CN117828907B - Dam break mathematical model for homogeneous dam and parameter determination method - Google Patents

Abstract

The present invention discloses a dam-break mathematical model and parameter determination method for a homogeneous dam, including: determining dam body characteristic parameters, dam material characteristic parameters, upstream water level-reservoir surface area relationship curve, time-flow curve and time-discharge curve; setting the allowable value of the upstream bottom elevation of the breach; calculating the initial steep slope elevation; setting the total calculation time and the time step of each calculation; calculating the breach flow, erosion, upstream reservoir water level change, upstream bottom elevation of the breach, breach top width change, breach bottom width change of each time step; judging whether shear failure occurs in the steep slope; updating data if shear failure occurs, continuing calculation if shear failure does not occur, judging whether the breach slope is stable, and finally outputting the calculation result. The dam-break mathematical model of the present invention takes into account factors such as soil moisture content, compaction degree, and clay content when calculating the steep slope erosion rate, so as to reduce the error of the prediction result.

Description

A dam-break mathematical model and parameter determination method for homogeneous dams

Technical Field

The invention belongs to the technical field of dam-break calculation of hydraulic engineering, and in particular to a dam-break mathematical model and parameter determination method for a homogeneous dam.

Background technique

The geometric shape of the homogeneous dam body and the physical and mechanical properties of the dam material are the key factors affecting the dam break process of the homogeneous dam. The differences in clay content and compaction degree of different dam materials will greatly affect the dam break process of the homogeneous dam. These parameters mainly affect the erosion rate of the homogeneous dam steep slope and the erosion resistance of the dam material when calculating the dam break mathematical model.

Existing literature: Zhong Qiming et al. [J]. Journal of Hydraulic Engineering, 2016, 47(12): 1519-1526, which discloses the research and application of the mathematical model of the overtopping and dam-break process of homogeneous earth dams. The _article adopts the steep slope erosion rate empirical model proposed by Tony L. Wahl in 1998 for the steep slope movement parameter CT . This model does not consider factors such as soil moisture content, compaction degree, and clay content. Studies have shown that these factors have an impact on the steep slope movement parameters, resulting in the poor performance of the mathematical model described in the paper in actual application and a small scope of application. Therefore, when predicting the breach flow process, there is a large error compared with the actual situation.

Summary of the invention

In order to overcome the technical problem that the parameter calculation in the existing homogeneous dam overtopping and dam-break mathematical model does not take into account factors such as soil moisture content, compaction degree, and clay content, resulting in large errors in the actual application of the mathematical model, the present invention provides a homogeneous dam overtopping and dam-break mathematical model and a parameter determination method taking into account factors such as soil moisture content, compaction degree, and clay content.

To achieve the above purpose, the technical measures taken by the present invention are:

A dam-break mathematical model and parameter determination method for a homogeneous dam, comprising the following steps:

Step 1: Determine the geometric characteristic parameters of the dam body, the physical and mechanical characteristic parameters of the dam material, the upstream water level-reservoir surface area relationship curve, the time-inflow curve and the time-discharge curve;

Step 2: Set the allowable value of the bottom elevation upstream of the breach;

The numerical value of the bottom elevation upstream of the breach is determined based on whether foundation erosion occurs. If foundation erosion does not occur, the value is 0; if foundation erosion occurs, the value should be less than 0.

Step 3, calculating the initial steep slope elevation;

Step 4: Set the total calculation time and the time step for each calculation;

Step 5, calculate the breach flow, erosion, upstream reservoir water level change, breach upstream bottom elevation, breach top width change and breach bottom width change at each time step;

Step 6, respectively calculate the upstream reservoir water pressure F1 _, the downstream tailwater pressure F2 , _the shear force _of the water flow on the upper part of the steep slope F3 _, the water weight on the upper part of the steep slope F4 , _the self-weight of the steep _{slope F5} , the friction force F6 on the steep _slope and the internal cohesion F7 _of the steep slope soil . If F1-F2+F3＞F6+F7 _, it is determined that _the steep slope has shear failure and _proceed to step 7, otherwise proceed to step 8;

Step 7, updating and calculating the upstream reservoir water level, the upstream bottom elevation of the breach, the top width of the breach, and the bottom width of the breach according to the change calculated in step 5;

Step 8, steep slope erosion calculation, update the calculated erosion amount, upstream reservoir water level, upstream bottom elevation of breach, breach top width and breach bottom width according to the change calculated in step 5;

Step 9, calculate ΔZ _w / Z _w and determine whether ΔZ _w / Z _w is less than ε , where ε is the calculation convergence condition and is taken as 0.0001 or adjusted according to the actual situation; if yes, proceed to step 10, otherwise return to step 5 to continue the calculation; Δ Z _w represents the change in the water level of the upstream reservoir, and Z _w represents the water level of the upstream reservoir;

Step 10, determine whether the breach slope is stable, if yes, determine whether the cumulative calculation time is greater than the set calculation time, if the cumulative calculation time is greater than or equal to the set total calculation time, then end the calculation, and output the calculation results of breach flow, erosion, upstream reservoir water level change, upstream bottom elevation change, breach top width change, and breach bottom width change;

If the accumulated calculation time is less than the set total calculation time, return to step 5 to calculate the next time step.

