Thermo-chemo-mechanical coupling in single-fluid flows through porous media

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LIBRARY MAFl~ENG CAMPUS CALL NO.:

2019

-07- 1 5

ACC.NO.: N".>RTH-WEST UNIVERSITY

Thermo-chemo-mechanical coupling in

single-fluid flows through porous media

by

Le

hlohonolo Jacob Phali (1 709804

1)

A thesis submitted in

fulfilm

ent of

the

requirements

for

t

he d

egree of

Do

ctor of Philosophy

in

App

lied

Mathematics

in

the

Department of

Mathematical

Science in the Faculty of Agriculture

,

Science

and

Technology at t

he

orth-West University.

O

ctober

2015

Sup

ervisor: P

rof M.T. Kambule

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Declaration

I Lehlohonolo Jacob Phali student number 17098041, declare that this thesis for the degree of Doctor of Philosophy in Applied Mathematics at North-West University, Mafikeng Campus, hereby submitted, has not previously been submitted by me for a degree at this or any other university, that this is my own work in design and execution and that all material contained herein has been duly acknowledged.

Signed:

Mr Lehlohonolo Jacob Phali

Date: ... .

This thesis has been submitted with my approval as a university supervisor and would certify that the requirements for the applicable Doctor of Philosophy degree rules and regulations have been fulfilled.

Signed: ... .

Prof M.T. Kambule

Date: ... .

This th sis has been submitted with my approval as a university co-supervisor and

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NWU

l

\ueRAl!Y

would certify that the requirements for the applicable Doctor of Philosophy degree rules and regulations have been fulfilled.

Signed: ... .

Prof E. Ebenso

Date: ... .

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Dedication

I dedicate this thesis to my whole family for their support and patience throughout my studies.

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Acknowledgements

I would like to thank my supervisor, the late Professor M. T. Kambulc for his guid-ance, motivation and support throughout my studies (may his soul rest in peace), my Co-Supervisor Professor E. E. Ebenso for his contribution, the North-West Uni-versity, Mafikeng campus, for the financial assistance and the Council for Scientific and Industrial Research (CSIR) for the Studentship towards my doctoral studies.

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Abstract

We consider a simple system, which captures thermal, chemical and mechanical ef-fects. In the beginning, we consider a system of a solid, fluid and an interface, each having species and thermodynamic properties, which allows for further extension. This approach presents a more general setting for the system's entropy inequality. The entropy production is associated with thermodynamic irreversibility that can be found in all fluid flow with heat and mass transfer processes. The approach we follow, with our slight variation was pioneered by Gray and Miller in their Thermody-namically Constrained Averaging Theory (TCAT) approach. This approach involves the averaging of Classical Irreversible Theory (CIT) from microscale to macroascale. From the inequality, together with further restrictions we simplify it to produce de-sired entropy fluxes using the Curie-Prigogine principle and the Onsager phenomeno-logical coefficients. Due to the dependence of fluid flow on the thermal conditions and flow conductivity, we modify Darcy's law to capture these effects. Depending on the temperature conditions and the flow forms we develop closed forms which are non-isothermal, locally isothermal and isothermal using the Darcy's law modifica-tions. We present thermo-chemo-mechanical closed forms in the form of single-phase flow with species and thermo-mechanical closed forms in the form of single-phase flow without species.

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Chapter 1 Introduction and definitions

1.1 Introduction

In the modeling of many chemically sensitive porous medium systems such as clay soils [25, 27], biological soft tissue [21, 22, 27], biopolymers [11, 33], biosynthesis in tissue engineering [32], nuclear waste disposal [23, 24], irrigation and fertilization, it is essential to know the chemical composition of each phase in the system. The need is to understand the effects of chemical reactions, temperature distribution, environmental and mechanical loads on the system. In fluid mechanics, fluid flow through porous media is the manner in which fluids behave when flowing through a porous medium, for example sponge or wood, or when filtering water using sand or another porous material. As commonly observed, some fluid flows through the media while some mass of the fluid is stored in the pores present in the media. The figure below is an example of how a porous media system looks like, under a unit area (Representative Elementary Volume, REV) where the particles represent the solid phase, water represents the fluid phase and q is the flux.

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Water

- -

Unit area

Solid

q

Figure 1.1

A lot of work has been done in this area of transport in porous media using Rational Thermodynamics (RT), which forms a basis to the Hybrid Mixture Theory. This theory was pioneered by Truesdell and Coleman (Jou et al. [25]). It entails the averaging theory and the use of constitutive th ory. In many cases isothermal mod-els were considered, the interface properties were assumed discontinuous and some models were postulated at macroscale (see Bluhm [8]). This imposed limitations on the ability to properly account for the contribution of microscale physical properties of the system to the macroscale and also the possible systematic extension of the theories used. A tremendous lot of work in the area of single-fluid flow has also been done using non-equilibrium thermodynamics, e.g. Bader and Kooi [35], Malusis et al. [37], Kaczmarek [25], Gray and Miller [16, 17, 18, 30, 31]. The justification for this approach is that changes brought about in soil systems create a system out of thermodynamical equilibrium and that transport processes such as diffusion of a solute in the soil need to be analyzed using non-equilibrium thermodynamics sec (Badder and Kooi [35]). This theory is posited at macroscale because according to Badder and Kooi it delivers a general framework for the macroscopic formulation of irreversible processes.

The entropy production is associated with thermodynamic irreversibility that can be found in all fluid fl.ow with heat and mass transfer processes. Entropy production

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destroy the available energy in a thermal system, hence, entropy generation mini-mization (EGM) is extremely necessary for efficient operation of thermal system and various engineering devices. Identifying various factors causing entropy production in fluid flow with heat and mass transfer in porous media will surely enhance im-prove design of engineering devices for efficient operation. Moreover, the inherent irreversibility in single-fluid flows through porous media with heat and mass transfer arise due to the exchange of momentum, energy and species concentration within the fluid, at solid boundaries and interface. Consequently, entropy production may occur as a result of fluid friction, heat and mass transfer in the direction of finite temperature and species concentration gradients [38, 39].

