Hashin-Shtrikman弹性模量边界的转换

Conversion of Hashin-Shtrikman elastic modulus bounds

当孔隙内介质为流体时,用经典的Hashin-Shtrikman弹性模量边界模型估算线弹性各向同性双相孔隙介质整体弹性模量将存在问题。首先基于整体介质与宏观各向同性组分满足的应力和应变组分关系,推导出整体介质弹性矩阵与组分弹性矩阵之间的显式关系,联合其应变关系,定义和分析了对应的两个系数张量,并据此导出了相关场变量位于统一坐标系时整体介质体积模量与剪切模量的关系;然后将经典的Hashin-Shtrikman弹性模量边界模型与上述关系式结合,得到4个新的整体介质弹性模量的估算公式,再与原始的Hashin-Shtrikman弹性模量边界进行了比较,最后针对饱和水纯净砂岩介质进行了试算。结果显示,孔隙介质整体与组分的应力应变、弹性矩阵以及弹性模量之间存在组分加权关系;新的弹性模量表达式计算值在组分弹性模量满足一定条件下位于原始的Hashin-Shtrikman边界内,这在一定程度上弥补了经典Hashin-Shtrikman边界模型对于整体弹性模量特别是孔隙度小于20%时剪切模量估算的不足。

When a rock is saturated with fluid, a satisfying estimation of the effective elastic modulus for the linear elastic isotropic two-phase porous rock cannot be given by the classical Hashin-Shtrikman elastic modulus bounds . First, based on the relations of stress and strain between the whole medium and its macroscopically isotropic constituents, an explicit relation between elastic matrices of the whole medium and its constituents was derived. Combining the explicit relation of elastic matrices with the strain＇s relation, two coefficient tensors were defined and analyzed to get a general relation between the bulk and shear modulus of the whole medium under the assumption that all the field variables are in a unified coordinate system. Secondly, combining these relations with the classical Hashin-Shtrikman elastic modulus bounds, four new expressions of elastic moduli of the whole medium were obtained and compared with the original bounds. Last, these formulae were applied in the computation of effective elastic moduli of clean sandstone saturated with pure water. The results show that the stress and strain, elastic matrices and elastic moduli of the whole medium can be given by the weighted average of those pertinent variables of constituents, and the elastic moduli obtained from the new formulae lie in the original Hashin-Shtrikman elastic modulus bounds, thus a better estimation of the effective elastic moduli of two-phase porous media can be found from these new bounds especially for shear modulus when the porosity of the medium is less than 20%.