Numerical Solutions to Infiltration Equation

Abstract

Unsaturated infiltration issues occur in many fields, such as rainfall-induced soil slope failures (Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020a; Wu et al. in Appl Math Model 80:408–425, 2020b; Jiang et al. in Eng Comput 38:1–14, 2022), solute migration simulation (Cross et al. in Adv Water Resour 136, 2020), and coal seam water injection and coalbed methane extraction (Liu et al. 2018; Wang et al. in J Comput Appl Math 367, 2020).

3.1 Introduction

Unsaturated infiltration issues occur in many fields, such as rainfall-induced soil slope failures (Wu et al. 2020a, b; Jiang et al. 2022), solute migration simulation (Cross et al. 2020), and coal seam water injection and coalbed methane extraction (Wang et al. 2020). Among them, the Richards’ equation (Richards 1931) is the basic governing equation for the numerical simulation of unsaturated infiltration. The effective and reliable numerical solution of the Richards’ equation is of great significance to scientific research and production in related fields. Before the advent of computers, the investigations on unsaturated infiltration mainly focused on the analytical solution of the infiltration equation, that is, the method of directly solving differential equations using relevant mathematical means. The studies on analytical solutions under certain conditions still have very important theoretical and practical significance. Analytical solutions to the Richards’ equation can be obtained under some simplified conditions (Broadbridge et al. 2017). For example, Srivastava and Yeh (1991) employed an exponential model describing the soil-water character curve (SWCC) to derive transient analytical solutions for one-dimensional infiltration in homogeneous and layered soils. Parlange et al. (1999) solved a complex series solution of the one-dimensional Richards equation in order to obtain the infiltration flux. Tracy (2006) proposed two-dimensional and three-dimensional analytical solutions to rainfall infiltration into homogeneous soils based on exponential function. Wu et al. (2016, 2020a) used the Laplace transform method to obtain an analytical solution considering the coupling of infiltration and deformation in unsaturated soils during rainfall, and used it to analyze the stability of infinite unsaturated slopes due to rainfall infiltration. However, the permeability coefficients and soil–water characteristic curves are very complex in reality, which leads to highly nonlinear partial differential equations analytical solutions of which are very difficult to obtain. Therefore, numerical methods are often developed to solve the Richards’ equation under common conditions (Zha et al. 2017; Ku et al. 2018; Zeng et al. 2018; Zhu et al. 2019; Eini et al. 2020).

With the development of computer techniques, numerical methods, namely finite difference method (FDM) (Liu et al. 2017), finite element method (FEM) (Crevoisier et al. 2009), and the finite volume method (FVM) (Liu 2017; Eymard et al. 2006), are increasingly used in infiltration analysis. The discretization of the time derivative is usually performed using the backward difference method (Pop and Schweizer 2011). For example, Patankar (1980) summarized FVM for numerical dis-cretization of heat transfer and fluid flow. Wang and Anderson (1982) introduced the application of finite difference and finite element methods for numerical simulation of groundwater infiltration and pollution propagation. Šimůnek et al. (2009) applied the finite element method to numerically solve the Richards equation and developed the commercial software Hydrus-1D. Zambra et al. (2011) constructed a finite volume method with high accuracy in space and time for solving nonlinear Richards equations. Ku et al. (2018) linearized the Richards’ equation and developed a numerical solution to the unsteady groundwater infiltration issue by the collocation Trefftz method. Zeng proposed a modified Richards’ equation and used FEM to more efficiently solve the variable saturated infiltration problem in heterogeneous soils. Chávez-Negrete et al. (2018) proposed an improved FDM combined with an adaptive step-size Crank–Nicolson method for solving the Richards’ equation. Li et al. (2022) established the convergence selection criteria of grid size and time step in finite element simulation of unsaturated infiltration, which can better enhance the numerical accuracy. Of course, other advanced numerical methods have certain advantages in terms of computational accuracy, computational efficiency or ease of implementation under certain conditions when solving variable saturated infiltration issues. These methods include hybrid finite element methods (Bergamaschi and Putti 1999), cellular automata methods (Mendicino et al. 2006), FDM with non-orthogonal grids (An et al. 2010), finite analysis methods (Zhang et al. 2016), meshless methods (Herrera et al. 2009; Ku et al. 2021), and Chebyshev spectral methods (Wu et al. 2020b). Table 3.1 lists the development of some classical numerical methods for solving Richards’ equation.

Table 3.1 Some studies on the numerical solution of Richards’ equation

In this chapter, numerical solutions of water infiltration equation in unsaturated soils are examined. Some improved methods such as fractional-order and Chebyshev spectral methods will be applied to investigate nonlinear infiltration in homogeneous and layered soils. The results will be compared with software simulation.

3.2 Numerical Solutions

3.2.1 Finite Difference Method

The FDM to solve Richards’ equation first divides the solution area into differential grids, replaces the continuous solution domain with a finite number of grid nodes, and stores the variables to be solved (pressure head, water content, etc.) in each grid node. The differential term in the Richards’ equation is replaced by the corresponding difference quotient, so that the partial differential equation is converted into an algebraic difference equation, and a differential equation system containing a finite number of unknown variables at discrete points is obtained, that is, a linear equation system. The solution of the linear equation system is obtained, and the numerical solution of the variables on the grid nodes is obtained. The principle is simple and easy to implement, and it is widely used. In Fig. 3.1, the distance along the z-axis is divided into N equal parts, the uniform grid step size is $\Delta z$, and the spatial derivative first-order central difference scheme can be expressed as:

$$\left( {\frac{\partial h}{{\partial z}}} \right)_{i} = \frac{{h^{*} \,_{i + 1} - h^{*} \,_{i - 1} }}{2\Delta z}$$

(3.1)

Fig. 3.1

Uniform grid and Chebyshev grid

where i represents the discrete grid nodes along the z-axis.

