Calculation method of nanoporous shale permeability considering transient water seepage

Abstract

The present invention provides a method for calculating the permeability of nanoporous shale that considers transient water seepage. Using a horizontally placed single capillary model with constant cross-section and ignoring the effects of gravity, the Navier-Stokes equations in a cylindrical coordinate system are simplified and solved to obtain a single capillary fluid velocity distribution model that considers transient flow and slippage length. A modified single-tube velocity model that considers transient flow and microscale effects is then obtained. In Laplace space, the capillary bundle model's water volume flow rate, porosity, cross-sectional area, and corresponding parameters in Darcy's law are set equal, and then simplified to obtain a porous media permeability model that considers transient flow, microscale effects, and pore size distribution. This method can describe water flow behavior over an entire time period, reflecting the temporal changes in water flow velocity, flow enhancement factor, and apparent liquid permeability.

Description

Technical Field

[0001] The present invention provides a method for calculating the permeability of nanoporous medium shale taking into account the transient seepage of water. Background Art

[0002] Apparent liquid permeability is a key parameter describing the flow of liquids in porous media. With the rapid development of shale oil in recent years, apparent liquid permeability has received significant attention as a crucial parameter in shale oil development. However, shales contain a high content of clay minerals, which swell when exposed to water, making experimental measurement of apparent liquid permeability difficult. Currently, theoretical calculations are mostly based on steady-state HP (Hagen-Poiseuille) laminar flow. However, any fluid flow is a transient process, with flow velocity being a function of time. Under constant velocity or constant pressure boundary conditions, the flow velocity ceases to vary with time only after a period of flow. Therefore, using steady-state HP theory to describe fluid flow neglects the flow behavior from the initial state to the steady state. However, transient HP theory effectively addresses this issue, encompassing both the period from the initial state to the steady state, when the flow velocity varies with time, and the period during the steady state, when the flow velocity remains constant. To reveal the transient flow of water in nanoporous shales, a model for the transient apparent liquid permeability (ALP) of water in mixed-wet porous media was developed by combining transient theory with the mechanism of water transport in nanoscale pores. This model considers both the transient flow characteristics of water at the nanoscale and the pore structure of shale porous media, and is an important supplement to the steady-state apparent liquid permeability model.

[0003] For any flow, its velocity is a transient process that changes continuously with time. In the pores of nano-scale porous media, it is worth exploring and studying how microscale effects (apparent viscosity and true slip length) affect the unsteady phase of the fluid. The study of transient HP flow is usually based on horizontal circular tubes. The most common boundary condition is that the fluid is initially stationary, and suddenly a constant pressure is applied, and the fluid begins to move. For such a flow process, the horizontal velocity derivative in the momentum equation cannot be ignored, which also brings difficulties to the solution of the momentum equation. Papanastasiou et al. solved the above momentum equation for the first time by the separation of variables method, and obtained an analytical solution to the transient HP flow velocity distribution, which provided a strong foundation for transient circular tube flow. Many researchers subsequently expanded and improved the transient HP equations based on this foundation, primarily developing transient velocity solutions for binary fluid mixtures, transient velocity solutions for moving boundaries, transient velocity solutions for second-order fluids, transient series velocity solutions for generalized Oldroyd-B fluids, transient velocity solutions for Oldroyd-B fluids in unbounded regions, transient velocity solutions for time-varying internal surface shear stress, and transient velocity solutions for periodic constant pressure boundaries. These studies laid a solid foundation for the development of transient flow in single tubes.

[0004] Water in nanotubes is a confined fluid, and its properties differ significantly from those of free fluids. In the free state, fluid parameters such as density and viscosity are simply functions of temperature and pressure, but under confinement, the fluid characteristics undergo dramatic changes. Experimental studies of carbon nanotube flow, molecular dynamics simulations, and LBM simulations have shown that as the diameter of the carbon nanotube increases, the configuration of water molecules in the nanotube gradually transitions from a single-file arrangement to an intermediate ordered arrangement, and finally to a configuration similar to that in the free state. This configuration causes the viscosity and density of the fluid near the wall to differ from those of the free fluid, resulting in radial inhomogeneity of fluid properties in the nanotube. Furthermore, this configuration significantly affects the slippage length of the wall fluid. To address this phenomenon, many researchers have proposed flow enhancement models that combine the effects of wall fluid viscosity and slippage length. These models consider the influence of tube diameter, wall fluid thickness, viscosity, and slippage length on the flow enhancement model. However, the viscosity and slip length of the fluid on the wall are closely related to the hydrophilicity or hydrophobicity of the pore wall. The aforementioned flow enhancement models do not address the relationship between these two factors and the wettability of the wall. To address this issue, Mattia et al. established a relationship between the flow enhancement factor and the liquid-wall interaction, surface diffusion, and contact angle. Wu et al. systematically studied the variations in the viscosity and slip length of water near surfaces of varying wettability. By analyzing a large amount of experimental and molecular simulation data, they established a relationship between the static contact angle and the viscosity and slip length of the water on the wall. This study preliminarily resolved the influence of wettability on the flow enhancement factor and further clarified the controversial issue of flow increase or decrease in molecular simulations and experiments. Other researchers have modified the flow enhancement factor to account for surface wettability. For example, Cui et al. modified the Mattia flow enhancement factor to account for mixed wettability, asphaltene, and viscosity enhancement, and applied it to oil transport. Fan also modified the Mattia flow enhancement model to account for roughness, studying the effect of the relative roughness of nanotubes on water flow. Wang modified Wu's viscosity and slip length models, further accounting for water flow in elliptical nanotubes, and established a velocity model and flow enhancement factor model for elliptical nanotubes. Research in this area is continuing to be reported, and our understanding of fluid flow in nanotubes is constantly improving.

