7 與式(2)的關係式，將力平衡方程式改寫為：
, 0
, ,
2
, + =
− + −
−
+ −
+ kki i t t i i
kk
i b
R p Q
R R nQ R u
N Q A
Nu γ γ ϑ 。 (22)
Biot 所引用之力學常數與常見之力學常數的關係如以下所示[5]：
G
N = ，
( )
GR N Q
A 2 1
2 = −
−
+ η ，
ν η ν
2 1
1
−
= − ， (23)
其中G為多孔介質之剪力模數（Shear Modulus）；ν 為多孔介質於排水情況下所測得之 波松比（Poisson’s Ratio）。茲考慮高溫水補注過程中，溫泉地層有形狀的改變，但總體 積無變化，即將多孔介質地層視為不可壓縮，如式(6)與式(7)所示。基於以上說明，引 用式(7)之關係式，並引用式(19)γt =
(
3A+2N)
αs+3Qαw、式(3)γt = 3(
Qαs +Rαw)
及式 (23)，則力平衡方程式(22)可進一步改寫為常見之型式：(
2 1)
, ,(
2 3)
, 0,kk + − kki− i− + s i+ i =
i G u p G b
Gu η λα ϑ 。 (24)
因係考慮於溫泉地層中，於座標原點處有一單點補注引致滲流現象，故若不考慮徹 體力b 的影響，則式(24)可以軸對稱圓柱座標表為： i
( )
1(
2 3)
01
2 2
2 =
∂
∂
− +
∂
− ∂
∂ −
− ∂ +
∇ G r
G r p G r u
ur η εr r λαs ϑ ， (25a)
( )
1(
2 3)
01
2 2 =
∂
∂
− +
∂
− ∂
∂
− ∂ +
∇ G z
G z p G
uz η εz λαs ϑ ， (25b)
其中 2
2 2
2 2 1
z r r
r ∂
+ ∂
∂ + ∂
∂
= ∂
∇ ；
z u r u r
ur r z
∂ +∂
∂ +
= ∂
ε ；Lame 常數
ν λ ν
2 1
2
= −G
。式(8)、式(16)、
式(25a)與式(25b)組成問題之基本方程式。
8
( ) ( ) ( ) ( )
{
, , , , , , , , , , ,} {
0,0,0,0}
lim0+ →
→ ur r z t uz r z t p r z t r z t
t ϑ 。 (27)
三、解析解
本文係以Laplace、Hankel 與 Fourier 積分轉換方法解析以上所述問題之數學模式，
可得出與時間有關之問題的解析解（Analytic Solution），該解析解亦可稱之為閉合解
（Closed-form Solution），如以下所示：
( )
−
−
−
−
= Ct
erfc R R r R
tr C t C R R
r t C R
tr C k G t Q z r
ur w w
1 3
1 1
2 2
1 3
1
2 2 exp 4
, 8
, π η π
γ
( )
−
+
−
− −
−
+ Ct
erfc R R r R
tr C t C R R
r t C C C C
C C R
tr C
C C G
Q
t h
1 3
1 1
2 2
1 1 3 2
3 1 3
2 3 1
2 2 exp 4
8π ηλ π
−
−
−
− Ct
erfc R R r R
tr C t C R R
r t C
3 3
3 3
2 2
3
2 2 exp 4
π
( )
−
−
−
− +
+ Ct
erfc R R r R
tr C t C R R
r t C R
tr G s C
3 3
3 3
2 2
3 3
3
2 2 exp 4
3
2 λα π ， (28a)
( )
−
−
−
−
= Ct
erfc R R z R
tz C t C R R
z t C R
tz C k G t Q z r
uz w w
1 3
1 1
2 2
1 3
1
2 2 exp 4
, 8
, π η π
γ
( )
−
+
−
− −
−
+ Ct
erfc R R z R
tz C t C R R
z t C C C C
C C R
tz C
C C G
Q
t h
1 3
1 1
2 2
1 1 3 2
3 1 3
2 3 1
2 2 exp 4
8π ηλ π
−
−
−
− Ct
erfc R R z R
tz C t C R R
z t C
3 3
3 3
2 2
3
2 2 exp 4
π
( )
−
−
−
− +
+ Ct
erfc R R z R
tz C t C R R
z t C R
tz G s C
3 3
3 3
2 2
3 3
3
2 2 exp 4
3
2 λα π ， (28b)
( ) ( )
−
+ −
=
t C erfc R t
C erfc R R C C C
C C Q
t C erfc R R k t Q z r p
t h w
w
3 1 1
3 2
3 1
1 2 2
1 2 4
1 , 4
, π πλ
γ ， (28c)
