(20 Marks) Show that the flow of water through a porous media (soil) is governed by the Laplace Equation:
∂²H/∂x² + ∂²H/∂y² + ∂²H/∂z² = 0
Use the conservation of water volume (net inflow = 0) for a control volume of soil and the constitutive law (Darcy's Law) of:
q = (qx, qy, qz) = (-k ∂H/∂x, -k ∂H/∂y, -k ∂H/∂z)
where q is the water flux vector (discharge per unit area); (qx, qy, qz) are the components of the flux vector in the coordinate directions; k is the hydraulic conductivity, H is the piezometric head defined as: H = p/γ + h where p is the pressure, γ is the specific weight of water and h is the elevation.
(20 Marks) Set up the mathematical boundary value problem (governing equation and all boundary conditions) that represents the flow of water through a pervious sand layer under a dam as pictured below. The sand layer may be assumed to have a rectangular shape in cross section (length = L in the x direction and depth = D in the y direction). The dam is very long so that water fluxes in the z direction (along the dam) may be neglected. The dam itself has a width of W centrally located on the top of the sand layer. You may assume that the piezometric head under the dam varies linearly from the upstream reservoir elevation of H1 down to the downstream water elevation. You may also assume that the material surrounding the sand block (upstream, downstream, and below) is also impervious.