ttim

Copyright (C), 2017, Mark Bakker.

TTim is a computer program for the simulation of transient multi-layer flow with analytic elements and consists of a library of Python scripts and FORTRAN extensions.

Mark Bakker, Delft University of Technology mark dot bakker at tudelft dot nl.

Submodules

• ttim.aquifer
• ttim.aquifer_parameters
• ttim.besselnumba
• ttim.besselnumba_old
• ttim.besselnumba_total
• ttim.besselnumbanew
• ttim.circareasink
• ttim.circinhom
• ttim.element
• ttim.equation
• ttim.fit
• ttim.inhom1d
• ttim.invlapnumba
• ttim.kuhlman_invlap
• ttim.linedoublet
• ttim.linedoublet1d
• ttim.linesink
• ttim.linesink1d
• ttim.model
• ttim.plots
• ttim.stripareasink
• ttim.trace
• ttim.version
• ttim.well

Attributes

__name__
__author__
__version__

Classes

CircAreaSink	Create a circular area-sink with uniform infiltration rate in aquifer layer 0.
Calibrate
Xsection3D	Cross-section inhomogeneity consisting of stacked aquifer layers.
XsectionMaq	Cross-section inhomogeneity consisting of stacked aquifer layers.
LeakyLineDoublet	Create a segment of a leaky wall, which is simulated with a line-doublet.
LeakyLineDoubletString	Create a string of leaky wall segements consisting of line-doublets.
LeakyLineDoublet1D	Leaky line doublet with specified resistance.
LineDoublet1DBase	LineDoublet1D Base Class.
HeadLineSink	Create a head-specified line-sink with optional width and resistance.
HeadLineSinkHo	HeadLineSink of which the head varies through time.
HeadLineSinkString	String of head-specified line-sinks with optional width and resistance.
LineSink	LineSink with non-zero and potentially variable discharge through time.
LineSinkDitchString	Create ditch consisting of a string of line-sink.
DischargeLineSink1D	Linesink1D with a specified discharge for each layer the linesink is in.
FluxDiffLineSink1D	1D flux-difference linesink element.
HeadDiffLineSink1D	1D head-difference linesink element.
HeadLineSink1D	1D head-specified linesink element.
LineSink1D	Linesink1D with a specified discharge.
LineSink1DBase	LineSink1D Base Class.
Model3D	Create a multi-layer model object consisting of many aquifer layers.
ModelMaq	Create model specifying a multi-aquifer sequence of aquifer-leakylayer-etc.
ModelXsection	Model class for cross-section models.
DischargeWell	Well with a specified discharge for each layer that the well is screened in.
HeadWell	Create a well with a specified head inside the well.
Well	Create a well with a specified discharge.
WellTest	Well Base Class.

Functions

timtrace(ml, xstart, ystart, zstart, tstartend, ...[, ...])	Compute a pathline by numerical integration of the velocity vector.
timtraceline(ml, xstart, ystart, zstart, tstart, delt, ...)
show_versions([optional])	Print the version of dependencies.

Package Contents

ttim.__name__ = 'ttim'

ttim.__author__ = 'Mark Bakker'

class ttim.CircAreaSink(model, xc=0, yc=0, R=0.1, tsandN=[(0, 1)], name='CircAreaSink', label=None)

Bases: ttim.element.Element

Create a circular area-sink with uniform infiltration rate in aquifer layer 0.

Infiltration rate in length / time, positive for water entering the aquifer.

Parameters:

• model (Model object) – model to which the element is added

• xc (float) – x-coordinate of center of area-sink

• yc (float) – y-coordinate of center of area-sink

• R (radius of area-sink)

• tsandN (list of tuples) – tuples of starting time and infiltration rate after starting time

• label (string or None (default: None)) – label of the area-sink

xc

yc

R

__repr__()

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

setflowcoef()

Separate function so that this can be overloaded for other types.

potinf(x, y, aq=None)

Can be called with only one x,y value.

disvecinf(x, y, aq=None)

Can be called with only one x,y value.

plot(ax=None)

Plot the element.

K1RI0r(rin, iaq, ipint)

I1RK0r(rin, iaq, ipint)

K1RI1r(rin, iaq, ipint)

I1RK1r(rin, iaq, ipint)

class ttim.Calibrate(model)

model

parameters

seriesdict

seriesinwelldict

set_parameter(name=None, layers=None, initial=0, pmin=-np.inf, pmax=np.inf, inhoms=None)

Set parameter to be optimized.

Parameters:

• name (str) – name can be ‘kaq’, ‘Saq’, ‘c’, ‘Sll’ or ‘kzoverkh’.

• layers (int or list of ints) – layer number(s) for which the parameter is set. If an integer is passed, parameter is associated with a single layer. If a list of layers is passed, layers must be consecutive and parameter is set for each layer from min(layers) up to and including max(layers).

