Question

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.

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8

Fig: 9

Fig: 10

Fig: 11

Fig: 12