In the present project we use the averaged Classical Irreversible Thermodynamics (CIT). The formalism of CIT was pioneered by Onsanger, Eckgart, Meixer and Pri-gogine (Jou et al. [24]). Its essence is the use of local equilibrium. In the recent years, Miller and Gray have extensively used a new CIT-based approach to develop closed forms for multiphase systems. In a series of papers they have laid a strong foundation for the development of models of flow and transport in porous media. They call their new method, the Thermodynamically Constrained Averaging The-ory (TCAT) approach. This approach entails the processes of; (i) averaging the balance equations for phase volumes, the interfaces and the common curves. It also involves the averaging of the CIT and the equilibrium conditions, from microscale to macroscale; (ii) constraining the entropy inequality by using Langrange multipli-ers together with the conservation equations and thermodynamic forms, similar to Liu [27] and (iii); simplifying the constrained inequality in order to obtain closed forms. In this approach the link between the micro-physics and the macro-physics is made very explicit, compared to the cases where RT is used. The heterogeneities of temperature, pressure and chemical potential are vividly revealed in the averag-ing volumes. The main difference between HMT and TCAT is that HMT is based on Rational Thermodynamics which does not properly account for the contribution of microscale physical properties of the system to the macroscale, whereas TCAT

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which is based on Classical Irreversible Thermodynamics, makes the link between the micro-physics and the macro-physics very explicit.

The present work was influenced by several papers in a series of papers by Miller and Gray where they used the TCAT approach. They developed components of the theory which will be needed to formulate TCAT-based macroscale models, in order to describe compositional multiphase, multispecies porous medium systems. They also formulated averaged Classical Irreversible Thermodynamics expressions for species in fluids, solids and interfaces. Using the TCAT approach the authors formulated a theory of multispecies flow and transport for single-fluid phase flow in porous media. One of the other papers which influenced the present project is the one by Kaczmarek [26]. His stated aim was to solve problems describing chemical consolidation and chemical stresses coupled with advective-dispersive transport in pore fluid. During the development of the model, he used averaged balance equations and averaged thermodynamic expressions from CIT. Interfacial jump conditions were used. The temperatures of different entities were considered the same. In order to increase the scope of use of the model and understand the impact of these restrictions, we found it desirable to relax them. The aim of the present project is to develop an extension to the TCAT approach by using this Averaged Classical Irreversible Thermodynamics approach which gives similar results for closed forms. Ours is a direct substitution method, which is a slight variation but equivalent to the Lagrange multipliers method in the TCAT approach. This involves the derivation of the entropy inequality for each entity, which are later put together to produce the system's entropy inequality. We also intend to extend the work that was done by Miller and Gray [30, 31], on the TCAT approach by deducing modifications of Darcy's law which were previously derived in RT, to account for different flows and use these modifications to come up with the closed forms.

We further extend the work to account not only for isothermal conditions as it was done previously but we also include the non-isothermal and locally isothermal cases for the isotropic case for single phase flow systems with species and without species.

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Where the isothermal case means that the system has a constant temperature across all entities, non-isothermal case means that the temperatures are not the same and locally isothermal case means the temperatures are the same. We also develop an extension to Kaczmarek ([26]) model.

1.2 System's definition

We consider a porous deformable continuum, consisting of an elastic solid matrix s, a viscous fluid

w,

an interface

ws

with thermodynamic properties and different tem-peratures for all entities. All three entities have N species each. Some species in an entity may have zero mass. We assume the existence of a repr sentative elementary volume (REV), denoted by D, which stays unchanged when we change position, size and shape.

We consider a system with chemical processes such as transport, homogeneous and heterogeneous chemical reactions. The heterogeneous reactions may be examples of a chemical reaction between the components of the fluid and those of the solid, adsorp-tion and desorption, dissolution and precipitation and the inj ction or withdrawal of a fluid from the porous medium.

1.2.1 Definitions

Within the REV a region occupied by a solid phase s, is denoted by

D

s,

the region occupied by the fluid phase w, is denoted by Dw and the interfacial region between the solid and fluid phases ws

, is

denoted by Dws· The index set of the entity qualifiers is given by

:J={

w,s,ws

}

and the index set of the phases only will be denoted by

J

p=

{

w,s

}

.

The set of entities in a representative elementary volume (REV), are given by the

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set

Considering the entity D1, let

r

₁be its boundary and define its closure by

D

1 = D1 U

r

1. The connected entity set for D1 is then given by

thus it is the set of entities which have a direct contact with entity D1. In our present case of a single phase flow

Ccw

= {Dws}

Ccs

=

{Dws}

C

ews

=

{D

w,

Ds},

Correspondingly,

J

ew

=

{

ws },

J

es

=

{

ws} and

J

ews

=

{

w, s} are their respective connected index sets.

1.2.2 Averaging and notation

We will use the averaging operator [30]

(1.1)

where Pi is a general property to be averaged and c is a weighting function. If c = 1, then c as a subscript on the left hand side of equation (1.1) is omitted. As a

first example, we consider the various geometric densities, E1, as averages by setting Pi= 1, then

l fol dr l

r

D1

E

=

(1)01,0

=

fodr

=

n

101 dr

=

n

·

(1.2) If l E

J

p,

then E1 is a volume fraction. If l

=

ws, then Ews is the specific interfacial area of the ws-interface.

A second example is an average of a microscopic function ft, over an entity of interest i.e. the case where

P

i

= J

i,

is given by

l fo1 f1dr l

l

f

=

U1)01,01

=

r d

=

D

f1dr.

Joi r t Oz

(1.3)

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Equations (1.2) and (1.3) give

(f1)ri

1,n

=

E:1,

/1.

The third example is when a microscale density function is used as a weighting function, thus calling it a mass weighted average

(1.4)

Equations (1.2) - (1.4) give (Pd1)n1,ri

=

E 1

/

jl.

In the case of averaging with weighting, which involves properties of chemical species

k in the entity l, we have

(1.5)

Using equations (1.2) - (1.5), we obtain (p1CktYkt)ri1,ri

=

E

1_p1

Ck1vk1. The use of the

double bar over a variable can be thought of as an indication that the macroscale variable is not just direct averaging. It is obtained as some other kind of average and/or combination of terms that has been specifically listed in defining the quantity; e.g.

(1.6) where t1 is the microscopic Cauchy stress tensor and ( v1 - v1) ( v1 - v1) is a second order tensor product of velocity fluctuations. The macroscopic stress tensor t I _i_s

thus an average of their weighted difference.

