Dupuit-Forchheimer Approximation Research Articles

Investigating steady unconfined groundwater flow using Physics Informed Neural Networks

A deep learning technique called Physics Informed Neural Networks (PINNs) is adapted to study steady groundwater flow in unconfined aquifers. This technique utilizes information from underlying physics represented in the form of partial differential equations (PDEs) alongside data obtained from physical observations. In this work, we consider the Dupuit–Boussinesq equation, which is based on the Dupuit–Forchheimer approximation, as well as a recent, more complete model derived by Di Nucci (2018) as underlying models. We then train PINNs on data obtained from steady-state analytical solutions and laboratory based experiments.Using PINNs, we predict phreatic surface profiles given different input flow conditions and recover estimates for the hydraulic conductivity from the experimental observations. We show that PINNs can eliminate the inherent inability of the Dupuit–Boussinesq equation to predict flows with seepage faces. Moreover, the inclusion of physics information from the Di Nucci and Dupuit–Boussinesq models constrains the solution space and produces better predictions than training on data alone. PINNs based predictions are robust and show a little effect from added noise in the training data. Furthermore, we compare the PINNs solutions obtained via the Di Nucci and Dupuit–Boussinesq flow models to examine the effects of higher order flow terms that are included in the Di Nucci formulation but are neglected by the Dupuit–Boussinesq approximation. Lastly, we discuss the effectiveness of using PINNs for examining groundwater flow.

Analytical solutions for flow and advective solute transport in unconfined watershed aquifers with depth-dependent hydraulic conductivity

We present analytical and semi-analytical flow and transport models for 2D (x–z vertical plane) flow and particle transport in stratified rock of sedimentary type or vertically non-uniform bedrock aquifers, which are essentially phreatic. The main focus is on the flow induced by the areal discharge in watershed aquifers in hard-rock formations (e.g. granite, gneiss), the weathering profiles of which comprise two or more stratiform zones differing in hydraulic conductivity, k, which thus can be represented by a decay function of the vertical coordinate, i.e., the depth, k(z). Several models of depth-dependent hydraulic conductivity (stepwise, power-law, exponential) have been tested to show the sensitivity of groundwater flow paths and ages (transit or residence time distributions) to various geological and hydraulic simplifications related to rock heterogeneity. The mathematical setup handles the Dupuit–Forchheimer approximation the validity of which, as well as the suggested solutions, were verified via comparison with numerical results obtained using MODFLOW–PMPATH packages.

Water table rise in urban shallow aquifer with vertically-heterogeneous soils: Girinskii’s potential revisited

ABSTRACT Seepage through an aquifer, the hydraulic conductivity of which varies vertically, is studied using the Dupuit-Forchheimer approximation (Girinskii’s potential) and numerically by FDM-MODFLOW and FEM-HYDRUS-2D. In urban water hydrology, the effect of compaction of the top stratum of an aquifer on the flow rate and the position of the water table (characterized by an integral quantity of the saturated/dry area) is analysed. The area of the saturated zone is evaluated as a function of the conductivity ratio of the two strata, and their thicknesses. Conductivity varying linearly and exponentially is also analytically studied. Boundary-value problems are solved for a linear (nonlinear) ordinary differential equation if the evaporation rate is constant (decreasing exponentially with depth). The MODFLOW and HYDRUS-2D simulations give consistent fields of the piezometric head, pressure head, and Darcian velocity, as well as the water table position, which may have a global minimum (watershed).

Analytical investigation of the exact groundwater divide between rivers beyond the Dupuit–Forchheimer approximation

AbstractThe groundwater divide is a key feature of river basins and significantly influenced by subsurface hydrological processes. For an unconfined aquifer between two parallel rivers or ditches, it has long been defined as the top of the water table based on the Dupuit–Forchheimer approximation. However, the exact groundwater divide is subject to the interface between two local flow systems transporting groundwater to rivers from the infiltration recharge. This study contributes a new analytical model for two‐dimensional groundwater flow between rivers of different water levels. The flownet is delineated in the model to identify groundwater flow systems and the exact groundwater divide. Formulas with two dimensionless parameters are derived to determine the distributed hydraulic head, the top of the water table and the groundwater divide. The locations of the groundwater divide and the top of the water table are not the same. The distance between them in horizontal can reach up to 8.9% of the distance between rivers. Numerical verifications indicate that simplifications in the analytical model do not significantly cause misestimates in the location of the groundwater divide. In contrast, the Dupuit–Forchheimer approximation yields an incorrect water table shape. The new analytical model is applied to investigate groundwater divides in the Loess Plateau, China, with a Monte Carlo simulation process taking into account the uncertainties in the parameters.

