Geotechnical News December 2010
37
GROUNDWATER
al, we can keep the large initial mesh
for the volumes where the
h
variations
are small, and generate finer meshes in
the volumes of high
h
variations (high
gradient zones). When examining the
second solution and the zones of high
variations, we may find that some lo-
cal refinements are still needed. Once
we are satisfied with our last refine-
ment and believe that further refine-
ment would add nothing, we should
not be satisfied with our belief, but
prove it. We must prepare a confirma-
tion mesh in which all elements will be
smaller (by half for example) of what
we thought would be our last mesh.
The confirmation mesh should give the
same results (heads, gradients, veloci-
ties, flow rates, etc.) as our last mesh.
If it is the case, then we have the proof
that we have designed and retained a
correct mesh. Note that the computing
time for the verification mesh may be
about four to nine times longer than
the time for our final and correct mesh.
Thus, we should avoid using the veri-
fication mesh for long transient prob-
lems (the computing time for this veri-
fication could take many hours) but use
it first for faster-to-solve steady-state
problems (which could then also serve
as initial conditions for the longer tran-
sient simulations).
Two simple rules to observe are:
(i) The higher the local variations in
h
(anywhere), gradient and
u
(un-
saturated zones), the finer the local
mesh;
(ii) The final solution must be indepen-
dent of the mesh size.
Conclusion
This short paper has examined the
case of a pumping well in an ideal
homogeneous confined aquifer, with
seven grids of 391 to 2672 elements.
The code used here easily converged
for this simple 2D case (immediate
convergence in two steps, relative
error on the modulus of the pore
pressure vector below 10
-6
), due to the
linearity of equations (confined fully
saturated aquifer, steady state) and the
small number of elements. However,
different numerical solutions were
obtained for different grid sizes. In
short, the finer the 2D grid, the more
accurate the numerical solution. It is
also preferable to model all geometric
details as accurately as possible. For
example, a pumping well is better
modelled using 4, 8, 16 or 32 nodes
located on its screen than using a single
node representing the well center (it is
then a well of infinitesimal diameter).
To complement this paper which fo-
cuses on large scale or regional studies,
two forthcoming papers will provide
a few examples for small scale stud-
ies with high local variations of the
hydraulic head, and for cases in which
unsaturated seepage plays a key role.
References
Chapuis, R.P., D. Chenaf, B. Bussière,
M. Aubertin, and R. Crespo, 2001.
Auser’s approach to assess numeri-
cal codes for saturated and unsatu-
rated seepage conditions. Canadian
Geotechnical Journal, 38(5): 1113–
1126.
Darcy, H. 1856. Les fontaines pub-
liques de la ville de Dijon. Victor
Dalmont, Paris.
Geo-slope
International.
2003.
SEEP/W User’s Guide, Version 5.
Calgary, Canada.
Richards, L.A. 1931. Capillary con-
duction of liquids through porous
medium. Physics, 1: 318–333.
Roache, P.J. 2009. Fundamentals of
verification and validation. Hermo-
sa Publishers, Socorro, NM
Roache, P.J. 1994. Perspective: Ameth-
od for uniform reporting of grid re-
finement studies. ASME Journal of
Fluids Engineering, 116: 405–413.