The spatial derivative second-order central difference format is expressed as:

$$\left( {\frac{{\partial^{2} h}}{{\partial z^{2} }}} \right)_{i} = \frac{{h^{*} \,_{i + 1} - 2h^{*} \,_{i} + h^{*} \,_{i - 1} }}{{\Delta z^{2} }}$$

(3.2)

Additionally, the first-order backward difference format of the time derivative is given by:

$$\left( {\frac{\partial h}{{\partial t}}} \right)_{j} = \frac{{h^{*j} - h^{*j - 1} }}{\Delta t}$$

(3.3)

where $\Delta t$ is the discrete time step, and j is the time node.

Substituting Eqs. (3.1)–(3.3) into the Richards’ equation, one can obtain (Zhu et al. 2022a):

$$\left( {\frac{{h_{i + 1}^{*j} - 2h_{i}^{*j} + h_{i - 1}^{*j} }}{{\Delta z^{2} }}} \right) + \alpha \left( {\frac{{h_{i + 1}^{*j} - h_{i - 1}^{*j} }}{2\Delta z}} \right) = c\left( {\frac{{h_{i}^{*j} - h_{i}^{*j - 1} }}{\Delta t}} \right)\quad 1 \le i \le N - 1$$

(3.4)

When the steady-state infiltration is incorporated, its finite difference format can be simplified to:

$$\left( {\frac{{h_{i + 1}^{*} - 2h_{i}^{*} + h_{i - 1}^{*} }}{{\Delta z^{2} }}} \right) + \alpha \left( {\frac{{h_{i + 1}^{*} - h_{i - 1}^{*} }}{2\Delta z}} \right) = 0$$

(3.5)

The system of linear equations formed by Eqs. (3.4)–(3.5) can be further written in matrix form:

$${\mathbf{Ah}}^{*} = {\mathbf{b}}$$

(3.6)

where A is a tridiagonal matrix of order (N − 1) × (N − 1); ${\mathbf{h}}^{*}$ and b are both column vectors of order (N − 1) × 1, and the first and last elements of vector b already contain boundary conditions. Equation (3.6) can be solved by relevant linear iterative methods, such as trigonometric decomposition method, Jacobi iterative method, Gauss–Seidel iterative method and relaxed iterative method.

3.2.2 Finite Volume Method

The finite volume method (FVM), also known as the control volume method, the basic idea is to divide the calculation area into a series of non-repetitive control volumes, and make a control volume around each grid point, and the differential equation is divided into a control volume. For each control volume integral, a system of linear equations to be solved is obtained. For the one-dimensional uniform grid coordinates in Fig. 3.1, the control volume here can be assumed to be $\Delta z \times 1 \times 1$, that is, the distance in the other directions except the z-axis is unit length. Furthermore, by integrating the Richards’ equation over the uniform control volume, one can obtain:

$$\int\limits_{i}^{i + \Delta z} {\int\limits_{t}^{t + \Delta t} {\frac{\partial \theta }{{\partial t}}{\text{d}}t{\text{d}}z} } = \int\limits_{t}^{t + \Delta t} {\int\limits_{i}^{i + \Delta z} {\frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\left( {\frac{\partial h}{{\partial z}} + 1} \right)} \right]{\text{d}}z{\text{d}}t} }$$

(3.7)

Furthermore, the discrete equation is written as:

$$\begin{aligned} & \frac{{K_{i + 1/2}^{j} (h_{i + 1}^{j} - h_{i}^{j} )}}{\Delta z} - \frac{{K_{i - 1/2}^{j} (h_{i}^{j} - h_{i - 1}^{j} )}}{\Delta z} + K_{i + 1/2}^{j} - K_{i - 1/2}^{j} \\ & \quad \quad = \Delta zC_{i}^{j - 1/2} \frac{{h_{i}^{j} - h_{i}^{j - 1} }}{\Delta t} \\ \end{aligned}$$

(3.8)

where $K_{i + 1/2}$ and $K_{i - 1/2}$ are the harmonic mean values of the permeability coefficients of adjacent nodes:

$$K_{i + 1/2} = \frac{{2K_{i} K_{i + 1} }}{{K_{i} + K_{i + 1} }}$$

(3.9)

$$K_{i - 1/2} = \frac{{2K_{i} K_{i - 1} }}{{K_{i} + K_{i - 1} }}$$

(3.10)

Equation (3.8) can be further simplified into the following matrix form:

$${\mathbf{A}}\left( {{\mathbf{h}}^{j} } \right){\mathbf{h}}^{j} = {\mathbf{b}}\left( {{\mathbf{h}}^{j} ,{\mathbf{h}}^{j - 1} } \right)$$

(3.11)

For the solution of Eq. (3.8), the Picard iteration method is usually used to solve it.

3.2.3 Finite Element Method

The finite element method (FEM) divides the solution domain into different elements, including triangular elements, rectangular elements, and quadrilateral elements. The equivalent integral equation of the problem can be obtained based on the variational principle and the orthogonalization principle of the weight function, and then the corresponding linear equation system can be obtained and solved iteratively. Usually, the Richards’ equation solved by FEM is more than two-dimensional, then the two-dimensional linearized Richards’ equation considering the x and z directions can be written as (Zhu et al. 2022b, c):

$$\frac{{\partial^{2} h^{*} }}{{\partial x^{2} }} + \frac{{\partial^{2} h^{*} }}{{\partial z^{2} }} + \alpha \frac{{\partial h^{*} }}{\partial z} = c\frac{{\partial h^{*} }}{\partial t}$$

(3.12)

As shown in Fig. 3.2, a rectangular element is used to divide the solution area. The number of discrete nodes along the x and z axes is N, then the discrete equation can be expressed in matrix form as follows:

$${\mathbf{Gh}}^{*} = - {\mathbf{P}}\frac{{{\mathbf{h}}^{*j} - {\mathbf{h}}^{*j - 1} }}{\Delta t}$$

(3.13)

