CN106383053A - Engineering mechanical parameter related brittleness index prediction method - Google Patents

Abstract

Belonging to the technical field of petroleum and natural gas exploration, the invention relates to an engineering mechanical parameter related brittleness index prediction method. The method includes the steps of: 1) core sampling design and sampling; 2) core sample mineral component testing; 3) testing of core sample Young's modulus and Poisson's ratio; 4) establishment of a mathematical model between rock mineral components and engineering mechanical parameters, and determining the relationship between the mineral components and the rock mechanical parameters; 5) calculation of the engineering mechanical parameters of all samples; and 6) calculation of the rock brittleness index. The method acquires the relationship between the rock mechanical parameters and mineral components and the relationship between the mineral content and the rock brittleness index through mathematical models. The method has the advantages of short test period, simple sampling operation and cost saving, can guide the fracturability evaluation of the rock stratum more efficiently, and provides technical support for oil and gas exploration and development.

Description

Brittleness index prediction method related to engineering mechanical parameters

Technical Field

The invention relates to the technical field of petroleum and natural gas exploration, in particular to a brittleness index prediction method related to engineering mechanical parameters.

Background

Rock brittleness is an important parameter in the evaluation of fracability during the development of tight oil. Engineering mechanical parameters representing rock brittleness are mainly Young modulus and Poisson ratio, and as the Young modulus and the Poisson ratio of a tested stratum must depend on core data and are expensive, generally, less data are obtained. In early studies, the brittleness index of rock samples was often calculated using the level of the main brittle mineral (quartz) content, or the brittle mineral combination (quartz feldspar, dolomite, calcite, etc.). On one hand, as the content of the minerals and the mineral combination and the engineering mechanical parameters do not establish necessary correlation, no good corresponding relation exists between the calculated brittleness index and the rock fracturing performance; on the other hand, engineering mechanical parameters are different among different minerals, and simple accumulation results among brittle minerals cannot be effectively applied to fracture evaluation of rocks.

Disclosure of Invention

In order to solve the problem of determining the brittleness index in the fracturing evaluation process of the compact oil and gas reservoir, the invention aims to provide a brittleness index prediction method which is related to engineering mechanics and is convenient and fast to calculate.

In order to realize the purpose of the invention, the invention adopts the following technical scheme:

a brittleness index prediction method related to engineering mechanical parameters is characterized by comprising the following steps:

1) sampling the core to obtain a core sample;

2) testing the mineral composition of the core sample;

3) testing the Young modulus and Poisson ratio of the core sample;

4) calculating the brittleness index of the core sample according to the Young modulus and the Poisson ratio;

5) establishing a mathematical model between the mineral composition and the brittleness index, and determining the relationship between the mineral composition and the brittleness index;

6) the brittleness index was calculated using the mineral composition.

Wherein, in the step 1), core samples used for composition test are uniformly distributed and taken out in the core section, and the core samples can be used for testing rock composition and engineering mechanical parameters.

Wherein in step 2), the composition of the core sample and its relative content are determined using X-ray diffraction (XRD).

In the step 3), a rock mechanics triaxial experiment is performed on the core sample to obtain a stress-strain curve of the core sample, so that the Young modulus and the Poisson ratio of the core sample are obtained.

In the step 4), calculating the brittleness index of the core sample according to the Young modulus and the Poisson ratio, wherein the calculation method comprises the following steps: according to the normalization results of the Young modulus and the Poisson ratio, defining the root mean square of the Young modulus and the Poisson ratio as a brittleness index;

Y_Brit＝(y_i-y_min)/(y_max-y_min)×100

B_Brit＝(b_max-b_i)/(b_max-b_min)×100

$B_{rit}=\sqrt{\frac{{Y_{Brit}}^2+{B_{Brit}}^2}{2}}$

in the formula: y is_BritNormalized young's modulus; y is_iAs a measure of Young's modulus, y_maxIs the maximum value of Young's modulus, y_minMinimum value of Young's modulus, B_BritA normalized Poisson's ratio; b_iIs a measure of poisson's ratio; b_maxIs the maximum poisson ratio; b_minIs the minimum poisson ratio; b is_ritIs a brittleness index and is dimensionless.

