nonisothermal consolidation in unsaturated soil
TRANSCRIPT
NONISOTHERMAL CONSOLIDATION IN UNSATURATED SOIL
By Thomas V. Edgar , 1 Member , ASCE, John D. Nelson,2
Fellow, ASCE, and David B. McWhorter3
ABSTRACT: Consolidation of unsaturated soil depends upon changes in stress-state variables resulting from applied stress and movement of pore water and gas. Water movement is highly dependent on heat flow, and much water movement takes place in the vapor phase. Consequently, water movement, soil deformation, and heat flow in an unsaturated soil mass are coupled phenomena. Equations are developed to describe water movement and deformation as functions of time, taking into account evaporation and temperature effects. In these equations, coordinate transforms are defined that-allow finite strain conditions to be included. A computer model was developed to solve these equations. Good correlation was observed between the model and results for standard cases of rigid, partially saturated flow and saturated consolidation. Examples are presented for drainage of single-and multiple-layered soil. An example is also presented for a uranium mill tailings profile comprising several layers with placement of a cover on the surface and taking into account both drainage and evaporation.
INTRODUCTION
Water movement, soil deformation, and heat flow in an unsaturated soil mass are coupled phenomena. A model has been developed that will predict water flux and soil deformation in a one-dimensional medium due to applied stress and both hydraulic and thermal gradients. The model considers mul-tilayered systems such as natural deposits comprising various strata or man-made structures such as tailings impoundments with covers.
Factors affecting the deformation of a soil include the total stress applied, the pressures of the liquid and the gas, the stress and moisture history, and the material properties. Furthermore, water may move both as a liquid and as a vapor. The phase change that takes place during evaporation within the pores and at the surface can alter the temperature of the profile and affect the flow. Therefore, the model takes into account heat flow as well as liquid, vapor, and air movement.
A set of.governing equations are derived based on a coordinate system that moves relative to the soil deformation. The use of this coordinate system accounts for the movement of the solid particles due to deformation. The governing equations have been solved for various boundary conditions using a finite difference model.
PREVIOUS RESEARCH
T w o bas ic me thods of analysis have been developed to describe the state of wate r in a soil . These are based on the the rmodynamic and the mecha -
'Asst. Prof., Civ. Engrg. Dept., Univ. of Wyoming, Laramie, WY 82071. 2Prof., Civ. Engrg. Dept., Colorado State Univ., Fort Collins, CO 80523. 3Prof., Agric. and Chem. Engrg. Dept., Fort Collins, CO. Note. Discussion open until March 1, 1990. To extend the closing date one month,
a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 10, 1987. This paper is part of the Journal of Geotechnical Engineering, Vol. 115, No. 10, October, 1989. ©ASCE, ISSN 0733-9410/89/0010-1351/$1.00 + $.15 per page. Paper No. 23932.
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nistic models of soil-water relationships. The work of Edlefson and Anderson (1943) was an application of thermodynamic principles progressing through consecutive equilibrium states. Further developments have applied the theory of irreversible processes to water flow problems [e.g., Taylor and Cary (1960), Letey (1968), and Olsen (1969)].
A model may represent either a rigid or a deforming soil structure. Theis (1935) developed a solution for a rigid model for transient water flow. Jacob (1940) showed that this solution also accounted for flow in slightly deforming confined aquifers. Terzaghi (1925, 1943) developed the conventional consolidation equation for isothermal flow in a saturated deforming medium. Biot (1941, 1955) developed a generalized theory for three-dimensional consolidation based on linearly elastic, anisotropic porous solids. McNabb (1960) took into account large strains and defined a functional form of the effective stress in terms of void ratio.
Gibson, England, and Hussey (1967) used a deformable coordinate system that was defined relative to an initial porosity condition. Gibson, Schiffman, and Cargill (1981) developed a solution for a thick deforming layer. Schiffman, Pane, and Gibson (1984) wrote the consolidation equation in terms of void ratio and considered nonlinear finite strain effects.
Raats and Klute (1968a, 1968b, 1969) developed a set of equations that describe the flow of water in a deforming medium using the theory of mixtures (Truesdell and Toupin 1960). Smiles and Rosenthal (1968) developed a relationship similar to that of McNabb (1960) for horizontal deformation, using the void ratio as the dependent variable.
The basic equation used to study the movement of water in partially saturated soils is the Richards' equation (1931). Philip and DeVries (1957) developed a vapor flux equation for rigid soils that was dependent on the gradient of the volumetric water content and the gradient of the temperature of the soil. Slegal and Davis (1977) and Baladi, Ayers, and Schoenhals (1981) derived similar equations and applied them to radial and spherical flow conditions. Dempsey (1978) used Philip and DeVries' (1957) theory to predict moisture change under pavements.
