# A New Analytical Model to Evaluate Uncertainty of Wellbore Collapse Pressure Based on Advantageous Synergies of Different Strength Criteria

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## Abstract

Considering the uncertainties of rock mechanical parameters, formation pressure and in situ stresses, the uncertainty of the wellbore collapse pressure should be evaluated. Before the uncertainty of evaluation, the collapse pressure model needs to be selected reasonably. In this paper, a new model was proposed to evaluate the collapse pressure, considering the advantageous synergies of different strength criteria. Especially, weight coefficients were introduced to represent the effect of different strength criteria on the collapse pressure, and were calculated by analytic hierarchy process. Then, an analytical method was proposed to address the uncertainty of the collapse pressure based on improved Rosenbluthe method, considering the new collapse pressure model. By means of the analytical method, the collapse pressure was obtained as the probability distribution under the condition that the uncertainties of input parameters were quantified based on well log data. More importantly, the analytical method was validated by Monte Carlo simulation. The results show that the probability distribution agrees very well between the analytical method and Monte Carlo simulation. Note that, the new collapse pressure model has the best matching for the probability distribution desired, which can be treated as the advantageous synergies of the new collapse pressure model.

## Keywords

Wellbore collapse pressure Uncertainty Probability distribution Analytical method Advantageous synergies## List of symbols

- \(A\)
Coefficient related to the internal friction angle and the cohesion in Modified Lade criterion

- \({A_1}\)
Coefficient related to the internal friction angle and the cohesion in Mogi-Coulomb criterion

- \(B\)
Coefficient related to the internal friction angle in Modified Lade criterion

- \({B_1}\)
Coefficient related to the internal friction angle in Mogi-Coulomb criterion

- \(c\)
Cohesion

- \({E_{\text{s}}}\)
The static elastic module

- \(g\)
The gravitational acceleration

- \({I_1}\)
The first stress invariant

- \({J_2}\)
The second deviatoric stress invariant

- \(m\)
Coefficient related to the internal friction angle

- \(k\)
Coefficient related to the internal friction angle and the cohesion

- \({P_0}\)
The drilling fluid pressure

- \({P_{\text{p}}}\)
The formation pressure

- \(P_{{\text{p}}}^{0}\)
The hydrostatic pressure

- \(x\)
The exponent constant

- \({y^*}\)
Coefficient used in improved Rosenbluthe method

- \(y_{{^{i}}}^{+}\)
Coefficient used in improved Rosenbluthe method

- \(y_{{^{i}}}^{ - }\)
Coefficient used in improved Rosenbluthe method

- \(z\)
Depth

- \(\alpha\)
Biot’s coefficient

- \(\Delta {t_{{\text{c\_measured}}}}\)
The measured compressive sonic transit time by well logging

- \(\Delta {t_{{\text{c\_normal}}}}\)
The normal compressive sonic transit time in shale obtained from normal trend line

- \({\varepsilon _x}\)
The tectonic strain along the horizontal maximum stress direction

- \({\varepsilon _y}\)
The tectonic strains along the horizontal minimum stress direction

- \({\eta _1}\)
Weight coefficient, representing the effect of Mohr–Coulomb criterion on the collapse pressure

- \({\eta _2}\)
Weight coefficient, representing the effect of Drucker–Prager criterion on the collapse pressure

- \({\eta _3}\)
Weight coefficient, representing the effect of modified Lade criterion on the collapse pressure

- \({\eta _4}\)
Weight coefficient, representing the effect of Mogi–Coulomb criterion on the collapse pressure

- \({\nu _{\text{s}}}\)
The static Poisson’s ratio

- \(\rho\)
The bulk density

- \({\sigma _{\text{H}}}\)
The horizontal maximum in situ stress

- \({\sigma _{\text{h}}}\)
The horizontal minimum in situ stress

- \({\sigma _{\text{v}}}\)
The vertical minimum in situ stress

- \(\sigma _{r}^{\prime }\)
The radial effective stress

- \(\sigma _{\theta }^{\prime }\)
The hoop effective stress

- \(\sigma _{z}^{\prime }\)
The vertical effective stress

- \(\varphi\)
The internal friction angle

## Notes

### Acknowledgements

The authors are very much indebted to the Projects Supported by PetroChina Innovation Foundation (2018D-5007-0309), the Fundamental Research Funds for the Central Universities (16CX02036A), and Applied Basic Research of Qindao (15-9-1-71-jch) for the financial support.

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