Implications of Molecular Thermal Fluctuations on Fluid Flow in

Jul 19, 2018 - The calculation assumes that the solid matrix in the porous media can be ... of the energy required to force fluids through porous medi...
0 downloads 0 Views 1MB Size
Download PDF
Subscriber access provided by UNIV OF NEW ENGLAND ARMIDALE

Fossil Fuels

Implications of Molecular Thermal Fluctuations on Fluid Flow in Porous Media and Its Relevance to Absolute Permeability Dag Chun Standnes Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b00478 • Publication Date (Web): 19 Jul 2018 Downloaded from http://pubs.acs.org on July 24, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 30

Energy & Fuels

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Implications of Molecular Thermal Fluctuations on Fluid Flow in Porous Media and Its Relevance to Absolute Permeability Dag Chun Standnes Equinor ASA, 5020 Bergen, Norway Institute for Energy Resources, University of Stavanger, NO-4068 Stavanger, Norway E-mail: [email protected], [email protected] ABSTRACT This paper is addressing the challenge related to the understanding of varying absolute permeability when forcing fluid (mostly water is considered) through a porous medium for varying system temperatures. It is assumed that the total energy dissipation splits into two contributions, a viscous and a thermal part. These two energy dissipation channels will exactly balance the supplied external energy. The thermal part has not been considered previously regarding forced water flow through porous media and it is substantiated theoretically by comparing it with the well-known Ludwig-Soret effect. It is based on a hypothesis that the flowing water is dissipating molecular kinetic energy to heat (thermal dissipation) due to an asymmetric spatial velocity distribution upand downstream the solid matrix. Based on theoretical calculations by Molina [Molina, M. Body Motion In a Resistive Medium at Temperature T. REVISTA MEXICANA DE FÍSICA. 2002, 48, 2, 132 – 134], an expression can be derived showing that thermal dissipation will increase proportional with the total inner surface area of the porous medium and the square root of absolute temperature. The calculation assumes that the solid matrix in the porous media can be considered as a large collection of large macroscopic “Brownian particles” being stagnant relative to the flowing water phase. The measured total energy dissipated in the porous medium equal to the supplied pressure, ∆P   , can be used to estimate two fit parameters, W and ∆P (weight factor with respect to viscous/thermal dissipation and a thermal dissipation efficiency factor), using a generalized version of Darcy’s law also accounting for changes in temperature reading ∆P_   =     μ T  ∗ + 1 − W  ∗! √T. Absolute permeability vs. temperature can then be calculated as )

K $%&'(&%  = K ∗ )*

! √+,

where K ∗ is a reference permeability. The latter expression was used

for two sets of experimental data reported in the literature by Aruna [Aruna, M. The effects of temperature and pressure on absolute permeability of sandstones. Ph.D. thesis Stanford University. 1976] and Weinbrandt at al. [Weinbrandt, R. M.; Ramey, H. R. Jr.; Cassé, F. J. The effect of temperature on Relative and Absolute Permeability of Sandstones. SPE Journal. 1975, 15, 5, 376 – 384]. The calculated results show excellent fit to the measured absolute permeability values corroborating the hypothesis proposed. The fractional magnitude of the total viscous and total thermal dissipation terms varied typically from 0.8 to 0.3 and from 0.2 to 0.7, respectively, in the temperature range 20 – 160 °C. Hence, total thermal dissipation constitutes a significant part of the energy required to force fluids through porous media even at ambient temperature which may have important consequences for processes where temperature is varying significantly in space and time. The reported data for absolute permeability vs. temperature in the literature shows large scatter and many hypotheses have been proposed to explain the data. Hence, further systematic empirical work investigating this challenge is very much encouraged using e.g. fluids with different molecular weights and polarity and porous media with different mineralogy in addition to performing molecular dynamics simulations. The current paper offers a theoretical explanation for the phenomenon observed related to reduction in absolute permeability for increasing system temperature which hopefully can contribute to improved test design and interpretation of experimental data.

