A Theoretical Framework for Modeling the

5 downloads 0 Views 3MB Size Report
ing seepage, and pressure solution. The new theory can be ... chanical mathematical modeling capacities that are able to address realistically the reactive multispecies ..... According to the Bowen (1982) definition, the total chemical potential ...
Original Research

A Theoretical Framework for Modeling the Chemomechanical Behavior of Unsaturated Soils Changfu Wei*

Physicochemical effects remain elusive in the theory of porous media. A continuum theory is presented for modeling the chemomechanical behavior of unsaturated soils with chemomechanical coupling. The new theory effectively explains many salient phenomena occurring in soils and describes the multiple coupled processes in the vadose zone.

A theoretical framework is presented for modeling the chemomechanical behavior of multiphase porous media, in general, and unsaturated soils, in particular, which can address skeletal deformation, fluid flow, heat conduction, solute diffusion, chemical reaction, and phase transition in a consistent and systematic way. A general expression is derived for the electrochemical potential of a fluid species with explicitly accounting for the effects of osmosis, capillarity, and adsorption. The equilibrium behavior of porous media is investigated, and the composition of pore water pressure is identified. Explicit formulations are developed for the effective stress and intergranular stress, with consideration of physicochemical effects. It is shown that the negative water pressure measured by a conventional transducer can be significantly different than the true pore water pressure. It is also theoretically revealed that, other than the soil water characteristic function, a new pressure (or potential) function accounting for the physicochemical effects is generally required in analyzing the coupled chemomechanical processes in unsaturated soils. The new theory is capable of effectively explaining many salient phenomena occurring in water-saturated porous media with a degree of saturation varying from an extremely low value to 100%, including Donnan osmosis, capillary fringe, air entry value, initial hydraulic head during seepage, and pressure solution. The new theory can be used to analyze the multiple coupled physical and chemical processes in the vadose zone. Abbreviations: AEV, air entry value; PWP, pore water pressure; REV, representative volume.

Changfu Wei, State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil mechanics, Chinese Academy of Sciences, Wuhan, Hubei 430071, P.R. China and College of Civil and Architectural Engineering, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China. *Corresponding author ([email protected]). Vadose Zone J. doi:10.2136/vzj2013.07.0132 Received 17 July 2013 Open access

© Soil Science Society of America

5585 Guilford Rd., Madison, WI 53711 USA. All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Chemomechanical behavior of porous media with multiphases and multispecies is of great interest in many diverse fields of science and engineering. To mention a few, this includes the industries of nuclear, hazardous, and municipal waste isolation; petroleum and gas extraction; technologies of methane gas hydrates exploitation; CO2 sequestration; and weak soil reinforcement, landform stability assessment, structural material durability, and weathering of rock masses. Understanding, controlling, and predicting the long-term effect of physicochemical processes on the mechanical performance of geomaterials is becoming an indispensable part of environmental impact assessment and performance assessment analysis. Thus, there is a clear need to develop comprehensive chemohydromechanical mathematical modeling capacities that are able to address realistically the reactive multispecies transport, multiphase flow, chemical reaction, phase transition, chemically induced deformation, and other related physicochemical processes in deformable soils. The coupling of multiple processes in porous media, including skeletal deformation, seepage, diffusion, and heat conduction, has been extensively investigated, and the poromechanic theory that is capable of describing these coupled physical processes has been very well developed (e.g., Coussy, 2004). Thus far, however, the behavior of porous media with multiphase and multispecies where physicochemical effects come into play remains elusive. Earlier research and practices were mainly concerned with reactive flow and transport, and numerous flow and transport models have been developed, without addressing mechanical issues (e.g., Spycher and Sonnenthal, 2003; Steefel et al., 2005; Xu and Pruess, 2001). With regard to the chemomechanical behavior of porous media, a few research studies have been performed with a focus on the chemoplasticity of fully saturated soils (e.g., Hueckel, 2002;

Vadose Zone Journal

Loret et al., 2004; Witteveen et al., 2013). In addition, the crossscale effect and multiprocesses coupling of multiphase systems has been an active field of research on experimental, theoretical, and numerical bases (e.g., Di Maio et al., 2002; Coussy, 2010). Over the past two decades, numerous efforts have been devoted to developing general theoretical frameworks for modeling multiphase and multispecies porous media. Among these, we mention the biological tissues-oriented theory of porous media developed by Huyghe and his coworkers (Huyghe and Janssen, 1997; Huyghe et al., 2004, 2007) and the hybrid theory of swelling porous media (Bennethum and Cushman, 1996a,b; 2002a,b; Bennethum et al., 1996, 2000; Bennethum, 2007, 2012). Through these efforts, general macroscopic governing equations have been developed that can be used to address the physicochemical and electrochemical coupling in multiphasic and multispecies systems. In addition, the significances and implications of pressures and potentials have been well addressed. Despite of this progress, a comprehensive theoretical framework is still lacking for modeling the chemomechanical behavior of unsaturated soils with variable saturation. Here our hypothesis is that when the saturation varies from 100% to a low extreme, the composition and concentration of a pore fluid are variable, resulting in intensive physicochemical effects in the soils. Indeed, for the unsaturated soils with low saturation, the pore water pressure (PWP) has not been very well defined, and in current practice, the methods for PWP measurement are used quite ambiguously, even without distinguishing the difference between the potential and the pressure. To overcome such difficulties, several fundamental issues have to be resolved, including: ʶʶ How do we reconcile the concepts of potentials and stresses (or pressures) in a unique framework that is used to address the coupled thermo-hydro-chemo-mechanical processes in unsaturated soils? ʶʶ How do we characterize the chemical potential of a species in the unsaturated soils with multispecies, where osmotic, capillary, and adsorptive effects are important? ʶʶ What is the explicit expression of the effective stresses when compositions and concentrations of pore water are variable? Solving the first problem depends largely on the answers to the last two issues. Based on fundamental thermodynamic principles and an averaging procedure, Nitao and Bear (1996) had rigorously derived mathematical formulations for the potentials of unsaturated soils and provided a complete set of governing equations for flow and transport processes in the soils. Although this framework describes the flow and transport processes in a multiphase system, it does not address the mechanical issues. As one of the fundamental concepts in the classical soil mechanics, the effective stress concept plays a crucial role in describing the mechanical behavior of saturated soils. After Terzaghi’s proposal

Journal Vadose Zone Journal

for it, intensive efforts have been made to extending the effective stress concept to unsaturated soil problems (e.g., Bishop, 1959; Kohgo et al., 1993; Khalili et al., 2004; Laloui and Nuth, 2009). Perhaps the most commonly used formulation of the effective stress for unsaturated soils is the one proposed by Bishop (1959). This equation states that the effective stress of unsaturated soils is equal to the total stress minus the averaged pore pressure that is the average of pore air and pore water pressure weighted by factor c. The validity of Bishop effective equation have been recently examined by Gray and Miller (2007), Gray and Schrefler (2007), Gray et al. (2009), and Nikooee et al. (2012), based on the principles of thermodynamics and a local averaging procedure with explicit consideration of interfacial effects. These authors have shown that Bishop’s effective stress can be theoretically recovered only if the interfacial tension terms are neglected. In a somewhat intuitive way, Lu and Likos (2006) and Lu et al (2010) have proposed a new formulation for the effective stress of unsaturated soils, which includes a stress term called the “suction stress.” Remarkably, the suction stress accounts for the effect of surface tension, negative pore pressure, electrochemical interactions, and other factors related to unsaturated soils. The applicability of the suction stress concept has been validated based on the shear strength properties of unsaturated soils extensively collected from the literature. Thus far, however, a solid theoretical basis has yet to be built for this concept. On the basis of the approach of the mixture hybrid theory, a detailed derivation of a comprehensive theoretical framework is presented here that can be used to describe the chemomechanical behavior of unsaturated soils with saturation varying from 100% to a low extreme value. Within this context, the effective stress concept is examined, and the general expression for the chemical potential of a species is developed. To shed new insights into the unsaturated soil behavior, equilibrium behavior of the soils is investigated in detail.

6 6Thermodynamic

Constraints

Preliminary The unsaturated soils under consideration are viewed as the porous continua composed of a solid matrix (s) with interconnected pores saturated with two immiscible fluids, namely, a liquid (l) and a mixed gas (g). Unless otherwise specified, symbol a or b represents a bulk phase, i.e., a or b = s, l, g, and f denotes a pore fluid (i.e., l and g). For the mathematical convenience, it is assumed that every bulk phase is composed of the same set of electrically charged species, denoted by i = 1, 2, ..., Z, respectively. Let C a i be the mass phase a, a, i.e., i.e., C a i = r a i / r a , where concentration of species i in phase ai r is the partial mass density of the i species in the a phase and r a ( = Sr a i ) is the intrinsic mass density of the bulk phase. With this notation, if a species is absent in a bulk phase, its concentration

p. 2 of 21

is zero, i.e., C a i = 0 . For convenience, all the other variables are summarized in the Appendix. Formally, the following theoretical derivation follows the procedure adopted by Bennethum and Cushman (1996b) and Bennethum et al. (1996, 2000), with the assumption that the interfaces among bulk phases possess no thermodynamic properties. To avoid redundancy, only key assumptions and general results are given, and useful details of derivation can be found in the abovecited references. Nonetheless, for convenience of reference, the balance equations are summarized in the Appendix.

Constitutive Assumptions The derivation starts with the assumptions regard to the specific Helmholtz free energy densities of three bulk phases, namely, As, A l , A g , respectively. To this end, we introduce the principle of phase separation (Passman et al., 1984), stating that the Helmholtz free energy of a phase depends solely on its independent state variables. Based on this principle, As can be assumed as a function of temperature T, volume fraction ns, intrinsic mass density r s, deformation gradient Fs and species mass concentration C s k (k = 1, 2, …, Z − 1), while A f as a function of T, n f, r f, and C f k (k = 1, 2, …, Z − 1). Here it is noted that among all C a i , only Z − 1 components are independent. As argued by Passman et al. (1984), use of the phase separation principle is justified by the fact that individual bulk phases are physically separated in porous media. In general, mass exchange may occur between the solid phase and the pore fluids. Thus, variables ns, r s, and Fs are independent (recalling the solid mass balance, Eq. [A2] in the Appendix). Kinematically, the variation of soil porosity can be additively decomposed into two components: one is due to the skeletal deformation and/or the solid grain compression and the other solely due to the mass exchange (via, say, dissolution and precipitation). As a consequence, variation of ns also has two contributions, one of which can be solely attributed to the mass exchange (denoted as nrs ). The evolution of nrs is governed by (see Appendix): D s nrs 1 = s Dt r

å

f =l , g

eˆsfs

[1]

where eˆsfs is the total mass exchange rate between the solid phase and pore fluid f.

Inclusion of Lagrangian strain tensor E s instead of deformation gradient Fs in Eq. [2] is due to the requirement of objectivity. Clearly, all the arguments in As and A f are independent. From Eq. [2] and [3], it is noted that the solid matrix is assumed to be elastic and the pore fluids are of Newtonian-type. Viscous effect of pore fluids can be addressed by assuming a dependence of A f on Ñv f. For simplicity, however, Ñv f is excluded in Eq. [3].

