Dupuit-Forchheimer Theory Research Articles

Second‐order two‐dimensional solution for the drainage of recharge based on Picard's iteration technique: A generalized Dupuit‐Forchheimer equation

Aquifer recharge is one of the most important problems in hydrology from both theoretical and practical points of view. One of the most widely accepted methods to deal with this problem is the use of the Dupuit‐Forchheimer theory. This theory assumes that the water table is almost horizontal, the vertical velocity is zero, and the horizontal velocity is uniform with depth. Surfaces of seepage are not considered. Despite these strong limitations the theory is applied, and success is frequently found in many cases despite its fundamental assumptions being violated. In this work an approximate 2‐D solution to the problem is sought on the basis of Picard's iteration technique, from which a second‐order differential equation for recharge problems is found. On the basis of this solution, a modified, analytical Dupuit‐Forchheimer (DF) ellipse is found which compares favorably with the full 2‐D solution of the problem. The analytical developments of this theory provide a generalized DF theory which permits as an outcome the analytical determination of the surface of seepage.

Steady-state water table height estimations with an improved pseudo-two-dimensional Dupuit-Forchheimer type model

Summary The study of unconfined steady aquifer flow is usually based on either the numerical integration of the Laplace equation or on its analytical solution using the complex variable theory. A further approach that uses Adomian’s method of decomposition yields simple analytical solutions in higher dimensions, does not require linearisation of the free-surface boundary condition and yields the elevation of the seepage face. A common approach is the introduction of simplified one-dimensional models that are often accurate enough for practical applications. However, the water table estimates derived by the so-called Dupuit-Forchheimer theory do not always fulfil the required accuracy. This work improves the Dupuit-Forcheimer hypotheses to obtain more precise results. For this purpose, the stream function of the groundwater flow net is formulated in natural, curvilinear coordinates. Next, an approximate one-dimensional model for the water table height is derived considering Darcy’s law, retaining the curved features of the flow net. The proposed model is a higher order Dupuit-Forchheimer type approach, which was favourably compared with 2D results for Polubarinova-Kochina’s rectangular dam problem and the drainage to symmetrically located ditches under steady-state conditions.

Slumping of groundwater mounds: revisiting the Polubarinova-Kochina theory

Decay or rise of the water table from a disturbed (mound or trough) position to a quiescent flat state is studied by a linear potential theory that does not rely on the Dupuit-Forchheimer vertical averaging but is a solution to the full Laplace equation. We consider an unconfined aquifer of high (infinite) thickness disturbed by a linear or point hydrodynamic dipole and assemblies of dipoles, which generate two- and three-dimensional seepage. Hydrologically, the dipoles mimic a channel (or circular-recharge basin), which generates the mound. The dipole ascends (descends) and the corresponding free surface, on which the isobaricity and kinematic conditions hold, slumps. A solvability condition, which stipulates no singularities in the seepage domain, is explicitly presented. The mound signal is defined as the time peak of the water table at any piezometer located away from the original recharge area. The flow net and isotachs prove the Bouwer caveat that the Dupuit-Forchheimer theory is specious if applied to high-thickness aquifers accommodating mounds originating from short infiltration events. The analytical value of the water table peak and the time of its arrival are compared with piezometric observations in recharge experiments conducted in a coastal aquifer of the United Arab Emirates, where the hydraulic conductivity is assessed from hydrographs. The inversely determined hydraulic conductivity fits well with those found from infiltration double-ring experiments and MODFLOW simulation.

Improving the Dupuit–Forchheimer groundwater free surface approximation

For steady two-dimensional free surface flow over a horizontal impervious base, the Dupuit–Forchheimer theory assumes that the vertical component of velocity is zero, even for non-zero accretion rate at the free surface. This is improved by assuming that the vertical velocity component is zero at the base, and is proportional to height above the base. This requires the piezometric head to depend linearly on the square of the height, and the two parameters in this relation can be fitted to the two boundary conditions at the free surface, to give an expression for the free surface slope in terms of accretion, free surface height, and the pressure integral. For problems in which the pressure integral is known explicitly, this first order of ordinary differential equation for the free surface height can be solved numerically. The solutions are more accurate than the Dupuit–Forchheimer expressions for the free surface, and much easier to calculate than numerical solutions to the full two-dimensional problem. Four examples are given, leading to some simple analytical approximations for quantities of interest.

Dupuit-Forchheimer theories for the shape of groundwater recharge mounds

The nonlinear Dupuit-Forchheimer theory and two linearized versions are considered as to their applicability for giving accurate results on recharge mound shapes beneath strip and square basins. Results on mound rises from the approximate theories are compared with those from the nonlinear theory to determine criteria for the use of the approximate theories.

