CN115901834A - Rock pore tortuosity fractal dimension prediction method based on nuclear magnetic resonance image - Google Patents

Rock pore tortuosity fractal dimension prediction method based on nuclear magnetic resonance image Download PDF

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CN115901834A
CN115901834A CN202211386093.7A CN202211386093A CN115901834A CN 115901834 A CN115901834 A CN 115901834A CN 202211386093 A CN202211386093 A CN 202211386093A CN 115901834 A CN115901834 A CN 115901834A
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pore
tortuosity
fractal dimension
capillary
rock
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王惠民
向雷
盛丹娜
盛金昌
罗玉龙
田佳丽
吴洪涛
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Hohai University HHU
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Abstract

The invention discloses a nuclear magnetic resonance image-based rock pore tortuosity fractal dimension prediction method, which is used for carrying out T on a rock sample after pretreatment 2 Testing the spectrum, and inverting to obtain a pore size distribution curve; acquiring an MRI image of a rock sample by using a nuclear magnetic resonance imaging module, and calculating a rock pore distribution fractal dimension based on the image; simplifying the rock pore space into capillaries with different pore sizes and tortuosity, and determining the number of the capillaries under different tortuosity; and according to the scaling relation between the pore distribution and the number of the capillaries, establishing the mathematical relation between the tortuosity fractal dimension and the pore distribution fractal dimension, and solving the tortuosity fractal dimension. The method adopts nuclear magnetic resonance technologyThe technology is an experimental means, directly obtains a pore distribution dimension representing a two-dimensional plane of a pore space through a nuclear magnetic image, takes the dimension as a target parameter and combines T 2 And (3) calculating a tortuosity fractal dimension by using the pore size distribution information of the map, and providing a basis for quantitatively describing the pore structure and permeability characteristics of the rock.

Description

Rock pore tortuosity fractal dimension prediction method based on nuclear magnetic resonance image
Technical Field
The invention belongs to the field of oil-gas exploration and development, and particularly relates to a rock pore tortuosity fractal dimension prediction method based on a nuclear magnetic resonance imaging technology.
Background
In the field of oil and gas exploration and development, a rock microscopic pore structure is a channel for gas-water two-phase flow to flow and diffuse, a tortuosity fractal dimension is used as a quantitative characterization parameter of the bending degree of the pore structure, the difficulty of fluid passing through a rock solid matrix is reflected, and the evaluation method is a basis for evaluating the oil and gas exploitation efficiency.
At present, the calculation of fractal dimension for rock pore tortuosity at home and abroad is mainly based on an empirical expression about macroscopic parameter porosity obtained from similar Sierspinski carpets, and characteristic parameters such as average pore radius, average tortuosity, capillary characteristic length and the like need to be determined. For a rock sample, the distribution of the pore structure of the rock sample is not an accurate self-similar model, the influence of the pore structure on the tortuosity of rock pores is discussed through a macroscopic parameter-porosity, and the influence of the arrangement mode of internal pores on the tortuosity of rock is neglected. Different pore distributions have a crucial influence on the fractal dimension of the tortuosity of the rock, and even rock samples with similar porosity can have different pore distributions and capillary tube numbers, so that the fractal dimension of the tortuosity of the rock pores is obviously different.
Disclosure of Invention
The invention aims to provide a rock pore tortuosity fractal dimension prediction method based on nuclear magnetic resonance images, which takes the nuclear magnetic resonance technology as an experimental means, directly obtains a pore distribution dimension representing a two-dimensional plane of a pore space through the nuclear magnetic resonance images and takes the pore distribution dimension as the dimensionTarget parameter, in combination with T 2 And (3) calculating the fractal dimension of tortuosity by using the pore size distribution information of the map, and accurately estimating the pore structure and permeability of different rocks.
In order to achieve the above purpose, the solution of the invention is:
a nuclear magnetic resonance image-based fractal dimension prediction method for rock pore tortuosity comprises the following steps:
step 1, pretreating a rock sample;
step 2, carrying out T on the rock sample 2 Testing the spectrum, and inverting to obtain a pore size distribution curve;
step 3, obtaining an MRI image of the rock sample by using a nuclear magnetic resonance imaging module, and calculating a rock pore distribution fractal dimension based on the image;
step 4, simplifying the rock pore space into capillaries with different pore sizes and tortuosity, and determining the number of the capillaries under different tortuosity;
and 5, establishing a mathematical relation between the tortuosity fractal dimension and the pore distribution fractal dimension according to the scaling relation between the pore distribution and the number of the capillaries, and solving the tortuosity fractal dimension.
