RESEARCH
CENTRE
BEHAVlOUR OF A CLAY BARRIER
DEVELOPED FOR USE IN A NUCLEAR
WASTE DISPOSAL VAULT
R.N. Yong, A.M.O. Mohamed, D M . Xu,
M.N.Gray and S.C.S, Cheung
GEO-ENVIRONMENTAL SERIES NO.91-4
2-August 1991
PSE
@
McGiII University
Montreal,Que Canada
F O R E W O R D
The f o l l o w i n g s e l e c t i o n o f p a p e r s o n t h e BEHAUIOUR OF A
CLAY BARRIER DEVELOPED FOR USE I N A NUCLEAR WASTE
DISPOSAL VAULT h a s r e c e n t l y b e e n p u b l i s h e d :
a n d Xu. D.M.
Yong, R.N.
"An I d e n t i f i c a t i o n T e c h n i q u e f o r E v a l u a t i o n o f
Phenomenological C o e f f i c i e n t s i n Coupled Flow
i n Unsaturated S o i l s "
I n t e r n a t i o n a l J o u r n a l f o r N u m e r i c a l and
A n a l y t i c a l M e t h o d s i n Geomechanics. 12:283-299.
1988.
a n d Xu, D.M.
Yong, R.N.,
Mohamed. A.M.O.
" C o u p l e d Heat-Mass T r a n s p o r t E f f e c t s o n M o i s t u r e
Redistribution Prediction i n Clay Barriers"
E n g i n e e r i n g G e o l o g y , 28:315-324.
1990.
Xu. D . M . a n d Yong. R.N.
Mohamed. A . M . O . .
" A p p l i c a t i o n o f an I d e n t i f i c a t i o n Technique t o
E v a l u a t e D i f f u s i o n Parameters i n a Coupled Flow'
P r o c e e d i n g s . IASTED I n t e r n a t i o n a l S y m p o s i u m o n M o d e l l i n g , S i m u l a t i o n and O p t i m i z a t i o n .
pp.103-106.
1990.
a n d Cheung, S.C.H.
Yong, R.N.,
Mohamed, A.M.O.
"Thermal Behaviour o f B a c k f i l l M a t e r i a l f o r a
N u c l e a r F u e l Waste D i s p o s a l V a u l t "
P r o c e e d i n g s , M a t e r i a l s R e s e a r c h S o c i e t y Symposium
176:649-656.
1990.
C h e u n g , S.C.H..
G r a y . M.N..
Yong. R.N.
and
Mohamed. A.M.O.
"The E f f e c t s o f M o i s t u r e C o n t e n t , S a l i n i t y and
Temperature on t h e Load-Bearing C a p a b i l i t y of
a Dense C l a y - B a s e d B a c k f i l l "
P r o c e e d i n g s . M a t e r i a l s R e s e a r c h S o c i e t y Symposium.
212:491-498.
1991.
Yong. R.N. a n d C h e u n g , S.C.H.
Xu. D.M..
Mohamed. A.M.O..
" E v a l u a t i o n o f Thermal C o n d u c t i v i t y o f B a c k f i l l
Material"
P r o c e e d i n g s , M a t e r i a l s R e s e a r c h S o c i e t y Symposium
212:507-513.
1991.
AN IDENTIFICATION TECHNIQUE FOR EVALUATION
OF PHENOMENOLOGICAL COEFFICIENTS IN COUPLED
FLOW IN UNSATURATED SOILS
RAYMOND
N.YONG' AND DA-MING XU'
Georechnical Research Cenrre. McGill Liniversiry. 817 Sherbrooke St. W . . Monrreal. PQ. Camdo, H3A 2K6
SUMMARY
To determine moisture movement and heat transfer through an unsaturated soil under temperature and
volumetnc Hater content gradients, i t is necessary to have knowledge of phenomenological coefficients of
the soil. However, in unsaturated flow. i.e. flow through unsaturated soil. these phenomenological coefficients are not constants, but vary with volumetric water content as well as temperature. In this paper. an
identification technique is proposed for evaluation of the phenomenological coefficients. The phenomenological coefficients are first assumed to be certain kinds of functions of volumetric water content and
temperature. The choice of the functional forms is based on an understanding of the physical situation, and
previous knowledge of water flow in the isothermal case. The constant parameters associated with the
functional forms are evaluated through the use ofthe identification technique. Once these phenomenological
coefficients are obtained as certain functions of the volumetric water content and the temperature for a
specified soil, analysis of coupled moisture flow and heat transfer in the unsaturated soil can proceed.
INTRODUCTION
The modelling of heat and mass transfer in unsaturated soils has been studied and well
reported.' - 4 A good review of some of the methods of solution for the more recognized
theoretical relationships can be found in the recent work by Thomas4-as part of the reporting of
his own studv of the ~ r o b l e mIt. is acknowledeed that im~lementationof the available models for
solution of real problems generally requires simplifying assumptions which tend 'to limit application of the chosen models. In this studv.
e d those models which
.. a method ofsolution is ~. r o ~ o s for
uses principles of irreversible thermodynamics in casting their governing relationships.
It is recalled that in using principles of irreversible thermodynamics to study the problem at
hand (i.e. heat and moisture movement in unsaturated soils), one recognizes that the situation
being considered is in a non-equilibrium state. Thermodynamic forces cause the constituent
components to move towards establishment of equilibrium by balancing the forces. The general
relationships between the rates of flow (fluxes J) and the thermodynamic forces X responsible for
the fluxes, can be described by a power series. For a state which is near equilibrium, the first
power in the series is generally used, i.e. J i = L i j X j ,where Lij represents the phenomenological
coefficients. This relationship is identified as the second postulate of t h e m ~ d y n a m i c s .By
~ and
large, prediction of heat and mass transport constitutes a major problem, not only because of the
-
.
William Scott Professor of Civil Engineering and Applied Mechanics, and Director
Postdoctoral Fellow.
0363-9061/88/030283-17SO8.50
0 1988 by John Wiley & Sons, Ltd.
Received 2 September 1986
284
R. N YONG A N D D-M. X U
complex sets of interaction of fluxes generated in the unsaturated soil, but also because of the
difficulties arising in ascribing and separating the various observed phenomena (from experiments) to the responsible sets of f l ~ x e s . ~ - ~
T o overcome the many difficulties and complexities in prediction, a combination of both
theoretical and experimental aspects of the problem needs to be utilized. In the present
consideration of heat and mass transpon as a coupled flow problem, the phenomenological
coefficients L,,(i=j) represent the moisture and heat transport coefficients, whereas L,,(i#j) are
the coupling coefficients which account for the effect of one type of flow on the other. Thus, a
knowledge of how these phenomenological coefficients vary with different initial moisture and
imposed temperature conditions would permit one to better appreciate the effects of these factors
on coupled heat and mass transport in the test soil being evaluated. Use is made of experimental
tests to provide the data for quantitative determination of the phenomenological coefficients. It is
acknowledged that coupled moisture and heat flow in a porous soil is very complex, and that a
quantitative theory to determine these phenomenological coefficients is affected by many factors.
In proposing the use of an identification technique (IT) for evaluation of these phenomenological
coefficients, it is suggested that the use of actual test data for evaluation of the phenomenological
coefficients identified from the analytical model provides an approximate solution technique, and
a means to avoid the difficulties heretofore experienced in rigorous solution of the problem.
FORMULATION
Consider an uusaturated rigid soil bar consisting of three phases (solid, liquid and gaseous), with
length L and cross-section AA, fixed at two ends, named A and B separately. This is shown in
Figure 1. Initially, i.e. at t =0, volumetric water content 6, defined as the volume of water divided
by the total soil bar volume, is uniformly distributed along the soil bar, 6(x, o)=6,. End A has
T,("C) and end B T,("C), and both ends have impervious (to water) boundaries. Thus water does
not flow out or into the soil bar from the two ends. This system is considered a closed system for
water (moisture) flow, but not for heat transfer.
The basic assumptions necessary to obtain coupled moisture flow and heat transfer equations
are as follows:
1. The unsaturated soil bar is considered as a porous rigid bar consisting of three phases: solid,
liquid and gaseous. This implies that the soil skeleton deformation is not considered.
2. Within the temperature range ( < 100•‹C)of the tests conducted, pressures deveIoped as a result
of vaporization processes are considered to be minimal, and can be ignored.
3. The total accumulation of moisture, due to the combined movements of vapour and liquid
water is considered as equivalent liquid water.
Figure I. Schematic diagram of a soil bar system
FLOW I N UNSATURATED SOILS
285
From the second postulate of irreversible thermodynamics, the governing equations can be
written as
Iae 1
where
Q
a
"
= - = f l ' u ~ dflux; i.e. fluid flow through per unit area per unit time,
21
2T
Q = :=heat
f
68
=,
CX
2T
=,
CX
GI
flux; i.e. heat flow through per unit area per unit time,
thermodynamic force due to water content gradient,
thermodynamic force due to temperature gradient,
0 = volumetric water content, 0 = B(x, t),
L,, = diffusion coefficient for fluid flow due to gradient of 8,
L,,= thermal conductivity coefficient for heat transfer due to gradient of T,
L,.,, L,,= coupling coefficients,
T = temperature, T = T(x, I).
t = time.
In unsaturated flow, the coefficients L , , , L,,, L,, and L,, are not constants, but vary as a
function of 0 and T. From mass and energy conservation, the following relationships can be
obtained:
where C,:=specific heat of the soil.
From equations (1) and (2), the governing equation can be obtained as follows:
For the boundaiy conditions, one obtains for T,
T(0, t) = T2 = const
T(1, t)= T, =const
T2>;T,
and for 0,
286
R. N. YONG AND D-M. XU
The initial conditions are given for T as
+
= T2 (TI - TZ)U(X)
where u(x) is a step function, and for 9,
B(x, 0) = 90
NON-DIMENSIONALIZATION
The above equations can be non-dimensionalized to yield more general equations. Let
Thus X E [0, L] changes to 5 E 10,
11, where the symbol E indicates 'belong
to'.
where t, is the time for the whole test when no further variations of 9 and Tare observed. Thus
changes to T E [0, I].
I E [0, I,]
B is a non-dimensional quantity,
8=9-9,
e
where is the differencebetween 9 and 9,. In this way, the initial condition for 9 can be written as
Let
Tis non-dimensional temperature. In this way, ends A and B have boundary conditions for T as
T(O,.r)= 1
and
T(1, T)=O
Using equations (SHll), the set of equations given as equations (3H7) can be written in nondimensionalized form as follows.
Governing equalion
FLOW IN UNSATURATED SOILS
where
Boundary conditions
For
T,
For
8,
Initial conditions
For
F,
For 8,
TRIAL FUNCTIONS FOR PHENOMENOLOGICAL COEFFICIEhTS
Recognizing that the phenomenological coefficients are dependent on 0 and T, the nondimensional form is written as
Lww
i,,, the diffusion coefficient associated with moisture transport due to volumetric water
content gradients, is similar to D(0), the coefficient of diffusivity. For isothermal conditions, Yong
and Warkentinlo have shown that D(0) varies significantly with 0, and that in general, D(8)can be
288
R N. YONG AND D.M. XU
written as
D@=K,e"'
where K , and K , are soil parameters to bc determined from calibration e ~ p e r i m e n t s . ' ~ - "
L,, vanes with T.
Taking note of the above, and assuming a linear dependency betwen C
, and T, one can write
Iww
L,,
=('II
+'I2
t)PJ
(20)
where a , , a , and a , are constanis to be determined lor a specified soil using the identification
technique to be described in the latier part or this paper.
.&, accounts for the effect of temperature gradient on moisrure flow, while 1,.
accounts for
the effect o l the volumetric water contenr gradient on heat transkr. They are identified as
coupling caeficients.
Moisture flow due 10 a temperature gradient has been ~nvestigatedexiensively. and several
mechanisms have t e n proposed to explain umhymoisture moves from the hot end to the cold
end.' - 3 TWObasic mechanisms lor the 'iner~unsaturated soil'are recognized, ie. vapour diffusion
and surface tension induced umaterflow. The total L,-, includes both effects.
When the soil bar is relatively dry, vapour diffusion may be dominant because the interconnecting pore spaces of the soil offer a relatively easy passage for vapour,to diKuse. When the soil is
relatively umet(but still unsaturatcd). surface tension induced moisture movernenl will dominate.
According to Phillip and de Vries13 the total coeficient oldiffusion, D,, is almost constant. D,,
which represents the dilkivity of moisturc due t o a temperature gradient, consists or D,,,, and
D, .,, as
D~ * = P
With the above reckoning, one can choose a Iunciional form of ,
C
D ~ = D ~ l i q +
As for L,,,
as follows:
i~ can be assumed that
L w = rl Lw,
where 7 is a parameter to be determined using the IT
L
T
i,, represents the thermal conductivity. Il LTT is large. heat will pass rhrou&
the soil quickly
from thc hot end to the c d d end. In the unsaturated soil it is expected ihat LTTmay change with
he volumnric water content 0 or g, and T. LTTcan b assumed as
CT,=a,+a,8+o,T
Summary
In summary, the relationships for the phenomenological c w f i c i e n ~ sare
(23)
FLOW I N U N S A T U R A T E D S O I L S
289
where a,(i= 1, 2, . . . , 9) and q are constants to be determined for a specified soil. For different
kinds o f soil, these constants are expected to have differentvalues, and will need to be determined
from laboratory test data input in conjunction with application of the IT.
It is noted that without the formulations (equation 24) it is not possible to determine Lww,
etc.,
using the system equations (12H17).The phenomenological coefficients cannot be determined
uniquely since there are four unknowns in the two equations in equation (12)(assuming 8 and T
are known at each point at any instance from the test data).
IDENTIFICATION TECHNIQUE
Basisjor identijicarion technique
From laboratory tests, using the controlled conditions and configuration shown in Figure 1 ,
the values of 0 and Tare recorded at differentpoints o f soil bar and for differenttime intervals.
W e thus obtain data as
and
where i, j represent the space and time intervals separately, e.g. for a soil bar consisting o f 10
elements, i= 1 I, and for total time r, which is divided into n intervals, j=n+ I .
The basic principle of the IT is to choose the best values of a,, a,, etc., such that the calculated
theoretical results of 8 and T from the system equations (12H17) would match closely with
measured values o f 8 and T obtained from the tests. Since the IT is an approximate method
associated with the test data for determination o f the system's properties, it is expected that exact
matching may not arise. There are at least three reasons why this non-exact matching can occur:
1. The system equations are approximate and do not fully describe the.physica1 process.
2. The chosen forms of the phenomenological coefficientsas functions of and Tare approximate. I f a larger parameter space involving more parameters a,, a,, etc., is chosen, the computed
results will be closer to the test data, but will require a considerably increased computational
effort.
