Equivalent plane strain solution
The theory of
consolidation of vertical drains under axisymmetric conditions has gained wide
acceptance in geotechnical engineering because of its simplicity. According to Hansbo (1981), for axisymmetric
flow the average degree of consolidation
U
¯
h
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGvbGbaebadaWgaaWcbaGaaKiAaaqabaaaaa@33C0@
on a horizontal plane at a depth z and at time t may be predicted from
|
|
U
¯
h
=1−exp(
−
8
T
h
μ
)
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGvbGbaebadaWgaaWcbaGaaKiAaaqabaGccqGH9aqpcaaIXaGaeyOeI0IaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisldaWcaaqaaiaaiIdacaWGubWaaSbaaSqaaiaajIgaaeqaaaGcbaGaeqiVd0gaaaGaayjkaiaawMcaaaaa@4052@
|
[1]
|
where
T
h
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGubWaaSbaaSqaaiaajIgaaeqaaaaa@33A7@
is the time factor, and the effect of smear
and well resistance (m) is given by
|
|
μ=ln(
n
s
)+(
k
h
k
′
h
)ln(
s
)−0.75+π(
2lz−
z
2
)
k
h
q
w
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@56C9@
|
[2]
|
In
the above,
n=
R/
r
w
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGUbGaeyypa0ZaaSGbaeaacaWGsbaabaGaamOCamaaBaaaleaacaqI3baabeaaaaaaaa@36C2@
, where R
is the radius of the influence zone of the drain and
r
w
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGYbWaaSbaaSqaaiaajEhaaeqaaaaa@33D4@
is the radius of the drain;
s=
r
s
/
r
w
,
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGZbGaeyypa0ZaaSGbaeaacaWGYbWaaSbaaSqaaiaajohaaeqaaaGcbaGaamOCamaaBaaaleaacaqI3baabeaakiaacYcaaaaaaa@38CD@
where
r
s
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGYbWaaSbaaSqaaiaajohaaeqaaaaa@33D0@
is the radius of the smear; l is the length of the drain having
one-way drainage or half this value for two-way drainage; z is the depth of the drain under consideration;
q
w
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGXbWaaSbaaSqaaiaajEhaaeqaaaaa@33D3@
is the discharge capacity of the drain; and
k
h
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaajIgaaeqaaaaa@33BE@
and
k
′
h
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaaqabaaaaa@33CA@
are the coefficients of horizontal
permeability outside and inside the smeared zone, respectively.
In
equation [2],
the final term represents the well resistance, which is a function of the drain
length l and depth z for
l>z
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGSbGaeyOpa4JaamOEaaaa@34A7@
and is inversely proportional to the drain
discharge capacity
(
q
w
)
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaadaqadaqaaiaadghadaWgaaWcbaGaaK4DaaqabaaakiaawIcacaGLPaaaaaa@3566@
.
Indraratna
and Redana (1997) showed that the degree of consolidation at a depth z in plane strain condition can be
represented by
|
|
U
¯
hp
=1−
u
¯
u
¯
0
=1−exp(
8
T
hp
μ
p
)
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGvbGbaebadaWgaaWcbaGaaKiAaiaajchaaeqaaOGaeyypa0JaaGymaiabgkHiTmaalaaabaGabmyDayaaraaabaGabmyDayaaraWaaSbaaSqaaiaaicdaaeqaaaaakiabg2da9iaaigdacqGHsislciGGLbGaaiiEaiaacchadaqadaqaamaalaaabaWaaWbaaSqabeaacqGHsislaaGccaaI4aGaamivamaaBaaaleaacaqIObGaaKiCaaqabaaakeaacqaH8oqBdaWgaaWcbaGaaKiCaaqabaaaaaGccaGLOaGaayzkaaaaaa@4982@
