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In this paper, the numerical solutions for groundwater flow in unsaturated layered soil using the Richards equation are presented. A linearisation process for the nonlinear Richards equation to deal with groundwater flow in unsaturated layered soil is derived. To solve one-dimensional flow in the unsaturated zone of layered soil profiles, flux cons...
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The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the...
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... where λ is the key parameter for solving the linearized Richards' equation in the conversion method, which can be defined as a constant λ = e αh d (Tracy 2006;Liu et al. 2015); h d is the pressure head value when the soil is dry. The relative 6 1 Background permeability coefficient is expressed as: ...
Heavy rainfall in extreme climates often causes natural disasters such as floods, landslides, and debris flows. Rainfall-induced slope instabilities are major geological natural disasters (Glade in Environ Geol 35:160–174, 1998; Dai et al. in Eng Geol 51:279–290, 1999; Iverson in Water Resour Res 36:1897–1910, 2000; Lee and Pradhan in Landslides 4:33–41, 2007; Li et al. in Landslides 13:1109–1123, 2016a; Li et al. in Ecol Eng 91:477–486, 2016b; Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020) that can result in considerable loss of life and damage to infrastructure. Extreme events such as storms, which are becoming more severe because of climate change, can trigger fatal landslides.
... Briggs et al. (2000) proposed a multi-grid correction method to quickly eliminate the iterative error in the iterative process, thereby obtaining a faster convergence rate. Recently, preconditioning methods can effectively reduce the condition number of the iterative matrix of linear equations, thereby improving the computational convergence rate (Benzi 2002;Zhu et al. 2022a, b, c), including the left preconditioning, right preconditioning, and two-side preconditioning (Liu et al. 2015). The preconditioning technique converts the original system of linear equations into a system that is easier to solve, thereby improving the convergence rate of the iterative process (Benzi 2002). ...
... The soil was assumed to be silt. The parameters of soil were described by Liu et al. (2015) and included θ s = 0.35, θ r = 0.14, α = 8 × 10 −3 , and k s = 9 × 10 −4 m/h. The boundary conditions are as follows: ...
The linear infiltration equations obtained by discretizing Richards’ equation need to be solved iteratively, including two approaches of linear and nonlinear iterations. The first method is to use numerical methods to directly numerically discretize Richards’ equations to obtain nonlinear ordinary differential equations and then use nonlinear iterative methods to iteratively solve, such as Newton’s method (Radu et al. in On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation. Numerical mathematics and advanced applications. Springer, pp 1194–1200, 2006), Picard method (Lehmann and Ackerer 1998), and the L -method (List and Radu 2016). The Picard method can be considered as a simplified Newton method, which linearly converges.
... Generally, RE first requires spatial discretization using numerical methods, such as finite difference method (FDM) (Liu et al., 2015;Zhu et al., 2019), finite element method (FEM) (Coombs and Augarde, 2020), mixed finite element method and finite volume method (FVM) (Eymard et al., 2006;Su et al., 2022). Additionally, the backward-Euler method is often used for time discretization (Wang and Schrefler, 1998;Pop and Schweizer, 2011). ...
... RE can be used to describe 1D rainfall infiltration in unsaturated soils, which is expressed as (Liu et al., 2015;Zha et al., 2017): ...
... In Fig. 1, the thickness of the soil is assumed to be 10 m. The model parameters includeα = 8 × 10 − 3 ,θ s = 0.35,θ r = 0.14, andK s = 1 × 10 − 4 m/h (Liu et al., 2015). The governing equation is shown in Eq. (1), the right side of which is equal to zero. ...
A modified Gauss-Seidel iterative method (MP(m)-GS) with multistep preconditioner is developed to solve partial differential equations of rainfall infiltration. The finite difference method (FDM) is applied to numerical dis-cretization. Furthermore, a system of linear equations is solved using an iterative scheme. For some unfavorable numerical conditions, such as large initial conditions, classical Picard iterative method usually has low accuracy and computational efficiency. Conventional linear iterative methods have a slower convergence rate, such as Gauss-Seidel (GS) and Jacobi iterative methods, particularly for small discrete space step size. Thus, MP(m)-GS is developed to simulate rainfall infiltration into unsaturated soils. The analytical solutions are employed to verify the proposed methods. The results indicate that the MP(m)-GS has higher computational efficiency and accuracy than the improved Picard methods. Compared with the conventional linear iterative methods GS and SOR, MP (m)-GS also demonstrates faster convergence rate and higher computation efficiency. The results show nice applications in modeling rainfall infiltration in unsaturated soils.
