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Underground Space 15 (2024) 312–330
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Research Paper

Coupling analysis method of grouting construction with

deformation response of adjacent existing tunnel
Ping-wei Jiang a, Zhi-hong Zhang a,⇑, Hong Zheng a, Jin-kun Huang b,c
a
Key Laboratory of Urban Security and Disaster Engineering of China Ministry of Education, Beijing University of Technology, Beijing 100124, China
b
China MCC17 Group Co. LTD, Maanshan, Anhui 243000, China
c
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 26 April 2023; received in revised form 19 July 2023; accepted 31 July 2023
Available online 21 October 2023

Abstract

In the new tunnel under close distance through the existing tunnel risk source, the grouting scheme developed to compensate for stra-
tum losses is still based on empirical methods, relying on the overburden thickness of existing tunnels. This can potentially lead to an
excessively high or low probability of uplift of existing tunnels. Proposing a coupled deformation analysis method between the grouting
construction and adjacent existing tunnels is of great theoretical significance for developing grouting schemes. In order to reasonably
limit the design parameters of grouting construction, based on the theory of fluid–solid coupling elastic pore-column expansion and
the theory of random media, the calculation method of stratum displacement which simultaneously considers the coexistence of grout
compaction expansion and permeability diffusion mode is derived, and the accuracy of the calculation method is verified by engineering
examples. The accuracy of this calculation method was verified through engineering examples. Combined with the deformation coordi-
nation condition, the existing tunnel is regarded as an elastic Euler-Bernoulli continuous beam, and the finite element coupling balance
equation of the interaction between the existing tunnel and the surrounding soil is obtained. Based on this, a coupling calculation model
of the grouting construction and the deformation response of the adjacent existing tunnel is established. Combined with three times of
grouting construction examples in the shield tunneling project of Beijing Metro Line 12 under the existing airport line, the reliability of
the coupling calculation model to determine the grouting construction parameters is verified. The calculation parameters in the coupling
calculation model have clear physical meanings, which can provide a theoretical basis for the grouting design of similar risk source
projects.

Keywords: Grouting; Close-range underpass; Tunnel; Random medium theory; Euler-Bernoulli continuous beam

1 Introduction underground pipelines, which seriously threatens the con-

struction safety (Hoek, 2001; Ye et al., 2015). With the
Grouting technology is the most commonly used con- increasing number of super-diameter shield continuous
struction method in solving the problems of compensating crossing existing line risk source project, most of the new
stratum loss and reinforcing weak strata. When using tunnels crossing existing tunnels in close proximity use
grouting technology to reinforce strata in the context of grouting technology to compensate for the settlement loss
complex environmental urban infrastructure construction, of existing tunnels. Currently, the grouting construction
it is very easy to induce damage to the structures around scheme cannot be integrated with the existing tunnel defor-
the road, adjacent municipal roads, and urban mation analysis. It relies solely on an empirical method
based on the existing tunnel cover thickness. This can
potentially lead to an excessively high or low probability
⇑ Corresponding author. of uplift of the existing tunnel (Agaiby & Grasso, 2017;
E-mail address: zhangzh2002@bjut.edu.cn (Z.-h. Zhang).

https://doi.org/10.1016/j.undsp.2023.07.005
2467-9674/Ó 2023 Tongji University. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 313

