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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 and group velocities of normal waves within the layer were derived. The presence of critical normal waves was established, similar to the appropriate concept of the wave theory in purely elastic wave guides near which the monotonous behavior of group velocities and fading coefficients disrupted. Thus, approximate analytical approach for the consideration of interaction between a layer (bed) with enclosing rocks and major parameters of flat monochromatic waves in the layer was used. Also, the escape of some vibration energy into the overlying and underlying rocks was considered.

      Bio determined the following boundaries of a low-frequency area where theoretic of results using both directions have been in a good agreement:

      where η and ρf are, respectively, fluid viscosity and density; and k and m are rock permeability and porosity, respectively.

Sound dispersion in a micro-nonuniform medium is usually tied with the relaxation processes. These processes are leading to value leveling of some thermodynamic parameter ξ which depends on the wave pressure (or tension) in the enclosing medium and in inclusions. This leveling usually occurs through processes of heat conduction and diffusion through the surface of inclusions and is described by a relaxation time ז_r. For the frequencies ω >> 1 ז_r, the dispersion law may look like follows [29]:

(2.1)

      where K is the frequency limit of the sound velocity; λ is the constant depending on the difference between the sound velocity limit values at high and low frequency and on ז_r; and K is the complex wave vector.

      A similar dispersion law is typical also for nonuniform porous media filled with a viscous liquid due to the sound wave dispersion on the surface of nonuniformities and their conversion to rapidly fading viscous waves [28].

Let us assume that prior to the time moment t = 0, there were no wave disturbances in the medium. Then, a flat wave spreading from the side of positive x-axis values, according to the dispersion law Equation (2.1), is described by the following causal first-order wave equation:

(2.2)

The last term in the left part of Equation (2.2) may be shown as φ_1/2(t)_* д_tu(t), where the asterisk indicates packing of two functions and

      The core of the packing operator Equation (2.2) is a particular case of a more common Abel core:

(2.3)

      where Г is the gamma function.

The core of Equation (2.3) kind may also be used for the description of more complex media. A singular function φ_α(t) at t = 0 is integratable if 0 ˂ α ˂ 1.

      Next, the authors exam the complete second order equation for flat single-dimension waves extracted from Equation (2.2) and generalized according to Equation (2.3) for mathematical modeling of wave spreading within micro-nonuniform media. The causal single-dimensional second-order wave equation is

(2.4)

2D

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