Uncertainty Evaluation of Safe Mud Weight Window Utilizing the Reliability Assessment Method

energies

Article Uncertainty Evaluation of Safe Mud Weight Window Utilizing the Reliability Assessment Method

Tianshou Ma 1,2,* , Tao Tang 1, Ping Chen 1,* and Chunhe Yang 2

1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China; [email protected] 2 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China; [email protected] * Correspondence: [email protected] (T.M.); [email protected] (P.C.); Tel.: +86-28-8303-2458 (T.M.); +86-28-8303-2707 (P.C.) Received: 8 January 2019; Accepted: 8 March 2019; Published: 12 March 2019

Abstract: Due to the uncertainty of formation properties and improper wellbore stability analysis methods, the input parameters are often uncertain and the required mud weight to prevent wellbore collapse is too large, which might cause an incorrect result. However, the uncertainty evaluation of input parameters and their influence on safe mud weight window (SMWW) is seldom investigated. Therefore, the present paper aims to propose an uncertain evaluation method to evaluate the uncertainty of SMWW. The reliability assessment theory was introduced, and the uncertain SMWW model was proposed by involving the tolerable breakout, the Mogi-Coulomb (MG-C) criterion and the reliability assessment theory. The influence of uncertain parameters on wellbore collapse, wellbore fracture and SMWW were systematically simulated and investigated by utilizing Monte Carlo simulation. Finally, the field observation of well SC-101X was reported and discussed. The results indicated that the MG-C criterion and tolerable breakout is recommended for wellbore stability analysis. The higher the coefficient of variance is, the higher the level of uncertainty is, the larger the impact on SMWW will be, and the higher the risk of well kick, wellbore collapse and fracture will be. The uncertainty of basic parameters has a very significant impact on SMWW, and it cannot be ignored. For well SC-101X, the SMWW predicted by analytical solution is 0.9921–1.6020 g/cm3, compared to the SMWW estimated by the reliability assessment method, the reliability assessment method tends to give a very narrow SMWW of 1.0756–1.0935 g/cm3 and its probability is only 80%, and the field observation for well kick and wellbore fracture verified the analysis results. For narrow SMWW formation drilling, some kinds of advanced technology, such as the underbalanced drilling (UBD), managed pressure drilling (MPD), micro-flow drilling (MFD) and wider the SMWW, can be utilized to maintain drilling safety.

Keywords: safe mud weight window; wellbore stability; reliability assessment method; uncertainty evaluation; collapse pressure; fracture pressure; pore pressure

1. Introduction Due to the exhaustion of conventional petroleum resources, much attention is being paid to some ultra-deep, unconventional and deep-water petroleum resources. To exploit these kinds of petroleum resources, more and more deep wells and ultra-deep wells are utilized [1,2]. The initial deep drilling was just made for petroleum exploration and production, but now is being applied to exploit deep geothermal energy and geo-resources, and to conduct international continental scientiﬁc drilling [1]. The initial oil well was only drilled to a depth of 19.8 m, while current oil wells can be drilled to more than 10,000 m, and the deepest well is the SG-3 well that was drilled to a depth of

Energies 2019, 12, 942; doi:10.3390/en12050942 www.mdpi.com/journal/energies Energies 2019, 12, 942 2 of 31 Energies 2019, 12, x FOR PEER REVIEW 2 of 29 deeper12,262 mthan on 6000–8000 the Kola Peninsula m. Deep and in Russiaultra-deep in 1984. drillin Currently,g usually mostencounters oil wells some can challenges, be drilled such deeper as thethan high-temperature 6000–8000 m. Deep and and high-pressure ultra-deep drilling (HTHP), usually wellbore encounters collapse, some wellbore challenges, fracture, such as lost the circulation,high-temperature gas kick, and and high-pressure blowout [1]. (HTHP), The reasons wellbore may collapse, be the uncertainty wellbore fracture, of formation lost circulation, pressure, thegas kick,uncertainty and blowout of formation [1]. The reasonslithology, may the be theunce uncertaintyrtainty of ofthe formation reservoir pressure, interface the uncertaintydepth, the uncertaintyof formation of lithology, completion the uncertaintydepth, and ofthe the interaction reservoir interfaceof multiple depth, factors. the uncertaintyIn order to of minimize completion or avoiddepth, the and above the interaction challenges of multiplethat encountered factors. In in order deep to and minimize ultra-deep or avoid drilling, the above drilling challenges engineers that needencountered to keep inthe deep wellbore and ultra-deep pressure within drilling, a drillingsafe window. engineers If the need safe to window keep the is wellbore converted pressure to the equivalentwithin a safe density, window. theIf safe the window safe window can be is convertedcalled a safe to mud the equivalent weight window density, (SMWW). the safe window The design can ofbe mud called weight a safe mudshould weight follow window the following (SMWW). principle: The design the of safe mud mud weight weight should should follow be the higher following than theprinciple: lowest thesafe safe mud mud weight weight and should lower bethan higher the highest than the safe lowest mud safe weight. mud In weight general, and the lower lowest than safe the mudhighest weight safe muddepends weight. on the In general,minimum the between lowest safe the mudpore weightpressure depends and collapse on the pressure, minimum while between the highestthe pore safe pressure mud andweight collapse depends pressure, on the while fracture the highest pressure safe [3–5]. mud weightThe important depends onfoundation the fracture of SMWWpressure should [3–5]. Thebe determined important foundation by wellbore of SMWWstability. should The wellbore be determined instability, by wellbore a classical stability. rock mechanicsThe wellbore problem, instability, is one a classical of the most rock mechanicscomplex problems problem, that is one encountered of the most during complex drilling problems and completion,that encountered which during cost the drilling drilling and industry completion, certainly which more cost thethan drilling $100 million industry per certainly year worldwide more than [5,6].$100 millionWellbore per instability year worldwide is recognized [5,6]. when Wellbore the instabilityhole diameter is recognized is markedly when different the hole from diameter the bit sizeis markedly and the differenthole does from not maintain the bit size its andstructural the hole integrity does not [3]. maintain In general, its structuralthe mechanical integrity failure [3]. occursIn general, when the wellbore mechanical stress failure concentrations occurs when exceed wellbore the stressrock strength, concentrations and the exceed mechanical the rock failure strength, of wellboreand the mechanical can be classified failure ofinto wellbore two main can betypes classiﬁed (Figure into 1): two(1) Compressive main types (Figure failure,1): (1)i.e., Compressive “tight hole” orfailure, “stuck i.e., pipe” “tight incidents, hole” or which “stuck are pipe” time-consuming incidents, which to solve are time-consuming and therefore expensive, to solve and an increased therefore boreholeexpensive, diameter an increased will occur borehole due diameterto brittle willfailure occur and due the tosubsequent brittle failure caving and of the the subsequent wellbore wall caving in brittleof the wellborerocks, a wallreduced in brittle borehole rocks, diameter a reduced which borehole occurs diameter in weak which (plastic) occurs shales, in weak sandstones, (plastic) shales, and salts;sandstones, and (2) and Tensile salts; failure, and (2) Tensilei.e., “lost failure, circulation” i.e., “lost or circulation” “mud loss” or problems, “mud loss” which problems, are potentially which are dangerous,potentially dangerous,thus representing thus representing a safety risk a that safety has risk to be that avoided. has to be avoided.

Figure 1. Typical wellbore instability. (a) Stable borehole; (b) Enlarged borehole; (c) Borehole breakout; Figure 1. Typical wellbore instability. (a) Stable borehole; (b) Enlarged borehole; (c) Borehole (d) Restricted borehole; (e) Borehole fracture. breakout; (d) Restricted borehole; (e) Borehole fracture. In order to investigate the wellbore stability, a large number of analysis methods has been In order to investigate the wellbore stability, a large number of analysis methods has been proposed, such as the elastic model, plastic model, elastoplastic model, poro-elastic model, proposed, such as the elastic model, plastic model, elastoplastic model, poro-elastic model, thermo-poro-elastic model, chemo-poro-elastic model and chemo-thermo-poro-elastic model [2–13]. thermo-poro-elastic model, chemo-poro-elastic model and chemo-thermo-poro-elastic model [2–13]. However, due to the uncertainty of formation lithology, the uncertainty of formation pressure, However, due to the uncertainty of formation lithology, the uncertainty of formation pressure, the the uncertainty of mechanical properties of rocks, and the unstable wellbore pressure, the input uncertainty of mechanical properties of rocks, and the unstable wellbore pressure, the input parameters of wellbore stability analysis never can be known precisely. In other words, the input parameters of wellbore stability analysis never can be known precisely. In other words, the input parameters are often uncertain, which might cause an incorrect result [14–30]. In order to quantify the parameters are often uncertain, which might cause an incorrect result [14–30]. In order to quantify influence of uncertain parameters on wellbore stability and SMWW, it’s necessary to utilize reliability the influence of uncertain parameters on wellbore stability and SMWW, it’s necessary to utilize assessment method. Morita [14] conducted an uncertainty analysis of borehole stability based on a reliability assessment method. Morita [14] conducted an uncertainty analysis of borehole stability statistical error analysis method. McLellan and Hawkes [15] indicated that the traditional models based on a statistical error analysis method. McLellan and Hawkes [15] indicated that the fail to account for the inherent variability of rock properties, as well as uncertain values for input traditional models fail to account for the inherent variability of rock properties, as well as uncertain parameters, a probabilistic technique was used to evaluate the risks of borehole instability or sand values for input parameters, a probabilistic technique was used to evaluate the risks of borehole production, and a spreadsheet-based Monte Carlo simulation tool was involved. Ottesen et al. [16] instability or sand production, and a spreadsheet-based Monte Carlo simulation tool was involved. present a new analysis method of wellbore stability based on the quantitative risk analysis (QRA) Ottesen et al. [16] present a new analysis method of wellbore stability based on the quantitative risk principles. De Fontoura [17] investigated three analytical methods for evaluating the influence of analysis (QRA) principles. De Fontoura [17] investigated three analytical methods for evaluating parameter uncertainties on wellbore stability, such as the first order second moment, first order the influence of parameter uncertainties on wellbore stability, such as the first order second moment, first order reliability model and statistical error analysis method, and contrasted with the

Energies 2019, 12, 942 3 of 31 reliability model and statistical error analysis method, and contrasted with the results generated by Monte Carlo method. Moos et al. [18] and Zoback [6] presented the use of QRA to formally account for the uncertainty in each input parameter to assess the probability of achieving a desired degree of wellbore stability at a given mud weight. Sheng et al. [19] indicated that because of uncertainty in some key influential parameters, these can lead to uncertainty of wellbore stability, a Monte Carlo uncertainty analysis technique was combined with a numerical geomechanical modelling method to develop a geostatistical approach to determine the uncertainty of wellbore stability. Al-Ajmi and Al-Harthy [20] calculated the value of drilling fluid pressure as a probability distribution by utilizing a probabilistic approach captures uncertainty in input variables through running a Monte Carlo simulation. Aadnøy [21], Aadnoy and Looyeh [4] also utilized QRA to assess the effects of the errors or uncertainties associated with the key data on the instability analysis. Zhang et al. [22] deduced a reliability calculation formula for collapse pressure in coal seam drilling by integration of the reliability theory and the Hoek-Brown criterion, and the Weibull distribution and Monte Carlo simulation was utilized in this study. Udegbunam et al. [23] investigated typical fracture and collapse models with respect to inaccuracies in the input data with a stochastic method. Kinik et al. [24,25] presented a mathematical model to estimate the true fracture pressure from the leak-off-test data, and the QRA was also involved to represent the probability-density distribution of fracture pressure. Gholami et al. [26] applied the QRA to consider the uncertainty of input parameters in determining of the mud weight window for different failure criteria, such as the Mohr-Coulomb (M-C), Hoek-Brown (H-B) and Mogi-Coulomb (MG-C) criterion. Plazas et al. [27] investigated a stochastic approach along with the conventional wellbore stability analysis enables to take into account uncertainty from input data. Although the reliability assessment method had been integrated with wellbore stability models, but most of the above studies still have some shortcomings: (1) The required mud weight to prevent wellbore collapse is too high [28,29], due to the fact traditional wellbore stability models assume that the critical failure appears at the highest point of stress concentration, in other words, it’s performed to yield no shear failure along the borehole wall [2,28]. In real drilling engineering, a tolerable breakout or an appropriate breakout width will not cause an unbearable collapse problem, and it can help lower the required mud weight to prevent collapse; once a tolerable breakout or an appropriate breakout width was involved, the SMWW can be therefore widened, and it’s beneficial to the narrow SMWW formations. However, most of the above studies did not involve a tolerable breakout or an appropriate breakout width, only Morita [14] and Aadnoy and Looyeh [4] investigated its influence on wellbore collapse. (2) Most of wellbore stability analysis that involved the reliability assessment method always uses the M-C criterion to determine the shear failure, while the M-C criterion ignored the influence of the intermediate principal stress, which makes the required mud weight to prevent wellbore collapse too conservative. Al-Ajmi and Zimmerman [7,8] introduced an analytical solution of collapse pressure in conjunction with a true triaxial criterion, i.e., the MG-C criterion, it overcomes the above defect. (3) Most of the above studies just focus on the uncertainty evaluation of wellbore collapse, and the uncertainty evaluation of SMWW is seldom investigated, and the influences of uncertain basic parameters on SMWW are also seldom investigated. Therefore, this study takes the tolerable breakout and the MG-C criterion into account to determine the required SMWW by using the reliability assessment method. Firstly, the basic principle of reliability assessment theory was introduced briefly. Secondly, the uncertain SMWW model was proposed by involving the tolerable breakout, the MG-C criterion and the reliability assessment method. Thirdly, the uncertainty of input parameters was investigated, and the uncertainty analysis of the equivalent mud weight of collapse pressure (EMWCP), the equivalent mud weight of fracture pressure (EMWFP), the equivalent mud weight of collapse pressure (EMWPP) and the SMWW were investigated. Finally, the field observation of well SC-101X was reported and discussed. Energies 2019, 12, 942 4 of 31

2. Reliability Assessment Theory Reliability calculations provide a means of evaluating the combined effects of uncertainties, and a means of distinguishing between conditions where uncertainties are particularly high or low [29]. “Reliability” as it is used in reliability theory is the probability of an event occurring or the probability of a “positive outcome”. Based on the theory of reliability assessment, we can sort the inﬂuencing factors into two types: (1) Loads Q, and (2) Resistances R [30]. The SMWW assessment needs to consider the inﬂuence of well kick, wellbore collapse, and wellbore fracture:

• For well kick, the loads Q denotes the pore pressure, while the resistances R denotes the wellbore pressure; and the basic random variables of loads and resistances of well control can be assumed as XQK = XQK1, XRK2, ··· , XQKn , XRK = {XRK1, XRK2, ··· , XRKn}. • For wellbore collapse, the loads Q denotes the collapse pressure that controlled by in-situ stress, pore pressure and rock properties; while the resistances R denotes the wellbore pressure and rock strength; and the basic random variables of loads and resistances of wellbore collapse can be assumed as XQC = XQC1, XRC2, ··· , XQCn , XRC = {XRC1, XRC2, ··· , XRCn}. • For wellbore fracture, the loads Q denotes the wellbore pressure; while the resistances R denotes the collapse pressure that controlled by in-situ stress, pore pressure, rock properties and rock strength; and the basic random variables of loads and resistances of wellbore fracture can be assumed as XQF = XQF1, XRF2, ··· , XQFn , XRF = {XRF1, XRF2, ··· , XRFn}.

