Change in filtration gradients during head water drawdown

. Changes in the water level in a reservoir lead to changes of filtration stream parameters in the soil structure. The position of the depression curve, value and direction of filtration gradients and speeds changes near the slope of the structure. Operation practice of waterworks in such conditions shows that emergencies are possible. Changes in filtration stream parameters can cause filtration deformations of soils and loss of slope stability of a soil dam. The article considers unsteady filtration in a homogeneous soil dam for five options of composing soils having various filtration coefficients. The reservoir drawdown rate is considered as an affecting factor. Using the numeric finite-element method and PLAXIS 2D software suite, filtration stream parameters were found for the considered options. Maximum filtration gradients near the upstream slope of the soil dam were found for various options of dam soil permeability and drawdown rate. As the reservoir draws down, directions and filtration gradients and rates change near the slope surface. Maximum gradients occur at various stages of drawdown depending on its speed and reservoir soil permeability. Maximum gradients are quite high, which shows the need for anti-filtration measures.


Introduction
Unsteady filtration issues are found in hydrotechnical practice in construction and operation of dams, when the reservoir is filled or drawn down, and when constructing hydraulic dams.
As the statistics shows, the second important factor (after water overflow over the dam crest) causing accidents in soil dams is filtration deformations of soils and concentrated filtration in the dam body [1][2].
Reduced water level before the dam changes filtration conditions of the soil structure: the position of the depression surface, filtration gradients, level and direction of filtration stream rates.These factors reduce retaining loads and, therefore, the slope stability [3][4].The reservoir drawdown leads to increased filtration gradients, which may cause filtration deformations in the soil mass or on the slope surface [5].Hydrotechnical practice shows multiple cases of accidents related with water level drop in the reservoir [6][7].The change in the structural stability and filtration conditions depend on several factors: soil permeability of the upstream fill, amount and rate of drawdown, upper slope gradient and some others [8].Currently, filtration calculations and calculations of filtration strength of soil dams are mandatory (SP 39.13330.2012Dams made of soil materials.Revised edition SNiP 2.06.05-84*).
Primary factors affecting the assessment of the filtration strength of the reservoir body soil and its stability is the magnitude and direction of head gradients used to find the filtration hydrodynamic load [6][7][8].
The upper slope stability must be evaluated when analyzing the soil dam for the drawdown scenario at a maximum possible reduction of the reservoir water level with the highest possible rate.It is not yet clear when the most dangerous state may occur during the drawdown in terms of slope stability and maximum head gradients.This article is dedicated to finding the most dangerous stages in the drawdown based on the solution of unsteady filtration.
Since the middle of the XX century, due to a complexity of solving the unsteady filtration task, hydraulic methods based on the Boussinesq equation with several assumptions were quite popular [9].As the calculation methods improved, new analytical [10] and hydraulic methods appeared [11][12].
The primary task in modern researches [13][14][15][16] is to find filtration components such as depression curve, effluent seepage gap height, filtration flow rate and gradient.This is achieved by using finite element method, finite volume method and boundary element method.
Modern software suites intended to solve filtration tasks are based on solving the primary differential equation of the filtration theory with known boundary conditions [10]: ; is the coefficient of water losses in soil., ,  µ Such calculations suggest water movement between particles with pores fully filled with water.Water moves from the area of higher pressure to the area of lower pressure.When finite elements, for which the obtained filtration head is less than their height position, is excluded from the computational domain, the depression curve position will be defined [15].Similar tasks of hydrology were considered in the papers of famous foreign [17][18][19][20][21] and Soviet-Russian scientists [22].
When solving filtration tasks in soil hydrology, the L.A. Richards [17] equation based on Darcy's equation is used to solve the tasks of unsteady filtration.One of the modifications of such equation in water transfer in vertical direction is represented as follows for the gravity pressure gradient of one: where Ψ is the capillary pressure of soil moisture (cmWC, k is the coefficient of hydraulic conductivity of soil (cm/day), a function of moisture pressure Ψ), µ is the coefficient of differential water capacity coefficient of soil (cmWC -1 ), t is the time (days), z is the coordinate along the vertical axis (cm).
Only numerical methods can be used to solve this equation.Coefficient are the function of capillary moisture pressure, but these µ = µ(Ψ)   =  (Ψ) functions have an empirical value [22][23], which leads to inaccurate results.The value of volumetric soil moisture is found in laboratory conditions and depends on capillary θ moisture pressure called the water retention curve.Ψ In 1980, Van Genuchten [20] used the Mualem method to propose a model for describing the hydrophysical properties of soils using the expressions of the Mualem-Van Genuchten model.This model is widely used in practice in numerical modeling of filtration processes [5][6].

Materials and methods
The subject of the research is homogeneous soil dam 24.0m high located on a non-permeable footing (Fig. 1).Slope gradients are 1:3 and 1:1.85 for upper and lower, respectively.For the dam bottom level of 0.00, the head water level is 22.0 m and the tail water level is 3.0 m.For calculations, the dam body material made of dispersive soils is taken: medium-grain sand, fine sand, sand clay and clay loam.Characteristics corresponding to these soil groups are given in Table 1.The study was carried out by the numerical method in PLAXIS 2D software suite with the PlaxFlow module allowing for filtration calculation, namely for unsteady filtration.The program calculates using the finite element method using the van Genuchten filtration computation model [20].The previous researches [24] showed high compatibility of results obtained using PLAXIS 2D and analytical, physical and other numerical methods.
After confirming the sufficient accuracy of the results using the selected method, the research was undertaken to find out the effects of the selected factors (dam slope gradients, filtration coefficients, reservoir drawdown rate) on the filtration gradient in the point where the filtration stream exits by the end of drawdown.

