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Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown
Computers and Geotechnics 56 (2014) 11–15
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown Kang-He Xie a, Da-Zhong Huang a,⇑, Yu-Lin Wang b, Yue-Bao Deng c a
Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China Department of Environment and Civil Engineering, Wuyi University, Wuyi Shan 354300, China c Institute of Geotechnical Engineering, Ningbo University, Ningbo 315211, China b
a r t i c l e
i n f o
Article history: Received 30 June 2013 Received in revised form 19 October 2013 Accepted 20 October 2013 Available online 9 November 2013 Keywords: Axisymmetric consolidation Groundwater drawdown Compressibility of fluid Laplace–Hankel transform Analytical solution
a b s t r a c t Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown was studied based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. Explicit analytical solutions are obtained in the Laplace–Hankel transform domain when groundwater drawdown is induced by a constant pumping well. Based on the solutions, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Land subsidence is a worldwide problem that has caused substantial damage to infrastructure and resulted in great economic losses [1,2]. The problem is more severe in regions with soft soils where groundwater is heavily exploited for industrial and domestic purposes [3]. Two large regions in China that suffer from severe subsidence problems are the Yangtze River Delta and the North China Plain, where thick, soft soil layers exist [3–6]. In each of these regions, as groundwater is pumped from the aquifer, the water level and pore water pressure are reduced, which leads to an increase in effective stress and consolidation of the soil layer and thus results in land subsidence. Because the permeability of an aquifer is usually much greater than that of a soil layer, the consolidation of a soil layer can be analyzed in two steps [7]. The first step is to compute or measure the groundwater drawdown in the aquifer. The second step is to compute the consolidation of the soil layer using the results of the first step as the boundary conditions. Several analytical solutions have been presented for the one-dimensional consolidation of a soil layer due to groundwater drawdown. Luo et al. [8] presented solutions for the one-dimensional consolidation of a soil layer due to constant groundwater drawdown. Li and Helm [9] ⇑ Corresponding author. E-mail address: [email protected] (D.-Z. Huang). 0266-352X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2013.10.004
presented solutions for the one-dimensional consolidation of a soil layer subjected to periodic groundwater fluctuation. Li [10] studied the one-dimensional consolidation of a nonlinear elastic soil layer subjected to periodic groundwater fluctuation. Teng et al. [11] studied the one-dimensional consolidation of a soil layer caused by constant groundwater drawdown with consideration of the body force effect. Tsai [12] studied the one-dimensional soil consolidation caused by constant groundwater drawdown with consideration of the viscosity effect. Liu et al. [13] presented solutions for the one-dimensional consolidation of a visco-elastic soil layer due to withdrawal of groundwater. In one-dimensional consolidation theory, groundwater drawdown is assumed to be widely and uniformly distributed in an aquifer. However, because there are different types of groundwater drawdown that correspond to different pumping conditions used in practice, it is necessary to study the consolidation of soil layers based on Biot’s three-dimensional consolidation theory [14,15]. On the other hand, the degree of saturation of a soil layer is usually not 100% because the soil may contain small bubbles. It has been reported that even a very small amount of gas in a soil dramatically increases the compressibility of the fluid [16]. Some researchers have studied the importance of fluid compressibility in the consolidation problem [16–20]. However, there have been no studies on the consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown.
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K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
Nomenclature ur uz r, h, z
radial displacement vertical displacement cylindrical coordinates stress component excess pore water pressure volumetric strain strain component shear modulus coefficient of permeability unit weight of water bulk modulus of pore fluid Poisson’s ratio
rij p
ev eij G k
cw M
m
t s n Jn a k2 Q H H2
This paper presents a study of the axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. When groundwater is pumped from an aquifer at a constant rate, explicit analytical solutions are obtained in the Laplace–Hankel transform domain. Based on the solutions for groundwater drawdown induced by a constant pumping well, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer.
