CN112461718A - Method for representing relationship between porosity and particle size distribution - Google Patents
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Abstract
The invention relates to a method for characterizing the relationship between porosity and particle size distribution of particles, which comprises the following steps: firstly, specifying a region range of two-dimensional particle accumulation simulation, wherein the length is L and the height is H; randomly generating a space abscissa x in a designated area range according to a uniform distribution method, wherein x is more than or equal to 0 and less than or equal to L, and assuming that all particles fall from the same height y and the height is high enough, namely y is more than H; secondly, establishing a particle size distribution method capable of accurately describing the particle size distribution rule of the particles; generating a series of particles to obtain a compact two-dimensional particle accumulation body; calculating the porosity of the two-dimensional particle accumulation body again; and finally, qualitatively and quantitatively describing the relation between the relative standard deviation and the porosity. The method adopts the periodic boundary, the condition that the porosity of the left and right boundaries is larger does not exist, the influence of the boundary on the porosity of the two-dimensional particle accumulation body is eliminated, and the precision of calculating the porosity of the accumulation body is improved.
Description
Technical Field
The invention relates to the technical field of rock porosity characterization, in particular to a method for characterizing the relationship between porosity and particle size distribution.
Background
The influence factors of the porosity are known, and the understanding of the flowing process of the porous medium is important in the fields of petroleum engineering, hydraulic engineering and the like. Porosity is a necessary parameter in many geological, engineered physical scenarios, including oil recovery, carbon dioxide sequestration, hydraulic systems, and contaminant transport. Therefore, many scholars have conducted extensive research into the characterization of porosity.
With high resolution digital imaging techniques, it has been possible to directly observe the pore structure of core samples. However, due to the complexity of the pore structure, it is difficult to calculate the porosity directly from the pore structure. The porosity calculation is generally carried out by experimental means, such as a sieve analysis method, but the experimental means is long in time consumption and high in energy input. With the development of high-performance computing technology, simulation of the rock particle deposition process has been feasible through computer simulation and development of particle accumulation programs. And the simulation method can greatly shorten the experimental time, visualize the dynamic accumulation process of the particles and characterize the relationship between the porosity and the accumulated particles.
The association between porosity and particle size or particle size distribution is generally accepted by the scholars, but there is no consistent conclusion about the qualitative, even quantitative, relationship between porosity and particle size distribution. The particle packing process or particle packing pattern determines the pore size between particles, and porosity is a visual depiction of particle packing, reflecting the extent of particle packing. However, the porosity is not only dependent on the particle packing pattern but also on the particle size distribution, particle shape, degree of classification, and the like. Therefore, qualitative and quantitative characterization of porosity is a more complex process.
Common methods for stacking spherical particles with equal diameters include cubic stacking, orthogonal stacking, rhombic stacking, quadrilateral stacking and the like, and the corresponding porosities are 47.6%, 39.5%, 26% and 30.2%, respectively. It can be seen that different packing patterns result in particle packing of different porosity. The actual rock particles are not of equal diameter, and the stacking mode is more diversified. In addition, the description of stacking characteristics with appropriate parameters is a key link in evaluating a stack. Average particle size is generally the simplest and convenient descriptive parameter, but evaluation of particle size alone is far from adequate. The particle size distribution can reflect the size condition of the particles and can also reflect the size ratio of the stacked particles. Thus, particle size distribution can be a key parameter in characterizing porosity. The average particle size of the particles, which is a description of the particle size, and the standard deviation of the particle size of the particles, which is a quantification of the degree of change in particle size, can be used to describe the particle size distribution. It is generally believed that on the one hand, large particles occupy much of the space between small particles and small pores, resulting in reduced porosity; on the other hand, large particles also result in small particles not being closely arranged together, resulting in increased porosity. In view of the two opposite effects of large particles on porosity, the scholars believe that both effects exist during particle packing, but the two conditions have different degrees of effect on final porosity. However, it is not clear how to quantify the relationship between particle size distribution and porosity.
Therefore, in order to qualitatively and quantitatively characterize the relationship between the porosity and the particle size distribution, and with attention paid to the rock particle stacking process in nature, it is necessary to provide a two-dimensional particle stacking simulation method for characterization of the porosity-particle size distribution relationship.
