HE Ji, CHEN Sheng-hong

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 430072 Wuhan, China, E-mail: jihechn@gmail.com

SHAHROUR Isam

Laboratoire Génie Civil et géo-Environnement, University Lille 1, 59650 Villeneuve dʹAscq, France

(Received February 1, 2012, Revised June 25, 2012)

A REVISED SOLUTION OF EQUIVALENT PERMEABILITY TENSOR FOR DISCONTINUOUS FRACTURES*

HE Ji, CHEN Sheng-hong

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 430072 Wuhan, China, E-mail: jihechn@gmail.com

SHAHROUR Isam

Laboratoire Génie Civil et géo-Environnement, University Lille 1, 59650 Villeneuve dʹAscq, France

(Received February 1, 2012, Revised June 25, 2012)

The equivalent permeability tensor is essential to the application of the equivalent porous media model in the numerical seepage simulation for fractured rock masses. In this paper, a revised solution of the equivalent permeability tensor is proposed to represent the influence of the fracture connectivity in discontinuous fractures. A correction coefficient is involved to reflect the complex seepage flow type through the rock bridge. This correction coefficient is back analyzed from single-hole packer tests, based on the Artificial Neural Network (ANN) back analysis and the Finite Element Method (FEM) seepage simulation. The limitation of this back analysis algorithm is that the number of single-hole packer tests should be equal or greater than the number of fracture sets, and three is the maximum number of the fracture sets. The proposed solution and the back analysis algorithm are applied in the permeability measurement and the seepage simulation for the Xiaowan arch dam foundation.

discontinuous fractures, permeability tensor, back analysis, packer test, Artificial Neural Network (ANN)

Introduction

The equivalent permeability tensor is essential to the application of the equivalent porous media model, which is widely used in the numerical seepage simulation for fractured rock masses. If fractures can be grouped into several sets with each set having a constant aperture, a uniform spacing and a dominant orientation, an analytical solution can be found for the equivalent permeability tensor formulated by fracture tensors as Eq.(1)[1]. This analytical solution implies that the rock mass permeability is obtained from the permeability superposition of individual fractures and intact rock matrices, by ignoring the complex interconnection of fractures and the flow transportation between fractures and rock matrices. Since this analytical solution is straightforward in theory and can represent the anisotropy of the rock mass permeability,it is widely used in engineering applications.

In the above analytical solution, the connectivity of fractures is not well considered, namely, the fractures are assumed to be continuous and throughgoing, which is not the case in the real situation. The fracture connectivity rate is a geological parameter describing the length ratio of fractures to rock bridges along the dominant orientation of fractures[2,3]. The fracture connectivity rate is a critical factor significantly influencing the rock mass permeability. For example, the fractures of large aperture and high density but with low connectivity would make small contributions to the permeability of the rock masses. Some studies were devoted to explain the effects of discontinuous fractures for the rock mass permeability[4-7]. However, it is still difficult to quantify the complex relationship between the fracture connectivity rate and the rock mass permeability so far. From a practical point of view, the introduction of the parameter of“connectivity factor” (i.e., ψiin Eq.(2), proposed by Oda[1]) into the traditional analytical solution of the equivalent permeability tensor (Eq.(1)) is an effective way to represent the influence of the fracture conne-ctivity. ψiis dimensionless and varies in the range of 0 ＜ ψi＜ 1 (ψi=1 for continuous fractures). Unfortunately, Oda did not further explain how to quantify this parameter.

