Environmental Engineering Reference
In-Depth Information
When in formula (
11.15
) the velocity v is replaced with the help of Darcy's Law
(
11.12
), one obtains:
r
K
f
rh ¼ Q
(11.16)
When the conductivity as a material parameter for a porous medium is known,
the differential (
11.16
) has a single unknown variable:
h,
the hydraulic or piezo-
metric head. For given boundary conditions, the differential (
11.16
) has to be
solved for
h
. A generalized version, which is very appropriate for 3D groundwater
flow, results if formulation (
11.14
) is taken into the continuity equation:
k
m
r p rgz
r
ð
Þ ¼ Q
(11.17)
where pressure
p
is the dependent variable.
For unsteady flow the generalization of (
11.16
) is given by:
S
@h
@t
¼r
K
f
rh
(11.18)
where
S
is the storage coefficient, which is dimensionless. It denotes the volume of
water released per unit area of aquifer and per unit drop in head. As far as a confined
aquifer (see Chap. 12) is concerned, the storage coefficient is a function of the
compressive qualities of water and matrix structures of the porous material. In 2D
horizontal models for the unconfined aquifer (see Chap. 12) the storage coefficient
is mainly determined by the change of the water column depth and thus may take
much higher values than in the confined aquifer.
Codes for modeling groundwater flow apply numerical methods to solve (
11.16
)
or (
11.17
) for the steady state, or (
11.18
) for the transient situation (Holzbecher
2002
). In a homogeneous porous medium the (
11.16
) becomes equivalent to the
Poisson equation. With the core version of MATLAB
numerical solutions can be
obtained for one-dimensional cases, and analytical solutions for one- and two-
dimensional cases. This will be presented in the next chapters.
®
11.4 Flow in Unsaturated Porous Media
One speaks of an unsaturated situation if there are two fluids filling the pore space
of a porous medium, a gaseous and an aqueous phase. Soil is the most important
environmental compartment, in which the unsaturated situation is met. The unsatu-
rated layer between the land surface on the top and the groundwater table at the
bottom is the unsaturated or vadose zone.