Fig. 3.2

2D infiltration model and rectangular element grid

where ${\mathbf{G}}$ represents the total conduction matrix on the left side of Eq. (3.12); ${\mathbf{P}}$ represents the storage matrix on the right side of Eq. (3.12); ${\mathbf{h}}^{*}$ is the array of the node pressure heads. Among them, the rectangular element (Fig. 3.2) composed of nodes i, j, m, and n only contributes to the Lth row of ${\mathbf{G}}$ (L = i, j, m, and n). The Lth row elements of matrix ${\mathbf{G}}$ in a rectangular element can be given by:

$$G_{L,i}^{e} = \int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {\left( {\frac{{\partial N_{i}^{e} }}{\partial x}\frac{{\partial N_{L}^{e} }}{\partial x} + \frac{{\partial N_{i}^{e} }}{\partial z}\frac{{\partial N_{L}^{e} }}{\partial z} - \alpha_{g} N_{i}^{e} N_{L}^{e} } \right)} {\text{d}}x{\text{d}}z}$$

(3.14a)

$$G_{L,j}^{e} = \int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {\left( {\frac{{\partial N_{j}^{e} }}{\partial x}\frac{{\partial N_{L}^{e} }}{\partial x} + \frac{{\partial N_{j}^{e} }}{\partial z}\frac{{\partial N_{L}^{e} }}{\partial z} - \alpha_{g} N_{j}^{e} N_{L}^{e} } \right)} {\text{d}}x{\text{d}}z}$$

(3.14b)

$$G_{L,m}^{e} = \int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {\left( {\frac{{\partial N_{m}^{e} }}{\partial x}\frac{{\partial N_{L}^{e} }}{\partial x} + \frac{{\partial N_{m}^{e} }}{\partial z}\frac{{\partial N_{L}^{e} }}{\partial z} - \alpha_{g} N_{m}^{e} N_{L}^{e} } \right)} {\text{d}}x{\text{d}}z}$$

(3.14c)

$$G_{L,n}^{e} = \int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {\left( {\frac{{\partial N_{n}^{e} }}{\partial x}\frac{{\partial N_{L}^{e} }}{\partial x} + \frac{{\partial N_{n}^{e} }}{\partial z}\frac{{\partial N_{L}^{e} }}{\partial z} - \alpha_{g} N_{n}^{e} N_{L}^{e} } \right)} {\text{d}}x{\text{d}}z}$$

(3.14d)

where $N_{L}^{e}$ represents the element basis function; and a and b are half the length and width of the rectangular element, respectively. $N_{L}^{e}$ can be formatted as:

$$N_{i}^{e} (x,z) = \frac{1}{4}\left( {1 - \frac{x}{b}} \right)\left( {1 - \frac{z}{a}} \right)$$

(3.15a)

$$N_{j}^{e} (x,z) = \frac{1}{4}\left( {1 + \frac{x}{b}} \right)\left( {1 - \frac{z}{a}} \right)$$

(3.15b)

$$N_{m}^{e} (x,z) = \frac{1}{4}\left( {1 + \frac{x}{b}} \right)\left( {1 + \frac{z}{a}} \right)$$

(3.15c)

$$N_{n}^{e} (x,z) = \frac{1}{4}\left( {1 - \frac{x}{b}} \right)\left( {1 + \frac{z}{a}} \right)$$

(3.15d)

Similarly, the Lth row elements of P can also be formed as:

$$P_{L,i}^{e} = c\int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {N_{i}^{e} N_{L}^{e} {\text{d}}x{\text{d}}z} }$$

(3.16a)

$$P_{L,j}^{e} = c\int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {N_{j}^{e} N_{L}^{e} {\text{d}}x{\text{d}}z} }$$

(3.16b)

$$P_{L,m}^{e} = c\int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {N_{m}^{e} N_{L}^{e} {\text{d}}x{\text{d}}z} }$$

(3.16c)

$$P_{L,n}^{e} = c\int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {N_{n}^{e} N_{L}^{e} {\text{d}}x{\text{d}}z} }$$

(3.16d)

Furthermore, the contributions of all rectangular elements are summed and assembled to form matrices G and P. Generally, Eqs. (3.14)–(3.16) are solved using Gaussian integration method (Wang and Anderson 1982). The Gaussian integration method makes the definite integral equal to a weighted sum over a finite number of points, and for a quadratic polynomial, the general expression for the double integral is as follows:

$$\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {f(\varepsilon ,\eta ){\text{d}}\varepsilon {\text{d}}\eta } } = f(\varepsilon_{1} ,\eta_{1} ) + f(\varepsilon_{2} ,\eta_{1} ) + f(\varepsilon_{1} ,\eta_{2} ) + f(\varepsilon_{2} ,\eta_{2} )$$

(3.17)

Among them, the four Gauss points are determined by $\varepsilon_{1} = - \,1/\sqrt 3$, $\varepsilon_{2} = 1/\sqrt 3$, $\eta_{1} = - \,1/\sqrt 3$, $\eta_{2} = 1/\sqrt 3$, and the ownership coefficients are all equal to 1. The integral of Eqs. (3.14)–(3.16) can be transformed into the form of Eq. (3.17) by the following variable transformation:

$$\varepsilon = \frac{x}{b}\,\,{\text{and}}\,\,\eta = \frac{z}{a}$$

(3.18)

Then, ${\text{d}}z = a{\text{d}}\varepsilon$, ${\text{d}}x = b{\text{d}}\eta$, the integral limit is from − 1 to 1, and the interpolation function (Eq. 3.15) in the coordinates $\left( {\varepsilon ,\eta } \right)$ is transformed into:

$$\overline{N}_{i}^{e} (\varepsilon ,\eta ) = \frac{1}{4}\left( {1 - \varepsilon } \right)\left( {1 - \eta } \right)$$

(3.19a)

$$\overline{N}_{j}^{e} (\varepsilon ,\eta ) = \frac{1}{4}\left( {1 + \varepsilon } \right)\left( {1 - \eta } \right)$$

(3.19b)

$$\overline{N}_{m}^{e} (\varepsilon ,\eta ) = \frac{1}{4}\left( {1 + \varepsilon } \right)\left( {1 + \eta } \right)$$

(3.19c)

$$\overline{N}_{n}^{e} (\varepsilon ,\eta ) = \frac{1}{4}\left( {1 - \varepsilon } \right)\left( {1 + \eta } \right)$$

(3.19d)

In summary, the integral transformation of Eqs. (3.14)–(3.16) is described as:

$$\int\limits_{ - a}^{a} {\int\limits_{ - b}^{b} {f(x,y){\text{d}}x{\text{d}}y} } = ab\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {f(\varepsilon ,\eta ){\text{d}}\varepsilon {\text{d}}\eta } }$$

(3.20)

where $f\left( {\varepsilon ,\eta } \right)$ is the transformation of $f\left( {x,z} \right)$ at the coordinate $\left( {\varepsilon ,\eta } \right)$.

