Dupuit-Forchheimer Assumption Research Articles

An analytical approach for urban groundwater transit time distributions accounting for the effect of stormwater infiltration system

The stormwater infiltration system (SIS) has been recognized as a feasible way to manage stormwater and restore streamflow. As a lumped representation of the flow and transport processes, the groundwater transit time distribution (TTD) is highly relevant to this restoration goal. However, the influence of the SIS on groundwater TTDs has rarely been studied. In this paper, we propose a novel analytical model for groundwater TTDs in regional urban aquifers accounting for the impact of the SIS. This analytical solution is developed for a homogeneous 2-D cross-section of an idealized unconfined aquifer following the Dupuit-Forchheimer assumption. The analytical solution is verified through the comparison with simulated TTDs using the Lagrangian particle tracking algorithm. The global sensitivity analysis reveals that the stormwater infiltration rate, the number of infiltration facilities, the size of the urban area, and the evapotranspiration are the most important factors for groundwater mean transit times (MTTs). Although groundwater discharge and MTT can be restored to the pre-urban one, groundwater TTDs will inevitably deviate from the pre-urban ones due to the alteration from diffuse recharge to infiltration-focused recharge, indicating that the urban groundwater flow regime cannot be fully restored to the pre-urban conditions. Overall, as a first-order approximation of the real-world urban groundwater system, the proposed analytical model provides decision-making support for urban water resources management and the design of the SIS.

An analytical solution of the tide-induced groundwater table overheight under a three-dimensional kinematic boundary condition

Time averaged groundwater table above the mean sea level has been observed in unconfined coastal aquifers. Several theoretical solutions were presented to investigate the groundwater table overheight (super-elevation) in response to tidal oscillations. Nevertheless, previous analytical solutions are generally limited to Dupuit-Forchheimer assumptions or a two-dimensional free surface model. In this paper, we introduced a more general condition that a three-dimensional kinematic boundary condition is considered. An exact analytical solution of the groundwater table overheight under the multi-tidal constituents was derived using the Fourier transform method. The solution shows the local groundwater table overheight is related to the mean aquifer thickness, tidal amplitude, damping coefficient as well as its positions in sand beaches. In addition, the analytical solution illustrates the simple linear superposition is not applicable to the multi-tidal components condition, whose groundwater table overheight becomes weak due to the non-linear effect of the three-dimensional kinematic boundary condition. In addition, concentrations were focused on the time averaged total flux density illustrating the groundwater exchange occurs both in the shore normal and shore parallel directions, which however is missing in the previous two-dimensional solution. Further cognitions on the groundwater circulation mechanism were also achieved, which originates from the large tidal amplitude region and fans out towards the small tidal amplitude region. These new findings are of great importance to extend our understandings of the groundwater hydrodynamics in unconfined coastal aquifers.

Response of Groundwater Levels in a Coastal Aquifer to Tidal Waves and Rainfall Recharge

Groundwater level in coastal aquifers is usually affected by tidal waves and rainfall recharge. Therefore, the objective of this study is to present a mathematical model to account for the effects of tidal waves and rainfall recharge simultaneously. The model is based on the Dupuit–Forchheimer assumptions and is separated into a tidal waves component and rainfall recharge component. A new more general analytical solution for the recharge component is acquired by the generalized integral transform technique. The beach slope, the inclination of an impermeable base of an aquifer, and any randomly distributed rainfall recharge are taken into account in the model. A new finding is that the highest fluctuation in groundwater levels might occur when the range of rainfall recharge is larger than the decay length.

Water-table fluctuation method for assessing aquifer recharge: application to Canadian aquifers and comparison with other methods

The water-table fluctuation (WTF) method is widely used to assess groundwater recharge but it is also criticized for certain limitations, namely that it is limited to calculating local recharge because most aquifers are relatively heterogeneous. This study aims at assessing groundwater recharge with the WTF method on a bigger spatial scale by comparing results from WTF with those obtained from different methods. The WTF method was thus applied to observation wells located in two aquifers in Eastern Canada for which groundwater recharge had previously been evaluated using several methods. Comparisons were conducted between WTF and other methods appropriate for regional assessments: (1) two water budget approaches, (2) hydrograph separation, and (3) an analytical regional solution based on Dupuit-Forchheimer assumptions. These showed that applying the WTF method to several observation wells and then calculating an average value can yield results that are comparable to those obtained on the scale of catchment areas. The study concludes that the WTF method can be used for the assessment of groundwater recharge at large regional scales if the aquifer is monitored by an appropriate network of observation wells.

