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Soils and Foundations 62 (2022) 101181
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Technical Paper
A simplified method for evaluating temperature effect on the behavior
of layered soil with a time-varying cylindrical heat source
Lujun Wang
Center for Hypergravity Experimental and Interdisciplinary Research, MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering,
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
Received 23 July 2021; received in revised form 2 June 2022; accepted 9 June 2022
Available online 1 July 2022
Abstract
The engineering application of cylindrical heat source usually influences the characteristics of the surrounding soil and induces significant coupling responses of temperature, seepage and stress fields. This paper presents a numerical investigation into the thermohydraulic-mechanical (THM) coupling behavior of layered soils surrounding a cylindrical heat source by combining the analytical layer
element method (ALEM) and the finite element method (FEM). The FEM is utilized to analyze the soil-cylindrical heat source interaction and the ALEM is used to obtain the fundamental solution of the layered soil. Then the THM coupling solution between the heat
source and the soil is obtained by the simplified FEM-ALEM coupling method. Detailed validation against experimental result, and verifications against semi-analytical solution and numerical results by Comsol Multiphysics are performed to confirm the robustness of the
present method, followed by extensive parametric studies to examine the effects of decaying half-life and type of the time-varying heat
source, and the soil layered characteristics on the behavior of soils. It’s evident that the present method is accessible and can be implemented via common programming language, e.g. Fortran.
Ó 2022 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Temperature effect; Layered soil; Simplified method; Consolidation; Time-varying load; Heat source
1. Introduction
Shallow geothermal energy is one of the most promising
options in addressing the energy shortage problem and has
become the fastest growing clean energy resource over the
world in recent years, contributing to the ever-increasing
number of ground source heat pump (GSHP) systems
(Brandl, 2006; Laloui et al., 2006; Faizal et al., 2019;
Loveridge et al., 2020). During the service period, the temperature variation of GSHP usually leads to various processes and changes in soil–water system including the
interaction between the thermal, hydraulic and mechanical
fields in the surrounding soils. The temperature gradient
Peer review under responsibility of The Japanese Geotechnical Society.
E-mail address: lujunwang@zju.edu.cn
and pore pressure gradient, in these engineering applications, cause the movement of pore water and conduction
of heat energy within the soils and then affect the engineering behavior of soils (Selvadurai and Suvorov, 2014;
Shahrokhabadi et al., 2020; Song et al., 2019; Wang and
Wang, 2020; Zagorščak et al., 2017).
Generally, a GSHP system can be taken as a special
form of heat source or sink for the ground. For the stable
temperature and high specific heat of the surrounding soils,
the energy efficiency of GSHP system is higher than that of
traditional air source heat pump (Self et al., 2013; Pan
et al., 2020). Ground heat exchanger (GHE) is one of the
key parts in GSHP system, whose thermal behavior and
heat transfer performance are significant to the operation
of GSHP system (Pan et al., 2020), and can be treated as
a vertical cylindrical heat source (VCHS) (Faizal et al.,
https://doi.org/10.1016/j.sandf.2022.101181
0038-0806/Ó 2022 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
L. Wang
Soils and Foundations 62 (2022) 101181
Nomenclature
c
C
E
E
G
K
n
p
s
t
T
u
uj
permeability coefficient (m/s)
specific heat capacity (J/kg°C)
Young’s modulus (kPa)
constitutive tensor depending on E and l
elastic shear modulus (kPa)
heat conduction coefficient (W/m°C)
Porosity of soil
pore fluid pressure (kPa)
Laplace transform parameter with respect to t
time (s)
temperature increment (°C)
solid desplacement (m)
displacement in j-th direction (m), j = r, z
a
af
l
j
q
f
n
e
r
cw
linear thermal expansion coefficient of solid matrix (/°C)
linear thermal expansion coefficient of pore fluid
(/°C)
Poisson’s ratio
thermal diffusivity coefficient (m2/s)
mass density (kg/m3)
Laplace transform parameter with respect to z
Hankel transform parameter with respect to r
strain stensor
total stress tensor (kPa)
pore fluid unit weight (kN/m3)
oped, which is usually different from that in engineering
practice.
Up to now, several methods or theories, such as the
finite layer method (FLM) (Small and Booker, 1986; Mei
et al., 2004; Bao et al., 2015) and the transfer matrix
method (TMM) (Pan, 1999; Ai et al., 2002; Zheng et al.,
2021), have been generally developed to investigate the
behavior of layered media. It is worth mentioning that
the stiffness matrix in FLM or TMM usually contains positive exponential functions, leading to the problem of computation overflow. To effectively avoid the computation
overflow problem, a newly developed method named the
analytical layer element method (ALEM) (Ai and Hu,
2015; Wang and Ai, 2018), whose matrices include none
of positive exponential function, is employed to obtain
the fundamental solution for the THM coupling problems
of layered soils. The ALEM has been utilized in investigating the response of layered thermoelastic media (Wang and
Ai, 2018) and poroelastic media (Ai and Hu, 2015; Wang
et al., 2020). The classical numerical methods, e.g. FEM
(Rotta Loria et al., 2015; Sutman et al., 2019) and the
boundary element method (BEM) (Smith and Booker,
1996; EI-Zein, 2006), are able to implement the THM coupling behaviors of layered soils, large requirement of computational storage is needed and the solution is highly
dependent on the discrete approach.
In this paper, a simplified method is developed by coupling the ALEM and FEM to evaluate the temperature
effects on the behavior of layered and normally consolidated soft soil subjected to a time-varying GHE. FEM is
employed to analyze the cylindrical heat source with
time-varying power output, and the ALEM (Ai and
Wang, 2015) is used to obtain the fundamental solution
of layered soil, then the THM coupling solution between
the heat source and soil is acquired by coupling FEM
and ALEM. Applicability and robustness of the simplified
coupling method are confirmed through extensive comparisons with available experimental data, semi-analytical
2019; Wang et al., 2019). Regarding the complex properties
of soils and loading characteristics of the vertical cylindrical heat source, distinctly different behavior of the surrounding soil can be expected, which greatly affects the
soil-VCHS interaction. This is required to comprehensive
understand the thermal response of the soil-VCHS system.
Thermo-mechanical soil–VCHS interaction has been investigated by different means, namely, numerical approach
(e.g. Dupray et al., 2014; Suryatriyastuti et al., 2014; Di
Donna and Laloui, 2015; Ozudogru et al., 2015;
Zagorščak et al., 2017; Sutman et al., 2019; Maiorano
et al., 2019; Moradshahi et al., 2020; Moradshahi et al.,
2021), analytical method (e.g. Loveridge et al., 2015;
Chen and McCartney, 2016; Pasten and Santamarina,
2014; Olgun et al., 2014), field test (e.g. Kalantidou et al.,
2012; Murphy and McCartney, 2014; Kramer et al.,
2015; Murphy and McCartney, 2015; Faizal et al., 2019),
and centrifuge test (e.g. McCartney and Rosenberg, 2011;
Stewart and McCartney, 2014; Goode III and
McCartney, 2015; Ng et al., 2015; Rotta Loria et al.,
2015; Wang et al., 2015).
