Multiscale Modelling in Porous Media

**Chapter 1**

**Abstract**

**1. Introduction**

**3**

Interfaces

*Paul Papatzacos*

Porous Flow with Diffuse

This chapter presents a model developed by the author, in publications dated from 2002 to 2016, on flow in porous media assuming diffuse interfaces. It contains five sections. Section 1 is an Introduction, tracing the origin of the diffuse interface formalism. Section 1 also presents the traditional compositional model, pointing out its emphasis on phases and questioning the concept of relative permeabilities. Section 2 presents the mass, momentum, and energy balance equations, for a multicomponent continuous fluid, in their most general form, at the pore level. The existence of constitutive equations with phase-inducing terms is mentioned, but the equations are not introduced at this level, and phases are not an explicit concern. Section 3 is about the averaging of the pore level equations inside a region

containing many pores. There is no explicit mention of phases and therefore not of relative permeabilities. Section 4 is the technical basis from which the constitutive equations of the model arise, and it is shown that many models can exist. Section 5

multicomponent, multiphase, and thermal flow in neutrally wetting porous media,

The model presented in this chapter was developed in Refs. [1–3]. Ref. [1] covers

The diffuse interface theory was initiated in 1893 in a paper by van der Waals

(see the translation by Rowlinson [5]) where he proposes to replace the old assumption of a surface between phases by the assumption of a continuous transition inside a thin interphase region, where certain quantities, notably the density, vary continuously. The core of his theory consists of a Helmholtz function modified by the addition of a term proportional to the squared gradient of the density, thus accounting for the energy stored in the region. *In the model presented in the following*

the one-component two-phase case, Ref. [2] is a generalisation to an arbitrary number of components and three phases (two liquids, one gas), Ref. [3] generalises to variable temperature. All three show applications to neutrally wetting media. The way wetting can be accounted for is discussed in general terms in Ref. [1]. A partial practical implementation, valid for incomplete wetting, is suggested in Ref. [4] but a fully satisfactory solution that accounts for total wetting (or capillary condensa-

introduces constitutive equations and presents a *minimal model* for

i.e., a model with a minimal amount of phenomenological parameters.

**Keywords:** flow in porous media, Marle averaging, diffuse interface,

multiphase flow, phase segregation, relative permeabilities

tion in adsorption terms) is still a matter for further research.

#### **Chapter 1**

## Porous Flow with Diffuse Interfaces

*Paul Papatzacos*

#### **Abstract**

This chapter presents a model developed by the author, in publications dated from 2002 to 2016, on flow in porous media assuming diffuse interfaces. It contains five sections. Section 1 is an Introduction, tracing the origin of the diffuse interface formalism. Section 1 also presents the traditional compositional model, pointing out its emphasis on phases and questioning the concept of relative permeabilities. Section 2 presents the mass, momentum, and energy balance equations, for a multicomponent continuous fluid, in their most general form, at the pore level. The existence of constitutive equations with phase-inducing terms is mentioned, but the equations are not introduced at this level, and phases are not an explicit concern. Section 3 is about the averaging of the pore level equations inside a region containing many pores. There is no explicit mention of phases and therefore not of relative permeabilities. Section 4 is the technical basis from which the constitutive equations of the model arise, and it is shown that many models can exist. Section 5 introduces constitutive equations and presents a *minimal model* for multicomponent, multiphase, and thermal flow in neutrally wetting porous media, i.e., a model with a minimal amount of phenomenological parameters.

**Keywords:** flow in porous media, Marle averaging, diffuse interface, multiphase flow, phase segregation, relative permeabilities

#### **1. Introduction**

The model presented in this chapter was developed in Refs. [1–3]. Ref. [1] covers the one-component two-phase case, Ref. [2] is a generalisation to an arbitrary number of components and three phases (two liquids, one gas), Ref. [3] generalises to variable temperature. All three show applications to neutrally wetting media. The way wetting can be accounted for is discussed in general terms in Ref. [1]. A partial practical implementation, valid for incomplete wetting, is suggested in Ref. [4] but a fully satisfactory solution that accounts for total wetting (or capillary condensation in adsorption terms) is still a matter for further research.

The diffuse interface theory was initiated in 1893 in a paper by van der Waals (see the translation by Rowlinson [5]) where he proposes to replace the old assumption of a surface between phases by the assumption of a continuous transition inside a thin interphase region, where certain quantities, notably the density, vary continuously. The core of his theory consists of a Helmholtz function modified by the addition of a term proportional to the squared gradient of the density, thus accounting for the energy stored in the region. *In the model presented in the following* *pages, the van der Waals theory is introduced at the upscaled level.* See Section 4. Generalising the van der Waals expression to *ν* chemical components [2], one obtains

$$\mathcal{F}^{\mathrm{F}} = \mathcal{F}^{\mathrm{bF}} + \sum\_{a=1}^{\nu} \frac{\mathbf{1}}{2} \Lambda^{a} |\nabla \mathcal{R}^{a}|^{2},\tag{1}$$

thermodynamical description of the fluid mixture involved is part of the core. The central purpose is to calculate the component densities, and other characteristic quantities such as fluid velocity and fluid temperature, as functions of space and time. If the approximation of constant temperature is valid, only the mass and

*Phases (and thereby relative permeabilities) do not take part in the formulation. They result from the solutions of the model equations, and are detected by rapid variations of densities, and by regions of approximately uniform densities.* They can be shown to

A *minimal model* is presented in Section 5, consisting of a minimal amount of

**A note on the appendices:** Some concepts are grouped in appendices for easy reference. Appendix 1, for example, lists all the assumptions the model is built on. **A note on wetting:** The problem of accounting for the wetting properties of the

: see Appendix 1, Assumption A0. Refs.

pore surface remains to be solved. There are two approaches to the problem: through boundary conditions to the Navier–Stokes equations at pore level, or through the theory of adsorption at the upscaled Darcy level. The first approach has been used in the publications considered here, and it is explained in Ref. [10] that the diffuse interface theory presented in Ref. [1] is consistent for neutral wetting,

**A note on notation:** Right-handed Cartesian coordinates ð Þ *x*1, *x*2, *x*<sup>3</sup> are assumed, the plane ð Þ *x*1, *x*<sup>2</sup> being horizontal, and axis *x*<sup>3</sup> pointing upwards. Any vector **<sup>A</sup>** has components *Ak* where *<sup>k</sup>* is 1, 2, or 3. In addition, the notation *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>=∂<sup>t</sup>* and *<sup>∂</sup><sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>=∂xk* is used. The summation convention applies to latin indexes *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>*, *<sup>l</sup>*: an index that is repeated in a term (as in *A jB <sup>j</sup>*, or *Ckk*) indicates summation

term (as in *Ai* ¼ *Bi*), means that the expression is valid for all values of the index in

The fluid is a mixture of *ν* chemical components, it is continuous, and if phases exist, there are interphase regions where quantities vary continuously, possibly rapidly. With assumptions A1 to A3 (Appendix 1) the balance equations for mass,

the set 1, 2, 3 f g. Symbols are otherwise defined when introduced.

momentum, and energy, are written below in a most general manner:

*<sup>∂</sup>t<sup>ε</sup>* <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ε</sup>vk* <sup>þ</sup> *<sup>j</sup>*

*α k*

*<sup>∂</sup>t*ð Þþ *<sup>ρ</sup>vi <sup>∂</sup>k*ð Þ¼ *<sup>ρ</sup>vivk* � *tki <sup>ρ</sup> fi*

For the theoretical basis of these equations see chapter 11 of the book by Hirschfelder *et al* [11]. Greek superscripts indicate the species so that *ρ<sup>α</sup>* is the mass

> *<sup>ρ</sup>* <sup>¼</sup> <sup>X</sup>*<sup>ν</sup> α*¼1

Further, **v** is the local velocity, defined as the total momentum divided by the

*<sup>k</sup>* � *tkivi*

*<sup>∂</sup>tρ<sup>α</sup>* <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup><sup>α</sup>vk* <sup>þ</sup> *<sup>i</sup>*

density of component *α*, while *ρ* is the total density:

*<sup>k</sup>*¼<sup>1</sup>*Ckk*). Note that an index that is repeated, but not in the same

� � <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> 1, … , *<sup>ν</sup>* , (3)

� � <sup>¼</sup> *<sup>ρ</sup> <sup>f</sup> <sup>k</sup>vk:* (5)

*<sup>α</sup>* is the non-convective mass current of component *α*,

*ρ<sup>α</sup>:* (6)

, (4)

exist in static equilibrium or steady state dynamical situations [1–3].

momentum balance equations are necessary.

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

*Porous Flow with Diffuse Interfaces*

i.e., for pore level wetting angles around 90<sup>∘</sup>

[2, 3] assume this limitation.

*<sup>j</sup>*¼<sup>1</sup>*<sup>A</sup> jB <sup>j</sup>*, or <sup>P</sup><sup>3</sup>

**2. Pore level equations**

total mass, both per volume; *ι*

with the property

**5**

parameters.

( P<sup>3</sup>

where *<sup>R</sup><sup>α</sup>* is the upscaled density of fluid component *<sup>α</sup>*, <sup>F</sup> is the Helmholtz function per unit volume, the superscript F referring to the fluid, while bF means "bulk fluid" and refers to fluid regions that are far away from interphase layers. The <sup>Λ</sup>*<sup>α</sup>* are constants (Assumption A7, Appendix 1). For later reference <sup>F</sup>bF and its differential are

$$\mathcal{F}^{\text{bF}} = -P^{\text{bF}} + \sum\_{a=1}^{\nu} \mathcal{C}^{\text{bF}a} \mathcal{R}^a, \qquad \mathbf{d} \mathcal{F}^{\text{bF}} = -\mathcal{S}^{\text{bF}} \mathbf{d}T + \sum\_{a=1}^{\nu} \mathcal{C}^{\text{bF}a} \mathbf{d} \mathcal{R}^a,\tag{2}$$

where *<sup>P</sup>*bF is the pressure in the bulk fluid, <sup>S</sup>bF the entropy per unit volume, and <sup>C</sup>bF*<sup>α</sup>* the chemical potential of component *<sup>α</sup>*, divided by its molar mass (the *<sup>C</sup>α=M<sup>α</sup>* of Refs. [2, 3]).

The van der Waals paper was followed, in 1901, by a paper by Korteweg [6] about the equations of motion of a fluid with large but continuous density changes, where Korteweg showed that, for such a fluid, the usual scalar pressure must be replaced by a symmetric second order tensor.

The van der Waals-Korteweg papers were apparently forgotten, then rediscovered in the nineteen seventies, the diffuse-interface method being introduced as a novel way to solve fluid mechanics problems in two-phase flow. For a review, see [7]. While van der Waals and Korteweg assumed that the gradients in Eq. (1) are small, modern advances have shown that this limitation can be lifted [7, 8].

Ref. [8] is, to the author's knowledge, the first formulation of flow in porous media with diffuse interfaces.<sup>1</sup> The purpose with such a novel formulation is to avoid some of the known weaknesses of the traditional compositional models treating multiple phase problems. The mathematical core of the compositional models used in reservoir engineering, consists of equations expressing mass balance for each chemical component. The distribution of the phases in the reservoir is essential to the formulation, and is determined at the beginning of each time step. A component, present in a phase, is transported with the Darcy velocity of the phase. Each phase-dependent Darcy velocity is written with a permeability that is modified by a multiplicative factor, the relative permeability. The above description emphasises two of the weaknesses of the traditional models. The first is the assumption that well-defined phases exist at all times. The second is the use of relative permeabilities, a concept that is "seriously questioned", as expressed by Adler and Brenner in a 1988 paper [9]. Concerning the understanding of relative permeabilities in the framework of the model presented here, see the comments following Eq. (100) below.

The mathematical core of the model (actually a family of models) to be presented consists of mass balance equations, one per chemical component, of a momentum balance equation, and of an entropy balance equation. The

<sup>1</sup> The author first became aware of this paper in 2015. There is one important difference with what is presented in the pages that follow. See the footnote in Section 4.

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

*pages, the van der Waals theory is introduced at the upscaled level.* See Section 4. Generalising the van der Waals expression to *ν* chemical components [2], one

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*α*¼1

where *<sup>R</sup><sup>α</sup>* is the upscaled density of fluid component *<sup>α</sup>*, <sup>F</sup> is the Helmholtz function per unit volume, the superscript F referring to the fluid, while bF means "bulk fluid" and refers to fluid regions that are far away from interphase layers. The <sup>Λ</sup>*<sup>α</sup>* are constants (Assumption A7, Appendix 1). For later reference <sup>F</sup>bF and its

1 2

<sup>C</sup>bF*<sup>α</sup>Rα*, dFbF ¼ �SbFd*<sup>T</sup>* <sup>þ</sup>X*<sup>ν</sup>*

where *<sup>P</sup>*bF is the pressure in the bulk fluid, <sup>S</sup>bF the entropy per unit volume, and <sup>C</sup>bF*<sup>α</sup>* the chemical potential of component *<sup>α</sup>*, divided by its molar mass (the *<sup>C</sup>α=M<sup>α</sup>*

The van der Waals paper was followed, in 1901, by a paper by Korteweg [6] about the equations of motion of a fluid with large but continuous density changes, where Korteweg showed that, for such a fluid, the usual scalar pressure must be

The van der Waals-Korteweg papers were apparently forgotten, then rediscovered in the nineteen seventies, the diffuse-interface method being introduced as a novel way to solve fluid mechanics problems in two-phase flow. For a review, see [7]. While van der Waals and Korteweg assumed that the gradients in

Eq. (1) are small, modern advances have shown that this limitation can be

Ref. [8] is, to the author's knowledge, the first formulation of flow in porous media with diffuse interfaces.<sup>1</sup> The purpose with such a novel formulation is to avoid some of the known weaknesses of the traditional compositional models treating multiple phase problems. The mathematical core of the compositional models used in reservoir engineering, consists of equations expressing mass balance for each chemical component. The distribution of the phases in the reservoir is essential to the formulation, and is determined at the beginning of each time step. A component, present in a phase, is transported with the Darcy velocity of the phase. Each phase-dependent Darcy velocity is written with a permeability that is modified by a multiplicative factor, the relative permeability. The above description emphasises two of the weaknesses of the traditional models. The first is the assumption that well-defined phases exist at all times. The second is the use of relative permeabilities, a concept that is "seriously questioned", as expressed by Adler and Brenner in a 1988 paper [9]. Concerning the understanding of relative permeabilities in the framework of the model presented here, see the comments

The mathematical core of the model (actually a family of models) to be presented consists of mass balance equations, one per chemical component, of a

<sup>1</sup> The author first became aware of this paper in 2015. There is one important difference with what is

momentum balance equation, and of an entropy balance equation. The

presented in the pages that follow. See the footnote in Section 4.

<sup>Λ</sup>*<sup>α</sup>* **<sup>∇</sup>***R<sup>α</sup>* j j<sup>2</sup>

, (1)

<sup>C</sup>bF*<sup>α</sup>*d*Rα*, (2)

*α*¼1

<sup>F</sup><sup>F</sup> ¼ FbF <sup>þ</sup>X*<sup>ν</sup>*

obtains

differential are

of Refs. [2, 3]).

lifted [7, 8].

following Eq. (100) below.

**4**

<sup>F</sup>bF ¼ �*P*bF <sup>þ</sup>X*<sup>ν</sup>*

*α*¼1

replaced by a symmetric second order tensor.

thermodynamical description of the fluid mixture involved is part of the core. The central purpose is to calculate the component densities, and other characteristic quantities such as fluid velocity and fluid temperature, as functions of space and time. If the approximation of constant temperature is valid, only the mass and momentum balance equations are necessary.

*Phases (and thereby relative permeabilities) do not take part in the formulation. They result from the solutions of the model equations, and are detected by rapid variations of densities, and by regions of approximately uniform densities.* They can be shown to exist in static equilibrium or steady state dynamical situations [1–3].

A *minimal model* is presented in Section 5, consisting of a minimal amount of parameters.

**A note on the appendices:** Some concepts are grouped in appendices for easy reference. Appendix 1, for example, lists all the assumptions the model is built on.

**A note on wetting:** The problem of accounting for the wetting properties of the pore surface remains to be solved. There are two approaches to the problem: through boundary conditions to the Navier–Stokes equations at pore level, or through the theory of adsorption at the upscaled Darcy level. The first approach has been used in the publications considered here, and it is explained in Ref. [10] that the diffuse interface theory presented in Ref. [1] is consistent for neutral wetting, i.e., for pore level wetting angles around 90<sup>∘</sup> : see Appendix 1, Assumption A0. Refs. [2, 3] assume this limitation.

**A note on notation:** Right-handed Cartesian coordinates ð Þ *x*1, *x*2, *x*<sup>3</sup> are assumed, the plane ð Þ *x*1, *x*<sup>2</sup> being horizontal, and axis *x*<sup>3</sup> pointing upwards. Any vector **<sup>A</sup>** has components *Ak* where *<sup>k</sup>* is 1, 2, or 3. In addition, the notation *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>=∂<sup>t</sup>* and *<sup>∂</sup><sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>=∂xk* is used. The summation convention applies to latin indexes *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>*, *<sup>l</sup>*: an index that is repeated in a term (as in *A jB <sup>j</sup>*, or *Ckk*) indicates summation ( P<sup>3</sup> *<sup>j</sup>*¼<sup>1</sup>*<sup>A</sup> jB <sup>j</sup>*, or <sup>P</sup><sup>3</sup> *<sup>k</sup>*¼<sup>1</sup>*Ckk*). Note that an index that is repeated, but not in the same term (as in *Ai* ¼ *Bi*), means that the expression is valid for all values of the index in the set 1, 2, 3 f g. Symbols are otherwise defined when introduced.

#### **2. Pore level equations**

The fluid is a mixture of *ν* chemical components, it is continuous, and if phases exist, there are interphase regions where quantities vary continuously, possibly rapidly. With assumptions A1 to A3 (Appendix 1) the balance equations for mass, momentum, and energy, are written below in a most general manner:

$$
\partial\_t \rho^a + \partial\_k \left(\rho^a \upsilon\_k + i\_k^a\right) = 0, \qquad (a = 1, \dots, \nu), \tag{3}
$$

$$
\partial\_t(\rho v\_i) + \partial\_k(\rho v\_i v\_k - t\_{ki}) = \rho f\_i,\tag{4}
$$

$$
\partial\_t \varepsilon + \partial\_k \left( \varepsilon v\_k + j\_k - t\_{ki} v\_i \right) = \rho f\_{\,k} v\_k. \tag{5}
$$

For the theoretical basis of these equations see chapter 11 of the book by Hirschfelder *et al* [11]. Greek superscripts indicate the species so that *ρ<sup>α</sup>* is the mass density of component *α*, while *ρ* is the total density:

$$
\rho = \sum\_{a=1}^{\nu} \rho^a. \tag{6}
$$

Further, **v** is the local velocity, defined as the total momentum divided by the total mass, both per volume; *ι <sup>α</sup>* is the non-convective mass current of component *α*, with the property

$$\sum\_{a=1}^{\nu} \mathbf{r}^a = \mathbf{0},\tag{7}$$

equal to 0 on the pore surface (denoted by Σ) and in the solid; likewise, any quantity with superscript S is equal to 0 on Σ and in the fluid. The quantities in the

expressing that the fluid velocity vanishes on <sup>Σ</sup>. Expressions of the type ð Þ *<sup>X</sup>* <sup>Σ</sup> for some *X*, used in this section and in the next, define the limit of the quantity *X* at Σ, along the line carrying the normal to Σ pointing towards the fluid. The existence of

As a preliminary to upscaling, Eqs. (3)–(5) are now explicitly written in terms of distributions, using the notation of Appendix 2 where *A* is replaced by F*α* or F, as appropriate, while *<sup>B</sup>* is replaced by S. As an example, *ρα*ð Þ **<sup>x</sup>**, *<sup>t</sup>* , is a distribution depending on space **x** and time *t*. It is continuous in time but not in space, being equal to *<sup>ρ</sup>*<sup>F</sup>*<sup>α</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* if **<sup>x</sup>** is inside a pore and to *<sup>ρ</sup>*<sup>S</sup>ð Þ **<sup>x</sup>**, *<sup>t</sup>* if **<sup>x</sup>** is in the rock. It is not defined

*<sup>ρ</sup>*<sup>F</sup> <sup>¼</sup> <sup>X</sup>*<sup>ν</sup> α*¼1

The generalised mass balance equation is obtained from Eq. (3), using

*<sup>α</sup>* ð Þ � **<sup>n</sup>**

From this equation one now gets, using **Table 1**, the equations that are sepa-

Turning to the generalised momentum balance equation, one must account for Assumption A4 (Appendix 1), implying that the momentum balance equations in the solid and on the pore surface have the form **0** ¼ **0**. Inside the pores, one simply

**<sup>v</sup>**<sup>F</sup> � �<sup>Σ</sup> <sup>¼</sup> **<sup>0</sup>**, (11)

*ρ*F*<sup>α</sup>:* (12)

<sup>Σ</sup> h i*δ*<sup>Σ</sup> <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> 1, … , *<sup>ν</sup> :* (13)

*<sup>∂</sup>tρ*<sup>S</sup> <sup>¼</sup> <sup>0</sup>*:* ð Þ in *<sup>S</sup>* (16)

, inð Þ *F :* (17)

(18)

*<sup>k</sup>* � *tkivi*

� � <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> 1, … , *<sup>ν</sup>* , inð Þ *<sup>F</sup>* (14)

<sup>F</sup>*<sup>α</sup>* � **<sup>n</sup>** � �<sup>Σ</sup> h i*δ*<sup>Σ</sup> <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> 1, … , *<sup>ν</sup>* , on ð Þ <sup>Σ</sup> (15)

first line of the table are not defined on Σ, but limit values are, as in

the normal implies some idealisation of the pore surface.

when **x** is on Σ but we assume that *ρ*<sup>F</sup>*<sup>α</sup>* � �<sup>Σ</sup> and *ρ*<sup>S</sup> � �<sup>Σ</sup> exist.

*α k* � � � � <sup>þ</sup> *<sup>ι</sup>*

> *<sup>k</sup>* þ *i* F*α k*

re-writes Eq. (4), with the superscript F on the density and velocity:

<sup>S</sup> � **<sup>n</sup>** � �<sup>Σ</sup> h i*δ*<sup>Σ</sup> <sup>¼</sup> *<sup>f</sup> <sup>k</sup>ρvk:*

*k v*F *<sup>i</sup>* � *tki* � � <sup>¼</sup> *<sup>ρ</sup>*<sup>F</sup> *fi*

The generalised energy balance equation is obtained from Eq. (5), using

<sup>þ</sup> *<sup>∂</sup><sup>k</sup> uvk* <sup>þ</sup>

1 2 *ρ*j j **v** 2 *vk* þ *j*

� � � �

rately valid inside the pores, on Σ, and in the solid:

<sup>3</sup>*∂tρ*<sup>F</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup>*<sup>F</sup>*αv*<sup>F</sup>

*∂<sup>t</sup> ρ*<sup>F</sup>*v*<sup>F</sup> *i* � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup>*<sup>F</sup>*v*<sup>F</sup>

> 1 2 *ρ*j j **v** 2

� � � �

<sup>F</sup> � **<sup>n</sup>** � �<sup>Σ</sup> � *<sup>J</sup>*

Eqs. (112), (113), and (11):

*<sup>∂</sup><sup>t</sup> <sup>u</sup>* <sup>þ</sup>

þ *J*

**7**

*ι*

Note finally that, according to Eq. (6),

*Porous Flow with Diffuse Interfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

Eqs. (112), (113), and (11):

*<sup>∂</sup>tρ<sup>α</sup>* f g <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup><sup>α</sup>vk* <sup>þ</sup> *<sup>i</sup>*

*<sup>J</sup>* is the non-convective energy current, and *tij* is the stress tensor; *<sup>ε</sup>* <sup>¼</sup> *<sup>u</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> *ρv*<sup>2</sup> is the total energy, *u* is the internal energy, both per unit volume, and **f** is an external force per unit mass of fluid, thus being species-independent. In the case of gravity

$$\mathbf{f} = \mathbf{g}, \quad \mathbf{g} = (0, 0, -\mathbf{g}), \quad \mathbf{g} = 9.81 \text{ m/s}^2. \tag{8}$$

It is convenient, for later use, to introduce the gravitational potential

$$W = -\mathbf{g} \cdot \mathbf{x}, \quad \text{whereby} \quad f\_i = -\partial\_i W. \tag{9}$$

The symmetry *tij* ¼ *tji* is assumed below. It is reminded that it expresses angular momentum balance for a fluid where no other torques exist than the one due to the external force **f**, and the one due the surface stress **n** � **t** acting on every surface element where the normal vector is **n**.

Within the van der Waals theory, one expects that *ι α <sup>k</sup>*, *Jk*, and *tki*, contain terms whose magnitudes are important in the interphase regions, but are otherwise negligible. No constitutive equations are introduced at the pore level, but it is mentioned for later reference, that for a simple one-phase fluid,

$$t\_{\vec{\eta}} = -p\delta\_{\vec{\eta}} + \theta\_{\vec{\eta}},\tag{10}$$

where *p* is the pressure and *θij* is the viscous stress tensor (symmetric and linear in the gradients of the *vi*, with coefficients of shear and bulk viscosity).

The upscaling, i.e., the averaging over many pores, is done in the next section by the method due to Marle [12]. This method assumes that the physical quantities that appear in the balance equations above are treated as distributions [12, 13], the underlying reason being that such quantities are discontinuous, and that one needs to average their partial derivatives. Taking *ρ* as an example, it is (i) a fluid density in a pore, (ii) a rock density in the rock matrix, (iii) undefined on the pore surface, and one needs to average *∂tρ* and *∂kρ*.

The physical quantities appearing in Eqs. (3) to (5) are listed in the first line of **Table 1**. Their values in the pores, or in the rock, are denoted with a superscript F, or S, as shown in the second and third lines of the table. In the third line, a missing entry indicates non-existence, the first two 0-values indicate no material transport in the solid, the third zero value follows from Assumption A4 (Appendix 1): as shown by Marle [12], this assumption implies that the momentum balance equations in the solid and on the pore surface have the form **0** ¼ **0**. It is important to keep in mind, especially when averaging, that any quantity with an F superscript is


**Table 1.**

*First line, left of the vertical: quantities appearing in Eqs. (3) and (4); right of the vertical: quantities only appearing in Eq. (5). Second and third lines: notation when specialising to the fluid and the solid. Concerning the missing entries and the three 0-values, see beginning of paragraph containing Eq. (11).*

X*ν α*¼1 *ι*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*<sup>J</sup>* is the non-convective energy current, and *tij* is the stress tensor; *<sup>ε</sup>* <sup>¼</sup> *<sup>u</sup>* <sup>þ</sup> <sup>1</sup>

the total energy, *u* is the internal energy, both per unit volume, and **f** is an external force per unit mass of fluid, thus being species-independent. In the case of gravity

**f** ¼ **g**, **g** ¼ ð Þ 0, 0, �*g* , *g* ¼ 9*:*81 m*=*s

The symmetry *tij* ¼ *tji* is assumed below. It is reminded that it expresses angular momentum balance for a fluid where no other torques exist than the one due to the external force **f**, and the one due the surface stress **n** � **t** acting on every surface

whose magnitudes are important in the interphase regions, but are otherwise negligible. No constitutive equations are introduced at the pore level, but it is mentioned

where *p* is the pressure and *θij* is the viscous stress tensor (symmetric and linear

The upscaling, i.e., the averaging over many pores, is done in the next section by the method due to Marle [12]. This method assumes that the physical quantities that appear in the balance equations above are treated as distributions [12, 13], the underlying reason being that such quantities are discontinuous, and that one needs to average their partial derivatives. Taking *ρ* as an example, it is (i) a fluid density in a pore, (ii) a rock density in the rock matrix, (iii) undefined on the pore surface,

The physical quantities appearing in Eqs. (3) to (5) are listed in the first line of **Table 1**. Their values in the pores, or in the rock, are denoted with a superscript F, or S, as shown in the second and third lines of the table. In the third line, a missing entry indicates non-existence, the first two 0-values indicate no material transport in the solid, the third zero value follows from Assumption A4 (Appendix 1): as shown by Marle [12], this assumption implies that the momentum balance equations in the solid and on the pore surface have the form **0** ¼ **0**. It is important to keep in mind, especially when averaging, that any quantity with an F superscript is

*α*

*First line, left of the vertical: quantities appearing in Eqs. (3) and (4); right of the vertical: quantities only appearing in Eq. (5). Second and third lines: notation when specialising to the fluid and the solid. Concerning*

*<sup>k</sup> ι* F*α*

*the missing entries and the three 0-values, see beginning of paragraph containing Eq. (11).*

In solid *ρ*<sup>S</sup> 00 0 *u*<sup>S</sup> *j*

in the gradients of the *vi*, with coefficients of shear and bulk viscosity).

It is convenient, for later use, to introduce the gravitational potential

element where the normal vector is **n**.

and one needs to average *∂tρ* and *∂kρ*.

**Generic** *ρ<sup>α</sup> ρ vk ι*

In fluid *ρ*<sup>F</sup>*<sup>α</sup> ρ*<sup>F</sup> *v*<sup>F</sup>

**Table 1.**

**6**

Within the van der Waals theory, one expects that *ι*

for later reference, that for a simple one-phase fluid,

*<sup>α</sup>* <sup>¼</sup> **<sup>0</sup>**, (7)

2

*<sup>W</sup>* ¼ �**<sup>g</sup>** � **<sup>x</sup>**, whereby *fi* ¼ �*∂iW:* (9)

*α*

*tij* ¼ �*pδij* þ *θij*, (10)

*<sup>k</sup> t*ki *f <sup>k</sup> u jk*

*<sup>k</sup> tki f <sup>k</sup> u*<sup>F</sup> *j*

F *k*

S *k*

*:* (8)

*<sup>k</sup>*, *Jk*, and *tki*, contain terms

<sup>2</sup> *ρv*<sup>2</sup> is

equal to 0 on the pore surface (denoted by Σ) and in the solid; likewise, any quantity with superscript S is equal to 0 on Σ and in the fluid. The quantities in the first line of the table are not defined on Σ, but limit values are, as in

$$\left(\mathbf{v}^{\mathrm{F}}\right)^{\Sigma} = \mathbf{0},\tag{11}$$

expressing that the fluid velocity vanishes on <sup>Σ</sup>. Expressions of the type ð Þ *<sup>X</sup>* <sup>Σ</sup> for some *X*, used in this section and in the next, define the limit of the quantity *X* at Σ, along the line carrying the normal to Σ pointing towards the fluid. The existence of the normal implies some idealisation of the pore surface.

Note finally that, according to Eq. (6),

$$
\rho^{\mathrm{F}} = \sum\_{a=1}^{\nu} \rho^{\mathrm{F}a}.\tag{12}
$$

As a preliminary to upscaling, Eqs. (3)–(5) are now explicitly written in terms of distributions, using the notation of Appendix 2 where *A* is replaced by F*α* or F, as appropriate, while *<sup>B</sup>* is replaced by S. As an example, *ρα*ð Þ **<sup>x</sup>**, *<sup>t</sup>* , is a distribution depending on space **x** and time *t*. It is continuous in time but not in space, being equal to *<sup>ρ</sup>*<sup>F</sup>*<sup>α</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* if **<sup>x</sup>** is inside a pore and to *<sup>ρ</sup>*<sup>S</sup>ð Þ **<sup>x</sup>**, *<sup>t</sup>* if **<sup>x</sup>** is in the rock. It is not defined when **x** is on Σ but we assume that *ρ*<sup>F</sup>*<sup>α</sup>* � �<sup>Σ</sup> and *ρ*<sup>S</sup> � �<sup>Σ</sup> exist.

