diff --git a/web/content/docs/benchmarks/liquid-flow/liquid-flow-theis-problem.md b/web/content/docs/benchmarks/liquid-flow/liquid-flow-theis-problem.md index f499ddfef5b4a9ba8247c42e77c7c8c583233e6b..6b4c796ecfc3c4d5a85951b2f1a191a478b51934 100644 --- a/web/content/docs/benchmarks/liquid-flow/liquid-flow-theis-problem.md +++ b/web/content/docs/benchmarks/liquid-flow/liquid-flow-theis-problem.md @@ -55,3 +55,184 @@ $$ where $\gamma\approx$ 0.5772 is the Euler-Mascheroni constant. For practical purposes, the simplest approximation of $W(u)$ was proposed as $W(u)=-0.5772-lnu$ for $u <$ 0.05. Other more exact approximations of the well function were summarized by R. Srivastava and A. Guzman-Guzman ## Results and evaluation + +The following figure compares the analytical solution, the result by ogs5, and + the result by ogs6 (labeled as `pressure`) within the range that satisfies + $u <$ 0.05. +{{< img src="../theis_comparison.png" >}} +The figure shows that there is a good match between the analytical solution and + the numerical solution obtained by using ogs5 or ogs6. + +<p>Some of the data of the above curves are given in the following table.</p> +<table> +<caption>Comparison of solutions (where <em>Distance</em> means the distance + from the position where the source term is applied).</caption> +<tbody> +<tr class="odd"> +<td style="text-align: left;">Distance</td> +<td style="text-align: left;">Analytic</td> +<td style="text-align: left;">OGS5</td> +<td style="text-align: left;">OGS6</td> +<td style="text-align: left;">Error</td> +<td style="text-align: left;">Error</td> +</tr> +<tr class="even"> +<td style="text-align: left;"></td> +<td style="text-align: left;">Solution (<span class="math inline"><em>h</em><sub><em>a</em></sub></span>)</td> +<td style="text-align: left;">(<span class="math inline"><em>h</em><sub>5</sub></span>)</td> +<td style="text-align: left;">(<span class="math inline"><em>h</em><sub>6</sub></span>)</td> +<td style="text-align: left;">(<span class="math inline">$$|\frac{h_5-h_a}{h_a}|$$</span>)</td> +<td style="text-align: left;">(<span class="math inline">$$|\frac{h_6-h_a}{h_a}|$$</span>)</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">0</td> +<td style="text-align: left;">12.8141</td> +<td style="text-align: left;">12.474</td> +<td style="text-align: left;">12.474</td> +<td style="text-align: left;">0.0265</td> +<td style="text-align: left;">0.0272</td> +</tr> +<tr class="even"> +<td style="text-align: left;">1.21799</td> +<td style="text-align: left;">8.9441</td> +<td style="text-align: left;">8.79341</td> +<td style="text-align: left;">8.79341</td> +<td style="text-align: left;">0.0168</td> +<td style="text-align: left;">0.0171</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">2.43597</td> +<td style="text-align: left;">7.48878</td> +<td style="text-align: left;">7.34717</td> +<td style="text-align: left;">7.34717</td> +<td style="text-align: left;">0.0189</td> +<td style="text-align: left;">0.0192</td> +</tr> +<tr class="even"> +<td style="text-align: left;">3.65396</td> +<td style="text-align: left;">6.59978</td> +<td style="text-align: left;">6.46176</td> +<td style="text-align: left;">6.46176</td> +<td style="text-align: left;">0.0209</td> +<td style="text-align: left;">0.0213</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">4.87195</td> +<td style="text-align: left;">5.94548</td> +<td style="text-align: left;">5.81072</td> +<td style="text-align: left;">5.81072</td> +<td style="text-align: left;">0.0226</td> +<td style="text-align: left;">0.0231</td> +</tr> +<tr class="even"> +<td style="text-align: left;">6.08994</td> +<td style="text-align: left;">5.43381</td> +<td style="text-align: left;">5.30267</td> +<td style="text-align: left;">5.30267</td> +<td style="text-align: left;">0.0241</td> +<td style="text-align: left;">0.0247</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">7.30792</td> +<td style="text-align: left;">5.00981</td> +<td style="text-align: left;">4.88283</td> +<td style="text-align: left;">4.88283</td> +<td style="text-align: left;">0.0253</td> +<td style="text-align: left;">0.0260</td> +</tr> +<tr class="even"> +<td style="text-align: left;">8.52591</td> +<td style="text-align: left;">4.65012</td> +<td style="text-align: left;">4.52793</td> +<td style="text-align: left;">4.52793</td> +<td style="text-align: left;">0.0262</td> +<td style="text-align: left;">0.0269</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">8.83041</td> +<td style="text-align: left;">4.56714</td> +<td style="text-align: left;">4.44623</td> +<td style="text-align: left;">4.44623</td> +<td style="text-align: left;">0.0264</td> +<td style="text-align: left;">0.0271</td> +</tr> +<tr class="even"> +<td style="text-align: left;">9.4394</td> +<td style="text-align: left;">4.4116</td> +<td style="text-align: left;">4.29344</td> +<td style="text-align: left;">4.29344</td> +<td style="text-align: left;">0.0267</td> +<td style="text-align: left;">0.0275</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">10.6574</td> +<td style="text-align: left;">4.12707</td> +<td style="text-align: left;">4.01501</td> +<td style="text-align: left;">4.01501</td> +<td style="text-align: left;">0.0271</td> +<td style="text-align: left;">0.0279</td> +</tr> +<tr class="even"> +<td style="text-align: left;">15.2248</td> +<td style="text-align: left;">3.28072</td> +<td style="text-align: left;">3.19698</td> +<td style="text-align: left;">3.19698</td> +<td style="text-align: left;">0.0255</td> +<td style="text-align: left;">0.0261</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">20.0968</td> +<td style="text-align: left;">2.61899</td> +<td style="text-align: left;">2.57517</td> +<td style="text-align: left;">2.57517</td> +<td style="text-align: left;">0.0167</td> +<td style="text-align: left;">0.0170</td> +</tr> +<tr class="even"> +<td style="text-align: left;">22.8373</td> +<td style="text-align: left;">2.31338</td> +<td style="text-align: left;">2.29626</td> +<td style="text-align: left;">2.29626</td> +<td style="text-align: left;">0.0074</td> +<td style="text-align: left;">0.0074</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">24.0553</td> +<td style="text-align: left;">2.18892</td> +<td style="text-align: left;">2.1846</td> +<td style="text-align: left;">2.1846</td> +<td style="text-align: left;">0.0019</td> +<td style="text-align: left;">0.0019</td> +</tr> +<tr class="even"> +<td style="text-align: left;">25.2732</td> +<td style="text-align: left;">2.07055</td> +<td style="text-align: left;">2.07958</td> +<td style="text-align: left;">2.07958</td> +<td style="text-align: left;">0.0043</td> +<td style="text-align: left;">0.0043</td> +</tr> +<tr class="odd"> +<td style="text-align: left;">25.5777</td> +<td style="text-align: left;">2.04164</td> +<td style="text-align: left;">2.05407</td> +<td style="text-align: left;">2.05407</td> +<td style="text-align: left;">0.0060</td> +<td style="text-align: left;">0.0060</td> +</tr> +<tr class="even"> +<td style="text-align: left;">29.8407</td> +<td style="text-align: left;">1.67134</td> +<td style="text-align: left;">1.73518</td> +<td style="text-align: left;">1.73518</td> +<td style="text-align: left;">0.0381</td> +<td style="text-align: left;">0.0367</td> +</tr> +</tbody> +</table> +<p>The analytical solutions are for an ideal problem with point wise pumping + term. While for the FEM analysis, the point wise source value is distributed to + the surface of a small hole around the source point in order to avoid the + singularity. One can see from the table that the precisions of the FEM + solutions are still acceptable with such transform of the point pumping + term.</p> diff --git a/web/content/docs/benchmarks/liquid-flow/theis_comparison.png b/web/content/docs/benchmarks/liquid-flow/theis_comparison.png new file mode 100644 index 0000000000000000000000000000000000000000..db9d03de6f923a536eb0fc83c70ecb0d70bc6bbe --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/theis_comparison.png @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8de6559ef5b271a073281187d4c5ba0a85f73f7b20afdab9b3e5432391117840 +size 9296