CONSTRUCTION AND RESEARCH OF FULL BALANCE ENERGY OF VARIATIONAL PROBLEM MOTION SURFACE AND GROUNDWATER FLOWS

Abstract Based on the laws of conservation of mass and momentum the basic equations of motion with unknown quantities velocity and piezometric pressure are written. These equations are supplemented with boundary and initial conditions describing the motion of compatible flows. Based on the laws of motion continuum, received conditions contact on the common border interaction of surface and groundwater flows. Variational problems formulated compatible flow. Energy norms of basic components of variational problem are analyzed. Correctness of constructing variational problem arising from construction of the energy system of equations that allow to investigate properties of the problem solution, its uniqueness, stability, dependence on initial data and more. Energy equation of motion of surface and groundwater flows are derived and investigated. It is shown that the total energy compatible flow depends on sources that are located inside the domain or on its border.


Introduction
An important role in studying the water cycle plays hydrological system. In general, research integrity of the system, taking into account all impacts, are complex and not always feasible problem for the study because only investigated some of the area involved in the water cycle [1][2][3] Highly likely part of the territory may be a watershed area (Fig. 1), which is characterized by similar climatic conditions and is influenced by such factors that affect the water movement. At the watershed may be an interaction between flow and located above and below water-bearing layers. Models of different dimensions are used in each layer to describe the water movement and their solutions are connected by boundary conditions [4][5][6].
We select in solid medium (liquid) moving surface layer ( ) 3 Ft R ∈ (Fig. 2)  t : x ,x ,x R , x x x,t x x ,x t .
Let's denote projection of its lower t : bases on the plane 12 0x x . The rest of the surface layer will be called the lateral surface layer ( ) Ft. Similarly denote part of fluid that moves in the soil, so the projection of the lower part will be written as Then, a layer of ground water

1. Initial boundary value problem of interaction of water flows
We formulate initial boundary problem of motion of surface and groundwater flows on the surface watershed considering boundary and initial conditions [7][8][9].
Find unknown quantities { } u, p, ϕ such that satisfy the following system of equations: { } -known function of sources of water influx; piezometric pressure; -velocity vector of fluid in the ground; q υ= ω , ω -volume porosity; FP nn =−   -vectors normal to the boundary area F Ω and P Ω in accordance; Boundary conditions [10,11]: where R -velocity of falling rain drops, 0 1 u , 0 2 u -horizontal components of velocity on the free where I -known function that describes the velocity of fluid flow through the surface P Λ .
Initial conditions:

Variational formulation of the problem of interaction of water flows
We introduce the following bilinear forms:

3. The properties of the components and norms of variational problem interaction water flows.
It should be noted that trilinear form

1. Equation balance energy of coupling water flow
Let's write variational equations for momentum   Computer Sciences and Mathematics As we see from (45), the total energy flow depends on the energy sources that are located within the region or within its boundaries.

Conclusions
On the basis of conservation laws basic equations and boundary and initial conditions are derived describing the compatible motion flow of surface and ground water with unknown values of velocity and piezometric pressure. Variational problems of compatible flow are formulated and the contact conditions on the common border are obtained based on the laws of motion continuum. Energy standards of basic components of variational problem are analyzed. Full energy equation of energy balance for coupling motion of surface and groundwater flows are constructed and studied that makes it possible to investigate the properties of solutions of the problem, such as stability, regularity, existence, convergence and so on.