Adding Multiple Wells

This notebook shows how a WellModel can be used to fit multiple wells with one response function. The influence of the individual wells is scaled by the distance to the observation point.

Developed by R.C. Caljé, (Artesia Water 2020), D.A. Brakenhoff, (Artesia Water 2019), and R.A. Collenteur, (Artesia Water 2018)

[1]:
import numpy as np
import pandas as pd
import pastas as ps
import matplotlib.pyplot as plt

ps.show_versions()
Python version: 3.7.8 | packaged by conda-forge | (default, Jul 31 2020, 02:37:09)
[Clang 10.0.1 ]
Numpy version: 1.18.5
Scipy version: 1.4.0
Pandas version: 1.1.2
Pastas version: 0.16.0b
Matplotlib version: 3.1.3

Load data from a Menyanthes file

Menyanthes is timeseries analysis software used by many people in the Netherlands. In this example a Menyanthes-file with one observation-series is imported, and simulated. There are several stresses in the Menyanthes-file, among which are three groundwater extractions with a significant influence on groundwater head.

Import the Menyanthes-file with observations and stresses.

[2]:
fname = '../data/MenyanthesTest.men'
meny = ps.read.MenyData(fname)

Get the distances of the extractions to the observation well. Extraction 1 is about two times as far from the observation well as extraction 2 and 3. We will use this information later in our WellModel.

[3]:
# Get distances from metadata
xo = meny.H["Obsevation well"]['xcoord']
yo = meny.H["Obsevation well"]['ycoord']
distances = []
extraction_names = ['Extraction 1', 'Extraction 2', 'Extraction 3']
for extr in extraction_names:
    xw = meny.IN[extr]["xcoord"]
    yw = meny.IN[extr]["ycoord"]
    distances.append(np.sqrt((xo-xw)**2 + (yo-yw)**2))
extraction_names = [name.replace(" ", "_") for name in extraction_names]  # replace spaces in names for Pastas
df = pd.DataFrame(distances, index=extraction_names, columns=['Distance to Observation well'])
df
[3]:
Distance to Observation well
Extraction_1 5076.464352
Extraction_2 2281.964490
Extraction_3 2783.783397

Then plot the observations, together with the diferent stresses in the Menyanthes file.

[4]:
# plot some series
f1, axarr = plt.subplots(len(meny.IN)+1, sharex=True, figsize=(14,7))
oseries = meny.H['Obsevation well']["values"]
oseries.plot(ax=axarr[0], color='k')
axarr[0].set_title(meny.H['Obsevation well']["Name"])
for i, (name, data) in enumerate(meny.IN.items(), start=1):
    data["values"].plot(ax=axarr[i])
    axarr[i].set_title(name)
plt.tight_layout(pad=0)
../_images/examples_010_multiple_wells.ipynb_7_0.png

Create a model with a separate StressModel for each extraction

First we create a model with a separate StressModel for each groundwater extraction. First we create a model with the heads timeseries and add recharge as a stress.

[5]:
ml = ps.Model(ps.TimeSeries(meny.H['Obsevation well']['values'], name="heads"))
INFO: Cannot determine frequency of series heads: freq=None. Resample settings are ignored and timestep_weighted_resample is used.

Get the precipitation and evaporation timeseries and round the index to remove the hours from the timestamps.

[6]:
IN = meny.IN['Precipitation']['values']
IN.index = IN.index.round("D")
IN.name = "prec"
IN2 = meny.IN['Evaporation']['values']
IN2.index = IN2.index.round("D")
IN2.name = "evap"

Create a recharge stressmodel and add to the model.

[7]:
sm = ps.StressModel2([IN, IN2], ps.Gamma, 'Recharge')
ml.add_stressmodel(sm)
INFO: Inferred frequency for time series prec: freq=D
INFO: Inferred frequency for time series evap: freq=D

Get the extraction timeseries.

[8]:
stresses = []
for name in extraction_names:
    stress = ps.TimeSeries(meny.IN[name.replace("_", " ")]['values'], name=name, settings='well')
    stresses.append(stress)
INFO: Cannot determine frequency of series Extraction_1: freq=None. Resample settings are ignored and timestep_weighted_resample is used.
INFO: Cannot determine frequency of series Extraction_2: freq=None. Resample settings are ignored and timestep_weighted_resample is used.
INFO: Cannot determine frequency of series Extraction_3: freq=None. Resample settings are ignored and timestep_weighted_resample is used.

Add each of the extractions as a separate StressModel.

[9]:
for stress in stresses:
    sm = ps.StressModel(stress, ps.Hantush, stress.name, up=False)
    ml.add_stressmodel(sm)
INFO: Time Series Extraction_1 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_2 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was sampled down to freq D with method timestep_weighted_resample.

