# A Prediction Method of Compacted Rock Hydraulic Permeability Based on the MGEMTIP Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. MGEMTIP Model

#### 2.2. Relationship between Permeability and Chargeability in the MGEMTIP Model

**Figure 2.**Different distribution types of perturbed medium. The rock is a combination of these two structures: (

**a**) the worse the connectivity of the polarization medium, the higher the chargeability; (

**b**) the higher the connectivity of the polarization medium, the lower the chargeability.

- (a)
- Part of the boundary does not provide polarization. Since the finite volume of the actual rock sample is different from that in the open-domain model of the actual exploration object, the boundary formed by part of the perturbation is in direct contact with the insulating material (air or rubber sleeve) during the test process, and no complete boundary of the perturbation is provided. According to Appendix A, the chargeability difference between the infinite boundary MGEMTIP and the finite boundary MGEMTIP is$$\Delta {\eta}_{t}^{\prime}={\eta}_{t}-{\eta}_{t}^{\prime}={\displaystyle {\sum}_{l=1}^{N}3{f}_{l}{\gamma}_{l}/(2{M}_{0})}={\displaystyle {\sum}_{l=1}^{N}{S}_{l}^{\mathrm{out}}{r}_{l}/(2V{M}_{0})}\ge 0$$
- (b)
- By approximating the rock structure, the effective boundary decreases and the equivalent radius of perturbations increases. According to Appendix B, the difference between the chargeability of the MGEMTIP of the complex perturbations and the chargeability of the finite boundary MGEMTIP is$$\Delta {\eta}_{t}^{\u2033}={\eta}_{t}^{\prime}-{\eta}_{t}^{\u2033}={\displaystyle {\sum}_{l=1}^{N}{S}_{l}^{\mathrm{f}}{r}_{l}(1-{\gamma}_{l})/(2V{M}_{0})}\ge 0$$

- (c)
- Limitation of the measured frequency band. According to the logarithmic form of the time constant in Formula (2)$$\mathrm{log}({\tau}_{l})=\mathrm{log}({r}_{l})+\mathrm{log}(2{\rho}_{l}+{\rho}_{0})-\mathrm{log}(2{\alpha}_{s}^{l})$$

**Figure 3.**The schematic diagram of the MGETIP and the chargeability change in the actual rock with the time constant under different conditions. The area enclosed by the chargeability (horizontal axis) corresponding to different time constants is the total chargeability. ${\eta}_{t}$, ${{\eta}^{\prime}}_{t}$, ${{\eta}^{\u2033}}_{t}$, and ${\eta}_{e}$, respectively, indicate the chargeability of the MGEMTIP model, the chargeability of the finite boundary model, the chargeability of the finite boundary complex structure model, and the measured chargeability. In the low-frequency band outside the actual measurement frequency band, the chargeability ${\eta}_{lowf}$ corresponding to large-scale structure still exists.

#### 2.3. Permeability Prediction Model

#### 2.4. Applicability Analysis of the Permeability Prediction Model

## 3. Physics Experiment

#### 3.1. Experimental Sample Information

^{−2}~10

^{4}Hz, saturated with 4% NaCl electrolyte solution, and the solution resistivity is 0.114 ohm-m under the test environment.

**Figure 4.**Sample mineral compositions. Pyrite and clay have high electrical conductivity, which has a significant impact on the electrical polarization or electrical conductivity of the rock.

No. | Lithology | Porosity/% | Permeability | Dry Resistivity | Conductive Mineral | Experimental Project | |
---|---|---|---|---|---|---|---|

/mD | /ohm-m | Clay/% | Pyrite/% | ||||

2-1-17 | Sandstone | 3.50 | 0.0230 | 5.70 × 10^{2} | 30 | 0 | Fitting samples with high clay |

