Author: Yared W. Bekele

Contents

• Background
• Volume Fraction Concept
• Kinematics
• Conservation Laws
  • Mass Balance
  • Linear Momentum Balance
  • Angular Momentum Balance
  • Energy Balance
• Thermodynamics and Phase Change
  • Entropy Inequality
  • Phase Transitions
• Conduction Laws
  • Darcy’s Law
  • Fourier’s Law
• Constitutive Laws
  • The Effective Stress Concept
  • Stress-Strain Relations
  • Compressibility of Phases
• Coupled Problems
  • Hydraulic Processes (\(p\) or \(p\text{-}v\) Formulation)
  • Hydro-Mechanical Processes (\(u\text{-}p\) Formulation)
  • Thermo-Hydro-Mechanical Processes (\(u\text{-}p\text{-}T\) Formulation)

Abstract

The fundamental concepts and formulations in porous media theory are presented. The discussion starts with a brief history of porous media theory. The volume fraction concept and kinematic relations are then presented. Conservation laws are derived in a general form which may be applicable to any porous material subjected to different physical processes. Thermodynamic formulations and phase transitions in porous media are briefly discussed. Formulations that complete the mathematical model for a given porous material, such as constitutive laws, are also discussed. The main porous materials of interest in this thesis are soils and specific conservation equations are derived for hydraulic, hydro-mechanical and thermo-hydro-mechanical processes by making the necessary assumptions for particular application needs.

Background

The study of porous materials is essential in various disciplines of science and engineering such as soil mechanics, oil and reservoir engineering, biomechanics, material sciences and chemical engineering. The porous materials studied under porous media theory can either occur naturally, such as soils and rocks, or may be artificial, such as sponges and synthetic polymers. The physical and mechanical properties of such materials have been studied experimentally and mathematically over several decades by a number of researchers. Modern porous media theory, which will be discussed in detail in later sections, developed over a long period of time to reach to what is known today. The historical development of the theory is documented in some publications such as de Boer and Ehlers (1988)¹ and de Boer (2003)².

There are some important milestones in the development of porous media theory, as presented in de Boer (1998)³. In the earliest stages of the theory, Leonhard Euler presented the first discussion on the geometry of a porous medium and indirectly contributed to the formulation of the axioms of continuum mechanics. The introduction of the volume fraction concept by Reinhard Woltman in the 19th century proved to be a fundamental contribution. This concept was further discussed by Achille Delesse. The development of mixture theory, a branch of porous media theory, was initiated by Adolph Fick. An important mathematical relation describing the motion of a fluid in a porous medium was derived by Henry Darcy, which we refer to today as Darcy’s law. In early 20th century, Paul Fillunger and Karl von Terzaghi made seminal contributions to the theory of liquid-saturated porous solids. Fillunger developed equations for uplift forces while studying the stresses acting on concrete and masonry gravity dams. Terzaghi also studied uplift forces independently of Fillunger and further made contributions to the understanding of capillary pressure, the concept of effective stresses and soil consolidation theory. The effective stress concept was discussed earlier by Fillunger. After mid 20th century, the theory of porous media started taking shape to become what we know today. Some of the contributions in this period include work on the theory of mixtures by Green and Naghdi (1967)⁴, without using the volume fraction concept, the study of fluid motion in porous media by Whitaker (1969)⁵, modeling of simultaneous heat, mass and momentum transfer by Whitaker (1977)⁶, the treatment of the mechanics of continuous porous media by Prevost (1980)⁷, a theory of immiscible and structured mixtures by Bedford and Drumheller (1983)⁸ and a generalized approach to the derivation of balance laws by Hassanizadeh (1986)⁹. Application of the theory of mixtures to the modeling of incompressible and compressible porous media was presented by Bowen (1980)¹⁰ and Bowen (1982) ¹¹, respectively. A discussion relating mixture theory and Biot’s approach to porous media theory can be found in Coussy et. al. (1986)¹².

In the following sections, the basic components of porous media theory are presented. The volume faction concept, a fundamental concept in porous media theory, is first discussed. The conservation laws of mass, momentum and energy are then presented in a general form for a porous medium with any number of constituents. The mechanics of phase change in a porous medium and how it affects the conservation laws is then treated. Constitutive equations that govern the constituents of the porous medium are required to complete the theory and these are discussed afterwards. These sections are presented in a brief way here and a detailed presentation of the topics may be referred from modern treatments of porous media theory such as in Coussy (2004)¹³, Vadasz (2008)¹⁴ and Ehlers and Bluhm (2013)¹⁵.

