The purpose of this work is to investigate typical fracture and collapse models with respect to accuracies in the input data. Uncertainties in the input data will be considered to show how they contribute to the cumulative uncertainties in model predictions. In this approach, the input parameters are assigned appropriate probability distributions. The distributions are then applied in the wellbore-stability models. By means of Monte Carlo simulation, the uncertainties are propagated and outputs, which follow a probability distribution, are generated.

## Introduction

Wellbore-stability analysis is necessary for safe drilling operations, especially now that oil and gas operators venture into more-challenging environments. A wide range of parameters is required for accurate study, many of which are subject to uncertainties caused by measurement errors. Error also can be introduced into data through the methods of interpretation used. Epistemic error, arising from imperfect human knowledge of a system, is another source of input uncertainties. Analytical models used for wellbore-stability analysis are also often associated with uncertainties. Mathematical modeling algorithms only try to approximate physical processes and are not true representations of the problems under study. The modelers should be aware of the imprecision and limitations of these physical models. Thus, output uncertainty stems from the variations in input data and uncertainties caused by wellbore-stability-modeling processes.

Expected values give no information about uncertainty. Deterministic estimation of the downhole pressure limits provides only single-point values that lack variability information. Instead, probability distributions can be used. With this approach, cumulative uncertainties in the output predictions can be quantified, leading to a more-informed decision.

## In-Situ Stress Field

For a given formation, the starting point in wellbore-stability analysis is the in-situ or pre-existing stress state. Knowledge of the stress state is key to handling borehole problems such as fracturing, lost circulation, collapse, and sand production. The in-situ stress state is normally assumed to coincide with vertical and horizontal directions. In relaxed depositional basins, the values of these horizontal stresses are usually lower than those of the vertical stress. The horizontal-stress magnitudes, however, may exceed those of the vertical stress in strongly tectonic regions.

A stress state can be defined as normal-fault, reverse-fault, or strike/slip-fault state of stress. The normal-fault stress state is assumed in this work. If the magnitudes of the three principal stresses and the direction of one of the stresses are known, then the stress state can be specified. The stress concentration is usually very high around the borehole wall. This effect decreases rapidly away from the hole. At a long distance from the wellbore, the principal in-situ stresses are undisturbed and lie along their in-situ directions.

## Wellbore Instabilities

Wellbore instabilities include such phenomena as breaking of intact rock around the wellbore because of high stress concentration or sudden temperature variations; loosening of rock fragments; and fracture extension from the wellbore into the formation, sometimes with a significant loss of drilling fluid. They also consist of such mechanisms as failure of rock around the borehole because of interaction with drilling fluid, squeezing of soft rocks such as salt and shales into the wellbore, and activation of pre-existing faults that intersect the wellbore.

A significant amount of breakouts resulting from mechanical failure around a wellbore can cause severe stability problems such as excessive torque and drag, sudden increase in bottomhole pressure, and stuck pipe. By application of good drilling procedures, cavings can be removed efficiently and safely. However, borehole instability remains the major cause of lost drilling time and lost downhole equipment.

M**echanisms of Wellbore Failure. Shear Failure. **The von Mises yield condition and the Mohr-Coulomb shear-failure criterion are the most commonly used hypotheses for evaluating rock shear failure. The von Mises condition considers the three principal in-situ stresses as active participants in wellbore compressive failure. Well planners frequently use the position of the stress state relative to the failure envelope as a yardstick for evaluating wellbore stability.

The Mohr-Coulomb model for shear failure neglects the intermediate principal stress but captures the effect of the directional strengths of shales. The maximum stress is the tangential stress, followed by the axial stress and the radial stress (well pressure). This model predicts a minimum well pressure that can cause wellbore collapse in the direction of least in-situ stress. Wellbore collapse is a result of shear failure of rock around a borehole. To prevent this from occurring during drilling, mud pressure must be such that it will effectively carry the load caused by the in-situ stresses around the wellbore.

* Tensile Failure. *Generally, rock formations are weak in tension. In most cases, the tensile strength of rock is set to zero on the premise that drilling-induced fractures initiate in flaws, joints, or pre-existing fractures around the wellbore.

The analysis of tensile failure involves application of the effective-stress concept. This implies that a formation fails in tension when the least effective principal stress exceeds the rock tensile strength. Increase in a wellbore pressure will cause the effective tangential stress to decrease. The effective radial stress remains constant, while the effective axial stress increases. At a certain well pressure, the value of the hoop stress becomes zero, and a vertical fracture initiates as the stress goes into tension. Thus, drilling-induced fractures are associated with minimum tangential stress. Critical fracture pressure is the well pressure beyond which a wellbore will fracture in tension.

