 Research
 Open Access
An approximate analytical solution of Richards equation with finite boundary
 Xi Chen^{1} and
 Ying Dai^{2}Email author
 Received: 14 June 2017
 Accepted: 25 October 2017
 Published: 9 November 2017
Abstract
We apply a series expansion technique to estimate the water content distribution and front position in finite boundary conditions. We derive an approximate analytical solution of the Richards equation (RE) for the horizontal infiltration problem. The solution is suitable for arbitrary hydraulic diffusivity in water infiltration. Compared with the finite element method, two examples in power law diffusivity and van Genuchten model are shown to test the accuracy of present approximation.
Keywords
 series expansion technique
 finite boundary
 approximate analytical solution
 Richards equation
 van Genuchten model
MSC
 35C10
 35K20
 76S05
1 Introduction
In these years, Heaslet and Alksne technique [4], perturbation technique [5], traveling wave method [6], and series method [7–10] are used to obtain an analytical solution of RE with semiinfinite boundaries. Fourier transformation, separation of variables [11, 12], and other approaches [13–19] are applied for some linear or linearized RE with finite boundaries (2). Nevertheless, there are few approximate analytical solutions of equations (1)(3) that can illustrate the longtime behavior for water infiltration in constant moisture content boundary conditions of equation (2), especially when the hydraulic diffusivity is an arbitrary function of θ. Our goal is deriving an approximate analytical solution of equations (1)(3).
Motivated by the methods and approaches mentioned, we focus on analyzing the water infiltration varying from a transient state to steady one and try to approximate the changes of water profile. By a series expansion technique we construct an approximate solution about space variable x and water profile to approximate the water content distribution for arbitrary diffusivity in RE. In addition, we analyze the relationship of time variable t and θ in a definite space coordinate.
2 Wetting front analysis
The solid lines in Figure 1 show the process of water infiltration varying from a transient flow to steady one. In this section, we construct an approximate analytical solution to simulate the infiltrate process.
3 Numerical simulation
In this section, two examples are shown to confirm the accuracy of the present method.
Example 1
(Powerlaw model)

\(n=1\): \(U _{1}=0.1445249313\).

\(n=2\): \(U _{1}=0.1667699993\), \(U _{2}=3.374544704\times10^{4}\).

\(n=3\): \(U _{1}=0.1571011460\), \(U _{2}=3.178898137\times10^{4}\), \(U _{3}=8.78800083\times10^{7}\).

\(n=4\): \(U _{1}=0.1620248146\), \(U _{2}=3.278527209\times10^{4}\), \(U _{3}=9.921998925\times10^{7}\), \(U_{4}=3.797520062\times10^{9}\).

\(n=5\): \(U _{1}=0.1589918636\), \(U _{2}=3.217156285\times10^{4}\), \(U _{3}=9.214216339\times10^{7}\), \(U_{4}=3.620798276\times10^{9}\), \(U_{5}=1.360893606\times10^{11}\).
Results of θ obtained from the FEM and fifthorder approximation
ϕ (mm/min ^{ 1/2 } )  Present method  FEM  Relative error (%) 

36.3899  0.5001  0.5  0.02 
20.1946  0.55  0.54266  1.353 
18.1551  0.6  0.594696  0.892 
14.8670  0.7  0.69924  0.109 
11.2396  0.8  0.8007  −0.087 
6.5490  0.9  0.900045  −0.005 
0  1  1  0 
The results of θ obtained from the present method and the FEM in \(\pmb{x_{{d}}=9\mbox{ mm}}\)
t (min)  Present method  FEM  Relative error (%) 

0.1276  0.505193  0.5027  0.4959 
0.2169  0.550959  0.562178  −1.9956 
0.2794  0.62067  0.6319  −1.7772 
0.3149  0.65244  0.6587  −0.9504 
0.3666  0.692165  0.69494  −0.3993 
0.4685  0.742965  0.743  −0.0047 
0.5337  0.76222  0.7638  −0.2069 
0.5765  0.77126  0.7737  −0.3154 
0.6353  0.780382  0.7841  −0.4742 
0.658  0.783064  0.7874  −0.5507 
0.6772  0.785114  0.7897  −0.5807 
Example 2
(van Genuchten model)

\(n=1\): \(U _{1}=3\mbox{,}266.070514\).

\(n=2\): \(U _{1}=3\mbox{,}600.557734\), \(U _{2}=2.796118274\times10^{9}\).
Results of θ obtained from the FEM and secondorder approximation
ϕ (mm/s ^{ 1/2 } )  2ndorder approximation  FEM  Relative error (%) 

0.9316  0.251  0.2529  −0.757 
0.8717  0.255  0.2594  −1.71 
0.8397  0.26  0.2656  −2.097 
0.7651  0.28  0.2853  −1.861 
0.7012  0.3  0.3033  −1.076 
0.483  0.35  0.3499  0.025 
0.3418  0.37  0.3697  0.09 
0  0.4  0.4  0 
We observe that the present approximations are closer to the FEM as order varies from 1 to 2 in Figure 6, and the maximal value of the relative error is −2.097% in 0.8397 mm/min^{1/2} for secondorder approximation in Table 3.
The results of θ obtained from the present method and the FEM in \(\pmb{x_{{d}}=1.5\mbox{ mm}}\)
t (s)  Present method  FEM  Relative error (%) 

3.58233  0.27141  0.2771  −2.05341 
4.211  0.29188  0.2929  −0.34824 
4.97556  0.30889  0.3075  0.452033 
5.2274  0.31535  0.314  0.429936 
6.026  0.32109  0.3208  0.090399 
7.036  0.32607  0.32878  −0.82426 
The present solution shown by dotted line approximates well with the FEM shown by solid lines in time simulation in Figure 9, and the maximum relative error is −2.0534% in Table 4.
4 Conclusion
In this paper, we analyzed the changes of wetting front position and derived an approximate analytical solution of RE with finite boundaries (2). The solid lines in Figure 1 can be approximated by the solution equation (17), which is a series solution. According to equation (17), the series solution in fact is an approximation for semiinfinite boundary with different initial values, which means that the process of water infiltration varying from transient flow to steady one in Figure 1 can be simulated by the solution of semiinfinite problem with a variable initial value. The presented examples for the power law and van Genuchten model demonstrate the accuracy of the present solution by comparing the present results with the results obtained by the FEM.
Declarations
Funding
The research has been supported by the Fundamental Research Funds for the Central Universities.
Authors’ contributions
Both authors contributed equally in this article. They read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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