Numerical studies on tunnel floor heave in swelling ground under humid conditions

International Journal of Rock Mechanics & Mining Sciences 55 (2012) 139–150

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Numerical studies on tunnel floor heave in swelling ground under humid conditions S.B. Tang n, C.A. Tang Institute for Rock Instability and Seismicity Research, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

abstract

Article history: Received 31 March 2011 Received in revised form 14 May 2012 Accepted 13 July 2012 Available online 3 August 2012

A humidity diffusion based numerical model is proposed to simulate the floor heave processes of swelling rock tunnel when it exposed to high humidity. The phenomenon of swelling in tunneling is treated as a humidity–mechanical coupled process, i.e. the stress redistribution as well the water vapor diffusion around the tunnel is taken into account. This allows one to model the observed floor heaves realistically without considering the complex chemical processes induced by water–rock interaction. Furthermore, the development of heave and pressure over the course of time can be studied. The swelling rock is considered as an elasto-plastic material with damage threshold, which allows one to predict the large heaves of a tunnel floor that are often observed in-situ. The relationship between the mechanical damage and humidity diffusion are discussed at the mesoscopic level. By studying the influence of parameters in the numerical model on the floor heave behavior, the time-dependent deformation and failure processes of tunnel under high humid condition are discussed in detail. The deformations at the floor level are larger than that of sidewalls interpreted here as a consequence of different humid boundary conditions. The numerical results provide a better understanding of timedependent behavior of floor heave of tunnel under the high humid condition. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.

Keywords: Numerical investigation Floor heave Tunnel Humidity diffusion

1. Introduction In the swelling rock engineering, water is one of an important factor causing the floor heave of tunnel because of the water adsorption by the flakey structure of the clay minerals [1]. Most of studies in laboratory tests as well as in-situ observations show that considerable pressures develop when preventing the swelling strains. For example [2], the swelling behaviors in a large ¨ number of tunnels in Baden-Wurttemberg (Southwestern Germany) have caused high heave in unreinforced tunnel floors and strong swelling pressure against inverts of tunnels with resisting supports. In severe cases in which it was attempted to prevent or impede swelling strains by means of a stiff tunnel lining, swelling pressures often developed that were high enough to lead to the destruction of the tunnel lining, even a 30 cm thick concrete support was destroyed during construction along large tunnel sections by shear failure [3]. To avoid such failures methods for computing the swelling pressure and the deformation development history were proposed in the past several decades. The first analytical approach to the swelling problem was given in [4], where the stability of tunnel masonry-work in connection with the construction of the

n

Corresponding author. Tel.: þ86 411 84708694; fax: þ 86 411 87315655. E-mail address: [email protected] (S.B. Tang).

lower Hauenstein tunnel in Switzerland was studied. And then the simplified methods of Einstein et al. [5], Grob [6] and Kova´ri et al. [7] were based upon a priori assumptions concerning the stresses or strains along the vertical symmetry axis beneath the tunnel floor. However, they did not predict the complete deformation and stress fields. By treating the swelling process on a continuum-mechanical basis, other researchers [8,9] proposed the complete stress–strain relations of swelling rocks. WittkeGattermann and Wittke [3] also proposed a constitutive law to describe the swelling phenomena in a realistic way and successfully applied to the conditions of the exploration gallery for the Freudenstein tunnel at the high-speed railway line Mannheim– Stuttgart. Furthermore, the movement of water and the chemical processes in the swelling rock should also be taken into consideration in the numerical model, in which the seepage flow equations must be considered simultaneously to the equations of stress analysis. Therefore, Anagnostou [10] proposed an improved computational model for tunnels in swelling rock that the swelling in the tunnel is analyzed as a thermo-hydro-chemomechanical (THCM) phenomenon using a theoretical framework founded on basic geochemical and thermodynamical principles, making it possible the formulation of consistent triggering swelling events as well as the identification of conditions conducing to either their evolution or exhaustion. It is known that the main reason for the swelling of rock is the suction of water by the clay minerals. However, most of the priori

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studies focused on the influence of free water on mechanical behavior of swelling rock. Another type of water, i.e. water vapor, is seldom taken into consideration in the tunnel stability studies. In fact, numerous studies indicated that humidity is also an important factor effect on the mechanical behavior of rocks. The static fatigue tests conducted on granite and anorthosite have shown that in a humid environment the long-term strengths of these crystalline igneous rocks could be less than 60 percent of their dry instantaneous strengths [11]. Such reduction in strength has implications for the design and construction of deep tunnels, mines and other underground installations. As the same as free water, the influence of humidity on mechanical behavior of rock by two important ways: mechanical and physico-chemical effect [12–13]: the lowering of the effective stress lowers the fracture strength, and the physico-chemical effect results in the mechanical properties of rock being time dependent which is a corrosive processes. For the swelling rock, in addition to the mechanical responses above mentioned, the humidity also causes the volume change. However, in the past several years, studies pay more attention on the effect of moisture or humidity on mechanical behavior of rock, hardly any of researches on the floor heave of tunnel under humid condition, especially the relationship between crack growth and the time-dependent mechanical behavior during the period of floor heave, even the crack growth processes at the humid condition have not been clarified yet. Therefore, it is important to further study the long-term behavior of floor heave under the humid condition to improve engineering construction. When the mechanism of floor heave is correctly understood can a mechanic model representing the actual state be established, and then effective numerical calculation can be performed. Therefore, this paper describes the development and implementation of a two-dimensional, plane strain finite element model to calculate both the stress and time-dependent deformation mechanical behavior of swelling rock under high humid condition, and then focus on the discussion of influence factors on floor heave of tunnel.

