An uncertainty analysis method is proposed for the assessment of the residual strength of a casing subjected to wear and non-uniform load in a deep well. The influence of casing residual stress, out-of-roundness and non-uniform load is considered. The distribution of multi-source parameters related to the residual anti extrusion strength and residual anti internal pressure strength of the casing after wear are determined using the probability theory. Considering the technical casing of X101 well in Xinjiang Oilfield as an example, it is shown that the randomness of casing wear depth, formation elastic modulus and formation Poisson’s ratio are the main factors that affect the uncertainty of residual strength. The wider the confidence interval is, the greater the uncertainty range is. Compared with the calculations resulting from the proposed uncertainty analysis method, the residual strength obtained by means of traditional single value calculation method is either larger or smaller, which leads to the conclusion that the residual strength should be considered in terms of a range of probabilities rather than a single value.

With the increase of deep well drilling and ultra-deep well drilling, the wear of the drill string to the inner wall of the technical casing has always being a problem that cannot be ignored. According to estimates, there exist casing wear issues and fracture or collapse accidents in more than 20 deep wells, including Yingke 1, Kecan 1, Dongqiu 5, Chegu 204, Yacheng 13-1-3, Bohai Caofeidian 18-2-1, Bozhong 13-1-12, Haoke 1, Quele 1, Yangxia 1 and Wushen 1 [

Studies have explored the residual strength of worn casing by using finite element method and theoretical analysis method. For instance, Jin et al. [

In deep wells, the movement of drill pipe is complex. As the lateral force acting on the casing wall increases, the friction and wear between the drill string and the casing during drilling are intensified. Therefore, this directly affects casing design strength and safety and reliability of well completion and production. El-Sayed et al. [

In fact, due to the randomness of process and formation parameters, the parameters affecting the casing strength of deep walls are also difficult to achieve accurately. Considering the uncertainty of casing load in deep wells and the randomness of casing geometric size and performance parameters [

The above investigations have gradually analyzed the effects of geometric parameters, performance parameters and process parameters on the residual strength of casing after wear. However, the above parameters have the characteristics of randomness, and there are few studies on the influence of parameter randomness on the residual strength of casing after wear under non-uniform load. Therefore, the influence of the randomness of multi-source parameters such as wear depth, residual stress and formation non-uniform load on the uncertainty of residual strength after casing wear is analyzed in this paper, and the residual strength after casing wear is quantitatively evaluated, so as to provide guidance on on-site casing safety selection and design.

Considering the effects of casing residual stress, out-of-roundness, unevenness and formation non-uniform load on casing strength, the residual collapse strength model and residual internal pressure strength model after casing wear can be obtained based on the casing wear efficiency model proposed by White et al. [

The residual anti-collapse model under non-uniform load after casing wear is:

The parameter description in

The residual internal pressure strength of casing after wear is:

where,

Casing residual collapse strength model and residual internal pressure strength model are functions of multi-source parameters such as wear parameters, casing performance parameters, casing geometric parameters and formation parameters. However, the multi-source parameters of deep well casings have the characteristics of randomness. According to a large number of on-site process parameters and data provided by the manufacturer, nonlinear fitting is performed according to the literature [

Through the correlation test of distribution type between the statistical results and the fitting curve [

Parameters | Normal distribution | Lognormal distribution | Weibull distribution | Voigt distribution |
---|---|---|---|---|

Outer diameter of casing | 0.50 | 0.88 | 0.67 | |

Wall thickness | 0.47 | 0.82 | 0.65 | |

Elastic modulus of casing | 0.57 | 0.86 | 0.53 | |

Poisson’s ratio of casing | 0.40 | 0.76 | 0.55 | |

Residual stress | 0.87 | 0.88 | 0.63 | |

Wear-thickness | 0.89 | 0.90 | 0.60 | |

Yield strength | 0.78 | 0.91 | 0.65 | |

Formation elastic modulus | 0.79 | 0.23 | 0.67 | |

Formation elastic modulus | 0.75 | 0.35 | 0.55 |

The fitting results show that the statistical laws of the above parameters are in good agreement with the normal distribution, and the correlation coefficients with the normal distribution are greater than 0.92, which is a significant strong correlation. The correlation coefficient of the normal distribution is greater than the average correlation coefficient of the lognormal distribution, Weibull distribution and Voigt distribution, so the statistical results of the above parameters fitted with the normal distribution.

