Seepage flow with free surface in fracture networks

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /2012wr011991, 2013 Seepage flow with free surface in fracture networks Qinghui Jiang, 1,2 Chi Yao, 2 Zuyang Ye, 2 and Chuangbing Zhou 1,2 Received 14 February 2012; revised 19 October 2012; accepted 12 November 2012; published 15 January [1] Crystalline rocks are often heterogeneous geological materials that contain numerous fractures of various attitudes and scales. Although considerable advances have been made in simulation of fluid flow through fractured media, our knowledge of seepage flow with free surfaces in fracture networks remains to be an outstanding issue. In this paper, the partial differential equations (PDEs) defined on the whole fracture network domain are formulated for free-surface seepage flow problems by an extension of Darcy s law. A variational inequality (VI) formulation is then presented, and the proof of the equivalence between the PDE and VI formulations is given. Since the boundary conditions involving the flux components in the PDE formulation become the natural conditions in the VI formulation, the difficulty of choosing the trial functions for numerical solutions is significantly reduced and the locations of seepage points can be easily determined. On the basis of the discrete VI, the corresponding numerical procedure for unconfined seepage analysis of discrete fracture network has been developed. The results from three typical examples demonstrate the validity and capability of the procedure for unconfined seepage problems involving complicated fracture networks. Citation: Jiang, Q., C. Yao, Z. Ye, and C. Zhou (2013), Seepage flow with free surface in fracture networks, Water Resour. Res., 49, doi: /2012wr Introduction [2] Fractured rocks are complicated geological media that have undergone a long process of geological evolution. There are numerous discontinuities with various attitudes and different scales. Generally, numerical models used in this study of steady seepage behaviors through fractured rocks can be divided into two categories: one is based on equivalent continuum approaches [e.g., Snow, 1969; Neuman, 1973; Hsieh and Neuman, 1985; Oda, 1985, 1986; Jackson, 2000] and the other is based on discrete fracture network (DFN) representations [e.g., Wilson and Witherspoon, 1974; Long et al., 1985; Dershowitz and Einstein, 1987; Cacas et al., 1990]. [3] In the equivalent continuum models, fractured rock is generalized as an equivalent porous medium, and it is usually assumed that seepage flow of underground water through fractured rock obeys Darcy s law. During the last three decades, great progress has been made in the numerical simulation of unconfined seepage problems based on continuum models. In addition to adaptive mesh methods, both intuitive methods and variational inequality (VI) methods with a fixed mesh have been well developed. The intuitive methods include the residual flow method [Desai and Li, 1983], the initial flow method [Zhang et al., 1988], 1 School of Civil Engineering, Wuhan University, Wuhan, China. 2 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China. Corresponding author: Q. Jiang, School of Civil Engineering, Wuhan University, Wuhan University, Wuhan , China. (jqh1972@yahoo. com.cn) American Geophysical Union. All Rights Reserved /13/2012WR and the permeability adjustment method [Bathe and Khoshgoftaar, 1979]. These methods usually adopt a heuristic iterative procedure to locate free surface and make the flow in the dry domain approximate zero. On the other hand, the VI methods [Kikuchi, 1977; Brezis et al., 1978; Alt, 1980; Oden and Kikuchi, 1980; Westbrook, 1985; Lacy and Prevost, 1987;Borja and Kishnani, 1991] are established on rigorous mathematical basis, and the external free surface is usually transformed into an internal boundary by constructing a new boundary problem defined on a fixed region. Once the problem is solved, the location of the free surface can be determined by the finite elements intersecting with the free surface. However, to the authors knowledge, only a few works have been carried out on the study of unconfined seepage problems in fracture networks, and almost all of them are extensions of the intuitive methods to fracture networks [Wang, 1993; Jing et al., 2001]. Since fractures in their examples are either small in amount or regularly arranged, the effectiveness of the intuitive methods needs to be validated further when involving complex fracture network. [4] In the DFN models, geological and hydraulic characteristics of each fracture can be explicitly taken into account, and heterogeneous spatial distribution of groundwater fluxes can also be described in an easy manner. For equivalent continuum models, however, these effects are difficult to account for, even though they are very important for performance assessments and design optimization in many rock engineering works, such as slopes, dam foundation, and nuclear repositories. Therefore, it is very essential to develop a sound numerical method for the unconfined seepage analysis in complex fracture networks, which is also the main purpose of this study. [5] TheVItheory[Kinderlehrer and Stampacchia, 1980] is an effective tool for solving the free-surface seepage problem. 176

