Applied Thermal Engineering 26 (2006) 1169–1175
www.elsevier.com/locate/apthermeng
Heat transfer analysis of ground heat exchangers
with inclined boreholes
Ping Cui
b
a,*
, Hongxing Yang b, Zhaohong Fang
a
a
Ground Source Heat Pump Research Center, Shandong University of Architecture and Engineering, Jinan, China
Department of Building Services Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
Received 18 August 2005; accepted 29 October 2005
Available online 9 December 2005
Abstract
Consisting of closed-loop of pipes buried in boreholes, ground heat exchangers (GHEs) are devised for extraction or injection of
thermal energy from/into the ground. Evolved from the vertical borehole systems, the configuration of inclined boreholes is considered in order to reduce the land plots required to install the GHEs in densely populated areas. A transient three-dimensional heat
conduction model has been established and solved analytically to describe the temperature response in the ground caused by a single
inclined line source. Heat transfer in the GHEs with multiple boreholes is then studied by superimposition of the temperature
excesses resulted from individual boreholes. On this basis, two kinds of representative temperature responses on the borehole wall
are defined and discussed. The thermal interference between inclined boreholes is compared with that between vertical ones. The
analyses can provide a basic and useful tool for the design and thermal simulation of the GHEs with inclined boreholes.
2005 Elsevier Ltd. All rights reserved.
Keywords: Ground-coupled heat pump; Inclined borehole; Ground heat exchanger; Heat conduction
1. Introduction
Due to reduced energy consumption and maintenance costs, ground-coupled heat pump (GCHP) systems, which use the ground as a heat source/sink, have
been gaining increasing popularity for space conditioning in buildings [1,2]. The efficiency of the GCHP systems is inherently higher than that of air source heat
pumps because the ground maintains a relatively stable
temperature throughout the year. The system is environment-friendly, producing less CO2 emission than the
conventional alternatives. The ground heat exchanger
(GHE) is devised for extraction or injection of heat
from/into the ground. These systems consist of a sealed
loop of pipes, buried in the ground and connected to a
*
Corresponding author. Tel.: +852 2766 4611; fax: +852 2774 6146.
E-mail addresses: [email protected] (P. Cui), behxyang@polyu.
edu.hk (H. Yang), [email protected] (Z. Fang).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.applthermaleng.2005.10.034
heat pump through which water/antifreeze is circulated.
The GCHP systems require a certain plot of ground for
installing the GHEs, which often becomes a significant
restriction against their applications in densely populated cities and towns. The vertical GHE is the most
popular design of GCHP systems currently employed,
since it requires less ground area than the horizontal
trench systems. These boreholes should be separated
by certain distances to ensure long term operation of
the system. Evolved from the vertical borehole systems,
inclined boreholes are considered as a favorable alternative to further reduce the land areas required for the
GHEs. The inclined boreholes can alleviate the thermal
interference among them in the ground while occupying
less land area on the ground surface than the vertical
GHEs.
Despite all the advantages of the GCHP systems,
commercial growth of the technology has been hindered
by higher capital cost of the system, of which a significant
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P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
Nomenclature
a
B
H
k
l
P
ql
rb
t
s
t0
x
y
z
ground thermal diffusivity (m2 s1)
space between boreholes (m)
borehole depth (m)
ground thermal conductivity (W m1 K1)
variable of the depth of borehole (m)
temperature at point P
heat flow per unit length of borehole (W m1)
borehole radius (m)
temperature (C)
integral parameter in Eq. (1)
ground far field temperature (C)
axial coordinate (m)
axial coordinate (m)
axial coordinate (m)
angle in cross-section circle
dimensionless temperature
time (s)
x
H
s
Subscripts
c
mean temperature of cross-section circle
e
temperature of borehole wall in ground heat
exchangers
i
base borehole
j
adjacent borehole
L
mean temperature over borehole depth L
r
representative temperature
x
temperature at the angle x of a cross-section
Greek symbols
a
inclined angle of borehole
b
direction angle of borehole
inclined borehole and introduce other complications
step by step. In a similar way to the vertical borehole
analysis [4–7], the inclined borehole buried in the ground
can be approximated as an inclined line source of finite
length in a semi-infinite medium. In the model, the
ground is regarded as a homogeneous semi-infinite medium; and its thermophysical properties do not change
with temperature; the boundary of the medium, i.e.
the ground surface, keeps a constant temperature all
the time t0 as its initial one throughout the period
concerned.
