There are two different forms of Richards equation that differ on how they deal with the non-linearity in the time-stepping term. Here we reproduce results from a seminal paper, and show the ease with which you can do this in SimPEG and Richards python packages.
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Richards Equation - Comparison to Ceilia et al. 1990
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| <p><strong>Objective:</strong></p> | |
| <p>There are two different forms of Richards equation that differ | |
| on how they deal with the non-linearity in the time-stepping term.</p> | |
| <p>The most fundamental form, referred to as the | |
| 'mixed'-form of Richards Equation [Celia <em>et al.</em>, 1990]</p> | |
| <p>$$\frac{\partial \theta(\psi)}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0 | |
| \quad \psi \in \Omega$$</p> | |
| <p>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 $\theta$.</p> | |
| <p>As noted in [Celia <em>et al.</em>, 1990] the 'head'-based form of Richards | |
| equation can be written in the continuous form as:</p> | |
| <p>$$\frac{\partial \theta}{\partial \psi}\frac{\partial \psi}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0 \quad \psi \in \Omega$$</p> | |
| <p>However, it can be shown that this does not conserve mass in the discrete formulation.</p> | |
| <p>Here we reproduce the results from Ceilia <em>et al.</em> (1990) demonstrating that the head-based formulation is not as good as the mixed-formulation.</p> | |
| <p><em>Celia, Michael A., Efthimios T. Bouloutas, and Rebecca L. Zarba. "<a href="http://www.webpages.uidaho.edu/ch/papers/Celia.pdf">A general mass-conservative numerical solution for the unsaturated flow equation.</a>" Water Resources Research 26.7 (1990): 1483-1496.</em></p> | |
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| In [1]: | |
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| <div class="highlight"><pre><span class="kn">from</span> <span class="nn">SimPEG</span> <span class="kn">import</span> <span class="o">*</span> | |
| <span class="kn">import</span> <span class="nn">Richards</span> | |
| </pre></div> | |
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| <pre> | |
| Warning: mumps solver not available. | |
| </pre> | |
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| <div class="highlight"><pre><span class="n">M</span> <span class="o">=</span> <span class="n">mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">40</span><span class="p">)])</span> | |
| <span class="n">M</span><span class="o">.</span><span class="n">setCellGradBC</span><span class="p">(</span><span class="s">'dirichlet'</span><span class="p">)</span> | |
| <span class="n">Ks</span> <span class="o">=</span> <span class="mf">9.4400e-03</span> | |
| <span class="n">E</span> <span class="o">=</span> <span class="n">Richards</span><span class="o">.</span><span class="n">Haverkamp</span><span class="p">(</span><span class="n">Ks</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">Ks</span><span class="p">),</span> <span class="n">A</span><span class="o">=</span><span class="mf">1.1750e+06</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">4.74</span><span class="p">,</span> | |
| <span class="n">alpha</span><span class="o">=</span><span class="mf">1.6110e+06</span><span class="p">,</span> <span class="n">theta_s</span><span class="o">=</span><span class="mf">0.287</span><span class="p">,</span> | |
| <span class="n">theta_r</span><span class="o">=</span><span class="mf">0.075</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="mf">3.96</span><span class="p">)</span> | |
| <span class="n">bc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">61.5</span><span class="p">,</span><span class="o">-</span><span class="mf">20.7</span><span class="p">])</span> | |
| <span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">nC</span><span class="p">)</span> <span class="o">+</span> <span class="n">bc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> | |
| <span class="k">def</span> <span class="nf">getFields</span><span class="p">(</span><span class="n">timeStep</span><span class="p">,</span><span class="n">method</span><span class="p">):</span> | |
| <span class="n">prob</span> <span class="o">=</span> <span class="n">Richards</span><span class="o">.</span><span class="n">RichardsProblem</span><span class="p">(</span><span class="n">M</span><span class="p">,</span><span class="n">E</span><span class="p">,</span> <span class="n">timeStep</span><span class="o">=</span><span class="n">timeStep</span><span class="p">,</span> <span class="n">timeEnd</span><span class="o">=</span><span class="mi">360</span><span class="p">,</span> | |
| <span class="n">boundaryConditions</span><span class="o">=</span><span class="n">bc</span><span class="p">,</span> <span class="n">initialConditions</span><span class="o">=</span><span class="n">h</span><span class="p">,</span> | |
| <span class="n">doNewton</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span> | |
| <span class="k">return</span> <span class="n">prob</span><span class="o">.</span><span class="n">field</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">Ks</span><span class="p">))</span> | |
| <span class="n">Hs_M10</span> <span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="s">'mixed'</span><span class="p">)</span> | |