The concrete content of the step 1 is that the rock sample is processed into a cylinder meeting the international mechanical society standard, and after the vacuum-pumping treatment, the pressure saturation is not less than 24 hours.
In the step 2, the rock sample is subjected to T by using the measuring system 2 The method comprises the following steps of (1) map testing, debugging a measurement system before testing, selecting an FID sequence to automatically search a center frequency and a hard pulse width to carry out initial debugging on the system; selecting CPMG sequence to perform T on rock sample 2 And (4) testing a map.
In the step 3, before the MRI image is acquired by the MRI module, parameter debugging is performed on the MRI module, an SE sequence is selected for imaging, and parameters are set.
In the step 3, before calculating the fractal dimension of the rock pore distribution, preprocessing the MRI image, including image splicing, performing filtering and denoising processing on the image by using ImageJ software, determining a threshold value by using a Matlab program, and performing binarization processing on the threshold value.
In the step 3, a box-dimension method is used for calculating a pore distribution fractal dimension D f Covering boxes with different side lengths on the surface of an MRI image respectively, counting the number N (d) of boxes containing pores, then continuously reducing the side length d of the boxes until the minimum box side length reaches a pixel, and according to a fractal scale relation ln [ N (d) ]]＝-D f And ln d, establishing linear fitting of the boxes with different sizes and the number of the boxes containing the pores in a logarithmic coordinate system, wherein the slope k of a fitting straight line is a fractal dimension for representing the distribution of the pore structure.
The specific process of the step 4 is as follows:
calculating the specific pore diameter r according to the following formula i Volume V of capillary i The following number of capillaries:
Figure BDA0003929847300000021
wherein V is a single specific aperture r i Volume of capillary tube of (2), V i To a specific aperture r i Total capillary volume of (a);
the volume of the capillary was calculated according to the following formula:
V＝πr 2 L t (r)
wherein r represents the capillary radius, L t (r) represents the length of the capillary with tortuosity;
Figure BDA0003929847300000031
wherein D is t Is a fractal dimension of tortuosity, and the change in the two-dimensional space is 1-2,L 0 Is the characteristic length of the capillary tube (2 r) 1-Dt (1-D) representing 2r t ) The power;
specific aperture r i Volume V of capillary i Number of capillaries i Expressed as:
Figure BDA0003929847300000032
the radius is greater than r i The number of capillaries is expressed as:
Figure BDA0003929847300000033
wherein n represents a pore diameter r i The number of capillaries.
The specific content of the step 5 is as follows:
according to the fractal theory, the number of capillaries with the capillary size larger than r and the pipe diameter r have the following power function relationship:
Figure BDA0003929847300000034
wherein D is f Is the fractal dimension of the pore distribution, r max Is the maximum value of the diameter of the capillary tube;
by the principle of NMR, the transverse relaxation time T of each NMR 2 Represents a specific capillary radius r, the signal amplitude of which represents the capillary radius r i Volume V of capillary i Transverse relaxation time T 2 The following relationship exists with respect to pore size:
Figure BDA0003929847300000035
where ρ is the relaxation rate, r is the pore radius, F S For geometric factors, with capillaries having tortuosity, F S ＝2；
Combining the above relations, obtaining:
Figure BDA0003929847300000041
wherein, T 2i Denotes the capillary radius r i A corresponding transverse relaxation time;
taking the logarithm of two sides of the above formula at the same time:
Figure BDA0003929847300000042
and substituting different tortuosity fractal dimensions into the formula for iterative calculation until the pore distribution fractal dimension is the same as the fractal dimension calculated by the box-counting dimension method, wherein the tortuosity fractal dimension is the characterization parameter of the tortuosity of the capillary.
After the scheme is adopted, the invention develops a lamella imaging experiment by utilizing a nuclear magnetic resonance imaging technology, determines the pore distribution fractal dimension by utilizing a box-counting dimension method, and further calculates the pore tortuosity fractal dimension by taking the pore distribution fractal dimension as a target parameter. Compared with the traditional method, the method obtains the tortuosity fractal dimension in a mode of combining experimental means with theoretical derivation, does not depend on characteristic values such as average pore radius, average tortuosity and the like, can quantitatively characterize the tortuosity difference caused by pore arrangement distribution, and provides a basis for quantitatively describing the pore structure and permeability characteristics of the rock.
