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Edited by: Antonio F. Miguel, University of Evora, Portugal

Reviewed by: Steffen Berg, Shell, Netherlands; Aydin Murat, Ontario Tech University, Canada

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We compute the fluid flow time correlation functions of incompressible, immiscible two-phase flow in porous media using a 2D network model. Given a properly chosen representative elementary volume, the flow rate distributions are Gaussian, and the integrals of time correlation functions of the flows are found to converge to a finite value. The integrated cross-correlations become symmetric, obeying Onsager's reciprocal relations. These findings support the proposal of a non-equilibrium thermodynamic description for two-phase flow in porous media.

Athermal fluctuations occur in a number of phenomena in nature and are important to biology, chemistry, and physics [

One area that seems not to have been analyzed in these terms is flow driven through porous media. Such flows are important for numerous geological and technical processes, for example, in oil production, CO_{2} sequestration, water transport in aquifers, or heterogeneous catalysis. An important class of porous media flows is the simultaneous flow of two immiscible fluids. In such a system, clusters of the two fluid phases, traveling through the porous medium, are constantly forced to split and recombine. Thus, the fluid configuration in the pore space changes, leading to fluctuations in the flow rate of each phase (fractional flow rate), as well as in the total flow rate. These fluctuations are of a mechanical nature and are different from but analogous to thermal fluctuations on the molecular level. The fluctuations appear on a mesoscopic scale much larger than the molecular scale of statistical thermodynamics, yet the mesoscopic scale that is defined by the pore sizes of the medium is very small compared to the overall system. In the most extreme cases, the pores can be in the nanosize regime, while the system of interest, for instance, in chalk oil reservoirs [

Our long-term aim is to find a non-equilibrium thermodynamic description for such flow systems. The challenge is then to define a suitable representative elementary volume (REV) where the essential assumption of local equilibrium, as expressed by the ergodic hypothesis, and microscopic reversibility hold. The statistical foundation of the theory was spelled out a long time ago [

Here, the aim is to move one step forward and examine the idea of time-reversal invariance or the microscopic reversibility of fluctuations [

In the present study, we use a dynamic pore network model to simulate immiscible two-phase flow in porous media. This model, introduced in the late nineties [

A crucial question that needs to be answered is whether the dynamic network model is ergodic or not. Flekkøy and Pride [

If one considers a steady state of immiscible two-phase flow, as we will in our model, we shall see that the fluctuations become Gaussian around a steady mean.

Hence, the concept of the REV at steady state is highly relevant and important for how we build a theory that can help us understand transport in porous media.

The transport of the two immiscible fluids through a two-dimensional porous medium is represented by a dynamical pore network model [_{ij} inside a link connecting nodes

Illustration of the network model. The network is occupied by two immiscible fluids (blue and gray colors). The equally long links (pores) have an hourglass-shaped structure and a random distribution of diameters.

Here, _{j} and _{i} are the pressures at nodes _{ij} is the capillary pressure, and _{ij} is the conductivity of the link. The links have an hourglass shape, and thus the capillary pressure is a function of the interface positions, _{ij}

Here, γ is the surface tension, _{ij} is the radius of link ij, and

The effect of the χ-function is to introduce zones of length β_{ij} at each end of the links where the pressure discontinuity of any interface is zero. The conductivity of the link, _{ij}, contains a geometrical factor and the effective viscosity of the link:

Here, _{ij} is the radius of the link, and the viscosity is defined as

with _{w, ij} being the saturation (i.e., the volume fraction) of the wetting phase in link _{i} are determined by solving the Kirchhoff equations. Further details of the model and solution methods can be found in Sinha et al. [_{ij} is calculated using Equation (1), and the positions of the interfaces are advanced with an appropriately small, constant time step of 10^{−5} s. A constant time step is used to facilitate a convenient calculation of the time-correlation functions. The simulations were started with a random distribution of the two liquid phases and were propagated at least 300,000 time steps to allow the system to reach a steady state. Statistics for the time correlation functions were collected for 9.7 million time steps. The length of a link in the network was set to 1 mm. We report results for each set of parameters as averages of at least 30 runs using different starting configurations of the two fluids. Volume flow rates and velocities refer to network averages. The properties of the steady-state flow are determined by the pressure drop across the system, Δ

We investigated time correlation functions and their long-time convergence for two choices of the parameter set μ_{w}, μ_{n}, and γ, which are viscosities of the wetting and the non-wetting phase and the surface tension, respectively.

Case (A) had viscosities ^{−3} – 10^{−2}. Here, the capillary number is defined as Ca = _{n}/γ, with

Case (B) had μ_{w} = 5μ_{n} (_{w} = 0.25, 0.5, and 0.75. Here, S_{w} is the volume fraction of the wetting phase of the total pore volume in the network.

