DOI: 10.1007/s11770-015-0483-3
Zoeppritz equation-based prestack inversion and
its application in fl uid identifi cation*
Huang Han-Dong1,2, Wang Yan-Chao♦1,2, Guo Fei3, Zhang Sheng1,2, Ji Yong-Zhen1, and Liu Cheng-Han4
Abstract: Prestack seismic inversion methods adopt approximations of the Zoeppritz equations to describe the relation between reflection coefficients and P-wave velocity, S-wave velocity, and density. However, the error in these approximations increases with increasing angle of incidence and variation of the elastic parameters, which increases the number of inversion solutions and minimizes the inversion accuracy. In this study, we explore a method for solving the refl ection coeffi cients by using the Zoeppritz equations. To increase the accuracy of prestack inversion, the simultaneous inversion of P-wave velocity, S-wave velocity, and density by using prestack large-angle seismic data is proposed based on generalized linear inversion theory. Moreover, we reduce the ill posedness and increase the convergence of prestack inversion by using the regularization constraint damping factor and the conjugate gradient algorithm. The proposed prestack inversion method uses prestack large-angle seismic data to obtain accurate seismic elastic parameters that conform to prestack seismic data and are consistent with logging data from wells.
Keywords: Prestack inversion, Zoeppritz equation, simultaneous inversion, fl uid identifi cation
Introduction
Prestack seismic inversion uses amplitude-versus-offset attributes in prestack seismic data of different
incident angles to obtain P- and S-wave velocity, density, and elastic moduli, and map the lateral variations in oil- and gas-bearing reservoirs. Therefore, prestack inversion is important for reservoir prediction and hydrocarbon detection in production and development of complex oil
and gas reservoirs.
Prestack inversion is primarily based on the Zoeppritz equations (Zoeppritz, 1919), which describe the energy distribution between reflection and transmission when elastic waves enter an elastic interface. However, the Zoeppritz equations cannot be directly applied to seismic exploration because they are complex and of high computational cost. Based on perturbation theory, Aki and Richards (1980) derived an approximation based on P-wave velocity, S-wave velocity, and density by
Manuscript received by the Editor December 22, 2014; revised manuscript received April 16, 2015.
*This research work is supported by the 973 Program of China (No. 2011CB201104 and 2011ZX05009) and the National Science and the Technology Major Project (No. 2011ZX05006-06).
1. State Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum-Beijing, Beijing 102249, China.2. College of Enhanced Oil Recovery, China University of Petroleum-Beijing, Beijing 102249, China.3. Research Institute, CNOOC Ltd.-Shenzhen, Guangzhou 510240, China.
4. School of Geophysics and Oil Resources, Yangtze University, Wuhan 434023, China.♦Corresponding author: Wang Yan-Chao (Email: wangyanchao86421@sina.com)© 2015 The Editorial Department of APPLIED GEOPHYSICS. All rights reserved.
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Zoeppritz equation-based prestack inversion
linearizing the Zoeppritz equations when the differences in elastic parameters across the reflecting interface were small. The Aki–Richards approximation laid the foundation for prestack inversion methods. Subsequently, many geophysicists simplifi ed the Zoeppritz equations. Shuey (1985) rearranged the Aki–Richards equation and obtained an approximation that refl ects the relation between the reflection coefficients and Poisson’s ratio across the reflecting interface. Shuey’s approximation has been widely used in AVO analysis and inversion of gas-bearing reservoirs (Castagna, 1998; Zhu, 2008). Zheng (1991) used power series to express the refl ection coeffi cients. Based on the Aki–Richards equation, Gray (1999) derived an equation for the relation between reflection coefficients and Lamé parameters. Russell et al. (2003, 2011) proposed a fluid indicating factor denoted by ρf (product of density and fluid) and a reflection coefficient approximation of porous fluid-saturated elastic medium. Meng (2004) used prestack nonlinear P- and S-wave velocity inversion, and obtained good results using the Aki–Richards approximation and generalized nonlinear inversion theory. Based on the Gray approximation, Wang (2007) proposed a method for wave impedance inversion that produced stable and accurate results. Based on the Russell approximation, Yin et al. (2010, 2013) derived and applied a new method for elastic impedance using the fluid factor f, Lamé parameters, and density.
