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Computers and Geotechnics 34 (2007) 532–538

Technical note

Sedimentation–consolidation of a double porosity material

Henry Wong a, Chin J. Leo b,*, J.M. Pereira c, Ph. Dubujet d

a Departement Genie Civil et Batiment (URA CNRS 1652), Ecole Nationale des Travaux Publics de l’Etat, 2 rue Maurice Audin,

69518 Vaulx en Velin, Franceb School of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney, NSW 1797, Australia

c Ecole Nationale des Ponts et Chaussees (ENPC), Institut Navier – CERMES, 6-8 av Blaise Pascal, 77455 Marne-la-Vallee cedex 2, Franced Ecole Nationale d’Ingenieur de St-Etienne (ENISE), 58, rue Jean PAROT – 42023 St-Etienne cedex 2, France

Received 13 October 2006; received in revised form 5 December 2006Available online 1 February 2007

Abstract

This paper studies the sedimentation–consolidation of a double porosity material, such as lumpy clay. Large displacements and finitestrains are accounted for in a multidimensional setting. Fundamental equations are derived using a phenomenological approach andnon-equilibrium thermodynamics, as set out by Coussy [Coussy, Poromechanics, Wiley, Chichester, 2004]. These equations particulariseto three non-linear partial differential equations in one dimensional context. Numerical implementation in a finite element code is cur-rently being undertaken.� 2006 Published by Elsevier Ltd.

Keywords: Sedimentation; Consolidation; Double porosity; Porous media

1. Introduction

It is quite common to find in porous materials an inter-weaving system of preferential pathways which facilitatesthe flow of fluid through the material. Fractured rock massis an evident example, and lumpy clays used in land recla-mation is another. In the earlier, the fractures are muchmore permeable to fluid flow than the intact host rock,while in the latter the inter-lump permeability of the clayis far higher than its intra-lump permeability. In thesematerials, the classic single porosity model for simulatingflow has been found to be deficient, hence the concept ofdouble porosity has been suggested by many investigators.

A double porosity model to describe flow in the frac-tured porous media in the petroleum industry was first pro-posed by Barenblatt et al. [2] and Warren et al. [21]. Thefractured porous media was deemed to consist of two over-lapping systems, one representing the fracture network andthe other the porous blocks hence giving rise to the term

0266-352X/$ - see front matter � 2006 Published by Elsevier Ltd.

doi:10.1016/j.compgeo.2006.12.001

* Corresponding author.E-mail address: c.leo@uws.edu.au (C.J. Leo).

‘‘double porosity’’. As the porous media was assumed tobe rigid, the coupling between fluid flow and mechanicaleffects was ignored. This coupling was later introduced byDuguid [6], Duguid et al. [7], Aifantis [1], Khaled [8], Wil-son et al. [22] and Khalili et al. [9] among others. The dou-ble porosity model was later adapted to the consolidationof lumpy clay in land reclamation works by Nogami et al.[14], who neglected the effects of self-weight, which is inprinciple the driving force of consolidation. Yang et al.[23] extended the theory to include the important effectsof self-weight. These authors, however, assumed small dis-placements and small strains, which would appear unreal-istic on account of the large change in porosity typicallyinvolved. The sedimentation process prior to consolidationwas also not considered.

The present paper proposes an extension of the modelsdeveloped previously to describe the sedimentation–consolidation behaviour of blocky soft porous materialssuch as lumpy clays based on the concept of double poros-ity. As such, the extension in the present paper unifies thesedimentation and consolidation processes of a doubleporosity material, and also includes formulation in Eulerian

H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538 533

and Lagrangian frameworks to properly take into accountthe finite strains and displacements which previous studieshave not undertaken. The model equations are obtainedthrough a general multidimensional approach based onthe principle of non-equilibrium thermodynamics and canbe easily specialised to the case of small strains. Quiteimportantly, the proposed model considers the critical cou-pling effect due to deformation compatibility between themacro and micro pores within the system (i.e. preferentialpathways and the porous lumps or blocks). This crucialcoupling has been suggested by Khalili et al. [10], Tuncayet al. [20], Loret et al. [13], Khalili et al. [11], Callariet al. [4], Pao et al. [16] among others. Khalili [12] has alsoshown that neglecting this cross coupling effect can lead tospurious numerical results particularly in the early timeresponse of the system.