Disclosure of Invention
Aiming at the problems in the prior art, the technical problems to be solved by the invention are as follows: the two-dimensional particle accumulation simulation method is reasonable and effective, can accurately reflect the deposition process of real rock particles and is used for representing the relationship between the porosity and the particle size distribution.
In order to solve the technical problems, the invention adopts the following technical scheme: a method for characterizing porosity in relation to particle size distribution, the method comprising the steps of:
s01: specifying a region range of two-dimensional particle accumulation simulation, wherein the length is L and the height is H; the units are dimensionless length units regardless of the specific units.
S02: randomly generating a space abscissa x in a designated area range according to a uniform distribution method, wherein x is more than or equal to 0 and less than or equal to L, and assuming that all particles fall from the same height y and the height is high enough, namely y is more than H; the height is sufficiently high that the initial height y of each particle is considered to be the same, and the height is sufficiently high, i.e. y > H, that there is no need to specify an ordinate.
S03: establishing a particle size distribution method capable of accurately describing a particle size distribution rule: therefore, the real proportional relation between large particles and small particles of the rock is effectively reflected and is used for generating the random radius of the subsequent particles.
S031: the mean value μ and standard deviation σ of the particle diameters of the two-dimensional particle stack to be generated are specified.
S032: taking the mean value mu and the standard deviation sigma as input parameters of a probability distribution function and a cumulative distribution function of normal distribution to obtain the probability distribution function
And cumulative distribution function
Where r represents the particle radius, σ represents the variance, erf represents the error function,
e denotes an exponential function, t denotes an argument, dt denotes the differential to t.
S033: performing weighted correction on the probability distribution function PDF (r) of S032 to obtain a weighted and corrected probability distribution function
Obtaining a weighted modified cumulative distribution function by integration
S034: the weighted and corrected cumulative distribution function c (r) described in S033 is used as a particle size distribution formula of particles in a particle stacking numerical simulation.
S04: randomly generating a probability value within the range of (0,1) according to uniform distribution in an equal probability manner, taking the probability value as the value of the weighted and corrected cumulative distribution function C (r), reversely deducing the probability value according to the weighted and corrected cumulative distribution function C (r) to obtain a random value as the particle radius r, and determining a circle (x, y, r) which is a two-dimensional particle in particle stacking numerical simulation according to the two elements of the particle radius r and a random abscissa x; the ordinate y is a value greater than the height of the specified stacking area, and the ordinate has no effect on the particle simulation as long as it is large enough.
S05: after a new particle (x, y, r) is generated, a particle fall and collision rule is defined, and the particle falls from a height:
if there are no existing particles directly below the falling particle, the particle falls on the lower boundary or on the ground, i.e. the particle has final coordinates of (x, r, r).
If there is a previously generated particle under the falling particle, the new particle collides with the previous particle, and if there is infinite friction between the particles, the new particle (x, y, r) collides with the particle (x)1,y1,r1) Is rotated to collide with the particles (x)1,y1,r1) The center of the circle is equal height, and y is equal to y1Thereafter, the newly formed particles (x, y, r) collide to a new location (x)2,y1And r) then falls again.
S06: repeating the falling and collision process of step S05, continuously updating the position coordinates of the newly generated particle until the particle falls on the ground, that is, y is 0 or the particle is stably accumulated on two other existing particles, that is, the falling and collision process of the particle is completed; wherein, when a new particle is stably stacked on two other existing particles, the coordinates of the two other existing particles are respectively assumed to be (x)3,y3,r3),(x4,y4,r4) The final position (x ', y') of the new particle satisfies the following equation:
s07: the left and right periodic boundaries are set for the particle packing process, thereby eliminating the influence of the boundaries on the porosity of the compact particle packing body. The cycle boundary means that when the particles (x, y, r) are in the falling or collision rotation process, if the region range of the left boundary is exceeded, the particles enter the designated region of the particle accumulation again from the right side, and the coordinate is (x, y, r)pY, r); similarly, if the particle is beyond the region of the right boundary, the particle enters the designated region of the particle pile again from the left side, and the coordinate is also (x)pY, r); the abscissa transformation formula of the left and right periodic boundaries is: x is the number ofpWhere mod (·) is a remainder function, the remainder of x divided by L is calculated.