Effect of depth-dependent hydraulic conductivity and anisotropy on transit time distributions

The paper presents exact closed-form analytical solutions describing a two-dimensional profile confined groundwater flow induced by areal recharge in a laterally bounded aquifer with anisotropic permeability, which decreases with depth. The mathematical formulation of the problem for a rectangular flow domain in terms of hydraulic head and stream function are free of the limitation of the Dupuit–Forchheimer assumption. Two different types of outflow boundary conditions associated with the aquifer discharge area, Dirichlet and Neumann, are explored. Analytical solutions with respect to stream function allow examining the distribution of recharge over depth (between contouring streamlines) for a variety of flow parameter combinations. The solutions are extended to allow the groundwater transit time distribution (TTD) to be calculated. It was found that the dependence of the transit time function on hydraulic conductivity anisotropy and depth-decay coefficients may exhibit non-monotonic behavior. The mathematical models introduced in the article are accompanied by computational simulations.

A 3D Numerical Study of Interface Effects Influencing Viscous Gravity Currents in a Parabolic Fissure, with Implications for Modeling with 1D Nonlinear Diffusion Equations

Although one-dimensional non-linear diffusion equations are commonly used to model flow dynamics in aquifers and fissures, they disregard multiple effects of real-life flows. Similarity analysis may allow further analytical reduction of these equations, but it is often difficult to provide applicable initial and boundary conditions in practice, or know the magnitude of effects neglected by the 1D model. Furthermore, when multiple simplifying assumptions are made, the sources of discrepancy between modeled and observed data are difficult to identify. We derive one such model of viscous flow in a parabolic fissure from first principals. The parabolic fissure is formed by extruding an upward opening parabola in a horizontal direction. In this setting, permeability is a power law function of height, resulting in a generalized Boussinesq equation. To gauge the effects neglected by this model, 3D Navier-Stokes multiphase flow simulations are conducted for the same geometry. Parameter variations are performed to assess the nature of errors induced by applying the 1D model to a realistic scenario, where the initial and boundary conditions can not be matched exactly. Numerical simulations reveal an undercutting effect observed in laboratory experiments, but not modeled when the Dupuit-Forchheimer assumption is applied. By selectively controlling the effects placed on the free surface in 3D simulations, we are able to demonstrate that free surface slope is the primary driver of the undercutting effect. A consistent lag and overshoot flow regime is observed in the 3D simulations as compared to the 1D model, based on the choice of initial condition. This implies that the undercutting effect is partially induced by the initial condition. Additionally, the presented numerical evidence shows that some of the flow behavior unaccounted for in the 1D model scales with the 1D model parameters.

Residence time distributions in non-uniform aquifer recharge and thickness conditions – An analytical approach based on the assumption of Dupuit-Forchheimer

Residence Times in aquifers result from their internal structure, from the hydrodynamic transport processes and from the recharge conditions to which they are exposed. Beyond the already known residence time distributions (RTD) for either constant aquifer thickness and/or uniform recharge, we investigate the effect of both distributed aquifer thickness and distributed recharge. We develop a semi-analytical approximation of the RTD for generic trapezoidal aquifers exposed to linearly-variable recharges. The solution is derived for a homogeneous 2D cross-sectional aquifer in steady-state conditions following the Dupuit-Forchheimer assumption according to which the vertical head gradients are much smaller than the horizontal head gradients. Close agreement with 2D numerical simulations demonstrates the relevance of the Dupuit-Forchheimer assumption to estimate RTDs as long as the aquifer thickness remains an order of magnitude smaller than the aquifer length. At equivalent aquifer volume, geometrical structure and recharge conditions result in non-trivial and complex RTD shapes that may be uniform, Gamma-like, power-law-like shapes as well as any intermediary shapes. The variety of RTD shapes encountered show the need to systematically include the aquifer structure and recharge conditions in the assessment of RTDs and for their subsequent use for problematics related to water quality. The semi-analytical approximation can be further used in a variety of aquifer systems in complement with other existing solutions as a Lumped Parameter Model for RTDs.

Analytical Solution of Linearized Boussinesq Equation for Unsteady Seepage Flow in Ditch-Drain Sloping Aquifer

Analytical Solution of Linearized Boussinesq Equation for Unsteady Seepage Flow in Ditch-Drain Sloping Aquifer This paper presents a generalized solution of linearized Boussinesq equation to predict unsteady flow of subsurface seepage induced by localized recharge over sloping ditch-drain aquifer. The mathematical model is based on Boussinesq equation with Dupuit-Forchheimer assumption in which the spatial location of the recharge basin is treated as an additional parameter. Analytical expressions for water head distribution in the aquifer and discharge rate into the ditches are obtained by solving the governing flow equation with eigenvalue-eigenfunction method. Upward and zero sloping cases are deduced from the main results by appropriately adjusting the slope parameter. A numerical example illustrates the combined influence of bed slope, recharge rate and spatial coordinate of the recharge basin on the dynamic profiles of phreatic surface.