In this paper, summation convention on repeated indi-ces is adopted unless otherwise stated. Positive stressesand strains are taken to mean tensile stresses and elonga-tions, while fluid pressures are taken as positive.

2. Main assumptions and fundamental equations

Following the same approach as in [3,15], a commonformulation at the macroscopic scale will be sought todescribe the hydromechanical phenomena inside both sed-imentation and consolidation zones. This macroscopicformulation will be developed on the averaged field quan-tities inside a typical Representative Elementary Volume(REV), using the framework set out by Coussy [5].Fig. 1 shows a typical configuration of a soft blocky por-ous system, such as lumpy clay, for which the presentmodel is applicable. Isolated and falling independentlyin the sedimentation zone, the clay lumps come into con-tact and interact with each other in the consolidationzone. In both zones, the system of clay lumps saturatedwith water represents a double porous network. At aspace scale an order of magnitude higher than the sizeof individual clay lumps (scale of a REV in Fig. 1), thematerial is idealised as a triphasic material, representedas the superposition of three continua [5], namely theintra-lump fluid phase ‘‘1’’, the inter-lump fluid phase

sea level

Se

X,xdredged claylum s

representative elementaryvolume (REV) in theconsolidation zone

(REV) in thesedimentationzone

intra-lvoids

Fig. 1. Conceptual representation of the sedim

‘‘2’’ and the solid skeleton ‘‘s’’. Eulerian porosities n1,n2 and ns = 1 � n1 � n2 are defined such that, within agiven elementary volume dXt in the actual configuration,the volume of intra-lump fluid, inter-lump fluid and solidskeleton are, respectively, given by n1dXt, n2dXt andnsdXt. The sum n = n1 + n2 gives the total Eulerianporosity. Transition from the sedimentation zone to theconsolidation zone is supposed to take place when theinter-lump porosity falls below a critical value. Each ofthe two fluid phases is assumed to form a continuous flownetwork and interact with each other through a massexchange term. Each of the three phases has an indepen-dent trajectory described by velocity fields v1, v2 and vs.Since we are mainly interested by the solid skeleton, thedescription will be focused on the movement of the solidparticles; the initial position at time t = 0 (Lagrange coor-dinates) of a skeleton particle is denoted by X and itsactual position at time t by x (Eulerian coordinates), thecorresponding displacement being U = x � X, with

vs ¼dsU

dt¼ _U ð1Þ

where ds/dt or a dot above a variable represents the mate-rial derivative with respect to the solid phase. We will useuppercase letters to denote vectorial operators in Lagrang-ian coordinates and small letters in Eulerian coordinates.Using this notation, the material derivative is defined as

_G ¼ dsGdt¼ oG

otþ vs � gradG ð2Þ

where grad is the gradient operator with respect to Euleriancoordinates x. The second order deformation gradient ten-sor is denoted by:

F ¼ 1þGradðUÞ ð3Þ

where 1 is the second order identity tensor and Grad thegradient operator with respect to Lagrangian coordinatesX. The Jacobian J of the transformation of the skeletonis defined as:

J ¼ detF;dsJdt¼ Jdivvs ð4Þ

a bed

inter-lumpvoids 2

consolidationzone

sedimentationzone

clear water zone

xc t

H

ump1)

x t

entation–consolidation in a lumpy clay.

