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Thermodynamics of ions and water transport in porous media
Journal of Colloid and Interface Science 307 (2007) 254–264 www.elsevier.com/locate/jcis
Thermodynamics of ions and water transport in porous media A. Revil 1 CNRS-CEREGE, Université Paul Cézanne, Département d’Hydrogéophysique et Milieux Poreux, BP 80, 13545 Aix-en-Provence Cedex 4, France Received 12 August 2006; accepted 30 October 2006 Available online 1 December 2006
Abstract The thermodynamic framework of Prigogine, de Groot, and Mazur is extended to study the transport of ions and water in thermoporoelastic materials assuming infinitesimal deformations. New expressions are developed for the first and second principles of nonequilibrium thermodynamics of multicomponent systems and a generalized power balance equation is derived. For porous materials, all the components cannot be treated on a symmetric basis. A Lagrangian framework associated with deformation of the solid phase is introduced and, in this framework, Curie’s principle is used to set up the form of the linear constitutive equations describing the transport of ions, water, and heat through the pore network. The material properties entering these equations were recently obtained by Revil and Linde [J. Colloid Interface Science 302 (2006) 682–694] using a volume-averaging approach based in the Nernst–Planck and Stokes equations. This provides a way to relate the material properties entering the constitutive equations to two textural parameters characterizing the topology of the pore space of the material (namely the tortuosity of the pore space and the permeability). The generalized power balance equation is used to derive the linear poroelastic constitutive equations (including the osmotic pressure) to describe the reversible contribution of deformation of the medium in response to ions and water transport through the connected porosity. © 2006 Elsevier Inc. All rights reserved. Keywords: Self-potential; Nonequilibrium thermodynamics; Porous material; Thermoporoelasticity; Dissipation
1. Introduction Transport of ions and the concomitant deformation of the skeleton of a charged porous composite, in response to change in the boundary conditions imposed on the system or the application of nonequilibrium thermodynamic forces, is a central subject in petrophysics, in colloidal chemistry, and in biology [1–10]. Recently, Revil and Linde [11] derived a new set of constitutive equations to model the transport of ions and water through a charged and water-saturated porous medium, assumed to be isotropic. The novelty of their approach is that all the material properties entering the constitutive equations are related to only two fundamental properties characterizing the topology of the solid and fluid mixture. These parameters are the tortuosity of the pore space (or the electrical formation factor, which that is equal to the tortuosity divided by the porosity) and the permeability. However, these constitutive equations E-mail address: [email protected]. 1 Also at CNRS-ANDRA/GDR FORPRO 0788, France.
0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.10.074
need to be inserted into a thermodynamic framework that includes mechanical deformation of the charged porous composite. Such a framework is developed in the present paper with a special focus on the reversible component of the deformation. A specific development for the irreversible contribution to deformation will be presented in a forthcoming work, but the thermodynamic framework developed below allows such a contribution. However, if the deformation is very small, the elastic contribution is likely to be the dominant mechanism of deformation, e.g., during the spreading of the ions through the connected porosity [10]. In this paper, I will use the term “thermodynamics” as referring to irreversible processes out of equilibrium and I will keep the term “thermostatics” to describe the equilibrium state, which is described in [11]. The porous material is formed by an isotropic skeleton of solid grains saturated by a multicomponent electrolyte comprising the solvent (water), the ions, and possibly neutral species. I consider below only small and reversible deformations and the mechanical constitutive equations will be linearized. I will assume that the system remains linear in the sense that the principle of superposition applies.
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
255
2. Thermodynamics of stressed multicomponent systems
The mass fractions (dimensionless) are defined by Ci = ρi /ρ. Equations (2) to (6) yield
In this section, I generalize the nonequilibrium thermodynamics approach of De Groot and Mazur [12]. De Groot and Mazur [12] developed a thermodynamic model of a multicomponent viscous fluid with hydrostatic stress, while in the present paper, I am interested in describing the thermodynamics of a multicomponent deformable porous continuum with a nonhydrostatic stress tensor. The porous body is broken up into a number of representative elementary volumes (REVs). These REVs are large enough to define thermodynamic potentials with statistical significance (e.g., the chemical potentials of the ions and the fluid pressure) and small enough so that in each REV, the gradients of the thermodynamic potentials of interest can be considered negligible. The porous body is not in equilibrium as thermodynamic potentials vary from REV to REV. However, the local equilibrium assumption implies that the specific entropy is the same function of extensive variables per unit mass as it is in equilibrium [13,14].
Dρ = −ρ∇ · v, (7) Dt DCi ρ (8) = −∇ · Jm i . Dt with the introduction of the specific volume v = 1/ρ, Eq. (7) yields a mass conservation equation for an observer moving with the barycentric center of mass motion, Dv = ∇ · v. (9) Dt In addition, it follows from Eqs. (4) to (6) that in the Lagrangian framework previously defined, the sum of all the fluxes is equal to zero: ρ
Q
Jm i = 0.
(10)
i=1
2.1. Continuity of mass
Another identity used later follows directly from the previous equations (see [12]),
I recall first, in an Eulerian framework (i.e., from the standpoint of a stationary observer), that the local conservation of mass for the component i is
Da ∂aρ =ρ − ∇ · (aρv), ∂t Dt where a can be a scalar, a vector, or a tensor.
∂ρi = −∇ · (ρi vi ), ∂t
(1)
where ρi (in kg m−3 ) represents the bulk density of partial component i (i ∈ {1, . . . , Q}), vi is its velocity (in m s−1 ), and t is time (in s). The law of mass conservation is obtained by summing the previous equations over all the constituents of the system, ∂ρ = −∇ · (ρv), ∂t Q ρ≡ ρi , ρv ≡
2.2. Continuity of charge I denote by qi the charge per unit mass of component i (qi = 0 if the species is neutral, water for instance). The total current density (in A m−2 ) is defined as the net amount of charge flowing per unit surface area and per unit time: Jm =
(2)
Q
ρ i qi vi .
(3) q=
Q Q 1 ρ i qi = Ci q i . ρ i=1
ρ i vi ,
(4)
i=1
where ρ is the total density of the porous system and is the sum of all the partial mass densities ρi of its components and v is the velocity of the barycentric center of mass. I introduce the substantial time derivative in a Lagrangian framework attached to the barycentric center of mass motion by ∂ D = + (v · ∇). (5) Dt ∂t I will define, in Section 4, another Lagrangian framework, attached this time to the mineral skeleton and in which the observable (measurable) fluxes are defined. The diffusion fluxes of the various species (ions, water, and grains) in the Lagrangian framework attached to the barycentric center of mass (superscript m) are defined by Jm i = ρi (vi − v).
(12)
i=1
The total charge per unit mass of the system q is defined by
i=1 Q
(11)
(6)
(13)
i=1
Using Eq. (6), the total current density is given by Jm = J m c + ρqv, Jm c ≡
Q
qi Jm i ,
(14) (15)
i=1 −2 where Jm c represents the conduction current (in A m ). Therefore the total current density is equal to the conduction current density plus a convective term ρqv. The porous medium or the colloidal suspension is also characterized by a global electroneutrality condition: the net charge of the grains is exactly counterbalanced by the net charge of the pore water. Consequently q = 0 and therefore the convective term is equal to zero in the Lagrangian framework attached to the barycentric center of mass. The continuity equation for the electrical charge is obtained by summing the continuity equations for the mass of the various
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charged components, Eqs. (8) and (15). This yields ρ
Dq = −∇ · Jm c, Dt
Using Eqs. (2), (7), and (20), I write the balance equation for the momentum density ρv, (16)
and therefore ∇ · Jm c = 0 if there is global electroneutrality in the system (q = 0). Neglecting the influence of the Lorentz force, neglecting electric and magnetic polarizations, and using the macroscopic electroneutrality condition (q = 0), the Maxwell stress tensor T¯¯ M and the Maxwell body force FM per unit mass are given by 1 2 ¯¯ ¯ ¯ TM =ε E⊗E− E I , (17) 2 ρFM = ρqE = 0, (18) where ε is the dielectric constant of the porous body, E ⊗ E represents a dyadic product (and therefore a second-order tensor, where ⊗ is the tensorial product), and I¯¯ is the unit dyadic with elements δij (Kronecker delta, δij = 1 if i = j and δij = 0 if i = j ). A force balance equation is given by ∇ · T¯¯ M = ρFM (= 0).
(19)
2.3. Force and momentum balance equations I first specify the assumptions made in developing the model. (1) The influence of the external magnetic field sources is ignored and the porous medium is assumed to be free of any magnetically polarizable particles. (2) Thermal free convection of the pore fluid is ignored. Only the case of scalar potential motion is considered in the present model. (3) I consider viscous-laminar flow of the pore fluid relative to the skeleton. The equation of motion of the system is Dv ρ = ∇ · σ¯¯ + ρF, Dt Q ρF ≡ ρi Fi ,
(20)
σ¯¯ =
σ¯¯ i ,
(21)
(22)
i=1
where σ¯¯ (expressed in N m−2 ) is the total (symmetric) Cauchy stress tensor (tension is considered positive). This stress tensor is equal to the sum of the peculiar (partial) stresses (positive for tension) of the individual components of the mixture, σ¯¯ i . The vector F is the total body force per unit mass exerted on the system and Dv/Dt represents the acceleration of the center of mass. The forces Fi entering Eq. (20) represent the gravity and the electrostatic body forces per unit mass exerted on component i. As these forces arise from scalar potentials ψi , Fi = −∇ψi (potential flow), and ∂ψi /∂t = 0, the total potential ψ of the system is defined by [12] ρψ =
Q i=1
2.4. Energy balance equation (first principle) From here, I generalize the approach of De Groot and Mazur [12] to thermoporoelastic media. It results from Eq. (20), a conservation equation for the kinetic energy of the barycentric center of motion, D 1 2 ρ (25) v = ∇ · (σ¯¯ · v) − σ¯¯ : ∇v + ρF · v, Dt 2 where the colon indicates a tensor dot product (a : b = a ij bij with the Einstein convention; note that the symbols “·” and “:” between tensors of various orders denote their inner product with simple or double contraction, respectively). This yields a conservation equation for the kinetic energy, ∂ 1 2 (26) ρv = −∇ · Wk − σ¯¯ : ∇v + ρF · v, ∂t 2 1 Wk = ρv2 v − σ¯¯ · v, (27) 2 where Wk is the kinetic energy flux in an Eulerian framework of reference (in J m−2 ). We denote as u the internal energy of the system. The internal energy flux vector Wu is the sum of the heat flux plus energy change due to diffusion and local stresses σ¯¯ i , Wu = Hm +
Q
ui I¯¯ − σ¯¯ i /ρi Jm i ,
(28)
i=1
i=1 Q
∂(ρv) = −∇ · (ρv ⊗ v − σ¯¯ ) + ρF. (24) ∂t The momentum flux associated with the momentum density ρv is (ρv ⊗ v − σ¯¯ ) and the term ρF therefore represents a source term of momentum.
where ui is the internal energy of species i, and Hm is the conductive heat flux including heat transported by mass diffusion (see [5]). A balance equation for the energy (kinetic plus internal energies) is therefore D 1 2 ρv + ρu = −∇ · Wu − σ¯¯ · v + ρT Π Dt 2 + ρv · F +
Q
Jm i · Fi ,
(29)
i=1
where Π (positive or negative) is the power per unit mass and temperature. The term ρT Π corresponds to the energy sources provided to the system by external sources and is not accounted for by [12]. This yields a local conservation equation for the internal energy, ∂ Jm (ρu) = −∇ · Wu + σ¯¯ : ∇v + ρT Π + i · Fi , ∂t Q
(30)
i=1
ρi ψi .
