Environmental Engineering Reference
In-Depth Information
In order to compute piezometric head from the potential, which is obtained as
solution of the potential equation, the (
14.13
)havetoberesolvedfor
h
. The result is:
= KðÞ
1
2
K H
2
'ðx; yÞþ
'
0
for the confined aquifer
p
2
hðx; yÞ¼
(14.15)
ð
'ðx; yÞ'
0
Þ=K
for the unconfined aquifer
Equations (11.16) and (11.17) are valid for porous media flow independent of the
number of spatial coordinates, i.e. in 1D, 2D and 3D. Under certain conditions these
may be reformulated as potential equations. From (11.6) can be derived that in case
of constant conductivity
K
the potential equation is valid for
' ¼ K h
in model
regions without sinks or sources. One may also use the pressure formulation (11.7)
to obtain a connection between potential and pressure:
k
m
r p rgz
' ¼
ð
Þ
(14.16)
14.3 Example: Groundwater Baseflow and Well
In the following example the formulae, derived for aquifers in the previous sub-
chapter, are applied to a system of wells in an aquifer. Input values are typical for
hydro geological set-ups. There are aquifer thickness, hydraulic conductivity and
baseflow in both coordinate directions. Moreover, there is a reference value for
piezometric head, which is to be valid at a very specific position of the model region.
Well coordinates and pumping rate are to be specified as well as the extension of
the model region:
% Baseflow
H = 5.;
% thickness [L]
h0 = 5.5;
% reference piezometric head [L]
K = 5.e-5;
% hydraulic conductivity [L/T]
Qx0 = 1.e-6;
% baseflow in x-direction [L^2/T]
Qy0 = 0;
% baseflow in y-direction [L^2/T]
% Well
x0 = 100;
% x-coordinate well position [L]
y0 = 0;
% y-coordinate well position [L]
Q = 1.e-4;
% pumping / recharge rate [L^3/T]
% Mesh
xmin = 0; xmax = 200;% min./max.
x-position of mesh [L]
ymin = -100; ymax = 100;% min./ max.
y-position of mesh [L]
% Reference point position in mesh
iref = 1; jref = 1;
xvec = linspace(xmin,xmax,100);
yvec = linspace(ymin,ymax,100);
[x,y] = meshgrid (xvec,yvec);
% mesh
r = sqrt((x-x0).*(x-x0)+(y-y0).*(y-y0));
% distances to well
phi = -Qx0*x + Qy0*y + (Q/(2*pi))*log(r);
% potential
phi0 = -phi(iref,jref) + K*H*h0 - 0.5*K*H*H;…
% reference potential
