Multiphase flow in deforming porous material

Computers and Geotechnics 31 (2004) 237–250

www.elsevier.com/locate/compgeo

Multiphase flow in deforming porous material

B.A. Schrefler *, F. Pesavento

Department of Structural and Transportation Engineering, University of Padova, via F. Marzolo 9, 35131 Padova, Italy

Abstract

Models for thermo-hydro-mechanical behaviour of saturated–unsaturated porous media are reviewed. The necessary balance

equations are derived using averaging theories. Constitutive equations are obtained using the Coleman–Noll procedure and ther-

modynamic equations for the model closure are introduced. A simplified form of the governing equations is then solved numerically

and the numerical properties are discussed. An example dealing with behaviour of concrete structures during tunnel fires concludes

the paper. The heat and mass transfer calculations in the tunnel needed as the input for the multiphase concrete model are also

shown. The behaviour of concrete under such situations, where very high temperatures are reached, can be satisfactorily simulated

only with an approach of the type presented here.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Multiphase; Interfaces; Concrete; High temperature; Damage; Fire; Tunnel; Radiation

1. Introduction

Multiphase porous media, i.e. porous media wherethe pores are filled by more than one fluid, are treated

here within the framework of averaging theories. In

particular the approach developed by Hassanizadeh

and Gray [1–5] is used. The isothermal case was shown

in Schrefler [6]. Here the approach is extended to non-

isothermal situations. It is recalled that interfaces be-

tween the constituents with their thermodynamic

properties are taken into account. In fact, fluids in aporous medium, such as gas and water, will remain

immiscible only if there are interfaces with non-zero

surface tension. If surface tension is zero, then capillary

pressure is zero too which means that the fluid pressure

will be equal all the time. Actually only averaging

theories such as that used here include explicitly in-

terfacial properties. As far as the constitutive rela-

tionships are concerned, limits to their form areobtained by systematic exploitation of the entropy in-

equality, following the procedure of Coleman and Noll

[7]. Particular attention will be paid to near equilibrium

results, which among others yield the well known laws

* Corresponding author. Tel.: +39-49-827-5611; fax: +39-49-827-

5604.

E-mail address: [email protected] (B.A. Schrefler).

0266-352X/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2004.01.005

of Darcy, Fick and Fourier, augmented by the con-

tributions of the interfaces. From the general mathe-

matical model a simplified one is extracted, which willbe solved numerically. Particular attention is focused

here on the boundary conditions: for a realistic simu-

lation of heat and mass transfer problems in deforming

porous media boundary conditions of the third type

are needed which include convective mass transfer and

convective and radiative heat transfer. The theoretical

developments in this paper are necessarily short. For a

full development the reader is referred to the papers byHassanizadeh and Gray [1–4], to Lewis and Schrefler

[8] and to Schrefler [6,9]. An example dealing with

behaviour of concrete structures during tunnel fires

concludes the paper. The heat and mass transfer cal-

culations in the tunnel needed as the input for the

multiphase concrete model are also shown. The be-

haviour of concrete under such situations, where very

high temperatures are reached, can be satisfactorilysimulated only with an approach of the type presented

here.

2. Macroscopic balance equations

For sake of brevity the microscopic balance equa-

tions for the constituents of the porous medium are

mail to: [email protected]

sw sg

238 B.A. Schrefler, F. Pesavento / Computers and Geotechnics 31 (2004) 237–250

omitted here as well as the kinematics. They can be

found e.g. in Lewis and Schrefler [8]. In this section the

macroscopic balance equations for mass, linear mo-

mentum and energy as well as the entropy inequality

are given, which have been obtained for the bulkmaterial of the phases and for the interfaces by sys-

tematically applying the averaging procedures to the

microscopic balance equations as outlined by Hassan-

izadeh and Gray [5]. The balance equations listed be-

low have been specialised for a deforming porous

material, where heat transfer and flow of water (liquid

and vapour) and of dry air are taking place, Lewis and

Schrefler [8]. The constituents are assumed to be im-miscible except for dry air and vapour, and chemically

non-reacting. The mixture of dry air and vapour

(moist air) will be simply called gas in the following.

All fluids are in contact with the solid phase. Disso-

lution of air in water is here neglected. Stress is defined

as tension positive for the solid phase, while pore

pressure is defined as compressive positive for the

fluids. In the averaging procedure volume density pa-rameters gp (volume fractions) appear which are

expressed in terms of commonly used variables in

multiphase flow.

For solid phase, gs ¼ 1� n where n ¼ dvw þ dvgð Þ=dvis porosity and dvp is the volume of constituent p within

a R.E.V. (representative elementary volume); for water

gw ¼ nSw, where Sw ¼ dvw= dvw þ dvgð Þ is the degree of

water saturation and for gas gg ¼ nSg with Sg ¼dvg= dvw þ dvgð Þ the degree of gas saturation. It follows

immediately that Sw þ Sg ¼ 1.

There appears also the specific surfaces of the inter-

faces aab, where the Greek letters refer to the bulk

phases involved. The inclusion of interface phenomena

which at first sight appear to be of secondary interest,

allow to treat the dependence of phase properties on

interface properties. At macroscale the system is mod-elled as the superposition of six continua: three phases

and three interfaces. At every spatial point average or

macroscopic properties are defined for each continuum

and the continua interact and exchange properties. Two

sets of balance equations are needed. One set is for the

bulk phases, the second is for the interfaces. The equa-

tions are listed next:

for solid

Ds 1� nð Þqs

Dtþ 1ð � nÞqs div vs ¼ esgs þ esws; ð1Þ

for liquid water

DwnSwqw

Dtþ nSwqw div vw ¼ ewgw þ ewsw; ð2Þ

for gas

DgnSgqg

Dtþ nSgqg div vg ¼ egwg þ egsg: ð3Þ

The mass source terms on the r.h.s. of Eqs. (1)–(3)

correspond to exchange of mass with interfaces sepa-

rating individual phases (phase changes) and couple

these equations with the corresponding balance equa-

tions written for the interfaces. These last ones may bewritten as

DabaabCab

Dtþ aabCab divwab ¼ �eaab � ebab þ eabwgs: ð4Þ

The last term in Eq. (4) describes mass exchange of the

interfaces with their contact line. Since we have three

phases composing the medium, there is only one contact

line. This contact line does not have thermodynamic

properties.The momentum balance equations for the bulk pha-

ses are

for solid

1ð � nÞqs Dsvs

Dt� div 1ðð � nÞtsÞ � 1ð � nÞqsg ¼ Ts

sg þ Tssw;