2-1-20 | Sandstone | 2.20 | 0.0750 | 2.50 × 10^{3} | 19.7 | 0 | |

201-6 | Shale | 5.40 | 0.4850 | 5.90 × 10^{2} | 27.8 | 2.2 | |

205-5 | Shale | 5.90 | 0.3810 | 9.40 × 10^{2} | 36.5 | 3.7 | |

WY-44V | Shale | 3.80 | 0.0400 | 8.50 × 10^{2} | 30.8 | 0.9 | |

WY-45V | Shale | 1.50 | 0.0350 | 8.30 × 10^{3} | 25.7 | 0.8 | |

WY-46V | Shale | 2.10 | 0.0300 | 7.80 × 10^{2} | 38.5 | 3.1 | |

WY-47V | Shale | 1.30 | 0.0500 | 1.00 × 10^{4} | 28.4 | 0.8 | |

WY-49V | Shale | 1.20 | 0.0580 | 2.30 × 10^{4} | 23.9 | 1 | |

WY-50V | Shale | 1.00 | 0.0660 | 3.40 × 10^{4} | 20.3 | 0.8 | |

WY-52V | Shale | 2.40 | 0.0270 | 1.10 × 10^{3} | 26.5 | 0.7 | |

2-1-10 | Limestone | 0.33 | 0.0420 | 3.10 × 10^{6} | 2 | 0 | Fitting samples with low clay |

2-1-14 | Limestone | 0.57 | 0.0300 | 1.40 × 10^{6} | 1 | 0 | |

2-1-18 | Dolomite | 1.40 | 0.1100 | 1.30 × 10^{6} | 1.5 | 3.1 | |

2035-7 | Limestone | 4.00 | 0.5440 | 7.30 × 10^{3} | 7 | 0 | |

2035-12 | Limestone | 5.40 | 0.1180 | 2.30 × 10^{7} | 2 | 0 | |

YH-2 | Dolomite | 2.97 | 1.2100 | 2.20 × 10^{5} | 1.3 | ||

AS-2 | Sandstone | 7.75 | 3.4500 | 8.16 × 10^{6} | 0 | 0 | |

AS-4 | Sandstone | 12.84 | 87.5000 | 1.08 × 10^{7} | 0 | 0 | |

AS-5 | Sandstone | 18.36 | 326.0000 | 1.11 × 10^{7} | 0 | 0 | |

W1 | Shale | 4.84 | 0.1100 | 1.80 × 10^{3} | 52 | 0 | Predicted samples with high clay |

W2 | Shale | 5.42 | 0.3160 | 1.28 × 10^{3} | 52.4 | 4.5 | |

W3 | Shale | 5.26 | 0.6260 | 3.61 × 10^{3} | 27.6 | 3.4 | |

W4 | Shale | 4.36 | 3.9900 | 2.24 × 10^{3} | 25.6 | 5.4 | |

W5 | Shale | 4.21 | 0.1310 | 4.39 × 10^{3} | 30.3 | 2.9 | |

XWX1-1 | Sandstone | 9.93 | 0.8019 | 4.23 × 10^{2} | 23.5 | 0 | |

XWX1-2 | Sandstone | 2.22 | 0.0712 | 1.26 × 10^{4} | 19 | 0 | |

XWX2A-1 | Sandstone | 7.88 | 0.2643 | 1.44 × 10^{4} | 18.6 | 2.3 | |

XWX2A-2 | Sandstone | 5.53 | 0.3578 | 4.22 × 10^{2} | 25 | 0 | |

XWX3-1 | Sandstone | 0.86 | 0.0045 | 9.91 × 10^{4} | 26.3 | 0 | |

XWX6-3 | Sandstone | 6.72 | 0.8354 | 8.31 × 10^{2} | 23.7 | 0 | |

XWX8-1 | Sandstone | 8.74 | 0.9193 | 1.45 × 10^{4} | 13.8 | 0 | |

LJ1-4C | Limestone | 0.18 | 0.0010 | 1.36 × 10^{5} | 0 | 5.7 | Predicted samples with low clay |

