# A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

^{−1}), k is the hydraulic conductivity (LT

^{−1}), and $\varphi $ is the potential head (L). Note that x, y and z denote the three principal axes of hydraulic conductivity.

_{s}is the specific storage (L

^{−1}) as the water volume released from a unit soil volume per unit declines in potential head [16], which is determined by the compressibility of solid skeleton and water. Generally, the water release as a result of the specific storage can be ignored in comparison to that of the gravitational specific yield. Thus, Equation (2) is reduced to

- (1)
- The initial condition [11]$${\left(\varphi \left(x,y,z,t\right)\right|}_{t=0}={\varphi}_{0}\left(x,y,z\right)$$
- (2)
- The potential head boundary S
_{1}[11]$$\varphi \left(t\right)=\overline{\varphi}\left(t\right)$$_{1}denotes the flow boundaries below the upstream and downstream water levels and are valued by the given water heads ${\overline{\varphi}}_{1}\left(t\right)$ and ${\overline{\varphi}}_{2}\left(t\right)$, respectively. - (3)
- (4)
- The Signorini-type seepage boundary S
_{3}

_{3u}with $\varphi \left(t\right)\le z,{q}_{n}\left(t\right)=0$ and the lower part below the transient free surface S

_{3l}with $\varphi \left(t\right)=z,{q}_{n}\left(t\right)\le 0$. Thus, the boundary condition on the potential seepage face can be unified as [11]

- (5)
- The transient free surface boundary ${S}_{f}$

## 3. The Line Element Method

#### 3.1. Equivalent Flow Velocity and Hydraulic Conductivity

_{l}and k

_{l}denote the flow velocity (LT

^{−1}) and hydraulic conductivity (LT

^{−1}) of the line elements.

_{l}(L

^{3}T

^{−1}) of the line elements yield

_{l}denotes the cross-section area of the line elements (L

^{2}).

_{x}, N

_{y}and N

_{z}are the numbers of line elements, which is obtained by:

_{x}, B

_{y}and B

_{z}are the spacings.

_{lx}

_{,}A

_{ly}and A

_{lz}are specified with a unit of area, Equation (14) can be simplified as

#### 3.2. Equivalent Continuity Equations

#### 3.3. Equivalent Transient Free Surface Boundary

#### 3.4. Unified Formulations and Numerical Procedure

## 4. Validations

#### 4.1. An Unconfined Aquifer

_{x}= k

_{y}= k

_{z}= 5 m/d and μ = 0.1. To evaluate the influence of mesh size and time step on the numerical accuracy and convergence, this aquifer is meshed by a 3D orthogonal line element with different mesh sizes (B

_{x}= B

_{y}= B

_{z}= 4 m, 2 m, 1 m) and simulated with various time steps ($\Delta t$ = 5 d, 1 d, 0.1 d).

_{x}= B

_{y}= B

_{z}= 1 m. The numerical results are almost consistent with the theoretical solutions and converge within three steps, which demonstrates that the proposed method is robust for different time steps.

#### 4.2. A Trapezoidal Dam

_{x}= k

_{y}= k

_{z}= 0.1 cm/s and μ = 0.4. The water level at the left boundary gradually increases from 0 cm to 15 cm.

#### 4.3. A Sand Flume

_{x}= k

_{y}= k

_{z}= 2.4 × 10

^{−3}cm/s and μ = 0.05. The water level at the left boundary gradually decreases from 1.0 m to 0.2 m after 48 min, while the right water level remains at 0.2 m. During numerical analysis, the drainage tunnel is specified as a Signorini-type seepage boundary. The comparisons of free surface locations after 18 min, 30 min and 48 min between the proposed line element and observations from Hu et al. [31] are presented in Figure 9b. The calculated free surface locations are very close to the measurements and are generally achieved within 6 iteration steps.

#### 4.4. A 3D Well

_{x}= k

_{y}= k

_{z}= 9.57 inch/min and μ = 0.3. The water level in the well decreases gradually from 48 inch to 12 inch at the initial time, while the water level of the peripheral boundary of the tank remains at 48 inch. Different from the uniform-sized element mesh in the above examples, the well model is divided into three domains meshed with non-uniformed sizes (coarse mesh 4 m, medium mesh 2 m and fine mesh 1 m) as shown in Figure 10a, in which the well with dense mesh is specified as a Signorini-type seepage boundary.

