{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input" ] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%config InlineBackend.figure_format = 'svg'\n", "import matplotlib as mpl\n", "mpl.rcParams['contour.negative_linestyle'] = 'solid'\n", "from myst_nb import glue\n", "\n", "import sys\n", "import os\n", "sys.path.append(os.path.abspath('../py_figures')) \n", "import diagrams_2d\n", "import inter_fig_chap7" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(content:chap7:well_near_coast)=\n", "# A well in uniform background flow near the coast" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The solutions in the previous section concerned the capture zone of a well in uniform background flow near a straight river. The critical discharge of the well was derived such that the well does not capture water from the river. In this section, the straight river is replaced by a straight and vertical coastal boundary and the maximum discharge of the well is determined such that the well does not pump any salt water. In other words, the maximum discharge is determined such that the interface does not reach the well. It turns out that this maximum discharge is quite a bit smaller than the maximum discharge for a well near a river ({numref}`content:chap7:well_near_river`).\n", "
     Consider a well in a coastal aquifer. The coastline is located along the line $x=0$. The well is located at $(x,y)=(-d,0)$, has a discharge $Q$, and a radius $r_w$. In absence of the well, the flow towards the coast is uniform and equal to $Q_x=U$. The aquifer is unconfined, the hydraulic conductivity is $k$ and the elevation of the bottom of the aquifer is $z_b$ with respect to sea level. The density of the seawater is $\\rho_s$. The sea is in direct contact with the aquifer (no seabed resistance), so that the head along the coastline is equal to sea level. The solution for the discharge potential is identical to the solution for a well near a river in direct contact with the aquifer (see Eq. {eq}`eq_dischargepot_7.3`). For this case of unconfined interface flow, the potential at the toe of the interface is (see Eq. {eq}`eq_phitoeunconfined4.2`)\n", "\n", "$$\n", "\\Phi_\\text{toe} = \\tfrac{1}{2}k\\frac{\\alpha+1}{\\alpha^2}z_b^2\n", "$$(eq_dischargepot_7.4)\n", "\n", "In the absence of pumping, the toe of the interface is located at (see Eq. {eq}`eq_xtoeconf4.1`)\n", "\n", "$$\n", "x_\\text{toe}=-\\frac{\\Phi_\\text{toe}}{U}\n", "$$(eq_interface_7.4)\n", "\n", "     First, consider the case that the discharge of the well is half the critical discharge for a well near a river (Eq. {eq}`eq_crdischarge_7.3`). A flow net is constructed showing that indeed the stagnation point is still far from the coast line for this value of $Q$ ({numref}`Figure %s `). The toe of the interface in the absence of pumping is plotted with a black dashed line, while the contour line representing $\\Phi_\\text{toe}$ for the case with pumping is plotted with a black solid line. \n", "It turns out that there are two contour lines where the potential is equal to $\\Phi_\\text{toe}$. One contour line runs from North to South and curves towards the well; this is the location of the toe of the interface during pumping. A second black contour is present around the well. This is because the discharge potential at the well is below $\\Phi_\\text{toe}$, but this does not mean that the well pumps saltwater. The black contour line around the well is not connected to the black contour line near the coast, which means that there is no way for the saltwater to reach the well. This is further illustrated in the right graph of {numref}`Figure %s `, where the potential is plotted as a function of $x$ along the line $y=0$. When converting the potential to a head, the values of the potential inside the closed black contour around the well must be converted to a head using the formula for unconfined flow rather than for unconfined interface flow." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "dischage of well: 1570.80 m^3/d\n" ] } ], "source": [ "# parameters\n", "rw = 0.3 # radius of well, m\n", "k = 40 # hydraulic conductivity, m/d\n", "zb = -20 # elevation of bottom of aquifer, m\n", "rhof = 1000 # density of fresh water, kg/m^3\n", "rhos = 1025 # density of salt water, kg/m^3\n", "U = 1 # uniform flow in x-direction, m^2/d\n", "d = 1000 # dischance of well from coast line, m\n", "Qcr = np.pi * U * d # critical discharge from well near river, m^3/d\n", "Q = Qcr / 2 # discharge of well, m^3/d\n", "print(f'dischage of well: {Q:.2f} m^3/d')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# solution\n", "alpha = rhof / (rhos - rhof) # alpha factor\n", "phitoe = 0.5 * k * (alpha + 1) / alpha ** 2 * zb ** 2\n", "xtoe = -phitoe / U # toe in the absence of pumping\n", "\n", "def potential(x, y, Q):\n", " r1sq = (x + d) ** 2 + y ** 2\n", " r2sq = (x - d) ** 2 + y ** 2\n", " return -U * x + Q / (4 * np.pi) * np.log(r1sq / r2sq) \n", "\n", "def streamfunc(x, y, Q, d=d):\n", " theta1 = np.arctan2(y, x + d)\n", " theta2 = np.arctan2(y, x - d)\n", " return -U * y + Q / (2 * np.pi) * (theta1 - theta2)\n", " \n", "xg, yg = np.meshgrid(np.linspace(-1200, 0, 120), np.linspace(-500, 500, 101))\n", "pot = potential(xg, yg, Q)\n", "psi = streamfunc(xg, yg, Q)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-output" ] }, "outputs": [ { "data": { 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic flow net and potential along y=0\n", "plt.subplot(121, aspect=1)\n", "plt.contour(xg, yg, pot, np.arange(-Qcr, Qcr, Qcr / 30), colors='C0')\n", "plt.contour(xg, yg, pot, [phitoe], colors='k')\n", "plt.contour(xg, yg, psi, np.arange(-Qcr, Qcr, Qcr / 30), colors='C1')\n", "plt.axvline(xtoe, color='k', linestyle='--')\n", "plt.subplot(122) # potential along y=0\n", "plt.plot(xg[0], pot[50], label=r'$\\Phi(x,0)$')\n", "plt.axhline(phitoe, ls='--', label=r'$\\Phi_{toe}$')\n", "plt.legend();" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-03-04T10:59:50.974681\n image/svg+xml\n \n \n Matplotlib v3.10.6, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams_2d.figdef2)\n", "plt.subplot(121, aspect=1)\n", "plt.title('(a)', loc='left')\n", "plt.contour(xg, yg, pot, np.arange(-Qcr, Qcr, Qcr / 30), colors='C0')\n", "cs = plt.contour(xg, yg, pot, [phitoe], colors='k')\n", "handle, _ = cs.legend_elements()\n", "plt.contour(xg, yg, psi, np.arange(-Qcr, Qcr, Qcr / 30), colors='C1')\n", "plt.plot(-d, 0, 'k.')\n", "line = plt.axvline(xtoe, color='k', linestyle='--')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('y (m)', labelpad=-12)\n", "plt.text(-500, 0, 'toe', rotation=90, va='center')\n", "plt.legend([handle[0], line], [r'$\\Phi_\\text{toe}$ with well', 'toe without well'])\n", "\n", "plt.subplot(122)\n", "plt.title('(b)', loc='left')\n", "plt.plot(xg[0], pot[50], label=r'$\\Phi(x,0)$')\n", "plt.axhline(phitoe, ls='--', label=r'$\\Phi_{toe}$')\n", "diagrams_2d.arrow((-1000, 150), (-1000, 300), 'well', ha='center', arrow='->')\n", "xsp = -np.sqrt(d ** 2 - Q * d / (np.pi * U))\n", "diagrams_2d.arrow((xsp, 270), (xsp, 420), 'stagnation point', ha='center', arrow='->')\n", "xtoe1 = -443\n", "diagrams_2d.arrow((xtoe1, 200), (xtoe1, 0), 'toe', ha='center', arrow='->')\n", "diagrams_2d.arrow((0, 0), (0, -200), 'coast', ha='center', arrow='->')\n", "plt.xlim(-500, -400)\n", "plt.xlim(-1200, 0)\n", "plt.xlabel('x (m)')\n", "plt.ylabel(r'$\\Phi$ (m$^3$/d)')\n", "plt.grid()\n", "plt.legend()\n", "plt.tight_layout()\n", "glue(\"fig_flownet_coast_7.4\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{glue:figure} fig_flownet_coast_7.4\n", "---\n", "name: fig_flownet_coast_7.4\n", "---\n", "A well in uniform background flow towards the coast for a case that $Q     The maximum discharge is found numerically. A function is written that takes the discharge $Q$ as input and returns the difference between the discharge potential at the stagnation point and $\\Phi_\\text{toe}$. This function equals zero for the maximum discharge. The `brentq` method is used to find $Q_\\text{max}$. Note