Question

An earthen dam is constructed on an impermeable bedrock layer. It is 85 m across,with conductivity and water levels above and below the dam shown in the figure and the current level of the water table shown in the black line. Assume that porosity is homogeneous, and that saturated hydraulic conductivity is homogeneous and isotropic.

A. Compute the volume of water (per 100 mwide strip of the dam, in m³/day) that passesinto the dam from the reservoir. Compute thisflow rate through a surface area that extendsvertically down from the edge of the surface ofwater in the reservoir. Do the same for the flowrate into the tailwaters. B. The water table within the dam is not at steady state, as indicated by your answers in part (A). Sketch a control volume and derive a governing equation (which will be a partial differential equation) for unsteady flow through the reservoir. Although the water table is not completely horizontal, assume all the flow lines are horizontal. Your final answer should be a partial differential equation in terms of the height of the water table in the dam, h(x,t).Remember that you can also simplify your answer using the chain rule for multiplication from calculus to get your answer in terms ofh²2. C. Assume that the water table has reached steady state (there are no changes with respect to time). Solve the partial differential equation for h(x), including calculating values for the integration constants using the two constant-head boundary conditions on both sides of the reservoir for the differential equation. Your final answer should be a function h(x), with x as the only remaining variable.

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