k
cw
time variable Laplace transform variable Hankel transform variable nth-order Bessel function of the first kind conductivity of the aquifer coefficient of permeability of the aquifer constant pumping rate thickness of the soil layer thickness of the aquifer
r2 p ¼
@ p ev þ @t M
ð6Þ
where k is the coefficient of permeability of the soil, cw is the unit weight of water, M is the bulk modulus (adjusted for porosity) of the pore fluid, and t is a time variable. 2.2. Boundary conditions
2. Mathematical model
Fig. 1 illustrates a groundwater drawdown in an aquifer. The aquifer is assumed to be rough and rigid, so there is no vertical or radial displacement at the bottom boundary of the soil layer. The upper boundary of the soil layer is permeable and stress-free. The boundary conditions can be expressed as
2.1. Governing equations
z ¼ 0 : rzz ¼ 0; rrz ¼ 0; p ¼ 0:
ð7Þ
It is assumed that the poroelastic layer is homogenous isotropic. The force equilibrium equations (with no body forces) specialized to axisymmetry are
z ¼ H : uz ¼ 0; ur ¼ 0; p ¼ cw hðr; tÞ:
ð8Þ
@ rrr @ rrz rrr rhh þ þ ¼0 @r @z r
3. Solutions
ð1Þ
@ rrz @ rzz rrz þ þ ¼0 @r @z r
ð2Þ
3.1. General solutions
where rrr, rhh, rzz are total normal stress components and rrz is shear stress component. Assuming the solid constituent is incompressible, the constitutive equations take the form
rij ¼ 2G eij þ
m 1 2m
ev dij pdij
r2
1 1 @ ev 1 @p ur þ ¼0 r2 1 2m @r G @r
^f ðn; z; sÞ ¼
Z
2
Z
0
f ðr; z; tÞ ¼
1 2pi
1
f ðr; z; tÞrJn ðnrÞest drdt
ð9Þ
0
Z
cþi1
ci1
Z
1
^f ðn; z; sÞnJ ðnrÞest dnds n
ð10Þ
0
O
h(r,t)
r
ð4Þ Soil layer
1 @ ev 1 @p r2 uz þ ¼0 1 2m @z G @z
1
ð3Þ
where the subscripts i and j can be r, h or z, G is the shear modulus, m is the Poisson’s ratio, p is the excess pore water pressure, dij is the Kronecker delta. The Navier-type equations for displacement and p are obtained by substituting Eq. (3) into Eqs. (1) and (2):
Laplace and Hankel transforms were employed to obtain the solutions. The nth-order Laplace–Hankel transform of f(r, z, t) and its inversion may be found for example in Sneddon [21]:
Total head distribution in the aquifer
H
ð5Þ
2
@ 1 @ @ where r2 ¼ @r 2 þ r @r þ @z2 , ur and uz are the radial and vertical z r displacement components, respectively, ev ¼ @u þ urr þ @u , is the vol@r @z umetric strain. It is assumed that the solid constituent is incompressible but the fluid constituent is compressible. Combined with Darcy’s law, the pore fluid mass conservation equation is given by
Aquifer z Fig. 1. Consolidation of a soil layer induced by groundwater drawdown in an aquifer.
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K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
where p and n are the parameters of the Laplace and Hankel transforms, respectively, and Jn() denotes the nth-order Bessel function of the first kind. From Eqs. (4) and (5), we obtain
r2 p ¼ 2gGr2 ev
ð11Þ
Using Eq. (3), the stress components in the transform domain can be obtained as follows: r^ rz 2G
1 ¼ 2M ½A1 coshðnzÞ þ A2 sinhðnzÞ þ nA5 coshðnzÞ þ nA6 sinhðnzÞ
M 2 nz½A1 sinhðnzÞ þ A2 coshðnzÞ 2gcGs nf½A3 coshðfzÞ þ A4 sinhðfzÞ
1m where g ¼ 12 m. Eliminating ev from Eqs. (6) and (11), we obtain
@ c r2 r2 p ¼ 0 @t where c ¼ c
2gGkM
w ðMþ2gGÞ
ð24Þ r^ zz
ð12Þ
2
d f2 dz2
!
2
d ^¼0 n2 p dz2
^ ¼ A1 sinhðnzÞ þ A2 coshðnzÞ þ A3 sinhðfzÞ þ A4 coshðfzÞ p
ð13Þ
ð14Þ
where A1, A2, A3, and A4 are constants to be determined from the boundary conditions. Taking the Laplace and Hankel transforms of Eq. (6), we obtain
! 2 b p d 2 b b n e þ p ¼ s v M dz2
cw
ð15Þ
Substituting Eq. (14) into Eq. (15), we obtain
^ev ¼
coshðfzÞ
Taking the Laplace and Hankel transforms of Eqs. (4) and (5), we obtain
ð17Þ
!
^ d 1 d^ev 1 dp ^z þ n2 u ¼0 1 2m dz G dz dz2
Using Eqs. (14), (19), (20), (24), and (25), the constants A1, A2, A3, A4, A5, and A6 can be determined from the boundary conditions (7) and (8). Expressions of A1 A6 are included in Appendix. Substituting the constants into Eqs. (14), (19), and (20), we can obtain explicit solutions in the transform domain. The vertical and radial displacement at the top surface of the soil layer in the transform domain are
^ z jz¼0 ¼ u
2gG þ M A1 2GMn
ð26Þ
^ r jz¼0 ¼ u
2gG þ M A2 2GMn
ð27Þ
ð18Þ
i
^ r ¼ ð121mÞM G1 u
z ½A1 2
3.2. Groundwater drawdown induced by a constant pumping well Fig 2 illustrates groundwater being pumped from an aquifer at a constant rate. Assuming that the radius of a well is infinitely small (i.e., a Theis well) [22], the groundwater drawdown in the aquifer is
hðr; tÞ ¼ coshðnzÞ þ A2 sinhðnzÞ
2gcGs n½A3 sin hðfzÞ þ A4 cos hðfzÞ þ A5 sinhðnzÞ þ A6 coshðnzÞ ð19Þ ^z ¼ u
h
1 G
þ ð121mÞM
þ 2gcGs f½A3
i
z ½A1 2
sin hðnzÞ þ A2 coshðnzÞ
Q 4pk2 H2
Z
1
u
ex dx x
ð29Þ
where Q is the pumping rate, k2 is the coefficient of permeability of r2 the aquifer, H2 is the thickness of the aquifer, u ¼ 4at is a parameter in the integral, and a is the conductivity of the aquifer. The Laplace–Hankel transform of Eq. (29) is
^¼ h
cos hðfzÞ þ A4 sinhðfzÞ þ A7 sinhðnzÞ þ A8 coshðnzÞ
ð28Þ
Eqs. (26)–(28) are the solutions in the Laplace–Hankel domain for the consolidation of a compressible poroelastic soil layer induced by general groundwater drawdown in the underlying aquifer.