Volume Fraction Concept

One of the most important concepts in the development of porous media theory is the volume fraction concept. This concept is used to a idealize a porous medium with multiple constituents as a homogeneous continuum. An important assumption in the volume fraction concept is that the solid constituent of the porous medium is considered as a reference volume such that only the fluid constituents can enter or leave the reference volume; see Bluhm and de Boer (1997)¹⁶.

Consider a porous medium composed of \(N\) constituents. Let \(V\) be the total volume of he porous medium and \(V^\alpha\) be the volume of phase \(\alpha\). Let \(\mathrm{d}V\) be a control volume element in the total volume and \(\mathbf x\) be its position vector in a global coordinate system at a given time \(t\). Similarly, let \(\mathbf r\) be the position vector of a microscopic volume element \(\mathrm{d} V_\mathrm{\beta}\) inside \(\mathrm{d}V\). The volume of constituent \(\alpha\) within the control volume can be determined by first defining a phase distribution function \(\chi^\alpha\) such that

Thus, the partial volume of constituent \(\alpha\) in the control volume can be written as

The volume fraction \(n^\alpha\) of phase \(\alpha\) can now be defined as

The position vector \(\mathbf r\) may be written in terms of the global position vector \(\mathbf x\) by introducing a local reference system \(\mathbf \xi\) with origin at \(\mathbf x\), such that \(\mathbf r = \mathbf x + \mathbf \xi\). The individual volumes of the constituents and their volume fractions satisfy the conditions

The volume fractions can now be used to describe the relationship between the densities of the constituents of the porous medium at microscopic and macroscopic levels. We denote the intrinsic real density of constituent \(\alpha\) by \(\rho_\alpha\) and the partial density at macroscale by \(\rho^\alpha\), following the notations according to Prevost (1980)⁷. The relationship between these two densities in terms of the volume fraction of the phase under consideration is given by

The total density of the mixture \(\rho\) is then the sum of the partial densities of all constituents i.e.

Kinematics

Kinematic relations describing the relative motions of the phases in a porous medium are briefly discussed here. These relations will be used later in deriving the conservation laws for a porous medium. A detailed presentation may be referred from Lewis and Schrefler (1998)¹⁷.

According to the volume fraction concept, a porous medium composed of \(N\) constituents is approximated as a homogeneous continuum. Thus, a material point defined by the position vector \(\mathbf x\) is simultaneously occupied by all phases. Let \(\mathbf X^\alpha\) be the reference position of phase \(\alpha\) at time \(t=t_\mathrm{o}\). The position of each material point \(\mathbf x^\alpha\) of phase \(\alpha\) at time \(t\) may be written in a Lagrangian description as

An Eulerian description of motion may be written for a non-singular Lagrangian description as

For a particle of phase \(\alpha\) with a defined path, Lagrangian descriptions of the elocity and acceleration are

Eulerian description of the velocity and acceleration may be derived by using equation \ref{e:xalpha} in \ref{e:vaalpha}. Given an Eulerian description of the velocity \(\mathbf v^\alpha(\mathbf x^\alpha,t)\), the Eulerian acceleration \(\mathbf a^\alpha\) may be derived by evaluating the time derivative of the velocity where the Lagrangian coordinates are held constant. That is, by applying the chain rule

For any differentiable function \(f^\alpha\) describing some physical property of phase \(\alpha\) in the porous medium, the material time derivative (also called convective derivative or Lagrangian derivative) is introduced to describe its rate of change relative to a chosen phase. If the Eulerian description of \(f^\alpha = f^\alpha (\mathbf x^\alpha,t)\) is given, its material time derivative with respect to a moving particle of phase \(\alpha\) is defined by

The material time derivative of \(f^\alpha\) with respect to a moving particle of another phase, say phase \(\beta\), is defined as

Equations \(\ref{e:mtdwrtalpha}\) and \(\ref{e:mtdwrtbeta}\) result in the relation

where

is the relative velocity of a particle of phase \(\beta\) with respect to phase \(\alpha\).The material time derivative may be applied to either a scalar or a vector quantity.

In porous media theory, we often require the rate of change of volume averaged quantities. For a vector physical property \(\mathbf f^\alpha\) of phase \(\alpha\) per unit volume, the total time derivative over a volume \(V\) is given according to Reynold’s transport theorem by

For a scalar physical property \(f^\alpha\) per unit volume, we have

Applying the divergence theorem, we get the expression

where \(\partial V\) is the boundary of the domain \(V\) and \(\mathbf n\) is the outward unit normal to the surface \(\mathrm{d} A\).

