Minimizing risk by determining a safe mud weight window for offshore wellbore construction

Wellbore stability problems cause many challenges in a drilling operation, such as pipe sticking, wellbore collapse, fluid loss and poor cement jobs. An engineer must minimize the risk of these problems during drilling operations. However, there is considerable uncertainty about different parameters, such as geomechanical rock properties of a drilled formation and data and parameters gathering are often incomplete.  

To ensure wellbore integrity, breakout and fracture geomechanical analysis was conducted to estimate a safe mud weight window (SMWW). The SMWW uncertainty evaluations of wellbore stability assessment for two failure criteria are compared: 1) Mohr-Coulomb and 2) Modified Lade criterion. We applied Monte Carlo simulations to investigate the uncertainty of the models, in addition to sensitivity analysis and confidence level analysis. The investigation outlined the advantages of including uncertainty evaluation when determining the optimum SMWW window, as compared to a classical deterministic calculation that was implemented at Norway’s Snorre field. The new work confirmed the capability of the proposed approach to solve a complex, nonlinear problem.

INTRODUCTION

Fig. 1. SMWW for wellbore stability, using Mohr-Coulomb and modified Lade criteria.and loads (QC) of wellbore collapse.

To derive an uncertainty solution in model simulation, the industry is focusing on verification and validation to mitigate risk. Geomechanical modeling consists of computing the stresses around the wellbore and comparing them to a failure model. To accomplish the objective, two different failure criteria were used and evaluated under uncertainty. Wellbore stability depends on various factors, such as geology, wellbore path, petrophysics and operational execution.¹ A primary result is the assessment of the allowable mud weight window (SMWW). The end-user also must decide whether a pessimistic or optimistic decision is supported, based on uncertainty considerations. 

Figure 1 illustrates the relationship of mud weight, wellbore stability, and various types of rock failure models. When the mud pressure is less than the pore pressure, the wellbore has splintering failure or spalling. When the mud pressure is less than the shear failure gradient, the wellbore has shear failure or breakout. If the mud pressure is too high, drilling-induced hydraulic fractures are generated, which may cause mud losses. To maintain wellbore stability, the mud weight should, therefore, be within an appropriate range. During drilling operations, the borehole wall stress and the mud pressure will balance the stresses and pressures that exist inside the formation. At high wellbore pressure, a tensile failure is most common. At low wellbore pressures, a shear failure leading to a collapsed wellbore is more likely. During these high/low pressure events, the wellbore often deforms and becomes oval or elliptical (wellbore breakout).

The basic stress model used is the so-called Kirsch equation, which assumes a pressure step at the wellbore wall, caused by an impermeable mud cake. However, during stimulation operations with water, there is no mud cake, and different equations apply. In general, the safe mud weight window is defined as the fracture limit, as the upper limit, and the collapse pressure (or pore pressure) as the lower limit. There are also various models from the simple linear elastic model, to more complex non-linear models. However, there is usually not sufficient data to justify more models that are complex. This lack of data also makes this an excellent candidate for uncertainty analysis.

PROBLEM STATEMENT

The selection of a distribution graph may differ, depending on data availability. Drilling is often used for the normal distribution. For data sets with evidence of mode or most likely value, it is recommended that triangular and uniform distributions be considered. For small samples, from which unrepresentative data points have been removed through rigorous analyses, uniform distributions are the preferred choice. If distribution parameters are known, then the distribution is defined. For example, the normal distribution is defined by its mean and standard deviation. 

The uniform distribution is defined by its minimum and maximum values, while the triangular distribution is defined by its minimum, most likely, and maximum values. Measures of dispersion, variance, standard deviation, and P10 to P90, show the extent to which a given data set spreads around the mean (or P50, for a symmetric distribution). The Monte Carlo simulations (MCS) are based on a procedure defined by Williamson (2006). It has four steps. First, select a failure criteria model:

•Non-penetrating Kirsch solution for wellbore fracturing

•Mohr-Coulomb for wellbore collapse

•Modified Lade criterion for wellbore collapse.

Table 1. Lower and upper limits for input variable

To provide a good representation of the wells’ stability around Snorre field in the North Sea, the below well data were selected because of problems with borehole instability as a consequence of low mud weight. After data gathering, we determined the lower and upper limits for input variables. The input parameters (now random variables) with assumed uncertainties are shown in Table 1 in a 1,700 m depth.

By using a range of possible values, instead of a single approximation, a realistic span can be created. When a model is based on ranges of estimates, the output of the model will also be in range a of estimation. 

