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Fetter, Ch. 4
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  1. 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

  2. Introduction • Groundwater possesses energy • in a variety of forms: • Mechanical • Thermal • Chemical

  3. 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

  4. Head in Water of Variable Density If z1 = z2 , hf = (ρp/ρf ) hpoint

  5. Head in Water of Variable Density Which way is water moving vertically— up or down? (see p.120)

  6. 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

  7. 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

  8. Equations of Groundwater Flow

  9. Equations of Groundwater Flow

  10. 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

  11. 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?

  12. Relationship of flow to gradient Flow nets in isotropic (A) and anisotropic (B) mediums

  13. Relationship of flow to gradient Flow nets in isotropic (A) and anisotropic (B) mediums

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. Refraction of Flow Lines Layer 1 K1 < K2 Layer 2

  27. 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 =

  28. 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

  29. 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

  30. Steady Flow in a Confined Aquifer Q = K (dh/dl) b(1)

  31. 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

  32. 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).

  33. Steady Flow in an Unconfined Aquifer Dupuit Assumptions

  34. 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

  35. 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

  36. 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

  37. Boundary conditions? Continuity? Unconfined Flow with Infiltration or Evaporation Q

  38. Unconfined Flow with Infiltration or Evaporation Q Q = [K (h12 - h22) / 2L ] + w(L/2 – x) Where is Q = 0?

  39. Unconfined Flow with Infiltration or Evaporation Q Solve for h at any x: hx2= h12– (h12 – h22) x/L + w/K(L –x) x

  40. Unconfined Flow with Infiltration or Evaporation Q d =[L/2] –[K/w (h12 – h22)]/2L

  41. Unconfined Flow with Infiltration or Evaporation Q If ratio K/w gets too big, then d <0 and no ground-water divide