2. Introduction of numerical model An important phenomenon of the mechanical behavior of swelling rock is time-dependent deformation [14]. The consequence of the time-dependent deformation of floor in swelling rock is the deterioration of rock strength and elastic modulus, resulting in the collapse of the tunnels. To study the timedependent deformation of rock, the rheology theories are broadly used in rock engineering [15–17]. The classic models of creep deformation are usually described by using viscoelasticity or viscoplasticity, which indicates that the time-dependent deformation is entirely attributed to the viscous effect. However, studies with the scanning electron microscope (SEM) suggested that the mechanisms of creep deformation and creep failure are the growth of microcracks nucleated at the pre-existing defects [18]. Shao et al. [19] also pointed out that the typical mechanisms of creep deformation can involve the sub-critical propagation of microcracks in hard rocks, pore collapse in highly porous rock, dissolution process due to chemical–mechanical coupling. It is clear that the classic approaches provide a mathematical description of creep, but do not take into account physical mechanisms [19], which indicate that it may be beneficial to study the timedependent behavior of rock (as well as floor heave) based on the progressive degradation of material structure in microscopic scale. In the viewpoint of classical fracture mechanics, a crack can propagate slowly even when the stress intensity factor is less than the critical level; this is called ‘‘subcritical crack growth’’. The main mechanism of subcritical crack growth is stress corrosion by

environment such as water and temperature [20–23]. Anderson and Grew [23] suggested that stress corrosion cracking plays an important role in the intrusion of magmas and in the transport of magmas upward through the lithosphere. According to the argument that subcritical crack growth is one of the main causes of time-dependent behavior in rocks [15,18,24–26], it can be concluded that stress corrosion is an important source of creep. Because water is one of the most important corrosive agents that influence rock performance, especially for soft rock, the investigation of floor heave by the influencing of water is vital for the timedependent behavior researches in the geological engineering. Furthermore, moisture or humidity is another form of water, which also behaves a time-dependent effect on the performance of rock. It has been shown the change of the relative humidity in air has strong effects on the crack growth in rocks, finally influencing its long-term behavior [27–28]. According to the above discussion, if the swelling rock tunnel is excavated under very humid conditions, in which the relative humidity at the free surface is higher than that in the surrounding rock, the humidity gradient becomes a driving force for moisture diffusion from high humidity zone to low humidity zone. It is known that the direct impacts of water invasion on surrounding rocks are the deterioration of rock strength and the expansion of rock volume. Both of these factors can potentially lead to an increase of deformation rate in rocks surrounding the tunnel, as well as result in stress field disturbance. Sometimes, damages or failures are inevitable. Because the humidity on the spatial distribution in surrounding rock is time-related which can be determined by humidity diffusion calculation, and the degradation of mechanical properties in the surrounding rock caused by humidity diffusion are also vary with spatial and time, both of them result in the time-dependent deformation. It can be concluded that the time-development swelling deformations can be modeled by the consideration of the humidity diffusion processes. In addition, because the water content in the floor is higher than any other part of the tunnel which results in great swelling and material properties degradation of swelling rock, it is easy to model the floor heave by considering the humidity movement and the corresponding mechanical responses. Furthermore, the swelling pressure can be large enough to lead to failures in the surrounding rock, accelerating the transportation of humidity on the surface of tunnel into the deep inside of surrounding rock, which means that the humidity diffusion and the mechanical response are coupled with each other, called humid-mechanical coupling process. Therefore, if a numerical model is aim to simulate the floor heave of tunnel under humid condition, it is important to consider the coupling among the behaviors of humidity diffusion processes, the degradation effect of humidity on physical properties of rock, the swell behavior of rock as it sucks up humidity, and the damage evolution processes of rock under the mechanical loading.

2.1. Heterogeneity consideration It is known that rock is a heterogeneous material at mesoscopic level. There are great different performances of rock between macroscopic and mesoscopic level because of its heterogeneity nature, such as the nonlinear deformation behaves at macroscopic level can be modeled by linearly elastic behaviors at mesoscopic level with heterogeneity consideration [29]. In the present model, to capture the heterogeneity of rock at the mesoscopic level, the mechanical properties of rock including the elastic modulus, the strength and the Poisson’s ratio are all assumed to conform to the Weibull distribution as defined by the

S.B. Tang, C.A. Tang / International Journal of Rock Mechanics & Mining Sciences 55 (2012) 139–150

φ ()

141

5.00×102 MPa

m=∞ m = 6.0 m = 3.0 m = 1.2

0



1.81×105 MPa

Fig. 1. Weibull distribution with different m index.

