Free Surface Groundwater Flow Research Articles

Towards a computationally efficient free-surface groundwater flow boundary condition for large-scale hydrological modelling

Shallow groundwater is a critical component of the terrestrial water cycle. It sustains baseflow in rivers, supplies root zones with soil moisture during dry periods, and directly influences the land-atmosphere exchange processes. Nonetheless, the integration of groundwater into large-scale hydrological models remains challenging. The most detailed way of representing groundwater dynamics is to incorporate three-dimensional, variably saturated flow processes in the subsurface representation of hydrological models. However, such detailed modelling is still a challenge for global hydrological applications, mainly due to its high computational demand. In this study, a free-surface boundary condition called the Groundwater Flow Boundary (GFB) is developed to represent groundwater dynamics in a more computationally-efficient manner than the full three-dimensional models do. We evaluate GFB using two synthetic test cases, namely an infiltration experiment and a tilted-v catchment, which focus on groundwater recharge and discharge processes, respectively. The simulation results from GFB are compared with a three-dimensional groundwater flow model and with an over-simplified approach using a free-drainage lower boundary condition to assess the impact of our assumptions on model results. We demonstrate that GFB is computationally more efficient compared to the three-dimensional model with limited loss in model performance when simulating infiltration and runoff dynamics.

Free surface groundwater flow solution using boundary collocation methods

In this paper, two meshless numerical algorithms are developed for the solution of two-dimensional steady-state diffusion equation that describes the stationary groundwater flow. The proposed numerical methods, which are truly meshless, quadrature-free and boundary only, are based on the method of fundamental solutions and singular boundary method respectively. The diffusion equation is transformed into a Poisson-type equation with a known fundamental solution. Numerical examples with moving boundary are presented and compared to the solutions obtained by the finite element method.

A variational principle for problems of steady-state ground water flow with a free surface

The problem of steady-state free surface ground water flow through a porous medium is represented in the form of a variational principle for a functional dependent on the flow domain. It is proved that on the region to be found this functional takes a minimum value. This variational principle is generalized to include the flow models that allow the presence of partially saturated zones in the medium. On simple examples it is shown how the variational formulation can be used for proving the existence or absence of solutions.

Development and evaluation of a numerical model for steady state interface and/or free surface groundwater flow

A 3D numerical procedure is developed to solve free surface and/or freshwater–saltwater interface groundwater flow problems under steady state conditions, using the Galerkin finite element technique and the Picard linearization method to solve nonlinear systems. The presented approach is an improvement of the Larabi and De Smedt (1997) scheme, and is showed to be robust, more efficient, more stable and more accurate than the latter. Benchmark tests involving different features are presented and discussed to validate the present approach.

New approximation for free surface flow of groundwater: capillarity correction

An existing capillarity correction for free surface groundwater flow as modelled by the Boussinesq equation is re-investigated. Existing solutions, based on the shallow flow expansion, have considered only the zeroth-order approximation. Here, a second-order capillarity correction to tide-induced watertable fluctuations in a coastal aquifer adjacent to a sloping beach is derived. A new definition of the capillarity correction is proposed for small capillary fringes, and a simplified solution is derived. Comparisons of the two models show that the simplified model can be used in most cases. The significant effects of higher-order capillarity corrections on tidal fluctuations in a sloping beach are also demonstrated.

Steepness expansion for free surface flows in coastal aquifers

Free surface flow of groundwater in aquifers has been studied since the early 1960s. Previous investigations have been based on the Boussinesq equation, derived from the non-linear kinematic boundary condition. In fact, the Boussinesq equation is the zeroth-order equation in the shallow-water expansion. A key assumption in this expansion is that the mean thickness of the aquifer is small compared with a reference length, normally taken to be the linear decay length. In this study, we re-examine the expansion scheme for free surface groundwater flows, and propose a new expansion wherein the shallow-water assumption is replaced by a steepness assumption. A comparison with experimental data shows that the new model provides a better prediction of water table levels than the conventional shallow-water expansion. The applicable ranges of the two expansions are exhibited.