( )
=
t C erfc R R t Q
z r
t h
2 3
1 , 4
, πλ
ϑ ， (28d)
9 式中係數C Gkw
η γ 2
1
1− = 、
( )
k Gk
C G s w αuγw η
γ α
λ 3
2 3
1 2
2− = + − 、
t
C m
=λ
−1
3 ；符號R= r2+z2 。所
研討出之解為與時間有關之閉合解，其可作為以數值分析方法探討其他相關問題之基 礎，因這些閉合解可據以檢驗數值分析技巧之正確性。
四、數值結果
本文於數值分析計算時，擬引用表一所示之花崗岩地層參數進行數值結果之研討。
本文係考慮溫泉地層為一均質等向之無限域，且將座標原點置於點補注源點位置上，故 所探討之問題不僅屬於軸對稱現象，亦為球對稱情況。因此於數值研討中，擬探討單點 補注水所引致之球對稱地層徑向位移uR，其與軸對稱情況下之地層徑向位移ur及地層 垂直位移uz的關係為uR = ur2+uz2 。
因係討論高溫補注水在花崗岩地層中之滲流所引致的地層力學行為變化問題，故考 慮單位時間內之補注水體積Qw =1m3 s，且其中所攜帶之熱能Qh =108J s。基於此，並 引用表一所示之地層參數，即可繪製出如圖三至圖五所示之數值結果。
表一 花崗岩之地質參數[7-9]
符號 參數值
補注水之單位重，γw 9,810N m3
花崗岩之滲透係數，k 10−12m s
花崗岩之剪力模數，G 18.7×109N m2
花崗岩之波松比，ν 0.27
固體介質之線性熱膨脹係數，αs 8.33×10−6 oC 補注水之線性熱膨脹係數，αw 6.67×10−5 oC 固體介質之密度，ρs 2,630kg m3
補注水之密度，ρw 1,000kg m3 固體介質之比熱，c s 775J kg⋅oC 補注水之比熱，c w 4,186J kg⋅oC 花崗岩之熱傳導係數，λt 2.79J s⋅m⋅oC
圖三是花崗岩地層因點補注水所引致之地層位移。由圖三(a)與圖三(b)知，愈接近點 補注水源，地層位移變化量愈大，且當時間t =1012秒時，其點補注水所引起的地層位移 已接近長期位移；理論模式中之地層長期位移完成時，所需時間為無限大。
10
(a) (b)
圖三 補注高溫水所引致花崗岩地層徑向位移uR = ur2+u2z
圖四 補注高溫水所引致花崗岩地層 超額孔隙水壓p
(
r,z,t)
圖五 補注高溫水所引致花崗岩地層 溫度變化量ϑ
(
r ,,z t)
11
由圖四得知，愈靠近點補注水源，其所引致超額孔隙水壓愈大；且在同一位置上，
花崗岩地層中之超額孔隙水壓會隨時間的增加而上升，當時間t =1012秒時，其超額孔隙 水壓亦已接近穩態情況。圖五係圖示地層之溫度變化量，由此圖可知，地層中各點之溫 度變化量均不相同，越接近點補注水源，其地層溫度變化越明顯；當時間t =1012秒時，
其地層溫度變化已接近長期條件下之結果。
五、結論
本文旨在探討點補注高溫水所引致之地層力學行為等之變化，數學模式中，係考慮 地層為均質等向之線彈性無限域飽和多孔介質，並以積分轉換方法加以解析。由研討結 果知：
1. 所研討出之解為與時間有關之閉合解，其可作為以數值分析方法探討其他相關問題之 基礎，因這些閉合解可據以檢驗數值分析技巧之正確性。
2. 由圖三(a)與圖三(b)知，愈接近點補注水源，地層位移變化量愈大，且當時間t=1012秒 時，其點補注水所引起的地層位移已接近長期位移；理論模式中之地層長期位移完成 時，所需時間為無限大。
3. 由圖四得知，愈靠近點補注水源，其所引致超額孔隙水壓亦愈大；且在同一位置上，
花崗岩地層中之超額孔隙水壓會隨時間的增加而上升，當時間t =1012秒時，其超額 孔隙水壓已接近穩態情況。
4. 圖五係圖示地層之溫度變化量，由此圖可知，地層中各點之溫度變化量均不相同，越 接近點補注水源，其地層溫度變化越明顯；當時間t =1012秒時，其地層溫度變化已 接近長期條件下之結果。
六、誌謝
本文係在國科會計畫NSC-94-2625-Z-216-001 補助下所完成，特此申謝。
參考文獻
1. 經濟部中央地質調查所，「臺灣溫泉地質網」，http://210.69.81.66/hotspring/，2006/9/15 瀏覽。
2. 經濟部水利署，「溫泉標準」，中華民國94 年 7 月 22 日經濟部經水字第 09404605610 號令，2005。
3. 周順安、蔣立為、張國強、曾鈞敏，「台灣溫泉水資源之調查及開發利用(2/4)」，經
12
濟部水利署專題研究計畫報告，共248 頁，2001。
4. Biot, M.A., “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid,” J.
Appl. Phys., Vol. 26, No. 2, pp. 182-185, 1955.
5. 呂 志 宗 ，「 熱 影 響 之 三 維 壓 密 問 題 之 研 究 」， 國 科 會 專 題 研 究 計 畫 報 告 ， NSC81-0410-E-216- 503，共 163 頁，1993。
