Thermally Induced Pressure Fluctuations in Single-Phase Fluid-Saturated Porous Media Described by the Fluctuation Dissipation Theorem

Abstract

The analogy between electrical and fluid flow resistance has been applied to describe the fluctuating pressure of thermal origin observed over single-phase fluid-saturated porous rock samples in equilibrium, i.e., in the absence of any macroscopic fluid flow. It is shown that fluctuating voltages in the electrical case, known as Johnson–Nyquist noise, and fluctuating pressures in the fluid case can be described by the fluctuation dissipation theorem (FDT). The average square of the fluctuating fluid pressure is therefore proportional to the generalized driving force, i.e., the thermal kinetic energy of the molecules (k_BT), the remaining resistance in the absence of any macroscopic flow called the thermal resistance R_T, and the bandwidth of the pressure fluctuations, Δf. The theoretical power spectral density (PSD) curves differ for the electrical and fluid cases being white in the former and red in the latter, i.e., it falls off as Constant/f² for increasing frequency. The calculated theoretical predictions are in general in good agreement with observed root mean square pressure values measured over a brine-saturated sandstone rock sample at 50 and 80 ℃ and air-saturated  chalk and sandstone  samples at 80 ℃, using reasonable values for the specific surface areas and bandwidths, although experiments indicate a stronger temperature dependency than predicted. Furthermore, the observed PSD curves are falling off as Constant/f² for frequencies above unity according to the prediction, i.e., they represent red noise. Pressure fluctuations or noise over single-phase fluid-saturated rock samples can be adequately characterized by the “fluid version” of the Johnson–Nyquist noise expression. The demonstration of red noise adds additional support to the existence and influence of the thermal resistance term previously introduced under dynamic conditions, because fluid behavior descriptions should be consistent at all levels. The results connect observations of macroscopic fluid behavior in porous media to actions occurring on the molecular level in a consistent way relying on the well-established explanation for Brownian motion and diffusion phenomena introduced by Einstein in 1905 and subsequent generalizations, i.e., the FDT. They may also serve as the basis for explaining non-thermal fluctuating pressures observed in two-phase flow through porous media under dynamic conditions. It is hypothesized that such fluctuations fundamentally are caused by interfacial capillary waves, which again are macroscopic manifestation of molecular thermal fluctuations, always present at the interface between two immiscible fluids. The consequences of the work in a more general sense are that new ways and methods for describing porous media flow in both single- and multiphase flow modes must be further developed since two rather than one resisting force mode is acting on every fluid phase present.

Appendices

Appendix A

The aim here is to derive an expression for the average fluctuating pressure squared over single-phase fluid-saturated rock samples using the FDT. The experimental setup is depicted in Fig. 2 where the instantaneous pressure difference generated by molecules-grain collisions on each side of the rock sample is measured with an accurate gauge. The average pressure squared, $\left\langle {p_{T}^{2} } \right\rangle$, follows directly from the FDT as for the electrical case. If measured over a period long compared to the extremely short duration for a collision between the fluid molecules and the solid matrix to occur (called ${\Delta}\text{t}$), it is given as (Callen and Welton 1951),

$$\left\langle {p_{T}^{2} } \right\rangle = \frac{2}{\pi }\mathop \int \limits_{0}^{\omega } E\left( {\omega ,T} \right)R_{F} {\text{d}}\omega ,$$

(7)

where the angular frequency, $\omega$, is given as, $\omega = 2\pi f$, where f is molecular-solid grain collision frequency. $E\left( {\omega ,T} \right)$ expresses the average energy of a harmonic oscillator of natural angular frequency, $\omega$. Furthermore, $R_{F}$ is the fluid flow resistance valid only when viscous resistance is acting, i.e., under dynamic conditions, described by Darcy’s law (Bear 1972; Darcy 1856),

$$p_{{{\text{Darcy}}}} = \frac{\mu L}{{AK}}q = R_{F} q = \frac{\mu L\phi }{K}u.$$

(8)