ACS Paragon Plus Environment

Energy & Fuels

Page 2 of 30

2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

NOMENCLATURE A – Cross-sectional area of rock sample (m2) Au – Unit area (= 1 m2) a – Radius of Brownian particle (m) CP – Confining pressure (PSI) D – Diffusion coefficient (m2/s) ./ – External force (N) F

F – Force due to Molina35 (N) g(u) – Maxwellian velocity probability density distribution kB – Boltmann’s constant (= 1.38·10-23 J/K) K – Absolute permeability (m2) K* – Absolute permeability for reference state (m2) K $%&'(&%  – Absolute permeability calculated based on generalized Darcy’s law (m2) L – Length of rock sample (m) m – Mass of water (fluid) molecule (kg) mB – Mass of Brownian particle (kg) N – Number of unit areas N =

 %& 1  2(3%' %% 456 7  8 1 2(3%' %% 56 

q – Volume flow rate (m3/s) ∆P – Differential pressure over rock sample (Pa)

∆P – Thermal dissipation efficiency factor (= ∆P =

9+*: ∗ => CD F < AB E ) ; G  ?@

(kg/msK1/2))

∆PF – Pressure loss due to thermal dissipation (Pa)

∆PF 1 − W – Total pressure loss due to thermal dissipation (Pa)

HF – Pressure loss due to thermal dissipation per unit area (= 1 m2) surface perpendicular to ∆P the direction of fluid movement (Pa) ∆P>12 – Pressure loss due to viscous dissipation ∆P ) (Pa)

∆P>12 W – Total pressure loss due to viscous dissipation (Pa)

∆P – Theoretical pressure loss accounting for variations in fluid viscosity only (no permeability reduction) (Pa)

ACS Paragon Plus Environment

Page 3 of 30

Energy & Fuels

3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

∆P_   – Total pressure loss according to generalized Darcy’s law splitting the total     measured pressure loss into viscous- and thermal dissipation (= μ T  ∗ + 1 − W  ∗! √T ) (Pa) R – Net resisting force (N) S – Specific surface area, internal surface area per bulk rock volume (m2/cm3) Sp – Specific surface area, internal surface area per pore volume of rock (m2/cm3) .S/t – Stochastic acceleration term describing the driving force for the Brownian particle caused by a

random force caused by thermal velocity fluctuations per mass (m/s2) T – Absolute temperature (K) t – Time (s) t – Temperature (°C)

.u//u – Pore scale macroscopic velocity vector/Pore scale macroscopic velocity magnitude (m/s)

V – Bulk volume of rock sample (m3) v – Darcy flux (m/s) .v/ F – Instantaneous microscopic water molecule velocity vector (m/s) .v/ M – Instantaneous microscopic velocity in x-direction (m/s)

VT – Average microscopic water molecule velocity at temperature T (thermal velocity = B

DE  ) (m/s) F

〈… 〉 – Time average Greek Letters ϵ – Coefficient of restitution. Parameter specifying the degree of momentum transfer in each collision ( – ) ϕ – Porosity ( – ) ϕv  – Biased Maxwellian velocity probability density for velocity in x-direction ( – )

ϕvT  – Maxwellian velocity probability density for velocity in y-direction ( – ) γ – Frictional/damping force in the Langevin equation (s-1) ρ – Fluid number density in 1 dimension (m-1)

μ – Fluid viscosity (Pas)

μW – Water viscosity (Pas)

1. INTRODUCTION

ACS Paragon Plus Environment

Energy & Fuels

Page 4 of 30

4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Absolute permeability is the ability for a porous medium to transmit fluids under influence of a given external pressure gradient1,2. It is of outermost importance in all areas in nature and technical applications where fluid transport processes through porous materials are occurring e.g. ground water hydrogeology, soil science, CO2 storage, reservoir engineering, soil mechanics and chemical engineering2. The absolute or intrinsic permeability K of a given porous medium is normally quantified using the law introduced in 1856 referred to as Darcy’s law3 later modified to its present form by Muskat4 by including fluid viscosity reading: K=

) +  

(1)