Wherein, in step 5), a mathematical model between the mineral composition and the brittleness index is established, and the relationship between the mineral composition and the brittleness index is determined. And, step 5) comprises the following substeps:

5a) the model assumes that: assuming that m core samples exist in the model, and each core sample contains n mineral components; the composition of the mineral constituents of the core sample is regarded as RⁿSample space, the mineral composition of the m core samples then forming a point set, the mineral composition matrix A of the sample space_mn：

$A_{mn}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right]$

At R^pIn space, a one-dimensional subspace F passing through the origin can be found₁Represents the one-dimensional subspace F₁Is directed by a unit vector u₁∈RⁿTo define; at F₁Upper i-th brittleness index valueCan be observed from the value M_i∈RⁿMapping to u from corresponding points₁To obtain F₁On the upper partThe coordinates of (a) are given by:

$B_{{\mathrm{rit}}_i}=M_i^T\frac{u_1}{\left|\right|u_1\left|\right|}=M_i^T{u}_1$

defining an optimal line F using a "least squares method₁: find u₁∈R^PTo minimize the following:

$\underset{i=1}{\overset{n}{\Σ}}\left|\right|M_i-B_{{\mathrm{rit}}_i}|{|}^2$

according to the Pythagorean theorem:minimization of the problem Equivalent to maximization

The problem becomes to find in | | | u₁Maximization under the constraint of 1 | | | >Obtaining:

$\left[\begin{array}{c}{B}_{{\mathrm{rit}}_1}\\ B_{{\mathrm{rit}}_2}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ B_{{\mathrm{rit}}_n}\end{array}\right]=\left[\begin{array}{c}{M}_1^T{u}_1\\ M_2^T{u}_1\\ \\ M_m^T{u}_1\end{array}\right]={\mathrm{Au}}_1$

this problem is re-expressed as: find when | | u₁U when 1 | |)₁∈RⁿTo maximize the quadratic term (Au)₁)^TAu₁) Or

$\underset{{u_1}^T{u}_1}{max}{u_1}^T{M}^T{\mathrm{Mu}}_1$

Mixing mineral component M_iTo a brittleness index B_ritIs treated as a multivariate linear regression model

B_rit＝β₁m₁+β₂m₂+…+β_nm_n

Formula (III) β₁β₂…β_nIs a regression coefficient

In the formula b₁b₂…b_nAre estimates of the regression coefficients.

5b) Parameter estimation

$\begin{array}{ccc}{B}_{rit}=\left[\begin{array}{c}{B}_{{\mathrm{rit}}_1}\\ B_{{\mathrm{rit}}_2}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ B_{{\mathrm{rit}}_n}\end{array}\right]& M=\left[\begin{array}{cccc}{m}_{11}& m_{12}& ...& m_{1n}\\ m_{21}& m_{22}& ...& m_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ m_{m1}& m_{m2}& ...& m_{mn}\end{array}\right]& B=\left[\begin{array}{c}{b}_1\\ b_2\\ \begin{array}{c}.\\ .\\ .\end{array}\\ b_m\end{array}\right]\end{array}$

B＝(M′M)^-1X′Y

5c) Hypothesis testing

Correlation coefficient-measure the degree of coincidence of regression equation and original data:

sum of the squares of total deviations SS:referred to as B_ritDispersion of (a). All areIs called B_ritSum of the squares of the total deviations SS:

SS_residthe sum of the squares of the residuals reflects the experimental valuesWith values calculated according to the regression equationThe smaller the total deviation, the better the regression effect; SS_RegressionThe regression sum of squares reflects M and B_ritB is caused by the linear relationship of_ritThe larger the magnitude of the change, the better the regression effect;

⑤r²correlation test

SS_RegressionThe larger, M and B_ritThe more important the regression relationship of r²The closer to 1, the other SS_residThe smaller the linear relationship, the better;

sixthly, significance test of the regression equation:

to test whether there is a significant linear relationship between the dependent and independent variables in the model, a statistic is constructed:

for a given significance level α, determine the rejection region F > F_α(k, n-k-1). And calculating a statistic value, and judging whether to reject the original hypothesis.

