Study on risk probability calculation of drilling complex and accidents.
Introduction
Because of the complex geological conditions and uncertainties in the drilling engineering Drilling engineering is a subset of petroleum engineering. It is primarily involved in the design and drilling of production and injection wells. The drilling engineer has the responsibility for the efficient penetration of the earth by a well bore, and for cementing of the steel , 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 A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. For example, a Bayesian network can be used to calculate the probability of a patient having a specific disease, given the prediction method, neural network neural network or neural computing, computer architecture modeled upon the human brain's interconnected system of neurons. Neural networks imitate the brain's ability to sort out patterns and learn from trial and error, discerning and extracting prediction method and so on[1]. In this paper, on the basis of the theory of the formation pressure prediction, the drilling complex and accidents were analyzed by the generalized stressstrength interference theory Interference theory refers to the idea that forgetting occurs because the recall of certain items interferes with the recall of other items. In nature, the interfering items are said to originate from an overstimulating environment. [2], 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 Noun 1. drilling fluid  a mixture of clays and chemicals and water; pumped down the drill pipe to lubricate and cool the drilling bit and to flush out the cuttings and to strengthen the sides of the hole drilling mud 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 decisionmaking methods are also different. In spite of the different risk mode, three elements should be considered [3]. 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 according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. 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 pore (por) a small opening or empty space. alveolar pores openings between adjacent pulmonary alveoli that permit passage of air from one to another. 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 The pressure coefficient is a dimensionless number used in aerodynamics and fluid mechanics, most often in the design and analysis of an airfoil. The relationship between the coefficient and the dimensional number is: , 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 valueadded 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 stressstrength 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 probability theory Branch of mathematics that deals with analysis of random events. Probability is the numerical assessment of likelihood on a scale from 0 (impossibility) to 1 (absolute certainty). 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 Monte Carlo Simulation A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables. 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 twodimensional 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 stra·tig·ra·phy n. The study of rock strata, especially the distribution, deposition, and age of sedimentary rocks. strat features; (3) the formation pressure at a depth in one layer has distribution characteristics. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (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 truncated adjective Shortened probability range of the cumulative probability, which ensure the higher credibility of the formation pressure at a depth. As can be seen, the conventional singlevalue 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 antikick drilling density in figure 2 (a), [[rho].sub.k.sup.(j)] is the antikick 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 singlevalued 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 antikick 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 [5], 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 singlevalued formation pressure curve using effective stress method Through a large number of experimental researches, the main factors affecting acoustic velocity in rock are lithology li·thol·o·gy n. 1. The gross physical character of a rock or rock formation. 2. The microscopic study, description, and classification of rock. , porosity porosity /po·ros·i·ty/ (poros´ite) the condition of being porous; a pore. po·ros·i·ty n. 1. The state or property of being porous. 2. 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 shale, sedimentary rock formed by the consolidation of mud or clay, having the property of splitting into thin layers parallel to its bedding planes. Shale tends to be fissile, i.e., it tends to split along planar surfaces between the layers of stratified rock. 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 theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. [6]: [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 Overburden pressure, lithostatic pressure, and vertical stress are terms that denote the pressure or stress imposed on a layer of soil or rock by the weight of overlying material. gradient; [G.sub.p] is the formation pressure gradient In atmospheric sciences (meteorology, climatology and related fields), the pressure gradient (typically of air, more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. . 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 hydrostatic pressure The pressure exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. Hydrostatic pressure increases in proportion to depth measured from the surface because of the increasing weight of fluid , 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 [7]: [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 Probability density function The function that describes the change of certain realizations for a continuous random variable. , and the distribution function of the formation pore pressure will be directly calculated by the union probability calculation method[9]. 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 stressstrength 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 de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the kick risk, the collapse risk, the leakage risk, the sticking risk and the leakage risk while shutin 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 annulus /an·nu·lus/ (an´ulus) pl. an´nuli [L.] anulus. an·nu·lus or an·u·lus n. pl. an·nu·lus·es or an·nu·li A circular or ringshaped structure. 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 antikick 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 antikick 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 antikick 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 In linear algebra, a row vector is a 1 × n matrix, that is, a matrix consisting of a single row: The transpose of a row vector is a column 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 Blowout The rapid sale of all shares in a new securities offering. See: hot issue. blowout The nearly immediate sale of a new security issue because of great investor demand. See also hot issue. 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 antikick drilling fluid density in figure 5(a), The equivalent drilling fluid density is the antileakage 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 (language, simulation) SIMulation ANalysis  (SIMAN) A simulation language, especially for manufacturing systems, developed by C. Dennis Pegden in 1983. ["Introduction to Simulation using SIMAN", C.D. Pegden et al, McGrawHill 1990]. 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 twodimensional 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 stressstrength 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 FiveYear National Science and Technology Major Project (2011ZX05021001, 2011ZX05005006), The 11th FiveYear National Key Technology R&D Program (NO. 2008BAB37B06). References [1] Zheng Xiaoping, Gao Jinji, Liu Mengting. Accident Prediction Theory and Method [M]. Fist Edition. Beijing: Tsinghua University Tsinghua University was established in Beijing in 1911 on the site of a former royal garden belonging to a prince, and was funded by an indemnity which Press, 2009. History [2] Zhang Hongcai. Research on StressStrength Interference Model Reliability Calculation Method [J]. Mechanical Design. 2001(6): 4547. [3] Chen Guohua. Risk Engineering [M]. Fist Edition. Beijing: National Defense Industry Press, 2007: 10. [4] Chen Tinggen, Guan guan: see curassow. Zhichuan. Drilling Engineering Theory and Technology [M]. Dongying: China University of Petroleum Press, 2000: 251253. [5] Ke Ke, Guan Zhichuan, Zhou Hang. An Approach to Determining Predrilling Formation Pore Pressure with Credibility for Deep Water Exploration Wells [J]. Journal of China University of Petroleum. 2009, 33(5): 6167. [6] Fan Honghai. New Methods for Prediction and Evaluation of Formation Pore Pressure [D]. China University of Petroleum, Doctoral Dissertation. 2001.5: 9299. [7] Eaton B A. The Equation for Geopressure Prediction from Well Logs. SPE SPE  Software Practice and Experience 544, 1975. [8] Wei Junhu. Mente Carlo Application in System Engineering [M]. Xi'an: Xi'an Jiaotong University Xi'an Jiaotong University (Simplified Chinese: 西安交通大学; Traditional Chinese: 西安交通大學 Press, 2007 [9] Mao Shisong, Cheng Yiming, Pu Xiaolong. Probability and Statistic statistic, n a value or number that describes a series of quantitative observations or measures; a value calculated from a sample. statistic a numerical value calculated from a number of observations in order to summarize them. [M]. Beijing: Higher Education Press, 2004. Wei Kai kai Noun NZ informal food [Maori] kai noun N.Z. (informal) food, grub (slang) provisions, fare, board, commons, eats (slang (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 Email: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) Leakagekilling 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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