6. Nowacki, W., Thermoelasticity, Pergamon Press, New York, 566p, 1986.
7. Rice, J.R. and M.P. Clear, “Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media with Compressible Constituents,” Rev. Geophys. Space Phys., Vol.
14, No. 2, pp. 227-241, 1976.
8. Britto, A.M., C. Savvidou, D.V. Madducks, M.J. Gunn and J.R. Booker, “Numerical and Centrifuge Modelling of Coupled Heat Flow and Consolidation Around Hot Cylinders Buried in Clay,” Geotechnique, Vol. 39, No. 1, pp. 13-25, 1989.
9. Ozisik, M.N., Heat Transfer, McGraw-Hill, New York, p. 751, 1985.
本 本 論 論 文 文 已 已 被 被 中 中 國 國 機 機 械 械 工 工 程 程
學 學 會 會 第 第 二 二 十 十 三 三 屆 屆 全 全 國 國 學 學 術 術
研 研 討 討 會 會 接 接 受 受 ， ， 正 正 由 由 會 會 議 議 籌 籌 備 備
處 處 排 排 版 版 印 印 刷 刷 中 中 ， ， 將 將 於 於
2 2 0 0 0 0 6 6 / / 1 1 1 1 / / 2 2 4 4 正 正 式 式 對 對 外 外 發 發 表 表 。 。
中國機械工程學會第二十三屆全國學術研討會論文集 崑山科技大學 台南、永康 中華民國九十五年十一月二十四日、二十五日 論文編號：XX-XXX
Analytic Solutions of Thermo-consolidation Due to a Point Heat Source Buried in a Porous Elastic Half-Space
John C.-C. Lu1, Feng-Tsai Lin2
1Department of Civil Engineering and Engineering Informatics, Chung Hua University
2Department of Naval Architecture, National Kaohsiung Marine University NSC Project No.：NSC94-2625-Z-216-001
Abstract
Based on Biot’s three-dimensional consolidation theory of porous media, analytical solutions of the transient thermo-consolidation deformation due to a point heat source buried in saturated isotropic porous elastic half-space are presented. Closed-form solutions of the horizontal and vertical displacements of the ground surface are obtained by using Laplace and Hankel integral transformations. In the analysis, case of isothermal pervious half-space boundary is studied.
The thermally consolidation as affected by the boundary conditions are illustrated and discussed.
Keywords：thermo-consolidation, point heat source, porous medium, closed-form solution.
1. Introduction
Heat source buried in the stratum leads to thermo-mechanical responses of fluid saturated porous medium. The heat source such as a canister of radioactive waste can cause temperature rise in the soil.