Kizilbash et al., 2017). Therefore, how to conduct coupling to verify the accuracy of the combined algorithm. Shi
analysis of grouting construction scheme and existing tun- et al. (2017) derived the ground deformation due to ground
nel deformation becomes an important construction prob- loss caused by shield tunnel construction based on
lem that needs to be solved in the continuous crossing Mindlin’s classical elasticity theory combined with random
existing line risk source project. medium theory, and applied the combined calculation
Back in 1994, Mair proposed the idea of using equal vol- model to a parallel tunnel construction example to verify
ume grouting to compensate for the lost volume of the stra- the applicability of the calculation method. The stochastic
tum and calculate ground uplift caused by subsurface medium theory was also successfully applied to ground
grouting in weak soils (Mair & Hight, 1994). El-Kelesh uplift prediction. Cai et al. (2019, 2022) established a pre-
et al. (2001) developed a mathematical model using the cir- diction model based on stochastic medium theory for
cular hole expansion method and the medium cone ground freezing and swelling analysis based on the forma-
destruction method to calculate surface uplift caused by tion process of frozen walls, and it was in good agreement
compression grouting based on the idea of isovolume com- with the measured data, which verified the applicability of
pensated loss stratigraphy. Schweiger et al. (2004) used the stochastic medium theory in ground uplift prediction. Guo
mirror image principle to derive a solution for displace- et al. (2015) applied the random media theory to consider
ment caused by grouting in a three-dimensional, elastic, surface expansion caused by simultaneous grouting com-
and semi-infinite space with a finite depth formation. paction expansion, grout infiltration, and grout shrinkage.
Cheong and Soga (2005) developed an experimental system They verified the reliability of the method by comparing
to study the mechanism of grouting-induced lifting and theoretical results with engineering practice results. How-
several numerical methods have been proposed based on ever, the relationship between grouting compaction expan-
the equal volume compensation stratigraphy idea sion, grout infiltration, and grouting radius was not
(Mooney et al., 2016; Cao et al., 2020; Nagel & Meschke, provided. Moreover, the grout diffusion was calculated
2011). In actual engineering grouting processes, the diffu- using the Newtonian fluid model, which differs from non-
sion process of grouting slurry is influenced by soil physical Newtonian fluids Rahman et al. (2015) (such as Bingham
properties, grouting pressure, and method. Various diffu- fluid or power-law fluid) that are commonly used for actual
sion modes of grout have no clear boundaries, and the dif- formation reinforcement.
fusion process can be summarized as a complex interplay Based on the fluid–solid coupling elastic pore-column
of infiltration, compaction dense, and splitting diffusion expansion theory, the power-law grout injection regional
modes (Bouchelaghem & Almosni, 2003; Foyo et al., soil stress field and displacement field solutions are derived.
2005; Ieronymaki et al., 2016). Under specific stratum con- It solved the problem of how to determine and calculate the
ditions and grouting conditions, only a dominant diffusion boundary between the grout compaction expansion zone
mode is used for diffusion. Therefore, to accurately assess and the grout permeability zone. And it is proposed to treat
the effect of grouting construction on adjacent structures, the existing tunnel as an elastic Euler-Bernoulli continuous
it is necessary to comprehensively consider the grout diffu- beam, and obtain the finite element coupling equilibrium
sion mode in the formation (Zhang et al., 2017; Song et al., equation of the interaction between the existing tunnel
2020). In addition, the random medium theory is a typical and the surrounding soil. The coupling calculation model
method for calculating the displacement of the strata, of grouting construction and the deformation response of
which can fully consider the diffusion process of the grout adjacent existing tunnels is established, and the reliability
compaction expansion and grout permeability coexistence of the calculation model is verified with engineering exam-
during the grouting process. ples. Finally, the engineering applicability of the calcula-
The stochastic medium theory, which was developed by tion model is summarized.
J. Litwiniszyn Axelrad (Axelrad, 1990), a Polish scholar,
was initially used to study surface displacement of coal 2 Computational modeling and solution
mining formations. The predicted results were verified
through rigorous mathematical derivation and sandbox 2.1 Basic assumptions
model tests. Liu (1993) studied and analyzed problems
related to surface deformation and movement generated Take the grouting section as a circular plane with the
by shallow surface excavation based on this theory, and initial stress of the formation as rb, as shown in Fig. 1.
summarized formulas for subsidence at each point of the Under the action of certain grouting pressure Pa, the
surface generated by excavation. Yang and Wang (2011) radius of grouting hole is Ra. The soil around the grouting
simplified the complex stochastic medium theory calcula- hole is squeezed, and the cavity generated after squeezing is
tion method and verified the effectiveness of the simplified filled by the grout, forming a compaction expansion zone
method in predicting surface displacement. Li et al. (2012) with a radius of ur. As the grouting process advances, the
combined stochastic medium theory with genetic algorithm grout begins to penetrate the squeezed soil. Until the end
to predict ground settlement caused by tunnel construction of grouting, the compaction expansion zone (zone Ⅰ) and
in mountainous areas, and compared the results with finite the grout infiltration zone (zone Ⅱ) are formed from the
element numerical calculations and field monitoring data center of the hole in the order of inward and outward.
314 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 1. Simplified calculation model of column grouting plane.

The displacement generated in the range of grout diffusion hydraulic gradient equation of grout in cylindrical diffusion
Rb during the grouting construction is the displacement of theory is shown in Eq. (3).
the grout compacted expansion zone. The overall diffusion
1n
1
radius is Rb. rr, rh are the radial stress and tangential stress Ke dp n
V ¼  ; ð2Þ
at one point of any soil body around the hole, respectively le dr
(Cai et al., 2019).
n
l Q
The following basic assumptions are made in response dp ¼  e rn dr: ð3Þ
K e 2ph
to the above discussion: (1) The soil is assumed to be an
isotropic, homogeneous, and elastic ideal material. (2) Both In Eqs. (2)–(3): V represents the grout percolation veloc-
the grout and soil particle skeleton are considered to be ity, while Ke and le denote the effective permeability and
incompressible. (3) The soil surrounding the grouting hole effective viscosity, respectively, Q represents the total vol-
is considered to be completely elastic prior to grouting. (4) ume of grout injection, and h is the design height of the col-
The effect of gravity on seepage and soil compression is umn grouting.
ignored, and no consideration is given to the effect of grout Integrating Eq. (3), the boundary conditions were taken
phase change. as the grouting hole wall (P in jr¼Ra ¼ P a ) and the grout
osmotic pressure at the calculated diffusion radius
(P in jr¼Rb ¼ P b ). The calculated grout seepage pressure field
2.2 Power law fluid penetration grouting theory
is as follows:
The power law quasi-plastic fluid exhibits the following Ra1n P b  R1n
b Pa þ r
1n
ðP a  P b Þ
P in ¼  ; ð4Þ
characteristics: its viscosity decreases as the shear rate R a  Rb
1n 1n

increases, a phenomenon known as shear thinning. The

rheological equation describing this behavior is as follows: where Pa and Pb are the grouting pressure at the hole wall
(Ra) and the calculated radius of diffusion (Rb),
s ¼ c 1 cn ; ð1Þ respectively.
The relationship among the differential grouting pres-
where s is shear stress; c1 is shear stress; c is shear rate; n is sure, diffusion radius, and grouting time (t) is as follows:
rheological exponent or power law exponent.
 
