Expressions for Ground Surface Displacements The focus of this study is on the horizontal and vertical

ELASTIC SUBSIDENCE SUBJECTED TO PERIODIC PUMPING

2. Mathematical Model

3.3 Expressions for Ground Surface Displacements The focus of this study is on the horizontal and vertical

displacements of the ground surface, z0, due to a periodic point sink. The transformed ground surface displacements u_r



^{0; ,} s



and u_z



^{0; ,} s



of the pervious half space are derived from equations (12a) and (12b) as follows:



^{0; ,}



⁰



^{1 2}

    

w r

Q cF s

u s exp h

Gk s

   



 

  



2 s

exp h

 c ^

 

   , (14a)



^{0; ,}



⁰



^{1 2}

    

w z

Q cF s

u s exp h

Gk s

   



 

  



2 s

exp h

 c ^

 

   . (14b) Applying the Laplace-Hankel inversion formulae lead to the following displacements:

     

, , 1 ; , ,

i st

r r

u r z t u z s J r e d ds



    



  



 

  ^ ^(15a)

     

, , 1 ; , .

i st

z z

u r z t u z s J r e d ds



    



  



 

  ^ ^(15b)

Using equations (15a)-(15b), the desired transient horizontal displacement u_r



r,0,t



and vertical displacement u_z



r,0,t



of the pervious ground surface due to a periodic point sink can be derived from equations (14a)-(14b) as follows:

     

 

2 2 3 2

, 0, 2 2 1

w r

r ct u r t Q

Gk r h



 

 

   

 

² ²

0 3

2 16 8

ct hr ct r h

 exp



    

  

 



2 2

0 1

8 8

r r

I I d

 

     

     

     

  

, (16a)

     

 

2 2 3 2

, 0, 2 2 1

w z

h ct u r t Q

Gk r h



 

 

   

 

2 2

2 2 3 2

0 2

ct h r h

ct erfc

r h

 

   

  





     

2 2

h r h

exp d

r h 

 

 

   

    

, (16b)

where ^I

 

^x is known as the modified Bessel function of the first kind of order  . The complementary error function is denoted as erfc x . The functions

 

^

 

^ct

and ^

 

^ct in equations (16a) and (16b) are defined as below:

       

2 2 1

2 2

sin sin

d d

n cT n ct

cT cT

ct ct

cT n cT cT

 







   





   

2 2

1 cos n cT^d 1 cos n ct

cT cT

  ^

   

     

 

  

  

, (17a)

     

2 2

1 sin ^d cos

n cT n ct

ct cT

cT n cT cT

 







   





   

2 2

1 cos n cT^d sin n ct

cT cT

  ^

  

   

  

  

, (17b)

in which ^d^

 

^t ^dt^{ }

 

^t . Solutions of consolidation deformations due to groundwater withdrawal at a constant strength are derived from equations (16a) and (16b) by taking an appropriate limit T_d  and using L’Hospital’s T rule. Carrying out the procedure, we obtain



^{, 0,}



^{2 2}



⁰¹





² ²



^{3 2}

Q ctr

u r t

Gk r h



 



   

 

² ²

0 3

2 16 8

ct ct hr r h

 exp



   

  

 



附 115

2 2

0 1

8 8

r r

I I d

 

     

     

     

  , (18a)

   

2 2

, 0, 2 2 1 4

w z

Q h ct r h

u r t exp

Gk r h ct



  

   

     

 

2 2

2 2 3 2 2

cth r h

erf r h ct

  

 

   

2 2

2 2 2

h r h

erfc r h ct

  

 

   

, (18b)

where ^{erf x}

 

is known as the error function and

 

erf x  erfc x .

The long-term ground surface horizontal and vertical displacements are found when t . Let

 

lim , 0,

r r

u^ u r t

  and z ^lim z



^{, 0,}



u^ u r t

  , then we have the closed-form solutions of long-term ground surface displacements.