534 H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538

In other words, a skeleton particle initially of volume dX0

will occupy a volume dXt = JdX0 at time t. The Green–Lagrange strain tensor, denoted by D, is related to J by

D ¼ ðtF � F� 1Þ=2; _J ¼ JðF�1 � tF�1Þ : _D ð5ÞThe movement of the fluid phases is described by two Eule-rian fluid mass fluxes relative to the solid skeleton, wa:

wa ¼ naqfðva � vsÞ; a ¼ 1; 2 ðno summation over aÞ ð6ÞWe will suppose in the sequel that the fluid filling up the in-tra-lump and inter-lump voids is incompressible so that thefluid mass density qf is constant. Based on the definition ofthe Jacobian J, we also define the Lagrangian porosities:

/a ¼ Jna ða ¼ 1; 2Þ; / ¼ /1 þ /2; / ¼ Jn ð7Þsuch that /adX0 = nadXt is the volume of fluid phase a indXt. The Lagrangian fluid mass fluxes Ma are defined suchthat Ma Æ NdA = wa Æ nda (which implies DivMa = Jdivwa),where da with unit normal n is the oriented material surfaceat time t which measures dA with unit normal N at t = 0.Recalling that nda = JtF�1 Æ NdA, we have

Ma ¼ JF�1 � wa ð8Þ

3. Mass balance equations

Taking into account the fluid incompressibility, the massbalance of fluid phases writes:

ona

otþ divðnavaÞ ¼ m!aJ�1q�1

f ;

a ¼ 1; 2 ðno summation over aÞ ð9Þ

where m!a dX0 dt is the fluid mass increase of phase a insidethe overall current volume dXt during time dt, due to ex-changes between the two fluid phases. Overall fluid massconservation implies:

m!1 þ m!2 ¼ 0 ð10Þ

The mass of solid skeleton contained inside an elementaryvolume dXt following the movement of solid skeleton isinvariant, hence:

ms dX0 ¼ qsðJ � /ÞdX0 ¼ qsð1� nÞdXt ¼ qsð1� /0ÞdX0

ð11Þ

Here ms and qs are, respectively, the solid mass content perunit initial volume and the density, taken to be constantsfor incompressible solid phase. We thus deduce that:

ms=qs ¼ J � / ¼ ð1� nÞJ ¼ ð1� /0Þ;J ¼ ð1� /0Þð1� nÞ�1 ¼ 1þ /� /0 ð12Þ

The fluid mass of phase a inside an elementary volume dXt

may be expressed via another Lagrangian variable asmadX0 = qf/adX0 = qfnadXt, hence:

ma ¼ qf/a ¼ Jqf na ð13Þ

is the mass content of fluid phase a per unit initial overallvolume. The fluid mass conservation relation (9) can alsobe expressed in terms of Lagrangian variables, using (7),(8) and (13):

_ma þDivMa ¼ m!a ð14Þ

4. Momentum balance equations

Neglecting acceleration terms, the momentum balancein terms of Eulerian variables writes:

divrþ qg� J�1m!2ðv2 � v1Þ ¼ 0;

q ¼ ð1� nÞqs þ nqf ð15Þ

where q is overall density of the mixture and g is the grav-ity. In Lagrangian variables, we have:

DivðF � pÞ þ ðms þ mfÞg� m!2ðv2 � v1Þ ¼ 0;

mf ¼ m1 þ m2 ð16Þ

Recall that the Piola–Kirchoff stress tensor p and Cauchystress tensor r are related by:

r ¼ 1

JF � p � tF ð17Þ

5. Thermodynamics analysis

Neglecting the acceleration terms (see [5,17] for details),the dissipation per unit current overall volume can bedecomposed into four components (skeleton, phasechange, mass and heat transfer):

u ¼ us þ u! þ ut þ uh

¼ r : gradðvsÞ þ la _maJ�1 � s _T � _w� wdivðvsÞ

þ m!2

Jl1 þ ðv1 � vsÞ2

2

!� l2 þ ðv2 � vsÞ2

2

!" #

þX

a

�sagradT � gradla þ g½ � � wa

þ � 1

Tq � gradT

� �P 0 ð18Þ

where la and sa are the mass-based chemical potential andentropy of fluid phase ‘‘a’’ and w the total free energy perunit current volume. Multiplying by the Jacobian J, anddenoting the Lagrangian densities S = Js, W = Jw andUa = Jua (i.e. per unit initial volume), the Lagrangianequivalence of (18) is

