# Graphic Template Establishment and Productivity Evaluation Model of Post-Fracturing Based on the Fluctuation Pattern of G-Function Curve

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{L}

_{1}. In the process of gradual pressure reduction in the fracture afterwards, the filtration loss of the fracturing fluid exists only in the main fracture due to the gradual closure of the natural fracture, and the filtration loss coefficient at this time is defined as C

_{L}

_{2}. Foreign scholars Mayer and Jacot [16] proposed the equation for the filtration loss coefficient of this process expressed as:

_{f}

_{0}is the pressure threshold when the natural fracture is closed (opened).

_{c}is the final closure pressure of all fractures and P

_{ci}is the formation pressure after all natural fractures are closed (only induced fractures are involved in the filtration).

## 3. Plate and Model Building

#### 3.1. Establishment of the G-Functional Graphic Template

#### 3.2. G-Function Yield Evaluation Model

_{ΔPt}of the inflection point at the first peak of the superposition derivative of the G-function, the slope k of the inflection point at the first peak of the superposition derivative of the G-function, the fluctuation number D of the G-function and the area S below the G-function.

_{ΔPt}and G-function was classified as the A

_{1}index to evaluate the flow of fluid in the main fracture. The wave number D of the G-function and the area S below the derivative curve of the G-function superposition were classified as A

_{2}indexes to evaluate the flow of fluid in branch fractures (natural fractures). The evaluation index Y of the G-function image was obtained by calculating the evaluation index extracted by an image recognition algorithm by formulating the evaluation standard formula.

- ${A}_{1}={V}_{1}\times 0.3+k\times 0.7$
- ${A}_{2}=D\times 0.5+S\times 0.5$
- ${V}_{{P}_{t}}=\Delta P/\Delta NolteG$
- $k=\Delta G\frac{dP}{dG}/\Delta NolteG$

## 4. Discussion

^{4}m

^{3}/d) and low initial production (≤10 × 10

^{4}m

^{3}/d) were verified graphically. Furthermore, the G-function curves of the high and low production sections of wells with production profile data were validated.

#### 4.1. G-Function Graphic Template Validation

^{4}m

^{3}/d) among the 24 wells are A2, B1, B2, B3, B4 and C1. The full-section G-function curves of these six wells were plotted separately and the G-function plates were used as the standard for fast matching as well as verification. Here, only the G-function curves for the full well section of well A2 are listed because of space limitations.

^{3}/day and a daily production of 32,145 m

^{3}, which is the best among the high production wells in this block.

^{3}/day were plotted with the full-section G-function as well as template matching according to the same method, and the results are shown in Table 3.

#### 4.2. Validation of the G-Function Yield Evaluation Model

_{ΔPt}, k, D and S. After that, we used the extracted parameters to substitute into Equation (11) for the calculation. Note that before calculating the capacity evaluation value Y, we needed to score the extracted parameters using the parameter scoring table in Table 1. The settlement results are shown in Table 5 and Table 6.

_{ΔPt}and k, which are used to evaluate the fluid flow in the main seam, were 0.63 and 0.8 respectively for the high-yield section. As a comparison, the average values of V

_{ΔPt}and k in the low-yielding section were 0.43 and 0.5, respectively, which were only 68.25% and 62.5% of those in the high-yielding section. In relation to the characteristics of the G-function curves of categories I and II in the G-function plot proposed by the author (fast rising and large upward convex area in the initial stage of the curve), they coincide with V

_{ΔPt}and k in the calculated results, which shows that the established G-function plot has some correlation with the capacity model.

