# Optimization Algorithm of Effective Stress Coefficient for Permeability

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{eff}is the effective stress (MPa), σ is the overlying stress or confining pressure (MPa), and P is the pore fluid pressure (MPa).

_{k}to distinguish it from the Biot coefficient).

_{k}is ESCP, which reflects the relative magnitude of the influence of pore fluid pressure and confining pressure on permeability.

## 2. Calculation Methods for the ESCP

#### 2.1. Cross-Plotting Method

#### 2.2. Response Surface Method

_{i}coefficients are determined from the fit.

_{i}coefficients are determined from the fit. Equation (6) can be derived by expanding Equation (7).

_{k}), Equations (6) and (7) can be written as follows:

#### 2.3. 3D Surface Fitting Method

## 3. Results

## 4. Discussion

#### 4.1. Comparison of the 3D Surface Fitting Method and Cross-Plotting Method

#### 4.1.1. Comparison of Three Fitting Functions

#### 4.1.2. Correlation Coefficient and RMSE

^{2}) and RMSE are listed in Table 3.

#### 4.1.3. Residual Error

#### 4.2. Comparison of the 3D Surface Fitting Method and Response Surface Method

#### 4.2.1. Correlation Coefficient and RMSE

#### 4.2.2. Residual Error

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Rock Sample | Model | R^{2} | RMSE | Calculation Results of 3D Surface Fitting Method | Calculation Results of Cross-Plotting Method |
---|---|---|---|---|---|