2 In the majority of existing VI methods for unconfined seepage analysis of continuum media, seepage points are singular and the issue of locating seepage points has not been solved theoretically. To eliminate the singularity of the seepage points, Zheng et al. [2005] proposed a VI method of Signorini type. This method constructed a new nonlinear boundary problem defined on the entire domain, imposed a boundary condition of Signorini type on the potential seepage surfaces to locate the seepage points, and adopted a penalized Heaviside function to improve the numerical stability. [6] In this paper, the VI method of Signorini type is extended to solve the unconfined seepage problem in fracture networks. For seepage in continuum media, the seepage field is continuous and the equation of continuity for any points in the whole domain of interest is the same. However, it is assumed that in DFN models flow only occurs in fractures based on the fact that the permeability of the rock matrix is often much lower than that of fractures and thus can be neglected. The flow field is restricted to the connected fracture network and separated by the rock matrix. The continuity equation at any points in a fracture segment is different from that at fracture intersections inside the fracture network. Therefore, the seepage analysis of the fracture network is based on fracture segments and intersections between fractures. By extending Darcy s law applied only on the fracture segments in the wet domain to the whole domain, a new nonlinear boundary value problem is established, which is defined on the whole fracture network domain. The boundary condition of the potential seepage face is specified as a complementary condition of Signorini type. To reduce the difficulty in selecting the trial functions for numerical solutions, a VI formulation for unconfined seepage analysis through the fracture network is then proposed. [7] The outline of this paper is as follows. In section 2, we describe the boundary value problem of free-surface seepage flow through fractured rocks. In section 3, the partial differential equation (PDE) formulation defined on the whole fracture network domain is established, which transforms the original external free boundary problem into an internal boundary. In section 4, an equivalent VI formulation is presented. The corresponding numerical procedure based on the discrete VI for unconfined seepage analysis of the fracture network is developed in section 5. Three numerical examples are discussed to illustrate the validity of the procedure in section 6. Section 7 reports the main conclusions from this work. 2. Statement of the Problem [8] It should be noted that a fracture network is not the same as a percolating network for groundwater. The main difference between them lies in the fact that there may be dead-end fractures, isolated fractures and singly unconnected fractures in the fracture system, which should be removed since they do not contribute to groundwater flow. In this sense, a fracture network needs to be regularized to become a connected graph, namely the percolating network. From the viewpoint of the graph theory [Reinhard, 2005], it indicates that in a connected graph there exists at least one connection between two arbitrary intersections. [9] A general illustration of unconfined steady seepage flow through a percolating network is shown in Figure 1. Figure 1. Illustration of seepage flow with free surface through fractured rocks. For the seepage flow problem with free surface, groundwater actually flows only in the wet domain w below a free surface G f. Obviously, the wet domain w will be determined if the free surface G f is located. Since the main aim of this study is to locate the free surface and seepage face, for simplicity, the possibility of capillary flow [Berkowitz, 2002] is not considered here. For seepage flow through porous media, free surface is a continuous surface where the pressure head is zero and the normal flux is also zero. However, for seepage flow through fractured rocks, since fluid flows directionally along the fracture network that is discrete in nature, a continuous free surface does not exist. Actually, thefreesurfaceg f here is defined by sequentially connecting the locations of fracture surfaces where water pressure is zero, as shown in Figure 1. Keep in mind that a free surface in a fracture network is not a flow line any more, which is different from the one defined in porous media. [10] Suppose that the upstream and downstream water levels are located at points A and D, respectively, segment BC is an impermeable boundary, E is the seepage point, and segment AE is the free surface in the fracture network. To establish the PDE formulation for the problem of freesurface seepage through the fracture network, the following assumptions are introduced: (1) the fluid is incompressible, (2) the intact rock matrix is impermeable and water flows only in the percolating network, and (3) the flow of water in the fractures follows the cubic law. [11] Take the fracture segment as a pair of smooth, parallel plates and set a local coordinate system for an arbitrary fracture segment ij, as shown in Figure 2. According to Darcy s law, the flow velocity within the fracture segment ij can be expressed as follows: Figure 2. v ij ¼ k ðin w Þ; (1) Local coordinate system for fracture segment ij. 177

