CN102930148B - Method for determining piping penetration coefficient based on random start - Google Patents

Abstract

The invention discloses a method for determining a piping penetration coefficient based on random start. The method for determining the penetration coefficient in the piping generation and development process is based on distribution densities or probability functions of a mean speed, a mean particles size of particles and a mean positional relation, based on parameters of mathematical expectation and the like, and based on combination of mechanical analysis. The method has significance for reasonably evaluating the dam safety, predicting an area where penetration damage possibly occurs and the form and reducing the losses caused by the penetration damage.

Description

A Method for Determination of Piping Permeability Coefficient Based on Random Start

technical field

The invention belongs to the technical field of groundwater seepage calculation, and in particular relates to a method for determining a piping seepage coefficient based on random start.

Background technique

In the past research on piping, attention was often focused on the study of the average characteristic (that is, the deterministic law), and the randomness of piping occurrence was not paid enough attention to. A considerable number of theoretical workers still used pure mechanical methods according to the inevitable phenomenon. Methods to conduct research without analyzing its randomness often fail to explain some actual phenomena. The randomness of piping can be understood from the following two aspects: 1) the randomness of the start of movable particles; 2) the randomness of the spatial distribution of soil properties.

The essence of piping is the process in which movable particles move and lose in pores under the drag of water flow, and the starting of particles in soil is accidental. In general, the size of a particle and its position in contact with other particles, as well as water velocity, are random variables that are difficult to determine in time. Therefore, the starting phenomenon has great chance. Even if it is known that the hydraulic critical condition of piping is in the starting state according to the previous judgment, it is difficult to determine the law of the number of starting particles because it is impossible to know the instantaneous velocity, diameter of starting particles, and their position and contact relationship at that time, that is, That is to say, if only the average permeation flow rate, the average particle size and the average position relationship of the particles are known, the number law is difficult to determine.

Contents of the invention

Purpose of the invention: In view of the problems existing in the calculation of the permeability coefficient during the piping development process, the purpose of the present invention is to propose a method for determining the permeability coefficient during the piping development process on the basis of considering the randomness of particle starting.

Technical solution: the present invention mainly calculates the randomness of the start of movable particles from the mesoscopic level, according to the distribution density or probability function of the average permeation velocity, the average particle size and the average positional relationship of the particles, and according to parameters such as mathematical expectation, Combined with the mechanical analysis, a method to determine the permeability coefficient during the piping development is proposed.

The method for determining the piping permeability coefficient based on random start according to the present invention is characterized in that it comprises the following steps:

(1) Divide the particles of non-cohesive piping soil into two types, skeleton particles and movable particles, and the critical particle size is calculated by formula (1):

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}{{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula: x _a is the critical particle size, if the particle size is larger than this, it is a skeleton particle, and if it is smaller, it is a movable particle; the coefficient _c1 is the shape coefficient of the particle; the coefficient _c2 is the coefficient related to the compactness of the soil (for the natural circle For shaped sand and gravel, c ₁ = 0.73, for rock debris, c ₁ = 1.0; coefficient c ₂ represents the compactness of soil, when it is the densest, c ₂ = 0, when it is slightly dense, c ₂ = 0.05, when it is the loosest, c ₂ =0.18); A _a and B _a are determined by formula (2) and formula (3) respectively:

${\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; dy</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; dy</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; dy</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

In the formula: y _a is the movable particle content corresponding to the critical particle size x _a ; y is the content smaller than a certain particle size x, and x(y) is the particle size corresponding to y;

The solution of x _a can adopt trial algorithm or iterative method: first assume the initial value x _a0 of x _a , calculate A _a and B _a from formula (2) and formula (3), and put A _a and B _a into formula ( 1) Obtain an x _a , record it as x _a1 , if the difference between x _a0 and x _a1 is small, take the average value of the two as x _a , otherwise use x _a1 to repeat the above steps;

(2) According to the critical particle size x _a calculated according to formula (1), the average particle size of each particle size group of movable particles smaller than x _a is:

${\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; D.</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; D.</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

In the formula: is the average particle diameter of the i movable particle group; D _(i-1) and D _(i) are respectively the representative particle diameters of the (i-1) and i particle size groups;

The particle number of the i-th movable particle group is:

${\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; D.</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

In the formula: N _i is the particle number of the i-th movable particle group; is the average particle size of the i-th movable particle group, determined by formula (4); M _i is the mass of the i-th movable particle group, which can be determined by the gradation curve; ρ _s is the density of movable particles;

(3) Considering the randomness of pore water velocity, assuming a normal distribution, the porosity of the soil after the loss of movable particles is:

${\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; E.</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; E.</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

In the formula: n′ is the porosity of the soil after the loss of movable particles; n is the original porosity; k is the grouping number of movable particles; E(N) _i is the expected value of the number of starting particles in the ith movable particle group , determined by formula (7); V _i is the initial pore flow velocity of the i-th motile group, determined by formula (8);

The expected value of the number of particles activated in the i-th motile group is:

In the formula: E(N) _i is the expected value of the number of starting particles in the i-th movable particle group; N _i is the number of movable particles in the i-th movable particle group, which is determined by formula (5); V _i is the i-th movable particle number Kinetochore group starting pore flow velocity is determined by formula (8); v' is the average pore flow velocity; σ is the standard deviation.