that the maximum discharge is just slightly more than half the critical discharge for a well near a river (Eq. {eq}`eq_crdischarge_7.3`). " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "maximum discharge: 1803.22 m^3/d\n", "Qmax as fraction of Qcr of Section 7.3: 0.57\n" ] } ], "source": [ "def func(Q):\n", " xs = -np.sqrt(d ** 2 - Q * d / (np.pi * U))\n", " return potential(xs, 0, Q) - phitoe\n", "\n", "from scipy.optimize import brentq\n", "Qmax = brentq(func, 1, np.pi * U * d)\n", "print(f'maximum discharge: {Qmax:.2f} m^3/d')\n", "print(f'Qmax as fraction of Qcr of Section 7.3: {Qmax / Qcr:.2f}')\n", "pot1 = potential(xg, yg, Qmax)\n", "psi1 = streamfunc(xg, yg, Qmax)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "     When pumping at slightly below the maximum discharge, the toe of the interface is still a significant distance from the well. When the discharge is increased to the maximum discharge, the toe of the saltwater suddenly breaks through to the well as the potential at the stagnation point drops below the potential at the toe; i.e., the head at the stagnation point drops below $-z_b/\\alpha$. An interactive figure is shown below ({numref}`fig_flownet_coast2_7.4`); the code is the same as for {numref}`Figure %s ` and is not repeated. The potential along the line $y=0$ shows that the maximum potential between the well and the coast is indeed $\\Phi_\\text{toe}$ when $Q = Q_\\text{max}$. 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# flow nets\n", "fig = plt.figure(figsize=diagrams_2d.figdef2)\n", "plt.subplot(121, aspect=1)\n", "plt.title('(a)', loc='left')\n", "plt.contour(xg, yg, pot1, np.arange(-2000, 2000, 100), colors='C0')\n", "cs = plt.contour(xg, yg, pot1, [phitoe], colors='k')\n", "handle, _ = cs.legend_elements()\n", "#cs.collections[0].set_label(r'$\\Phi_{toe}$ with well')\n", "plt.contour(xg, yg, psi1, np.arange(-2000, 2000, 100), colors='C1')\n", "line = plt.axvline(xtoe, color='k', linestyle='--', label='toe without well')\n", "plt.plot(-d, 0, 'k.')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('y (m)', labelpad=-12)\n", "plt.legend([handle[0], line], [r'$\\Phi_\\text{toe}$ with well', 'toe without well'])\n", "\n", "plt.subplot(122)\n", "plt.title('(b)', loc='left')\n", "plt.plot(xg[0], pot1[50], label=r'$\\Phi(x,0)$')\n", "plt.axhline(phitoe, ls='--', label=r'$\\Phi_{toe}$')\n", "plt.xlim(-1200, 0)\n", "plt.xlabel('x (m)')\n", "diagrams_2d.arrow((-1000, 150), (-1000, 300), 'well', ha='center', arrow='->')\n", "xsp = -np.sqrt(d ** 2 - Qmax * d / (np.pi * U))\n", "diagrams_2d.arrow((xsp, 200), (xsp, 350), 'stagnation point', ha='center', arrow='->')\n", "plt.ylabel(r'$\\Phi$ (m$^3$/d)')\n", "plt.grid()\n", "plt.legend()\n", "plt.tight_layout()\n", "glue(\"fig_flownet_coast2_7.4\", fig, display=False)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/javascript": [ "'use strict';\n", "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " const force = true;\n", "\n", " if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n", " root._bokeh_onload_callbacks = [];\n", " root._bokeh_is_loading = undefined;\n", " }\n", "\n", "const JS_MIME_TYPE = 'application/javascript';\n", " const HTML_MIME_TYPE = 'text/html';\n", " const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " const CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " const script = document.createElement(\"script\");\n", " node.appendChild(script);\n", " }\n", "\n", " /**\n", " * Handle when an output is cleared or removed\n", " */\n", " function handleClearOutput(event, handle) {\n", " function drop(id) {\n", " const view = Bokeh.index.get_by_id(id)\n", " if (view != null) {\n", " view.model.document.clear()\n", " Bokeh.index.delete(view)\n", " }\n", " }\n", "\n", " const cell = handle.cell;\n", "\n", " const id = cell.output_area._bokeh_element_id;\n", " const server_id = cell.output_area._bokeh_server_id;\n", "\n", " // Clean up Bokeh references\n", " if (id != null) {\n", " drop(id)\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n", " cell.notebook.kernel.execute(cmd_clean, {\n", " iopub: {\n", " output: function(msg) {\n", " const id = msg.content.text.trim()\n", " drop(id)\n", " }\n", " }\n", " });\n", " // Destroy server and session\n", " const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n", " cell.notebook.kernel.execute(cmd_destroy);\n", " }\n", " }\n", "\n", " /**\n", " * Handle when a new output is added\n", " */\n", " function handleAddOutput(event, handle) {\n", " const output_area = handle.output_area;\n", " const output = handle.output;\n", "\n", " // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n", " if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n", " return\n", " }\n", "\n", " const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n", "\n", " if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n", " toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n", " // store reference to embed id on output_area\n", " output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n", " }\n", " if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n", " const bk_div = document.createElement(\"div\");\n", " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", " const script_attrs = bk_div.children[0].attributes;\n", " for (let i = 0; i < script_attrs.length; i++) {\n", " toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n", " toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n", " }\n", " // store reference to server id on output_area\n", " output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n", " }\n", " }\n", "\n", " function register_renderer(events, OutputArea) {\n", "\n", " function append_mime(data, metadata, element) {\n", " // create a DOM node to render to\n", " const toinsert = this.create_output_subarea(\n", " metadata,\n", " CLASS_NAME,\n", " EXEC_MIME_TYPE\n", " );\n", " this.keyboard_manager.register_events(toinsert);\n", " // Render to node\n", " const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n", " render(props, toinsert[toinsert.length - 1]);\n", " element.append(toinsert);\n", " return toinsert\n", " }\n", "\n", " /* Handle when an output is cleared or removed */\n", " events.on('clear_output.CodeCell', handleClearOutput);\n", " events.on('delete.Cell', handleClearOutput);\n", "\n", " /* Handle when a new output is added */\n", " events.on('output_added.OutputArea', handleAddOutput);\n", "\n", " /**\n", " * Register the mime type and append_mime function with output_area\n", " */\n", " OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n", " /* Is output safe? */\n", " safe: true,\n", " /* Index of renderer in `output_area.display_order` */\n", " index: 0\n", " });\n", " }\n", "\n", " // register the mime type if in Jupyter Notebook environment and previously unregistered\n", " if (root.Jupyter !== undefined) {\n", " const events = require('base/js/events');\n", " const OutputArea = require('notebook/js/outputarea').OutputArea;\n", "\n", " if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n", " register_renderer(events, OutputArea);\n", " }\n", " }\n", " if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n", " root._bokeh_timeout = Date.now() + 5000;\n", " root._bokeh_failed_load = false;\n", " }\n", "\n", " const NB_LOAD_WARNING = {'data': {'text/html':\n", " \"
\\n\"+\n", " \"

\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"

\\n\"+\n", " \"
    \\n\"+\n", " \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n", " \"
  • use INLINE resources instead, as so:
    • \\n\"+\n", " \"
\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
