# Study on risk probability calculation of drilling complex and accidents.

Introduction

Because of the complex geological conditions and uncertainties in the drilling engineering, the engineering and technical programs can't be adapt to the down hole conditions timely, which easily causes the drilling complex and accidents. Therefore the drilling risk analysis is particularly important. At present, most of the drilling risk analysis employ the experience or qualitative risk assessment methods, which rely largely on the experience and the judgment of the experts or the analyst, and the disadvantage is subjective. Regional history statistical methods are mostly used to calculate the drilling risk probability, which lacks theoretical foundation. The common accident prediction methods are as follows: regression prediction method, time series prediction method, Markov prediction method, gray prediction method, Bayesian network prediction method, neural network prediction method and so on. In this paper, on the basis of the theory of the formation pressure prediction, the drilling complex and accidents were analyzed by the generalized stress--strength interference theory , and the risk probability and its vulnerable sections were determined. Which has guiding significance for the design of casing program and the choice of the drilling fluid density.

1 Drilling engineering risk

1.1 Drilling engineering risk concept

To different industries and different objects, the definition of risk mode is different, which leads to risk analysis, evaluation and decision-making methods are also different. In spite of the different risk mode, three elements should be considered . Firstly the kind of risk mode must be ensured in special environment. Secondly the risk rate should be known when the complex happens. Thirdly the consequences should be clear. Based on the theory, the definition of drilling engineering risk is given below.

The risk of drilling engineering is the possibility of the drilling complex and accidents, which are caused by some factors, including a variety of uncertainties, the unreasonable design, the technology level and construction technology and so on.

According to the concept and characteristics of the risk of drilling engineering, there are three problems to be solved: (1)To determine drilling accident classification and its failure mode; (2)To determine the probability of the risk of accidents; (3)To determine the accidents consequences.

[FIGURE 1 OMITTED]

1.2 The risk type and failure mode in the drilling engineering

1.2.1 The safety drilling fluid density window

The safe density window of drilling fluid is the base of safe drilling design. The pore pressure, collapse pressure and fracture pressure of formation compose the safe density window of drilling fluid under the static condition. To establish safe constraint condition, the geologic and engineering factors should be considered during drilling construction. According to the practical drilling condition, the boundary conditions of the safe drilling fluid density window are listed in Table 1.

[TABLE 3 OMITTED]

Where, [S.sub.b] is the swabbing pressure coefficient, g/[cm.sup.3]; [S.sub.g] is the fluctuation pressure coefficient, g/[cm.sup.3]; [DELTA][rho] is the additional drilling fluid density, g/[cm.sup.3]; [S.sub.f] is the safety value-added for the formation fracture pressure, g/[cm.sup.3]; [S.sub.c] is the circulation pressure loss coefficient, g/[cm.sup.3]; [S.sub.k] is the allowable amount of kick, g/[cm.sup.3]; [DELTA]P is the allowable amount of sticking, MPa ; [h.sub.pmax] is the depth of the maximum formation pressure in the open hole section, m ; h is the well depth, m

In order to ensure the drilling safety, the drilling fluid density must meet the following conditions:

max {[[rho].sub.k], [[rho].sub.cd]} [less than or equal to] [[rho].sub.d] [less than or equal to] min {[[rho].sub.L], [[rho].sub.sk], [[rho].sub.cu]}, [[rho].sub.kick] [less than or equal to] [[rho].sub.kl] (1)

Where, [[rho].sub.d] is the drilling fluid density used in drilling; [[rho].sub.kick] is the drilling fluid density handling the drilling complex.

1.2.2 Risk types and risk failure mode in the drilling engineering

From the safe density window of drilling fluid, the major drilling risks under the pressure constraint condition are kick risk, collapse risk, leakage risk, stick risk and leakage risk while killing. According to the generalized stress-strength interference theory, the drilling risk modes are as follows:
```Table 2 Risk types and risk failure mode in the drilling engineering

Drilling
Drilling complex and Risk complex and
accidents mode accidents

Kick risk [[rho].sub.d] < [[rho].sub.k] Stick risk
Collapse risk [[rho].sub.d] < [[rho].sub.c] Leakage risk
Leakage risk [[rho].sub.kcik] > [[rho].sub.L]
while killing

Drilling complex and Risk
accidents mode

Kick risk [[rho].sub.d] > [[rho].sub.sk]
Collapse risk [[rho].sub.d] > [[rho].sub.L]
Leakage risk
while killing
```

2 Risk probability analysis of the drilling engineering risk

When the drilling risk type and mode have been confirmed, the tasks of risk analysis are the calculation of risk probability and the analysis of accidents results. Risk probability is the probability of the complex and accidents in the drilling process. Risk probability analysis could ensure the risk probability and its vulnerable sections.