Here $p_{{{\text{Darcy}}}}$ is the pressure required to force the fluid through the rock sample, q is the measurable volume flow rate, u is the interstitial fluid pore flux, $u = q/\left( {A\phi } \right)$, $\mu$ is the fluid viscosity, L is the length of the rock sample, and K is the absolute permeability. All terms in Eq. (8) are, however, solely related to deterministic macroscopic quantities of which none exhibit any fluctuations, and the only flow resistance as specified on the right-hand side is obviously due to viscous resistance. Hence, Eq. (8) must therefore be generalized to also account for the effect of the thermal kinetic energy of the molecules, i.e., the random fluid flux, before applied in Eq. (7). Such generalizations are commonly performed by writing the macroscopic fluid pore flux, u in Eq. (8), as the sum of a deterministic term representing the common fluid movement plus a random fluctuating term representing the random flux movement of individual fluid molecules (Chapman and Cowling 1953, Hauge 1970; Schekochihin 2020). Hence,

$$u = v + \left\langle w \right\rangle ,$$

(9)

where v is the deterministic fluid flux equal to the conventional flux, u, used in Darcy’s law since the latter also is deterministic without any fluctuations and w is the random flux term. The random flux is always present under both static and dynamic conditions and has the ensemble average properties, $\left\langle w \right\rangle = 0$ and $\left\langle w \right\rangle ^{2} \ne 0$. Furthermore, the external pressure required to force the fluid through the medium called, $p_{{{\text{Pump}}}}$, is conventionally balanced by the deterministic viscous pressure term, $p_{{{\text{Vis}}}}$ only, as in Darcy’s law. But as previously discussed in Introduction, the effect of the random thermal fluctuations, i.e., the thermal resistance force, must also be accounted for under dynamic conditions. The pump pressure can therefore in a similar way to Eq. (9) be written as the sum of a deterministic viscous and a random fluctuating pressure term,

$$p_{{{\text{Pump}}}} = p_{{{\text{Vis}}}} + \left\langle {p_{T} } \right\rangle .$$

(10)

${p_{T} }$ represents the random fluctuating pressure which can be measured under static conditions using a sensitive gauge as demonstrated herein and becomes the thermal resistance mode under dynamic conditions. Its average square represents the smallest pressure fluctuations always present in the system as it is caused by the ever-present random thermal kinetic energy of the fluid molecules meaning that it has the ensemble average properties, $\left\langle {p_{T} } \right\rangle = 0$, and $\left\langle {p_{T}^{2} } \right\rangle > 0$. Darcy’s law in Eq. (8) can hence be generalized by replacing, u, on the right-hand side with, $v + \left\langle w \right\rangle$ as in Eq. (9) and $p_{{{\text{Darcy}}}}$, by, $p_{{{\text{Pump}}}}$ from Eq. (10). Squaring both sides and taking the square root gives,

$$p_{{{\text{Pump}}}} = \sqrt {p_{{{\text{Vis}}}}^{2} + \left\langle {p_{T}^{2} } \right\rangle } = \frac{{\mu L\phi }}{K}\sqrt {v^{2} + \left\langle {w^{2} } \right\rangle } .$$

(11)

The cross terms disappear due to the properties, $2vw = 2p_{{{\text{Vis}}}} p_{T} = 0$. The fluid flow resistance, which is the pre-factor in the very right-hand side expression, includes the absolute permeability K, which must be generalized to account for both viscous and thermal resistances (Standnes 2018). The relative influence of the two resistance modes is given as (Standnes 2021),

$$K = K^{*} \frac{\mu \left( T \right)}{{W\mu \left( T \right) + \left( {1 - W} \right)\Delta P_{T} \sqrt T }}$$

(12)

where $K^{*}$ is a reference permeability, which is fixed once it has been measured at a given temperature. The generalized permeability expression in Eq. (12) was constructed so that the permeability term in Darcy’s law can be replaced directly with its right-hand side (Standnes 2018). W is a weight factor between the two resistance modes being zero if only thermal resistance is considered, and $\Delta P_{T}$ is a thermal dissipation efficiency factor. The weight factor was introduced as a fit parameter (Standnes 2018), since neither the viscous nor the thermal resistance force currently can be calculated analytically for flow through porous media due to the complexity of the pore structure. The complexity makes it, for example, currently impossible to calculate the viscous resistance when single-phase fluids are flowing through porous media using the Navier–Stokes equation due to the irregularity of the boundary conditions. Inserting Eq. (12) for the absolute permeability into Eq. (11) gives,

$$p_{{{\text{Pump}}}} = \sqrt {p_{{{\text{Vis}}}}^{2} + \left\langle {p_{T}^{2} } \right\rangle } = \left[ {\frac{{L\phi }}{{K^{*} }}W\mu + \frac{{L\phi }}{{K^{*} }}\left( {1 - W} \right)\Delta P_{T} \sqrt T } \right]\sqrt {v^{2} + \left\langle {w^{2} } \right\rangle }$$

(13)

The left side of Eq. (13) states that the pump pressure required to force the fluid through the rock sample is balanced by the sum of the viscous and the thermal resistance pressures. The right-hand side contains the corresponding resistance expressions, viscous and thermal, in the first parenthesis, multiplied with the square root of the sum of the macroscopic and random fluid flux terms squared, respectively. Only the macroscopic flux term contributes under dynamic conditions and only the random term under static conditions. But the effect of the latter is always present in terms of either a flow resistance (thermal resistance) or a fluctuation as it is fundamentally generated by the ever-present thermal kinetic energy of the fluid molecules. Hence, the intertwined relationship between these two obeys the FDT (Kubo 1966). Both resistance terms increase proportional to the macroscopic flux in magnitude under dynamic conditions (Standnes 2021).

