# Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Brief Overview of Homotopy-Based Methods

#### 2.1. Homotopy Analysis Method

#### 2.2. Homotopy Perturbation Method

#### 2.3. Optimal Homotopy Asymptotic Method

## 3. Governing Equation and Solution Methodologies

#### 3.1. Theis Solution

#### 3.2. Homotopy-Based Solutions

#### 3.2.1. HAM-Based Solution

#### 3.2.2. HPM-Based Solution

#### 3.2.3. OHAM-Based Solution

## 4. Results and Discussion

#### 4.1. Validation of the Well Function’s Approximations

#### 4.2. Numerical Convergence and Validation of the HAM-Based Solution

^{3}s

^{−1}, $T=0.0023$ m

^{2}s

^{−1}, and $S=7.5\times {10}^{-4}$. Using these parameter values, we assessed the HAM-based solution by calculating the squared residual errors for different orders of approximations. The squared residual errors for different orders of approximation are plotted in Figure 5, where it is seen that the error decreases with the increasing order of approximation. Thus, the numerical convergence was established, and the choice of operators and parameters was validated. For a quantitative assessment, the numerical results are also reported in Table 1, along with the computational time taken by the computer to produce the corresponding order of approximations. It can be seen from the table that even though HAM is an analytical series approximation technique, it still does not involve time complexity. On the other hand, for the selected case, we compared the Theis solution (with the integral computed using the MATLAB script ‘integral’, which uses the global adaptive quadrature rule [24]) and the 10th-order HAM-based approximate solution in Figure 6. It may be noted that the same method was used for the numerical solution related to HPM and OHAM. An excellent agreement is found between the computed and observed values. Additionally, for a comparative idea, 4th-, 7th-, and 10th-order approximations were considered and compared with the Theis solution, as shown in Table 2. It can be observed from the table that the higher the order of approximation, the better the accuracy. All computations were performed using the BVPh 2.0 package developed by [25]. A flowchart containing the steps of HAM for the present problem is provided in Figure 7. The theoretical convergence analysis is provided in Appendix A.

#### 4.3. Validation of HPM-Based Solution

#### 4.4. Validation of OHAM-Based Solution

#### 4.5. Comparison between Different Approximations

^{3}s

^{−1}, $T=0.0023$ m

^{2}s

^{−1}, and $S=7.5\times {10}^{-4}$. Moreover, for each of the cases, we computed the well function numerically using ‘integral’ of MATLAB to obtain the main solution. The series Equation (35) with 40 terms, Equation (38) with 10 terms, 10th-order HAM-based solution, four-term HPM solution, and three-term OHAM solution were considered. Importantly, it may be noted that the numerical values of solutions are very small, which can make the computations ill-posed or produce numerical instabilities. To that end, logarithmic form for the error was considered. Specially, we checked the performances of the approximations by calculating the percentage error as PE (%) = 100 $\times \frac{\left(\mathrm{ln}{W}_{num}-\mathrm{ln}{W}_{apprx}\right)}{\mathrm{ln}{W}_{num}}$, where ${W}_{num}$ and ${W}_{apprx}$ are the values of $W\left(v\right)$ obtained from the Theis solution and the corresponding approximation, respectively. The percentage errors were calculated for the approximations and compared, as shown in Figure 12. It can be seen that among the series approximations, HAM- and OHAM-based approximations provide accurate approximations for the problem. On the other hand, the HPM-based solution is shown only within a small domain, as the solution provides accurate values there. Further, series approximation given by [12,13,14] are reasonably accurate within the domain. The different homotopy-based methods provide solutions that are valid within a certain range of the domain. The HAM- and OHAM-based approximations are more accurate and valid for larger domains, as they contain convergence-control parameters, which monitor the rate and radius of convergence of the series solutions. Further, the OHAM-based solution is more preferable due to its ability to provide an accurate approximation with just two–three terms of the series. Finally, it is concluded that while the homotopy-based methods do not produce as accurate solutions as do the closed-form formulae available in the literature, they are better than the series expansions and also may be improved further using different sets of base functions, linear operators, and initial approximations. It may also be noted that the proposed study differs from the existing empirical formula-based work from the viewpoint of its derivation, which starts from the governing differential equation. Therefore, the approach is flexible to use when the flow configuration is different, and the model parameters vary.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Convergence Theorems

