smear_zones¶

Function summary¶

`_g`(r_rw, re_rw[, nflow, nterms])	Non-darcian equal strain radial consolidation term
`_gbar`(r_rw, re_rw[, nflow, nterms])	Non-darcian equal strain radial consolidation term
`_kapx`(n, s, kap)	Value of kap=kh/ks for overlap part of intersecting linear smear zones
`_sx`(n, s)	Value of s=r/rw marking the start of overlapping linear smear zones
`back_calc_drain_spacing_from_eta`(eta, …[, muw])	Back calculate the required drain spacing to achieve a given eta
`drain_eta`(re, mu_function, args, *kwargs)	Calculate the vertical drain eta parameter for a specific smear zone
`k_linear`(n, s, kap, si)	Permeability distribution for smear zone with linear permeability
`k_overlapping_linear`(n, s, kap, si)	Permeability smear zone with overlapping linear permeability
`k_parabolic`(n, s, kap, si)	Permeability distribution for smear zone with parabolic permeability
`mu_constant`(n, s, kap)	Smear zone parameter for smear zone with constant permeability
`mu_ideal`(n, *args)	Smear zone permeability/geometry parameter for ideal drain (no smear)
`mu_linear`(n, s, kap)	Smear zone parameter for smear zone linear variation of permeability
`mu_overlapping_linear`(n, s, kap)	Smear zone parameter for smear zone with overlapping linear permeability
`mu_parabolic`(n, s, kap)	Smear zone parameter for parabolic variation of permeability
`mu_piecewise_constant`(s, kap[, n, kap_m])	Smear zone parameter for piecewise constant permeability distribution
`mu_piecewise_linear`(s, kap[, n, kap_m])	Smear zone parameter for piecewise linear permeability distribution
`mu_well_resistance`(kh, qw, n, H[, z])	Additional smear zone parameter for well resistance
`non_darcy_beta_constant`(n, s, kap[, nflow, …])	Non-darcian flow smear zone permeability/geometry parameter for smear zone with constant permeability.
`non_darcy_beta_ideal`(n[, nflow, nterms])	Non-darcian flow smear zone permeability/geometry parameter for ideal drain (no smear).
`non_darcy_beta_piecewise_constant`(s, kap[, …])	Non-darcian flow smear zone permeability/geometry parameter for smear zone with piecewise constant permeability.
`non_darcy_drain_eta`(re, iL, gamw, …)	For non-Darcy flow calculate the vertical drain eta parameter
`non_darcy_u_piecewise_constant`(s, kap, si[, …])	Pore pressure at radius for piecewise constant permeability distribution
`re_from_drain_spacing`(sp[, pattern])	Calculate drain influence radius from drain spacing
`scratch`()	scratch pad for testing latex markup for docstrings
`u_constant`(n, s, kap, si[, uavg, uw, muw])	Pore pressure at radius for constant permeability smear zone
`u_ideal`(n, si[, uavg, uw, muw])	Pore pressure at radius for ideal drain with no smear zone
`u_linear`(n, s, kap, si[, uavg, uw, muw])	Pore pressure at radius for linear smear zone
`u_parabolic`(n, s, kap, si[, uavg, uw, muw])	Pore pressure at radius for parabolic smear zone
`u_piecewise_constant`(s, kap, si[, uavg, uw, …])	Pore pressure at radius for piecewise constant permeability distribution
`u_piecewise_linear`(s, kap, si[, uavg, uw, …])	Pore pressure at radius for piecewise constant permeability distribution

Module listing¶

Smear zones associated with vertical drain installation.

Smear zone permeability distributions etc.

geotecha.consolidation.smear_zones.back_calc_drain_spacing_from_eta(eta, pattern, mu_function, rw, s, kap, muw=0)[source]¶

Back calculate the required drain spacing to achieve a given eta

eta = 2 / re**2 / (mu + muw)

eta is used in radial consolidation equations u= u0 * exp(-eta*kh/gamw*t)

Parameters:

eta : float: eta value.
pattern : [‘Triangle’, ‘Square’]: Drain installation pattern.
mu_function : obj: The mu_funtion to use. e.g. mu_ideal, mu_constant, mu_linear, mu_overlapping_linear, mu_parabolic, mu_piecewise_constant, mu_piecewise_linear.
rw : float: Drain/well radius.
s : float or 1d array_like of float: Ratio of smear zone radius to drain radius (rs/rw). s can only be a 1d array is using a mu_piecewise function
kap : float or 1d array_like of float: Ratio of undisturbed horizontal permeability to permeability at in smear zone (kh / ks) (often at the drain-soil interface). Be careful when defining s and kap for mu_piecewise_constant, and mu_piecewise_linear because the last value of kap will be used at the influence drain periphery. In general the last value of kap should be one, representing the start of the undisturbed zone.
muw : float, optional: Well resistance mu term, default=0.

Returns:

sp : float: Drain spacing to get the required eta value
re : float: Drain influence radius
n : float: Ratio of drain influence radius to drain radius, re/rw

Notes

When using mu_piecewise_linear or mu_piecewise_constant only define s and kap up to the start of the undisturbed zone. re will be varied.

For anyting other than mu_overlapping_linear do not trust any returned spacing that gives an n value less than the extent of the smear zone.

geotecha.consolidation.smear_zones.drain_eta(re, mu_function, *args, **kwargs)[source]¶

Calculate the vertical drain eta parameter for a specific smear zone

eta = 2 / re**2 / (mu+muw)

eta is used in radial consolidation equations u= u0 * exp(-eta*kh/gamw*t)

Parameters:

re : float: Drain influence radius.
mu_function : obj or string: The mu_funtion to use. e.g. mu_ideal, mu_constant, mu_linear, mu_overlapping_linear, mu_parabolic, mu_piecewise_constant, mu_piecewise_linear. This can either be the function object itself or the name of the function e.g. ‘mu_ideal’.
muw : float, optional: Well resistance mu term, default=0.
*args, **kwargs : various: The arguments to pass to the mu_function.

Returns:

eta : float: Value of eta parameter

Examples

>>> drain_eta(1.5, mu_ideal, 10)
0.563178340433...
>>> drain_eta(1.5, 'mu_ideal', 10)
0.5631783404334...
>>> drain_eta(1.5, mu_constant, 5, 1.5, 1.6, muw=1)
0.4115837724144...

geotecha.consolidation.smear_zones.k_linear(n, s, kap, si)[source]¶

Permeability distribution for smear zone with linear permeability

Normalised with respect to undisturbed permeability. i.e. if you want the actual permeability then multiply by whatever you used to determine kap.