The percolation motion equation of the power-law rhe- 1 le u n  1n 
Pa  Pb ¼ Rb  Ra1n R2n
b ; ð5Þ
ological model of grout is shown in Eq. (2), and the radial 1  n K e 2t
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 315

where u is the porosity. In Eqs. (12)–(14):k1n ¼ nððn1Þlnþ2Þ

, k2n ¼
ðR1n
a þRb Þðn3Þðl1Þ
1n

nððn1Þl1Þ nð2l1Þ
,k3n ¼ R1n þR1n ðn3Þðl1Þ.
2.3 Radius of compaction expansion zone ðR1n
a þRb Þðn3Þðl1Þ
1n
ð a b Þ

The equilibrium differential equation satisfied by the soil

stress, considering the volumetric effect of the injected 2.4 Calculation model of grouting reinforcement stratum
grout, is as follows: response
@rr rr  rh @P in
þ þn ¼ 0; ð6Þ 2.4.1 Ground displacement due to compaction expansion
@r r @r
(O1p-x1)
where n is the pore water pressure coefficient. The response of grout construction and stratigraphy can
The geometric equation is shown in Eq. (7). be modeled as the sum of the responses of individual
dur ur micro-elements. The overall coordinate system (X, Y, Z)
er ¼ ; eh ¼ : ð7Þ
dr r is used to describe the stratum, while a local coordinate sys-
tem (u, v, w) is used for the grout diffusion region. As illus-
Constitutive relation is as follows:
trated in Fig. 2, the micro-element body of infinitesimal


1  l2 l width, length, and thickness in the compaction expansion
er ¼ rr  r ; ð8Þ
E 1l h zone is defined as dudvdw. Based on the stochastic medium

 theory, the starting point for analysis is taken as the begin-
1  l2 l ning of grout construction. The uplift value in the Z-
eh ¼ rh  r ; ð9Þ
E 1l r direction, denoted as Ww, at a formation coordinate point
(X, Y, Z) caused by the grouting of a unit micro-element in
where er and eh are the radial strain and tangential strain,
the compaction expansion zone after time t can be calcu-
respectively; ur is the radial displacement; l is the Poisson’s
lated according to Eq. (15). Similarly, the displacement
ratio; E is the modulus of elasticity.
value in the X-direction, denoted as Uu, can be obtained
According to the relationship of equations of elastic
using Eq. (16).
mechanics Eqs. (7)–(9), the joint collation is obtained as
The uplift of the strata and horizontal displacement are
follows:
shown in Eq. (15) and Eq. (16), respectively (Guo et al.,
d2 ur 1 dur v n ur 2015).
þ þ Ar  2 ¼ 0: ð10Þ
dr2 r dr E r ZZZ
tan2 b
ðlþ1Þð2l1Þ W w ðX ; Y ; Z Þ ¼
In Eq. (10): v ¼ , A¼ n ðR
P a P b Þðn1Þ
1n .
ðZ  wÞ2
Xx
ðl1Þ ð 1n a þRb Þ (
h i
)
ptan2 b
Solving Eq. (10) yields:  exp  2
ðX  uÞ þ ðY  vÞ2
dudvdw; ð15Þ
ðZ  wÞ2
C2 v A
ur ¼ C 1 r þ  rnþ2 : ð11Þ ZZZ
r E ð n  1Þ ð n  3Þ ðX  uÞtan2 b
Uu ðX ; Y ; Z Þ ¼
C1 and C2 in Eq. (11) are coefficients to be determined. Xx ðZ  wÞ2
( )
ptan2 b h i
Combining with the boundary conditions
(rr jr¼Ra ¼ P a ,rr jr¼1 ¼ rb ) to determine the integration con-  exp  2
ðX  uÞ þ ðY  vÞ 2
dudvdw; ð16Þ
stants, the stress distribution in the elastic region is ðZ  w Þ2
obtained as
where b is the angle of influence for ground uplift (Cai
R2p
rr ¼ ðrb  P a Þ þ rb et al., 2019).
r2
  Assuming that the grouting materials is uniformly dis-
k1n ðP a  P b Þ Rpnþ3  rnþ3 tributed in the formation after injection, with uniform dif-
 ; ð12Þ fusion radii of Ra and ur before and after injection,
r2
respectively. The volume of the compaction grout cavity
R2p after injection is X1  x1, and the radial expansion value
rh ¼ ðrb  P a Þ þ rb
r2 is equal to ur minus Ra. The displacement of the formation
 
caused by the compaction grout body is shown in Eq. (17).
ðP a  P b Þ k1n Rpnþ3  k2n rnþ3
þ ; ð13Þ ZZZ
r2 tan2 b
W w1 ðX ; Y ; Z Þ ¼
ðl þ 1Þ h   2
X1 x1 ðZ  wÞ
ur ¼ ðP a  P b Þ k1n Rpnþ3  k3n rnþ3 ( )
Er i ð14Þ ptan2 b h 2 2
i
 exp  ðX  uÞ þ ðY  vÞ dudvdw: ð17Þ
þð1  2lÞr2 rb þ ðrb  P a ÞR2a : ðZ  w Þ
2
316 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 2. Strata response caused by arbitrary micro-element.