_{x}= 10k

_{y}= k

_{z}= 9.57 in/min and k

_{x}= k

_{y}= 10k

_{z}= 9.57 in/min, respectively. The comparisons of free surface variation with time from the isotropic and anisotropic I and II cases are shown in Figure 11. The corresponding 3D depression cones of the three cases after 5 min are presented in Figure 12.

_{y}can lead to the increase in flow resistance and the slight hydraulic gradient in the y direction. Conversely, the flow paths in the x direction are more unimpeded in terms of seepage flow, and the hydraulic gradients are more substantial. Therefore, the corresponding 3D depression cone of anisotropic I is elliptical with different curvature at the x and y directions.

_{z}leads to the increase in flow resistance in the z direction, while the seepage flow predominantly occurs in the horizontal plane, and the hydraulic gradient becomes mild. Similar to the isotropic case, the depression cone in the anisotropic II case is also circular because of the permeability symmetry k

_{x}= k

_{y}but with smaller curvature. Note that this special effect cannot be characterized by the two-dimensional seepage analysis [37].

## 5. Conclusions

**F**of the line element model, there is no complicated area integral and no confirmation of the surface shape; only the position of the intersection between the line element and free surface is required to calculate the value of ${\mathit{N}}^{\mathrm{T}}{u}_{f}\mathit{N}$. In addition, the element mesh around the free surface is fixed without updating the timestep and iteration number. The proposed method can be extended to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zhou, C.; Chen, Y.; Jiang, Q.; Lu, W. A generalized multi-field coupling approach and its application to stability and deformation control of a high slope. J. Rock Mech. Geotech. Eng.
**2011**, 3, 193–206. [Google Scholar] [CrossRef] - Jiang, Q.; Qi, Z.; Wei, W.; Zhou, C. Stability assessment of a high rock slope by strength reduction finite element method. Bull. Eng. Geol. Environ.
**2015**, 74, 1153–1162. [Google Scholar] [CrossRef] - Zhang, Y.; Zhang, Z.; Xue, S.; Wang, R.; Xiao, M. Stability analysis of a typical landslide mass in the Three Gorges Reservoir under varying reservoir water levels. Environ. Earth Sci.
**2020**, 79, 42. [Google Scholar] [CrossRef] - Ye, Z.; Fan, X.; Zhang, J.; Sheng, J.; Chen, Y.; Fan, Q.; Qin, H. Evaluation of connectivity characteristics on the permeability of two-dimensional fracture networks using geological entropy. Water Resour. Res.
**2021**, 57, e2020WR029289. [Google Scholar] [CrossRef] - Ye, Z.; Yang, J.; Xiong, F.; Huang, S.; Cheng, A. Analytical relationships between normal stress and fluid flow for single fractures based on the two-part Hooke’s model. J. Hydrol.
**2022**, 608, 127633. [Google Scholar] [CrossRef] - Asghar, Z.; Ali, N.; Waqas, M.; Nazeer, M.; Khan, W.A. Locomotion of an efficient biomechanical sperm through viscoelastic medium. Biomech. Model Mechanobiol.
**2020**, 19, 2271–2284. [Google Scholar] [CrossRef] - Asghar, Z.; Ali, N.; Javid, K.; Waqas, M.; Dogonchi, A.S.; Khan, W.A. Bio-inspired propulsion of micro-swimmers within a passive cervix filled with couple stress mucus. Comput. Methods Programs Biomed.
**2020**, 189, 105313. [Google Scholar] [CrossRef] - Javid, K.; Waqas, M.; Asghar, Z.; Ghaffari, A. A theoretical analysis of Biorheological fluid flowing through a complex wavy convergent channel under porosity and electro-magneto-hydrodynamics Effects. Comput. Methods Programs Biomed.
**2020**, 191, 105413. [Google Scholar] [CrossRef] - Neuman, S.; Witherspoon, P. Analysis of nonsteady flow with a free surface using the finite element method. Water Resour. Res.
**1971**, 7, 611–623. [Google Scholar] [CrossRef] - Desai, C.S.; Lightner, J.G.; Somasundaram, S. A numerical procedure for 3D transient free surface seepage. Adv. Water Resour.
**1983**, 6, 175–181. [Google Scholar] [CrossRef] - Chen, Y.; Hu, R.; Zhou, C.; Li, D.; Rong, G. A new parabolic variational inequality formulation of Signorini’s condition for non-steady seepage problems with complex seepage control systems. Int. J. Numer. Anal. Methods Geomech.
**2011**, 35, 1034–1058. [Google Scholar] [CrossRef] - Rafiezadeh, K.; Ataie-Ashtiani, B. Transient free-surface seepage in 3D general anisotropic media by BEM. Eng. Anal. Bound. Elem.
