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CN106338778B - A kind of shale petrofacies continuous prediction method based on well logging information - Google Patents

A kind of shale petrofacies continuous prediction method based on well logging information Download PDF

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CN106338778B
CN106338778B CN201610724474.XA CN201610724474A CN106338778B CN 106338778 B CN106338778 B CN 106338778B CN 201610724474 A CN201610724474 A CN 201610724474A CN 106338778 B CN106338778 B CN 106338778B
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蒋裕强
蒋婵
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Abstract

本发明公开了一种基于测井信息的页岩岩相连续预测方法,所述基于测井信息的页岩岩相连续预测方法是先采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,找出能反映页岩岩相类型的测井响应特征,运用人工神经网络技术,利用测井曲线对页岩岩相类型进行连续性预测。本发明克服了单凭岩石样品取样分析建立页岩岩相,容易丢失页岩的连续性属性及其差异,导致页岩属性认识差异的问题;采用测井信息预测页岩岩相的方法,是一种成本低且快速、有效的方法。

The invention discloses a method for continuously predicting shale lithofacies based on well logging information. The method for continuously predicting shale lithofacies based on well logging information is to first use the analysis and testing of core wells and thin-section data to analyze the shale lithofacies. Determine the type of shale, find out the logging response characteristics that can reflect the type of shale lithofacies, and use artificial neural network technology to predict the continuity of shale lithofacies types by using logging curves. The invention overcomes the problem that the shale lithofacies is easily lost based on rock sample sampling and analysis, and the continuity attribute and its difference of the shale are easily lost, resulting in differences in understanding of shale attributes; the method of predicting shale lithofacies by using logging information is A low-cost, fast and effective method.

Description

一种基于测井信息的页岩岩相连续预测方法A continuous prediction method for shale lithofacies based on logging information

技术领域technical field

本发明属于页岩气开发技术领域,尤其涉及一种基于测井信息的页岩岩相连续预测方法。The invention belongs to the technical field of shale gas development, in particular to a method for continuously predicting shale lithofacies based on logging information.

背景技术Background technique

页岩岩相划分多从成因角度划分,对页岩气有利开发层段不适用。The division of shale lithofacies is mostly done from the perspective of genesis, which is not applicable to intervals that are favorable for the development of shale gas.

岩石样品取样分析建立页岩岩相,容易丢失页岩的连续性属性及其差异,导致页岩属性认识差异,用测井解释方法也存在判定失误较多、不同的解释人员解释结果差异的问题。Rock sample sampling analysis to establish shale lithofacies is easy to lose the continuity attributes and differences of shale, resulting in differences in the understanding of shale attributes. There are also many judgment errors and differences in interpretation results by different interpreters when using logging interpretation methods .

发明内容Contents of the invention

本发明的目的在于提供一种基于测井信息的页岩岩相连续预测方法,旨在解决岩石样品取样分析建立页岩岩相,容易丢失页岩的连续性属性及其差异,导致页岩属性认识差异的问题。The purpose of the present invention is to provide a continuous prediction method for shale lithofacies based on well logging information, aiming at solving the problem of establishing shale lithofacies through sampling analysis of rock samples, which is easy to lose the continuity attribute and its difference of shale, resulting in shale attribute The problem of recognizing differences.

本发明是这样实现的,一种基于测井信息的页岩岩相连续预测方法,所述基于测井信息的页岩岩相连续预测方法是利用测井信息,采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,然后根据页岩岩相与测井响应特征之间的关系,运用人工神经网络技术,建立利用测井参数预测页岩岩相类型的模型;对空间页岩岩相类型进行连续性预测。The present invention is realized in this way, a method for continuously predicting shale lithofacies based on well logging information. The method for continuously predicting shale lithofacies based on well logging information is to utilize the well logging information and adopt the analytical tests and The type of shale lithofacies is judged by thin section data, and then according to the relationship between shale lithofacies and logging response characteristics, artificial neural network technology is used to establish a model for predicting shale lithofacies types by using logging parameters; Continuous prediction of lithofacies types.

