The Interface Between Fresh and Salt Groundwater in Horizontal Aquifers: The Dupuit–Forchheimer Approximation Revisited

Abstract

We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75-82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit-Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit-Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit-Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.

Modelling and optimisation of water management in sloping coastal aquifers with seepage, extraction and recharge

“…If we ignore terms which are of order O( ) and smaller, equation ( 13) is reduced to the classical Dupuit-Forchheimer approximation [7,8,9,10,11,12], which we integrate (using ( 5)) to get p = ĥ − ẑ. ( 14)…”

“…In the past three decades, there have been many studies (both laboratory experiments and mathematical modelling) on water management in coastal aquifers. The most notable mathematical strategy for modelling the aquifer water table is the Dupuit-Forchheimer approximation [7,8,9,10,11,12], which neglects the pressure variation in the vertical direction and considers the aquifer to be long and thin. In this approximation the Darcy equations for flow in porous media are integrated across the aquifer depth to obtain a nonlinear partial differential equation for the water table height.…”

“…Strack uses the method of images to find an analytical solution, assuming the water is recharged and extracted at point sources and sinks, respectively. In [21] a streamfunction-vortex formulation is used instead which models the sharp interface with a distribution of vortices [9]. Other sharp-interface models use conformal mapping techniques in the hodograph plane [22,23,24].…”

Existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow in asymptotically flat porous media

“…In flat water aquifers, this assumption is called Dupuit-Forchheimer approximation. For example, it is utilized in [13] to derive a nonlocal differential equation that describes the movement of a sharp interface between fresh and salt groundwater.…”

Analytical Solution for Interface Flow to a Sink With an Upconed Saline Water Lens: Strack's Regimes Revisited

“…A considerable progress has been achieved in combination of analytical and numerical modeling of 2‐D and 3‐D steady and transient intrusion‐upconing problems, with abrupt and dispersing interfaces, including flow regimes when both fresh and saline liquids are mobile (see e.g., Bakker & Schaars, ; Eeman, ; Haitjema, ; Houben & Post, ; Van Duijn & Schotting, ; Zhang & Hocking, ; Zhang et al, ).…”