We report first that the fluctuations in flow velocities are Gaussian when a suitable representative volume (REV) is chosen. We proceed to give the structure of the time correlation functions for the REV. The results for what we will call from now the Green–Kubo coefficients for the network follow from this.

In case (A), the resistance is determined by the positions of the interfaces in the links only, as the two phases have the same viscosity. In case (B), the resistance to flow in link

A typical example of fluctuations in the total volume flow

Fluctuations of the volumetric flow Q for case A (see text). The pressure gradient was 100 kPa/m. During the time span shown, 1.3 s, a volume twice the total volume of the network has passed. The inset shows the same quantity over a short time interval.

Distribution of network-averaged instantaneous total (wetting and non-wetting) fluid velocity for network size 30 x 20 (case A). _{0} is the total average velocity and sgn is the sign function.

Distributions of network-averaged instantaneous fluid velocities for two network sizes (30 × 20, 60 × 40).

Ideally, the system size of the simulation is sufficiently large and representative of the statistical ensemble. In this, the entropy and other thermodynamic properties are proportional to the system size (i.e., they are extensive [^{2} of the fluctuations is proportional to the system size, i.e., the area A. This relation is plotted in

Scaling of the variance σ^{2} of the flow velocity size distributions with system size.

With a well-defined REV, and with Gaussian fluctuations established, we can proceed to define the time correlation functions _{RS} for the fluctuating quantities

where the brackets 〈⋯ 〉 indicate ensemble averages.

The fluctuation from the mean, δ

and

^{−3} s) there is a slower, logarithmic decay which is more pronounced for larger values of γ (between 1 and 100 ms). These two regimes are followed by a slow long-timescale decay. The rapid decay appears on the timescale that corresponds to the time necessary to evolve the flow by one average link volume. As shown in

Dependence of (scaled) time correlation function 〈

The regime of the logarithmic decay is within the time of evolving the flow by the total volume of the network and is more pronounced for higher surface tensions and thus higher capillary pressures in the pores. Hence, the decay corresponds to parts of the flow that are slow-moving or frustrated. These are the regimes of interest here. They contain the relative movements of the two flows in terms of their mutual displacement.

It is interesting to note some similarities with time correlation functions of glass [

We attempted a fit of ^{−3} s and in some cases the flat top (at times < 10^{−4} s) are not well-described. Fit parameters for the different choices of γ are summarized in

Fit of function F(t) (Equation 9) to the autocorrelation function of the total volume flow for case (A) with γ = 25 mN/m (Δ

Fitting parameters for

_{1} |
_{2} |
|||||
---|---|---|---|---|---|---|

30 | 0.28 | 0.17 | 0.39 | 22 | 1.40 | 0.52 |

25 | 0.21 | 0.23 | 0.34 | 6.6 | 1.75 | 0.36 |

20 | 0.13 | 0.50 | 0.49 | 0.059 | 2.69 | 0.16 |

15 | 0.15 | 0.22 | 0.39 | 0.096 | 2.28 | 0.16 |

10 | 0.11 | 0.043 | 0.31 | 0.26 | 2.09 | 0.15 |

^{6}/s^{2}]·10^{−2} (_{1}, τ_{2})

The Green–Kubo method employs integrals of suitable time correlation functions C_{RS} (as defined in Equation 6) to compute coefficients, _{RS}:

The convergence of the integral over the time correlation function of the total flow is shown in

Convergence of the integrated time correlation function [case (A), γ = 30 mN/m and 30 × 20 links]. The colored lines represent individual trajectories; the thicker black line is the average of all trajectories.

We computed the integrals for the autocorrelation and cross-correlation functions of the wetting and non-wetting phases,

with the indexes

Integrated time correlation functions for case (A) and three different settings of the pressure drop across the network.

Λ_{ww} [cm^{6}/s]·10^{−4} |
0.46 | 0.72 | 1.59 |

Λ_{nn} [cm^{6}/s]·10^{−4} |
0.69 | 1.51 | 1.88 |

Λ_{wn} [cm^{6}/s]·10^{−4} |
−0.11 | −0.35 | −0.73 |

Λ_{nw} [cm^{6}/s]·10^{−4} |
−0.10 | −0.32 | −0.70 |

_{ij} are obtained from Equation (11). The uncertainties for Λ_{i, j} are estimated to be less than 21%

Volumetric flow rates Q and fluid velocities v for case (A) and three different settings of the pressure drop across the network.

