A method for predicting elastic modulus based on MT framework and bimodal pore aspect ratio distribution

Abstract

This invention discloses an elastic modulus prediction method based on the Mori-Tanaka framework and a bimodal pore aspect ratio distribution, belonging to the field of rock physics modeling technology. By introducing a bimodal probability distribution model to describe the aspect ratios of soft and hard pores, and considering the high formation stress and concentrated pore aspect ratio distribution within the rock in the deep saline aquifer experimental area, the polarization factor under different aspect ratio conditions is derived based on the Mori-Tanaka model, strengthening the correlation between elastic modulus and pore aspect ratio. Furthermore, a complete prediction system is constructed by combining the Mori-Tanaka (MT) framework and the Gassmann fluid substitution framework, respectively. This system can more realistically approximate the statistical distribution law of pore morphology in actual rocks under the probabilistic framework, overcoming the systematic errors caused by single-value assumptions, thereby improving the reliability of the model's prediction of macroscopic rock properties. This provides an efficient and accurate theoretical tool for reservoir prediction research in deep saline sandstone formations.

Description

Technical Field

[0001] This invention relates to the field of rock physics modeling technology, specifically to a method for predicting elastic modulus based on the MT framework and bimodal pore aspect ratio distribution. Background Technology

[0002] Rock physics inversion involves observing the stress response of representative rock samples in different directions, constructing a unified rock physics model to achieve an effective fit between observed and simulated data, and thus obtaining the inversion results of the subsurface rock model parameters. Deep sandstone within saline aquifers typically contains pore spaces of different scales, including near-spherical hard pores at the microscale and flattened soft pores at the mesoscale. The geometric characteristics of these pores, especially their aspect ratio, significantly influence the elastic properties of the rock, thereby affecting the propagation velocity and attenuation of seismic waves.

[0003] To predict the equivalent elastic modulus of porous rocks, scholars have developed various theoretical rock physics models. The Mori-Tanaka model, based on mean-field theory, is suitable for predicting the equivalent modulus of ellipsoidal inclusions with low porosity and can consider the average interaction between inclusions (Mori and Tanaka, 1973). For spherical hard pores, this model provides a concise analytical expression. Later, Benveniste et al. derived the Mori-Tanaka model to a porous inclusion model (Benveniste, 1987), but did not further derive it for pore distribution. The differential equivalent medium theory (DEM) simulates composite materials by progressively adding inclusions, can handle pores with arbitrary aspect ratios, and implicitly considers the interaction between pores (Berryman, 1980). However, DEM typically requires numerical solution of differential equations, resulting in a large computational burden and difficulty in obtaining analytical forms. Furthermore, the Gassmann equation is widely used to convert dry rock modulus to saturated rock modulus, but it only applies to low-frequency (fluid pressure equilibrium) conditions and cannot describe high-frequency jet flow effects.

[0004] Pores in real rocks often have different aspect ratios, and their distribution frequently exhibits probabilistic statistical characteristics, concentrating within a certain range and possessing a dominant mean. Existing studies have approximated this by assuming a single aspect ratio, but have failed to fully utilize the true statistical information of pore geometry (Xu and White, 1995; Keys and Xu, 2002). Furthermore, for predicting high-frequency saturated rock moduli, existing methods largely rely on complex numerical simulations or lack physically transparent analytical expressions, making them difficult to rapidly apply to the interpretation of real-world data.

[0005] Therefore, how to establish a unified method that can consider the probability distribution of pore aspect ratio, analytically express dry rocks and high and low frequency saturation moduli, and at the same time take into account the interaction between pores and computational efficiency has become a technical problem that urgently needs to be solved in the field of rock physics. Summary of the Invention

[0006] To address the aforementioned problems, this invention proposes a method for predicting the elastic modulus of rocks with multi-scale porosity. By introducing a bimodal distribution to describe the aspect ratio of soft and hard pores, the polarization factor under different aspect ratios is derived based on the Mori-Tanaka model. Furthermore, a complete prediction system is constructed by combining the Mori-Tanaka (MT) framework and the Gassmann fluid substitution framework, respectively, providing an efficient and accurate theoretical tool for reservoir prediction research of deep sandstone in saline aquifers.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: A method for predicting the elastic modulus based on the MT frame and bimodal porosity aspect ratio distribution, specifically including the following steps: S10: Based on the pore structure of deep sandstone in the saline aquifer, the pores are divided into soft pores and hard pores. Probability distribution models for soft pores and hard pores are established separately, and then unified into a bimodal distribution model. S20: Measurement of longitudinal wave velocity in sandstone via ultrasonic testing transverse wave velocity The bulk modulus and shear modulus of deep sandstone in saline aquifers were calculated. The mineral composition of deep sandstone in saline aquifers was determined by XRD. The equivalent bulk modulus of the mineral matrix was calculated using Voigt-Reuss-Hill averaging. and shear modulus ; S30: Establish a petrological model of deep sandstone in a saline aquifer with multi-peak pore distribution, and calculate the pore shape factor and polarization tensor; S40: Construct a model containing a hard-porous matrix, using the mineral matrix obtained in S20 as the substrate, and treating all pores within the single-peak distribution of hard pores in S10 as inclusions; considering the contribution of hard pores of all aspect ratios to the modulus, calculate the equivalent medium bulk modulus containing only dry hard pores. and shear modulus ; S50: Construct a dry rock model with bimodal aspect ratio pores and calculate the equivalent bulk modulus and shear modulus of the dry rock containing both soft and hard pores. S60: Calculate the deep pore fluid modulus of the saline aquifer based on the Batzle-Wang fluid model; S70: Establish saturated rock models under high and low frequency limits.