It is worth noting that the previous studies assume the
soil is homogeneous media. However, due to long term
geological sedimentation processes, natural soils are normally distributed in layers and necessary to take the layered
characteristics into consideration. Thus, it is necessary to
extend the research scope from homogeneous soil to layered ones. Therefore, several studies for examining the
behavior of layered soil surrounding GHE are developed
by experimental studie (Guo et al., 2018; Li et al., 2019;
Pan et al., 2020), numerical method (Florides et al., 2013;
Luo et al., 2014) and analytical method (Erol and
Francois, 2018; Pan et al., 2020; Wang et al., 2015; Zhou
et al., 2016; Wang, 2022). In engineering practice, the temperature of GSHP is usually periodical variation with time
(Laloui et al., 2006; Selvadurai and Suvorov, 2014;
Shahrokhabadi et al., 2020). Several heat source models,
treating the source intensity as constant, have been devel2
L. Wang
Soils and Foundations 62 (2022) 101181
div r ¼ 0
solutions and numerical results using Comsol Multiphysics. Extensive parametric studies are performed to
investigate the effects of temperature increment, heat
source’s type and layered characteristic on the soil behavior. The main innovations of this paper can be summarized
as follows: (1) the ALEM imposes no restriction on the
layer thickness and no limitation on the number of layers,
and avoids the exponential overflow and computational
instability caused by ill-conditioned matrices; (2) temperature and excess pore pressure responses of homogeneous
and multilayered soil surrounding a vertical GHE are compared; (3) the time-dependent behavior of layered soils subjected to a time-varying heat source is explored; (4) a
generalized cylindrical heat source model, which relates
to solid cylindrical heat source in energy piles, tubular
and hollow cylindrical heat source in GSHP systems, is
developed to study the responses of soils in different engineering applications.
ð1Þ
where div denotes the divergence.
According to the generalized thermoelastic Hooke’s law
and the effective stress principle, the constitutive equations
can be written as (Biot, 1956; Laloui, et al., 2006).
div r ¼ div ðE : eÞ bgradT gradp
ð2Þ
where grad represents the gradient and e ¼ grad u,
b ¼ 2Gagð1 þ lÞ, here g ¼ 1=ð1 2lÞ and G ¼ E=2ð1 þ lÞ.
Considering Fourier’s law and Darcy’s law, the total
heat flow QT and the pore water flow Qf along z- direction
are given as.
Z t
@T
dt
ð3aÞ
K
QT ¼ @z
0
Z t
c@p
Qf ¼ dt
ð3bÞ
0 cw @z
2. Methodology
Considering Fourier’s law and energy balance equation,
one can obtain the heat conduction equation as.
2.1. Fundamental solution of layered soils with a heat source
@T
¼ divðjgradT Þ
@t
ð4Þ
where j ¼ K=qC.
According to mass balance equation and Darcy’s law,
the continuity equation of seepage is derived as (Ai and
Wang, 2018).
@ev
@T
c
¼ div
þ au
gradp
ð5Þ
@t
cw
@t
Illustrated in Fig. 1 is a cylindrical source embedded in a
multi-layered saturated soil system. The heat source
changes temperature field by transferring the heat to the
surrounding soil and subsequently influences the excess
pore pressure (EPP) dissipation and the soil deformation.
To evaluate the temperature effect of the heat source on
the behavior of layered soil, ALEM is firstly employed to
obtain the soil fundamental solution, and then based on
the principle of FEM, a simplified method by coupling
the ALEM and FEM is developed to solve this problem.
For saturated normally consolidated homogeneous isotropic soft soils, the equilibrium equation for the thermal
consolidation problem ignoring the body force in the cylindrical coordinate system is (Biot, 1956; Booker and
Savvidou, 1985).
r
z
þ urr þ @u
.
where au ¼ 3að1 nÞ þ 3naf and ev ¼ @u
@r
@z
To solve the previous partial differential equations
(PDEs), the integral transform approach is utilized to convert them into ordinary differential equations (ODEs). The
mth-order Laplace-Hankel transform of f ðr; z; tÞ and its
inversion are (Sneddon, 1972).
Z þ1
f m ðn; z; tÞ ¼
f ðr; z; tÞJ m ðnrÞrdr
ð6aÞ
0
Z
þ1
f m ðn; z; sÞ ¼
est f m ðn; z; tÞds
ð6bÞ
0
where the overbars ‘‘–” and ‘‘” are hereafter used to
denote Hankel transform (HT) and Laplace transform
(LT) of a given function, respectively.
Taking the application of LT and HT to Eqs. (1)–(5),
the PDEs are transformed into ODEs, one obtains (Ai
and Wang, 2015).
ð7Þ
K ðn; z; sÞ ¼ ½ A1 ðn; z; sÞ A2 ðn; z; sÞ Rðn; 0; sÞ
where
"
#T
K ðn; f; sÞ ¼ ur ðn; f; sÞ; uz ðn; f; sÞ; p ðn; f; sÞ; T ðn; f; sÞ
"
Fig. 1. A layered soil system with a cylindrical heat source in the
cylindrical coordinate system.
,
#T
Rðn; 0; sÞ ¼ ur ðn; 0; sÞ; uz ðn; 0; sÞ; p ðn; 0; sÞ; T ðn; 0; sÞ; rrz ðn; 0; sÞ; rz ðn; 0; sÞ; Qf ðn; 0; sÞ; QT ðn; 0; sÞ
3
,
L. Wang
2
where A1 ðn; z; sÞ and A2 ðn; z; sÞ refer to matrices of order
4 4.
From Eq. (7), the following expression can be obtained.
3
2
6 K ðn; 0; sÞ 7
5¼
4
K ðn; z; sÞ
I44
044
A1 ðn; z; sÞ A2 ðn; z; sÞ
The
2
here 044 and I44 , respectively, denote 0 matrix and the
unit matrix of order 4 4.
Taking the application of HT and LT to Eqs. (2) and
(3), one obtains.
3
6 C ðn; 0; sÞ 7
5¼
4 C ðn; z; sÞ
044
B1 ðn; z; sÞ
"
3
6 K ðn; 0; sÞ 7
4
5
K ðn; z; sÞ
and
C ðn; z; sÞ
I44
Rðn; 0; sÞ ¼ Nðn; z; sÞ Rðn; 0; sÞ
B2 ðn; z; sÞ
between
6 C ðn; 0; sÞ 7
6 K ðn; 0; sÞ 7
4 5 ¼ Uðn; z; sÞ 4 5
ð10Þ
K ðn; z; sÞ
1
where Uðn; z; sÞ ¼ Nðn; z; sÞ Mðn; z; sÞ is a 8 8 stiffness
matrix establishing the relationships between the generalized stresses and displacements in the transform domain
of a single layer, as shown in Fig. 2. The elements of
Uðn; z; sÞ are listed in Appendix II and detailed derivation
can refer to the work by Ai and Wang (2015).
ð9aÞ
here
relationship
3
6 C ðn; 0; sÞ 7
4 5 linked by the coefficients vector Rðn; 0; sÞ
C ðn; z; sÞ
can be derived as.
2 3
2
3
Rðn; 0; sÞ ¼ Mðn; z; sÞ Rðn; 0; sÞ
ð8Þ
2
Soils and Foundations 62 (2022) 101181
#T
C ðn; z; sÞ ¼ rrz ðn; z; sÞ; rz ðn; z; sÞ; Qf ðn; z; sÞ; QT ðn; z; sÞ
2.2. Simplified method for layered soil
ð9bÞ
where B1 ðn; z; sÞ and B2 ðn; z; sÞ denote matrices of order
4 4.
A layered soil system, consisting of n layers with different material properties and thicknesses, embedded with a
cylindrical source is illustrated in Fig. 3. The thickness of
the ith layer is hi ¼ H i H i1 , here H i and H i1 denote
the depths from the surface to the bottom and top of the
ith layer. Applying Eq. (10) to the ith layer produces.
2 3
2
3
6 C ðn; H i1 ; sÞ 7
6 K ðn; H i1 ; sÞ 7
4 5 ¼ UðiÞ 4 5
C ðn; H i ; sÞ
ð11Þ
K ðn; H i ; sÞ
here UðiÞ ¼ Uðn; hi ; sÞ is the element of the ith layer.
As illustrated in Fig. 3(a) and (b), the length and radius
of the source are L and r0 , respectively. The total heat flow
Fig. 2. The generalized stresses and displacements for a single soil layer.