• initial (float, optional) – initial value for the parameter (the default is 0)

• pmin (float, optional) – lower bound for parameter value (the default is -np.inf)

• pmax (float, optional) – upper bound for paramater value (the default is np.inf)

• inhoms (str, list) – string with name of inhomogeneity, list with string names of inhomogeneities or list of inhomogeneities inhomogeneity(ies) for which the parameter is set. If a string is passed, parameter is associated with a single inhomogeneity. If a list of strings of inhomogeneities (or list of inhomogeneities) is passed, parameter is set for each inhomogeneity in the list. This allows linking of parameters across inhomogeneities.

set_parameter_by_reference(name=None, parameter=None, initial=0, pmin=-np.inf, pmax=np.inf)

Set parameter to be optimized.

Parameters:

• name (str) – parameter name

• parameter (np.array) – array reference containing the parameter to be optimized. must be specified as reference, i.e. w.rc[0:]

• initial (float, optional) – initial value for the parameter (the default is 0)

• pmin (float, optional) – lower bound for parameter value (the default is -np.inf)

• pmax (float, optional) – upper bound for paramater value (the default is np.inf)

series(name, x, y, layer, t, h, weights=None)

Method to add observations to Calibration object.

Parameters:

• name (str) – name of series

• x (float) – x-coordinate

• y (float) – y-coordinate

• layer (int) – layer number, 0-indexed

• t (np.array) – array containing timestamps of timeseries

• h (np.array) – array containing timeseries values, i.e. head observations

seriesinwell(name, element, t, h)

Method to add observations to Calibration object.

Parameters:

• name (str) – name of series

• element (element object with headinside function)

• t (np.array) – array containing timestamps of timeseries

• h (np.array) – array containing timeseries values, i.e. head observations

residuals(p, printdot=False, weighted=True, layers=None, series=None)

Method to calculate residuals given certain parameters.

Parameters:

• p (np.array) – array containing parameter values

• printdot (bool, optional) – print dot for each function call

Returns:

array containing all residuals

Return type:

np.array

residuals_lmfit(lmfitparams, printdot=False)

fit_least_squares(report=True, diff_step=0.0001, xtol=1e-08, method='lm')

fit_lmfit(report=False, printdot=True, **kwargs)

residuals_leastsq(logparams, printdot=False)

fit_leastsq(report=True, diff_step=0.0001, xtol=1e-08)

fit(report=False, printdot=True, **kwargs)

rmse(weighted=True, layers=None)

Calculate root-mean-squared-error.

Returns:

return rmse value

Return type:

float

topview(ax=None, layers=None, labels=True)

Plot topview of model with calibration points.

Parameters:

ax (matplotlib.axes.Axes, optional) – axes to plot on (the default is None, which creates a new figure)

xsection(ax=None, labels=True)

Plot cross-section of model with calibration points.

Parameters:

ax (matplotlib.axes.Axes, optional) – axes to plot on (the default is None, which creates a new figure)

class ttim.Xsection3D(model, x1, x2, kaq=1, z=(4, 3, 2, 1), Saq=0.001, kzoverkh=0.1, poraq=0.3, topboundary='conf', phreatictop=False, topres=0, topthick=0, topSll=0, toppor=0.3, tsandhstar=None, tsandN=None, name=None)

Bases: Xsection

Cross-section inhomogeneity consisting of stacked aquifer layers.

Vertical resistance is computed from vertical hydraulic conductivity and the anisotropy factor.

Parameters:

• model (Model) – Model to add the cross-section to, usually an instance of ModelXsection.

• x1 (scalar) – x-coordinate of the left boundary of the cross-section.

• x2 (scalar) – x-coordinate of the right boundary of the cross-section.

• kaq (array) – Hydraulic conductivities of the aquifers.

• z (array) – Elevations of the tops and bottoms of the layers.

• Saq (array) – Specific storage of the aquifers.

• kzoverkh (scalar) – Ratio of vertical hydraulic conductivity to horizontal hydraulic conductivity.

• poraq (array) – Porosities of the aquifers.

• topboundary (str) – Type of top boundary. Can be ‘conf’ for confined, ‘semi’ for semi-confined or “leaky” for a leaky top boundary.

• phreatictop (bool) – If true, interpret the first specific storage coefficient as specific yield., i.e. it is not multiplied by aquifer thickness.

• topres (scalar) – Resistance of the top boundary. Only used if topboundary is ‘leaky’.

• topthick (scalar) – Thickness of the top boundary. Only used if topboundary is ‘leaky’.

• topSll (scalar) – Specific storage of the top boundary. Only used if topboundary is ‘leaky’.

• toppor (scalar) – Porosity of the top boundary. Only used if topboundary is ‘leaky’.