Drawings
FIG. 1 is a flow chart of the present invention;
figure 2 is a schematic of the pore size distribution curves for 4 sandstone samples;
FIG. 3 is a schematic view of a capillary model with tortuosity;
FIG. 4 is a schematic representation of a slice position of a rock sample;
figure 5 is an image of the pore distribution of the different lamellae of the sandstone sample 1;
FIG. 6 is a schematic representation of MRI image processing;
figure 7 is the pore distribution fractal dimension for 4 sandstone samples;
FIG. 8 is a determination of a tortuosity fractal dimension in conjunction with a pore distribution fractal dimension;
fig. 9 shows fractal dimensions of tortuosity calculated by different methods.
Detailed Description
The technical solution and the advantages of the present invention will be described in detail with reference to the accompanying drawings.
As shown in fig. 1, the invention provides a method for predicting a fractal dimension of rock pore tortuosity based on a nuclear magnetic resonance image, which is described below by taking sandstone as an example and comprises the following steps:
step 1, processing a sandstone sample into a cylinder of 50 x 100mm according to the standard of the international mechanical society for samples obtained by sampling rock cores of an engineering target area, vacuumizing the sandstone sample by using a vacuum saturator, and then pressurizing and saturating the sandstone sample for not less than 24 hours; in this example, a total of 4 sandstone samples were collected;
step 2, performing T on the sandstone sample by using the measuring system 2 Performing spectrum testing and inverting to obtain a pore size distribution curve, which can be matched with the graph shown in FIG. 2;
step 3, acquiring an MRI image of the sandstone sample by using a nuclear magnetic resonance imaging module, and calculating a rock pore distribution fractal dimension based on the image;
step 4, simplifying the rock pore space into capillaries with different pore sizes and tortuosity, and determining the number of the capillaries under different tortuosity, wherein a model diagram is shown in FIG. 3;
and 5, establishing a mathematical relation between the tortuosity fractal dimension and the pore distribution fractal dimension according to the scaling relation between the pore distribution and the number of the capillaries, and solving the tortuosity fractal dimension.
In the step 2, T of the sandstone sample is obtained by using the measuring system 2 Before the map, a measurement system needs to be debugged in an early stage, specifically, a computer control system is opened, and an FID sequence is selected to automatically find the center frequency and the hard pulse width so as to debug the system in the early stage; selecting CPMG sequence to perform T on rock sample 2 And (3) performing spectrum testing, and determining the CPMG parameters as follows for the sandstone type: the magnetic field intensity is 0.3T, the central frequency is 12MHz, the echo interval is 0.2ms, the number of echoes is 12000, the accumulation frequency is 64, and the waiting time is 3000ms;
in step 3, before the MRI image is acquired by the MRI module, parameter debugging needs to be performed on the MRI module, specifically, an SE sequence is selected for imaging, and the parameters are set as follows: the repeated waiting time is 3000ms, the accumulation times is 64, the axial direction of the rock is taken as the imaging direction, the set layer thickness is 5mm, the interlayer spacing is 5mm, the rock is divided into 9 lamella layers for imaging experiments, the lamella positions of 4 sandstone samples are respectively shown in fig. 4, and the pore distribution of different lamella layers of the sandstone sample 1 is shown in fig. 5.
In the step 3, before calculating the fractal dimension of the rock pore distribution, the MRI image is preprocessed, including image splicing, the image is filtered and denoised by ImageJ software, a threshold value is determined by Matlab program and binarized, and the preprocessed slice image is as shown in fig. 6.
In the step 3, a box-dimension method is used for determining a pore distribution fractal dimension D f Covering boxes with different side lengths on the MRI image surface respectively, counting the number N (d) of boxes containing pores (namely containing pore pixels), then continuously reducing the side length d of the boxes until the minimum box side length reaches pixels, and according to a fractal scale relation ln [ N (d) ]]＝-D f And ln d, establishing linear fitting of the boxes with different sizes and the number of the boxes containing the pores in a logarithmic coordinate system, wherein the slope k of a fitting straight line is a fractal dimension for representing the distribution of the pore structure. The fitting results are shown in fig. 7, and the fractal dimensions of pore distribution of 4 sandstone samples are 1.3405, 1.3318, 1.3592 and 1.3262, respectively.
