Internal Erosion Suite

Blanket Theory Case 8

Case 8 has a semi-pervious top stratum on both the riverside and landside of the levee, and a partially penetrating seepage barrier (for example, sheet pile wall or slurry wall) into the pervious substratum is at the centerline of the levee. The pervious substratum is divided into four zones to apply the method of fragments as shown in Figure due to the partially penetrating seepage barrier. Since the top strata are semi-pervious, the effective seepage entry distance from the riverside levee toe must be calculated based on the type of seepage entrance (riverside boundary condition), and the effective seepage exit distance from the landside levee toe must be calculated based on the type of seepage exit (landside boundary condition).

Method of Analysis

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

Levee Geometry

This levee geometry is the same as in Case 7, but there is a partially penetrating seepage barrier into the pervious substratum is at the centerline of the levee. Figure illustrates a blanket extending infinitely landward (Case 8a). Figure illustrates an open seepage exit (Case 8b). Figure illustrates a seepage block (impervious boundary) beyond the landside levee toe that prevents any seepage exit into the pervious foundation landside of the seepage block (Case 8c).

There is an additional input for the length of the embedment of the seepage barrier into the pervious substratum (s). Values greater than or equal to the thickness of the pervious stratum (d) have an orange background. The input is illustrated in Figure .

Figure: BT Case 8a: Basic levee geometry for infinitely long landside blanket.

Figure: BT Case 8b: Basic levee geometry for open seepage exit.

Figure: BT Case 8c: Basic levee geometry for seepage block.

Figure: Step 2 of BT Case 8 worksheet: Levee geometry.

Pervious Substratum Characterization

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

Riverside Blanket (Top Stratum) Characterization

The riverside blanket (top stratum) characterization is the same as in Case 5.

Landside Blanket (Top Stratum) Characterization

The landside blanket (top stratum) characterization is the same as in Case 6.

Blanket Theory Assumptions

The check of BT assumptions is the same as in Case 7.

Seepage Characterization

Step 7 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.

$$Q_{S} = k_{h,f}H\frac{d}{x_{1}+2d(K/K')+x_3}$$

where:

k_h,f = horizontal permeability of pervious substratum
H = net hydraulic head on the levee
d = thickness of the pervious substratum
x₁ = effective seepage entrance from the riverside levee toe
x₃ = effective seepage exit from the landside levee toe
K / K′ = ratio of the complete elliptic integral of the first kind to the complementary integral

The value of K / K′ is obtained from a table of complete elliptical integrals of the first kind. In Table, the modulus (m) is calculated using Equation.

$$m = \cos(\frac{\pi s}{2d})\sqrt{\tanh^{2}({\frac{\pi b}{2d}}) + \tan(\frac{\pi s}{2d})}$$

where:

s = length of embedment of the seepage barrier into the pervious substratum
b = half base width of the levee = L₂ / 2

Table: Complete elliptical integrals of the first kind.