(23)
Wu = ρuv + Hm +
Q
ui I¯¯ − σ¯¯ i /ρi Jm i .
i=1
(31)
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
Equation (30) generalizes Eq. (34) of Ref. [12] to thermoporoelastic bodies. A local conservation equation for the potential energy is given by ∂ Jm (ρψ) = −∇ · Wψ − ρv · F − i · Fi , ∂t Q
(32)
i=1
Wψ = ρψv +
N
ψ i Jm i ,
(33)
i=1
where Wψ is the potential energy flux vector in an Eulerian framework of reference. The total energy e of the system per unit mass is the sum of the kinetic energy, potential energy, and internal energy per unit mass. It is given by e = v2 /2 + ψ + u, which defines the internal energy u per unit mass of the system. The internal energy corresponds to the energy associated with the short-range interactions and motions of the individual components of the porous mixtures. The internal energy is dependent on the temperature and density (pressure) of the system as discussed later. The balance equation for the total energy is ∂ (ρe) = −∇ · We + ρT Π, ∂t We = Wk + Wu + Wψ , We = ρev − σ¯¯ · v + Hm +
(34) (35)
Q i=1
σ¯¯ i (ui + ψi )I¯¯ − · Jm i , ρi
(36)
where We represents the total energy flux (energy per unit mass per unit surface area and per unit time). This energy flux includes several contributions (1) a convective term ρev − σ¯¯ · v, (2) a conductive heat flux Hm , and (3) a diffusive energy flux. Equations (34) and (36) generalize Eqs. (31) and (33) of Ref. [12]. Subtracting the total energy given by Eq. (34) to the sum of the kinetic plus potential energy given by Eq. (29), respectively, and using Eq. (36) I obtain a balance equation for the internal energy u, Du = −∇ · Wu + σ¯¯ : ∇v + ρT ΠT , Dt Q
ΠT ≡ Π + Jm (ρT ), i · Fi
(37)
ρ
(38)
i=1
where ΠT is the total power supplied to the system including the influence of the body forces. The stress tensor is decomposed into a reversible elastic component (superscript e) and a dissipative component (superscript d) corresponding to irreversible e d mechanisms of deformation, σ¯¯ = σ¯¯ + σ¯¯ (following [13]). Using Eqs. (31), (37), and (38), I obtain a balance equation for the thermal energy, Du DQ e d = + v σ¯¯ : ∇v + v σ¯¯ : ∇v + v Dt Dt DQ ρ = −∇ · Hm . Dt
Q
Equation (39) replaces Eq. (39) of De Groot and Mazur [12]. The parameter Q is here the heat per unit of mass in the system and v ≡ 1/ρ the specific volume (see Section 2.1). The terms of the left-hand side of Eq. (39) represent the rate of gain of internal energy per unit mass. These terms are (1) the rate of internal energy input by conduction per unit mass DQ/Dt, (2) the reversible rate of internal energy increase per unit mass e by compaction v σ¯¯ : ∇v, (3) the irreversible rate of internal energy increase per unit mass due to dissipation associated with d irreversible mechanisms of deformation v σ¯¯ : ∇v, (4) the irreversible rate of internal energy increase per unit mass by the internal motion of the various components forming the system, and finally (5) the power per unit mass and temperature supe plied to the system. The term v σ¯¯ : ∇v can be either positive or negative depending upon whether the porous system is expanding or contracting. This term is a reversible mode of exchange of energy between mechanical and internal energy. The term d v σ¯¯ : ∇v is always positive and represents an irreversible degradation of mechanical to thermal energy. For practical applications, it is convenient to recast the thermal energy equation in terms of temperature and heat capacity rather than in terms of internal energy. I first specify an equation of state for the internal energy u(Ci , v, T ) of the porous system as a function of the pressure and the temperature, Dv DT ∂u + ∂T v,Ci Dt T ,Ci Dt N ∂u DCi + , ∂Ci v,T Dt i=1 ∂P Dv Du = −P + T Dt ∂T T Dt Du = Dt
∂u ∂v
DT ui ∇ · Jm −v i , Dt
(41)
Q
(42)
i=1
where I have use an additional equation of state for the average pressure P (v, T ) defined below by Eq. (45) and the conservation equations for the various components of the system. The term ui = (∂u/∂Ci )T ,v is the partial internal energy per unit mass of component i, and Cv ≡ (∂u/∂T )v (in J kg−1 ◦ C−1 ) represents the heat capacity (specific heat) of the system measured at constant volume per unit mass. The stress tensors σ¯¯ and σ¯¯ i are decomposed into their isotropic (P and pi ) and deviatoric (S¯¯ and s¯¯ i ) components, ¯¯ σ¯¯ = −P I¯¯ + S, σ¯¯ i = −pi I¯¯ + s¯¯ i , P=
Q
(43) (44)
pi ,
(45)
s¯¯ i .
(46)
i=1
S¯¯ = (40)
+ Cv
Fi · Jm i + T Π, (39)
i=1
257
Q i=1
258
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
For the pore water, the partial deviatoric stress is null. Combining Eqs. (39), (40), and (42) yields
N DT ∂P m m ρCv ui Ji − T ∇ · v + DS = −∇ · H − Dt ∂T v i=1
+
Q
Fi · Jm i + ρT Π,
(47)
where Ψi = ui − T si is the specific Helmholtz free energy of component i and si is the specific entropy of component i. Replacing the time derivative of the internal energy in the Gibbs– Duhem equation by its expression given by Eq. (37) yields
Q ¯ m Ds m ¯ ¯ ρT ui I − σ¯ i /ρi Ji = −∇ · H + Dt i=1
i=1 d where Dv/Dt = v∇ · v. The term = σ¯¯ : ∇v 0 (in −3 −1 J m s ) is the component of the total dissipation function associated with the dissipative deformation processes (see below, Section 2.5). Equation (47) states that the temperature of the system changes because of heat conduction, deformation effects, heating associated with irreversible deformation (e.g., viscous, plastic, or viscoplastic deformations), friction, and power supplied to the system. Equation (47) is written in a more familiar form as D (ρCv T ) = −∇ · H + QS , (48) Dt N m H≡H − (49) ui Jm i ,
+ σ¯¯ : ∇v + e
DS
i=1
∂P QS ≡ − T ∂T +
Q
∇ · v + DS + v
Fi · Jm i
Q
Jm ¯¯ : ∇v i · Fi − σ e
i=1
−
Q
(ui − T si )I¯¯ − σ¯¯ i /ρi ∇Jm i + ρT Π.
(53)
i=1
After long, but straightforward, algebraic manipulations, Eq. (53) can be recast as a local continuity equation for entropy, Ds = −∇ · Sm + Θ + ρΠ, Dt Q Hm m m si Ji , + S = T ρ
(54) (55)
i=1
T Θ = −Hm ∇ ln T − ui · Jm i
Q
si Jm i · ∇T
i=1
i=1
−
+ ρT Π,
(50)
i=1
where H is the heat flux (in J m−2 s−1 ) and QS term (in J m−3 s−1 ).
Q
is the heat source
Q
d ∇ · μ¯¯ i − Fi Jm ¯¯ : ∇v, i +σ
(56)
i=1
where Θ is the entropy source (inner entropy production per unit volume and unit time). 3. Dissipation function
2.5. The entropy balance equation (second principle) Prigogine [14] showed that the classical definition of entropy in thermostatics applies to a system in the vicinity of equilibrium. Different authors (e.g., [13]) agree that this assumption extends to a very wide class of phenomena, some of them being far from equilibrium. I denote by s the entropy per unit mass of the system. The entropy is related to the internal energy u, to the specific volume v, and to the mass fraction Ci according to the Euler relationship of thermostatics. In linear nonequilibrium thermodynamics, the Gibbs–Duhem relationship is assumed to hold, Du ¯ e Ds ¯¯ i : ∇Jm =ρ − σ¯ : ∇v − μ ρT i , Dt Dt Q
(51)
i=1
where the terms P (Dv/Dt) and μi (DCi /Dt) appearing in the classical formulation of the Gibbs–Duhem equation of Prigogine [14] and De Groot and Mazur [12] have been replaced e ¯¯ i : ∇Jm , respectively, to account for the by −v(σ¯¯ : ∇v) and μ i tensorial nature of the elastic stress and chemical potential tensors. The chemical potential tensor of species i is defined by [13] ¯¯ i = Ψi I¯¯ − σ¯¯ i /ρi = (ui − T si )I¯¯ − σ¯¯ i /ρi , μ
(52)
In Section 2, the results developed by De Groot and Mazur [12] for a viscous fluid were generalized to a multi-component system including the influence of the full stress tensor and both elastic and nonreversible deformations. These equations were written in a Lagrangian framework attached to the barycentric center of mass of the system, which comprises Q components. These components are (1) N components located into the pore water (anions, cations, and neutral species), (2) the pore water itself (i.e., the neutral solvent), and (3) the grains (so Q = N + 2). The balance equation for the internal energy u for an open porous medium (first principle) is given by Eq. (37) with Eqs. (28) and (38). The local balance law for entropy (second principle) for an open system is given by Eqs. (54) to (56). The stress tensor is divided into a reversible elastic component (superscript e) and a dissipative component (superscript d) corresponding to irreversible mechanisms of deformation; i.e., e d σ¯¯ = σ¯¯ + σ¯¯ [13]. For an open porous material, the entropy production can be divided into two contributions, Ds/Dt = (Ds/Dt)e + (Ds/Dt)i [15]. This expression is just an alternative form of the entropy balance equation; δse is the entropy exchanged between the system and its surrounding and δsi is the entropy produced in the system itself. The term (Ds/Dt)e can be either positive or negative, while Θ = ρ(Ds/Dt)i 0,
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
the inner entropy production of the system, is necessarily positive. In linear nonequilibrium thermodynamics, the total dissipation of the system is equal to the product of the temperature with the entropy production term, D = T Θ, and D 0 because Θ 0 and T > 0. This yields D = −Hm ∇ ln T −
Q
si Jm i · ∇T
i=1
−
Q
∇ · μ¯¯ i − Fi Jm ¯¯ : ∇v, i +σ d
(57)
i=1
D = −Sm ∇T −
Q
S ∇ · μ¯¯ i − Fi Jm i +D ,
(58)
i=1
where I have replaced the heat flow (divided by the temperad ture) by the entropy flux and where D S + σ¯¯ : ∇v represents the dissipation component associated with irreversible deformation processes (e.g., with viscoplastic deformation). This dissipation D S will be described in detail in a forthcoming paper. In Section 2, I derived a general formulation for the two principles of thermodynamics in a Lagrangian framework attached to the barycentric center of mass of the system. Therefore all the components are treated in a symmetrical fashion and the equations have been stated in the framework associated with this barycentric center of mass. However, a practical Lagrangian framework to interpret laboratory measurements is the one associated with deformation of the skeleton (e.g., [1]). This yields a new Lagrangian derivative associated with the velocity of the solid phase (subscript s) according to ∂ d = + vs · ∇. (59) dt ∂t The pore fluid (solvent plus ions and dissolved neutral species) is written with the subscript f. The fluid is assumed to be ideal, which implies that the peculiar stresses of the components of the pore electrolyte are assumed to satisfy the Dalton’s law of ideal mixtures, i.e., σ¯¯ i = −pi I¯¯, where pi is the partial pressure of component i. I write as p the pore fluid pressure of a reservoir in contact with the charged porous medium and as p¯ the fluid pressure of the pore fluid in the charged porous medium (e.g., [11]). The pore fluid pressure p is defined thermodynamically as the pressure of an external fluid in local contact with the system and in thermodynamic equilibrium with it [11]. A direct consequence of the previous assumptions is that the chemical potential of the components of the pore water are considered as scalars as discussed further below. I denote as φρi , φρw , and (1 − φ)ρs the partial mass densities of component i, solvent, and solid phase, respectively (φ is the porosity). I assume that the forces acting on both the pore fluid and the mineral matrix are always in balance and thus the acceleration of the barycentric center of mass is neglected. Obviously the time scale to reach mechanical equilibrium is always much shorter than the time scale to reach the (thermostatic) equilibrium state. This mechanical balance assumption yields
259
ρF + ∇ · σ¯¯ = 0 (Appendix A). Prigogine’s theorem states that in the mechanical equilibrium state, the barycentric velocity v entering into the definition of the diffusion flow in the dissipation function can be replaced by another arbitrary velocity. This theorem is extremely important for determining the dissipation function in the newly defined Lagrangian framework. In this mineral matrix framework, the measurable fluxes are defined by Ji ≡ ρ(±) φ(vi − vg ) = Jm i − ρi φ(vg − v),
(60)
Jw ≡ ρw φ(vw − vg ) = Jm w ρw φ(vg − v), m 0 = Js − ρs (1 − φ)(vs − v), S = Sm − ρs(vs − v), d d d σ¯¯ : ∇vs = σ¯¯ : ∇v − (vs − v)∇ · σ¯¯ ,
(61) (62) (63) (64)
where the subscripts i correspond to the component i, respectively, the subscript w corresponds to water, and the subscript g corresponds to the mineral grains. The theorem of Prigogine can be generalized by combining the Gibbs–Duhem relationship, the assumption of mechanical equilibrium, and the dissipation function written in the new Lagrangian framework as shown in Appendix A. The peculiar body forces per unit mass entering the dissipation function, see Eq. (58), include the electrical force plus the gravity force. For the components i dissolved in water, i ∈ (1, . . . , N), Fi = −qi ∇ψ/ρi + g ≈ −qi ∇ψ/ρi ,
(65)
while for water Fw = g. In these equations, qi is the charge of species i, qi = (±e)Zi , e is the elementary charge (1.6 × 10−19 C), and Zi is the valence of species i. Note that both forces are derived from a scalar potential as required by the theory developed in Section 2 (the gravitational acceleration is related to the gravitational potential ϕ = gz by g = −∇ϕ, z positive upward). The effective potentials per unit mass of the ions and water are scalars defined by μˆ i = μR i + μi + qi ψ/ρi + gz,
(66)
μˆ w = μR w
(67)
+ μw + gz,
R R respectively, where μR i , μw , and μg are the chemical potentials in a stress-free reference state [11]. Combining Prigogine’s theorem (see Appendix A) and the definition of the effective chemical potentials yields a new form of the dissipation function,
D=−
N
Ji ∇ μˆ i − Jw ∇ μˆ w − S∇T + D S .
