# Multifractal Characteristics and Classification of Tight Sandstone Reservoirs: A Case Study from the Triassic Yanchang Formation, Ordos Basin, China

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## Abstract

**:**

_{min}), generalized dimension parameter (D

_{max}), information dimension (D

_{1}), and correlation dimension (D

_{2}) were in good correlations with the porosity and permeability, which can well characterize the pore structure and reservoir heterogeneity of the study area, while the others didn’t respond well. Meanwhile, there also were good relationships between these multifractal and MICP parameters.

## 1. Introduction

## 2. Methodology

#### 2.1. Geological Setting

#### 2.2. Experiments

#### 2.3. Multifractal Analysis Theory

^{−k}(k = 0, 1, 2, …, j). The MICP data can be segmented in N(ε) part within scale ε:

#### 2.4. Mercury Injection Capillary Pressure (MICP) Theory

_{50}, R

_{50}, $\overline{r}$, P

_{d}, S

_{max}and W

_{e}are introduced. P

_{50}is median pressure, MPa, referring to the capillary pressure corresponding to the mercury saturation 50%. R

_{50}is the median radius, μm, pore radius corresponding to median pressure P

_{50}. $\overline{r}$ is the average throat radius, μm, indicating the pore structure distribution. The three front parameters, P

_{50}, R

_{50}and $\overline{r}$ all reflect the physical properties of rock pores. P

_{d}, S

_{max}and W

_{e}are three MICP parameters, respectively, referring to displacement pressure (MPa), maximum mercury saturation (%) and efficiency of mercury withdrawal (%). The latter three parameters, P

_{d}, S

_{max}and W

_{e}reflect the difficulty of mercury injection and mercury withdrawal, are also the embodiment of the complexity of the pore structure and clay content [52,53].

## 3. Results and Discussion

#### 3.1. Porosity, Permeability and MICP Data

_{d}ranges from 0.1289 MPa to 5.218 MPa, showing a good negative correlation with $\overline{r}$. The sensitivity to the permeability of different parameters listed in Table 1 is diverse. Among them, only the average pore radius $\overline{r}$ and displacement pressure P

_{d}show a good correlation (or negative correlation) with permeability (${R}^{2}>0.7$), while the correlations between permeability and the other petrophysical parameters are less the 0.25. $\overline{r}$ and P

_{d}are good parameters to reflect the heterogeneity of tight sandstone in the study area. The maximum capillary pressure is only 49.871 Mpa, the maximum mercury saturation S

_{max}of almost all samples reaches over 70%. This ensures that our experimental mercury injection curves can reflect most pore volume, some microporous information still will be omitted. When studying the reservoirs with lower permeability, the maximum capillary pressure should be higher than 100 MPa [54]. The mercury removal efficiency of the samples is about 30%. The efficiency of mercury withdrawal W

_{e}is often influenced by physical properties, the type and content of clay. In general, the better the physical properties are, the greater mercury withdrawal efficiency is [55]. The values of P

_{d}and W

_{e}shows that the pore throats are fine and have poor connectivity and low percolation capacity.

#### 3.2. Classification of the Pore Structure

#### 3.3. Pore Characteristics of Reservoirs with Different Pore Structure Types

#### 3.4. Multifractal Spectrum Parameters

_{−}

_{10}represents D

_{min}and D

_{10}represents D

_{max}. D

_{0}, D

_{1}and D

_{2}also are important dimension spectrum parameters, with the meanings of the capacity dimension parameter, information dimension parameter and correlation dimension parameter, respectively.

_{min}and α

_{max}represent α

_{−10}and α

_{10}, respectively. At the same time, α

_{min}is a subset of the maximum probability, while α

_{max}is corresponding to a subset of the minimum probability. When q < 0, $f(\alpha )$ increases as $\alpha $ increasing. Contrary to when q < 0, $f(\alpha )$ decreases as $\alpha $ increases. The singularity strength range is defined as $\mathsf{\Delta}\alpha $=$\alpha $

_{max}−$\alpha $

_{min}. $\mathsf{\Delta}\alpha $ is used to describe the complexity and heterogeneity of multifractal objects. With the heterogeneity of the pore structure increasing, $\mathsf{\Delta}\alpha $ increases. The parameter Δ$f$ is defined as the difference between the value of $f({\alpha}_{max})$ and $f({\alpha}_{min})$, which is equal to the ratio of the minimum value to the maximum value of multifractal singularity spectrum [4].