Substituting Eqs. (14) and (16) into Eqs. (17) and (18), we obtain
h
þ
1 . 2G
^ ¼ A1 sinhðnzÞ þ A2 ½coshðnzÞ coshðfzÞ þ A3 sinhðfzÞ p ð16Þ
! 2 d 1 1 2 ^¼0 ^r n n^ev þ np u 1 2m G dz2
where M2 ¼
2g1 2M
The excess pore water pressure in the soil layer in the transform domain is
A1 A2 A3 A4 sinhðnzÞ coshðnzÞ þ sinhðfzÞ þ M M 2gG 2gG
2
þ 2gcGs n2 ½A3 sinhðfzÞ þ A4 coshðfzÞ þ A7 n coshðnzÞ þ A8 n sinhðnzÞ ð25Þ
where f2 ¼ n2 þ cs. The solution of Eq. (13) is
k
1 ¼ M 2 nz½A1 coshðnzÞ þ A2 sinhðnzÞ þ 2M ½A1 sinhðnzÞ þ A2 coshðnzÞ
.
Taking the Laplace and Hankel transforms of Eq. (12), we obtain
!
2G
Q 1 2pk2 H2 s s=a þ n2
ð30Þ
ð20Þ where A5, A6, A7, and A8 are constants to be determined from boundary conditions. The Laplace–Hankel transform of volumetric strain is
^ev ¼ nu ^r þ
O Soil layer
^z du dz
r
h(r,t) Total head distribution in the aquifer
ð21Þ
Substituting Eqs. (16), (19), and (20) into Eq. (21), we obtain Q
M 1 A1 ¼ nA5 þ nA8
ð22Þ
M 1 A2 ¼ nA6 þ nA7 where M 1 ¼
gþ1 22M
Aquifer
z
H2
ð23Þ
1 2G
.
Fig. 2. Groundwater drawdown induced by a constant pumping well.
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K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
The solutions for groundwater drawdown induced by a constant pumping well can be obtained by substituting Eq. (30) into Eqs. (26)–(28).
-0.06 -0.05
χ =5 χ =10
-0.04
Solutions in the time domain can be achieved by inversion of the Laplace transform and the Hankel transform. Talbot’s algorithm has been used for the numerical inversion of the Laplace transform [23]. Based on the solutions for groundwater drawdown induced by a constant pumping well, we analyzed the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer. The following dimensionless quantities are defined:
u'r
4. Results and discussion
χ =20 χ =∞
T =0.1
-0.03 -0.02
T =0.05
-0.01 0.00 0
2
4
6
8
10
r'
4pk2 H2 G 4pk2 H2 G 2pk2 H2 p u ; u0r ¼ u ; p0 ¼ ¼ ; cw QH z cw QH r cw Q M 2gGk 2gGkt r z T¼ v¼ ; a¼ ; r 0 ¼ ; z0 ¼ : cw a K H H cw H2 u0z
K¼
T =0.2
Fig. 4. The influence of the compressibility of the pore fluid on the radial displacement distribution.
2ð1 þ mÞ G; is the bulk modulus of the elastic solid skeleton: 3ð1 2mÞ
0.2 0.0
4.1. Vertical displacement of the soil layer
-0.2
Fig. 3 shows the influence of fluid compressibility on the vertical displacement distribution of the soil layer for m = 0.4, a = 0.01, T = 0.05, 0.1, 0.2 and v varying from 5 to 1. The vertical displacement is greatest in the center of the layer and becomes smaller along the radial direction. The vertical displacement changes rapidly near the pumping well. The vertical displacement becomes larger as time increases. The less compressible the fluid is, the greater the vertical displacement is.
-0.4
-0.8
χ =5
-1.0
χ =∞
p'
χ =2
-1.2 -1.4 0.01
0.1
1
10
T
Fig. 4 shows the influence of fluid compressibility on the radial displacement distribution of the soil layer for m = 0.4, a = 0.01, T = 0.05, 0.1, 0.2 and v varying from 5 to 1. A negative value indicates the direction opposite the radial direction. The radial displacement reaches the maximum value where r0 1 and then decreases in the radial direction. The radial displacement becomes larger as time increases. The less compressible the fluid is, the greater the radial displacement is. The position of the maximum radial displacement is not influenced by the fluid compressibility.
0.00
Fig. 5. The influence of the compressibility of the pore fluid on the development of excess pore pressure.
4.3. Excess pore water pressure in the soil layer Fig. 5 shows the influence of fluid compressibility on the development of excess pore pressure for r0 = 0, z0 = 1/3 for m = 0.4, a = 0.01, and v varying from 1 to 1. At the early stage of consolidation, the less compressible the fluid is, the greater the rate of the consolidation is. The influence of fluid compressibility disappears gradually with the increase of time. The final excess pore water pressure is not influenced by the fluid compressibility. 5. Conclusions
T =0.05
Analytical solutions have been obtained in the Laplace–Hankel transform domain for the axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown. Based on the explicit solutions for groundwater drawdown induced by a constant pumping well, we investigated the influence of fluid compressibility on the consolidation behavior of the soil layer. The results include the following:
0.10 T =0.1
u'z
χ =1
-1.6 1E-3
4.2. Radial displacement of the soil layer
0.05
-0.6
χ =5
0.15
χ =10 χ =20
T =0.2
0.20
χ =∞
0.25 0.30 0
1
2
3
4
5
r' Fig. 3. The influence of fluid compressibility on the settlement distribution.
