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##### Fetter, Ch. 4

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**Fetter, Ch. 4**• 4.1 Introduction • 4.2 Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 Darcy’s Law • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Introduction**• Groundwater possesses energy • in a variety of forms: • Mechanical • Thermal • Chemical**Fetter, Ch. 4**• 4.1 Introduction • 4.2Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Head in Water of Variable Density**If z1 = z2 , hf = (ρp/ρf ) hpoint**Head in Water of Variable Density**Which way is water moving vertically— up or down? (see p.120)**Head in Water of Variable Density**hfresh = 5,500 ft ρ = 1,000 kg/m3 5,000 ft ρ = 1,100 kg/m3 hpoint =5,000 ft Z =0 ft**Fetter, Ch. 4**• 4.1 Introduction • 4.2 Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Fetter, Ch. 4**• 4.1Introduction • 4.2Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Relationship of flow to gradient**Hydraulic conductivity ellipse: technique for determining effect of anisotropy on flow Look at result: is flow deflected towards or away from highest K?**Relationship of flow to gradient**Flow nets in isotropic (A) and anisotropic (B) mediums**Relationship of flow to gradient**Flow nets in isotropic (A) and anisotropic (B) mediums**Flow Nets**El. = 200 ft A F B G H E D C 30 ft El. = 200 ft 30 ft A F B G H E D C**Fetter, Ch. 4**• 4.1Introduction • 4.2Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**El. = 200 ft**A F B G H E C 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F B El. = 170 ft G H E 195 D C 190 185**El. = 200 ft**A F B G H E D C 30 ft El. = 200 ft 30 ft A F B El. = 170 ft G H E 195 D C At point C: ht = 192.5 ft z = 128 ft hp = 64.5 ft p = 64.5 ft x 62.4 pcf = 4,025 psf 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F B El. = 170 ft G H E full stream tube ns= 2.2 nd = 6 ns / nd = 2.2 / 6 = 0.37 Q = K grad h A Q for square #1 = Q1 Q1 = K (grad h) A full stream tube 195 D C partial stream tube 190 185**El. = 200 ft**A • Flow net properties: • Flow and equipotential lines form squares, • Each stream tube carries equal flow. • Partial flow tubes are allowed. 30 ft El. = 200 ft 30 ft A F B El. = 170 ft G H E 195 D C 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F B El. = 170 ft G H E full stream tube number of tubes (ns)= 2.2 number of drops (nd ) = 3 x 2 = 6 ns / nd = 2.2 / 6 = 0.37 H = 200 ft – 170 ft = 30 ft full stream tube 195 D C partial stream tube 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F B El. = 170 ft G H E full stream tube ns= 2.2 nd = 6 ns / nd= 2.2 / 6 = 0.37 Q = K grad h A Q through square A = QA QA = K [ (H / nd) / L ] (L * W) QA = K H W / nd L L 195 D C A L partial stream tube L 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F 3 B 2 1 El. = 170 ft G H E B full stream tube Since flow is steady state, flow is continuous through the tube: QA = QB = QC = Q And since flow in all full tubes is equal: QA = QB = QC =Q1 = Q2 = Q = K H W / nd Total flow under dam is sum of all tubes: ΣQ = Q1 + Q2 + Q3 195 D C A partial stream tube 1 C 2 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F 3 B 2 1 El. = 170 ft G H E B full stream tube Total flow under dam is sum of all tubes: ΣQ = Q1 + Q2 + Q3 = nf * Q ΣQ = nf(K H W / nd) ΣQ = K H W (nf/ nd) 195 D C A partial stream tube 1 C 2 190 185**Flow Nets**El. = 200 ft 30 ft El. = 200 ft 30 ft A Each stream tube carries equal flow F 3 B 2 1 El. = 170 ft G H E B full stream tube Total flow under this dam: ΣQ = K H W (nf / nd ) ΣQ = K (30 ft) W (0.37) If K = 100 ft/day and W = 1000 ft: ΣQ = (100 ft/d) (30 ft) (1000 ft) (0.37) ΣQ = 11.1 x 106 ft3/day, About 11 million cubic feet per day 195 D C A partial stream tube 1 C 2 190 185**Fetter, Ch. 4**• 4.1 Introduction • 4.2Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Refraction of Flow Lines**Layer 1 K1 < K2 Layer 2**Refraction of Flow Lines**Continuity, or steady-state flow—demands that Q1 = Q2 Note that: a = b cosσ1 and c = b cosσ2 K1 tan σ1 K2 tan σ2 =**Fetter, Ch. 4**• 4.1 Introduction • 4.2 Mechanical Energy • 4.3 Hydraulic Head • 4.4 Head in Water of Variable Density • 4.5 Force Potential and Hydraulic Head • 4.6 When is Darcy’s Law Applicable? • 4.7 Equations of Groundwater Flow • 4.8 Solutions of Flow Equations • 4.9 Gradient of Hydraulic Head • 4.10 Relationship of Flow to Gradient • 4.11 Flow Lines and Flow Nets • 4.12 Refraction of Flow Lines • 4.13 Steady Flow in Confined Aquifers • 4.14 Steady Flow in Unconfined Aquifers**Steady Flow in a Confined Aquifer**Q = K (dh/dL) A Q = K (dh/dL) b(1) Q = K [(h1-h2)/L] b QL/(Kb) = h1-h2 h2 = h1 - QL/(Kb) Solve for h at any x: hX= h1 - Qx/(Kb) (Linear equation) X Q Q**Steady Flow in a Confined Aquifer**Q = K (dh/dl) b(1)**Steady Flow in an Unconfined Aquifer**Continuity, or steady-state flow—demands that Q1 = Q2 At any x, Q = K h dh/dL But, h=h(x) Thus, dh/dL = dh/dx dh/dx= dh/dx (x) Q Q**Steady Flow in an Unconfined Aquifer**Dupuit Assumptions • Hydraulic gradient is equal to the slope of the water table. • Stream (flow) lines are horizontal. • Equipotential lines are vertical (follows from #2).**Steady Flow in an Unconfined Aquifer**Dupuit Assumptions**Steady Flow in an Unconfined Aquifer**Continuity, or steady-state flow—demands that Q1 = Q2 At any x, Q = K h dh/dL But, h=h(x) Thus, dh/dL = dh/dx dh/dx= dh/dx (x) Q Q**Q = K h (dh/dx)**∫ Q dx= ∫ h dh Q x = h2/2 + C At x = 0, h = h1 At x = L, h = h2 Q (L – 0) = (h22 – h12 )/2 QL = K (h22 – h12 )/2 h22 = h12- 2QL/K (Dupuit Equation, not linear ) Steady Flow in an Unconfined Aquifer Q Q**Steady Flow in an Unconfined Aquifer**h22 = h12- 2QL/K (Dupuit Equation, not linear ) Solve for h at any x: hx2= h12 - 2Qx/K Q = K (h12 - h22) / 2L Q Q**Boundary conditions?**Continuity? Unconfined Flow with Infiltration or Evaporation Q**Unconfined Flow with Infiltration or Evaporation**Q Q = [K (h12 - h22) / 2L ] + w(L/2 – x) Where is Q = 0?**Unconfined Flow with Infiltration or Evaporation**Q Solve for h at any x: hx2= h12– (h12 – h22) x/L + w/K(L –x) x**Unconfined Flow with Infiltration or Evaporation**Q d =[L/2] –[K/w (h12 – h22)]/2L**Unconfined Flow with Infiltration or Evaporation**Q If ratio K/w gets too big, then d <0 and no ground-water divide