Threshold Pressure in Non-Darcian Flow Derived from the Langevin Equation and Fluctuation Dissipation Theorem: Generalized Darcy’s Law

Abstract

This work addresses the explanation for the threshold pressure sometimes observed in porous media flow leading to so-called non-Darcian flow where a certain external pressure is required to initiate macroscopic flow. Using the Langevin equation, it is first shown that the rapid fluctuating random pressure spikes on each side of the grains in porous media caused by molecular collisions between fluid molecules and the mineral surfaces generate a root mean square (RMS) pressure value even in the absence of macroscopic flow. The RMS pressure can be interpreted as a zero-rate resistance pressure term, which will act as a threshold resistance pressure over which the externally supplied pressure must exceed to initiate macroscopic flow. Using the fluctuation dissipation theorem, the RMS threshold pressure value is quantified, and its magnitude is mainly governed by the specific surface area of the medium and the thermal kinetic energy of the fluid molecules, i.e., the system temperature. Measurable threshold pressures are therefore expected to occur in media with high specific surface areas which is in line with empirical observations. Plotting Darcy velocity vs. pressure gradient for different threshold gradients shows good agreement with measured data. The presence of the zero-rate pressure resistance term further supports the existence of a thermal resistance term previously introduced under dynamic flow conditions. In the latter case, the re-active viscous resistance term must also be included to obtain the total flow resistance. A generalized form of Darcy’s law is therefore proposed, which accounts for the small pressure fluctuations observed in the absence of macroscopic flow, and therefore contains two resistance terms under dynamic conditions, i.e., viscous- and thermal resistances. Another implication of the nature and presence two resistance terms is also discussed with respect to flow resistance during aqueous vs. gaseous flow, essentially the Klinkenberg effect. This work connects porous media flow description, more specifically the mechanism underlying the thermal resistance mode, to well-known phenomena which require considerations about the molecular nature of fluids, e.g., the fluid-particle phenomenon called Brownian motion. It is shown that the fluid-porous medium system in the static case essentially is Brownian motion in the limit of large size porous media. The effect of the molecular collisions is, however, averaged out in this case and therefore insufficient to generate “Brownian motion” but can instead be measured and used to characterize the system. Such an interpretation is supported experimentally by the ever-present small pressure fluctuations of thermal origin, which show red or Brownian noise behavior.

Appendix

The average of the fluctuating random pressure squared is according to the FDT generated as the product of the resistance in the absence of macroscopic flow and the average energy of a harmonic oscillator integrated over all angular frequencies $\omega$ for the fluctuating fluid molecules (Callen and Welton 1951),

$$\langle \Delta {p}_{\mathrm{R}}^{2}\rangle =\frac{2}{\pi }{\int }_{0}^{\omega }E(\omega ,T){R}_{F}d\omega ,$$

(17)

where $E\left(\omega ,T\right)=\frac{1}{2}\mathrm{\hslash }\omega +\frac{\mathrm{\hslash }\omega }{{e}^{\frac{\mathrm{\hslash }\omega }{{k}_{B}T}-1}}$ is the average energy of the oscillator and $\hslash$ is Planck’s constant divided by $2\uppi$. The term takes the equipartition value, ${k}_{B}T$, at high temperatures so $E\left(\omega ,T\right) \sim {k}_{B}T$ at ambient conditions upon neglecting the zero-point energy term, $\frac{1}{2}\mathrm{\hslash }\omega$. Equation (17) can then be integrated directly if the flow resistance ${R}_{F}$ is constant. Using the substitution, $d\omega =2\pi df,$ therefore gives,

$$\langle {\Delta p}_{R}^{2}\rangle \approx 4{k}_{B}T\bullet {R}_{F}\Delta f,$$

(18)

where $\Delta f$ is the bandwidth of the molecular oscillations. The flow resistance, ${R}_{F}$, is defined by writing Darcy’s law on the same form as Ohm’s law for electrical resistances (Standnes et al. 2022). Hence, it is given as the proportional coefficient in the equation,

$$\Delta p={R}_{F}q=\frac{\mathrm{\mu L}}{\mathrm{AK}}q,$$

(19)

where $q$ is the measurable volume flow rate. ${R}_{F}$ must, however, be generalized compared to the conventional Darcy’s law as previously mentioned (Eq. (3)) because of the presence of the thermal resistance mode. Using the expression for the permeability in Eq. (3) with a weight factor $W=0$ since no macroscopic flow is assumed together with the thermal efficiency factor expression, $\Delta {P}_{T}=\frac{4}{3\pi }\frac{{K}^{*}}{\upphi }\left[\frac{{\mathrm{SV}}_{B}}{2\widehat{\mathrm{A}}}\right]\frac{\rho (1+\varepsilon )}{L}\sqrt{\frac{8{k}_{B}}{m}}$ (Standnes 2021), gives,

$${R}_{F}\left(W=0\right)={R}_{T}=\frac{\mathrm{\mu L}}{\mathrm{AK}}=\frac{8\sqrt{2}}{3\pi }\frac{\rho \left(1+\varepsilon \right)}{\mathrm{A\phi }}\left[\frac{{\mathrm{SV}}_{B}}{2\widehat{\mathrm{A}}}\right]\sqrt{\frac{{k}_{B}T}{m}},$$

(20)

where ${R}_{T}$ is the resistance in the absence of viscous resistance because $W=0$ under static conditions. Inserting Eq. (20) into Eq. (18) hence gives for the average fluctuating pressure squared according to the FDT,

$$\langle {\Delta p}_{R}^{2}\rangle =\frac{32\sqrt{2}}{3\pi }{k}_{B}T\frac{\rho (1+\varepsilon )}{\mathrm{A\phi }}\left[\frac{{\mathrm{SV}}_{B}}{2\widehat{\mathrm{A}}}\right]\sqrt{\frac{{k}_{B}T}{m}}\Delta f$$

(21)

The threshold value expression is then given as, ${\Delta p}_{\mathrm{THR}}=\sqrt{\langle \Delta {p}_{R}^{2}\rangle }$, which can be compared to the instantaneous measured fluctuations converted to an RMS value given by Eq. (14).