Rock Mech Rock Eng (2011) 44:491–496 DOI 10.1007/s00603-011-0149-2
TECHNICAL NOTE
Back Analysis of Equivalent Permeability Tensor for Fractured Rock Masses from Packer Tests Ji He • Sheng-hong Chen • Isam Shahrour
Received: 3 November 2010 / Accepted: 7 April 2011 / Published online: 21 April 2011 Ó Springer-Verlag 2011
Keywords Fractured rock Permeability tensor Back analysis Packer test Artificial neural network
1 Introduction Equivalent porous media model is widely used in numerical seepage analysis for fractured rock masses. The key issue for this model application is to evaluate equivalent permeability tensor by following methods: -Geometric method. This method measures the geometry of dominant fracture sets (Priest 1993; Wines and Lilly 2002), and formulates the equivalent permeability tensor by fracture tensor (Oda 1985). It is straightforward in theory and can represent the anisotropy of rock masses. However, the geometric parameters are difficult to be measured precisely, especially the fracture aperture. -Experimental method. Single-hole packer test (Fig. 1, Singhal and Gupta 1999), triple hydraulic probe (Louis and Maini 1970), and cross-hole test (Hsich and Neuman 1985; Hsich et al. 1985) are the main in situ tests used in practice. The rock permeability can be carried out from these test results by analytical solutions. In situ tests can provide exact information of field, but they are rather costly and time-consuming. The single-hole packer test is relatively J. He (&) S. Chen State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 430072 Wuhan, China e-mail:
[email protected] I. Shahrour Laboratoire de Me´canique de Lille (UMR 8107), Universite´ des Sciences et Technologies de Lille, Cite´ Scientifique, 59656 Villeneuve D’Ascq, France
simple and cheap among them. Unfortunately, it is under the assumption of rock permeability isotropy. -Back analysis method. This method numerically simulates the seepage field in rock masses, and inversely calculates rock mass permeability by back analysis algorithm, such as Linear Regression, Genetic Algorithm (Goldberg and Richardson 1987), and Artificial Neural Network (ANN) (Werbos 1994; Mukhopadhyay 1999). The use of back analysis results in larger volumes of rock masses being tested. However, an anisotropic permeability tensor has six independent components at least. Traditional back analysis method is still difficult to deal with all of them. A new approach is proposed in this paper by combining the above three methods. The main procedures are as follows: firstly, the spacing and orientation of dominant fracture sets are measured in field; secondly, a group of single-hole packer tests are conducted (experimental method), and their numerical simulations are implemented using the Finite Element Method (FEM); thirdly, from the numerical and in situ packer test results, the apertures of dominant fracture sets are back analyzed by the ANN (back analysis method); finally, the anisotropic equivalent permeability tensor is formulated by fracture tensor from the obtained parameters of fracture geometry (geometric method). In this way, the use of back analysis for fracture aperture avoids the difficulty in measurement, and the use of single-hole packer test in back analysis deals with the limitation of permeability isotropy.
2 Formulation of the Equivalent Permeability Tensor All fractures in rock masses are grouped into sets according to site investigation. Each set has constant aperture, uniform spacing and dominant orientation. The rock mass
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unknown parameter and needs for back analysis, while the other fracture geometric parameters are directly measured in field.
3 Back Analyses by the Artificial Neural Network
Fig. 1 Sketch of single-hole packer test
permeability is contributed from individual fractures and intact rocks. Ignoring the interaction of fractures, the equivalent permeability tensor of fractured rock masses is formulated by fracture tensor as (Oda 1985): 2 3 1 ðnix Þ2 nix niy nix niz n 3 X gai 6 7 ½K ¼ 1 ðniy Þ2 niy niz 5 4 niy nix 12b m i i¼1 niz nix niz niy 1 ðniz Þ2 2 3 kr 5 ð1Þ kr þ4 kr where, n is the total number of fracture sets; i is the serial number of fracture sets; ai is the fracture aperture; bi is the fracture spacing; nix ; niy ; niz are the direction cosines of fracture normal; kr is the permeability coefficient of intact rock; m is the kinematical viscosity coefficient; g is the gravity acceleration. Fracture aperture is rather tiny but significantly sensitive to fracture permeability, so it is difficult but has to be measured precisely. Therefore, to determine the rock mass permeability ½K in this study, ai is treated as the only
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Artificial neural network (ANN) (Werbos 1994; Mukhopadhyay, 1999) is a mathematical model that simulates the structure and function of biological neural network. It consists of an interconnected group of artificial neurons usually with one input layer, one or several hidden layers, and one output layer (Fig. 2). The ANN is often used as a nonlinear data modeling tool to fit the complex relationship between its inputs and outputs through ‘‘training’’ phase. The outputs can be predicted for any input using the trained ANN. In this study, the injected flow rate Qj of each singlehole packer test and the aperture ai of each fracture set are respectively defined as the input and output of the ANN (Fig. 2). To train the ANN, a series of numerical simulations are implemented for the conducted single-hole packer tests by the FEM. In these simulations, a variety of ai are supposed and the relevant Qj are simulated. These ai and Qj are formed as samples to train the ANN for fitting their relationship. The exact ai can be back analyzed by inputting in situ observed Qj into the trained ANN. Qj is the only observed data for each single-hole packer test. It means that in order to back analyze ai from Qj , the number of single-hole packer tests should be equal or more than the number of fracture sets. Moreover, as to avoid the linear correlation of permeability contributed from different fracture sets, three fracture sets are the maximum for the back analysis. Similarly, as to avoid the linear
Fig. 2 Sketch of the ANN structure for back analysis. Qj denotes the injected flow rate of the jth single-hole packer test; ai denotes the aperture of the ith fracture set; Qj and ai are respectively defined as the input and output of the ANN
Back Analysis of Equivalent Permeability Tensor
correlation of Qj , the packer tests should have different directions.
