# Methods of Decline Curve Analysis for Shale Gas Reservoirs

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## Abstract

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## 1. Introduction

## 2. DCA Models and Applications

#### 2.1. Origins of DCA

#### 2.2. Arps Decline Model

#### 2.3. Modified Hyperbolic Decline Model

#### 2.4. Power Law Exponential Decline Model

#### 2.5. Stretched Exponential Decline Model

#### 2.6. Duong Model

#### 2.7. Logistic Growth Model

#### 2.8. Extended Exponential DCA Model

#### 2.9. Fractional Decline Curve Model

#### 2.10. Probabilistic Decline Curve Model

#### 2.11. Comparisons of DCA Models with Field Data

## 3. Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Shale samples. (

**a**) Outcrop of natural fractures in Woodford Shale, Oklahoma, USA [3]; (

**b**) High-resolution images of Montery shale, California, USA [4]. A, B, C/D are the scan images under different resolutions for the same shale core sample. A, B, and C are for the same core sample, under different resolutions, 10μm, 3 μm and 2 μm respectively. D is for another core sample under 2 μm resolution. Arrows in B and C mark pores less than 80 nm. Arrows in D mark carbonate micrite.

**Figure 4.**Comparisons of four DCA models [12]: (

**a**) Barnett shale gas well 314 results comparison; (

**b**) Eagle Ford shale gas Well 204 comparison.

**Figure 5.**Comparison results for two well groups from Barnett shale [13], where EOH stands for the end of history used in the matching, and EOP is the end of production data. In both cases 36 months of historical data were used for matching. (

**a**) Comparison results for an 81-well Denton County group; (

**b**) comparison results for a 127-well Wise County group.

**Figure 6.**Comparison results for one well from Fayetteville shale. The red dots are the original field data. (

**a**) The matching results for monthly flow rate; (

**b**) the EUR results of DCA models.

Index | Methods | Year | Decline Curve Expressions | References |
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1 | Arps Model | 1945 | $q={q}_{i}{(1+\frac{bt}{{a}_{i}})}^{-1/b}$ | [20] |

2 | Modified Hyperbolic Decline Model | 1988 | $q(t)=\{\begin{array}{c}{q}_{i}{(q+b{D}_{1}t)}^{-1/b};\left(D<{D}^{*}\right)\\ {q}_{i}{e}^{-{D}_{2}t};\left(D\ge {D}^{*}\right)\end{array}$ | [26] |

3 | Power Law Exponential Decline Model | 2008 | $q(t)={\widehat{q}}_{i}{e}^{\left[-{D}_{\infty}t-{\widehat{D}}_{i}{t}^{\widehat{n}}\right]}$ | [12,27,28,29,30,32] |

4 | Stretched Exponential Decline Model | 2009 | $q={q}_{i}{e}^{-{(\frac{t}{\tau})}^{n}}$ | [33] |

5 | Duong Model | 2011 | $q={q}_{i}{t}^{-m}{e}^{\frac{a}{1-m}({t}^{1-m}-1)}$ | [6,12,28,29,35,36] |

6 | Logistic Growth Analysis Model | 2011 | $q(t)=\frac{Kn\widehat{a}{t}^{n-1}}{{(\widehat{a}+{t}^{n})}^{2}}$ | [28,29,39] |

7 | Extended Exponential Decline Curve | 2016 | $\frac{\mathrm{ln}\frac{q}{{q}_{i}}}{t}={\beta}_{l}+{\beta}_{e}{e}^{-{t}^{n}}$ | [40] |

8 | Fractional Decline Model | 2016 | $q=m{E}_{\alpha ,1}\left(-\lambda {t}^{\alpha}\right)=m{\displaystyle {\displaystyle \sum}_{k=0}^{\infty}}\frac{{(-\lambda {t}^{\alpha})}^{k}}{\mathsf{\Gamma}\left(\alpha k+1\right)}$ | [6,35] |

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Tan, L.; Zuo, L.; Wang, B. Methods of Decline Curve Analysis for Shale Gas Reservoirs. *Energies* **2018**, *11*, 552.
https://doi.org/10.3390/en11030552

**AMA Style**

Tan L, Zuo L, Wang B. Methods of Decline Curve Analysis for Shale Gas Reservoirs. *Energies*. 2018; 11(3):552.
https://doi.org/10.3390/en11030552

**Chicago/Turabian Style**

Tan, Lei, Lihua Zuo, and Binbin Wang. 2018. "Methods of Decline Curve Analysis for Shale Gas Reservoirs" *Energies* 11, no. 3: 552.
https://doi.org/10.3390/en11030552