A method for obtaining colloid deposition rate using a microscopic periodic pore structure model

Abstract

The present invention discloses a method for obtaining colloid deposition rate by using a microscopic periodic pore structure model, determining the size of the periodic pore structure model according to the actual porous medium characteristics, and correcting the model size according to the porosity and flow rate; using computational fluid dynamics software to obtain the flow field in the pore and the medium wall distance field; obtaining the colloid motion trajectory based on Lagrangian particle tracking, counting the number of deposited particles, and calculating the colloid deposition rate in the pore; and proposing a method for calculating the colloid deposition rate according to three periodic pore models by using a weighted average method to represent the complex pore structure. The simulation results are compared with the predicted values, proving the reliability of the simulation results. The proposed method can deeply study the influence of pore structure on colloid deposition, so as to be applied to projects such as nano-colloid remediation of soil and groundwater pollution.

Description

Technical Field

[0001] The invention belongs to the field of groundwater pollution migration and restoration, and in particular relates to a method for obtaining colloid deposition rate by adopting a microscopic periodic pore structure model. Background Art

[0002] The migration and deposition of colloidal particles are widely present in nature and engineering. For example, when special colloidal particles are injected into aquifers to repair groundwater pollution, the repair effect is very limited due to the deposition and clogging characteristics of the colloids. Understanding the deposition behavior of colloids in porous media will help people formulate reasonable pollution prevention and control strategies. It is of great significance to study the migration and deposition behavior of colloids from the microscopic mechanism level.

[0003] Previous studies generalized porous media into a single pore structure, such as the classic Happel model and the contraction tube model, and then used theoretical analysis or numerical simulation to study the deposition rate of colloids in the pores. In fact, due to the contact and stacking effect of the medium particles, even if the size of the medium particles is uniform, the pore structure will be spatially non-uniform. Most previous studies ignored the effect of complex pore structures on colloid deposition, resulting in partial deviations between simulation results and experiments. Summary of the invention

[0004] Purpose of the invention: The purpose of the present invention is to provide a method for obtaining the colloid deposition rate using a microscopic periodic pore structure model. According to the stacking of regular spherical medium particles, three periodic pore structures are proposed, and the model structure is adjusted according to the porosity of the porous medium, and a method of using a combination of multiple periodic structures to represent the porous medium is adopted. The comparison of the simulation results of the three pore structures and the theoretical prediction values ​​proves the effectiveness and reliability of the present invention.

[0005] Technical solution: A method of obtaining colloid deposition rate using a microscopic periodic pore structure model of the present invention comprises the following steps:

[0006] Step 1, constructing a periodic pore model of porous media, including a sphere-in-cube structure model I, a body-centered structure model II, and a face-centered structure model III;

[0007] Step 2: Based on the periodic pore model of porous media, the flow field is solved by the Stokes equation, and the distance field between the pores of the porous media and the surface of the media is solved by the Eikonal equation;

[0008] Step 3, simulating the migration trajectory of the colloid based on the Lagrangian method;

[0009] Step 4: Calculate the deposition rate of the colloid and verify it.

[0010] Furthermore, step 1 is specifically as follows: assuming that the porous medium particles are of equal size and are stacked according to a periodic repetitive rule, the pore morphology is divided into three types according to the stacking structure of the porous medium, namely, a sphere-in-cube structure model I, a body-centered structure model II, and a face-centered structure model III; the position and spacing of the medium particles are determined according to the porosity of the porous medium to ensure that the porosity of the model is consistent;

[0011] The true porosity of model I is:

[0012]

[0013] The mathematical expressions of the true porosity of Model II and Model III are:

[0014]

[0015] Where: ε is the porosity of the model or porous medium; r c is the radius of the medium particle; Δd ​​is the distance between the centers of two adjacent medium particles and the particle surfaces; for model I, the porosity calculated according to formula (1) is always 0.476, so the cube is shrunk into a cylinder to ensure that the porosity of the porous medium can be changed arbitrarily; for models II and III, Δd is calculated in reverse according to formula (2);

[0016] First, a solid model of medium particle accumulation is established, and then the solid medium particles are dug out, leaving only the pore space. Due to the differences in the size of the model inlet cross-section, the cross-sectional flow rate is calculated based on the total inlet flow to ensure the consistency of the flow field. Model I uses the Darcy flow rate U ∞ , while models II and III use the average inter-pore velocity and the fully developed flow inlet boundary; the inter-pore velocity of models II and III is calculated according to the flow rate, that is:

[0017]

[0018] Where: U ∞ is the Darcy velocity; u is the average flow velocity between pores.

