Geology Reference
In-Depth Information
Flow Rule
Time-dependant analysis requires assumptions to be made concerning the
form of fluid flow within the fracture network. The general form of fluid
flow assumed in each fracture is based on an analytical solution, known as
the “cubic law”, for fluid flow between approximately parallel surfaces
(Witherspoon et al., 1980). For ground water purposes at shallow depth, in
unconfined or semi-confined situations, a linear form is used. Transmissivity
is proportional to the permeability and to the fracture thickness. It is also
assumed that fractures are filled with a porous material of storativity
S
.
However transmissivities and storativities are modified to account for the
effects of pressure changes between fracture surfaces when the fracture
becomes desaturated. Thus, in FRACAS, the volumetric flux (m
3
s
-1
) in the
x-direction through a length
l
(m) of a fracture has the form:
3
-
aglFdh
dx
Q
=
(1)
12*
where
a
is the hydraulic aperture (m) of the fracture,
g
is acceleration due
to gravity (ms
-2
), * is the kinematic viscosity (m
2
s
-1
),
dh
/
dx
is the hydraulic
head gradient driving flow through the fracture, and
F
is a dimensionless
function dependent on effective pressure.
Analytical and empirical expressions for
F
, in which
F
decreases as the
effective pressure becomes negative (
F
= 1 under saturated conditions) are
presented in the literature. As an example we adopt the Van Genuchten
formalism: the water content
+
and the fluid pressure
,
are linked by equation
(2), where the residual water content
and
k
are adjustable parameters
to be calibrated. +
s
is the water content at the saturation (, = 0) and l = 1-
1/k. In this case the
F
factor is derived as a power function of the water
content, according to the Brooks and Corey formulation (3).
+
r
,
++
1
1( )
r
+
e
=
(2)
++
k
,
s
r
!
"
F
(,)= +
e
v
(3)
The FRACAS geometry underlying the fluid flow law is illustrated here
below. For this geometry, the volumetric flux from the centre of fracture
i
to the centre of fracture
j
may be approximated by equation (4), where the
geometric mean fracture hydraulic conductivity
k
ij
(m
3
s
-1
) is defined as:
hh
kkLL
(
)
i
j
i
j
i
j
Q
ij
=
k
,
k
(4)
ij
ij
LL
kLLk
i
j
j
i
j
i
in which, based on equation (1), the fracture hydraulic conductivities are
defined as:
