avo modelling.pdf

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38 CSEG RECORDER January 2004

Introduction

Zoeppritzs equations with ray tracing and full elastic

wave equation with finite difference method (FDM) are

two most commonly used techniques in AVO modeling.

The main difference between these two techniques is the

former calculates the primary-only reflectivities and the

latter calculates particle displacements in subsurface.

Zoeppritz modeling has the advantages of being fast and

easy for identifying p rimary reflections. Elastic wave equ a-

tion modeling, however, accounts for direct waves,

primary and multiple-reflection waves, converted waves,

head wav es, as well as diffraction waves. It overcomes the

shortcomings of the ray tracing ap proach that breaks dow n

in many cases such as at the edges where the calculated

amplitude is infinite or in the shadow zones where the

amplitude is zero. Baker (1989) summarized the advan-

tages using wave equation modeling in the cases of

complex geology or complex wave p henom ena as: a) auto-

matic generation of diffractions, critical refraction and

multiples; b) more accurate amplitudes and waveforms,

especially in the pr esence of small structur es and thin beds;

and c) no missing of seismic events re g a rdless of

complexity. For structured plays, elastic wave equation

modeling is especially useful in both imaging and AVO

analysis.

Based on Zoeppritz modeling and elastic wave equation

modeling, different methodologies in AVO modeling are

developed to take into account of th e issues in association

with data acquisition, processing and interpretation. This

pap er, Part 2 of AVO Modeling in Seismic Processing and

Interpretation, reviews Zoeppritz equations and elastic

wave equations and discusses pros and cons of the

meth odologies in AVO mod eling.

Zoeppritz equations and elastic wave

equations

For a two-layer interface, in total sixteen Zoeppr itz equations

describe the energy partitioning of reflected and transmitted

waves (Aki and Richards, 1980). The commonly u sed equ a-

tions are: an incident P-wave is reflected as a P-wave (P-P) or

a converted S-wave (P-SV). Table 1 lists these two equations.

Where a1, b1 and r1 are P- and S-wave velocities and density

for the layer above and a2, b2 and r2 are for the layer below;

i1 and i2 are reflected and transmitted angles for P-wave; and

j1 and j2 are the reflected and transm itted angles for S-waves.

It can be seen that they have comp lex forms

In AVO modeling, Zoepp ritz equations are used to generate

exact solutions. AVO attributes are then calculated based on

the simplified Zoeppritz equations that have been

discussed in Par t 1 of this p ap er (Li et al., 2003). In AVO

attribute inversion, least squares method (L2) or absolute

deviation minim ization (L1) method s that fits the observed

amp litud es to a linear equation are often u sed. For examp le,

Shueys equation yields estima tes of zero-offset reflectivity

and grad ient; and Fattis equation gives estimates of P- and

S-reflectivities.

Table 2 lists 2-D elastic wave equ ations. The equations of

motion are derived first based on Newtons law and

expressed by stress compon ents x, z an d zx. The form

expressed by d isplacements, ux and uz is derived th rough

using strain-displacement relations and stress-strain rela-

tions. In Table 2, Sx and Sz are source components. To

numerically stimulate wave propagation in subsurface, the

equations of motion ar e often transformed into finite differ-

ence equations with given boun dar y conditions. Kelly et al.

(1976) gave a detailed description of 2-D finite difference

AVO Modeling in Seismic Process ingand InterpretationII. MethodologiesYongyi Li, Jonathan Downton, and Y ong X u, Core Laboratories Reservoir Technologies Division

Calgary, Canada

Continued on Page 39

Table 1. Zoeppritz equations for reflected P wave and converted S-wave.


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equation s. For 3-D case, one m ay refer to th e finite difference

equations given by Jastram and Tessmer (1994). In finite differ-

ence calculation, explicit scheme approach is often used. It

calculates pa rticle motion for a sp ace location at an a dva nced

time exclusively from the motion th at is already d etermined for

previous time. In AVO mod eling, a single shot gather with a flat-

layered mod el can be generated for the cases where the stru c-

tural effect is not the major concern. In stru ctured ar eas, a series

of shot gathers are generated based on a geological model and

these shot gathers are the input for further processing.

AVO modeling methodologies

Different approaches can be taken in conducting an AVO

modeling in terms of the objective. Single interface modeling,

single gather modeling, 2D stratigraphic modeling and 2D

elastic wave equation modeling are most commonly used and

are to be discussed.

