DTIC ADA278154: Growth Condition of an Ice Layer in Frozen Soils Under Applied Loads. 2. Analysis.

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by  Defense Technical Information Center

AD-A278 154
DTIC
ELECTE
AR 1319M D
4
Growth Conditioo-fb anIce Layer
in Frozen Soils Under Applied Loads
2: Analysis
Yoshisuke Nakano and Kazuo Takeda January 1994
94 4 13 002
The results of an experimental study on the steady growth condition aoa
segregated ice layer under various applied pressures were presented In Part
1. Using the data obtained, we evoluate the accuracy of the model MI, and
the predicted steady growth condition is found to be in good agreement with
the condition found empirically. The concept of segregation potential Intro-
duced by Konrad and Morgenstern Int he early 1980s Is examined based on
M .M is found to be consistent with the empirical data that were used to
1 1
support their segregation potential theory.
Cover, 4paroAis for test~ng Ice growth Ins oils under app~ledloads. (PhYoki by
K. Takeda.)
For convision of Simetrcunitsto U.SIBrtsh custmary units omeasurment
consult Sbntin dP~ricte forLUse of Mhe inenatmina Systm of Units (SI), ASTM
Standard E380-89a, published by t*e Amnerican Society for Testing and Mater-
Ials, 1916 Race S., Philadelp'ila, Pa. 19103.
CRREL Report 94-1
US Army Corps
of Engineers
Cold Regions Research &
Engineering Laboratory
Growth Condition of an Ice Layer
in Frozen Soils Under Applied Loads
2: Analysis
Yoshisuke Nakano and Kazuo Takeda January 1994
NTIS CRA&I
DTIC TAB
Unannounced 0
Ju6tification, .................
By
Dist ebutionj
Availability Codes
Avail andjor
Dist Special
Prepmed for V17C QUALT WSPECM 3
OFFICE OF THE CHIEF OF ENGINEERS
Aoproved for pubic rekeae; disfrlbuffon Isu rnimted.
PREFACE
This report was prepared by Dr. Yoshisuke Nakano, Chemical Engineer, of the Applied
Research Branch, Experimental Engineering Division, U. S. Army Cold Regions Research
and Engineering Laboratory, and by Dr. Kazuo Takeda of the Technical Research Institute,
Konoike Construction Co., Ltd., Konohana, Osaka, Japan. Funding for Dr. Nakano's re-
search was provided by DA Project 4A161102AT24, Research in Snow, Ice and Frozen Ground,
Task SC, Work Unit F01, PhysicalP rocessesi n FrozenS oil. Dr. Takeda's experimental work was
funded by Konoike Construction Company.
The authors thank Dr. Virgil Lunardini and Dr. Y.C. Yen of CRREL for their technical
review of this report.
The contents of this report are not to be used for advertising or promotional purposes.
Citation of brand names does not constitute an official endorsement or approval of the use
of such commercial products.
ii
CONTENTS Page
Preface ......................................................................................................................................... ii
Nomenclature ............................................................................................................................. iv
Introduction ................................................................................................................................ 1
Properties of M, ......................................................................................................................... 4
M odel M and segregation potential ...................................................................................... 8
1
Results of data analysis ............................................................................................................. 11
1. Steady growth condition ................................................................................................. 11
2. Dependence of y* on T .................................................................................................. 13
3. Dependence of Tj' on ................................................................................................ 15
Discussion and conclusions ................................................................................................... 17
Literature cited ........................................................................................................................... 18
Abstract ....................................................................................................................................... 19
ILLUSTRATIONS
Figure
1. A steadily growing ice layer in a freezing soil .............................................................. 1
2. An essential frozen fringe R ................................................... 2
1 ......................................
3. Temperature gradients a, and a .............................................. 3
0 ................................ . ..
4. Trajectories a(t) that approximately describe the condition of freezing at the
formation of the final ice lens .................................................................................... 9
5. Values of SP vs. the values of the average temperature gradient ........................... 10
0
6. Values of y* vs. %0u nder various applied pressures a ............................................ 12
7. Average values y* vs. a ................................................................................................. 12
8. Values of y*vs. the temperature -Ti under various applied pressures a ............ 14
9. Values of -1 vs. the mass flux of water f(cid:127)0 under various applied pressures ( .... 15
10. Values of-Ti* vs. the flux fto under various applied pressures a ............................ 17
TABLES
Table
1. Calculated values Y* and the average measured values Y* under various applied
pressures a ..................................................................................................................... 12
2. Sum mary of data analysis with a = 48.7 kPa .............................................................. 13
moii
NOMENCLATURE
a function defined by eq lla
ae function defined by eq 27f
ai constant where i = 0,1, ... 3
ai function defined by eq 25d and 25e, where i =0, 1
A constant
b function defined by eq 1lb
h, constant where i = 1,2.
