# A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Development of the Line-Element Model

#### 2.1. Steady Free Surface Flow in Three-Dimensional Porous Media

_{x}, v

_{y}and v

_{z}denote the flow velocity in the x, y and z (or 1, 2 and 3) directions, respectively:

_{x}, k

_{y}and k

_{z}denote the saturated hydraulic conductivity in the x, y and z (or 1, 2 and 3) directions, respectively; $\varphi =p/(\rho g)+z$ is the total head of water; and $p/(\rho g)$ is the water pressure head.

_{1}, S

_{2}, S

_{3}and S

_{4}are the water head, flux, seepage face and free surface boundaries, respectively.

#### 2.2. Equivalent Flow Velocity, Hydraulic Conductivity and Continuity Equation

_{lx}and a cross-section area A

_{lx}, the flow velocity v

_{lx}and flow rate Q

_{x}through the line elements in the x direction can also be described by Darcy’s law:

_{y}and N

_{z}are the number of rows and columns of the x-directional line elements in the control volume.

_{y}and B

_{z}are the row and column distances, as shown in Figure 2c, and are equal to Δy/N

_{y}and Δz/N

_{z}, respectively. For convenience, in mathematics, A

_{ij}is nominated as a unit of area, and Equation (10) is reduced to:

_{ly}, v

_{lz}and hydraulic conductivity k

_{ly}, k

_{lz}can be derived as:

_{x}is the x-directional distance.

_{ly}= v

_{lz}= 0 in the x-directional line elements. Thus, the continuity Equation (1) in the x, y and z directions can be simplified as:

#### 2.3. Unified Formulations and Boundary Conditions

## 3. Finite Line-Element Method

**B**is the one-dimensional geometric matrix in the local coordinate system, and η is the iteration step. The iterative process should satisfy the convergence condition as below:

## 4. Validations

#### 4.1. A Rectangular Dam with Tailwater

_{x}= B

_{y}= B

_{z}= 0.1 m, and the penalty parameter λ is valued as 1 × 10

^{−10}. The water head distribution and free surface locations are shown in Figure 3b,c, respectively. For comparison, the numerical predictions from Borja and Kishnani [9] and Bardet and Tobita [21] are also plotted. The results from Borja and Kishnani have a considerable discrepancy near the downstream because of singular seepage points. The free surface predicted by the proposed line-element method agrees well with that from Lacy and Prevost and Bardet and Tobita.

_{x}= B

_{y}= B

_{z}= 0.1 m, 0.25 m, 0.5 m) and penalty parameters (λ = 1 × 10

^{−10}, 0.01, 0.1, 0.25, 0.5) are considered. The free surface locations from different mesh sizes are plotted in Figure 4; the three curves are almost identical except that the seepage point from B

_{x}= B

_{y}= B

_{z}= 0.5 m is higher than others. As a result, the influence of mesh size is not notable on the distribution of free surface but on seepage point.

^{−4}m

^{2}/s. The relative error between the numerical and analytical results decreases with the decrease in λ. It is found that when λ is equal to or less than 0.1, the calculated free surface location, discharge per unit width and iteration step is independent of λ. Therefore, λ = 0.1 is used for other illustrated examples. Note that the error tolerance is set as 0.001, which can guarantee numerical accuracy and efficiency.

#### 4.2. A right Trapezoidal Dam

_{x}= B

_{y}= B

_{z}= 0.1 m, 0.2 m, 0.25 m), and the related predictions of free surface are shown in Figure 6c. In addition, the finite element results based on the continuum model from Lacy and Prevost [8] and Zheng et al. [10] are also presented in Figure 6b for comparison. There is good agreement between the proposed line-element approach and the continuum-based finite element methods. In this right trapezoidal dam, the proposed line-element model is also not sensitive to the mesh size and can achieve an accurate solution in six iterations.

#### 4.3. A Left Bank Abutment Slope of Kajiwa Dam in Southwestern China

^{2}with an annual average flow of 101 m

^{3}/s. This concrete face rockfill dam is 171 m in height, and the crest elevation is 2885 m. The installed capacity is 452.4 MW, and the total storage capacity is 374.5 million m

^{3}at the normal water level of 2850 m.