1. Single interface modeling

One of the ad vantages of single interface mod eling is freedom

from tu ning. This method is often u sed to show theoretical AVO

responses. For examp le, Rutherford and Williams (1989) used

this meth od to classify AVO typ es for a sha le-gas interface, and

Shuey (1985) used it to comp are betw een the exact Zoeppr itz

solution (Aki and Richard , 1980) and the solu tions from h is

simplified equations. To demonstrate single interface AVO

mod eling three exam ples are selected and sh own in Figure 1.

Figure 1a shows the AVO responses for a case of shale overlying

porous gas sand; where the results from tw o- and three-term

Shueys equations are compared to evaluate the error intro-

duced by the tru ncation of the third term. The maximu m an d

minimu m amp litud e locations and inflection point location as

indicated are calculated from the three-term equ ation. Figure 1b

show s an examp le of the anisotr opic effect of shale for a shale-

sand interface (VTI medium ). It can be seen that w ith increasing

January 2004CSEG RECORDER 39

Article ContdAVO Mod eling in Seismic Pr ocessing and Inter pr etat ionII. MethodologiesContinued from Page 38


Table 2. 2-D Elastic wave equations.

Figure 1. Single int erface AVO modeling for isotropic, vertical transversal and horizontal transversal media.


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anisotropy, the deviation of the amp litud e from the isotropic

case increases. A third examp le (Figure 1c) shows the am plitude

variation for horizontal transverse mediu m or fractured rocks

(HTI medium ). In the fracture plane, the r ock is considered

isotropic and its AVO response is the same as that of isotropic

case. In the plane p erpen dicular to the fracture p lane, the AVO

response has d ifferent character due to anisotropic effect. Fitting

the amp litud e surface using an AVO equation such as Rger s

(1996), the orientation of fractures can be solved and fracture

density can be calculated ba sed on th e grad ients parallel and

perpendicular to the fractures.

It is worth emphasizing that the anisotropic parameters, ,

and are required in an anisotropic AVO modeling. However,

to determine these parameters often encounters difficulties,

especially in in-situ conditions. Measur ements of ro c k

anisotropic properties for clastic rocks have been conducted

mainly in laboratory (e.g., Wang, 2002; Johnston and

Christensen, 1995; Vernik and Liu, 1995) and in situ through

c rosshole seismic or VSP (e.g., A r m s t rong et al. 1994;

Winterstein and Paulsson, 1989), and seismic refraction (Leslie

and Lawton, 1999). Li (2001) derived the relat ionsh ips for calcu-

lating anisotropic parameters in clastic rocks using conven-

tional well logs. In fractured reservoirs, shear wav e anisotropic

information can be m easured in-situ by cross-dipole logs based

on sh ear w ave sp litting (e.g., Esmersoy et al.). Since complete

anisotropic information often can not be obtained, anisotropic

AVO modeling is still facing challenges.

2. Single gather modeling

Single cmp gather modeling is the most commonly used AVO

modeling approach. Both Zoeppritz equations with ray tracing

or full wave elastic equation modeling can be used. In singlecmp gather modeling, well logs are often used to construct the

geological mod els. The geological models ar e flat-layered and

structure effect is thus not considered. Zoeppritz equations

modeling calculate primary reflections. Elastic modeling

provides additional information for transmission loss, multi-

ples and converted w ave. These two m ethods are often used at

the same time for calibration and stud ying the waveform s other

than prim ary reflections. Since elastic waves can not be isolated

from each other in elastic wave equation modeling Kennetts

meth ods (1979) may be u sed to calculate single mod e of elastic

waves such as multiples or converted wave. There are several

other approaches to conduct wave equation modeling. Readers

interested may refer to a review on seismic modeling by

Carcione et a l. (2002).

F i g u re 2 shows the cmp gathers generated by Zoeppritz

mod eling and finite difference method , respectively. It can be

seen that Zoeppr itz modeling gives clean primary reflections

(Figure 2a). In the gather of elastic modeling, inter-bedded mu lti-

ples and converted wav e energy can be observed (Figure 2b).

This is helpful to determine an appropriate data processing

strategy to attenuate the noise and pr eserve the primary reflec-

tions. In elastic mod eling, a specific event such as conv erted

wave or multiple may be isolated through smoothing out a

segmen t of the logs correspon ding to a sp ecific layer. Acoustic

modeling may also be incorp orated w ith elastic mod eling to

identify converted w ave energy.