The generalised mass balance equation is obtained from Eq. (3), using Eqs. (112), (113), and (11):

$$\left\{\partial\_{\mathbf{t}}\rho^{a}\right\} + \left\{\partial\_{\mathbf{k}}\left(\rho^{a}v\_{\mathbf{k}} + \mathbf{i}\_{\mathbf{k}}^{a}\right)\right\} + \left[\left(\mathbf{l}^{a}\cdot\mathbf{n}\right)^{\Sigma}\right]\delta\_{\Sigma} = \mathbf{0}, \qquad (a = \mathbf{1}, \ldots, \nu). \tag{13}$$

From this equation one now gets, using **Table 1**, the equations that are separately valid inside the pores, on Σ, and in the solid:

$$\partial \partial\_t \rho^{\rm Fe} + \partial\_k \left( \rho^{\rm Fe} v\_k^{\rm F} + i\_k^{\rm Fe} \right) = 0, \quad (a = 1, \dots, \nu), \quad (\text{in } F) \tag{14}$$

$$\left[\left(\mathbf{t}^{\mathrm{Fa}} \cdot \mathbf{n}\right)^{\Sigma}\right] \delta\_{\Sigma} = \mathbf{0}, \qquad (a = \mathbf{1}, \ldots, \nu), \quad (\text{on } \Sigma) \tag{15}$$

$$
\partial\_t \rho^\mathcal{S} = \mathbf{0}. \quad \text{(in } \mathcal{S}\text{)}\tag{16}
$$

Turning to the generalised momentum balance equation, one must account for Assumption A4 (Appendix 1), implying that the momentum balance equations in the solid and on the pore surface have the form **0** ¼ **0**. Inside the pores, one simply re-writes Eq. (4), with the superscript F on the density and velocity:

$$
\partial\_t \left( \rho^\mathbf{F} v\_i^\mathbf{F} \right) + \partial\_k \left( \rho^\mathbf{F} v\_k^\mathbf{F} v\_i^\mathbf{F} - t\_{ki} \right) = \rho^\mathbf{F} f\_i, \quad \text{(in } F\text{)}. \tag{17}
$$

The generalised energy balance equation is obtained from Eq. (5), using Eqs. (112), (113), and (11):

$$\begin{split} \left\{ \partial\_{t} \left( u + \frac{1}{2} \rho |\mathbf{v}|^{2} \right) \right\} &+ \left\{ \partial\_{k} \left( u \nu\_{k} + \frac{1}{2} \rho |\mathbf{v}|^{2} \nu\_{k} + j\_{k} - t\_{ki} \nu\_{i} \right) \right\} \\ &+ \left[ \left( \mathbf{J}^{\mathrm{F}} \cdot \mathbf{n} \right)^{\Sigma} - \left( \mathbf{J}^{\mathrm{S}} \cdot \mathbf{n} \right)^{\Sigma} \right] \delta\_{\Sigma} = f\_{k} \rho \nu\_{k} . \end{split} \tag{18}$$

Using **Table 1**, one then obtains:

$$\partial\_t \left( u^{\mathbf{F}} + \frac{\mathbf{1}}{2} \boldsymbol{\rho}^{\mathbf{F}} \left| \mathbf{v}^{\mathbf{F}} \right|^2 \right) + \partial\_k \left( u^{\mathbf{F}} v\_k^{\mathbf{F}} + \frac{\mathbf{1}}{2} \boldsymbol{\rho}^{\mathbf{F}} \left| \mathbf{v}^{\mathbf{F}} \right|^2 v\_k^{\mathbf{F}} + f\_k^{\mathbf{F}} - t\_{ki} v\_i^{\mathbf{F}} \right) = f\_k \boldsymbol{\rho}^{\mathbf{F}} v\_k^{\mathbf{F}}, \quad \text{(in } \mathbf{F} \text{)}$$

$$\bf{(19)}$$

*∂<sup>t</sup> ρ*F*v*<sup>F</sup> *i*

Then

denoted *I*

defined by

**9**

*α*

Σ). It is not defined as the average of *ι*

*ρ*<sup>F</sup> *fi*

� � <sup>∗</sup> *<sup>m</sup>* � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup>*F*v*<sup>F</sup>

*Porous Flow with Diffuse Interfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

*k v*F

convolution operations in the equations above.

component *α* adsorbed at the pore surface:

*<sup>i</sup>* � *tki* � � <sup>∗</sup> *<sup>m</sup>* � � ¼ � ð Þ *nktki*

Note that the last term on the right-hand side of the last equation originally is

� � <sup>∗</sup> *<sup>m</sup>*, but *fi* can be taken out of the convolution integral because of Assumption A5 (Appendix 2). The definitions of averaged (or upscaled) quantities now follow (step 3). Porosity Φ is defined first, then follow definitions suggested by the

**Porosity** Let a function *χ*ð Þ **x** be 1 when **x** is in F, and let it be 0 otherwise.

**Species adsorption** The left-hand side of Eq. (26) defines the amount, *K*Σ*<sup>α</sup>*, of

*δ*Σ

*<sup>ρ</sup>*<sup>S</sup> <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � <sup>Φ</sup> *<sup>R</sup>*<sup>S</sup>

**Density of component** *α* The first term on the left-hand side of Eq. (25)

Note that Eq. (12) implies that the averaged total fluid density, *R*, is

**Fluid velocity** The first term on the lef-hand side of Eq. (28) has the convolution of a product of two term, where the average of one of them is known from Eqs. (32) and (33). The averaged fluid velocity, denoted *Vi* (without the F

*<sup>R</sup>* <sup>¼</sup> <sup>X</sup>*<sup>ν</sup> α*¼1

superscript since there is no velocity in S or on Σ) is then defined by

*ρ*<sup>F</sup>*v*<sup>F</sup> *i*

*ρ*<sup>F</sup>*<sup>α</sup>v*<sup>F</sup> *<sup>k</sup>* þ *ι* F*α k*

**Diffusive mass current of component** *α* The second convolution on the lefthand side of Eq. (25) is used to define the upscaled diffusive current in the fluid,

F*α*

convection and diffusion contribute to it [12]. Keeping in mind the constraint that the averaged equations should have the same form as the original ones, *I*

� � <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>Φ</sup>*R<sup>α</sup>Vk* <sup>þ</sup> <sup>Φ</sup>*<sup>I</sup>*

*<sup>k</sup>* (without the F superscript since there are no diffusive currents in S or on

**Density of solid** Eq. (27) suggests defining the solid density *R*<sup>S</sup> by:

Note that, according to definition (22), Φ can depend on **x**.

*ι* <sup>F</sup>*<sup>α</sup>* � **<sup>n</sup>** � �<sup>Σ</sup>

suggests defining the density *R<sup>α</sup>* of fluid component *α* by

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup>*<sup>S</sup> <sup>∗</sup> *<sup>m</sup>* � � <sup>¼</sup> 0, (27)

*χ* ∗ *m* ¼ Φ*:* (29)

*:* (31)

� � <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>K</sup>*Σ*α:* (30)

*<sup>ρ</sup>*<sup>F</sup>*<sup>α</sup>* <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>R</sup>α:* (32)

*<sup>R</sup><sup>α</sup>* <sup>¼</sup> *<sup>ρ</sup>*<sup>F</sup> <sup>∗</sup> *<sup>m</sup>:* (33)

� � <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>Φ</sup>*RVi:* (34)

*<sup>k</sup>* because both pore level effects of

*α*

*α <sup>k</sup>* is

*<sup>k</sup>:* (35)

� � <sup>∗</sup> *<sup>m</sup>* <sup>þ</sup> *fi <sup>ρ</sup>*<sup>F</sup> <sup>∗</sup> *<sup>m</sup>* � �*:* (28)

<sup>Σ</sup>*δ*<sup>Σ</sup>

$$\left[\left(\mathbf{J}^{\mathrm{F}} \cdot \mathbf{n}\right)^{\Sigma} - \left(\mathbf{J}^{\mathrm{S}} \cdot \mathbf{n}\right)^{\Sigma}\right] \delta\_{\Sigma} = \mathbf{0}, \quad \text{(on } \Sigma\text{)}\tag{20}$$

$$
\partial\_t u^\mathbf{S} + \partial\_k f\_k^\mathbf{S} = \mathbf{0}, \quad \text{(in } \mathbf{S}). \tag{21}
$$

#### **3. Averaging**

The Marle averaging process [12] is followed in all essentials, except in the assumption that well-defined phases exist, separated by interphase surfaces. The averaging volume is a sphere of radius *r*, large when compared to a pore radius, small when compared to a linear dimension of the reservoir. A *<sup>C</sup>*<sup>∞</sup> function *<sup>m</sup>*ð Þ **<sup>x</sup>** is introduced, somewhat flat around ∣**x**∣ ¼ 0 and equal to zero for ∣**x**∣ ≥*r*, normalised so that its integral over all space is equal to 1. (See Eq. (17) in Ref. [12] for an example of such a function.) Given any function of space and time, *f*ð Þ **x**, *t* , its average *F*ð Þ **x**, *t* is obtained by the convolution

$$F(\mathbf{x},t) \equiv (\boldsymbol{f} \ast \boldsymbol{m})(\mathbf{x},t) = \int\_{\mathbb{R}^{\mathcal{J}}} \boldsymbol{f}(\mathbf{y},t) m(\mathbf{x} - \mathbf{y}) d\mathbf{y},\tag{22}$$

where the integration is over all of space. The convolution ensures that *F* is *C*<sup>∞</sup> [12].

The averaged balance equations are differential equations in the averaged quantities (averaged densities, velocities, ...). These equations are established by a threestep process. Step 1: the generalised equations for mass, momentum, and energy balance are each in turn convoluted with *m*. Step 2: the following rules [12] are applied, allowing to take the differential operators out of the averaging convolutions:

$$\left\{\partial\_t f^Z\right\} \ast m = \partial\_t \left(f^Z \ast m\right),\tag{23}$$

$$\left\{\partial\_i f^Z\right\} \ast m = \partial\_i \left(f^Z \ast m\right) - \varepsilon^Z \left[\left(f^Z\right)^\Sigma n\_i \delta\_\Sigma\right] \ast m,\tag{24}$$

where *f* <sup>Z</sup> � *<sup>f</sup>* <sup>Z</sup>ð Þ **<sup>x</sup>**, *<sup>t</sup>* and Z is F, F*α*, or S; *<sup>ε</sup>*<sup>Z</sup> is 1 if Z is F or F*α*, �1 if Z is S. Eq. (11) is also applied at step 2. Step 3: the remaining convolutions are used to define averaged quantities, where the following constraints should be obeyed: (i) except for the definition of porosity, the way to define the averages is suggested by the equations obtained after completion of step 2; (ii) the averaged equations have essentially the same forms as Eqs. (3) to (5).

Differential equations in the averaged quantities result from the three steps. It is shown below that the mass and momentum balance equations should be treated together, and that the energy equation can be treated as an addition.

The averaging of the mass and momentum balance equations follows. Steps 1 and 2 are applied to Eqs. (14) to (17), and lead to:

$$\partial\_t \left( \rho^{\rm Fa} \ast m \right) + \partial\_k \left( \left( \rho^{\rm Fa} v\_k^{\rm F} + l\_k^{\rm Fa} \right) \ast m \right) = \left( \left( \mathbf{t}^{\rm Fa} \cdot \mathbf{n} \right)^{\Sigma} \delta\_{\Sigma} \right) \ast m,\tag{25}$$

$$\left(\left(\mathfrak{l}^{\text{Fa}} \cdot \mathfrak{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast m = 0,\tag{26}$$

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

Using **Table 1**, one then obtains:

<sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>u</sup>*F*v*<sup>F</sup>

<sup>F</sup> � **<sup>n</sup>** � �<sup>Σ</sup> � *<sup>J</sup>* <sup>S</sup> � **<sup>n</sup>** � �<sup>Σ</sup> h i

*J*

average *F*ð Þ **x**, *t* is obtained by the convolution

*∂i f* <sup>Z</sup> n o

essentially the same forms as Eqs. (3) to (5).

and 2 are applied to Eqs. (14) to (17), and lead to:

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup>*<sup>F</sup>*<sup>α</sup>* <sup>∗</sup> *<sup>m</sup>* � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>ρ</sup>*<sup>F</sup>*<sup>α</sup>v*<sup>F</sup>

*F*ð Þ� **x**, *t* ð Þ *f* ∗ *m* ð Þ¼ **x**, *t*

*∂t f* <sup>Z</sup> n o

<sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>∂</sup><sup>i</sup> <sup>f</sup>*

*k* þ 1 2 *ρ*<sup>F</sup> **v**<sup>F</sup> � � � � 2 *v*F *<sup>k</sup>* þ *J* F *<sup>k</sup>* � *tkiv*<sup>F</sup> *i*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*<sup>∂</sup>tu*<sup>S</sup> <sup>þ</sup> *<sup>∂</sup>kJ*

S

The Marle averaging process [12] is followed in all essentials, except in the assumption that well-defined phases exist, separated by interphase surfaces. The averaging volume is a sphere of radius *r*, large when compared to a pore radius, small when compared to a linear dimension of the reservoir. A *<sup>C</sup>*<sup>∞</sup> function *<sup>m</sup>*ð Þ **<sup>x</sup>** is introduced, somewhat flat around ∣**x**∣ ¼ 0 and equal to zero for ∣**x**∣ ≥*r*, normalised so that its integral over all space is equal to 1. (See Eq. (17) in Ref. [12] for an example of such a function.) Given any function of space and time, *f*ð Þ **x**, *t* , its

> ð 3

The averaged balance equations are differential equations in the averaged quantities (averaged densities, velocities, ...). These equations are established by a threestep process. Step 1: the generalised equations for mass, momentum, and energy balance are each in turn convoluted with *m*. Step 2: the following rules [12] are applied, allowing to take the differential operators out of the averaging convolutions:

<sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>∂</sup><sup>t</sup> <sup>f</sup>*

<sup>Z</sup> ∗ *m* � �

Eq. (11) is also applied at step 2. Step 3: the remaining convolutions are used to define averaged quantities, where the following constraints should be obeyed: (i) except for the definition of porosity, the way to define the averages is suggested by the equations obtained after completion of step 2; (ii) the averaged equations have

together, and that the energy equation can be treated as an addition.

*ι* <sup>F</sup>*<sup>α</sup>* � **<sup>n</sup>** � �<sup>Σ</sup>

Differential equations in the averaged quantities result from the three steps. It is shown below that the mass and momentum balance equations should be treated

The averaging of the mass and momentum balance equations follows. Steps 1

*δ*Σ � �

*<sup>k</sup>* þ *ι* F*α k* � � <sup>∗</sup> *<sup>m</sup>* � � <sup>¼</sup> *<sup>ι</sup>*

<sup>Z</sup> ∗ *m* � �

<sup>Z</sup> � �<sup>Σ</sup>

*niδ*<sup>Σ</sup> � �

<sup>F</sup>*<sup>α</sup>* � **<sup>n</sup>** � �<sup>Σ</sup>

*δ*Σ � �

∗ *m* ¼ 0, (26)

� *<sup>ε</sup>*<sup>Z</sup> *<sup>f</sup>*

<sup>Z</sup>ð Þ **<sup>x</sup>**, *<sup>t</sup>* and Z is F, F*α*, or S; *<sup>ε</sup>*<sup>Z</sup> is 1 if Z is F or F*α*, �1 if Z is S.

where the integration is over all of space. The convolution ensures that *F* is

� �

<sup>¼</sup> *<sup>f</sup> <sup>k</sup>ρ*F*v*<sup>F</sup>

*δ*<sup>Σ</sup> ¼ 0, on ð Þ Σ (20)

*<sup>f</sup>* **<sup>y</sup>**, *<sup>t</sup>* � �*<sup>m</sup>* **<sup>x</sup>** � **<sup>y</sup>** � �d**y**, (22)

, (23)

∗ *m*, (24)

∗ *m*, (25)

*<sup>k</sup>* ¼ 0, in S ð Þ*:* (21)

*<sup>k</sup>* , in F ð Þ

(19)

*<sup>∂</sup><sup>t</sup> <sup>u</sup>*<sup>F</sup> <sup>þ</sup>

**3. Averaging**

*C*<sup>∞</sup> [12].

where *f*

**8**

<sup>Z</sup> � *<sup>f</sup>*

1 2 *ρ*<sup>F</sup> **v**<sup>F</sup> � � � � 2

� �

$$
\partial\_t \left( \rho^S \ast m \right) = 0,\tag{27}
$$

$$\partial\_t \left( \left( \rho^\mathrm{F} \boldsymbol{v}\_i^\mathrm{F} \right) \ast \boldsymbol{m} \right) + \partial\_k \left( \left( \rho^\mathrm{F} \boldsymbol{v}\_k^\mathrm{F} \boldsymbol{v}\_i^\mathrm{F} - \mathfrak{t}\_{ki} \right) \ast \boldsymbol{m} \right) = - \left( \left( \boldsymbol{n}\_k \boldsymbol{t}\_{ki} \right)^\Sigma \delta\_\Sigma \right) \ast \boldsymbol{m} + f\_i \left( \rho^\mathrm{F} \ast \boldsymbol{m} \right). \tag{28}$$

Note that the last term on the right-hand side of the last equation originally is *ρ*<sup>F</sup> *fi* � � <sup>∗</sup> *<sup>m</sup>*, but *fi* can be taken out of the convolution integral because of Assumption A5 (Appendix 2). The definitions of averaged (or upscaled) quantities now follow (step 3). Porosity Φ is defined first, then follow definitions suggested by the convolution operations in the equations above.

**Porosity** Let a function *χ*ð Þ **x** be 1 when **x** is in F, and let it be 0 otherwise. Then

$$
\chi \* m = \Phi.\tag{29}
$$

Note that, according to definition (22), Φ can depend on **x**.

**Species adsorption** The left-hand side of Eq. (26) defines the amount, *K*Σ*<sup>α</sup>*, of component *α* adsorbed at the pore surface:

$$\left(\left(\mathbf{r}^{\mathrm{Fa}} \cdot \mathbf{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast m = K^{\Sigma a}. \tag{30}$$

**Density of solid** Eq. (27) suggests defining the solid density *R*<sup>S</sup> by:

$$
\rho^{\mathcal{S}} \ast m = (\mathbb{1} - \Phi) \mathcal{R}^{\mathcal{S}}.\tag{31}
$$

**Density of component** *α* The first term on the left-hand side of Eq. (25) suggests defining the density *R<sup>α</sup>* of fluid component *α* by

$$
\rho^{\text{Fe}} \ast \mathcal{m} = \mathcal{R}^a. \tag{32}
$$

Note that Eq. (12) implies that the averaged total fluid density, *R*, is

$$R = \sum\_{a=1}^{\nu} R^a = \rho^{\text{F}} \ast m. \tag{33}$$

**Fluid velocity** The first term on the lef-hand side of Eq. (28) has the convolution of a product of two term, where the average of one of them is known from Eqs. (32) and (33). The averaged fluid velocity, denoted *Vi* (without the F superscript since there is no velocity in S or on Σ) is then defined by

$$(\rho^\mathcal{F}v\_i^\mathcal{F}) \ast m = \Phi RV\_i. \tag{34}$$

**Diffusive mass current of component** *α* The second convolution on the lefthand side of Eq. (25) is used to define the upscaled diffusive current in the fluid, denoted *I α <sup>k</sup>* (without the F superscript since there are no diffusive currents in S or on Σ). It is not defined as the average of *ι* F*α <sup>k</sup>* because both pore level effects of convection and diffusion contribute to it [12]. Keeping in mind the constraint that the averaged equations should have the same form as the original ones, *I α <sup>k</sup>* is defined by

$$\left(\rho^{\mathrm{Fa}}v\_k^{\mathrm{F}} + \mathfrak{l}\_k^{\mathrm{Fa}}\right) \ast m = \Phi R^a V\_k + \Phi I\_k^a. \tag{35}$$

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

Note that, summing this equation over *α* from 1 to *ν*, and using previous results, one gets

$$\sum\_{a=1}^{\nu} I\_k^a = \mathbf{0}.\tag{36}$$

Internal energy per unit volume of solid

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

*j* S *<sup>k</sup>* ∗ *m* ¼ *J*

*δ*Σ

*δ*Σ

*j* <sup>S</sup> � **<sup>n</sup>** <sup>Σ</sup>

*j* <sup>F</sup> � **<sup>n</sup>** <sup>Σ</sup>

<sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>Φ</sup>U<sup>F</sup>*Vk* <sup>þ</sup>

Using these definitions in Eqs. (43) to (45) one obtains:

*<sup>∂</sup><sup>t</sup>* ð ÞU <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> <sup>þ</sup> *<sup>∂</sup>kJ*

*<sup>∂</sup><sup>t</sup>* <sup>Φ</sup>U<sup>F</sup> <sup>þ</sup> *<sup>∂</sup><sup>k</sup>* <sup>Φ</sup>U<sup>F</sup>*Vk* <sup>þ</sup> *<sup>J</sup>*

*α*

**4. Basis for constitutive equations**

<sup>þ</sup> *<sup>∂</sup><sup>k</sup>* <sup>Φ</sup>U<sup>F</sup>*Vk* <sup>þ</sup>

1 2

S

Eq. (52) contains a redundancy in the form of a balance equation for kinetic energy. This equation can be obtained directly by multiplying both sides of Eq. (42) with *Vi*, summing over *i*, and using Eq. (41). Subtracting the equation thus obtained

> F *k* ¼ �*Q*<sup>F</sup>!<sup>S</sup> � <sup>F</sup><sup>F</sup>

primarily, the densities *R<sup>α</sup>*, the velocity components *Vi* and the temperature *T*. Assuming that the approximation of constant *T* is valid, one needs only focus on the mass and momentum balance equations, (39) to (42). One sees in this case that one

In practical application, the upscaled balance equations will be used to calculate,

*<sup>k</sup>*, the *Tij*, and the F*<sup>i</sup>* in terms of *T* and the *R<sup>α</sup>*. If the

<sup>Φ</sup>*R*j j **<sup>V</sup>** <sup>2</sup>

S

Internal energy current of solid

*Porous Flow with Diffuse Interfaces*

Solid to fluid energy transfer

Fluid to solid energy transfer

Internal energy current of fluid

1 2

<sup>Φ</sup>*R*j j **<sup>V</sup>** <sup>2</sup>

¼ �*Q*<sup>F</sup>!<sup>S</sup> þ Φ*RVk f <sup>k</sup>:* ð Þ in F

*u*<sup>F</sup>*v*<sup>F</sup> *<sup>k</sup>* þ 1 <sup>2</sup> *<sup>ρ</sup>*<sup>F</sup> **<sup>v</sup>**<sup>F</sup> 2 *v*F *<sup>k</sup>* þ *j* F *<sup>k</sup>* � *tkiv*<sup>F</sup> *i*

*<sup>∂</sup><sup>t</sup>* <sup>Φ</sup>U<sup>F</sup> <sup>þ</sup>

from Eq. (52), one gets

needs expressions for the *I*

**11**

Internal energy per unit volume of fluid

*<sup>u</sup>*<sup>F</sup> <sup>þ</sup> 1 <sup>2</sup> *<sup>ρ</sup>*<sup>F</sup> **<sup>v</sup>**<sup>F</sup> 2 <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>Φ</sup>U<sup>F</sup> <sup>þ</sup>

*<sup>u</sup>*<sup>S</sup> <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � <sup>Φ</sup> <sup>U</sup>S, (46)

<sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>Q</sup>*S!F, (48)

<sup>∗</sup> *<sup>m</sup>* ¼ �*Q*<sup>F</sup>!S, (49)

1 2

> 1 2

*Vk* þ *J* F *<sup>k</sup>* � *TkiVi*

*Q*<sup>F</sup>!<sup>S</sup> þ *Q*<sup>S</sup>!<sup>F</sup> ¼ 0*:* ð Þ on Σ (53)

<sup>Φ</sup>*R*j j **<sup>V</sup>** <sup>2</sup>

<sup>Φ</sup>*R*j j **<sup>V</sup>** <sup>2</sup>

*<sup>k</sup>* ¼ �*Q*<sup>S</sup>!<sup>F</sup>*:* ð Þ in S *:* (54)

, (50)

(51)

(52)

*Vk* þ *J* F *<sup>k</sup>* � *TkiVi:*

*kVk* <sup>þ</sup> *Tji<sup>∂</sup> jVi:* (55)

*<sup>k</sup>*, (47)

**Stress tensor** The second convolution on the left-hand side of Eq. (28) suggests defining *Tki*, the upscaled version of *tki*, by

$$\left(\rho^{\rm F}v\_{k}^{\rm F}v\_{i}^{\rm F} - t\_{ki}\right) \ast m = \Phi RV\_{i}V\_{k} - T\_{ki}.\tag{37}$$

Note that *tki* ¼ *tik* implies *Tki* ¼ *Tik*.

**Frictional force per unit volume** The first convolution on the right-hand side of Eq. (28) suggests defining the frictional force per unit volume F<sup>F</sup> *<sup>i</sup>* by

$$\left( \left( n\_k t\_{ki} \right)^{\Sigma} \delta\_{\Sigma} \right) \* m = \mathfrak{F}\_i^{\mathrm{F}}.\tag{38}$$

The upscaled mass and momentum balance equations now follow, in the following order: mass balance for the solid, mass balance at the pore surface, mass balance for the fluid in the pores, and momentum balance for the fluid in the pores:

$$\partial\_t \left[ (\mathbf{1} - \Phi) \mathcal{R}^S \right] = \mathbf{0},\tag{39}$$

$$K^{\Sigma a} = \mathbf{0}, \quad (a = \mathbf{1} \dots \boldsymbol{\nu}), \tag{40}$$

$$
\partial\_t(\Phi \mathcal{R}^a) + \partial\_k \left( \Phi \mathcal{R}^a V\_k + \Phi I\_k^a \right) = 0, \quad (a = 1 \dots \nu), \tag{41}
$$

$$
\partial\_t(\Phi RV\_i) + \partial\_k(\Phi RV\_k V\_i - T\_{ki}) = \mathfrak{F}\_i^{\mathrm{F}} + \Phi R f\_i. \tag{42}
$$

The first equation states that porosity and solid density do not vary with time (consistently with Assumption 4 (Appendix 2) but can vary in space. The second equation states that adsorption is negligibly small, for any component, consistent with Assumption A0 (Appendix 2). The remaining two equations determine the *ν* component densities and the three velocity components. This means that the components of the diffusive mass current, of the stress tensor, and of the frictional force must be provided. (Constitutive equations).

The averaging of the energy balance equations now follows. Steps 1 and 2 are applied to Eqs. (19) to (21), giving:

$$\begin{split} & \left[ \partial\_{t} \left[ \left( u^{\text{F}} + \frac{1}{2} \rho^{\text{F}} \middle| \mathbf{v}^{\text{F}} \right|^{2} \right) \* m \right] + \partial\_{k} \left[ \left( u^{\text{F}} v\_{k}^{\text{F}} + \frac{1}{2} \rho^{\text{F}} \middle| \mathbf{v}^{\text{F}} \right|^{2} v\_{k}^{\text{F}} + f\_{k}^{\text{F}} - t\_{ki} v\_{i}^{\text{F}} \right) \* m \right] \\ & = \left[ \left( \mathbf{J}^{\text{F}} \cdot \mathbf{n} \right)^{\Sigma} \delta\_{\Sigma} \right] \* m + f\_{k} \left( \rho^{\text{F}} v\_{k}^{\text{F}} \* m \right), \end{split} \tag{43}$$

$$\left(\left(\mathbf{J}^{\mathrm{F}} \cdot \mathbf{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast m - \left(\left(\mathbf{J}^{\mathrm{S}} \cdot \mathbf{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast m = \mathbf{0},\tag{44}$$

$$
\partial\_t \left( u^{\mathbb{S}} \ast m \right) + \partial\_k \left( \mathbb{J}\_k^{\mathbb{S}} \ast m \right) = - \left[ \left( \mathbb{J}^{\mathbb{S}} \cdot \mathbf{n} \right)^{\Sigma} \delta\_{\Sigma} \right] \ast m. \tag{45}
$$

As in the case of Eq. (28), Assumption A4 (Appendix 2) has been used to take *f <sup>k</sup>* out of the convolution on the right-hand side of Eq: (43). Six definitions are introduced below, built on the definitions that were introduced in connection with the averaging of the mass and momentum balance equations. An underscore indicates the defined quantity.

Internal energy per unit volume of solid

$$
u^{\mathbb{S}} \ast m = (\mathbb{1} - \Phi) \underline{\mathcal{U}^{\mathbb{S}}}.\tag{46}$$

Internal energy current of solid

Note that, summing this equation over *α* from 1 to *ν*, and using previous results,

**Stress tensor** The second convolution on the left-hand side of Eq. (28) suggests

**Frictional force per unit volume** The first convolution on the right-hand side

<sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>F</sup><sup>F</sup>

*α k*

The first equation states that porosity and solid density do not vary with time (consistently with Assumption 4 (Appendix 2) but can vary in space. The second equation states that adsorption is negligibly small, for any component, consistent with Assumption A0 (Appendix 2). The remaining two equations determine the *ν* component densities and the three velocity components. This means that the components of the diffusive mass current, of the stress tensor, and of the frictional force

The averaging of the energy balance equations now follows. Steps 1 and 2 are

*<sup>k</sup>* þ 1 2 *ρ*<sup>F</sup> **v**<sup>F</sup> � � � � 2 *v*F *<sup>k</sup>* þ *J* F *<sup>k</sup>* � *tkiv*<sup>F</sup> *i*

*<sup>k</sup>* <sup>∗</sup> *<sup>m</sup>* � � ¼ � *<sup>J</sup>*

As in the case of Eq. (28), Assumption A4 (Appendix 2) has been used to take *f <sup>k</sup>*

<sup>S</sup> � **<sup>n</sup>** � �<sup>Σ</sup>

∗ *m* � *J*

S

out of the convolution on the right-hand side of Eq: (43). Six definitions are introduced below, built on the definitions that were introduced in connection with the averaging of the mass and momentum balance equations. An underscore

� �

*δ*Σ � �

<sup>S</sup> � **<sup>n</sup>** � �<sup>Σ</sup>

*δ*Σ h i

� �

<sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>u</sup>*<sup>F</sup>*v*<sup>F</sup>

*<sup>k</sup>* <sup>∗</sup> *<sup>m</sup>* � �,

*δ*Σ � �

<sup>Σ</sup>*δ*<sup>Σ</sup> � �

The upscaled mass and momentum balance equations now follow, in the following order: mass balance for the solid, mass balance at the pore surface, mass balance for the fluid in the pores, and momentum balance for the fluid in the pores:

*<sup>∂</sup>t*ð Þþ <sup>Φ</sup>*RVi <sup>∂</sup>k*ðΦ*RVkVi* � *Tki*Þ ¼ <sup>F</sup><sup>F</sup>

� � <sup>∗</sup> *<sup>m</sup>* <sup>¼</sup> <sup>Φ</sup>*RViVk* � *Tki:* (37)

*<sup>∂</sup><sup>t</sup>* ð Þ <sup>1</sup> � <sup>Φ</sup> *<sup>R</sup>*<sup>S</sup> � � <sup>¼</sup> 0, (39) *<sup>K</sup>*Σ*<sup>α</sup>* <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> <sup>1</sup> … *<sup>ν</sup>* , (40)

� � <sup>¼</sup> 0, ð Þ *<sup>α</sup>* <sup>¼</sup> <sup>1</sup> … *<sup>ν</sup>* , (41)

*<sup>i</sup>* þ Φ*Rfi*

*<sup>k</sup>* ¼ 0*:* (36)

*<sup>i</sup>* by

*:* (42)

∗ *m*

∗ *m* ¼ 0, (44)

∗ *m:* (45)

(43)

*<sup>i</sup> :* (38)

X*ν α*¼1 *I α*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

defining *Tki*, the upscaled version of *tki*, by

Note that *tki* ¼ *tik* implies *Tki* ¼ *Tik*.

must be provided. (Constitutive equations).

∗ *m*

<sup>∗</sup> *<sup>m</sup>* <sup>þ</sup> *<sup>f</sup> <sup>k</sup> <sup>ρ</sup>*<sup>F</sup>*v*<sup>F</sup>

*<sup>∂</sup><sup>t</sup> <sup>u</sup>*<sup>S</sup> <sup>∗</sup> *<sup>m</sup>* � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup> <sup>J</sup>*

*J* <sup>F</sup> � **<sup>n</sup>** � �<sup>Σ</sup>

applied to Eqs. (19) to (21), giving:

� �

*δ*Σ h i

indicates the defined quantity.

1 2 *ρ*<sup>F</sup> **v**<sup>F</sup> � � � � 2

� �

*<sup>∂</sup><sup>t</sup> <sup>u</sup>*<sup>F</sup> <sup>þ</sup>

<sup>F</sup> � **<sup>n</sup>** � �<sup>Σ</sup>

¼ *J*

**10**

*ρ*F*v*<sup>F</sup> *k v*F *<sup>i</sup>* � *tki*

of Eq. (28) suggests defining the frictional force per unit volume F<sup>F</sup>

ð Þ *nktki*

*<sup>∂</sup><sup>t</sup>* <sup>Φ</sup>*R<sup>α</sup>* ð Þþ *<sup>∂</sup><sup>k</sup>* <sup>Φ</sup>*R<sup>α</sup>Vk* <sup>þ</sup> <sup>Φ</sup>*<sup>I</sup>*

one gets

$$
\overline{J\_k^S} \ast m = \underline{J\_k^S},\tag{47}
$$

Solid to fluid energy transfer

$$\left(\left(\mathbf{j}^{\mathbb{S}} \cdot \mathbf{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast m = \underline{Q\_{\mathbb{S}\to\mathbb{F}}},\tag{48}$$

Fluid to solid energy transfer

$$\left(\left(\mathbf{j}^{\mathrm{F}} \cdot \mathbf{n}\right)^{\Sigma} \delta\_{\Sigma}\right) \ast \boldsymbol{m} = \underline{\mathbf{Q}\_{\mathrm{F} \to \mathbf{S}}},\tag{49}$$

Internal energy per unit volume of fluid

$$\left(\boldsymbol{u}^{\mathrm{F}} + \frac{1}{2}\boldsymbol{\rho}^{\mathrm{F}}\left|\mathbf{v}^{\mathrm{F}}\right|^{2}\right) \ast \boldsymbol{m} = \boldsymbol{\Phi}\underline{\boldsymbol{\mathcal{U}}}^{\mathrm{F}} + \frac{1}{2}\boldsymbol{\Phi}\boldsymbol{R}|\mathbf{V}|^{2},\tag{50}$$

Internal energy current of fluid

$$\left(\mathbf{u}^{\mathrm{F}}\boldsymbol{v}\_{k}^{\mathrm{F}} + \frac{\mathbf{1}}{2}\boldsymbol{\rho}^{\mathrm{F}}|\mathbf{v}^{\mathrm{F}}|^{2}\boldsymbol{v}\_{k}^{\mathrm{F}} + \boldsymbol{j}\_{k}^{\mathrm{F}} - \mathbf{t}\_{kl}\boldsymbol{v}\_{i}^{\mathrm{F}}\right) \ast \boldsymbol{m} = \boldsymbol{\Phi}\boldsymbol{\mathcal{U}}^{\mathrm{F}}\boldsymbol{V}\_{k} + \frac{\mathbf{1}}{2}\boldsymbol{\Phi}\boldsymbol{\mathcal{R}}|\mathbf{V}|^{2}\boldsymbol{V}\_{k} + \underline{\boldsymbol{J}}\_{k}^{\mathrm{F}} - T\_{ki}\boldsymbol{V}\_{i}.\tag{51}$$

Using these definitions in Eqs. (43) to (45) one obtains:

$$\begin{aligned} &\partial\_t \left( \Phi \mathcal{U}^{\mathrm{F}} + \frac{1}{2} \Phi \mathcal{R} |\mathbf{V}|^2 \right) + \partial\_k \left( \Phi \mathcal{U}^{\mathrm{F}} V\_k + \frac{1}{2} \Phi \mathcal{R} |\mathbf{V}|^2 V\_k + f\_k^{\mathrm{F}} - T\_{ki} V\_i \right) \\ &= -Q\_{\mathrm{F} \to \mathcal{S}} + \Phi \mathcal{R} V\_k f\_k \quad \text{ (in } \mathcal{F} \text{)} \end{aligned} \tag{52}$$

$$Q\_{\text{F}\to\text{S}} + Q\_{\text{S}\to\text{F}} = \mathbf{0}. \quad (\text{on } \Sigma) \tag{53}$$

$$
\partial\_t \left[ (\mathbf{1} - \Phi) \mathcal{U}^{\mathbb{S}} \right] + \partial\_{\mathbf{k}} f\_k^{\mathbb{S}} = -Q\_{\mathbb{S} \to \mathbb{F}}. \quad (\text{in } \mathbb{S}). \tag{54}
$$

Eq. (52) contains a redundancy in the form of a balance equation for kinetic energy. This equation can be obtained directly by multiplying both sides of Eq. (42) with *Vi*, summing over *i*, and using Eq. (41). Subtracting the equation thus obtained from Eq. (52), one gets

$$
\partial\_t \left( \Phi \mathcal{U}^{\mathcal{F}} \right) + \partial\_k \left( \Phi \mathcal{U}^{\mathcal{F}} V\_k + f\_k^{\mathcal{F}} \right) = -Q\_{\mathcal{F} \to \mathcal{S}} - \mathfrak{F}\_k^{\mathcal{F}} V\_k + T\_{j\mathcal{i}} \partial\_j V\_i. \tag{55}
$$

#### **4. Basis for constitutive equations**

In practical application, the upscaled balance equations will be used to calculate, primarily, the densities *R<sup>α</sup>*, the velocity components *Vi* and the temperature *T*. Assuming that the approximation of constant *T* is valid, one needs only focus on the mass and momentum balance equations, (39) to (42). One sees in this case that one needs expressions for the *I α <sup>k</sup>*, the *Tij*, and the F*<sup>i</sup>* in terms of *T* and the *R<sup>α</sup>*. If the

approximation of constant temperature is not valid, one must first distinguish between the temperatures of the solid and the fluid, and one needs an equation for the energy transfer in case of a temperature difference. One introduces the simplifying Assumption A6 (see Ref. [12] and Appendix 1), from which it follows that just one additional equation is needed for calculating *T*ð Þ **x**, *t* . Such an equation usually describes the evolution of either total energy or total entropy. In either case, expressions are needed for the currents *J* S *<sup>k</sup>* and *J* F *<sup>k</sup>* . (Expressions for the energy transfers *Q*S!<sup>F</sup> and *Q*F!<sup>S</sup> are unnecessary since they cancel when taking the sum of the solid and fluid energies). Most of what is needed is obtained in Section 5 by applying the theory of irreversible processes, starting from the evolution equation for entropy, although it seems that some preliminary work is unavoidable to directly obtain an expression for the pressure tensor *Pij* that replaces the usual scalar pressure.

