Page - 1199 - in Book of Full Papers - Symposium Hydro Engineering

3. Seepage flow through Earth Dams Seepage flow of water through porous media depends on the soil media, type of flow, properties of liquid and hydraulic gradient. Seepage piping account for approximately 40% of all earth dam failures. Different methods have been developed to solve seepage problems, these methods can be classified as analytical, experimental and a numerical methods. Ground water flows in the direction of decreasing potential energy caused by differences in pressure and elevation. A common measure of this potential energy is the total head, 𝛟 which is simply the sum of pressure head and elevation head. The volume rate of flow per unit area is directly proportional to the rate of change of head as given by the differential form of Darcy's Law. The following general governing equation for seepage through earth dams can be considered as [1, 3]: 𝛛 𝛛 𝐱 [𝐊 𝐱 𝛛 𝐇 𝛛 𝐱 ] + 𝛛 𝛛 𝐲 [𝐊 𝐲 𝛛 𝐇 𝛛 𝐲 ] + 𝛛 𝛛 𝐳 [ 𝐤 𝐳 𝛛 𝐇 𝛛 𝐳 ] = 𝐒 𝝏 𝒉 𝝏 𝒕 In Ω ……………. (1) where, Kx, Ky and Kz are the coefficient of permeability in x, y, z directions, respectively, S is specific yield and 𝐇 = p/γw + z = total fluid head, P = pressure, γw = unit weight of water and z = elevation head. Equation (1) is known as Laplace’s equation which is considered as the governing equation for groundwater there dimensions flow through aquifers. For an isotropic, homogeneous aquifer under steady state conditions, Equation (2) can be simplified to the following equation: 𝝏 𝟐 𝐇 𝝏 𝒙 𝟐 + 𝝏 𝟐 𝐇 𝝏 𝒛 𝟐 = 𝟎 In Ω ………………………………………………….. (2) It is assumed that; the soil media is homogeneous, isotropic, and physically stable; the pressure is atmospheric everywhere on the water table (phreatic surface); the flow of ground water through the flow domain is steady and the hydraulic conductivity through the dam body is constant everywhere. 3.1 Boundary Conditions Figure (1) shows a schematic representation for the problem statement and boundary conditions for a typical earth dam. These boundary conditions can be briefly summarized as follows: 3.1.1 Entrance Surface (Γ1) The upstream boundary surface (Γ1) is the entrance surface at which the percolation of reservoir water through the media starts. This surface is considered as an equipotential line which is known as Dirchlet Condition for a prescribed head as: 𝐇 (x,y) = H1 ……………………………………………………………….. (3) 3.1.2 Phreatic Surface (Γ2) The boundary surface (Γ2) is the phreatic surface of the flow through the dam. This boundary is considered as a stream line. The phreatic surface, although it is considered as a boundary condition, its location and its profile are unknown a priori. For the unknown phreatic surface the boundary condition is: 𝐇 (x, y) = y ……………………………………………………………….... (4. a) ∂𝐇 ∂n = 0 …………………………………………………………............ (4. b) where n is the normal directions to the boundary (Γ2). 3.1.3 Exit Surface (Seepage surface) (Γ3) The boundary surface (Γ3) is the exit surface or seepage surface. This boundary is considered as an isobar at which the pressure along it is atmospheric, hence, the boundary condition along such a surface is: 𝐇 (x, y) = y ………………………………………...................................... (5) The geometry of the seepage surface is known, except its upper limit the exit point, which is laying on the unknown phreatic surface. The location of this point is a part of the required solution. 3.1.4 Dam Foundation Boundary (Γ4) The boundary surface (Γ4) is the dam foundation boundary, which is also known as Neumann Condition. In the present study, this boundary is assumed to be impervious i.e. this surface prevents the flow of water across it such that: ∂𝐇 ∂n = 0 ………………………………………………………………………… (6) where n is the normal directions to the boundary (Γ4). 4. Stability of Earth Dam Slopes Every soil mass which has a slope at its end is subjected to shear stresses on internal surfaces or failure plans in the soil mass near the slope. This is due to the gravitational forces that try to pull down parts of the soil mass adjoining the slope. Several models and analytical techniques have been developed to determine the critical slip surface and the associated factor of safety such as method of slices. The factor of safety is: F.S. = 𝝉 𝒇 𝝉 ………………………………………………………………… (7) 1199