Unsteady Seepage Through Random Fracture Network in Rock Mass CHAI Junrui
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Unsteady Seepage Through Random Fracture Network in Rock Mass CHAI Junrui
Physical and Numerical Simulation of Geotechnical Engineering 1st Issue, Sep. 2010 Unsteady Seepage Through Random Fracture Network in Rock Mass CHAI Junrui1, 2, HE Yang2, LI Kanghong2, LIU Zhao2 1. College of Civil and Hydroelectric Engineering, China Three Gorges university, Yichang, Hubei Province, People’s Republic of China, 443002 2. College of Hydroelectric Engineering, Xi’an University of Technology, Xi’ an, Shaanxi Province, People’s Republic of China, 710048 ABSTRACT:It is of great importance to study the unsteady fluid flow in the fracture network in rock mass and to simulate the seepage field in the fractured rock mass. The random distribution of the fracture network in rock mass is simulated by the Monte-Carlo method. The numerical program is developed based on the mathematical model of unsteady seepage in the fracture network. A case study is carried out by means of the program. It can be concluded that (1) The hysteresis of the hydraulic head distribution in the unsteady seepage field occurs in the case when the boundary condition is varying; (2) The hysteresis of the hydraulic head distribution in the unsteady seepage field is more obvious on the condition of the randomly variable apertures than the uniform apertures; (3) The seepage discharge is centralized in the larger-aperture channel; and (4) The hydraulic head distribution in the unsteady seepage field can be effected greatly by the variable apertures when the boundary hydraulic head conditions of the upstream and the downstream are varying greatly. KEYWORDS: Random fracture network, Unsteady seepage, Monte-Carlo method, Variable apertures 1 INTRODUCTION Seepage analysis is very important in hydraulic engineering, especially in high dam engineering. Seepage stability has more and more obvious effect on the safety and economy of the project [1-5]. The natural characteristics of the structure surface of rock mass fractures in the space can be shown by the geometry characteristics which include the orientation, configuration, scale, aperture and so on. These geometry characteristics have direct effect on the permeability of fractured media, such as the quantity, orientation and distribution. In engineering practice, we can investigate the nature of rock mass in the project area, and then take some appropriate measures to make sure the project safety. In most cases, the permeability of rock matrix is so weak that it can be ignored, and the fluid flows only along the connected fracture network. There are different types of fracture network, formed as the result of different mechanical reasons and different scales. Because the fractures resistant to water exist in the system, the connecting characteristic of fractures is very poor, which leads to the discontinuous distribution of fluid flows in fracture net work. This system is called as the unconnected system. For this reason, seepage in rock mass can be considered as seepage through the connected fracture network[6-9]. In this paper, the distribution of fractures in rock mass is simulated by the statistical characteristics of the geometric elements of fractures and the Monte-Carlo method. Then analysis of unsteady seepage through random fracture network in rock mass is carried out. Finally the hydraulic head and seepage discharge distributions in fractures in rock mass varying with time are obtained and studied, from which the hysteresis of the hydraulic head © ST. PLUM-BLOSSOM PRESS PTY LTD distribution and the centralized channel flow effect are observed obviously. 2 RANDOM SIMULATED METHOD 2.1 FRACTURE NETWORK BY THE MONTE-CARLO Principle of the Monte-Carlo method The Monte-Carlo method [10] for solving the mathematical problem is from the contrary procedures. If the probability distribution satisfies a certain mathematical equation, the trials of random sample are made to produce the random variables, and the average values of the results are taken as the appropriate estimate solution of mathematical equation [8, 9]. In the same way, we can use the Monte-Carlo method to simulate the fracture network. In the process of random simulation, it is the most important thing to produce the random numbers which have the regular distribution. At present, the linear congruent method is an important and popular one to refine the random numbers. Because it has a simple algorithm which can be understood easily, the linear congruent method is performed easily and has the good statistic law. The iteration equation of the multiple congruent method is as follows [11]: (1) xi axi 1 cm o dm In which a is the non-negative factor, c is the non-negative increment, m is the module, mod m is the remainder which get after division m . So we can Unsteady Seepage Through Random Fracture Network in Rock Mass DOI: 10. 