多重孔岩石微分等效介质模型及其干燥情形下的解析近似式

经典的微分等效介质(DEM)理论可用于确定多孔介质的弹性性质,但由于缺乏多重孔DEM方程,其估计的多重孔岩石的等效弹性模量依赖于包裹体(即不同孔隙纵横比的孔或缝)的添加顺序.本文首先从Kuster-Toksöz理论出发建立了Zimmermann和Norris两种形式的多重孔DEM方程.Norris形式的多重孔DEM方程预测的等效弹性模量总是位于Hashin-Shtrikman上下限内,而Zimmermann形式的多重孔DEM方程有时会越界.然后,通过使用干燥岩石模量比的解析近似式,对两个相互耦合的Norris形式DEM方程进行解耦得到干燥多重孔岩石的体积和剪切模量解析式.用全DEM方程的数值解对解析近似式的有效性进行了测试,解析公式的计算结果在整个孔隙度分布区间与数值解吻合良好.对实验室测量数据在假设岩石含有双重孔隙的情形下用双重孔DEM解析公式对岩石的弹性模量进行了预测,结果表明,解析式准确地预测了弹性模量随孔隙度的变化.双重孔(即软、硬孔)DEM解析模型可用来反演各孔隙类型的孔隙体积比,它可以通过实验室测量与理论预测之间的平方误差最小反演得到.砂岩样品的反演结果揭示,软孔的孔隙体积百分比与粘土含量没有明显的相关性.

A differential effective medium model of multiple-porosity rock and its analytical approximations for dry rock

The classic differential effective medium (DEM) theory can be used to determine the elastic properties of the porous medium, but the final elastic properties of multiple-porosity rock depend on the added order of the different pore-type inclusions due to the lack of DEM equations for multiple-porosity rock. This paper first derives the differential equations of both Zimmermann's and Norris's versions for multiple-porosity rock from the Kuster-Toksöz theory. The elastic moduli predicted by the DEM equations of Norris's version never violate Hashin-Shtrikman bounds while those predicted by the DEM equations of Zimmermann's version violate bounds in some cases. Then, we derive analytical solutions of the bulk and shear moduli for dry rock from the differential equations of Norris's version by applying an analytical approximation for dry-rock modulus ratio, in order to decouple these equations. The validity of these analytical approximations is tested by integrating the full DEM equation numerically. The analytical formulae give good estimates of the numerical results over the whole porosity range. The analytical formulae have been used to predict the elastic moduli for the sandstone experimental data by assuming that the porous rock contains dual-porosity of both cracks and pores. The results show that they can accurately predict the variations of elastic moduli with porosity for dry sandstones. We also apply nonlinear global optimization algorithm to find the best estimate for the pore volume percentage of both cracks and pores by minimizing the error between theoretical predictions and experimental measurements based on the dual-porosity DEM analytical model. The inversions for the sandstone experimental data show that there is no direct correlation between the crack volume percentage and clay volume.