3.2.4 Numerical Approximation to the Fractional-Time Richards’ Equation

Fractional Richards’ equations can be further divided into space fractional, time fractional, and space–time fractional partial differential equations. The time fractional Richards’ equation is expressed as follows:

$$C(h)\frac{{\partial^{\gamma } h}}{{\partial t^{\gamma } }} = \frac{\partial }{\partial z}\left[ {K(h)\left( {\frac{\partial h}{{\partial z}} + 1} \right)} \right]$$

(3.21)

where $\gamma$ is the order of the time derivative. When fractional order $\gamma = 1$, Eq. (3.21) degenerates into the classical Richard equation.

The Caputo fractional derivative with respect to the function f(x) can be defined as (Pachepsky et al. 2003):

$${}_{0}^{C} D_{x}^{\gamma } f(x) = \frac{1}{\Gamma (n - \gamma )}\int\limits_{0}^{x} {\frac{{f^{(n)} (\tau )}}{{(x - \tau )^{\gamma - n + 1} }}{\text{d}}\tau }$$

(3.22)

where ${}_{0}^{C} D_{x}^{\gamma }$ represents the $\gamma$-order Caputo fractional derivative; $\Gamma$ is the gamma function, and n is a positive integer (representing an integer derivative). In addition, the Riemann–Liouville (R-L) fractional derivative can be defined as (Su 2014):

$${}_{0}^{RL} D_{x}^{\gamma } f(x) = \frac{1}{\Gamma (n - \gamma )}\frac{{{\text{d}}^{n} }}{{{\text{d}}x^{n} }}\int\limits_{0}^{x} {\frac{f(\tau )}{{(x - \tau )^{\gamma - n + 1} }}{\text{d}}\tau }$$

(3.23)

where ${}_{0}^{RL} D_{x}^{\gamma }$ represents the $\gamma$-order R-L fractional derivative.

It can be seen from Eqs. (3.22)–(3.23) that in the definition of R-L fractional derivative, the fractional integral is first obtained and then the integer derivative is obtained, while the definition of Caputo derivative is to first obtain the integer derivative, and then the fractional integral is calculated. There is a great connection between the two, and there are also essential differences, which are mainly reflected in the actual physical models. In the process of actually solving the initial value problem of differential equations, Caputo derivatives are more widely used and have more physical background than R-L derivatives.

For the solution of Eq. (3.21), the finite difference method is used for numerical discretization, and meshing is conducted. Let $\eta$ and $\Delta z$ be the time and space steps, respectively. The simulation time is divided into M equal parts, and the z-axis is divided into N equal parts:

$$t_{m} = m\eta ,\quad m = 0,1, \ldots M$$

(3.24)

$$z_{i} = i\Delta z,\quad i = 0,1, \ldots N$$

(3.25)

For the left-hand fractional derivative term of Eq. (3.21), the Caputo fractional derivative is defined as:

$$\frac{{\partial^{\gamma } h\left( {z,t} \right)}}{{\partial t^{\gamma } }} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int\limits_{0}^{t} {\frac{{\partial h\left( {z,\tau } \right)}}{\partial \tau }\frac{{{\text{d}}\tau }}{{\left( {t - \tau } \right)^{\gamma } }}}$$

(3.26)

The integral of the pressure head derivative in Eq. (3.26) can be directly approximated by the numerical differential equation, which is deduced as follows:

$$\begin{aligned} \frac{{\partial^{\gamma } h\left( {z,t_{m} } \right)}}{{\partial t^{\gamma } }} & = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int\limits_{0}^{{t_{m} }} {\frac{{h^{\prime}\left( \tau \right){\text{d}}\tau }}{{\left( {t_{m} - \tau } \right)^{\gamma } }}} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\sum\limits_{j = 0}^{m - 1} {\int\limits_{{t_{j} }}^{{t_{j + 1} }} {\frac{{h^{\prime}\left( {t_{m} - \tau } \right){\text{d}}\tau }}{{\tau^{\gamma } }}} } \\ & \approx \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\sum\limits_{j = 0}^{m - 1} {\frac{{h\left( {t_{m} - t_{j} } \right) - h\left( {t_{m} - t_{j + 1} } \right)}}{\eta }\int\limits_{{t_{j} }}^{{t_{j + 1} }} {\tau^{ - \gamma } {\text{d}}\tau } } \\ & = \frac{{\eta^{ - \gamma } }}{{\Gamma \left( {2 - \gamma } \right)}}\sum\limits_{j = 0}^{m - 1} {\left( {h_{m - j} - h_{m - j - 1} } \right)\left[ {(j + 1)^{1 - \gamma } - j^{1 - \gamma } } \right]} \\ \end{aligned}$$

(3.27)

Equation (3.27) is further simplified as:

$$\frac{{\partial^{\gamma } h\left( {z,t_{m} } \right)}}{{\partial t^{\gamma } }} = \frac{{\eta^{ - \gamma } }}{{\Gamma \left( {2 - \gamma } \right)}}\sum\limits_{j = 0}^{m - 1} {b_{j}^{\left( \gamma \right)} \left( {h_{m - j} - h_{m - j - 1} } \right)}$$

(3.28)

where $b_{j}^{\left( \gamma \right)} = \left( {j + 1} \right)^{1 - \gamma } - j^{1 - \gamma }$.