#### Appendix A.1. Convergence Theorem of HAM-Based Solution

**Theorem**

**A1.**

**Proof.**

**Theorem**

**A2.**

**Proof.**

#### Appendix A.2. Convergence Theorem of OHAM-Based Solution

**Theorem**

**A3.**

**Proof.**

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**Figure 2.**Comparison between numerical solution and 10, 20, and 40 terms of the series Equation (35) for $W\left(v\right)$.

**Figure 3.**Comparison between numerical solution and 2, 5, and 100 terms of the series Equation (38) for $W\left(v\right)$.

**Figure 4.**Comparison between numerical solution, 40 terms of the series Equation (35), and 10 terms of the series Equation (38) for $W\left(v\right)$.

**Figure 5.**Squared residual error (${\Delta}_{m}$) versus different orders of approximations ($m$) of the HAM-based solution for the selected case.

**Figure 6.**Comparison between the Theis solution and 10th-order HAM-based solution for the selected case.

**Figure 8.**Comparison between the Theis solution and the four terms of HPM-based solution for the selected case.

**Figure 10.**Comparison between the Theis solution and the three terms of OHAM-based solution for the selected case.

**Figure 12.**Percentage errors of the approximations: (

**a**) Equation (35) with 40 terms, (

**b**) Equation (38) with 10 terms, (

**c**) Equation (39) [12], (

**d**) Equation (42) [13], (

**e**) Equation (45) [14], (

**f**) Equation (64) (10th-order HAM solution), (

**g**) Equation (78) (four-term HPM solution), and (

**h**) Equation (87) (three-term OHAM solution).

**Table 1.**Squared residual error (${\Delta}_{m}$) and computational time versus different orders of approximation ($\mathrm{m}$) for the selected case.

$\mathbf{Order}\text{}\mathbf{of}\text{}\mathbf{Approximation}\text{}\left(\mathbf{m}\right)$ | $\mathbf{Squared}\text{}\mathbf{Residual}\text{}\mathbf{Error}\text{}\left({\mathbf{\Delta}}_{\mathit{m}}\right)$ | Computational Time (s) |
---|---|---|

2 | 7.15 $\times $ 10^{−4} | 0.212 |

4 | 5.23 $\times $ 10^{−5} | 1.135 |

6 | 1.98 $\times $ 10^{−5} | 2.481 |

8 | 1.01 $\times $ 10^{−5} | 5.074 |

10 | 8.86 $\times $ 10^{−6} | 6.843 |

12 | 6.23 $\times $ 10^{−6} | 10.149 |

$\mathit{u}$ | Numerical Solution | HAM-Based Approximation | ||
---|---|---|---|---|

4th Order | 7th Order | 10th Order | ||

0.1 | 2.523 $\times $ 10^{−1} | 3.009 $\times $ 10^{−1} | 2.821 $\times $ 10^{−1} | 2.726 $\times $ 10^{−1} |

1 | 3.036 $\times $ 10^{−2} | 3.995 $\times $ 10^{−2} | 3.548 $\times $ 10^{−2} | 3.305 $\times $ 10^{−2} |

2 | 6.768 $\times $ 10^{−3} | 9.302 $\times $ 10^{−3} | 8.011 $\times $ 10^{−3} | 7.228 $\times $ 10^{−3} |

3 | 1.806 $\times $ 10^{−3} | 2.560 $\times $ 10^{−3} | 2.146 $\times $ 10^{−3} | 1.886 $\times $ 10^{−3} |

4 | 5.230 $\times $ 10^{−4} | 7.603 $\times $ 10^{−4} | 6.213 $\times $ 10^{−4} | 5.377 $\times $ 10^{−4} |

5 | 1.589 $\times $ 10^{−4} | 2.359 $\times $ 10^{−4} | 1.881 $\times $ 10^{−4} | 1.627 $\times $ 10^{−4} |