Furthermore, in step 3, the initial steep slope elevation, i.e., the initial value of the upstream bottom elevation of the breach, is calculated, specifically including:

Z _b =( l _down -l _n ) sin( α _x )

Where: Frn is the Froude number; _Qb is the breach flow, ^m3 /s _; Bt is the initial breach top width _, m, the breach width of the downstream slope takes the _same value; C is the Xie Cai coefficient; Zw is the _upstream reservoir water level, m; Zb is the elevation of the upstream bottom of the breach, m. g is the gravity acceleration, m/ ^s2 ; ldown is the length of the downstream dam slope, m; ln is the length from the vertex of the downstream dam slope _{to the} initial steep slope position, m; αx is the slope angle of the downstream dam _slope , °;

The calculation method of Xie Cai coefficient is:

Where: n is the Manning roughness coefficient, n = (d ₅₀ ) ^1/6 /12, d represents the particle size;

The value _of αx is a fixed value, which is the basic data of the homogeneous dam. Bt is _the initial breach top width, which is generally an assumed value and can be selected according to the actual situation. The values of _ZW and _Zb are given as the initial values of the calculation. The initial values of the calculation are the actual conditions at the time of dam breach, and the subsequent values are calculated by the program.

The breach flow is calculated using the water balance formula:

Where: Q is the total discharge flow, ^m3 /s; Qb is the breach flow, ^m3 /s; _Qspill is the spillway outflow, ^m3 /s, corresponding to the discharge value at each moment in the time-discharge array, and is set to zero when there is no spillway outflow; _{Qin is} the inflow flow, ^m3 /s, corresponding to the inflow value at each moment in the time-inflow array; Aw _is the upstream reservoir area, ^m2 ; Zw is the upstream reservoir _water level, m; t is time, s.

Furthermore, when there is only the upstream water level-reservoir capacity relationship curve, a quadratic curve fitting is used:

Among them, A _w is the upstream reservoir area, m ² ; Z _w is the upstream reservoir water level, m; V _w is the upstream reservoir capacity, m ³ ; a ₁ , b ₁ and c ₁ are all fitting coefficients, among which a ₁ and b ₁ need to retain 8 decimal places.

Furthermore, if there is a lack of upstream water level-reservoir capacity and upstream water level-reservoir surface area relationship curves, the following formula is used for calculation:

When the reservoir area and storage capacity of _a certain water level of a reservoir are known, the two coordinate points ( Z _w1 , A _w1 ) and ( Z _w1 , V _w1 ) are used to calculate the values of a2 and b2 . When only the total storage capacity is available, b2 _is taken as 2.0, and _{the value of a2 is} obtained by substituting ( Z _w2 , V _w1 ) into A _W = a ₂ Z _w ² .

Furthermore, the weir flow formula is used to calculate the breach flow in step 5:

In the formula: σs is the downstream tailwater inundation coefficient; Bd _is the breach bottom width, m; Zw is the upstream reservoir water level, m; Zb _{is the} upstream bottom elevation _of the breach, m; β is the breach _slope ratio, °; when ₍ Zd - Zb )/( Zw _- Zb ) < _0.8 , it is free outflow, and _{when (Zd} - Zb ₎ /( Zw _- Zb ₎ ≥ 0.8, it is submerged outflow; Zd _is the downstream tailwater level, m.

Furthermore, when calculating the erosion amount in step 5, the trace erosion rate is expressed as d x /d t :

Where: C _h is the steep slope erosion parameter related to the dam material characteristics, which is the key factor controlling the dam break process of homogeneous dams; q is the flow rate per unit width of the breach, m ³ /s; h is the steep slope height.

Furthermore, the calculation formula of the steep slope erosion _parameter Ch is:

In the formula: k is the multivariate nonlinear regression fitting coefficient, w is the moisture content of the dam material when the homogeneous dam breaks, w _opt is the optimal moisture content of the homogeneous dam material, w/w _opt represents the water holding state of the homogeneous dam material, c 1 is the corresponding estimated coefficient; ρ _d is the dry density of the homogeneous dam, kg/m ³ ; ρ _dm is the maximum dry density of the dam material, kg/m ³ , ρ _d / ρ _dm is the compaction degree of the homogeneous dam, c 2 is the corresponding estimated coefficient; ρ _w is the density of water, kg/m ³ _, ρ _d / ρ _w represents the relative density of the dam material with water as the standard substance, c 3 is the corresponding estimated coefficient; c _c is the clay content of the dam material, ranging from 0 to 1, c 4 is the corresponding estimated coefficient; the corresponding parameter values are k =-2.931, c 1=0.130, c 2=0.315, c 3=2.052, c 4=0.117.