[0005] Based on the research on the flow enhancement factor of single nanotube fluids, many scholars are also paying attention to the seepage of fluids in nanoscale porous media. Most of the research in this area is based on the nanocapillary bundle model, which usually equates the pore network in the porous medium to a parallel capillary bundle. By considering the pore size distribution in the porous medium, the flow of fluid in a single nanotube is upgraded to the seepage of fluid in the porous medium, and the apparent liquid permeability calculation model (APL) is obtained. However, due to the extremely complex pore network in the porous medium, the parallel capillary bundle model cannot well represent the complex pore network of the porous medium. To solve this problem, fractal geometry methods are applied to upgrade the flow of fluid in a single nanotube to the seepage of fluid in porous media. This method can take into account the effects of pore size distribution, pore throat tortuosity, and pore wall plasticity, and has achieved good practical results.

[0006] The single-tube flow model, single-tube velocity model, and shale apparent liquid permeability model proposed in existing technical solutions are mainly based on the steady-state HP theory. Therefore, the established models cannot reflect the changes in fluid velocity, fluid flow, and apparent liquid permeability over time.

[0007] The flow of fluid in a single nanotube is affected by microscale effects. Existing technical solutions fail to reflect the impact of microscale effects on the fluid velocity and flow rate in the unstable stage, and cannot consider the impact of microscale effects on the velocity and flow stabilization time.

[0008] The apparent liquid permeability model established by the existing technical solution cannot consider the influence of the pore size distribution on the apparent liquid permeability in the unstable stage, and cannot consider the influence of the pore size on the apparent permeability stabilization time. Summary of the Invention

[0009] The present invention aims to solve the above problems and proposes a permeability calculation method based on transient HP theory, which takes transient flow, microscale effects and pore size distribution into consideration.

[0010] The technical solution of the present invention is:

[0011] A method for calculating the permeability of nanoporous shale considering transient water seepage is as follows:

[0012] Transient flow is common in any flow, and the fluid velocity is a function of time. For constant velocity or constant pressure boundary conditions, in the initial stage of flow, the fluid velocity changes continuously with time. As time goes by, the fluid velocity gradually reaches stability, at which time the time increases but the fluid velocity no longer changes. Permeability is generally measured by applying constant pressure boundary conditions at both ends of the core. In order to derive the apparent liquid permeability model, the flow of water in a single tube under constant pressure boundary conditions must be considered first. In the process of fluid flow under constant pressure boundary, due to the influence of the boundary layer, the fluid velocity profile gradually transitions from a slug type to a parabolic type, and then reaches stability, and the fluid flow becomes a fully developed stable flow, such as Figure 1 .

[0013] Step 1: To describe the transient flow of fluid in the capillary, a horizontally placed single capillary model with constant cross-section and ignoring the effect of gravity is used, and the pressure gradient is constant. After considering the above factors, the NS equations in the cylindrical coordinate system are simplified to the following form:

[0014]

[0015] Where: u is the fluid velocity in the horizontal direction, m / s;

[0016] r is the radial distance from the center of the tube, m;

[0017] ρ is the fluid density, kg / m 3 ;

[0018] μ is the fluid viscosity, Pa·s;

[0019] Δp is the pressure difference, Pa;

[0020] L is the length of the tube, m;

[0021] Solve equation (1) and consider the slip boundary conditions, introduce the following initial boundary conditions

[0022]

[0023] Where: λ is the slip length, m;

[0024] R is the radius of the tube, m;

[0025] t is time, s;

[0026] Taking Laplace transformation of equations (1) and (2) with respect to t, we can obtain the Bessel function of the imaginary quantity. Under the initial and boundary conditions, we can obtain the solution of the equation as follows:

[0027]

[0028] Where: is a single capillary fluid velocity distribution model considering transient flow and slip length, m / s 2 ;

[0029] s is the Laplace space variable, unit, dimensionless;

[0030] I0 is the 0th order first kind imaginary quantity Bessel function, I1 is the 1st order first kind imaginary quantity Bessel function;

[0031] Equation (3) is a single capillary fluid velocity distribution model that takes into account transient flow and slippage length.