Considering the two wellbore-failure mechanisms, the absence of sufficient well pressure capable of supporting the load caused by high stress concentration around the wellbore can lead to a wellbore collapse, but an excessive mud weight will cause borehole fracturing, sometimes with a loss of drilling fluid into the formation.

## Wellbore-Stability Modeling

Both numerical simulators and analytical models are used for wellbore-stability analyses. These tools do not provide accurate descriptions of geological processes, mainly because of limited human knowledge of subsurface strata. However, on the basis of geomechanical principles, drilling engineers can estimate fracturing and collapse pressures by use of mathematical approximations, which describe the relationships among input variables.

In this work, a vertical-well configuration is considered. The formation around the wellbore is assumed to be linearly elastic. Therefore, complex material behavior such as nonlinear elasticity or elastoplasticity is not treated. If an inclined wellbore is assumed, the in-situ stress field must reflect borehole inclination and direction.

## Uncertainty in the Input Data

Measurement and interpretation errors are the major causes of input-parameter uncertainties. This section briefly presents some important parameters affecting wellbore stability and their assumed measurement or prediction uncertainties.

Pore pressure can be estimated with direct measurements. For a very-low-permeability rock such as shale, indirect methods, which use drilling data and well logs, are used. If a nonpermeable barrier exists over an interval, a discontinuous pore-pressure profile is expected. Therefore, higher uncertainty is associated with the indirect pore-pressure measurements than with the direct estimations.

We have mentioned that the determination of the in-situ stress state is crucial to wellbore-stability analyses; stress magnitudes will ultimately affect the accuracy of the model predictions. These should be considered uncertain parameters because there are no existing methods to measure the stresses accurately. However, the in-situ stresses can be estimated by use of several methods. The overburden stress is calculated by integrating the bulk density of drilled cuttings over the depth interval, with values obtained every 30 m. At greater depths, density or sonic logs are used to estimate overburden stress.

The minimum horizontal stress can be estimated with leakoff-test (LOT) data, by interpreting the slope change (deviation from linearity) on an LOT plot when pressure drops after a mud pump is stopped. In a relaxed depositional environment, equal horizontal stresses are normally assumed. The value of maximum horizontal stress is more difficult to estimate with direct methods. With the inversion technique, an improved accuracy in the estimations of both magnitude and direction of the two horizontal stresses can be obtained. In addition, the rock-mechanical properties such as cohesive rock strength and rock-friction angle are often derived from indirect measurements by interpreting sonic logs. There is higher uncertainty in the estimation of cohesive rock strength than in the estimation of internal friction.

For a discussion of the example case and simulation results, please see the complete paper.

## Sensitivity Analysis

A sensitivity analysis is conducted to ascertain input factors that are most responsible for output variability and that require further research. A modeler can thereby justify whether input-parameter estimates are accurate enough for a model to give reliable predictions. If not, more work will be directed toward improving the estimations of these uncertain parameters.

The sensitivity analysis is required to understand how the model predictions respond to changes in input variables, thereby complementing the analyses presented so far in this work. This allows determination of the parameters that contribute most to the cumulative uncertainties in the critical fracture- and collapse-pressure predictions. The results from the analyses will be useful when calibrating the models against offset-well data.

For a full discussion of the differential method used in this analysis, please see the complete paper.

**Monte Carlo Sensitivity Analysis. **In this analysis, the input parameter that is considered uncertain assumes a probability distribution, while other parameters are treated as fixed factors. This is done to quantify the individual contribution of each parameter to the cumulative output variances. The base-case scenario, in which all the input data are treated as random variables, serves as a standard for measuring the degree of the output variability caused by the uncertainty in a given parameter. In the probabilistic approach, only triangular distribution will be considered.

In each run, both the uncertain parameter and fixed factors are applied in the wellbore-stability models. The outputs, which follow a probability distribution, are generated after 600,000 simulations. In the results, the fracture-pressure distribution for the minimum-horizontal-stress uncertainty has the highest spread or variance compared with other input parameters; therefore, minimum horizontal stress is the most influential factor. For the collapse pressure, the sensitivity determined for the cohesive rock strength shows that that parameter is the most important input factor responsible for the uncertainty in the pressure.

This article, written by *JPT* Technology Editor Chris Carpenter, contains highlights of paper SPE 166788, “Uncertainty Evaluation of Wellbore-Stability-Model Predictions,” **John Emeka Udegbunam, Bernt Sigve Aadnøy, **SPE, and **Kjell Kåre Fjelde,** SPE, University of Stavanger, prepared for the 2013 SPE/IADC Middle East Drilling Technology Conference and Exhibition, Dubai, 7–9 October. The paper has not been peer reviewed.