First, the steep slope erosion rate dx/dt has an impact on the breach flow. The larger the steep slope erosion rate dx/dt, the larger the breach flow. The steep slope erosion rate dx/dt is also affected by the steep slope erosion parameter Ch , which is positively correlated. The steep slope erosion parameter _{Ch is} affected by the soil moisture content, compaction degree, and clay content, and they are all negatively correlated. Therefore, the increase of soil moisture content, compaction degree, and clay content will reduce the breach flow. If these factors are not considered, the calculation accuracy of the breach flow will be reduced.

Furthermore, in step 5, the following formula is used to calculate the change in the top width and bottom width of the breach:

In the formula: Δ B _t is the increment of the breach top width, m; Δ B _d is the increment of the breach bottom width, m; n _loc represents the location of the breach, n _loc = 1 means that the breach is located at the dam abutment and can only develop in one direction; n _loc = 2 means that the breach is located in the middle of the dam crest and can develop to both sides; Δ z _b is the increment of the breach depth, m.

Furthermore, in step 10, it is determined whether the breach slope is stable, that is, whether the driving force Fd of the sliding wedge _is greater than the anti-sliding force Fr. If _so , it indicates that the breach slope is unstable. Fd and Fr _{are respectively} expressed as:

Where: Ws _is the weight of the sliding wedge, kg; θ is the slope angle _of the breach slope after instability, °; γb is the bulk density of the soil, N/ ^m3 ; Hs is the height of the breach slope, m; φ is the internal friction angle _of the soil, °; c is the cohesion of the soil, kPa; β is the slope ratio of the breach slope.

Beneficial effects of the present invention:

The dam-break mathematical model of the present invention takes into account factors such as soil moisture content, compaction degree, clay content, etc. when calculating the steep slope erosion rate, thereby reducing the error of the prediction result.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a schematic diagram of the initial breach formation of a homogeneous dam overtopping;

Figure 2 shows the initial scouring pit location of the homogeneous dam overtopping and collapse;

Figure 3 is a schematic diagram of the formation of the initial scouring pit during the overtopping and collapse of a homogeneous dam;

Figure 4 is a schematic diagram of the formation of a homogeneous dam overtopping and breach steep slope;

Figure 5 is a schematic diagram of the overtopping and erosion of the steep slope of the homogeneous dam;

Figure 6 is a schematic diagram of shear failure stress on a steep slope;

Figure 7 is a schematic diagram of the development process of the breach at the top of a homogeneous dam;

Figure 8 is a schematic diagram of the development process of the breach on the downstream slope of the homogeneous dam;

Figure 9 is a schematic diagram of the stability analysis of the breach slope of a homogeneous dam;

Figure 10 is a model calculation flow chart;

Figure 11 is the water level-storage capacity relationship curve of Youfangqu Dam No. 1;

Figure 12 shows the water level-reservoir capacity relationship curve for Changhangou Dam No. 1 and No. 2;

Figure 13 is a diagram showing the calculation results of the dam-break flow of the No. 1 dam in Youfang Channel;

Figure 14 is a diagram showing the calculation results of the dam-break flow of Changhangou No. 1 Dam;

Figure 15 shows the calculation results of the dam break flow of Changhangou No. 2 Dam.

Detailed ways

The present invention is described in detail below with reference to the accompanying drawings.

The method was written into a program using computer language, and the dam breach cases of Youfangqu No. 1, Changhangou No. 1 and Changhangou No. 2 were selected to explain the calculation process of the mathematical model of the method.

As shown in FIG10 , the mathematical model calculation process of the present method includes:

Step 1, determine and input the geometric characteristic parameters of the dam body, the physical and mechanical characteristic parameters of the dam material, the upstream water level-reservoir surface area relationship curve, the time-inflow curve and the time-discharge curve, as shown in Table 1 below;

Table 1 Foundation parameters of three breached dams

Among them, w/wopt represents the water holding state of homogeneous dam materials; ρd /ρdm _is the compaction degree _of homogeneous dam construction, kd _is the erosion coefficient _of dam materials; τc is the starting shear stress of dam materials; Ch is the steep _slope erosion parameter.

