Groundwater Flow Equations Groundwater Hydraulics Daene C. McKinney

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

Control Volume Control volume of dimensions  x,  y,  z Completely saturated with a fluid of density  [M/L 3 ] Discharge q [L 3 /T] Mass flux  q [M/T] x y z Mass flux inMass flux out Control volume

Mass Flux Mass in - Mass out= Mass flux mass flux in mass flux out Mass flux Mass inMass out Mass flux =

Mass Flux Fluid mass in the volume: Continuity: Mass flux = change of mass mass flux in mass flux out Get rid of DV

Aquifer Storage Water compressibilityAquifer compressibility Now, put it back into the continuity equation

Continuity Equation r constant with x Storage coefficient Darcy’s Law

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 HeadSpecific dischargeDischarge

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 Darcy’s Law

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

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

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

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

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

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

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