#### 3.5. Multifractal Characteristics of Pore Structure in Tight Sandstones

_{max}and D

_{max}, while Type III pore structure always has the highest values of $\tau $

_{min}and D

_{min}. In the multifractal spectrum $\alpha ~f(\alpha )$, the highest points of $f(\alpha )$ are almost the same. With the changes of pore structure types, the left and right hook both change a lot, which indicates the heterogeneity of pore throat distribution of different types of reservoirs.

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2},$f$

_{max},$f$

_{min}, Δα and △$f$, were obtained from the analysis of the multifractal spectrum $\alpha ~f(\alpha )$ and the generalized dimension spectrum $q~{D}_{q}$. Table 4 lists the multifractal parameters of three pore structure types. The average values of α

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2}and $f$

_{min}increase from Type I to Type III reservoir, while the average values of $f$

_{max}and △$f$ decrease. In Table 4, among all the multifractal parameters, α

_{min}, D

_{max}, D

_{1}, D

_{2}and $f$

_{max}can be used to type the different pore structure types.

#### 3.6. The Relationship between Multifractal Parameters and the Porosity and Permeability

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2},$f$

_{max},$f$

_{min}, Δα and △$f$ were used to classify the pore structure types. Figure 9 shows that the parameters α

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2}, and Δα are negatively correlated with porosity and permeability, whereas $f$

_{max}and △$f$ are negatively correlated with the porosity and permeability. As Figure 9 shows, some multifractal parameters, such as α

_{min}, D

_{max}, D

_{1}, and D

_{2}are in strong correlations with porosity and permeability, with the absolute values of the correlation coefficients higher than 0.57.

_{max}, D

_{min},$f$

_{max},$f$

_{min}, Δα and △$f$ display weak correlations with porosity and permeability, with correlation coefficients ranging from 0.0112 to 0.5097. Although these parameters are different in different types of reservoirs, the correlation coefficients are still not high enough, which might relate to the pore throat tortuosity and different cementation type. In conclusion, multifractal parameters, α

_{min}, D

_{max}, D

_{1}, and D

_{2}are great indicators of the porosity and permeability in tight sandstone reservoirs.

#### 3.7. The Relationship between Multifractal Parameters and Pore Structure Parameters

_{50}and P

_{d}are positively correlated with α

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2},$f$

_{min}and Δα, but negatively correlated with $f$

_{max}and △$f$, with logarithmic curve fitting. $\overline{r}$ is negatively correlated with α

_{max}, α

_{min}, D

_{max}, D

_{min}, D

_{1}, D

_{2},$f$

_{min}and Δα, but positively correlated with $f$

_{max}and △$f$, with logarithmic curve fitting. The correlation coefficients between small pore proportion and large proportion to multifractal parameters are good but totally opposite, as their contributions to the permeability are different. Combined with the preceding conclusions, the average pore radius $\overline{r}$ and the displacement pressure P

_{d}are good parameters to reflect the heterogeneity of tight sandstone in the study area; the multifractal parameters α

_{min}, D

_{max}, D

_{1}and D

_{2}display strong correlations with porosity and permeability. The multifractal parameters α

_{min}, D

_{max}, D

_{1}and D

_{2}are also in good correlation with both the average pore radius $\overline{r}$ and the displacement pressure P

_{d}.

## 4. Conclusions

_{d}are sensitive and effective parameters. They can be used to reflect the heterogeneity of tight sandstone in the study area.

