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Two analytical solutions using segregation variable method to calculate the hydraulic head under steady and unsteady flow conditions based on Tóth’s classical model were developed. The impacts of anisotropy ratio, hydraulic conductivity (

Tóth [

Furthermore, the physical mechanisms of groundwater flow system have been explained by Tóth [

A progression of groundwater flow system investigations has been conducted based on the assumption that the medium is isotropic and homogeneous [

Based on a critical review of the literature, one can see that most of the previous studies focused on the steady-state flow of groundwater. However, the groundwater flow is usually transient in the fields. Models of steady-state groundwater flow can result in significant limitations in applications. Besides, the prevailing analytical solutions of unsteady groundwater flow are essentially targeted on determining well flow issues. There are few specific reports about analytical solutions for unsteady nested groundwater flow systems [

In most of the previous studies, the media were assumed to be homogeneous and isotropic [

Idealized cross section of a drainage basin’s valley flank of the sinusoidal water table [

The governing equation of the potential function for a 2D steady-state flow field considering the effect of anisotropy can be described as follows:

The boundary conditions can be written as

The following dimensionless variables are defined:

The mathematical model can be solved by using the segregation variable method. The details can be found in Appendix

If we set

Equation (

The dimensionless hydraulic head,

Distribution of hydraulic head (line with dates) and velocity in basin with different anisotropy ratio ((a)

The area of equal velocity region decreases significantly with the anisotropy ratio (the contour interval is 0.003). The penetrating depth of groundwater flow systems decreases with the horizontal hydraulic conductivity, or the horizontal flow velocity increases with the horizontal hydraulic conductivity. Additionally, stagnant zones can be found from the simulating results. They are located at the places that have opposite direction for the streamlines (stagnant zone-1) and at both sides at the bottom in the field (stagnant zone-2). The stagnant zone-1 moves toward ground surface as

According to Wang and Wan [

The initial condition and boundary conditions can be written as^{−1}) is specific yield and

Similarly, some additional dimensionless variables can be defined:

The potential function can be obtained using segregation variable technique (the detailed derivation can be found in Appendix

Equation (

A MATLAB program was developed to calculate (^{−1}, and

Distribution of hydraulic head (line with dates) and velocity in basin at different periods: (a)

Time series of hydraulic head in different nodes of the upper boundary.

The hydraulic heads at the right side are significantly higher than those at the left side if a sinusoidally undulating water table is set at

Sensitivity analysis is a tool to analyze the impact of the input parameters on the results of a model. It is essential to analyze the sensitivity of parameters in order to improve the accuracy of groundwater flow models and to reduce errors induced from the uncertainty of hydrogeological parameters [

In unsteady groundwater flow systems, hydraulic head and specific yield can have great impacts on the evolution process of unsteady flow. The normalized sensitivity method [

The values of

Time series of hydraulic head of the midpoint (

On the basis of Tóth’s classical model, analytical solution of hydraulic head containing hydraulic conductivity under steady and unsteady flow conditions is obtained. From this study, the following conclusions can be drawn:

For the steady flow, the area of equal velocity region becomes much smaller with a larger anisotropy ratio, and the penetrating depth of groundwater flow systems becomes smaller. Stagnant zones locate at the places which have opposite directions for the streamlines and both sides at the bottom of the field. Finally, when

For the transient flow, when time is large enough, the ultimate distribution of unsteady flow is consistent with the steady groundwater flow system model. The closer it is to the right side, the faster it is for the flow to approach steady state.

For the transient flow, a relative increase (e.g., 10%, 50%, and 100%) in

Analytical solution of hydraulic head for (

Let

One can obtain two ordinary differential equations:

According to the left and right boundary conditions,

Considering the bottom boundary condition,

When

When

Using the condition (

General solution for (

Using the condition (

Let

When

General solution for (

According to boundary conditions, one can get that

General solution for (

Using the condition (

Therefore,

Combined with the upper boundary condition, one can obtain

Then one can obtain

Analytical solution of hydraulic head for (

Let

One can obtain three ordinary differential equations:

According to the boundary conditions, one can get

When

When

Using the condition (

When

General solution for (

Therefore,

General solution for (

Therefore,

General solution for (

Derived from the principle of superposition,

So,

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is sponsored by National Natural Science Foundation of China (Grants nos. 41272258, 41372253, 41521001, and U1403282), National Basic Research Program of 973 Program (Grant no. 2010CB428802), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant no. CUG140503). The authors also would like to thank the editor and the anonymous reviewer for rendering valuable comments and suggestions for improving this paper.