Understanding Groundwater Flow in Aquifers

Groundwater flow in aquifers, whether confined or leaky, involves complex concepts like aquifer transmissivity, storativity, conductivity, Darcy's law, and continuity equations. Focusing on horizontal flow approximation and utilizing formulas for transmissivity and conductivity allows us to study the movement of water through different types of aquifers.

• Groundwater Flow
• Aquifers
• Transmissivity
• Darcys Law
• Hydrogeology

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1. Flow in Aquifers 1 Confined Aquifer Flow Groundwater Hydraulics Daene C. McKinney 1

2. Summary Confined Aquifer Flow Continuity Equation Steady Horizontal Flow Transmissivity 2

3. Horizontal Flow Approximation Previously used CV approach to develop continuity equation for flow in porous media. Now we want to focus on groundwater flow in aquifers (confined and leaky to begin). Flow in aquifers is essentially horizontal. Instead of considering flow as three-dimensional, with = (?,?,?,?) we may treat the problem in terms of an average head, = (?,?,?) Where the average is taken along a vertical line extending from the bottom to the top of the aquifer 3

4. Aquifer Transmissivity Transmissivity (T) Discharge through entire thickness of an aquifer per unit width per unit head gradient Hydraulic gradient = 1 m/m ? = ?? ? 1 ? 0 Transmissivity, T, volume of water flowing an area 1 m x b under hydraulic gradient of 1 m/m ? = ? ? ?? 1 m b 1 m Storativity (S) Volume of water released from (or added to) storage in the aquifer per unit horizontal area per unit decline (or rise) of the average (over the vertical) piezometric head in the aquifer 1 m Conductivity, K, volume of water flowing an area 1 m x 1 m under hydraulic gradient of 1 m/m ? = ??? 4

5. Flow in a Confined Aquifer Darcy s law ? = ?? ? ??,??= ?? ? ??) Ground surface (??= ?? Head in confined aquifer Confining Layer ??? ? b 2,? h qx ???+ ? Confined aquifer b 2,? K y x Bedrock ??? ? 2,? ???+ ? ? ? + ???,? ? ???,?+ ? ? ? 2,? 2 2 = S ? ? ? + ? ? 5

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

7. Flow in a Leaky Aquifer Ground surface Water table in unconfined aquifer ?? Head in confined aquifer Leaky Confining Layer K b h0 ??? ? b Leakage from above 2,? ???+ ? h qx Confined aquifer 2,? b K y x Bedrock Datum, z = 0 ??? ? 2,? ???+ ? ? ? + ???,? ? ???,?+ ? ? ? + ?? ? ? ? 2,? 2 2 = S ? ? ? + ? ? 7

8. Flow in a Leaky Aquifer ??? ? 2,? ???+ ? ? ? + ???,? ? ???,?+ ? ? ? + ?? ? ? ? 2,? 2 2 = S ? ? ? + ? ? Divide by ?, ?, ? and take limit as ?, ?, ? 0,and ??= ? ? 0 ? ??= ? ? ?? ? ?? ? ?? ? ?? 0 ? = S? ?? + ?? + ? ?? 8

9. Continuity Equation in a Confined Aquifer Homogeneous aquifer (K = constant) ?2 ??2+?2 0 ? = S? ? + ? ??2 ?? Steady flow ?2 ??2+?2 0 ? ? + ? = 0 ??2 No leakage ?2 ??2+?2 ??2= 0 One-dimensional flow ?2 ??2= 0 9

10. Well in a Leaky Aquifer ? ? ?? ? ?? ? ?? ? ?? 0 ? ??? ? ?? = S? ?? + ?? + ? ?? ?=1 ? ? ?? = 0,?? ? ?? Dirac delta function 1,?? ? = ?? 10

11. Notation ? ?? ? ?? ? ?? Gradient operator Rectangular coordinates =? ? +? ? +? ? = ? ?? ?? ?? =? ? +1 ? ? +? Cylindrical coordinates ? ? ?? ? ?? ?? Divergence Rectangular coordinates ??+??? ? ? =??? ??+??? ?? ? ? =1 ? ??? ?? +1 ??? ??+??? Cylindrical coordinates ? ? ?? Laplacian Rectangular coordinates = ?2? =?2 +?2 +?2 ? ? ??2 ??2 ??2 Cylindrical coordinates ?2 +?2 = ?2? =1 ? ?? ?? +1 ? ? ?2 ??2 ??2 ? ?? 11

12. Groundwater Notation ? ?? ? ?? ? ?? ?? 0 0 0 0 0 ?? ? =? ?? ? +? ?? ? +? ? = ?? 0 ? = ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ?? ?? 0 0 0 0 0 ?? ? ?? ? + ?? ? ?? ? + ?? ? ?? ? ?? = ?? 0 ?? = = ?? ?? ? ? +? ? +? ? ?? ? + ?? +? ?? ? ?? ? + ?? ? ?? ? ?? ? ? ?? = ?? ?? ?? ? ?? ? ?? ? ?? ? ?? ? ?? = ?? + ?? ?? 12

13. 1-D Steady Flow in a Confined Aquifer Find the head in a homogeneous, confined aquifer with steady flow from left to right ? ?? ? ?? ? ?? ? ?? 0 ? = S? ?? + ?? + ? Governing equation ?? ?2 ??2= 0 Simplifications: steady, 1-D flow, T = Constant, No leakage Ground surface ? = ?+ ? ? h(x) ? Confining Layer Head in the aquifer Confined aquifer Qx hA hB b K z y x Bedrock L 13

14. Steady Flow in a Confined Aquifer L = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, ?= 0.35 Find: head, specific discharge, and average velocity h(x)= hA+hB-hA q= -KhB-hA x =100-0.02xm L L = -(20 m/d)80-100 1000 Ground surface = 0.4 m/day v =q Confining Layer f Confined aquifer Qx K=2-m/d hA=100m hB=80m =1.14 m/day b z y x Bedrock L=1000m 14

15. Summary Confined Aquifer Flow Continuity Equation Steady Horizontal Flow Transmissivity 15