The derivation of the pressure tensor is given below, followed by the derivation of the evolution equation for the total entropy. *It is essential to use expression (1) in both derivations.*<sup>2</sup>

#### **4.1 The pressure tensor**

One considers the upscaled fluid, consisting of a mixture of *ν* components in a container with surface ∂Ω and volume Ω, inside a large bath at uniform and constant temperature *T*. One looks for conditions of equilibrium in the presence of gravity. The fluid has the Helmholtz free energy density given by Eq. (1), and it is assumed that the bounding surface is neutrally wetting so that there is no energy stored on ∂Ω. The total energy stored in the fluid is

$$F = \int\_{\Omega} \left[ \mathcal{F}^{\mathcal{F}} + \mathcal{W} \sum\_{a=1}^{\nu} \mathcal{R}^a \right] d\Omega,\tag{56}$$

*<sup>∂</sup>kPik* � *fi* <sup>¼</sup> <sup>0</sup>*:* (59)

*Tij* ¼ �Φ*Pij* þ Θ*ij:* (61)

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>iR<sup>α</sup>* ð Þ *<sup>∂</sup>kR<sup>α</sup>* ð Þ*:* (60)

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>iR<sup>α</sup>* ð Þ<sup>d</sup> *<sup>∂</sup>iR<sup>α</sup>* ð Þ*:* (62)

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>iR<sup>α</sup>* ð Þ<sup>d</sup> *<sup>∂</sup>iR<sup>α</sup>* ð Þ*:* (63)

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ<sup>d</sup> *<sup>∂</sup>kR<sup>α</sup>* ð Þ*:* (64)

� � <sup>¼</sup> *<sup>Q</sup>*, (65)

*<sup>δ</sup>ik* <sup>þ</sup>X*<sup>ν</sup> α*¼1

where

<sup>C</sup>bF*<sup>α</sup>*

obtaining

term.

**13**

*Pik* <sup>¼</sup> *<sup>P</sup>*bF �X*<sup>ν</sup>*

*α*¼1

**4.2 The entropy evolution equation**

to get the entropy equation for the fluid.

the assumption that the Λ*<sup>α</sup>* are constants,

<sup>d</sup>F<sup>F</sup> ¼ �SbFd*<sup>T</sup>* <sup>þ</sup>X*<sup>ν</sup>*

<sup>d</sup>F<sup>F</sup> ¼ �SFd*<sup>T</sup>* <sup>þ</sup>X*<sup>ν</sup>*

*<sup>T</sup>*dS<sup>F</sup> <sup>¼</sup> <sup>d</sup>U<sup>F</sup> �X*<sup>ν</sup>*

equation), then with *Vi∂<sup>i</sup>* (second equation):

generalise Eq. (10) by setting

*Porous Flow with Diffuse Interfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

Λ*<sup>α</sup>R<sup>α</sup>*∇<sup>2</sup>

*<sup>R</sup><sup>α</sup>* �X*<sup>ν</sup> α*¼1

presence of the Λ's is exclusively due to their presence in Eq. (60).

*α*¼1

*α*¼1

*α*¼1

entropy (including fluid and solid). Such an equation must be of the form

*<sup>∂</sup><sup>t</sup>* <sup>Φ</sup>S<sup>F</sup> <sup>þ</sup> ð ÞS <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup>* <sup>Φ</sup>S<sup>F</sup>*Vk* <sup>þ</sup> *Pk*

" #

1 2

<sup>Λ</sup>*<sup>α</sup>* **<sup>∇</sup>***R<sup>α</sup>* j j<sup>2</sup>

The first equation obviously generalises the classical **∇***P* ¼ *f*, and it is natural to

*Tij* and *Pij* are symmetric, so that Θ*ij* ¼ Θ*ji*. Eqs. (60) and (61) are used in the long calculation that lead to the evolution equation for the entropy. They are also used to get expressions (99) and (100) for the modified Darcy equation, where the

Eqs. (1) and (2), together with Eqs. (41), (54), (55), (60), and (61), are needed

Taking the differential of Eq. (1) and using Eq. (2), one obtains, keeping in mind

<sup>C</sup>bF*<sup>α</sup>*d*R<sup>α</sup>* <sup>þ</sup>X*<sup>ν</sup>*

<sup>C</sup><sup>F</sup>*<sup>α</sup>*d*R<sup>α</sup>* <sup>þ</sup>X*<sup>ν</sup>*

*α*¼1

*α*¼1

This equation leads immediately to two conclusions. There is no additional entropy, and no additional chemical potentials due to large density gradients, since: (i) <sup>S</sup>F, defined as *<sup>∂</sup>*F<sup>F</sup>*=∂T*, is equal to <sup>S</sup>bF; (ii) <sup>C</sup><sup>F</sup>*<sup>α</sup>*, defined as *<sup>∂</sup>*F<sup>F</sup>*=∂R<sup>α</sup>*, is equal to

. The differential of <sup>F</sup><sup>F</sup> can now be re-written with a simpler notation:

To obtain dS<sup>F</sup> in terms of dU<sup>F</sup> and dF<sup>F</sup> one differentiates <sup>U</sup><sup>F</sup> <sup>¼</sup> *<sup>T</sup>*S<sup>F</sup> þ FF,

<sup>C</sup><sup>F</sup>*<sup>α</sup>*d*R<sup>α</sup>* �X*<sup>ν</sup>*

This expression is now used to construct an evolution equation for the total

where <sup>S</sup><sup>S</sup> is the entropy of the rock, *Pk* is a diffusive current, and *<sup>Q</sup>* is a source

Eq. (64) directly gives the two equations that follow, by replacing d with *∂<sup>t</sup>* (first

*α*¼1

where *W* is the gravitational potential (see Eqs. (8) and (9)).

One now looks for the conditions the *R<sup>α</sup>* satisfy when *F* is at its minimum, given that the total mass, Ð <sup>Ω</sup>*R*dΩ, is constant. This is minimisation with constraint, a standard problem in variational calculus. It is easily found that

$$\frac{\partial \mathcal{F}^{\rm bF}}{\partial \mathcal{R}^a} + \mathcal{W} - \kappa - \Lambda^a \nabla^2 \mathcal{R}^a = \mathbf{0}, \quad (\text{in} \quad \mathfrak{Q} \ ), \tag{57}$$

$$\mathbf{n} \cdot \nabla R^a = \mathbf{0}, \quad (\text{on} \quad \partial \Omega), \tag{58}$$

where *κ* is a Lagrange multiplier, possibly a function of *T* but not of the *R<sup>α</sup>*. See Ref. [14] for the technique and the theorems involved: essentially, the expression on the left-hand side of Eq. (57) must be continuous. Note that Eq. (58) expresses the non-wetting property of the outer boundary: the density neither increases nor decreases along the normal.

The next step consists in multiplying Eq. (57) with *∂iR<sup>α</sup>* and summing over *α*. Each term of the resulting equation can then be re-written as a gradient (first term), or as a sum of a gradient and the component of a force (sum of second and third terms), or as a sum of a gradient and a divergence (fourth term). The result is the following expression:

<sup>2</sup> This is where the present paper and Ref. [8] differ most.

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

$$
\partial\_k P\_{ik} - f\_i = \mathbf{0}.\tag{59}
$$

where

approximation of constant temperature is not valid, one must first distinguish between the temperatures of the solid and the fluid, and one needs an equation for the energy transfer in case of a temperature difference. One introduces the simplifying Assumption A6 (see Ref. [12] and Appendix 1), from which it follows that just one additional equation is needed for calculating *T*ð Þ **x**, *t* . Such an equation usually describes the evolution of either total energy or total entropy. In either case,

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S *<sup>k</sup>* and *J* F

transfers *Q*S!<sup>F</sup> and *Q*F!<sup>S</sup> are unnecessary since they cancel when taking the sum of the solid and fluid energies). Most of what is needed is obtained in Section 5 by applying the theory of irreversible processes, starting from the evolution equation for entropy, although it seems that some preliminary work is unavoidable to directly obtain an expression for the pressure tensor *Pij* that replaces the usual scalar

The derivation of the pressure tensor is given below, followed by the derivation of the evolution equation for the total entropy. *It is essential to use expression (1) in*

One considers the upscaled fluid, consisting of a mixture of *ν* components in a

<sup>F</sup><sup>F</sup> <sup>þ</sup> *<sup>W</sup>*X*<sup>ν</sup>*

" #

One now looks for the conditions the *R<sup>α</sup>* satisfy when *F* is at its minimum, given

where *κ* is a Lagrange multiplier, possibly a function of *T* but not of the *R<sup>α</sup>*. See Ref. [14] for the technique and the theorems involved: essentially, the expression on the left-hand side of Eq. (57) must be continuous. Note that Eq. (58) expresses the non-wetting property of the outer boundary: the density neither increases nor

The next step consists in multiplying Eq. (57) with *∂iR<sup>α</sup>* and summing over *α*. Each term of the resulting equation can then be re-written as a gradient (first term), or as a sum of a gradient and the component of a force (sum of second and third terms), or as a sum of a gradient and a divergence (fourth term). The result is the

*α*¼1 *Rα*

<sup>Ω</sup>*R*dΩ, is constant. This is minimisation with constraint, a

container with surface ∂Ω and volume Ω, inside a large bath at uniform and constant temperature *T*. One looks for conditions of equilibrium in the presence of gravity. The fluid has the Helmholtz free energy density given by Eq. (1), and it is assumed that the bounding surface is neutrally wetting so that there is no energy

stored on ∂Ω. The total energy stored in the fluid is

*<sup>∂</sup>*FbF

<sup>2</sup> This is where the present paper and Ref. [8] differ most.

*F* ¼ ð Ω

where *W* is the gravitational potential (see Eqs. (8) and (9)).

standard problem in variational calculus. It is easily found that

*<sup>∂</sup>R<sup>α</sup>* <sup>þ</sup> *<sup>W</sup>* � *<sup>κ</sup>* � <sup>Λ</sup>*<sup>α</sup>*∇<sup>2</sup>

*<sup>k</sup>* . (Expressions for the energy

dΩ, (56)

*<sup>R</sup><sup>α</sup>* <sup>¼</sup> 0, inð Þ <sup>Ω</sup> , (57)

**<sup>n</sup>** � **<sup>∇</sup>***R<sup>α</sup>* <sup>¼</sup> 0, on ð Þ <sup>∂</sup><sup>Ω</sup> , (58)

expressions are needed for the currents *J*

pressure.

*both derivations.*<sup>2</sup>

**4.1 The pressure tensor**

that the total mass, Ð

decreases along the normal.

following expression:

**12**

$$P\_{ik} = \left[P^{\text{bF}} - \sum\_{a=1}^{\nu} \Lambda^a \mathbb{R}^a \nabla^2 \mathbb{R}^a - \sum\_{a=1}^{\nu} \frac{1}{2} \Lambda^a |\nabla \mathbb{R}^a|^2\right] \delta\_{ik} + \sum\_{a=1}^{\nu} \Lambda^a (\partial\_i \mathbb{R}^a)(\partial\_k \mathbb{R}^a). \tag{60}$$

The first equation obviously generalises the classical **∇***P* ¼ *f*, and it is natural to generalise Eq. (10) by setting

$$T\_{\vec{\eta}} = -\Phi P\_{\vec{\eta}} + \Theta\_{\vec{\eta}}.\tag{61}$$

*Tij* and *Pij* are symmetric, so that Θ*ij* ¼ Θ*ji*. Eqs. (60) and (61) are used in the long calculation that lead to the evolution equation for the entropy. They are also used to get expressions (99) and (100) for the modified Darcy equation, where the presence of the Λ's is exclusively due to their presence in Eq. (60).

#### **4.2 The entropy evolution equation**

Eqs. (1) and (2), together with Eqs. (41), (54), (55), (60), and (61), are needed to get the entropy equation for the fluid.

Taking the differential of Eq. (1) and using Eq. (2), one obtains, keeping in mind the assumption that the Λ*<sup>α</sup>* are constants,

$$\mathbf{d}\mathcal{F}^{\mathrm{F}} = -\mathcal{S}^{\mathrm{bF}}\mathrm{d}T + \sum\_{a=1}^{\nu} \mathcal{C}^{\mathrm{bF}a}\mathrm{d}\mathcal{R}^{a} + \sum\_{a=1}^{\nu} \Lambda^{a}(\partial\_{i}\mathcal{R}^{a})\mathbf{d}(\partial\_{i}\mathcal{R}^{a}).\tag{62}$$

This equation leads immediately to two conclusions. There is no additional entropy, and no additional chemical potentials due to large density gradients, since: (i) <sup>S</sup>F, defined as *<sup>∂</sup>*F<sup>F</sup>*=∂T*, is equal to <sup>S</sup>bF; (ii) <sup>C</sup><sup>F</sup>*<sup>α</sup>*, defined as *<sup>∂</sup>*F<sup>F</sup>*=∂R<sup>α</sup>*, is equal to <sup>C</sup>bF*<sup>α</sup>* . The differential of <sup>F</sup><sup>F</sup> can now be re-written with a simpler notation:

$$\mathbf{d}\mathcal{F}^{\mathrm{F}} = -\mathcal{S}^{\mathrm{F}}\mathrm{d}T + \sum\_{a=1}^{\nu} \mathcal{C}^{\mathrm{Fa}}\mathrm{d}\mathcal{R}^{a} + \sum\_{a=1}^{\nu} \Lambda^{a}(\partial\_{i}\mathcal{R}^{a})\mathrm{d}(\partial\_{i}\mathcal{R}^{a}).\tag{63}$$

To obtain dS<sup>F</sup> in terms of dU<sup>F</sup> and dF<sup>F</sup> one differentiates <sup>U</sup><sup>F</sup> <sup>¼</sup> *<sup>T</sup>*S<sup>F</sup> þ FF, obtaining

$$\mathcal{T}\mathbf{d}\mathcal{S}^{\mathrm{F}} = \mathbf{d}\mathcal{U}^{\mathrm{F}} - \sum\_{a=1}^{\nu} \mathcal{C}^{\mathrm{F}a} \mathbf{d}\mathcal{R}^{a} - \sum\_{a=1}^{\nu} \Lambda^{a}(\partial\_{k}\mathcal{R}^{a})\mathbf{d}(\partial\_{k}\mathcal{R}^{a}).\tag{64}$$

This expression is now used to construct an evolution equation for the total entropy (including fluid and solid). Such an equation must be of the form

$$
\partial\_t \left[ \Phi \mathcal{S}^{\mathcal{F}} + (\mathbf{1} - \Phi) \mathcal{S}^{\mathcal{S}} \right] + \partial\_k \left( \Phi \mathcal{S}^{\mathcal{F}} V\_k + P\_k \right) = Q,\tag{65}
$$

where <sup>S</sup><sup>S</sup> is the entropy of the rock, *Pk* is a diffusive current, and *<sup>Q</sup>* is a source term.

Eq. (64) directly gives the two equations that follow, by replacing d with *∂<sup>t</sup>* (first equation), then with *Vi∂<sup>i</sup>* (second equation):

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$$T\partial\_t \mathcal{S}^{\mathcal{F}} = \partial\_t \mathcal{U}^{\mathcal{F}} - \sum\_{a=1}^{\nu} \mathcal{C}^{\mathcal{F}a} \partial\_t \mathcal{R}^a - \sum\_{a=1}^{\nu} \Lambda^a (\partial\_k \mathcal{R}^a) \partial\_t (\partial\_k \mathcal{R}^a) . \tag{66}$$

*<sup>T</sup>*dS<sup>S</sup> <sup>¼</sup> <sup>d</sup>U<sup>S</sup>

*<sup>T</sup>∂<sup>t</sup>* ð ÞS <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> � � ¼ �*∂kJ*

1

X*<sup>ν</sup> α*¼1

determinable inside the model. Similarly, the currents *J*

**5. Constitutive equations and the minimal model**

product of a force (explicit or generalised) and a current. F<sup>F</sup>

*TQ* ¼ �ð Þ *Kk=<sup>T</sup> <sup>∂</sup>kT* <sup>þ</sup> <sup>Θ</sup>*ji<sup>∂</sup> jVi* � <sup>F</sup><sup>F</sup>

*<sup>T</sup> <sup>∂</sup>kKk* <sup>¼</sup> *<sup>∂</sup><sup>k</sup>*

*<sup>∂</sup><sup>t</sup>* <sup>Φ</sup>S<sup>F</sup> <sup>þ</sup> ð ÞS <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> � � <sup>þ</sup> *<sup>∂</sup><sup>k</sup>* <sup>Φ</sup>S<sup>F</sup>*Vk* <sup>þ</sup>

Taking the sum of Eqs. (69) and (73) one sees that *Q*S!<sup>F</sup> cancels, and that the right-hand side consists of a sum of scalar products and of a divergence, say *∂kKk*. A last step remains because the left-hand side has a multiplicative factor *T* (see Eqs. (69) and (73)). Dividing both sides by *T* does not affect the scalar products but it modifies the divergence, at least if one assumes that *T* is variable, to *∂kKk=T*. This

> *Kk T* � �

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ *<sup>R</sup>α∂iVi* <sup>þ</sup> *<sup>∂</sup>iI*

*Q*<sup>F</sup>!<sup>S</sup> and *Q*<sup>S</sup>!<sup>F</sup> do not appear in the entropy equation, and are thus not

summed, so that they are not determined individually inside the model. (See the

The source term in the entropy equation plays a central role in what follows. It has been written as a sum of scalar products. Each term of this sum is the scalar

while the gradient of some quantity (temperature, velocity component, ...) is a generalised force. The theory of irreversible processes states that linear relations exist (at least for processes not far from equilibrium), between forces and currents. The coefficients, called phenomenological coefficients, are parameters whose signs must be such that the source term cannot be negative, to ensure against a decrease of the entropy when the system is isolated. The linear relations just mentioned are constitutive equations, and it is implied that the phenomenological coefficients must be provided as input. For various reasons, some coefficients can be put equal

þ *Kk*

which implies

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*DOI: http://dx.doi.org/10.5772/intechopen.95474*

and, using (54):

is easily remedied by writing

One finally obtains

*Kk* ¼ *J* F *<sup>k</sup>* þ *J* S *<sup>k</sup>* � Φ

where

next section.)

**15**

, (71)

*<sup>k</sup>* � *Q*S!F*:* (73)

*<sup>T</sup>*<sup>2</sup> *<sup>∂</sup>kT:* (74)

¼ *Q*, (75)

*<sup>k</sup>:* (76)

*<sup>k</sup>∂kA<sup>α</sup>:* (77)

*<sup>k</sup>* only appear as

*<sup>k</sup>* is an explicit force,

*Kk T*

> X*ν α*¼1

X*ν α*¼1 *I α* *AαI α*

� �

*α i* � � � <sup>Φ</sup>

*kVk* � Φ

F *<sup>k</sup>* and *J* S

*<sup>T</sup>∂<sup>t</sup>* ð ÞS <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> � � <sup>¼</sup> *<sup>∂</sup><sup>t</sup>* ð ÞU <sup>1</sup> � <sup>Φ</sup> <sup>S</sup> � �, (72)

S

$$T\partial\_i \left(\mathcal{S}^{\mathrm{F}} V\_i\right) = T\mathcal{S}^{\mathrm{F}}\partial\_i V\_i + V\_i \partial\_i \mathcal{U}^{\mathrm{F}} - \sum\_{a=1}^{\nu} \mathcal{C}^{\mathrm{F}a} V\_i \partial\_i \mathcal{R}^a - V\_i \sum\_{a=1}^{\nu} \Lambda^a (\partial\_k \mathcal{R}^a)(\partial\_i \partial\_k \mathcal{R}^a) . \tag{67}$$

Since it is <sup>Φ</sup>S<sup>F</sup> that is needed to get Eq. (65), one needs to multiply the two equations above with Φ and then commute Φ with *∂<sup>t</sup>* and *∂i*. The same commutations are necessary on the right-hand sides of the above equations, so as to get <sup>Φ</sup>U<sup>F</sup> and thus allow the use of Eq. (55). Assumption A8 (Appendix 2) has been added to Assumption A4 so as to avoid the proliferation of *∂i*Φ-terms in the minimal model to be presented. With this simplification in place, and remembering to replace *Tij* in Eq. (55) with the expression obtained from Eqs. (60) and (61), one obtains, after elementary but somewhat long calculations:

$$\begin{split} T\left[\partial\_{t}(\boldsymbol{\Phi}\boldsymbol{\mathcal{S}}^{\mathrm{F}}) + \partial\_{i}(\boldsymbol{\Phi}\boldsymbol{\mathcal{S}}^{\mathrm{F}}\boldsymbol{V}\_{i})\right] &= -\,\partial\_{t}\boldsymbol{\mathcal{J}}\_{k}^{\mathrm{F}} + \,\mathcal{Q}\_{\boldsymbol{\mathcal{S}}\rightarrow\mathrm{F}} - \,\mathfrak{J}\_{k}^{\mathrm{F}}\boldsymbol{V}\_{k} + \,\Theta\_{ji}\partial\_{j}\boldsymbol{V}\_{i} \\ &+ \,\boldsymbol{\Phi}\sum\_{a=1}^{\nu}\mathcal{C}^{\mathrm{F}a}\partial\_{i}\boldsymbol{I}\_{i}^{a} + \,\boldsymbol{\Phi}\sum\_{a=1}^{\nu}\Lambda^{a}(\partial\_{k}\boldsymbol{R}^{a})\partial\_{k}\partial\_{i}\boldsymbol{I}\_{i}^{a} \\ &+ \,\boldsymbol{\Phi}\sum\_{a=1}^{\nu}\Lambda^{a}[\boldsymbol{R}^{a}(\partial\_{i}\boldsymbol{V}\_{i})\partial\_{k}\partial\_{k}\boldsymbol{R}^{a} + (\partial\_{k}\boldsymbol{R}^{a})\partial\_{k}(\boldsymbol{R}^{a}\partial\_{i}\boldsymbol{V}\_{i})]. \end{split} \tag{68}$$

Keeping in mind that one is looking for an entropy equation of the form of Eq. (65), and in anticipation of using the methods of irreversible processes, one now considers each term, or group of terms, on the right-hand side above and writes it either as a divergence, or as the sum of a divergence and a scalar product. Most terms on the first line already have the required form. The general term in the first sum on the second line is easily transformed as required, the general term in the second sum also, although with somewhat more work. As to the third line, it is easily seen to be a sum of divergences. One then gets:

$$\begin{aligned} T\left[\partial\_t \left(\boldsymbol{\Phi} \boldsymbol{\mathcal{S}}^{\mathrm{F}}\right) + \partial\_t \left(\boldsymbol{\Phi} \boldsymbol{\mathcal{S}}^{\mathrm{F}} V\_i\right)\right] &= \partial\_k \left[ -J\_k^{\mathrm{F}} + \boldsymbol{\Phi} \sum\_{a=1}^{\nu} \boldsymbol{A}^a I\_k^a + \boldsymbol{\Phi} \sum\_{a=1}^{\nu} \boldsymbol{\Lambda}^a (\partial\_k \boldsymbol{\mathcal{R}}^a) \left(\boldsymbol{\mathcal{R}}^a \partial\_i V\_i + \partial\_i I\_i^a\right) \right] \\ &+ Q\_{\mathbb{S}\to \mathbb{F}} - \mathfrak{F}\_k^{\mathrm{F}} V\_k + \Theta\_{\mathbb{H}} \partial\_j V\_i - \boldsymbol{\Phi} \sum\_{a=1}^{\nu} I\_i^a \partial\_i \boldsymbol{A}^a, \end{aligned} \tag{69}$$

where *<sup>A</sup><sup>α</sup>* ¼ C<sup>F</sup>*<sup>α</sup>* � <sup>Λ</sup>*<sup>α</sup>*∇<sup>2</sup> *R<sup>α</sup>*. Note that *A<sup>α</sup>* occurs twice on the right-hand side of Eq. (69), once as *A<sup>α</sup>*, and once as *∂iA<sup>α</sup>*, both times multiplying *I α <sup>k</sup>* in a sum over *α*. Eq. (36) then implies that one can modify the above expression of *A<sup>α</sup>* by the addition of any expression that does not depend on *α*. To conform with the notation of Ref. [2], and especially Ref. [3], one then sets

$$A^a = \mathcal{C}^{Fa} + \mathcal{W} - \Lambda^a \nabla^2 R^a. \tag{70}$$

as the expression to substitute on the right-hand side of Eq. (69).

The entropy equation for the solid is much easier to obtain because of Assumption A4 (Appendix 2). Indeed, Eq. (64) is replaced by

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$$T\mathbf{d}\mathcal{S}^{\mathcal{S}} = \mathbf{d}\mathcal{U}^{\mathcal{S}},\tag{71}$$

which implies

*<sup>T</sup>∂t*S<sup>F</sup> <sup>¼</sup> *<sup>∂</sup>t*U<sup>F</sup> �X*<sup>ν</sup>*

� � <sup>¼</sup> *<sup>T</sup>*S<sup>F</sup>*∂iVi* <sup>þ</sup> *Vi∂i*U<sup>F</sup> �X*<sup>ν</sup>*

elementary but somewhat long calculations:

� � � � ¼ � *<sup>∂</sup>kJ*

easily seen to be a sum of divergences. One then gets:

*<sup>T</sup> <sup>∂</sup><sup>t</sup>* <sup>Φ</sup>S<sup>F</sup> � � <sup>þ</sup> *<sup>∂</sup><sup>i</sup>* <sup>Φ</sup>S<sup>F</sup>*Vi*

*<sup>T</sup> <sup>∂</sup><sup>t</sup>* <sup>Φ</sup>S<sup>F</sup> � � <sup>þ</sup> *<sup>∂</sup><sup>i</sup>* <sup>Φ</sup>S<sup>F</sup>*Vi*

**14**

� � � � <sup>¼</sup> *<sup>∂</sup><sup>k</sup>* �*<sup>J</sup>*

*<sup>T</sup>∂<sup>i</sup>* <sup>S</sup><sup>F</sup>*Vi*

*α*¼1

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*α*¼1

F

Keeping in mind that one is looking for an entropy equation of the form of Eq. (65), and in anticipation of using the methods of irreversible processes, one now considers each term, or group of terms, on the right-hand side above and writes it either as a divergence, or as the sum of a divergence and a scalar product. Most terms on the first line already have the required form. The general term in the first sum on the second line is easily transformed as required, the general term in the second sum also, although with somewhat more work. As to the third line, it is

þ Φ X*ν α*¼1 C<sup>F</sup>*<sup>α</sup> ∂iI α <sup>i</sup>* þ Φ

þ Φ X*ν α*¼1

> F *<sup>k</sup>* þ Φ

<sup>þ</sup>*Q*<sup>S</sup>!<sup>F</sup> � <sup>F</sup><sup>F</sup>

"

Eq. (69), once as *A<sup>α</sup>*, and once as *∂iA<sup>α</sup>*, both times multiplying *I*

Assumption A4 (Appendix 2). Indeed, Eq. (64) is replaced by

of Ref. [2], and especially Ref. [3], one then sets

X*ν α*¼1

where *<sup>A</sup><sup>α</sup>* ¼ C<sup>F</sup>*<sup>α</sup>* � <sup>Λ</sup>*<sup>α</sup>*∇<sup>2</sup>*R<sup>α</sup>*. Note that *<sup>A</sup><sup>α</sup>* occurs twice on the right-hand side of

Eq. (36) then implies that one can modify the above expression of *A<sup>α</sup>* by the addition of any expression that does not depend on *α*. To conform with the notation

*<sup>A</sup><sup>α</sup>* ¼ C<sup>F</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>W</sup>* � <sup>Λ</sup>*<sup>α</sup>*∇<sup>2</sup>

as the expression to substitute on the right-hand side of Eq. (69). The entropy equation for the solid is much easier to obtain because of

*A<sup>α</sup>I α k* þΦ X*ν α*¼1

*kVk* <sup>þ</sup> <sup>Θ</sup>*ji<sup>∂</sup> jVi* � <sup>Φ</sup>

*<sup>k</sup>* <sup>þ</sup> *<sup>Q</sup>*<sup>S</sup>!<sup>F</sup> � <sup>F</sup><sup>F</sup>

Since it is <sup>Φ</sup>S<sup>F</sup> that is needed to get Eq. (65), one needs to multiply the two equations above with Φ and then commute Φ with *∂<sup>t</sup>* and *∂i*. The same commutations are necessary on the right-hand sides of the above equations, so as to get <sup>Φ</sup>U<sup>F</sup> and thus allow the use of Eq. (55). Assumption A8 (Appendix 2) has been added to Assumption A4 so as to avoid the proliferation of *∂i*Φ-terms in the minimal model to be presented. With this simplification in place, and remembering to replace *Tij* in Eq. (55) with the expression obtained from Eqs. (60) and (61), one obtains, after

<sup>C</sup>F*<sup>α</sup>∂tR<sup>α</sup>* �X*<sup>ν</sup>*

*α*¼1

X*ν α*¼1

*kVk* <sup>þ</sup> <sup>Θ</sup>*ji∂jVi*

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ*∂k∂iI*

<sup>Λ</sup>*<sup>α</sup> <sup>R</sup><sup>α</sup>*ð Þ *<sup>∂</sup>iVi <sup>∂</sup>k∂kR<sup>α</sup>* <sup>þ</sup> *<sup>∂</sup>kR<sup>α</sup>* ð Þ*∂<sup>k</sup> <sup>R</sup><sup>α</sup>* ð Þ *<sup>∂</sup>iVi* ½ �*:*

*α i*

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ *<sup>R</sup><sup>α</sup>∂iVi* <sup>þ</sup> *<sup>∂</sup>iI*

X*ν α*¼1 *I α <sup>i</sup> <sup>∂</sup>iA<sup>α</sup>*,

*α*

*R<sup>α</sup>:* (70)

(68)

*α i*

(69)

� �#

*<sup>k</sup>* in a sum over *α*.

X*ν α*¼1

<sup>C</sup>F*<sup>α</sup>Vi∂iR<sup>α</sup>* � *Vi*

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ*∂<sup>t</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ*:* (66)

<sup>Λ</sup>*<sup>α</sup> <sup>∂</sup>kR<sup>α</sup>* ð Þ *<sup>∂</sup>i∂kR<sup>α</sup>* ð Þ*:* (67)

$$T\partial\_t\left[ (\mathbf{1} - \Phi)\mathcal{S}^S \right] = \partial\_t\left[ (\mathbf{1} - \Phi)\mathcal{U}^S \right],\tag{72}$$

and, using (54):

$$T\partial\_t\left[ (\mathbf{1} - \Phi)\mathcal{S}^{\mathcal{S}} \right] = -\partial\_k f^{\mathcal{S}}\_k - Q\_{\mathcal{S}\to\mathcal{F}}.\tag{73}$$

Taking the sum of Eqs. (69) and (73) one sees that *Q*S!<sup>F</sup> cancels, and that the right-hand side consists of a sum of scalar products and of a divergence, say *∂kKk*. A last step remains because the left-hand side has a multiplicative factor *T* (see Eqs. (69) and (73)). Dividing both sides by *T* does not affect the scalar products but it modifies the divergence, at least if one assumes that *T* is variable, to *∂kKk=T*. This is easily remedied by writing

$$\frac{1}{T}\partial\_k K\_k = \partial\_k \left(\frac{K\_k}{T}\right) + \frac{K\_k}{T^2} \partial\_k T. \tag{74}$$

One finally obtains

$$
\partial\_t \left[ \Phi \mathcal{S}^{\mathcal{F}} + (\mathbf{1} - \Phi) \mathcal{S}^{\mathcal{S}} \right] + \partial\_k \left( \Phi \mathcal{S}^{\mathcal{F}} V\_k + \frac{K\_k}{T} \right) = Q,\tag{75}
$$

where

$$K\_k = f\_k^\mathrm{F} + f\_k^\mathrm{S} - \Phi \sum\_{a=1}^\nu \Lambda^a (\partial\_k \mathcal{R}^a) \left( R^a \partial\_i V\_i + \partial\_i I\_i^a \right) - \Phi \sum\_{a=1}^\nu A^a I\_k^a. \tag{76}$$

$$T\mathcal{Q} = -(K\_k/T)\partial\_k T + \Theta\_{\bar{\mathcal{H}}}\partial\_{\bar{\mathcal{J}}}V\_i - \mathfrak{F}\_k^F V\_k - \Phi \sum\_{a=1}^{\nu} I\_k^a \partial\_k A^a. \tag{77}$$

*Q*<sup>F</sup>!<sup>S</sup> and *Q*<sup>S</sup>!<sup>F</sup> do not appear in the entropy equation, and are thus not determinable inside the model. Similarly, the currents *J* F *<sup>k</sup>* and *J* S *<sup>k</sup>* only appear as summed, so that they are not determined individually inside the model. (See the next section.)

#### **5. Constitutive equations and the minimal model**

The source term in the entropy equation plays a central role in what follows. It has been written as a sum of scalar products. Each term of this sum is the scalar product of a force (explicit or generalised) and a current. F<sup>F</sup> *<sup>k</sup>* is an explicit force, while the gradient of some quantity (temperature, velocity component, ...) is a generalised force. The theory of irreversible processes states that linear relations exist (at least for processes not far from equilibrium), between forces and currents. The coefficients, called phenomenological coefficients, are parameters whose signs must be such that the source term cannot be negative, to ensure against a decrease of the entropy when the system is isolated. The linear relations just mentioned are constitutive equations, and it is implied that the phenomenological coefficients must be provided as input. For various reasons, some coefficients can be put equal

to zero, one often recurring reason being the belief that they are negligible in the physical situation considered. Thus there is a family of models, each member being characterised by its set of non-zero phenomenological coefficients.

The previously mentioned minimal model (see the text after Eq. (67)), is the model that contains the least possible number of non-zero phenomenological coefficients. "Least possible" means that one must obey the constraints that exist for these coefficients (Onsager symmetry, isotropy, sign) and arbitrarily setting some of them equal to zero is not always possible. A certain amount of trial and error is also required to avoid unduly reducing the model's predictive power.