The loads and resistances can be assumed as: ( Q = QX = QX , X , ··· , X k Qk Qk1 Rk2 Qkn (1) Rk = R(XRk) = R(XRk1, XRk2, ··· , XRkn) where Qk and Rk are the loads and resistances; k denotes the subscript, k can value for K, C and F for well kick, wellbore collapse, and wellbore fracture, respectively. The margin of safety, M, is the difference between the resistance and the load. The margin of safety for well control, wellbore collapse and wellbore fracture can be expressed as [30,31]:

Mk = Rk − Qk (2) where Mk is the margin of safety for factor k. In general, the loads and resistances are two independent random variables, and the reliability and failure probability can be expressed as [30,31]:

Prk = P(Mk > 0) = P[(Rk − Qk) > 0] (3)

Pf k = P(Mk < 0) = P[(Rk − Qk) < 0] (4) where Prk is the reliability of factor k; Pfk is the failure probability of factor k. There is a relationship between the reliability and failure probability:

Prk + Pf k = 1 (5)

If we assume that the probability density function of Qk and Rk are fQ(Qk) and fR(Rk), respectively. Figure2 shows the interference of the probability density function of Qk and Rk, we just take the assessment of wellbore collapse as an example, the well control and wellbore fracture is similar. The overlapping zone denotes the probability of failure. The smaller overlapping zone, the wellbore will be more reliable, i.e., the risk of wellbore collapse will be lower; the bigger overlapping zone, Energies 2019, 12, 942 5 of 31 the wellbore will be more unreliable, i.e., the risk of wellbore collapse will be higher. The reliability and failure probability can be expressed as [31]:

( R ∞ R ∞ R ∞ Prk = P(Mk > 0) = 0 f (Mk)dMk = 0 0 fQ(Mk + Qk) fR(Rk)dRkdMk R 0 R 0 R ∞ (6) P = P(M < 0) = f (M )dM = fQ(M + Q ) fR(R )dR dM f k k −∞ k k −∞ −Mk k k k k k where fQ(Qk)Energies and f2019R(R, 12k,) x areFOR PEER the REVIEW probability density function of Qk and Rk, respectively.5 of 29

Figure 2. ProbabilityFigure 2. Probability densities densities for thefor the resistances resistances (R (RC)C and) and loads loads (QC) of ( wellboreQC) of collapse. wellbore collapse.

3. Modeling3. for Modeling Safe Mud for Safe Weight Mud Weight Window Window Utilizing Utilizing Reliability Reliability Assessment Assessment

3.1. Stress Distribution on the Borehole Wall 3.1. Stress Distribution on the Borehole Wall In order to determine the stress distribution on the wall of the vertical borehole, several basic In orderassumptions to determine are made the to stresspropose distributionthis model [3–5]: on(1) The the material wall ofof deep the formation vertical is borehole, linear elastic, several basic assumptionshomogeneous are made to and propose isotropic; this (2) The model mechanical [3–5 ]:model (1) Thecan be material simplified of as deepa plane-strain formation problem; is linear elastic, (3) The formation rock obeys the small deformation assumption; (4) In-situ stress state (ΗH, Ηh, Ηv), homogeneouspore and pressure isotropic; (pp), strength (2) The parameters mechanical (c, Π) and model material can parameters be simpliﬁed (v) are known. as a Based plane-strain on the problem; (3) The formationstress distribution rock obeys around the the small wellbore, deformation and considering assumption; the effects of flow (4) induced In-situ stress, stress the statestress (σH, σh, σv), components on the wall of the borehole (r = R) can be expressed as [5], pore pressure (pp), strength parameters (c, ϕ) and material parameters (v) are known. Based on the σ = p stress distribution around the wellbore,° rm and considering the effects of ﬂow induced stress, the stress °σθσθσ=−+− ++ ® θ p ()()1 2cos2 1 2cos2 components on the wall of the borehole (mHhr = R) can be expressed as [5]: (7) °σσ=−() σσ − θ ¯° zv2cos2v Hh  where Ηv, ΗH and Ηh are the vertical, maximum and minimum horizontal in-situ stresses respectively,  σr = pm MPa; pm is the wellbore pressure, MPa; v is the Poisson’s ratio; Ό is the angle of circumference, (°); Ηr, σ = −p + (1 − 2 cos 2θ)σ + (1 + 2 cos 2θ)σ (7) ΗΌ and Ηz are the stressθ componentsm on the borehole wall,H MPa. h  For a vertical well,σz = accordingσv − 2 vto( theσH Equation− σh) cos (1), 2tangentialθ stress (ΗΌ) and axial stress (Ηz) are functions of the angle Ό, and the tangential and axial stresses will vary sinusoidally, as shown in Figure 3. Both tangential and axial stresses reach the highest magnitude when Ό = Δ/2 or 3Δ/2, i.e., where σv, σH and σh are the vertical, maximum and minimum horizontal in-situ stresses respectively, Όmax = Δ/2 or 3Δ/2. ◦ MPa; pm is the wellbore pressure, MPa; v is the Poisson’s ratio; θ is the angle of circumference, ( ); σr, σθ and σz are the stress components on the borehole wall, MPa. σ r For a vertical well, according to the Equation (1),σ tangential stress (σθ) and axial stress (σz) are θ σ functions of the angle θ, and the tangential and axialz stresses will vary sinusoidally, as shown in

Figure3. Both tangential and axial stresses reach the highest magnitude when θ = π/2 or 3π/2, i.e., π π θmax = /2 or 3 /2.

Wellbore stress / MPa / stress Wellbore Circumference angle / (°) Figure 3. The stress distribution on the borehole wall. Energies 2019, 12, x FOR PEER REVIEW 5 of 29

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

3. Modeling for Safe Mud Weight Window Utilizing Reliability Assessment

3.1. Stress Distribution on the Borehole Wall In order to determine the stress distribution on the wall of the vertical borehole, several basic assumptions are made to propose this model [3–5]: (1) The material of deep formation is linear elastic, homogeneous and isotropic; (2) The mechanical model can be simplified as a plane-strain problem; (3) The formation rock obeys the small deformation assumption; (4) In-situ stress state (σH, σh, σv), pore pressure (pp), strength parameters (c, φ) and material parameters (v) are known. Based on the stress distribution around the wellbore, and considering the effects of flow induced stress, the stress components on the wall of the borehole (r = R) can be expressed as [5]: σ = p  rm σθσθσ=− +()() − + +  θ pmHh1 2 cos 2 1 2 cos 2 (7) σσ=−() σσ − θ  zv2cos2v Hh where σv, σH and σh are the vertical, maximum and minimum horizontal in-situ stresses respectively, MPa; pm is the wellbore pressure, MPa; v is the Poisson’s ratio; θ is the angle of circumference, (°); σr, σθ and σz are the stress components on the borehole wall, MPa. For a vertical well, according to the Equation (1), tangential stress (σθ) and axial stress (σz) are functions of the angle θ, and the tangential and axial stresses will vary sinusoidally, as shown in FigureEnergies 20193. Both, 12, 942 tangential and axial stresses reach the highest magnitude when θ = π/2 or 3π/2,6 ofi.e., 31 θmax = π/2 or 3π/2.

100 σ 95 r σ θ σ 90 z

75 Wellbore stress MPa / Wellbore 48 47 0 90 180 270 360 Circumference angle / (°) Figure 3. TheThe stress stress distribution on the borehole wall. Energies 2019, 12, x FOR PEER REVIEW 6 of 29 3.2. Breakout Width Model

3.2. Breakout Width Model As shown in Figure4, the traditional method predicts the collapse pressure using the critical failureAs point, shown and in theFigure critical 4, the failure traditional point is method located atpredicts the point theA collapse: pressure using the critical failure point, and the critical failure point is located at the point A: θ = θ = 0.5π or θ = θ = 1.5π (8) c θθmax == πc θθmax π cc=max 0.5 or = max 1.5 (8) where θ is the angle of critical failure point B,(◦); θ the angle of critical failure point A,(◦). where θcc is the angle of critical failure point B, (°); θmaxmax the angle of critical failure point A, (°).

FigureFigure 4. 4. TheThe required required cohesion cohesion to to keep wellbore stability (Modified (Modiﬁed from [[2]).2]).

However, the breakout width model as assumessumes a tolerable breakout width (2 ω)) to calculate the collapse pressure, i.e., the critical failure point occurs at the pointpoint BB [[2,28]:2,28]: θθ±± ω πω θθ ±± ω πω cc==0.5or==1.5max max (9) θc = θmax ± ω = 0.5π ± ω or θc = θmax ± ω = 1.5π ± ω (9) where ω is the half of the tolerable breakout width, (°). ◦ whereInω general,is the half the of breakout the tolerable occurs breakout as a width,series of ( ). successive spalls along the direction of the minimumIn general, principal the horizontal breakout occursstress and as aresults series from of successive shear failure spalls subparallel along the to directionthe free wall of the of theminimum borehole principal [2,6,28,32–37]. horizontal As stress shown and in results Figure from 5, where shear failure①, ② subparalleland ③ represent to the free the wall breakout of the zonesborehole during[2,6, 28deepening.,32–37]. As For shown a tolerable in Figure breakout,5, where the 1 ,wellbore 2 and 3 collapserepresent accident the breakout could zonesnot happen, during because the borehole will form a stable shape after several breakouts. In general, when the breakout width does not exceed a tolerable value, the wellbore breakouts are usually expected to increase in depth, but not in width, resulting in a stable borehole after several breakouts [2,6]. The measured breakout widths also compared very well with those predicted by the simple theory [6]. Thus, the given point B (θc = θmax ± ω) can be replaced by the traditional critical failure point A to calculate the collapse pressure for a given tolerable breakout. Zoback [6] indicated that the tolerable breakout width (2ω) to prevent borehole collapse can be set as 90° for a vertical borehole. Once the breakout width (2ω) exceeds the allowable breakout width, the borehole cannot maintain its stability. Substituting Equation (9) into Equation (7), the stress components of the critical failure point B can be rewritten as: σ = rmp  σθωσθωσ=− + −() + + + () +  θ pmHh1 2 cos 2max 1 2 cos 2 max (10) σσ=−()() σσ − θ + ω  zv2cos2v Hh max

Energies 2019, 12, 942 7 of 31 deepening. For a tolerable breakout, the wellbore collapse accident could not happen, because the borehole will form a stable shape after several breakouts. In general, when the breakout width does not exceed a tolerable value, the wellbore breakouts are usually expected to increase in depth, but not in width, resulting in a stable borehole after several breakouts [2,6]. The measured breakout widths also compared very well with those predicted by the simple theory [6]. Thus, the given point B (θc = θmax ± ω) can be replaced by the traditional critical failure point A to calculate the collapse pressure for a given tolerable breakout. Zoback [6] indicated that the tolerable breakout width (2ω) to prevent borehole collapse can be set as 90◦ for a vertical borehole. Once the breakout width (2ω) exceedsEnergies the 2019 allowable, 12, x FOR breakoutPEER REVIEW width, the borehole cannot maintain its stability. 7 of 29

(a) (b)

FigureFigure 5. Schematic5. Schematic representation representation of breakoutof breakout growth growth (Reproduced (Reproduced from from [6]). [6]). (a) ( Thea) The theoretical theoretical modelmodel after after the formationthe formation of wellbore of wellbore breakouts, breakouts, wellbore wellbore breakouts breakouts deepen deepen but do but not do widen; not (widen;b) The (b) measuredThe measured breakout breakout widths comparedwidths compared very well very with we thosell with predicted those predicted by the simple by the theory. simple theory.

