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The scientific investigation on frost heave goes back to the 1920s [l]: Taber observed the formation of successive ice layers (ice lenses) as a freezing front moved ...
!2tm~T1CAL COMPUTER MODELLING

PERGAMON

Mathematical

and Computer

Modelling

37 (2003) 595-602 www.elsevier.nl/locate/mcm

Freezing Processes in Porous Media: Formation of Ice Lenses, Swelling of the Soil F. TALAMUCCI di Matematica “II. Din? Viale Morgagni, 67/a 50134 Firenze, Italy [email protected] Dipartimento

Abstract-when moist porous soil is freezing, a volume expansion is generally observed. The volume increase is mainly due to a water migration process from the base of the soil up to the freezing front, which separates the lower unfrozen part from the upper frozen one. The coupled heat-mass transfer process is accompanied, under particular conditions, to the formation of pure ice segregated layers. Conversely, if the freezing process is too fast or the overburden pressure acting on the column of soil is relevant, no macroscopic accumulation of ice is observed. It is generally accepted that a thin transitionregion (frozen fringe), where water and ice coexist in the porous space, separates the unfrozen from the frozen parts of the soil. The strong interest on ground freezing is motivated by at least two reasons: preventing frost damages produced on roads pavements, pipelines, or other structures, and predicting the effects of artificial-freezing techniques for tunneling or underground constructions. We are going to present a mathematical model of frost heave where water is driven through the porous space by the coupling of a pressure and a chemical gradient. In particular, the interest is focused on detecting which are the boundary values for temperature (or thermal flux) that determine the process of lens formation or frost penetration, once the properties of the soil are known. @ 2003 Elsevier Science Ltd. All rights reserved.

Keywords-Porous

media,Phasechange,Frostheave,Ice lensing.

1. THE

PHYSICAL

PROCESS

When fine-grained moist soil is subjected to a freezing process, a volume expansion can be generally observed.

The increase of volume is not only due to the different densities of water

and ice, but mainly to a water migration process from the unfrozen part of the soil towards the freezing front. The phenomenon is known as frost heave, and it is the result of strongly coupled heat and mass transfer process occurring in the porous soil. The scientific investigation on frost heave goes back to the 1920s [l]: Taber observed the formation of successive ice layers (ice lenses) as a freezing front moved downward when a column of saturated clay was frozen from the top down. The final configuration of the sample of soil consists of a sequence of pure ice layers alternating with regions of completely frozen soil. Later on, many experiments were performed in order to understand the mechanism of frost heave. The heave amount of a frozen specimen can be very considerable: in the tests of [2], for instance, the final height of a specimen of Kanto loam is more than three times the initial one. If the freezing 0895-7177/03/t - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(03)00053-O

Typeset

by &&-‘I@

F. 'L‘ALAMIJCCI

596

process is too rapid or if the overburden pressure acting on the sample of soil is remarkable. formation of lens may not occur:

water in the porous space freezes and ice is kept within the

porous matrix, even though the porosity of the soil may be affected by the passage of the freezing front. We refer to such a process as frost penetration. The most relevant factors controlling

the process of frost heave are the freezing rate. the

property of the soil (grain size, dry density, permeability,

. . . ), and the overburden

Typically, smaller grain size (clays, fine silts) corresponds to higher frost susceptibility

pressure. and more

dramatic heaves experienced by the soil. The large interest

in frost heave is motivated

by at least two reasons:

on one hand it is

important to prevent frost heave damages produced on road pavements [3], pipelines, and building foundations in regions with periodical or permanent freezing, and on the other hand, techniques of artificial freezing for consolidating

soils during tunneling

[4] and underground.constructions,

preventing collapses, have been more and more used in recent years. In the frame of the latter situation,

we quote that artificial

freezing is adopted also for paralysing

soil pollution

(rapid

freezing if a pollutant is solved in the soil water, slow freezing in case of cleaning contaminated soils). The physical problem can be outlined as in Figure 1.

coolin

+

+

temperature

or thermal ~&LX top of soil

9++++

heave of soil

frozen soil soil grains and ice

POROUS

MEDIUM

yireezingm

change of phase

unfrozen soil soil grains and water

ttt

water migration base of soil

Figure 1. The physical situation.