[0008] More preferably, step S10 includes the following steps: S101: Due to formation compaction, the porosity distribution in the deeper saline layer is more concentrated than in the shallower layers. Considering the distribution of pores in the rock as a continuous random distribution and a bounded closed set, according to the extreme value theorem... exist There must exist a maximum value point on the above. , so that: in, Let D be the aspect ratio of the pores, and D be the domain of the function, which is a closed interval. Based on the probability statistics that can find the maximum aspect ratio distribution of soft and hard pores, a bimodal pore aspect ratio model with a truncated Gaussian distribution is established to represent the aspect ratio of soft pores. Aspect Ratio of Hard Pores They are modeled as truncated Gaussian distributions respectively. : in These represent the average aspect ratio distributions of soft and hard pores, respectively. The standard deviations of the aspect ratio distributions for soft and hard pores are respectively; S102: Will The combination of these two factors can be represented as the aspect ratio distribution of the entire rock's pores, i.e., a truncated Gaussian bimodal distribution. .

[0009] More preferably, in step S20, the longitudinal wave velocity of sandstone is measured by ultrasonic experiment. transverse wave velocity The bulk modulus and shear modulus of deep sandstone in the saline aquifer were calculated; the mineral composition of the deep sandstone in the saline aquifer was determined by XRD, and the equivalent bulk modulus of the mineral matrix was calculated using the Voigt-Reuss-Hill average. and shear modulus Specifically, it includes the following steps: S201: Measurement of longitudinal wave velocity in sandstone mass via ultrasonic testing transverse wave velocity Calculate the bulk modulus and shear modulus of deep sandstone in the saline aquifer: The ultrasonic experiment involves placing a rock sample between two ultrasonic probes of an anisotropic ultrasonic testing instrument. One probe emits ultrasonic waves, while the other probe receives the waves after they pass through the rock. The anisotropic ultrasonic testing instrument is then placed in a triaxial pressure chamber, which is a steel container with axial pressure applied at both ends and confining pressure applied on the sides, for measurement. The arrival time and waveform of the waves are analyzed, and the longitudinal and transverse wave velocities of the isotropic rock are calculated. The bulk modulus of the deep sandstone in the saline aquifer was calculated based on the measured transverse and longitudinal wave characteristics and bulk density of the sandstone. and shear modulus : Among them, volume density It is obtained from the mass / volume of the sample, i.e. g / cc; S202: The mineral composition of deep sandstone in the saline aquifer was determined by XRD, and the equivalent bulk modulus of the mineral matrix was calculated using the Voigt-Reuss-Hill average. and shear modulus : Based on the mineral component content determined by XRD and the corresponding mineral bulk modulus and shear modulus obtained from tables, the bulk modulus of the deep sandstone matrix in the saline aquifer is calculated using the Voigt-Reuss-Hill model. and shear modulus ; This represents the upper limit of the bulk modulus of the mineral components. Lower limit of bulk modulus of mineral components This represents the upper limit of the shear modulus of the mineral component. This represents the lower limit of the shear modulus of the mineral components. , Let J be the bulk modulus and shear modulus of the j-th component. Let be the volume fraction of the i-th component.

[0010] In a further preferred embodiment, the specific process of establishing a petrological model of deep sandstone in a saline aquifer with a multi-peaked pore distribution and calculating the pore shape factor and polarization tensor in step S30 is as follows: Based on Eshelby's inclusion theory, within the Mori-Tanaka framework, this study addresses each aspect ratio covered by the bimodal distribution in S10. Calculate their shape factors respectively. and the corresponding polarization tensor in The polarization factor is expressed as a formula for bulk modulus and shear modulus. , This represents the corresponding positional component within the fourth-order tensor. Since the pore morphology discussed in this model is ellipsoidal, in the x1-x2-x3 coordinate system, the two minor axes of the ellipsoid lie on x1 and x2 respectively, and... = That is, the tensor we are looking for is a symmetric matrix. = Only the upper half of the matrix needs to be derived; in For integrals related to aspect ratio: obtain polarization factor , The aspect ratio can be expressed as: ; Among them, shape factor It can be represented as: in The base Poisson's ratio can be expressed as: The equivalent bulk modulus of the base. The base is the equivalent shear modulus.