_{min}, D

_{max}, D

_{1}, and D

_{2}are great indicators of the heterogeneity of reservoirs, they are also in good correlation with the average pore radius $\overline{r}$ and the displacement pressure P

_{d}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Thin section images and scanning electron microscopy (SEM) images of the tight sandstone in the study area: (

**a**) intergranular pores and feldspar dissolution pore; (

**b**) intergranular pores and enriched mica; (

**c**) the surface of the rock particles covered with padded chlorite; (

**d**) feldspar dissolved pore.

**Figure 4.**The mercury injection capillary pressure (MICP) curves of different types of reservoirs: (

**a**) the pore throat size distributions of different types; (

**b**) the mercury capillary pressure curves of different types.

**Figure 5.**Different pore proportions of different reservoir types: (

**a**) different pore proportion of Type I; (

**b**) different pore proportion of Type II; (

**c**) different pore proportion of Type III.

**Figure 6.**Relationships of the different pore proportion of different pore structure types: (

**a**) relationships of large pore proportion of different pore structure types; (

**b**) relationships of medium pore proportion of different pore structure types; (

**c**) relationships of small pore proportion of different pore structure types.

**Figure 7.**Multifractal analysis of the MICP data in the tight sandstone. (

**a**) the mass exponent $\tau (q)$; (

**b**) the generalized dimension spectrum $q~{D}_{q}$; (

**c**) the multifractal spectrum $\alpha ~f(\alpha )$.

**Figure 8.**Multifractal spectrum of MICP data in tight sandstone samples with different types of reservoirs. (

**a**) the mass exponent $\tau (q)$ of different types; (

**b**) the generalized dimension spectrum $q~{D}_{q}$ of different types; (

**c**) the multifractal spectrum $\alpha ~f(\alpha )$ of different types.

**Figure 9.**The cross plots between multifractal parameters and the porosity & permeability. (

**a**) the cross plots between α

_{max}, α

_{min}and porosity; (

**b**) the cross plots between D

_{max}, D

_{min}and porosity; (

**c**) the cross plots between D

_{1}, D

_{2}and porosity; (

**d**) the cross plots between $f$

_{max},$f$

_{min}and porosity; (

**e**) the cross plot between △$f$ and porosity; (

**f**) the cross plot between Δα and porosity; (

**g**) the cross plots between α

_{max}, α

_{min}and permeability; (

**h**) the cross plots between D

_{max}, D

_{min}and permeability; (

**i**) the cross plots between D

_{1}, D

_{2}and permeability; (

**j**) the cross plots between $f$

_{max},$f$

_{min}and permeability; (

**k**) the cross plot between △$f$ and permeability; (

**l**) the cross plot between Δα and permeability.

**Figure 10.**The cross plots between multifractal parameters and MICP parameters. (

**a**) the cross plots between α

_{max}, α

_{min}and P

_{50}; (

**b**) the cross plots between α

_{max}, α

_{min}and average pore radius; (

**c**) the cross plots between α

_{max}, α

_{min}and P

_{d}; (

**d**) the cross plots between α

_{max}, α

_{min}and small pore proportion; (

**e**) the cross plots between α

_{max}, α

_{min}and medium pore proportion; (

**f**) the cross plots between α

_{max}, α

_{min}and large pore proportion; (

**g**) the cross plots between D

_{max}, D

_{min}and P

_{50}; (

**h**) the cross plots between D

_{max}, D

_{min}and average pore radius; (

**i**) the cross plots between D

_{max}, D

_{min}and P

_{d}; (

**j**) the cross plots between D

_{max}, D

_{min}and small pore proportion; (

**k**) the cross plots between D

_{max}, D

_{min}and medium pore proportion; (

**l**) the cross plots between D

_{max}, D

_{min}and large pore proportion.

