COUPLED FLUID FLOW AND ELASTO-PLASTIC DEFORMATION MODEL OF ROCK MASSES WITH FINITE-LENGTH JOINTS AND ITS APPROACH IN XILUODU HYDRO PROJECT

Sheng Jinchang, Su Baoyu, Zhao Jian

College of water conservancy and hydropower engineering,

Hohai University, Nanjing, China

Abstract: Jointed rock masses is characterized as equivalent continuum medium. A coupled fluid flow and elasto-plastic deformation model of rock masses with finite-length joints is presented, which emphasized the mechanical deformation behavior and its relation with permeability of finite-length joints. As an example, the model and FEM analysis is applied in seepage control analysis of Xiluodu hydraulic project. The results show that the coupled effect has unfavorable influence on the safety of dam and work of underground grottos.

Keywords: jointed rock masses, seepage field, stress field, coupled, finite-length joint

1  INTRODUCTION

Rock masses are characterized by the existence of distributed joints. The joints are often viewed as long and infinite ones in coupled fluid flow and stress analysis. In fact, joints in rock masses are not infinite and cut through the REV (Representative Element Volume). If joints are simplified as infinite ones, the bounding action of the matrix around joints is neglect, the displacement of joints may be enlarged, and so is the affect of stress on permeability. So the coupled action should be considered in the coupled fluid flow and stress analysis model.  To solve the above question, the complete coupled fluid flow and elasto-plastic deformation analysis model, which is based on reference 1, is established by characterization of jointed rock masses as equivalent continuum media. The proposed model for jointed rock masses is implemented into a four-freedom element analysis program to analyze seepage control optimization design of Xiluodu hydraulic project.

2  CONTROL EQUATIONS OF COUPLED ANALYSIS

Jointed rock masses are generalized as equivalent continuum medium on both mechanical behavior and permeability. Coupled fluid flow and deformation analysis is formulated by taking four-freedom complete coupled model. The detailed control equations (seepage continuity equation, seepage Darcy’s law, static equilibrium equations, elasto-plastic constitutive relations of jointed rock masses, and etc.) follow mainly the procedure in Sheng (2000) and Sheng (1999). Following, the constitutive relations of a joint and jointed rock masses are given first, then a simplified method is introduced to improve the elasto-plastic constitutive model of finite-length joints, and the influence of stress on permeability of finite-length joints is discussed.

2.1  The constitutive relation of joint

A strong non-linear relation exists between the deformation and stress of a joint. The aperture of a joint affects directly the permeability of the joint. The constitutive relation of a joint is treated as elasto-plastic model. The elasto-plastic constitutive relation of a joint is^[2]:

                                                     (1)

2.2  The elasto-plastic constitutive relation of jointed rock masses

An average method is adopted here. Constitutive relation of jointed rock masses is established based on the average stress and average strain of REV. The average stress and average strain of REV can be defined as follows:

                                                  (2)

where V refers to the volume of REV.

By taking constitutive relation of a joint into the formula below, the elasto-plastic constitutive relation of jointed rock masses is:

                                                 (3)

2.3  A simplified mtehod to calculate finite-length joint

If the increments of the traction acting on the joints are expressed in terms of the increment of the average stress, the complete incremental constitutive relation is obtained. Depending on the joint geometry in the REV and the mechanical properties of the joint and the surrounding matrix, the average incremental stresses over the REV by^[4,5]

                                              (4)

Where  is called the “joint stress concentration tensor”(SCT)^[4], and  are the average incremental stresses over the REV in the local coordinate system. Note that the SCT should depend on the joint geometry as well as the mechanical properties of both the joint and the matrix.