ð5Þ

for water

nSwqw Dwvw

Dt� div nSwtwð Þ � nSwð Þqwg ¼ Tw

wg þ Twws; ð6Þ

for gas

nSgqg Dgvg

Dt� div nSgtg

� �� nSg� �

qgg ¼ Tggw þ Tg

gs; ð7Þ

where ta is the partial stress tensor which is symmetric.

The r.h.s. terms in Eqs. (5)–(7) describe supply of

momentum from the interfaces, i.e. related to phase

changes. Analogous balance equations can be written

for the momentum of the three interfaces:

aabCab Dabwab

Dt� div aabsab

� �� aabCabgab

¼ � Taab

�þ eaabv

a;s�þ Tb

ab

�þ ebabv

b;s�

þ eaab�

þ ebab�wab;s þ sabwgs; ð8Þ

where sab is the surface stress tensor, which is also

symmetric.The last r.h.s. term corresponds to momentum supply

from the contact line �wgs� to the ab interface.

The energy balance equation for the bulk phases may

be written as follows

for solid

1ð � nÞqs DsEs

Dt� 1ð � nÞts : grad vs � div 1ðð � nÞqsÞ

� 1ð � nÞqshs ¼ Qs þ Qs ; ð9Þ

B.A. Schrefler, F. Pesavento / Computers and Geotechnics 31 (2004) 237–250 239

for water

nSwqw DwEw

Dt� nSwtw : grad vw � div nSwqwð Þ � nSwqwhw

¼ Qwws þ Qw

wg; ð10Þ

for gas

nSgqg DgEg

Dt� nSgtg : grad v

g � div nSgqg� �

� nSgqghg

¼ Qggs þ Qg

gw: ð11Þ

The source terms in Eqs. (9)–(11) describe supply of

heat to bulk phase from the interfaces, related to phase

changes. The energy balance equations for the three

interfaces read

aabCabDabEab

Dt�aabsab : gradwab�div aabqab

� ��aabCabhab

¼� Qaab

hþ Ta

ab � va;abþ eaab Ea;ab�

þ1=2 va;ab� �2�i

� Qbab

hþ Tb

ab � vb;abþ ebab Eb;ab�

þ1=2 vb;ab� �2�iþ Qab

wgs;

ð12Þ

where Ea;ab ¼ Ea � Eab.

The terms in square brackets in Eq. (12) describe theenergy supply from the bulk phase to the interface, energy

associated with momentum supply and energy related to

mass supply because of phase changes. The last r.h.s. term

is supply of heat to the interface from the contact line.

We assume that entropy fluxes are due solely to heat

input and the entropy external source terms are due only

to external energy sources. Thus, the entropy balance

may be expressed for the bulk phases as followsfor solid

1ð � nÞqs Dsks

Dt� div 1ð

�� nÞ q

s

hs

�� 1ð � nÞqs h

s

hs

¼ Ussg þ Us

sw þ Ks; ð13Þ

for water

nSwqw Dwkw

Dt� div nSw

qw

hw

� �� nSwqw h

w

hw

¼ Uwwg þ Uw

ws þ Kw; ð14Þ

for gas

nSgqg Dgkg

Dt� div nSg

qg

hg

� �� nSgqg h

g

hg

¼ Uggw þ Ug

gs þ Kg: ð15Þ

The two first terms in r.h.s. of Eqs. (13)–(15) describe

the entropy supply to the bulk phases from the inter-

faces, while the last one is the rate of net production of

entropy in the bulk phase.

Similarly, for the interfaces we have the following

three entropy balance equations

aabCab Dabkab

Dt� div aab

qab

hab

� �� aabCab h

ab

hab

¼ � Uaab

�þ eaabk

a;ab�� Ub

ab

�þ ebabk

b;ab�þ Uab

wgs þKab:

ð16Þ

The terms in parentheses in the r.h.s. of Eq. (16) are

supply of entropy from the interfaces and resulting from

mass supply (phase change), the last but one accounts

for entropy supply to the interface from the contact lineand the last one is the rate of net production of entropy

in the interface.

The terms related to exchange of mass, momentum,

energy and entropy between interfaces via the contact

lines must satisfy some restrictions, because the contact

lines do not possess any thermodynamic properties as

already stated. Thus, the following relations holdXab

eabwgs ¼ 0;

Xab

sabwgs

�þ eabwgsw

ab�¼ 0;

Xab

Qabwgs

hþ sabwgs � wab þ eabwgs Eab

�þ 1=2 wab

� �2�i ¼ 0;

Xab

Uabwgs

�þ eabwgsk

ab�¼ 0:

ð17Þ

3. The second law of thermodynamics

The balance laws must be supplemented with the

second law of thermodynamics, which states that forany process the rate of net entropy production must be

non-negative

K ¼ Ks þ Kw þ Kg þX

ab¼gs;gw;sw

Kab P 0; ð18Þ

where Kp is the rate of net production of entropy in the

bulk phases and interfaces and ab ¼ gs, gw, sw refer to

the interfaces between gas and solid, gas and water and

water and solid, respectively.