Ecohydrology of Scottish peatlands

ABSTRACTMires are terrestrial ecosystems which conserve organic production in the form of peat because their soils are waterlogged. Scotland's damp climate makes it rich in mires, with fens, raised mires and blanket mires occurring widely. Intact examples are mostly treeless due to wind exposure. They have two functional layers of soil: a shallow acrotelm overlying the catotelm or peat deposit proper. The acrotelm is the main region of diagenesis and contains the water table, which lies close to the surface for most of the time. Evapotranspiration depends on water table depth and surface vegetation. In the catotelm, water transmission is described by Darcy's law with sufficient exactness to allow prediction of the water table profile. An analysis of seepage using Dupuit–Forchheimer theory predicts a hemi-elliptical profile whose curvature is governed by the water balance in dry years and which, in its turn, closely controls the overall shape of the mire. Water flow is more rapid in the acrotelm, where it creates a regular series of different soil physical regimes which are linked with distinct combinations of surface topography and vegetation arranged sequentially from the centre of a mire to its edge. Such regular arrangements are most clearly seen in the Flowe Country of E Sutherland and Caithness. Acrotelm structure also permits variation in flow rate with depth, so that the amplitude of water table oscillations is narrowly confined and the mire thus protected against both desiccation and sheet flow.

Steady state perched groundwater mounds on thick sublayers

Perched mounds that develop beneath a strip recharge basin are considered using the potential theory for a saturated flow. The mounds are assumed to develop upon an aquitard or sublayer whose thickness is large enough so that the vertical velocity at the base of the mound does not vary with distance from the centerline of the basin. A finite difference technique was used to solve the potential theory, and 20 mound profiles were determined for K/KL = 10, 50, 100, 500 and R/K = 0.2, 0.35, 0.50, 0.65, 0.80. K/KL is the ratio of the permeabilities, and R is the recharge rate. These profiles are compared to those based on the approximate Dupuit‐Forchheimer (DF) theory, and a criterion for the range of validity of the DF theory for predicting the maximum mount thickness H0 is derived. It is found that for a sufficiently large value of K/KL, which depends on R/K and the desired accuracy, the DF theory is adequate. For smaller values of K/KL the potential theory must be used. Equipotential lines and velocity distributions are presented for a typical case where the potential and DF mound profiles are quite different.

Simple Equations For Hooghoudt Equivalent Depth

Due to its simplicity and relative accuracy as compared with other more elaborate and hence more complicated methods, the Hooghoudt solution to the drainage design problem is widely used by the practicing engineer. In addition, it is the notion of equivalent depth that Hooghoudt introduced, which makes his analysis more important. The equivalent depth has gained considerable significance in the treatment of the transient drainage problem based on the Dupuit-Forchheimer theory. For calculations on a computer in solving the transient flow problems, the computation of the equivalent depth requires the storage in the computer memory of an appreciable size table (maximum 300 values). For a faster determination of the equivalent depth, and to avoid storage of a table in the computer memory, approximate formulas have been introduced. After the detailed reworking of the solution of Hooghoudt equations, it was considered appropriate to proceed with finding simple algebraic equations which approximate the equivalent depth in a more satisfactory way. This is the subject of the present note.

Shapes of steady state perched groundwater mounds

A potential theory flow solution for the potential function, stream function, and shape of the water table is given for a class of steady state two‐ and three‐dimensional perched groundwater mounds formed under a long rectangular recharge basin or under a circular recharge basin. The solutions are done by a Gram‐Schmidt method and a simple iteration scheme. The mounds are formed in a stratum of conductivity k1 overlying a perching stratum of much lower conductivity k2. Capillary fringe effects are neglected. The recharge rate is R. Potential theory mound heights are compared with those given by the Dupuit‐Forchheimer (DF) theory. For the cases computed (R/k2 = 10, 100, and 500 and k1/k2 = 50,100, 500, 600, 750, and 1000) the DF theory gives apex heights of mounds correct to better than 7% for two‐ dimensional mounds. For three‐dimensional mounds the DF theory gives in one case a mound height that is 69% too low (for R/k2 = 500 and k1/k2 = 600) and in another case a mound height that is 28% too low (for R/k2 = 500 and k1/k2 = 1000). Profiles of the computed mounds are graphed, and examples of use of the graphs in applications are given. Sample flow nets are presented.