For finite difference discretization on the right side of Eq. (3.21), one obtains:

$$\begin{aligned} \frac{\partial }{\partial z}\left[ {K\left( h \right)\left( {\frac{\partial h}{{\partial z}} + 1} \right)} \right] & = \frac{{K_{i + 1/2}^{m + 1} (h_{i + 1}^{m + 1} - h_{i}^{m + 1} )}}{{\Delta z^{2} }} - \frac{{K_{i - 1/2}^{m + 1} (h_{i}^{m + 1} - h_{i - 1}^{m + 1} )}}{{\Delta z^{2} }} \\ & \quad + \frac{{K_{i + 1/2}^{m + 1} - K_{i - 1/2}^{m + 1} }}{\Delta z} \\ \end{aligned}$$

(3.29)

$K_{i + 1/2}$ and $K_{i - 1/2}$ can be expressed as:

$$K_{i + 1/2} = \frac{{2K_{i} K_{i + 1} }}{{K_{i} + K_{i + 1} }}$$

(3.30)

$$K_{i - 1/2} = \frac{{2K_{i} K_{i - 1} }}{{K_{i} + K_{i - 1} }}$$

(3.31)

Combining Eqs. (3.28) and (3.29), the discrete format of the fractional-time Richards’ equation can be obtained as follows:

$$\begin{aligned} & \frac{{\eta^{ - \gamma } }}{{\Gamma \left( {2 - \gamma } \right)}}C(h)_{i}^{m - 1/2} \sum\limits_{j = 0}^{m - 1} {b_{j}^{\left( \gamma \right)} \left( {h_{m - j,i} - h_{m - j - 1,i} } \right)} \\ & \quad = \frac{{K_{i + 1/2}^{m} (h_{i + 1}^{m} - h_{i}^{m} )}}{{\Delta z^{2} }} - \frac{{K_{i - 1/2}^{m} (h_{i}^{m} - h_{i - 1}^{m} )}}{{\Delta z^{2} }} + \frac{{K_{i + 1/2}^{m} - K_{i - 1/2}^{m} }}{\Delta z} \\ \end{aligned}$$

(3.32)

where $C(h)_{i}^{m - 1/2}$ represents the average value of the specific moisture capacity ($C(h)_{i}^{m}$) at the previous time step and the specific moisture capacity ($C(h)_{i}^{m - 1,k}$) of the current iteration step ($k \ge 1$) of the current time step.

Equation (3.32) can be simplified into matrix form Ax = b, and then iteratively solved by Picard method. In order to evaluate the fitting effect of the proposed time fractional model, two indicators are selected, namely the root mean square error (RMSE) and the relative error (RE):

$${\text{RSE}} = \left[ {\frac{1}{N - 1}\sum\limits_{i = 1}^{N - 1} {\left( {\frac{{h_{i} - h_{i}^{*} }}{{h_{i}^{*} }}} \right)^{2} } } \right]^{0.5}$$

(3.33)

$${\text{RE}} = 100 \times \left[ {\frac{1}{N - 1}\sum\limits_{i = 1}^{N - 1} {\left( {\left| {\frac{{h_{i} - h_{i}^{*} }}{{h_{i}^{*} }}} \right|} \right)} } \right]$$

(3.34)

where $h_{i}$ is the numerical solution of Richards’ equation, and $h_{i}^{*}$ is the exact solution of Richards’ equation. The smaller the values of the two errors, the higher the computational accuracy of the proposed approach.

3.3 An Improved Method for Non-uniform Spatial Grid

In the numerical solution of Richards’ equation, FDM can be used for numerical discretization and iterative solution. However, to obtain a more reliable numerical solution, the space step size of conventional uniform grid (Fig. 3.1) is often small, particularly under some unfavorable numerical conditions, such as infiltration into dry and layered soils with greatly different permeability coefficients, this makes the calculation time-consuming and even the accuracy cannot be improved very well. The realization of unstructured space grids and dynamic grid methods in some studies is often complicated and inappropriate (Chávez-Negrete et al. 2018; Dolejší et al. 2019). Therefore, this chapter proposes an improved FDM numerical discretization process using a non-uniform Chebyshev space grid, which is compared with the traditional uniform space grid. This method can provide a certain reference for the numerical simulation of unsaturated infiltration.

• (1) Uniform grid method

The Richards’ equation is directly numerically discretized by the finite difference method of uniform grid, and one can gain (Zhu et al. 2020):

$$\begin{aligned} & \frac{{\left[ {K_{i + 1/2}^{j} (h_{i + 1}^{j} - h_{i}^{j} ) - K_{i - 1/2}^{j} (h_{i}^{j} - h_{i - 1}^{j} )} \right]}}{{\Delta z^{2} }} + \frac{{K_{i + 1/2}^{j} - K_{i - 1/2}^{j} }}{\Delta z} \\ & \quad = \frac{{C_{i}^{j - 1/2} \left( {h_{i}^{j} - h_{i}^{j - 1} } \right)}}{\Delta t} \\ \end{aligned}$$

(3.35)

Comparing Eq. (3.29), it can be found that the mathematical meaning of Eqs. (3.35) and (3.29) is consistent. When its steady-state infiltration is considered, the FDM format can be simplified as (Zhu et al. 2020):

$$\frac{{\left[ {K_{i + 1/2} (h_{i + 1} - h_{i} ) - K_{i - 1/2} (h_{i} - h_{i - 1} )} \right]}}{{\Delta z^{2} }} + \frac{{K_{i + 1/2} - K_{i - 1/2} }}{\Delta z} = 0$$

(3.36)

• (2) Improved Chebyshev space grid method

Because the pressure head in the unsaturated infiltration problem often has a large change at the boundary and the soil layer interface, a finer mesh is usually required for densification. However, the uniform grid generates too many computing grid nodes in the process of densification, and the computation is time-consuming and even not accurate enough. Then, a Chebyshev grid coordinate is proposed as (Wu et al. 2020b; Zhu et al. 2020):

$$z_{i} = \cos \left( {i\pi /N} \right) \times \frac{L}{2} + \frac{L}{2},\quad i = N,N - 1, \ldots ,0$$

(3.37)

where L is the thickness of the soil layer.

