2. 7.Effective medium Upscaling problem Backus averaging
O’Doherty-Anstey approximation
Reuss and Voigt models
Bio-Gassmann model
Hertz-Mindlin model

3. Upscaling problem
Does seismic wave see the
thin-layering?

4. Upscaling problem From microscopic to macroscopic scale
From pore (graine) scale (millimeters)
From log-scale (centimeteers)

5. Upscaling problem
Traditionally, upscaling has meant upscaling of reservoir petrophysical properties and flow parameters dedicated for reservoir fluid flow simulation. However, due to the progresses mentioned above, there is a need to extend the concept of upscaling of geological models, for rock physics properties, seismic modelling and analysis. For instance, in 4D history matching, the need for up and downscaling might differ from the traditional concept of upscaling.

6. Backus averaging
Sequential Backus Averaging is a method of averaging the properties of a stack of thin layers so they are similar to average properties of a single thick layer.
Figure 7.1. The Backus averaging scheme

7. Backus averaging
The advantage of Sequential Backus Averaging is that no artificial "blocks" are introduced into the geology during the upscaling of the well-log data. In this example the density log is blocky, but the compressional- and shear-wave velocity logs have gradational tops and appear thicker. Blocking would distort the amplitudes. Furthermore, if blocking were based solely upon either the density or the sonic curves, the result would be wrong for the other curve.
Figure 7.2. The Backus averaging versus blocking averaging

8. Backus averaging Thin beds appear thinner at oblique incidence angles.
Figure 7.3. The thin beds

9. Backus averaging Figure 7.4. Adjusting of averaging operator
At nonnormal incidence, the averaging operator must be adjusted to include the apparent bed thinning.
Figure 7.4. Adjusting of averaging operator

10. Backus averaging Figure 7.5. AVO signatures from different models
The offset synthetic shows differing AVO signatures for the same elastic property contrasts, associated with step-functions, blocky beds, and gradational interfaces.
Figure 7.5. AVO signatures from different models

11. Backus averaging
(7.1)

12. Backus averaging
(7.2)
(7.3)
How many combinations of the stiffness coefficients enter these matrices?

13. Backus averaging
(7.4)
(7.5)

14. The effective vertical velocity from Backus averaging
(7.6)
Stovas and Arntsen, 2003

15. Layering
Figure 7.6. The layering effect (each model computed by compression and
doubling of the previous one)

16. Reflection-transmission versus layering and contrast
Figure 7.7. The reflection (bottom)and transmission (top) responses with
different contrasts (to the right is 4 times larger).

17. Binary medium (multiples)
Figure 7.8. Multiples contribution into
the reflection response

18. Propagation versus contrast
Figure 7.9. Transmission from thin layer model (change in r due to change in r only)

19. Effective properties versus net-to-gross
Figure Effective properties from Backus averaging in a binary medium
Stovas, Landro and Avseth, 2004

20. Turbidite sequence from Ainsa basin
Figure Turbidite system as an example of binary medium

21. Binary medium
(7.7)
(7.8)
(7.9)

22. Binary medium (7.10) (7.11) The propagator matrix is not unitary
(7.12)

23. Binary medium From the characteristic equation (7.13)
we compute the eigenvalues
(7.14)

24. Binary medium The propagating regime with complex eigenvalues
and the blocking regime with real eigenvalues:
(7.15)

25. Propagating and blocking regimes
Figure Re a as a function of frequency versus layering and contrast.
Filled low frequency area relates to an effective medium, next coming gap
relates to transition medium. The interchanging of these zones is repeatable.

26. Velocity limits The time average limit means that
the pulse width is much less than
the propagation time through the cycle)
(7.16)
The effective medium limit can be
computed assuming phases being
small (low frequency limit)
(7.17)
(7.18)
The geometrical average limit

27. Velocity limits versus volume fraction
Figure Velocity versus fraction. The larger reflection coefficient the more deviation
between time-average and effective medium velocities. The position for maximum
difference between them moving to high values of volume fraction with r increase.