[0011] In a further preferred embodiment, in step S40, a hard-porous matrix model is constructed, using the mineral matrix obtained in S20 as the substrate, and all pores within the single-peak distribution of hard pores in S10 are considered as inclusions; considering the contribution of hard pores of all aspect ratios to the modulus, the equivalent medium bulk modulus containing only dry hard pores is calculated. and shear modulus The process is as follows: in, The bulk modulus of dry rock containing only hard pores. To calculate the polarization factor of the bulk modulus of hard pores, For hard porosity fraction, The equivalent bulk modulus of the base. The bulk modulus of the hard hole; in, The shear modulus of dry rock containing only hard pores. The polarization factor for the hard pore shear modulus. The base equivalent shear modulus, For the shear modulus of a hard hole; Will and Substituting the aspect ratio Gaussian distribution of hard pores : The formula for bulk modulus can be expressed as: The formula for shear modulus can be expressed as: .

[0012] In a further preferred embodiment, in step S50, a dry rock model with bimodal aspect ratio pores is constructed, and the equivalent bulk modulus and shear modulus of the dry rock containing both soft and hard pores are calculated. The specific steps are as follows: S40 obtained As a new substrate, the aspect ratio Gaussian distribution of soft pores in S10 is substituted. Calculate the equivalent bulk modulus of dry rock containing both soft and hard pore distributions. and shear modulus ; in, For soft pores, aspect ratio To calculate the polarization factor for the bulk modulus and shear modulus of soft pores, These are the bulk modulus and shear modulus of the soft pores.

[0013] More preferably, in step S60, calculating the deep pore fluid modulus of the saline aquifer based on the Batzle-Wang fluid model includes the following steps: S601: Calculate the pore fluid density of pure water , can be represented as: in, For ground temperature, ; Formation pressure, MPa; S602: The density of pore fluid in a saline aquifer can be calculated as follows: in, Density of saline water, g / cc; Density of pure water, g / cc; Mineralization; For temperature, ; Formation pressure, MPa; S603: Calculation of the bulk modulus of pore fluid in a saline aquifer From the definition of elastic modulus, we can obtain: in, The velocity of sound in salt water is m / s; It can be represented as: in, For matrix coefficients, These are the empirical parameter table values ​​for the Batzle-Wang model.

[0014] In a further preferred embodiment, step S70, establishing the saturated rock model under high and low frequency limits, specifically includes the following steps: S701: Fluid substitution of pores yields saline-saturated pores, wherein the bulk modulus of saline water is... With a shear modulus of 0, the pore fluid is in a non-relaxed state under the high-frequency limit and can be considered as an inclusion; repeat steps S40 and S50: Using the mineral matrix in S20 as a substrate, fluid-saturated unimodal hard pores were added to calculate the rock modulus containing saline-water-saturated hard pores. Then, using the above medium as a substrate, fluid-saturated soft pores were added to obtain the saturated rock bulk modulus under the high-frequency limit. and shear modulus Simulate the elastic response of pore fluids in a non-relaxed framework; S702: Calculate the saturated rock modulus under the low-frequency limit: Calculate the dry rock modulus obtained in S50. and By applying the Gassmann equation for fluid substitution, the bulk modulus of saturated rock under the low-frequency limit was obtained. : in, Total porosity The bulk modulus of the mineral matrix. The bulk modulus of saline water and the shear modulus in the low-frequency limit. .

[0015] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention introduces a bimodal probability distribution model to describe the aspect ratio of soft and hard pores. It takes into account the high formation stress in the deep experimental area of ​​the saline layer and the concentrated distribution of the aspect ratio of pores inside the rock, thus better characterizing the internal pore structure of sandstone. The elastic modulus value calculated by the model is in high agreement with the experimentally measured value.

[0016] 2. This invention, based on the Mori-Tanaka model, explicitly derives the polarization factors P and Q for different aspect ratios under an oblate spheroid shape, strengthening the correlation between elastic modulus and pore aspect ratio. Unlike existing models that numerically integrate empirical formulas, this invention constructs a pore aspect ratio distribution model combined with the MT model for faster elastic modulus calculation, filling the gap in the traditional MT model's characterization of pore morphology changes. Furthermore, it combines the Mori-Tanaka framework and the Gassmann fluid substitution framework to construct a complete prediction system, enabling a more realistic approximation of the statistical distribution of pore morphology in actual rocks within a probabilistic framework. This overcomes the systematic errors caused by single-valued assumptions, thereby improving the reliability of the model's prediction of macroscopic rock properties.

[0017] 3. The parameters of this invention are flexible and widely applicable. They can be adjusted according to the statistical results of actual rock samples or well logging interpretation for different regions and different lithologies. Moreover, the calculation process is clear and easy to implement in engineering. Attached Figure Description

[0018] Figure 1 This is a diagram of a bimodal distribution model of pores.

[0019] Figure 2 This is a graph showing the distribution of bulk modulus and shear modulus of dry rock as a function of the average aspect ratio of soft and hard pores. Figure 2 (a) is a graph showing the effect of the aspect ratio of soft pores on the bulk modulus of dry rock. Figure 2 (b) is a graph showing the effect of the aspect ratio of soft pores on the shear modulus of dry rock. Figure 2 (c) is a graph showing the effect of the aspect ratio of hard pores on the bulk modulus of dry rock. Figure 2 (d) is a graph showing the effect of the aspect ratio of hard pores on the shear modulus of dry rock.