## 5. Conclusions

- (1)
- Based on the theoretical basis of the G-function, the G-function graphic template and the corresponding fracture pattern were proposed by analyzing the corresponding G-function curve patterns of different blocks in combination with production, and verified according to high- and low-production wells with high accuracy, which can effectively provide feedback on the field fracturing effect and guide the subsequent fracturing construction.
- (2)
- Based on the morphological characteristics of the G-function curve, a G-function production evaluation mathematical model based on four indicators is proposed and validated using high- and low-producing sections of shale gas wells with production profile data. From the validation of the production capacity model, it can be seen that the average value of the production evaluation Y of the high-producing section is higher than that of the low producing section by 75.9%, which is much larger than that of the low-producing section, which shows the accuracy and adaptability of this model.
- (3)
- The work intensity at the fracturing site is high and the construction time is long, so it is difficult to have time to evaluate the fracturing effect after completing a section of fracturing construction and to continue the fracturing operation in the next section. Therefore, the author proposes a G-function plate for evaluating the post-fracturing effect, according to which the G-function curve can be quickly matched with the plate after finishing the drawing of the G-function curve to judge the fracture pattern formed after fracturing the section, which is convenient for guiding the efficient construction of fracturing work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The main morphological chart of G-function. (

**a**)Type I, (

**b**) type II, (

**c**) type III, (

**d**) type IV.

**Figure 2.**Fracture morphology corresponding to different types of G-function ((

**a**–

**d**) correspond to the crack patterns corresponding to the first, second, third and fourth types of G-functions respectively).

**Figure 4.**Example of calculation process. (

**a**) Area Map, (

**b**) grayscale, (

**c**) binarization, (

**d**) Canny edge detection and Hoffman recognition, (

**e**) angle recognition and slope calculation.

**Figure 5.**G-function curve of A2 well section. (

**1**–

**27**are the G-function curves for all segments of shale gas well A2).

**Figure 6.**Example of Class 4 G-function for A2 well. (

**a**) section 2 fracturing, (

**b**) section 4 fracturing, (

**c**) section 23 fracturing, (

**d**) section 25 fracturing.

Evaluating Indicator | Interval Score | ||||
---|---|---|---|---|---|

0 | 0.25 | 0.5 | 0.75 | 1 | |

k | <15 | 15 ≤ k < 30 | 30 ≤ k < 45 | 45 ≤ k < 60 | ≥60 |

V_{ΔPt} | <10 | 10 ≤ V_{ΔPt} < 20 | 20 ≤ V_{ΔPt} < 30 | 30 ≤ V_{ΔPt} < 40 | ≥40 |

D | <4 | 4 ≤ D < 6 | 6 ≤ D < 8 | 8 ≤ D < 10 | ≥10 |

S | <0.2 | 0.2 ≤ S < 0.3 | 0.3 ≤ S < 0.4 | 0.4 ≤ S < 0.5 | ≥0.5 |

Block | Serial Number | Well Number | Young’s Modulus (GPa) | Poisson Ratio | Testing Production (10^{4} m^{3}) | Daily Gas Production (m^{3}) | Accumulated Production (10^{4} m^{3}) |
---|---|---|---|---|---|---|---|

I | 1 | A1 | 37.4 | 0.23 | 14.36 | 10,862 | 2716.9 |

2 | A2 | 38.38 | 0.2 | 32.8 | 32,145 | 1237.78 | |

3 | A3 | 42.1 | 0.21 | 12.78 | 23,644 | 2684.84 | |

II | 4 | B1 | 38.8 | 0.225 | 34.28 | 34,656 | 1309.79 |

5 | B2 | 51.66 | 0.21 | 31.7 | 56,705 | 2047.5 | |

6 | B3 | 38.8 | 0.2 | 22.7 | 22,114 | 544.18 | |

7 | B4 | 42.6 | 0.21 | 20.9 | 22,266 | 779.54 | |

III | 8 | C1 | 35.5 | 0.2 | 19.6 | 16,590 | 583.53 |

9 | C2 | 44 | 0.195 | 9.01 | 18,868 | 668.11 | |

10 | C3 | 44 | 0.17 | 7.9 | 17,318 | 705.07 |

Category | Well Number | I | II | III | IV | Proportion of Class I and II % |
---|---|---|---|---|---|---|