After introducing the balance laws into Eq. (18) and

substituting the internal energy by the Helmholtz free

energy, which is defined for the bulk phases as

Aa ¼ Ea � haka; a ¼ s;w; g ð19Þand for the interfaces as

Aab ¼ Eab � habkab; ab ¼ gw;ws; gs ð20Þan appropriate form of the entropy inequality (18) is

obtained [5]. The balance equations of mass, momentum

and energy must be supplemented by constitutive


equations describing the behaviour of individual phases.

In total there are 30 equations and the same number of

independent variables can be chosen as basic indepen-

dent fields. These field quantities or/and combinations

and space and time derivatives of them that are objectivecan enter as independent constitutive variables. Their

choice should be based on the expected behaviour of the

medium, as well as they should account macroscopically

for the microstructure due to the interfaces. For this last

reason for instance volume fractions, their gradients, the

specific surfaces of the interfaces and their gradients

may be added to the list of primary variables. This

augments accordingly the list of dependent variables toeliminate the ensuing equation deficit. The independent

variables are function of time and space. The list of in-

dependent variables chosen here is qa, Cab, va;s, wab;s, Ea,

ha, gradha, hab, gradhab, n, gradn, Sa, gradSa, aab,gradaab.

The remaining variables appearing in the balance

equations must be expressed in terms of the primary

unknowns and their derivatives. The equation deficit iseliminated by also requiring constitutive forms for some

of the time derivatives, (here of porosity, the degree of

water saturation and of specific surfaces of the interfaces

[5]) and by thermodynamic equilibrium equations. For

the list of dependent variables, for which constitutive

relations are needed, see [6].

Helmholtz free energy for the bulk phases is assumed

to have the following functional form which is particu-larly simple, but sufficient for our purpose

Aw ¼ Aw qw; hw; Swð Þ; ð21Þ

Ag ¼ Ag qg; hg; Sg� �

; ð22Þ

As ¼ As qs; hs;Es; Swð Þ; ð23Þ

and for the interfaces

Aab ¼ Aab Cab; hab; aab; Sw� �

;

where ab ¼ gw;ws; gs: ð24Þ

All remaining dependent variables are allowed to

depend on the complete set of independent variables

given above. Note that the assumptions (21)–(24) departfrom the principle of equipresence.

According to the principle of admissibility, the con-

stitutive postulates relating dependent to independent

variables must not violate the balance laws and the en-

tropy inequality. These requirements are satisfied using

the procedure proposed by Coleman and Noll [7]. The

rather lengthy transformations of the entropy inequality

necessary are omitted here. They can be found in [5,9].Following the procedure of Coleman–Noll the fol-

lowing non-equilibrium results are obtained:

ka ¼ � oAa

oha; a ¼ w; g; s; ð25Þ

kab ¼ � oAab

ohab; ab ¼ gw;ws; gs; ð26Þ

tw ¼ �pwI; ð27Þ

tg ¼ �pgI; ð28Þ

ts ¼ tse � psI; ð29Þ

where tse ¼ qs Fsð ÞT � oAs

oEs � Fs, is the effective stress tensor

of the solid phase, and ps qs; hs;Es; Swð Þ ¼ qsð Þ2 oAs

oqs the

thermodynamic pressure of the solid phase,

sab ¼ cabI; ð30Þwhere cab is the surface tension.

Some additional information can be obtained, whenexamining the system under consideration at equilib-

rium state where there is no relative movement of phases

and interfaces, degree of saturation with water (thus also

with gas) and porosity are constant, the phases and in-

terfaces have the same uniform temperature. At these

conditions the total rate of entropy production K equals

to zero, i.e. reaches its minimum value. Thus the nec-

essary and sufficient conditions for K to be at minimumat equilibrium are

oKozk

� �eq

¼ 0; k ¼ 1; . . . ; 46 ð31Þ

and

o2Kozk ozm

� �eq

be positive semi-definite;

k;m ¼ 1; . . . ; 46: ð32Þ

Application of restriction (31) to the entropy in-

equality allows to obtain the following main relationsvalid at equilibrium

psð Þeq ¼ Sgpg þ Swpw; ð33Þ

pc ¼ pgð � pwÞeq; ð34Þ

qsð Þeq ¼ qwð Þeq ¼ qgð Þeq ¼ 0; ð35Þ

qgwð Þeq ¼ qwsð Þeq ¼ qgsð Þeq ¼ 0: ð36Þ

The definition of capillary pressure, Eq. (34), shows

that pc depends on independent variables as follows

pc ¼ pc Sw; n; awg; aws; h; qw; qg;Cwg;Cws;Cgsð Þ: ð37Þ

At equilibrium, pc is given by Eq. (34). The capillary

pressure is commonly assumed to be a function of

Sw; qw, and qg; n and h. Eq. (37) shows that also inter-

facial areas and surface densities play a role. Cases

where surface densities may change and affect capillary

pressure are when surfactants are present and changethe character of the interfaces.


Eqs. (35) and (36) indicate that at equilibrium there is

no heat transfer within the phases and interfaces, what is

true for a wide class of practical problems, where a state

of ‘‘local thermal equilibrium’’ or, more general, ‘‘local

thermodynamic equilibrium’’ is assumed. Finally, atequilibrium the Gibbs free energy per unit mass for each

phase and interface will be equal.

For subsequent discussions attention is restricted

more to the mechanical aspect of the problem and less to

its thermal aspect. It is assumed that temperature dif-

ferences between phases at a macroscopic point are

negligible also near equilibrium which is acceptable for a

large class of problems. For these cases, a state of localthermal equilibrium prevails such that all phases and

interfaces will have the same temperature h at a point,

which although may still vary in space.

