Modeling Overland Flow

1
Modeling Overland Flow Mauricio Santillana and
Clint Dawson
OVERLAND FLOW EQUATION The main scaling
assumption in overland flow modeling is the fact
that frictional forces are dominant, since the
water depth can be extremely shallow (in the
order of centimeter or millimeters). For such
scenario, the acceleration terms in the 2-D SW
momentum equations are neglected leading to a
relationship between the horizontal velocities
and the surface water slope. The 2-D SW
continuity equation combined with the free
surface boundary condition become a doubly
nonlinear degenerate parabolic equation given by
ABSTRACT The objectives of this investigation are
to provide a detailed analysis of the equations
of overland flow using Mannings formula and
Chézys formula and to develop a two dimensional
mathematical model capable of simulating the
shallow water flow, characteristic of overland
flow, using Finite Element Techniques. The latter
will be computationally implemented and verified.
t 30 seconds
Rain duration t
t 20 seconds
MOTIVATION This investigation is part of a
project whose objectives are to develop, analyze,
and implement numerical algorithms which honor
the scales of ground and surface water flows, and
which accurately model the coupling between these
flow regimes. The goal of this work is to
thoroughly understand an important surface water
flow model used to simulate overland flow.
Understanding the assumptions and limitations of
this model from a physical and a mathematical
perspective will lead to the application of
appropriate coupling strategies. Overland flow
is a term that is used mostly to describe the
shallow movement of water across land surfaces
both when rainfall has exceeded the infiltration
rate of the grounds surface (Hurton Overland
Flow), and when the entire soil column becomes
completely saturated and water exfiltrates at the
surface (Dunne Overland Flow). The main
assumption in overland flow is that the fluid
motion is dominated by gravity and balanced by
the boundary shear stress. Bodies of water such
as wetlands, see Figure 1 (a), where water flows
through vegetated surfaces, have been
successfully modeled using the differential
equation governing overland flow, often referred
to as the diffusive wave approximation of the
Shallow Water (SW) equations.
The diffusive wave approximation of the Shallow
Water Equations
Depth h H - z (meters)
t 10 seconds
X coordinate (meters)
Figure 2. Comparison of the numerically
calculated depth profiles at the cessation of
rainfall with Iwagakis experimental results on a
three plane cascade. Solid lines are numerical
results and dashed lines experimental results.
where
and
See Figure 1. (b)
Water Surface
THEORETICAL ANALYSIS A detailed study of
existence, some regularity results, a comparison
result, uniqueness and nonnegativity of weak
solutions of this equation is presented for the
zero Dirichlet initial/boundary value problem in
Alonso, Santillana and Dawson (2007). The
properties of the Galerkin method as a means to
approximate the solution of this equation, such
as stability and a priori error estimates, are
obtained in Santillana and Dawson (2007).
h(x,y)
H(x,y)
Land Surface
8 m.
8 m.
8 m.
z(x,y)
24 m.
y
NUMERICAL EXPERIMENTS In order to verify the
performance of this model to simulate real life
experiments, we decided to set and
Such a choice in parameters corresponds to
Mannings friction formula. The main reason why
these parameters are a good starting point is the
existence of experimental data in order to
validate results. A 1-D Model was implemented
numerically using both, a semi-implicit in time
continuous Galerkin and a discontinuous Galerkin
formulation in order to reproduce the results of
a set of laboratory experiments performed by
Iwagaki (1955). These experiments were designed
to produce unsteady flows in a channel 24m long.
The channel was divided into three sections of
equal length and different slopes (?
0.02,0.015, 0.01). See Figure 3. During
experiments, three different rainfall intensities
(0.108, 0.064 and 0.80 cm/s) were simultaneously
applied to each section. The profiles of the
numerical examples are shown in Figure 2. The
numerical results matched reasonably well with
the experimental results. Both formulations
showed similar performance.
Datum
Figure 3. Iwagakis laboratory set up. Unsteady
water flow generated by rainfall with different
durations (10, 20, and 30 seconds).
x
(b)
(a)

• REFERENCES
• R. Alonso, M. Santillana, and C. Dawson,
  Analysis of the diffusive wave approximation of
  the Shallow Water equations, SIAM Journal on
  Mathematical Analysis, in review, 2007
• V. Aizinger and C. Dawson, A discontinuous
  Galerkin method for two-dimensional flow and
  transport in shallow water, Advances in Water
  Resources, 25, pp. 67-84, 2002.
• Daugherty, R., Franzini, J. and Finnemore, J.,
  Fluid Mechanics with Engineering Applications,
  McGraw-Hill, 1985.
• Feng, Ke, Molz, F.J., A 2-D diffusion based,
  wetland flow model, Journal of Hydrology 196,
  230-250, 1997.
• Iwagaki, Y., Fundamental Studies of runoff
  analysis by characteristics. Bulletin 10,
  Disaster Prevention Res. Inst., Kyoto University,
  Kyoto, Japan, 25pp, 1955.
• M. Santillana and C. Dawson. Analysis of the
  continuous Galerkin formulation to solve
• the diffusive wave approximation of the shallow
  water equations, in preparation, 2007
• Turner, A.K., Chanmeesri, N., Shallow Flow of
  Water Through Non-Submerged Vegetation,
  Agricultural Water Management 8, 375 385, 1984.
  
• Vreugdenhil, C.B., Numerical Methods for
  Shallow-Water Flow, Kluwer Academic Publishers,
  1994.
• Zhang, W. and Cundy, T.W., Modeling of two
  dimensional flow. Water Resources Res. 25,
  2019-2035,1989.

Figure 1. (a) Diagram of ground and surface
water. Source www.agr.gc.ca (b) Land surface
elevation (Bathymetry) is given by z, water
surface elevation by H, and depth by h.
SURFACE WATER MODELS Models for surface water
flows are derived from the incompressible, three
dimensional Navier-Stokes equations, which
consist of momentum equations for the three
velocity components , and a
continuity equation Depending on the physics of
the flow, scaling arguments are used in order to
obtain effective equations for the problem in
hand. In Shallow Water Theory, the main scaling
assumptions are that the vertical length scale as
well as the vertical velocity are small relative
to the horizontal scales. These reduces the third
momentum equation into the hydrostatic pressure
approximation and leaves
us with two effective momentum equations in the
horizontal direction. Upon vertical integration,
we can obtain the 2-D shallow water momentum
equations. A third equation comes from combining
the depth averaged continuity equation with the
free surface boundary condition.
FUTURE WORK A 2-D Local Discontinuous Galerkin
formulation is currently being implemented. The
performance of such model still needs to be
tested. The next steps in the investigation are
two 1) The first one is to couple the 2-D LDG
overland flow model with a 2-D / 3-D SWE model.
Such model has been already implemented by Vadym
Aizinger and Clint Dawson as an improvement to
UTBEST ( University of Texas Bays and Estuaries
Simulator ). 2) The second one is to couple such
models with a two phase flow model solved using a
Discontinuous Galerkin formulation developed by
Shuyu Sun and Mary Wheeler.