Step 2: Set the allowable value of the bottom elevation upstream of the breach, and all three dam bodies are set to 0;

Step 3, calculating the initial steep slope elevation;

As shown in Figure 1, the initial breach of the homogeneous dam is formed with the initial steep slope elevation:

Z _b =( l _down -l _n )sin( α _x )

As shown in Figures 2 and 3, water flowing over the dam top and eroding the downstream dam slope is the initial stage of the homogeneous dam overtopping and dam failure. The velocity of the overtopping water flowing through the downstream dam slope gradually increases, which will cause the formation of an initial scouring pit on the downstream dam slope. The initial scouring pit is formed at point B, where the steep slope of the downstream slope is formed:

Where: Frn is the Froude number; _Qb is the breach flow, ^m3 /s _; Bt is _the initial breach top width, m, and the downstream slope breach width takes the same value; C is the Xie Cai coefficient; Zw is the upstream _reservoir water level, m; Zb is the upstream bottom elevation of the breach, m _. g is the gravity acceleration, m/ ^s2 ;

The calculation method of Xie Cai coefficient is:

In Figure 2, dn _is the water flow depth on _the downstream dam slope, m; αs _is the slope angle of the upstream dam slope, °; αx is the slope angle of the downstream dam slope, °; ldown is the length of the _downstream dam slope, m; ln _is the length from the vertex of the downstream dam slope to the initial steep slope position, m.

The breach flow is calculated using the water balance formula:

Where: Q is the total discharge flow, ^m3 /s; Qb is the breach flow, ^m3 /s; _Qspill is the spillway outflow, ^m3 /s, corresponding to the discharge value at each moment in the time-discharge array, and is set to zero when there is no spillway outflow; _{Qin is} the inflow flow, ^m3 /s, corresponding to the inflow value at each moment in the time-inflow array; Aw _is the upstream reservoir area, ^m2 ; Zw is the upstream reservoir _water level, m; t is time, s.

The above model calculation uses the water level-reservoir area relationship curve A _w - Z . However, when there is only the water level-reservoir capacity relationship curve V _w - Z as shown in Figures 11 and 12 , a quadratic curve fitting can be used for V _w - Z :

a1 _, b1 _{and c1} are _{all fitting coefficients, where a1} and _b1 need to be rounded to 8 decimal places.

If there is a lack of water level-reservoir capacity and water level-reservoir surface area relationship curves, the following formula can be used for calculation:

When the reservoir area and storage capacity of a certain _water level of the reservoir are known, the two coordinate points ( Z _w1 , A _w1 ) and ( Z _w1 , V _w1 ) can be used to calculate the values of a2 and b2 . Z _w1 is the specific value of Z _w when the reservoir area and storage capacity are known, _and A _w1 is the specific value of A _w when the reservoir area and storage capacity are known. When only the total storage capacity (the storage capacity corresponding to the verification water level) is available, _the recommended value of b2 is 2.0. Substitute ( Z _w2 , V _w1 ) into A _W = a ₂ Z _w ² to obtain the value _{of a2} . Z _w2 is the specific value of Z _w when only the total storage capacity is available.

This method is a method of using other known parameters to quickly and accurately obtain water level and reservoir area when there is no relationship between water level and reservoir area, and finally obtain A _w - Z _w data for flow calculation.

Step 4: Set the total calculation time and the time step for each calculation;

Step 5, calculate the breach flow, erosion, upstream reservoir water level change, downstream tailwater level change, breach top width change and breach bottom width change at each time step;

During the overtopping and dam-breaking process of a homogeneous dam, the weir flow formula is used to calculate the breach flow:

In the formula: σs _{is the} downstream tailwater inundation coefficient, _and the value _of σs is shown in Table 2; Bd is _the breach bottom width, m; Zw is the upstream reservoir water level, m; Zb is _the upstream bottom elevation of the breach, m; β is the breach slope ratio, ° _; when ( Zd - Zb )/ (Zw _- Zb ₎ < 0.8, it is free outflow, and when (Zd _{- Zb} ) / (Zw _{- Zb} ) ≥ 0.8, it is submerged outflow; Zd _is the downstream tailwater level, m.

Table 2 Submergence coefficient σs _value table

As shown in Figures 3, 4, and 5, after the initial scour pit is formed on the downstream slope, it will erode upstream in the form of a steep slope. The free surface of the steep slope is upright. The erosion rate can be expressed as dx / dt :

In the formula: C _h is the steep slope erosion parameter related to the dam material, which is the key factor controlling the dam failure process of the homogeneous dam and is calculated using the steep slope erosion parameter model of the present invention. In the figure, H represents the height of the dam body.

The process of establishing the calculation model _of steep slope erosion parameter Ch includes:

(1) The multivariate nonlinear regression analysis method is used to consider the nonlinear relationship between the clay content c _c , water content w and dry density ρ _d of the dam material and the erosion parameters. The expression of the nonlinear regression model is:

_Where : n = 4, Xi ₌ ( x1 _, x2 , x3 _, x4 )= ( w / wopt _, ρd _/ ρdm _, ρd / _{ρw , cc )} ;

.