If the system is considered ‘‘near’’ equilibrium some

additional simplifications may be obtained, as far as

the constitutive functions are concerned. In such situ-

ations a linear dependence of constitutive functions

(describing thermodynamic flows) on primary variables(thermodynamic potentials) may be postulated. Some

of these linear relations are widely used in practice, like

for example Darcy�s law or Fick�s law for fluid flow

and Fourier�s law. For a complete description of the

procedure of linearization see [9]. The fact that the

constitutive assumptions are to be linearized does not

influence any conclusion that may be drawn concerning

the equilibrium state of the system (i.e. only the dy-namic state will be influenced, Gray and Hassanizadeh

[10,11]).

As pointed out in Section 1, for the model closure

state equations are still needed. Moist air (gas) in the

pore system is assumed to be a perfect mixture of two

ideal gases, dry air and water vapour. The equation of

perfect gas is hence valid

pga ¼ qgahR=Ma pgw ¼ qgwhR=Mw; ð38Þ

qg ¼ qga þ qgw; pg ¼ pga þ pgw;

Mg ¼qgw

qg

1

Mw

�þ qga

qg

1

Ma

��1

;ð39Þ

where R is the universal gas constant and Ma the molar

mass of the constituents.

The equation of state of water is of the formpw ¼ pw qw; T ; Swð Þ [10,11].

4. Heat and mass transfer in deforming partially saturated

geomaterials

From the equations listed in the previous sections a

model for heat and mass transfer in partially saturatedgeomaterials will now be established. The system of

governing equations of this model will then be solved

numerically. It is assumed that the system is near equi-

librium as explained in the previous sections. Compared

to the general theory outlined in the previous sections

the model is simple, for instance it does not consider the

balance equations for the interfaces. However comparedto the models currently in use in the geomechanics

community, it is rather advanced. It may be considered

as a first step in the effort to obtain such thermody-

namically based numerical models. Its complexity can

then be augmented at will following the equations given

in the theoretical section.

4.1. Simplified field equations

The model is built in the following way. From the

linearised equations Fick�s law and Fourier�s law are

chosen neglecting interfacial terms. As far as Darcy�sequation is concerned, the linear momentum balance

equation for the fluid phases is used directly with an

appropriate constitutive assumption for the momentum

exchange term in the form

gaqa ta ¼ �Ragavsa þ pa gradga; ð40Þwhere Ra is given by

Ka ¼ ga Raa

� ��1 ¼ k

lpqa; ga; Tð Þ: ð41Þ

These equations are supplemented by the mass bal-

ance equations for solid (1), water (2), vapour and gas

(3), the sum of the linear momentum balance equation

for the constituents (5)–(7), and the sum of the energy

balance equation for the constituents (9)–(11), which has

been transformed into an enthalpy balance (see [8]).

Further, for the fluid stresses, the solid pressure and thecapillary pressure the respective equilibrium values are

taken, Eqs. (27), (28), (33) and (34). The effective stress

is assumed in the form:

t ¼ 1ð � nÞtse � I Sgpg�

þ Swpw�: ð42Þ

Finally the state equations for water and gas, Eqs.

(38) and (39) of Section 3 are used and the capillary

pressure saturation relationship (37) (with a much sim-

pler functional dependence suggested by experimentalobservations pc ¼ pc Sw; hð Þ). Actually, the inverse of this

function is used because of the choice of the primary

variables.

Vapour pressure has to be expressed as function of

the relevant primary variables capillary pressure and

temperature. This is obtained by means of the Kelvin–

Laplace equation, which is a thermodynamic relation at

equilibrium (see [12,13]).

pgw

pgws¼ exp

pcMw

qwRh

� �; ð43Þ

where the vapour saturation pressure pgws is obtained

from the Clausius–Clapeyron equation.


Some rearrangements of these equations (see [8]) yield

the following field equations. The macroscopic mass

balance equations are:

for the solid phase

1� nqs

Dsqs

Dt�Dsn

Dtþ 1ð � nÞdiv vs ¼ 0; ð44Þ

for dry air

� nDsSwDt

�bs 1ð � nÞSgDsTDt

þ Sgdivvsþ Sgn

qga

Ds

DtMa

hRpga

� �

� 1

qgadiv qgMaMw

M2g

Dg gradpga

pg

� �" #þ 1

qgadiv nSgqgavgs

� �¼ 0; ð45Þ

for the water species, i.e. liquid water and vapour to-

gether

n qwð � qgwÞDsSwDt

� bswg

DsTDt

þ qgwSg�

þ qwSw�

� div vs þ nqwSwKw

Dspw

Dtþ Sgn

Ds

DtMw

hRpgw

� �

� div qg MaMw

M2g

Dg gradpgw

pg

� �" #

þ div qgw kkrg

lg½

� gradpg þ qg gð � as � agsÞ�

�

þ div qw kkrw

lw½

� gradpw þ qw gð � as � awsÞ�

�¼ 0; ð46Þ

where

bswg ¼ bs 1ð � nÞ Sgqgw�

þ qwSw�þ nbwq

wSw; ð47Þ

with bp the thermal expansion coefficients.

The linear momentum balance equation for fluids is

gpvps ¼ kkrp

l½ � gradpp þ qp gð � as � apsÞ�; ð48Þ

and for the multiphase medium

� qas � nSwqw aws½ þ vws � grad vw� � nSgqg

� ags½ þ vgs � grad vg� þ div tþ qg ¼ 0: ð49Þ

Finally the enthalpy balance for the multiphase

medium may be written as

qCp

� �eff

oTot

þ qwCwp v

w�

þ qgCgpv

g�� gradT

� div veff gradTð Þ ¼ � _mDHvap; ð50Þ

where

qCp

� �eff

¼ qsCsp þ qwC

wp þ qgC

gp ;

veff ¼ vs þ vw þ vg;

DHvap ¼ H gw � Hw;

ð51Þ

with Hp the specific enthalpy and Cpp the heat capacity.