3. The tests will likely involve errors o f measurement, e.g. inexact maintenance o f boundary and
initial conditions during the tests.
Crirerion
Different criteria can be specified to obtain 'the best chosen values o f parameters a,, a,, etc.'
The criterion adopted herein for determination o f these parameters and hence the evaluation of
the phenomenological coefficientsis described as follows: Suppose we have m parameters a, ( i = 1,
2,. . . , m) and q to be determined from the test data (in our case, i=9, i.e. a,, a,, a,, a,, a,, a,, a,,
a,, a,). In the test, B and Tare at 1 number o f points o f the soil bar o f length L (they couId be
equally spaced),and n differenttime intervals. Thus we know Tijand Oij(i = 1,2,. . . 1 + I and j = 1,
2,. . . n+ 1). The non-dimensional quantities o f gij and Ti, are therefore obtained from equations
(8H11)as functions o f 5 and r. Based on the test data, a8/ar, &/a(, aT/ac, etc., can be obtained by
using the finite difference formuIae. In the end we obtain S set of 'full data'-where 'full data'
indicates that the values o f 8 and Tand their first partial derivatives are known. S might be less
than (n+ I)(!+1). Normally S > m, implying that the data sets are more than, or at least equal to,
the number o f parameters to be determined.
290
R. N. YONG AND D-M. XU
If S = m , parameters a; will be uniquely determined by solving a set of algebraic equations
resulting from equations (12H17) after inclusion of the test data and the derivatives into the
equations. In this instance, we can solve directly for a, since the number of unljnown ai is equal to
the number of equations. If S>m, the unknowns are less than the number of equations. Since a
direct solution for a, to satisfy all the equations is not possible, a search for the best ai is
implemented using the least-squares technique.
Integrating equation (12) with respect to ( from 0 to t i , we obtain:
and
I;(
Using boundary condition (15a), equation (26) becomes
I
5'
I
ci
The term
88
,d(
8s
Td;={.ww(8T),4iT).
07
ae
(28)
a1
<i
represents the non-dimensional accumulating water per unit time.
07
Due to the limitations of the assumed phenomenological coefficients, equations'(27) and (28)
cannot be satisfied exactly. T o proceed with the solution, we define
ae
(=Ti
(29)
r=zj
If tiand r j are specified and since the values of Qij and Tijare obtained as test data, I,, etc. will
be seen to depend only on the parameters a,(k= 1, 2, . . . , m b a s noted in equations (24):
(&A,= M i j h )
(30)
In a similar way, define
r=r,
<=o
c = r,
=( & ~ ) i j ( ~ k )
(31)
The accuracy of solution for (E,),~and ( E , ) ~will
~ be dependent on the choice of a,. Defining the
FLOW I N UNSATURATED SOILS
total error function
E
29 1
as
where the summation is with respect to the whole data set, and since E is positive definite, it is
noted that the best choice of a, is one which renders E a minimum value, i.e.
Equation (34) is a necessary but not sufficient condition of (33). The numerical method to
obtain the best choices of a, will be dealt with in the next section. Equation (33) in association
with equations (29H31) constitute the criterion for the IT.
It is noted that
and ( E , ) ~ ~ are regarded equally, without the involvement of any weighting
functions, and that direct use of the test data is made without actually solving the differential
equations. Obviously, if the system equations and the asssumed phenomenological function
forms closely represent the physical system performance, the theoretical solutions of a and Twill
show good accord with the test data.
NUMERICAL METHOD
While the general procedure for determination of the a, values has been described in the previous
section, the detailed manner in which such is done is still dependent on the adopted numerical
method. In this section, the numerical method for determination of a, is described. Two separate
steps are needed: arrangement of the test data, and use of test data for analysis using the gradient
method.
Arrangement of the test data
From the test or tests, since 0 and T a r e measured a t different points of soil bar and different
time intervals, T,, T, and 0, are known, and tKus 0 and d a r e also known.
{(corresponding to co-ordinate x) and T (corresponding to time t ) form a c-Tplane. If 0 and T
are measured at equal space and time intervals, the solid points in Figure 2 will show where and
when 0 and T a r e measured.
Figure 2. Measured Ti, and Gi, in <-rplane
292
R. N. YONG AND D-M. X U
The first derivatives a@/&, etc., can be evaluated by using the finite difference relationship:
The values of 8, T, dwar, d8/d<, dT/d< obtained can now be stored in the computer program or
are arranged in tabular format for easy visual check.
To apply the gradient method for determination of a,, the numerical method uses two mean
procedures: (i) find a good guess value for a, and q, k = 1,2, . . . , m; (ii) continue to revise a, and q
values (using the gradient method) until the final a, and q values are reached.
(i) Guess values oja,. Denote the guess values of a, and q as a: and qo. For the kind of problem
depicted in Figure 1 the guess values a: can be given as
From test data, a:, a: and a): can be directly estimated. Since the temperature redistribution
stabilizes in a much shorter time than the volumetric water content 0, q0 can be chosen to be zero.
(ii) Gradienl melhod. When the guess values a:(k= I, 2, . . . , m) are obtained, E can then be
evaluated from equations (22), (29) and (31). This represents the global error. Let each one of a,
change to a: +Aa:. Obtaining d~/da,from the computer calculation, the next values of a,, named
a;, will be given as
where A is a small step value. The new E can be evaluated from the a; va1ues.E wiU be less in
general because it changes along a negative gradient. This procedure is repeated until E cannot be
improved further. The last values of a; will be the best chosen values of a, The detailed technique
is best described in the flow chart shown in Figure 3.
Following through the procedure shown in the flow chart, the various values of a,(k = 1 . . . m)
will be obtained. Thus, the non-dimensionalized phenomenological coefficients given as equations (24) can now be fully evaluated, i.e.
From the solution of equations (24), the values of the phenomenological coefficients L,,,
L,,, L,, (equations 13a-l3d) can be obtained.
L,,,
APPLICATION
T o demonstrate application of the IT, a set of test results obtained by Cassel, Nielsen and
Biggar,l2 in their study of soil-water movement in response to imposed temperature gradients, is
294
R. N . YONG A N D D-M.XU
used. The soil used was a fine sandy loam which was oven-dried and packed in 20 cm long and
7.60 cm diameter glass cylinders. The cylinders were totally insulated, and temperature gradients
were established lengthwise along the cylinders through the end-plates which could be maintained in heated or cooled status via circulation of preheated or precooled water. Figure 4 shows
the temperature redistribution along the length of a test sample (test B1) originally at a uniform
temperature of 2 N o C . The temperatures at the left end (T,)and right end (T,) were maintained at
18.60•‹Cand 8.7SoC, respectively. As noted in the graph, temperature distribution between the
'hot' kft end and the 'cool' right end changed significantly in the first 24 hours and reached a near
equilibrium state after about 5-6 hours. This contrasts with the time taken (352 hours) for the
soil-water redistribution curves shown in Figure 5 to reach equilibrium distribution. As indicated
in the figure and affirmed by Cassel er a1.,I2 moisture movement occurred over a period of at least
352 hours, and no further measurable redistribution in soil-water occurred after that time period
for test B1. Note that the nonlinear equilibrium temperature distribution shown in Figure 4 is a
result of heat loss above ambient from the system, and has been thoroughly discussed by the
authors.
The information given in Figure 5 shows that, as time progresses, the 'hot' left end becomes
drier whereas the 'cool' right end becomes wetter. Working with the graphical data as presented,
and noting that the 'final' temperature distribution is reached significantly before the first set of
soil-water content redistribution is available (44 hours versus 76 hours), the 'final' temperature
-
..
0.22 hours
0.28
1.02
o 2.35.
4.50
o
0
-
Thermal diffusivity is s h o w n
for e a c h c a l c u l a t e d curve
DISTANCE (cm)
Figure 4. Initial and transient soil temperature distribution for run BI. go was 0.077 cm3/cm3; TI.,,.=23,4"C; T, and TR
were 18.60 and 8.75-C, respectively. Solid lines were calculated using a Fourier series solution of the one-dimensional heal
flow equation (from Cassel el a/.") (Note:With reference to the analysis developed in this paper, T,= T, and T R = T , )
FLOW I N UNSATURATED SOILS
o
0
76 h o u r s
186
352
5
10
15
D I S T A N C E (cm)
Figure 5. Initla1 and transient soil-water content distributions for soil column BI with an average water conlent of
0,077 cm3/cm'. Initially. the soil column was at a uniform temperature of 23.4"C;thereafter, at distances 010 and 20 crn.
the temperature was maintained at 18.60 and 8.75"C, respectively (from Cassel el a/.'')
profile is used to calculate L,,, and L,, (equations 13a and 13b). The values for a , to a,, as
required in equations (20) and (21), are obtained from the computer program developed (flow
chart in Figure 3). The results obtained show:
L,., =(0.0157 + 0.0307 T) eQ7'6a
(384
Substituting equations (38a) and (38b) in equations (l3a) and (13b), and using equations (10) and
(1 I), the values for L,, and L,, can be obtained-as shown in Figures 6 and 7, respectively.
These show that L,, is not particularly sensitive to changes in 0, but responds well to changes in
T. On the other hand, L,, shows some sensitivity to 0 and more so to T.Note that L,, and L,,
correspond respectively to the apparent isothermal diffusivity D: and the apparent thermal
diffusivity DT coefficients (at the temperature T) used by Cassel et a l . l Z
It is useful to compare the prediction for L,, and LwT with the corresponding coefficients
obtained by Cassel et al.12 From the test data obtained, using the basic relations given by Philip
and de Vries,13 Cassel et a l . l z obtained values for apparent isothermal diffusivity and apparent
thermal diffusivity which they designated as Dpd and Dpd for comparison with their own
calculations of D r and D r . Table I shows the comparison of the values for these coefficients. Note
that they cannot be directly measured, but are obtained through application of a model which
'describes' the physics of the test. Thus, one can observe differences in reporting of values for these
coefficients-not only because of possible model differences, but also perhaps due to numerical
solution techniques.
Comparing the values of L,, and L,, with the other corresponding coefficients-as shown in
Table I-it is seen that the values for L,,=D,
accord well with both the P h i l i p 4 e Vries and
Cassel et 01. values. All three computations for DT show good correspondence. The same cannot
R. N. YONG AND D-M. X U
Table I. Comparison of calculated coefficients of isothermal and thermal diffusivily using experimental measurements from Cassel er al."
Distribution
along
column
(cm)
Mean
temperature
("c)
Mean water
content
0=cm1/cm3
Cassel et 01."
D,'
(cm2/day)
Philipde Vriesl'
~Prcd.
(cm2/day)
Yong-Xu
(this study)
Lww=Do
(cm2/day)
Cassel pr a1."
D
(cm2/day "C)
:
Philip-de Vries"
~ P
r.m d
(cm2/day "C)
Yong-Xu
(this study)
LWT=DT
(cmz/day "C)
.n
5
k t said lor all three compu~aiionsTor D,. By and large, L,, = D, is seen to be from five t o Iun
tirncs highcr t h a n ihe P h i l i ~ j cVrics values. Both of these accord nciiher umith Cassel er 01. nor
rollow the trend lor D,. The reasons for the disagmment in D, v;ducs lor D i and Dptu have bccn
wcll discussed by <:assel er ~ l . , ~and
' will not be repatcd t~crc.
A method using the IT
to t'valuatr the phcnomenologicaI cmHicidnts o i soils is proposed. This
approxlmatc rnethod oCer*alua~~on
requires the use of both lheore~icalknoalcdge and actual test
daia- i n combination. For soil systems ~ h i c hconsist olsolids, liquid and gaseous phases. I he use
of purc physical models requires too many physical paramcters Tor description o l phenomenologicill bchaviour. IFrcquenrly, theorctical predictions based on such kinds o i models do not agree
c o l ac~ual
with itctull icst data. The proposcd IT rnethod described in this study rclies on ~ h use
tcsf data lo cvaluate rhe phenon~enologicalcoefficients drrivcd from application of the principles
nl irrcvzrsible thermodynamics, and i s thus much morc appropriate for engineering use.
[:sing acrual tesl data lrnm thc stud! reported by Cassel P I u / . . rhe
~ ~ IT is uscd lo compute ihe
apparunt isothcmaI difiusivity (L..,.
D,I and the apparcnt thermal difhsivil! [I,,.,- U , ) for
various values *sf 8 and T.Comparison between thcsz values and those computed b!: Cassel e:
nl." and the P h i l i d c Vricr:' values calculated show good agreement Tor D, hu1 no1 for D,
FIo*evcr. the D, comparisons for all three computational emcis art: widcl: varying, excepi tha:
the trends shown by P h i l i p l e VriesL3a p p a r to accord with the rrends Cnr L,.,,: U, and nor
with ('assel c: ul." Thc overall comparison shows that the IT does provide a rclativcly simple
m a n s ior using horh theoretically dcvclopcd relationships for the coeliicients and actual tcsr data
in rcaluatinn of these coeficirnts- - which cannot be rnrasurtd dirwtly, but cornpufrd from itw
r hrorerical model.
This stud!' has k e n supported br, a gran1 from the Natural Scienws and Engincermg Research
Council ol (Yanada (NSEKC) under Grant No. A-882.
APPENDIX I: KOTATIOK
- Pararncrers involved in phcnomcnolngical eficients.
=kit end of soil bar.
= R i g h t cnd ol soil bar.
-. Specific hcat for soil.
= Diffusivity of a n unsaturatcd soil due to a moisture gradiznl
in isnihcrmal condition.
=Introduced in Relerunce 2, which indicaies ihi total diffusion
due to a tempzraturc p d i e n t .
= Introduced in Refcrcnce 2. which indicates the diffusion in
liquid water form due to a temperature gradient.
= In~roduccdin Reference 2, which indicates rhe difiusion in
raponzcd water form duc rn a tcrnperature gradient
= F ; u r n k r of poinrs of soil bar o l length L.
= Lcngth of soil bar.
= Phenomenological coefficirn~s.
FLOW I N UNSATURATED SOILS
= Non-dimensional phenomenological matrix, see (18).
=Corresponding non-dimensional phenomenological coefficients
=Parameters.
=Number of time intervals.
=Fluid flux.
=Time.
=Temperature.
= Non-dimensional temperature, defined in (1 1).
=Discretized non-dimensional temperature.
=Space co-ordinate for one-dimensional soil bar.
=Total error function, defined in (30).
= Error function, defined in (31).
= Error function, defined in (29).
=Parameter involved in coupling phenomenological coefficients.