U ¼ Us þ U! þ Ut þ Uh ¼ p : _Dþ la _ma � S _T � _W

þ m!2 l1 þ ðv1 � vsÞ2

2

( )� l2 þ ðv2 � vsÞ2

2

( )" #

þX

a

�saGradT �Gradla þ g � F½ � �Ma

þ � 1

TQ �Grad

� �P 0 ð19Þ

H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538 535

where we have accounted for 2 _D ¼ tF � gradvs þ tgradvsð Þ�F and p = JF�1 Æ r Æ tF�1. From now on, we will assumeisothermal conditions dT = 0 (hence uh = 0). We willadopt herein the classic decoupling of the various compo-nents of dissipation and will require each of them to beindependently non-negative. Recall the following classicthermodynamics relations of fluids:

wa ¼ ea � Tsa; la ¼ wa þ P a=qf ;

dea ¼ �P adð1=qfÞ þ T dsa ð20Þ

Here, Pa, wa and ea are, respectively, the fluid pressure, thefree and internal energy per unit mass of fluid phase ‘‘a’’.From (20), we deduce that:

dwa ¼ �P ad1

qf

� �� sadT ; dla ¼

dP a

qf

� sadT ð21Þ

Under isothermal conditions, dT = 0, we get:

dla ¼ dP a=qf ð22ÞIn view of the last result, the non-negativity of ut can besatisfied by the generalised Darcy’s law:

wa ¼ �ka½gradP a � qfg� ð23Þwhere ka = Ka/qa, and Ka (in m/s) is the classic hydraulicconductivity. The non-negativity of u!(ouU!), neglectingthe relative velocities and on account of (22), leads to theinequality:

m!2

qf

ðP 1 � P 2ÞP 0 ð24Þ

which can be satisfied by the following phenomenologicalrelation:

m!2 ¼ �m!1 ¼ xðP 1 � P 2Þ ð25Þwhere the positive constant x is the exchange coefficient.Note that the above relation has already been suggestedby [5,10,23] in an empirical manner. Our theoretical devel-opment shows that this relation can be derived naturallyfrom thermodynamic principles. We now decompose thetotal free energy into the sum of the skeleton free energyand the fluid free energies W = Ws + mawa. Substituting thisinto the expression of the dissipation above, and on ac-count of (20), we get:

Us ¼ p : _Dþ P a_/a � _Ws P 0 ð26Þ

This form suggests that the natural independent variablesof Ws are D, /a and v, so that:

Us ¼ p� oWs

oD

� �: _Dþ

Xa

P a �oWs

o/a

� �_/a �

oWs

ov_v P 0

ð27Þwhere v is the internal variable used to describe irreversibil-ities. This leads to the classic state equations:

p ¼ oWs

oD; P a ¼

oWs

o/a

ð28Þ

and the remaining dissipation term:

Us ¼ �oWs

ov_v P 0 ð29Þ

Assuming reversible behaviour so that Us = 0 and thedependence on the internal variables v disappears, the sec-ond order cross derivatives being independent of the orderof differentiation leads to the Maxwell equations:

oP a

oDij¼ o2Ws

oDijo/a

¼ o2Ws

o/aoDij¼ opij

o/a

¼ Baij ð30Þ

Differentiating pij and Pa then gives:

_pij ¼ L0ijkl_Dkl þ Ba

ij_/a ð31Þ

_P a ¼ Baij

_Dij þ rak_/k ð32Þ

with L0ijkl ¼ o2Ws

oDijoDkl; rak ¼ o2Ws

o/ao/k.

Using the Legendre transformation Gs = Ws � Pa/a andsubstituting into (26), we can show that:

_pij ¼ Lijkl_Dkl � ba

ij_P a ð33Þ

_/a ¼ baij

_Dij þ Rak_P k ð34Þ

Lijkl ¼o2Gs

oDijoDkl; � o/a

oDij¼ o2Gs

oDijoP a¼ opij

oP a¼ �ba

ij;

Rak ¼ �o2Gs

oP aoP kð35Þ

For classic isotropic behaviour, we have Lijkl = l(dikdjl +dildjk) + kdijdkl and ba

ij ¼ badij, hence:

