small. Consequently, the magnitude of reflection amplitude change with frequency is small.
We have generated synthetic traces for porous layered media by
using the package OASES 共Stern et al., 1985; Schmidt and Tango,
1986兲. Detailed information about the package is given by Schmidt
共2004兲. The OASES package has been used for studying elastic and
poroelastic wave propagation by academia 共Gurevich et al., 1997;
Gelinsky et al., 1998兲 and by the oil industry.
The traces in Figure 3a are the model 1 impulse responses from
the OASES program convolved with 15-, 25-, 35-, 45-, and 55-Hz
Ricker wavelets. In Figure 3b, the NI magnitude for model 1 is predicted by equation 16 for the same five frequencies and then plotted
against frequency. The analytic prediction is represented by the
dashed line. The maximum value of the reflections was normalized
according to the analytic value at 55 Hz. In Figure 3b, the normalized magnitudes from the numerical modeling are plotted as a solid
line. In general, the results from numerical modeling agree with the
analytic predictions. The amplitudes increase slightly toward higher
frequencies because of velocity-dispersion effects.
In Figure 4, model 2 impulse responses from the OASES program
convolved with the five Ricker wavelets are displayed. As predicted
in the analytic study, we expect a phase shift from low to high frequency. This is observed in the numerical-modeling result 共Figure
4兲. Below the dashed line in the figure, the onset of the wavelet
changes from an apparent negative at 15 Hz to positive at 55 Hz.
This indicates a phase distortion dependent on the frequency band.
For model 3, the velocity-dispersion effect is significant because
the unconsolidated sands have large porosity and permeability. In
Time (s)
Low-frequency dim-out reservoirs
N45
0.104
0.103
0.102
0.101
5
25
45
65
Frequency (Hz)
Figure 3. 共a兲 Model 1 impulse responses convolved with 15-, 25-,
35-, 45-, and 55-Hz Ricker wavelets; 共b兲 maximum magnitudes
from numerical modeling 共solid line兲 and analytic predictions
共dashed line兲 versus frequency. Magnitudes from numerical modeling are normalized according to the analytic prediction at 55 Hz.
WAVE-PROPAGATION NUMERICAL MODELING
Wave-propagation numerical modeling was performed on the
three reservoir models to demonstrate the dispersion effects on amplitudes in porous media. In each model, the porous reservoir has
1-m-thick layers that have the same rock frame, but fully brine-saturated thin layers alternate with fully gas-saturated layers. The parameters of the three models are the same as the models in the previous section. As before, above the porous reservoir is a half-space of
nondispersive shale. The source is 800 m above the top of the reservoir, and the receiver is 990 m above.
Figure 4. Model 2 impulse responses convolved with 15-, 25-, 35-,
45-, and 55-Hz Ricker wavelets. Below the dashed line, the onset of
the wavelet changes from an apparent negative at 15 Hz to positive
at 55 Hz.
Downloaded 25 Feb 2010 to 129.7.72.244. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
N46
Wavelet center frequency (Hz)
Time (s)
0.58
0.60
0.62
0.64
0.66
0.68
b)
15 25 35 45 55
兩MIN (magnitudes)兩
a)
Ren et al.
0.28
0.26
0.24
0.22
5
25
45
65
Frequency (Hz)
Figure 5. 共a兲 Model 3 impulse responses convolved with 15-, 25-,
35-, 45-, and 55-Hz Ricker wavelets; 共b兲 maximum magnitudes
from numerical modeling 共solid line兲 and analytic predictions
共dashed line兲 versus frequency. Magnitudes from numerical modeling are normalized according to analytic prediction at 15 Hz.
Figure 5a, the model 3 impulse responses from the OASES program
after convolution with the Ricker wavelets are displayed. The maximum magnitude value of the reflection was normalized according to
the maximum absolute value predicted by equation 16 and then plotted against frequency in Figure 5b. Again, in general, the results
from numerical modeling agree with the analytic predictions that the
magnitudes increase toward lower frequencies. This theoretical
mechanism predicts low-frequency bright spots.
DISCUSSION
The following arguments might arise for this study. First, the reservoir with alternating thin layers of porous gas and wet sands could
not exist in nature. The reason we chose this model is that it is easy to
perform numerical modeling and can represent a patchy-saturation
model 共Dutta and Seriff, 1979兲, which is reasonable to expect in nature. Second, for all examples that we selected, most variation in the
NI magnitude occurred within the frequency range of 0.01– 15 Hz.