Watertable dynamics under capillary fringes: experiments and modelling

Watertable heights and total moisture content were measured in a sand column where the piezometric head at the base (“the driving head”) varied as a simple harmonic with periods in the range from 14.5 min to 6.5 h. The watertable height h( t) responded very closely to the driving head compared with the predictions of previous analytical and numerical models. The total moisture quantified as an equivalent, saturated height h tot( t) varied very little compared with the watertable height. Neither h( t) nor h tot( t) deviated significantly from simple harmonics when the driving head was simple harmonic. This indicates that non-linear effects are weak and hence that analysis based on linear solutions have fairly broad applicability. When h( t) and h tot( t) are simple harmonic, the ratio n d=[d h tot/d t]/[d h/d t] is a constant in the complex formalism. Its magnitude | n d| is the usual effective porosity while its argument accounts for the phase shift which is always observed between h( t) and h tot( t). Within the current range of experiments this dynamic, effective porosity n d appears to be almost independent of the forcing frequency, i.e., it is a function of the soil and its compaction only. Introducing the complex n d enables analytical solution for the watertable height in the column which is simpler and more consistently accurate over a range of frequencies than previous models including Richard’s equation with van Genuchten parameters corresponding to the measured water retention curve. The complex n d can be immediately adopted into linear watertable problems in 1 or 2 horizontal dimensions. Compared with “no fringe solutions”, this leads to modification of the watertable behaviour which is in agreement with experiments and previous models. The use of a complex n d to account for the capillary fringe in watertable models has the advantage, compared with previous models, e.g., [Parlange J-Y, Brutsaert W. A capillary correction for free surface flow of groundwater. Water Resour Res 1987; 23(5):805–8.] that the order of the differential equations is lower. For example, the linearised Boussinesq equation with complex n d is still of second order while the Parlange and Brutsaert equation is of third order. The extra work of calculating the imaginary part of the initially complex solution is insignificant compared to dealing with higher order equations. On the basis of the presently available data it also seems that the “complex n d approach” is more accurate. This is to be expected since the complex n d accounts implicitly for hysteresis while the Green–Ampt model does not. With respect to linear watertable waves, accounting for the capillary fringe through the complex n d is a very simple extension since the determination of wave numbers already involves complex numbers.

A moving mesh algorithm for 3-D regional groundwater flow with water table and seepage face

A numerical algorithm is described for solving the free-surface groundwater flow equations in 3-D large-scale unconfined aquifers with strongly heterogeneous conductivity and surface recharge. The algorithm uses a moving mesh to track the water-table as it evolves according to kinematic and seepage face boundary conditions. Both steady-state and transient algorithms are implemented in the SECO-Flow 3-D code and demonstrated on stratigraphy based on the Delaware Basin of south-eastern New Mexico.

A capillarity correction for free surface flow of groundwater

The Boussinesq equation describes aquifer movement when capillarity effects are unimportant. A simple correction term to the equation is proposed here to take those effects into account. The correction can be quite significant for relatively short times. Comparison with experimental observations for a sudden drawdown shows that the present theory greatly improves the Boussinesq equation.

Unsteady Shallow Groundwater Flow Over a Curved Impermeable Boundary

The asymptotic method which produced the new Dressler equation for curved open channel hydraulic flow is employed here to obtain analogous equations for unsteady, two‐dimensional, free surface groundwater flow over an arbitrarily curved bed. The result is a pair of partial differential equations: the first linear, stemming from Darcy's Law, and the second nonlinear, expressing mass conservation. The hydraulic conductivity and porosity can be functions of position. The flow component above and parallel to the bed behaves as a vortex in the small with centrifugal acceleration but with no centrifugal term in the pressure. For the special case of a plane sloping bed our equations reduce to the Wooding‐Chapman extension of the Dupuit‐Forchheimer equations. Our two‐dimensional flow equations are generalized to a special case of a nonisotropic medium and to include the effect of a variable recharge rate. All the systems are parabolic, thus describing diffusion, and can be integrated numerically as readily as the simpler equations for plane beds.

Regionalized drought flow hydrographs from a mature glaciated plateau

The drought or base flow characteristics of six basins in the Finger Lakes region are obtained by considering for each available record the lower envelope of |dQ/dt| as a function of Q, where Q is the flow rate. This procedure avoids the uncertainty regarding a proper time reference after each rainfall event, and it eliminates the effects of evapotranspiration. The results suggest that among several expressions, Boussinesq's nonlinear solution of free surface groundwater flow is best suited to parameterize the observed hydrographs. The obtained parameters can be related to the basin characteristics, viz., drainage area and the total stream length, in accordance with relationships derived on the basis of the Dupuit‐Boussinesq aquifer model. This result allows the determination of drought flow parameters for ungaged sites within the region.

Linearized solutions of free-surface groundwater flow with uniform recharge

The movement of the water table caused by uniform recharge (spreading) is considered here. The flow is saturated and the aquifer is unconfined and underlain by an impervious bottom. For a two-dimensional flow the recharge applies to a finite interval, and for a three-dimensional flow it applies to a circle and a rectangle. The solutions are derived by an expansion in a small parameter which linearizes the exact equations of flow [Dagan, 1964a]. Solutions of different cases are obtained by numerical integration. The investigation of the second-order term η2 representing the mound's crest for a two-dimensional deep flow shows that the first-order solution η1 may represent accurately the exact solution if the ratio R = R′/K is sufficiently small. For a shallow flow (c = c′/L ≪ 1), Dupuit assumptions may be adopted so as to give simpler equations of flow.

Some effects of the unsaturated zone on nonsteady free-surface groundwater flow as studied in a scaled granular model

Symmetrical free-surface flow of groundwater to outflow channels is studied in a scaled granular model employing screened sand and a spirit-water mixture. Differences between observed and computed hydrographs are analyzed in a combined study of unsaturated and saturated flow.