Influence of water level duration on dike breach triggering, focusing on system behaviour hazard analyses in lowland rivers

Figures & data

Figure 1. Probabilistic framework schematisation.

Figure 2. Top left map show case study location within the Netherlands. Indicated on larger map are the breach locations including the ones analysed in detail, as well as dike rings, river branches and the protection standards (failure probabilities of the embankment) associated with each dike trajectory.

Figure 3. Example fragility surface for piping mechanism. Line in red shows original fragility curve, for which a duration of 48 h is considered most applicable by the experts. The cyan line shows the resulting threshold due to a sampling value of .7.

Table 1. Factors to adjust fragility curves for certain durations, as suggested by three different dike fragility experts, and overall averages.

Expert	1 h	24 h	168 h	720 h
Overtopping
A	0.005	0.2	0.8	0.9
B	0.1	1.3	2	2
C	1	10	50	50
Average	0.37	3.83	17.6	17.63
Piping
A	0.0005	0.05	0.8	0.9
B	0.2	1	2	2.5
C	0.01	1	10	20
Average	0.07	0.68	4.27	7.8
Macrostability
A	0.02	0.1	0.6	0.9
B	0.7	1.3	1.8	2
C	0.01	1	10	20
Average	0.24	0.8	4.13	7.63

Figure 4. Return period water levels for Scenarios 0, 1 and 2, at locations P1_r, W6_l, L19_1 and I13_l. These locations can be seen on a map in Figure 2. Note that the x-axis scales are not the same per location. Scenarios 0, 1 and 2 represent no system behaviour, system behaviour dependent on water level and system behaviour dependent on water level and duration, respectively.

Figure 5. Relative change in failure probability of the overall trajectories when system behaviour is implemented, scenarios 1 (top) and 2 (bottom) Scenarios 1 and 2 represent system behaviour dependent on water level and system behaviour dependent on water level and duration, respectively.

Table 2. Averaged breaching data for scenarios at selected location. Data relates to the 12,000 simulations performed for each scenario.

Attribute	Scenario	Value
Location	–	P1_r (48_1)	W6_l (41_2)	L19_1 (16_3)	I13_r (53_1)
Number of breaches	1	251	961	388	1014
	2	318	1067	497	1167
Average volume when breached (m^3)	1	93,500 m^3	335,000 m^3	375,000 m^3	7230 m^3
	2	72,100 m^3	309,000 m^3	324,000 m^3	4410 m^3
Average breach width (m)	1	200 m	360 m	316 m	59 m
	2	185 m	353 m	310 m	63 m

Figure 6. Expected breaches and breach volumes per return period event in the system.

Figure 7. Return period discharges for Scenarios 0, 1 and 2, at locations P1_r, W6_l, L19_1 and I13_l. These locations can be seen on a map in Figure 2. Note that the x-axis scales are not the same per location Scenarios 0, 1 and 2 represent no system behaviour, system behaviour dependent on water level and system behaviour dependent on water level and duration, respectively.

Figure A1. Breach locations used in the model and associated dike sections.

Table A1. Traject location failure probabilities are given in return period years. The protection standard is shown against the failure probabilities calculated with and without system behaviour for each dike breaching scenario. The values are rounded for clarity.

Trajectory Location	Associated Breach locations	Failure probability in return period years
		Protection Standard	Sc0 (Water level trigger)	Sc0 (Water level and duration trigger)	Sc1	Sc2
15_1	L16_r, L18_r, L20_r, L14_r, L21_r, L13_r	30,000	26,500	28,200	38,100	44,800
15_2	L23_r	30,000	23,200	50,100	23,200	56,500
16_1	W15_r, W17_r, W13_r, W14_r	30,000	30,900	50,500	368,000	437,000
16_2	L24_l	10,000	10,000	12,000	10,000	12,000
16_3	L19_l,	10,000	9880	8960	12,200	11,400
16_4	L15_l, L17_l, L12_l	10,000	11,200	9580	13,200	11,800
24_3	W12_l, W16_l	10,000	10,700	17,500	29,200	37,200
38_1	W11_l, W9_l	10,000	10,200	18,700	22,300	32,800
41_1	W5_l	10,000	7960	8290	8380	9030
41_2	W7_l, W6_l	3000	2460	2700	2550	2830
42_1	W1_l, W3_l	3000	3100	2970	3220	3080
43_1	L11_l, L10_l	10,000	9730	8430	12,700	10,600
43_2	L8_l, L7_l, L5_l	3000	2750	2490	2880	2640
43_3a	L4_l, L2_l, P2_l, W2_r	10,000	9810	9310	11,700	10,900
43_4a	W4_r	10,000	8280	8330	8680	9020
43_5	W10_r, W8_r	10,000	10,100	9140	12,800	11,400
44_1	L9_r	10,000	11,000	9390	12,900	11,100
45_1	L6_r	30,000	24,100	28,100	30,600	33,900
47_1	I2_l, I1_l, L1_r, L3_r	30,000	26,500	28,200	38,100	44,800
48_1	P3_r, P1_r, R2_r, R1_r	30,000	23,200	50,100	23,200	56,500
48_2	I3_r	30,000	30,900	50,500	368,000	437,000
49_2	I4_r	10,000	10,000	12,000	10,000	12,000
50_1	I6_r	10,000	9880	8960	12,200	11,400
51_1	I8_r, I7_r	10,000	11,200	9580	13,200	11,800
52_1	I15_l	10,000	10,700	17,500	29,200	37,200
52_2	I12_l, I9_l, I10_l, I11_l	10,000	10,200	18,700	22,300	32,800
52_3	I5_l	10,000	7960	8290	8380	9030
53_1	I13_r	3000	2460	2700	2550	2830
53_2	I14_r, I16_r	3000	3100	2970	3220	3080