The initial pore flow velocity of the i-th movable particle group is:

In the formula: V _i is the initial pore flow velocity of the i-th movable particle group; is the average particle size of the i-th movable particle group, determined by formula (4); ρ _s and ρ _w are the densities of the movable particle and the fluid respectively; μ is the viscosity coefficient of the fluid; θ is the line connecting the centers of the two particles the angle with the vertical direction; is the angle between the dragging direction of the water flow and the horizontal direction, and g is the acceleration of gravity;

(4) Divide the movable particles into k particle groups from fine to coarse, then the remaining particle mass of the i particle group The value of i from 1 to k is calculated to calculate the remaining mass of each particle group, and according to the calculation results, the updated soil grading curve after the loss of movable particles is obtained. Put the calculation result of formula (6) and the d _ω determined by the updated grading curve into formula (9) to obtain the permeability coefficient K of the soil after the loss of movable particles:

$\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 780</span>}\frac{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

In the formula: K is the soil permeability coefficient after the loss of movable particles; d _ω is the equivalent particle size, which is determined according to formula (10):

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

In the formula: d _i is the average particle size of the i-th grade particle size, is the particle weight corresponding to the particle size class of d _i , and d ₁ is the particle size of the first group of movable particles.

Compared with the prior art, the present invention has the beneficial effects that: the permeability coefficient distribution calculated by the method of the present invention through the above-mentioned process is more accurate and reliable due to the consideration of the randomness of rock and soil particles starting, and the result is more accurate and reliable through sub-area The calculation can finally obtain the change process of piping development. It is of great significance to reasonably evaluate the safety of dams, predict the possible areas and forms of seepage damage, and reduce the loss caused by seepage damage.

Description of drawings

Fig. 1 is the particle grading curve of the sand layer calculated in Example 1 of the present invention;

Fig. 2 is the embankment model partition and the unit division figure that the embodiment of the present invention 1 calculates;

Fig. 3 is the water head contour map that occurs when the piping outlet is calculated at 30m away from the foot of the embankment in embodiment 1 of the present invention;

Fig. 4 is the hydraulic gradient contour diagram of the region near the piping outlet calculated in Embodiment 1 of the present invention;

Fig. 5 is a contour diagram of the hydraulic gradient of the embankment after the calculation of the permeability coefficient of the region with a gradient greater than 0.2 in Embodiment 1 of the present invention.

Detailed ways

The technical solutions of the present invention will be described in detail below, but the protection scope of the present invention is not limited to the embodiments.

Embodiment 1: In this embodiment, seepage calculation is carried out by taking a large embankment as an example.

The levee adopts a homogeneous embankment structure, ^{which is formed by accumulation of homogeneous clay with a permeability coefficient of 1×10 -8 m} /s. layer, the permeability coefficient is 1.5×10 ^-4 m/s. The physical indexes of the sand layer are shown in Table 1, where C _c and C _u are the curvature coefficient and unevenness coefficient of the particle gradation curve, G _s is the relative density of soil particles, n is the initial porosity, d ₁₀ , d ₃₀ and d ₆₀ They are the diameters of soil particles corresponding to 10%, 30% and 60% of mass cumulative percentages smaller than a certain particle size. See Fig. 1 and Fig. 2 for particle gradation and calculation partition, and Fig. 3 and Fig. 4 for the distribution of hydraulic gradient after the emergence of piping outlets. The range where the hydraulic gradient is greater than 0.2 is taken as the calculation range, and the calculation result is shown in Figure 5.

Table 1 physical index of sand layer in the embodiment of the present invention 1

C _c C _u G _s no d ₁₀ (mm) d ₃₀ (mm) d ₆₀ (mm) 1.29 7.60 2.68 0.38 0.113 0.355 0.62

According to the inventive method, each step calculation result is as follows:

(1) Take c ₁ =0.73, c ₂ =0.05, calculate through step (1) x _a =0.5mm,

(2) Divide the movable particles into 3 groups, which are 0-0.075mm, 0.075mm-0.25mm, and 0.25mm-0.5mm. See Table 2 for the calculation results of particle size grouping, average particle size, mass content, and the number of particles in each group.