4.2. Initial and boundary conditions

The initial conditions specify the full fields of gas

pressure, capillary or water pressure, temperature, dis-

placements and velocities

pg ¼ pg0; pc ¼ pc0; T ¼ T0; u ¼ u0; _u ¼ _u0; at t ¼ t0:

ð52ÞThe boundary conditions are formulated for vol-

ume-averaged quantities, which are continuous fields.

Thus they concern an arbitrary boundary of the anal-

ysed space domain which corresponds to the ‘‘macro-

scopic’’ external surface of the porous body coinciding

with its fixed boundary considered during volumeaveraging.

The boundary conditions can be imposed values on

Cp or fluxes on Cqp, where the boundary C ¼ Cp [ Cq

p.

The imposed values on the boundary for gas pressure,

capillary or water pressure, temperature and displace-

ments are

pg ¼ pg on Cg; pc ¼ pc on Cc;

T ¼ T on CT ; u ¼ u on Cu:ð53Þ

The volume averaged flux boundary conditions for

water species and dry air mass balance equations and

the energy conservation equation, to be imposed at the

interface between the porous media and the surrounding

fluid are as follows

qga�vg�

� qg�vwv�� n ¼ qga on Cq

g;

qwv�vg�

þ qw�vw þ qg�vwv�� n

¼ bc qwv�

� qwv1�þ qwv þ qw on Cq

c ;

� qw�vwDhvap�

� veff rT�� n

¼ ac Tð � T1Þ þ r0e T 4�

� T 41�þ qT on Cq

T ; ð54Þ

where n is the unit vector, perpendicular to the surface

of the porous medium, pointing toward the surrounding

gas, qgw1 and T1 are, respectively, the mass concentration

of water vapour and temperature in the undisturbed gas

phase distant from the interface, ac, bc, r0 and e are

convective heat and mass transfer coefficients, the

Boltzmann constant and the emissivity, while qga, qwv, qw

and qT are the imposed dry air flux, imposed vapour

flux, imposed liquid flux and imposed heat flux,

respectively.

The convective term on the r.h.s. of the last of

Eq. (54) corresponds to Newton�s law of cooling and

describes the conditions occurring in most practical sit-

uations at the interface between a porous medium and

the surrounding fluid (air in this case).The traction boundary conditions for the displace-

ment field are


t � n ¼ ^t on Cqu; ð55Þ

where ^t is the imposed traction.

4.3. Numerical solution

The governing equations of the model are discretisedin space by means of the finite element method [14,15].

The unknown variables are expressed in terms of their

nodal values as,

pg tð Þ ¼ Np �pg tð Þ; pc tð Þ ¼ Np �p

c tð Þ;T tð Þ ¼ Nt

�T tð Þ; u tð Þ ¼ Nu �u tð Þ:ð56Þ

The variational or weak form of the model equations,

including also the ones required to complete the model,

was obtained in [8,16] by means of Galerkin�s method(weighted residuals), and can be written in the following

concise discretised matrix form,

Cij xð Þ oxot

þ Kij xð Þx ¼ f i xð Þ; ð57Þ

with

Kij ¼

Kgg Kgc Kgt 0

Kcg Kcc Kct 0

Ktg Ktc Ktt 0

Kug Kuc Kut Kuu

26664

37775;

Cij ¼

Cgg Cgc Cgt Cgu

0 Ccc Cct Ccu

0 Ctc Ctt Ctu

0 0 0 0

26664

37775; f i ¼

fg

fc

ft

fu

8>>><>>>:

9>>>=>>>;: ð58Þ

The time discretization is accomplished through afully implicit finite difference scheme (backward differ-

ence),

Wi xnþ1ð Þ ¼ Cij xnþ1ð Þ xnþ1 � xn

Dtþ Kij xnþ1ð Þxnþ1

� f i xnþ1ð Þ ¼ 0; ð59Þ

where superscript �i� (i ¼ g; c; t; u) denotes the state var-

iable, n is the time step number and Dt the time step

length.

The equation set (59) is solved by means of a

monolithic Newton–Raphson type iterative procedure[8,16]:

Wi xknþ1

� �¼ � oWi

ox

��Xknþ1

Dxknþ1;

xkþ1nþ1 ¼ xk

nþ1 þ Dxknþ1; ð60Þ

where k is the iteration index and the Jacobian matrix is

defined by

oWi

ox

��xknþ1

¼

oWg

o�pgoWg

o�pcoWg

o�T

oWg

o�uoWc

o�pgoWc

o�pcoWc

o�T

oWc

o�uoWt

o�pgoWt

o�pcoWt

o�T

oWt

o�uoWu

o�pgoWu

o�pcoWu

o�T

oWu

o�u

266666666664

377777777775

��x¼xk

nþ1

: ð61Þ

4.4. Constitutive model for thermo-chemical and mechan-

ical deterioration of concrete at high temperature

During heating, concrete at high temperature is ex-

posed to complicated physical and chemical transfor-

mations [17,18], causing changes of its inner structure,

what has also a sensible influence on the materialproperties. From a practical point of view, among the

most important macroscopic consequences of these

processes are concrete dehydration and crack develop-

ment, resulting in a significant decrease of mechanical

properties of concrete at high temperatures. The mate-

rial stress – strain behaviour in such conditions is highly

non-linear and depends not only on the history of me-

chanical loading during heating, but also on the tem-perature and dehydration degree [18].

For these reasons a parameter describing the degree

of the thermally induced material deterioration process

advancement has been introduced, similarly as done by

Gerard et al. [19] and Nechnech et al. [20]. It is called

thermo-chemical damage, V , because it accounts for

changes of material stiffness, both due to thermally in-

duced micro-cracks, caused mainly by stresses at micro-and meso-level, (e.g. resulting from different thermal

expansion coefficients of cement paste and aggregate,

and from local increase of dehydration products� vol-ume), and due to decrease of concrete strength proper-

ties caused by the dehydration process (thus related to

the Cdehydr value).

