Continuity Equation for Groundwater Flow

Groundwater Flow EquationsGroundwater Hydraulics

Daene C. McKinney

1SummaryGeneral Groundwater FlowControl Volume AnalysisGeneral Continuity EquationConfined Aquifer FlowContinuity EquationIntegrate over vertical dimensionTransmissivityContinuityExamplesUnconfined Aquifer FlowDarcy LawContinuity EquationExamplesControl VolumeControl volume of dimensions Dx, Dy, Dz Completely saturated with a fluid of density rxyzMass flux inMass flux out

Control volume3Mass FluxMass flux = Mass in - Mass out:

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

4Aquifer Storage

Continuity Equation

Confined Aquifer Flow7Horizontal Aquifer FlowMost aquifers are thin compared to horizontal extentFlow is horizontal, qx and qyNo vertical flow, qz = 0 Average properties over aquifer thickness (b)

Ground surfaceBedrockConfined aquiferQxKxyzhHead in confined aquiferConfining Layerb

8Aquifer TransmissivityTransmissivity (T) Discharge through thickness of aquifer per unit width per unit head gradientProduct of conductivity and thicknessHydraulic gradient = 1 m/mPotentiometric SurfaceConfining BedConfined Aquiferb1 m1 m1 mTransmissivity, T, volume of water flowing an area 1 m x b under hydraulic gradient of 1 m/mConductivity, K, volume of water flowing an area 1 m x 1 m under hydraulic gradient of 1 m/m

9Continuity EquationContinuity equation

Darcys Law

Continuity

Ground surfaceBedrockConfined aquiferQxKxyzhHead in confined aquiferConfining Layerb

Radial Coordinates10Example Horizontal FlowConsider steady flow from left to right in a confined aquifer Find: Head in the aquifer, h(x)

Ground surfaceBedrockConfined aquiferQxKxyzhBConfining LayerbhALsteady flowHead in the aquifer

11Example Horizontal FlowL = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, f = 0.35Find: head, specific discharge, and average velocity

Ground surfaceBedrockConfined aquiferQxK=2-m/dxyzhB=80mConfining LayerbhA=100mL=1000m12Unconfined Aquifer Flow13Flow in an Unconfined AquiferDupuit approximations Slope of the water table is smallVelocities are horizontalGround surfaceBedrockUnconfined aquiferWater tableDxQxKhxyz

14Steady Flow in an Unconfined Aquifer1-D flow

Steady State

K = constant

hFlowhAhBWater TableGround SurfaceBedrockLx

15Steady Flow in an Unconfined AquiferK = 10-1 cm/secL = 150 mhA = 6.5 mhB = 4 mx = 150 mFind Q

hFlowhA=6.5mhB=4mWater TableGround SurfaceBedrockL=150mx

K=0.1cm/s16SummaryGeneral Groundwater FlowControl Volume AnalysisGeneral Continuity EquationConfined Aquifer FlowContinuity EquationIntegrate over vertical dimensionTransmissivityContinuityExamplesUnconfined Aquifer FlowDarcy LawContinuity EquationExamples