Adoption of Extended Dupuit–Forchheimer Assumptions to Non-Darcy Flow Problems

Seepage flow is one of the most practical problems in the field of civil, geology, hydrology and agricultural engineering. Nonlinear flow through permeable deposits takes place when the mean size of the aggregates is coarse. In this paper, an analytical solution is presented for unconfined fully developed turbulent flow under “extended Dupuit–Forchheimer (D–F)” assumptions. A dimensionless form of the solution is therefore presented, and then to show the accuracy of the solution, its results are compared with new experimental data and a Darcy-based solution. According to the results of the present study, it may be concluded that the proposed analytical solution could be used to analyze satisfactorily water surface profiles and normal depth in such sloping permeable porous media under fully developed turbulent flow condition.

Simulating water flow in variably saturated soils: a comparison of a 3D model with approximation-based formulations

In hydrological models, variably saturated flow is often described using the Richards equation, either in a fully three-dimensional (3D) implementation or using a quasi-3D framework based on the 1D Richards equation for vertical flow and a flow-approximation for the other two dimensions. However, it is unclear in which configuration or under which boundary conditions these approximations can produce adequate estimates. In this study, two formulations with a quasi-3D approach are benchmarked against a fully 3D model (HYDRUS-3D). The formulations are: the Real-time Integrated Basin Simulator + VEGetation Generator for Interactive Evolution (tRIBS + VEGGIE) model that uses the Dupuit–Forchheimer assumption and the Tethys & Chloris (T&C) model that implements the kinematic approach. Effects of domain slope, hillslope size, event size and initial moisture conditions on simulated runoff and soil moisture dynamics are examined in event-based simulations at the hillslope scale. The Dupuit–Forchheimer assumption (tRIBS-VEGGIE) produces deviations from the HYDRUS-3D solutions only for simulations with initially dry soil. Using the kinematic approach (T&C) results in deviations from the 3D solution primarily for the small hillslope domain in combination with a gentle slope angle. This applies especially to the partition between subsurface and surface runoff production, with T&C being biased towards the latter. For all other cases investigated, the simpler formulations provide reasonable approximations of the 3D model.

Groundwater response to tidal fluctuation and rainfall in a coastal aquifer

Summary This paper presents an analytical solution generated by linearizing the one-dimensional non-linear Boussinesq equation to characterize the variation of groundwater level induced by tidal waves and rainfall in a coastal unconfined sloping aquifer. The area of the coastal aquifer is divided into 2 zones – the coastal zone of rainfall and the inland zone of non-rainfall. The derivation of the solution is based on the Dupuit–Forchheimer assumptions of a wide range of problems in unconfined groundwater flow and the analytical solutions of coastal and inland water tables of both zones are obtained. The results indicate that the effects of the bottom slope of the aquifer and the beach slope on the groundwater fluctuation are substantial in the unconfined aquifer. Large rainfall intensity with tidal waves will enhance the fluctuation significantly, but less effect on the fluctuation is found for the cases of steeper sloping bottom under the same rainfall condition.

Nonlinear Inversion of an Unconfined Aquifer: Simultaneous Estimation of Heterogeneous Hydraulic Conductivities, Recharge Rates, and Boundary Conditions

A new inverse method is developed to simultaneously estimate heterogeneous hydraulic conductivities, source/sink rates, and unknown boundary conditions for steady-state flow in an unconfined aquifer. Unlike objective function-based techniques, the new method does not optimize any data-model misfits. Instead, its formulation is developed by honoring physical flow principles as well as observation data at sampled locations. Under the Dupuit–Forchheimer assumption of negligible vertical flow, accuracy and stability of the new method are demonstrated using synthetic heterogeneous aquifer problems with increasingly complex flow: (1) aquifer domains without source/sink effects; (2) aquifer domains with a point sink (a pumping well operating under a constant discharge rate); (3) aquifer domains with constant or spatially variable recharge; (4) aquifer domains with constant or spatially variable recharge undergoing single-well pumping. For all problems, inversion yields stable solutions under increasing head measurement errors (up to $$\pm $$ 10 % of the total head variation in a problem), although accuracy of the estimated parameters degrades with the increasing errors. The inverse method is successfully tested on problems with high hydraulic conductivity contrasts—up to 10,000 times between the maximum and minimum values. In inverting several heterogeneous problems, if the aquifer is assumed homogeneous with a constant recharge rate, physically meaningful parameter estimates (i.e., equivalent conductivities and mean recharge rates) can be determined. Alternatively, if the inverse parameterization contains spurious parameters, inversion can identify such parameters, while the simultaneous estimation of non-spurious parameters is not affected. The method obviates the well-known issues associated with model “structure errors”, when inverse parameterization either simplifies or complexifies the true parameter field.