• tsandhstar (list of tuples) – list containing time and water level pairs for the hstar boundary condition.

• tsandN (list of tuples) – list containing time and infiltration pairs for the infiltration boundary condition.

• name (str) – Name of the cross-section.

class ttim.XsectionMaq(model, x1, x2, kaq=1, z=(1, 0), c=(), Saq=0.001, Sll=0, poraq=0.3, porll=0.3, topboundary='conf', phreatictop=False, tsandhstar=None, tsandN=None, name=None)

Bases: Xsection

Cross-section inhomogeneity consisting of stacked aquifer layers.

Parameters:

• model (Model) – Model to add the cross-section to, usually an instance of ModelXsection.

• x1 (float) – x-coordinate of the left boundary of the cross-section.

• x2 (float) – x-coordinate of the right boundary of the cross-section.

• kaq (array) – Hydraulic conductivities of the aquifers.

• z (array) – Elevations of the tops and bottoms of the layers.

• c (array) – Resistance of the leaky layers.

• Saq (array) – Specific storage of the aquifers.

• Sll (array) – Specific storage of the leaky layers.

• poraq (array) – Porosities of the aquifers.

• porll (array) – Porosities of the leaky layers.

• topboundary (str) – Type of top boundary. Can be ‘conf’ for confined, ‘semi’ for semi-confined or “leaky” for a leaky top boundary.

• phreatictop (bool) – If true, interpret the first specific storage coefficient as specific yield., i.e. it is not multiplied by aquifer thickness.

• tsandhstar (list of tuples) – list containing time and water level pairs for the hstar boundary condition.

• tsandN (list of tuples) – list containing time and infiltration pairs for the infiltration boundary condition.

• name (str) – Name of the cross-section.

class ttim.LeakyLineDoublet(model, x1=-1, y1=0, x2=1, y2=0, res='imp', order=0, layers=0, label=None, addtomodel=True)

Bases: LineDoubletHoBase, ttim.equation.LeakyWallEquation

Create a segment of a leaky wall, which is simulated with a line-doublet.

The specific discharge through the wall is equal to the head difference across the wall divided by the resistance of the wall.

Parameters:

• model (Model object) – Model to which the element is added

• x1 (scalar) – x-coordinate of fist point of line-doublet

• y1 (scalar) – y-coordinate of fist point of line-doublet

• x2 (scalar) – x-coordinate of second point of line-doublet

• y2 (scalar) – y-coordinate of second point of line-doublet

• res (scalar or string) – if string: ‘imp’ for an impermeable wall (same as res = np.inf) if scalar: resistance of leaky wall

• order (int (default is 0)) – polynomial order of potential jump along line-doublet (head jump if transmissivity is equal on each side of wall)

• layers (scalar, list or array) – layer(s) in which element is placed if scalar: element is placed in this layer if list or array: element is placed in all these layers

• label (str or None) – label of element

See also

LeakyLineDoubletString

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.LeakyLineDoubletString(model, xy=[(-1, 0), (1, 0)], res='imp', order=0, layers=0, label=None)

Bases: ttim.element.Element, ttim.equation.LeakyWallEquation

Create a string of leaky wall segements consisting of line-doublets.

Parameters:

• model (Model object) – Model to which the element is added

• xy (array or list) – list or array of (x,y) pairs of coordinates of end-points of the segements in the string

• res (scalar or string) – if string: ‘imp’ for an impermeable wall (same as res = np.inf) if scalar: resistance of leaky wall

• order (int (default is 0)) – polynomial order of potential jump along line-doublet (head jump if transmissivity is equal on each side of wall)

• layers (scalar, list or array) – layer(s) in which element is placed if scalar: element is placed in this layer if list or array: element is placed in all these layers

• label (str or None) – label of element

See also

LeakyLineDoublet

res = 'imp'

order = 0

ldlist = []

nld

__repr__()

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

potinf(x, y, aq=None)

Returns array (nunknowns,nperiods).

disvecinf(x, y, aq=None)

Returns array (nunknowns,nperiods).

plot(ax=None)

Plot the element.

class ttim.LeakyLineDoublet1D(model, xld=0, res='imp', layers=0, label=None)

Bases: LineDoublet1DBase, ttim.equation.LeakyWallEquation

Leaky line doublet with specified resistance.

Parameters:

• model (Model object) – model to which the element is added

• xld (float) – x-coordinate of the line doublet

• res (float) – resistance of the line doublet

• layers (int, array or list) – layer (int) or layers (list or array) in which line doublet is located

• label (string or None (default: None)) – label of the element

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.LineDoublet1DBase(model, xld=0, tsandbc=[(0, 0)], res='imp', layers=0, type='', name='LineDoublet1DBase', label=None)

Bases: ttim.element.Element

LineDoublet1D Base Class.

