2 7.Effective medium Upscaling problem Backus averaging O’Doherty-Anstey approximationReuss and Voigt modelsBio-Gassmann modelHertz-Mindlin model
3 Upscaling problemDoes seismic wave see thethin-layering?
4 Upscaling problem From microscopic to macroscopic scale From pore (graine) scale (millimeters)From log-scale (centimeteers)
5 Upscaling problemTraditionally, 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 averagingSequential 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 averagingThe 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
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 gaprelates 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 thanthe propagation time through the cycle)(7.16)The effective medium limit can becomputed assuming phases beingsmall (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 deviationbetween time-average and effective medium velocities. The position for maximumdifference between them moving to high values of volume fraction with r increase.
28 Stack of binary layersThe 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)
31 Stack of binary layersPropagating regime(7.27)(7.28)
32 Stack of binary layersBlocking regime(7.29)(7.30)
33 Stack of binary layersFigure 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 layersFigure Amplitude C as a function of frequency versus layering and contrast(the gaps relates to the extremely large values).
35 Stack of binary layersFigure Transmission response versus layering and contrast.Note the difference between TRT (transmission time for time average medium) andTEM (transmission time for effective medium). Weak transmission for r=0.87 andModel M16 is due to the wavelet spectrum is in the blocking regime, see Figure 7.12)
36 Stack of binary layersFigure Reflection response versus layering and contrast.
37 Stack of binary layersFigure Transmission (solid line) and reflection (dotted line) amplitudesas a function of frequency versus layering and contrast.
38 Phase velocityFigure Phase velocity as a function of frequency versus layering and contrast.The effective medium is the low frequency part (around effective medium limit), thetransition medium is for dramatical increase in velocity and time average medium isfor oscillating part around time average velocity limit. Note that for small r, the widthof 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.
44 O’Doherty-Anstey approximation The ODA result for the retarded transmissivity causedby propagation through a set of layers is:(7.35)where N is the number of layers and R+(z) is the causal halfof the normalized autocorrelation of the reflectivity functionin a z-transform notationz-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 increasewith triaxial pressure increase
51 O’Doherty-Anstey approximation Figure Contrubution of first-order multiples into PP transmission (left) andPS 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 transmissionattenuation due to scattering
57 O’Doherty-Anstey approximation The transmission phase(7.52)also consists of two termsthe time-average termthe 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 modelIsostress 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 twodisctinct 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 modelIsostrain model (the load-bearing domain), porosity is less than critical porosity(7.59)
63 Thin-layer modelFigure Snapshot for a thin layer model (f=30Hz)
64 Reuss averagingFigure Snapshot for an effective Reuss model (f=30Hz)
65 Voigt averagingFigure Snapshot for an effective Voigt model (f=30Hz)
66 Average slownessFigure Snapshot for an effective average slowness model (f=30Hz)
67 Average velocityFigure Snapshot for an effective average velocity model (f=30Hz)
68 Backus averageFigure Snapshot for an effective Backus model (f=30Hz)
69 Bio-Gassmann modelBiot (1956): frequency dependent velocities of saturated rocks in terms of the dry rock propertiesGassmann (1951): the low frequency limit of Biot equationsAssumptions and limitations:- Rock is isotropic- All minerals making up rock have same bulk and shear moduli- Fluid-bearing rock is completely saturatedBiot equations can be extended to VTI medium
70 Gassmann model P-wave velocity S-wave velocity Density (7.60)(7.61)P-wave velocityS-wave velocityDensityBulk modulus of solid frameworkShear wave modulusIntrinsic modulus of solid matrixSaturated fluid bulk modulusPorosity
71 Gassmann model Fluid density Matrix density Oil density Water density (7.62)Fluid bulk modulus(7.63)Fluid density(7.64)Fluid densityMatrix densityOil densityWater densityOil bulk modulusWater nulk modulus
72 Hertz-Mindlin modelThe 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 ratioShear modulusPorosityAverage number of contacts per grainHydrostatic 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-MindlinFigure Relative (to the initial model) changes in P-wave velocity, S-wave velocityand density versus water saturation and effective pressure changes.Within the Hertz-Mindlin model density does not change with pressure.
76 Gassmann-MindlinFigure 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-MindlinFigure The behavior of the PP and PS reflection coefficients with changingeffective pressure. The initial model reflection coefficients are plotted by circles.The curves are sampled in the change in effective pressure of Gpa.
78 Gassmann-MindlinFigure Stacked PP reflection coefficient versus saturation and pressure
79 Gassmann-MindlinFigure Stacked PS reflection coefficient versus saturation and pressure