On prognosis of salt regime
The study of the diffusion of salt water in the soil of Lapidusoia and Verigin. Calculation of leaching rates. Initial and boundary conditions for salt distribution. Analysis of the influence of gravity, pressure and soil resistance on water filtration.
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Язык  английский 
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MSUEE, Problem Research Laboratory,
УДК 631.6
On prognosis of salt regime
Dimitrios Papadopoulos
Sydney, Australia
Equation of movement of salt and water
In Lapidus [10] and N. N. Verigin [2] the movement of salt water in soil under conditions of full saturation of pores was studied. The dissolution of salt contained in soil was also taken into account.
It was considered that the movement of solute is determined by the forces of gravity, pressure, diffusion and resistance to movement. Forces of inertia and elasticity were considered negligible.
In the movement of solutes the main forces are the ones that determine the filtration of water (gravity, pressure, resistance). In the process of dissolution the main factor is diffusion. Taking into account both kinds of forces allows a more accurate description of the process of movement of salt in soil. This was the basic principle during the derivation of the equation.
.(1)
where c = c(x, у, t)  concentration of salt in (x,у) at the moment t; D'  coefficient of convective diffusion;  components of filtration velocity;  coefficient of salt exchange; м  soil porosity; с_{s}  concentration of saturation.
If the space coordinates are measured in meters and time in days, the dimensions of v are m/day, diffusion D'  m^{2}/day,  1/day.
Equation (1) can be interpreted as follows: The change of concentration in the point (x,y) at the moment t is equal to the influx as a result of the difference in concentration in the solute (diffusion part of the equation), transfer of salts by the moving water (convective part) and dissolution of the salt contained in the soil [1].
In the onedimensional case (e.g. leaching at some distance of the drains) equation takes the form [1]
(1)
where  water velocity in the pores; D = D?/м  convective diffusion coefficient;  coefficient of solubility.
Theoretical and experimental research show that under conditions of strong presence of weakly soluble salts (gypsum) and significant velocities, the last member in (1) might play the most important role [1]. As shown in [7] the solution of (1) in this case gives the formula of Volobuev [3].
Under conditions of highly soluble salts and very small amounts in the hard phase (e.g. Chlorine) equation (1) satisfactorily describes the natural processes without the last member г(с_{s}с) [1]
.(2)
Note that D in this case takes into account the specifics of movement of solutes in a porous environment (the socalled longitudinal and transverse effects) and is not equal to the normal diffusion coefficient in an immobile solute [4].
When solving equation (2) the following boundary conditions are often applied:
; (3)
. (4)
Condition (3) describes the salt balance on the surface (the change of concentration increases with the water velocity and decreases with the increase of D, and is proportional to the difference of concentration in soilwater and leaching water).
Condition (4) describes the fact that as x increases, the concentration of salt does not change with x (although it still remains a function of t). S.F. Averyanov [1] and A.I. Golovanov [4] have solved (2) with boundary conditions (3), (4) and homogeneous initial distribution of salt. Their solutions are very useful for the calculation of leaching rates. L.M. Reks [8] has solved (2) with boundary conditions (3), (4) , and stepwise and linear initial distributions. Reks' solution is especially useful for the prognosis of watersalt regime of irrigated lands for several consecutive years. In practice, the initial distribution of salt is given as a stepfunction and Reks' solution takes this fact into account. But this positive side of the solution is at the same time its weakness, since the real distribution of salt is continuous and there are no infinite gradients in nature. Despite the fact that many experiments show that Reks' solution describes the process of movement of salt and water satisfactorily, it would be very useful for theoretical analysis (and practical calculations) to have an exact solution that takes into account the continuous nature of the initial distribution. It would also be useful to have a solution with the solubility г being not zero.
Analytical solution for a limited layer
The onedimensional equation and the initial and boundary conditions in a layer [0, L] are:
; (5)
; (6)
; (7)
c(x, 0) = ш(x). (8)
As shown in [11] the new variable transforms (5) and (6)…(8) to the form:
,(9)
Where
,(10)
and
; (11)
; (12)
. (13)
In [9, Parabolic Equations, Problem 37] a solution of the equation (9) is given for somewhat more complex conditions than (11)…(13).
Since a very wide class of functions can be approximated by a polynomial [12], one could assume that an initial condition given by a polynomial of degree k will describe any initial distribution of salt:
.(14)
The solution given in [9] for our conditions can be presented as:
;(15)
;(16)
(17)
where л_{n} =z_{n} /L  positive roots of the equation
;(18)
And
;(19)
;(20)
;(21)
.(22)
Integrals (21), (22) are given in [6, 2.663, 2.667]. The complexity of formulae (15)…(22) lends itself to the use of computers. In this case the integral (20) can be calculated using one of the known algorithms (see for a example [13]).
Prognosis of the watersalt regime
As it was mentioned earlier, L. M. Reks' solution is especially useful for the prognosis of the saltwater regime for several consecutive periods. The same can be said for the formulae (15)…(22). The «algorithm» of the prognosis process is as follows:
1. After the salt sampling and statistical analysis of the data, the initial distribution of salt is established on a level of confidence q (our research with A.I. Golovanov [5] has shown that the level of confidence q should be 0.1 (i.e. the probability p that the content of salt in the layer (hi, h_{i+1}) does not exceed the chosen value c_{i} should be ? 0.9).
Let the initial distribution be (h_{0},с_{0}), (h_{1}, с_{1}), ..., (h_{к}, с_{к}).
2. Using formula (15)…(22), the contents c(x, t) ?in x_{0}=h_{0}, x_{1}=h_{1}, …, x_{k}=h_{k} are calculated (the values of v, D, t are given for each period).
3. The found values are taken as the new initial distribution and the calculations using formulae (15)…(22) are repeated as many times as the number of periods. The possible change of the values of v, D, t and c_{w} should be taken into account.
The number of periods includes both  irrigation and winter periods. The estimate of rainfall in winter can be done in various ways. Parameters v and D are assumed to be known for each period. In the example below the results of the calculations using formulae (15)…(22) are shown. The calculations were performed using a program written in C (the program prognoz.exe is available at the Problem Research Laboratory of MSUEE and can be provided by Professor V.V. Shabanov). diffusion salt water verigin filtration
Example
The following example is totally hypothetical and is given as a demonstration of the solution only. The initial distribution in four points is as follows:
x: 0.0, 0.3, 0.6, 1.0
c(x,0): 1.0, 0.8, 0.8, 1.0
This distribution can be approximated by the polynomial
P(x) = 1.000 ? 1.071x + 1.468x^{2} ? 0.397x^{3}.
Prognosis of salt regime for 4 periods
Х 
Initial 
Period 1 
Period 2 
Period 3 
Period 4 