=Volumetric moisture content.
=Initial uniform volumetric moisture content.
=Difference between 0 and 0;.
=Discretized non-dimensional volumetric moisture content.
=Non-dimensional co-ordinate, see (8); and
= ~ o n ~ d i m e n s i o n time,
a l see (9).
REFERENCES
1. S. A. Taylor and J . W. Cary.'Linear equation for simuitaneous flow of matter and energy in a continuous soil system',
Soil Sci., 28 167-172 11964).
2. J. W. C & ~ , ' S O moislure
~I
&ansport due to thermal gradients: practical aspects', Soil Sci., 30,428-433 (1966).
3. V. Dakshanmurthy and D. G. Fredlund,'A mathematical model for predicting moisture flow in an unsaturated soil
under hydraulic and temperature gradients', Water Resources Res., 17, 714-722 (1981).
4. H. R. Thomag 'Modelling two-dimensional heat and moisture transfer in unsaturated soils inchdins gravity effects',
In!. j. numer. anal. merhods geomech., 9. 573-588 (1985).
5. L. Onsager. 'Reciprocal relation in irreversible processes. 11'. Phys. Rer-., 38, 2265-2279 (1931).
6. R. L. Rollins and M. G. Spangler, 'Movement of soil moisture under a thermal gradient'. Highway Research Board,
Proc. #33, pp. 492-508 (1954).
7. R. S. Chahal, ' E k t of temperature and trapped air on the energy status of water in porous media', Soil Sci., 98,
107-1 12 (1964).
8. C. H. M. Van Bavel. 'Gaseous diffusion and porosity in porous media', Soil Sci., 73, 91-104 (1952).
9. W. L. Hutcheson,'Moisture flow reduced by thermal gradients within unsaturated soils', Highway Research Board.
Special Report 40. 113-133 (1958).
10. R. N. Yong and B. P. Warkentin, Soil Properties and Behaoiour, Elsevier, Amsterdam, 1975,449 pp.
11. R. N.Yong 'The swelling of a montmorillonite clay at elevated temperatures'. Proc. Third Asian Regional Conf. on
Soil Mechanics and Foundation Eng. Haifa, pp. 12G128 (1967).
12. D. K. Cassel. D. R. Nielsen and J. W. Biggar, 'Soil-water movement in response to imposed temperature gradients',
Soil Sci.. 33, 493-500 (1969).
13. J. R. Philip and D. A. de Vries. 'Moisture movement in porous materials under temperature gradients', EOS Trans.,
AGU. 38(2), 222-232 (1957).
Engineering Geology, 28 (1990) 315-324
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Coupled Heat-Mass Transport Effects on Moisture
Redistribution Prediction in Clay Barriers
U Y M O N D N. YONG, ABDELMOHSEN 0. MOHAMED and DA-MING XU
Ceotechnical Research Centre, McGill University, Montreal, Que. H3A 2K6 (Canada)
(Accepted for publication February 14. 1989)
ABSTRACT
Yong. R.N., Mohamed, A.-M.O. and Xu, D.-M.. 1990. Coupled heat-mass transport effects on
moisture redistribution prediction in clay barriers. Eng. Geol., 28: 315-324.
The issues associated with coupled transient heat and mass transfer in unsaturated clay barriers
surrounding a buried heat source are considered, and the effects ofdriving forces in both the liquid and
vapour phases are discussed. The reasons for the inability of the model of Philip and de Vries (1957) to
accurately predict the redistribution of moisture in unsaturated clay barriers are explained The model
of Yong and Xu (1988) is used to predict the moisture redistribution of non-swelling unsaturated clay
barriers. The large variations of the moisture values predicted by the two models are also discussed.
The general problem of transient heat and mass transfer arises from the irnposition of a thermal gradient onto a compacted clay barrier. In a partly saturated soil
system, such as that represented by the bufferlbackfill material currently being
considered a s a barrier for a nuclear fuel waste isolation vault (Lopez et al., 1984;
Yong et al., 1988), imposition of a thermal gradient on the system evokes both
coupled heat and mass transfer. The nature of the coupling and the characteristics of the transfer processes will be influenced not only by the initial conditions
that identify the state of the material, but also by the boundary conditions, e.g.,
access to or denial of water, and restraints concerning volume change.
This presentation addresses one particular set of issues: coupled heat and
mass transfer. Other issues such a s boundary conditions, initial conditions
such a s locked-in stresses, vapour transport, resultant volume change and
swelling pressure, etc., need to be mentioned a s pertinent items to be studied in
detail a t a later date. For now, the coupled heat-mass transfer problem is
considered to be the first step in a system study to be conducted before
embarking on the study that incorporates the other issues.
MECHANISMS AND PROCESSES
It is accepted that the ability to predict temperature redistribution in a soil
mass under thermal gradients in controlled laboratory test conditions is fairly
0013-79521901$03.50
(
1990 Elsevier S c ~ e n c ePublishers B.V
316
R.N.Y O X ET AL.
good (Selvadurai, 1988; Yong and Xu, 1988). In a soil system with very low
permeability it is not uncommon to assume that mass transfer (i.e., water
movement) lags considerably behind heat transfer, thus permitting the mass
transfer effects on the thermal coe5cients to be ignored. Experiments show
that T reaches equilibrium in a matter of hours, for a test on a clay soil bar of
about 20 cm long, whereas 8 reaches equilibrium only after more than one
month, where T and 0 are temperature and volumetric water content, respectively. We can safely assume that 8 does not affect T significantly.
In situations where a proper accounting of the redistribution of moisture
content is important-especially in view of access to outside water and also to
changes in the local swel1ing.pressures-it becomes important to determine
the rate and degree of moisture redistribution. Moisture movement in unsaturated soils due to coupled heat and mass transfer is less well understood
than moisture movement in saturated soils. This is because the mechanism and
mathematical description associated with saturated flow considers moisture
movement to occur only in the liquid phase, whereas in unsaturated soils the
flow takes place in both the liquid and vapour phase, requiring complex
formulations to account for coupled interactive effects.
The following elaboration of the problem describes the various complex
interactions and phenomena that reflect processes occurring in a non-swelling
soil; the added complexities of swelling pressure and volume change need to be
considered, but will not be considered a t the present time.
In the vapour phase, water (vapour) molecules diffuse towards the lowertemperature ("cold") region in a soil specimen subject to the thermal gradients, and may condense into liquid, thus causing a change in the soil moisture
potential in the "cold" region. This change contributes to the already existing
soil-moisture potential gradients developed by the thermally induced. difference in liquid-water energy status, leading to circulation in the liquid phase of
the soil specimen.
Besides the vapour diffusion process resulting from thermally induced
vapour concentration gradients, moisture in the vapour phase may be transported through thermo-diffusion and Knudsen flow. The relative kinetic
motion of molecules of various substances (such as air and vapour in the soil
pore space) in a mixture placed under non-isothermal conditions (Grew and
Ibbs, 1952) is identified as thermo-diffusion, whereas Knudsen flow, or slippage
flow, is said to occur in air-filled pores when the pore radii are nearly equal to
the mean free path length of water vapour molecules (Carman, 1956). It is
useful to note that the mean free path of the water vapour molecules is about
0.1 pm a t normal temperature and pressure, and that this increases inversely
with pressure (Jennings et a]., 1952). Another mechanism of interest is the
thermal moisture transfer induced by a volume change of entrapped and
dissolved air bubbles when temperature is varied (Nerpin and Globus, 1969).
In partly saturated clays, a common assumption is that thermally induced
liquid flow results from thermo-capillary film flow under temperature-induced
surface tension gradients, and that this occurs in the direction of decreasing
temperature (Bouyoucos, 1915; Philip and de Vries, 1957; Derjaguin and
MOISTURE REDISTRIBUTIOX PREDlCTlOS IN CLAY BARRIERS
315
Melnikova, 1958; Globus and Mogilevsky, 1969). Whilst variation of capillary
suction with temperature may result because of the temperature-dependency
of surface tension (Philip and de Vries, 1957), studies by Gardner (1955) and
Wilkinson and Klute (1962) indicate that the effect of temperature change on
matric suction can be greater than that expected from surface tension alone.
The discrepancy has been explained in terms of induced changes in the radii of
the liquid-air interfacial menisci in unsaturated soil pores following volume
changes of both dissolved and entrapped air bubbles (Peck, 1960; Chahal, 1964,
1965).
There are a t least four possible reasons why water flows in the liquid phase
under the influence of a thermal madient. First. a s noted above for unsaturated soils, the surface tension of water against air increases as the
temperature drops, and so moisture in unsaturated soil could flow from the
warm regions to the cool region under the influence of a surface tension
gradient. Soil moisture suction also increases as the temperature drops, which
could contribute to the moisture flow. This is the basis of the thermally
induced liquid flow equation developed by Philip and de Vries (1957). However,
as a second possibility, there might be some flow from the cold region to the
warm region resulting from the difference in specific heat content between the
liquid layer adsorbed in the solid surface and the specific heat content of
the liquid layer in the pores. Furthermore, the third possibility is a net motion
generated by random kinetic energy changes associated with the hydrogen
bond distribution that develops under a thermal gradient, and so another
transfer can be developed (Cary, 1966). A fourth possibility is the flow that
results from thermally induced osmotic gradients, e.g., spontaneous diffusion
of dissolved salts through a solution from warmer areas into colder areas (i.e.,
Soret effect) (Cary, 1966).
The fundamental question that remains is: How significant is each and every
one of these mechanisms/processes? Undoubtedly, composition, initial conditions and boundary conditions must be considered a s important partners in
the development of any answer to the question. In the final analysis, all the
interactive processes will produce coupled relationships that control the
distribution of heat and moisture. In an analysis the coupling effects and
the means to model them realistically must be considered.
-
MODELLING FOR PREDICTION
The model generally used a t the present time by many researchers is the
Philip-de Vries model which pays attention to the soil water potential in the
determination of diffusivity. Application of the Philip-de Vries model (1957) is
used to calculate and predict moisture movement under temperature gradients
in a n essentially uncoupled mode, i.e., vapour and liquid fluxes due to
temperature and moisture gradients are uncoupled. Additionally, certain
assumptions are needed with respect to (1) the validity of linear superposition
of vapour and liquid fluxes in obtaining total moisture and thermal diffusivity
coefficients, and (2) soil moisture potential and temperature dependencies. By
318
R.N. YONG ET AL.
and large, it is assumed that the soil moisture potential is independent of
temperature, although this is highly questionable and has been shown to be in
serious error for swelling soils.
The P h i l i p d e Vries model permits calculations to accommodate vapour
transfer effects. These are reflected in both the diffusion coefficients dealing
with moisture movement and the coe5cients dealing with temperature. In a
closed o r partly closed system, these terms can become important. However,
the difficulty encountered is the ability to measure vapour transfer, particularly if a shutdown experiment is allowed to equilibrate to some temperature
to permit handling of samples originally subjected to high temperatures. The
intervening period after shutdown and prior to sample examination allows the
sample to equilibrate to some extent and also permits the internal vapour to
dissipate or retransform to the liquid phase. The use of in situ gauges to
measure any of these type of events is attractive but is restricted by the
availability of reliable tools.
The above observations and measurements made after the ex~erimentswere
finished, are used in total mass balance analysis which does not show mass
conservation (Shah et al., 1984; Yong et al., 1988). Depending upon the
temperature gradient and the length of the experiment (time) a large loss in
moisture is evident-presumably through vapour transfer 'at the boundaries.
The three main concerns can be summarized a s follows: (1) inability to predict
moisture redistribution under temperature gradients (to explain actual
measurements); (2) questions about the role of vapour transfer; and (3) coupled
heat mass transfer, but uncoupled sets of calculations.
The above three points serve as the main motivation for developing the
analysis using the postulates or irreversible thermodynamics-as reported by
Yong and Xu (1988). In the development, a n identification technique is used to
permit calculation of the phenomenological coefficients, which are coupled in
a n attempt to resolve the theoretical prediction using the end-point measurements of temperature and moisture c o ~ t e n t The
.
modelling procedure considers a no-volume change situation and acknowledges that heat and mass
transfer are coupled; the calculation is performed accordingly.
The formula used to describe moisture flow within a soil column and the
temperature distribution along the soil column is given by the following
equations in non-dimensional form:
Governing eq.:
where: 0 =volumetric moisture content; T = temperature; C, = specific heat for
soil; Lww=diffusion coefficient for moisture flow due to a gradient of 0;
LTT= thermal conductivity coefficient for heat transfer due to a gradient of T;
L,,=coupling
coefficient accounting for effect of temperature gradient on
MOISTLIRE REDISTRIBUTION PREDICTION IN CLAY BARRIERS
3 19
moisture flow; and LTw=coupling coefficient accounting for effect of water
content gradient on heat transfer.
t J = time for whole test to reach equilibrium, T2=temperature a t the'hot end,
T, =temperature at the cold end, and L = length of the sample.
Boundary conditions, for l?
T(0, T) = 1
(2a)
T(1, 7)=0
(2b)
For 8:
Initial conditions, for T:
R S YOSG ET AL.
For
8:
The basic principle behind the identification technique used to determine
the phenomenological coefficients relies on matching experimentally obtained
end values for 9 and T a t various time sets in a controlled test with predicted
values. The predicted values are obtained from "guess" functions for L,,
L,,,
L,, and L,,, the phenomenological coefficients, where the "guess" functions
are written in terms of 0 and T. Insight into the physical behaviour of the
system is critical to being able to "guess" the type of function needed to
describe each of the phenomenological coefficients. The criterion established
to accept matching was determined by using the gradient method.
The guess relationships used for the phenomenological coefficients were:
Lww=(a, + a2T)e"lR
L,,=a,+a,T+
a,@
Lrw=~Lwr
LTT=a, + a,B+ a,T
where ai ( i = 1,2,...,9) and
under test.
v are constants to be determined
for the soil column
DISCUSSION
To test the capability of the Yong-Xu model for predicting heat and maw transfer, the calculated coefficients of isothermal and thermal diffusivity were compared with experimental measurements in a study reported by Cassel et al. (1969).
The soil used was a fine sandy loam (non-swelling soil) which was oven-dried
and packed in glass cylinders 200 rnm long and 76 rnrn in diameter. The cylinders
were totally insulated and temperature gradients were established lengthwise
along the cylinders through the end plates which could be maintained in a
heated or cooled status via circulation of 'preheated or precooled water.
