In addition, ρ is the density, K is the bulk modulus, µ is the shear modulus, λ is Lamé’s coefficient, E is Young’s modulus, and v is Poisson’s ratio.

In this section, we use the symbol s to represent the frequency, or Fourier transform variable, following the terminology of Bracewell (1965). When x refers to time, the symbol f is often used for frequency instead of s; when x refers to space, the symbol k is often used for spatial frequency instead of s. Do not confuse this with the use of s when defining the Laplace transform (Section 1.3) or S when defining the analytic signal (Section 1.2).

Spheres are often used as idealized representations of grains in unconsolidated and poorly consolidated sands. They provide a means of quantifying geometric relations, such as the porosity and the coordination number, as functions of packing and sorting. Using spheres also allows an analytical treatment of mechanical grain interactions under stress. Recent research also started to reveal how the properties of the granular medium change as particle shapes differ from perfect spheres.

Nur et al. (1991, 1995) and other workers have championed the simple, if not obvious, idea that the P and S velocities of rocks should trend between the velocities of the mineral grains in the limit of low porosity and the values for a mineral–pore-fluid suspension in the limit of high porosity.

If we wish to predict the effective dielectric permittivity ε of a mixture of phases theoretically, we generally need to specify: (1) the volume fractions of the various phases, (2) the dielectric permittivity of the various phases, and (3) the geometric details of how the phases are arranged relative to each other.

Biot (1956) derived theoretical formulas for predicting the frequency-dependent seismic velocities of saturated rocks in terms of the dry-rock properties. His formulation incorporates some, but not all, of the mechanisms of viscous and inertial interaction between the pore fluid and the mineral matrix of the rock.

It was established experimentally by Darcy () that the fluid flow rate in a fluid-saturated porous medium is linearly related to the pressure gradient by the following equation

If we wish to predict theoretically the effective elastic moduli of a mixture of grains and pores, we generally need to specify: (1) the volume fractions of the various phases, (2) the elastic moduli of the various phases, and (3) the geometric details of how the phases are arranged relative to each other. If we specify only the volume fractions and the constituent moduli, the best we can do is predict the upper and lower bounds (shown schematically in Figure 4.1.1).

In an isotropic, linear elastic material, the stress and strain are related by Hooke’s law as follows (e.g., Timoshenko and Goodier, 1934)

Introduction

Seismic reflections depend on the contrast of the P- and S-wave velocity and density between strata in the subsurface, while the velocity and density depend on the lithology, porosity, rock texture, pore fluid, and stress. These two links, one between the rock’s structure and its elasticity and the other between the elasticity and signal propagation, form the physical foundation of the seismic-based interpretation of rock properties and conditions. One approach to interpreting seismic data for the physical state of rock is forward modeling. Lithology, porosity, stress, pore pressure, and the fluid in the rock, as well as the reservoir geometry, are varied, the corresponding elastic properties are calculated, and then synthetic seismic traces are generated.

These synthetic traces are compared to real seismic data: full gathers; full stacks; and/or angle stacks. The underlying supposition of such interpretation is that if the seismic response is similar, the properties and conditions in the subsurface that give rise to this response are similar as well. Systematically conducted perturbational forward modeling helps create a catalogue (field guide) of seismic signatures of lithology, porosity, and fluid away from well control and, by so doing, sets realistic expectations for hydrocarbon detection and monitoring and optimizes the selection of seismic attributes in an anticipated depositional setting.