6 | 4.983 $\times $ 10^{−5} | 7.530 $\times $ 10^{−5} | 5.870 $\times $ 10^{−5} | 5.150 $\times $ 10^{−5} |

7 | 1.598 $\times $ 10^{−5} | 2.453 $\times $ 10^{−5} | 1.871 $\times $ 10^{−5} | 1.689 $\times $ 10^{−5} |

8 | 5.213 $\times $ 10^{−6} | 8.111 $\times $ 10^{−6} | 6.066 $\times $ 10^{−6} | 5.688 $\times $ 10^{−6} |

9 | 1.723 $\times $ 10^{−6} | 2.713 $\times $ 10^{−6} | 1.993 $\times $ 10^{−6} | 1.955 $\times $ 10^{−6} |

10 | 5.753 $\times $ 10^{−7} | 9.160 $\times $ 10^{−7} | 6.617 $\times $ 10^{−7} | 6.814 $\times $ 10^{−7} |

**Table 3.**Comparison between four terms of the HPM-based approximation and Theis solution for the selected case.

$\mathit{u}$ | Numerical Solution | Four Terms of the HPM-Based Approximation |
---|---|---|

0.1 | 2.523 $\times $ 10^{−1} | 3.135 $\times $ 10^{−1} |

0.3 | 1.253 $\times $ 10^{−1} | 1.533 $\times $ 10^{−1} |

0.5 | 7.747 $\times $ 10^{−2} | 9.362 $\times $ 10^{−2} |

0.7 | 5.173 $\times $ 10^{−2} | 6.246 $\times $ 10^{−2} |

0.9 | 3.601 $\times $ 10^{−2} | 4.346 $\times $ 10^{−2} |

1.1 | 2.574 $\times $ 10^{−2} | 3.097 $\times $ 10^{−2} |

1.3 | 1.875 $\times $ 10^{−2} | 2.237 $\times $ 10^{−2} |

1.5 | 1.384 $\times $ 10^{−2} | 1.563 $\times $ 10^{−2} |

1.7 | 1.033 $\times $ 10^{−2} | 8.139 $\times $ 10^{−3} |

2.0 | 6.768 $\times $ 10^{−3} | 1.298 $\times $ 10^{−2} |

**Table 4.**Comparison between three terms of the OHAM-based approximation and Theis solution for the selected case.

$\mathit{u}$ | Numerical Solution | Three Terms of OHAM-Based Approximation |
---|---|---|

0.1 | 2.523 $\times $ 10^{−1} | 2.466 $\times $ 10^{−1} |

1 | 3.036 $\times $ 10^{−2} | 3.143 $\times $ 10^{−2} |

2 | 6.768 $\times $ 10^{−3} | 6.539 $\times $ 10^{−3} |

3 | 1.806 $\times $ 10^{−3} | 2.112 $\times $ 10^{−3} |

4 | 5.230 $\times $ 10^{−4} | 8.502 $\times $ 10^{−4} |

5 | 1.589 $\times $ 10^{−4} | 3.066 $\times $ 10^{−4} |

6 | 4.983 $\times $ 10^{−5} | 1.000 $\times $ 10^{−4} |

7 | 1.598 $\times $ 10^{−5} | 3.013 $\times $ 10^{−5} |

8 | 5.213 $\times $ 10^{−6} | 8.461 $\times $ 10^{−6} |

9 | 1.723 $\times $ 10^{−6} | 2.283 $\times $ 10^{−6} |

10 | 5.753 $\times $ 10^{−7} | 5.978 $\times $ 10^{−7} |

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**MDPI and ACS Style**

Kumbhakar, M.; Singh, V.P. Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer. *Mathematics* **2023**, *11*, 1652.
https://doi.org/10.3390/math11071652

**AMA Style**

Kumbhakar M, Singh VP. Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer. *Mathematics*. 2023; 11(7):1652.
https://doi.org/10.3390/math11071652

**Chicago/Turabian Style**

Kumbhakar, Manotosh, and Vijay P. Singh. 2023. "Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer" *Mathematics* 11, no. 7: 1652.
https://doi.org/10.3390/math11071652