Since the direct solution is not suitable for solving the nonlinear regression model, based on the principle of least squares method, the Gauss-Newton iteration method is adopted, and the Taylor series expansion is used to approximately replace the nonlinear regression model. By setting the initial value of each estimated coefficient and performing multiple iterative solutions, the regression coefficients are constantly approaching the optimal regression coefficients of the nonlinear regression model, and finally the residual sum of squares of the original model is minimized.

(2) The model prediction accuracy is evaluated using the goodness of fit R2 and root mean square _error Erms ^. The evaluation method is:

Where: SSE is the sum of squared errors; SST is the sum of squared deviations; n is the number of case groups used to establish the model, and n = 4 in this model.

Where: i = the number of dependent variables (i.e. the number of measured data sets), y _mi is the measured value of the i -th dependent variable; is the predicted value of the ith dependent variable; /> is the average of all measured values of the dependent variable; R2 ranges from 0 to 1. When R2 is closer to 1, the degree ^of fit between the predicted ^value and the measured value is higher.

The present invention uses 45 groups of experimental data with measured data. Due to limited space, only part of the experimental data is shown in Table 3. According to the specific method for establishing the steep erosion parameter model mentioned above, the steep erosion parameter calculation model is finally fitted as follows, achieving a goodness of fit R ² =0.873 and _{a root mean square error Erms} = 0.569.

In the formula: k is the multivariate nonlinear regression fitting coefficient, w is the moisture content of the dam material when the homogeneous dam breaks, w _opt is the optimal moisture content of the homogeneous dam material, w/w _opt represents the water holding state of the homogeneous dam material, c 1 is the corresponding estimated coefficient; ρ _d is the dry density of the homogeneous dam, kg/m ³ , ρ _dm is the maximum dry density of the dam material, kg/m ³ , ρ _d / ρ _dm is the compaction degree of the homogeneous dam, c 2 is the corresponding estimated coefficient; ρ _w is the density of water, kg/m ³ ; ρ _d / ρ _w represents the relative density of the dam material with water as the standard substance, c 3 is the corresponding estimated coefficient; c _c is the clay content of the dam material, ranging from 0 to 1, c 4 is the corresponding estimated coefficient; the corresponding parameter values are k =-2.931, c 1=0.130, c 2=0.315, c 3=2.052, c 4=0.117.

Table 3 Partial test data

Compared with the existing steep slope erosion model, the results are shown in Table 4. The goodness of fit R ² of the model is close to 1 and _{the root mean square error Erms} is relatively small, indicating that the overall prediction accuracy of the model is high.

Table 4 Comparison of prediction accuracy of different models

When the dam material is in an unsaturated state, its "steep slope" and breach slope angle are close to vertical; similarly, for dam bodies with high clay content, the breach slope angle is also steep, and its breach slope angle β can be regarded as 90°. During the dam breach process, the expansion of the top and bottom widths of the dam crest breach and the bottom bed erosion have the following relationship, as shown in Figure 7:

The following formula is used to calculate the change in the top width and bottom width of the breach:

In the formula: Bt is _the breach _top width, ΔBt is _the breach top _width increment, m; Bd is the breach bottom width, ΔBd _is the breach bottom width increment, m; nloc represents the breach location, nloc = 1 means that the breach is located at the dam abutment and can only develop in one direction; _nloc = 2 means that the breach is located in _the middle of the dam crest and can develop to both sides; Δzb _is the breach depth increment, m.

For breaches on the downstream slope of a homogeneous dam, a correction coefficient that takes breach characteristics into account is introduced to calculate the top and bottom widths of the breach on the downstream slope. The model assumes that the expansion of the top and bottom widths of the breach on the downstream slope has the following relationship with the erosion of the bottom bed, as shown in Figure 8:

Where: Bd is _the breach bottom width, m; Bdown is the breach top width _of the downstream slope, m; bdown _{is the} breach bottom width _of the downstream slope, m; nloc is the breach location parameter (the breach is 2 if it is located in the middle of the dam body and 1 if it is located at the dam shoulder); zb is the upstream bottom elevation of the breach, m; γ is the slope angle of the breach side slope on the downstream slope, °; cb _is the correction coefficient.

