Method for measuring different-scale pore rebound coefficients of shale based on pressure-pressure low-field nuclear magnetic resonance experiment

Abstract

The invention discloses a method for measuring different-scale pore rebound coefficients of shale based on a pressure-pressure low-field nuclear magnetic resonance experiment, and relates to the technical field of shale reservoir physical property measurement, which comprises the following steps: carrying out low-field nuclear magnetic resonance measurement on a saturated water shale sample under different confining pressures; obtaining a nuclear magnetic resonance transverse relaxation time spectrogram of the saturated water shale sample under different confining pressures and the area of each spectrum peak in the spectrogram; acquiring a pore size distribution curve graph of the saturated water shale sample according to the area of each spectral peak in the spectrogram and the relationship between the transverse relaxation time and the pore size; and based on the area of each spectral peak in the spectrogram, calculating a total pore average pore rebound coefficient and pore rebound coefficients of different scales in combination with a pore rebound coefficient conversion formula. The method can be used for rapidly, nondestructively and finely measuring the pore rebound coefficient of the shale, and is simple and convenient to operate.

Description

Technical Field

[0001] This invention relates to the field of shale reservoir property measurement technology, and in particular to a method for measuring the porosity resilience coefficient of shale at different scales based on low-field nuclear magnetic resonance experiments with overburden pressure. Background Technology

[0002] Shale porosity, as a crucial factor in reservoir evaluation, is a vital carrier of methane occurrence, a direct target of tectonic alteration, and a key factor in the success or failure of fracturing for enhanced production. The degree to which shale pore volume changes during tectonic uplift depends on the shale pore resilience coefficient, a critical parameter characterizing shale reservoir features and central to understanding the dynamic adjustment and accumulation mechanism of shale gas. Conventional testing methods (gas-phase and liquid-phase methods) provide qualitative and quantitative characterization of shale micropore-fracture features under varying confining pressures. However, they cannot conduct pore-fracture studies under sustained geological conditions, limiting the detailed characterization of the sensitivity of shale pore-fractures to confining pressure changes. In contrast, low-field nuclear magnetic resonance (NMR) testing under overburden pressure offers advantages such as speed and non-destructive testing, and allows for continuous analysis of the same sample under different confining pressures, reducing errors caused by the heterogeneity of shale samples. Furthermore, the mechanisms of action of pores and fractures of different pore sizes in shale during desorption, diffusion, and seepage differ, and the degree of rebound adjustment under tectonic uplift stress varies. As shale gas exploration and development research deepens, it is imperative to conduct detailed studies on the dynamic evolution of shale reservoirs and their porosity and permeability under tectonic uplift conditions. Conventional shale pore structure testing methods cannot accurately depict these details, while overburden low-field nuclear magnetic resonance (NMR) measurements can detect and reflect these details, thus laying a solid foundation for refined shale reservoir research. Summary of the Invention

[0003] To address the technical problems existing in the prior art, this invention proposes a method for measuring the porosity resilience coefficient of shale at different scales based on low-field nuclear magnetic resonance experiments with overburden pressure.

[0004] To achieve the above objectives, this invention provides a method for measuring the porosity resilience coefficient of shale at different scales based on low-field nuclear magnetic resonance experiments with overburden pressure, comprising: Low-field nuclear magnetic resonance (NMR) measurements were performed on saturated water shale samples under different confining pressures to obtain the transverse relaxation time spectra of the NMR samples under different confining pressures and the area of ​​each peak in the spectra. By analyzing the area of ​​each spectral peak in the spectrum and utilizing the relationship between transverse relaxation time and pore size, the pore size distribution curve of the saturated water shale sample can be obtained. Based on the area of ​​each spectral peak in the spectrum, and combined with the pore resilience coefficient conversion formula, the average pore resilience coefficient of total pores and the pore resilience coefficients of different scales are calculated.

[0005] Preferably, the method for obtaining the saturated water shale sample is as follows: The shale sample was cut into a preset shape along the parallel bedding direction using wire cutting. The cut shale sample was dried and placed in a vacuum pump to be evacuated. It was then subjected to negative pressure saturation water and normal pressure saturation water for different preset time lengths to obtain the saturated water shale sample.

