On the solution of transient free-surface flow problems in porous media by finite-difference methods

doi:10.1016/0022-1694(71)90005-9

Journal of Hydrology

Volume 12, Issue 3, February 1971, Pages 177-210

https://doi.org/10.1016/0022-1694(71)90005-9 Get rights and content

Abstract

Numerical solutions are obtained for a few initial-boundary-value problems in free-surface, saturated liquid flow through porous media. The “exact” differential equations governing the problems have been approximated by finite-difference equations, and the resulting system of algebraic equations is solved by using an automatic digital computer. Only two-dimensional, homogeneous and isotropic flow models — including infiltration — are dealt with, but in principle there is little difficulty in extending the models to nonhomogeneous and anisotropic media. Though also possible in principle, an extension to three dimensions will demand a drastic increase in the necessary storage space of the computer.

The question of computational stability of the difference schemes adopted is considered.

Good agreement with some results of other calculations is found for the two distinct models that have been studied — one “ditch drainage” and one earth dam model.

References (22)

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The classical, two dimensional, dam problem was solved analytically and independently by Davison in 1932 and Hamel in 1934 and involved double integrals. In 1938 Polubarinova-Kochina produced solutions involving single integrals and in 1940 showed that the Davison-Hamel solutions could be reduced to ones obtained by her method. Missing from the conversion was a linkage between scale constants which is given in this paper. Initial calculations by these authors were difficult because of evaluations of elliptic integrals and numerical quadratures. As a free boundary problem it is now a test case for a multitude of numerical methods. Surveying analytical and numerical approaches, it is found that both are limited in accuracies and are overwhelmingly concerned with the location of the free boundary and its seepage point. Analytical expressions are here given for the gradients near the three critical points of free boundary entry and exit and the intersection of seepage face and tailwater. New numerical results to six decimal places are given for tables and contours on and within the dam boundary for potential and stream functions and orthogonal potential gradients.
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Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.
Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face
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Citation Excerpt :
For some simple configurations, approximate methods to compute the seepage face height can be found in [28–33]. In the field of numerical models, the seepage face problem is analyzed, among others, in [34–42]. Within the framework of extended Boussinesq equations (in the above-mentioned meaning), this note proposes a new set of one-dimensional model equations including vertical effects and seepage face.
This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.
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A groundwater investigation including several pumping tests has been carried out by Thames Water Utilities Limited (TWUL) to improve the understanding of the distribution of hydraulic properties of the Chalk in the Swanscombe area of Kent in south-eastern England. The pumping test behaviour is complicated by: the fractured condition of the Chalk, simultaneous pumping from adjacent boreholes, and variable pumping rates during the test. In addition, the groundwater flow system is complicated by quarrying of the Chalk. Analytical solutions for pumping test analysis fail to represent these complex flow processes and cannot reproduce the observed time-drawdown curves. A layered cylindrical grid numerical model has been applied to the results of the Swanscombe pumping test. This model can represent the heterogeneity of the aquifer and the detailed flow processes close to the abstraction borehole such as well storage, seepage face and well losses. It also includes a numerical representation of the moving water-table using a grid that deforms to eliminate numerical instabilities. The analyses of the test results demonstrate that they are significantly influenced by fracture flow, which needs to be included to improve the simulation of the groundwater system; not withstanding this, the layered cylindrical grid numerical model reproduced many of the features in observed time-drawdown, which allowed an assessment of the hydraulic characteristics of the aquifer as well as the investigation of the impact of quarries on the test results. This has demonstrated that the numerical model is a powerful tool that can be used to analyse complex pumping tests and aid to improvement of the conceptual understanding of a groundwater system.
Drainage of recharge to symmetrically located downstream boundaries with special reference to seepage faces
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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.
Nonlinear programming approach for a transient free boundary flow problem
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On the solution of transient free-surface flow problems in porous media by finite-difference methods

Abstract

Theory of ground water movement

Free surface flow in homogeneous porous medium

Trans. Amer. Soc. Civil Engrs.

Linearized solution of unsteady deep flow towards an array of horizontal drains

J. Geophys. Res.

Physical principles of water percolation and seepage

Second order linearized theory of free-surface flow in porous media

La Houille Blanche

Linearized solutions of free-surface groundwater flow with uniform recharge

J. Geophys. Res.

Initial motion problem in porous media

The falling water table in tile and ditch drainage

Water-table recession in tile-drained land

J. Geophys. Res.

Theory of infiltration

Advances in Hydroscience

Relaxation methods