The spatial velocity gradient of phase \(\alpha\),

can be decomposed into symmetric and skew-symmetric parts as

where \(\mathbf D^\alpha\) and \(\mathbf W^\alpha\) are the rate of deformation and spin tensors, respectively, which may be expressed as

Conservation Laws

The various physical processes within a porous medium induced by external and internal factors are described mathematically using onservation laws, which describe the evolution of physical parameters relevant to the processes. The main conservation laws for a porous medium are mass, momentum and energy balance equations.

The kinematic equations described in the previous section are used to derive the balance equations for a given porous medium. The derivation of these balance equations considers internal and external factors affecting the state of the porous medium and the interactions between the phases in the medium. This is required to ensure that the individual balance equations of the phases result in the balance equation of the whole mixture when superimposed; see Hassanizadeh and Gray (1980)¹⁸. In the following sections, the balance equations of mass, momentum and energy are derived for an individual phase in a porous medium and for the mixture as a whole. The presentation follows the works of de Boer (2006)¹⁹ and Lewis and Schrefler (1998)¹⁷.

Mass Balance

The law of conservation of mass for phase \(\alpha\) requires that the rate of change of mass be equal to any other mass of that phase being added to or leaving from the system, internally from other constituents or externally from other sources. The rate of change of mass \(M^\alpha\) of phase \(\alpha\) over a domain \(V\) is described by

and mass conservation requires this rate to be balanced with all mass exchanges among other phases, i.e.

where the second term in the above equation represents the sum of mass exchanges per unit volume from all phases \(\beta\) to phase \(\alpha\). Applying Reynold’s transport theorem to the rate of change of \(M^\alpha\), a generalized mass balance equation for phase \(\alpha\) can be written as

The general mass balance equation for a porous medium with \(N\) constituents is then obtained by summation of the individual mass balance equations for each phase i.e.

The mass exchange term between the phases has the constraint

when summed over all the \(N\) constituents of the mixture, reducing the overall mass balance equation to

In equation \ref{e:masbalalpha}, the motion of a particle of phase \(\alpha\) is expressed relative to the same phase. In practice, it is more convenient to take one of the phases (usually the solid phase) as a reference to describe the motion of all other phases. The individual mass balance equations for these phases can then be modified by introducing their relative velocity with respect to the reference phase.

Linear Momentum Balance

The balance of linear momentum for a given phase \(\alpha\) requires that the material time derivative its momentum \(\mathbf P^\alpha\) be in equilibrium with the sum of all internal interaction forces and external forces. The rate of change of momentum over a domain \(V\) is described by

and the balance of linear momentum requires

where the second term represents the sum internal momentum exchanges over time to phase \(\alpha\) from all other phases \(\beta\) and \(\mathbf F^\alpha\) represents external forces. The external forces involve body forces \(rho^\alpha \mathbf b^\alpha\) acting on the constituents over the volume \(V\) and surface forces \(\mathbf t^\alpha\) acting on the boundary \(\partial V\) i.e.

Cauchy’s stress tensor \(\mathbf \sigma^\alpha\) and the surface tractions \(\mathbf t^\alpha\) of phase \(\alpha\) are related by

where \(\mathbf n\) is an outward unit normal vector on the boundary. Using \ref{e:transporttheorem}, applying the divergence theorem and considering the mass balance principle, the linear momentum balance equation for phase \(\alpha\) becomes

The overall linear momentum balance equation for the mixture with \(N\) constituents is obtained by summation of the individual phase equations i.e.

The sums of the partial stresses, body forces and acceleration can be represented by their total equivalents:

The internal exchange of momentum between the phases is required to satisfy the constraint

Using \ref{e:partialandtotalsums} and \ref{e:momexchangeconstraint} in \ref{e:genmombal}, the linear momentum balance equation for the mixture becomes

For a static condition (\(\mathbf a = \mathbf 0\)), this reduces to

Angular Momentum Balance

The balance of angular momentum or moment of momentum states that the material time derivative of the angular momentum is equal to the moments of all external forces where the moments are referred to a certain fixed point. The angular momentum \(\mathbf H^\alpha\) of phase \(\alpha\) is given by

where \(\mathbf x\) is a position vector to the fixed point. The moment of the external forces using \ref{e:externalforces} is given by

Angular momentum balance requires

where the moment of the exchanged momentum between the phases is omitted. Simplifying the above equation by applying the mass balance and momentum balance principles derived earlier gives

where \(\mathbf I\) is the identity tensor. The above equation requires

and this is satisfied if

The total stress as the sum of the partial stresses consequently must satisfy

Thus, the conservation of angular momentum proves that Cauchy’s stress tensor is symmetric. Note that if the angular momentum exchange between the phases is assumed to be non-zero, the stress tensor would not be symmetric, requiring the introduction of additional rotational degrees of freedom.