Forward modeling and AVO inversion methods mentioned above use approximations of the Zoeppritz equations and have two evident disadvantages: 1) the precision of the inversion solution decreases with increasing angle of incidence, as Figure 1 shows that the precision of the approximations (e.g., Aki–Richards) decreases with increasing incident angle, especially when the angle of incidence is larger than 20°. 2) most approximations make use of medium–small-angle seismic data and neglect large-angle data. Consequently, the accuracy of the S-wave velocity and density inversion is low, which hinders fluid identification. In
this study, we derive the P-wave reflection coefficients and corresponding partial derivatives by using Zoeppritz equations directly, and construct a three-parameter sensitivity matrix. We also propose an improved prestack inversion method, which combines the prestack simultaneous inversion and conjugate gradient algorithm, to obtain more accurate elastic parameters for fl uid identifi cation and mapping of the lateral variations in reservoirs.
21.510.50-0.5
0
20
4060Incldent angle (°)
80
ZoeppritzAki & richardsThree term shueyTwo term shuey
Fig.1 Precision analysis of the Zoeppritz equation
and approximations.
Zoeppritz equation-based prestack
inversion method
We first analyze the effect of the P-and S-wave velocity, and density on the P-wave reflection coeffi cients and inversion precision, and then discuss the prestack seismic simultaneous inversion algorithm and its stability problem.
Sensitivity analysis of the seismic elastic parameter
Using the AVO models proposed by Rutherford and Williams (1989) and Castagna (1997), we construct the four AVO models listed in Table 1 by using the practical
Table 1 AVO model parameters
ModelModel IModel IIModel IIIModel IV
Lithology
Shale
Gas-bearing sandstone Shale
Gas-bearing sandstone Shale
Gas-bearing sandstone Shale
Gas-bearing sandstone
Vp (m/s)30204060254026802450182034501920
Vs (m/s)14552530112016157858251570925
ρ (kg/cm3)23002400230021002200190024002000
200
P-wave reflection coefficientHuang et al.
logging data in YX area in Shengli Oilfi eld and calculate the variation of the P-wave refl ection coeffi cients when the P- and S-wave velocity and density of gas-bearing sandstone increase or decrease 10% using the Zoeppritz equation, as shown in Figures 2 and 3. Subsequently, we analyze the sensitivity of the reflection coefficients to the abovementioned parameters and reach the following conclusions.
First, we show the sensitivity of the reflection coefficients to the P- and S-wave velocities when the wave velocities and density increase or decrease and the incident angle is less than 50° (see Figure 2). The reflection coefficient sensitivity to density slightly differs in all four models but we conclude that model I produces the most precise inversion results when we use the Zoeppritz equation and the AVO characteristics in prestack inversion.
Second, in all models and for increasing or decreasing velocities and incident angles of 0°–50°, the variations of the reflection coefficients are most obvious for the P-wave velocity, which suggests that this incident angle range is sensitive to P-wave velocity. Besides, the sensitivity of the reflection coefficients to P-wave velocity increases with increasing incident angle. Consequently, the prestack inversion produces precise data for P-wave velocity. Next, the sensitivity of the reflection coefficient to density is relatively high when the incident angle range is 0°–20°; however, the sensitivity decreases with increasing incident angle. Finally, the sensitivity of the reflection coefficient to S-wave velocity is the lowest when the incident angle range is 0°–20°. Nevertheless, the sensitivity increases with increasing incident angle. The sensitivity of the reflection coefficient to S-wave velocity is higher than
Model IModel IIModel IIIModel IV
Variation of reflection coefficientVariation of reflection coefficientVariation of reflection coefficientVariation of reflection coefficient10
0.250.250.250.25
0.20.20.20.20.15
0.150.150.15
0.10.10.10.1
0.050.050.050.05
00
1020304050
0
01020304050
00
20304050
00
1020304050
Incident angle (°)Incident angle (°)
P-wave velocity decreaseIncident angle (°)
S-wave velocity decreaseDensity decreaseIncident angle (°)
Fig.2 Sensitivity analysis of the P-wave refl ection coeffi cients for decreasing velocities and density.
Model I
0.90.8
0.20.18
Model II
0.20.18
Model III
0.20.18
Model IV
Variation of reflection coefficientVariation of reflection coefficient0.70.60.50.40.30.20.100
10203040Incident angle (°)
50
Variation of reflection coefficientVariation of reflection coefficient0
10203040Incident angle (°)
0.160.140.120.10.080.060.040.0200
10203040Incident angle (°)
50
0.160.140.120.10.080.060.040.020
50
0.160.140.120.10.080.060.040.0200
10203040Incident angle (°)
50
P-wave velocity decreaseS-wave velocity decreaseDensity decreaseFig.3 Sensitivity analysis of the P-wave refl ection coeffi cient for increasing velocities and density.