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Chapter

2 Averaging of systems

In this chapter we adopt the equations which have been averaged by Gray and Hassanizadeh [15] using microscale equations and the localization approach. Hence -forth, subscripts on quantities will indicate a microscale and superscript will denote a macroscale quantity. The averaging volume will represent a point at the macroscale. To capture the fractional contribution of each entity to a volume of interest (REV), we include the geometric densities c1, which according to equation (1.2), give the ratio of the l entity to that of the REV. Note that in a biphasic system of a solid

s

and fluid w, the relationship of the two volume fractions is €8

+

cw

=

1. We now list

the macroscale balance equations, present the microscale origins of the macroscale

variables and use the same notation as in the TCAT series.

2.1 System

of

macroscale

balance

equations

(a) Conservation of mass equations for phases and the interface We have the equations l l - ~ k-+l - € p I : d 1

+

L..,; M , for l E

J

=

{

w,s,ws

}

,

kE:Jc1 8 (2.1)

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where p1 is the macroscale density of the l-th entity, I is the rank two unit tensor. Furthermore,

is the rate of strain tensor,

is the material derivative referenced to the macroscopic velocity of the l-th

entity, where Vt is the microscale velocity for entity l and v1

=

(v1)n1,n1,p₁is the mass averaged intrinsic velocity. The superscript T denotes the transpose of the rank two tensor,

ws ➔ l l ➔ ws

M = - M = (p1n1 · (vws - v_{1)) 11}w,,n for l E J,,

=

{

w, s} represents mass exchange from the ws-interface to the l-phase, where Vws is the microscopic interface velocity and Pt is the microscopic density of the l-th

phase.

(b) Conservation of mass equations for species

Since we intend to model chemical processes then we need to consider mass

balance laws of species. The equations should account for concentrations, diffusive-dispersion and non-advective fluxes.

For each species i in the entity l, the mass fraction is given by

il

Cii

=

E._ and satisfies ri

Each diffusive-dispersive velocity is

and the non-advective flux, which accounts for dispersive and diffusive trans-port, is given by

N

Jii

=

c1 p1Ciiuii and satisfies

L

Jii = 0.

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For each i-th species and for l E

J

the mass balance equation is then given by D1(ct/cii) Dt - - - ik ➔ it

=

- Etptcitv . Vt - V . Jit

+

€tril

+

L M kE.:lc1 (2.2)

where pit is the macroscopic species density, ril are the macroscopic homoge

-neous chemical or biological reaction and

N ik-,it k ➔ t

is the macroscopic heterogeneous chemical reaction such that

I::

i

M

=

M .

(c) Conservation of linear momentum equations for phases and the interface

We have the equations N -Etptv[I : di+ V. (cttt) - L ElptCiigil " ( ~ ik-,it k-tt) + ₀ ₀ Mv + T kE.:lc1 i for l E

J

(2.3)

where the macroscale l-phase stress tensor is obtained from the microscopic quantities as indicated

where ( Vt - v1) ( Vt - v1) is a tensor product of velocity fluctuations, tt is the microscopic Cauchy stress tensor and thus t t is the average of a weighted difference of the two tensors. The momentum transfer due to mass exchange is

ws-,k

Mv - k-tws M V

and the macroscopic momentum exchange vector is related to the microscopic quantities as follows

wT k - kTws

= (

Ilk . [ tk - pk(Vws - vk) ( Vk - vie)]) nw,,n

for k E

Jp

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(d) Conservation of angular momentum equations for phases and the interface In this case we assume that our system is non-polar, hence the Cauchy stress tensor is symmetric, i.e.

-

(

-)

T

ti

=

tl ' for l E

J.

( e) Conservation of total energy equations for phases and the interface We have the following equations

for l E

J

(2.4)

where h1 is the external energy source, or after simplification, the equations are

(2.5)

The above macroscopic quantities are obtained from the microscale counte r-parts as follows El T (E1)n1,n \ ~ 1 ( v1 - v 1 ) . ( v1 - v 1 )) ni,n E

1 +c'p'

C

'

/

'

+

K

I+l

)

11

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/ q

1

+ ( El - El + _!(v1- v1) · (v1 - v1)) P1(v1 - v1)) \ Pt E1p1 2 !11,!1

+

\ t

(,Pu

-

,t,")

p,C.,(

v

,

-

v

1) -

K

J

p,

(

v

, -

v

1) +t

1

·

(v1 - v1)) !11,!1 ws-tk k-tws Me - Me for k E

Jp

ws-tk k-tws ws-tk Tc Tv - Tv

=

T ·V for k E

J

p

ws-tk k-tws Tc Q - Q

=

(nk · qk + Ilk· tk(Yk - V

))n

w.,rl

+ / (Ek - E! kk) Pk(Yws - vk) · Ilk) \ Pk p l1ws,n

+ / · f) 'l/Jik - 'l/Jik)pkCik(Yws - vk) · Ilk)

\ i l1ws,n

+ / t(vk - vTc) · (vk - vTc)Pk(Yws - Yk) ·Ilk)

\ l1ws,n

- ( K jpk(Yws - vk) ·Ilk) for k E

.J,,.

To further simplify the total energy equation (2.5), we substitute the mass equations (2.1), (2.2) and the momentum equation (2.3). After some amount of manipulations, we obtain the following simplified equation:

= N

El ~ iws-til ws-tl

+E1h1

+

-1-1 L.t M

+

Q , Ep i

for l E :J

(f) Balance of entropy equations for phases and the interface We have the following equations

~ : l

=

-r/

I : d

I

+ v7 . ( El

cpl

)

+

El bl +

L

cM:

+ k;l)

+

J\

t

,

kE:Tc1

12

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\

NWU

J

LIBRA8_Y

for l E

.J

,

(2.

7)

where

ry1

=

(ri1)0.1,n is the entropy density, r.p1

=

(r.p1)0.₁,n1 - (ri1(v1 - v

1_))n

1,n1 is the entropy flux, b1

=

(61)0.1,0.1 is the external entropy source,

A 1

₌

_(A1)0.

