# Determination of minimum and maximum stress profiles using wellbore failure evidences: a case study—a deep oil well in the southwest of Iran

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## Abstract

The main objective of this paper is estimating the horizontal stresses and calibration of the log-derived horizontal stress profiles in deep oil wells according to their wall failure evidences, including both compressive (breakouts) and tensile failures (drilling-induced tensile fractures). Estimation of the horizontal stress profiles using well logs is one of the standard methods in the oil industry. Another method for estimating horizontal stresses is analyzing failure evidences in the wellbore wall. By integrating these two methods, a practical strategy was followed in this research to determine the horizontal stress profiles. By this strategy, minimum and maximum horizontal stress profiles are determined in such a way that the stress concentration at the wellbore wall at the tensile fracture identified depths, exceeds the formation tensile strength, while at the breakouts identified depths it exceeds the compressive strength of the formation. An advantage of this procedure is that does not require to measure in situ stresses. Also, due to the presence of a large number of breakouts and the induced fractures detected in different zones of a deep wellbore, log-derived stress profile calibration is done using the stress state in different zones that causes to increase the accuracy and reliability of the obtained horizontal stress profile. The proposed solution was applied to determine the horizontal stress profiles in a deep oil well in the southwest of Iran as a real case.

## Keywords

Stress profile Wellbore failure Well logs Formation strength Tectonic strains## Introduction

Orientation and magnitude of principal stresses in hydrocarbon fields in are considered as a very important information in Petroleum Engineering. The stress state in the earth’s crust controls stress concentration around a wellbore and therefore plays a critical role in wellbore instability, well stimulation operations, fluid flow in fractured reservoirs, and sand production (Bell 1996; Willson et al. 1999). Knowledge of the stress state is not only important in the petroleum industry, but it is also vital in geotechnical applications and other solid earth sciences. Thus, widespread studies have been conducted and different measurement methods and theories or empirical relationships have been established for estimating the stress state in the earth’s crust. The underground stress state, with a simplifying assumption, can be described by three mutually orthogonal principal stress components, i.e., a vertical stress (*S* _{v}) and two horizontal principal stresses (*S* _{H} and *S* _{h}). *S* _{v} is a factor due to weight of the overburden, and its magnitude can be calculated using overburden density log (McGarr and Gay 1978). According to Zoback et al. (2003), hydraulic fracturing test, leak-off test (LOT), and measurement of pressure while drilling (PWD) are three common methods for the least principal stress measurement in the deep wells. The magnitude of minimum horizontal stress (*S* _{h}) in the normal and strike-slip faulting regimes can be measured using the mentioned methods. Measurement of the maximum horizontal stress (*S* _{H}) using the mentioned methods is not directly possible (Zoback 2010) and is required to use theoretical and empirical relationships to estimate *S* _{H} based on *S* _{h} and *S* _{v} values (Binh et al. 2011; Zang et al. 2012). Also, other than being expensive and time-consuming, these tests cannot provide continuous horizontal stress profiles (Song and Hareland 2012; Sone and Zoback 2014). Therefore, development of the theoretical relationships for predicting stress state in deep oil wells is always of interest for the oil industry. In the following sections, first, the conventional techniques for determining horizontal stress profiles using the well logs are introduced. A discussion of the relationship between in situ stresses and wall rock strength at formation depths of breakouts and drilling-induced fractures will be given after that. Next, using a mathematically based strategy, the log-derived horizontal stress profiles are calibrated using in situ stress conditions that it can be estimated based on the observed wellbore wall failures at different depths. Finally, the horizontal stress profiles of a deep oil well in the southwest of Iran are determined using the introduced strategy and their results will be checked by analyzing compressive and tensile wellbore failures.

## Determination of stress profiles using the well logs (log-derived stress profile)

*v*is Poisson’s ratio,

*S*

_{V}is vertical total stress,

*α*is Biot’s poroelastic constant, and

*P*

_{P}is pore pressure.

*σ*

_{tect}is calibration or additional tectonic stress.

By adding the tectonic stress term to Eq. 1, the horizontal stress profile can be displaced and its conformity with measured in situ stresses can be increased. The adding tectonic stress value is determined through difference between the measured value of stress at a specific depth and the stress calculated by Eq. 1 at the same depth.