In Fig. 3.1, the Chebyshev grid nodes are only highly refined at the interface, greatly reducing the number of grid nodes. Furthermore, the FDM discrete format combined with the Chebyshev grid can be expressed as:

$$\begin{aligned} & \frac{1}{{\delta z_{i} }}\left[ {\frac{{K_{i + 1/2}^{j} (h_{i + 1}^{j} - h_{i}^{j} )}}{{\Delta z_{i + 1} }} - \frac{{K_{i - 1/2}^{j} (h_{i}^{j} - h_{i - 1}^{j} )}}{{\Delta z_{i} }}} \right] + \frac{{K_{i + 1/2}^{j} - K_{i - 1/2}^{j} }}{{\delta z_{i} }} \\ & \quad = C_{i}^{j - 1/2} \frac{{h_{i}^{j} - h_{i}^{j - 1} }}{\Delta t} \\ \end{aligned}$$

(3.38)

where $\Delta z_{i}$ represents the unequal spacing between the Chebyshev grid nodes; $\delta z_{i}$ is the unequal spacing of the computing nodes (Fig. 3.1).

Equation (3.38) can also be simplified into a matrix form such as Eq. (3.13). Due to the nonlinear relationship between permeability coefficient and water content, the coefficient matrix A needs to be repeatedly evaluated by nonlinear iteration method after numerical discretization. Among them, Picard method is a more classical and practical nonlinear iterative method. Based on the Chebyshev grid discretization format of the Richards’ equation, a program for unsaturated infiltration was developed using the MATLAB (R2014a) language.

3.3.1 Validation Example

This test describes one-dimensional transient unsaturated infiltration in homogeneous unsaturated soil (Ku et al. 2018; Zhu et al. 2022a), the soil thickness L = 10 m, exponential model parameters are: $\alpha$ = 1 × 10⁻⁴, θ_s = 0.50, θ_r = 0.11, and the saturated permeability coefficient k_s = 2.5 × 10⁻⁸ m/s. Furthermore, the boundary conditions can be given by:

$$h\left( {z = 0} \right) = h_{{\text{d}}}$$

(3.39)

$$h\left( {z = 10} \right) = 0$$

(3.40)

To verify the computational accuracy of the proposed method, the following error indicators are used in the comparative analysis, namely the root mean square error (RSE), the relative error (RE), and maximum relative error (MRE):

$${\text{RSE}} = \left[ {\frac{1}{N - 1}\sum\limits_{i = 1}^{N - 1} {\left( {\frac{{h_{i} - h_{i}^{*} }}{{h_{i}^{*} }}} \right)^{2} } } \right]^{0.5}$$

(3.41)

$${\text{RE}} = 100 \times \left[ {\frac{1}{N - 1}\sum\limits_{i = 1}^{N - 1} {\left( {\left| {\frac{{h_{i} - h_{i}^{*} }}{{h_{i}^{*} }}} \right|} \right)} } \right]$$

(3.42)

$${\text{MRE}} = 100 \times \left[ {\max \left| {\frac{{h_{i} - h_{i}^{*} }}{{h_{i}^{*} }}} \right|_{i = 1}^{N - 1} } \right]$$

(3.43)

where $h_{i}$ is the numerical solution of Richards’ equation and $h_{i}^{*}$ is the exact solution of Richards’ equation. The smaller the values of the three errors, the higher the computational accuracy of the proposed method. The total simulation time was set to 5 h, the number of discrete nodes was set to 100, 150, and 200, and the time steps were set to 0.01, 0.02, and 0.04 h, respectively.

In Fig. 3.3, the numerical solutions are derived using two grid methods under the conditions of $\Delta t$ = 0.01 h and N = 200 and compared with the analytical solutions. The numerical solution obtained by the uniform grid method has a large deviation from the exact solution, particularly after t > 2 h (Fig. 3.3a), while the numerical solution obtained by the Chebyshev grid method is in good agreement with the analytical solution (Fig. 3.3b).

Fig. 3.3

Comparison of numerical and analytical solutions obtained by different grid methods

Figure 3.4a represents the maximum relative error (MRE) obtained by different grid methods at different time steps when the number of nodes N is 100. The MRE obtained by the Chebyshev grid method ranges from 0.6 to 3.5%, and it first decreases and then increases with the increase of time. When t < 4 h, the MRE decreases with decreasing time step. The MRE obtained by the uniform grid method ranges from 1.3 to 49%, particularly when t > 1 h, the MRE increases over time and is much larger than that obtained by the Chebyshev grid.

Fig. 3.4

Comparison of the maximum relative error of different grid methods under different numerical conditions

Figure 3.4b shows the MRE obtained by different grid methods under different numbers of discrete grid nodes when the time step $\Delta t$ = 0.01 h. It can be found that the MRE obtained by the uniform grid as time increases. In addition, the MRE of the two methods has a decreasing trend with the increase of the number of grid nodes N.

In Table 3.2, the RSE of both methods and the RE of the uniform grid decrease with the increase of N, while the RE of the Chebyshev grid increases with increasing N. However, it can be seen from the numerical value that the RSE of the Chebyshev grid is nearly 100 times different from the uniform grid method, and the RE is more than 10 times different from the uniform grid method. This test indicates that the proposed Chebyshev grid method is not limited by the number of grid nodes to improve the accuracy, that is, the proposed method achieves higher numerical accuracy with fewer discrete nodes, and has a smaller computational cost.

Table 3.2 Numerical accuracy at t = 5 h

3.3.2 Unsaturated Infiltration in Layered Soils

The mathematical model is shown in Fig. 3.5. The model parameters of the two-layer soils are set to θ_s = 0.35, θ_r = 0.14, $\alpha$ = 8 × 10⁻³. The thickness of soil layer 1 and soil layer 2 are both set to 5 m, the number of discrete nodes in each layer is 40, and the boundary conditions are consistent with Sect. 3.3.1, where h_d = − 10³ m.

Fig. 3.5

Chebyshev grid of two layers in unsaturated soil

In layered soils, due to the influence of different soil types, different combinations have a great influence on unsaturated infiltration. Table 3.3 lists the saturated permeability coefficients in different saturated soils. In order to further verify the applicability of the proposed grid method, it is assumed that the saturated permeability coefficient of soil layer 1 is 10⁻¹ m/s, and the saturated permeability coefficient of the second layer gradually changes from coarse sandy soil to clay in Table 3.3, the saturated permeability coefficient of the second layer ranges from 10⁻² to 10⁻⁹ m/s.