5503/J. PNSGE. 2010. 01.005 call this method as the remainder expression of the initial value x0 ,the m .Given crosses with each other or with the boundary to form the connected fracture network [9]. Because the fractures produced by this method are strongly random, we directly number the junctions and elements of the fracture network. A case study of simulating fracture network: On a dam site, there is a 20 m 20 m area with three groups of fractures. The probability distributions of the geometric parameters of fractures in rock mass can be obtained from the joint statistics and category, which are shown as Table 1. The fracture network simulated by the Monte-Carlo method is shown as Figure 1. The simulated results demonstrate clearly the relationships and distribution of the fracture network in rock mass. Figure 2 is the connecting fracture network, which is got by deleting the unconnecting fractures in Figure 1. Thus, we can use these results to analyze effectively the unsteady seepage through the fracture network in rock mass. average random number can be calculated by the iteration step. In fact, the distribution of the fracture network in rock mass is so complex that it is difficult to describe it by the investigation method. The process to simulate the fracture network is as follows: (1) The fractures of outcrop rock should be investigated, including the spacing, aperture, tendency, dip angle, trace length and so on; (2) The probability distribution functions of fracture geometric elements are analyzed and established by statistics; (3) The fracture network is simulated from its probability distribution functions by the Monte-Carlo method. The practical statistical data is used to simulate the fracture network, the unconnected fractures may occur. In this situation, the independent fractures along which fluid can not flow through must be deleted. Thus, every fracture Table 1 The input data for Monte-Carlo simulation Group of joints Dip angle (°) 1 2 3 Average value 90 180 45 Standard deviation 5.7 5.8 6.4 Average value 3.4 4.1 4.3 Average value 2.3 3.5 2.1 Standard deviation 0.2 0.3 0.4 Average value 1.75 2.98 3.63 Average value 0.00025 0.00034 0.00041 Standard deviation 0.00002 0.00001 0.00003 Normal distribution Trace(m) Negative exponential distribution Spacing distance(m) Homogeneous distribution Discontinous distance (m) Negative exponential distribution Aperture(m) Normal distribution A D A 4 1 5 6 5 8 13 11 14 10 11 16 16 6 7 12 10 7 9 4 17 D 3 2 1 2 15 12 13 17 20 14 15 19 18 21 18 23 22 21 24 22 28 29 58 54 39 36 30 33 52 38 53 20 25 24 23 31 56 41 40 42 43 59 19 26 27 2534 26 32 37 35 38 35 34 48 36 55 37 46 2739 28 40 3141 29 33 32 44 30 44 61 50 45 51 77 62 65 47 63 48 46 49 66 64 54 68 55 70 90 69 88 60 71 72 69 62 C B Figure 1 The fracture network simulated by the Monte-Carlo method 3 NUMERICAL SIMULATION OF UNSTEADY SEEPAGE THROUGH RANDOM FRACTURE NETWORK 3.1 Numerical program model and 86 59 79 83 58 87 89 63 70 90 91 93 64 84 85 66 92 B 95 56 61 5280 50 51 81 57 79 73 78 80 74 87 53 65 67 68 C Figure 2 The connected fracture network The network system method based on discontinuous media should be used for seepage through fracture network in rock mass [12]. The point where two fractures cross is taken as a node and the fracture between two nodes is regarded as a linear element. The volume of water flowing into or out of the same node is equal to the variation of water storage volume in each node. So the model for 2-D unsteady seepage through fracture network can be obtained computational 30 Physical and Numerical Simulation of Geotechnical Engineering 1st Issue, Sep. 2010 as follows[13, 14], which should be combined with the initial and boundary conditions. in which D is the storage matrix of fracture network, I is the unit matrix. According to equation (3), the hydraulic heads at t t can be determined by the computational hydraulic heads at t . Area is studied, and then the quantity of flow on boundary can be obtained. The computational program for unsteady seepage through random fracture network in rock mass is based on equations (2) and (3). From a simple example verification, the maximal error between the numerical solution and the theoretical one is 0.02%. Thus, the effectiveness of the computational program has been verified. N N' dH i q j w j i Qi d i j 1 j 1 dt i (2) 1,2,3,, N in which q j j 1,2, , N is the flow discharge through node i along the line element j ; j is