Constraints by the Second Law According to the second law of thermodynamics (see Appendix), it is necessary that ha =

¶A a ¶T

[4]

Z æ ln r s i t s = t s - p s 1- å ççç i ç mi i =1 è Z æ ln r f i t f =- p f 1- å ççç i è mi i =1 ç

ö÷ ÷÷÷1 ø

[5]

ö÷ ÷÷1 ÷ø

[6]

mˆ a k 1 = æn ö æ1 ö n ö æ1 m a k - m a Z -l ççç k - Z ÷÷÷1 + ççç u a k × u a k ÷÷÷1-ççç u a Z × u a Z ÷÷÷1 çè m k m Z ÷ø è 2 ø è2 ø

[7]

where h a is the entropy density; t s the intrinsic intergranular stress tensor, transited through grain–grain contacts; p a the thermodynamic pressure of a phase; 1 is the unit isotropic tensor with components of d ij (i.e., Kronecker’s delta); m a i the electrochemical potential tensor of a i species; mˆ a k the relative chemical potential of species a k; u a i (= v a i - v a ) the diffusion velocity; n i the valence number of species i; mi the molar mass of species i; and l is a Lagrangian multiplier, associated with the electrical neutrality. Variables t s, p a , and m a i are defined by ¶A s s t (F ) ¶E s

ts = rsFs

2

p a = (r a )

¶A a ¶r a

m a i = A a i 1-

tai ra i

[8] [9] [10]

Hence, it is proposed herein that s

A =A

s

(T , nrs , r s , E s , C s k );

A f = A f (T , n f , r f , C

fk

k = 1,2,..., Z -1

[2]

m a = A a 1-

); f = l , g ; k = 1,2,..., Z -1 [3]

Vadose Zone Journal Journal

According to the Bowen (1982) definition, the total chemical potential tensor of a bulk phase is given by ta ra

[11]

It is straightforward to prove that

p. 3 of 21

TLD = å

Z

ma = åC a i ma i

a

[12] ai

ai

ai

f

ai

s Z æ lC si n i ö÷ ÷÷ l = G s l - t m s - å ççç rs è mi ÷ø i =1 ç

[13]

Z æ lC si n i ö÷ ÷÷ l = G f l m s - å ççç ÷ø ç m i i =1 è

[14]

ai

 be the chemical potential tensor of a species. m  Let m related to m a i through ln i 1 mi

ai

can be

[15]

According to the classical definition of electrochemical potential (Atkins and dePaula, 2002), it is immediately apparent that multiplier l is equal to F x , where F is Faraday’s constant, and x is the local electric potential. Dropping all the diffusion terms, one can cast Eq. [7] into  m 1= m

ak

 -m

aZ

[16]

The residual dissipation inequality is given by Eq. [A46] in the Appendix. To address chemical reactions in the porous media, consider several stoicheiometric chemical reactions (including dissolution–precipitation between the two bulk phases) occur in the pores, through which the solid matrix and the pore liquid (water) exchange mass. These chemical reactions can be generally represented by å a å iZ=1 k aJ [a i ] = 0 , where [a i] represents the i molar concentration [mol/m3] of species a i, and k aJ represents i the reaction coefficients of the Jth reaction ( J = 1, 2, ..., M). If a species (say a i) does not participate in the reaction, k aJ = 0 ; i otherwise, k aJ i > 0 for a reactant, and k aJ i < 0 for a product. Let pˆ J be the rate of the Jth reaction, then

(

i

b¹a

Z

i

M

=-å k aJ i mi pˆ J

æ

b¹a

i =1 èb¹a

i

i

M Z ö÷ J J ÷÷÷ =-å å k a i m i pˆ ø÷ J =1 i =1

(

)

Z -1

where d sa = 1, if a = s; otherwise, d sa = 0; pmf = n f r f prs = n s r s

¶A f

[20]

¶n f

s ¶A s s s ¶A =r n ¶nrs ¶nr

[21]

pmf is the configurational pressure, which is sometimes called the swelling pressure for expansive soils (Bennethum and Weinstein, 2004; Bennethum, 2012); prs represents the effort (with the same unit as pressure) required to break the chemical bonds of the solid material in the non-deformed condition.

Thermodynamic Equilibrium Inequality [19] includes a set of independent variables {zm}with components ÑT, D s n f /Dt, v f,s , pˆ J and (na r a k u a k ) . It is straightforward to prove that the sufficient and necessary conditions for the porous media to attain equilibrium are na

å T qa = 0

[22]

p f − p s − p mf = 0

[23]

a

(p

f m

-p

f

)Ñ n é

Z

åå kaJ mi êêê i

ë

f

m

Z æ ö ln f f ÷ + å Tˆ f b - å ççç i Ñ(n f r i ) ÷÷ = 0 ÷ø çm b¹ f i =1 è i ai

:1

3

æ t s : 1 p s ö÷ ù ç ú + çç + r ÷÷÷d sa ú = 0 çç 3r s r s ÷ø úû è

[24] [25]

Ñ×(m a k - m a Z ) = 0

[18]

Equation [22] is the condition for thermal equilibrium, Eq. [23, 24, and 26] are the conditions for hydromechanical equilibrium, and Eq. [25] is the conditions for chemical equilibrium. Remarkably, for a system in equilibrium, all the thermal, hydromechanical, and chemical equilibrium conditions must be satisfied.

Substituting these two equation in Eq. [A46], one obtains the residual inequality as

Vadose Zone Journal Journal

)

[17]

J =1

a a = å ççç å eˆab + rˆa å eˆab ç

f

- å å éêÑ×(m a k - m a Z )ùú ×(na r a k u a k ) ³ 0 ë û a k=1

a i =1

a + rˆa å eˆab

s

) DDnt

[19] é Z æ öù ˆ f - çç ln i Ñ(n f r f i )÷÷ú × v f , s - å êê pmf - p f Ñn f + å T ÷ f b åç ú ÷øú ç b¹ f f êë i =1 è m i û é m a i : 1 æ t s : 1 p s ö ù üï M ì Z ï ï ÷ + å íåå k aJ i mi êê + ççç s + rs ÷÷d sa úú ïýpˆ J ÷ø ï a i=1 çè 3r 3 r J =1ï ê úû ïïþï ë ï î

Here, the diffusion terms including (u × u ) and (u Ä u ) have been neglected, since these terms are generally small compared to the magnitude of m a i . In addition, one has

ˆak

(

+ å p f - p s - pmf

i =1

 ai = mai m

ï a Z é a i æ a i 1 a i a i ÷ö a i ù ü ï na ì ï íq - å ê r çççm + (u × u )1÷÷× u ú ï ý×ÑT êë è úû ïï ï ø 2 T ï = 1 i î þ

[26]

p. 4 of 21

Several common results can be deduced from Eq. [25]. First, for a general chemical reaction occurring in pore fluids, d sf = 0, and Eq. [2.25] yields Z

åå k Jf i mi (m f : 1) = 0 i

f i=1

[27]

Now consider a species j that can exchange its mass among the solid, the pore liquid, and the pore gas. Symbolically, this exchanging process can be represented as [ s j ] Û [l j ] Û [ g j ] . For exchange [l j ] Û [ g j ] , one can assume k lJ =-k gJ = 1 , and it follows from j j Eq. [27] that lj

m :1 = m

gj

:1

[28]

and for [ s j ] Û [ f j ] ( f = l or g), setting k sJj =-k Jf j = 1 , one obtains m s i : 1 çæ t s : 1 prs ÷÷ö m f i : 1 + çç s + s ÷÷ = 3 3 çè 3r r ÷ø

[29]

An equation similar to Eq. [29] was derived by Bennethum et al. (1996, 2000), although the breaking energy term (i.e., prs / r s ) appears here for the first time. a

Let m j be the electrochemica l potentia l, and then a a m j = (m j : 1)/3 . Equation [28] states that the chemical potential of any species in a multiphasic system in equilibrium is continuous across the interfaces between bulk phases, which is consistent with the classical result of thermodynamics (Atkins and dePaula, 2002). Equation [29] is a new result, implying that the chemical equilibrium between the solid matrix and a pore fluid can be intervened by the intrinsic intergranular stress. In the left-hand side of Eq. [29], (m s i : 1)/3 accounts mainly for the elastic energy of the solid matrix, the second term is the elastic energy of the solid material, and the third term represents the energy required to break the chemical bonds of the solid material. ˆ f , from the With Eq. [24], eliminating the exchange terms, T fb linear momentum balance equations of pore fluids (Eq. [A6] in the Appendix), one can derive n f r f a f + n f Ñp f + pmf Ñn f Z é æ ln öù + å êê n f r f i Ñççç i ÷÷÷úú - n f r f b = 0 çè m i ÷øú i =1 êë û

[30]

where af is the acceleration of the fluid. Using the state equations for Gf, pf, , pmf and m f, Eq. [30] can be cast into Z

a f + å C f i Ñ× m f i +h f ÑT - b = 0 i =1

Vadose Zone Journal Journal

[31]

From Eq. [26], it follows that Ñ× m a k = Ñ× m a Z (k = 1, 2, ..., Z − 1). Noting that a f = 0 and ÑT = 0 at static equilibrium, one obtains Ñ× m f Z - b = Ñ× m f k - b = 0

[32]

Integrating Eq. [32] yields

m f i - sym[ b Ä ( x - x 0 )] = c1

[33]

where i = 1, 2, ..., Z; sym( ) represents the symmetrical part of a second-order tensor; x and x 0 are the spatial coordinates of the point of interest and the reference point, respectively; c is independent of spatial coordinate x. Clearly, at equilibrium, the sum of electrochemical and gravitational potentials is spatially uniform for any species.

Linear Dissipative Processes In analyzing the behavior of a porous medium, it is crucial to characterize its evolution process. It is assumed herein that during its evolution process, the porous medium deviates only slightly from thermodynamic equilibrium. With this assumption, and according to Inequality [19], it is assumed that Z é ù 1 q a - å r a i êm a i + (u a i × u a i )1ú× u a i =-k q ×ÑT ê úû 2 ë i =1

p f - p s - pmf = q f

D s n f Dt

[34] [35]

Z

æ

ö

i Ñ(n f r f )÷÷÷ =-x f × v f , s [36] ( pmf - p f )Ñn f + å Tˆ ff b - åçèççç ln m ø÷ b¹ f

Z

i =1

i

i

é ma i :1 æ t s :1 p s ö ù ÷ + ççç s + rs ÷÷d sa úú = V J pˆ J ç 3 r ÷ø úû è 3r êë

åå kaJ mi êê

[37]

Ñ×(m a k - m a Z ) =-za k ×(na r a k u a k )

[38]

a i =1

i

where k q, q f, xf, zJ, and za k (k = 1, 2, ..., Z − 1) are material coefficients. Here, for simplicity, all the cross effects have been neglected. Equation [34] is the generalized Fourier law for porous media with multiphase and multispecies; Eq. [35] describes the rate of the fluid volume fraction; Eq. [36] represents the constraint on the linear momentum exchange between bulk phases; Eq. [37] accounts for the mass exchange processes; Eq. [38] is the generalized Fick’s law for porous media with multiphase and multispecies. It is remarkable that the second law does not exert any control on ˆ f i . This result is the linear momentum exchange of species, i.e., T fb different from the previous results (e.g., Bennethum and Cushman,

p. 5 of 21

1996b; Bennethum et al., 1996, 2000). It is clear that within this context porous media are treated as highly reactive ones in the sense of Bataille and Kestin (1977). Substituting Eq. [34–38] in Eq. [19] yields Z -1

T L D = ÑT × k q ×ÑT + å å (na r a k u a k )× za k ×(na r a k u a k ) +å v f

f ,s

×x f × v

f ,s

a k=1 M

(

+ å V J (pˆ ) + å q f D n / Dt J =1

J 2

f

s

f

2

)

[39] ³0

Coefficient tensor k q, xf, and za k are symmetric and positively definite, and the scalar coefficients VJ and q f are non-negative.

6 Chemical

and Electrochemical Potentials

As discussed above, the chemical potential, m f i , (or electrochemical potential, m f i ) of the species in the pore fluids plays a crucial role in characterizing the behavior of a porous medium with multiphase and multispecies. In the following, general expressions of m f i and m f i are developed for the pore fluids.

Decomposition of the Specific Helmholtz free energy of pore fluids

In Eq. [3], A f is assumed to be a function of T, r f, and C f k (k = 1, 2, ..., Z − 1), as well as n f. The dependence of A f on n f accounts for the effect of the surface forces associated with interfaces, implying that a solution in the pores differs from that in the free bulk state in that the former is under the influence of the surface forces (Nitao and Bear, 1996). These surface forces include the surface tension on the interfaces and the adsorptive forces stemming from the physicochemical interactions among different bulk phases, including electrostatic forces, Van der Waals attraction, double-layer repulsion, and so on. In general, porous media (e.g., geomaterials) are electrically charged. Hence, it is expected that significant physicochemical interactions occur among mineral surfaces, water dipoles, and electrically charged species. These interactions can modify the potential energy of pore fluids, so that a fluid in the pores has a potential energy different from the fluid free of the surface forces at the same thermodynamic condition (temperature, mass density, and mass fractions). To gain insight into the character of the specific Helmholtz free energy defined by Eq. [3], we adopted a procedure by Nitao and Bear (1996) to analyze the effect of surface forces on the energy potential. Consider a representative volume (REV) of a pore fluid at point x, at temperature T, average mass density r f and mass fraction C f i . Consider also a reservoir R( x ) outside the porous medium domain containing the same fluid as the REV at the same thermodynamic conditions (T, r f, and C f i ). The reservoir is at the same reference elevation as the REV. It is important to note that, unlike the fluid in the reservoir, the pore fluid is subjected to the surface forces.

Vadose Zone Journal Journal

Now consider a system consisting of an infinitesimal amount of fluid in R( x ) . Let the system be transported, in a thermodynamically reversible manner, from R( x ) to the same sate as the REV. To maintain constant r f and T during the movement, the same amount of surface forces as in the REV has to be applied to the system so that the system (the transported fluid) has the same potential field as in the REV. As a consequence, the system changes by a pressure Dp f and amount of specific heat Dq (= TDh f ). In applying the surface force, some amount of work must be externally input to the system. According to the principle of energy conservation, under the condition of constant T and r f, the variation of specific internal energy, E f, of the system is given by DE f = −Dw f + TDh f, or equivalently, E f − Eˆ f = −Dw f + T(h f − hˆ f ), where Dw f is the specific work done against the surface forces, and Eˆ f and hˆ f are the specific internal energy and specific entropy of the system before applying the surface forces, that is, in R( x ) . Let W f be the surface energy potential, and W f = − Dw f < 0. Then, introducing the Legendre transformation (Eq. [A23] in the Appendix), one obtains A f (T , n f , r f , C

fk

) = Aˆ f (T , r f , C

fk

) +W f (T , n f )

[40]

where A f and Aˆ f are the specific Helmholtz free energies of the pore fluid in the REV and the fluid in R( x ) , respectively. Remarkably, function Aˆ f as defined is equal to the specific Helmholtz free energy of the system, i.e., the fluid in R( x ) . Hence, once Aˆ f is defined, as a function of the same T, r f , and C f i as in R ( x) , W f can be fully specified as a function of on T and nf only. Decomposition of energy potential has usually been exercised in addressing the effect of pore water films on the behavior of unsaturated porous media (e.g., Coussy, 2004, 2010). It is noted, however, that unlike the previous approaches (Coussy, 2004, 2010), here the surface energy potential W f is considered part of the free energy potential of the pore fluid, which is subjected to the surface forces.