进一步,所述基于测井信息的页岩岩相连续预测方法包括以下步骤:Further, the method for continuously predicting shale lithofacies based on logging information includes the following steps:

步骤1,采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,将页岩岩相划分为富有机质硅质页岩、富有机质碳酸盐质页岩、富有机质黏土质页岩、贫有机质硅质页岩、贫有机质碳酸盐质页岩、贫有机质黏土质页岩、贫有机质灰岩等7种类型,用A、B、C、D、E、F、G表示这7种类型,将其数量化以后用7维行向量表示,如某个取样点判定为其中某一类,则在这一类上的取值为1,而在其它类上的取值为0,例如某一取样点经判定属于A类,则用向量[1 0 0 0 0 0 0]表示,属于B类,则用向量[0 1 0 0 0 0 0]表示;然后将测井曲线与对应深度的页岩岩相类型进行对比,找出几种能反映页岩岩相类型的常规测井曲线,作为预测页岩岩相类型的测井响应特征;Step 1: Determine the type of shale lithofacies using the analytical tests and thin-section data of the core well, and divide the shale lithofacies into organic-rich siliceous shale, organic-rich carbonate shale, and organic-rich clay shale. There are seven types of rock, organic-poor siliceous shale, organic-poor carbonate shale, organic-poor clay shale, and organic-poor limestone, which are represented by A, B, C, D, E, F, and G. There are 7 types, which are quantified and represented by 7-dimensional row vectors. If a sampling point is determined to be one of the categories, the value of this category is 1, while the value of other categories is 0 , for example, if a certain sampling point is judged to belong to class A, it will be represented by vector [1 0 0 0 0 0 0], and if it belongs to class B, it will be represented by vector [0 1 0 0 0 0 0]; then the log curve and Compare the shale lithofacies types at corresponding depths, and find several conventional logging curves that can reflect the shale lithofacies types, as the logging response characteristics for predicting shale lithofacies types;

对每种页岩岩相类型找出与之对应的若干个测井取样点,作为建立预测模型的学习样本;For each type of shale lithofacies, find several corresponding well logging sampling points as learning samples for establishing the prediction model;

利用所有取样点的页岩岩相类型以及与深度对应的测井资料,以P个测井参数[X1,X2,…,XP]T作为输入,以对应的页岩岩相类型作为输出,建立预测页岩岩相类型的人工神经网络模型;Using the shale lithofacies types of all sampling points and the logging data corresponding to the depth, P logging parameters [X 1 ,X 2 ,…,X P ] T are used as input, and the corresponding shale lithofacies types are used as Output, establish artificial neural network model to predict shale lithofacies type;

步骤2,运用人工神经网络技术,对空间页岩岩相连续性预测,根据预测结果,画出该井段的页岩岩相剖面图,进而确定出页岩气的有利开发层段;Step 2: Use artificial neural network technology to predict the continuity of spatial shale lithofacies, draw the shale lithofacies profile of the well section according to the prediction results, and then determine the favorable development intervals of shale gas;

步骤3,根据预测结果,画出该井段的页岩岩相剖面图,进而确定出页岩气的有利开发层段。Step 3, draw the shale lithofacies profile of the well section according to the prediction results, and then determine the favorable intervals for shale gas development.

进一步,所述模型(以A类为例)为:Further, the model (taking Class A as an example) is:

式中:In the formula:

m测井取样点,用XAi=[XAi1,XAi2,…,XAip]T表示第i个取样点的P个测井参数;For m logging sampling points, use X Ai = [X Ai1 , X Ai2 ,..., X Aip ] T to represent the P logging parameters of the i-th sampling point;

i=模式号;i = mode number;

m=训练模式总数;m = total number of training patterns;

XAi=类型A的第i训练模式;X Ai = the i-th training pattern of type A;

σ=平滑参数;σ = smoothing parameter;

P=度量空间的维数;P = dimensionality of the metric space;

X=要预测类型的某个点的参数。X = parameter at a certain point of the type to be predicted.

