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)

The Middle Mesozoic Drift and Cooling Phase begins with the main phase of sea floor spreading, slowly but steadily opening the Gulf of Mexico basin. Initially hypersaline conditions resulted in basin-wide deposition of an original thickness of 4 km of evaporites (halite and updip anhydrite), called the Louann Salt, which likely formed with episodic seawater influx from the Atlantic Ocean. Strontium seawater analysis suggests 170 Ma as a proxy age for the Louann Salt. The arid eolian Norphlet Formation is subsequently deposited, followed by marine carbonates, evolving from ramp microbalites (Smackover) to platform margin reef systems of the Haynesville and Cotton Valley. Rafting apart of the Smackover and Norphlet in the northeast Gulf of Mexico began in this phase, possibly associated with oceanic crustal cooling which created a dip slope to the south and west. This set up a major new petroleum province which is host to several new giant oil discoveries. Periods of reduced bottom circulation resulted in at least two phases of source rock development, in the Oxfordian and Tithonian stages, that are linked to petroleum generation for both conventional and unconventional plays.

The Middle Miocene marked the emergence of the Appalachian uplands as a significant sediment source to the Gulf of Mexico. As a result, the Tennessee River joined the Mississippi in creating the dominant fluvial/deltaic depocenter. At the same time, supply from western interior uplands decreased. Two Miocene deposodes and multiple eustatically modulated high-frequency Pliocene—Pleistocene deposodes are recorded in northern Gulf stratigraphy. The continental slope wedge prograded onto the shallow Sigsbee salt, initiating canopy deformation and rapid basinward canopy advance. Salt-encased minibasins created rugose slope topography with multiple, efficient sediment traps. Nonetheless, large volumes of sediment bypassed the continental slope and constructed a series of large, long-lived abyssal plain fans. A narrow coastal plain and shelf prograded along the western Gulf margin. Extensional growth faulting was compensated basinward by compressional faulting and folding above Paleogene detachments. In the Sureste, the river-fed, prograding continental margin and ongoing basement deformation mobilized salt of the Campeche Salt Basin.