3.3.Substituting Collaspe Pressure Equation Model (9) for into M-C Equation Criterion (7), the stress components of the critical failure point B can be rewritten as: 3.3.1. M-C Criterion  σr = pm If considering the conventional effective stress concept, the detail derivation process of the σθ = −pm + [1 − 2 cos 2(θmax + ω)]σH + [1 + 2 cos 2(θmax + ω)]σh (10) M-C criterion can refer to Appendix A, and it can be expressed as [7]:  σz = σv − 2v(σH − σh) cos 2(θmax + ω) σσ=+ 13Cq (11) 3.3. Collaspe Pressure Model for M-C Criterion where C and q are given as: 3.3.1. M-C Criterion ϕ  =−−2cosc ()α Cqp1 p If considering the conventional effective stress1sin− concept,ϕ the detail derivation process of the M-C  (12) criterion can refer to AppendixA, and it can be1sin+ expressedϕ as [7]: q =  1sin− ϕ σ1 = C + qσ3 (11) where φ is the friction angle, (°); c is the cohesive strength, MPa; σ1 and σ3 are the major and minor whereprincipalC and stress,q are given MPa; as:α is the Biot’s coefficient; pp is the pore pressure, MPa. ( 2c cos ϕ C = 1−sin ϕ − (q − 1)αpp 3.3.2. Collapse Pressure 1+sin ϕ (12) q = 1−sin ϕ According to Equation (7), the radial stress is perpendicular to the borehole wall, i.e., it’s a ◦ whereprincipalϕ is the stress. friction In angle,these models ( ); c is thethere cohesive are three strength, permutations MPa; σ1 ofand theσ 3threeare the principal major and stresses minor that principal stress, MPa; α is the Biot’s coefﬁcient; p is the pore pressure, MPa. need to be investigated in order to determinep the required mud pressure [19]: (1) σz ≥ σθ ≥ σr, (2) σθ ≥ σz ≥ σr, and (3) σθ ≥ σr ≥ σz. In the fields, the case where σθ ≥ σz ≥ σr is the most commonly encountered stress state corresponding to borehole collapse [19]. Substituting σ1 = σθ, σ2 = σz and σ3 = σr into Equations (11) and (12), the collapse pressure model with a tolerable breakout width can be expressed as: −++++−()θωσ () θωσ 1 2 cos 2maxHh 1 2 cos 2 max C p = (13) c 1 + q

where pc is the collapse pressure, MPa. If set the critical failure point θc = θmax ± ω =π/2 or 3π/2, Equation (13) can be reduced to the traditional model: 3σσ−−C p = Hh (14) c 1+ q

Energies 2019, 12, 942 8 of 31

3.3.2. Collapse Pressure According to Equation (7), the radial stress is perpendicular to the borehole wall, i.e., it’s a principal stress. In these models there are three permutations of the three principal stresses that need to be investigated in order to determine the required mud pressure [19]: (1) σz ≥ σθ ≥ σr, (2) σθ ≥ σz ≥ σr, and (3) σθ ≥ σr ≥ σz. In the ﬁelds, the case where σθ ≥ σz ≥ σr is the most commonly encountered stress state corresponding to borehole collapse [19]. Substituting σ1 = σθ, σ2 = σz and σ3 = σr into Equations (11) and (12), the collapse pressure model with a tolerable breakout width can be expressed as: [1 − 2 cos 2(θ + ω)]σ + [1 + 2 cos 2(θ + ω)]σ − C p = max H max h (13) c 1 + q where pc is the collapse pressure, MPa. If set the critical failure point θc = θmax ± ω =π/2 or 3π/2, Equation (13) can be reduced to the traditional model: 3σ − σ − C p = H h (14) c 1 + q

3.3.3. Equivalent Mud Weight of Collapse Pressure (EMWCP)

The collapse pressure (pc) can be predicted by using Equation (13) or Equation (14), and the equivalent mud weight (EMW) of collapse pressure (EMWCP) can be expressed as: p EMWCP = c (15) gTVD where EMWCP is the equivalent mud weight of collapse pressure, g/cm3; g is the gravitational acceleration, g = 9.8 m/s2; TVD is the true vertical depth, km. According to Equations (13)–(15), the true vertical depth (TVD), the gravitational acceleration (g) and the critical failure point A (θmax) are usually certain, while the maximum horizontal stress (σH), minimum horizontal stress (σh), tolerable breakout width (ω), pore pressure (pp), strength parameters (c, ϕ) and Biot’s coefﬁcient (α) are usually uncertain. The basic random variables of loads and resistances of wellbore collapse can be expressed as: ( X = σ , σ , p , α, ω QC H h p (16) XRC = {c, ϕ, pm}

The loads and resistances of wellbore collapse can be expressed as: ( Q = EMWCP = Q(X ) = Qσ , σ , p , α, ω C QC H h p (17) RC = EMWWP = R(XRC) = R(c, ϕ, pm)

Substituting Equations (13)–(16) into Equation (17), the reliability and failure probability can be determined by using the reliability assessment theory.

3.4. Collaspe Pressure Model for MG-C Criterion

3.4.1. MG-C Criterion If considering the conventional effective stress concept, the detail derivation process of the MG-C criterion can refer to AppendixB, and the MG-C criterion can be expressed as [2,7]:

1 2 2 0 0 I1 − 3I2 = a + b I1 − σ2 − 2αpp (18) Energies 2019, 12, 942 9 of 31

0 0 where I1, I2, a and b are given as: ( I = σ + σ + σ 1 1 2 3 (19) I2 = σ1σ2 + σ2σ3 + σ3σ1

( a0 = 2c cos ϕ (20) b0 = sin ϕ where σ2 is the intermediate principal stress, MPa; I1 is the ﬁrst stress tensor invariant, MPa; I2 is the second stress tensor invariant, MPa.

3.4.2. Collapse Pressure According to Equation (7), the radial stress is perpendicular to the borehole wall, i.e., it’s a principal stress. In these models there are three permutations of the three principal stresses that need to be investigated in order to determine the required mud pressure [19]: (1) σz ≥ σθ ≥ σr, (2) σθ ≥ σz ≥ σr, and (3) σθ ≥ σr ≥ σz. In the ﬁelds, the case where σθ ≥ σz ≥ σr is the most commonly encountered stress state corresponding to borehole collapse [19]. Substituting σ1 = σθ, σ2 = σz and σ3 = σr into Equations (18)–(20), the collapse pressure model with a tolerable breakout width can be expressed as:

1 1 1 n 2 o 2 p = A − 12a0 + b0A − 2αp − 3(A − 2B)2 (21) c 2 6 p where A, B, a0 and b0 are given as: ( A = [1 − 2 cos 2(θ + ω)]σ + [1 + 2 cos 2(θ + ω)]σ max H max h (22) B = σv − 2v(σH − σh) cos 2(θmax + ω)

( a0 = 2c cos ϕ (23) b0 = sin ϕ

0 0 If set the critical failure point θ = θmax ± ω = π/2 or 3π/2, substituting θ = θmax ± ω = π/2 or 3π/2 into Equation (22), the collapse pressure model can degrade into the traditional model: ( A = 3σ − σ H h (24) B = σv + 2v(σH − σh)

3.4.3. Equivalent Mud Weight of Collapse Pressure (EMWCP)

The collapse pressure (pc) can be predicted by using Equation (21), and the EMWCP can be expressed as: p EMWCP = c (25) gTVD According to Equations (21)–(25), the TVD, the gravitational acceleration (g) and the critical failure point A (θmax) are usually certain, while the vertical stress (σv), maximum horizontal stress (σH), minimum horizontal stress (σh), Poisson’s ratio (v), tolerable breakout width (ω), pore pressure (pp), strength parameters (c, ϕ) and Biot’s coefﬁcient (α) are usually uncertain. The basic random variables of loads and resistances of wellbore collapse can be expressed as: ( X = σ , σ , σ , p , v, α, ω QC H h v p (26) XRC = {c, ϕ, pm} Energies 2019, 12, 942 10 of 31

The loads and resistances of wellbore collapse can be obtained: ( Q = EMWCP = Q(X ) = Qσ , σ , σ , p , v, α, ω C QC H h v p (27) RC = EMWWP = R(XRC) = R(c, ϕ, pm)

Substituting Equations (21)–(26) into Equation (27), the reliability and failure probability can be determined by using the reliability assessment theory.

3.5. Facture Pressure Model

3.5.1. Tensile Failure Criterion When the tensile stress exceeds the tensile strength, the wellbore fracture will occur. In general, the maximum tensile stress criterion can be used to determine the wellbore fracture. If considering the conventional effective stress concept, the tensile failure criterion can be written as:

σ3 − αpp + St = 0 (28) where St is the tensile strength, in MPa.

3.5.2. Facture Pressure According to Equation (10), only the hoop stress can obtain the negative value, and the minor ◦ ◦ ◦ ◦ principal stress (σ3) occurs at θ = 0 or 180 . Substituting θ = 0 or 180 into Equation (10), the minor principal stress (σ3) can be expressed as:

σ3 = 3σh − σH − pm − αpp (29)

Substituting Equation (29) into Equation (28), the fracture pressure model can be obtained:

p f = 3σh − σH − αpp + St (30) where pf is the fracture pressure, MPa.

3.5.3. Equivalent Mud Weight of Fracture Pressure (EMWFP)

The fracture pressure (pf) can be predicted by using Equation (30), and the EMW of fracture pressure (EMWFP) can be expressed as:

p f EMWFP = (31) gTVD where EMWFP is the equivalent mud weight of fracture pressure, g/cm3. According to Equations (30) and (31), the TVD and the gravitational acceleration (g) are usually certain, while the maximum horizontal stress (σH), minimum horizontal stress (σh), pore pressure (pp), strength parameters (St) and Biot’s coefﬁcient (α) are usually uncertain. The basic random variables of loads and resistances of wellbore fracture can be expressed as: ( XQF = {pm} (32) XRF = σH, σh, pp, St, α

The loads and resistances of wellbore fracture can be obatined: ( QC = EMWWP = Q(XQF) = Q(pm) (33) RC = EMWFP = R(XRF) = R σH, σh, pp, St, α Energies 2019, 12, 942 11 of 31

Substituting Equations (30)–(32) into Equation (33), the reliability and failure probability can be determined by using the reliability assessment theory.

3.6. Equivalent Mud Weight of Pore Pressure (EMWPP) The equivalent mud weight of pore pressure (EMWPP) also can be obtained by:

pp EMWPP = (34) gTVD where EMWPP is the equivalent mud weight of pore pressure, g/cm3. According to Equation (34), the TVD and the gravitational acceleration (g) are usually certain, while the pore pressure (pp) is usually uncertain. The basic random variables of loads and resistances of wellbore fracture can be expressed as: ( X = p QK p (35) XRK = {pm}

The loads and resistances of wellbore fracture can be obtained: ( Q = EMWPP = Q(X ) = Qp C QK p (36) RC = EMWWP = R(XRK) = R(pm)

Substituting Equations (34) and (35) into Equation (36), the reliability and failure probability can be determined by using the reliability assessment theory.

3.7. SMWW Model In order to avoid or minimize wellbore instability problems and keep wellbore safety, the wellbore overﬂow, wellbore collapse and wellbore fracture should be avoided. The design of mud weight should follow the following principle: safe mud weight should higher than the lowest safe mud weight and lower than the highest safe mud weight. In general, the lowest safe mud weight needs to determine by using the pore pressure and collapse pressure, while the highest safe mud weight needs to determine by using the fracture pressure, and the SMWW can be determined by [5]:

max{EMWPP, EMWCP}< SMWW < min{EMWFP} (37) where SMWW is the safe mud weight window, g/cm3. Thus, in order to quantitatively assess the reliability of SMWW, we also should determine the reliability of the lowest safe mud weight by integrating the reliability between EMWPP and EMWCP, and the reliability of the highest safe mud weight is just determined by EMWFP.

4. Results and Discussion To investigate the inﬂuence of uncertain parameters on wellbore stability and SMWW, take the basic parameters of well SC-101X as an example, the TVD of the formation is 2008–2432 m, and the basic parameters are shown in Table1. For the calculation of wellbore stability and SMWW, Monte Carlo simulation was utilized with 10,000 iterations. Energies 2019, 12, 942 12 of 31

Table 1. Basic parameters for SMWW analysis.

No. Parameter Distribution Mean Std Dev Var P5 P95 1 Vertical stress/MPa Normal 54.80 2.74 5% 50.29 59.30 2 Maximum horizontal stress/MPa Normal 43.87 8.77 20% 29.40 58.30 3 Minimum horizontal stress/MPa Normal 30.91 3.09 10% 25.82 35.99 4 Pore pressure/MPa Normal 21.39 2.14 10% 17.86 24.90 5 Biot’s coefﬁcient Normal 0.95 0.05 5% 0.87 1.00 6 Cohesive strength/MPa Normal 18.14 3.63 20% 12.17 24.09 7 Internal friction angle/(◦) Normal 35.00 7.00 20% 23.48 46.50 8 Tensile strength/MPa Normal 6.00 1.20 20% 4.02 7.97 9 Poisson’s ratio Normal 0.25 0.05 20% 0.17 0.33 10 Breakout width/(◦) Normal 40.00 4.00 10% 33.42 46.55 Note: P5 and P95 denote the lower and upper limit within the conﬁdence interval of 5–95%.

4.1. Uncertainty of Basic Parameters The basic parameters of wellbore stability analysis are uncertain, and these basic parameters are not exact values but the average estimates [19], and the average values themselves are known to have flaws. The real values are usually variable and following some distribution. As shown in Table1, the mean value, standard deviation (Std Dev), variance (Var) of each parameter are presented. The mean of the input data were interpreted and identified from logging data. Regarding the choice of random variable distributions, some of the probability distributions normally used in wellbore stability analysis include the uniform distribution, triangular distribution, lognormal distribution, Gamma distribution, Beta distribution, and Weibull distribution [30]. From an engineering perspective, the normal distribution is the preferred choice for wellbore stability problems, due to the distribution functions are mostly normally distributed [38,39]. It should be noticed that the random variable distributions used in this work are based on assumption, but the choice of random variable distributions should be revisited. Regarding the uncertainties in the input data, according to the engineering experience of Aadnøy [21], measurement and interpretation errors are the major sources of input parameter uncertainties [21,23]. Currently, the uncertainties are only “educated guesses” with possibility for future improvements, and the assumed uncertainties in both measurement and interpretation results were listed in Table2. Although the uncertainties in the input data were based on assumption, the orders of magnitudes of the input data were assumed correct. On this basis, Monte Carlo simulation was utilized with 10,000 iterations. The simulating results of the probability density histogram for basic parameters were shown in Figure6. It is clearly shown that the uncertain distribution of each parameter satisfies the normal distribution, the higher the coefficient of variance is, the higher the level of uncertainty will be, and the wider the distribution of parameter values will be. The 90% confidence interval (from 5 to 95%, i.e., P5–P95) of each parameter was also listed in Table1. For example, the 90% confidence interval is 36.64–51.07 MPa for the estimation of maximum horizontal stress, while it’s just 50.29–59.30 MPa, 28.35–33.44 MPa, 17.86–24.90 MPa for the estimations of vertical stress, minimum horizontal stress and pore pressure, respectively. It is clearly shown that the estimation of maximum horizontal stress has a higher confidence interval of P5 and P95, due to its high level of uncertainty. Energies 2019, 12, 942 13 of 31

