A new squirt coefficient to simulate P‐wave attenuation

Abstract

In double‐permeability models, the squirt coefficient representing the fluid mass equation came from mathematical parameterization that was inconsistent with physics, resulting in the prediction of an infinitely large squirt coefficient at an infinitely high frequency. This was a drawback of the models, as an infinitely large squirt flux rate means unrealistic infinite large velocity and kinematic energy (velocity and energy must be bounded). This paper is motivated by solving the problem. At first, when P‐wave compresses a porous rock, the contact of grains (the normal of which is the wave direction) is very compliant, which generates a strong squirt transverse to the wave direction. In contrast, the squirt along the wave direction is relatively small. Based on such a transverse squirt mechanism, a new squirt coefficient is derived, which yields a vanishing value at infinite high frequency. Secondly, the squirt coefficient is put into a previous model to predict velocity and attenuation of ultrasonic P‐wave. Two core samples (Berea sandstone and Boise sandstone) are used for illustration; both the measured velocity and quality factor are well simulated. It is notable that there is an interesting shoulder on the velocity dispersion curves before the asymptotical behaviour of velocity at high frequency.

Monochromatic Wave of Fluid Pressure in a Porous Rock

“…An issue is how to validate the dynamic permeability (Equation 10) experimentally. Using the measured fast P‐wave for validation is not a good way, because the dominant mechanism of fast P‐wave is the pore‐scale squirt rather than dynamic permeability (Budiansky and O'Connell 1976; Mochizuki 1982; Jones and Nur 1983; Li et al 2021; Li 2022). Fortunately, S‐wave in fluid saturated rocks has no squirt (Li 2020).…”

“…The cause is that when fast P‐wave compresses a porous rock, fluid will squirt from compliant contact of grains to the main pore space (Murphy et al 1984; Dvorkin et al 1995; Johnson 2001; Pride et al 2004). Using a double porosity model, Li et al (2021) and Li (2022) showed that the pore‐scale squirt is the dominant mechanism of fast P‐wave which cannot be simulated by models of single porosity such as Biot (1956a, 1956b) theory. Whether for fast P‐wave or for slow P‐wave, the contact of grains invariably has a larger volumetric strain than the main pore space has (with the consideration of the relatively rigid solid material).…”

Ultrasonic P-Wave to Acquire Parameters of Boise Sandstone

“…More recently, an effort of modifying the constitutional relations in [16][17][18] was made by [21] to automatically yield the Gassmann [2] velocity at the lowfrequency end. Later on, the squirt coefficient from mathematical parameterization in [16][17][18] was upgraded by [22] based on fluid mechanics. P-wave is associated with pushing/pulling stress onto the surfaces of a rock unit, thus changing the rock-unit volume and the pore pressure [1][2][3][4][5][6].…”

“…Based upon the intrapore squirt, Li et al [21] and Li [22] developed a model of P-wave in isotropic rock saturated with fluid. For Berea sandstone (a classic sandstone with relatively uniform diameter of grain), the model [21,22] accurately simulated ultrasonically measured velocity and attenuation [30] of the sandstone saturated with water.…”

“…King [31] well measured velocity of ultrasonic 2 Geofluids P-wave in Boise sandstone both dry and saturated. In this paper, the squirt model [21,22] shall be used to simulate the saturated velocity and to predict the viscous squirt attenuation of Boise sandstone in [31]. The motivation is to confirm the applicability of the model [21,22] to Boise sandstone and to acquire its average parameters.…”