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Zoeppritz equation-based prestack inversion
the sensitivity to density when the incident angle is greater than 30°, which shows that large-angle prestack seismic data are also sensitive to S-wave velocity. Hence, we can improve the precision of the S-wave velocity inversion by using large-angle prestack seismic data.
From the above, it can be seen that medium- and small-angle seismic data reveal more about P-wave velocity and density. Nevertheless, large-angle seismic wave data provide signifi cant information about P- and S-wave velocity as well. Thus, we could improve the precision and stability of the inversion parameters by using the empirical formula between velocity and density (Huang, 2011) and large-angle prestack seismic data in prestack constrained inversion. Figure 1 shows that the precision of the approximations (e.g., Aki–Richards) decreases with increasing incident angle, especially at wide incident angles, the precision is lower than that of the Zoeppritz equation. Therefore, prestack inversion directly based on the Zoeppritz equation can make the most of wide-angle prestack seismic data; furthermore, constraints to wave velocities and density promote stable prestack inversion solutions and improve the inversion precision of S-wave velocity and density as well.
where subscript l is the serial number of sampling points and n is the sample number. We substitute equation (2)
rst partial to equation (1) and derive ∆Vp, ∆Vs, and the fi
derivative of ∆ρ as follows:
wSin1wSi
2('d¦'Vpl¦'Vsl
wVpll 0wVsll 0
n1
wSin1wSi
0,)¦¦'Ul
VUwwl 0ll 0pl
n1
(3)
n1
wSin1wSi
2('d¦'Vpl¦'Vsl
wVpll 0wVsll 0
wSin1wSi
0,)¦¦'Ul
UVwwl 0ll 0sl
n1
(4)
wSin1wSi
2('d¦'Vpl¦'Vsl
wVpll 0wVsll 0
n1
wSin1wSi)¦ 0,¦'Ul
UUwwl 0ll 0l
n1
(5)wSi
,
l 0wVpl
n1
where 'di DVp,Vs,USVp,Vs,U. Because ¦ii
n1
Improved prestack inversion algorithm
Conventional prestack inversion results are characterized
by strong nonlinearity and multiple solutions. Thus, we use the conjugate gradient algorithm. Moreover, to minimize the effect of noise, we use regularization to constrain the simultaneous prestack seismic inversion. The following objective function is used in the inversion (Huang, 2013)
I(m) S(m)'Domin, (1)
2
n1
wSwSi
and ¦i are not all zero, we obtain equation ¦l 0wUll 0wVsl
(6) from equations (3) to (5)
wSn1wSn1wS
'd¦'Vpl 0. (6)¦'Vsl¦'Ul
wVpll 0wVsll 0wUll 0In order to test the effect of Zoeppritz equation based
prestack inversion, we conduct simultaneous inversion of small-, medium-, and large-angle (≥30°) seismic data, which decreases the number of inversion solutions and increases the stability of prestack inversion. Assuming three small-, middle-, and large-angle gather datasets D1, D2 and D3, we write
n1
wS°G(Vp) ¦l 0wVpl°°n1
wS°
®G(Vs) ¦. (7)
wVl 0pl°
°n1
°G(U) ¦wS°l 0wUl¯
n1
where m = (Vp, Vs, ρ) and D is the angle gather. S(m)
∆
is the expected seismic response; it is expressed as the convolution of the P-wave reflection coefficient R(m) calculated using the Zoeppritz equations and
seismic wavelet it is expressed as the convolution of the seismic wavelet W and the P-wave refl ection coeffi cient calculated using the Zoeppritz equations.
We expand S(m)∆ as Taylor series and omit the second-order derivatives
wSn1wSn1wS
,S(m) S(m)¦'Vpl¦'Vsl¦'Ul
wVpll 0wVsll 0wUll 0
'
n1
202
(2)
We substitute equation (7) to equation (6) and obtain
Huang et al.
'd1G1(Vp)'VpG1(Vs)'VsG1(U)'U 0 °®'d2G2(Vp)'VpG2(Vs)'VsG2(U)'U 0. (8)°'dG(V)'VG(V)'VG(U)'U 0
33pp3ss3¯
Subscripts 1, 2, and 3 in equation (8) correspond to small-, medium-, and large-angle stack gathers. The matrix form of equation (8) is
(3) Fix the recursion equation of the local linear optimization
'mk 'mk1ak1Pk1.