The usual vector notation is used for tensors of order 1, i.e., a bold faced letter is used when the subscript can be suppressed. Vector forces are here denoted **X** (with components *Xi*), and vector currents are denoted **Y** (with components *Yi*), with a superscript to discriminate between the different currents and forces in an alphabetic order aa shown below. (*Y* has been chosen instead of the traditional *J* so as to avoid confusion with the currents related to the energy equations.) The usual tensor conventions are assumed for latin subscripts (see "A note on notation" in the Introduction). Concerning the second order tensors Θ*ji* and *∂ jVi*, on the right-hand side of Eq. (77), one sets <sup>Δ</sup>*ji* � *<sup>∂</sup> jVi* and, referring to Appendix 3, especially to Eq. (115), one writes

$$
\Delta\_{\vec{\eta}} = X^A \delta\_{\vec{\eta}} + X^B\_{\vec{\eta}} + \varepsilon\_{\vec{\eta}k} \tilde{X}\_k,\tag{78}
$$

$$
\Theta\_{\vec{\eta}} = Y^A \delta\_{\vec{\eta}} + Y^B\_{\vec{\eta}}, \tag{79}
$$

*YC <sup>i</sup>* <sup>¼</sup> *<sup>L</sup>CA*

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*Y<sup>D</sup> <sup>i</sup>* <sup>¼</sup> *<sup>L</sup>DA*

*Yα <sup>i</sup>* <sup>¼</sup> *<sup>L</sup><sup>α</sup><sup>A</sup>*

scripts, denoted say, *LMN*

**5.1 The viscosity tensor Θ***ij*

the Onsager symmetry:

Then *<sup>Y</sup><sup>A</sup>* <sup>¼</sup> *<sup>Y</sup><sup>B</sup>*

**5.2 Vector currents and forces**

being second order tensors:

**17**

*<sup>i</sup> <sup>X</sup><sup>A</sup>* <sup>þ</sup> *<sup>L</sup>CB*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

*<sup>i</sup> <sup>X</sup><sup>A</sup>* <sup>þ</sup> *<sup>L</sup>DB*

*<sup>i</sup> <sup>X</sup><sup>A</sup>* <sup>þ</sup> *<sup>L</sup><sup>α</sup><sup>B</sup>*

*LMN* ð Þ *<sup>n</sup>* <sup>¼</sup> *LNM*

*ikl X<sup>B</sup>*

*ikl X<sup>B</sup>*

*iklX<sup>B</sup>*

ð Þ *<sup>n</sup>* , obeys

concerning subscripts are not needed in what follows.

*ij* ¼ 0, so that

*YC <sup>i</sup>* <sup>¼</sup> *LCC ik X<sup>C</sup>*

*Y<sup>D</sup> <sup>i</sup>* <sup>¼</sup> *LCD ik X<sup>C</sup>*

> *<sup>k</sup>* <sup>þ</sup> *LD<sup>α</sup> ik X<sup>D</sup>*

*Yα <sup>i</sup>* <sup>¼</sup> *LC<sup>α</sup> ik X<sup>C</sup>*

and the source term of the entropy equation reduces to

*TQ* <sup>¼</sup> **<sup>X</sup>***<sup>C</sup>* � **<sup>Y</sup>***<sup>C</sup>* <sup>þ</sup> **<sup>X</sup>***<sup>D</sup>* � **<sup>Y</sup>***<sup>D</sup>* <sup>þ</sup>X*<sup>ν</sup>*

System (82) now reduces to linear relations between vectors, the coefficients

*<sup>k</sup>* <sup>þ</sup> *<sup>L</sup>CD ik X<sup>D</sup>*

*<sup>k</sup>* <sup>þ</sup> *<sup>L</sup>DD ik X<sup>D</sup>*

> *<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*kl* <sup>þ</sup> *LCC ik X<sup>C</sup>*

*kl* <sup>þ</sup> *LDC ik X<sup>C</sup>*

*kl* <sup>þ</sup> *<sup>L</sup><sup>α</sup><sup>C</sup> ik X<sup>C</sup>*

*<sup>k</sup>* <sup>þ</sup> *LCD ik X<sup>D</sup>*

*<sup>k</sup>* <sup>þ</sup> *LDD ik X<sup>D</sup>*

*<sup>k</sup>* <sup>þ</sup> *<sup>L</sup><sup>α</sup><sup>D</sup> ik X<sup>D</sup>*

In these expressions, the *L* are the phenomenological coefficients. They are independent of the generalised forces, but they can depend on the temperature and the component densities. They are tensors, their orders being equal to the number of subscripts. They obey the Onsager relations [11, 15]: any coefficient with *n* sub-

where the *n* subscripts are the same but not necessarily in the same order. Details

With hindsight, one knows that the upscaled viscosity tensor is not required, so that one is justified in setting equal to zero all the *L*'s in the first two lines of the system of Eqs. (82), and also setting to zero all the *L*'s related to the zeroed ones by

*<sup>L</sup>AZ* <sup>¼</sup> *<sup>L</sup>BZ* <sup>¼</sup> *LZA* <sup>¼</sup> *LZB* <sup>¼</sup> 0, for any Z*:* (84)

*α*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*Lαβ ik <sup>X</sup><sup>β</sup>* *LC<sup>β</sup> ik <sup>X</sup><sup>β</sup>*

*LD<sup>β</sup> ik <sup>X</sup><sup>β</sup>*

Θ*ij* ¼ 0, (85)

**<sup>X</sup>***<sup>α</sup>* � **<sup>Y</sup>***<sup>α</sup>:* (86)

*<sup>k</sup>* (87)

*<sup>k</sup>* (88)

*<sup>k</sup>*, ð Þ *α* ¼ 1 … *ν* , (89)

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*LC<sup>β</sup> ik <sup>X</sup><sup>β</sup> k*

*LD<sup>β</sup> ik <sup>X</sup><sup>β</sup> k*

*<sup>k</sup>*, ð Þ *α* ¼ 1 … *ν :*

*Lαβ ik <sup>X</sup><sup>β</sup>*

ð Þ *<sup>n</sup>* , ð Þ *<sup>M</sup>* 6¼ *<sup>N</sup>* <sup>¼</sup> *<sup>A</sup>*, *<sup>B</sup>*,*C*, *<sup>D</sup>*, 1, … , *<sup>ν</sup>* , (83)

where *X<sup>A</sup>* and *X<sup>B</sup> ij* are found by replacing *Z* by Δ in Eqs. (115). Note that *Y<sup>B</sup> ij*, being symmetric, has no antisymmetric part.

Referring now to the vectors on the right-hand side of Eq. (77), one introduces the following notation:

$$X\_k^C = -\partial\_k T \qquad \qquad Y\_k^C = K\_k / T$$

$$X\_k^D = -\mathfrak{F}\_k^F \qquad \qquad Y\_k^D = V\_k \tag{80}$$

$$X\_k^a = -\Phi \partial\_k A^a, \qquad \qquad Y\_k^a = I\_k^a.$$

Using Eq. (116), one easily finds that

$$T\mathbf{Q} = \mathbf{X}^A Y^A + \mathbf{X}\_{\vec{\eta}}^B Y\_{\vec{\eta}}^B + \mathbf{X}^C \cdot \mathbf{Y}^C + \mathbf{X}^D \cdot \mathbf{Y}^D + \sum\_{a=1}^{\nu} \mathbf{X}^a \cdot \mathbf{Y}^a. \tag{81}$$

Note that *Q* is a proper scalar since the source of entropy is independent of the coordinate system, and does not change sign when the coordinate system changes handedness. The right-hand side of Eq. (81) is then a sum of scalar products of proper tensors, the only pseudo vector, *X*~ *<sup>k</sup>*, having dropped out.

Referring to the first sentence of this section, one writes the currents as linear combinations of the forces:

$$\begin{aligned} Y^A &= L^{AA}X^A + L^{AB}\_{kl}X^B\_{kl} + L^{AC}\_k X^C\_k + L^{AD}\_k X^D\_k + \sum\_{\beta=1}^{\nu} L^{A\beta}\_k X^\beta\_k \\ Y^B\_{ij} &= L^{BA}\_{ij}X^A + L^{BB}\_{ijk}X^B\_{kl} + L^{BC}\_{ijk}X^C\_k + L^{BD}\_{ijk}X^D\_k + \sum\_{\beta=1}^{\nu} L^{B\beta}\_{ijk}X^\beta\_k \end{aligned} \tag{82}$$

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to zero, one often recurring reason being the belief that they are negligible in the physical situation considered. Thus there is a family of models, each member being

The previously mentioned minimal model (see the text after Eq. (67)), is the model that contains the least possible number of non-zero phenomenological coefficients. "Least possible" means that one must obey the constraints that exist for these coefficients (Onsager symmetry, isotropy, sign) and arbitrarily setting some of them equal to zero is not always possible. A certain amount of trial and error is

The usual vector notation is used for tensors of order 1, i.e., a bold faced letter is used when the subscript can be suppressed. Vector forces are here denoted **X** (with components *Xi*), and vector currents are denoted **Y** (with components *Yi*), with a superscript to discriminate between the different currents and forces in an alphabetic order aa shown below. (*Y* has been chosen instead of the traditional *J* so as to avoid confusion with the currents related to the energy equations.) The usual tensor conventions are assumed for latin subscripts (see "A note on notation" in the Introduction). Concerning the second order tensors Θ*ji* and *∂ jVi*, on the right-hand side of Eq. (77), one sets <sup>Δ</sup>*ji* � *<sup>∂</sup> jVi* and, referring to Appendix 3, especially to

*ij* <sup>þ</sup> *<sup>ε</sup>ijkX*<sup>~</sup> *<sup>k</sup>*, (78)

*ij*, (79)

*ij*,

(80)

(82)

**<sup>X</sup>***<sup>α</sup>* � **<sup>Y</sup>***<sup>α</sup>:* (81)

characterised by its set of non-zero phenomenological coefficients.

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also required to avoid unduly reducing the model's predictive power.

<sup>Δ</sup>*ij* <sup>¼</sup> *<sup>X</sup><sup>A</sup>δij* <sup>þ</sup> *<sup>X</sup><sup>B</sup>*

<sup>Θ</sup>*ij* <sup>¼</sup> *<sup>Y</sup><sup>A</sup>δij* <sup>þ</sup> *<sup>Y</sup><sup>B</sup>*

Referring now to the vectors on the right-hand side of Eq. (77), one introduces

*ij* are found by replacing *Z* by Δ in Eqs. (115). Note that *Y<sup>B</sup>*

*YC*

*Y<sup>D</sup> <sup>k</sup>* ¼ *Vk*

*Yα <sup>k</sup>* ¼ *I α k:*

*ij* <sup>þ</sup> **<sup>X</sup>***<sup>C</sup>* � **<sup>Y</sup>***<sup>C</sup>* <sup>þ</sup> **<sup>X</sup>***<sup>D</sup>* � **<sup>Y</sup>***<sup>D</sup>* <sup>þ</sup>X*<sup>ν</sup>*

Note that *Q* is a proper scalar since the source of entropy is independent of the coordinate system, and does not change sign when the coordinate system changes handedness. The right-hand side of Eq. (81) is then a sum of scalar products of

Referring to the first sentence of this section, one writes the currents as linear

*<sup>k</sup>* <sup>þ</sup> *<sup>L</sup>AD <sup>k</sup> X<sup>D</sup>*

*<sup>k</sup>* <sup>þ</sup> *LBD ijk X<sup>D</sup>*

*kl* <sup>þ</sup> *LAC <sup>k</sup> X<sup>C</sup>*

*kl* <sup>þ</sup> *LBC ijk X<sup>C</sup>* *<sup>k</sup>* ¼ *Kk=T*

*α*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*<sup>k</sup>* <sup>þ</sup>X*<sup>ν</sup> β*¼1

*L<sup>A</sup><sup>β</sup> <sup>k</sup> <sup>X</sup><sup>β</sup> k*

*LB<sup>β</sup> ijkX<sup>β</sup> k*

Eq. (115), one writes

where *X<sup>A</sup>* and *X<sup>B</sup>*

the following notation:

being symmetric, has no antisymmetric part.

Using Eq. (116), one easily finds that

*<sup>Y</sup><sup>A</sup>* <sup>¼</sup> *LAAXA* <sup>þ</sup> *LAB*

*ij <sup>X</sup><sup>A</sup>* <sup>þ</sup> *LBB*

*TQ* <sup>¼</sup> <sup>3</sup>*XAYA* <sup>þ</sup> *<sup>X</sup><sup>B</sup>*

combinations of the forces:

*YB ij* <sup>¼</sup> *LBA*

**16**

*X<sup>C</sup>*

*X<sup>D</sup> <sup>k</sup>* ¼ �F<sup>F</sup> *k*

*Xα*

*ijY<sup>B</sup>*

proper tensors, the only pseudo vector, *X*~ *<sup>k</sup>*, having dropped out.

*kl X<sup>B</sup>*

*ijklX<sup>B</sup>*

*<sup>k</sup>* ¼ �*∂kT*

*<sup>k</sup>* ¼ �Φ*∂kA<sup>α</sup>*,

$$Y\_i^C = L\_i^{CA}X^A + L\_{ikl}^{CB}X\_{kl}^B + L\_{ik}^{CC}X\_k^C + L\_{ik}^{CD}X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{C\beta}X\_k^{\beta}$$

$$Y\_i^D = L\_i^{DA}X^A + L\_{ikl}^{DB}X\_{kl}^B + L\_{ik}^{DC}X\_k^C + L\_{ik}^{DD}X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{D\beta}X\_k^{\beta}$$

$$Y\_i^a = L\_i^{aA}X^A + L\_{ikl}^{dB}X\_{kl}^B + L\_{ik}^{aC}X\_k^C + L\_{ik}^{AD}X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{d\beta}X\_k^{\beta}, \quad (a = 1 \dots \nu).$$

In these expressions, the *L* are the phenomenological coefficients. They are independent of the generalised forces, but they can depend on the temperature and the component densities. They are tensors, their orders being equal to the number of subscripts. They obey the Onsager relations [11, 15]: any coefficient with *n* subscripts, denoted say, *LMN* ð Þ *<sup>n</sup>* , obeys

$$L\_{(n)}^{\rm MN} = L\_{(n)}^{\rm NM}, \quad (\mathbf{M} \neq \mathbf{N} = \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}, \mathbf{1}, \dots, \nu), \tag{83}$$

where the *n* subscripts are the same but not necessarily in the same order. Details concerning subscripts are not needed in what follows.

#### **5.1 The viscosity tensor Θ***ij*

With hindsight, one knows that the upscaled viscosity tensor is not required, so that one is justified in setting equal to zero all the *L*'s in the first two lines of the system of Eqs. (82), and also setting to zero all the *L*'s related to the zeroed ones by the Onsager symmetry:

$$\mathbf{L}^{\mathbf{AZ}} = \mathbf{L}^{\mathbf{BZ}} = \mathbf{L}^{\mathbf{ZA}} = \mathbf{L}^{\mathbf{ZB}} = \mathbf{0}, \quad \text{for any } \mathbf{Z}. \tag{84}$$

Then *<sup>Y</sup><sup>A</sup>* <sup>¼</sup> *<sup>Y</sup><sup>B</sup> ij* ¼ 0, so that

$$
\Theta\_{\vec{\eta}} = \mathbf{0},
\tag{85}
$$

and the source term of the entropy equation reduces to

$$TQ = \mathbf{X}^C \cdot \mathbf{Y}^C + \mathbf{X}^D \cdot \mathbf{Y}^D + \sum\_{a=1}^{\nu} \mathbf{X}^a \cdot \mathbf{Y}^a. \tag{86}$$

#### **5.2 Vector currents and forces**

System (82) now reduces to linear relations between vectors, the coefficients being second order tensors:

$$Y\_i^C = L\_{ik}^{CC} X\_k^C + L\_{ik}^{CD} X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{C\beta} X\_k^{\beta} \tag{87}$$

$$Y\_i^D = L\_{ik}^{CD} X\_k^C + L\_{ik}^{DD} X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{D\beta} X\_k^{\beta} \tag{88}$$

$$Y\_i^a = L\_{ik}^{Ca} X\_k^C + L\_{ik}^{Da} X\_k^D + \sum\_{\beta=1}^{\nu} L\_{ik}^{a\beta} X\_k^\beta, \quad (a = 1 \dots \nu), \tag{89}$$

where Onsager symmetry is accounted for. Remembering that *Y<sup>α</sup> <sup>i</sup>* ¼ *I α <sup>k</sup>*, Eq. (36) leads to three constraints on the *L*-coefficients of Eq. (89):

$$\sum\_{a=1}^{\nu} L\_{ik}^{Ca} = \sum\_{a=1}^{\nu} L\_{ik}^{Da} = \sum\_{a=1}^{\nu} L\_{ik}^{a\beta} = \mathbf{0}. \tag{90}$$

<sup>ℓ</sup>*CC* <sup>&</sup>gt;0, <sup>ℓ</sup>*DD* <sup>&</sup>gt;0, <sup>ℓ</sup>*αβ* <sup>&</sup>lt; 0, ð Þ *<sup>α</sup>*<sup>&</sup>lt; *<sup>β</sup> :* (96)

<sup>ℓ</sup>*CC* <sup>¼</sup> *<sup>k</sup>=T*, implying *Ki* ¼ �*k∂iT*, (97)

Φ*η*

*<sup>i</sup>* <sup>¼</sup> *Vi=*ℓ*DD*, and it is shown below that

� �, (98)

*<sup>∂</sup>iP*bF � *Rfi*

*α*¼1

*KR* Φ*η ∂i*∇<sup>2</sup>

Λ*αRα∂i*∇<sup>2</sup>

*Rα*

*:* (99)

, and the material

*R:* (100)

, (101)

Using the notation of expressions (80) in the first of expressions (93) one gets *Ki* proportional to *∂iT*. Now *Ki=T* is an entropy transport by diffusion, so that *Ki* is

where *k*>0 is the thermal conductivity of the averaged solid–fluid system.

, implies *Vi* <sup>¼</sup> *<sup>K</sup>*

*<sup>∂</sup>iP*bF � *Rfi* <sup>þ</sup> *<sup>R</sup>*ð*∂tVi* <sup>þ</sup> *Vk∂kVi*Þ �X*<sup>ν</sup>*

away from interphase regions. In this expression, *η* is the viscosity of the pore fluid, and *K* is the absolute permeability. As already mentioned, the phenomenological coefficients can be functions of the component densities and of the temperature, and it is known that *η* is such a function (see Section 3.4 in Ref. [3]). The possibility of letting *K* be such a function is not used in the minimal model. To prove the implication (98), one uses Eq. (42), with the upscaled stress tensor given

" #

The sum over *α* vanishes in the bulk fluid and one is left with the Darcy formula with an additional term, proportional to the material derivative of the velocity. One can carry out order of magnitude estimates of the three first terms in the square brackets above, in the manner of Section 4.1 of Ref. [3]. Using the numerical values given in the Appendix of the same reference, one easily finds that, if the gradient of

derivative term is of order 10�9. Neglecting the material derivative, one obtains the Darcy formula, modified by terms that only become significant inside the interphase regions. Specialised to a one component fluid, this modified Darcy formula is:

> *<sup>∂</sup>iP*bF � *Rfi* � � <sup>þ</sup> <sup>Λ</sup>

It is shown in Ref. [4] that the added non-Darcy term can, in some well-defined flow types, produce a relative permeability when its numerical contribution is taken away as an added term, then put back as a multiplicative factor to the Darcy term. However, it is concluded in Ref. [4] that relative permeabilities cannot capture the

Given below is another version of *Vi*, that follows by application of the Gibbs-Duhem equation, obtained by differentiating <sup>F</sup>bF and using the expression of dFbF

> *<sup>R</sup><sup>α</sup>∂iA<sup>β</sup>* þ S<sup>F</sup>*∂iT* " #

X*ν α*¼1

These coefficients are determined below.

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*DOI: http://dx.doi.org/10.5772/intechopen.95474*

The second of expressions (93) gives F<sup>F</sup>

<sup>ℓ</sup>*DD* <sup>¼</sup> *<sup>K</sup>* Φ2 *η*

by Eqs. (60), (61), and (85). Using Eq. (41) one obtains:

pressure is of order 1, then the gravity term *Rfi* is of order 10�<sup>3</sup>

*Vi* ¼ � *<sup>K</sup>* Φ*η*

> *Vi* ¼ � *<sup>K</sup>* Φ*η*

full complexity of two-phase flow.

(Eqs. (2)):

**19**

See Ref. [3].

the choice

*Vi* ¼ � *<sup>K</sup>* Φ*η*

heat transport by diffusion. To recover Fourier's law one sets

The linear combinations above show the possibilities of constructing models where interactions between thermal conduction, fluid flow, and mass diffusion are quantified by choosing the *L* coefficients. Also, thermal conduction and permeability can be modelled by second order tensors, through tensors *LCC ik* and *LDD ik* .

However, if one limits oneself to the minimal model where: (i) the diffusive entropy current, *Ki=T*, is only due to the *∂kT* force, (ii) the F<sup>F</sup> *<sup>k</sup>* frictional force is only due to fluid velocity, *Vi*, (iii) each non-convective mass current, *I α <sup>i</sup>* , is only due to the gradients of the *A<sup>β</sup>* , then one requires

$$L\_{ik}^{CD} = \mathbf{0}, \quad \text{and} \quad L\_{ik}^{Ca} = L\_{ik}^{Da} = \mathbf{0} \quad \text{for all} \quad a,\tag{91}$$

not violating the constraints in Eqs. (90).

Note that it is not possible to simplify the model to the extent that *all* crosscouplings are eliminated, since that would imply that *Lαβ ik* are zero except when the superscripts are equal: such a matrix would not obey the third constraint in (90). One could of course set *Lαβ ik* ¼ 0 for all *α* and *β*, but that would eliminate all the non-convective mass currents from the model, and probably make it useless.

In the minimal model one can add a fourth requirement to the three above: the upscaled fluid and the upscaled medium are isotropic. Then further simplifications result since the remaining second order tensors, *LCC ik* , *LDD ik* , and *<sup>L</sup>αβ ik* , that are properties of the solid and the fluid, must be invariant under rotations of the coordinate axes. Such tensors are called isotropic and it can be shown that an isotropic second order tensor is proportional to the Kronecker delta (see Ref. [15]). Isotropy thus introduces the following restrictions:

$$L\_{ik}^{CC} = \ell^{CC} \delta\_{ik}, \qquad L\_{ik}^{DD} = \ell^{DD} \delta\_{ik}, \qquad L\_{ik}^{a\beta} = \ell^{a\beta} \delta\_{ik}, \tag{92}$$

and one gets

$$\mathbf{Y}^{\mathrm{C}} = t^{\mathrm{CC}} \mathbf{X}^{\mathrm{C}}, \qquad \mathbf{X}^{D} = \left(\mathbf{1}/t^{\mathrm{DD}}\right) \mathbf{Y}^{D}, \qquad \mathbf{Y}^{a} = \sum\_{\beta=1}^{\nu} t^{a\beta} \mathbf{X}^{\beta}. \tag{93}$$

The source term of the entropy equation is now

$$\mathbf{TQ} = \boldsymbol{\epsilon}^{\text{CC}} \left| \mathbf{X}^{\text{C}} \right|^2 + \left( \mathbf{1}/\boldsymbol{\epsilon}^{\text{DD}} \right) \left| \mathbf{Y}^{\text{D}} \right|^2 + \sum\_{a=1}^{\nu} \sum\_{\beta=1}^{\nu} \boldsymbol{\epsilon}^{a\beta} \mathbf{X}^a \cdot \mathbf{X}^\beta. \tag{94}$$

Keeping in mind that the ℓ*αβ*-matrix is symmetric, and that the sum of all elements on the same line or the same column is zero, it can be shown that

$$\sum\_{a=1}^{\nu} \sum\_{\beta=1}^{\nu} \ell^{a\beta} \mathbf{X}^a \cdot \mathbf{X}^\beta = -\sum\_{a,\beta, a<\beta} \ell^{a\beta} \left| \mathbf{X}^a - \mathbf{X}^\beta \right|^2. \tag{95}$$

According to the expression for the source of entropy above, the remaining phenomenological coefficients must then satisfy

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

where Onsager symmetry is accounted for. Remembering that *Y<sup>α</sup>*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*LD<sup>α</sup> ik* <sup>¼</sup> <sup>X</sup>*<sup>ν</sup> α*¼1

The linear combinations above show the possibilities of constructing models where interactions between thermal conduction, fluid flow, and mass diffusion are quantified by choosing the *L* coefficients. Also, thermal conduction and permeabil-

However, if one limits oneself to the minimal model where: (i) the diffusive

*ik* <sup>¼</sup> *LD<sup>α</sup>*

Note that it is not possible to simplify the model to the extent that *all* cross-

superscripts are equal: such a matrix would not obey the third constraint in (90).

In the minimal model one can add a fourth requirement to the three above: the upscaled fluid and the upscaled medium are isotropic. Then further simplifications

*ik* <sup>¼</sup> <sup>ℓ</sup>*DDδik*, *<sup>L</sup>αβ*

non-convective mass currents from the model, and probably make it useless.

properties of the solid and the fluid, must be invariant under rotations of the coordinate axes. Such tensors are called isotropic and it can be shown that an isotropic second order tensor is proportional to the Kronecker delta (see Ref. [15]).

**<sup>Y</sup>***<sup>C</sup>* <sup>¼</sup> <sup>ℓ</sup>*CC***X***<sup>C</sup>*, **<sup>X</sup>***<sup>D</sup>* <sup>¼</sup> <sup>1</sup>*=*ℓ*DD* � �**Y***<sup>D</sup>*, **<sup>Y</sup>***<sup>α</sup>* <sup>¼</sup> <sup>X</sup>*<sup>ν</sup>*

<sup>þ</sup> <sup>1</sup>*=*ℓ*DD* � � **<sup>Y</sup>***<sup>D</sup>* �

<sup>ℓ</sup>*αβ***X***<sup>α</sup>* � **<sup>X</sup>***<sup>β</sup>* ¼ � <sup>X</sup>

Keeping in mind that the ℓ*αβ*-matrix is symmetric, and that the sum of all elements on the same line or the same column is zero, it can be shown that

According to the expression for the source of entropy above, the remaining

� � � 2 þX*<sup>ν</sup> α*¼1

*α*, *β*, *α*<*β*

*Lαβ*

*ik* ¼ 0 for all *α* and *β*, but that would eliminate all the

*ik* , *LDD*

*ik* , and *<sup>L</sup>αβ*

*β*¼1

X*ν β*¼1

<sup>ℓ</sup>*αβ* **<sup>X</sup>***<sup>α</sup>* � **<sup>X</sup>***<sup>β</sup>* � � � � 2

ℓ*αβ***X***<sup>β</sup>*

<sup>ℓ</sup>*αβ***X***<sup>α</sup>* � **<sup>X</sup>***<sup>β</sup>*

leads to three constraints on the *L*-coefficients of Eq. (89):

*LC<sup>α</sup> ik* <sup>¼</sup> <sup>X</sup>*<sup>ν</sup> α*¼1

ity can be modelled by second order tensors, through tensors *LCC*

due to fluid velocity, *Vi*, (iii) each non-convective mass current, *I*

, then one requires

entropy current, *Ki=T*, is only due to the *∂kT* force, (ii) the F<sup>F</sup>

*ik* <sup>¼</sup> 0, and *<sup>L</sup><sup>C</sup><sup>α</sup>*

couplings are eliminated, since that would imply that *Lαβ*

result since the remaining second order tensors, *LCC*

Isotropy thus introduces the following restrictions:

*ik* <sup>¼</sup> <sup>ℓ</sup>*CCδik*, *<sup>L</sup>DD*

The source term of the entropy equation is now

� � � 2

*TQ* <sup>¼</sup> <sup>ℓ</sup>*CC* **<sup>X</sup>***<sup>C</sup>* �

X*ν α*¼1

X*ν β*¼1

phenomenological coefficients must then satisfy

*LCC*

and one gets

**18**

the gradients of the *A<sup>β</sup>*

One could of course set *Lαβ*

*LCD*

not violating the constraints in Eqs. (90).

X*ν α*¼1

*<sup>i</sup>* ¼ *I α*

*ik* ¼ 0*:* (90)

*ik* and *LDD*

*ik* are zero except when the

*ik* , that are

*ik* <sup>¼</sup> <sup>ℓ</sup>*αβδik*, (92)

*:* (93)

*:* (94)

*:* (95)

*α*

*ik* ¼ 0 for all *α*, (91)

*ik* .

*<sup>i</sup>* , is only due to

*<sup>k</sup>* frictional force is only

*<sup>k</sup>*, Eq. (36)

$$t^{\mathcal{C}C} > 0, \qquad t^{\mathcal{D}D} > 0, \qquad t^{a\beta} < 0, \quad (a < \beta). \tag{96}$$

These coefficients are determined below.

Using the notation of expressions (80) in the first of expressions (93) one gets *Ki* proportional to *∂iT*. Now *Ki=T* is an entropy transport by diffusion, so that *Ki* is heat transport by diffusion. To recover Fourier's law one sets

$$t^{\text{CC}} = k/T,\quad \text{imlying}\quad K\_i = -k\partial\_i T,\tag{97}$$

where *k*>0 is the thermal conductivity of the averaged solid–fluid system. See Ref. [3].

The second of expressions (93) gives F<sup>F</sup> *<sup>i</sup>* <sup>¼</sup> *Vi=*ℓ*DD*, and it is shown below that the choice

$$\mathcal{E}^{\rm DD} = \frac{K}{\Phi^2 \eta}, \quad \text{implies} \quad V\_i = \frac{K}{\Phi \eta} \left( \partial\_i \mathcal{P}^{\rm bF} - R \mathcal{f}\_i \right), \tag{98}$$

away from interphase regions. In this expression, *η* is the viscosity of the pore fluid, and *K* is the absolute permeability. As already mentioned, the phenomenological coefficients can be functions of the component densities and of the temperature, and it is known that *η* is such a function (see Section 3.4 in Ref. [3]). The possibility of letting *K* be such a function is not used in the minimal model. To prove the implication (98), one uses Eq. (42), with the upscaled stress tensor given by Eqs. (60), (61), and (85). Using Eq. (41) one obtains:

$$\Delta V\_i = -\frac{K}{\Phi \eta} \left[ \partial\_i P^{\text{bF}} - R^f\_{\text{f}} + R(\partial\_l V\_i + V\_k \partial\_k V\_i) - \sum\_{a=1}^{\nu} \Lambda^a R^a \partial\_l \nabla^2 R^a \right]. \tag{99}$$

The sum over *α* vanishes in the bulk fluid and one is left with the Darcy formula with an additional term, proportional to the material derivative of the velocity. One can carry out order of magnitude estimates of the three first terms in the square brackets above, in the manner of Section 4.1 of Ref. [3]. Using the numerical values given in the Appendix of the same reference, one easily finds that, if the gradient of pressure is of order 1, then the gravity term *Rfi* is of order 10�<sup>3</sup> , and the material derivative term is of order 10�9. Neglecting the material derivative, one obtains the Darcy formula, modified by terms that only become significant inside the interphase regions. Specialised to a one component fluid, this modified Darcy formula is:

$$W\_i = -\frac{K}{\Phi \eta} \left[ \partial\_i P^{\text{bF}} - Rf\_i \right] + \Lambda \frac{KR}{\Phi \eta} \partial\_i \nabla^2 R. \tag{100}$$

It is shown in Ref. [4] that the added non-Darcy term can, in some well-defined flow types, produce a relative permeability when its numerical contribution is taken away as an added term, then put back as a multiplicative factor to the Darcy term. However, it is concluded in Ref. [4] that relative permeabilities cannot capture the full complexity of two-phase flow.

Given below is another version of *Vi*, that follows by application of the Gibbs-Duhem equation, obtained by differentiating <sup>F</sup>bF and using the expression of dFbF (Eqs. (2)):

$$V\_i = -\frac{K}{\Phi \eta} \left[ \sum\_{a=1}^{\nu} R^a \partial\_i A^\beta + \mathcal{S}^{\text{F}} \partial\_i T \right],\tag{101}$$

where *A<sup>α</sup>* is given by Eq. (70). Using now the notation of expressions (79) in the third of expressions (93) one gets the diffuse mass current of component *α*:

$$I\_k^a = -\Phi \sum\_{\beta=1}^{\nu} t^{\alpha \beta} \partial\_k A^\beta,\tag{102}$$

*f*

obtained from statistical physics, as shown in Ref. [16].

*<sup>α</sup>* <sup>¼</sup> <sup>d</sup>*u<sup>α</sup>* <sup>þ</sup> *<sup>P</sup>*d*v<sup>α</sup>*, one obtains

<sup>0</sup> � *Ts<sup>α</sup>*

where volume, temperature, and pressure are *v<sup>α</sup>*

equations of the thermal model. Assuming *s*

d d*T*

**A.1 Appendix 1: Assumptions**

**A1:** There are no sources or sinks

**A4:** The solid is perfectly rigid

**A2:** There is no loss of energy by radiation

in terms of the experimentally measurable function *c<sup>α</sup>*

d*s <sup>α</sup>* <sup>¼</sup> *<sup>c</sup> α*

Integrating the last two displayed equations, one obtains *f*

<sup>0</sup> � *RGT* ln *<sup>v</sup><sup>α</sup>*

*vα* 0 þ ð*T T*<sup>0</sup>

the right-hand sides of Eqs. (104) and (107), one obtains an expression for *<sup>T</sup>* ln <sup>V</sup>*<sup>α</sup>*

Concerning examples of numerical solutions of the equations of the minimal model, see Ref. [2] for phase segregation, and for coning at uniform temperature;

*α*

*P*0

fact the derivative with respect to *<sup>T</sup>* of *<sup>T</sup>* ln <sup>V</sup>*<sup>α</sup>* that is needed in the differential

*RGT* ln <sup>V</sup>*<sup>α</sup>* <sup>½</sup> ð Þ *<sup>T</sup>* � ¼ *RG* ln *eRGT*<sup>0</sup>

see Ref. [3] for an injection-production situation at variable temperature.