Energies 2019, 12, x FOR PEER REVIEW 12 of 29 Table 2. Assumed uncertainties in both measurement and interpretation results. Table 2. Assumed uncertainties in both measurement and interpretation results. Uncertainty in Uncertainty in No. Parameter No. Parameter Uncertainty in MeasurementMeasurement UncertaintyInterpretation in Interpretation 1 Vertical stress 5% 5–10% 1 Vertical stress 5% 5–10% 2 Maximum horizontal stress / 10–20% 2 Maximum horizontal stress / 10–20% 3 Minimum3 horizontal Minimum stress horizontal stress 5% 5% 5–10% 5–10% 4 Pore4 pressure Pore pressure 2–5% 2–5% 5–30% 5–30% 5 Biot’s5 coefficient Biot’s coefﬁcient / / 5–10% 5–10% 6 Cohesive6 strength Cohesive strength / / 10–50% 10–50% 7 Internal7 friction Internal angle friction angle / / 10–30% 10–30% 8 Tensile8 strength Tensile strength / / 10–50% 10–50% 9 Poisson’s9 ratio Poisson’s ratio / / 20–50% 20–50%

0.16 0.05 Samples 10000 Samples 10000 0.04 0.12 0.03 0.08 0.02 0.04 0.01 Probability density Probability density 0.00 0.00 45 50 55 60 65 10 20 30 40 50 60 70 80 Vertical stress (MPa) Maximum horizontal stress (MPa)

(a) Vertical stress (b) Maximum horizontal stress 0.15 0.20 Samples 10000 Samples 10000 0.12 0.15 0.09 0.10 0.06 0.05 0.03 Probability density Probability density 0.00 0.00 20 25 30 35 40 14 16 18 20 22 24 26 28 Minimum horizontal stress (MPa) Pore pressure (MPa)

(c) Minimum horizontal stress (d) Pore pressure 8 0.12 Samples 10000 Samples 10000 0.10 6 0.08 4 0.06 0.04 2 0.02 Probability density Probability density 0 0.00 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 5 1015202530 Biot coefficient Cohesive strength (MPa)

(e) Biot’s coefficient (f) Cohesive strength 0.06 0.35 Samples 10000 Samples 10000 0.05 0.30 0.25 0.04 0.20 0.03 0.15 0.02 0.10 0.01 0.05 Probability density Probability density 0.00 0.00 10 20 30 40 50 60 2345678910 Internal friction angle / (°) Tensile strength (MPa)

(g) Internal friction angle (h) Tensile strength

Figure 6. Cont.

Energies 2019, 12, x FOR PEER REVIEW 13 of 29 Energies 2019, 12, x FOR PEER REVIEW 13 of 29 Energies 2019, 12, 942 14 of 31 Figure 6. Cont. Figure 6. Cont. 8 0.12 8 Samples 10000 0.12 Samples 10000 Samples 10000 0.10 Samples 10000 6 0.10 6 0.08 0.08 4 0.06 4 0.06 0.04 2 0.04 2 0.02 Probability density 0.02Probability density

Probability density 0 Probability density 0.00 0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 25 30 35 40 45 50 55 0.10 0.15 0.20Poisson's 0.25 ratio 0.30 0.35 0.40 25 30 35Breakout 40 width 45 (°) 50 55 Poisson's ratio Breakout width (°) (i) Poisson’s ratio (j) Breakout width (i) Poisson’s ratio (j) Breakout width Figure 6. Probability density histogram of basic parameters. Figure 6. Figure 6. ProbabilityProbability density density histogram histogram of of basic basic parameters. parameters. 4.2. Uncertainty Analysis of EMWCP 4.2. Uncertainty Analysis of EMWCP 4.2.1. EMWCP Probability Analysis forfor bothboth M-CM-C andand MG-CMG-C CriteriaCriteria 4.2.1. EMWCP Probability Analysis for both M-C and MG-C Criteria EMWCP means the equivalent mud weight of collapsecollapse pressure, and there are two types of collapseEMWCP model, means i.e., thethe M-Cequivalent model andmud MG-C weight model.mode of col. llapse We comparedcompared pressure, thethe and simulatingsimulating there are resultsresults two types betweenbetween of collapseM-C modelmodel model, andand i.e., MG-CMG-C the M-C model, model, model the the results and results MG-C are are shown mode shownl. in We Figurein Figurecompared7, the 7, statisticsthe the statistics simulating both both for results M-C for M-C model between model and M-CMG-Cand modelMG-C model andmodel was MG-C collected was model, collected in Tablethe inresults3 .Table It is clearlyare 3. shownIt is found clearly in thatFigure found the 7, same thatthe uncertainty statisticsthe same both uncertainty in basicfor M-C parameters inmodel basic andresultsparameters MG-C in differentmodel results was in answers differentcollected for answers M-Cin Table model for 3. M-CIt and is modelclearly MG-C and model.found MG-C that The model. the analytical same The uncertaintyanalytical solution ofsolution in the basic M-C of 3 3 parameterscriterionthe M-C (0.6354criterion results g/cm in(0.6354 different) is g/cm obviously answers3) is obviously higher for M-C than higher model that than ofand the thatMG-C MG-C of themodel. criterion MG-C The (0.5195criterion analytical g/cm (0.5195 solution), andg/cm theof3 ), 3 3 3 theM-Cand M-C criterionthe criterion M-C tendscriterion (0.6354 to calculate tends g/cm )to ais highercalculate obviously mean a higherhigher EMWCP thanmean (0.6438 that EMWCP of g/cm the MG-C)(0.6438 than thatcriterion g/cm of3 the) than(0.5195 MG-C that criteriong/cm of the), 3 3 and(0.5538MG-C the criterion g/cmM-C criterion). This(0.5538 depends tends g/cm 3to). on Thiscalculate the conservativedepends a higher on the nature mean conservative of EMWCP the M-C nature criterion,(0.6438 of g/cmthe due M-C to) than the criterion, M-C that criterionof due the to 3 MG-Cignoredthe M-C criterion the criterion inﬂuence (0.5538 ignored ofg/cm the intermediate).the This influence depends principal of on the the stress,intermediateconservative which makesnatureprincipal theof the requiredstress, M-C which criterion, EMW makes to preventdue tothe thewellborerequired M-C criterion collapseEMW to isignored prevent too conservative. the wellbore influence However,collapse of the is theintermediate too MG-C conservative. criterion principal overcameHowever, stress, the thewhich defect MG-C makes of thecriterion M-Cthe requiredcriterion.overcame EMW Inthe other defectto words,prevent of the the wellbore M-C M-C criterion. criterion collapse under-predictsIn isot hertoo words,conservative. rock the strengthM-C However, criterion andestimates the under-predicts MG-C the criterion EMWCP rock overcametoostrength conservative, andthe estimatesdefect while of the the EMWCPM-C MG-C criterion. criterion too conservative, In rightly other predicts words, while rock thethe M-CMG-C strength criterion criterion and estimatesunder-predicts rightly predicts the EMWCP rock rock strengthmuchstrength more and and exactly.estimates estimates the the EMWCP EMWCP too much conservative, more exactly. while the MG-C criterion rightly predicts rock strength and estimates the EMWCP much more exactly. 2.5 100 2.5 100 2.0 M-C 80 2.0 M-C MG-C 80 MG-C 1.5 M-C 60 1.5 M-C MG-C 60 MG-C 1.0 40 1.0 40 0.5 20 Probability density Probability 0.5 20 Probability density Probability

0.0 0 Cumulative probability (%)

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60 Cumulative probability (%) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.43 1.6 Equivalent mud weight (g/cm3 ) Equivalent mud weight (g/cm ) Figure 7. Probability histogram of EMWCP both for M-C andand MG-CMG-C model.model. Figure 7. Probability histogram of EMWCP both for M-C and MG-C model. Table 3.3. The statistics ofof EMWCPEMWCP bothboth forfor M-CM-C andand MG-CMG-C model.model. Table 3. The statistics of EMWCP both for M-C and MG-C model. EMWCPEMWCP StatisticsStatistics EMWCP Statistics M-CM-C MG-C MG-C 3 M-C MG-C P5/(g/cm3 ) 0.3221 0.2317 3 0.3221 0.2317 P5/(g/cmP5/(g/cm ) )3 0.3221 0.2317 P95/(g/cm3 ) 0.9825 0.9063 P95/(g/cm3 ) 0.9825 0.9063 P95/(g/cm ) 3 0.9825 0.9063 Mean/(g/cm3 ) 0.6438 0.5538 Mean/(g/cm3 ) 0.6438 0.5538 AnalyticalMean/(g/cm solution/(g/cm) 33) 0.64380.6354 0.55380.5195 Analytical solution/(g/cm3 ) 0.6354 0.5195 Analytical solution/(g/cm3 ) 0.6354 0.5195 StdStd Dev/(g/cm Dev/(g/cm3)) 0.20290.2029 0.2011 0.2011 Std Dev/(g/cm3) 0.2029 0.2011

Energies 2019, 12, 942 15 of 31 Energies 2019, 12, x FOR PEER REVIEW 14 of 29

AsAs shownshown in in Figure Figure8, due8, due to the to influence the influence of intermediate of intermediate principal principal stress for stress the MG-C for the criterion, MG-C thecriterion, rock strength the rock increase strength firstly increase and thenfirstly decrease, and then but decrease, the rock but strength the rock is much strength higher is much than thathigher of thethan M-C that criterion. of the M-C Moreover, criterion. the standardMoreover, deviation the standard in the M-Cdeviation criterion in isthe slightly M-C criterion higher than is slightly that of thehigher MG-C than criterion, that of the which MG-C also criterion, reveals thatwhich the also level reveals of uncertainty that the level estimated of uncertainty by MG-C estimated criterion by is lesserMG-C than criterion that ofis thelesser M-C than criterion. that of Furthermore,the M-C criterion. the EMWCP Furthermore, with probabilitythe EMWCP (P95) with denotes probability the minimum(P95) denotes possible the minimum EMW with possible probability EMW of 95%,with whichprobability defines of the95%, likelihood which defines of preventing the likelihood wellbore of collapsepreventing as awellbore function ofcollapse EMW, itas cana function be obtained of EMW, from the it cumulativecan be obtained probability from curves.the cumulative The P95 EMWCPprobability estimated curves. byThe the P95 M-C EMWCP criterion estimate is 0.9825d by g/cm the M-C3, while criterion the P95 is 0.9825 EMWCP g/cm estimated3, while the by theP95 MG-CEMWCP criterion estimated is 0.9063 by the g/cm MG-C3. The criterion minimum is 0.9063 possible g/cm EMW3. The estimated minimum by thepossible reliability EMW assessment estimated methodby the reliability is obviously assessment higher than method that ofis theobviously analytical higher solution. than that In the of followingthe analytical section, solution. we just In usethe thefollowing MG-C modelsection, to we analyze just use the the EMWCP. MG-C model to analyze the EMWCP.

FigureFigure 8. PlotPlot of ofstrength strength criteria criteria in {σ in1-σ {2σ} 1space-σ2} space(Modified (Modiﬁed from [40]). from (a [)40 M-C]). criterion; (a) M-C ( criterion;b) MG-C (criterion.b) MG-C criterion.

4.2.2.4.2.2. InfluenceInfluence ofof BreakoutBreakout WidthWidth AsAs mentionedmentioned inin thethe Introduction,Introduction, thethe requiredrequired EMWCPEMWCP predictedpredicted byby traditionaltraditional modelmodel isis tootoo conservative.conservative. TheThe traditionaltraditional modelsmodels assumeassume thatthat thethe criticalcritical failurefailure appearsappears atat thethe highesthighest pointpoint ofof stressstress concentration,concentration, inin otherother words,words, it’sit’s performedperformed toto yieldyield nono shearshear failurefailure alongalong thethe borehole wall. However,However, thethe real real drilling drilling engineering engineering allows allows a tolerable a tolerable breakout breakout or an or appropriate an appropriate breakout breakout width, whichwidth, dowhich not causedo not an unbearablecause an unbearable collapse problem. collapse Theproblem. breakout The width breakout model width was involvedmodel was to overcomeinvolved thisto overcome problem, butthis itproblem, is seldom but investigated it is seldom bytraditional investigated models. by traditional The influence models. of mean The breakout width was simulated, and the results were shown in Figure9. For different breakout situation influence◦ of mean◦ breakout◦ width◦ was simulated,◦ and◦ the results were shown in Figure 9. For ofdifferentω = 0 , breakoutω = 10 , ωsituation= 20 , ωof= ω30 = 0°,, ω ω= =40 10°,and ω =ω 20°,= 50 ω =, the30°, P95 ω = EMWCP40° and ω estimated = 50°, the byP95 the EMWCP MG-C 3 3 3 3 3 3 criterionestimated is by 1.1471 the MG-C g/cm criterion, 1.1341 g/cmis 1.1471, 1.0629 g/cm g/cm3, 1.1341, 0.9755 g/cm3, g/cm 1.0629, g/cm 0.90623, 0.9755 g/cm g/cmand3 0.8593, 0.9062 g/cm g/cm3 respectively.and 0.8593 g/cm The3 relationship respectively. between The relationship the EMWCP between and breakout the EMWCP width wereand breakout fitted and width expressed were asfitted follows: and expressed as follows: = =×−×+×+× −6 −−−363ωωω− × −4 422 + × − 33 + ( 22 == ) EMWCPEMWCPP95 P954.6858 4.685810 10ω 3.8040 3.804010 10ω 1.6000 1.600010 ω 1.14831.1483 (RR 0.9982)0.9982 −6 3 −4 2 −4 2 EMWCPP90 = 3.1797=×−×+×+× 10 ω−−−63−ωωω2.6884 × 10 ω 42+ 3.8431 × 10 4ω + 1.0313 (R 2 == 0.9990) EMWCPP90 3.1797−6 103 2.6884 10−4 2 3.8431 10−4 1.0313 (R 2 0.9990) (38) EMWCPP85 = 2.5424 × 10 ω − 2.1160 × 10 ω − 4.6500 × 10 ω + 0.9596 (R = 0.9998) −−−63 42 4 2 (38) EMWCP=×−×−×+ 2.5424−6 103 ωωω 2.1160 10−4 2 4.6500 10−4 0.9596 (R2 = 0.9998) EMWCPP80 =P852.5092 × 10 ω − 2.2044 × 10 ω + 4.3316 × 10 ω + 0.8966 (R = 0.9997) =×−×+×+−−−63ωωω 42 4 2 = EMWCPP80 2.5092 10 2.2044 10 4.3316 10 0.8966 (R 0.9997) 3 where EMWCPP95 is the minimum possible EMWCP with probability of 95%, g/cm ; EMWCPP90 where EMWCPP95 is the minimum possible EMWCP with probability3 of 95%, g/cm3; EMWCPP90 is is the minimum possible EMWCP with probability of 90%, g/cm ; EMWCPP85 is the minimum the minimum possible EMWCP with probability3 of 90%, g/cm3; EMWCPP85 is the minimum possible possible EMWCP with probability of 85%, g/cm ; EMWCPP80 is the minimum possible EMWCP with 3 probabilityEMWCP with of 80%, probability g/cm3. of 85%, g/cm ; EMWCPP80 is the minimum possible EMWCP with probability of 80%, g/cm3.

