Mathematical Model Basic Equations

The soil mass is considered as a homogeneous isotropic full space with a vertical axis of symmetry. The mechanical, seepage and thermal properties of the stratum are treated as isotropic. Therefore, the constitutive behaviors of the elastic soil skeleton for linear axisymmetric deformation in the cylindrical coordinates

(

r ,,θ z

)

are:

( )

r u G r u

G _{r r}

rr ν

ν ν

σ ν

2 1

2 2

1 1 2

+ −

∂

∂

−

= −

′

ν βϑ

ν ₋

∂

∂ + −

z u

G _z

2 1

2 , (1a)

( )

r u G

r u

G _{r r}

ν ν ν

σ_θθ ν

2 1

1 2 2

1 2

− + −

∂

∂

= −

′

ν βϑ

ν −

∂

∂ + −

z u

G _z

2 1

2 , (1b)

r u G r u

G _{r r}

zz ν

ν ν

σ ν

2 1

2 2

1 2

+ −

∂

∂

= −

′

( )

_βϑ

ν

ν −

∂

∂

− + −

z u

G _z

2 1

1

2 , (1c)



 





∂ +∂

∂

= ∂

′ r

u z G u^{r z}

σrz , (1d) where σ^rr′ , σ_θθ′ , σ_zz′ , and σ_rz′ are the effective stress components. The shear stress components σ_r′ and _θ σ_θ′ vanish _z with a vertical axis of symmetry. The quantity ϑ measures the temperature change of the soil mass. The variables u_r and u_z are displacements in the radial and axial directions, respectively. In addition, the parameters G and ν are the shear modulus and Poisson’s ratio for the solid skeleton when the material is deformed under drained condition (i.e., the fluid pressure remains constant). The thermal expansion factor β is defined as

( ) ( ) ( )

_s

s G

Gα ν ν λ α

β =2 1+ 1−2 = 2 +3 . Here, λ is the Lame constant of the isotropic soil mass and α_s denotes the linear thermal expansion coefficient of the isotropic soil skeleton.

According to Terzaghi’s effective stress concept, the total stress τ_ij of a saturated porous material is given by

( )

(

^u

)(

_u

)

^ij

ij

ij p

B δ

ν ν

ν σ ν

τ − +

− −

= ′

1 2 1

3 , (2)

where ν_u is the undrained Poisson’s ratio of the poroelastic soil. The excess pore fluid pressure p is positive for compression, and δ_ij is the Kronecker

delta.

The total stress must satisfy the equilibrium relations of τ_ij,j+b_i =0. By equations (1a)-(1d) and Terzaghi’s effective stress concept (2), the equilibrium equations for axisymmetric deformation with vanishing body forces

(

b_i =0

)

can be expressed in terms of displacements u , _i temperature change of the soil mass ϑ , and excess pore fluid pressure p as below:

r p r

Gu r u G

G _r ^r

∂

− ∂

∂ −

∂ + −

∇ ε α

ν ²

2

2 1

=0

∂

− ∂ r

β ϑ , (3a)

2 0 1

2 =

∂

− ∂

∂

− ∂

∂

∂ + −

∇ z z

p z

u G

G _z ε α β ϑ

ν ^{, (3b)}

where ∇² =∂² ∂r² +1r⋅∂ ∂r+∂² ∂z² is the differential operator. The volume strain

z u r u r

u_r ∂ + _r +∂ _z ∂

∂

ε = and the

parameter α =3

(

ν_u −ν

) ( [

B1−2ν

)(

1+ν_u

) ]

. Besides of equations (3a) and (3b), two other equations for the four variables u_r, uz , p and ϑ are obtained from the conservations of mass and energy:

( )

[

⁻

]

^{+ =0}

⋅

∇ nv_f v_s q_f , (4)

=0 +

⋅

∇

− h q_h , (5)

where n is the porosity of the porous medium. The quantities v_f and v are _s the velocities of fluid and solid, respectively.

The values h, q and _f q are the heat h

flux vector, the internal/external fluid and

heat sources, respectively.

The isotropic pore fluid flow and thermal flow are assumed to be governed by Darcy’s law and Fourier’s law as follows:

( )





 





∂ +∂

∂

− ∂

=

− _{r z}

w s

f z

p r

p

nv v k i i

γ , (6)



 





∂ +∂

∂

− ∂

= _{t r z}

z r i i

h λ ϑ ϑ , (7)

where k denotes the permeability of the isotropic soils. The value γ_w is the unit weight of pore water. The quantity λ_t is the thermal conductivity of the soil mass.