Energy Balance

The law of conservation of energy, the first law of thermodynamics, requires that the rate of change of the internal and kinetic energies be balanced by the rate of mechanical work and heat. For phase \(\alpha\) in a porous medium, this is written mathematically as

where \(E^\alpha\) is the internal energy, \(K^\alpha\) is the kinetic energy, \(E^{\beta \alpha}\) is the rate of internal energy exchange from all other phases \(\beta\) to phase \(\alpha\), \(W^\alpha\) is the rate of mechanical energy or work and \(H^\alpha\) is the rate of heat energy. For a given domain \(V\), we have

where \(e^\alpha = e^\alpha(\mathbf x,t)\) is the specific internal energy \(\alpha, h^\alpha = h^\alpha(\mathbf x,t)\) is the partial energy source and \(\mathbf q^\alpha = \mathbf q^\alpha(\mathbf x,t)\) is the partial heat flux vector.

Using \ref{e:energies} in \ref{e:thermodynamics}, simplifying the material time derivatives of the integrals and utilizing the mass and linear momentum balance equations in \ref{e:masbalalpha} and \ref{e:mombalalpha}, the energy balance equation for phase \(\alpha\) becomes

where \(Q^\alpha = \rho^\alpha h^\alpha\) is the heat supply to phase \(\alpha\). The overall energy balance equation is obtained by summation of the individual energy balance equations of the constituents of the porous medium i.e.

If the momentum exchange between the phases is assumed to be zero, the energy balance equation reduces to

The total heat supply and total heat flux can be expressed as

and a constraint on the energy exchange between the phases is introduced as

This reduces the energy balance equation to

Thermodynamics and Phase Change

The second law of thermodynamics (entropy inequality) is used to state the restrictions on constitutive equations. It follows from the conservation of energy (first law of thermodynamics) with the introduction of the absolute temperature.

Entropy Inequality

The assumption of entropy inequality for each phase \(\alpha\) is a sufficient but restrictive condition. For the existence of dissipation mechanisms within the porous medium, an entropy inequality for all the constituents is both a necessary and sufficient condition, de Boer (2006)¹⁹.

For phase \(\alpha\), let

be its entropy where \(s^\alpha\) is the specific entropy. With \(T^\alpha\) as the absolute temperature of phase \(\alpha\), the entropy inequality for all the constituents of the porous medium is expressed as

Performing the material time derivative in the above equation using the transport theorem, utilizing the mass balance equation and the divergence theorem, the local form of the entropy inequality becomes

The entropy inequality may also be written as a function of the Helmholtz free energy

together with the energy balance in \ref{e:enebal}. The inequality in \ref{e:entropyineq} considers a general case where the constituents have a different absolute temperature \(T^\alpha\). It may be simplified for the case when all the constituents have the same absolute temperature \(T = T^\alpha\).

Phase Transitions

The physical processes in a porous medium sometimes involve the transition of one phase into another such as phase change from liquid to vapor, liquid to solid or solid to liquid. These types of phase changes are referred to as first-order transitions, see de Boer and Bluhm (1999)²⁰, whereas other transitions such as from a super fluid to ordinary fluid (e.g. helium) are called second-order transitions. The discussion here focuses on first-order transitions. We take a closer look here at the effect of phase transitions in a porous medium.

The balance equations in the conservation laws section are presented in a general form such that the effect of phase transition(s) on the conserved quantities can be considered. Consider a two-phase porous medium where phase transition from one phase to another occurs, e.g. freezing/melting. The exchange of quantities during phase change remains constrained. For instance, the mass balance equations for the individual constituents, according to \ref{e:masbalalpha}, are given by

wherein the sum on the mass exchange term is omitted since we have only one phase contributing to the mass of another. The constraint in \ref{e:massconstraint} implies

Phase transitions in porous media have been studied in the past with several simplifying assumptions. The need for a rigorous mathematical and thermodynamical treatment of such problems was clear. Phase change from liquid to gas was given a detailed thermodynamic treatment, for example, by de Boer and Kowalski (1995)²¹ and de Boer (1995)²², showing that the phase transition is governed by the difference of the chemical potentials of the constituents. A review of various phase change phenomena in porous media such as freezing/melting, boiling, drying/evaporation and condensation was presented by Yortsos and Stubos (2001)²³. A detailed thermodynamic approach to various phase change problems can be found in Fremond (2012)²⁴.

