饱和多孔碳酸盐岩弹性模量新模型

M. A. Ismail, F. Morgan
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摘要

波速在理论上由牛顿-拉普拉斯方程定义,它将波速V与弹性模量M和密度ρ之比的平方根联系起来。因此,该方程表明速度与密度成反比。然而,通过不同岩石的纵波速度的现场测量和实验室实验表明,速度与加德纳数值近似所述的密度的四次方成正比。为了澄清理论和观测之间的明显差异,利用威利时间平均方程和牛顿-拉普拉斯方程推导出弹性模量M的新表达式。新导出的弹性模量M的表达式提供了M对密度的大约9次方的依赖关系,随后得到了观测到的速度对密度的4次方的依赖关系。此外,Gardner的方程被修改以准确地获得密度范围内的速度(从1g /cm3到大约3g /cm3)。结果在实际速度和密度测井数据上进行了验证。结果验证了弹性模量的推导表达式和加德纳方程的推广形式。
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New Model for the Elastic Modulus for Saturated Porous Carbonate Rocks
The wave velocity is defined theoretically by the Newton-Laplace equation, which relates the wave velocity, V, to the square root of the ratio of the elastic modulus, M, and density, ρ. Therefore, the equation indicates that the velocity is inversely proportional to density. However, the in-situ field measurements and laboratory experiments of compressional wave velocity through different rocks show otherwise, where the velocity is directly proportional to approximately the 4th power of density as stated by Gardner's numerical approximation. To clarify the apparent contrast between theory and observations, a new expression for the elastic modulus, M, is derived using Wyllie's time average equation and the Newton-Laplace equation. The new derived expression of the elastic modulus, M, provides dependence of M on density to approximately the 9th power, which subsequently results with the observed dependence of velocity on the 4th power of density. In addition, Gardner's equation is modified to accurately obtain the velocity over range of densities (from 1 g/cm3 to around 3 g/cm3). The findings are tested on real velocity and density well-log data. The results validate the derived expression of the elastic modulus as well as the generalized form of Gardner's equation.
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