Wherein in step 6), b is determined according to step 5)₁，b₂…b_nAnd calculating the brittleness index of the core sample.

Compared with the prior art, the brittleness index calculation method related to engineering mechanical parameters has the following beneficial effects:

1) the method solves the problems encountered in the engineering fracturing process from the angle of engineering mechanical tests. The fracturing performance of the core is directly related to the engineering mechanical property, and the test result can be directly used for making a fracturing scheme and is an intuitive expression of the fracturing performance. The problem is solved from the perspective of engineering mechanics, and the practicability of the invention is increased.

2) And searching for determinants of engineering mechanical parameters according to the mineral component content of the rock sample. The engineering mechanical properties of rock are determined by the mineral composition and relative content thereof. The content of minerals with different engineering mechanical properties contributes differently to the brittleness of the rock. The mineral with high Young modulus and small Poisson ratio has higher brittleness index. Through the establishment of the mathematical model, the relation between the engineering mechanical property and the mineral content in the research area can be found out.

3) Engineering mechanical tests are more expensive than analysis of mineral composition in terms of sample requirements and time periods. The invention utilizes a small amount of engineering mechanical tests and combines with the conventional test project achievements to efficiently and quickly obtain the brittleness index of the research area, thereby saving the cost on the basis of ensuring the data validity.

Drawings

FIG. 1 is a model map of the mapping between mineral content and friability index.

FIG. 2 photo of rock mechanics triaxial pressure test core sample before fracturing.

FIG. 3 is a photo of a fractured rock mechanical triaxial pressure test core sample.

FIG. 4 rock brittleness index calculated by different methods for a well.

Detailed Description

The brittleness index calculation method of the present invention is further described below with reference to specific examples to help those skilled in the art to more completely, accurately and deeply understand the inventive concept and technical scheme of the present invention; it is to be understood that the description in specific embodiments is intended to be illustrative, and not restrictive, of the scope of the invention, which is defined by the appended claims.

Example 1

The brittleness index prediction method of the embodiment comprises the following steps:

1. designing and sampling a core sample;

the sample is ensured to be consistent in depth according to sampling requirements of XRD whole rock X-ray diffraction analysis and engineering mechanical tests.

2. Testing mineral components of the core sample;

XRD diffraction test analysis of the sample shows that the rock mineral composition in the compact rock stratum is various, and the main components are quartz, calcite, analcite, clay, dolomite, feldspar, pyrite and the like. The mineral composition test data are shown in table 1:

table 1 XRD analysis of mineral content

Sample numbering	Depth (m)	Horizon	Quartz crystal	Calcite	Analcime (analcime)	Clay	Dolomite	Feldspar	Pyrite
										1	2939.53	Ek2 1	13	2	48	18	2	15	2
2	2953.31	Ek2 1	16	3	27	13	18	20	3
										3	2955.57	Ek2 1	16	5	29	14	8	21	7
4	2966.03	Ek2 1	14	18	9	7	39	12	1
										5	2974.07	Ek2 1	10	5	14	11	52	7	1
6	3025.10	Ek2 2	17	13	24	11	6	28	1
										7	3032.09	Ek2 2	16	14	14	12	28	15	1
8	3032.26	Ek2 2	14	5	16	10	43	11	1
										9	3059.18	Ek2 2	14	8	28	13	23	13	1
10	3187.22	Ek2 3	15	6	6	20	26	26	1
										11	3197.13	Ek2 3	21	8	6	20	18	18	9
12	3209.06	Ek2 3	18	7	0	25	11	38	1
										13	3234.94	Ek2 3	19	0	9	15	41	16	0
14	3239.22	Ek2 3	14	0	26	33	13	13	1
										15	3297.81	Ek2 3	14	10	8	10	48	9	1
16	3310.64	Ek2 4	28	14	21	12	5	19	1
										17	3315.08	Ek2 4	14	13	17	15	27	13	1
18	3318.20	Ek2 4	36	15	6	9	9	25	0
										19	3354.35	Ek2 4	40	8	0	3	0	49	0
20	3360.54	Ek2 4	39	14	0	2	0	45	0
										21	3376.96	Ek2 4	37	12	0	3	0	48	0
22	3385.59	Ek2 4	43	4	0	10	0	0	0