**A0:** None of the chemical species completely wets the rock

**A3:** There are no chemical reactions between the chemical components

**A5:** The external force per unit volume *fi* is approximately constant in the

*f*

one can write

where *c<sup>α</sup>*

*f*

where *u<sup>α</sup>*

**A. Appendices**

averaging volume

**21**

id*:*gas,*<sup>α</sup>* <sup>¼</sup> *<sup>u</sup><sup>α</sup>*

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*DOI: http://dx.doi.org/10.5772/intechopen.95474*

<sup>0</sup> and *s α*

obeying *T*d*s*

id*:*gas,*<sup>α</sup>* <sup>¼</sup> *RGT* ln <sup>V</sup>*<sup>α</sup>*ð Þ *<sup>T</sup>*

It is briefly shown below that <sup>V</sup>*<sup>α</sup>* can also be obtained from an expression of

id*:*gas,*<sup>α</sup>* in terms of the heat capacity of the component. Assuming *Pv<sup>α</sup>* <sup>¼</sup> *RGT*, one easily finds that, with *T* and *v<sup>α</sup>* as independent variables, the molar internal energy *u<sup>α</sup>* (the "id.gas" superscript is dropped for simplicity) is independent of *v<sup>α</sup>*, and that

*<sup>V</sup>* is the molar heat capacity of component *α*. The molar entropy, *s*

<sup>d</sup>*u<sup>α</sup>* <sup>¼</sup> *<sup>c</sup> α*

where *<sup>v</sup><sup>α</sup>* is the molar volume, *RG* is the gas constant, *<sup>e</sup>* <sup>¼</sup> exp 1ð Þ, and <sup>V</sup>*<sup>α</sup>*ð Þ *<sup>T</sup>* is a function of temperature with the dimension of a volume per mole. <sup>V</sup>*<sup>α</sup>* depends on the atomic masses, the principal moments of inertia of the molecule, .... It can be

*ev<sup>α</sup>* , (104)

*<sup>V</sup>*ð Þ *T* d*T*, (105)

*<sup>V</sup>*ð Þ *<sup>T</sup>* <sup>d</sup>*T=<sup>T</sup>* <sup>þ</sup> *RG*d*vα=vα:* (106)

*α <sup>V</sup> T*<sup>0</sup> ð Þd*T*<sup>0</sup>

0, *<sup>T</sup>*0, and *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *RGT*0*=v<sup>α</sup>*

<sup>1</sup> � *<sup>T</sup> T*0 � �*<sup>c</sup>*

<sup>0</sup> ¼ 0, one obtains

*cα <sup>V</sup> T*<sup>0</sup> ð Þ *<sup>T</sup>*<sup>0</sup> <sup>d</sup>*T*<sup>0</sup>

� ð*T T*<sup>0</sup>

<sup>0</sup> are the internal energy and the entropy at a reference state

*α*,

id*:*gas,*<sup>α</sup>* as *<sup>u</sup><sup>α</sup>* � *Ts<sup>α</sup>*:

*<sup>V</sup>*. As stated in Ref. [16], it is in

, (107)

*:* (108)

0. Equating

where it is reminded that the ℓ*αβ*-matrix is symmetric, its off diagonal elements are negative, and the elements of any line (or column) sum to zero. In addition, the elements can be functions of the densities. This last turns out to be extremely useful because the *<sup>∂</sup>k*CF*<sup>β</sup>* behave as 1*=R<sup>β</sup>* when *<sup>R</sup><sup>β</sup>* goes to zero, which is not acceptable when the differential equations of the model are solved numerically. In the minimal model one then sets

$$\mathcal{E}^{a\beta} = -a \left[ \mathcal{R}^a \mathcal{R}^\beta \right]^2, \qquad (a \neq \beta), \tag{103}$$

using the same positive number *a* for all elements. Each diagonal element ℓ*αα* is then the negative sum of the off diagonal elements of line *α*. For an example of such a matrix in the case *ν* ¼ 3, see Refs. [2, 3].

#### **5.3 Closing details**

It is easy to see that the first two restrictions displayed in Eqs. (92) do not implicate any other assumptions done in the minimal model so that the restrictions can be lifted, either singly or together, thus allowing thermal conductivity and/or permeability to be represented by a second order tensor when experiments indicate that such upgrading is required.

A non-thermal version of the model consists of *ν* mass balance equations, see Eqs. (41), where the Darcy velocity is given by Eq. (101) and the mass diffusion velocities by Eqs. (102) and (103), the auxiliary variables being defined by Eqs. (70). Fot the thermal version of the model one needs to re-write the entropy equation, Eq. (75), in terms of temperature: see Section 2.3 of Ref. [3].

Start and boundary conditions must be supplied for the numerical solutions of the differential equations of the model. Special attention must be taken with the boundary conditions since the equations are of the fourth degree in the space variables. See Ref. [3] for a detailed presentation.

The transport coefficients, *k* and *η*, of Eqs. (97) and (98), are needed as functions of *T* and of the *R<sup>α</sup>*. See Ref. [3] and references given there.

The central thermodynamical function of the model is the Helmholtz function, especially in the bulk, introduced by Eq. (1). It is calculated from the equation of state of the mixture considered, which must be van der Waals or related (Redlich-Kwong, ...) so that, for temperatures less than the critical, regions of unstable fluid insure the existence of interphase regions; association terms must be included for the polar molecules of the mixture (see Ref. [16] and references given there).

It is shown in Ref. [16] that, independently of the equation of state that is chosen, the Helmholtz function contains the sum of the Helmholtz functions of the components, each considered as a gas where molecular interactions are neglected (ideal). Each ideal gas Helmholtz function contains a function of *T* that drops out under differentiation when *T* is assumed constant, but is important for the thermal model in accounting for the energy stored in the internal degrees of freedom of the molecules. The Helmholtz function of one mole of an ideal gas of component *α* is (see Refs. [2, 16] and references given there):

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

where *A<sup>α</sup>* is given by Eq. (70). Using now the notation of expressions (79) in the

ℓ*αβ∂kA<sup>β</sup>*

, (102)

, ð Þ *α* 6¼ *β* , (103)

X*ν β*¼1

where it is reminded that the ℓ*αβ*-matrix is symmetric, its off diagonal elements are negative, and the elements of any line (or column) sum to zero. In addition, the elements can be functions of the densities. This last turns out to be extremely useful because the *<sup>∂</sup>k*CF*<sup>β</sup>* behave as 1*=R<sup>β</sup>* when *<sup>R</sup><sup>β</sup>* goes to zero, which is not acceptable when the differential equations of the model are solved numerically. In the minimal

using the same positive number *a* for all elements. Each diagonal element ℓ*αα* is then the negative sum of the off diagonal elements of line *α*. For an example of such

It is easy to see that the first two restrictions displayed in Eqs. (92) do not implicate any other assumptions done in the minimal model so that the restrictions can be lifted, either singly or together, thus allowing thermal conductivity and/or permeability to be represented by a second order tensor when experiments indicate

A non-thermal version of the model consists of *ν* mass balance equations, see Eqs. (41), where the Darcy velocity is given by Eq. (101) and the mass diffusion velocities by Eqs. (102) and (103), the auxiliary variables being defined by Eqs. (70). Fot the thermal version of the model one needs to re-write the entropy

Start and boundary conditions must be supplied for the numerical solutions of the differential equations of the model. Special attention must be taken with the boundary conditions since the equations are of the fourth degree in the space

The central thermodynamical function of the model is the Helmholtz function, especially in the bulk, introduced by Eq. (1). It is calculated from the equation of state of the mixture considered, which must be van der Waals or related (Redlich-Kwong, ...) so that, for temperatures less than the critical, regions of unstable fluid insure the existence of interphase regions; association terms must be included for the polar molecules of the mixture (see Ref. [16] and references given there). It is shown in Ref. [16] that, independently of the equation of state that is chosen, the Helmholtz function contains the sum of the Helmholtz functions of the components, each considered as a gas where molecular interactions are neglected (ideal). Each ideal gas Helmholtz function contains a function of *T* that drops out under differentiation when *T* is assumed constant, but is important for the thermal model in accounting for the energy stored in the internal degrees of freedom of the molecules. The Helmholtz function of one mole of an ideal gas of component *α* is

The transport coefficients, *k* and *η*, of Eqs. (97) and (98), are needed as

equation, Eq. (75), in terms of temperature: see Section 2.3 of Ref. [3].

functions of *T* and of the *R<sup>α</sup>*. See Ref. [3] and references given there.

variables. See Ref. [3] for a detailed presentation.

(see Refs. [2, 16] and references given there):

**20**

third of expressions (93) one gets the diffuse mass current of component *α*:

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*I α <sup>k</sup>* ¼ �Φ

<sup>ℓ</sup>*αβ* ¼ �*a R<sup>α</sup>R<sup>β</sup>* � �<sup>2</sup>

model one then sets

**5.3 Closing details**

a matrix in the case *ν* ¼ 3, see Refs. [2, 3].

that such upgrading is required.

$$f^{\text{id.gas},a} = R\_G T \ln \frac{\mathcal{V}^a(T)}{ev^a},\tag{104}$$

where *<sup>v</sup><sup>α</sup>* is the molar volume, *RG* is the gas constant, *<sup>e</sup>* <sup>¼</sup> exp 1ð Þ, and <sup>V</sup>*<sup>α</sup>*ð Þ *<sup>T</sup>* is a function of temperature with the dimension of a volume per mole. <sup>V</sup>*<sup>α</sup>* depends on the atomic masses, the principal moments of inertia of the molecule, .... It can be obtained from statistical physics, as shown in Ref. [16].

It is briefly shown below that <sup>V</sup>*<sup>α</sup>* can also be obtained from an expression of *f* id*:*gas,*<sup>α</sup>* in terms of the heat capacity of the component. Assuming *Pv<sup>α</sup>* <sup>¼</sup> *RGT*, one easily finds that, with *T* and *v<sup>α</sup>* as independent variables, the molar internal energy *u<sup>α</sup>* (the "id.gas" superscript is dropped for simplicity) is independent of *v<sup>α</sup>*, and that one can write

$$\mathbf{d}u^{a} = c\_{V}^{a}(T)\mathbf{d}T,\tag{105}$$

where *c<sup>α</sup> <sup>V</sup>* is the molar heat capacity of component *α*. The molar entropy, *s α*, obeying *T*d*s <sup>α</sup>* <sup>¼</sup> <sup>d</sup>*u<sup>α</sup>* <sup>þ</sup> *<sup>P</sup>*d*v<sup>α</sup>*, one obtains

$$\mathbf{ds}^a = \mathbf{c}\_V^a(T)\mathbf{d}T/T + \mathbf{R}\_G\mathbf{d}\boldsymbol{\nu}^a/\boldsymbol{\nu}^a. \tag{106}$$

Integrating the last two displayed equations, one obtains *f* id*:*gas,*<sup>α</sup>* as *<sup>u</sup><sup>α</sup>* � *Ts<sup>α</sup>*:

$$f^{\text{id.gas},a} = u\_0^a - Ts\_0^a - R\_G T \ln \frac{\nu^a}{v\_0^a} + \int\_{T\_0}^{T} \left(1 - \frac{T}{T'}\right) c\_V^a(T') \text{d}T',\tag{107}$$

where *u<sup>α</sup>* <sup>0</sup> and *s α* <sup>0</sup> are the internal energy and the entropy at a reference state where volume, temperature, and pressure are *v<sup>α</sup>* 0, *<sup>T</sup>*0, and *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *RGT*0*=v<sup>α</sup>* 0. Equating the right-hand sides of Eqs. (104) and (107), one obtains an expression for *<sup>T</sup>* ln <sup>V</sup>*<sup>α</sup>* in terms of the experimentally measurable function *c<sup>α</sup> <sup>V</sup>*. As stated in Ref. [16], it is in fact the derivative with respect to *<sup>T</sup>* of *<sup>T</sup>* ln <sup>V</sup>*<sup>α</sup>* that is needed in the differential equations of the thermal model. Assuming *s α* <sup>0</sup> ¼ 0, one obtains

$$\frac{d}{dT}[R\_G T \ln \mathcal{V}^a(T)] = R\_G \ln \frac{eR\_G T\_0}{P\_0} - \int\_{T\_0}^{T} \frac{c\_V^a(T')}{T'} \mathbf{d}T'.\tag{108}$$

Concerning examples of numerical solutions of the equations of the minimal model, see Ref. [2] for phase segregation, and for coning at uniform temperature; see Ref. [3] for an injection-production situation at variable temperature.

#### **A. Appendices**

#### **A.1 Appendix 1: Assumptions**

**A0:** None of the chemical species completely wets the rock

**A1:** There are no sources or sinks

**A2:** There is no loss of energy by radiation

**A3:** There are no chemical reactions between the chemical components

**A4:** The solid is perfectly rigid

**A5:** The external force per unit volume *fi* is approximately constant in the averaging volume

**A6:** At each point, the difference between the solid and fluid temperatures is negligible

**A7:** The Λ*<sup>α</sup>* are constant numbers inside the porous medium

**A8:** The porosity Φ is uniform in space

#### **A.2 Appendix 2: Derivatives of distibutions – Formulas**

The following is a set of formulas for the space derivatives of distributions having discontinuities across a given surface Σ. For proofs, see [12, 13].

Consider a function *f*ð Þ **x**, *t* symbolising a physical quantity, continuous in time but dicontinuous in space across a surface Σ that divides space in two regions, called A-side and B-side; *f* ¼ *f <sup>B</sup>* on the B-side, *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup> <sup>A</sup>* on the A-side. Σ has a normal vector **n** at each point, pointing towards A:

$$f(\mathbf{x},t) = \begin{cases} f^{\mathcal{A}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ f^{\mathcal{B}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma, \end{cases} \tag{109}$$

One now defines the following regular distributions:

$$\{\partial\_{\mathbf{k}}f(\mathbf{x},t)\} = \begin{cases} \partial\_{\mathbf{k}}f^{\mathcal{A}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ \partial\_{\mathbf{k}}f^{\mathcal{B}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma, \end{cases} \tag{110}$$

$$\{\partial\_{\varnothing}f(\mathbf{x},t)\} = \begin{cases} \partial\_{t}f^{A}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ \partial\_{t}f^{B}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma. \end{cases} \tag{111}$$

Then [12, 13].

$$
\partial\_{\varnothing}f = \{\partial\_{\varnothing}f\},
\tag{112}
$$

If *Zij* is a tensor, then *Z* is a scalar, *Z*^ is a symmetric traceless tensor, and *Z*~*<sup>k</sup>* is a pseudo-vector. A most useful property is as follows: if *Uij* and *Vij* are two tensors

*UijVij* <sup>¼</sup> <sup>3</sup>*UV* <sup>þ</sup> *<sup>U</sup>*^ *ijV*^ *ij* <sup>þ</sup> <sup>2</sup>*U*<sup>~</sup> *iV*<sup>~</sup> *<sup>i</sup>:* (116)

then

*Porous Flow with Diffuse Interfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

**Author details**

Paul Papatzacos

**23**

University of Stavanger, Stavanger, Norway

provided the original work is properly cited.

\*Address all correspondence to: paul.papatzacos@lyse.net

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

$$
\partial\_{\mathbb{k}}f = \{\partial\_{\mathbb{k}}f\} + \left[ \left( \left. f^{\mathbb{A}} \right|^{\Sigma} - \left( \left. f^{\mathbb{B}} \right|^{\Sigma} \right) \right] n\_{\mathbb{k}} \delta\_{\Sigma}, \tag{113}
$$

where *δ*<sup>Σ</sup> is a surface-Dirac distribution, its action h i *δ*Σ, *φ* on a so-called test function *φ* [13] being defined as:

$$
\langle \delta\_{\Sigma}, \rho \rangle = \int\_{\Sigma} \rho \, \mathbf{d} \Sigma. \tag{114}
$$

#### **A.3 Appendix 3: Formulas concerning tensors**

Let *Zij* be an arbitrary second order tensor. It can be shown (see Ref. [15]) that the following expression has general validity:

$$Z\_{i\bar{j}} = Z\delta\_{i\bar{j}} + \hat{Z}\_{i\bar{j}} + \varepsilon\_{i\bar{j}k}\tilde{Z}\_k, \text{where} \begin{cases} Z = \frac{1}{3}Z\_{kk}, \\\\ \hat{Z}\_{i\bar{j}} = \frac{1}{2}\left(Z\_{i\bar{j}} + Z\_{j\bar{i}}\right) - \frac{1}{3}Z\_{kk}\delta\_{i\bar{j}}, \\\\ \hat{Z}\_k = \frac{1}{2}\varepsilon\_{klm}Z\_{lm}. \end{cases} \tag{115}$$

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

**A6:** At each point, the difference between the solid and fluid temperatures is

The following is a set of formulas for the space derivatives of distributions

Consider a function *f*ð Þ **x**, *t* symbolising a physical quantity, continuous in time but dicontinuous in space across a surface Σ that divides space in two regions, called

*<sup>A</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the A‐side of <sup>Σ</sup>

ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the B‐side of <sup>Σ</sup>,

*<sup>A</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the A‐side of <sup>Σ</sup>

*<sup>B</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the B‐side of <sup>Σ</sup>,

*<sup>A</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the A‐side of <sup>Σ</sup>

*<sup>B</sup>*ð Þ **<sup>x</sup>**, *<sup>t</sup>* **<sup>x</sup>** on the B‐side of <sup>Σ</sup>*:*

� *f <sup>B</sup>* � �<sup>Σ</sup> � �

*<sup>A</sup>* � �<sup>Σ</sup>

where *δ*<sup>Σ</sup> is a surface-Dirac distribution, its action h i *δ*Σ, *φ* on a so-called test

ð Σ

Let *Zij* be an arbitrary second order tensor. It can be shown (see Ref. [15]) that

*<sup>Z</sup>* <sup>¼</sup> <sup>1</sup>

8 >>>>>>><

>>>>>>>:

*<sup>Z</sup>*^*ij* <sup>¼</sup> <sup>1</sup>

*<sup>Z</sup>*~*<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

<sup>3</sup> *Zkk*,

<sup>2</sup> *Zij* <sup>þ</sup> *Zji* � � � <sup>1</sup>

<sup>2</sup> *<sup>ε</sup>klmZlm:*

h i *δ*Σ, *φ* ¼

*<sup>∂</sup>tf* <sup>¼</sup> f g *<sup>∂</sup>tf* , (112)

*nk δ*Σ, (113)

*φ* dΣ*:* (114)

<sup>3</sup> *Zkkδij*,

*<sup>A</sup>* on the A-side. Σ has a normal vector

(109)

(110)

(111)

(115)

**A7:** The Λ*<sup>α</sup>* are constant numbers inside the porous medium

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

**A.2 Appendix 2: Derivatives of distibutions – Formulas**

having discontinuities across a given surface Σ. For proofs, see [12, 13].

*<sup>B</sup>* on the B-side, *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>*

**A8:** The porosity Φ is uniform in space

negligible

A-side and B-side; *f* ¼ *f*

Then [12, 13].

**22**

**n** at each point, pointing towards A:

f g *<sup>∂</sup>kf*ð Þ **<sup>x</sup>**, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup><sup>k</sup> <sup>f</sup>*

f g *<sup>∂</sup>tf*ð Þ **<sup>x</sup>**, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup><sup>t</sup> <sup>f</sup>*

function *φ* [13] being defined as:

*<sup>f</sup>*ð Þ¼ **<sup>x</sup>**, *<sup>t</sup> <sup>f</sup>*

(

(

*f B*

One now defines the following regular distributions:

*∂<sup>k</sup> f*

*∂t f*

**A.3 Appendix 3: Formulas concerning tensors**

the following expression has general validity:

*Zij* <sup>¼</sup> *<sup>Z</sup>δij* <sup>þ</sup> *<sup>Z</sup>*^*ij* <sup>þ</sup> *<sup>ε</sup>ijkZ*~*k*, where

*<sup>∂</sup>kf* <sup>¼</sup> f g *<sup>∂</sup>kf* <sup>þ</sup> *<sup>f</sup>*

(

If *Zij* is a tensor, then *Z* is a scalar, *Z*^ is a symmetric traceless tensor, and *Z*~*<sup>k</sup>* is a pseudo-vector. A most useful property is as follows: if *Uij* and *Vij* are two tensors then

$$U\_{\vec{\eta}}V\_{\vec{\eta}} = \Im UV + \hat{U}\_{\vec{\eta}}\hat{V}\_{\vec{\eta}} + 2\tilde{U}\_i\tilde{V}\_i. \tag{116}$$

#### **Author details**

Paul Papatzacos University of Stavanger, Stavanger, Norway

\*Address all correspondence to: paul.papatzacos@lyse.net

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Papatzacos P: Macroscopic two-phase flow in porous media assuming the diffuse-interface model at pore level. Transport in Porous Media 2002;49: 139–174. DOI: 10.1023/A: 1016091821189

[2] Papatzacos P: A model for multiphase and multicomponent flow in porous media, built on the diffuse-interface assumption. Transport in Porous Media 2010;82:443–462. DOI: 10.1007/ s11242-009-9405-2

[3] Papatzacos P: A model for multiphase, multicomponent, and thermal flow in neutrally wetting porous media, built on the diffuseinterface assumption. Journal of Petroleum Science and Engineering 2016;143:141–157. DOI: 10.1016/j. petrol.2016.02.027

[4] Papatzacos P, Skjæveland SM: Diffuse-interface modelling of twophase flow for a one-component fluid in a porous medium. Transport in Porous Media 2006;65:213–236. DOI: 10.1007/ s11242-005-6081-8

[5] van der Waals JD: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Journal of Statistical Physics 1979;20:200–244. DOI: 10.1007/BF01011514

[6] Korteweg DJ: Sur la forme que prennent les equations etc, Archives Neerlandaises des Sciences Exactes et Naturelles, 1901;II:6–20. DOI: Not known

[7] Anderson DM, McFadden GB, Wheeler AA: Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 1998;30:139–165. DOI: 10.1146/annurev.fluid.30.1.139

[8] Ganesan V, Brenner H: A diffuseinterface model of two-phase flow in porous media. Proceedings of the Royal Society London A 2000;456:731–803. DOI: 10.1098/rspa.2000.0537

**Chapter 2**

Multiscale Modeling of

Porous Media

*Kieu Hiep Le*

**1. Introduction**

**25**

**Abstract**

Non-Isothermal Fluid Transport

To preserve the product quality as well as to reduce the logistics and storage cost, drying process is widely applied in the processing of porous material. In consideration of transport phenomena that involve a porous medium during drying, the complex morphology of the medium, and its influences on the distribution, flow, displacement of multiphase fluids are encountered. In this chapter, the recent advanced mass and energy transport models of drying processes are summarized. These models which were developed based on both pore- and continuum-scales, may provide a better fundamental understanding of non-isothermal liquid–vapor transport at both the continuum scale and the pore scale, and to pave the way for designing, operating, and optimizing drying and relevant industrial processes.

**Keywords:** drying model, multiscale – modeling, porous media, pore-network

In both natural systems (i.e. clays, aquifers, oil and gas reservoirs, plants, and biological tissues) and industrial systems (i.e. fuel cells, concrete, textiles, polymer composites, capillary heat pipe, and paper, etc.), porous media are often encountered. From a morphological point of view, porous media is composed of a persistent solid matrix and an interconnected void space that can be occupied by fluid phases. In several porous systems, the void space is initially filled by liquid water. To maintain the product quality, prolong the storage time as well as reduce the logistics cost, the liquid water is needed to be removed by using various drying techniques. During the drying process, the complex morphology of the media and its influences on the capillary flow, liquid – vapor phase change under thermal effects are faced. These processes are complex by themselves [1, 2]. Additionally, it should be noted that the industrial dryers consume approximately 12% of the total energy used in manufacturing processes [3]. For several industrial sectors (i.e. tissue, food, and agriculture) the energy consumption ratio of drying may reach 33%. As a complicated, important, and high energy consumption process, the dryer designing, and operation should not be done by trial and error. A better fundamental understanding of this drying process at the pore- and continuous scales (i.e. understanding the influence of porous structure on the drying behavior, drying

model, continuum model, diffusion model, upscaling strategy

Involved in Drying Process of

[9] Adler PM, Brenner H: Multiphase phase flow in porous media. Annual Review of Fluid Mechanics 1988;20:35– 59. DOI: 10.1146/annurev. fl.20.010188.000343

[10] Papatzacos P, Skjæveland SM: Relative permeability from thermodynamics. Society of Petroleum Engineers Journal 2004;9:213–236. DOI: 10.2118/87674-PA

[11] Hirschfelder JO, Curtiss CF, Bird RB: Molecular theory of gases and liquids, John Wiley and Sons, Inc. New York, Chapman and Hall, Limited, London, 1954. DOI: 10.1002/ pol.1955.120178311

[12] Marle CM: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. International Journal of Engineering Science 1982;20:643–662. DOI: 10.1016/0020-7225(82)90118-5

[13] Appel W: Mathematics for physics and physisists, Princeton University Press 2007. ISBN-13: 978–0–691-13102-3

[14] Weinstock R: Calculus of Variations, Dover 1974. ISBN-13: 0– 486–63069-2

[15] Reichl LE: A Modern Course in Statistical Physics, 2nd Edition, John Wiley and Sons 1998. ISBN: 0–471– 59520-9

[16] Papatzacos P. The Helmholtz free energy of pure fluid substances and fluid mixtures. In: Proceedings of the 8th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, (HEFAT2011); 11–13 July 2011; Mauritius.

#### **Chapter 2**

**References**

1016091821189

s11242-009-9405-2

petrol.2016.02.027

s11242-005-6081-8

[5] van der Waals JD: The

DOI: 10.1007/BF01011514

known

**24**

thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Journal of Statistical Physics 1979;20:200–244.

[6] Korteweg DJ: Sur la forme que prennent les equations etc, Archives Neerlandaises des Sciences Exactes et Naturelles, 1901;II:6–20. DOI: Not

[7] Anderson DM, McFadden GB, Wheeler AA: Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 1998;30:139–165. DOI:

10.1146/annurev.fluid.30.1.139

[8] Ganesan V, Brenner H: A diffuseinterface model of two-phase flow in

[3] Papatzacos P: A model for multiphase, multicomponent, and thermal flow in neutrally wetting porous media, built on the diffuseinterface assumption. Journal of Petroleum Science and Engineering 2016;143:141–157. DOI: 10.1016/j.

[4] Papatzacos P, Skjæveland SM: Diffuse-interface modelling of twophase flow for a one-component fluid in a porous medium. Transport in Porous Media 2006;65:213–236. DOI: 10.1007/

139–174. DOI: 10.1023/A:

[1] Papatzacos P: Macroscopic two-phase flow in porous media assuming the diffuse-interface model at pore level. Transport in Porous Media 2002;49:

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

porous media. Proceedings of the Royal Society London A 2000;456:731–803.

[9] Adler PM, Brenner H: Multiphase phase flow in porous media. Annual Review of Fluid Mechanics 1988;20:35–

[10] Papatzacos P, Skjæveland SM: Relative permeability from

thermodynamics. Society of Petroleum Engineers Journal 2004;9:213–236. DOI:

[11] Hirschfelder JO, Curtiss CF, Bird RB: Molecular theory of gases and liquids, John Wiley and Sons, Inc. New York, Chapman and Hall, Limited, London, 1954. DOI: 10.1002/

[12] Marle CM: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. International Journal of Engineering Science 1982;20:643–662. DOI: 10.1016/0020-7225(82)90118-5

[13] Appel W: Mathematics for physics and physisists, Princeton University Press 2007. ISBN-13: 978–0–691-13102-3

[14] Weinstock R: Calculus of Variations, Dover 1974. ISBN-13: 0–

[15] Reichl LE: A Modern Course in Statistical Physics, 2nd Edition, John Wiley and Sons 1998. ISBN: 0–471–

[16] Papatzacos P. The Helmholtz free energy of pure fluid substances and fluid mixtures. In: Proceedings of the 8th International Conference on Heat Transfer, Fluid Mechanics and

Thermodynamics, (HEFAT2011); 11–13

July 2011; Mauritius.

486–63069-2

59520-9

DOI: 10.1098/rspa.2000.0537

59. DOI: 10.1146/annurev. fl.20.010188.000343

10.2118/87674-PA

pol.1955.120178311

[2] Papatzacos P: A model for multiphase and multicomponent flow in porous media, built on the diffuse-interface assumption. Transport in Porous Media 2010;82:443–462. DOI: 10.1007/

## Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of Porous Media

*Kieu Hiep Le*

### **Abstract**

To preserve the product quality as well as to reduce the logistics and storage cost, drying process is widely applied in the processing of porous material. In consideration of transport phenomena that involve a porous medium during drying, the complex morphology of the medium, and its influences on the distribution, flow, displacement of multiphase fluids are encountered. In this chapter, the recent advanced mass and energy transport models of drying processes are summarized. These models which were developed based on both pore- and continuum-scales, may provide a better fundamental understanding of non-isothermal liquid–vapor transport at both the continuum scale and the pore scale, and to pave the way for designing, operating, and optimizing drying and relevant industrial processes.

**Keywords:** drying model, multiscale – modeling, porous media, pore-network model, continuum model, diffusion model, upscaling strategy

#### **1. Introduction**

In both natural systems (i.e. clays, aquifers, oil and gas reservoirs, plants, and biological tissues) and industrial systems (i.e. fuel cells, concrete, textiles, polymer composites, capillary heat pipe, and paper, etc.), porous media are often encountered. From a morphological point of view, porous media is composed of a persistent solid matrix and an interconnected void space that can be occupied by fluid phases. In several porous systems, the void space is initially filled by liquid water. To maintain the product quality, prolong the storage time as well as reduce the logistics cost, the liquid water is needed to be removed by using various drying techniques. During the drying process, the complex morphology of the media and its influences on the capillary flow, liquid – vapor phase change under thermal effects are faced. These processes are complex by themselves [1, 2]. Additionally, it should be noted that the industrial dryers consume approximately 12% of the total energy used in manufacturing processes [3]. For several industrial sectors (i.e. tissue, food, and agriculture) the energy consumption ratio of drying may reach 33%. As a complicated, important, and high energy consumption process, the dryer designing, and operation should not be done by trial and error. A better fundamental understanding of this drying process at the pore- and continuous scales (i.e. understanding the influence of porous structure on the drying behavior, drying

time determination, and energy consumption prediction) may help to design better products and processes.

In this chapter, a review of the recent advanced model in a selection of drying phenomena involved in porous media is illustrated. We discussed the state-of-theart numerical methods as complementary ways to get more insight. The future challenges and the hint at solutions to accommodate for them are also given.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

In 1950s, a new modeling approach to describe the transport process in the porous medium was proposed by Fatt. In his model, the void space of the porous media was modeled by a network of cylindrical tubes. In individual tubes, the mass transfer equation has been written and the discrete mass conservation equations are written for each pore node. This model has been named as pore network model. Afterward, several pore-scale models, such as the direct numerical approach, the volume of fluid approach, and method, have been presented in literature based on the idea of pore network model. Since the transport phenomena are considered directly at the pore-scale, the better fundamental understanding on the interaction between solid structure and fluid flow can be provided. As discussed in Section 1, the mentioned interaction can be upscaled to determine the macroscopic transport coefficients of the continuous models such as the relative permeabilities of the

In this section, we focus on the non – isothermal drying pore network models

Based on the invasion percolation algorithm, several non-isothermal pore network models were developed to simulate the hot air drying process [2, 7–9]. These models accounted capillary, gravity effects, viscous effects, and the transport of vapor by diffusion in the gas phase under the thermal effect. Recently, the pore network model has been applied to describe the non-isothermal mixture vapor liquid water transport during the superheated steam drying process [10]. The features of this pore network model is the fully treatment of condensed liquid by introducing the newly formulated liquid invasion rules in a two-dimensional domain. In this chapter, this two-dimensional pore network model is extended to three-dimensional model in this work to simulate the transport processes inside a capillary porous medium undergoing superheated steam drying with two different heating modes: (i) the convective heating at the top surface (convective- heating mode) and (ii) the simultaneous convective and conductive heating at the top and

Due to the absence of diffusion in the gas phase, the water vapor is transported by convective mechanism only and the Hagen-Poiseuille law is applied to calculate

> *ij* <sup>8</sup>*υvL pi* � *<sup>p</sup> <sup>j</sup>*

where the vapor mass flow rate in the cylindrical throat connecting pores i and j is denoted by *M*\_ *<sup>v</sup>*,*ij*, pi and pj present the pressure of the vapor phase at pores I and j, respectively. The length and radius of throat are respectively indicated by *L* and *r*. The kinematic viscosity of the vapor phase is denoted by *υv*. Under the quasi-steady assumption, the mass conservation of water vapor in a empty/partially pore is

(1)

and their application in estimating the macroscopic transport coefficients.

**2.1 Pore network model for superheated steam drying**

bottom surfaces (contact-heating mode) (c.f. **Figure 2**).

the vapor mass flow rate between two adjacent pores [10].

written as

**27**

*<sup>M</sup>*\_ *<sup>v</sup>*,*ij* <sup>¼</sup> *<sup>π</sup>r*<sup>4</sup>

**2. Pore-scale models**

liquid–gas mixture, the liquid diffusivity.

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

The drying process of porous media may be considered at different scales. At different scales, the approaches used to describe the transport processes may be quite different [4]. For instance, in a drying plant, thousands or millions of drying products can be dried simultaneously and the state of the drying agent evolves over time and space. Thus, the intra-sample distribution of process parameters should be lumped to avoid the extremely expensive computational cost. As a result, the lumped model is the suitable model that should be implemented in drying plant simulation. Coming to the sample drying process, the internal transport process is taken into account. The heat, mass, and momentum conservation equations are developed based on the first physics principles. In a control volume, the process properties are assumed to be homogenous and isotropic and the impact of microheterogeneity on the transport process is omitted. The macro-scale models often result in a system of partial differential equations. Both globe and local drying behavior of the sample can be well predicted by using the continuous models if the size of the domain is large enough. For a thin layer of the porous medium, where micro-heterogeneity plays a role, the continuous models are not valid. Due to the interaction between the solid skeleton and the fluid phases, the transport equations should be derived for individual pores. Since the transport phenomena are directly investigated at the pore-scale, the impact of changes at the pore or microstructure such as wettability, pore shape, pore size distribution, and pore structure on drying behavior has been effectively interpreted by pore-scale models [5, 6]. One should notes that these models are not standing individually. The results of pore scale model can be upscaled to generate the effective parameters of macro scale models which can be reduced to the lumped models. An example multiscale modeling approach for wood particle drying process is presented in **Figure 1**.

#### **Figure 1.**

*Multiscale modeling approach for porous particle drying process.*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

In this chapter, a review of the recent advanced model in a selection of drying phenomena involved in porous media is illustrated. We discussed the state-of-theart numerical methods as complementary ways to get more insight. The future challenges and the hint at solutions to accommodate for them are also given.

#### **2. Pore-scale models**

time determination, and energy consumption prediction) may help to design better

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

The drying process of porous media may be considered at different scales. At different scales, the approaches used to describe the transport processes may be quite different [4]. For instance, in a drying plant, thousands or millions of drying products can be dried simultaneously and the state of the drying agent evolves over time and space. Thus, the intra-sample distribution of process parameters should be lumped to avoid the extremely expensive computational cost. As a result, the lumped model is the suitable model that should be implemented in drying plant simulation. Coming to the sample drying process, the internal transport process is taken into account. The heat, mass, and momentum conservation equations are developed based on the first physics principles. In a control volume, the process properties are assumed to be homogenous and isotropic and the impact of microheterogeneity on the transport process is omitted. The macro-scale models often result in a system of partial differential equations. Both globe and local drying behavior of the sample can be well predicted by using the continuous models if the size of the domain is large enough. For a thin layer of the porous medium, where micro-heterogeneity plays a role, the continuous models are not valid. Due to the interaction between the solid skeleton and the fluid phases, the transport equations should be derived for individual pores. Since the transport phenomena are directly investigated at the pore-scale, the impact of changes at the pore or microstructure such as wettability, pore shape, pore size distribution, and pore structure on drying behavior has been effectively interpreted by pore-scale models [5, 6]. One should notes that these models are not standing individually. The results of pore scale model can be upscaled to generate the effective parameters of macro scale models which can be reduced to the lumped models. An example multiscale modeling

approach for wood particle drying process is presented in **Figure 1**.

products and processes.