All LineDoublet1D elements are derived from this class

Parameters:

• model (Model object) – Model to which the element is added

• xld (float) – x-coordinate of the line doublet

• tsandbc (list of tuples) – list of tuples of the form (time, bc) for boundary conditions

• res (float) – resistance of the line doublet

• layers (int, array or list) – layer (int) or layers (list or array) in which line doublet is located

• type (string) – type of element, “g” for given, “v” for variable and “z” for zero.

• name (string) – name of the element

• label (string, optional) – label of the element

tiny = 1e-08

nparam

xld

__repr__()

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

setflowcoef()

Separate function so that this can be overloaded for other types.

potinf(x, y=0, aq=None)

Can be called with only one x value.

disvecinf(x, y=0, aq=None)

Can be called with only one x value.

abstractmethod changetrace(xyzt1, xyzt2, aq, layer, ltype, modellayer, direction, hstepmax)

plot(ax=None)

Plot the element.

class ttim.HeadLineSink(model, x1=-1, y1=0, x2=1, y2=0, tsandh=[(0, 1)], res=0, wh='H', layers=0, label=None, addtomodel=True)

Bases: LineSinkBase, ttim.equation.HeadEquation

Create a head-specified line-sink with optional width and resistance.

Inflow per unit length of line-sink is computed as:

\[\sigma = w(h_{aq} - h_{ls})/c\]

where \(c\) is the resistance of the bottom of the line-sink, \(w\) is the width over which water enters the line-sink, \(h_{aq}\) is the head in the aquifer at the center of the line-sink, \(h_{ls}\) is the specified head inside the line-sink Note that all that matters is the conductance term \(w/c\) but both are specified separately

Parameters:

• model (Model object) – Model to which the element is added

• x1 (scalar) – x-coordinate of fist point of line-sink

• y1 (scalar) – y-coordinate of fist point of line-sink

• x2 (scalar) – x-coordinate of second point of line-sink

• y2 (scalar) – y-coordinate of second point of line-sink

• tsandh (list or 2D array of (time, head) values or string) – if list or 2D array: pairs of time and head after that time if ‘fixed’: head is fixed (no change in head) during entire simulation

• res (scalar (default is 0)) – resistance of line-sink

• wh (scalar or str) – distance over which water enters line-sink if ‘H’: (default) distance is equal to the thickness of the aquifer layer (when flow comes mainly from one side) if ‘2H’: distance is twice the thickness of the aquifer layer (when flow comes from both sides) if scalar: the width of the stream that partially penetrates the aquifer layer

• layers (scalar, list or array) – layer(s) in which element is placed if scalar: element is placed in this layer if list or array: element is placed in all these layers

• label (str or None) – label of element

See also

HeadLineSinkString

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.HeadLineSinkHo(model, x1=-1, y1=0, x2=1, y2=0, tsandh=[(0.0, 1.0)], order=0, layers=0, label=None, addtomodel=True)

Bases: LineSinkHoBase, ttim.equation.HeadEquationNores

HeadLineSink of which the head varies through time.

May be screened in multiple layers but all with the same head

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.HeadLineSinkString(model, xy=[(-1, 0), (1, 0)], tsandh=[(0, 1)], res=0, wh='H', layers=0, label=None)

Bases: LineSinkStringBase, ttim.equation.HeadEquation

String of head-specified line-sinks with optional width and resistance.

Inflow per unit length of line-sink is computed as:

\[\sigma = w(h_{aq} - h_{ls})/c\]

where \(c\) is the resistance of the bottom of the line-sink, \(w\) is the width over which water enters the line-sink, \(h_{aq}\) is the head in the aquifer at the center of the line-sink, \(h_{ls}\) is the specified head inside the line-sink Note that all that matters is the conductance term \(w/c\) but both are specified separately

Parameters:

• model (Model object) – Model to which the element is added

• xy (array or list) – list or array of (x,y) pairs of coordinates of end-points of line-sinks in string

• tsandh (list or 2D array of (time, head) values or string) – if list or 2D array: pairs of time and head after that time if ‘fixed’: head is fixed (no change in head) during entire simulation

• res (scalar (default is 0)) – resistance of line-sink

• wh (scalar or str) – distance over which water enters line-sink if ‘H’: (default) distance is equal to the thickness of the aquifer layer (when flow comes mainly from one side) if ‘2H’: distance is twice the thickness of the aquifer layer (when flow comes from both sides) if scalar: the width of the stream that partially penetrates the aquifer layer

• layers (scalar, list or array) – layer(s) in which element is placed if scalar: element is placed in this layer if list or array: element is placed in all these layers

• label (str or None) – label of element

See also

HeadLineSink

x

y

nls

tsandh = [(0, 1)]

res

wh = 'H'

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.LineSink(model, x1=-1, y1=0, x2=1, y2=0, tsandQ=[(0, 1)], res=0, wh='H', layers=0, label=None, addtomodel=True)

Bases: LineSinkBase

LineSink with non-zero and potentially variable discharge through time.

