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Groundwater Flow Equations Groundwater Hydraulics Daene C. McKinney

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Summary General Groundwater Flow – Control Volume Analysis – General Continuity Equation Confined Aquifer Flow – Continuity Equation – Integrate over vertical dimension – Transmissivity – Continuity – Examples Unconfined Aquifer Flow – Darcy Law – Continuity Equation – Examples

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Control Volume Control volume of dimensions x, y, z Completely saturated with a fluid of density x y z Mass flux inMass flux out Control volume

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Mass Flux Mass flux = Mass in - Mass out: Fluid mass in the volume: Continuity: Mass flux = change of mass mass flux in mass flux out

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Aquifer Storage

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Continuity Equation

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Confined Aquifer Flow

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Horizontal Aquifer Flow Most aquifers are thin compared to horizontal extent – Flow is horizontal, q x and q y – No vertical flow, q z = 0 – Average properties over aquifer thickness (b) Ground surface Bedrock Confined aquifer QxQx K x y z h Head in confined aquifer Confining Layer b

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Aquifer Transmissivity Transmissivity (T) – Discharge through thickness of aquifer per unit width per unit head gradient – Product of conductivity and thickness Hydraulic gradient = 1 m/m Potentiometric Surface Confining Bed Confined Aquifer b 1 m Transmissivity, T, volume of water flowing an area 1 m x b under hydraulic gradient of 1 m/m Conductivity, K, volume of water flowing an area 1 m x 1 m under hydraulic gradient of 1 m/m

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Continuity Equation Continuity equation Darcy’s Law Continuity Ground surface Bedrock Confined aquifer QxQx K x y z h Head in confined aquifer Confining Layer b Radial Coordinates

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Example – Horizontal Flow Consider steady flow from left to right in a confined aquifer Find: Head in the aquifer, h(x) Ground surface Bedrock Confined aquifer QxQx K x y z hBhB Confining Layer b hAhA L steady flow Head in the aquifer

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Example – Horizontal Flow L = 1000 m, h A = 100 m, h B = 80 m, K = 20 m/d, = 0.35 Find: head, specific discharge, and average velocity Ground surface Bedrock Confined aquifer QxQx K=2-m/d x y z h B =80m Confining Layer b h A =100m L=1000m

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Unconfined Aquifer Flow

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Flow in an Unconfined Aquifer Dupuit approximations – Slope of the water table is small – Velocities are horizontal Ground surface Bedrock Unconfined aquifer Water table xx QxQx K h x y z

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Steady Flow in an Unconfined Aquifer 1-D flow Steady State K = constant h Flow hAhA hBhB Water Table Ground Surface Bedrock L x

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Steady Flow in an Unconfined Aquifer K = cm/sec L = 150 m h A = 6.5 m h B = 4 m x = 150 m Find Q h Flow h A =6.5m h B =4m Water Table Ground Surface Bedrock L=150m x K=0.1cm/s

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Summary General Groundwater Flow – Control Volume Analysis – General Continuity Equation Confined Aquifer Flow – Continuity Equation – Integrate over vertical dimension – Transmissivity – Continuity – Examples Unconfined Aquifer Flow – Darcy Law – Continuity Equation – Examples

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