A method for evaluating wellbore instability risk using Nataf transformation to achieve parameter correlation

Abstract

The present invention discloses a method for evaluating the risk of wellbore instability by using Nataf transformation to realize parameter correlation. First, the geomechanical parameter curve of the target reservoir is obtained based on well logging data and indoor core experiments, and the reservoir with the same lithology is used as the parameter sample data at the depth. In addition, considering the non-normal correlation problem existing in the parameters themselves, the K-S test and Pearson linear correlation coefficient are first used to establish the goodness-of-fit of the characteristic distribution function within the parameters and the correlation between the parameters; then the Monte-Carlo method and the Nataf transformation method are effectively combined to statistically calculate the wellbore instability pressure under the linear elastic theory. Finally, a quantitative evaluation of the wellbore instability risk of a directional well is achieved. This method can more objectively consider the true correlation between geomechanical parameters, improve the efficiency of simulation analysis, and realize the risk evaluation of wellbore collapse and rupture of any well, providing theoretical support for pre-drilling prediction and wellbore instability.

Description

Technical Field

[0001] The present invention relates to a wellbore instability risk assessment method for realizing parameter correlation by utilizing Nataf transformation, and belongs to the technical field of (ultra) deep oil and gas drilling and completion. Background Art

[0002] Effective analysis of the risk of wellbore instability in directional wells plays an important role in the exploration of oil and gas resources in complex deep formations. As an important indicator for the smooth implementation of the drilling process, wellbore stability is not only related to the strength characteristics of the formation rock and the state of the wellbore surrounding rock, but is also directly affected by factors such as in situ stress and wellbore trajectory design. However, the uncertainty and non-independence of geomechanical parameters greatly increase the difficulty of predicting instability pressure and evaluating wellbore stability. Therefore, the present invention effectively combines the Nataf transformation method with the Monte-Caro method to perform sampling simulation on geomechanical parameters, thereby realizing the wellbore instability risk evaluation of any well. This method is simple and easy to implement, can effectively retain the characteristics of the original parameter correlation, and can well realize the uncertainty analysis and risk assessment of formation instability pressure. Summary of the Invention

[0003] The present invention mainly overcomes the deficiencies in the prior art and proposes a wellbore instability risk assessment method that utilizes Nataf transformation to achieve parameter correlation.

[0004] The present invention provides a technical solution to solve the above technical problems: a method for evaluating wellbore instability risk using Nataf transformation to achieve parameter correlation, comprising the following steps:

[0005] Step 1: Combine well logging data and indoor core experiments to obtain geomechanical parameter samples of the target reservoir;

[0006] The geomechanical parameters include: mechanical parameters of formation rock, rock strength parameters, and ground stress and pore pressure values; the mechanical parameters of formation rock include elastic modulus and Poisson's ratio; the rock strength parameters include cohesion, internal friction angle and tensile strength.

[0007] Step 2: Based on the linear elastic theory and the Mohr-Coulomb criterion and the tensile failure criterion, an evaluation model for the instability of the wellbore is established;

[0008] Based on the linear elastic theory, the Mohr-Coulomb criterion and the tensile failure criterion are combined to establish an arbitrary wellbore instability evaluation model, which specifically includes:

[0009] The effective stress of the surrounding rock at the wellbore wall of any well is obtained from the linear elastic medium theory:

[0010]

[0011] In the formula The effective stress of the wellbore in the cylindrical coordinate system is solved as follows, in MPa; its characteristic roots correspond to the effective principal stresses σ1′, σ2′, σ3′ of the wellbore;

[0012]

[0013] Where v is Poisson's ratio, dimensionless; P w is the liquid column pressure, MPa; θ is the wellbore angle, °; σ xx , σ yy , σ zz and σ xz , σ xy , σ yz are the normal stress and shear stress in the wellbore coordinate system, MPa; α is the Biot coefficient, dimensionless; P p is the pore pressure, MPa;

[0014] The Mohr-Coulomb criterion is:

[0015]

[0016] Where, is the maximum shear stress, MPa; C is the cohesive force, MPa; is the internal friction angle, °; MPa;

[0017] The tensile rupture criterion is:

[0018] σ′3=-S t

[0019] Where: S t is the tensile strength of rock, MPa;

[0020] By known well inclination and azimuth Combined with the Mohr-Coulomb criterion and the tensile failure criterion, the instability pressure of the wellbore and the periphery of the wellbore is calculated, and the maximum value of the Mohr-Coulomb calculation result at the periphery of the wellbore is taken as the collapse pressure P b The minimum value of the tensile failure criterion calculation result is taken as the rupture pressure P f :

[0021] P b =max((f b (θ0),…,f b (θ i ),…f b (θ 360 ))

[0022] P f =min(f f(θ0),…,f f (θ i ),…f f (θ 360 ))

[0023] Where, P b is the collapse pressure, MPa; P f is the burst pressure, MPa; θ i The corresponding angle when the well circumference is 0 to 360 degrees with a certain step length, °; f b (θ) and f f (θ) represents the collapse pressure and fracture pressure at the wellbore angle θ, respectively, in MPa.

[0024] Step 3: Use the KS test method to perform a characteristic distribution fit goodness test on the parameter samples in step 1, and quantify the parameters into the original parameter uncertainty space composed of the characteristic distribution function;

[0025] The KS test method is:

[0026] Dn=max|Sn(x)-F0(x)|

[0027] Where Sn(x) is the empirical cumulative probability distribution function; F0(x) is the specific cumulative probability distribution function, hereinafter referred to as the characteristic distribution; Dn is the KS test value. The smaller the Dn value, the closer it is to the assumed distribution condition.

[0028] The original parameter uncertainty space is: Based on the KS test results, each parameter is represented by a characteristic distribution with high goodness of fit, and these characteristic distributions construct the original parameter uncertainty space.

[0029] Step 4: Based on the data sample in step 1, obtain the Pearson linear correlation coefficient between the parameters and construct a coefficient matrix;

[0030] In step 4, the Pearson linear correlation coefficients between parameters are obtained to construct a coefficient matrix, including:

[0031]

[0032]

[0033] In the formula, m represents the number of samples, n represents the number of parameters, and X j Represents parameter X i and parameter X j The mean of; O represents the correlation coefficient matrix; parameter r ij Represents parameter X i and parameter X j Pearson linear correlation coefficient.

[0034] Step 5: According to the Nataf transformation method, the constructed original non-normal distribution coefficient matrix is ​​transformed into the normal distribution space;

[0035] In step 5, the Nataf transformation method includes: given n-dimensional correlated random variables X, X = [x1, x2, ..., x n ] T , and its corresponding distribution function, then:

[0036]

[0037] Where Φ(·) represents the cumulative distribution function of the standard normal distribution; Φ -1 (·) corresponds to the inverse function of the standard normal distribution; f i (x i ) and F i (x i ) are parameters x i The probability distribution function and cumulative distribution function of y i is the parameter x of the original uncertainty space i The values ​​are converted to normal distribution by Nataf; μ and σ are the mean and standard deviation of the corresponding parameters; r′ ij is the original correlation coefficient r ij Correlation coefficient after Nataf transformation.

[0038] Step 6: Generate sampling data in an independent normal space based on the Monte-Carlo method, and convert the sample data into a normal correlation distribution space using the singular value decomposition method;

[0039] In step 6, the Monte-Carlo method is a method of performing repeated sampling under a known distribution function to generate sample data, and then estimating the target by importing the sample data into the research model.

[0040] Step 7: Combine steps 5 and 6 to project the sample data back to the original non-normal space;

[0041] In step 7, the singular value decomposition method is: given n-dimensional independent normal distribution random variables Z = [z1, z2, ..., z n ] T , then the normal correlation random variable Y=[y1,y2,…,y n ] T The following conversions exist:

[0042]

[0043] Where U is the correlation coefficient r′ij The unitary matrix, By the correlation coefficient r′ ij The singular value decomposition of is generated, that is: Among them It is a diagonal matrix composed of singular values ​​from large to small;

[0044] In step 7, projecting the sampled data back to the original non-normal space includes using the Monte-Carlo method described in step 6 to generate a large number of n-dimensional independent standard normal distribution random numbers, and combining the Nataf transformation method and singular value decomposition method described in steps 5 and 6 to map back to the original parameter uncertainty space described in step 3.