Table 3.3 Typical permeability coefficient values for saturated soils

It can be seen from Fig. 3.6a that the numerical solutions obtained by the uniform grid and the Chebyshev grid are approximate, and both can well simulate the unsaturated infiltration process in case 2. Figure 3.6b depicts the numerical results of case 4. The ratio of the saturated permeability coefficients of the upper and lower soil layers is in the order of 10⁴. The uniform grid cannot accurately represent the pressure head at the interface, but the numerical solution obtained by the Chebyshev grid can accurately describe the pressure head variation at the interface. In Fig. 3.6c, the Chebyshev grid can also accurately describe the variations in the pressure head at the interface under case 6 (Table 3.4).

Fig. 3.6

Comparison of numerical solutions under different cases

Table 3.4 Permeability coefficient for cases 1–8

Similar to the two-layer soil, the proposed improved grid method is applied to the three-layer unsaturated soils. The mathematical model is described in Fig. 3.7, where L₁ = L₃ = 4 m and L₂ = 2 m. The number of discrete nodes in each layer of soil is 40, the parameter θ_s is set to 0.46, the three-layer unsaturated soil permeability coefficient is listed in Table 3.5, and the boundary conditions can be expressed as: h(z = 0) = 0, h(z = 10 m) = − 1000 m. Other numerical conditions are consistent with the two-layered soils.

Fig. 3.7

One-dimensional infiltration model of three-layer unsaturated soil

Table 3.5 Permeability coefficient for cases 9–13

In Fig. 3.8, the proposed Chebyshev grid method can better characterize the pressure head change between the two interfaces in case 11 compared to the uniform grid method. This test further demonstrates that the proposed Chebyshev grid method can obtain more reliable numerical solutions with fewer grid nodes under some unfavorable infiltration conditions, particularly when the saturated permeability coefficient changes greatly in layered soils.

Fig. 3.8

Comparison of numerical solutions for case 11

3.4 Application of Chebyshev Spectral Method to Richards’ Equation

3.4.1 Chebyshev Spectral Method

The Chebyshev spectral method (CSM) was developed by Gottlieb et al. (1978). It is widely used to solve numerical problems expressed by partial differential equations. The advantage of the CSM is the use of non-uniform mesh discretization and the Chebyshev differentiation matrix to improve the accuracy of the numerical simulation. In Fig. 1.2, the N + 1 Chebyshev point coordinates in the interval [− 1, 1] can be expressed as:

$$z_{j} = \cos \left( {j\pi /N} \right),\quad j = 0,1, \ldots N$$

(3.44)

The first-order derivative of interpolation function $p^{\prime}\left( {\mathbf{z}} \right)$ can be written as:

$$p^{\prime}\left( {\mathbf{z}} \right) = {\mathbf{D}}_{N} {\mathbf{h}}$$

(3.45)

where ${\mathbf{z}}$ is represented as the N + 1-dimensional vector $\left( {z_{0} ,z_{1} , \ldots ,z_{N} } \right)^{{\text{T}}}$, ${\mathbf{h}}$ represents the N + 1 dimensional vector $\left( {h_{0} ,h_{1} , \ldots ,h_{N} } \right)^{{\text{T}}}$ composed of the pressure head, and ${\mathbf{D}}_{N}$ is an $\left( {N + 1} \right) \times \left( {N + 1} \right)$ Chebyshev differentiation matrix.

The expression of each element in Chebyshev differentiation matrix ${\mathbf{D}}_{N}$ for any number of nodes (N + 1) is given by:

$$\left( {{\mathbf{D}}_{N} } \right)_{00} = \frac{{2N^{2} + 1}}{6},\quad \left( {{\mathbf{D}}_{N} } \right)_{NN} = - \frac{{2N^{2} + 1}}{6},$$

(3.46a)

$$\left( {{\mathbf{D}}_{N} } \right)_{jj} = \frac{{ - z_{j} }}{{2\left( {1 - z_{j}^{2} } \right)}},\quad j = 1, \ldots ,N - 1$$

(3.46b)

$$\left( {{\mathbf{D}}_{N} } \right)_{ij} = \frac{{c_{i} }}{{c_{j} }}\frac{{\left( { - 1} \right)^{i + j} }}{{z_{i} - z_{j} }},\quad (i \ne j)i,j = 0, \ldots ,N,$$

(3.46c)

where $\left( {{\mathbf{D}}_{N} } \right)_{ij}$ represents the element in row i + 1 and column j + 1 of Chebyshev differentiation matrix ${\mathbf{D}}_{N}$ for the first derivative and,

$$c_{i} = \left\{ {\begin{array}{*{20}l} {2,} & {i = 0,N} \\ {1,} & {i = 1, \ldots ,N - 1} \\ \end{array} } \right.$$

(3.47)

To obtain the Chebyshev differentiation matrix for the second derivative, the matrix can be directly squared to ${\mathbf{D}}_{N}$. Similarly, Chebyshev differentiation matrices for higher-order derivatives can be obtained as follows:

$$\frac{\partial }{\partial z} \to {\mathbf{D}}_{N}$$

(3.48a)

$$\frac{{\partial^{2} }}{{\partial z^{2} }} \to {\mathbf{D}}_{N}^{2}$$

(3.48b)

$$\frac{{\partial^{3} }}{{\partial z^{3} }} \to {\mathbf{D}}_{N}^{3}$$

(3.48c)

Because ${\mathbf{D}}_{N}$ is obtained in the interval [− 1, 1], it is necessary to scale ${\mathbf{D}}_{N}$ to other intervals, that is,

$$\frac{\partial }{\partial z} \to \frac{2}{L}{\mathbf{D}}_{N}$$

(3.49a)

$$\frac{{\partial^{n} }}{{\partial z^{n} }} \to \left( {\frac{2}{L}{\mathbf{D}}_{N} } \right)^{n}$$

(3.49b)

Through the above process, the partial differential equation including the RE can be easily and efficiently solved. Thus, the governing equation (Eq. 2.1) can be rewritten as:

$$\frac{1}{c}{\mathbf{D}}_{N}^{2} {\mathbf{h}}^{*} + \frac{{\alpha_{g} }}{c}{\mathbf{D}}_{N} {\mathbf{h}}^{*} = \frac{{\partial {\mathbf{h}}^{*} }}{\partial t}$$

(3.50)

In Eq. (3.50), Eq. (2.1) has been transformed into an ordinary differential equation (ODE) by introducing Chebyshev differentiation matrices. In this book, the variable-step four- and fifth-order Runge–Kutta method is used to solve Eq. (3.50), which is the ODE solver in MATLAB. Meanwhile, the 2D linear RE can also be solved using the same method.