The thermo-chemical damage parameter, V , is de-

fined in terms of the experimentally determined evolu-tion of Young�s modulus of mechanically undamaged

material (i.e. heated to a given temperature, without any

additional mechanical load), E0, expressed as a function

of temperature,

V ¼ 1� E0ðT ÞE0ðTaÞ

; ð62Þ

where Ta ¼ 20 �C is room temperature.

Mechanical damage of concrete is considered fol-

lowing the scalar isotropic model by Mazars [21,22]. In

this model, the damaged material at given temperature,

T , is supposed to behave elastically and to remain iso-

tropic. Its Young�s modulus at this temperature, EðT Þ,can be obtained from the value of mechanically un-

damaged material at the same temperature, E0ðT Þ


[21,22], and mechanical damage parameter, d, being a

measure of cracks� volume density in the material,

EðT Þ ¼ 1ð � dÞE0ðT Þ: ð63ÞA total effect of the mechanical and thermo-chemical

damages, to which the material is exposed at the same

time, is multiplicative, i.e. the total damage parameter,

D, is defined by the following formula,

D ¼ 1� EðT ÞE0ðTaÞ

¼ 1� EðT ÞE0ðT Þ

E0ðT ÞE0ðTaÞ

¼ 1� 1ð � dÞ � 1ð � V Þ: ð64Þ

Therefore, the classical effective stress concept, is

modified to take into account both the mechanical and

thermo-chemical damage, so a further reduction of re-

sistant section area due to thermo-chemical degradation

is added to that caused by the mechanical damage, i.e.

the section reduction by cracking:

~r ¼ rS~S¼ r

1� dð Þ 1� Vð Þ ; ð65Þ

where S and ~S mean total and resistant area of thedamaged material, r is the tensor of nominal stress and~r the tensor of ‘‘modified’’ effective stress (in the sense of

Mazars [21,22]). This formulation of damage model is

implemented by means of a non-local technique. For

further details, see [23,24].

5. Behaviour of concrete in tunnel fires

We extend here the procedure outlined in Sections

2–4 to deal with the behaviour of concrete walls and

ceilings in case of tunnel fires. The recent fires which

occurred in major European tunnels (Channel [25,26],

Mont-Blanc, Great Belt Link, Tauern, St.Gotthard,

etc.) emphasise the serious hazards they represent and

the impact they may have on human beings, economyand repair costs. Extensive and heavy damage in the

concrete elements were observed. The structural damage

can be attributed to two main factors, namely, spalling

of concrete and excessive temperatures attained in both

concrete and steel components leading to serious loss of

load bearing capacity. The temperature field inside the

wall depends on the incident heat fluxes (convective and

radiative), related to location and type of fire, air-flowrate through the tunnel and smoke properties.

The numerical simulation of such complex situations

requires advanced numerical models which allow to

simulate the fire scenario in the tunnel and the real

physical behaviour of the concrete structure in such se-

vere conditions. The behaviour of concrete can be sat-

isfactorily simulated only with a multiphase porous

media approach such as that used in this paper. In case ofhigh temperatures the following mass transport mecha-

nisms take place in concrete: for capillary water we have

darcian type flow and the thermodynamic force is the

water pressure gradient; for adsorbed water there is dif-

fusive flow and the force is the water concentration

gradient. For chemically adsorbed water there is no

transport, but it acts as source or sink term. Water va-pour presents both darcian flow under gas pressure

gradient and diffusive flow under water vapour concen-

tration gradient. Similarly dry air presents darcian flow,

also under gas pressure gradient and diffusive flow under

dry air concentration gradient. All these phenomena can

be modelled by the equations shown in Sections 3 and 4.

Further there are phase changes of chemical and physical

nature: dehydration, evaporation and desorption whichare endothermic processes, while hydration, condensa-

tion and adsorption are exothermic ones. All these re-

quire appropriate exchange terms in the balance

equations, which have been introduced in [13,16,24,27].

As far as the fire scenario is concerned, this represents a

boundary condition for the concrete wall.

Prediction of the thermal field within a tunnel in

presence of fire is quite complex, since combustion givesway to a combination of heat transfer phenomena (by

convection, radiation and conduction) with strong in-

teractions among them. As well known, in the reaction

zone chemical energy is converted into internal energy of

the products thus leading to convective movements of

the fluid, as well as short wave radiation; at the same

time mutual radiation takes place among the walls and

to the other parts of the fluid, according to their tem-peratures and radiative properties. A complete proce-

dure for the computer simulation of such complex

phenomena is described in detail in [28]. Here, our at-

tention is focused on the combustion and thermal ra-

diation models which have been applied in the numerical

example presented in this work.

Nowadays, several combustion models are available

for fire simulation in enclosures: among them, the so-called ‘‘volumetric heat source (VHS)’’ [29], appears to

be quite simple when compared to other methods (e.g.

the ‘‘Eddy break-up’’ or the ‘‘presumed probability

density function (prePDF)’’ models [30]), specifically

developed to take into account some chemical aspects of

the reactions. VSH method, instead, does not consider

any chemical reaction and simply assumes a given spa-

tial and temporal distribution of the equivalent heatgeneration H per unit volume:

H ¼ f ðr; tÞ; ð66Þwhere r is coordinate vector, a t – time. Usually, H is

assumed to be constant throughout a given volume Vrepresenting the zone of reaction and therefore, being

QðtÞ the overall heat generation:

H ¼ QðtÞ=V : ð67ÞDespite of its simplicity, this model appears to be

reliable enough for enclosure fires, as shown by the


experimental validations reported in [29], and can

therefore be applied with satisfactory confidence.

