1. INTRODUCTION
In coastal areas, tidal dynamics causes periodic fluctuation of the groundwater response in coastal aquifers. The dynamic behavior of the groundwater flow associated with the fluctuation of the piezometric head plays a crucial role in coastal hydrology (Volker & Zhang, 2001; Chuang et al., 2010; Asadi-Aghbolaghi et al., 2012). Finding out the interaction between the tidal dynamics as well as the groundwater level may be useful for investigating coastal hydrogeology in large scale which is of importance to solve various coastal hydrogeological problems, such as deterioration of the marine environment, seawater intrusion, and the stability of coastal engineering structures with rapidly development in coastal areas (Zhao et al., 2019; Li et al., 2019). Accordingly, considerable research has been devoted to understanding the tide–induced groundwater level fluctuations in a coast aquifer (Ataie-Ashtiani et al., 2001; Chuang & Yeh, 2011; Huang et al., 2015). Analytical studies have been found to be common for understanding tidal variation affecting groundwater level in coastal areas (Li & Jiao, 2002; Chen et al., 2016; Fang et al., 2018). Sun (1997) solved an analytical solution of groundwater response to tidal–induced boundary condition in an estuary. Jiao and Tang (1999) studied the groundwater response that consisted of unconfined and confined aquifers with a semi–permeable layer. Jiao & Tang (2001) further proposed an exact solution for the groundwater response in a leaky confined aquifer. Jeng et al. (2002) discussed the tidal propagation in a coupled unconfined and confined coastal aquifer system. However, due to the mathematical complexity, it is a great challenge for finding exact analytical solutions of governing equations for different boundary conditions. As a result, the developed analytical solutions are usually limited and are restricted to simple problems with specific boundary conditions.
Numerical approaches, such as the finite element method (FEM) and the finite difference method (FDM), approximate equations in discrete steps, both in time and in space (Liao et al., 2018). The advantage of these approaches is no limit to the complexity of various coastal hydrogeological problems. However, the accuracy cannot be as accurate as the analytical method due to numerical errors (Ku et al., 2019). Despite the success of several numerical methods for finding the solution, the traditional approaches need mesh generation, and hence tedious time calculation, to give an approximation to the solution (Liu et al., 2019). There is therefore still a need on numerical scheme in the development of newly advanced approaches.
Meshfree approaches appear as a rival alternative to mesh generation techniques. The conception of meshfree approaches is to solve a problem by using arbitrary or unstructured points which are collocated in the problem domain without using any meshes or elements. The utilization of meshfree approach to acquire approximate solutions is advantageous for problem domain involving arbitrary geometry (Chang et al., 2016; Kołodziej & Grabski, 2018). The collocation Trefftz method (CTM) is one of meshfree approaches for dealing with boundary value problem (BVP) where solutions can be described as a combination of general solutions which satisfy the governing equation (Trefftz, 1926; Kita & Kamiya, 1995; Li et al., 2008). The CTM may release the difficulties for finding exact solutions of governing equations and remains the characteristics of high accuracy due to the adoption of the general solutions (Ciałkowki & Grysa, 2010; Grysa & Maciejewska, 2013; Grysa et al., 2014). The original CTM is limited to homogeneous and stationary BVPs. Lately, a spacetime meshfree method based on Minkowski spacetime for the modeling of transient subsurface flow problems has been developed by Ku et al. (2018).
In this study, we present a pioneering work for numerical solutions of tide–induced groundwater response in a coast aquifer using the spacetime meshfree method. This paper presents the numerical solutions of tide–induced groundwater response using the spacetime collocation approach (SCA). The newly developed SCA is based on the Trefftz method. The Trefftz basis functions have to satisfy the governing equation of the tide–induced groundwater response in a coastal confined aquifer with an estuary tidal–induced boundary. The phase as well as the amplitude of tide can then vary with time and position. The separation of variables is utilized to formulate the solutions to be the Trefftz functions. For the discretization of the domain, arbitrary collocation points are assigned on the spacetime domain boundaries. The SCA is validated for several numerical examples with analytical solutions. The comparison of the results and accuracy for the SCA with the time–marching FDM is carried out. In addition, the SCA is adopted to evaluate the tidal and groundwater piezometer data at the Xing–Da port, Kaohsiung, Taiwan.
2. PROBLEM STATEMENT
The governing equation of one–dimensional tide–induced groundwater response in a coastal confined aquifer is expressed as follows.
in . (1)
In the preceding equations, denotes the spacetime domain, hdenotes the groundwater head in the confined aquifer, t denotes time, S denotes the storage coefficient of the confined aquifer,T denotes the transmissivity of the confined aquifer, Ldenotes the leakage coefficient, where , K and b denote the vertical permeability and thickness of the semi–permeable layer.