Table A2. Verheij van der Knaap parameters used in the breach growth formulation.

Parameter name	Description	Range	Default value	Unit
f1	Empirical factor for breach width	0.5–5	1.3	–
f2	Empirical factor for breach width	0.01–1	0.04	–
Uc	Critical flow velocity	0.1–10	0.2	m/s
B0	Initial breach width	–	20	m
g	Acceleration due to Gravity	–	9.81	m/s^2

Table A3. Parameters of the calculated curves for each location. The sigma values for Overtopping, Piping and Macrostability are given in the first 3 columns. The final column shows the difference between the mu values calculated for scenario 2.

Location	Sigma values for mechanisms (m)			Difference between Mu values; Sc2–Sc1
	Overtopping	Piping	Macro-stability
L9_r	0.16	0.8	1.03	–0.08
L16_r	0.12	3.5	1.21	–1.07
L18_r	0.08	1.12	1.29	–0.15
L23_r	0.12	1.46	1.39	–0.39
L20_r	0.16	1.25	1.29	–0.16
L14_r	0.16	0.99	1.25	–0.11
L21_r	0.07	1.61	1.33	–0.34
L13_r	0.13	0.87	0.97	–0.06
I14_r	0.14	1.48	0.85	–0.36
I16_r	0.1	1.48	0.75	–0.31
I13_r	0.2	0.95	0.72	–0.18
I15_l	0.12	0.68	0.71	–0.13
I12_l	0.1	0.78	0.8	–0.15
I9_l	0.12	0.72	0.85	–0.1
I5_l	0.17	0.65	0.7	–0.14
I8_r	0.17	0.58	0.67	–0.07
I7_r	0.16	0.51	0.68	–0.07
I6_r	0.22	0.62	0.38	–0.04
I4_r	0.13	1.75	0.35	–0.41
I3_r	0.2	0.88	0.84	–0.13
P3_r	0.09	1.08	0.73	–0.08
P1_r	0.18	1.33	0.86	–0.12
R2_r	0.5	1.02	0.73	–0.08
R1_r	0.25	1.28	0.75	–0.15
I2_l	0.13	1.38	1.05	–0.13
I1_l	0.09	1.61	1.24	–0.14
L1_r	0.23	1.29	1.12	–0.08
L3_r	0.17	1.14	1.17	–0.16
L11_l	0.12	1.1	0.49	–0.06
L10_l	0.23	0.55	0.43	–0.02
L8_l	0.17	0.55	1.16	–0.05
L7_l	0.19	0.82	1.33	–0.07
L5_l	0.13	0.59	1.19	–0.02
L4_l	0.12	0.74	0.7	–0.04
L2_l	0.15	0.72	0.79	–0.02
P2_l	0.1	1.01	0.61	–0.08
W10_r	0.15	0.73	0.97	0.01
W8_r	0.24	0.84	1.27	–0.02
W4_r	0.18	0.93	1.94	–0.02
W2_r	0.12	1.09	2.79	–0.04
W1_l	0.17	1.36	0.86	–0.12
W3_l	0.14	1.83	0.87	–0.26
W7_l	0.15	1.41	1.53	–0.19
W6_l	0.17	1	1.49	–0.08
W5_l	0.11	1.28	0.56	–0.12
W11_l	0.25	0.72	1.55	–0.3
W9_l	0.14	0.65	1.41	–0.25
L15_l	0.13	0.8	0.68	–0.06
L24_l	0.1	0.77	1.44	–0.54
W15_r	0.2	0.2	0.2	–0.11
L17_l	0.15	2.02	0.54	–0.31
L6_r	0.18	0.77	1.11	–0.05
W12_l	0.13	0.11	0.11	–0.09
W16_l	0.15	0.14	0.14	–0.21
W17_r	0.14	0.16	0.16	–0.02
W13_r	0.07	0.16	0.16	–0.05
L19_l	0.14	0.88	0.49	–0.04
L22_l	0.15	1.13	0.47	–0.14
L12_l	0.07	1.01	0.57	–0.07
W14_r	0.19	0.2	0.2	–0.03
I10_l	0.11	0.84	1.07	–0.12
I11_l	0.11	0.88	1.26	–0.13