Linearized AVO and Poroelasticity for HRS9. Brian Russell, Dan Hampson and David Gray 2011

Transcription

1 Linearized AO and oroelasticity for HR9 Brian Russell, Dan Hampson and David Gray 0

2 Introduction In this talk, we combine the linearized Amplitude ariations with Offset (AO) technique with the Biot-Gassmann theory of poroelasticity. We first review the theory of linearized AO. We then review the theory of poroelasticity. We next derive a linearized AO equation which is related to the poroelastic properties of the reservoir. Finally, we will apply this equation to both model and real data examples.

3 Mode Conversion of an Incident -wave Incident -wave,, i Reflected -wave φ r r Reflected -wave R () Consider an interface between two different geological formations, shown on the left. An incident -wave on the boundary produces and reflected and transmitted waves.,, φ t t Transmitted -wave This is called mode conversion, and we wish to compute the amplitudes of each ray. Transmitted -wave 3

4 The Zoeppritz Equations 4 Zoeppritz (99) derived the amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the layer boundary, which gives four equations with four unknowns. The solution is as follows: cos sin cos sin sin cos sin cos cos cos cos sin sin cos sin cos cos sin cos sin φ φ φ φ φ φ φ φ φ φ T T R R

5 The Bortfeld-Aki-Richards (B-A-R) Equation Bortfeld (960), Richards and Frasier (976) and Aki and Richards (980) derived a linearized version of the Zoeppritz equations, given as: R ( ) a b c where: a sec, b 4 γ sin, c 0.5 γ sin, and γ. 5

6 The Linearized Approximation Layer alues Incident -wave,,,, i Reflected -wave φ r φ t r t Reflected -wave R () Transmitted -wave Averages Differences Transmitted -wave In the linearized approximation, we use averages and differences of the parameters (p), and assume that p/p < 0.. 6

7 Wiggins version of Bortfeld-Aki-Richards A numerically equivalent, but algebraically reformulated, version of the Bortfeld-Aki-Richards equation was derived by Wiggins (983) and is written: R ( ) a R b G 0 c C where: a b c, sin sin, tan. R G C 0 the 4 γ γ p p the curvature. intercept, the gradient, 7

8 mith and Gidlow (987), Gidlow et al. (99) and Fatti et al. (994) reformulated to linearized zero-offset, -wave and density reflectivity. Their equation is written: D R c R b R a R ) ( The mith-gidlow-fatti (-G-F) equation. and,, sin tan, 8sin, tan γ γ D R R c b a 8

9 Comparing the three equations Although all three of the previous equations give exactly the same value for at a given angle, they each have advantages: The intercept and gradient terms from the Wiggins equation require a knowledge of angle but not γ. Also, this equation can be used for qualitative crossplot analysis. The other two formulations both require a knowledge of angle and γ, but give us estimates of physical parameters. The /, / and / terms from the original B-A-R equation can be used to invert for and velocity and density (Buland and Omre (003)). The R 0, R 0 and R D terms from the -G-F equation can be used to invert for and impedance and density (immons and Backus (996), Hampson et al. (005)). 9

10 Intercept vs Gradient Analysis Top: Crossplot with interpreted zones. Bottom: Zones on seismic, where: ink Top of Gas Yellow Base of Gas Blue Hard streak 0

11 huey s version of Bortfeld-Aki-Richards where:.,, /, ) ( ) ( 0 0 σ σ σ σ σ σ σ σ σ σ R D D D R G huey (985) rewrote the Bortfeld-Aki-Richards equation using,, and σ, and only the form of the gradient term (G) is changed: C c b G R a R 0 ) (

12 huey s Equation Instead of simply using algebra to re-arrange terms, huey made use of the differential form given by: σ σ This means that huey s equation will give slightly different values than the Bortfeld-Aki-Richards equation. As shown in the next slide, the values are close for small changes (i.e. the p/p terms approximately 0. or less).

13 Gas and Model Aki-Richards vs huey Amplitude Angle (degrees) This figure shows a comparison between the Aki-Richards and huey equations for a typical gas sand. A-R Top A-R Base huey Top huey Base 3

14 The elastic constant formulation o far we have considered only parameterizations of the linearized Zoeppritz equations that involve -wave velocity, -wave velocity, oisson s ratio and density. However, we know that the velocities are functions of more fundamental constants, as given here: λ µ K (4 / 3) µ, µ where: λ the first Lamé constant, µ the second Lamé constant, or shear modulus, and K the bulk modulus, or reciprocal of compressibility. Note that: K λ 3 µ 4

15 The elastic constants We can extract the three terms µ, λ and K from the density and velocities using the following equations: µ λ µ K µ 3 The first two equations above are the basis for the lambda-mu-rho (LMR) method, although in practice we use and -impedance instead of velocity, which leads to λ and µ rather than λ and. 5

16 AO and the elastic constants Gray et al. (999) used an approach similar to huey (985) to re-formulate the Aki-Richards equation for the elastic constants. That is, they used the differential forms shown below: µ µ λ µ µ λ λ λ K K K K 6

17 AO and the elastic constants Using the differential forms shown on the previous slide (and a lot of algebra), Gray et al. (999) derived two new equations, one for λ, µ and, and one for K, µ and : µ µ γ γ µ µ γ λ λ γ sec 4 sin sec 3 sec 3 4 ) ( sec 4 sin sec sec 4 ) ( K K R R These equations are given on page 44 of the textbook by Avseth et al. Note the similarity of the two equations. 7