Permeability is linear with value 1/kap at the drain soil interface i.e. at s=1 k=k0=1/kap. for si>s, permeability=1.

Parameters:	n : float Ratio of drain influence radius to drain radius (re/rw). s : float Ratio of smear zone radius to drain radius (rs/rw). kap : float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks). si : float of ndarray of float Normalised radial coordinate(s) at which to calc the permeability i.e. si=ri/rw.
Returns:	permeability : float or ndarray of float Normalised permeability (i.e. ki/kh) at the si values.

Notes

Linear distribution of permeability in smear zone is given by:

\[\begin{split}\frac{k_h^\prime\left({r}\right)}{k_h}= \left\{\begin{array}{lr} \frac{1}{\kappa} \left({A\frac{r}{r_w}+B}\right) & s\neq\kappa \\ \frac{r}{\kappa r_w} & s=\kappa \end{array}\right.\end{split}\]

where \(A\) and \(B\) are:

\[A=\frac{\kappa-1}{s-1}\]

\[B=\frac{s-\kappa}{s-1}\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.k_overlapping_linear(n, s, kap, si)[source]¶

Permeability smear zone with overlapping linear permeability

Normalised with respect to undisturbed permeability. i.e. if you want the actual permeability then multiply by whatever you used to determine kap.

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float Ratio of drain influence radius to drain radius (re/rw). s : float Ratio of smear zone radius to drain radius (rs/rw). kap : float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks). si : float of ndarray of float Normalised radial coordinate(s) at which to calc the permeability i.e. si=ri/rw
Returns:	permeability : float or ndarray of float Normalised permeability (i.e. ki/kh) at the si values.

Notes

When \(n>s\) the permeability is no different from the linear case.

When \(n\leq (s+1)/2\) then all the soil is disturbed and the permeability everywhere is equal to \(1/\kappa\).

When \((s+1)/2<n<s\) then the smear zones overlap. the permeability for \(r/r_w<s_X\) is given by:

\[\begin{split}\frac{k_h^\prime\left({r}\right)}{k_h}= \left\{\begin{array}{lr} \frac{1}{\kappa} \left({A\frac{r}{r_w}+B}\right) & s\neq\kappa \\ \frac{r}{\kappa r_w} & s=\kappa \end{array}\right.\end{split}\]

In the overlapping part, \(r/r_w>s_X\), the permeability is given by:

\[k_h(r)=\kappa_X/\kappa\]

where \(A\) and \(B\) are:

\[A=\frac{\kappa-1}{s-1}\]

\[B=\frac{s-\kappa}{s-1}\]

\[\kappa_X= 1+\frac{\kappa-1}{s-1}\left({s_X-1}\right)\]

\[s_X = 2n-s\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.k_parabolic(n, s, kap, si)[source]¶

Permeability distribution for smear zone with parabolic permeability

Normalised with respect to undisturbed permeability. i.e. if you want the actual permeability then multiply by whatever you used to determine kap.

Permeability is parabolic with value 1/kap at the drain soil interface i.e. at s=1 k=k0=1/kap. for si>s, permeability=1.

Parameters:	n : float Ratio of drain influence radius to drain radius (re/rw). s : float Ratio of smear zone radius to drain radius (rs/rw). kap : float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks). si : float of ndarray of float Normalised radial coordinate(s) at which to calc the permeability i.e. si=ri/rw
Returns:	permeability : float or ndarray of float Normalised permeability (i.e. ki/kh) at the si values.

Notes

Parabolic distribution of permeability in smear zone is given by:

\[\frac{k_h^\prime\left({r}\right)}{k_h}= \frac{\kappa-1}{\kappa} \left({A-B+C\frac{r}{r_w}}\right) \left({A+B-C\frac{r}{r_w}}\right)\]

where \(A\), \(B\), \(C\) are:

\[A=\sqrt{\frac{\kappa}{\kappa-1}}\]

\[B=\frac{s}{s-1}\]

\[C=\frac{1}{s-1}\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, Rohan, and Buddhima Indraratna. 2006. ‘Vertical Drain Consolidation with Parabolic Distribution of Permeability in Smear Zone’. Journal of Geotechnical and Geoenvironmental Engineering 132 (7): 937-41. doi:10.1061/(ASCE)1090-0241(2006)132:7(937).

geotecha.consolidation.smear_zones.mu_constant(n, s, kap)[source]¶

Smear zone parameter for smear zone with constant permeability

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float or ndarray of float Ratio of drain influence radius to drain radius (re/rw). s : float or ndarray of float Ratio of smear zone radius to drain radius (rs/rw) kap : float or ndarray of float. Ratio of undisturbed horizontal permeability to smear zone horizontal permeanility (kh / ks).
Returns:	mu : float smear zone permeability/geometry parameter

Notes

The \(\mu\) parameter is given by:

\[\mu=\frac{n^2}{\left({n^2-1}\right)} \left({\ln\left({\frac{n}{s}}\right) +\kappa\ln\left({s}\right) -\frac{3}{4}}\right)+ \frac{s^2}{\left({n^2-1}\right)}\left({1-\frac{s^2}{4n^2}} \right) +\frac{\kappa}{\left({n^2-1}\right)}\left({\frac{s^4-1}{4n^2}} -s^2+1 \right)\]

where:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability.

References

[1]	Hansbo, S. 1981. ‘Consolidation of Fine-Grained Soils by Prefabricated Drains’. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

geotecha.consolidation.smear_zones.mu_ideal(n, *args)[source]¶

Smear zone permeability/geometry parameter for ideal drain (no smear)

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float or ndarray of float Ratio of drain influence radius to drain radius (re/rw). args : anything args does not contribute to any calculations it is merely so you can have other arguments such as s and kappa which are used in other smear zone formulations.
Returns:	mu : float Smear zone permeability/geometry parameter.

