The modeling of transient flow in unsaturated soils for rainfall-induced landslides using a novel spacetime collocation method is presented. A numerical solution obtained in the spacetime coordinate system is approximated by superpositioning Trefftz basis functions satisfying the linearized Richards equation for collocation points on the spacetime domain boundary. The Gardner exponential model is adopted to derive the linearized Richards equation to describe the soil-water characteristic curve in unsaturated soils. To deal with the rainfall-induced landslides, the infinite slope stability analysis coupled with the proposed meshless method with the consideration of the fluctuation of time-dependent matric suction is developed. The proposed method is validated for several test problems. Application examples of transient modeling of flow for rainfall-induced landslides in homogenous unsaturated soils are also conducted. Numerical results demonstrate that the proposed method is highly accurate to deal with transient flow in unsaturated soils for rainfall-induced landslides. In addition, it is found that the numerical method using the Richards equation with the Gardner model may provide a promising solution for different soil textures.
In recent decades, global climate change impacts have increased the frequency of severe meteorological phenomena such as heavy rainfall events [
In the past, theoretical models have been developed to deal with slope stability assuming a steady or quasi-steady water table associated with an infinite-slope stability analysis [
Numerical approaches to the modeling of various unsaturated geofluid phenomena using the mesh-based methods, such as the finite difference method (FDM) or the finite element method (FEM), have been studied extensively [
In this paper, a novel spacetime collocation method for modeling of transient flow in unsaturated soils for rainfall-induced landslides is presented. For the modeling of transient flow in unsaturated soils, we proposed a pioneering work based on the proposed boundary-type meshless method and utilized the spacetime coordinate system instead of that in the original Euclidean space [
The generalized Richards equation represents the movement of water in saturated and unsaturated soils, which can be expressed as
This is also known as the variably saturated flow equation to describe the groundwater flow in saturated and unsaturated zones simultaneously [
The definition of the elevation head in an unsaturated inclined slope.
Considering only the unsaturated flow, (
The specific moisture capacity function can be defined by
Considering an one-dimensional transient flow problem, the duration of the rainfall events is far shorter than the transmission time of pore water pressure in the
To normalize the hydraulic conductivity of unsaturated soils with respect to its saturated hydraulic conductivity, we obtained
The high nonlinearity of physical behavior of unsaturated soils is rooted in the SWCC. The SWCC represents the relationship between water content and matric suction in unsaturated soils, which is important for the seepage analysis in the vadose zone. Many mathematical equations have been proposed to describe the SWCC. The Gardner and van Genuchten models of SWCC are commonly found in analytical and numerical solutions for flow problems, respectively [
For the porous media including gilat loam and sandy loam, the Gardner model exhibits good match with the van Genuchten model [
Adopting the Gardner exponential model, the linearized Richards equation for the one-dimensional transient Richards equation can be derived as follows [
In this study, the proposed spacetime collocation method rooted from the CTM. The Trefftz method is first proposed by the German mathematician Erich Trefftz [
In this paper, we proposed a novel spacetime collocation method, named the spacetime collocation method (SCM) [
We assume that time and the speed of light are absolute physical quantities that play the role of the independent variable such that the spacetime coordinate system is an
Illustration of the proposed collocation scheme.
Conventional collocation scheme for one-dimensional transient flow problem
Collocation points of one-dimensional transient flow problem in the spacetime domain
Considering the one-dimensional spacetime domain,
The initial condition can be described as
The transient pressure head of the linearized Richards equation [
The process of the separation of variables may be used by having the transient solution as
Substituting the above equation into (
Each term in the above equation must be a constant for a nonzero solution. Accordingly, we obtained the following equations.
The general solution for
A system of simultaneous linear equations can then be obtained as
The first verification example is an one-dimensional transient subsurface flow problem in homogenous unsaturated soil. The unsaturated soil is initially dry until water begins to infiltrate the soil. The ponding on the ground surface is then maintained holding the pressure head to zero. This is known as the one-dimensional Green-Ampt problem [
The initial condition was the soil in dry condition where
The boundary conditions of the top and bottom boundaries are the pressure head-type boundary condition, which can be expressed as follows.
The analytical transient solution of this example can be found as follows [
The
Finally, the transient solution
In this example, there is one-dimensional in space and one-dimensional in time. It is clear that the spacetime domain is a rectangular shape. We adopted 600 boundary collocation points and a source point. The number of boundary points in space domain and in time domain was considered to be 200 and 400, respectively. The Dirichlet boundary values were given on boundary collocation points which collocated on three sides of the domain except the right side which is the final time solution to be found. The order of the basis function depicted in (
The profiles of the computed results from the proposed method and the FDM on different times were then compared with the exact solution, as shown in Figure
Comparison of results for solving one-dimensional Green-Ampt problem.
Comparison of the absolute error of the computed results and the FDM.
The second example under investigation is the transient modeling of one-dimensional flow in a homogenous unsaturated inclined slope. A column of soil is initially dry until water begins to infiltrate the soil. A pool of water at ground surface is then maintained holding the pressure head to zero. The thickness of the soil is 10 (m) and the slope angle is 20 degrees, as shown in Figure
Schematic illustration of homogenous unsaturated inclined slope.
The initial condition was the soil in dry condition maintained as
The transient analytical solution can be expressed as follows [
In this study, the modeling of one-dimensional transient flow in the homogenous unsaturated inclined slope is conducted. To validate the applicability of the proposed Gardner model, we selected two different soils which are sandy soil and silty loam for this example. The data sets from Brooks and Corey [
Unsaturated soil parameters used in the verification example 2.