A diagram of the physical model for a single inclined
line source is illustrated in Fig. 1. The coordinate of the
top of the line source at the ground surface is (x0, y0); its
length is H; the inclined angle of the line source with the
z-axis is denoted by a; the direction of the inclination is
b; and the heating rate per length of the line source is ql.
In order to solve this problem, a virtual line-sink with
ds
-q l
t0
virtual line sink
α
o
x
r_
portion is attributed to the GHEs. Thus, it is crucial to
work out appropriate and validated tools, by which the
thermal behaviour of the GCHP systems can be assessed
and then, optimised in technical and economical aspects.
However, the thermal analysis on the GHE with inclined
boreholes is extremely difficult for engineering applications, for it has to be treated as transient and threedimensional. Few studies, therefore, have been carried
out on the GHE with inclined boreholes due to complexity of its heat transfer analysis, except some qualitative
discussions from a Swedish researcher [3] who did some
numerical simulation on the heat conduction of inclined
boreholes in a specific GHE. However, the numerical
solution of transient three-dimensional heat transfer is
too computationally intensive to be applied generally
in engineering designs.
On the basis of our previous studies on heat transfer
of GHEs with vertical boreholes, a model has been
established and solved analytically to describe the temperature response in the ground caused by a single
inclined line source. Heat transfer in the GHE with multiple boreholes can then be studied by superimposition
of the temperature excesses resulted from individual
boreholes. The main objective of this paper is to provide
a practical algorithm for engineers to design or analyze
the GCHPs with inclined boreholes.
(x0,y ,0)
β
0
P (x,y,z)
α
r+
s
In order to develop the theoretical model of inclined
GHEs, a basic and simple case is to study a single
ds
y
H
2. Heat transfer analysis of an inclined line source
of finite length
q
l
line source
z
Fig. 1. The geometry of a finite line source in a semi-infinite medium.
P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
the same length H but a negative heating rate ql is set
on symmetry to the boundary as shown in Fig. 1. Thus,
the temperature rise at time s in a random point
P(x, y, z) in the semi-infinite medium can be deduced [8]:
tðx; y; z; sÞ t0
¼
ql
4kp
Z
0
H
8
<erfc 2prþffiffiffi
as
:
rþ
erfc
rffiffiffi
p
2 as
r
9
=
ds;
;
ð1Þ
where
1171
3.1. Temperature response on the cross-section circle
of the borehole
In order to simplify the problem, set the coordinate of
the top of the line source as x0 = y0 = 0 and b = 0 (i.e.
the line source lies in the plane xoz) as the direction of
inclination has no effect on the temperature rise for a
single borehole. The cross-section at the borehole depth
l is a tilted circle, whose coordinates can be conveniently
expressed by the following parameters:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
rþ ¼ ðx x0 s sin a cos bÞ þ ðy y 0 s sin a sin bÞ þ ðz s cos aÞ ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r ¼ ðx x0 s sin a cos bÞ2 þ ðy y 0 s sin a sin bÞ2 þ ðz þ s cos aÞ2 .
Introducing the following dimensionless variables:
x
y
z
x0
y
X ¼ ; Y ¼ ; Z ¼ ; X0 ¼ ; Y0 ¼ 0 ;
H
H
H
H
H
as
ðtðx; y; z; sÞ t0 Þ4kp
Fo ¼ 2 ; Hp ¼
;
ql
H
the dimensionless temperature excess caused by a single
inclined line source can be expressed as a function of the
following dimensionless variables:
Hp ¼ f ðX ; Y ; Z; X 0 ; Y 0 ; a; b; FoÞ.
ð2Þ
Once the temperature response to a single inclined
line source is determined, it can be superimposed in
space to obtain the temperature response to multiple
inclined line sources. For a GHE with n boreholes, of
which the parameters of the ith borehole is distinguished
as (x0i, y0i, ai, bi, Hi), the dimensionless temperature rise
at the point P(x, y, z) can be obtained according to the
linear superposition principle:
n
X
H¼
f ðX ; Y ; Z; X 0i ; Y 0i ; ai ; bi ; FoÞ.
