1、Applications ofGeophysical Inversion and ImagingPart 4 AVO Modelling and AnalysisIntroductionIn our previous section on rock physics we discussed fluid effects on P and S-wave velocity,and density.We then looked at post-stack inversion,its advantages and limitations.Next,we considered the recording
2、of S-wave data.In this section,we will first review the basic principles of AVO and show its relationship with P and S-wave velocity.We will then look at the AVO response of two simple models,one wet and one gas-saturated.We will then look at various AVO modeling schemes,including full wave equation
3、 modeling and anisotropic modeling.“Bright spots”Recall that in the section on inversion,we showed the“bright spot”shown above,and pointed out that in the 1970s this would have been interpreted as a gas sand.3The AVO methodBut“bright spots”can be caused by lithologic variations as well as gas sands.
4、This lead geophysicists in the 1980s to start looking at pre-stack seismic data.The amplitude increase with offset shown here is an example of a Class 3 sand,as we will discuss later.4Reflected P-wave=RP(q1)Reflected SV-wave=RS(q1)Transmitted P-wave=TP(q1)Incident P-waveTransmitted SV-wave=TS(q1)VP1
5、,VS1,r1VP2,VS2,r2q11q1q22Consider the interface between two geologic horizons of differing P and S-wave velocity and density and an incident P-wave at angle q q1.This will produce P and S reflected and transmitted waves,as shown above.Mode Conversion5But how do we utilize mode conversion?There are a
6、ctually two ways:(1)Record the converted S-waves using three-component receivers(in the X,Y and Z directions).This was discussed in the last chapter.(2)Interpret the amplitudes of the P-waves as a function of offset,or angle,which contain implied information about the S-waves.This is called the AVO(
7、Amplitude versus Offset)method.In the AVO method,we can make use of the Zoeppritz equations,or some approximation to these equations,to extract S-wave type information from P-wave reflections at different offsets.Utilizing Mode Conversion6The Zoeppritz EquationsZoeppritz derived the amplitudes of th
8、e reflected and transmitted waves using the conservation of stress and displacement across the layer boundary,which gives four equations with four unknowns.Inverting the matrix form of the Zoeppritz equations gives us the exact amplitudes as a function of angle:11111211222112211112211122122111222111
9、1221122112cos2sincossin2sin2cos2sin2cos2cos2cos2cos2sinsincossincoscossincossinqqqrrrrrrrrqqqqqPSPPPSSPSPSPSSPSPSPVVVVVVVVVVVVVVVTTRRThe Aki-Richards EquationThe is a linearized approximation to the.The initial form(Richards and Frasier,1976)separated the velocity and density terms:SSPPPVVcbVVaRrrq)
10、(where:,sin4,sin25.0,cos2122222qqqPSPSVVcVVba.2,2,2,2121212121212tiSSSSSSPPPPPPandVVVVVVVVVVVVqqqrrrrrrWiggins version of Aki-RichardsA totally equivalent form was derived by Wiggins.He separated the equation into three reflection terms,as follows:qqqq222sintansin)(CBARPwhere:curvature.21gradient,24
11、21intercept,21220pPPSSSPSpPpPPVVCVVVVVVVVBVVRArrrrTo see why the A term in the linearized approximation is approximately equal to the zero-offset reflection coefficient,recall that in Part 2 we showed that:Reflectivity approximation.,212120PPPPPPPPPVZZZ ZZ ZZ=RrThis leads to:.212ln2ln2ln0rrrPPPPPVVV
12、ZR10.and,21,sin2tan21,sin8,tan1:where ,)(02222222200rrrrqqqqqDSSSPSPSDSPRVVRVVfVVedfReRdRRFattis version of Aki-RichardsAnother equivalent form was derived by Fatti et al.They separated the equation into three reflection terms,as follows:A Summary of the Aki-Richards Eq.All three forms of the Aki-Ri
13、chards equation consist of the sum of three terms,each term consisting of a weight multiplied by an elastic parameter(i.e.a function of VP,VS or r).Here is a summary:Note that the weighting terms a,b,c and d,e,f contain the squared VP/VS ratio as well as q.However,in the Wiggins et al.formulation,th