S08: repeating the steps S02-S07 to generate a series of particles, setting the height of the generated particle accumulation body as h, finishing the process of repeatedly generating the particles when one particle exists in the two-dimensional particle accumulation body and the vertical coordinate of the circle center of the particle is larger than the height h, and stopping continuously generating new particles to obtain a compact two-dimensional particle accumulation body; since each particle is finally stably stacked on other particles or a few particles are directly stacked on the ground at the position of y-0, the two-dimensional particle stack generated by the simulation of the above process is a compact stack.
S09: the porosity was calculated for the two-dimensional particle packing generated at S08.
S10: repeating the steps S01-S09 to design different particle size mean values muiStandard deviation σiCombining to generate N two-dimensional particle stacks, and calculating the porosity phi corresponding to each particle stackiWherein, i ═ 1,2, 3.., N;
s11: in relative standard deviation
As a parameter for describing the particle size distribution of the particles, the relative standard deviation of each of the stacks was calculated
Wherein, i is 1,2, 3.
S12: qualitatively and quantitatively describing the relationship between the relative standard deviation and porosity as
The a, the b and the c are coefficients which are determined by a nonlinear regression method.
As an improvement, the method for calculating the porosity in S09 is as follows: from the bottom boundary of the two-dimensional particle accumulation body, upwards intercepting accumulation body segments with a certain height h ', wherein h' is less than h, thereby eliminating the influence of no particle accumulation on the upper part of the accumulation body on the porosity; and calculating the porosity phi of the accumulation body segment, and repeating the steps to completely calculate the porosity of the two-dimensional particle accumulation body.
Compared with the prior art, the invention has at least the following advantages:
1. the two-dimensional particle accumulation method provided by the invention adopts the periodic boundary, the condition that the porosity of the left and right boundaries is larger does not exist, the influence of the boundary on the porosity of the two-dimensional particle accumulation body is eliminated, and the precision of calculating the porosity of the accumulation body is improved.
2. The conventional particle size distribution method is difficult to accurately describe the particle size distribution rule of particles, and cannot effectively reflect the proportional relation between large particles and small particles in a real rock sample. For example, a uniform distribution determines that particles with different radii have the same probability of appearing, unlike the actual case. The normal distribution determines that the number of particles with a certain radius is subject to the normal distribution, but the volume of the particles with a certain radius is not subject to the normal distribution, which causes that the proportion of the small particles in the generated particle stack is seriously smaller, and the real proportion relation of the small particles is difficult to reflect. The inventive particle size distribution method can be used as the rule of particle size distribution in particle accumulation simulation, and the method performs weighted correction on normal distribution, ensures that the volume of particles with a certain radius obeys normal distribution (if the two-dimensional particle accumulation simulation is performed, the area of the particles with a certain radius obeys normal distribution), can remarkably improve the proportion of small particles in an accumulation body, and accords with the particle distribution condition of a real rock sample.
3. The two-dimensional particle stacking method provided by the invention generates random radius according to the established particle size distribution method, so that the influence of various stacking modes on the porosity is indirectly considered due to the non-uniform diameter and random particle radius, and the method is more consistent with the process of real rock particle stacking.
4. The two-dimensional particle accumulation simulation method for porosity-particle size distribution relation representation, provided by the invention, is wide in application range, is completely based on code development, and has high innovation and flexibility.
Drawings
FIG. 1 is a flow chart of a two-dimensional particle packing simulation method for porosity-particle size distribution relationship characterization;
FIG. 2 is a schematic illustration of a two-dimensional particle produced according to the particle size distribution method set forth in the examples of the present invention;
FIG. 3 is a schematic diagram showing a first case of a particle falling collision rule in the embodiment of the present invention;
FIG. 4 is a diagram showing a second case of a particle falling collision rule in the embodiment of the present invention;
fig. 5 is a schematic diagram of the solution of the final position coordinates of the new particle when the new particle is stably stacked on the other two existing particles in the embodiment of the present invention.
FIG. 6 is a schematic diagram of the left and right side cycle boundaries in an embodiment of the present invention.
Fig. 7 is a diagram of a two-dimensional particle packing model generated according to the proposed two-dimensional particle packing simulation method in an embodiment of the present invention.
Fig. 8 is a graph for studying the qualitative relationship between the porosity and the mean and the relative standard deviation of the particle size of particles based on the generated particle packing model according to an embodiment of the present invention.
Fig. 9 is a graph of the quantitative relationship of the porosity with the relative standard deviation, based on the generated two-dimensional particle packing model, in the example of the present invention.
