ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 1, pp. 42–53. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © A. Yu. Belyaev, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 1, pp. 49–64.
Proof of Dupuit’s Assumption for the Free Boundary Problem
in an Inhomogeneous Porous Medium
A. Yu. Belyaev
Water Problems Institute, Russian Academy of Sciences, Moscow, Russia
Received November 17, 2016
Abstract—For free boundary problems describing steady groundwater ﬂows, the asymptotic be-
havior of solutions is studied in the situation where the scale in one of the spatial directions is
much less than that in the other directions. The convergence of solutions to a certain limit is
proved. Properties of the limit solution agree with the assumption known as Dupuit’s assumption in
engineering applications, which customarily serves as a basis for constructing approximate models
of groundwater ﬂows in thin aquifers.
Keywords: variational inequality, Darcy’s law, dam problem.
In the study of stationary groundwater ﬂows, there arises a free boundary problem; in a traditional
setting, this is a second-order linear elliptic equation with redundant boundary conditions in the ﬂow
domain. A part of the boundary of the domain is called the free surface; its position is not known
in advance and has to be determined. On the free surface, both Dirichlet and Neumann boundary
conditions are imposed, while on each of the other parts, only one of these conditions is given. Because
of the extra boundary conditions, the elliptic problem is overdetermined in the domain of the ﬂow, and
the sought-for position of the free surface must ensure the consistency of the system.
The ﬁrst problem of this kind was studied at the physical level of rigor in the middle of the nineteenth
century by J. Dupuit. He considered the steady ﬁltration of water through an earthﬁll dam between two
reservoirs ﬁlled to diﬀerent levels and found an explicit expression for the position of the upper boundary
of the ﬂow zone. In the mathematical literature, the term “dam problem” is often used for all problems of
this class independently of their engineering interpretation.
The explicit expression found by Dupuit is an approximate, rather than exact, solution of the problem.
To construct this approximate solution, Dupuit assumed that, in the situation where the vertical size of
the domain in the dam problem is much less than its horizontal size, the ﬂow zone is located under the
graph z = h(x) of an unknown function h( · ) of the horizontal coordinate x, and the pressure of water
in the ﬂow is distributed hydrostatically, i.e., linearly increases with the depth. Using this assumption,
Dupuit derived an ordinary second-order equation for h( · ) of the form (T (h)h
=0, from which h(x)
is found explicitly. The approach based on Dupuit’s assumption has been extensively applied to simplify
many engineering problems on groundwater ﬂows; in the technical literature, it is called the hydraulic
approximation or Boussinesq’s approximation (for nonstationary problems).
Formally, Dupuit considered a whole family of problems with a small parameter ε determining the
diﬀerence between the vertical and horizontal scales, rather than a single free boundary problem, and his
assumption refers to the asymptotic behavior of solutions as ε → 0. Therefore, the problem of justifying
this assumption reduces to the question on the convergence of solutions. A justiﬁcation of Dupuit’s
assumption at the physical level of rigor can be found in virtually any textbook on groundwater theory
(see, e.g., , ), but there exist no mathematical proofs, except a few examples in which solution have
been found explicitly or almost explicitly. The purpose of this paper is to ﬁll this gap.