Fig. 3. Diagram of a cylindrical heat source buried in layered soil.
4
L. Wang
of the heat source is QT 0 and it is uniformly distributed
along the length. The cylindrical heat source is divided into
N 0 subsegments of micro heat source along the depth and
each micro heat source is replaced by a uniform micro
plane heat source in the middle of the subsegment whose
heat flow is qzm ðtÞ. Assuming the length of the micro heat
source is Dl yields.
L¼
N0
X
Dl
where P ðn; H i ; sÞ is the general micro external load vector in LT and HT domain. The detailed expressions for dif
ferent types of P ðn; H i ; sÞ are provided in Appendix I.
Considering the continuity conditions of the soil layers’
interfaces and merging the analytical layer elements, the
relationship between generalized stresses and displacements for the behavior of multilayered soil embedded with
a cylindrical heat source can be established. For the case
where the heat source is embedded in an arbitrary depth,
as shown in Fig. 3(a), the global stiffness matrix is assembled as.
ð12aÞ
m¼1
QT 0 ¼ pr20
N0 Z
X
m¼1
mDl
qz ðtÞdz ¼
N0
X
qzm ðtÞ
Soils and Foundations 62 (2022) 101181
ð12bÞ
m¼1
ðm1ÞDl
Eq. (12b) presents the expression of the total heat flow
of a cylindrical heat source. For a hollow heat source
whose external and internal radii are r1 and r2 respectively,
the expression of the total heat flow is.
N 0 Z mDl Z r2
N0
X
X
QT 0 ¼ 2pr
qz ðtÞdrdz ¼
qzm ðtÞ
ð12cÞ
m¼1
ðm1ÞDl
r1
m¼1
ð15aÞ
here qz ðtÞ denotes the density of heat flow along the depth.
Supposing that the surface is free and the bottom at
z ? 1 is fixed and impermeable, produces.
rzz ðn; 0; sÞ ¼ rrz ðn; 0; sÞ ¼ u ðn; 0; sÞ ¼ 0
For the case where the surface of a cylindrical heat
source is flush with the surface of layered soil, as shown
in Fig. 3(b), the global stiffness matrix is assembled as.
ð13aÞ
ur ðn; z; sÞ ¼ uz ðn; z; sÞ ¼ Qf ðn; z; sÞ ¼ QT ðn; z; sÞ ¼ 0;
as z ! 1
ð13bÞ
It’s worth noting that Eqs. (13a) and (13b) can be
regarded as a general form of boundary conditions. For
other application of practice, the boundary conditions presented here are available by replacing the corresponding
expression in Eqs. (13a) and (13b) with the needed condi
tions, e.g., when the bottom is permeable, Qf ðn; z; sÞ is just
needed replaced by u ðn; z; sÞ ¼ 0 in Eq. (13b).
Considering the continuity condition between any two
adjacent layers without a heat source, the boundary condition between the (j + 1)th layer and jth layer can be given
as.
þ
C n; H þ
;
s
¼
C
n;
H
;
s
;
K
n;
H
;
s
¼
K
n;
H
;
s
j
j
j
j
ð15bÞ
According to Eqs. (15a) and (15b) and considering the
boundary conditions, solution for the behavior of layered
saturated soil with a time-varying heat source in LT and
HT domain can be obtained. To obtain the solutions in
physical domain, the Talbot algorithm (Talbot, 1979) is utilized to inverse LT and the Gauss–Legendre quadrature (Ai
et al., 2002; Wang and Ai, 2018) is employed to inverse HT.
Regarding a heat pump system, the number of the cylindrical heat source is ns , and each heat source is divided into
N s subsegments of micro heat sources. The general load–
displacement relationship of a single heat source is given
as that of Eq. (15), which can be rewritten as.
ð14aÞ
where H þ
j and H j are the depth H j of the ðj þ 1Þth layer
and jth layer; respectively:
At the interface plane z ¼ H i , the continuity conditions
of a micro plane heat source are.
C n; H þ
i ; s ¼ C n; H i ; s þ P ðn; H i ; sÞ;
ð14bÞ
K n; H þ
i ; s ¼ K n; H i ; s
KjT ¼ Uj
1
CjT
ð16Þ
here CjT and KjT denote global vectors of the nodal displacement and interaction force of soil by the jth heat source,
1
respectively. Uj denotes the global flexibility matrix of
5
L. Wang
layered soil surrounding a cylindrical heat source by Pan
et al. (2020). The present solution and the experiment result
provided by Pan et al. (2020) are compared to confirm the
method, as shown in Fig. 4. The soil properties are listed in
Table 2. The surface temperature boundary condition is
23.5 °C, before starting the experiment, the soil was left
to stabilize until its initial average temperature was 23.5 °
C. The length and diameter of the cylindrical heat source
are 200 mm and 6 mm, respectively. The soil depth was a
half length greater than the length of the heat source to
minimize the bottom boundary effect on the temperature
in the short term test. Reasonably good agreement can be
observed between the present results and the experiment
data by Pan et al. (2020). It’s worth noting that the slight
difference between the two results may be owing to the
effects of surface boundary condition and pore water, while
the trends are the same.
soils, which is obtained by ALEM as shown in Eq. (11) and
Appendix II. The global matrix equations of the heat pump
system can be obtained by applying Eq. (16) to each single
heat source in the heat pump system, which is expressed as.
KjT ¼ Uj
1
CjT
Soils and Foundations 62 (2022) 101181
ð17Þ
where KTG and CTG are the global column vectors of order
ns ðN s þ 1Þ 1. Based on Eq. (17), the relationship among
the nodal temperature and nodal displacement of the heat
pump system can be established and then further obtain the
solutions.
It’s known that the discretization of the cylindrical heat
source is a key factor for the accuracy and stability of the
solution. Four values of discretization number N were discussed: N = 10, 16, 20, 50. The relative error is defined as
jT T j
ee ¼ NT 50 100%, here T N is the dimensionless tempera50
ture when the cylindrical heat source is divided into N subsegments. As shown in Table 1, ee is as high as 9% between
N = 10 and N = 50, ee decreases with the increase of N. ee is
less than 0.51% when N = 20. On the consideration of the
computational efficiency and accuracy, N = 20 is chosen in
the following examples.
The main process for the presented method can be summarized as: (1) based on the basic equations for THM coupling problems, the relationship for the state vectors of a
single layer is described by an analytical layer element,
which is obtained using LT and HT; (2) dividing the cylindrical heat source into a series of subsegments of micro
heat sources along the depth, each micro heat source is
replaced by a uniform micro plane heat source in the middle of the subsegment; (3) considering the initial and continuity conditions of the multilayered soil, the solution for
the behavior of layered soil with a cylindrical heat source
is obtained in the transform domain by coupling ALEM
and FEM; (4) the actual solution in the physical domain
is further obtained by inversing LT and HT.
2.3. Validation and verification
(1) Case 1: compared with experiment
A laboratory-scale experiment has been carried out to
investigate the variation of temperature filed of a twoTable 1
The relative error with different discretization number N.
N
r=r0
10
16
20
50
ee
0.42766
9.243
0.40494
3.439
0.39347
0.509
0.39148
0.000
4
T N
ee
0.16983
3.908
0.17365
1.748
0.17598
0.426
0.17673
0.000
10
T N
ee
0.06166
5.416
0.06349
2.602
0.06488
0.469
0.06519
0.000
1
T N
Fig. 4. Comparison of temperature distribution along depth surrounding
a cylindrical source in a two-layered soil.
