The power-law distribution of the rock pore structure: inversion and analysis of laboratory velocity-pressure data

Linking the pore structure distributions and velocity-pressure relationship is important for understanding geological processes and in situ stress conditions. During the pressure loading, the pressure-velocity relationship is controlled by variation of pore-aspect-ratio spectrum distribution (pore-aspect-ratio vs. porosity) with pressure: on the one hand, the “flat crack” (aspect ratio) will be squeezed and closed, causing the large velocity variation in the low-pressure regimes; on the other hand, the shape and volume of the “round pore” (aspect ratio) are less sensitive to pressure, controlling the smaller velocity change at high pressures. The above pore-structure variation process and its effect on the velocity-pressure relationship allow us to invert and analyze the distribution characteristics from the laboratory velocity-pressure data.  

The power-law distribution has been universally applied to characterize the length and aperture distribution in natural fractures; however, it is still not well understood whether the pore-aspect-ratio spectrum distribution also follows the power law and how this power-law distribution relates to other power-law distributions of fractures. In this study, we examine the universality of the power-law relationship for the pore aspect ratio spectrum distribution and show that this distribution is the natural outcome of the power-law behavior of the length and aperture distribution in the rock fracture network.

We first collected laboratory ultrasonic velocity-pressure data for 52 rock samples with various porosities and lithologies. We then used a power law assumption to link the pore aspect ratio and porosity. The power-law exponents (hereafter referred to as the E factor) of the power-law distributions were successfully obtained by inverting the compiled velocity-pressure datasets. The inversion results showed a universal relationship between the E factor and porosity. We also demonstrated that the power-law distribution of the pore-aspect-ratio spectrum and the universal E factor-porosity trend can be explained using the length and aperture distribution characteristics of the microcrack system of rocks. This universal E factor-porosity trend provides a link between porosity and sensitivity of seismic velocities to pressure. Hence, our work not only emphasizes the relevance of natural fracture fractal distributions for a broad range of lithologies but also provides the parameterized method for relating the velocity-pressure relationship of any rock to porosity variation.