14、is term is in the elastic parameter B.cba,EquationWeightsElastic ParametersAki-RichardsWiggins et al.Fatti et al.SSPPVVVV,rrqqq222sintan,sin,1CBA,00DSPRRRfed,12A physical interpretation of the three equations is as follows:(1)Since the seismic trace consists of changes in impedance rather than veloc
15、ity or density independently,the original form of the Aki-Richards equation is rarely used.(2)The A,B,C formulation of the Aki-Richards equation is very useful for extracting empirical information about the AVO effect(i.e.A,which is called the intercept,B,called the gradient,and C,called the curvatu
16、re)which can then be displayed or cross-plotted.Explicit information about the Vp/Vs ratio is not needed in the weights.(3)The Fatti et al.formulation gives us a way to extract quantitative information about the P and S reflectivity which can then be used for pre-stack inversion.The terms RP0 and RS
17、0 are the linearized zero-angle P and S-wave reflection coefficients.Physical interpretation13Let us now see how to get from the geology to the seismic using the second two forms of the Aki-Richards equation.We will do this by using the two models shown below.Model A consists of a wet,or brine,sand,
18、and Model B consists of a gas-saturated sand.Wet and Gas Models(a)Wet model(b)Gas modelVP1,VS1,r r1VP2,VS2,r r2VP1,VS1,r r1VP2,VS2,r r214In the section on rock physics,we computed values for wet and gas sands using the Biot-Gassmann equations.The computed values were:Wet:VP2=2500 m/s,VS2=1250 m/s,r2
19、=2.11 g/cc,s2 =0.33 Gas:VP2=2000 m/s,VS2=1310 m/s,r2=1.95 g/cc,s2=0.12 Values for a typical shale are:Shale:VP1=2250 m/s,VS1=1125 m/s,r1=2.0 g/cc,s1=0.33This gives us the following values at the top of the sand/shale zone:Model Values25.0 and ,054.0,105.0,105.02PSSSPPVVVVVVrrWet Sand:328.0 and ,025.
20、0,152.0,118.02PSSSPPVVVVVVrrGas Sand:15Exercise 4-1Compute the intercept,A,the gradient term,B,and the curvature,C,for the top and base of both the wet model and the gas model,given the parameters on the previous page.Note that the base values are the negative of the top due to symmetry:Wet Model To
21、p:A=B=C=Gas Model Top:A=B=C=Wet Model Base:A=B=C=Gas Model Base:A=B=C=Exercise 4-2For both the next exercise and a later exercise on anisotropy effects in AVO,you will need to compute the following trigonometric functions of four angles.Compute them here for later use.Angle0153045sin2qtan2qsin2q*tan
22、2qExercise 4-3Using the A,B,and C terms for the wet and gas models that were computed in Exercise 1,work out the values for R(q)at angles of 0o,15o,30o,and 45oin the table below.Then,plot the results on graph paper,with and without the third term,as a function of q.Top:Base:Wet ModelGas ModelAngle(d
23、egrees)R(q)R(q)-term3R(q)R(q)-term30153045Wet ModelGas ModelAngle(degrees)R(q)R(q)-term3R(q)R(q)-term30153045Wet Model AVO Curves0.0RP(q)q0.10.2-0.1-0.210o20o30o40o50o19Gas Model AVO Curvesq0.0RP(q)0.10.2-0.1-0.210o20o30o40o50o20Wet Model AVO Curvesq0.0RP(q)10o20o30o40o50o0.10.2-0.1-0.2Wet Sand Base
24、Wet Sand Top21Gas Model AVO Curvesq0.0RP(q)10o20o30o40o50o0.10.2-0.1-0.2Gas Sand TopGas Sand Base22This figure on the right shows the computed AVO curves for the top and base interfaces of the gas sand using all three terms(A,B,and C)in the Aki-Richards equation,and then only the first two terms(A a
25、nd B).Note the deviation of the two above 25 degrees.Gas Model AVO CurvesThis figure on the right shows the computed AVO curves for the top and base interfaces of the wet sand using all three terms(A,B,and C)in the Aki-Richards equation,and then only the first two terms(A and B).Note the deviation o