\"}};\n", "\n", " function display_loaded(error = null) {\n", " const el = document.getElementById(null);\n", " if (el != null) {\n", " const html = (() => {\n", " if (typeof root.Bokeh === \"undefined\") {\n", " if (error == null) {\n", " return \"BokehJS is loading ...\";\n", " } else {\n", " return \"BokehJS failed to load.\";\n", " }\n", " } else {\n", " const prefix = `BokehJS ${root.Bokeh.version}`;\n", " if (error == null) {\n", " return `${prefix} successfully loaded.`;\n", " } else {\n", " return `${prefix} encountered errors while loading and may not function as expected.`;\n", " }\n", " }\n", " })();\n", " el.innerHTML = html;\n", "\n", " if (error != null) {\n", " const wrapper = document.createElement(\"div\");\n", " wrapper.style.overflow = \"auto\";\n", " wrapper.style.height = \"5em\";\n", " wrapper.style.resize = \"vertical\";\n", " const content = document.createElement(\"div\");\n", " content.style.fontFamily = \"monospace\";\n", " content.style.whiteSpace = \"pre-wrap\";\n", " content.style.backgroundColor = \"rgb(255, 221, 221)\";\n", " content.textContent = error.stack ?? error.toString();\n", " wrapper.append(content);\n", " el.append(wrapper);\n", " }\n", " } else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(() => display_loaded(error), 100);\n", " }\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls == null || js_urls.length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " root._bokeh_is_loading = css_urls.length + js_urls.length;\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", "\n", " function on_error(url) {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " for (let i = 0; i < css_urls.length; i++) {\n", " const url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " }\n", "\n", " for (let i = 0; i < js_urls.length; i++) {\n", " const url = js_urls[i];\n", " const element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.8.0.min.js\"];\n", " const css_urls = [];\n", "\n", " const inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {\n", " }\n", " ];\n", "\n", " function run_inline_js() {\n", " if (root.Bokeh !== undefined || force === true) {\n", " try {\n", " for (let i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", "\n", " } catch (error) {throw error;\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " } else if (force !== true) {\n", " const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\n", " }\n", " }\n", "\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n", " run_inline_js();\n", " } else {\n", " load_libs(css_urls, js_urls, function() {\n", " console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", " run_inline_js();\n", " });\n", " }\n", "}(window));" ], "application/vnd.bokehjs_load.v0+json": "'use strict';\n(function(root) {\n function now() {\n return new Date();\n }\n\n const force = true;\n\n if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n root._bokeh_onload_callbacks = [];\n root._bokeh_is_loading = undefined;\n }\n\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n const NB_LOAD_WARNING = {'data': {'text/html':\n \"
\\n\"+\n \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\n\"+\n \"
    \\n\"+\n \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n \"
  • use INLINE resources instead, as so:
    • \\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded(error = null) {\n const el = document.getElementById(null);\n if (el != null) {\n const html = (() => {\n if (typeof root.Bokeh === \"undefined\") {\n if (error == null) {\n return \"BokehJS is loading ...\";\n } else {\n return \"BokehJS failed to load.\";\n }\n } else {\n const prefix = `BokehJS ${root.Bokeh.version}`;\n if (error == null) {\n return `${prefix} successfully loaded.`;\n } else {\n return `${prefix} encountered errors while loading and may not function as expected.`;\n }\n }\n })();\n el.innerHTML = html;\n\n if (error != null) {\n const wrapper = document.createElement(\"div\");\n wrapper.style.overflow = \"auto\";\n wrapper.style.height = \"5em\";\n wrapper.style.resize = \"vertical\";\n const content = document.createElement(\"div\");\n content.style.fontFamily = \"monospace\";\n content.style.whiteSpace = \"pre-wrap\";\n content.style.backgroundColor = \"rgb(255, 221, 221)\";\n content.textContent = error.stack ?? error.toString();\n wrapper.append(content);\n el.append(wrapper);\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(() => display_loaded(error), 100);\n }\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error(url) {\n console.error(\"failed to load \" + url);\n }\n\n for (let i = 0; i < css_urls.length; i++) {\n const url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (let i = 0; i < js_urls.length; i++) {\n const url = js_urls[i];\n const element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.8.