Bi ith constituent of the mixture. Subscripts i = 1, 2, and 3 are used to denote unfrozen water, ice
and soil minerals, respectively
c, heat capacity of the ith constituent
d unit of time (day)
di density of the ith constituent
e void ratio
f, mass flux of the ith constituent relative to that of soil minerals where i 1, 2
flo mass flux of water in the unfrozen part of the soil
I function defined by eq 16b
k thermal conductivity of a frozen fringe defined by eq 9a
k thermal conductivity of the unfrozen part of the soil
0
k, thermal conductivity of an ice layer
kf) limiting value of kd efined by eq 9c
/C hydraulic conductivity in the unfrozen part of the soil
/Ki empirical function defined by eq 4a where i = 1, 2
K1i limiting value of Ki as x approaches n, while x is in RI, i = 1,2
Ko limiting value of Ki as x approaches no while x is in RI, i = 1,2
1
L latent heat of fusion of water, 334 J g-4
m location of the free end of the column
M4 name of a model where i = 1, 2,3
n boundary in Ro
t% boundary with i = 0, 1 where no denotes the boundary where T = 0°C and n1 the interface
between an ice layer and a frozen fringe
n boundary between RIO and R ,
10 1
Po gravity term, 0.098 [kPa/cm]
Pi pressure of the ith constituent where i = 1, 2
P value of P at no
10 1
P1. value of P at n
1
P value of P at n,
21 2
iv
r rate of heave
R unfrozen part of the soil
0
R, frozen fringe
RI part of the soil bounded by no and n10
0
RI, essential frozen fringe
R ice layer
2
Rm region in the diagram of temperature gradients where an ice layer melts
R, region in the diagram of temperature gradients where the steady growth of an ice layer occurs
Ra boundary between R, and R.
Rs* boundary between Rm and R,
Ru region in the diagram of temperature gradients where the steady growth of an ice layer does
not occur
S property of a given soil
SP segregation potential defined by eq 2a
0
t time
T temperature
T temperature at n,
1
T defined by eq 41
10
TI, calculated value of T from the measured temperature profile in R
1 2
T* empirically determined value of Tj
1
T1* temperature at n, when eq i holds true
To constant
(T')a average temperature gradient in R
1
AiT defined by eq 3c
x spatial coordinate
y variable defined by eq 17c
Y variable defined by eq 38b
y. variable defined by eq 3a
a(t) trajectory [a (t), %(t)] in the diagram of temperature gradients
1
or absolute value of the temperature gradient at no
0
at, absolute value of the limiting temperature gradient as x approaches n, while x is in R, defined
2
by eq6
af absolute value of the temperature gradient near n, in R
2
ot. absolute value of the temperature gradient near no in Ro
P defined by eq 11c
Y constant, 1.12 MPa OC-1
y, defined by eq 24b
8 thickness of a frozen fringe
8* defined by eq 27b
Se thickness of an essential frozen fringe defined by eq 27a
80 defined by eq 13c
v
n defined by eq 9b
p composition of the soil
p, bulk density of the ith constituent
a effective pressure defined by eq I and 16c
% empirical function of T defined by eq 14a
%0i valueof(cid:127) at T= T,
*(cid:127) empirical function of T defined by eq 14b
*,1 value of * at T = T
1 1
p variable defined by eq 21b
4p, variable defined by eq 25f
o) dimensionless quantity defined by eq.18a throiugh 18d where i =0, L..., 3
i subscript denotes the ith constituent of the mixture consisting of unfrozen water (i f 1), ice (i
= 2) and soil minerals (i = 3)
* superscript used to indicate the value of any variable evaluated when a point (ap,o r) in the
diagram of temperature gradients is on Rs
superscript used to indicate the value of any variable evaluated when a point (at, %) in the
diagram of temperature gradients is on Rs *
vi
Growth Condition of an Ice Layer in
Freezing Soils Under Applied Loads
2. Analysis
YOSHISUKE NAKANO AND KAZUO TAKEDA
INTRODUCTION
In this report we will consider the one-directional steady growth of an ice layer. Let the freezing
process advance from the top down and the coordinate x be positive upwards, with its origin fixed
at some point in the unfrozen part of the soil. A freezing soil in this problem may be considered to
consist of three parts: the unfrozen partRe, the frozen fringeR and the ice layer R, as shown in Figure
1 2
1.T he physical properties of parts Re and R are well understood but our knowledge on the physical
2
properties and the dynamic behavior of part R, does not appear sufficient for engineering applica-
tions.
