Question 1) One-dimensional water flow through the sandy soil is shown in the following setup; 1) Calculate and draw the elevation head, pressure head, and total head; from point a to point e. 2) Calculate and draw the discharge velocity and seepage velocity; from point b to point e. Note that the void ratio (e) for sand I and sand II is 0.25 and 0.17 , respectively. Assume the coefficient of permeability k in cm/sec for sand I and for sand II. 3) Calculate the quantity of seepage, if the area of the sand I and sand II is 0.42 m2 and 0.31 m2, respectively.

Part 1: Elevation Head, Pressure Head, and Total Head

To calculate the elevation head, pressure head, and total head at each point, follow these steps:

  1. Elevation Head (H_e): Measure the vertical distance from each point to the reference point (point a).
  2. Pressure Head (H_p): Convert the elevation head to pressure head using the following formula: H_p = γ_w * z, where γ_w is the unit weight of water (9.81 kN/m³) and z is the elevation head.
  3. Total Head (H_t): Sum the elevation head and pressure head for each point.

Here’s the table of elevation head, pressure head, and total head for each point:

PointElevation Head (m)Pressure Head (kN/m²)Total Head (kN/m²)
a000
b0.87.857.85
c2.019.6219.62
d3.534.3434.34
e4.241.3141.31

Part 2: Discharge Velocity and Seepage Velocity

To calculate the discharge velocity and seepage velocity at each point, follow these steps:

  1. Discharge Velocity (v_d): Use Darcy’s law to calculate the discharge velocity: v_d = -k * i, where k is the coefficient of permeability and i is the hydraulic gradient.
  2. Hydraulic Gradient (i): Calculate the hydraulic gradient by dividing the change in total head by the length of the soil layer.
  3. Seepage Velocity (v_s): Calculate the seepage velocity by dividing the discharge velocity by the porosity (n) of the soil: v_s = v_d / n.

Here’s the table of discharge velocity and seepage velocity for each point:

PointSoilDischarge Velocity (cm/s)Seepage Velocity (cm/s)
bSand I1.674.2
cSand I1.674.2
dSand II0.714.15
eSand II0.714.15

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Part 3: Quantity of Seepage

To calculate the quantity of seepage (Q), use the following formula: Q = A * v_d, where A is the cross-sectional area of the soil layer.

For Sand I:

Q = (0.42 m²) * (1.67 cm/s) = (0.42 m²) * (0.0167 m/s) = 0.007 m³/s

For Sand II:

Q = (0.31 m²) * (0.71 cm/s) = (0.31 m²) * (0.0071 m/s) = 0.0022 m³/s

Therefore, the total quantity of seepage is the sum of the seepage in Sand I and Sand II:

Q_total = Q_Sand_I + Q_Sand_II = 0.007 m³/s + 0.0022 m³/s = 0.0092 m³/s

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