Sample 1 | quadratic polynomial | 0.9206 | 0.003083 | 1.551 | 0.509 |

Sample 1 | exponential | 0.9085 | 0.003121 | 1.507 | 0.509 |

Sample 1 | power law | - | - | - | 0.509 |

Sample 2 | quadratic polynomial | 0.9333 | 0.002921 | 1.539 | 0.612 |

Sample 2 | exponential | 0.9085 | 0.003121 | 1.507 | 0.612 |

Sample 2 | power law | - | - | - | 0.612 |

**Table A2.**Stress sensitivity test data (rock sample D13−4 from Xiao [18]).

σ/MPa | P/MPa | k/mD | σ/MPa | P/MPa | k/mD |
---|---|---|---|---|---|

51.00 | 26.02 | 0.21032 | 45.00 | 10.35 | 0.19216 |

51.00 | 22.46 | 0.18967 | 45.00 | 14.35 | 0.19428 |

51.00 | 18.15 | 0.17284 | 45.00 | 18.44 | 0.20139 |

51.00 | 14.01 | 0.15984 | 45.00 | 22.41 | 0.20740 |

51.00 | 10.24 | 0.14886 | 45.00 | 26.50 | 0.23058 |

51.00 | 6.01 | 0.14468 | 39.00 | 26.58 | 0.42405 |

51.00 | 10.09 | 0.14444 | 39.00 | 22.49 | 0.37336 |

51.00 | 14.23 | 0.14734 | 39.00 | 18.47 | 0.33524 |

51.00 | 18.24 | 0.15184 | 39.00 | 14.50 | 0.30324 |

51.00 | 22.31 | 0.16027 | 39.00 | 10.34 | 0.27396 |

51.00 | 26.08 | 0.17022 | 39.00 | 6.32 | 0.25667 |

45.00 | 26.51 | 0.30610 | 39.00 | 10.31 | 0.26015 |

45.00 | 22.41 | 0.27366 | 39.00 | 14.45 | 0.27361 |

45.00 | 18.31 | 0.23324 | 39.00 | 18.42 | 0.28018 |

45.00 | 14.18 | 0.21248 | 39.00 | 22.53 | 0.30107 |

45.00 | 10.22 | 0.19993 | 39.00 | 26.54 | 0.32863 |

45.00 | 6.24 | 0.19301 |

**Table A3.**Stress sensitivity test data (rock sample D15−2 from Xiao [18]).

σ/MPa | P/MPa | k/mD | σ/MPa | P/MPa | k/mD |
---|---|---|---|---|---|

12.00 | 10.19 | 0.06359 | 42.00 | 19.81 | 0.01131 |

17.00 | 10.19 | 0.03808 | 47.00 | 19.79 | 0.00962 |

22.00 | 10.12 | 0.02416 | 52.00 | 19.73 | 0.00871 |

27.00 | 10.05 | 0.01733 | 47.00 | 19.77 | 0.00915 |

32.00 | 9.98 | 0.01350 | 42.00 | 19.82 | 0.01035 |

37.00 | 9.92 | 0.01140 | 37.00 | 19.89 | 0.01254 |

42.00 | 9.87 | 0.00998 | 32.00 | 19.96 | 0.01671 |

47.00 | 9.85 | 0.00896 | 27.00 | 20.12 | 0.02621 |

52.00 | 9.82 | 0.00819 | 22.00 | 20.13 | 0.04885 |

47.00 | 9.34 | 0.00846 | 27.00 | 25.13 | 0.04658 |

42.00 | 9.85 | 0.00904 | 32.00 | 25.20 | 0.02961 |

37.00 | 9.93 | 0.00980 | 37.00 | 25.00 | 0.01979 |

32.00 | 9.96 | 0.01094 | 42.00 | 24.92 | 0.01412 |

27.00 | 10.00 | 0.01318 | 47.00 | 24.83 | 0.01130 |

22.00 | 10.07 | 0.01813 | 52.00 | 24.75 | 0.00964 |

17.00 | 10.15 | 0.02907 | 47.00 | 24.79 | 0.01036 |

12.00 | 10.19 | 0.05745 | 42.00 | 24.91 | 0.01247 |

17.00 | 15.18 | 0.06110 | 37.00 | 25.01 | 0.01640 |

22.00 | 15.11 | 0.03592 | 32.00 | 25.06 | 0.02511 |

27.00 | 15.07 | 0.02268 | 27.00 | 25.19 | 0.04535 |

32.00 | 14.98 | 0.01650 | 32.00 | 30.08 | 0.04340 |

37.00 | 14.90 | 0.01315 | 37.00 | 30.05 | 0.02762 |

42.00 | 14.87 | 0.01095 | 42.00 | 29.93 | 0.01842 |

47.00 | 14.76 | 0.00934 | 47.00 | 29.85 | 0.01311 |

52.00 | 14.73 | 0.00857 | 52.00 | 29.75 | 0.01039 |

47.00 | 14.73 | 0.00898 | 47.00 | 29.81 | 0.01196 |

42.00 | 14.77 | 0.00966 | 42.00 | 29.91 | 0.01542 |

37.00 | 14.84 | 0.01169 | 37.00 | 30.00 | 0.02328 |

32.00 | 14.88 | 0.01365 | 32.00 | 30.08 | 0.04093 |

27.00 | 14.97 | 0.01880 | 37.00 | 35.04 | 0.04064 |

22.00 | 15.09 | 0.03031 | 42.00 | 34.96 | 0.02616 |

17.00 | 15.13 | 0.05882 | 47.00 | 34.84 | 0.01635 |

22.00 | 20.25 | 0.05688 | 52.00 | 34.71 | 0.01247 |

27.00 | 20.11 | 0.03019 | 47.00 | 34.79 | 0.01508 |

32.00 | 20.03 | 0.01948 | 42.00 | 35.00 | 0.02260 |

37.00 | 19.92 | 0.01455 | 37.00 | 35.11 | 0.03903 |

**Table A4.**Accuracy and calculation results of 3D surface fitting method (rock sample D13−4 and D15−2).

Rock Sample | Model | R^{2} | RMSE | Calculation Results of 3D Surface Fitting Method | Calculation Results of Response Surface Method |
---|---|---|---|---|---|