Fig. 10 is a graph comparing a conventional boundary (left) and a periodic boundary (right).
Detailed Description
In order that those skilled in the art may better understand the present invention. The present invention is described in further detail below.
In order to further illustrate the effectiveness of the technical method, the two-dimensional particle stacking simulation method provided by the invention is further illustrated by taking the characterization of the porosity-particle size distribution relation of the two-dimensional particle stacking body as an example.
Referring to fig. 1, the method comprises the steps of:
s01: the area range for the two-dimensional particle packing simulation was specified as being 800 long and 1000 high, regardless of the specific unit.
S02: in the designated area, randomly generating a space abscissa x according to the uniform distribution in the range of 0 to 800, namely x is more than or equal to 0 and less than or equal to 800. Given that all particles fall from the same height high enough, there is no need to specify the ordinate.
S03: a particle size distribution method capable of accurately reflecting the proportional relation between large particles and small particles and describing the particle size distribution rule is established and used for generating the random radius of subsequent particles.
The method specifically comprises the following steps:
s031: specifying a particle size mean value mu and a standard deviation sigma of a two-dimensional particle accumulation body to be generated;
s032: taking the mean value mu and the standard deviation sigma as input parameters of a probability distribution function and a cumulative distribution function of normal distribution to obtain the probability distribution function
And cumulative distribution function
Wherein r represents the particle radius, σ2Representing the variance, erf represents the error function,
e denotes an exponential function, t denotes an argument, dt denotes the differential to t.
S033: performing weighted correction on the probability distribution function PDF (r) of S032 to obtain a weighted and corrected probability distribution function
Obtaining a weighted modified cumulative distribution function by integration
S034: the weighted and corrected cumulative distribution function c (r) described in S033 is used as a particle size distribution formula of particles in a particle stacking numerical simulation.
S04: according to the uniform distribution, a probability value is randomly generated in the range of (0,1) with equal probability, and is used as the value of the weighted and corrected cumulative distribution function c (r), according to the weighted and corrected cumulative distribution function c (r), a random value can be obtained by reverse extrapolation as the particle radius r, and a circle (x, y, r), namely a two-dimensional particle in the particle stacking numerical simulation, is determined according to the particle radius r and a random abscissa x (the ordinate y is a value larger than the height of the specified stacking area, and the ordinate has no influence on the particle simulation if the ordinate is large enough, which is not discussed here), and the like, which is shown in fig. 2.
S05: after a new particle (x, y, r) is generated, particle fall and collision rules are defined. The particles fall from a height, and generally divided into two cases: if there are no existing particles directly below the falling particle, the particle falls on the lower boundary or on the ground, i.e. the particle has final coordinates of (x, r, r), which is the case with reference to fig. 3; if there are previously generated particles under the falling particles, the new particles may collide with the previous particles. Assuming infinite friction between particles, a new particle (x, y, r) will collide with a particle (x)1,y1,r1) Is rotated to collide with the particles (x)1,y1,r1) The center of the circle is equal height, and y is equal to y1The newly formed particles (x)Y, r) to a new position (x)2,y1R) and then falls again, see fig. 4;
s06: repeating the falling and collision process of step S05, continuously updating the position coordinates of the newly generated particle until the particle falls on the ground, that is, y is 0 or the particle is stably accumulated on two other existing particles, that is, the falling and collision process of the particle is completed; wherein, when a new particle is stably stacked on two other existing particles, the coordinates of the two other existing particles are respectively assumed to be (x)3,y3,r3),(x4,y4,r4) The final position (x ', y') of the new particle satisfies the following equation, see fig. 5:
s07: the left and right periodic boundaries are set for the particle packing process, thereby eliminating the influence of the boundaries on the porosity of the compact particle packing body. The cycle boundary means that when the particles (x, y, r) are in the falling or collision rotation process, if the region range of the left boundary is exceeded, the particles enter the designated region of the particle accumulation again from the right side, and the coordinate is (x, y, r)pY, r), i.e. (x + L, y, r). Similarly, if the particle is beyond the region of the right boundary, the particle enters the designated region of the particle pile again from the left side, and the coordinate is also (x)pY, r), i.e. (x-L, y, r). The abscissa transformation formula of the left and right periodic boundaries is: x is the number ofpWhere mod (·) is a residue function, the remainder of x divided by L is calculated, see fig. 6.