Energies 2019, 12, 942 16 of 31 Energies 2019, 12, x FOR PEER REVIEW 15 of 29

100 1.2 95% ω=0° P95 1.1 P90 ω=10° 80 P85 ω ) =20° 3 P80 ω=30° 1.0 60 ω=40° ω=50° 0.9 40 0.8

Probability (%) 20 0.7 EMWCP (g/cm EMWCP

0 0.6 0.0 0.5 1.0 1.5 2.0 0 1020304050 Equivalent mud weight (g/cm3) Breakout width ω (°)

(a) (b)

FigureFigure 9.9. InﬂuenceInfluence ofof breakoutbreakout onon EMWCP.EMWCP. ( a(a)) The The cumulative cumulative probability probability of of EMWCP; EMWCP; ( b(b)) EMWCP EMWCP variesvaries withwith breakoutbreakout width.width.

ItIt isis clearlyclearly shownshown thatthat thethe EMWCPEMWCPP95,, EMWCP EMWCPP90P90, EMWCP, EMWCPP85P85 andand EMWCP EMWCPP80 P80estimatedestimated by the by theMG-C MG-C criterion criterion always always decline decline with with breakout breakout width width in in nonlinear. nonlinear. In In other other words, thethe EMWCPEMWCP declinesdeclines withwith breakout width, but but not not the the higher higher the the breakout breakout width width the the better better wellbore wellbore stability. stability. A Amore more accurate accurate definition deﬁnition of of collapse collapse is is that that when when the the breakout breakout width or angle exceedsexceeds aa criticalcritical limit,limit, suchsuch thatthat the the remaining remaining unaffected unaffected section section of the wellboreof the wellbore wall can nowall longer can upholdno longer the surroundinguphold the highsurrounding stresses andhigh ﬂows stresses inward and causing flows inward bore enlargement, causing bore subsequent enlargement, collapse subsequent and complete collapse failure and ofcomplete the wellbore failure [21 of]. the Zoback wellbore [6] indicated [21]. Zoback that the[6] indicated tolerable breakoutthat the tolerable width (2 ωbreakout) to prevent width borehole (2ω) to ◦ collapseprevent borehole is approximately collapse is 90 approximatelyfor a vertical 90° borehole. for a vertical Once borehole. the breakout Once width the breakout (2ω) exceeds width the(2ω) allowableexceeds the breakout allowable width, breakout the borehole width, the will borehole cannot maintain will cannot itself maintain stability. itself stability.

4.2.3.4.2.3. InfluenceInfluence ofof MeanMean ValueValue AccordingAccording toto EquationsEquations (16),(16), (18)(18) andand (22),(22), thethe EMWCPEMWCP usuallyusually dependsdepends onon thethe verticalvertical stressstress (σ ), the maximum horizontal stress (σ ), the minimum horizontal stress (σ ), the pore pressure (σvv), the maximum horizontal stress (σH),H the minimum horizontal stress (σh), theh pore pressure (pp), (thepp), Biot’s the Biot’s coefficient coefficient (α), ( αthe), the strength strength parameters parameters (c, ( cφ, ),ϕ ),the the Poisson’s Poisson’s ratio ratio ( (vv),), and thethe tolerabletolerable breakoutbreakout widthwidth ((ωω).). The influence influence of the mean valuevalue ofof eacheach basicbasic parameterparameter isis aa keykey aspectaspect toto EMWCP.EMWCP. We We simulated simulated the the influence influence of different of different mean valuemean of value each basicof each parameter basic parameter on the cumulative on the probabilitycumulative of probability EMWCP, andof EMWCP, the results and were the shownresults inwere Figure shown 10, wherein Figure the 10, mean where value the of mean each basicvalue parameterof each basic was parameter changed ± 10%was fromchanged base value.±10% from It is clearly base value. found thatIt is theclearly vertical found stress, that pore the pressure, vertical Biot’sstress, coefficient, pore pressure, and cohesiveBiot’s coefficient, strengthhad and a cohe verysive strong strength impact had on thea very cumulative strong impact probability on the of EMWCP,cumulative while probability the other of parameters EMWCP, while had a the lesser other impact. parameters had a lesser impact. AsAs shownshown in in Figure Figure 10 10j,j, it showedit showed the the impact impact of meanof mean value value change change on EMWCP, on EMWCP, all of all those of basicthose parametersbasic parameters were changedwere changed±10% ±10% from basefrom value,base value, but the but corresponding the corresponding variation variation of EMWCP of EMWCP gave differentgave different answers. answers. The ranking The ranking of the influencingof the influenc degreeing ofdegree mean of value mean change value is change the cohesive is the strengthcohesive >strength pore pressure > pore ≈pressureBiot’s coefficient≈ Biot’s coefficient > vertical > stressvertical > stress maximum > maximum horizontal horizontal stress > stress breakout > breakout width >width internal > internal friction anglefriction > minimumangle > minimum horizontal horizontal stress > Poisson’s stress > ratio.Poisson’s In general, ratio. theIn general, EMWCP the is positivelyEMWCP is correlated positively with correlated the vertical with stress,the vertical maximum stress, horizontalmaximum stress,horizontal pore stress, pressure pore and pressure Biot’s coefficient;and Biot’s coefficient; while it is negativelywhile it is negatively correlated corr withelated the minimum with the minimum horizontal horizontal stress, cohesive stress, strength,cohesive internalstrength, friction internal angle, friction Poisson’s angle, ratioPoisson’s and breakoutratio and width.breakout width.

Energies 2019, 12, 942 17 of 31 Energies 2019, 12, x FOR PEER REVIEW 16 of 29

100 100

80 80

σ =49.32MPa σ =39.48MPa 60 v 60 H σ =54.80MPa σ =43.87MPa v H σ =60.28MPa σ =48.26MPa 40 v 40 H

Probability (%) 20 Probability (%) 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (a) Vertical stress (b) Maximum horizontal stress

100 100

80 80

σ =27.82MPa p =19.25MPa 60 h 60 p σ =30.91MPa p =21.39MPa h p σ =34.00MPa p =23.53MPa 40 h 40 p

Probability (%) 20 Probability (%) 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (c) Minimum horizontal stress (d) Pore pressure

100 100

80 80

60 α=0.86 60 c=16.33MPa α=0.95 c=18.14MPa α=1.00 c=19.95MPa 40 40

Probability (%) Probability 20 (%) Probability 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (e) Biot’s coefficient (f) Cohesive strength

100 100

80 80

60 ϕ=31.50° 60 v=0.23 ϕ=35.00° v=0.25 ϕ=38.50° v=0.28 40 40 Probability (%) Probability Probability (%) 20 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 Equivalent mud weight (g/cm3) Equivalent mud weight (g/cm ) (g) Internal friction angle (h) Poisson’s ratio 100 0.66 σ σ σ v H h 0.64 α ω pp ϕ 80 ) 0.62 c v 3 0.60 ω 60 =36.00° 0.58 ω=40.00° ω=44.00° 0.56 40 0.54 0.52 Probability (%) 20 EMWCP (g/cm 0.50 0.48 0 0.46 0.00 0.25 0.50 0.75 1.00 1.25 1.50 -10 -5 0 5 10 3 Change from base value (%) Equivalent mud weight (g/cm ) (i) Breakout width (j) Impact of mean value change on EMWCP

FigureFigure 10. 10.Inﬂuence Influence ofof meanmean value on on cumu cumulativelative probability probability of of EMWCP. EMWCP.

Energies 2019, 12, x FOR PEER REVIEW 17 of 29

4.2.4. Influence of Variance Due to the uncertainty of basic parameters, not only the mean value of input parameter affect the EMWCP, but also the uncertainties also affect the EMWCP. In general, the variance represented the degree of uncertainty for a single input parameter. The influences of variance on the cumulative Energiesprobability2019, 12of, 942 EMWCP were simulated, and the results were shown in Figure 11, where18 ofthe 31 variance of input parameter was changed from 0.0 to 0.4. It is clearly found that the vertical stress, maximum horizontal stress, pore pressure, Biot’s coefficient, cohesive strength had a very strong 4.2.4. Influence of Variance impact on the cumulative probability of EMWCP, while the other parameters had a lesser impact. WhenDue the tovariance the uncertainty changes from of basic 0.0 to parameters, 0.4, all of the not cumulative only the meanprobability value of of EMWCP input parameter changed, butaffect the the corresponding EMWCP, but variation also the of uncertainties cumulative probability also affect gave the EMWCP.different Inanswers. general, For the the variance vertical stress,represented maximum the degree horizontal of uncertainty stress, minimum for a single horizont inputal parameter. stress, internal The influencesfriction angle of variance and Poisson’s on the ratio,cumulative with the probability increase of of EMWCP variance, were the simulated,cumulative and probability the results curves were showngradually in Figure moved 11 ,onto where right. the varianceBut for the of pore input pressure, parameter Biot’s was coefficient, changed fromcohesi 0.0ve to strength 0.4. It isand clearly breakout found width, that the with vertical the increase stress, ofmaximum variance, horizontal the 50–100% stress, of the pore cumulative pressure, Biot’sprobability coefficient, curves cohesive gradually strength moved had onto a right, very strongwhile theimpact 0–50% on of the the cumulative cumulative probability probability of curves EMWCP, gradually while themoved other onto parameters left. In fact, had we a lesserusually impact. focus Whenon the the50–100% variance of changesthe cumulative from 0.0 toprobability 0.4, all of curv the cumulativee, thus, with probability the increase of EMWCP of variance, changed, the cumulativebut the corresponding probability variationcurves gradually of cumulative moved probability onto right. gave In differentother words, answers. the risk For of the wellbore vertical collapsestress, maximum increases horizontalwith increase stress, of variance. minimum On horizontal the whole, stress, the ranking internal of friction the influencing angle and degree Poisson’s of varianceratio, with is thethe increasecohesive of strength variance, > theBiot’s cumulative coefficient probability > pore pressure curves > gradually vertical stress moved > ontomaximum right. horizontalBut for the stress pore pressure,> breakout Biot’s width coefficient, > internal cohesive friction strengthangle > minimum and breakout horizontal width, stress with the> Poisson’s increase ratio.of variance, Moreover, the 50–100% we usually of the focus cumulative on the probabilityhigh cumulative curves probability gradually movedof the ontocurves, right, and while we just the extracted0–50% of thethe cumulativeP95 EMWCP probability to reveal curves their influences gradually, movedthe results onto were left. In shown fact, we in usuallyFigure focus11j. It onis clearlythe 50–100% found of that the the cumulative P95 EMWCP probability increase curve, with the thus, variance, with the and increase the ranking of variance, of the the rate cumulative of change probabilityis the cohesive curves strength gradually > vertical moved stress onto > right. Biot’s In coefficient other words, > pore the riskpressure of wellbore > maximum collapse horizontal increases withstressincrease > internal of friction variance. angle On > thebreakout whole, width theranking > minimum of the horizontal influencing stress degree > Poisson’s of variance ratio. is the cohesiveMoreover, strength a sensitivity > Biot’s coefficient analysis >was pore carried pressure out > for vertical the EMWCP, stress > maximumthe results horizontal were shown stress in >Figure breakout 12. The width results > internal showed friction clearly angle that > minimumthe influence horizontal ranking stress of the > Poisson’s basic parameters ratio. Moreover, is the wecohesive usually strength focus on> Biot’s the highcoefficient cumulative > pore probability pressure > ofvertical the curves, stress > and maximum we just horizontal extracted thestress P95 > breakoutEMWCP towidth reveal > their internal influences, friction the angle results > were minimum shown inhorizontal Figure 11 j.stress It is clearly> Poisson’s found thatratio. the It P95 is consistentEMWCP increase with the with results the variance, of Figure and 11. the From ranking the ofrisk the analysis rate of change point isof the view, cohesive we should strength be > verticalconcerned stress with > Biot’sthe top coefficient five factors > pore which pressure impact > maximum the EMWCP horizontal the most; stress and > internalthese factors friction are angle the >cohesive breakout strength, width > Biot’s minimum coefficient, horizontal pore stresspressu >re, Poisson’s vertical ratio.stress and maximum horizontal stress.

100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) Probability 20 (%) Probability 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (a) Vertical stress (b) Maximum horizontal stress 100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) 20 Probability (%) 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (c) Minimum horizontal stress (d) Pore pressure

Figure 11. Cont. Energies 2019, 12, x FOR PEER REVIEW 18 of 29 Energies 2019, 12, 942 19 of 31 Figure 11. Cont.

100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) 20 Probability (%) 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (e) Biot’s coefficient (f) Cohesive strength 100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) Probability 20 (%) Probability 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (g) Internal friction angle (h) Poisson’s ratio 100 1.2 σ σ σ v H h p α ω 80 1.1 p ) φ

3 C v Var=0.00 60 Var=0.10 1.0

Var=0.20 (g/cm Var=0.30 40 Var=0.40 P95 0.9

Probability (%) Probability 20 0.8 EMWCP 0 0.7 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.0 0.1 0.2 0.3 0.4 3 Var Equivalent mud weight (g/cm ) (i) Breakout width (j) Impact of variance on P95 EMWCP

FigureFigure 11. 11. InfluenceInﬂuence of of variance variance on on probability probability of of EMWCP. EMWCP.