(1) The vertical and radial displacement become larger as time increases. The less compressible the fluid is, the greater the vertical and radial displacement are. The position of the maximum radial displacement is not influenced by the fluid compressibility.
K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
(2) At the early stage of consolidation, the less compressible the fluid is, the greater the rate of the consolidation is. The influence of fluid compressibility disappears gradually with the increase of time. The final excess pore water pressure is not influenced by the fluid compressibility. Acknowledgements This research is supported by the National Natural Science Foundation of China (Nos. 51278453 and 51179170) and Doctoral Fund of Ministry of Education of China (No. 20120101110029), and the supports are gratefully acknowledged. Appendix A. Expressions for A1 A8 are shown as follows: n ^ 1 1 A1 ¼ 2cMn cwB h coshðHfÞ coshðHnÞ fðGsg 2cMn2 Þ½1 cosh ðHfÞcosh ðHnÞ þðGsg þ 2cMn2 þ MsÞn tanhðHfÞ tanhðHnÞ 1
þ2GMsgff½1 coshðHnÞcosh ðHfÞM 1 þ Hn½n tanhðHfÞ f tanhðHnÞM 2 g
o
^
A2 ¼ 2cGMsgn cwB h fn coshðHnÞ sinhðHfÞ f coshðHfÞ sinhðHnÞ þ2MM1 coshðHnÞ½n sinhðHfÞ f sinhðHnÞ þ2HnMM2 ½f f coshðHfÞ coshðHnÞ þ n sinhðHfÞ sinhðHnÞg n ^ A3 ¼ 2Gsg cwB h Gsg cMn2 þ cMn½n coshðHfÞ coshðHnÞ f sinhðHfÞ sinhðHnÞ n 2 þ2M cMn2 coshðHnÞ½coshðHfÞ coshðHnÞM1 2MM1 M 2 Gsgcosh ðHnÞ þHMn2 M 2 ½cf coshðHnÞ sinhðHfÞ þ cn coshðHfÞ sinhðHnÞ þ 2GHsgM 2 gg
n 2 B ¼ sinhðHfÞ 2G2 s2 g2 2cMn2 ½2Gsg þ cMðf2 þ n2 Þsinh ðHnÞ o 2 þ8G2 M 2 s2 g2 M 2 ½cosh ðHnÞM 1 þ H2 n2 M 2 þ 2cMfnðGsg cMn2 Þ ½2 sinhðHnÞ coshðHfÞ sinhð2HnÞ þ 8GM2 M2 sgcHfn2 ½coshðHnÞ coshðHfÞ The remaining constants can be determined from A1, A2, A3 and B c 1 c 2gG þ M fA3 A1 ; A6 ¼ nþ A2 ; A7 A4 ¼ A2 ; A5 ¼ 2gGs 2Mn 2gGs 2GMn c 1 2gG þ M c A2 ; A8 ¼ ¼ n A1 fA3 : 2gGs 2Mn 2GMn 2gGs
15
References [1] Galloway DL, Burbey TJ. Review: regional land subsidence accompanying groundwater extraction. Hydrogeol J 2011;19:1459–86. [2] Xue YQ, Zhang Y, Ye SJ, Wu JC, Li QF. Land subsidence in China. Environ Geol 2005;48:713–20. [3] Hu RL, Yue ZQ, Wang LC, Wang SJ. Review on current status and challenging issues of land subsidence in China. Eng Geol 2004;76:65–77. [4] Chai JC, Shen SL, Zhu HH, Zhang XL. Land subsidence due to groundwater drawdown in Shanghai. Geotechnique 2004;54(2):143–7. [5] Chai JC, Shen SL, Zhu HH, Zhang XL. 1D analysis of land subsidence in Shanghai. Lowland Technol Int 2005;7(1):33–41. [6] Li CJ, Tang XM, Ma TH. Land subsidence caused by groundwater exploitation in the Hangzhou–Jiaxing–Huzhou plain, China. Hydrogeol J 2006;14:1652–65. [7] Xu YS, Shen SL, Cai ZY, Zhou GY. The state of land subsidence and prediction approaches due to groundwater withdrawal in China. Nat. Hazards 2008;45:123–35. [8] Luo GY, Pan H, Cao H, Yin XL. Analysis of settlements caused by decompression of confined water. Rock Soil Mech 2004;25(Supp. 2):196–200 (in Chinese). [9] Li J, Helm DC. Using an analytical solution to estimate the subsidence risk caused by ASR applications. Environ Eng Geosci 2001;7(1):67–79. [10] Li J. A nonlinear elastic solution for 1-D subsidence due to aquifer storage and recovery applications. Hydrogeol J 2003;11(6):646–58. [11] Teng CM, Tsai TL, Huang LH. Effects of body force on transient poroelastic consolidation due to groundwater pumping. Environ Geol 2008;54: 1507–16. [12] Tsai TL. Viscosity effect on consolidation of poroelastic soil due to groundwater table depression. Environ Geol 2009;57:1055–64. [13] Liu JC, Lei GG, Mei G X. One-dimensional consolidation of visco-elastic aquitard due to withdrawal of deep-groundwater. J Central South Univ 2012;19(1):282–6. [14] Biot MA. General theory of three-dimensional consolidation. J Appl Phys 1941;12:155–64. [15] Wang HF. Theory of linear poroelasticity with applications to geomechanics and hydrogeology. New Jersey: Princeton University Press; 2000. p. 215– 229. [16] Chen GJ. Consolidation of multilayered half space with anisotropic permeability and compressible constituents. Int J Solids Struct 2004;41(16):4567–86. [17] Carter JP, Booker JR. Analysis of pumping a compressible pore fluid from a saturated elastic half space. Comput Geotech 1987;4(1):21–42. [18] Booker JR, Carter JP. Withdrawal of a compressible pore fluid from a point sink in an isotropic elastic half space with anisotropic permeability. Int J Solids Struct 1987;23(3):369–85. [19] Ai ZY, Wu C, Han J. Transfer matrix solution for three-dimensional consolidation of a multi-layered soil with compressible constituents. Int J Eng Sci 2008;46(11):1111–9. [20] Rani S, Kumar R, Singh SJ. Consolidation of an anisotropic compressible poroelastic clay layer by axisymmetric surface loads. Int J Geomech 2011;11(1):65–71. [21] Sneddon IN. The use of integral transform. New York: McGraw-Hill; 1972. [22] Boonstra J. Well hydraulics and aquifer tests. In: Delleur JW, editor. The handbook of groundwater engineering. Boca Raton: CRC press; 1999. [23] Talbot A. The accurate numerical inversion of Laplace transforms. IMA J Appl Math 1979;23(1):97–120.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown Kang-He Xie a, Da-Zhong Huang a,⇑, Yu-Lin Wang b, Yue-Bao Deng c a
Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China Department of Environment and Civil Engineering, Wuyi University, Wuyi Shan 354300, China c Institute of Geotechnical Engineering, Ningbo University, Ningbo 315211, China b
a r t i c l e
i n f o
Article history: Received 30 June 2013 Received in revised form 19 October 2013 Accepted 20 October 2013 Available online 9 November 2013 Keywords: Axisymmetric consolidation Groundwater drawdown Compressibility of fluid Laplace–Hankel transform Analytical solution
a b s t r a c t Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown was studied based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. Explicit analytical solutions are obtained in the Laplace–Hankel transform domain when groundwater drawdown is induced by a constant pumping well. Based on the solutions, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Land subsidence is a worldwide problem that has caused substantial damage to infrastructure and resulted in great economic losses [1,2]. The problem is more severe in regions with soft soils where groundwater is heavily exploited for industrial and domestic purposes [3]. Two large regions in China that suffer from severe subsidence problems are the Yangtze River Delta and the North China Plain, where thick, soft soil layers exist [3–6]. In each of these regions, as groundwater is pumped from the aquifer, the water level and pore water pressure are reduced, which leads to an increase in effective stress and consolidation of the soil layer and thus results in land subsidence. Because the permeability of an aquifer is usually much greater than that of a soil layer, the consolidation of a soil layer can be analyzed in two steps [7]. The first step is to compute or measure the groundwater drawdown in the aquifer. The second step is to compute the consolidation of the soil layer using the results of the first step as the boundary conditions. Several analytical solutions have been presented for the one-dimensional consolidation of a soil layer due to groundwater drawdown. Luo et al. [8] presented solutions for the one-dimensional consolidation of a soil layer due to constant groundwater drawdown. Li and Helm [9] ⇑ Corresponding author. E-mail address: [email protected] (D.-Z. Huang). 0266-352X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2013.10.004
presented solutions for the one-dimensional consolidation of a soil layer subjected to periodic groundwater fluctuation. Li [10] studied the one-dimensional consolidation of a nonlinear elastic soil layer subjected to periodic groundwater fluctuation. Teng et al. [11] studied the one-dimensional consolidation of a soil layer caused by constant groundwater drawdown with consideration of the body force effect. Tsai [12] studied the one-dimensional soil consolidation caused by constant groundwater drawdown with consideration of the viscosity effect. Liu et al. [13] presented solutions for the one-dimensional consolidation of a visco-elastic soil layer due to withdrawal of groundwater. In one-dimensional consolidation theory, groundwater drawdown is assumed to be widely and uniformly distributed in an aquifer. However, because there are different types of groundwater drawdown that correspond to different pumping conditions used in practice, it is necessary to study the consolidation of soil layers based on Biot’s three-dimensional consolidation theory [14,15]. On the other hand, the degree of saturation of a soil layer is usually not 100% because the soil may contain small bubbles. It has been reported that even a very small amount of gas in a soil dramatically increases the compressibility of the fluid [16]. Some researchers have studied the importance of fluid compressibility in the consolidation problem [16–20]. However, there have been no studies on the consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown.
