On a fully coupled numerical methods for 2D surface-subsurface
flow hydrological model
E. Cordano (firstname.lastname@example.org), G. Formetta (email@example.com), R. Rigon (firstname.lastname@example.org)
1 Department of Civil and Environmental Engineering - CUDAM, University of Trento, Italy
Abstract Time (s) 103 104 105 2*105 Conclusions
The core of hydrological and land surface models is the We implemented a robust conservative model accounts
budget of extensive quantities like mass (of water, soil, for nonlinearities, drying and wetting, above all. The
sediment particles or other chemical component), H scheme is conservative and mass budget was verified for
momentum and energy in a representative finite volume. a mountain catchment in the central Alps (Saldur Creek/
This usually derives from discretization of the appropriate Val Mazia, South Tyrol, Italy) and resulted verified for all
partial differential equations. In this contribution some experiments made up to a precision of 10-8 .
mathematical characteristics of these equations are The coupling of surface/groundwater flow is a powerful
emphasized and a method for the integration of this type tool to capture dunnian saturation overland flow and
of equation based on a new numerical scheme which describe seepage and spring phenomena.
evolves from Brugnano and Casulli, 2008 and Casulli, The method works for any kind of mesh (squared or
2009, is shown. unstructured).
In this work we present a catchment-scale, coupled,
surface-water/groundwater model with structured or Future Work
unstructured grid. The surface water flow and overland
The next steps are:
runoff are solved by integrating the shallow water
Fig. (3): Subsurface water depth h (bottom) and surface water depth H
equation in 2D, whereas Boussinesq equation (BEq), (top) simulated a sub-catchment of Saldur Creek after 10^3 , 10^4, 2 10^4, • to implement the unsaturated zone processes and the
based on Darcy’s law, is solved for groundwater flow. 10^5, 2 10^5 seconds since the beginning of the drainage processes
model’s applications at several spatial scales;
Both equations are derivations of the fluid flow Navier-
• simulate large watersheds like the Adige basin (North-
Stokes Equation and they are solved in a similar way. The
Eastern Italy, 110000 km2) whose Saldur Creek is a sub-
equations are discretized in a coarse grid which, however, Tab. (2): Parameters of catchment;
accounts for sub-grid variations of topography and soil the simulation
properties. Wetting and drying of both the subsurface and
• extends to a 3D representation of subsurface flow
surface storage are obviously modelled.
(Richards’ Equation) which has a similar structure of BEq
The model can also take into account the presence of
unsaturated zones above water table (e.g. Cordano and The method and 3D investigations on lateral flow in mountain
hillslopes and small catchments;
Rigon, 2008, Hilberts et al, 2005). The methods used fall The mass balance equation (1) coupled with the
under the theorem demonstrated by Brugnano and momentum balance (2) and (3) leads to a weakly- • use netCDF format data for input/output as a possibility
Casulli, 2008, and Casulli, 2009, and thus, their nonlinear system whose solution is unique and can be for the user;
convergence to the exact solution are guaranteed. found with a Newton-like iterative scheme (Casulli, 2009).
However, implementation problems can occur in case of
• improve the communicability with the R console in order
steep hillslopes or complex topography, and these (4) to use R’s geostatistical tools for analyzing the data in
problems are discussed and solved with applications to
Alpine catchments. The T operator represents the connectivities among the
The models are implemented as a Free Software and cells whereas the function V is the water volume in a cell • merge with land-surface distributed models, like the
coupled with a Geographical Information System (GIS). and is an increasing function of the total head; it is Distributed Hydrological model GEOtop;
Further details of this work are available on http:// defined by the integral on the cells of water thickness
www.geotop.org/cgi-bin/moin.cgi/Boussinesq#preview.q multiplied by porosity. The vector η is the unknown. • provide the model as OMS 3.0 components
Fig. 1 shows a schematic view of a cell where bedrock
and terrain surfaces are variable (complex topography)
whereas water level is constant under hydrostatic Tab. (1): List of symbols References
hypothesis. The Digital Elevation Model of the basin
The model’s equation (terrain and bedrock) are analyzed at two different grid The numerical experiment Brugnano L., V. Casulli (2008), Iterative solution of piecewise linear
systems. SIAM Journal on Scientific Computing, vol 30 pp. 463-472
The model predicts water surface elevation and water resolutions (see tab. 2): the coarser one (COARSE) is the The model was run for a drainage situation in Ramudeltal, Casulli V. (2009), A high-resolution wetting and drying algorithm for
content by solving the discretized water mass balance in dimension of the cell in Fig. 1 and is utilized to discretize a mountain subcatchment in the Saldur Creek basin in the free-surface hydrodynamics, International Journal for Numerical
a cell: (1), the finer (FINE) is the resolution of topography and Methods in Fluids, vol. 60 pp. 391 - 408, DOI10.1002/fld.1896;
Ötztal Alps (South Tyrol, Italy), already utilized by Cordano
(1) soil properties distributed within the coarse cell. and Rigon (2009). The catchment elevations ranges Casulli V., Walters R. A., "An Unstructured Grid, Three-Dimensional
Fig. 2 shows the topology between the cells (in this case between 1800 and 3330 meters above sea level, a soil Model based on the Shallow Water Equations". International journal
where the subsurface velocity is given by Darcy’ law: a rectangular mesh is represented) for numerical methods in fluids, 2000, v. 3, p. 331-348.
depth equal to 2.5 m is assumed uniformly distributed for
(2) the test. The process starts with an initial water depth of Cordano, E., R. Rigon (2008), A perturbative view on the subsurface
water pressure response at hillslope scale, Water Resour. Res.,
20 cm from the lowest point of the bedrock of each cell. A
and the surface velocity from discretized momentum 44,W05407doi10.1029/2006WR005740.
Fig. (1): Graphical drainage boundary condition is applied at the outlet of the
equation (Casulli and Walters,2000): rappresentation of the Cordano E.,R. Rigon (2009) “A Solver of the 2D Boussinesq Equation
cells basin. Tab. 2 contains the timestep and grid resolution
with Sub-Grid parameterization of terrain and soil characteristics” ,
(3) American Geophysical Union, Fall Meeting 2009, abstract
with: Fig. 3 shows the maps of surface water depth H and A. G. J. Hilberts, P. A. Troch, and C. Paniconi (2005), Storage‐
subsurface water depths h at several time instants (103, dependent drainable porosity for complex hillslopes, Water Resour.
Res., 41, W06001, doi:10.1029/2004WR003725
104, 2 104, 105, 2 105 seconds) .
Initially there is no surface runoff but only subsurface flow. Acknowledgment
Fig. (2): Grid After 104 seconds, water exfiltrates in the convergent
zones and surface flow is enhanced in the hollow. We acknowledge prof. V. Casulli for his comments and suggestions
All the symbols are described in Table 1.