3 where ¼ z þ p=r w is the total head, p is the pressure head, z is the vertical coordinate, r w is the water unit weight, and k ij is the permeability of the fracture segment ij and can be obtained by the cubic law [Snow, 1965;Witherspoon et al., 1980]. [12] The flow of water in fracture segment ij is determined by the equation of continuity: ¼ 0 ðin w Þ: (2) [13] In this study, the endpoints of fractures and the intersections between fractures are termed as nodes. Assuming that there are m i fracture segment converge at node i. Due to the fact that node i itself cannot store water, the algebraic sum of flux at node i should be null according to the principle of mass conservation, i.e., applied to fracture segments in the wet domain is extended to the whole fracture network domain ( ¼ w [ d ). The extended Darcy s law can be written as follows: v ij ¼ k þ v 0; (8) in which v 0 is the initial velocity of flow and is employed to counteract the virtual seepage velocity in d, where H( z) is a Heaviside function, v 0 ¼ Hð zþk ; (9) Hð zþ ¼ 0 1 if if z ðin w Þ <z ðin d Þ : (10) b ij v ij ¼ 0 ðin w Þ: (3) where b ij is the hydraulic aperture of fracture ij. [14] For the unconfined seepage flow problem, groundwater actually flows only in the wet domain w below a free surface G f. The water flow in the wet domain w should satisfy the continuity equations (2) and (3) as well as the following boundary conditions: [15] 1. The condition on the water head boundary i ¼ / ðnode i 2 G ¼ AB þ CDÞ; (4) where / is the known head on G. For the upstream and downstream faces, / equals to H and h, respectively. [16] 2. The condition on the flux boundary q ij ¼ b ij v ij ¼ 0 ðnode i 2 G q ¼ BCÞ; (5) where G q is the impermeable boundary. [17] 3. The condition on the free-surface boundary c ¼ z c ; q ic j w ¼ q cj j d ¼ 0 ðfracture ij across G f ¼ AEÞ; (6) where c stands for the intersection point of fracture ij and the free surface. [18] The condition on the seepage face boundary i ¼ z i ; q ij 0 ðnode i 2 G 0 s ¼ DEÞ: (7) 3. PDE Formulation on the Whole Fracture Network Domain [19] For unconfined seepage problem in a continuum media, Zheng et al. [2005] has proposed a VI method by generalizing Darcy s law to the whole domain and imposing the boundary conditions of Signorini type on the potential seepage surfaces. Based on the idea of the VI formulation of Signorini type, a boundary value problem defined on the whole fracture network is established. Darcy s law only [20] For the seepage through the fracture network, the groundwater always flows out of the domain along the outlet fractures at the potential seepage surface G s ¼ AGFD. The boundary condition of Signorini type [Kinderlehrer and Stampacchia, 1980] at G s ¼ AGFD can be written as follows: i z i ; q ij 0 ðnode i 2 G ð i z i Þq ij ¼ 0 s ¼ AGFDÞ: (11) [21] For the restricted potential function on the wet domain w to be the solution to the original problem, it still demands that on the interface G f ¼fðx; zþj ¼ zg between the dry domain d and the wet domain w, satisfies the flux equilibrium condition: q ic j w ¼ q cj j d ¼ 0onG f ¼ AE: (12) [22] Condition (12) on G f is necessary for the PDE formulation of this problem, which remains with the internal free boundary condition. [23] From the above discussions, the PDE formulation defined on the entire fracture network domain can be concluded, i.e., seeking a function that satisfies the controlling equations (2), (3), and (8) as well as both the external boundary conditions (4), (5), and (11) and the internal freesurface boundary condition (12). [24] The total water head defined on the entire fracture network domain has the following properties: (1) the maximum and minimum values of are the elevations H and h of the water level on the upstream and downstream boundaries, respectively, as shown in Figure 1; (2) divides the entire domain into two subdomains, namely, the wet domain w ¼fðx; zþjðx; zþ z; ðx; zþ 2g and the dry domain d ¼fðx; zþjðx; zþ < z; ðx; zþ 2g. The proof is demonstrated in Appendix A. 4. VI Formulation on the Whole Fracture Network Domain [25] Since there are an unknown internal free surface G f and a Signorini complementary condition on the seepage 178