4 Procedures of the Back Analysis 4.1 Measure the geometry of fracture sets All fractures in rock masses are grouped into three sets, and their orientations and spacings are measured in field. 4.2 Conduct the single-hole packer tests in field The number of single-hole packer tests should be equal or more than that of fracture sets. The directions of their test bore-holes need to be different. Each packer test requires at least one test segment to provide test data. If there are more than one test segment, their observed injected flow rates are averaged for the general representation of nearby rocks. The available in situ test results should display a linear relation between test pressure and injected flow rate, so as to ensure the laminar flow state in fractures. However, packer test often meets problems in practice, such as measurement error, test pressure loss, turbulence effect and fracture aperture deformation due to high test pressure (Singhal and Gupta 1999). These problems lead to bias from linear relation in test results. In order to evaluate the bias, a straight line is often fitted from the test results using the Least Square Method (LSM) (Fig. 3). If the test results seem close to the fitted straight line, it implies that the test results have nearly linear relation and they are available for the back analysis in this study.
Fig. 3 Test pressure-injected flow rate diagram of packer tests. Point symbols denote the in situ observed injected flow rate of packer tests (ZK13-4 and ZK13-3 in Xiaowan Project) under the test pressures of 0.3 Mpa, 0.6 Mpa and 1.0 Mpa. The solid and dash straight lines are fitted from these test results using the Least Square Method
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4.3 Establish the finite element model for single-hole packer test According to the diameter, orientation and test segment location of test bore-hole, a finite element model is established for each single-hole packer test. The simulation region of this model is defined as a cylinder to represent the rock masses around the test bole-hole (Fig. 4). The model is built in the coordinate system of x-axis being east, y-axis being north, and z-axis being upright. The hydraulic boundary condition for this model is set as follows and showed in an axial section (Fig. 1): GFE is applied with test pressure head, AB and BC are set as overflow boundaries, AG and DC are impervious, and the lengths of ED and AB are assumed equal to the length of GF (test segment). The Finite Element Method (FEM) has been well used in numerical seepage analysis for anisotropic domain with complex boundary condition (Zienkiewicz et al. 2005). In this study, the boundary of free surface is approached by iteration using the Residual Flow Procedure (RFP) (Desai 1976). 4.4 Prepare the samples for training the ANN 1. Determine the number of samples and the range of sample variables (i.e., the range of fracture apertures). It is noted that increasing the sample number can improve back analysis accuracy, but that will also lead to more difficulty in the ANN convergence. Moreover, the range of fracture apertures should cover the exact aperture to enable reliable back analysis.
Fig. 4 Sketch of the finite element model for a single-hole packer test
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packer tests are implemented by the FEM simulation. Their information is listed in Table 1. The geometric and permeable information of the rock mass is listed in Table 2. It is further pretended that the fracture apertures in Table 2 are unknown parameters and need to be back analyzed. In the back analysis, the range of fracture apertures is set as 1.5E-5 to 2.5E-5 m for Fracture set 1, 1.875E-5 to 3.125E-5 for Fracture set 2 and 2.25E–5 to 3.75E-5 for Fracture set 3 (i.e., 25% deviation from the apertures in Table 2). The sample number is set as 30. The ANN hidden unit number is set as 8. The back analyzed apertures are listed in Table 3, in which their relative errors are compared with the desired apertures (Table 2). The maximum relative error is 2.47%. It implies that the back analysis algorithm can provide accurate results if the sample number and the ANN hidden unit number are appropriate. The equivalent permeability tensor calculated from the back analyzed apertures is also listed in Table 3.
2. Uniformly choose a group of apertures in the range, arrange them by the concept of Uniform Design (Fang et al. 2000), and calculate the equivalent permeability tensors relative to these apertures using Eq. 1. 3. Simulate the packer tests by the FEM with the equivalent permeability tensors calculated in Step (2), and compute their injected flow rates. 4. Form the samples with the chosen apertures and the computed injected flow rates for training the ANN. 4.5 Train the ANN with the samples The ANN is specified by hidden layer number, hidden unit number, momentum factor and learning rate. In this study, the hidden layer number is set as 1, the momentum factor is set as 0.2 and the learning rate is set as 0.8. The hidden unit number affects the precision of back analysis significantly. However, there is no exact way to identify its optimal value, so trial-and-error management is used in this study for minimizing the ANN convergence error.