The risk probability could be calculated by the probability theory analysis or the simulation method. For a simple model or a model with less random variable, the distribution function of the drilling risks could be calculated directly through the probability theory analysis, and then the risk probability would be determined from the distribution function. For a complex model or a model with more random variable, it's difficult to carry out theoretical analysis, but the Monte Carlo simulation method would easily calculate the risk probability. In the article, the risk probability was calculated by the Monte Carlo simulation method. In order to facilitate the calculation and improve the computing speed, the formation pressure matrix was proposed; meanwhile its construction method was given in the article.

2.1 Formation pressure matrix

Formation pressure matrix is a two-dimensional array, which is constituted by the well depth and the formation pressure with discrete or continuous distribution characteristics (as shown in formula 2). Formation pressure matrix includes pore pressure matrix, collapse pressure matrix and fracture pressure matrix. Characteristics of the formation pressure matrix: (1) which is formation parameters; (2) which contains stratigraphic features; (3) the formation pressure at a depth in one layer has distribution characteristics.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

At a depth, the row vectors of the formation pressure matrix are the formation pressure value satisfying more than one cumulative probability; the column vectors are the formation pressure value satisfying a cumulative probability. For [F.sup.x.sub.ij]: x signifies the formation pore pressure, the formation collapse pressure or the formation fracture pressure; i is the formation depth; j is the cumulative probability. ([j.sub.min], [j.sub.max]) (0 [less than or equal to] [j.sub.min] [less than or equal to] 1;0 [less than or equal to] [j.sub.max] [less than or equal to] 1) is the truncated probability range of the cumulative probability, which ensure the higher credibility of the formation pressure at a depth. As can be seen, the conventional single-value formation pressure curve is just a special case of the formation pressure matrix with a cumulative probability.

The row vectors of the formation pressure have certain distribution area, which could make the actual formation pressure values contained in the array by the greatest degree. Therefore, the formation pressure matrix could accurately describe the formation pressure distribution.

2.2 The influence of formation pressure and drilling fluid density on the drilling risk probability

As can be seen from the safety drilling fluid density window and the drilling risk mode, the drilling fluid density and the formation pressure curve is an important factor of the drilling risk, which directly determine whether the drilling risk happen.

Take kick risk for example, [[rho].sub.k] is the anti-kick drilling density in figure 2 (a), [[rho].sub.k.sup.(j)] is the anti-kick drilling fluid density with j cumulative probability in figure 2 (b), [[rho].sub.d] is the actual drilling fluid density used in the drilling construction. According to the drilling risk mode: (1) In figure 2 (a), Because of the single-valued pressure curve, the formation pressure is also a single value. When [[rho].sub.d] is greater than [[rho].sub.k.sup.(h)], the kick risk wouldn't occur; when [[rho].sub.d] is smaller than [[rho].sub.k.sup.(h)], the kick risk would happen. (2) In figure 2 (b), Based on the concept of formation pressure matrix, [[rho].sub.k] (h) (the lower limit value of the anti-kick drilling fluid) is a distribution zone at well depth of h. When the actual drilling fluid density [[rho].sub.d] is greater than the max value [[rho].sub.k] (h, [j.sub.max]) of the formation pressure zone, the kick risk wouldn't occur; when [[rho].sub.d] is smaller than the min value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the formation pressure zone, the kick risk would happen. Within the distribution zone, the kick risk probability depends on the cumulative probability of [[rho].sub.k.sup.(h)] greater than [[rho].sub.d]. Other drilling risks are similar to the kick risk, which wouldn't repeat due to space limitation.

[FIGURE 2 OMITTED]

2.3 Formation pressure matrix construction method

On the basis of the calculation method of the formation pressure profile with credibility , a calculation method of the formation pressure matrix was built by combining Eaton method with Effective stress method. The calculation steps of the formation pore pressure matrix are as follows:

1. Calculating the single-valued formation pressure curve using effective stress method

Through a large number of experimental researches, the main factors affecting acoustic velocity in rock are lithology, porosity and vertical effective stress. If the lithology is homogeneous, the acoustic velocity is mainly the function of porosity and vertical effective stress. For the normal compaction or the undercompaction shale strata in original loading stress state, the porosity is the function of the vertical effective stress. Therefore the acoustic velocity is mainly the function of the vertical effective stress for shale formation. After obtained the effective stress using the acoustic velocity dates, the formation pressure could be calculated by the Effective stress theorem :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Where, V is the acoustic velocity, km/s; [sigma] is the vertical effective stress, MPa; a, k, B, d are the simple method model coefficients; [G.sub.0] is the overburden pressure gradient; [G.sub.p] is the formation pressure gradient.

