A triple porosity scheme for fluid/solid substitution: theory and experiment

Abstract

Quantifying the effects of pore‐filling materials on elastic properties of porous rocks is of considerable interest in geophysical practice. For rocks saturated with fluids, the Gassmann equation is proved effective in estimating the exact change in seismic velocity or rock moduli upon the changes in properties of pore infill. For solid substance or viscoelastic materials, however, the Gassmann theory is not applicable as the rigidity of the pore fill (either elastic or viscoelastic) prevents pressure communication in the pore space, which is a key assumption of the Gassmann equation. In this paper, we explored the elastic properties of a sandstone sample saturated with fluid and solid substance under different confining pressures. This sandstone sample is saturated with octadecane, which is a hydrocarbon with a melting point of 28°C, making it convenient to use in the lab in both solid and fluid forms. Ultrasonically measured velocities of the dry rock exhibit strong pressure dependency, which is largely reduced for the filling of solid octadecane. Predictions by the Gassmann theory for the elastic moduli of the sandstone saturated with liquid octadecane are consistent with ultrasonic measurements, but underestimate the elastic moduli of the sandstone saturated with solid octadecane. Our analysis shows that the difference between the elastic moduli of the dry and solid‐octadecane‐saturated sandstone is controlled by the squirt flow between stiff, compliant, and the so‐called intermediate pores (with an aspect ratio larger than that of compliant pore but much less than that of stiff pores). Therefore, we developed a triple porosity model to quantify the combined squirt flow effects of compliant and intermediate pores saturated with solid or viscoelastic infill. Full saturation of remaining stiff pores with solid or viscoelastic materials is then considered by the lower embedded bound theory. The proposed model gave a reasonable fit to the ultrasonic measurements of the elastic moduli of the sandstone saturated with liquid or solid octadecane. Comparison of the predictions by the new model to other solid substitution schemes implied that accounting for the combined effects of compliant and intermediate pores is necessary to explain the solid squirt effects.

A Seismic‐Frequency Laboratory Study of Solid Substitution in Bentheim Sandstone

“…Finally, let us note, that the results of this study are at variance with ultrasonic data on the same rock (Sun, Gurevich, Lebedev, Glubokovskikh, Mikhaltsevitch, et al, ; Sun, Gurevich, Lebedev, Glubokovskikh, Squelch, et al, ). In particular, Sun, Gurevich, Lebedev, Glubokovskikh, Mikhaltsevitch, et al () found that the Ciz‐Shapiro model underestimated both bulk and shear moduli of the rock saturated with octadecane and explained this discrepancy by the effect of stiffening of compliant pores (grain‐to‐grain contacts).…”

“…Finally, let us note, that the results of this study are at variance with ultrasonic data on the same rock (Sun, Gurevich, Lebedev, Glubokovskikh, Mikhaltsevitch, et al, 2018;Sun, Gurevich, Lebedev, Glubokovskikh, Squelch, et al, 2018). In particular, Sun, Gurevich, Lebedev, Glubokovskikh, Mikhaltsevitch, et al (2018) found that the Ciz-Shapiro model underestimated both bulk and shear moduli of the rock saturated with octadecane and explained this discrepancy by the effect of stiffening of compliant pores (grain-to-grain contacts). In contrast, the low-frequency measurements for both bulk and shear moduli in the present study are consistent with the Ciz-Shapiro model, which suggests that the effect of stiffening of the compliant pores in presence of solid is negligible.…”

“…Note that the sharp rise of the bulk moduli with pressure, observed at low confining pressures (Figure 5a), can be explained by the effect of squeezing of the compliant pores in the sample. In more detail, the effect of the complaint pores on the pressure dependence of the bulk modulus of Bentheim sandstone is discussed in Sun, Gurevich, Lebedev, Glubokovskikh, Mikhaltsevitch, et al (2018).…”

A solid/fluid substitution scheme constrained by pore-scale numerical simulations

“…However, the Gassmann theory has an important restriction: it assumes that the pressure in the pore fill is spatially uniform. Thus for rocks filled with solid substances or high-viscosity or non-Newtonian fluids, the Gassmann theory is invalid as the shear compliance of the pore fill impedes the pressure communication between different pores (Makarynska et al 2010;Saxena & Mavko 2014;Glubokovskikh et al 2016;Sun et al 2018). Ciz & Shapiro (2007, referred to as C&S model) derived an approximate extension of the Gassmann equation to a solid pore fill.…”

“…Ciz & Shapiro (2007, referred to as C&S model) derived an approximate extension of the Gassmann equation to a solid pore fill. However, recent studies (Saxena & Mavko 2014;Glubokovskikh et al 2016;Sun et al 2018) show that the C&S model gives a lower bound to the elastic moduli of porous and fractured rocks saturated with a solid or high-viscosity liquid because it also assumes that stress throughout the pore fill is spatially uniform. This assumption should be accurate if all pores are of approximately the same shape, but can be violated if the pore space contains pores with a range of shapes and compliances.…”