*ε*

_{x}and

*ε*

_{y}are tectonic strains in horizontal plane,

*E*is the Young’s modulus,

*α*

_{t}is thermal coefficient of expansion, and Δ

*T*is temperature change (temperature at a particular depth minus the ambient surface temperature). Blanton and Olson (1999), for the sake of simplification, assumed that the horizontal strain in one direction is equal to zero (i.e.,

*ε*

_{x}or

*ε*

_{y}= 0). In Eqs. 3 and 4 in addition to tectonic strain, the effect of temperature was considered in the calculation of horizontal stresses. There are two advantages in using the tectonic strain instead of the tectonic stress. The first one is to increase the accuracy of the minimum horizontal stress (

*S*

_{h}), and the second one is possibility of calculating the maximum horizontal stress (

*S*

_{H}) profile using well logs.

To achieve a log-derived stress profile with acceptable accuracy, at the least, the stress state in one point of well should be defined to calibrate the stress profile, which is required to carry out an in situ stress test in the well. Because the in situ stress measurement data are not available for many of deep oil wells, calibration of the log-derived horizontal stress profile, and reaching to a reliable estimation of the horizontal stress values are not possible for all drilled wells.

## Wellbore failures analysis

*μ*is the coefficient of frictional sliding on an optimally oriented pre-existing fault.

*S*

_{1}and

*S*

_{3}are the maximum and minimum principal stresses which can be corresponded to

*S*

_{H},

*S*

_{h}, or

*S*

_{v}using Anderson’s faulting theory. Then, the range of allowable values of horizontal stresses is limited based on non-catastrophic failures of the wellbore with respect to the rock strength of the wellbore wall. The most common non-catastrophic borehole failures are breakouts that formation will be lead to enlargement of the wellbore cross section (Bell and Gough 1979). The enlargement of the wellbore is caused by the development of intersecting conjugate shear planes, which cause pieces of the borehole wall to spall off (Reinecker et al. 2003). Therefore, breakouts can be detected using the (four-arm) caliper log tool that provides two diameters of the borehole cross section (Reinecker et al. 2003). The Formation mechanism of the breakout was first discussed by Gough and Bell (1981) and later expanded by Zoback et al. (1985). Breakouts are formed in symmetric zones around the borehole at the azimuth of least horizontal principal stress (

*θ*= 90°, 270°) where the circumferential stress (\(\sigma_{\theta \theta }\)) is the largest and exceeds the formation compressive strength.

*P*is the difference between the wellbore pressure (mud weight,

*P*

_{m}) and pore pressure (

*P*

_{P}), and

*Nϕ*is calculated using rock friction angle (

*ϕ*).

*σ*

_{ θθ }can be calculated as:

*σ*

^{∆T }represents the thermal stresses arising from the difference between the mud temperature and formation temperature (∆

*T*), and ∆

*P*is the difference between the wellbore pressure (mud weight,

*P*

_{m}) and pore pressure (

*P*

_{P}).

*T*

_{0}) of the formation (Aadnoy 1990):

*σ*

_{ θθ }) around a vertical borehole is minimum in the S

_{H}direction (

*θ*= 0, 180) and can be calculated as:

## Calibration of log-derived stress profiles using wellbore failure evidences

For estimating the maximum and minimum horizontal stresses, both mentioned methods have some limitations in practice. Determining the log-derived stress profiles using Blanton and Olson (1999) method (Eqs. 3, 4) the stress state must be known in a depth of the well that is not available for many deep oil wells. Also, for determination of a particular range of horizontal stresses using analysis of the wellbore failure evidences, both breakout and induced tensile fracture must be available at the same depth of the wellbore. Thus, application of this method becomes limited, as in many cases, the breakouts and induced tensile fractures are not formed at same depths.

*S*

_{h}and

*S*

_{H}in Eqs. 8 and 10 that, respectively, the stress condition required to form the breakout and induced tensile fractures by the equal defined terms from Eqs. 3 and 4, Eqs. 11 and 12 can be obtained as:

*S*

_{h}is the horizontal stress arising from the overbidding loading that is calculated using Eq. 1.