When the joints are all long enough and cut through the REV,  is simply given by unit matrix. When the joints are not infinite, it is not possible to have general analytical solution for the SCT. In order to provide the stress concentration tensor for general situation, a simplified method has to be developed.  Let us consider a REV that contains one joint only (see Fig.1a). The matrix is isotropic and linearly elastic. The average incremental stress of the REV is denoted by  and the average incremental stress over the joint is denoted by . The problem is decomposed into a homogeneous problem, a problem of an open slit with non-zero traction on it, and the joint itself. In the homogeneous problem, an infinitely extended solid under zero stresses at infinity has the traction  on the slit surface. The condition to be satisfied in this decomposition is that the incremental displacement of the joint due to the incremental traction on the joint  equals the displacement jump of the slit in the sub-problem due to the incremental traction . Since the relationship between the incremental displacement jump of the joint and the incremental traction is given by the constitutive equations of the joint presented in equation (2), the task is to find the relationship between the displacement jump of the slit and the traction in the sub-problem. As a modeling of the problem, the remaining material of the sub-problem can be replaced by three springs as show in Fig.1e. The relationship between the incremental displacement jumps and the incremental traction on the slit is characterized by three tangential stiffness , , and  which are called the normal system stiffness and the shear system stiffness, respectively.

Fig. 1  Decomposition and simplification of the closed joint problem

2.4  The relationship between permeability and stress of joint

Suppose the joints within REV can be generalized as N groups, among which the unit normal vector component in group k is , the average aperture is , the communicating rate of joint is , the average linear density is , then the seepage tension of equivalent continuum medium is:

                                         (5)

Where, varies with stress, which can be calculated by equation (1).

3  ENGINEERING APPLICATION IN XILUODU HYDRO PROJECT

There are many bedding disturbed belts, intraformational disturbed belts and several sets of joints in rock masses. Its permeability is largely affected by such factors as geologic structure, weathering, unloading and carst formation. There are mainly six groups of predominant joint system in the basalt rock masses of the dam site. But in different sections rarely do three groups of joint emerge. The underground grottos are set in basalt rock masses that are more than 200m below the normal pool level 600m, sitting on the upstream side of the dam axis. The key retaining structure is concrete arch dam that is 285 meters. The dam body and dam shoulder will bear huge load after impoundment of the reservoir. Therefore, how will the interaction between seepage field of the rock masses and stress field affect the sliding resistance of rock stricture planes? This is an important issue that deserves our serious attention, for the failure of Malpasset arch dam ⁽¹⁾ is closely related to such a problem.

3.1  Calculation scope and calculation parameters

This paper takes the rock masses of left bank as the object of study. Owing to the problem of boundary conditions, calculation scope is very large. The distance between upstream and downstream boundaries is 4km. The distance perpendicular to the axis of river is about 1.5 km (referring to Fig.2). The wall of grottos is assumed to be non-permeable boundary. The upstream water level in the calculation is taken as 600m, a downstream one 382.4m. The dam is supposed to be the non-permeable material. There are four groups of joints, its mechanical parameters and geometry & physics parameters are shown in Table 1 and Table 2.

               Table 1  Mechanical parameters of various materials in Xiluodu Dam

         Table 2  Geometrical and physical parameters of various joint groups for the left-side

3.2  Calculation results and analysis

The following schemes plan to be applied for comparative analysis so as to facilitate analyzing the interaction of coupled seepage and deformation.

No.1 The uncoupled analysis of the dam site is calculated as per the seepage calculating model and seepage tension defined previously.

No.2 The coupled case between seepage field and stress field can be calculated by simulation of the impoundment process of the reservoir.

Fig.2  Left bank calculation area and the sketch showing the location of various profiles

Fig.3 shows the profile drawing of equipotential line in uncoupled and coupled cases. (For every profile, its location in the plan is shown as in Fig.2). It can be seen from Fig.3 that: the hydraulic pressure in the upstream generally tends to be higher than the one calculated by coupled method while the same one in the down stream is smaller than the one by uncoupled method. At the top of grottos the hydraulic pressure by uncoupled (No.1) method is smaller than the one by coupled method (No.2). The hydraulic pressure increases considerably at the top of grottos, being about 32m because stress of upper rock masses is larger than the one of lower rock masses. The results by coupled method are unfavorable to the stability of rock around the grottos. So, such factors must be attended to in the design.

Fig.4 to Fig.7 show the comparison of equipotential line of normal stress and shear tress of joint in No.1 and No.2 cases (in which solid line showing the results by coupled method in No.2 case while dotted line displaying the one by uncoupled method in No.1 case). It can be seen that the normal stress of joints grows smaller due to the rising of water level and shear stress somehow increases. But the stress of horizontal joint changes distinctly (as shown in Fig. 4 to Fig.5) and the stress of steep joints hardly changes, neither does the shear stress.