A Generic Expression for Chemical Potential To derive a generic formulation of chemical potential, consider a species in the pore f luid, denoted as f i . Consider a small representative volume, dV, of a deforming porous medium, whose initial volume before deformation is dV0. Apparently, dV can be related to dV0 through dV = JdV0, where J is the Jacobian of the deformation gradient, F, of the solid matrix, i.e., J = det(F) > 0. Hence, the total specific energy of the fluid with regard to dV0 is Jn f r fA f. Let M f i be the specific mass of a species f i with regard to dV0, i.e., M f i = Jn f r f i . According to its very definition, the chemical potential of species f i can be defined as (e.g., Bennethum and Cushman, 2002b) fi

m =

(

¶ Jn f r f A f

(

¶ M

fi

)

)

= A f +r f f

J , n ,T

¶A f

¶r f i

[41] J , n f ,T

p. 6 of 21

equal to porosity n; pmf / r f is considered as a function of T and n f. Equation [48] can be recast into

Substituting Eq. [40] in Eq. [41], one obtains m f i = mˆ f i +W f

[42a]

where mˆ f i is the chemical potential of a species in the bulk fluid free of surface forces, and ¶Aˆ f mˆ f i = Aˆ f +r f ¶r f i

[42b] J , n f ,T ,...

According to its classical definition (Atkins and dePaula, 2002), chemical potential mˆ f i is explicitly given by mˆ f i (T , p f , C

fj

) = m Åf i (T , p f ) +

RT f ln a f i (T , p f , m j ) mi

[43]

where a f i is the activity of the species that is a function of T, p f, and molar fraction m f i ; R is the universal gas constant; m Åf i the chemical potential of the species f i at the pure state. Activity a f i is defined in such a way that it approaches m f i as m fi ® 0 for a solute species, and m f i ® 1 for the solvent. m f i is related to f C i through m fi =C

fi

é Z êm i å C ê j =1 ë

(

fj

ù -1 /m j ú ú û

)

[44]

Hence, the chemical potential can be expressed as m f i (T , p f , C

fj

,n f )= RT f m Åf i (T , p f ) + ln a f i (T , p f , m j ) +W f (T , n f ) mi

[45]

and the electrochemical potential is given by m f i (T , p f , C

fj

,n f )=

m Åf i (T , p f ) +

F xn i RT f ln a f i (T , p f , m j ) +W f (T , n f ) + mi mi

[46]

To gain more insights into the composition and character of W f, recall the state equation for pmf , i.e., Eq. [20]. Using Eq. [40], one obtains ¶W f ¶n f

=

pmf

[47]

nfrf

Integrating Eq. [47] yields W f (T , n f ) = W0f (T ) + ò

n n0f

f

é p f (T , n f ) ù ê m ú f ê ( n f r f ) ú dn êë úû

[48]

[49]

where

A f (T , n0f ) = W0f (T ) - ò

n0f

0+

C f (T , n f ) = ò

nf

0+

é p f (T , n f ) ù ê m ú f ê ( n f r f ) ú dn ëê ûú

[50]

é p f (T , n f ) ù ê m ú f ê ( n f r f ) ú dn ëê ûú

[51]

where 0+ is a small positive quantity, approaching zero. For the pore liquid, 0+ represents the smallest fluid volume fraction at which the porous medium is considered “dry.” Assuming the dependence of A f on n0f is to highlight the fact that the surface energy potential depends on the porosity only through the fluid volume fraction, n f, which is equal to n0f multiplied by the degree of saturation, Sf . Noticeably, W0f (T ) in Eq. [48] can be viewed as the total surface energy potential of the pore fluid fully occupying the pore space in the porous medium, and -W0f (T ) is equal to the work required to overcome both the adsorptive and capillary forces in transporting, in a thermodynamically reversible way, the fluid from a reservoir to fully saturate the porous medium at the same T, r f, and C f k . Because the integral in Eq. [50] represents the surface energy potential associated with the capillary forces, it is clear from Eq. [50] that A f is indeed the surface energy potential accounting only for the adsorptive forces in a fully saturated porous medium. As described by Eq. [49], the total surface energy potential has two contributions, i.e., A f and C f. In the literature (e.g., Tuller et al., 1999), the surface energy (or matric) potential is usually decomposed into two components, which accounts for the effect of adsorptive forces and the effect of capillary forces, respectively. As implied by Eq. [49], it is generally difficult, if not impossible, to clearly distinguish A f and C f, since both account for the effect of the adsorptive forces. Nevertheless, it is clear that only C f depends on the degree of saturation (through n f ) and accounts for the effects of both the adsorptive and capillary force, whereas A f is associated only with the effect of adsorptive forces in the porous media at full saturation.

6 Equilibrium

where W0f (T ) = W f (T , n0f ) ; n0f is the fluid volume fraction at full saturation, and for the wetting fluid, n0f is approximately

Vadose Zone Journal Journal

W f (T , n f ) = A f (T , n0f ) +C f (T , n f )

fi

Properties

f

Because m is a function of T, p f, n f, and C j , enforcing equilibrium equations, Eq. [27–29] and Eq. [33] may yield rich colligative properties such as osmotic pressure, undercooling, superheating, and Kelvin effect in porous media. These properties f describe the correlations among T, p f, n f, and C j . Because of

p. 7 of 21

its central importance, the Donnan osmotic phenomenon is discussed in detail in the following, In addition, a detailed analysis of different procedures for measuring the negative pore water pressure in unsaturated soils is also given.

Donnan Osmotic Phenomenon Consider a soil layer in the vadose zone, in which a groundwater observation well (denoted as W) is drilled down below the water table, as schematically shown in Fig. 1. The soil solution in the observation well keeps in contact, and in equilibrium, with the pore water in the soil. In the following, to Fig. 1. Schematic illustration of osmotic phenomena. Due to the electrically charged distinguish the water solutions in the well and in nature of soils, the equilibrium solution in the well is different from the pore water in that the soil pores, they are called “equilibrium solution” they have different concentrations. The steady-state pressure profiles in the right-hand side ll ) and the true highlight the difference between the equilibrium solution pressure ( pW and “pore water,” respectively. Suppose that the soil W ll). pore water pressure (p 33 has a fixed charge density of cfix fix [mol/m ], which represents the total number of electric charges m−, Y n+), one has fixed in a unit volume of the soil. For the purpose For ions g (= X X m− Y n+ of illustration, it is assumed that the fixed charges are negative and p AlAl - p0l0l RT llgg ll llgg llgg that both the soil solution and the pore water are composed of a ll ll T p C T p ( , , ) ( , ) ln a AAgg [57] m = m + + Å A 0 A Å A 0 A l m− n+ l m− n+ g g solvent (H22O) and two charged species (X m gg (X  and Y Y  ), whose r ÅÅ valence numbers are −m and n, respectively. ll ll m BBgg (T , p ll , C gg , n ll ) = Consider Point A in the equilibrium solution in the well, which [58] F x n gg [58] p ll - p0l0l RT ll ll m ÅÅgg (T , p0l0l ) + + ln a gg +W ll + is at the same elevation as Point B in the soil. The condition of llgg m gg m gg r ÅÅ electrical neutrality requires that in the equilibrium solution, ll

ll

nn+ +

m m-

nC AAYY n+ mC AAXX mc = = l0l0 mYY nn++ m XX mm-r AA

[52]

where c00 [mol/m33] is the total number of negative charges in a unit volume of equilibrium solution. In the pore water, ll

ll

m nn+ m+ c fix nC YY mC XX fix = + ll ll mYY nn++ m XX mm-nr

[53]

(

[54]

According to Eq. [46], the chemical potential of the solvents in the well and the soil are ll

ll

m AAHH22OO = m ÅÅHH22OO (T , p0l0l ) + ll

ll

m BBHH22OO = m ÅÅHH22OO (T , p0l0l ) +

( p AlAl - p0l0l ) ll

r ÅÅHH22OO ( p ll - p0l0l ) ll r ÅÅHH22OO

ll

nn

é

ll mm-- nn X X

mù ll nn++ m Y

ll

Y

[59]

where P (= p ll - p AlAl ) is the generalized osmotic pressure, and O(P, W ll) is given by nm XX mm--

(P +r

ll mm-- ll X X Å Å W

mmYY nn++

)+ RT r (P +r

ll nn++ ll YY Å Å W

)

[55]

O(P, W ll ) =

RT ll ln a HH22OO +W ll mHH22OO

[56]

By substituting Eq. [55] and [56] in Eq. [54], one can derive

where r ÅÅHH22OO is the mass density of pure water, which is assumed to be constant; p0l0l is the pressure of pure water at some reference ll state; m ÅÅHH22OO is the chemical potential of pure water.

Vadose Zone Journal Journal

mù ll nn++ m YY A A

) (a ) úûú = ln êëê(a ) (a ) úûú +O(P, W )

RT ll ln a AAHH22OO mHH22OO

+

+

ll

Using Eq. [54], [57], and [58], it is straightforward to prove that é ll mm-ln ê a AAXX ê ë

At equilibrium, according to Eq. [33], m lAlAii = m lBlBii , i = H 22O, X mm--, Y nn++

ll

where r ÅÅgg and m ÅÅgg are the mass density and chemical potential, ll respectively, of species g at the pure state. Here, r ÅÅgg is assumed to be constant. It is also noted that the local electric potential is zero in the well, where ions are uniformly distributed in the equilibrium solution.

mHH22OO ll

RT r HH22OO

ll mRT r ÅÅXX m-

(P +r

llH H22O O

æ a llHH22OO ö÷ ç W = ln çç AlAl ÷÷÷ çç a HH22OO ÷÷ è ø ll

)

ll nn++ YY Å Å

[60]

[61]