[0006] Preferably, obtaining the relationship between the lateral relaxation time and the aperture includes: (1); In the formula, The surface relaxation time is expressed in milliseconds (ms). Surface relaxation rate, in nm / ms; Pore ​​specific surface area, in cm² 2 / g; S The pore area; V Pore ​​volume; The pore geometry factor; Where is the pore radius; Rearranging equation (1), we obtain the relationship between surface relaxation time and pore diameter: (2); In the formula, The pore diameter; Let the conversion coefficient The relationship between the lateral relaxation time and the aperture is obtained: (3); In the formula, This represents the lateral relaxation time.

[0007] Preferably, the average conversion coefficient is used to study the pore size of the shale, and the pore size distribution curve of the saturated water shale sample is obtained.

[0008] Preferably, the formula for converting the porosity resilience coefficient is: (4); In the formula, The average porosity resilience coefficient is the total porosity. The stress difference is the confining pressure applied during the experiment, expressed in MPa. The change in total pore volume of shale under staged confining pressure conditions is expressed in cm³. 3 ; The total pore volume of the sample under the simulated maximum burial depth conditions, in cm³. 3 .

[0009] Preferably, the method for calculating the average porosity rebound coefficient of the total porosity by using the peak area of ​​the transverse relaxation time under different confining pressure conditions is as follows: (5); In the formula, The average porosity resilience coefficient is the total porosity. This represents the stress value of the maximum confining pressure applied during the experiment, expressed in MPa. For the first i The stress value under the confining pressure is expressed in MPa. The peak area of ​​the transverse relaxation time measured under the maximum confining pressure stress during the experiment is dimensionless. and The first i The pore volume and transverse relaxation time spectrum peak area under the confining pressure are dimensionless.

[0010] Preferably, calculating the resilience coefficient of the pores at different scales includes: Based on the pore size distribution curve of the saturated water shale sample and the pore classification method proposed by the International Union of Theoretical and Applied Chemistry, the pores are divided into micropores, mesopores and macropores. The rebound coefficients of pores at different scales are calculated, and the rebound coefficients of micropores, mesopores and macropores are calculated based on the rebound coefficients of pores at different scales.

[0011] Preferably, the method for calculating the resilience coefficient considering pore sizes is as follows: The relationship between total pore volume and pore components at each scale is obtained as follows: (6); In the formula, This represents the total pore volume of shale, in cm³. 3 ; This refers to the total volume of micropores in shale, expressed in cm³. 3 ; This refers to the total volume of shale mesopores, expressed in cm³. 3 ; This refers to the total volume of macropores in shale, expressed in cm³. 3 ; Differentiating both sides of equation (6), we obtain equation (7): (7); In the formula, This is the effective stress value; Divide both sides of equation (7) by the total pore volume of the sample under the simulated maximum burial depth conditions. Equation (8) is obtained: (8); Substituting equation (6) into equation (5) yields the resilience coefficient considering pore sizes: (9); (10); In the formula, , as well as These represent the pore volumes of micropores, mesopores, and macropores under the maximum effective stress in the experiment, respectively, in cm³. 3 ; , as well as These represent the resilience coefficients of shale micropores, mesopores, and macropores under varying effective stress, respectively, in MPa. -1 , is the average pore resilience coefficient of total pores.

[0012] Preferably, the method for calculating the resilience coefficients of the micropores, mesopores, and macropores is as follows: (11); (12); (13); In the formula, , , These represent the spectral peak areas of the transverse relaxation times of micropores, mesopores, and macropores measured under the maximum confining pressure stress during the experiment, respectively, and are dimensionless. Representing the first i Micropore, mesopore, and macropore volumes under confining pressure, in cm³ 3 ; , Representing the first i The spectral peak area of ​​the transverse relaxation time of micropores, mesopores, and macropores under confining pressure is dimensionless.