In the step 4, the specific method for determining the number of the capillaries under different tortuosity comprises the following steps:
specific aperture r i Volume V of capillary i The number of capillaries is given by the following relationship:
Figure BDA0003929847300000061
wherein V is a single specific aperture r i Volume of capillary tube of (2), V i To a specific aperture r i Total capillary volume of (2).
Calculate the volume of the capillary:
V＝πr 2 L t (r) (2)
wherein r represents the capillary radius, L t (r) represents the length of the capillary with tortuosity.
Figure BDA0003929847300000062
Wherein D is t Is a fractal dimension of tortuosity, and the change in the two-dimensional space is 1 to 2,L 0 Is the characteristic length of the capillary tube, (2 r) 1-Dt (1-D) representing 2r t ) To the power.
Substituting the formula (2) and the formula (3) into the formula (1) can obtain the specific aperture r i Volume V of capillary i Number of capillaries i Can be expressed as:
Figure BDA0003929847300000063
the radius is larger than r i The number of capillaries can be expressed as:
Figure BDA0003929847300000064
wherein n represents a pore diameter r i The number of capillaries;
the specific content of the step 5 is as follows:
according to the fractal theory, the number of capillaries with the capillary size larger than r and the pipe diameter r have the following power function relationship:
Figure BDA0003929847300000071
wherein D is f Is the fractal dimension of the pore distribution, r max Is the maximum tube diameter of the capillary tube.
By NMR principle, transverse relaxation time T of each NMR 2 Represents a specific capillary radius r, the signal amplitude of which represents the capillary radius r i Capillary tube bodyProduct V i Transverse relaxation time T 2 The following relationship exists with respect to pore size:
Figure BDA0003929847300000072
where ρ is the relaxation rate, r is the pore radius, F S For geometric factors, with capillaries having tortuosity, F S And taking 2.
Combining the above relationships, one can obtain:
Figure BDA0003929847300000073
wherein, T 2i Denotes the capillary radius r i A corresponding transverse relaxation time;
the logarithm is taken at the same time on two sides of the formula (8):
Figure BDA0003929847300000074
and (3) substituting different tortuosity fractal dimensions into formula (9) for iterative calculation until the pore distribution fractal dimension is the same as the fractal dimension calculated by the box-counting dimension method, wherein the tortuosity fractal dimension is the characterization parameter of the tortuosity of the capillary. The fractal dimension of tortuosity is determined by combining the fractal dimension of pore distribution, and the result of iterative calculation is shown in fig. 8. The fractal dimensions of tortuosity of the 4 sandstone samples are 1.5271, 1.5215, 1.4719 and 1.5406 respectively. The pore structure characteristic data of the sandstone sample calculated by the embodiment is shown in table 1:
TABLE 1 sandstone sample pore structure characteristics
Figure BDA0003929847300000081
FIG. 9 is a graph showing comparison of fractal dimensions of tortuosity based on average tortuosity by the present invention and other researchers at present.
In summary, the invention provides a sandstone pore tortuosity fractal dimension prediction method based on a nuclear magnetic resonance imaging technology, which considers the relationship between the number of capillaries under different tortuosity and the pore distribution, establishes a new method for determining the sandstone pore tortuosity fractal dimension based on the nuclear magnetic resonance imaging technology, and comprises the following steps: acquiring a rock pore size distribution curve and a lamella image by taking a nuclear magnetic resonance technology as an experimental means; preprocessing the nuclear magnetic image, and obtaining a pore distribution fractal dimension by using a box-of-dimensions method; and then, the fractal dimension of the sandstone tortuosity is obtained by iteration by taking the pore distribution fractal dimension as a target parameter and combining the fractal scalar relationship between the number of capillaries under different tortuosity and the pore diameter. The fractal dimension obtained by the method can describe the tortuosity difference caused by different pore arrangement modes in the rock, and provides a basis for quantitatively describing the pore structure and permeability of the rock.
The above embodiments are only for illustrating the technical idea of the present invention, and the technical idea of the present invention is not limited thereto, and any modifications made on the basis of the technical solution according to the technical idea of the present invention fall within the protective scope of the present invention.