(68)
i=1
I use now Curie’s principle (see [16]) to set up the form of the linear constitutive equations for the fluxes. This principle states that for an isotropic system, the coupling of tensors, whose orders differ by an odd number, does not occur. From this principle, it follows that I must consider separately scalars, vectors, and second-rank tensors in the dissipation function. I will focus below only on the vectorial fluxes entering the dissipation function. The vectorial contribution to the dissipation function
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A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
is D=−
N
Ji ∇ μˆ i − Jw ∇ μˆ w − H∇T /T ,
(69)
i=1
where I have replaced the entropy flux S by H/T , where H is the heat flux, including a convective contribution. The seepage velocity U is U=
N
Ji /ρi + Jw /ρw ,
(70)
i=1
ρw U ≈ J w .
(71)
Note that the definitions of the Darcy velocity and the heat flux are not unique. The gradients of the effective chemical potentials of the pore water and ions are kb T ∇ p¯ ∇ ln C¯ w + − g − sw ∇T , ρw ρw kb T qi ∇ μˆ i = ∇ ln C¯ i − E − si ∇T , ρi ρi ∂ μˆ i , si ≡ ∂T Ci ,p,ψ ∇ μˆ w =
(72)
and two textural parameters, the formation factor and the permeability. The continuity equations can be written as a function of volumetric fractions per unit initial volume and therefore involved the determinant J of the deformation gradient tensor discussed in Section 4. The (N + 2) generalized continuity equations associated with the N -form of the constitutive equations are ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ Ji ρi φ ρ i Ri d ⎢ ··· ⎥ ⎢ ··· ⎥ ⎢···⎥ J∇ · ⎣ ⎦ = − ⎣ (78) ⎦, ⎦+⎣ Jw ρ w Rw ρw φ dt H QS ρCv T where Ri and Rw are the production rates of ions i and water, respectively, within the controlled volume per unit mass of the porous medium (in mol m−3 s−1 ) and QS is here the intrinsic heat source (bulk rate of heat production, expressed in J m−3 s−1 ); see Eq. (50). 4. Mechanical equations
(73)
4.1. Generalized power balance equation
(74)
In order to discuss the influence of the reversible (elastic) component of the stress tensor in the present model, it is necessary to first derive a generalized power balance equation for the specific Helmholtz free energy of the porous composite. Taking the conservation equation for the internal energy and for the entropy of the system in a Lagrangian framework attached to the barycentric center of mass of the system yields Du Ds ρ −T = −∇ · Wu − T Sm + σ¯¯ : ∇v Dt Dt (79) − Sm ∇T + ρT ΠT − D, N N σ¯¯ i m ¯ m μ¯ i Ji , Ψi I¯¯ − Ji = Wu − T Sm = (80) ρi
where p is the pore water pressure of a reservoir locally in equilibrium with the pore water of the porous medium, E is the electrical field, and si is the specific entropy of pore fluid component i. The swelling pressure π is defined as π ≡ −kb T ln C¯ w and p¯ ≡ p +π −ρb gz (in Pa) is the effective pore fluid pressure. Therefore the gradients of the effective chemical potentials of the water and ions per unit mass are given by ∇ μˆ w = ∇p/ρw − sw ∇T ,
(75)
∇ μˆ i = ∇ μ˜ i /ρi − si ∇T ,
(76)
respectively, and μ˜ i represents the electrochemical potentials of species i per unit volume. The linear nonequilibrium thermodynamic theory implies linear relationships between the flows and the generalized driving forces that cause these flows. This yields a form of the constitutive equations that I call “the N -form,” ⎡ ⎤ ⎤ ⎡ Ji ∇ μˆ i ⎢···⎥ ¯ ⎢ ··· ⎥, (77) ⎣ ⎦ = −N¯ ⎣ ⎦ Jw ∇ μˆ w H ∇T /T where N¯¯ is an (N + 2) square matrix of material transport properties. Inserting the linear phenomenological equations within the dissipation function shows that the entropy production is a homogeneous quadratic function of the independent forces entering Eq. (77). The second law of thermodynamics states that this expression is definitely positive (D 0). This yields Nij = Nj i (Onsager’s reciprocity) (N¯¯ is symmetric), Nii 0, and Nij2 Nii Njj (i = j ) [15]. By volume-averaging the local Nernst–Planck and Stokes equations, Revil and Linde [11] provide the necessary relationships between the material properties entering the matrix N and both the constituent properties
i=1
i=1
where the total Helmholtz free energy and the specific Helmholtz free energy per unit mass of the component i are defined by Ψ = u − T s and Ψi = ui − T si , respectively. This yields the generalized power balance equation N
DΨ DT m ρ μ¯¯ i Ji + σ¯¯ : ∇v +s = −∇ · Dt Dt i=1
− S : ∇T + ρT ΠT − D, m
(81)
which is the fundamental equation required to formulate the thermoporoelastic constitutive equations. The term dΨ + s dT represents part of the energy externally supplied to the porous medium and stored (if dΨ + s dT > 0) or extracted (if <0) during the time interval dt and converted in heat. Introducing the expression of the dissipation function, Eq. (58) into Eq. (81), yields N DΨ DT e ρ (82) ¯¯ : ∇v. μ¯¯ i : ∇Jm +s =− i +σ Dt Dt i=1
The Helmholtz free energy corresponds to the potential associated with the entropy of the porous system.