The term periodic boundary means that when the particles generated by simulation exceed the left and right boundaries of the accumulation region, the particles can cross the boundaries and enter the accumulation region again from the other side. The method has the advantages that the periodic boundary can ensure that the particles are tightly packed and arranged, and the influence of the traditional boundary on the porosity of the particle packed body is eliminated. The period boundary can be regarded as a finite region to complete the stacking in an infinite range, namely although the stacking region is only L in length, the period boundary can be regarded as an infinite region approximately, so that the range of particle stacking simulation is indirectly expanded, and the reliability of the porosity of the stacking body is improved. If no period boundary is applied, tight packing cannot be guaranteed on the left side and the right side of the particle accumulation body, in other words, the porosity at the boundary is abnormally increased due to the fact that no period boundary is applied, as shown in fig. 10, the traditional boundary generates abnormally large porosity, and further the porosity of the whole particle accumulation body is not of reference significance, and the calculated porosity of the accumulation body is abnormally large compared with the real situation, so that the accuracy of subsequent porosity-particle size relation representation is seriously influenced. While applying periodic boundaries avoids the creation of larger voids at the left and right boundaries.
In order to accurately characterize the relationship between porosity and particle size distribution, it is a prerequisite that the porosity of a particle stack be reasonably and accurately calculated. If a period boundary is applied, the range of the region simulating the pile-up can be indirectly expanded within a limited pile-up simulation region length L, the period boundary can approximate the pile-up region length to be infinitely long, and the pile-up range is large enough, the calculation of the porosity is more accurate. On the other hand, periodic boundaries can eliminate the effect of traditional boundaries on porosity. The traditional left and right side boundaries limit the particles generated by simulation in the whole accumulation area, so that the porosity of the particles at the left and right boundaries is abnormally increased, and the calculation of the porosity of the whole accumulation body is interfered. Therefore, the application of the periodic boundary ensures that the stack can be tightly packed and the porosity calculation is more reasonable and accurate.
In addition, S03 provides a particle size distribution method, which ensures that the particle size area generated by simulation follows normal distribution, rather than number, and the particle size distribution method is closer to the real situation of the particle size distribution of natural rock particles, and ensures that the particle accumulation simulation can truly and approximately reflect the real rock particle accumulation process. Then, on the basis of the method, a two-dimensional particle stacking simulation method (including period boundaries) is completely proposed for calculation of subsequent porosity and relational characterization of particle size distribution of the porosity particles by using the particle size distribution method.
S08: repeating steps S02-S07, a series of particles are generated. Assuming that the height h of the generated particle stack is 750, when a particle exists in the two-dimensional particle stack, and the vertical coordinate of the center of the particle is greater than the height h, the process of repeatedly generating particles is completed, and the generation of new particles is stopped, because each particle is finally stably stacked on other particles or a few particles are directly stacked on the ground y equal to 0, the two-dimensional particle stack generated through the simulation of the above process is a compact stack, see fig. 7.
S09: for the two-dimensional particle packing generated at S08, the porosity is calculated using a suitable method. From the bottom boundary, upward intercepting a stacking body segment with a certain height h ', h' being less than h, thereby eliminating the influence of no particle stacking on the upper part of the stacking body on the porosity. The porosity φ of the stack segments was calculated.
S10: repeating S01-S09, designating different particle size mean values mu and standard deviations sigma, and designing different particle size mean value and standard deviation combinations, wherein mu ranges from 20 to 40 and the interval is 1; σ starts from 0 to σmaxThe interval is also 1, and the number of the first,
int (·) is an integer function, and there are 235 (μ, σ) combinations, so that N is 235 two-dimensional particle stacks are generated, and the porosity Φ i corresponding to each particle stack is calculatediWherein, i ═ 1,2, 3.
S11: in relative standard deviation
As a parameter for describing the particle size distribution of the particles, the relative standard deviation of each of the stacks was calculated
Wherein, i is 1,2, 3.
