A study on multple time lapse seismic avo inversion

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-HuaKey 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 theinjection-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. Theamplitude 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 ofoil 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 causedby 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 todiscriminate 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 Zoeppritzequation 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 Zoeppritzequation solution is very complex, which makes it difficult to be applied in real production. Many scholarssimplified the equations in different approximate forms[4∼6], and using these forms the seismic AVO inversioncan 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 quantitativelyinterpreted by time-lapse seismic AVO inversion. The P wave AVO inversion has achieved success in somedegree, and in the same time, it has lots of shortcomings. With the progress of seismic technique in recentyears[8,9], multi-component seismic data can be acquired in scientific research and real production[10∼13], and itis possible to study converted wave AVO in multi-component seismic data[14∼16]. In this paper, the feasibilityto 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: [email protected]

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 isconducted to discriminate the variations of oil saturation and effective pressure, to realize the seismic dataquantitative interpretation.

2 FEABILITY OF TIME-LAPSE SEISMIC AVO INVERSION

The feasibility of time-lapse seismic AVO inversion to discriminate oil saturation and effective pressurevariations 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 changescan be calculated by the constructed rock physical relationship[17] based on the data measured in laboratoryand in the S oil field (Fig. 1 and Fig. 2). Fig. 1 and Fig. 2 show that the variations of oil saturation and effectivepressure affect P wave and S wave velocities by different rules. Time-lapse seismic simulation can be conductedby solving Zoeppritz equation precisely[18], and then the AVO curves can be analyzed before and after theproduction. The Zoeppritz equation can be written as the following for P wave incidence[4].

A1 cos θ1 −B1 sinλ1 + A2 cos θ2 + B2 sinλ2 = A0 cos θ1, (1a)

A1 sin θ1 + B1 cos λ1 −A2 sin θ2 + B2 cos λ2 = −A0 sin θ1, (1b)

A1Z1 cos 2λ1 −B1W1 sin 2λ1 −A2Z2 cos 2λ2 −B2W2 sin 2λ2 = −A0Z1 cos 2λ1, (1c)

A1γ1W1 sin 2θ1 + B1W1 cos 2λ1 + A2γ2W2 sin 2θ2 −B2W2 cos 2λ2 = A0γ1W1 sin 2θ1. (1d)

In the formulae, A1, A2 are the reflection amplitude and transmission amplitude of P wave, A0 incident Pwave amplitude, B1, B2 are the reflection amplitude and transmission amplitude of S wave, ρi medium density,γi = βi/αi, Zi = ρiαi,Wi = ρiβi, i = 1, 2 and Zi,Wi are wave impedance.

Fig. 1 Velocity of P wave and S wave varied

with oil saturation

Fig. 2 Velocity of P wave and S wave varied

with effective pressure

According to the real situation of S oil field, AVO curves of reservoir upper interface have been simulatedrespectively when oil saturation and effective pressure varied (Fig. 3 and Fig. 4). Fig. 3 and Fig. 4 show thatP-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 pressurevaried. P-S wave AVO curves change insigificantly when oil saturation varied, but change obviously at largeincident angles when effective pressure varied. In conclusion, quantitative interpretation of the variations of oilsaturation and effective pressure by multiple time-lapse seismic AVO inversion is feasible, because P-P and P-Swave AVO curves have different varying rules when oil saturation and effective pressure vary.

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 complexsolution, so it is necessary to do reasonable simplification to them. In 1980, Aki and Richards[3] presented thesimplified Zoeppritz equations in isotropic medium. At small incident angles the approximate P wave and Swave reflection formulas for P wave incidence can be written as follows.

RP−P(θ) =A + B sin2 θ + C sin4 θ, (2)

RP−S(θ) =E sin θ + F sin3 θ, (3)

where

A =12

(∆VP

VP+

∆ρ

ρ

),

B =12

(∆VP

VP− 4η2 ∆VS

VS− 2η2 ∆ρ

ρ

),

C =12

∆VP

VP,

E =− 2η∆VS

VS−(

η +12

)∆ρ

ρ,

F =η

[(12

+3η

4

)∆ρ

ρ+ (1 + 2η)

∆VS

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


lower formations respectively, θ1 and θ2 incident angle and transmission angle respectively. In real reflectionseismic 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 anddensity of reservoir are respectively, VP2, VS2 and ρ2 before development, V ′

P2, V′S2 and ρ′2 after development.

Based on the simplified Zoeppritz equations above, the reflections of P-P wave and P-S converted wave can bewritten as follows.