12
K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
Nomenclature ur uz r, h, z
radial displacement vertical displacement cylindrical coordinates stress component excess pore water pressure volumetric strain strain component shear modulus coefficient of permeability unit weight of water bulk modulus of pore fluid Poisson’s ratio
rij p
ev eij G k
cw M
m
t s n Jn a k2 Q H H2
This paper presents a study of the axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. When groundwater is pumped from an aquifer at a constant rate, explicit analytical solutions are obtained in the Laplace–Hankel transform domain. Based on the solutions for groundwater drawdown induced by a constant pumping well, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer.
k
cw
time variable Laplace transform variable Hankel transform variable nth-order Bessel function of the first kind conductivity of the aquifer coefficient of permeability of the aquifer constant pumping rate thickness of the soil layer thickness of the aquifer
r2 p ¼
@ p ev þ @t M
ð6Þ
where k is the coefficient of permeability of the soil, cw is the unit weight of water, M is the bulk modulus (adjusted for porosity) of the pore fluid, and t is a time variable. 2.2. Boundary conditions
2. Mathematical model
Fig. 1 illustrates a groundwater drawdown in an aquifer. The aquifer is assumed to be rough and rigid, so there is no vertical or radial displacement at the bottom boundary of the soil layer. The upper boundary of the soil layer is permeable and stress-free. The boundary conditions can be expressed as
2.1. Governing equations
z ¼ 0 : rzz ¼ 0; rrz ¼ 0; p ¼ 0:
ð7Þ
It is assumed that the poroelastic layer is homogenous isotropic. The force equilibrium equations (with no body forces) specialized to axisymmetry are
z ¼ H : uz ¼ 0; ur ¼ 0; p ¼ cw hðr; tÞ:
ð8Þ
@ rrr @ rrz rrr rhh þ þ ¼0 @r @z r
3. Solutions
ð1Þ
@ rrz @ rzz rrz þ þ ¼0 @r @z r
ð2Þ
3.1. General solutions
where rrr, rhh, rzz are total normal stress components and rrz is shear stress component. Assuming the solid constituent is incompressible, the constitutive equations take the form
rij ¼ 2G eij þ
m 1 2m
ev dij pdij
r2
1 1 @ ev 1 @p ur þ ¼0 r2 1 2m @r G @r
^f ðn; z; sÞ ¼
Z
2
Z
0
f ðr; z; tÞ ¼
1 2pi
1
f ðr; z; tÞrJn ðnrÞest drdt
ð9Þ
0
Z
cþi1
ci1
Z
1
^f ðn; z; sÞnJ ðnrÞest dnds n
ð10Þ
0
O
h(r,t)
r
ð4Þ Soil layer
1 @ ev 1 @p r2 uz þ ¼0 1 2m @z G @z
1
ð3Þ
where the subscripts i and j can be r, h or z, G is the shear modulus, m is the Poisson’s ratio, p is the excess pore water pressure, dij is the Kronecker delta. The Navier-type equations for displacement and p are obtained by substituting Eq. (3) into Eqs. (1) and (2):
Laplace and Hankel transforms were employed to obtain the solutions. The nth-order Laplace–Hankel transform of f(r, z, t) and its inversion may be found for example in Sneddon [21]:
Total head distribution in the aquifer
H
ð5Þ
2
@ 1 @ @ where r2 ¼ @r 2 þ r @r þ @z2 , ur and uz are the radial and vertical z r displacement components, respectively, ev ¼ @u þ urr þ @u , is the vol@r @z umetric strain. It is assumed that the solid constituent is incompressible but the fluid constituent is compressible. Combined with Darcy’s law, the pore fluid mass conservation equation is given by
Aquifer z Fig. 1. Consolidation of a soil layer induced by groundwater drawdown in an aquifer.
13
K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
where p and n are the parameters of the Laplace and Hankel transforms, respectively, and Jn() denotes the nth-order Bessel function of the first kind. From Eqs. (4) and (5), we obtain
r2 p ¼ 2gGr2 ev
ð11Þ
Using Eq. (3), the stress components in the transform domain can be obtained as follows: r^ rz 2G
1 ¼ 2M ½A1 coshðnzÞ þ A2 sinhðnzÞ þ nA5 coshðnzÞ þ nA6 sinhðnzÞ
M 2 nz½A1 sinhðnzÞ þ A2 coshðnzÞ 2gcGs nf½A3 coshðfzÞ þ A4 sinhðfzÞ
1m where g ¼ 12 m. Eliminating ev from Eqs. (6) and (11), we obtain
@ c r2 r2 p ¼ 0 @t where c ¼ c
2gGkM
w ðMþ2gGÞ
ð24Þ r^ zz
ð12Þ
2
d f2 dz2
!
2
d ^¼0 n2 p dz2
^ ¼ A1 sinhðnzÞ þ A2 coshðnzÞ þ A3 sinhðfzÞ þ A4 coshðfzÞ p
ð13Þ
ð14Þ
where A1, A2, A3, and A4 are constants to be determined from the boundary conditions. Taking the Laplace and Hankel transforms of Eq. (6), we obtain
! 2 b p d 2 b b n e þ p ¼ s v M dz2
cw
ð15Þ
Substituting Eq. (14) into Eq. (15), we obtain
^ev ¼
coshðfzÞ
Taking the Laplace and Hankel transforms of Eqs. (4) and (5), we obtain
ð17Þ
!