p. 8 of 21

Comparing Eq. Eq. [60] [60] and and [61], [61], one one can can see see that that O(P, O(P, W Wlll)) isis aa small small Comparing quantity, which which is negligible in in Eq. Eq. [59]. [59]. For For aaa dilute dilute solution, solution, quantity, quantity, which isis negligible negligible in Eq. [59]. For dilute solution, lll mmm--lll mmm--lll lll nnn+++ and aaaYYYnnn+++ » Hence,one one obtains ...Hence, obtains Eq. [60] and [61], one can see that O(P, W l) is a small and Hence, one obtains »m mAAAXXX ,,,and aaaAAAXXX » » m »m mYYY Comparing » m l) is a small Comparing Eq. [60] and [61], one can see that quantity, O(P, Wwhich is negligible in Eq. [59]. For a dilute solution, l l l l n+ m nnn nnn m m m m m X m X ml l l l l l l l l l l l quantity, Comparing which Eq. [60] and in [61], Eq. one can a» dilute that solution, , and W l)aisY an+small aFor m AO(P, » m Y . Hence, one obtains nnn+ m m m- is negligible m- [59]. + + + +see XXXm XXXm YYYnnn+ YYYA [62] [62] [62] m m » m m m m » m m m » m m l mlm l l mn+ A AA A A A Y n+ which is»negligible . Hence, [60] in Eq. and one [59]. [61], obtains For one can a dilute see that solution, O(P, W l) is a small a AX »quantity, m AX , and aComparing m Y Eq. l ml ml n+ l n+ n n m m X X Y Y l m- one l ma- dilute lobtains l solution, , and a which . Hence, in Eq. a A » m A quantity, » m is negligible n+ [59]. For X X Y n+ [62] ll+nnn++Y+ » m m l ml mlll lmm l lm m m n + n A A n n m m Y Y Y Xl X l m- This ThislYequation can be solved for and CY . Hence, provided that be ,,,provided ccc000,,, ,for andC onethat obtains This equation can be solved and C provided that acan »solved m C aXXXY and C » mC nequation + A A lY n+for X X m[62] m m » m Am m l ,,mand lare andn nnn lllare specified.l XAm- n lY n+ m ccfix n+ specified. X fix [62] l l n+ mfix mY » nm A m A n m l ml n+ m Thisl equation l can be solved for C X m- and C Y , provided that c0, mY n+ [62] l nA+X mX ml X Ym» m m A Remarkably, unless the soiland hasno no fixed charges (i.e.,cccc0fix zero), Remarkably, unless has charges (i.e., This equation can be solved forthe C cfixYfixed , and , provided n l are specified. that , isisiszero), Remarkably, unless the soil has no fixed charges (i.e., zero), C soil fix fix l n+ l mlllllliii l i i Y i X Care generally different than C . This phenomenon is usually isisisspecified. generally . This phenomenon is usually cfix, andThis nC equation can different be solved than for and C , provided that c 0, C generally different than C . This phenomenon is usually CC AA A l n+ l ml Y X called Donnan’s effect (Mitchell and Soga, 2005), which can be no fixed called effect (Mitchell and Soga, which can be cfix , andDonnan’s n areThis specified. equation can be solved Remarkably, for C2005), unless and the C soil , provided has thatcharges c 0, (i.e., cfix is zero), li li l attributed tocthe the existence of the fixed charges in the porous media. attributed existence of the fixed charges in the porous media. Remarkably, unlessto the soil , and has n no are fixed specified. charges C is (i.e., generally c is different zero), than C . This phenomenon is usually attributed to the existence of the fixed charges in the porous media. A fix fix li Theexample example discussed here represents a special case, in which only The discussed represents a special case, in which only C l i is generally Remarkably, different unless than thehere C soil . has This no phenomenon called fixed charges Donnan’s is (i.e., usually effect c is (Mitchell zero), and Soga, 2005), which can be The example discussed here represents a special case, in which only A fix li li m− n+ m− n+ two solutes (i.e., ionsX Xm− andSoga, Yn+ ) exist in the water solution and X  Y  (i.e., ions and YY ) exist in the water solution and called Donnan’s Ctwo issolutes generally effect Remarkably, (Mitchell different and than unless C the attributed . 2005), This soil has phenomenon which to no the fixed can existence charges be is usually of (i.e., the fixed c is charges zero), in the porous media. two solutes (i.e., ions X and ) exist in the water solution and A fix li li the soil (at Point B) is fully saturated. In a more general case, where the (at Point B) is saturated. In a more general case, where attributed called to soil the Donnan’s existence C effect of is generally thefully (Mitchell fixed charges different and Thein Soga, example than the porous 2005), C Adiscussed . This media. which phenomenon here can represents be is usually a special case, in which only m− n+) exist thediscussed waterto solution consists ofaeffect multiple species, can be expected the water solution consists multiple species, itititcan be The example attributed the called here existence represents Donnan’s of of the special fixed two (Mitchell charges case, solutes inin which (i.e., and the ions porous Soga, only X expected 2005), media. and Y which caninbethe water solution and the water solution consists of multiple species, can be expected m− n+)the that the mass concentrations ofinspecies species are different infully the pores that the mass concentrations of in the two solutes The (i.e., example ions X discussed attributed and Yhere to represents exist existence the theasoil water special ofare the (atdifferent solution Point fixed case,charges B) inand is which inpores only saturated. the porousIn media. a more general case, where that the mass concentrations of species are different in the pores llliii m− n+ ii than inB) the well; that is, C . than in the well; that . the soil (at two Point solutes is(i.e., The fully ions example saturated. X is, and discussed YaC ) the exist here general water represents in the solution case, water where a special solution consists case, and of multiple in which species, only it can be expected than in the well; that is, . CllliIn ¹ Cmore ¹ C ¹ C AAA m− n+ the waterthe solution soil (at Point consists twoB)solutes of is fully multiple (i.e., saturated. ions species, Xthat Initand the a can more mass Ybegeneral )expected concentrations exist case, in thewhere water ofsolution species are anddifferent in the pores li i Itfollows follows from Eq. [61] that It from [61] that that thethe mass water concentrations solution theEq. soil consists (at of Point species of multiple B)are is than fully different species, saturated. in theinwell; itthe can Inpores that abemore expected is, Cgeneral . where It follows from Eq. [61] that ¹ C Alcase, li li than in the thatwell; the mass that the is, concentrations of consists species are of multiple differentspecies, in the pores it can be expected Cwater ¹ Csolution A. l li lllHHH222iOO than in lthe that the is, mass concentrations . It follows of from species Eq. are [61] different that in the pores C ¹ C ll well; llthat l l l P= = ppp -pppAAA = =P PDDD-r -rÅÅÅ OW Wl A [63] P [63] P = = P -r W [63] li li It follows from Eq. [61] than thatin the well; that is, C ¹ C A . l It follows from Eq. [61] that P = p l - p Al = P D -r ÅH2O W l [63] l H2Donnan where PD is the Donnan osmotic pressure, given by P is the osmotic pressure, given by It follows from Eq. [61] that where P is the Donnan osmotic pressure, given by l where l l O D -r P = p - p A = PD W [63] D Å lH O l P = p l - p AllllHHH= [63] lll OOO 2ööö W æææ-r O OPD 222O RTrrrÅÅÅ RT RT ççççççalaaAAAHHH222Å l H2O Pl is the Donnan osmotic pressure, given by where l÷ ÷÷÷÷÷÷÷÷= D PDDD= = lnçpçç [64] P ln [64] P= p P -r W [63] P = ln [64] D Å ççç lllHHH OOOA÷÷÷÷÷ where P D is the Donnan given by mHHH222OOO osmotic m m Fig. Fig.2. 2.Relations Relationsamong amongP PDD andnnnlllin inaaaclayey clayeysoil soilsaturated saturatedwith with Fig. 2. Relations among P and in clayey soil saturated with ø èçèçèçaaa 222 ø÷÷ø÷÷pressure, l H2O ö l H2O fix æ D,,,ccc000,,,cccfix fix,,,and RTgiven r Å by çç a A ÷÷ where P D is the Donnan osmotic pressure, NaCl solution. (A) P vs. c for various cccfix under NaCl solution. (A) P vs. c for various underthe thefully fullysaturated saturated NaCl solution. (A) P vs. c for various under the fully saturated D 0 D 0 fix D 0 fix ÷ P = ln [64] l l H2O æ H2O ÷ö çç l H O ÷ D condition; condition; (B) (B) P PDD vs. nnnlll for for various various ccc000 at at cccfix = 100 100 [mol/m [mol/m333].].]. The The condition; (B) P vs. for various at = 100 [mol/m The RTEquation rÅ ç a[63] 2given Equation [63] general expression form the osmotic pressure in isisisaaageneral expression for the where P is the Donnan osmotic pressure, Equation [63] general expression for the osmotic pressure in fix D vs. fix = a pressure H 2osmotic O Fig. 2. Relations among ,equal c0, cfix, and nl in a clay èç[64] ø÷÷ byin PD = ln çç l HAl2O ÷÷÷ æ l HD O ö solutions are assumed as ideal, so that the activity of a species solutions are assumed as ideal, so that the activity of a speciesPisisisDequal solutions are assumed as ideal, so that the activity of a species equal 2 ç ÷ H O RT r a ç ÷ 2 porous media, implying that P has two contributions, namely, the media, implying that P has two contributions, namely, theFig. 2. Relations mporous NaCl PD vs. c0 for various cfix und among PD, c0, cfix, and nl in a clayey soilsolution. saturated(A) with to toits itsmolar molarfaction. faction. to its molar faction. PHD2O= èç aÅ lnø÷ çç Al ÷÷÷l H2O æ l H2O ö [64] condition; (B) PD vs. nl for various c0 at cfix = ççpressure ÷ H2Or ÷ RT a ç Donnan osmotic pressure (P ) and the pressure induced by the Donnan (P ) and the pressure induced by the Equation [63] is a general expression for the osmotic pressure in Donnan osmotic pressure (P ) and the pressure induced by the mHosmotic NaCl solution. (A) P vs. c for various c under the fully saturated l Å A ÷ ÷ a O ççD D D among 0 Fig. 2. Relations PD, c0, cfixfix, and solutions n in a clayey saturated with è lllHHH OOO ølll lnD 2 P = ÷÷ [64] aresoil assumed l for various 3 The as ideal, so that the activity 222 condition; (B) solution. PD vs. n(A) atvarious cfix = 100 [mol/m ç l Hporous surface -rexpression W2O)))...Figure Figure depicts the dependence ofP PDD (((D -r 2222Odepicts the dependence of Equationsurface [63] is forces aforces general for osmotic pressure implying in that has twoNaCl contributions, namely, thec0 among surface forces -r W Figure depicts the dependence of P PRelations cits under then].fully saturated l in a clayey Å Å ÅmW H D D vs. c0 for fix Fig. 2. P , c , c , and soil saturated with èç athe ø÷÷ media, to molar faction. fix solutionscondition; are assumed(B) as P ideal,vs.sonthat activity of ca0 species equal 3]. The l forthe llalin various c0vs.D at =various 100is[mol/m onccc000,,implying ,cccfix , and n in a clayey soil saturated with a NaCl solution (i.e., on , and n a clayey soil saturated with a NaCl solution (i.e., porous media, Equation [63] is that general P has two expression contributions, Donnan for the osmotic osmotic namely, pressure the pressure (P in ) and the pressure induced by the on , and n in a clayey soil saturated with a NaCl solution (i.e., D fix NaCl solution. (A) P c for c under the fully saturated D fix fix D 0 fix to its molar faction. l H2O l lll ideal, ll assumed solutionsppplare soxxxthat activity of ac species equal l for 3]. The ++ −− +,, Cl −). P vs. the nIn various cfixis 100 [mol/m =condition; +rasof (x(B) = pppAlAlAl +r bbb((P = +r xxD H22media, O,pressure NaEquation Cl(P ).DIt It)[63] canisP beahas seen that P could be significant, where derivation, use isis=made made of ii == H Na can be seen that P be significant, where derivation, use of Donnanporous osmotic implying that and the general pressure surface two contributions, expression induced forces (-r by for namely, the W osmotic ) . Figure the pressure 2 depicts inthe W dependence DCC 0 at AA W W faction. AC))) .. In Åthe D D 2O, D could tolllits molar l H2O l solutions are assumed as ideal, so that the activity of a species is equal lll lll... ) and l of depending on ,.ccFigure cfix and2(P and depending ,,,and nnndepicts surface forces Donnan (-rosmotic porous Wccc00)0,,pressure media, implying on thethe that c0dependence , cpressure P, has andtwo ninduced incontributions, P a clayey by soil the saturated namely, the solution (i.e., depending on and and =rrralllbbNaCl b(((xxx000) and pppAwith = xxxits = »rrrÅÅÅHHH222OOO.... rrrlll » » Å on AA A A A))molar D fix D fix fix to faction. l H2O l l l l +, Cl on c0, cfixsurface , and nlforces in a clayey Donnan (-r Åsoil Wosmotic saturated ) . Figure pressure with 2i depicts =aH NaCl (P the )Na solution and dependence the−(i.e., ).pressure It can of Pbe induced by the P D could be significant, where pW = p A +r b( x A - xC ) . In deriva 2O,D D seen that l H2O l l l l l H2O lll l l l + − l l l l =Dp A +r b( x A xC ) p.ppW Consider anpoint point (denoted ascould C) inaon the zone, differs from by amount amount of P. IfxaaaAconventional Clearly, Consider (denoted as in the Clearly, ppp differs from If i = H2O, on Na c0, c,fix Cl, and ). aIt can in abe clayey forces seen soil (that -rsaturated depending W )C) . with Figure be significant, NaCl 2 cdepicts cvadose solution the dependence nzone, (i.e., . where pW of P In derivation, use made and of r l » r Å . Consider asurface point (denoted as C) in the vadose zone, Clearly, differs from by amount of P. If conventional p A = r of b( xP. )conventional ÅP D W W by 0is0,vadose fix, and l l l l l l l + − l l H O -x ) 2 p = p +r b ( x whose vertical coordinate is x (Fig. 1). At equilibrium, transducer (say a tensiometer) is used to measure the “pore water isa clayey x that 1). Atbeequilibrium, (sayAar tensiometer) to measure use the is “pore water depending i whose = on H2cO, cNa and , Cl c0n,coordinate ).c.fixIt, and can nbeinseen soil P Dsaturated could with significant, a NaCl . In derivation, made of p Asolution = r where btransducer ( x(i.e., » r Å A . Cis used W) and 0 -xA C 0, vertical fix, on C C (Fig. l llliii +l llliii l H2Ol lll l l l p l +r −l differs p = b ( x x ) +bbb(((xxxAAon = m + b ( x x ) , where m assumes a form pressure” in a soil, the measured value is indeed equal to p .In In amount mmmlAlAliAii + = m + b ( x x ) , where m assumes a form pressure” in a soil, the measured value is indeed equal to p . depending i c=x0xx,000H c)))fix O, , and Na n , . Cl ). Consider It can be seen a point that P (denoted could be significant, as C) in the where vadose and zone, Clearly, . p . In derivation, from use by is made of of P + = m + b ( x x ) , where m assumes a form pressure” in a soil, the measured value is indeed equal to p . In p = r b ( x x ) r » r W A A C 0 W C 0 W C 0 W AC A 0 A C C C C C C 2 D Å llliii l H2O l lll l l a conventional l . vadose l differs similar to mmmdepending , x is the vertical coordinate of the groundwater the p is called the solution pressure” to following, “equilibrium similar to , x is the vertical coordinate of the groundwater the following, p is called the “equilibrium solution pressure” to Consider a point (denoted on as c C) , c in , whose and the n vertical zone, coordinate Clearly, is x (Fig. p 1). from At equilibrium, by amount “equilibrium of and transducer P. If (say . a tensiometer) is used to mea similar to , x is the vertical coordinate of the groundwater the following, p is called the solution pressure” to p = r b ( x x ) r » r W W A W 0 A BBB 000 0 fix C Å li lll ll ithe table,and and the gravitational acceleration. highlight the factfrom that can bepressure” measured using conventional table, bbbathe gravitational itititxisisis0vadose straightforward fact that pused be measured using aconventional whose vertical Consider coordinate point (denoted is x C acceleration. (Fig. asm1). +At bNow in (Now xequilibrium, )straightforward = m Cl i +zone, b( xtransducer ,highlight where (say am ptensiometer) assumes aisform by to amount measure of theP. in“pore aIfsoil, a water the measured value is inde table, and the gravitational acceleration. Now straightforward highlight the fact that ppW can be measured using aaconventional conventional Athe C - x 0 )Clearly, W W can Cdiffers AC) li li li li lll l l l l l l to prove that the true pore water pressure at point C is given by transducer. Remarkably, both p and p are mechanical pressures, transducer. Remarkably, bothpisispfrom and prove that pore atmpoint C is by transducer. Remarkably, both and are pressures, m A + b(whose xto x 0vertical )= mConsider +coordinate btrue ( xC ax 0 point )water , where is xsimilar (denoted (Fig. m C assumes to1). as At, xaC) equilibrium, is in thegiven the vertical pressure” vadose coordinate transducer zone, in a soil, of the (say Clearly, measured groundwater a tensiometer) p differs value used indeed thepto following, pW measure bymechanical amount to the pW .“pore isIn ofcalled P.water Ifthe a conventional “equilibrium s AW Wequal C the B Cpressure 0form li l l =b, (xpppx0AlAlAlAis+P +P +r Because point C is arbitrarily arbitrarily and thus canthe beused used toadescribe describe the mechanical behavior ofbe = C is and thus they can be the mechanical behavior of similar to mplpApiCmClCll+ -the whose x 0+r )vertical =lllbm b(xxxCCC)coordinate table, , where the andpoint groundwater isbmthe xCl iC gravitational assumes (Fig. 1).a form Atacceleration. the equilibrium, following, pressure” Now pW itin isthey isstraightforward acalled transducer soil, the measured “equilibrium (sayto tensiometer) value highlight solution is indeed isthe pressure” used fact equal tothat measure to pW .can In the “pore measured water u = +P +r bb(vertical ((ClxixxA+ ))-... xBecause Because point C is arbitrarily and thus they can be used to describe the mechanical behavior of 0 )of A Acoordinate B li li l l l and chosen, one obtains general expression for the true pore water unsaturated soils. In constitutive modeling, however, it is critical chosen, expression for the true pore water unsaturated soils. In constitutive modeling, however, it is critical table, andsimilar b the gravitational toone m lBiobtains mobtains , lAxi 0+isbacceleration. (the xaaaAgeneral vertical - x0 ) = Now coordinate mto + prove it b is ( straightforward x that of x the the ) , groundwater where true pore m water assumes highlight pressure the a the form following, fact at point that pressure” C p is is can given called in be by a measured soil, the “equilibrium the transducer. measured using a conventional solution Remarkably, value is pressure” indeed both equal to p to p In m chosen, one general expression for the true pore water unsaturated soils. In constitutive modeling, however, it is critical 0 W W .are C C C li l l l l l l pand pressure asthesimilar to distinguish which pressure is used. Unfortunately, such an pressure as to distinguish which pressure is used. Unfortunately, such an to provetable, that and the true bas pore gravitational water to m pressure acceleration. , x is the at p point vertical = Now p C +P it is coordinate is given +r straightforward b ( by x of the x transducer. ) groundwater . Because highlight Remarkably, point the C the fact is both following, arbitrarily that p can p be are is measured and called mechanical thus the they using “equilibrium pressures, can a conventional be used solution to describe pressure” the mec to pressure to distinguish which pressure is used. Unfortunately, such an C A A C W W B 0 l l l l l important issue traditionally has been overlooked. important issue traditionally has been pC = p Ato+P prove +rthat b( xtable, the xand .b Because thewater gravitational point pressure chosen, C acceleration. one at is point arbitrarily obtains CNow isa given general it is by straightforward and expression thustransducer. theyfor canthe beRemarkably, true used highlight pore to describe water the both fact pthe that and mechanical unsaturated pWoverlooked. are canmechanical be behavior soils. measured Inofconstitutive pressures, using a conventional modeling, h A -true C ) pore l l chosen, one pCl lll= obtains p Allll +P ato+r general prove b( x that expression - xthe ) true . Because for pressure pore the water true point as pore pressure C water is arbitrarily at pointunsaturated C is given and thus by soils.they In constitutive transducer. can be usedRemarkably, modeling, to describe tohowever, distinguish both the mechanical pl and it is critical pwhich behavior are pressure mechanical of is used. pressures, Unfo A C W [65] [65] [65] = pppW +P l ppp = = +P W W +P l Determination of Surface Energy Potential Determination of Surface Potential pressurechosen, as one obtains pC = paAl general +P +rexpression b( x A - xCfor ) . the Because true pore point water Ctoisdistinguish arbitrarily unsaturated whichsoils. and pressure thus In constitutive they is used. can be Unfortunately, modeling, important used Energy to describe however, issue such traditionally theitan mechanical is criticalhasbehavior been overloo of gggis For unsaturated soils, the effect of surface forces on A is generally A  For unsaturated soils, the effect of surface forces on A generally pressure as chosen, one obtains a general expression for the true important pore to water issue distinguish traditionally unsaturated which has pressure been soils. overlooked. In is used. constitutive Unfortunately, modeling, such however, an it is critical For unsaturated soils, the effect of surface forces on A is generally l [65] p l = pW +P Determination ofHence, Surface Ener small, and andissue the gaseous phasewhich behaves as an ideal ideal mixedUnfortunately, gas. Hence, small, the phase behaves as an mixed gas. pressure as important togaseous traditionally distinguish has been pressure overlooked. is used. such an l l [65] p = pW +P Determinationimportant of Surface Energy Potential issue traditionally For unsaturated has beensoils, overlooked. the effect of surface forc l l [65] p = pW +P Determination Surface Energy Potential For unsaturated soils, the effectof of surface forces small, onand A gthe is generally gaseous phase behaves as an ide l l [65] p = p +P W g isp. Vadose Zone Zone Journal Journal p. of21 21 Journal Vadose 999of p. of 21 of mixed Surface Potential small, and Forthe unsaturated gaseousDetermination phase soils,behaves the effect as an of surface ideal forcesgas. onHence, AEnergy generally