3. Testing engineering mechanical parameters (Young modulus and Poisson ratio) of a core sample;

the poisson ratio and the rock compressive strength of the rock are obtained by measuring the longitudinal and transverse deformation of the rock test piece in a regular shape under the action of triaxial pressure.

Using a servo-controlled rock mechanics triaxial experimental setup, the core sample as shown in fig. 2 was placed in the pressure chamber of a triaxial autoclave, a certain lateral pressure (σ _3 ═ 25MPa) was applied, and then a vertical pressure (σ _1) was applied until the core sample was destroyed as shown in fig. 3. The instrument simultaneously measures axial and radial stress-strain data to respectively obtain axial and radial stress-strain relations. The Young's modulus and Poisson's ratio of each sample were obtained from the stress-strain relationship, as shown in Table 2.

TABLE 2 Young's modulus and Poisson's ratio for core samples

Sample numbering	Depth before homing (m)	Horizon	Confining pressure (MPa)	Poisson ratio	Young's modulus (GPa)
						1	2939.53	Ek2 1	25	0.379	15.62
2	2953.31	Ek2 1	25	0.417	13.96
						3	2955.57	Ek2 1	25	0.309	19.33
4	2966.03	Ek2 1	25	0.313	23.3
						5	2974.07	Ek2 1	25	0.265	23.62
6	3025.10	Ek2 2	25	0.326	21.85
						7	3032.09	Ek2 2	25	0.193	27.97
8	3032.26	Ek2 2	25	0.279	30.82
						9	3059.18	Ek2 2	25	0.274	26.19
10	3187.22	Ek2 3	25	0.341	14.59
						11	3197.13	Ek2 3	25	0.226	22.15
12	3209.06	Ek2 3	25	0.402	9.64
						13	3234.94	Ek2 3	25	0.271	28.1
14	3239.22	Ek2 3	25	0.277	13.72
						15	3297.81	Ek2 3	25	0.184	21.61
16	3310.64	Ek2 4	25	0.259	31.57
						17	3315.08	Ek2 4	25	0.213	25.78
18	3318.20	Ek2 4	25	0.198	43.72
						19	3354.35	Ek2 4	25	0.206	22.14
20	3360.54	Ek2 4	25	0.197	26.78
						21	3376.96	Ek2 4	25	0.238	25.92
22	3385.59	Ek2 4	25	0.311	26.89

3. Calculating the brittleness index of the sample according to the Young modulus and the Poisson ratio;

using the calculation formula of brittleness index:

Y_Brit＝(y_i-y_min)/(y_max-y_nin)×100

B_Brit＝(b_max-b_i)/(b_max-b_min)×100

$B_{rit}=\sqrt{\frac{{Y_{Brit}}^2+{B_{Brit}}^2}{2}}$

in the formula: y is_BritIs the normalized young's modulus; y is_maxIs the maximum value of Young's modulus; y is_minIs the Young's modulus minimum; b is_BritIs the normalized Poisson's ratio; b_maxIs the maximum poisson ratio; b_minIs the minimum poisson ratio; b is_ritIs a brittleness index and is dimensionless.

From table 2, it can be found that:

y_min＝9.64；y_max＝43.72；b_max＝0.417；b_min＝0.184

determining brittleness index Y of samples No. 1-22_Brit1As shown in table 3.