The variation coefficient is an index to measure the degree of randomness of a parameter. The selection of the main randomness parameter can reduce the dimensionality and also lower the computational workload. The variation coefficient can be defined as [

The statistical results show that the variation coefficients of wear depth, formation elastic modulus and formation Poisson’s ratio (strip width as shown in

where,

Since the randomness of elastic modulus, Poisson’s ratio and wear thickness are the main factors impacting the uncertainty of residual collapse strength of casing after wear, they can be regarded as three-dimensional random variables that affect the uncertainty of residual collapse strength of casing. Let

Therefore,

Then,

The partial derivative of

Let the probability density function of (

where,

According to the probability theory, the probability density function of

Since the wear depth, the formation elastic modulus and the formation Poisson’s ratio are normally distributed, the normal distribution function can be substituted into

Integrating

Similarly, the residual internal pressure strength after casing wear can be expressed as:

where,

Then:

The partial derivative of

Therefore, the probability density function of residual collapse strength after casing abrasion can be written as:

The corresponding cumulative probability function is:

According to

According to

When the cumulative probabilities of casing residual strength are u and v (V > U), the set elements, and the corresponding residual strength are:

where,

Consequently, the distribution interval of casing residual strength with the confidence level (

Based on the above calculation method of residual strength after wear, the residual strength of casing after wear of more than 10 wells in the southern edge of Xinjiang Oilfield was calculated and analyzed. Taking the technical casing of well X101 as an example, the structure of well X101 is shown in

Well section (m) | Wear depth (mm) | Mean value of non-uniformity coefficient | Percentage reduction of casing collapse strength (%) | Percentage reduction of casing internal pressure strength (%) | Whether the casing has an accident or not |
---|---|---|---|---|---|

0–500 | 2.1–2.8 | 1.2 | 15 | 18 | None |

500–1000 | 1.8–2.7 | 1.35 | 16 | 15 | None |

1000–1500 | 1.0–1.3 | 1.35 | 24 | 18 | None |

1500–2500 | 1.1–2.3 | 1.21 | 16 | 13 | None |

2500–3500 | 0.9–1.5 | 1.25 | 9 | 13 | None |

3500–4900 | 1.1–1.3 | 1.25 | 8 | 9 | None |

4900–5300 | 0.5–0.8 | 1.37 | 19 | 6 | Collapse |

5300–5900 | 0.0–0.7 | 1.15 | 3 | 3 | None |

The residual internal pressure strength after casing wear and tear is mainly affected by the wear depth. Therefore, the residual internal pressure strength of the casing is roughly negatively correlated with the wear depth (compared with

As shown in

It can also be seen from

Therefore, compared with a single curve, the residual strength interval with certain reliability information is more concerned with the residual strength value that may exist at any well depth. For engineering designers, it is more practical and effective to grasp the possible range of residual strength than to understand a single value. Designers can formulate corresponding emergency standby scheme according to the interval results. Thus, the impact of accidents on the drilling process can be reduced.

In this paper, an uncertainty analysis approach is proposed for evaluating the residual strength of a casing subjected to wear and non-uniform loads in deep wells. The three aspects can be summarized as follows:

Considering the influence of casing geometric, casing performance, drilling process and formation parameters on the residual strength after casing wear, the variation coefficients of multi-source parameters are analyzed. It is established that the randomness of wear depth, formation elastic modulus and formation Poisson’s ratio are the main factors affecting the uncertainty of collapse strength after casing wear. The multi-source parameter sensitivity parameter analysis can reduce the dimension of the residual strength model after casing wear. The statistical results indicate that the sensitivity random parameters primarily obey the normal distribution.

The probability density function and cumulative probability function of residual collapse strength and residual internal pressure strength after casing wear are derived by the probability theory. The confidence interval expression method of residual strength after casing wear in deep wells with a specific probability can be achieved.

The analysis and calculation of several wells in the southern edge of Xinjiang oilfield demonstrate that compared with the traditional single value calculation method, the uncertainty calculation method of residual strength can determine a more reasonable safety threshold of casing strength. This can provide a more powerful basis for the selection of ultra-deep well casing.

Thanks are due to Qi Liu for assistance with the experiments and to Meng Li for valuable discussion.

where, _{unweff}_{s}_{c}_{H} is the maximum horizontal _{h} is the minimum horizontal _{c}_{R} is residual stress of casing, MPa; _{y} is the casing yield strength, MPa;