# FLOW: Richards: 1D: Celia1990#

There are two different forms of Richards equation that differ on how they deal with the non-linearity in the time-stepping term.

The most fundamental form, referred to as the ‘mixed’-form of Richards Equation Celia1990

$\frac{\partial \theta(\psi)}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0 \quad \psi \in \Omega$

where $$\theta$$ is water content, and $$\psi$$ is pressure head. This formulation of Richards equation is called the ‘mixed’-form because the equation is parameterized in $$\psi$$ but the time-stepping is in terms of $$\theta$$.

As noted in Celia1990 the ‘head’-based form of Richards equation can be written in the continuous form as:

$\frac{\partial \theta}{\partial \psi} \frac{\partial \psi}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0 \quad \psi \in \Omega$

However, it can be shown that this does not conserve mass in the discrete formulation.

Here we reproduce the results from Celia1990 demonstrating the head-based formulation and the mixed-formulation.

/home/vsts/conda/envs/simpeg-test/lib/python3.8/site-packages/discretize/operators/differential_operators.py:1762: UserWarning:



import matplotlib.pyplot as plt
import numpy as np

import discretize
from SimPEG import maps
from SimPEG.flow import richards

def run(plotIt=True):
M = discretize.TensorMesh([np.ones(40)])
params = richards.empirical.HaverkampParams().celia1990
k_fun, theta_fun = richards.empirical.haverkamp(M, **params)
k_fun.KsMap = maps.IdentityMap(nP=M.nC)

bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]

def getFields(timeStep, method):
timeSteps = np.ones(int(360 / timeStep)) * timeStep
prob = richards.SimulationNDCellCentered(
M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method=method,
)
prob.time_steps = timeSteps
return prob.fields(params["Ks"] * np.ones(M.nC))

Hs_M010 = getFields(10, "mixed")
Hs_M030 = getFields(30, "mixed")
Hs_M120 = getFields(120, "mixed")

if not plotIt:
return
plt.figure(figsize=(13, 5))
plt.subplot(121)
plt.plot(40 - M.gridCC, Hs_M010[-1], "b-")
plt.plot(40 - M.gridCC, Hs_M030[-1], "r-")
plt.plot(40 - M.gridCC, Hs_M120[-1], "k-")
plt.ylim([-70, -10])
plt.title("Mixed Method")
plt.xlabel("Depth, cm")
plt.legend(
(r"$\Delta t$ = 10 sec", r"$\Delta t$ = 30 sec", r"$\Delta t$ = 120 sec")
)
plt.subplot(122)
plt.plot(40 - M.gridCC, Hs_H010[-1], "b-")
plt.plot(40 - M.gridCC, Hs_H030[-1], "r-")
plt.plot(40 - M.gridCC, Hs_H120[-1], "k-")
plt.ylim([-70, -10])
plt.xlabel("Depth, cm")
plt.legend(
(r"$\Delta t$ = 10 sec", r"$\Delta t$ = 30 sec", r"$\Delta t$ = 120 sec")
)

if __name__ == "__main__":
run()
plt.show()


Total running time of the script: ( 0 minutes 6.779 seconds)

Estimated memory usage: 17 MB

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