D13−4 | power law | 0.8952 | 0.02397 | 0.3388 | 0.25920 |

D13−4 | quadratic polynomial | 0.8948 | 0.02442 | 0.3537 | 0.25920 |

D13−4 | exponential | 0.8924 | 0.02428 | 0.3518 | 0.25920 |

D15−2 | power law | 0.9741 | 0.002434 | 0.9002 | 0.6497 |

D15−2 | quadratic polynomial | 0.9428 | 0.003613 | 0.7978 | 0.6497 |

D15−2 | exponential | 0.9367 | 0.003828 | 0.7767 | 0.6497 |

Reference | Year | Rock Type | Fitting Model | ||
---|---|---|---|---|---|

Exponential | Power Law | Quadratic Polynomial | |||

Reyes [31] | 2002 | shale | exponential | ||

Chalmers [32] | 2012 | shale | exponential | ||

Dong [23] | 2013 | shale | power law | ||

Xiao [18] | 2013 | sandstone | exponential | power law | quadratic polynomial |

Dou [33] | 2016 | - | exponential | ||

Lu et al. [34] | 2020 | sandstone | power law | ||

Zheng et al. [35] | 2020 | shale | exponential |

## References

- Terzaghi, K. The shearing resistance of saturated soils and the angle between planes of shear. First Int. Conf. Soil Mech.
**1936**, 1, 54–59. [Google Scholar] - Biot, M.A. Theory of Deformation of a Porous Viscoelastic Anisotropic Solid. J. Appl. Phys.
**1956**, 27, 459–467. [Google Scholar] [CrossRef] - Biot, M.A.; Willis, D.G. The Elastic Coefficients of the Theory of Consolidation. J. Appl. Mech. Trans. ASME
**1957**, 24, 594–601. [Google Scholar] [CrossRef] - Nur, A.; Byerlee, J.D. An Exact Effective Stress Law for Elastic Deformation of Rock with Fluids. J. Geophys. Res.
**1971**, 76, 6414–6419. [Google Scholar] [CrossRef] - Yu, B.; Liu, C.; Zhang, D.; Zhao, H.; Li, M.; Liu, Y.; Yu, G.; Li, H. Experimental Study on the Anisotropy of the Effective Stress Coefficient of Sandstone Under True Triaxial Stress. J. Nat. Gas Sci. Eng.
**2020**, 84, 103651. [Google Scholar] [CrossRef] - An, C.; Killough, J.; Xia, X. Investigating the Effects of Stress Creep and Effective Stress Coefficient on Stress-Dependent Permeability Measurements of Shale Rock. J. Pet. Sci. Eng.
**2021**, 198, 108155. [Google Scholar] [CrossRef] - Ghabezloo, S.; Sulem, J.; Guedon, S.; Martineau, F. Effective stress law for the permeability of a limestone. Int. J. Rock Mech. Min. Sci.
**2009**, 46, 297–306. [Google Scholar] [CrossRef] [Green Version] - Meng, F.B.; Li, X.; Baud, P.; Wong, T.F. Bedding Anisotropy and Effective Stress Law for the Permeability and Deformation of Clayey Sandstones. Rock Mech. Rock Eng.
**2020**, 123, 4707–4729. [Google Scholar] [CrossRef] - Wang, Y.; Meng, F.; Wang, X.; Baud, P.; Wong, T.-F. Effective Stress Law for the Permeability and Deformation of Four Porous Limestones. J. Geophys. Res.