4 face in the aforementioned PDE formulation, it is very difficult to find a trial function for a numerical solution. To make both condition (12) on G f and the flux component of condition (11) on G s become natural boundary conditions, we seek a VI formulation that is mathematically equivalent to the PDE formulation. [26] Given a trial function set, U VI ¼f j i ¼ /; on G ; i z i ; on G s g; (13) the VI formulation equivalent to the PDE formulation can be represented as follows: find a function in U VI, so that for 8 2 U VI, there holds ð; Þ ð Þ; (14) where (, ) and ( ) take the following form: ð; Þ X ð Þ X Þ b ij k ij dl; (15) b ij v 0 dl; (16) [27] Proofs of the mathematical equivalence between the VI formulation and the PDE formulation are given in Appendix B. 5. Numerical Solution by Finite Element Method 5.1. Finite Element Formulation [28] The fracture segments in the fracture network are modeled as line elements in finite element method (FEM). The nodes of the element are just the two endpoints of the fracture segment. By using linear interpolation, the hydraulic potential function of the fracture segment ij can be written as follows: ¼ N i i þ N j j ; (17) where i and j are the total water head values of nodes i and j, respectively; N i and N j are the shape functions with N i ¼ 1 l/ and N j ¼ l/. [29] Adopting finite element approximation to equation (17), the discrete form of the VI is stated as follows: seek a water head vector / rþ1 in U h VI, such that for 8 2 Uh VI, there always holds B j ; (22) U h VI ¼f j 2 Rn ; i ¼ / i ; for i 2 G ; i z i ; for i 2 G s g; (23) where r denotes the iteration step; B is the geometric matrix of the line element ij; K e and K stand for the local element and global hydraulic conductivity matrices, respectively; q is the virtual flux matrix caused by the initial flow rate v 0 ; and K e and K denote the local element and global penalized hydraulic conductivity matrices, respectively Penalized Heaviside Function [30] When solving the VI in the discrete form (equation (18)), numerical instability may occur due to the discontinuity of the Heaviside function (equation (10)). The main reason lies in the fact that the Heaviside function is a discontinuous step function, which may cause oscillation of numerical integration when fracture elements intersect the free surface. [31] To overcome the potential instability of the FE solution process, Heaviside function in equation (18) is substituted by a continuous penalized Heaviside function, as is shown in Figure 3. The penalized Heaviside function is written as follows: 8 1 if z >< ð zþ H ð zþ ¼ if < z < ; >: 2 0 if z (24) where the penalty parameter is defined as half of the width of transition layer from the wet domain to the dry domain. Obviously, when approaches zero, the penalized Heaviside function H ð zþ tends toward the original Heaviside function Hð zþ. [32] In reality, for unconfined water seepage, there is a transition layer (termed as capillary fringe) between the wet and dry domains due to the capillary pressure effects, where water pressure decreases to zero gradually. For seepage in continuum media, Zheng et al. [2005] adopted a penalized Heaviside function with two parameters " 1 and " 2 to overcome the numerical instability of FE solution. The parameter " 1 is defined as the width of the transition layer above the free surface and specified as the vertical distance between the lowest integration point and the lowest node. While, the parameter " 2 is defined as the width of ðw / rþ1 Þ T K/ rþ1 ðw / rþ1 Þ T q r ; (18) in which K ¼ X Z K e ; K e ¼ B T b ij k ij Bdl; (19) q r ¼ X Z B T b ij v r 0 dl ¼ K r ; (20) K ¼ X K e ; Ke ¼ Z Hð r zþb T b ij k ij Bdl; (21) Figure 3. Penalized Heaviside function H. 179

5 the transition layer below the free surface and specified as the vertical distance between the highest integration point and the highest node. When complicated drainage structures are deployed in a dam, Chen et al. [2008] found that the penalized Heaviside function proposed by Zheng et al. [2005] is too strict to obtain the converged solution and suggested to adopt an adaptive procedure for progressively relaxing the penalized Heaviside function to improve the numerical stability. In our algorithm, the width of the transition layer above or below the free surface is taken as the mean length of fracture elements. For specially complicated cases where dramatic downfall of the free surface occurs in the complex fracture network, the value of can be properly relaxed to obtain good convergence by using the adaptive procedure proposed by Chen et al. [2008]. [33] For unconfined seepage problem in fractured network, the whole domain can be divided into three subdomains for FE analysis, i.e., the domain 0 w that consists of the fracture elements absolutely located in the wet domain; the domain 0 d that consists of the fracture elements absolutely located in the dry domain; and the domain t that consists of the fracture elements located in the transition zone from the wet domain to the dry domain. The fracture elements located in the transition domain t can generally be classified into four types. The penalized hydraulic conductivity matrices for the four types of fracture elements are summarized in Table Illustrative Examples [34] The FE solution algorithm of the unconfined seepage analysis for the fracture network is implemented into a computer program named FracSeep. Three typical examples are solved using the program to illustrate the validity and capability of the VI formulation established in this study A Homogeneous Rectangular Dam [35] Consider a homogeneous rectangular dam with a height of 12 m and a width of 10 m, as shown in Figure 4. The bottom boundary of the dam is assumed to be impermeable. The water levels of the upstream and downstream surfaces are 10 and 2 m, respectively. An artificial fracture network system formed by two orthogonal sets of fractures with constant hydraulic apertures, and constant spacing is employed to simulate the homogeneous isotropic medium. Figure 4. Rectangular dam with tail water. The equivalent hydraulic aperture of the fracture can be expressed as follows: b ¼ð12Bk=gÞ 1=3 ; (25) where k is the permeability coefficient of the dam material and B is the spacing between two neighboring fractures. Specifically in this example, by choosing k ¼ ms 1 and B ¼ 0.2 m, we then obtain b ¼ 0.1 mm. [36] For this problem, the empirical solution of the free surface [Zhou et al., 1996] is as follows: z ¼ð100 8xÞ 1=2 : (26) [37] Figure 4 shows the locations of the free surfaces from the empirical solution and the proposed method, respectively. We can observe from Figure 4 that there is good agreement between these two methods. [38] To validate the discharge evaluated by the proposed VI method, Dupuit s formula is employed. According to Table 1. Penalized Hydraulic Conductivity Matrices for the Four Types of Fracture Elements Located in the Transition Zone a Types Descriptions Conditions Element-Penalized Hydraulic Conductivity Matrix K e 1 Node i of the fracture element ij is located in the wet domain 0 w, while node j is located in the dry i z i and j z j domain 0 d 2 Both nodes i and j of the fracture element ij are located in the transition domain t i i j j 3 Node i of the fracture element ij is located in the i z i and < j z j < wet domain 0 w, while node j is located in the transition domain t 4 Node i of the fracture element ij is located in the < i z i <and j z j transition domain t, while node j is located in the wet domain 0 w yj b yi yj ij k ij B T B 2 yi yj 4 b ij k ij B T B ð yjþ 2 4ðyi yjþ b ij k ij B T B 2yi 4yj 2 y 2 i 4ðyi yjþ b ij k ij B T B a y i and y j denote the pressure head at the nodes i and j, respectively; denotes half of the width of the transition layer; k ij is the permeability of facture element ij; and and b ij denote the length and the hydraulic aperture of fracture element ij, respectively. 180

6 Dupuit s formula, the total discharge (per unit width) of a homogeneous dam can be expressed as follows: q ¼ k h h 2 ; (27) 2L where h 1 and h 2 represent the water levels of the upstream and downstream faces, respectively, and L denotes the width of the dam. [39] The discharge of the dam calculated by Dupuit s formula is m 3 s 1, while the numerical solution based on the fracture network seepage analysis is m 3 s 1. The error between these two results is only 1.79%, showing that the numerical solution is reasonable and reliable DFN Model From DECOVALEX-2011 Project [40] To evaluate the hydraulic permeability and hydromechanical behavior of the fractured rock, the international cooperative research project DECOVALEX [Tsang et al., 2004] for radioactive waste disposal provided a complicated DFN model 20 m 20 m in size, which contains 7996 fractures (52,540 fracture elements) and 29,814 nodes in task C of the fifth stage (DECOVALEX-2011), as shown in Figure 5. The statistical parameters of fractures that generate the DFN model are from the field geological survey at the Sellafield area, Cambria, England [Min et al., 2004]. The trace lengths of the fractures obey a truncated powerlaw distribution with a fractal dimension of 2.2 and a density of 4.6 m 2. The fracture orientations are assumed to obey the Fisher distribution. The geometric parameters of the identified four sets of fractures are listed in Table 2. [41] As a challenging example, we employ this DFN model to perform an unconfined seepage analysis. Suppose the head is 10 m on the left boundary, 20 m on the right, and the bottom is impermeable. To analyze the influence of hydraulic apertures of the fractures on fluid flow behavior of the fracture network, two cases are considered: (1) the apertures of all fractures are constant and equal to 65 lm and (2) fracture apertures obey the lognormal distribution Figure 5. Geometry of fracture system in the DFN model with 20 m 20 m size [adapted from Min et al., 2004]. Table 2. Fracture Parameters Used for DFN Generation [Min et al., 2004] Joint Set Dip/Dip Direction Fisher Constant Fracture Density (m 2 ) Mean Trace Length (m) 1 8/ / / / and are related to trace lengths of the fractures by the following equation [Baghbanan and Jing, 2007]: l ¼ lmin D þ gðb iþ gðb in Þ 1=D ðlmax D l D min gðb im Þ gðb in Þ Þ ; (28) where l min is the minimum fracture