Let us consider a deep point heat source of strength Q located at the position of

( )

0,0 and neglect the action of fluid source. Substituting equations (6)-(7) into equations (4)-(5) lead to

1 =0



 





∂ +∂

∂ + ∂

∂

∂^{2 2}

2 2

w z

p r p r r

p k

γ , (8)



 





∂ +∂

∂ + ∂

∂

∂

2 2 2

2 1

z r r

t r

ϑ ϑ λ ϑ

( ) ( )

0

2 =

+ r z

r Q δ δ

π , (9)

where δ

( )

x is the Dirac delta function.

Equations (3a), (3b), (8) and (9) constitute the basic governing equations of the steady state axisymmetric thermoelastic consolidation of a saturated isotropic porous medium.

Boundary Conditions

The effect of point heat source vanishes at infinity

(

z→± ∞

)

. Therefore, the boundary conditions for the full space are

represented as:

( ) ( ) ( ) ( ) {

u_r r z u_z r z p r z r z

}

zlim , , , , , ,ϑ ,

±∞

→

{

0,0,0,0

}

= . (10)

The basic equations (3a), (3b), (8), (9) and corresponding boundary conditions (10) constitute the mathematical model of the presented study.

3. Analytic Solution

The governing partial differential equations (3a), (3b), (8) and (9) can be reduced to ordinary differential equations by performing appropriate Hankel transforms (Sneddon, 1951) with respect to the radial coordinate r :

( )

p

G dz

u u d

dz

d _z

r ~ ~

1

~ 2 2 ²

2

2 ηξ  − η− ξ +αξ



 



 −

( )

~ _0,

2 1 1

2 =

−

+ + ξϑ ν

α ν _s

(11a)

( )

dz p d u G dz

d dz

u d

r z ~

2 ~ 1 ~

2 ₂ ²

2 ξ α

η ξ

η  −



 



 −

+

−

( )

^~ _0,

2 1 1

2 =

−

− +

dz

s dϑ ν

α

ν (11b)

~ 0

2 2

2  =



 



 − p

dz d k

w

γ ξ , (11c)

( )

0 2

2 ~

2

2  + =



 



 − Q z

dz d

t δ

ϑ π ξ

λ , (11d)

where ξ is Hankel transform parameter;

(

ν

) (

ν

)

η= 1− 1−2 ; and the symbols u~_r, u~_z, p~ , ϑ^~ are defined as

( )

; ⁼

∫

0^∞

( ) ( )

, 1

~ z ru r z J r dr

u_r ξ _r ξ , (12a)

( )

; ⁼

∫

0^∞

( ) ( )

, 0

~ z ru r z J r dr

u_z ξ _z ξ , (12b)

( )

; ⁼

∫

0^∞

( ) ( )

, 0

~ z rp r z J r dr

p ξ ξ , (12c)

( )

; ⁼

∫

0^∞

( ) ( )

, 0

~ z ξ rϑ r z J ξr dr

ϑ , (12d)

in which J_n( )x represents the first kind of Bessel function of order n. The Hankel inversions formula are denoted by

( )

r,z ⁼

∫

0^∞ξu~

( ) ( )

z;ξ J1 ξr dξ

u_{r r} , (13a)

( )

r,z ⁼

∫

0^∞ξu~

( ) ( )

z;ξ J0 ξr dξ

u_{z z} , (13b)

( )

r,z ⁼

∫

0^∞ξ~p

( ) ( )

z;ξ J0 ξr dξ

p , (13c)

( )

⁼

∫

0^∞ ~

( ) ( )

; 0

, ξϑ ξ ξ ξ

ϑ r z z J r d . (13d)

Taking Hankel transform for the far boundary conditions at z→±∞, Eq. (10), yield the transformed boundary conditions as following

( ) ( ) ( ) ( )

{

^{~ ;ξ ,~ ;ξ ,~ ;ξ ,ϑ~ ;ξ}

}

lim u_r z u_z z p z z

z→±∞

{

0,0,0,0

}

= . (14)

where u~_r , u~_z , p~ and ϑ^~ follow the definitions of Eqs. (12a)-(12d).

