1. Flow to Wells – 2 Steady flow to a well in an unconfined aquifer
Groundwater Hydraulics
Daene C. McKinney

2. Summary Notation Steady flow to a well in an unconfined aquifer
Steady flow to a well in an unconfined aquifer with recharge

3. Flow to a Well in an Unconfined Aquifer

4. Groundwater Notation Unconfined aquifer
𝛻∙𝐾ℎ𝛻ℎ= 𝜕 𝜕𝑥 𝐾 𝑥 ℎ 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝐾 𝑦 ℎ 𝜕ℎ 𝜕𝑦 = 𝑆 𝑦 𝜕ℎ 𝜕𝑡
General equation,
rectangular coordinates
ℎ 𝜕ℎ 𝜕𝑥 = 𝜕 ℎ 2 𝜕𝑥
𝛻∙ 𝐾 2 𝛻 ℎ 2 = 𝜕 𝜕𝑥 𝐾 𝑥 2 𝜕 ℎ 2 𝜕𝑥 + 𝜕 𝜕𝑦 𝐾 𝑦 2 𝜕 ℎ 2 𝜕𝑦 = 𝑆 𝑦 𝜕ℎ 𝜕𝑡
Steady flow, homogeneous,
isotropic aquifer
𝛻 2 ℎ 2 = 𝜕 2 ℎ 2 𝜕 𝑥 𝜕 2 ℎ 2 𝜕 𝑦 2 =0

5. Groundwater Notation Unconfined aquifer
𝛻∙ 𝐾 2 𝛻 ℎ 2 = 1 𝑟 𝜕 𝜕𝑟 𝐾 𝑟 2 𝑟 𝜕 ℎ 2 𝜕𝑟 𝑟 2 𝜕 𝜕𝜃 𝐾 𝜃 2 𝑟 𝜕 ℎ 2 𝜕𝜃 = 𝑆 𝑦 𝜕ℎ 𝜕𝑡
General equation,
radial coordinates
𝛻 2 ℎ 2 = 𝑑 𝑑𝑟 𝑟 𝑑 ℎ 2 𝑑𝑟 =0
Steady flow, homogeneous,
isotropic aquifer

6. Steady Flow to a Well in an unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Pre-pumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
Governing equation
in radial coordinates
𝛻 2 ℎ 2 = 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕 ℎ 2 𝜕𝑟 𝑟 2 𝜕 2 ℎ 2 𝜕 𝜃 2 = 2 𝑆 𝑦 𝐾 𝜕ℎ 𝜕𝑡
Assumptions: Steady state flow,
radial symmetry (isotropic medium),
𝑑 𝑑𝑟 𝑟 𝑑 ℎ 2 𝑑𝑟 =0

7. Steady Flow to a Well in an Unconfined Aquifer
𝑑 𝑑𝑟 𝑟 𝑑 ℎ 2 𝑑𝑟 =0
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Pre-pumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
𝑟 𝑑 ℎ 2 𝑑𝑟 = 𝐶 1
Darcy’s law
𝑄=𝐴𝑞= 2𝜋𝑟ℎ 𝐾 𝑑ℎ 𝑑𝑟 =𝜋𝐾𝑟 𝑑 ℎ 2 𝑑𝑟
𝑟 𝑑 ℎ 2 𝑑𝑟 = 𝑄 𝜋𝐾 =𝐶 1
𝑑 ℎ 2 = 𝑄 𝜋𝐾 𝑑𝑟 𝑟
ℎ 2 = 𝑄 𝜋𝐾 𝑙𝑛 𝑟 + 𝐶 2

8. Steady Flow to a Well in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Pre-pumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
ℎ 2 = 𝑄 𝜋𝐾 𝑙𝑛 𝑟 + 𝐶 2
Boundary condition:
ℎ= ℎ 1 @ 𝑟= 𝑟 1
ℎ 1 2 = 𝑄 𝜋𝐾 𝑙𝑛 𝑟 1 + 𝐶 2
𝐶 2 = ℎ 1 2 − 𝑄 𝜋𝐾 𝑙𝑛 𝑟 1
ℎ 2 = 𝑄 𝜋𝐾 𝑙𝑛 𝑟 + ℎ 1 2 − 𝑄 𝜋𝐾 𝑙𝑛 𝑟 1
ℎ 2 = ℎ 𝑄 𝜋𝐾 𝑙𝑛 𝑟 𝑟 1

