Density, Porosity, Seismic Velocities, Reflectivity, Geomechanical Engineering Properties

Density

Density, σ, is the intrinsic unit mass of a material, defined as the mass of one cubic meter with units of kilograms per cubic meter (kg/m3). However, the old cgs units of grams-per-cubic-centimeter are still widely used. One gm/cc is equivalent to 1,000 kg/m3.

Porosity

Porosity, saturation, and density are related as a function of porosity 𝝓 as: (32)

in which subscripts f and m refer to the formation and the matrix, respectively, and S is the fractional water saturation. A water density of 1,000 kg/m3 is assumed. Figure 238 shows the density ranges of common materials. Figure 238. Density ranges in common materials.

Seismic Velocities (VS, VP)

If the ground is stressed by an explosion or a hammer blow, it generates three fundamental types of elastic waves: P (primary, push-pull) waves; S (secondary, shear) waves, and surface waves. The P and S waves propagate through the body of the earth; the surface waves can exist only close to the free surface.

Only P and S are discussed in this section. P waves are characterized by having a particle motion in the direction of propagation, whereas S waves have particle motion transverse to the direction of propagation. P waves are the faster of the two, with velocities typically 50% higher than those for associated S waves. The wave velocities Vp and Vs are related to the elastic moduli (Young's modulus (E), Poisson's ratio (ν), and Bulk modulus (k)) and the density ρ as follows. (33)

Since liquids have no shear rigidity, shear waves cannot propagate through them. Velocities have SI units of meters per second, sometimes also expressed as kilometers per second or meters per millisecond.

The P-wave velocity of a water/competent rock mixture obeys the following relationship (Wyllie's Equation) reasonably well up to porosities of 0.35. (34)

where S is the slowness (1/V), and subscripts f ,w, and m stand for formation, water, and matrix, respectively. Assuming a matrix Vp of 5,950 m/sec for sandstone, and a water velocity of 1,500 m/sec we get the porosity dependence shown in figure 239. Figure 239. P-wave velocities as a function of porosity. Valid for
competent rock only. Overestimates velocity for soft sediments.

Reflectivity

The cause of seismic reflections is contrasts in the seismic impedance (ρV) across a boundary (ρ is the density of the rock, V1 and V2 are seismic velocities on either side of the boundary). In particular, for waves at normal incidence, the ratio R of reflected-to-incident amplitude is given by: (35)

Geomechanical (Engineering) Properties

Seismic velocities can be related to standard geotechnical properties. For example, Poisson's ratio ν can be found from: (36)

Figures 240 and 241 show the S and P wave velocities of seismic waves for a number of different rock types. Figure 240. S-wave velocity ranges for common materials. Figure 241. P wave velocity ranges for common materials.