CC BY
8
Fayziev Khomitkhan, doctor of technical sciences, professor, Tashkent Institute of Architecture and Construction E-mail: [email protected] Rakhimov Sherzod Abduvakhobjonovich,
senior lecturer,
Tashkent Institute of Architecture and Construction E-mail: [email protected] Baymatov Shakhriddin Khushvaktovich,
senior lecturer,
Tashkent Institute of Architecture and Construction E-mail: [email protected]
CALCULATION OF UNSTEADY FILTRATION IN EARTH DAMS BY THE FINITE DIFFERENCE METHOD
Abstract: This article presents the solutions to the problem of unsteady filtering in earth dams using the finite difference method. An algorithm for solving the problem of unsteady filtering has been developed and a calculation program in the C# language has been compiled. The capabilities of the developed methods, the program for calculating unsteady filtering and the assessment of the reliability of the results obtained are illustrated byan example of filtration problems.
Keywords: earth dams, filtration, unsteady filtering, abrupt drawdown in water level in the upstream, rate of drawdown in water level, finite difference method.
All solutions to the problems of unsteady filtering solved by analytical, hydraulic and analog methods are performed under certain restrictions: the simplest types of dams are considered; the drawdown in water level in each particular case is taken as instantaneous; an unsteady flow is considered as a succession of steady states. Naturally, these techniques are not equivalent to each other. Each of them has its own limits of application and provides the necessary accuracy within these limits only. Therefore, in general cases, they serve only for estimates in the first approximation. Some of them, more accurate solutions, are hardly ever used because of the complexity of the dependencies obtained.
As is known [1], at horizontal aquiclude and constant filtration coefficient of a medium, the nonlinear Boussinesq equation for a one-dimensional problem has the following form [1].
h _ Kr dt / dx where h - is the hydrodynamic thrust;
x - is the abscissa of the depression curve point; t - is the current time;
At present the problem of harmful effects of filtering flow in dams erected of local materials is very urgent. Especially this concerns the structures of seasonal operation of reservoirs with large-amplitude oscillations in the upstream level. Changes in the upstream and downstream levels cause variations in the position of depression surface and the parameters of filtration flow (filtration gradients, rates and costs). Analysis of the operation of upper wedge of the dam at rapid drawdown of the reservoir has shown that its stability is significantly reduced due to the effect of hydrodynamic forces directed towards the upstream.
The problem of unsteady filtering is one of the most difficult in the field of filtration theory. For the first time the general equations of unsteady filtering have been proposed by N. E. Zhukovsky [1]. However, due to considerable complexity of the solution, these equations have been used much later.
A simplified derivation of the equation of unsteady filtering has been proposed by J. Boussinesq [1]. His assumption is that the horizontal rates of filtration flow are constant along the vertical direction; this formed the basis of the hydraulic theory of unsteady filtering. Further development of the problem of calculating unsteady filtering has been conducted by P. Ya. Polubarinova - Kochina, V. I. Aravin and S. N. Numerov, N. N. Verigin, V. S. Lukyanov, V. M. Shestakov, V. P. Nedriga, A. L. Mozhevitdinov and A. G. Suleymanov, L. N. Rasskazov, N. A. Aniskin, F. B. Abutaliev, Kh. Fayziev, S. N. Babakaev et al.
^ dh h — v dx
(l)
KT, fJ. - are the coefficients of filtration and filter loss in soil.
To solve equation (l), several approximate methods have been proposed, including the linearization method. It is described in detailed in [1].
At present, a qualitatively higher level of solving such problems is achieved using numerical methods and, first of all, using the finite difference method.
Below, we consider the solutions of the Boussinesq equation by numerical-finite-difference method. Here equations (l) are considered together with the following initial and boundary conditions:
(2)
h lt=0 = K
(1- x,)(h(0,t) x=0 -hp)+ s £ ix__0 = o (3)
(l-«2 )(h{L,t) x=l -hp ) £/*=L = 0 (4)
where hp is an average hydrodynamic thrust, x and t are the variables; and <x.2 are the corresponding parameters equal to zero or 1 for the boundary conditions of the 1st and 2nd kinds. Problems (l) - (4) are called the boundary value problems of the mathematical physics.