**2014**, 46, 51–66. [Google Scholar] [CrossRef] - Akhtar, S.; Hussain, Z.; Nadeem, S.; Najjar, I.R.; Sadoun, A.M. CFD analysis on blood flow inside a symmetric stenosed artery: Physiology of a coronary artery disease. Sci. Prog.
**2023**, 106, 00368504231180092. [Google Scholar] [CrossRef] [PubMed] - Akhtar, S.; Hussain, Z.; Khan, Z.A.; Nadeem, S.; Alzabut, J. Endoscopic balloon dilation of a stenosed artery stenting via CFD tool open-foam: Physiology of angioplasty and stent placement. Chin. J. Phys.
**2023**, 85, 143–167. [Google Scholar] [CrossRef] - Lin, S.; Cao, X.; Zheng, H.; Li, Y.; Li, W. An improved meshless numerical manifold method for simulating complex boundary seepage problems. Comput. Geotech.
**2023**, 155, 105211. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York, NY, USA, 1972. [Google Scholar]
- Xu, Z.H.; Ma, G.W.; Li, S.C. A graph-theoretic pipe network method for water flow simulation in a porous medium: GPNM. Int. J. Heat Fluid Flow
**2014**, 45, 81–97. [Google Scholar] [CrossRef] - Afzali, S.H.; Monadjemi, P. Simulation of flow in porous media: An experimental and modeling study. J. Porous Media
**2014**, 17, 469–481. [Google Scholar] [CrossRef] - Abareshi, M.; Hosseini, S.M.; Sani, A.A. Equivalent pipe network model for a coarse porous media and its application to two-dimensional analysis of flow through rockfill structures. J. Porous Media
**2017**, 20, 303–324. [Google Scholar] [CrossRef] - Moosavian, N. Pipe network modelling for analysis of flow in porous media. Can. J. Civ. Eng.
**2019**, 46, 1151–1159. [Google Scholar] [CrossRef] - Ye, Z.; Qin, H.; Chen, Y.; Fan, Q. An equivalent pipe network model for free surface flow in porous media. Appl. Math. Model.
**2020**, 87, 389–403. [Google Scholar] [CrossRef] - Ye, Z.; Fan, Q.; Huang, S.; Cheng, A. A 1D line element model for transient free surface flow in porous media. Appl. Math. Comput.
**2021**, 392, 125747. [Google Scholar] - Ren, F.; Ma, G.; Wang, Y.; Fan, L. Pipe network model for unconfined seepage analysis in fractured rock masses. Int. J. Rock Mech. Min. Sci.
**2016**, 88, 183–196. [Google Scholar] [CrossRef] - Wei, W.; Jiang, Q.; Ye, Z.; Xiong, F.; Qin, H. Equivalent fracture network model for steady seepage problems with free surfaces. J. Hydrol.
**2021**, 603, 127156. [Google Scholar] [CrossRef] - Chen, Y.; Zhou, C.; Zheng, H. A numerical solution to seepage problems with complex drainage systems. Comput. Geotech.
**2008**, 35, 383–393. [Google Scholar] [CrossRef] - Carman, P.C. Fluid flow through granular beds. Chem. Eng. Res. Des.
**1937**, 15, S32–S48. [Google Scholar] [CrossRef] - Liu, R.; Jiang, Y.; Li, B.; Yu, L. Estimating permeability of porous media based on modified Hagen-Poiseuille flow in tortuous capillaries with variable lengths. Microfluid. Nanofluid.
**2016**, 20, 120. [Google Scholar] [CrossRef] - Sheng, J.; Huang, T.; Ye, Z.; Hu, B.; Liu, Y.; Fan, Q. Evaluation of van Genuchten-Mualem model on the relative permeability for unsaturated flow in aperture-based fractures. J. Hydrol.
**2019**, 9, 315–324. [Google Scholar] [CrossRef] - Oden, J.T.; Kikuchi, N. Theory of variational inequalities with applications to problems of flow through porous media. Int. J. Eng. Sci.
**1980**, 18, 1173–1284. [Google Scholar] [CrossRef] - Zhang, W. Calculation of Unsteady Flow of Groundwater and Evaluation of Groundwater Resources; Science Press: Beijing, China, 1983. [Google Scholar]
- Hu, R.; Chen, Y.F.; Zhou, C.B.; Liu, H.H. A numerical formulation with unified unilateral boundary condition for unsaturated flow problems in porous media. Acta Geotech.
**2017**, 12, 277–291. [Google Scholar] [CrossRef] - Hall, H. An investigation of steady flow toward a gravity well. Houille Blanche
**1955**, 10, 8–35. [Google Scholar] [CrossRef] - Cooley, R. Some new procedures for numerical solution of variably saturated flow problems. Water Resour. Res.
**1983**, 19, 1271–1285. [Google Scholar] [CrossRef] - Clement, T.; Wise, W.; Molz, F. A physically based, two-dimensional, finitedifference algorithm for modeling variably saturated flow. J. Hydrol.
**1994**, 161, 71–90. [Google Scholar] [CrossRef] - Ward, A.L.; Zhang, Z.F. Effective Hydraulic Properties Determined from Transient Unsaturated Flow in Anisotropic Soils. Vadose Zone J.
**2007**, 6, 913–924. [Google Scholar] [CrossRef] - Zhang, Z.F. Relationship between Anisotropy in Soil Hydraulic Conductivity and Saturation. Vadose Zone J.
**2014**, 13, 1–8. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Transient free surface flow in a 3D soil dam. (