ð3Þ
8
x ¼ l sin a þ rb cos a cos x;
>
>
<
y ¼ rb sin x;
>
>
:
z ¼ l cos a rb sin a cos x;
ð4Þ
where rb denotes the radius of the borehole, x means the
angle between the radius passing the concerned point on
the circle and the specific radius at the plane xoz
(0 6 x < 2p). Substituting Eq. (4) into Eq. (1), the temperature response at the concerned point in the circle
can be derived. Accordingly, the dimensionless expression is given as follows:
1
0
r1þ
r
p
ffiffiffiffiffi
p1ffiffiffiffiffi
erfc
erfc
1B
2
Fo
2
Fo C
B
CdS;
Hx ðL;Rb ;Fo;a;xÞ ¼
@
A
r1þ
r1
0
Z
ð5Þ
where
i¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
ðL sin a þ Rb cos a cos x S sin aÞ þ ðRb sin xÞ þ ðL cos a Rb sin a cos x S cos aÞ ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ðL sin a þ Rb cos a cos x S sin aÞ2 þ ðRb sin xÞ2 þ ðL cos a Rb sin a cos x þ S cos aÞ2 ;
r1þ ¼
r1
Rb ¼ rb =H ; L ¼ l=H ; S ¼ s=H .
3. Temperature response on the inclined borehole wall
In design and simulation of the GHEs in GCHP
systems, a characteristic temperature response on the
borehole wall is usually required [9]. However, the
temperature responses on the inclined borehole wall
perimeter of any cross-section perpendicular to its axis
are unequal and vary with the borehole depth because
the heat transfer of the inclined line source is threedimensional.
Here S is an integral variable, i.e. the relative distance
between a point on the wall and the top of the borehole,
and L means the relative depth of the concerned crosssection circle.
According to the symmetry of the temperature field
on the plane xoz, the dimensionless integral mean temperature along the circle can be presented as follows:
Z
1 p
Hc ðL; Rb ; Fo; aÞ ¼
Hx dx.
ð6Þ
p 0
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P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
and the relative errors between the two temperatures
are less than 0.001%. Therefore, the representative temperature Hir, which is easier to be computed, can be used
to replace the mean temperature of the cross-section circle on the borehole wall in engineering applications.
3.3. Representative temperature of the borehole wall
There are usually two characteristic borehole wall
temperatures in practice. One is the integrated mean
temperature along the borehole depth:
Z 1
Hi ðRb ; Fo; aÞ ¼
Hc dL.
ð7Þ
0
Fig. 2. Temperature profiles along the borehole depth with different
relative radii.
For fixed parameters of Rb, a and Fo, the borehole
wall mean temperature varies with its depth, which is
computed and plotted in Fig. 2. The variation of the
temperature on the inclined borehole wall is almost similar to that on the vertical borehole (see Ref. [9]), which
illustrates that drilling the inclined borehole is unnecessary for a GHE with a single borehole.
3.2. Representative temperature of the cross-section circle
For a specific cross-section circle of the borehole, the
dimensionless temperature at any point on the circle is
only a function of x. The radius of the borehole is minor
compared with its depth, which means the thermal influence of the boundary condition (ground surface) on the
circle may be negligible. Hence, the temperatures variation along the circle is substantially little. Take the middle cross-section (i.e. l = H/2) as an example,
computation shows that the temperature (named as
Hir) of the point on the circle with x = p/2, i.e. its coordinate is (H sin a/2, rb, H cos a/2), can be recommended
to the representative temperature instead of its mean
temperature (Hc). Table 1 lists Hir and Hc for different
conditions.
From Table 1, it is noted that the representative temperatures are almost equal to the mean temperatures
with different Fo and a for a borehole with Rb = 0.001,
Table 1
Comparisons
(Rb = 0.001)
of
the
representative
and
mean
The integrated temperature is regarded as a more
reasonable characteristic temperature. However, it is
too complicated and, therefore, inconvenient to be
employed directly in engineering applications. A more
common scheme is to use the temperature at the middle
of the borehole wall as its representative temperature.
Thus, the representative temperature Hir of the middle
section circle can be employed instead of the integral
mean temperature Hi over the borehole depth. Both
Hir and Hi are the function of a, Rb, Fo, so that the relative error, D ¼ Hir =Hi 1, is related with these three
variables. The variations of the relative errors with a,
Rb, Fo are illustrated in Figs. 3–5, respectively, where
the data concerned can be used for engineering
calculations.
As shown in the figures, the representative temperature Hir is higher than the mean temperature. The maximum relative error between them may be as high as
10.6% in the studied ranges (a 6 30, Rb 6 0.005), which
is still acceptable in most practices as in the similar situations recommended for the vertical borehole GHEs.