... Hayek (2016) proposed a general analysis model of one-dimensional transient vertical infiltration. Although their actual applicability is limited, these are good references for theoretical analysis (Zambra et al. 2012;Liu et al. 2015;Zeng et al. 2018;Zhu et al. 2021). ...
... (7)-(8) may be simpler than that of Eq. (1). To acquire the approximate solution of Eq. (7), FDM is used for the numerical discretization (Liu et al. 2015). For a 1D unsaturated flow problem, the second-order partial derivatives in Eq. (7) only consider the z-direction. ...
... To verify the effectiveness of the proposed method, we present an example in which a 1D steady-state unsaturated flow is simulated in homogeneous porous media (Liu et al. 2015). The mathematical model is shown in Fig. 1, where the soil thickness (L) is 10 m; the other parameters are = 8 × 10 −3 , s = 0.35 , r = 0.14 , and K s = 1 × 10 −4 m/h. ...
Richards’ equation is often used in unsaturated flow problems and has a wide range of applications. In the numerical solution, the Richards’ equation is linearized first, and then the finite difference method is used for numerical discretization and iterative calculation. The traditional iterative methods such as Jacobi, Gauss–Seidel (GS) and SOR iterative methods have a slower convergence rate, especially when the discrete space step is small and the time step is large. Therefore, we adopt the integral correction method and the multistep preconditioner to improve the traditional iterative methods and propose an improved Gauss–Seidel iterative method (ICMP(m)-GS) with multistep preconditioner based on the integral correction method to solve the linear equations derived from linearized Richards’ equation. Through examples of unsaturated flow, convergence rate and computational accuracy of the proposed algorithm are validated by comparing the traditional methods and analytical solutions. The results show that the proposed ICMP(m)-GS can greatly improve the ill-condition of linear equations. Compared with the conventional methods GS, SOR and a single improvement method, ICMP(m)-GS has a faster convergence rate, higher calculation efficiency and calculation accuracy. The application example shows that the proposed method can also well simulate the time-varying law of pressure head in the rainfall infiltration of unsaturated soil slopes, and has a nice application effect.
... Due to the complexity, the Richards equation considering the SWCC in unsaturated soils is usually solved using the numerical methods. Mesh-based numerical techniques such as the finite difference method and the finite element method are well documented and typically used to solve the unsaturated flow equation in the past [20][21][22]. ...
... The unsaturated flow in soils can be described by the following variably saturated flow equation [21] ...
In this article, the modeling of infiltration--induced landslides, in unsaturated soils using the ra-dial basis function (RBF) method, is presented. A novel approach based on the RBF method is proposed to deal with the nonlinear hydrological process in the unsaturated zone. The RBF is first adopted for curve fitting to build the representation of the soil water characteristic curve (SWCC) that corresponds to the best estimate of the relationship between volumetric water con-tent and matric suction. The meshless method with the RBF is then applied to solve the nonline-ar Richards equation with the infiltration boundary conditions. Additionally, the fictitious time integration method is adopted in the meshless method with the RBF for tackling the nonlineari-ty. To model the stability of the landslide, the stability analysis of infinite slope coupled with the nonlinear Richards equation considering the fluctuation of transient pore water pressure is developed. The validation of the proposed approach is accomplished by comparing with exact solutions. The comparative analysis of the factor of safety using the Gardner model, the van Genuchten model and the proposed RBF model is provided. Results illustrate that the RBF is advantageous for reconstructing the SWCC with better estimation of the relationship than con-ventional parametric Gardner and van Genuchten models. We also found that the computed safety factors significantly depend on the representation of the SWCC. Finally, the stability of landslides is highly affected by matric potential in unsaturated soils during the infiltration pro-cess.