Thermal radiation inside a tunnel with fire is quite

complex since participating media (including smoke,

soot, liquid droplets, ash and dust particles) are usu-ally present in a wide range of temperatures and

compositions. Again this presents a boundary condi-

tion for the structural problem as indicated in the third

of Eq. (54). The ‘‘Monte Carlo’’ statistical method is

used to obtain this flux [31]. In principle, a point in the

domain of the function is selected at random and,

sampling from the probability distributions that model

the physical process, the mapping of this to a pointin the co-domain is calculated. After many repetitions,

the correspondence between areas in the domain and

the co-domain may be described statistically with in-

creasing confidence.

In this regard, a computer model named SMOKE 2.1

has been specifically written for tunnel fires, as a de-

velopment of previous works in the field of radiative

heat exchanges with non-participating medium [32–34].Basically, the space enclosed by the tunnel as well as

its surfaces are discretized into a suitable number of

elements. For each of them, assuming the temperature

to be known, the radiative emission can be determined

as follows, being r0 the Stefan–Boltzmann coefficient.

For surface elements with area A, total emissivity e andabsolute temperature TS, the well known Stefan–Boltz-

mann�s law leads to the following expression for thepower emitted by the surface:

qr ¼ r0eAT 4S : ð68Þ

Instead, for fluid elements with volume V and

absolute temperature Tf the radiative emission can be

approximated as follows [35]:

qr � 4r0eVT 4f ; ð69Þ

where e is an equivalent emissivity depending on

temperature and chemical composition of the fluid, as

well as on the so-called ‘‘mean beam length’’ L of the

element:

e ¼ f ðTf ; L; cÞ; ð70Þwhere c means concentration vector describing compo-sition of gasses, taking into account the possible pres-

ence of solid and liquid particles.

According to Eckert [36], the mean beam length can

be approximated as

L � 3:6V =A: ð71ÞAssuming a lambertian behaviour for both gas and

surfaces, the application of the ‘‘Monte Carlo’’ ap-

proach to the problem under consideration can be car-

ried out by generating at the centre of each element(surfaces or volumes) a certain number N of rays with

randomly selected direction and the following initial

power qb:

qb ¼ qr=N : ð72ÞThen, each ray is traced throughout the tunnel in

order to locate its interception with a wall and the

resulting segment is subdivided into intervals of suit-able length Dx. At each increment Dx the power of the

ray is progressively reduced, because its energy is

partly absorbed and partly scattered. Being ðqbÞx the

ray power at a given location x, the absorbed com-

ponent ðqaÞDx, to be added to the heat generation of

the fluid, will be:

ðqaÞDx ¼ aDxðqbÞx; ð73Þwhere aDx is the absorption coefficient of the fluid over

Dx. Instead, the scattered component, to be assigned toone of several auxiliary radiant sources located along

the path, will be:

ðqaÞDx ¼ qDxðqbÞx; ð74Þwhere qDx is the scattering coefficient of the fluid over

Dx. For the application of this procedure both aDx andqDx must be known.

Unfortunately, information on scattering phenomena

in tunnel fires is limited and qDx must be determined by areasonable assumption.

Also the evaluation of aDx is quite complex, espe-

cially because the selective behaviour of most fluids

makes aDx not constant. However, in tunnel fires where

significant stratification usually occurs and the ab-

sorbing fluid is confined in the hot upper layer, an

estimation of aDx of the smoke can be obtained as

follows. For an uniform volume with ‘‘mean beamlength’’ L, the overall absorption coefficient aL is given

in [35] as a function of its chemical composition, its

temperature Tf and the temperature T0 of the emitting

source:

aL ¼ f ðTf ; T0; cÞ: ð75ÞTherefore, it can be shown that a reasonable esti-

mation of aDx, for the smoke with average temperature

at a given distance from the main source, is

aDx ¼ 1� ð1� aLþDxÞ=ð1� aLÞ; ð76Þ

where in this case L is the overall length of the ray path

across the smoke (starting from the main source).

When the ray reaches a wall, absorption and reflec-

tion will be treated in accordance with the surface ra-

diative properties and the reflected component will beemitted by an additional auxiliary source placed on the

wall. Then, the same process is applied to each auxiliary

source, until the rays leave the system or their powers

become negligible. By repeating this procedure several

times with different sets of rays initiated at the main

sources, the radiative behaviour of the system becomes

increasingly clear and the radiation absorbed by the

various elements (surfaces or volumes) can consequentlybe determined.

Fig. 2. Discretization used in 2D calculation of the top stressed section

of the tunnel (section of the fire). Points indicate the tunnel zones

considered in the Figs. 3–8.

Table 1

Properties for ordinary concrete at 20 �C

Parameter Symbol Value

Porosity n [)] 0.1

Intrinsic permeability K[m2] 10�18

Apparent density q [kg/m3] 2585

Specific heat C [J/kgK] 940

Thermal conductivity v [W/mK] 1.67

Young�s modulus E [GPa] 33

Poisson�s coefficient m [)] 0.20

Compression strength fc [MPa] 30


5.1. Case study

The model described in the previous paragraphs has

been applied to a fire scenario for which some experi-

mental measurements were available [37].Since the measurements already provided the tem-

peratures of the fluid in several points inside the tunnel,

Fig. 1, CFD computations were not necessary.

As far as thermal radiation was concerned, the fol-

lowing assumptions were made. The fluid was divided

into two layers, the lower one (up to 3.17 m from the

road) made of fresh air at ambient temperature, the

upper one (from 3.17 m up to the vault) consisting ofsmoke with 20% CO2, 10% H2O (percentages by vol-

ume) and a scattering coefficient (for short wave only)

qDx ¼ 0:006, with Dx ¼ 0:1 m.

Considering the plume of fire at a temperature

T0 ¼ 1200 �C and emitting a radiative power qr ¼ 2:5MW, the absorbed components at the walls (the quan-

tities absorbed by the fluid are of no use in this case)

were calculated with the assumed values of reflectanceq ¼ 0:25 for vault (of the dark grey colour) and q ¼ 0:15for the asphalt road pavement [38].