Notes

The \(\mu\) parameter is given by:

\[\mu=\frac{n^2}{\left({n^2-1}\right)} \left({\ln\left({n}\right)-\frac{3}{4}}\right)+ \frac{1}{\left({n^2-1}\right)}\left({1-\frac{1}{4n^2}} \right)\]

where:

\[n = \frac{r_e}{r_w}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius

References

[1]	Hansbo, S. 1981. “Consolidation of Fine-Grained Soils by Prefabricated Drains”. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

geotecha.consolidation.smear_zones.mu_linear(n, s, kap)[source]¶

Smear zone parameter for smear zone linear variation of permeability

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float or ndarray of float Ratio of drain influence radius to drain radius (re/rw). s : float or ndarray of float Ratio of smear zone radius to drain radius (rs/rw). kap : float or ndarray of float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks).
Returns:	mu : float Smear zone permeability/geometry parameter.

Notes

For \(s\neq\kappa\), \(\mu\) is given by:

\[\mu=\frac{n^2}{\left({n^2-1}\right)} \left[{ \ln\left({\frac{n}{s}}\right) -\frac{3}{4} +\frac{s^2}{n^2}\left({1-\frac{s^2}{4n^2}}\right) -\frac{\kappa}{B}\ln\left({\frac{\kappa}{s}}\right) +\frac{\kappa B}{A^2 n^2}\left({2-\frac{B^2}{A^2 n^2}} \right)\ln\left({\kappa}\right) -\frac{\kappa\left({s-1}\right)}{A n^2} \left\{ 2 +\frac{1}{n^2} \left[ {\frac{A-B}{A}\left({\frac{1}{A}}-\frac{s+1}{2} \right)} -\frac{s+1}{2} -\frac{\left({s-1}\right)^2}{3} \right] \right\} }\right]\]

and for the special case \(s=\kappa\), \(\mu\) is given by:

\[\mu=\frac{n^2}{\left({n^2-1}\right)} \left[{ \ln\left({\frac{n}{s}}\right) -\frac{3}{4} +s-1 -\frac{s^2}{n^2}\left({1-\frac{s^2}{12n^2}}\right) -\frac{s}{n^2}\left({2-\frac{1}{3n^2}}\right) }\right]\]

where \(A\) and \(B\) are:

\[A=\frac{\kappa-1}{s-1}\]

\[B=\frac{s-\kappa}{s-1}\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.mu_overlapping_linear(n, s, kap)[source]¶

Smear zone parameter for smear zone with overlapping linear permeability

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float or ndarray of float Ratio of drain influence radius to drain radius (re/rw). s : float or ndarray of float Ratio of smear zone radius to drain radius (rs/rw). kap : float or ndarray of float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks).
Returns:	mu : float Smear zone permeability/geometry parameter.

See also

mu_linear: \(\mu\) for non-overlapping smear zones
mu_ideal: \(\mu\) for ideal drain with no smear zone

Notes

The smear zone parameter \(\mu\) is given by:

\[\begin{split}\mu_X = \left\{\begin{array}{lr} \mu_L\left({n,s,\kappa}\right) & n\geq s \\ \frac{\kappa}{\kappa_X}\mu_L \left({n, s_X,\kappa_x}\right) & \frac{s+1}{2}<n<s \\ \frac{\kappa}{\kappa_X}\mu_I \left({n}\right) & n\leq \frac{s+1}{2} \end{array}\right.\end{split}\]

where \(\mu_L\) is the \(\mu\) parameter for non_overlapping smear zones with linear permeability, \(\mu_I\) is the \(\mu\) parameter for no smear zone, and:

\[\kappa_X= 1+\frac{\kappa-1}{s-1}\left({s_X-1}\right)\]

\[s_X = 2n-s\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.mu_parabolic(n, s, kap)[source]¶

Smear zone parameter for parabolic variation of permeability

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:	n : float or ndarray of float Ratio of drain influence radius to drain radius (re/rw). s : float or ndarray of float Ratio of smear zone radius to drain radius (rs/rw). kap : float or ndarray of float Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh/ks).
Returns:	mu : float Smear zone permeability/geometry parameter

Notes

The smear zone parameter \(\mu\) is given by:

\[\mu = \frac{n^2}{\left({n^2-1}\right)} \left({ \frac{A^2}{n^2}\mu_1+\mu_2 }\right)\]

where,

\[\mu_1= \frac{1}{A^2-B^2} \left({ s^2\ln\left({s}\right) -\frac{1}{2}\left({s^2-1}\right) }\right) -\frac{1}{\left({A^2-B^2}\right)C^2} \left({ \frac{A^2}{2}\ln\left({\kappa}\right) +\frac{ABE}{2}+\frac{1}{2}-B -\left({A^2-B^2}\right)\ln\left({\kappa}\right) }\right) +\frac{1}{n^2C^4} \left({ -\left({\frac{A^2}{2}+B^2}\right) \ln\left({\kappa}\right) +\frac{3ABE}{2}+\frac{1}{2}-3B }\right)\]

\[\mu_2= \ln\left({\frac{n}{s}}\right) -\frac{3}{4} +\frac{s^2}{n^2}\left({1-\frac{s^2}{4n^2}}\right) +A^2\left({1-\frac{s^2}{n^2}}\right) \left[{ \frac{1}{A^2-B^2} \left({ \ln\left({\frac{s}{\sqrt{\kappa}}}\right) -\frac{BE}{2A} }\right) +\frac{1}{n^2C^2} \left({ \ln\left({\sqrt{\kappa}}\right) -\frac{BE}{2A} }\right) }\right]\]

where \(A\), \(B\), \(C\) and \(E\) are:

\[A=\sqrt{\frac{\kappa}{\kappa-1}}\]

\[B=\frac{s}{s-1}\]

\[C=\frac{1}{s-1}\]

\[E=\ln\left({\frac{A+1}{A-1}}\right)\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, Rohan, and Buddhima Indraratna. 2006. ‘Vertical Drain Consolidation with Parabolic Distribution of Permeability in Smear Zone’. Journal of Geotechnical and Geoenvironmental Engineering 132 (7): 937-41. doi:10.1061/(ASCE)1090-0241(2006)132:7(937).

geotecha.consolidation.smear_zones.mu_piecewise_constant(s, kap, n=None, kap_m=None)[source]¶

Smear zone parameter for piecewise constant permeability distribution

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:

s : list or 1d ndarray of float: Ratio of segment outer radii to drain radius (r_i/r_0). The first value of s should be greater than 1, i.e. the first value should be s_1; s_0=1 at the drain soil interface is implied.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability in each segment kh/khi.
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soilpermeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.