6
L. Wang
Soils and Foundations 62 (2022) 101181
(2) Case 2: compared with FLM
Another example is presented to compare the present
solution with those by FLM (Savvidou and Booker,
1989). In the example, a solid cylindrical heat source with
a length of h and a radius of r0 is embedded in the homogeneous soil, the soil thickness is 200h and the depth from
the soil surface to the heat source top is 100h, here
h ¼ 20r0 . The soil parameters and normalized variables
are taken as: l ¼ 0:25, s ¼ jt=r0 , T N ¼ 0:02387QT 0 t=r0 ,
QT 0 denotes the total heat flow of the heat source. All the
adopted parameters are the same as those given by
Savvidou and Booker (1989). Fig. 5 illustrates the temperature variation curves at the mid-depth of the source with
time, and a good match between the present results and the
results by FLM (Savvidou and Booker, 1989) can be
observed as well.
Fig. 6. Comparison of EPP along z-direction surrounding a cylindrical
source in a two-layered soil.
(3) Case 3: compared with FEM
Comsol Multiphysics) is divided into 9720 quadrilateral
elements, the temperature change and infiltration condition
at the surface boundary of the model are 0 °C and permeable, the bottom boundary conditions are insulated and
impermeable. Here c1 ¼ c2 , c1 and c2 are the permeability
coefficients of the two layers, the normalized variable pN
2G
is defined as pN ¼ vT N , here v ¼ 12l
½au ð1 lÞ að1 þ lÞ
and T N ¼ 0:02387q0 t=r0 . Reasonable agreement is observed
between the present results and those by FEM (by Comsol
Multiphysics). The present method has advantage in solving more complex coupled THM problems, as it solves
the problems by coupling the ALEM and FEM approach
and has no restriction on the number of layers and the
layer thickness.
Furthermore, an example is given to investigate the
response of a two-layered soil with an buried cylindrical
heat source, as shown in Fig. 6, The length and radius of
the heat source are h and r0 respectively, the total thickness
of the soil is 200h and the depth from the soil surface to the
heat source top is 100h, here h ¼ 20r0 . The FEM model (by
Table 2
Soil properties.
Property
Layer
Layer 1
Layer 2
Mass density (kg/m3)
Heat conduction coefficient (W/m°C)
Heat capacity (J/kg°C)
Thickness (mm)
1835
1.09
1657
100
1989
2.12
1558
200
3. Results and discussions
This section focuses on the soil response in term of EPP
and vertical displacement induced by a time-varying cylindrical heat source. Three scenarios are considered, i.e.,
time-varying heat source, different types of heat source
and a three-layered soil with a cylindrical heat source.
The temperature change and infiltration condition at the
surface boundary of the model are 0 °C and permeable,
the bottom boundary conditions are insulated and impermeable. Values for parameters are provided in Table 3,
the effect of one parameter, the others are determined by
default values. Permeability and heat conduction coefficient of the soil are 0:83 108 m=s and 1w/m°C, respectively. Poisson’s ratio is chosen as 0.25. The thickness of
the soil is 1000m and the length of the cylindrical heat
source is h ¼ 10m, h ¼ 20r0 , the source is in the middle of
the soil. The total initial strength of the heat source is q0 .
The normalized parameters of s, T N and pN are chosen as
those in Section 2.3.
Fig. 5. Comparison of temperature over time surrounding a cylindrical
heat source in a homogeneous soil.
7
L. Wang
Soils and Foundations 62 (2022) 101181
Table 3
Soil properties for parametric study.
Parameters
Normalized half-life of the decaying heat source, t
Heat source types
Number of layers
3.1. Behavior of saturated soil with a time-varying heat
source
Other values
0.01, 0.1
Solid cylindrical
3
0.001, 1, 10, 1
Tubular, hollow cylindrical
1
perature tends to be stable eventually for the source with
constant strength (t ¼ 1), consistent with the observations made in previous works (e.g., Small and Booker,
1986; Ai and Wang, 2017). Fig. 8 illustrates that EPP
achieves its peak value at early time and tends to be stable
finally. It is observed that negative EPP appear at later time
for the case of t ¼ 0:01, which is related to the cooling of
the decaying heat source.
In this subsection, the calculated point is at the middepth of the source, and r=r0 ¼ 1. Fig. 7 shows the time
variation of temperature induced by decaying heat sources
with different half-lives, here a normalized parameter t is
given to reflect the decaying half-life.
t ¼ j=xL2
Default values
ð18Þ
3.2. Effect of heat source types
The temperature increases to the peak before decreasing
to zero in the final stage. The larger the decaying half-life is,
the higher the peak value will achieve. Moreover, the tem-
Figs. 9 and 10 illustrate the time variations of temperature and EPP of the point at r=r0 ¼ 1 and z=h ¼ 0:5
Fig. 7. Time variation of temperature induced by decaying heat sources
with different half-life.
Fig. 9. Time variation of temperature subjected to three types of heat
source.
Fig. 8. Time variation of EPP induced by decaying heat sources with
different half-life.
Fig. 10. Time variation of EPP subjected to three types of heat source.
8
L. Wang
3.3. Behaviors of layered soil with a cylindrical source
induced by three types of heat source, e.g. solid cylindrical,
tubular and hollow cylindrical source. Fig. 9 indicates that
the curves corresponding to the three cases are quite different from each other in the initial stage s 1. When the
radii are the same, the temperature variation caused by
the tubular source is larger than that caused by the solid
or hollow ones. Additionally, the temperature increases
over time and almost unaffected by the heat source type
when s > 1. From Fig. 10, it is observed that the curves
of EPP increase along time firstly and then tend to converge and achieve the maximum. The curves for the three
types are nearly overlapped after EPP achieve the maximum values, and the difference in EPP is quite small. It
is probably attributed to the combined effect of the increase
of EPP induced by the increasing temperature and the dissipation of EPP induced by seepage flow.
In practical engineering, natural soils are mostly layered
media owing to long periods of sedimentation process. The
effect of stratification on the behavior of a typical threelayered soil system with a no-decaying cylindrical source
is discussed here, the thickness relationship of three layers
is h1 : h2 : h3 ¼ 3 : 4 : 3. Owing to long term geological sedimentation processes, soils are normally distributed in layers, various permeability and heat conduction coefficients
in different layers should be considered, the relationships
of permeability and heat conduction coefficient among
the layers are provided in Table 4, other parameters can
refer to Table 3. Four cases are investigated here, in case
1, the soil is homogeneous medium; in case 2, permeability
coefficients are changed, while other parameters are the
same; in case 3, heat conduction coefficients are changed;
in case 4, both permeability coefficients and heat conduction coefficients are changed.
Figs. 11 and 12 illustrate the distributions of temperature increment and EPP along depth in a three-layered soil
surrounding a solid cylindrical heat source at different s,
respectively. Fig. 11 shows that the temperature changes
almost the same in the relatively early stage (s ¼ 0:1) for
Table 4
Parameter relationships among different layers.
Parameters
Case 1
Case 2
Case 3
Case 4
c1:c2:c3
K1:K2:K3
1:1:1
1:1:1
1:3:9
1:1:1
1:1:1
1:1.2:1.5
1:3:9
1:1.2:1.5
Soils and Foundations 62 (2022) 101181
Fig. 11. Distribution of temperature increment along depth at r=r0 ¼ 1 in a three-layered soil.