**Table 1.**The porosity, permeability and mercury injection capillary pressure (MICP) parameters of samples.

No. | Porosity (%) | K (mD) | P_{50} (MPa) | R_{50} (µm) | $\overline{\mathit{r}}(\mu \mathbf{m})$ | P_{d} (MPa) | S_{max} (%) | W_{e} (%) | Large Pore (%) | Medium Pore (%) | Small Pore (%) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 12.2 | 0.635 | 21.8635 | 0.0336 | 0.1362 | 1.6577 | 75.49 | 32.3168 | 0 | 36.47 | 39.02 |

2 | 14.1 | 9.987 | 4.5277 | 0.1624 | 1.4537 | 0.1341 | 72.59 | 37.2628 | 33.47 | 19.46 | 19.66 |

3 | 12.8 | 1.553 | 4.6343 | 0.1586 | 0.7140 | 0.3210 | 76.47 | 29.1339 | 25.96 | 27.38 | 23.13 |

4 | 13.3 | 1.318 | 7.9185 | 0.0928 | 0.5350 | 0.4078 | 73.78 | 30.1309 | 16.23 | 32.9 | 24.65 |

5 | 14.5 | 2.951 | 5.9248 | 0.1241 | 0.7622 | 0.2800 | 78.42 | 31.2342 | 27.2 | 24.23 | 26.99 |

6 | 13.4 | 1.285 | 9.3323 | 0.0788 | 0.5089 | 0.4364 | 74.11 | 30.1099 | 16.36 | 31.5 | 26.25 |

7 | 13.5 | 1.151 | 4.9514 | 0.1485 | 0.5617 | 0.3976 | 82.15 | 32.5816 | 22.36 | 30.91 | 28.88 |

8 | 15.0 | 5.365 | 5.8377 | 0.1259 | 0.9252 | 0.2202 | 74.77 | 32.7690 | 27.18 | 24.22 | 23.37 |

9 | 13.6 | 1.244 | 4.5437 | 0.1618 | 0.6162 | 0.4111 | 75.79 | 29.7635 | 23 | 30.77 | 22.02 |

10 | 13.6 | 1.515 | 4.0092 | 0.1834 | 0.6537 | 0.4114 | 76.13 | 30.5637 | 26.91 | 27.55 | 21.67 |

11 | 13.4 | 2.088 | 4.0521 | 0.1814 | 0.8023 | 0.3387 | 72.23 | 28.0463 | 30.56 | 23.35 | 18.32 |

12 | 5.7 | 0.108 | 29.7888 | 0.0247 | 0.0629 | 5.2180 | 64.38 | 31.5089 | 0 | 10.58 | 53.8 |

13 | 13.6 | 1.107 | 3.2311 | 0.2275 | 0.5251 | 0.3982 | 81.24 | 29.9167 | 15.25 | 42.66 | 23.33 |

14 | 13.6 | 1.107 | 3.9024 | 0.1884 | 0.3257 | 0.7812 | 79.67 | 33.3898 | 2.33 | 53.54 | 23.8 |

15 | 13.8 | 2.585 | 2.5664 | 0.2864 | 0.8658 | 0.1289 | 84.58 | 28.2272 | 30.89 | 28.38 | 25.31 |

16 | 14.0 | 0.486 | 6.0551 | 0.1214 | 0.1679 | 1.4097 | 84.15 | 31.8996 | 0 | 52.21 | 31.94 |

17 | 10.2 | 0.392 | 6.9278 | 0.1061 | 0.1420 | 1.9421 | 88.75 | 36.6435 | 0 | 50.19 | 38.56 |

18 | 9.7 | 0.156 | 13.7947 | 0.0533 | 0.0979 | 2.1185 | 75.9 | 20.6429 | 0 | 27.61 | 48.29 |

19 | 7.9 | 0.168 | 9.9355 | 0.0740 | 0.1976 | 0.9940 | 79.53 | 28.3013 | 0.91 | 43.11 | 35.51 |

20 | 10.8 | 0.271 | 10.0876 | 0.0729 | 0.1335 | 1.9989 | 81.85 | 31.1925 | 0 | 44.17 | 37.68 |

21 | 10.3 | 0.395 | 9.9605 | 0.0738 | 0.1609 | 1.5684 | 68.03 | 20.1933 | 0.43 | 40.5 | 27.1 |

22 | 11.7 | 0.800 | 4.5632 | 0.1611 | 0.3004 | 0.7287 | 80.32 | 29.9648 | 3.82 | 52.64 | 23.86 |