26、f the two above 25 degrees.Wet Model AVO CurvesParameters for ABC and Fatti eqs.Her is a comparison of the results from the ABC and Fatti equations for the top of the sands(because of symmetry in this example,the base of sand values are simply these values multiplied by-1):025.0,063.0,071.0059.0,242
27、.0,071.000DSPRRRCBAGas Sand:Wet Sand:053.0,079.0,079.0054.0,079.0,079.000DSPRRRCBANote that A and B have the same polarity for the gas sand and opposite polarity for the wet sand,whereas RP0 and RS0 have opposite polarity for the gas sand and the same polarity for the wet sand.The reason for this wi
28、ll be clear later.25Aki-Richards valuesHere are computed values for the ABC and Fatti versions of the Aki-Richards equation at angles of 0,30 and 60 degrees:0o Gas30o Gas60o Gas0o Wet30o Wet60o Wet1st Term 2nd Term 3rd Term2nd Term 3rd Term1st TermRP(q)ABC MethodFatti MethodAngle/Sand-0.0710.079-0.0
29、710.07900000000-0.0710.079-0.1370.064-0.071-0.0710.0790.079-0.060-0.006-0.095-0.0426x10-5-0.0200.0050.106-0.040-0.002-0.181-0.385-0.133-0.285-0.1250.025-0.0600.1190.318-0.119-0.0610.13826There was a lot of information in the last slide,but the key points are:(1)The individual terms in each approach
30、are different,but the sum is always identical.(2)For an angle of zero degrees,the second two terms in both methods are equal to zero,and the scalar on the first term in the Fatti method is equal to one.(3)In the ABC method,the first term is always the zero offset reflection coefficient,but this is t
31、rue only at zero angle in the Fatti method.(4)The third term makes less of a contribution to the sum in the Fatti method than in the ABC method.(5)The next slides will show the results at all angles.Summary of the ABC and Fatti methods27This figure on the right shows the AVO curves computed using th
32、e Zoeppritz equations and the two and three term ABC equation,for the gas sand model.Notice the strong deviation for the two term versus three term sum.Note:On the next four plots,the curves have been calculated as a function of incident angle and scaled to average angle.Zoeppritz vs ABC Gas SandZoe
33、ppritzABC method:two termABC method:three term28This figure on the right shows the AVO curves computed using the Zoeppritz equations and the two and three term ABC equation,for the wet sand model.Again,notice the strong deviation for the two term versus three term sum.Zoeppritz vs ABC Wet SandZoeppr
34、itzABC method:two termABC method:three term29This figure on the right shows the AVO curves computed using the Zoeppritz equations and the two and three term Fatti equation,for the gas sand model.Notice there is less deviation between the two term and three term sum than with the ABC approach.Zoeppri
35、tz vs Fatti Gas SandZoeppritzFatti method:two termFatti method:three term30This figure on the right shows the AVO curves computed using the Zoeppritz equations and the two and three term Fatti equation,for the wet sand model.As in the gas sand case,there is less deviation between the two term and th
36、ree term sum than with the ABC approach.Zoeppritz vs Fatti Wet SandZoeppritzFatti method:two termFatti method:three term31This final computed synthetic seismogram is shown above on the right,where the log curves are on the left.Notice that the sand is thin enough that the wavelets from the top and b
37、ottom of the layer“tune”together.The final synthetic seismogram32Ostranders Paper(1984)was one of the first to write about in gas sands and proposed a simple two-layer model which encased a low impedance,low Poissons ratio sand,between two higher impedance,higher Poissons ratio shales.This model is
38、shown in the next slide.Ostranders model worked well in the Sacramento valley gas fields.However,it represents only one type of (Class 3)and the others will be discussed in the next section.Ostranders ModelOstrander(1984)wrote the classic paper on AVO.His model is shown above.Notice that the model c
