Geology Reference
In-Depth Information
DOUBLE POROSITY MODEL USING WARREN
AND ROOT METHOD
Due to the fact that blocks separated by fractures compose hard rock aquifers,
one method allowing describing such behaviour is proposed. This method
derives from the conceptual double porosity model developed by Barenblatt
et al. (1960). The concept (Fig. 4) supposes a confined aquifer constituted
by two media: the fractures, transmissive but poorly capacitive, and the
matrix, capacitive but poorly transmissive. Each medium is characterised by
its hydraulic properties.
K
f
and
S
f
are the permeability and the storage
coefficient of the fracture medium respectively and
K
m
and
S
m
are the
permeability and the storage coefficient of the matrix respectively. The flow,
radial to the pumping well, is only controlled by the transmissivity of fractures
(flow from matrix to pumping well is nil,
K
f
>>
K
m
) and the fractures
network drains the matrix where the flow is stationary (spatial variation of
the hydraulic head is neglected). The expression of the drawndown is:
Q
T
s
(
r
,
t
)=
4
F
(
u*
, ,');
f
Tt
f
u*
=
2
(
S
S
)
r
f
m
K
K
m
r
2
=
and
f
S
f
=
S
S
f
m
= 4
n
(
n
+ 2)/
l
2
shape factor, parameter characteristic of the geometry of the fractures and
aquifer matrix where
n
(dimensionless) is the number of fracture sets (
n
=
1, 2, 3; see Fig. 4),
l
the width of matrix blocs (in metre),
where
is interporosity flow coefficient (dimensionless),
, a factor, for
early time analysis equal to zero and for late time analysis equal to 1/3
(orthogonal system,
n
= 2, 3) or 1 (
n
= 1),
r
a radial distance from the
pumping well,
T
f
the fractures transmissivity (
T
f
=
H
K
f
;
H
: aquifer thickness),
Q
the pumping rate and
t
the time.
This method is well illustrated at pumping well IFP-16. The logarithmic
derivatives of drawdowns in IFP-16 have the typical shape of a double
porosity aquifer (Fig. 5a): (i) well effects and flow trough fractures to pumping
well, (ii) transitional period: the “U” illustrates the contribution of the matrix
flow through fractures to the pumping (note that in most cases the “U” of