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
4.2. Mechanical constitutive relationships In the following, I assume that the porous medium has a thermoporoelastic behavior. In addition, I assume small deformations for which the whole theory is linear and obeys the superposition principle. I denote as Ω0 (the volume bounded by the surface ∂Ω0 ) the Lagrangian reference configuration for the porous medium, as X the vector locating an element of the porous grain skeleton in this reference configuration, and as Ω(t) the Eulerian configuration at time t and as x(X, t) the position of the same material element at time t in the Eulerian configuration. The spatial position of the material element is given by x(X, t) = X + u(X, t), where u is the solid displacement. Deformation of the skeleton is characterized by the deformation gradient tensor P¯¯ by dx(X, t) ≡ P¯¯ dX. The Green–Lagrange strain tensor e¯¯ and the second Piola–Kirchhoff stress tensor T¯¯ (both symmetric) are defined by (e.g., [2,4]) 1 e¯¯ ≡ P¯¯ T P¯¯ − I¯¯ , 2 −1 T¯¯ ≡ J P¯¯ −1 σ¯¯ P¯¯ T ,
(83) (84)
dV J ≡ det P¯¯ = , dV0
(85)
where P¯¯ T represents the transposed matrix of P¯¯ , A¯¯ = P¯¯ T P¯¯ is the right Cauchy–Green tensor, B¯¯ = A¯¯ −1 = P¯¯ −1 P¯¯ T is the Piola deformation tensor, and J denotes the Jacobian of the deformation gradient tensor (J > 0) and is also equal to the ratio between the volume of the elementary volume in the current configuration dV divided by the volume this element occupied in the reference state dV0 . Tension is considered positive. In addition, Euler’s formula is (e.g., [3]) dJ = J ∇ · vs . (86) dt From Eq. (81), written now in the Lagrangian framework defined in Section 3, I have d(ρΨ ) + ρΨ ∇ · vs dt N+1 dT e = Tr(σ¯¯ ∇vs ) − μi ∇ · Ji − ρs , dt
(87)
i=1
where the sum over i is extended to the N + 1 components of the pore water (N components plus the solvent, i.e., the water molecules). Using Eq. (86), I obtain d(ρΨ ) d(JρΨ ) =J + ρΨ ∇ · vs . (88) dt dt The mass balance equation for species i is dmi = −J ∇ · Ji , dt where mi = J φρi . Equations (87) to (89) yield ¯ N+1 dmi d(JρΨ ) de¯ dT μi = Tr T¯¯ + − Jps . dt dt dt dt i=1
(89)
261
The time derivative of the Helmholtz free energy per unit mass of the pore fluid, ρf Ψf , is obtained using the chain of algebraic manipulations Ψf = −pvf +
N+1
μi C¯ if ,
(91)
μi ρi ,
(92)
i=1
ρf Ψf = −p +
N+1 i=1
d(ρf Ψf ) dp dρi dμi μi ρi =− + + , dt dt dt dt N+1
N+1
i=1
i=1
(93)
where vf is the specific volume of the pore fluid in the porous medium and C¯ if = ρi /ρf is the mass fraction of pore fluid component i. The Gibbs–Duhem relationship in the pore fluid is ρf sf δT − δp +
N+1
ρi δμi + Q¯ V δ ϕ¯ + ρf gδz = 0,
(94)
i=1
where sf is the entropy per unit mass of the pore fluid, Q¯ V is the net excess of electrical charge in the connected porosity of the porous medium (e.g., [6], [11], and [17]), and ϕ¯ is the local electrical potential (position-dependent) in the pore fluid of the body. The osmotic pressure in the connected porosity of the porous continuum is given by (e.g., [6,17]) ϕ¯ π=
¯ V dϕ¯ . Q
(95)
0
This effective pressure includes gravitational and osmotic effects. In addition, state equations for the density and entropy per unit mass of the pore fluid are (e.g., [2]) 1 1 = 0 1 − βf (p¯ − p¯ 0 ) + αf θ , ρf ρ f Cpf p¯ − p¯ 0 sf − sf0 = θ − αf , T0 ρf0 1 1 ∂ρf ≡ , βf ≡ Kf ρf ∂ p¯ T 1 ∂ρf αf ≡ − , ρf ∂T p Cvf − Cpf =
T0 αf2 ρf0 βf
(96) (97) (98) (99) (100)
,
where θ = T − T0 , Cpf and Cvf are the fluid mass heat at constant pressure and volume, respectively, βf is the isothermal bulk compressibility of the pore fluid, and αf is the cubic thermal dilatation coefficient of the pore fluid. I write now the Gibbs– Duhem relationship, Eq. (94), in a more customary form, ρf sf δT − δ p¯ +
N+1
ρi δμi = 0,
(101)
i=1
(90)
dT dp¯ dμi ρi − + = 0. dt dt dt N+1
ρf sf
i=1
(102)
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A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
This yields N+1 dT d(ρf Ψf ) dρi μi = + ρf sf . dt dt dt
(103)
i=1
I denote as vp = J φ the specific pore volume of the porous medium in the deformed state per unit referential volume. I obtain dvp d(J φρf Ψf ) d(ρf Ψf ) = vp + ρf Ψf , dt dt dt
N+1 dρi dT d(J φρf Ψf ) μi = vp + ρf sf dt dt dt i=1
N+1 dvp + −p¯ + μi ρi , dt
(104)
(105)
i=1
and in addition, I have N +1
μi
i=1 N +1 i=1
N+1 d(vp ρi ) dmi μi = , dt dt
(106)
N+1 N+1 dvp dmi dρi μi ρi μi v p = + . dt dt dt
(107)
i=1
μi
i=1
i=1
I now determine the free energy of the mineral matrix per unit initial volume of the porous medium, W . This quantity is obtained by removing from the free energy of the saturated porous medium the free energy of the pore fluid. It corresponds to the elastic energy of the porous body. This yields W = J (ρΨ − φρf Ψf ) and therefore ¯ dvp de¯ dT dW = Tr T¯¯ + p¯ − J (ρs − φρf sf ) . (108) dt dt dt dt I assume that the potential W depends only on the external variables e, ¯¯ vp , and T , i.e., W = W (e, ¯¯ vp , T ). The differentiation of W yields deij ∂W ∂W dW = + dt ∂eij vp ,T dt ∂vp eij ,T dvp dT ∂W × + , (109) dt ∂T eij ,vp dt ∂W T¯¯ = (110) , ∂ e¯¯ vp ,T ∂W p¯ = (111) , ∂vp eij ,T ∂W S − mf sf = − (112) , ∂T eij ,vp where S ≡ Jρs is the specific entropy per unit referential volume and mf ≡ Jρf φ = vp ρf is the specific mass of water per unit referential volume. In linear thermoporoelasticity, this yields the following constitutive equations for charged porous materials: mf ¯¯ (113) T¯¯ − T¯¯ 0 = 4 C¯ : e¯¯ − M B¯¯ 0 − Aθ, ρf
mf p¯ − p¯ 0 = −M B¯¯ : e¯¯ + M 0 + ρf0 Lθ, ρf ¯ S − S0 − mf sf0 = A¯ : e¯¯ − Lmf + Cv θ/T0 .
(114) (115)
Due to the global electroneutrality of the porous medium, there are no electrostatic terms in the constitutive equations, except for the osmotic pressure, and therefore I recover the classical form of the mechanical constitutive equations given by [2]. The terms T¯¯ 0 , p¯ 0 , and S0 characterize the thermostatic state. If we consider a stress-free thermostatic configuration, T¯¯ 0 = 0. I denote as 4 C¯ the (fourth-order) stiffness tensor of undrained isothermic elastic moduli, A¯¯ and B¯¯ are symmetric second-order tensors, M is the Biot modulus, and Cv is the volume heat capacity per unit of initial volume in an isodeformation undrained experiment. They are defined by ∂Tij ∂Tkl = , Cij kl ≡ (116) ∂ekl mf ,T ∂eij mf ,T 0 ∂ p¯ , M ≡ ρf (117) ∂mf eij ,T ∂Tij Bij ≡ − (118) , ∂ p¯ eij ,T ∂Tij , Aij ≡ − (119) ∂T eij ,mf ∂S 0 L ≡ sf − (120) , ∂T eij ,mf ∂(S − mf sf0 ) Cv ≡ T0 (121) . ∂T eij ,mf The tensor T0 A¯¯ is the tensor of undrained strain latent heats and T0 L represents the isodeformation latent heat of variation in fluid mass content. The tensor B¯¯ is the Biot stress tensor, which reduces to the Biot coefficient ζ in the isotropic case. In the infinitesimal transformation assumption and for isotropic cases, the Green strain tensor e¯¯ is replaced by the infinitesimal strain tensor ε¯¯ and the Piola–Kirchhoff stress tensor T¯¯ can be replaced by the Cauchy stress tensor σ¯¯ . Thus, the constitutive equations of linear thermoporoelasticity of charged porous materials are 2 mf δij σij = K − μ εkk + 2μεij − ζ M 3 ρf0 − Kαb θ δij , p¯ − p¯ 0 = −ζ Mεkk + M
mf ρf0
(122)
+ Cv θ,
S − S0 − ρf φsf0 = αb Kεkk − αm M
mf ρf0
(123)
+
Cv θ, T0
(124)
where K is the undrained isothermal bulk modulus, μ is the shear modulus (which remains the same for a drained or an undrained experiment), αb is the cubic undrained thermal dilatation coefficient, ζ is the Biot coefficient, and αm T0 represents the cubic isodeformation latent heat of fluid pressure variation, αm = (ζ − φ0 )αg + φ0 αf (Coussy [2, p. 91]). The
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
stress tensor obeys the mechanical equilibrium condition discussed in Appendix A. Appropriate boundary conditions can be used to close the system of equations.
d ¯¯ s − Fs − ∇ · σ¯¯ , + ρw φ(∇μw − Fw ) + ρs (1 − φ) ∇ · μ
In the present work, the constitutive equations of transport derived in [11] are encapsulated into a more fundamental thermodynamic theory. This theory accounts for the reversible and irreversible components of the deformation of the porous skeleton and is applicable to a multicomponent electrolyte filling a charged porous material. We paid special attention to small thermoporoelastic deformation, as we are mainly interested in deformation effects associated with the transport of ions and water. The next step will be to show that in such a system, any thermodynamic disturbance is the source of electromagnetic signals recordable away from the source using a network of very sensitive nonpolarizing electrodes and magnetic sensors (see [6] and [18], for example). These signals can be inverted to recover the transport properties of the medium using algorithms derived initially in electroencephalography. Such an application will be presented in a forthcoming work. In this paper, we have neglected several mechanisms of transport like (a) the hydrodynamic dispersion of heat and solutes through the porous material, (b) the transition between viscous to inertial laminar flow at high Reynolds numbers, and (c) free convection of the pore fluid. All these aspects require specific developments that will be incorporated into future works, including nonelastic contributions to deformation.
(A.2) N (∇μi − Fi )Ji − (∇μw − Fw )Jw D s ≡ −S∇T − i=1 d + σ¯¯ : ∇vs ,
(A.3)
Ds
where is the total dissipation function written in the Lagrangian framework associated with deformation of the solid phase. In order to demonstrate the validity of Prigogine’s theorem, D m = D s in Eq. (A.2), I can use the force balance equation written under the assumption of mechanical equilibrium, N
ρi φFi + ρw φFw + ρs (1 − φ)Fs + ∇ · σ¯¯ = 0,
(A.4)
i=1 e d and where σ¯¯ = σ¯¯ + σ¯¯ . In addition, the Gibbs–Duhem relationship of thermostatic yields N
e ρs∇T + ∇ · σ¯¯ + φ ρi ∇μi + ρw ∇μw i=1
¯¯ s = 0. + ρs (1 − φ)∇ · μ
(A.5)
Equations (A.4), (A.5), and (A.2) yield immediately to Prigogine’s theorem. I now discuss the form of the force balance equation. Using Eqs. (21) and (A.4), the force balance equation is
Acknowledgments This work is supported by the CNRS (The French National Research Council), ANDRA (The French National Agency for Radioactive Waste management), the GDR FORPRO (Research action Contribution FORPRO-2006/12A), and ANR ECCOPNRH (Project POLARIS). I especially appreciated the supports from Scott Altmann, Daniel Coelho, and Joël Lancelot and I thank the two referees for their constructive comments and Dominique Gibert for fruitful discussions.
The dissipation function written in a Lagrangian framework attached to the barycentric center of mass motion is
m (∇μi − Fi )Jm i − (∇μw − Fw )Jw
i=1
d ¯¯ s − Fs Jm − ∇ ·μ ¯¯ : ∇v. s +σ
ρF + ∇ · σ¯¯ = 0.
(A.6)
Using Eqs. (21) and (65) and the electroneutrality condition of the whole system, ρq = 0, where q is given by Eq. (13), I obtain, ρg + ∇ · σ¯¯ = 0.
(A.7)
References
Appendix A. Extension of Prigogine’s theorem
N
I now write the fluxes in the new Lagrangian framework associated with deformation of the grain skeleton (superscript s), N m s D = D − (vs − v) ρs∇T + ρi φ(∇μi − Fi ) i=1
5. Concluding statements
D m = −Sm ∇T −
263
(A.1)
[1] M.A. Biot, J. Appl. Phys. 12 (1941) 155–164. [2] O. Coussy, Mechanics of Porous Continua, Wiley, New York, 1995, 455 pp. [3] J.M. Huygue, J.D. Janssen, Int. J. Eng. Sci. 35 (1997) 793–802. [4] M.E. Levenston, S.R. Eisenberg, A.J. Grodzinsky, Int. J. Solids Structures 35 (34–35) (1998) 4999–5019. [5] J.J. Nitao, J. Bear, Water Resour. Res. 32 (1996) 225–250. [6] A. Revil, P. Leroy, K. Titov, J. Geophys. Res. 110 (2005), doi:10.1029/ 2004JB003442, B06202. [7] Y. Zhou, R.K.N.D. Rajapakse, J. Graham, Int. J. Solids Structures 36 (1999) 4841–4868. [8] M.A. Murad, J.H. Cushman, Transp. Por. Media 28 (1997) 69–108.