-Solid Earth
**2018**, 123, 4707–4729. [Google Scholar] [CrossRef] - Xiao, W.; Bernabe, Y.; Evans, B.; Mok, U.; Zhao, J.; Ren, X.; Chen, M. Klinkenberg Effect and Effective Pressure for Gas Permeability of Tight Sandstones. J. Geophys. Res.-Solid Earth
**2019**, 124, 1412–1429. [Google Scholar] [CrossRef] - Berryman, J.G. Effective Stress for Transport Properties of Inhomogeneous Porous Rock. J. Geophys. Res.-Solid Earth
**1992**, 97, 17409–17424. [Google Scholar] [CrossRef] [Green Version] - Robin, P.Y.F. Note on Effective Pressure. J. Geophys. Res.
**1973**, 78, 2434–2437. [Google Scholar] [CrossRef] - Bernabe, Y. Comparison of The Effective Pressure Law for Permeability and Resistivity Formation Factor in Chelmsford Granite. Pure Appl. Geophys.
**1988**, 127, 607–625. [Google Scholar] [CrossRef] - Bernabe, Y. The Effective Pressure Law for Permeability in Chelmsford Granite and Barre Granite. Int. J. Rock Mech. Min. Sci.
**1986**, 23, 267–275. [Google Scholar] [CrossRef] - Yang, C.; Liu, J. Petroleum rock mechanics: An area worthy of focus in geo-energy research. Adv. Geo-Energy Res.
**2021**, 5, 2. [Google Scholar] [CrossRef] - Wang, F.; Gong, R.; Huang, Z.; Meng, Q.; Zhang, Q.; Zhan, S. Single-phase inflow performance relationship in stress-sensitive reservoirs. Adv. Geo-Energy Res.
**2021**, 5, 10. [Google Scholar] [CrossRef] - Walsh, J.B. Effect of Pore Pressure and Confining Pressure on Fracture Permeability. Int. J. Rock Mech. Min. Sci.
**1981**, 18, 429–435. [Google Scholar] [CrossRef] - Xiao, W.L. The Study on Non-Linear Effective Stress for Permeability in Low-Permeability Rocks. Ph.D. Thesis, Southwest Petroleum University, Chengdu, China, 2013. [Google Scholar]
- Jones, F.O., Jr. A Laboratory Study of the Effects of Confining Pressure on Fracture Flow and Storage Capacity in Carbonate Rocks. J. Pet. Technol.
**1975**, 27, 21–27. [Google Scholar] [CrossRef] - Bernabe, Y. The Effective Pressure Law for Permeability During Pore Pressure and Confining Pressure Cycling of Several Crystalline Rocks. J. Geophys. Res.-Solid Earth Planets
**1987**, 92, 649–657. [Google Scholar] [CrossRef] - Xiao, W.L.; Jiang, L.; Li, M.; Zhao, J.Z.; Zheng, L.L.; Li, X.F.; Zhang, Z.P. Effect of Clay Minerals on the Effective Pressure Law in Clay-rich Sandstones. J. Nat. Gas Sci. Eng.
**2015**, 27, 1242–1251. [Google Scholar] [CrossRef] - Li, M.; Xiao, W.L.; Bernabe, Y.; Zhao, J.Z. Nonlinear Effective Pressure Law for Permeability. J. Geophys. Res.-Solid Earth
**2014**, 119, 302–318. [Google Scholar] [CrossRef] - Dong, J.J.; Hsu, J.Y.; Wu, W.J.; Shimamoto, T.; Hung, J.H.; Yeh, E.C.; Wu, Y.H.; Sone, H. Stress-dependence of the Permeability and Porosity of Sandstone and Shale From TCDP Hole-A. Int. J. Rock Mech. Min. Sci.
