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Analytical Zero Inertia Modeling

Faculty of Forestry, Geo- and Hydro Sciences, Institute of Hydrology and Meteorology, Chair of Hydrology. Andy Philipp, Gerd H. Schmitz, Rudolf Liedl. Analytical Zero Inertia Modeling. For the Robust Portrayal of Rainfall Runoff Processes. Water 2010, Québec City, 7/6/2010.

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Analytical Zero Inertia Modeling

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  1. Faculty of Forestry, Geo- and Hydro Sciences, Institute of Hydrology and Meteorology, Chair of Hydrology Andy Philipp, Gerd H. Schmitz, Rudolf Liedl Analytical Zero Inertia Modeling For the Robust Portrayalof Rainfall Runoff Processes Water 2010, Québec City, 7/6/2010

  2. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Contents • 01 Motivation and Fields of Application • 02 ZI Model Approach and Solution Procedure • 03 Application 1: Wave Translation in Non-Prismatic Channels Under Significant Infiltration • 04Application 2: Surface Flow on Hillslopes • 05Discussion and Outlook A. Philipp: Zero Inertia Modeling

  3. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Motivation and Fields of Application 01 Introduction • ZI model (or ~ diffusive wave) approach in between full hydrodynamic process description and kinematic wave approximation A. Philipp: Zero Inertia Modeling

  4. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Motivation 01 Introduction • Typically: numerical solution procedures • Computationally demanding • Coupling of models (for example to account for infiltration) is expensive • Model optimization is expensive • Numerical inconveniencies • initially dry channels, free boundary problems • fast surges • weak process dynamics (standing waves, sheet flow, small to zero slopes) •  Desired: robust and accurate solution of a comprehensive process description without the aforementioned inconveniencies Analytical ZI model A. Philipp: Zero Inertia Modeling

  5. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Usually, ZI equations incorporate no separation of lateral mass inflow and momentum contribution • We introduce such a separation to account e.g. for infiltration •  Extended ZI equations (including lateral inflow/outflow q term [e.g. infiltration] and qu/gA term  lateral momentum contribution) • Equations should be solved analytically with respect to natural/irregular cross sections A. Philipp: Zero Inertia Modeling

  6. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Arbitrarily varying cross sections can be described by using analytically profile functions  relations for h and R • Geometry parameters p1…p4 (least squares fitted) • p1 and p3 account for some kind of common geometry property(e.g. become 1 for rectangular cross section), p2 and p4 are dependant on x A. Philipp: Zero Inertia Modeling

  7. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • Momentum appears on left and right hand side of momentum eq. • Approach for the analytical solution • Assuming a location for a transient momentum representative cross section •  Inflow boundary can be considered as a kind of momentum representative cross section • Multiplying the momentum equation by R2β Momentum (which is set equal to zero in kinematic wave analysis) is represented only by the right hand side of the momentum eq. A. Philipp: Zero Inertia Modeling

  8. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • The solution of the system of partial differential continuity equation and momentum equation requires the specification of initial and boundary conditions • At the fixed upstream boundary, the boundary condition is • At the downstream moving boundary (at x = xtip) the following conditions have to be satisfied (surge flow case) • Initial condition (for the surge flow case) A. Philipp: Zero Inertia Modeling

  9. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • Considering the upstream boundary condition, a direct integration of the continuity equation yields Q(x,t) as… where the integrand can be obtained by differentiating the momentum equation • This system solves analytically the initially discussed modified ZI equations and satisfies the boundary conditions • And accounts for feedback of flow and infiltration over an initially dry bed across an extending domain (q term in the continuity equation) • Solution of the non-linear problem firstly requires the evaluation of the position of the advancing wave front and the wetted cross sectional area at the inflow boundary • Introducing N observation points and setting x = 0 in the momentum equation leads to an highly convergent iterative procedure A. Philipp: Zero Inertia Modeling

  10. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Wave Translation in Non-PrismaticChannels Under Significant Infiltration 03 Application 1 • Prismatic (parabolic) test case A. Philipp: Zero Inertia Modeling

  11. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Wave Translation in Non-PrismaticChannels Under Significant Infiltration 03 Application 1 • Non prismatic test case A. Philipp: Zero Inertia Modeling

  12. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Wave Translation in Non-Prismatic ChannelsUnder Significant Infiltration 03 Application 1 • Oman test case – Wadi Ahin; with significant infiltration A. Philipp: Zero Inertia Modeling

  13. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Wave Translationin Non-Prismatic ChannelsUnder Significant Infiltration 03 Application 1 • Oman test case – Artificial Groundwater Rech. A. Philipp: Zero Inertia Modeling

  14. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Oman test case – Standing wave solution; weak process dynamics 03 Application 1 A. Philipp: Zero Inertia Modeling