The thermodynamic equilibrium between the phases during phase change is described by the Clausius-Clapeyron equation. This equation can be derived based on the equilibrium requirement between the chemical potentials of the two phases; see Loch (1978)²⁵ for the specific case of freezing and melting. For two phases \(\alpha\) and \(\beta\) undergoing phase transition, their chemical potentials are required to be in equilibrium, i.e. \(\mu^\alpha = \mu^\beta.\) The chemical potentials are a function of the specific entropies, the specific volumes, the pressures and the temperatures of the phases. In terms of these quantities, we have

where \(s^\alpha\) and \(s^\beta\) are the specific entropies of the phases, \(v^\alpha\) and \(v^\beta\) their specific volumes, \(p^\alpha\) and \(p^\beta\) the pressures and \(T\) is the temperature, assuming the same temperature for both phases. The equation above may be rearranged to give

In terms of the densities \(\rho^\alpha = 1/v^\alpha\) and \(\rho^\beta = 1/v^\beta\), we have

The change in entropy between the phases under constant pressure and temperature may be expressed as

where \(L\) is the specific latent heat which depends on the type of phase change that occurs e.g. latent heat of fusion for melting. Using this in \ref{e:clausius1} gives

Conduction Laws

The equations governing the flow of fluid and heat in a porous medium are given by Darcy’s law and Fourier’s law, respectively. We refer to these laws here as conduction laws, as used in Coussy (2004)¹³.

Darcy’s Law

The specific discharge of fluid in a porous medium was shown to be proportional to the head loss based on experiments by Henry Darcy in 1857, Verruijt (2013)²⁶. Darcy’s law may be derived for fully saturated porous media from the fluid momentum balance equation, see for instance Whitaker (1986)²⁷.

Darcy’s law relates the flow of fluid in a porous medium with respect to the solid phase as

where \(n^\mathrm{f}\) is the volume fraction of the fluid, \(\mathbf v^\mathrm{f}\) is the fluid velocity and \(\mathbf v^\mathrm{s}\) is the solid elocity. A more general form of Darcy’s law, considering the relative permeability of a phase in a medium where the permeability varies, is given by

where \(k_\mathrm{r}\) is the relative permeability coefficient varying between 0 and 1, \(\mu\) is the viscosity of the fluid, \(\mathbf \kappa\) is the intrinsic permeability matrix of the material, \(p^\mathrm{f}\) is the fluid pressure, \(\mathbf b\) represents body forces, \(\mathbf a^s\) is the acceleration of the solid phase and \(\mathbf a^{\mathrm{f} \mathrm{s}}\) is the relative acceleration between the fluid and solid phases. The tortuosity of the pores is sometimes considered in the Darcy equation. If the inertia terms are neglected, Darcy’s law reduces to

It is common to express Darcy’s law in terms of the hydraulic conductivity matrix \(\mathbf k\) instead of the intrinsic permeability matrix \(\mathbf \kappa\), where the two are related by

in which \(g\) is the acceleration due to gravity. This results in

wherein we have kept the same relative coefficient \(k_\mathrm{r}\) for hydraulic conductivity that varies with the porosity and degree of fluid saturation of the porous medium.

Fourier’s Law

Thermal conduction in a porous medium is expressed using Fourier’s law which states that the rate of heat flow is proportional to the negative gradient of the temperature. Fourier’s law for the heat flux \(\mathbf q\) is given by

where \(\mathbf \lambda\) is the effective thermal conductivity matrix of the porous medium and \(T\) is the temperature. For isotropic thermal conductivity in a medium, the above equation becomes

where \(\lambda\) is the effective thermal conductivity coefficient which is a function of the individual thermal conductivities of the constituents of the porous medium and their volume fractions. Fourier’s law has a similar form both at the microscopic and macroscopic levels, Coussy (2004)¹³.

Constitutive Laws

The balance equations presented in Conservation Laws are valid for any porous medium. To complete these balance equations for a specific material, a constitutive model needs to be introduced, de Souza Neto et al. (2011)²⁸. We first introduce the effective stress concept before discussing the constitutive equations required to complete the description.

The Effective Stress Concept

The effective stress concept is widely used in porous media applications, especially in soil mechanics. The historical development of this concept is documented in de Boer and Ehlers (1990)²⁹. The concept was already conceived and studied by scientists by the end of the 19th century. A significant development with a mathematical background came later in early 20th century from the significant contributions of Paul Fillunger and especially Karl von Terzaghi; see Skempton (1960)³⁰.