The initial condition for solving Equation (1) can be expressed as
, (2)
where denotes the initial total head. The boundary conditions of Equation (1) are given as
on , (3)
on , (4)
where n is the normal direction, is the spacetime Dirichlet boundary, denotes the spacetime Neumann boundary, and denote the Dirichlet and Neumann boundary conditions, respectively.
3. THE SPACETIME COLLOCATION APPROACH
The spacetime collocation scheme begins with the consideration of transient Trefftz basis functions which are also the general solutions of the governing equation. Thus, it may be necessary to derive general solutions for the tide–induced groundwater flow problem in a confined aquifer. To establish the transient Trefftz functions for this problem, the separation of variables is adopted by considering the transient solutions of the tide–induced groundwater flow problem.
. (5)
For simplicity, the following equations are considered.
, and . (6)
Inserting Equation (6) into Equation (1) can obtain the following equation.
in . (7)
Dividing by on both sides in Equation (7), we yield the equation as follows.
, and , (8)
where is a separation constant. To evaluate the eigenvalue of Equation (8), the p constant is introduced to ensure this constant to be positive or negative value. Hence, we may consider , , and . The formulation of Trefftz functions for one–dimensional groundwater flow problems are expressed as follows.
Considering , we may obtain the following solutions.
, and , (9)
where , and are constants. Substituting Equation (9) into Equation (5) may acquire the following equation.
, (10)
where and are unknown coefficients to be determined.
Considering , we may yield . We may therefore obtain the following equation.
, (11)
where is constant. The following solution may then be obtained.
, (12)
where and are unknown coefficients to be determined.
Considering , we may yield the possible solutions.
, and , (13)
where , and are constants. Substituting Equation (13) into Equation (5) may yield the following equation.
, (14)
where and are unknown coefficients to be evaluated.
Considering , we may yield the following solutions.
, and , (15)
where , and are constants. Substituting Equation (15) into Equation (5), we may obtain
, (16)
where and are unknown coefficients to be evaluated. According to superposition theorem, the transient solution of this problem is described as the linear combination of the general solutions. Consequently, we may yield the following series expansion.
, (17)
where is the order of the Trefftz basis functions, , , , , , , and are unknowns to be found. From Equation (17), we may also obtain the complete Trefftz basis functions, T , as follows.
(18)
For the flux boundary, the Neumann boundary condition is given as
. (19)
To evaluate the coefficients in Equations (17) and (19), the collocation method is utilized. We may assign collocation points on the Dirichlet and Neumann boundary using Equation (17) and (19). Then, the system of linear algebraic equations may be assembled as follows.
. (20)
Finally, the equation can be written as follows.
, (21)
where is a matrix from the basis functions, is a vector, is a vector,J is the boundary point number, Q is the terms of the basis function, which can be defined as , are time, are spatial coordinates, are unknowns to be evaluated, are Dirichlet boundary data, and are the boundary point number in time and space domain, respectively. The unknown coefficients are determined for solving Equation (21). To obtain the tide–induced groundwater response for the spacetime domain, the inner collocation points in the spacetime domain have to be collocated. The tide–induced groundwater response at inner collocation points can then be computed by Equation (21) for the spacetime domain. The schematic diagram for the one–dimensional tide–induced groundwater problems is depicted in Figure 1.
Instead of utilizing the original Euclidean space, we adopt a collocation scheme based on spacetime coordinates to model the transient tide–induced groundwater problem. On the basis of the spacetime collocation scheme, time is considered to be an independent variable. Figure 2 (a) indicates a one–dimensional transient tide–induced groundwater problem. From Figure 2 (a), denotes the space domain, denotes the space dimension of the problem, and denotes the total elapsed time. It is clear that this problem is one–dimensional in both time as well as space. Based on the spacetime collocation scheme, the original one–dimensional space domain turns into a spacetime domain in two–dimensions, as shown in Figure 2 (b). Hence, both the boundary and initial values can be provided on the spacetime boundary. Due to the inaccessible final time boundary data, the spacetime collocation scheme transforms a one–dimensional transient tide–induced groundwater problem into an inverse BVP in two–dimensions.
4. NUMERICAL EXAMPLES
To investigate the effectiveness and accuracy of the proposed method, five numerical examples are conducted. The objective of each numerical example is depicted in Table 1. In the numerical example 1 and 2, we conduct the sensitivity analysis and results comparison with the analytical solution. In the numerical example 3 and 4, the accuracy of the proposed method is compared with that of the conventional time–marching FDM. Furthermore, we consider an application example in numerical example 5, where a time series of real tidal fluctuation data recorded at Xing–Da port by the Ministry of Economic Affairs Water Resources Department are used to study tide–induced groundwater flow problem.