((

(

)) (( )) ((

)) (( )) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )(

) (

)(

)

)

small, and theFor gaseous unsaturated phase behaves soils, the aseffect an ideal of surface mixed gas. forces Hence, on A g is generally

A g is assumed to be a function of T, p g, and C g only. According to Eq. [23], p g − p s = 0, and p l - p s - pml = 0 . It follows that p l - p g = pml

[66]

By substituting Eq. [65] in Eq. [66] and introducing Eq. [47] and [63], one derives nl r l

¶W l ¶n l

l

+r ÅH2O W l = - s M +P D g

[67]

l - pW )

where s M (= p is the matric suction or the soil water l characteristic function. For a dilute solution, r l » r ÅH2O . By integrating Eq. [67] from nl to n0l , one obtains n l r l W l (T , n l ) = n0l r l W l (T , n0l ) + ò

n0l

n

l

é s (T , n l ) -P (T , n l )ù dn l D êë M úû

[68]

Noticeably, while sM is a function of T and n l , P D can be considered as a function of T and nl only to the extent that both c 0 l and cfix are specified. In general, P D is a function of T, nl, and C i , which has yet to be specified. Such a development goes beyond the scope of this paper and may become a topic for future research. It is remarkable that Eq. [67] also applies to the fully saturated condition, at which nr l

¶W l (T , n) l +r ÅH2O W l (T , n) = P D (T , n) - s M (T , n) ¶n

[69]

Here it is recalled that porosity n is equal to n0l . Equation [69] is the evolution equation for W l in fully saturated deforming soils. At infinitesimal deformation, the derivative term is negligible, so l that r ÅH2O W l (T , n) = P D (T , n) - s M (T , n) . Equation [63] yields P(T, n) = sM(T, n). According to the very definition of the air entry value (AEV) of a soil, sM(T, n) simply equals to the AEV. Hence, at full saturation, the generalized osmotic pressure is equal to the air entry value of the soil.

Measurement of Negative Pore Water Pressure In the following, three typical types of the techniques for l measuring the negative equilibrium solution pressure (i.e., pW ) are introduced, including the tensiometer, the vapor equilibrium technique, and the axis-translation technique. Briefly discussed here are the working principles of these techniques, and for a detailed analysis of their advantages and limitations, interested readers can refer to related references (e.g., Ridley and Wray, 1996; Lu and Likos, 2004).

Vadose Zone Journal Journal

Tensiometer A tensiometer consists of a ceramic filter with a high air entry value, a strain gauged diaphragm, and a small reservoir. The reservoir is located behind the ceramics and at the front of the strain gauged diaphragm. Before measurement, the ceramics and the reservoir have to be fully saturated. Then the tensiometer is inserted into the point of interest in the soil, at which the pore water contacts with the water in the reservoir of the tensiometer (i.e., the equilibrium solution) through the ceramics. Measurement begins after equilibrium is achieved. The working principle of a tensiometer is quite similar to that of a traditional PWP. Unlike the traditional PWP transducer, however, a tensiometer includes a high air-entry value ceramics, which is used to prevent the air enters into the water reservoir in the tensiometer during the measurement. The pressure measured l by a tensiometer is indeed pW , as discussed above. The true pore water pressure can be recovered using Eq. [65]. Due to the water cavitation, a tensiometer can only be applied to measure a negative pore water pressure down to −100 kPa.

Axis-Translation Technique To prohibit the water cavitation, the so-called axis-translation technique can be applied. Consider a representative volume of the unsaturated soil exposed to an ambient air pressure p g, at which the equilibrium solution pressure and the true pore water pressure l are pW and pl, respectively. To avoid the cavitation of pore water, g g p is increased to p P during the measurement. As a consequence, the true pore water pressure rises to p Pl . Now consider a reservoir containing the soil solution in equilibrium with the pore water, and l the solution pressure in the reservoir (denoted as pW P ) remains l constant during the measurement. For convenience, pW P can be chosen as the atmospheric pressure, patm. According to Hilf (1956), if the saturation remains unchanged, the difference between the air pressure and pore water pressure is l l constant, i.e., p g - pW = p pg - pW P . Hence, one has l l pW = p g - p Pg + pW P 0

[89b]

l where ( pW ) LL¢ is the equilibrium solution pressure at plane LL¢ and s M (T , n0l ) equals the AEV. As a consequence, l l ( p l ) LL¢ = ( pW ) LL¢ +P LL ' = ( pW ) LL¢ + AEV = 0 , that is, the true pore water pressure at xLL’ is zero.

The steady-state pressure profiles in the vadose zone deduced from Eq. [88] and [89] are illustrated in Fig. 1, which highlights the l l difference between pl and pW . Remarkably, the pW profile is different from the hydrostatic pressure profile, which is the case, since the gradients of T or C l k generally exist in the vadose zone.

Capillary Relaxation For the gaseous phase, p g » 0 . Using Eq. [35] and [47], one m obtains p g - p l =-n l r l

¶W l D s nl -q l Dt ¶n l

[90]

Assuming that Eq. [63] and [65] are valid in transient conditions, one can derive l s M = p g - pW = P- n l r l

¶W l ¶n l

-q l

D s nl Dt

[91]

Equation [90] and [91] is the evolution equation of nl that can be used to address the dynamic effect of capillarity (Hassanizadeh et al., 2002; O’Carroll et al., 2005; Manthey et al., 2008). It has been shown that the dynamic effect of capillarity can be attributed to local fluid flow in locally heterogeneous porous media (Wei and Muraleetharan, 2006, 2007).

D sC f k + n f r f v f , s ×ÑC f k -Ñ× éê z-f k1 × Ñ×(m f k - m f Z ) ùú = ë û Dt M é Z [93] ù å êêC f k å k Jf i mi -k Jf k mk úú pˆ J J =1 ë i =1 û

(

nfrf

(

)

)

where pˆ J can be determined once if the chemical reaction is specified.

