Equations utilized for low-frequency and ultrasonic velocity measurements after Mavko et al. (2009). The abbreviation |$\sigma _{\rm ax}$| stands for the vertical stress, |$\varepsilon _{\rm ax}$| and |$\varepsilon _{\rm rad}$| for the axial and radial strain, respectively and |$\rho _b$| for the bulk density at dry conditions
| Parameter . | Low-frequency . | Ultrasonic velocity . |
|---|---|---|
| Young’s modulus E | |$\frac{\sigma _{\rm ax}}{\varepsilon _{\rm ax}}$| | |$\rho _b V^2_S \left(\frac{3V^2_P - 4V^2_S}{V^2_P - V^2_S}\right)$| |
| Poisson’s ratio |$\nu$| | |$\frac{\varepsilon _{\rm rad}}{\varepsilon _{\rm ax}}$| | |$\frac{V^2_P-2V^2_S}{2(V^2_P-V^2_S)}$| |
| Bulk modulus K | |$\frac{E}{3(1-2\nu )}$| | |$\rho _b \left( V^2_P -\frac{4}{3} V^2_S \right)$| |
| Shear modulus G | |$\frac{E}{2(1+\nu )}$| | |$V^2_S \rho _b$| |
| Parameter . | Low-frequency . | Ultrasonic velocity . |
|---|---|---|
| Young’s modulus E | |$\frac{\sigma _{\rm ax}}{\varepsilon _{\rm ax}}$| | |$\rho _b V^2_S \left(\frac{3V^2_P - 4V^2_S}{V^2_P - V^2_S}\right)$| |
| Poisson’s ratio |$\nu$| | |$\frac{\varepsilon _{\rm rad}}{\varepsilon _{\rm ax}}$| | |$\frac{V^2_P-2V^2_S}{2(V^2_P-V^2_S)}$| |
| Bulk modulus K | |$\frac{E}{3(1-2\nu )}$| | |$\rho _b \left( V^2_P -\frac{4}{3} V^2_S \right)$| |
| Shear modulus G | |$\frac{E}{2(1+\nu )}$| | |$V^2_S \rho _b$| |
Equations utilized for low-frequency and ultrasonic velocity measurements after Mavko et al. (2009). The abbreviation |$\sigma _{\rm ax}$| stands for the vertical stress, |$\varepsilon _{\rm ax}$| and |$\varepsilon _{\rm rad}$| for the axial and radial strain, respectively and |$\rho _b$| for the bulk density at dry conditions
| Parameter . | Low-frequency . | Ultrasonic velocity . |
|---|---|---|
| Young’s modulus E | |$\frac{\sigma _{\rm ax}}{\varepsilon _{\rm ax}}$| | |$\rho _b V^2_S \left(\frac{3V^2_P - 4V^2_S}{V^2_P - V^2_S}\right)$| |
| Poisson’s ratio |$\nu$| | |$\frac{\varepsilon _{\rm rad}}{\varepsilon _{\rm ax}}$| | |$\frac{V^2_P-2V^2_S}{2(V^2_P-V^2_S)}$| |
| Bulk modulus K | |$\frac{E}{3(1-2\nu )}$| | |$\rho _b \left( V^2_P -\frac{4}{3} V^2_S \right)$| |
| Shear modulus G | |$\frac{E}{2(1+\nu )}$| | |$V^2_S \rho _b$| |
| Parameter . | Low-frequency . | Ultrasonic velocity . |
|---|---|---|
| Young’s modulus E | |$\frac{\sigma _{\rm ax}}{\varepsilon _{\rm ax}}$| | |$\rho _b V^2_S \left(\frac{3V^2_P - 4V^2_S}{V^2_P - V^2_S}\right)$| |
| Poisson’s ratio |$\nu$| | |$\frac{\varepsilon _{\rm rad}}{\varepsilon _{\rm ax}}$| | |$\frac{V^2_P-2V^2_S}{2(V^2_P-V^2_S)}$| |
| Bulk modulus K | |$\frac{E}{3(1-2\nu )}$| | |$\rho _b \left( V^2_P -\frac{4}{3} V^2_S \right)$| |
| Shear modulus G | |$\frac{E}{2(1+\nu )}$| | |$V^2_S \rho _b$| |
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