AS-6 | Sandstone | 15.80 | 247.0000 | 9.57 × 10^{6} | 0 | 0 | |

AS-7 | Sandstone | 16.50 | 331.4000 | 1.25 × 10^{7} | 0 | 0 | |

AS-8 | Sandstone | 9.70 | 12.6000 | 7.64 × 10^{6} | 0 | 0 |

#### 3.2. Theoretical Chargeability and Measured Chargeability

## 4. Results and Discussion

#### 4.1. Fitting Relationship

#### 4.2. Prediction Results

#### 4.3. Relationship with K-C Model

#### 4.4. Application of the Permeability Prediction Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**The schematic diagram of the effective medium theory. A heterogeneous medium contains many different types of perturbed media, each of which contains resistivity ${\rho}_{l}$, effective radius ${r}_{l}$, and surface-polarizability coefficient ${\alpha}_{s}^{l}$. The resistivity of a heterogeneous medium is equivalent to the CR of an effective medium through the effective-medium approach.

**Figure 5.**SEM of reservoir shale. The secondary pyrite particles in clay minerals are aggregated, and the particles as a whole are about 0.01 mm in scale.

**Figure 6.**Relationship between measured permeability and relative chargeability difference of rock samples. The samples include 30 new samples and 6 samples from Tong et al. [30].

**Figure 7.**Fitting results from three prediction models for 36 samples. The difference of R

^{2}is smaller.

**Figure 8.**Prediction results of three models based on (

**a**) high-clay predicted samples and (

**b**) low-clay predicted samples; (

**a**) the three prediction models based on high clay rocks have poor prediction results for low clay rocks (D > 1), while the $\eta $-prediction for high clay samples is best ($D=0.2825$ ); (

**b**) the $\eta $ -prediction model based on low clay rocks has a worst prediction result for low clay rocks ($D=1.0238$ ) and a best prediction result for high clay rocks ($D=0.6677$ ).

No. | ${\mathsf{\rho}}_{\mathit{s}}(\mathbf{\Omega}\xb7\mathbf{m})$ | ${\mathsf{\eta}}_{\mathit{e}}$ | Main Conductive Medium | Main Polarization Medium | Spatial Correlation | ${\mathsf{\eta}}_{\mathit{t}}$ | ||
---|---|---|---|---|---|---|---|---|

Medium | ${\mathit{V}}_{\mathit{m}}$ | Medium | ${\mathit{V}}_{\mathit{n}}$ | |||||

2-1-17 | 66.42 | 9.49% | Clay | 28.95% | Solution | 3.50% | 0 | 15.75% |

2-1-20 | 44.02 | 4.37% | Clay | 19.27% | Solution | 2.20% | 0 | 9.90% |

201-6 | 25.79 | 5.33% | Clay | 26.30% | Pyrite | 2.08% | 1 | 35.61% |

205-5 | 19.24 | 7.20% | Clay | 36.50% | Pyrite | 3.48% | 1 | 42.93% |

WY-44V | 20.37 | 10.22% | Clay | 29.63% | Pyrite | 1.15% | 1 | 17.53% |

WY-45V | 132.24 | 10.15% | Clay | 25.31% | Pyrite | 0.69% | 1 | 12.26% |

WY-46V | 69.52 | 28.61% | Clay | 37.69% | Pyrite | 3.23% | 1 | 38.57% |

WY-47V | 138.89 | 8.23% | Clay | 28.03% | Pyrite | 1.09% | 1 | 17.43% |

WY-49V | 168.93 | 11.95% | Clay | 23.61% | Pyrite | 1.19% | 1 | 22.59% |

WY-50V | 194.76 | 10.08% | Clay | 20.10% | Pyrite | 0.59% | 1 | 13.30% |

WY-52V | 44.18 | 9.61% | Clay | 25.86% | Pyrite | 0.88% | 1 | 15.28% |

2-1-10 | 2134.65 | 5.13% | HRM | 97.68% | Solution/Clay | 2.32% | 0 | 10.46% |

2-1-14 | 2878.45 | 5.60% | HRM | 98.44% | Solution/Clay | 1.56% | 0 | 7.04% |

2-1-18 | 594.43 | 15.18% | HRM | 94.06% | Solution/Clay/ Pyrite | 5.94% | 0 | 26.71% |