Fig.1 Sketch of a single-hole packer test

In this paper, a simple transformation formula is proposed to calculate the connectivity factor from the fracture connectivity rate. In this formula, a correction coefficient is involved to reflect the complex seepage flow type through the rock bridge. This correction coefficient needs to be back analyzed from the results of single-hole packer tests[8,9](Fig.1). The back analysis algorithm is similar to that proposed in authorsʼ previous paper[10], in which the Finite Element Method (FEM) is adopted in the packer test simulation and the Artificial Neural Network (ANN) is used as a back analysis tool. The back analysis there was to obtain the fracture aperture under the assumption that all fractures are continuous, but the back analysis here is to obtain the correction coefficient of discontinuous fractures. The connectivity factor can be calculated from the back analyzed correction coefficient using the transformation formula, and then the exact equivalent permeability tensor can be obtained by using Eq.(2).

where n is the total number of fracture sets, i is the serial number of fracture sets, aiis the fracture aperture, biis the fracture spacing,,,are the direction cosines of the fracture normal, kris the permeability coefficient of intact rock, v is the kinematic viscosity coefficient, g is the gravitation acceleration, and ψiis the connectivity factor.

1. Transformation formula from fracture connectivity rate to connectivity factor

To quantify the influence of the fracture connectivity in the equivalent permeability tensor, a simple transformation formula is established to obtain the connectivity factor. The assumptions in deriving this formula are as follows:

(1) The permeability coefficient of a fracture kfsatisfies the assumption of the Parallel Plate Model

(2) The flow path length in the fracture is Lf= ηL, the flow path length in the rock bridge is Lr= (1-η) L, and the entire length of the flow path in the rock mass is L = Lf+ Lr, where η is the fracture connectivity rate (0 ＜ η ≤100%).

(3) The permeability of the rock bridge is isotropic, and it is equal to the permeability coefficient of the intact rock kf.

Fig.2 Sketch of the flow path in the rock bridge

(4) The flow path in the rock bridge may follow Path 1, Path 2, Path 3 (Fig.2) or even extend to adjacent fractures. The exact flow path is very complicated and difficult to be identified[11]. From a practical point of view, the flow path is approximately determined as Path 1, and the flow path width in the rockbridge is assumed to be m times wider than the fracture aperture α for correcting the flow path approximation. m is, therefore, named in this paper the“correction coefficient”.

The velocity in the fracture is

The velocity in the rock bridge is

The equivalent velocity in the rock mass is

where ΔΦfis the hydraulic potential loss in the fracture, ΔΦris the hydraulic potential loss in the rock bridge, keqis the equivalent permeability coefficient of the rock mass along the fracture plane.

The seepage flow rate in the fracture and in the rock bridge should remain constant, i.e.,auf= maur=aueq. According to Eqs.(4)-(6), the equivalent permeability coefficient of a discontinuous fracture is formulated as

From Eq.(7), the transformation formula to obtain the connectivity factor from the connectivity rate is

Substituting Eqs.(3) and (8) into Eq.(2) and assuming that each fracture set has a uniform correction coefficient mi, the revised solution of the equivalent permeability tensor [K] for discontinuous fractures is obtained. Apparently, miis the only unknown parameter in [K], since other parameters can be directly measured in the field.

To find the value of m, a rock mass containing a single fracture is numerically analyzed by the Finite Element Method (FEM). The numerical model is designed as a square (Fig.2) with the side length equal to the fracture spacing b (i.e. AB = BC = b). The hydraulic boundary conditions are set as follows: the hydraulic potential on AD is set as b, the hydraulic potential on BC is set as zero, and both AB and DC are impervious. In this way, the flow rate through the fracture can be measured from the seepage field simulated by the FEM and the relevant m can be calculated using Eq.(7). Table 1 lists the value of m relative to various fracture apertures and connectivity rates. The intact rock permeability coefficient remains 1.0×10-5m/s. Figure 3 shows the seepage flow type near the rock bridge, when the fracture aperture is 0.005 m. The results show that m depends on the dimension and the distribution of fractures significantly.

Table 1 Numerical calculation of the correction coefficient

Fig.3 Velocity distribution near the rock bridge

It is time-consuming and even difficult to conduct numerical analyses to calculate m, especially, for the fractures with a complicated distribution in the real project. In this study, the back analysis method is used to estimate m based on the ANN from singlehole packer test results. Furthermore, since the packer tests can provide some rather exact information of thefield, the back analysis combined with the packer tests can compensate the bias in the geological measurements (e.g., the fracture aperture measurement, the fracture connectivity rate measurement) and finally we may come up with a relatively accurate equivalent permeability tensor for fractured rock masses.