Mass Exchange and Chemical Reactions As discussed in earlier sections, the mass exchange processes among various bulk phases can be addressed through specifying several typical stoicheiometric chemical reactions. To derive a general equation governing the mass exchange and chemical reactions, we cast Eq. [37] into the Arrhenius’ type as æ E J ÷ö pˆ J = G J A J expççç a ÷÷ çè RT ÷ø

[94]

where EaJ is the activation energy of reaction J; GJ is a coefficient accounting for the rate of reaction; A J is the chemical affinity of the reaction, given by é

Z

æ t s :1

ps ö

ù

A J = åå k aJ i mi êêm a i + ççç s + rs ÷÷÷d sa úú çè 3r r ÷ø úû a i =1 êë

[95]

where m a i = (m a i : 1)/3 ; d sa = 1 if a = s, and otherwise d sa = 0. In soils, the solid grains are generally coated by the aqueous phase (i.e., the pore water). Therefore, the reactions of main concern here may include vaporization–condensation between the pore water and pore gas, pressure solution–precipitation between the solid grains and the pore water, and the chemical reactions in pore fluids.

Vaporization–Condensation Process For a vaporizing–condensing processes, it follows from Eq. [95] that l

A J = m H2O -m

g H2O

[96]

Species Diffusion

In a vaporizing process, A J > 0 , whereas A J < 0 in a condensing process. Using Eq. [56] and [72], it is straightforward to prove that

Species diffusion occurs only in pore fluids. Multiplying Eq. [A2] by C f i , and then subtracting the resultant from Eq. [A1], it follows after some algebraic manipulations that

A J =-

D sC f k + n f r f v f , s ×ÑC f k +Ñ× n f r f k u f k = Dt æ ö çç fk fk ÷ ÷÷-C f k å eˆ f ˆ ˆ e r + å ç fb fb ÷÷ çèb¹ f b¹ f ø

nfrf

(

)

Substituting Eq. [17], [18], and [38] in Eq. [92], one obtains

Journal Vadose Zone Journal

[92]

æ H r ö÷ s RT ln ççç r ÷÷- l M mH2O çè H W ÷ø r ÅH2O

[97]

where H r (= pvg / p0g ) is the relative humidity of the pore gas, g r and H W (= pvW / p0g ) is the relative humidity of the equilibrium solution at T and pg (as in the pore gas). At equilibrium, -

RT0 æç H 0r ln ç r mH2O ççè H W

ö÷ s M0 ÷÷÷- l H O = 0 ø rÅ 2

[98]

p. 14 of 21

Here, subscript 0 denotes the equilibrium state. Linearizing Eq. [97] and using Eq. [98], one obtains

A J »-

RT0 DH r mH2O H 0r

-

Ds M

[99]

l

r ÅH2O

Pressure Solution–Precipitation Pressure dissolution and precipitation occur only between the solid matrix and the pore water in unsaturated soils. Without loss of generality, consider a group of chemical reaction processes, which can be represented by Z

k sJm [ s m ] + å k lJi [l i ] Û 0

[100]

i =1

4

æ t s :1

ps öù

Z

[101] si

where is a function of (or nr), defined by Eq. [21]; Dm and li Dm represents the deviations of the chemical potentials from their equilibrium counterparts. The chemical potential of a species in the solid phase is generally dominated by the volumetric strain energy of the solid skeleton. Thus, one can write Dm s i »

p ¶m s i De v » sI De v ¶e v r

[102]

(= −(ts : 1)/3) is the intergranular pressure. Noting that

where pI De v = DpI/K, where K is the secant bulk modulus of the solid matrix, Eq. [101] can be approximated as éæ p ö Dp Dp s I -1÷÷÷ s I + s r ø r r êëè K

A J » k sJi m i êêççç

ù ú ú úû

[103]

where all the terms including Dm l i are dropped out, since they are generally much smaller than the bracketed term. As implied by Eq. [103], a dissolving–precipitating process in the pores can be intervened by the intergranular pressure applied to the porous media. In general, however, the applied pressure and temperature do not exert simple controls on the pressure solution, since prs is also involved in. For dissolution, A J > 0 , whereas A J < 0 for precipitation. In the early beginning of the dissolution, few chemical bonds break, and thus Dprs » 0 . From Eq. [103], one can show that for the dissolution to occur, it requires that pI > K. Hence, there exists a threshold of the applied pressure, below which

Journal Vadose Zone Journal

Z

[104]

i=1

A J = k sJi mi êêDm s i +Dççç s + rs ÷÷÷ úú + å k lJi mi Dm l i r ÷ø ûú i=1 èç 3r ëê nrs

The chemical affinity of a chemical reaction in a pore fluid is

A J = å k Jf i mi Dm f i

Accordingly, the chemical affinity is given by

prs

Variable prs is defined by Eq. [21]. Due to the current shortage of experimental data, it is impossible to develop here a realistic constitutive equation for prs . As a first approximation, one can assume Dprs = QDnrs based on the linearization of Eq. [21], where Q is a positive material parameter. Provided that the mass exchange rates are given by Eq. [18], Dnrs can be determined by integrating Eq. [1].

Chemical Reactions in a Pore Fluid

As an instance, the quartz dissolution is represented by [ sSiO2 ]+2[l H2O ] Û 4[l H+ ] +[l SiO2- ] .

é

no pressure dissolution can occur. Clearly, as the pressure increases, more and more chemical bonds will break, i.e., prs becomes more and more negative, and the dissolution will cease at A J = 0 . These theoretical results are consistent with experimental observations (e.g., Taron and Elsworth, 2010).

Particularly, in the pore water, the chemical potential is given by li m li = m Å (T , p0l ) +

p l - p0l li rÅ

+

F xn i RT ln a l i +W l + mi mi

[105]

During a chemical reaction, both the mass and the number of electric charges are conserved. Hence, substituting Eq. [105] in Eq. [104], one obtains æ

Z

ö

Z

k lJ

Z

A J = çççå k lJi u l i ÷÷÷Dp l + RT0 å l i Da l i + RDT å k lJi ln a0l i i çè i=1 [106] ø÷ i =1 a0 i =1

where u l i is the molar volume of species li. Noticeably, for a specified Dp l , the first term in the right-hand side of Eq. [106] could be positive, zero, or negative, depending l on the specific type of chemical reaction. Because p l = pW +P , one can conclude that pressurization and the surface forces in the pores could have different controls on the chemical reactions in porous media.

6 Summary

and Conclusions

Developed here is a continuum theory of unsaturated soils, in which the soils are viewed as the porous media composed of multiphase and multispecies. Within the proposed framework, various coupling processes in unsaturated soils, including skeletal deformation, heat conduction, pore fluid flow, diffusion, phase transition, chemical reactions, and capillary relaxation, are addressed in a systematic and consistent way. The conditions for the thermal, mechanical, and chemical equilibrium of unsaturated soils are established, and the driving forces for various dissipative

p. 15 of 21

processes are identified and characterized. A general expression is developed for the chemical potential of a species in a fluid phase, which takes into account the thermal, pressure, osmotic, capillary, adsorptive, and electrostatic effects of unsaturated soils. The behavior of unsaturated soils at equilibrium is investigated by enforcing the equilibrium conditions with incorporation of the proposed chemical potential formula. In particular, the Donnan osmotic phenomenon is discussed in detail, providing deep insights into the conception, composition, and measurement of pore water pressure, and a general formulation is developed for calculating the osmotic pressure. It is found that the true pore water pressure has three contributing components, including the equilibrium solution pressure, Donnan osmotic pressure, and the pressure induced by the surface forces. While the first component can be measured based on the conventional methods, the last two components cannot be directly measured. In addition, the experimental techniques used to measure the equilibrium solution pressure are discussed, and it is theoretically shown that the axistranslation technique can be applied, without water cavitation, down to a negative pressure much lower than expected. Formulations of effective and intergranular stress tensors are developed for both fully saturated and unsaturated soils. It is suggested that it is the intergranular stress, instead of the effective stress, that controls the chemomechanical behaviors of both unsaturated and saturated soils. The new theory provides solid theoretical evidence for the recent development of the suction stress concept. In particular, a closed-form equation is developed for the suction stress, which can be used to describe the chemomechanical behavior of unsaturated soils. A complete set of general governing equations are developed for addressing the chemomechanical behavior of unsaturated porous media in general, and unsaturated soils in particular, with a saturation ranging from extremely low to 100%. Remarkably, it is shown that other than the soil water characteristic function, another pressure (or potential) potential) function function P(T , n l , C l k ) (or l P D or W ) is generally required in analyzing the coupled chemomechanical processes in unsaturated soils. This conclusion is consistent with the recent development of the suction stress concept. The new theory is capable of explaining many salient phenomena occurring in water-saturated porous media, including Donnan osmosis, capillary fringe, air entry value, initial hydraulic head during seepage, pressure solution, and more.

Acknowledgments

This research was funded by one of National Basic Research Programs of China under Grant 2012CB026102, the National Science Foundation of China (NSFC) under Grants 51239010 and 11372078, and the Natural Science Foundation of Guangxi under Grant 2011 GXNSFE018004.

Vadose Zone Journal Journal

66Appendix Notation Ñ Da /Dt l g l s ai a ab Aa

a fi b

C

ai

a eˆab ai eˆab

Ea

E ai Eˆ a i

Es fi Fs ha Ga Hr

ˆi a i

m fi mi

Spatial gradient operator [L−1] material derivative following the motion of a phase [T−1] unit tensor, with components of d ij pore gas pore liquid solid matrix species i in a bulk phase bulk phase

a–b interface specific Helmholtz free energy of a phase [L2 T−2] activity of species f i external supply of linear momentum [L T−2] mass fraction of species a i total mass exchange rate from a–b interface to a phase [M L−3 T−1] mass exchange rate of species a i from a–b interface to a phase [M L−3 T−1] specific internal energy of a phase [L2 T−2] specific internal energy of a i species [L2 T−2] production rate of specific internal energy of species a i [M L−1 T−2] Lagrangian stain tensor species i in pore fluid f deformation gradient of the solid skeleton external supply of energy to a phase [L2 T−3] specific Gibbs free energy of a phase [L2 T−2] relative humidity production rate of linear momentum of species a i [M L−2 T−2] molar fraction of species f i molar mass of species i [M/mol]

n nr na

porosity porosity change due to chemical reaction volume fraction of a phase volume fraction change of solid due to chemical reaction

p pa

intergranular pressure [M L−1 T−2] thermodynamic pressure of a phase [M L−1 T−2] configurational pressure of fluid f [M L−1 T−2]

nrs

pmf l pW

peq qa a Qˆ ab

equilibrium solution pressure [M L−1T−2] equivalent pore pressure [M L−1 T−2] heat flux vector of a phase [M T−3] supply of heat to a phase from a–b interface [M L−1 T−3]

sM Sf

matric suction [M L−1 T−2] degree of saturation of f fluid mass production rate of species a i [M L−3 T−1]

R T

universal gas constant absolute temperature [q] supply of momentum to a phase from a–b interface [M L−2 T−2]

rˆa i

a Tˆab ai Tˆab

supply of momentum to a i species from a–b interface [M L−2 T−2]

ta

Cauchy stress tensor of a phase [M L−1 T−2]

p. 16 of 21

uai

diffusion velocity of species a i [L T−1]

va

velocity of a phase [L T−1]

v f,s ni x

A

J

relative velocity of fluid f with respect to the solid [L T−1] valence number of species i spatial coordinate [L] chemical affinity

Af Cf

potential associated with adsorptive force [L2 T−2] potential associated with capillary force [L2 T−2] Faraday’s constant

ha L

specific entropy of a phase [L2 T−2 q −1] production rate of entropy [L2 T−3] reaction coefficient of f i species

F

k Jf i ra

r

ai

intrinsic mass density of a phase [M L−3] partial mass density of a i species [M L−3]

r aÅi

mass density of species a i in pure state

mai

electrochemical potential of a i species [L2 T−2]

m aÅi

electrochemical potential of a i species in pure state [L2 T−2]

m

ai

electrochemical potential tensor of a i species [L2 T−2]

m a i

chemical potential of a i species [L2 T−2]

 ai m

chemical potential tensor of a i species [L2 T−2]

mˆ a i

relative chemical potential of a i species [L2 T−2] surface energy potential of fluid f [L2 T−2]

Wf pˆ J

rate of chemical reaction [T−1]

P PD

osmotic pressure [M L−1 T−2] Donnan osmotic pressure [M L−1 T−2]

fˆ aabi

entropy exchange rate of a i

fˆ aab

entropy exchange rate of a phase [M L2 T−3 q −1]

s

s ¢T s ¢I ss ts

species [M L2 T−3 q −1]

total Cauchy stress tensor [M L−1 T−2] Terzaghi effective stress tensor [M L−1 T−2] intergranular stress tensor [M L−1 T−2] suction stress [M L−1 T−2] intrinsic intergranular stress [M L−1 T−2]

Balance Equations and the Second Law

Da i a a i (n r )+ na ra i Ñ× v a i = å eˆaba i + r a i Dt b¹a

[A1]

Summing Eq. [A1] over i (= 1, 2, ..., Z) yields the mass balance equation for a bulk phase; that is, Da a a (n r ) + na ra Ñ× v a = å eˆaba i Dt b¹a

[A2]

Z ai a where eˆab , and va is the velocity the mass center of = å i=1 eˆab the a phase, defined by Z

Z

i =1

i =1

v a = å ra i v a i ra = å C a i v a i

[A3]

Mass conservation requires that Z

å rˆa

i

i =1

Z æ ö÷ ai ai = 0, åå ççç å eˆab + rˆa i ÷÷ = å å eˆab =0 ÷ ÷ ç ø a b¹a a i =1 èb¹a

Linear Momentum Balance The linear momentum balance equation for a species is given by na r a i

Da i v a i ˆ a i + ˆi a i -Ñ×(na t a i ) - na r a i b = å T ab Dt b¹a

[A5]

Summing up Eq. [A5] over i (= 1, 2, ..., Z) yields the linear momentum balance equation for a bulk phase; that is, na r a

Da v a ˆa -Ñ×(na t a ) - na r a b = å T ab Dt b¹a

[A6]

where ta , the Cauchy stress tensor of a phase, is given by Z

Z

i =1

i =1

(

a ai ˆa = t a = å (t a i -r a i u a i Ä u a i ), T u å Tˆ aba + eˆab ab

)

[A7]

u a i ( = v a i - v a ) is the diffusion velocity. Equation [A5] and [A6] are restricted by

The balance equations and the second law of thermodynamics, originally developed by Hassanizadeh and Gray (1979a,b) and Gray and Hassanizadeh (1989) for multiphase porous media based on a local averaging procedure, and later generalized by Bennethum and Cushman (1996a; 2002a) to multispecies porous media, are introduced here. In the following derivations, no thermodynamic properties are assigned to the interfaces between bulk phases, and thus only the balance equations of species and bulk phases are introduced.