(17)
ªG1(Vp)G1(Vs)G1(U)ºª'Vpºª'd1º«»» «'d». (9)'V «G2(Vp)G2(Vs)G2(U)»«s«»«2»«G3(Vp)G3(Vs)G3(U)»«¼«¬'d3»¼¬¼¬'U»We then calculate the equation
G'm 'd, (10)where 'm ª¬'Vp,'Vs,ǼUº¼ is the model parameters correction matrix.
Based on the regularization constraint principle, we transform equation (10) into the damping regular equation
TTT
ª¬GGOBBº¼'m G'd, (11)
T
We set the convergence conditions and use the
correction of the iteratively calculated model parameters to obtain the P-wave velocity, S-wave velocity, and density.
Sensitivity matrix
Matrix G in equation (10) is referred to as the inversion sensitivity matrix and it is a Jacobian matrix. Zoeppritz equations describe the relation between reflection coefficients and elastic parameters. Hence, a sensitivity matrix based directly on a well-defined equation is critical to the inversion algorithm. (1) Matrix construction The sensitivity matrix G is
ªG1(Vp)G1(Vs)G1(U)º«»
G «G2(Vp)G2(Vs)G2(U)». (18)
«G3(Vp)G3(Vs)G3(U)»¬¼Each element is a partitioned matrix that according to
the convolution model is
n1
wSiwRiwRi
GWW(V)(il1)(il)¦°p
wVpl1wVpll 0wVpl
°
n1°wSiwRiwR°
W(il1)W(il)i,®G(Vs) ¦wVsl1wVsll 0wVsl°
°n1
wSwRiwR
°G(U) ¦i W(il1)W(il)i
wUl1wUl°l 0wUl¯
(19)
where C = GG + λBB is the regularization factor, λ is the damping factor, and B is the identity matrix. Using the conjugate gradient algorithm, we solve the generalized linear inversion equation mentioned above by taking the following steps.
(1) Fix the search direction (conjugate vector)
T
T
Pk GckEkPk1,where
(12)
c Gc (13)k Gk1Dk1CPk1,
ccc Ek (Gck,Gk)/(Gk1,Gk1),and when k = 0,
T
Gc0 P0 G'dC'm0. (15)
(14)
where subscript i denotes the ith sampling point.
From equations (18) and (19), we obtain the local G(Vp) elements by solving
wRiwR
and i. By analogy, wVPl1wVPl
(2) Fix the modified step a of the model parameters
correction matrix ∆m
the sensitivity matrix G is worked out.
The Zoeppritz equation is expressed as (Zoeppritz, 1919)
AR B, (20)where
c (16) ak1 Gck1,Gk1/Pk1,CPk1.
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Zoeppritz equation-based prestack inversion
ªsinD1«
«cosD1«
A «cos2E1 «
««
«sin2D1¬ªRppº«R»
R=«ps»,
«Tpp»«»«¬Tps»¼
cosE1
sinE1
Vs1
sin2E1Vp1
cos2E1
sinD2cosD2U2Vp2cos2E2
U1Vp1
Vp1Vs1U2Vs22Vp1
sin2D2VU1Vs2p21
º
»»»U2Vs2
sin2E2», (21)U1Vp1»
»
U2Vp1Vs1»
cos2E2»2
U1Vs1
¼
cosE2sinE2
ªsinD1º«cosD»
1»B «, (22)
«cos2E1»«»sin2D¬1¼
then substitute to equation (19) to solve the elements of
the sensitivity matrix G.
(2) Derivation of the matrix elements
The derivatives of the reflection coefficient with respect to the three parameters are
wRwAwBwRwAwB11AR+AR+()() °VVVVVVwwwwwwpppppp°llll1l1l1°wAwBwRwAwB°wRR+R+ A1( A1()),®wwwwwwVVVVVVslslsl1sl1sl1°sl°wRwAwBwRwAwB° A1( A1(R+R+))°wwwwwwUUUUUUlll1l1l1¯l ection coeffi cient that we want and Rpp is the P-wave refl
to solve.