9
L. Wang
Soils and Foundations 62 (2022) 101181
Fig. 12. Distribution of EPP along depth at r=r0 ¼ 1 in a three-layered soil.
of soil have significant effects both on the variations of temperature and EPP.
different cases, then the differences appear and get larger
with time. The curves of case 1 and case 2 are almost coincident, this is due to that the effects of permeability on temperature changes can almost be neglected. The heat
conduction coefficients relationship in case 3 and case 4
are the same and the corresponding curves are well coincident, which is different from those of case 1 and case 2. It
can be learned that the temperature change differences
would be larger when the heat conductivity difference
between the soil layers is larger. Owing to the various permeability in different layers, Fig. 12 shows sudden changes
of EPP at the interfaces of adjacent layers. It is interesting
that the EPP at layer 1 is obviously larger than that at layers 2 and 3 for cases 2 and 4 for the permeability of layer 1
is the smallest one. With the same permeability, the EPP of
case 2 is slightly larger than that of case 4, and the EPP of
case 1 is slightly larger than that of case 3. A possible reason is that extra pore pressure dissipating induced by the
heat conduction results in changing of EPP. In the early
stages (s ¼ 0:1; 1; 10), the EPP difference caused by the different permeability gradually increases with time, however,
the EPP difference decreases, since the pore water dissipation is gradually in dominant in long term. Consequently,
it can be reasonably to conclude that the layered properties
4. Summary
This paper develops a simplified method to examine the
temperature effect on the behavior of layered soil subjected
to a time-varying cylindrical heat source. Comparisons
with experimental result, semi-analytical solution and
numerical result by Comsol Multiphysics are given to confirm the robustness of the method, followed by extensive
parametric studies to investigate the effects of decaying
half-life and types of the heat source, and the soil layered
characteristics on the soil behavior. The main findings are
drawn as follows.
(1) Under the decaying heat source’s influence, the temperature increases to the peak at first and then
decreases to zero over time. The peak value increases
with the decaying half-life. EPP achieves its peak
value early and then tends to be stable. Due to the
cooling of the decaying heat source, the negative
EPP appears at later time.
10
L. Wang
Soils and Foundations 62 (2022) 101181
Fig. 13. Time-varying heat sources: (a) a constant heat source, (b) a decaying heat source.
Table 5
Expressions for different types of general micro external loads.
P ðn; H i ; sÞ
Constant heat source
Time-varying heat source
Type of heat source
Solid cylindrical source
qm (r = r0)
Tubular soure qm
(r = r0)
Hollow cylindrical source qm
(rext = r1, rint = r2)
h
iT
0Þ
0; 0; 0; qmprJ 10ðnr
ns2
h
iT
J 1 ðnr0 Þ
0; 0; 0; prqm0 nsðsþxÞ
h
iT
0 ðnr 0 Þ
0; 0; 0; qm J2ps
2
h
iT
qm
r2 J 1 ðnr2 Þr1 J 1 ðnr1 Þ
0; 0; 0; pns
2
r2 r2
T
m J 0 ðnr 0 Þ
½0; 0; 0; q2psðsþxÞ
T
qm
1 J 1 ðnr 1 Þ
½0; 0; 0; pnsðsþxÞ
ðr2 J 1 ðnrr22Þr
Þ
r21
2
2
1
coelastic correspondence principle, the long-time
dependent behaviors of clayey soil caused by creep
and consolidation can be described based on the
developed explicit solution.
(2) The temperature change caused by a tubular heat
source is larger than that caused by a solid or a hollow ones when the radii of the heat source are the
same. The temperature increases over time almost
unaffected by the heat source type when s > 1. The
EPP first increases along time and then tends to converge and achieves the maximum. The EPP curves are
nearly overlapped after achieve the maximum.
(3) The layered characteristics show significant influences
on the variations of temperature and EPP of the soil.
A sharp gradient of EPP forms near the interface
owing to the significant difference in permeability of
three layers. The thermal interaction is a significant
process that should be considered as refined calculations of temperature and EPP of GHE installed in
layered soils.
(4) This paper develops a fundamental solution for THM
coupling behaviors of isotropic, poroelastic and layered soils, and several extensions can be improved
in the future. Firstly, the anisotropic mechanical
and thermal characteristics of natural soils can be
considered by replacing the governing equations
(2)–(5) with the corresponding equation of anisotropic soils. Secondly, with the aid of typical viscoelastic
models (e.g., Maxwell, Merchant) and the elastic–vis-
Declaration of Competing Interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the Zhejiang Provincial
Natural
Science
Foundation
of
China
(No.
LY21E080026) and the National Natural Science Foundation of China (No. 52078458).
Appendix I.
The expressions for different types of
P ðn; H i ; sÞ (Fig. 13) are listed in Table 5.
Here qm denotes the strength of the mth heat source,
x ¼ ln2=c, c is the half-life of a decay heat source.
11
L. Wang
Soils and Foundations 62 (2022) 101181
Appendix II.
(a) When j ¼ cðk þ 2GÞ=cw U11 ¼ U55 ¼ 2Gsdna2 b1 ðna1 d 1 d 2 g1 2Gjnd 1 d 5 f 1 d 3 y 4 Þ=‘,
U12 ¼ U21 ¼ U56 ¼ U65 ¼ 2Gn2 ða1 ð2Gsdja2 b1 d 1 ðdd 4 d 5 d 2 f 2 Þ þ nd 2 l3 Þ
,
2Gjnð2dd 5 f 1 g1 þ f 21 g3 þ dd 25 g4 ÞÞ=‘
U13 ¼ U31 ¼ U57 ¼ U75 ¼ 2Gdjna2 ðnðd 1 ðf 1 g1 þ d 5 g4 Þ d 3 ðdd 5 g1 þ f 1 g3 ÞÞ sa1 a2 b1 d 1 f 4 Þ=‘,
U14 ¼ U58 ¼ Gnðd 4 ða1 d 2 ðd2 ng1 g2 þ l2 l4 sx1 y 3 Þ 2Gd2 jnd 5 f 1 g2 2Gdjnd 25 l2 Þþ
,
dð4sza1 a22 x1 y 3 na1 d 22 g1 l1 þ nd 2 ð2Gjd 5 f 1 l1 þ g2 y 4 ÞÞÞ=d‘d 2
U15 ¼ U51 ¼ 4Gsdna1 a2 b1 y 4 =‘,
U16 ¼ U61 ¼ U25 ¼ U52 ¼ 4Gsdn2 a1 a2 b1 ða1 d 2 g1 2Gjd 5 f 1 Þ=‘,
U17 ¼ U71 ¼ U35 ¼ U53 ¼ 4Gdjna1 a2 y 3 =‘,
U18 ¼ U54 ¼ 2Gna2 ðd2 ng2 ð2Gjd 5 f 1 a1 d 2 g1 Þ þ sa1 x1 y 3 ðd 2 zdd 4 Þ þ l2 y 4 Þ=d‘d 2 ,
U22 ¼ U66 ¼ 2Gsna1 a2 b1 ð2Gjnf 1 f 4 þ d 2 ðdnd 1 g1 d 3 l5 ÞÞ=‘,
U23 ¼ U32 ¼ U67 ¼ U76 ¼ 2Gdjna2 ðd 1 y 3 d 3 y 2 Þ=‘,
U24 ¼ U68 ¼ Gnða1 d 22 ðl1 l5 dn2 g1 g2 Þ 2Gjnd 4 f 1 ðdf 1 g2 þ d 5 l2 Þ 4szda1 a22 x1 y 2 þ
,
d 2 ð2Gdjn2 d 5 f 1 g2 þ 2Gjnf 21 l1 þ a1 d 4 ðng1 l2 dg2 l5 þ sx1 y 2 ÞÞÞ=d‘d 2
U26 ¼ U62 ¼ 4Gsna1 a2 b1 ð2Gjnf 21 þ a1 d 2 l5 Þ=‘,
U27 ¼ U72 ¼ U36 ¼ U63 ¼ 4Gdjna1 a2 y 2 =‘,
U28 ¼ U64 ¼ 2Gna2 ð2Gdjnf 21 g2 þ 2Gjnd 5 f 1 l2 þ a1 ðd 2 ðdg2 l5 ng1 l2 Þ þ sx1 y 2 ðzdd 4
,
d 2 ÞÞÞ=d‘d 2
U33 ¼ U77 ¼ djð2Gnðs2 a1 a22 b1 d 1 d 2 d 3 þ jnðdnd 25 g1 f 1 f 2 g1 þ d 5 ðnf 1 g3 f 2 g4 ÞÞÞ þ a1 d 4 y 1 Þ=s‘b1 ,
U34 ¼ U78 ¼ ð8Gs3 zdna21 a32 b1 d 1 d 2 x1 2Gna2 ðd2 jnf 4 g1 g2 þ s3 a21 b1 d 1 d 22 d 4 x1 Þ þ 4szda1 a22 x1 y 1
,
d 2 ðsa1 d 4 x1 y 1 þ 2Gdjnðnf 1 g2 l4 þ l1 y 2 ÞÞ þ 2Gjnd 4 ðd2 d 5 g2 l5 l2 y 3 ÞÞ=2sd‘b1 d 2
U37 ¼ U73 ¼ 2dja1 a2 ð2Gs2 na1 a2 b1 d 1 d 2 þ y 1 Þ=s‘b1 ,
U38 ¼ U74 ¼ a2 ðsa1 x1 ðd 2 zdd 4 Þð2Gs2 na1 a2 b1 d 1 d 2 þ y 1 Þ þ 2Gjnðl2 y 3 d2 g2 y 2 ÞÞ=sd‘b1 d 2 ,
U41 ¼ U42 ¼ U43 ¼ U45 ¼ U46 ¼ U47 ¼ 0, U44 ¼ U88 ¼ Kdd 4 =sd 2 ,
U48 ¼ U84 ¼ 2Kda2 =sd 2 , U81 ¼ U82 ¼ U83 ¼ U85 ¼ U86 ¼ U87 ¼ 0.
