# Use of the Kalman Filter for the Interpretation of Aquifer Tests Including Model and Measurement Errors

## Abstract

**:**

## 1. Introduction

^{1}T

^{−1}), the hydraulic transmissivity ($T$, L

^{2}T

^{−1}), particularly for horizontal flow, the storage coefficient ($S$, L

^{3}L

^{−3}), the specific storage (Ss, L

^{−1}), and the hydraulic diffusivity (D, L

^{2}T

^{−1}), among others [2].

## 2. Materials and Methods

#### 2.1. Cooper–Jacob Solution for the Interpretation of Aquifer Tests

- (a)
- The flow in the aquifer follows Darcy’s law.
- (b)
- The aquifer is homogeneous, isotropic and of infinite areal extent.
- (c)
- The piezometric surface before pumping is horizontal.
- (d)
- Water is instantaneously removed from storage upon a decline in head.
- (e)
- The pumping well is fully penetrating, and the aquifer is of uniform thickness with horizontal bottom, and therefore flow is radial-horizontal everywhere within the aquifer to the well.
- (f)
- The discharge rate from the pumping well is constant.
- (g)
- The diameter of the pumping well is infinitesimally small, meaning that storage within it can be neglected.
- (h)
- Well losses are neglected.

- $s=\mathrm{drawdown}\mathrm{in}\mathrm{the}\mathrm{observation}\mathrm{well}\mathrm{located}\mathrm{at}\mathrm{a}\mathrm{distance}\mathrm{r}\mathrm{from}\mathrm{the}\mathrm{pumping}\mathrm{well}[\mathrm{L}].$
- $Q=\mathrm{constant}\mathrm{flow}\mathrm{rate}[{\mathrm{L}}^{3}{\mathrm{T}}^{-1}].$
- $T=\mathrm{aquifer}\mathrm{transmissivity}[{\mathrm{L}}^{2}{\mathrm{T}}^{-1}].$
- $W(u)=\mathrm{well}\mathrm{function}.$
- $u=\mathrm{auxiliar}\mathrm{function}.$$$u=\frac{{r}^{2}S}{4Tt}$$
- $S=\mathrm{storage}\mathrm{coefficient}[{\mathrm{L}}^{3}{\mathrm{L}}^{-3}].$
- $t=\mathrm{ellapsed}\mathrm{time}\mathrm{from}\mathrm{the}\mathrm{start}\mathrm{of}\mathrm{pumping}[\mathrm{T}].$
- $r=\mathrm{distance}\mathrm{between}\mathrm{the}\mathrm{pumping}\mathrm{and}\mathrm{the}\mathrm{observation}\mathrm{well}[\mathrm{L}].$

#### 2.2. Development of Cooper–Jacob Solution for the Kalman Filter

#### 2.3. Adaptation to the Kalman Filter

#### 2.4. Optimization

#### 2.5. Case Study

- (a)
- The aquifer test “Oude Korendijk” presented in [34].

- (b)
- The aquifer test of Todd and Mays corresponding to exercise 4.4.2 of [4].

## 3. Results

- -
- Initial values: $T=100{\mathrm{m}}^{2}{\mathrm{d}}^{-1}$ and $S=0.00001{\mathrm{m}}^{3}{\mathrm{m}}^{-3}$.
- -
- The prior estimate errors covariance matrix $[\begin{array}{cc}0.25& 5\\ 5& 100\end{array}]$
- -
- The model errors covariance matrix $[\begin{array}{cc}0.0001& 0.001\\ 0.001& 0.1\end{array}]$
- -
- The variance of the drawdown measurement error (VDME) = 0.01 ${\mathrm{m}}^{2}$.

#### 3.1. The Aquifer Test “Oude Korendijk”

#### 3.2. The Aquifer Test of Todd and Mays

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison between the measured drawdowns, simulated with the Kalman Filter (VDME = 0.01 ${\mathrm{m}}^{2}$) and the Cooper–Jacob solution (the aquifer test “Oude Korendijk”).

**Figure 3.**Comparison between the measured drawdowns, simulated with the Kalman filter (VDME = 1.00 ${\mathrm{m}}^{2}$), and the Cooper–Jacob solution (the aquifer test “Oude Korendijk”).

**Figure 4.**Comparison between the measured drawdowns and the Cooper–Jacob solution using $T=420{\mathrm{m}}^{2}{\mathrm{d}}^{-1},$ $S=0.00016{\mathrm{m}}^{3}{\mathrm{m}}^{-3}$ (the aquifer test “Oude Korendijk”).