The solid skeleton and pore fluid expand due to the heat source, and the volume increase of pore fluid is greater than that of the voids of solid matrix. This leads to an increase in pore fluid pressure and a reduction in effective stress. Therefore, thermal failure of soil will occur as a result of losing shear resistance due to the decrease in effective stress.
Attention is focused on the analytical solutions of the ground surface displacements of an isotropic stratum due to a point heat source. The response of the soil was satisfactorily modeled by assuming it as a thermoelastic porous continuum [1]. It suggested that linear theory was adequate for a repository design based on technical conservatism. For example, Hueckel and Peano [2] 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. Given these modest temperature increments, Hollister et al. [3] observed that any significant non-linear behavior and/or plastic deformation of the soil would be confined to a relatively small volume of soil around the waste canister itself. In this case, a linear model can provide a reasonable approximation to the assessment of a proposed design [4].
Governing equations of a fluid-saturated poroelastic solid in an isothermal quasi-static state were developed by Biot [5, 6]. Booker and Savvidou [1, 7, 8] derived an extended Biot theory including the thermal effects and presented solutions of thermo-consolidation around the spherical and point heat sources. In their solutions, the flow properties are considered as isotropic or transversely isotropic, whereas the elastic and thermal properties of the soils are treated as isotropic.
Nevertheless, the above studies did not include thermoelastic displacements of the ground surface due to a buried point heat source.
In this study, the soil mass is modeled as an isotropic saturated elastic half-space of porous medium.
Case of isothermal pervious half-space boundary is investigated. With the help of Mathematica tools, the transient horizontal and vertical displacements of the ground surface due to a point heat source of constant strength are obtained by using Laplace and Hankel integral transforms. Results are illustrated and compared to provide better understanding of the time dependent ground surface displacements due to a point heat source.
2. Mathematical Model 2.1 Basic Equations
Figure 1 shows a point heat source buried in a saturated porous stratum at a depth h . The porous soil mass is considered as a homogeneous isotropic thermoelastic half-space. The constitutive stress behaviors of the elastic soil skeleton are:
( )
ij ij s ij
ij
ij G G p
G ϑδ δ
ν α εδ ν
ν ε ν
σ −
−
− + + −
= 1 2
1 2 2
1
2 2 , (1)
Here, σ , ij ε and ϑ are the total stress components, ij strain components and temperature increment measured from the reference state of the porous medium, respectively; ε is the volume strain of the porous medium; δ is the Kronecker delta. The excess pore ij water pressure p is positive for compression. The constants ν , G and α are the Poisson’s ratio, s shear modulus, and linear thermal expansion coefficient of the skeletal materials, respectively. The strains ε ij
中國機械工程學會第二十三屆全國學術研討會論文集 崑山科技大學 台南、永康 中華民國九十五年十一月二十四日、二十五日 論文編號：XX-XXX and displacement components ui are given by the
linear law
(
ij ji)
ij =12 u, +u,
ε , (2) The total stress components must satisfy the equilibrium equations:
,j+ i=0
ij b
σ , (3) where bi denote the body forces. By using Eqs. (1) and (2), the equilibrium equations for axially symmetric problem without body forces bi can be expressed in terms of displacements ui, excess pore water pressure p and temperature change of the thermoelastic half-space ϑ in cylindrical coordinates (r,θ,z) as follows:
( ) 0
2 1
1 2 2
1 2
2 =
∂
∂
−
− +
∂
−∂
∂ −
∂ + −
∇ r
G r p r Gu r u G
G r r s ϑ
ν α ν ε
ν ,
(4a)
( ) 0
2 1
1 2 2
1
2 =
∂
∂
−
− +
∂
−∂
∂
∂ + −
∇ z
G z p z u G
G z s ϑ
ν α ν ε
ν , (4b)
where the Laplacian operator ∇2 can be expressed as
2 2 2
2
2 =∂ ∂r +1r∂ ∂r+∂ ∂z
∇ , while the volume
strain of the porous medium ε can be denoted as z
u r u r
ur ∂ + r +∂ z ∂
∂
ε= .