Step 6, when the steep slope develops upstream to a certain extent, under the action of the upstream head pressure and the shear stress of the overtopping water flow, the steep slope undergoes shear failure and collapses. The force analysis of the damaged wedge is shown in Figure 6. The forces on the wedge mainly include _the upstream reservoir water pressure F1 , the downstream _tail water pressure F2 , the shear force of the water flow on the upper part of the steep slope F3 _{, the} water _{weight on} the upper part of the steep _slope F4 , the self-weight of the steep slope F5 , _the friction force F6 on the steep slope and the internal cohesion of the steep slope soil F7 _. If F1 - F2 + F3 ＞ F6 + _F7 , it is determined that the steep _slope has undergone shear failure _and proceed to step 7, otherwise proceed to step 8;

, where Z _d ＜Z _h , F ₂ =0;

In the formula: Bd is the bottom width _of the _breach , m; ρw is the water density, kg/ ^m3 ; g is _the acceleration of gravity, m/ ^s2 ; Zw is the upstream reservoir water level, _m ; Zh is the bottom elevation downstream of the breach, m; _Zb is the bottom elevation upstream of the breach, m; Zd is the downstream tailwater level _, m; n is the Manning roughness coefficient; Qb is the breach flow, ^m3 /s; _Aw is the cross-sectional _area of the breach water flow; R is the hydraulic radius at the breach, which is the ratio of the water-passing cross-sectional area to the wetted perimeter; ldt is the length of the top of the scarp, m; _{ldb is} the length of the bottom of the scarp, m; h is the height of the scarp, m; φ is the internal friction angle of the soil, °; c is the cohesion of the soil, kPa.

Step 7: After the wedge collapses, the erosion mode of the remaining dam body by the overtopping water flow changes from steep slope erosion to surface erosion, that is, the elevation of the breach bottom continues to decrease. Since the dam building materials are usually viscous, the erosion rate of the dam material can be calculated by the following formula:

d Z _b /d t = k _d ( τ _b - τ _c )

In the formula: d Z _b /d t is the breach bottom bed erosion rate, m/s; k _d is the dam material erosion coefficient; τ _b is the water flow shear stress, Pa; τ _c is the starting shear stress of the dam material, Pa.

τ _c and k _d are key factors affecting breach development and breach flow evolution, and their values are of great significance to improving the prediction accuracy of the model. The calculation method of k _d is as follows:

In the formula: γ _w is the bulk density of water, N/m ³ ; γ _d is the dry bulk density of soil, N/m ³ ; c _c is the clay content, %.

The water flow shear stress τ _b can be calculated using the Manning formula:

In the formula: ρw is the density of water, kg/ ^m3 ; g is the acceleration of gravity, m/ ^s2 ; n is the Manning roughness coefficient at the _breach ; Aw _is the cross-sectional area of the breach water flow, ^m2 ; R is the hydraulic radius of the breach, m.

According to the change calculated in step 5, the upstream reservoir water level, the upstream bottom elevation of the breach, the top width of the breach and the bottom width of the breach are updated and calculated;

_Zw = Zw _{-1 +} ΔZw

Z _b =Z _b-1 + ΔZ _b

Bt ₌ Bt _{- 1} + ΔBt

B _d = B _d-1 + Δ B _d

In the formula, _Zw represents the upstream reservoir water level at the nth calculation, Zw _-1 represents the upstream reservoir water level at the n- 1th calculation _, ΔZw represents the change in the upstream reservoir water level between the n-1th and n -1th calculations; _Zb represents the upstream bottom elevation of the breach at the nth calculation, Zb _-1 represents the upstream bottom elevation of the breach at the n-1th calculation, _ΔZb represents the change in the upstream bottom elevation of the breach between the n-1th and n- 1th calculations; Bt represents the top width of the breach at the nth calculation, _{Bt -1} represents the top width of the breach at the n-1th calculation, ΔBt _represents the change in the top width of the breach between the n-1th and n- 1th calculations; Bd represents the bottom width of the breach at the nth calculation, Bd _- 1 represents the bottom width of the breach at the n-1th calculation, ΔBd _represents the change in the bottom width of the breach between the n-1th and n- _1th calculations.

Step 8: The breach at the dam crest continues to erode, and the erosion amount, upstream reservoir water level, upstream bottom elevation of the breach, breach top width, and breach bottom width are updated based on the change calculated in step 5;

xn ₌ xn _-1 + Δx

_Zw = Zw _{-1 +} ΔZw

Z _b =Z _b-1 + ΔZ _b

Bt ₌ Bt _{- 1} + ΔBt

B _d = B _d-1 + Δ B _d

Step 9, calculate ΔZ _w / Z _w and determine whether ΔZ _w / Z _w is less than ε . ε is the calculation convergence condition and is set to 0.0001 or adjusted according to the actual situation. If ΔZ _w / Z _w is less than ε , proceed to step 10, otherwise return to step 5 to continue the calculation.

