Permeability measurements using oscillatory flows

Abstract

We describe a versatile apparatus for measuring the permeability of porous materials using oscillatory flows. The permeabilities are measured by an original spectral analysis of the pressure and fluid-displacement signals. The measurements are shown to be in very good agreement with classical drainage experiments performed on the same device. Our apparatus and methodology will be useful if small fluid displacements are required, for example in reactive porous media.

Graphic abstract

Appendices

Appendix A: Drainage experiments

In this appendix, we describe complementary drainage experiments, performed using essentially the same setup as was used for the oscillatory experiments. These experiments were used to compare the results with a classical technique for which the flow is unidirectional and driven by gravity. Indeed, the versatility of the setup allows us to compare both measurements without changing the configuration of glass beads. The drainage experiments were performed after each oscillating run to compare both measurements accurately.

Experiments A reservoir is placed near the top of the flow cell and communicates with it through a siphon. This allow us to increase the volume of water flowing through the porous medium significantly and to perform reliable experiments. The diameter of the upper cylinder (cyl. B in Fig. 1) is actually smaller than that of the other cylinder to leave enough room for the siphon. The piston is fixed during the experiments and the only relevant recording with the acquisition card is the pressure signal. The drainage starts when the draining tap at the bottom of the flow cell is open (see Fig. 1). The flow is then unidirectional and the flux is measured with a weighing scale (Ohaus) placed at the outlet of the draining tap. The scales are connected to a computer which records the measurement with an acquisition frequency of 2.9 Hz. The signals are then analyzed using the procedure described in the next paragraph.

Processing The pressure balance and the analysis principle remain the same as the ones described in Sect. 3.1. However, since the flows are not periodic for these experiments, we do not compute $\varPi$ by subtracting the mean value of $\varDelta P(t)$, but by subtracting the hydrostatic pressure difference recorded when there is no flow. Contrary to the oscillating measurements, the sampling rates of the signal V(t) (measured with the scales) is much smaller than the sampling rate of the acquisition of $\varPi (t)$. We interpolate V(t) with a cubic spline on the time intervals of $\varPi (t)$. Since differentiating V(t) would increase the uncertainties of the measurements, we rather integrate $\varPi (t)$ numerically to fit the signal of V(t). Indeed, integrating Eq. 10 leads to $\int _0^t{\varPi (t')dt'}=R_H. V(t)$ which allows us to measure $R_H$ with a single parameter fit.

Fig. 9

Measurements of the hydraulic resistance from drainage experiments performed with 1 mm diameter glass beads (E). Top: fit of the integrated signal $(1/R_H)\int _0^t \varPi$ (magenta line) to the measurements of V(t) (blue dots) with $R_H$ as a fitting parameter. We find $R_H=2.3.10^8 \mathrm{Pa.s.m}^{-3}$. Bottom: validation of the fit by comparing the values of Q(t) computed with numerical differentiation of V(t) (blue dots) with the scaled signal of $\varPi (t)/R_H$ (magenta line)

The result of the processing is shown in Fig. 9. The fit is plotted in the top graph and we check afterwards the good match of the values of $\varPi (t)/R_H$ and Q(t) by differentiating the raw data of V(t) (bottom graph). Note that the fit is performed on the points recorded between 1.25 s after the opening and 1.25 s before the locking of the draining tap. In the example showed in Fig. 9, the standard error associated with $R_H$ was calculated using Matlab was five orders of magnitude below its value. The main source of uncertainty associated with this measurement actually comes from the data recorded by the scales, which leads to a relative precision of the order of $3\%$ for these measurements.

Appendix B: Analysis of the signal harmonics

The verification of the linearity between $\varPi (t)$ and Q(t) presented in Sect. 4.2 could also be done in the frequency domain. Since both signals are periodic with a period $1/f_0=1$ s, they can be easily decomposed in harmonics of frequencies $i.f_0$ and whose amplitudes are denoted $\pi _i$ and $q_i$ respectively by FFT. Equation 10 implies the relationship

$$\begin{aligned} \pi _i = R_H q_i. \end{aligned}$$

(18)

The spectral analysis shown in Sect. 3.2.2 enables the identification of the harmonics of the pressure and flux signals and therefore a check on relationship 18. However, since the signals are subject to some noise of a peak to peak amplitude of 9 mV, we only considered here the harmonics whose amplitudes were greater or equal to 10 mV, and whose frequencies were smaller than 25 Hz as the harmonics are better defined in this frequency range (see Fig. 5). The results shown in Fig. 10 gather all the data collected per porous medium. The results indicate that the linearity between the harmonics $\pi _i$ and $q_i$ is indeed verified as a consequence of Darcy’s law in the Fourier space.

Fig. 10

Comparison of the amplitudes of the harmonics of frequencies $i.f_0$ of the pressure ($\pi _i$) and flow rate ($q_i$) signals. All the harmonics gather on the same straight line for each set of packed beads as a consequence of Darcy’s law in the frequency space. The legend indicates the values of the linear regression coefficient computed for each porous medium