A synthetic cmp gather can be tied to a recorded seismic cmp

gather at a well location to validate seismic response of know n

reservoir conditions. The reservoir conditions can be altered

and the corresponding synthetic gathers can be calculated to

examine the predicted seismic responses. The often-varied

parameters for reservoir conditions are: thickness, porosity,

water saturation, fluid type, and lithology. Contrast between

lithological units is also of interest as it influences AVO

responses as well. Further, frequency bandwidth and seismic

wavelet are other parameters to be tested. NMO stretching and

offset-depend ent tuning may also be of interest. A variety of

AVO attributes can be calculated from the synthetic modeled

gathers. The most commonly used attributes are angle stacks,

P- and S-impedance reflectivities, fluid factor, and gradient.

Cross-plots of these attributes can be analyzed for meaningful

relationships. Random and coherent noise can be added to

stud y their effects on AVO attribute inversion.

When frequ ency or w avelet at reservoir level is given, correct

rock physical proper ty inpu t becomes essential in single cmp

Article Contd

40 CSEG RECORDERJanuary 2004


AVO Mod eling in Seismic Pr ocessing and Inter pr etat ionII. MethodologiesContinued from Page 39

Figure 2. a) Zoeppritz equation modeling and b) elastic wave equation modeling.


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gather m odeling. Complete P- and S-wav e velocity and d ensity

log curv es from sur face are the basic inpu t. The procedure for

obtaining correct rock properties for input involves log data QC,

log data corrections, and rock physical property prediction.Quick results may be prod uced w hen dip ole sonics are avail-

able. Before to do so, corrections on log d ata m ay require either

for prod ucing missing log segments or editing log curves. Tying

to recorded seismic is an imp ortant QC step. The modeling

result may help in identifying whether am plitudes have been

properly recovered in da ta processing. The correct inpu t mod el,

especially in those areas withou t d ipole sonics, requires infor-

mation from geology, petrophysics, and even engineering.

Converted wave modeling aims on the analysis for multi-

component seismic data that contains amplitude variation with

offset information for both comp ressional and converted wav es.

The separated P-wave d ata (P-P) and converted wave d ata (P-S)

can be used to p erform P-wave AVO analysis and convertedwave AVO analysis. The P-Pdata can incorporated with P-S data

in AVO reflectivity inversion. Its adva ntages w ere sum ma rized

by Larson (1999), wh ich includ es: better signal-to-noise ratio d ue

to larger am oun t of data; better S-reflectivity or S-imp edan ce

estimates du e to that converted wave m ore depends on shear

impedan ce contrast; and better elastic parameter estimates w hen

P-P contrasts are weak or P-P data has low signal-to noise ratio.In add ition, density information can be d etermined by u sing

converted wave d ata (e.g., Jin et al., 1999). This may be typ ically

useful for the areas wh ere the d ensity contrasts are considered

better than P-impedance and S-impedance contrasts such as oil

sand plays in the WCSB (Dumitresu e et al., 2003). Figure 3 shows

an example of AVO modeling using P-Pand P-S Zoeppritz equa-

tions with ray tracing, where the travel time of converted wav e

gather is corrected to P-wave travel time.

3. Two-dimensional stratigraphic modeling

2-D stratigraphic modeling has the advantage of taking into

accoun t lateral va riations in geology. The geological section to

be modeled may come from seismic interpretation or can beconstructed u sing well logs as control points. The lateral varia-

tion can be structure, reservoir thickness, porosity, lithology,

fluid type, or fluid satu ration. Based on the geological mod el of


Article Contd



Figure 3. Zoeppritz equation modeling for P wave a) and Converted wave b).

Figure 4. 2-D stratigraphic AVO modeling with control of four wells: a) P-impedance; b) S-impedance; c) lr; and d) (lr-mr)


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the subsurface rock properties, cmp gathers are calculated and

then p rocessed as a 2-D line.

An examp le for a gas play w ith well control at four locations is

shown in Figure 4 (Li et al., 2000). In this example, the target

zone changes from silt sand, gas sand to brine-saturated sand .

In total, 100 cmp g athers w ere generated . P- and S-reflectivity

were extracted using Fattis equation and then inverted into P-

and S-imped ance sections. The an d sections w ere calcu -

lated. For this case, we can see th e d ifference in sensitivities of

the inverted elastic rock properties in response to the gas: P-

and S-impedances are unable to delineate the reservoir but lr

an d do. The cmp gathers at silt, gas sand, and wet well

locations are shown in Figure 5. Class I AVO anomaly can be

seen in the gather corresponding to the gas well.


Article Contd



Figure 5. CMP gathers at the silt, sand, and gas well locations of Figure 4.

Figure 6. Acoustic and elastic modeling and pre-stack time and depth migration results for a layered structured model.