^ d 1 d^ev 1 dp ^z þ n2 u ¼0 1 2m dz G dz dz2
Using Eqs. (14), (19), (20), (24), and (25), the constants A1, A2, A3, A4, A5, and A6 can be determined from the boundary conditions (7) and (8). Expressions of A1 A6 are included in Appendix. Substituting the constants into Eqs. (14), (19), and (20), we can obtain explicit solutions in the transform domain. The vertical and radial displacement at the top surface of the soil layer in the transform domain are
^ z jz¼0 ¼ u
2gG þ M A1 2GMn
ð26Þ
^ r jz¼0 ¼ u
2gG þ M A2 2GMn
ð27Þ
ð18Þ
i
^ r ¼ ð121mÞM G1 u
z ½A1 2
3.2. Groundwater drawdown induced by a constant pumping well Fig 2 illustrates groundwater being pumped from an aquifer at a constant rate. Assuming that the radius of a well is infinitely small (i.e., a Theis well) [22], the groundwater drawdown in the aquifer is
hðr; tÞ ¼ coshðnzÞ þ A2 sinhðnzÞ
2gcGs n½A3 sin hðfzÞ þ A4 cos hðfzÞ þ A5 sinhðnzÞ þ A6 coshðnzÞ ð19Þ ^z ¼ u
h
1 G
þ ð121mÞM
þ 2gcGs f½A3
i
z ½A1 2
sin hðnzÞ þ A2 coshðnzÞ
Q 4pk2 H2
Z
1
u
ex dx x
ð29Þ
where Q is the pumping rate, k2 is the coefficient of permeability of r2 the aquifer, H2 is the thickness of the aquifer, u ¼ 4at is a parameter in the integral, and a is the conductivity of the aquifer. The Laplace–Hankel transform of Eq. (29) is
^¼ h
cos hðfzÞ þ A4 sinhðfzÞ þ A7 sinhðnzÞ þ A8 coshðnzÞ
ð28Þ
Eqs. (26)–(28) are the solutions in the Laplace–Hankel domain for the consolidation of a compressible poroelastic soil layer induced by general groundwater drawdown in the underlying aquifer.
Substituting Eqs. (14) and (16) into Eqs. (17) and (18), we obtain
h
þ
1 . 2G
^ ¼ A1 sinhðnzÞ þ A2 ½coshðnzÞ coshðfzÞ þ A3 sinhðfzÞ p ð16Þ
! 2 d 1 1 2 ^¼0 ^r n n^ev þ np u 1 2m G dz2
where M2 ¼
2g1 2M
The excess pore water pressure in the soil layer in the transform domain is
A1 A2 A3 A4 sinhðnzÞ coshðnzÞ þ sinhðfzÞ þ M M 2gG 2gG
2
þ 2gcGs n2 ½A3 sinhðfzÞ þ A4 coshðfzÞ þ A7 n coshðnzÞ þ A8 n sinhðnzÞ ð25Þ
where f2 ¼ n2 þ cs. The solution of Eq. (13) is
k
1 ¼ M 2 nz½A1 coshðnzÞ þ A2 sinhðnzÞ þ 2M ½A1 sinhðnzÞ þ A2 coshðnzÞ
.
Taking the Laplace and Hankel transforms of Eq. (12), we obtain
!
2G
Q 1 2pk2 H2 s s=a þ n2
ð30Þ
ð20Þ where A5, A6, A7, and A8 are constants to be determined from boundary conditions. The Laplace–Hankel transform of volumetric strain is
^ev ¼ nu ^r þ
O Soil layer
^z du dz
r
h(r,t) Total head distribution in the aquifer
ð21Þ
Substituting Eqs. (16), (19), and (20) into Eq. (21), we obtain Q
M 1 A1 ¼ nA5 þ nA8
ð22Þ
M 1 A2 ¼ nA6 þ nA7 where M 1 ¼
gþ1 22M
Aquifer
z
H2
ð23Þ
1 2G
.
Fig. 2. Groundwater drawdown induced by a constant pumping well.
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K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
The solutions for groundwater drawdown induced by a constant pumping well can be obtained by substituting Eq. (30) into Eqs. (26)–(28).
-0.06 -0.05
χ =5 χ =10
-0.04
Solutions in the time domain can be achieved by inversion of the Laplace transform and the Hankel transform. Talbot’s algorithm has been used for the numerical inversion of the Laplace transform [23]. Based on the solutions for groundwater drawdown induced by a constant pumping well, we analyzed the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer. The following dimensionless quantities are defined:
u'r
4. Results and discussion
χ =20 χ =∞
T =0.1
-0.03 -0.02
T =0.05
-0.01 0.00 0
2
4
6
8
10
r'
4pk2 H2 G 4pk2 H2 G 2pk2 H2 p u ; u0r ¼ u ; p0 ¼ ¼ ; cw QH z cw QH r cw Q M 2gGk 2gGkt r z T¼ v¼ ; a¼ ; r 0 ¼ ; z0 ¼ : cw a K H H cw H2 u0z
K¼
T =0.2
Fig. 4. The influence of the compressibility of the pore fluid on the radial displacement distribution.