[0013] Compared with the prior art, the present invention has the following advantages and technical effects: The overburden nuclear magnetic resonance measurement method provided by this invention can not only measure the average porosity rebound coefficient of total porosity (i.e., the total porosity rebound coefficient of shale), but also simultaneously measure the porosity rebound coefficients of micropores, mesopores and macropores, laying the foundation for further research on shale reservoirs. Attached Figure Description

[0014] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart illustrating a method for measuring the resilience coefficient of shale pores at different scales based on low-field nuclear magnetic resonance experiments under overburden pressure, according to an embodiment of the present invention. Figure 2 This is a graph showing the variation of shale pore distribution under different effective stresses in the TB-1 sample of this invention. Figure 3 This is a graph showing the variation of shale pore distribution under different effective stresses in the SH-1 sample of this invention. Figure 4 This is a graph showing the variation of shale pore distribution under different effective stresses in the DFA-1 sample of this invention. Figure 5 This is a dynamic variation diagram of the pore resilience coefficient of the TB-1 sample in an embodiment of the present invention. Figure 6 This is a dynamic variation diagram of the pore resilience coefficient of the SH-1 sample in an embodiment of the present invention. Figure 7 This is a dynamic change diagram of the pore resilience coefficient of the DFA-1 sample in an embodiment of the present invention. Detailed Implementation

[0015] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0016] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0017] This invention proposes a method for measuring the resilience coefficient of shale pores at different scales based on low-field nuclear magnetic resonance experiments with overburden pressure, such as... Figure 1 ,include: Low-field nuclear magnetic resonance (NMR) measurements were performed on saturated water shale samples under different confining pressures to obtain the transverse relaxation time spectra of the NMR samples under different confining pressures and the area of ​​each peak in the spectra. By analyzing the area of ​​each spectral peak in the spectrum and utilizing the relationship between transverse relaxation time and pore size, the pore size distribution curve of the saturated water shale sample can be obtained. Based on the area of ​​each spectral peak in the spectrum, and combined with the pore resilience coefficient conversion formula, the average pore resilience coefficient of total pores and the pore resilience coefficients of different scales are calculated.

[0018] The method of this invention primarily utilizes low-field nuclear magnetic resonance (NMR) technology to measure the porosity resilience of shale. For water-saturated shale samples, NMR detects the hydrogen nuclei in the water. Hydrogen-containing substances ( 1 The relaxation behavior of H) in shale reservoirs is usually influenced by the combined effects of fluid properties, pore properties, and pore structure, while the relaxation time (H) T 2 It can effectively analyze hydrogen-containing substances. 1 H) relaxation process in shale reservoirs. Relaxation time collected by the instrument ( T 2 The pore volume is directly proportional to the water content in the sample, allowing us to determine the volume of various pores within the sample. Furthermore, the rebound deformation of pores during the uplift process in shale samples allows us to obtain the nuclear magnetic resonance signal amplitude under different confining pressures, thus enabling us to determine the shale's pore rebound coefficient.

[0019] Furthermore, the method for obtaining saturated water shale samples is as follows: The shale sample was cut into a preset shape along the parallel bedding direction using wire cutting. The cut shale sample was dried and placed in a vacuum pump to be evacuated. It was then subjected to negative pressure saturation water and normal pressure saturation water for different preset time lengths to obtain the saturated water shale sample.

[0020] Specifically, Sample pretreatment: In this embodiment, the shale sample was collected from the Longmaxi Formation in the Sichuan Basin. To avoid artificial cracks that may be caused by manual drilling of the column sample, the shale sample was wire-cut along the bedding direction to a cylindrical sample with a diameter of 25 mm and a length of 50 mm. Then, the shale cylindrical sample was placed in a vacuum pump and evacuated first with negative pressure saturated water for 4 hours, and then with normal pressure saturated water for 48 hours to obtain a saturated water shale sample.

[0021] Experimental procedure: Set the relevant parameters of the nuclear magnetic resonance analyzer, and then measure the nuclear magnetic signals of shale samples sequentially under different confining pressures (such as 0, 5, 10, 15, 20 MPa).