Returns:

mu : float: Smear zone permeability/geometry parameter

Notes

The smear zone parameter \(\mu\) is given by:

\[\mu = \frac{n^2}{\left({n^2-1}\right)} \sum\limits_{i=1}^{m} \kappa_i \left[{ \frac{s_i^2}{n^2}\ln \left({ \frac{s_i}{s_{i-1}} }\right) -\frac{s_i^2-s_{i-1}^2}{2n^2} -\frac{\left({s_i^2-s_{i-1}^2}\right)^2}{4n^4} }\right] +\psi_i\frac{s_i^2-s_{i-1}^2}{n^2}\]

where,

\[\psi_{i} = \sum\limits_{j=1}^{i-1}\kappa_j \left[{ \ln \left({ \frac{s_j}{s_{j-1}} }\right) -\frac{s_j^2-s_{j-1}^2}{2n^2} }\right]\]

and:

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{hi}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the outer radius of the ith segment, \(k_h\) is the undisturbed horizontal permeability in the ith segment, \(k_{hi}\) is the horizontal permeability in the ith segment

References

[1]	Walker, Rohan. 2006. ‘Analytical Solutions for Modeling Soft Soil Consolidation by Vertical Drains’. PhD Thesis, Wollongong, NSW, Australia: University of Wollongong. http://ro.uow.edu.au/theses/501

[2]	Walker, Rohan T. 2011. ‘Vertical Drain Consolidation Analysis in One, Two and Three Dimensions’. Computers and Geotechnics 38 (8): 1069-77. doi:10.1016/j.compgeo.2011.07.006.

geotecha.consolidation.smear_zones.mu_piecewise_linear(s, kap, n=None, kap_m=None)[source]¶

Smear zone parameter for piecewise linear permeability distribution

mu parameter in equal strain radial consolidation equations e.g. u = u0 * exp(-8*Th/mu)

Parameters:

s : list or 1d ndarray of float: Ratio of radii to drain radius (r_i/r_0). The first value of s should be 1, i.e. at the drain soil interface.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability at each value of s.
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soilpermeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.

Returns:

mu : float: Smear zone permeability/geometry parameter.

Notes

With permeability in the ith segment defined by:

\[\frac{k_i}{k_{ref}}= \frac{1}{\kappa_{i-1}} \left({A_ir/r_w+B_i}\right)\]

\[A_i = \frac{\kappa_{i-1}/\kappa_i-1}{s_i-s_{i-1}}\]

\[B_i = \frac{s_i-s_{i-1}\kappa_{i-1}/\kappa_i}{s_i-s_{i-1}}\]

the smear zone \(\mu\) parameter is given by:

\[\mu = \frac{n^2}{n^2-1} \left[{ \sum\limits_{i=1}^{m}\kappa_{i-1}\theta_i + \Psi_i \left({ \frac{s_i^2-s_{i-1}^2}{n^2} }\right) +\mu_w }\right]\]

where,

\[\begin{split}\theta_i = \left\{ \begin{array}{lr} \frac{s_i^2}{n^2}\ln \left[{\frac{s_i}{s_{i-1}}}\right] -\frac{s_i^2-s_{i-1}^2}{2n^2} -\frac{\left({s_i^2-s_{i-1}^2}\right)^2}{4n^4} & \textrm{for } \frac{\kappa_{i-1}}{\kappa_i}=1 \\ \frac{\left({s_i^2-s_{i-1}^2}\right)}{3n^4} \left({3n^2-s_{i-1}^2-2s_{i-1}s_i}\right) & \textrm{for }\frac{\kappa_{i-1}}{\kappa_i}= \frac{s_i}{s_{i-1}} \\ \begin{multline} \frac{s_i}{B_i n^2}\ln\left[{ \frac{\kappa_i s_i}{\kappa_{i-1}s_{i-1}}}\right] -\frac{s_i-s_{i-1}}{A_in^2} \left({1-\frac{B_i^2}{A_i^2n^4}}\right) \\-\frac{\left({s_i-s_{i-1}}\right)^2}{3A_in^2} \left({2s_i+s_{i-1}}\right) \\+\frac{B_i}{A_i^2 n^4}\ln\left[{ \frac{\kappa_{i-1}}{\kappa_i}}\right] \left({1-\frac{B_i^2}{A_i^2n^2}}\right) \\+\frac{B_i}{2A_i^2 n^4} \left({ 2s_i^2\ln\left[{ \frac{\kappa_{i-1}}{\kappa_i}}\right] -s_i^2 + s_{i-1}^2 }\right) \end{multline} & \textrm{otherwise} \end{array}\right.\end{split}\]

\[\Psi_i = \sum\limits_{j=1}^{i-1}\kappa_{j-1}\psi_j\]

\[\begin{split}\psi_i = \left\{ \begin{array}{lr} \ln\left[{\frac{s_j}{s_{j-1}}}\right] - \frac{s_j^2- s_{j-1}^2}{2n^2} & \textrm{for } \frac{\kappa_{j-1}}{\kappa_j}=1 \\ \frac{\left({s_j - s_{j-1}}\right) \left({n^2-s_js_{j-1}}\right)}{s_jn^2} & \textrm{for }\frac{\kappa_{j-1}}{\kappa_j}= \frac{s_j}{s_{j-1}} \\ \begin{multline} \frac{1}{B_i}\ln\left[{\frac{s_j}{s_{j-1}}}\right] +\ln\left[{\frac{\kappa_{j-1}}{\kappa_j}}\right] \left({\frac{B_j}{A_j^2n^2}-\frac{1}{B_j}}\right) \\-\frac{s_j-s_{j-1}}{A_j^2n^2} \end{multline} & \textrm{otherwise} \end{array}\right.\end{split}\]

and:

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{ref}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the radius of the ith radial point, \(k_{ref}\) is a convienient refernce permeability, usually the undisturbed horizontal permeability, \(k_{hi}\) is the horizontal permeability at the ith radial point

References

Derived by Rohan Walker in 2011 and 2014. Derivation steps are the same as for mu_piecewise_constant in appendix of [1] but permeability is linear in a segemetn as in [2].