Philip (1969) extended the theory developed by Smiles and Rosenthal (1968) to account for flow of water in partially saturated deforming soils. Sposito (1973) showed indirectly that the overburden potential defined by Philip was related to an effective stress equation. Narasimhan et al. (1977, 1978a, 1978b) developed a model for water flow in a partially saturated deforming soil by use of an extension of Richards' equation, and by using Bishop's (1960) effective stress equation as the basic stress-state variable.
Barden, Madedor, and Sides (1969) proposed a constitutive law for unsaturated soils in terms of two independent stress-state variables. Fredlund and Morgenstern (1976) defined three stress-state variables for unsaturated soils, only two of which are independent. Fredlund and Hasan (1979) considered the volume changes and fluid flow due to changes in those stress-state variables. Dakshanamurthy and Fredlund (1981a,b) extended the model to include conductive heat flow. Lloret and Alonso (1980) modeled consolidation of swelling and collapsing soils in terms of the same stress-state variables.
GOVERNING EQUATIONS
This paper presents an alternative finite-strain formulation that defines a more general Lagrangian material coordinate in terms of the movement of
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the solid phase. This defines a pair of coordinate transforms that may be used to convert from a rigid reference frame through which the solids flow to a deforming coordinate system that moves with the solid particles. The system of governing equations is defined for three-phase flow during which the soil can deform due to changes in either the effective or capillary stresses. Since the energy flow is also defined using this coordinate system, both conductive heat flow and convective heat flow associated with phase changes of the water within the pore space can be modeled. This permits investigation of the effects of evaporation on consolidation. A computer model has been developed for this system, and examples are presented for saturated and unsaturated drainage, application of a surface cover, and evaporation in a deformable soil.
The system considered in developing the governing equations comprises an element of soil containing a solid phase and two fluid phases—liquid and gas. Each phase is separated by an interface. All phases are macroscopically continuous. The gas phase consists of dry air and vapor, both of which behave as ideal gases. The liquid can be any wetting fluid, but for purposes herein it is generally considered to be water. The only chemical change considered is the phase change between the liquid and its vapor.
Coordinate System Two types of reference volumes are used in this analysis. The first type
is an elemental volume of constant dimension that is fixed in space. As the soil deforms, all three phases pass through the sides of the element. In the second type, the dimensions of the base of the element are constant, and the vertical dimension is defined by a solid coordinate system in which the individual particles are assumed to stay in the same position relative to each other as the soil deforms. Thus, the solid coordinate system deforms with the solid phase. Znidarcic and Shiffman (1982) have shown that the original derivation of the consolidation equation by Terzaghi was based on a similar type of deformable coordinate system.
A solid coordinate r is defined which is a function of the height z above a zero reference datum, the time t, and the void ratio e of the material. It is defined by the following:
dr 8z
and
dr
dt
1 (1) 1 + e(z,t)
1 + e(z,t) (2)
where vs — the velocity of the solid phase at elevation z. A solid particle located at point z, has a solid coordinate rt defined by
(3)
Fig. 1 shows the relationship between rx and zt for a soil with a constant void ratio. The first integral on the right-hand side of Eq. 3 is a measure of the volume of the solid from elevation zero to elevation zx at time tx. The
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-̂
-
2
*"
I
Void
Solid
e
r
r
( l+e) r
Soil Element
Phase Diagram
FIG. 1. z and r Coordinates
second integral on the right-hand side determines the volume of solid that has passed the reference datum during the period from time zero to time tx. Eq. 1 has been used by several other researchers (McNabb 1960; Smiles and Rosenthal 1968; Philip 1969; DeSimone and Viggiani 1976; Monte and Kri-zek 1976).
Eqs. 1 and 2 can be used to define a pair of coordinate transforms. If P is any property, then
dP_
dz
and
dP
dr
dr
dz
1 dP
1 + e dr
dP
dt
dP = , dr
dr
z3z
dP + — , ^
vs dP dP
1 + e dr dt
(4)
(5)
These may be used as operators to transform equations from the fixed reference system to the solid coordinate system. An important aspect of this coordinate system is that it does not rely on the presence of an initial space coordinate that must be selected initially (Schiffmann et al. 1988).
Mass Balance Equations Fig. 2 shows the soil, consisting of the solid, the liquid, and the gas phases.