Results
Homogeneous earth dam (Figure 1) was modeled with various dam body materials as per Table 1.According to calculations for each of the variants, the depression curve position was found on days 4, 12 and 18 after the start of drawdown.Figure 2 shows the depression curve position by the need of drawdown (on day 18) for all variants.As Figure 3 shows, for clay soils with the filtration coefficient , the change К  ≤1 * 10 −2 in the position of the depression curve occurs only near the upper slope.Figure 4 gives maximum heights of the depression curve in section 4-4 (Fig. 1) corresponding to Figure 3.The previous researches studied the effects of some factors on the maximum filtration gradient [25].Such factors were considered as dam soil filtration coefficients, reservoir drawdown rate, dam upper slope gradient.According to the results, charts were built for changes in maximum filtration gradients allowing for preliminary evaluation of the filtration strength of soil and calculation of parameters of reverse filters if necessary.
In the cases under consideration [25], the maximum filtration gradients were found near the lower drawdown level at the time when it ended.The maximum gradients were observed near the surface of the upper slope at elevations of 4.00 to 8.00 m (except for variants with the filtration coefficients K f = 0.001 m/day).As an example, Fig. 4 shows calculation at the end of drawdown for the variant with a medium-grained sand (K f = 10.0 m/day) and clay loam (K f = 0.001 m/day).We can see that in the second case the filtration stream discharge takes almost the entire surface of the slope, which may necessitate measures to protect it.When considering the distribution fields of the filtration gradient in vector representation for the variant with K f = 10.0 m/day (Figure 5), we see that the maximum is in the point located at the head water level after drawdown.Applicable designing standards for soil dams provide requirements for filtration calculation and stability analysis of the upper slope taking into account the unsteady filtration at the maximum possible water level reduction into the reservoir against maximum design levels at the maximum possible drawdown rate.The primary results of filtration calculation of the upper slope are as follows: defining the depression curve position, filtration flow rate and average filtration gradient.The operation practice of water retaining structures shows hazard of so-called reverse erosion caused by significant filtration rates and gradients.To find the maximum filtration gradients against the drawdown level and possible stages of their occurrence, additional studies were conducted.
To find maximum filtration gradients, the dam cross-section (Fig. 1) with the dam body materials given in Table 1 were considered.The drawdown by 18 m was modeled, with the head water level fall rates of 0.25 m/day, 1 m/day, 2 m/day.For each meter of drawdown, the maximum filtration gradients near the upper edge were recorded.According to the numerical modeling, dependency charts of the filtration gradient and drawdown height were obtained (Figs.6-8).As the results show, the most dangerous moments characterized by the maximum filtration gradients are found at intermediate drawdown stages rather than at the end of drawdown to minimal levels (Figs.6-8).Figures 9-10 give isofields of filtration gradients and locations of maximum gradients for 2 calculation variants at drawdown rates if 1 and 0.25 m/day.
In the considered cases, the difference between the peak filtration gradient (Fig. 8) and filtration gradient by the end of drawdown was 29%.

Conclusion
1.The maximum filtration gradients for drawdown are recorded at the filtration stream exit to the upper slope.The maximum filtration gradients are quite high (0.3 to 0.9 depending on the soils and drawdown rates), which shows the need to evaluate the filtration strength of dam soils.2. Designing must account for the entire drawdown period by finding the point in time corresponding to the maximum filtration gradient.For the considered variants, the maximum gradients are obtained at the drawdown stage to elevations between 12.0 and 9.0 m.
function in the considered area changing upon  =  (, , , ) coordinates and in time ; are filtration coefficients in the directions of ,

Fig. 2 .
Fig. 2. Calculation of depression curve position for various filtration coefficients of the dam body material.

Fig. 3 .
Fig. 3. Dependence of the depression curve point position in section 4-4 for each variant of filtration coefficients for drawdown rate of 1 m/day.

Fig. 4 .
Fig. 4. Isofields of filtration gradient in the upper slope zone of the earth dam at the time of complete reservoir drawdown.

Fig. 5 .
Fig. 5. Vector representation of the filtration gradient in the dam body after drawdown.

Fig. 6 .
Fig. 6.Chart of changes in maximum filtration gradient as the drawdown progresses for drawdown rate of 0.25 m/day and various filtration coefficients.

Fig. 7 .
Fig. 7. Chart of changes in maximum filtration gradient as the drawdown progresses for drawdown rate of 1.00 m/day and various filtration coefficients.

Fig. 8 .
Fig. 8. Chart of changes in maximum filtration gradient as the drawdown progresses for drawdown rate of 2.00 m/day and various filtration coefficients.

Fig. 9 .
Fig. 9. Drawdown moment when the maximum filtration gradient is recorded for a drawdown rate of 1.00 m/day and filtration coefficient of 0.001 m/day (the filtration gradient is shown by isofields).

Fig. 10 .
Fig. 10.Drawdown moment when the maximum filtration gradient is recorded for a drawdown rate of 0.25 m/day and filtration coefficient of 0.001 m/day (the filtration gradient is shown by isofields).

Table 1 .
Soil characteristics of the dam body.