[0020] Figure 3 This is a graph showing the variation of dry rock bulk modulus and shear modulus with total porosity.

[0021] Figure 4 This is a graph showing the variation of bulk modulus with total porosity in deep saturated sandstone in saline aquifers under high and low frequency limits.

[0022] Figure 5 This is a graph showing the variation of shear modulus with total porosity in deep saturated sandstone in a saline aquifer under high-frequency limits.

[0023] Figure 6The left figure shows the variation of bulk modulus and shear modulus of dry rock with soft pore density; Figure 6 The right figure shows the variation of saturated rock bulk modulus and shear modulus with soft pore density at high and low frequencies.

[0024] Figure 7 This is a graph showing the change in bulk modulus of deep sandstone in a saline aquifer with increasing axial and confining pressure. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The embodiments described in this invention are only some, not all, of the embodiments. Based on the described embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention. In this embodiment, the saline sandstone layer in the study area is at a depth of more than 800m. The mass of the sandstone sample collected is 62.715g, the diameter is 25mm, the length is 50mm, and the bulk density is 2.56g / cc.

[0026] Specifically, the following steps are included: S10: Classify sandstone pore structures in deep saline aquifers and establish probability distribution models for different types of pores; S101: Pores are classified into two categories: soft pores (microfractures, intergranular fissures) and hard pores (intergranular pores, solution pores). Due to formation compaction in deeper saline layers, the aspect ratio distribution of pores is more concentrated than in shallower layers. The distribution of pores in the rock is considered a continuous random distribution, and is a bounded closed set. Based on the extreme value theorem... exist There must exist a maximum value point on the above. , so that: in, Let be the aspect ratio of the pores, and D be the domain of the function, a closed interval.

[0027] Based on the finding of at least two maximum aspect ratio distributions through probability statistics (i.e., the concentrated distribution range of aspect ratios for soft and hard pores), a bimodal pore aspect ratio model with a truncated Gaussian distribution is established to represent the aspect ratio of soft pores. Aspect Ratio of Hard Pores They are modeled as truncated Gaussian distributions respectively. : 3) in These represent the average aspect ratio distributions of soft and hard pores, respectively. The standard deviations of the aspect ratio distributions for soft and hard pores are respectively; S102: Will The combination of these two factors can be represented as the aspect ratio distribution of the pores throughout the rock, such as... Figure 1 As shown, this is a truncated Gaussian bimodal distribution. (4); S20: Measurement of longitudinal wave velocity in sandstone via ultrasonic testing transverse wave velocity The bulk modulus and shear modulus of deep sandstone in the saline aquifer were calculated; the mineral composition of the deep sandstone in the saline aquifer was determined by XRD, and the equivalent bulk modulus of the mineral matrix was calculated using the Voigt-Reuss-Hill average. and shear modulus ; S201: Measurement of longitudinal wave velocity in sandstone mass via ultrasonic testing transverse wave velocity Calculate the bulk modulus and shear modulus of deep sandstone in the saline aquifer: The ultrasonic experiment involves placing a rock sample between two ultrasonic probes of an anisotropic ultrasonic testing instrument. One probe emits ultrasonic waves, while the other probe receives the waves after they pass through the rock. The anisotropic ultrasonic testing instrument is then placed in a triaxial pressure chamber, which is a steel container with axial pressure applied at both ends and confining pressure applied on the sides, for measurement. The arrival time and waveform of the waves are analyzed, and the longitudinal and transverse wave velocities of the isotropic rock are calculated. Table 1. Simulated transverse and longitudinal wave velocities of saline sandstone in a study area under different stress depths. In Table 1, the vertical stress caused by the axial pressure at the formation depth is calculated using the empirical formula for the average pressure gradient. The vertical stress caused by the overlying rock strata The depth of the strata; As shown in Table 1, under a burial depth of 800m, the vertical stress (axial pressure) calculated according to the above formula (5) is 15-20MPa; the confining pressure is the pressure that the surrounding rocks exert uniformly on the rock. Referring to the geostress characteristics of deep saline aquifers in similar sedimentary basins, the minimum horizontal stress (confining pressure) is usually slightly lower than the vertical stress, with a difference of about 5MPa. Therefore, the confining pressure is set at 10-15MPa in this experiment. The saline aquifer sandstone section in the research area of ​​this patent is deeper than 800m, and the sandstone samples collected are near 800m. According to the above formula, the actual longitudinal and transverse wave velocities correspond to an axial pressure of 15-20MPa, and the confining pressure is the measured value within the range of 10-15MPa. The bulk modulus of deep sandstone in the saline aquifer was calculated based on the measured transverse and longitudinal waves of the sandstone. and shear modulus : ; Among them, volume density It is obtained from the mass / volume of the sample, i.e. g / cc; S202: The mineral composition of deep sandstone in a saline aquifer in a certain study area was determined by X-ray diffraction (XRD). The volume fraction of quartz was 60.1%, feldspar was 20.8%, and clay was 19.1%. S203: Calculate the equivalent bulk modulus of the mineral matrix using the Voigt-Reuss-Hill average. and shear modulus : Based on the mineral component content determined in step S202 and the corresponding mineral bulk modulus and shear modulus obtained from the table, the bulk modulus of the deep sandstone matrix in the saline aquifer is calculated by substituting it into the Voigt-Reuss-Hill model. and shear modulus ; Table 2 shows the mineral composition and mineral modulus values ​​of saline sandstone in a certain study area. Substituting the values ​​into the Voigt-Reuss-Hill model, the bulk modulus of the deep sandstone mineral matrix in the saline aquifer was calculated. and shear modulus ; This represents the upper limit of the bulk modulus of the mineral components. Lower limit of bulk modulus of mineral components This represents the upper limit of the shear modulus of the mineral component. This represents the lower limit of the shear modulus of the mineral components. , Let J be the bulk modulus and shear modulus of the j-th component. Let be the volume fraction of the i-th component; S30: Establish a petrological model of deep sandstone in a saline aquifer with multi-peak pore distribution, and calculate the pore shape factor and polarization tensor; S301, based on Eshelby's inclusion theory and within the Mori-Tanaka framework, addresses each aspect ratio covered by the bimodal distribution in S10. Calculate their shape factors respectively. and the corresponding polarization tensor in The polarization factor is expressed as a formula for bulk modulus and shear modulus. , This represents the corresponding positional component within the fourth-order tensor. Since the pore morphology discussed in this model is ellipsoidal, in the x1-x2-x3 coordinate system, the two minor axes of the ellipsoid lie on x1 and x2 respectively, and... = That is, the tensor we are looking for is a symmetric matrix. = Only the upper half of the matrix needs to be derived; in For integrals related to aspect ratio: obtain polarization factor , The aspect ratio can be expressed as: Among them, shape factor It can be represented as: in The base Poisson's ratio can be expressed as: The equivalent bulk modulus of the base. The base is the equivalent shear modulus.