[1]	(1, 2) Walker, Rohan. 2006. ‘Analytical Solutions for Modeling Soft Soil Consolidation by Vertical Drains’. PhD Thesis, Wollongong, NSW, Australia: University of Wollongong. http://ro.uow.edu.au/theses/501

[2]	(1, 2) Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.mu_well_resistance(kh, qw, n, H, z=None)[source]¶

Additional smear zone parameter for well resistance

Parameters:

kh : float: The normalising permeability used in calculating kappa for smear zone calcs. Usually the undisturbed permeability i.e. the kh in kappa = kh/ks
qw : float: Drain discharge capacity. qw = kw * pi * rw**2. Make sure the kw used has the same units as kh.
n : float: Ratio of drain influence radius to drain radius (re/rw).
H : float: Length of drainage path.
z : float, optional: Evaluation depth. Default = None, in which case the well resistance factor will be averaged.

Returns:

mu : float: mu parameter for well resistance

Notes

The smear zone parameter \(\mu_w\) is given by:

\[\mu_w = \frac{k_h}{q_w}\pi z \left({2H-z}\right) \left({1-\frac{1}{n^2}}\right)\]

when \(z\) is None then the average \(\mu_w\) is given by:

\[\mu_{w\textrm{average}} = \frac{2k_h H^2}{3q_w}\pi \left({1-\frac{1}{n^2}}\right)\]

where,

\[n = \frac{r_e}{r_w}\]

\[q_w = k_w \pi r_w^2\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(k_h\) is the undisturbed horizontal permeability, \(k_w\) is the drain permeability

References

[1]	Hansbo, S. 1981. ‘Consolidation of Fine-Grained Soils by Prefabricated Drains’. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

geotecha.consolidation.smear_zones.non_darcy_beta_constant(n, s, kap, nflow=1.0001, nterms=20, *args)[source]¶

Non-darcian flow smear zone permeability/geometry parameter for smear zone with constant permeability.

beta parameter is in equal strain radial consolidation equations with non-Darcian flow.

Parameters:

n : float or ndarray of float: Ratio of drain influence radius to drain radius (re/rw).
s : float or ndarray of float: Ratio of smear zone radius to drain radius (rs/rw)
kap : float or ndarray of float.: Ratio of undisturbed horizontal permeability to smear zone horizontal permeanility (kh / ks).
nflow : float, optional: non_darcian flow exponent
nterms : int, optional: Number of terms to use in series
args : anything: args does not contribute to any calculations it is merely so you can have other arguments such as s and kappa which are used in other smear zone formulations.

Returns:

beta : float: Smear zone permeability/geometry parameter.

See also

_g: used in this function.
_gbar: used in this function.

Notes

\[\beta = \frac{1}{N^2-1} \left({ 2\overline{g}\left({N}\right) -2\overline{g}\left({1}\right) -g\left({1}\right) \left({N^2-1}\right) }\right)\]

\(g\left({y}\right)\) and \(\overline{g}\left({y}\right)\) are described in the _g and _gbar functions respectively.

\[n = \frac{r_e}{r_w}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius.

References

[1]	Hansbo, S. 1981. “Consolidation of Fine-Grained Soils by Prefabricated Drains”. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

[2]	Walker, R., B. Indraratna, and C. Rujikiatkamjorn. “Vertical Drain Consolidation with Non-Darcian Flow and Void-Ratio-Dependent Compressibility and Permeability.” Geotechnique 62, no. 11 (November 1, 2012): 985-97. doi:10.1680/geot.10.P.084.

Examples

>>> non_darcy_beta_ideal(20, 1.000001, nterms=20)
2.2538...
>>> non_darcy_beta_ideal(np.array([20, 10]), 1.000001, nterms=20)
array([2.253..., 1.578...])
>>> non_darcy_beta_ideal(15, 1.3)
2.618...
>>> non_darcy_beta_ideal(np.array([20, 15]), np.array([1.000001,1.3]), nterms=20)
array([2.253..., 2.618...])

geotecha.consolidation.smear_zones.non_darcy_beta_piecewise_constant(s, kap, n=None, kap_m=None, nflow=1.0001, nterms=20, *args)[source]¶

Non-darcian flow smear zone permeability/geometry parameter for smear zone with piecewise constant permeability.

beta parameter is in equal strain radial consolidation equations with non-Darcian flow.

Parameters:

s : list or 1d ndarray of float: Ratio of segment outer radii to drain radius (r_i/r_0). The first value of s should be greater than 1, i.e. the first value should be s_1; s_0=1 at the drain soil interface is implied.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability in each segment kh/khi.
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soil permeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.
nflow : float, optional: non_darcian flow exponent
nterms : int, optional: Number of terms to use in series

Returns:

beta : float: Smear zone permeability/geometry parameter.

Notes

The non-darcian smear zone parameter \(\beta\) is given by:

\[\beta = \frac{1}{\left({n^2-1}\right)} \sum\limits_{i=1}^{m} \kappa^{1/n}_i \left[{ 2\overline{g}\left({s_i}\right) -2\overline{g}\left({s_{i-1}}\right) }\right] +\psi_i \left({s_i^2-s_{i-1}^2}\right)\]

where,

\[\psi_{i} = \sum\limits_{j=1}^{i-1}\kappa^{1/n}_j \left[{ g\left({s_j}\right) -g\left({s_{j-1}}\right) }\right]\]

and:

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{hi}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the outer radius of the ith segment, \(k_h\) is the undisturbed horizontal permeability in the ith segment, \(k_{hi}\) is the horizontal permeability in the ith segment

References

None because it is new.

Examples

>>> mu_piecewise_constant([1.5, 3, 4],[2, 3, 1], n=5)
2.2533...
>>> non_darcy_beta_piecewise_constant(s=np.array([1.5, 3, 4]),
... kap=np.array([2, 3, 1]), n=5, nflow=1.000001)
2.2533...

geotecha.consolidation.smear_zones.non_darcy_drain_eta(re, iL, gamw, beta_function, *args, **kwargs)[source]¶

For non-Darcy flow calculate the vertical drain eta parameter

eta = 2 / (re**2 * beta**nflow * (rw * gamw)**(nflow-1) * nflow * iL**(nflow-1))

nflow will be obtained from the **kwargs. rw will be back calculated from the n parameter (n=re/rw) which is usually the first of the *arg parameters or one of the **kwargs

Note that eta is used in radial consolidation equations: [strain rate] = (u - uw)**n * k / gamw * eta Compare with the Darcian case of (eta terms are calculated differerntly for Darcy and non-Darcy cases): [strain rate] = (u - uw) * k / gamw * eta

Note that non_darcy_drain_eta only uses the exponential portion of the Non-Darcian flow relationship. If hydraulic gradients are greater than iL then the flow rates will be overestimated.