Moreover, a sensitivityCohesive strength analysis was carried out for the EMWCP, the results were shown in Figure 12. The results showedBiot coefficient clearly that the influence ranking of the basic parameters is the cohesive strength > Biot’s coefficientPore > pressure pore pressure > vertical stress > maximum horizontal stress > breakout width > internal frictionVertical angle stress > minimum horizontal stress > Poisson’s ratio. It is consistent with the results of FigureMax 11 horizontal. From stress the risk analysis point of view, we should be concerned with the top five factors which impactBreakout width the EMWCP the most; and these factors are the cohesive strength, Biot’s coefficient, poreInternal pressure, friction angle vertical stress and maximum horizontal stress. Min horizontal stress Poisson's ratio

0.44 0.48 0.52 0.56 0.60 0.64 0.68 Mean EMWCP (g/cm3)

Figure 12. Sensitivity tornado for EMWCP.

Energies 2019, 12, x FOR PEER REVIEW 18 of 29

Figure 11. Cont.

100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) 20 Probability (%) 20

0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (e) Biot’s coefficient (f) Cohesive strength 100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) Probability 20 (%) Probability 20

3 C v Var=0.00 60 Var=0.10 1.0

Var=0.20 (g/cm Var=0.30 40 Var=0.40 P95 0.9

Probability (%) Probability 20 0.8 EMWCP 0 0.7 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.0 0.1 0.2 0.3 0.4 3 Var Equivalent mud weight (g/cm ) (i) Breakout width (j) Impact of variance on P95 EMWCP Energies 2019, 12, 942 20 of 31 Figure 11. Influence of variance on probability of EMWCP.

Cohesive strength Biot coefficient Pore pressure Vertical stress Max horizontal stress Breakout width Internal friction angle Min horizontal stress Poisson's ratio

0.44 0.48 0.52 0.56 0.60 0.64 0.68 Mean EMWCP (g/cm3) Energies 2019, 12, x FOR PEER REVIEW 19 of 29 FigureFigure 12. 12. SensitivitySensitivity tornado tornado for for EMWCP. EMWCP. 4.3. Uncertainty Analysis of EMWFP 4.3. Uncertainty Analysis of EMWFP

4.3.1. EMWFP Probability Analysis 4.3.1. EMWFP Probability Analysis EMWFP means the equivalent mud weight of fracture pressure, the simulating results is shownEMWFP in Figure means 13. theThe equivalent mean EMWFP mud weight estimated of fracture by the pressure,reliability the assessment simulating method results isis shown1.6021 3 ing/cm Figure3, and 13 the. The analytical mean EMWFPsolution estimated(1.6020 g/cm by3) theis very reliability close to assessment the mean EMWFP method isestimated 1.6021 g/cm by the, 3 andreliability the analytical assessment solution method. (1.6020 The g/cmprobability) is very decreases close towith the increase mean EMWFP of EMW, estimated that means by thethe reliabilityhigher the assessment EMW, the method. higher the The risk probability of wellbor decreasese fracture with is. increaseWe should of EMW, determine that means a proper the higherprobability the EMW,to obtain the the higher corresponding the risk of wellboreEMWFP, fracturesuch as is.the WeEMWFP should with determine probability a proper (P95) probabilitydenotes the to maximum obtain the correspondingpossible EMW EMWFP,with probability such as the of EMWFP95%, which with defines probability the (P95)likelihood denotes of thepreventing maximum wellbore possible fracture EMW withas a probabilityfunction of of EMW, 95%, whichit can deﬁnes be obtained the likelihood from the of preventingcumulative wellboreprobability fracture curves. as As a function shown ofin EMW,Figure it 13, can the be obtainedEMWFP fromestimated the cumulative by the reliability probability assessment curves. 3 Asmethod shown is in0.6120 Figure g/cm 13,3, the 0.8386 EMWFP g/cm3 estimated, 0.9800 g/cm by the3 and reliability 1.0834 g/cm assessment3 for the methodprobability is 0.6120 of 95%, g/cm 90%,, 3 3 3 0.838685% and g/cm 80%, respectively. 0.9800 g/cm Theand maximum 1.0834 g/cm possiblefor EMW the probability estimated ofby 95%,the reliability 90%, 85% assessment and 80% respectively.method is obviously The maximum lower possiblethan that EMW of the estimated analytical by thesolution reliability (1.6020 assessment g/cm3). methodIn other is words, obviously the 3 loweruncertainty than thatof basic of the parameters analytical results solution in a (1.6020 lower EMWFP, g/cm ). Inand other the risk words, of wellbore the uncertainty fracture ofbecome basic parametersmuch higher. results in a lower EMWFP, and the risk of wellbore fracture become much higher.

0.7 100 0.6 80 0.5 0.4 60

0.3 40 0.2 20

Probability density Probability 0.1

0.0 0 (%) probability Cumulative 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 Equivalent mud weight (g/cm ) Figure 13. Probability density and cumulative probability for EMWFP.

4.3.2.4.3.2. InfluenceInfluence ofof MeanMean ValueValue AccordingAccording toto Equations Equations (30)–(33), (30)–(33), the EMWFPthe EMWFP is usually is usually depended depended on the maximum on the horizontalmaximum stress (σ ), the minimum horizontal stress (σ ), the pore pressure (p ), the Biot’s coefficient (α), and the horizontalH stress (σH), the minimum horizontalh stress (σh), thep pore pressure (pp), the Biot’s tensile strength (S ). The influence of the mean value of each basic parameter is a key aspect to coefficient (α), andt the tensile strength (St). The influence of the mean value of each basic parameter EMWFP.is a key Weaspect simulated to EMWFP. the influence We simulated of different the mean infl valueuence of of each different basic parameter mean value on theof cumulativeeach basic probabilityparameter ofon EMWFP,the cumulative and the resultsprobability were of shown EMWFP, in Figure and 14the, whereresults the were mean shown value in of eachFigure basic 14, where the mean value of each basic parameter was changed ±10% from base value. It is clearly found that the maximum horizontal stress, minimum horizontal stress, pore pressure and Biot’s coefficient had a very strong impact on the cumulative probability of EMWFP, while the other parameters had a lesser impact. As shown in Figure 14f, it showed the impact of mean value change on EMWFP, all of those basic parameters were changed ±10% from base value, but the corresponding variation of EMWFP gave different answers. The ranking of the influencing degree of mean value change is the minimum horizontal stress > the maximum horizontal stress > the Biot’s coefficient ≈ the pore pressure > the tensile strength. The EMWFP is positively correlated with the minimum horizontal stress and the tensile strength, while it is negatively correlated with the maximum horizontal stress, the Biot’s coefficient and the pore pressure.

Energies 2019, 12, 942 21 of 31 parameter was changed ±10% from base value. It is clearly found that the maximum horizontal stress, minimum horizontal stress, pore pressure and Biot’s coefﬁcient had a very strong impact on the Energiescumulative 2019, 12 probability, x FOR PEER of REVIEW EMWFP, while the other parameters had a lesser impact. 20 of 29

100 100

80 80

σ =39.48MPa σ =27.82MPa 60 H 60 h σ =43.87MPa σ =30.91MPa H h σ =48.26MPa σ =34.00MPa 40 H 40 h

Probability (%) Probability 20 (%) Probability 20

0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (a) Maximum horizontal stress (b) Minimum horizontal stress

100 100

80 80

p =19.25MPa α=0.86 60 p 60 p =21.39MPa α=0.95 p α=1.00 p =23.53MPa 40 p 40

Probability (%) Probability 20 (%) Probability 20

0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (c) Pore pressure (d) Biot’s coefficient 100 2.2 σ σ H h pp α S 80 2.0 0 t ) 3 1.8 S=5.40MPa 60 t S=6.00MPa t 1.6 S=6.60MPa 40 t 1.4

Probability (%) Probability 20 1.2 EMWFP (g/cm

0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -10 -5 0 5 10 3 Change from base value (%) Equivalent mud weight (g/cm ) (e) Tensile strength (f) Impact of mean value change on EMWFP

FigureFigure 14. InfluenceInﬂuence of of mean mean value value on on cu cumulativemulative probability of EMWFP.

As shown in Figure 14f, it showed the impact of mean value change on EMWFP, all of those 4.3.3. Influence of Variance basic parameters were changed ±10% from base value, but the corresponding variation of EMWFP gaveSimilarly, different answers.not only Thethe rankingmean value of the of influencing each basic degreeparameter of mean affects value the change EMWFP, is the but minimum also the uncertaintyhorizontal stress can >also the maximumaffect the horizontalEMWFP. In stress gene >ral, the Biot’sthe variance coefficient represented≈ the pore the pressure degree > theof uncertaintytensile strength. for basic The EMWFP parameter. is positively We simulated correlated the with influence the minimum of variance horizontal on the stress cumulative and the probabilitytensile strength, of EMWFP, while itand is negativelythe results correlatedwere shown with in Figure the maximum 15, where horizontal the variance stress, of each the Biot’sbasic parametercoefficient andwas thechanged pore pressure. from 0.0 to 0.4. It is clearly found that the maximum horizontal stress and the minimum horizontal stress had a very strong impact on the cumulative probability of EMWFP, while4.3.3. Influencethe other ofparameters Variance had a lesser impact. When the variance changed from 0.0 to 0.4, all of the cumulativeSimilarly, notprobability only the of mean EMWFP value changed. of each With basic the parameter increase affects of varian thece, EMWFP, the 50–100% but also of the cumulativeuncertainty canprobability also affect curves the EMWFP. gradually In general, moved theonto variance left, while represented the 0–50% the degree of the of uncertaintycumulative probabilityfor basic parameter. curves gradually We simulated moved the influenceonto right. of In variance fact, we on usually the cumulative focus on probability the 50–100% of EMWFP, of the cumulative probability curve, thus, with the increase of variance, the cumulative probability curves gradually moved onto left. In other words, the risk of wellbore fracture increases with increase of variance. On the whole, the ranking of the influencing degree of variance is the minimum horizontal stress > the maximum horizontal stress > the Biot’s coefficient > the pore pressure > the tensile strength. Moreover, we usually focus on the high cumulative probability of the curves, and we just extracted the P95 EMWFP to reveal their influences, the results were shown in Figure 15f. It

Energies 2019, 12, 942 22 of 31 and the results were shown in Figure 15, where the variance of each basic parameter was changed from 0.0 to 0.4. It is clearly found that the maximum horizontal stress and the minimum horizontal stress had a very strong impact on the cumulative probability of EMWFP, while the other parameters had a lesser impact. When the variance changed from 0.0 to 0.4, all of the cumulative probability of EMWFPEnergies 2019 changed., 12, x FOR WithPEER REVIEW the increase of variance, the 50–100% of the cumulative probability curves21 of 29 gradually moved onto left, while the 0–50% of the cumulative probability curves gradually moved ontois clearly right. found In fact, that we the usually P95 focusEMWFP on thedecrease 50–100% with of the the variance, cumulative and probability the ranking curve, of the thus, rate with of thechange increase is also of minimum variance, the horizontal cumulative stress probability > the maximum curves gradually horizontal moved stress onto> the left. Biot’s In othercoefficient words, > the pore risk ofpressure wellbore > the fracture tensile increases strength. with increase of variance. On the whole, the ranking of the influencingMoreover, degree a sensitivity of variance analysis is the minimumwas carried horizontal out for the stress EMWFP, > the maximumthe results horizontal were shown stress in >Figure the Biot’s 16. The coefficient results >showed the pore clearly pressure that >the the ranking tensile strength.of the basic Moreover, parameters we usuallyis the minimum focus on thehorizontal high cumulative stress > the probability maximum of horizo the curves,ntal stress and we> the just Biot’s extracted coefficient the P95 ≈ the EMWFP pore pressure to reveal > their the influences,tensile strength. the results It is consistent were shown with in Figure the results 15f. It of is Figure clearly 15. found From that the the risk P95 analysis EMWFP point decrease of view, with thewe variance,should be and concerned the ranking with of the the top rate two of change factors is which also minimum impact the horizontal EMWFP stress the most; > the maximumand these horizontalfactors are stressthe minimum > the Biot’s horizontal coefficient stress > the and pore the pressuremaximum > thehorizontal tensile strength.stress.

100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) Probability 20 (%) Probability 20

0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (a) Maximum horizontal stress (b) Minimum horizontal stress

100 100

80 80 Var=0.00 Var=0.00 60 Var=0.10 60 Var=0.10 Var=0.20 Var=0.20 Var=0.30 Var=0.30 40 Var=0.40 40 Var=0.40

Probability (%) Probability 20 (%) Probability 20

0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 3 Equivalent mud weight (g/cm ) Equivalent mud weight (g/cm ) (c) Pore pressure (d) Biot’s coefficient 100 1.0

80 0.5 ) 3 Var=0.00 60 Var=0.10 0.0 Var=0.20 (g/cm σ Var=0.30 H 40 P95 -0.5 σ Var=0.40 h pp α 0 Probability (%) Probability 20 -1.0 St EMWFP 0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00.10.20.30.4 3 Var Equivalent mud weight (g/cm ) (e) Tensile strength (f) Impact of variance on P95 EMWFP

Figure 15. InfluenceInﬂuence of of variance variance on on cumu cumulativelative probability of EMWFP.

Energies 2019, 12, 942 23 of 31

Moreover, a sensitivity analysis was carried out for the EMWFP, the results were shown in Figure 16. The results showed clearly that the ranking of the basic parameters is the minimum horizontal stress > the maximum horizontal stress > the Biot’s coefﬁcient ≈ the pore pressure > the tensile strength. It is consistent with the results of Figure 15. From the risk analysis point of view, we should be concerned with the top two factors which impact the EMWFP the most; and these factors Energiesare the 2019 minimum,, 12,, xx FORFOR horizontal PEERPEER REVIEWREVIEW stress and the maximum horizontal stress. 22 of 29

Min horizontal stress

Max horizontal stress

Pore pressure

Biot coefficient

Tensile strength

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 33 MeanMean EMWFPEMWFP (g(g/c/cm ))

FigureFigure 16. 16. SensitivitySensitivity tornado tornado for for EMWFP. EMWFP.