1,n is the entropy production of the l-phase,

k--tl ws--tl

Mr, is the entropy exchange due to the exchange of mass, <I> is the entropy exchange from thews-interface to the l-phase and the overall entropy exchange term is

= N

T/ l "\""' iws--til w.s--tl

-l-l L.,, M

+

<l>

=

(n1 .

[<pi+

T/t(Yws - v1)])nw,,O.·

cp i We also have

and

2.2 Thermodynamic systems

To demonstrate the link between microphysics and macrophysics, we summarize the averaging of the microscale thermodynamic system to the macroscale and determine the material derivatives of the internal energies. We assume Classical irreducible thermodynamic for our system at the microscale. The summaries for the fluid phase and the interface are extracted from the detailed account by Gray

[

1 4].

The summary for the solid phase, is from Gray and Schrefler

[

1

9 ].

2.2.1 Th

e

fluid pha

se

volume with

s

pecies

The microscale internal energy of the fluid phase w, is

N

Ew

=

Ew(T/w, PwCiw)

=

0wT/w

+

L

µ;wPwCiw - Pw,

13

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where ''lw is microscalc entropy per volume, PwCiw is microscale mass per unit volume

of fluid phase w and Pw is the microscale pressure in the w-phase. The microscale

absolute temperature and chemical potential are respectively given by

0w = 8Ew 8"7111 and 8Ew µw = 8pwCiw

The differential of the internal energy and its corresponding Gibbs-Duhem equations are given by N Bwd"lw

+

L

µiwd(pwCiw), (2.9) and N 'f/wd0w - dpw

+

L

PwCiwdµiw 0 (2.10)

respectively. While assuming that temperature, chemical potential and pressure

are inhomogeneous in the averaging volume, we proceed to average the microscopic

internal energy to the macroscale level. The process yields the following internal energy for the w-phase

In cases where the external gravitational potential is influencing the system [14], then the integrals in expression (2.11) are not necessarily zero. If they are, then we obtain explicit macroscopic definitions of absolute temperature, chemical potential and thermodynamic pressure in term of their microscale counterparts as follows:

(2.12)

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(2.13)

(2.14) Due to the definitions (2.12), (2.13) and (2.14), the macroscale internal energy e x-pression (2.11), then reduces to

N

Efii

=

e

w

r?

+

:z=µiwEwpwc iw - pwEw. ₍₂_.1₅₎

Therefore the functional dependence of the internal energy is thus

(2.16)

If the microscale temperature, chemical potential and pressure arc homogeneous in the averaging volume, then the macroscale internal energy Ew would be independent of them.

Assuming inhomogeneity in the averaging volume, we obtain the following macroscale

material derivative of the internal energy of the fluid. For the full details see Gray

[14]. We have

Dt

(2.17)

2.2.2 The

solid

phase

volume

with

species

The reader is referred to Gray and Schrefler [19] for a detailed account of the present summary.

Assumption: The solid phase is elastic.

The microscale internal energy of the solid phase with species is given by

N Cs " Cs Es

=

Es(rJs, PsCis, -.- )

=

0srJs

+

L..t µisPsCis

+

Cls : -.- , Js i Js 15 (2.18)

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where 'f/s is the entropy per volume, PsCis is the mass per volume and Cs/.js is the strain. Using these microscopic extensive variables one can determine the corre-sponding intensive variables of the absolute temperature and the chemical potential of the i-th species, as follows:

respectively. The differential of the microscopic internal energy for the solid phase together with its corresponding Gibbs-Duhem equation are given respectively by

and

By the averaging of the extensive variables, we obtain: internal energy:

(2.19)

(2.20)

(2.21)

Subtracting the integral of the Gibbs-Duhem from equation (2.21) and further s im-plifying, we obtain (2.22) entropy: (2.23) mass: (2.24) 16

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and strain:

(2.25)

In the case where the external gravitational potential is influencing the system, the integrals are not necessarily zero. If they are, then we obtain definitions of the macroscale intensive variables as follows:

absolute temperature: (2.26) chemical potential: (2.27) and stress: (2.28) With these definitions, equation (2.22) then becomes

(2.29)

After some considerable manipulations of equation (2.29), we obtain the material

derivative of the macroscale internal energy of the solid (see

[

19 ])

Dt

(2.30)

where 'v xx is the deformation tensor.

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2.2.3 The

interface with species

We integrate the respective microscale quantities, expressed per unit area, over the interface within the REV, Gray [14] to obtain the following: internal energy: entropy: mass: velocity: and specific surface: vws

A

r

Pwsds, Jnws l

i

ws wsn PwsVwsds € p nws

The microscale internal energy per unit area is given by

By averaging we obtain the macroscale internal energy per volume

where the macroscale intensive variables are given as follows: temperature: 18 (2.31) (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38)

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chemical potential:

and interfacial tension:

ws 1

1·

d

I

=

~ /WS s.

Ews 3 G

flws

The derivative of the macroscale surface internal energy is (see Gray [14])

Dt - "v . [

A

1 w

s

nsns . (vws - Vws)bws - ,ws)ds] - dws :

[A

1

nsns(,ws - ,ws)ds]

fl

ws

+

[A

1 w

s

bws - ,ws)("v'. ns)ns. (vws - Vws)cls] - "v0ws.

[A

1 w

s

nsns. (vws -vws)77ds] N - "v

L

/1,iws · [

A

1 ws

Il8Il8 · (vws - Vw 8 )PwsCiwsds] - "v,ws . [

A

1w

s

nsns . (vws - Vws)ds] ' where the material derivative restricted to the interface is given by

a'

at

Dt

a

at+

Yws . Ilwllw . "v and "v'

=

(I - Ilwilw) . "v. 19 (2.39) (2.40) (2.41)

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2.3 Equilibrium conditions

In this subsection we give a summary of the averaged equilibrium conditions. Their derivations from microscale to macroscale are detailed in Gray [13] and Gray and

Schrefler [19]. These results were arrived at using calculus of variations and are an extension of the work of Boruvka [7] and Boruvka and eumann [8].

