An irrigation pond is built from an impermeable concrete cylinder installed in a hole with deep(saturated) clay soils lying at the bottom the pond. The clay has a hydraulic conductivity of 1x10-3 m/day. During summer, evaporation leads to a loss of 5 mm/day from the pond.The pond contains 50 cm depth of water at the beginning of summer. Assume that the pressure in the pore water below the pond is zero everywhere, and that it is the flux of water in the soil (regardless of the stored depth of water in the pond) that drives the drainage from the pond. A. Write an expression for the amount of time (At ) it takes for the pond to dry out, in terms of the change in storage (AS) and the vertical fluxes out of the pond control volume (E and 9₂), which you should define as fluxes leaving the control volume (i.e., both fluxes will be positive, meaning E points in the +z direction and q₂ in the -z direction). B. How long does it take for the pond to dryout? C. The fact that water was flowing vertically in(A) and (B) means that we do not have hydrostatic pressure within the soil. Assuming uniformity in the x and y directions and the z-axis oriented upward, in what range must the

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.

The goal of this question is to use Darcy's Law and the concept of a control volume to write a water balance for flow in porous media in three dimensions. The multiple parts of the question are intended to help you through the steps of this derivation. The control volume is a cube with dimensionsAx x Ay × Az [L³]. It contains a homogeneous and isotropic porous medium with porosity and saturated hydraulic conductivity Ksat [L/T]. The porous medium is saturated by fluid with a constant density p,although we won't need to assume saturation until part (F). Water is flowing through the control volume at steady state. The volumetric flux of water per unit area [L/T] can be denoted by q=qxi+qyj+qzk, where the bold type indicates a vector quantity, and i, j and k are the unit vectors in the x, y and z directions respectively. For a single direction, e.g. the x direction, the flux entering the control volume is qx (x), and the flux leaving the control volume is qx (x + Ax) A. Sketch the 3D control volume and the components of the volumetric water flux in each direction. Label all relevant dimensions and show a coordinate axis. B. Write down expressions (in units ofmass/time) for the mass flux entering the control volume in the x direction; the mass flux leaving the control volume in the x direction; the net mass flux in the x direction; and ■ the net mass fluxes in the y and zdirections. C. Write down the change of the mass stored in the control volume (in units of mass/time) interms of volumetric water content (do not assume saturation). D. Using the results from (B) and (C), write out the mass balance for the control volume accounting for mass fluxes in x, y, and z directions, in units of mass/time. E. Divide the answer to (D) by the density p and the volume Ax x Ay x Az. Apply the definition of a derivative to represent the rates of change in space and time in differential form. F. Apply the criteria that the porous media is saturated (VWC=0 = p). Given the definition of the control volume, this implies steady state(the amount of stored water is not changing over time). Using Darcy's Law for saturated flow, re-write your answer to part (E) in terms of the hydraulic gradient in the x, y and z directions and Ksat. Simplify the equation.