_pij ¼ ð2ldikdjl þ kdijdklÞ _Dkl � ba_P adij ð36Þ

_/a ¼ ba_Dmm þ Rak

_P k ð37Þwhere ba is a priori positive as a positive pressure incrementshould lead to an algebraic stress reduction in (36), due tothe traction-positive sign convention. Through the Max-well symmetry relation (35), this leads to the consistent re-sult that an overall volume increase will lead to increase inporosities according to (37). Note that constitutive rela-tions (36) and (37) are also consistent with the previouswork of Khalili et al. [10] under small strains. Their analyt-ical formulae therefore furnish a first estimate for theparameters ba and Rak under large strains. Eq. (36) canbe rewritten as:

ðpij þ baP adij_Þ ¼ Lijkl

_Dkl ¼ ð2ldikdjl þ kdijdklÞ _Dkl ð38Þsuggesting the following effective stress:

p0ij ¼ pij þ baP adij ð39Þ

6. One dimensional consolidation under oedometricconditions

In one dimensional oedometric conditions, only the dis-placement in the direction x1 = x (X1 = X) is non-zero. Sim-ilarly only the strain component D11 = D is non-zero.Denoting r11 = r and p11 = p, we deduce from (17) and (8)that r = Jp and wa = Ma. Neglecting the momentum contri-bution m!a(v2 � v1), Eqs. (14)–(16) and (23) simplifies to:

536 H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538

_ma þoMa

oX¼ m!a ð40Þ

orox� qg ¼ 0 ð41Þ

oðJpÞoX

� ðms þ mfÞg ¼ 0 ð42Þ

wa ¼ Ma ¼ �kaoP a

oxþ qfg

� �¼ �ka

oP a

oXJ�1 þ qfg

� �;

a ¼ 1; 2 ðno summation over aÞ ð43Þ_/a ¼ ba

_Dþ Rak_P k ð44Þ

Following [5], the constitutive law for the skeleton can bewritten as:

p0 ¼ pþ baP a ¼ E-ð/Þ ð45Þwhere E is a reference oedometric modulus. We will assumein the sequel that the clay lumps come into contact witheach other and consolidation starts when the porosity /falls below a critical value /c. Hence, -(/) will be zerofor / > /c, and starts to climb up when / falls below /c.Eqs. (1), (4), (5) and (12) allow to deduce the followingrelations in one dimensional cases:

D ¼ 1

2ðJ 2 � 1Þ; _D ¼ J _J ; J ¼ 1þ oU

oX;

_J ¼ o2U

oXot;

oJoX¼ o

2U

oX 2; / ¼ oU

oXþ /0;

o/oX¼ o2U

oX 2þ o/0

oXð46Þ

These relations together with (37) allow J, /a, na to be ex-pressed in terms of U. Substitution of (13), (25), (43) and(46) into (40), we get two equations:

qf b1 1þ oUoX

� �o

2UoXot

þ R11_P 1 þ R12

_P 2

� �

� o

oXk1

oP 1

oX1þ oU

oX

� ��1

þ qfg

!" #¼ �xðP 1 � P 2Þ

ð47Þ

qf b2 1þ oUoX

� �o

2UoXot

þ R21_P 1 þ R22

_P 2

� �

� o

oXk2

oP 2

oX1þ oU

oX

� ��1

þ qfg

!" #¼ xðP 1 � P 2Þ

ð48Þ

A third equation is obtained by developing the momentumbalance equation (42), using (45):

1þ oUoX

� �E

o-ð/ÞoX

� oðbaP aÞoX

� �

þ E-ð/Þ � baP að Þ o2U

oX 2� ms þ qf

oUoXþ /0

� �� �g ¼ 0

ð49Þ

The three partial differential equations (47)–(49), togetherwith the adequate boundary conditions, constitute a

three-fields-problem on (U,P1,P2). Dependency of ka onthe Lagrangian porosity /a (say ka = kad(/a)) can easilybe accounted for, using:

okað/aÞoX

¼ ka0

ddð/aÞd/a

o/a

oX

¼ ka0

ddð/aÞd/a

ba

o2U

oX 2þ o/0

oX

� �ð50Þ

It is more usual to make ka depend on the Eulerian poros-ity na (say ka = ka0d(na)), the calculations will be slightlymore complicated but no theoretical difficulties arise