[0022] Furthermore, the relationship between the transverse relaxation time and the aperture is obtained, including: (1); In the formula, The surface relaxation time is expressed in milliseconds (ms). Surface relaxation rate, in nm / ms; Pore ​​specific surface area, in cm² 2 / g; S The pore area; V Pore ​​volume; The pore geometry factor; Where is the pore radius; Rearranging equation (1), we obtain the relationship between surface relaxation time and pore diameter: (2); In the formula, The pore diameter; Let the conversion coefficient The relationship between the lateral relaxation time and the aperture is obtained: (3); In the formula, This represents the lateral relaxation time.

[0023] Furthermore, the average conversion coefficient was used to study the pore size of the shale, and the pore size distribution curve of the saturated water shale sample was obtained.

[0024] Specifically, previous studies on shale and conversion coefficients Extensive research has shown that shale typically contains interconnected cylindrical pores. In this embodiment, the pore geometry factor... The value is taken as 2, while the conversion coefficient of the Wufeng-Longmaxi Formation shale in the Sichuan Basin is... The values ​​range from 40.48 nm / ms to 50 nm / ms, with an average value of 45.35 nm / ms. Therefore, this embodiment uses the average conversion factor to study the pore size of shale and obtains the pore size distribution curve of the shale sample.

[0025] Furthermore, the formula for converting the porosity resilience coefficient is: (4); In the formula, The average porosity resilience coefficient is the total porosity. The stress difference is the confining pressure applied during the experiment, expressed in MPa. The change in total pore volume of shale under staged confining pressure conditions is expressed in cm³. 3 ; The total pore volume of the sample under the simulated maximum burial depth conditions, in cm³. 3 .

[0026] Furthermore, the average porosity rebound coefficient of the total porosity is calculated by using the peak area of ​​the transverse relaxation time under different confining pressures. The method is as follows: (5); In the formula, The average porosity resilience coefficient is the total porosity. This represents the stress value of the maximum confining pressure applied during the experiment, expressed in MPa. For the first i The stress value under the confining pressure is expressed in MPa. The peak area of ​​the transverse relaxation time measured under the maximum confining pressure stress during the experiment is dimensionless. and The first i The pore volume and transverse relaxation time spectrum peak area under the confining pressure are dimensionless.

[0027] Specifically, the experiment can obtain the nuclear magnetic resonance of the sample under different confining pressures. T 2 The spectrum and the area of ​​each peak in the spectrum. Because... T 2 Relaxation time can effectively reflect pore volume; therefore, the average pore resilience coefficient under different confining pressures can be obtained. T 2 The peak area is calculated.

[0028] Furthermore, the resilience coefficient of pores at different scales was calculated, including: Based on the pore size distribution curve of the saturated water shale sample and the pore classification method proposed by the International Union of Theoretical and Applied Chemistry, the pores are divided into micropores, mesopores and macropores. The rebound coefficients of pores at different scales are calculated, and the rebound coefficients of micropores, mesopores and macropores are calculated based on the rebound coefficients of pores at different scales.

[0029] In this embodiment, the range of different pore sizes is: micropores <2nm, mesopores 2~50nm and macropores >50nm.

[0030] Furthermore, the method for calculating the resilience coefficient considering pore sizes is as follows: The relationship between total pore volume and pore components at each scale is obtained as follows: (6); In the formula, This represents the total pore volume of shale, in cm³. 3 ; This refers to the total volume of micropores in shale, expressed in cm³. 3 ; This refers to the total volume of shale mesopores, expressed in cm³. 3 ; This refers to the total volume of macropores in shale, expressed in cm³. 3 ; Differentiating both sides of equation (6), we obtain equation (7): (7); In the formula, This is the effective stress (confining pressure) value; Divide both sides of equation (7) by the total pore volume under the maximum confining pressure. Equation (8) is obtained: (8); Substituting equation (6) into equation (5) yields the resilience coefficient considering pore sizes: (9); (10); In the formula, , as well as These represent the pore volumes of micropores, mesopores, and macropores under the maximum effective stress in the experiment, respectively, in cm³. 3 ; , as well as These represent the resilience coefficients of shale micropores, mesopores, and macropores under varying effective stress, respectively, in MPa. -1 , is the average pore resilience coefficient of total pores.

