Probability estimation method for elastic modulus of rock mass

Abstract

The invention discloses a rock mass elastic modulus probability estimation method, which comprises the following steps: S1, acquiring a compression wave velocity Vp of a rock on the basis of field test investigation and statistics for a to-be-estimated hole section area of a rock mass mechanical parameter; s2, determining priori knowledge about the rock elasticity modulus; s3, constructing a Bayesian calculation framework; s4, using an adaptive MH-MCMC algorithm to iterate relevant parameters in the relation equation under a Bayesian calculation framework, generating posterior probability samples corresponding to the rock elasticity moduli, and determining probability distribution of the rock elasticity moduli; and S5, determining the value of the elastic modulus of the actual engineering rock mass based on the H-B criterion, and obtaining the probability distribution of the on-site rock mass parameters. According to the method provided by the invention, the probability range of the rock mass mechanical parameters can be efficiently estimated, the method is used for guiding the formulation of solutions considering various conditions during actual construction, and the method has better economic benefits and has important guiding significance for engineering practice.

Description

Technical Field

[0001] The present application belongs to the technical field of engineering geology and rock mechanics in geotechnical engineering, and in particular to a probabilistic estimation method of rock elastic modulus. Background Art

[0002] Acquisition of rock mass mechanical parameters: Long-term stability of rock mass and disaster prediction during the excavation and unloading process of rock mass engineering are the focus of engineering construction support work. Accurate determination of rock mass elastic modulus is an important rock mass mechanical parameter for conducting engineering rock mass stability analysis and on-site engineering support design optimization. Rock elastic modulus is one of the physical and mechanical properties of rock, which refers to the ratio of stress to strain in the elastic range of rock. The elastic modulus of rock varies with the type, composition, particle size, water content, and pore size of the rock. Due to factors such as size effect and joints and fissures, the elastic modulus of rock cannot be directly applied on site. Under the influence of geological activities, occurrence state, and human activities, there is generally a certain uncertainty in the elastic modulus of rock mass in a specific site. This uncertainty includes physical uncertainty, measurement uncertainty, and model conversion uncertainty. Probability distribution is usually used in geotechnical engineering to characterize the uncertainty of rock and soil.

[0003] At present, there are some rock mass mechanical parameter probability estimation methods based on a large amount of on-site geological survey work, such as Patent 1 (CN 115600398B) A large cave rock mass parameter probability estimation method based on Monte Carlo simulation), etc. However, for projects under complex geological conditions, due to the high cost of on-site tests and limited geological survey sites, it is often difficult to conduct a large number of related mechanical tests, and a large amount of effective rock mass elastic modulus information is often difficult to obtain, so these methods are not applicable.

[0004] Problems in current research: (1) In the study of probabilistic estimation methods for rock mass parameters, the spatial variability of rock mass parameters caused by complex geological conditions has not been well considered, and there is a lack of comprehensive and detailed geological survey work for the supporting projects. In particular, for complex engineering rock mass, the uncertainty analysis of its mechanical parameters does not fully consider the joint properties and geological environment of the rock mass. (2) For projects with good site conditions, a large amount of detailed geological survey work is the basis for revealing the spatial variability of rock mass mechanical parameters. However, for projects with complex geological conditions (such as deep buried tunnels, large underground caverns, and severely weathered rock slopes), there are no conditions for conducting sufficient geological surveys. How to use field and indoor and outdoor test data to obtain the distribution of elastic modulus parameters of rocks or rock masses is an urgent problem to be solved. (3) Most of the existing tests are based on parameters obtained from indoor and outdoor tests. When there is less data, the use of certain values ​​is not representative and universal.

[0005] Due to the complexity of the rock mass and its geological environment, the spatial variability of rock mass mechanical parameters and the influence of human subjectivity, the statistical determination process is full of great uncertainty, which makes the estimated rock mass mechanical parameters inaccurate. Summary of the invention

[0006] The purpose of this application is to overcome the problems of the prior art and disclose a probabilistic estimation method for rock elastic modulus. This application adopts uncertainty analysis theory to determine rock mechanical parameters. The probability-based parameter determination method is more instructive for practical engineering applications.