[0045] Step 8: Combine the mapping data, count the instability pressure at each time, estimate the cumulative distribution curve, and finally achieve the risk assessment of the wellbore instability model in step 2.

[0046] In step 8, the cumulative distribution curve shows the cumulative probability that the instability pressure value is less than or equal to a certain value; the corresponding horizontal axis represents the value of the random variable instability pressure, and the vertical axis represents the cumulative probability.

[0047] The present invention has the following beneficial effects: the present invention proposes a method for evaluating the risk of wellbore instability by using Nataf transformation to realize parameter correlation. The method is simple and easy to implement, retains the original parameter correlation, and realizes the risk evaluation of wellbore instability with uncertainty of geomechanical parameters under effective consideration of parameter correlation, providing a scientific basis for the analysis of wellbore stability in drilling in complex medium formations. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a flowchart of the present invention. DETAILED DESCRIPTION

[0049] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0050] like Figure 1 As shown, the present invention provides a wellbore instability risk assessment method using Nataf transformation to achieve parameter correlation, comprising the following steps:

[0051] Step 1: Combine logging data and indoor core experiments to obtain geomechanical parameter samples of the target reservoir. Geomechanical parameters mainly include: mechanical parameters of formation rocks, rock strength parameters, and ground stress and pore pressure values.

[0052] Among them, the mechanical parameters of the formation rock include elastic modulus and Poisson's ratio; the rock strength parameters include cohesion, internal friction angle and tensile strength; the pore pressure value refers to the pore pressure of the original formation.

[0053] For the above parameters, their mechanical parameters and strength parameters can be obtained through triaxial compression tests of rocks; ground stress can be obtained through acoustic emission tests; and pore pressure can be obtained by statistically analyzing existing pore pressure data.

[0054] Step 2: Based on linear elasticity theory combined with the Mohr-Coulomb criterion and the tensile failure criterion, an evaluation model for the instability of the wellbore of any well is established;

[0055] The effective stress of the surrounding rock at the wellbore wall of any well is obtained from the linear elastic medium theory:

[0056]

[0057] In the formula The effective stress of the wellbore in the cylindrical coordinate system is calculated as follows, in MPa; its characteristic roots correspond to the effective principal stresses σ1′, σ2′, and σ3′ of the wellbore.

[0058]

[0059] Where v is Poisson's ratio, dimensionless; P w is the liquid column pressure, MPa; θ is the wellbore angle, °; σ xx ,σ yy ,σ zz and σ xz , σ xy , σ yz are the normal stress and shear stress in the wellbore coordinate system, MPa; α is the Biot coefficient, dimensionless; P p is the pore pressure, MPa.

[0060] The normal stress and shear in the borehole coordinate system described in the present invention should be obtained by in situ stress conversion, which mainly includes: conversion between the in situ stress coordinate system and the geodetic coordinate system (due north), and conversion between the borehole coordinate system and the geodetic coordinate system:

[0061]

[0062]

[0063]

[0064]

[0065] Where: ω b The angle between the projection line of the actual wellbore axis on the horizontal plane and the true north direction, referred to as the well inclination azimuth, γb is the well inclination angle, that is, the angle between the wellbore axis and the plumb line. i is the angle between the horizontal maximum ground stress direction and the north direction; γ i is the angle at which the overburden pressure deviates from the vertical direction

[0066] The Mohr-Coulomb criterion is:

[0067]

[0068] Where, is the maximum shear stress, MPa; C is the cohesive force, MPa; is the internal friction angle, °; MPa.

[0069] The tensile rupture criterion is:

[0070] σ′3=-S t

[0071] Where: S t is the tensile strength of rock, MPa.