2.4.2 Displacement of the formation due to the permeability W w ðX ; Y ; Z Þ ¼ W w1 ðX ; Y ; Z Þ þ W w2 ðX ; Y ; Z Þ ð24Þ

stress (X2  x2)
Similarly, the horizontal displacement of the formation,
At the depth w, the displacement of the micro-element
caused by (O1x1), which represents the grout expansion
under the action of the penetration stress increment is as
zone, and (X2  x2), which represents the grout infiltration
in Guo et al. (2015):
zone, can be obtained and is shown in Eq. (25).
av Dp
dsðu; v; wÞ ¼ dw; ð18Þ U u ðX ; Y ; Z Þ ¼ U u1 ðX ; Y ; Z Þ þ U u2 ðX ; Y ; Z Þ; ð25Þ
1 þ e0
where e0 is the initial void ratio; av is the compressibility where
coefficient; Dp is the effective stress increment. U u1 ðX ; Y ; Z Þ ¼ U u1X ðX ; Y ; Z Þ cos a
The unit seepage force (D) in the direction of seepage
þ U u1Y ðX ; Y ; Z Þ sin a; ð26Þ
per unit volume of soil is as follows:
Z ZZ ( )
ðj  uÞtan2 b ptan2 b h i
D ¼ cc i; ð19Þ U u1j ðX ; Y ; Z Þ ¼ 2
exp  2
ðX  uÞ2 þ ðY  vÞ2 dudvdw;
X1 x1 ð Z  wÞ ðZ  wÞ
where cc = qcg is the grout gravity, and i is the average ð27Þ
hydraulic gradient. It is considered that no loss of grout Z ZZ
pressure occurs in the compaction expansion zone. U u2j ðX ; Y ; Z Þ ¼ U uj ðX ; Y ; Z Þdsðu; v; wÞdudvdw:
X2 x2
ðP a  P b Þ ð28Þ
i¼ ; ð20Þ
c c ð Rb  u r Þ
In Eq. (26), U u1X ðX ; Y ; Z Þ and U u1Y ðX ; Y ; Z Þ are the
ðP a  P b ÞðH  w Þ
Dp ¼ cc iðH  wÞ ¼ : ð21Þ components of the horizontal displacement caused by the
Rb  u r microcellular body in the X and Y directions, respectively.
Substituting Eq. (21) into Eq. (18), the uplift in the for- a is the angle with the positive direction of the X-axis.
mation induced by permeability is shown in Eq. (22). j = X, Y.
Equations (24) and (25) provide estimates for the verti-
av ðP a  P b ÞðH  wÞ
dsðu; v; wÞ ¼ dw ð22Þ cal and horizontal displacements of the strata resulting
ð Rb  u r Þ ð 1 þ e 0 Þ from the combined effects of grout compaction and soil
Substituting Eq. (22) into Eq. (15), the uplift of the soil infiltration during grouting construction. The radii of the
within the penetration range X2  x2 is shown in Eq. (23). compaction expansion zone, denoted as ur, and the infiltra-
ZZZ tion zone, referred to as Rb, are both functions of the
W w2 ðX ; Y ; Z Þ ¼ W w ðX ; Y ; Z Þdsðu; v; wÞdudvdw grouting pressure Pa and time t. The volumes of the two
X2 x2 areas in the column grouting form can be expressed as
ð23Þ Z t Z 2pZ h
X1  x1 ¼ ur ðP a ; tÞdtdadh; ð29Þ
As a result, the vertical displacement of the formation, 0 0 0
caused by (X1  x1), which represents the grout expansion Z tZ 2pZ h
zone, and (X2  x2), which represents the grout infiltration X2  x2 ¼ ½Rb ðP a ; tÞ  ur ðP a ; tÞdtdadh: ð30Þ
zone, can be obtained and is shown in Eq. (24). 0 0 0
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 317

Table 1
Calculation parameter table.
tanb H (m) Ra (m) Rb (m) Dr (m) n
0.839 5 0.05 0.6 0.0101 0.5
Pa (MPa) Pb (MPa) e0 av (MPa1) E (MPa) l
0.5 0.01 0.661 0.18 15 0.2

Fig. 3. Flow chart of indoor grouting test.

2.4.3 Verification of calculation model of stratum response in In this paper, the calculation method of the compaction
grouting construction expansion zone is determined by the elastic fluid–solid cou-
The computational model in this paper is compared with pling theory. A power-law grout body with a common
the monitoring data of the engineering application from engineering water-cement ratio (W/C) of 0.7 is selected,
Guo et al. (2015), while the consolidation shrinkage effect and ur = 0.0101 m is calculated. The calculated results
of the grout is considered with reference to this literature. are consistent with the results obtained in the laboratory
The case study involved land subsidence caused by ground- of literature Guo et al. (2015). And it can be seen from
water seepage on Shaohuai Road in the K1302 section of Fig. 4 that the actual field monitoring values are consistent
Hukun Highway, where grouting lifting was employed to with the calculated values in this paper at the maximum
address the settlement. Calculation parameters are summa- displacement height of ground uplift caused by grouting
rized in Table 1. Among them, the calculation example is construction. The ground displacement obtained from the
measured by combining indoor and outdoor experiments calculation model basically matches the field monitoring
to obtain a grouting compaction expansion zone of data numerically, which verifies the reliability of the calcu-
Dr = 0.01 m. The flow of the indoor test is shown in lation method of the compaction expansion zone proposed
Fig. 3, and the calculation results are shown in Fig. 4. in this paper.
318 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 4. Comparison of calculation results.

Fig. 5. Vertical free displacement of soil at the tunnel node position.