The sample of moist fine-grained soil saturated with water is subjected to a freezing temperflux) at the upper surface. The base of the soil is permeable to transfer of

ature (or thermal

liquid from the underground. Both the advancement of the frozen soil on the unfrozen part (heat transfer effects) and the migration of water from the base towards the intermediate region where the change of phase takes place (mass transfer effects) determine the volume expansion. As the frozen part advances, if particular transversal

thermal

and mechanics

break of the soil in proximity of the intermediate

conditions

are verified, a

region may occur and the growth

of an ice lens takes place. The growth of the lens is fed by the upward water migration. As a first approximation,

we may believe that frozen and unfrozen soils are separated

by a

sharp interface: in this case, the change of phase occurs only at that surface. However, a careful observation of the phenomenon shows the existence of a thin transitional zone, called frozen fringe, where water and ice coexist in the porous space and where change of phase takes place. It is comprised between the 0” C isotherm (freezing front) and the lower boundary of the totally frozen soil.

Freezing Processes in Porous Media

2. MODELLING As a general

feature,

along the vertical the liquid

we assume

direction

fringe is assumed

model).

occur and the porous

the mathematical

using a continuum

approach

We denote

by E the porosity

of the mixture

is partitioned

In what follows, indexes in the process

Q~(z, t), k = w,i,s,

in the process

Moreover,

matrix

vary appreciably

no chemical

in the unfrozen

model,

we start

(for details,

reactions

only

between

soil and in the frozen

with the heat and mass conservation

we refer to [5]).

of the soil and by Y the ice volumetric into water

are the temperature densities

content.

The unit of volume (1 - E)

((1 - v)E), ice (vs), and soil grains

w, i, s refer to water,

the specific

k(z, t, T), the specific volume

involved

Equations

In order to formulate

involved

PROBLEM

to be undeformable.

2.1. Conservation

equations,

the quantities

(one-dimensional

and the soil grains

THE

597

ice, and soil grains,

respectively.

T(z, t), the volumetric

velocity

The quantities

of each component

k = w,i,s,

pk(T),

heat of each component

Q(T),

the thermal conductivity of the soil k = w,i, s, and the latent heat per unit

L.

The mass conservation g

equation

is

((1 - ~)%lJ + &Vi) + & (P&J

+ Pi%) =

0,

water

+ ice, (I)

= 0, The Rankine-Hugoniot

condition

at any interface

bwl~ + PiQi]]- g [[(l where [[xl] = x+ - xEnergy conservation

= lim,,i+ is

C(Y) g

x - lim,,i-

where C(Y, T) = (1 - v)se,c, At any interface i we have

h Constitutive

2.2. The

most

relevant

i writes v (2))

&P,

+

(2)

&z/pi]] = 0,

x.

+ (wuqu + wiqi + w,qs) g

+ ;

$ (EV)+ g Vi)((Ci -

+ pi

soil grains,

(3) c,)T - L) = 0,

+ EVQ~C~+ (1 - E)Q~C, is the equivalent

(ccw- ci)T + L) ($ [[(I - v)&]] -

[[q,,,]]) =

heat capacity.

[[-kg]] .

(4)

Equations physical

and chemical

processes

occur

volumetric content v varies in the porous space because heat transfer processes are strictly related to the interfacial

within

the frozen

fringe.

The

ice

of the change of phase. Mass and effects between the tree constituents

(water, ice, and soil grains). The physical and chemical mechanisms involved are at the present not yet evident, mainly owing to the experimental difficulties of measuring the quantities entering the process. The lack of a common understanding of the basic phenomena motivates number of different models and theories on ice segregation proposed in the literature. The following questions are crucial in order to formulate a model for frost heave. (1) Which

is the equation

Formally,

for the water flux in the frozen fringe? aZ p = &I,,,, T), chemical potential we can write qw = - 9,

the large

[6]. Hence,

F. TALAMUCCI

While

it is a common

point

the unfrozen

soil (Darcy’s

controversial.

In the literature,

of view that

Ki(p,,

the question

about

law),

T)

we find two different

E Kc,

water

Kz(p,,

T)

= 0 within

flux in the frozen

fringe

in

positions:

(i) Ks I 0 [7], and (ii) Ks > 0 [8,9].