This is caused mainly by the thickness selected for the thin layers. As
the thickness of the thin layer decreases, the relaxation frequency
moves toward higher frequencies 共Carcione and Picotti, 2006兲,
which means that the rapid variation in NI magnitude also moves toward higher frequencies. Third, the mesoscopic mechanism is not
the only mechanism that causes dispersion; there might be other
mechanisms such as heterogeneity and squirt flow. These mechanisms would be superimposed and could either enhance or reduce
the overall dispersion. However, the key message of this study is not
about what the mechanisms are; it is about different characteristics
of the variation of the NI magnitude versus frequency when dispersion occurs. As long as high dispersion and attenuation occur in reservoir zones, we expect amplitude changes with frequency, which in
general can be characterized by our three classes.
CONCLUSIONS
We studied characteristics of the magnitude and phase angle of NI
from an interface between a nondispersive medium and a dispersive
medium. Using Biot’s poroelastic theory, we performed numerical
modeling for three reservoir models. According to the characteristics of the amplitude-versus-frequency curves for the interface between a nondispersive and a dispersive medium, we divided amplitude-versus-frequency responses into three classes.
One class consists of a low-frequency dim-out reservoir that has a
higher impedance than the overburden medium at zero frequency.
Reflection amplitude decreases toward the lower frequencies because of velocity dispersion.
A second class consists of a phase-shift reservoir that has a slightly lower impedance than that of the overburden medium at zero frequency. At lower frequencies, the reflection has a small negative value, whereas it becomes positive at higher frequencies.
A third class consists of a low-frequency bright-spot reservoir that
has a much lower impedance than that of the overburden medium at
zero frequency. Reflection amplitude increases toward lower frequencies because of velocity dispersion.
ACKNOWLEDGMENTS
We thank the Department of Ocean Engineering of the Massachusetts Institute of Technology for permission to use the software OASES. We are grateful to Dr. Andrey Bakulin, Dr. Hua-Wei Zhou, and
anonymous reviewers for their constructive comments.
APPENDIX A
THE NI IN ATTENUATIVE MEDIA
The formula for the complex reflection and transmission coefficients are derived for a harmonic P-wave propagating along the
z-direction in an attenuative medium. It is assumed that the particle
displacement UP0 will decrease exponentially with the distance traveled. It is expressed as
冋冉 冊 册
UP0 ⳱ exp i␻ t ⳮ
z
ⳮ ␣z ,
VP
共A-1兲
where VP and ␣ are the frequency-dependent phase velocity and attenuation coefficient and ␻ is angular frequency. The equation of
motion is
␳
⳵ 2Uz ⳵ ␴ zz
,
⳱
⳵ t2
⳵z
共A-2兲
where Uz and ␴ zz are particle displacement and stress, respectively,
and ␳ is the medium density. In attenuative media, the stress ␴ zz and
strain ␧zz are related by the complex elastic constants ␭* and ␮* as
共Trapeznikova, 1985兲
␴ zz ⳱ 共␭* Ⳮ 2␮*兲⑀ zz,
共A-3兲
where ␧zz is the strain 共␧zz ⳱ ⳵ Uz / ⳵ z兲. The constants ␭ and ␮ are
equivalent to the Lamé constants in pure elastic media and are related to the complex elastic constants as ␭* ⳱ ␭ Ⳮ i␭⬘ and ␮* ⳱ ␮
Ⳮ i␮⬘. Losses resulting from imperfect elasticity are described by ␭⬘
and ␮⬘.
Inserting equation A-3 into equation A-2 yields the wave equation
2
⳵ 2U z ␭ * Ⳮ 2 ␮ * ⳵ 2U z
2 ⳵ Uz
⳱
⳱
c
,
⳵ z2
⳵ t2
␳
⳵ z2
共A-4兲
which is fulfilled by equation A-1 if
1
1
i␣
⳱ ⳮ .
c VP
␻
共A-5兲
When a P-wave propagates normal to a plane boundary between
two attenuative media, the P-wave is reflected and transmitted at the
Downloaded 25 Feb 2010 to 129.7.72.244. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
Amplitude-versus-frequency variations
boundary. Subscripts 1 and 2 represent the incident and transmitted
media, respectively. In medium 1, the particle displacement of the
incident P-wave is given by equation A-1. The particle displacement
of the reflected P-wave is given by
冋冉
* exp i␻ t Ⳮ
UP1 ⳱ RPP
冊 册
z
Ⳮ ␣ 1z .