According to Darcy’s law, the governing equation of the conservation of mass can be expressed as
0
2 3 =
∂ + ∂
∂ + ∂
∂ +∂
∇
− t t
n p p t
k
u w
α ϑ ε β
γ , (5)
where k and n are the permeability and porosity of the porous medium, respectively; β and γ are the w compressibility and unit weight of pore water, respectively; αu= 1( −n)αs+αw and α is the w coefficient of linear thermal expansion of the pore water.
Consider a point heat source of constant heat generation rate H that is located at point ( )0,h . The uncoupled governing equation in axially symmetry is obtained from the conservation of energy and heat conduction law as following
( ) ( ) ( ) 0 2
2 − − =
∂ + ∂
∇
− r z h u t
r H m t
t δ δ
π ϑ ϑ
λ , (6)
in which λ is the coefficient of heat conduction; t
( ) s s w
wc n c
n
m= ρ +1− ρ and cw, cs are the specific heats of the pore water and skeletal materials, while ρ , w ρ are their densities; s δ( )x and u( )t are Dirac delta and Heaviside unit step functions, respectively.
Eqs. (4a), (4b), (5) and (6) constitute the basic
governing equations of the time-dependent axially symmetric thermoelastic responses of a saturated porous medium.
2.2 Basic Boundary Conditions and Initial Conditions
Consider the half-space surface, z = 0, is a traction-free, isothermal and pervious boundary for all time t≥0. The mathematical statements of the boundary conditions are:
(r,0,t)=0, zz(r,0,t)=0,p(r,0,t)=0,
rz σ
σ and ϑ(r,0,t)=0, (7)
The boundary conditions at z→∞ due to the effect of the point heat source must vanish at any time.
Assuming no initial change in displacements, temperature and seepage for the thermally poroelastic medium, the initial conditions at time t=0 of the mathematical model due to a point heat source can be treated as
(r,z,0)=0, u (r,z,0)=0, p(r,z,0)=0,
ur z
( , ,0) 0.
and ϑ r z = (8) The transient ground surface displacements can be derived from the differential equations (4a), (4b), (5) and (6) with the boundary conditions at z=0 and
∞
→
z , and initial conditions at time t=0.
3. Analytic Solution 3.1 Figures and tables
The governing partial differential equations (4a), (4b), (5) and (6) can be reduced to ordinary differential equations by performing appropriate Laplace and Hankel transforms [9] with respect to the time variable
t and the radial coordinate r:
( ) p
G dz
u u d
dz
d z
r ~ 1 ~
1
~ 2 2 2
2
2 ηξ − η− ξ + ξ
−
( ) ~ 0, 2
1 1
2 =
−
+ + ξϑ
ν α ν s
(9a)
( )
dz p d u G dz
d dz
u d
r 2 ~z 1 ~
1 ~
2 2 2
2 −
−
+
− ξ η ξ
η
( ) ~ 0,
2 1 1
2 =
−
− +
dz
s dϑ
ν α
ν (9b)
p s dz n
u u d s dz p
d
k z
r w
~ ~
~
~
2 2
2 ξ ξ β
γ +
+
+
−
−
,
~ 0 3 =
+ αusϑ (9c)
( ) 0, 2
~
2 ~
2
2 + − − =
−
− z h
s ms H
dz d
t δ
ϑ π ϑ ξ
λ (9d)
where ξ and s are Hankel and Laplace transform
中國機械工程學會第二十三屆全國學術研討會論文集 崑山科技大學 台南、永康 中華民國九十五年十一月二十四日、二十五日 論文編號：XX-XXX parameters; η=(1−ν) (1−2ν); and the symbols u~ , r
u~ , z p~ , ϑ~ are defined as
( ; , ) { ( , , )} ( ) ,
~
0 1
∫
∞= rL u r zt J r dr s
z
ur ξ r ξ (10a)
( ; , ) { ( , , )} ( ) ,
~
0 0
∫
∞= rL u r zt J r dr s
z
uz ξ z ξ (10b)
( ; , ) { ( , , )} ( ) ,
~
0 0
∫
∞= rL pr zt J r dr s
z
p ξ ξ (10c)
( ; , ) { ( , , )} ( ) ,
~
0 0
∫
∞= rL r zt J r dr s
zξ ϑ ξ
ϑ (10d)
in which Jn( )x represents the first kind of Bessel function of order n. The Laplace transformations with respect to ur, uz, p and ϑ are denoted by
( )
{ , , } ( , , ) ( )exp ,
∫
0∞ −= u r z t st dt t
z r u
L r r (11a)
( )
{ , , } ( , , ) ( )exp ,
∫
0∞ −= u r zt st dt t
z r u
L z z (11b)
( )
{ , , } ( , , ) ( )exp ,
∫
0∞ −= pr zt st dt t
z r p
L (11c)
( )
{ , , } ( , , ) ( )exp .