[0007] The purpose of this application is achieved through the following technical solutions:

[0008] A probabilistic estimation method for rock mass elastic modulus, the probabilistic estimation method for rock mass elastic modulus comprising:

[0009] S1: For the tunnel section where the rock mass mechanical parameters are to be estimated, based on field test investigation and statistics, obtain the rock compression wave velocity V p ;

[0010] S2: Determine the prior knowledge about the rock elastic modulus;

[0011] S3: Building a Bayesian computing framework;

[0012] S4: Using the adaptive MH-MCMC algorithm under the Bayesian calculation framework to iterate the relevant parameters in the relationship equation, generate the posterior probability samples corresponding to each rock elastic modulus, and determine the probability distribution of the rock elastic modulus;

[0013] S5: Determine the value of the elastic modulus of the actual engineering rock mass based on the HB criterion and obtain the probability distribution of the on-site rock mass parameters.

[0014] According to a preferred embodiment, step S2 comprises:

[0015] S21: Determine the rock elastic modulus E s The distribution form of

[0016] S22: Constructing rock elastic modulus E s and compression wave velocity V p Uncertainty regression model.

[0017] According to a preferred embodiment, in step S21, the rock elastic modulus E s The distribution form of is defined as:

[0018] E s =exp(μ N +σ N z)

[0019] Among them, z is a random variable that follows a standard normal distribution, and the parameter μ N and σ N is ln(E s )’s mean and standard deviation.

[0020] According to a preferred embodiment, in step S22,

[0021] ln(V p )=(a′μ N +b′)+a′σ N z+ε

[0022] Among them, a′ and b′ are the relationship parameters fitted when establishing the logarithmic model, ε is used to express the difference between the output variable of the regression model and the test results, that is, the uncertainty of the conversion, ln(V p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and

[0023] According to a preferred embodiment, step S3 includes: using the total probability formula to construct E s -V p The uncertainty regression model is transformed and expressed as:

[0024]

[0025] P(E s |Data) is the probability distribution of rock elastic modulus based on prior information and field test data, P(E s |μ,σ) is the probability distribution of elastic modulus defined by mean and standard deviation, and P(μ,σ|Data) is the rock elastic modulus E s The joint probability distribution of the mean and standard deviation of , P(μ,σ) is the prior information of the project. a′ and b′ are used to establish E s With V p The relationship parameters of the established uncertainty regression relationship model;

[0026] Where ln(V p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and i represents the i-th test sample, then ln(V pi ) represents the compression wave velocity of the i-th test sample, n s Represents the total number of test samples, μ min and μ max YesE s Minimum and maximum values ​​of the mean, σ min and σ max YesEs The minimum and maximum values ​​for the standard deviation.

[0027] According to a preferred embodiment, step S4 comprises:

[0028] Candidate samples of the Markov chain are generated from the Gaussian distribution, the algorithm and input parameters used in the MCMC sampling calculation are determined, the Bayesian estimation of the rock mechanics parameters is performed, and several calculation samples are calculated to obtain the probability distribution of the rock elastic modulus.

[0029] According to a preferred embodiment, the sampling process of adjusting the algorithm parameters is performed by a nonlinear heuristic algorithm and an adaptive adjustment strategy;

[0030] Among them, nonlinear heuristic algorithms include particle swarm optimization, simulated annealing algorithm, genetic algorithm, and differential evolution algorithm;

[0031] The adaptive adjustment strategy is implemented by adaptively changing the covariance of the proposed distribution, thereby changing the spatial extension and directionality of the proposed distribution to adaptively approach the target distribution.

[0032] According to a preferred embodiment, step S5 includes: determining the value of the elastic modulus of the actual engineering rock mass based on the HB criterion, comparing it with the average value of the rock elastic modulus in step S4, calculating the reduction coefficient ε, thereby reducing the rock mechanical parameters as a whole and obtaining the probability distribution of the on-site rock mass parameters.