2ð1 þ mÞ G; is the bulk modulus of the elastic solid skeleton: 3ð1 2mÞ
0.2 0.0
4.1. Vertical displacement of the soil layer
-0.2
Fig. 3 shows the influence of fluid compressibility on the vertical displacement distribution of the soil layer for m = 0.4, a = 0.01, T = 0.05, 0.1, 0.2 and v varying from 5 to 1. The vertical displacement is greatest in the center of the layer and becomes smaller along the radial direction. The vertical displacement changes rapidly near the pumping well. The vertical displacement becomes larger as time increases. The less compressible the fluid is, the greater the vertical displacement is.
-0.4
-0.8
χ =5
-1.0
χ =∞
p'
χ =2
-1.2 -1.4 0.01
0.1
1
10
T
Fig. 4 shows the influence of fluid compressibility on the radial displacement distribution of the soil layer for m = 0.4, a = 0.01, T = 0.05, 0.1, 0.2 and v varying from 5 to 1. A negative value indicates the direction opposite the radial direction. The radial displacement reaches the maximum value where r0 1 and then decreases in the radial direction. The radial displacement becomes larger as time increases. The less compressible the fluid is, the greater the radial displacement is. The position of the maximum radial displacement is not influenced by the fluid compressibility.
0.00
Fig. 5. The influence of the compressibility of the pore fluid on the development of excess pore pressure.
4.3. Excess pore water pressure in the soil layer Fig. 5 shows the influence of fluid compressibility on the development of excess pore pressure for r0 = 0, z0 = 1/3 for m = 0.4, a = 0.01, and v varying from 1 to 1. At the early stage of consolidation, the less compressible the fluid is, the greater the rate of the consolidation is. The influence of fluid compressibility disappears gradually with the increase of time. The final excess pore water pressure is not influenced by the fluid compressibility. 5. Conclusions
T =0.05
Analytical solutions have been obtained in the Laplace–Hankel transform domain for the axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown. Based on the explicit solutions for groundwater drawdown induced by a constant pumping well, we investigated the influence of fluid compressibility on the consolidation behavior of the soil layer. The results include the following:
0.10 T =0.1
u'z
χ =1
-1.6 1E-3
4.2. Radial displacement of the soil layer
0.05
-0.6
χ =5
0.15
χ =10 χ =20
T =0.2
0.20
χ =∞
0.25 0.30 0
1
2
3
4
5
r' Fig. 3. The influence of fluid compressibility on the settlement distribution.
(1) The vertical and radial displacement become larger as time increases. The less compressible the fluid is, the greater the vertical and radial displacement are. The position of the maximum radial displacement is not influenced by the fluid compressibility.
K.-H. Xie et al. / Computers and Geotechnics 56 (2014) 11–15
(2) At the early stage of consolidation, the less compressible the fluid is, the greater the rate of the consolidation is. The influence of fluid compressibility disappears gradually with the increase of time. The final excess pore water pressure is not influenced by the fluid compressibility. Acknowledgements This research is supported by the National Natural Science Foundation of China (Nos. 51278453 and 51179170) and Doctoral Fund of Ministry of Education of China (No. 20120101110029), and the supports are gratefully acknowledged. Appendix A. Expressions for A1 A8 are shown as follows: n ^ 1 1 A1 ¼ 2cMn cwB h coshðHfÞ coshðHnÞ fðGsg 2cMn2 Þ½1 cosh ðHfÞcosh ðHnÞ þðGsg þ 2cMn2 þ MsÞn tanhðHfÞ tanhðHnÞ 1
þ2GMsgff½1 coshðHnÞcosh ðHfÞM 1 þ Hn½n tanhðHfÞ f tanhðHnÞM 2 g
o
^
A2 ¼ 2cGMsgn cwB h fn coshðHnÞ sinhðHfÞ f coshðHfÞ sinhðHnÞ þ2MM1 coshðHnÞ½n sinhðHfÞ f sinhðHnÞ þ2HnMM2 ½f f coshðHfÞ coshðHnÞ þ n sinhðHfÞ sinhðHnÞg n ^ A3 ¼ 2Gsg cwB h Gsg cMn2 þ cMn½n coshðHfÞ coshðHnÞ f sinhðHfÞ sinhðHnÞ n 2 þ2M cMn2 coshðHnÞ½coshðHfÞ coshðHnÞM1 2MM1 M 2 Gsgcosh ðHnÞ þHMn2 M 2 ½cf coshðHnÞ sinhðHfÞ þ cn coshðHfÞ sinhðHnÞ þ 2GHsgM 2 gg
n 2 B ¼ sinhðHfÞ 2G2 s2 g2 2cMn2 ½2Gsg þ cMðf2 þ n2 Þsinh ðHnÞ o 2 þ8G2 M 2 s2 g2 M 2 ½cosh ðHnÞM 1 þ H2 n2 M 2 þ 2cMfnðGsg cMn2 Þ ½2 sinhðHnÞ coshðHfÞ sinhð2HnÞ þ 8GM2 M2 sgcHfn2 ½coshðHnÞ coshðHfÞ The remaining constants can be determined from A1, A2, A3 and B c 1 c 2gG þ M fA3 A1 ; A6 ¼ nþ A2 ; A7 A4 ¼ A2 ; A5 ¼ 2gGs 2Mn 2gGs 2GMn c 1 2gG þ M c A2 ; A8 ¼ ¼ n A1 fA3 : 2gGs 2Mn 2GMn 2gGs
15
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