  15. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Surface Flow on Hillslopes 04 Application 2 • Sheet flow as a kind of representative model of what is happening when surface runoff occurs • Important for modeling runoff concentration/flood prediction • Problems • Surface roughness • Low flow depthsweak process dynamics • Methodology • Setting up the analyticalZI model for specifichydraulic situation • Explicitly expressing Q(x,t),H(x,t) • Synthetic test catchment andmodel comparison (KW, HD, ZI) A. Philipp: Zero Inertia Modeling

  16. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Surface Flow on Hillslopes 04 Application 2 • Test catchment, parameters, boundary conditions (SCHMID [1986]) A. Philipp: Zero Inertia Modeling

  17. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Surface Flow on Hillslopes 04 Application 2 • Mass conservation; comparison of the three models (HD, KW, ZI) for a quasi stationary test case; qualitative comparison of computational demand A. Philipp: Zero Inertia Modeling

  18. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Surface Flow on Hillslopes 04 Application 2 • Q(t) @ x = 80 m, Boundary Condition Scenarios 1 and 2 Stepwise changing boundary conditions! A. Philipp: Zero Inertia Modeling

  19. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Discussion and Outlook 05 Discussion and Outlook • Reliable and robust portrayal of a wide range of flow processes • No numerical inconveniencies and straightforward solution • Easy model coupling and handling • Computational efficiency • Fields of application • Surface runoff; spatially distributed hydrological modeling • Surge flow problems • Flow with high process interaction (e.g. significant infiltration or groundwater interaction) • Outlook • Integration in spatially distributed hydrological model  robustness, computational efficiency, reduction of uncertainties • Account for falling limb of the hydrograph A. Philipp: Zero Inertia Modeling

  20. Thank You For Your Attention! • Further Reading:Philipp, A., G. H. Schmitz, R. Liedl: An Analytical Model of Surge Flow in Non-Prismatic Permeable Channels and Its Application in Arid Regions. Journal of Hydraulic Engineering. May 2010, Volume 136, Number 5, Pages 290 to 298. • Gerd H. Schmitz, Rudolf Liedl, Andy PhilippInstitute of Hydrology and MeteorologyDresden University of Technology A. Philipp: Zero Inertia Modeling

  21. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Surface Flow on Hillslopes 04 Application 2 • H(t) @ x = 80 m, Boundary Condition Scenarios 1 and 2 A. Philipp: Zero Inertia Modeling

  22. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Basics 02 ZI Model Approach • Assumptions: parabolic water level surface, free moving boundary A. Philipp: Zero Inertia Modeling

  23. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Extended ZI equations (including lateral q term  within mass balanceas q term [e.g. infiltration] and qu/gA term  lateral momentum contribution; could be omitted) A. Philipp: Zero Inertia Modeling

  24. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Multiplying the momentum equation by R2βyields • Inflow boundary can be considered as a kind of momentum representative cross section • Momentum (which is set equal to zero in kinematic wave analysis) is represented by the right hand side of the above eq. A. Philipp: Zero Inertia Modeling

  25. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Momentum is represented by the transient amount of the momentumat x = 0  right hand side no longer depends explicitly on x and can be rearranged: • Right hand side may be regarded as measure of transient amount of momentum at x = 0, covering contributions from bottom slope, friction and infiltration through the bottom A. Philipp: Zero Inertia Modeling

  26. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Model Equations 02 ZI Model Approach • Arbitrarily varying cross sections can be described by using analytically profile functions  relations for h and R • Geometry parameters p1…p4 (least squares fitted) • p1 and p3 account for some kind of common property of channel geometry (e.g. become 1 for rectangular cross section), p2 and p4 are dependant on x A. Philipp: Zero Inertia Modeling

  27. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • The solution of the system of partial differential continuity equation and momentum equation requires the specification of initial and boundary conditions • At the fixed upstream boundary, the BC isRestriction: falling hydrographs cannot be used due to ZI assumptions! • At the downstream moving boundary (at x = xtip) the following conditions have to be satisfied (surge flow case) A. Philipp: Zero Inertia Modeling

  28. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • Initial condition for the surge flow case • Using the (p1…4-)parameterized expression for the hydraulic radius, the momentum equation reads A. Philipp: Zero Inertia Modeling

  29. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • Considering the upstream boundary condition, a direct integration of the continuity equation yields Q(x,t) as (considering the aforementioned restrictions)… where the integrand can be obtained by differentiating the momentum equation • This system solves the initially discussed ZI equations and satisfies the boundary conditions • AND accounts for feedback of flow and infiltration over an initially dry bed across an extending domain (q term in the continuity equation) A. Philipp: Zero Inertia Modeling

  30. MOTIVATION | METHODOLOGY | APPLICATION | DISCUSSION AND OUTLOOK Solution Procedure 02 ZI Model Approach • Solution of the non-linear problem firstly requires the evaluation of the position of the advancing wave front and the wetted cross sectional area at the inflow boundary • Introducing N observation points and setting x = 0 in the momentum equation and further considerable mathematical calculus leads to the iterative procedure: A. Philipp: Zero Inertia Modeling

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