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**Groundwater Flow Equations**

Groundwater Hydraulics Daene C. McKinney

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**Summary General Groundwater Flow Confined Aquifer Flow**

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

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**Control Volume Control volume of dimensions Dx, Dy, Dz**

Completely saturated with a fluid of density r x y z Mass flux in Mass flux out Control volume

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**Mass Flux Mass flux = Mass in - Mass out: Fluid mass in the volume:**

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

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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, qx and qy No vertical flow, qz = 0 Average properties over aquifer thickness (b) Ground surface Bedrock Confined aquifer Qx 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 Qx 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 Qx K x y z hB Confining Layer b hA L steady flow Head in the aquifer

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**Example – Horizontal Flow**

L = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, f = 0.35 Find: head, specific discharge, and average velocity Ground surface Bedrock Confined aquifer Qx K=2-m/d x y z hB=80m Confining Layer b hA=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 Dx Qx K h x y z

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**Steady Flow in an Unconfined Aquifer**

1-D flow Steady State K = constant h Flow hA hB Water Table Ground Surface Bedrock L x

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**Steady Flow in an Unconfined Aquifer**

K = 10-1 cm/sec L = 150 m hA = 6.5 m hB = 4 m x = 150 m Find Q h Flow hA=6.5m hB=4m Water Table Ground Surface Bedrock L=150m x K=0.1cm/s

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**Summary General Groundwater Flow Confined Aquifer Flow**

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

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Figure from Hornberger et al. (1998) Darcy’s data for two different sands.

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