To formulate Eq. (13) in a way which capture both the static and the dynamic cases, and at the same time is similar to Ohm’s law so it obeys linear response theory, volume flow rates, q and $q_{T}$, are introduced such that, $v = q/A\phi$ and $w = q_{T} /A\phi$. Hence, Eq. (13) can in the general dynamic case be written,

$$p_{{{\text{Pump}}}} = \sqrt {p_{{{\text{Vis}}}}^{2} + \left\langle {p_{T}^{2} } \right\rangle } = \left[ {\frac{L}{{AK^{*} }}W\mu + \frac{L}{{AK^{*} }}\left( {1 - W} \right)\Delta P_{T} \sqrt T } \right]\sqrt {q^{2} + \left\langle {q_{T}^{2} } \right\rangle } .$$

(14)

Equation (14) is equal to the generalized form of Darcy’s law derived from the Langevin equation presented in Standnes (2022). In the static case, however, when all the terms, W, v, and $p_{{{\text{Vis}}}}^{2}$, are equal to zero, Eq. (14) becomes,

$$p_{{{\text{Pump}}}} = \sqrt {\left\langle {p_{T}^{2} } \right\rangle } = \frac{L}{{AK^{*} }}\Delta P_{T} \sqrt T \sqrt {\left\langle {p_{T}^{2} } \right\rangle } .$$

(15)

Hence, although the pump pressure formally is equal zero, it is still fluctuating due to the random fluid volume rate, $q_{T}$. To further develop Eq. (15) to obtain a “microscopic” expression for the flow resistance in terms of measurable quantities to be used in the FDT expression, the thermal efficiency factor, $\Delta P_{T}$, is given as (Standnes 2021),

$$\Delta P_{T} = \frac{4}{3\pi }\frac{{K^{*} }}{\phi }\left[ {\frac{{SV_{B} }}{{2\hat{A}}}} \right]\frac{{\rho \left( {1 + \varepsilon } \right)}}{L}\sqrt {\frac{{8k_{B} }}{m}} .$$

(16)

The inner surface area of the rock sample, SV_B, is divided by a factor two because the total surface area in the rock sample can always be divided into two parts, the up- and downstream part (Standnes 2021). These two areas are equal and can be approximated with the total inner surface area divided by two when neglecting the very small area which separate the up- and downstream parts.

The relevant resistance term to consider here in the static case is then equal to the pre-factor in front of the $\sqrt {\left\langle {q_{T}^{2} } \right\rangle }$ term in Eq. (15). Using Eq. (16) in the pre-factor in Eq. (14), the expression for R_F in Eq. (7) becomes,

$$R_{F} \left( {W = 0} \right) = R_{T} = \frac{8\sqrt 2 }{{3\pi }}\frac{{\rho \left( {1 + \varepsilon } \right)}}{A\phi }\left[ {\frac{{SV_{B} }}{{2\hat{A}}}} \right]\sqrt {\frac{{k_{B} T}}{m}} .$$

(17)

As mentioned,

$$E\left( {\omega ,T} \right) = \frac{1}{2}\hbar \omega + \frac{\hbar \omega }{{e^{{\frac{\hbar \omega }{{k_{B} T}}}} - 1}},$$

(18)

in Eq. (7) expresses the average energy of a harmonic oscillator where $\hbar$ is Planck’s constant divided by 2π. The condition, $k_{B} T \gg \hbar \omega$, is fulfilled at ambient conditions (Kittel and Kroemer 1980), so $E\left( {\omega ,T} \right)$ obtains the equipartition value, $\sim k_{{\text{B}}} T$, upon neglecting the zero-point energy term, $\frac{1}{2}\hbar \omega$ (Callen and Welton 1951). The final “molecular” expression for the average fluctuating pressure squared, $p_{T}^{2}$, in Eq. (7), therefore reads by performing the integral recalling that $\omega = 2\pi f$,

$$\left\langle {p_{T}^{2} } \right\rangle = 4k_{{\text{B}}} T \cdot R_{T} \cdot \Delta f = 4k_{{\text{B}}} T \cdot \frac{{8\sqrt 2 }}{{3\pi }}\frac{{\rho \left( {1 + \varepsilon } \right)}}{{A\phi }}\left[ {\frac{{SV_{B} }}{{2\hat{A}}}} \right]\sqrt {\frac{{k_{{\text{B}}} T}}{m}} \cdot \Delta f$$