In the normal compaction formation, the vertical effective stress is calculated by equation (4).

[sigma] = [G.sub.o]-[alpha][G.sub.h] (4)

Where, [G.sub.h] is the hydrostatic pressure, MPa.

According to the equation (3) and (4), the acoustic velocity in the normal compaction strata could be calculated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Where, [V.sub.n] is the acoustic velocity in the normal compaction strata, km/s.

(2) Determination of the Eaton index and its distribution

Substituting the formation pore pressure, the actual acoustic velocity and the acoustic velocity under the normal compaction conditions for the corresponding coefficient in the Eaton formula :

[G.sub.p] = [G.sub.o]-([G.sub.o]-[G.sub.h]) [(V/[V.sub.n]).sup.n] (6)

Then the Eaton index n would be calculated as:

n = ln([G.sub.o]-[G.sub.h])/[G.sub.o]-[G.sub.h)/ln(V/[V.sub.n]) (7)

(3) Determination of the formation pore pressure matrix

After obtaining the Eaton index data set, there are two ways to build the formation pressure matrix. (1) Dealing with the anomalous values of the Eaton index vector properly, then substituting the Eaton index into the Eaton formula for Monte Carlo simulation, finally the formation pore pressure matrix would be determined by statistically analyzing the results of Monte Carlo simulation; (2) Making statistical analysis directly of the Eaton index vector, then calculating the probability density function, and the distribution function of the formation pore pressure will be directly calculated by the union probability calculation method. Then, according to the accuracy requirements, constructing the column vectors of the formation pore pressure corresponding with cumulative probability. The formation pore pressure matrix would be set up by these vectors.

On the basis of the formation pore pressure matrix, with the conventional formation fracture and collapse pressure calculation methods, the fracture pressure matrix and the collapse pressure matrix would be built by the similar simulation method.

2.4 Drilling engineering risk probability calculation

On the basis of the analysis of the drilling risk mode, the formation pressure and the drilling fluid density, by the generalized stress-strength interference theory, the drilling risks probability calculation formula are as shown in Table 3.

Where, [R.sub.k](h), [R.sub.c](h), [R.sub.sk](h), [R.sub.L](h), [R.sub.KL](h) separately denote the kick risk, the collapse risk, the leakage risk, the sticking risk and the leakage risk while shut-in at the well depth of h; [[rho].sub.d] is the actual drilling fluid density, g/[cm.sup.3]; [[rho].sub.kick] is the annulus pressure gradient while shutting in the well, g/[cm.sup.3].

Taking the kick risk for example, the calculation steps are as follows:

1. Calculate the lower limit value of the anti-kick drilling fluid density [[rho].sub.k]

Based on the formation pressure matrix concept, the formation pore pressure matrix is shown as equation (8). For scientific computing, the design coefficient matrixes are built.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

(1) When the design factors are single value

Swab pressure coefficient matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Additional drilling fluid density matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the lower limit value of anti-kick drilling fluid density can be calculated by equation (9).

[[rho].sub.k] = [P.sub.p] + [S.sub.b] + [[DELTA].sub.[rho]] = [([P.sup.p.sub.ij] + a + b).sub.mxn] (9)

(2) When the design factors are discrete value or in line with continuous distribution

Swab pressure coefficient matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Additional drilling fluid density matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the lower limit value of anti-kick drilling fluid density can be calculated by equation (10).

[[rho].sub.k] = [P.sub.p] + [S.sub.b] + [DELTA][rho] = [([P.sup.p.sub.ij] + [a.sub.ij] + [b.sub.ij]).sub.mxn] (10)

(2) Calculate the kick risk probability at the depth of h

At the depth h, [[rho].sub.k.sup.(h)] is a row vector, the distribution function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] could be obtained by statistical analysis of [[rho].sub.k.sup.(h)] (as shown in figure 3), and then the kick risk probability could be calculated by equation (11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[FIGURE 4 OMITTED]

4 Example analysis

Take X well as an example to carry out risk probability analysis. Through careful inspection to the well history data, the process of the drilling complex and accidents is as follows: The outlet flows of the drilling fluid raised when drilling to 3382m, then the field engineers increased the drilling fluid density to prevent the blowout occur. As the pump flows is too large in the moment, which leads to the leakage accident at 3412m. The drilling complex and accidents at the scene were simulated using the risk assessment method established in the article, the analysis results were as figure 5. The equivalent drilling fluid density is the anti-kick drilling fluid density in figure 5(a), The equivalent drilling fluid density is the anti-leakage drilling fluid density in figure 5(b), the blue line is the minimum level of the truncated cumulative probability, the black line is the maximum level of the truncated cumulative probability, the red line is the actual drilling fluid density.