trace length; l max is the maximum trace length; b im and b in stand for the upper and lower limits of fracture aperture, respectively; and D denotes the fractal dimension. In this equation gðb i Þ¼erf½ðln b i b i log Þ= p ffiffiffi 2 Š, thetermhilog and are the first and second moments of the lognormal distribution of fracture apertures, respectively; and erf[ ] is the error function. [42] Figure 6 shows the location of free surface and normalized flow rate distribution in the DFN model under two different aperture conditions. Flow rate in each fracture is normalized with respect to the total flow rate. When the fracture system has a constant hydraulic aperture, the fluid flow rate distribution pattern inside the DFN model is relatively uniform and the free surface is gently depressed. When fracture apertures follow the lognormal distribution and are correlated with fracture trace lengths, the flow pathways are dominated more by the fractures with larger aperture values. Meanwhile, influenced by the random distribution of fracture aperture, the free surface is sharply dropped at local positions. Figure 7 shows the flow rates normalized with respect to the mean flow rate in the outlet fractures along the left vertical boundary. Compared with the results from the constant aperture distribution, change in the flow rates along the left vertical boundary is more abrupt under the correlated aperture-trace length distribution A Fractured Rock Slope [43] A fractured rock slope with a height of m and a width of m is shown in Figure 8. To bring down the underground water level in the slope and improve its stability, four layers of drainage tunnels are deployed in the slope. The drainage tunnels have a rectangular section with a height of 3.5 m and a width of 3 m [Zhang, 1999]. [44] The bedrock of the slope mainly consists of fresh or slightly weathered plagioclase granite. The permeability coefficient of the intact rock is of the magnitude of cm s 1, which can be considered as impermeable compared with the fractures from the practical point of view. Large numbers of fractures of grades IV and V exist in the rock slope and form a fracture network, which controls the groundwater flow in the slope. According to the field geological survey, four sets of fractures are identified, and their statistical parameters are listed in Table 3 [Zhang, 1999]. [45] According to the geometrical parameters and statistic distribution of the fractures given in Table 3, the Monte 181

7 JIANG ET AL.: SEEPAGE FLOW IN FRACTURE NETWORKS Figure 7. ized flow rates in each fracture intersecting the left vertical boundary of the model: (a) constant apertures of 65 lm and (b) correlated aperture-trace length of fractures. Flow rates are normalized with respect to the mean flow rates (total flow rate divided by the number of fractures) in the boundary. Figure 6. Location of free surface and normalized flow distribution in the DFN model: (a) constant apertures of 65 lm and (b) correlated aperture-trace length of fractures. Flow rates are normalized with respect to the total flow rates. Carlo method is employed to generate a DFN. The isolated fractures are removed and dead-end fractures are trimmed, which results in a percolating network with 28,903 fracture elements and 15,851 nodes, as shown in Figure 8. To the authors knowledge, there is no analytical solution available in the existing literature for unconfined seepage analysis involving such a complicated fracture network especially when considering drainage tunnels. The calibration of the fracture network generated by the Monte Carlo method is not considered in this research, since the main focus is to verify the validity of the proposed method and its capability in solving free-surface seepage problems in complicated fracture networks. [46] Suppose that heads of the right and left boundaries of the slope are 200 m and 62.2 m, and the bottom boundary is impermeable. The other boundaries, including the drainage tunnels located in the slope, are taken as the seepage faces satisfying the Signorini type complementary condition. [47] Shown in Figure 9 is the normalized flow rate distribution inside the slope with and without drainage tunnels. It can be observed that the seepage flow behaviors within the fracture network are heterogeneous. Comparison between Figures 9a and 9b shows that when the drainage tunnels are enabled, the free surface is obviously lowered, which leads to a significant decrease of seepage pressure on the seepage Figure 8. Geometry of a fractured rock slope with drainage tunnels. 182

