Tight carbonate reservoir is characterized by low degree of pores development, high degree of filling, various types of reservoir space, and complex pores structure[15–16]. For the porous reservoir, pores size distribution parameters and pores size spectrum calculation are done according to the electrical imaging bottom layer conductivity data matrix, to carry out quantitative calculation of reservoir parameters.

## 2.1 Porosity calculation method

Porosity calculation method is done based on the electrical imaging. The conventional porosity calculation methods are volume model and core calibration method. The volume model method is used to obtain the logging response value of the rock skeleton, while core calibration method is used to obtaine the correlation between porosity and logging curve. For the former, the response value of rock skeleton is not fixed because of the diversity of lithology in the study area; thus, it is not applicable. For the latter, there is a good correlation between porosity and the average resistivity of electrical imaging (Fig. 6) based on the logging and coring data used in this study. Therefore, this paper considers using the average resistivity imaging to calculate the formation porosity.

In Fig. 4, the relationship between the average resistivity of electrical imaging logging and porosity is

$$\varphi =0.3856{{R}_{a}}^{-0.366},{R}^{2}=0.6963,$$

1

where\(\varphi\) is the porosity in \(\text{\%}\); \({R}_{a}\) is the mean value of the electrical imaging resistivity, \(\varOmega .m\); \({\text{R}}^{2}\) is determination coefficient.

The mean value of the resistivity is obtained using the arithmetic mean value of electrical imaging resistivity matrix, as shown in Eq. (2).

$${R}_{a}=\frac{\sum _{1}^{360}{R}_{i}}{360},$$

2

where \({R}_{i}\) represents each resistivity data in the electrical imaging resistivity matrix,\(\varOmega .m\),while 360 is the electrical imaging resistivity data obtained by Schlumberger Techlog through “Full Image Computation”.

It is found from the Eq. (2) that porosity can be represented by the average resistance calculated by electrical imaging.

## 2.2 Pores diameter calculation method

Electrical imaging resistivity is determined by porosity, mud filtrate resistivity, and formation cementation degree. Cementation degree is quantitatively characterized by cementation index, which reflects pores type, i.e., pores size, to a certain extent. Therefore, based on the formation factor formula, the cementation index of reservoir is calculated by the average resistivity and porosity of electrical imaging. Moreover, the cementation index and pores diameter are calibrated in homogeneous porous formation to establish the relationship between cementation index and pores diameter before calculating the pores diameter.

The relationship between cementation index, formation resistivity, and porosity is obtained using the formation factor formula shown in Eq. (3)

$$m=\frac{log10\left(a{R}_{mf}\right)-log10\left({R}_{a}\right)}{log10\left(\varphi \right)},$$

3

where \(m\) is cementation index; \({R}_{mf}\) is the mud filtrate resistivity,\(\varOmega .m\); \(a\) is the lithology coefficient with value 1; \(a\) is the cementation index.

The calculation steps of mud resistivity \({R}_{mf}\) are as follows:

(1) Calculate the mud resistivity at 18 degrees Celsius:

$${R}_{m18}={R}_{m0}\times (1+0.023\times ({T}_{0}-18\left)\right),$$

4

where \({R}_{m18}\) is the resistivity of the mud at 18\(\text{℃}\),\(\varOmega .m\); \({R}_{m0}\) is the resistivity of surface mud,\(\varOmega .m\); \({T}_{0}\) is the ground temperature, \(\text{℃}\).

(2) The mud resistivity under formation temperature is calculated by Eq. (5 ~ 6):

$${T}_{1}=G\times \text{D}\text{e}\text{e}\text{p}/100,$$

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$${R}_{m}=\frac{{R}_{m18}}{1+0.023\times \left({T}_{1}-18\right)},$$

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where \({T}_{1}\) is the formation temperature, \(\text{℃}\); \(G\) is the isothermal gradient; \(\text{D}\text{e}\text{e}\text{p}\) is the depth of the well, m; \({R}_{m}\) is the resistivity of mud, \(\varOmega .m\);

(3) Calculation of the mud filtrate resistivity is

$${R}_{mf}=C\times {R}_{m}^{1.07},$$

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(4) The coefficient C in Eq. (7) is calculated by mud density using the equation

$$C=0.769\times {DEN}_{m}^{2}-3.086\times {DEN}_{m}+3.437,$$

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where \({DEN}_{m}\) is the mud density measured on the ground, \(g/{cm}^{3}\).

According to the characteristics of tight carbonate reservoir in Ordos Basin, the tight dolomite formation with porosity less than 3% in the fifth member of Ma formation in the east of Ordos Basin (which has a small amount of intergranular pores, small pores diameter, and relatively homogeneous) is selected as the fine intergranular pores scale formation. The dolomite formation in the fifth member of Ma formation in the east of paleo uplift is selected as the medium-sized intergranular solution pores scale formation, while Mawu in the east of paleo uplift is selected as the stratum with the largest pores size. The cross plot of cementation index and pores diameter is shown in Fig. 5.