|
[3]
|
where
u
¯
0
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWG1bGbaebadaWgaaWcbaGaaKimaaqabaaaaa@33A8@
is the initial pore pressure,
u
¯
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWG1bGbaebaaaa@32C1@
is the pore pressure at time t (average values),
T
hp
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGubWaaSbaaSqaaiaajIgacaqIWbaabeaaaaa@34A2@
is time factor in plane strain, and
|
|
μ
p
=[
α+β
k
hp
k
′
hp
+θ(
2lz−
z
2
)
]
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacqaH8oqBdaWgaaWcbaGaaKiCaaqabaGccqGH9aqpdaWadaqaaiabeg7aHjabgUcaRiabek7aInaalaaabaGaam4AamaaBaaaleaacaqIObGaaKiCaaqabaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaiaajchaaeqaaaaakiabgUcaRiabeI7aXnaabmaabaGaaGOmaiaadYgacaWG6bGaeyOeI0IaamOEamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@4B9F@
|
[4]
|
where
k
hp
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaajIgacaqIWbaabeaaaaa@34B9@
and
k
′
hp
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaiaajchaaeqaaaaa@34C5@
are the undisturbed horizontal and
corresponding smear zone permeabilities, respectively. Ignoring higher order terms, the geometric
parameters a
and b
and the flow term q
are given by
|
|
α=
2
3
−
2
b
s
B
(
1−
b
s
B
+
b
s
2
3
B
2
)
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacqaHXoqycqGH9aqpdaWcaaqaaiaaikdaaeaacaaIZaaaaiabgkHiTmaalaaabaGaaGOmaiaadkgadaWgaaWcbaGaaK4CaaqabaaakeaacaWGcbaaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaamOyamaaBaaaleaacaqIZbaabeaaaOqaaiaadkeaaaGaey4kaSYaaSaaaeaacaWGIbWaa0baaSqaaiaajohaaeaacaqIYaaaaaGcbaGaaG4maiaadkeadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaaaa@46DC@
|
[5]
|
|
|
β=
1
B
2
(
b
s
−
b
w
)
2
+
b
s
3
B
3
(
3
b
w
2
−
b
s
2
)
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacqaHYoGycqGH9aqpdaWcaaqaaiaaigdaaeaacaWGcbWaaWbaaSqabeaacaaIYaaaaaaakmaabmaabaGaamOyamaaBaaaleaacaqIZbaabeaakiabgkHiTiaadkgadaWgaaWcbaGaaK4DaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaWcaaqaaiaadkgadaWgaaWcbaGaaK4CaaqabaaakeaacaaIZaGaamOqamaaCaaaleqabaGaaG4maaaaaaGcdaqadaqaaiaaiodacaWGIbWaa0baaSqaaiaajEhaaeaacaqIYaaaaOGaeyOeI0IaamOyamaaDaaaleaacaqIZbaabaGaaKOmaaaaaOGaayjkaiaawMcaaaaa@4CEB@
|
[6]
|
|
|
|
|
|
and
|
|
θ=
2
k
hp
2
k
′
hp
B
q
z
(
1−
b
w
B
)
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacqaH4oqCcqGH9aqpdaWcaaqaaiaaikdacaWGRbWaa0baaSqaaiaajIgacaqIWbaabaGaaKOmaaaaaOqaaiqadUgagaqbamaaBaaaleaacaqIObGaaKiCaaqabaGccaWGcbGaamyCamaaBaaaleaacaWG6baabeaaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaWgaaWcbaGaaK4DaaqabaaakeaacaWGcbaaaaGaayjkaiaawMcaaaaa@4541@
|
[7]
|
where
q
z
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGXbWaaSbaaSqaaiaajQhaaeqaaaaa@33D6@
is the equivalent plane strain discharge
capacity of the drain, and dimensions
B,
b
s
, and
b
w
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGcbGaaiilaiaaykW7caWGIbWaaSbaaSqaaiaajohaaeqaaOGaaiilaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaamOyamaaBaaaleaacaqI3baabeaaaaa@3D93@
are defined in Figure 1.