[0032] Step 2: The flow of water in shale porous media has many peculiarities. Since the pores in shale porous media are mostly at the nanometer level, and the hydrophilicity of organic kerogen and inorganic clay minerals in the shale matrix varies greatly, the interaction forces between water molecules and different pore walls vary greatly. For hydrophilic pore walls, there is a strong interaction force between water and the pore wall, causing the viscosity of water molecules near the pore wall to be much higher than the viscosity of free water in the middle of the pore, and the slippage between water molecules near the pore wall and the pore wall is small. For hydrophobic walls, the interaction force between water and the pore wall is weak, causing the viscosity of water molecules near the pore wall to be lower than the viscosity of free water in the middle of the pore, and the slippage between water molecules near the pore wall and the pore wall is large. The single-tube fluid slippage length proposed by Wu, which takes into account the difference in wettability of the tube wall, can well describe the effect of walls with different wettability on fluids.

[0033]

[0034] Where: l st is the slip length considering the wettability of the tube wall, m; θ is the contact angle, degrees;

[0035] The water viscosity in a single tube is a function of the tube diameter and can be characterized by weighting the cross-sectional area by the wall water viscosity and the free water viscosity.

[0036]

[0037] Where: μ R is the effective viscosity, Pa·s;

[0038] μ w is the viscosity of the wall water in Pa·s; μ b is the viscosity of free water, Pa·s;

[0039] A i is the cross-sectional area of the wall area, m2; A t is the total cross-sectional area of the pipe diameter;

[0040] In order to consider the effect of wettability on the viscosity of wall water, Wu proposed the relationship between wall water and free water:

[0041]

[0042] Viscosity and slip corrections are performed on Equation (3) using Equations (4) to (6) (using μ R Replace μ, l st Replace λ) to obtain a single-tube velocity model that considers transient flow and microscale effects;

[0043]

[0044] Where: It is a modified single tube velocity model considering transient flow and microscale effects.

[0045] Step 3:

[0046] Integrating equation (7) with respect to r from 0-R, we can obtain the single-pipe water volume flow rate considering transient flow and microscale effects:

[0047]

[0048] Where: is the single pipe water volume flow rate considering transient flow and microscale effects, m 3 / s;

[0049] Comparing it with the steady-state no-slip HP model in Laplace space, the flow enhancement factor considering transient flow and microscale effects is obtained as

[0050]

[0051] Where: ε is the flow enhancement factor considering transient flow and microscale effects, dimensionless;

[0052] is the slip-free Hagen-Poiseuille volume flow rate, m 3 .

[0053] Step 4: Expand the fluid flow in a single tube to a porous medium, use the non-equal diameter capillary bundle model to equate the pore network of the porous medium, and use the pore size distribution frequency function of the porous medium to describe the heterogeneous pore size distribution of the nano-scale porous medium;

[0054]

[0055] Where: f is the pore size distribution frequency, dimensionless; σ is the deviation coefficient, dimensionless; ω is the mean expectation, m;

[0056] Multiplying equation (8) by f(R) and integrating R from the smallest pore size to the largest pore size yields the water volume flow rate in the porous medium: for

[0057]

[0058] Where: is the water volume flow rate of the capillary bundle model;

[0059] N is the number of capillaries, dimensionless;

[0060] R max is the maximum pore size of the porous medium, m; R min is the minimum pore size of the porous medium, m;

[0061] The relationship between the porosity of the capillary bundle model and the number of capillaries Substituting into equation (11), the simplified water volume flow rate of the capillary bundle model considering transient flow, microscale effect and pore size distribution in Laplace space is obtained: for

[0062]

[0063] Where: is the porosity of the capillary bundle model, dimensionless;

[0064] A is the cross-sectional area of the capillary bundle model, m 2 ;

[0065] In Laplace space, the water volume flow rate, porosity, cross-sectional area of the capillary bundle model and the corresponding parameters in Darcy's law are equal to obtain the following formula:

[0066]

[0067] After simplifying Equation (13), we can obtain the porous media permeability model considering transient flow, microscale effect and pore size distribution:

[0068]

[0069] Where: In order to consider the apparent liquid permeability model of transient flow, microscale effect and pore size distribution, the Laplace numerical inversion method and numerical integration method are used for calculation;

[0070] Among them, the analysis of formula (14) yields:

[0071] (1) When the pore size heterogeneity is not considered, the integral and f(R) in Equation (14) are removed, and Equation (14) degenerates into the apparent liquid permeability model considering transient flow and microscale effects:

[0072]

[0073] Where: An apparent liquid permeability model that takes transient and microscale effects into account;

[0074] (2) When the slip length is not considered, Equation (15) degenerates into the apparent liquid permeability model that considers transient flow and viscosity affected by the wettability of the microtube wall.

[0075]

[0076] Where: It is an apparent liquid permeability model that takes into account transient flow and apparent viscosity;

[0077] (3) When the effect of microtubule wettability on water viscosity is not considered, Equation (16) degenerates into an apparent liquid permeability model that only considers transient flow.

[0078]

[0079] Where: The apparent liquid permeability model considers transient flow;

[0080] (4) When the time variation is not considered or the time tends to infinity, Equation (17) degenerates into the intrinsic permeability model of the homogeneous capillary bundle:

[0081]

[0082] Where: k ins is the intrinsic permeability model of a homogeneous capillary bundle.