VP1
冋冉
冊 册
z
ⳮ ␣ 2z ,
VP2
共A-7兲
• continuity of normal displacement components: 兺1Uz ⳱ 兺2Uz
• continuity of normal stress components: 兺1␴ zz ⳱ 兺2␴ zz
Inserting equations A-1, A-6, and A-7 into the first boundary condition and setting z ⳱ 0 and t ⳱ 0, we have
*
* .
TPP
⳱ 1 ⳮ RPP
共A-8兲
Inserting equation A-3 into the second boundary condition and
then replacing the strain with displacements, we have
共␭1* Ⳮ 2␮1*兲
冋冉
冊 冉
i␻
* i␻ Ⳮ ␣
ⳮ
ⳮ ␣ 1 ⳮ RPP
1
VP1
VP1
冉
*
⳱ 共␭2* Ⳮ 2␮2*兲TPP
ⳮ
冊
i␻
ⳮ ␣2 .
VP2
冊册
共A-9兲
Multiplying the left side of equation A-9 by ␳ 1 / ␳ 1 and right side by
␳ 2 / ␳ 2 yields
冋冉
冊 冉
冊
i␻
i␻
ⴱ
ⳮ ␣ 1共␻ 兲 ⳮ RPP
Ⳮ ␣ 1共 ␻ 兲
ⳮ
VP1共␻ 兲
VP1共␻ 兲
冉
*
⳱ ␳ 2c22TPP
ⳮ
i␻
ⳮ ␣ 2共 ␻ 兲 .
VP2共␻ 兲
冊册
共A-10兲
Equations A-8 and A-10 fully define the complex reflection and
transmission coefficients of the P-wave. Solving the two equations,
we have
* ⳱
RPP
␻ 共␳ 2VP2 ⳮ ␳ 1VP1兲 ⳮ iVP1VP2共␳ 2␣ 1 ⳮ ␳ 1␣ 2兲
, 共A-11兲
␻ 共␳ 2VP2 Ⳮ ␳ 1VP1兲 ⳮ iVP1VP2共␳ 2␣ 1 Ⳮ ␳ 1␣ 2兲
*
TPP
⳱
2␳ 1VP1共␻ ⳮ iVP2␣ 2兲
. 共A-12兲
␻ 共␳ 2VP2 Ⳮ ␳ 1VP1兲 ⳮ iVP1VP2共␳ 2␣ 1 Ⳮ ␳ 1␣ 2兲
After additional algebra, the moduli and phase angles of complex
reflection and transmission coefficients are given by
冋
共A-14兲
TPP共␻ 兲 ⳱ 2␳ 1VP1
and TPP are reflection and transmission coefficients of the
PP
P-wave.
At the boundary, the two continuity conditons are
RPP ⳱
2␳ 1VP1␳ 2VP2␻ 共␣ 2VP2 ⳮ ␣ 1VP1兲
,
2 2
2
2
2
␳ 2VP2共VP1
␣ 21 Ⳮ ␻ 2兲 ⳮ ␳ 21VP1
共VP2
␣ 22 Ⳮ ␻ 2兲
⫻
冑
2
VP2
␣ 22 Ⳮ ␻ 2
2 2
VP2共␳ 2␣ 1 Ⳮ ␳ 1␣ 2兲2
␻ 2共VP2␳ 2 Ⳮ VP1␳ 1兲2 Ⳮ VP1
,
共A-15兲
*
where R*
␳ 1c21
␺ RPP共␻ 兲 ⳱ tanⳮ1
共A-6兲
In medium 2, the displacement of the transmitted P-wave Up2 is
*
UP2 ⳱ TPP
exp i␻ t ⳮ
N47
2 2
␻ 2共␳ 2VP2 ⳮ ␳ 1VP1兲2 Ⳮ VP1
VP2共␳ 2␣ 1 ⳮ ␳ 1␣ 2兲2
2 2
␻ 2共␳ 2VP2 Ⳮ ␳ 1VP1兲2 Ⳮ VP1
VP2共␳ 2␣ 1 Ⳮ ␳ 1␣ 2兲2
册
1/2
,
共A-13兲
␺ TPP共␻ 兲 ⳱ tanⳮ1
␳ 2VP2␻ 共␣ 1VP1 ⳮ ␣ 2VP2兲
.
2
␳ 2VP2共␣ 1VP1␣ 2VP2 Ⳮ ␻ 2兲 Ⳮ ␳ 1VP1共␣ 22VP2
Ⳮ ␻ 2兲
共A-16兲
If medium 1 is pure elastic, which means ␣ 1 ⳱ 0 and VP1 is constant,
the reflection coefficient and its phase angle become equations 16
and 17 in the main text.
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