velocity_hydraulic_gradient

Class summary

DarcyFlowModel([k]) Darcian flow model
HansboNonDarcianFlowModel([klinear, kstar, …]) Hansbo non-darcian flow model
OneDimensionalFlowRelationship Base class for defining velocity vs hydraulic gradient relationships

Module listing

Relationships between velocity and hydraulic gradient

class geotecha.constitutive_models.velocity_hydraulic_gradient.DarcyFlowModel(k=1.0)[source]

Bases: geotecha.constitutive_models.velocity_hydraulic_gradient.OneDimensionalFlowRelationship

Darcian flow model

Parameters:
k : float, optional

Darcian permeability. Default k=1.

Notes

Darcian flow is described by:

\[v = ki\]

Methods

dv_di(hyd_grad, **kwargs) Slope of velocity vs hydraulic gradient relationship
i_from_v(velocity, **kwargs) Hydraulic gradient from velocity
plot_model(**kwargs) Plot the velocity-hydraulic gradient relationship
v_and_i_for_plotting(**kwargs) Velocity and hydraulic gradient that plot the relationship
v_from_i(hyd_grad, **kwargs) Velocity from hydraulic gradient
vdrain_strain_rate(eta, head, **kwargs) Vertical drain strain rate as based on the eta method”“
dv_di(hyd_grad, **kwargs)[source]

Slope of velocity vs hydraulic gradient relationship

Parameters:
hyd_grad : float

Hydraulic gradient.

**kwargs : any

Any keyword arguments are ignored.
Returns:
slope : float

slope of velocity-hydraulic gradient relationship at hyd_grad.

Examples

>>> a = DarcyFlowModel(k=3)
>>> a.dv_di(8.0)
3.0
>>> a.dv_di(-8.0)
-3.0
>>> a.dv_di(np.array([5.0, 8.0]))
array([3., 3.])
i_from_v(velocity, **kwargs)[source]

Hydraulic gradient from velocity

Parameters:
v : float

Flow velocity.

**kwargs : any

Any keyword arguments are ignored.
Returns:
hyd_grad : float

Hydraulic gradient.

Examples

>>> a = DarcyFlowModel(k=3)
>>> a.i_from_v(24.0)
8.0
>>> a.i_from_v(-24.0)
-8.0
>>> a.i_from_v(np.array([15, 24]))
array([5., 8.])
non_Darcy = False
v_and_i_for_plotting(**kwargs)[source]

Velocity and hydraulic gradient that plot the relationship

Parameters:
npts : int, optional

Number of points to return. Default npts=100.

xmin, xmax : float, optional

Range of x (i.e. hydraulic gradient) values from which to return points. Default xmin=0, xmax=50.
Returns:
x, y : 1d ndarray

npts permeability, and void ratio values between xmin and xmax.
v_from_i(hyd_grad, **kwargs)[source]

Velocity from hydraulic gradient

Parameters:
hyd_grad : float

Hydraulic gradient.

**kwargs : any

Any keyword arguments are ignored.
Returns:
v : float

Flow velocity.

Examples

>>> a = DarcyFlowModel(k=3)
>>> a.v_from_i(8.0)
24.0
>>> a.v_from_i(-8.0)
-24.0
>>> a.v_from_i(np.array([5.0, 8.0]))
array([15., 24.])
vdrain_strain_rate(eta, head, **kwargs)[source]

Vertical drain strain rate as based on the eta method”“

[strain rate] = head * self.k * eta

Parameters:
eta : float

Value of vertical drain geometry, peremability parameter. This value should be calculated based on Darcy’s law (see geotecha.consolidation.smearzones.drain_eta)

head : float

Hydraulic head driving the flow. For vertical drains this is usually the difference between the average head in the soil and the head in the drain.

**kwargs : any

Any additional keyword arguments are ignored.
Returns:
strain_rate : float

Strain rate based on eta method.

See also

geotecha.consolidation.smearzones
Functions to determine eta.