Large-productivity Well | A2 | 12 | 9 | 3 | 3 | 77.78 |

B1 | 9 | 7 | 5 | 2 | 80 | |

B2 | 9 | 7 | 4 | 3 | 69.6 | |

B3 | 8 | 11 | 2 | 1 | 76.2 | |

B4 | 7 | 6 | 1 | 1 | 87.7 | |

Stripper Well | C1 | 5 | 11 | 4 | 1 | 76.2 |

C2 | 3 | 5 | 15 | 0 | 34.8 | |

C3 | 0 | 5 | 8 | 10 | 23.8 |

Category | Well Number | I | II | III | IV | Proportion of Class I and II % |
---|---|---|---|---|---|---|

High-yield section | D1 | 3 | 3 | 0 | 0 | 100 |

D2 | 3 | 4 | 0 | 0 | 100 | |

D3 | 2 | 1 | 0 | 0 | 100 | |

D4 | 3 | 2 | 0 | 0 | 100 | |

D5 | 2 | 0 | 1 | 1 | 80 | |

Low-yield section | D1 | 1 | 0 | 1 | 1 | 33.3 |

D2 | 1 | 0 | 1 | 2 | 25 | |

D3 | 0 | 1 | 1 | 1 | 33.3 | |

D4 | 1 | 0 | 1 | 1 | 33.3 | |

D5 | 0 | 0 | 0 | 3 | 0 |

Well Number | Stage Number | Pressure Drop Rate | Initial Slope | Number of Fluctuations | Area under the Curve | V_{ΔPt} Score | k Score | D Score | S Score | Evaluation Value Y Score |
---|---|---|---|---|---|---|---|---|---|---|