2035-7 | 78.65 | 4.94% | HRM | 89.28% | Solution/Clay | 10.72% | 0 | 48.24% |

2035-12 | 42.77 | 2.29% | HRM | 92.71% | Solution/Clay | 7.29% | 0 | 32.81% |

YH-2 | 131.18 | 1.44% | HRM | 95.77% | Solution/Clay | 4.23% | 0 | 19.04% |

AS-2 | 18.06 | 6.77% | HRM | 92.25% | Solution | 7.75% | 0 | 34.88% |

AS-4 | 1.04 | 2.89% | HRM | 87.16% | Solution | 12.84% | 0 | 57.77% |

AS-5 | 0.26 | 2.83% | HRM | 81.64% | Solution | 18.36% | 0 | 82.64% |

W1 | 16.99 | 4.57% | Clay | 49.48% | Solution | 4.84% | 0 | 21.78% |

W2 | 12.27 | 5.89% | Clay | 49.56% | Pyrite | 4.50% | 1 | 40.86% |

W3 | 24.17 | 7.09% | Clay | 26.15% | Pyrite | 3.40% | 1 | 58.51% |

W4 | 28.94 | 7.56% | Clay | 24.48% | Pyrite | 5.40% | 1 | 99.25% |

W5 | 45.09 | 7.97% | Clay | 29.02% | Pyrite | 2.90% | 1 | 44.96% |

XWX1-1 | 25.01 | 6.61% | Clay | 23.50% | Solution | 9.93% | 0 | 44.69% |

XWX1-2 | 51.57 | 5.63% | Clay | 19.00% | Solution | 2.22% | 0 | 9.99% |

XWX2A-1 | 23.69 | 2.58% | Clay | 18.60% | Solution | 7.88% | 0 | 35.46% |

XWX2A-2 | 21.44 | 3.76% | Clay | 25.00% | Solution | 5.53% | 0 | 24.89% |

XWX3-1 | 818.59 | 3.62% | Clay | 26.30% | Solution | 0.86% | 0 | 3.87% |

XWX6-3 | 39.84 | 2.74% | Clay | 23.70% | Solution | 6.72% | 0 | 30.24% |

XWX8-1 | 193.20 | 3.68% | Clay | 13.80% | Solution | 8.74% | 0 | 39.33% |

LJ1-4C | 16946.92 | 24.90% | HRM | 94.13% | Solution/Pyrite | 5.87% | 0 | 26.41% |

AS-6 | 6.46 | 8.36% | HRM | 84.20% | Solution | 15.80% | 0 | 71.10% |

AS-7 | 1.66 | 5.05% | HRM | 83.50% | Solution | 16.50% | 0 | 74.25% |

AS-8 | 8.34 | 7.54% | HRM | 90.30% | Solution | 9.70% | 0 | 43.65% |

No. | F | ${\mathsf{\sigma}}^{\u2033}$ | ${\mathit{m}}_{\mathit{e}}(\mathbf{mS}/\mathbf{m})$ |
---|---|---|---|