[0028] S40: Construct a model containing a hard-porous matrix, using the mineral matrix obtained in S20 as the substrate, and treating all pores within the single-peak distribution of hard pores in S10 as inclusions; considering the contribution of hard pores of all aspect ratios to the modulus, calculate the equivalent medium bulk modulus containing only dry hard pores. and shear modulus The formula is expressed as follows: in, The bulk modulus of dry rock containing only hard pores. To calculate the polarization factor of the bulk modulus of hard pores, For hard porosity fraction, The equivalent bulk modulus of the base. The bulk modulus of the hard hole; in, The shear modulus of dry rock containing only hard pores. The polarization factor for the hard pore shear modulus. The base equivalent shear modulus, For the shear modulus of a hard hole; Will and Substituting the aspect ratio Gaussian distribution of hard pores : The formula for bulk modulus can be expressed as: The formula for shear modulus can be expressed as: ; S50: Construct a dry rock model with bimodal pore distribution and calculate the equivalent bulk modulus of the dry rock containing both soft and hard pore distributions. and shear modulus ; S501: Obtained from S40 As a new substrate, the aspect ratio Gaussian distribution of soft pores in S10 is substituted. Calculate the equivalent bulk modulus of dry rock containing both soft and hard pore distributions. and shear modulus ; in, For soft pores, aspect ratio To calculate the polarization factor for the bulk modulus and shear modulus of soft pores, These are the bulk modulus and shear modulus of the soft pores.

[0029] Figure 2 This is a graph showing the distribution of bulk modulus and shear modulus of dry rock as a function of the average aspect ratio of soft and hard pores. Figure 2 (a) is a graph showing the effect of the aspect ratio of soft pores on the bulk modulus of dry rock. Figure 2 (b) is a graph showing the effect of the aspect ratio of soft pores on the shear modulus of dry rock. Figure 2 (c) is a graph showing the effect of the aspect ratio of hard pores on the bulk modulus of dry rock. Figure 2(d) is a graph showing the effect of the aspect ratio of hard pores on the shear modulus of dry rock. As can be seen from the graph, as the aspect ratio of soft pores increases, the bulk modulus of dry rock decreases monotonically. The shear modulus first increases with the aspect ratio of soft pores, reaches its maximum peak value near 0.046, and then shows a monotonically decreasing trend. As the aspect ratio of hard pores increases, the bulk modulus and shear modulus of dry rock decrease monotonically, and the rate of decrease gradually increases. The changes in bulk modulus and shear modulus of dry rock with porosity are as follows: Figure 3 As shown, under low porosity conditions, the relationship can be approximated as linear. With constant porosity, the bulk modulus of dry rock decreases monotonically with the increase of the average aspect ratio of soft pores, and the decay becomes positive. The shear modulus of dry rock decreases monotonically with the increase of the average aspect ratio of soft pores, and the decay becomes positive.

