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

Summary Confined Aquifer Flow Continuity Equation Steady Horizontal Flow Transmissivity

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)

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

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

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

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

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

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

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

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 + 𝜕 2 𝜕 𝑦 2 + 𝜕 2 𝜕 𝑧 2 𝛻∙𝛻 = 𝛻 2 𝑨= 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕 𝜕𝑟 + 1 𝑟 2 𝜕 2 𝜕 𝜃 2 + 𝜕 2 𝜕 𝑧 2

Groundwater Notation 𝛻ℎ= 𝜕ℎ 𝜕𝑥 𝒊 + 𝜕ℎ 𝜕𝑦 𝒋 + 𝜕ℎ 𝜕𝑧 𝒌 = 𝜕ℎ 𝜕𝑥 𝜕ℎ 𝜕𝑦 𝜕ℎ 𝜕𝑧 𝑲= 𝐾 𝑥 0 0 0 𝐾 𝑦 0 0 0 𝐾 𝑧 𝐾𝛻ℎ= 𝐾 𝑥 0 0 0 𝐾 𝑦 0 0 0 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

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

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

Summary Confined Aquifer Flow Continuity Equation Steady Horizontal Flow Transmissivity