!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~
(