Really only used for testing.

class ttim.LineSinkDitchString(model, xy=[(-1, 0), (1, 0)], tsandQ=[(0, 1)], res=0, wh='H', layers=0, Astorage=None, label=None)

Bases: LineSinkStringBase, ttim.equation.MscreenDitchEquation

Create ditch consisting of a string of line-sink.

The total discharge for the string is specified and divided over the line-sinks such that the head at the center inside each line-sink is equal. A width and resistance may optionally be specified.

Inflow per unit length of line-sink is computed as:

\[\sigma = w(h_{aq} - h_{ls})/c\]

where \(c\) is the resistance of the bottom of the line-sink, \(w\) is the width over which water enters the line-sink, \(h_{aq}\) is the head in the aquifer at the center of the line-sink, \(h_{ls}\) is the specified head inside the line-sink Note that all that matters is the conductance term \(w/c\) but both are specified separately

Parameters:

• model (Model object) – Model to which the element is added

• xy (array or list) – list or array of (x,y) pairs of coordinates of end-points of line-sinks in string

• tsandQ (list or 2D array of (time, discharge) values) – if list or 2D array: pairs of time and discharge after that time

• res (scalar (default is 0)) – resistance of line-sink

• wh (scalar or str) – distance over which water enters line-sink if ‘H’: (default) distance is equal to the thickness of the aquifer layer (when flow comes mainly from one side) if ‘2H’: distance is twice the thickness of the aquifer layer (when flow comes from both sides) if scalar: the width of the stream that partially penetrates the aquifer layer

• layers (scalar, list or array) – layer(s) in which element is placed if scalar: element is placed in this layer if list or array: element is placed in all these layers

• label (str or None) – label of element

nls

Astorage = None

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.DischargeLineSink1D(model, xls=0, tsandq=[(0, 1)], res=0, wh='H', layers=0, label=None)

Bases: LineSink1DBase

Linesink1D with a specified discharge for each layer the linesink is in.

Parameters:

• model (Model object) – model to which the element is added

• x (float) – x-coordinate of the linesink

• tsandq (list of tuples) – tuples of starting time and specific discharge after starting time

• res (float) – resistance of the linesink

• layers (int, array or list) – layer (int) or layers (list or array) in which linesink is located

• label (string or None (default: None)) – label of the linesink

Examples

Example of an infinitely long linesink that pumps with a specific discharge of 100 between times 10 and 50, with a specific discharge of 20 between times 50 and 200, and zero speficic discharge after time 200.

>>> DischargeLineSink1D(ml, tsandq=[(10, 100), (50, 20), (200, 0)])

class ttim.FluxDiffLineSink1D(model, xls=0, layers=0, label=None, aq=None)

Bases: LineSink1DBase, ttim.equation.FluxDiffEquation

1D flux-difference linesink element.

Used to ensure continuity of flux in a cross-section model, e.g. at the boundary of an inhomogeneity.

Parameters:

• model (Model object) – Model to which the element is added

• xls (float) – x-coordinate of the linesink

• layers (int, array or list) – layer (int) or layers (list or array) in which linesink is located

• label (string, optional) – label of the element

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

plot(ax=None)

Plot the element.

class ttim.HeadDiffLineSink1D(model, xls=0, layers=0, label=None, aq=None)

Bases: LineSink1DBase, ttim.equation.HeadDiffEquation

1D head-difference linesink element.

Used to ensure continuity of head in a cross-section model, e.g. at the boundary of an inhomogeneity.

Parameters:

• model (Model object) – Model to which the element is added

• xls (float) – x-coordinate of the linesink

• layers (int, array or list) – layer (int) or layers (list or array) in which linesink is located

• label (string, optional) – label of the element

• aq (Aquifer object) – aquifer in which the element is located

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

plot(ax=None)

Plot the element.

class ttim.HeadLineSink1D(model, xls=0, tsandh=[(0, 1)], res=0, wh='H', layers=0, label=None)

Bases: LineSink1DBase, ttim.equation.HeadEquation

1D head-specified linesink element.

Parameters:

• model (Model object) – Model to which the element is added

• xls (float) – x-coordinate of the linesink

• tsandh (list of tuples) – list of tuples of the form (time, head) for head conditions

• res (float) – resistance of the linesink

• wh (string or float) – wetted perimeter of the linesink, “H” for aquifer height, “2H” for 2x aquifer height (two-sided flow) or specify any float value

• layers (int, array or list) – layer (int) or layers (list or array) in which linesink is located

• label (string, optional) – label of the element

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.LineSink1D(model, xls=0, tsandq=[(0, 1)], res=0, wh='H', vres=0.0, wv=1.0, layers=0, label=None)

Bases: LineSink1DBase, ttim.equation.MscreenEquation

Linesink1D with a specified discharge.