[0033] The aforementioned main scheme of the present application and its further options can be freely combined to form multiple schemes, all of which are schemes that can be adopted and claimed for protection in the present application. After understanding the scheme of the present application, those skilled in the art can understand that there are multiple combinations based on the prior art and common knowledge, all of which are technical schemes to be protected by the present application, and they are not exhaustively listed here.

[0034] Beneficial effects of this application:

[0035] This application introduces the probability theory method and uses the distribution of elastic modulus (Es) to describe the deformation characteristics of rock mass, which is more reasonable than using a certain value (such as the average value of test data). The rock block compression wave velocity Vp is used to evaluate the elastic modulus Es of the rock. In the process of establishing the statistical model, the conversion uncertainty of the model is quantitatively considered, and prior knowledge is introduced to supplement the information contained in the model, so that this method has obvious advantages under small sample conditions.

[0036] This application makes full use of the improved MH-MCMC algorithm, and based on the constructed probability model, quickly generates a large number of samples of rock elastic modulus that meet the target distribution form, and gives the probability distribution range of rock and on-site rock elastic modulus in detail through investigation and statistics. The method provided by this application can efficiently estimate the probability range of rock mechanical parameters, which can be used to guide the formulation of solutions that take into account various situations when actually constructing, has better economic benefits, and has important guiding significance for engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a schematic flow chart of the probabilistic estimation method of rock elastic modulus in this application. DETAILED DESCRIPTION

[0038] The following describes the embodiments of the present application through specific examples, and those skilled in the art can easily understand other advantages and effects of the present application from the contents disclosed in this specification. The present application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present application. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict.

[0039] Aiming at the characteristics of large-scale engineering structures and construction, and taking into account the spatial variability of complex geological conditions and the influence of human subjectivity, a probabilistic estimation method for rock elastic modulus is provided to solve the problem of insufficient samples when estimating rock mechanical parameters based on traditional methods. The probabilistic estimation method for rock elastic modulus specifically includes the following steps.

[0040] Step 1: For the tunnel section where the mechanical parameters of the large-scale engineering rock mass are to be estimated, the input parameters of the rock are obtained based on field test investigation and statistics. The input parameters include the rock compression wave velocity V p .

[0041] Step 2: Determine the prior knowledge about the rock elastic modulus. The prior knowledge includes the following contents.

[0042] Step 2.1 Determine the rock elastic modulus E s The distribution form of rock elastic modulus E s It basically obeys the log-normal distribution, and its distribution can generally be defined as formula (1):

[0043] E s =exp(μ N +σ N z) (1)

[0044]

[0045] Among them, z is a random variable that follows a standard normal distribution, and the parameter μ N and σ N is ln(E s ) can be calculated according to formula (2) and formula (3), where μ and σ are the rock elastic modulus E s The mean and standard deviation of .

[0046] Step 2.2: Construct the rock elastic modulus E s and compression wave velocity V p The uncertainty regression model can be expressed by the following relationship:

[0047] ln(V p )=(a′μ N +b′)+a′σ N z+ε (4)

[0048] Among them, a′ and b′ are the relationship parameters fitted when establishing the logarithmic model, ε is used to express the difference between the output variable of the regression model and the test results, that is, the uncertainty of the conversion, ln(V p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and Among them, z is a random variable that follows a standard normal distribution, and the parameter μ N and σ N is ln(E s ) can be calculated according to formula (2) and formula (3).

[0049] Step 3: Build a Bayesian computing framework:

[0050] Step 3.1: Use the total probability formula to construct E s -V p The uncertainty regression model is transformed into the following formula:

[0051] P(E s |Data)=∫P(E s |μ,σ)×P(μ,σ|Data)dμdσ (5)

[0052] In the formula, P(E s |Data) is the probability distribution of rock elastic modulus based on prior information and field test data, P(E s |μ,σ) is the probability distribution of elastic modulus defined by mean and standard deviation, and P(μ,σ|Data) is the rock elastic modulus E s The joint probability distribution of the mean and standard deviation of , P(μ,σ) is the prior information of the project.