Step 10, determine whether the breach slope is stable, if yes, determine whether the cumulative calculation time is greater than the set calculation time, if the cumulative calculation time is greater than or equal to the set total calculation time, then end the calculation, and output the calculation results of breach flow, erosion, upstream reservoir water level change, upstream bottom elevation change, breach top width change, and breach bottom width change;

As the depth of the breach at the dam crest continues to increase, the breach slope may become unstable. The limit equilibrium method is used to analyze the slope stability, as shown in Figure 9. When the driving force Fd of the sliding wedge _is greater _than the anti-sliding force Fr , the slope becomes unstable, that is:

Fd _＞ Fr _​

Where: Fd and Fr _{can be} expressed as:

Where: Ws _is the weight of the sliding wedge, kg; θ is the slope angle _of the breach slope after instability, °; γb is the bulk density of the soil, N/ ^m3 ; Hs is the height of the breach slope, m; φ is the internal friction angle _of the soil, °; c is the cohesion of the soil, kPa; β is the slope ratio of the breach slope.

The above steps are used to calculate the characteristic parameters such as breach peak flow Q _p , breach peak flow occurrence time T _p , breach top width B _t , etc. The calculation results are shown in Table 5.

Table 5 Verification results of three groups of measured cases

The results show that the dam-break model can accurately predict the breach flow, and the prediction accuracy error of the breach peak flow is within ±5%. For the prediction of the peak flow occurrence time of the breach of Youfangqu Dam No. 1 and Changhangou Dam No. 1, the calculation error of the model is less than ±10%, while for the prediction of Changhangou Dam No. 2, the relative error is within ±20%. Due to the lack of measurement data of the breach top width of Changhangou Dam No. 1, no comparison is made here, indicating that the relative error of the prediction result of the breach width of this model is within ±35%.

Claims (3)

1. A dam break mathematical model and parameter determining method for a homogeneous dam is characterized by comprising the following steps:

Step 1, determining geometrical characteristic parameters of a dam body, physical and mechanical characteristic parameters of a dam material, an upstream water level-reservoir surface area relation curve, a time-incoming flow curve and a time-draining curve;

step 2, setting a bottom elevation allowable value of the upstream of the crumple;

Step 3, calculating an initial abrupt bank elevation, namely an initial value of the upstream bottom elevation of the crumple, specifically comprising:

Z_b=(l_down-l_n)sin(α_x)

Wherein: fr _n is the Froude number; q _b is the flow of the crumple, m ³/s;B_t is the top width of the initial crumple, and the width of the crumple on the downstream slope takes the same value; c is the thank you coefficient; z _w is the upstream reservoir level, m; z _b is the upstream bottom elevation of the crumple, m; g is gravity acceleration, m/s ²;l_down is the downstream dam slope length, m; l _n is the length of the peak of the downstream dam slope from the initial steep bank position, m; alpha _x is the slope angle of the downstream dam slope;

Wherein the thank you coefficient calculation mode is:

Wherein: n is a Manning roughness coefficient, n= (d ₅₀)^1/6/12, d represents particle size;

Wherein the flow of the crumple is calculated by adopting a water balance formula:

Wherein: q is total leakage flow, m ³/s;Q_b is breach flow, m ³/s;Q_spill is spillway outflow flow, m ³/s corresponds to the leakage value in each moment in the time-leakage array, and zero is set when no spillway flows; q _in is the warehouse-in flow, m ³/s, and corresponds to the incoming flow value at each moment in the time-incoming flow array; a _w is the upstream reservoir face area, m ²;Z_w is the upstream reservoir water level, m; t is time, s;

when there is only an upstream water level-reservoir capacity relationship curve, a quadratic curve fit is used:

wherein A _w is the upstream reservoir surface area, m ²;Z_w is the upstream reservoir water level, m; v _w is upstream reservoir capacity, m ³;a₁、b₁ and c ₁ are fitting coefficients, where a ₁ and b ₁ require 8 bit decimal places to be preserved;

If the upstream water level-reservoir capacity and the upstream water level-reservoir surface area relation curve are absent, the following formula is adopted for calculation:

When the area of a certain water level of the reservoir and the reservoir capacity are known, calculating the values of a ₂ and b ₂ by adopting two coordinate points (Z _w1,A_w1）、（Z_w1,V_w1), and when the total reservoir capacity is only available, taking the value of b ₂ as 2.0, and carrying (Z _w2,V_w1) into A _W=a₂Z_w ² to calculate the value of a ₂; z _w1 is the specific value of Z _w under the condition that the area of the warehouse surface and the storage capacity are known, A _w1 is the specific value of A _w under the condition that the area of the warehouse surface and the storage capacity are known, and Z _w2 is the specific value of Z _w under the condition that the total storage capacity is only available

Step4, setting the calculated total duration and the time step of each calculation;

Step 5, calculating the flow rate, erosion amount, upstream reservoir water level variation, upstream bottom elevation, top width variation and bottom width variation of the crumple openings in each time step;

And calculating the flow of the burst orifice by adopting a weir flow formula:

Wherein: σ _s is the downstream tailwater inundation coefficient; b _d is the bottom width of the crumple, m; z _w is the upstream reservoir level, m; z _b is the upstream bottom elevation of the crumple, m; beta is the slope ratio of the side slope of the crumple; when (Z _d-Z_b)/(Z_w-Z_b) is less than 0.8, the flow is free, and when (Z _d-Z_b)/(Z_w-Z_b) is more than or equal to 0.8, the flow is submerged; z _d is the downstream tailstock level, m;

the trace-source erosion rate when calculating erosion amount is expressed as dx/dt:

Wherein: c _h is a steep bank erosion parameter related to dam material characteristics, and is a key factor for controlling the dam breaking process of the homogeneous dam; q is the single-wide flow of the crumple, m ³/s; h is the height of the steep bank;

the calculation formula of the steep bank erosion parameter C _h is as follows:

；

Wherein: k is a multiple nonlinear regression fitting coefficient, w is the water content of the dam material when the homogeneous dam breaks, w _opt is the optimal water content of the homogeneous dam material, w/w _opt is characterized as the water holding state of the homogeneous dam material, and c1 is a corresponding estimation coefficient; ρ _d is the dam dry density of the homogeneous dam, kg/m ³,ρ_dm is the maximum dry density of the dam material, kg/m ³,ρ_d/ρ_dm is the dam compaction degree of the homogeneous dam, and c2 is the corresponding estimation coefficient; ρ _w is the density of water, kg/m ³ _,ρ_d/ρ_w is characterized as the relative density of dam material with water as standard substance, c3 is the corresponding estimation coefficient; c _c is the content of the dam adhesive particles, the value is 0-1, and c4 is the corresponding estimation coefficient; the corresponding parameter takes the value of k= -2.931, c1=0.130, c2=0.315, c3= 2.052, c4=0.117;

Step 6, respectively calculating the upstream reservoir water pressure F ₁, the downstream tail water pressure F ₂, the water flow shearing force F ₃ on the upper part of the abrupt bank, the water weight F ₄ on the upper part of the abrupt bank, the dead weight F ₅ of the abrupt bank, the friction force F ₆ born by the abrupt bank and the cohesive force F ₇ in the soil body of the abrupt bank, if F ₁- F₂+ F₃＞F₆+ F₇ is carried out, judging that the abrupt bank is subjected to shearing damage, and carrying out step 7, otherwise, carrying out step 8;

Step 7, updating and calculating the upstream reservoir water level, the upstream bottom elevation of the crumple opening, the top width of the crumple opening and the bottom width of the crumple opening according to the variable quantity calculated in the step 5;

step 8, calculating erosion of the abrupt bank, and updating and calculating the erosion quantity, the water level of the upstream warehouse, the elevation of the upstream bottom of the crumple, the top width of the crumple and the bottom width of the crumple according to the variable quantity obtained by the calculation in the step 5;

Step 9, calculating DeltaZ _w / Z_w, judging whether DeltaZ _w / Z_w is smaller than epsilon, wherein epsilon is a calculation convergence condition, and taking 0.0001 or adjusting according to actual conditions; if yes, step 10 is carried out, otherwise, step 5 is returned to continue calculation; Δz _w represents the amount of change in the upstream reservoir water level, and Z _w represents the upstream reservoir water level;

Step 10, judging whether the side slope of the crumple is stable, if so, judging whether the accumulated calculation time is longer than the set calculation time length, if so, ending the calculation, and outputting calculation results of the crumple flow, the erosion amount, the upstream reservoir water level change amount, the upstream bottom elevation change amount, the crumple top width change amount and the crumple bottom width change amount;

And (5) returning to the step (5) to calculate the next time step if the accumulated calculation time is smaller than the set total calculation time.

2. The dam-break mathematical model and parameter determining method for homogeneous dam according to claim 1, wherein the calculating of the variation of the top width of the breach and the variation of the bottom width of the breach in step 5 adopts the following formula:

Wherein: Δb _t is the top width increment of the crumple, m; Δb _d is the base width increment of the crumple, m; n _loc represents the position of the breach, n _loc =1 represents that the breach is positioned on the dam abutment, and the breach can only develop towards one direction; n _loc =2 indicates that the breach is positioned in the middle of the dam crest, and the breach can develop to two sides; Δz _b is the vent depth increment, m.

3. The dam break mathematical model and parameter determining method for homogeneous dam according to claim 1, wherein in step 10, it is determined whether the breach slope is stable, i.e. whether the driving force F _d of the sliding wedge is greater than the anti-sliding force F _r, if so, it is indicated that the breach slope is unstable; f _d and F _r are respectively expressed as:

Wherein: w _s is the weight of the sliding wedge, kg; θ is the slope angle after the slope of the crumple is unstable; gamma _b is the soil volume weight, N/m ³;H_s is the height of the side slope of the breach, m; phi is the internal friction angle of the soil body; c is the cohesive force of the soil body, kPa; beta is the slope ratio of the ulcer slope.