the ' connecting linear element; w j j 1,2, …, N is ' the quantity of supply on each linear element; source (or sink) of node node i ; S di i 2 i ; Hi Qi is the 3.2 A case study is the hydraulic head at N' b l j 1 j j , where Si 3.2.1 Fracture network with uniform apertures Here we use the fracture network simulated by the Monte-Carlo method in section 2.2 to carry out analysis of unsteady seepage. The fracture network can be meshed into 68 nodes and 86 line elements shown as Figure 2. The aperture of each line element is 0.001 m , the depth is 1 m . is the elastic storage coefficient of the fracture which is connected with i ; b j , l j are the element j , respectively. node aperture and length of line 5 The coefficient of water reserve is S = 6.9 10 . The initial and boundary conditions are as follows for six cases. If the vertical water supply was ignored, the equation (2) can be rewritten as follows [15]. D D t t t G t I H t H Table 2 Case (3) The initial and boundary conditions for six cases Initial condition Boundary condition 1 H AB 100 m , H CD 20m , Q AC Q BD 0 H AB descending with 1m / s 2 H AB 100 m , H CD 20m , Q AC Q BD 0 H AB descending with 1m / h 3 H AB 100 m , H CD 20m , Q AC Q BD 0 H AB descending with 1m / d 4 H AB 20m , H CD 20m , Q AC Q BD 0 H AB ascending with 1m / s 5 H AB 20m , H CD 20m , Q AC Q BD 0 H AB ascending with 1m / h 6 H AB 20m , H CD 20m , Q AC Q BD 0 H AB ascending with 1m / d The computational program for unsteady seepage through random fracture network in rock mass is used to calculate the seepage fields under the above six cases. The typical hydraulic head contours of seepage fields at the initial moment and each moment under case 3 are shown as Figure 4 to Figure 7. According to Figure 4 to Figure 7, the seepage field in rock mass is varying with the boundary conditions and time. From Figure 7 we can see clearly that the hydraulic heads inside are higher than hydraulic heads nearby the boundary. When the hydraulic head on the boundary descends suddenly, the hydraulic heads inside need time to reach the steady condition. So the hysteresis of the hydraulic head is caused by the change of hydraulic head sometimes. The boundary hydraulic heads vary so fast that the hydraulic heads inside have no time to keep up with it, making the hydraulic head inside higher than the boundaries. This is an extremely disadvantage factor to the stability of rock mass. 31 Unsteady Seepage Through Random Fracture Network in Rock Mass DOI: 10. 5503/J. PNSGE. 2010. 01.005 20 20 18 18 16 16 14 14 12 10 Y(m) Y(m) 12 10 8 8 6 6 4 4 2 0 0 2 4 6 8 10 12 14 16 18 2 20 X(m) 0 0 2 4 6 8 10 12 14 16 18 20 X(m) Figure 4 The contour of hydraulic head at initial time Figure 5 The contour of hydraulic head after falling for 20 days 20 20 18 18 16 16 14 14 12 Y(m) Y(m) 12 10 10 8 8 6 6 4 4 2 2 0 0 2 4 6 8 10 12 14 16 18 0 0 20 2 4 X(m) 6 8 10 12 14 16 18 20 X(m) Figure 6 The contour of hydraulic head after falling Figure 7 The contour of hydraulic head after falling for 60 days for 80 days 3.2.2 Fracture network with randomly variable apertures As the above example, the engineering condition, the basic assumption and the geometry elements are also the same. It is only that the apertures of fracture network are randomly variable. The aperture of each fracture is also simulated by the Monte-Carlo method. The apertures of fractures accord with the normal distribution shown as Table 1. Some typical apertures which is produced randomly are shown as Table 3. The depth of fracture is also 1 m . The coefficient of water reserve is also S 6.9 10 5 . The initial and boundary conditions are also shown as Table 2 for six cases. The typical hydraulic head contours of seepage fields at the initial moment and each moment under case 3 are shown as Figure 8 to Figure 11. 32 Physical and Numerical Simulation of Geotechnical Engineering 1st Issue, Sep. 2010 The typical random apertures Group of joints X1(m) Y1 (m) X2 (m) Y2 (m) Aperture(m) 1 16.637861 16.056965 16.784464 20.000000 0.000237 1 14.411842 0.000000 14.353835 0.877495 0.000260 1 14.632769 3.217984 14.655147 5.145730 0.000209 1 15.177883 10.160059 14.196506 20.000000 0.000245 2 18.126020 15.724996 3.532354 11.015261 0.000343 2 20.000000 12.260617 5.337903 13.779658 0.000340 2 3.158185 12.188874 0.000000 12.541636 0.000331 2 20.000000 11.872809 17.372505 11.721912 0.000340 3 17.957096 10.186799 19.529942 12.373300 0.000418 3 4.682361 0.000000 5.974410 1.592494 0.000487 3 7.727613 2.568667 20.000000 14.521752 0.000352 3 0.000000 0.940207 4.478462 5.853339 0.000413 20 20 18 18 16 16 14 14 12 12 Y(m) Y(m) Table 3 10 10 8 8 6 6 4 4 2 2 0 0 2 4 6 8 10 12 14 16 18 0 0 20 2 4 6 8 X(m) Figure 8 The contour of hydraulic head at initial time 14 16 18 20 20 18 18 16 16 14 14 12 12 Y(m) Y(m) 12 Figure 9 The contour of hydraulic head after falling for 20 days 20 10 10 8 8 6 6 4 4 2 2 0 0 10 X(m) 2 4 6 8 10 12 14 16 18 0 0 20 X(m) 2 4 6 8 10 12 14 16 18 20 X(m) Figure 10 The contour of hydraulic head after falling for 60 days Figure 11 The contour of hydraulic head after falling for 80 days According to Figure 8 to Figure 11, the seepage field in rock mass is varying with the boundary conditions and time. It can be clearly shown from Figure 11 that the hydraulic heads inside are higher than hydraulic head nearby the 33 Unsteady Seepage Through Random Fracture Network in Rock Mass DOI: 10. 