8 JIANG ET AL.: SEEPAGE FLOW IN FRACTURE NETWORKS Table 3. Parameters of Fractures and Probability Models [Zhang, 1999] Dip ( ) Trace Length (m) Aperture (mm) Fracture Set Mean Values Variance Probability Model Mean Values Variance Probability Model Mean Values Variance Probability Model Mean Spacing (m) Lognormal Lognormal Lognormal Lognormal surface. Although the flow rate distribution in the deeper locations of the slope has no obvious variation, the flow rates of the fractures on the downstream seepage face decrease obviously and the flow rates of the fractures near the drainage tunnels increase significantly. [48] The flow rates per unit width through the upstream and downstream surfaces and out of the drainage tunnels are listed in Table 4. The total flow rate out of the slope is m3 s 1 without the drainage tunnels. After the drainage tunnels are enabled, the total flow rate out of the slope is m3 s 1, with the flow rates from the drainage tunnels being m3 s 1 and the flow rates at the seepage surface being m3 s 1. The total flow rate out of the slope is markedly increased, while the flow rate at the downstream seepage surface is significantly decreased. Therefore, the drainage tunnels play an important function for draining the slope and lowering the water level of groundwater. It can be further observed from Table 4 that the flow rates out of the drainage tunnels are gradually increased from the upstream to downstream. This indicates that the positions of drainage tunnels have a prominent influence on the effect of the drainage tunnels in drainage, which is very important for design optimization of the drainage system and safety assessment of slope engineering. 7. Figure 9. Locations of the free surfaces and normalized flow distribution in the slope: (a) without drainage tunnels and (b) with drainage tunnels. Flow rates are normalized with respect to the total flow rates. Conclusions [49] The problem of groundwater seepage through fractured rock is complex due to the existences of free surfaces and seepage surfaces. In this study, the PDEs defined on the whole fracture network are formulated for free-surface seepage problems. A VI formulation equivalent to the PDE formulation is then proposed, and corresponding numerical procedure has been developed. Based on this work, the following main conclusions can be drawn: [50] 1. By specifying the seepage face boundary as a Signorini-type condition, the proposed VI method can effectively eliminate the singularity of seepage points. Based on the VI method, a numerical procedure for unconfined seepage analysis of fracture networks is developed by using line elements in FEM to simulate the fractures. The effectiveness of the procedure is verified by comparison of numerical results and the empirical solutions from a homogeneous rectangular dam. [51] 2. Analysis of the DFN model from the DECOVALEX project indicates that when the aperture variation among fractures is considered, seepage flow becomes more heterogeneous than in the case of constant aperture. This example also exhibits the capability of the proposed model to account for very complicated fracture networks. [52] 3. The proposed model is applied to predict the freesurface seepage behavior through a fractured rock slope with four drainage tunnels. The redistribution of flow rate after the installation of drainage system is accurately modeled, and the effect of each drainage tunnel is precisely analyzed. The simulation in this paper may provide a new insight for 183

9 Table 4. Flow Rates per Unit Width Through the Upstream and Downstream Surfaces and out of the Drainage Tunnels (Unit: 10 5 m 3 s 1 ) Conditions Upstream Surface Drainage Tunnels Downstream Surface Without drainage tunnels With drainage tunnels performance assessments and design optimization of complex drainage system. Appendix A: Proofs of the Two Properties of the Total Water Head / [53] Property 1. The maximum and minimum values of are the elevations H and h of the water level on the upstream and downstream boundaries, respectively. [54] Property 2. divides the entire domain into two subdomains, namely, the wet domain w ¼fðx; zþjðx; zþ z; ðx; zþ 2g and the dry domain d ¼fðx; zþjðx; zþ < z; ðx; zþ 2g. [55] The proof is as follows. For an arbitrary fracture segment ij, first, only reaches its maximum or minimum values at node i or j, because varies linearly along the fracture segment ij according to the continuity equation = ¼ 0. Second, if reaches its maximum (or minimum) value at node i, then there is q ij > 0 (or q ij < 0). [56] For the whole fracture network, it follows that cannot reach its maximum value at the nodes inside the fracture network, nodes on the impermeable boundary G q, and the potential seepage boundary G s.if reaches its maximum value at a certain node i inside, then, for all the fractures that connected to node i, there is q ij > 0, which contradicts the mass conservation equation (3). If arrives in its maximum value at a certain node i 0 located on the impermeable boundary G q ¼ BC or the potential seepage boundary G s ¼ AGFD, then there is q i 0 j > 0, which contradicts the boundary condition (5) or (11). Definitely, will not arrive in its maximum value on the downstream water head boundary G ¼ CD. Hence, can get its maximum value H only on the upstream water head boundary G ¼ AB. [57] Similarly, it is not possible for to get its minimum value at the nodes located inside the fracture network or at the nodes on the impermeable boundary G q.if reaches