Theory behind my model
From GeoMod
My model is based primarily on the LISFLOOD-FP Model. The model is a two-dimensional hydrodynamic model specifically designed to simulate floodplain inundation in a computationally efficient manner over complex topography. It is capable of simulating grids up to 106 cells for dynamic flood events and can take advantage of new sources of terrain information from remote sensing techniques such as airborne laser altimetry and satellite interferometric radar. The model predicts water depths in each grid cell at each time step, and hence can simulate the dynamic propagation of flood waves over fluvial, coastal and estuarine floodplains. It is a non-commercial, research code developed as part of an effort to improve our fundamental understanding of flood hydraulics, flood inundation prediction and flood risk assessment.
History in Development of LISFLOOD-FP
These models treat the floodplain as a series of discrete storage cells, with the flow between cells calculated explicitly using analytical flow formulae such as the Manning equation. The basic principle behind such codes is that, with the exception of catastrophic floods (e.g. dam-break scenarios), two-dimensional flow over inundated plains is most often a slow, shallow phenomenon where local free surface slopes are very small. As such, it can be hypothesised that floodplain flows are primarily influenced by bed roughness rather than topographically-induced velocity gradients, allowing the inertial terms to be dropped from the dynamic governing equations of de St Venant (Cunge et al., 1980).
This simplified mathematical description was first incorporated in the numerical model of Zanobetti et al. (1968, 1970) for the simulation and prediction of seasonal flooding episodes across the Mekong Delta, Cambodia . Covering an area of 50,000 km 2, the two-dimensional model consisted of 350 polygonal cells chosen to represent discrete flood storage compartments on the floodplain. Flow between cells was calculated according to Manning/Strickler or weir-type formulae (Cunge, 1975).
In 1975-76 a one-dimensional inertial (de St Venant) model of looped channel flow was integrated with an inertia-less storage cell algorithm, and solved implicitly using the finite difference Preissmann (1961) scheme. Thus the storage cell approach first applied to the Mekong Delta was now linked to a more accurate representation of in-channel flow processes and provided the earliest example of a coupled 1D-2D models. A full description of all early algorithmic and computational details can be found in Cunge et al. (1980).
During the 1980s and 1990s software and computing developments led researchers to attempt to integrate coupled 1D-2D models within Geographical Information Systems using a variety of cell geometries including hexagons (Estrela and Quintas, 1994), regular grids (Bechteler et al., 1994; Bladé et al. 1994) or whole floodplain units (Romanowicz et al., 1996). However, such applications lacked the topography and validation data necessary to capitalize on the software and modeling developments. Only recently has the wide availability of fine spatial resolution topographic data sets from such sources as airborne laser altimetry (LiDAR) and model validation data from satellite and airborne radar systems allowed storage cell modeling to become a realistic possibility.
LISFLOOD-FP was developed in 1999 in a joint effort between the University of Bristol and the EU Joint Research Centre to capitalize on these new data sources (Bates and De Roo, 2000). The model was designed to be the simplest physically plausible representation capable of simulating dynamic flooding, thereby allowing large areas to be modelled at fine spatial resolution (10-100m cell sizes). The model was designed to work on a regular Cartesian grid to allow ready integration with available GIS data sets and was written in a dynamic GIS language, PCRASTER, to further facilitate this link. It has since been re-coded in c++ for computational efficiency. Following, many similar models have appeared in the literature including Venere and Clausse (2002), Dhondia and Stelling (2002), Coe et al., (2002), Bradbrook et al. (2004) and Yu and Lane (in press a and b).
Model Structure:
Introduction
LISFLOOD-FP is a coupled 1D/2D hydraulic model based on a raster grid. Effectively, flooding is treated using an intelligent volume-filling process based on hydraulic principles and embodying the key physical notions of mass conservation and hydraulic connectivity.
Assumptions
• Channel flow can be represented by either the kinematic or diffusion wave approximations.
• We assume the channel to be wide and shallow, so the wetted perimeter is approximated by the channel width.
• For plain flooding and out-of-bank flow we assume that flow can be treated using a series of storage cells discretized as a raster grid.
• Flow between storage cells can be calculated using analytical uniform flow formulae (the Manning equation or a weir equation). This yields an approximate solution of a diffusion wave in two dimensions.
• There is no exchange of momentum between main channel and floodplain flows, only mass.
• We assume flow to be gradually varied.
• The model uses standard SI units for length (metres), time (seconds), flux (volume per time in m3s-1) etc.
Channel flow
Channel flow is handled using a one-dimensional approach that is capable of capturing the downstream propagation of a floodwave and the response of flow to free surface slope, which can be described in terms of continuity and momentum equations as:
where Q is the volumetric flow rate in the channel, A the cross sectional area of the flow, q the flow into the channel from other sources (i.e. from the floodplain or possibly tributary channels), S 0 the down-slope of the bed, n the Manning’s coefficient of friction, P the wetted perimeter of the flow, and h the flow depth. For problems with no channels present this function can simply be switched off. The term in brackets in Eq. 2 is the diffusion wave term, which forces the channel flow to respond to both the bed slope and the free surface slope. This can be switched on or off in the model to enable both kinematic and diffusion wave approximations to be tested. Eq's 1 and 2 are discretized using finite differences and a fully implicit scheme for the time dependence.
Floodplain flow
When bankfull depth is exceeded, water is transferred from the channel to the overlying floodplain grid. Floodplain flows are similarly described in terms of continuity and momentum equations, discretized over a grid of square cells, which allows the model to represent 2-D dynamic flow fields on the floodplain. We assume that the flow between two cells is simply a function of the free surface height difference between those cells:
where h i,j is the water free surface height at the node (i,j), Dx and Dy are the cell dimensions, n is the effective grid scale Manning’s friction coefficient for the floodplain, and Qx and Qy describe the volumetric flow rates between floodplain cells. Qy is defined analogously to Qx. The flow depth, h flow, represents the depth through which water can flow between two cells, and is defined as the difference between the highest water free surface in the two cells and the highest bed elevation (this definition has been found to give sensible results for both wetting cells and for flows linking floodplain and channel cells).
Adaptive time stepping
The model time step is set by the user. However, too large a time step can result in ‘chequerboard’ oscillations in the solution which rapidly spread and amplify, rendering the simulation useless. Ironically, these oscillations occur most readily in areas with low free surface gradients, where we might expect obtaining a solution to be easiest. For this reason latest versions of the code (Version 2.0 and above) include an adaptive time step option based on an analysis of the above equations and an analogy to a diffusion system to determine the optimum time step to maintain stability.