For periodic pumping:

   

 

2 2 2 2

lim4 2 1

r t

hrf t u Q

Gk r h r h h



 



 

    , (19a)



₀

  

2 2

lim4 2 1

z t

Q hf t

u Gk r h



 



 

  . (19b)

For pumping at a constant strength:



⁰



² ²



² ²



4 2 1

w r

Q hr

u Gk r h r h h



 

  

    , (20a)



⁰



² ²

4 2 1

w z

Q h

u Gk r h



 

 

  . (20b)

The maximum long-term ground surface horizontal displacement and vertical displacement of the half space due to a point fluid sink at a constant strength are derived from equations (20a) and (20b) by letting

1.272

r h h and r0 , respectively. Here, (1 5) 2 1.618

   is known as the golden ratio [17,18]. For the point fluid sink at a constant strength problem, the maximum horizontal displacement u_r_max and vertical displacement u_z_max of the ground surface derived from equations (20a) and (20b) are

 

⁰

^ ^

max 2.5

1 2 1

, 0, 4

r r

u u h Q

 

  

     , (21a)

 

max

0, 0, 1 2

z z

u u Q

 



    . (21b)

The maximum ground surface horizontal displacement is approximately 30% of the maximum ground surface vertical displacement, i.e.,

max 2.5 max

1 0.3003

u ^ ^ ^{at r} ^h. (22)

The value r h is derived when dur



r^{, 0,}



dr is equal to zero. It is noticed from equations (21a) and (21b) that the maximum long-term ground surface horizontal displacement and settlement for pumping at a constant strength is not directly dependent on the pumping depth h of the point sink.

4. Numerical Results

The long-term elastic consolidation subjected to pumping at a constant strength is numerically analyzed. The profiles of normalized horizontal and vertical displacements at the ground surface z0 for isotropic saturated half space are shown in Figures 3 and 4, respectively. The ground surface reveals significant horizontal displacement. The example in Figure 3 shows that the maximum ground surface horizontal displacement is about 30% of the maximum ground settlement at

1.272 r h h.

Figure 3. Normalized horizontal displacement profile at the ground surface z = 0 for pumping at a constant strength.

附 116

Figure 4 shows that the ground surface vertical displacement is around 61.8% of the maximum ground surface vertical displacement at radial variable r h, where the maximum ground surface horizontal displacement occurred.

Figure 4. Normalized settlement profile at the ground surface z = 0 for pumping at a constant strength.

5. Conclusions

Closed-form solutions of the elastic consolidation due to periodic pumping from pervious saturated elastic half space were obtained by using Laplace-Hankel transformations. Transient ground surface horizontal displacement and settlement of the half space aquifer were investigated. The solutions can be used to evaluate numerical models and numerical simulations of the elastic consolidation settlement near the sink point. The results show:

1. The maximum long-term ground surface horizontal displacement and vertical displacement of the half space due to a point fluid sink at a constant strength are derived from equations (20a) and (20b) by letting

1.272

r h h and r , respectively. 0

2. It is noticed from equations (21a) and (21b) that the maximum long-term ground surface horizontal displacement and settlement are independent on the pumping depth h of the point sink at a constant strength for the isotropic elastic half space.

Acknowledgements

This work is supported by the National Science Council of Republic of China through grant NSC100-2221-E-216-025.

References

[1] J.F. Poland, Guidebook to studies of land subsidence due to ground-water withdrawal (Paris: Unesco, 1984).

[2] M.A. Biot, General theory of three-dimensional consolidation, Journal of Applied Physics, 12(2), 1941, 155-164.

[3] M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics, 26(2), 1955, 182-185.

[4] J.R. Booker & J.P. Carter, Analysis of a point sink embedded in a porous elastic half space, International Journal for Numerical and Analytical Methods in Geomechanics, 10(2), 1986, 137-150.

[5] J.R. Booker & J.P. Carter, Long term subsidence due to fluid extraction from a saturated, anisotropic, elastic soil mass, Quarterly Journal of Mechanics and Applied Mathematics, 39(1), 1986, 85-97.

[6] J.R. Booker & J.P. Carter, Elastic consolidation around a point sink embedded in a half-space with anisotropic permeability, International Journal for Numerical and Analytical Methods in Geomechanics, 11(1), 1987, 61-77.

[7] J.R. Booker & J.P. Carter, Withdrawal of a compressible pore fluid from a point sink in an isotropic elastic half space with anisotropic permeability, International Journal of Solids and Structures, 23(3), 1987, 369-385.

[8] G.J. Chen, Analysis of pumping in multilayered and poroelastic half space, Computers and Geotechnics, 30(1), 2002, 1-26.