∫
0∞ −= r zt st dt t
z r
L ϑ ϑ (11d)
Taking Hankel and Laplace transforms for the boundary conditions at z = 0, Eq. (7), yield the transformed boundary conditions as following
,
~ 0
~r − uz= dz
u
d ξ (12a)
( 1) ~ 0,
~z+ − ur = dz
u
d η ξ
η (12b) ,
~ =0
p (12c) ,
~ =0
ϑ (12d) where u~r, u~z, p~ and ϑ~
follow the definitions of Eqs. (10a)-(10d). In this manipulation, the boundary conditions at z→∞ are needed to perform the integral transformations.
The transformed transient ground surface displacements due to a point heat source can be found from the transformed differential equations (9a)-(9d) and the transformed boundary conditions at z = 0 and
∞
z→ as below:
( ) ( ) − (− )
= − h
s c Gm s H
ur a ξ
η
ξ π exp
1 2 , 2
;
~ 0
2
− +
− h
c s s
cb
1 2
2exp ξ
, exp
2 2
2
− +
+ h
c s s
cc ξ (13a)
( ) ( ) (− )
= − h
s c Gm s H
uz a ξ
η
ξ π exp
1 2 , 2
;
~ 0
2
− +
+ h
c s s
cb
1 2
2 exp ξ
, exp
2 2
2
− +
− h
c s s
cc ξ (13b)
in which
( ) ,
2 1
1 2
3 1
ν α ν
−
− +
= s
a
G c
c c (14a)
( 2 1),
3 2 1
c c c cb c
= − (14b)
( ) ( ) ,
2 1
1 2
1 2 3
2 1
ν α ν
−
− +
= − s
c
G c c c
c
c c (14c)
where ca+cb=cc and
1, 2
2
1= +
β η
η
γ Gn
G c k
w
(15a)
2 m,
c =λt (15b)
(1 ) (1 ) .
3 1
3
s u
w
c k
α ν α ν
ν
γ − + +
= − (15c)
Using the Hankel inversions formula defined as following
( , , ) {~( ; , )} ( ) ,
0 1
∫
∞ −1= ξL u zξ s J ξr dξ t
z r
ur r (16a)
( , , ) {~( ; , )} ( ) ,
0 1 0
∫
∞ −= ξL u zξ s J ξr dξ t
z r
uz z (16b)
in which the Laplace inversions are defined as
( )
{ } ~( ; , ) ( )exp , 2
, 1
~ ;
1
∫
−+∞∞− = i
i r
r u z s st ds
s i z u
L α
α ξ
ξ π (17a)
( )
{ } ~( ; , ) ( )exp . 2
, 1
~ ;
1
∫
−+∞∞− = i
i z
z u z s stds
s i z u
L α
α ξ
ξ π (17b)
Using integral transform handbook [10], and integral inversions listed in Eqs. (16a)-(16b) and (17a)-(17b), the transient horizontal and vertical displacements ur(r,0,t) and uz(r,0,t) of the ground surface due to a point heat source of constant strength H are obtained as follows:
( ) ( )
( )
− +
= − 32
2
1 2
2 , 2
0
, h r
tr c Gm
t H r
ur a
η π
( )
∫
− − + − b ct h r
hr exp t c c
c 1
0
2 2 3
1
1 8
2
16τ τ
τ
τ τ
τ r d
r I
I
−
⋅
8 8
2 1 2 0
中國機械工程學會第二十三屆全國學術研討會論文集 崑山科技大學 台南、永康 中華民國九十五年十一月二十四日、二十五日 論文編號：XX-XXX
( )
∫
− − + + c ct ct hrexp h r c
c 2
0
2 2 3
2
2 8
2
16τ τ
τ
8 , 8
2 1 2
0
−
⋅ τ
τ
τ r d
r I
I (18a)
(r,0,t)=2 (2H−1)Gm
(
h2c+rth2)
23uz a
η π
+ + −
+ ct
r exp h
t c r h
h c cb
1 2 2 1
2 2
1 π 4
( )
+
− + + +
t c
r erfc h
r h
h r
h th c
1 2 2
2 2 2
23 2
1
2 2
+ + −
− ct
r exp h
t c r h
h c cc
2 2 2 2
2 2
2 π 4
( )
2 2 2 .2 2 2
2 2 2 3 2
2
+
− + + +
t c
r erfc h
r h
h r
h th c
(18b) The long-term ground surface horizontal and vertical displacements can be found as following by letting t→∞:
( ,0, ) 4 (2 1) ,
1 2
1 RR
hr c c c c Gm r H
ur b c
−
= −
∞ π η (19a)
( ,0, ) 4 (2 1) .