In eq. 1 q is volume flow rate, ∆P the external pressure applied, A the cross-sectional area for flow, L the length of the porous medium and μ the viscosity of the injected fluid. It will be assumed herein that the fluid flowing through the porous medium is liquid water if otherwise not specified. The word fluid will, however, be used when naturally referring to general cases like other liquids or gases. Darcy’s law gives the volume rate of water through a porous rock sample when imposing an external pressure loss over the sample. The coefficient quantifying the relationship is the absolute permeability of the medium. Measuring all quantities on the right-hand side in eq. 1 allows for calculation of this quantity. Darcy’s law is an empirical law formulated in a similar way as many other natural laws expressing a linear relationship between an imposed external force and the response of the system. Other examples are Ohm’s law stating the relationship between imposed voltage and the responding current through electrical resistors and Fourier’s law for heat conduction driven by differences in temperature. Some common features relating these laws will be mentioned later. The absolute permeability is normally attributed as an intrinsic property of the porous medium at hand following the statement by Muskat4 from 1937 saying that the absolute permeability, "is thus a constant determined only by the structure of the medium in question and is entirely independent of the nature of the fluid”. Theoretical arguments challenging this statement related to energy dissipation for fluids flowing though porous media is the topic of the current paper. Klinkenberg5 reported actually as early as 1941 that the absolute permeability of a medium is dependent on the fluid used for the measurement. He was particularly addressing the differences due to liquid and gas transport2. It should be mentioned that Muskat acknowledged the contribution by Klinkenberg and admitted that the influence of slip on gas flow had been overlooked for at least 10 years in the petroleum industry5 (Discussion section). Further measurements of absolute permeabilities in the 1970s in particularly using different liquids at different temperatures revealed significant variations in absolute permeabilities (see6-13 and references therein). Absolute permeability decreased typically with increasing temperature in the range 20 – 200 °C by 50 – 80 %. The motivation for performing such permeability measurements was driven by researchers investigating the use of hot water for power generation and the impact of elevated temperature on the mechanical properties of natural porous rocks. Since then, several studies have confirmed that absolute permeability in many cases is decreasing with increasing temperature8,12,14-16 but contradictory results showing minor response have also been published8,12. Hence, it is therefore still no common consensus about the impact heating actually has on the absolute permeability of porous rock samples12. To the knowledge of the author, no universal explanation has been put forward connecting all these slightly scattered and somewhat contradictory observations. The subject area is extremely complex as variations in absolute permeability vs. temperature also is affected by the stress state of the porous material under investigation9, mobilization of particles or if rock-fluid interactions are taking place in addition to several other effects discussed later. A comprehensive review summarizing contribution from different works performed in the area has relatively recently been published by Rosenbrand et al.8 The article is referring to the most important work performed and different hypotheses put forward attempting to corroborate the observed decrease in absolute permeability vs. increasing

ACS Paragon Plus Environment

Page 5 of 30

Energy & Fuels

5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

temperature. The most important mechanisms mentioned for observing reduced permeability for increasing temperature were (a) thermal expansion (b) increased compressibility (c) mineral dissolution/precipitation (d) changes of the electrical double layer indirectly causing changes in effective porosity (e) particle mobilization due to changes in the surface charge of the minerals which could cause permeability reduction by plugging of the particles in downstream pore throats8,16. Additionally, Aruna7 proposed strong reactions between the Silica matrix and water leading to chemisorption and reduced effective flow areas as another possible mechanism. Rosenbrand et al.8 analyzed their own empirical data obtained using Berea sandstone cores in terms of the quantity referred to as SP which is specific surface area per bulk volume rock S divided by porosity (SP is expressing the internal surface area per unit pore volume rock). The advantage of such an approach according to Rosenbrand et al.8 is that it opens for the possibility to compare results related to the phenomenon of reduced permeability between rock samples with different lithology. Hence, the authors reported their absolute permeability reduction in the Berea cores at elevated temperatures as increase in internal surface area per unit pore volume. None of the effects deduced from the hypotheses referred (a) – (d) above were large enough to account for the observed variations in absolute permeability vs. temperature changes. The authors therefore concluded that the presence of Kaolinite in the Berea cores was (reference to hypothesis (e)) the most likely explanation for the observed reduction in absolute permeability for tests performed in the temperature range 23 – 80 °C. Many of the above-mentioned effects are certainly important for the processes taking place particularly when heating natural porous media systems where e.g. large variations in mineral compositions, clay content, ability to adsorb fines, rock strength etc. are frequently occurring and could impact absolute permeability9,12 upon heating. The aim of the current paper is to propose a hypothesis for the reduction of absolute permeability vs. increased temperature which has not been discussed in the literature so far to the knowledge of the author (not considered e.g. by Bear2). Based on the hypothesis, a generalized version of Darcy’s law will be derived so its consequences can be compared with measured data. The hypothesis is not related to any of the abovementioned effects meaning that these are assumed absent when comparing measured variation of absolute permeability vs. temperature to the generalized version of Darcy’s law developed herein. It is related to an effect solely related to molecular thermal fluctuations in the fluid when being forced through a porous medium by an external energy source. The absolute permeability of a porous medium is determined by energy conservation where the external power supply required to force the fluid through the medium in steady-state equal to the measured pressure difference between inlet and outlet of the rock sample multiplied with the volume flow rate is exactly balanced by the energy dissipated into heat (assume incompressible horizontal flow). The external energy supply is assumed to dissipate into two channels either along the fluid – solid matrix interface usually referred to as viscous dissipation2 (which includes energy required for deformations of the fluid) and at the same interface as an additional friction channel due to thermal fluctuations of the molecules constituting the fluid which will be referred to as thermal dissipation. The latter channel has not been considered for liquid porous media flow. The proposed hypothesis will furthermore be substantiated by theoretical arguments by referring to its similarity to the Ludwig-Soret effect observed for Brownian particles under influence of a temperature gradient. Darcy’s law accounts only for viscous dissipation and is hence unable to account for other energy dissipation channels than the indirect effect due variation of fluid viscosity vs. temperature. The hypothesis proposed states that the decrease in absolute permeability for increasing system temperature is due to higher thermal energy dissipation rate at elevated temperature caused by increased momentum transfer from the fluctuating water molecules propagating through the porous medium to the solid matrix. On a very fundamental level it is therefore a consequence of the Fluctuation-Dissipation Theorem (FDT)17,18 which will be described briefly in the next section. The hypothesis will be used to formulate a generalized version of Darcy’s law accounting for thermal dissipation in addition to viscous dissipation. The results using the generalized version of Darcy’s law