[0072] By knowing the in-situ stress distribution of the formation and the inclination angle of any well and azimuth The instability pressure around the wellbore can be calculated by combining the Mohr-Coulomb criterion and the tensile failure criterion. The maximum value of the Mohr-Coulomb calculation result around the wellbore is taken as the collapse pressure P b The minimum value of the tensile failure criterion calculation result is taken as the rupture pressure P f :

[0073] P b =max((f b (θ0),…,f b (θ i ),…f b (θ 360 ))

[0074] P f =min(f f (θ0),…,f f (θ i ),…f f (θ 360 ))

[0075] Where, P b is the collapse pressure, MPa; P f is the burst pressure, MPa; θ i The corresponding angle when the well circumference is 0 to 360 degrees with a certain step length, °; f b (θ) and ff (θ) represents the collapse pressure and fracture pressure at the wellbore angle θ, MPa

[0076] Step 3: Use the KS test method to perform a characteristic distribution fitting goodness test on the parameter samples in step 1, and quantify the parameters into the original parameter uncertainty space composed of the characteristic distribution function.

[0077] The KS test method is:

[0078] Dn=max|Sn(x)-F0(x)|

[0079] Where Sn(x) is the empirical cumulative probability distribution function; F0(x) is a specific cumulative probability distribution function (hereinafter referred to as characteristic distribution), such as normal distribution, uniform distribution, etc.; and Dn is the KS test value. The smaller the Dn value, the closer it is to the assumed distribution condition.

[0080] The characteristic distribution includes known distribution types such as normal distribution, lognormal distribution, triangular distribution, extreme value distribution, uniform distribution, gamma distribution and Cauchy distribution.

[0081] The original uncertainty space includes: the KS test result represents each parameter with a characteristic distribution with high goodness of fit, and the original parameter uncertainty space constructed by these characteristic distributions.

[0082] Step 4: Based on the data sample in step 1, obtain the Pearson linear correlation coefficient between the parameters and construct the coefficient matrix. This mainly includes using the parameters obtained in step 1 to perform the following Pearson linear correlation coefficient calculation and matrix construction:

[0083]

[0084]

[0085] In the formula, m represents the number of samples, n represents the number of parameters, and X j Represents parameter X i and parameter X j The mean of; O represents the correlation coefficient matrix; parameter r ij Represents parameter X i and parameter X j Pearson linear correlation coefficient.

[0086] Step 5: According to the Nataf transformation method, the constructed original non-normal distribution coefficient matrix is ​​transformed into the normal distribution space;

[0087] The Nataf transformation method is to obtain the random variable X of n-dimensional related geomechanical parameters (X = [x1, x2, ..., x n ] T ) and its characteristic distribution function with high KS fitting goodness of fit, then:

[0088]

[0089] Where Φ(•) represents the cumulative distribution function of the standard normal distribution; Φ -1 (•) corresponds to the inverse function of the standard normal distribution; f i (x i ) and F i (x i ) are parameters x i The probability distribution function and cumulative distribution function of y i is the parameter x of the original uncertainty space i The values ​​are converted to normal distribution by Nataf; μ and σ are the mean and standard deviation of the corresponding parameters; r′ ij is the original correlation coefficient r ij Correlation coefficient after Nataf transformation.

[0090] Step 6: Generate sampling data in an independent normal space based on the Monte-Carlo method, and convert the sample data into a normal correlation distribution space using the singular value decomposition method;

[0091] The Monte-Carlo method is a method of generating sample data through a large number of repeated samplings under the condition of a known distribution function, and estimating the target by importing the sample data into the research model.

[0092] The singular value decomposition method is: given n-dimensional independent normal distribution random variables Z = [z1, z2, ..., z n ] T Then the normal correlation random variable Y=[y1,y2,…,y n ] T The following conversions exist:

[0093]

[0094] Where U is the correlation coefficient r′ ij The unitary matrix, By the correlation coefficient r′ ij The singular value decomposition of is generated, that is: Among them It is a diagonal matrix composed of singular values ​​from large to small.

[0095] Therefore, the sample data generated by a large number of repeated sampling under the known distribution function conditions described by the Monte-Carlo method is to generate a large number of sampling sample data under the independent standard normal distribution, and then the singular value decomposition method is used to convert the data into the correlated normal distribution space.