Employing the representative temperature on the middle
section as the characteristic temperature of the whole
borehole can greatly simplify the process of the calcula-
temperatures
Fo
a = 10
a = 30
Hir
Hc
Hir
Hc
0.01
0.1
1
10.002
12.048
12.662
10.002
12.048
12.662
10.022
12.012
12.550
10.022
12.012
12.550
Fig. 3. Relative error between Hir and Hi vs. Fo.
P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
0
1173
(x0 i ,y0 i ,0)
o
x
(x0 j ,y0 j ,0)
βj
βi
l
y
αi
borehole i
αj
borehole j
z
Fig. 6. A schematic of two inclined boreholes.
Fig. 4. Relative error between Hir and Hi vs. Rb.
Substituting Eq. (8) into Eq. (1), the temperature response at any axial distance l of the ith borehole caused
by the adjacent jth one can be derived accordingly.
Therefore, the dimensionless expression can be rearranged as a function of the following variables:
Hij;L ¼ fL ðX 0i ; Y 0i ; L; ai ; bi ; X 0j ; Y 0j ; aj ; bj ; FoÞ.
ð9Þ
A representative temperature rise on the borehole
wall caused by its adjacent one is also required to characterize the thermal interference between them. As the
same with a single inclined borehole, there are two characteristic temperatures. One is the mean temperature
rise, which can be obtained by integrating along the
depth of the concerned borehole:
Z 1
Hij ¼
Hij;L dL.
ð10Þ
0
Fig. 5. Relative error between Hir and Hi vs. a.
tion in engineering practice. Of course, more reasonable
integrated mean temperature defined in Eq. (7) may also
be used with sophisticated computer software.
4. Thermal interference among inclined boreholes
In engineering practices the radius of a borehole (typically from 0.05 m to 0.1 m) is much smaller than the
space between adjacent boreholes, which is usually above
3 m. Thus, the temperature response on the concerned
borehole wall caused by its adjacent boreholes can be
approximately treated as the response on its axis since
its radius can be ignored. Take two boreholes (referred
to as i and j) as an example, and suppose the borehole i
is the one concerned, and j is its adjacent one as shown
in Fig. 6. The coordinates of the point at any axial distance l of the ith borehole may be presented as
8
>
< x ¼ x0i þ l sin ai cos bi ;
y ¼ y 0i þ l sin ai sin bi ;
ð8Þ
>
:
z ¼ l cos ai .
The other is a representative temperature response at
the middle of the concerned borehole wall. The processes
of the two methods, however, are both a little complicated for practical applications of GHEs with multiple
inclined boreholes. Again, while it is possible to compute
such temperature rises according to Eq. (1) or Eq. (10)
with appropriate computer software, an approximate
approach is recommended to calculate the thermal interference between adjacent inclined boreholes on the basis
of computations and comparison, i.e. the thermal interference between two inclined boreholes may be estimated
as that of two relevant vertical boreholes disposed at a
distance between the middle points of the inclined boreholes. Thus the temperature rise on the ith borehole
caused by the jth one can be obtained by the two supposed vertical boreholes, which is recommended as a
new representative temperature (Hij,r). Involving a twodimensional process, the expression of Hij,r is much simpler, and can be found in references [5,6].
Now take an example of the two inclined boreholes
with the following parameters: X0i = 0, Y0i = Y0j = 0,
ai = aj = 20, and the value of X0j is obviously equal
to the relative space B* between them. Fig. 7 presents
the comparisons of the representative and the mean
temperatures vs. Fo in different inclining directions. It
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P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
Fig. 7. Comparisons of Hij and Hij,r.
shows that the thermal influence between boreholes
increases with Fo and gradually approaches constant
when Fo is large enough. Though the maximum relative
error resulted from the approximated approach reaches
nearly 17% when Fo = 1.5 for B* = 0.05 and a = 20, the
temperature rise caused by the heat source in adjacent
boreholes is much smaller than that caused by the heat
source in the concerned borehole itself due to the significant difference. Furthermore, the representative temperature is higher than the mean temperature, which can
result in a conservative design for the GHEs. Therefore,
the approximate method can be acceptable for engineering applications.
where the function means the maximum of the temperature responses among the n boreholes.
