Modelling of hillslope storage under temporally varied rainfall recharge

Ping-Cheng Hsieh¹ ; Tzu-Ting Huang¹

¹ Department of Soil and Water Conservation, National Chung Hsing University, 145 Xingda Rd., South Dist., Taichung City 40227, Taiwan

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 9.

Résumé

Water storage inside hillslopes is a crucial issue of environment and water resources. This study separately built a numerical model and an analytical model employing a hillslope-storage equation to simulate the water storage in a sloping aquifer response to recharge. The variable width of hillslope was hypothetically represented by an exponential function to categorize the hillslope into three types: uniform, convergent, and divergent. An integral transform technique was introduced to derive the analytical solution whereas a finite difference method was employed for the numerical modelling. As a result, under the same scenario a gap existed between the two solutions to distinct forms of the water storage equation, and the gap decreases with a falling recharge rate for convergent hillslopes. Moreover, all outflows gradually approached one value based on different hillslopes under the same accumulative recharge amount for six typical rainfall recharge patterns. Particularly, while the recharge stops, the outflow decreases and then mildly rises for a long time for convergent hillslope because of the slow water release near the upstream boundary where the storage water is relatively abundant due to the widest width.

Détail

DOI : 10.1051/mmnp/2023009

Bibliographie

[1] M.G. Anderson, T.P. Burt The role of topography in controlling throughflow generation 1978 331 344

[2] W. Brutsaert The unit response of groundwater outflow from a hillslope 1994 2759 2763

[3] W. Chen, W.H. Wang, K.A. Nguyen Soil erosion and deposition in a Taiwanese watershed using USPED 2022 3397

[4] E. Childs Drainage of groundwater resting on a sloping bed 1997 1256 1263

[5] D.N. Dralle, G.F. Boisramé, S.E. Thompson Spatially variable water table recharge and the hillslope hydrologic response: Analytical solutions to the linearized hillslope Boussinesq equation 2014 8515 8530

[6] I.S. Evans, An integrated system of terrain analysis and slope mapping. University of Durham, England, UK (1979).

[7] Y. Fan, R. Bras Analytical solutions to hillslope subsurface storm flow and saturation overland flow 1998 921 927

[8] D.P. Genereux, H.F. Hemond, P.J. Mulholland Spatial and temporal variability in streamflow generation on the West Fork of Walker Branch Watershed 1993 137 166

[9] S. Issak, M.A. Ashraf Impact of soil erosion and degradation on water quality: a review 2017 1 11

[10] B.S. Lin, C.K. Chen, K. Thomas, C.K. Hsu, H.C. Ho Improvement of the K-factor of USLE and soil erosion estimation in Shihmen reservoir watershed 2019 355

[11] J.J. Mcdonnell A rationale for old water discharge through macropores in a steep, humid catchment 1990 2821 2832

[12] M.P. Mosley Streamflow generation in a forested watershed, New Zealand 1979 795 806

[13] E.M. O’Loughlin Prediction of surface saturation zones in natural catchments by topographic analysis 1986 794 804

[14] M.N. Ozisik, Boundary Value Problems of Heat Conduction. Dover Publications, INC., New York, USA (1968).

[15] M. Ranjram, J.R. Craig Use of an efficient proxy solution for the hillslope-storage Boussinesq problem in upscaling of subsurface stormflow 2021 e2020WR029105

[16] S. Sahoo, B. Sahoo, S.N. Panda Hillslope-storage Boussinesq model for simulating subsurface water storage dynamics in scantily-gauged catchments 2018 219 234

[17] S. Sahoo, B. Sahoo Modelling the variability of hillslope drainage using grid-based hillslope width function estimation algorithm 2019 71 78

[18] C.W. Shu Total-variation-diminishing time discretizations 1988 1073 1084

[19] D.S. Shih, Analysis of typhoon and rainfall patterns in Taiwan. Master thesis, Department of Civil Engineering, National Cenal University, Zhongli, Taiwan, 2001 (in Chinese).

[20] R.C. Swanson, E. Turkel On central-difference and upwind schemes 1992 292 306

[21] P.A. Troch, E.E. Van Loon, A. Hilberts Analytical solutions to a hillslope-storage kinematic wave equation for subsurface flow 2002 637 649

[22] P.A. Troch, C. Paniconi, E.E. Van Loon The hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response 2003 1316

[23] P.A. Troch, A.H. Van Loon, A.G.J. Hilberts Analytical solution of the linearized hillslope-storage Boussinesq equation for exponential hillslope width functions 2004 W08601

[24] N.E.C. Verhoest, P.A. Troch Some analytical solutions of the linearized Boussinesq equation with recharge for a sloping aquifer 2000 793 800

[25] K.B. Wartalska, M. Kazmierczak Nowakowska and A. Kotowski, Analysis of hyetographs for drainage system modeling 2020 149

[26] R. Woods, L. Rowe The changing spatial variability of subsurface flow across a hillside 1996 51 86

[27] R. Woods, M. Sivapalan, J. Robinson Modeling the spatial variability of subsurface runoff using a topographic index 1997 1061 1073

[28] M.C. Wu, P.C. Hsieh Improved solutions to the linearized Boussinesq equation with temporally varied rainfall recharge for a sloping aquifer 2019 826