(2) Are

T and water pressure

temperature

A common

point

of view is that

holds at the phase equilibrium: equilibrium

values,

If y and p,

are dependent

phase equilibrium (water,

- (Apt/pi) models

process

fringe? generalized

= (LAT*/Tc),

equation

where * denotes

[7]), the previous

[7]. On the other hand,

are independent

the

and the previous

equation

is claimed

in the so-called

equation

nonequi-

holds only at the

[lo-121.

are the most relevant

stituents

in the frozen

Clausius-Clapeyron

and To is 273 K.

(equilibrium

the freezing

T and p,

models,

(3) Which

(Apk/p,)

pi is the ice stress,

to hold even during librium

p, independent

the so-called

interfacial

ice, and soil grains)

effects due to the contacts

among

the three

con-

of the frozen fringe?

p, - p, = 4(v). In other In capillary models [13-151, water-ice effects are dominant: models [16,17], water-soil grains effects prevail. This leads to the empirical law v = v(T).

(4) Which

is the mechanism

Main

theories

of ice segregation?

are Terzaghi’s

where P is the overburden is the neutral existence water

stress,

the break

of a segregation

tends

stress

pressure

somewhere

(5) Is

boundary

ice at rest with respect

(it is assumed stress,

[19]). In other

that

models

p, = P and the lens starts

[20], the stress to form simply

to the porous

gions

Research

statements

and Engineering

based

on

when the

matrix?

review of frost heave models,

we investigated,

balance

of the frozen fringe vanishes.

3. A QUASI-STEADY The model

+ (1 - x)PZ

of ce [18]), the

Ki is so small at T = T, that

For some modelers, ice can move through the porous space by means this case, a common position is qz = qz(t) (rigid ice assumption [7]). For a more complete

P = a,, + ce,

and on = xp,

to the vanishing

T, (the permeability

temperature

to accumulate

idea

of the soil is ascribed

the top of the frozen fringe is simply speed of the upper

partition

P, oe is the effective

In

we refer to [5].

MODEL

on the experimental

Laboratory,

of regelation.

Hanover,

FOR FROST data

performed

HEAVE by Nakano

NY, U.S. A), is based

(Cold

Rr-

on the following

[11,17].

The variation of temperature with respect to time is slow (quasi-steady assumption). T and p, are independent in the frozen fringe. The value of p, at the top of the frozen

(1:;

fringe is a given positive (iii) The empirical functions

constant 0 which is essentially the overburden pressure. K1 and Kz appearing in the water flux law depend only on 7’;

they are strictly positive, increasing functions, and vanishing for very low temperatures. < 0, v(0) = 0. (iv) Interfacial effects are described by the function v = v(T), 0 5 v < 1, w No capillarity effects are present. and (4 Pore ice in the frozen fringe is at rest with respect to the porous matrix (no regelation) the frozen soil is simply shifted up during the growth of the lens, without any appreciable (vi)

deformation. The quantities Ic,, lcf, ki (thermal conductivities), pw, pi, pS (specific densities), and L (latent heat per unit volume) are constant. On the other hand, the thermal conductivity

Freezing Processes

in the frozen fringe Icjj depends a function 3.1.

of v (hence of T).

The Mathematical

the conservation

above lead to the following

(isotherm

T. Empirically,

we will assume

c,,, M

kjj

is determined

as

ci .

Problem

It can be seen [5] that stated

on the temperature

Moreover,

599

in Porous Media

T = 0), zs(t) (upper

equations

system,

boundary

(l)-(4)

together

where the unknown

quantities

of the frozen fringe),

perature at the top of the frozen fringe T,y(t) = T(zs(t), and the water pressure pW(KS, t) :

with assumptions are the boundaries

(i)-(vi) z~(t)

and ZT(~) (top of the soil), the tem-

t), the volumetric

water discharge

qw(t),

G.(t) ZF(t)

-

Z.dt)

= &

kffhl)

s0

+1(t) q&!(t) = (1 - w)es(~)

(f-3

d71, - hQo(Q J%JJ

p&(t)

Ts(t)

=

1

K2(7))

a+ J 0

puqw(t)

+

_

&(I

-

Ks)(Pi

kff(rl)Qdt)

qw(t) = -K1(Ts(t))

-40) = 4

-

d17

ho(t)

Kl(77)

(7)

+

z~(0)