We extract the derivative of equation (20) with respect to M, after assuming that the model parameters across the refl ecting interface are M = (Vpl+1, Vpl, Vsl+1, Vsl, ρl+1, ρl),
wAwRwB
RA . (23)wMwMwM
Then, the derivative of R with respect to M is
(25)and according to Snell’s law, if the transmission angle
α2, the refl ection angle β1, and the transmission angle β2 of the converted S-wave are replaced with the incident
cient matrixangle α1, we obtain the coeffi
wRwAwB
). (24) A1(R+
wMwMwMBased on equation (24), we solve the derivative of Rpp
with respect to Vpl+1, Vpl, Vsl+1, Vsl, ρl+1, and ρl, which we
ª«sinD1«««cosD1««A ««Vs122«122sinD1vp1«««sin2D1««¬1Vs12Vp21sinD12Vp2sinD1Vp1Vp22V2p1Vs1sinD1Vp11Vs1Vp1Vs12V22p112sin2D12Vs12sinD1V2p1sinD1Vs2U2Vp(122sin2D1)VpU1Vp2211Vp1Vs12sinD1U2Vs2sinD1Vp2212sin2D12VpU1Vs2211º»12sinD1Vp1»»»Vs2sinD1»Vp1»»,(26)22»Vs2U2Vs2sinD122Dsin11»Vp21U1Vp21»»2U2Vp1Vs2Vs2»2D(12sin)»1Vs2U1Vs12»2¼Vs222sinD1ªº«»cosD1«»2», (27) B «Vs12«122sinD1»Vp1«»«»sin2D¬¼1wBwAwAwAwBwB
, , , , , and and
wUwVpwVswUwVpwVs
then substitute them to equation (25). Subsequently, the partial derivatives of the refl ection coeffi cients with respect to P-wave velocity, S-wave velocity, and density We derive
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are obtained and substituted to equation (19). Thus, each element in the sensitivity matrix is determined.
0
5
10
Angle of incide (degre)1520253035
40
45
50
15401540Time (ms)1580160016201640
Control for inversion stability
When we use equation (11) in the inversion of
prestack seismic traces, the damping factor vector in the regularization constraint vector λ is equivalent to a compromise parameter. The estimated resolution of the inversion solution increases with decreasing λ, whereas the estimated resolution and deviation of the inversion solution decreases with increasing λ. Therefore, λ is crucial to the stability and resolution of the inversion.In this study, we assume that the variance C∆m of the model parameters correction matrix ∆m and variance σ2 of the seismic data noise ∆d–G(m)·∆m are random sequences with zero mean; hence, the damping factor is Oki V2i/C'mi i 1,2,\",length(D), (28)where k refers to the number of iterations. Equation (28) describes the damping factor. The latter is a function of the number of iterations only, which improves the antinoise properties.
Fig.4 Prestack seismic angle gather with a small amount of
random noise.
and the Zoeppritz equation, where angle higher than the critical angle is not included.
According to Figure 5, the comparison of the inversion using the Zoeppritz and Aki–Richard equations suggest that P-wave velocity and density inversion results based on the Zoeppritz equation are more accurate and the proposed inversion algorithm is feasible. Moreover, the inversion results for the S-wave velocity are not as good because the synthetic P-wave seismic records lack suffi cient information for the S waves, which affects the stability of the S-wave velocity inversion.
Model computations
One-dimensional model validation algorithm
Figure 4 shows the P-wave seismic record of an actual well, which is synthesized using a 30 Hz Ricker wavelet
P-wave velocity (m/s)
30001540155015601570Time (ms)Time (ms)Two-dimensional model validation algorithm
Based on the oil-reservoir distribution and rock physical parameters of the work area, we built a geological model that is shown in Figure 6 and with the parameters listed in Table 2.
S-wave velocity (m/s)4500
16001540155015601570
Time (ms)Density (kg/m3)
2400
220015401550156015701580159016001610162016301640
2300
2400
2500
2600
35004000180020002200
1580159016001610162016301640
Initial model
1580159016001610162016301640
Well log dataAki & richards formula inversionZoeppritz equation inversion
Fig.5 Comparison of the inversion results.
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Zoeppritz equation-based prestack inversion
Table 2 Parameters of the two-dimensional geological model
LithologyMudstone
Gas-bearing sandstone Oil-bearing sandstoneWater-bearing sandstone
Tight sandstone
Vp (m/s)21001670190020002200
Vs (m/s)8001160111010601200
ρ (kg/m3)21001990205020702200
Figure 7 is a synthetic stacked seismic section at incident angles of 30–40˚. To use it in prestack inversion, two data points were selected at common depth points (CDP) of 60 and 120 to constrain the seismic inversion. Moreover, the interpretation of the seismic section is also used as a constraint to improve the reliability of the inversion.
