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Wave Spreading Patterns in the Porous Media 2.1 Spread of Vibration in Reservoir
ОглавлениеWave 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 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 זr. For the frequencies ω >> 1 זr, the dispersion law may look like follows [29]:
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 זr; 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:
The last term in the left part of Equation (2.2) may be shown as φ1/2(t)* дtu(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:
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 φα(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
2D and 3D scalar wave equations of the medium model under review is easy to write down for an isotropic case by way of replacing the operator δxx with the dimension-appropriate Laplace operator. These equations obviously describe the wave spreading in some viscous-elastic medium. The right part of Equation (2.4) is different from zero at the availability of diffuse sources. The fundamental solution of Equation (2.4) must satisfy initial conditions
where δ(x) is Dirac delta function.
This fundamental solution enables the presentation of a general solution Equation (2.4) in the form of modified Duhamel integral, and in and of itself, it describes a wave impulse excited by an instantaneous point source. As Equation (2.4) is asymptotic at ω >> 1/τr, this model is applicable only near the wave impulse front during the time period or at a distance (C ∞ = 1) smaller than זr (which at accepted dimensionless units equals one).
The wave impulse front that emerged at the moment t = 0 and at the point x = 0 reaches the points +x and −x at the moment t = |x|. Solution of the problem Equations (2.4) and (2.5) represents the wave after this moment if to record t = |x| + τ and 0 ˂ τ ˂ τr=1:
where the function fα(ζ) may be represented through an inverse Laplace transform:
Its expansion in a series is
where H(ζ) is Heaviside step function. In a special case at α = ½, the series (2.8) converges to
Due to nonuniform convergence of the series Equation (2.8), may also be found asymptotic formula for fα(ζ). It is determined using the saddle point method:
This correlation becomes equal to the exact expression Equation (2.9) at α = 1/2.
The function fα(ζ) defined by the equality Equation (2.7) converges to δ(ζ) at λ → 0. There is the equivalent representation of the last term in expression Equation (2.6) for cores of Equation (2.3) type. The packing φα (ζ)* fα(ζ) may be computed explicitly for this case and the left part of Equation (2.6) may then be written in a form more convenient for computation as it no longer includes the second integral. The function t = (|x| + τ|x|) is smooth and is faster tending to zero with the approach to the front corresponding to τ = 0 than nay power of τ, remaining infinitely differentiable which is obvious from (2.10).
The trivariate Green function for the medium model under review represents a solution of the spherically symmetric Cauchy problem [30]:
where
The solution of Equations (2.11) and (2.12) has the following format:
where it is assumed that
(2.14)
The Green’s function (2.13) behind the front (τ ˂˂ αr) as presented in space-time has the following format:
According to the theory of generalized functions, tends to δ(τ)/2 at λ → 0. The Green function Gα(t,r) shown in Equation (2.15) converges in this case to a simple Green function of the classical wave differential equation, i.e., to . The solution at α = 1/2 may be presented in a simple explicit format:
(2.16)
where
(2.17)
(2.18)
(2.19)
As two independent polarization types are available in a solid isotropic medium, the total displacement vector u(r,t) should be presented in the form of in-plane ul and lateral u r displacements:
(2.20)
Each of these displacements may be directed by integrodifferential equation of Equation (2.11) type but with different multiplier before the terms including time derivatives due to the differences in the high frequency sound velocity limit for these two wave types. The integral terms creep cores also differ (displacement creep function and triaxial creep function). However, their time correlation is identical if identical processes lead to local relaxation. For this reason, the methods designed for scalar equations may be applied with every scalar component of every polarization displacement vector simply by scaling the derived solutions.
