The seismic responses caused by different reservoir parameter variations are numerically simulated, and then the feasibility of discriminating different reservoir parameters and realizing quantitative interpretation using time-lapse seismic AVO technique is ensured. Based on Aki and Richards’ simplified AVO equation, the formula of P-P wave and P-S wave for time-lapse seismic AVO was derived in details. According to the rock physical model of S oil field and the formula acquired, the multiple time-lapse seismic AVO inversion equations are achieved to discriminate the changes of oil saturation and effective pressure. It is shown by simulated data experiment that the time-lapse seismic AVO inversion is feasible, and the formula derived in this paper is effective to discriminate the changes of oil saturation and effective pressure, and to improve the precision of time-lapse seismic interpretation.

1. CHINESE JOURNAL OF GEOPHYSICS Vol.48, No.4, 2005, pp: 974∼981
A STUDY ON MULTIPLE TIME-LAPSE SEISMIC AVO INVERSION
LI Jing-Ye CHEN Xiao-Hong HAO Zhen-Jiang RUI Zhen-Hua
Key Laboratory for Hydrocarbon Accumulation Mechanism, Ministry of Education,
China University of Petroleum, Beijing 102249, China
Abstract The seismic responses caused by different reservoir parameter variations are numerically simulated,
and then the feasibility of discriminating different reservoir parameters and realizing quantitative interpretation
using time-lapse seismic AVO technique is ensured. Based on Aki and Richards’ simplified AVO equation, the
formula of P-P wave and P-S wave for time-lapse seismic AVO was derived in details. According to the rock
physical model of S oil field and the formula acquired, the multiple time-lapse seismic AVO inversion equations
are achieved to discriminate the changes of oil saturation and effective pressure. It is shown by simulated data
experiment that the time-lapse seismic AVO inversion is feasible, and the formula derived in this paper is effective
to discriminate the changes of oil saturation and effective pressure, and to improve the precision of time-lapse
seismic interpretation.
Key words Time-lapse seismic, AVO inversion, Rock physics, Numerical simulation, Quantitative interpretation.
1 INTRODUCTION
Time-lapse seismic reservoir monitoring technique has been widely applied to monitor the reservoir varia-
tions caused by production, then help to search dead oil area, to ensure new well location, and to optimize the
injection-extraction scheme for higher recovery ratio[1] . But during the production with water or gas injection,
the seismic responses can vary with the oil saturation change caused by oil extraction and water or gas injection,
on the other hand, the pressure system change in reservoir also can cause the variation of seismic responses. The
amplitude difference of post-stack time-lapse seismic data is the final processing data with higher resolution,
but the stacked data lose a lot of important information and make it difficult to discriminate the variations of
oil saturation and effective pressure for their coupled seismic responses[2] .
Thus, the quantitative characterization of reservoir is much needed during the reservoir development.
Laboratory and field data show that the variations of velocities of P wave and S wave and density caused
by the changes of oil saturation, effective pressure and other reservoir parameters are different. Therefore,
time-lapse seismic AVO (Amplitude Versus Offset) simulation and inversion is a most potential method to
discriminate the variations of oil saturation and effective pressure and realize the time-lapse seismic data quan-
titative interpretation[3] . Since 1980, a lot of geophysicists have been studying seismic AVO, and the Zoeppritz
equation is the basic theory[4] , which precisely describes the relations between amplitude, incident angle, den-
sity and velocities, and makes it possible to predict the lithology by seismic data only. But the full Zoeppritz
equation solution is very complex, which makes it difficult to be applied in real production. Many scholars
simplified the equations in different approximate forms[4∼6] , and using these forms the seismic AVO inversion
can be conducted. At present, P wave AVO inversion has been applied as a routine method in oil prediction.
In recent years, Landro[7] presented that the oil saturation and effective pressure changes can be quantitatively
interpreted by time-lapse seismic AVO inversion. The P wave AVO inversion has achieved success in some
degree, and in the same time, it has lots of shortcomings. With the progress of seismic technique in recent
years[8,9] , multi-component seismic data can be acquired in scientific research and real production[10∼13] , and it
is possible to study converted wave AVO in multi-component seismic data[14∼16] . In this paper, the feasibility
to discriminate different reservoir parameter variations by multiple time-lapse AVO inversion is proved. Then,
P-P wave and P-S wave time-lapse AVO equation is derived in details from Aki and Richards’ approximate
E-mail: ljy3605@sina.com