2
1 R
h c c c c Gm r H
uz b c
−
− −
=
∞ π η (19b)
where R= h2+r2 and R1= h2+r2+h . The long-term maximum ground surface vertical displacement uzmax due to the buried point heat source can be derived from Eq. (19b) by letting r=0 as below:
(0,0, ) 4 (2 1) .
2 1
max
−
− −
=
∞
= c
c c c Gm u H
uz z b c
η
π (20)
4. Numerical Results
Of particular interest is the thermo-consolidation of the stratum at each stage of the consolidation process.
Defining the average thermo-consolidation ratio U as following:
( ) .
ion, consolidat
thermo max.
term -long
, 0 , , at time ion consolidat
-thermo
max z z
u t r u
U = t (21)
Following Ma and Hueckel [11, 12] and Bai and Abousleiman [13], the selected representative parameters are listed in Table 1 to verify the proposed solutions.
The constants ci(i=1,2,3,a,b,c) can be derived as shown in Table 2 on the basis of the parameters listed
in Table 1 together with Eqs. (14a)-(14c) and (15a)-(15c).
Based on the parameters listed in Table 1, Figure 2 displays the average thermo-consolidation ratio U.
Note that initially the settlement appeared and then the heave phenomenon is observed on the ground surface as shown in Figure 2 at r h=0 and r h=2 , respectively. The maximum transient ground surface vertical displacement is about 260% of the long-term ground surface vertical displacement at r h=0 in this model. The average thermo-consolidation ratio U at
=0 h
r initially increases rapidly but the rate of heave phenomenon then slows. Since U approaches 1 asymptotically, theoretically thermo-consolidation is never achieved.
The profiles of normalized vertical and horizontal displacements at the ground surface z=0 are shown in Figures 3 and 4, respectively. The ground surface has significant horizontal displacement. For example, Figure 4 shows that the long-term maximum surface horizontal displacement is about 30% of the long-term maximum ground surface displacement. Figures 3 and 4 also concluded that the transient thermo-consolidation deformations are much larger than the long-term thermally consolidation displacements.
5. Conclusions
Closed-form solutions of the transient thermo-consolidation due to a point heat source from isothermal pervious elastic half-space were obtained by using Laplace and Hankel transformations. The study investigates not only the vertical settlement, but also the ground surface horizontal displacement.
Based on the numerical results, we found that the long-term maximum surface horizontal displacement is about 30% of the long-term maximum surface vertical displacement. From the average thermo-consolidation ratio U and vertical displacements at the ground surface, we found that initially the settlement appeared and then the heave phenomenon is observed on the ground surface in this model. It is concluded that transient ground surface displacements are important and the horizontal displacement should be properly considered for better prediction of the transient thermo-consolidation induced by a buried heat source.
6. Acknowledgment
This work is supported by the National Science Council of Taiwan, R.O.C., through grant NSC94-2625-Z-216-001.
7. References
1. Booker, J.R. and Savvidou, C., Consolidation Around a Point Heat Source, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 9, No. 2, pp. 173-184, 1985.
2. Hueckel, T. and Peano, A., Some Geotechnical Aspects of Radioactive Waste Isolation in Continental Clays, Computers and Geotechnics, Vol.