... Generally, RE first needs to use numerical methods for spatial discretization, including finite difference method (FDM) [16,17], finite element method [18,19], and finite volume method [20,21]. For time discretization, a backward-Euler finite-difference stepping scheme is usually adopted [22,23]. ...
... In addition, the relaxation factor w in the SOR method directly affects the convergence rate, but determining the optimum value of the relaxation factor is difficult. Due to some shortcomings of conventional iteration methods, increasing attention has focused on improving their convergence rate [16,31,[39][40][41][42][43]. ...
... Liu [46] used a two-sided preconditioning conjugate gradient method to solve an ill-posed linear system. Additionally, the newly developed dynamical Jacobian-inverse free method was developed to overcome the non-convergence for solving the infiltration problem in layered soils [16]. An improved Jacobi iterative method with two-side equilibration was presented to obtain the numerical solution of groundwater flow in unsaturated soils [17]. ...
This paper studies the potential of using the successive over-relaxation iteration method with polynomial preconditioner (P(m)-SOR) to solve variably saturated flow problems described by the linearized Richards' equation. The finite difference method is employed to numerically discretize and produce a system of linear equations. Generally, the traditional Picard method needs to re-evaluate the iterative matrix in each iteration, so it is time-consuming. And under unfavorable conditions such as infiltration into extremely dry soil, the Picard method suffers from numerical non-convergence. For linear iterative methods, the traditional Gauss-Seidel iteration method (GS) has a slow convergence rate, and it is difficult to determine the optimum value of the relaxation factor w in the successive over-relaxation iteration method (SOR). Thus, the approximate optimum value of w is obtained based on the minimum spectral radius of the iterative matrix, and the P(m)-SOR method is extended to model underground water flow in unsaturated soils. The improved method is verified using three test examples. Compared with conventional Picard iteration, GS and SOR methods, numerical results demonstrate that the P(m)-SOR has faster convergence rate, less computation cost, and good error stability. Besides, the results reveal that the convergence rate of the P(m)-SOR method is positively correlated with the parameter m. This method can serve as a reference for numerical simulation of unsaturated flow.
... The RE generally requires spatial discretization using numerical methods, such as finite difference method (FDM) [16,18], finite element method (FEM) [19] and finite volume method (FVM) [20]. A backward-Euler finite-difference stepping scheme is usually adopted for time discretization [21]. ...
... Furthermore, the vertex-centered finite-difference grid and the backward Euler finite-difference stepping scheme are used for the numerical discretization of Eq. (18). The matrix format of the standard Picard iteration (PI) method can be written as follows [26]: ...
... unsaturated soil parameters are g = 8 × 10 −3 , s = 0.35 , r = 0.14 , and K s = 1 × 10 −4 m/h [18]. ...
The Hermitian and skew-Hermitian splitting iteration method (HSS) is commonly an effective linear iterative method for solving sparse non-Hermite positive definite equations. However, it is time-consuming to solve linear equations. Hence, inexact Hermitian and skew-Hermitian splitting iteration approaches with multistep preconditioner (PIHSS(m)) are proposed for analyzing underground water flow. For unsaturated porous media, an exponential model is adopted to linearize the Richards equation. The governing equations are discretized using the finite element method to produce a system of linear equations. Furthermore, the inexact Hermitian and skew-Hermitian splitting iteration methods (IHSS) and PIHSS(m) are used to solve the linear equations. The results show that PIHSS(m) can effectively solve the 1D unsaturated flow problem and 2D transient drainage problem in partially and completely saturated soils. The IHSS has higher numerical accuracy than the classical methods such as Picard method and Gauss-Seidel iterative method. Compared with IHSS, PIHSS(m) achieves faster convergence rate and higher computational efficiency, particularly for solving groundwater flow problems with high grid density. Additionally, the numerical results reveal that PIHSS(m) has excellent acceleration, that is, at least 50% acceleration compared with the IHSS.
... Due to the nonlinear nature of RE, the analytical solution is difficult to obtain, and it is often obtained based on an exponential form (Srivastava and Yeh 1991;Tracy 2006). Thus, different numerical methods were employed to solve RE (Zha et al. 2013;Zeng et al. 2018), including finite volume method (FVM), finite difference method (Liu et al. 2015), and finite element method (Pop et al. 2004). Wang and Anderson (1982) introduced the practical application of finite difference and finite element to the numerical simulation of groundwater seepage and pollution propagation. ...