Limitations of Dupuit-Forchheimer Theory in Drainage

Limitations of Dupuit-Forchheimer Theory in Drainage

Exact theory for the shape of the free water surface about a well in a semiconfined aquifer

An exact theory is developed for the height of the water table about a well penetrating to a barrier of a semiconfined aquifer. The flow is steady state, and the free surface is a streamline. The problem is a classic one for which the Dupuit-Forchheimer theory happens to give an exact formula for the well discharge but fails to give the proper shape of the free surface. The height of the free surface, at any point, is obtained through an equation in which the height occurs both linearly and in terms of an infinite series of transcendental functions. The equation is solvable by iteration. The theory starts with the development of a potential function. Any capillary fringe is neglected. The potential function is obtained by assuming that a fictitious flow medium occupies the space of the cone of depression. The lower boundary of the fictitious flow medium has the same boundary conditions as the upper boundary of the actual flow medium. The problem is important in well pollution. The theoretical methods may also be applied to the free surface problem when the free surface is not a streamline but receives steady flux.

Mixing-layer flows in a saturated porous medium

The steady two-dimensional seepage flow by gravitational convection, of a fluid of density ρ1surrounded by a fluid of density ρ0(≠ρ1) at rest, leads to a potential problem from which the shape of the interface can be determined. When the two fluids are slightly miscible the interface is replaced by a mixing layer, and it is shown that the first-order Prandtl equations for the flow in the layer possess an exact similarity solution. The profile across the layer is of the same form as the profile of the laminar incompressible half-jet with one fluid at rest, and a formula is obtained for the scale of mixing-layer thickness as a function of distance downstream. Three examples are discussed.(a) Flow of fluid of density ρ1from a horizontal line source. When (ρ1> ρ0) a stagnation point exists above the source, and the fluid ultimately descends in a vertical column of width proportional to the source strength. At the stagnation point, the mixing-layer thickness is finite and is proportional to the square root of the radius of curvature of the interface. At a sufficiently great distance downstream, the thickness increases as the square root of the distance, as in the straight laminar half-jet. These results have been tested experimentally in a Hele-Shaw cell.(b) Symmetrical flow of an ascending column of fluid (ρ1< ρ0) about an obstacle in the form of a finite horizontal strip. The column reforms after passing the obstacle, and the mixing-layer thickness returns to the value corresponding to an unobstructed vertical half-jet. The flow has been produced experimentally.(c) Flow in a lens of fresh water overlying salt water, with inflow due to precipitation, as in a two-dimensional Ghyben-Herzberg lens. Here the potential flow solution is calculated approximately by means of Dupuit-Forchheimer theory. In the steady-state solution the thickness of the mixing layer between fresh and saline water is found to be finite and, as in (a), proportional to the square root of the radius of curvature of the lens.

Seepage of steady rainfall through soil into drains

Theoretical formulas for the height of all points of an arch‐shaped water table are derived for steady rainfall seeping into homogeneous soil drained by tubes or ditches. The soil is underlain by an impermeable layer and the drains are taken to be equally spaced and equally deep. The solution is based on exact mathematical procedures, but utilizes a physical assumption that the height of the water table as measured from a reference plane passing through the lowest points of the water table is small compared to the depth of the impermeable layer. For tube drains running half full and when the impermeable layer is at great depth, the theoretical height H of the water table midway between drain tubes takes the particularly simple form, H = (2sR/πk)ln(2s/πγ) where H is height above the reference plane, 2r is the diameter of the drain tube, 2s is the spacing between drain tubes, R is the rainfall rate, and k the hydraulic conductivity. Flow nets for tube and ditch drainage are presented; a table for making practical computations has been prepared. The theory is compared with field data of Kirkham and De Zeeuw; good agreement was obtained. Finally, the interesting point is brought out that the Dupuit‐Forchheimer (DF) theory emerges from the general theory for the special case of very large ratio of drain spacing to height of water saturated soil and for an essentially flat water table. Thus the analysis presents a rational foundation for DF theory which has heretofore been lacking.

The Seepage of Water Through Dams with Vertical Faces

The method of hodographs, recently developed by Hamel for the analysis of ground-water flow systems containing free surfaces, is applied numerically to six cases of the seepage of water through dams in which the ratio of the inflow fluid head to the width of the dam at the base varied from 2.04 to 1.08. In one case the velocity distribution was computed both along the inflow and outflow faces, as well as along the base of the dam. The pressure and velocity distributions along the bases and the velocity distribution along the inflow faces were found for 5 other cases, in 4 of which the outflow fluid head was zero. The heights of the surfaces of seepage and the total fluxes were calculated for all six cases. These calculations show that the Dupuit-Forchheimer theory of the free surface and velocity distributions in gravity flow systems is definitely wrong, and that the extension of the empirical results of Wyckoff, Botset and Muskat, obtained with radial gravity flow systems, to linear systems is unjustified. The fluxes, however, given by this extension or the Dupuit-Forchheimer theory turn out to be in close agreement with those calculated by the exact theory. The evaluation of the fundamental integral over the argument of the modular elliptic function involved in the calculations is extended beyond the range given by Hamel and Gunther so as to include all real values of the modular elliptic function.