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rowanc1
93 lines
2.8 KiB
ReStructuredText
93 lines
2.8 KiB
ReStructuredText
.. _api_Richards:
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Richards Equation
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*****************
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There are two different forms of Richards equation that differ
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on how they deal with the non-linearity in the time-stepping term.
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The most fundamental form, referred to as the
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'mixed'-form of Richards Equation [Celia et al., 1990]
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.. math::
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\frac{\partial \theta(\psi)}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0
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\quad \psi \in \Omega
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where theta is water content, and psi is pressure head.
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This formulation of Richards equation is called the
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'mixed'-form because the equation is parameterized in psi
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but the time-stepping is in terms of theta.
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As noted in [Celia et al., 1990] the 'head'-based form of Richards
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equation can be written in the continuous form as:
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.. math::
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\frac{\partial \theta}{\partial \psi}\frac{\partial \psi}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0
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\quad \psi \in \Omega
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However, it can be shown that this does not conserve mass in the discrete formulation.
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Here we reproduce the results from Ceilia et al. (1990):
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.. plot::
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from SimPEG import *
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from simpegFLOW import Richards
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import matplotlib.pyplot as plt
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M = Mesh.TensorMesh([np.ones(40)])
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M.setCellGradBC('dirichlet')
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params = Richards.Empirical.HaverkampParams().celia1990
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model = Richards.Empirical.Haverkamp(M, **params)
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bc = np.array([-61.5,-20.7])
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h = np.zeros(M.nC) + bc[0]
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def getFields(timeStep,method):
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prob = Richards.RichardsProblem(M,model, timeStep=timeStep, timeEnd=360,
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boundaryConditions=bc, initialConditions=h,
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doNewton=False, method=method)
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return prob.fields(params['Ks'])
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Hs_M10 = getFields(10., 'mixed')
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Hs_M30 = getFields(30., 'mixed')
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Hs_M120= getFields(120.,'mixed')
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Hs_H10 = getFields(10., 'head')
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Hs_H30 = getFields(30., 'head')
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Hs_H120= getFields(120.,'head')
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plt.figure(figsize=(13,5))
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plt.subplot(121)
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plt.plot(40-M.gridCC, Hs_M10[-1],'b-')
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plt.plot(40-M.gridCC, Hs_M30[-1],'r-')
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plt.plot(40-M.gridCC, Hs_M120[-1],'k-')
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plt.ylim([-70,-10])
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plt.title('Mixed Method')
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plt.xlabel('Depth, cm')
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plt.ylabel('Pressure Head, cm')
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plt.legend(('$\Delta t$ = 10 sec','$\Delta t$ = 30 sec','$\Delta t$ = 120 sec'))
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plt.subplot(122)
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plt.plot(40-M.gridCC, Hs_H10[-1],'b-')
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plt.plot(40-M.gridCC, Hs_H30[-1],'r-')
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plt.plot(40-M.gridCC, Hs_H120[-1],'k-')
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plt.ylim([-70,-10])
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plt.title('Head-Based Method')
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plt.xlabel('Depth, cm')
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plt.ylabel('Pressure Head, cm')
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plt.legend(('$\Delta t$ = 10 sec','$\Delta t$ = 30 sec','$\Delta t$ = 120 sec'))
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plt.show()
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Richards
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========
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.. automodule:: simpegFLOW.Richards.Empirical
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:show-inheritance:
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:members:
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:undoc-members:
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