[0030] S60: Calculate the deep pore fluid modulus of the saline aquifer based on the Batzle-Wang fluid model. S601: Calculate the pore fluid density of pure water , can be represented as: in, For ground temperature, ; Formation pressure, MPa; S602: The density of pore fluid in a saline aquifer can be calculated as follows: in, Density of saline water, g / cc; Density of pure water, g / cc; Mineralization; For temperature, ; Formation pressure, MPa; S603: Calculation of the bulk modulus of pore fluid in a saline aquifer From the definition of elastic modulus, we can obtain: in, The velocity of sound in salt water is m / s; It can be represented as: in, For matrix coefficients, These are the empirical parameter table values ​​for the Batzle-Wang model.

[0031] S70: Establishing saturated rock models under high and low frequency limits S701: Obtain saline-saturated pores by fluid substitution of pores (the bulk modulus of saline water) (Shear modulus is 0) Under the high-frequency limit, the pore fluid is in a non-relaxed state and can be regarded as an inclusion; repeat steps S40 and S50: Using the mineral matrix in S20 as a substrate, fluid-saturated unimodal hard pores were added to calculate the rock modulus containing saline-water-saturated hard pores. Then, using the above medium as a substrate, fluid-saturated soft pores were added to obtain the saturated rock bulk modulus under the high-frequency limit. and shear modulus To simulate the elastic response of pore fluids in a non-relaxed framework.

[0032] S702: Calculate the saturated rock modulus under the low-frequency limit: Calculate the dry rock modulus obtained in S50. and By applying the Gassmann equation for fluid substitution, the bulk modulus of saturated rock under the low-frequency limit was obtained. : in, Total porosity The bulk modulus of the mineral matrix. The bulk modulus of saline water and the shear modulus in the low-frequency limit. .

[0033] In particular, in actual implementation, the parameters of the MoriTanaka model need to be set according to the actual geological conditions of the study area. The parameters used in this embodiment are shown in Table 3. Table 3 Input parameters of the Mori-Tanaka model Figure 4 This graph shows the variation of bulk modulus with total porosity in deep saturated sandstone within a saline aquifer in a certain study area at high and low frequency limits. At high frequencies, the model is in a non-relaxed state, with no fluid flow between pores and some support provided to the pores. The bulk modulus is significantly higher than that at low frequencies. In this case, the pores are isolated from each other, and the bulk modulus decreases monotonically with increasing porosity. The low-frequency curve indicates that the model is in a relaxed state overall, with the fluid within the pores flowing completely to equilibrium under external stress. The bulk modulus decreases monotonically with increasing porosity, and the rate of change is greater than that of the high-frequency modulus.

[0034] Figure 5 This is a graph showing the variation of shear modulus with porosity in deep saturated sandstone in a saline aquifer within a certain study area at high frequency limits. The saturated shear modulus decreases with increasing porosity at high frequencies. At low frequencies, the fluid cannot transmit shear stress, and the model relaxes under low-frequency conditions. The shear modulus of saturated rock is consistent with that of dry rock.

[0035] Figure 6 The left figure shows the variation of dry rock modulus with soft pore density. The bulk modulus and shear modulus of dry rock decrease monotonically as the soft pore density increases. Figure 6 The right figure shows the variation of bulk modulus and shear modulus of saturated rock with soft pore density at high and low frequencies. At both high and low frequency limits, the bulk modulus and shear modulus of saturated rock decrease monotonically with increasing soft pore density. Pore fluid only affects volumetric deformation and does not affect shear deformation, so its contribution to shear modulus is zero. At the low frequency limit, the fluid flows fully, and the shear modulus of high-frequency saturated rock coincides with the shear modulus of low-frequency saturated rock as a function of soft pore volume fraction.

[0036] Figure 7 The graph shows the change in bulk modulus of deep sandstone samples from saline aquifers with increasing axial pressure (blue) and confining pressure (red). Under low pressure, the bulk modulus increases significantly with increasing pressure. After the axial pressure reaches 25 MPa and the confining pressure reaches 20 MPa, the rate of change of the curve decreases rapidly, showing a convergence trend. This process can be divided into two stages. Before the axial pressure reaches 25 MPa, the soft pores inside the rock sample gradually close under the action of external stress, and the bulk modulus of the rock sample increases due to the reduced average influence of soft pores on the background modulus. After the axial pressure reaches 25 MPa, the soft pores inside the rock sample are basically closed, and the hard pores, due to their aspect ratio being close to 1, provide some support to the external stress, and the overall modulus change tends to stabilize.

[0037] Will Figure 7 Comparison with model prediction results ( Figures 2-6 Deep sandstone layers in saline aquifers often exhibit low porosity and a bimodal distribution of aspect ratio.

[0038] The sandstone used in this example is buried at a depth of more than 800m, and the effective stress it experiences can reach 15-20MPa according to laboratory simulation. The measured bulk modulus of 33.053-34.384 is highly consistent with the model's predicted low porosity value of 32.73-33.25.