=H >

(8)

Pw)k!T(t)~

+

4w(t)ZF(t)

(9)

Ko

>

+“‘f$)’

'

t, + ;f($$;;

kuao(t),

b,

(10) (11)

%lJtZs(%t) &3(t)= 0,

(12)

az

k?(t) IO,

(13)

f%LJtzs~t) >

0

az

(14)

-’ 0 5 ZF(t) 5 &S(t) 5 z?“(t),

(15)

%0(t)L 0. Conditions the process

(16)

(12)-( 14) are the most peculiar of the model, as they allow us to discriminate between of lens formation and frost penetration: whenever is(t) vanishes, a lens is growing

at the height z = zs; whenever is(t) < 0, the freezing front is advancing and the water pressure gradient must vanish. For more details about this mechanism, we refer to [5,20]. The given quantities and kff (T), the specific v (vs = ITS)), and Ko. properties (cf. [5,17,21]):

In particular,

0 < v(T) I: v(0) = 1, k, = kjj(0)

k,, kf , are the thermal conductivities heat L, the empirical functions K1, Kz, Kl(T), Kz(T), and kjf(T) have the following

and functions of (6)-(16) densities p,,, , pi, the latent

I kjj

v(T),

v’(T)

< 0,

< kj,

“;j(T)

< 0,

Kl(T)

> 0,

K;(T)

L 0,

h(T)

>

K;(T)

> 0,

0,

TliIflooV(T) -+ TEm/jj(T)

= 1, = kj,

Kl(T)

= 0,

Tlimm Kz(T)

= 0.

lim

T-+-CO

The initial data are the position of the lower boundary of the frozen soil b and the initial height of the soil H. The thermal boundary conditions are the given thermal fluxes -kuao and -kpl at the extremities

of the soil.

F. TALAMUCCI

600

The main aim of the mathematical penetration, thermal

melting)

.

conductivity,

The interesting the physical

model is to predict

the kind of process

for a given soil with specific properties ) and for given boundary

point consists

process

develops

of outlining

(dry density,

conditions

different

(lens formation, permeability,

frost

porosity.

cyo and ~1.

regions in an (~0, al)-plane

in each of which

in a specific way.

We call C={(~O,CY~), a0 2 0, al 2 0 I3! solution

(z~,zs,z~,T~,q~,p~)

of (6)-(16)

such that

is SEO},

~={(cEo,cu~),

(z~,zs,z~,Ts,q~,p~)

of (6)-(16)

such that

is < O}.

a0 2 0, cy1 L 0 1 El! solution

The sets L and 3 correspond penetration,

respectively,

In order

to investigate

which is proved

to the regions

on the (a~, crl)-plane

C and 3,

the regions

we start

in [22], as well as all the remaining

PROPOSITION 3.1. Let us call I(6)

sets (which

The temperature sius-Clapeyron 3.2.

T, corresponds

the following

stated

= CJ+ j”(K2(q)/Kl(q))

will be described

generalized

with

results

NJ E (-cq O), then L = .F = 0. If, on the contrary, there exists T, < 0 such that unlimited

where lens formation

and frost

occur. preliminary

dv > 0, V’6 E (-oo,O).

Z(T,)

result,

below. IfI

> 0,

= 0, then L and 3 are nonempty

below) on the (a~, al)-plane.

to the phase-equilibrium

temperature

appearing

in the Clau-

equation.

The Set L (Lens Formation)

As shown in [22], in order to detect

the solvability

of system

duce the variable C((YO,CY~)= kfc~ - k,cuo/lp,k,oo, straight-line directions T,

=

(6)-(16)

0 5 c < Kz(O)/Ic,,

{(ao, cq) : cro > 0, kj"1

= k,ao(l

it is convenient

to intro-

and the corresponding

+ p&z)}.