Fig.4 Sketch of the ANN structure for back analysis

2. Back analysis by ANN

The ANN[12,13]is a mathematical model for simulating the structure and the functions of biological neural networks. It is an adaptive system (usually with one input layer, one or several hidden layers, and one output layer) (as shown in Fig.4) based on the information flowing through the network. The ANN is often used as a back analysis method to fit the complex relationship between its inputs and outputs through “training” phase, and then to predict the outputs for any input by the trained ANN.

The single-hole packer test, the triple packer test[14]and the cross-hole test[15,16]are the main in-situ tests used in practice for rock mass permeability measurements. The single-hole packer test is most widely used due to its simplicity and low cost. However, each single-hole packer test concerns with only one observed data, i.e., the injected flow rate, so its application should be under the assumption that the rock mass permeability is isotropic. In this paper, multiple single-hole packer tests combined with the ANN back analysis can avoid this limitation. The injected flow rate of each single-hole packer test Qjand the correction coefficient of each fracture set miare defined as the inputs and the outputs of the ANN, respectively (Fig.4). To train the ANN, a series of numerical simulations are implemented for the single-hole packer tests by the FEM. In these simulations, a variety of miare assumed, and then the relevant Qjis simulated by the FEM with the equivalent permeability tensor calculated according to Eqs.(2), (3) and (8). These miand Qjare formed as samples to train the ANN for fitting their relationship. The exact miis back analyzed by inputting the in-situ observed Qjinto the trained ANN. With the back analyzed mi, the exact anisotropic equivalent permeability tensor can be finally calculated.

To make the back analysis valid, the number of Qjshould be equal or greater than the number of mi, namely the number of single-hole packer tests should be equal or greater than the number of fracture sets. To avoid the linear correlation of the permeability contributed from each fracture set, three is the maximum number of the fracture sets for the back analysis. Similarly, to avoid the linear correlation of Qj, the packer tests should be conducted in different directions.

3. Procedures of back analysis

3.1 Investigate fracture geological information and conduct single-hole packer tests

The geological information of each fracture set is investigated, including the orientation, the spacing, the aperture and the connectivity rate. Single-hole packer tests are carried out in the field and the test results are collected. The available test results should have a linear test pressure/injected flow rate relationship to reflect the laminar flow type they have.

3.2 Establish the finite element models and the boundary conditions

The finite element models are built for the singlehole packer tests (Fig.5), according to the diameters, the orientations and the test segment locations of their test bore-holes. The hydraulic boundary conditions are set as shown in Fig.1 (in an axial section): GFE is applied with the test pressure head, AB and BC are possible overflow boundaries, AG and DC are impervious, and the lengths of ED and AB are assumed equal to the length of GF (i.e., the length of the test segment). The FEM is widely used for the seepage analysis in the anisotropic domain with complex boundary conditions[17]. The seepage steady analysis based on the FEM is conducted here for the singlehole packer test simulations. The boundary of the free surface in this study is approached by iterations using the initial flow method[18].

Fig.5 Sketch of the finite element model for a single-hole packer test

3.3 Prepare the samples for training the ANN

The sample number and the range of sample variables (i.e., the range of the correction coefficients)are determined. The values of the correction coefficients are uniformly selected in the range, and arranged by the Uniform Design[19]. The packer tests are simulated by the FEM with the equivalent permeability tensors, which are computed from the arranged correction coefficients. The injected flow rates are calculated from the simulated packer tests. These injected flow rates and the relevant correction coefficients form the samples to train the ANN.

Table 2 Geological information of the dam foundation

3.4 Back analyze the correction coefficient

The in-situ observed injected flow rates of the packer tests are used as the inputs taken into the trained ANN. The ANN outputs are the exact correction coefficients back analyzed. The exact equivalent permeability tensor is finally calculated using Eqs.(2), (3) and (8).