本发明提供的基于测井信息的页岩岩相连续预测方法,克服了单凭岩石样品取样分析建立页岩岩相,容易丢失页岩的连续性属性及其差异,导致页岩属性认识差异的问题。本发明的页岩岩相的连续性预测对于优选页岩气有利层段和有利勘探区块有重要的意义,例如富含有机质的层段和富含有机质的区块才是有利的层段和区块。传统的页岩岩相的识别是采用岩芯分析以及薄片鉴定的方法,由于取芯井较少且取芯成本太高而且分析化验项目较多薄片鉴定又费时所以不利于页岩岩相在纵向上及横向(平面)上的的预测,本发明考虑采用测井信息预测页岩岩相的方法,是一种成本低且快速、有效的方法,例如一口井取芯加上分析化验费用至少要100万元以上,而一口井测井费用只需10万元左右。本发明根据页岩岩相与测井响应特征之间的关系,运用人工神经网络技术,建立利用测井参数预测页岩岩相模型;对空间页岩岩相进行连续性预测,不仅极大的提高了效率,而且极大的节约了成本。The continuous prediction method of shale lithofacies based on well logging information provided by the present invention overcomes the problem that the shale lithofacies is established solely based on rock sample sampling and analysis, which easily loses the continuity attribute and its difference of shale, which leads to differences in understanding of shale attributes question. The continuity prediction of shale lithofacies in the present invention is of great significance for optimizing favorable intervals and exploration blocks of shale gas, for example, intervals rich in organic matter and blocks rich in organic matter are favorable intervals and blocks. The traditional identification of shale lithofacies is by core analysis and thin section identification. Because there are few core wells, the cost of coring is too high, and there are many analytical and laboratory items, thin section identification is time-consuming, so it is not conducive to the longitudinal identification of shale lithofacies. For prediction on the upper and lateral (plane), the present invention considers the method of using logging information to predict shale lithofacies, which is a low-cost, fast and effective method. For example, a well coring plus analysis and testing costs at least More than 1 million yuan, while the cost of logging a well is only about 100,000 yuan. According to the relationship between shale lithofacies and logging response characteristics, the present invention uses artificial neural network technology to establish a shale lithofacies prediction model using logging parameters; to predict the continuity of spatial shale lithofacies, not only greatly Efficiency is improved, and cost is greatly saved.

附图说明Description of drawings

图1是本发明实施提供的基于测井信息的页岩岩相连续预测方法的流程图。Fig. 1 is a flow chart of the continuous prediction method of shale lithofacies based on well logging information provided by the implementation of the present invention.

具体实施方式detailed description

为了使本发明的目的、技术方案及优点更加清楚明白,以下结合实施例,对本发明进行进一步详细说明。应当理解,此处所描述的具体实施例仅仅用以解释本发明,并不用于限定本发明。In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the examples. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.

本发明选择有机质含量和矿物组成两大参数为页岩岩相划分的依据,先按有机质含量是否超过2%,划分出富有机质和贫有机质页岩;再按黏土、石英及长石含量、碳酸盐矿物含量的多少建立7种页岩岩相类型的划分标准,本发明利用测井信息与通过岩石分析化验资料鉴定的页岩岩相之间的关系,运用人工神经网络技术,建立预测模型,开展空间页岩岩相的连续性预测。克服了单凭岩石样品取样分析建立页岩岩相,容易丢失页岩的连续性属性及其差异,从而导致页岩属性认识上的差异。In the present invention, the two parameters of organic matter content and mineral composition are selected as the basis for the division of shale lithofacies. Firstly, organic matter-rich and organic matter-poor shale is divided according to whether the organic matter content exceeds 2%; How much salt mineral content to establish the classification standard of 7 kinds of shale lithofacies types, the present invention utilizes the relationship between logging information and shale lithofacies identified through rock analysis test data, and uses artificial neural network technology to establish a prediction model , to carry out the continuity prediction of spatial shale lithofacies. Overcoming the establishment of shale lithofacies based on rock sample sampling and analysis, it is easy to lose the continuity attributes and differences of shale, which leads to differences in the understanding of shale attributes.

以下结合图1对本发明作出进一步的说明。The present invention will be further described below in conjunction with FIG. 1 .