(17)

of Kz and kff, for any c E (0, Kz(O)/k,), there exists exactly one temperature T:(c) < 0 such that Kz(Ti(c)) = ckrf(T;(c)). F or a g iven pair (~0, CY~),T:(c) is the temperature = 0 on ZS. It can also be seen that there exists exactly corresponding to the critical case %

By the properties

one cl E (O,Kz(O)/k,)

with the property

.&krr(#G($) drl. Again, for the properties Ti(cro)

< 0 such that

of kff,

%(T~(ao))

Z(T;(cl))

- clJ(Tz(cl))

for any given CQ > 0 there = -kua&,

exists

Ix (@a~))

C2 = { (ao, M) I 1 (T;(c))

J(0)

=

one temperature The critical

case

region is empty). in the angle @ = { (cro, cyl) 1 0 5 c 5 cl}. More

precisely, C is the nonempty set bounded by dL = { ( ao, ~1) I kf”l where &, is defined by Z(T,) - r = -k,&ob and Cl = { (QO,~

exactly

= k kff(q) dv.

w h ere Jo(e)

TS = T!j(ao) corresponds to .ZF = 0 (the unfrozen As is shown in Figure 2, the region L is contained

= 0, where we defined

- c.7 (T&o)) - c kuaobK&7

=

= kuao,

QO 2 60) u Cl u CZ

O},

(T;(c))

+ K;‘%

(T;(c))

= 0}

are two curves on the (CEO,crl)-plane contained in p. An important property of them is that any straight line defined by (17), 0 < c < cl, matches each curve in exactly one point, respectively, Pi(c) and Pi in Figure 2. Moreover, we see that PI(O) 5 ( Go,kuGo/kf), PI(Y) 3 Pz(s) = (&,&I) where &O and 61 are defined by -k,&b = J‘i,(TJ:(cl)), k& 7 k,&o(l + pwLcl). A further property of the curves Cl and Cz, which is useful in order to understand the qualitative shape of the region L, is the following: IPI - O( increases for c increasing from 0 to cl, IPz(c) - 01 decreases

for c increasing

from 0 to cl, lim,,o+

IPz(c) - 01 = +oo.

Freezing Processes

601

in Porous Media

I

GO

60

Figure 2. The regions L (lens formation) and 3 (frost penetration) on the (ao, al)plane. The curve C2 is the boundary in common to the two regions; the region 3 is bounded on the left-hand side by the straight line an = ~50. The two parts 30 and 31 are related, respectively, to the cases of vanishment of the front speed and to the faster freezing process.

The

properties

listed

Figure 2. On the boundary

above

allow us to sketch

aC, the process

the profiles

of lens formation

develops

of the two curves, in a special

as plotted

in

way. As a matter

of

fact, if (cye,crr) E TO, QO 2 50,

then Q,,, = 0, T = T,,

phase equilibrium;

if

(~o,w)

E Cl,

then ZF = 0, TS = Tt(ao),

no unfrozen

if

(a~,

E C2,

then

al)

TS

= T:(c(oo,

al)),

+,(b,

dz

4

=

0,

flux is driven thermal

The following data performed

results

predicted

with constant

(1) the freezing

thermal

3.3.

The

Set

Performing

3

(Frost

a similar

gradient.

with the experimental

of the frozen

fringe

are constant,

and the linear growth of ZT is given by q(t)

smaller

by

fluxes (~0, or) [21]:

Ts and the thickness

temperature

b - @GO)-l%(G); (2) the heave rate is constant kh~o)lb%d%; (3) the region C becomes

by the model are in good correspondence

soil;

as the overburden

pressure

2.0 =

= H + pzut(kfcxl -

0 increases.

Penetration) analysis

for the case of frost penetration,

we see that

the set F is the

region on the ((~0, or)-plane delimited by 83 = {(a~, CY~): CYO= ho, CY~2 &I} U C2. The straight line c = cl separates 3 into two regions where the process of frost penetration develops

in different

ways.

Indeed,

we define the angles ,L30= {(as, ai)

: Icp~ < k,cxo 5 k,oo(l

: +1 > ~&o(1+ b&d). +-&J&)1, Pl = {( Qo,W) We see that if (~0, or) E 55~ = F n fi,-,,then there exists a time tf (finite or infinite) is(t) = 0; if (oo,crr) E 31 = 3npl, then there exists a finite time tsuch that lim,,,,

such that ZF(~ = 0.

In the former occurrence

at t = tf (transition

according

to

I

r:(c) Ko(l

T-S(O)

The latter Following

process).

The two possibilities

- hT(Y))kff(Y)E

vwo)2wY) -

case corresponds

the soil zg (faster

p(y)dy =

of lens formai.ion =- x

o(:cIrr iuc~rt~l~

(