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

A kind of brittleness index Forecasting Methodology related to engineering mechanics parameter

Technical field

A kind of the present invention relates to technical field of oil and gas exploration, it is more particularly related to and engineering forces Learn the related brittleness index Forecasting Methodology of parameter.

Background technology

In fine and close oil development process, rock brittleness is an important parameter in compressibility evaluation procedure.Represent rock The engineering mechanics parameter of stone fragility is mainly Young's modulus and Poisson's ratio, because formation testing Young's modulus and Poisson's ratio must be according to Bad rock core information, and expensive, and general acquisition data is less.In the research of early stage, often use main brittle mineral (stone English) content or brittle mineral combine the height of (quartzy feldspar, dolomite, calcite etc.) content to calculate the crisp of rock sample Sex index.On the one hand, due to not setting up inevitable phase between the content of mineral and mineral assemblage and engineering mechanics parameter Guan Xing, calculates and does not have good corresponding relation between brittleness index and rock compressibility；Between another aspect different minerals Engineering mechanics parameter is different, and what between brittle mineral, simple accumulation result can not be effectively applied to rock can pressure break Property evaluate.

Content of the invention

In order to solve the problems, such as the determination of brittleness index in fine and close oil and gas reservoir compressibility evaluation procedure, the purpose of the present invention It is to provide the brittleness index Forecasting Methodology of a kind of related to engineering mechanics and convenient and swift calculating.

In order to realize foregoing invention purpose, present invention employs technical scheme below：

A kind of brittleness index Forecasting Methodology related to engineering mechanics parameter is it is characterised in that comprise the following steps：

1) boring sample obtains core sample；

2) mineralogical composition of core sample is tested；

3) Young's modulus and Poisson's ratio of core sample is tested；

4) brittleness index of core sample is calculated according to Young's modulus and Poisson's ratio；

5) set up the Mathematical Modeling between mineralogical composition and brittleness index, determine the pass between mineralogical composition and brittleness index System；

6) mineralogical composition is utilized to calculate brittleness index.

Wherein, in step 1) in, it is uniformly distributed in rock core section and take out the core sample being used for composition test, and rock Core sample can be used rock composition and the test of engineering mechanics parameter.

Wherein, in step 2) in, composition and its relative amount of core sample is determined using X-ray diffraction method (XRD).

Wherein, in step 3) in, rock mechanics triaxial tests are adopted to core sample, obtains the ess-strain of core sample Curve, thus obtain Young's modulus and the Poisson's ratio of core sample.

Wherein, in step 4) in, the brittleness index of core sample, its computational methods are calculated according to Young's modulus and Poisson's ratio As follows：According to the normalization result of Young's modulus and Poisson's ratio, defining the root mean square of the two is 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 formula：Y_BritYoung's modulus for normalizing；y_iFor the measured value of Young's modulus, y_maxFor Young's modulus maximum, y_min For Young's modulus minimum of a value, B_BritPoisson's ratio for normalizing；b_iMeasured value for Poisson's ratio；b_maxFor Poisson's ratio maximum；b_minFor Poisson's ratio minimum of a value；B_ritFor brittleness index, dimensionless.

Wherein, in step 5) in, set up the Mathematical Modeling between mineralogical composition and brittleness index, determine mineralogical composition and fragility Relation between index.And, step 5) include following sub-step：

5a) model hypothesis：Have m core sample in hypothesized model, in each core sample, have n kind mineralogical composition； The mineralogical composition composition of core sample is considered as a RⁿSample space, then the mineralogical composition of m core sample constitutes a point Collection, the mineralogical composition matrix A of sample space_mn：

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

In R^pIn space, an one-dimensional subspace F passing through initial point can be found₁, represent this one-dimensional subspace F₁Straight line Direction is by a unit vector u₁∈RⁿTo define；In F₁Upper i-th brittleness index valueCan be by observed value M_i∈Rⁿ U is mapped to by respective point₁And obtain, F₁OnCoordinate be given by：

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

Use the optimum line F of " least square method " definition₁：Find u₁∈R^PTo minimize following formula：

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

According to Pythagorean theorem：Minimization problem formula It is equivalent to maximize

Problem is changed into finding | | u₁| |=1 constraint is lower to be maximizedObtain：

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

This problem is expressed as again：Find and work as | | u₁| | u when=1₁∈RⁿTo maximize quadratic term (Au₁)^TAu₁) or

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

Mineralogical composition M_iTo brittleness index B_ritMapping be considered as multiple linear and return to model

B_rit=β₁m₁+β₂m₂+…+β_nm_n

β in formula₁β₂…β_nFor regression coefficient

B in formula₁b₂…b_nEstimate for regression coefficient.