**Figure 1.**

**26**

*Multiscale modeling approach for porous particle drying process.*

In 1950s, a new modeling approach to describe the transport process in the porous medium was proposed by Fatt. In his model, the void space of the porous media was modeled by a network of cylindrical tubes. In individual tubes, the mass transfer equation has been written and the discrete mass conservation equations are written for each pore node. This model has been named as pore network model. Afterward, several pore-scale models, such as the direct numerical approach, the volume of fluid approach, and method, have been presented in literature based on the idea of pore network model. Since the transport phenomena are considered directly at the pore-scale, the better fundamental understanding on the interaction between solid structure and fluid flow can be provided. As discussed in Section 1, the mentioned interaction can be upscaled to determine the macroscopic transport coefficients of the continuous models such as the relative permeabilities of the liquid–gas mixture, the liquid diffusivity.

In this section, we focus on the non – isothermal drying pore network models and their application in estimating the macroscopic transport coefficients.

#### **2.1 Pore network model for superheated steam drying**

Based on the invasion percolation algorithm, several non-isothermal pore network models were developed to simulate the hot air drying process [2, 7–9]. These models accounted capillary, gravity effects, viscous effects, and the transport of vapor by diffusion in the gas phase under the thermal effect. Recently, the pore network model has been applied to describe the non-isothermal mixture vapor liquid water transport during the superheated steam drying process [10]. The features of this pore network model is the fully treatment of condensed liquid by introducing the newly formulated liquid invasion rules in a two-dimensional domain. In this chapter, this two-dimensional pore network model is extended to three-dimensional model in this work to simulate the transport processes inside a capillary porous medium undergoing superheated steam drying with two different heating modes: (i) the convective heating at the top surface (convective- heating mode) and (ii) the simultaneous convective and conductive heating at the top and bottom surfaces (contact-heating mode) (c.f. **Figure 2**).

Due to the absence of diffusion in the gas phase, the water vapor is transported by convective mechanism only and the Hagen-Poiseuille law is applied to calculate the vapor mass flow rate between two adjacent pores [10].

$$
\dot{M}\_{v,\dot{\imath}} = \frac{\pi r\_{\dot{\imath}}^4}{8\nu\_v L} \left(p\_i - p\_j\right) \tag{1}
$$

where the vapor mass flow rate in the cylindrical throat connecting pores i and j is denoted by *M*\_ *<sup>v</sup>*,*ij*, pi and pj present the pressure of the vapor phase at pores I and j, respectively. The length and radius of throat are respectively indicated by *L* and *r*. The kinematic viscosity of the vapor phase is denoted by *υv*. Under the quasi-steady assumption, the mass conservation of water vapor in a empty/partially pore is written as

**Figure 2.** *Two different heating modes used in the pore network simulations.*

$$
\sum\_{j} \dot{M}\_{v, \vec{\mu}} - \sum\_{j} \pi r\_{t, \vec{\mu}} \, ^2 \dot{m}\_{\text{evp}/\text{con}, \vec{\mu}} - \pi r\_{p, i} \, ^2 \dot{m}\_{\text{evp}/\text{con}, i} = \mathbf{0} \tag{2}
$$

Due to the non-uniform pore size distribution, the liquid within the network is transported under capillary action. Thus, the large pores/throats are preferentially emptied or filled to maintain the lowest liquid pressure according to Eq. 4. As a result, the liquid phase disintegrates into several liquid clusters. Since the liquid mass balance must be satisfied in the entire network as well as for each liquid cluster, the morphological information of liquid cluster is required. The Hoshen– Kopelman algorithm is applied for labeling the liquid clusters.

$$p\_l = p\_v - \frac{2\sigma(T)\cos\theta}{r\_{p/t}}\tag{3}$$

balance equation written for the control volume around pore i in time step Δt is

*Heat balance at a vapor pore. The arrows indicate the direction of the heat flux towards each face of the control*

*The 2D sketch of liquid invasion rules: Liquid invasion from single liquid throat to its neighboring pore (a), liquid invasion from fully filled liquid throat in a liquid cluster to its neighboring empty pore (b), liquid*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

*redistribution from fully filled pore in a liquid cluster to its neighboring empty throat (c).*

*<sup>L</sup>* <sup>¼</sup> *<sup>ρ</sup>eff*,*iceff*,*iVi*

The convective heat transfer boundary condition presented in Eq. 8 is applied for both heating modes. At the bottom of the network, perfect thermal insulation (Eq. 9) is imposed for the convective heating mode, whereas uniform constant

> � � *y*¼*Ly*

*∂T ∂y* � � � � *y*¼0 <sup>Δ</sup>*<sup>t</sup>* ð Þ *Ti*,*t*þΔ*<sup>t</sup>* � *Ti*

•*n* ¼ *α Tsteam* � *T*j

2 *m*\_ *evp=con*,*<sup>i</sup>*

*y*¼*Ly* � � (8)

¼ 0 (9)

(7)

*m*\_ *evp=con*,*ji* þ Δ*Hevpπrp*,*<sup>i</sup>*

*Ti*,*t*þΔ*<sup>t</sup>* � *T <sup>j</sup>*,*t*þΔ*<sup>t</sup>*

þΔ*Hevp*<sup>X</sup>

temperature (Eq. 10) is set for the contact-heating mode.

" #

*j πrt*,*ij* 2

þ Δ*Hevpρlvl*

discretized as

**Figure 4**

*volume.*

**29**

**Figure 3.**

X *j*

� *Acv*,*ijλeff*,*ij*

*λeff ∂T ∂z* � � � � *y*¼*Ly*

To fully model the phase transition including both evaporation and condensation, in addition the emptying and refilling events, the liquid invasion represented in **Figure 3** is introduced in the present model. The time steps for the emptying and refilling events of meniscus pores or throats are computed by

$$\Delta t\_{\text{empty},cn/st,p/t} = \frac{V\_{p/t} \rho\_l(T) \mathbf{S}\_{p/t}}{\sum\_{c \le \text{tr}} \dot{\mathbf{M}}\_{\text{exp}/con}} \text{ and } \Delta t\_{\text{refill},cn/st,p/t} = -\frac{V\_{p/t} \rho\_l(T) \left(\mathbf{1} - \mathbf{S}\_{p/t}\right)}{\sum\_{c \le \text{tr}} \dot{\mathbf{M}}\_{\text{exp}/con}} \tag{4}$$

The time step for the first two liquid invasion events is calculated by

$$
\Delta t\_{\rm in,cn/st} = -\frac{V\_{p/t,p}\rho\_l(T)}{\sum\_{cn/st} \dot{M}\_{evp/con}} \tag{5}
$$

The pore/throat saturations after the capillary re-equilibration event are computed by

$$\mathbf{S}\_{t,ih} = \mathbf{1} \text{ and } \mathbf{S}\_{p,h} = \frac{V\_{p,h} - V\_{t,ih}}{V\_{p,h}} \tag{6}$$

The thermal energy supplied to a control volume (c.f. **Figure 4**) lead to a change of the enthalpy in the control volume: the control volume temperature increases, and the phase transition occurs. Based on a fully implicit scheme, the energy

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

#### **Figure 3.**

X *j*

**Figure 2.**

*<sup>Δ</sup>tempty*,*cn=st*,*p=<sup>t</sup>* <sup>¼</sup> *Vp=<sup>t</sup>ρl*ð Þ *<sup>T</sup> Sp=<sup>t</sup>* <sup>P</sup>

computed by

**28**

*<sup>M</sup>*\_ *<sup>v</sup>*,*ij* �<sup>X</sup> *j πrt*,*ij* 2

*Two different heating modes used in the pore network simulations.*

Kopelman algorithm is applied for labeling the liquid clusters.

refilling events of meniscus pores or throats are computed by

*cn=stM*\_ *evp=con*

*m*\_ evp*=*con,*ji* � *πrp*,*<sup>i</sup>*

Due to the non-uniform pore size distribution, the liquid within the network is transported under capillary action. Thus, the large pores/throats are preferentially emptied or filled to maintain the lowest liquid pressure according to Eq. 4. As a result, the liquid phase disintegrates into several liquid clusters. Since the liquid mass balance must be satisfied in the entire network as well as for each liquid cluster, the morphological information of liquid cluster is required. The Hoshen–

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*pl* <sup>¼</sup> *pv* � <sup>2</sup>*σ*ð Þ *<sup>T</sup>* cos *<sup>θ</sup>*

To fully model the phase transition including both evaporation and condensation, in addition the emptying and refilling events, the liquid invasion represented in **Figure 3** is introduced in the present model. The time steps for the emptying and

The time step for the first two liquid invasion events is calculated by

*<sup>Δ</sup>tin*,*cn=st* ¼ � *Vp=t*,*ρl*ð Þ *<sup>T</sup>* P

The pore/throat saturations after the capillary re-equilibration event are

*St*,*ih* <sup>¼</sup> 1 and*Sp*,*<sup>h</sup>* <sup>¼</sup> *Vp*,*<sup>h</sup>* � *Vt*,*ih*

The thermal energy supplied to a control volume (c.f. **Figure 4**) lead to a change of the enthalpy in the control volume: the control volume temperature increases, and the phase transition occurs. Based on a fully implicit scheme, the energy

*rp=<sup>t</sup>*

and *<sup>Δ</sup>t*refill,*cn=st*,*p=<sup>t</sup>* ¼ � *Vp=<sup>t</sup>ρl*ð Þ *<sup>T</sup>* <sup>1</sup> � *Sp=<sup>t</sup>*

*cn=stM*\_ *evp=con*

*Vp*,*<sup>h</sup>*

P

2

*m*\_ *evp=con*,*<sup>i</sup>* ¼ 0 (2)

� �

*cn=stM*\_ *evp=con*

(3)

(4)

(5)

(6)

*The 2D sketch of liquid invasion rules: Liquid invasion from single liquid throat to its neighboring pore (a), liquid invasion from fully filled liquid throat in a liquid cluster to its neighboring empty pore (b), liquid redistribution from fully filled pore in a liquid cluster to its neighboring empty throat (c).*

#### **Figure 4**

*Heat balance at a vapor pore. The arrows indicate the direction of the heat flux towards each face of the control volume.*

balance equation written for the control volume around pore i in time step Δt is discretized as

$$\begin{split} \sum\_{j} -A\_{\text{ev}, \text{ji}} \lambda\_{\text{eff}, \text{ji}} \frac{T\_{i, t + \Delta t} - T\_{j, t + \Delta t}}{L} &= \frac{\rho\_{\text{eff}, i} c\_{\text{eff}, i} V\_i}{\Delta t} (T\_{i, t + \Delta t} - T\_i) \\ + \Delta H\_{\text{ev}} \sum\_{j} \pi r\_{t, \text{ji}} \,^2 \dot{m}\_{\text{ev}/\text{con}, \text{ji}} + \Delta H\_{\text{ev}} \pi r\_{p, i} \,^2 \dot{m}\_{\text{ev}/\text{con}, i} \end{split} \tag{7}$$

The convective heat transfer boundary condition presented in Eq. 8 is applied for both heating modes. At the bottom of the network, perfect thermal insulation (Eq. 9) is imposed for the convective heating mode, whereas uniform constant temperature (Eq. 10) is set for the contact-heating mode.

$$\left[\lambda\_{\rm eff} \frac{\partial T}{\partial \mathbf{z}}\Big|\_{\mathbf{y}=L\_{\gamma}} + \Delta H\_{\rm evp} \rho\_{\rm I} v\_{\rm l}\Big|\_{\mathbf{y}=L\_{\gamma}}\right] \bullet \mathbf{n} = a \left(T\_{\rm steam} - \left.T\right|\_{\mathbf{y}=L\_{\gamma}}\right) \tag{8}$$

$$\left.\frac{\partial T}{\partial \mathbf{y}}\right|\_{\mathbf{y}=\mathbf{0}} = \mathbf{0} \tag{9}$$

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$$\left.T\right|\_{\mathcal{Y}=L\_{\mathcal{Y}}} = T\_{\text{bottom}}\tag{10}$$

**2.2 Effective parameters of continuous model assessed from the pore network**

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

As mentioned in the previous section, the heat and mass transfer inside the porous domain can be simulated by using pore network models. However, since the pore network model considers the non-isothermal fluid flow in the pore individually, the number of the discrete equations of the model increases with the size of the domain. Thus, the pore network model cannot be used to simulate the macroscale drying process of porous media which is consists of billions of pores. To describe the drying process of macroscopic porous material, the continuous models developed based on the macroscopic effective transport parameters should be used. These effective parameters are often correlated from a serial of experimental data. To provide a fundamental understanding of the link between the transport coefficient and the fluid transport mechanisms, the transport coefficients are revisited by using the pore network simulation results. For example, the influence of network saturation on the non-local equilibrium effect, the effective moisture transport is investigated in the works of Kharaghani et al. [11–13] for the isothermal drying process. The incorporation of these effective parameters in the highly non-isothermal drying process is still an open issue. In the future, the impact of non-isothermal pore scale

fluid transport on the effective parameter should be taken into account.

derived based on the first physical principles using effective parameters.

In continuum-scale models, the porous media is assumed isotropic and homogeneous. Both solid structure and fluid properties are averaged in a representative elementary volume (REV) based on the volume averaging technique. The size of REV should not large enough to avoid the fluctuations of the morphological properties due to micro heterogeneity [14, 15]. The energy and mass conservation equations are

In diffusion model, the mixture of liquid water and vapor water flow is considered as single phase named moisture and the gradient of moisture content is solely

**modeling**

*Impact of heating mode on the drying kinetic curve.*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 6.**

**3. Continuum-scale models**

**3.1 Diffusion model**

**31**

The simulations are carried out for a 10 � 10 � 10 pore network with throat radius distribution 100 � 10 μm, pore radius distribution 250 � 10 μm and uniform distance between two adjacent pores L = 1000 μm. Because of the high computational demand required for a full simulation of drying, only one realization of network was considered here. The solid skeleton of the network is made of typical glass with thermal conductivity λ<sup>s</sup> = 1 W/mK. The steam velocity at the network surface is 5 m/s. For the contact-heating mode, a uniform temperature boundary condition Tbottom = 105°C is imposed at the network bottom (c.f. Eq. 10), whereas the impermeable heat transfer boundary condition is set for the convective heating mode (c.f. Eq. 9). To depict the influence of heating mode on drying characteristics, the evolution of drying rate and of the local network saturation over network saturation obtained for these two heating modes are shown in **Figures 5** and **6**.

Since a large amount of thermal energy was supplied to the network with the contact-heating mode, the drying rate remains high and drying time is thus short compared to the network with the convective-heating mode. Also, since the surface temperature of the network with the contact-heating mode reaches boiling temperature fast, the surface condensation period is short and thus the amount of condensed liquid at the network surface is less, compared to the convective-heating mode. Moreover, it can be seen that the semi-constant drying rate period obtained from the network with the contact-heating mode is long compared to the convective heating model. This tendency can be explained by the temperature dependency of the surface tension; a lower surface tension obtained at high temperature leads to a lower capillary pressure and a higher liquid pressure at the liquid–vapor-solid interface. The pores/throats near the bottom of the network with the contactheating mode are preferentially emptied. As a result, the vapor fingers appear markedly, and the bottom of the network is dried completely when the middle and top zones are still partly wet. The impact of vapor fingering on the phase distribution is shown in **Figure 5**. More number of liquid clusters is obtained with the contact-heating mode compared to the convective-heating mode. On the other hand, the network with the convective-heating mode is dried evenly from the top to the bottom zone. Since the top and middle zones remain saturated for long time, an extended quasi-constant drying rate period is obtained for the network with the contact-heating mode (c.f. **Figure 6**).

**Figure 5.** *Impact of heating mode on the local saturation evolution over network saturation.*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 6.** *Impact of heating mode on the drying kinetic curve.*

*T*j

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contact-heating mode (c.f. **Figure 6**).

*Impact of heating mode on the local saturation evolution over network saturation.*

**Figure 5.**

**30**

The simulations are carried out for a 10 � 10 � 10 pore network with throat radius distribution 100 � 10 μm, pore radius distribution 250 � 10 μm and uniform distance between two adjacent pores L = 1000 μm. Because of the high computational demand required for a full simulation of drying, only one realization of network was considered here. The solid skeleton of the network is made of typical glass with thermal conductivity λ<sup>s</sup> = 1 W/mK. The steam velocity at the network surface is 5 m/s. For the contact-heating mode, a uniform temperature boundary condition Tbottom = 105°C is imposed at the network bottom (c.f. Eq. 10), whereas the impermeable heat transfer boundary condition is set for the convective heating mode (c.f. Eq. 9). To depict the influence of heating mode on drying characteristics, the evolution of drying rate and of the local network saturation over network saturation obtained for these two heating modes are shown in **Figures 5** and **6**. Since a large amount of thermal energy was supplied to the network with the contact-heating mode, the drying rate remains high and drying time is thus short compared to the network with the convective-heating mode. Also, since the surface temperature of the network with the contact-heating mode reaches boiling temperature fast, the surface condensation period is short and thus the amount of condensed liquid at the network surface is less, compared to the convective-heating mode. Moreover, it can be seen that the semi-constant drying rate period obtained from the network with the contact-heating mode is long compared to the convective heating model. This tendency can be explained by the temperature dependency of the surface tension; a lower surface tension obtained at high temperature leads to a lower capillary pressure and a higher liquid pressure at the liquid–vapor-solid interface. The pores/throats near the bottom of the network with the contactheating mode are preferentially emptied. As a result, the vapor fingers appear markedly, and the bottom of the network is dried completely when the middle and top zones are still partly wet. The impact of vapor fingering on the phase distribution is shown in **Figure 5**. More number of liquid clusters is obtained with the contact-heating mode compared to the convective-heating mode. On the other hand, the network with the convective-heating mode is dried evenly from the top to the bottom zone. Since the top and middle zones remain saturated for long time, an extended quasi-constant drying rate period is obtained for the network with the

*<sup>y</sup>*¼*Ly* ¼ *Tbottom* (10)

#### **2.2 Effective parameters of continuous model assessed from the pore network modeling**

As mentioned in the previous section, the heat and mass transfer inside the porous domain can be simulated by using pore network models. However, since the pore network model considers the non-isothermal fluid flow in the pore individually, the number of the discrete equations of the model increases with the size of the domain. Thus, the pore network model cannot be used to simulate the macroscale drying process of porous media which is consists of billions of pores. To describe the drying process of macroscopic porous material, the continuous models developed based on the macroscopic effective transport parameters should be used. These effective parameters are often correlated from a serial of experimental data. To provide a fundamental understanding of the link between the transport coefficient and the fluid transport mechanisms, the transport coefficients are revisited by using the pore network simulation results. For example, the influence of network saturation on the non-local equilibrium effect, the effective moisture transport is investigated in the works of Kharaghani et al. [11–13] for the isothermal drying process. The incorporation of these effective parameters in the highly non-isothermal drying process is still an open issue. In the future, the impact of non-isothermal pore scale fluid transport on the effective parameter should be taken into account.

#### **3. Continuum-scale models**

In continuum-scale models, the porous media is assumed isotropic and homogeneous. Both solid structure and fluid properties are averaged in a representative elementary volume (REV) based on the volume averaging technique. The size of REV should not large enough to avoid the fluctuations of the morphological properties due to micro heterogeneity [14, 15]. The energy and mass conservation equations are derived based on the first physical principles using effective parameters.

#### **3.1 Diffusion model**

In diffusion model, the mixture of liquid water and vapor water flow is considered as single phase named moisture and the gradient of moisture content is solely driving force of moisture transport. Since the water diffusion flow leads to the evolution of water concentration in the control volume, the mass conservation of water is derived as

$$\frac{\partial \rho\_0 X}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \cdot \left[ -D\_{\text{eff}} \frac{\partial}{\partial \mathbf{x}} (\rho\_0 X) \right] = \mathbf{0}.\tag{11}$$

The validated diffusion model of wood drying is implemented in a CFD model is

*The evolution of drying history during optimization process for Dref determination (a) and experimental and*

(L = 900 mm, D = 200 mm, and H = 25 mm) is presented in **Figure 8**. The air with a temperature of 140°C and moisture content of 7 g ram vapor/kg dry air flows into the dryer with a velocity of 0.1 m/s. The initial moisture and temperature of wood are 0.61 kg water/kg dried solid and 20°C. The 2D velocity profile of drying agent is presented in **Figure 8** and the evolution of moisture content of plate number (1), (2), (3), and (4) are plotted together with the mean moisture content of all plates in **Figure 9**. The results indicate a noticeable moisture content maldistribution of wood plates which should be remedied by wood plate disturbing during the drying

In the 1970s, the simultaneous heat, mass, and momentum transfer for nonisothermal drying process is proposed by Whitaker based on the volume averaging technique. In this model, all pore-level mechanisms for heat and mass transfer are considered. The liquid flow due to capillary forces, vapor and gas flow due to convection and diffusion are taken into account in this model. Using the volume averaging technique, the properties of fluid and solid phases such as velocity, density, pressure are averaged in REV. Afterward, the macroscopic differential equations were defined in terms of average field quantities. The detail descriptions of Whitaker's model can be found in Vu et al. [20, 21]. In this section, the mass and

used to investigate the drying behavior of a pilot dryer. The sketch of dryer (L = 1000 mm, D = 3000 mm, and H = 500 mm) where 30 plates of wood

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

process.

**Figure 8.**

**33**

**Figure 7.**

**3.2 Whitaker's continuum model**

*numerical mean moisture content evolutions over time (b).*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

energy conservation equations are briefly recalled.

*Pilot dryer model and velocity profile inside the dryer.*

In Eq. 11, the apparent density of the dry porous medium where the void volume is taken into account is computed as *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> *Ms <sup>V</sup>* (kg dry solid/m3 ), *<sup>X</sup>* <sup>¼</sup> *Ml Ms* (kg water/kg dry solid) is the dry-based moisture content of the wet solid. *Deff* (m2 /s) is the effective diffusivity of moisture in the porous medium. Similarly, the energy conservation equation is written as the change of energy density caused by the enthalpy flow which consists of contributions from diffusive heat flow and heat conduction, namely,

$$\frac{\partial}{\partial t} \left[ (\rho\_0 c\_{p,t} + \rho\_0 c\_{p,l} X) T \right] - \nabla \cdot \left[ D\_{\rm eff} c\_{p,l} T \nabla (\rho\_0 X) \right] - \nabla \cdot \left[ \lambda\_{\rm eff} \nabla (T) \right] = \mathbf{0}. \tag{12}$$

In Eq. 12, *cp*,*<sup>s</sup>* and *cp*,*<sup>l</sup>* (J/kg.K) denote the specific heat capacity of solid and liquid water, respectively; *λeff* (W/m.K) denotes the effective thermal conductivity of the porous medium.

To perform the diffusional model simulation, the effective thermal conductivity and effective moisture diffusivity needed to be known. Several methods such as fitch method, laser flash, hot-disk method and hot-wire method, can be used to measure the effective thermal conductivity directly whereas measurement of effective moisture diffusivity is rather more complex. Generally, the effective diffusivity of wetted porous materials often decreases during the water dehydration process since the liquid water is strongly captured in the small pores under a higher capillary action compared to large pores [16, 17]. Recently, Khan et al. [18] reported that the moisture diffusivity for isotropic porous materials, therein food products, can be described as a function of moisture content as

$$D\_{\rm eff} = D\_{\rm ref} \left( \frac{\mathbf{1} + \mathbf{X}\_0}{\mathbf{1} + \mathbf{X}} \frac{\rho\_l (\mathbf{1} + \mathbf{X}) + \rho\_0 \mathbf{X}}{\rho\_l (\mathbf{1} + \mathbf{X}\_0) + \rho\_0 \mathbf{X}\_0} \right)^2,\tag{13}$$

where *D*ref (m2 /s) is the reference diffusivity, *X0* is the initial moisture content. The question raised here is how to determine the reference effective diffusivity *D*ref. Instead of measurement directly, it can be computed numerically by an optimization routine using invert method. The objective optimizing function *g Deff* � � is the sum of square of difference between numerical moisture content *Xi*,*sim* and experimental *Xi*, exp as

$$\lg\left(D\_{\text{eff}}\right) = \sum\_{i=1:nnd} \left(X\_{\text{exp}}(t\_i) - X\_{sim}(t\_i)\right)^2,\tag{14}$$

where *t*<sup>i</sup> denotes the time interval *i* of the data sampling. By minimizing the value of objective, the numerical moisture evolution is fitted to the experimental observation and the value of effective diffusivity is obtained. For an example, this method is applied for single wood particle drying process in superheated steam environment. The detail description of the geometrical information, initial and boundary conditions is presented in Le et al. [19]. The effective diffusivity of 6.633 � <sup>10</sup>�<sup>9</sup> <sup>m</sup><sup>2</sup> /s determined using the experimental data observed at a drying temperature of 140°C can be used reliable for a large range of drying temperature as shown in **Figure 7**. It implies that the moisture diffusivity model presented in Eq. 13 can be used resonable for wooden porous media.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 7.**

driving force of moisture transport. Since the water diffusion flow leads to the evolution of water concentration in the control volume, the mass conservation of

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*<sup>∂</sup><sup>x</sup>* � �*Deff*

diffusivity of moisture in the porous medium. Similarly, the energy conservation equation is written as the change of energy density caused by the enthalpy flow which consists of contributions from diffusive heat flow and heat conduction, namely,

*∂ <sup>∂</sup><sup>x</sup>* ð Þ *<sup>ρ</sup>*0*<sup>X</sup>* � �

In Eq. 11, the apparent density of the dry porous medium where the void volume is

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup>*0*cp*,*<sup>s</sup>* <sup>þ</sup> *<sup>ρ</sup>*0*cp*,*lX* � �*<sup>T</sup>* � � � <sup>∇</sup> � *Deffcp*,*lT*∇ð Þ *<sup>ρ</sup>*0*<sup>X</sup>* � � � <sup>∇</sup> � *<sup>λ</sup>eff* <sup>∇</sup>ð Þ *<sup>T</sup>* � � <sup>¼</sup> <sup>0</sup>*:* (12)

In Eq. 12, *cp*,*<sup>s</sup>* and *cp*,*<sup>l</sup>* (J/kg.K) denote the specific heat capacity of solid and liquid water, respectively; *λeff* (W/m.K) denotes the effective thermal conductivity of the

To perform the diffusional model simulation, the effective thermal conductivity and effective moisture diffusivity needed to be known. Several methods such as fitch method, laser flash, hot-disk method and hot-wire method, can be used to measure the effective thermal conductivity directly whereas measurement of effective moisture diffusivity is rather more complex. Generally, the effective diffusivity of wetted porous materials often decreases during the water dehydration process since the liquid water is strongly captured in the small pores under a higher capillary action compared to large pores [16, 17]. Recently, Khan et al. [18] reported that the moisture diffusivity for isotropic porous materials, therein food products, can be

> *ρl*ð Þþ 1 þ *X ρ*0*X ρl*ð Þþ 1 þ *X*<sup>0</sup> *ρ*0*X*<sup>0</sup>

/s) is the reference diffusivity, *X0* is the initial moisture content. The

*X*expð Þ� *ti Xsim*ð Þ*ti* � �<sup>2</sup>

/s determined using the experimental data observed at a drying

� �<sup>2</sup>

*<sup>V</sup>* (kg dry solid/m3

¼ 0*:* (11)

*Ms* (kg water/kg dry

/s) is the effective

, (13)

� � is the sum of square of

, (14)

), *<sup>X</sup>* <sup>¼</sup> *Ml*

*∂ρ*0*X ∂t* þ *∂*

solid) is the dry-based moisture content of the wet solid. *Deff* (m2

taken into account is computed as *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> *Ms*

described as a function of moisture content as

*Deff* ¼ *Dref*

*g Deff*

can be used resonable for wooden porous media.

using invert method. The objective optimizing function *g Deff*

� � <sup>¼</sup> <sup>X</sup>

1 þ *X*<sup>0</sup> 1 þ *X*

question raised here is how to determine the reference effective diffusivity *D*ref. Instead of measurement directly, it can be computed numerically by an optimization routine

where *t*<sup>i</sup> denotes the time interval *i* of the data sampling. By minimizing the value of objective, the numerical moisture evolution is fitted to the experimental observation and the value of effective diffusivity is obtained. For an example, this method is applied for single wood particle drying process in superheated steam environment. The detail description of the geometrical information, initial and boundary conditions is presented in Le et al. [19]. The effective diffusivity of

temperature of 140°C can be used reliable for a large range of drying temperature as shown in **Figure 7**. It implies that the moisture diffusivity model presented in Eq. 13

difference between numerical moisture content *Xi*,*sim* and experimental *Xi*, exp as

*i*¼1:*end*

water is derived as

*∂*

porous medium.

where *D*ref (m2

6.633 � <sup>10</sup>�<sup>9</sup> <sup>m</sup><sup>2</sup>

**32**

*The evolution of drying history during optimization process for Dref determination (a) and experimental and numerical mean moisture content evolutions over time (b).*

The validated diffusion model of wood drying is implemented in a CFD model is used to investigate the drying behavior of a pilot dryer. The sketch of dryer (L = 1000 mm, D = 3000 mm, and H = 500 mm) where 30 plates of wood (L = 900 mm, D = 200 mm, and H = 25 mm) is presented in **Figure 8**. The air with a temperature of 140°C and moisture content of 7 g ram vapor/kg dry air flows into the dryer with a velocity of 0.1 m/s. The initial moisture and temperature of wood are 0.61 kg water/kg dried solid and 20°C. The 2D velocity profile of drying agent is presented in **Figure 8** and the evolution of moisture content of plate number (1), (2), (3), and (4) are plotted together with the mean moisture content of all plates in **Figure 9**. The results indicate a noticeable moisture content maldistribution of wood plates which should be remedied by wood plate disturbing during the drying process.

#### **3.2 Whitaker's continuum model**

In the 1970s, the simultaneous heat, mass, and momentum transfer for nonisothermal drying process is proposed by Whitaker based on the volume averaging technique. In this model, all pore-level mechanisms for heat and mass transfer are considered. The liquid flow due to capillary forces, vapor and gas flow due to convection and diffusion are taken into account in this model. Using the volume averaging technique, the properties of fluid and solid phases such as velocity, density, pressure are averaged in REV. Afterward, the macroscopic differential equations were defined in terms of average field quantities. The detail descriptions of Whitaker's model can be found in Vu et al. [20, 21]. In this section, the mass and energy conservation equations are briefly recalled.

**Figure 8.** *Pilot dryer model and velocity profile inside the dryer.*

**Figure 9.** *Moisture content evolution over time of wood.*

Mass conservation equation of liquid water

$$\frac{\partial}{\partial t}(\rho\_l \varepsilon\_l) + \nabla \cdot (\rho\_l v\_l) + \dot{m}\_v = 0 \tag{15}$$

calculated as *Kr*,*<sup>l</sup>* <sup>¼</sup> *<sup>ε</sup><sup>l</sup>*

bound and free water:

Tref = 0°C.

**Figure 10.**

**35**

*with different drying temperature (d).*

*ψ* <sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 10**, indicates the validity of the model.

The mass conservation equation of liquid can be rewrite as

*<sup>∂</sup><sup>t</sup> <sup>ε</sup><sup>l</sup>*,*<sup>f</sup> <sup>ρ</sup><sup>l</sup>* <sup>þ</sup> *<sup>ρ</sup><sup>b</sup>*

*∂*

and *Kr*,*<sup>g</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ε</sup><sup>l</sup>*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*ψ* <sup>2</sup>

The Whitaker model's is suitable for the porous media made by impermeable solid phase. This model is applied to describe the drying process of wood material [23]. A good agreement between experimental and numerical data, shown in

For porous media structured by permeable solid, the liquid water is not only transported in the void space, but it also diffuses in the solid phase. In these hygroscopic porous materials, the liquid water is available in the form of both

where *Ml*, *Ml*,*<sup>f</sup>* and *Mf*,*<sup>b</sup>* denote the mass of total, free and bound liquid water, respectively. As a result, the total moisture content is the sum of the free and bound moisture content *X* ¼ *Xl*,*<sup>f</sup>* þ *Xf*,*<sup>b</sup>*. It should be noted that the bound water is mainly accumulated inside the cells of cellular porous products. A small amount of liquid water is located in small pores inside the solid matrix. In Le et al. [24], the bound water is assumed to be accumulated in the small pores of the macro-solid matrix. Thus, the bound water is removed by both local evaporation and liquid diffusion.