$m^2$	K / K′	$m^2$	K / K′	$m^2$	K / K′	$m^2$	K / K′	$m^2$	K / K′
0.000	0.000	0.17	0.706	0.43	0.938	0.69	1.198	0.95	1.828
0.001	0.325	0.18	0.716	0.44	0.946	0.70	1.211	0.96	1.901
0.002	0.349	0.19	0.726	0.45	0.955	0.71	1.224	0.97	1.992
0.003	0.366	0.20	0.735	0.46	0.964	0.72	1.237	0.98	2.123
0.004	0.379	0.21	0.745	0.47	0.973	0.73	1.251	0.990	2.347
0.005	0.389	0.22	0.754	0.48	0.982	0.74	1.265	0.991	2.381
0.006	0.398	0.23	0.763	0.49	0.991	0.75	1.279	0.992	2.418
0.007	0.406	0.24	0.773	0.50	1.000	0.76	1.294	0.993	2.461
0.008	0.413	0.25	0.782	0.51	1.009	0.77	1.310	0.994	2.510
0.009	0.420	0.26	0.791	0.52	1.018	0.78	1.326	0.995	2.568
0.01	0.426	0.27	0.800	0.53	1.028	0.79	1.343	0.996	2.639
0.02	0.471	0.28	0.808	0.54	1.037	0.80	1.360	0.997	2.731
0.03	0.502	0.29	0.817	0.55	1.047	0.81	1.377	0.998	2.860
0.04	0.526	0.30	0.826	0.56	1.057	0.82	1.397	0.999	3.081
0.05	0.547	0.31	0.834	0.57	1.066	0.83	1.416	0.9991	3.115
0.06	0.565	0.32	0.843	0.58	1.076	0.84	1.439	0.9992	3.152
0.07	0.582	0.33	0.852	0.59	1.087	0.85	1.462	0.9993	3.195
0.08	0.598	0.34	0.860	0.60	1.098	0.86	1.484	0.9994	3.244
0.09	0.612	0.35	0.869	0.61	1.107	0.87	1.511	0.9995	3.302
0.10	0.625	0.36	0.877	0.62	1.118	0.88	1.538	0.9996	3.373
0.11	0.638	0.37	0.886	0.63	1.129	0.89	1.567	0.9997	3.465
0.12	0.650	0.38	0.895	0.64	1.140	0.90	1.600	0.9998	3.594
0.13	0.662	0.39	0.903	0.65	1.151	0.91	1.634	0.9999	3.814
0.14	0.674	0.40	0.911	0.66	1.162	0.92	1.672	1	∞
0.15	0.684	0.41	0.920	0.67	1.174	0.93	1.718
0.16	0.695	0.42	0.929	0.68	1.186	0.94	1.770

To interpolate intermediate values of K / K′ for m² between 0.9999 and 1 in Table, the maximum number that can be stored by Excel of 1.79769313486232E308 is used for K / K′ = ∞. If the seepage barrier is fully penetrating (s ≥ d), m is 1; K / K′ = ∞; and Q_s = 0.

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

Likelihood of Heave/Blowout at Landside Toe

Step 8 calculates the excess hydraulic head at the landside levee toe (h_o) using Equation.

$$h_{o} = H\frac{x_3}{x_{1}+2d(K/K')+x_{3}}$$

where:

H = net hydraulic head on the levee
d = thickness of the pervious substratum
x₁ = effective seepage entrance from the riverside levee toe
x₃ = effective seepage exit from the landside levee toe

The vertical seepage exit gradient at the landside levee toe (i_x) is calculated the same as in Case 6, and the factor of safety against heave/blowout (based on vertical seepage gradients) at the landside levee toe (FS_vg) is calculated the same as in Case 2. The format for the tabular and graphical output is the same as in Case 2.

Likelihood of Heave/Blowout at Given Distance from Landside Toe

If the blanket extends infinitely landward, Step 9 calculates the excess hydraulic head at a distance x from the landside levee toe (h_x) using Equation.

$$h_{x} = h_{o}e^{-c_{bl}x}$$

where:

h_o = excess hydraulic head at the landside levee toe
x = distance from the landside levee toe

If a seepage block exists landward of the landside levee toe that prevents any seepage exit into the pervious foundation landside of the seepage block, Step 9 calculates the excess hydraulic head at a distance x from the landside levee toe (h_x) using Equation 40.

$$h_{x} = \frac{\cosh(c_{bl}(L_{3}-x))}{\cosh(c_{bl}L_3)}$$

where:

h_o = excess hydraulic head at the landside levee toe
c_bl = constant for the landside blanket
L₃ = distance from the landside levee toe to the seepage block for the “seepage block (impervious boundary)” boundary condition
x = distance from the landside levee toe

If an open seepage exit exists landward of the landside levee toe, Step 9 calculates the excess hydraulic head at a distance x from the landside levee toe (h_x) using Equation.

$$h_{x} = h_{o}\frac{sinh(c_{bl}(L_{3}-x))}{cosh(c_{bl}L_3)}$$

where:

h_o = excess hydraulic head at the landside levee toe
c_bl = constant for the landside blanket
L₃ = distance from the landside levee toe to the open seepage exit for the “open seepage exit” boundary condition
x = distance from the landside levee toe

The vertical seepage exit gradient at a distance x from the landside levee toe (i_v,x) is calculated the same as in Case 6, and the factor of safety against heave/blowout (based on vertical seepage gradients) at a distance x from the landside levee toe (FS_vg,x) is calculated the same as in Case 2. The format for the tabular and graphical output is the same as in Case 2.