[0039] The calculation results from the examples demonstrate that the method of this invention has significant advantages in calculating the modulus of deep sandstone in saline aquifers, especially suitable for situations where the formation stress in deep saline aquifers is high and the internal pore aspect ratio distribution of the rock is concentrated. Traditional MT models only consider single inclusions with specific pore shapes, and the variation in pore aspect ratio can only be obtained through numerical integration of empirical formulas, resulting in complex calculations and failing to fully characterize the internal pore structure of the rock. This method improves the model's realism in characterizing pore structure by establishing a probability model of pore aspect ratio distribution. While representing the distribution of soft and hard pores, it also derives an explicit solution for the model modulus with respect to pore aspect ratio under the Mori-Tanaka framework using the Eshelby tensor, improving the model's computational efficiency and parameter correlation. Furthermore, it establishes a flattened elliptical pore inclusion model for all aspect ratios, enabling accurate calculation of the dry rock modulus and the saturated rock modulus under high and low frequency limiting conditions.

[0040] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention, and within the spirit and principles of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for predicting the elastic modulus based on an MT frame and a bimodal pore aspect ratio distribution, characterized in that, Specifically, the steps include the following: S10: Based on the pore structure of deep sandstone in the saline aquifer, the pores are divided into soft pores and hard pores. Probability distribution models for soft pores and hard pores are established separately, and then unified into a bimodal distribution model. S20: Measurement of longitudinal wave velocity in sandstone via ultrasonic testing transverse wave velocity The bulk modulus and shear modulus of deep sandstone in saline aquifers were calculated. The mineral composition of deep sandstone in saline aquifers was determined by XRD. The equivalent bulk modulus of the mineral matrix was calculated using Voigt-Reuss-Hill averaging. and shear modulus ; S30: Establish a petrological model of deep sandstone in a saline aquifer with multi-peak pore distribution, and calculate the pore shape factor and polarization tensor; S40: Construct a model containing a hard-porous matrix, using the mineral matrix obtained in S20 as the substrate, and treating all pores within the single-peak distribution of hard pores in S10 as inclusions; considering the contribution of hard pores of all aspect ratios to the modulus, calculate the equivalent medium bulk modulus containing only dry hard pores. and shear modulus ; S50: Construct a dry rock model with bimodal aspect ratio pores and calculate the equivalent bulk modulus and shear modulus of the dry rock containing both soft and hard pores. S60: Calculate the deep pore fluid modulus of the saline aquifer based on the Batzle-Wang fluid model; S70: Establish saturated rock models under high and low frequency limits.

2. The method for predicting the elastic modulus based on the MT frame and bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S10, the probability distribution models for soft pores and hard pores are established separately, and then unified into a bimodal distribution model, including the following steps: S101: Due to formation compaction, the porosity distribution in the deeper saline layer is more concentrated than in the shallower layers. Considering the distribution of pores in the rock as a continuous random distribution and a bounded closed set, according to the extreme value theorem... exist There must exist a maximum value point on the above. , so that: in, Let D be the aspect ratio of the pores, and D be the domain of the function, which is a closed interval. Based on the probability statistics that can find the maximum aspect ratio distribution of soft and hard pores, a bimodal pore aspect ratio model with a truncated Gaussian distribution is established to represent the aspect ratio of soft pores. Aspect Ratio of Hard Pores They are modeled as truncated Gaussian distributions respectively. : in These represent the average aspect ratio distributions of soft and hard pores, respectively. The standard deviations of the aspect ratio distributions for soft and hard pores are respectively; S102: Will The combination of these two factors can be represented as the aspect ratio distribution of the entire rock's pores, i.e., a truncated Gaussian bimodal distribution. 。 3. The method for predicting the elastic modulus based on the MT frame and bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S20, the longitudinal wave velocity of sandstone is measured by ultrasonic experiment. transverse wave velocity The bulk modulus and shear modulus of deep sandstone in the saline aquifer were calculated; the mineral composition of the deep sandstone in the saline aquifer was determined by XRD, and the equivalent bulk modulus of the mineral matrix was calculated using the Voigt-Reuss-Hill average. and shear modulus Specifically, it includes the following steps: S201: Measurement of longitudinal wave velocity in sandstone mass via ultrasonic testing transverse wave velocity : The ultrasonic experiment involves placing a rock sample between two ultrasonic probes of an anisotropic ultrasonic testing instrument. One probe emits ultrasonic waves, while the other probe receives the waves after they pass through the rock. The anisotropic ultrasonic testing instrument is then placed in a triaxial pressure chamber, which is a steel container with axial pressure applied at both ends and confining pressure applied on the sides, for measurement. The arrival time and waveform of the waves are analyzed, and the longitudinal and transverse wave velocities of the isotropic rock are calculated. The bulk modulus of the deep sandstone in the saline aquifer was calculated based on the measured transverse and longitudinal wave characteristics and bulk density of the sandstone. and shear modulus : Among them, volume density It is obtained from the mass / volume of the sample, i.e. g / cc; S202: The mineral composition of deep sandstone in the saline aquifer was determined by XRD, and the equivalent bulk modulus of the mineral matrix was calculated by combining the Voigt-Reuss-Hill average. and shear modulus : Based on the mineral component content determined by XRD and the corresponding mineral bulk modulus and shear modulus obtained from tables, the bulk modulus of the deep sandstone matrix in the saline aquifer is calculated using the Voigt-Reuss-Hill model. and shear modulus ; This represents the upper limit of the bulk modulus of the mineral components. Lower limit of bulk modulus of mineral components This represents the upper limit of the shear modulus of the mineral component. This represents the lower limit of the shear modulus of the mineral components. , Let J be the bulk modulus and shear modulus of the j-th component. Let be the volume fraction of the i-th component.