Additionally, also the mutual radiation between the

walls and smoke was calculated assuming the walls as

black bodies. Smoke was calculated assuming the walls

as black bodies. The convective heat transfer was esti-

mated from the correlations given in [39]. Heat transfer

was estimated from the correlations given in [39]. Heattransfer coefficient was equal to 27 W/m2 K in the zone

affected by the plume.

At this point the thermo-structural model was applied

and the temperatures and stresses were calculated (see

Fig. 2 for the discretization employed in the calculations).

The tunnel has been built of an ordinary concrete

whose properties are reported in Table 1.

It has been assumed that the fire acted with maximumenergy for a time span of 7.5 min maintaining stationary

Fig. 1. Schematic representation of the longitudinal and cross sections of the

the instrumented sections at 0, 20 and 50 m from the flames.

conditions for fire. In the real fire test, the firemen began

to extinguish the fire after 5 min. The extra 2.5 min have

been simulated to analyse a further evolution of the

hygro-thermo-mechanical behaviour of the concrete

vault, assuming that the heat fluxes are the same as

during the first 5 min of fire. The resulting distributions

of temperature, relative humidity, vapour pressure, as

tunnel, with the position of fire, smoke and maximum temperatures in

Fig. 4. Relative humidity distributions at time stations t ¼ 5 min and

t ¼ 7:5 min (a) and time histories at three points of tunnel located at

different heights above the pavement.


well as mechanical-, thermo-chemical- and total damage

at three locations: A, B, C of the central tunnel cross-

section (see Fig. 2), located at different heights above the

tunnel pavement, h ¼ 6:03, 5.15 and 1.5 m, for two time

stations, t ¼ 5 min (the end of the real fire test) andt ¼ 7:5 min (the end of the simulations), are shown in

Figs. 3(a), 4(a), 5(a), 6(a), 7(a) and 8(a). Because of the

short time of exposition and the total thermal power

involved during the test, only a superficial 6-cm layer of

the vault has been affected by the phenomenon and

shown in the graphs. In Figs. 3(b), 4(b), 5(b), 6(b), 7(b)

and 8(b), the time evolutions of the analysed physical

quantities in the points lying 8 mm from the vault�ssurface (except of the temperatures, where the surface

values are shown) in the locations A, B and C are pre-

sented. It is worth to underline that two of them, i.e. A

and B are situated in upper part of the tunnel, filled with

the smoke, what influenced significantly the heat radia-

tion and the resulting temperature distribution (Fig. 3).

It is possible to note the sharp desaturation process af-

fecting the layer of concrete close to the heated surface(Fig. 4) and the thermo-diffusion inside the pores of the

material. A significant increase of the vapour pressure in

this layer is observed, Fig. 5.

No visible vault deterioration has been observed

during the experimental test. However, the numerical

calculations showed that because of the temperatures

reached, a considerable thermo-chemical damage is

Fig. 5. Vapour pressure distributions at time stations t ¼ 5 min and

t ¼ 7:5 min (a) and time histories at three points of tunnel located at

different heights above the pavement.

Fig. 3. Temperature distributions at time stations t ¼ 5 min and t ¼ 7:5

min (a) and time histories at three points of tunnel located at different

heights above the pavement.

Fig. 6. Thermo-chemical damage parameter distributions at time sta-

tions t ¼ 5 min and t ¼ 7:5 min (a) and time histories at three points of

tunnel located at different heights above the pavement.

Fig. 7. Mechanical damage parameter distributions at time stations

t ¼ 5 min and t ¼ 7:5 min (a) and time histories at three points of

tunnel located at different heights above the pavement.

Fig. 8. Total damage parameter distributions at time stations t ¼ 5 min

and t ¼ 7:5 min (a) and time histories at three points of tunnel located

at different heights above the pavement.


present in the layer of concrete directly exposed to fire,

in the three considered locations, Fig. 7(a). This is

mainly due to the high temperatures reached, about 630

K after 5 min and 690 K after 7.5 min (Fig. 3), whichinduce thermo-chemical changes in the microstructure

of the material in terms of porosity and permeability.

Also mechanical features of concrete are affected by

these processes (i.e. dehydration and cracking) and this

leads to a damaging of material. A thermal dilatation of

the surface vault layer, while the internal part remained

at the initial temperature, caused a considerable me-

chanical damage of the surface layer of the vault, Fig. 6,what contributes to the total concrete damage, Fig. 8.

In presence of a mechanical and thermo-chemical

damage of the concrete, i.e. when the thermal front

deeply enters into the wall causing differential thermal

dilatation for a larger depth of material, significant

values of pore vapour pressure can produce a separation

of superficial layers of concrete, phenomenon known as

spalling [16,23,24,27]. With the increase of the fire du-ration, a risk of this phenomenon significantly increases.

6. Conclusions

In this paper a thermodynamically consistent model

for heat and mass transfer in deforming porous media,


including phase changes, has been developed. Hybrid

mixture theory has been used for this purpose and in-

terface properties have been included. From this theo-

retical model a simplified numerical model has been

extracted. This model is used for simulating the behav-iour of heated concrete in tunnel fires. Significant phe-

nomena such as spalling and thermal diffusion have

been evidenced. This application shows, together with

other not reported here, that the model is extremely

versatile.

Acknowledgements

The authors are grateful to Prof. P. Brunello from the

Dipartimento di Costruzione dell�Architettura, Univer-

sit�a IUAV (Venice, Italy), for his valuable contribution.

This work has been partially funded by the UE project‘‘UPTUN – Cost-Effective, Sustainable and Innovative

Upgrading Methods for Fire Safety in Existing Tunnels’’.

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Documents

Multiphase flow in deforming porous material