[0083] The technical effects of the present invention are:

[0084] (1) The proposed transient model can describe the flow behavior over the entire time period and reflect the temporal changes of flow velocity, flow enhancement factor, and apparent liquid permeability;

[0085] (2) The variation characteristics of the flow velocity and flow enhancement factor of confined water in nanoscale pores under the action of microscale effects during the entire flow stage (including transient and steady-state stages) were revealed, and the influence of microscale effects on the time it takes for confined water flow to reach steady-state was clarified;

[0086] (3) The influence of microscale effects and the PSD of porous media on APL during the entire flow stage was revealed, and the influence of the average pore size and standard deviation of PSD on the time it takes for APL to reach stability was clarified. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1: Schematic diagram of water flow in nanopores with hydrophilic pore walls, mixed wettability walls, and hydrophobic pore walls from the initial time to the stable time.

[0088] Figure 2 : Comparison of the velocity profile model in Eq. (7) with the Papanastasiou model and the HP model.

[0089] Figure 3 : Comparison between the single-tube velocity model (7) and the Qing Wang model considering transient flow and microscale effects.

[0090] Figure 4 : Comparison of the model prediction data in this invention with experimental and MD simulation data in the literature;

[0091] (a) Comparison of the flow stabilization time predicted by equation (8) with experimental and MD simulation data;

[0092] (b) Comparison of the flow enhancement factor predicted by the model (9) with the experimental and MD simulation data under steady state.

[0093] Figure 5 : The flow enhancement factor changes with pore radius and contact angle at different times;

[0094] (a) t = 1 × 10 -12 s. (b) t = 1 × 10 -10 s. (c)t=1×10 -8 s. (d)t=1×10 -6 s.

[0095] Figure 6 : Variation of stability time with pore radius at different contact angles.

[0096] Figure 7 : Changes of permeability enhancement factor at different times under different wettability of pore walls;

[0097] (a) Hydrophilic pore wall at a contact angle of 30°;

[0098] (b) Hydrophobic pore wall at a contact angle of 150°.

[0099] Figure 8 : Changes in pore radius distribution frequency under different cases;

[0100] (a) Fixed average pore size with increasing standard deviation;

[0101] (b) The standard deviation is fixed and the average pore size increases.

[0102] Figure 9 : Effect of PSD parameters on ALP under hydrophilic surface (contact angle of 30 degrees);

[0103] (a) Effect of average pore size on APL;

[0104] (b) Effect of standard deviation on APL.

[0105] Figure 10 : Effect of PSD parameters on ALP under hydrophobic surface (contact angle of 150 degrees);

[0106] (a) Effect of average pore size on APL;

[0107] (b) Effect of standard deviation on APL.

[0108] Figure 11 : Effect of PSD parameters on ALP stabilization time. DETAILED DESCRIPTION

[0109] Example 1 - Comparison of the single-tube velocity model (7) considering transient flow and microscale effects with the Papanastasiou model and the non-slip steady-state HP velocity model. The porous media permeability model (14) in the article considering transient flow, microscale effects and heterogeneous pore size distribution is established based on the capillary bundle model. Therefore, the accuracy of the velocity distribution and flow rate of water flow in a single capillary is the basis for the effectiveness of transient permeability. To this end, the single-tube velocity model (7) considering transient flow and microscale effects is first verified.

[0110] Equation (7) is compared with the analytical solution of incompressible transient HP flow in a circular tube under constant pressure boundary conditions proposed by Papanastasiou and the non-slip steady-state HP velocity model. Since the Papanastasiou model does not consider the microtube wall viscosity and slip, in order to make comparisons under the same conditions, the slip length l in Equation (7) is replaced by st Set to 0, viscosity is set to the viscosity of free water. Calculated data is density 1000kg / m 3 , viscosity is 0.001Pa·s, pore radius is 8×10 -9 m, and the pressure gradient is 1×10 11 pa / m; In the process of calculating the analytical solution of transient HP flow, the roots of the Bessel function of the first kind of imaginary quantity of order 0 are involved. These roots are expressed as Calculation results are shown in Figure 2 ; Four time points were selected to compare the calculation results of the two models. At each time point, the single tube velocity model (7) considering transient flow and microscale effects is highly consistent with the calculation results of the Papanastasiou transient velocity model. In addition, in the initial stage, that is, when the time is 1×10 -11s, the velocity distributions calculated by the above two models are both slug-type. As time goes on, the velocity distribution gradually changes from slug-type to parabola-type, which is consistent with the velocity distribution change within the inlet section of the circular pipe flow. Finally, at a time of 5×10 -11 s, the results of Equation (7) and the Papanastasiou model coincide with those of the non-slip HP model, indicating that at 5×10 -11 At the time point of s seconds, the unsteady flow turns into a steady flow. This is consistent with the results of the numerical solution of the NS equations. The above comparison shows that the single-tube velocity model (7) considering transient flow and microscale effects is accurate.