Parameters for SWCC | Soil type | |
---|---|---|
Sandy soil | Silty loam | |
Saturated hydraulic conductivity (m/h) | ||
Pore size distribution parameter (1/m) | ||
Saturated water content | 0.50 | 0.46 |
Residual water content | 0.11 | 0.14 |
We adopted 375 boundary collocation points and a source point. The number of boundary points in space domain and in time domain was considered to be 125 and 250, respectively. The Dirichlet boundary values were given on boundary collocation points which collocated on three sides of the domain using the analytical solution for the problem. The order of the basis function depicted in (
The fitting curves of the SWCC for two different types of soil are illustrated in Figure
Fitting curves of the SWCC.
Sandy soil
Silty loam
Result comparison with the exact solution for sandy soil and silty loam.
Sandy soil
Silty loam
The absolute error of the computed results for sandy soil and silty loam.
Sandy soil
Silty loam
Since the proposed spacetime collocation method is advantageous in solving problems of irregular geometry, the third verification example is the two-dimensional transient subsurface flow problem in homogenous unsaturated soil enclosed by an irregular boundary. With a two-dimensional simply connected domain, the governing equation can be expressed as follows.
The unsaturated soil parameters are the same as in verification example 1. The total simulation time is two hours. The initial condition is the unsaturated soil in dry condition maintained as pressure head to be –100 (m), which can be described as follows.
The two-dimensional irregular boundary under consideration can be defined as the following parametric equation.
The boundary conditions are assumed to be the Dirichlet boundary condition. The Dirichlet boundary data is applied using the following exact solution [
The transient numerical solution can be obtained using the following equations [
We utilized the spacetime coordinate system to model the transient two-dimensional flow in unsaturated soil. There is two-dimensional in space and one-dimensional in time in this two-dimensional transient analysis. Consequently, the spacetime domain is transformed into a three-dimensional irregular object domain, as shown in Figure
Spacetime boundary collocation points of the two-dimensional transient flow in unsaturated soils.
The order of the basis function for this example was 10. We collocated 2162 boundary collocation points and a source point. The number of boundary points in the space domain and in the time domain was considered to be 162 and 2000, respectively. We adopted 1325 inner points uniformly placed inside the three-dimensional spacetime domain to obtain the computed results of the pressure head at different times. Figure
Comparison of computed results with the exact solution for the modeling of two-dimensional transient flow in unsaturated soils.
The absolute error of the computed results for the modeling of two-dimensional transient flow in unsaturated soils.
The application example under investigation is a one-dimensional, transient, unsaturated flow problem in a homogeneous inclined slope. In this example, the thickness of the soil is assumed to be 10 (m) and the slope angle is 33 degrees. The governing equation is depicted as (
Unsaturated soil parameters used in the comparison example.
This study | van Genuchten model | Iverson’s model | |
---|---|---|---|
Slope angle (degree) | 33 | ||
Soli thickness (m) | 10 | ||
Simulation time (h) | 2 | ||
Saturated hydraulic conductivity (m/h) | |||
Other parameters | |||
Notation:
To examine the applicability of the Gardner model, we compare the Gardner model with the van Genuchten model and measured data [
From Figure
Fitting results of the SWCC using van Genuchten and Gardner models.
Comparison of computed pressure heads.
This study
van Genuchten model
Iverson’s model for
Iverson’s model for
To deal with the rainfall-induced landslides, the infinite slope stability analysis coupled with the proposed meshless method with the consideration of the fluctuation of time-dependent matric suction is developed. The failure mechanism of infinite slopes mainly involves colluvium, weathered rock formations, or shallow failure and planar slides in bedrock alternations located at thin depth below the ground level. The infinite slope theory is commonly used to evaluate the slope stability while the sliding soil thickness is far less than the slope height. The factor of safety (
Figure
Comparison of the computed
This study
van Genuchten model
Iverson’s model for
Iverson model’s for
A novel spacetime collocation method for modeling of transient flow in unsaturated soils for rainfall-triggered landslides is developed. This pioneering work is based on the boundary-type meshless method and provides a promising solution for modeling the transient flow in unsaturated soils. The validity of the proposed method is established for a number of unsaturated flow problems. The fundamental concepts and the construct of the proposed method are clearly addressed in detail. Findings from this study are drawn as follows.
The pioneering work in this study is the first successful attempt to solve the transient flow in unsaturated soils for an inclined slope using the novel spacetime collocation method. For the modeling of the transient flow in unsaturated soils, we proposed an innovative concept that one may utilize the spacetime coordinate system instead of that in the original Euclidean space. Consequently, both the initial and boundary conditions can be treated as boundary conditions on the spacetime domain boundary. The unsaturated flow problems can then be transformed into the inverse boundary value problem. Therefore, the one-dimensional problems can be solved without using the traditional time-marching scheme. Since the SWCC is a major factor that influences the nonlinear physical relationship of unsaturated soils, we adopted the Gardner model to formulate the linearized Richards equation. For the numerical modeling of transient flow, we proposed a pioneering work using the proposed boundary-type meshless method in which the hydrological model coupled with the infinite slope stability analysis with the consideration of the SWCC can be used to model rainfall-induced landslides. From the results of verification examples, it is found that the computed numerical results agree well with the analytical transient solution. Results from the numerical examples demonstrate that highly accurate numerical solutions with the error in the order of
The data used to support the findings of this study are included within the article. The data supporting Figures
The authors declare that they have no conflicts of interest.
This study was partially supported by the National Science Council under project Grant MOST 106- 621-M-019-004-MY2. The authors thank the National Science Council for the generous financial support.