# Information Entropy Evolution for Groundwater Flow System: A Case Study of Artificial Recharge in Shijiazhuang City, China

^{1}

^{2}

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## Abstract

**:**

_{01}= 0.749, the grey correlation grade between groundwater flow system information entropies and groundwater withdrawal series is γ

_{02}= 0.814, as the groundwater withdrawal is the main driving force of groundwater flow system entropy variation; based on the numerical simulation results, information entropy increased with artificial recharge, and a smaller recharge water volume would enhance the information entropy drastically, but then doubled water would not increase the information correspondingly, which could be useful to assess the health state of groundwater flow systems.

## 1. Introduction

## 2. Study Area

^{2}and the depth of the groundwater level to the ground surface at the center of the cone of depression was 52.40 m in 2006. Groundwater overdrafting has greatly impacted the local groundwater flow system [19,20].

## 3. Methodology

#### 3.1. Field Entropy Methodology

- (1)
- Calculate the areas between each contour line and the highest groundwater level contour line;
- (2)
- Calculate the elevation difference between each groundwater contour line and the lowest groundwater level contour line;
- (3)
- Assign the total area as A, the difference between the highest and the lowest groundwater level contour line as B, x = a/A, y = b/B as the two coordinates respectively, the Strahler integral values and information entropies could be acquired according to a series of (a, b), which can draw the Strahler integral curve.

#### 3.2. Grey Correlation Analysis

_{0}

_{i}(k) is the sequence of deviation of the reference sequence ${x}_{0}^{*}(k)$ from the sequence ${x}_{i}^{*}(k)$ for comparison:

## 4. Results and Discussion

#### 4.1. Entropy Variation during 1960–2005

_{i}S represents the system entropy production during irreversible processes, and it always increases the system entropy is positive; d

_{e}S is the system entropy flow provided by the environment, when the material and energy exchange between systems and their environment is conducive to the system ordering, d

_{e}S is negative, so when the total entropy decreases, or d

_{e}S is positive, the total entropy increases as a result.

_{e}S, which is input from the environment, offset the entropy changes within the groundwater flow system d

_{i}S, the groundwater flow system information entropy change dS = 0, and information entropy values remained in steady state. After the 1970s, due to the rapid socio-economic development with the resulting rapid increase in demand for water resources, groundwater excavation has greatly changed the regional groundwater flow system structures and function. Although the increased hydraulic gradient induces groundwater recharge, groundwater flow system information entropy d

_{i}S increases as a result, but groundwater excavation with human activity makes the information entropy output d

_{e}S larger and the d

_{i}S, groundwater flow system information entropy declines continually; when the d

_{i}S equals the d

_{e}S, which is inspired by increasing excavation, the groundwater flow system reaches a balanced state again, and a dissipative structure forms. Dissipative structure theory and information entropy theory could be applied in the study of groundwater flow system information evolution.

- (1)
- Between 1960 and 1965, groundwater excavation in Shijiazhuang City is not as high as today’s and abundant precipitation made the regional groundwater flow system be in a balanced state between groundwater recharge and discharge. There was little fluctuation of groundwater level, close to the natural groundwater flow state, and the system is steady, dS ≈ 0, and the groundwater flow system information entropies vary around −0.13;
- (2)
- Between 1965 and 1980, with the rapid social and economic development, groundwater excavation increased year by year, combined with decreased precipitation, causing an increasing imbalance between groundwater recharge and discharge, a regional cone of depression had formed, expanded and deepened rapidly, and these factors broke the previous steady-state, changing the original groundwater flow state greatly; the entropy change d
_{e}S increased over d_{i}S, and since dS < 0, the groundwater flow system information entropies decreased to −0.30; - (3)
- Between 1980 and 1995, concerning with the environmental geological problems inspired by groundwater excavation, a planned groundwater excavation at Shijiazhuang City was put into effect, and this effectively reduced the rates of groundwater level decline, slowing down the extended speed of the cone of depression, and the local flooding effectively recharged the groundwater in 1988, alleviating the long-term negative balance state; the groundwater flow system information entropy change d
_{e}S decreased less than d_{i}S, and since dS > 0, the groundwater flow system information entropies increased to −0.20; - (4)
- Between 1995 and 2005, although the groundwater excavation varied to a small degree, the successive years of drought, decreasing the groundwater recharge, and the reinforcement project of the Huangbizhuang secondary dam reducing the effective recharge of groundwater further, made the groundwater flow system information entropy change d
_{e}S increase over d_{i}S again, and the groundwater flow systems information entropy was dramatically reduced to −0.35.

_{i}S and d

_{e}S. However, excavation and mismanagement of the groundwater resource, and groundwater flow system information entropies show a decreasing trend, and short-term fluctuations of groundwater recharge or discharge will result in partial entropy fluctuations.