_{1}r

^{3}) and metamorphic quartz sandstone, carbonaceous slate and phyllitic slate from the fourth member (O

_{1}r

^{4}) of the Rengong Formation of the lower Ordovician system. Above 2830 m, the slope surface is covered by a large area of residual slope deposits and mainly consists of the glacial fluvial (fglQ3), eluvial (el plus dlQ4), colluvial (col plus dlQ4) and diluvial (plQ4) deposits.

^{−6}m/s and 1.5 × 10

^{−10}m/s, respectively. The water level along the mountain side is 2980 m, and along the river side, it is 2850 m. The bottom and lateral boundaries perpendicular to the river are impermeable, and the residual boundary is specified as a potential seepage boundary.

_{x}= B

_{y}= B

_{z}= 5 m, 10 m, 20 m). As shown in Figure 9, by comparing the three kinds of mesh size, it can be seen that the results from B

_{x}= B

_{y}= B

_{z}= 5 m and B

_{x}= B

_{y}= B

_{z}= 10 m are almost overlapped and agree well with the continuum model. In contrast, the deviation of B

_{x}= B

_{y}= B

_{z}= 20 m is pronounced. In order to obtain higher calculation accuracy and consume less calculation time, the mesh size of B

_{x}= B

_{y}= B

_{z}= 10 m is selected in this paper.

_{x}= B

_{y}= B

_{z}= 10 m, as shown in Figure 10a, with line elements and nodes. For comparison, the tetrahedron element mesh (Figure 10b) is also presented to model the three-dimensional steady free surface flow based on the parabolic variational inequality algorithm of Signorini’s condition proposed by Zheng et al. [10] and Chen et al. [1], and the details are as follows:

**B**is the partial derivatives matrix of

**N**, N

_{i}is the shape function and m is the node number of volume element.

## 5. Conclusions

**K**in the line-element model is accurate, while that of the continuum model is approximate subject to Gaussian points. Therefore, the proposed line-element model can obtain accurate solutions in a few steps, especially for complicated engineering applications, and the numerical difficulty is greatly decreased.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The continuum model with cubic packing of grains. (

**b**) The pore space occupied by water flow. (

**c**) The line-element model in three-dimensional space.

**Figure 3.**(

**a**) Rectangular dam with mesh size. (

**b**) Water head distribution from the line-element method. (

**c**) Free surface locations from different methods.

**Figure 6.**(

**a**) Right trapezoidal dam with mesh size. (

**b**) Free surface locations between the proposed line-element approach and the continuum-based finite element method. (

**c**) Free surface locations from different mesh sizes.

**Figure 9.**The free surface locations from different profiles of different mesh sizes: (

**a**) y = 50 m; (

**b**) y = 200 m.

**Figure 10.**(

**a**) The line-element mesh. (

**b**) The tetrahedron element mesh of the left bank abutment slope of Kajiwa Dam.

**Figure 11.**The contours of water pressure head from (

**a**) the line-element model and (

**b**) the tetrahedron element model.

λ | Analytical (m^{2}/s) | Numerical (m ^{2}/s) | Iteration Steps | Relative Error |
---|---|---|---|---|

1 × 10^{−10} | 1.111 × 10^{−4} | 1.100 × 10^{−4} | 8 | 0.99 |

0.01 | 1.100 × 10^{−4} | 8 | 0.99 | |

0.10 | 1.100 × 10^{−4} | 8 | 0.99 | |

0.25 | 1.099 × 10^{−4} | 8 | 1.08 | |

0.50 | 1.092 × 10^{−4} | 7 | 1.71 |

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**MDPI and ACS Style**

Yuan, Q.; Yin, D.; Chen, Y.
A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media. *Water* **2023**, *15*, 1030.
https://doi.org/10.3390/w15061030

**AMA Style**

Yuan Q, Yin D, Chen Y.
A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media. *Water*. 2023; 15(6):1030.
https://doi.org/10.3390/w15061030

**Chicago/Turabian Style**

Yuan, Qianfeng, Dong Yin, and Yuting Chen.
2023. "A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media" *Water* 15, no. 6: 1030.
https://doi.org/10.3390/w15061030