23 | 10.5 | 0.451 | 3.5488 | 0.2071 | 0.4010 | 0.4524 | 86.62 | 27.8590 | 11.59 | 49.73 | 25.3 |

24 | 10.5 | 0.279 | 6.7273 | 0.1093 | 0.1505 | 1.4602 | 88.1000 | 30.7172 | 0 | 50.65 | 37.45 |

No. | Quartz (%) | Feldspar (%) | Mica (%) | Chlorite (%) | Iron Calcite (%) | Main Particle Size (μm) | Sorting | Grinding Roundness | Cementation Type |
---|---|---|---|---|---|---|---|---|---|

1 | 18 | 42 | 23 | 5 | 2 | 0.10–0.30 | M | A | chlorite thin film |

3 | 22 | 53 | 5 | 4 | 6 | 0.2–0.5 | M | A | chlorite thin film |

4 | 20 | 54 | 7 | 4 | 2 | 0.15–0.5 | M | SA-A | chlorite thin film |

7 | 21 | 55 | 3 | 4 | 7 | 0.15–0.4 | M | A | pore-chlorite thin film |

9 | 23 | 57 | 0 | 6 | 1 | 0.1–0.3 | M-G | SA-A | chlorite thin film |

11 | 22 | 55 | 5 | 6.5 | 1 | 0.1–0.32 | M-G | SA-A | chlorite thin film |

12 | 15 | 50 | 7 | 0 | 18 | 0.10–0.35 | M | A | pore |

13 | 20 | 60 | 3 | 6 | 2 | 0.10–0.30 | M-G | A | chlorite thin film |

15 | 21 | 56 | 5 | 5 | 3 | 0.10–0.35 | M | A | pore-chlorite thin film |

16 | 22 | 55 | 6 | 7 | 3 | 0.05–0.15 | G | A | pore-chlorite thin film |

20 | 30 | 39 | 8 | 2 | 1 | 0.05–0.20 | M | SA | chlorite thin film-pore |

21 | 23 | 53.5 | 3 | 3 | 1.5 | 0.2–0.5 | M | SA | chlorite thin film-pore |

22 | 22 | 48 | 8.5 | 3 | 2 | 0.12–0.38 | M | A | pore-chlorite thin film |

23 | 22 | 55 | 6 | 3 | 2 | 0.16–0.28 | G | A | pore |

24 | 22 | 52 | 6 | 3 | 0 | 0.12–0.24 | G | A | pore-chlorite thin film |

Type | Porosity (%) | K (Md) | P_{50} (MPa) | $\overline{\mathit{r}}(\mu \mathbf{m})$ | P_{d} (MPa) | Large Pore (%) | Medium Pore (%) | Small Pore (%) |
---|---|---|---|---|---|---|---|---|

Type I | 12.8–15 | 1.151–9.987 | 2.57–5.92 | 0.56–1.45 | 0.13–0.41 | 22.36–33.47 | 19.46–30.91 | 18.32–28.88 |

13.81 | 3.160 | 4.56 | 0.82 | 0.29 | 27.50 | 26.25 | 23.26 | |

Type II | 10.5–13.6 | 0.451–1.318 | 3.23–9.33 | 0.30–0.54 | 0.40–0.78 | 2.33–16.36 | 31.5–53.54 | 23.33–26.25 |

12.68 | 1.011 | 5.42 | 0.43 | 0.53 | 10.93 | 43.83 | 24.53 | |

Type III | 5.7–14 | 0.108–0.635 | 6.06–29.79 | 0.06–0.20 | 0.99–5.22 | 0–0.91 | 10.58–52.21 | 27.1–53.8 |

10.3 | 0.369 | 11.97 | 0.15 | 1.91 | 0.52 | 40.81 | 37.32 |

Type | α_{max} | α_{min} | D_{max} | D_{min} | D_{1} | D_{2} | $\mathit{f}\mathit{max}$ | $\mathit{f}\mathit{min}$ | △α | $\mathit{f}$ |
---|---|---|---|---|---|---|---|---|---|---|

