Derivation of the Nonlinear Shallow Water Equations Over Unsaturated Porous Media: A Discontinuous Galerkin Method for Coarse-Grained Beaches

Abstract

This work examines the physics of free-surface flow and groundwater flow within a coupled model. Coupled models for such phenomena are not clearly justified, and there is a lack of precision in the derivation of such models. The primary objective of this work is to derive a coupled model of the shallow water equations (SWE) and Richards’ equation (RE) using asymptotic considerations. The numerical coupling approach chosen for the unified model will be described as a parallel coupling. Additionally, numerical considerations regarding how to solve this model using the discontinuous Galerkin (DG) methods will be provided. Furthermore, the exchange of information between the two models, which are time-synchronized, will be explained. The solution of RE coupled with SWE, following the described procedure and implemented using the DG formulation, is integrated into RIVAGE (an in-house numerical code based on the DG method). This implementation is then tested on a numerical problem and validated against an experimental benchmark.

Keywords

asymptotic analysis shallow water equations Richards equation unsaturated porous media discontinuous Galerkin two-way coupling

Get full access to this article

View all access options for this article.

References

Anderson

Woessner

Hunt

(2015). Applied ground water modeling: Simulation of flow and advective transport .

Beavers

G. S.

Joseph

D. D.

(1967). Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30(1), 197–207. https://doi.org/10.1017/S0022112067001375

Cardenas

M. B.

Zlotnik

V. A.

(2003). Three-dimensional model of modern channel bend deposits. Water Resources Research, 39(6). https://doi.org/10.1029/2002WR001383

Caviedes-Voullième

Murillo

Garcia-Navarro

(2012). Numerical simulation of groundwater- surface interactions by external coupling of the 3D Richards equation and the full 2D shallow-water equations.

Clément

J.-B.

(2021). Numerical simulation of flows in unsaturated porous media by an adaptive discontinuous Galerkin method: Application to sandy beaches. PhD thesis, Université de Toulon.

Clément

J.-B.

Golay

Ersoy

Sous

(2020). Adaptive discontinuous Galerkin method for Richards equation. In Topical problems of fluid mechanics 2020, (pp. 27–34), Prague, Czech Republic. Institute of Thermomechanics, AS CR, v.v.i. https://hal.science/hal-02507326, DOI: https://doi.org/10.14311/TPFM.2020.004.

Clément

J.-B.

Golay

Ersoy

Sous

(2021). An adaptive strategy for discontinuous Galerkin simulations of Richards’ equation: Application to multi-materials dam wetting. Advances in Water Resources, 151, 103897. https://doi.org/10.1016/j.advwatres.2021.103897

Dawson

(2008). A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow. Computational Geosciences, 12(4), 451–472. https://doi.org/10.1007/s10596-008-9085-y

Delestre

(2010). Simulation du ruissellement D’eau De pluie sur des surfaces agricoles. PhD thesis, Université d’Orléans.

10.

Delpierre

Rattez

Soares-Frazao

(2023). Finite-volume coupled surface-subsurface flow modelling in earth dikes. Journal of Hydraulic Research, 61(5), 754–763. https://doi.org/10.1080/00221686.2023.2246936

11.

Discacciati

Miglio

Quarteroni

(2002). Mathematical and numerical models for coupling surface and groundwater flows. Applied Numerical Mathematics, 43(1), 57–74. https://doi.org/10.1016/S0168-9274(02)00125-3

12.

Discacciati

Quarteroni

(2009). Navier–Stokes/Darcy coupling: Modeling, analysis, and numerical approximation. Revista Matemática Complutense, 22, 315–426. https://doi.org/10.5209/rev_REMA.2009.v22.n2.16263

13.

Dong

Zhang

Bai

(2013). A hybrid coupled model of surface and subsurface flow for surface irrigation. Journal of Hydrology, 500, 62–74. https://doi.org/10.1016/j.jhydrol.2013.07.018

14.

Ersoy

Lakkis

Townsend

(2021). A Saint–Venant model for overland flows with precipitation and recharge. Mathematical and Computational Applications, 26(1), 1. https://doi.org/10.3390/mca26010001

15.