following probability density function:    m  m a m1 a jðaÞ ¼ exp

ao ao

ao

1.26×104 MPa ð1Þ

where a is a given mechanical property (such as the strength or elastic modulus), a0 is the scale parameter and m is the shape parameter which defines the shape of the distribution function. In the present study, the parameter m defines the degree of material homogeneity, is thus referred to as the homogeneity index instead. This approach of assigning material properties has been shown to be effective to reflect the heterogeneity of rock [29]. Fig. 1 shows the Weibull distribution of parameter a. It can be seen that the mechanical property of meso-elements are distributed in a more and more narrow range around parameter a0 with an increase in homogeneity index, indicating that the mechanical property such as strength is more homogeneous with a higher homogeneity index, and vice versa. To obtain a numerical sample which the mechanical properties of meso-elements following the Weibull distribution law, the sample is discretized into a large number of small elements at the beginning of numerical modeling, where the small elements are called meso-elements. And then a series of random numbers represented by o1, o2y, on (where n represent number of mesoscopic elements) ranging from 0 to 1 are generated based on the Monte-Carlo method. Finally, the mechanical properties of each element ai can be obtained as   1=m 1 ai ¼ ao ln ð2Þ 1oi The numerical samples with homogeneity index m¼1.5 and 5 are shown in Fig. 2, where both of them contain 200  200 elements and have the same mechanical properties except for homogeneity index. The gray shadings of different degrees represent different values of the elastic modulus, with lighter shadings indicating higher values of the elastic modulus, and vice-versa. It can be seen from Fig. 1 that a greater homogeneity index leads to more elements with the elastic modulus similar to the scale parameter a0 (¼50 GPa in Fig. 2), and values change on a smaller scale range from 12.6 GPa to 73.6 GPa in Fig. 1a. However, with a smaller homogeneity index means that the elastic modulus change on a larger scale which range from 500 MPa to 181 GPa in Fig. 2a.

7.36×104 MPa Fig. 2. Numerical samples of rock with homogeneity index (a) m¼ 1.5 and (b) m ¼5. The different degrees of gray shading reflect the distribution of elastic modulus over the specimen, with lighter shadings indicating higher values of the elastic modulus, and vice-versa.

taken into account. The moisture content of each point in surrounding rock will change with time, as defined by the mass transport function. However, it is difficult to determine the change of humidity in the surrounding rock under the humid condition because it depend on many factors such as the permeability of rock, original fractures and porous of rock, stress state and time factors, etc. Philip and de Vries [30] pointed out that Boayoucos was the first to carry out a detail experimental study on moisture transport in soil in 1915, and Lewis reported a simple but easily accepted theory model which assumed that the moisture movement in porous material conforms to Fick’s law in 1921. Luikov [31] and Kim and Lee [32] also pointed out that moisture transport in porous bodies can be modeled by the Fick’s law. Moisture transport is usually coupled with heat transfer, and the phenomenological theory developed by Philip and DeVries [33] yields the following differential equation which governs moisture movement in the unsaturated porous solid under the combined action of thermal and moisture gradients:

2.2. Humidity diffusion equations

@y @K ¼ rðDy ryÞ þ rðDT rTÞ þ y @t @z

In the swelling rock engineering, besides of the effects of the initial stress and the excavation, the action of water must also be

where y is the volumetric moisture content; T is the temperature (1C); r is the gradient or del operator; DT is the thermal moisture

ð3Þ

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diffusivity, Dy is the isothermal moisture diffusivity; Ky is the unsaturated hydraulic conductivity. The terms Dy and DT are made up of two components related to the liquid and vapor phases. The last term on the right-hand side of Eq. (3) accounts for gravitational effects. If temperature changes in the excavation tunnel are small, and the gravitational effect of moisture is neglected, Eq. (3) can be written as a pure diffusion equation: @y ¼ rðDy ryÞ @t

ð4Þ

In a certain range, moisture content behaves linearly with relative humidity. Then another form of Eq. (4) is given by the following partial differential equation: @h ¼ rðDh rhÞ @t

ð5Þ

where h is relative humidity, t represents time, Dh is the humidity diffusion coefficient which depends on the humidity degree and damage evolution. The boundary conditions governing Eq. (5) is divided into two categories: an essential (Dirichlet) boundary condition and a natural (Neumann or Robin) boundary condition. The essential boundary condition for the humid field can be written as h¼h

for xi A S1

ð6Þ

and the natural boundary condition for the moisture loss from region S2 can be expressed in the form: Dh

@h ¼ g h ðhhen Þ @n

for xi A S2

ð7Þ

where h is the relative humidity on the surface of tunnel (S1), gh is the moisture exchange coefficient and hen is the relative humidity of the environment in the tunnel. Furthermore, the description of a problem will be complete when suitable initial conditions are imposed on relative humidity (h) i.e., h ¼ ho

for xi A O:

ð8Þ

2.3. Degradation of mechanical properties with humidity The increase in humidity definitely causes the degradation of rock mechanical properties. In fact, early in 1960s, researchers have taken note that the role of water on rock is not only the impact on the effective stresses, but also the reduction in rock strength [34–35]. Laijtai et al. [36] studied the time-dependent deformation in rock caused by water. Compared with air-dry state, the elastic modulus for saturated granodiorite was reduced by about 40%, and the ratio between the transversal and axial deformation was increased by more than 10%. Colback and Wiid [37] were among the first to investigate strength reduction in sandstones due to moisture content. They maintained that the reduction in strength was due primarily to a reduction in tensile strength, the liquid reducing the surface free energy of the rock and therefore the energy of fracture propagation, so leading to reduced strength and less brittle deformation. Hawkins and McConnell [38], and Bell and Culshaw [39] upheld this concept. The degree of sensitivity to moisture content of mechanical behavior, such as strength, elastic modulus, Poisson’s ration, depends on the physical properties of rock, the stress state, moisture, weight and other factors. A large number of experimental results show that for rocks in coal mining, such as shale, mudstone and siltstone, the uniaxial compressive strength and elastic modulus behave a linear degradation with water content [40]. Va´sa´rhelyi [41] analyzed the published data and showed

that there is a linear correlation between the dry and fully saturated uniaxial compressive strengths. However, Hawkins and McConnell [38] carried out tests to determine the influence of the water content on the strength of fifteen sandstones. They found that the relationship between water content and uniaxial compressive strength could be described by an exponential equation. In this paper, the impacts of humidity on the mechanical properties (strength, elastic modulus and Poisson’s ratio) are considered as the following relationship: 8 Xo for ho hcri <   X¼ ð9Þ hhcri or h o hcri : X o 1 1hcri o where X and X0 represent the current and initial mechanical properties, respectively. h is the current relative humidity and hcri is a critical humidity value, which means that as the humidity is lower than hcri, the change of the mechanical properties of rock is less obvious and can be neglected, but as the humidity is higher than hcri, the degradation of material properties by humidity must to be considered. Parameter o is the degradation coefficient for mechanical properties with humidity. 2.4. Swelling of rock There are numeroius theories and methods to study the swelling behavior of swelling rock. According to the simplest continuum-mechanical model, swelling is considered as a pure stress-analysis problem. However, the stress-analysis proves to be insufficient for a realistic modeling of the observed behavior because it predicts swelling not only in the tunnel floor, but also at the crown and walls [7]. An improved computational model for tunnel in swelling rock should take the movement of water into consideration, which solves the equations of seepage flow and the stress analysis simultaneously, i.e. coupled hydraulic–mechanical model. Biot’s poroelasticity theory provides the simplest model for a hydraulic–mechanical process. However, the numerical simulations show that the modeling of larger floor heaves is not possible on the basis of elasticity theory, which is because that neglecting the limits to rock strength leads to inadmissible stress fields, as well as to a severe underestimation of swelling deformation [10]. The swelling processes of swelling rock not only contain hydro-mechanical behavior, but also include complex chemical phenomena, indicating that a consideration of the anhydrite–water–gypsum system requires a coupled hydraulic– mechanical–chemical model [10]. However, it is difficult to deal with the chemical action in the water–rock system in the numerical modeling. For the computational rock engineering, the main concern is the change of mechanical parameters of rock over the time of wetting processes, as shown in Eq. (9), which simplifies the complex mechanical–chemical computation. As the tunnel subjected to wetting by the humidity, the swelling behavior result in large floor heaves and an absence of comparable deformations at the walls and crown. There are two main factors causing the change of stress and strain fields in the surrounding swelling rock in tunnel due to the action of wetting. One is because the freely dilated deformation of rock is limited by both internal and external restraints, and the other is the softening of partial wetting rock. By the movement of moisture, the moisture content of each point in the surrounding rock will change with time, resulting in the stress and strain fields also change along with the movement of moisture. There are three main processes as the swelling rock subjected to wetting which like that of the thermal behavior of materials [42]: (1) the humidity (thermal) transport within the swelling rock (materials) leads to the change of a humidity (thermal) field; (2) the dilation and softening of rock mass (materials) are caused by the change

S.B. Tang, C.A. Tang / International Journal of Rock Mechanics & Mining Sciences 55 (2012) 139–150

of a humidity (thermal) field; (3) the change of stress and displacement fields is caused by the dilation and softening. Furthermore, these three processes are coupled with each other. It should be mentioned that the dilation and softening caused by wetting in a swelling rock are similar to those by the thermal effects of a material, but the physical mechanism is different. The increase of temperature causes the increase of molecule kinetic energy, resulting in the changes of the volume and the physical properties. However, those changes under the wetting condition is caused by the increasing of the hydrated velum thickness in the swelling rock and the decreasing of the binding force among molecules, more complex than that of thermal condition. It can be observed from above that the mechanical processes of swelling rock under wetting condition can be modeled as that of thermal behavior of material, in which the first process is modeled by Eqs. (5)–(8), and the softening of swelling rock in the second process is modeled by Eq. (9). The dilation of rock as it sucks up humidity in the second process can be modeled as that of thermal expansion, which is given by [42]