[0111] Example 2—Comparison of the single-tube velocity model (7) considering transient flow and microscale effects with the Qing Wang model

[0112] In Example 1, the slip length l in formula (7) is st = 0, and the viscosity is set to the viscosity of free water. Therefore, the correctness of Equation (7) under the microscale effect is not verified. In order to verify the accuracy of Equation (7) under the condition of considering the microscale effect, it is compared with the Qing Wang model. The calculated data are shown in Tables 1 and 2. Since Equation (7) is a transient model, and Qing Wang is based on steady state, Equation (7) is used at 9×10 -11 The calculation result of s (9×10 -11 s) and compared with the QingWang model. Figure 3 It can be seen that for the three different sets of calculated data, the degree of agreement between Equation (7) and the Qing Wang model is very high, indicating that Equation (7) is accurate in considering the single-tube velocity model of microscale effects.

[0113] Table 1 Qing Wang model calculation data

[0114] r(m) <![CDATA[r b (m)]]> <![CDATA[μ w( (Pa·s)]]> <![CDATA[μ b (Pa·s)]]> Δp / L (Pa· / m) <![CDATA[l st (m)]]> case1 <![CDATA[3.00×10 -9 ]]> <![CDATA[2.3×10 --9 ]]> 0.0011 0.001 <![CDATA[1×10 +11 ]]> <![CDATA[1.00×10 -9 ]]> case2 <![CDATA[7.00×10 -9 ]]> <![CDATA[6.30×10 -9 ]]> 0.0011 0.001 <![CDATA[1×10 +11 ]]> <![CDATA[2.00×10 -9 ]]> case 3 <![CDATA[1.00×10 -8 ]]> <![CDATA[9.30×10 -9 ]]> 0.0011 0.001 <![CDATA[1×10 +11 ]]> <![CDATA[3.00×10 -9 ]]>

[0115] Table 2 Calculation data of formula (7)

[0116] r(m) δ(m) μ(Pa·s) Δp / L (Pa· / m) <![CDATA[l st (m)]]> case1 <![CDATA[3.00×10 -9 ]]> <![CDATA[7.00×10 -10 ]]> 0.00105 <![CDATA[1×10 +11 ]]> <![CDATA[1.00×10 -9 ]]> case2 <![CDATA[7.00×10 -9 ]]> <![CDATA[7.00×10 -10 ]]> 0.00105 <![CDATA[1×10 +11 ]]> <![CDATA[2.00×10 -9 ]]> case 3 <![CDATA[1.00×10 -8 ]]> <![CDATA[7.00×10 -10 ]]> 0.00105 <![CDATA[1×10 +11 ]]> <![CDATA[3.00×10 -9 ]]>

[0117] Example 3 - Comparison of the water flow stabilization time predicted by the single-pipe water volume flow model (8) considering transient flow and microscale effects and the water flow stabilization time predicted by molecular simulation

[0118] The time required for water to reach stability in nanotubes is generally 10 -9s order of magnitude, it is difficult for experimental methods to capture such a short time. In order to verify the accuracy of the time when the water flow reaches stability as predicted by formula (8), the prediction results of molecular simulation are used for comparison. A sign that the fluid flows in the microtube and reaches a stable state is that the number of fluid molecules in the microtube no longer increases. The time when the number of water molecules predicted by the Tao J molecular simulation model reaches stability is compared with the time when the water volume flow rate reaches stability calculated by formula (8), as shown in Table 3. Under four different wetting conditions, the calculation results of formula (8) and the Tao J molecular simulation model are very consistent, as shown in Table 3. Figure 4 a, indicating that equation (8) is accurate, where the error bars represent the error in contact angle.

[0119] Table 3 Stability time verification example

[0120]

[0121] Among them, Tao J specifically refers to the paper Tao J, Song X, Bao B, et al. The Role of Surface Wettability on Water Transport through Membranes[J]. Chemical Engineering Science, 2020, 219: 115602.

[0122] Example 4 - Comparison of the calculated results of the flow enhancement factor model considering transient flow and microscale effects with the molecular simulation model and experimental results. A large number of scholars have studied the flow enhancement factor in nanotubes and have accumulated a wealth of experimental and molecular simulation data. The formula (9) in this paper is a single nanotube flow model under transient conditions, which is the basis for the subsequent permeability model. In order to verify its accuracy, the calculation results of formula (9) (time is 9×10 -11 s, formula (9) reaches stability) is compared with the flow enhancement factor in the literature (molecular simulation or experimental results). The comparison samples are 21, including 12 experimental results and 9 molecular simulation results, Table 4. Figure 4 As can be seen from equation (b), under different nanoporous materials and pore diameters, the flow enhancement factor spans four orders of magnitude (1-10,000). The calculated results of equation (9) are in good agreement with the molecular simulation and experimental results in most literature, with the error bars due to the uncertainty of the contact angle. This demonstrates that equation (9) can reflect the fundamental physical phenomena of water flow in nanotubes / microtubes and is reliable as the basis for the subsequent apparent liquid permeability model, equation (9).