[0019] Furthermore, in step 2, the flow field is solved by the Stokes equation as follows:

[0020] The formula for solving the flow field of Stokes equation is as follows:

[0021]

[0022] Where: P is the fluid pressure, μ is the fluid viscosity coefficient; u is the flow velocity; the upper boundary of the model is set to a constant flow velocity, and the lower boundary is zero pressure.

[0023] Furthermore, in step 2, the distance field between the pores of the porous medium and the surface of the medium is solved by the Eikonal equation as follows:

[0024] Let the distance between the colloid surface and the medium surface be h, define a minimum distance σ = 0.5nm, and freeze the colloid when h < σ. In order to accurately calculate h in numerical calculation, it is necessary to obtain the distance field between the pore and the medium surface. Considering that there will be multiple medium particle surfaces in the pore structure, the Eikonal equation is used to obtain the distance field. Let the distance δ on the medium surface be w =0, and other boundaries are Modified Eikonal equation form:

[0025]

[0026] Where: is the symbol of the Nabla operator; δ w is the wall distance; is the reciprocal of the wall distance; is the normal vector pointing to the colloidal particle from the nearest wall; s is the smoothness factor;

[0027] The final separation distance between the colloid and the medium surface is:

[0028] h=δ w,p -r p (9)

[0029] Where: w,p is the distance between the center coordinate of the colloid and the wall.

[0030] Furthermore, step 3 is specifically as follows: simulating the movement of colloids based on the Lagrangian method:

[0031]

[0032] The specific expression of particle force is:

[0033] F D =6πμr p (uv) (11)

[0034]

[0035] Where: m is the mass of the colloid; m * is the virtual mass of the colloid in the fluid; v is the particle velocity vector; F is the force applied to the colloid, F D is the drag force, F G is the effective gravity, F L,Saffman is the lift, F B is the random Brownian force, F Pis the pressure gradient force; r p is the colloid radius; ρ p and ρ f are the densities of colloid and fluid respectively; R(0,1) is a Gaussian random number vector with a mean value of 0 and a variance of 1; k B =1.38×10 -23 J / K, is the Boltzmann constant; T is the absolute temperature; Δt is the time step of the numerical calculation; Formula (11) adopts the Stokes drag law and is applicable to the relative Reynolds number Re r =d p ρ f The condition of ||Uv|| / μ<<1, that is, the condition of slow flow rate and small colloid size.

[0036] Furthermore, step 4 is specifically as follows:

[0037] The colloid deposition rate was calculated using the following formula:

[0038] η=N dep / N tot (16)

[0039] Where: η is the colloid deposition rate; N dep is the number of particles finally deposited; N tot is the number of particles released from the upstream inlet;

[0040] For the simulation results of the three pore structures, the analytical solution and the nonlinear regression prediction equation were used for comparative verification.

[0041] Furthermore, the expression of the analytical solution is:

[0042]

[0043] The nonlinear regression prediction equation is expressed as:

[0044]

[0045] Where: γ=(1-ε) 1 / 3 and A s is the porosity related variable; N A is the attraction number; N G is the gravity number; N pe is the Peclet number; N R is the intercept number, both are dimensionless variables, and the specific expressions are as follows:

[0046]

[0047] Where: A 123 is the Hamaker constant for pairwise additivity; kB =1.38×10 -23 J / K is the Boltzmann constant; T is the absolute temperature; U ∞ is the average inlet velocity; D = k B T / (6πμr p ) is the colloid diffusion coefficient; ρ p is the colloid density; ρ f is the fluid density; μ is the fluid dynamic viscosity coefficient.

[0048] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0049] The effect of the heterogeneity of pore structure in porous media on colloid deposition is considered. Three representative periodic pore models are proposed to simulate colloid deposition, and the simulation results are weighted averaged to characterize the complex pore conditions of porous media, which solves the problem of using a single pore model to represent porous media in the past.

[0050] It can simultaneously consider the coupled mechanisms of diffusion, gravity deposition and interception. The colloidal particles are allowed to stagnate at a given separation distance to simulate the deposition state of the colloidal particles. This method can simultaneously meet the Brownian (particle size less than 2μm) and non-Brownian (particle size greater than 2μm) colloidal deposition simulations, and is more unified in method, overcoming the assumption that Brownian colloids only consider diffusion deposition and non-Brownian colloids only consider gravity sedimentation and interception.

[0051] The present invention has strong applicability and expansibility. The present invention adopts an analytical method and a modified Eikonal equation to calculate the distance between the colloid and the medium surface, which expands the applicability of the present method and can be further applied to the conditions where the medium surface is rough, there is electrostatic force or surface physical and chemical heterogeneity.