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

4. the degree that coefficient correlation-measurement regression equation is consistent with initial data：

Total sum of squares of deviations SS：Referred to as B_ritDeviation.AllDeviation square sum Referred to as B_ritTotal sum of squares of deviations SS：

SS_residResidual sum of squares (RSS) reflects experiment valueWith the value by regression equation calculationTotal deviation, It is less, shows that regression effect is better；SS_ReturnRegression sum of square reflects M and B_ritLinear relationship and cause B_ritChange is big Little, it is bigger, shows that regression effect is better；

⑤r²Correlation test

SS_ReturnBigger, M and B_ritRegression relation more important, r²It is closer to 1, another SS_residLess, linear relationship is better；

6. the significance test of regression equation：

In order to whether there is significant linear relationship between the dependent variable in testing model and independent variable, construct statistic：

For given level of significance α, determine region of rejection F ＞ F_α (k, n-k-1).Calculate statistics value, and judge whether to refuse null hypothesis.

Wherein, in step 6) in, according to step 5) the middle b determining₁, b₂…b_n, calculate the brittleness index of core sample.

Compared with prior art, the brittleness index computational methods related to engineering mechanics parameter of the present invention have Following beneficial effect：

1) solve the problems, such as to be met engineering fracturing process from the angle of engineering mechanics test.Rock core can pressure break Property is to have directly to contact with its engineering mechanics property, and its result of the test can be used directly to the formulation of Fracturing Project, is to press The performance directly perceived of fragility.From the angle solve problem of engineering mechanics, increased the practicality of the present invention.

2) pass through the mineralogical composition content of rock sample, find the decisive factor of engineering mechanics parameter.The engineering forces of rock Learn property, be the mineralogical composition internal by it and its relative amount number determined.The mineral of different engineering mechanics property, Its content is how many, and the contribution done of fragility for rock is different.Young's modulus is high, the little mineral of Poisson's ratio, its brittleness index Higher.By the foundation of Mathematical Modeling, the relation between research area's engineering mechanics property and mineral content can be found out.

3) engineering mechanics test in the requirement of sample and will be higher than to required for mineralogical composition analysis on the time cycle Expense.The present invention utilizes a small amount of engineering mechanics test, in conjunction with conventionally test project achievement, efficiently quick or must study the crisp of area Sex index, on the basis of ensureing data validity, cost-effective.

Figure of description

Fig. 1 is the mapping model figure between mineral content and brittleness index.

Photo before Fig. 2 rock mechanics triaxial compression test core sample pressure break.

The post-fracturing photo of Fig. 3 rock mechanics triaxial compression test core sample.

The rock brittleness index that Fig. 4 well distinct methods calculate.

Specific embodiment

Below in conjunction with specific embodiment, brittleness index computational methods of the present invention are further elaborated, to help this The professional and technical personnel in field has more complete, more accurate and deep understanding to the inventive concept of the present invention, technical scheme；Need Statement is to be all exemplary in the description in specific embodiment, and is not meant to limiting the scope of the invention, The interest field of the present invention is defined by the claim limiting.

Embodiment 1

The brittleness index Forecasting Methodology of the present embodiment, it comprises the following steps：

1. boring sample design and sampling；

Sample according to XRD total rock X diffraction analysis and engineering mechanics test sampling requirement it is ensured that sample is one in depth Cause.

2. core sample mineralogical composition test；

The XRD diffraction experiment analysis being carried out by sample, in tight stratum, rock forming mineral forms various, main component For quartz, calcite, analcime, clay, dolomite, feldspar, pyrite etc..Its mineralogical composition test data such as table 1：

Table 1 XRD analysis mineral become content

Sample number into spectrum	Depth (m)	Layer position	Quartz	Calcite	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. core sample engineering mechanics parameter (Young's modulus and Poisson's ratio) test；

By measuring the deflection of rock sample vertical and horizontal under triaxial pressure effect of regular shape, thus trying to achieve The Poisson's ratio of rock and compressive strength of rock.

Using SERVO CONTROL rock mechanics triaxial tests device, core sample as shown in Figure 2 is placed on three axle high pressure The pressure of kettle is indoor, applies certain lateral pressure (σ _ 3=25MPa), then applies pressure at right angle (σ _ 1), until such as Fig. 3 institute Show so that core sample destroys.Instrument measures axial direction and radial stress strain data simultaneously, respectively obtains axial direction and radial stress Strain stress relation.By stress-strain relation, obtain Young's modulus and the Poisson's ratio of each sample, as shown in table 2.