eh ¼ bðhÞDh

ð10Þ

where eh is humidity strain, and b(h) is the swelling coefficient which depends on humidity status, Dh is the humidity change variable. The stress caused by the changing of humidity in the swelling rock is called humidity stress, as shown in Eq. (10). Bai and Li [43] theoretically proved this type of humidity stress field based on the thermodynamics, suggesting that the total strain under the condition of the coupling of stress and humid field can be decomposed into elastic strain and swelling strain. Therefore, according to the thermal stress, the third process mentioned above can be modeled by humid-mechanical coupling stress, i.e. the constitutive and equilibrium equations of swelling rock under the coupling of external loading and humidity stress is given by   1þn n skk dij þ bDh eij ¼ esij þ ehij ¼ sij þ  ð11Þ E 1þ n where i, j, k¼1, 2; sij and eij are the stress and strain components, E and v are the elastic modulus and Poisson’s ratio, respectively, esij and ehij are the elastic strain and the humid strain, respectively. According to Eqs. (5), (9) and (11), the humidity diffusion induced stress evolution can be calculated by a coupling process. The large deformation at the floor, i.e. floor heave, also can be modeled by a simple manner without considering the complex chemical behavior in the swelling rock as it sucking water. Furthermore, the humid stress can be large enough to cause failures around the surrounding rock of tunnel. It is necessary to consider the damage behavior of rock at this situation, which is described below.

2.5. Damage process for the failure behavior First, the sample is discretized into a large number of small elements, and the numerical model will automatically assigns properties to each element as Weibull distribution function descript by Eqs. (1) and (2). And then, finite element method (FEM) is used to perform the humidity diffusion process and stress distribution analysis. After that, the stress filed will be examined using failure criterion, i.e. Mohr–Coulomb criterion with tensile cut-off, and the latter being the preferential one. If the stress states of each meso-element reach the threshold determined by failure criteria, damage in the meso-element is occurred. For the damage processing, continuum damage mechanics (CDM) has been extensively applied to model the progressive degradation of materials caused by microcracking [44].

143

The classical CDM can be described as following E ¼ ð1dÞE0

ð12Þ

where d represents the damage variable, which ranges from zero for the undamaged material to one for fully damaged; E and E0 are the elastic modulus of the damaged and the undamaged material, respectively. It must be pointed out that the element and its damage are assumed to be isotropic and elastic, and therefore E, E0 and d are all scalars. For each element, the stress–strain curve of each element is assumed to be linear elastic until the attainment of a damage threshold, followed by strain softening. As mentioned above that there are two types of damage in the presented numerical model, i.e. tensile and shear damages. When the tensile stress of a mesoelement is higher than its maximum tensile strength, i.e. s3 r st, tensile damage occurs. And the corresponding damage variable is given by 8 e o et0 > <0 Ze d ¼ 1 et0 et0 r e o etu ð13Þ > :1 e Z etu where Z is the residual strength coefficient (0  1), i.e., Z ¼ str/st and str is the residual strength for damaged element. Et0 is the ultimate elastic strain, i.e., et0 ¼ st/E0, and E0 is the elastic modulus for undamaged element. Etu is the maximum tensile strain, i.e. when the tensile strain for an element reaches etu, the element completely loses bearing capacity. However, when the shear stress on an element exceeds its shear strength, shear damage occurs. And the mechanical damage variable d is defined as ( 0 e o ec0 d¼ ð14Þ 1 Zeec0 e Z ec0 where Z has the same definition as for tensile damage, ec0 is the maximum principal strain when the maximum principal compressive stress reaches the uniaxial compressive strength of the element, i.e., ec0 ¼ sc/E0, and sc is the compressive strength of the element. Furthermore, the damage evolution not only softens the mechanical properties, but also accelerates the humidity diffusion in the surrounding rock. In order to take account this type of effect on the humidity diffusion processes, the relationship between damage and the humidity diffusion coefficient is given by 8 D d¼0 > < h0 Dh ¼ xDh,d 0 od o dcri ð15Þ > :D d Z dcri h,cri where Dh0 is the initial humidity diffusion coefficient of rock, dcri is a critical damage value which means that as the damage is equal or larger than dcri, the humidity diffusion coefficient suddenly increase to a constant value Dh,cri. Dh,d is the humidity diffusion coefficient at the time that the element first damage, and x is a mutation coefficient of diffusion.

3. Modeling strategy Under the high humid condition of underground engineering, elastic modulus and strength will degrade with humidity increasing. The increased humidity also results in swelling of rock. Therefore, damage may occur during the humidity diffusion process. To model the floor heave and damage processes caused by humidity diffusion, the model shown in Fig. 3 is adopted as an example, where points A and B are at the center of floor and right

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250

Overburden loading

Weight Lateral Pressure

Lateral Pressure

Point B

Displacement (mm)

200

150

Point B Point A

100

50

0

Point A

0

20

40 60 Time (day)

80

100

Fig. 4. Relationships between the time and the displacements at Point A and B.

