Understanding Unconfined Aquifer Flow: Theory and Equations

Unconfined aquifer flow occurs above the water table with complexities in solving governing equations. Approximate solutions are derived for steady horizontal flow and Dupuit approximations in this type of aquifer.

• Aquifers
• Groundwater
• Hydraulics
• Flow
• Unconfined

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1. Flow in Aquifers 2 Unconfined Aquifer Flow Groundwater Hydraulics Daene C. McKinney 1

2. Summary Unconfined Aquifer Flow Continuity Equation Steady Horizontal Flow Steady Horizontal Flow with Infiltration Drains 2

3. Unconfined Aquifer Flows bounded above by a water table occur in unconfined aquifers Thickness of capillary fringe above the water table is assumed to be much smaller than the saturated domain below the water table The water table is a nonlinear boundary and makes the exact solution of the governing equations almost impossible Approximate solutions can be obtained based on a linearization of the boundary conditions or the nonlinear continuity equation. 3

4. Flow in an Unconfined Aquifer Dupuit approximations Slope of the water table is small Velocities are horizontal Considers a streamtube bounded by two streamlines: water table and impervious bottom Ground surface Water table Qx h(x) K z y Unconfined aquifer x Bedrock w ??= ?? ? ?? 4

5. Continuity Equation in an Unconfined Aquifer N (accretion, infiltration) ??? ? 2,? ???+ ? ? ? 2,? Water table + ???,? ? ???,?+ ? ? ? 2 2 = ?? ? ? ? + ? ? ??? ? h 2,? ???+ ? 2,? y x 5

6. Continuity Equation in an Unconfined Aquifer ??? ? 2,? ???+ ? ? ? + ???,? ? ???,?+ ? ? ? 2,? 2 2 = ?? ? ? ? + ? ? Divide by ?, ?, ? and take limit as ?, ?, ? 0 ? = ? ? (??= ?? ? ??? ??+??? ? ?? ??,? = ?? ? + ? = ?? ??) ?? ? ?? ?? ? ? ?? ?? ? ? ?? + + ? = ?? ?? ?? ? 2 ?? ? 2 ?? ,? = ?? (??= ?? ? 2 ?? ? 2 ?? ?? 2 ? ?? ?? 2 ? ?? ? ?? ) + + ? = ?? 2 2 6

7. Continuity Equation in an Unconfined Aquifer Homogeneous aquifer (K = constant) ?2 2 ??2 +?2 2 ??2 ? 2 ? ?? + ? = ?? Steady flow ?2 2 ??2 +?2 2 ??2 ? 2 + ? = 0 No accretion (infiltration) ?2 2 ??2 +?2 2 ??2 = 0 One-dimensional flow ?2 2 ??2 = 0 7

8. 1-D Steady Flow in an Unconfined Aquifer Find the head in a homogeneous, unconfined aquifer with steady flow from left to right ? 2 ?? ? 2 ?? ?? 2 ? ?? ?? 2 ? ?? ? ?? Governing equation + + ? = ?? ?2 2 ??2 Simplifications: steady, 1-D flow, K = Constant, No accretion = 0 2 2 ? ? ? Ground Surface 2+ 2? = ? ? Water Table Head in the aquifer ? 2 ?? ??= ?? Flow hA 2 h hB 2 2 ? ? ? = ?? ? Bedrock L 2 x 8

9. 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 Ground Surface Water Table Flow hA=6.5m h K=0.1cm/s hB=4m Bedrock L=150m x 2 2 ? ? 42 6.52 150 ?2/? = 7.56?3 ? = ?? ? = 86.4 ?/? ?/? 2 2 9

10. 1-D Steady Flow in an Unconfined Aquifer with Infiltration Find the head in a homogeneous, unconfined aquifer with steady flow from left to right ? 2 ?? ? 2 ?? ?? 2 ? ?? ?? 2 ? ?? ? ?? Governing equation + + ? = ?? ?2 2 ??2 +2? Simplifications: steady, 1-D flow, K = Constant, With accretion ?= 0 N, Infiltration 2= ? Ground Surface ??2+ ?1? + ?2 Water Table hMax 2 2 ? ? h ? ? +? Flow 2+ 2= ? hA ? ? ? ? hB Bedrock L x 10

11. Steady flow in an unconfined aquifer with infiltration ? 2 ?? ??= ?? N, Infiltration 2 Ground Surface Water Table 2 2 ? ? ? = ?? ? ? ? 2? 2 ? hMax h Flow hA hB Bedrock L x ???????=? ? 2 2 ? 2+ ? 2?? 11

12. Example (Embankment) N, Infiltration Given L = 3000 m K = 20 m/day hA = 30 m hB = 20 m N = 500 mm/yr Ground Surface Water Table hMax h Flow hA hB Bedrock L x Find: Flow to the streams and shape of water table 2 2 ? ? ?(?) = ?? ? ? ? 2? 2 ? 12

13. Example (Cont.) N, Infiltration (202-302 3000 Q =-20 +0.00137 20 3000) 2 Ground Surface =-0.39m3/day at x = 0 Water Table hMax (202-302 3000 Q =-20 -0.00137 20 h 3000) Flow hA 2 =-20 hB (-1.6667-0.2055) 2 Bedrock L x =18.72m3/day at x = L +3000 -500 3000 20 xdivide= 2*0.00137 = 2716.5m (or 283.5m from right boundary) 2 13

14. Unconfined Flow in an Aquifer with Horizontal Stratification Observation Well A Observation Well B Ground Surface ? = ?1?? ? ?? ?? ?2 ? Water Table K2 ??? = ?1?? ?2 ? ? hA hB K1 a Bedrock L x ?=? ? ??? = ?1? ? ?2 ? ? ?=0 ? ? 2 2 ? = ?1? ? ? ?2 ? 2 ? ? ? 2 ? ? ? ? 14

15. Example ?= 98.6 ?; ?= 97.2 ?;? = 20 ? ?1= 2.2 ?/???;?2= 16.2 ?/???;? = 400 ? 2 2 ? = ?1? ? ? ?2 ? 2 ? ? ? 2 ? ? ? ? 97.22 2 98,62 2 ? = 2.2 20 97.2 98.6 16.2 20 97.2 20 98.6 400 400 ? = 4.51?3 Observation Well A Observation Well B ???/? Ground Surface Water Table K2 hA hB K1 a Bedrock L 15 x

16. Example (Drains) Subsurface drains are often used to prevent the water table elevation from increasing to an elevation where it would waterlog the roots of crops or bring unwanted salts and other chemicals in to the root zone. Steady, 1-D flow , homogeneous, isotropic aquifer, w/infiltration d2h2 ( ) dx2 N, Infiltration =-2N K Ground Surface Water Table h2=Nx 2 K(L- x)+hd hMax hd = h2 at x =L 2 hMax 2 Lower Confining Layer L =NL2 4K x 2 +hd 16

17. Summary Unconfined Aquifer Flow Continuity Equation Steady Horizontal Flow Steady Horizontal Flow with Infiltration Drains 17