[0031] Specifically, the method for calculating the resilience coefficients of the micropores, mesopores, and macropores is as follows: (11); (12); (13); In the formula, , , These represent the spectral peak areas of the transverse relaxation times of micropores, mesopores, and macropores measured under the maximum confining pressure stress during the experiment, respectively, and are dimensionless. Representing the first i Micropore, mesopore, and macropore volumes under confining pressure, in cm³ 3 ; , Representing the first i The spectral peak area of ​​the transverse relaxation time of micropores, mesopores, and macropores under confining pressure is dimensionless.

[0032] To more clearly illustrate the technical solution of the present invention, specific embodiments are provided below for description: (1) Experimental materials: three shale cylinders with a diameter of 25 mm and a length of 50 mm (numbered TB, DFA and SH respectively), and distilled water.

[0033] (2) Experimental instruments: MacroMR12-150H-1 nuclear magnetic resonance imaging system from Numai Testing Company was used for the experiment; Test parameter settings: Experimental test temperature is 25℃; Echo interval (TE) is 0.1ms; Repeat sampling wait time (TW) is 3s; Number of echoes (NECH) is 10000; Number of scans is 64; Number of iterations is 10000.

[0034] (3) Experimental procedure: 1) Pretreatment: To avoid artificial cracks that may be caused by manual drilling of column samples, the shale samples were wire-cut into cylindrical samples with a diameter of 25 mm and a length of 50 mm along the direction parallel to the bedding. The shale samples were dried at 100℃ for 24 h, and then the shale cylindrical samples were placed in a vacuum pump and evacuated. First, they were saturated with water under negative pressure for 4 h, and then saturated with water under normal pressure for 48 h to obtain saturated water shale samples.

[0035] 2) Experimental procedure: Set the relevant parameters of the overburden nuclear magnetic resonance analyzer, and then measure the nuclear magnetic signals of shale samples (TB, DFA and SH) in sequence under different confining pressures (0, 5, 10, 15 and 20 MPa respectively).

[0036] 3) Obtaining the pore size distribution curve of shale samples: The experiment yielded nuclear magnetic resonance (NMR) results for samples (TB, DFA, and SH) under different confining pressures. T 2 The spectrum is obtained, and the pore size distribution curves of different samples are obtained by combining the formulas (1)-(3).

[0037] 4) Calculation of the porosity resilience coefficient of shale samples: The experiment yielded nuclear magnetic resonance (NMR) results for samples (TB, DFA, and SH) under different confining pressures. T 2 The spectrum and the area of ​​pores at different scales in the spectrum are shown in Table 1, and the rebound coefficients of pores at different scales are shown in Table 2. Based on the peak areas of pores at different scales under different confining pressures and combined with formulas (4)-(13), the average rebound coefficient of total pores, the rebound coefficient of micropores, the rebound coefficient of mesopores and the rebound coefficient of macropores can be calculated. Figure 2 This is a graph showing the variation of shale porosity distribution under different effective stresses in the TB-1 sample. Figure 3 This is a graph showing the variation of shale porosity distribution under different effective stresses in the SH-1 sample. Figure 4 This is a graph showing the variation of shale pore distribution under different effective stresses in the DFA-1 sample. Figure 5 This is a dynamic variation graph of the porosity resilience coefficient of the TB-1 sample. Figure 6 This is a dynamic variation graph of the porosity resilience coefficient of the SH-1 sample. Figure 7 This is a dynamic variation graph of the porosity resilience coefficient of the DFA-1 sample.

[0038] Then, by mathematically fitting the experimental values ​​of the pore resilience coefficients at different scales corresponding to various pressure values, the mathematical expressions for the pore resilience coefficients of shale at different scales can be obtained. The specific results are as follows: ① Mathematical model of microporous resilience coefficient: (14); In the formula, The micropore porosity is... , , All are constants. These are the coefficients of the springback equation, dimensionless. This is the effective stress.