After the tedious manipulation, the closed-form solutions of thermoelastic consolidation deformation, temperature change of the soil mass, excess pore water pressure, and effective stresses due to a

point heat source buried in an isotropic elastic full space can be obtained by using Hankel inversions and (1a)-(1d) as follows:

( )

( )

R r u Q

t

r πλs ν

ν α

−

= +

1 8

1 , (15a)

( )

( )

R z u Q

t

z πλs ν

ν α

−

= + 1 8

1 , (15b)

=0

p , (15c) R

Q

t

1 4πλ

ϑ = , (15d)

( )

(

−

)

^_^{ + }_^

− +

′ = 1 ²₃

1 4

1

R r R QG

t

rr πλs ν

ν

σ α , (15e)

( )

( )

R

QG

t

s 1

1 4

1 ν πλ

ν σ_θθ α

−

− +

′ = , (15f)

( )

(

−

)

^_^{ + }_^

− +

′ = 1 ²₃

1 4

1

R z R QG

t s

zz πλ ν

ν

σ α , (15g)

( )

(

1

)

³

4 1

R rz QG

t s

rz πλ ν

ν σ α

−

− +

′ = , (15h)

where the symbol R= r²+z² . From these solutions, the excess pore fluid pressure disappears under the steady state thermoelastic consolidation.

On the basis of equations (15a)-(15h), we found that all field quantities are functions of the distance from the heat source and are proportional to the linear thermal expansion coefficient. However, they are inversely proportional to the thermal conductivity. The solutions show that the shear modulus does not have influence on displacements and temperature change of the soils.

4.Numerical Results

Figure 1. Stresses on soil element due to a deep point heat source Q.

Referring to the soil element in Figure 1, the numerical results of the effective stress distributions of σ ′_ij are summarized in Figures 2-5. The assumed Poisson’s ratio of the soil mass is ν =0.3, and the effective stresses were normalized by the factor QGα_s 4πλ_t . As shown in the figures 2-5, the isobaric contours of effective stresses of soil mass σ^rr′ , σ_θθ′ , σ_zz′ , and

σrz′ near the point heat source are compressive, and it helps to prevent thermal failure of soil.

High-level radioactive waste generates heat, and it leads to temperature increase for the soil surrounding the canister. Hueckel and Peano (1987) indicated that European guidelines require that temperature increments in the soil close to the heat

source should not exceed 80°C while the temperature increments at the ground surface is limited to less than 1°C. In the previous studies, the heat outputs generated by canisters are assumed to be 224 W/m (Ma and Hueckel, 1993), 325 W/m (Ma and Hueckel, 1992), or 1000 W/m (Smith and Booker, 1996).

On the basis of equations (15a)-(15h), Table 1 gives the normalized values of the derived analytical solutions. The selected representative parameters of isotropic soil, Boom clay, are listed in Table 2 to verify the proposed solutions. At the distance of r = 0~10 m and z = 0.3 m away from the heat source corresponding to the parameters listed in Table 2, the displacements, effective stresses and temperature increases of the stratum due to a deep point heat source are shown in Table 3. The maximum radial displacement u_r, vertical displacement u_z, radial stress σ_rr′ , hoop stress σ_θθ′ , vertical stress σ_zz′ , shear stress σ_rz′ and temperature increase ϑ of the stratum shown in Table 3 are 1.86 mm, 1.86 mm,

−1.35 MPa, −1.24 MPa, −2.48 MPa, −0.48 MPa and 51°C, respectively. The temperature increase is below 1°C at the soil 16 m away from the point heat source.

Besides, the ratio of the radial normal stress σrr′ to the vertical normal stress σ′_zz ranges from 0.5 to 2.0, and the ratio of the hoop normal stress σ_θθ′ to the vertical normal stress σ_zz′ ranges from 0.5 to 1.0.

Figure 2. Isobaric contours of normalized effective radial stress −σ_rr′

[

QGα_s 4πλ_t

]

due to a point heat source Q.

Figure 3. Isobaric contours of normalized effective hoop stress −σ_θθ′

[

QGα_s 4πλ_t

]

due to a point heat source Q.

Figure 4. Isobaric contours of normalized effective vertical stress −σ_zz′

[

QGα_s 4πλ_t

]

due to a point heat source Q.

Figure 5. Isobaric contours of normalized effective stress −σ_rz′

[

QGα_s 4πλ_t

]

due to

a point heat source Q.

5. Conclusions

The closed-form solutions of thermoelastic consolidation due to a point heat source buried in an isotropic elastic full space were obtained using the Hankel transformation technique. The results were examined by the numerical results. All field quantities are functions of the distance from the heat

source and are proportional to the linear thermal expansion coefficient. However, they are inversely proportional to the thermal conductivity. The solutions also show that the shear modulus does not have influence on displacements and temperature change of the soils.

