Geoscience Reference
In-Depth Information
q = vn
(4.9)
Local flow of groundwater is related to the local gradient according to Darcy's
law
17
q = -k dh/ds
or
q
x
= -k
h/
x, q
y
= -k
h/
y, q
z
= -k
h/
z
(4.10)
where
k
[m/s] is the hydraulic permeability. By measuring or calculating
dh/ds
everywhere in the soil one can establish the groundwater flow pattern and by using
(4.7) the local pore pressure
u
.
Method of squares
For 2D flow in a homogeneous soil (
k
is constant), Darcy's law can be expressed
by the stream function
or the so-called flow-potential
= k
, (here
is written
in stead of
h
)
q
x
=
k
/
x =
/
x =
/
y
(4.11a)
q
y
=
k
/
y =
/
y =
/
x
(4.11b)
2
Mass conservation
q
x
/
x
q
y
/
y
= 0 leads then to
= 0. Because
is a
scalar, it is single-valued, which implies
(
/
x)/
y =
(
/
y)/
x
. Using
2
(4.11), this yields
are scalar harmonic functions, either of
which can be used to characterise a groundwater flow field.
Equation (4.11) holds for any orthogonal system and by rotating
(x,y)
to
(s,n)
,
where
s
is the direction of the flow vector
q
(Fig 4.1b), one obtains
= 0. Thus,
and
q
s
= q =
/
s =
/
n
and
q
n
=
0
=
/
n =
/
s
(4.12)
Interpretation of
/
n
= 0 implies
= constant along
n
-direction.
/
s
= 0
leads to
= constant along
s
-direction. Therefore, the flow field can be visualised
by a curved orthogonal network with lines
= constant (equipotential lines) and
lines
= constant (flow lines), see Fig 4.1b.
Interpretation of
/
s =
/
n
(in 4.12) leads for finite increments to
,
everywhere in the flow field. Thus, each potential flow field possesses a unique
value
/
s =
/
n
. Choosing locally equal steps, i.e.
s =
n
, gives
=
, everywhere, which only depends on the sketched square flow net, i.e.
the choice of the step
s
. This is the magic of the method of squares.
can be found from boundary conditions. If from one
boundary with known potential head
The unique value
1
(that boundary is an equipotential) there are
17
If the fluid density varies, equilibrium is covered by the law:
q
= -(
/
)(
u
/
s
-
w
), where
is the intrinsic permeability and
the dynamic pore fluid viscosity. For constant density:
q
= -(
w
/
)(
u
/
w
s
-1) = -k
h
/
s
.
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