[0053] Among them, the rock elastic modulus Es It obeys the log-normal distribution, expressed as:

[0054]

[0055] In the formula, μ N and σ N is ln(E s )’s mean and standard deviation.

[0056]

[0057] In the formula, P(E s |Data) is the probability distribution of rock elastic modulus based on prior information and test data, P(E s |μ,σ) is the probability distribution of elastic modulus defined by mean and standard deviation, and P(μ,σ|Data) is the rock elastic modulus E s The joint probability distribution of the mean and standard deviation of , P(μ,σ) is the prior information of the project.

[0058] Where P(μ,σ) can be represented by a uniform distribution over a large range and is defined as:

[0059]

[0060] In the formula, μ min and μ max YesE s Minimum and maximum values ​​of the mean, σ min and σ max are the minimum and maximum values ​​of the standard deviation. Both values ​​need to be determined by field researchers based on engineering experience.

[0061] The calculation method of P(Data|μ,σ) is defined as:

[0062]

[0063] The simplified calculation formula is as follows:

[0064]

[0065] Where: P(E s |Data) is the probability distribution of rock elastic modulus based on prior information and field test data, P(E s |μ,σ) is the probability distribution of elastic modulus defined by mean and standard deviation, and P(μ,σ|Data) is the rock elastic modulus E s The joint probability distribution of the mean and standard deviation of , P(μ,σ) is the prior information of the project. a′ and b′ are used to establish E s With V pThe relationship parameters of the uncertainty regression relationship model established. p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and i represents the i-th test sample, then ln(V pi ) is the compression wave velocity of the ith test sample. s Represents the total number of test samples. N and σ N is ln(E s )’s mean and standard deviation, μ min and μ max YesE s Minimum and maximum values ​​of the mean, σ min and σ max YesE s The minimum and maximum values ​​of the standard deviation need to be determined by field researchers based on engineering experience.

[0066] Step 4: Under the Bayesian calculation framework, the improved adaptive MH-MCMC algorithm is used to iterate the relevant parameters in the relationship equation to generate a large number of posterior probability samples corresponding to the rock elastic modulus, and then determine the probability distribution of the rock elastic modulus.

[0067] Specifically, it includes: selecting a suitable initial state, generating candidate samples of the Markov chain from the Gaussian distribution; determining the algorithm and input parameters used in the MCMC sampling calculation, performing Bayesian estimation of rock mechanics parameters, and calculating a large number of calculation samples to obtain the probability distribution of the rock elastic modulus.

[0068] Preferably, the sampling process of the algorithm parameters is adjusted through a nonlinear heuristic algorithm and an adaptive adjustment strategy, thereby improving the sampling efficiency, accelerating the parameter transfer, and then deducing the posterior distribution of the parameters.

[0069] The nonlinear heuristic algorithm includes particle swarm algorithm, simulated annealing algorithm, genetic algorithm, differential evolution algorithm, etc. The adaptive adjustment strategy is implemented by adaptively changing the covariance of the proposed distribution, thereby changing the spatial extension and directionality of the proposed distribution to adaptively approach the target distribution.

[0070] Further, we select a suitable proposed distribution, such as uniform distribution or Gaussian distribution, and set a suitable initial state, and set the probability density function PDF f(E n *|E n-1 ), where n represents the nth state of the Markov chain and the superscript * represents the candidate state. The average value of the n-1th state is E n-1 The acceptance ratio r of the parameters is calculated according to the Metropolis-Hastings (MH) algorithm.a , its expression is shown in formula (11) and formula (12):

[0071]

[0072]

[0073] Where I is the posterior probability density, and the integral of I is calculated using grid approximation. The corresponding function is as follows:

[0074]