Before development

RP−P0(θ) =12

(∆VP

VP+

∆ρ

ρ

)+

[12

∆VP

VP− 4

(VS

VP

)2 ∆VS

VS− 2

(VS

VP

)2 ∆ρ

ρ

]sin2 θ +

12

∆VP

VPsin4 θ, (4)

RP−S0(θ) =[−2

VS

VP

∆VS

VS−(

VS

VP+

12

)∆ρ

ρ

]sin θ +

{VS

VP

[(12

+34

VS

VP

)∆ρ

ρ+(

1 + 2VS

VP

)∆VS

VP

]}sin3 θ. (5)

After development

RP−P1(θ) =12

(∆V ′

P

V ′P

+∆ρ′

ρ′

)+

[12

∆V ′P

V ′P

− 4(

V ′S

V ′P

)2 ∆V ′S

V ′S

− 2(

V ′S

V ′P

)∆ρ′

ρ′

]sin2 θ +

12

∆V ′P

V ′P

sin4 θ, (6)

RP−S1(θ) =[−2

V ′S

V ′P

∆V ′S

V ′S

−(

V ′S

V ′P

+12

)∆ρ′

ρ′

]sin θ +

{V ′

S

V ′P

[(12

+34

V ′S

V ′P

)∆ρ′

ρ′+(

1 + 2V ′

S

V ′P

)∆V ′

S

V ′P

]}sin3 θ.(7)

In the formula

V ′P =

VP1 + V ′P2

2=

VP1 + VP2 + ∆V PSP2

2= VP

(1 +

∆V PSP2

2VP

),

∆V ′P = V ′

P2 − VP1 = VP2 + ∆V PSP2 − VP1 = ∆VP + ∆V PS

P2 ,

V ′S =

VS1 + V ′S2

2=

VS1 + VS2 + ∆V PSS2

2= VS

(1 +

∆V PSS2

2VS

),

∆V ′S = V ′

S2 − VS1 = VS2 + ∆V PSS2 − VS1 = ∆VS + ∆V PS

S2 ,

ρ′ =ρ1 + ρ′2

2=

ρ1 + ρ2 + ∆ρPS2

2= ρ

(1 +

ρPS2

2ρ

),

∆ρ′ = ρ′2 − ρ1 = ρ2 + ∆ρPS2 − ρ1 = ∆ρ + ∆ρPS

2 .

∆V PSP2 ,∆V PS

S2 and ∆ρPS2 are integrated differences of the P wave and S wave velocities and densities respec-

tively before and after development caused by oil saturation and effective pressure variation. During the real

development,∆V PS

P2

2VP� 1 is reasonable, therefore

∆V ′P

V ′P

=∆VP + ∆V PS

P2

VP

(1 +

∆V PSP2

2VP

) ∼= ∆VP + ∆V PSP2

VP. (8)

In the same way, under the conditions of∆V PS

S2

2VS� 1 and

∆ρPS2

2ρ� 1, the following formulae can be

obtained

∆V ′S

V ′S

∼=∆VS + ∆V PS

S2

VS,

∆ρ′

ρ′∼=

∆ρ + ∆ρPS2

ρ, (9)

V ′S

V ′P

=VS

(1 +

∆V PSS2

2VS

)VP

(1 +

∆V PSP2

2VP

) ≈ VS

VP. (10)


After the replacement in formulae (6) and (7) by formulae (8∼10), the following reflection equations canbe gained

RP−P1(θ) =12

(∆VP

VP+

∆ρ

ρ+

∆V PSP2

VP+

∆ρPS2

ρ

)+

[12

∆VP

VP− 4

(VS

VP

)2 ∆VS

VS− 2

(VS

VP

)2 ∆ρ

ρ+

12

∆V PSP2

VP

−4(

VS

VP

)2 ∆V PSS2

VS− 2

(VS

VP

)2 ∆ρPS2

ρ

]sin2 θ +

12

(∆VP

VP+

∆V PSP2

VP

)sin4 θ, (11)

RP−S1(θ) =[−2

VS

VP

∆VS

VS−(

VS

VP+

12

)∆ρ

ρ− 2

VS

VP

∆V PSS2

VS−(

VS

VP+

12

)∆ρPS

2

ρ

]sin θ +

{VS

VP

[(12

+34

VS

VP

)∆ρ

ρ

+(

1 + 2VS

VP

)∆VS

VS

]+

VS

VP

[(12

+34

VS

VP

)∆ρPS

2

ρ+(

1 + 2VS

VP

)∆V PS

S2

VS

]}sin3 θ. (12)

Subtracting formulae (4) and (5) from (11) and (12) respectively, the reflection difference equations canbe written as follows