Solve the model.

Note the use of ps.LmfitSolve. This is because of an issue concerning optimization with small parameter values in scipy.least_squares. This is something that may influence models containing a WellModel (which we will be creating later) and since we want to keep the models in this Notebook as similar as possible, we’re also using ps.LmfitSolve here.

[10]:
ml.solve(solver=ps.LmfitSolve)
INFO: Time Series Extraction_1 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_2 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was extended to 1950-05-01 00:00:00 by adding 0.0 values.
INFO: There are observations between the simulation timesteps. Linear interpolation between simulated values is used.
Fit report heads                       Fit Statistics
========================================================
nfev     645                    EVP                93.69
nobs     2844                   R2                  0.93
noise    True                   RMSE                0.23
tmin     1960-04-28 12:00:00    AIC                19.97
tmax     2015-06-29 09:00:00    BIC               109.26
freq     D                      Obj                22.83
warmup   3650 days 00:00:00     ___
solver   LmfitSolve             Interpolated         Yes

Parameters (15 were optimized)
========================================================
                    optimal     stderr     initial  vary
Recharge_A      1686.680335    ±19.54%  210.498526  True
Recharge_n         1.050298     ±3.71%    1.000000  True
Recharge_a       908.878934    ±27.43%   10.000000  True
Recharge_f        -1.488694    ±14.46%   -1.000000  True
Extraction_1_A    -0.000872  ±3598.74%   -0.000178  True
Extraction_1_a  9056.164653   ±958.60%  100.000000  True
Extraction_1_b     3.117841  ±2053.76%    1.000000  True
Extraction_2_A    -0.000059    ±35.46%   -0.000086  True
Extraction_2_a  2103.789349   ±203.32%  100.000000  True
Extraction_2_b     0.011679   ±236.92%    1.000000  True
Extraction_3_A    -0.000021    ±39.92%   -0.000170  True
Extraction_3_a  1069.202458   ±186.69%  100.000000  True
Extraction_3_b     0.003923   ±246.13%    1.000000  True
constant_d        11.141739     ±4.98%    8.557530  True
noise_alpha       49.583766     ±8.48%    1.000000  True

Parameter correlations |rho| > 0.5
========================================================
Recharge_A     Recharge_a      0.86
               Extraction_2_A -0.59
Recharge_n     Recharge_a     -0.68
Recharge_a     Extraction_2_A -0.55
Recharge_f     constant_d     -0.99
Extraction_1_A Extraction_1_a  0.97
               Extraction_1_b -1.00
               Extraction_2_a -0.62
               Extraction_2_b  0.60
Extraction_1_a Extraction_1_b -0.98
               Extraction_2_a -0.60
               Extraction_2_b  0.57
Extraction_1_b Extraction_2_a  0.67
               Extraction_2_b -0.65
Extraction_2_A Extraction_2_a  0.84
               Extraction_2_b -0.89
Extraction_2_a Extraction_2_b -0.99
Extraction_3_A Extraction_3_a  0.67
               Extraction_3_b -0.84
Extraction_3_a Extraction_3_b -0.94

Visualize the results

Plot the decomposition to see the individual influence of each of the wells.

[11]:
ml.plots.decomposition();
../_images/examples_010_multiple_wells.ipynb_21_0.png

We can calculate the gain of each extraction (quantified as the effect on the groudnwater level of an extraction of ~1000 m3/d).

[12]:
for i in range(len(extraction_names)):
    name = extraction_names[i]
    sm = ml.stressmodels[name]
    p = ml.get_parameters(name)
    gain = sm.rfunc.gain(p) * 1e6 / 365.25
    print("{0}: gain = {1:.2f} m / million m^3/year".format(name, gain))
    df.at[name, 'gain StressModel'] = gain
Extraction_1: gain = -0.05 m / million m^3/year
Extraction_2: gain = -0.27 m / million m^3/year
Extraction_3: gain = -0.13 m / million m^3/year

Create a model with a WellModel

We can reduce the number of parameters in the model by including the three extractions in a WellModel. This WellModel takes into acount the distances from the three extractions to the observation well, and assumes constant geohydrological properties. All of the extractions now share the same response function, scaled by the distance between the extraction well and the obervation well.

First we delete the existing StressModels with the well-data.

[13]:
for name in extraction_names:
    ml.del_stressmodel(name)

We have all the information we need to create a WellModel: - timeseries for each of the extractions, these are passed as a list of stresses - distances from each extraction to the observation point, note that the order of these distances must correspond to the order of the stresses.