[0075] Among them, m μ,E , m σ,E , respectively correspond to the number of grid intervals of the corresponding parameters, and the calculation formula is shown in formula (14):

[0076]

[0077] In the formula, Δμ E , Δσ E Corresponding to μ E , σ E The grid interval length is . Therefore, the jth interval The expression of is shown in formula (15):

[0078]

[0079] Generate a random number from a uniform distribution in the range of [0,1]. If the acceptance rate is greater than the random number, the candidate sample will be accepted as the nth sample. Otherwise, update the next sampling point according to the heuristic algorithm, select the point that maximizes the likelihood function as the initial state, select the sampling point according to the heuristic algorithm, and calculate the acceptance rate again. If the acceptance rate is greater than the random number, the candidate sample will be accepted as the nth sample, otherwise the n-1th sample will be saved as the nth sample.

[0080] Step 5: Determine the elastic modulus of the rock mass. Specifically, the elastic modulus of the actual engineering rock mass is determined based on the HB criterion (Hoek-Brown), and compared with the average value of the rock elastic modulus in step 4, and the reduction factor ε is calculated, so as to reduce the rock mechanical parameters as a whole and obtain the probability distribution of the on-site rock mass parameters.

[0081] This application introduces the probability theory method and uses the distribution of elastic modulus (Es) to describe the deformation characteristics of rock mass, which is more reasonable than using a certain value (such as the average value of test data). The rock block compression wave velocity Vp is used to evaluate the elastic modulus Es of the rock. In the process of establishing the statistical model, the conversion uncertainty of the model is quantitatively considered, and prior knowledge is introduced to supplement the information contained in the model, so that this method has obvious advantages under small sample conditions.

[0082] This application makes full use of the improved MH-MCMC algorithm, and based on the constructed probability model, quickly generates a large number of samples of rock elastic modulus that meet the target distribution form, and gives the probability distribution range of rock and on-site rock elastic modulus in detail through investigation and statistics. The method provided in this application can efficiently estimate the probability range of rock mechanical parameters, which can be used to guide the formulation of solutions that take into account various situations during actual construction, has better economic benefits, and has important guiding significance for engineering practice.

[0083] Glossary

[0084] Bayesian principle:

[0085] The Bayesian principle, also known as the Bayesian theorem, is a basic theorem in probability theory that describes how to update the probability of an event based on existing evidence. The Bayesian method can naturally handle uncertainty and changing information, and can continuously update the probability based on new evidence, making the results more flexible and practical.

[0086] The mathematical expression of Bayes' theorem is:

[0087]

[0088] Where: P(A|B) is the conditional probability of event A occurring given event B. P(B|A) is the conditional probability of event B occurring given event A. P(A) is the prior probability of event A. P(B) is the total probability of event B, which can be calculated using the total probability formula.

[0089] MCMC Sampling Method:

[0090] Markov Chain Monte Carlo (MCMC) is a class of algorithms for sampling from complex probability distributions. MCMC methods are widely used in Bayesian statistics because they can efficiently sample from posterior distributions (even if these distributions are difficult to compute directly).

[0091] The core idea of ​​MCMC is to construct a Markov chain so that its stable distribution is the target distribution. Through the random walk method, MCMC can generate samples over a long period of time to approximate the target distribution.

[0092] Common MCMC sampling algorithms include:

[0093] 1. Metropolis-Hastings algorithm: This is the basic algorithm of MCMC, which generates candidate samples through the proposal distribution and decides whether to accept them based on the acceptance rate.

[0094] 2. Gibbs Sampling: When the conditional distribution of the target distribution is easy to sample, each variable can be sampled in turn. Only one variable is updated at a time, keeping the other variables unchanged.

[0095] 3.Hamiltonian Monte Carlo (HMC): By introducing momentum variables, Hamiltonian dynamics in physics is used to improve sampling efficiency, which is suitable for high-dimensional space.