(19)

Equation (19) is the fluid flow analog to the electrical Johnson–Nyquist expression in Eq. (1). It should be noted that the cross-sectional area term, A, apparently makes Eq. (19) depending on system size. It is only apparent as the “resistance term” is defined as the pre-factor to the volume rate in Eq. (10). The volume rates were introduced to make the static and the dynamic cases consistent and comparable and aligned with linear response theory. Hence, if the more physically correct fluid flux term, w, had been used, the area term would have disappeared, and the final expression would only contain intensive quantities independent of system size.

To find an analytical expression for the PSD curve using the Wiener-Khinchin theorem, an expression for the autocorrelation function is required (Hauge 1970). It should therefore be noticed that eq. (A13) formally is the value of a white noise autocorrelation function for the time-varying pressure signal, p_T(t), where the delta function has been replaced by a function with a small but finite width, $\Delta$t (Grassia 2001, Hauge 1970). A common way to do this is to replace it with a so-called sinc function (Woodward and Davies 1952), i.e., $\text{sin}\frac{[\pi(t = t^{{\prime }})]}{[\pi(t = t^{{\prime }})]}$. Eq. (A13) is therefore derived from an autocorrelation function, which produces a finite frequency range, so infinite energies are avoided. It has the form

$$\left\langle {p_{\text{T}}(t)p_{\text{T}}(t^{\prime}) } \right\rangle \approx 4k_{{\text{B}}} {\text{T}} \cdot \frac{8\sqrt 2 }{{3\pi }}\frac{{\rho \left( {1 + \varepsilon } \right)}}{A\upphi }\left[ {\frac{{SV_{B} }}{{2\hat{A}}}} \right]\sqrt {\frac{{k_{{\text{B}}} {\text{T}}}}{m}} \cdot \text{sinc} \left[\frac{2(t-t^{\prime})}{\Delta t}\right] .$$

(20)

Equation (20) will therefore be used when deriving the PSD expression for agitation of fluid molecules in the medium in Appendix B.

Appendix B

The PSD curve is derived from the white noise pressure signal autocorrelation function in eq. (A14) using the Wiener-Khinchin theorem, i.e., by taking its Fourier transform (Hauge 1970). In the frequency domain, an integration is equivalent to division by a factor, $2{\pi}\text{if}$ Since the PSD of the red noise signal, Sp_Rp_R(f) is the integral of the white noise PSD (Kuo 1996, Milotti 2002), we have the relationship, $\text{{Sp}}_\text{{R}}\text{{p}}_\text{{R}}\text{(f)}=|\text{{Sp}}_\text{{T}}\text{{p}}_\text{{T}}\text{(f)}/4{\pi}^2\text{f}^2|$ , from eq. (3a). SP_TP_T is here equal eq. (A14) without the $\text{sinc} \left[\frac{2(t-t^{\prime})}{\Delta t}\right]$ function since its Fourier transform is a constant with width determined by the width of the time signal ${\Delta}\text{t}$. The PSD for the measured pressure fluctuations, Sp_Rp_R(f), is then given as,

$$S_{xx} \left( f \right) = \left| {\hat{x}\left( f \right)} \right|^{2} ,$$

(21)

$$S_{{p_{T} p_{T} }} = \left[ {k_{{\text{B}}} T} \right] \cdot \left[ {\frac{32\sqrt 2 }{{3\pi }}\frac{{\rho \left( {1 + \varepsilon } \right)}}{A\phi } \cdot \left( {\frac{{SV_{B} }}{{2\hat{A}}}} \right) \cdot \sqrt {\frac{{k_{{\text{B}}} T}}{m}} } \right],$$

(22)

$$S_{{p_{R} p_{R} }} \left( f \right) = \left| {\frac{{S_{{p_{T} p_{T} }} }}{{\left[ {i2\pi f} \right]^{2} }}} \right| = k_{{\text{B}}} T \cdot \left[ {\frac{8\sqrt 2 }{{3\pi^{3} }}\frac{{\rho \left( {1 + \varepsilon } \right)}}{A\phi } \cdot \left( {\frac{{SV_{B} }}{{2\hat{A}}}} \right) \cdot \sqrt {\frac{{k_{{\text{B}}} T}}{m}} } \right] \cdot \frac{1}{{f^{2} }},$$

(23)

which falls off as, ~ Constant/f², and therefore behaves like red noise.