The permeability coefficient of a concrete slab rockfill dam (CFRD) was calculated in this paper on the basis of the equivalent quasicontinuum model for the percolation of crackintensive face. This calculation helped simplify the establishment of a finite element model and improve the efficiency of calculating the seepage of dams. Moreover, an inversion algorithm based on particle swarm optimization and support vector machine was proposed and applied. Comparison of the permeability coefficients produced from the two methods showed minimal difference. On this basis, the seepage field of the dam was analyzed. The analysis and field monitoring data reveal that the proposed algorithm is of high application value, which lays a foundation for future studies on the seepage properties of CFRD with cracks.
Seepage has been a major concern in dam engineering since the face rockfill dam emerged. Design deficiencies in the dam body and antiseepage, damage to the face from squeezing and face cracks, and similar factors will result in leakage and unsteady seepage. Once the face is seriously damaged, its antiseepage function will be weakened; consequently, seepage discharge will increase and flow through the bedding and transition areas, resulting in cracks and collapse of the face [
Some scholars have used the equivalent continuum method (ECM) and the discrete medium method to solve the problem of fissured flow through CFRD cracks [
Cracks on the concrete slab may be caused either by construction and water pressure or by temperature, thereby affecting the permeability coefficient, which is changeable due to the load from water pressure and the dead loads of the cushion, transition layer, and main rockfill. At this time, the permeability coefficient of the concrete slab, cushion, transition layer, and main rockfill can be calculated with monitoring data, which is crucial for improving the analysis on the 3D seepage field and providing an objective assessment of real seepage conditions. At present, numerous scholars have studied the permeability coefficient of earthrock or corewall dam on the basis of practical monitoring data. Inverse analysis on the permeability coefficient of earthrock dam was conducted by Wang et al. [
In summary, a method based on PSO and SVM for calculating the permeability coefficient was proposed in this paper from monitoring data of seepage. The algorithm was applied in practice, and the result was compared with that from an equivalent continuum model. The comparison revealed minimal difference between the two methods. The seepage properties were analyzed, and onspot monitoring data showed that the method proposed is of high application value. Therefore, this study paves the way for further investigations on the seepage properties of CFRD with cracks.
The problem of seepage through cracks on a single face or through a joint can be regarded as the seepage through cracks on parallel faces. In reality, the length of one crack is considerably longer than its width. Assuming that no water exchange exists along the crack, the mathematical model will be as follows [
The governing equation for incompressible fluid is as follows:
On the basis of Figure
Water flow through equivalentwidth cracks.
Suppose that, at the initial moment, steady flow is present, the initial condition of (
Boundary condition satisfies the following equation:
Through equations (
The previous equations describe the process of solving the hydraulic characteristics of unsteady flow. For steady flow, the following equation can be obtained:
Its boundary condition is the second equation in equation (
The pressure head on one crack surface is
Then, the unit discharge of steady flow inside the crack is as follows:
Several of the existing CFRDs have numerous cracks; thus, this study is crucial. Given the properties of seepage along crackintensive faces, assuming that numerous cracks are densely distributed is reasonable. That is, most of the cracks traverse the whole crack along the perpendicular direction, such that the seepage mostly penetrates the cracks and the seepage flows are laminar.
With these assumptions, the seepage analysis model can be developed, as shown in Figure
Seepage analysis model.
A dense seepage may help create a connected potential surface between the crackintensive face and cushion, such that piezometer heads downstream are approximately equal, and all piezometer heads upstream are
When
A crackintensive face is seen as a macroscopic quasicontinuum medium, and its seepage coefficient is regarded as an equivalently uniform seepage coefficient. Despite the tedious and complicated preparation, the calculation efficiency was greatly improved because the distribution features and locations of cracks are irrelevant. Therefore, the face can be divided into meshes and each unit seepage coefficient is
The SVM was used to develop the model and find a nonlinear mapping [
The insensitivity function
The optimum regression hyperplane satisfies that all sample points are close to the hyperplane, and the distance between them are within
Insensitivity function
In common regression fitting, the fitting error is considered, which means that the error of some fitting points can exceed
At this moment, the constraint conditions are as follows:
The solving process for equation (
Kernel function was used to replace the dot product in the previous equation, which will be changed into the following equation:
When
Kernel function is a mapping function formed in a highdimensional space when the input vector goes through a nonlinear transformation, so it determines the nonlinearity of the SVM. That is, different kernel functions require different algorithms. The kernel function, which can precisely reflect the distribution characteristics of samples, will greatly improve the nonlinearity of the SVM.
In this paper, radialbasis kernel function was applied:
Loss function is used to assess the inconsistent degree between
In summary, the SVM is a learning machine for searching a nonlinear mapping function (equation (
Particle swarm optimization, proposed by Eberhart and Kennedy in 1995, is a swarm intelligent optimization [
First, suppose that group
Second, the fitness degree for
Third, the location and velocity of each particle can be renewed during iteration by tracking the individual and group extreme values, as follows:
Finally, the optimum solution is ascertained once the conditions are satisfied.
With monitoring data on seepage discharge and PSOSVM, the inversion was performed, as shown in Figure
Flow of the permeability coefficient inversion.
First, the value range of the permeability coefficient was determined on the basis of real engineering. The orthogonal test was adopted to generate the testing program combinations used in the finite element model.
Second, the 3D finite element was used to calculate the testing program in each group. The seepage discharge in each group was computed. Then, the two results will be used as learning samples in the inversion model.
Penalty factor C and kernel parameter
Fourth, on the basis of the inversion model, the least error between the calculated and measured values of seepage discharge was used as the objective function; then, the best combination of the permeability coefficient was determined through PSO.
The objective function is as follows:
The constraint condition is as follows:
The equivalent continuum model was used to calculate the equivalent permeability coefficient of one face of CFRD; the inversion method based on PSSBVM was adopted in the inversion of the coefficient. The seepage fields identified from the two methods were compared; monitoring data verified the feasibility of the model and inversion method.
The highest body of one CFRD is 120.0 m, and the elevation of its top is 760.00 m; the top extends as long as 259.8 m from east to west and the face is as thick as
In all studies on CFRDs, no regional seepage discharge has been measured [
Distribution of measuring weirs.
Monitoring data of seepage discharge.
Figure
Onsite inspection of regional cracks.
Distribution of all cracks.
Parameters of cracks.