[0123] Table 4 Enhancement factor verification examples

[0124]

[0125] Among them, Thomas and McGaughey specifically refers to the paper Thomas John A, McGaughey Alan JH. Reassessing fast water transport through carbon nanotubes. [J]. Nano letters, 2008, 8 (9);

[0126] Secchi, et al., 2016. Specifically, the paper Secchi, Eleonora, Marbach, et al. Massive radius-dependent flow slippage in carbon nanotubes[J]. Nature, 2016, 537(7619): 210–213;

[0127] Thomas and McGaughey specifically refer to the paper Thomas JA, McGaughey AJ Water flow in carbon nanotubes: Transition to subcontinuum transport[J]. Phys Rev Lett, 2009102(18):184502;

[0128] Mohamed, et al, 2019, specifically the paper Shaat M, Zheng Y. Fluidity and phase transitions of water in hydrophobic and hydrophilic nanotubes[J]. Scientific Reports, 2019, 9(1).

[0129] Holt, et al, 2006, specifically the paper Jason K. Holt, Hyung Gyu Park, Yinmin Wang, Michael Stadermann, Alexander B. Artyukhin, Costas P. Grigoropoulos, Aleksandr Noy, Olgica Bakajin. Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes[J]. Science, 2006, 312(5776).

[0130] Example 5 - Effect of pore size and contact angle on transient flow enhancement factor and its stabilization time In order to understand the evolution of the transient flow enhancement factor (Equation (9)) at different time periods, the flow enhancement factor at different time points in a single tube was calculated. Figure 5 . At a time of 1×10 -12 When the contact angle is very small, the fluid flow in a single tube with a very small radius can be fully developed. When the flow time is short and R is very small, the contact angle has a certain influence on the flow enhancement factor, and the flow enhancement factor is mostly less than 1. As time increases (1×10 -10 s), the water flow is fully developed in the single tube with a slightly larger R, and the contact angle has a more obvious effect on the flow enhancement factor. With different contact angles, the flow enhancement factor increases or decreases, but the flow enhancement factor of the single tube with a larger radius is still less than 1; when the time is 1×10 -8 s, except for the water flow in the single tube with a contact angle of 170 degrees, the water flow in other cases was fully developed; when the time was 1×10 -6 At s, the fluid flow in all single tubes reaches stability. At this time, the flow enhancement factor is exactly the same as the result of the reference [], which shows that the transient flow enhancement factor formula (9) proposed in this paper is an important supplement to the Wu model.

[0131] From the above analysis, we can see that the contact angle has a certain influence on the time it takes for the flow enhancement factor to reach stability. The specific influence results are shown in Figure 6 From the overall trend, it can be seen that the time it takes for the flow enhancement factor to reach stability is controlled by the pore size and contact angle, and increases with the increase of the single tube radius and contact angle. When the contact angle is less than 90 degrees, the time it takes for the flow enhancement factor to reach stability remains almost unchanged as the contact angle increases, indicating that under hydrophilic conditions, the time it takes for water flow to reach a steady state is mainly controlled by the pore size and has nothing to do with the contact angle. When the contact angle is greater than 90 degrees, the time it takes for water flow to reach stability becomes significantly longer as the contact angle increases. When the contact angle is 170 degrees, the time it takes for water flow to reach stability is about 100 times that when the contact angle is 150 degrees. This indicates that under hydrophobic conditions, the time it takes for water flow to reach stability is mainly controlled by the contact angle, followed by the pore size.

[0132] Example 6 - Effects of Apparent Viscosity, Slip Length, and PSD on Apparent Liquid Permeability The following discusses the effects of heterogeneous viscosity, slip length, and pore size heterogeneity on permeability. For ease of description, the permeability enhancement η is defined to represent the ratio of the permeability calculation results under different conditions to the intrinsic permeability formula (18). Specific results are shown in Figure 7 For the liquid apparent permeability model (17) that only considers transient flow, it is only related to time and average pore size. The pore size in the calculation is given as a fixed value of 3×10 -9 m, so Equation (17) only changes with time. When the time is 1×10-13 s to 8.1×10 -12 s, the ratio of the calculated result of formula (17) to the intrinsic permeability is less than 1. This is due to the unsteady flow in the initial stage. -12 After s, the flow reaches stability, and Equation (17) is equal to the result of the calculation of the intrinsic permeability. Figure 7 a, 7b. For formula (16), when the contact angle is 30 degrees, the value of η after stabilization is less than 1. This is due to the strong adsorption of water on the pore wall, which increases the viscosity of water in the pore wall area. However, when the contact angle is 150 degrees, the value of η after stabilization is slightly greater than 1. This is due to the decrease in the viscosity of the wall water. For formula (15), when the contact angle is 30 degrees, the value of η after stabilization is less than 1, indicating that the contribution of slippage to permeability is very small. When the contact angle is 150 degrees, the value of η after stabilization can reach about 40, indicating that the contribution of slippage to permeability is very significant. For formula (14), the pore size distribution range used in the calculation is 0-100nm. When the contact angle is 30 degrees, η is about 1.5, and when the contact angle is 150 degrees, η can reach 55, indicating that although the average pore size is 3×10 -9 m, but when the heterogeneous distribution of pore size is considered, a part of large pores will be added, which contributes more to the permeability; in addition, for the strongly hydrophobic pore wall, the slippage phenomenon is also obvious when the pore size is large.