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[9] M.A. Murad, J.H. Cushman, Int. J. Eng. Sci. 38 (2000) 517–564. [10] L. Loret, A. Gajo, F.M.F. Simoes, Eur. J. Mech. A Solids 23 (2004) 763– 782. [11] A. Revil, N. Linde, J. Colloid Interface Sci. 302 (2006) 682–694, doi:10.1016/j.jcis.2006.06.051. [12] S.R. De Groot, P. Mazur, Non-equilibrium Thermodynamics, Dover, New York, 1984, 510 pp. [13] C. Truesdell, Rational Thermodynamics, McGraw–Hill, New York, 1969.
[14] I. Prigogine, Etude Thermodynamique des Phénomènes Irreversibles, Desoer, Liège, 1947. [15] L. Onsager, Phys. Rev. 37 (1931) 405–426. [16] P. Curie, Oeuvres, Gauthier-Villars, Paris, 1908. [17] A. Revil, P.J. Leroy, J. Geophys. Res. 109 (2004), doi:10.1029/ 2003JB002755, B03208. [18] A. Revil, H. Schwaeger, L.M. Cathles, P. Manhardt, J. Geophys. Res. B 104 (9) (1999) 20033–20048.
Thermodynamics of ions and water transport in porous media A. Revil 1 CNRS-CEREGE, Université Paul Cézanne, Département d’Hydrogéophysique et Milieux Poreux, BP 80, 13545 Aix-en-Provence Cedex 4, France Received 12 August 2006; accepted 30 October 2006 Available online 1 December 2006
Abstract The thermodynamic framework of Prigogine, de Groot, and Mazur is extended to study the transport of ions and water in thermoporoelastic materials assuming infinitesimal deformations. New expressions are developed for the first and second principles of nonequilibrium thermodynamics of multicomponent systems and a generalized power balance equation is derived. For porous materials, all the components cannot be treated on a symmetric basis. A Lagrangian framework associated with deformation of the solid phase is introduced and, in this framework, Curie’s principle is used to set up the form of the linear constitutive equations describing the transport of ions, water, and heat through the pore network. The material properties entering these equations were recently obtained by Revil and Linde [J. Colloid Interface Science 302 (2006) 682–694] using a volume-averaging approach based in the Nernst–Planck and Stokes equations. This provides a way to relate the material properties entering the constitutive equations to two textural parameters characterizing the topology of the pore space of the material (namely the tortuosity of the pore space and the permeability). The generalized power balance equation is used to derive the linear poroelastic constitutive equations (including the osmotic pressure) to describe the reversible contribution of deformation of the medium in response to ions and water transport through the connected porosity. © 2006 Elsevier Inc. All rights reserved. Keywords: Self-potential; Nonequilibrium thermodynamics; Porous material; Thermoporoelasticity; Dissipation
1. Introduction Transport of ions and the concomitant deformation of the skeleton of a charged porous composite, in response to change in the boundary conditions imposed on the system or the application of nonequilibrium thermodynamic forces, is a central subject in petrophysics, in colloidal chemistry, and in biology [1–10]. Recently, Revil and Linde [11] derived a new set of constitutive equations to model the transport of ions and water through a charged and water-saturated porous medium, assumed to be isotropic. The novelty of their approach is that all the material properties entering the constitutive equations are related to only two fundamental properties characterizing the topology of the solid and fluid mixture. These parameters are the tortuosity of the pore space (or the electrical formation factor, which that is equal to the tortuosity divided by the porosity) and the permeability. However, these constitutive equations E-mail address: [email protected]. 1 Also at CNRS-ANDRA/GDR FORPRO 0788, France.
0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.10.074
need to be inserted into a thermodynamic framework that includes mechanical deformation of the charged porous composite. Such a framework is developed in the present paper with a special focus on the reversible component of the deformation. A specific development for the irreversible contribution to deformation will be presented in a forthcoming work, but the thermodynamic framework developed below allows such a contribution. However, if the deformation is very small, the elastic contribution is likely to be the dominant mechanism of deformation, e.g., during the spreading of the ions through the connected porosity [10]. In this paper, I will use the term “thermodynamics” as referring to irreversible processes out of equilibrium and I will keep the term “thermostatics” to describe the equilibrium state, which is described in [11]. The porous material is formed by an isotropic skeleton of solid grains saturated by a multicomponent electrolyte comprising the solvent (water), the ions, and possibly neutral species. I consider below only small and reversible deformations and the mechanical constitutive equations will be linearized. I will assume that the system remains linear in the sense that the principle of superposition applies.
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
255
2. Thermodynamics of stressed multicomponent systems
The mass fractions (dimensionless) are defined by Ci = ρi /ρ. Equations (2) to (6) yield
In this section, I generalize the nonequilibrium thermodynamics approach of De Groot and Mazur [12]. De Groot and Mazur [12] developed a thermodynamic model of a multicomponent viscous fluid with hydrostatic stress, while in the present paper, I am interested in describing the thermodynamics of a multicomponent deformable porous continuum with a nonhydrostatic stress tensor. The porous body is broken up into a number of representative elementary volumes (REVs). These REVs are large enough to define thermodynamic potentials with statistical significance (e.g., the chemical potentials of the ions and the fluid pressure) and small enough so that in each REV, the gradients of the thermodynamic potentials of interest can be considered negligible. The porous body is not in equilibrium as thermodynamic potentials vary from REV to REV. However, the local equilibrium assumption implies that the specific entropy is the same function of extensive variables per unit mass as it is in equilibrium [13,14].
Dρ = −ρ∇ · v, (7) Dt DCi ρ (8) = −∇ · Jm i . Dt with the introduction of the specific volume v = 1/ρ, Eq. (7) yields a mass conservation equation for an observer moving with the barycentric center of mass motion, Dv = ∇ · v. (9) Dt In addition, it follows from Eqs. (4) to (6) that in the Lagrangian framework previously defined, the sum of all the fluxes is equal to zero: ρ
Q
Jm i = 0.
(10)
i=1
2.1. Continuity of mass
Another identity used later follows directly from the previous equations (see [12]),
I recall first, in an Eulerian framework (i.e., from the standpoint of a stationary observer), that the local conservation of mass for the component i is
Da ∂aρ =ρ − ∇ · (aρv), ∂t Dt where a can be a scalar, a vector, or a tensor.
∂ρi = −∇ · (ρi vi ), ∂t
(1)
where ρi (in kg m−3 ) represents the bulk density of partial component i (i ∈ {1, . . . , Q}), vi is its velocity (in m s−1 ), and t is time (in s). The law of mass conservation is obtained by summing the previous equations over all the constituents of the system, ∂ρ = −∇ · (ρv), ∂t Q ρ≡ ρi , ρv ≡
2.2. Continuity of charge I denote by qi the charge per unit mass of component i (qi = 0 if the species is neutral, water for instance). The total current density (in A m−2 ) is defined as the net amount of charge flowing per unit surface area and per unit time: Jm =
(2)
Q
ρ i qi vi .
(3) q=
Q Q 1 ρ i qi = Ci q i . ρ i=1
ρ i vi ,
(4)
i=1
where ρ is the total density of the porous system and is the sum of all the partial mass densities ρi of its components and v is the velocity of the barycentric center of mass. I introduce the substantial time derivative in a Lagrangian framework attached to the barycentric center of mass motion by ∂ D = + (v · ∇). (5) Dt ∂t I will define, in Section 4, another Lagrangian framework, attached this time to the mineral skeleton and in which the observable (measurable) fluxes are defined. The diffusion fluxes of the various species (ions, water, and grains) in the Lagrangian framework attached to the barycentric center of mass (superscript m) are defined by Jm i = ρi (vi − v).
(12)
i=1
The total charge per unit mass of the system q is defined by
i=1 Q
(11)
(6)
(13)
i=1
Using Eq. (6), the total current density is given by Jm = J m c + ρqv, Jm c ≡
Q
qi Jm i ,
(14) (15)
i=1 −2 where Jm c represents the conduction current (in A m ). Therefore the total current density is equal to the conduction current density plus a convective term ρqv. The porous medium or the colloidal suspension is also characterized by a global electroneutrality condition: the net charge of the grains is exactly counterbalanced by the net charge of the pore water. Consequently q = 0 and therefore the convective term is equal to zero in the Lagrangian framework attached to the barycentric center of mass. The continuity equation for the electrical charge is obtained by summing the continuity equations for the mass of the various
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A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
charged components, Eqs. (8) and (15). This yields ρ
Dq = −∇ · Jm c, Dt
Using Eqs. (2), (7), and (20), I write the balance equation for the momentum density ρv, (16)
and therefore ∇ · Jm c = 0 if there is global electroneutrality in the system (q = 0). Neglecting the influence of the Lorentz force, neglecting electric and magnetic polarizations, and using the macroscopic electroneutrality condition (q = 0), the Maxwell stress tensor T¯¯ M and the Maxwell body force FM per unit mass are given by 1 2 ¯¯ ¯ ¯ TM =ε E⊗E− E I , (17) 2 ρFM = ρqE = 0, (18) where ε is the dielectric constant of the porous body, E ⊗ E represents a dyadic product (and therefore a second-order tensor, where ⊗ is the tensorial product), and I¯¯ is the unit dyadic with elements δij (Kronecker delta, δij = 1 if i = j and δij = 0 if i = j ). A force balance equation is given by ∇ · T¯¯ M = ρFM (= 0).
(19)
2.3. Force and momentum balance equations I first specify the assumptions made in developing the model. (1) The influence of the external magnetic field sources is ignored and the porous medium is assumed to be free of any magnetically polarizable particles. (2) Thermal free convection of the pore fluid is ignored. Only the case of scalar potential motion is considered in the present model. (3) I consider viscous-laminar flow of the pore fluid relative to the skeleton. The equation of motion of the system is Dv ρ = ∇ · σ¯¯ + ρF, Dt Q ρF ≡ ρi Fi ,
(20)
σ¯¯ =
σ¯¯ i ,
(21)
(22)
i=1
where σ¯¯ (expressed in N m−2 ) is the total (symmetric) Cauchy stress tensor (tension is considered positive). This stress tensor is equal to the sum of the peculiar (partial) stresses (positive for tension) of the individual components of the mixture, σ¯¯ i . The vector F is the total body force per unit mass exerted on the system and Dv/Dt represents the acceleration of the center of mass. The forces Fi entering Eq. (20) represent the gravity and the electrostatic body forces per unit mass exerted on component i. As these forces arise from scalar potentials ψi , Fi = −∇ψi (potential flow), and ∂ψi /∂t = 0, the total potential ψ of the system is defined by [12] ρψ =
Q i=1
2.4. Energy balance equation (first principle) From here, I generalize the approach of De Groot and Mazur [12] to thermoporoelastic media. It results from Eq. (20), a conservation equation for the kinetic energy of the barycentric center of motion, D 1 2 ρ (25) v = ∇ · (σ¯¯ · v) − σ¯¯ : ∇v + ρF · v, Dt 2 where the colon indicates a tensor dot product (a : b = a ij bij with the Einstein convention; note that the symbols “·” and “:” between tensors of various orders denote their inner product with simple or double contraction, respectively). This yields a conservation equation for the kinetic energy, ∂ 1 2 (26) ρv = −∇ · Wk − σ¯¯ : ∇v + ρF · v, ∂t 2 1 Wk = ρv2 v − σ¯¯ · v, (27) 2 where Wk is the kinetic energy flux in an Eulerian framework of reference (in J m−2 ). We denote as u the internal energy of the system. The internal energy flux vector Wu is the sum of the heat flux plus energy change due to diffusion and local stresses σ¯¯ i , Wu = Hm +
Q
ui I¯¯ − σ¯¯ i /ρi Jm i ,
(28)
i=1
i=1 Q
∂(ρv) = −∇ · (ρv ⊗ v − σ¯¯ ) + ρF. (24) ∂t The momentum flux associated with the momentum density ρv is (ρv ⊗ v − σ¯¯ ) and the term ρF therefore represents a source term of momentum.
where ui is the internal energy of species i, and Hm is the conductive heat flux including heat transported by mass diffusion (see [5]). A balance equation for the energy (kinetic plus internal energies) is therefore D 1 2 ρv + ρu = −∇ · Wu − σ¯¯ · v + ρT Π Dt 2 + ρv · F +
Q
Jm i · Fi ,
(29)
i=1
where Π (positive or negative) is the power per unit mass and temperature. The term ρT Π corresponds to the energy sources provided to the system by external sources and is not accounted for by [12]. This yields a local conservation equation for the internal energy, ∂ Jm (ρu) = −∇ · Wu + σ¯¯ : ∇v + ρT Π + i · Fi , ∂t Q
(30)
i=1
ρi ψi .