RP−P1(θ)−RP−P0(θ) =12

(∆V PS

P2

VP+

∆ρPS2

ρ

)+

(12

∆V PSP2

VP− 4

(VS

VP

)2 ∆V PSS2

VS

−2(

VS

VP

)2 ∆ρPS2

ρ

)sin2 θ +

12

∆V PSP2

VPsin4 θ, (13)

RP−S1(θ)−RP−S0(θ) =[−2

VS

VP

∆V PSS2

VS−(

VS

VP+

12

)∆ρPS

2

ρ

]sin θ

+{

VS

VP

[(12

+34

VS

VP

)∆ρPS

2

ρ+(

1 + 2VS

VP

)∆V PS

S2

VS

]}sin3 θ. (14)

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 thevariations 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 canbe calculated that the relationship between water saturation variation and reservoir density change is alsolinear[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 showsthat the relationships between effective pressure variation and the P wave and S wave velocity changes arenonlinear, but in the range of 15MPa to 25MPa, that is the possible variation range of effective pressure in Soil field, the relationships are linear approximately. While in this range, the data measured in laboratory showthat the effect of effective pressure variation on density can be ignored. In conclusion, the relative variationrelationship between seismic parameters and oil saturation and effective pressure can be written as follows,

∆V PSP2

VP= a∆S + b∆P,

∆V PSS2

VS= c∆S + d∆P,

∆ρPS2

ρ= e∆S. (16)

The values of a, b, c, d and e can be calculated by the constructed rock physical relationships. Based on theformulae (16), (13) and (14), the following equations can be derived

∆RP−P(θ) =L + M sin2 θ + N sin4 θ, (17)

∆RP−S(θ) =P sin θ + Q sin3 θ, (18)


where

L =12(a + e)∆S +

12b∆P,

M =

[12a− 4

(VS

VP

)2

c− 2(

VS

VP

)2

e

]∆S +

[12b− 4

(VS

VP

)2

d

]∆P,

N =12a∆S +

12b∆P,

P =(−2

VS

VPc− VS

VPe− 1

2e

)∆S − 2

VS

VPd∆P,

Q =(

12

VS

VPe +

34

V 2S

V 2P

e +VS

VPc + 2

V 2S

V 2P

c

)∆S +

(VS

VPd + 2

V 2S

V 2P

d

)∆P.

Therefore, the equations for calculation of oil saturation and effective pressure variations can be written asu11 u12

u21 u22

u31 u32

u41 u42

u51 u52

[

∆S

∆P

]=

L

M

N

P

Q

, (19)

where

u11 =12(a + e),

u12 =12b,

u21 =12a− 4

(VS

VP

)2

c− 2(

VS

VP

)2

e,

u22 =12b− 4

(VS

VP

)2

d,

u31 =12a,

u32 =12b,

u41 =− 2VS

VPc− VS

VPe− 1

2e,

u42 =− 2VS

VPd,

u51 =12

VS

VPe +

34

V 2S

V 2P

e +VS

VPc + 2

V 2S

V 2P

c,

u52 =VS

VPd + 2

V 2S

V 2P

d.

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 setcan be solved by the method of Monte Carlo, such as simulated annealing algorithm, genetic algorithms and soon, through optimization in whole limited range. Then quantitative reservoir characterization can be conductedon time-lapse seismic data[21].


5 EXPERIMENTS ON SIMULATED DATA

The P wave velocity, S wave velocity and density of upper cap formation are constant before and afterproduction, VP1=2.896km/s, VS1=1.410km/s, ρ1=2.250g/cm3 respectively, rock porosity 31%. The values ofa, 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 andafter reservoir production. Based on the differences of P-P wave reflections and P-S converted wave reflectionsbefore and after production, the values of the parameters L,M,N,P and Q can be calculated by the methodof minimum mean-square value generalized reversion. Then by the same method, the equation set (19) can besolved 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 productionsof 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 caneffectively suppress the uncertain factors in time-lapse seismic data, improve the inversion precision, obtain morereliable oil saturation and effective pressure variations, and make the seismic data interpretation progress fromqualitative analysis to quantitative characterization. From the point of reservoir management, the quantitativereservoir parameters variation data are much valuable, which is an effective assistance both for monitoring thewell production and for planning to drill new water-injection wells in new formations. One point should beemphasized that the amplitude-keep process and repeatability process to the time-lapse seismic data are thekeys 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’ simplifiedAVO ones, so the AVO inversion algorithms for Aki and Richards’ approximations also can be extended to thetime-lapse seismic AVO inversion conveniently.

ACKNOWLEDGMENTS

We thank National Natural Science Foundation of China for supporting the project (40174037). We thankNational High Technique Scheme of China (863) for supporting the project (2003AA602110-2).

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