.. _use-case-flood-model: A flood model ============= Introduction ------------ The following figure presents a dyke protecting industrial facilities. When the river level exceeds the dyke height, flooding occurs. The model is based on a crude simplification of the 1D hydrodynamical equations of Saint-Venant under the assumptions of uniform and constant flow rate and large rectangular sections. .. figure:: ../_static/flooding_section.png :align: center :alt: flooding section :width: 50% Flooding section Four independent random variables are considered: - :math:Q: flow rate :math:[m^3 s^{-1}] - :math:K_s: Strickler :math:[m^{1/3} s^{-1}] - :math:Z_v: downstream height :math:[m] - :math:Z_m: upstream height :math:[m] When the Strickler coefficient increases, the riverbed generates less friction. The model depends on four parameters: * the height of the dyke: :math:H_d = 3 :math:[m], * the altitude of the river banks: :math:Z_b = 55.5 :math:[m], * the river length: :math:L = 5000 :math:[m], * the river width: :math:B = 300 :math:[m]. The altitude of the dyke is: .. math:: Z_d = Z_b + H_d The slope :math:\alpha of the river is assumed to be close to zero, which implies: .. math:: \alpha = \frac{Z_m - Z_v}{L}, if :math:Z_m \geq Z_v. The water depth is: .. math:: H = \left(\frac{Q}{K_s B \sqrt{\alpha}}\right)^{0.6}, for any :math:K_s, Q>0. The flood altitude is: .. math:: Z_c = H + Z_v. The altitude of the surface of the water is greater than the altitude of the top of the dyke (i.e. there is a flood) if: .. math:: S = Z_c - Z_d is greater than zero. The following figure presents the model with more details. .. figure:: ../_static/flooding_section_detail.png :align: center :alt: flooding section details :width: 50% Flooding section detail If we substitute the parameters into the equation, we get: .. math:: S = \left(\frac{Q}{300 Ks \sqrt{(Zm-Zv)/5000}}\right)^{3/5} +Zv-58.5. We assume that the four inputs have the following distributions: - :math:Q ~ Gumbel(mode=1013, scale=558), :math:Q > 0 - :math:K_s ~ Normal(mu=30.0, sigma=7.5), :math:K_s > 0 - :math:Z_v ~ Uniform(a=49, b=51) - :math:Z_m ~ Uniform(a=54, b=56) Moreover, we assume that the input random variables are independent. We want to estimate the flood probability: .. math:: P_f = P(S>0). References ---------- * Iooss B, LemaĆ®tre P (2015) A review on global sensitivity analysis methods. In: Meloni C., Dellino G. (eds) Uncertainty management in Simulation-Optimization of Complex Systems: Algorithmsand Applications, Springer * Baudin M., Dutfoy A., Iooss B., Popelin AL. (2015) OpenTURNS: An Industrial Software for Uncertainty Quantification in Simulation. In: Ghanem R., Higdon D., Owhadi H. (eds) Handbook of Uncertainty Quantification. Springer Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import flood_model as flood_model >>> # Load the use case flood model >>> fm = flood_model.FloodModel() API documentation ----------------- See :class:~openturns.usecases.flood_model.FloodModel. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_data_analysis/sample_analysis/images/thumb/sphx_glr_plot_compare_unconditional_conditional_histograms_thumb.png :alt: :ref:sphx_glr_auto_data_analysis_sample_analysis_plot_compare_unconditional_conditional_histograms.py .. raw:: html
.. toctree:: :hidden: /auto_data_analysis/sample_analysis/plot_compare_unconditional_conditional_histograms .. raw:: html
.. only:: html .. figure:: /auto_calibration/bayesian_calibration/images/thumb/sphx_glr_plot_bayesian_calibration_flooding_thumb.png :alt: :ref:sphx_glr_auto_calibration_bayesian_calibration_plot_bayesian_calibration_flooding.py .. raw:: html
.. toctree:: :hidden: /auto_calibration/bayesian_calibration/plot_bayesian_calibration_flooding .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_flood_model_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_flood_model.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_flood_model .. raw:: html
.. only:: html .. figure:: /auto_calibration/least_squares_and_gaussian_calibration/images/thumb/sphx_glr_plot_calibration_flooding_thumb.png :alt: :ref:sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_flooding.py .. raw:: html
.. toctree:: :hidden: /auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_flooding