Table I (columns 5, 6) and Fig.1 show the isothermal water diffusivity (D,)
predicted by the Philip-de Vries model a s discussed and calculated by Cassel
et al. (1969) with the isothermal water diffusivity (L,,)
calculated by the
Yong-Xu model. As noted, there is a difference of a t least one order of
magnitude between the values calculated by the two models, and the Yong-Xu
model shows a decrease in the isothermal water diffusivity with decreasing
temperature. Note that the Cassel predictions (Table 1, column 4) vacillate
between plus and minus values, and do not correspond t o the other findings.
Fig.2 compares the thermal water diffusivity (D,) predicted by the Philip-de
Vries model as calculated by Cassel et al. (1969) against the thermal water
diffusivity (L,,) calculated by the Yong-Xu model, (Table I, columns 8,9). The
results show that D, decreases slightly with decreasing temperature in
321
MOISTURE REDISTRIBUTION PREDICTION IN CIAY BARRIERS
TABLE I
Comparison of calculated coefficients of isothermal and thermal diffusivity using experimental
measurements from Cassel et al. (1969)
Disuibulion
along
column
(cm)
Mem
Mean
C z s e l e l al.
water
Dl
Philip
De Vriee
Yong-Xu
(this study)
C a u e l et al.
temperature
("3
content
(em2/day)
Fd
L,, = D,
(cm'iday "C)
(crn'lday)
(em2/day)
B (em'/em')
-
Z
..-9>
'
.
2
'-
a
f
0
"
I
-
Philip
de Vries
Yong.Xu
(this study)
q"'
Lw, = D.
(em2/day C ) (emZ/day 'C)
Yong-Xu ?;lode1
m
n ?
' - E
"J
2-
D:
-$
1.00
.75
.so .25
-
,.
Philip-de Vries Model
..
"
*
0.00
I
I
2
6
L
8
10 12
L
I
14 16
18
20
Distance Along Colurnn~lO(rnm1
Fig.1. Variations of isothermal water diffusivity along soil column.
contrast to L,,, which shows a large decrease with decreasing temperature.
The values converge in the midsection of the test sample. The Cassel values
again demonstrate an up-down trend.
CONCLUDING REMARKS
Moisture flow due to temperature gradient is transported in both vapour and
liquid phases. The driving forces in the vapour phases are: (1) a thermally
induced vapour concentration gradient; (2) thermo-capillary due to a temperature-induced surface tension gradient; (3) thermo-diffusion due to relative
kinetic motion of molecules; (4) Knudsen flow or slippage flow in air-filled
pores; and ( 5 ) volume change of entrapped and dissolved air bubbles. The
R . S . YONC ET AL.
-.>
Z
0
,0250Y o n ~ - X uModel
0
3
22
u
2
.0175.O 150
1
Philip-de Vries Model
L
'2
,01251
D ~ s t a n c eAlong Column .lO(mm)
Fig.2. Variations of thermal water diffusivity along soil column.
driving forces in the liquid phase are: (1) a thermally induced moisture
concentration gradient or thermo-self diffusion; (2) thermo-capillary; and (3)
thermo-osmosis.
The general expression for D, ( = D o , +D,,) and D,(= D,,+D,,)
in the
Philip-de Vries model, where the second subscripts I and u refer to liquid and
vapour, respectively, includes some of the above driving forces, while the rest
of the driving forces are missing due to the difficulties arising in establishing
suitable relationships. The following assumptions were used for those driving
forces that are included: (1) Darcy's law is valid; (2) soil water fitential does
not depend on temperature; (3) relative humidity depends on temperature; and
(4) there is no overall volume change. The validity of Darcy's law is doubtful
for unsaturated flow in which there is a change in pore geometry where the
overall volume remains constant. In addition, the soil moisture potential is
assumed t o be independent of temperature, although this is highly questionable and has been shown to be in serious error for swelling soils.
The Yong-Xu coupled flux model compares well with the Philip-de Vries
(uncoupled) model insofar a s computations/predictions of D, (L,,) are concerned. In that regard, we could argue that the coupled model "works". Thus,
noting t h a t the coefficients L,, and L,, calculated by the Yong-Xu model are
based on the use of an identification technique in conjunction with actual
experimental measurements, we are reassured that the values for L,,, which
correspond closely to D,, reflect the actual test situation and material
"properties". However, we cannot help but note the differences in D, and L,,.
Thus, the question that arises is whether the one order of magnitude differences predicted in D, (L,,) by the Yong-Xu model in contrast to the Philip-de
Vries model reflects the coupling effect not considered in the Philip-de Vries
model. Is it possible that the coupling set of calculations would produce a one
order of magnitude difference in calculations for the isothermal diffusivity
coefficient? Obviously, more work needs to be performed to obtain a better
MOISTURE REDISTRIBUTION PREDICTION IN CLAY BARRIERS
323
appreciation of the coupling affect. The consequences of a large difference in
D,/L,, values can only lead to problems in determining the "real" sets of soil
thermal properties.
The other sets of concern are:
(1) The present comparative study using Cassel's data where the calculations are made for a non-swelling soil. How does this relate to a swelling buffer/backfill material where the swelling component is indeed fairly
large?
(2) The role of locked-in stresses and associated volume change or developed
swelling pressures (local) within the system as moisture transfer progresses.
These have not been built into the analyses. How influential are they with
respect to the coupled or uncoupled sets of calculations for both isothermal
and thermal diffusivity coefficients?
These sets of concerns are not insignificant. However for the present, we can
only march systematically and begin examining of the transfer, where the
ability to measure such phenomena is severely restricted by equipment
capabilities and restrictions. With the two models (Yong-Xu and Philip-de
Vries), we a t least have the capability to pursue the problem further to more
fully delineate not only the coupled/uncoupled effect, but also the interrelationship between physical models (e.g., Philip-de Vries) and mixed (Yong-Xu)
models.
ACKNOWLEDGEMENTS
This study was supported by both AECL through contract research, and by
NSERC (through Grant No. A-882). Acknowledgement is made to Dr. M.N.
Gray and Dr. S. Cheung of AECL and Dr. P. Boosinsuk of the GRC McGill for
invaluable input and advice. This paper was approved by AECL for publication.
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Carman, P.C., 1956. Flow of Gases Through Porous Media. Butterworths. London. England.
Cassel, D.K.. Nielsen, D.R. and Biggar, J.W., 1969. Soil water movement in response to imposed
temperature gradients. Soil Sci. Soc. Am. Proc., 33: 493-500.
Chahal, R.S., 1964. Effect of temperature on the energy status of water in porous media. Soil Sci..
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Chahal, R.S., 1965. Effect of temperature and trapped air on matric suction. Soil Sci., 100: 262-265.
Derjaguin, B.V. and Melnikova, N.K., 1958. Mechanism of Moisture Equilibrium and Migration in
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Gardner, R.. 1955. Relation of temperature to moisture tensionof soil. Soil Sci.. 72: 257-265.
Globus. A.M. and Mogilevsky, B.M.. 1969. Thermal Transfer of Liquid in Porous Media. Spec,
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Grew, K.E. and Ibbs. T.L.. 1952. Thermal Diffusion in Gases. Cambridge U n ~ v Press.
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Jennings, J.E.. Heyman. P.R.B. and Wolpert. L., 1952. Some laboratory studies of the migration of
moisture in soils under temperature gradients. South Afr. Natl. Build. Res. Inst. Bull., 9,
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Lopez. R.S., Cheung. S.C.H. and Dixon, D.A., 1984. The Canadian Program for Sealing Underground Nuclear Fuel Waste Vaults. Can. Geotech. J.. 21: 593-596.
Nerpin, S.V. and Globus, A.M., 1969. Influence of temperature on the transport of water. Proc.
Wageningen Symp. Water in the Unsaturated Zone, UNESCOIInt. Assoc. Sci. Hydrol..
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Peck, A.J., 1960. Changes of moisture tension with temperature and air pressure: theoretical. Soil
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Philip, J.R. and de Vries, D.A., 1957. Moisture movement in porous materials under temperature
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Selvadurai, A.P.S., 1988. Thermal Peiformance of a Buffer Material Developed for use in a Nuclear
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Shah, D.J., Ramsey, J.W. and Wang, M.. 1984. An experimental determination of the heat and mass
transfer coefficients in moist unsaturated soils. Int. J. Heat Mass Transfer, Z7:1075-1085.
Wilkinson. G.E. and Klute, A. 1962. The temperature effect on equilibrium energy status of water
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M O D E L L I N G , SIMlLATION AND OPTIMIZATION
~ontteae.
May
1990
APPLICATION O F A N IDENTIFICATlON T E C H N I Q U E T O EVALUATE
D I F F U S I O N P A R A M E T E R S IN A C O U P L E D F L O W
XIOHAHED.
A.M.O..Xu.. D.M.. YOHC. R.N..
GEOTECHSICAL
RZSEMCHCENTRE. MCGILLUNIVERSITY.
MONTREAL.
CANADA.
AND
C ~ c u t i c S.C.B.
.
ATOMIC ENERGYOF CANADALIMITED.PINAWA.MANITOBA
ABSTRACT
This paper f o c w on the c x p e r i m a t d study and the ?henomenological evaluation of the a r a m e k n governing coupled heat
and moistmwemcnt in b d a d e r i a l . A o n e d i m m s i o d
m t d-tion
is d to examine the heat induced moisture
m o v r m d in s buffa msLeria compacted at s p x S e d optimum
moi3tufc c o n M t and madmum dry demity. %ad &a of onedimcpdonal mu are used ta determine the time dependent moistdhuibution that develops ar a 4 1 of the heating action.
T h e aperimmral r s u l t s of the ~ l p r a t v r eand moist- &lributionr M d to develop the materid p K a m e t a gweming
coupled heat and moistmovcmmt within the buffer m d d .
The model as well as BLgulation technique for coupled heat
and moisture m w m m t in porous media p r o w by Yong and
Xu [lj is used in the minimiration technique for evaluation of the
phenomecolo.$cd diffusion parameters. The relatiomhips of t h e
p a r a m e t a in an indated column of buffa materid with one end
subjated to c l e w e d tempcratof 100.C are p r c ~ r e d .Work is
in p r o g a s to justif?. and g m a d i r e t h e relationships with w:
iaus b i t i d and boundKv conditiom. Hmce omdiction of the hent
induced moist>e mov&mt within bufTa & & I in a r e p i t o r y
envimnmat 4 be d a n c e d .
by Yong and Xu [ I ] and Yong et d . [3j is uwd to cdcvlate t h ~
&ion
parametm.
EXPERIMENTAL APPARATUS
A schematic d e i g n of the t a t equipmmt is shin Fig. 1
can be equipped with a motrolled tcmprrar- &n
pkLe ta p d & the& porential and a pressvre
cell to mearurc &e reaction pressure ta indicate w a t a movarm:
d primetrm by s
with time. The r p d m m is inrulared ~ o m irr
ri 'd PVC Liner to miaimize the volumc &anr of t h e , s p d m e u
T& specimen and PVC Liner are encwed wit 111 a n g d suppor:
of concrete and s t 4 amund the perimeren and steel rnd plates
A cmiing system is installed inside the concrete u d boctom enc
p l a t e to control the remperature around the perimeter. and hencc
the t e m p r a t u r e gradicnrs. A ring - type heate: is itlctalled ithe heating plat? in order ra conirol rhe temperature input. Thc
t e m p n t u r e of the ring is monitored and ~anrmUedby the thermdogic control unit. T h s unit rnhancu rhc ability to caicu1a:e
the t o t d input heat quantity. .4 tempraturc monitoring w.i: is
Each end of the r p d m e n
KEY WORDS
Difhios heat, moisture, buffer, transport, minimization
INTRODUCTION
The Canadiaa Sudear Fuei Waste Management program is
-kg
the concept of d i s p i n g of -re
in a vault excavated
at a depth of 500 to 1000 m below the ground s u r k e in the plutonic rock of the CaDadian Shield [Zj. Beddo the naturally low
pcrmefbility rock, a nvmbet of mginured barriers would be wed
ta 1-1
radionudide release from the r o l i W wasre. T h e are
the corrosion raistant c m t a i n a , the b d a and the b a & l L The
containem d be pkced in b o d o l a drilled into the BOOIS of onpl-ent
room and wparsted born the h a t rocL by a bentonite
- sand buffer matmid. The remainder of the vault will be m e d
with sn earth-.
d a y based b d .
-
-
The eneral prabGm u n d a investigation adthe pcrformance af%uffermateridw d e r an imposed thermd gradient, which
in turn induces coupled h a t and m- Bm in the system The nature of the coupling and the characteristic of the 3 m p m c a s a will
be iduenced not only by the initid conditiom (e.g.. volumetric mt a content and temperature) but a h by the boundary conditiom
(e.g., access to mrer intake or volume change).
....
Pi."
.Ui
This paper addrmer the experimenrd study and e d u a t i o n
of the materid p a r a m n m which govrming the process of coupled
heat and mars Bow in the buffer materid. A one- dimensional
t a t s cod-tion
is d to examine the transient temperature
and moisture Bow in a buffer materid compacted a maximum dry
d a i r y and optimum moist- content. Several &a of one - &mensiod tests are used LO investigate the nature of the traruient
process of both (cmperature and moisture movonent under different imposed temperature gadienu. The mcasurcd tranrimt emperature and moisture distributiam are u d to cdcukte the diffusion p s r a m e t m that g o m heat and moisture mwancnt w i t h
the buffer materid. An identification technique which dacribed
I.0.W
-
_n
Y. .I...
CI-.
I-,
1
L..Y
.-.-
Y..
..-i .li
Y..
U....,I.. " h i
a
.
-m
I.,
..r_
m
Lli
.-<
used to r d the v a r i u g thwocouples i n s d e d by am !Xq and
convertin the thermocouple readings from millivolt to &men celsius and kplaying them on s digital voltmeter. The -rimmtal
apparatus and 11s Mmciated components are dacribed in detnil by
Mohamed ct al. 141.
EXPEWMENTAL PROCEDURE
BuKer Material
The h d e r m a m i d s e l d for t h investhption uar alaboratory prepared mixture of d u m bentonite ( A v o d ) end graded
lndusmin slLca rand sl s proportion of 50/50 by dry wdght. The
d e t d e d wmpositiond study of the bentonite has been reported by
Quigley 15;. The minng solution used aar a "reference" synthetic
tbc pundwater (CCW) with a rrclpc pvm by Abry et d.
6 The Lqud Lmtt o!the A v o 4 clay
uar 2S4 0% and the
plastx ~ m was
t 40.7%. whle the gram rue d~rtributiona . 3%~
rand. 12% sdt, aod 35% clay. The spdfic proportions of the mixt m were cborm becsl~geof their potentially attrxtivc physical,
chemical and m&anicd pmpcnies for the required p a i o r m c e
criteria adopted in the Canadian nuclear fud warre disposd pro.