Each phase can flow into or out of the element at velocities of vs, vw, and vg. The subscripts s, w, and g refer to the solid, liquid, and gas phases respectively. The gas phase consists of a mixture of several gases. That component which comprises the vapor from the liquid phase will be termed the "vapor"; all other gases collectively will be termed "air." The subscripts (i and a will refer to the vapor and air, respectively.
The Darcy (or volumetric) flux Jw is the volume of water that passes a cross-sectional area per unit time. As such, it has the units of velocity and is numerically equal to the seepage velocity. Since a portion of the cross-sectional area contains solids, only the area that contains water is available for flow. The superficial or average velocity of water in the pores v„ can be determined by dividing the volumetric flux by the porosity in saturated
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d.
nsPs">.
nviPvi vw, out
out
TTT n 9 P* \ > u t
TZL
dz
ns/°s
®
Solid
( I ) ( 0 nw/°w
Liquid
M
n J U i ( I ) n g / y jng/>a
i
Vapor! Air
Gas
n n s />s v s , .
n g/°g v g i l n w^w v Wj n
ns + n w + n g = I
" ' M a s s per Unit Total Volume
(a)
dr
/°w Jw, out
(2)
s
-®
Solid
A>wJw:.
(2) P 8
Liquid
o e{ZK e ( 2 )
1 ^ 8 ! ^ e g
f t t , t f t
Vapor i Air
Gas i i
i—hn
/>0J"out
ftJfli,
( 2 )Mass per Unit Solid Volume
(b)
FIG. 2. Reference Volume for Mass Balance: (a) Fixed Coordinate System; (b) Deforming Coordinate System
soils or by the product of the degree of saturation and the porosity (Sn = nw) in unsaturated soils. The flux, and thus the superficial velocity, must be measured relative to the solid particles since it is the interaction between the water and the solid phase that is described by Darcy's law. In a rigid soil, the velocity of the solid phase is zero, and the flux can be measured relative to any fixed plane. In deforming soils, the flux must be determined relative to the moving solid phase, i.e.
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J w
— = v„ - vs nw
Similarly for the gas phase
J* - = vg-vs
(6)
(7)
The mass flux of the gas is described by the baricentric velocity of the mixture of the vapor and air
PgJg = P^Jy, + PaJa (8)
where p^ and p„ = the partial densities of the vapor and air.
Solid Phase As shown in Fig. 2(a), the mass of solid inside an element with constant
total volume is nsps, where ns = the volume fraction of the solid; and ps = the solid mass density. The mass flowing into or out of the element is equal to the product of the solid mass density ps, the velocity of the solid vs, and the area of solid at the element face (nsdxdy). The mass balance equation for the solid phase is
dnsps dnspsvs
dt dz
Because the solid phase is incompressible, Eq. 9 becomes
dn, dn.v, — + —— = 0 dt dz
(9)
(10)
This may be converted to a form containing the void ratio e, rather than the solid porosity, to produce
1 be dvs
dz
v. de
1 + e dt 1 + e dz = 0 (11)
where e is defined in terms of the total void space. By converting to the solid coordinate system using the coordinate trans
forms (Eqs. 4 and 5), Eq. 11 becomes
1
1 + e
or
—v„ de de
.1 + e dr dt
1
1 + e 1 + e
1 de
1 + e dr 0.
de dvs
dt dr
(12)
(13)
This indicates that the void ratio change that occurs in a deforming element is due to the velocity difference between the solid particles that define its surfaces.
Fluid Phases The mass balance equation for the liquid phase is similar to that for the
solid, except that the amount of liquid changing phase to or from the vapor
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state must also be considered. If M = the mass amount of liquid per unit volume that is being removed from the liquid phase per unit time due to evaporation, then the liquid mass balance equation in the fixed coordinate system is
3p^ + d^n^ + M = o
dt dz
If condensation is taking place, M is negative. An equation similar to Eq. 14 may be written for the vapor component
of the gas phase, except M must be given the opposite sign from the liquid equations to indicate that the vapor is exchanging a mass equal and opposite to that of the liquid phase:'
j V j ; + tv * M. _ M = Q ( 1 5 )
dt dz
An equation similar to Eq. 15 may be written for the air component of the gas. Since phase change of the air is not considered, a mass generation term is not required. Therefore
dP°"s + dpangva = Q
dt dz
Equations for each component of the fluid phase in the solid coordinate system can be developed by transforming Eqs. 14-16 using Eqs. 4 and 5 to produce:
Liquid:
J^Jl + -^-^ + (1 + e)M' = 0 (17) dt dr
ENGINEERING SOCIETIES LIBRARY Vapor:
Sp^s , dpi,J] + dt dr
Air:
d + ^ = o OGT-2-1989 (18>
dpae, + dpj. = o (19) dt dr
where e„, eg, and M' = the component volumes {Vw and Vg) and the mass of water transferred between the liquid and vapor phases divided by the volume of solid, respectively, i.e., M' = M/{\ + e). Eqs. 18 and 19 may be added together to produce an equation for the entire gas phase:
Gas:
5p„e„ dp„J„ _^_5 + J^A _ (i + e)M< = o (20)
dt dr
Eqs. 17 and 20 correspond to the flows shown in Fig. 2(b).