At equilibrium all velocities are constant and equal; i.e.

vw

=

v5

=

vws

=

constant. (2.42)

This constraint leads to

= =

ds = dws = 0. (2.43)

The macroscale temperatures are also constant and equal

= =

ew

=

es

=

ews

=

constant (2.44)

and hence their gradients vanish;

= =

v es

=

v ews

=

0. (2.45)

At equilibrium, the sum of chemical and gravitational potential are related as follows:

and as a result where µ•ws

+

1/)iws 18 _._,_,_Ts _\ _C_{s )}

1 \

_t_s_:

I)

µ

+

'/.I

+

Cls : - - - - -Ps]s n n SI S Ps 3 n S) n S constant 2 -:-Cls : (V xx'V xx), ]s 20 (2.46) (2.47) (2.48)

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and 'iJ xx'vxx represents a special of tensor product. The geometric variables are constant at equilibrium and hence their corresponding material derivatives vanish;

Dt Dt (2.49)

Within the solid phase the equilibrium condition is

=

o

.

(2.50) Equations (2.47), (2.50) and the Gibbs-Duhem equation to deduce the equilibrium

condition

(2.51)

The expression for the balance of normal stress at the fluid-solid interface is

(2.52) while the lateral stress at the solid surface obeys the equilibrium condition

(n S · t S · I'),., Hw, s,'"' Hws

=

0, (2.53)

where I' = I - nsns is the surface unit tensor.

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Chapter 3 Entropy production

From the equations (2.17), (2.30) and (2.41) we obtain the derivatives of the entropy

densities of the fluid phase, solid phase and the interface. From these we determine

the entropy production of each phase volume and the interface. In order to facilitate

easy reading of this section, most of the tedious calculations have been carried out

in the appendices A, B, C and D.

3.1 Fluid phase

entropy

production

In this subsection we begin with the derivative of the entropy density of the fluid

phase w and finally derive the entropy production for w. For the fluid phase we have

the following equation from the material derivative of the internal energy, equation

(2.17):

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To obtain the entropy production from equation (3.1), we perform the simplification in appendix A.

From (9) in the appendix, the expression for the entropy production of the fluid

phase is

ws--tw

Q ws--tw

+

-

_

--

<P

ew

3.2 Solid phase entropy production

(3.2)

We now consider the material derivative of the internal energy of the solid phase, (2.30). We use it to determine the entropy production of the solid phase.

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To determine the entropy production from equation (3.3), most of the simplification

is detailed in appendix B.

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(3.4)

3.3 Interface

entropy

production

In developing thermodynamic relations for the interface at the microscale, the mate-rial derivatives should be restricted to the interface. We therefore use the following interface restricted derivatives.

where

8'

at

Dlws Dt 8 [}t

+

Yws · Ilwllw · 'iJ and 'v'

=

(I - nwnw) · 'v.

For the interface, we use the equation of the material derivative for the internal

energy of the interface (2.41)

~ Dwsgws - N

(

µ

i:s

)

DWS(cwspwsciws) ews Dt

L

ews Dt i 1ws Dwscws ews Dt 25

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N 1

L

n iws { ( ( ws) C ) }

+----;;-

V µ · llwllw · Yws - V Pws iws flw,,fl 0111s . i (3.5)

To determine the entropy production for the interface from equation (3.5), we give

details of the simplification in appendix C.

The result from appendix C, is thus the following expression of the entropy

produc-tion A_WS ( N D's (•'• .1,iw,) = ) 1 · 'f'iws - <p · D's JC'/j/

+

e

w

s

L

PwsCiws Dt - Pws Dt 1. .Ows,n 26

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N iw]\s-,il / -_ ( V 1 -,s -_ vws_ ,s _) [ ( El = = ) N iws-,i]\/ ll

+

~ ~

_

vl,s . _ _ _

+

~ ---=-- _ T/ l ~ _ L L Bl 2 L g,iis L c:l pl lE:lv i lE:lv i l

g

w

s (

(

s) ) +---;;-v' · Ilwllw · Vws - V T/ws flws,O gws N

+

_---;;-1

L "

_v

(

_µiws

+

._'f'1,iws) _·( _Il_w_ll_w_·( _V_w_s_- _Vs) _P_{ws iws}C ) _{n n}

0ws . ws, t 1

n

(

s)

)

1 ( ) ds

+---;;-

V • Ilwllw · Vws - V "fws flws,0

+ ---;;-

Ilwllw"fws flws,0 : 0~

e

~

- l= ((v''nw)nw · (vws - V8hws)- (3.6) gws

3.4 Entropy inequality

The second law of thermodynamics states that the entropy production is non-negative

i.e A

=

As+ A

w

+

A_

w

s

2'.

0. The aim in this subsection is to derive the system's total

entropy production and to write it as a sum of flux-force products, such that each

factor of the product is zero at equilibrium, as follows

A=

L

F

0t.

x

0t+

LF

.e

· X

.e

+

L F-y: X-y

Ot .B 'Y

The first product sum on the right hand side of the equality sign is a product of

scalars, the second is a dot product of vectors and the last is double contraction of

second order tensors. Addition of (3.2), (3.4), (3.6) and some simplifications yield

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1 \ N Ds ([µit

+

'tµit] - [µii+

· t/JilJ

)

D8 Kl ) } ~ C - --'---'---- E ~wsbws

+---;;-

L., Pt il Dt - P l ~ - C 01 i n n _I_, l [\ D's (ews _ ews) N D's (µiws _ µiws))

+

ews 'rlws Dt

+

L

PwsCiws Dt l Ows,n 1 \ N D's ( 1Piws - 1/Jiws) D's Kws) Ewshws+]

+

~ ~ PwsCiws D - Pws D E ews L., t l t nws,!1

+

€:

(

t

s

- e

)

:

a

s+ €:s

(r,;;'s

-

,ws

1)

:

aws es f)ws

-

L

[Elql, '7 (

1 =

)

+

t

Jil, '7 ( µii~ 'lj;il) -

t

(11

F

~

1/Jil) Elritl

lE:1 0 l i 0 l i 0 l

~

( 1 · 1 ) [ws--tl -- ws--+l ( El _ (vf,s - vws,s)) N . ·]

+

L., ---;;- - ~ T ·Vl,s

+

Q

+ _

+

Vl,s . - - ' - - - ~ ~ iws--til 01 ews Elpl 2 L.,, M lE:fp i N ~ ~ ( 1 1 ) -,-iws--til Vw,s { _ _ L., L., ---;;- _ ~ µil M

+~.

