Internal Erosion Suite

Blanket Theory Case 5

Case 5 has a semi-pervious top stratum on only the riverside of the levee. The pervious substratum is divided into three zones to apply the method of fragments as shown in Figure. The difference between Cases 3 and 5 is that there is an impervious top stratum on the riverside for Case 3 while the top stratum is semi-pervious for Case 5. Therefore, the effective seepage entry distance from the riverside levee toe must be calculated based on the type of seepage entrance (riverside boundary condition).

Method Analysis

The method of analysis is the same as in Case 1.

Levee Geometry

Step 2 characterizes the levee geometry. The input includes the levee crest elevation, landside levee toe elevation, and base width of levee (L₂). Use the drop-down list to select the riverside boundary condition using a drop-down list. The options for the type of seepage entrance include:

• No riverside borrow pits or seepage blocks

• Riverside borrow pit (open seepage entrance) that penetrates the blanket and extends to the pervious substratum

• A seepage block (impervious boundary) between the riverside levee toe and the river that prevents any seepage entrance into the pervious foundation riverside of the seepage block

For “no riverside borrow pits or seepage blocks,” the input for L₁ is the distance from the riverside levee toe to the river. For “riverside borrow pit (open seepage entrance),” the input for L₁ is the distance from the riverside levee toe to the borrow pit. For “seepage block (impervious boundary),” the input for L₁ is the distance from the riverside levee toe to the seepage block. The input is illustrated in Figure.

Figure: Step 2 of BT Case 5 worksheet: Levee geometry.

Pervious Substratum Characterization

The pervious stratum characterization is the same as in Case 1.

Riverside Blanket (Top Stratum) Characterization

The selection in step 1 affects the input for step 4, and cells that do not apply have a gray background. These cells are not used in subsequent calculations even if data is present. The input includes the transformed thickness (z_br) and transformed vertical permeability (k_v,b_r) of the riverside blanket.

Use the drop-down list to select the method of estimating the transformed vertical permeability of the riverside blanket. Input the transformed vertical permeability directly or calculate it using an input permeability ratio (k_h,f/k_v,b_r).

For deterministic analysis, input only the most likely values. The mean values used for subsequent calculations are the most likely (or mode) values. Figure illustrates the deterministic input. Figure illustrates the deterministic input for permeability ratio as input.

Figure: Step 4 of BT Case 5 worksheet: Deterministic input.

Figure: Step 4 of BT Case 5 worksheet: Deterministic input for permeability ratio as input.

For probabilistic analysis without using @RISK, input the minimum and maximum values in addition to the most likely value, and triangular distributions represent the random variables. The mean values used in subsequent calculations are the average of the minimum, most likely, and maximum values. Figure illustrates the probabilistic input without using @RISK. Figure illustrates the probabilistic input without using @RISK for permeability ratio as input.

Figure: Step 4 of BT Case 5 worksheet: Probabilistic without using @RISK.

Figure: Step 4 of BT Case 5 worksheet: Probabilistic input without using @RISK for permeability ratio as input.

For probabilistic analysis using @RISK, input the minimum, most likely, and maximum values of the random variables, and use an @RISK formula for a triangular distribution in the third column as a default. Alternatively, input a valid @RISK distribution in lieu of this default formula, and the user-specified input displays in the fourth column. The mean values used in subsequent calculations are the mean for the @RISK distribution entered in the third column. Figure illustrates the probabilistic input using @RISK. Figure illustrates the probabilistic input without using @RISK for permeability ratio as input.

Figure: Step 4 of BT Case 5 worksheet: Probabilistic input using @RISK.

Figure: Step 4 of BT Case 5 worksheet: Probabilistic input using @RISK for permeability ratio as input.

If no riverside seepage block (impervious boundary) exists, the effective seepage entry distance from the riverside levee toe (x₁) is calculated using Equations 18 and 19.

$$x_{1} = \frac{tanh(c_{br}L_{1})}{c_{br}}$$

where:

L₁ = distance from the riverside levee toe to the river for the “no riverside borrow pits or seepage blocks” boundary condition or the distance from
the riverside levee toe to the borrow pit for “riverside borrow pit (open seepage entrance)” boundary condition
c_br = constant for the riverside blanket

$$c_{br} = \sqrt{ \frac{ k_{v,br} }{ k_{h,f} \cdot z_{br} \cdot d } }$$

where:

k_h,f = horizontal permeability of pervious substratum
d = thickness of the pervious substratum
k_v,br = transformed vertical permeability of the riverside blanket (top substratum)
z_br = transformed thickness of the riverside blanket (top substratum)

If a seepage block exists between the riverside levee toe and the river that prevents any seepage entrance into the pervious foundation riverside of the seepage block, the effective seepage entry distance from the riverside levee toe (x₁) is calculated using Equation.

$$x_{1} = \frac{1}{c_{br}tanh(c_{br}L_1)}$$

where:

L₁ = distance from the riverside levee toe to the seepage block for the “seepage block (impervious boundary)” boundary condition

The effective seepage length plus width of the levee is calculated using Equation.

$$S = x_{1} + L_2$$

where:

L₂ = base width of the levee
x₁ = effective seepage entry distance from the riverside levee toe for selected boundary condition

The riverside boundary condition selection in step 2 affects the equation for _x₁ _used in step 4, and cells that do not apply have a gray background.

Blanket Theory Assumptions

Step 5 checks the BT (method of fragments) assumptions against the input parameters to ensure essentially vertical equipotential lines, vertical flow through the blanket and horizontal flow through the pervious foundation, and semi-pervious blanket behavior. For permeability ratios greater than about 1,000 to 4,000, the semi-pervious blanket is effectively impervious, and Case 3 is more appropriate than Case 5. For deterministic analysis, the assumptions are checked for the most likely values of the random variables. For probabilistic analysis, the assumptions are checked for the mean values of the random variables. Values outside of the model assumptions have an orange background. Figure illustrates the check of BT assumptions.

Figure: Step 5 of BT Case 5 worksheet: BT assumptions.

Seepage Characterization

Step 6 calculates the net hydraulic head on the levee (H) the same as in Case 1. The flow or seepage per unit length of the levee (Q_s) is calculated using Equation.

$$Q_{S} = k_{h,f}H\frac{d}{x_{1} + L_{2} + 0.43d}$$

where:

k_h,f = horizontal permeability of pervious substratum
H = net hydraulic head on the levee
L₂ = base width of the levee
d = thickness of the pervious substratum
x₁ = effective seepage entrance from the riverside levee toe

The format for the tabular output is the same as in Case 1.

Likelihood of Heave/Blowout at Landside Toe

Since the excess hydraulic head at the landside levee toe is zero, vertical seepage exit gradients are not computed for this case.

Likelihood of Heave/Blowout at Given Distance from Landside Toe

Since the excess hydraulic head at the landside levee toe and at any distance x from the landside levee toe are zero, vertical seepage exit gradients at any distance x from the landside levee toe are not computed for this case.