(23)
Wu = ρuv + Hm +
Q
ui I¯¯ − σ¯¯ i /ρi Jm i .
i=1
(31)
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
Equation (30) generalizes Eq. (34) of Ref. [12] to thermoporoelastic bodies. A local conservation equation for the potential energy is given by ∂ Jm (ρψ) = −∇ · Wψ − ρv · F − i · Fi , ∂t Q
(32)
i=1
Wψ = ρψv +
N
ψ i Jm i ,
(33)
i=1
where Wψ is the potential energy flux vector in an Eulerian framework of reference. The total energy e of the system per unit mass is the sum of the kinetic energy, potential energy, and internal energy per unit mass. It is given by e = v2 /2 + ψ + u, which defines the internal energy u per unit mass of the system. The internal energy corresponds to the energy associated with the short-range interactions and motions of the individual components of the porous mixtures. The internal energy is dependent on the temperature and density (pressure) of the system as discussed later. The balance equation for the total energy is ∂ (ρe) = −∇ · We + ρT Π, ∂t We = Wk + Wu + Wψ , We = ρev − σ¯¯ · v + Hm +
(34) (35)
Q i=1
σ¯¯ i (ui + ψi )I¯¯ − · Jm i , ρi
(36)
where We represents the total energy flux (energy per unit mass per unit surface area and per unit time). This energy flux includes several contributions (1) a convective term ρev − σ¯¯ · v, (2) a conductive heat flux Hm , and (3) a diffusive energy flux. Equations (34) and (36) generalize Eqs. (31) and (33) of Ref. [12]. Subtracting the total energy given by Eq. (34) to the sum of the kinetic plus potential energy given by Eq. (29), respectively, and using Eq. (36) I obtain a balance equation for the internal energy u, Du = −∇ · Wu + σ¯¯ : ∇v + ρT ΠT , Dt Q
ΠT ≡ Π + Jm (ρT ), i · Fi
(37)
ρ
(38)
i=1
where ΠT is the total power supplied to the system including the influence of the body forces. The stress tensor is decomposed into a reversible elastic component (superscript e) and a dissipative component (superscript d) corresponding to irreversible e d mechanisms of deformation, σ¯¯ = σ¯¯ + σ¯¯ (following [13]). Using Eqs. (31), (37), and (38), I obtain a balance equation for the thermal energy, Du DQ e d = + v σ¯¯ : ∇v + v σ¯¯ : ∇v + v Dt Dt DQ ρ = −∇ · Hm . Dt
Q
Equation (39) replaces Eq. (39) of De Groot and Mazur [12]. The parameter Q is here the heat per unit of mass in the system and v ≡ 1/ρ the specific volume (see Section 2.1). The terms of the left-hand side of Eq. (39) represent the rate of gain of internal energy per unit mass. These terms are (1) the rate of internal energy input by conduction per unit mass DQ/Dt, (2) the reversible rate of internal energy increase per unit mass e by compaction v σ¯¯ : ∇v, (3) the irreversible rate of internal energy increase per unit mass due to dissipation associated with d irreversible mechanisms of deformation v σ¯¯ : ∇v, (4) the irreversible rate of internal energy increase per unit mass by the internal motion of the various components forming the system, and finally (5) the power per unit mass and temperature supe plied to the system. The term v σ¯¯ : ∇v can be either positive or negative depending upon whether the porous system is expanding or contracting. This term is a reversible mode of exchange of energy between mechanical and internal energy. The term d v σ¯¯ : ∇v is always positive and represents an irreversible degradation of mechanical to thermal energy. For practical applications, it is convenient to recast the thermal energy equation in terms of temperature and heat capacity rather than in terms of internal energy. I first specify an equation of state for the internal energy u(Ci , v, T ) of the porous system as a function of the pressure and the temperature, Dv DT ∂u + ∂T v,Ci Dt T ,Ci Dt N ∂u DCi + , ∂Ci v,T Dt i=1 ∂P Dv Du = −P + T Dt ∂T T Dt Du = Dt
∂u ∂v
DT ui ∇ · Jm −v i , Dt
(41)
Q
(42)
i=1
where I have use an additional equation of state for the average pressure P (v, T ) defined below by Eq. (45) and the conservation equations for the various components of the system. The term ui = (∂u/∂Ci )T ,v is the partial internal energy per unit mass of component i, and Cv ≡ (∂u/∂T )v (in J kg−1 ◦ C−1 ) represents the heat capacity (specific heat) of the system measured at constant volume per unit mass. The stress tensors σ¯¯ and σ¯¯ i are decomposed into their isotropic (P and pi ) and deviatoric (S¯¯ and s¯¯ i ) components, ¯¯ σ¯¯ = −P I¯¯ + S, σ¯¯ i = −pi I¯¯ + s¯¯ i , P=
Q
(43) (44)
pi ,
(45)
s¯¯ i .
(46)
i=1
S¯¯ = (40)
+ Cv
Fi · Jm i + T Π, (39)
i=1
257
Q i=1
258
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
For the pore water, the partial deviatoric stress is null. Combining Eqs. (39), (40), and (42) yields
N DT ∂P m m ρCv ui Ji − T ∇ · v + DS = −∇ · H − Dt ∂T v i=1
+
Q
Fi · Jm i + ρT Π,
(47)
where Ψi = ui − T si is the specific Helmholtz free energy of component i and si is the specific entropy of component i. Replacing the time derivative of the internal energy in the Gibbs– Duhem equation by its expression given by Eq. (37) yields
Q ¯ m Ds m ¯ ¯ ρT ui I − σ¯ i /ρi Ji = −∇ · H + Dt i=1
i=1 d where Dv/Dt = v∇ · v. The term = σ¯¯ : ∇v 0 (in −3 −1 J m s ) is the component of the total dissipation function associated with the dissipative deformation processes (see below, Section 2.5). Equation (47) states that the temperature of the system changes because of heat conduction, deformation effects, heating associated with irreversible deformation (e.g., viscous, plastic, or viscoplastic deformations), friction, and power supplied to the system. Equation (47) is written in a more familiar form as D (ρCv T ) = −∇ · H + QS , (48) Dt N m H≡H − (49) ui Jm i ,
+ σ¯¯ : ∇v + e
DS
i=1
∂P QS ≡ − T ∂T +
Q
∇ · v + DS + v
Fi · Jm i
Q
Jm ¯¯ : ∇v i · Fi − σ e
i=1
−
Q
(ui − T si )I¯¯ − σ¯¯ i /ρi ∇Jm i + ρT Π.
(53)
i=1
After long, but straightforward, algebraic manipulations, Eq. (53) can be recast as a local continuity equation for entropy, Ds = −∇ · Sm + Θ + ρΠ, Dt Q Hm m m si Ji , + S = T ρ
(54) (55)
i=1
T Θ = −Hm ∇ ln T − ui · Jm i
Q
si Jm i · ∇T
i=1
i=1
−
+ ρT Π,
(50)
i=1
where H is the heat flux (in J m−2 s−1 ) and QS term (in J m−3 s−1 ).
