Acoustic and Vibrational Enhanced Oil Recovery. George V. Chilingar

Acoustic and Vibrational Enhanced Oil Recovery - George V. Chilingar


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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).

      where the function (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

      The trivariate Green function for the medium model under review represents a solution of the spherically symmetric Cauchy problem [30]:

      where image

      where it is assumed that

      (2.14)image

      (2.16)image

      where

      (2.17)image

      (2.18)image

      (2.19)Скачать книгу