2-1-17 | 517.55 | 11.5221 | 1.6323 |

2-1-20 | 366.77 | 8.1300 | 1.0921 |

201-6 | 211.22 | 17.2002 | 2.4788 |

205-5 | 155.18 | 22.1860 | 4.0711 |

WY-44V | 158.51 | 31.5545 | 5.4056 |

WY-45V | 1041.73 | 5.5821 | 0.7590 |

WY-46V | 502.12 | 7.9912 | 3.1579 |

WY-47V | 1118.21 | 3.5422 | 0.5801 |

WY-49V | 1305.63 | 4.7706 | 0.6958 |

WY-50V | 1530.63 | 3.5658 | 0.5247 |

WY-52V | 351.57 | 14.2435 | 2.0736 |

2-1-10 | 17659.59 | 0.1741 | 0.0257 |

2-1-14 | 23601.09 | 0.1397 | 0.0219 |

2-1-18 | 4471.45 | 1.7568 | 0.2402 |

2035-7 | 645.38 | 5.2304 | 0.7891 |

2035-12 | 362.71 | 9.1927 | 0.7271 |

YH-2 | 1126.95 | 0.9363 | 0.1428 |

AS-2 | 148.17 | 20.2161 | 3.4935 |

AS-4 | 8.86 | 371.2225 | 26.9525 |

AS-5 | 2.23 | 1867.0926 | 104.8977 |

W1 | 142.32 | 14.3595 | 2.5664 |

W2 | 101.51 | 23.6047 | 4.5197 |

W3 | 197.77 | 17.5357 | 2.7247 |

W4 | 235.79 | 16.7108 | 2.4150 |

W5 | 366.06 | 11.4582 | 1.6260 |

XWX1-1 | 205.90 | 0.7644 | 2.4675 |

XWX1-2 | 428.40 | 0.3601 | 1.0310 |

XWX2A-1 | 202.57 | 0.3506 | 1.0630 |

XWX2A-2 | 181.25 | 0.4065 | 1.6862 |

XWX3-1 | 6930.47 | 0.0128 | 0.0426 |

XWX6-3 | 340.21 | 0.2538 | 0.6682 |

XWX8-1 | 1634.67 | 0.4037 | 0.1837 |

LJ1-4C | 125430.11 | 0.0052 | 0.0110 |

AS-6 | 52.37 | 8.4709 | 11.8593 |

AS-7 | 13.87 | 12.0579 | 28.8854 |

AS-8 | 68.09 | 1.9752 | 8.3591 |

Prediction Model | Fitted Samples | Model Parameters | Coefficient of Determination R^{2} | Geometric Mean Error D | |||
---|---|---|---|---|---|---|---|

a | b | c | Low-Clay | High-Clay | |||

${k}^{\ast}=\frac{a{({\eta}_{t}-{\eta}_{e})}^{b}}{{\eta}_{e}^{c}}$ | High-clay | 0.0580 | 0.8136 | 0.8983 | 0.8613 | 1.1887 | 0.2825 |

Low-clay | 5.1725 | 1.9503 | 0.5212 | 0.7915 | 1.0238 | 0.6677 | |

${k}^{\ast}=\frac{a}{{F}^{b}{{\sigma}^{\u2033}}^{c}}$ | High-clay | 3.27 × 10^{6} | 2.1705 | 1.9321 | 0.6015 | 1.1237 | 1.6848 |

Low-clay | 3.72 × 10^{4} | 1.5361 | 0.5382 | 0.9581 | 0.6773 | 1.1681 | |

${k}^{\ast}=\frac{a}{{F}^{b}{m}_{e}^{c}}$ | High-clay | 4.12 × 10^{3} | 1.6994 | 1.2995 | 0.6180 | 1.2440 | 0.5491 |

Low-clay | 1.14 × 10^{3} | 1.1445 | 0.1652 | 0.9547 | 0.8327 | 0.9795 |

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**MDPI and ACS Style**

Tong, X.; Yan, L.; Xiang, K.
A Prediction Method of Compacted Rock Hydraulic Permeability Based on the MGEMTIP Model. *Minerals* **2023**, *13*, 281.
https://doi.org/10.3390/min13020281

**AMA Style**

Tong X, Yan L, Xiang K.
A Prediction Method of Compacted Rock Hydraulic Permeability Based on the MGEMTIP Model. *Minerals*. 2023; 13(2):281.
https://doi.org/10.3390/min13020281

**Chicago/Turabian Style**

Tong, Xiaolong, Liangjun Yan, and Kui Xiang.
2023. "A Prediction Method of Compacted Rock Hydraulic Permeability Based on the MGEMTIP Model" *Minerals* 13, no. 2: 281.
https://doi.org/10.3390/min13020281