# Critical Drawdown Pressure of Depleted Reservoir

## Authors

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DOI: 10.1007/s40098-013-0071-5

- Cite this article as:
- Yan, C., Deng, J., Lai, X. et al. Indian Geotech J (2014) 44: 101. doi:10.1007/s40098-013-0071-5

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## Abstract

For long time production, most oilfields have entered the later development stage, and the pore pressure is seriously depleted. The pressure depletion will affect in situ stress, and change the stress state around the borehole, and then the affect the critical drawdown pressure to cause sand production. The theoretical formula of two horizontal in situ stress changes with pore pressure is obtained based on generalized Hoek’s law, and then the stress distribution formula around the borehole in pressure depleted reservoir is established. The calculation model of sand production critical bottom hole flowing pressure in depleted reservoir is established using the Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively. The influence of pressure depletion on critical drawdown pressure is analyzed. The results show that: with the pore pressure decreasing, the horizontal in situ stress and critical drawdown pressure become smaller; the error of prediction model based on Mogi–Coulomb criterion is the minimum, and it is more in line with the actual field data; the prediction model based on Mohr–Coulomb criterion is conservative but most safe. The establishment of prediction model provides a guidance for actual production decisions in pressure depleted reservoir.

### Keywords

Pressure depletionIn-situ stressSand productionCritical drawdown pressureRock strength criterion## Introduction

Sand production, which is the failure of formation due to in situ stress and fluid flow, is widespread during production of oil and gas sandstone reservoirs [20]. A great deal of work has been down in the general area of sand production (e.g. [13, 17, 18, 23]). Some models view sand production as a mixed hydro mechanical process [5, 15]. Some others base their sanding model solely on mechanical stability [3, 21]. For brittle rock with high strength, sand production occur only mechanical failure happens [24]. With the development of oil and gas, if the formation energy can not effectively supply, the reservoir pressure will deplete, which will increase the risk of sand production [12]. Sand production of depleted reservoir has been researched [1, 16, 22, 24], but hadn’t considered the influence of pressure depletion on in situ stress. Mohr–Coulomb criterion and Drucker–Prager criterion are the mostly used failure criterions for sanding prediction [4, 11, 14]. But the Mohr–Coulomb criterion is too conservative while the Drucker–Prager criterion tends to be unsafe [6, 25]. Mogi–Coulomb criterion has been proved more suitable for evaluate borehole breakouts [2, 26]. This paper analyzes the effect of pressure depletion on in situ stress, and critical bottom hole flowing pressure model in pressure depleted reservoir is established based on Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively.

## The Effect of Pressure Depletion on In-situ Stress

_{p}is the original pore pressure; \( \alpha \) is effective stress coefficient.

_{p1}, keep the overburden pressure unchanged, and then the constitutive relationship in horizontal direction satisfies the following equation:

## Borehole Stress Analysis

## Critical Bottom Hole Flowing Pressure Prediction Model

In the open-hole wells, the formation generally have high strength, only the formation is failure, it is likely to cause sand production [8]. Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion are selected in this paper for researching on critical bottom hole flowing pressure in pressure depleted reservoir.

### Based on the Mohr–Coulomb Criterion

When \( P_{P1} = P_{P} \), Eq. (9) will be simplified to the critical bottom hole flowing pressure prediction model at the initial development stage of the oilfield without depletion.

### Based on the Drucker–Prager Criterion

### Based on the Mogi–Coulumb Criterion

## Analysis

When \( P_{P1} = P_{P} \), Eq. (28) will be simplified to the model at the initial development stage of the oilfield.

Counting parameters

\( \sigma_{H} \) (MPa) | \( \sigma_{h} \) (MPa) | \( \sigma_{v} \) (MPa) | μ | \( \alpha \) | \( f \) | \( \phi \) (°) | \( C \) (MPa) |
---|---|---|---|---|---|---|---|

54 | 48 | 42 | 0.25 | 0.7 | 0.2 | 37 | 12 |

When the reservoir pressure decreases to 5.5 MPa, the critical drawdown pressure based on Mohr–Coulomb criterion is reduced to zero. When the reservoir pressure continues to decrease, the critical drawdown pressure based on Mogi–Coulomb criterion and Drucker–Prager criterion also gradually decreases to zero. At this time, no matter how much the drawdown pressure is, sand production will occur. Even the strength of consolidated sandstone is high, when the reservoir pressure decreases to a certain extent, sand production problems may be faced. If production continues, sand control measures must be taken.

## Case Study

Comparison of calculation results of critical drawdown pressure

Well No. | Critical drawdown pressure (MPa) | Critical drawdown pressure by testing (MPa) | ||
---|---|---|---|---|

Mohr–C | D–P | Mogi–C | ||

A | 6.7 | 9.4 | 8.0 | 8.7 |

B | 5.7 | 8.2 | 6.8 | 6.2 |

C | 6.2 | 8.9 | 7.6 | 7.9 |

D | 5.3 | 7.4 | 6.4 | 5.7 |

Average relative error (%) | 14.7 | 20.7 | 8.6 |

It can be seen from the comparison, the critical drawdown pressure based on Mogi–C has the minimum error with the actual value; it is followed by that based on Mohr–C; the critical drawdown pressure based on D–P has the maximum error. But the calculated results based on Mohr–C are lower than the actual values, and it indicates that the critical drawdown pressure based on Mohr–C is the safest.

The effect of intermediate principal stress is important in rock strength [19], so Mohr–C is too conservative, due to ignoring the strengthening effect of the intermediate principal stress. The influence of intermediate principal stress is considered in D–P, but the strengthening effects of the intermediate principal stress and minimum principle stress are the same in this criterion. The D–P has been proved to overestimate the intermediate principal stress effect and predict a higher strength than reality [10, 6]. The Mogi–C neither ignores the strengthening effect of intermediate principal stress, as is done by Mohr–C, nor does it predict a strength as unrealistically high as does the D–P. The Mogi–C is shown to accurately model laboratory failure data on a range of different rocks types [2, 26], so the critical drawdown pressure based on Mogi–C is the closest to the actual value.

## Conclusions

- (1)
The in situ stress change law along with the reservoir pressure depletion is derived, and based on this the dynamic stress distribution around borehole is obtained for the pressure depleted reservoir in the development process.

- (2)
The critical bottom hole flowing pressure calculation models in pressure depleted reservoir are established based on Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively.

- (3)
The critical drawdown pressure based on Drucker–Prager criterion is the maximum, and that based on Mogi–Coulomb criterion is less, and that based on Mohr–Coulomb criterion is the minimum. The reservoir pressure depletion will lead to critical drawdown pressure decrease, but the decreasing rate is less than the pore pressure.

- (4)
The critical drawdown pressure based on Mogi–Coulomb criterion is the closest to the actual value, but that based on Mohr–Coulomb criterion is the safest.

## Acknowledgments

The authors gratefully acknowledge the support of Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51221003), National Natural Science Foundation Project of China (Grant No. 51174219) and National Oil and Gas Major Project (Grant No. 2011ZX05009-005).