2. Li J Y et al.: A Study on Multiple Time-Lapse Seismic AVO Inversion 975
equations[3] . At last, according to the real reservoir situations, multiple time-lapse seismic AVO inversion is
conducted to discriminate the variations of oil saturation and effective pressure, to realize the seismic data
quantitative interpretation.
2 FEABILITY OF TIME-LAPSE SEISMIC AVO INVERSION
The feasibility of time-lapse seismic AVO inversion to discriminate oil saturation and effective pressure
variations is decided by their different time-lapse seismic responses. After the analysis of rock characteristics,
the relations between P wave and S wave velocity variations and oil saturation and effective pressure changes
can be calculated by the constructed rock physical relationship[17] based on the data measured in laboratory
and in the S oil field (Fig. 1 and Fig. 2). Fig. 1 and Fig. 2 show that the variations of oil saturation and effective
pressure affect P wave and S wave velocities by different rules. Time-lapse seismic simulation can be conducted
by solving Zoeppritz equation precisely[18] , and then the AVO curves can be analyzed before and after the
production. The Zoeppritz equation can be written as the following for P wave incidence[4] .
A1 cos θ1 − B1 sin λ1 + A2 cos θ2 + B2 sin λ2 = A0 cos θ1 , (1a)
A1 sin θ1 + B1 cos λ1 − A2 sin θ2 + B2 cos λ2 = −A0 sin θ1 , (1b)
A1 Z1 cos 2λ1 − B1 W1 sin 2λ1 − A2 Z2 cos 2λ2 − B2 W2 sin 2λ2 = −A0 Z1 cos 2λ1 , (1c)
A1 γ1 W1 sin 2θ1 + B1 W1 cos 2λ1 + A2 γ2 W2 sin 2θ2 − B2 W2 cos 2λ2 = A0 γ1 W1 sin 2θ1 . (1d)
In the formulae, A1 , A2 are the reflection amplitude and transmission amplitude of P wave, A0 incident P
wave amplitude, B1 , B2 are the reflection amplitude and transmission amplitude of S wave, ρi medium density,
γi = βi /αi , Zi = ρi αi , Wi = ρi βi , i = 1, 2 and Zi , Wi are wave impedance.
Fig. 1 Velocity of P wave and S wave varied Fig. 2 Velocity of P wave and S wave varied
with oil saturation with effective pressure
According to the real situation of S oil field, AVO curves of reservoir upper interface have been simulated
respectively when oil saturation and effective pressure varied (Fig. 3 and Fig. 4). Fig. 3 and Fig. 4 show that
P-P wave AVO curves change obviously for both bigger and smaller incident angles when oil saturation varied,
while the change is obvious at small incident angles and weak at large incident angles when effective pressure
varied. P-S wave AVO curves change insigificantly when oil saturation varied, but change obviously at large
incident angles when effective pressure varied. In conclusion, quantitative interpretation of the variations of oil
saturation and effective pressure by multiple time-lapse seismic AVO inversion is feasible, because P-P and P-S
wave AVO curves have different varying rules when oil saturation and effective pressure vary.

3. 976 Chinese J. Geophys. Vol.48, No.4
Fig. 3 P-P wave and P-S converted wave reflection coefficient varied with oil saturation
Fig. 4 P-P wave and P-S converted wave reflection coefficient varied with effective pressure
3 TIME-LAPSE SEISMIC AVO THEORY
It is hard to conduct petrophysics analysis by AVO inversion using Zoeppriz equations for their complex
solution, so it is necessary to do reasonable simplification to them. In 1980, Aki and Richards[3] presented the
simplified Zoeppritz equations in isotropic medium. At small incident angles the approximate P wave and S
wave reflection formulas for P wave incidence can be written as follows.
RP−P (θ) =A + B sin2 θ + C sin4 θ, (2)
3
RP−S (θ) =E sin θ + F sin θ, (3)
where
1 ∆VP ∆ρ
A= + ,
2 VP ρ
1 ∆VP ∆VS ∆ρ
B = − 4η 2 − 2η 2 ,
2 VP VS ρ
1 ∆VP
C = ,
2 VP
∆VS 1 ∆ρ
E = − 2η − η+ ,
VS 2 ρ
1 3η ∆ρ ∆VS
F =η + + (1 + 2η) ,
2 4 ρ VS
VP =(VP1 + VP2 )/2, ∆VP = VP2 − VP1 , VS = (VS1 + VS2 )/2, ∆VS = VS2 − VS1 ,
ρ =(ρ1 + ρ2 )/2, ∆ρ = ρ2 − ρ1 , η = VS /VP , θ = (θ1 + θ2 )/2.
In above formulae, VP1 , VP2 , VS1 , VS2 , ρ1 and ρ2 are P and S wave velocities and densities of upper and