To evaluate the performance of the proposed method, an L₂ norm is used to indicate the error of solution $h^{*}$:

$$L_{2} \left( {h^{*} } \right) = \left\| {h^{*,a} (z,t) - h^{*} (z,t)} \right\|_{2}$$

(3.51)

where $h^{*,a} (z,t)$ and $h^{*} (z,t)$ are one-dimensional (1D) analytical and numerical solutions, respectively. Simultaneously, the numerical and analytical solutions with spatial and temporal grids can produce an absolute error defined as follows:

$${\text{err}}_{A} \left( {h^{*} } \right) = h^{*,a} (z,t) - h^{*} (z,t).$$

(3.52)

3.4.2 Validation Example

Test 1 represents transient infiltration in homogeneous unsaturated soils. In the mathematical model, the soil thickness (L) is assumed to be 10 m. The model parameters are θ_s = 0.50, θ_r = 0.11, $\alpha$ = 1 × 10⁻⁴, and k_s = 9 × 10⁻⁵ m/h (Zhu et al. 2019). The governing equation is described as Eq. (2.1).

When water infiltrates into a soil mass, ponding on the ground maintains the pressure head at zero (Green and Ampt 1911). The upper and lower boundary conditions can be expressed as:

$$h\left( {z = 0,t} \right) = h_{{\text{d}}}$$

(3.53)

$$h\left( {z = L,t} \right) = 0$$

(3.54)

The effect of $\Delta z$ and N on the results of Test 1 was analyzed. The time step was set to 0.01 h. Different numbers of nodes or grid sizes, that is, N/$\Delta z$ = 20/0.5 (m), N/$\Delta z$ = 40/0.25 (m), and N/$\Delta z$ = 80/0.125 (m) were selected to investigate the influence of the different methods on $L_{2} (h^{*} )$. The total simulation time was 5 h. Simultaneously, three different initial conditions h₀ were chosen to evaluate their influence on the numerical accuracy and stability of the FDM and CSM.

In Fig. 3.9, $L_{2} (h^{*} )$ of the CSM were consistently smaller than that of the conventional FDM. The numerical accuracy increased as the mesh was refined, and the accuracy of the traditional FDM was significantly affected by the initial conditions. However, the proposed method (CSM) was less affected, and the numerical accuracy was stable in the order of 10⁻⁶–10⁻⁷ in this example. The result indicates that the CSM was more robustness than the FDM. Figure 3.10 demonstrates the comparison of the computed pressure head profiles with different grid sizes at t = 2 h. Compared with the analytical solution, it can be easily seen that the numerical accuracy of the CSM was less influenced by the grid sizes than that of the FDM (Fig. 3.10). That is, the CSM achieved better numerical results with fewer mesh nodes.

Fig. 3.9

Comparison of the calculation accuracy for solving 1D transient infiltration with different grid sizes and initial conditions using different methods: a h₀ = − 10 m; b h₀ = − 1000 m; c h₀ = − 100,000 m

Fig. 3.10

Comparison of the computed profiles of the pressure head with different grid sizes at t = 2 h: a $\Delta z = 0.5$ m; b $\Delta z = 0.25$ m; c $\Delta z = 0.125$ m

The pressure head profiles were obtained from the computed results over time under h₀ = − 100,000 m (Fig. 3.11). The number of nodes or grid sizes was 40/0.25 (m). Figure 3.11 illustrates that the numerical solutions of the CSM agree well with the analytical solutions over time, whereas the numerical solutions of the FDM have larger errors than the analytical solutions. In Fig. 3.12, the minimum absolute errors of the proposed CSM and the FDM are approximately 10⁻¹¹ and 10⁻⁶, respectively. Consequently, the extended CSM is characterized by higher accuracy than traditional FDM with fewer nodes.

Fig. 3.11

Comparison of the calculation results (pressure head) from different methods when h₀ = − 100,000 m

Fig. 3.12

Absolute error in the results computed using the CSM and FDM relative to the analytical solution for the 1D transient infiltration at different simulated times

3.5 Conclusions

This chapter first summarizes the commonly used spatial numerical discrete methods of Richards’ equation, including FDM, FVM, and FEM. Furthermore, this chapter proposes an improved FDM numerical discretization process using a non-uniform Chebyshev space grid and compares it with the numerical results and analytical solutions obtained from a conventional uniform space grid. Additionally, based on the non-uniform Chebyshev grid method, the Chebyshev spectral method is proposed to solve the Richards’ equation, and the following conclusions can be obtained:

• (1) The proposed Chebyshev grid method, while keeping the number of discrete nodes unchanged, uses a cosine function to ingeniously densify the interfaces on both sides and then numerically solve it to obtain a more reliable numerical solution. The numerical results indicate that the numerical solutions obtained by this method are in good agreement with the analytical solutions under some unfavorable numerical conditions, such as infiltration into a dry soil. At the same time, the results illustrate that the proposed Chebyshev grid method can obtain higher numerical accuracy with a smaller number of nodes than the conventional uniform grid, and the computational cost is small.

• (2) The Chebyshev spectral method (CSM) based on the discretization of Chebyshev grid is extended to simulate unsaturated infiltration in porous media. Chebyshev differential matrix can construct n-order Chebyshev differential matrix conveniently and quickly. This is the first time that the CSM has been used successfully to solve 1D and 2D transient infiltration problems in unsaturated soils. Compared with the traditional FDM, the CSM was more efficient, and it achieved higher accuracy with a lower-resolution grid. The CSM was not sensitive to the initial conditions.