Therefore, the equation of pores diameter is obtained as follows:

$${D}_{p}=126573{e}^{-3.138m},{R}^{2}=0.6363,$$

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where \({D}_{a}\) is the pores diameter, \(\mu m\).

## 2.3 Calculation method of pores diameter spectrum

From the Eqs. (3) and (8), 360 pores diameters can be calculated for each depth point to obtain the pores diameter matrix data of the whole well section. However, we prefer to study the difference between pores size distribution instead of the pores data matrix. Therefore, the pores data is divided into five intervals:10− 1 ~ 100、100 ~ 101、101 ~ 102、102 ~ 103、and 103 ~ 104. Further, each interval is divided into five intervals, resulting to 25 intervals, which are divided to count 360 pores diameter data. See Table 1 for the specific division rules of the interval.

Table 1

Statistical table of pores diameter interval

Index interval | Pores diameter range | Index range | Pores diameter range |

10 | -1.00 | 0.10 | 10 | 1.60 | 39.81 |

10 | -0.80 | 0.16 | 10 | 1.80 | 63.10 |

10 | -0.60 | 0.25 | 10 | 2.00 | 100.00 |

10 | -0.40 | 0.40 | 10 | 2.20 | 158.49 |

10 | -0.20 | 0.63 | 10 | 2.40 | 251.19 |

10 | 0.00 | 1.00 | 10 | 2.60 | 398.11 |

10 | 0.20 | 1.58 | 10 | 2.80 | 630.96 |

10 | 0.40 | 2.51 | 10 | 3.00 | 1000.00 |

10 | 0.60 | 3.98 | 10 | 3.20 | 1584.89 |

10 | 0.80 | 6.31 | 10 | 3.40 | 2511.89 |

10 | 1.00 | 10.00 | 10 | 3.60 | 3981.07 |

10 | 1.20 | 15.85 | 10 | 3.80 | 6309.57 |

10 | 1.40 | 25.12 | 10 | 4.00 | 10000.00 |

To compare the pores diameter distribution between different depths, divide the above statistics by the sum to get the pores diameter distribution component of each interval.

The pores size spectrum is the distribution range of pores diameter and the pores component of each pores size range. Based on the pores size spectrum, the component porosity of five types of pores are obtained by counting the porosity of micro-pores, fine pores, medium pores, coarse pores, and pores diameter interval. These five types of pores are classified according to the pores diameter in the industry standard of cast thin section identification. Based on the industry standard, the pores diameter classification of the five types of pores is shown in Table 2.

Table 2

Dimension table of carbonate reservoir space

Pores type | Pores subclass | Pores diameter(µm) |

Pores | Micro pores | < 10 |

Small pores | 10 ~ 100 |

Medium pores | 100 ~ 500 |

Coarse pores | 500 ~ 2000 |

Hole | Hole | > 2000 |

If the pores size distribution component is multiplied by pores, the sum of each interval is porosity (decimal), and the value of each interval is porosity component.

## 2.4 Calculation of permeability by pores composition method

Tight carbonate reservoirs in Ordos Basin can increase pores diameter and throat width. The permeability contribution rate of different pores components is unique. The hole contribution rate is high, whereas the micropores contribution rate is low. The larger the pores size, the greater the permeability contribution. Therefore, the relationship between porosity and permeability of different pores components can be established, and the reservoir permeability can be calculated by linear combination.

Further, the vuggy reservoir is developed in Mawu on the east side of paleo uplift. The rock samples of vuggy reservoir are selected to analyze the porosity and permeability, while the porosity permeability relationship of vuggy reservoir is established as the relationship of coarse porosity & hole and permeability.

The relationship between permeability and porosity of coarse pores & hole is as follows:

$${K}_{1}=1.1769{e}^{25.61{\varphi }_{1}} and {R}^{2}=0.667,$$

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where, \({\varphi }_{1}\) is the porosity of coarse pores and hole, \(\%\); \({K}_{1}\) is the permeability partly contributed by the coarse pores and hole, \(mD\).

Furthermore, intergranular porosity reservoir is developed in Mawu reservoir in the east of paleo uplift. Rock samples of intergranular porosity and intergranular solution porosity reservoir are selected to analyze porosity, permeability, and establishes porosity permeability relationship, which is regarded as porosity permeability relationship of medium pores and small pores, as shown in Fig. 7.

The relationship between the permeability and medium pores & small pores is as follows:

$${K}_{2}=0.0049{e}^{38.194{\varphi }_{2}}, {R}^{2}=0.4509,$$

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where, \({\varphi }_{2}\) is the porosity of medium pores & small pores, \(\%\); \({K}_{2}\) is the permeability, partly contributed by the medium pores & small pores, \(mD\).

The contribution of micropores porosity to permeability is below 0.05\(mD\), which is low. So, the contribution of micropores porosity to permeability is not considered.

Therefore, the formula for the component porosity and the permeability is

$$K=0.005{\text{e}}^{37.68\text{*}{\varphi }_{2}}+1.1769{\text{e}}^{25.61\text{*}{\varphi }_{1}},$$

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where \(K\) is the reservoir permeability, \(mD\).