'7(€wpw) _ Ew pw'7 Kw_ 'rlws90ws lE:fp i 01 0ws 0w E N _ L Ewpwciw'7(µiw

+

1/Jiw)- WST W _

t

iw~7iw (vw,s _ v'ws,s)} i i 2 1 = V-ws,s - { ws--tl N iws--til (vf,s _ vws,s) +~(nwnw,ws)nws,n : dws

+--;;-

·

~ T

+

~ ~ M 2 ews ews L., L., L., lEJ,, lE:fp i

-

\7

(

cw• 7w') - cw' pw' \7

K

f' -

l

''

\7

0 ,;;'

-

t

cw' pw• c;w,

\7

(µ ;w,

+

,;,;w,

)

}

N iws--til

_

LL

1 [

(Kk +µii+ 1/Jil) _ (Kfs

+

µiws

+

1/Jiws)] lE:fp i 0

1 \

[

.

C

s

]

) )

~

c 1 /

1

f,s ----;;- Ils ·ts· Ils - as . - . Ils · (vs - Vws - L., --;;-g · V es Js Ows,11 LE:1 f)l - l_ (ns ·ts· I'· (v0 - vs))n n - (Pw(Vws - vs) · ns)n • n (~ - __;_)

a

s

,

;,

WS1 • 'W I 0w 0ws

+

l= / [\/ · ts - '7 a 8 :

~

s

]

· (

V

s -

vs)) es \ Js n.,n - 1= ([Pw

+

Ils. ts. Ils

+

/WS '7'ns - Pwsgws. ns](vws - vs. ns))nw.,11 ews - ( 1 1 ) -(n _s. t . _s n _S(v _WS- vs) . n _S)n n _H_{ws ,~}_' --=- - ---=- -0 -es ws 28

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N

+

_----;;-1

L

'7_V( _µiws

+

._If'1,iws) _.( _Il_w_ll_w_.( _Y_ws_- _Vs) _P_{ws iws}C ) _nws_,_n gws . i 1 '7}(1'tls ( ( s) ) A 0

+----;;-

V E . Ilwllw . Yws - V Pws nws,n

=

~

.

gws (3.7)

A detailed simplification of equation (3.7) is given in appendix D. The result from appendix D is

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N

-'v (cws,ws

(

I

-

c ws)) - L Ewspwsciws

(

I

-

c ws) 'v(µiws

+

'lj;iws)

iws-tis

M

where c ws is the geometric orientation tensor for the ws interface defined by [15]

A close analysis of equation (3.8), shows that it is not completely in the flux-force

form. The last three terms need closer scrutiny of their behavior at equilibrium. These are the sum involving the homogenous reaction factor ril and the averages

involving the factor (vs - v5). At equilibrium sum µIT+ 'lj;IT is a constant for l E

{w, ws }. For this sum to be regarded as a conjugate force, it must be zero at

equilibrium. If so, then we have 'v µIT

=

gii at equilibrium. In the case of the solid phase we will have

1/t

..

:

I

)

/

C

s

)

Ps

\-3-

n.,n. - \ c, : PsJs n.,n. We will also regard the two averages involving (vs - v9

), as negligible. The equation

(40)

(30) then reduces to the following production of entropy, which is similar to the

TCAT inequalities.

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(3.9)

3.5 Closure relation approximations

From equation (3.9), we obtain our closure relation approximations of entropy fluxes and entropy sources. These are fluxes expressed as first order Taylor expressions, near equilibrium and expanded in terms of all the forces in the system. For higher order systems, higher order Taylor expansions may be considered for these fluxes.

The approximations must always satisfy the inequality (3.9). They should be e x-pressive of the physics under consideration.

3.5.1 Entropy

fluxe

s

and external entropy sources

We consider a macroscopically simple system for the entropy flux and the entropy source terms. This means that entropy flux and the entropy source terms will be

expressed without any additional constitutive variables, made up of the non-zero conjugate forces. Note that in complex systems, there will be non-zero constitutive variable from the forces.

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3.5.1.1 Entropy fluxes

Fl'Uid and interface entropy fl'Uxes

The entropy fluxes for the fluid and the interface are given in terms of the heat flux and mass flux as follows

lE{w,ws}.

(3.10)

Solid phase entropy flux

The solid s phase entropy flux has and additional contribution from the microscopic

stress tensor, the Green's deformation tensor and the velocity fluctuations as shown below

3.5.1.2 External entropy sources

Solid and fluid phase sources

The external entropy sources, for both the solid s and the fluid w phases, are given by the expression

for l E

Jp-

(3.12)

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In the above entropy source, we see contributions from the external heat source, material derivative of temperature fluctuations, kinetic energy due to velocity fluc

-tuations, fluctuations of the sum of gravitational and chemical potential.

Interface entropy source

For the interface ws, the external entropy source is derived from similar quantities as those in the phase expressions, the difference is that the material derivatives in

this case, are restricted to the interface; i.e.

1

ews

D

's

K

fs

)

- Pws Dt rlws,O (3.13) In Rational thermodynamic [4-10, 20-23, 32,33] the relationships between entropy

flux and heat flux or the relationship between external entropy source and the e

x-ternal heat source are usually assumed to be simple [11]; viz. Eringen's assumptions and this is where temperature gets to be introduced into the system. Equations (3.9)-(3.12) are extensions of Eringen's assumptions.

3.6 Representation of other fluxes

In this project we focus our attention on the first order Taylor expansion or linear

combinations. In the linear Taylor representation of isotropic tensors (Truesdell and

Noll 1965) (see Jou et al. [24]), fluxes and forces of different tensorial rank do not

couple. The independence of the processes of different order tensors is often referred

to as the 'Curie principle'. We now collect the fluxes and forces in terms of their

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tensorial rank. The scalar fluxes are listed in the following set Fls

=

_{iwsM ➔iw ; M ; iws-tis ( O"s Cs 1 )

~

iws---tiw} - - : - . - -ns . ts . lls L..,, !VI Ps Js Ps l1ws,l1ws i where i

=

1, 2, ... , N l E

Jr,.