3 Interaction among new tunnel, existing tunnel and the tunnel does not exist, w1 represents the vertical free dis-
surrounding soil placement vector of the soil at the tunnel node location
caused by external load (Klar et al. 2005; Lin et al. 2019).
The longitudinal dimension of the existing tunnel is
w1 ¼ ½w1 ð1Þ; w1 ð2Þ;   ; w1 ðiÞ;   ; w1 ðnÞ; w1 ðn þ 1Þ ð31Þ
much larger than its transverse dimension, thereby
exhibiting force characteristics equivalent to those of a In Eq. (31), w1 represents the vertical free displacement
homogeneous elastic Euler-Bernoulli continuous beam. of the soil at the existing tunnel node caused by external
To model this structure, it is divided into N sections of load, while wi(i) denotes the vertical free displacement of
two-node four-degree-of-freedom elastic beam units along the soil at the i-th existing tunnel node.
the longitudinal direction, with a total of N + 1 nodes, as In reality, interactions between the existing tunnel and
shown in Fig. 5. Grouting construction will induce a free the surrounding soil are inevitable. On one hand, the stiff-
displacement field in the surrounding soil. Assuming that ness of the tunnel impedes the free movement of the soil,
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 319

which in turn constrains soil deformation. On the other Neglecting the axial deformation of the existing tunnel,
hand, the tunnel itself will experience a corresponding dis- the distributed force F(z) on the continuous beam unit can
placement as a result of this interaction force. The interac- be transformed into the equivalent nodal force Fw(i),
tion force and displacement between the tunnel and its Fw(i + 1) and the equivalent nodal bending moment Mw(i),
surrounding soil will continually adjust until a new equilib- Mw(i + 1), as shown in Fig. 6. When the unit length l is suffi-
rium state is achieved. ciently small, Mw(i) and Mw(i + 1) are approximately 0.
Considering the existing tunnel as the object, the equilib- Therefore, the distributed force F(z) acting on the existing
rium state of the tunnel is disrupted by the vertical dis- tunnel can be expressed as a nodal force vector Fw.
tributed force from the surrounding soil, denoted as F(z), F w ¼ ½F w ð1Þ; F w ð2Þ;   ; F w ðiÞ;   ; F w ðnÞ; F w ðn þ 1Þ ð33Þ
which induces a resulting vertical displacement w(z). This
displacement can be transformed into a nodal displacement In Eq. (33): Fw is the force vector acting on the vertical
vector w. node of the existing tunnel, and Fw(i) is the vertical force
acting on the i-th node on the existing tunnel.
w ¼ ½wð1Þ; wð2Þ;   ; wðiÞ;   ; wðnÞ; wðn þ 1Þ ð32Þ The overall stiffness matrix K that expresses the relation-
ship between nodal vertical force and nodal vertical displace-
In Eq. (32): w is the vertical displacement vector of the ment of the continuous beam can be derived from the nodal
existing tunnel node caused by the external load F(z). w(i) four degrees of freedom continuous beam unit stiffness
is the vertical displacement of the i-th existing tunnel node matrix Ke. Subsequently, Eq. (34) exhibits the finite element
caused by the external load F(z). equation for force equilibrium of the existing tunnel.

Fig. 6. Analysis schematic for the vertical force and displacement of existing tunnel.

Fig. 7. Plane view of the underpass (1∶2500).

320 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 8. Sectional view of the underpass (1∶500).

Fw ¼ K  w ð34Þ Gij can be determined using the basic Mindlin solution for a
concentrated load located inside the semi-elastic space
With the surrounding soil as the object, the force vector body, with horizontal force Fx at coordinates (0, 0, c)
from the existing tunnel fw acts on the surrounding soil in and vertical force Fz at any point (x0 ; y 0 ; z0 ). Equations
equilibrium, resulting in nodal soil displacement vector is (36) and (37) give the soil vertical displacements wFx and
as follows: wFz, respectively (Zhou et al., 2018).
w2 ¼ G  f w ; ð35Þ 0
F xy0 z  c ð3  4mÞðz0  cÞ 6cz0 ðz0 þ cÞ
wF x ¼ þ 
where w2 is the displacement of the soil around the existing 16pGð1  mÞ R31 R32 R52
tunnel under the action of fw. G denotes the flexibility 4ð1  mÞð1  2mÞ
matrix for soil nodes, where Gij corresponds to the vertical þ ;
R2 ð R2 þ z 0 þ c Þ
free displacement of node i due to a vertical unit force act-
ing on node j in the soil continuum medium. The value of ð36Þ
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 321

Table 2
Calculation parameters of 3 times grouting formation.
Parameters Pa (MPa) Pb (MPa) e0 E(MPa) l n tanb av (MPa1) ur (m)
1st 0.20 0.01 0.85 15 0.20 0.6 0.45 0.15 0.0112
2nd 0.35 0.01 0.65 18 0.20 0.6 0.75 0.15 0.0230
3rd 0.30 0.15 0.55 19 0.20 0.6 0.75 0.15 0.0312

Table 3
Calculation parameters of existing tunnel.
Parameters EI (Pa) A(m2) Poisson’s ratio Cohesion (Pa) Angle of internal friction(°) Dilation angle(°)
Existing tunnel 9.11  1010 9.8 0.30 2  106 45 30

"
Fz
2
3  4m 8ð1  mÞ  ð3  mÞ ðG  K þ E Þ  w ¼ w 1 ð41Þ
wF z ¼ þ
16pGð1  mÞ R1 R2 In Eq. (41), E denotes the unit matrix. Figure 6 illus-
# trates that the deformation and forces of the existing tunnel
ðz0  cÞ2 ð3  4mÞðz0 þ cÞ2  2cz0 6cz0 ðz0 þ cÞ2
þ þ þ : are no longer influenced by the grouting construction out-
R31 R32 R52 side the L0 grouting area dimensions. As a result, the
ð37Þ boundary conditions are as follows:

In Eqs. (36) and (37): c is the distance from the point of wðL0 =2Þ ¼ 0; wðL0 =2Þ ¼ 0
; ð42Þ
action of the concentrated force to the origin (O, X, Y, Z). wh ðL0 =2Þ ¼ 0; wh ðL0 =2Þ ¼ 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
R1 ¼ x02 þ y 02 þ ðz0  cÞ , and R2 ¼ x02 þ y 02 þ ðz0 þ cÞ . where w and wh represent the vertical displacement and
G and m are taken as the weighted average of the shear deflection of the Euler-Bernoulli continuous beam nodes,
modulus and Poisson’s ratio of the overlying soil layer, respectively. The displacement vector w for the existing
respectively. The obtained Gij from the above equation tunnel nodes is determined by the one-dimensional discrete
approaches infinity, indicating that the formula is unsuit- finite element, the cubic interpolation fitting method of
able for calculating the displacement caused by the unit spline can be employed to obtain the continuous function
nodal force at the node itself. As a result, instead of using w(z) for the longitudinal displacement distribution of the
circular center displacement, the mean value of circumfer- existing tunnel. Using primary, secondary, and tertiary
ential vertical displacement is utilized. For contact node i, derivatives, the longitudinal deformation curvature, bend-
which divides the Euler-Bernoulli continuous beam N ing moment, and shear force distribution functions of the
sections with a length of l as the radius to form the circum- existing tunnel can be derived as follows:
ference, the circumference is divided into m nodes 8
>
> wh ðzÞ ¼ dwdzðzÞ
according to Eq. (38). < 2
M ðzÞ ¼ EI d dzw2ðzÞ ; ð43Þ
1 Xm
>
>
Gii ¼ Gs;m ð38Þ : 3
m s¼1 F s ðzÞ ¼ EI d dzw3ðzÞ

The mechanical equilibrium condition of the interaction where wh ðzÞ represents the distribution function for curva-
between the existing tunnel and the soil body is shown in ture of the existing tunnel deformation, M ðzÞ denotes the
Eq. (39). distribution function for bending moment, and F s ðzÞ stands
for the distribution function for shear force.
F w ¼ f w ð39Þ
However, it should be noted that Eq. (43) is only cap-
If the existing tunnel and soil meet the displacement able of characterizing the elastic deformation of the exist-
compatibility condition, then the deformation coupling ing tunnel, and fails to account for its plastic
equation between the two can be expressed as Eq. (40). deformation zone. Thus, to address this issue, a
two-dimensional finite element plane model of the existing
w ¼ w1 þ w2 ð40Þ
tunnel is established. The continuous function w(z) for lon-
By combining Eqs. (32), (33), (38) and (39), Eq. (41) can gitudinal displacement distribution of the tunnel is applied
be derived as the finite element equilibrium equation for the to unit nodes as a known displacement boundary condi-
longitudinal deformation characteristics of the existing tion. The corresponding constitutive equation is then
tunnel. employed to determine the plastic deformation zone of
322 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 9. Calculation flow chart.

the existing tunnel resulting from the grouting of the pipe piece at 6.4 m, and width of the pipe piece at
construction. 1.2 m. The interval tunnel passes over existing rail transit
airport line obliquely at approximately 60° angle, with a
4 Example verification vertical distance of 4.089 m. Due to high risk level, the
underpass interval tunnel construction is challenging,
4.1 Project example requiring strict settlement control measures. Specifically,
the following aspects pose significant challenges: (1)
In order to confirm the accuracy of the presented cou- crossing construction at a small angle with the turning
pled model and comprehensively evaluate the impact of radius of 350–400 m, long and wide crossing distance;
grouting on the deformation of the existing tunnel, three (2) surface of the crossing section being the capital
grouting operations were selected as comparative exam- airport expressway without any reinforcement treatment
ples. These works were chosen based on the settlement measures or ground monitoring conditions; (3) the sec-
deformation caused by the new tunnel underpass. As tion’s left and right curves cross the construction path
shown in Fig. 7, the new Line 12 interval between Xibahe twice consecutively, which poses a risk; (4) the settlement
Station and Sanyuanqiao Station starts from Xibahe Sta- control requirement for the existing line structure is
tion in the west and is laid along the East Road of the 3 mm, which is exceptionally challenging to control.
North Third Ring Road to the southeast. The total dis- The main strata crossed by the tunnel under the rail tran-
tance of this interval is 1481.0 m, with a line spacing of sit airport line include powder clay layer ④, fine sand
17.2–35.0 m. Shield method is utilized for construction, powder layer ⑤, and powder clay layer ⑥, as depicted
with the shield blade diameter at 6.68 m, outer diameter in Fig. 8.
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 323

Fig. 10. Lifting curve and displacement cloud chart of existing tunnel caused by grouting.

4.2 Calculation 0.5 mm. Column grouting was employed, with a column
height (h) of 2.56 m, radius of the grouting pipe (Ra) at
Upon completion of the new tunnel construction under 0.05 m, grout design diffusion radius (Rb) of 0.5 m, grout-
the existing tunnel, significant vertical settlement deforma- ing volume of 0.3 m3, and grouting pressure < 0.35 MPa.
tion occurs in the existing tunnel. To control this deforma- Before each grouting operation, field borehole tests were
tion, a grout filling construction method was used three utilized to readjust the parameters, fully considering the
times to compensate for ground loss. The settlement mon- changes in soil physical properties due to the solidification
itoring values of the existing tunnel are presented in of the grout within the grouting area. The calculated
Fig. 11, which clearly depicts the effectiveness of grouting parameters for the three replenishment strata and the exist-
in compensating for stratum loss. Following the second ing tunnel are shown in Tables 2 and 3, respectively. EI is
replenishment of grout, the maximum value of ground the bending stiffness of the existing tunnel section, and A is
compensation was observed to lift the existing tunnel by the cross-sectional area of the existing tunnel. The calcula-
324 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 11. Field measured data.

