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2
Wave Spreading Patterns in the Porous Media
2.1 Spread of Vibration in Reservoir
Wave spreading in the media with fading is best studied for the conditions of low amplitude flat waves within a boundless and uniform porous medium saturated with a viscous fluid. Usually, two directions are identified in these studies.
The first direction involves construction of idealized models of porous medium. In such models, the solid phase is viewed as a system of variously packed grains with fluid-filled voids. It is assumed that the effect of a relative motion of the fluid and rock matrix on the wave spreading is negligible. This assumption is substantiated for low frequencies although exact frequency criteria of its applicability are absent. It is believed that this approach is applicable in seismology and seismic exploration as it allows for an approximate computation of major parameters of the elastic vibration field for idealized porous medium models with a certain grain packing at the assigned thermodynamic conditions.
Another study direction is based on mechanics of a continuous medium and on thermodynamics of irreversible processes. This direction was first described by Frenkel [24] and subsequently expanded by Bio and Rahmatullin [18], Nikolayevsky [15], Nigmatullin [14], and others. This direction presents substantially broader opportunities. In describing a porous medium by a set of thermodynamic variables (“observed” mechanical and concealed parameters), it is possible to determine various dissipative functions, to evaluate the system behavior in time and the relaxation effects. The elastic constants in the equations of wave spreading, according to the conformity principle, may be replaced by operators, and this way, various processes of absorption and dispersion may be accounted for. For instance, it may be processes associated with surface effects, dissipative phenomena directly in the solid phase or liquid, etc. Within a framework of this approach, temperature effects may also be considered, effects of porous medium compressibility changes at changing of frequency and other relaxation processes resulting in fading of the elastic waves.
This theory gives the fading coefficient