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.8.0.min.js\"];\n const css_urls = [];\n\n const inline_js = [ function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\nfunction(Bokeh) {\n }\n ];\n\n function run_inline_js() {\n if (root.Bokeh !== undefined || force === true) {\n try {\n for (let i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n\n } catch (error) {throw error;\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "
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const Q = slider.value;\\n\\n // Find nearest index\\n let idx = 0;\\n let bestDiff = Infinity;\\n for (let i=0; i1000.0 m\\u00b3/d          |         1200.0 m\\u00b3/d          |         1400.0 m\\u00b3/d          |         1600.0 m\\u00b3/d          |         1803.2 m\\u00b3/d\"}}]}}]}};\n", " const render_items = [{\"docid\":\"dca583e3-aa0b-483c-9388-ad9250722fb0\",\"roots\":{\"p2824\":\"a791e5c1-31c5-498b-a8b5-e86fcb8cde74\"},\"root_ids\":[\"p2824\"]}];\n", " void root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " let attempts = 0;\n", " const timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "p2824" } }, "output_type": "display_data" } ], "source": [ "inter_fig_chap7.fig_7_8()" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{raw} html\n", "\n", "```\n", "\n", "```{figure} ../py_figures/1x1_pixel.png\n", "---\n", "name: fig_flownet_coast2_7.4\n", "---\n", "A well in uniform background flow towards the coast; flow net (left) and potential along $y=0$ (right). The red cross is the location of the stagnation point.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{exercise}\n", ":label: ex_stagnation_7.4\n", "\n", "The head and interface position for $Q=Q_\\text{max}$ is shown in {numref}`Figure %s `. Compute the location of the stagnation point and the head at the well for this case.\n", "\n", "
\n", " Solution \n", "\n", "$h_w=-1.73$ m \\\n", "$x_s= -652.7$ m\n", "\n", "
\n", "\n", "```" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-03-04T10:59:53.187971\n image/svg+xml\n \n \n Matplotlib v3.10.6, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from numpy.lib.scimath import sqrt as csqrt\n", "xs = -csqrt(d ** 2 - Qmax * d / (np.pi * U))\n", "\n", "def pot2h(pot, k, alpha, zb):\n", " phitoe = 0.5 * k * (alpha + 1) / alpha**2 * zb**2\n", " Cu = -0.5 * k * (alpha + 1) * zb**2 / alpha\n", " h = np.zeros(len(pot))\n", " h[pot < phitoe] = np.sqrt(2 * pot[pot < phitoe] / (k * (alpha + 1)))\n", " h[pot >= phitoe] = np.sqrt(2 * (pot[pot >= phitoe] - Cu) / k) + zb\n", " return h\n", "\n", "x = np.hstack((np.linspace(-1200, -d - rw, 100), np.linspace(-d + rw, 0, 200)))\n", "pot = potential(x, 0, Qmax)\n", "h = np.zeros(len(pot))\n", "phitoe = 0.5 * k * (alpha + 1) / alpha**2 * zb**2\n", "Cu = -0.5 * k * (alpha + 1) * zb**2 / alpha\n", "h[x < xs] = np.sqrt(2 * (pot[x < xs] - Cu) / k) + zb\n", "h[x > xs] = pot2h(pot[x > xs], k, alpha, zb)\n", "zeta = np.nan * np.ones(len(x))\n", "zeta[x > xs] = -40 * h[x > xs]\n", "\n", "fig = plt.figure(figsize=diagrams_2d.figdef2)\n", "plt.subplot(111, aspect=10)\n", "plt.plot(x, h, zorder=10)\n", "plt.plot(x, zeta, diagrams_2d.C1dark, zorder=10)\n", "diagrams_2d.arrow((xs, zb), (-700, -17), 'toe', '->')\n", "plt.axhline(zb, color='k')\n", "plt.plot([-d, -d], [zb, 0], 'k--')\n", "plt.plot([0, 0], [zb, 2], 'k:')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('z (m) - VE=10x')\n", "plt.grid()\n", "plt.tight_layout()\n", "glue(\"fig_interface_position_7.4\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "" ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.7" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false } }, "nbformat": 4, "nbformat_minor": 4 }