Fig. 12. Longitudinal deformation curvatures of tunnel.

tion flow is demonstrated in Fig. 9, where an elastoplastic 4.3 Calculation results

principal model based on Mohr–Coulomb theory was cho-
sen to analyze the elastic–plastic deformation of the exist- Referring to Tables 2 and 3, the parameters were calcu-
ing tunnel. lated as per the calculation flow illustrated in Fig. 9. The
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 325

Fig. 13. Longitudinal bending movements of tunnel.

Fig. 14. Longitudinal shearing forces of tunnel.

relationship between the three times grouting operations compensation for settlement loss in the left line of the exist-
and the response of the existing tunnel is depicted in ing tunnel was neglected when the right shield crossed the
Fig. 10. Figure 11 shows the field monitoring data, where airport line. Therefore, the settlement distribution along
326 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

Fig. 15. Longitudinal deformation curvatures of tunnel. Fig. 17. Longitudinal shearing forces of existing tunnel.

small in magnitude, and the existing tunnel does not exhibit

any plastic deformation. The curvature, bending moment,
and shear force of the existing tunnel were obtained by
interpolating the displacement boundary conditions three
times, as shown in Figs. 12–14.
From Fig. 12, it is evident that the longitudinal defor-
mation curvature of the existing tunnel caused by the 1st,
2nd, and 3rd grout fill operations are 9.01  106,
3.73  105, and 2.01  105, respectively. The corre-
sponding locations are 20, 18, and 16 m from the centerline
of the underpass tunnel, with corresponding radii of curva-
ture at 110 987, 26 809, and 49 751 m, respectively. The
radius of curvature of the metro tunnel deformation influ-
enced by adjacent constructions should not be less than 15
000 m, indicating that the curvature meets the requirements
for protecting the existing tunnel. The point with zero cur-
vature represents the reverse bend point of the deformation
curve. The position of the reverse bend point appears to be
about 30 m (5 times the excavation length) below the tun-
Fig. 16. Longitudinal bending movements of existing tunnel.
nel midline, indicating that the effect of overall tunnel bulge
deformation is approximately 10 times the length of the
the centerline of the new tunnel is symmetrically grouting area.
distributed. As depicted in Fig. 13, the maximum bending moment is
Upon comparison of Figs. 10 and 11, it is evident that located directly above the centerline of the underpass tun-
the maximum uplift values of the existing tunnel caused nel during the three grouting processes, with maximum val-
by the three compensation grouting operations are 0.143, ues of 144.480, 688.105, and 376.389 kNm, respectively.
0.501, and 0.264 mm, respectively. The maximum uplift Due to the characteristics of the existing shield tunnel tube
values monitored in the field are 0.119, 0.461, and sheet lining construction, after longitudinal bending of the
0.234 mm, respectively. The calculation errors are tunnel, the neutral axis will shift, causing the annular tube
16.85%, 7.92%, and 14.23%, respectively. These calculation sheet to be tensioned and opened, with the tension being
errors fully satisfy the engineering design construction borne by the bolts at the opening. Therefore, the longitudi-
within an acceptable order of magnitude, thus verifying nal bending resistance of the existing tunnel is related to the
the reliability of the coupled calculation model presented moment of inertia of the section, longitudinal bolt stiffness,
in this paper. This model can provide a theoretical basis number of bolts, and bolt inclination angle. In the litera-
for accurately lifting and correcting settlement under exist- ture of Zhou et al. (2018), the maximum bending moment
ing tunnels. Furthermore, the uplift value is numerically corresponding to the bolt under yielding tensile stress was
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 327

Fig. 18. Lifting curve and displacement cloud chart of existing tunnel caused by grouting replenishment.

Fig. 19. Grouting causes the plastic zone of the existing tunnel.