Freeze

R. A.

Harlan

R. L.

(1969). Blueprint for a physically-based, digitally-simulated hydrologic response model. Journal of Hydrology, 9(3), 237–258. https://doi.org/10.1016/0022-1694(69)90020-1

16.

Furman

(2008). Modeling coupled surface–subsurface flow processes: A review. Vadose Zone Journal, 7(2), 741–756. https://doi.org/10.2136/vzj2007.0065

17.

Girault

Rivière

(2009). DG approximation of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition. SIAM Journal on Numerical Analysis, 47(3), 2052–2089. https://doi.org/10.1137/070686081

18.

Layton

W. J.

Schieweck

Yotov

(2002). Coupling fluid flow with porous media flow. SIAM Journal on Numerical Analysis, 40(6), 2195–2218. https://doi.org/10.1137/S0036142901392766

19.

Marche

(2007). Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European Journal of Mechanics - B. Fluids, 26(1), 49–63. https://doi.org/10.1016/j.euromechflu.2006.04.007

20.

Mikelic

Jäger

(2000). On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM Journal on Applied Mathematics, 60(4), 1111–1127. https://doi.org/10.1137/S003613999833678X

21.

Morita

Yen

B. C.

(2000). Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows. International Journal for Numerical Methods in Fluids, 32(8), 921–957. https://doi.org/10.1002/(SICI)1097-0363(20000430)32:8<921::AID-FLD993=3.0.CO;2-3

22.

Pietro

D. A. D.

Ern

(2012). Mathematical aspects of discontinuous galerkin methods.

23.

Pons

(2018). Modélisation des tsunamis: Propagation et impact PhD thesis, Université de Toulon. Springer.

24.

Poussel

Ersoy

Golay

(2024). Wetting and drying treatments with mesh adaptation for shallow water equations using a Runge–Kutta discontinuous Galerkin method. Working paper or preprint. https://hal.science/hal-04676186.

25.

Pringle

W. J.

Yoneyama

Mori

(2016). Two-way coupled long wave—RANS model: Solitary wave transformation and breaking on a plane beach. Coastal Engineering, 114, 99–118. https://doi.org/10.1016/j.coastaleng.2016.04.011

26.

Richards

L. A.

(1931). Capillary conduction of liquids through porous mediums. Physics, 1(5), 318–333. https://doi.org/10.1063/1.1745010

27.

Richardson

L. F.

(1922). Weather prediction by numerical process. Cambridge University Press.

28.

Rivière

Yotov

(2005). Locally conservative coupling of Stokes and Darcy flows. SIAM Journal on Numerical Analysis, 42(5), 1959–1977. https://doi.org/10.1137/S0036142903427640

29.

Saffman

P. G.

(1971). On the boundary condition at the surface of a porous medium. Studies in Applied Mathematics, 50(2), 93–101. https://doi.org/10.1002/sapm197150293

30.

Steenhauer

Pokrajac

O’Donoghue

(2012). Numerical model of swash motion and air entrapment within coarse-grained beaches. Coastal Engineering, 64, 113–126. https://doi.org/10.1016/j.coastaleng.2012.01.004

31.

Steenhauer

Pokrajac

O’Donoghue

Kikkert

G. A.

(2011). Subsurface processes generated by bore-driven swash on coarse-grained beaches. Journal of Geophysical Research: Oceans, 116(C4). https://doi.org/10.1029/2010JC006789

32.

Todd

D. K.

(1980). Groundwater hydrology. Geological Magazine, 118(4), 442. https://doi.org/10.1017/S0016756800032477

33.

VanderKwaak

J. E.

Loague

(2001). Hydrologic-response simulations for the R-5 catchment with a comprehensive physics-based model. Water Resources Research, 37(4), 999–1013. https://doi.org/10.1029/2000WR900272

34.

Yeh

G.-T.

Cheng

H.-P.

Cheng

Lin

Martin

(1998). A numerical model simulating water flow and contaminant and sediment transport in WAterSHed systems of 1-D stream-river network, 2-D overland regime, and 3-D Subsurface Media (WASH123D: Version 1.0).