(3.14)

The set of scalar conjugate forces is

(3.15)

The following is the set which lists vector fluxes;

+cw PW gw

+

WT W

+ ( y

W,S

~

v

ws,s

)

t

iwivtw;

L

w:pt

t lEJp

+

L

(y

l

,

s

-2

V

ws,s

)

L

iwMil - v'(cws,ws(I - Gws)) - Ews pwsgws

lE.:Tv i

N

- Ews pws(I - Gws)v' Kjs -

L

E·,us pwsci,iis(I - Gws)v'(µiws

+

'l/Jiws)

_ 17ws(I _ Gws)v'ews;

J

D

}.

where i

=

1, 2, ... , N l E

J

35

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The vector conjugate forces are listed in the set

vws,s.

_,

_wher_e _i

₌

₁_,₂_,_.._._,_N _l_E_,J

(3.17)

The list of rank two tensor fluxes are

(3.18)

The rank two tensor forces are,

Cft (3.19)

3.6 .

1 Cross-coupled linear representation

Linear representations of cross-coupled fluxes have to satisfy equation (3.9) and r

e-spond to a particular desired physical system. 3.6.1.1 Rank two fluxes

In the rank two tensor fluxes, we have the stress tensors driven by strain rate tensors which contribute viscous or frictional effects:

= = = = tw

+

pwl

=

A11 : dw

+

A12: d8

+

A13: dws, (3.20) (3.21) and (3.22)

The coefficients Ai, i

=

1, 2, 3 are rank 4 tensors. These are partial derivatives of

the flux with respect to the strain rate tensor and then evaluated at equilibrium; e.g.

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at equilibrium. The other coefficients are determined in a similar manner.

3.6.1.2 Rank one fluxes

The rank one tensor fluxes are driven by temperature gradients which contribute Fourier heat conduction effects, relative velocities which contribute the drag force effect and chemical and gravitational gradients which contribute chemo-mechanical

effects: (i) Heat fluxes: N q

1 =

B

n

·

'701

+

B

1

2

·

vr,:s

+

LB/

_{3 ·}

'v

(

i

+v/1)

l

E

{

w,

ws

}

and Esqs - / (ts - as :

~

s

r)

·

(vs - V8))

= B

s1 · 'i.705 \ Js n.,n N

+

LB

~

2 .

'v

(/s

+

'1/Jis).

(ii) Mass fluxes:

N Jil = Bili· 'i.70

1 +

B

1 2i ·

v1'8

+

L

B

Si ·

'v

(µkl+ '1/Jkl) , k (3.23) (3.24) (3.25) (3.26)

where i

=

1, 2, ... , N and l E .:]. In the mass fluxes, '70s expresses the heat conduction effects, vr,:s express ultrafiltration or pore pressure gradients and

(µkl+ '1/Jkl) represent multicomponent diffusion [26]. (iii) Momentum exchange fluxes

N

Ew pw'v

K

j

+

L

Ew pwciw'v(µiw

+

'1/Jiw)

+

,,-/n'vew - 'v(Ewpw)

+

Ew pwgw

(47)

+

ws,;w

+

(

ylii,s

~

y

WS

,

S

)

t

iwx;rw

=

B41 . 'V

ew +

B42 . vw,s i N

+

L

B~3 . 'V (µkw

+

'1/)kw) ,(3.27) k and

L

wT l

+

L

(yl,s

~

Vws,s)

t

iwMil - 'V(Ews,ws(l - Gws)) - Ewspwsgws

lE:lp lE:lp i

N

-Ews pws(l - c ws)'V(Kfs) -

L

Ews pwsci,iis(l - c ws)'V(µiws

+

'lj;iw.s)

N

-17'1.US(l _ c ws)'i:Jews

=

B51 . 'i:Jews

+

B52 . vws,s

+

L

B~3 . 'i:J (µkws

+

'1/)kws) .

k

(3.28)

In the momentum exchange fluxes, v1,s represent viscous drag and 'V(µki

+

'lj;ki)

express chemo-osmotic effect and also depend on gravity. To simplify calcula -tions we let 'VCki

=

'V(µki

+

'l/Jki).

3.6.1.3 Rank zero fluxes

In the rank zero tensor fluxes are driven by the sum of the chemical and gravitational potentials and temperature differences.

(a) Homogeneous chemical reaction

r"

~

D

k

,

(

µ'';,

,p"'

)

+

D

f

₂

[

U<

~

+ µiw

+

,t/Jiw) _ (Kf

+

µiws

+

,t/Jiws)]

Di {

(K

s

is

.,ii

)

(

a

s

Cs

1 )

+

13 E

+

µ

+

'11

+

- :

-

.

-

_- _n_s_·_t_s_·_Il_s

Ps Js Ps S1ws,S1ws

- ( !(1

i5

+

_µiws

+

'lj;iws)

}

_wh_e_r_e

i

=

1, 2, ... , N, l E

J

(b) Heterogeneous chemical reaction

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and

(c) The heat exchange fluxes are driven by temperature difference, contributing Newton's cooling effects as follows:

'"'(,t +

E~w

t

'

'"Af""

+

v

"

• ',

[

wT w

+

v

"

• '

~

v"''•'

t

iw'iJ'"'

]

(3.29) and (3.30) ( c) Balance of normal stress at interface (3.31) 39

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Chapter 4 Thermo-chemo-mechanical closed

forms

4 .1

Background

In this chapter we will develop a theory on single-phase-fluid with species. Sev-eral authors have written on this subject, just to mention a few, Hassanizadeh [23],

looked at concentration on nuclear repository using Hybrid Mixture Theory which is based on Rational Thermodynamics. Kaczmarek looked at chemically sensitive soils under isothermal conditions. Malusis et. al. looked at semi-permeable membrane behavior basing his theory on non-equilibrium thermodynamics at macroscale level

(eg. landfills). Gray and Miller also looked at a similar model in one of their the r-modynamically constrained averaging theory papers [31]. Badder and Kooi looked at the theories of osmosis in ground water basing their theory on non-equilibrium thermodynamics at macroscale, all the above authors looked at the isothermal case. Extending on the above theories, our approach will differ as follows:

( a) We move from most general development to the specific by imposing restri c-tions. The future relaxation of the restrictions will allow for further develo

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ment of the theory.