Based on the numerical results obtained

for the isotropic thermo-consolidation, the effective stresses of soil mass are compressive. At the distance of r = 0~10 m and z = 0.3 m away from the heat source corresponding to the parameters listed in Table 2, the obtained maximum radial displacement u_r, vertical displacement u_z, radial stress σ_rr′ , hoop stress σ_θθ′ , vertical stress σ_zz′ , shear stress σ_rz′ and temperature increase ϑ of the stratum are

1.86 mm, 1.86 mm, -1.35 MPa, -1.24 MPa, -2.48 MPa, -0.48 MPa and 51°C, respectively. The ratio of the radial normal stress σ_rr′ to the vertical normal stress σ′_zz ranges from 0.5 to 2.0, and the ratio of the hoop normal stress σ_θθ′ to the vertical normal stress σ_zz′ ranges from 0.5 to 1.0.

The results are helpful for the repository design of radioactive waste.

Table 1. Normalized values of the displacements, effective stresses and temperature change of the isotropic soil due to a deep point heat source Q

z

r ( )

( ^ν)

πλ ν α

− + 1 8

1

t s r

Q

u ( )

( ^ν)

πλ ν α

− + 1 8

1

t s z

Q

u ( )

( )z QG

t s

rr

ν πλ

ν α

σ

− +

− ′

1 4

1 ( )

( )z QG

t s

ν πλ

ν α

σ_θθ

− +

− ′

1 4

1 ( )

( )z QG

t s

zz

ν πλ

ν α

σ

− +

− ′

1 4

1 ( )

( )z QG

t s

rz

ν πλ

ν α

σ

− +

− ′

1 4

1

z Q πλt

ϑ

4 0.0 0.0000 1.0000 1.0000 1.0000 2.0000 0.0000 1.0000 0.5 0.4472 0.8944 1.0733 0.8944 1.6100 0.3578 0.8944 1.0 0.7071 0.7071 1.0607 0.7071 1.0607 0.3536 0.7071 1.5 0.8321 0.5547 0.9387 0.5547 0.7254 0.2560 0.5547 2.0 0.8944 0.4472 0.8050 0.4472 0.5367 0.1789 0.4472 2.5 0.9285 0.3714 0.6916 0.3714 0.4226 0.1281 0.3714 3.0 0.9487 0.3162 0.6008 0.3162 0.3479 0.0949 0.3162 3.5 0.9615 0.2747 0.5287 0.2747 0.2955 0.0726 0.2747 4.0 0.9701 0.2425 0.4708 0.2425 0.2568 0.0571 0.2425 4.5 0.9762 0.2169 0.4237 0.2169 0.2271 0.0459 0.2169 5.0 0.9806 0.1961 0.3847 0.1961 0.2037 0.0377 0.1961 5.5 0.9839 0.1789 0.3520 0.1789 0.1846 0.0315 0.1789 6.0 0.9864 0.1644 0.3244 0.1644 0.1688 0.0267 0.1644 6.5 0.9884 0.1521 0.3006 0.1521 0.1556 0.0229 0.1521 7.0 0.9899 0.1414 0.2800 0.1414 0.1442 0.0198 0.1414 7.5 0.9912 0.1322 0.2620 0.1322 0.1345 0.0173 0.1322 8.0 0.9923 0.1240 0.2462 0.1240 0.1259 0.0153 0.1240 8.5 0.9932 0.1168 0.2321 0.1168 0.1184 0.0136 0.1168 9.0 0.9939 0.1104 0.2195 0.1104 0.1118 0.0121 0.1104 9.5 0.9945 0.1047 0.2082 0.1047 0.1058 0.0109 0.1047 10.0 0.9950 0.0995 0.1980 0.0995 0.1005 0.0099 0.0995

Table 2. Selected representative parameters (Ma and Hueckel, 1992)

Symbol Value Unit

G 100 MPa

Q 325 J/s

αs _{1.17 × 10}^{−4 o}_C⁻¹

λt _1.69 J/(s⋅m⋅°C)

ν 0.35 Dimensionless

Note: The strength of the point heat source Q that simulates the high-level radioactive waste is obtained from an assumed 1 m length line heat source with the heat output strength of 325 W/m.

Table 3. Typical values of the displacements, stresses and temperature increase of the isotropic soil due to a deep point heat source Q at the distance of r = 0~10 m and z = 0.3 m

away from the heat source corresponding to the parameters listed in Table 2

r (m) z

(m) ^r

u (m)

uz

(m)

σrr′ (Pa)

σθθ′ (Pa)

σzz′ (Pa)

σrz′ (Pa)