Parameters:

re : float: Drain influence radius.
iL : float: Limiting hydraulic gradient beyond which flow follows Darcy’s law.
gamw : float: Unit weight of water. Usually gamw=10 kN/m**3 or gamw=9.807 kN/m**3.
beta_function : obj or string: The non_darcy_beta function to use. e.g. non_darcy_beta_ideal non_darcy_beat_constant, non_darcy_piecewise_constant. This can either be the function object itself or the name of the function e.g. ‘non_darcy_beta_ideal’.
*args, **kwargs : various: The arguments to pass to the beta_function.

Returns:

eta : float: Value of eta parameter for non-Darcian flow

Examples

>>> non_darcy_drain_eta(re=1.5, iL=10, gamw=10,
... beta_function='non_darcy_beta_ideal', n=15, nflow=1.3, nterms=20)
0.09807...
>>> non_darcy_drain_eta(1.5, 10, 10,
... 'non_darcy_beta_ideal', 15, nflow=1.3, nterms=20)
0.09807...

>>> non_darcy_drain_eta(re=1.5, iL=10, gamw=10,
... beta_function='non_darcy_beta_ideal', n=np.array([20.0, 15.0]),
... nflow=np.array([1.000001, 1.3]), nterms=20)
array([0.3943..., 0.0980...])

geotecha.consolidation.smear_zones.non_darcy_u_piecewise_constant(s, kap, si, uavg=1, uw=0, muw=0, n=None, kap_m=None, nflow=1.0001, nterms=20)[source]¶

Pore pressure at radius for piecewise constant permeability distribution

Warning

muw must always be zero. i.e. no well resistance (It exists to have the same inputs as u_piecewise_constant.

Parameters:

s : list or 1d ndarray of float: Ratio of segment outer radii to drain radius (r_i/r_0). The first value of s should be greater than 1, i.e. the first value should be s_1; s_0=1 at the drain soil interface is implied.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability in each segment kh/khi.
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter. Default = 0
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soilpermeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.
nflow : float, optional: non_darcian flow exponent
nterms : int, optional: Number of terms to use in series

Returns:

u : float of ndarray of float: Pore pressure at specified si.

Notes

non_darcy_u_piecewise_constant() The pore pressure in the ith segment is given by:

\[u_i(y) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \kappa^{1/n}_i \left({ g\left({y}\right) -g\left({s_{i-1}}\right) }\right) +\psi_i }\right]+u_w\]

where,

\[\psi_{i} = \sum\limits_{j=1}^{i-1}\kappa^{1/n}_j \left[{ g\left({s_j}\right) -g\left({s_{j-1}}\right) }\right]\]

and:

\(g\left({y}\right)\) is described in the _g function

\[y = \frac{r}{r_0}\]

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{hi}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the outer radius of the ith segment, \(k_h\) is the undisturbed horizontal permeability in the ith segment, \(k_{hi}\) is the horizontal permeability in the ith segment

References

none because it is new.

Examples

>>> u_piecewise_constant([1.5, 3,], [2, 3], 1.6, n=5, kap_m=1)
array([0.4153...])
>>> non_darcy_u_piecewise_constant([1.5, 3,], [2, 3], 1.6, n=5, kap_m=1,
... nflow=1.0000001)
array([0.4153...])
>>> non_darcy_u_piecewise_constant([1.5, 3,], [2, 3], 1.6, n=5, kap_m=1,
... nflow=1.3)
array([0.3865...])

geotecha.consolidation.smear_zones.re_from_drain_spacing(sp, pattern='Triangle')[source]¶

Calculate drain influence radius from drain spacing

Parameters:	sp : float Distance between drain centers. pattern : [‘Triangle’, ‘Square’], optional Drain installation pattern. default = ‘Triangle’.
Returns:	re : float drain influence radius

Notes

The influence radius, \(r_e\), is given by:

\[\begin{split}r_e = \left\{\begin{array}{lr} S_p \frac{1}{\sqrt{\pi}}=S_p\times 0.564189583 & \textrm{square pattern}\\ S_p \sqrt{\frac{\sqrt{3}}{2\pi}}=S_p\times 0.525037567 & \textrm{triangular pattern} \end{array}\right.\end{split}\]

References

Eta method is described in [1].

[1]	(1, 2) Walker, Rohan T. 2011. ‘Vertical Drain Consolidation Analysis in One, Two and Three Dimensions’. Computers and Geotechnics 38 (8): 1069-77. doi:10.1016/j.compgeo.2011.07.006.

geotecha.consolidation.smear_zones.scratch()[source]¶: scratch pad for testing latex markup for docstrings

geotecha.consolidation.smear_zones.u_constant(n, s, kap, si, uavg=1, uw=0, muw=0)[source]¶

Pore pressure at radius for constant permeability smear zone

Parameters:

n : float: Ratio of drain influence radius to drain radius (re/rw).
s : float: Ratio of smear zone radius to drain radius (rs/rw).
kap : float: Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks).
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter.

Returns:

u : float or ndarray of float: Pore pressure at specified si

Notes

The uavg is calculated from the eta method. It is not the uavg used when considering the vacuum as an equivalent surcharge. You would have to do other manipulations for that.

Noteing that \(s_i=r_i/r_w\), the radial pore pressure distribution in the smear zone is given by:

\[u^\prime(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \kappa\left({ \ln\left({s_i}\right) -\frac{1}{2n^2}\left({s_i^2-1}\right) }\right) +\mu_w }\right]+u_w\]

The pore pressure in the undisturbed zone is:

\[ \begin{align}\begin{aligned}u(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \ln\left({\frac{s_i}{s}}\right) -\frac{1}{2n^2}\left({s_i^2-s^2}\right) +\kappa\left[{ \ln\left({s}\right) -\frac{1}{2n^2}\left({s^2-1}\right) }\right] +\mu_w }\right]+u_w\\where:\end{aligned}\end{align} \]

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Hansbo, S. 1981. ‘Consolidation of Fine-Grained Soils by Prefabricated Drains’. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

geotecha.consolidation.smear_zones.u_ideal(n, si, uavg=1, uw=0, muw=0)[source]¶

Pore pressure at radius for ideal drain with no smear zone

Parameters:

n : float: Ratio of drain influence radius to drain radius (re/rw).
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter

Returns:

u : float or ndarray of float: Pore pressure at specified si

Notes

The uavg is calculated from the eta method. It is not the uavg used when considering the vacuum as an equivalent surcharge. You would have to do other manipulations for that.