Core sample brittleness index of table 31-22

Sample numbering	Depth before homing (m)	YBrit1	YBrit2	Sample numbering	Depth before homing (m)	YBrit1
							1	2939.53	16.93	7.51	12	3209.06	3.22
2	2953.31	6.34	0.00	13	3234.94	58.41
							3	2955.57	37.39	21.83	14	3239.22	36.03
4	2966.03	42.36	24.00	15	3297.81	67.56
							5	2974.07	53.13	32.23	16	3310.64	66.08
6	3025.10	37.44	20.80	17	3315.08	67.46
							7	3032.09	74.96	47.34	18	3318.20	97.00
8	3032.26	60.69	35.04	19	3354.35	63.62
							9	3059.18	54.97	32.56	20	3360.54	72.36
10	3187.22	23.57	13.11	21	3376.96	62.30
							11	3197.13	59.34	37.68	22	3385.59	48.05

4. Mathematical model between engineering mechanical parameters and brittleness index

The brittle mineral content is an important factor affecting the brittleness and fracturability of fine-grained sedimentary rocks. The higher the content of quartz and other brittle minerals is, the stronger the rock brittleness is, induced cracks are easy to form in the hydraulic fracturing process, and a complex crack network is formed, thereby being beneficial to the exploitation of compact oil. Brittleness index in the narrow sense B_rit1Quartz mineral content is used to represent:

generalized brittleness index B_rit2Comprising quartz, feldspar, dolomite, calcite, pyrite and analcite:

in the classical brittleness index and the generalized brittleness index, coefficients of minerals such as quartz are set to 1, and in order to make a model comparable, coefficients of quartz are set to 1 first, and then coefficients of other minerals are calculated respectively in the process of performing regression analysis.

The mathematical model established by the step 5 in the invention content part of the specification

In the formula b₁b₂…b_nAre estimates of the regression coefficients.

Wherein,

$\begin{array}{ccc}{B}_{rit}=\left[\begin{array}{c}{B}_{{\mathrm{rit}}_1}\\ B_{{\mathrm{rit}}_2}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ B_{{\mathrm{rit}}_n}\end{array}\right]& M=\left[\begin{array}{cccc}{m}_{11}& m_{12}& ...& m_{1n}\\ m_{21}& m_{22}& ...& m_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ m_{m1}& m_{m2}& ...& m_{mn}\end{array}\right]& B=\left[\begin{array}{c}{b}_1\\ b_2\\ \begin{array}{c}.\\ .\\ .\end{array}\\ b_m\end{array}\right]\end{array}$

B＝(M′M)^-1X′Y

and establishing a model suitable for local brittleness index evaluation by using the mathematical model. The mineral components of samples nos. 1 to 22 and Brit3 were introduced into the SPSS database using SPSS multivariate statistical software, and subjected to multivariate linear regression to obtain the following models:

by comparing the mineral composition with the brittleness index B_rit3Multiple linear regression analysis of (a) to (b) to obtain an evaluation model of friability index for the study area (table 4). The DW value of a regression model of the regression equation is 1.984, is close to 2, and shows that the rock mineral components are independent; r value of the regression model is0.903，R²A value of 0.815, indicating a brittleness index B_rit3There is a good linear relationship with the mineral composition, and the model passes the significance hypothesis. The regression equation between brittleness index and mineral composition in the study area is:

B_rit3quartz +0.629 dolomite +0.521 feldspar +0.25 calcite +0.204 pyrite +0.18 analcite +0.021 clay.

5. And calculating the rock brittleness index by using the mineral components.

The total 1219 total rock X-ray diffraction analysis data of a certain well core-taking section are calculated according to the classical brittleness index, the generalized brittleness index and the brittleness index calculation model established in the step 5, and the classical brittleness index, the generalized brittleness index and the regression equation brittleness index are respectively calculated, and the result is shown in FIG. 4.