8" 121.
F-
Sample Preparation
The buffer matcrid spedfied above was mixed with appracimarely 19% granitic groundwater by weight. Sampla w e stored
in a humid room for a period of one week to rereach equilibrium. To
achieve a constant dry density of 1.6 1.7 Mg/ml. samples were
divided into 6~ layers and a known quantity ol wet materid uas
compacted rraticdly in each la>=. TLcrmocoupla were placed E&
ter wmpaction inside the soil sample at different poriuons dong
the length of the samples Samples were left fur a period of 24
h o r n to reach equilibrium after the installation procedures w r e
wmpletcd.
-
Samples were then heated ax one end by a heater a t a constant
remperatm of 100•‹C. A schematic diof the heating cell is
shwan in Kg. 1. The perimeter of the sample. - i.e. the concrete
section around the sample. war set h? using a cooling ?stem to
-1
it to a k n o ~ mtemperature. Temperature m e a r ~ ~ l l l e nar
rs a
function of space and time w m talien to delerminc the ternperamre
prodler don the sampla dvrin heating. Furthermore. tests were
into 9
ended at dikerent tima and tfe sampla were -timed
portions to d c t m i n c the moisture distribution dong their length.
EXPERIMENTAL RESULTS
T a t Nos. 1 to 4 in Table 1 were performed at the same initial
and bound- conditians. The onlr difTamce betwem t h e t v o a
of experime& is the time 2 whi& the t a t war stopped and ihe
samples were mtioned to measure their volvmetric aatcr content.
~~~~
~
~
Tea No. 3 was 6 t o p ~ adirectly
l
aftcr the remperature distribution
d e d equilibrium, end the samples vae sectioned into 9 portions
to measure their volumetric wakx wntmt. Each portion aar 80.3
mm in dismelcr end 9.66 mm long. Their urah no noticeable
variation in the volumetric water wntmt distribution due to the
time rslrm to: (1) remove the sample from the mould. (2) cut
Visualinspection
it into 9 portions, end (3) take m-urrmmts.
of earh scction after cutting indicated that the water content uas
distributed uniformly in the radial direction. Therefore, timc at
first m e d was inaeaxd to 0.294
which the water contmt
dsya.
The ianpcrature distributions w a c similar ar a function of
sp- cad time in all the experiments, and a typical rempaature
dis(ribution p m d l s from T a t KO. 1 is discussed Figure 2 shthe tanpaatm distribution pro& for T a t No. 1 at differen;
time in&,
where it can he .em that the temperature distributions d c d s d y st& aftcr 0.107 d a y .
The volumetric water content distribution ar a function of
in Fig. 3. The volumetric w s t a wnspacc and time is p-ed
t m t was la+ a t the hot end due to the imposed heat flux at this
boundary which dmve the moisture away from the hot end in a
longitudinel direction. The I k t measl~ableprofile war achieved
aft- 0.294 days. A f k appro%ima(ely 3 d y , the ,volumetric nmter
m t m t decreased at the hearer side to 11 o and m a d to 40%
a t the wld end. It is appropriate to note that all the uperimentd
results show a m a a balance.
ANALYSIS O F EXPERIMENTAL DATA
This section focuses on the evaluation d the diffusion parameters governi the coupled heat and moistvrc movement in buffer
mataid h
d on the actud experimental data presented in Figs.
2 and 3. The rnethodologv adopted is that proposed by Yong and
Xu [I]. The one - dimensional coupled heat and mass flow equation
t a k a the forms:
where 0 = the volumetric
wntcnt. B = B(s.1); T = temperat-,
T = T(r,t);
Ltvw = diffusion d c i e n t for h i d flow
due to gradient of 8; Lrr = thermal conductivity coefficient for
heat t r a d e r due to gradieit of T Lwj-,LTn, = wupling c o d fidents arising from tempaatrre and moisture; i = timc; C, =
specific heat of sol.
The boundary conditions corresponding to the experiments
are given by
T = TI
ot
r = 0;
and
T = T2
ai r = f
(3)
Parameter Identiecation
mr
--.
In order to idmtify the msrerid parameM from Lhe expai m d data, it ia assvmcd that the nondimenriod parametas
can be expressed in termp of 8' and T' and a set of unknown coefBaents a. (i = 1.2. ...,10). The particular rep-tations
suggested
by Yong and Xu 1.51 take the forms:
L',,
Li,.T = a,
f tI"(b,.l
o
= (a1 + a , ~ ' ) c " ~
(13)
Ljw = a ~ o L b r
M
(14)
un
at
1
=0
and
r=
+ 4' + aoT'
(15)
These r e p m t a t i o n c are not meant to be unique. They are
propabed bared on the extensive obwnations made during to the
e x p r i m m t d study of heat and m o i s t w Bow in porous media.
The parameter identification can now. be reduced lo the determi.
nation of the unknown c d c i e n t r a , (i=1.2, 10) Lorn the known
e x p k t d d u e s of T' and 8' ar a function of r and C.
WIM Fran The mema E!amml 0
F b 3 Vdvnsbls Wsler Cmlarl Dl~lrlbullmms s Fmctlm
01 W
I ud l~l m s
as + Lwr-aT = 0
ar
ar
+ asT' + a&.
n w
LjT = a ,
Lww-
(12)
....
e
(4)
Integrating Eqs. (7) and (8) with respect to ( from 0 ro (,, we
obtain
The initid condirionc corresponding to the upriments are
,=,
givm by
1=0
@a)
T=Tl
at,O<r<t
(5b)
e=e0
ot o < z < t
(5~)
T=Ti
at
=e-eo;r
d
ae- + L -ar
K
ac
=L
-
t
T - Tl
,r = -;(=
T, - T,
t,
= -.
l
f
a(
and
f
1
'8
a ( L ~ Wae-T ) + -a( L ~ - )a?..
a ac
c ac ac
-a~'
=-
(8)
z d (
m
- ( L > ~as.
a(- + rn-) K
Ljw = tt(i-1 - TdLrw; L b = L L (9b)
~
PCm
PCv
by:
in nondimenriond form are giThe boundzy
(ce)ij =
.
ap
ond T'=O
at
(=I
as.
l F l r = r , d c - ( L ~ ~ -a(
+Lb~-)
(18)
BT'
K
~ c = c (19)
~ , ~ ~ ~
In order to delerminr the totd m o r obtained h m the valumetric moisture content and the tempaature dirtributionc o v a all
l~cationsand for dl times, an error norm c can be defined
= Crcco,:,
(=O;
o
and
where
at
$
:I=
Due to the limitations of the arrumed nondimeariond p h c
nomyologicd parametem (Egr. (12) to (15)), Eqs. (17) and (18)
are not exactly zcm. The corresponding error terms for the volumetric water content (ce) and temperature (qJcan be ddned ar:
(7)
ac
ar
as-
~ " $ ~ ( - ( L ~ ~ -a(+ L ~ ~ - ) I , = < , = O (171
(6)
where : 8' = nondimmsional volumetric w a r n content; T =
nondimzwiond temperature; i f = hnal time.
T h e governing equations are
T'=l
(16 b)
Applying of the moisture conrent boundary condirions gives:
where: T,= temperature at bat end: T2 = temperature at cooler
end: P = lengthofthesample: & = initid volumetric water content.
Eqs. (1) and (2) can be pr-d
in a nondimensiond form
by using the following repnesentationc.
B
1
(164
(108)
+ (.r,:,1
(21)
i.j
For least square emor, the equationr required for determining
ai are obtained Lom the conditions
~ h e ' i n i t i dconditions in nandimensiond form are given by
T'=l
at
(=O;
and T ' = O
at
O<Cs1
(lla)
Furthermore. Eqs. (19) and (20) in arrodation with the condition represented by Eq. ? constitute the criterion for the i d a tification technique. The solution of the resultant nonlinear set
of equations is tb adopt a solut~wascheme based on the gradient
method where the ini:id atimatc oi thecocfficicnts af (1=1.2..... 10)
is adjusted i1era:ively. i.e.
where r is the iteration number and A is a small increment a p
plied to the coe%iuents at the rl* iteration. The iteration continues
with a!'' (i = 1.2 ....,10) until A< < 0, whue Ac = d"" - el').
The increment A"+') is assigned a specific d u e 1/10A(". .4t the
end of an iteration, the new c d d e n t s will give a small magnirude o! c . The procedure is repeated unti! Ac is emluated to a
predetemrined order o! accuracy~
APPLICATION
The transient tunperature and volumetric water content p r e
6la are used in =sodation r i t n Eqs. 19 to 23 to calculate the
parameters a; ( i = 1.2....10). The fo!louring results are obrained in
a nondimensional form
L;YH, = 0.05i2 + O.O?ST. + 0.04268.
(14a)
REFERENCES
[I] Yon RN.. and Xu, D M . "An Identillcation Technique
for k v a l u a t i o n o f P h e n o m e n o l o i c d Coefecients in Coupled Flow .in U n s a t u r a t e d s o b Int. J. for Numer. and
Andy. Methods in Geornechanics, Val. 12. (1965) pp. 263299.
The calculated phenomenological parameters at steady state
conditions are listed in Table 2 ar a function of distance and volumetric warer content. The isothermal warer diffusivity ( L w w )
increares as the volumetric water content increases, u!hile the
thermal water diffusivity ( L n e T ) dccre-5 ar the vollunetric aata content increases. Furthermorn, the thermal diffusivit). ( L r r )
increases with increaring volumetric aater content up to 8 = 35%
and then decreares. These results fallour the same trend as those
reported in literature 111. 1t should be nored that the relationship
of the parameters (Eq. 23) are evaluared for a given bound* condition at higher temperature. Work is in progress to justify t h e
relationships vith lau,er temperaturs.
CONCLUSION
This study outlines an experimentd technique that can be
used to study the neat induced moisture movement nitnin a buffer
material. The one - dimensional tests conducted on buffer material
can be used to assess the time - dependent movement of moisture
and m p e r a t u r e due ro an externally imposed thermal gradient.
The experimental results for the time-depmdent temperature and
moisture distribution as well as the identification technique can be
used to calculate the functional relationships of diffusion parameters arsociated with the mathematical model describing coupled
heat and moisture movement within the buffer material. These
relationships are determined for a given initid and boundary conditions. The d d i t y of t h a e relationships needs to be verified for
variou initid and b o u n d q mditions. This work is now under
p m g a s a t the Geotechnical Research Centre, McGill University.
ACKNOWLEDGEMENTS
This stud? was supported by Atomic Energy of Canada Limited (AECL) through a mearch contract.
121 Lopez. R.S.. Cheung. S.C.H., and Dixon. D.A. &'TheCanad i a n P r o g r a m for Sealing Underground Xuclear Fuel
W a s t e Vaults" Canadian Geotechnical J. Vol 21. So. 3,
(1964) pp. 593-596.
Yong. R.X., Mohamed. A.M.O. and Xu. D.51. "Coupled Heatmass l k a n s p o r t EKect o n Moisture Redistribution Prediction i n Clay Barriers" P m ' l i n g s . Anificd clay Barriers for High Level Radioactive WastcRepsitories E%. R
Pusch. Or anized by Swedish Nuclear and 1IHs.e manage
Xuclear Energy Agency. (1966) pp. 47-57.
ment with ~ E C D
[4]'Mohamed. A.M.O.. Yong. R N . , Capruscio, F.. and Cheung. S.C.H. "A Coupled H e a t and.Moisture Flow Testi n g Celln Paper prepared far Geotechnical Testing J . . .ASTM
(1990).
I
-.
[5] Quigley, R.M. "Quantitative Mineralogy a n d Prelimin a r y P o r e W a t e r C h e m i s t r y o f Candidate Buffer a n d
Backfill Materials for N u c l e a r Waste Disposal Vault"
Atomic E n e r a o f CanadaLimited, Report AECL 762i. Pinawz
Manitoba (1964).
16) Abry. D.RM., Abry, RG.F.. Ticknor, I<.\'.. and Vandergraaf.
T.T. "Procedure t o D e t e r m i n e Sorption Coefficients of
Radionuclides o n Rock C o u p o n s U n d e r Static Conditions" Atomic E n e r a of Canada Lirnired. Technical Record
TR-169, Pinau.4 Manitoba (1962).
T E M A L BEEAVIOUR OF BACRFILL HATERIAL FOR A NLICLFM
FUEL WASTE DISPOSAL VAULT
R.N.YONG and A.M.O.MOBIUIED
Geotechnical Research Centre, McGill University,
817 Sherbrooke St., V., Montreal, Quebec, Canada E3A 2K6
S.C.E.cEEuNG
Atomic Energy O f Canada Limited, Vhiteshell Research Establishment
Pinawa, Manitoba, Canada, ROE 1LO
The concept of disposing of used nuclear fuel in engineered rock
formations is being studied in theTanadian Nuclear Puel Vaste Management
Program. After the used fuel is emplaced in the vault, the vault would be
backfilled. The backfill has to satisfy a number of engineering requirements. A reference backfill with satisfactory hydraulic, thermal and
mechanical properties has already been selected.
As the used fuel in the waste containers decays, heat will be generated
and this heat will raise the temperature of the backfill material. The
performqnce of the reference backfill material was evaluated over the
temperature range 20-100•‹C. This paper addresses the results of experiments
on thermal response and pressure development in the backfill for the period
shortly after vault closure.
INTRODUCTION
The Canadian Nuclear Puel Vaste Management Program is assessing the
concept of dispsing of waste in a vault excavated at a depth of 500 to
1000 m below the ground surface in plutonic rock of the Canadian Shield 111.
Besides the natural low permeability rock, a number of engineered-barriers
are used to limit radionuclide release from the solidified waste. These are
the corrosion-resistant container, the buffer and the backfill. The
containers will be placed in boreholes drilled into the floors of emplacement rooms and separated from the host rock by bentonite-sand buffer
material. The remainder of the vault will be filled with an earthen,claybased backfill.
Various backfill materials have been studied 121. The reference
material is a mixture of crushed granite aggregates and Lake Agassiz clay.
It has been found that a well-graded mixture with a m a x i m aggregate size
of 19 mm and 25% clay content compacted to a dry density of 2.2 ll&/mf has
satisfac- tory engineering properties, such as low hydraulic conductivity,
some swelling capacity, sufficient load bearing capacity and acceptable
thermal conductivity [2,3j.
