Geoscience Reference
In-Depth Information
The case that now the finite difference value
N
is equal to the wellhead
w
, is
only correct when the well radius is
r
w
=
d
exp(
/
2) = 0.208
d
and this is not
necessarily the case for an arbitrary grid size
d
. The best grid size near the well is
d
=
4.81
r
w
. If
d
is much larger, which is usually the case in FD-modelling, the
resulting
N
will be significantly larger than the real wellhead
w
. A separate
correction step has to be added to the numerical result.
2
r
w
(
)
ln
)
(9.22)
w
N
d
N
d
Although the numerical finite difference calculation is accurate, the choice of a
linear polynomial causes the result in singular points to be inaccurate. Similar
behaviour is to be expected in the numerical analysis of mechanical problems at
places of high discontinuity.
Biot effect
The storage term of a unit bulk volume can be formulated, as follows (Verruijt)
S =
(
n
)
f
+
)
sf
(
2
n
))
du +
2
d
(9.23)
with
)
f
the fluid compressibility,
)
sf
the fluid induced grain compressibility,
2
= 1
(1
n
)
)
ss
/
)
is the Biot coefficient,
)
ss
the grain-to-grain contact compressibility, and
)
the soil bulk compressibility. Disregarding the term with
)
sf
and adopting
Geertsma's suggestion
2
= 1-(
)
sf
/
)
), leads to the commonly applied formula for
S
,
which however is only consistent if
ss
.
For uniform grain compression, the micro-elastic behaviour can be expressed by
)
sf
= (1-
n
)
)
)
the local average fluid stress,
w
the displacement, and
R
the
grain radius. The local micro-elastic behaviour at the grain-to-grain contacts is
expressed by
=
w/R
, with
sf
=
w/R
, with
r
the grain contact area radius. For the relative grain
contact
r
/
R
holds 0 <
r/R
< 1.
)
ss
ss
is related to the grain contact area. This problem
is known as Herz' problem. Because of geometric constraints
)
)
ss
is non-linear.
Elaboration
46
of Herz' solution yields (
)
2
= 0.055
w/R
and
r/R
= 4.28
)
)
.
sf
sf
Furthermore,
sf
for
r/R
= 0.12, which implies that e.g. for
n
= 0.25 and 16%
relative grain contact Geertsma's suggestion holds.
A constant overburden and an effective stress concept (4.1a), according to
)
ss
=
)
'
=
-(1-
)
sf
/
)
)
u
, yields the following vertical strain variation (one-dimensional)
d
z
= -
)
v
d
' =
)
v
((1
-
)
sf
/
)
)du-d
)
=
)
v
(1
-
)
sf
/
)
)
du
(9.24)
46
Timosheko & Goodier (1970) Theory of Elasticity:
E =
)
sf
1
;
P =
r
2
; 3/2 = 0.388
(
PE
2
/
R
2
)
1/3
;
w
= 1.23 (
P
2
/
E
2
R
)
1/3
;
r
= 1.11 (
PR
/
E
)
1/3
; /
E
= 0.211(
w
/
R
)(
ER
2
/
P
)
1/3
;
P
=
r
2
= (1.11(
PR
/
E
)
1/3
)
2
;
P
1/3
= 3.86
(
R
/
E
)
2/3
;
/
E
)
2
= 0.055
w
/
R
. For
d(/
E
)/d(
w
/
R
) = 1, )
ss
= )
sf
at
/
E
= 0.027 ;
r
= (
P
/) ;
r
/
R
= 4.28
/
E
=0.12
/
E
= 0.211(
w
/
R
)(
ER
2
/
P
)
1/3
= 0.211(
w
/
R
)(
ER
2
/(3.86
(
R
/
E
)
2/3
))
1/3
; (
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