its minimum value at a certain node i inside, then, for all the fractures that connected to node i, we have q ij < 0, which contradicts the mass conservation equation (3). If gets its minimum value at a certain node i 0 located on the impermeable boundary G q ¼ BC, we have q i0 j < 0, which contradicts the boundary condition (5). Again, if reaches its minimum value at the node i located on the boundary G s ¼ AGFD, we have q ij < 0. According to the complementary boundary condition (11), we have i ¼ z i under this condition. However, z i > h; thus, can get its minimum value h only on the downstream water head boundary G ¼ CD. [58] According to the above discussion and the definition of the potential function, ¼ z þ p=r w, one can conclude that p > 0 at the nodes located on the boundaries AB, BC, and CD, see Figure 1. In the fractures near the boundary ABCD, the potential function varies linearly; therefore, there exists a domain, written as w, in the fracture network where water pressure p is positive. Analogously, for the nodes located on the boundary AGF, we have p=r w ¼ z < H z < 0. Therefore, there exists a domain, written as d, in the fracture network where water pressure p has a negative value. Till now, the aforementioned two properties of have been verified. Appendix B: Proofs of Equivalence Between the PDE and the VI [59] To prove the equivalence between the PDE and VI formulations, we first expand ; ð Þ ð Þ via integration by parts: ; ð Þ ð Þ¼ X ð b ij k ij b ijv 0 dl ¼ X Z ð Þb ij v ij 0 þ ð Þb ij dl ¼ X X Z ð i i Þb ij v ij j j bij v ij þ ð Þb ij dl ¼ Xn ð i i Þb ij v ij þ X Z ð Þb ij i¼1 dl; (B1) where n stands for the total number of nodes in the whole domain. B1. Proof for PDE ) VI [60] If is the solution to the PDE formulation, then by substituting the continuity equations (2) and (3) and the external boundary condition (4), equations (5) and (11) into equation (B1), combined with the fact that 8 2 U VI,we obtain the following equation: ; ð Þ ð Þ ¼ X ð i i Þq ij i2g s ¼ X i2g s ¼ X i2g s ð i z i Þq ij X i2g s ð i z i Þq ij 0: ð i z i Þq ij (B2) 184

10 [61] Therefore, is also the solution to the VI formulation. B2. Proof for VI ) PDE [62] Suppose is the solution to VI formulation. From equations (14) and (B1) and the fact that both and belong to VI, i.e., they satisfy the water head boundary condition on G, we have the following equation: ; ð Þ ð Þ ¼ X ð i i Þb ij v ij þ X i2 in þ X Z ð Þb ij 0; i2g q[g s ð i i Þb ij v ij (B3) where in denotes all the nodes inside the fracture network. [63] Taking ¼ þ 1 and ¼ 1, respectively, where 1 is any function that equals to zero at all the nodes on the entire fracture network domain, the continuity equation (2) can be derived. [64] Using equation (2), equation (B3) can be reduced to the following form: ; ð Þ ð Þ ¼ X ð i i Þb ij v ij þ X i2 in i2g q ð i i Þb ij v ij Notice that the second term on the right side of equation (B6) is a constant. Thus, if the above inequality holds for 8 2 VI, then the first term on the right side of equation (B6) must be nonnegative, i.e., X i2g s ð i z i Þq ij 0: (B7) Since is any function that satisfies z on G s, we obtain the following equation: q ij 0 ðnode i 2 G s Þ: (B8) [69] Of course, we can also take ¼ z in equation (B6), which leads to the following equation: X i2g s ðz i i Þq ij 0: (B9) Combining equations (B8) and (B9) with the condition of i z i on G s, we can obtain the complementary condition on G s as follows: ð i z i Þq ij ¼ 0 ðnode i 2 G s Þ: (B10) þ X i2g s ð i i Þb ij v ij 0: (B4) To deduce the inner boundary condition on G f, the integration on in equation (B1) is now represented as the sum of the integrations in subdomain w and d. [65] Taking ¼ þ 2 and ¼ 2, respectively, in equation (B4), with 2 being any function that equals to zero at the nodes located on boundaries G, G q, and G s, the continuity equation (3) at any nodes inside the fracture network can be obtained. [66] Using equation (3), equation (B4) can be simplified as follows: ; ð Þ ð Þ ¼ X ð i i Þq ij w þ X ð i i Þq ij d þ X ð Þb ij w Z dl þ X ð d Z Þb ij dl 0: (B11) ; ð Þ ð Þ ¼ X ð i i Þb ij v ij þ X i2g q i2g s ð i i Þb ij v ij 0: (B5) [70] Using the results of above derivations, the following inequality holds for 8 2 U VI. ; ð Þ ð Þ ¼ X ð c c c2g f Þ q ci j w þ q cj j d [67] Taking ¼ þ 3 and ¼ 3, respectively, in equation (B5), in which 3 is any function that becomes null at the nodes located on boundaries G and G s, the flux boundary condition (5) on G q can be obtained. [68] Using condition (5), equation (B5) can be simplified as follows: ; ð Þ ð Þ ¼ X ð i z i Þq ij X i2g s i2g s ð i z i Þq ij 0: (B6) þ X i2g s ð i i Þq ij 0: (B12) [71] Taking ¼ þ 4 and ¼ 4, respectively, in equation (B12), with 4 being an arbitrary function that becomes null at the nodes located on boundaries G and G s, then we get the flux equilibrium condition (12) on G f. [72] Up to here, all the equations and boundary conditions in the PDE formulation have been derived. Hence, the 185

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