where
d2 ¼ n2 þ s=j, b1 ¼ k þ 2G, c1 ¼ c=cw , x1 ¼ b þ b1 au , x2 ¼ b b1 au , k ¼ 2Gl=ð1 2lÞ, a1 ¼ ezn , a2 ¼ ezd , d 1 ¼ 1 a21 ,
d 2 ¼ 1 a22 , d 3 ¼ 1 þ a21 , d 4 ¼ 1 þ a22 , d 5 ¼ a2 d 3 a1 d 4 , f 1 ¼ na1 d 2 da2 d 1 , f 2 ¼ na2 d 1 da1 d 2 , f 3 ¼ d2 a1 d 4 n2 a2 d 3 ,
f 4 ¼ dd 1 d 4 nd 2 d 3 , g1 ¼ sza2 b1 d 1 þ 2Gjnd 5 , g2 ¼ sza1 d 2 x1 þ 2djb1 d 5 au , g3 ¼ sza2 b1 d 3 þ 2Gjf 2 , g4 ¼ szda2 b1 d 3 2Gjnf 1 , g5 ¼ szda1 d 4 x1 2jnb1 f 1 au , g6 ¼ sa1 d 2 x1 þ szda1 d 4 x1 þ 2djb1 f 2 au ,
l1 ¼ sba1 d 2 þ g5 sa1 b1 d 2 au , l2 ¼ ng5 sna1 d 2 x1 2sb1 f 1 au , l3 ¼ dg21 g3 g4 , l4 ¼ sa2 b1 d 1 ng3 , l5 ¼ sda2 b1 d 1 þ ng4 ,
y 1 ¼ s2 da22 b21 d 21 þ n2 l3 ,
y 2 ¼ nf 1 g1 þ d 5 l5 ,
y 3 ¼ dnd 5 g1 f 1 l4 ,
y 4 ¼ 2Gdjnd 25 a1 d 2 l4 ,
‘ ¼ 2Gs2 na21 a2 b1 d 1 d 22 þ
a1 d 2 y 1 2Gjnðdd 5 y 2 þ f 1 y 3 Þ.
(b) When j–cðk þ 2GÞ=cw .
U11 ¼ U55 ¼ 2Gscna2 ðnd 1 l5 d 4 y 6 Þ=‘
U12 ¼ U21 ¼ U56 ¼ U65 ¼ 2Gn2 ð2Gnwf 21 g3 þ a1 b1 d 2 ðsca2 d 1 g3 þ nl1 Þ þ sca2 d 4 l5
þ2Gcwd 7 ðnf 1 g1 þ y 3 ÞÞ=‘
U13 ¼ U31 ¼ U57 ¼ U75 ¼ 2Gcnwa2 ðd 1 y 3 þ d 4 y 5 Þ=‘,
U14 ¼ U58 ¼ 2Gsnðd 6 ð2Gcnwd 7 l6 þ a1 b1 d 2 y 4 Þ þ wa1 a3 f 7 x1 y 5 þ cnd 3 ðg5 l5 þ
g4 y 6 ÞÞ=ðc2 d2 Þw‘a3 d 3
U15 ¼ U51 ¼ 4Gscna1 a2 y 6 =‘, U16 ¼ U61 ¼ U25 ¼ U52 ¼ 4Gscn2 a1 a2 l5 =‘
U17 ¼ U71 ¼ U35 ¼ U53 ¼ 4Gcnwa1 a2 y 5 =‘
U18 ¼ U54 ¼ 4Gsnð2Gcnwd 7 l6 þ a1 ðb1 d 2 y 4 þ wf 6 x1 y 5 ÞÞ=ðd2 c2 Þw‘d 3 ,
U22 ¼ U66 ¼ 2Gsna1 a2 ð2Gnwf 1 f 8 þ b1 d 2 ðcnd 1 g1 d 4 l4 ÞÞ=‘,
U23 ¼ U32 ¼ U67 ¼ U76 ¼ 2Gcnwa2 ðd 4 y 3 þ d 1 y 5 Þ=‘,
U24 ¼ U68 ¼ 2Gsnðd 3 ð2Gcn2 wd 7 f 1 g4 þ 2Gnwf 21 g5 þ a1 b1 d 2 ðsca2 d 1 g5 þ nl3 ÞÞþ
,
d 6 ð2Gnwf 1 l6 a1 b1 d 2 y 1 Þ wa1 a3 f 7 x1 y 3 Þ=ðd2 c2 Þw‘a3 d 3
U26 ¼ U62 ¼ 4Gsna1 a2 ð2Gnwf 21 þ a1 b1 d 2 l4 Þ=‘,
12
L. Wang
Soils and Foundations 62 (2022) 101181
U27 ¼ U72 ¼ U36 ¼ U63 ¼ 4Gcnwa1 a2 y 3 =‘,
U28 ¼ U64 ¼ 4Gsnð2Gnwf 1 l6 a1 ðb1 d 2 y 1 þ wf 6 x1 y 3 ÞÞ=ðc2 d2 Þw‘d 3 ,
U33 ¼ U77 ¼ cc1 ð2Gnwðf 2 y 3 þ nd 7 y 5 Þ þ a1 b1 d 5 q1 Þ=s‘,
U34 ¼ U78 ¼ ð2Gnðsc2 a2 d 1 d 3 d 7 g5 þ cnd 3 ðf 1 ðg4 l2 g1 g5 Þ þ d 7 l3 Þ d 6 ðcd 7 y 1 þ f 1 y 4 ÞÞ
,
þa1 a3 f 7 x1 q1 Þ=ðc2 d2 Þ‘a3 d 3
2
U37 ¼ U73 ¼ 2cwa1 a2 q1 =s‘, U38 ¼ U74 ¼ ð4Gnðcd 7 y 1 þ f 1 y 4 Þ 2a1 f 6 x1 q1 Þ=ðc2 d Þ‘d 3 ,
U41 ¼ U42 ¼ U43 ¼ U45 ¼ U46 ¼ U47 ¼ 0, U44 ¼ U88 ¼ Kdd 6 =sd 3 ,
U48 ¼ U84 ¼ 2Kda3 =sd 3 , U81 ¼ U82 ¼ U83 ¼ U85 ¼ U86 ¼ U87 ¼ 0.