[0052] In summary, this method is suitable for conditions that are closer to real groundwater seepage and colloid deposition, achieving accurate simulation results, breaking through the limitations of simplified analytical solutions and indoor experiments, and thus assisting and guiding engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 A flowchart of the steps for implementing the present invention;

[0054] Figure 2 There are three periodic pore structure models;

[0055] Figure 3 Schematic diagram of colloidal particle trajectory and deposition position;

[0056] Figure 4 Comparison between the colloid deposition rate of the periodic pore structure model and the predicted value. DETAILED DESCRIPTION

[0057] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0058] The specific steps and processes of the present invention are shown in Figure 1 The method comprises the following steps:

[0059] Step 1: Construct a periodic pore model

[0060] It is assumed that the porous media particles are of equal size and are stacked according to a periodic repetitive pattern. According to the stacking structure of the porous media, the pore morphology is divided into three types: sphere-in-cube structure (model I), body-centered structure (model II), and face-centered structure (model III), as shown in Figure 2 .

[0061] The location and spacing of the media particles are determined according to the porosity of the porous media to ensure consistent porosity across the model.

[0062] The true porosity of model I is:

[0063]

[0064] The mathematical expressions of the true porosity of models II and III are:

[0065]

[0066] Where: ε is the porosity of the model or porous medium; r c is the radius of the medium particle; Δd ​​is the distance between the centers of two adjacent medium particles and the particle surfaces. For model I, the porosity calculated according to formula (1) is always 0.476, so the cube is shrunk into a cylinder to ensure that the porosity of the porous medium can be changed arbitrarily. For models II and III, Δd is calculated in reverse according to formula (2).

[0067] First, a solid model of medium particle accumulation is established, then the medium particle entity is dug out, and finally the pore space is left. The final pore structure is as follows Figure 2 As shown in Figure 1, due to the different sizes of the model inlet cross-sections, in order to ensure the consistency of the flow field, the cross-sectional velocity is calculated based on the total inlet flow. Model I uses the Darcy velocity U ∞ , while models II and III use the average inter-pore velocity and adopt the fully developed flow inlet boundary. The inter-pore velocity of models II and III is calculated based on the flow rate, that is:

[0068]

[0069] Where: U ∞ is the Darcy velocity; u is the average flow velocity between pores.

[0070] Step 2: Flow and distance field simulation

[0071] In most cases in nature, groundwater flows slowly and is laminar. On the other hand, the concentration of colloids in nature is small, and the effect on the flow field can be ignored. Therefore, the flow field is solved first, and then the movement of colloids in the pores is simulated. The Stokes equation is used to solve the flow field:

[0072]

[0073] Where: P is the fluid pressure, μ is the fluid viscosity coefficient, and u is the flow rate. The upper boundary of the model is set to a constant flow rate, and the lower boundary is set to zero pressure.

[0074] Let the distance between the colloid surface and the medium surface be h, define a minimum distance σ = 0.5nm, and freeze the colloid when h < σ. In order to accurately calculate h in numerical calculation, it is necessary to obtain the distance field between the pore and the medium surface. Considering that there may be multiple medium particle surfaces in the pore structure of the present invention, the Eikonal equation is used to obtain the distance field. Let the distance δ on the medium surface be w =0, and other boundaries are Modified Eikonal equation form:

[0075]

[0076] Where: is the symbol of the Nabla operator; δ w is the wall distance; is the reciprocal of the wall distance; is the normal vector pointing to the colloidal particle from the nearest wall; s is the smoothing factor, which is usually taken as 0.05.

[0077] The final separation distance between the colloid and the medium surface is:

[0078] h=δ w,p -r p (9)

[0079] Where: w,p is the distance between the center coordinate of the colloid and the wall, which is obtained by sampling the distance field at the colloid position.

[0080] Step 3: Simulation of colloid migration trajectory

[0081] The flow rate is further transmitted to the colloid particles, and the movement of the colloid is simulated according to the Lagrangian method:

[0082]

[0083] The specific expression of particle force is:

[0084] F D =6πμrp (uv) (11)

[0085]

[0086] Where: m is the mass of the colloid; m * is the virtual mass of the colloid in the fluid (for spherical colloids, it is about the mass of 1 / 2 colloid volume replacing the fluid); v is the particle velocity vector; F is the force applied to the colloid, F D is the drag force, F G is the effective gravity, F L,Saffman is the lift, F B is the random Brownian force, F P is the pressure gradient force; r p is the colloid radius; ρ p and ρ f are the densities of colloid and fluid respectively; R(0,1) is a Gaussian random number vector with a mean value of 0 and a variance of 1; k B =1.38×10 -23 J / K is the Boltzmann constant; T is the absolute temperature; Δt is the time step of the numerical calculation. Formula (11) adopts the Stokes drag law and is applicable to the relative Reynolds number Re r =d p ρ f The condition of ||Uv|| / μ<<1, that is, the condition of slow flow rate and small colloid size.