1  7.98  0.05  0.30  14  48.89  0.31  0.31 
2  8.51  0.22  0.34  15  53.01  0.37  0.24 
3  12.89  0.17  0.29  16  56.64  0.41  0.43 
4  13.15  0.16  0.34  17  56.81  0.29  0.31 
5  14.47  0.12  0.21  18  77.42  0.28  0.34 
6  17.96  0.24  0.27  19  77.53  0.07  0.29 
7  26.39  0.33  0.23  20  86.77  0.40  0.28 
8  29.90  0.37  0.27  21  86.96  0.35  0.31 
9  30.62  0.48  0.24  22  87.42  0.28  0.18 
10  34.21  0.23  0.46  23  90.34  0.38  0.26 
11  34.60  0.46  0.33  24  90.85  0.26  0.29 
12  40.55  0.43  0.21  25  95.11  0.16  0.44 
13  43.93  0.05  0.30  26  106.75  0.13  0.27 
Figure
3D finite element model of CFRD.
In the model, the following boundary conditions were mainly considered:
Bedrock boundary: the basic depth of area analyzed was set at 120 m. Undrained boundary was adopted as the intercepting boundary in this analysis.
Upstream and downstream boundaries of the dam area: monitoring data on the geological conditions of the dam and the underground water distribution were used as references to assess the seepage situation to fill the underground water in the dam bed. For the comparisons of seepage discharges, the flow of underground water along the water flow at the bed is believed to be negligible. Therefore, the flow exchange at the boundary is zero, which indicates undrained boundary.
Known boundary of the water level: to avoid the influence of rainfall, the water level without rainfall for a long time was used upstream. The bed below the water level upstream and downstream was set as the known boundary of constant head.
Potential overflow boundary: seepage may happen at the dam body above the water level at the upstream bed and at the rock surface on both banks. Thus, in the calculations, seepage was regarded as potential overflow boundary, and the final iteration result was used to see whether seepage occurred.
Bank bounadary: as the level of underground water at both banks is higher than the water head at the bed, the overall seepage flows to the bed as supplement seepage. Hence, the boundary conditions on both banks were considered water head boundaries.
Upstream water level and rainfall are two major factors influencing seepage discharge, and Figure
Monitoring data of upstream water level and rainfall.
Water level and rainfall during the steady period (2012101 to 20121031).
Table
Permeability coefficients to be inverted.
Material  Permeability coefficient (m/s) 

Concrete slab  1 × 10^{−11}∼1 × 10^{−9} 
Cushion  1 × 10^{−6}∼1 × 10^{−5} 
Transition layer  1 × 10^{−4}∼1 × 10^{−3} 
Major rockfill  0.0001∼0.01 
Level of each factor.