A simplified equation of approximate interface profile in stratified coastal aquifers

In this study, we provide a simplified equation with a new concise set of parameters defining the approximate interface profile in stratified coastal aquifers. We derive the solution directly by applying the Dupuit-Forchheimer approximation at the outset, and mathematically prove its sameness for the uniform flow in a vertical plane with the interface profile computed using the comprehensive discharge potential theory by Strack and Ausk (2015). We present a surprising result that interface elevation in the case with a seepage and outflow face at the coastal boundary turns out to be same as in which water table and interface coincides with the coastline. The interface profile is a function of the transmissivity-field centroid elevation and the total transmissivity of the freshwater-flow domain above the interface. We found that the boundary potential at the coast in the discharge-potential theory solution represents the first moment of the transmissivity. We also analyzed the differences in the interface profiles in aquifers with different hydraulic-conductivity contrasts for two sets of boundary conditions—constant head and constant flux. For a given-flux boundary, interface profiles in aquifer configurations with different conductivity contrasts converge in the upper part of the aquifer and diverge at the aquifer bottom, leading to different toe-positions. Conversely, for a given-head boundary, interface profiles in aquifer configurations with different conductivity contrasts significantly differ in the upper part of the aquifer, while converge at the aquifer bottom resulting in the same toe-position. We provide explicit solutions for the toe-position and freshwater discharge in unconfined aquifers, an extension to our previous work for confined aquifers. Insights from the newly identified parameters can also potentially help in reducing the uncertainty in the estimation of the real-world SWI extent and optimize the requirement of the extent of the hydraulic conductivity field characterization.

Modelling of 2-D seepage from aquifer towards stream via clogged bed: The toth-trefftz legacy conjugated

Baseflow-type interaction between a river and adjacent/subjacent aquifer across a thin clogging layer of fine sediments controls the dynamics of surface-pore water resources, quality of both waters, seepage induced erosion of the river bed and other hydrological phenomena. Transient 2D phreatic flow from an unconfined aquifer into a river with a thin low-permeable cake (hydraulic skin) is approximated by a sequence of steady states, each of which assumes the water table to be horizontal. The scalar and vector fields of piezometric head, stream function and Darcian velocity are found from analytical solution of the Dirichlet and Robin (linear combination of the velocity potential and its normal derivative) boundary value problems for the piezometric head (harmonic function). The time-shrinking Tothian “unit basins” are a half-strip, half-plane or rectangle. Stream banks are assumed to be horizontal or vertical segments. Cross-flow from the aquifer into the stream is controlled by the aquifer-skin conductivity ratio and the stages of the river and adjacent aquifer. The head and cross-flux on the interface (Robin's boundary) is shown to vary along this line and therefore even for vertical river banks the Dupuit–Forchheimer approximation is not strictly valid. Numerical simulations in HYDRUS2D are reasonably close to the analytical results. Early-stage drawdown of a rectangular cake due to a sudden drop of the water level on the river side and formation of a seepage face is analysed with potential applications to stability of earth dams.

Analytic solutions for fresh groundwater lenses floating on saline water under desert dunes: The Kunin-Van Der Veer legacy revisited

A fully-saturated lens of steady fresh groundwater floating in a homogeneous and isotropic desert sandy aquifer is analytically studied based on a hydrological model by Kunin and interface solution by Van Der Veer. A static saline groundwater is beneath the lens. A phreatic surface of moving fresh water inside the lens is partially recharged (either naturally or by managed aquifer recharge) from the vadose zone and partially exfiltrates to it. A spatially focused recharge and intensive evapotranspiration preserve a steady downward-upward topology of fresh water motion. In terms of the 1-D Dupuit-Forchheimer approximation in a horizontal in-lens saturated flow a boundary value problem (BVP) for an ODE for the Strack potential is solved. The shape of the water table and, based on the Ghijben-Herzberg assumption, the interface are found. The total volume of the positive-pore pressure water flowing within the lens is evaluated. Constant infiltration and evaporation rates as well as evaporation linearly decreasing with depth of the water table (counted from the ground surface) are considered. The case of 2-D flow is tackled by the Toth model. A triangular analytic element approximates a half of the flow domain and consists of an isobaric side and two no-flow sides. Conformal mapping of this triangle onto a reference plane and solution of the Dirichlet BVP in a half-plane deliver the distribution of infiltration-exfiltration intensity along the water table, total flow rate and locus of the hinge point. A mathematically more cumbersome approximation of the flow domain assumes the water table to be a tilted straight line but the interface to be found as a free boundary. Solution of the corresponding BVP uses a curvilinear triangle in the hodograph plane.