*The distribution of moisture content (a), temperature (b), vapor pressure (c) over drying time for wood particle drying at 120 °C and the comparison between experimental and numerical mean moisture content obtained*

(J/kg) are the specific enthalpy of solid, liquid, vapor and air, respectively. Using the assumption of constant specific heat capacity, the specific enthalpy of these components can be calculated respectively by *hs* <sup>¼</sup> *cp*,*<sup>s</sup> <sup>T</sup>* � *Tref* , *hl* <sup>¼</sup> *cp*,*<sup>l</sup> <sup>T</sup>* � *Tref* , *ha* <sup>¼</sup> *cp*,*<sup>a</sup> <sup>T</sup>* � *Tref* and *hv* <sup>¼</sup> *cp*,*<sup>v</sup> <sup>T</sup>* � *Tref* <sup>þ</sup> *<sup>Δ</sup>hevp Tref* at reference temperature

[21, 22]. In Eq. 19, *hs*, *hl*, *hv* and *ha*

*Ml* ¼ *Ml*,*<sup>f</sup>* þ *Mf*,*<sup>b</sup>* (21)

<sup>þ</sup> <sup>∇</sup> � ð Þþ *<sup>ρ</sup>lv*l,f <sup>þ</sup> *<sup>ρ</sup>bvb <sup>m</sup>*\_ *<sup>v</sup>* <sup>¼</sup> <sup>0</sup> (22)

Mass conservation equation of water vapor

$$\frac{\partial}{\partial t} \left( \rho\_v \varepsilon\_{\rm g} \right) + \nabla \cdot \left( \rho\_v v\_{\rm g} \right) - \nabla \cdot \left( \rho\_{\rm g} D\_{\rm eff} \nabla \left( \frac{\rho\_v}{\rho\_{\rm g}} \right) \right) - \dot{m}\_v = 0 \tag{16}$$

Mass conservation of water in both vapor and liquid phases (sum of Eqs. 15 and 16)

$$\frac{\partial}{\partial t} \left( \rho\_l \mathbf{e}\_l + \rho\_v \mathbf{e}\_\mathbf{g} \right) + \nabla \cdot \left( \rho\_l \mathbf{v}\_l + \rho\_v \mathbf{v}\_\mathbf{g} \right) - \nabla \cdot \left( \rho\_\mathbf{g} D\_{\text{eff}} \nabla \left( \frac{\rho\_v}{\rho\_\mathbf{g}} \right) \right) = \mathbf{0} \tag{17}$$

Mass conservation of air in the gas phase

$$\frac{\partial}{\partial t} \left( \rho\_a \varepsilon\_\mathfrak{g} \right) + \nabla \cdot \left( \rho\_a v\_\mathfrak{g} \right) - \nabla \cdot \left( \rho\_\mathfrak{g} D\_{\mathfrak{eff}} \nabla \left( \frac{\rho\_a}{\rho\_\mathfrak{g}} \right) \right) = \mathbf{0} \tag{18}$$

Energy conservation equation

$$\begin{split} \frac{\partial}{\partial t} \left( \rho\_s \varepsilon\_b h\_t + \rho\_l \varepsilon\_l h\_l + \rho\_v \varepsilon\_g h\_v + \rho\_a \varepsilon\_g h\_a \right) + \nabla \cdot \left[ \rho\_l h\_l \upsilon\_l + (\rho\_v h\_v + \rho\_a h\_a) \upsilon\_g \right] \\ - \nabla \cdot \left[ h\_a \rho\_g D\_{\text{eff}} \nabla \left( \frac{\rho\_a}{\rho\_\mathcal{g}} \right) \right] - \nabla \cdot \left[ h\_v \rho\_g D\_{\text{eff}} \nabla \left( \frac{\rho\_v}{\rho\_\mathcal{g}} \right) \right] - \nabla \cdot \left[ \lambda\_{\text{eff}} \nabla (T) \right] = \mathbf{0} \end{split} \tag{19}$$

In these equations, *ρl*, *ρ<sup>v</sup>* and *ρ<sup>s</sup>* (kg/m3) are the mass density of liquid, vapor and air, respectively. *<sup>ε</sup><sup>l</sup>* <sup>¼</sup> *Vl <sup>V</sup>*, *<sup>ε</sup><sup>g</sup>* <sup>¼</sup> *Vg <sup>V</sup>* are the volume fraction of the liquid and gas phases. The porosity of the medium is computed as *ψ* ¼ *ε<sup>l</sup>* þ *ε<sup>g</sup>* and *ε<sup>s</sup>* ¼ 1 � *ψ* denotes the volume fraction of the solid phase. The superficial velocity of the liquid and gas phased *vl* and *vg* (m/s) are calculated as

$$v\_l = -\frac{KK\_{r,l}}{\mu\_l} \nabla p\_l \text{ and } v\_\mathbf{g} = -\frac{KK\_{r,\mathbf{g}}}{\mu\_\mathbf{g}} \nabla p\_\mathbf{g} \tag{20}$$

where *K* (m<sup>2</sup> ) is the absolute permeability of the porous medium. *Kr*,*<sup>l</sup>* and *Kr*,*<sup>v</sup>* are the relative permeabilities of the liquid and gas phases, which are respectively *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

calculated as *Kr*,*<sup>l</sup>* <sup>¼</sup> *<sup>ε</sup><sup>l</sup> ψ* <sup>2</sup> and *Kr*,*<sup>g</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ε</sup><sup>l</sup> ψ* <sup>2</sup> [21, 22]. In Eq. 19, *hs*, *hl*, *hv* and *ha* (J/kg) are the specific enthalpy of solid, liquid, vapor and air, respectively. Using the assumption of constant specific heat capacity, the specific enthalpy of these components can be calculated respectively by *hs* <sup>¼</sup> *cp*,*<sup>s</sup> <sup>T</sup>* � *Tref* , *hl* <sup>¼</sup> *cp*,*<sup>l</sup> <sup>T</sup>* � *Tref* , *ha* <sup>¼</sup> *cp*,*<sup>a</sup> <sup>T</sup>* � *Tref* and *hv* <sup>¼</sup> *cp*,*<sup>v</sup> <sup>T</sup>* � *Tref* <sup>þ</sup> *<sup>Δ</sup>hevp Tref* at reference temperature Tref = 0°C.

The Whitaker model's is suitable for the porous media made by impermeable solid phase. This model is applied to describe the drying process of wood material [23]. A good agreement between experimental and numerical data, shown in **Figure 10**, indicates the validity of the model.

For porous media structured by permeable solid, the liquid water is not only transported in the void space, but it also diffuses in the solid phase. In these hygroscopic porous materials, the liquid water is available in the form of both bound and free water:

$$\mathbf{M}\_l = \mathbf{M}\_{l\underline{f}} + \mathbf{M}\_{\underline{f},b} \tag{21}$$

where *Ml*, *Ml*,*<sup>f</sup>* and *Mf*,*<sup>b</sup>* denote the mass of total, free and bound liquid water, respectively. As a result, the total moisture content is the sum of the free and bound moisture content *X* ¼ *Xl*,*<sup>f</sup>* þ *Xf*,*<sup>b</sup>*. It should be noted that the bound water is mainly accumulated inside the cells of cellular porous products. A small amount of liquid water is located in small pores inside the solid matrix. In Le et al. [24], the bound water is assumed to be accumulated in the small pores of the macro-solid matrix. Thus, the bound water is removed by both local evaporation and liquid diffusion. The mass conservation equation of liquid can be rewrite as

$$\frac{\partial}{\partial t} \left( \varepsilon\_{l\,f} \rho\_l + \overline{\rho}\_b \right) + \nabla \cdot \left( \rho\_l v\_{l,\rm f} + \overline{\rho\_b v\_b} \right) + \dot{m}\_v = \mathbf{0} \tag{22}$$

#### **Figure 10.**

*The distribution of moisture content (a), temperature (b), vapor pressure (c) over drying time for wood particle drying at 120 °C and the comparison between experimental and numerical mean moisture content obtained with different drying temperature (d).*

Mass conservation equation of liquid water

Mass conservation equation of water vapor

� � <sup>þ</sup> <sup>∇</sup> � *<sup>ρ</sup>vvg*

� � <sup>þ</sup> <sup>∇</sup> � *<sup>ρ</sup>lvl* <sup>þ</sup> *<sup>ρ</sup>vvg*

� � <sup>þ</sup> <sup>∇</sup> � *<sup>ρ</sup>avg*

Mass conservation of air in the gas phase

*∂ <sup>∂</sup><sup>t</sup> <sup>ρ</sup>aε<sup>g</sup>*

Energy conservation equation

" # !

phased *vl* and *vg* (m/s) are calculated as

�<sup>∇</sup> � *haρgDeff*<sup>∇</sup> *<sup>ρ</sup><sup>a</sup>*

air, respectively. *<sup>ε</sup><sup>l</sup>* <sup>¼</sup> *Vl*

where *K* (m<sup>2</sup>

**34**

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup>sεshs* <sup>þ</sup> *<sup>ρ</sup>lεlhl* <sup>þ</sup> *<sup>ρ</sup>vεghv* <sup>þ</sup> *<sup>ρ</sup>aεgha*

*ρg*

*<sup>V</sup>*, *<sup>ε</sup><sup>g</sup>* <sup>¼</sup> *Vg*

*vl* ¼ � *KKr*,*<sup>l</sup> μl*

*∂ <sup>∂</sup><sup>t</sup> <sup>ρ</sup>vε<sup>g</sup>*

*Moisture content evolution over time of wood.*

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup>lε<sup>l</sup>* <sup>þ</sup> *<sup>ρ</sup>vε<sup>g</sup>*

*∂*

**Figure 9.**

*∂*

*∂*

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*<sup>∂</sup><sup>t</sup> <sup>ρ</sup><sup>l</sup>* ð Þþ *<sup>ε</sup><sup>l</sup>* <sup>∇</sup> � ð Þþ *<sup>ρ</sup>lvl <sup>m</sup>*\_ *<sup>v</sup>* <sup>¼</sup> <sup>0</sup> (15)

*ρg*

! !

*ρg*

� �

*<sup>V</sup>* are the volume fraction of the liquid and gas phases.

*μg*

� <sup>∇</sup> � *<sup>λ</sup>eff*∇ð Þ *<sup>T</sup>* � � <sup>¼</sup> <sup>0</sup>

∇*pg* (20)

! !

*ρg*

� *m*\_ *<sup>v</sup>* ¼ 0 (16)

¼ 0 (17)

¼ 0 (18)

(19)

! !

� � � <sup>∇</sup> � *<sup>ρ</sup>gDeff* <sup>∇</sup> *<sup>ρ</sup><sup>v</sup>*

Mass conservation of water in both vapor and liquid phases (sum of Eqs. 15 and 16)

� � � <sup>∇</sup> � *<sup>ρ</sup>gDeff* <sup>∇</sup> *<sup>ρ</sup><sup>v</sup>*

� � � <sup>∇</sup> � *<sup>ρ</sup>gDeff* <sup>∇</sup> *<sup>ρ</sup><sup>a</sup>*

*ρg*

� � <sup>þ</sup> <sup>∇</sup> � *<sup>ρ</sup>lhlvl* <sup>þ</sup> ð Þ *<sup>ρ</sup>vhv* <sup>þ</sup> *<sup>ρ</sup>aha vg*

� <sup>∇</sup> � *hvρgDeff*<sup>∇</sup> *<sup>ρ</sup><sup>v</sup>*

" # !

In these equations, *ρl*, *ρ<sup>v</sup>* and *ρ<sup>s</sup>* (kg/m3) are the mass density of liquid, vapor and

<sup>∇</sup>*pl* and *vg* ¼ � *KKr*,*<sup>g</sup>*

) is the absolute permeability of the porous medium. *Kr*,*<sup>l</sup>* and *Kr*,*<sup>v</sup>*

The porosity of the medium is computed as *ψ* ¼ *ε<sup>l</sup>* þ *ε<sup>g</sup>* and *ε<sup>s</sup>* ¼ 1 � *ψ* denotes the volume fraction of the solid phase. The superficial velocity of the liquid and gas

are the relative permeabilities of the liquid and gas phases, which are respectively

For the superheated steam drying process of rice seed, the bound water diffusivity can be calculated as

$$
\overline{D}\_b = \overline{D}\_{b, \text{ref}} \left( \frac{\rho\_w + \frac{X}{1+X} \rho\_s}{\rho\_w + \frac{X\_0}{1+X\_0} \rho\_s} \right)^2 \text{with} \, \overline{D}\_{b, \text{ref}} = 0.71 \times 10^{-9} \,\text{m}^2/\text{s} \tag{23}
$$

The internal structure of rice seed obtained by μ-CT and the comparison between the experimental observations and numerical results are presented in **Figure 11**. It indicates the validity of the proposed model.

Beside the porous material made by permeable solid phase, we consider the drying process of cellular plant product (i.e. fruits and vegetables) which is comprised of several types of cells such as parenchyma, collenchyma, and solute-conducting cells shown in **Figure 12** [25, 26]. In these materials, the water is mainly accumulated in the cell space. For several cellular product; i.e. eggplant, cucumber; the amount of intracellular water make up more than 95% of total water [27]. During the drying process, both intracellular and extracellular water is removed. The extracellular water is transported to the medium surface due to both capillary flow and internal evaporation. The changing of extracellular water content leads to the different of water potential between the cell space and intercellular void space which is the driving force of the intracellular water transport across the cell membrane as shown in **Figure 12**. A advance heat and mass transfer model for superheated steam drying process of cellular porous material was developed in Le et al. [28]. The conservation equations of extracellular and intracellular water are recalled as

$$
\rho\_l \frac{\partial \varepsilon\_{l,\text{ex}}}{\partial t} + \rho\_l \nabla . (v\_{l,\text{ex}}) = -\dot{m}\_v + j\_{w,1} A\_v \frac{\varepsilon\_{l,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{24}
$$

In Eqs. 24–26, *<sup>ε</sup>l*,*ex* <sup>¼</sup> *Vl*,*ex*

**Figure 12.**

tion flux is denoted by *m*\_ *<sup>v</sup>*(kg/m<sup>3</sup>

*removal (reprinted from [28] with the permission).*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

in a unit volume. The term *Av <sup>ε</sup>l*,*ex*

diffusive water flux through the cell walls *j*

*<sup>∂</sup> <sup>ρ</sup><sup>v</sup>* ð Þ *<sup>ε</sup><sup>v</sup>*,*ex ∂t*

*ρl ∂ε<sup>l</sup>*,*ex ∂t*

*ρl ∂ε<sup>l</sup>*,in *∂t* þ *j <sup>w</sup>*,1*Av*

the cell-matrix reads

where *<sup>ε</sup><sup>l</sup>*,*in* <sup>¼</sup> *Vl*,*in*

membrane, i.e. *j*

**37**

control volume reads [26, 28].

*ρcp <sup>∂</sup><sup>T</sup> ∂t*

*<sup>w</sup>*,1 and *j*

(Pa) and water conductivity of the cell membrane as

�*j <sup>w</sup>*,1*Av*

the water in the intercellular void space can be written as

þ *ρl*∇*:*ð Þþ *vl*,*ex*

*ε<sup>l</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

> *ε<sup>l</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

*<sup>V</sup>* denotes the volume fraction of cell water. Using the local thermal equilibrium assumption, the heat balance equation of the

<sup>þ</sup> *cp*,*<sup>l</sup>ρlvl*,*ex* <sup>þ</sup> *cp*,*<sup>v</sup>ρvvv*,*ex* <sup>∇</sup> � *<sup>T</sup>*

In conservation equation system, the diffusive water fluxes through the cell

These diffusive fluxes are determined by using the concept of the water potential *ϕ*

<sup>¼</sup> <sup>∇</sup> � *<sup>λ</sup>eff* <sup>∇</sup>*<sup>T</sup>* � <sup>Δ</sup>*hevp <sup>m</sup>*\_ *<sup>v</sup>* <sup>þ</sup> *<sup>j</sup>*

*<sup>V</sup>* , *<sup>ε</sup>v*,*ex* <sup>¼</sup> *Vv*,*ex*

water vapor in the intercellular void space, respectively. The volumetric evapora-

the medium, which is the exchange area between cells and intercellular void space

liquid, where *ψ* ¼ *ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex* denotes the intercellular porosity of the medium. The

mass conservation equation for the extracellular water vapor can thus be written as

After rearranging of Eqs. 24 and 25, the overall mass conservation equation for

� *j <sup>w</sup>*,2*Av*

We assumed that the water diffusion across the cell membrane is the mechanism of cell water removal, the mass conservation equation for intracellular water inside

> þ *j <sup>w</sup>*,2*Av*

*<sup>∂</sup> <sup>ρ</sup><sup>v</sup>* ð Þ *<sup>ε</sup><sup>v</sup>*,*ex ∂t*

þ ∇*:*ð Þ¼ *ρvvv*,*ex m*\_ *<sup>v</sup>* þ *j*

/m<sup>3</sup>

*<sup>w</sup>*,2 (kg/m<sup>2</sup>

*<sup>w</sup>*,2*Av*

s). *Av* (m<sup>2</sup>

*Pictorial representation of extracelluar liquid water (in blue) and intracellular liquid water (in green)*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*<sup>V</sup>* are the volume fractions of liquid water and

*ε<sup>v</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

þ ∇*:*ð Þ *ρvvv*,*ex*

*ε<sup>v</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

*ε<sup>v</sup>*,*ex <sup>ε</sup><sup>l</sup>*,*ex* <sup>þ</sup> *<sup>ε</sup><sup>v</sup>*,*ex* (30)

*<sup>w</sup>*,2*Av*

*<sup>w</sup>*,2 (red and violet arrows in **Figure 12**), need to be known.

¼ 0

*ε<sup>v</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

*<sup>ε</sup>l*,*ex*þ*εv*,*ex*refers to the cell surface area wetted by the

) is the volumetric specific area of

s) serve as vapor sources. The

(27)

(28)

¼ 0 (29)

$$\frac{\partial(\rho\_v \varepsilon\_{v,\text{ex}})}{\partial t} + \nabla.(\rho\_v v\_{v,\text{ex}}) = \dot{m}\_v + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{25}$$

$$\begin{split} \rho\_l \frac{\partial \varepsilon\_{l,\infty}}{\partial t} + \rho\_l \nabla . (v\_{l,\infty}) + \frac{\partial (\rho\_v \varepsilon\_{v,\infty})}{\partial t} + \nabla . (\rho\_v v\_{v,\infty}) \\ -j\_{w,1} A\_v \frac{\varepsilon\_{l,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} - j\_{w,2} A\_v \frac{\varepsilon\_{v,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} = 0 \end{split} \tag{26}$$

**Figure 11.**

*The morphology of rice seed (a) and the numerical temperature and moisture content evolution over time obtained from diffusion simulations (b).*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 12.**

For the superheated steam drying process of rice seed, the bound water diffu-

The internal structure of rice seed obtained by μ-CT and the comparison between the experimental observations and numerical results are presented in

þ *ρl*∇*:*ð Þ¼� *vl*,*ex m*\_ *<sup>v</sup>* þ *j*

þ ∇*:*ð Þ¼ *ρvvv*,*ex m*\_ *<sup>v</sup>* þ *j*

*<sup>∂</sup> <sup>ρ</sup><sup>v</sup>* ð Þ *<sup>ε</sup><sup>v</sup>*,*ex ∂t*

� *j <sup>w</sup>*,2*Av*

*The morphology of rice seed (a) and the numerical temperature and moisture content evolution over time*

þ *ρl*∇*:*ð Þþ *vl*,*ex*

*ε<sup>l</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex* *<sup>w</sup>*,1*Av*

*<sup>w</sup>*,2*Av*

þ ∇*:*ð Þ *ρvvv*,*ex*

*ε<sup>v</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

*ε<sup>l</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

*ε<sup>v</sup>*,*ex ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex*

¼ 0

Beside the porous material made by permeable solid phase, we consider the drying process of cellular plant product (i.e. fruits and vegetables) which is comprised of several types of cells such as parenchyma, collenchyma, and solute-conducting cells shown in **Figure 12** [25, 26]. In these materials, the water is mainly accumulated in the cell space. For several cellular product; i.e. eggplant, cucumber; the amount of intracellular water make up more than 95% of total water [27]. During the drying process, both intracellular and extracellular water is removed. The extracellular water is transported to the medium surface due to both capillary flow and internal evaporation. The changing of extracellular water content leads to the different of water potential between the cell space and intercellular void space which is the driving force of the intracellular water transport across the cell membrane as shown in **Figure 12**. A advance heat and mass transfer model for superheated steam drying process of cellular porous material was developed in Le et al. [28]. The conservation equations of

with*Db*,*ref* <sup>¼</sup> <sup>0</sup>*:*<sup>71</sup> � <sup>10</sup>�9m2

*=*s (23)

(24)

(25)

(26)

sivity can be calculated as

*Db* ¼ *Db*,*ref*

*<sup>ρ</sup><sup>w</sup>* <sup>þ</sup> *<sup>X</sup>* <sup>1</sup>þ*<sup>X</sup> <sup>ρ</sup><sup>s</sup>*

!<sup>2</sup>

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*<sup>ρ</sup><sup>w</sup>* <sup>þ</sup> *<sup>X</sup>*<sup>0</sup> <sup>1</sup>þ*X*<sup>0</sup> *<sup>ρ</sup><sup>s</sup>*

**Figure 11**. It indicates the validity of the proposed model.

extracellular and intracellular water are recalled as

*ρl ∂ε<sup>l</sup>*,*ex ∂t*

*ρl ∂ε<sup>l</sup>*,*ex ∂t*

**Figure 11.**

**36**

*obtained from diffusion simulations (b).*

*<sup>∂</sup> <sup>ρ</sup><sup>v</sup>* ð Þ *<sup>ε</sup><sup>v</sup>*,*ex ∂t*

> �*j <sup>w</sup>*,1*Av*

*Pictorial representation of extracelluar liquid water (in blue) and intracellular liquid water (in green) removal (reprinted from [28] with the permission).*

In Eqs. 24–26, *<sup>ε</sup>l*,*ex* <sup>¼</sup> *Vl*,*ex <sup>V</sup>* , *<sup>ε</sup>v*,*ex* <sup>¼</sup> *Vv*,*ex <sup>V</sup>* are the volume fractions of liquid water and water vapor in the intercellular void space, respectively. The volumetric evaporation flux is denoted by *m*\_ *<sup>v</sup>*(kg/m<sup>3</sup> s). *Av* (m<sup>2</sup> /m<sup>3</sup> ) is the volumetric specific area of the medium, which is the exchange area between cells and intercellular void space in a unit volume. The term *Av <sup>ε</sup>l*,*ex <sup>ε</sup>l*,*ex*þ*εv*,*ex*refers to the cell surface area wetted by the liquid, where *ψ* ¼ *ε<sup>l</sup>*,*ex* þ *ε<sup>v</sup>*,*ex* denotes the intercellular porosity of the medium. The diffusive water flux through the cell walls *j <sup>w</sup>*,2 (kg/m<sup>2</sup> s) serve as vapor sources. The mass conservation equation for the extracellular water vapor can thus be written as

$$\frac{\partial(\rho\_v \varepsilon\_{v,\text{ex}})}{\partial t} + \nabla.(\rho\_v v\_{v,\text{ex}}) = \dot{m}\_v + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{27}$$

After rearranging of Eqs. 24 and 25, the overall mass conservation equation for the water in the intercellular void space can be written as

$$\begin{split} \rho\_l \frac{\partial \varepsilon\_{l, \text{ex}}}{\partial t} + \rho\_l \nabla . (v\_{l, \text{ex}}) + \frac{\partial (\rho\_v \varepsilon\_{v, \text{ex}})}{\partial t} + \nabla . (\rho\_v v\_{v, \text{ex}}) \\ -j\_{w, 1} A\_v \frac{\varepsilon\_{l, \text{ex}}}{\varepsilon\_{l, \text{ex}} + \varepsilon\_{v, \text{ex}}} - j\_{w, 2} A\_v \frac{\varepsilon\_{v, \text{ex}}}{\varepsilon\_{l, \text{ex}} + \varepsilon\_{v, \text{ex}}} = \mathbf{0} \end{split} \tag{28}$$

We assumed that the water diffusion across the cell membrane is the mechanism of cell water removal, the mass conservation equation for intracellular water inside the cell-matrix reads

$$\rho\_l \frac{\partial \varepsilon\_{l,\text{in}}}{\partial t} + j\_{w,1} A\_v \frac{\varepsilon\_{l,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} = \mathbf{0} \tag{29}$$

where *<sup>ε</sup><sup>l</sup>*,*in* <sup>¼</sup> *Vl*,*in <sup>V</sup>* denotes the volume fraction of cell water.

Using the local thermal equilibrium assumption, the heat balance equation of the control volume reads [26, 28].

$$\begin{split} \left< \rho \varepsilon\_{p} \right> & \frac{\partial T}{\partial t} + \left( c\_{p,l} \rho\_{l} v\_{l,\infty} + c\_{p,v} \rho\_{v} v\_{v,\infty} \right) \nabla \cdot T \\ &= \nabla \cdot \left( \lambda\_{\text{eff}} \nabla T \right) - \Delta h\_{\text{exp}} \left( \dot{m}\_{v} + j\_{w,\mathfrak{L}} A\_{v} \frac{\varepsilon\_{v,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} \right) \end{split} \tag{30}$$

In conservation equation system, the diffusive water fluxes through the cell membrane, i.e. *j <sup>w</sup>*,1 and *j <sup>w</sup>*,2 (red and violet arrows in **Figure 12**), need to be known. These diffusive fluxes are determined by using the concept of the water potential *ϕ* (Pa) and water conductivity of the cell membrane as

**Figure 13.**

*Simulated and experimental evolution of the normalized total, intracellular and extracellular moisture content over time for potato drying in superheated steam environment at 180°C (replotted from [28] with the permission).*

$$j\_{w,1} = -\rho\_l k\_p \left(\phi\_{w,in} - \phi\_{l, \text{ex}}\right) \tag{31}$$

scales shall be carried out. For example, several effective macroscopic parameters such as the capillary pressure curve or the relative permeabilities may be obtained from a post-processing of the Monte-Carlo simulation results of pore-scale modeling. The moisture diffusivity under the thermal effect also should be revisted and compared with the insothermal drying process. This upscaling statergy can help on

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

This work is funded by the VIETNAMESE MINISTRY OF EDUCATION AND

School of Heat Engineering and Refrigeration, Hanoi University of Science and

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: hiep.lekieu@hust.edu.vn

both the product and system design, operation, and optimization.

TRAINING (MOET) under research project B2021-BKA-12.

**Acknowledgements**

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Author details**

Technology, Hanoi, Vietnam

provided the original work is properly cited.

Kieu Hiep Le

**39**

$$j\_{w,2} = -\rho\_l k\_p \left(\phi\_{w,in} - \phi\_{v,\infty}\right) \tag{32}$$

The water potential *ϕ* is computed as the difference of the total energy of one kilogram water molecules compared to liquid water molecule energy at the free liquid – water interface at 1 bar [25]. For the intracellular water, the water potential *ϕ<sup>w</sup>*,*in*, which was empirically determined in [29] as a function of the moisture content *ϕ<sup>w</sup>*,*in* ¼ *f X*ð Þ *in* . The potential of liquid water in the intercellular void space *ϕ<sup>l</sup>*,*ex* is calculated as the capillary pressure, i.e. *ϕ<sup>l</sup>*,*ex* ¼ �*pc* . The water vapor potential *ϕ<sup>v</sup>*,*ex* is computed from the gas theory based on the Gibbs free energy [30].

$$\phi\_{v,\text{ex}} = \frac{\tilde{R}T\rho\_l}{\tilde{M}\_w} \ln \frac{p\_{v,\text{ex}}}{p\_{v,\text{sat}}} \tag{33}$$

This model has been used to describe the superheated steam drying process of potato. As can be seen in **Figure 13**, the experimental observations can be reflected fairly by numerical results. It implies the predictive potential of the proposed model. Furthermore, the extracellular moisture is removed very soon when the dry process commences. The drying process is mainly driven by the dehydration of intracellular water by the diffusion mechanism. Although the Whitaker's continuum approach is effectively used to predict the drying behavior, the parameters of the continuum model is seemingly difficult to be measured. Thus, in the future, with the help of pore-scale model, these effective parameters should be theoretically extracted from the pore level simulations.

#### **4. Conclusions**

In this chapter, some recent advance models on heat and mass transfer during non-isothermal drying process of porous media have been reviewed. It indicates that the drying process at pore-scale and macro-scale has been thoroughly investigated. However, a system study on the bridges between the pore-, macro- and plant *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

scales shall be carried out. For example, several effective macroscopic parameters such as the capillary pressure curve or the relative permeabilities may be obtained from a post-processing of the Monte-Carlo simulation results of pore-scale modeling. The moisture diffusivity under the thermal effect also should be revisted and compared with the insothermal drying process. This upscaling statergy can help on both the product and system design, operation, and optimization.

### **Acknowledgements**

*j*

*j*

*ϕ<sup>l</sup>*,*ex* is calculated as the capillary pressure, i.e. *ϕ<sup>l</sup>*,*ex* ¼ �*pc*

extracted from the pore level simulations.

**4. Conclusions**

**38**

**Figure 13.**

*permission).*

*<sup>w</sup>*,1 ¼ �*ρlkp ϕ<sup>w</sup>*,*in* � *ϕ<sup>l</sup>*,*ex*

*Simulated and experimental evolution of the normalized total, intracellular and extracellular moisture content over time for potato drying in superheated steam environment at 180°C (replotted from [28] with the*

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*<sup>w</sup>*,2 ¼ �*ρlkp ϕ<sup>w</sup>*,*in* � *ϕ<sup>v</sup>*,*ex*

The water potential *ϕ* is computed as the difference of the total energy of one kilogram water molecules compared to liquid water molecule energy at the free liquid – water interface at 1 bar [25]. For the intracellular water, the water potential *ϕ<sup>w</sup>*,*in*, which was empirically determined in [29] as a function of the moisture content *ϕ<sup>w</sup>*,*in* ¼ *f X*ð Þ *in* . The potential of liquid water in the intercellular void space

*ϕ<sup>v</sup>*,*ex* is computed from the gas theory based on the Gibbs free energy [30].

*<sup>ϕ</sup><sup>v</sup>*,*ex* <sup>¼</sup> *RT*<sup>~</sup> *<sup>ρ</sup><sup>l</sup> M*~ *<sup>w</sup>*

This model has been used to describe the superheated steam drying process of potato. As can be seen in **Figure 13**, the experimental observations can be reflected fairly by numerical results. It implies the predictive potential of the proposed model. Furthermore, the extracellular moisture is removed very soon when the dry process commences. The drying process is mainly driven by the dehydration of intracellular water by the diffusion mechanism. Although the Whitaker's continuum approach is effectively used to predict the drying behavior, the parameters of the continuum model is seemingly difficult to be measured. Thus, in the future, with the help of pore-scale model, these effective parameters should be theoretically

In this chapter, some recent advance models on heat and mass transfer during non-isothermal drying process of porous media have been reviewed. It indicates that the drying process at pore-scale and macro-scale has been thoroughly investigated. However, a system study on the bridges between the pore-, macro- and plant

ln *pv*,*ex pv*,*sat*

(31)

(32)

. The water vapor potential

(33)

This work is funded by the VIETNAMESE MINISTRY OF EDUCATION AND TRAINING (MOET) under research project B2021-BKA-12.

### **Author details**

Kieu Hiep Le School of Heat Engineering and Refrigeration, Hanoi University of Science and Technology, Hanoi, Vietnam

\*Address all correspondence to: hiep.lekieu@hust.edu.vn

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] M. Prat, Recent advances in porescale models for drying of porous media, Chemical Engineering Journal 86 (2002) 153–164.

[2] V.K. Surasani, T. Metzger, E. Tsotsas, Consideration of heat transfer in pore network modelling of convective drying, International journal of heat and mass transfer 51 (2008) 2506–2518.

[3] A.S. Mujumdar, Handbook of Industrial Drying, CRC Press, 2014.

[4] T. Defraeye, Advanced computational modelling for drying processes – A review, Applied Energy 131 (2014) 323–344.

[5] M. Sahimi, Flow and transport in porous media and fractured rock: From classical methods to modern approaches, 2nd rev. and enlarged ed. ed., Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim, op. 2011.

[6] T. Sochi, Pore-Scale Modeling of Non-Newtonian Flow in Porous Media, PhD dissertation, London, 2007.

[7] S. Taslimi Taleghani, M. Dadvar, Two dimensional pore network modelling and simulation of nonisothermal drying by the inclusion of viscous effects, International Journal of Multiphase Flow 62 (2014) 37–44.

[8] V.K. Surasani, T. Metzger, E. Tsotsas, Drying Simulations of Various 3D Pore Structures by a Nonisothermal Pore Network Model, Drying Technology 28 (2010) 615–623.

[9] V.K. Surasani, T. Metzger, E. Tsotsas, Influence of heating mode on drying behavior of capillary porous media: Pore scale modeling, Chemical engineering science 63 (2008) 5218–5228.

[10] K.H. Le, A. Kharaghani, C. Kirsch, E. Tsotsas, Discrete pore network modeling

of superheated steam drying, Drying Technology 35 (2017) 1584–1601.

appropriate effective diffusivity for different food materials, Drying Technology 35 (2017) 335–346.

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

Isothermal Drying of Plant Cellular Materials Based on the Pore Network Approach, Drying Technology 32

[27] A. Halder, A.K. Datta, R.M. Spanswick, Water transport in cellular tissues during thermal processing, AIChE J. 57 (2011) 2574–2588.

Continuum-scale modeling of superheated steam drying of cellular plant porous media, International journal of heat and mass transfer 124

A moisture transfer model for isothermal drying of plant cellular materials based on the pore network approach, Drying Technology 32 (2014)

[28] K.H. Le, E. Tsotsas, A. Kharaghani,

[29] B. Xiao, J. Chang, X. Huang, X. Liu,

[30] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed., Academic Press, Amsterdam,

(2014) 1071–1081.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

(2018) 1033–1044.

1071–1081.

Boston, 2009.

[19] K.H. Le, T.T.H. Tran, E. Tsotsas, A. Kharaghani, Superheated steam drying of single wood particles: Modeling and comparative study with hot air drying, Chem. Eng. Technol. 44 (2021) 114–123.

[20] H.T. Vu, E. Tsotsas, Mass and Heat Transport Models for Analysis of the Drying Process in Porous Media: A Review and Numerical Implementation, International Journal of Chemical Engineering 2018 (2018) 1–13.

[21] H.T. Vu, E. Tsotsas, A Framework and Numerical Solution of the Drying Process in Porous Media by Using a Continuous Model, International Journal of Chemical Engineering 2019

[22] K.H. Le, N. Hampel, A. Kharaghani, A. Bück, E. Tsotsas, Superheated steam drying of single wood particles: A characteristic drying curve model deduced from continuum model simulations and assessed by

experiments, Drying Technology 36

[23] A. Kharaghani, K.H. Le, T.T.H. Tran, E. Tsotsas, Reaction engineering approach for modeling single wood particle drying at elevated air temperature, Chemical engineering science 199 (2019) 602–612.

[24] N. Hampel, K.H. Le, A. Kharaghani, E. Tsotsas, Continuous modeling of superheated steam drying of single rice grains, Drying Technology 37 (2019)

[25] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed.

[26] B. Xiao, J. Chang, X. Huang, X. Liu,

ed., Academic Press, [Place of publication not identified], 2009.

A Moisture Transfer Model for

(2019) 1–16.

(2018) 1866–1881.

1583–1596.

**41**

[11] A. Attari Moghaddam, A. Kharaghani, E. Tsotsas, M. Prat, Kinematics in a slowly drying porous medium: Reconciliation of pore network simulations and continuum modeling, Physics of Fluids 29 (2017) 22102.

[12] X. Lu, A. Kharaghani, E. Tsotsas, Transport parameters of macroscopic continuum model determined from discrete pore network simulations of drying porous media: Throat-node vs. throat-pore configurations, Chemical engineering science 223 (2020) 115723.

[13] X. Lu, E. Tsotsas, A. Kharaghani, Insights into evaporation from the surface of capillary porous media gained by discrete pore network simulations, International journal of heat and mass transfer 168 (2021) 120877.

[14] P. Perré, The proper use of mass diffusion equations in drying modeling: Introducing the drying intensity number, Drying Technology 33 (2015) 1949–1962.

[15] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying, in: Advances in Heat Transfer Volume 13, Elsevier, 1977, pp. 119–203.

[16] E. Tsotsas, A.S. Mujumdar (Eds.), Modern Drying Technology, Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim, Germany, 2007.

