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The Study of the Drainage of Ground Water by the Boundary Element Method

The Study of the Drainage of Ground Water by the Boundary Element Method. Adrian Carabineanu Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy University of Bucharest, Department of Mathematics Bucharest, ROMANIA. Introduction.

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The Study of the Drainage of Ground Water by the Boundary Element Method

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  1. The Study of the Drainage of Ground Water by the Boundary Element Method Adrian Carabineanu Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy University of Bucharest, Department of Mathematics Bucharest, ROMANIA

  2. Introduction • The paper deals with the study of the steady motion of underground water determined by rains falling during a long period with a constant rate of flow. We consider that the ground consists of a layer of constant thickness bounded from below by a horizontal impermeable bed. Above the impermeable bed there is a saturated region where the pores from the ground are completely filled by water. The saturated region is separated by a free surface from a unsaturated region where the pores are filled partially by water and partially by air. The water coming from the rain flows vertically in the unsaturated region until it meets the free separation surface.

  3. Introduction

  4. Introduction • Usually the underground water is drained by a network of drains having the same shape. One also considers that the distance between every two neighboring drains is the same. Therefore in the plane where we study the filtration flow we have to consider two families of of axes of symmetry, perpendicular to the straight impermeable bed : the axes from one family (I) lie at equal distances to two neighboring drains and the axes from the other family (II) are just the axes of symmetry of the drains (figure 1). Because of the symmetry properties it suffices to study the motion of the underground water in a domain delimited by an axis of symmetry from family (I) and by another axis of summetry from family (II). The scope of the paper is to determine (taking into account the rate of rain flow and the geometry of the porous ground layer) the shape of the free surface and subsidiary the hydrodynamic spectrum (the streamlines, the equipotential lines and the velocity field).

  5. The case of the drains consisting of narrow rectangular channels • Darcy’s law equation ofcontinuity

  6. The case of the drains consisting of narrow rectangular channels • the streamfunction the Cauchy – Riemann equations the complex potential

  7. Boundary conditions AO and OC are streamlines On DE and EA we impose Setting we get In the portion of the drain filled with water the pressure is equal to the hydrostatic pressure stands for the level of the water in the drain Let q denote the constant rate of rainfall

  8. Dimensionless variables we take as reference dimensional quantities k and the hydraulic conductivity on the free surface EA on the wet surface DE on the surface of alimentation CD on the impermeable bed OC and on the axis of symmetry AO

  9. An iterative procedure for solving the problem of drains consisting of rectangular narrow channels by means of the boundary element method • We start from Cauchy’s formula is approximated by a polygon consisting of segments (panels) linear interpolation

  10. An iterative procedure for solving the problem of drains consisting of rectangular narrow channels by means of the boundary element method • Taking we introduce the coefficients we obtain the algebraic system Introducing the notations we deduce, separating the real terms from the imaginary ones:

  11. An iterative procedure for solving the problem of drains consisting of rectangular narrow channels by means of the boundary element method The equations of the system (30) – (31) are not independent and in the sequel we shall choose among them independent equations At the first iteration we shall consider that the domain in which we study the filtration flow is

  12. First iteration On the impermeable OC bed we consider the nodes On the wall CE of the drain we take the nodes On the free surface AE we consider the nodes and on the axis of symmetry AE

  13. First iteration • Taking into account the boundary conditions we shall form the system of equations

  14. Second iteration • After solving the system we calculate the position of the nodes from the free surface, corresponding to the second iteration, For the domain of motion corresponding to the second iteration, the system is replaced by

  15. Next iterations • We perform new iterations (according to the procedure presented at the second iteration) until, beginning from the p - th iteration the shape of the free surface does not change any more

  16. Example

  17. Drains consisting of circular tubes We apply an iterative procedure similar to the procedure for rectangular drains

  18. Drains consisting of circular tubes. Example

  19. References • [1] Brebbia, C.A., Telles, J. C. F., Wrobel, L. C., 1984: Boundary Element Techniques, Springer-Verlag. • [2] France, p. W., Parekh, C. J., Peters, J. C., Taylor C.,1971: Numerical Analysis of Free Surface Seepage Problems, J. Irrigation Drainage Div. ASCE 97, 165-179. • [3] Pietraru, V., 1977: Seepage computation, Ceres Publishers, Bucharest.

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