[GRAPHIC OMITTED]

[GRAPHIC OMITTED]

As can be seen from the simulation analysis results, because the formation pressure is relatively large, meanwhile the drilling fluid density is relatively low at the depth of 3200m~3400m, which leads to the higher kick risk probability. The higher kick risk is consistent with the increasing outlet flows of the drilling fluid at 3382m.

When the outlet flows of the drilling fluid raised, the field engineers increased the drilling fluid density to prevent the blowout occur. Because the heavier drilling fluid density is close to the formation fracture pressure when drilling to 3412m, the leakage risk increases accordingly. Meanwhile, for the better fracture development at the strata, the leakage accident happened. Comparing the well history data with the risk analysis results, the risk assessment results are basically in line with the engineering practice.

5 Conclusions

1. The concept of the drilling engineering risk was proposed in the article. Risk probability and risk consequence are the main content of the risk analysis of the drilling engineering.

2. The concept of the formation pressure matrix was proposed in the article, which would facilitate the calculation and improve the computing speed. Formation pressure matrix is a two-dimensional array, which is constituted by the well depth and the formation pressure with discrete or continuous distribution characteristics. Formation pressure matrix is a comprehensive reflection of the uncertainty of the formation pressure. Formation pressure matrix includes pore pressure matrix, collapse pressure matrix and fracture pressure matrix.

3. Based on the generalized stress-strength interference theory and the criterion of the safe drilling fluid window, the risk mode of the drilling complex and accidents was established, meanwhile, the calculation method of the risk probability was given.

4. Taking X well as an example to carry out risk probability analysis, the risk assessment results are basically in line with the engineering practice.

6 Acknowledgements

This work was financially supported by the National Basic Research Program of China (973 Program) (NO.2010CB226706), The 12th Five-Year National Science and Technology Major Project (2011ZX05021-001, 2011ZX05005-006), The 11th Five-Year National Key Technology R&D Program (NO. 2008BAB37B06).

References

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 Zhang Hongcai. Research on Stress-Strength Interference Model Reliability Calculation Method [J]. Mechanical Design. 2001(6): 45-47.

 Chen Guohua. Risk Engineering [M]. Fist Edition. Beijing: National Defense Industry Press, 2007: 10.

 Chen Tinggen, Guan Zhichuan. Drilling Engineering Theory and Technology [M]. Dongying: China University of Petroleum Press, 2000: 251-253.

 Ke Ke, Guan Zhichuan, Zhou Hang. An Approach to Determining Pre-drilling Formation Pore Pressure with Credibility for Deep Water Exploration Wells [J]. Journal of China University of Petroleum. 2009, 33(5): 61-67.

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Wei Kai (1), Guan Zhichuan (1), Ke Ke (2) and Zhao Tingfeng (1)

(1) College of Petroleum Engineering, China University of Petroleum, Qingdao266580, China;

(2) Sinopec Research Institute of Petroleum Engineering, Beijing100101, China

E-mail:upcweikai@163.com
```Table 1 Boundary conditions of the safe drilling fluid density window

Safe drilling fluid density Boundary conditions

Kick lower limit value of drilling [[rho].sub.k](h) = [P.sub.p](h)
fluid density [[rho].sub.k] (h) + [S.sub.b] + [DELTA][rho]

Borehole collapse lower limit value [[rho].sub.c](h) =
of drilling fluid density [[rho].sub.c](h) + [S.sub.b]
[[rho].sub.c] (h)

Stick upper limit value of drilling [[rho].sub.sk](h) =
fluid density [[rho].sub.sk] (h) [P.sub.p](h) [DELTA]P/h x 0.0098

Leakage upper limit value of [[rho].sub.L](h) =
drilling fluid density [P.sub.f](h)-[S.sub.g]-[S.sub.c]
[[rho].sub.L] (h)

Leakage-killing upper limit value [[rho].sub.kl](h) =
of drilling fluid density [P.sub.f](h)-[S.sub.g]-[S.sub.k]
[[rho].sub.kl] (h) [h.sub.pmax]/h
```
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