[17] P. Suvarnakuta, S. Devahastin, A.S. Mujumdar, A mathematical model for low-pressure superheated steam drying of a biomaterial, Chemical Engineering and Processing: Process Intensification 46 (2007) 675–683.

[18] M.I.H. Khan, C. Kumar, M.U.H. Joardder, M.A. Karim, Determination of *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

appropriate effective diffusivity for different food materials, Drying Technology 35 (2017) 335–346.

**References**

153–164.

[1] M. Prat, Recent advances in porescale models for drying of porous media, Chemical Engineering Journal 86 (2002)

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

of superheated steam drying, Drying Technology 35 (2017) 1584–1601.

[12] X. Lu, A. Kharaghani, E. Tsotsas, Transport parameters of macroscopic continuum model determined from discrete pore network simulations of drying porous media: Throat-node vs. throat-pore configurations, Chemical engineering science 223 (2020) 115723.

[13] X. Lu, E. Tsotsas, A. Kharaghani, Insights into evaporation from the surface of capillary porous media gained by discrete pore network simulations, International journal of heat and mass

[14] P. Perré, The proper use of mass diffusion equations in drying modeling: Introducing the drying intensity number, Drying Technology 33 (2015)

[15] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying, in: Advances in Heat Transfer Volume 13,

[16] E. Tsotsas, A.S. Mujumdar (Eds.), Modern Drying Technology, Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim,

[17] P. Suvarnakuta, S. Devahastin, A.S. Mujumdar, A mathematical model for low-pressure superheated steam drying of a biomaterial, Chemical Engineering and Processing: Process Intensification

[18] M.I.H. Khan, C. Kumar, M.U.H. Joardder, M.A. Karim, Determination of

Elsevier, 1977, pp. 119–203.

Germany, 2007.

46 (2007) 675–683.

transfer 168 (2021) 120877.

1949–1962.

[11] A. Attari Moghaddam, A. Kharaghani, E. Tsotsas, M. Prat, Kinematics in a slowly drying porous medium: Reconciliation of pore network simulations and continuum modeling, Physics of Fluids 29 (2017) 22102.

[2] V.K. Surasani, T. Metzger, E. Tsotsas, Consideration of heat transfer in pore network modelling of convective drying, International journal of heat and mass transfer 51 (2008) 2506–2518.

[3] A.S. Mujumdar, Handbook of Industrial Drying, CRC Press, 2014.

computational modelling for drying processes – A review, Applied Energy

[5] M. Sahimi, Flow and transport in porous media and fractured rock: From

approaches, 2nd rev. and enlarged ed. ed., Wiley-Vch Verlag GmbH & Co.

[6] T. Sochi, Pore-Scale Modeling of Non-Newtonian Flow in Porous Media, PhD dissertation, London, 2007.

[7] S. Taslimi Taleghani, M. Dadvar, Two dimensional pore network modelling and simulation of nonisothermal drying by the inclusion of viscous effects, International Journal of Multiphase Flow 62 (2014) 37–44.

[8] V.K. Surasani, T. Metzger, E. Tsotsas, Drying Simulations of Various 3D Pore Structures by a Nonisothermal Pore Network Model, Drying Technology 28

[9] V.K. Surasani, T. Metzger, E. Tsotsas, Influence of heating mode on drying behavior of capillary porous media: Pore scale modeling, Chemical engineering

[10] K.H. Le, A. Kharaghani, C. Kirsch, E. Tsotsas, Discrete pore network modeling

science 63 (2008) 5218–5228.

(2010) 615–623.

**40**

[4] T. Defraeye, Advanced

classical methods to modern

KGaA, Weinheim, op. 2011.

131 (2014) 323–344.

[19] K.H. Le, T.T.H. Tran, E. Tsotsas, A. Kharaghani, Superheated steam drying of single wood particles: Modeling and comparative study with hot air drying, Chem. Eng. Technol. 44 (2021) 114–123.

[20] H.T. Vu, E. Tsotsas, Mass and Heat Transport Models for Analysis of the Drying Process in Porous Media: A Review and Numerical Implementation, International Journal of Chemical Engineering 2018 (2018) 1–13.

[21] H.T. Vu, E. Tsotsas, A Framework and Numerical Solution of the Drying Process in Porous Media by Using a Continuous Model, International Journal of Chemical Engineering 2019 (2019) 1–16.

[22] K.H. Le, N. Hampel, A. Kharaghani, A. Bück, E. Tsotsas, Superheated steam drying of single wood particles: A characteristic drying curve model deduced from continuum model simulations and assessed by experiments, Drying Technology 36 (2018) 1866–1881.

[23] A. Kharaghani, K.H. Le, T.T.H. Tran, E. Tsotsas, Reaction engineering approach for modeling single wood particle drying at elevated air temperature, Chemical engineering science 199 (2019) 602–612.

[24] N. Hampel, K.H. Le, A. Kharaghani, E. Tsotsas, Continuous modeling of superheated steam drying of single rice grains, Drying Technology 37 (2019) 1583–1596.

[25] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed. ed., Academic Press, [Place of publication not identified], 2009.

[26] B. Xiao, J. Chang, X. Huang, X. Liu, A Moisture Transfer Model for

Isothermal Drying of Plant Cellular Materials Based on the Pore Network Approach, Drying Technology 32 (2014) 1071–1081.

[27] A. Halder, A.K. Datta, R.M. Spanswick, Water transport in cellular tissues during thermal processing, AIChE J. 57 (2011) 2574–2588.

[28] K.H. Le, E. Tsotsas, A. Kharaghani, Continuum-scale modeling of superheated steam drying of cellular plant porous media, International journal of heat and mass transfer 124 (2018) 1033–1044.

[29] B. Xiao, J. Chang, X. Huang, X. Liu, A moisture transfer model for isothermal drying of plant cellular materials based on the pore network approach, Drying Technology 32 (2014) 1071–1081.

[30] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed., Academic Press, Amsterdam, Boston, 2009.

Section 2

Mathematical Models in

Porous Media and Solutions

**43**

Section 2

## Mathematical Models in Porous Media and Solutions

**Chapter 3**

**Abstract**

*Twinkle R. Singh*

differential equations

**1. Introduction**

**45**

Study on Approximate Analytical

This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as *vi* ¼ �*vn*, has been considered here main aim, here is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* is chosen as a dependent variable while *x and t* are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of

**Keywords:** Variational Iteration method (VIM), Adomian decomposition method (ADM), Laplace Adomian decomposition method (LADM), nonlinear partial

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2–10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11–28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations

Method with Its Application

Arising in Fluid Flow

our method is illustrated by different numerical.

**Chapter 3**

## Study on Approximate Analytical Method with Its Application Arising in Fluid Flow

*Twinkle R. Singh*

#### **Abstract**

This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as *vi* ¼ �*vn*, has been considered here main aim, here is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* is chosen as a dependent variable while *x and t* are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of our method is illustrated by different numerical.

**Keywords:** Variational Iteration method (VIM), Adomian decomposition method (ADM), Laplace Adomian decomposition method (LADM), nonlinear partial differential equations

#### **1. Introduction**

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2–10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11–28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations

**44**

(PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29–46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic systems.

The mathematical condition for imbibition phenomenon is given by Scheidegger

Where *vi* & *vn* are the seepage velocities of injected & native liquids respectively.

*vn* ¼ �*vi*

*ki* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> *i kn* ¼ 1 � *αSn*, α ¼ 1*:*11

Where *ki* & *kn* denotes fictitious relative permeability. *Si* & *Sn* denotes satura-

According to the Darcy's law, the basic equations of the phenomenon as; [78]

*δn <sup>K</sup>*

> þ *∂vi*

þ *∂vn*

Combing equations (1)-(5) and using the relation for capillary pressure as,

*KD S*ð Þ*<sup>i</sup> <sup>β</sup> <sup>∂</sup>Si*

*<sup>δ</sup>nki*þ*δikn* and <sup>β</sup> being small capillary pressure coefficient.

*<sup>φ</sup>L*<sup>2</sup> , 0<sup>≤</sup> *<sup>x</sup>*<sup>≤</sup>

*∂x*

*LSio*

Where v*<sup>i</sup>* and v*<sup>n</sup>* are the seepage velocities, k*<sup>i</sup>* and kare the relative permeabilities *δ<sup>i</sup>* and *δ<sup>n</sup>* are the kinematic viscosities (which are constants), p*<sup>i</sup>* and p*<sup>n</sup>* are pressure of the injected and native liquid respectively, *φ* and *K* are the porosity and the permeability of the homogeneous porous medium; S*<sup>i</sup>* is the saturation of the injected liquid; p*<sup>c</sup>* is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the

*∂pi ∂x*

*∂pn ∂x*

*vi* ¼ �*vn* (3) *pc* ¼ *pn* � *pi* (4)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (5)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (6)

¼ 0 (7)

*<sup>B</sup> :* (8)

(1)

(2)

*vi* ¼ � *ki δi <sup>K</sup>*

*vn* ¼ � *kn*

*φ ∂Si ∂t* 

*φ ∂Sn ∂t* 

þ *∂ ∂x*

*<sup>L</sup>* , *<sup>θ</sup>* <sup>¼</sup> *Lt*

It is assumed is that an average value of D(S*i*) =*D* (S*i*)

*<sup>ξ</sup>* <sup>¼</sup> *<sup>x</sup>*

The relation between relative permeability and phase-saturation,

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

tions of injected and native liquids respectively.

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

**3. Mathematical structure of the model**

imbibition face x=0.

p*c*=*β Si* [70], we get,

Where D(S*i*) = *kikn*

Eq. (7), becomes;

**47**

Using the transformation,

*φ ∂Si ∂t* 

[78]); viz,

Same way Laplace Adomian's Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50–56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57–75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov [56] solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the solution of Falkner-Skan equation.

In the solution procedure of VIM; many repeated computations and computations of the unneeded forms, which take more time and effort beyond it, so a modification has been shown to reduce these unneeded forms.

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have analyzed it for homogeneous porous medium.

In this Present investigated model, Imbibition takes place over a small part of a large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of countercurrent imbibition phenomenon as *vi* ¼ �*vn*, Our purpose is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* has been chosen as a dependent variable while *x* and *t* are chosen as independent variable.

#### **2. Imbibition phenomenon**

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the imbibed fluid, is higher.

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow DOI: http://dx.doi.org/10.5772/intechopen.97548*

The mathematical condition for imbibition phenomenon is given by Scheidegger [78]); viz,

$$v\_n = -v\_i$$

Where *vi* & *vn* are the seepage velocities of injected & native liquids respectively. The relation between relative permeability and phase-saturation,

$$k\_i = \mathbb{S}\_i^3$$

$$k\_n = \mathbf{1} - a\mathbf{S}\_n, \alpha = \mathbf{1}.\mathbf{1}\mathbf{1}$$

Where *ki* & *kn* denotes fictitious relative permeability. *Si* & *Sn* denotes saturations of injected and native liquids respectively.

#### **3. Mathematical structure of the model**

(PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29–46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator

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equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic

Same way Laplace Adomian's Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50–56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57–75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov [56] solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the

In the solution procedure of VIM; many repeated computations and computa-

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have

In this Present investigated model, Imbibition takes place over a small part of a

tions of the unneeded forms, which take more time and effort beyond it, so a

large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of countercurrent imbibition phenomenon as *vi* ¼ �*vn*, Our purpose is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* has been chosen as a dependent

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the

modification has been shown to reduce these unneeded forms.

variable while *x* and *t* are chosen as independent variable.

systems.

solution of Falkner-Skan equation.

**2. Imbibition phenomenon**

imbibed fluid, is higher.

**46**

analyzed it for homogeneous porous medium.

According to the Darcy's law, the basic equations of the phenomenon as; [78]

$$v\_i = -\left(\frac{k\_i}{\delta\_i}\right) K \frac{\partial p\_i}{\partial \mathbf{x}} \tag{1}$$

$$v\_n = -\left(\frac{k\_n}{\delta\_n}\right) K \left.\frac{\partial p\_n}{\partial \mathbf{x}}\right|\_{\mathbf{x}}\tag{2}$$

$$
v\_i = -v\_n\tag{3}$$

$$p\_c = p\_n - p\_i \tag{4}$$

$$
\rho \left( \frac{\partial \mathbf{S}\_i}{\partial t} \right) + \frac{\partial v\_i}{\partial \mathbf{x}} = \mathbf{0} \tag{5}
$$

$$
\rho \left( \frac{\partial \mathbf{S}\_n}{\partial t} \right) + \frac{\partial v\_n}{\partial \mathbf{x}} = \mathbf{0} \tag{6}
$$

Where v*<sup>i</sup>* and v*<sup>n</sup>* are the seepage velocities, k*<sup>i</sup>* and kare the relative permeabilities *δ<sup>i</sup>* and *δ<sup>n</sup>* are the kinematic viscosities (which are constants), p*<sup>i</sup>* and p*<sup>n</sup>* are pressure of the injected and native liquid respectively, *φ* and *K* are the porosity and the permeability of the homogeneous porous medium; S*<sup>i</sup>* is the saturation of the injected liquid; p*<sup>c</sup>* is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the imbibition face x=0.

Combing equations (1)-(5) and using the relation for capillary pressure as, p*c*=*β Si* [70], we get,

$$\log\left(\frac{\partial \mathbf{S}\_i}{\partial t}\right) + \frac{\partial}{\partial \mathbf{x}} \left[ \mathbf{K} \mathbf{D}(\mathbf{S}\_i) \beta \left(\frac{\partial \mathbf{S}\_i}{\partial \mathbf{x}}\right) \right] = \mathbf{0} \tag{7}$$

Where D(S*i*) = *kikn <sup>δ</sup>nki*þ*δikn* and <sup>β</sup> being small capillary pressure coefficient. It is assumed is that an average value of D(S*i*) =*D* (S*i*) Using the transformation,

$$\xi = \frac{\varkappa}{L} \quad , \theta = \frac{Lt}{\varrho L^2} , \quad 0 \le \varkappa \le \frac{LS\_{io}}{B} . \tag{8}$$

Eq. (7), becomes;

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$$\left(\frac{\partial \mathbf{S}\_i}{\partial \theta}\right) + \beta \overline{D}(\mathbf{S}\_i) \frac{\partial^2 \mathbf{S}\_i}{\partial \xi^2} = \mathbf{0}$$

*∂Si <sup>∂</sup><sup>θ</sup>* ¼ �*βD S*ð Þ*<sup>i</sup> ∂Si* 2 *∂ξ*<sup>2</sup>

$$\frac{\partial \mathbf{S}\_i}{\partial \theta} = \varepsilon \frac{\partial \mathbf{S}\_i^{\;2}}{\partial \xi^2} \text{ Where } \varepsilon = -\beta \overline{D}(\mathbf{S}\_i) \tag{9}$$

By the Hopf-Cole transformation [79, 80] equation (9) reduces to the Burger's equation.

$$\mathbf{S}\_{i\_{\theta}}^{\*} + \mathbf{S}\_{i}^{\*} \mathbf{S}\_{i\_{\xi}}^{\*} = \varepsilon \mathbf{S}\_{i}^{\*} \mathbf{g}\_{\xi} \tag{10}$$

To solve equation (10) by VIM, substituting in equation (14) by

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

And can obtain the following variational iteration formula:

*in* � ð *θ*

0

*S* ∗

*S* ∗ *in<sup>τ</sup>* <sup>þ</sup> *<sup>S</sup>* <sup>∗</sup>

Using (14), the approximate solutions *Un*ð Þ *x*, *t* are obtained by substituting;

*<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, 0 *<sup>S</sup>* <sup>∗</sup>

<sup>1</sup> *θ*; *where β*<sup>0</sup>

<sup>1</sup> *<sup>θ</sup>* <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> 1 *θ*2

*S* ∗

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

Approximate solutions are given below;

*S* ∗ *<sup>i</sup>*<sup>3</sup> <sup>¼</sup> *<sup>S</sup>* <sup>∗</sup> *i*0 *e <sup>ξ</sup>* � *<sup>β</sup>*<sup>1</sup> 2 *θ*2 2! <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> 2 *θ*3 3!

1.VIM can contain a series solution not exactly like ADM.

2.VIM needs many modifications to overcome the wasted time in the repeated

*<sup>θ</sup> S* <sup>∗</sup> *<sup>i</sup> S* <sup>∗</sup> *<sup>i</sup><sup>ξ</sup>* � *<sup>ε</sup><sup>S</sup>* <sup>∗</sup>

*Si*ð Þ¼ *ξ*, 0 *e*

*<sup>i</sup>*<sup>1</sup> ð Þ¼ *<sup>ξ</sup>*, *<sup>θ</sup> <sup>β</sup>*<sup>0</sup>

12 *θ*3 <sup>3</sup> � *εβ*<sup>1</sup> 2 *θ*2 2

*ξ*

1 *θ*

To overcome these problems, following ADM and LADM is suggested.

*<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, *<sup>θ</sup> <sup>L</sup>*�<sup>1</sup>

*S* ∗

*<sup>i</sup>*<sup>2</sup> ð Þ¼ *<sup>ξ</sup>*, *<sup>θ</sup> <sup>β</sup>*<sup>0</sup>

*S* ∗

From the analysis we can observed is this:

calculations and unneeded terms.

And recursive relation is:

Then:

**49**

Now applying ADM to equation (10); we get

*S* ∗

*S* ∗ *<sup>i</sup>*<sup>1</sup> <sup>¼</sup> *<sup>S</sup>* <sup>∗</sup> *i*0 *e <sup>ξ</sup>* � *<sup>β</sup>*<sup>0</sup>

> *S* ∗ *<sup>i</sup>*<sup>2</sup> <sup>¼</sup> *<sup>S</sup>* <sup>∗</sup> *i*0 *e <sup>ξ</sup>* � *<sup>β</sup>*<sup>0</sup>

*<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>S</sup>* <sup>∗</sup>

& g(x,t) = 0

Similarly,

And so on … .. Notes on VIM *RUn* ¼ �*U*<sup>2</sup>

*NUn* ¼ *Un*ð Þ *Un xx*

*nx*

*in <sup>S</sup>* <sup>∗</sup> *in* � � *ξ* � *<sup>ε</sup><sup>S</sup>* <sup>∗</sup> *inξξ* � �*d<sup>τ</sup>* (16)

*i*0 *e*

<sup>1</sup> <sup>¼</sup> *<sup>S</sup>* <sup>∗</sup> <sup>2</sup> *<sup>i</sup>*<sup>0</sup> *e*

<sup>2</sup> *where <sup>β</sup>*<sup>1</sup>

*<sup>ξ</sup>* (17)

<sup>2</sup>*<sup>ξ</sup>* � *<sup>ε</sup><sup>S</sup>* <sup>∗</sup> *i*0 *e*

*<sup>i</sup>ξξ* h i (18)

*<sup>ξ</sup>* � �

<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*<sup>0</sup> <sup>1</sup> *S* <sup>∗</sup> *i*0 *e ξ*

With the condition

*S* ∗ *<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, 0 *<sup>S</sup>* <sup>∗</sup> *i*0 *e <sup>ξ</sup>* at time *θ* = 0 and *ξ* > 0

#### **3.1 Solution of the Burger's equation by variational iteration method**

To add the basic concepts of VIM, considering the below mentioned nonlinear partial differential equations:

$$\begin{aligned} Lu(\mathbf{x},t) + Ru(\mathbf{x},t) + Nu(.,t) &= \mathbf{g}(\mathbf{x},t), \\ u(\mathbf{x},0) &= e^{\mathbf{x}} \end{aligned} \tag{11}$$

Where *<sup>L</sup>* <sup>¼</sup> *<sup>∂</sup> ∂t* � �, *R* is a linear operator which has partial derivatives with respect to x, Nu(x,t) is a nonlinear term and g(x,t) is an inhomogeneous term.

As per the VIM [6, 7];

$$U\_{n+1}(\mathbf{x}, t) = U\_n(\mathbf{x}, t) + \int\_0^t \lambda \{LU\_n + \overline{RU\_n} + \overline{NU\_n} - \mathbf{g}\} d\mathbf{x} \tag{12}$$

Where *λ* is called a general Lagrange multiplier [81, 82] which can be identified optimally via vatiational theory, *RUn* and *NUn* are considered as restricted variations,

i.e. *δRUn* ¼ 0, *δNUn* ¼ 0 calculating variation with respect to *Un*;

$$\begin{aligned} \lambda'(\boldsymbol{\pi}) &= \mathbf{0} \\ \mathbf{1} + \lambda(\boldsymbol{\pi})\_{\boldsymbol{\pi}=t} &= \mathbf{0} \end{aligned} \tag{13}$$

The Lagrange multiplier, therefore, can be considered as *λ*=-1. Now, substituting the multiplier in (12), then

$$U\_{n+1}(\mathbf{x}, t) = U\_n - \int\_0^t \{L(U\_n) + R(U\_n) + N(U\_n) - \mathbf{g}\} d\tau \tag{14}$$

$$\mathcal{S}\_{i\_0}^\* + \mathcal{S}\_i^\* \mathcal{S}\_{i\_\xi}^\* = \varepsilon \mathcal{S}\_i^\* \, \_{\xi\xi} \tag{15}$$

With the constrain

*S* ∗ *<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, 0 *<sup>S</sup>* <sup>∗</sup> *i*0 *e <sup>ξ</sup>* at time *θ* = 0 and *ξ* > 0 *Study on Approximate Analytical Method with Its Application Arising in Fluid Flow DOI: http://dx.doi.org/10.5772/intechopen.97548*

To solve equation (10) by VIM, substituting in equation (14) by

$$RU\_{\pi} = -U\_{\pi\_{\pi}}^2$$

$$NU\_{\pi} = U\_{\pi}(U\_n)\_{\infty}$$

& g(x,t) = 0

*∂Si ∂θ* � �

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*∂Si* 2

*S* ∗ *<sup>i</sup><sup>θ</sup>* <sup>þ</sup> *<sup>S</sup>* <sup>∗</sup> *<sup>i</sup> S* <sup>∗</sup>

**3.1 Solution of the Burger's equation by variational iteration method**

to x, Nu(x,t) is a nonlinear term and g(x,t) is an inhomogeneous term.

ð*t*

0

optimally via vatiational theory, *RUn* and *NUn* are considered as restricted

i.e. *δRUn* ¼ 0, *δNUn* ¼ 0 calculating variation with respect to *Un*;

*λ*0 ð Þ¼ *τ* 0 <sup>1</sup> <sup>þ</sup> *λ τ*ð Þ*<sup>τ</sup>*¼*<sup>t</sup>* <sup>¼</sup> <sup>0</sup>

The Lagrange multiplier, therefore, can be considered as *λ*=-1.

ð*t*

0

*S* ∗ *<sup>i</sup><sup>θ</sup>* <sup>þ</sup> *<sup>S</sup>* <sup>∗</sup> *<sup>i</sup> S* <sup>∗</sup>

*<sup>ξ</sup>* at time *θ* = 0 and *ξ* > 0

Where *λ* is called a general Lagrange multiplier [81, 82] which can be identified

*<sup>i</sup><sup>ξ</sup>* <sup>¼</sup> *<sup>ε</sup><sup>S</sup>* <sup>∗</sup>

*<sup>ξ</sup>* at time *θ* = 0 and *ξ* > 0

*u x*ð Þ¼ , 0 *e*

*Un*þ<sup>1</sup>ð Þ¼ *x*, *t Un*ð Þþ *x*, *t*

Now, substituting the multiplier in (12), then

*Un*þ<sup>1</sup>ð Þ¼ *x*, *t Un* �

With the constrain

*i*0 *e*

*<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, 0 *<sup>S</sup>* <sup>∗</sup>

*S* ∗

**48**

*∂Si <sup>∂</sup><sup>θ</sup>* <sup>¼</sup> *<sup>ε</sup>*

*∂Si*

equation.

*S* ∗

*<sup>∂</sup><sup>θ</sup>* ¼ �*βD S*ð Þ*<sup>i</sup>*

With the condition

*i*0 *e*

partial differential equations:

*∂t*

As per the VIM [6, 7];

*<sup>i</sup>* ð Þ¼ *<sup>ξ</sup>*, 0 *<sup>S</sup>* <sup>∗</sup>

Where *<sup>L</sup>* <sup>¼</sup> *<sup>∂</sup>*

variations,

*∂Si* 2 *∂ξ*<sup>2</sup>

þ *βD S*ð Þ*<sup>i</sup>*

By the Hopf-Cole transformation [79, 80] equation (9) reduces to the Burger's

To add the basic concepts of VIM, considering the below mentioned nonlinear

*Lu x*ð Þþ , *t Ru x*ð Þþ , *t Nu*ð Þ¼ , *t g x*ð Þ , *t* ,

� �, *R* is a linear operator which has partial derivatives with respect

*<sup>i</sup><sup>ξ</sup>* <sup>¼</sup> *<sup>ε</sup><sup>S</sup>* <sup>∗</sup>

*∂*2 *Si <sup>∂</sup>ξ*<sup>2</sup> <sup>¼</sup> <sup>0</sup>

*<sup>∂</sup>ξ*<sup>2</sup> Where *<sup>ε</sup>* ¼ �*βD*ð Þ <sup>S</sup>*<sup>i</sup>* (9)

*<sup>x</sup>* (11)

*<sup>λ</sup> LUn* <sup>þ</sup> *RUn* <sup>þ</sup> *NUn* � *<sup>g</sup>* � �*d<sup>τ</sup>* (12)

f g *L U*ð Þþ *<sup>n</sup> R U*ð Þþ *<sup>n</sup> N U*ð Þ� *<sup>n</sup> g dτ* (14)

*<sup>i</sup> ξξ* (15)

(13)

*<sup>i</sup> ξξ* (10)

And can obtain the following variational iteration formula:

$$\mathcal{S}\_{i\_{\
u+1}}^{\*} = \mathcal{S}\_{i\_{\pi}}^{\*} - \int\_{0}^{\theta} \left\{ \mathcal{S}\_{i\_{n\_{\pi}}}^{\*} + \mathcal{S}\_{i\_{n}}^{\*} \left( \mathcal{S}\_{i\_{n}}^{\*} \right)\_{\xi} - \varepsilon \mathcal{S}\_{i\_{n\_{\pi}^{\pm}}}^{\*} \right\} d\tau \tag{16}$$

Using (14), the approximate solutions *Un*ð Þ *x*, *t* are obtained by substituting;

$$\mathbf{S}\_{i}^{\*}\left(\xi,\mathbf{0}\right) = \mathbf{S}\_{i\_{0}}^{\*}\mathbf{e}^{\xi} \tag{17}$$

Approximate solutions are given below;

$$\mathcal{S}\_{i\_1}^\* = \mathcal{S}\_{i\_0}^\* e^{\xi} - \beta\_1^0 \theta; \quad \text{where} \ \beta\_1^0 = \left(\mathcal{S}\_{i\_0}^{\*^2} e^{2\xi} - \varepsilon \mathcal{S}\_{i\_0}^\* e^{\xi}\right)^2$$

$$\mathcal{S}\_{i\_2}^\* = \mathcal{S}\_{i\_0}^\* e^{\xi} - \beta\_1^0 \theta + \beta\_1^1 \frac{\theta^2}{2} \quad \text{where } \beta\_1^1 = \beta\_1^0 \mathcal{S}\_{i\_0}^\* e^{\xi}$$

Similarly,

$$\mathcal{S}\_{i\_3}^\* = \mathcal{S}\_{i\_0}^\* e^{\xi} - \beta\_2^1 \frac{\theta^2}{2!} + \beta\_2^2 \frac{\theta^3}{3!}$$

And so on … .. Notes on VIM From the analysis we can observed is this:


To overcome these problems, following ADM and LADM is suggested. Now applying ADM to equation (10); we get

$$\mathcal{S}\_{i}^{\*}\left(\xi,\theta\right) = L\_{\theta}^{-1} \left[\mathcal{S}\_{i}^{\*}\mathcal{S}\_{i\_{\xi}}^{\*} - \varepsilon \mathcal{S}\_{i\_{\xi\xi}}^{\*}\right] \tag{18}$$

And recursive relation is:

$$\mathcal{S}\_i(\xi, \mathbf{0}) = e^{\xi}$$

Then:

$$\mathcal{S}\_{i\_1}^\*(\xi, \theta) = \beta\_1^0 \theta$$

$$\mathcal{S}\_{i\_2}^\*(\xi, \theta) = \beta\_{12}^0 \frac{\theta^3}{3} - \varepsilon \beta\_2^1 \frac{\theta^2}{2}$$

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$$\mathcal{S}\_{i\_3}^\*(\xi, \theta) = \beta\_{123}^0 \frac{\theta^4}{4} - \varepsilon \beta\_3^1 \frac{\theta^3}{3}$$

and so on …

Now, applying (LADM) Laplace transform with respect to t on both sides of (10);

$$\begin{aligned} \mathcal{S}\_i^\* \left( \mathbf{x}, t \right) &= L^{-1} \left[ \frac{\mathbf{1}}{\mathfrak{s}} L \left[ \mathcal{S}\_{i\_0}^\* \mathcal{S}\_{i\_{0\_\xi}}^\* - \varepsilon \mathcal{S}\_{i\_{0\_\xi}}^\* \right] \right] \\ & \quad \mathcal{S}\_{i\_1}^\* = \beta\_1^0 e^{\xi} \theta \end{aligned}$$

$$\begin{aligned} \mathcal{S}\_{i\_2}^\* &= \left( \beta\_1^{0^2} e^{2\xi} - \varepsilon \beta\_1^0 e^{\xi} \right) \frac{\theta^2}{2!} \\ \mathcal{S}\_{i\_3}^\* &= \left( \beta\_0^3 - \varepsilon \beta\_1^3 \right) \frac{\theta^3}{3!} \end{aligned}$$

And so on …

#### **4. Interpretation**

It is concluded that for the non linear partial differential equation of imbibitions phenomenon in oil recovery process, through graphs, it has been observed that the

**Figure 3.**

**Figure 4.** *Plot of Saturation S* <sup>∗</sup>

**Figure 5.** *Plot of Saturation S* <sup>∗</sup>

**51**

*<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for ADM Solution.*

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

*<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for LADM Solution.*

*3-Dimensional VIM Solution.*

**Figure 1.** *Plot of Saturation S* <sup>∗</sup> *<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for VIM Solution.*

**Figure 2.** *Plot of Saturation S* <sup>∗</sup> *<sup>i</sup>* ð Þ *ξ*, *θ versus θ for VIM Solution.*

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow DOI: http://dx.doi.org/10.5772/intechopen.97548*

*S* ∗

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>L</sup>*�<sup>1</sup> <sup>1</sup>

*S* ∗ *<sup>i</sup>*<sup>2</sup> <sup>¼</sup> *<sup>β</sup>*0<sup>2</sup> 1 *e*

*<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for VIM Solution.*

*<sup>i</sup>* ð Þ *ξ*, *θ versus θ for VIM Solution.*

*S* ∗ *<sup>i</sup>*<sup>3</sup> <sup>¼</sup> *<sup>β</sup>*<sup>3</sup>

*S* ∗

and so on …

And so on …

**4. Interpretation**

**Figure 1.** *Plot of Saturation S* <sup>∗</sup>

**Figure 2.** *Plot of Saturation S* <sup>∗</sup>

**50**

*<sup>i</sup>*<sup>3</sup> ð Þ¼ *<sup>ξ</sup>*, *<sup>θ</sup> <sup>β</sup>*<sup>0</sup>

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

123 *θ*4 <sup>4</sup> � *εβ*<sup>1</sup> 3 *θ*3 3

Now, applying (LADM) Laplace transform with respect to t on both sides of (10);

<sup>2</sup>*<sup>ξ</sup>* � *εβ*<sup>0</sup> 1 *e <sup>ξ</sup>* � � *θ*<sup>2</sup>

> <sup>0</sup> � *εβ*<sup>3</sup> 1 � � *<sup>θ</sup>*<sup>3</sup>

It is concluded that for the non linear partial differential equation of imbibitions phenomenon in oil recovery process, through graphs, it has been observed that the

h i � �

2!

3!

*s L S* <sup>∗</sup> *<sup>i</sup>*<sup>0</sup> *<sup>S</sup>* <sup>∗</sup> *i*0*ξ* � *<sup>ε</sup><sup>S</sup>* <sup>∗</sup> *i*0*ξξ*

*S* ∗ *<sup>i</sup>*<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*<sup>0</sup> 1 *e ξ θ*

**Figure 4.** *Plot of Saturation S* <sup>∗</sup> *<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for ADM Solution.*

**Figure 5.** *Plot of Saturation S* <sup>∗</sup> *<sup>i</sup>* ð Þ *ξ*, *θ versus ξ for LADM Solution.*

**Figure 6.** *Plot of Saturation S* <sup>∗</sup> *<sup>i</sup>* ð Þ *ξ*, *θ versus θ for LADM Solution.*

**Figure 7.** *3-Dimensional LADM Solution*

saturation of injected water during imbibition, increases and it is noted that LADM gives faster accuracy compare to VIM and ADM (**Figures 1**–**7**).

**Author details**

Twinkle R. Singh

**53**

Applied Mathematics and Humanities Department, Sardar Vallabhbhai National

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Institute of Technology (SVNIT), Surat, Gujarat, India

provided the original work is properly cited.

\*Address all correspondence to: twinklesingh.svnit@gmail.com

#### **5. Conclusions**

The VIM, the ADM and the LADM are successfully applied to Burger's equation. The results which are obtained by ADM are a powerful mathematical tool to solve nonlinear partial differential equation. It has been noted that this method is reliable and requires fewer computations; and scheme LADM gives better and very faster accuracy in comparison with VIM.

#### **Acknowledgements**

The authors are thankful to Applied Mathematics and Humanities Department of S. V. National Institute of Technology, Surat for the encouragement and facilities. *Study on Approximate Analytical Method with Its Application Arising in Fluid Flow DOI: http://dx.doi.org/10.5772/intechopen.97548*

### **Author details**

saturation of injected water during imbibition, increases and it is noted that LADM

The VIM, the ADM and the LADM are successfully applied to Burger's equation. The results which are obtained by ADM are a powerful mathematical tool to solve nonlinear partial differential equation. It has been noted that this method is reliable and requires fewer computations; and scheme LADM gives better and very faster

The authors are thankful to Applied Mathematics and Humanities Department of S. V. National Institute of Technology, Surat for the encouragement and facilities.

gives faster accuracy compare to VIM and ADM (**Figures 1**–**7**).

*<sup>i</sup>* ð Þ *ξ*, *θ versus θ for LADM Solution.*

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

**5. Conclusions**

*3-Dimensional LADM Solution*

**Figure 7.**

**Figure 6.** *Plot of Saturation S* <sup>∗</sup>

accuracy in comparison with VIM.

**Acknowledgements**

**52**

Twinkle R. Singh

Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat, Gujarat, India

\*Address all correspondence to: twinklesingh.svnit@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Section 3