9. Example – Steady Flow to a Well in an Unconfined Aquifer
Given:
Q = 300 m3/hr
Two observation wells,
1. h1 = 40 r1 = 50 m
2. h2 = 43 r2 = 100 m
Find: K
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Prepumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
ℎ 2 = ℎ 𝑄 𝜋𝐾 𝑙𝑛 𝑟 𝑟 1
ℎ 2 2 = ℎ 𝑄 𝜋𝐾 𝑙𝑛 𝑟 2 𝑟 1

10. Steady flow to a Well in an Unconfined Aquifer with Recharge
1 𝑟 𝜕 𝜕𝑟 𝐾 2 𝑟 𝜕 ℎ 2 𝜕𝑟 +𝑊=0
𝑑 𝑑𝑟 𝑟 𝑑 ℎ 2 𝑑𝑟 + 2𝑊 𝐾 𝑟=0
𝑟 𝑑 ℎ 2 𝑑𝑟 + 𝑊 𝐾 𝑟 2 = 𝐶 1
𝑑 ℎ 2 𝑑𝑟 = 𝐶 1 𝑟 − 𝑊 𝐾 𝑟
radius of influence

11. Steady flow to a Well in an Unconfined Aquifer with Recharge
𝑑 ℎ 2 𝑑𝑟 = 𝐶 1 𝑟 − 𝑊 𝐾 𝑟
Boundary condition 1: No flow across radius of influence
𝑞=−𝐾ℎ 𝑑ℎ 𝑑𝑟 =− 𝐾 2 𝑑 ℎ 2 𝑑𝑟 =0;
𝑑 ℎ 2 𝑑𝑟 𝑟= 𝑟 0 = 𝐶 1 𝑟 0 − 𝑊 𝐾 𝑟 0 =0
radius of influence
𝐶 1 = 𝑊 𝐾 𝑟 0 2

12. Steady flow to a Well in an Unconfined Aquifer with Recharge
𝑑 ℎ 2 𝑑𝑟 = 𝐶 1 𝑟 − 𝑊 𝐾 𝑟
𝐶 1 = 𝑊 𝐾 𝑟 0 2
Boundary condition 2: Recharge within radius of influence is captured by the well
𝑄=𝐴𝑊= 𝜋 𝑟 0 2 𝑊
𝑟 0 2 = 𝑄 𝜋𝑊
radius of influence
𝐶 1 = 𝑄 𝜋𝐾
𝑑 ℎ 2 𝑑𝑟 = 𝑄 𝜋𝐾 1 𝑟 − 𝑊 𝐾 𝑟

13. Steady flow to a Well in an Unconfined Aquifer with Recharge
𝑑 ℎ 2 𝑑𝑟 = 𝑄 𝜋𝐾 1 𝑟 − 𝑊 𝐾 𝑟
𝑑 ℎ 2 = 𝑄 𝜋𝐾 𝑑𝑟 𝑟 − 𝑊 𝐾 𝑟𝑑𝑟
ℎ 2 = 𝑄 𝜋𝐾 ln 𝑟 − 𝑊 2𝐾 𝑟 2 + 𝐶 2
Boundary condition 3: Recharge within radius of influence is captured by the well
radius of influence
ℎ= ℎ 0 @ 𝑟= 𝑟 0
ℎ 0 2 = 𝑄 𝜋𝐾 ln 𝑟 0 − 𝑊 2𝐾 𝑟 𝐶 2
ℎ 2 = ℎ 𝑄 𝜋𝐾 𝑙𝑛 𝑟 𝑟 0 − 𝑊 2𝐾 𝑟 2 − 𝑟 0 2

14. Example - Steady flow to a Well in an Unconfined Aquifer with Recharge
25 cm diameter well to maintain lowered water table
K = 1x10-5 m/s (0.864 m/d)
Bottom of aquifer is horizontal at 20 m bgs
Water table is 1 m bgs
Unconfined aquifer is recharged at W = 0.06 m/d
Water table must be lowered 3 m over site r
Drawdown at corners must be at least 3 m
r=35.35 m
ℎ 2 = ℎ 𝑄 𝑤 𝜋𝐾 𝑙𝑛 𝑟 𝑟 0 − 𝑊 2𝐾 𝑟 2 − 𝑟 0 2
16 2 = 𝜋 𝑟 𝜋𝐾 𝑙𝑛 𝑟 0 − (0.864) − 𝑟 0 2
Solve by iteration: r0 = 70 m
𝑄 𝑤 =𝜋 𝑟 0 2 𝑊=𝜋 = 𝑚 3 /𝑑

15. Summary Notation Steady flow to a well in an unconfined aquifer
Steady flow to a well in an unconfined aquifer with recharge