This section describes a number of benchmarks for which an exact analytical solution can be found in the literature.

1.1. Study Case described in "Technisch Rapport Voorland Zettingsvloeiing"

Description

The example given in "Annex A - Case Study" of the Deltares report 1200503-001-GEO-0004 "Concept Technisch Rapport Voorland Zettingsvloeiing" of G.A. van den Ham & Co is used.
In this example, both global and detailed checks are completed on the basis of a (fictitious) dike section, which with regard to geometry and soil structure is typical of the Southwest Delta.
The dike has a height of NAP+5 m, a crest width of 3 m and a slope of 1:3.
The foreshore begins to imaginary toe of the dike at an elevation of NAP and is 60 m wide. The toe of the trench is NAP-15 m and has a slope of 1:6. The dike section is 800 m long. The phreatic level is at NAP-2 m.
The soil profile is as follows:

• from NAP+3.5 m to NAP+1 m: peat
• from NAP+1 m to NAP-5 m: silty clay
• from NAP-5 m to NAP-18 m: moderately to loosely compacted sand (Calais)
• from NAP-18 m to NAP-30 m: densely compacted sand

For the Detailed check, the following input is used:

• Required probability of failure of the dike: 1:4000 year
• Percentage probability of failure by liquefaction: 1 %
• Maximum allowable retrogression length: L_allowable = 60 m (length of the foreland)
• Considered dike length: 800 m
• Mean value of the area ratio (c = A₂/A₁): 1.4

Benchmarks results

The details of the calculation can be found in annex A of the report. The main results are given in the table below. The global check fails but the detailed check passes.

Global check - Step 1a:

Thickness of the sensible to liquefaction layer: H_vw = 25 m

Channel depth: H = 15 m

Marge = 2 H_vw + 1.5 (H - H_vw) = 35 m

Slope of the observation profile: 1:15

Assessment level: Z_beoord = -10 m

X_Ssign = X_{dike toe at river} - marge - (Z_voorland - Z_beoord) * slope = -15 m

X_Szv = 30 m

X_Ssign > X_Szv => Flow slide would not lead to damage on levee => Global check passes.

Detailed check: see table below: Probability of flood damage by liquefaction (= 8.10 × 10^-11) < Allowable probability of failure (2.50 × 10^-6) => Detailed check passes.

Overall check: as Global and Detailed check pass, the Overall check passes. 

D-Flow Slide results

D-Flow Slide needs extra input parameters compared to the Study Case:

• Ψ_5m =  -0.03
• D₅₀ = 180 μm and D₁₅ = 130 μm for Calais sand
• D₅₀ = 160 μm and D₁₅ = 110 μm for Compacted sand 
• According to the values of D₅₀, the sand type is set to "Very fine sand"
• Cohesive layers factor = 0.2
• Migration velocity = 0.0001 mm/year

Results of benchmark 1-1 for the Global Check

Results of benchmark 1-1 for the Detailed Check

1.2. Determination of the steepest possible breaching profile (step 7 of the Global check)

Description

The local critical profile given in Table A.4.2a from CUR113b is modeled in D-Flow Slide for both type of sand (medium fine and coarse sand). 

Different cases are considered:

Benchmarks results

The inputted channel slope is given in the tables below per case, and compared with the allowed slope (from CUR table) for the breaching step 7 of the Global check.

Case A:

The surface line coincides with the critical (local) profile given in Table A.4.2a from CUR113b as can be seen in the table below.

The soil has grain diameters of D₅₀ = 200.1 μm and D₁₅ = 100.1 μm, which corresponds to "Medium fine" sand type.

The local and average checks for all the points along the channel slope fail, so the Global check for Step 7 - Breaching fails.

Case B:

The surface line is steeper than the critical (local) profile given in Table A.4.2a from CUR113b (local slope are 1:X-0.1m instead of 1:X).

The soil has grain diameters of D₅₀ = 200.1 μm and D₁₅ = 100.1 μm, which corresponds to "Medium fine" sand type.

The local and average checks for all the points along the channel slope fail, so the Global check for Step 7 - Breaching fails.

Case C:

The surface line is softer than the critical (local) profile given in Table A.4.2a from CUR113b (local slope are 1:X+0.1m instead of 1:X).

The soil has grain diameters of D₅₀ = 200.1 μm and D₁₅ = 100.1 μm, which corresponds to "Medium fine" sand type.

The local and average checks for all the points along the channel slope pass, so the Global check for Step 7 - Breaching passes.

Case D:

The surface line is softer than the critical (local) profile given in Table A.4.2a from CUR113b, but is steeper than the critical average profile at one point.

The soil has grain diameters of D₅₀ = 200.1 μm and D₁₅ = 100.1 μm, which corresponds to "Medium fine" sand type.

The local and average checks for all the points along the channel slope pass except for one point for which the average check fails, so the Global check for Step 7 - Breaching fails.

Case E:

The surface line coincides with the critical (local) profile given in Table A.4.2a from CUR113b .

The soil has grain diameters of D₅₀ = 500.1 μm and D₁₅ = 250.1 μm, which corresponds to "Coarse sand/Gravel" soil type.

The local check for all the points along the channel slope fails, so the Global check for Step 7 - Breaching fails.

Case F:

The surface line is steeper than the critical (local) profile given in Table A.4.2a from CUR113b (local slope are 1:X-0.1m instead of 1:X).

The soil has grain diameters of D₅₀ = 500.1 μm and D₁₅ = 250.1 μm, which corresponds to "Coarse sand/Gravel" soil type.

The local and average checks for all the points along the channel slope fail, so the Global check for Step 7 - Breaching fails.

Case G:

The surface line is softer than the critical (local) profile given in Table A.4.2a from CUR113b (local slope are 1:X+0.35m instead of 1:X).

The soil has grain diameters of D₅₀ = 500.1 μm and D₁₅ = 250.1 μm, which corresponds to "Coarse sand/Gravel" soil type.

The local and average checks for all the points along the channel slope pass, so the Global check for Step 7 - Breaching passes.

Case H:

The surface line is softer than the critical (local) profile given in Table A.4.2a from CUR113b, but is steeper than the critical average profile.

The soil has grain diameters of D₅₀ = 500.1 μm and D₁₅ = 250.1 μm, which corresponds to "Coarse sand/Gravel" soil type.

The local and average checks for all the points along the channel slope pass except for one point for which the average check fails, so the Global check for Step 7 - Breaching fails.

Case I:

The surface line is softer than the critical (local) profile given in Table A.4.2a from CUR113b (local slope are 1:X+0.1m instead of 1:X), but the channel is deeper than 40m.

The soil has grain diameters of D₅₀ = 200.1 μm and D₁₅ = 100.1 μm, which corresponds to "Medium fine" sand type.

The local and average checks for all the points along the channel slope pass but the channel is deeper than 40m (which is not covered by the CUR table), so the Global check for Step 7 - Breaching fails.

D-Flow Slide results

Results of benchmark 1-2 for the Global Check - Step 7 (Breaching)