In the vault, the backfill will be heated by the vaste containers up to
a maximum temperature of 100•‹C. This paper describes laboratory studies of
thermal response and pressure development in backfill subjected to various
temperature gradients in a closed system, i.e. a no-moisture-flux boundary
at the external boundary of the backfill. The closed system is used to
simulate the hydraulic condition of the rock for the period soon after vault
closure. During this period, hydraulic flow from the surrounding rock to
the backfill is likely to be low due to slow re-establishment of the
groundwater table to its natural condition. The backfill could undergo
thermal drying. with the potential for shri-e
and craddng that ray
impair its t h e d perfomance. The development of backfill pressure under
elevated temperatures is not clear and needs to be studied.
In this study, ve have used a heating cell designed for one-dimensional
heat flov in the backfill vith the capability of providing various
temperature gradients vith a maximum temperature of up to 100•‹C.
The heating cell consists of up to 3 cylindrical aluminum tubes.Each
tube has an internal diameter of 152.4 mm, a height of 116.45 mm and a
thickness of 12.7mm.The dimensions of the tube are similar to those used for
the Modified Proctor Compaction test (ASTH 01557). This tube vas selected
because it could accommodate the large-size aggregates in the backfill, the
high forces used in compaction,and minimize the heat flov through the
backfill in the radial direction. Figure l(a) shovs the schematic diagram
of the assembled cell. The length of the test specimen can be changed using
different numbers of tubes. A PVC (Polyvinyl Chloride) liner vas attached
to the inside of the tube to insulate the specimen. The cell vas capped at
both ends by an aluminum plate capable of alloving fluid to enter the test
specimen. In this study, no fluid vas actually permitted to enter, simulating a possible condition in the vault at a time shortly after closure A
neoprene gasket vas installed betveen tubes and at end plates to prevent
moisture from escaping.
lhenocwple
assemblhs&
25.4mm spacings
type
51mm
mreaded rod
PVC lmer
9 5mm ?tuck
neoprene O-ring
alwninum
12.7mm thick
pressure honsducer
porws stone
Figure l(a)
Heating Cell Test Apparatus
Temperature gradients in the backfill vere imposed by heating one of
the end plates vith a 124 vatt (ring-type element) heater. A copper plate
vas housed in the plate to transmit the heat to the backfill. The heater
supply vas automatically controlled to maintain the constant elevated
temperature required at the end plate. Each time the heater vas activated,
the time totalizer vould record the total time of heater supply output, from
vhich the cumulative thermal energy vas calculated. The temperatures vithin
the backfill vere monitored by type T (Copper-Constantan) thermocouples,
each encased in a stainless steel sheath. Temperature-compensated pressure
transducers vere mounted at the end plates to monitor the change of pressure
exerted by the backfill. In this paper, ve vill refer to the heater end
plate as the hot end and the other end as the cold end. In some tests vith
shorter specimen lengths, the backfill pressure at the middle of the
specimen vas also monitored by the transducer housed at the inside periphery
of the tube. The effect of differential thermal expansion betveen the
transducer and aluminum used in the tube and the end plate vas accounted for
in the measurement.
After the backfill specimen vas placed in the cell and the instrumentation vas assembled, the cell vas insulated vith layers of pipe wrap,
expanded polystyrene and fiberglass to minimize system heat losses (see
Figure l(b)).
Figure l(b)
Insulation for Heating Cell Test Apparatus
UATERIALS AND SPECIUEN PREPARATION
The backfill material used vas a mixture of granite aggregate and Lake
Agassiz clay. The aggregate component consists of granodiorite crushed to
various screen sizes (21. The granodiorite consists mainly of quartz,
feldspar, plagioclase, microcline, biotite and chlorite, vhereas montmorillonite, illite, quartz, kaolinite and feldspar make up the clay. The fluid
used vas a synthetic granitic groundvater(GGU).
The quantitative analysis
of the tvo materials and the chemical composition of GGU are reported [4,5].
The appropriate proportions of aggregates and the povdered clay (25% on
veight basis) vere mixed vith an amount of GGU solution corresponding to the
optimum vater content (about 8%). The moisture in the mixture vas alloved
to equilibrate in a sealed container for 2 veeks. The mixture vas then
compacted into the assembled tubes to a dry density of - 2.2 Ug/m3 folloving
the Uodified Proctor compaction procedure. After compaction, the test tubes
vere attached to the end plates. Thermocouples vere installed in the
specimen by drilling pilot holes vith a diameter less than that of the
thermocouple to ensure tight contact. Once the instrumentation was
installed, the test cell was insulated as s h o w in Figure l(b).
The heating control unit was then set to the desired heater
temperature. Throughout the experiment, the test cell was laid on its side
to alleviate the influence of backfill self-weight on pressure and the
effect of gravity on moisture transport.
RESULTS AND DISCUSSION
.
Seven tests vere conducted vith two different specimen lengths at
heater temperatures of 50•‹C to 90•‹C. The lengths vere 116.4 mm for Test
Nos.1 to 5, and 349.2 mm for Test Nos. 6 6 7. The results are presented
below.
Thermal Response
Figure 2 shows typical time-dependent temperature profiles in the
specimen and the aluminum tubes from Test No. 7. In this test, the heater
temperature was 90•‹C. A large temperature difference existed between the
two ends of the specimen at the early stages of the test. The temperature
gradient in the longitudinal direction of the backfill was highly nonlinear. Subsequently, the temperature difference and the non-linearity of
the gradient decreased vith time. After -1439 min, the temperature profile
did not change any further, indicating that thermal equilibrium was reached
At steady state, the temperature gradient was almost linear.
Tube
Specimen
Time (min.)
A = 31
8 = 153
C = 340
D = 1439 - 7878
Figure 2 Temperature Profiles of the Specimen and Tube
for Test #7
The temperature profiles of the aluminum tubes followed trends similar
to those of the specimen. However, the magnitude and linearity of the
temperature gradient are much less than those of the specimen. Uoreover,
the temperature of the tube at the hot end did not coincide vith the pre-set
heater temperature. This is likely due to heat loss associated vith the end
boundary. Thus, a uniform temperature distribution could not be maintained
in the heater plate, and a temperature difference of
20•‹C vas observed.
-
The time-dependent cumulative heater supply energy curve for Test No.7
is shovn in Figure 3. The straight-line portion of the curve indicates the
quantity of heat required to maintain the thermal equilibrium of the heating
cell system at steady state. The extrapolation of the straight line portion
of the curve back to the energy axis gives the quantity of heat used to
raise the heat content of the system to the equilibrium thermal condition.
The time for the system to reach steady-state is indicated by the start of
the straight line portion of the curve and corresponds closely to the time
required for the specimen to reach thermal equilibrium.
A
---
Legend
ENERGY
Ternperoture
Pressure
TIME (minutes)
Figure 3 Ueasured Cumulative Energy, Pressure and Temperature at Two
Ends of the Reference Backfill vith Time for Test #7.
Similar thermal responses vere observed for other tests. Eovever, the
time required for the shorter specimen to reach steady-state temperature vas
about one-third (- 400 min) of that of the longer specimen. The steadystate temperature gradients of the tests are shovn in Table 1, and indicate
the feasibility of the cell to impose various gradients vith either various
heater temperatures for the same specimen length or different lengths vith
the same temperature. The difference betveen the pre-set heater temperature
and the hot-end temperature vas observed to be lover vhen the heater
temperature vas lover. The rate of cumulative energy consumption at steadystate vas found to increase linearly vith heater temperatures for Test
Nos.1-5, shoving that the effect of non-uniform temperature in the heater
plate on heat transfer is insignificant. In particular, the intercept of
the curve at the heater temperature axis indicates a zero rate at room
temperature, shoving the consistency of the measurement.
TABLE I
Measured Steady-state Temperature Gradients,T and Maximum Pressures, P,
in the Reference Backfill vith Beater ~empera?ures.
Test No.
Beater
Temp. (OC)
T,(OC/m)
Rot end
P(KPa)
Middle
Cold end
Pressure Development
When the backfill is subjected to elevated temperatures, it vill tend
to expand. If the expansion is not restricted, it vill expand as the
temperature increases. If the expansion is restricted, an expansion
pressure will develop, which will increase as the temperature increases.
The expansion pressure also depends on the extent of expansion; the more the
expansion is allowed, the less the pressure vill be. In this study, the
expansion pressure vas found to depend on temperature, heat induced moisture
movement, frictional forces between the backfill materials and betveen the
tube and the backfill, and the thermal expansion betveen the tube and the
backfill.
Figure 3 shovs the time-dependent pressure development and thermal
responses of the backfill at the hot and cold ends of the heating cell for
Test No.?. The pressures at both ends increased similarly in response to a
250 min. Aftervards, the
rise in temperature at the hot end for up to
pressure began to decrease slowly at the hot end, vhile the temperature
remained constant. This vas not the case at the cold end vhere the pressure
increased further at a constant rate until the end temperature reached a
steady state. Once steady state vas reached, the pressure increased more
slovly with time. The results suggest that the pressures developed at both
ends before 250 min are related to the local thermal expansion of the
aggregates at the hot end since the pressure at the hot end does not
increase vith further heating. At the cold end, the pressure increase
before steady state is related to the cumulative thermal expansion of all
the aggregates in the direction of lover temperature; othervise the pressure
at the cold end should be less than that at the hot end. The difference of
pressure at the tvo ends is likely balanced by the frictional forces in the
backfill and at the tube-backfill interface.
-
Once the end temperature becomes constant, the thermal expansion
pressure should not change. Bovever, the pressure vas shovn to change
slovly vith time. Such a change is associated vith heat-induced moisture
movement. It has been shovn that moisture migrates from hot to cold regions
in unsaturated soil under a temperature gradient 161. Consequently, ve
expect that the backfill vill become drier at the hot end and vetter at the
cold end vith time relative to its initial moisture condition. In the
backfill, the moisture is mainly associated vith the clay component. The
volume occupied by clay vater vill increase at the cold end and decrease at
the hot end. As a result, the pore space available in the backfill for
thermal expansion vill increase at the hot end and decrease at the cold end.
The expansion pressures at the hot and cold ends vill decrease and increase,
respectively. The moisture content profile at the end of the experiment has
been determined by destructive testing. The results do not shov a clear
trend of moisture migration because of non-representative sampling arising
from non-homogeneous material properties. Hovever, the moisture content vas
- 3% lover at the hot end and 1% higher at the cold end than the bounding
values (caused by material variation) of the initial vater content in the
backfill. Prom these results, it can be concluded that the continuing
change of pressure after steady state is reached reflects moisture migration
from the hot to the cold region,and the time required for moisture
equilibrium is much longer than that required to reach thermal equilibrium.
Although moisture transfer is still occuring in the backfill after thermal
equilibrium is achieved, the straight-line portion of the cumulative energy
curve remains practically unaffected,shoving that the effect of moisture
redistribution vithin the backfill on the heat transfer process is
insignificant.
-
The development of backfill pressure in other tests varied vith
location. In general, the developed expansion pressure vas found to respond
to temperature changes and moisture changes in the backfill, as can be
expected. The maximum presures for the tests are summarized in Table 1.
The results do not shov the higher pressure expected at higher heater
temperatures at both ends. This reflects that the pressure also depends on
the boundary constraints of the cell, the frictional forces and possibly the
contact of the aggregate vith the transducer. For example, the higher
pressure at the cold end for Test No. 2 (compared to Test No. 5) possibly
indicates that the longitudinal thermal expansion associated vith the tube
at the ends is much smaller in Test No. 2 than in Test No. 5. It is also
possible that the transfer of thermal expansion pressure from the aggregates
to the cold end is resisted by frictional forces. The zero- and 75- kPa
pressures at the middle for tests Nos. 2 and 4 may reflect that the thermal
expansion of the backfill is equal to and greater, respectively, than that
of the tube in the radial direction. The zero pressure measured in.the test
can also be due to a lack of aggregate contact vith the transducer.
The above results shov that the backfill pressure under non-isothermal
conditions depends on temperature ranges and gradients, moisture content,
aggregate distribution, material and cell friction, and cell constraints. In
general, the pressure at the cold end is much higher. This implies a tight
contact at the backfill-rock interface since a lover rock temperature is
expected. A maximum pressure of 100 kPa vas measured in these tests.
This pressure may not be representative for the upper bound limit in the
vault since the thermal expansion characteristics of the host rock
surrounding the backfill are different. The backfill pressure under vault
conditions vill need further study.
-
CONCLUSIONS
In this study, ve examined the thermal responses and pressure
development in the backfill for the period soon after vault closure using a
one-dimensional heat flov cell. The folloving main conclusions vere dravn:
1.
Thermal equilibrium is reached before moisture equilibrium.
2.
Hoisture migrates from a hot to a cold region under a temperature
gradient.
3.
The heat transfer process is not significantly affected by moisture
redistribution.
4.
Thermal expansion increases the backfill pressure. The extent to vhich
the backfill pressure increases depends on temperature, material and
rock vall friction and host rock constraints.
ACKNOVLEDGEHENT
The authors of McGill University acknovledge the financial support and
the permission for publication given by Atomic Energy of Canada Limited.
REFERENCES
V.T.Eancox, 2nd Int. Conf. on Rad. Vaste Hgt. Vpg. Man. 1-9, Canadian
Nuclear Society, (1986).
R.N.Yong, P.Boonsinsuk, G.Vong, Can. Geotec. J. 23(2),
E.S.Radhakrishna, K.C.Lau,
1766-1785 (1984).
216-228 (1986).
A.M.Cravford, J. Geotech. Eng. (ASCE) 110,
R.M. Ouigley, AECL report-7827, Atomic Energy of Canada Ltd., (1984).
D.R.M. Abry, R.C.F. Abry, K.V. Ticknor, T.T. Vandergraff,
Record TR-189, Atomic Energy of Canada Limited (1982).
Technical
J.V.Carey, Soil Sci. Soc. Amer., Proc.30,428-433 (1966).
*
Technical Records are unrestricted, unpublished reports available from
SDDO, Atomic Energy of Canada Limited Research Company, Chalk River,
Ontario, Canada KOJ 1JO.
491
YaL Re% Soc. Symp. Proc. Vol. 212. G I 9 9 1 Materials Research Socksty
THE EFFECTS OF UOISTURE CONTENT. SALINITY AND TEUPERATURE ON THE LOADBEARING CAPABILITY OF A DENSE CLAY-BASED BACKFILL
STEVEN. C.E. CEEUNG*, U.N. GRAY., R.N. YONG" AND A.H.O. UOBA~ED*.
'AECL Research. Whiteshell Laboratories, Pinava, Manitoba, Canada ROE 1U)
Geotechnical Research Centre, UcGill University. HontrCal, QuCbec,
Canada E3A 2K6
..