Q
is the heat source
Q
d ∇ · μ¯¯ i − Fi Jm ¯¯ : ∇v, i +σ
(56)
i=1
where Θ is the entropy source (inner entropy production per unit volume and unit time). 3. Dissipation function
2.5. The entropy balance equation (second principle) Prigogine [14] showed that the classical definition of entropy in thermostatics applies to a system in the vicinity of equilibrium. Different authors (e.g., [13]) agree that this assumption extends to a very wide class of phenomena, some of them being far from equilibrium. I denote by s the entropy per unit mass of the system. The entropy is related to the internal energy u, to the specific volume v, and to the mass fraction Ci according to the Euler relationship of thermostatics. In linear nonequilibrium thermodynamics, the Gibbs–Duhem relationship is assumed to hold, Du ¯ e Ds ¯¯ i : ∇Jm =ρ − σ¯ : ∇v − μ ρT i , Dt Dt Q
(51)
i=1
where the terms P (Dv/Dt) and μi (DCi /Dt) appearing in the classical formulation of the Gibbs–Duhem equation of Prigogine [14] and De Groot and Mazur [12] have been replaced e ¯¯ i : ∇Jm , respectively, to account for the by −v(σ¯¯ : ∇v) and μ i tensorial nature of the elastic stress and chemical potential tensors. The chemical potential tensor of species i is defined by [13] ¯¯ i = Ψi I¯¯ − σ¯¯ i /ρi = (ui − T si )I¯¯ − σ¯¯ i /ρi , μ
(52)
In Section 2, the results developed by De Groot and Mazur [12] for a viscous fluid were generalized to a multi-component system including the influence of the full stress tensor and both elastic and nonreversible deformations. These equations were written in a Lagrangian framework attached to the barycentric center of mass of the system, which comprises Q components. These components are (1) N components located into the pore water (anions, cations, and neutral species), (2) the pore water itself (i.e., the neutral solvent), and (3) the grains (so Q = N + 2). The balance equation for the internal energy u for an open porous medium (first principle) is given by Eq. (37) with Eqs. (28) and (38). The local balance law for entropy (second principle) for an open system is given by Eqs. (54) to (56). The stress tensor is divided into a reversible elastic component (superscript e) and a dissipative component (superscript d) corresponding to irreversible mechanisms of deformation; i.e., e d σ¯¯ = σ¯¯ + σ¯¯ [13]. For an open porous material, the entropy production can be divided into two contributions, Ds/Dt = (Ds/Dt)e + (Ds/Dt)i [15]. This expression is just an alternative form of the entropy balance equation; δse is the entropy exchanged between the system and its surrounding and δsi is the entropy produced in the system itself. The term (Ds/Dt)e can be either positive or negative, while Θ = ρ(Ds/Dt)i 0,
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
the inner entropy production of the system, is necessarily positive. In linear nonequilibrium thermodynamics, the total dissipation of the system is equal to the product of the temperature with the entropy production term, D = T Θ, and D 0 because Θ 0 and T > 0. This yields D = −Hm ∇ ln T −
Q
si Jm i · ∇T
i=1
−
Q
∇ · μ¯¯ i − Fi Jm ¯¯ : ∇v, i +σ d
(57)
i=1
D = −Sm ∇T −
Q
S ∇ · μ¯¯ i − Fi Jm i +D ,
(58)
i=1
where I have replaced the heat flow (divided by the temperad ture) by the entropy flux and where D S + σ¯¯ : ∇v represents the dissipation component associated with irreversible deformation processes (e.g., with viscoplastic deformation). This dissipation D S will be described in detail in a forthcoming paper. In Section 2, I derived a general formulation for the two principles of thermodynamics in a Lagrangian framework attached to the barycentric center of mass of the system. Therefore all the components are treated in a symmetrical fashion and the equations have been stated in the framework associated with this barycentric center of mass. However, a practical Lagrangian framework to interpret laboratory measurements is the one associated with deformation of the skeleton (e.g., [1]). This yields a new Lagrangian derivative associated with the velocity of the solid phase (subscript s) according to ∂ d = + vs · ∇. (59) dt ∂t The pore fluid (solvent plus ions and dissolved neutral species) is written with the subscript f. The fluid is assumed to be ideal, which implies that the peculiar stresses of the components of the pore electrolyte are assumed to satisfy the Dalton’s law of ideal mixtures, i.e., σ¯¯ i = −pi I¯¯, where pi is the partial pressure of component i. I write as p the pore fluid pressure of a reservoir in contact with the charged porous medium and as p¯ the fluid pressure of the pore fluid in the charged porous medium (e.g., [11]). The pore fluid pressure p is defined thermodynamically as the pressure of an external fluid in local contact with the system and in thermodynamic equilibrium with it [11]. A direct consequence of the previous assumptions is that the chemical potential of the components of the pore water are considered as scalars as discussed further below. I denote as φρi , φρw , and (1 − φ)ρs the partial mass densities of component i, solvent, and solid phase, respectively (φ is the porosity). I assume that the forces acting on both the pore fluid and the mineral matrix are always in balance and thus the acceleration of the barycentric center of mass is neglected. Obviously the time scale to reach mechanical equilibrium is always much shorter than the time scale to reach the (thermostatic) equilibrium state. This mechanical balance assumption yields
259
ρF + ∇ · σ¯¯ = 0 (Appendix A). Prigogine’s theorem states that in the mechanical equilibrium state, the barycentric velocity v entering into the definition of the diffusion flow in the dissipation function can be replaced by another arbitrary velocity. This theorem is extremely important for determining the dissipation function in the newly defined Lagrangian framework. In this mineral matrix framework, the measurable fluxes are defined by Ji ≡ ρ(±) φ(vi − vg ) = Jm i − ρi φ(vg − v),
(60)
Jw ≡ ρw φ(vw − vg ) = Jm w ρw φ(vg − v), m 0 = Js − ρs (1 − φ)(vs − v), S = Sm − ρs(vs − v), d d d σ¯¯ : ∇vs = σ¯¯ : ∇v − (vs − v)∇ · σ¯¯ ,
(61) (62) (63) (64)
where the subscripts i correspond to the component i, respectively, the subscript w corresponds to water, and the subscript g corresponds to the mineral grains. The theorem of Prigogine can be generalized by combining the Gibbs–Duhem relationship, the assumption of mechanical equilibrium, and the dissipation function written in the new Lagrangian framework as shown in Appendix A. The peculiar body forces per unit mass entering the dissipation function, see Eq. (58), include the electrical force plus the gravity force. For the components i dissolved in water, i ∈ (1, . . . , N), Fi = −qi ∇ψ/ρi + g ≈ −qi ∇ψ/ρi ,
(65)
while for water Fw = g. In these equations, qi is the charge of species i, qi = (±e)Zi , e is the elementary charge (1.6 × 10−19 C), and Zi is the valence of species i. Note that both forces are derived from a scalar potential as required by the theory developed in Section 2 (the gravitational acceleration is related to the gravitational potential ϕ = gz by g = −∇ϕ, z positive upward). The effective potentials per unit mass of the ions and water are scalars defined by μˆ i = μR i + μi + qi ψ/ρi + gz,
(66)
μˆ w = μR w
(67)
+ μw + gz,
R R respectively, where μR i , μw , and μg are the chemical potentials in a stress-free reference state [11]. Combining Prigogine’s theorem (see Appendix A) and the definition of the effective chemical potentials yields a new form of the dissipation function,
D=−
N
Ji ∇ μˆ i − Jw ∇ μˆ w − S∇T + D S .
(68)
i=1
I use now Curie’s principle (see [16]) to set up the form of the linear constitutive equations for the fluxes. This principle states that for an isotropic system, the coupling of tensors, whose orders differ by an odd number, does not occur. From this principle, it follows that I must consider separately scalars, vectors, and second-rank tensors in the dissipation function. I will focus below only on the vectorial fluxes entering the dissipation function. The vectorial contribution to the dissipation function
260
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
is D=−
N
Ji ∇ μˆ i − Jw ∇ μˆ w − H∇T /T ,
(69)
i=1
where I have replaced the entropy flux S by H/T , where H is the heat flux, including a convective contribution. The seepage velocity U is U=
N
Ji /ρi + Jw /ρw ,
(70)
i=1
ρw U ≈ J w .
(71)
Note that the definitions of the Darcy velocity and the heat flux are not unique. The gradients of the effective chemical potentials of the pore water and ions are kb T ∇ p¯ ∇ ln C¯ w + − g − sw ∇T , ρw ρw kb T qi ∇ μˆ i = ∇ ln C¯ i − E − si ∇T , ρi ρi ∂ μˆ i , si ≡ ∂T Ci ,p,ψ ∇ μˆ w =
(72)
and two textural parameters, the formation factor and the permeability. The continuity equations can be written as a function of volumetric fractions per unit initial volume and therefore involved the determinant J of the deformation gradient tensor discussed in Section 4. The (N + 2) generalized continuity equations associated with the N -form of the constitutive equations are ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ Ji ρi φ ρ i Ri d ⎢ ··· ⎥ ⎢ ··· ⎥ ⎢···⎥ J∇ · ⎣ ⎦ = − ⎣ (78) ⎦, ⎦+⎣ Jw ρ w Rw ρw φ dt H QS ρCv T where Ri and Rw are the production rates of ions i and water, respectively, within the controlled volume per unit mass of the porous medium (in mol m−3 s−1 ) and QS is here the intrinsic heat source (bulk rate of heat production, expressed in J m−3 s−1 ); see Eq. (50). 4. Mechanical equations
(73)
4.1. Generalized power balance equation
(74)
In order to discuss the influence of the reversible (elastic) component of the stress tensor in the present model, it is necessary to first derive a generalized power balance equation for the specific Helmholtz free energy of the porous composite. Taking the conservation equation for the internal energy and for the entropy of the system in a Lagrangian framework attached to the barycentric center of mass of the system yields Du Ds ρ −T = −∇ · Wu − T Sm + σ¯¯ : ∇v Dt Dt (79) − Sm ∇T + ρT ΠT − D, N N σ¯¯ i m ¯ m μ¯ i Ji , Ψi I¯¯ − Ji = Wu − T Sm = (80) ρi
where p is the pore water pressure of a reservoir locally in equilibrium with the pore water of the porous medium, E is the electrical field, and si is the specific entropy of pore fluid component i. The swelling pressure π is defined as π ≡ −kb T ln C¯ w and p¯ ≡ p +π −ρb gz (in Pa) is the effective pore fluid pressure. Therefore the gradients of the effective chemical potentials of the water and ions per unit mass are given by ∇ μˆ w = ∇p/ρw − sw ∇T ,
(75)
∇ μˆ i = ∇ μ˜ i /ρi − si ∇T ,
(76)
respectively, and μ˜ i represents the electrochemical potentials of species i per unit volume. The linear nonequilibrium thermodynamic theory implies linear relationships between the flows and the generalized driving forces that cause these flows. This yields a form of the constitutive equations that I call “the N -form,” ⎡ ⎤ ⎤ ⎡ Ji ∇ μˆ i ⎢···⎥ ¯ ⎢ ··· ⎥, (77) ⎣ ⎦ = −N¯ ⎣ ⎦ Jw ∇ μˆ w H ∇T /T where N¯¯ is an (N + 2) square matrix of material transport properties. Inserting the linear phenomenological equations within the dissipation function shows that the entropy production is a homogeneous quadratic function of the independent forces entering Eq. (77). The second law of thermodynamics states that this expression is definitely positive (D 0). This yields Nij = Nj i (Onsager’s reciprocity) (N¯¯ is symmetric), Nii 0, and Nij2 Nii Njj (i = j ) [15]. By volume-averaging the local Nernst–Planck and Stokes equations, Revil and Linde [11] provide the necessary relationships between the material properties entering the matrix N and both the constituent properties
i=1
i=1
where the total Helmholtz free energy and the specific Helmholtz free energy per unit mass of the component i are defined by Ψ = u − T s and Ψi = ui − T si , respectively. This yields the generalized power balance equation N
DΨ DT m ρ μ¯¯ i Ji + σ¯¯ : ∇v +s = −∇ · Dt Dt i=1
− S : ∇T + ρT ΠT − D, m
(81)
which is the fundamental equation required to formulate the thermoporoelastic constitutive equations. The term dΨ + s dT represents part of the energy externally supplied to the porous medium and stored (if dΨ + s dT > 0) or extracted (if <0) during the time interval dt and converted in heat. Introducing the expression of the dissipation function, Eq. (58) into Eq. (81), yields N DΨ DT e ρ (82) ¯¯ : ∇v. μ¯¯ i : ∇Jm +s =− i +σ Dt Dt i=1
The Helmholtz free energy corresponds to the potential associated with the entropy of the porous system.