Table 2 core sample Young's modulus and Poisson's ratio

Sample number into spectrum	Depth (m) before playback	Layer position	Confined pressure (MPa)	Poisson's 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. sample brittleness index is calculated according to Young's modulus and Poisson's ratio；

Computing formula using 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 formula：Y_BritFor the Young's modulus under normalizing；y_maxFor Young's modulus maximum；y_minFor Young's modulus minimum of a value； B_BritFor the Poisson's ratio under normalizing；b_maxFor Poisson's ratio maximum；b_minFor Poisson's ratio minimum of a value；B_ritFor brittleness index, immeasurable Guiding principle.

Can find from table 2：

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

Obtain brittleness index Y of 1-22 sample_Brit1, as shown in table 3.

Table 3 1-22 core sample brittleness index

Sample number into spectrum	Depth (m) before playback	YBrit1	YBrit2	Sample number into spectrum	Depth (m) before playback	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. the Mathematical Modeling between engineering mechanics parameter and brittleness index

Brittle mineral content be impact pulveryte fragility and can pressure break ability key factor.The brittle minerals such as quartz Content is higher, and rock brittleness is stronger, easily forms induced fractures, form complicated fracture network during fracturing, from And be conducive to the exploitation of fine and close oil.Brittleness index B of narrow sense_rit1Represented with quartz mineral content：

Brittleness index B of broad sense_rit2Comprise quartz, feldspar, dolomite, calcite, pyrite and analcime：

The brittleness index of classical brittleness index and broad sense is all set to 1 the coefficient of the mineral such as quartz, in order that model tool There is comparability, during carrying out regression analysis, first quartzy coefficient is set to 1, then calculate other more respectively The coefficient of mineral.

The Mathematical Modeling set up using specification Summary step 5

B in formula₁b₂…b_nEstimate for regression coefficient.

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

Using this Mathematical Modeling, set up and be applied to local brittleness index evaluation model.Using SPSS multivariate statistics software, The mineralogical composition of 1-22 sample, Brit3 are directed respectively in SPSS database, carry out multiple linear regression calculating, obtain following Model：

By to mineralogical composition and brittleness index B_rit3Multiple linear regression analysis obtain study area's brittleness index evaluation Model (table 4).The DW value of the regression model of regression equation is 1.984, close to 2, illustrates it is mutual between rock mineral composition Independent；The R value of regression model is 0.903, R²It is worth for 0.815, show brittleness index B_rit3Exist very well and mineralogical composition between Linear relationship, this model assumed by conspicuousness.Research area's brittleness index and mineralogical composition between regression equation be：

B_rit3=quartzy+0.629 dolomite+0.521 feldspar+0.25 calcite+0.204 pyrite+0.18 analcime+ 0.021 clay.

5. utilize mineralogical composition to calculate rock brittleness index.

Certain well coring section has 1219 total rock X diffraction analysis data, according to classical brittleness index, broad sense brittleness index and The brittleness index computation model set up in above-mentioned steps 5, calculates its classical brittleness index, broad sense brittleness index respectively and returns Return equation brittleness index, result is as shown in Figure 4.

Classical brittleness index is distributed between 0-0.5, and average 0.2；Broad sense brittleness index is distributed between 0-1, averagely 0.85；This brittleness index is distributed between 0-1, and average 0.50.Classical brittleness index and broad sense brittleness index have in use There is certain defect, classical brittleness index is less than normal, broad sense fragility value is bigger than normal, when judging its engineering properties, is often ignored Or over-evaluate its compressibility.The brittleness index being gone out using regression equation calculation can preferably describe the crisp of pulveryte Property feature, has great importance for preferred engineering dessert.

Table 4 research area brittleness index evaluation model

Model collects^{O, d}

A. predictive variable：Clay, calcite, pyrite, dolomite, analcime, feldspar

B. for by the recurrence (no intercept model) of initial point, R side can measure near (being explained by recurrence) initial point because Variable sex ratio in variable.For the model comprising intercept it is impossible to this is compared with R side.

C. dependent variable：Brittleness index

D. by the linear regression of initial point

Coefficient^{A, b}

A. dependent variable：Brittleness index

B. by the linear regression of initial point

For the ordinary skill in the art, specific embodiment is simply exemplarily described to the present invention, Obviously the present invention implements and is not subject to the restrictions described above, as long as employ method of the present invention design entering with technical scheme The improvement of the various unsubstantialities of row, or the not improved design by the present invention and technical scheme directly apply to other occasions , all within protection scope of the present invention.