... The thickness (L) of the soil layer is 10 m. Gardner model is used to represent the soil-water characteristic curve of unsaturated soils, and the parameters are a = 1 × 10 −4 /m, θ s = 0.50, θ r = 0.11, and K s = 9 × 10 −5 m/s (Liu et al. 2015). The initial condition is assumed to be h(z,t=0)=h d . ...
The Richards equation is widely used in the simulation of unsaturated flow and related fields. In the numerical solution process, the finite volume method can be used to carry out numerical discretization, and the Picard method is used for iterative solution. However, in order to obtain a more reliable numerical solution, the space step size of the conventional uniform grid is often small. Especially under some unfavorable numerical conditions, such as infiltration into dry soil and layered soil with very different hydraulic conductivity, this will be very time-consuming and even slow to converge. Thus, combined with the non-uniform grid in the form of Chebyshev, an improved Picard method based on the non-uniform two-grid correction scheme (NTG-PI) is proposed to model 1D unsaturated flow in porous media. Through three examples of unsaturated flow under unfavorable conditions, the proposed scheme was verified. The results show that, compared with the conventional Picard method, NTG-PI can obtain higher numerical accuracy with a smaller number of nodes, and also has a faster convergence rate. This method can provide a certain reference for the numerical simulation of unsaturated flow.
... Several numerical approaches based on the meshebased methods to the modeling of flow movement in unsaturated porous media have been proposed, such as the boundary element method [26], the finite difference method (FDM) [20,21], and the finite element method [31,33]. Even though the success of meshebased methods is effective and easily implemented for dealing with unsaturated flow problems, limitations still remain while utilizing the meshebased methods including meshegeneration for complex geometries, or short time interval for the numerical convergence. ...
... We therefore compare the computed results with that of the FDM, as depicted in Fig. 9. The results agree very well with those of the FDM [20]. ...
This paper presents a study for solving unsaturated flow in heterogeneous porous media using the meshless method with the radial basis function (RBF). For modeling the nonlinear hydrological process in unsaturated zone, an exponential model is introduced in the Richards equation such that we may obtain the linearized Richards equation. We adopt the multiquadric function as the RBF in the meshless method for solving the linearized Richards equation. For simulating the unsaturated flow problems in layered heterogeneous soils, the flux and the head must satisfy the continuity condition at the interface. Several examples are carried out for modeling the hydrological process in multi–layered unsaturated soils. The results demonstrate that we only discretize by a set of points without tedious mesh generation and significantly enhance the applicability for solving unsaturated flow problems, especially in heterogeneous multi–layered soils with extreme physical property contrasts.
... In recent years, the modeling of flows in saturated and unsaturated porous media has attracted considerable interest [1][2][3]. The analysis of saturated and unsaturated flows is crucial in practical engineering applications, especially in problems pertaining to soil science, hydrogeology, and geotechnical engineering [4][5][6]. Therefore, the modeling of saturated and unsaturated flows has emerged as a crucial research topic. ...
In this paper, we propose a novel meshless approach that involves using space–time polyharmonic radial polynomial basis functions for modeling saturated and unsaturated flows in porous media. In this study, space–time polyharmonic radial polynomial basis functions were developed in the space–time domain using a meshless collocation method. This domain contains three sets of collocation points, namely the inner, source, and boundary points, for the spatial and temporal discretization of the governing equation. Because the initial and boundary data are accessible space–time boundaries, the solutions of groundwater flows problems are approximated by solving the inverse boundary value problem in the space–time domain without using the conventional time-marching scheme. Saturated and unsaturated flow problems were investigated to demonstrate the robustness of the proposed method. The results obtained using the proposed approach were compared with those obtained using the conventional polyharmonic spline radial basis function. The proposed space–time polyharmonic radial polynomial basis functions obtained highly accurate solutions. Moreover, in solving saturated and unsaturated flow problems, the accuracy and stability of the proposed functions were higher than those of the conventional time-marching scheme.