Parameters:

• model (Model object) – model to which the element is added

• x (float) – x-coordinate of the linesink

• tsandq (list of tuples) – tuples of starting time and specific discharge after starting time

• res (float) – resistance of the linesink

• layers (int, array or list) – layer (int) or layers (list or array) in which linesink is located

• label (string or None (default: None)) – label of the linesink

Examples

Example of an infinitely long linesink that pumps with a specific discharge of 100 between times 10 and 50, with a specific discharge of 20 between times 50 and 200, and zero specific discharge after time 200.

>>> LineSink1D(ml, tsandq=[(10, 100), (50, 20), (200, 0)])

nunknowns

vres

wv = 1.0

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.LineSink1DBase(model, xls=0, tsandbc=[(0, 1)], res=0, wh='H', layers=0, type='', name='LineSink1DBase', label=None, aq=None, inhomelement=False)

Bases: ttim.element.Element

LineSink1D Base Class.

All LineSink1D elements are derived from this class

Parameters:

• model (Model object) – Model to which the element is added

• xls (float) – x-coordinate of the line sink

• tsandbc (list of tuples) – list of tuples of the form (time, bc) for boundary conditions

• res (float) – resistance of the line sink

• wh (string or float) – wetted perimeter of the linesink, “H” for aquifer height, “2H” for 2x aquifer height (two-sided flow) or specify any float value

• layers (int, array or list) – layer (int) or layers (list or array) in which line sink is located

• type (string) – type of element, “g” for given, “v” for variable and “z” for zero.

• name (string) – name of the element

• label (string, optional) – label of the element

• aq (Aquifer object) – aquifer in which the element is located

• inhomelement (boolean) – set to True if element is part of an inhomogeneity

tiny = 1e-08

nparam

xls

res

wh = 'H'

aq = None

__repr__()

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

setflowcoef()

Separate function so that this can be overloaded for other types.

potinf(x, y=0, aq=None)

Can be called with only one x value.

disvecinf(x, y=0, aq=None)

Can be called with only one x,y value.

abstractmethod changetrace(xyzt1, xyzt2, aq, layer, ltype, modellayer, direction, hstepmax)

plot(ax=None)

Plot the element.

class ttim.Model3D(kaq=1, z=[4, 3, 2, 1], Saq=0.001, kzoverkh=0.1, poraq=0.3, topboundary='conf', phreatictop=False, topres=0, topthick=0, topSll=0, toppor=0.3, tmin=1, tmax=10, tstart=0, M=10, timmlmodel=None)

Bases: TimModel

Create a multi-layer model object consisting of many aquifer layers.

The resistance between the layers is computed from the vertical hydraulic conductivity of the layers.

Parameters:

• kaq (float, array or list) – hydraulic conductivity of each layer from the top down if float, hydraulic conductivity is the same in all aquifers

• z (array or list) – elevation of top of system followed by bottoms of all layers from the top down bottom of layer is automatically equal to top of layer below it if topboundary=’conf’: length is number of layers + 1 if topboundary=’semi’: length is number of layers + 2 as top of leaky layer on top of systems needs to be specified

• Saq (float, array or list) – specific storage of all aquifers layers if float, sepcific storage is same in all aquifers layers if phreatictop is True and topboundary is ‘conf’, Saq of top aquifer is phreatic storage coefficient (and not multiplied with the layer thickness)

• kzoverkh (float) – vertical anisotropy ratio vertical k divided by horizontal k if float, value is the same for all layers length is number of layers

• topboundary (string, 'conf' or 'semi' (default is 'conf')) – indicating whether the top is confined (‘conf’) or semi-confined (‘semi’). currently only implemented for ‘conf’

• topres (float) – resistance of top semi-confining layer, only read if topboundary=’semi’

• topthick (float) – thickness of top semi-confining layer, only read if topboundary=’semi’

• phreatictop (boolean) – the storage coefficient of the top aquifer layer is treated as phreatic storage (and not multiplied with the aquifer thickness)

• tmin (scalar) – the minimum time for which heads can be computed after any change in boundary condition.

• tmax (scalar) – the maximum time for which heads can be computed.

• tstart (scalar) – time at start of simulation (default 0)

• M (integer (default 10)) – the number of terms to be used in the numerical inversion algorithm. 10 is usually sufficient. If drawdown curves appear to oscillate, more terms may be needed, but this seldom happens.