okaðnaÞoX

¼ ka0ð/aÞddðnaÞ

dna

ona

oX

¼ ka0

ddðnaÞdna

baJ�2 ð1� /0Þo2UoX 2

þ Jo/0

oX

� �ð51Þ

One example of the relative permeability function d(na) isthe Kozeny–Carman relation [5]:

dðnaÞ ¼na

na0

� �31� na0

1� na

� �2

ð52Þ

applicable to granular materials where na0 is the value ofthe Eulerian porosity at a given reference state (at whichka = ka0). As for soft clayey materials, the following func-tion between void ratio and logarithmic relative permeabil-ity often applies:

ea � ea0 ¼ cklog10ðdÞ ð53Þwhere ck is slope of the ea � log10(ka) relationship (e.g.Tavenas et al. [18]), ea0 is the void ratio corresponding tona0, and the void ratio and the porosity are related by thefollowing expression: na = ea/(1 + ea).

7. Sedimentation zone

In the sedimentation zone, the clay lumps are not in con-tact and therefore the intra-lump voids do not form a con-tinuous flow network. The double porosity modeldescribed above cannot simulate this regime consistently.In keeping with the sedimentation analysis of [5], we willsuppose:

(H1) No consolidation of clay lumps takes place in the sed-imentation zone and the intra-lump voids remainconstant, in other words /1 = /10. Hence M1 =w1 = 0.

(H2) v1 = vs, the clay lumps move as undeformable solidgrains through the inter-lump fluid. They appear atthe macroscopic scale as grains falling like rigidbodies with apparent density:

q0s ¼ð1� /10 � /20Þqs þ qf/10

1� /20

ð54Þ

(H3) The effective stress within clay lumps is zero, whichimplies P1 = P2, noted P. There is no contact betweenclay lumps hence the overall effective stress is alsozero, thus r 0 = r + P = 0, implying:

H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538 537

r ¼ �P ð55ÞThe mass conservation of each phase {s, 1,2} with no massexchange implies:

oð1� n1 � n2Þot

þ o

oxðð1� n1 � n2ÞvsÞ ¼ 0;

on1

otþ o

oxðn1v1Þ ¼ 0;

on2

otþ o

oxðn2v2Þ ¼ 0 ð56Þ

Their sum gives: ovG/ox = 0 with vG = (1 � n)vs + n1v1

+ n2v2 implying that vG is constant inside the sedimenta-tion zone. The same conclusion can be reached by addingthe three mass conservation equations in the consolidationzone. In this latter zone, the mass exchange between fluidphases 1 and 2 is non-zero but they cancel out each other,leading to the same conclusion that vG is constant in theconsolidation zone. Since at the bottom (X = x = 0), allvelocities vanish, we therefore conclude that vG = 0 every-where (see for example [3]). Moreover, Eqs. (40)–(42) holdwith m!a ¼ 0, while (43) only holds for a = 2. Combining(41) and (55) gives:

oPox¼ �qg ð57Þ

Substitution into (43) for a = 2 yields:

w2 ¼ M2 ¼ k2ðq� qfÞg ¼ k2ð1� nÞðqs � qfÞg ð58ÞThe time derivative of (13), on account of (H1) and (12)leads to:

_m2 ¼ qf_/2 ¼ qf

_/ ¼ qfð1� /0Þds

dt1

1� n

� �ð59Þ

Substitution of (58) and (59) into (40), with m!2 ¼ 0 finallyleads to the equation sought:

qfð1� /0Þo

ot1

1� n

� �X¼cte

þ ðqs � qfÞgo

oXfk2ð1� nÞg ¼ 0

ð60ÞBy inspection, we find that if /2 is independent of spaceand time, then _/ ¼ _J ¼ 0. This implies that J = 1, andn2 = /2 are constant, so is k2, hence the above equationis identically satisfied. Therefore, /2 = cte is a particularsolution in the sedimentation zone. This would indeed be

