Vol.:(0123456789) 1 3 Environmental Earth Sciences (2018) 77:447 https://doi.org/10.1007/s12665-018-7646-5 ORIGINAL ARTICLE Development of a groundwater/surface ﬁnite-element ﬂow model: application to the Barcés watershed P. Vellando 1 · R. Juncosa 1 · F. Padilla 1 · H. García‑Rábade 1 Received: 11 January 2018 / Accepted: 18 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A ﬁnite-element model has been developed in MATLAB to solve the groundwater ﬂow in natural watersheds. The present model incorporates some eﬃcient tools that evaluate the natural conditions of the water catchments including an evaluation of the free water ﬂow based on the continuity of mass. The resulting model does not include the veriﬁcation of the continuity of momentum nor a turbulence model, but still provides a proper overall assessment of the ﬂow in the watershed, that can be reﬁned with a further turbulent assessment of the free surface ﬂow. The model has been veriﬁed through its comparison with some benchmark problems showing to be both accurate and numerically eﬃcient. Finally, it has been used in the resolution of some practical cases that show its capabilities with a special focus on the Barcés river watershed, intensely monitored by this research group in the last decades. Keywords Groundwater/surface ﬂow · Finite elements · Natural watersheds Introduction In the natural ﬂow of water in nature, water drops ﬂow indis- tinctly over or underneath the ground surface. The ground- water ﬂow takes places as a slow motion through a porous media in which the Reynolds number can be assumed to be small and results into a laminar ﬂow. Groundwater ﬂow can be assessed through the Darcy equation, that simply takes the ﬂux q (discharge per unit area) as proportional to the hydraulic gradient (− ∇h), with h being the water head. The ﬂow of water at the free surface, in contrast, is fast and should be assumed as turbulent; the conservation of momen- tum should be taken into account together with the conserva- tion of mass; this way leads to the Navier–Stokes equations. Many authors have developed plenty of numerical models giving solution to the joint surface and groundwater ﬂow. Interesting details on the interaction between both ﬂows can be seen in Sophocleous (2002) and Furman (2008). Some examples of coupled surface/groundwater models can be found in Querner (1997), Jobson and Harbaugh (1999), Morita and Yen (2002), Panday and Huyakorn (2004), Gun- duz and Aral (2005), Kollet and Maxwell (2006), Padilla et al. (2008), and Camporese et al. (2009), mainly using a diﬀusion wave approximation to the Saint–Venant equations. Other models use the kinematic wave approximation to the Saint–Venant equations, as the ones proposed by Graham and Refsgaard (2001) and Hussein and Schwartz (2003), or the plain conservation of mass for the surface ﬂow as in Merrit and Konikow (2000) and Prudic et al. (2004). Some authors such as Tung (1985) and Padilla et al. (2008) have tried to incorporate concepts taken from classic hydrology, such as the Muskingum method, to assess the free surface ﬂow in computational models in which the governing equa- tion is the groundwater ﬂow equation (i.e., Darcy and con- tinuity equations). These models ﬁnally result into the deﬁ- nition of a modiﬁed conductivity that can be used through some numerical adaptation and that, in most of the cases, requires a further calibration work to adjust it. Diﬀerent models have attempted to solve this kind of problems through diﬀerent numerical approaches, the most common of which are ﬁnite diﬀerences, ﬁnite volumes, and ﬁnite elements. There exist coupled ﬁnite-diﬀerence mod- els by Yuan et al. (2008), Sparks (2004), and Liang et al. (2007), who use the shallow water equations and a 2D saturated model for groundwater through a MacCormack ﬁnite-diﬀerence scheme, or the three-dimensional one by * P. Vellando pvellando@udc.es 1 E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de La Coruña, Campus de Elviña, 15071 La Coruña, Spain