#### 4.2. Factor Analysis for Entropy Evolution

_{01}= 0.749, and the grey correlation grade between groundwater flow system information entropies and groundwater withdrawal is γ

_{02}= 0.814, as the groundwater excavation is the main driving force of groundwater flow system information entropy variation.

#### 4.3. Entropy Evolution to Artificial Recharge

^{4}m

^{3}per year and 8.72 × 10

^{4}m

^{3}per year, respectively, according to the extreme weather conditions in the Han River Basin (transferred water source area). Three scenarios with different weather conditions are proposed as follows (which are also discussed in detail in [20]):

- Scenario 1: there is no artificial recharge with the present groundwater abstraction status. The groundwater level in 2020 could be predicted by the established groundwater simulation model as shown in Figure 3A.
- Scenario 2: under scarcity conditions in both the Huangbizhuang Groundwater Reservoir (HGR) and Hanjiang Reservoir (HR), the quantity that could be used to recharge is 4.64 × 10
^{4}m^{3}/a, and the artificial recharge lasts for seven years, from 2013 to 2020, with the local groundwater level enhanced as shown in Figure 3B. - Scenario 3: under abundance conditions in both HGR and HR, and the quantity that could be used to recharge is 8.72 × 10
^{4}m^{3}/a, and the artificial recharge lasts for seven years, from 2013 to 2020, with the local groundwater level enhanced as shown in Figure 3C.

## 5. Conclusions

- (1)
- Groundwater flow systems have typical characteristics of dissipative structures, and their evolution can be described by information entropy, entropy input from the environment and the internal system entropy variation determines the direction of groundwater flow system evolution;
- (2)
- Groundwater system entropies generally showed a decreasing trend in Shijiazhuang City, where the entropy variation of the groundwater flow systems can be divided into four stages: steady entropy period (1960–1965), decreasing entropy period (1965–1980), increasing entropy period (1980–1995) and second entropy decreasing period (1995–2005);
- (3)
- The correlation grade between groundwater flow system information entropies and precipitation is γ
_{01}= 0.749, the correlation grade between groundwater flow system information entropies and groundwater withdrawal is γ_{02}= 0.814, as groundwater excavation is the main driving force of groundwater system entropy variation; - (4)
- Based on the numerical simulation results, information entropy increased with artificial recharge, and a smaller recharge water volume would enhance the information entropy drastically, but then doubling the water would not increase the information correspondingly, which could be useful to assess the health state of groundwater flow systems.
- (5)
- This study attempts to connect the information entropy theory and groundwater flow system and perform a preliminary exploration to explain the meaning of groundwater flow system entropy. It should be firmly believed that groundwater flow systems can be described with information entropy language, which should be useful for groundwater resource evaluation and management.

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Groundwater level contour maps. (

**A**–

**C**) are groundwater level contour map in scenarios 1, 2 and 3, respectively.

**Table 1.**Information entropy of groundwater system and average precipitation, average groundwater abstraction between 1960 and 2005.

Year | Precipitation [mm] | Groundwater Excavation [10^{8} m^{3}] | Entropy Variation [ΔH] |
---|---|---|---|

1960–1965 | 677.12 | 0.6208 | −0.0004 |

1965–1970 | 437.52 | 0.7269 | −0.00328 |

1970–1975 | 475.38 | 1.5872 | −0.01156 |

1975–1980 | 519.56 | 2.0378 | −0.01884 |

1980–1985 | 415.64 | 3.3076 | 0.00768 |

1985–1990 | 486.78 | 3.1351 | 0.01198 |

1990–1995 | 479.72 | 6.6624 | −0.00314 |

1995–2000 | 520.7 | 6.072 | −0.01 |

2000–2005 | 469.74 | 4.9622 | −0.0207 |

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**MDPI and ACS Style**

Xu, W.; Du, S.
Information Entropy Evolution for Groundwater Flow System: A Case Study of Artificial Recharge in Shijiazhuang City, China. *Entropy* **2014**, *16*, 4408-4419.
https://doi.org/10.3390/e16084408

**AMA Style**

Xu W, Du S.
Information Entropy Evolution for Groundwater Flow System: A Case Study of Artificial Recharge in Shijiazhuang City, China. *Entropy*. 2014; 16(8):4408-4419.
https://doi.org/10.3390/e16084408

**Chicago/Turabian Style**

Xu, Wei, and Shanghai Du.
2014. "Information Entropy Evolution for Groundwater Flow System: A Case Study of Artificial Recharge in Shijiazhuang City, China" *Entropy* 16, no. 8: 4408-4419.
https://doi.org/10.3390/e16084408