C ¼ dX c

dt¼

k1ð/10Þ oP 1

oX J�1 þ qfg� �

þ k2ð/2cÞ oP 2

oX J�1 þ qfg� �

þ k20ð1� n0Þðqs � qfÞgqfð/20 � /2cÞ

ð67Þ

the case if initially the clay lumps are dispersed in a homo-geneous manner, with a uniform porosity /20. We willrestrict ourselves hereafter to this particular case in orderto compare the present formulation with previous works.With vG = 0 and (H2), we deduce that vs = �n2(v2 � vs).The right hand member can be explicitly determined using(6) and (58). This allows to deduce the solid skeleton veloc-

ity and subsequently the displacement in the sedimentationzone:

vseds ¼ �k20

ð1� n0Þqf

ðqs � qfÞg; U ¼ vseds t ð61Þ

In other words, the upper boundary of the sedimentationzone goes down with a constant speed of vsed

s , so that:

xtðtÞ ¼ xðH ; tÞ ¼ H þ vseds t ð62Þ

Above xt(t) is the clear water zone in which no solid parti-cle is present, hence the fluid pressure inside the sedimenta-tion zone xc(t) < x < xt(t) is given by:

P 1 ¼ P 2 ¼ P ¼ qf gðH � xtðtÞÞ þ q0gðxtðtÞ � xÞ ð63Þwith q0 = (1 � /0)qs + /0qf the initial overall density of thedouble porous media.

8. Sedimentation–consolidation interface

The sedimentation and consolidation zones are sepa-rated by a moving interface xc(t) of which the Lagrangiancounterpart is Xc(t). The interface advancement speed isgoverned by a jump condition which results from fluidmass conservation [5]:

sM � mfCtX c¼ 0 ð64Þ

where C is the Lagrangian speed of advancement of the dis-continuity and sfbX = f(X+) � f(X�) denotes the jump of afunction across the position X. Classical results like[3,5,15,19] show that /2 has a jump-discontinuity. In ourcase, /1 should be continuous as the effective stress ‘‘just’’starts to become non-zero at X�c so as to induce consolida-tion of the clay lumps. On the sedimentation side Xþc :

M ¼ M2 ¼ k20ð1� n0Þðqs � qfÞg;

mf ¼ qf/ ¼ qfð/10 þ /20Þ ð65Þ

while on the consolidation side X�c :

M ¼ M1 þM2; Ma ¼ �kað/aÞoP a

oXJ�1 þ qfg

� �;

mf ¼ qf/ ¼ qfð/10 þ /2cÞ ð66Þ

Substitution of (65) and (66) into (64) yields the Lagrang-ian advancement speed of the interface:

9. Continuity and boundary conditions

Eqs. (47)–(49) need to be solved with adequate bound-ary conditions. At the interface xc(t), the three unknowns(U,P1,P2) are continuous, given by (61) and (63). Atthe bottom X = x = 0, the impervious base conditionwrites:

538 H. Wong et al. / Computers and Geotechnics 34 (2007) 532–538

oP a

oXJ�1 þ qf g ¼ 0 ða ¼ 1; 2Þ ð68Þ

while the displacement U must satisfy:

U ¼ 0 ð69ÞOwing to the highly non-linear character of the system ofequations involved, they have to be solved numerically.

10. Conclusions

A theoretical model describing the sedimentation–con-solidation of a soft double porosity material such as lumpyclay has been introduced starting from a non-equilibriumthermodynamics approach. The formulation includes thecoupling effect due to deformation compatibility betweenthe macro and micro pores within the double porosity sys-tem. Both Eulerian and Lagrangian descriptions are usedin this model, taking into account large displacementsand finite strains. Equations of general validity areobtained in a multi-dimensional setting, before particula-rising to the case of one dimension. In this case, the theo-retical developments result in a system of three partialdifferential equations to describe the large strain consolida-tion and one partial differential equation to describe thesedimentation, with an additional equation on the interfaceposition. All equations are highly non-linear and thereforecan only be solved numerically. Their numerical implemen-tation into a finite element code is on-going. We leave thedetails of the variational formulation, finite element dis-cretisation and numerical results to a later paper.

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