Q [cm^{3}/s] |
0.308 | 0.869 | 1.565 |

Q_{w} [cm^{3}/s] |
0.139 | 0.401 | 0.723 |

Q_{n} [cm^{3}/s] |
0.169 | 0.467 | 0.841 |

v [m/s] | 0.037 | 0.104 | 0.188 |

v_{w} [m/s] |
0.033 | 0.096 | 0.174 |

v_{n} [m/s] |
0.041 | 0.113 | 0.203 |

v_{n}-v_{w} [m/s] |
0.007 | 0.017 | 0.028 |

Integrated time correlation functions for case (B) and three different choices for the saturation.

_{w} |
|||
---|---|---|---|

Λ_{ww} [cm^{6}/s]·10^{−4} |
0.011 | 0.006 | 0.001 |

Λ_{nn} [cm^{6}/s]·10^{−4} |
0.212 | 0.113 | 0.020 |

Λ_{wn} [cm^{6}/s]·10^{−4} |
−0.046 | −0.024 | −0.005 |

Λ_{nw} [cm^{6}/s]·10^{−4} |
−0.048 | −0.027 | −0.005 |

_{ij} are obtained from Equation (11). Uncertainties for Λ_{i, j} are estimated to be less than 21%

Volumetric flow rates Q and fluid velocities v for case (B) and three different choices for the saturation.

_{w} |
|||
---|---|---|---|

Q [m^{3}/s] |
0.763 | 0.503 | 0.359 |

Q_{w} [cm^{3}/s] |
0.137 | 0.208 | 0.248 |

Q_{n} [cm^{3}/s] |
0.626 | 0.295 | 0.110 |

v [m/s] | 0.92 | 0.61 | 0.44 |

v_{w} [m/s] |
0.066 | 0.050 | 0.040 |

v_{n} [m/s] |
0.10 | 0.072 | 0.055 |

v_{n}-v_{w} [m/s] |
0.034 | 0.022 | 0.015 |

In both cases (A) and (B), we find that the integral cross-correlations obey the Onsager reciprocal relation, Λ_{ij} = Λ_{ji}, within the statistical error in the simulations. This result is new for a meso-level description like the one used here and is encouraging for the overall aim; to create a non-equilibrium thermodynamic description for the macroscopic level. The finding applies to a well-defined REV, for which we have a Gaussian distribution of fluctuations, analogous to the corresponding distribution on the molecular level.

It is interesting that the cross coefficients are all negative. This makes sense for network flow, where one component cannot advance faster (on average) than the mean flow unless the other component advances slower (on average).

The Λ_{ij}s for case (B), where the surface tension is zero, show an extreme limit property, because the cross-correlation functions obey Λ_{ww}Λ_{nn} − Λ_{wn}Λ_{nw} ≈ 0. A singular matrix of coefficients is the essence of complete coupling of the two fluids' flows; they are linearly dependent. For all choices of saturation,

The value of ζ for case (B) can be deduced by looking at the coefficients in ^{2} ≈ 20 or ζ = 4.5 ± 0.5 for all _{w}. The value is close to the ratio of fluid viscosities, which will describe the dissipation.

All coefficients in

To investigate the origin of the Onsager symmetry in more detail, we examined the cross-correlations in _{AB}(τ) = _{BA}(τ). This equality is illustrated in

Cross-correlation functions of wetting (Q_{w}) and non-wetting (Q_{n}) flows. The upper panels

Our investigation of time correlation functions has revealed interesting parallels between the time correlation functions of two immiscible fluids in a porous media, those observed for glass and stress-yield fluids, and those for molecular fluctuations. A network with incompressible fluids has been used as a model for the porous medium, but the findings should not be restricted to this. We have been able for the first time to find Onsager symmetry in athermal fluctuations on the meso level. The symmetry of the coefficients implies time-reversal invariance or microscopic reversibility of fluctuations also on the meso level, in agreement with recent experimental [_{AB}(τ) = _{BA}(τ), holding for all timescales except very short timescales.

We found that the structure of the time correlation functions depends on the surface tension. Integrals over auto- and cross-correlation functions of an REV were found to converge, and the integrals of the cross-correlation functions essentially obeyed Onsager's symmetry. The coefficients obtained in this manner may have a relation to porous medium permeabilities. Further research on time correlation functions to compute the transport properties of immiscible-two phase flow is therefore encouraged.

One may ask whether the dynamic pore network model we have used really reflects the dynamics of real porous media or whether ergodicity has been built into it somehow, ergodicity that is not there in real porous media. Apart from doing experiments that will answer this question, one may repeat the measurements here but based on other models that differ substantially from ours, such as the Lattice Boltzmann Method.

The datasets generated for this study are available on request to the corresponding author.

MW conceptualized the study, carried out the simulation, and wrote the first draft. MG assisted with the computational work. All authors contributed to data analysis, and in developing the theory as well as shaping the manuscript to its final form.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to thank Prof. Daan Frenkel for helpful discussions of this work. We thank the reviewers for providing us with exceptionally good and challenging reviews.