Examples

>>> a = DarcyFlowModel(k=3)
>>> a.vdrain_strain_rate(eta=2.5, head=4)
30.0
class geotecha.constitutive_models.velocity_hydraulic_gradient.HansboNonDarcianFlowModel(klinear=None, kstar=None, n=None, i0=None, iL=None)[source]

Bases: geotecha.constitutive_models.velocity_hydraulic_gradient.OneDimensionalFlowRelationship

Hansbo non-darcian flow model

Various combinations of parameters can be used to specify the model. Basically three parameters are required, one of which must be kstar or klinear. If n==1 or i0=0, or iL=0 then the model will reduce to Darcy flow model with peremability equal to klinear (or kstar if klinear is not defined).

Parameters:
klinear : float, optional

This is the slope of the linear portion of the v-i relationship for i>iL. klinear = kstar * n * iL**(n-1). Default klinear=None. klinear and kstar cannot both be None, you must use one or the other.

kstar : float, optional

Permability coefficient in the power law, v=kstar*i**n, flow section, i<iL. Default kstar=None. klinear and kstar cannot both be None.

n : float, optional

Exponent for non-Darcian portion of flow. n must be greater than or equal to 1. Default n=None.

i0 : float, optional

x-axis intercept of the linear portion of the flow law. Default i0=None. i0 must be greater or equal to zero.

iL : float, optional

Limiting hydraulic gradient, beyond which v-i relationship is linear. iL = i0 * n / (n -1). Default iL=None.

Notes

Hansbo’s Non-darcian flow relationship is defined by:

\[\begin{split}v=\left\{\begin{array}{lr} k^{\ast}i^{n} & i<i_{L} \\ k_{\rm{linear}} \left({i-i_0}\right) & i\geq i_{L} \end{array}\right.\end{split}\]

where,

\[k_{\rm{linear}} = k^{\ast}ni_L^\left(n-1\right)\]
\[i_L=\frac{i_0 n}{\left(n-1\right)}\]

Examples

These examples show the various ways to define the model: >>> p = dict(kstar=2, n=1.3, iL=16.2402611294, i0=3.7477525683, klinear=6) >>> a = HansboNonDarcianFlowModel(kstar=p[‘kstar’], n=p[‘n’], iL=p[‘iL’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(kstar=p[‘kstar’], n=p[‘n’], i0=p[‘i0’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(kstar=p[‘kstar’], iL=p[‘iL’], i0=p[‘i0’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(kstar=p[‘kstar’], iL=p[‘iL’], klinear=p[‘klinear’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… >>> a = HansboNonDarcianFlowModel(kstar=p[‘kstar’], i0=p[‘i0’], klinear=p[‘klinear’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(n=p[‘n’], iL=p[‘iL’], klinear=p[‘klinear’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(n=p[‘n’], i0=p[‘i0’], klinear=p[‘klinear’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6 >>> a = HansboNonDarcianFlowModel(iL=p[‘iL’], i0=p[‘i0’], klinear=p[‘klinear’]) >>> print(a) kstar = 2 n = 1.3 iL = 16.240… i0 = 3.747… klinear = 6

If the behaviour is not as you expect when i0=0, iL=0, n==1 then use very small values of i0 or iL, n just greater than 1.

Methods

dv_di(hyd_grad, **kwargs) Slope of velocity vs hydraulic gradient relationship
i_from_v(velocity, **kwargs) Hydraulic gradient from velocity
plot_model(**kwargs) Plot the velocity-hydraulic gradient relationship
v_and_i_for_plotting(**kwargs) Velocity and hydraulic gradient that plot the relationship
v_from_i(hyd_grad, **kwargs) Velocity from hydraulic gradient
vdrain_strain_rate(eta, head, **kwargs) Vertical drain strain rate as based on the eta method”“
dv_di(hyd_grad, **kwargs)[source]

Slope of velocity vs hydraulic gradient relationship

Parameters:
hyd_grad : float

Hydraulic gradient.

**kwargs : any

Any keyword arguments are ignored.
Returns:
slope : float

slope of velocity-hydraulic gradient relationship at hyd_grad.

Examples

>>> a = HansboNonDarcianFlowModel(kstar=2, n=1.3, iL=16.2402611294)
>>> a.dv_di(8.0)
4.851...
>>> a.dv_di(-8.0)
-4.851...
>>> a.dv_di(np.array([8.0, 20.0]))
array([4.851..., 6...])
i_from_v(velocity, **kwargs)[source]

Hydraulic gradient from velocity

Parameters:
v : float

Flow velocity.