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
4.2. Mechanical constitutive relationships In the following, I assume that the porous medium has a thermoporoelastic behavior. In addition, I assume small deformations for which the whole theory is linear and obeys the superposition principle. I denote as Ω0 (the volume bounded by the surface ∂Ω0 ) the Lagrangian reference configuration for the porous medium, as X the vector locating an element of the porous grain skeleton in this reference configuration, and as Ω(t) the Eulerian configuration at time t and as x(X, t) the position of the same material element at time t in the Eulerian configuration. The spatial position of the material element is given by x(X, t) = X + u(X, t), where u is the solid displacement. Deformation of the skeleton is characterized by the deformation gradient tensor P¯¯ by dx(X, t) ≡ P¯¯ dX. The Green–Lagrange strain tensor e¯¯ and the second Piola–Kirchhoff stress tensor T¯¯ (both symmetric) are defined by (e.g., [2,4]) 1 e¯¯ ≡ P¯¯ T P¯¯ − I¯¯ , 2 −1 T¯¯ ≡ J P¯¯ −1 σ¯¯ P¯¯ T ,
(83) (84)
dV J ≡ det P¯¯ = , dV0
(85)
where P¯¯ T represents the transposed matrix of P¯¯ , A¯¯ = P¯¯ T P¯¯ is the right Cauchy–Green tensor, B¯¯ = A¯¯ −1 = P¯¯ −1 P¯¯ T is the Piola deformation tensor, and J denotes the Jacobian of the deformation gradient tensor (J > 0) and is also equal to the ratio between the volume of the elementary volume in the current configuration dV divided by the volume this element occupied in the reference state dV0 . Tension is considered positive. In addition, Euler’s formula is (e.g., [3]) dJ = J ∇ · vs . (86) dt From Eq. (81), written now in the Lagrangian framework defined in Section 3, I have d(ρΨ ) + ρΨ ∇ · vs dt N+1 dT e = Tr(σ¯¯ ∇vs ) − μi ∇ · Ji − ρs , dt
(87)
i=1
where the sum over i is extended to the N + 1 components of the pore water (N components plus the solvent, i.e., the water molecules). Using Eq. (86), I obtain d(ρΨ ) d(JρΨ ) =J + ρΨ ∇ · vs . (88) dt dt The mass balance equation for species i is dmi = −J ∇ · Ji , dt where mi = J φρi . Equations (87) to (89) yield ¯ N+1 dmi d(JρΨ ) de¯ dT μi = Tr T¯¯ + − Jps . dt dt dt dt i=1
(89)
261
The time derivative of the Helmholtz free energy per unit mass of the pore fluid, ρf Ψf , is obtained using the chain of algebraic manipulations Ψf = −pvf +
N+1
μi C¯ if ,
(91)
μi ρi ,
(92)
i=1
ρf Ψf = −p +
N+1 i=1
d(ρf Ψf ) dp dρi dμi μi ρi =− + + , dt dt dt dt N+1
N+1
i=1
i=1
(93)
where vf is the specific volume of the pore fluid in the porous medium and C¯ if = ρi /ρf is the mass fraction of pore fluid component i. The Gibbs–Duhem relationship in the pore fluid is ρf sf δT − δp +
N+1
ρi δμi + Q¯ V δ ϕ¯ + ρf gδz = 0,
(94)
i=1
where sf is the entropy per unit mass of the pore fluid, Q¯ V is the net excess of electrical charge in the connected porosity of the porous medium (e.g., [6], [11], and [17]), and ϕ¯ is the local electrical potential (position-dependent) in the pore fluid of the body. The osmotic pressure in the connected porosity of the porous continuum is given by (e.g., [6,17]) ϕ¯ π=
¯ V dϕ¯ . Q
(95)
0
This effective pressure includes gravitational and osmotic effects. In addition, state equations for the density and entropy per unit mass of the pore fluid are (e.g., [2]) 1 1 = 0 1 − βf (p¯ − p¯ 0 ) + αf θ , ρf ρ f Cpf p¯ − p¯ 0 sf − sf0 = θ − αf , T0 ρf0 1 1 ∂ρf ≡ , βf ≡ Kf ρf ∂ p¯ T 1 ∂ρf αf ≡ − , ρf ∂T p Cvf − Cpf =
T0 αf2 ρf0 βf
(96) (97) (98) (99) (100)
,
where θ = T − T0 , Cpf and Cvf are the fluid mass heat at constant pressure and volume, respectively, βf is the isothermal bulk compressibility of the pore fluid, and αf is the cubic thermal dilatation coefficient of the pore fluid. I write now the Gibbs– Duhem relationship, Eq. (94), in a more customary form, ρf sf δT − δ p¯ +
N+1
ρi δμi = 0,
(101)
i=1
(90)
dT dp¯ dμi ρi − + = 0. dt dt dt N+1
ρf sf
i=1
(102)
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A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
This yields N+1 dT d(ρf Ψf ) dρi μi = + ρf sf . dt dt dt
(103)
i=1
I denote as vp = J φ the specific pore volume of the porous medium in the deformed state per unit referential volume. I obtain dvp d(J φρf Ψf ) d(ρf Ψf ) = vp + ρf Ψf , dt dt dt
N+1 dρi dT d(J φρf Ψf ) μi = vp + ρf sf dt dt dt i=1
N+1 dvp + −p¯ + μi ρi , dt
(104)
(105)
i=1
and in addition, I have N +1
μi
i=1 N +1 i=1
N+1 d(vp ρi ) dmi μi = , dt dt
(106)
N+1 N+1 dvp dmi dρi μi ρi μi v p = + . dt dt dt
(107)
i=1
μi
i=1
i=1
I now determine the free energy of the mineral matrix per unit initial volume of the porous medium, W . This quantity is obtained by removing from the free energy of the saturated porous medium the free energy of the pore fluid. It corresponds to the elastic energy of the porous body. This yields W = J (ρΨ − φρf Ψf ) and therefore ¯ dvp de¯ dT dW = Tr T¯¯ + p¯ − J (ρs − φρf sf ) . (108) dt dt dt dt I assume that the potential W depends only on the external variables e, ¯¯ vp , and T , i.e., W = W (e, ¯¯ vp , T ). The differentiation of W yields deij ∂W ∂W dW = + dt ∂eij vp ,T dt ∂vp eij ,T dvp dT ∂W × + , (109) dt ∂T eij ,vp dt ∂W T¯¯ = (110) , ∂ e¯¯ vp ,T ∂W p¯ = (111) , ∂vp eij ,T ∂W S − mf sf = − (112) , ∂T eij ,vp where S ≡ Jρs is the specific entropy per unit referential volume and mf ≡ Jρf φ = vp ρf is the specific mass of water per unit referential volume. In linear thermoporoelasticity, this yields the following constitutive equations for charged porous materials: mf ¯¯ (113) T¯¯ − T¯¯ 0 = 4 C¯ : e¯¯ − M B¯¯ 0 − Aθ, ρf
mf p¯ − p¯ 0 = −M B¯¯ : e¯¯ + M 0 + ρf0 Lθ, ρf ¯ S − S0 − mf sf0 = A¯ : e¯¯ − Lmf + Cv θ/T0 .
(114) (115)
Due to the global electroneutrality of the porous medium, there are no electrostatic terms in the constitutive equations, except for the osmotic pressure, and therefore I recover the classical form of the mechanical constitutive equations given by [2]. The terms T¯¯ 0 , p¯ 0 , and S0 characterize the thermostatic state. If we consider a stress-free thermostatic configuration, T¯¯ 0 = 0. I denote as 4 C¯ the (fourth-order) stiffness tensor of undrained isothermic elastic moduli, A¯¯ and B¯¯ are symmetric second-order tensors, M is the Biot modulus, and Cv is the volume heat capacity per unit of initial volume in an isodeformation undrained experiment. They are defined by ∂Tij ∂Tkl = , Cij kl ≡ (116) ∂ekl mf ,T ∂eij mf ,T 0 ∂ p¯ , M ≡ ρf (117) ∂mf eij ,T ∂Tij Bij ≡ − (118) , ∂ p¯ eij ,T ∂Tij , Aij ≡ − (119) ∂T eij ,mf ∂S 0 L ≡ sf − (120) , ∂T eij ,mf ∂(S − mf sf0 ) Cv ≡ T0 (121) . ∂T eij ,mf The tensor T0 A¯¯ is the tensor of undrained strain latent heats and T0 L represents the isodeformation latent heat of variation in fluid mass content. The tensor B¯¯ is the Biot stress tensor, which reduces to the Biot coefficient ζ in the isotropic case. In the infinitesimal transformation assumption and for isotropic cases, the Green strain tensor e¯¯ is replaced by the infinitesimal strain tensor ε¯¯ and the Piola–Kirchhoff stress tensor T¯¯ can be replaced by the Cauchy stress tensor σ¯¯ . Thus, the constitutive equations of linear thermoporoelasticity of charged porous materials are 2 mf δij σij = K − μ εkk + 2μεij − ζ M 3 ρf0 − Kαb θ δij , p¯ − p¯ 0 = −ζ Mεkk + M
mf ρf0
(122)
+ Cv θ,
S − S0 − ρf φsf0 = αb Kεkk − αm M
mf ρf0
(123)
+
Cv θ, T0
(124)
where K is the undrained isothermal bulk modulus, μ is the shear modulus (which remains the same for a drained or an undrained experiment), αb is the cubic undrained thermal dilatation coefficient, ζ is the Biot coefficient, and αm T0 represents the cubic isodeformation latent heat of fluid pressure variation, αm = (ζ − φ0 )αg + φ0 αf (Coussy [2, p. 91]). The
A. Revil / Journal of Colloid and Interface Science 307 (2007) 254–264
stress tensor obeys the mechanical equilibrium condition discussed in Appendix A. Appropriate boundary conditions can be used to close the system of equations.
d ¯¯ s − Fs − ∇ · σ¯¯ , + ρw φ(∇μw − Fw ) + ρs (1 − φ) ∇ · μ
In the present work, the constitutive equations of transport derived in [11] are encapsulated into a more fundamental thermodynamic theory. This theory accounts for the reversible and irreversible components of the deformation of the porous skeleton and is applicable to a multicomponent electrolyte filling a charged porous material. We paid special attention to small thermoporoelastic deformation, as we are mainly interested in deformation effects associated with the transport of ions and water. The next step will be to show that in such a system, any thermodynamic disturbance is the source of electromagnetic signals recordable away from the source using a network of very sensitive nonpolarizing electrodes and magnetic sensors (see [6] and [18], for example). These signals can be inverted to recover the transport properties of the medium using algorithms derived initially in electroencephalography. Such an application will be presented in a forthcoming work. In this paper, we have neglected several mechanisms of transport like (a) the hydrodynamic dispersion of heat and solutes through the porous material, (b) the transition between viscous to inertial laminar flow at high Reynolds numbers, and (c) free convection of the pore fluid. All these aspects require specific developments that will be incorporated into future works, including nonelastic contributions to deformation.
(A.2) N (∇μi − Fi )Ji − (∇μw − Fw )Jw D s ≡ −S∇T − i=1 d + σ¯¯ : ∇vs ,
(A.3)
Ds
where is the total dissipation function written in the Lagrangian framework associated with deformation of the solid phase. In order to demonstrate the validity of Prigogine’s theorem, D m = D s in Eq. (A.2), I can use the force balance equation written under the assumption of mechanical equilibrium, N
ρi φFi + ρw φFw + ρs (1 − φ)Fs + ∇ · σ¯¯ = 0,
(A.4)
i=1 e d and where σ¯¯ = σ¯¯ + σ¯¯ . In addition, the Gibbs–Duhem relationship of thermostatic yields N
e ρs∇T + ∇ · σ¯¯ + φ ρi ∇μi + ρw ∇μw i=1
¯¯ s = 0. + ρs (1 − φ)∇ · μ
(A.5)
Equations (A.4), (A.5), and (A.2) yield immediately to Prigogine’s theorem. I now discuss the form of the force balance equation. Using Eqs. (21) and (A.4), the force balance equation is
Acknowledgments This work is supported by the CNRS (The French National Research Council), ANDRA (The French National Agency for Radioactive Waste management), the GDR FORPRO (Research action Contribution FORPRO-2006/12A), and ANR ECCOPNRH (Project POLARIS). I especially appreciated the supports from Scott Altmann, Daniel Coelho, and Joël Lancelot and I thank the two referees for their constructive comments and Dominique Gibert for fruitful discussions.
The dissipation function written in a Lagrangian framework attached to the barycentric center of mass motion is
m (∇μi − Fi )Jm i − (∇μw − Fw )Jw
i=1
d ¯¯ s − Fs Jm − ∇ ·μ ¯¯ : ∇v. s +σ
ρF + ∇ · σ¯¯ = 0.
(A.6)
Using Eqs. (21) and (65) and the electroneutrality condition of the whole system, ρq = 0, where q is given by Eq. (13), I obtain, ρg + ∇ · σ¯¯ = 0.
(A.7)
References
Appendix A. Extension of Prigogine’s theorem
N
I now write the fluxes in the new Lagrangian framework associated with deformation of the grain skeleton (superscript s), N m s D = D − (vs − v) ρs∇T + ρi φ(∇μi − Fi ) i=1
5. Concluding statements
D m = −Sm ∇T −
263
(A.1)
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