Claims (7)

1. a kind of brittleness index Forecasting Methodology related to engineering mechanics parameter is it is characterised in that comprise the following steps：

1) boring sample obtains core sample；

2) mineralogical composition of core sample is tested；

3) Young's modulus and Poisson's ratio of core sample is tested；

4) brittleness index of core sample is calculated according to Young's modulus and Poisson's ratio；

5) set up the Mathematical Modeling between mineralogical composition and brittleness index, determine the relation between mineralogical composition and brittleness index；

6) mineralogical composition is utilized to calculate brittleness index.

2. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：In step 1) in, in rock core section all The core sample being used for composition test is taken out in even distribution, and core sample can be used rock composition and engineering mechanics ginseng The test of number.

3. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：In step 2) in, using X-ray diffraction Method determines composition and its relative amount of core sample.

4. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：In step 3) in, core sample is adopted Use rock mechanics triaxial tests, obtain the stress-strain diagram of core sample, thus obtaining Young's modulus and the pool of core sample Pine ratio.

5. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：In step 4) in, according to Young's modulus Calculate the brittleness index of core sample with Poisson's ratio, its computational methods is as follows：Normalization knot according to Young's modulus and Poisson's ratio Really, defining the root mean square of the two is 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 formula：Y_BritYoung's modulus for normalizing；y_iFor the measured value of Young's modulus, y_maxFor Young's modulus maximum, y_minFor poplar Family name's modulus minimum of a value, B_BritPoisson's ratio for normalizing；b_iMeasured value for Poisson's ratio；b_maxFor Poisson's ratio maximum；b_minFor Poisson Compare minimum of a value；B_ritFor brittleness index, dimensionless.

6. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：Step 5) include following sub-step：

5a) model hypothesis：Have m core sample in hypothesized model, in each core sample, have n kind mineralogical composition；Rock core The mineralogical composition composition of sample is considered as a RⁿSample space, then the mineralogical composition of m core sample constitutes a point set, sample The mineralogical composition matrix A in product space_mn：

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

In R^pIn space, an one-dimensional subspace F passing through initial point can be found₁, represent this one-dimensional subspace F₁Straight line direction By a unit vector u₁∈RⁿTo define；In F₁Upper i-th brittleness index valueCan be by observed value M_i∈RⁿBy phase Should put and be mapped to u₁And obtain, F₁OnCoordinate be given by：

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

Use the optimum line F of " least square method " definition₁：Find u₁∈R^PTo minimize following formula：

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

According to Pythagorean theorem：Minimization problem formula It is equivalent to maximize

Problem is changed into finding | | u₁| |=1 constraint is lower to be maximizedU₁∈Rⁿ, obtain：

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

This problem is expressed as again：Find and work as | | u₁| | u when=1₁∈RⁿTo maximize quadratic term (Au₁)^TAu₁) or

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

Mineralogical composition M_iTo brittleness index B_ritMapping be considered as multiple linear and return to model

B_rit=β₁m₁+β₂m₂+…+β_nm_n

β in formula₁β₂…β_nFor regression coefficient

B in formula₁b₂…b_nEstimate for regression coefficient.

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

1. the degree that coefficient correlation-measurement regression equation is consistent with initial data：

Total sum of squares of deviationsReferred to as B_ritDeviation.AllDeviation square sum be referred to as B_ritTotal sum of squares of deviations SS：

SS_residResidual sum of squares (RSS) reflects experiment valueWith the value by regression equation calculationTotal deviation, it is got over Little, show that regression effect is better；SS_ReturnRegression sum of square reflects M and B_ritLinear relationship and cause B_ritThe size of change, It is bigger, shows that regression effect is better；

②r²Correlation test

SS_ReturnBigger, M and B_ritRegression relation more important, r²It is closer to 1, another SS_residLess, linear relationship is better；

3. the significance test of regression equation：

In order to whether there is significant linear relationship between the dependent variable in testing model and independent variable, construct statistic：

For given level of significance α, determine region of rejection F ＞ F_α(k, n-k- 1).Calculate statistics value, and judge whether to refuse null hypothesis.

7. brittleness index Forecasting Methodology according to claim 1 it is characterised in that：In step 6) in, according to step 5) in The b determining₁, b₂…b_n, calculate the brittleness index of core sample.