[0133] Example 7 - Effect of PSD parameters on permeability and its stabilization time

[0134] From the above analysis, it can be seen that after considering the heterogeneous distribution of pore size, the permeability value increases to a great extent. In order to clarify the influence of the pore size distribution frequency function on the permeability and its stabilization time, two schemes are given. Scheme 1 is that the average pore size is fixed and the deviation coefficient changes from small to large. Figure 8 a. Table 5; Scheme 2 is that the deviation coefficient is a fixed value, and the average pore size changes from small to large. Figure 8 b. Table 6. For hydrophilic surfaces, as the deviation coefficient increases, the permeability also increases significantly. This is because as the deviation coefficient increases, the pore size distribution range becomes wider, which also means that a part of large pores and small pores will increase at the same time, but the increase in large pores will be greater than that in small pores. Figure 9 a, so the permeability increases; in addition, with the increase of the average pore radius, the permeability also increases significantly. This is because the average pore diameter increases, the proportion of large pores increases, and the permeability increases accordingly. Figure 9 b. For hydrophobic surfaces, the permeability increases with the increase of the deviation coefficient and the average pore size. Figure 10 , the reason is the same as that of hydrophilic surface.

[0135] from Figure 9 and Figure 10 It can be seen that under different pore size distribution conditions, the time for the permeability to reach stability is also different (the length of the horizontal line after stability). Therefore, the time for the permeability to reach stability in the above cases is statistically analyzed, such as Figure 11 From the overall trend, as the deviation coefficient and average pore diameter increase, the time it takes for the permeability to stabilize also increases. Furthermore, the larger the contact angle, the longer the time it takes for the permeability to stabilize, which is consistent with the conclusion drawn in Section 2. The increase in the average pore diameter causes a greater increase in the stabilization time than the deviation coefficient, indicating that the increase in the number of macropores caused by the increase in the average pore diameter is greater than the deviation coefficient.

[0136] Table 5 Effect of deviation coefficient on permeability (Scheme 1)

[0137] <![CDATA[r ave (×10 -9 m)]]> f(dimensionless) <![CDATA[μ(×10 -9 m)]]> σ case1 15 0.1303 2.748 0.2 case2 15 0.0848 2.798 0.3 case 3 15 0.0614 2.868 0.4

[0138] Table 6 Effect of average pore size on permeability (Scheme 2)

[0139] <![CDATA[r ave (×10 -9 m)]]> f(dimensionless) <![CDATA[μ(×10 -9 m)]]> σ case1 10 0.0921 2.463 0.4 case2 15 0.0614 2.868 0.4 case 3 20 0.0460 3.156 0.4

Claims

1. A method for calculating the permeability of nanoporous shale considering transient water seepage, characterized by: The method is as follows: Step 1: Consider the flow of water in a single tube under constant pressure boundary conditions; adopt a horizontally placed single capillary tube model with constant cross-section and ignoring the influence of gravity, simplify and solve the NS equations in the cylindrical coordinate system to obtain a single capillary fluid velocity distribution model that considers transient flow and slip length; The velocity distribution model of a single capillary fluid considering transient flow and slip length is as follows: Where: is a single capillary fluid velocity distribution model considering transient flow and slip length, m / s 2 ; ρ is the fluid density, kg / m 3 ; s is the Laplace space variable, dimensionless; Δp is the pressure difference, Pa; L is the length of the tube, m; μ is the fluid viscosity, Pa·s; r is the radial distance from the center of the tube, m; R is the radius of the tube, m; λ is the slip length, m; I0 is the 0th order first kind imaginary total Bessel function, I1 is the 1st order first kind imaginary total Bessel function; Step 2: Perform viscosity and slip corrections on the single capillary fluid velocity distribution model that considers transient flow and slip length to obtain a corrected single tube velocity model that considers transient flow and microscale effects; The modified single tube velocity model considering transient flow and microscale effects is specifically: Where: To modify the single tube velocity model considering transient flow and microscale effects; μ R is the apparent viscosity, Pa·s; Step 3: By integrating the radius r from 0-R for the modified single-tube velocity model considering transient flow and microscale effects, the single-tube fluid volume flow rate considering transient flow and microscale effects is obtained; Step 4: Use the non-equal diameter capillary bundle model to equate the pore network of the porous medium and use the pore size distribution frequency function of the porous medium to describe the pore size distribution of the nano-scale porous medium. The water volume flow rate of the capillary bundle model is obtained by multiplying the single-tube fluid volume flow rate considering transient flow and microscale effects by the pore size distribution frequency function of the porous medium and then integrating R from the smallest pore size to the largest pore size. In Laplace space, the water volume flow rate, porosity, cross-sectional area of the capillary bundle model and the corresponding parameters in Darcy's law are made equal and then simplified to obtain a porous media permeability model that takes into account transient flow, microscale effects and pore size distribution. The porous media permeability model considering transient flow, microscale effects and pore size distribution is as follows: Where: A porous media permeability model that considers transient flow, microscale effects, and pore size distribution; μ b is the viscosity of free water, Pa·s; is the porosity of the capillary bundle model, dimensionless; R max is the maximum pore size of the porous medium, m; R min is the minimum pore size of the porous medium, m; l st is the slip length considering the effect of pipe wall wettability, m; f(R) is the pore size distribution frequency function of the porous medium.

2. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 1, characterized in that: The pore size distribution frequency function of the porous medium is specifically: Where: σ is the deviation coefficient of pore size distribution; ω is the average expectation of pore size distribution.

3. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 2, characterized in that: The apparent liquid permeability model considering transient flow and microscale effects is: Where: It is an apparent liquid permeability model that takes transient and microscale effects into account.

4. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 2, characterized in that: The apparent liquid permeability model considering transient flow and apparent viscosity is: Where: It is an apparent liquid permeability model that takes transient flow and apparent viscosity into account.

5. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 2, characterized in that: The apparent liquid permeability model considering transient flow is: Where: is the apparent liquid permeability model considering transient flow.

6. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 2, characterized in that: Without considering the time variation, or when time tends to infinity, the intrinsic permeability model of the homogeneous capillary bundle is Where: k ins is the intrinsic permeability model of a homogeneous capillary bundle.

7. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 2, characterized in that: The specific implementation process of step 1 is as follows: During the flow of fluid under constant pressure boundary, due to the influence of boundary layer, the velocity profile of fluid gradually changes from slug type to parabola type, and then reaches stability, and the fluid flow becomes fully developed stable flow; In order to describe the transient flow of fluid in the capillary, a horizontally placed single capillary model with equal cross-section and ignoring the influence of gravity is adopted, and the pressure gradient is constant. After considering the above factors, the NS equations in the cylindrical coordinate system are simplified to the following form: Where: u is the fluid velocity in the horizontal direction, m / s; r is the radial distance from the center of the tube, m; Δp is the pressure difference, Pa; L is the length of the tube, m; Solve equation (1) and consider the slip boundary conditions, introduce the following initial boundary conditions Taking Laplace transformation of equations (1) and (2) with respect to t, we get the Bessel equation of imaginary quantity. The solution of the equation is obtained under the initial and boundary conditions: Where: is a single capillary fluid velocity distribution model considering transient flow and slip length, m / s 2 .

8. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 7, characterized in that: The specific implementation process of step 2 is as follows: Viscosity and slip corrections to the velocity distribution model of a single capillary fluid considering transient and slip length Where: l st is the slip length considering the wettability of the tube wall, m; θ is the contact angle, degrees; in Where: μ R is the apparent viscosity, Pa·s; μ w is the viscosity of the wall water in Pa·s; A i is the cross-sectional area of the wall region, m 2 ; A t is the total cross-sectional area of the pipe diameter, m 2 ; The modified single tube velocity model considering transient flow and microscale effects is: Where: To modify the single tube velocity model considering transient flow and microscale effects.

9. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 8, characterized in that: The specific implementation process of step 3 is as follows: The modified single-tube velocity model considering transient flow and microscale effects is integrated from 0 to R to obtain the single-tube water volume flow rate considering transient flow and microscale effects. Where: is the single pipe water volume flow rate considering transient flow and microscale effects, m 3 / s.

10. The method for calculating the permeability of nanoporous shale considering transient water seepage according to claim 9, characterized in that: The specific implementation process of step 4 is as follows: The volume flow rate in a single tube is extended to porous media, the pore network of the porous media is equivalent to the non-equal diameter capillary bundle model, and the pore size distribution frequency function of the porous media is used to describe the pore size distribution of the nano-scale porous media. Where: f is the pore size distribution frequency, no steel; σ is the deviation coefficient, no steel; ω is the average expectation, m; Multiplying equation (8) by f(R) and integrating R from the smallest pore size to the largest pore size yields the water volume flow rate of the capillary bundle model: for Where: is the water volume flow rate of the capillary bundle model, m 3 ; N is the number of capillaries, dimensionless; R max is the maximum pore size of the porous medium, m; R min is the minimum pore size of the porous medium, m; the relationship between the porosity of the capillary bundle model and the number of capillaries is Substituting into formula (11), the simplified water volume flow rate of the capillary bundle model considering transient flow, microscale effect and pore size distribution in Laplace space is: Where: is the porosity of the capillary bundle model, dimensionless; A is the cross-sectional area of the capillary bundle model, m 2 ; In Laplace space, the water volume flow rate, porosity, cross-sectional area of the capillary bundle model and the corresponding parameters in Darcy's law are equal, and simplified to obtain the porous media permeability model considering transient flow, microscale effects and pore size distribution. Where: A porous media permeability model considering transient flow, microscale effects, and pore size distribution.

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