Noteing that \(s_i=r_i/r_w\), the radial pore pressure distribution is given by:

\[u(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \ln\left({\frac{r}{r_w}}\right) -\frac{(r/r_w)^2-1}{2n^2} +\mu_w }\right]+u_w\]

where:

\[n = \frac{r_e}{r_w}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius.

References

[1]	Hansbo, S. 1981. ‘Consolidation of Fine-Grained Soils by Prefabricated Drains’. In 10th ICSMFE, 3:677-82. Rotterdam-Boston: A.A. Balkema.

geotecha.consolidation.smear_zones.u_linear(n, s, kap, si, uavg=1, uw=0, muw=0)[source]¶

Pore pressure at radius for linear smear zone

Parameters:

n : float: Ratio of drain influence radius to drain radius (re/rw).
s : float: Ratio of smear zone radius to drain radius (rs/rw).
kap : float: Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks).
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter.

Returns:

u : float or ndarray of float: Pore pressure at specified si.

Notes

The uavg is calculated from the eta method. It is not the uavg used when considering the vacuum as an equivalent surcharge. You would have to do other manipulations for that.

Noteing that \(s_i=r_i/r_w\), the radial pore pressure distribution in the smear zone is given by:

\[u^\prime(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \kappa\left({\frac{1}{B}\ln\left({s_i}\right) +\left({\frac{B}{A^2n^2}-\frac{1}{B}}\right) \ln\left({B+As_i}\right) +\frac{1-s_i}{An^2} }\right) +\mu_w }\right]+u_w\]

The pore pressure in the undisturbed zone is:

\[u(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \ln\left({\frac{s_i}{s}}\right) -\frac{s_i^2-s^2}{2n^2} +\kappa \left[{ \frac{1}{B}\ln\left({s}\right) +\left({\frac{B}{A^2n^2}-\frac{1}{B}}\right) \ln\left({\kappa}\right) +\frac{1-s}{An^2} }\right] +\mu_w }\right]+u_w\]

for the special case where \(s=\kappa\) the pore pressure in the undisturbed zone is:

\[u^\prime(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ s\frac{\left({n^2-s_i}\right) \left({s_i-1}\right)}{n^2s_i} +\mu_w }\right]+u_w\]

The pore pressure in the undisturbed zone is:

\[u(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \ln\left({\frac{s_i}{s}}\right) +s-1+\frac{s}{n^2} -\frac{s_i^2-s^2}{2n^2} +\mu_w }\right]+u_w\]

where:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

If \(s=1\) or \(\kappa=1\) then u_ideal will be used.

References

[1]	Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

geotecha.consolidation.smear_zones.u_parabolic(n, s, kap, si, uavg=1, uw=0, muw=0)[source]¶

Pore pressure at radius for parabolic smear zone

Parameters:

n : float: Ratio of drain influence radius to drain radius (re/rw).
s : float: Ratio of smear zone radius to drain radius (rs/rw).
kap : float: Ratio of undisturbed horizontal permeability to permeability at the drain-soil interface (kh / ks).
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter.

Returns:

u : float of ndarray of float: Pore pressure at specified si.

Notes

The uavg is calculated from the eta method. It is not the uavg used when considering the vacuum as an equivalent surcharge. You would have to do other manipulations for that.

Noteing that \(s_i=r_i/r_w\), the radial pore pressure distribution in the smear zone is given by:

\[u^\prime(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \frac{\kappa}{\kappa-1}\left\{{ \frac{1}{A^2-B^2} \left({ \ln\left({s_i}\right) -\frac{1}{2A} \left[{ \left({A-B}\right)F +\left({A+B}\right)G }\right] }\right) +\frac{1}{2n^2AC} \left[{ \left({A+B}\right)F +\left({A-B}\right)G }\right] }\right\} +\mu_w }\right]+u_w\]

The pore pressure in the undisturbed zone is:

\[u(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \ln\left({\frac{s_i}{s}}\right) -\frac{s_i^2-s^2}{2n^2} +A^2 \left[{ \frac{1}{A^2-B^2} \left({ \ln\left({s}\right) -\frac{1}{2}\left[{ \ln\left({\kappa}\right) +\frac{BE}{A}}\right] }\right) +\frac{1}{2n^2C^2} \left({\ln\left({\kappa}\right) -\frac{BE}{A}}\right) }\right] +\mu_w }\right]+u_w\]

where \(A\), \(B\), \(C\), \(E\), \(F\), and \(G\) are:

\[A=\sqrt{\frac{\kappa}{\kappa-1}}\]

\[B=\frac{s}{s-1}\]

\[C=\frac{1}{s-1}\]

\[E=\ln\left({\frac{A+1}{A-1}}\right)\]

\[F(r/r_w) = \ln\left({\frac{A+B-Cs_i}{A+1}}\right)\]

\[G(r/r_w) = \ln\left({\frac{A-B+Cs_i}{A-1}}\right)\]

and:

\[n = \frac{r_e}{r_w}\]

\[s = \frac{r_s}{r_w}\]

\[\kappa = \frac{k_h}{k_s}\]

\(r_w\) is the drain radius, \(r_e\) is the drain influence radius, \(r_s\) is the smear zone radius, \(k_h\) is the undisturbed horizontal permeability, \(k_s\) is the smear zone horizontal permeability

References

[1]	Walker, Rohan, and Buddhima Indraratna. 2006. ‘Vertical Drain Consolidation with Parabolic Distribution of Permeability in Smear Zone’. Journal of Geotechnical and Geoenvironmental Engineering 132 (7): 937-41. doi:10.1061/(ASCE)1090-0241(2006)132:7(937).

geotecha.consolidation.smear_zones.u_piecewise_constant(s, kap, si, uavg=1, uw=0, muw=0, n=None, kap_m=None)[source]¶

Pore pressure at radius for piecewise constant permeability distribution

Parameters:

s : list or 1d ndarray of float: Ratio of segment outer radii to drain radius (r_i/r_0). The first value of s should be greater than 1, i.e. the first value should be s_1; s_0=1 at the drain soil interface is implied.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability in each segment kh/khi.
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soilpermeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.

Returns:

u : float of ndarray of float: Pore pressure at specified si.