The classical brittleness index is distributed between 0 and 0.5, and the average is 0.2; the generalized brittleness index is distributed between 0 and 1, and the average value is 0.85; the brittleness index of the time is distributed between 0 and 1, and the average value is 0.50. The classical brittleness index and the generalized brittleness index have certain defects in use, the classical brittleness index is small, the generalized brittleness value is large, and the classical brittleness index and the generalized brittleness index are often ignored or overestimated for the compressibility of the alloy when the engineering property of the alloy is judged. The brittleness index calculated using the regression equation can better describe the brittleness characteristics of fine grained sedimentary rock, and is of great significance for the preferred engineered dessert.

TABLE 4 model for evaluation of brittleness index in research area

Model summary^o，d

a. Prediction variables: clay, calcite, pyrite, dolomite, analcite, feldspar

b. For regression through the origin (no intercept model), the R-side can measure the proportion of variability in the dependent variable near the origin (explained by the regression). For models containing an intercept, this cannot be compared to the R-square.

c. Dependent variable: index of brittleness

d. Linear regression through the origin

Coefficient of performance^a，b

a. Dependent variable: index of brittleness

b. Linear regression through the origin

It is obvious to those skilled in the art that the present invention is not limited to the above embodiments, and it is within the scope of the present invention to adopt various insubstantial modifications of the method concept and technical scheme of the present invention, or to directly apply the concept and technical scheme of the present invention to other occasions without modification.

Claims (7)

1. A brittleness index prediction method related to engineering mechanical parameters is characterized by comprising the following steps:

1) sampling the core to obtain a core sample;

2) testing the mineral composition of the core sample;

3) testing the Young modulus and Poisson ratio of the core sample;

4) calculating the brittleness index of the core sample according to the Young modulus and the Poisson ratio;

5) establishing a mathematical model between the mineral composition and the brittleness index, and determining the relationship between the mineral composition and the brittleness index;

6) the brittleness index was calculated using the mineral composition.

2. The brittleness index prediction method of claim 1, wherein: in the step 1), core samples used for composition testing are uniformly distributed and taken out in the core section, and the core samples can be used for testing rock compositions and engineering mechanical parameters.

3. The brittleness index prediction method of claim 1, wherein: in step 2), the composition of the core sample and its relative content were determined using X-ray diffraction.

4. The brittleness index prediction method of claim 1, wherein: in the step 3), a rock mechanics triaxial experiment is carried out on the core sample to obtain a stress-strain curve of the core sample, so that the Young modulus and the Poisson ratio of the core sample are obtained.

5. The brittleness index prediction method of claim 1, wherein: in the step 4), calculating the brittleness index of the core sample according to the Young modulus and the Poisson ratio, wherein the calculation method comprises the following steps: according to the normalization results of the Young modulus and the Poisson ratio, defining the root mean square of the Young modulus and the Poisson ratio as a brittleness index;

Y_Brit＝(y_i-y_min)/(y_max-y_min)×100

$\begin{array}{c}{B}_{Brit}=\left(b_{\max }-b_i\right)/\left(b_{\max }-b_{\min }\right)\×100\\ B_{rit}=\sqrt{\frac{{Y_{Brit}}^2+{B_{Brit}}^2}{2}}\end{array}$

in the formula: y is_BritNormalized young's modulus; y is_iAs a measure of Young's modulus, y_maxIs the maximum value of Young's modulus, y_minMinimum value of Young's modulus, B_BritA normalized Poisson's ratio; b_iIs a measure of poisson's ratio; b_maxIs the maximum poisson ratio; b_minIs the minimum poisson ratio; b is_ritIs a brittleness index and is dimensionless.