300 250

Fig. 3. Numerical model for swelling rock tunnel in which Point A and B are at the location of center of floor and sidewall, respectively.

sidewall, and are used to observe or measure the deformation at these locations. The model size is large enough to ignore the boundary effect. The horizontal width and the vertical height are all 200 m. The inversed U-shaped tunnel is in the center of model. The radius of the inversed U-shaped is 16 m and the height is 12 m. The model is discretized into 500  500 (250,000) elements. According to the practical experience and mining theory, the upper boundary is loaded by the gravity stress of overburden rock. And the confining stresses are applied on the left and right boundaries. The bottom of numerical model is set to vertical displacement constraint. The initial humidity of the model is 20%. And the humid boundary condition at the floor of tunnel is 100%. However, the roof and the sidewalls of tunnel keep on 50% humid condition. The initial elastic modulus, the Poisson’s ratio and the density of surrounding rock is 1000 MPa, 0.25 and 2000 kg/m3, respectively. The initial humidity diffusion coefficient is 10  9 m2/s, and the initial relative humidity of surrounding rock is 20%. Other parameters are described in the following sections. In order to study the influence of swelling, the degradation of mechanical properties by humidity, and the coefficient of lateral pressure on floor heave behavior, different rock properties and loading conditions are used in the numerical simulation. Furthermore, the damage processes, i.e. the crack initiation, propagation and coalescence will also be simulated to study the mechanism of floor heave at the high humid condition.

4. Results and discussion 4.1. Effect of elastic modulus degradation by humidity on floor heave of tunnel To study the effect of elastic modulus degradation by humidity on floor heave of tunnel, the parameter o in Eq. (9) is set to 0.3, 0.6 and 0.8, respectively, but with other parameters remaining unchanged. Fig. 4 shows the relationships between time and the displacements at Point A and B shown in Fig. 3. It can be seen that

Displacement (mm)

Vertical displacement constraint 200 150 ω=0.3 ω=0.6 ω=0.8

100 50 0 0

20

40 60 Time (day)

80

100

Fig. 5. Relationships between the time and the displacements at point A with different degradation coefficients of elastic modulus.

the deformation at the beginning of excavation rapidly increase over time. If the tunnel maintaining the relatively high humidity condition, moisture will continuously diffuse into the surrounding rock, which results in further increase of the deformation of surrounding rock of tunnel. However, the change rates of displacement become smaller and smaller. The displacements at the location Point A tend to be stable after 30 day of excavation. For the underground engineering, the water consumption for construction results in a lot of water accumulation on the floor, causing great degradation of elastic modulus and strength of the floor strata. Therefore, the transportation of humidity in the floor strata enhanced its deformation capacity, which can be seen from Fig. 4 that the deformation at point A is larger than that of point B. This explains why bottom heave is likely to occur in humid tunnel, especially for swelling soft rock. Fig. 5 shows the time-dependent displacement of point A with different degradation coefficient of elastic modulus. It can be seen that with the increase of degradation coefficient of elastic modulus, the floor heave of the tunnel becomes more serious. The largest displacements under the three conditions are 161, 205 and 296 mm, respectively. The floor heave at the state of o ¼0.8 is almost two times larger than that at the state of o ¼0.3. It has been recognized that the maximum degradation of elastic modulus of red sandstone and granodiorite are 60% and 20% as they in saturated state, i.e. o ¼ 0.6 and 0.2, respectively. Therefore, for the red sandstone floor, the humidity diffusion is significantly influencing the floor heave of

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145

20%

100% 1 day

20 day

100 day 0MPa

19.6MPa 1 day

20 day

100 day

Fig. 6. Humidity diffusion processes (upper three figures) and associated change of shear stress distribution (the lower three figures) in the surrounding rock. The gray colors represent the magnitude of relative humidity and stress, with lighter colors indicating higher values, and vice-versa.

4.2. Effect of the coefficient of lateral pressure on floor heave of tunnel Tectonic stress is caused by the movement of the earth’s crust. For deep excavations, the level of tectonic stress is greater than the vertical stress. Therefore, to study the influence of coefficient of lateral pressure on floor heave of tunnel, the ratio of l ¼ sH/sv is set to 0.5, 1.0, 1.5 and 2.0, respectively, where sH and sv are the

350 300 Displacement (mm)

tunnel which is necessary to taken some strengthen method such as grooving in the corners or both sides of tunnel [45]. Fig. 6 shows the humidity diffusion processes in the surrounding rock and the associated change of shear stress distribution. It can be seen that, as moisture diffuses from the surface to the rock at a greater depth, the stress distribution around the tunnel also change. The increasing in humidity results in degradation of rock elastic modulus and the stress concentration transfer to the deep of surrounding rock. For the underground engineering, it is commonly recognized that no rheological phenomena occur before excavation. It is because the geological body has experienced evolutions over millions of years, and reached a state of equilibrium in a stable geological environment. However, when the geological body is excavated, the open surface is exposed to the external high humid condition. And as the humidity in the external environment is higher than that of surrounding rock, the surrounding rock absorbs moisture from the environment. In the early stage of excavation, the environmental humidity disturbs the surrounding rock only on the surface of excavated space. And as time elapses, the humidity at the surface spread into the surrounding rock and lead to additional time-related deformation as shown in Fig. 4. It indicates that at the high humid condition, the time-dependent deformation is not only caused by stress, but also affected by environmental factors such as humidity. Furthermore, the deformation caused by the environmental factors is not only a time function but also a function of spatial coordinates. It is significantly different from the stress caused time-dependent deformation. The results shown in Fig. 4 provide a better understanding of time-dependent behavior of floor heave of tunnel under the high humid condition.

