# A Calculation Model of the Dimensionless Productivity Index Based on Non-Piston Leading Edge Propulsion Theory in Multiple Oilfield Development Phases

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of a Calculation Model of the Dimensionless Productivity Index

#### 2.1. Formula for Leading Edge Propulsion Distance

_{D}is the dimensionless productivity index, Q

_{t}is the liquid production (m

^{3}/s), Q

_{o}is oil production (m

^{3}/s), Q

_{w}is the water production (m

^{3}/s), μ

_{o}is oil phase viscosity (mPa·s), μ

_{w}is the water phase viscosity (mPa·s), K

_{ro}is the oil phase relative permeability (10

^{−3}μm

^{2}), K

_{rw}is the relative permeability of the water phase (10

^{−3}μm

^{2}), K

_{o}is the oil phase permeability (10

^{−3}μm

^{2}), K

_{w}is the water phase permeability (10

^{−3}μm

^{2}), K is the reservoir permeability (10

^{−3}μm

^{2}), P is the pressure difference along the displacement direction (MPa), L is the band length (m), ΔP is the pressure difference (MPa), and A is the sectional area (m

^{3}).

_{w}is the water cut (%), S

_{w}is the water saturation (%), and v

_{t}is the seepage velocity of the total liquid yield (m/s).

#### 2.2. Calculation Formula of Water Flooding-Polymer Flooding-Binary Compound Flooding in Multi-Development Phases

_{w}is the water phase migration velocity (m/s) and v

_{o}is the oil phase migration velocity (m/s).

_{c}is capillary pressure (Mpa), and P

_{o}and P

_{w}are oil and water phase pressure, respectively (10

^{−1}Mpa).

_{o}and K

_{w}are f

_{w}-related functions, which can be expressed as:

_{w}is obtained by deriving the water saturation S

_{w}:

^{3}).

#### 2.2.1. Water Flooding Stage

_{1}is the leading edge radius of the water flooding stage:

_{w}is the aqueous phase viscosity (mPa·s) and K

_{ro}/K

_{rw}is the fluid-to-water ratio.

#### 2.2.2. Polymer Flooding Stage

_{2}is the leading edge radius of the polymer flooding state,

_{w}is the polymer phase viscosity (mPa·s) .

#### 2.2.3. Binary Compound Flooding Stage

_{3}is the leading edge radius of the binary compound flooding stage,

_{r}is the binary phase viscosity ( mPa·s) and K

_{ro}/K

_{rr}is the oil binary mobility ratio.

## 3. Variation Rule of Dimensionless Productivity Index: Case Study of Block W of JZ9-3 Oilfield

#### 3.1. Oilfield Overview

#### 3.2. Comparative Analysis of the Dimensionless Productivity Index

#### 3.3. Factors Affecting of the Dimensionless Producyivity Index

#### 3.3.1. Water Saturation

#### 3.3.2. Water–Oil Viscosity Ratio

## 4. Conclusions

- In this study, we established a calculation model of the dimensionless productivity index suitable for the multiple development phases of oilfields, including water flooding, polymer flooding, and binary compound flooding, and we established an index suitable for medium and low permeability reservoirs. The calculation results of the dimensionless productivity index model were compared with the actual statistical results; the calculation error rate of the dimensionless productivity index for the three oilfield development phases were 4.67%, 17.65%, and 18.50%, respectively. The average error rate was 10.38% in the overall development phase.
- In the development processes of water flooding, polymer flooding, and binary compound flooding, the dimensionless productivity index shows a trend of first rising, then falling, and finally stabilizing with the increase in PV injection. Before the effects of polymer injection occur, the dimensionless productivity index increased steadily and the increase rate was 71.75% monthly, and the oil displacement effect of the reservoir worsened. After the polymer injection became effective, the dimensionless productivity index steadily decreased, the decrease rate was 52.63% monthly, and the effect of oil displacement of the reservoir improved. After injection into the binary compound system, the dimensionless productivity index further decreased at a rate of 41.65% monthly, and the oil displacement effect of the reservoir was further improved. The method can be used to evaluate the law of the production fluid of the oil well after the polymer injection and binary compound injection.
- Under constant pressure conditions, the dimensionless productivity index is positively and linearly related with water saturation; under the same PV number, the dimensionless productivity index was larger, and the water saturation was higher. The change in the dimensionless productivity index was more drastic. During the stages of water flooding, ineffective polymer flooding, effective polymer flooding, and binary compound flooding, the average increase, and decrease ranges of the dimensionless productivity index were 71.42%, 6.24%, −63.64%, and −42.31%, respectively.
- Under constant pressure conditions, the dimensionless productivity index has a positively correlated linear relationship with the water–oil viscosity ratio; under the same PV number, the dimensionless productivity index was larger, and the water–oil viscosity ratio was higher. In the stages of water flooding, ineffective polymer flooding, effective polymer flooding, and binary compound flooding, the average increases and decreases of the dimensionless productivity index were 54.27%, 42.31%, −55.34%, and −65.20%, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the leading edge propulsion of (

**a**) water flooding, (

**b**) polymer flooding, and (

**c**) binary compound flooding stages in the multi-development phases of an oilfield.

**Figure 3.**Comparison of the change in the dimensionless productivity index of the actual statistics and the new calculation model with the pore volume (PV) number injected.

**Figure 4.**The variation in the dimensionless productivity index with the PV number injected under different water saturations.

**Figure 5.**The variation in the dimensionless productivity index with the PV number injected under different water–oil viscosity ratios.

Parameter | Value |
---|---|

Porosity (%) | 0.29 |

Well spacing (m) | 350 |

Permeability (μm^{2}) | 1.50 |

Capillary pressure (Pa) | 0.20 |

Formation pressure (MPa) | 16.43 |

Water injection pore volume multiple (PV) | 0.64 |

Polycondensation pore volume multiple (PV) | 1.00 |

Note binary pore volume multiple (PV) | 1.60 |

Ground water viscosity (mPa·s) | 0.80 |

Underground crude oil viscosity (mPa·s) | 7.00 |

Polymer viscosity (mPa·s) | 9.89 |

Polymer–surfactant binary compound viscosity (mPa·s) | 12.40 |

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**MDPI and ACS Style**

Huang, B.; Liang, Q.; Fu, C.; He, C.; Song, K.; Liu, J.
A Calculation Model of the Dimensionless Productivity Index Based on Non-Piston Leading Edge Propulsion Theory in Multiple Oilfield Development Phases. *Processes* **2019**, *7*, 821.
https://doi.org/10.3390/pr7110821

**AMA Style**

Huang B, Liang Q, Fu C, He C, Song K, Liu J.
A Calculation Model of the Dimensionless Productivity Index Based on Non-Piston Leading Edge Propulsion Theory in Multiple Oilfield Development Phases. *Processes*. 2019; 7(11):821.
https://doi.org/10.3390/pr7110821

**Chicago/Turabian Style**

Huang, Bin, Qiaoyue Liang, Cheng Fu, Chunbai He, Kaoping Song, and Jinzi Liu.
2019. "A Calculation Model of the Dimensionless Productivity Index Based on Non-Piston Leading Edge Propulsion Theory in Multiple Oilfield Development Phases" *Processes* 7, no. 11: 821.
https://doi.org/10.3390/pr7110821