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Total Soil Since the volume and the density of the solid are assumed to be constant
within the transformed reference volume, i.e., (dpjdt) = 0, Eqs. 17 and 20 may be combined to yield
dR _ d(ps + pwe„ + pges) d(pwJw + pgJs) V^ 1 /
at at ar where R = the mass of the soil in the reference element divided by the volume of solid within the element. If density of the liquid and the gas remain constant, then the density terms may be removed from Eqs. 17 and 20. If those equations are added, an equation relating the change in volume to the volumetric fluxes is obtained
de dJw 3J„ — = (22) dt dr dr
Energy Balance Equation The internal energy of the soil may be formally described using the prin
ciples of continuum mechanics coupled with the theory of mixtures (Trues-dell and Toupin 1960). The procedure by which this is accomplished is as follows (Edgar 1983):
1. The momentum balance must be defined for each phase, considering the momentum diffusion of each of the fluids relative to the solid reference.
2. The kinetic energy balance is obtained by multiplying the momentum balance equation for each phase by the phase velocity.
3. The total energy balance equations are defined for each phase, considering heat flux, the work performed by the deformation of each phase, and any sources or sinks of heat.
4. The potential energy balance equations are derived for each phase. 5. The internal energy balance equations are obtained by subtracting the ki
netic and potential energy equations for each phase from the total energy equation for that phase. By assuming that the phase temperatures at a point are equal, the internal energy equation for each phase may be added together to form the internal energy balance equation for the soil.
Using these procedures for a deforming soil, the internal energy balance equation for the soil is
d(psU* + pweJJ* + pgegU*) d(PwJwU* + pgJgU*g)
dt dr
aq de deg uwJw dnw ugJg dng = + Rh — (CT - uw) 1- (ug - uw) 1 H
dr dt dt n„ dr n2 dr
Jw J a H F' +
1 l s—w ' nw n„ F'g_w - p,AHg-jLl + e)M' = 0 (23)
where U*, U*, and U* = the specific internal energies of the solid, liquid, and gas, respectively; q = the heat flux per unit area; R = the mass of the
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soil per volume of solid; h = the rate of heat generated per unit mass of solid; AHg-w = the enthalpy of vaporization; and F''s~„ and Fg-W are the internal shear forces developed between the solid and liquid and the liquid and gas, respectively. The balance equation indicates that the change in internal energy of the system with respect to time will be due to: (1) The convective flux of energy associated with the fluids; (2) the flov of heat across the system boundaries; (3) the addition or subtraction of a heat source or sink; (4) the power developed by the effective stress, causing the entire reference volume to deform; (5) the power developed by the capillary stress, causing the relative locations of the liquid-gas interface to deform; (6) the relative difference in power developed by the liquid flux on differing liquid areas on the two boundary, surfaces; (7) the relative difference in power developed by the gas flux on the differing gas areas on the boundary surfaces; (8) the power required to overcome the shearing resistance of the liquid and solid; (9) the power required to overcome the shearing resistance of the liquid and gas; and (10) the enthalpy change of the system due to the vaporization of the liquid.
If the fluid fluxes were zero, the change in internal energy would be due solely to heat flow, generation of heat within the system, the power due to volume deformation, and evaporation. In most cases, the power terms reduce to zero or are very small in comparison to the heat flux or enthalpy change; therefore, they may be neglected except in specific instances.
Constitutive Relationships The principal solution variables are the liquid pressure u„, the gas pressure
ug, and the temperature T. All other variables, including void ratio and total stress, may be determined through constitutive relationships using these three independent variables or the geometry of the system.
Constitutive relationships utilized in the solution of the equations include the following.
Void Ratio e Fredlund and Morgenstern (1977) showed that for unsaturated soils, the
void ratio can be expressed in terms of two appropriate stress-state variables. The void ratio relationship used was
Ae = CCA log (a - uw) - C,„A log (us - uw) (24)
where Ae = the change in void ratio of the soil corresponding to changes in the effective stress and the soil suction; Cc = the compression index; and Cm = a similar term relating the void ratio to the soil suction. In the computer model, stress hysteresis was accounted for by adjusting the slope of the rebound curves to a fraction of the slope of the virgin curve.