250 200 150 100

λ=0.5 λ=1.0 λ=1.5 λ=2.0

50 0 0

20

40 60 Time (day)

80

100

Fig. 7. Relationship between the time and displacements at point A with different confine pressures, where l is the coefficient of lateral pressure.

horizontal and vertical stress of the model, and l is called as coefficient of lateral pressure. Fig. 7 shows the modeling results of floor heave at different sH/sv level. It can be seen that the displacement at point A in the tunnel increase with the increasing of sH/sv. The maximum displacement is 137 mm as l ¼0.5. And when the coefficient of lateral pressure reach to 2.0, the maximum displacement increase to 339 mm, which is about 2.5 times than that of l ¼0.5. For the inverted U-shaped opening, the tensile stresses zone in the roof and floor are found when the opening is subjected to uniaxial compression [46]. It suggests that the remote cracks are possible formed in the roof and floor near the sides of the opening by the action of tensile stress. For this situation, the water on the floor is much easier infiltrating into the surrounding rock, resulting larger swelling deformation, i.e. floor heave. With the increasing of lateral pressure, the stresses in the floor gradually change to compressive stresses, and the largest value of stresses occurs at re-entrants between the floor and the sidewalls. The generation of compressive stress reduces the permeability of the rock, which is helpful to prevent the diffusion of humidity into the rock. However, the compressive stress in the floor is able to cause

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great deformation upward into the opening, still resulting in floor heave. It should be mentioned that, as the lateral pressure increase to a certain degree of stress level, a tensile stress zone is observed in the sidewall of the opening, which should be paid more attention of failures.

swelling properties of rock [47]. Furthermore, applying floor bolt is an effective method to prevent the floor heave of tunnel. 4.4. Effect of damage processes on floor heave of tunnel The discussion mentioned above did not consider the damage of tunnel during the humidity diffusion process. However, the

4.3. Effect of swelling coefficient on floor heave of tunnel

750 1 day 10 day 30 day 60 day 70 day

600 Displacement (mm)

The swell of rock is an important parameter influence the floor heave of tunnel. In general, the rock strength degrades with the increasing of water content, but with a small volume change. However, for some swelling rocks, because of the containing of montmorillonite, severe volume increased after soaking, resulting great heave of floor. Fig. 8 shows the numerical results of floor heave with different swelling coefficients, i.e. the parameter b in Eq. (10). It can be seen that the displacement of floor heave significantly enhanced with the increase of swelling coefficient. All of the deformations at point A with different swelling coefficients increase rapidly at the beginning of 25 day excavation. And then with the continue exposure of tunnel into the high humid condition, the growth of deformations under the four situation begin to decrease and gradually becoming convergence. Since the humidity diffusion coefficient of rock is affected by the excavation, due to newly produced cracks, the humidity diffuse in the surface layer of floor rock with very rapid speed, but with very slow speed in the undisturbed surrounding rock. This is the reason for rapid increasing of deformation at point A at the beginning excavation and then gradually decreasing to converge. The larger value of the swelling coefficient indicates the clearer of this phenomenon. The results also indicate that the maximum floor heave of b ¼0.009 is 1.8 times greater than that of b ¼0. To inhibition of the expansion of rock, it can use the physical, chemical and mechanical methods or combination of these methods. When the floor accumulate water, it is important to drain it to reduce to the maximum water content of surrounding rock on the bottom of tunnel. This is because that if there is no water supply in floor strata would decrease the water content of rock and consequently reduce the floor heave of tunnel. In addition, it is useful to pour some quicklime on the floor strata, which not only reduces moisture content in the swelling rock by absorbing water, but also produces Ca2 þ ion which will exchange with the cations in the montmorillonite so as to change the

450

300

150

0 0

3

6 9 Distance (m)

12

15

Fig. 9. Displacement evolution along the floor of tunnel.

20%

300 100%

Displacement (mm)

250 200 150 100

β=0 β=0.006 β=0.009 β=0.009

50 0 0

20

40 60 Time (day)

80

100

Fig. 8. Relationships between the time and displacements at point A with different swelling coefficients.

Fig. 10. Humidity distribution after 70 day.

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after 1 day

after 10 day

after 30 day

after 60 day

after 65 day 20%

147

after 70 day magnitude of humidity

100%

Fig. 11. Humidity diffusion processes during floor heave associated with failure evolution.

strength of surrounding rock degrades with the increasing of water content. Under the compression of the tectonic stress and the gravity stress, the surrounding rock, especially at the floor of tunnel, is easy damage or failure. The damage evolution not only softens the mechanical properties, but also accelerates the humidity diffusion in the surrounding rock which is shown in Eq. (15), resulting in grate floor heave. This is a coupling process of humidity diffusion and damage. In this section, to study the damage effect on floor heave of tunnel, the initial strength of surround rock is set to a low value, i.e. 20 MPa.