**2010**, 47, 1141–1157. [Google Scholar] [CrossRef] - Shi, Y.L.; Wang, C.Y. Pore Pressure Generation in Sedimentary Basins: Overloading Versus Aquathermal. J. Geophys. Res.-Solid Earth Planets
**1986**, 91, 2153–2162. [Google Scholar] [CrossRef] - David, C.; Wong, T.F.; Zhu, W.L.; Zhang, J.X. Laboratory Measurement of Compaction-induced Permeability Change in Porous Rocks: Implications for the Generation and Maintenance of Pore Pressure Excess in the Crust. Pure Appl. Geophys.
**1994**, 143, 425–456. [Google Scholar] [CrossRef] - Yin, S.X.; Wang, S.X. Rock permeability at different scales and the relationship between stress and its mechanism. Sci. CHINA (SERIES D)
**2006**, 5, 472–480. [Google Scholar] - Qiao, L.P.; Wang, Z.C.; LI, S.C. Effective Stress Law for Permeability of Tight Gas Reservoir Sandstone. Chin. J. Rock Mech. Eng.
**2011**, 30, 1422–1427. [Google Scholar] - Zhao, J.; Xiao, W.; Li, M.; Xiang, Z.; Li, L.; Wang, J. The Effective Pressure Law for Permeability of Clay-Rich Sandstones. Pet. Sci.
**2011**, 8, 194–199. [Google Scholar] [CrossRef] [Green Version] - McKee, C.R.; Bumb, A.C.; Koenig, R.A. Stress-dependent Permeability and Porosity of Coal and Other Geologic Formations. SPE Form. Eval.
**1988**, 3, 81–91. [Google Scholar] [CrossRef] - Zhang, T.; Li, Z.; Adenutsi, C.D. A new model for calculating permeability of natural fractures in dual-porosity reservoir. Adv. Geo-Energy Res.
**2017**, 1, 7. [Google Scholar] [CrossRef] [Green Version] - Reyes, L.; Osisanya, S.O. Empirical Correlation of Effective Stress Dependent Shale Rock Properties. J. Can. Pet. Technol.
**2002**, 41, 47–53. [Google Scholar] [CrossRef] - Chalmers, G.R.L.; Ross, D.J.K.; Bustin, R.M. Geological Controls on Matrix Permeability of Devonian Gas Shales in the Horn River and Liard Basins, Northeastern British Columbia, Canada. Int. J. Coal Geol.
**2012**, 103, 120–131. [Google Scholar] [CrossRef] - Dou, H.G.; Zhang, H.J.; Yao, S.L.; Zhu, D.; Sun, T.; Ma, S.Y.; Wang, X.L. Measurement and evaluation of the stress sensitivity in tight reservoirs. Pet. Explor. Dev.
**2016**, 43, 1022–1028. [Google Scholar] [CrossRef] - Lu, R.B.; Hu, L.; Wang, W.J.; Yang, L.; Zhang, Q. Stress Sensitivity Characterization and Field Application in High Temperature-Pressure Gas Reservoir. Spec. Oil Gas Reserv.
**2020**, 27, 108–113. [Google Scholar] - Zheng, Y.X.; Liu, J.J.; Liu, Y.C.; Shi, D.; Zhang, B.H. Experimental Investigation on the Stress-Dependent Permeability of Intact and Fractured Shale. Geofluids
**2020**, 2020, 8897911. [Google Scholar] [CrossRef]

