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.