[9] G.J. Chen, Steady-state solutions of multilayered and cross-anisotropic poroelastic half-space due to a point sink, International Journal of Geomechanics, 5(1), 2005, 45-57.

[10] W. Kanok-Nukulchai & K.T. Chau, Point sink fundamental solutions for subsidence prediction, Journal of Engineering Mechanics, ASCE, 116(5), 1990, 1176-1182.

[11] J.-Q. Tarn & C.-C. Lu, Analysis of subsidence due to a point sink in an anisotropic porous elastic half space, International Journal for Numerical and Analytical Methods in Geomechanics, 15(8), 1991, 573-592.

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[12] J. C.-C. Lu & F.-T. Lin, The transient ground surface displacements due to a point sink/heat source in an elastic half-space, Geotechnical Special Publication No. 148, ASCE, 2006, 210-218.

[13] J. C.-C. Lu & F.-T. Lin, Modelling of consolidation settlement subjected to a point sink in an isotropic porous elastic half space, Proceedings of the 17th IASTED International Conference on Applied Simulation and Modelling, Corfu, Greece, 2008, 141-146.

[14] J. C.-C. Lu & F.-T. Lin, Analysis of transient ground surface displacements due to an impulsive point sink in an elastic half space, Proceedings of the IASTED International Conference on Environmental Management and Engineering, Banff, Alberta, Canada, 2009, 211-217.

[15] I.N. Sneddon, Fourier transforms (New York:

McGraw-Hill, 1951, 48-70).

[16] A. Erdelyi, W. Magnus, F. Oberhettinger & F.G.

Tricomi, Tables of integral transforms (New York:

McGraw-Hill, 1954).

[17] M. Livio, The golden ratio: The story of phi, the world’s most astonishing number (New York: Broadway Books, 2002).

[18] R.A. Dunlap, The golden ratio and Fibonacci numbers (Singapore: World Scientific Publishing, 1997).

Nomenclature

b i Body forces (Pa/m)

c Parameter, ck n_w(m²/s)

 

erf x Error function (Dimensionless)

 

erfc x Complementary error function (Dimensionless)

 

f t Periodic function (Dimensionless)

 

F s Laplace-Hankel transforms of ^{f t} ^(m²^s)

G Shear modulus of the isotropic porous aquifer (N/m²)

h Depth of the periodic sink point (m)

r, z

i i Unit vector in r/z direction (Dimensionless)

 

I_ x Modified Bessel function of the first kind of order  (Dimensionless)

 

J_ x Bessel function of the first kind of order

 (Dimensionless)

k Permeability of the isotropic porous aquifer (m/s)

n Porosity of the porous aquifer (Dimensionless)

p Excess pore water pressure (N/m²)

p Laplace-Hankel transforms of p (Ns) q Rate of water extracted from the saturated

porous aquifer per unit volume (1/s) Q 0 Strength of the periodic pumping (m³/s)



^r^{, ,}^ ^z



Cylindrical coordinates system (m, radian, m)

s Laplace transform parameter (s^-1) t Time variable (s)

T Pumping period (s)

T d Duration of groundwater withdrawal in one period T (s)

r, z

u u Radial/axial displacement of the porous aquifer (m)

r, z

u u  Laplace-Hankel transforms of u u (m_r _z ³s)

max, max

r z

u u Maximum long-term ground surface horizontal/vertical displacement (m)

r , z

u^ u^ Long-term ground surface

horizontal/vertical displacement (m)

w, s

v v Velocity of pore water/solid matrix (m/s)

 Compressibility of pore water (m²/N)

w Unit weight of groundwater (N/m³)

 

 Dirac delta function (m^-1)

ij Kronecker delta (Dimensionless)

 Volume strain of the porous aquifer (Dimensionless)

ij Strain components of the porous aquifer (Dimensionless)

 Parameter, ^^{ }



¹ ^

 

^{1 2}^ ^



(Dimensionless)

 Poisson’s ratio of the isotropic porous aquifer (Dimensionless)

 Hankel transform parameter (m^-1)

ij Total stress components of the porous aquifer (N/m²)

 Golden ratio, 1.6180339887...

(Dimensionless)

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附錄 9

參與本計畫案之兼任研究助理謝適任同學的碩士論文：單井抽水所引致軸對稱彈性沉陷之研究

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在文檔中典型抽水所引致三維壓密沉陷解析 (頁 130-135)