It is hereby certified that the work included in the thesis entitled "A 2-D Coupled Surface and Subsurface Flow Model for Piedmont Area River Flow Simulation" by Sudarshan Patowary, Roll Number 11610413, student in the Department of Civil Engineering, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy has been carried out under my supervision and that this work has not been submitted elsewhere for a degree. Pinakeswar Mahanta, Professor, Department of Mechanical Engineering, Indian Institute of Technology Guwahati, for their keen interest, valuable suggestions and guidance provided to complete this work as members of the PhD Committee.
List of figure
77 Figure 6.9 (c) Sensitivity analysis of volume for permeability 77 Figure 6.9 (d) Sensitivity analysis of infiltration for permeability 78 Figure 6.10 Groundwater level height for different values of α 79.
List of Table
List of Notation
Introduction
- Purpose of the study
- Organisation of the thesis
The existence of a Piedmont area in a river bed is one of the critical parameters of many that can significantly influence the river flow scenario. Therefore, in this chapter a short literature overview is given of the different numerical techniques for solving the above equations.
Literature Review
- Introduction
- Previous works on unsteady free surface flow model
- Concluding remark on free surface flow model
- Previous work on infiltration and ground water model
They proposed a modified version of Galerkin finite element scheme for solving the modified form of the Saint-Venant equation. The explicit finite volume upwind method was used for solving the main part of the model.
Governing Equation
Introduction
- Governing equation for unsteady free surface flow model
- Governing equation for infiltration model
- Governing equation for ground water model
- Coupling of all the governing equation
- Conclusion
The lower slope of the channel is small, i.e. the flow depth measured perpendicular to the channel bottom or measured vertically is approximately the same. The governing equation of unsteady free surface flow, i.e. the continuity and momentum equations in the conservation form, taking into account lateral flow, has been used by different investigators (Chow, et al. 1988, Strelkoff 1970, Choudhry H.M. 2008). 3.1; ( )t is the time-dependent effective suction head; s, o are the saturated and initial soil moisture content, respectively; ks is the saturated hydraulic conductivity.
F* and f* are obtained by solving the Richards equations (Reeder et al. 1980) as follows. The movement of infiltrated water through subsurface layers forms another important aspect of the runoff process. Since the flow field under the riverbed is not rectangular, for the numerical solution of Richard's equation, it is necessary to transform the equation into a rectangular domain.
This chapter discussed the governing equations for the development of a mathematical model that considers the groundwater recharge area.
Solution of Governing Equation
- Introduction
- Numerical formulation of free surface flow model by finite difference method In Saint-Venant as well as in some other non-linear partial difference equations, among
- Numerical solution of Richards equation
- Conclusion
A detailed sketch about the space and time discretization of the finite difference method is shown in figure 4.1. In implicit finite-difference schemes, the spatial partial derivatives and the coefficient are replaced in terms of values at the unknown time level. The solution obtained from a finite difference scheme has a dissipative error if the main term of the truncation errors in the scheme has even derivatives and the solution has dispersive errors if the main term has odd derivatives.
Finite difference and finite elements are the most common numerical methods for solving nonlinear equations. Gottardi and Venutelli (1993) derived the finite difference approximation of the 1-D Richards equation using the ADI scheme. In this equation, the finite difference equation is implicit in the z direction and explicit in the x direction and is referred to as the z sweep.
An implicit finite difference Beam and Warming scheme was used to solve the shallow water equation in combination with Green-Ampt.
Model Development and Validation
Introduction
Development of free surface flow model .1 Initial condition
An additional boundary condition, the so-called intermediate boundary condition, is also necessary if we look at the lateral outflow and inflow of water in the main stream. The discharge hydrography as shown in figure 5.1 has been taken as the upstream boundary condition for the hypothetical river reach. When water level is available on any downstream control path, the downstream boundary condition can be replaced by flow depth on that control path.
This intermediate boundary condition is solved by the positive characteristic equation and Crew's equation as discussed earlier in case of downstream boundary condition. For 2-D flow, one more boundary condition is required which is known as solid boundary which has no flow through it. The reflective boundary condition is implemented by creating a dummy cell at the end of the surface, as shown in fig 5.2.