Liquid Void Ratio ew
The liquid void ratio is equal to the product of the void ratio and the degree of saturation. The degree of saturation can be related to the capillary pressure (ug — uw) by various relationships (Brooks and Corey 1964; Fink and Jackson 1973; Gillum et al. 1979). In this solution, an equation of the Brooks-Corey (1964) type was used. Thus
(w„ — uw)d S = ^ — (25)
. (ug - uj _
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where the numerator is the capillary displacement pressure, the denominator is the actual capillary stress or soil suction, and p = an exponent that fits the curve to experimental data. Some hysteretic effects are included in the solution model to allow for the usual case where the soil will not reach complete saturation on imbibition [e.g., Corey (1977)].
Flux Laws
Liquid Volumetric Flux Jw
The volumetric flux of the liquid is approximated by Darcy's law. Using the solid coordinate transform, the flux may be approximated by
dh, Kw dh, K„skwrdh, J„ = -Kw —' = - -' = '^^ —' (26)
dz 1 + e dr 1 + e dr
where h, = total head; Kw = the actual hydraulic conductivity; and Kws and kwr are defined as follows.
The term Kws defines the Kozeny-Carmen relationship (Lambe and Whitman 1969) between the saturated hydraulic conductivity Kwo at known void ratio ewo and any other value of void ratio e such that
i \ . nit iV v.?
. 3 . ,
e \ 1 + ew
1 + (27)
The term k„r is the relative conductivity, defined as the ratio between the actual conductivity Kw and the saturated conductivity Kws. It is equal to one when the soil is saturated and, when unsaturated, it is equal to (Corey, 1977)
kwr = 5(2+33)/p ( 2 g )
where S = the degree of saturation.
Gas Volumetric Flux Jg
The gas volumetric flux was determined in a manner similar to the liquid flux except that the relative conductivity was that for the non-wetting phase, i.e.
1 + e
where
kgr = (1 - Sf{\ - 5) ( 2 + p ) / p (30)
Vapor Mass Flux p^J^ The vapor component of the gas flux is due to convection and diffusion.
The convective mass flux p ^ is
Pv-J* = PuJg (31)
and the diffusive mass flux p^J^ is
pX=-Ddf^-^-^ (32) dz 1 + e dr
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T The diffusion coefficient D was determined using Jury and Letey's (1979) modification of the Philip-deVries' theory (1957).
Heat Flux q The conductive heat flux is found using Fourier's law
dT -k dT q= -k— = (33)
dz 1 + e dr
where k = the thermal conductivity; and T = the temperature. DeVries (1963) proposed a thermal conductivity model for k based on an analogy to electrical flux through a fluid with suspended particles. It has been well verified (Wierenga et al. 1969; Sepaskah and Boersma 1979) and was used here.
MODEL DEFORM
The governing equations, as well as the constitutive relationships, were solved using finite difference modeling. The soil profile was divided into slices of thickness hr, which maintained a constant thickness even though the soil underwent large deformation.
The model uses the implicit solution technique and solves the equations in order of liquid balance, gas balance, and heat balance. An option is available to solve any combination of liquid balance and gas or heat or vapor balance. A detailed description of the model is given in Edgar (1983).
Model Validation Two standard cases were investigated to validate the model. The first case
compared the ability of the program to model the partially saturated infiltration problem described by Philip (1957a,b). The second case investigated the ability of the program to model the classical Terzaghi solution for consolidation of a saturated soil.
Comparison to Philip's Solution Philip (1957a,b) developed a numerical technique that determines the depth
of infiltration of water into a rigid soil with time. He used the Yolo light clay tested by Moore (1939) for his example. The initial condition specified a uniform liquid void ratio e„ of 0.47. At time t = 0, e„ was raised to 0.98 at the surface, and water was allowed to infiltrate into the soil.
The solid and dashed lines in Fig. 3 show the results of Philip's analysis through 40.5 days. The results of the model simulation are shown as the data points. Both the depth of infiltration and the shape of the saturation curve are in close agreement, especially at short times. At 40.5 days, the predicted depth to the wetting front agrees with Philip's solution within two centimeters. The depths at which 50% saturation was reached (ew = 0.73) agreed within approximately 3% (190 cm versus 196 cm).