The main idea behind effective stress in a porous medium is separating the stress that effectively causes solid deformation, hence the name, from all other stresses in the mixture. If we consider a porous medium composed of two phases, solid (s) and fluid (f), the total stress as the sum of the partial stresses is given according to \ref{e:partialandtotalsums} by

The partial stresses corresponding to the fluid phase, using the volume fraction concept, can be written us

where \(n^\mathrm{f}\) is the volume fraction of the fluid and \(\mathbf \sigma_\mathrm{f}\) is the pore fluid stress. However, a distinction should be made between the partial stress of the solid phase \(\mathbf \sigma^\mathrm{s}\) and the effective stress \(\mathbf \sigma^\mathrm{\prime}\); this has been discussed, for example, by Prevost (1980)⁷. The partial stress of the solid phase is given by

where \(n^\mathrm{s} \mathbf \sigma_\mathrm{f}\) takes into account the effect of the pore fluid stress on the individual solid grains of the porous medium. It is assumed, in the above equation, that the contact areas between the solid grains are negligible such that the pore fluid surrounds each grain. Each solid grain is subjected to intergranular forces that are in excess of the pore fluid stress and this is characterized by the effective stress \(\mathbf \sigma^\mathrm{p}rime\). The total stress for a fluid-saturated porous medium is then given by

Stress-Strain Relations

A constitutive equation for the solid phase relates the effective stress and the strain. The constitutive stress-strain relation in a general form may be expressed as

where \(\mathbf D = \mathbf D(\mathbf \sigma^\prime, \mathbf \varepsilon, \dot{\mathbf \varepsilon})\) is the constitutive tangent tensor, \(\mathrm{d} \mathbf \varepsilon\) is the total strain increment, \(\mathrm{d} \mathbf \varepsilon^\mathrm{c}\) is the creep strain increment, \(\mathrm{d} \mathbf \varepsilon^\mathrm{s}_\mathrm{v}\) is the volumetric strain increment and \(\mathrm{d} \mathbf \varepsilon^\mathrm{o}\) considers all other strain increments not directly associated with the effective stress. Let \(p\) represent the average fluid pressure from all the fluid phases in the porous medium. This pressure induces a hydrostatic stress distribution in the solid phase, thus causing a purely volumetric strain given in incremental form by

where \(K_\mathrm{s}\) is the bulk modulus of the solid skeleton. The effective stress relation in \ref{e:effstress} is usually modified by a corrective parameter known as Biot’s coefficient, \(\alpha\), Lewis and Schrefler (1998)¹⁷. The modified effective stress equation (keeping the same notation for the modified effective stress) reads

where \(\mathbf I\) is the identity tensor. It can be shown that Biot’s coefficient takes the value

where \(K_\mathrm{o}\) is the overall bulk modulus of the porous medium. We can now simply write

The constitutive tangent tensor takes various forms depending on the type of stress-strain relation employed. Some examples of constitutive laws include linear elasticity, thermoelasticity, elastoplasticity and viscoplasticity. In this thesis, we focus on linearly elastic and thermoelastic constitutive laws.

The tangent stiffness for linear elasticity (for stresses and strains in Voigt notation) is given by

where

with \(\boldsymbol{\mathit{1}}\) being a matrix of ones, \(E\) is the Young’s modulus of the soil and \(\nu\) is the Poisson’s ratio. For thermoelastic material models, these parameters are usually defined as a function of temperature and/or other temperature dependent parameters.

Compressibility of Phases

For compressible phases in a porous medium, the density of a phase may be a function of pressure, temperature and other factors. We look at the equations of state for water, the most common fluid in a porous medium, and solid skeleton. See Lewis and Schrefler (1998)¹⁷.

The equation of state for water is given by

where \(\rho_\mathrm{w}\) and \(\rho_{\mathrm{w}\mathrm{o}}\) are the current and initial densities, \(\alpha_\mathrm{w}\) is the thermal expansion coefficient, \(K_\mathrm{w}\) is the bulk modulus and \(p^\mathrm{w}\) and \(p^\mathrm{w}_\mathrm{o}\) are the current and initial pore water pressures. Performing Taylor series expansion of \ref{e:eqofstatewater0} and retaining first order terms gives

which can then be used to derive

For a compressible solid phase, the material time derivative of the density may be derived from the mass balance of the solid:

The solid density is a function of the average pressure on the solid from all other phases \(p^s\), the temperature \(T\) and the first invariant of the effective stress \(\text{tr}~\mathbf \sigma^\prime\). The variation of the solid density may then be written as

where \(\alpha_\mathrm{s}\) is the thermal expansion coefficient of the solid. The first term on the right hand side of the equation above represents the volumetric strain of the solid. This strain may be negligible for soils but significant for materials such as rock. Introducing a constitutive equation for the first invariant of the effective stress and using Biot’s coefficient from \ref{e:Biotscoeff}, an alternative form of the solid compressibility may be written as

Coupled Problems

Different types of physical processes may occur in a porous medium depending on the type of the material and its application. Typical examples include hydraulic, mechanical, thermal and chemical processes. Two or more physical processes often occur in a porous medium leading to what we refer to as coupled processes. We focus on hydraulic, hydro-mechanical and thermo-hydro-mechanical phenomena in soils.