4. Li J Y et al.: A Study on Multiple Time-Lapse Seismic AVO Inversion 977
lower formations respectively, θ1 and θ2 incident angle and transmission angle respectively. In real reflection
seismic exploration, the simplification to full Zoepporitz equations is reasonable.
The P wave velocity, S wave velocity and density of upper cap formation are VP1 , VS1 and ρ1 respectively,
and their values are constant before and after reservoir development. The P wave velocity, S wave velocity and
density of reservoir are respectively, VP2 , VS2 and ρ2 before development, VP2 , VS2 and ρ2 after development.
Based on the simplified Zoeppritz equations above, the reflections of P-P wave and P-S converted wave can be
written as follows.
Before development
2 2
1 ∆VP ∆ρ 1 ∆VP VS ∆VS VS ∆ρ 1 ∆VP
RP−P0 (θ) = + + −4 −2 sin2 θ + sin4 θ, (4)
2 VP ρ 2 VP VP VS VP ρ 2 VP
VS ∆VS VS 1 ∆ρ VS 1 3 VS ∆ρ VS ∆VS
RP−S0 (θ) = −2 − + sin θ + + + 1+2 sin3 θ. (5)
VP VS VP 2 ρ VP 2 4 VP ρ VP VP
After development
2
1 ∆VP ∆ρ 1 ∆VP VS ∆VS VS ∆ρ 1 ∆VP
RP−P1 (θ) = + + −4 −2 sin2 θ + sin4 θ, (6)
2 VP ρ 2 VP VP VS VP ρ 2 VP
VS ∆VS VS 1 ∆ρ VS 1 3 VS ∆ρ V ∆VS
RP−S1 (θ) = −2 − + sin θ + + + 1+2 S sin3 θ.(7)
VP V S VP 2 ρ VP 2 4 VP ρ VP VP
In the formula
PS PS
VP1 + VP2 VP1 + VP2 + ∆VP2 ∆VP2
VP = = = VP 1 + ,
2 2 2VP
PS PS
∆VP = VP2 − VP1 = VP2 + ∆VP2 − VP1 = ∆VP + ∆VP2 ,
PS PS
VS1 + VS2 VS1 + VS2 + ∆VS2 ∆VS2
VS = = = VS 1 + ,
2 2 2VS
PS PS
∆VS = VS2 − VS1 = VS2 + ∆VS2 − VS1 = ∆VS + ∆VS2 ,
ρ1 + ρ2 ρ1 + ρ2 + ∆ρPS
2 ρPS
ρ = = =ρ 1+ 2 ,
2 2 2ρ
∆ρ = ρ2 − ρ1 = ρ2 + ∆ρPS − ρ1 = ∆ρ + ∆ρPS .
2 2
∆VP2 , ∆VS2 and ∆ρPS are integrated differences of the P wave and S wave velocities and densities respec-
PS PS
2
tively before and after development caused by oil saturation and effective pressure variation. During the real
PS
∆VP2
development, 1 is reasonable, therefore
2VP
PS PS
∆VP ∆VP + ∆VP2 ∼ ∆VP + ∆VP2 .
= PS = (8)
VP ∆VP2 VP
VP 1 +
2VP
PS
∆VS2 ∆ρPS
2
In the same way, under the conditions of 1 and 1, the following formulae can be
2VS 2ρ
obtained
PS
∆VS ∼ ∆VS + ∆VS2 ∆ρ ∼ ∆ρ + ∆ρPS
2
= , = , (9)
VS VS ρ ρ
PS
∆VS2
VS 1 +
VS 2VS VS
= PS
≈ . (10)
VP ∆VP2 VP
VP 1+
2VP