• timmlmodel (optional instance of a solved TimML model) – a timml model may be included to add steady-state flow

name = 'Model3D'

class ttim.ModelMaq(kaq=[1], z=[1, 0], c=[], Saq=[0.001], Sll=[0], poraq=[0.3], porll=[0.3], topboundary='conf', phreatictop=False, tmin=1, tmax=10, tstart=0, M=10, timmlmodel=None)

Bases: TimModel

Create model specifying a multi-aquifer sequence of aquifer-leakylayer-etc.

Parameters:

• kaq (float, array or list) – hydraulic conductivity of each aquifer from the top down if float, hydraulic conductivity is the same in all aquifers

• z (array or list) – elevation tops and bottoms of the aquifers from the top down leaky layers may have zero thickness if top=’conf’: length is 2 * number of aquifers if top=’semi’: length is 2 * number of aquifers + 1 as top of leaky layer on top of systems needs to be specified

• c (float, array or list) – resistance of leaky layers from the top down if float, resistance is the same for all leaky layers if top=’conf’: length is number of aquifers - 1 if top=’semi’: length is number of aquifers

• Saq (float, array or list) – specific storage of all aquifers if float, sepcific storage is same in all aquifers if phreatictop is True and topboundary is ‘conf’, Saq of top aquifer is phreatic storage coefficient (and not multiplied with the layer thickness)

• Sll (float, array or list) – specific storage of all leaky layers if float, sepcific storage is same in all leaky layers if phreatictop is True and topboundary is ‘semi’, Sll of top leaky layer is phreatic storage coefficient (and not multiplied with the layer thickness)

• topboundary (string, 'conf' or 'semi' (default is 'conf')) – indicating whether the top is confined (‘conf’) or semi-confined (‘semi’)

• phreatictop (boolean) – the storage coefficient of the top model layer is treated as phreatic storage (and not multiplied with the aquifer thickness)

• tmin (scalar) – the minimum time for which heads can be computed after any change in boundary condition.

• tmax (scalar) – the maximum time for which heads can be computed

• tstart (scalar) – time at start of simulation (default 0)

• M (integer) – the number of terms to be used in the numerical inversion algorithm. 10 is usually sufficient. If drawdown curves appear to oscillate, more terms may be needed, but this seldom happens.

• timmlmodel (optional instance of a solved TimML model) – a timml model may be included to add steady-state flow

name = 'ModelMaq'

class ttim.ModelXsection(naq=1, tmin=1, tmax=10, tstart=0, M=10, timmlmodel=None)

Bases: TimModel

Model class for cross-section models.

Parameters:

• naq (integer) – number of aquifers

• tmin (float) – the minimum time for which heads can be computed after any change in boundary condition.

• tmax (float) – the maximum time for which heads can be computed.

• tstart (float, optional) – time at start of simulation (default 0)

• M (integer, optional) – the number of terms to be used in the numerical inversion algorithm. 10 is usually sufficient.

• timmlmodel (timml.Model) – a timml model may be included to add a steady-state flow result to the computed solution.

elementlist = []

elementdict

vbclist = []

zbclist = []

gbclist = []

tmin = 1

tmax = 10

tstart = 0

M = 10

aq

name = 'TimModel'

modelname = 'ml'

timmlmodel = None

plots

plot

check_inhoms()

Check if number of aquifers in inhoms matches number of aquifers in model.

initialize()

ttim.timtrace(ml, xstart, ystart, zstart, tstartend, tstartoffset, tstep, nstepmax=100, hstepmax=10, silent=False, correctionstep=True)

Compute a pathline by numerical integration of the velocity vector.

Pathline is broken up in sections for which starting times are provided. Pathline is computed from first starting time + offset until second starting time, then continued from second starting time + offset until third starting time, etc.

Parameters:

• model (Model object) – model

• xstart (float) – x-coordinate of starting location of pathline

• ystart (float) – y-coordinate of starting location of pathline

• zstart (float) – z-coordinate of starting location of pathline

• tstartend (list) – list of starting times of pathline. last entry is the ending time.

• tstartoffset (float or list) – time after starting time when pathline is started. value or list. if this value is smaller than tmin it may cause problems after change in boundary conditions

• tstep (scalar or list) – maximum time step for each step along pathline. Either one value for all sections, or list with values for each section.

• nstepmax (integer) – maximum number of steps per section

• hstepmax (float or integer) – maximum length of horizontal step

• silent (boolean) – parameter to indicate if message should be printed to the screen for each section of the pathline

• correctionstep (boolean) – parameter to indicate if a correction step (Euler’s method) should be taken. Taking a correction step is more accurate, especially for curved pathlines.

Returns:

result –

• xyzt : 2D array with four columns: x, y, z, t along pathline

• message : list with text messages of each section of the pathline

• status : numerical indication of the result. Negative is likely undesirable.