To solve equation (l) with initial and boundary conditions (2) - (4) the finite difference method [2] is used. Transfer to the following dimensionless variables:
, * h * x
h =—; x =—; hcp L
. t * KAp
t =—; a =--
t u
m r*
h = h * • hp; x = x * / L
cp
t = t * •tm; a= a * Substituting these dimensionless quantities into equation (l) the following dimensionless differential equations convenient for solving are obtained:
d2 (h* • hcp) a d(h* • hcp)
d(x* • L)2 a d(t* • tm)
hcp dh hcp dh - =a-
L dx
In the last equation we denote-- =---; — =-
t dt
m
d2h* aL h dt *
1
aL t„
dx *2 tm ot x m then the following dimensionless differential equations are obtained
d 2h2 = 1
dx *2 " x dt * (5)
To solve equation (5) with initial and boundary conditions (2) - (4), a grid is built:
x-i = x-i_ + Ax, x0 =0, xN =1, Ax =1
n
i = 1, n -1; tk = tk-1 +At, t0 = 0, tk = 1, At = -
k
In further calculations, for convenience, the variables are written without an asterisk. The first order derivatives are approximated as follows:
dh « hi+1 -h- + 0(Ax), t = huh. + 0(Ax), dk 2 Ax v ' dk Hx v '
dh h -h . . x
— -^ + 0 (Ax)
dk Ax v ' To approximate the second order derivative, the Taylor
formula is used:
h(!+i)=h+L-" Ax+ dh
dh dx2 dh dx2
/ x = 0
(Ax )
+ ••• (6)
,(Ax )2
+ ••
(7)
i-' ' dx x=0 dx2 2
Summing up equations (6) and (7) we get: h +1 - 2h' + h_l _ d^h (Ax)2 _ dx2 •
In equation (5), the derivatives of the first and second order are replaced by finite difference relation: h+1 - 2hi + h,-1 _ 1 h, - h
(Ax )2
x At Ax2
h+i- 2h + h-i = — (( - h )
, Ax % Ax 2r h;i, -2h--h =--h,
xAt ' xAt
2--
Ax
2^
x At
h + k-i =-
Ax2
h.
xAt '
Introducing some notation the following systems of algebraic equations are obtained:
afr+i - ^^ + ch-i =-dl (8)
i = 1, n -1
„ , „ Ax2 Ax2 r
where a{ = ci = 1, bi = 2--,di =-hi,
xAt xAt
i = 1, n -1
Then, equations (8) are solved by the sweep method using the following recurrent formulas:
ho = Aoh1 + Bo:__(9)
h1 = Aihi+1 + B, i = 1, n-1 (10)
where A0, B0,A{, B{ are the sweep coefficients.
If in (3) a1 = 0, then the sweep coefficients are determined as follows:
If A0 = 0, B0 = h0, P0 = A0h1 + B0, , and a1 = 1 then at i = 1 :
dh
d2h
hi = ho + — /x=oAx ^^r /x=
(Ax )2
dx
dx2
+ ...
dh
hi - ho - 0,5 ( - 2hi + ho ) = — X=o Ax = 0
2hi - 2h0 - h2 + 2hi - h0 = 0 4hi - 3h0 - h2 = 0, h2 = 4hi - 3h0 aih2 - bihi + cih0 = -di
(11) (12)
At i = 1, substituting (ll) into equation (12) the dependencies for the sweep coefficients are obtained: a1 (4h1 - 3h0 ) - b1h1 + c1h0 = -d1 4a1h1 - 3a1h0 - b1h1 + c 1h0 = -d1
(4a1 - b1 )h1 -(3a1 - c1 )h0 = -d,
(3a1 - c 1 )h0 = (4a1 - b1 ) + d1.
h = ^a-h hi+ di
3al - c 1 3al - cl
ho = AA + B 0. Determine the sweep coefficients.
a=^acK, B=- di
3a1 - c1 3a1 - c 1 In the general case, the sweep coefficients are determined by the following formula:
A =-
d
B
c B , + d . -
i = -L, i = i, n -1 (13)
1 h - cA-1 1 h - cA-1 If from initial conditions (4) a2 = 0, then hN = 1. If a2 = 1, then:
h - h -dh/ Ax/ M+
N-1 nN ~ x-N A - 2 'x-N „
dx dx 2
hN - hN-1 + 0,5 (hN - 2hN-1 + hN-2 ) = 0
2hN - 2hN-1 + hN - 2hN-1 + hN-2 = 0
3h - 4h - h = 0
N N-1 N-2 U
With the following recurrent formulas: h --a h + B
N-1 N-2'lN ^ "N-1 ,
hN-2 = AN-2hN-1 + BN-2 = AN-2 {AN-1hN + BN-1 ) + BN-2 = = AN-2 AN-1hN + AN-2BN-1 + BN-2 , 3hN - 4hN-1 + AN-2 AN-1hN + AN-2BN-1 + BN-2 = 0 , 3hN - 4hN-1hN + AN-2AN-1hN = 4BN-1 + AN-2BN-1 - BN-2 (3 - 4hN-1 + AN-2AN-1 )K = 4BN-1 + AN-2BN-1 - BN-2
At i = N for hn the following formulas are obtained:
h = 4Bn-1 — AN-2BN-1 — BN-2 (14)
3 - 4AN-1 + AN-2 AN-1 If a1 = 0, ,thenh0 and ax = I,
. _ 4a1 - b1 B _ d1 A — -, B0 ~
0 O 0 0
3aL - cl 3al - cL
At i = i, n — 1 the sweep coefficients are determined by (8):
aht+ - btht+ct (A _xht+B - ) = -dt ah+i- bA+cA,-ih, + ciB,-i + di =0, aKi- b+cA-i+cA-i+d =0 > b - crA-i= aKi+c>B>-i+d> >
h = d' h + AL+Ä i k - cA,.-i i+1 k - cA,.-i
A = dr , B = crBr-1 + dr
1 - cA-1 . k - cA-1
i = 1, n -1.