GEOMECHANICAL MODEL DESIGN

A geomechanical model reveals the mechanical behavior of rock and wellbore, and is used to better manage drilling programs. Mechanical earth models (MEMs) are constructed for wells. Models describe rock elastic and strength properties; in-situ stresses and pore pressure as a function of depth are established. The MEMs consisted of continuous profiles of the following rock mechanical data and parameters along the well trajectories:

• Mechanical stratigraphy, the differentiation of clay-supported rock from grain-supported rock
• Formation elastic properties, including dynamic and static Young’s modulus and Poisson’s ratio
• Rock strength parameters, including unconfined compressive strength (UCS), friction angle and tensile strength
• Pore pressures and leak-off tests (LOT)
• In-situ stress state, including the azimuth of the minimum horizontal stress, magnitudes of vertical stress, minimum and maximum horizontal stresses.

Probability distribution. The interference of the probability density function of Qk (loads denotes the pore pressure) and Rk (the resistances R denotes the wellbore pressure), using assessment of wellbore collapse are depicted in Fig. 2. The overlapping zone denotes the probability of failure. The smaller overlapping zone indicates the wellbore will be more reliable, that is, the risk of wellbore collapse will be lower. Conversely, a larger overlap indicates an increased risk of wellbore collapse.

Fig. 2. Probability densities for the resistances (RC) and loads (QC) of wellbore collapse.

To demonstrate the power of the MCS procedure, we provide several examples taken from North Sea field data. The simulated real data in the columns belong to North Sea fields—which represent the different input quantities—are now linked, row-by-row, according to the functional relationship for fracture and collapse pressure. The outputs and histogram from this set of calculations depict the possible amounts, which refer to the measured and the mean, which provide an estimate of the measured, and the distribution of, values. The data in the output (measured) column can now be evaluated further. Some analysis of outputs includes:

• Plotting a frequency histogram for input data, using the Excel chart function, such as σH, σh, Po, α, τo.
• Analysis of the shape of the distribution base on visual control of the frequency graph.
• Perfect presentation of statistics including mean, mode, median and standard deviation (standard uncertainty) using standard Excel statistical functions.
• A best practice is to copy the output into another column and sort from smallest to largest, exclude the lowest 2.5% and highest 2.5% of values (based on row number) to give a 95% coverage interval of pressure. The Excel PERCENTILE function can be applied to determine the required coverage of interval boundaries.
• Skewness and kurtosis: these statistics could provide additional support when considering the shape of the output as its closeness to normality or when determining the coverage interval.

SHEAR FAILURE CRITERION

The Mohr Coulomb failure model only uses maximum and minimum principal stress.² The failure model can be described as:

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Implemented calculation (deterministic). First, we calculate Mohr-Coulomb parameters to generate deterministic input data. We are using the statistical solution method in this part, based on the Mohr-Coulomb failure criterion, because we can define the required strength properties. We also must define a friction angle range and pore pressure, too. In this calculation, the single-point estimates of the geopressures will be too optimistic to produce a good SMWW. This may lead to drilling problems. However, it is virtually impossible to analyze the associated risk and uncertainty, based on these fixed input data. The only way to achieve this is by running stochastic simulations.³

Implemented calculation (deterministic) let σh = 1.5 sg σH= 1.8 sg; Po = 1.05 sg; α = 30°; τo = 0.5 sg. Pwf=3 *σH-σh– Po = 3*1.5-1.4-1.05=1.65sg =13.91ppg Pwc=1/2 * (3 σH-σh) *(1-sin α) +Po*sin α- τo*cos α= 1/2 * (3 *1.8-1.5) *(1-sin 30) +1.05*sin 30- .5*cos 30=1.076sg=9ppg

Fig. 3. SMWW workflow of wellbore stability.

At first, we ran the simulation with 5,000 data generation, with uncertainty outlined in Table 1 for each parameter. After running the Excel program, it was determined that the results were not realistic and could not be used for practical purposes in the field. Additionally, in the overlap area, data indicated there was collapse and fracture in the borehole, which is not possible. We needed to provide the most reliable set of results for use as input for methods of analysis for drilling engineering. The most realistic SMWW results can be predicted, using the above-mentioned equation base for workflow of wellbore stability, using Mohr-Coulomb criteria, Fig. 3.

We can now calculate a histogram plot of collapse and fracture pressure for 20,000 Monte Carlo trials, generated by Excel with an uncertainty in estimation (± 5%) for all input data. The distribution for the entire input variable is normal. Maximum and minimum collapse and fracture pressure, for 20,000 input data determined, are shown in Fig. 4. Looking at the new histogram, it is relatively simple to determine that there is overlap, but it is in a smaller range, as compared to the previous overlap interval. 