In the following sections, we discuss how the general conservation laws are used to derive the governing equations for the specific processes under consideration. To illustrate this, we assume a fluid-saturated biphasic porous medium with solid (s) and fluid (f) phases. Based on the volume fraction concept and \eqref{e:partialdensities}, the partial densities for the solid and fluid phases, \(\rho^\mathrm{s}\) and \(\rho^\mathrm{f}\), are given by

where \(n^\mathrm{s}\) and \(n^\mathrm{f}\) are the volume fractions of the solid and fluid, respectively, and \(\rho_\mathrm{s}\) and \(\rho_\mathrm{f}\) are their real densities at the microscopic level. For a fluid-saturated porous medium, we have \(n^\mathrm{s} + n^\mathrm{f} = 1.\) For simplicity in the following sections, we set \(n^\mathrm{f} = n\) and \(n^\mathrm{s} = 1-n\). We will refer to \(n\) as the porosity of the material.

Hydraulic Processes (\(p\) or \(p\text{-}v\) Formulation)

Hydraulic processes in a porous medium are mainly described by the law of conservation of mass. The mass balance equation for the fluid phase according to \eqref{e:masbalalpha} is

in which we have ignored any mass exchange between the solid and fluid phases. Considering the flow of the fluid with respect to the solid phase, we write the material time derivative of \(\rho^\mathrm{f}\) according to \eqref{e:mtdwrtbetarel} as

which, according to \eqref{e:mtdwrtbeta}, becomes

wherein we have used \(\mathbf v^{\mathrm{s} \mathrm{f}} = \mathbf v^\mathrm{s} - \mathbf v^\mathrm{f}\). From \eqref{e:partialdensities_sf}, we have

The porosity remains constant if there are no mechanical deformations i.e. \(\frac{\partial n}{\partial t} = 0\). With this, and neglecting any spatial variations in the fluid density (i.e. \(\nabla \rho^\mathrm{f} = 0\)), \eqref{e:masbalfluid1} reduces to

If the fluid is assumed to be incompressible, \eqref{e:masbalfluid2} further reduces to

The fluid velocity according to Darcy’s law in this case is

with no mechanical deformations (\(\mathbf v^\mathrm{s} = \mathbf 0\)). Using \eqref{e:permandhydcond} and \eqref{e:Darcyslaw}, the fluid mass balance equation for a constant hydraulic conductivity (\(k_\mathrm{r}=1\)) can be written as

where \(\gamma_\mathrm{f}\) is the unit weight of the fluid. The only flow driving forces considered here are the fluid pressure \(p^\mathrm{f}\) and the body forces \(\mathbf b\). The equation above governs the steady-state flow of fluid in a porous medium and may be solved for the pressure as the unknown, resulting in the so called \(p\)-formulation.

For a compressible fluid, the temporal variation of the fluid density needs to be taken into account. For instance, if our fluid is water, from the equation of state for water in \eqref{e:eqofstatewater} we may write

where \(K_\mathrm{f}\) is the bulk modulus of the fluid and isothermal conditions are assumed. Using \eqref{e:compressiblefluid} in \eqref{e:masbalfluid2}, we get

wherein we have used \(\mathbf w = n \mathbf v^\mathrm{f}\). Sometimes, the above equation is solved together with Darcy’s law for the pressure \(p^\mathrm{f}\) and Darcy’s velocity \(\mathbf w\) as the unknowns. The equations to be solved simultaneously in this case are

which results in the so called \(p\text{-}v\) formulation.

Hydro-Mechanical Processes (\(u\text{-}p\) Formulation)

Hydro-mechanical processes couple the flow of fluid in a porous medium with solid deformation. The governing equations describing the coupling are derived by superposition of the individual linear momentum and mass balance equations of the solid and fluid phases.

From \eqref{e:masbalfluid1} and \eqref{e:fluiddensitydt}, with no spatial variations in fluid density, the mass balance equation for the fluid phase is given by

The mass balance equation for the solid phase, based on , is given by

From \eqref{e:partialdensities_sf} we have \(\rho^\mathrm{s} = (1-n) \rho_\mathrm{s}\). Using this in the above equation, and neglecting spatial variations in the porosity and the solid density, gives

Dividing \eqref{e:masbalfluidHM} by \(\rho_\mathrm{f}\) and \eqref{e:masbalsolidHM} by \(\rho_\mathrm{s}\) and adding the two equations gives the overall mass balance equation for the fluid-saturated porous medium as