D1 | 9 | 33.3 | 36 | 4 | 0.354 | 0.75 | 0.5 | 0.25 | 0.75 | 0.5525 |

10 | 21.41 | 46.2 | 6 | 0.397 | 0.5 | 0.75 | 0.5 | 0.75 | 0.66 | |

19 | 17.05 | 46.27 | 7 | 0.284 | 0.25 | 0.75 | 0.5 | 0.5 | 0.57 | |

20 | 21.36 | 30.69 | 6 | 0.198 | 0.5 | 0.5 | 0.5 | 0.25 | 0.4625 | |

22 | 39.12 | 117.53 | 11 | 0.300 | 0.75 | 1 | 1 | 0.5 | 0.8725 | |

23 | 30.6 | 93.64 | 8 | 0.086 | 0.75 | 1 | 0.75 | 0 | 0.76 | |

D2 | 3 | 18.05 | 30.37 | 7 | 0.226 | 0.25 | 0.5 | 0.5 | 0.5 | 0.4475 |

4 | 36.39 | 91.74 | 6 | 0.396 | 0.75 | 1 | 0.5 | 0.75 | 0.835 | |

5 | 23.34 | 36.32 | 2 | 0.435 | 0.5 | 0.5 | 0 | 0.75 | 0.4625 | |

6 | 27.18 | 49.01 | 4 | 0.326 | 0.5 | 0.75 | 0.25 | 0.5 | 0.585 | |

8 | 11.64 | 20 | 6 | 0.253 | 0.25 | 0.25 | 0.5 | 0.5 | 0.325 | |

10 | 30.51 | 45.89 | 6 | 0.392 | 0.75 | 0.75 | 0.5 | 0.75 | 0.7125 | |

11 | 27.82 | 31.95 | 4 | 0.362 | 0.5 | 0.5 | 0.25 | 0.75 | 0.5 | |

D3 | 13 | 86.73 | 404.03 | 10 | 0.169 | 1 | 1 | 1 | 0.25 | 0.8875 |

14 | 26.39 | 62.19 | 6 | 0.336 | 0.5 | 1 | 0.5 | 0.5 | 0.745 | |

15 | 69.09 | 295.58 | 6 | 0.110 | 1 | 1 | 0.5 | 0 | 0.775 | |

D4 | 3 | 16.88 | 31.5 | 5 | 0.338 | 0.25 | 0.5 | 0.25 | 0.5 | 0.41 |

7 | 58.07 | 161.71 | 9 | 0.670 | 1 | 1 | 0.75 | 1 | 0.9625 | |

9 | 80.67 | 137.82 | 7 | 0.203 | 1 | 1 | 0.5 | 0.5 | 0.85 | |

10 | 74.99 | 95.04 | 6 | 0.121 | 1 | 1 | 0.5 | 0 | 0.775 | |

11 | 35.65 | 65.71 | 3 | 0.625 | 0.75 | 1 | 0 | 1 | 0.7975 | |

D5 | 15 | 23.57 | 52.9 | 8 | 0.140 | 0.5 | 0.75 | 0.75 | 0 | 0.585 |

21 | 35.69 | 83.18 | 6 | 0.509 | 0.75 | 1 | 0.5 | 1 | 0.8725 | |

10 | 17.25 | 36.85 | 3 | 0.167 | 0.25 | 0.5 | 0 | 0.25 | 0.335 |

Well Number | Stage Number | Pressure Drop Rate | Initial Slope | Number of Fluctuations | Area under the Curve | ΔPt Score | k Score | D Score | S Score | Evaluation Value Y Score |
---|---|---|---|---|---|---|---|---|---|---|

D1 | 12 | 23.42 | 39.71 | 4 | 0.532 | 0.5 | 0.5 | 0.25 | 1 | 0.388 |

18 | 11.72 | 23.8 | 5 | 0.234 | 0.25 | 0.25 | 0.25 | 0.5 | 0.213 | |

21 | 23.39 | 78.59 | 6 | 0.217 | 0.5 | 1 | 0.5 | 0.5 | 0.67 | |

D2 | 25 | 8.71 | 15.25 | 2 | 0.176 | 0 | 0.25 | 0 | 0.25 | 0.123 |

26 | 32.8 | 78.28 | 7 | 0.365 | 0.75 | 1 | 0.5 | 0.75 | 0.723 | |

27 | 36 | 58.87 | 4 | 0.530 | 0.75 | 0.75 | 0.25 | 1 | 0.563 | |

D3 | 1 | 33.62 | 38.91 | 5 | 0.944 | 0.75 | 0.5 | 0.25 | 1 | 0.44 |

2 | 23.03 | 91.28 | 9 | 0.202 | 0.5 | 1 | 0.75 | 0.5 | 0.708 | |

3 | 31.16 | 140.26 | 7 | 0.352 | 0.75 | 1 | 0.5 | 0.75 | 0.723 | |

D4 | 1 | 30 | 30.7 | 4 | 0.184 | 0.75 | 0.5 | 0.25 | 0.25 | 0.44 |

2 | 12.34 | 19.76 | 4 | 0.279 | 0.25 | 0.25 | 0.25 | 0.5 | 0.213 | |

5 | 10.28 | 15.15 | 4 | 0.165 | 0.25 | 0.25 | 0.25 | 0.25 | 0.213 | |

D5 | 4 | 6.35 | 12.27 | 4 | 0.125 | 0 | 0 | 0.25 | 0 | 0.038 |

5 | 11.28 | 18.21 | 6 | 0.164 | 0.25 | 0.25 | 0.5 | 0.25 | 0.25 | |

6 | 12.33 | 14.25 | 5 | 0.159 | 0.25 | 0 | 0.25 | 0.25 | 0.09 |

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## Share and Cite

**MDPI and ACS Style**

Li, S.; Liu, D.; Du, C.; Ma, P.; Li, M.; Gao, H.
Graphic Template Establishment and Productivity Evaluation Model of Post-Fracturing Based on the Fluctuation Pattern of G-Function Curve. *Processes* **2023**, *11*, 1657.
https://doi.org/10.3390/pr11061657

**AMA Style**

Li S, Liu D, Du C, Ma P, Li M, Gao H.
Graphic Template Establishment and Productivity Evaluation Model of Post-Fracturing Based on the Fluctuation Pattern of G-Function Curve. *Processes*. 2023; 11(6):1657.
https://doi.org/10.3390/pr11061657

**Chicago/Turabian Style**

Li, Sichen, Dehua Liu, Chengsheng Du, Pan Ma, Minxuan Li, and Hailiang Gao.
2023. "Graphic Template Establishment and Productivity Evaluation Model of Post-Fracturing Based on the Fluctuation Pattern of G-Function Curve" *Processes* 11, no. 6: 1657.
https://doi.org/10.3390/pr11061657