*Модель фильтрации сквозь однородную пористую среду*

*Инженерный журнал: наука и инновации*

# 9·2016 11

**Filtration model through a homogeneous porous medium**

©

A.A. Gurchenkov

1,2

, M.V. Nosov

1

1

Institution of the Russian Academy of Sciences Dorodnicyn Computing Centre

of RAS, Moscow, 119333, Russia

2

Bauman Moscow State Technical University, Moscow, 105005, Russia

*The study considers a model of vertical water transfer in soil; describes the water trans-*

*fer process by one-dimensional nonlinear parabolic equation. The diffusion coefficient*

*and the soil hydraulic conductivity included in the equation is calculated by van Genuch-*

*ten formulas widely used in practice. An important model component is the evaporation*

*from the soil surface. The study formulates the problem of determining evaporation as an*

*optimal problem — the one, in which the phase variables are the soil moisture values at*

*different depths, and control is the desired evaporation. The mean-square soil moisture*

*values deviation from some prescribed values derived from calculations based on the*

*hydrological models is the objective function. We solve the numerical optimization by the*

*steepest descent method; the objective function gradient is calculated using the fast au-*

*tomatic differentiation method (FAD).*

*Keywords:*

*the steepest descent method, the objective function, the optimal control prob-*

*lem, a method of fast automatic differentiation.*

REFERENCES

[1]

Genuchten

M.Th., van.

*Soil Science Society of America Journal*

, 1980, vol. 44,

pp. 892–898.

[2]

Ayda-Zade K.R., Evtushenko Yu.G.

*Matematicheskoe modelirovanie — Math*

*modeling*

, 1989, vol. 1, pp. 121–139.

[3]

Griewank A. On automatic differentiation

*. Mathematical Programming: Recent*

*Developments and Applications.*

Iri M., Tanabe K

*.,*

ed. Tokyo, Kluwer

Academic Publ., 1989, pp. 83–108.

[4]

Griewank A., Corliss G. F., ed.

*Automatic Differentiation of Algorithms. Theory,*

*Implementation and Application.*

Philadelphia, Society for Industrial and

Applied Mathematics (SIAM) Publ., 1991, pp. 238–245.

[5]

Evtushenko Yu.G.

Automatic differentiation viewed from optimal control

theory.

*Automatic Differentiation of Algorithms. Theory, Implementation and*

*Application.*

Griewank A., Corliss G.F., ed. Philadelphia, Society for Industrial

and Applied Mathematics (SIAM) Publ., 1991, pp. 25–30.

[6]

Evtushenko Yu.G.

*Optimization methods and software,*

1998, vol. 9, pp. 45–75.

[7]

Griewank A.

*Evaluating Derivatives.*

Philadelphia, Society for Industrial and

Applied Mathematics (SIAM) Publ.

*,*

2000, pp. 43–49.

[8]

Karmanov V.G.

*Matematicheskoe programmirovanie*

[Mathematical

programming]. Moscow, Fizmatlit Publ., 2004.

[9]

Abakarov

A.Sh., Sushkov Yu.A.

*Trudy FORA — Works of the Adygea Republic*

*Physical Society,*

2004, no. 1, pp. 154–160.

[10]

Takha Khemdi A.

*Vvedenie v issledovanie operatsiy*

[Operations research: an

Introduction]

*.*

8

th

ed., Moscow, Williams Publ., 2007, 912 p.

[11]

Plotnikov

A.D.

*Matematicheskoe*

*programmirovanie*

[Mathematical

programming]. Proc. of the express-course

*.*

Minsk, Novoe znanie Publ.,

2006,

171 p.