Rearranging the above equation gives

where \(\mathbf w = n (\mathbf v^\mathrm{f} - \mathbf v^\mathrm{s})\) is Darcy’s velocity. Assuming isothermal conditions and water as the fluid, the compressibility of the fluid is given by \eqref{e:compressiblefluid} and from \eqref{e:compressiblesolid} we get for the solid phase

where for a single fluid phase we have set \(p^\mathrm{s} = p^\mathrm{f}\). Equation \eqref{e:masbalHM2}, based on \eqref{e:compressiblefluid} and \eqref{e:compressiblesolidHM}, can be rewritten as

For incompressible solid and fluid phases (\(1/K_\mathrm{s} = 1/K_\mathrm{f} = 0\) and \(\alpha=1\)), the mass balance equation reduces to

The linear momentum balance equations for the solid and fluid phases, according to \eqref{e:mombalalpha}, assuming a static condition and no momentum exchange between the phases, are given by

Summation of the two equations above and use of \eqref{e:partialandtotalsums}, \eqref{e:totalstress_sf} and \eqref{e:effstress2}, gives the momentum balance equation for the whole medium as

wherein the total density of the porous medium is

For coupled hydro-mechanical (HM) processes in a porous medium, the mass balance equation in \eqref{e:masbalHM} and the linear momentum balance equation in \eqref{e:mombalHM} can be solved simultaneously for the displacement and pressure field variables. This results in what is known as a \(u\text{-}p\) formulation. Sometimes these equations are solved simultaneously with Darcy’s equation to obtain the displacement, pressure and Darcy’s velocity, leading to the so-called \(u\text{-}p\text{-}v\) formulation.

Thermo-Hydro-Mechanical Processes (\(u\text{-}p\text{-}T\) Formulation)

Thermo-hydro-mechanical (THM) processes in a porous medium couple heat transfer and fluid flow with the deformation of solid. The governing equations describing the coupling are derived by superposition of the individual linear momentum, mass and energy balance equations of constituent phases.

We consider a fluid-saturated porous medium again. The linear momentum balance equation in this case is the same as in \eqref{e:mombalHM}. The overall mass balance equation is derived from \eqref{e:masbalHM2} by considering non-isothermal conditions for the compressibilities of the solid and fluid phases. Based on \eqref{e:eqofstatewater} and \eqref{e:compressiblesolid}, we get

Using \eqref{e:compressibleTHM} in \eqref{e:masbalHM2} and simplifying gives the overall mass balance equation for a fluid-saturated porous medium subjected to THM coupled processes as

where \(\alpha_\mathrm{o} = n\alpha_\mathrm{w} + (\alpha-n)\alpha_\mathrm{s}\) is the overall thermal expansion coefficient of the porous medium.

The energy balance equation for a single phase, neglecting any mass, momentum and energy exchanges between the solid and fluid phases, is obtained from \eqref{e:enebalalpha} as

The individual energy balance equations for the solid and fluid phases are then written as

Viscous dissipation may be significant in applications where the material flows at high rates e.g. due to injection or molding in polymer processing. For the applications of interest in this thesis, it may be neglected. With this assumption and taking all material time derivatives with respect to the solid phase, using \eqref{e:mtdwrtbetarel}, the individual balance equations become

The specific internal energies for the solid and fluid phases as a function of temperature are given by

where \(c_\mathrm{s}\) and \(c_\mathrm{f}\) are their respective specific heat capacities. Applying these to \eqref{e:enebalTHM0} gives

Summation of the individual balance equations gives the overall energy balance equation for the porous medium as

wherein we have implied

for the total heat flux and heat supply, respectively.

Defining the effective heat capacity of the medium as

and noting that \(\mathbf w = n (\mathbf v^\mathrm{f} - \mathbf v^\mathrm{s})\) is Darcy’s velocity, \eqref{e:enebalTHM2} is further simplified to obtain the energy balance equation for the medium as

The material time derivative of the temperature with respect to the solid phase is given by

For the applications of interest in this thesis, the convective heat flux in the solid phase is usually negligible i.e. \(\nabla T \cdot \mathbf v^\mathrm{s} \approx\) 0. With this assumption and using Fourier’s law, \eqref{e:fourierslaw}, in \eqref{e:enebalTHM3}, the energy balance equation for the medium becomes

For THM coupled processes in a porous medium, the linear momentum, mass and energy conservation laws in \eqref{e:mombalHM}, \eqref{e:masbalTHM} and \eqref{e:enebalTHM}, respectively, may be solved simultaneously for the displacement, pressure and temperature field variables. This results in what is usually referred to as a \(u\text{-}p\text{-}T\) formulation.

References