RRI Model
Kushiro Wetland, Japan | Part I: Workflow Optimization (48 Workflows × 3 Resolutions) → Part II: Binary Extraction & Box-Counting Fractal Analysis
Model Structure Overview
Rainfall-Runoff-Inundation (RRI) model is a two-dimensional model capable of simulating rainfall-runoff and flood inundation simultaneously (Sayama et al., 2012, Sayama et al., 2015a, Sayama et al., 2015b). The model deals with slopes and river channels separately. At a grid cell in which a river channel is located, the model assumes that both slope and river are positioned within the same grid cell. The channel is discretized as a single line along its centerline of the overlying slope grid cell. The flow on the slope grid cells is calculated with the 2D diffusive wave model, while the channel flow is calculated with the 1D diffusive wave model. For better representations of rainfall-runoff-inundation processes, the RRI model simulates also lateral subsurface flow, vertical infiltration flow and surface flow. The lateral subsurface flow, which is typically more important in mountainous regions, is treated in terms of the discharge-hydraulic gradient relationship, which takes into account both saturated subsurface and surface flows. On the other hand, the vertical infiltration flow is estimated by using the Green-Ampt model. The flow interaction between the river channel and slope is estimated based on different overflowing formulae, depending on water-level and levee-height conditions.
To learn more about the RRI Model see https://www.pwri.go.jp/icharm/research/rri/rri_top.html.
Governing Equations of RRI Model
A method to calculate lateral flows on slope grid-cells is characterized as “a storage cell-based inundation model” (e.g. Hunter et al. 2007). The model equations are derived based on the following mass balance Equation 1 and momentum Equation 2 and Equation 3 for gradually varied unsteady flow.
Shallow Water Equations
The governing equations are:
\[ \frac{\partial h}{\partial t} + \frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} = r - f \tag{1}\]
\[ \frac{\partial q_x}{\partial t} + \frac{\partial (u q_x)}{\partial x} + \frac{\partial (v q_x)}{\partial y} = - g h \frac{\partial H}{\partial x} - \frac{\tau_x}{\rho_w} \tag{2}\]
\[ \frac{\partial q_y}{\partial t} + \frac{\partial (u q_y)}{\partial x} + \frac{\partial (v q_y)}{\partial y} = - g h \frac{\partial H}{\partial y} - \frac{\tau_y}{\rho_w} \tag{3}\]
where \(h\) is the height of water from the local surface, \(q_x\) and \(q_y\) are the unit width discharges in \(x\) and \(y\) directions, \(u\) and \(v\) are the flow velocities in \(x\) and \(y\) directions, \(r\) is the rainfall intensity, \(f\) is the infiltration rate, \(H\) is the height of water from the datum, \(ρ_w\) is the density of water, \(g\) is the gravitational acceleration, and \(τ_x\) and \(τ_y\) are the shear stresses in \(x\) and \(y\) directions. The second terms of the right side of Equation 2 and Equation 3 are specified as follows.
\[ \frac{\tau_x}{\rho_w} = \frac{\tau_b}{\rho_w} \frac{u}{\sqrt {u^2 + v^2}} \tag{4}\]
\[ \frac{\tau_y}{\rho_w} = \frac{\tau_b}{\rho_w} \frac{v}{\sqrt {u^2 + v^2}} \tag{5}\]
where \(\tau_b\) is the bed shear stress defined as
\[ \tau_b = \rho_w ghi \tag{6}\]
where \(i\) is the friction slope. Substitution of Equation 6 and the Manning’s formula into Equation 4 and Equation 5 yields
\[ \frac{\tau_x}{\rho_w} = \frac{gn^2 u \sqrt{u^2 + v^2}}{h^{1/3}} \tag{7}\]
\[ \frac{\tau_y}{\rho_w} = \frac{gn^2 v \sqrt{u^2 + v^2}}{h^{1/3}} \tag{8}\]
where \(n\) is the Manning’s roughness coefficient. Using the relation of \(\sqrt{u^2 + v^2} = \frac{1}{n} i^{1/2} h^{2/3}\) and \(q_x = uh\) and \(q_y = vh\) for Equation 7 and Equation 8, we obtain
\[ \frac{\tau_x}{\rho_w} = ngh^{-2/3} i^{1/2} q_x \tag{9}\]
\[ \frac{\tau_y}{\rho_w} = ngh^{-2/3} i^{1/2} q_y \tag{10}\]
and \(i\) is defined as
\[ i = \sqrt{ \left( \frac{\partial H}{\partial x} \right)^2 + \left( \frac{\partial H}{\partial y} \right)^2 } \tag{11}\]