含孔隙、裂隙岩石的高频体积模量

High-frequency bulk moduli of fluid-saturated cracked porous rocks

前人大量的实验观测证实了体波在流体饱和多孔岩石中传播时存在速度随频率变化的现象.体积模量是决定纵波波速的重要参数之一,研究多孔岩石的高频体积模量对解释超声实验测得的纵波波速具有重要意义.考虑岩石中软孔隙(比如裂隙)的影响,Mavko和Jizba提出了一种修正的Gassmann公式,计算液体饱和多孔岩石的高频体积模量.但Mavko和Jizba提出的计算公式不是通过严格推导得出的,有些过程带有一定猜测性.至今没有一个基于严格推导得到的高频体积模量表达式,人们对多孔岩石高频体积模量物理内涵的认识仍存在不足.本文基于双重孔隙介质理论推导含孔隙、裂隙的流体饱和岩石的高频体积模量表达式,它与Brown-Korringa公式具有相同的形式,只需要将原Brown-Korringa公式中的排水体积模量、固体基质体积模量和孔隙体积模量分别替换为高频时的有效排水体积模量、固体基质体积模量和孔隙体积模量.由于在高频时裂隙与孔隙间没有流体交换,含有流体的裂隙相当于是固体基质的一部分,因此高频有效固体基质的体积模量小于纯固体的体积模量;如果裂隙内流体压强不影响孔隙内流体含量变化,本文得到的高频有效排水体积模量的表达式与Gurevich基于Sayers-Kachanov方法得到的完全一致.数值计算表明:裂隙含量越高,利用Mavko-Jizba公式计算出的高频体积模量与本文公式的偏差越大;高频有效孔隙体积模量总是近似等于纯固体的体积模量,不像高频有效固体基质体积模量那样随裂隙含量的增加而显著降低.

Many experimental data have confirmed velocity dispersion phenomena for body waves in fluid saturated porous rocks. Bulk modulus is a critical parameter for compressional wave velocity, so studing the high-frequency bulk modulus is of importance for ultrasonic compressional wave velocity measurement. Considering the soft pores such as cracks in rocks, Mavko and Jizba proposed a modified Gassmann equation to calculate the liquid saturated bulk modulus for high-frequency condition. However, the equation proposed by Mavko and Jizba is not derived through a rigorous derivation. And there is not an exact expression for high-frequency bulk modulus up to now. Upon above existing problems, we derive an exact expression of high-frequency bulk modulus for fluid saturated cracked porous rocks based on double-porosity model. The derived expression has the same form as Brown-Korringa equation. In the high-frequency limit, there is no time for fluid mass exchange between cracks and pores, and the fluid in cracks become a part of the host, so that the high-frequency effective solid bulk modulus is less than the pure solid bulk modulus. If fluid pressure in cracks has no effect on the fluid content variation in pores, the derived high-frequency drained bulk modulus reduces to Gurevich＇s result which derived from Sayers-Kachanov method. Numerical results show that the deviation of high-frequency saturated bulk modulus between Mavko-Jizba＇s equation and this derived expression increases as crack density increases. And, the high-frequency effective pore bulk modulus is always approximate to pure solid bulk modulus. In contrast, the high-frequency effective solid bulk modulus decreases obviously as crack density increases.