To solve Eqs. 11 and 12, they must be assumed equal to zero or equal to a certain amount. Breakout incidence in a wellbore wall indicates that Eq. 11 is equal or greater than zero. Also, the occurrence of induced fracture suggests that Eq. 12 is equal or less than zero. These equations can be considered approximately equal to zero at both end points of breakouts and induced fractures. Therefore, the data obtained from these points are used for solving Eqs. 11 and 12. Hence, in Eqs. 11 and 12, tectonic strain parameters of *ε* _{x} and *ε* _{y} are the only unknown, using two calibration points, these equations will be solved and these two unknown parameters can be determined. With replacing elastic properties of the wall rock and the other parameters for each end point of a wellbore failure, an equation will be obtained. However, considering the inaccuracies and errors in involved data, the use of more calibration points can be caused to increase the accuracy level of the results. In many cases, there are several breakouts and induced fractures at different depths of a wellbore and, therefore, more than two equations can be obtained. Hence, to improve the accuracy level of the estimated tectonic strain parameters, the least-squares approach was used in this study.

## Applications and validation of the results

As shown in Fig. 1, the orientation of borehole breakouts that indicate azimuth of *S* _{h} is about 43° and the orientation of induced fractures that indicate azimuth of *S* _{H} is about 133°.

Using FMI images, 24 points, i.e., 12 breakouts and 12 induced fractures, were selected in the studied interval. On the basis of the proposed solution, the optimum values for *ε* _{x} and *ε* _{y} (two unknown parameters) with a minimum error in these 24 points are 5.1e−4 and −6.8e−4, respectively.

*W*

_{bo}):

*θ*

_{b}=

*π*−

*W*

_{bo}.

*µ*(Handin 1969; Nelson et al. 2005). FMI images of the studied well indicate the presence of a breakout with an average width of 46° at depth of 4143 m, but there is not induced fracture in this depth. The possible stress ranges in the absence or presence of induced fracture (dash line) are shown in Fig. 5. The dash-dot line shows the magnitude of

*S*

_{H}(for a given value of

*S*

_{h}) that is required to cause breakouts with a width of 46° (

*θ*= 67) by using Eq. 13 and considering

*P*

_{P}= 47 MPa, ∆

*P*= 3.8 MPa, and UCS = 75.9 MPa. The effect of wellbore cooling was disregarded, because there was no significant temperature difference between the drilling mud and the formation in the studied interval. However, as stated by Zoback et al. (2003), even for the large temperature differences (∆

*T*) such as 25 °C, significant changes do not exert on the estimated stresses.

As shown in Fig. 5, the horizontal stresses derived from the calibrated stress profile at the mentioned depth, fall within the possible range of stresses (see study carried out by Zoback et al. 2003).

As displayed in Fig. 6, there is an acceptable compliance between the calculated and the actual breakout widths that indicates accuracy level of the estimated horizontal stresses.

## Conclusions

In this research, a practical strategy for calibration of the log-derived stress profile using failure evidences of the wells was developed. In this strategy, as discussed earlier, the minimum and maximum horizontal stress profiles were determined. In the proposed approach, values of two tectonic strains in horizontal plane (*ε* _{x} and *ε* _{y}) used for calibration of log-derived stress profiles are assumed constant in the studied interval and are determined based on stress conditions in the detected failure evidence at the wellbore wall. Therefore, using this strategy, the magnitude of tectonic strain in two directions (*ε* _{x} and *ε* _{y}) can be determined, and it is not required to consider one of these parameters equal to zero. Considering the proposed method, the maximum and minimum horizontal stress profiles of a deep oil well in the southwest of Iran were calibrated based on wellbore failure evidences detected at different depths. Comparing the obtained values of horizontal stresses at various depths of the studied interval using this method and other techniques indicates the reliability of the estimated stress profiles. The proposed technique in this study can be used in the other projects with similar conditions.

## Notes

### Acknowledgements

The authors would like to express their appreciation to the National Iranian South Oil company for providing filed data and also permission of using data. Moreover, the authors really appreciate Dr. A. Moradi and Dr. Armaghani for his technical help and advice.

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