18 λ or K? Russell et al. (003) asked the question: For the porous reservoir rock, which term is more applicable, λ or K? As we showed, it doesn t matter when each term is expanded for porous media. We thus replaced these terms with a more general term f, which reduces to either λ or Κ. The theory was developed by Biot (94) for λ and µ, and Gassmann (95) for K and µ. A good summary is found in Krief et al. (990). 8

19 General equation for -wave velocity By equating Biot and Gassmann s formulations, the general equation for -wave velocity can be written: _ f s f fluid/porosity term α M ( α is the Biot coefficient and s dry skeleton term 4 Kdry µ λdry µ 3 M (often called the fluid K pore ) modulus) 9

20 Extracting the fluid term Using the seismic velocities and density, we can extract the fluid term as shown below: f c( ) f s cµ The constant c must be chosen so that the term s cµ is equal to zero. This gives us the following relationship: c dry γ dry 0

21 hysical meaning of the fluid term Noting that µ and dividing both sides of the previous equation through by this term, we find that: f µ γ γ dry dry As expected, this implies that the greater the difference in the two velocity ratios, the stronger the fluid effect. Conversely, when the two velocity ratios are identical, (i.e. for non-porous rocks) the fluid term equals zero.

22 Table of values Here is a table of values for the various ratios: (p/s)^ p/s σdry Kdry/µ λdry/µ (3) () () Note in the above table that () corresponds to K-µ, () to λµ and (3) to a poroelastic clean sand.

23 The generalized form Using the generalized equation for -wave velocity, we can re-formulate the Aki-Richards equation using the differential form shown below: µ µ f f f f The new equation is shown in the next slide. 3

24 A generalized formulation The re-formulation of the Aki-Richards equation using f, µ and is given by: where: µ µ ) ( c b f f a R dry dry dry dry c b a and, sec 4 sin sec 4 sec 4 4 γ γ γ γ γ γ γ 4

25 ome observations The following comments can be made about the general formulation: If we substitute γ dry into the previous formulation, we obtain the Gray et al. (999) expression for λ, µ,. If we substitute γ dry 4/3 into the previous formulation, we obtain the Gray et al. (999) expression for K, µ,. ince we never have a situation in which γ /γ dry <, the scaling coefficient for the fluid term will always be positive or zero. The fluid term equals zero if we are dealing with a dry or non-porous rock. For a constant γ dry s/µ, µ/µ is identical to s/s. 5

26 A model example On the right are the modeled AO curves in which the top layer is a wet sand and the bottom layer is a gas sand. The Aki-Richards and f- µ- curves are very close. γ dry.333 for each layer, but the µ and K values varied. 6

27 The general 3-term formulation Note that all of the equations we have discussed can be written in the same form: R ( ) ap bp cp3, where : and p, a, b,and c are functions p, and p 3 are functions of and of, /,, σ, (dry or wet), f, or µ. A summary of all the equations is shown on the next slide. We can implement a weighted least-squares approach to extract the p terms from pre-stack gathers. 7

28 Three term summary Need to know Able to compute Method a b c p p p 3 Wiggins R G (,, ) 0 huey B-A-R, γ, γ R 0 G( ),σ, -G-F γ, γ, R 0 R 0 f-µ-, γ, γ dry, γ, γ dry f f µ µ 8

29 Real data study Input gathers We applied the f-µ- method to a Class 3 gas sand from Alberta. The super-gathers are shown above, with the zone of interest highlighted. ince the far angle is at 30 o, the density term extraction is considered unreliable. 9

30 Real data study Fluid result Here is the fluid extraction (f/f ) with a picked event at the zero-crossing of the gas sand. We used a dry velocity ratio squared of

31 A real data study rock skeleton result Here is the rock skeleton extraction (µ/µ ) with a picked event at the zero-crossing of the gas sand. 3

32 f/f vs µ/µ results Here is a comparison of the delta fluid result (top) with the delta shear modulus result (bottom). Note the change in polarity at the gas sand when comparing the two results. 3

33 Conclusions In this talk, we combined the linearized Amplitude ariations with Offset (AO) technique with the Biot-Gassmann theory of poroelasticity. This gave us a way to extract fluid and skeleton effects from a reservoir using pre-stack angle gathers, from a knowledge of the dry and urated velocity ratios. One caution is that it is not clear what dry means for rocks such as shales and fractured carbonates. More research is needed. 33

34 References Biot, M. A., 94, General theory of three-dimensional consolidation, Journal of Applied hysics,, Bortfeld, R., 96, Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves: Geophysical rospecting, 09, Buland, A. and Omre, H, 003, Bayesian linearized AO inversion: Geophysics, 68, Fatti, J. L., ail,. J., mith, G. C., trauss,. J. and Levitt,. R., 994, Detection of gas in sandstone reservoirs using AO analysis: A 3-D seismic case history using the geostack technique: Geophysics, 59, Gassmann, F., 95, Uber die Elastizitat poroser Medien, ierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, 96,

35 References Gray, F., Chen, T. and Goodway, W., 999, Bridging the gap: Using AO to detect changes in fundamental elastic constants, 69th Ann. Int. Mtg: EG, Hampson, D.., Russell, B. H. and Bankhead, B., 005, imultaneous inversion of pre-stack seismic data, 75th Ann. Internat. Mtg.: oc. of Expl. Geophys., Richards,. G. and Frasier, C. W., 976, cattering of elastic waves from depth-dependent inhomogeneities: Geophysics, 4, Russell, B., Hedlin, K., Hilterman, F. and Lines, L., 003, Fluid-property discrimination with AO: A Biot-Gassmann perspective: Geophysics, 68, immons, J.L. and Backus, M.M., 996, Waveform-based AO inversion and AO prediction-error: Geophysics, 6,

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