6 Engineering application 4.6 Back analyze the exact apertures Xiaowan Hydropower Project is located at the Lancang River in the Yunnan Province of China. Its foundation contains three dominant fracture sets. Their geometric information and the intact rock permeability are listed in Table 4. A series of single-hole packer tests are conducted, and ZK13-4, ZK13-3 and ZK107 are the three of them (Fig. 5 and Table 5). The in situ test results of ZK13-4 and ZK13-3 are displayed in Fig. 3. The difference between the test results and their fitted lines seems little from this figure. It means that the test results are available for the back analysis. Since ZK107 is fixed with the test pressure of 0.48 MPa during test, it cannot provide such figure.
Inputting the in situ observed injected flow rate of each packer test into the trained ANN, the outputs are the exact apertures of fracture sets. Substituting these apertures into Eq. 1 the equivalent permeability tensor is finally obtained.
5 Verifications This section is to verify the back analysis algorithm established in Sect. 4. It is assumed that a rock mass contains three sets of fractures and has three single-hole packer tests conducted nearby in different directions. These
Table 1 Information of the packer tests (Verifications) Packer test
Test segment depth (m)
Diameter (m)
Dip direction
Dip angle
Test pressure (MPa)
In situ observed injected flow rate (m3/s)
1
10–15
0.075
N60°W
0°
0.6
9.32E–6
N30°E
60°
0.6
1.06E–5
90°
0.6
1.03E–5
2 3
Table 2 Geometric and permeable information of the rock mass (Verifications)
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Fracture set
Dip direction
Dip angle
Aperture (m)
Spacing (m)
Intact rock permeability coefficient (m/s)
1
S60°E
90°
2.0E-5
0.5
1.0E–9
2
S30°W
30°
2.5E-5
0.55
3
N30°E
60°
3.0E-5
2.14
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Table 3 Back analyzed apertures and equivalent permeability tensor (Verifications) Fracture
Equivalent permeability tensor (m/s)
Set
Aperture (m)
Relative error %
1
1.98E-5
-1.20
2.44E-8
-5.43E-10
1.99E-9
2
2.56E-5
2.47
-5.43E-10
2.38E-8
3.45E-9
3
3.07E-5
2.26
1.99E-9
3.45E-9
1.89E-8
According to site investigation, the aperture of Fracture set 3 is much wider than that of Fracture sets 1 and 2. The range of fracture apertures in the back analysis are set as 5E–6 to 1E-5 m for Fracture sets 1 and 2, and 5E–5 to 2E-4 m for Fracture set 3. The sample number is chosen as 30. The ANN hidden unit number is defined as 17. The back analyzed apertures and equivalent permeability tensor are listed in Table 6. These results have been used in designing the seepage control of Xiaowan Project.
7 Conclusions In this paper, a new approach is established to back analyze the equivalent permeability tensor for fractured rock masses, based on the fracture tensor formulation, the packer test simulation and the ANN back analysis. If the
Table 4 Geometric and permeable information of the rock masses (Engineering application)
Fracture set 1
Fig. 5 Sketch of the packer test locations (Unit: m). The arrows denote the directions of test bore-holes. The bore-hole of ZK107 is vertical. F denotes fault
Table 6 Back analyzed apertures and equivalent permeability tensor (Engineering application) Fracture
Equivalent permeability tensor (m/s)
Set
Aperture (m)
1
9.37E-6
9.65E-8
-1.17E-9
-1.27E-8
2
9.10E-6
-1.17E-9
1.02E-7
-2.57E-9
3
5.11E-5
-1.27E-8
-2.57E-9
7.56E-8
sample number and the ANN hidden unit number are appropriate, the back analysis error is small and acceptable (2.47% in this paper). The limitation of this approach is that the number of packer tests must be equal or more than that of fracture sets, and three fracture sets are the maximum.
Orientation description
Dip direction
Dip angle
Spacing (m)
Intact rock permeability coefficient (m/s)
Along river
S85°E
88°
0.5
6.8E-8
N10°E
80°
0.55
N79°E
25°
2.14
Steep dip 2
Across river Steep dip
3
Along river Gentle dip
Table 5 Information of the packer tests (Engineering application)
Packer test
Test segment No.
Depth (m)
Diameter (m)
Dip direction
Dip angle
Test pressure (MPa)
In situ observed injected flow rate (m3/s)
ZK13-4
2
8.56 - 13.69
0.075
N80°W
5°
0.6
3.28E-05
ZK13-3
2
8.36 - 13.49
0.075
S10°W
7°
0.6
1.59E-05
ZK107
6
51.81 - 61.98
0.056
90°
0.48
1.33E-04
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496 Acknowledgments This work is granted by the National Science Foundation of China (NSFC: 51079109).
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