基于线性近似的微分等效介质方程解耦方法

岩石物理理论在弹性模量与速度估算中发挥着重要作用。干岩石骨架弹性模量是计算饱和岩石弹性模量和速度的关键参数,微分等效介质理论(DEM)是计算干岩石骨架等效模量的理论依据,但由于其方程是耦合的,很难得到精确解析解,通常只能求取近似数值解。基于前人研究成果,假设干岩石骨架模量比是孔隙度的线性函数,利用极化因子表达式,将耦合微分等效方程解耦为常微分方程,构建了一种新的干岩石骨架等效模型;与微分方程数值解做对比,可知本文导出的解析解能有效计算干岩石骨架等效模量;利用Han实验数据和实际井资料做测试,结果表明用该模型可有效估算岩石纵横波速度。

A decoupling approach for differential equivalent equations based on linear approximation

Rock physics theory plays an important role in the estimation of elastic modulus and velocity. Dry rock skeleton elastic modulus is the key parameter for calculating saturated rock elastic modulus and velocity. Differential equivalent medium theory is the theoretical basis for calculating dry rock skeleton equivalent modulus. Because of the coupled equations, it is difficult to obtain its exact analytical solution and we usually only obtain an approximate numerical solution. Based on predecessors' research results, we assume that dry rock skeleton modulus ratio is a linear function of porosity. With the polarization factor expressions, we decouple the coupled differential equivalent equations to ordinary differential equations, and build a new dry rock skeleton equivalent model. Analytical solution derived in this paper can efficiently calculate the equivalent modulus of dry rock skeleton compared with numerical simulations. In the end, tests on Han's experimental data and real logging data show that the proposed approach can effectively estimate rock P- and S-waves' velocity.