Notes

The pore pressure in the ith segment is given by:

\[u_i(r) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \kappa_i\left({\ln\left({\frac{r}{r_{i-1}}}\right) -\frac{r^2/r_0^2-s_{i-1}^2}{2n^2}}\right) +\psi_i+\mu_w }\right]+u_w\]

where,

\[\psi_{i} = \sum\limits_{j=1}^{i-1}\kappa_j \left[{ \ln \left({ \frac{s_j}{s_{j-1}} }\right) -\frac{s_j^2-s_{j-1}^2}{2n^2} }\right]\]

and:

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{hi}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the outer radius of the ith segment, \(k_h\) is the undisturbed horizontal permeability in the ith segment, \(k_{hi}\) is the horizontal permeability in the ith segment

References

[1]	Walker, Rohan. 2006. ‘Analytical Solutions for Modeling Soft Soil Consolidation by Vertical Drains’. PhD Thesis, Wollongong, NSW, Australia: University of Wollongong. http://ro.uow.edu.au/theses/501

[2]	Walker, Rohan T. 2011. ‘Vertical Drain Consolidation Analysis in One, Two and Three Dimensions’. Computers and Geotechnics 38 (8): 1069-77. doi:10.1016/j.compgeo.2011.07.006.

geotecha.consolidation.smear_zones.u_piecewise_linear(s, kap, si, uavg=1, uw=0, muw=0, n=None, kap_m=None)[source]¶

Pore pressure at radius for piecewise constant permeability distribution

Parameters:

s : list or 1d ndarray of float: Ratio of radii to drain radius (r_i/r_0). The first value of s should be 1, i.e. at the drain soil interface.
kap : list or ndarray of float: Ratio of undisturbed horizontal permeability to permeability at each value of s.
si : float of ndarray of float: Normalised radial coordinate(s) at which to calc the pore pressure i.e. si=ri/rw.
uavg : float, optional = 1: Average pore pressure in soil. default = 1. when `uw`=0 , then if uavg=1.
uw : float, optional: Pore pressure in drain, default = 0.
muw : float, optional: Well resistance mu parameter.
n, kap_m : float, optional: If n and kap_m are given then they will each be appended to s and kap. This allows the specification of a smear zone separate to the specification of the drain influence radius. Default n=kap_m=None, i.e. soilpermeability is completely described by s and kap. If n is given but kap_m is None then the last kappa value in kap will be used.

Returns:

u : float or ndarray of float: Pore pressure at specified si.

Notes

With permeability in the ith segment defined by:

\[\frac{k_i}{k_{ref}}= \frac{1}{\kappa_{i-1}} \left({A_ir/r_w+B_i}\right)\]

\[A_i = \frac{\kappa_{i-1}/\kappa_i-1}{s_i-s_{i-1}}\]

\[B_i = \frac{s_i-s_{i-1}\kappa_{i-1}/\kappa_i}{s_i-s_{i-1}}\]

The pore pressure in the ith segment is given by:

\[u_i(s) = \frac{u_{avg}-u_w}{\mu+\mu_w} \left[{ \sum\limits_{i=1}^{m}\kappa_{i-1}\phi_i + \Psi_i +\mu_w }\right]+u_w\]

where,

\[\begin{split}\phi_i = \left\{ \begin{array}{lr} \ln\left[{\frac{s}{s_{i-1}}}\right] - \frac{s^2- s_{i-1}^2}{2n^2} & \textrm{for } \frac{\kappa_{i-1}}{\kappa_i}=1 \\ \frac{\left({s - s_{i-1}}\right) \left({n^2-ss_{i-1}}\right)}{sn^2} & \textrm{for }\frac{\kappa_{i-1}}{\kappa_i}= \frac{s_i}{s_{i-1}} \\ \begin{multline} \frac{1}{B_i}\ln\left[{\frac{s}{s_{i-1}}}\right] +\ln\left[{A_is+B_i}\right] \left({\frac{B_i}{A_i^2n^2}-\frac{1}{B_i}}\right) \\-\frac{s-s_{i-1}}{A_i^2n^2} \end{multline} & \textrm{otherwise} \end{array}\right.\end{split}\]

\[\Psi_i = \sum\limits_{j=1}^{i-1}\kappa_{j-1}\psi_j\]

\[\begin{split}\psi_i = \left\{ \begin{array}{lr} \ln\left[{\frac{s_j}{s_{j-1}}}\right] - \frac{s_j^2- s_{j-1}^2}{2n^2} & \textrm{for } \frac{\kappa_{j-1}}{\kappa_j}=1 \\ \frac{\left({s_j - s_{j-1}}\right) \left({n^2-s_js_{j-1}}\right)}{s_jn^2} & \textrm{for }\frac{\kappa_{j-1}}{\kappa_j}= \frac{s_j}{s_{j-1}} \\ \begin{multline} \frac{1}{B_i}\ln\left[{\frac{s_j}{s_{j-1}}}\right] +\ln\left[{\frac{\kappa_{j-1}}{\kappa_j}}\right] \left({\frac{B_j}{A_j^2n^2}-\frac{1}{B_j}}\right) \\-\frac{s_j-s_{j-1}}{A_j^2n^2} \end{multline} & \textrm{otherwise} \end{array}\right.\end{split}\]

and:

\[n = \frac{r_m}{r_0}\]

\[s_i = \frac{r_i}{r_0}\]

\[\kappa_i = \frac{k_h}{k_{ref}}\]

\(r_0\) is the drain radius, \(r_m\) is the drain influence radius, \(r_i\) is the radius of the ith radial point, \(k_{ref}\) is a convienient refernce permeability, usually the undisturbed horizontal permeability, \(k_{hi}\) is the horizontal permeability at the ith radial point

References

Derived by Rohan Walker in 2011 and 2014. Derivation steps are the same as for mu_piecewise_constant in appendix of [1] but permeability is linear in a segemetn as in [2].

[1]	(1, 2) Walker, Rohan. 2006. ‘Analytical Solutions for Modeling Soft Soil Consolidation by Vertical Drains’. PhD Thesis, Wollongong, NSW, Australia: University of Wollongong. http://ro.uow.edu.au/theses/501

[2]	(1, 2) Walker, R., and B. Indraratna. 2007. ‘Vertical Drain Consolidation with Overlapping Smear Zones’. Geotechnique 57 (5): 463-67. doi:10.1680/geot.2007.57.5.463.

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