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4. Two-dimensional elastic wave equation modeling

Recently, there is renewed interest in 2-D and 3-D elastic

mod eling for AVO analysis in stru cturally complex areas. This

is because seismic imaging alone does not describe a reservoir

completely. Elastic modeling will help to solve some specific

issues of AVO analysis in structurally complex areas. Thisincludes correctly positioning the signals and the amplitudes,

and d etermining th e angles of reflection at a comm on image or

reflection point. In comparison to ray tracing method, elastic

mod eling genera tes the most realistic synthetic data.

Conducting 2-D and 3-D full wave elastic modeling by using

f i n i t e d i ff e rence m etho d takes significant time. Recent

advances in computing power have made single shot gather

modeling trivial and 2-D modeling practical. Current major

players in 2-D and 3-D elastic modeling include Los Alamos

National Lab, Lawrence Livermore N ational Lab, University of

Houston, and Stanford University. One recent accomplishment

is converting the Marmousi model to an elastic version and

generating shot gathers using full wave elastic modeling(Martin et a l., 2002).

While 2-D stratigraphic modeling does not take the complete

wave-field and structure effect into account, 2-D elastic wave

equation modeling does. Another difference is that 2-D elastic

wave equation modeling generates shot gathers instead of cmp

gathers. The synthetic data n eeds to be treated as recorded d ata

and taken through a processing sequence. In 2-D elastic

modeling, field acquisition parameters can be used and 3-

comp onent data can be generated.

To demonstrate explicitly all the waveforms from elastic

modeling and to make a comparison between acoustic and

elastic mod eling, a layered geological mod el is stud ied (Li et al.,

2003). The P-wave velocity model was initially used by Kelly et

al. (1982) to demonstrate acoustic wave equation modeling in

seismic imaging and interpretation. Here, this model is

converted into elastic that consists of P-wave v elocity, S-wave

velocity, and density models. The reservoir at the top of the

anticline is mod eled as gas-charged with Vp/ Vs ratio equal to

1.5. The m aximum dip of the layers is about 20 degrees. Figure

6b shows the snapshots from both acoustic and elastic

modeling at a shot location on the right wing of the anticline.

The waves can be identified include reflected compressional

wave (PP) and converted S-wave (PS). The shot gathers from

acoustic modeling and elastic mod eling at th e same location are

shown in Figure 6c. It can be seen that, except for the converted

waves, the amplitudes of the same events are different as well.This can be observed at the events of PP1, PP2, and PP3. It

implies that elastic modeling data is required for true ampli-

tude migration. The pre-stack time migrated and depth

migrated results for this example are shown in Figure 6d.

Figure 7 show s an examp le of elastic modeling using a m odel

built by well logs. This mod el has dip s as high as 45 degrees. A

gas-charged reservoir is embedded in the top of the anticline, and

another gas-charged reservoir is located at the fau lt. It is expected

the effect of structure on amplitudes is significant for this type of

stru cture. One hundred and ninety-one synthetic shot gathers

were generated w ith a source interval of 50 m an d m aximu m

offset of 6000 m. The r eceiver sta tion interv al is 25 m. The shot

gathers located at the top of the anticline, between the top of theanticline and the fault, and at the fault are shown in Figure 7b.

Figure 8 shows a pre-stack depth migration example using the

data generated by elastic wave equation modeling. This study

is typically for a structurally complex model in Mackenzie

Delta, Canada (Xu, 2003). In processing the common image

gathers after pre-stack depth migration were used. It can be

seen that the reservoir is well characterized by the fluid factor

stack, which would not typically be considered in interpreta-

tion of real data. Consequ ently, for a structur e play, an a priori

stud y may h elp to determ ine if AVO analysis is feasible or what

kind of wor kflow can achieve optima l solution.

Conclusions

Single interface modeling, single cmp gather modeling, 2-D

stratigraphic modeling, and 2-D elastic wave equation

mod eling have their pros and cons. The suitability of a meth od

is determined by the objectives of a project. Regardless the

methodologies, input model and calibration after the modeling




Figure 7. Elastic modeling for a 2-D structure model constructed by well logs: a) P-wave velocity; and b) selected shot gathers.


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are the key for successful application of AVO mod eling in d ata

processing and interpretation. It demonstrates that 2-D elastic

modeling is an effective tool for studying AVO in structurally

complex plays. It also demon strates the fact that AVO mod eling

provides significant information in seismic data acquisition,

processing and interpretation.

Acknowledgements

The authors would like to thank Core Laboratories Reservoir

Technologies Division for supporting this work. The authors

are also like to acknowledge the helps from and discussions

with Bob Somerv ille, Huimin Gu an, and Lu iz Loures.

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Figure 8. a) P-wave velocity model, and b) fluid stack based on PSDM data for a typical structure play.

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