Riesenkampf's vortex solution revisited for 2-D commingling of groundwater in a three-layered aquifer: Vertical-inclined-horizontal seepage in aquitard

Using exact and explicit analytical solutions and MODFLOW simulations we show that downstream of an unconformity (transition from an aquifuge layer to a homogeneous aquitard) groundwater seeps at varying angles with respect to the layering. As a generalization of the Anderson (2003) two-layered composite, a steady, 2-D Darcian flow in a three-layered aquifer is studied. This flow is generated by different inlet piezometric heads in thick upper and lower strata and a cross-flow through an aquitard sandwiched between them. Analytically, a line vortex combined with a dipole at infinity describes commingling between the strata with refraction (continuity of head and normal flux component) along the upper and lower boundaries of the aquitard. The Fourier method by Riesenkampf (1940) gives explicit expressions for the specific discharge vector fields in the three media. MODFLOW models finite lengths composites of rectangular and octagonal shapes. The Dupuit–Forchheimer approximation is illustrated to oversimplify the flow topology.

Corrected interface flow model for seawater intrusion in confined aquifers: relations to the dimensionless parameters of variable-density flow

Sharp-interface (or interface) flow models with Dupuit-Forchheimer approximation are widely used to assess, to first order, an aquifer’s vulnerability to seawater intrusion (SWI) and to evaluate sustainable management options for coastal groundwater resources at the screening level. Recognising that interface flow models overestimate SWI, corrections have been proposed to account for the neglected mixing and also for the outflow through a finite gap. These corrections, however, were introduced in the context of specific studies and may not be generally applicable as proposed. The interface model is revisited, placing its corrections in the context of variable-density flow (VDF) theory, by expressing them in terms of the dimensionless parameters governing VDF in schematised (aspect ratio = thickness/length) homogeneous confined coastal aquifers: the coupling parameter (α), a Peclet number (Pe), and the dispersivities ratio (rα). Interfaces are compared to the 50%-salinity lines of VDF numerical solutions and regression equations are developed for estimating the outflow gap and for correcting the length of the interface (terminating with a blunted edge); the dispersion correction, which modifies the interface curvature, is restated with a variable exponent. The corrections for dispersion and for the interface length appear to be the most effective; an outflow gap is important only at small α values (strong advection relative to vertical flow due to density differences). These concepts are applied successfully to calculate the interface position in the lowermost confined sub-unit of the Coastal Plain aquifer of Israel, as an estimate of SWI.

Vertically integrated flow in stratified aquifers

We present a set of continuous discharge potentials that can be used to determine the vertically integrated flow in stratified aquifers. The method applies to cases where the boundaries are vertical and either the hydraulic head is given, or the boundary is a seepage face, or the integrated discharge is given. The approach is valid for cases of given recharge through the upper and/or lower boundaries of the aquifer. The method is valid for any values of hydraulic conductivity; there are no limitations of the contrast for the method to be valid. The flows in the strata may be either confined or unconfined, and locally perched conditions may exist, but the effect of capillarity is not included. The hydraulic head is determined by applying the Dupuit-Forchheimer approximation. The main advantage of the approach is that very complex conditions in stratified aquifer systems, including locally perched conditions and extremely complex flow systems can be treated in a relatively straight forward approach by considering only the vertically integrated flow rates. The approach is particularly useful for assessing groundwater sustainability, as a model to be constructed prior to developing a fully three-dimensional numerical model.

A correction for Dupuit–Forchheimer interface flow models of seawater intrusion in unconfined coastal aquifers

Interface flow models that use the Dupuit-Forchheimer (DF) approximation for assessing the freshwater lens and the seawater intrusion in coastal aquifers lack representation of the gap through whic ...

