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Unformatted text preview: EAS 44600 Groundwater Hydrology Lecture 10: Equations of Groundwater Flow Dr. Pengfei Zhang Equations of Groundwater Flow Groundwater flow can be described by partial differential equations where the spatial coordinates, x , y , z , and time, t , are the independent variables. Let us consider flow within a representative elementary volume (R.E.V.) of an aquifer (Figure 101). In this R.E.V. groundwater can enter or exit any face of the cube. The mass of groundwater entering the cube minus the mass of groundwater exiting the cube equals the accumulation (or depletion) of groundwater in the cube. A quick dimensional analysis shows that the rate of mass input or output of groundwater through the R.E.V. is the product of ρ q A : T M L T L L L M A q w REV REV w w w = ⋅ ⋅ = ⋅ ⋅ 2 2 3 3 ρ (101) where ρ is the density, q is the specific discharge, and A is the crosssectional area of the R.E.V. Input zdirection Across dx dy face: dy dx q z ⋅ ⋅ ρ Input ydirection Across dx dz face: dz dx q y ⋅ ⋅ ρ Input xdirection Across dy dz face: dz dy q x ⋅ ⋅ ρ Output ydirection Output xdirection Across dx dz face: Across dy dz face: dz dx dy y q q y y ⋅ ⋅ ∂ ∂ + ) ( ρ ρ dz dy dx x q q x x ⋅ ⋅ ∂ ∂ + ) ( ρ ρ dz dy dx Output zdirection Across dx dy face: dy dx dz z q q z z ⋅ ⋅ ∂ ∂ + ) ( ρ ρ Figure 101. Schematic diagram showing groundwater input and output through an R.E.V. 101 If the R.E.V. has dimensions of dx , dy , and dz (Figure 101), the mass balance of water moving through the R.E.V. in the xdirection is equal to the input, dy dz q x ⋅ ⋅ ⋅ ρ , minus the output, dy dz dx x q q x x ⋅ ⋅ ∂ ⋅ ∂ + ⋅ ) ( ρ ρ , where the differential term in the output is the change in groundwater flux during transport through the R.E.V. in the xdirection across the distance dx . The rate of accumulation (or depletion) of water in the cube is equal to ) ( dz dy dx n t ⋅ ⋅ ⋅ ⋅ ∂ ∂ ρ , as indicated by the dimensional analysis: T M L L L L M T dz dy dx n t w REV REV w w w = = ⋅ ⋅ ⋅ ⋅ ∂ ∂ 3 3 3 3 1 ) ( ρ (102) So, mathematically speaking, the conservation of mass is: ) ( ) ( ) ( ) ( dz dy dx n t dy dx dz z q q dz dx dy y q q dz dy dx x q q dy dx q dz dx q dz dy q z z y y x x z y x ⋅ ⋅ ⋅ ⋅ ∂ ∂ = ⋅ ⋅ ∂ ∂ + − ⋅...
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This note was uploaded on 12/29/2010 for the course EAS 44600 taught by Professor Pengfeizhang during the Spring '10 term at CUNY City.
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