A New Package in MODFLOW to Simulate Unconfined Groundwater Flow in Sloping Aquifers

The nonhorizontal-model-layer (NHML) grid system is more accurate than the horizontal-model-layer grid system to describe groundwater flow in an unconfined sloping aquifer on the basis of MODFLOW-2000. However, the finite-difference scheme of NHML was based on the Dupuit-Forchheimer assumption that the streamlines were horizontal, which was acceptable for slope less than 0.10. In this study, we presented a new finite-difference scheme of NHML based on the Boussinesq assumption and developed a new package SLOPE which was incorporated into MODFLOW-2000 to become the MODFLOW-SP model. The accuracy of MODFLOW-SP was tested against solution of Mac Cormack (1969). The differences between the solutions of MODFLOW-2000 and MODFLOW-SP were nearly negligible when the slope was less than 0.27, and they were noticeable during the transient flow stage and vanished in steady state when the slope increased above 0.27. We established a model considering the vertical flow using COMSOL Multiphysics to test the robustness of constrains used in MODFLOW-SP. The results showed that streamlines quickly became parallel with the aquifer base except in the narrow regions near the boundaries when the initial flow was not parallel to the aquifer base. MODFLOW-SP can be used to predict the hydraulic head of an unconfined aquifer along the profile perpendicular to the aquifer base when the slope was smaller than 0.50. The errors associated with constrains used in MODFLOW-SP were small but noticeable when the slope increased to 0.75, and became significant for the slope of 1.0.

Groundwater Flow in Sloping Aquifer under Localized Transient Recharge: Analytical Study

AbstractThis paper analyzes the groundwater flow system in an unconfined downward-sloping aquifer of semi-infinite extent in response to localized transient recharge. The aquifer is in contact with a water body of constant water level at one end and receives localized transient recharge from a recharge basin of finite width. The mathematical model is based on the Boussinesq equation with Dupuit-Forchheimer assumption, in which the spatial coordinate of the recharge basin is treated as an additional parameter. Analytical solutions of the linearized Boussinesq equation are obtained using the Laplace transform technique by dividing the aquifer in a three-zone system containing both Dirichlet and Neumann boundary conditions at the hypothetical interfaces. Upward- and zero-sloping cases are deduced from the main results by appropriately adjusting the slope parameter. To assess the validity and efficiency of the linearization method, the nonlinear Boussinesq equation is also solved using a fully explicit predic...

Boussinesq equation-based model for flow in drainage layer of highway

In order to study the water flow in the drainage layer of highway under steady-state condition, one-dimensional (1D) Boussinesq equation-based model with Dupuit-Forchheimer assumption was established and the semi-analytical solutions to predict the water-table height were presented. In order to validate the model, a two-dimensional (2D) saturated flow model based on the Laplace equation was applied for the purpose of the model comparison. The water-table elevations predicted by 1D Boussinesq equation-based model and 2D Laplace equation-based model match each other well, which indicates that the horizontal flow in drainage layer is dominated. Also, it validates the 1D Boussinesq equation-based model which can be applied to predict the water-table elevation in drainage layer. Further, the analysis was conducted to examine the effect of infiltration rate, hydraulic conductivity and slope of drainage layer on the water-table elevation. The results show that water-table falls down as the ratio of Is to K decreases and the slope increases. If the aquifer becomes confined by the top of drainage layer due to the larger ratio of Is to K or smaller slope, the solution presented in this work can also be applied to approximate the water-table elevation in unconfined sub-section as well as hydraulic head in the confined sub-section.