A typical configuration of the GHEs with inclined
boreholes is the rectangle pattern in practical applications. Fig. 8 depicts a layout of 20 boreholes on the
ground surface in four rows with five boreholes each
(4 · 5). The six inner boreholes are vertical, with relative
space B1 ¼ 0:1 and the peripheral boreholes are inclined
outward with the equal tilting angle and the relative
space B2 of 0.05. The temperature responses of such a
GHE with different tilting angles are plotted in Fig. 9.
The curve for a = 0, i.e. the vertical GHE, is also plotted in Fig. 9 for comparison purpose. The temperature
rises for a = 10 and a = 20 are obviously 16.3% and
27.3% lower, respectively, than that in the vertical
GHE case when Fo = 5.0.
Consider a GHE in a (2 · 5) rectangle pattern in two
different cases of vertical and inclined boreholes with
identical spacing. The tilting angle of each borehole is
20 outward in the case of inclined boreholes. The dimensionless temperature responses of the cases are presented
in Fig. 10 with different relative borehole space B*. Both
the temperature responses of the inclined and the vertical
GHEs increase with decrease in the space between boreholes. However, the thermal influence of the boreholes in
5. Temperature response on the borehole wall in GHEs
with multiple inclined boreholes
U-tubes in the multiple boreholes of GHEs are generally joined in parallel configurations. The temperature
responses on each borehole caused by its adjacent boreholes are obviously different, which mainly depends on
the spacing and geometric disposal of the boreholes.
Hence, a base borehole needs to be found out, which
has the highest temperature rise or the worst heat transfer condition among all the boreholes, as the benchmark
temperature rise on the borehole wall in a GHE.
For each borehole, its temperature response on the
borehole wall to heating of the GHE consists of two
parts: the primary temperature rise due to the line source
(U-tube) in the borehole itself and the second one
caused by the rest boreholes in the GHE. Thus, the temperature response on the base borehole wall in a GHE
with n boreholes can be expressed as follows:
0
1
n
X
B
C
He ¼ maxðHi Þ ¼ max B
H
þ
Hij;r C
ir
@
A;
j¼1
j6¼i
Fig. 8. An inclined GHE with different spaces in rectangle.
ð11Þ
Fig. 9. Comparisons of He between different GHEs (4 · 5).
P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175
Fig. 10. Comparisons of He between different GHEs (2 · 5).
the inclined GHE is much less than that in the vertical
one, especially in the case of B* = 0.03, where the
inclined GHE (2 · 5) with a = 20 may gain 35% reduction in the temperature rise compared to that in the case
of vertical boreholes.
The curves in Figs. 9 and 10 show that the thermal
interference between boreholes can be subdued considerably by either expanding the planar spacing among
boreholes or deviating the boreholes away from each
other along their depth. Hence, drilling inclined boreholes can be a favorable alternative to minimize the temperature rise of the GHEs in case of limited ground area
to install the GHEs.
6. Conclusions
This paper presents an extensive analysis of the heat
transfer of the inclined GHEs from a basic case of a single inclined borehole to practical cases of the GHEs with
multiple boreholes of mixed vertical and inclined ones.
For a single inclined borehole, the model of an inclined
line source with finite length in a semi-infinite medium is
developed to describe the heat conduction process in
GHEs, especially for long-term operation. The representative temperature of a specific point on the middle
cross-section circle of the borehole wall is recommended
for the design of GHEs instead of using its integral mean
temperature along the cross-section circle.
For multiple inclined boreholes, an expression is presented to determine the thermal interference between
adjacent boreholes. Meanwhile, an approximate method
is proposed to calculate the thermal interference
between two inclined boreholes where two vertical bore-
1175
holes are supposed to substitute for the inclined ones.
The representative temperature obtained from the
approximate method is recommended for engineering
designs.
Comparisons between the GHEs of typical rectangular patterns with inclined or vertical boreholes show that
the temperature rise on the borehole wall of the inclined
GHE can be 10–35% lower than that of the vertical
GHE for long-term performance in commonly encountered conditions in engineering practice. Thus, inclusion
of inclined boreholes in the GHE configuration can
improve its thermal performance especially for the
GCHP systems with imbalanced annual loads and limited land allowance to install the GHE. The benefit from
drilling inclined boreholes is evident in such situations.
Acknowledgements
The authors wish to thank the financial supports
from the Natural Science Foundation of China (Project
No. 50476040) and the Hong Kong Research Grants
Council (RGC) (Project No. 530204).
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