Acoustic and Vibrational Enhanced Oil Recovery. George V. Chilingar

wave equations of the medium model under review is easy to write down for an isotropic case by way of replacing the operator δ_xx 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

(2.5)

      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/τ_r, this model is applicable only near the wave impulse front during the time period or at a distance (C ∞ = 1) smaller than ז_r (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 ˂ τ_r=1:

(2.6)

      where the function fα(ζ) may be represented through an inverse Laplace transform:

(2.7)

      Its expansion in a series is

(2.8)

      where H(ζ) is Heaviside step function. In a special case at α = ½, the series (2.8) converges to

(2.9)

Due to nonuniform convergence of the series Equation (2.8), may also be found asymptotic formula for f_α(ζ). It is determined using the saddle point method:

(2.10)

This correlation becomes equal to the exact expression Equation (2.9) at α = 1/2.

The function f_α(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 φ_α (ζ)* f_α(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]:

(2.11)

(2.12)

      where

The solution of Equations (2.11) and (2.12) has the following format:

(2.13)

      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:

(2.15)

According to the theory of generalized functions, tends to δ(τ)/2 at λ → 0. The Green function G_α(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) Скачать книгу