The influence of pore structure in reservoir sandstone on dispersion properties of elastic waves

The pore structure of reservoir sandstone can significantly influence its elastic properties (e. g. elastic wave velocities), and also determine fluid-flow related wave dispersion and attenuation. In the common velocity dispersion models, only the compliant pore or crack with a fixed aspect ratio and concentration has been taken into account, while the fixed aspect ratio is considered as the average value of the rock, which is not completely realistic in reservoir rock. Because rock usually contains compliant pore ( crack) showing a distribution of aspect ratios. This study presents a procedure to determine the pore aspect distribution of compliant pore (crack) from the pressure dependence of velocities. Based on the pore aspect distribution of compliant pore, a new method is suggested to extend the existing squirt flow model to consider the complex pore structure, especially when the aspect ratio has a relatively wide distribution. Based on micro-structure of a reservoir sandstone and the non-linear feature of velocities as a function of effective pressure, the pore system of the sandstone can be ideally classified into two groups, i. e. stiff pores with an aspect ratio larger than 0. 01 and soft pores with an aspect ratio smaller than 0. 01. In light of the poroelasticity theory, an analytic expression for the minimal initial aspect ratio is deduced under the assumption that the shape of soft pores is spheroidal. With this equation, a method to invert the distribution of the aspect ratio and corresponding pore volumes is presented by using the measured ultrasonic velocities as a function of pressure. Using an iterative procedure to add soft pore with different aspect ratios into the rock frame, the existing squirt fluid model of Gurevich is extended to consider complex pore structure of reservoir sandstones, especially when the aspect ratio has a relatively wide distribution. With the pore aspect ratio distribution, the extended Gurevich's squirt-flow model is used to compute the wave velocities and attenuation as functions of frequency as well as pressure. When considering aspect ratio distribution of crack pore in the reservoir rock, the overall dispersion curve shows rapid velocity increase around a relatively wide squirt-flow relaxation frequency range of 1 similar to 104 Hz, which covers the typical seismic and sonic logging frequencies, indicating the mechanism of a continuous relaxation spectrum of the complex pore system. Compared with typical velocity dispersion curves based on a single aspect ratio squirt fluid model, the dispersion curves of the sandstone with a relatively wide distribution of aspect ratios do not show the low-frequency and middle-frequency range. This implies that for the rock samples at low pressure, it is not always profitable to employ Gassmann's equation alone to predict the water-saturated velocities at typical seismic exploration frequencies. With the increasing pressure, velocity dispersion of the Gurevich's squirt-flow model and the extended Gurevich's squirt-flow model based on the aspect ratio distribution will decrease. To illustrate the validation of the extended Gurevich's squirt-flow model, we compare predictions of our squirt model with laboratory measurement of two water-saturated sandstones at ultrasonic frequency of 700 kHz. We observe that the new model based on the pore structure is more accurate in predicting the pressure dependence of compression and shear velocities for the water-saturated sample than the Gassmann's equation and Gurevich's squirt-flow model. Our extended Gurevich's squirt-flow model is consistent with Gassmann's equation at low-frequency limit, and also with the Mavk-Jizba model at high-frequency. Through this study, we suggest that the squirt flow may be still important even in the seismic frequency band, and can cause apparent velocity deviation from the predictions based on Gassmann's equation. Thus, our work can be considered as an important extension of the existing squirt fluid models.