If i = 1, then
A = ai B = ciB0 +d
b1 c1A0 b1 c1A0 The values of A0, B0 are determined from boundary conditions (3).
If i = 2, then A2 =-a2-, B2 = cB + d
b2 c2 Ai b2 c2 Ai
A = n -1A-1
b , - c A -,
n-1 n-1 n-2
Bn 1 =
c ,B t + d i
n-1 n-2 n-1
b , - c A "!
n-1 n-1 n-2
Thus, the sweep coefficients are determined by a straight sweep from initial and boundary conditions (2) (3). The values of the function
h - AhM + Bt, A N - i,i .
h = A{hi+1 + B{, i = i,n-i.
are determined by the inverse sweep of the boundary condition (4).
The error ofthe problem approximation is of O = (h2 +t) order where h = max (Ax ), Ax = i/ n , the computational scheme is unconditionally stable in time and is reduced to solving the problem at h and t tending to zero [8].
On the basis of the stated method of solving the boundary value problem [3] and numerous experiments on its testing on a computer, a working algorithm of calculation is developed.
Calculation program in C# language is aimed to perform the following computational operations:
1. Start the program.
2. Input of initial data.
3. Input of initial value of the thrust.
Organization of the first stage of the cycle in time (t = 0,
t = t + 0.5t).
1. Calculation of coefficients of difference equations
a{b,ciudi (i = i,n-i).
2. Calculation of coefficients of direct sweep E0, B0.
3. Determination of coefficients of direct sweep at the remaining points (( = i,n — i j.
4. Calculation of H{ ati = n.
5. Calculation of the field of thrust Hn - (i = n - i,i).
6. Replacement of old values with new ones
H = H (i = ÖN).
7. Check of the time condition (t < T). If the time condition is met, this is the end of the program, and if not, the control is transferred to 4.
8. End of the program.
The developed methods and calculation program allow solving the problems of unsteady filtration in earth dams without drainage and with various types of drainage systems
(drainage prism, tubular drainage, layered drainage etc.) with impermeable and permeable bases.
The possibilities of the developed methods, the program for calculating unsteady filtering and the assessment of the reliability of the results obtained are illustrated below using the example of filtration problems [4].
A homogeneous earth dam with permeable base is considered. The initial depth of the upstream is h1(0) = 27 m; the laying of the upstream is m1 = 3; the laying of the downstream is m = 2; the filtration coefficient of soil of the dam body and the base is KT = K = 0.44 m/day; the coefficient of water loss in soil of the dam body and the base is y = 0.2; the rate of water level drawdown in the reservoir is v = 0.6 m/day, the
level of the downstream is assumed unchanged and equal to h2(0) = 9m.
Solution of the filtration problem for the steady-state regime is taken as initial condition at the depths of the upstream and downstream 27 m and 9 m, respectively. Results of solving this non-stationary problem by the method of finite differences and its comparison with the V. Shestakov method are presented in (Fig. l).
Thus, the results obtained using the FDM well agree qualitatively and quantitatively with the results of the methods compared. Differences in the estimates of the depression surface in the calculated sections do not exceed the permissible limits.
Figure 1. Results of calculations of unsteady filtering: 1 - initial depression curve at t = 0; 2 - positions of the depression surface by the V. Shestakov method at t = 20 days; 3 - positions of the depression surface by the FDM at t = 20 days
References:
1. Polubarinova-Kochina P. Ya. Theory of Groundwater Movement.- Moscow: Nauka, 1977.- 664 p.
2. Samarsky A. A. Introduction to the theory of difference schemes.- Moscow, Nauka, 1971.- 552 p.
3. Fayziev Kh., Babakaev S., Khazhiev I., Normatov M., Akhmedov I. Numerical Solution of a Boundary Value Problem of Unsteady Filtration in Homogeneous Earth Dams by the Finite Difference Method. Architecture. Building. Design.-No. 3. 2013.- P. 52-56.
4. Fayziev Kh., Babakayev S., Khazhiev I. O. Calculation of Unsteady Filtration in Homogeneous Earth Dams with Permeable Base. Vestnik. TIIZHT.-No. 1. 2014.- P. 21-25.