本发明实施例的基于测井信息的页岩岩相连续预测方法包括以下步骤:The method for continuously predicting shale lithofacies based on well logging information in the embodiment of the present invention includes the following steps:

采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,将页岩岩相划分为富有机质硅质页岩、富有机质碳酸盐质页岩、富有机质黏土质页岩、贫有机质硅质页岩、贫有机质碳酸盐质页岩、贫有机质黏土质页岩、贫有机质灰岩等7种类型,为了叙述方便,以下用A、B、C、D、E、F、G表示这7种类型,将其数量化以后用7维行向量表示,如某个取样点判定为其中某一类,则在这一类上的取值为1,而在其它类上的取值为0,例如某一取样点经判定属于A类,则用向量[1 0 0 0 0 0 0]表示,属于B类,则用向量[0 1 0 0 0 0 0]表示,……,以此类推。The shale lithofacies types were judged by the analytical tests and thin section data of cored wells, and the shale lithofacies were divided into organic-rich siliceous shale, organic-rich carbonate shale, organic-rich clay shale, and organic-poor clay shale. There are seven types of organic siliceous shale, organic-poor carbonate shale, organic-poor clay shale, and organic-poor limestone. For the convenience of description, A, B, C, D, E, F, and G are used below Represent these 7 types, quantify them and express them with 7-dimensional row vectors. If a certain sampling point is determined to be one of the categories, the value of this category is 1, while the value of other categories is is 0, for example, if a certain sampling point is judged to belong to class A, it will be represented by vector [1 0 0 0 0 0 0], and if it belongs to class B, it will be represented by vector [0 1 0 0 0 0 0], ..., with And so on.

然后将测井曲线与对应深度的页岩岩相类型进行对比,找出几种能反映页岩岩相类型的常规测井曲线(参数),如声波AC、伽马HCGR、密度DEN、电阻率RT…等P个测井参数,作为预测页岩岩相类型的测井响应特征(参数)。以下为了叙述方便用向量X=[X1,X2,…,XP]T表示P个测井参数。Then compare the logging curve with the shale lithofacies type at the corresponding depth to find several conventional logging curves (parameters) that can reflect the shale lithofacies type, such as acoustic wave AC, gamma HCGR, density DEN, resistivity P logging parameters such as RT... are used as logging response features (parameters) for predicting shale lithofacies types. For the convenience of description, the vector X=[X 1 ,X 2 ,...,X P ] T is used to represent the P logging parameters.

对每种页岩岩相类型找出与之对应的若干个测井取样点,作为建立预测模型的学习样本,如A类有m个测井取样点,则用XAi=[XAi1,XAi2,…,XAip]T表示A类的第i个取样点的P个测井参数。然后利用所有取样点的页岩岩相类型以及与深度对应的测井资料,以P个测井参数[X1,X2,…,XP]T作为输入,以对应的页岩岩相类型作为输出,作为建立预测页岩岩相类型的人工神经网络模型的学习样本。For each shale lithofacies type, find out several corresponding well logging sampling points as learning samples for establishing a prediction model. If there are m logging sampling points in type A, use X Ai = [X Ai1 , X Ai2 ,...,X Aip ] T represents the P logging parameters of the i-th sampling point of type A. Then, using the shale lithofacies types of all sampling points and the logging data corresponding to the depth, P logging parameters [X 1 ,X 2 ,…,X P ] T are used as input, and the corresponding shale lithofacies types As an output, it is used as a learning sample for establishing an artificial neural network model for predicting shale lithofacies types.

根据不同的页岩岩相类型所对应的测井响应特征以及分布范围,利用7种页岩岩相类型所对应的学习样本采用概率神经网络方法建立页岩岩相类型的预测模型,如A类的预测模型为:According to the logging response characteristics and distribution ranges corresponding to different shale lithofacies types, use the learning samples corresponding to 7 shale lithofacies types to establish a prediction model of shale lithofacies types with the method of probability neural network, such as type A The prediction model for is:

式中:In the formula:

i=模式号。i = mode number.

m=训练模式总数。m = total number of training patterns.

XAi=类型A的第i训练模式(取样点)。X Ai =i-th training pattern (sampling point) of type A.

σ=平滑参数。σ = smoothing parameter.

P=度量空间的维数。P = dimensionality of the metric space.

X=要预测类型的某个点的参数。X = parameter at a certain point of the type to be predicted.