The energy balance equation for a species is given by

Mass Balance

na r a i

The mass balance equation for a species is given by

[A4]

Z

(Tˆ aba + eˆaba v a ) +(Tˆ abb + eˆabb v b ) = 0, å(ˆi a i

i

i

i

i

i =1

i

)

+ rˆa i v a i = 0 [A8]

(Tˆ aba + eˆaba v a ) +(Tˆ abb + eˆabb v b ) = 0

[A9]

i

Energy Balance Da i E a i - na t a i : sym(Ñv a i ) +Ñ×(na q a i )- na r a i ha i Dt

ai = å Qˆ ab + Eˆ a i

[A10]

b¹a

Journal Vadose Zone Journal

p. 17 of 21

Summing up Eq. [A10] yields the energy balance equation for a bulk phase Da E a - na t a : sym(Ñv a ) +Ñ×(na q a )- na r a ha Dt

na r a i

ai = å Qˆ ab

[A11]

b¹a

where Z

E = åC a

æ a i 1 a i a i ö÷ çE + u ×u ÷ èçç ø÷ 2

ai

i =1

[A12]

é ù æ ö 1 q a = å ê q a i - t a i × u a i +r a i ççç E a i + u a i × u a i ÷÷÷u a i ú ê úû è ø 2 i =1 ë Z

Z

(

ha = å C a i ha i + b × u a i i =1

)

[A13]

Z é ai a ˆ a i × u a i + eˆa i çæ E a i - E a + 1 u a i × u a i ÷÷öùú [A15] = å ê Qˆ ab +T Qˆ ab ab ab ç ÷øú çè ê 2 û i =1 ë

Equation [A10] and [A11] are restricted by

Z

é

å êê Eˆ a

i

i =1 ë

æ öù 1 + ˆi a i × v a i + rˆa i ççç E a i + v a i × v a i ÷÷÷ú = 0 úû è ø 2

é ˆa ˆa a æ a 1 a a ÷öù a ç ê Q ab + Tab × v + eˆab E + v × v ÷÷ú + êë èçç øúû 2

é ˆb ˆ b × v b + eˆb æç E b + 1 v b × v b ö÷÷ùú = 0 ê Q ab + T ab ab ç êë èç ø÷úû 2

[A22]

It has been well recognized that, in deriving a general thermomechanical theory of porous media, it is more convenient to use the Helmholtz free energy instead of the internal energy function. By Legendre transformation, the Helmholtz free energy density for a species and a bulk phase are defined, respectively, as A a i = E a i -ha i T , A a = E a -haT

[A23]

Using Eq. [A12] and noting that ha = å i ha i , one can derive

[A14]

é ˆ ai ˆ ai ai öù ai æ çç E a i + 1 v a i × v a i ÷÷ú + ê Q ab + Tab × v + eˆab êë èç ø÷úû 2 é ˆ bi ˆ bi bi öù bi æ çç E b i + 1 v b i × v b i ÷÷ú = 0 ê Q ab + Tab × v + eˆab ÷øú ç êë è 2 û

(fˆ aab + eˆaba ha ) +(fˆ abb + eˆabb hb ) = 0

[A16]

[A17]

Z æ ö 1 A a = å C a i ççç A a i + u a i × u a i ÷÷÷ è ø 2 i =1

[A24]

Remarkably, Aa includes a contribution due to the diffusion of the species. Replacing E a i in Eq [A19] by A a i and Aa , one can derive the following expression for the second law, æ Da A a DaT ö÷ ÷÷ + å na t a : sym(Ñv a ) T L =-å na r a ççç +ha çè Dt Dt ø÷ a a Z é ù ïü æ ö na ïì 1 - å íïq a + å ê t a i × u a i -r a i ççç A a i + u a i × u a i ÷÷÷u a i ú ýï×ÑT ê úû ïï ï è ø 2 T i =1 ë a îï þ æ ö 1 f f ,s , , s s a a a a ˆ ÷ - å å T f b × v - å å ççç A + v × v ÷÷ eˆab è ø 2 [A25] f b¹ f a b¹a Z é æ ö ù 1 + åå na ê t a i -r a i ççç A a i + u a i × u a i ÷÷÷1ú : sym(Ñu a i ) ê è ø úû 2 ë a i =1 Z é æ æ ööù 1 + åå êÑ×(na t a i )-Ñççna r a i ççç A a i + u a i × u a i ÷÷÷÷÷÷ú × u a i ê ç è ø è øúû 2 a i =1 ë ³0

[A18]

Additive Decomposition of Porosity

The Second Law of Thermodynamics

Introducing the mass balance equation of the solid phase (Eq. [A2]), one can derive

The second law of thermodynamics requires that the rate of total entropy production L in the unsaturated soils due to dissipation be nonnegative; that is,

ù 1 D s n éê (1- n) D s r s =ê s + (1- n)Ñ× v s úú - s Dt êë r Dt úû r

L= Z

1 åå T a i =1

ai ai é æ ai ai ê-na r a i çç D E -T D h ê çç Dt Dt è êë

na q a i ai ×ÑT + å Qˆ ab -T fˆ aabi T b¹a

(

÷ö÷ + na t a i : sym Ñv a i ( ) [A19] ÷÷ø ù + Eˆ a i -T hˆ a i úú ³ 0 úû

) (

)

The entropy balance requires that

(

fˆ aabi Z

ai ai + eˆab h

å (h i =1

ˆ ai

ai

) +(

+ rˆ h

bi fˆ ab

ai

bi bi + eˆab h

)= 0

Vadose Zone Journal Journal

)= 0

[A20]

å

b= l , g

eˆssb

[A26]

It is clear that the variation of porosity includes two contributions, which are induced by mechanical actions (including solid material compression and skeletal deformation) and mass exchanges, respectively. The variation of the porosity due to chemical reactions is described by D s nr D s nrs 1 ==- s Dt Dt r

å

b= l , g

eˆssb

[A27]

where the subscript “r” denotes the chemical reactions. [A21] p. 18 of 21

Constraints of the Second Law Substituting Eq. [2] and [3] into Inequality [A25] can yield an expression of TL, which is a linear function of DaT/ DT, Ñva , ÑT, ak a v f,s, eˆab , å b¹a eˆab + rˆa k (k = 1, 2, ..., Z − 1), Ñu a i and u a i (i = 1, 2, ..., Z). These variables are not independent, because u a i ’s and Ñu a i ’s are correlated. Namely, Z

å na r a u a i

i

=0

[A28]

i =1

Taking the spatial gradient on both sides, one obtains Z

å na ra Ñu a i

Z

+ å u a i ÄÑ(na r a i ) = 0

i

i =1

[A29]

i =1

In addition, the species are electrically charged. The electrical neutrality requires that the total number of charges in the porous media be zero. According to this requirement, one has Z

æn

ååççççè mi na ra i

a i =1

i

÷÷ö = 0 ÷÷ø

[A30]

where n i and m i are the valence number and molar mass of charged species a i, respectively. ni < 0 (or > 0) for an anion (or a cation), and ni = 0 for a species with zero charge. Applying operator Ds/Dt to Eq. [A30], and introducing Eq. [A2], one can prove that ìï é æ öù üï ïn ê ai ç ai ÷ a ai a,s a ai ai ÷ú ïï êÑ×(n r v ) +Ñ×(n r u )-ççç å eˆab + rˆ ÷÷÷ú ýï = 0 [A31] èb¹a øúû ïþï i =1 ï îï i êë

Z -1 æ ln ö 1 a - å çççG a + v a , s × v a , s - å C a k mˆ a k + Z ÷÷÷ å eˆab ç m Z ÷ø b¹a 2 a è k=1 Z -1 é ö÷ æn n öù æ ak - å å êêmˆ a k +l ççç k - Z ÷÷÷úú ççç å eˆab + r a k ÷÷ ÷ èç m k m Z ÷øûú èçb¹a ø÷ a k=1 ëê é ak 1 ak ak ù êm + (u × u )1-mˆ a k 1ú [A32] Z -1 ê ú 2 ú : sym(Ñu a k ) - å å na r a k ê æn ú ê aZ n ö a k=1 ú ê-G -l ççç k - Z ÷÷÷1 ê ú çè m k m Z ø÷ ë û

é ù 1 - å na r a Z êm a Z + (u a Z × u a Z )1- Ga Z ú : sym(Ñu a Z ) ê úû 2 ë a ïìï é a a k a k 1 a k a k ù ˆ a k ïü Ñ×ê n r (m + (u × u )1ú -m Ñ(na r a k ) ïïï Z -1ï ï êë úû 2 ïý× u a k - å å ïí æ n k n Z ö÷ ï a ak aZ aZ a ak a ak ï a k=1 ï ç ÷Ñ(n r )ïï ïï-n r p - G ×Ñ(n r ) -l çç ïï çè m k m Z ÷ø÷ îïï þï ïìï é a a Z æ a Z 1 a Z a Z ö÷ù ïüï ïÑ× ê n r çççm + (u × u )1÷÷úú ïï a è øû ý× u Z 2 - å ïí ëê ï a a a ï a ï ïï-n r Z p Z - Ga Z ×Ñ(na r a Z ) ïïï î þ ³0

where l is a scalar, pa Z a vector, Ga Z a symmetric second-order tensor, and 2

p a = (r a )

s ¶A a s s s ¶A , = r F (F s ) t t ¶r a ¶E s

m a k = A a k 1-

Z

åå ïíïmi a

The constraints represented by Eq. [A28], [A29], and [A31] can be removed by using the Lagrange-multiplier method. To this end, the right-hand sides of Eq. [A28], [A29], and [A31] are first multiplied by multipliers pa Z , Ga Z , and l, respectively, and then added into the right-hand side of Eq. [A25]. Finally, introducing Eq. [2] and [3], one can derive æ ¶A a ö Da T T L c =-å na r a ççç +ha ÷÷÷ ÷ø Dt èç ¶T a ì a Z é a æ a 1 a a ö a ùï ü na ï -å ï íq - å ê r i çççm i + (u i × u i )1÷÷÷× u i ú ï ý×ÑT ê ú ï è ø T 2 ûï i =1 ë a ï ï î þ si ö ù é Z æ ln r ÷ + n s êê t s + p s 1- t s + åççç i ÷÷1úú × sym(Ñv s ) ç m i ÷ø ú i =1 è ëê û si ö ù é Z æ ln r ÷ + å n f êê t f + p f 1 + å ççç i ÷÷1úú × sym Ñv f ç m i ÷ø ú f i =1 è ëê û

(

(

+å p - p f

f

s

- pmf

)

)

D s nrs Ds n f - prs Dt Dt

é Z æ öù ˆ f - çç ln i Ñ(n f r f i )÷÷ú × v f , s - å êê pmf - p f Ñn f + å T åççè m ÷÷ú fb øúû i f êë i =1 b¹ f

(

)

Vadose Zone Journal Journal

pmf = n f r f prs = n s r s

ta k

[A34]

ra k

¶A f

[A35]

¶n f

¶A s ¶nrs

G a = Aa +

[A33]

[A36]

pa

[A37]

ra

¶A a mˆ a k = ¶C a k

[A38]

In a general thermodynamic process, variables D aT/Dt, Ñv a , and Ñu a i can vary arbitrarily. Hence, for Inequality [A32] to be constantly satisfied, it requires that ¶A a +ha = 0 ¶T

[A39]

Z æ ln r s i t s + p s 1 - t s + å ççç i ç mi i =1 è Z æ ln r f i t f + p f 1 + å ççç i è mi i =1 ç

÷ö ÷÷÷1 = 0 ø

ö÷ ÷÷1 = 0 ÷ø

[A40] [A41]

p. 19 of 21

æn æ1 ö n ö m a k + ççç u a k × u a k ÷÷÷1 -mˆ a k 1 - Ga Z -l ççç k - Z ÷÷÷1 = 0 [A42] è2 ø èç m k m Z ø÷

æ1 ö m a Z + ççç u a Z × u a Z ÷÷÷1- Ga Z = 0 è2 ø

[A43]

[A44]

Bennethum, L.S., and T. Weinstein. 2004. Three pressures in porous media. Transp. Porous Media 54(1):1–34. doi:10.1023/A:1025701922798

Bennethum, L.S. 2012. Macroscopic flow potentials in swelling porous media. Transp. Porous Media 94:47–68. Bishop A. W. 1959. The principle of effective stress. Teknisk Ukeblad 39:859–863. Bowen, R.M. 1982. Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6):697–735.

and from Eq. [A42] that mˆ a k 1 =

Coussy, O. 2004. Poromechanics. John Wiley & Sons, New York.

