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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)

import os
import numpy as np
import pandas as pd
import pastas as ps
import matplotlib.pyplot as plt

ps.show_versions()

Python version: 3.11.6
NumPy version: 1.26.4
Pandas version: 2.2.2
SciPy version: 1.13.0
Matplotlib version: 3.8.4
Numba version: 0.59.1
LMfit version: 1.3.1
Latexify version: Not Installed
Pastas version: 1.5.0


Set the coordinates of the extraction wells and calculate the distances to the observation well.

# Specify coordinates observations
xo = 85850
yo = 383362

# Specify coordinates extractions
relevant_extractions = {
"Extraction_2": (83588, 383664),
"Extraction_3": (88439, 382339),
}

# calculate distances
distances = []
for extr, xy in relevant_extractions.items():
xw = xy[0]
yw = xy[1]
distances.append(np.sqrt((xo - xw) ** 2 + (yo - yw) ** 2))

df = pd.DataFrame(
distances,
index=relevant_extractions.keys(),
columns=["Distance to observation well"],
)
df

Distance to observation well
Extraction_2 2282.070989
Extraction_3 2783.783397

Read the stresses from their csv files

# read oseries
"data_notebook_10/Observation_well.csv", index_col=0, parse_dates=[0]
).squeeze()
oseries.name = oseries.name.replace(" ", "_")
stresses = {}
for fname in os.listdir("data_notebook_10"):
os.path.join("data_notebook_10", fname), index_col=0, parse_dates=[0]
).squeeze()
stresses[fname.strip(".csv").replace(" ", "_")] = series


Then plot the observations, together with the diferent stresses.

# plot timeseries
f1, axarr = plt.subplots(len(stresses.keys()) + 1, sharex=True, figsize=(10, 8))
oseries.plot(ax=axarr[0], color="k")
axarr[0].set_title(oseries.name)

for i, name in enumerate(stresses.keys(), start=1):
stresses[name].plot(ax=axarr[i])
axarr[i].set_title(name)


## 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.

# create model
ml = ps.Model(oseries)


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

prec = stresses["Precipitation"]
prec.index = prec.index.round("D")
prec.name = "prec"
evap = stresses["Evaporation"]
evap.index = evap.index.round("D")
evap.name = "evap"


Create a recharge stressmodel and add to the model.

rm = ps.RechargeModel(prec, evap, ps.Exponential(), "Recharge")


Modify the extraction timeseries.

extraction_ts = {}

for name in relevant_extractions.keys():
# get extraction timeseries
s = stresses[name]

# convert index to end-of-month timeseries
s.index = s.index.to_period("M").to_timestamp("M")

# resample to daily values
new_index = pd.date_range(s.index[0], s.index[-1], freq="D")
s_daily = ps.ts.timestep_weighted_resample(s, new_index, fast=True).dropna()
name = name.replace(" ", "_")
s_daily.name = name

# append to stresses list
extraction_ts[name] = s_daily


Add each of the extractions as a separate StressModel.

for name, stress in extraction_ts.items():
sm = ps.StressModel(stress, ps.Hantush(), name, up=False, settings="well")


Solve the model.

ml.solve()

Fit report Observation_well       Fit Statistics
================================================
nfev    18                     EVP         94.41
nobs    2844                   R2           0.94
noise   True                   RMSE         0.21
tmin    1960-04-28 12:00:00    AICc     -8801.40
tmax    2015-06-29 09:00:00    BIC      -8736.01
freq    D                      Obj         63.90
warmup  3650 days 00:00:00     ___
solver  LeastSquares           Interp.       Yes

Parameters (11 optimized)
================================================
optimal     initial  vary
Recharge_A      1518.480732  210.498526  True
Recharge_a       795.348855   10.000000  True
Recharge_f        -1.265597   -1.000000  True
Extraction_2_A    -0.000109   -0.000086  True
Extraction_2_a  1286.807840  100.000000  True
Extraction_2_b     0.032393    1.000000  True
Extraction_3_A    -0.000043   -0.000171  True
Extraction_3_a   264.109078  100.000000  True
Extraction_3_b     0.827891    1.000000  True
constant_d        10.702163    8.557530  True
noise_alpha        0.005010    1.000000  True

INFO: Time Series 'Extraction_3' was extended in the past to 1950-05-01 00:00:00 by adding 0.0 values.
INFO: There are observations between the simulation time steps. Linear interpolation between simulated values is used.


### Visualize the results#

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

ml.plots.decomposition();


We can calculate the gain of each extraction (quantified as the effect on the groundwater level of a continuous extraction of ~1 Mm$$^3$$/yr).

for name in relevant_extractions.keys():
sm = ml.stressmodels[name]
p = ml.get_parameters(name)
gain = sm.rfunc.gain(p) * 1e6 / 365.25
print(f"{name}: gain = {gain:.3f} m / Mm^3/year")
df.at[name, "gain StressModel"] = gain

Extraction_2: gain = -0.299 m / Mm^3/year
Extraction_3: gain = -0.119 m / Mm^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 account 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 observation well.

First we create a new model and add recharge.

ml_wm = ps.Model(oseries, oseries.name + "_wm")
rm = ps.RechargeModel(prec, evap, ps.Gamma(), "Recharge")


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 parameters accordingly.