**b**) The homogeneous media with arrayed spherical grains. (

**c**) The 3D orthogonal network of tubes.

**Figure 3.**The orthogonal line element mesh. The dimensions of this aquifer are 500 m in length, 12 m in height and 8 m in width. Different from the solid element (tetrahedron and hexahedron), for which that the flow domain is mapped using the nodes and lines.

**Figure 4.**(

**a**) Locations of transient free surface versus time. (

**b**) Iteration steps versus time for different mesh sizes (B

_{x}= B

_{y}= B

_{z}= 4 m, 2 m, 1 m).

**Figure 5.**(

**a**) Locations of transient free surface versus time. (

**b**) Iteration steps versus time for various time steps ($\Delta t$ = 5 d, 1 d, 0.1 d). The mesh size is fixed as B

_{x}= B

_{y}= B

_{z}= 1 m.

**Figure 6.**The orthogonal line element mesh. The dimensions of the trapezoidal dam are 34.0 cm in length, 26.2 cm in height and 2.4 cm in width.

**Figure 7.**(

**a**) Locations of transient free surface versus time. (

**b**) Iteration steps versus time for various mesh sizes (B

_{x}= B

_{y}= B

_{z}= 1.2 cm, 0.6 cm, 0.3 cm). The time step is fixed as 1 min.

**Figure 8.**(

**a**) Locations of transient free surface versus time. (

**b**) Iteration steps versus time for various time steps ($\Delta t$ = 1 min, 0.5 min, 0.25 min). The mesh size is set as 0.6 cm.

**Figure 9.**(

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

**b**) Locations of transient free surface versus time. The dimensions of this flume are 1 m in length, 1.2 m in height and 0.1 m in width. The mesh size of 0.02 m and the time step of 0.5 min are used in this sand flume model.

**Figure 10.**(

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

**b**) Locations of steady free surface. The well is 48 inch in height, and the radii of the well and total sand tank are 4.8 inch and 76.8 inch. The well model is divided into three domains meshed with non-uniformed sizes (coarse mesh 4 m, medium mesh 2 m, fine mesh 1 m).

**Figure 11.**Free surface locations versus time: (

**a**) isotropic case, (

**b**) anisotropic I in x direction, (

**c**) anisotropic I in y direction, (

**d**) anisotropic II, (

**e**) comparison at 5 min.

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**MDPI and ACS Style**

Chen, Y.; Yuan, Q.; Ye, Z.; Peng, Z.
A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media. *Water* **2023**, *15*, 3072.
https://doi.org/10.3390/w15173072

**AMA Style**

Chen Y, Yuan Q, Ye Z, Peng Z.
A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media. *Water*. 2023; 15(17):3072.
https://doi.org/10.3390/w15173072

**Chicago/Turabian Style**

Chen, Yuting, Qianfeng Yuan, Zuyang Ye, and Zonghuan Peng.
2023. "A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media" *Water* 15, no. 17: 3072.
https://doi.org/10.3390/w15173072