ABSTRACT
To predict the mechanical performance of earthen backfill in a nuclear
fuel vaste disposal vault, the effects of temperature and moisture content
on the backfill load-deformation characteristics must be knovn. Uodified
California Bearing Ratio (CBR) tests vere w e d to obtain these data. The
results show that the load-bearing capability of thebackfill is not likely
affected by vater salinity.
The load-bearing capability decreases vith
increasing temperature and vater content or decreasing dry density. The
effects of temperature, dry density, and moisture transport processes on
load-bearing capability were found to be relatively small compared to those
of moisture content. Under the expected Canadian vault environment, the
backfill shpuld retain sufficient bearing capacity to satisfy its
mechanical design function, vhich is principally to limit expansion of
svelling buffer material.
The concept of disposing of nuclear fuel vaste in a vault excavated
at a depth of 500 to 1000 m below the ground surface of plutonic rock in
the Canadian Shield is being assessed as part of the Canadian Nuclear Fuel
Vaste Uanagement Program 1
In addition to natural lov-permkability
rock, a number of engineered barriers are used to limit radionuclide
releases from the solidified vaste. Corrosion-resistant containers will be
placed in boreholes drilled into the floors of emplacement rooms and
separated from the host rock by a bentonite-sand buffer material.
The
remainder of the vault vill be filled with an earthen, clay-based backfill
(see Figure 1).
Figure 1.
Engineered barriers and components of a nuclear
fuel waste disposal vault.
A mixture of crushed granite aggregates and Lake Agassiz clay has been
proposed as a reference backfill material. A mixture of 75% of vell-graded
aggregates vith a maximum particle size of 19 mm and 25% clay can be
compacted to a dry density of 2.2 Hg/m3. At this density the backfill has
satisfactory engineering properties such as lov hydraulic conductivity,
some svelling capacity, and acceptable thermal conductivity 121.
The backfill in the vault vill be heated to a maximum temperature of
100•‹C by the radioactive decay in the vaste containers [I]. The induced
temperature gradients can cause the initial moisture in the backfill to
move. The backfill closer to the heat source can dry. Further from the
heat source the backfill vill vet due to vater movement dovn the
temperature gradient and influx of vater from the rock mass.
This
condition of a drier;hotter
backfill near the vaste containers and a
vetter, cooler material some distance from the containers is likely to be
most extreme soon after vault closure vhen the groundvater table has yet to
return to its natural condition. In the longer term,the temperatures in
the backfill vill become more uniform and the backfill vill become vater
saturated as the groundvater pressure increases. Under this vide range of
conditions the backfill is required to restrain the bentonite-based
svelling buffer material from extruding from the emplacement borehole.
This paper describes laboratory studies of the effects of dry density,
vater salinity, temperature and moisture transport processes (drying and
vetting) on the load-deformation properties of backfill. The ability of
the backfill to resist the svelling pressures of the buffer is discussed.
HATERIALS AND EXPERIHENTATION
Materials
The backfill material tested vas a reference mixture of 75 vt.% of
granite aggregate and 25 vt.% Lake Agassiz clay 121.
The aggregate
component consists of granodiorite, crushed to various screen sizes. The
granodiorite consists mainly of quartz, feldspar, plagioclase, microcline,
biotite and chlorite.
Hontmorillonite, illite, quartz, kaolinite and
feldspar make up the clay.
Three types of solution - deionized distilled vater (DW), granitic
vater (GW) and standard Canadian Shield saline vater (SW) - vere used to
study the effects of porevater salinity on the load-deformation
characteristics of the backfill. The compositions of GW and SW solution
are given elsevhere [3].
GW solution vhich is slightly saline (ionic
strength of -0.3 rnol/L), vas used for studies of the effects of temperature
and moisture transport processes on the load-deformation characteristics.
The appropriate proportions of dry aggregate and povdered clay vere
mixed thoroughly vith various amounts of solution corresponding to vater
contents ranging from 7 to 10%. The moisture in the mixture vas alloved to
equilibrate in a sealed container for one day before the backfill vas
compacted in a cylindrical mold in accordance vith procedures specified for
California Bearing Ratio (CBR) tests in ASTH-D-1883 141.
Experimentation
Three test series vere performed to determine the effects of vater
salinity, temperature and vater transport processes on the load-deformation
properties of the compacted backfill.
Test Series No. 1
These tests vere designed to determine the effects of dry density and
vater salinity on the load-deformation properties of the backfill.
The backfill material was mixed with the three different types of
solution prior to compaction. The compacted dry density varied from 2 to
2.3 ng/m3 due to differences in the mixing moisture content and vater
salinity.
The compacted backfill vas alloved to soak for 4 days by
immersing the assembled mold in water. Figure 2(a) shovs the arrangement
under vhich the compacted specimens were soaked. Two perforated end plates
vere used as end caps on the compacted specimen to confine the material yet
allov access of water to the backfill. The bottom end plate vas fixed
vhile the top end plate vas alloved to move under a weight of 2.2 kg during
the uptake of vater. It should be noted that thermocouples are not used in
these tests. Specimens vere soaked with the same water used for mixing.
After soaking;the
mold vas taken out of the water and measurements
vere taken to determine the dry density.
The top end plate vas then
removed and the specimen was tested in accordance with the CBR loadpenetration test procedures specified in ASTn n-lRg?.
Figure 2a.
Equipment set up for
California Bearing Ratio
(CBR) test vi th no
vater uptake.
Test Series No. 2
These tests vere designed to study the effects of moisture transport
processes on the load-deformation properties of backfill. Tvo sets of
tests vere carried out.
The first set of tests vas designed to determine the effects of drying
on the load-deformation properties of the backfill. The mixed backfill
material containing only GW vater vas compacted into the CBR mold as in
test series No. 1. Folloving compaction, the mold vithout the top end
plate vas placed inside an oven, and was heated to various temperatures up
to 100•‹C. In some tests, vater vas allowed to evaporate from the top of
the backfill as i t vas heated.
These specimens referred to here have
unsealed conditions. For the other tests, water was not alloved to escape
during heating and the specimens are referred to as having sealed
conditions. To ensure that the backfill reached the oven temperature,
thermocouples vere installed in the backfill during compaction (see Figure
2a) and the temperature profiles vere recorded as a function of time. The
backfill vas found to reach the oven temperature in less than one day. A
heating period of tvo days vas used. After removal from the oven, the
specimen vas alloved to cool to room temperature and the load-penetration
test performed as in test series No. 1.
The second set of tests vas designed to determine the effects of
revetting on the load-deformation properties of the backfill.
After
heating for tvo days in the oven, the backfill vas cooled and then exposed
to vater uptake in the oven. During the second heating cycle, the backfill
vas sealed at the top vith a non-perforated end plate and vas alloved to
take up vater from its base through a perforated end plate. The rate of
vater uptake from a vater source outside the oven could be monitored in a
burette (Figure 2b).
Once there was no further change in the reading of
the burrette, the mold vas disconnected from the water source and taken out
of oven to cool to room temperature before conducting a load-penetration
test.
1I/
Figure 2b.
Equipment set up for
California Bearing
Ratio (CBR) test
vith vater uptake.
Test Series No. 3
This series of tests was designed to study the effects of temperature
on the load-deformation properties of the backfill.
The experimental procedures vere similar to those in Test Series No.
2, except that the backEill in the oven vas sealed vith non-perforated end
plates both at the top and the bottom for the drying phase, and the loadpenetration test was performed vith the backfill temperature maintaine4 at
the oven temperature. Immediately after removal from the oven, the mold
vas insulated and placed on a firebrick heated to the oven temperature.
The load-penetration test vas effected soon after the assembly was
completed. The temperature of the backfill during the load-penetration
test was found to drop less than Z•‹C. As in series No. 2, some specimens
vere heated vith access to water through a perforated base plate.
RESULTS AND DISCUSSION
In total, 36 tests were conducted in Test Series No. 1 to 3. The
moisture content profiles and dry density of the backfill vere determined
at the end of the tests in all three test series. The backfill column was
sectioned into three parts to allow the moisture content to be determined
by oven drying at 105OC.
In test series No. 1, all specimens vere
virtually saturated vith vater during the load-penetration test. In test
series No. 2, the moisture content of the unsealed specimen after heating
was found to decrease from the top to the bottom. In test series No. 2 and
3, redistribution of water in the sealed specimen after heating vas
generally observed. The moisture content was generally found to be higher
in the middle of the specimen. For the revetted specimens, a uniform
moisture content vas found. The trends of the results are summarized as
follovs.
1.
Load-Deformation Characteristics
Figure 3 shovs typical load-penetration curves for the backfill. The
data vere obtained during the heating phase at three different oven
temperatures in Test Series No. 3. The dry densities and moisture content
profiles are similar for all specimens.
The resistance to penetration
i . . the force required to penetrate a given distance into the backfill)
increased linearly vith penetration up to 3 to 4 m m and then began to level
off. In all cases it reached a maximum value, the load-bearing capacity,at
In the load-penetration tests,
penetrations greater than 10 mm.
penetrations of up to 20 mm vere used. To study the relative performance
of the backfill under the different test conditions, a resistance force
termed as load-beafing capability vas used. This is defined as the load
required for a penetration of 5 mm in the backfill. Analysis of a number
of tests shoved the effects of test variables on load-bearing capacity to
be similar in proportion to those on load-bearing capability. Load-bearing
capability vas alvays significantly less than load-bearing capacity.
Figure 3.
Resistance to
penetration in the
backfill as a
function of
temperature from the
specimens vith
heating only of Test
Series No. 3. r,
and V/C are the dry
density and vater
content, respectively.
2.
Effects of Temperature
-
The results in Finure 3 shov that the load-bearing capability
decreases vith increasing temperature.
It is noted that the moisture
distribution in these specimens is fairly uniform.
The same trend is
observed in revetted specimens of series No. 3. A number of mechanisms may
be involved. The load-bearing capability reflects the shearing resistance
of the backfill. Temperature can affect both the cohesion and the friction
components of the strength envelope of the backfill.
The increase of
temperature vill decrease surface tension (increase pore vater pressure).
This results in decreasing the effective confining stresses and thereby
decreases strength.
3
Effects of Dry density and Water Salinity
Figure 4 shovs the relationship betveen the load-bearing capability
expressed by the California Load Bearing Ratio and the backfill dry density
determined from the vater-saturated specimens.
The results from specimens
mixed and soaked in the three different solutions are shovn. The loadbearing capability increases vith increasing dry density. The effects of
vater salinity appear to be negligible.
Figure 4.
Relationship between the
California Load-Bearing
Ratio, CBR, and dry
density, y, of the
backfill vith three
different types of
solutions from Test
Series No. 1. (Symbols
are defined in the text.)
4. Effects of Vater Redistribution
Figure 5 shows the relationship between the load-bearing capability
and the averaged moisture content of the backfill specimen from Test Series
No. 1 to 3.
In these tests, the factors affecting the load-bearing
capability are the water content, the dry density, temperature and vatertransport processes such as drying and revetting. The results tend to show
that the load-bearing capability primarily depends on the moisture content
and increases as the water content decreases.
Although the data-are
limited in the lower moisture range, the trend of greater shear strength or
70
GU
-
.-.
* nmrg-u.-a
.
rn.ln.
."d
YUmp
(-..-I
a-fi
DM..,(..MI
0-0
('."
r m m l--.I4
r-r
50
-
,- 'I0 -
rn I,mq
(l,
I
,
..".l!!rq
Figure 5.
-
Relationship between
load-bearing
capability and
averaged water
content, w/c of the
backfill from Test
Series No. 2 and
No. 3.
load-bearing capability vith lover moisture content is commonly observed.
The effect appears most significant in systems vith moisture conten*y belov
8%.
Analyses of the data from test series No.2 shoved that the loaddeformation characteristics of the backfill also depend on the distribution
of moisture in the backfill. The moisture content distribution of both the
sealed and unsealed specimens can be affected during heating. This factor
largely accounts for the variation in the load-bearing capability at a
moisture content of approximately 7 X , as shovn in Figure 5.
SYNTHESIS OF RESULTS
The backfill in the proposed Canadian disposal vault is likely to be
subjected to drying and revetting processes. The dry density, yd, of the
backfill may vary from 2 to 2.2 Ug/m3. This density range allovs for
uncertainties about the practical limits of the mixing process and volume
change in the backfill. The moisture content of the backfill at saturation
vill range from 12% at yd=2 Ug/m3 to 8% at yd=2.2 Ug/m3. Figure 5 shovs
that the load-bearing capability above a moisture content of 8% ranges from
2 to 7 HPa.
At a vater content less than 8X, the load-bearing capability
is higher than at saturation. Consequently, only the mechanical performance of the backfill at saturation needs to be considered further.
In the room in vhich the vaste is to be emplaced, the backfill is
required to support its ovn veight and to resist the svelling pressure from
the buffer. The self weight of the backfill will yield approximately
0.1 UPa. This is small compared to the swelling pressure from the buffer
vhich vill increase as the buffer takes up vater to reach a maximum value
of 2 UPa at saturation [ 5 , 6 ] . The backfill in the vault is expected to be
saturated much earlier than the buffer 151. This is due to the combined
effects of temperature gradients and the higher permeability of the
By neglecting the frictional resistance of the buffer
backfill [ 2 , 6 ] .
against the rock vall of the emplacement borehole, the total upthrust of
2 UPa from the buffer can be assumed to act against the backfill.
This
pressure is vithin the range of the load-bearing capability ( 2 to 7 UPa) of
the backfill measured at a penetration depth of 5 mm. At this pene'tration,
the load-bearing capability is much less than the minimum load-bearing
capacity of the backfill and punching shear of the buffer into the backfill
is unlikely.
To detail the time-dependent deformation of the backfill, the heatinduced moisture movement and the load-deformation characteristics need to
be known as functions of the moisture content in the backfill, temperature,
geometry of the emplacement room and water supply conditions at the rock
backfill interface boundaries. Due to the site-specific nature of these
variables and the effects of scale on the backfill material performance,
full-scale experiments are needed to conclusively confirm the likely extent
to which the buffer vill penetrate the backfill. In this respect, AECL is
conducting experiments in its Underground Research Laboratory to confirm
that the backfill vill effectively limit extrusion of the buffer.
CONCLUSIONS
Based on the above studies, the folloving conclusions can be dravn for
Canadian reference backfill material:
The load-bearing capability will likely not be significantly affected
by vater salinity.
The load-deformation characteristics depend largely on water content
and its distribution.
(3) The effects of temperature, dry density and vater-transport processes
on load-bearing capabilities are small compared to the effects of
moisture content belov 8%.
(4) The backfill should be able to resist the punching shear of the buffer
vithout mechanical failure.
ACKNOWLEDGEMENT
This paper is jointly funded by AECL Research and Ontario Bydro under the
auspices of the CANDU Ovners Group.