5. 978 Chinese J. Geophys. Vol.48, No.4
After the replacement in formulae (6) and (7) by formulae (8∼10), the following reflection equations can
be gained
PS 2 2
1 ∆VP ∆ρ ∆VP2 ∆ρPS
2 1 ∆VP VS ∆VS VS ∆ρ 1 ∆VP2PS
RP−P1 (θ) = + + + + −4 −2 +
2 VP ρ VP ρ 2 VP VP VS VP ρ 2 VP
2 PS 2
VS ∆VS2 VS ∆ρPS
2 1 ∆VP PS
∆VP2
−4 −2 sin2 θ + + sin4 θ, (11)
VP VS VP ρ 2 VP VP
PS
VS ∆VS VS 1 ∆ρ VS ∆VS2 VS 1 ∆ρPS
2 VS 1 3 VS ∆ρ
RP−S1 (θ) = −2 − + −2 − + sin θ + +
VP VS VP 2 ρ VP VS VP 2 ρ VP 2 4 VP ρ
VS ∆VS VS 1 3 VS ∆ρPS
2 VS PS
∆VS2
+ 1+2 + + + 1+2 sin3 θ. (12)
VP VS VP 2 4 VP ρ VP VS
Subtracting formulae (4) and (5) from (11) and (12) respectively, the reflection difference equations can
be written as follows
PS 2
1 ∆VP2 ∆ρPS
2
PS
1 ∆VP2 VS PS
∆VS2
RP−P1 (θ) − RP−P0 (θ) = + + −4
2 VP ρ 2 VP VP VS
2
VS ∆ρPS
2
PS
1 ∆VP2
−2 sin2 θ + sin4 θ, (13)
VP ρ 2 VP
PS
VS ∆VS2 VS 1 ∆ρPS
2
RP−S1 (θ) − RP−S0 (θ) = −2 − + sin θ
VP VS VP 2 ρ
VS 1 3 VS ∆ρPS
2 VS PS
∆VS2
+ + + 1+2 sin3 θ. (14)
VP 2 4 VP ρ VP VS
Formulae (13) and (14) above are the time-lapse seismic AVO equations derived in the paper.
4 TIME-LAPSE SEISMIC AVO INVERSIONS
According to the constructed relationship of petrophysics in S oil field, the relationships between the
variations of P wave and S wave velocities and oil saturation change is linear in the range of 20% to 80%
of oil saturation, which has been shown in Fig. 1. Based on the following substance balance Eq.(15), it can
be calculated that the relationship between water saturation variation and reservoir density change is also
linear[19,20]
ρsat = (1 − Φ)ρs + Φρo + Φ(ρw − ρo )Sw . (15)
ρsat is density of saturated rock, Φ rock porosity, whose value is constant before and after development, ρs , ρo
and ρw densities of sandstone frame, oil and water respectively, Sw water saturation in reservoir. Fig. 2 shows
that the relationships between effective pressure variation and the P wave and S wave velocity changes are
nonlinear, but in the range of 15MPa to 25MPa, that is the possible variation range of effective pressure in S
oil field, the relationships are linear approximately. While in this range, the data measured in laboratory show
that the effect of effective pressure variation on density can be ignored. In conclusion, the relative variation
relationship between seismic parameters and oil saturation and effective pressure can be written as follows,
PS PS
∆VP2 ∆VS2 ∆ρPS
2
= a∆S + b∆P, = c∆S + d∆P, = e∆S. (16)
VP VS ρ
The values of a, b, c, d and e can be calculated by the constructed rock physical relationships. Based on the
formulae (16), (13) and (14), the following equations can be derived
∆RP−P (θ) =L + M sin2 θ + N sin4 θ, (17)
3
∆RP−S (θ) =P sin θ + Q sin θ, (18)

6. Li J Y et al.: A Study on Multiple Time-Lapse Seismic AVO Inversion 979
where
1 1
L = (a + e)∆S + b∆P,
2 2
2 2 2
1 VS VS 1 VS
M = a−4 c−2 e ∆S + b−4 d ∆P,
2 VP VP 2 VP
1 1
N = a∆S + b∆P,
2 2
VS VS 1 VS
P = −2 c − e − e ∆S − 2 d∆P,
VP VP 2 VP
2 2
1 VS 3 VS VS VS VS V2
Q= e+ 2 e + V c + 2 V 2 c ∆S + d + 2 S d ∆P.
2
2 VP 4 VP P P VP VP
Therefore, the equations for calculation of oil saturation and effective pressure variations can be written as
   
u11 u12 L
u u22  M 
 21  ∆S  
 u31 u32  =  N , (19)
   