28. Stack of binary layers
The propagator matrix can be represented by the eigenvalue decomposition
(7.19)
(7.20)
(7.21)

29. Stack of binary layers Product of M cycles (7.22)
Transmission and reflection response
(7.23)

30. Stack of binary layers Propagating regime Blocking regime (7.24)
(7.25)
(7.26)

31. Stack of binary layers
Propagating regime
(7.27)
(7.28)

32. Stack of binary layers
Blocking regime
(7.29)
(7.30)

33. Stack of binary layers
Figure cos a as a function of frequency versus layering and contrast
(blue line is for the reference time average medium).
Stovas and Ursin, 2005

34. Stack of binary layers
Figure Amplitude C as a function of frequency versus layering and contrast
(the gaps relates to the extremely large values).

35. Stack of binary layers
Figure Transmission response versus layering and contrast.
Note the difference between TRT (transmission time for time average medium) and
TEM (transmission time for effective medium). Weak transmission for r=0.87 and
Model M16 is due to the wavelet spectrum is in the blocking regime, see Figure 7.12)

36. Stack of binary layers
Figure Reflection response versus layering and contrast.

37. Stack of binary layers
Figure Transmission (solid line) and reflection (dotted line) amplitudes
as a function of frequency versus layering and contrast.

38. Phase velocity
Figure Phase velocity as a function of frequency versus layering and contrast.
The effective medium is the low frequency part (around effective medium limit), the
transition medium is for dramatical increase in velocity and time average medium is
for oscillating part around time average velocity limit. Note that for small r, the width
of transition zone is narrow comparing with high r case.

39. Transition from effective to time average medium
Critical wavelength-spacing ratio:
l/d=3 (Helbig, 1984)
l/d=5-8 (Carcione et al., 1991)
l/d=10 (Marion et al., 1992, 1994)

40. Transition from effective to time average medium
(7.31)
(7.32)
(7.33)
(7.34)

41. Transition from effective to time average medium
Figure Effective, transition and time average medium (volume fraction 0.5)
versus contrast.

42. O’Doherty-Anstey approximation
Plane waves are normally incident on a sequency of horizontal layers. If the layers are lossless the shape of the frequency spectrum of the reflection response depends on the reflection coefficient series. The law of dependence can be found by solving the wave equation for the boundary and initial conditions of the seismic experiment. The O’Doherty-Anstey formula is an approximation to this law, and its validity would imply a lowpass spectrum of the reflection/transmission response if the reflectivity power spectrum has a highpass trend.

43. O’Doherty-Anstey approximation

44. O’Doherty-Anstey approximation
The ODA result for the retarded transmissivity caused
by propagation through a set of layers is:
(7.35)
where N is the number of layers and R+(z) is the causal half
of the normalized autocorrelation of the reflectivity function
in a z-transform notation
z-transform:
(7.36)

45. O’Doherty-Anstey approximation
its Fourier representation
(7.37)
(7.38)

46. O’Doherty-Anstey approximation
Now recall that reflectivity is a differential process,
and if the elastic parameters are stationary in time, then
(7.39)
and our first, scaling, coefficient goes to zero leaving,
(7.40)

47. O’Doherty-Anstey approximation
Figure Examples of submillimetric fine layering from Beringen coal mine:
Top – coarse sedimentary rock (sandstone),
Bottom – fine sedimentary rock (shaly siltstone)

48. O’Doherty-Anstey approximation
Figure Thin micrograph of Rotliegend Sandstone (at 2990m depth).
Left – laminated structure due to differences in grain size and packing.
Right – details of two laminae, upper: coarser grained laminae with intergranular pores, lower: finer grained laminae with partly filled inrergranular space by detrital clays and dolomite.

49. O’Doherty-Anstey approximation
Figure Thin section micrographs. Scale = 0.25 mm.
Single lamina of very fine-grained, poorly sorted quarz sandstone in shale (2570m depth).
Two laminae of very fine-grained, well sorted quartz arenit interlaminated wirh sandy shale (2920m depth).
Laminated, very fine grained sandstone and interbedded silty shale (2650m depth)

50. O’Doherty-Anstey approximation
Figure Thin section of Rotliegend sandstone (left) and P-wave increase
with triaxial pressure increase

51. O’Doherty-Anstey approximation
Figure Contrubution of first-order multiples into PP transmission (left) and
PS reflection (right).
(7.40)
Stovas and Ursin, 2004

52. O’Doherty-Anstey approximation
The propagator matrix for the stack of N layers (see eq. 7.7)
(7.41)
(7.42)
(7.44)
(7.43)
Stovas and Arntsen, 2003

