Kochubey O. O., Yevdokymov D. V., Kriachunenko O. L., Deriy V. S.

Oles Honchar Dnipropetrovsk National University


Last years due to food crisis further development of agricultural industry has become one of the most important aim of Humanity. Development of agricultural industry is impossible without correspondent development of agricultural science. For example, a progress in plant-growing completely depends on success in agricultural science, but the last is connected with understanding of biological, chemical, mechanical and thermal processes in soils. Biological and chemical processes are investigated intensively using all opportunities of modern science. As for filtration flows and mass transfer processes in soils, which must be investigated mainly by mathematical tools, the unnecessary simplified mathematical models are used in this field. For example, roots of plants suck up water, that stimulates a specific filtration flows. However, cross-section area of plant root is negligibly small in comparison with reference size of whole domain. As a result, multiscale problems must be formulated as correct mathematical models of filtration in soils. In fact, multiscale problems have never used for this aim because of mathematical and computational difficulties are intrinsic to them. The similar situation takes place with sorption of salts dissolved in water by particle of rigid frame of soils, that was not considered properly. It is evident, that to improve the situation, new generalized mathematical models and algorithms of numerical solution must be proposed. The present work is to be considered as an attempt to provide some progress in this direction.

Generally speaking, there are two mathematical models of mass transfer in soil filtration model applied for water motion in saturated soil, and diffusive model by A.V. Lykov for unsaturated media. Accuracy and applicability of the second model requires additional special investigations and discussions. For the sake of simplicity, the following consideration will be restricted by the filtration model in plane case.

Let us consider a domain формула of saturated porous media. Let there is filtration flow due to some external cautions and there are roots of plants which are perpendicular to a plane формула. The roots of plants create locally decreased pressure due to osmotic effect and, as a result, they suck up water. Cross-section areas of roots of plants are negligibly small. Then the filtration flow is described by following equation:

формула, (1)

формула,   (2)

where формула is pressure in porous media; формула is number of roots of plants; формула are intensities of sinks; формула is Dirac's delta-function; формула are Cartesian coordinates, формула is point of domain формулаформула is position of the root of plant; формула is velocity of the filtration flow; формула is filtration coefficient. Correspondent boundary conditions are prescribed on the boundary формула of the domain формула for equation (1). Intensity of sink формула depends on osmotic pressure, created by correspondent root of plant and local "pressure in infinity". The last term means the pressure in position of the sink, if the sink is absent there. The simplest dependence of формула on pressure is, of course, linear, what provides completely linear filtration problem. However, such non-linear dependences are possible too, that leads to evident mathematical and computational difficulties.

Presence of delta- functions in the formulation of problem (1) makes it multiscale. As a result, the most popular finite difference and finite element methods can not be applied for its numerical solution. Thus boundary element method is the only tool, that can be used here. To apply a boundary element method, let us transform the initial governing differential equation to boundary integral equation:

формула  (3)

where формула is fundamental solution of Laplace equation; формула is function of boundary shape.

Equetion (3) can be solved by usual boundary element algorithm with only difference in last term in right hand side of equation if формула linear depends on формула, equation (3) can be approximated by system lineal algebraic equations, including additional equations for determinations of unknown values формула In the case of non-linear dependence, equations for формула are non-linear and require a special iteration process to handle them.

Let us consider some substance transfer by filtration flow. It is described by following equation

формула, (4)

where формула is concentration of transferred substance; формула is filtration velocity; формула is diffusion coefficient; the second term in right hand ride of equation (4) формула describes sorption process, that is this term represents distributed sources (sinks); the last term in right hand side of equation (4) corresponds to the mentioned effect of roots of plants.

Equation (4) with necessary initial and boundary conditional is multiscale problem, but this case requires completely another approach. The following spitting

формулаформула  (5)

gives an opportunity to avoid computational difficulties. Concentration формула corresponds to distributed sources and it is described by transfer equation like (4) without the last term. Concentration формула field can be calculated by usual finite deferens method, for example, explicit “up-wined” scheme is used in the present work:

формула (6)

Concentration формулаcorresponds to localized sources (sinks) and it can be analyzed by potential theory methods. For example, this field can be approximated by sum of fundamental solution of Oseen’s equation.

The proposed approach is illustrated by several examples of numerical calculations, which conforms its workability and effectiveness.