**Figure 1.**Fitting curves of experimental data of rock samples 1 and 2. (

**a**) Sample 1 was fitted by a quadratic polynomial; (

**b**) sample 1 was fitted by the exponential model; (

**c**) sample 2 was fitted by a quadratic polynomial; (

**d**) sample 2 was fitted by the exponential model.

**Figure 2.**Experimental data of rock sample D13−4 was fitted by a quadratic polynomial. (

**a**) Set the ESCP as 1; (

**b**) set the ESCP as 0.3537.

**Figure 3.**Longitudinal section of 3D fitting surface for the experimental data of rock sample D15−2. (

**a**) Sample D15−2 was fitted by the exponential model (ESCP was set as 0.7978 calculated by the 3D surface fitting method); (

**b**) sample D15−2 was fitted by the exponential model (ESCP was set as 0.6479 calculated by the response surface method).

**Figure 4.**Residual error distribution diagram (rock sample 1). (

**a**) The cross-plotting method (quadratic polynomial fitting rock sample 1); (

**b**) the cross-plotting method (exponential fitting rock sample 1); (

**c**) the 3D surface fitting method (quadratic polynomial fitting rock sample 1); (

**d**) the 3D surface fitting method (exponential fitting rock sample 1).

**Figure 5.**Residual error distribution diagram (rock sample 2). (

**a**) The cross-plotting method (quadratic polynomial fitting rock sample 2); (

**b**) the cross-plotting method (exponential fitting rock sample 2); (

**c**) the 3D surface fitting method (quadratic polynomial fitting rock sample 2); (

**d**) the 3D surface fitting method (exponential fitting rock sample 2).

**Figure 6.**Residual error distribution diagram (rock sample D13−4). (

**a**) The response surface method (quadratic polynomial fitting rock sample D13−4); (

**b**) the response surface method (exponential fitting rock sample D13−4); (

**c**) the 3D surface fitting method (quadratic polynomial fitting rock sample D13−4); (

**d**) the 3D surface fitting method (exponential fitting rock sample D13−4).

**Figure 7.**Residual error distribution diagram (rock sample D15−2). (

**a**) The cross-plotting method (quadratic polynomial fitting rock sample D15−2); (

**b**) the cross-plotting method (exponential fitting rock sample D15−2); (

**c**) the 3D surface fitting method (quadratic polynomial fitting rock sample D15−2); (

**d**) the 3D surface fitting method (exponential fitting rock sample D15−2).

σ/MPa | P/MPa | k/mD | |
---|---|---|---|

Sample 1 | Sample 2 | ||

10 | 2 | 0.00533 | 0.00631 |

10 | 5 | 0.0113 | 0.0142 |

10 | 8 | 0.019 | 0.0182 |

20 | 4 | 0.0042 | 0.00324 |

20 | 8 | 0.00932 | 0.0073 |

20 | 12 | 0.0151 | 0.0155 |

20 | 16 | 0.0309 | 0.03 |

30 | 5 | 0.00487 | 0.0024 |

30 | 10 | 0.00551 | 0.00421 |

30 | 15 | 0.0103 | 0.0115 |

30 | 20 | 0.015 | 0.014 |

30 | 25 | 0.0307 | 0.0313 |

ESCP was obtained by the cross-plotting method | 0.509 | 0.612 |

**Table 2.**Comparison of the effects of confining and pore fluid pressures on permeability (samples 1 and 2).

σ/MPa | P/MPa | k/mD | ∆σ/MPa | ∆k/mD | |||
---|---|---|---|---|---|---|---|

Sample 1 | Sample 2 | Sample 1 | Sample 2 | Sample 1 | Sample 2 | ||

10 | 5 | 0.0113 | 0.0142 | 20 | 20 | 0.00643 | 0.0118 |

30 | 5 | 0.00487 | 0.0024 | - | - | - | - |

30 | 25 | 0.0307 | 0.0313 | 20 | 20 | 0.02583 | 0.0289 |

Method | Sample | Model | ESCP | R^{2} | RMSE |
---|---|---|---|---|---|

3D surface fitting | 1 | Quadratic polynomial | 1.551 | 0.9206 | 0.003083 |

1 | Exponential | 1.551 | 0.9085 | 0.003121 | |

2 | Quadratic polynomial | 1.539 | 0.9333 | 0.002921 | |

2 | Exponential | 1.539 | 0.9085 | 0.003121 | |

Cross-plotting | 1 | Quadratic polynomial | 0.509 | 0.2066 | 0.009189 |

1 | Exponential | 0.509 | 0.0928 | 0.009321 | |

2 | Quadratic polynomial | 0.612 | 0.2955 | 0.008951 | |

2 | Exponential | 0.612 | 0.1494 | 0.009026 |

Method | Sample | Model | ESCP | R^{2} | RMSE |
---|---|---|---|---|---|

3D surface fitting | D13−4 | Quadratic polynomial | 0.3537 | 0.8948 | 0.02442 |

D13−4 | Exponential | 0.3518 | 0.8924 | 0.02428 | |

D15−2 | Quadratic polynomial | 0.7767 | 0.9367 | 0.003828 | |

D15−2 | Exponential | 0.7978 | 0.9428 | 0.003613 | |

response surface | D13−4 | Quadratic polynomial | 0.2592 | 0.8845 | 0.02515 |

D13−4 | Exponential | 0.2592 | 0.8824 | 0.02497 | |

D15−2 | Quadratic polynomial | 0.6497 | 0.9193 | 0.004293 | |

D15−2 | Exponential | 0.6497 | 0.9185 | 0.004282 |

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**MDPI and ACS Style**

Zhang, X.; Liu, J.; Song, J.
Optimization Algorithm of Effective Stress Coefficient for Permeability. *Energies* **2021**, *14*, 8345.
https://doi.org/10.3390/en14248345

**AMA Style**

Zhang X, Liu J, Song J.
Optimization Algorithm of Effective Stress Coefficient for Permeability. *Energies*. 2021; 14(24):8345.
https://doi.org/10.3390/en14248345

**Chicago/Turabian Style**

Zhang, Xiaolong, Jianjun Liu, and Jiecheng Song.
2021. "Optimization Algorithm of Effective Stress Coefficient for Permeability" *Energies* 14, no. 24: 8345.
https://doi.org/10.3390/en14248345