Squirt flow in porous media saturated by Maxwell-type non-Newtonian fluids

Mechanical waves, which are commonly employed for the noninvasive characterization of fluid-saturated porous media, tend to induce pore-scale fluid pressure gradients. The corresponding fluid pressure relaxation process is commonly referred to as squirt flow and the associated viscous dissipation can significantly affect the waves' amplitudes and velocities. This, in turn, implies that corresponding measurements contain key information about flow-related properties of the probed medium. In many natural and applied scenarios, pore fluids are effectively non-Newtonian, for which squirt flow processes have, as of yet, not been analyzed. In this work, we present a numerical approach to model the attenuation and modulus dispersion of compressional waves due to squirt flow in porous media saturated by Maxwell-type non-Newtonian fluids. In particular, we explore the effective response of a medium comprising an elastic background with interconnected cracks saturated with a Maxwell-type non-Newtonian fluid. Our results show that wave signatures strongly depend on the Deborah number, defined as the relationship between the classic Newtonian squirt flow characteristic frequency and the intrinsic relaxation frequency of the non-Newtonian Maxwell fluid. With larger Deborah numbers, attenuation increases and its maximum is shifted towards higher frequencies. Although the effective plane-wave modulus of the probed medium generally increases with increasing Deborah numbers, it may, however, also decrease within a restricted region of the frequency spectrum.

Figure 1

Schematic illustration of (a) the probed porous medium; (b) a REV of such medium, which contains a set of interconnected orthogonal cracks; and (c) the oscillatory relaxation test employed to obtain the frequency-dependent overall stress and strain of the medium. Note that the side length $L$ of the REV is considered to be much smaller than the prevailing wavelengths $\lambda$.

Figure 2

Normalized inverse quality factor ${\overline{Q}}_p^{-1}$ as a function of the normalized frequency $\omega /{\omega }_c$ for the model shown in Fig. 1. We illustrate the results considering a Newtonian pore fluid (blue line with circles) and Maxwell-type non-Newtonian fluids for different Deborah numbers (colored solid lines). Dashed black lines denote the high-frequency asymptotic behavior of the Newtonian and non-Newtonian fluids.

Figure 3

Real part of the normalized effective bulk modulus $\text{Re}\left\{\overline{H}\right\}$ as a function of the normalized frequency $\omega /{\omega }_c$ for the model illustrated in Fig. 1. We illustrate the results considering a Newtonian pore fluid (blue line with circles) and non-Newtonian fluids with different Deborah numbers (colored solid lines).

Figure 4

Relationship between the normalized maximum frequency ${\omega }_{\max }/{\omega }_c$ and the Deborah number $\chi$. We compare the values obtained from the numerical estimations (black circles) and the analytical approximation given by Eq. (26) (red dashed line).