5503/J. PNSGE. 2010. 01.005 boundaries. This is the hysteresis of hydraulic head which is just as the example in Section 3.2.1. Comparing Figure 7 with Figure 11, we can see that the hydraulic head inside on the variable apertures is higher than that on the uniform apertures. When the apertures are considered as random distribution, the hysteresis of the hydraulic head is more obvious, comparing with the uniform apertures. Comparing Figure 4-7 with Figure 8-11, the maximal relative difference of hydraulic head at the same node at the same time is 18.17%, which is resulted only from the randomly variable apertures. When the hydraulic head difference between upstream and downstream is great, the seepage field was effected greatly by random apertures; when the hydraulic head difference is small, the seepage field was little effected by random apertures. In addition, the hydraulic heads and flow rates of typical nodes and elements are shown as Table 4. From it, the centralized channel flow effect can be seen clearly. If the hydraulic gradient is the same, the flow rate is proportional to the cube of the aperture. Thus, at the node of the fracture network, the most of the flow rate flows along the fracture with larger aperture. So, it can be shown that the fracture network model can reflect the centralized channel flow effect. Table 4 The hydraulic heads and flow rates of typical nodes and elements (after falling for 20 days) Flow rate ( 106 m3 s ) Aperture (m) Hydraulic head (m) Element Node 18 39 49 4 32.29768 18 21 22 23 0.0002448 0.0004239 0.0004239 0.0002448 18.99125 24.18184 -6.81789 -1.77417 38.34599 29 53 54 55 0.0002448 0.0003446 0.0003446 0.0002448 7.71453 23.3983 -13.2056 -6.51999 21.38649 51 74 76 77 0.0003989 0.0004456 0.0003989 0.0004456 6.78837 -10.48982 -3.92423 13.37318 CONCLUSION ACKNOWLEDGEMENTS The fracture network model in rock mass hydraulics is employed in this paper to simulate 2D unsteady seepage field in the discontinuous and anisotropic rock mass. From the case study, the conclusions can be followed as follows. (1) The Monte-Carlo method is feasible to simulate the fracture network in rock mass, which is based on the fracture geometric element statistical law. (2) It can be shown from the case study that analysis of unsteady seepage reflects more effectively the seepage field in practice. The hysteresis of the hydraulic head distribution in the unsteady seepage field occurs in all the six cases when the boundary condition is varying. (3) The hysteresis of the hydraulic head distribution in the unsteady seepage field is more obvious on the condition of the randomly variable apertures than the uniform apertures; and the hydraulic head distribution in the unsteady seepage field can be effected greatly by the variable apertures when the boundary hydraulic head conditions of the upstream and the downstream are varying greatly. (4) The seepage discharge is centralized in the larger-aperture channel (fracture), i.e., the centralized channel flow effect can be shown in the case study. The financial support from the Project 50579092 sponsored by National Natural Science Foundation of China (NSFC), Research Fund 20096118110007 for the Doctoral Program of Higher Education of China, the Project 106-220331 sponsored by the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCS) by State Education Ministry (SEM), the Research Project 03JK098 sponsored by Shaanxi Provincial Education Department (SNED), the Project 2004ABB012 sponsored by Hubei Provincial Science and Technology Department (HBSTD), the Project 603108, 603402 sponsored by China Three Gorges University (CTGU) and the Scientific Innovation Project 106-210303, 220275 sponsored by Xi’an University of Technology (XAUT) is gratefully acknowledged. REFERENCES [1]. [2]. [3]. 34 Mao Changxi. Seepage Analysis and Control. China Hydropower Press: Beijing, 2003 (in Chinese) Chai Junrui. Seepage Theory for Dam Engineering. Tibet People Press: Lasa, 2001 (in Chinese) Chai Junrui. 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