Fig.6 Sketch of the single-hole packer test positions (m). The arrows denote the test bore-hole directions and ZK107 is vertical. F denotes a fault

4. Engineering applications

The Xiaowan Hydropower Project is located at the Lancang River in Yunnan Province of China. It has a double curvature concrete arch dam of 292 m high, with an installed generating capacity of 4 200 megawatts (MW). There are three dominant fracture sets in the dam foundation with the geological information listed in Table 2. The tiny fractures in the rock masses can not be measured as clearly as the dominant fracture sets. Their permeability contribution is put into the intact rock permeability coefficient in Table 2. In order to study the permeability of the dam foundation, a series of single-hole packer tests are conducted. ZK13-4, ZK13-3 and ZK107 are the three of them in different directions as shown in Fig.6 and Table 3. The arrows denote the test bore-hole directions and F denotes a fault in Fig.6. It is noted that ZK107 is vertical. These packer test results are used to back analyze the equivalent permeability tensor of the rock masses in the dam foundation.

Preparing the samples to train the ANN, the range of correction coefficients is set as 10 to 200 for fracture sets 1 and 2, and 3 000 to 6 000 for fracture set 3. To make the back analysis reliable, the range should include the exact correction coefficients. However, the exact correction coefficients are unknown and need for the back analysis, so the range is adjusted by the trial and error method in this study. In addition, more sample number can improve the back analysis accuracy, but that might lead to difficulties of the ANN convergence in training. For a balance, the sample number is chosen as 30. Therefore, 30 groups of the correction coefficients are uniformly selected in their ranges and arranged by the Uniform Design. 30 groups of the injected flow rates are calculated from the packer test simulations using the FEM, with the equivalent permeability tensors computed from the arranged correction coefficients. These correction coefficients and injected flow rates form 30 samples to train the ANN. With the in-situ observed injected flow rates (Table 3) as the inputs taken into the trained ANN, the exact correction coefficients and the exact equivalent permeability tensor are back analyzed and listed in Table 4. The results are used to conduct the seepage analysis for this project.

Figure 7 is the finite element model of the dam foundation before excavation with 337 041 elements and 219 187 nodes. We use a coordinate system with x-axis pointing to east, y-axis to north and z-axis upright. The groundwater level is found to be about 40 m lower than the mountaintop on both sides of the river, which is used as the hydraulic boundary condition onthe left and right bank boundaries (Fig.7). The river level is 988m, set as the hydraulic boundary condition on the river bed. The upstream, downstream and bottom boundaries of the model are impermeable. Through the FEM simulation, the seepage field in the dam foundation is obtained.

Table 3 Information of the packer tests

Table 4 Back analysis results

Fig.7 Finite element model of the Xiaowan Project foundation

Figure 8 shows the hydraulic potential contours in the dam foundation, where the dash line denotes the future location of the arch dam. Figure 9 shows the cross river section at y = 200 m, where the dash lines denote the free surface. These figures indicate that the groundwater flows down into the river with a high level, and the right bank has a higher water table than the left bank. The simulated results can provide some general and detailed groundwater information for the seepage control design of the project.

Fig.8 Hydraulic potential contours in the dam foundation (m)

Fig. 9 Hydraulic potential contours in the cross river section at y = 200 m (m)

5. Conclusion

In this paper, a revised solution of the equivalent permeability tensor is proposed to represent the influence of the fracture connectivity for discontinuous fractures. It involves a correction coefficient to reflect the complex seepage flow type through the rock bridge. This coefficient is back analyzed from singlehole packer tests, based on the ANN back analysis and the FEM seepage simulation. The limitation of this method is that the number of single-hole packer tests should be equal or greater than the number of fracture sets, and three is the maximum number of the fracture sets. This proposed solution is successfully used in the permeability measurement and the seepage simulation for the Xiaowan arch dam foundation.

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10.1016/S1001-6058(11)60295-3

* Project supported by the National Natural Science Foundation of China (Grant No. 51079109).

Biography: HE Ji (1982-), Male, Ph. D.