0.0 
1000 
0,521 
0,087 
0,107 
0,080 

0.1 
0,551 
0,105 
0,113 
0,084 

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
0,800 1000 
0,580 0,606 0,630 0,650 0,668 0,682 0,693 0,699 0,702 
0,124 0,143 0,162 0,180 0,197 0,212 0,224 0,232 0,235 
0,119 0,125 0,130 0,134 0,138 0,141 0,144 0,145 0,146 
0,089 0,093 0,096 0,100 0,102 0,105 0,106 0,107 0,108 

Parameters D, v, t, c_{w} are:
г=0, D = 0.005 m^{2}/day for all periods;
v = 0.003, 0.010, 0.003, 0.003 (m/day);
t = 90, 100, 80, 90 (days);
c_{w} = 0, 0, 0.001, 0;
Table shows the results of the calculations.
Conclusion
Prognosis of the salt regime with or without the salt exchange (solid phase) can be performed using formulae (15)…(22) of the solution of parabolic equation given in [9].
References
1. Averyanov S.F. Preduprezhdenie zasoleniya oroshaemykh zemel …(“Prevention of salinity…”), in Oroshaemoe zemledelie evropeiskoi chasti SSSR, Moskva: Kolos, 1965.
2. Verigin N.N. Nekotorye voprosy physicokhimicheskoi gydrodynamiki… (“Some issues of physicochemical hydrodynamics…”), Izv. AN SSSR STN, 1953 №10.
3. Volobuev V.R. On the Leaching Rates, GiM, 1959. №12.
4. Plyusnin I.I., Golovanov A.I..Meliorativnoe pochvovedenie //Ameliorative Soil Science, Moskva, 1983.
5. Golovanov A. Papadopoulos On the Calculated Probability of Salt Content, SUEE, 2004.
6. Gradshtein I.S., Ryzhik I.M. Tables of Integrals..., 1996, eBook.
7. Papadopoulos D. On Determination of Leaching Rates, Gydrotehnika I Melioratsiya, 1973. №7.
8. Reks L.M. O prognoze zasoleniya pochv posle promyvok (“On prognosis of salinity…”), Pochvovedenie, 1969. №7/
9. Budak B.M., Samarskii A.A., Tikhonov. A.N. A Collection of Problems in Mathematical Phisics, New York, 1988/
10. Lapidus L., Amundson L.R. Mathematics of Adsorption in Beds Journal of Physical Chemistry. v.56/8. 1952 984988.
11. Tikhonov A .N., Samarskii A. Equations of Mathematical Physics, New York, 1990.
12. Korn Granino, Korn Teresa. “Mathematical Handbook for Scientists and Engineers, Dover, 2000.
13. Press et al W H. Numerical Recipes in C, Cambridge, 1988.
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