[0087] Step 4: Deposition rate calculation and verification

[0088] Computational fluid dynamics was used to simulate the flow field, distance field, and particle tracking. A sufficient number of particles were released at the inlet to ensure the accuracy of the simulation results. After the simulation, the colloid deposition rate was calculated using the following formula:

[0089] η=N dep / N tot (16)

[0090] Where: η is the colloid deposition rate; N dep is the number of particles finally deposited; N tot is the number of particles released from the upstream inlet.

[0091] For the simulation results of the three pore structures, two prediction results were used for comparison and verification: (1) the analytical solution proposed by Yao et al. (1971); (2) the prediction equation obtained by Kamai et al. (2015) based on nonlinear regression of limited experimental results. The analytical solution expression is:

[0092]

[0093] The regression equation is expressed as:

[0094]

[0095] Where: γ=(1-ε) 1 / 3 and As are porosity-related variables; N A is the attraction number; N G is the gravity number; N pe is the Peclet number; N R is the intercept number. They are all dimensionless variables, and the specific expressions are as follows:

[0096]

[0097] Where: A 123 is the Hamaker constant for pairwise additivity; k B =1.38×10 -23 J / K is the Boltzmann constant; T is the absolute temperature; U ∞ is the average inlet velocity; D = k B T / (6πμr p ) is the colloid diffusion coefficient; ρ p is the colloid density; ρ f is the fluid density; μ is the fluid dynamic viscosity coefficient.

[0098] In a simulation example, the model parameters are: U ∞ =4.63×10 -6 m / s, T = 298K, medium particle diameter d c =500μm,ρ p =1050kg / m 3 , colloid diameter d p ∈[0.04,16]μm, total number of released particles N tot = 5000. The trajectories of some colloidal particles in the pore structure of model II during the simulation are shown in Figure 2. Figure 3 As shown in Figure 2, the deposited particles stay near the surface of the medium. The colloid deposition rate obtained by using a uniform medium particle spacing and inlet flow rate is shown in Figure 2. Figure 4 As shown in Figure 2, after correcting the porosity and inlet flow rate, the colloid deposition rate is significantly more accurate. The simulation results of the three different pore structures (models I to III) are different, indicating that the pore structure affects the colloid deposition rate. p For the case of >2 μm, the colloid deposition rates simulated by the three pore structures are consistent with the experimental regression predictions, which indicates that for colloids of larger sizes, the pore structure has little effect on the deposition rate.

[0099] For d pFor the case of ≤2μm, a weighted average method considering the simulation results of three pore structures is proposed, and the average colloidal particle deposition rate of porous media is calculated using the following mathematical expression:

[0100]

[0101] Where: weighting coefficient γ1+γ2+γ3=1, which can be obtained by fitting according to the test results; η I , η II and η III They are the simulation results of colloid deposition rate in three periodic pore models proposed by the present invention. When formula (19) is applied to a specific indoor test, γ1~γ3 can reflect the inherent characteristics of porous media.

Claims

1. A method for obtaining colloid deposition rate using a microscopic periodic pore structure model, characterized in that: The steps include: Step 1, constructing a periodic pore model of porous media, including a sphere-in-cube structure model I, a body-centered structure model II, and a face-centered structure model III; Step 2, the above three models based on the porous medium periodic pore model are respectively solved by the Stokes equation to obtain the flow field, and the Eikonal equation is used to obtain the distance field between the porous medium pores and the medium surface; Step 3, simulating the migration trajectory of the colloid based on the Lagrange equation of particle motion; Step 4: After calculating the flow field, distance field and particle migration trajectory simulation of fluid mechanics, calculate and verify the colloid deposition rates of the three models in the porous medium periodic pore model, and take the weighted average of the colloid deposition rates of the three models to calculate the average colloid particle deposition rate of the porous medium periodic pore model.

2. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 1, characterized in that: Step 1 is as follows: assuming that the porous medium particles are of equal size and are stacked according to a periodic repetitive pattern, the pore morphology is divided into three types according to the stacking structure of the porous medium, namely, sphere-in-cube structure model I, body-centered structure model II, and face-centered structure model III; the position and spacing of the medium particles are determined according to the porosity of the porous medium to ensure that the porosity of the model is consistent; The true porosity of model I is: The mathematical expressions of the true porosity of Model II and Model III are: Where: ε is the porosity of the model or porous medium; r c is the radius of the medium particle; Δd ​​is the distance between the centers of two adjacent medium particles and the particle surfaces; for model I, the porosity calculated according to formula (1) is always 0.476, so the cube is shrunk into a cylinder to ensure that the porosity of the porous medium can be changed arbitrarily; for models II and III, Δd is calculated in reverse according to formula (2); First, a solid model of medium particle accumulation is established, and then the solid medium particles are dug out, leaving only the pore space. Due to the differences in the size of the model inlet cross-section, the cross-sectional flow velocity is calculated based on the total inlet flow to ensure the consistency of the flow field. Model I uses the Darcy velocity U ∞ , while models II and III use the average inter-pore velocity and the fully developed flow inlet boundary; the inter-pore velocity of models II and III is calculated according to the flow rate, that is: Where: U ∞ is the Darcy velocity; u is the average flow velocity between pores.

3. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 1, characterized in that: In step 2, the flow field is solved by the Stokes equation as follows: The formula for solving the flow field of Stokes equation is as follows: Where: P is the fluid pressure, μ is the fluid viscosity coefficient; u is the flow velocity; the upper boundary of the model is set to a constant flow velocity, and the lower boundary is zero pressure.

4. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 1, characterized in that: In step 2, the distance field between the pores of the porous medium and the surface of the medium is solved by the Eikonal equation as follows: Let the distance between the colloid surface and the medium surface be h, define a minimum distance σ = 0.5nm, and freeze the colloid when h < σ. In order to accurately calculate h in numerical calculation, it is necessary to obtain the distance field between the pore and the medium surface. Considering that there will be multiple medium particle surfaces in the pore structure, the Eikonal equation is used to obtain the distance field. Let the distance δ on the medium surface be w =0, and other boundaries are Modified Eikonal equation form: Where: is the symbol of the Nabla operator; δ w is the wall distance; is the reciprocal of the wall distance; is the normal vector pointing to the colloidal particle from the nearest wall; s is the smoothness factor; The final separation distance between the colloid and the medium surface is: h=δ w,p -r p (9) Where: w,p is the distance between the center coordinate of the colloid and the wall, r p is the colloid radius.

5. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 1, characterized in that: Step 3 is as follows: Use Newton's second law to simulate the movement of colloids: The specific expression of particle force is: F D =6πμr p (u-v) (11) Where: m is the mass of the colloid; * is the virtual mass of the colloid in the fluid; v is the particle velocity vector; F is the force applied to the colloid, F D is the drag force, F G is the effective gravity, F L,Saffman is the lift, F B is the random Brownian force, F P is the pressure gradient force; r p is the colloid radius; ρ p and ρ f are the densities of colloid and fluid respectively; R(0,1) is a Gaussian random number vector with a mean value of 0 and a variance of 1; k B =1.38×10 -23 J / K, is the Boltzmann constant; T is the absolute temperature; Δt is the time step of the numerical calculation; Formula (11) adopts the Stokes drag law and is applicable to the relative Reynolds number Re r =d p ρ f The condition of ||Uv|| / μ<<1, that is, the condition of slow flow rate and small colloid size.

6. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 1, characterized in that: Step 4 is as follows: The colloid deposition rate was calculated using the following formula: n=N dep / N tot (16) Where: η is the colloid deposition rate; N dep is the number of particles finally deposited; N tot is the number of particles released from the upstream inlet; For the simulation results of the three pore structures, the analytical solution and the nonlinear regression prediction equation were used for comparative verification.

7. The method for obtaining colloid deposition rate using a microscopic periodic pore structure model according to claim 6, characterized in that: The expression of the analytical solution is: The nonlinear regression prediction equation is expressed as: Where: γ=(1-ε) 1 / 3 and A s is the porosity related variable; N A is the attraction number; N G is the gravity number; N pe is the Peclet number; N R is the intercept number, both are dimensionless variables, and the specific expressions are as follows: Where: A 123 is the Hamaker constant for pairwise additivity; k B =1.38×10 -23 J / K is the Boltzmann constant; T is the absolute temperature; U ∞ is the average inlet velocity; D = k B T / (6πμr p ) is the colloid diffusion coefficient; ρ p is the colloid density; ρ f is the fluid density; μ is the fluid dynamic viscosity coefficient.

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