[0096] Step 7: Combine steps 5 and 6 to project the sample data back to the original non-normal space;

[0097] The method of projecting the sampled data back to the original non-normal space is to use the Monte-Carlo method and singular value decomposition described in step 6 to generate a large number of n-dimensional correlated normally distributed random numbers; and then combine the Nataf transformation method described in step 5 to map the data back to the original parameter uncertainty space described in step 3.

[0098] Step 8: Combine the mapping data, count the instability pressure at each time, estimate the cumulative distribution curve, and finally achieve the risk assessment of the wellbore instability model in step 2.

[0099] The detailed process described in Step 8 includes entering the mapping data into the calculation model for wellbore instability pressure (collapse pressure and fracture pressure) for any well established in Step 2; calculating the pressure values ​​calculated for each set of mapping data; and displaying the cumulative probability of the instability pressure being less than or equal to a certain value using a cumulative distribution curve, where the horizontal axis of the cumulative distribution curve represents the value of the random variable instability pressure and the vertical axis represents the cumulative probability. For ease of identification, the quantified collapse pressure and fracture pressure are plotted on the same cumulative distribution chart, allowing for rapid assessment of wellbore instability risk.

[0100] The above description does not limit the present invention in any form. Although the present invention has been disclosed through the above embodiments, it is not intended to limit the present invention. Any technician familiar with the profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. A method for evaluating wellbore instability risk using Nataf transformation to achieve parameter correlation, characterized in that: The following steps are involved: Step 1: Combine well logging data and indoor core experiments to obtain geomechanical parameter samples of the target reservoir; Step 2: Based on the linear elastic theory and the Mohr-Coulomb criterion and the tensile failure criterion, an evaluation model for the instability of the wellbore is established; Step 3: Use the KS test method to perform a characteristic distribution fit goodness test on the parameter samples in step 1, and quantify the parameters into the original parameter uncertainty space composed of the characteristic distribution function; Step 4: Based on the data sample in step 1, obtain the Pearson linear correlation coefficient between the parameters and construct a coefficient matrix; Step 5: According to the Nataf transformation method, the constructed original non-normal distribution coefficient matrix is ​​transformed into the normal distribution space; Step 6: Generate sampling data in an independent normal space based on the Monte-Carlo method, and convert the sample data into a normal correlation distribution space using the singular value decomposition method; The singular value decomposition method is: given n-dimensional independent normal distribution random variables Z = [z1, z2, ..., z n ] T , then the normal correlation random variable Y=[y1,y2,…,y n ] T The following conversions exist: Where U is the correlation coefficient r′ ij The unitary matrix, By the correlation coefficient r′ ij The singular value decomposition of is generated, that is: Among them It is a diagonal matrix composed of singular values ​​from large to small; Step 7: Combine steps 5 and 6 to project the sample data back to the original non-normal space; Including, using the Monte-Carlo method described in step 6 to generate a large number of n-dimensional independent standard normal distribution random numbers, and combining the Nataf transformation method and singular value decomposition method described in steps 5 and 6 to map back to the original parameter uncertainty space described in step 3; Step 8: Combine the mapping data and bring them into the calculation model for the wellbore instability pressure of any well established in step 2. Count the instability pressure at each time and estimate the cumulative distribution curve. Finally, the risk assessment of the wellbore instability model in step 2 is achieved to facilitate the rapid assessment of the wellbore instability risk.

2. The method for evaluating wellbore instability risk by utilizing Nataf transformation to realize parameter correlation according to claim 1, characterized in that: In step 1, the geomechanical parameters include: mechanical parameters of the formation rock, rock strength parameters, and ground stress and pore pressure values; the mechanical parameters of the formation rock include elastic modulus and Poisson's ratio; the rock strength parameters include cohesion, internal friction angle and tensile strength.