090095010001050Time (ms)11001150120012501300
20406080
CDP100
120140160180200
Fig.6 Two-dimensional geological model: gas (yellow), oil (red), water (blue), tight sandstone
(turquoise), and mudstone (gray).
10
900
20
30
40
5060
w60
70
CDP
8090100110120
w120
130140150160170180190200
950.0950.0
1000
1000.01000.0
1050.01050.0
Time (ms)1100
1100.01100.0
1150.01150.0
1200
1200.01200.0
1250.01250.0
1300
Fig.7 Simulation wells and the interpretation of the structure of the original seismic profi les.
The red and blue logging curves represent the P-wave velocity and S-wave velocity, respectively.
Figures 8–10 show the P-wave velocity, S-wave
velocity, and density profiles. The inversion results of the three parameters basically are consistent with the 206
model results. This suggests that the proposed prestack inversion algorithm is reliable.
Huang et al.
10900
2030405060w607080CDP
90100110120w120130140150160170180190200950.0950.01000
1000.01000.0Time (ms)1050.01050.01100.01100
1100.01150.01150.01200
1200.01200.01250.01250.0167017011732176417951826185718881919195119822013204420752106213821692200Fig.8 P-wave velocity inversion section. The red logging curve is the P- wave velocity.
10900
2030405060w607080CDP
90100110120w120130140150160170180190200950.0950.01000
1000.01000.0Time (ms)1050.01050.01100.01100
1100.01150.01150.01200
1200.01200.01250.01250.0800824847871894918941965988101210351059108211061129115311762200Fig.9 S-wave velocity inversion section. The red logging curve is the S-wave velocity.
Application
To test the applicability of the proposed prestack inversion algorithm to actual data, we consider the Pinghu formation (T30−T40), which comprises delta deposits affected by tides. The target stratum is a relatively thin and well-developed sand body that pinches out laterally, which makes the reservoir extremely irregular. The seismic data have high signal-to-noise ratio. As shown in Figure 11, the basic frequency of the
target stratum is about 30 Hz and the frequency band is around 10−50 Hz. The wave is reflected at 45˚. In the inversion, we use prestack angle gathers of 2°−15°, 15°−28°, and 28°−42°.
First, we build a three-dimensional stratigraphic model according to the structural interpretation to constrain the seismic trace extrapolation. Figure 12 shows the P-wave velocity, S-wave velocity, and density inversion results near the FD-1 well and logging data. The inversion results are in good agreement with the logging data.
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Zoeppritz equation-based prestack inversion
10900
2030405060w6070CDP
8090100110120w120130140150160170180190200950.0950.01000
1000.01000.0Time (ms)1050.01050.01100.01100
1100.01150.01150.01200
1200.01200.01250.01250.0190020022015202720392052206420762089210121142126213821512163217521882200Fig.10 Density inversion section. The red logging curve is the density.
Line2170CDP10442600
219010442210104422301043225010422270104122901041FD-13281.23381.23478.73574.5B33284.03390.93488.93585.13678.33778.93875.03970.22800Time (ms)3669.73763.83859.63959.53000
4060.54157.04259.44367.6Amplitude1.000.750.500.250010203040506070Ferquency (Hz)80901001101203200
4464.4Fig.11 Frequency spectrum analysis of the target stratum in the prestack seismic data.
P-wave velocity (m/s)180038005800InversionWell28502850S-wave velocity (m/s)90019002900InversionWell2850Density (kg/m3)210024002700InversionWell290029002900Time (ms)Time (ms)29502950Time (ms)2950300030003000305030503050Fig.12 Comparison of the inversion results for the FD-1 well and logging data.
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After analyzing the core data for the FD-1and BD3 wells, we fi nd that the P-wave velocity in the sandstone and mudstone is 2700–4300 m/s and 3300–5100 m/s, respectively. Moreover, the oil and gas reservoir is characterized by low density, low Poisson’s ratio, and low Lamé parameters. We use cross plots of these parameters and the P-wave velocity obtained with the prestack inversion to identify the fluid type in the sandstone reservoirs.