Permeability coefficient (m/s)  

Concrete slab  Cushion  Transition layer  Major rockfill  
1  1 × 10^{−11}  1 × 10^{−6}  1 × 10^{−4}  0.0001 
2  6 × 10^{−11}  3.5 × 10^{−6}  3.5 × 10^{−4}  0.0005 
3  1 × 10^{−10}  6 × 10^{−6}  6 × 10^{−4}  0.001 
4  5.5 × 10^{−9}  8.5 × 10^{−6}  8.5 × 10^{−4}  0.005 
5  1 × 10^{−9}  1 × 10^{−5}  1 × 10^{−3}  0.01 
Calculation conditions and results of the finite element algorithm.
Test  Permeability coefficient (m/s)  Calculated seepage discharge (L/s)  

Concrete slab  Cushion  Transition layer  Major rockfill  
1  1 × 10^{−11}  1 × 10^{−6}  1 × 10^{−4}  0.0001  2.63 
2  1 × 10^{−11}  3.5 × 10^{−6}  3.5 × 10^{−4}  0.0005  2.91 
3  1 × 10^{−11}  6 × 10^{−6}  6 × 10^{−4}  0.001  2.95 
4  1 × 10^{−11}  8.5 × 10^{−6}  8.5 × 10^{−4}  0.005  3.01 
5  1 × 10^{−11}  1 × 10^{−5}  1 × 10^{−3}  0.01  3.04 
6  6 × 10^{−11}  1 × 10^{−6}  3.5 × 10^{−4}  0.001  2.92 
7  6 × 10^{−11}  3.5 × 10^{−6}  6 × 10^{−4}  0.005  3.01 
8  6 × 10^{−11}  6 × 10^{−6}  8.5 × 10^{−4}  0.01  3.05 
9  6 × 10^{−11}  8.5 × 10^{−6}  1 × 10^{−3}  0.0001  2.68 
10  6 × 10^{−11}  1 × 10^{−5}  1 × 10^{−4}  0.0005  2.93 
11  1 × 10^{−11}  1 × 10^{−6}  6 × 10^{−4}  0.01  3.02 
12  1 × 10^{−10}  3.5 × 10^{−6}  8.5 × 10^{−4}  0.0001  2.68 
13  1 × 10^{−10}  6 × 10^{−6}  1 × 10^{−3}  0.0005  2.94 
14  1 × 10^{−10}  8.5 × 10^{−6}  1 × 10^{−4}  0.001  2.98 
15  1 × 10^{−10}  1 × 10^{−5}  3.5 × 10^{−4}  0.005  3.04 
16  5.5 × 10^{−10}  1 × 10^{−6}  8.5 × 10^{−4}  0.0005  2.91 
17  5.5 × 10^{−10}  3.5 × 10^{−6}  1 × 10^{−3}  0.001  2.98 
18  5.5 × 10^{−10}  6 × 10^{−6}  1 × 10^{−4}  0.005  3.05 
19  5.5 × 10^{−10}  8.5 × 10^{−6}  3.5 × 10^{−4}  0.01  3.08 
20  5.5 × 10^{−10}  1 × 10^{−5}  6 × 10^{−4}  0.0001  2.7 
21  1 × 10^{−9}  1 × 10^{−6}  1 × 10^{−3}  0.005  3.01 
22  1 × 10^{−9}  3.5 × 10^{−6}  1 × 10^{−4}  0.01  3.07 
23  1 × 10^{−9}  6 × 10^{−6}  3.5 × 10^{−4}  0.0001  2.70 
24  1 × 10^{−9}  8.5 × 10^{−6}  6 × 10^{−4}  0.0005  2.96 
25  1 × 10^{−9}  1 × 10^{−5}  8.5 × 10^{−4}  0.001  3.00 
Initial parameters were set as 200 times for the group iteration and 20 for the number of groups. The permeability coefficient of each material was calculated through the inversion algorithm, which is shown in Table
Inversion results of the permeability coefficient (m/s).
Concrete slab  Cushion  Transition layer  Major rockfill 