æn ö æ1 ö n ö æ1 m a k - m a Z -l ççç k - Z ÷÷÷1 + ççç u a k × u a k ÷÷÷1-ççç u a Z × u a Z ÷÷÷1 çè m k m Z ø÷ è 2 ø è2 ø

[A45]

Using Eq. [A27] and Eq. [A39–A43], one obtains the residual dissipation inequality as TLD = å a

na ìïï a Z é a i çæ a i 1 a i a i ÷ö a i ù üïï íq - å ê r ççm + (u × u )1÷÷× u ú ý×ÑT ê è úû ïï ø 2 T ïîï i =1 ë þ

(

+ å p f - p s - pmf f

Bennethum, L.S., and J.H. Cushman. 2002b. Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics: II: Constitutive theory. Transp. Porous Media 47:337–362. doi:10.1023/A:1015562614386

Bennethum, L.S. 2007. Theory of flow and deformation of swelling porous materials at the macroscale. Comput. Geotech. 34:267–278. doi:10.1016/j.compgeo.2007.02.003

It follows from Eq. [A43] that æ1 ö Ga Z º m a Z + ççç u a Z × u a Z ÷÷÷1 è2 ø

electroquasistatics: I: Macroscale field equations. Transp. Porous Media 47:309–336. doi:10.1023/A:1015558130315

s

) DDtn

(

)

Di Maio, C., T. Huechel, and B. Loret, editors. 2002. Chemo-mechanical coupling in clays: From Nano-scale to engineering application. Proceedings of the Workshop, Maratea. Swets & Zeitlinger, Lisse. Gens, A., E.E. Alonso, J. Suriol, and A. Lloret. 1995. Effect of structure on the volumetric behavior of a compacted soils. In: Proc. 1st Int. Conf. on Unsaturated Mechanics, p. 83–88. Gray, W.G., and S.M. Hassanizadeh. 1989. Averaging theorems and averaged equations for transport of interface properties in multiphase systems. Int. J. Multiphase Flow 15(1):81–95.

f

é Z æ öù ˆ f - çç ln i Ñ(n f r f i )÷÷ú × v f , s - å êê pmf - p f Ñn f + å T ÷ú f b åç ç ø÷ûú b¹ f f ëê i =1 è m i Z -1 æ p s ö÷ æ ln ö 1 a - å çççG a + v a , s × v a , s - å C a k mˆ a k + Z ÷÷÷ å eˆab -ççç rs ÷÷ å eˆsfs ç çè r ÷ø f 2 m Z ÷ø b¹a k=1 a è Z -1 é ö÷ æn n öù æ ak - å å êêmˆ a k +l ççç k - Z ÷÷÷úú ççç å eˆab + rˆa k ÷÷ ÷ ç ÷÷ø m m ç è ø k Z ûú èb¹a a k=1 ëê

Coussy, O. 2010. Mechanics and physics of porous solids. 1st ed. John Wiley & Sons, West Sussex.

Gray, W.G., and B.A. Schrefler. 2007. Analysis of the solid phase stress tensor in multiphase porous media. Int. J. Numer. Anal. Methods Geomech. 31(4):541–581.

[A46]

Z -1

- å å éêÑ×(m a k - m a Z )ùú ×(na r a k u a k ) ³ 0 ë û a k=1

References

Gray, W.G., and C.T. Miller. 2007. Consistent thermodynamic formulations for multiscale hydrologic systems: Fluid pressures. Water Resour. Res. 43. doi:10.10029/2006WR005811. Gray, W.G., B.A. Schrefler, and F. Pesavento. 2009. The solid phase stress in porous media mechanics and the Hill-Mandel condition. J. Mech. Phys. Solids 57:539–554. doi:10.1016/j.jmps.2008.11.005 Hassanizadeh, S.M., and W.G. Gray. 1979a. General conservation equations for multiphase systems: I. Averaging procedure. Adv. Water Resour. 2:131–203. doi:10.1016/0309-1708(79)90025-3 Hassanizadeh, S.M., and W.G. Gray. 1979b. General conservation equations for multiphase systems: 2. Mass, momenta, energy and entropy equations. Adv. Water Resour. 2:192–203. doi:10.1016/03091708(79)90035-6 Hassanizadeh, S.M., M.A. Celia, and H.K. Dahle. 2002. Dynamic effect in capillary pressure-saturation relationship and its impact on unsaturated flow. Vadose Zone J. 1:38–57. doi:10.2136/vzj2002.3800

Atkins, P.W., and J.D. De Paula. 2002. Physical chemistry. 7th ed. Oxford Univ. Press, Oxford, UK.

Hilf, J.W. 1956. An investigation of pore-water pressure in compacted cohesive soils. Ph.D thesis. Technical Memorandum 654. U.S. Department of the Interior Bureau of Reclamation, Denver, CO.

Baker, K., and S. Frydman. 2009. Unsaturated soil mechanics: Critical review of physical foundations. Eng. Geol. 106:26–39. doi:10.1016/j. enggeo.2009.02.010

Hueckel, T. 2002. Reactive plasticity for clays during dehydration and rehydration. Part 1: Concept and options. Int. J. Plast. 18:281–312. doi:10.1016/S0749-6419(00)00099-1

Bataille, J., and J. Kestin. 1977. Thermodynamics of mixtures. J. NonEquilibrium Thermodyn. 2:49–65.

Huyghe, J.M., and J.D. Janssen. 1997. Quadriphasic mechanics of swelling incompressible porous media. Int. J. Eng. Sci. 35(8):793–802. doi:10.1016/S0020-7225(96)00119-X

Bennethum, L.S., and J.H. Cushman. 1996a. Multisace, hybrid mixture theory for swelling systems: I: Balance laws. Int. J. Eng. Sci. 34(2):125– 145. doi:10.1016/0020-7225(95)00089-5 Bennethum, L.S., and J.H. Cushman. 1996b. Multisace, hybrid mixture theory for swelling systems: II. Constitutive theory. Int. J. Eng. Sci. 34(2):147–169. doi:10.1016/0020-7225(95)00090-9 Bennethum, L.S., M.A. Murad, and J.H. Cushman. 1996. Clarifying mixture theory and the macroscale chemical potential for porous media. Int. J. Eng. Sci. 34(14):1611–1621. doi:10.1016/S00207225(96)00042-0 Bennethum, L.S., M.A. Murad, and J.H. Cushman. 2000. Macroscale thermodynamics and the chemical potential for swelling porous media. Transp. Porous Media 39(2):187–225. doi:10.1023/A:1006661330427 Bennethum, L.S., and J.H. Cushman. 2002a. Multicomponent, multiphase thermodynamics of swelling porous media with

Journal Vadose Zone Journal

Huyghe, J.M., R. Van Loon, and F.T.P. Baaijens. 2004. Fluid-solid mixtures and electrochemomechanics: The simplicity of Lagrangian mixture theory. Comput. Appl. Math. 23(2–3):235–258. doi:10.1590/S010182052004000200008 Huyghe, J.M., M.M. Molenaar, and F.P. Baajens. 2007. Poromechanics of Compressible Charged Porous Media Using the Theory of Mixtures. J. Biomech. Eng. 129(5):776–785. doi:10.1115/1.2768379 Khalili, N., F. Geiser, and G.E. Blight. 2004. Effective stress in unsaturated soils, A review with new evidence. Int. J. Geommechanics 4(2):115– 126. doi:10.1061/(ASCE)1532-3641(2004)4:2(115) Kohgo, Y., M. Nakano, and T. Miyazaki. 1993. Theoretical aspects of constitutive modeling for unsaturated soils. Soils Found. 33(4):49–63. doi:10.3208/sandf1972.33.4_49 Laloui, L., and M. Nuth. 2009. On the use of generalized effective stress in the constitutive modeling of unsaturated soils. Comput. Geotech. 36:20–23. doi:10.1016/j.compgeo.2008.03.002

p. 20 of 21

Loret, B., A. Gajo, and F.M.F. Simoes. 2004. A note on the dissipation due to generalized diffusion with electro-chemo-mechanical coupling in heteroionic clays. Eur. J. Mech. Solids 23:763–782. doi:10.1016/j. euromechsol.2004.04.004

Ridley, A.M., and W.K. Wray. 1996. Suction measurement: A review of current theory and practices—State of the art report. In: E. E. Alonso and P. Delage, editors, Unsaturated soils, Proc. 1st Int. Conf. on Unsaturated Soils, Paris. Balkema, Rotterdam. p. 1293–1322.

Lu, N., and W.J. Likos. 2004. Unsaturated soil mechanics. 1st ed. John Wiley & Sons, New York.

Spycher, N.F., and E.L. Sonnenthal. 2003. Fluid flow and reactive transport around potential nuclear waste emplacement tunnels at Yucca Mountain, Neveda. J. Contam. Hydrol. 62–63(0):653–673.

Lu, N., and W.J. Likos. 2006. Suction stress characteristic curve for unsaturated soil. J. Geotech. Geoenviron. Eng. 132(2):131–142. doi:10.1061/(ASCE)1090-0241(2006)132:2(131) Lu, N., J.W. Godt, and D.T. Wu. 2010. A closed-form equation for effective stress in unsaturated soil. Water Resour. Res. 46. doi:10.1029/2009WR008646. Manthey, S., S. M. Hassanizadeh, R. Helmig, and R. Hilf. 2008. Dimensional analysis of two-phase flow including a rate-independent capillary pressure-saturation relationship. Adv. Water Resour. 31:1137–1150, doi:10. 1016/j.advwartres.2008.01.021. Mitchell, J.K., and K. Soga. 2005. Fundamentals of soil behavior. 3rd ed. John Wiley & Sons, New York. Nikooee, E., G. Habibagahi, S. M. Hassanizadeh, and A. Ghhramani. 2012. Effective stress in unsaturated soils: A thermodynamic approach on the interfacial energy and hydromechanical coupling. Transp. Porous Med. doi:10.1007/s11242-0012-0093-y.

Steefel, C.I., D.J. DePaolo, and P.C. Lichtner. 2005. Reactive transport modeling: An essential tool and a new paradigm for the earth sciences. Earth Planet. Sci. Lett. 240:539–558. Taron, J., and D. Elsworth. 2010. Constraints on compaction rate and equilibrium in the pressure solution creep of quartz aggregates and fractures: Controls of aqueous concentration. J. Geophys. Res. 115. doi:10.1029/2009JB007118. Tuller, M., O. Dani, and L.M. Dudley. 1999. Adsorption and capillary condensation in porous media: Liquid retention and interfacial configurations in angular pores. Water Resour. Res. 35:1949–1964. Wei, C.-F., and K.K. Muraleetharan. 2006. Acoustical characterization of fluid-saturated porous media with local heterogeneities. Theory and application. Int. J. Solids Struct. 43:982–1008. Wei, C.-F., and K.K. Muraleetharan. 2007. Linear viscoelastic behavior of porous media with nonuniform saturation. J. Eng. Sci. 45:698–715.

Nitao, J.J., and J. Bear. 1996. Potentials and their roles in transport in porous media. Water Resour. Res. 32(2):225–250. doi:10.1029/95WR02715

Witteveen, P., A. Ferrari, and L. Laloui. 2013. An experimental and constitutive investigation on the chemo-mechanical behavior of a clay. Geotechnique 63(3):244–255. doi:10.1680/geot.SIP13.P.027

O’Carroll, D.M., T.J. Phelan, and L.M. Abriola. 2005. Exploring dynamic effect in capillary pressure in multistep outflow experiments. Water Resour. Res. 41. doi:10.1029/2005WR004010

Xu, T., and K. Pruess. 2001. Modeling multiphase non-isothermal fluid flow and reactive geochemical transport in variably saturated fracture rocks: 1. Methodology. Am. J. Sci. 301:16–33. doi:10.2475/ ajs.301.1.16

Passman, S.L., J.W. Nunziato, and E.K. Walsh. 1984. A theory of multiphase mixtures. Appendix C of Rational Thermodynamics, 2nd ed. Sringer-Verlag, New York.

Vadose Zone Journal

p. 21 of 21