**Figure 5.**Comparison between the measured drawdowns, simulated with the Kalman filter (VDME = 0.01 ${\mathrm{m}}^{2}$) and the Cooper–Jacob solution (the aquifer test of Todd and Mays).

**Table 1.**Aquifer test data from the aquifer test “Oude Korendijk” [34].

Time after Pumping Started (min) | Drawdown (m) | Time after Pumping Started (min) | Drawdown (m) |
---|---|---|---|

0.10 | 0.040 | 18.00 | 0.680 |

0.25 | 0.080 | 27.00 | 0.742 |

0.50 | 0.130 | 33.00 | 0.753 |

0.70 | 0.180 | 41.00 | 0.779 |

1.00 | 0.230 | 48.00 | 0.793 |

1.40 | 0.280 | 59.00 | 0.819 |

1.90 | 0.330 | 80.00 | 0.855 |

2.33 | 0.360 | 95.00 | 0.873 |

2.80 | 0.390 | 139.00 | 0.915 |

3.36 | 0.420 | 181.00 | 0.935 |

4.00 | 0.450 | 245.00 | 0.966 |

5.35 | 0.500 | 300.00 | 0.990 |

6.80 | 0.540 | 360.00 | 1.007 |

8.30 | 0.570 | 480.00 | 1.050 |

8.70 | 0.580 | 600.00 | 1.053 |

10.00 | 0.600 | 728.00 | 1.072 |

13.10 | 0.640 | 830.00 | 1.088 |

**Table 2.**Aquifer test data from the aquifer test of Todd and Mays [4].

Time after Pumping Started (min) | Drawdown (m) | Time after Pumping Started (min) | Drawdown (m) |
---|---|---|---|

1 | 0.2 | 24 | 0.72 |

1.5 | 0.27 | 30 | 0.76 |

2 | 0.3 | 40 | 0.81 |

2.5 | 0.34 | 50 | 0.85 |

3 | 0.37 | 60 | 0.9 |

4 | 0.41 | 80 | 0.93 |

5 | 0.45 | 100 | 0.96 |

6 | 0.48 | 120 | 1 |

8 | 0.53 | 150 | 1.04 |

10 | 0.57 | 180 | 1.07 |

12 | 0.6 | 210 | 1.1 |

14 | 0.63 | 240 | 1.12 |

18 | 0.67 |

**Table 3.**Hydraulic parameters determined for data of the aquifer test “Oude Korendijk” using different interpretation procedures.

Parameter | Theis Procedure * | Cooper-Jacob Procedure * | Şen Procedure * | AquiferWin (Theis Solution) | KF-Based Proposed Procedure (VDME = 0.01 ${m}^{2}$) | KF-Based Proposed Procedure (VDME = 0.01 ${m}^{2}$) |
---|---|---|---|---|---|---|

$T({\mathrm{m}}^{2}{\mathrm{d}}^{-1})$ | 342–418 | 375–401 | 342–420 | 480.67 | 510.59 | 505.76 |

$S({\mathrm{m}}^{3}{\mathrm{m}}^{-3})$ | 0.00017 | 0.00022–0.00017 | 0.00016–0.0002 | 0.000112 | 0.000089 | 0.000088 |

**Table 4.**Hydraulic parameters determined for data of the aquifer test of Todd using different interpretation procedures.

Parameter | Theis Procedure * | Cooper–Jacob Procedure * | AquiferWin (Theis Solution) | KF-Based Proposed Procedure (VDME = 0.01 ${m}^{2}$) |
---|---|---|---|---|

$T({\mathrm{m}}^{2}{\mathrm{d}}^{-1})$ | 1110 | 1144 | 1138.17 | 1180.43 |

$S({\mathrm{m}}^{3}{\mathrm{m}}^{-3})$ | 0.000206 | 0.000193 | 0.00019 | 0.00017 |

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Júnez-Ferreira, H.E.
Use of the Kalman Filter for the Interpretation of Aquifer Tests Including Model and Measurement Errors. *Water* **2022**, *14*, 522.
https://doi.org/10.3390/w14040522

**AMA Style**

Júnez-Ferreira HE.
Use of the Kalman Filter for the Interpretation of Aquifer Tests Including Model and Measurement Errors. *Water*. 2022; 14(4):522.
https://doi.org/10.3390/w14040522

**Chicago/Turabian Style**

Júnez-Ferreira, Hugo Enrique.
2022. "Use of the Kalman Filter for the Interpretation of Aquifer Tests Including Model and Measurement Errors" *Water* 14, no. 4: 522.
https://doi.org/10.3390/w14040522