B类的预测模型为:The prediction model for class B is:

G类的预测模型为:The prediction model of class G is:

将要预测页岩岩相类型的井段的M个测井参数按深度顺序逐个代入上述模型,算出fA(X)、fB(X)、…、fG(X),找出最大值1,例如若fA(x)=1,则将该点归入A类,以次类推。Substituting the M logging parameters of the well section to predict the type of shale lithofacies into the above model one by one in order of depth, calculate f A (X), f B (X), ..., f G (X), and find the maximum value 1 , for example, if f A (x)=1, then the point is classified into class A, and so on.

根据预测结果,画出该井段的页岩岩相剖面图,进而确定出页岩气的有利开发层段。According to the prediction results, the shale lithofacies profile of the well section is drawn, and then the favorable intervals for shale gas development are determined.

以上所述仅为本发明的较佳实施例而已,并不用以限制本发明,凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等,均应包含在本发明的保护范围之内。The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present invention should be included in the protection of the present invention. within range.

Claims (2)

1.一种基于测井信息的页岩岩相连续预测方法,其特征在于,所述基于测井信息的页岩岩相连续预测方法是先采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,然后建立依据测井响应特征预测页岩岩相类型的模型;运用人工神经网络技术,对空间页岩岩相类型进行连续性预测;1. A continuous prediction method for shale lithofacies based on well logging information, characterized in that, the continuous prediction method for shale lithofacies based on well logging information is to first use the analysis and testing of core wells and thin section data to analyze the shale Determine the type of lithofacies, and then establish a model to predict the type of shale lithofacies based on the logging response characteristics; use artificial neural network technology to continuously predict the type of lithofacies in space; 所述基于测井信息的页岩岩相连续预测方法包括以下步骤:The method for continuously predicting shale lithofacies based on logging information includes the following steps: 步骤一,采用取芯井的分析化验和薄片资料对页岩岩相类型进行判定,将页岩岩相划分为富有机质硅质页岩、富有机质碳酸盐质页岩、富有机质黏土质页岩、贫有机质硅质页岩、贫有机质碳酸盐质页岩、贫有机质黏土质页岩、贫有机质灰岩7种类型,用A、B、C、D、E、F、G表示这7种类型,将其数量化以后用7维行向量表示,如某个取样点判定为其中某一类,则在这一类上的取值为1,而在其它类上的取值为0,当某一取样点经判定属于A类,则用向量[1 0 0 0 0 0 0]表示,属于B类,则用向量[0 1 0 0 0 0 0]表示;然后将测井曲线与对应深度的页岩岩相类型进行对比,找出几种能反映页岩岩相类型的常规测井曲线,作为预测页岩岩相类型的测井响应特征;Step 1: Determine the type of shale lithofacies by using the analytical tests and thin-section data of cored wells, and divide the shale lithofacies into organic-rich siliceous shale, organic-rich carbonate shale, and organic-rich clay shale. rock, organic-poor siliceous shale, organic-poor carbonate shale, organic-poor clay shale, and organic-poor limestone. After quantifying it, it is represented by a 7-dimensional row vector. If a certain sampling point is determined to be one of the categories, the value of this category is 1, while the value of other categories is 0. When a certain sampling point is determined to belong to category A, it is represented by vector [1 0 0 0 0 0 0], and if it belongs to category B, it is represented by vector [0 1 0 0 0 0 0]; Compare the depth of shale lithofacies types, and find several conventional logging curves that can reflect the shale lithofacies types, as the logging response characteristics for predicting shale lithofacies types; 步骤二,对每种页岩岩相类型找出与之对应的若干个测井取样点,作为建立预测模型的学习样本;Step 2, for each type of shale lithofacies, find a number of corresponding logging sampling points as learning samples for establishing a prediction model; 利用所有取样点的页岩岩相类型以及与深度对应的测井资料,以P个测井参数[X1,X2,…,XP]T作为输入,以对应的页岩岩相类型作为输出,建立预测页岩岩相类型的人工神经网络模型;Using the shale lithofacies types of all sampling points and the logging data corresponding to the depth, P logging parameters [X 1 ,X 2 ,…,X P ] T are used as input, and the corresponding shale lithofacies types are used as Output, establish artificial neural network model to predict shale lithofacies type; 步骤三,运用人工神经网络技术,对空间页岩岩相连续性预测,根据预测结果,画出该井段的页岩岩相剖面图,进而确定出页岩气的有利开发层段。Step 3: Use artificial neural network technology to predict the continuity of spatial shale lithofacies, draw the shale lithofacies profile of the well section according to the prediction results, and then determine the favorable intervals for shale gas development. 2.