The relative standard deviation is used to characterize the particle size distribution of the particles, mainly for the following reasons: in general, the mean μ of the particle size of the particles and the standard deviation σ of the particle size of the particles are the two parameters that most commonly describe the particle size distribution, but by qualitative correlation, it is found that the porosity does not followThe change in the mean value μ of the particle size, i.e. simply changing the mean value of the particle size (i.e. the size of the particles) has no effect on the porosity. The porosity is greatly influenced by the standard deviation sigma of the particle size of the particles, the standard deviation of the particle size of the particles also represents the sorting degree, and the porosity is obviously changed along with the different sorting degrees of the particle size. However, the standard deviation σ of the particle diameters is not comparable to the mean μ of the particle diameters, and for example, the size of a stack of small particles is very different from the size of the small particles, and the particle size distribution is very uneven, in which case the mean of the particle diameters is small but the relative standard deviation is large. The other group of particles are large and small, the distribution is uniform, but the particles are large, the average value of the particle sizes of the particles is large, and the relative standard deviation is small. In this case, there is no way to compare the two stacks of particles by relative standard deviation alone. Therefore, the relative standard deviations cannot be compared with each other independently of the mean particle size of the particles. Therefore, the present patent proposes using the relative standard deviation
As a descriptive parameter of the particle size distribution for the subsequent determination of the quantitative relationship.
S12: the relationship between porosity and relative standard deviation is qualitatively described and the results are graphically shown in fig. 8. Observing and analyzing the qualitative result chart, the parabolic relation exists between the porosity and the permeability, and the relation between the porosity and the permeability is quantitatively expressed, referring to fig. 9, the relation expression is
And determining coefficients a, b and c as-0.1611, 0.06882 and 0.1828 respectively through secondary regression, thereby completing qualitative and quantitative characterization of the porosity and the particle size distribution of the particles.
How to fit the quantitative relationships a, b, c is solved:
by repeating the experiment a plurality of times, i.e. a plurality of particle packing simulation experiments, the relative standard deviation of the particle size of the packed particles of each group and the corresponding porosity, i.e. each scatter point in fig. 9, can be obtained. By using the scatter points (specific data) and using nonlinear regression fitting, a functional relation closest to the distribution trend of the scatter diagram can be obtained, namely specific values of three coefficients of the functional relation a, b and c are determined or solved.
On the basis of the particle stacking numerical model generated by the method, the qualitative and quantitative relation between the porosity and the particle size distribution of the particles is researched, and the fact that the porosity is irrelevant to the mean value of the particles and relevant to the relative standard deviation can be seen. The larger the relative standard deviation is, the more uneven the particle size distribution is, the smaller the small particles block the pores formed by the large particles, and the smaller the porosity is; the smaller the relative standard deviation, the better the particle sorting, the tighter the particle packing, and the smaller the porosity as well. In the embodiment of the invention, the quantitative relation between the two is a parabolic relation, and a relation formula of the two can be determined through nonlinear regression analysis. Therefore, the two-dimensional particle stacking simulation method for the porosity-particle size distribution relation characterization provided by the invention is proved to be feasible.
The working principle of the method is as follows:
the present invention first fixes the range of the particle accumulation body by specifying the region (L × H) of the two-dimensional particle accumulation simulation. And establishing a particle size distribution method C (r) capable of reflecting the real particle size distribution of the rock particles, defining the random coordinates and the radius of the generated particles, and generating a two-dimensional particle (x, y, r). And then, establishing a falling and collision rule of the particles, calculating final stable coordinates (x ', y', r) after the particles fall or collide, and applying left and right side boundary conditions to the two-dimensional particle accumulation area to eliminate the influence of the boundary on the porosity. This process is repeated to produce a close-packed two-dimensional stack of particles. Finally, different combinations of mean values mu and standard deviations sigma are designed to generate a series of two-dimensional particle accumulation bodies, and the porosity phi of each accumulation body is calculated respectively and used for follow-up statistical study of qualitative and quantitative relations between the porosity and particle size distribution of particles.
Finally, the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered in the claims of the present invention.