Table 2 Grouping of movable particles in Example 1 of the present invention.

Particle size range mm 0-0.075 0.075-0.25 0.25-0.5 The average particle size 0.038mm 0.163mm 0.375mm % by mass 6.2 11 25.2 Number of particles in 1cm ³ 1338610 30091 5661

(3) Calculate n'=0.5 through step (3).

(4) After step (4) d _ω =0.449mm, K=0.35.

(5) By adopting the permeability coefficient K=0.35, the contour map of the hydraulic gradient can be calculated and obtained in the range where the hydraulic gradient is greater than 0.2.

As stated above, while the invention has been shown and described with reference to certain preferred embodiments, this should not be construed as limiting the invention itself. Various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (2)

1. A method for determining the piping permeability coefficient based on random start, characterized in that it may further comprise the steps: (1) Divide the particles of non-cohesive piping soil into two types, skeleton particles and movable particles, and the critical particle size is calculated by formula (1): ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>\frac{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}{{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ In the formula: x _a is the critical particle size, which is larger than the particle size, which is the skeleton particle, and smaller than it, which is the movable particle; the coefficient _c1 is the shape coefficient of the particle; the coefficient _c2 is the coefficient related to the compactness of the soil; A _a and B _a is determined by formula (2) and formula (3): ${\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; dy</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; dy</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; dy</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ In the formula: y _a is the movable particle content corresponding to the critical particle size x _a ; y is the content smaller than a certain particle size x, and x(y) is the particle size corresponding to y; (2) According to the critical particle size x _a calculated according to formula (1), the average particle size of each particle size group of movable particles smaller than x _a is: ${\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; D.</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; D.</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ In the formula: is the average particle diameter of the i movable particle group; D _(i-1) and D _(i) are respectively the representative particle diameters of the (i-1) and i particle size groups; The particle number of the i-th movable particle group is: ${\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 6</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}{{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; D.</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ In the formula: N _i is the particle number of the i-th movable particle group; is the average particle size of the i-th movable particle group, determined by formula (4); M _i is the mass of the i-th movable particle group, which can be determined by the gradation curve; ρ _s is the density of movable particles; (3) Considering the randomness of pore water velocity, assuming a normal distribution, the porosity of the soil after the loss of movable particles is: ${\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; E.</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; E.</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$ In the formula: n′ is the porosity of the soil after the loss of movable particles; n is the original porosity; k is the grouping number of movable particles; E(N) _i is the expected value of the number of starting particles in the ith movable particle group , determined by formula (7); V _i is the initial pore flow velocity of the i-th motile group, determined by formula (8); The expected value of the number of particles activated in the i-th motile group is: In the formula: E(N) _i is the expected value of the number of starting particles in the i-th movable particle group; N _i is the number of movable particles in the i-th movable particle group, which is determined by formula (5); V _i is the i-th movable particle number Kinetochore group starting pore flow velocity is determined by formula (8); v' is the average pore flow velocity; σ is the standard deviation; The initial pore flow velocity of the i-th movable particle group is: In the formula: V _i is the initial pore flow velocity of the i-th movable particle group; is the average particle size of the i-th movable particle group, determined by formula (4); ρ _s and ρ _w are the densities of the movable particle and the fluid respectively; μ is the viscosity coefficient of the fluid; θ is the line connecting the centers of the two particles the angle with the vertical direction; is the angle between the dragging direction of the water flow and the horizontal direction, and g is the acceleration of gravity; (4) Divide the movable particles into k particle groups from fine to coarse, then the remaining particle mass of the i particle group Take the value of i from 1 to k to calculate the remaining mass of each particle group, and obtain the updated soil grading curve after the loss of movable particles according to the calculation results; Put the calculation result of formula (6) and the d _ω determined by the updated grading curve into formula (9) to obtain the permeability coefficient K of the soil after the loss of movable particles: $\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 780</span>}\frac{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$ In the formula: K is the soil permeability coefficient after the loss of movable particles; d _ω is the equivalent particle size, which is determined according to formula (10): $\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}}{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$ In the formula: d _i is the average particle size of the i-th grade particle size, is the particle weight corresponding to the particle size class of d _i , and d ₁ is the particle size of the first group of movable particles. 2. the method for determining the piping permeability coefficient based on random start according to claim 1 is characterized in that: in the step (1), the solution of the critical particle size x _a can adopt a trial and error method or an iterative method: first assume the initial value of x _a value x _a0 , calculate A _a and B _a from formula (2) and formula (3), put A _a and B _a into formula (1), get a x _a , denote it as x _a1 , if x _a0 and x If the difference between _a1 is small, take the average value of the two as x _a , otherwise use x _a1 to repeat the above steps.