6. The brittleness index prediction method of claim 1, wherein: step 5) comprises the following substeps:

5a) the model assumes that: assuming that m core samples exist in the model, and each core sample contains n mineral components; the composition of the mineral constituents of the core sample is regarded as RⁿSample space, the mineral composition of the m core samples then forming a point set, the mineral composition matrix A of the sample space_mn：

$A_{mn}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right]$

At R^pIn space, a one-dimensional subspace F passing through the origin can be found₁Represents the one-dimensional subspace F₁Is directed by a unit vector u₁∈RⁿTo define; at F₁Upper i-th brittleness index valueCan be observed from the value M_i∈RⁿMapping to u from corresponding points₁To obtain F₁On the upper partThe coordinates of (a) are given by:

$B_{{\mathrm{rit}}_i}=M_i^T\frac{u_1}{\left|\right|u_1\left|\right|}=M_i^T{u}_1$

defining an optimal line F using a "least squares method₁: find u₁∈R^PTo minimize the following:

$\underset{i=1}{\overset{n}{\Σ}}\left|\right|M_i-B_{{\mathrm{rit}}_i}|{|}^2$

according to the Pythagorean theorem:minimization of the problem Equivalent to maximization

The problem becomes to find in | | | u₁Maximization under the constraint of 1 | | | >U of (a)₁∈RⁿObtaining:

$\left[\begin{array}{c}{B}_{{\mathrm{rit}}_1}\\ B_{{\mathrm{rit}}_2}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ B_{{\mathrm{rit}}_m}\end{array}\right]=\left[\begin{array}{c}{M}_1^T{u}_1\\ M_2^T{u}_1\\ M_m^T{u}_1\end{array}\right]={\mathrm{Au}}_1$

this problem is re-expressed as: find when | | u₁U when 1 | |)₁∈RⁿTo maximize the quadratic term (Au)₁)^TAu₁) Or

$\underset{{u_1}^T{u}_1}{\max }{u_1}^T{M}^T{\mathrm{Mu}}_1$

Mixing mineral component M_iTo a brittleness index B_ritIs treated as a multivariate linear regression model

B_rit＝β₁m₁+β₂m₂+…+β_nm_n

Formula (III) β₁β₂…β_nIs a regression coefficient

In the formula b₁b₂…b_nAre estimates of the regression coefficients.

5b) Parameter estimation

$\begin{array}{ccc}{B}_{rit}=\left[\begin{array}{c}{B}_{{\mathrm{rit}}_1}\\ B_{{\mathrm{rit}}_2}\\ \begin{array}{c}.\\ .\\ .\end{array}\\ B_{{\mathrm{rit}}_n}\end{array}\right]& M=\left[\begin{array}{cccc}{m}_{11}& m_{12}& ...& m_{1n}\\ m_{21}& m_{22}& ...& m_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ m_{m1}& m_{m2}& ...& m_{mn}\end{array}\right]& B=\left[\begin{array}{c}{b}_1\\ b_2\\ \begin{array}{c}.\\ .\\ .\end{array}\\ b_m\end{array}\right]\end{array}$

B＝(M′M)^-1X′Y

5c) Hypothesis testing

Correlation coefficient-measure the degree of coincidence of the regression equation with the original data:

sum of squares of total deviationsReferred to as B_ritDispersion of (a). All areIs called B_ritSum of the squares of the total deviations SS:

SS_residthe sum of the squares of the residuals reflects the experimental valuesWith values calculated according to the regression equationThe smaller the total deviation, the better the regression effect; SS_RegressionThe regression sum of squares reflects M and B_ritB is caused by the linear relationship of_ritThe larger the magnitude of the change, the better the regression effect;

②r²correlation test

SS_RegressionThe larger the size of the tube is,m and B_ritThe more important the regression relationship of r²The closer to 1, the other SS_residThe smaller the linear relationship, the better;

③ the significance test of the regression equation:

to test whether there is a significant linear relationship between the dependent and independent variables in the model, a statistic is constructed:

for a given significance level α, determine the rejection region F > F_α(k, n-k-1). And calculating a statistic value, and judging whether to reject the original hypothesis.

7. The brittleness index prediction method of claim 1, wherein: in step 6), according to b determined in step 5)₁，b₂…b_nAnd calculating the brittleness index of the core sample.