Create the WellModel and add to the model.

w = ps.WellModel(list(extraction_ts.values()), "WellModel", distances)


Solve the model.

As we can see, the fit with the measurements (EVP) is similar to the result with the previous model, with each well included separately.

ml_wm.solve()

Fit report Observation_well        Fit Statistics
=================================================
nfev    34                     EVP          93.46
nobs    2844                   R2            0.93
noise   True                   RMSE          0.23
tmin    1960-04-28 12:00:00    AICc     -13674.59
tmax    2015-06-29 09:00:00    BIC      -13621.08
freq    D                      Obj          11.53
warmup  3650 days 00:00:00     ___
solver  LeastSquares           Interp.        Yes

Parameters (9 optimized)
=================================================
optimal     initial  vary
Recharge_A   1395.489980  210.498526  True
Recharge_n      1.001207    1.000000  True
Recharge_a    911.305670   10.000000  True
Recharge_f     -1.999997   -1.000000  True
WellModel_A    -0.000327   -0.000756  True
WellModel_a   526.326243  100.000000  True
WellModel_b   -16.286063  -15.674262  True
constant_d     12.085909    8.557530  True
noise_alpha    56.711253    1.000000  True

Warnings! (1)
=================================================
Parameter 'Recharge_f' on lower bound: -2.00e+00

INFO: Time Series 'Extraction_3' was extended in the past to 1950-05-01 00:00:00 by adding 0.0 values.
INFO: There are observations between the simulation time steps. Linear interpolation between simulated values is used.


### Visualize the results#

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

ml_wm.plots.decomposition();


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

ml_wm.plots.stacked_results(
figsize=(10, 8),
stacklegend=True,
stackcolors={"Extraction_2": "C1", "Extraction_3": "C2"},
);


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

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

wm = ml_wm.stressmodels["WellModel"]
for i, name in enumerate(relevant_extractions.keys()):
# get parameters (note use of stressmodel for this)
p = wm.get_parameters(model=ml_wm, istress=i)
# calculate gain
gain = wm.rfunc.gain(p) * 1e6 / 365.25
name = wm.stress[i].name
print(f"{name}: gain = {gain:.3f} m / Mm^3/year")
df.at[name, "gain WellModel"] = gain

Extraction_2: gain = -0.240 m / Mm^3/year
Extraction_3: gain = -0.164 m / Mm^3/year


Calculate gain as function of radial distance for and plot the result, including the estimated uncertainty.

r = np.logspace(3, 3.6, 101)

# calculate gain and std error vs distance
params = ml_wm.get_parameters(wm.name)
gain_wells = wm.rfunc.gain(params, r=wm.distances.values) * 1e6 / 365.25
gain_vs_dist = wm.rfunc.gain(params, r=r) * 1e6 / 365.25
gain_std_vs_dist = np.sqrt(wm.variance_gain(ml_wm, r=r)) * 1e6 / 365.25

fig, ax = plt.subplots(1, 1, figsize=(10, 3))
ax.plot(r, gain_vs_dist, color="C0", label="gain")
ax.plot(
wm.distances,
gain_wells,
color="C3",
marker="o",
mfc="none",
label="wells",
ls="none",
)
ax.fill_between(
r,
gain_vs_dist - 2 * gain_std_vs_dist,
gain_vs_dist + 2 * gain_std_vs_dist,
alpha=0.35,
label="CI 95%",
)
ax.axhline(0.0, linestyle="dashed", color="k")
ax.legend(loc=(0, 1), frameon=False, ncol=3)
ax.grid(visible=True)
ax.set_ylabel("gain [m / (Mm$^3$/yr)]");


## Compare individual StressModels and WellModel#

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

df.style.format("{:.4f}")

Distance to observation well gain StressModel gain WellModel
Extraction_2 2282.0710 -0.2994 -0.2403
Extraction_3 2783.7834 -0.1188 -0.1643

Visually compare the two models, including the calculated contribution of the wells.

Note that there is some extra code at the bottom to calculate two step responses for the “WellModel” model, for comparison purposes with the “2-wells” model.

# give models descriptive name
ml.name = "2_wells"
ml_wm.name = "WellModel"

# plot well stresses together on same plot:
smdict = {0: ["Recharge"], 1: ["Extraction_2", "Extraction_3", "WellModel"]}

# comparison plot
mc = ps.CompareModels([ml, ml_wm])
mosaic = mc.get_default_mosaic(n_stressmodels=2)
mc.plot(smdict=smdict)

sumwells = ml.get_contribution("Extraction_2") + ml.get_contribution("Extraction_3")
mc.axes["con1"].plot(
sumwells.index, sumwells, ls="dashed", color="C0", label="sum 2_wells"
)
mc.axes["con1"].legend(loc=(0, 1), frameon=False, ncol=4)

# remove WellModel response for r=1m and add response twice, scaled with actual
# distances, for comparison with the two responses from the first model
mc.axes["rf1"].lines[-1].remove()  # remove original step response
for istress in range(2):
# get parameters and distance for istress
p = ml_wm.stressmodels["WellModel"].get_parameters(istress=istress)
# calculate step
step = ml_wm.get_step_response("WellModel", p=p)
# plot step
mc.axes["rf1"].plot(step.index, step, color="C1")
# recalculate axes limits
mc.axes["rf1"].relim()