ϑ (°C) _zz

rr

σ σ

′

′ σzz

σ_θθ

′

′

0.0 0.3 0.000000 0.001859 −1,239,572 −1,239,572 −2,479,144 0 51.0 0.50 0.50 0.1 0.3 0.000588 0.001764 −1,293,558 −1,175,961 −2,234,327 −352,788 48.4 0.58 0.53 0.2 0.3 0.001031 0.001547 −1,348,736 −1,031,386 −1,745,423 −476,024 42.4 0.77 0.59 0.3 0.3 0.001315 0.001315 −1,314,765 −876,510 −1,314,765 −438,255 36.1 1.00 0.67 0.4 0.3 0.001487 0.001116 −1,219,739 −743,743 −1,011,491 −356,997 30.6 1.21 0.74 0.5 0.3 0.001594 0.000957 −1,106,692 −637,755 −806,572 −281,362 26.2 1.37 0.79 0.6 0.3 0.001663 0.000832 −997,836 −554,354 −665,224 −221,741 22.8 1.50 0.83 0.7 0.3 0.001709 0.000732 −900,813 −488,291 −564,061 −176,795 20.1 1.60 0.87 0.8 0.3 0.001741 0.000653 −816,826 −435,243 −488,903 −143,094 17.9 1.67 0.89 0.9 0.3 0.001764 0.000588 −744,776 −391,987 −431,186 −117,596 16.1 1.73 0.91 1.0 0.3 0.001781 0.000534 −682,967 −356,188 −385,598 −98,034 14.7 1.77 0.92 1.5 0.3 0.001823 0.000365 −476,850 −243,100 −252,450 −46,750 10.0 1.89 0.96 2.0 0.3 0.001839 0.000276 −363,711 −183,879 −187,925 −26,975 7.6 1.94 0.98 2.5 0.3 0.001846 0.000222 −293,282 −147,689 −149,786 −17,471 6.1 1.96 0.99 3.0 0.3 0.001850 0.000185 −245,463 −123,342 −124,563 −12,212 5.1 1.97 0.99 4.0 0.3 0.001854 0.000139 −184,897 −92,708 −93,226 −6,914 3.8 1.98 0.99 5.0 0.3 0.001856 0.000111 −148,215 −74,241 −74,507 −4,438 3.1 1.99 1.00 6.0 0.3 0.001857 0.000093 −123,648 −61,901 −62,056 −3,087 2.5 1.99 1.00 7.0 0.3 0.001858 0.000080 −106,054 −53,076 −53,173 −2,271 2.2 1.99 1.00 8.0 0.3 0.001858 0.000070 −92,837 −46,451 −46,517 −1,739 1.9 2.00 1.00 9.0 0.3 0.001858 0.000062 −82,546 −41,296 −41,342 −1,375 1.7 2.00 1.00 10.0 0.3 0.001859 0.000056 −74,307 −37,170 −37,204 −1,114 1.5 2.00 1.00

6. Acknowledgement

The work is supported by the National Science Council of the Republic of China through grant NSC94-2625-Z-216-001.

References

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Nomenclature

bi body forces (Pa/m)

B Skempton’s pore pressure coefficient (Dimensionless)

G shear modulus of the porous medium when the material is deformed under drained condition (Pa)

h heat flux vector (J/s⋅m²)

z r i

i , unit vector parallel to the radial/vertical direction (Dimensionless)

k permeability of the porous medium (m/s) n porosity of the porous medium

(Dimensionless)

p excess pore fluid pressure, positive for compression (Pa)

qf internal/external fluid source (s⁻¹) qh internal/external heat source (J/s⋅m³) Q strength of the point heat source (J/s)

(r,θ,z) cylindrical coordinates (m, radian, m) R parameter, R= r²+z² (m)

z r u

u, radial/axial displacement of the porous medium (m)

s f v

v , velocity of fluid/solid (m/s)

α parameter, α =3(νu−ν) (

[

B1−2ν)(1+νu)

]

(Dimensionless)

αs linear thermal expansion coefficient for solid skeleton of the porous medium (^oC⁻¹)

β thermal expansion factor of the porous medium, β =(2G+3λ)α_s (Pa/°C) γw unit weight of pore water (N m³) δ(x) Dirac delta function (m⁻¹)

δij Kronecker delta (Dimensionless)

ε dilatation of the porous medium (Dimensionless)

ϑ temperature change of the porous medium (°C)

λ Lame constant (Pa)

λt thermal conductivity of the isotropic porous medium (J/s⋅m⋅°C)

µ parameter, µ= λ_tr λ_tz (Dimensionless)

ν Poisson’s ratio for the solid skeleton when the material is deformed under drained condition (Dimensionless)

νu undrained Poisson’s ratio of the porous medium (Dimensionless)

σij′ effective stress components of the porous medium (Pa)

τij total stress components of the porous medium (Pa)

在文檔中 坡地災害潛勢、觀測與預警技術之研究（二）-子計畫： (頁 190-200)