3. The method for evaluating wellbore instability risk by utilizing Nataf transformation to realize parameter correlation according to claim 1, characterized in that: In step 2, an arbitrary wellbore instability evaluation model is established by combining the Mohr-Coulomb criterion and the tensile failure criterion under the linear elastic theory, specifically including: The effective stress of the surrounding rock at the wellbore wall of any well is obtained from the linear elastic medium theory: In the formula The effective stress of the wellbore in the cylindrical coordinate system is solved as follows, in MPa; its characteristic roots correspond to the effective principal stresses σ1′, σ2′, σ3′ of the wellbore; Where v is Poisson's ratio, dimensionless; P w is the liquid column pressure, MPa; θ is the wellbore angle, °; σ xx ,σ yy ,σ zz and σ xz , v xy ,σ yz are the normal stress and shear stress in the wellbore coordinate system, MPa; α is the Biot coefficient, dimensionless; P p is the pore pressure, MPa; The Mohr-Coulomb criterion is: Where, is the maximum shear stress, MPa; C is the cohesive force, MPa; is the internal friction angle, °; MPa; The tensile failure criterion is: σ′3=-S t Where: S t is the tensile strength of rock, MPa; By known well inclination and azimuth Combined with the Mohr-Coulomb criterion and the tensile failure criterion, the instability pressure of the wellbore and the periphery of the wellbore is calculated, and the maximum value of the Mohr-Coulomb calculation result at the periphery of the wellbore is taken as the collapse pressure P b The minimum value of the tensile failure criterion calculation result is taken as the rupture pressure P f : P b =max(f b (θ0),…,f b (i i ),…f b (i 360 )) P f =min(f f (θ0),…,f f (i i ),…f f (i 360 )) Where, P b is the collapse pressure, MPa; P f is the burst pressure, MPa; θ i The corresponding angle when the well circumference is 0 to 360 degrees with a certain step length, °; f b (θ) and f f (θ) represents the collapse pressure and fracture pressure at the wellbore angle θ, respectively, in MPa.

4. The method for evaluating wellbore instability risk by utilizing Nataf transformation to realize parameter correlation according to claim 1, wherein: In step 3, the KS test method is: Dn=max|Sn(x)-F0(x)| Where Sn(x) is the empirical cumulative probability distribution function; F0(x) is the specific cumulative probability distribution function, hereinafter referred to as the characteristic distribution; Dn is the KS test value. The smaller the Dn value, the closer it is to the assumed distribution condition.

5. The method for evaluating wellbore instability risk by realizing parameter correlation using Nataf transformation according to claim 1, characterized in that: In step 3, the original parameter uncertainty space is: each parameter is represented by a characteristic distribution with high goodness of fit according to the KS test result, and these characteristic distributions construct the original parameter uncertainty space.

6. The method for evaluating wellbore instability risk by realizing parameter correlation using Nataf transformation according to claim 1, characterized in that: In step 4, the Pearson linear correlation coefficients between parameters are obtained to construct a coefficient matrix, including: In the formula, m represents the number of samples, n represents the number of parameters, and Represents parameter X i and parameter X j The mean of; O represents the correlation coefficient matrix; parameter r ij Represents parameter X i and parameter X j Pearson linear correlation coefficient.

7. The method for evaluating wellbore instability risk by realizing parameter correlation using Nataf transformation according to claim 1, characterized in that: In step 5, the Nataf transformation method includes: given n-dimensional correlated random variables X, X = [x1, x2, ..., x n ] T , and its corresponding distribution function, then: Where Φ(·) represents the cumulative distribution function of the standard normal distribution; Φ -1 (·) corresponds to the inverse function of the standard normal distribution; f i (x i ) and F i (x i ) are parameters x i The probability distribution function and cumulative distribution function of y i is the parameter x of the original uncertainty space i The values ​​are converted to normal distribution by Nataf; μ and σ are the mean and standard deviation of the corresponding parameters; r′ ij is the original correlation coefficient r ij Correlation coefficient after Nataf transformation.

8. The method for evaluating wellbore instability risk by using Nataf transformation to achieve parameter correlation according to claim 1, characterized in that: In step 6, the Monte-Carlo method is a method of performing repeated sampling under a known distribution function to generate sample data, and then estimating the target by importing the sample data into the research model.

9. The method for evaluating wellbore instability risk by realizing parameter correlation using Nataf transformation according to claim 1, characterized in that: In step 8, the cumulative distribution curve shows the cumulative probability that the instability pressure value is less than or equal to a certain value; the corresponding horizontal axis represents the value of the random variable instability pressure, and the vertical axis represents the cumulative probability.

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