53. O’Doherty-Anstey approximation
Determinant of propagator matrix
(7.45)
For binary medium
(7.46)

54. O’Doherty-Anstey approximation
The elements of the total propagator matrix
(7.47)
(7.48)

55. O’Doherty-Anstey approximation
The transmission and reflection response:
(7.49)
(7.50)

56. O’Doherty-Anstey approximation
The transmission amplitude
(7.51)
consists of two two terms:
attenuation due to transmission
attenuation due to scattering

57. O’Doherty-Anstey approximation
The transmission phase
(7.52)
also consists of two terms
the time-average term
the scattering term

58. O’Doherty-Anstey approximation
The phase velocity
(7.53)
The zero-frequency limit
(7.54)

59. O’Doherty-Anstey approximation
With approximation of the type
(7.55)
and transmission amplitude 7.51 we obtain
(7.56)
and zero-frequency limit 7.54 becomes
(7.57)

60. O’Doherty-Anstey approximation
Figure The phase velocity and transmission amplitude versus frequency.
The comparison between exact, weak-contrast and O’Doherty-Anstey approximation.
Note the low frequency range, less than 5 Hz.

61. Reuss model
Isostress model (valid for suspensions, with the fluid phase load-bearing), porosity is greater than critical porosity.
(7.58)
The critical porosity separates the mechanical and acoustic behavior into two
disctinct domains.
For porosity less the critical one the mineral grains are load-bearing.
For porosity larger the critical one the sediment becomes a suspension.

62. Voigt model
Isostrain model (the load-bearing domain), porosity is less than critical porosity
(7.59)

63. Thin-layer model
Figure Snapshot for a thin layer model (f=30Hz)

64. Reuss averaging
Figure Snapshot for an effective Reuss model (f=30Hz)

65. Voigt averaging
Figure Snapshot for an effective Voigt model (f=30Hz)

66. Average slowness
Figure Snapshot for an effective average slowness model (f=30Hz)

67. Average velocity
Figure Snapshot for an effective average velocity model (f=30Hz)

68. Backus average
Figure Snapshot for an effective Backus model (f=30Hz)

69. Bio-Gassmann model
Biot (1956): frequency dependent velocities of saturated rocks in terms of the dry rock properties
Gassmann (1951): the low frequency limit of Biot equations
Assumptions and limitations:
- Rock is isotropic
- All minerals making up rock have same bulk and shear moduli
- Fluid-bearing rock is completely saturated
Biot equations can be extended to VTI medium

70. Gassmann model P-wave velocity S-wave velocity Density
(7.60)
(7.61)
P-wave velocity
S-wave velocity
Density
Bulk modulus of solid framework
Shear wave modulus
Intrinsic modulus of solid matrix
Saturated fluid bulk modulus
Porosity

71. Gassmann model Fluid density Matrix density Oil density Water density
(7.62)
Fluid bulk modulus
(7.63)
Fluid density
(7.64)
Fluid density
Matrix density
Oil density
Water density
Oil bulk modulus
Water nulk modulus

72. Hertz-Mindlin model
The Hertz-Mindlin model (Mindlin, 1949) can be used to describe the properties of precompacted granular rocks

73. Hertz-Mindlin model Poisson’s ratio Shear modulus Porosity
(7.65)
(7.66)
Poisson’s ratio
Shear modulus
Porosity
Average number of contacts per grain
Hydrostatic confinig pressure

74. Gassmann-Mindlin Figure 7.32. The vertical P-wave and
S-wave velocities versus water saturation and effective pressure changes.
Stovas and Landro, 2005

75. Gassmann-Mindlin
Figure Relative (to the initial model) changes in P-wave velocity, S-wave velocity
and density versus water saturation and effective pressure changes.
Within the Hertz-Mindlin model density does not change with pressure.

76. Gassmann-Mindlin
Figure The behavior of the PP and PS reflection coefficients with changing water saturation. The initial model reflection coefficients are plotted by circles.
The curves are sampled in the water saturation change of 0.2.

77. Gassmann-Mindlin
Figure The behavior of the PP and PS reflection coefficients with changing
effective pressure. The initial model reflection coefficients are plotted by circles.
The curves are sampled in the change in effective pressure of Gpa.

78. Gassmann-Mindlin
Figure Stacked PP reflection coefficient versus saturation and pressure

79. Gassmann-Mindlin
Figure Stacked PS reflection coefficient versus saturation and pressure