Note: the WellModel only works with a special version of the Hantush response function called HantushWellModel. This is because the response function must support scaling by a distance \(r\). The HantushWellModel response function has been modified to support this. The Hantush response normally takes three parameters: the gain \(A\), \(a\) and \(b\). This special version accepts 4 parameters: it interprets that fourth parameter as the distance \(r\), and uses it to scale the \(A\) and \(b\) parameters accordingly.

Create the WellModel and add to the model.

[14]:
w = ps.WellModel(stresses, ps.HantushWellModel, "Wells", distances, settings="well")
ml.add_stressmodel(w)
WARNING: It is recommended to use LmfitSolve as the solver when implementing WellModel. See https://github.com/pastas/pastas/issues/177.
INFO: Time Series Extraction_2 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_1 was sampled down to freq D with method timestep_weighted_resample.

Solve the model.

We are once again using ps.LmfitSolve. The user is notified about the preference for this solver in a WARNING when creating the WellModel (see above).

As we can see, the fit with the measurements (EVP) is the same as before.

[15]:
ml.solve(solver=ps.LmfitSolve)
INFO: Time Series Extraction_2 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was sampled down to freq D with method timestep_weighted_resample.
INFO: Time Series Extraction_3 was extended to 1950-05-01 00:00:00 by adding 0.0 values.
INFO: Time Series Extraction_1 was sampled down to freq D with method timestep_weighted_resample.
INFO: There are observations between the simulation timesteps. Linear interpolation between simulated values is used.
Fit report heads                     Fit Statistics
======================================================
nfev     283                    EVP              93.68
nobs     2844                   R2                0.94
noise    True                   RMSE              0.23
tmin     1960-04-28 12:00:00    AIC               8.06
tmax     2015-06-29 09:00:00    BIC              61.63
freq     D                      Obj              23.02
warmup   3650 days 00:00:00     ___
solver   LmfitSolve             Interpolated       Yes

Parameters (9 were optimized)
======================================================
                  optimal   stderr       initial  vary
Recharge_A   1.296245e+03  ±17.40%  2.104985e+02  True
Recharge_n   1.002833e+00   ±3.65%  1.000000e+00  True
Recharge_a   8.534953e+02  ±28.45%  1.000000e+01  True
Recharge_f  -2.000000e+00  ±13.14% -1.000000e+00  True
Wells_A     -2.182820e-04  ±48.43% -8.609196e-05  True
Wells_a      6.959607e+02  ±32.20%  1.000000e+02  True
Wells_b      5.322245e-08  ±61.14%  8.749377e-08  True
constant_d   1.219999e+01   ±4.79%  8.557530e+00  True
noise_alpha  5.467031e+01   ±8.28%  1.000000e+00  True

Parameter correlations |rho| > 0.5
======================================================
Recharge_A Recharge_a  0.81
           Wells_a    -0.51
Recharge_n Recharge_a -0.72
Recharge_a Recharge_f -0.60
           Wells_a    -0.53
           constant_d  0.77
Recharge_f constant_d -0.95
Wells_A    Wells_a     0.93
           Wells_b    -1.00
Wells_a    Wells_b    -0.93

Visualize the results

Plot the decomposition to see the individual influence of each of the wells

[16]:
ml.plots.decomposition();
../_images/examples_010_multiple_wells.ipynb_31_0.png

Plot the stacked influence of each of the individual extraction wells in the results plot

[17]:
ml.plots.stacked_results(figsize=(10, 10));
../_images/examples_010_multiple_wells.ipynb_33_0.png

Get parameters for each well (including the distance) and calculate the gain. The WellModel reorders the stresses from close to the observation well, to far from the observation well. We have take this into account during the post-processing.

The gain of extraction 1 is lower than the gain of extraction 2 and 3. This will always be the case in a WellModel when the distance from the observation well to extraction 1 is larger than the distance to extraction 2 and 3.

[18]:
wm = ml.stressmodels["Wells"]
for i in range(3):
    p = wm.get_parameters(model=ml, istress=i)
    gain = wm.rfunc.gain(p) * 1e6 / 365.25
    name = wm.stress[i].name
    print("{0}: gain = {1:.2f} m / million m^3/year".format(name, gain))
    df.at[name, 'gain WellModel'] = gain
Extraction_2: gain = -0.23 m / million m^3/year
Extraction_3: gain = -0.17 m / million m^3/year
Extraction_1: gain = -0.04 m / million m^3/year

Compare individual StressModels and WellModel

Compare the gains that were calculated by the individual StressModels and the WellModel.

[19]:
df
[19]:
Distance to Observation well gain StressModel gain WellModel
Extraction_1 5076.464352 -0.045164 -0.044959
Extraction_2 2281.964490 -0.273452 -0.233410
Extraction_3 2783.783397 -0.129116 -0.169794