05 groundwater flow equations

hidraULICA

1. Groundwater Flow Equations Groundwater Hydraulics Daene C. McKinney

2. 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

3. Control Volume Control volume of dimensions D x , D y , D z Completely saturated with a fluid of density r x y z Mass flux in Mass flux out Control volume

4. Mass Flux Mass flux = Mass in - Mass out: mass flux in mass flux out Mass flux Mass in Mass out

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

6. Aquifer Storage Water compressibility Aquifer compressibility Chain rule Now, put it back into the continuity equation

7. Continuity Equation

8. Confined Aquifer Flow

9. 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 Q x K x y z h Head in confined aquifer Confining Layer b

10. 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 1 m 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

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

12. 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 Q x K x y z h B Confining Layer b h A L steady flow

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

14. Unconfined Aquifer Flow

15. Flow in an Unconfined Aquifer Dupuit approximations Slope of the water table is small Velocities are horizontal Ground surface Bedrock Unconfined aquifer Water table D x Q x K h x y z

16. Steady Flow in an Unconfined Aquifer 1-D flow Steady State, K = constant Find h ( x ) h Flow h A h B Water Table Ground Surface Bedrock L x

17. Steady Flow in an Unconfined Aquifer K = 10 -1 cm/sec L = 150 m h A = 6.5 m h B = 4 m x = 150 m Find h(x), Q h Flow h A =6.5m h B =4m Water Table Ground Surface Bedrock L =150m x K =0.1cm/s

18. 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

19. Groundwater Flow Equations Examples

20. 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 Q x K x y z h B Confining Layer b h A L steady flow Head in the aquifer

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

22. Steady Flow in an Unconfined Aquifer 1-D flow Steady State, K = constant h Flow h A h B Water Table Ground Surface Bedrock L x

23. Steady Flow in an Unconfined Aquifer K = 10 -1 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