Semianalytical series solutions for three-dimensional groundwater-surface water interaction

A semianalytical grid-free series solution method is presented for modeling 3-D steady state free boundary groundwater-surface water exchange in geometrically complex stratified aquifers. Continuous solutions for pressure in the subsurface are determined semianalytically, as is the location of the water table surface. Mass balance is satisfied exactly over the entire domain except along boundaries and interfaces between layers, where errors are shown to be acceptable. The solutions are derived and demonstrated on a number of test cases and the errors are assessed and discussed. This accurate and grid-free scheme can also be a helpful tool for providing insight into lake-aquifer and stream-aquifer interactions. Here it is used to assess the impact of lake sediment geometry and properties on lake-aquifer interactions. Various combinations of lake sediment are considered and the appropriateness of the Dupuit-Forchheimer approximation for simulating lake bottom flux distribution is investigated. In addition, the method is applied to a test problem of surface seepage flows from a complex topographic surface; this test case demonstrated the method's efficacy for simulating physically realistic domains.

Analytical solution for a phreatic groundwater fall: the Riesenkampf and Numerov solutions revisited

Steady, two-dimensional Darcian flow in a homogeneous isotropic unconfined aquifer, bounded from below by a rectangular wedge representing bedrock, is studied by the theory of holomorphic functions. A triangle of the complex potential domain is mapped onto a circular triangle in the hodograph plane with the help of an auxiliary variable. A full potential theory results in closed-form integral representations for the complex potential and complex velocity, from which the flow rate and free surface are calculated using computer algebra built-in functions. This solution, uniformly valid in the whole flow domain, is compared with simpler approximate ones, retrieved from an analytical archive. Two flow zones are distinguished: a tranquil subdomain where the Dupuit-Forchheimer approximation is suitable and a nappe (a subdomain with a rapidly changing Darcian velocity and steep slope of the phreatic surface) where the Numerov or Polubarinova-Kochina solutions, in terms of the full potential model, are available. Approximations in the two zones are conjugated by matching the positions of the water table and the flow rates, which eventually agree well with the obtained comprehensive solution.

Understanding and modelling drainage of railway ballast

Railway track drainage has a significant effect on both the response of a rail to loading and the durability of the ballast and sub-ballast. Although various standards and design methods exist for the surface drainage there is limited information on sub-surface drainage. In particular there is a paucity of analytical or numerical studies of drainage of track foundations. The present study re-evaluates results of a laboratory-based investigation of track drainage. A time-variant numerical model, based on the Dupuit–Forchheimer approximation, was developed to reproduce the experimental results. The model was also used to show the importance of the slope of subgrade and the ballast/sub-ballast permeability ratio in reducing the water retention time in the track support layers. The numerical model has practical applications as it can be used to evaluate groundwater conditions beneath the track for rainfall storm return periods and can thus be used to design track drainage.

Drainage of recharge to symmetrically located downstream boundaries with special reference to seepage faces

Summary The seepage face is an important feature of the drainage process when recharge occurs to a permeable region with lateral outlets. Examples of the formation of a seepage face above the downstream water level include agricultural land drained by open ditches and the ballast beneath railway tracks. The first part of the paper considers vertical section two-dimensional analytical and numerical solutions for recharge to permeable regions above horizontal impermeable bases having two symmetrically positioned downstream boundaries with zero or specified external depths of water. For vertical downstream boundaries with seepage faces above the downstream water level, empirical relationships are developed relating the seepage face height on the downstream boundary to the downstream discharge. The same problems are also analysed using the one-dimensional Dupuit–Forchheimer approximation. Improved estimates of the entire water-table profile are achieved when the downstream boundary head is specified as a function of the downstream discharge using relationships derived from the two-dimensional solutions, instead of the customary assumption that the water table is drawn down to the ditch water level. Finally, time-variant simulations, based on the Dupuit–Forchheimer approximation, are introduced for a single layer and for a system of two horizontal layers having different hydraulic conductivities.

