30
Geotechnical News • September 2012
GROUNDWATER
Influence of element size in numerical studies of seepage:
Unsaturated zones, steady-state
Robert P. Chapuis
Between the water table and the
ground surface, there is an unsaturated
zone. Here, the hydraulic conductiv-
ity
K
can vary by several orders of
magnitude over a small suction range,
and thus over short vertical distances.
When using a numerical code, small
elements are needed to accurately
model groundwater seepage and
contaminant migration. This paper
examines the influence of element size
on the numerical solution for a simple
example under steady-state conditions.
It suggests a rule-of-thumb to select
the size or height of finite elements.
Background
A first companion paper (Chapuis
2010) explained what happens with
large meshes and large elements, espe-
cially when trying to study a regional
groundwater problem. A second com-
panion paper (Chapuis 2012) provided
basic rules for assessing the influence
of small details especially in engineer-
ing projects This paper is the third of
a series dealing with the influence of
element size.
The example in this paper is that of
a vertical column under steady-state
conditions. This is a 1D (one-dimen-
sional) problem, which can be solved
using a variety of numerical codes,
but only those solving the complete
conservation equation (Richards
1931), and not a simplified or modi-
fied equation. However, each code has
its own ways of treating the Darcy and
conservation equations, to select ele-
ment sizes and time steps, using more
or less automatic procedures. These
internal features of each numerical
code are not discussed hereafter. The
results of this paper were obtained
using the finite element code Seep/W
(Geo-slope 2003, 2007), which was
shown to give reliable results when
correctly used (Chapuis et al. 2001).
Similar results can be obtained using
any code, although built-in specific
procedures of a code may obscure our
understanding of convergence and
oscillation issues
for unsaturated
seepage.
The exact
numerical study
of unsaturated
seepage is more
difficult than
that of saturated
seepage. This
happens because
seepage equa-
tions involve
the volumetric
water content
θ(u)
and unsatu-
rated hydraulic
conductivity
K(u
)
versus pore water
pressure
u
, which
are described
by highly non-
linear functions.
Of course, a
few numerical
codes still make
simplifications,
and either use
constants for
these functions,
or strongly lin-
earized equations
for highly non-
linear physical
phenomena that
require highly
non-linear equations for their correct
description. To obtain correct solu-
tions (e.g., Chapuis and Dénes 2008;
Chapuis et al. 2005), the mesh must be
refined vertically in the vadose zone
to avoid convergence problems and to
overcome the difficulties associated
Figure 1. Unsaturated functions for volumetric water
content
θ
and hydraulic conductivity K. These functions
are used in the 1D column example.