**kwargs : any

Any keyword arguments are ignored.
Returns:
velocity : float

Flow velocity.

Examples

>>> a = HansboNonDarcianFlowModel(kstar=2, n=1.3, iL=16.2402611294)
>>> a.i_from_v(29.858)
8.0...
>>> a.i_from_v(-29.858)
-8.0...
>>> a.i_from_v(np.array([29.858, 97.514]))
array([ 8.0..., 20.0...])
non_Darcy = True
v_and_i_for_plotting(**kwargs)[source]

Velocity and hydraulic gradient that plot the relationship

Parameters:
npts : int, optional

Number of points to return. Default npts=100.

xmin, xmax : float, optional

Range of x (i.e. hydraulic gradient) values from which to return points. Default xmin=0, xmax=50.
Returns:
x, y : 1d ndarray

npts permeability, and void ratio values between xmin and xmax.
v_from_i(hyd_grad, **kwargs)[source]

Velocity from hydraulic gradient

Parameters:
hyd_grad : float

Hydraulic gradient.

**kwargs : any

Any keyword arguments are ignored.
Returns:
v : float

Flow velocity.

Examples

>>> a = HansboNonDarcianFlowModel(kstar=2, n=1.3, iL=16.2402611294)
>>> a.v_from_i(8.0)
29.857...
>>> a.v_from_i(-8.0)
-29.857...
>>> a.v_from_i(np.array([8.0, 20.0]))
array([29.857..., 97.513...])
vdrain_strain_rate(eta, head, **kwargs)[source]

Vertical drain strain rate as based on the eta method”“

[strain rate] = head**self.nflow * self.klinear * gamw**(nflow - 1) * eta

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

Parameters:
eta : float

Value of vertical drain geometry, peremability parameter. This value should be calculated based on Hansbo’s non-Darcian flow model (see geotecha.consolidation.smearzones.non_darcy_drain_eta)

head : float

Hydraulic head driving the flow. For vertical drains this is usually the difference between the average head in the soil and the head in the drain.

gamw : float, optional

Unit weight of water. Note that this gamw must be consistent with the value used to determine eta. Default gamw=10.

**kwargs : any

Any additional keyword arguments, other than ‘gamw’, are ignored.
Returns:
strain_rate : float

Strain rate based on eta method.

See also

geotecha.consolidation.smearzones
Functions to determine eta.

Examples

>>> a = HansboNonDarcianFlowModel(klinear=2, n=1.3, iL=16.2402611294)
>>> a.vdrain_strain_rate(eta=0.1, head=4, gamw=10)
2.419...
class geotecha.constitutive_models.velocity_hydraulic_gradient.OneDimensionalFlowRelationship[source]

Bases: object

Base class for defining velocity vs hydraulic gradient relationships

Attributes:
non_Darcy: True

A class variable indicating if model has a non-Darcian flow relationship. Can be overridden in subclasses. Basically the Darcian flow equations are simple and it might be simpler to bypass the use of a OneDimensionalFlowRelationship.

Methods

dv_di(hyd_grad, **kwargs) Slope of velocity vs hydraulic gradient relationship
i_from_v(velocity, **kwargs) Hydraulic gradient from velocity
plot_model(**kwargs) Plot the velocity-hydraulic gradient relationship
v_and_i_for_plotting(**kwargs) Velocity and hydraulic gradient that that plot the relationship
v_from_i(hyd_grad, **kwargs) Velocity from hydraulic gradient
vdrain_strain_rate(eta, head, **kwargs) Vertical drain strain rate as based on the eta method
dv_di(hyd_grad, **kwargs)[source]

Slope of velocity vs hydraulic gradient relationship

i_from_v(velocity, **kwargs)[source]

Hydraulic gradient from velocity

non_Darcy = True
plot_model(**kwargs)[source]

Plot the velocity-hydraulic gradient relationship

v_and_i_for_plotting(**kwargs)[source]

Velocity and hydraulic gradient that that plot the relationship

v_from_i(hyd_grad, **kwargs)[source]

Velocity from hydraulic gradient

vdrain_strain_rate(eta, head, **kwargs)[source]

Vertical drain strain rate as based on the eta method