Claims (2)
1. The method for characterizing the relationship between the porosity and the particle size distribution of the particles is characterized by comprising the following steps: the method comprises the following steps:
s01: specifying a region range of two-dimensional particle accumulation simulation, wherein the length is L and the height is H;
s02: randomly generating a space abscissa x in a designated area range according to a uniform distribution method, wherein x is more than or equal to 0 and less than or equal to L, and assuming that all particles fall from the same height y and the height is high enough, namely y is more than H;
s03: establishing a particle size distribution method capable of accurately describing a particle size distribution rule:
s031: specifying a particle size mean value mu and a standard deviation sigma of a two-dimensional particle accumulation body to be generated;
s032: taking the mean value mu and the standard deviation sigma as input parameters of a probability distribution function and a cumulative distribution function of normal distribution to obtain the probability distribution function
And cumulative distribution function
Where r represents the particle radius, σ represents the variance, erf represents the error function,
e represents an exponential function, t represents an independent variable, and dt represents the differential to t;
s033: performing weighted correction on the probability distribution function PDF (r) of S032 to obtain a weighted and corrected probability distribution function
Obtaining a weighted modified cumulative distribution function by integration
S034: taking the weighted and corrected cumulative distribution function C (r) of S033 as a particle size distribution formula of particles in the particle stacking numerical simulation;
s04: randomly generating a probability value within the range of (0,1) according to uniform distribution in an equal probability manner, taking the probability value as the value of the weighted and corrected cumulative distribution function C (r), reversely deducing the probability value according to the weighted and corrected cumulative distribution function C (r) to obtain a random value as the particle radius r, and determining a circle (x, y, r) which is a two-dimensional particle in particle stacking numerical simulation according to the two elements of the particle radius r and a random abscissa x;
s05: after a new particle (x, y, r) is generated, a particle fall and collision rule is defined, and the particle falls from a height:
if there are no existing particles directly below the falling particle, the particle falls on the lower boundary or on the ground, i.e. the particle has final coordinates of (x, r, r);
if there is a previously generated particle under the falling particle, the new particle collides with the previous particle, and if there is infinite friction between the particles, the new particle (x, y, r) collides with the particle (x)1,y1,r1) Is rotated to collide with the particles (x)1,y1,r1) The center of the circle is equal height, and y is equal to y1Thereafter, the newly formed particles (x, y, r) collide to a new location (x)2,y1And r) then falls again;
s06: repeating the falling and collision process of step S05, continuously updating the position coordinates of the newly generated particle until the particle falls on the ground, that is, y is 0 or the particle is stably accumulated on two other existing particles, that is, the falling and collision process of the particle is completed; wherein, when a new particle is stably stacked on two other existing particles, the coordinates of the two other existing particles are respectively assumed to be (x)3,y3,r3),(x4,y4,r4) The final position (x ', y') of the new particle satisfies the following equation:
s07: setting left and right side periodic boundaries for the particle accumulation process, wherein the periodic boundaries are that when the particles (x, y, r) are in the falling or collision rotation process and if the region range of the left side boundary is exceeded, the particles enter the designated region of the particle accumulation again from the right side, and the coordinate is (x)pY, r); similarly, if the particle is beyond the region of the right boundary, the particle enters the designated region of the particle pile again from the left side, and the coordinate is also (x)pY, r); the abscissa transformation formula of the left and right periodic boundaries is: x is the number ofp(x) mod (L), where mod (·) is a remainder function, calculating the remainder of x divided by L;
s08: repeating the steps S02-S07 to generate a series of particles, setting the height of the generated particle accumulation body as h, finishing the process of repeatedly generating the particles when one particle exists in the two-dimensional particle accumulation body and the vertical coordinate of the circle center of the particle is larger than the height h, and stopping continuously generating new particles to obtain a compact two-dimensional particle accumulation body;
s09: for the two-dimensional particle packing generated at S08, porosity was calculated:
s10: repeating the steps S01-S09 to design different particle size mean values muiStandard deviation σiCombining to generate N two-dimensional particle stacks, and calculating the porosity phi corresponding to each particle stackiWherein, i ═ 1,2, 3.., N;
s11: in relative standard deviation
As a parameter for describing the particle size distribution of the particles, the relative standard deviation of each of the stacks was calculated
Wherein, i ═ 1,2, 3.., N;
s12: qualitatively and quantitatively describing the relationship between the relative standard deviation and porosity as
The a, the b and the c are coefficients which are determined by a nonlinear regression method.
2. A method for characterizing porosity as a function of particle size distribution as claimed in claim 1, wherein: the method for calculating the porosity in S09 is as follows: and (3) from the bottom boundary of the two-dimensional particle accumulation body, upwards intercepting an accumulation body segment with a certain height h ', wherein h' is less than h, calculating the porosity phi of the accumulation body segment, and repeating the steps to completely calculate the porosity of the two-dimensional particle accumulation body.
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