Figure 15 shows the P-wave velocity inversion data for wells FD-1 and B3. We clearly see the pattern and distribution, and pinch-out characteristics of the low-45%40%35%30%25%20%15%10%5%0%2.5%0.0%4.5%0.0%0.0%2.0%2.9%11.2%8.4%4.4%1.2%0.5%2.1%1.0%0.0%0.8%0.3%0.1%24.3%21.5%17.5%15.1%37.7%velocity sandstone in the Pinghu formation. Figure 16 shows the inversion results for the fluid-sensitive Poisson’s ratio. The results show that the gas and water-bearing zones agree with prediction in Figure 15 and is the gas and water reservoirs encountered in the wells FD-1 and B3. Meanwhile, The FD-1 Log also shows the gas layer encountered by drilling at 3565m corresponds to low value zone (Poisson’s ratio value is from 0.255 to 0.3) in the inverted Poisson’s ratio section in Figure 16 and the B3 log shows that the water-beating sandstone at 3644 m corresponds to high value zone (the poisson’s ratio value is from 0.31 to 0.34). The gas-bearing layer
78Lame coefficient* Density6858483828180.270.280.290.300.310.32Poisson’s ratio
Oil-gas layerWater layerOil- bearing water layer41.9%SandstoneMudstone2700-29002900-31003100-33003300-35003500-37003700-39003900-41004100-43004300-45004500-47004700-49004900-51000.330.340.35P-wave velocity distribution (m/s)
Fig.13 P-wave velocity distribution in the sandstone and
mudstone.
Fig.14 Cross plot of the product of density and Lamé
parameters vs Poisson’s ratio.
Line2170CDP1044260021901044FD-1221010442230104322501042B322701041229010412800Time (ms)3000Sandstonepinchout3200Low-velocitysandstone26002769293831063275344436133781395041194288445646254794496351315300Fig.15 Velocity inversion section for wells FD-1 and B3.
209
Zoeppritz equation-based prestack inversion
and water-bearing layer both belong to same set of sand bodies. Similarly, a gas-rich reservoir is encountered at 4062 m and 4110 m, respectively. These high-precision
2170104426002190104422101044CDP22301043predictions suggest that the proposed prestack inversion algorithm is reliable.
22501042227010412290104126003565.6-3568.2 m gas layer3644.0-3653.5 m water layer28003810.1-3822.6 m gas layer2800Time (ms)4051.3-4056.8 m gas layer4099.1-4105.9 m gas layer30004062.0-4076.9 m gas layer30004110.1-4122.6 m gas layer4094.1-4107.2 m gas layer4142.3-4152.0 m gas layer320032000.2550.2610.2670.2740.2800.2860.2920.2990.3050.3110.3170.3240.3300.3360.3420.3490.355Fig.16 Poisson’s ratio inversion results for wells FD-1 and B3 and logging data for gamma
rays (left) and P-wave velocity (right).
Conclusions
P-wave reflection coefficients are most sensitive to P-wave velocity, whereas the sensitivity to density decreases and the sensitivity to S-wave velocity increases with increasing angle of incidence. Unlike the Zeoppritz equations, the accuracy of the Aki–Richards approximation decreases with increasing angle of incidence. Consequently, prestack inversion based on the Zeoppritz equations takes advantage of large-angle seismic data and thus improves the inversion results for elastic parameters. Finally, we fi nd good agreement between logging data and inversion results, suggesting that the proposed method can be used in oil and gas exploration.
Haiming for their assistance in the prestack inversion algorithm and also to thank the editor and anonymous reviewers for their helpful comments.
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Acknowledgements
The authors wish to thank Liu Hongchang and Xu 210
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Professor Huang Han-Dong Ph.D. received his M.S. from
China University of Petroleum (Huadong) in Applied Geophysics in 1994 and his Ph.D. in Geodetection and Information Technology in 2000 from Chengdu University of Technology. He is presently in the Enhanced Oil Recovery Institute of China University of Petroleum (Beijing). He is primarily interested in methods that
combine geology and geophysics.