where
c2 ¼ n2 þ s=w, d2 ¼ n2 þ s=j, w ¼ b1 c1 , b1 ¼ k þ 2G, c1 ¼ c=cw , x1 ¼ b þ b1 au ,
a1 ¼ ezn , a2 ¼ ezc , a3 ¼ ezd , d 1 ¼ 1 a21 , d 2 ¼ 1 a22 , d 3 ¼ 1 a23 , d 4 ¼ 1 þ a21 , d 5 ¼ 1 þ a22 , d 6 ¼ 1 þ a23 ,
d 7 ¼ ða2 a1 Þð1 a1 a2 Þ, d 8 ¼ ða3 a2 Þð1 a2 a3 Þ, d 9 ¼ ða3 a1 Þð1 a1 a3 Þ, f 1 ¼ ca2 d 1 na1 d 2 , f 2 ¼ na2 d 1 ca1 d 2 ,
f 3 ¼ da1 d 3 na3 d 1 , f 4 ¼ da3 d 1 na1 d 3 , f 5 ¼ da2 d 3 ca3 d 2 , f 6 ¼ da3 d 2 ca2 d 3 , f 7 ¼ dd 2 d 6 cd 3 d 5 , f 8 ¼ nd 2 d 4 cd 1 d 5 ,
g1 ¼ sza2 d 1 þ 2Gnc1 d 7 ,
g2 ¼ szca2 d 4 þ 2Gnc1 f 1 ,g3 ¼ sza2 d 4 þ 2Gc1 f 2 ,
g4 ¼ ba1 c1 d 8 wa3 d 7 au þ ja2 d 9 au ,
g6 ¼ bna1 c1 f 6 dwa3 f 1 au þ cja2 f 4 au ,
l1 ¼ cg21 g2 g3 ,
l2 ¼ sa2 d 1 ng3 ,
g5 ¼ ba1 c1 f 5 þ wa3 f 2 au þ ja2 f 3 au ,
l3 ¼ cng1 g4 þ g2 g5 , l4 ¼ sca2 d 1 þ ng2 , l5 ¼ 2Gwd 7 f 1 þ a1 b1 d 2 g1 , l6 ¼ df 1 g4 d 7 g6 , y 1 ¼ ng1 g6 dg4 l4 , y 2 ¼ l4 cng3 ,
y 3 ¼ nf 1 g1 d 7 l4 ,
y 4 ¼ cdng1 g4 þ g6 l2 ,
y 5 ¼ cnd 7 g1 þ f 1 l2 ,
y 6 ¼ 2Gcnwd 27 a1 b1 d 2 l2 ,
q1 ¼ n2 l1 þ sa2 d 1 y 2 ,
2
2
‘ ¼ 2Gnwð2cnd 7 f 1 g1 þ f 1 l2 cd 7 l4 Þ þ a1 b1 d 2 q1 .
Faizal, M., Bouazza, B., McCartney, J.S., Haberfield, C., 2019. Effects of
cyclic temperature variations on thermal response of an energy pile
under a residential building. J. Geotech. Geoenviron. Eng., ASCE 145
(10), 04019066.
Florides, G.A., Christodoulides, P., Pouloupatis, P., 2013. Single and
double U-tube ground heat exchangers in multiple-layer substrates.
Appl. Energy 102, 364–373.
Goode III, J.C., McCartney, J.S., 2015. Centrifuge modeling of boundary
restraint effects in energy foundations. J. Geotech. Geoenviron. Eng.,
ASCE 141 (8), 04015034.
Guo, Y., Zhang, G., Liu, S., 2018. Investigation on the thermal response
of full-scale PHC energy pile and ground temperature in multi-layer
strata. Appl. Therm. Eng. 143, 836–848.
Kalantidou, A., Tang, A.M., Pereira, J.-M., Hassen, G., 2012. Preliminary
study on the mechanical behaviour of heat exchanger pile in physical
model. Géotechnique 62 (11), 1047–1051.
Kramer, C.A., Ghasemi-Fare, O., Basu, P., 2015. Laboratory thermal
performance tests on a model heat exchanger pile in sand. Geotech.
Geol. Eng. 33 (2), 253–271.
Laloui, L., Nuth, M., Vulliet, L., 2006. Experimental and numerical
investigations of the behaviour of a heat exchanger pile. Int. J. Numer.
Anal. Meth. Geomech. 30 (8), 763–781.
Li, W.X., Li, X.D., Du, R.Q., Wang, Y., Tu, J.Y., 2019. Experimental
investigations of the heat load effect on heat transfer of ground heat
exchangers in a layered subsurface. Geothermics 77, 75–82.
Loveridge, F., McCartney, J.S., Narsilio, G.A., Sanchez, M., 2020. Energy
geostructures: A review of analysis approaches, in situ testing and
model scale experiments. Geomech. Energy Environ. 22 100173.
Loveridge, F., Olgun, C.G., Brettmann, T., Powrie, W., 2015. The thermal
behaviour of three different auger pressure grouted piles used as heat
exchangers. Geotech. Geol. Eng. 33, 273–289.
Luo, J., Rohn, J., Bayer, M., Priess, A., Xiang, W., 2014. Analysis on
performance of borehole heat exchanger in a layered subsurface. Appl.
Energy 123 (15), 55–65.
Maiorano, R.M.S., Marone, G., Russo, G., Girolamo, L.D., 2019.
Experimental behavior and numerical analysis of energy piles.
Proceedings of the XVII ECSMGE-2019. https://doi.org/10.32075/
17ECSMGE-2019-0819.
Mei, G.X., Yin, J.H., Zai, J.M., Yin, Z.Z., Ding, X.L., Zhu, G.F., Chu, L.
M., 2004. Consolidation analysis of a cross-anisotropic homogeneous
elastic soil using a finite layer numerical method. Int. J. Numer. Anal.
Meth. Geomech. 28 (2), 111–129.
References
Ai, Z.Y., Hu, Y.D., 2015. Multi-dimensional consolidation of layered
poroelastic materials with anisotropic permeability and compressible
fluid and solid constituents. Acta Geotech. 10, 263–273.
Ai, Z.Y., Wang, L.J., 2015. Axisymmetric thermal consolidation of
multilayered porous thermoelastic media due to a heat source. Int. J.
Numer. Anal. Meth. Geomech. 39, 1912–1931.
Ai, Z.Y., Wang, L.J., 2017. Thermal performance of stratified fluid-filled
geomaterials with compressible constituents around a deep buried
decaying heat source. Meccanica 52, 2769–2788.
Ai, Z.Y., Wang, L.J., 2018. Precise solution to 3D coupled thermohydromechanical problems of layered transversely isotropic saturated
porous media. Int. J. Geomech., ASCE 18 (1), 04017121.
Ai, Z.Y., Yue, Z.Q., Tham, L.G., Yang, M., 2002. Extended Sneddon and
Muki solutions for multilayered elastic materials. Int. J. Eng. Sci. 40,
1453–1483.
Bao, H.Y., Hou, X.M., Yang, C., 2015. Finite layer method to calculate
the dynamic compliance of a foundation. 7th International Conference
on Measuring Technology and Mechatronics Automation
(ICMTMA), 2015(1), 496–498..
Biot, M.A., 1956. Thermoelasticity and irreversible thermodynamics. J.
Appl. Phys. 27 (3), 240–253.
Booker, J.R., Savvidou, C., 1985. Consolidation around a point heat
source. Int. J. Numer. Anal. Meth. Geomech. 9, 173–184.