39、onsists of a low acoustic impedance gas sand encased between two shales34Synthetic from Ostranders Model(a)Well log responses for the model.(b)Synthetic seismic.Notice that Ostranders model produces an increase in amplitude on the pre-stack synthetic gather.35AVO Curves from Ostrander(a)Response fro
40、m top of model to 45o.(Note that the transmitted P-wave amplitude is shifted to plot within the data range).(b)Response from base of model to 45o.36Ostranders case study-stackOstranders case study is from the Sacramento basin.The stack above has“bright spots”at locations A,B,and C,but only A and B a
41、re due to gas.37Ostranders case study Supergathers from locations A,B,and C.Note that locations A and B show amplitude increases with offset but C does not.(A)(B)(C)38Shueys EquationShuey(1985)rewrote the Aki-Richards equation using VP,r r,and s s,writing the basic form the same way:2)1(121)1(2ssssD
42、DAB12122,/:ssssssrrPPPPVVVVDwhereOnly the gradient is different than in the Aki-Richards expression,and is given by:qqqq222sintansin)(CBARShuey vs Aki-Richards In this course,we have been using a modeled gas sand and wet sand example.Using Shueys equation for this example,we get the following compar
43、ison with the answers in exercise 6-1:B(Aki-Richards)B(Shuey)Gas Sand Top:-0.242-0.252 Wet Sand Top:-0.079-0.079Why do we get the same values in the wet case but not in the gas case?Gas Sand ModelAki-Richards vs Shuey-0.250-0.200-0.150-0.100-0.0500.0000.0500.1000.1500.2000.250051015202530354045Angle
44、(degrees)AmplitudeA-R TopShuey TopA-R BaseShuey BaseThis figure showsa comparisonbetween the Aki-Richards and Shuey equationsfor the gas sand we just considered.Shuey vs Aki-Richards Multi-layer AVO modelingMulti-layer modeling in the consists first of creating a stack of N layers,generally using we
45、ll logs,and defining the thickness,P-wave velocity,S-wave velocity,and density for each layer.You must then decide what effects are to be included in the model:primaries only,converted waves,multiples,or some combination of these.42The Possible Modelled EventsThe following example,taken from Simmons
46、 and Backus(AVO Modeling and the locally converted shear wave,Geophysics 59,p1237,August,1994),illustrates the effect of wave equation modeling.The figure above shows the modelling options.43The Oil Sand ModelSimmons and Backus used the thin bed oil sand model shown above.44Response to various algor
47、ithms(A)Primaries-only Zoeppritz,(B)+single leg shear,(C)+double-leg shear,(D)+multiples,(E)Wave equation solution,(F)Linearized approximation.Simmons and Backus(1994)45Primary and Converted WavesSimmons and Backus(1994)Zoeppritz,primaries onlyZoeppritz,primaries+single leg conversionsAki-Richards,p
48、rimariesSingle leg conversions46Logs from a real data exampleThe logs shown above come from a real data example in the Colony sand that we will look at in the next section.Models from a Real Data Example(a)Full elastic wave.(b)Zoeppritzequation.(c)Aki-Richardsequation.AVO ModelingBased on AVO theory
49、 and the rock physics of the reservoir,we can perform AVO modeling,as shown above.Note that the model result is a fairly good match to the offset stack.P-waveDensityS-wavePoissons ratioSyntheticOffset StackAnisotropy and AVOSo far,we have considered only the isotropic case,in which earth parameters
50、such as velocity do not depend on seismic propagation angle.In the next few slides,we will discuss anisotropy,in particular the case of Transverse Isotropy with a vertical symmetry axis,or VTI.We will then see how anisotropy affects the AVO response.Finally,we will look at this effect on our origina