[0039] Since this embodiment only conducted relevant experiments on three shale samples from the Longmaxi Formation, Section 1, in different regions, there are certain errors. However, it has certain reference value for the study of pore rebound. In addition, the correlation of the fitting equations of the micropore rebound coefficients of each sample in this embodiment is greater than 0.99, indicating a strong correlation. The coefficient weighted average of the three groups of samples yielded A1=0.0070; B1=-25.00; C1=-0.0071.

[0040] ② Mathematical model of mesoporous resilience coefficient: (15); In the formula, The first pore rebound sensitivity factor is dimensionless. , , All are constants.

[0041] In this embodiment, the fitting equation for the mesoporous pore resilience coefficient has a certain correlation (R). 2 (All are greater than 0.49). The coefficient-weighted average of the three groups of samples was calculated to obtain A2=-0.34; B2=1.55; C2=0.0022.

[0042] ③ Mathematical model of macroporous springback coefficient: (16); In the formula, , , All are constants. The second pore rebound sensitivity factor is dimensionless.

[0043] The fitting equations for the macropore resilience coefficients of each sample showed a strong correlation (R0). 2 (All are greater than 0.97). The coefficient-weighted average of the three groups of samples was calculated to obtain A3=-0.0058; B3=8.75; C3=0.024.

[0044] Table 1 Table 2 The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for measuring the rebound modulus of shale at different scales of pores based on the measurement of the overburden pressure low-field nuclear magnetic resonance experiment, characterized in that, The application relates to a method for calculating the pore elastic coefficient of a saturated water shale sample. The method comprises the following steps: The method comprises the following steps: The method comprises the following steps:

2. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 1, characterized in that, The method comprises the following steps: The method comprises the following steps:

3. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 1, characterized in that, The method comprises the following steps: （1）； wherein is the surface relaxation time in ms; is the surface relaxation rate in nm / ms; is the pore specific surface area in cm 2 / g; S is the pore area; V is the pore volume; is the pore geometry factor; is the pore radius; The method comprises the following steps: （2）； In the formula, Dp is the pore diameter; Let the conversion coefficient obtaining the relationship between the transverse relaxation time and the pore size: （3）； In the formula, is the transverse relaxation time.

4. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 3, characterized in that, The method comprises the following steps:

5. The method for measuring the coefficient of resilience of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with the pressure loading according to claim 1, characterized in that, The method comprises the following steps: （4）； In the formula, is the total pore average pore modulus, is the stress difference of the applied confining pressure during the experiment, in MPa; is the total pore volume change of the shale under the confining pressure of the stage, in cm 3 ; is the total pore volume of the sample under the condition of the maximum burial depth simulated by the experiment, in cm 3 .

6. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 5, characterized in that, The method comprises the following steps: （5）； wherein is the total porosity average pore modulus, is the stress value of the maximum confining pressure applied during the experiment, in MPa; is the stress value at the i confining pressure, in MPa; is the peak area of the transverse relaxation time measured at the stress value of the maximum confining pressure applied during the experiment, dimensionless; and are the pore volume and the peak area of the transverse relaxation time at the i confining pressure, respectively, dimensionless.

7. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 6, characterized in that, The method comprises the following steps: The method comprises the following steps:

8. The method for measuring the rebound modulus of different scale pores of shale based on the low-field nuclear magnetic resonance experiment with overpressure according to claim 7, characterized in that, The method comprises the following steps: The method comprises the following steps: （6）； wherein is the total volume of shale micropores, in cm 3 ; is the total volume of shale micropores, in cm 3 ; is the total volume of shale mesopores, in cm 3 ; is the total volume of shale macropores, in cm 3 ; The method comprises the following steps: （7）； In the formula, is the effective stress value; Dividing both sides of equation (7) by the total pore volume of the sample at the maximum burial depth simulated in the experiment gives equation (8): （8）； The method comprises the following steps: （9）； （10）； wherein, , and represent the micropore, mesopore, and macropore pore volume at the experimental maximum effective stress, respectively, in cm 3 ; , and are the effective stress change dependent modulus of the micropore, mesopore, and macropore pore volume, respectively, in MPa -1 , is the total pore average pore modulus.

9. 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