Claims (8)

1. A rock pore tortuosity fractal dimension prediction method based on nuclear magnetic resonance images is characterized by comprising the following steps:
step 1, pretreating a rock sample;
step 2, carrying out T on the rock sample 2 Testing the spectrum, and inverting to obtain a pore size distribution curve;
step 3, obtaining an MRI image of the rock sample by using a nuclear magnetic resonance imaging module, and calculating a rock pore distribution fractal dimension based on the image;
step 4, simplifying the rock pore space into capillaries with different pore sizes and tortuosity, and determining the number of the capillaries under different tortuosity;
and 5, establishing a mathematical relation between the tortuosity fractal dimension and the pore distribution fractal dimension according to the scaling relation between the pore distribution and the number of the capillaries, and solving the tortuosity fractal dimension.
2. The method of claim 1, wherein: the concrete content of the step 1 is that the rock sample is processed into a cylinder meeting the international mechanical society standard, and after the vacuum-pumping treatment, the cylinder is pressurized and saturated.
3. The method of claim 1, wherein: in the step 2, the rock sample is subjected to T by using the measuring system 2 The method comprises the following steps of (1) map testing, debugging a measurement system before testing, selecting an FID sequence to automatically search a center frequency and a hard pulse width to carry out initial debugging on the system; selecting CPMG sequence to perform T on rock sample 2 And (4) testing a map.
4. The method of claim 1, wherein: in the step 3, before the MRI image is acquired by the MRI module, parameter debugging is performed on the MRI module, an SE sequence is selected for imaging, and parameters are set.
5. The method of claim 1, wherein: in the step 3, before calculating the fractal dimension of the rock pore distribution, preprocessing the MRI image, including image splicing, performing filtering and denoising processing on the image by using ImageJ software, determining a threshold value by using a Matlab program, and performing binarization processing on the threshold value.
6. The method of claim 1, wherein: in the step 3, a box-dimension method is used for calculating a pore distribution fractal dimension D f Covering boxes with different side lengths on the surface of an MRI image respectively, counting the number N (d) of boxes containing pores, then continuously reducing the side length d of the boxes until the minimum box side length reaches a pixel, and according to a fractal scale relation ln [ N (d) ]]＝-D f lnd, and establishing linear fitting of boxes with different sizes and the number of boxes containing pores in a logarithmic coordinate system, wherein the slope k of a fitting straight line is a fractal dimension for representing pore structure distribution.
7. The method of claim 1, wherein: the specific process of the step 4 is as follows:
calculating the specific pore diameter r according to the following formula i Volume V of capillary i The following number of capillaries:
Figure FDA0003929847290000021
wherein V is a single specific aperture r i Volume of capillary tube of (1), V i To a specific aperture r i Total capillary volume of (a);
the volume of the capillary was calculated according to the following formula:
V＝πr 2 L t (r) (2)
wherein r represents the capillary radius, L t (r) represents the length of the capillary with tortuosity;
Figure FDA0003929847290000022
wherein D is t Is a fractal dimension of tortuosity, and the change in the two-dimensional space is 1-2,L 0 Is the characteristic length of the capillary tube, (2 r) 1-Dt (1-D) representing 2r t ) The power;
when the formula (2) and the formula (3) are substituted into the formula (1), the aperture r is specified i Capillary volume V i Number of capillaries i Expressed as:
Figure FDA0003929847290000023
the radius is greater than r i The number of capillaries of (a) is expressed as:
Figure FDA0003929847290000024
wherein n represents a pore diameter r i The number of capillaries.
8. The method of claim 1, wherein: the specific content of the step 5 is as follows:
according to a fractal theory, the number of capillaries with the capillary size larger than r and the pipe diameter r have the following power function relationship:
Figure FDA0003929847290000031
wherein D is f Is the fractal dimension of the pore distribution, r max Is the maximum value of the diameter of the capillary tube;
by the principle of NMR, the transverse relaxation time T of each NMR 2 Represents a specific capillary radius r, the signal amplitude of which represents the capillary radius r i Volume V of capillary i Transverse relaxation time T 2 The following relationship exists with respect to pore size:
Figure FDA0003929847290000032
where ρ is the relaxation rate, r is the pore radius, F S For geometric factors, with capillaries having tortuosity, F S ＝2；
Combining the above relations, obtaining:
Figure FDA0003929847290000033
wherein, T 2i Denotes the capillary radius r i A corresponding transverse relaxation time;
the simultaneous logarithm of both sides of equation (8) is:
Figure FDA0003929847290000034
and (3) substituting different tortuosity fractal dimensions into formula (9) for iterative calculation until the pore distribution fractal dimension is the same as the fractal dimension calculated by the box-counting dimension method, wherein the tortuosity fractal dimension is the characterization parameter of the tortuosity of the capillary.
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