4.4.4.4. Uncertainty Uncertainty Analysis Analysis of of EMWPP EMWPP EMWPPEMWPP means means the the equivalent equivalent mud mud weight weight of of pore pore pressure, pressure, the the simulating simulating results results is shown is shown in 3 Figurein Figure 17. 17The. The mean mean EMWPP EMWPP estimated estimated by the by thereliability reliability assessment assessment method method is 0.9921 is 0.9921 g/cm g/cm33,, andand, 3 thetheand analyticalanalytical the analytical solutionsolution solution (0.9921(0.9921 (0.9921 g/cmg/cm g/cm33)) isis alsoalso) is alsothethe same. thesame. same. TheThe The probabilityprobability probability increasesincreases increases withwith with increaseincrease increase ofof EMW,of EMW, that that means means the the higher higher the the EMW, EMW, the the lower lower the the risk risk of ofwell well kick kick is.is. We We should should determine determine a propera proper probability probability to obtain to obtain the thecorresponding corresponding EMWPP, EMWPP, such such as the as EMWPP the EMWPP with probability with probability (P95) denotes(P95) denotes the minimum the minimum possible possible EMW EMW with with probabi probabilitylitylity ofof of95%,95%, 95%, whichwhich which definesdefines deﬁnes thethe the likelihoodlikelihood likelihood ofof preventing well well kick kick as as a function a function of EMW, of EMW, it can it be can obtained be obtained from the from cumulative the cumulative probability probability curves. Ascurves. shown As in shownFigure 17, in Figurethe EMWPP 17, the estimated EMWPP by estimated thethe reliabilityreliability by the assessmentassessment reliability methodmethod assessment isis 1.15521.1552 method g/cmg/cm is33,, 3 3 3 3 1.11921.1552 g/cmg/cm33,, , 1.0949 1.11921.0949 g/cmg/cmg/cm33 ,andand 1.0949 1.07561.0756 g/cm g/cmg/cmand33 1.0756forfor thethe g/cm probabilityprobabilityfor the ofof probability 95%,95%, 90%,90%, of 95%,85%85% and 90%,and 80%80% 85% respectively.and 80% respectively. The minimum The minimum possible possible EMW EMWestimated estimated by the by thereliability reliability assessment assessment method method is is 3 obviouslyobviously higherhigher thanthan thatthat of of the the analytical analytical solution solution (0.9921 (0.9921 g/cm g/cm).33).). In InIn other otherother words, words,words, the thethe uncertainty uncertaintyuncertainty of ofpore pore pressure pressure results results in ain lower a lower EMWPP, EMWPP, and and the the risk risk of wellof well kick kick become become much much higher. higher.

4.5 100 4.0 3.5 80 3.0 60 2.5 2.0 40 1.5 1.0 20 Probability density Probability Probability density Probability 0.5 Cumulative probability (%) probability Cumulative 0.0 0 (%) probability Cumulative 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 33 Equivalent mud weight (g/cm ))

FigureFigure 17. 17. ProbabilityProbability density density and and cumulative cumulative probability probability for for EMWPP. EMWPP.

AccordingAccording to to Equations Equations (34)–(36), (34)–(36), the the EMWP EMWPPP only only depends depends on the on thepore pore pressure pressure (ppp).). TheThe (pp). influenceinfluenceThe inﬂuence ofof thethe of meanmean the mean valuevalue value andand variancevariance and variance ofof porepore of porepressurepressure pressure isis thethe is keykey the aspectaspect key aspect toto EMWPP.EMWPP. to EMWPP. WeWe simulated the influence of differentt meanmean valuevalue ofof porepore pressurepressure onon thethe cumulativecumulative probabilityprobability ofof EMWFP, and the results were shown in Figure 18,, wherewhere thethe meanmean valuevalue ofof porepore pressurepressure waswas changed ±10% from base value. It is clearly found that the uncertain pore pressure had a very strong impact on the cumulative probability of EMWPP. The EMWPP is positively correlated with thethe meanmean valuevalue ofof porepore pressure.pressure. Moreover,Moreover, wewe alalso simulated the influence of variance on the cumulative probability of EMWPP, and the results were shown in Figure 19, where the variance of pore pressure was changed from 0.0 to 0.4. When the variance changed from 0.0 to 0.4, the 50–100% of the cumulative probability curves gradually moved onto right, while the 0–50% of the cumulative probability curves gradually moved onto left. In fact,, wewe usuallyusually focusfocus onon thethe 50–100%50–100%

Energies 2019, 12, 942 24 of 31

We simulated the influence of different mean value of pore pressure on the cumulative probability of EMWFP, and the results were shown in Figure 18, where the mean value of pore pressure was changed ±10% from base value. It is clearly found that the uncertain pore pressure had a very strong impact on the cumulative probability of EMWPP. The EMWPP is positively correlated with the mean value of pore pressure. Moreover, we also simulated the influence of variance on the cumulative probability of EMWPP, and the results were shown in Figure 19, where the variance of pore pressure was changed Energies 2019, 12, x FOR PEER REVIEW 23 of 29 fromEnergies 0.0 2019 to, 0.4.12, x WhenFOR PEER the REVIEW variance changed from 0.0 to 0.4, the 50–100% of the cumulative probability23 of 29 curves gradually moved onto right, while the 0–50% of the cumulative probability curves gradually of the cumulative probability curve, thus, with the increase of variance, the cumulative probability movedof the cumulative onto left. In probability fact, we usually curve, focusthus, onwith the th 50–100%e increase of of the variance, cumulative the probabilitycumulative curve,probability thus, curves gradually moved onto right. In other words, the risk of well kick linearly increased with withcurves the gradually increase ofmoved variance, onto the right. cumulative In other probability words, the curves risk graduallyof well kick moved linearly onto increased right. In otherwith increase of variance, as shown in Figure 19b. words,increase the of riskvariance, of well as kick shown linearly in Figure increased 19b. with increase of variance, as shown in Figure 19b. 100 100 80 80 60 60 40 40 Pp=19.25MPa Probability (%) Probability 20 Pp=19.25MPaPp=21.39MPa Probability (%) Probability 20 Pp=21.39MPaPp=23.53MPa 0 Pp=23.53MPa 00.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.253 1.50 Equivalent mud weight (g/cm3) Equivalent mud weight (g/cm ) Figure 18. Influence of mean value of pore pressure on cumulative probability for EMWPP. FigureFigure 18. 18. InfluenceInfluence of of mean mean value value of of pore pore pressure pressure on cumulative probability for EMWPP. 100 1.7 1.7 100 1.6 P95 80 1.6 P95P90

) 1.5 P85 80 3 P90

) 1.5 P85P80 60 3 1.4 1.4 P80 60 Var=0.00 1.3 1.3 40 Var=0.00Var=0.10 1.2 Var=0.20 40 Var=0.10 1.2 Var=0.20Var=0.30 1.1

Probability (%) 20 Var=0.30Var=0.40 1.1 EMWCP (g/cm EMWCP Probability (%) 20 Var=0.40 1.0

EMWCP (g/cm EMWCP 1.0 0 0.9 00.00 0.25 0.50 0.75 1.00 1.25 1.50 0.9 0.00.10.20.30.4 0.00 0.25Equivalent 0.50 mud 0.75 weight 1.00 (g/cm 1.253) 1.50 0.00.10.20.30.4Var Equivalent mud weight (g/cm3) Var (a) (b) (a) (b) Figure 19.19. InﬂuenceInfluence ofof porepore pressurepressure variancevariance onon EMWPP.EMWPP. ((aa)) InﬂuenceInfluence ofof variancevariance ofof porepore pressure;pressure; Figure 19. Influence of pore pressure variance on EMWPP. (a) Influence of variance of pore pressure; ((bb)) EMWCPEMWCP variesvaries withwith variance.variance. (b) EMWCP varies with variance. 4.5. Uncertainty Analysis of SMWW 4.5. Uncertainty Analysis of SMWW On thethe basisbasis ofof thethe uncertaintyuncertainty analysis of EMWCP, EMWFPEMWFP and EMWPP, thethe cumulativecumulative On the basis of the uncertainty analysis of EMWCP, EMWFP and EMWPP, the cumulative probabilityprobability ofof SMWWSMWW cancan bebe obtained,obtained, and the resultsresults were shownshown in Figures 2020 andand 2121 andand TableTable4 4.. probability of SMWW can be obtained, and the results were shown in Figures 20 and 21 and Table 4. Figure 2020 shows shows the the interference interference of theof probabilitythe probab densityility density function function of EMWCP, of EMWCP, EMWFP EMWFP and EMWPP, and Figure 20 shows the interference of the probability density function of EMWCP, EMWFP and theEMWPP, overlapping the overlapping zone denotes zone the probabilitydenotes the of wellboreprobability instability. of wellbore The smaller instability. overlapping The smaller zone, EMWPP, the overlapping zone denotes the probability of wellbore instability. The smaller theoverlapping wellbore zone, will be the more wellbore reliable, will i.e., be themore risk reliable, of wellbore i.e., the instability risk of wellbore will be lower.instability The will bigger be overlapping zone, the wellbore will be more reliable, i.e., the risk of wellbore instability will be overlappinglower. The bigger zone, overlapping the wellbore zone, will the be morewellbore unreliable, will be more i.e., the unreliable, risk ofwellbore i.e., the risk instability of wellbore will lower. The bigger overlapping zone, the wellbore will be more unreliable, i.e., the risk of wellbore beinstability higher. will be higher. instability will be higher. 4.0 100 4.0 100 3.5 EMWPP 3.5 EMWPPEMWCP 80 3.0 EMWCPEMWFP 80 3.0 EMWFP 2.5 60 2.5 2.0 60 2.0 1.5 40 1.5 40 1.0 1.0 20 Probability density 0.5 20 Probability density 0.5

0.0 0 (%) probability Cumulative

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.00 (%) probability Cumulative 0.0 0.5 1.0 1.5 2.0 2.53 3.0 Equivalent mud weight (g/cm3) Equivalent mud weight (g/cm ) Figure 20. Probability density histogram of SMWW. Figure 20. Probability density histogram of SMWW.

Energies 2019, 12, x FOR PEER REVIEW 23 of 29

of the cumulative probability curve, thus, with the increase of variance, the cumulative probability curves gradually moved onto right. In other words, the risk of well kick linearly increased with increase of variance, as shown in Figure 19b.

100

Pp=19.25MPa Probability (%) Probability 20 Pp=21.39MPa Pp=23.53MPa 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 3 Equivalent mud weight (g/cm ) Figure 18. Influence of mean value of pore pressure on cumulative probability for EMWPP.

100 1.7 1.6 P95 80 P90

) 1.5 P85 3 P80 60 1.4 Var=0.00 1.3 40 Var=0.10 1.2 Var=0.20 Var=0.30 1.1 Probability (%) 20 Var=0.40

EMWCP (g/cm EMWCP 1.0 0 0.9 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00.10.20.30.4 Equivalent mud weight (g/cm3) Var

(a) (b)

Figure 19. Influence of pore pressure variance on EMWPP. (a) Influence of variance of pore pressure; (b) EMWCP varies with variance.

4.5. Uncertainty Analysis of SMWW On the basis of the uncertainty analysis of EMWCP, EMWFP and EMWPP, the cumulative probability of SMWW can be obtained, and the results were shown in Figures 20 and 21 and Table 4. Figure 20 shows the interference of the probability density function of EMWCP, EMWFP and EMWPP, the overlapping zone denotes the probability of wellbore instability. The smaller overlapping zone, the wellbore will be more reliable, i.e., the risk of wellbore instability will be lower. The bigger overlapping zone, the wellbore will be more unreliable, i.e., the risk of wellbore Energies 2019, 12, 942 25 of 31 instability will be higher.

4.0 100 3.5 EMWPP EMWCP 80 3.0 EMWFP

2.5 60 2.0 1.5 40 1.0 20 Probability density 0.5

0.0 0 (%) probability Cumulative 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3 Equivalent mud weight (g/cm ) Energies 2019, 12, x FOR PEER REVIEWFigure 20. Probability density histogram of SMWW. 24 of 29

100 95% 90% EMWPP 85% 80 EMWCP EMWFP

Probability (%) Probability 20

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 Equivalent mud weight (g/cm ) FigureFigure 21. 21. CumulativeCumulative probability probability of of SMWW. SMWW.

Table 4. The statistics for EMWPP, EMWCP, EMWFP and SMWW. Table 4. The statistics for EMWPP, EMWCP, EMWFP and SMWW.