To screen the data for input parameters, let’s review the algorithm outlined in Fig. 3. Data screening should be performed prior to the Monte Carlo procedure. Often, data screening procedures are so tedious that they are skipped. However, after an analysis produces unanticipated results, the data are then scrutinized. The program needed to encompass the entire data screening process. When we saw overlap in the histograms and scatter plots, we were able to verify most of our data assumptions before beginning the actual analysis. We offer two cases of executing a proposed procedure, using an Excel programing base on statistical methods, Fig. 5. 

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Using the lowest 2.5% and highest 2.5% of values (based on row number) produces 95% coverage interval of pressure for Case 1, between 1.19 sg and 1.37 sg, Fig. 6.

Implemented calculation (deterministic) for Case 2 we let σh = σH= 1.4 sg; Po = 1.05 sg; α = 30°; τo = 0.0 sg. Pwf=2*1.4-1.05=1.75sg Pwc=1.4*(1-sin30) +1.05*sin30=1.225sg

Based on uncertainty in estimation (± 5%) for all input data, these calculations are matched with the Monte Carlo simulations, Fig. 7.

Stochastic prediction case 2. The lowest 2.5% and highest 2.5% of values (based on row number) provides a 95% coverage interval of pressure for Case 2 between 1.32 sg and 1.44 sg, Fig. 8 and Fig. 9.

MODIFIED LADE CRITERIA

The Lade criterion for failure of frictional materials is given by Ewy⁴

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The quantities sigma 1, sigma 2, and sigma 3 are the three principal stresses, S1 and η are material constants, and P0 is pore pressure. At first, we put maximum induction stress in modified lade and calculated at a lower borehole pressure. Accordingly, we used the same  procedure and calculation for Mohr Coulomb. The analysis indicated the mud window for collapse is between 0.77 sg and 1.05 sg, and collapse can occur. The mud window for fracture is between 1.32 sg and 2.03 sg, and fracture can also occur. When the SMWW is between 1.05 sg and 1.31 sg, it is extremely unlikely a fracture or collapse will occur (predicted to 95% confidence). This range of values have a very good correlation with the SMWW case study presented above.

CONCLUSIONS

Fig. 4. Unrealistic results base N 20,000 and uncertainty.

Selecting the appropriate safe mud weight window is critical in offshore operations, to ensure the safe and economic delivery of a high-quality wellbore. Making selections from alternative well designs can lead to enhanced well stability, lower capital costs and a reduction in drilling time. However, typically there is a tradeoff of these benefits; with certain designs leading to lower costs and others leading to higher productivity or higher risk during drilling operations in the long term.

Borehole stability research required a large amount of field data, which are not always available, especially in exploratory drilling. The Monte Carlo method is widely used in engineering for sensitivity analysis, and it contributed to quantitative risk analysis applied to analyze the uncertainty of the wellbore stability.

The results show that an SMWW can be calculated between the implemented calculation (deterministic) and the Monte Carlo method. This documented that the proposed method could satisfy the drilling engineering application.

Drilling techniques, such as the UBD, and MPD with narrow SMWW, can be used to manage wellbore pressure, so that wellbore stability can be maintained. Also, an accurate SMWW analysis is a good method to prevent other drilling problems, such as lost circulation. 

Fig. 5. Histogram of collapse and fracture for Case 1.

Fig. 6. Range of values for safe mud weight management in margin well (Case 1 with stochastic calculation).

Fig. 7. Range of values for safe mud weight management in margin well for Case 2 deterministic calculation.

Fig. 8. Histogram of collapse and fracture for Case 2.

Fig. 9. Importance of mud weight management in margin well, in Case 2 (stochastic).

REFERENCES

1. Aadnoy, B.S., “Modern well design,” second edition, CRC Press/Balkema, P.O. Box 447, 2300 AK Leiden, The Netherlands, 2010.
2. Aadnoy, B.S., and A.K. Hansen, “Bounds on in-situ stress magnitudes improve wellbore stability analyses,” Journal of Petroleum Science and Engineering, pp. 115 – 120, 2005.
3. Udegbunam, J.E., B.S. Aadnoy and K.K. Fjelde, “Uncertainty evaluation of wellbore stability model predictions, Journal of Petroleum Science and Engineering, 124, 254–263, 2014.
4. Ewy, R.T., “Wellbore stability prediction by use of a modified Lade criterion,” SPE Drilling and Completion, 14:85–91, 1999.