Type I | 1.18–1.24 | 0.14–0.28 | 0.17–0.32 | 1.11–1.15 | 0.56–0.70 | 0.31–0.53 | 0.22–0.50 | −0.01–0.00 | 0.90–1.06 | 0.22–0.50 |

1.21 | 0.21 | 0.24 | 1.13 | 0.63 | 0.42 | 0.37 | 0.00 | 0.99 | 0.37 | |

Type II | 1.16–1.90 | 0.26–0.46 | 0.29–0.49 | 1.11–1.71 | 0.67–0.80 | 0.48–0.67 | 0.00–0.59 | 0.00–0.20 | 0.79–1.54 | −0.01–0.55 |

1.33 | 0.36 | 0.39 | 1.23 | 0.71 | 0.56 | 0.29 | 0.07 | 0.97 | 0.23 | |

Type III | 0.99–2.14 | 0.34–0.79 | 0.40–0.82 | 0.91–1.92 | 0.71–0.92 | 0.58–0.93 | −0.28–0.19 | −0.19–0.55 | 0.2–1.80 | −0.30–0.45 |

1.66 | 0.48 | 0.53 | 1.50 | 0.77 | 0.69 | −0.07 | 0.08 | 1.17 | 0.05 |

**Table 5.**The correlation coefficients between multifractal parameters and pore structure parameters.

α_{max} | α_{min} | D_{max} | D_{min} | D_{1} | D_{2} | $\mathit{f}\mathit{max}$ | $\mathit{f}\mathit{min}$ | Δα | $\mathit{f}$ | |
---|---|---|---|---|---|---|---|---|---|---|

Porosity | −0.278 | −0.580 | 0.618 | −0.264 | −0.737 | −0.730 | 0.393 | −0.054 | −0.024 | 0.224 |

K | −0.261 | −0.682 | −0.714 | −0.238 | −0.780 | −0.831 | 0.510 | −0.101 | −0.011 | 0.316 |

P_{50} | 0.192 | 0.443 | 0.476 | 0.188 | 0.439 | 0.514 | −0.189 | 0.047 | 0.012 | −0.062 |

$\overline{r}$ | −0.328 | −0.686 | −0.718 | −0.302 | −0.712 | −0.804 | 0.582 | −0.114 | −0.028 | 0.360 |

P_{d} | 0.312 | 0.635 | 0.664 | 0.290 | 0.649 | 0.742 | −0.523 | 0.099 | 0.028 | −0.316 |

L-pore | −0.383 | −0.512 | −0.547 | −0.359 | −0.598 | −0.660 | 0.572 | −0.060 | −0.069 | 0.373 |

M-pore | 0.164 | 0.006 | 0.006 | 0.155 | 0.032 | 0.030 | −0.225 | −0.012 | 0.133 | −0.205 |

S-pore | 0.149 | 0.642 | 0.680 | 0.132 | 0.653 | 0.735 | −0.363 | 0.118 | 0.000 | −0.171 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Z.; Mao, Z.; Shi, Y.; Wang, D.
Multifractal Characteristics and Classification of Tight Sandstone Reservoirs: A Case Study from the Triassic Yanchang Formation, Ordos Basin, China. *Energies* **2018**, *11*, 2242.
https://doi.org/10.3390/en11092242

**AMA Style**

Jiang Z, Mao Z, Shi Y, Wang D.
Multifractal Characteristics and Classification of Tight Sandstone Reservoirs: A Case Study from the Triassic Yanchang Formation, Ordos Basin, China. *Energies*. 2018; 11(9):2242.
https://doi.org/10.3390/en11092242

**Chicago/Turabian Style**

Jiang, Zhihao, Zhiqiang Mao, Yujiang Shi, and Daxing Wang.
2018. "Multifractal Characteristics and Classification of Tight Sandstone Reservoirs: A Case Study from the Triassic Yanchang Formation, Ordos Basin, China" *Energies* 11, no. 9: 2242.
https://doi.org/10.3390/en11092242