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

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

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
ℎ=ℎ(𝑥,𝑦,𝑧,𝑡)
ℎ = ℎ (𝑥,𝑦,𝑡)

4. Aquifer Transmissivity
Transmissivity (T)
Discharge through entire thickness of an aquifer per unit width per unit head gradient
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
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
𝑇= 𝐾 𝑏
𝐾 = 1 𝑏 0 𝑏 𝐾 𝑧 𝑑𝑧
𝑆= 𝑆 𝑠 𝑏

5. Flow in a Confined Aquifer
Darcy’s law
𝒒=−𝑇𝜵ℎ
( 𝑞 𝑥 =− 𝑇 𝑥 𝜕ℎ 𝜕𝑥 , 𝑞 𝑦 =− 𝑇 𝑦 𝜕ℎ 𝜕𝑦 )
Ground surface
Head in confined aquifer
Confining Layer b
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦
𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦 qx h
Confined aquifer b K
Bedrock
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦 − 𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦 ∆𝑦∆𝑡+ 𝑞 𝑦 𝑥,𝑦− ∆𝑦 2 − 𝑞 𝑦 𝑥, 𝑦+ ∆𝑦 2 ∆𝑥∆𝑡
=S∆𝑥∆𝑦 ℎ 𝑡+∆𝑡 −ℎ 𝑡

6. Continuity Equation in a Confined Aquifer
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦 − 𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦 ∆𝑦∆𝑡+ 𝑞 𝑦 𝑥,𝑦− ∆𝑦 2 − 𝑞 𝑦 𝑥, 𝑦+ ∆𝑦 2 ∆𝑥∆𝑡
=S∆𝑥∆𝑦 ℎ 𝑡+∆𝑡 −ℎ 𝑠
Divide by ∆𝑥, ∆𝑦, ∆𝑡 and take limit as ∆𝑥, ∆𝑦, ∆𝑡 →0
𝒒=−𝑇𝜵ℎ
( 𝑞 𝑥 =− 𝑇 𝑥 𝜕ℎ 𝜕𝑥 , 𝑞 𝑦 =− 𝑇 𝑦 𝜕ℎ 𝜕𝑦 )
− 𝜕 𝑞 𝑥 𝜕𝑥 + 𝜕 𝑞 𝑦 𝜕𝑦 =S 𝜕ℎ 𝜕𝑡
𝜕 𝜕𝑥 𝑇 𝑥 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝑇 𝑦 𝜕ℎ 𝜕𝑦 =S 𝜕ℎ 𝜕𝑡

7. Flow in a Leaky Aquifer 𝑞 𝑧 𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦 b 𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦
Ground surface
Bedrock
Confined aquifer qx K h
Head in confined aquifer b
Water table in unconfined aquifer h0 b’ K’
Datum, z = 0
Leaky Confining Layer
Leakage from above
𝑞 𝑧 b
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦
𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦 − 𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦 ∆𝑦∆𝑡+ 𝑞 𝑦 𝑥,𝑦− ∆𝑦 2 − 𝑞 𝑦 𝑥, 𝑦+ ∆𝑦 2 ∆𝑥∆𝑡+ 𝑞 𝑧 ∆𝑥∆𝑦∆𝑡
=S∆𝑥∆𝑦 ℎ 𝑡+∆𝑡 −ℎ 𝑠

8. Flow in a Leaky Aquifer
𝑞 𝑥 𝑥− ∆𝑥 2 ,𝑦 − 𝑞 𝑥 𝑥+ ∆𝑥 2 ,𝑦 ∆𝑦∆𝑡+ 𝑞 𝑦 𝑥,𝑦− ∆𝑦 2 − 𝑞 𝑦 𝑥, 𝑦+ ∆𝑦 2 ∆𝑥∆𝑡+ 𝑞 𝑧 ∆𝑥∆𝑦∆𝑡
=S∆𝑥∆𝑦 ℎ 𝑡+∆𝑡 −ℎ 𝑠
Divide by ∆𝑥, ∆𝑦, ∆𝑡 and take limit as ∆𝑥, ∆𝑦, ∆𝑡 →0, and
𝑞 𝑧 =− 𝐾 ′ 𝜕 ℎ ′ 𝜕𝑧 =𝐾′ ℎ 0 −ℎ 𝑏′
𝜕 𝜕𝑥 𝑇 𝑥 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝑇 𝑦 𝜕ℎ 𝜕𝑦 +𝐾′ ℎ 0 −ℎ 𝑏′ =S 𝜕ℎ 𝜕𝑡