[0096] The above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. A probabilistic estimation method for rock mass elastic modulus, characterized in that: The probability estimation method of the rock mass elastic modulus includes: S1: For the tunnel section where the rock mass mechanical parameters are to be estimated, based on field test investigation and statistics, obtain the rock compression wave velocity V p ; S2: Determine the prior knowledge about the rock elastic modulus; S3: Building a Bayesian computing framework; S4: Using the adaptive MH-MCMC algorithm to iterate the relevant parameters in the relationship equation under the Bayesian calculation framework, the posterior probability samples corresponding to each rock elastic modulus are generated to determine the probability distribution of the rock elastic modulus; S5: Determine the value of the elastic modulus of the actual engineering rock mass based on the HB criterion and obtain the probability distribution of the on-site rock mass parameters.

2. The probabilistic estimation method of rock mass elastic modulus according to claim 1, characterized in that: Step S2 includes: S21: Determine the rock elastic modulus E s The distribution form of S22: Constructing the rock elastic modulus E s and compression wave velocity V p Uncertainty regression model.

3. The probabilistic estimation method of rock mass elastic modulus according to claim 2, characterized in that: In step S21, the rock elastic modulus E s The distribution form of is defined as: E s =exp(μ N +σ N from) Among them, z is a random variable that follows a standard normal distribution, and the parameter μ N and σ N is ln(E s )’s mean and standard deviation.

4. The probabilistic estimation method of rock mass elastic modulus according to claim 3, characterized in that: In step S22, ln(V p )(a′µ N +b′)+a′σ N z+ε Among them, a′ and b′ are the relationship parameters fitted when establishing the logarithmic model, ε is used to express the difference between the output variable of the regression model and the test results, that is, the uncertainty of the conversion, ln(V p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and 5. The probabilistic estimation method of rock mass elastic modulus according to claim 4, characterized in that: Step S3 includes: using the total probability formula to construct E s -V p The uncertainty regression model is transformed and expressed as: P(E s |Data) is the probability distribution of rock elastic modulus based on prior information and field test data, P(E s |μ,σ) is the probability distribution of elastic modulus defined by mean and standard deviation, and P(μ,σ|Data) is the rock elastic modulus E s The joint probability distribution of the mean and standard deviation of , P(μ,σ) is the prior information of the project. a′ and b′ are used to establish E s With V p The relationship parameters of the established uncertainty regression relationship model; Where ln(V p ) is a Gaussian random variable, whose mean and standard deviation are a′μ N +b′ and i represents the i-th test sample, then ln(V pi ) represents the compression wave velocity of the i-th test sample, n s Represents the total number of test samples, μ min and μ max YesE s Minimum and maximum values ​​of the mean, σ min and σ max YesE s The minimum and maximum values ​​for the standard deviation.

6. The probabilistic estimation method of rock mass elastic modulus according to claim 4, characterized in that: Step S4 includes: Candidate samples of Markov chain are generated from Gaussian distribution, the algorithm and input parameters used in MCMC sampling calculation are determined, Bayesian estimation of rock mechanical parameters is performed, and several calculation samples are calculated to obtain the probability distribution of rock elastic modulus.

7. The probabilistic estimation method of rock mass elastic modulus according to claim 6, characterized in that: The sampling process of adjusting algorithm parameters through nonlinear heuristic algorithms and adaptive adjustment strategies; Among them, nonlinear heuristic algorithms include particle swarm optimization, simulated annealing algorithm, genetic algorithm, and differential evolution algorithm; The adaptive adjustment strategy is implemented by adaptively changing the covariance of the proposed distribution, thereby changing the spatial extension and directionality of the proposed distribution to adaptively approach the target distribution.

8. The probabilistic estimation method of rock mass elastic modulus according to claim 1, characterized in that: Step S5 includes: The value of the elastic modulus of the actual engineering rock mass is determined based on the HB criterion, and compared with the average value of the rock elastic modulus in step S4, and the reduction coefficient ε is calculated, so as to reduce the rock mechanical parameters as a whole and obtain the probability distribution of the on-site rock mass parameters.

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