StatisticsStatistics EMWPPEMWPP EMWCP EMWCP EMWFP EMWFP SMWW SMWW P80/(g/cmP80/(g/cm3) 3) 1.07561.0756 0.8083 0.8083 1.0935 1.0935 1.0756–1.0935 1.0756–1.0935 P85/(g/cmP85/(g/cm3) 3) 1.09491.0949 0.8487 0.8487 0.9791 0.9791 No SMWW No SMWW P90/(g/cmP90/(g/cm3) 3) 1.11931.1193 0.9053 0.9053 0.8240 0.8240 No SMWW No SMWW 3 P95/(g/cmP95/(g/cm) 3) 1.15521.1552 0.9063 0.9063 0.6144 0.6144 No SMWW No SMWW 3 P100/(g/cmP100/(g/cm) 3) 1.37341.3734 1.5160 1.5160 −0.8036−0.8036 No SMWW No SMWW 3 Mean/(g/cmMean/(g/cm) 3) 0.99210.9921 0.5538 0.5538 1.6021 1.6021 0.9921–1.6021 0.9921–1.6021 3 StdStd Dev/(g/cm Dev/(g/cm) 3) 0.09920.0992 0.2060 0.2060 0.6013 0.6013 / / Analytical solution/(g/cm3) 0.9921 0.5195 1.6020 0.9921–1.6020 Analytical solution/(g/cm3) 0.9921 0.5195 1.6020 0.9921–1.6020

FigureFigure 2121 showsshows the the cumulative cumulative probability probability of EMWCP, of EMWCP, EMWFP EMWFP and EMWPP, and EMWPP, and the statisticsand the statisticsall for EMWCP, all for EMWCP, EMWFP andEMWFP EMWPP and wereEMWPP collected were incollected Table4 .in It Table is clearly 4. It foundis clearly that found when that the whenuncertainty the uncertainty of basic parameters of basic parameters is considered, is considered, the mean EMWCP the mean is 0.5538 EMWCP g/cm is3 0.5538, and the g/cm P953, EMWCPand the P95is 0.9063 EMWCP g/cm is3, while0.9063 the g/cm analytical3, while solution the analytical (0.5195 solution g/cm3) is (0.5195 obviously g/cm too3) lowis obviously to maintain too wellbore low to maintainstability, thatwellbore is, the collapsestability, risk that of is, the the wellbore collapse is actuallyrisk of the much wellbore higher thanis actually that of traditionalmuch higher analysis. than thatSimilarly, of traditional when the analysis. uncertainty Similarly, of basic when parameters the uncertainty is considered, of basic the meanparameters EMWPP is isconsidered, 0.9921 g/cm the3, meanand the EMWPP P95 EMWPP is 0.9921 is 1.1552 g/cm3 g/cm, and3 the, while P95 the EMWPP analytical is 1.1552 solution g/cm (0.99213, while g/cm the3 )analytical is obviously solution lower, (0.9921that is, g/cm the kick3) is riskobviously of the welllower, is actuallythat is, the much kick higher risk of than the well that is of actually traditional much analysis. higher Moreover,than that ofwhen traditional the uncertainty analysis. ofMoreover, basic parameters when the is uncertainty considered, of the basic mean parameters EMWFP isis 1.6021considered, g/cm 3the, and mean the EMWFPP95 EMWFP is 1.6021 is 0.6144 g/cm3 g/cm, and the3, while P95 EMWFP the analytical is 0.6144 solution g/cm3, (1.6020while the g/cm analytical3) is obviously solution (1.6020 lower, thatg/cm is,3) isthe obviously fracture risklower, of the that wellbore is, the isfracture actually risk much of the higher wellbore than that is actually of traditional much analysis. higher than that of traditionalAccording analysis. to the Equation (37), the lowest safe mud weight of SMWW should be depended on the highAccording of EMWCP to the and Equation EMWPP, (37), and the the lowest highest safe safe mud mud weight weight of of SMWW SMWW should should be be depended depended on on the high of EMWCP and EMWPP, and the highest safe mud weight of SMWW should be depended on the EMWFP, that is, the lowest safe mud weight should be the EMWPP, while the highest safe mud weight should be the EWMCP. As shown in Figure 21 and Table 4, when the cumulative probability is 95%, the P95 EMWFP (0.6144 g/cm3) is obviously lower than both of EMWPP (1.1552 g/cm3) and EMWCP (0.9063 g/cm3), there is no SMWW. When the cumulative probability is 90%, the P90 EMWFP (0.8240 g/cm3) is obviously lower than both of EMWPP (1.1193 g/cm3) and EMWCP (0.9053 g/cm3), there is still no SMWW. When the cumulative probability is 85%, the P85 EMWFP (0.9791 g/cm3) is obviously lower than EMWPP (1.0949 g/cm3) but higher than EMWCP (0.8487 g/cm3), that is, there is still no SMWW. When the cumulative probability is 80%, the P80 EMWFP (1.0935 g/cm3) is obviously higher than both of EMWPP (1.0756 g/cm3) and EMWCP (0.8083 g/cm3), there is a SMWW of 1.0756–1.0935 g/cm3, but it is very narrow SMWW. The SMWW predicted by analytical solution is 0.9921–1.6020 g/cm3, compared to the SMWW estimated by the reliability assessment method, the SMWW predicted only by mean value is very wide to maintain wellbore stability, but the reliability assessment method tends to give a narrow SMWW of 1.0756–1.0935 g/cm3 and its probability is only 80%, that is, it is very difficult to drill without any wellbore instability problems.

Energies 2019, 12, 942 26 of 31 the EMWFP, that is, the lowest safe mud weight should be the EMWPP, while the highest safe mud weight should be the EWMCP. As shown in Figure 21 and Table4, when the cumulative probability is 95%, the P95 EMWFP (0.6144 g/cm3) is obviously lower than both of EMWPP (1.1552 g/cm3) and EMWCP (0.9063 g/cm3), there is no SMWW. When the cumulative probability is 90%, the P90 EMWFP (0.8240 g/cm3) is obviously lower than both of EMWPP (1.1193 g/cm3) and EMWCP (0.9053 g/cm3), there is still no SMWW. When the cumulative probability is 85%, the P85 EMWFP (0.9791 g/cm3) is obviously lower than EMWPP (1.0949 g/cm3) but higher than EMWCP (0.8487 g/cm3), that is, there is still no SMWW. When the cumulative probability is 80%, the P80 EMWFP (1.0935 g/cm3) is obviously higher than both of EMWPP (1.0756 g/cm3) and EMWCP (0.8083 g/cm3), there is a SMWW of 1.0756–1.0935 g/cm3, but it is very narrow SMWW. The SMWW predicted by analytical solution is 0.9921–1.6020 g/cm3, compared to the SMWW estimated by the reliability assessment method, the SMWW predicted only by mean value is very wide to maintain wellbore stability, but the reliability assessment method tends to give a narrow SMWW of 1.0756–1.0935 g/cm3 and its probability is only 80%, that is, it is very difﬁcult to drill without any wellbore instability problems. In this situation, the traditional over-balanced drilling (OBD) maybe cannot drill successfully, but there is an available mud weight window (0.8083–1.0935 g/cm3) to avoid wellbore collapse and fracture, we maybe can utilize underbalanced drilling (UBD), managed pressure drilling (MPD), or micro-ﬂow drilling (MFD) to conquer well kick, and we can drill successfully. Another choice should be wider the SMWW, the only way is to prevent lost circulation.

4.6. Field Observation Report In the design of well SC-101X, the traditional OBD was designed to drill in the 2008–2432 m section, the density of water-based drilling mud was designed as 1.07 g/cm3 by considering the impact of the surge and swab pressures. In real drilling operation, the actual drilling mud density was also 1.07 g/cm3. When this well was drilled at the depth of 2022 m, the driller operates drilling rig to pull the drill string up and down to prepare making a connection, the mud man found drilling mud loss, the average loss rate of drilling mud was approximately 3.67 m3/h, and the total loss volume of drilling mud was approximately 10.5 m3. In this process, the drilling mud was added some lost circulation materials (LCMs) to against drilling mud loss, the LCMs formula was given as follows: 4% GZD-B + 5% GZD-C + 6% GZD-D. After multiple plugging, the pressure-bearing capacity was tested, and the lost pressure was enhanced to 1.19 g/cm3. The drilling mud density of 1.07 g/cm3 was still used to drill. But when this well was drilled at the depth of 2067 m, the driller operates drilling rig to pull the drill string up and down to prepare making a connection, the mud man found a gas kick. The well was shut-in to prepare well kill. After testing the shut-in standpipe pressure, the killing mud density was determined as 1.13 g/cm3. During well killing, the drilling mud loss occurred again with a low average loss rate of 3.5 m3/h. After the well was killed successfully, and considering the higher pore pressure of the lower formation, the use of MPD was considered. Before drilling using MPD, the drilling mud was again added some lost circulation materials (LCMs), and the LCMs formula was given as follows: 5% GZD-O + 5% GZD-A + 10% GZD-B + 10% GZD-C + 14% GZD-D + 4% FDJ-1 + 3% Nutshell + 3% JD-5. After multiple plugging, the pressure-bearing capacity was tested, and the lost pressure was enhanced to 1.37 g/cm3. In the process of MPD, the operating parameters were listed in Table5, and finally, the section of 2067–2432 m of well SC-101X was drilled successfully. To sum up, the field observation for well kick and wellbore fracture is consistent with analysis results of SMWW that estimated by uncertain model. Thus, the present model was therefore verified, and it is much more accurate than the traditional analytical solution. Energies 2019, 12, 942 27 of 31

Table 5. Operating parameters of MPD.

No. Parameters Value 1 Well depth (m) 2067–2432 2 EMWPP (g/cm3) 1.08 3 EMWCP (g/cm3) 1.09 (Enhanced to 1.37) 4 Mud displacement (L/s) 20–25 5 Mud density (g/cm3) 1.05–1.16 6 Pressure control during drilling (MPa) 0–1 7 Pressure control during pump-stopping (MPa) 2–3

5. Conclusions On the basis of the reliability assessment theory, the uncertain EMWCP model was proposed by involving the tolerable breakout and the MG-C criterion, the uncertain EMWFP and EMWPP model was also proposed, and the uncertain SMWW model was determined by EMWCP, EMWPP and EMWFP. Finally, taking the basic parameters of well SC-101X as an example, the inﬂuence of uncertain basic parameter on SMWW was systematically investigated and discussed. The main conclusions can be drawn as follows:

(1) For wellbore stability analysis, the uncertain distribution of each basic parameter satisfies the normal distribution, the higher the coefficient of variance is, the higher the level of uncertainty will be, and the larger the impact on wellbore stability will be. (2) The EMWCP predicted by the M-C criterion is obviously higher than that of the MG-C criterion, because of the MG-C criterion rightly predicts rock strength and estimates the EMWCP much more exactly. Thus, the MG-C criterion is recommended for wellbore stability analysis. (3) The breakout width has a very significant impact on EMWCP, and the EMWCP always decline with breakout width in nonlinear. The tolerable breakout is recommended for wellbore stability analysis, and the tolerable breakout width (2ω) of 45◦ is recommended for a vertical well. (4) The EMWCP estimated by the uncertain model is obviously higher than that of analytical model, that is, the collapse risk of the wellbore is heightened by the uncertain basic parameters. The cumulative probability of wellbore collapse gradually increases with increasing of variance, and the influencing degree of uncertain basic parameter is the cohesive strength > Biot’s coefficient > pore pressure > vertical stress > maximum horizontal stress > breakout width > internal friction angle > minimum horizontal stress > Poisson’s ratio. (5) The EMWPP estimated by the uncertain model is obviously higher than that of mean pore pressure, that is, the kick risk is also heightened by the uncertain pore pressure. The cumulative probability of well kick gradually also increases with increasing of variance for pore pressure. (6) The EMWFP estimated by the uncertain model is obviously lower than that of analytical model, that is, the fracture risk of the wellbore heightened by the uncertain basic parameters. The cumulative probability of wellbore fracture gradually increases with increasing of variance, and the influencing degree of uncertain basic parameter is the minimum horizontal stress > maximum horizontal stress > Biot’s coefficient > pore pressure > tensile strength. (7) For the SC-101X well, the SMWW predicted by analytical solution is 0.9921–1.6020 g/cm3 and very wide, compared to the SMWW estimated by the uncertain model, the P80 SMWW estimated by the uncertain model is only 1.0756–1.0935 g/cm3 and very narrow, that is, there is a low possibility to drill without any wellbore instability problems. The field observation for well kick and wellbore fracture verified the analysis results of SMWW that estimated by uncertain model, and both lost circulation prevention and MPD were utilized to maintain drilling safety. (8) To drill the formation with narrow SMWW, some kinds of advanced drilling technology, such as the UBD, MPD and MFD, can be utilized to control the wellbore pressure accurately, so that the Energies 2019, 12, 942 28 of 31

wellbore stability can be maintained. Meanwhile, another choice should be wider the SMWW, the only way is to prevent lost circulation.

Author Contributions: Conceptualization, T.M. and P.C. Investigation, T.M. and T.T. Methodology, T.M. Writing—original draft, T.M. and T.T. Writing—review and editing, P.C. and C.Y. Funding: This research was funded by the National Natural Science Foundation of China (Grant Nos. 41874216 and 51604230), the China Postdoctoral Science Foundation (Grant Nos. 2017T100592 and 2016M600626), and the Young Elite Scientists Sponsorship Program by CAST (2017QNRC001). Acknowledgments: This work was supported by the Program of Introducing Talents of Discipline to Chinese Universities (111 Plan) (Grant No. D18016). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

HTHP High-temperature and high-pressure QRA Quantitative risk analysis M-C Mohr-coulomb MG-C Mogi-coulomb H-B Hoek-brown TVD True vertical depth EMWCP Equivalent mud weight of collapse pressure EMWFP Equivalent mud weight of fracture pressure EMWPP Equivalent mud weight of pore pressure EMW Equivalent mud weight Std Dev Standard deviation Var Variance SMWW Safe mud weight window UBD Underbalanced drilling MPD Managed pressure drilling MFD Micro-ﬂow drilling

Appendix A M-C Criterion The M-C criterion can be expressed by using principal stress [7],

σ1 = C0 + qσ3 (A1) where C0 and q are given as, ( 2c cos ϕ C0 = 1−sin ϕ 1+sin ϕ (A2) q = 1−sin ϕ ◦ where ϕ is the friction angle, ( ); c is the cohesive strength, MPa; σ1 and σ3 are the major and minor principal stress, MPa. If considering the conventional effective stress concept, the M-C criterion can be rewritten as,

σ1 − αpp = C0 + q σ3 − αpp (A3) where α is the Biot’s coefﬁcient; pp is the pore pressure, MPa. Rearranging Equation (A3), the M-C criterion can be expressed as [7],

σ1 = C + qσ3 (A4) where C is given as, C = C0 − (q − 1)αpp (A5) Energies 2019, 12, 942 29 of 31

Appendix B MG-C Criterion The MG-C criterion can be expressed as [2,7],

τoct = a + bσm,2 (A6) where τoct, σm,2, a and b are given as,

 h i 1  = 1 ( − )2 + ( − )2 + ( − )2 2 τoct 3 σ1 σ2 σ2 σ3 σ3 σ1 (A7)  1 σm,2 = 2 (σ1 + σ3) ( √ a = 2 2 c cos ϕ 3√ (A8) 2 2 b = 3 sin ϕ where a is the intersection of the line on τoct-axis; b is the inclination of the line; σ2 is the intermediate principal stress, MPa. If considering the conventional effective stress concept, the MG-C criterion can be rewritten as,

1 2 2 0 0 I1 − 3I2 = a + b I1 − σ2 − 2αpp (A9)

0 0 where I1, I2, a and b are given as, I = σ + σ + σ 1 1 2 3 (A10) I2 = σ1σ2 + σ2σ3 + σ3σ1 a0 = 2c cos ϕ (A11) b0 = sin ϕ

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