ACS Paragon Plus Environment

Energy & Fuels

6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

will then be compared to empirical data reported in the literature where absolute permeability has been measured vs. changes in system temperature. The rest of this paper is organized in the following way. The main features of Brownian motion as a background for the molecular fluctuations occurring on the microscopic level and the consequences therefrom are outlined first in the next theory section. The thermal dissipation effect is then shown to be caused by forced movement of water through the porous medium giving rise to an additional channel for energy dissipation. A split of the total resistance to flow through a porous rock sample into a viscous and a thermal dissipation term is then proposed, the latter being directly dependent on the absolute temperature of the system. The Results and discussion section first presents a generalized version of Darcy’s law accounting for both viscous and thermal dissipation where only the latter contribution is a function of the system temperature (after accounting for fluid viscosity variation vs. temperature in the viscous term). The absolute permeability calculated using the generalized version of the Darcy equation is compared to observed experimental data from the literature followed by a discussion of the results. Finally, conclusions and suggestions for more testing are presented in the last section.

2. THEORY 2.1. Brownian motion and the FDT. The discovery of what is normally referred to as Brownian motion was reported in 1828 by the botanist Robert Brown19 although Jan Ingenhousz (more known for discovering the photosynthesis) reported the same phenomenon already in 1784 observing irregular movements of fine powders of charcoal floating in alcohol (Abbott et al.20). Brown observed irregular motion of small pollen grains moving in water under the microscope. Albert Einstein21 explained the observation in 1905 as a consequence of molecular action on the microscopic scale as the pollen grains are changing direction due to collision with water molecules moving randomly in thermal equilibrium. Einstein furthermore realized that the thermal fluctuations of the water molecules generating a random force responsible for the irregular motion of the pollen grains also would cause a resisting damping force to the same movement of the seeds as they collide with the water molecules. Hence, there should be a relationship between the two forces as they have the same origin, i.e. water molecules fluctuating in thermal equilibrium18. In fact, his famous relationship between the diffusion coefficient D and the absolute temperature T (D = k Z T/Friction, kB – Boltzmann’s constant) is an early version of the FDT. A similar process is also occurring in electrical circuits discovered by Johnson22 in 1928. Johnson could measure a random fluctuating voltage over a resistor also called thermal noise even in the absence of an external voltage. In this case, thermal energy is causing random damping of the moving electrons generating the fluctuating voltage and hence a random oscillating current. The phenomenon was explained by Nyquist23 in 1928 showing that the thermal fluctuations generating the thermal noise also is the cause for the dissipation of electrical energy to heat (Joule heating) when an external voltage is applied over the resistance. It will be shown later that the dissipation of electrical energy also bears similarity to the thermal dissipation effect as both increases with increasing temperature which in the former case is due to increased inelastic collision frequency between the electrons propagating through the conduction bands and the metallic ions in the lattice experiencing increasing vibration at elevated temperatures. In the latter case it is caused by a combination of increasing molecular collision frequency and momentum transfer from the fluctuating water molecules to the solid matrix. In both cases, transfer of kinetic energy to heat i.e. increased friction or resistance to flow is the ultimate effect. The FDT referred to above was, however, not formally derived before 1951 by Callen and Welton17 from more fundamental principles applying standard perturbation theory for the timedependent Schrödinger equation. It applies to both classical and quantum mechanical systems. It states (the qualitative version of the theorem referred to herein. For the quantitative version see