At a given
stress level, to maintain the same degree of consolidation at each time step,
the average degree of consolidation for both axisymmetric
(
U
¯
h
)
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaadaqadaqaaiqadwfagaqeamaaBaaaleaacaqIObaabeaaaOGaayjkaiaawMcaaaaa@3553@
and equivalent plain strain
(
U
¯
hp
)
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaadaqadaqaaiqadwfagaqeamaaBaaaleaacaqIObGaaKiCaaqabaaakiaawIcacaGLPaaaaaa@364E@
conditions are made equal, hence
|
|
U
¯
h
=
U
¯
hp
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGvbGbaebadaWgaaWcbaGaaKiAaaqabaGccqGH9aqpceWGvbGbaebadaWgaaWcbaGaaKiAaiaajchaaeqaaaaa@37DB@
|
[8]
|
Combining
equations [3]
and [8]
with the original Hansbo (1981) theory (equation [1]) defines
the time factor ratio by the following equation:
|
|
T
hp
T
h
=
k
hp
k
h
R
2
B
2
=
μ
p
μ
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaadaWcaaqaaiaadsfadaWgaaWcbaGaaKiAaiaajchaaeqaaaGcbaGaamivamaaBaaaleaacaqIObaabeaaaaGccqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGaaKiAaiaajchaaeqaaaGcbaGaam4AamaaBaaaleaacaqIObaabeaaaaGcdaWcaaqaaiaadkfadaahaaWcbeqaaiaaikdaaaaakeaacaWGcbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9maalaaabaGaeqiVd02aaSbaaSqaaiaajchaaeqaaaGcbaGaeqiVd0gaaaaa@4647@
|
[9]
|
Indraratna
and Redana (1997) showed that if the radius of the axisymmetric influence zone
of a single drain (R) were taken to
be the same as the half-width (B) in
plain strain, then the relationship between
k
hp
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaajIgacaqIWbaabeaaaaa@34B9@
and
k
′
hp
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaaceWGRbGbauaadaWgaaWcbaGaaKiAaiaajchaaeqaaaaa@34C5@
is given by
|
|
k
hp
=
k
h
[
α+β
k
hp
k
′
hp
+θ(
2lz−
z
2
)
]
[
ln(
n
s
)+(
k
h
k
′
h
)ln(
s
)−0.75+π(
2lz−
z
2
)
k
h
q
w
]
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaajIgacaqIWbaabeaakiabg2da9maalaaabaGaam4AamaaBaaaleaacaqIObaabeaakmaadmaabaGaeqySdeMaey4kaSIaeqOSdi2aaSaaaeaacaWGRbWaaSbaaSqaaiaajIgacaqIWbaabeaaaOqaaiqadUgagaqbamaaBaaaleaacaqIObGaaKiCaaqabaaaaOGaey4kaSIaeqiUde3aaeWaaeaacaaIYaGaamiBaiaadQhacqGHsislcaWG6bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaabaWaamWaaeaaciGGSbGaaiOBamaabmaabaWaaSaaaeaacaWGUbaabaGaam4CaaaaaiaawIcacaGLPaaacqGHRaWkdaqadaqaamaalaaabaGaam4AamaaBaaaleaacaqIObaabeaaaOqaaiqadUgagaqbamaaBaaaleaacaqIObaabeaaaaaakiaawIcacaGLPaaaciGGSbGaaiOBamaabmaabaGaam4CaaGaayjkaiaawMcaaiabgkHiTiaaicdacaGGUaGaaG4naiaaiwdacqGHRaWkcqaHapaCdaqadaqaaiaaikdacaWGSbGaamOEaiabgkHiTiaadQhadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaadaWcaaqaaiaadUgadaWgaaWcbaGaaKiAaaqabaaakeaacaWGXbWaaSbaaSqaaiaajEhaaeqaaaaaaOGaay5waiaaw2faaaaaaaa@7250@
|
[10]
|
If
both the smear and well resistance are ignored, then the simplified ratio of
plane strain to axisymmetric horizontal permeability
k
h
MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaajIgaaeqaaaaa@33BE@
is represented by
|
|
k
hp
k
h
=
0.67
[
ln(
n
)−0.75
]
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqabaqaaeaadaaakeaadaWcaaqaaiaadUgadaWgaaWcbaGaaKiAaiaajchaaeqaaaGcbaGaam4AamaaBaaaleaacaqIObaabeaaaaGccqGH9aqpdaWcaaqaaiaaicdacaGGUaGaaGOnaiaaiEdaaeaadaWadaqaaiGacYgacaGGUbWaaeWaaeaacaWGUbaacaGLOaGaayzkaaGaeyOeI0IaaGimaiaac6cacaaI3aGaaGynaaGaay5waiaaw2faaaaaaaa@4519@
|
[11]
|
If
the effect of well resistance is ignored, the permeability in the smear zone
can be isolated by neglecting the final terms of the denominator and numerator
in equation [10]
to give
|
|
k
′
hp
k
hp
=
β
k
hp
k
h
[
ln(
n
s
)+(
k
h
k
′
h
)ln(
s
)−0.75
]−α
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@58C3@
|
[12]
|