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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 values underestimated by two or three orders of magnitude at low frequencies compared with the values measured in real media. However, this discrepancy may be imaginary. It may be explained by that the preconditions of the theory relatively boundless and uniform nature of the media do not match the observation conditions at low frequencies when, due to the need of using large measurement bases, absorbing properties of a rock massif with a characteristic dimension no smaller the wave length is evaluated. Thus, the studied medium may not be viewed as uniform relative to its physical properties, and the standard mechanism of viscous friction becomes insufficient (at least in the low frequency area) for the description of elastic waves’ fading patterns in saturated porous media.
To confirm this, an attempt was undertaken, remaining within the framework of this theory, to evaluate the effect of accidental nonuniformities in the medium by introducing a transformation energy exchange mechanism between different wave types [12]. The predicted absorption coefficients and their correlation with frequency well agreed with the experimental data. A more general approach was based on a well-developed procedure of averaging differential equations with rapidly oscillating coefficients [7]. The obtained results enabled a substantial expansion of applicability boundaries of the fading transformational mechanism for a sufficiently broad spectrum of nonuniformity values existing in the real media. This allowed explaining most substantial discrepancy between the theoretical conclusions and experimental data, which indicate the permanent measured experimental fading decrement within a wide frequency range. Also explained was experimentally observed low increase in wave spread velocity with increasing frequency. Frenkel-Bio-Nikolayevsky equations described a linear approximation of wave spreading. With an increase of the source vibration amplitude, the appearance in the medium of nonlinear effects resulting in the formation of stable wave fronts, increase of vibration amplitude far from the source and other phenomena is possible.
Another factor affecting the wave spreading is various boundaries and rock stratification in the productive reservoirs. Experimental observations showed that in a porous medium saturated with oil, gas, or water, reflection coefficients from various fluids’ contact surfaces are comparable with reflection coefficients by geological boundaries separating different lithologies [13]. When boundaries are available with clearly expressed reflecting properties, interactions appear and energy exchange between different wave types.
At certain conditions, the medium stratification results in resonance, wave phenomena [27]. The influence of separation boundaries on wave spreading in the saturated porous media was studied by several authors. The analytical description of the reflection and refraction processes at the boundary of such media is complex. That is why only cases of flat wave normal incidence on the separation boundary of two media were studied or cases of an arbitrary incidence angle on the free surface (contact with vacuum). The problem of a flat wave at-angle incidence on the boundary corresponding with the contact of different fluids (oil, gas, and water) in a porous medium was numerically solved by Berson [1]. Insufficient knowledge of reflection and refraction processes for this case was due to a great number of parameters in the Frenkel-Bio-Nikolayevsky equations, the parameters characteristic for any specific situation. A special problem was selection of the source data for possible calculations as it was necessary, within several calculation situations, to identify specific features of wave transformation processes at the boundaries.
Within the framework of this theory, flat harmonic vertically and horizontally polarized waves within isotropic saturated layer limited by absolutely rigid semi-space were also studied [7]. Analytic formats of the fading coefficients were derived. Analytic formats for phase