ACS Paragon Plus Environment

Page 6 of 30

Page 7 of 30

Energy & Fuels

7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Kubo18) that to a process where dissipation of molecular kinetic energy into heat occurs there must correspond another process where thermal energy (heat) generates kinetic energy. The former is demonstrated in the case of an object falling through a liquid with a constant velocity where the reduction in potential energy is balanced by viscous dissipation at the fluid-object interface caused by numerous collisions between water molecules and the object. The latter is exemplified in the case of the Brownian particle which is moving randomly due momentum transfer from the random fluctuating water molecules. The fundamental principle to both these processes is of outermost importance when studying irreversible processes in thermodynamics. If an external disturbance on a system in equilibrium is not too large (linear response theory, see e.g. Chandler24); the response of the system to the disturbance is similar to the system’s internal response to the fluctuations occurring continuously also in thermal equilibrium. Hence, Brownian particles moving through a fluid will experience an average frictional force (on the macroscopic level) also refer to as a damping force responsible for dissipation of molecular kinetic energy into heat. The same macroscopic average damping force caused by microscopic molecular fluctuations is also responsible for the resistance and hence the energy dissipation for forced movement of particles driven by the influence of an external force. For spherical particles, the damping force is described by the famous Stokes’ law25 derived by Stokes from the celebrated continuum Navier-Stokes equation in 1850 (presented 1850 and published 1856). It will be shown in the next section that this frictional force is corresponding to the usual viscous dissipation term accounted for by Darcy’s law. Additionally, it will also be demonstrated that there is another channel available for energy dissipation i.e. conversion of molecular kinetic energy to heat which could be significant when considering forced movement of fluids through porous media. The latter channel is presently not accounted for in Darcy’s law and is corresponding to the thermal dissipation term hypothesized to be the physical cause for observing reduced absolute permeability for increased system temperatures. 2.2. The Langevin equation and implications of thermal fluctuations on porous media flow – The thermal dissipation effect. The irregular Brownian motion exhibited by a small particle residing in water (could be pollen grains as in Brown’s case) is normally described semi-phenomenological using the Langevin equation26 reading27: ./

(



= −γu ./ + S./t

(2)

where .u/ is velocity of the Brownian particle, γ is the friction/damping coefficient and .S/t a stochastic time-dependent acceleration term due to random collisions between the particle and the fluctuating water molecules. For a spherical Brownian particle, the friction/damping term γ is given by Stokes’ law25: γ=

aG)b % FE

(3)

where a is the particle radius and mB the mass of the Brownian particle. The origin of the macroscopic friction force is on the microscopic scale caused by rapidly fluctuating water molecules which appears as Stokes’ law (eq. 3) on the macroscopic scale. This law can in fact also be derived using a time-correlation formula for γ as demonstrated by Robert Zwanzig28 assuming symmetrical distribution of energy dissipation (shear stress and pressure) on the surface of the sphere independent of flow direction. Hence, the friction responsible for the energy dissipation is only dependent on the viscosity of water on the macroscopic scale. Now, considering forced movement (external perturbation of the system) of Brownian particles through water due to an external force .F/ (e.g. electric charged Brownian particles under the influence of an external electrical field), the Langevin equation takes the form:

ACS Paragon Plus Environment

Energy & Fuels

Page 8 of 30

8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

./

(



.c/

= −γu ./ + S./t + F

(4)

E

Eq. 4 contains the same resistance/damping force (γ term) as for the free Brownian particle with no dependency on temperature except indirectly through the water viscosity term (eq. 3). Hence, ./