211
面处波阻抗差异、储层内部波的频散与衰减，以及 顶底界面波的调谐与干涉。模拟结果表明，渗透率 的增加显著降低纵波速度，使其在高、低频弹性极 限之间发生频散。储层速度频散与层状构造共同导 致反射系数的频变现象。在储层与围岩波阻抗接近 的情况下，地震响应对渗透率变化具有敏感性，对 于不同储层厚度，当围岩为高速页岩时，反射波叠 加振幅随渗透率增加而增加；当围岩为低速页岩时， 叠加振幅随渗透率增加而降低。 关键词：斑块饱和模型，频散与衰减，渗透率，传播 矩阵，AVO 基于Zoeppritz方程的叠前地震反演方法研究及其在 流体识别中的应用 //Zoeppritz equation-based prestack inversion and its application in fluid identification,黄捍 东口，王彦超1，2，郭飞3，张生丨,2，纪永祯1，刘承汉4， APPLIED GEOPHYSICS, 2015，12(2)，P. 199-211. 摘要：地震低频信息能够提高分辨率与成像精度， 改善反演质量，甚至直接进行油气检测，需要对其 进行有效保护与拓展。对于子波而言，缺失低频信 息会导致主瓣幅度降低、第一旁瓣幅度增加，并出 现次级旁瓣呈周期震荡衰减的现象；从合成地震 记录和典型地质模型来看，低频缺失会产生假同相 轴，造成分辨率提高的假象，且模型不同位置的特 征存在一定差异；对缺失低频的模型数据进行波阻 抗反演，会造成构造失真、岩性改变的假象，特别 是高陆构造和薄互层。针对缺失低频的地震资料， 本文还研究了基于压缩感知与稀疏约束的拓频方 法，开发了相应的模块，并对实际CIP道集进行处 理，取得了较好的应用效果。 关键词：地震子波，正演模拟，低频拓展，压缩感 知，稀疏约束 微地震定位的加权弹性波干涉成像法//Weighted-elastic-wave interferometric imaging of microseismic source location,李喬丨.2,陈浩丨，王秀明1，APPLIED GEOPHYSICS, 2015，12(2)，P. 221-234. (1.油气资源与探测国家重点实验室（中国石油大学 (北京)），北京102249; 2.中国石油大学提高釆收率 研究院，北京10224% 3.中海石油深圳分公司研究院， 广州510240; 4.长江大学地球物理与石油资源学院， 湖北武汉430100) 摘要：现阶段的叠前地震反演技术中用于描述反射 系数与纵、横波速度和密度之间的关系几乎完全是 Zoeppritz方程的近似式，由于这些近似公式在大角 度和弹性参数变化剧烈时误差较大，这不仅降低了 反演解的精度，而且增加了叠前反演的多解性。本 文探索了直接利用Zoeppritz方程求解精确反射系数 的理论方法，并基于广义线性反演理论详细推导了 基于叠前大角度地震资料的纵、横波速度和密度三 参数同步反演算法，同时在反演过程引入正则化约 束阻尼因子和共轭梯度算法，有效降低了反演的不 适定性和提高了反演收敛性。理论模型试算和实际 工区应用表明，本文提出的反演方法能够有效利用 大角度（一般入射角〉30°)的叠前地震数据，获得 更精确的地震弹性参数反演结果，并且反演结果忠 实于地震资料，与井吻合较好。 关键词：叠前反演，Zoeppritz方程，高精度，流体 识别 地震低频信息缺失特征分析及拓频方法研究//Low-frequency data analysis and expansion,张军华、 张彬彬、张在金、梁鸿贤2，葛大明2，APPLIED GEOPHYSICS, 2015, 12(2), P. 212-220. (1.中国石油大学（华东）地学院，青岛266580; 2.胜利油田物探研究院，东营257022) (1.中国科学院声学研究所，北京100190; 2.中国 科学院大学，北京100049) 摘要：震源定位是微地震监测关键技术之一。本文 提出用于微地震定位的弹性波和加权弹性波�WEW) 干涉成像方法。该方法在保证定位精度的同时，还 可避免震源假象。通过各向同性水平层介质状模型 的数值试验，初步表明该方法可适应低信噪比微震 信号、速度随机扰动、较稀疏的检波器分布等情况， 并在速度模型存在一定的系统误差时也仍保持较高 的定位精度。由于干涉成像方法不需要进行初至拾 取，定位效率相对传统走时方法也得到了提高。釆 用二维断层模型试算 Nnumerical results of using a two-dimensional fault model,表明方法还能实现多震 源定位，且比逆时成像有更高的定位精度。 关键词：微地震监测，震源定位，弹性波，干涉偏移， 逆时成像 广义炮检距地震菲涅耳带横向叠加波场研究//Lateral wave-field stacking of seismic Fresnel zones for the generalized-offset case,田楠，范廷恩，王宗俊，蔡文 涛，APPLIED GEOPHYSICS, 2015,12(2), P. 234-243. (中海油研究总院，北京100027) 摘要：便于不同观测系统的统一，本文定义了广义 炮检距概念，给出了空间平界面广义炮检距不同阶 地震菲涅耳带表达式。基于波动理论，推导出了广 275
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