Consolidation of Saturated Soil (Terzaghi's Solution) A second profile was used to test the model for consolidation of a satu
rated soil. The profile consisted of a three-meter layer of clay overlain by two meters of sand. The soils were considered to be saturated initially. Water was allowed to drain out the top only. The profile was first allowed to con-
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2.5
2.0
1.5
w 1.0
0.5
166 min
Philip's Solution
° DEFORM Solution
"0.4 0.5 0.6 0.7 0.8 0.9 I Liquid Void Ratio - ew
FIG. 3. Long-Term Infiltration Curves for Yolo Light Clay
solidate under its own weight. When full consolidation had been reached, a surface load of 250 kPa was applied. The predicted change in elevation of the top of the clay layer over time is shown in Fig. 4. The coefficient of consolidation cv varied with void ratio and hydraulic conductivity. For each model element, it varied by as much as 100% during the consolidation process.
The standard Casagrande curve fitting method was used to determine an effective average value for c„ for the clay layer. This value was c, = 2.3 X 10-5 m2/s. The predicted rate of settlement of the clay using this value of cv in Terzaghi's solution (1943) is shown as the solid line in Fig. 4. The agreement between Terzaghi's solution and the DEFORM solution is good.
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Cv « 2.35 x l O " B mVsec
Terzaghi Theory
o DEFORM
^ 3 * 5 3 ^
' I i i i i_i "'"O.OI O.I 1.0 6.0
Time Since Appl icat ion of 2 5 0 kPa Load (days)
FIG. 4. Elevation of Top of Clay versus Time for Saturated Consolidation
Example Solutions Example solutions were developed to demonstrate the model for a wider
set of conditions. Two soil profiles are discussed herein. The mechanical properties for the soils used in these examples were based on typical values observed for uranium mill tailings having 40% sand and 60% slimes (Sherry 1982).
Baseline Case The baseline profile consists of a 6.0-m thick layer of a sand-slurry mix
ture deposited at its maximum void ratio. To achieve this initial void ratio over the entire height, it is assumed that the initial effective and capillary stresses are equal to a value of 1.0 kPa throughout the profile. At time t = 0, the water pressure is set equal to zero at the base, and drainage occurs. The initial values of soil parameters are listed in Table 1.
The predicted liquid void ratio profile at various times is shown in Fig. 5. Three distinct stages occur during drainage of the deforming profile. During the initial stage of drainage, the soil remains saturated. The volume of water that drains out equals the decrease in void volume that is caused by an increase in the effective and capillary stresses as drainage occurs. The initial stage is shown in Fig. 5 from 0.0-0.9 days.
The second stage begins when the capillary pressure of the water at the surface reaches the displacement pressure. At this time, the soil becomes unsaturated. There is a break in the 1.9 day curve at an elevation of about 3.4 m. This point represents the front above which the soil is unsaturated and below which the soil is saturated. The unsaturated front moves down the profile until it is slightly above the elevation that corresponds to the displacement pressure of the soil.
The last stage of drainage occurs more slowly as the total head decreases to its hydrostatic value. The rate of drainage is controlled by the gradient of the total head of the liquid at any location and the hydraulic conductivity
c.co
n r\r\
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TABLE 1. Properties of Example Soil Layers
Soil number
(1)
Initial height (m)
(2)
Void Ratio Coefficients"
e„b
(3) cc
(4) (5)
Saturation Coefficients0
(ug - uw)d (kPa) (6)
P (7)
Kj (m/s) (8)
(a) Baseline
1 6.0 0.92 0.19 0.02 6.9 0.28 1 x 10"5
(b) Three Layer
1 2 3
3.0 1.0 1.0
0.92 2.27 0.92
0.19 0.61 0.19
0.02 0.23 0.02
6.9 6.9 6.9
0.28 0.17 0.28
1 x 10"5
1 x 10~5
1 x 10"6
(c) Four Layer
1 2 3 4
2.4 0.6 0.8 2.0
0.92 2.27 0.92 0.50
0.19 0.61 0.19 0.10
0.02 0.23 0.02 0.05
6.9 6.9 6.9
20
0.28 0.17 0.28 0.20
1 x 10~5
1 x 10~4
1 x lfr5
1 x 10"8
"Eq. 24. be„ = void ratio when (CJ - uw) and (us - «,„) = 1.0. cEq. 25. dEq. 27.
at that point. Two factors reduce the hydraulic conductivity during this process. First, as the stress-state variables increase, the void ratio decreases, and the saturated hydraulic conductivity decreases. Secondly, the relative hydraulic conductivity decreases as the degree of saturation decreases.
During the total drainage period, 77% of the settlement occurs while the soil is still saturated and 15% occurs while the unsaturated front moves downward. Only 8% of the settlement occurs during the last stage of drainage.
