Basic Geophysical Concepts 14
Body wave velocities have form: velocity= V P = V S = V E = K + (4 /3)µ ρ µ ρ E ρ = λ + µ ρ where ρ density K bulk modulus = 1/compressibility µ shear modulus λ Lamé's coefficient E Young's modulus ν Poisson's ratio M P-wave modulus = K + (4/3) µ modulus density P wave velocity S wave velocity E wave velocity Moduli from velocities: µ = ρv S M = ρv P In terms of Poisson's ratio we can also write: K = ρ V P 4 3 V S E = ρv E ( ) V P V = 1 v S (1 v) V E V = ( 1+ v)(1 v) P (1 v) v = V P V S (V P V S ) = V E V S V S Relating various velocities: V P V S = 4 V E V S 3 V E V S V E V S = 3 V P V 4 S V P V 1 S 15
We usually quantify Rock Physics relations in terms of moduli and velocities, but in the field we might look for travel time or Reflectivity ρ 1 V 1 ρ V The reflection coefficient of a normally-incident P- wave on a boundary is given by: where ρv is the acoustic impedance. Therefore, anything that causes a large contrast in impedance can cause a large reflection. Candidates include: Changes in lithology Changes in porosity Changes in saturation Diagenesis R = ρ V ρ 1 V 1 ρ V +ρ 1 V 1 16
AVO Amplitude Variation with Offset V P1, V S1, ρ 1 θ 1 φ 1 Reflected S-wave Deepwater Oil Sand Incident P-wave Reflected P-wave Transmitted P-wave φ θ V P, V S, ρ Transmitted S-wave N.4 Recorded CMP Gather Synthetic In an isotropic medium, a wave that is incident on a boundary will generally create two reflected waves (one P and one S) and two transmitted waves. The total shear traction acting on the boundary in medium 1 (due to the summed effects of the incident an reflected waves) must be equal to the total shear traction acting on the boundary in medium (due to the summed effects of the transmitted waves). Also the displacement of a point in medium 1 at the boundary must be equal to the displacement of a point in medium at the boundary. 17
AVO - Aki-Richards approximation: P-wave reflectivity versus incident angle: Intercept R(θ) R 0 + 1 + 1 V P V P R 0 1 V P V P V P V P + ρ ρ Gradient V S ρ V P ρ + V S V S sin θ [ tan θ sin θ] In principle, AVO gives us information about Vp, Vs, and density. These are critical for optimal Rock Physics interpretation. We ll see later the unique role of P- and S-wave information for separating lithology, pressure, and saturation. 18
Seismic Amplitudes Many factors influence seismic amplitude: Source coupling Source radiation pattern Receiver response, coupling, and pattern Scattering and Intrinsic Attenuation Sperical divergence Focusing Anisotropy Statics, moveout, migration, decon, DMO Angle of Incidence Reflection coefficient Source Rcvr 19
Intervals or Interfaces? Crossplots or Wiggles? Rock physics analysis is usually applied to intervals, where we can find fairly universal relations of acoustic properties to fluids, lithology, porosity, rock texture, etc. Interval Vp vs. Phi Interval Vp vs. Vs In contrast, seismic wiggles depend on interval boundaries and contrasts. This introduces countless variations in geometry, wavelet, etc. A B 0
Impedance vs. depth Convolutional Model Reflectivity Stanford Rock Physics Laboratory - Gary Mavko Normal Incidence Seismic Rock properties in each small layer Derivatives of layer properties Convolve With wavelet Smoothed image of derivative of impedance Normal incidence reflection seismograms can be approximated with the convolutional model. Reflectivity sequence is approximately the derivative of the impedance: R(t) 1 d dt ln ( ρv ) Seismic trace is smoothed with the wavelet: S(t) w(t) R(t) Be careful of US vs. European polarity conventions! 1
Inversion Stanford Rock Physics Laboratory - Gary Mavko Two quantitative strategies to link interval rock properties with seismic: Forward modeling Inversion We have had great success in applying rock physics to interval properties. For the most part, applying RP directly to the seismic wiggles, requires a modeling or inversion step. We often choose a model-based study, calibrated to logs (when possible) to Diagnose formation properties Explore situations not seen in the wells Quantify signatures and sensitivities
The Rock Physics Bottleneck At any point in the Earth, there are only 3 (possibly 4) acoustic properties: Vp, Vs, density, (and Q). No matter how many seismic attributes we observe, inversions can only give us three acoustic attributes Others yield spatial or geometric information. Seismic Attributes Traveltime Vnmo Vp/Vs Ip,Is Ro, G AI, EI Q anisotropy etc Acoustic Properties Vp Vs Density Q Reservoir Properties Porosity Saturation Pressure Lithology Pressure Stress Temp. Etc. 3
Problem of Resolution Log-scale rock physics may be different than seismic scale 4
Seismic properties (velocity, impedance, Poisson Ratio, etc) depend on pore pressure and stress Units of Stress: 1 bar = 10 6 dyne/cm = 14.50 psi 10 bar = 1 MPa = 10 6 N/m 1 Pa = 1 N/m = 1.45 10-4 psi = 10-5 bar 1000 kpa = 10 bar = 1 MPa Stress always has units of force/area Mudweight to Pressure Gradient 1 psi/ft = 144 lb/ft 3 = 19.4 lb/gal =.5 kpa/m 1 lb/gal = 0.05 psi/ft 5