| <span class="n">Hs_M30</span> <span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="s">'mixed'</span><span class="p">)</span> | |
| <span class="n">Hs_M120</span><span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">120</span><span class="p">,</span><span class="s">'mixed'</span><span class="p">)</span> | |
| <span class="n">Hs_H10</span> <span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="s">'head'</span><span class="p">)</span> | |
| <span class="n">Hs_H30</span> <span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="s">'head'</span><span class="p">)</span> | |
| <span class="n">Hs_H120</span><span class="o">=</span> <span class="n">getFields</span><span class="p">(</span><span class="mi">120</span><span class="p">,</span><span class="s">'head'</span><span class="p">)</span> | |
| </pre></div> | |
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| <pre> | |
| NewtonRoot stopped by maxIters. norm: 5.1942e-05 | |
| NewtonRoot stopped by maxIters. norm: 1.0531e-06 | |
| NewtonRoot stopped by maxIters. norm: 1.1545e-06 | |
| NewtonRoot stopped by maxIters. norm: 2.2226e-06 | |
| NewtonRoot stopped by maxIters. norm: 2.6211e-06 | |
| </pre> | |
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| <div class="highlight"><pre><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">13</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span> | |
| <span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_M10</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'b-'</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_M30</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'r-'</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_M120</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'k-'</span><span class="p">)</span> | |
| <span class="n">title</span><span class="p">(</span><span class="s">'Mixed Method'</span><span class="p">);</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'Depth, cm'</span><span class="p">);</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'Pressure Head, cm'</span><span class="p">)</span> | |
| <span class="n">legend</span><span class="p">((</span><span class="s">'dt = 10 sec'</span><span class="p">,</span><span class="s">'dt = 30 sec'</span><span class="p">,</span><span class="s">'dt = 120 sec'</span><span class="p">))</span> | |
| <span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_H10</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'b-'</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_H30</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'r-'</span><span class="p">)</span> | |
| <span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_H120</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="s">'k-'</span><span class="p">)</span> | |
| <span class="n">title</span><span class="p">(</span><span class="s">'Head-Based Method'</span><span class="p">);</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'Depth, cm'</span><span class="p">);</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'Pressure Head, cm'</span><span class="p">)</span> | |
| <span class="n">legend</span><span class="p">((</span><span class="s">'dt = 10 sec'</span><span class="p">,</span><span class="s">'dt = 30 sec'</span><span class="p">,</span><span class="s">'dt = 120 sec'</span><span class="p">))</span> | |
| </pre></div> | |
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| <pre> | |
| <matplotlib.legend.Legend at 0x10fb352d0> | |
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| In [4]: | |
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| <div class="highlight"><pre><span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">clim</span><span class="p">,</span> <span class="n">tlt</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span> | |
| <span class="n">ax</span><span class="o">.</span><span class="n">cla</span><span class="p">()</span> | |
| <span class="n">linem</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_M10</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="s">'r-'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> | |
| <span class="n">lineh</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mi">40</span><span class="o">-</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,</span> <span class="n">Hs_H10</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="s">'b-'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> | |
| <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s">'mixed'</span><span class="p">,</span><span class="s">'head'</span><span class="p">))</span> | |
| <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">'Time %4.f seconds'</span><span class="o">%</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="mi">10</span><span class="p">))</span> | |
| <span class="n">M</span><span class="o">.</span><span class="n">video</span><span class="p">(</span><span class="n">Hs_H10</span><span class="p">,</span><span class="n">function</span><span class="p">,</span><span class="n">colorbar</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span> | |
| </pre></div> | |
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| Out[4]:</div> | |
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| " type="video/mp4"> | |
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| </video> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> |
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