Temperature Effects The variation in temperature in real soils will cause a complex series of
effects to occur. The nature of these effects are often unclear, and in some cases, conflicting reports of thermal effects have been presented. Thermal effects on the hydraulic conductivity and the pressure-saturation relationship were neglected. Those factors investigated included the effect of temperature on vapor density, vapor flux, and the amount and rate of evaporation. The effect of evaporation on temperature in the soil is described in the following.
Evaporation In the previous case, evaporation was not considered. The baseline case
was used in an analysis to demonstrate the effects of evaporation. The model DEFORM considers evaporation to be a diffusion process. The
relative humidity at the soil surface is determined by the temperature and the capillary stress acting in the top element. The relative humidity of the atmosphere is separated from the surface value by a distance referred to as the diffusion length. This produces a vapor density gradient which, when
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4 -
§ 3
*The Soil is m
Saturated Below This Point
yy?%
y T ^ . 9 days / 1 1 1 1 1 . 1 1 I 1 1 I i
\ \ \ \ \
! / \ / i / 0 .9 days
4.5 days / /
» /
i / \i i
\ 11 \ ' / 82.0 days\ \ j T = 0 0
A 1 A i s \ / N \ . . / \ \*J
\
i
i !
i i !
^ *
i
0.2 0.4 0.6 0.8 Liquid Void Ratio - ew
FIG. 5. Drainage Profiles for Baseline Case
multiplied by the diffusion coefficient of vapor in air, determines the vapor flux from the soil surface into the atmosphere.
The actual value of diffusion length is a function of the surface roughness, the wind velocity, the relative humidity and temperature of the atmospheric air, vegetation, degree of saturation and temperature of the surface, the salt and chemical concentrations at the surface, and many other factors. This would obviously be a difficult parameter to determine analytically. Thus, for a particular situation, it is recommended that the value of diffusion length be determined by running the program with various values of diffusion length until an average evaporation rate is determined that matches the estimated or actual evapotranspiration rate.
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160
140-
E E 120
100
8 0
6 0
4 0
20
" -
-
-/
J*
i 1
1 1
1 1 1
1 1
1 1
1
1 1 / I /
I / 1 /
1 /
I /
' / J>
/ ' ! / / / /
/
y ._
" " " / • /
/ / / /
/ / / /
/ / / / f / /
/ /
/ /
Dif fus ion Length above Soil =0 .005 m n i l i . _ I L _ l . f ^ _ : t ^ ^ i n
Dif fus ion Length above Soli =0 .020 m
i i i i i i i
20 3 0 4 0 50 6 0 Time Since Star t of Drainage (days)
7 0 80
FIG. 6. Cumulative Evaporation as Function of Time for Different Values of Diffusion Length
Fig. 6 shows the effects of different diffusion lengths on the evaporation rates. The curves may be divided into two parts. The first part has a steep slope and represents the evaporation that takes place while the profile is draining and the surface layer is still not air-dry. When the diffusion length is short, the evaporation rate is high and the water evaporates faster than it can flow upward in the liquid state. Thus, the surface dries quickly, as is shown when the diffusion length is 0.005 m. As the diffusion length becomes longer, the liquid is able to replenish the evaporated water in the top layer for a longer period of time. Thus, while the initial evaporation rate is not as great with the longer diffusion length, the flux continues longer, and more water can evaporate during the first portion of the evaporation, as is shown for a diffusion length of 0.020 m.
The second part of the curve represents a much lower evaporation rate, which occurs after the top layer has dried to equilibrium with the atmosphere. At this stage, the vapor must diffuse through the top soil layer since the hydraulic conductivity of the surface soil is essentially zero. The principal factor that determines the vapor flux rate is the distance between that point where the vapor density is controlled by the soil and that point where it is controlled by the atmosphere. This distance is the diffusion length. As before, as the value of diffusion length decreases, the greater the flux rate increases. It can be seen that the maximum cumulative evaporation is relatively insensitive to the value of diffusion length.
The effects of evaporation on soil temperature are shown in Fig. 7. The soil profile has an initial uniform temperature of 15° C. The temperatures at the upper and lower surfaces are maintained at 15° C, while the temperature in the soil is allowed to change. The evaporation causes the soil to cool, with a cooling front that progresses down the profile as vapor flow occurs deeper in the soil. The maximum temperature drop occurs at approximately 36 days, corresponding to the time at which the soil surface dries and the
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2.3 days
7.3 days \
36.2 days \ \ ^ ^
\ X A 58.4 days__\ / /» \
Y1 / \ \ 82.0 days. __
~
1 1
\ 1
\
* 1
\ 1
\ 1
{ \ A
\ \
\ \
\
1 i " ) l I I I 1 I I
0 4 8 12 16 20 Temperature (°C)
FIG. 7. Effect of Evaporation on Temperature in Soil Profile with Surface and Base Temperature Equal to 15° C
evaporation rate decreases. After that time, the heat loss due to evaporation becomes much less, and heat flow enables the soil temperature to increase.