104.5 MNm for the Nanjing shield tunnel. Based on engi- achieve the desired reinforcement effect, while excessive
neering analogy, it can be concluded that the three times pressure will inevitably cause significant stratigraphic dis-
grouting constructions in this project were within the stable placement and damage to the existing strata structure.
range. For instance, when Beijing Metro Line 17 underpassed Bei-
As shown in Fig. 14, the longitudinal shear force of the jing Metro Line 6, the grouting construction caused local
existing tunnel caused by the 1st–3rd fill grout is 24.048, arch deformation of 126 mm in the roadbed, limiting sub-
116.580, and 65.221 kN, respectively. The corresponding way speed to 10 km/h in this section. The curvature, bend-
locations are 18, 12, and 12 m from the centerline of the ing moment, and shear force of the existing tunnel of Line
underpass tunnel, respectively. During the project, if the 6 were calculated using Eq. (43), as shown in Figs. 15 to 17.
sum of the frictional force and shear strength of joint bolts Figures 15–17 show that the displacement boundary
on the longitudinal contact surface of the tube sheet is condition generates a radius of curvature of 136.98 m,
insufficient to resist the additional shear force of the tunnel maximum positive and negative bending moments of
section, the tube sheet is prone to misalignment and dislo- 72.41 MNm and 92.20 MNm respectively, and maxi-
cation. In severe cases, this may lead to destabilization mum positive and negative shear forces of 5.56 MN and
damage. According to literature (Zhou et al., 2018), the 5.56 MN respectively. These values exceed the thresholds
shear capacity of shield tunnel section bolts (M36 type proposed in the literature for radius of curvature and shear
8.8 grade) is about 4.9–6.4 MN. It can be concluded that force, and are close to the bending moment threshold.
there is enough reserve shear capacity in the three-times Therefore, this model can serve as a reference for grouting
grouting construction processes of this calculation case. parameter qualification.
Although Eq. (43) is based on elastic beam theory, the
5 Design process of grouting construction scheme grouting construction parameters are limited to ensure
the stability of existing tunnels but do not consider the
The construction of a new tunnel can disturb the sur- plastic deformation that can occur during the grouting pro-
rounding environment, resulting in settlement, deforma- cess. Deformation cracks resulting from the plastic dis-
tion, and additional stress on existing buildings. placement of the existing tunnel can lead to difficulties in
Controlling the impact of new lines on existing lines and later water leakage prevention and reinforcement corrosion
ensuring the normal operational safety of existing lines treatments, significantly decreasing the durability of the
during construction is a major problem in shield construc- structure. Therefore, after obtaining grouting parameters
tion research. Grouting is the main measure taken to pre- through the coupled calculation model, an elastoplastic
vent and correct existing tunnel deformation during analysis is necessary to prevent large plastic deformations
construction. The key to solving this problem is to limit that may affect the tunnel’s future durability (Gan et al.,
the grouting construction parameters using a reasonable 2022; Ng et al., 2016). This paper applies the existing tun-
calculation model. A small grouting pressure cannot nel displacement boundary conditions of Line 6 to the cal-
328 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330

affected by the bottom lifting displacement and subject to

tensile damage, increasing the likelihood of tube sheet
cracking. To reduce plastic deformation, support can be
added inside the existing tunnel at the midline, the grouting
range can be expanded in the span direction of the existing
tunnel, or the top of the existing tunnel can be reinforced
with grouting.
This paper proposes a comprehensive analysis method
that takes into consideration the coupling of existing tunnel
deformation and grouting construction. Figure 20 illus-
trates the coupling calculation model guiding the develop-
ment process of the grouting construction scheme. Firstly,
the pre-defined construction parameters are obtained
according to the thickness of the existing tunnel covering
soil based on the general situation of the project. Through
this method, the mutual coupling between existing struc-
tural deformation and grouting construction can be
achieved. Constantly adjust the grouting design parameters
to ensure the grouting construction and the stability
requirements of the existing tunnel structure. At the same
time, combined with the elasto-plastic analysis, the durabil-
ity performance of the existing tunnel structure is also
guaranteed under the premise of ensuring the stability
requirements of the existing tunnel structure, which is more
reasonable to guide the design and construction.

6 Conclusions

(1) Based on the theory of fluid–solid coupling elastic

hole column expansion and considering the rheological
properties of grout, the solutions of soil stress field and dis-
placement field in the power law grouting area are derived.
Furthermore, by incorporating the random medium the-
ory, a calculation method that considers both grout com-
paction expansion and infiltration-diffusion modes at the
same time has been established for predicting stratum dis-
placement. The grouting compaction expansion zone deter-
mined by the calculation method in this paper and
Fig. 20. Design and calculation process of grouting scheme. measured by the test are both 0.01 m. And the actual mon-
itoring value in the field is consistent with the calculated
value in this paper at the maximum displacement height
of ground uplift caused by grouting construction, which
culation model and analyzes the vertical displacement verifies the accuracy of the calculation method.
cloud and plastic deformation distribution cloud, as shown (2) A coupled calculation model was developed to ana-
in Figs. 18 and 19. lyze the deformation response between grouting construc-
Figures 18 and 19 depict the vertical deformation and tion and adjacent tunnels using finite element analysis of
plastic distribution resulting from Line 170 s grouting con- the interaction between the tunnel and surrounding soil.
struction in the existing tunnel. The imposed displacement After calculation, the maximum uplift of the existing tun-
boundary condition causes a noticeable plastic strain zone nel caused by three compensation groutings is 0.143,
in the No.6 shield tunnel, with concentrated plastic defor- 0.501, and 0.264 mm, respectively, and the calculation
mation appearing at the midline position and approxi- errors are 16.85%, 7.92%, and 14.23%, respectively. Under
mately 18 m from it in the new No.17 tunnel. this order of magnitude, the calculation error fully satisfies
Specifically, the shield tube sheet in the middle line position the engineering design and construction. This verifies the
has experienced bottom lifting displacement, causing shear reliability of the coupling calculation model in this paper,
damage to the shield tube sheet and bolts and potentially which can provide a theoretical basis for the precise lifting
leading to tube sheet misalignment. At about 18 m from and correction of the settlement of the existing tunnel
the center line, the shield tube sheet and bolts have been under the tunnel.
 P.-w. Jiang et al. / Underground Space 15 (2024) 312–330 329

(3) The coupled calculation model was used to analyze a Bouchelaghem, F., & Almosni, A. (2003). Experimental determination of
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Liu, B. C. (1993). Ground surface movements due to underground
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appeared to influence the work reported in this paper. Mair, R.J. and Hight, D.W. (1994), Compensation Grouting, World
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Foundation of China (Grant No. 52130905). on surface settlements. Tunnelling and Underground Space Technology,
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Lessons learnt from unusual ground settlement during double-o-tube