4. The method for predicting the elastic modulus based on the MT frame and bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S30, the specific process of establishing a petrological model of deep sandstone in a saline aquifer with multi-peak pore distribution and calculating the pore shape factor and polarization tensor is as follows: Based on Eshelby's inclusion theory, within the Mori-Tanaka framework, this study addresses each aspect ratio covered by the bimodal distribution in S10. Calculate their shape factors respectively. and the corresponding polarization tensor in The polarization factor is expressed as a formula for bulk modulus and shear modulus. , This represents the corresponding positional component within the fourth-order tensor. Since the pore morphology discussed in this model is ellipsoidal, in the x1-x2-x3 coordinate system, the two minor axes of the ellipsoid lie on x1 and x2 respectively, and... = That is, the tensor we are looking for is a symmetric matrix. = Only the upper half of the matrix needs to be derived; in For integrals related to aspect ratio: obtain polarization factor , The aspect ratio can be expressed as: ； Among them, shape factor It can be represented as: in The base Poisson's ratio can be expressed as: The equivalent bulk modulus of the base. It is the equivalent shear modulus of the base.

5. The method for predicting the elastic modulus based on the MT frame and bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S40, a hard-porous matrix model is constructed, using the mineral matrix obtained in S20 as the substrate, and all pores within the single-peak distribution of hard pores in S10 are considered as inclusions; considering the contribution of hard pores of all aspect ratios to the modulus, the equivalent medium bulk modulus containing only dry hard pores is calculated. and shear modulus The process is as follows: in, The bulk modulus of dry rock containing only hard pores. To calculate the polarization factor of the bulk modulus of hard pores, For hard porosity fraction, The equivalent bulk modulus of the base. The bulk modulus of the hard hole; in, The shear modulus of dry rock containing only hard pores. The polarization factor for the hard pore shear modulus. The base equivalent shear modulus, For the shear modulus of a hard hole; Will and Substituting the aspect ratio Gaussian distribution of hard pores : The formula for bulk modulus can be expressed as: The formula for shear modulus can be expressed as: 。 6. The method for predicting the elastic modulus based on an MT frame and a bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S50, a dry rock model with bimodal aspect ratio pores is constructed, and the equivalent bulk modulus and shear modulus of the dry rock containing both soft and hard pores are calculated. The specific steps are as follows: S40 obtained As a new substrate, the aspect ratio Gaussian distribution of soft pores in S10 is substituted. Calculate the equivalent bulk modulus of dry rock containing both soft and hard pore distributions. and shear modulus ; in, For soft pores, aspect ratio To calculate the polarization factor for the bulk modulus and shear modulus of soft pores, These are the bulk modulus and shear modulus of the soft pores.

7. The method for predicting the elastic modulus based on an MT frame and a bimodal pore aspect ratio distribution according to claim 1, characterized in that, In step S60, the deep pore fluid modulus of the saline aquifer is calculated based on the Batzle-Wang fluid model, including the following steps: S601: Calculate the pore fluid density of pure water , can be represented as: in, For ground temperature, ; Formation pressure, MPa; S602: The density of pore fluid in a saline aquifer can be calculated as follows: in, Density of saline water, g / cc; Density of pure water, g / cc; Mineralization; For temperature, ; Formation pressure, MPa; S603: Calculation of the bulk modulus of pore fluid in a saline aquifer From the definition of elastic modulus, we can obtain: in, The velocity of sound in salt water is m / s; It can be represented as: in, For matrix coefficients, These are the empirical parameter table values ​​for the Batzle-Wang model.

8. The method for predicting the elastic modulus based on the MT frame and bimodal pore aspect ratio distribution according to claim 1, characterized in that, Step S70, establishing the saturated rock model under high and low frequency limits, specifically includes the following steps: S701: Fluid substitution of pores yields saline-saturated pores, wherein the bulk modulus of saline water is... With a shear modulus of 0, the pore fluid is in a non-relaxed state under the high-frequency limit and can be considered as an inclusion; repeat steps S40 and S50: Using the mineral matrix in S20 as a substrate, fluid-saturated unimodal hard pores were added to calculate the rock modulus containing saline-water-saturated hard pores. Then, using the above medium as a substrate, fluid-saturated soft pores were added to obtain the saturated rock bulk modulus under the high-frequency limit. and shear modulus Simulate the elastic response of pore fluids in a non-relaxed framework; S702: Calculate the saturated rock modulus under the low-frequency limit: Calculate the dry rock modulus obtained in S50. and By applying the Gassmann equation for fluid substitution, the bulk modulus of saturated rock under the low-frequency limit was obtained. : in, Total porosity The bulk modulus of the mineral matrix. The bulk modulus of saline water and the shear modulus in the low-frequency limit. .

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