Fig. 9 shows the displacement evolution along the floor of tunnel after excavation. It indicates that the displacement of floor increase with time elapsed. The analytical solution shows that the largest displacement occurred at the middle of floor. However, due to the effect of heterogeneous feature and damage of rock, the largest displacement not happened at the middle point of floor as that of analytical solution but a certain deviation, which is shown in Fig. 9. It also can be seen from Fig. 9 that the changing rate of displacement at the beginning of excavation is larger than that of 30 day later. This phenomenon is due to the humidity at the

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beginning of excavation keep on a high diffusion rate, causing the mechanical properties of rocks in the floor strata also with a higher weakening speed and then resulting in the floor keep a higher deformation rate. During this period, the damage in the floor strata also plays an important role accelerating the deformation. However, as time has experienced 30 day, the decreased humidity diffusion of the floor rock results in the decreasing of deformation rate. Fig. 10 shows the humidity distribution as time experienced 70 day. It can be seen from the failure pattern of the tunnel that plenty of failures occurred at the floor of tunnel, however, only a few failures produced on the two sides and the roof of the tunnel, which means that the humidity diffusion induced failures most likely to occur in the floor strata of the tunnel, and should be paid more attention. Fig. 11 shows the humidity diffusion processes with failure evolution during the diffusion of humidity. Because the initial humidity at the floor is higher than that of roof and the two sides of tunnel, there is a larger humidity gradient in the floor strata than that in the sidewall rock, resulting in humidity change more rapidly in the floor strata than any other zone in the tunnel. It can be seen from Eq. (9) that elastic modulus and strength of rock at the bottom of tunnel degrade significantly because of more moisture content. The degradation of elastic modulus denotes that with the same loading, the floor would generate a greater deformation. And the degradation of strength means that the rock

at the bottom of tunnel is much easier to fail, such as shown in Fig. 11 as the time experienced ten days after excavation of the tunnel. With the humidity diffusion, material properties in a deep range of surrounding rock decreases with the increase in humidity, accelerating the floor heave of tunnel. Fig. 12 shows the shear stress distribution as time after seventy days excavation of the tunnel. It can be seen from this figure that high shear stress concentrated around the tunnel. And higher tensile stress occurred in vicinity of bottom of tunnel, leading to failure occurred firstly on the surface of floor. Fig. 13 shows the shear stress evolution processes and the failure occurrences after the time of excavation. With failures gradually transmit from the surface of tunnel floor to deep rock, the floor heave get more and more serious. As the underground engineering exposed to high humid condition, the constraint of floor released, which denotes that the floor would generate deformation to the free surface. In addition, the floor still needs to bear the weight of overlying strata. Moreover, the humidity diffusion processes also play an important role in the deformation of tunnel floor. These three types of deformation consists the overall deformation of floor heave. It can be seen from the simulated results that, for the engineering under high humid condition, humidity is the major factor for its time-dependent deformation. This type of deformation would ultimately lead to failure of surrounding rock and the floor heave. If the underground engineering undergoes many times of floor heave, it is more difficult to maintain its stability.

0 MPa 5. Conclusions

14.2MPa

Fig. 12. Shear stress distribution after 70 day. The gray levels represent the magnitude of shear stress, in which the lighter of the gray denotes higher shear stress.

As the underground engineering exposed to high humid condition, the free surface subjected to the action of humidity, and the mechanical properties of surrounding rock degrade with the humidity increase. Based on the point of view of humidity diffusion, the effect of humidity on stability of the floor heave is discussed in this paper. When a tunnel is exposed to external environment with high humidity, humidity is the major factor for its time-dependent behaviors. The change in humidity not only results in humidity stresses in rock but also significant degradation of rock mechanical properties, such as elastic modulus and strength. Stress disturbance due to humidity change leads to different degrees of damage in rock and in turn enhances the humidity diffusion capacity. This process would further disturbance on humidity gradient and leads to new humidity diffusion process and additional deformation in the tunnel floor. The higher humid condition at the floor of tunnel is an important factor that causes the floor heave behavior. The studies on floor heave of tunnel under high humid condition found that: (1) The deformation of floor heave at the beginning of excavation rapidly increases over time. However, with the further diffusion of humidity in surrounding rock, the change rates of displacement become smaller and smaller, and finally tend to be stable. Furthermore, the increase in humidity results in degradation of elastic modulus of surrounding rock and the stress concentration transfer to the deep of surrounding rock. (2) The floor heave of tunnel increase with the increasing of sH/sv, which indicates that the cavity shape should pay attention to the in-situ rock stress. (3) The floor heave of tunnel is significantly enhanced by the increasing of swelling coefficient. However, the convergent deformation rate decreases with the increasing of swelling coefficient. The larger value of the swelling coefficient indicates the clearer of this phenomenon. (4) The modeling of damage processes with stress redistribution indicates that humidity diffusion in underground engineering

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0 MPa

after 1 day

after 10 day

after 30 day

after 60 day

after 65 day

after 70 day

magnitude of shear stress

149

14.2MPa

Fig. 13. Shear stress evolution processes during floor heave.

greatly degrade the strength of rock and subsequently result in great deal of fractures. The initiation, propagation and coalescence of fractures lead to the displacement of floor greatly increased. (5) Under the high humid condition, the time-dependent deformation of rock is not only caused by stress, but also affected by environmental factors such as humidity. Furthermore, the deformation caused by the environmental factors is not only a time function, but also a function of spatial coordinates,

which significantly different from the stress caused timedependent deformation. Acknowledgment This project was financially supported by the National Basic Research Program of China (973 Program, Grant No. 2011CB013 503), the National Natural Science Foundation of China (51121005, 51004020), the China Postdoctoral Science Foundation funded

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