Brandl, H., 2006. Energy foundations and other thermos-active ground
structures. Géotechnique 56 (2), 81–122.
Chen, D., McCartney, J.S., 2016. Calibration parameters for load transfer
analysis of energy piles in uniform soils. Int. J. Geomech., ASCE 17 (7),
04016159.
Dupray, F., Laloui, L., Kazangba, A., 2014. Numerical analysis of
seasonal heat storage in an energy pile foundation. Comput. Geotech.
55 (1), 67–77.
Di Donna, A., Laloui, L., 2015. Numerical analysis of the geotechnical
behaviour of energy piles. Int. J. Numer. Anal. Meth. Geomech. 39 (8),
861–888.
EI-Zein, A., 2006. Laplace boundary element model for the thermoelastic
consolidation of multilayered media. Int. J. Geomech., ASCE 6 (2),
136–140.
Erol, S., Francois, B., 2018. Multilayer analytical model for vertical
ground heat exchanger with groundwater flow. Geothermics 71, 294–
305.
13
L. Wang
McCartney, J.S., Rosenberg, J.E., 2011. Impact of heat exchange on side
shear in thermo-active foundations. Geo-Frontiers 2011, 488–498.
Moradshahi, A., Khosravi, A., McCartney, J.S., Bouazza, A., 2020. Axial
load transfer analyses of energy piles at a rock site. Geotech. Geol.
Eng. 38, 4711–4733.
Moradshahi, A., Faizal, M., Bouazza, A., McCartney, J.S., 2021. Effect of
nearby piles and soil properties on the thermal behaviour of a fieldscale energy pile. Can. Geotech. J. 38, 4711–4733.
Murphy, K.D., McCartney, J.S., 2014. Thermal borehole shear device.
Geotech. Test. J., ASTM 37 (6), 1040–1055.
Murphy, K.D., McCartney, J.S., 2015. Seasonal response of energy
foundations during building operation. Geotech. Geol. Eng. 33, 343–
356.
Ng, C.W.W., Shi, C., Gunawan, A., Laloui, L., Liu, H., 2015. Centrifuge
modelling of heating effects on energy pile performance in saturated
sand. Can. Geotech. J. 52 (8), 1045–1057.
Olgun, C.G., Ozudogru, T.Y., Abdelaziz, S.L., Senol, A., 2014. Long-term
performance of heat exchanger pile groups. Acta Geotech. 10 (5), 553–
569.
Ozudogru, T.Y., Olgun, C.G., Arson, C.F., 2015. Analysis of friction
induced thermo-mechanical stresses on a heat exchanger pile in
isothermal soil. Geotech. Geol. Eng. 33, 357–371.
Pan, A., McCartney, J.S., Lu, L., You, T., 2020. A novel analytical
multilayer cylindrical heat source model for vertical ground heat
exchangers installed in layered ground. Energy 200 117545.
Pasten, C., Santamarina, J., 2014. Thermally induced long-term displacement of thermoactive piles. J. Geotech. Geoenviron. Eng., ASCE 140
(5), 06014003.
Pan, E., 1999. Green’s functions in layered poroelastic half-spaces. Int. J.
Numer. Anal. Meth. Geomech. 23 (13), 1631–1653.
Rotta Loria, A.F., Gunawan, A., Shi, C., Laloui, L., Ng, C.W.W., 2015.
Numerical modelling of energy piles in saturated sand subjected to
thermo-mechanical loads. Geomech. Energy Environ. 1, 1–15.
Savvidou, C., Booker, J.R., 1989. Consolidation around a heat source
buried deep in a porous thermoelastic medium with anisotropic flow
properties. Int. J. Numer. Anal. Meth. Geomech. 13 (1), 75–90.
Self, S.J., Reddy, B.V., Rosen, M.A., 2013. Geothermal heat pump
systems: status review and comparison with other heating options.
Appl. Energy 101, 341–348.
Selvadurai, A.P.S., Suvorov, A.P., 2014. Thermo-poromechanics of a
fluid-filled cavity in a fluid-saturated geomaterial. Proc. R. Soc. A
Math. Phys. Eng. Sci. 470 (2163), 20130634.
Shahrokhabadi, S., Cao, D.T., Vahedifard, F., 2020. Thermal effects on
hydromechanical response of seabed-supporting hydrocarbon pipelines. Int. J. Geomech., ASCE 20 (1), 04019143.
Small, J.C., Booker, J.R., 1986. Behaviour of layered soil or rock
containing a decaying heat source. Int. J. Numer. Anal. Meth.
Geomech. 10 (5), 501–509.
Soils and Foundations 62 (2022) 101181
Smith, D.W., Booker, J.R., 1996. Boundary element analysis of linear
thermoelastic consolidation. Int. J. Numer. Anal. Meth. Geomech. 20
(7), 457–488.
Sneddon, I.N., 1972. The Use of Integral Transform. McGraw-Hill, New
York.
Song, Z., Liang, F.Y., Chen, S.L., 2019. Thermo-osmosis and mechanocaloric couplings on THM responses of porous medium under point
heat source. Comput. Geotech. 112, 93–103.
Stewart, M.A., McCartney, J.S., 2014. Centrifuge modeling of soil–
structure interaction in energy foundations. J. Geotech. Geoenviron.
Eng., ASCE 140 (4), 04013044.
Suryatriyastuti, M.E., Mroueh, H., Burlon, S., 2014. A load transfer
approach for studying the cyclic behavior of thermo-active piles.
Comput. Geotech. 55, 378–391.
Sutman, M., Olgun, C.G., Laloui, L., 2019. Cyclic load–transfer approach
for the analysis of energy piles. J. Geotech. Geoenviron. Eng., ASCE
145 (1), 04018101.
Talbot, A., 1979. The accurate numerical inversion of Laplace transforms.
IMA J. Appl. Math. 23 (1), 97–120.
Wang, L.J., 2022. An analytical model for 3D consolidation and creep
process of layered fractional viscoelastic soils considering temperature
effect. Soils Found. 62 (2) 101124.
Wang, L.J., Ai, Z.Y., 2018. Transient thermal response of a multilayered
geomaterial subjected to a heat source. KSCE J. Civ. Eng. 22 (9),
3292–3301.
Wang, L.J., Liu, X.T., Wan, L., Wang, L.H., 2020. Semianalytical
solution for evaluating viscoelastic consolidation of saturated soils
with overlying dry layers. Int. J. Geomech., ASCE 20 (9), 04020142.
Wang, L.J., Wang, L.H., 2020. Semianalytical analysis of creep and
thermal consolidation behaviors in layered saturated clays. Int. J.
Geomech., ASCE 20 (4), 06020001.
Wang, L.J., Zhu, B., Chen, Y.M., Kong, D.Q., Chen, R.P., 2019. Coupled
consolidation and heat flow analysis of layered soils surrounding
cylindrical heat sources using a precise integration technique. Int. J.
Numer. Anal. Meth. Geomech. 43, 1539–1561.
Wang, W., Regueiro, R., McCartney, J.S., 2015. Coupled axisymmetric
thermoporo-elasto-plastic finite element analysis of energy foundation
centrifuge experiments in partially saturated silt. Geotech. Geol. Eng.
33 (2), 373–388.
Zagorščak, R., Sedighi, M., Thomas, H.R., 2017. Effects of thermoosmosis on hydraulic behavior of saturated clays. Int. J. Geomech.,
ASCE 17 (3), 04016068.
Zheng, L., Deng, T., Liu, Q.J., 2021. Buckling of piles in layered soils by
transfer matrix method. Int. J. Struct. Stab. Dyn. 21 (8), 2150109.
Zhou, G.Q., Zhou, Y., Zhang, D.H., 2016. Analytical solutions for two
pile foundation heat exchanger models in a double-layered ground.
Energy 112 (1), 655–668.
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