Predicting Collector Well Yields with MODFLOW

Groundwater flow models are commonly used to design new wells and wellfields. As the spatial scale of the problem is large and much local-scale detail is not needed, modelers often utilize two-dimensional (2D) or quasi three-dimensional models based on the Dupuit-Forchheimer assumption. Dupuit models offer a robust set of tools for simulating regional groundwater flow including interactions with surface waters, the potential for well interference, and varying aquifer properties and recharge rates. However, given an assumed operating water level or drawdown at a well screen, Dupuit models systematically overpredict well yields. For design purposes, this discrepancy is unacceptable, and a method for predicting accurate well yields is needed. While published methods exist for vertical wells, little guidance is available for predicting yields in horizontal screens or collector wells. In plan view, a horizontal screen has a linear geometry, and will likely extend over several neighboring cells that may not align with rows or columns in a numerical model. Furthermore, the model must account for the effects of converging three-dimensional (3D) flow to the well screens and hydraulic interference among the well screens; these all depend on the design of a specific well. This paper presents a new method for simulating the yield of angled or horizontal well screens in numerical groundwater flow models, specifically using the USGS code MODFLOW. The new method is compared to a detailed, 3D analytic element model of a collector well in a field of uniform flow.

Numerical Simulation of Groundwater Table Falling in Horizontal and Sloping Aquifers by Differential Quadrature Method (DQM)

Since a nonlinear partial differential equation was developed by Boussinesq based on Darcy’s law and the Dupuit-Forchheimer assumption, it has played an essential role in the development of simulation models for the solution of various flow problems in porous media. To solve Boussinesq’s equation, both analytical and numerical solutions, such as finite difference (FD) or finite element (FE), have been sought. The differential quadrature method (DQM) is a new numerical method frequently used by researchers to solve partial differential equations. In this paper, DQMs have been coupled with explicit, implicit, and Crank-Nicholson FD and applied to solve Boussinesq’s equation for a drainage problem. This case has been solved previously both analytically and numerically, including by DQM, by other researchers. Those who had employed DQM had linearized Boussinesq’s equation first and solved the linearized form implicitly. In this work, the nonlinear form of Boussinesq’s equation has been solved using three models based on DQM for solving the dimensionless form and on one approach for solving the dimensional form of Boussinesq’s equation. The obtained results and their degree of accuracy are compared with available experimental and numerical data found in the literature, and on that basis, it is concluded that DQM generates accurate results, is very easy to formulate and operate, does not need large mesh size, and is very time efficient.

Coupled Surface-Subsurface Model for Simulating Drainage from Permeable Friction Course Highways

Permeable friction course (PFC) is a porous asphalt pavement placed on top of a regular impermeable roadway. Under small rainfall intensities, drainage is contained within the PFC layer; but under higher rainfall intensities, drainage occurs both within and on top of the porous pavement. A computer model—the permeable friction course drainage code (Perfcode)—is developed to study this two-dimensional unsteady drainage process. Given a hyetograph, geometric information regarding the roadway layout, and hydraulic properties of the PFC media, the model predicts the variation of water depth within and on top of the PFC layer through time. The porous layer is treated as an unconfined aquifer using Darcy’s law and the Dupuit-Forchheimer assumptions. Surface flow is modeled using the diffusion wave approximation to the Saint-Venant equations. A mass balance approach is used to couple surface and subsurface phases. Straight and curved roadway geometries are accommodated via a curvilinear grid. The model is valida...

Estimating hydraulic conductivity for the Martian subsurface based on drainage patterns — A case study in the Mare Tyrrhenum Quadrangle

Hydraulic conductivity K, as the coefficient of proportionality in Darcy's Law, is critical in understanding the past Martian hydrologic cycle, climate, and landform evolution. However, K and its spatial variability on Mars are thus far poorly constrained due to lack of accessibility. Using an innovative method based on surface drainage dissection patterns, which has been successfully tested in the Oregon Cascades on Earth, we estimated K in the Mare Tyrrhenum Quadrangle on Mars. The basic assumption is that under long-term dynamic equilibrium conditions, the overall dissection pattern in a watershed as reflected in drainage density is controlled by the interplay among surface runoff, groundwater flow, topography, and aquifer properties. K is calculated following a derivative of Darcy's Law under DuPuit–Forchheimer assumptions with drainage density D, valley depth d, recharge rate R, and aquifer thickness H as inputs. The results are consistent with the published K values and reveal spatial variability.

Estimating hydraulic conductivity from drainage patterns—A case study in the Oregon Cascades

This study introduces a new method for estimating hydraulic conductivity based on the concept of effective groundwater drainage length and DuPuit-Forchheimer assumptions. The effective groundwater drainage length is related to the surface drainage dissection patterns (as expressed in drainage density) forming over long periods of time. Application of the new method to the Oregon Cascades yielded hydraulic conductivity values similar to those documented in the literature. This method represents an effective and efficient way of estimating hydraulic conductivity for regions where the interplay among surface drainage, groundwater, and topography has established a steady-state dynamic equilibrium. It also provides a theoretically sound approach for extrapolating limited local measurements to a large region and revealing the spatial variation of hydraulic conductivity.