REFERENCES
V.T. Bancox, Proceedings of the 2nd International Conference on
Radioactive Waste Management, Winnipeg, Manitoba, pp. 1-9, Canadian
Nuclear Society, Toronto, 1986.
R.N. Yong, P. Boonsinsuk and G. Vong, Can. Geotech. J., 23(2),
1986.
216-228
D.R.M. Abry, R.C.F. Abry, K.V. Ticknor and T.T. Vandergraaf, Procedure
to Determine Sorption Coefficients of Radionuclide on Rock Coupons
Under Static Conditions, Atomic Energy of Canada Limited Technical
Record TR-189, 1982. Unpublished report available from S D W , AECL
Research, Chalk River Laboratories, Chalk River, Ontario KOJ 1JO.
American Society for Testing and Materials, CBR (California Bearing
Ratio) of Laboratory-Compacted Soils, D 1883-87, ASTU, Philadelphia,
PA, 1987.
R. Pusch, L. BBrgesson and S. Knutsson, Buffer Mass Test- Improved
Models for Water Uptake and Redistribution in the Heater Holes and
Tunnel Backfill, Stripa Project Report, TR 83-05, Svensk
KarnbrZnslef6rsorjning AB, Stockholm, Sveden, 1983.
S.C.H. Cheung, M.N. Gray and D.A. Dixon, Coupled Processes Associated
vith Nuclear Waste Repositories 30, pp. 393-407, Academic Press, Inc.,
1987.
w t k r . Sot. S y m p Proc. Vol.
212.
: $991 H a i r r i a l s
Research -lev
CVALUATIOS OF THERMAL CONDUCTIVITY
OF BACKFILL MATERIAL
D . N XU,A . h I . 0 . !vlOHALf.IED, .43D R.Pi YOSG
G e o t e c k c d Rrrsrilrt3 Ccnrrc, 4IcGill University, 517 Shrrbrrnkr St., Lt... Liontreal, Quelwc, C ~ u e d nH3A ? K G
S.C.H. CHEUKG
AECL h c h ~ . ' h i t d i e ULaborarories, Pinawa, Manitoba , C m a d a ROE 1U1
ABSTRACT
A serirs o i tcsrs haw been performed at the Gmtechnical Research Centre on
bzc!&l material u d e i imposed tcmpcraturc gradients. It has heen dcmonstrated
that the water content distribution during thc transient p r o c a s d m not affect thc
trmperzturr distribution. Thc T r i d Function Tcchnjque (TFT)has bcen applied
to calculate thc thermal conductivity of the &
b!ac.l
materid. Using these values
a:d the proposed method of solu:ion, the t e u ~ p e r a t u r ep r o 6 h have becn calculated.
The co~npariscrnbetwwr: the :hcoreticaUy prcdicrcd a3d t h c c x p e r i m e n t d y measured temperature profiles shows good agreement cxccpt in t h c vicinity of t h c heating
bomdarr., =here a t l i n :twr:~:d boundary layer e x i s t s . The vaporization process in
this +in layer contributes to the turbulent effect thn: van hr niodrled by using a
thermal shock condition.
INTHODUCTLON
Safe and p c m m c n t disposal of radioactive n-u:t: rcquircs isolation of a number
of diveme r h c m j d clcmcn:s from :hc crwiro:~ntm: for a :ong time. The C u a d j u
approach in:n zuclcnr Iucl waste railp
r wn t.i s ~ L s p o s din a vault rined: deep into
p1u:onir rock of thc C a r l a d j a Shhild .1:. Llcsidcs :hc natural lo\v-permcahility rock, a
r.zmber of en+npered bnrricrr, arr uscd to limit :ac!ionuc:ide relrase from :he solidified
waste. These a-e the corroc;ion resistmt t:ont~rier.:he buffer md the bi~cMll.T h e
ron:ainc~-5 4: be plawd in borrholes drilled into the floors of e q h c c z e n ! r m z 5
and sep&-awd from ttir hos: rock by ber.tonitc-sand buffc: n n t e r i d . The rcnainder
of the mul: wil: bt. lillcd with au eiuthcn. clay-b&ed bnckW.
I: is knon-I:t h a t :he tberrnd conductivity of a soil is n f m c t i o n of its moisture
rontm:. With t h e existenrr of trmpera:urc gradien:~in n soil mcdium. :hc moisture
in rtir ::dim wil: r&nr:bl;te itself to new equilibrium values to conform wit11 the
applied :emprraturc. g a d j e n t . Subwqucntly, thc ocrv mois:urc content tiistributio:~
will affcc: t h e t e x p x a t u r e distribution in :he soil nedwm. This proccss i i l u t r a t m
:he coupling e&t between the :enlperature and moisture distribution in a porous
nwdiuni.
The &ec: of ~ n o i s t u mr d i ~ : r i b u t i o r ~01: t h e imthrrrxal and t h c r m d d i h s i r i t i c s
ir: buffer material d u e to t e r u p r r a t x e ~ a d i w i t tias
s k :i:lvestigatetl.
~
.in ide11tifu:ntioa technique a= developed to calcu1a:e the diffusion parameters in a ful: couplj~ig
process [1.3.41.However, i n iul unsaturaed backfill material.. sit~r:rir.ois:urc djffwivit? is very low (in the order of 10-8rn7/s to 10-:'rn'/s) and moisturc movement takes
a much longer time to reach nil cyuiiibrium stale $1 in c o r n p i s o n to tcmpcraturc
equilibrium. one car. so:ve tt:e problc: bu i g n o r z g ~~~~~~~~u-ntrr :IIO\-~:I:~:I: cffrc:~.
This is called rhr serr.i-coupkg process.
Fro111a consideration of the : y of ~p r o b l e m a!orementioned. ir is apparcnr that
i:i
there is a need for a sir.pliEd tectniyce for calcx!ating the thermal co~itl~cti\-it:transiem process. Ir: :his srt;dy, a Trial F c r ~ c t l o ~Techniqw
:
(TFTj is dcvc:oped to
obtain :1 ) m m d y t i c a l solx:iorl of t L e terr.peratcre disrribxion. and i 2 ; a calru1a:ien
of t h e : h e m a l co~idxctivity.
Svstern Description
An insulated one-dimensional soil column with length L, fixed at both ends A
and B, is shown in Fig. 1. Initially the temperature To and volumetric moisture
content 0, of the soil column are uniform. At time t = 0, end A is heated by an
electric heater, and its te~nperatureis raised with time; thus, heat transfers along the
soil column from A to B. The temperature at end A. T A , is gradually increased
from initial temperature To to the find design temperature, T,. The rate of increase in
TA depends on the test conditions. The interface between end A and soil is hereafter
referred to as heating boundary. The temperature at end B is allowed to seek its own
level in contact with its immediate environment, i.e., room temperature. Thus before
the heat front reaches end B, the temperature TB is maintained at TB = To. When
the heat front reaches end B, TB gradually increases with time t. The interface
between end B and soil is hereafter referred to as unheated boundary.
Figure 1
S o i l Column S y s t e m
Governing E q u a t i o n s
The formula used to describe the temperature distribution within an unsaturated
soil column is given by the following equation in one-dimensional flow:
where: X = thermal conductivity, C, = specific heat, T = temperature, and s =
distance. The boundary and initial conditions for the case considered in this study
are as follows:
T ( z , O ) = T , for O < z < L
(4)
where: a is a positive number that can be determined experimentally from the
difference between the final temperatures at both ends of the test sample.
Furthennore. Eq. (1) can be presented in a nondimensional form as follows:
where:
T' = ( T - To)/(Tl - To)
A. = X/C"
(6~)
(6d)
The boundary and initial conditions in a nondimensional form are as follows:
and
T*(E,0) = 0
(9)
It should be noted that a' can be obtained from a. .Analysis of test data shows
that a' falls within the range of 0.1 to 0.15.
TRIAL FUXCTION TECHNIQUE (TFT)
Analysis of heat transfer along the soil colunm requires that one distinguishes
between movement of the heat front in the following cases: (1) before it reaches the
unheated boundary (stage l ) , and ( 2 ) when it has reached the unheated boundary
(stage 2). If T, is used to denote the time taken for the heat front to arrive at the
boundary, then the first stage occurs at time T 5 TO whereas second stage occurs at
T > T,. This is shown in Fig. 2.
In the first stage, the temperature TB at boundary B is the same as its initial
value. since the heat front has not arrived at this boundary. With time 7 . the heat
front S ( T ) progresses further into the soil column. In the meantime, the temperature
T.4 a t the heated side -4 also gradually increases because of imposed conditions. as
sho~vnin Fig. 2a. In the second stage, the whole soil column has been thermally
disturbed and the temperature profile continues to develop with time 7 . The temperature TB at B boundary gradually increases until the temperature profile reaches
the final equilibrium stace.
F i r s t Stage-Before H e a t Front Reaches t h e U n h e a t e d B o u n d a r y
The trial function is assumed as follows.
where S(T)is a generalized coordinate in the trial function and
.4 check of the trial function shows that
1. At E = O.Ta(O.T ) = T;(T), i.e. the boundary condition at end -4 is satisfied.
2. .At
= S(T).T.[S(T),T]= 0. This shows that before the heat front reaches
distance S(T).i.e.. the temperature at S( T ) is still unchanged.
3. \Vhen ,E 2 S ( T )thermal
.
disturbance has not occurred.
<
(a) First Stage: Before Heat
Front Reaches the Unheated
Boundary
Figure 2
(b) Second Stage: After Heat
Front Reaches the Unheated
Boundary
Stages of Heat Transfer
%
4. The assumed function for T ' ( ( : 7 ) and its deri~ative
are continuous for two
rexions (( < S ( T )and ( 2 S ( r ) ) ,thus the spatial gradient of temperature is
continuous.
5. When T = O,S(T) = 0 and mA(r) = 0 , thus T ' ( < ,T ) = 0 for the whole soil
column. indicating that the initial condition is automatically satisfied.
-
can
) be simulated
In order to proceed with the solution, the temperature ~ . A ( T
by a function. From test data, an eqonential function appears to be appropriate to
and T :
specify the relation between &.A(T)
where cl is an exponential parameter, which is used to characterize how fast
mA(r)increases. Using Eqs. ( I ? ) , (10) and (5), the final analytical solution for S ( T )
takes the following form 161:
-
-
It is easy to verify that when T
0. S ( T ) 0. Once the c l value is known,
S ( T )is uniquely determined by r. When S ( T )= 1. the corresponding TO is the time
taken for the heat front to reach another boundary.
Second Stage-After H e a t Front Reaches t h e U n h e a t e d B o u n d a r v
In a similar way, one can assume the following trial function to specify the
temperature profiles at the second stage as:
where mB(r),which is to be determined, specifies the theoretical temperature at
end B. This trial function automatically satisfies the boundary and initial conditions,
and the continuous spatial temperature gradients.
The final analytical solution of ~ B ! T is) given by 16):
Q B ( 7 ) = (1 - a')
+ rle-clr' - [ q- ( 1 - a*)].-37'
(15)
where:
T is the time from the beginning of the test. It should be noted that T' besins
from the time the heat front reaches the unheated boundary. Once @ B ( T )is obtained,
the temperature profile'can be determined by using Eq. ( 1 4 ) .
COMPARISON BETWEEN THEORY AND TEST DATA
For comparing the theoretical predictions with actual measured values, laboratory tests have been conducted to study heat and moisture movement in the backfill
for almost six months. The detailed test arrangement and procedure can be found
in Yong, et al. [ 5 ] . Using the length of the soil specimen of 316.20 mm, the heat
input a t end A raised the temperature from 22•‹C to 90•‹C. The temperatures were
recorded by thermocouples at different times at different positions z along the soil
colunln. T h e experimental data are shown in Table 1.
TmLE 1
Measured Temperature a s a Function of Time Along t h e S o i l Column
The one-dimensional temperature can be easily calculated through the use of
Eq. ( 6 c ) . From the above data. some significant features are noted: ( 1 ) a t about
55 min boundar! B begins to change its temperature noticeably. and (2) in the
vicinity of the heating boundary -4,the temperature changes sharply. This abrupt
change indicates a thermal boundary layer. a thin cransition layer exists adjacent to
the heating boundary. with heat transport porperties much lower than those of soil
column. For calculation purposes. one can take t,+5 min. and neglect the thermal
boundary layer effect. One needs to use an estens~ontechnique to revise the data
in order to obtain a reasonable temperature increase at the heating boundary. By
using the analytical model, represented by Eqs. ( 1 2 ) and ( 1 5 ) . and experimentally
determined results which are listed in Table 1, the following parameters are obtained:
Through Eq. (61)). A' is given by:
The theoretical temperature profiles calculated with the procedures developed
are shown in Fig. 3 together with the test data obtained from the experiments
conducted. The results show good agreement betn-een theory and test data except
in the vicinity of the heating boundary, where a thin thermal boundary layer occurs.
The theoretical predictions and test results show that the transfer at the first stage
occurs in a very short time period, (one hour or so), in comparison with the second
stage (one day or so). After about a day. the temperature profiles tend to stay in
equ~librium.
-9
-
.8
-
.7
..
1.0
-6
Figure
Predicted
6
.\
t = Minutes
II
A
B
C
-
3
.
Experimental
6
12
20
C o m p a r i s o n Between P r e d i c t e d a n d E x p e r i m e n t a l
Normalized Temperature P r o f i l e s f o r C = 1 and a = 0 . 1
CONCLUDING REMARKS
The TFT permits one to determme the thermal diffusivity of unsaturated backfill
materm1 in transient state. which for the tests conducted show the unsaturated soil
test column to be A' = 7.15 mmL/s. The calculated thermal diffusivity a t steadystate condition of the same test based on the ciosed form solution technique reported
equals 4.996 mm2/s. Furthermore, the thermal conductivity of a mixture of
i a o crushed granite and 25%) Avonlea bentonite compacted to a dry density of 1.8
hIg/m3 at 14% moisture was calculated by Radhakrishna [S] to be 1.S IY/m.OC. Using
. calculated thermal diffusivity
a specific heat equals to S.26 x 10W5 C a l / ~ n m ' . ~ Cthe
is 5.2 mm2/s.
IJJJ~]
This 5:ut!y is Cnar:cia:l!: ~ u p p n r t cby~ thr C'madiari S u c h : Furl bi.'a\tr > I a : i a g r wen: Proqrzrn which is jointly f ~ i d c i A t o r ~ i i cE n r r o
~f Canada L i x i t e d (..1ECL I
and Ontario Hydro :~:lde-the a~:spiccsof thrr CASDr owners g r o ' ~ ; ) .