( equation 4 at steady-state ( = 0, no acceleration effects considered assuming creeping flow) is for time-scales much longer than the typical collision frequency between the Brownian particle and the water molecules given as: .c/ FE

− γu ./ = 0

(5)

since the time average of the stochastic force is zero (〈S./t〉 = 0. Hence, the external energy supply is balanced solely by the viscous dissipation term (see e.g. eq. 2 in Tartakovsky et al.29) i.e. the γ term in eq. 3 for spherical Brownian particles. The situation described above related to eq. 5 corresponds to a case where Brownian particles are moving with constant macroscopic velocity .u/ through a stagnant water phase (except for the molecular fluctuations). The energy supplied by the external force F./ is balanced by the viscous dissipation term γ only. Applying such an approach for forced flow of water through porous media, however, requires the assumption depicted in Fig. 1 where instead water is forced through numerous large macroscopic stagnant “Brownian particles” representing the porous matrix with the same constant relative macroscopic velocity .u/. Hence, in this picture, which will be used herein, the energy dissipation will be equal to the case where the Brownian particle is moving relative to the water phase since only the relative velocity u./ in eq. 5 is significant. Now, calculating the viscous dissipation term analytically for fluid flow through porous media is extremely challenging due to the complex boundary conditions involved and it can normally only be determined experimentally using Darcy’s law eq. 1. Hence, the application of Darcy’s law automatically attributes all supplied external energy to viscous dissipation only as previously noted. It is here relevant to mentioned that the approach of using the Langevin equation for modeling of porous medium flow has been performed previously although for other purposes than the approach outlined here. Tartakovsky et al.29 studied e.g. transport of tagged particles through a porous medium using a stochastic Langevin equation with the aim of separate the effect of advective and diffusive mixing. They used an equation identical to eq. 5 and added a random noise term to be able to account for deviations from the smooth flow paths predicted by the Darcy-scale continuum flow equations. Maier et al.30 used a stochastic differential equation to describe particle positions representing tracer movement in porous media. Zhou et al.31 used a Langevin type equation to study the propagation of fluid fronts through porous media.

At this point, it is crucial to observe that an additional channel for energy dissipation caused by differences in relative velocity between the water molecules colliding with the up- and downstream part of the rock matrix due to the forced water movement will occur, see Fig. 2 (asymmetrical velocity distribution). This effect not yet considered for water flow through porous media to the knowledge of the author is, however, crucial for the direct dependence of absolute permeability vs. changes in system temperature according to the hypothesis proposed. Because water is forced through the porous medium from left to right in Fig. 2 with macroscopic velocity u./, the solid matrix will experience higher force on the left-hand side (water molecules will hit the matrix with velocity .u/ + .v/ F ) than on the right-hand side (water molecules will hit the matrix with velocity – .v/F + u ./) caused by differences in instantaneous velocities and collision frequencies of the fluctuating water

ACS Paragon Plus Environment

Page 9 of 30

Energy & Fuels

9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

molecules on the microscopic scale. This asymmetric momentum transfer inducing an additional net resisting force is therefore responsible for the energy dissipation channel referred to as the thermal dissipation effect as it will be shown to increase in magnitude with the square root of the absolute temperature (the term is independent of fluid viscosity). It is hence a direct implication of the impact of forced movement of random fluctuating water molecules in thermal equilibrium on the microscopic level inducing a net pressure loss i.e. dissipation of molecular kinetic energy to heat (FDT) along the water - solid matrix interface. To be able to quantify the thermal dissipation effect theoretically, it is useful to first study the approach applied by Tenenbaum et al.32 when modeling heat transfer and conduction through walls and liquids. They extended an approach originally proposed by Lebowitz and Spohn33 to investigate the microscopic molecular mechanisms underlying Fourier’s macroscopic law for heat conduction using molecular dynamics (MD) simulations. The original model presented by Lebowitz and Spohn33 is addressing heat conduction in Lorentz gases assuming a Maxwellian molecular probability density velocity distribution. The results obtained by Tenenbaum et al.32, however, also demonstrated the applicability of this approach for describing transfer and conduction of heat from a hot to a cold wall through an intermediate liquid. The movements of the molecules in liquid state were in general assumed to follow a Maxwellian velocity probability density distribution ϕ4vT 7 given Fh6

F

by expressions on the form ϕ4vT 7 = B?GD  exp