(a) No.1 Uncoupled case                (b) No.2 Coupled case

Fig.3  Distribution of equipotential line in coupled and uncoupled cases(Ⅰ-Ⅰcross section)

Fig.4  Comparison of normal stress in 4th joint            Fig.5  Comparison of shear stress in 4th joint

Fig.6  Comparison of normal stress in 1st joint       Fig.7  Comparison of shear stress in 1st joint

Fig.8 shows the equipotential line of seepage field of uncoupled and coupled cases in the dam foundation. There exist distinct difference in the distribution of equipotential line for seepage field between coupled and uncoupled cases. The equipotential line of seepage field calculated by coupled method tends to go downstream. Since there is great hydraulic pressure carried by the upstream side of the dam, the compressive stress of rock masses in the upstream of the dam is caused to go down accordingly. So the aperture becomes large as compared to that before impoundment and permeability increases. However, the compressive stress increases, which is carried by the joint of rock masses in the downstream dam abutment and dam base. As a result, there comes the opening of the upstream joint while the downstream joint closes.

Fig.9 to Fig.12 show the comparative distribution of normal stress and shear stress of two groups of joints of joint in dam base (the dotted line indicates the uncoupled results while the solid line gives that of coupled method). The 1st group displays the steep joints and the 4th group illustrates horizontal joints. As seen from the figures, the normal stress of joints of upstream is smaller of coupled case than that of uncoupled case whereas it is the contrary case in the downstream. But the normal stress of the horizontal joints changes much more than that of the steep joints. (see Fig.9 and Fig.11). The shear stress by coupled method increases too. However, the compressive pressure of joints close to dam abutment and dam base increases substantially and joints close, hence permeability goes small. Judging from the above analysis, coupled action makes the equipotential line of seepage field close to dam abutment and dam base point to downstream. The uplift pressures of the dam abutment and dam base go up and the normal stress of joints in down stream fall. So coupled action will bring undesirable effects on the stability of the dam.

(a) No.1 Uncoupled case                 (b) No.2 Coupled case

Fig.8  Distribution of equipotential line of coupled and uncoupled cases (1-1 cross section)

                        Fig.9 Comparison of normal stress in 4th joint                    Fig.10  Comparison of shear stress in 4th joint

Fig.11  Comparison of normal stress in 1st joint        Fig.12  Comparison of shear stress in 1st joint

4  CONCLUSIONS

(1) Jointed rock masses is generalized as equivalent continuum medium, the control equations of complete coupled fluid flow and elasto-plastic deformation are formulated, which emphasized the mechanical deformation behavior and its relation with permeability of finite-length joints. The continuum model offers a powerful analytical method. (2) The relationship between rock permeability and stress environment should be taken into consideration, for it brings about large difference in calculation results. Because the upstream of the dam body bears huge hydraulic pressure, the compressive pressure of upstream rock masses is caused to go down and the joints to burst open. At the same time, the joint plane in downstream closes and permeability decreases while shear stress increases. So the coupled fluid flow and deformation should be carried out in designing large hydraulic structures with high water head. (3) Since coupled analysis involves both the knowledge of geological structure of joint rock masses and knowing of its mechanical behavior and rock seepage rule as well, a large amount of calculation work is required to be done if taking into account the complex structures of underground grottos group. Therefore, the research work needs further perfection and developing. This model itself requires further testing perfection and revision.

References

[1]  Research of the failure of Malpasset arch dam, External Water Conservancy and Hydropower,1980,11.

[2]  Sheng Jinchang, Su Baoyu, etc: J. of Hohai University, 1999,27(5),27.

[3]  Sheng Jinchang, Su Baoyu, etc: Chinese J. of Rock Mech. and Eng., 2000,19(3),304.

[4]  Cai M.,Horri H.: Mech. Mater., 1992, 13(3): 217-26.

[5]  Cai,M., H.Horii: Int. J. of rock Mech. and Min. Eng. Sci. and Geomech. Abstr., 1993,30(4): 351-359.