It has been shown empirically (Radd and Oertle 1973, Takashi et al. 1981) that there is a unique
temperature Tj* at n for a given pressure of ice P at n and a given pressure of water P in Re when
1 21 1 1
an existing ice layer neither grows nor melts and the mass flux of waterf in R vanishes. This temper-
1 1
ature Tj** at the phase equilibrium of water is given as
a = P - Pi = -YTj'*, if f,=O (1)
21
where yis a constant with the value of 1.12 MPa °C-1, and
a and P21 are often referred to as the effective pressure m
and the overburden pressure, respectively. Equation 1 is
often called the generalized Clausius-Clapeyron equa-
tion, whichis attributed to Edlefsen and Anderson (1943). 2
Konrad and Morgenstem (1980, 1981, 1982) empiri-
cally found that the rate of water intakeflo at the forma- n T=T1
tion of the final ice lens is proportional to the average
temperature gradient (T'). in the frozen fringe. This may
be written as R,
x
flo -SPO(V). (2a) n T=TTo=0-
where a prime denotes differentiation with respect to x.
The positive proportionality factor SP0 is termed the seg- 0 Ro
regation potential, which is a property of a given soil.
Konrad and Morgenstem also found empirically that n
SP is a decreasing function of both the applied pressure
0
Fa nd the suction of water (-P ) at no. We will write this Figure 1. Schematic of a steadily growing
10
dependence as ice layer in af reezing soil.
erally depend on the temperature Tand the composition of the soil. We will describe such functional
dependence of /(cid:127) (i = 1,2) as
KI = KI(T, p) (4d)
K = K(T, p) (4e)
2 2
where p symbolically denotes the composition of the soil that is uniquely determined by the bulk
densities of unfrozen water Pl, ice P2 and soil minerals P.
3
Nakano (1990) obtained the exact mathematical solution to the problem of a steadily growing ice
layer based on the model MI. Analyzing the behavior of this solution, we showed that M, is
consistent with eq 1 (Nakano 1990) and eq 3b (Nakano and Takeda 1991). We also showed (Nakano
and Takeda 1991) that M, can accurately describe the steady growth condition of an ice layer under
negligible applied pressures.
The steady growth condition of an ice layer with
or without applied pressure is the region &~bound-
ed by a curve R* and a straight line R** in the dia-
gram of temperature gradients as shown in Figure R-
3. The region & is defined as
(k1/hk)cIt > a3> k, (k0.+ LbK2z))-! a, (5) EO
where k, and ko are the thermal conductivities of R
2
and Ro, respectively, oh is the absolute value of the
temperature gradient at no and a, is the limiting
value of the temperature gradient at n, in R de-
2
fined as
al =-limr(x). (6) al
x-+n1
x in R2 Figure3 . Temperatureg radientsa , and or
In eq 5, L is the latent heat of fusion of water, b is a
function of the thickness 6 of R defined by eq 73c and 73e in Nakano (1990), and K is the limiting
1 21
value of K asxapproaches n, when x is in R and an asterisk denotes that K* is the value of K when
2 1 21
a point (a ,oa) is on R* in the diagram of temperature gradients.
1
In Figure 3 we will refer to the region as R. where %X> (k/ko)CXj'o <0 and an ice layer is melting.
1
The boundary R * is given as
ao = (ku/ko)a I on R**. (7a)
An existing ice layer neither grows no meltsf 0 vanishes and eq I holds true on Rt *. The boundary
R* is given as
ao = ki(ko + LbKl*,)4,a on R*. (7b)
It is easy to see from eq 5 that the steady growth condition of a given soil is uniquely determined by
a, and %. Nakano (1990) showed that all physical variables such asf , TI, 6, etc., are also uniquely
10
determined by and %for given hydraulic conditions, and applied pressureo . The hydraulic con-
1 0
dition in our tests is specified by the distance 80 between no and n where the pressure Pu of water
is kept at the atmospheric pressure. Therefore, any point in Rs is uniquely specified by a, and o%fo r
given , PU and o. We will write this as
0
3