  ∆P  
 u41 u42  P 
u51 u52 Q
where
1
u11 = (a + e),
2
1
u12 = b,
2
2 2
1 VS VS
u21 = a − 4 c−2 e,
2 VP VP
2
1 VS
u22 = b − 4 d,
2 VP
1
u31 = a,
2
1
u32 = b,
2
VS VS 1
u41 = − 2 c − e − e,
VP VP 2
VS
u42 = − 2 d,
VP
2
1 VS 3 VS VS V2
u51 = e+ 2 e+ c + 2 S c,
2
2 VP 4 VP VP VP
VS V2
u52 = d + 2 S d.
2
VP VP
The Eq.(19) is an overdetermined set, and it should have solutions in theory. But for real seismic data,
the effective signals are often polluted by all kinds of noises, which make the equation set have bad stability.
Fortunately, for the real reservoirs, their parameters variation is limited in certain range, so the equation set
can be solved by the method of Monte Carlo, such as simulated annealing algorithm, genetic algorithms and so
on, through optimization in whole limited range. Then quantitative reservoir characterization can be conducted
on time-lapse seismic data[21] .

7. 980 Chinese J. Geophys. Vol.48, No.4
5 EXPERIMENTS ON SIMULATED DATA
The P wave velocity, S wave velocity and density of upper cap formation are constant before and after
production, VP1 =2.896km/s, VS1 =1.410km/s, ρ1 =2.250g/cm3 respectively, rock porosity 31%. The values of
a, b, c, d and e in Eq.(16) can be calculated through constructed rock physical relations in S oil field. The time-
lapse seismic data for multiple AVO inversion are acquired by seismic response numerical simulation before and
after reservoir production. Based on the differences of P-P wave reflections and P-S converted wave reflections
before and after production, the values of the parameters L, M, N, P and Q can be calculated by the method
of minimum mean-square value generalized reversion. Then by the same method, the equation set (19) can be
solved for variations of water saturation and effective pressure. The inversion results are shown in Table 1.
Table 1 Results of variations of water saturation and effective pressure
by multiple time-lapse seismic AVO inversion
Water saturation Effective pressure True Inversion Error of True Inversion Error of
(%) (MPa) saturation saturation saturation pressure pressure pressure
Before After Before After variation variation variation variation variation variation
production production production production (%) (%) (%) (MPa) (MPa) (MPa)
50 50 17 22 0.00 2.318 2.318 5.00 5.27256 0.27256
20 70 17 17 50.00 47.303 2.697 0.00 0.19306 0.19306
30 70 17 22 40.00 37.266 2.734 5.00 4.63952 0.36048
The errors between true values and inversion ones are caused by the approximations in rock physical rela-
tions and other approximations during the AVO inversion equations’ derivation. Fortunately in real productions
of oil fields, the minor errors are reasonable and acceptable.
6 CONCLUSIONS AND SUGGESTION
Multiple time-lapse seismic AVO inversion integrates P wave information and S wave information, so it can
effectively suppress the uncertain factors in time-lapse seismic data, improve the inversion precision, obtain more
reliable oil saturation and effective pressure variations, and make the seismic data interpretation progress from
qualitative analysis to quantitative characterization. From the point of reservoir management, the quantitative
reservoir parameters variation data are much valuable, which is an effective assistance both for monitoring the
well production and for planning to drill new water-injection wells in new formations. One point should be
emphasized that the amplitude-keep process and repeatability process to the time-lapse seismic data are the
keys for the technique. Therefore time-lapse seismic process is very important job in prophase. The time-
lapse seismic AVO inversion equations derived in the paper are similar with the Aki and Richards’ simplified
AVO ones, so the AVO inversion algorithms for Aki and Richards’ approximations also can be extended to the
time-lapse seismic AVO inversion conveniently.
ACKNOWLEDGMENTS
We thank National Natural Science Foundation of China for supporting the project (40174037). We thank
National High Technique Scheme of China (863) for supporting the project (2003AA602110-2).
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