  • -2 : reached maximum number of steps before reaching maximum time

  • -1 : starting z value not inside aquifer

  • +1 : reached maximum time

  • +2 : reached element

  • +3 : flows out of top of aquifer

Return type:

dictionary with three items

ttim.timtraceline(ml, xstart, ystart, zstart, tstart, delt, tmax, nstepmax=100, hstepmax=10, correctionstep=True, silent=False)

ttim.__version__ = '0.8.0'

ttim.show_versions(optional=True)

Print the version of dependencies.

Parameters:

optional (bool, optional) – Print the version of optional dependencies, by default False

Return type:

None

class ttim.DischargeWell(model, xw=0, yw=0, tsandQ=[(0, 1)], rw=0.1, res=0, layers=0, label=None)

Bases: WellBase

Well with a specified discharge for each layer that the well is screened in.

This is not very common and is likely only used for testing and comparison with other codes. The discharge must be specified for each screened layer. The resistance of the screen may be specified. The head is computed such that the discharge \(Q_i\) in layer \(i\) is computed as

\[Q_i = 2\pi r_wH_i(h_i - h_w)/c\]

where \(c\) is the resistance of the well screen and \(h_w\) is the head inside the well.

Parameters:

• model (Model object) – model to which the element is added

• xw (float) – x-coordinate of the well

• yw (float) – y-coordinate of the well

• tsandQ (list of tuples) – tuples of starting time and discharge after starting time

• rw (float) – radius of the well

• res (float) – resistance of the well screen

• layers (int, array or list) – layer (int) or layers (list or array) where well is screened

• label (string or None (default: None)) – label of the well

Examples

Example of a well that pumps with a discharge of 100 between times 10 and 50, with a discharge of 20 between times 50 and 200, and zero discharge after time 200.

>>> Well(ml, tsandQ=[(10, 100), (50, 20), (200, 0)])

class ttim.HeadWell(model, xw=0, yw=0, rw=0.1, tsandh=[(0, 1)], res=0, layers=0, label=None)

Bases: WellBase, ttim.equation.HeadEquation

Create a well with a specified head inside the well.

The well may be screened in multiple layers. The resistance of the screen may be specified. The head is computed such that the discharge \(Q_i\) in layer \(i\) is computed as

\[Q_i = 2\pi r_wH_i(h_i - h_w)/c\]

where \(c\) is the resistance of the well screen and \(h_w\) is the head inside the well.

Parameters:

• model (Model object) – model to which the element is added

• xw (float) – x-coordinate of the well

• yw (float) – y-coordinate of the well

• rw (float) – radius of the well

• tsandh (list of tuples) – tuples of starting time and discharge after starting time

• res (float) – resistance of the well screen

• layers (int, array or list) – layer (int) or layers (list or array) where well is screened

• label (string (default: None)) – label of the well

nunknowns

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

class ttim.Well(model, xw=0, yw=0, rw=0.1, tsandQ=[(0, 1)], res=0, rc=None, layers=0, wbstype='pumping', label=None)

Bases: WellBase, ttim.equation.WellBoreStorageEquation

Create a well with a specified discharge.

The well may be screened in multiple layers. The discharge is distributed across the layers such that the head inside the well is the same in all screened layers. Wellbore storage and skin effect may be taken into account. The head is computed such that the discharge \(Q_i\) in layer \(i\) is computed as

\[Q_i = 2\pi r_wH_i(h_i - h_w)/c\]

where \(c\) is the resistance of the well screen and \(h_w\) is the head inside the well.

Parameters:

• model (Model object) – model to which the element is added

• xw (float) – x-coordinate of the well

• yw (float) – y-coordinate of the well

• rw (float) – radius of the well

• tsandQ (list of tuples) – tuples of starting time and discharge after starting time

• res (float) – resistance of the well screen

• rc (float) – radius of the caisson, the pipe where the water table inside the well flucuates, which accounts for the wellbore storage

• layers (int, array or list) – layer (int) or layers (list or array) where well is screened

• wbstype (string) – ‘pumping’: Q is the discharge of the well ‘slug’: volume of water instantaneously taken out of the well

• label (string (default: None)) – label of the well

hdiff = None

nunknowns

wbstype = 'pumping'

initialize()

Initialize the element.

Initialization of terms that cannot be initialized before other elements or the aquifer is defined.

As we don’t want to require a certain order of entering elements, these terms are initialized when Model.solve is called The initialization class needs to be overloaded by all derived classes

setflowcoef()

Separate function so that this can be overloaded for other types.

class ttim.WellTest(model, xw=0, yw=0, tsandQ=[(0, 1)], rw=0.1, res=0, layers=0, label=None, fp=None)

Bases: WellBase

Well Base Class.

All Well elements are derived from this class

fp = None

setflowcoef()

Separate function so that this can be overloaded for other types.