2.52 × 10^{−10}  5.11 × 10^{−6}  5.57 × 10^{−4}  4.34 × 10^{−2} 
Hydrograph of particle swarm iteration.
To verify the reasonability of the inversion results, prior analysis was conducted on seven sets of seepage discharge data observed when the water level was steady with minimal rainfall (no rainfall in at least 15 days). The calculated value was revealed to be close to the measured value. Table
Comparisons between calculated and monitoring values.
Date  Upstream water (m)  Calculated seepage discharge (L/s)  Monitoring seepage discharge (L/s)  Calculated datamonitoring data/monitoring data (%) 

20041219  753.09  3.23  3.1  4.02 
20081220  738.76  3.46  3.7  6.49 
2009116  740.63  3.1  2.86  4.98 
2009127  737.55  3.03  3.15  3.81 
2010116  749.95  3.67  3.98  7.79 
20111224  745.56  3.48  3.7  5.95 
20121015  738.18  3.04  3.32  8.43 
On the basis of the designed data and seepage parameters from inversion, the seepage properties in cases of normal water level, designed flood level, and check flood level were calculated. Table
Dam body materials and permeability coefficient
Material  Concrete slab  Cushion  Transition layer  Major rockfill  Bedrock  Toe slab  Peripheral joints  Curtain  Rock  Parapet wall 


2.85 × 10^{−10}  5.11 × 10^{−6}  5.57 × 10^{−4}  4.34 × 10^{−2}  1.5 × 10^{−7}  1 × 10^{−10}  1 × 10^{−10}  5 × 10^{−8}  1.5 × 10^{−7}  1 × 10^{−10} 
In the analysis, the calculation conditions are normal water level (755.00 m), designed flood level (756.2 m), and check flood level (759.1 m).
Figures
Typical section contour line of seepage pressure in case of the normal water level (m): (a) contour line of seepage pressure of river bed section and (b) contour line of seepage pressure of dam axis section.
Typical section contour line of seepage pressure in case of the designed flood level (m): (a) contour line of seepage pressure of river bed section and (b) contour line of seepage pressure of dam axis section.
Typical section contour line of seepage pressure in case of the check flood level (m): (a) contour line of seepage pressure of river bed section and (b) contour line of seepage pressure of dam axis section.
Despite there are 206 cracks on the concrete slab, as shown in Figure
There is 20% difference between the results from the two methods, and it is hard to say which method is more accurate, but when there is only monitoring data of seepage discharge, the PSOSVM inversion method could be more appropriate; conversely, if there are no monitoring data of seepage discharge, the equivalent quasicontinuum model should be adopted. When using the equivalent quasicontinuum model, parameters of all cracks should be measured, which usually takes a lot material resources and time. Most hydropower stations only carry out cracks detection once every few years. Conversely, the monitoring data of seepage discharge usually can be got easily by measuring weirs. So, the PSOSVM inversion method can be suitable and justified for most cases.
Various shapes of cracks may appear on the face of the CFRD during construction or storage due to its thickness and stiffness difference relative to the cushion, which is harmful to its antiseepage system. Considering the important role of concrete slabs in the antiseepage system of the CFRD, the quality of these faces must be under strict control to reduce the damage caused by seepage from cracks, and checking the cracks regularly is necessary.
The permeability coefficient of the CFRD was calculated in the equivalent continuum medium model, which simplifies the construction of the finite element model and improves the calculation efficiency. The inversion algorithm based on PSOSVM proved satisfactory in inverting the permeability coefficient with monitoring data. With the coefficients from the two methods, seepage discharge was identified, which was relatively close to the monitoring value, indicating that the two methods are feasible and applicable. This study lays the necessary foundation for further studies on the seepage features of CFRD cracks.
Conversely, measuring the regional seepage discharge can help identify the accurate discharge at each part, thereby safeguarding a secure operation of the monitor, narrowing down the searching range for accidents in an emergency, avoiding blindness, and facilitating the testing of the antiseepage effect on both banks, the waterstop effect of peripheral joints, and the working conditions of concrete slabs.
The data on cracks used to support the findings of this study are available from the corresponding author upon request. The other data used to support the findings of this study are included within the article.
The authors declare no conflicts of interest.
This study was funded by the National Key R&D Program of China (2016YFC0401601, 2017YFC0804607, and 2018YFC0407104), National Natural Science Foundation of China (Grant nos. 51739003, 51479054, 51779086, 51579086, 51379068, 51579083, 51579085, and 51609074), Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (YS11001), Jiangsu Natural Science Foundation (Grant no. BK20160872), Special Project Funded by National Key Laboratory (20165042112), Key R&D Program of Guangxi (AB17195074), and Central University Basic Research Project (2017B11114 and 2018B25514).