Development and verification of a 3-D integrated surface water–groundwater model

Summary

Coupled modelling of surface and subsurface systems is a valuable tool for quantifying surface water–groundwater interactions. In the present paper, the 3-D non-steady state Navier–Stokes equations, after Reynolds averaging and with the assumption of a hydrostatic pressure distribution, are for the first time coupled to the 3-D saturated groundwater flow equations in an Integrated suRface watEr–grouNdwater modEl (IRENE). A finite-difference method is used for the solution of the governing equations of IRENE. A semi-implicit scheme is used for the discretisation of the surface water flow equations and a fully implicit scheme for the discretisation of the groundwater flow equations. The two sets of equations are coupled at the common interface of the surface water and groundwater bodies, where water exchange takes place, using Darcy’s law. A new approach is proposed for the solution of the coupled surface water and groundwater equations in a simultaneous manner, in such a fashion that gives computational efficiency at low computational cost. IRENE is verified against three analytical solutions of surface water–groundwater interaction, which are chosen so that different components of the model can be tested. The model closely reproduces the results of the analytical solutions and can therefore be used for analysing and predicting surface water–groundwater interactions in real-world cases.

Introduction

Surface water and groundwater are not isolated components of the hydrologic cycle. Instead, all surface water bodies (e.g. streams, lakes, wetlands and estuaries) are often hydraulically connected to groundwater and the interaction between them affects both their quantity and quality. Depending on the relative surface water and groundwater levels, surface water bodies can gain water and pollutants from groundwater systems or vice versa. Winter et al., 1998, Sophocleous, 2002 present comprehensive overviews of surface water–groundwater interactions in relation to climate, landform and hydrogeology. However, surface water and groundwater are usually considered as two separate systems and are analysed independently. This separation is partly due to the different time scales, which apply in surface water and groundwater flows and partly due to the difficulties in measuring and modelling their interactions (Winter et al., 1998). In the past few decades environmental awareness has increased and with it the realization that effective water resources management requires a realistic and detailed description of the interactions between surface water and groundwater.

Until the advent of computers, much of the literature on the interaction between surface water and groundwater was concerned with analytical solutions. Analytical solutions to interaction problems are important because apart from their application being simple and straightforward, they also provide a means of validating the mathematics of a numerical solution. An analytical solution also gives good insight into the dependence of the solution variable on the state variables (Singh, 2004). The limitation of the analytical solutions is that they are applicable to simplified or idealized cases. For complex interaction problems mathematical models need to be used.

There is a number of models available that simulate the interactions between surface water and groundwater. A summary of widely used integrated surface water–groundwater models is presented in Table 1. These models can be distinguished based on the type of equations and the spatial dimension (1-D, 2-D or 3-D) used to describe surface water and groundwater flows and on the coupling method of the surface water and the groundwater components. In the published integrated surface water–groundwater models, surface water flow is usually described by the 1-D Saint Venant equations (e.g. Swain and Wexler, 1996) or the 2-D shallow water equations (e.g. Sparks, 2004, Liang et al., 2007). Further simplified equations, such as the diffusion and kinematic wave approximations to the Saint Venant equations, are also employed for 2-D overland flow and 1-D stream flow (e.g. Jobson and Harbaugh, 1999, Vanderkwaak, 1999, Hussein and Schwartz, 2003, Panday and Huyakorn, 2004, Morita and Yen, 2002, Gunduz and Aral, 2005). Groundwater flow is considered as 2-D saturated, 3-D saturated or variably saturated. There are cases, however, e.g. in estuaries, wetlands and lakes, where it is desirable to use a 3-D model to describe surface water flow. To the authors’ knowledge there is no integrated surface water–groundwater model available, which considers 3-D surface water and 3-D groundwater flow.

Considering the coupling method, most integrated surface water–groundwater models are based on the idea of solving for common parameters linking surface water and groundwater flows in an iterative or non-iterative manner (e.g. Swain and Wexler, 1996, Jobson and Harbaugh, 1999, Bradford and Katopodes, 1998). The iterative approach involves subsequent solutions of the surface water and groundwater flow equations, at each time step, until the difference between the subsequent solutions for the common parameters falls within a specified tolerance. The non-iterative approach involves the solution of the surface water and groundwater flow equations in succession without iteration. Iterative models generally require more computational time than non-iterative models due to the subsequent solutions of the surface water and groundwater flow equations within a time step. On the other hand, the iterative improvement of the solution provides more accurate results compared to the non-iterative approach (Gunduz and Aral, 2005). Recently, a few studies on integrated surface water–groundwater modelling have been reported in the literature (e.g. Vanderkwaak, 1999, Panday and Huyakorn, 2004, Gunduz and Aral, 2005), where the surface water and groundwater flow equations are solved simultaneously at each time step. This coupling approach is faster than the iterative approach since it does not require iterative solutions of the surface water and groundwater flow equations. It is also more accurate than the iterative approach because there is no need to check for convergence of the solution parameters between iterations (Gunduz and Aral, 2005).

In this paper, coupling of the 3-D surface water flow and 3-D saturated groundwater flow in an Integrated suRface watEr–grouNdwater modEl (IRENE) is presented. IRENE consists of a 3-D surface water flow sub-model, namely FLOW-3DL (e.g. Stamou et al., 2007a, Stamou et al., 2007b) and a 3-D saturated groundwater flow sub-model. FLOW-3DL has been refined to a new version where a semi-implicit scheme is used to solve the governing surface water flow equations. The refined version of FLOW-3DL is also capable of simulating wetting and drying processes, which is important in modelling surface water flow, especially when it is coupled with groundwater flow. Unlike most of the other published integrated models, where the surface water and groundwater flow sub-models are linked vertically (e.g. Swain and Wexler, 1996, Jobson and Harbaugh, 1999), the 3-D character of IRENE permits detailed linking of surface water flow with groundwater flow in areas adjacent and below the surface water body. One of the reasons that 3-D surface water flow and 3-D groundwater flow have never before been coupled is the computational cost of such an approach. Here, the surface water and groundwater flow equations of IRENE are coupled at the common interface of the surface water and groundwater bodies using Darcy’s law and are solved simultaneously within a common structure, which guarantees computational efficiency at a low computational cost. In the coupling method, both mass and momentum transfers between the surface water and groundwater domains are considered.

The paper is organised as follows: the governing equations of the surface water and the groundwater flow sub-models of IRENE are described in “Mathematical formulation”; the numerical methods used for the discretisation of the surface water and groundwater equations and the coupling method of the surface water and groundwater flows are explained in “Numerical methodology”; in “Model verification”, the model is verified against three analytical solutions of surface water–groundwater interaction; finally, conclusions are discussed in “Conclusions”.