Truncating Cross-Sectional Groundwater Models under Wetlands

Cross-sectional models, which represent two-dimensional flow in the vertical plane, tend to have problematic aspect ratios since the aquifer thickness is often small compared to the lateral extent of the flow domain. For that reason, the model domain is usually limited to the immediate area of interest, for instance the aquifer section underneath a dam. We propose a Cauchy boundary condition to represent flow from remote wetlands that are left out of the truncated model. The resistance to flow inherent to such a boundary depends on the aquifer properties and the resistance to flow through the wetland bottom. While the Cauchy boundary condition is based on the Dupuit-Forchheimer approximation to flow underneath the remote wetlands, the error appears to be negligible (less than 0.6%) for most practical cases, including flow in stratified aquifers. For the case of multiple aquifers underneath the wetlands, the total flow in the truncated model can be a few percent in error, which is typically acceptable for most engineering applications. The approach is illustrated with an application near a levee-borrow canal setting in the Florida Everglades.

Dupuit-Forchheimer Analyses of Steady-State Water-Table Heights due to Accretion in Drained Lands Overlying Undulating Sloping Impermeable Beds

The Dupuit-Forchheimer approximation is used in an investigation of steady-state water-table heights due to accretion in ditch-drained lands resting on an undulating impermeable bed that slopes away from a peak midway between drainage ditches toward a lower level at the drains. Analytical expressions are obtained for the water-table profiles assuming both horizontal flow and flow parallel to the impermeable base. These are compared with numerical results obtained for the Laplace solution of the flow problem that show the equipotentials to be better approximated as being normal to the base than vertical. There is good agreement for large slopes between the water-table heights obtained assuming one-dimensional flow parallel to the sloping base and the two-dimensional numerical results. Poorer agreement is obtained as the slope becomes less with results approaching those given by assuming horizontal flow which always results in underestimates. At small accretion rates agreement is obtained with both Dupuit-Forchheimer analyses and the Laplace solution. The maximum height of the water table above the base decreases and is closer to the drain as the slope increases.

Analytical solution to 2D problem for an anticline-diverted brine flow with a floating hydrocarbon trap

Explicit rigorous solution to a steady-state J. Bear (Dynamics of Fluids in Porous Media. Elsevier, New York, 1972) problem of brine flow in an anticline homogeneous isotropic rock with a gas cap under the anticline crest is obtained by the methods of complex analysis. The stagnant hydrocarbon volume is separated from the subjacent moving brine by a sharp interface, which is a free boundary “hanging” in the formation with the loci of the anticline roof-attachment and roof-detachment points, as well as of an inflexion point, a priori unknown. Mathematically, a conformal mapping of the complex potential strip and integral representation of the Hilbert problem for the inversed complex Darcian velocity are used to obtain the physical coordinate, complex velocity and complex potential as functions of an auxiliary variable. The interfaces are plotted for various incident brine flow rates, angles of dipping anticline flanks and gas pressure. For a gas trap comparisons with the interface calculated by the Dupuit–Forchheimer approximation are carried out.

Vertically Integrated Flows, Discharge Potential, and the Dupuit‐Forchheimer Approximation

We summarize observations made by various researchers regarding the discharge potential, vertically integrated flows, and the Dupuit-Forchheimer approximation. If a regional flow model is formulated in terms of the discharge potential and the boundary conditions can be written in terms of the discharge potential and the boundary discharges, then the discharges are correct throughout the model domain, without embracing any additional simplifying assumptions.

Hydrology and dissolved organic carbon biogeochemistry in an ombrotrophic bog

AbstractAt the Mer Bleue bog, Ontario, Canada, DOC export measured at the basin outflow was −8·3 ± 3·7 g C m−2yr−1, and DOC loading via precipitation was estimated to be 1·5 ± 0·7 g C m−2yr−1. Discharge and DOC export calculated using a Dupuit–Forchheimer approximation compared well (within 1 g C m−2yr−1) to outflow estimates of DOC export, and confirmed that outflow measurements were a suitable proxy for DOC seepage at the peatland margins. DOC export was 12% of the magnitude of the residual carbon sink measured at the peatland. The [DOC] across groundwater transects decreased with depth, and [DOC] sampled below 0·75 m depths remained fairly constant over the study period. However, [DOC] exported through the acrotelm (0 to 0·45 m peat depth) was variable, ranging from 40 mg l−1after snowmelt to 70 mg l−1during the growing season. Fluorescence analysis revealed that exported DOC was ‘allochthonous‐like’, whereas DOC in the catotelm (deeper layers of peat) became more ‘autochthonous‐like’ with depth. A conceptual model is developed to summarize the hydrological processes and controls which affect DOC biogeochemistry at the Mer Bleue. Copyright © 2001 John Wiley & Sons, Ltd.