9. Continuity Equation in a Confined Aquifer
Homogeneous aquifer (K = constant)
Steady flow
No leakage
One-dimensional flow
𝑇 𝜕 2 ℎ 𝜕 𝑥 𝜕 2 ℎ 𝜕 𝑦 2 +𝐾′ ℎ 0 −ℎ 𝑏′ =S 𝜕ℎ 𝜕𝑡
𝑇 𝜕 2 ℎ 𝜕 𝑥 𝜕 2 ℎ 𝜕 𝑦 2 +𝐾′ ℎ 0 −ℎ 𝑏′ =0
𝜕 2 ℎ 𝜕 𝑥 𝜕 2 ℎ 𝜕 𝑦 2 =0
𝑑 2 ℎ 𝑑 𝑥 2 =0

10. Well in a Leaky Aquifer Dirac delta function
𝜕 𝜕𝑥 𝑇 𝑥 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝑇 𝑦 𝜕ℎ 𝜕𝑦 +𝐾′ ℎ 0 −ℎ 𝑏′ ± 𝑤=1 𝑊 𝑄 𝑤 𝛿 𝒙− 𝒙 𝑤 =S 𝜕ℎ 𝜕𝑡
𝛿 𝒙− 𝒙 𝑤 = 0, 𝑖𝑓 𝒙 ≠ 𝒙 𝑤 1, 𝑖𝑓 𝒙= 𝒙 𝑤

11. Notation Gradient operator Divergence Laplacian
𝛻 = 𝜕 𝜕𝑥 𝒊 + 𝜕 𝜕𝑦 𝒋 + 𝜕 𝜕𝑧 𝒌 = 𝜕ℎ 𝜕𝑥 𝜕ℎ 𝜕𝑦 𝜕ℎ 𝜕𝑧
Gradient operator
Rectangular coordinates
Cylindrical coordinates
Divergence
Laplacian
𝛻 = 𝜕 𝜕𝑟 𝒓 + 1 𝑟 𝜕 𝜕𝜃 𝜃 + 𝜕 𝜕𝑧 𝒌
𝛻∙𝑨= 𝜕 𝐴 𝑥 𝜕𝑥 + 𝜕 𝐴 𝑦 𝜕𝑦 + 𝜕 𝐴 𝑟 𝜕𝑧
𝛻∙𝐴= 1 𝑟 𝜕 𝑟 𝐴 𝑟 𝜕𝑟 + 1 𝑟 𝜕 𝐴 𝜃 𝜕𝜃 + 𝜕 𝐴 𝑟 𝜕𝑧
𝛻∙𝛻 = 𝛻 2 𝑨= 𝜕 2 𝜕 𝑥 𝜕 2 𝜕 𝑦 𝜕 2 𝜕 𝑧 2
𝛻∙𝛻 = 𝛻 2 𝑨= 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕 𝜕𝑟 𝑟 2 𝜕 2 𝜕 𝜃 𝜕 2 𝜕 𝑧 2

12. Groundwater Notation
𝛻ℎ= 𝜕ℎ 𝜕𝑥 𝒊 + 𝜕ℎ 𝜕𝑦 𝒋 + 𝜕ℎ 𝜕𝑧 𝒌 = 𝜕ℎ 𝜕𝑥 𝜕ℎ 𝜕𝑦 𝜕ℎ 𝜕𝑧
𝑲= 𝐾 𝑥 𝐾 𝑦 𝐾 𝑧
𝐾𝛻ℎ= 𝐾 𝑥 𝐾 𝑦 𝐾 𝑧 𝜕ℎ 𝜕𝑥 𝜕ℎ 𝜕𝑦 𝜕ℎ 𝜕𝑧 = 𝐾 𝑥 𝜕ℎ 𝜕𝑥 𝐾 𝑦 𝜕ℎ 𝜕𝑦 𝐾 𝑧 𝜕ℎ 𝜕𝑧 = 𝐾 𝑥 𝜕ℎ 𝜕𝑥 𝒊 + 𝐾 𝑥 𝜕ℎ 𝜕𝑦 𝒋 + 𝐾 𝑥 𝜕ℎ 𝜕𝑧 𝒌
𝛻∙𝐾𝛻ℎ= 𝜕 𝜕𝑥 𝒊 + 𝜕 𝜕𝑦 𝒋 + 𝜕 𝜕𝑧 𝒌 ∙ 𝐾 𝑥 𝜕ℎ 𝜕𝑥 𝒊 + 𝐾 𝑥 𝜕ℎ 𝜕𝑦 𝒋 + 𝐾 𝑥 𝜕ℎ 𝜕𝑧 𝒌
= 𝜕 𝜕𝑥 𝐾 𝑥 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝐾 𝑦 𝜕ℎ 𝜕𝑦 + 𝜕 𝜕𝑧 𝐾 𝑧 𝜕ℎ 𝜕𝑧

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

14. Steady Flow in a Confined Aquifer
L = 1000 m, hA = 100 m, hB = 80 m, K = 20 m/d, 𝜙= 0.35
Find: head, specific discharge, and average velocity
Ground surface
Bedrock
Confined aquifer Qx
K=2-m/d x y z
hB=80m
Confining Layer b
hA=100m
L=1000m

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