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Unformatted text preview: 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #4: Continuity and Flow Nets Equation of Continuity • Our equations of hydrogeology are a combination of o Conservation of mass o Some empirical law (Darcy’s Law, Fick’s Law) • Develop a control volume, rectangular parallelepiped, REV (Representative Elementary Volume) z y x dx dy dz x, y, z q mass inflow rate – mass outflow rate = change in mass storage q x = specific discharge in xdirection (volume flux per area) at a point x,y,z L 3 /L 2T Consider mass flow through plane yz at (x,y,z) x ρ dy dz L/T M/L 3 L L = M/T Rate of change of mass flux in the xdirection per unit time per crosssection is ∂ ] [ ρ dydz q dx dy dz x, y, z x ∂ x x  dx/2 x x + dx/2 1.72, Groundwater Hydrology Lecture Packet 4 Prof. Charles Harvey Page 1 of 13 mass flow into the entry plane yz is ∂ dx ] [ ρ dydz q − [ ρ q ] dydz x x ∂ x 2 And mass flow out of the exit plane yz is ∂ dx ] [ ρ dydz q + [ ρ q ] dydz x x ∂ x 2 In the xdirection, the flow in minus the flow out is ∂ ] − [ ρ dxdydz q x ∂ x Similarly, the flow in the ydirection through the plane dxdz (figure on left) dy dz x, y, z dx dy dz x, y, z dx ⎡ ⎤ ⎡ ⎤ ∂ ⎢ [ ρ dxdz q − ∂ [ ρ q y ] dy dxdz ⎥ − ⎢ [ ρ dxdz q + ∂ [ ρ q y ] dy dxdz ⎥ − = ∂ y [ ρ dxdydz q ⎣ y ] ] y ] ∂ y 2 ⎦ ⎣ y ∂ y 2 ⎦ for the net ymass flux. Similarly, we get for the net mass flux in the zdirection: ∂ ] − [ ρ dxdydz q z ∂ z The total mass flux (flow out of the box) is ⎡ ∂ [ ρ q ] ∂ [ ρ q y ] − ∂ [ ρ q ] ⎤ ⎥ dxdydz x − z ⎢ − ⎣ ∂ x ∂ y ∂ z ⎦ 1.72, Groundwater Hydrology Lecture Packet 4 Prof. Charles Harvey Page 2 of 13 x Let’s consider time derivative = 0 (Steady State System) ⎡ ∂ [ ρ q x ] ∂ [ ρ q y y ] ∂ [ ρ q z z ] ⎤ = = + ∂ ⎥ ⎦ ∂ dxdydz ∂ M t − ⎢ ⎣ ∂ + ∂ How does Darcy’s Law fit into this? h z ∂ ∂ ∂ • For anisotropy (with alignment of coordinate axes and tensor principal h y ∂ ∂ h x K = − q K − = K − = q q z zz ∂ y yy x xx r − = • Assuming constant density for groundwater q h K ∇ directions), or + ∂ [ q y y ∂ ] ⎡ ⎤ ∂ [ ∂ q x x ] ∂ [ ] q z z ∂ ρ ⎢ ⎣ 0 − = + ⎥ ⎦ substituting for q, ⎡ ⎤ ⎥ ⎦ ⎛ ⎞ ⎟ ⎟ + ⎠ h x K ∂ ∂ h y K ∂ ∂ h z K ∂ ∂ ∂ ∂ ∂ ⎛ x ⎜ ∂ ⎞ ⎟ + ⎠ z ⎛ ⎜ ∂ ⎞ ⎟ ⎠ ρ ⎢ 0 = ( − )( − ) y ∂ ⎝ ⎜ ⎜ x y z ⎝ ⎝ ⎣ Steadystate flow equation for heterogeneous, anisotropic conditions: ⎡ ⎤ ⎞ ⎟ + ⎠ ∂ ⎛ y ∂ ⎝ ⎜ ⎜ K y ⎞ h x K ∂ ∂ h y ∂ ⎟ ⎟ + ∂ ⎠ h z ∂ ⎞ ⎟ ∂ ∂ ∂ x ⎛ ⎜ ∂ z ⎛ ⎜ ∂ = K ⎢ ⎥ ⎦ x z ⎝ ⎝ ⎠ ⎣ For isotropic, homogeneous conditions (K is not directional) ∂ 2 h ∂ 2 h ∂ 2 h ⎤ ⎡ 0 = K + + ⎢ ⎣ z ⎥ ∂ ⎦ x ∂ 2 y ∂ 2 2 This is the “diffusion equation” or “heatflow equation” ∂ 2 ∂ 2 h ∂ 2 h ⎤...
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This note was uploaded on 09/13/2010 for the course CE 451 taught by Professor Lee during the Spring '07 term at USC.
 Spring '07
 Lee

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