(b) To develop the entropy inequality for our system we will average non-equilibrium thermodynamics (Classical Irreversible Thermodynamics), from microscale to

macrascale. In so doing we wish to capture the influence of micro-physics in macro-physics (see chapter 2 and 3). This approach was pioneered by Miller

and Gray in their TCAT papers.

( c) Due to the influences of the strain rate, velocity and the pressure on the flow we consider the modifications of Darcy's law.

(d) Several fluid-flow phenomena like osmosis are influenced by thermal conditions.

We consider non-isothermal and locally isothermal cases.

4.2 Cross coupling

Moving away from the most general we come to the more specific by bringing in restrictive assumptions. This will simplify the equations we will arrive at. Therefore

in this chapter we will be deducing the results for the flow of transport for single

fluid phase flow with species from the inequality ( equation 3.9) in chapter 3. We will be cross coupling thermal, chemical and mechanical fluxes. Before we do that here are the assumptions made to the system to suit the type of system we will be

modeling.

4.2.1 Secondary assumptions A

(i) Assume equation (3.9), entropy inequality form

(ii) o mass exchange between both the phases and chemical reactions within a

phase

(ii) Negligible velocity fluctuations (iv) No interfacial effects

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(v) Local and advective acceleration are negligible

(vi) Fluid is macroscopically viscous

(v

ii

)

System is isotropic

For example, landfill leacheage, clay soils [Malusis and Shackelford] [Badder and Kooi], fluid flow through nuclear disposal sites, etc.

4.2.2

Full

fluxes

From the entropy inequality (3.9), we have the following fluxes:

4.2.2.1 Simplification of fluxes

The fluxes that follow will be represented by cross-coupled forces so as to represent

various desired effects. For the fluid which is assumed to be macroscopically viscous,

the stress tensor has the following second order form

= =

tw

=

-pwl

+

A : dw.

( 4.1)

From the isotropic condition A

=

2kµI, where I is the fourth order unit tensor and kµ is a constant.

The heat fluxes are given by these first order tensor forms

(4.2)

(4.3)

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The momentum exchange flux is given by these first order tensor forms

- - - s~w =

c'u pw'v(µw

+

'lj/iJ)

+

rJw'vew _ 'v(cwpw)

+

cw pwg,--,;

+

T

=

Ri 'v0w

+

R₂

v

w

,s

N

+

L Bkl'vCkT (4.4)

Where R1 and R2 are proportionality second order tensors.

The heat exchange fluxes are given by these scalar forms

Q

= S1

---;;- - ---;;-

=

-

Q

s➔w ( 1 1 ) w➔s

ew es

k

4.3 Closed forms from the entropy inequality

(4.5)

To arrive at closed forms we first need to consider modifications of Darcy's law. These laws are influenced by permeability tensor, thermal conditions, osmotic effects and strain rate.

4 .3.1

Modified

Darc

y's

law

From equation (4.4) and Gibbs-Duhem relation

s-,w =

Ew'vpw _ E:wpw'v'l/Jw _ 'v(cwpw)

+

E:wpw

+

g

w+

T

=

Ri 'v0w _ R₂

v

w,s

N

+

L Bkl'vCkT (4.6)

k

Since

g

w

=

-

'v'l/Jw, then equation ( 4.6) reduces to

N

cw'vpw - 'v(cwpw)+ s-;t

=

R1 'v0w - R2vw,s

+

L Bkl'vckT (4.7) k

From the conservation of momentum equation (2.3), we have

(4.8)

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for the fluid and

(4.9)

for the solid. From equations ( 4.8) and ( 4.9), the expression for the total stress is therefore given by (4.10) From (4.1), (4.7) and (4.8), we have N

R

2 .

vill,s

=

2kµ''v . (~wdw) - E.w ( 'vpw - pwgw)

+

R1. vew

+

L

Bktv ckT

(4

.11)

k

The modified Darcy's law is thus given by

vw,s 2kµR21. V. (cwdw) - cwR21. ( 'vpw - pwgw) + R21R1. 'v0w N

+R;-1

L

Bktv ckT (4.12)

k

This expresses the fact that the fluid flow is influenced by fluid viscosity, pressure gradient and temperature gradient.

(a) For viscous fluids, equation (4.12) becomes this Brinkman type equation

E.WVW,S

=

2kµc.WR;-1 . 'v. (cwdw) _ (cw)2R;-l . ('vpw _ pwgw)

N

+cwR;-lR1. v ew + cwR;-1

L

BktvckT k

(4.13)

Rajagopal [32], derived a similar equation for the isothermal case, using Mixture

Theory.

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(b) Constant conductivity: For a certain lower range of flow rates and inviscid fluids, equation (4.12) becomes the standard Darcy's equation

€wVw,§

=

- K . (v'pw - pwgw)

+

EWR21R1. v1ew

+

N

€wR

₂

1

L

Bk1y1cH

k

(4.14)

where K

=

(c:w)2R₂1 defines the hydraulic conductivity (permeability) tensor. Since we are considering isotropic cases, then K

=

KI, where I is the second order unit tensor and J( is the hydraulic conductivity parameter.

(c) Velocity dependent conductivity: For inviscid fluids, higher flow rates and the case

where R2

=

(a+ b[vw,s[)I in the hydraulic conductivity tensor K as in (b) above,

equation ( 4.12) becomes this Forchheimer type equation

€'Wv'W,§

=

- K . (v'pw - pwgw)

+

c:WR21R1. v1ew

+

N

cwR;-1

L

Bkly1ckT.

k

(4.15) Bennethum and Giorgi [3], derived a similar equation for the isothermal case using Hybrid Mixture Theory.

( d) Pressure dependent conductivity: In this case of invicid fluids and where the hydraulic conductivity depends on the fluid pressure i.e.

where the parameter K(pw) is a scalar function of the fluid pressure, equation (4.12) becomes - J<(pw)(v'pw - pwgw)

+

cwR;-lR1. v'0w

+

N €wR;-l

L

Bkly1ckT_ k (4.16) Rojagopal [32], derived a similar equation in the isothermal case using Mixture The

-ory. Because of the permeability, equation ( 4.13) can be classified into constant,

velocity dependent and pressure dependent permeability. Equations ( 4.13) - ( 4.16)

Thermo-chemo-mechanical coupling in single-fluid flows through porous media

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