Because the Piedmont region is developed at the foot of a hill, it is assumed that water infiltrated through the recharge area moves only along the river.
Validation of the model
This infiltrated water moves as groundwater under the riverbed and after some time it rejoins the main stream downstream of the river. The remaining ground surface was covered to prevent evaporation loss, due to the symmetry only one side (here the right side) of the flow domain needs to be modelled. 5.6, the modeled part of the flow domain is 3.00 m × 2.00 m, with no-flow boundaries at the bottom and on the left side (taking into account the symmetry).
At the ground surface, the constant flux of 3.55 mday-1 is applied over the left 0.50 m from the top of the modeled domain. Transient positions of the water table are compared with the experimental results reported by Vauclin et al. The assumptions made in the governing equation are also maintained in the development of the model.
From the comparison of the results, it was found that there is a good match between them.
Model Application
- Introduction
- Model application on hypothetical case
- Results obtained from hypothetical case
- Field application of the model
- conclusion
From this figure, we observed that due to the presence of the piedmont area, the height of the water surface downstream of the piedmont area decreases. Compression of the graph reveals that the height of the water surface downstream of the recharge area decreases with increasing values of hydraulic conductivity 'k'. From this figure, we have seen that due to the presence of the piedmont zone, the top of the depth hydrograph decreases.
It can be seen from this figure that as the hydraulic conductivity increases, the top of the discharge hydrograph decreases. This section presents the results obtained using the developed model in the field. It can be seen from this figure that due to the presence of the piedmont area, the height of the water surface d/s of the piedmont area decreases.
From the comparison of these two figures it is observed that due to the presence of piedmont zone the peak of the depth hydrography decreases by 13% and 10% for 200 years and 10 years return period respectively. From this figure it can be seen that the peak of the discharge hydrograph decreases by 9.49 % and 6.47% for the return period 10 years and 200 years respectively due to the presence of piedmont zone. Double ring infiltrometer test was carried out in the field to determine the infiltration characteristics of the river.
Conclusion, Discussion and Scope of Future Study
Introduction
Conclusion and discussion
From the literature it can be seen that the researchers had widely used the finite difference and the finite element scheme for the solution of the same. From the validation of the model it is found that there is a good agreement between them and they can be used to develop a coupled model. After validating the model, it is applied to the hypothetical river stretch where such a piedmont area is considered in the computational domain.
After detailed analysis of the model in hypothetical cases, the model is applied to a tributary of the Brahmaputra River where such a piedmont zone has been reported (Goswami et al.1996). To know the infiltration characteristics of the river, double ring infiltrometer test is conducted in the field. Using a double ring infiltrometer, hydraulic conductivity is calculated at the soil surface (up to a depth of 20 cm from the surface).
In the absence of detailed geological data, the underground soil profile of the Piedmont zone is considered homogeneous with an average permeability characteristic.
Scope of future study
Local people also confirmed that in most of the nearby wells, the water table remains at the same depth during the rainy season. For obtaining design rainfall intensity of the study area, IDF curve generated using precipitation data from a tea garden located 25 km away from the study area was used. Since the primary focus of the field application is to establish model applicability in the actual field, model is implemented with the above input information, but an error due to data insufficiency may well be there.
Results obtained from different subsurface soil formation concluded that unsaturated flow depends on the subsurface soil layer. In our field application, we considered the soil profile under the riverbed to be homogeneous. So in the future, underground soil investigations can be carried out for application of the model in the field with more reliability.
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2010) "finite-element algorithm for modeling variably saturated flows." J. 2004) "Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels."
Paper Presented and Published
2004) "Flow of Saturated and Unsaturated Water in Inclined Porous Media." Environmental Model Assessment. 2009) "A Generalized Richards Equation for Modeling Surface/Subsurface Flow." J. Brutsaert (1971) "A Functional Iteration Technique for the Solution of the Richards Equation Applied to Two-Dimensional Infiltration Problems". Marin (2005) "Determining Basin and Infiltration Status in Layered Soils Under Intermittent Rainfall". J. 2004) "Backward conservative scheme for Saint Venant's equations.".
Paper under review