如权利要求1所述的基于测井信息的页岩岩相连续预测方法,其特征在于,所述A类的模型为:2. the shale lithofacies continuous prediction method based on logging information as claimed in claim 1, is characterized in that, the model of described A class is: <mrow> <msub> <mi>f</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mrow> <mi>p</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>&amp;sigma;</mi> <mi>p</mi> </msup> </mrow> </mfrac> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mi>exp</mi> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>A</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>A</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msup> <mi>&amp;sigma;</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>;</mo> </mrow> <mrow><msub><mi>f</mi><mi>A</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo>(</mo><mn>2</mn><mi>&amp;pi;</mi><mo>)</mo></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>&amp;sigma;</mi><mi>p</mi></msup></mrow></mfrac><mfrac><mn>1</mn><mi>m</mi></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mi>exp</mi><mo>&amp;lsqb;</mo><mo>-</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>A</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow><mi>T</mi></msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>A</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msup><mi>&amp;sigma;</mi><mn>2</mn></msup></mrow></mfrac><mo>&amp;rsqb;</mo><mo>;</mo></mrow> 式中:In the formula: M为测井取样点,用XAi=[XAi1,XAi2,…,XAip]T表示第i个取样点的P个测井参数;M is the logging sampling point, and X Ai = [X Ai1 , X Ai2 ,..., X Aip ] T represents the P logging parameters of the i-th sampling point; i=模式号;i = mode number; m=训练模式总数;m = total number of training patterns; XAi=类型A的第i训练模式;X Ai = the i-th training pattern of type A; σ=平滑参数;σ = smoothing parameter; P=度量空间的维数;P = dimensionality of the metric space; X=要预测类型的某个点的参数;X = the parameter of a certain point of the type to be predicted; B类的模型为:The model of class B is: <mrow> <msub> <mi>f</mi> <mi>B</mi> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mrow> <mi>p</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>&amp;sigma;</mi> <mi>p</mi> </msup> </mrow> </mfrac> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mi>exp</mi> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>B</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>B</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msup> <mi>&amp;sigma;</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>;</mo> </mrow> <mrow><msub><mi>f</mi><mi>B</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo>(</mo><mn>2</mn><mi>&amp;pi;</mi><mo>)</mo></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>&amp;sigma;</mi><mi>p</mi></msup></mrow></mfrac><mfrac><mn>1</mn><mi>m</mi></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mi>exp</mi><mo>&amp;lsqb;</mo><mo>-</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>B</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow><mi>T</mi></msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>B</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msup><mi>&amp;sigma;</mi><mn>2</mn></msup></mrow></mfrac><mo>&amp;rsqb;</mo><mo>;</mo></mrow> G类的模型为:The model of class G is: <mrow> <msub> <mi>f</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mrow> <mi>p</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>&amp;sigma;</mi> <mi>p</mi> </msup> </mrow> </mfrac> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mi>exp</mi> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>X</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msup> <mi>&amp;sigma;</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>.</mo> </mrow> 2 <mrow><msub><mi>f</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo>(</mo><mn>2</mn><mi>&amp;pi;</mi><mo>)</mo></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>&amp;sigma;</mi><mi>p</mi></msup></mrow></mfrac><mfrac><mn>1</mn><mi>m</mi></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mi>exp</mi><mo>&amp;lsqb;</mo><mo>-</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow><mi>T</mi></msup><mrow><mo>(</mo><mi>X</mi><mo>-</mo><msub><mi>X</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msup><mi>&amp;sigma;</mi><mn>2</mn></msup></mrow></mfrac><mo>&amp;rsqb;</mo><mo>.</mo></mrow> 2
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