Three-Layer Case Tailings are generally hydraulically deposited. This frequently causes the
tailings profiles to comprise different layers with widely differing properties, rather than a single thick homogeneous layer. A three-layer profile was analyzed, in which the top and bottom layers were the same material as the baseline soil. The middle layer corresponds to a slimes material tested by Sherry (1982). (Slimes refers to that portion of the tailings finer than a 200 mesh sieve.) Values of initial soil parameters for all layers are listed in Table
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~ 4 E
5 3
292
t ' 0 . 0
* ln Eoch Layer, the Soil is
Saturated Below this Point
Layer I
0.5 1.0 1.5 Liquid Void Ratio - e w
2.0 2.5
FIG. 8. Drainage Profiles for Four-Soil Case
1. The middle layer was finer grained and more compressible than the other two.
Water can drain out the base and evaporate from the surface. The results are shown in Fig. 8. Each profile shows the distinct liquid void ratio associated with the three soil layers. In each soil layer, the zone below the point marked with an asterisk is saturated, and the zone above the point is unsaturated. At 2.3 days, the surface is drying out, and at 267 days, the dry surface crust is very evident. The lower layer starts to desaturate at 2.3 days. The effect that the middle, less permeable layer has on retarding drainage is evident. The lower portion of the upper layer is still saturated, while the upper portion of the lower layer is not.
After 267 days, the profile reaches equilibrium. The surface dries to a liquid void ratio of 0.035, forcing the vapor to diffuse through the top crust. The middle layer still contains considerably more water than the surrounding layers. The shape of the liquid void ratio curve is dependent on the upward flow of the liquid due to evaporation and the combined loadings of the effective and capillary stresses.
The lower soil layer has undergone a considerable reduction in height. Even so, the liquid void ratio of this layer is similar to the profiles shown for the single-soil case.
Application of Soil Cover This case examines application of a cover to the three-soil case at the end
of the drainage period described previously (267 days). The initial soil properties are given in Table 1. The initial conditions for the lowest three layers correspond to the conditions at 267 days for the previous case. The cover is 2.0 m thick, with a dry unit weight of 17.33 kN/m3 and a water content
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of 0.160. Both drainage and evaporation are allowed. The results are shown in Fig. 8. Again, asterisks indicate the dividing
point between the saturated and unsaturated zones for each layer. The drainage in the lower layers occurs quickly and reaches near-equilibrium conditions in 25 days.
There is an increase in liquid void ratio at the surface of the original three-layer profile when evaporation is discontinued due to the cover. However, the liquid void ratio at the top of the cover decreases due to evaporation. It follows the same pattern as did the surface in the previous case.
There was a general decrease in the amount of liquid in the profile due to the applied load by the cover. Overall, there was a total deformation of approximately 120 mm. .
CONCLUSION
A combined theory of soil deformation, mass, and heat flow has been developed that allows a soil to undergo large deformations while being subjected to a wide variety of natural environmental boundary conditions. This derivation has been based on the concept of the solid coordinate system, in which the characteristic lengths in the soil are defined by the spacing of the soil particles.
Data presented on uranium tailing material indicate that it undergoes significant deformation subsequent to deposition. This deformation, and the secondary changes it causes on the soil properties, such as compressibility and hydraulic conductivity, is not determined in standard, partially saturated flow models. The deformation also affects the long-term equilibrium water content of the soil, which can alter the gas emission rate and other geo-technical considerations.
Credibility of the model was established by the agreement between the results obtained using DEFORM and those for classical solutions for rigid solids and for consolidation of saturated soils. The results obtained for hypothetical examples are reasonable and describe well the nature of equilibrium water-content profiles and settlement in actual tailings impoundments (Nelson and Davis 1987; Chen et al. 1988). Comparison of results predicted by the model DEFORM with actual measurements of pore-water pressure in large-scale laboratory columns has indicated that within the ability to determine actual soil properties, the analyses can predict actual performance (Ko-murka 1985).
ACKNOWLEDGMENTS
This work was performed as part of the Department of Energy Contract Number DE-AC04-82AL19453, entitled Colorado State University Study of Mechanical Stability/Contaminant Migration. Additional support has been provided by the Civil Engineering Departments at Colorado State University and the University of Wyoming.
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