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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 10-1). 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 (10-1) where ρ is the density, q is the specific discharge, and A is the cross-sectional area of the R.E.V. Input z-direction Across dx- dy face: dy dx q z ⋅ ⋅ ρ Input y-direction Across dx- dz face: dz dx q y ⋅ ⋅ ρ Input x-direction Across dy- dz face: dz dy q x ⋅ ⋅ ρ Output y-direction Output x-direction 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 z-direction Across dx- dy face: dy dx dz z q q z z ⋅ ⋅ ∂ ∂ + ) ( ρ ρ Figure 10-1. Schematic diagram showing groundwater input and output through an R.E.V. 10-1 If the R.E.V. has dimensions of dx , dy , and dz (Figure 10-1), the mass balance of water moving through the R.E.V. in the x-direction 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 x-direction 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 (10-2) 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.
- Spring '10