{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "remove-input" ] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from myst_nb import glue\n", "%config InlineBackend.rc = {'figure.dpi': 300}\n", "%config InlineBackend.figure_format = 'svg'\n", "import sys\n", "sys.path.append(\"../py_figures\")\n", "import diagrams_2d_chap9 as diagrams_2d\n", "import inter_fig_chap9" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(content:chap9.2:periodic_wells)=\n", "# A well with a periodic discharge\n", "\n", "The discharge of a well may vary through time periodically. For example, in an aquifer storage and recovery (ASR) system, water is injected into the aquifer during times of abundant water availability and pumped out of the aquifer during times of water scarcity. For an ASR well, the discharge varies around zero, for example sinusoidally as\n", "\n", "$$\n", "Q(t) = A \\sin(2 \\pi t / \\tau)\n", "$$(eq_Qsin9.2)\n", "\n", "where $A$ and $\\tau$ are the amplitude and period of the sinusoidal discharge. The boundary condition at the well is approximated as\n", "\n", "$$\n", "\\lim_{r \\to 0} -2\\pi r Q_r = A\\sin(2\\pi t / \\tau)\n", "$$(eq_bc_well9.2)\n", "\n", "while the boundary condition at infinity is ({eq}`eq_bctheis29.1`). There is no initial condition: The well has been pumping with the sinusoidal discharge (Eq. {eq}`eq_Qsin9.2`) for a long time. Flow in the aquifer is governed by the diffusion equation ({eq}`eq_deq_rad_sym_transient_flow9.0`).\n", "
      The solution for the stated problem is obtained using separation of variables: The solution is written as a function of the radial distance $r$ multiplied by a function of the time $t$. The analysis is easiest conducted in terms of complex variables. The head solution corresponding to boundary condition ({eq}`eq_Qsin9.2`) is the imaginary part of the complex solution ({cite:alp}`bruggeman1999analytical`):\n", "\n", "$$\n", "h = -\\frac{A}{2 \\pi T} \\Im\\left[\\text{K}_0\\left(\\frac{r}{\\mu}\\sqrt{i}\\right)\\text{e}^{2\\pi i t/\\tau}\\right] \n", "$$(eq_hsinwell9.2)\n", "\n", "where the characteristic length $\\mu$ is defined as\n", "\n", "$$\n", "\\mu = \\sqrt{\\frac{T}{S}\\frac{\\tau}{2\\pi}} \n", "$$(eq_char_L_mu9.2)\n", "\n", "Note that $\\mu$ has the dimension of length. The corresponding radial component of the discharge vector is\n", "\n", "$$\n", "Q_r = -\\frac{A}{2 \\pi} \\Im\\left[ \\frac{\\sqrt{i}}{\\mu} \\text{K}_1\\left(\\frac{r}{\\mu}\\sqrt{i}\\right)\\text{e}^{2\\pi i t/\\tau}\\right] \n", "$$(eq_radial_dis9.2)\n", "\n", "The sinusoidal head variation is largest at the well and decreases with distance from the well. Discharge variations with a short period damp out at a shorter distance from the well than variations with a longer period. It is noted that taking the real part of the complex function in Eq. ({eq}`eq_hsinwell9.2`) results in the solution for the case that the sine function in Eq. ({eq}`eq_Qsin9.2`) is replaced by a cosine function.\n", "
      In the example below, the solution is plotted for a sinusoidal well discharge with a period of 1 year in an unconfined aquifer where the transmissivity is again approximated as constant. The function `kv` of the `scipy.special` package is used to compute the modified Bessel functions of complex argument, because the functions `k0` and `k1` only accept real arguments. In the left graph of {numref}`Figure %s `, the head variation is plotted at the well and at a distance $r=2\\mu$. In the right graph of {numref}`Figure %s `, the total flow in the aquifer $Q_\\text{tot}=2\\pi rQ_r$ is plotted at the well and at $r = 6\\mu$. Both the amplitude of the head and of the total discharge decrease with the distance to the well, but the amplitude of the head decreases much faster. The amplitude of the head decreases to less than 5% of the amplitude of the well at $r = 2 \\mu$, while the amplitude of the total flow decreases to less than $0.05A$ at $r = 6 \\mu$." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "characteristic length mu: 241.02 m\n" ] } ], "source": [ "# parameters\n", "T = 100 # transmissivity, m^2/d\n", "S = 1e-1 # storage coefficient, -\n", "tau = 365 # period, d\n", "A = 200 # amplitude of discharge function, m^3/d\n", "rw = 0.3 # radius of well, m\n", "mu = np.sqrt(T * tau / (S * 2 * np.pi))\n", "print(f'characteristic length mu: {mu:.2f} m')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# solution\n", "from scipy.special import kv\n", "def hsinwell(r, t):\n", " hcomplex = -A / (2 * np.pi * T) * kv(0, r * np.sqrt(1j) / mu) * \\\n", " np.exp(2 * np.pi * 1j * t / tau)\n", " return hcomplex.imag\n", "\n", "def Qrsinwell(r, t):\n", " Qrcomplex = -A / (2 * np.pi) * np.sqrt(1j) / mu * \\\n", " kv(1, r * np.sqrt(1j) / mu) * np.exp(2 * np.pi * 1j * t / tau)\n", " return Qrcomplex.imag" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-03-04T21:13:02.528690\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic plot\n", "t = np.linspace(0, 2 * tau, 100)\n", "plt.subplot(121)\n", "plt.plot(t, hsinwell(rw, t), label='$r=r_w$')\n", "plt.plot(t, hsinwell(2 * mu, t), label='$r=2\\\\mu$')\n", "plt.legend()\n", "plt.subplot(122)\n", "plt.plot(t, Qrsinwell(rw, t) * 2 * np.pi * rw, label='$r=r_w$')\n", "plt.plot(t, Qrsinwell(6 * mu, t) * 2 * np.pi * 6 * mu, 'C2', label='$r=6\\\\mu$')\n", "plt.legend();" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-03-04T21:13:03.011207\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", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=diagrams_2d.figdef2)\n", "t = np.linspace(0, 2 * tau, 100)\n", "plt.subplot(121)\n", "plt.plot(t, hsinwell(rw, t), label='$r=r_w$')\n", "plt.plot(t, hsinwell(mu, t), label='$r=\\\\mu$')\n", "plt.plot(t, hsinwell(2 * mu, t), label='$r=2\\\\mu$')\n", "plt.xticks(np.arange(0, 2 * 365 + 1, 365/2), np.arange(0, 2.01, 0.5))\n", "plt.xlabel('time (year)')\n", "plt.ylabel('head (m)')\n", "plt.title('(a)', loc='left')\n", "plt.legend()\n", "plt.grid()\n", "plt.subplot(122)\n", "plt.plot(t, Qrsinwell(rw, t) * 2 * np.pi * rw, label='$r=r_w$')\n", "plt.plot(t, Qrsinwell(3 * mu, t) * 2 * np.pi * 3 * mu, 'C3', label='$r=3\\\\mu$')\n", "plt.plot(t, Qrsinwell(6 * mu, t) * 2 * np.pi * 6 * mu, 'C4', label='$r=6\\\\mu$')\n", "plt.xticks(np.arange(0, 2 * 365 + 1, 365/2), np.arange(0, 2.01, 0.5))\n", "plt.xlabel('time (year)')\n", "plt.ylabel('$Q_{tot}$ (m$^3$/d)')\n", "plt.title('(b)', loc='left')\n", "plt.legend()\n", "plt.grid()\n", "plt.tight_layout()\n", "glue('fig_sine_var', fig, display=False)" ] }, { "cell_type": "code", "execution_count": 6, "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", "
\n" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "(function(root) {\n", " function embed_document(root) {\n", " const docs_json = {\"5ea57b65-b9a4-4352-b6ce-10931f036589\":{\"version\":\"3.8.0\",\"title\":\"Bokeh Application\",\"config\":{\"type\":\"object\",\"name\":\"DocumentConfig\",\"id\":\"p1139\",\"attributes\":{\"notifications\":{\"type\":\"object\",\"name\":\"Notifications\",\"id\":\"p1140\"}}},\"roots\":[{\"type\":\"object\",\"name\":\"Column\",\"id\":\"p1138\",\"attributes\":{\"children\":[{\"type\":\"object\",\"name\":\"Row\",\"id\":\"p1137\",\"attributes\":{\"children\":[{\"type\":\"object\",\"name\":\"Figure\",\"id\":\"p1020\",\"attributes\":{\"width\":390,\"height\":292,\"x_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p1022\"},\"y_range\":{\"type\":\"object\",\"name\":\"Range1d\",\"id\":\"p1030\",\"attributes\":{\"start\":-2.4,\"end\":2.4}},\"x_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1031\"},\"y_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1032\"},\"title\":{\"type\":\"object\",\"name\":\"Title\",\"id\":\"p1023\",\"attributes\":{\"text\":\"(a)\"}},\"outline_line_color\":\"black\",\"renderers\":[{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1062\",\"attributes\":{\"data_source\":{\"type\":\"object\",\"name\":\"ColumnDataSource\",\"id\":\"p1008\",\"attributes\":{\"selected\":{\"type\":\"object\",\"name\":\"Selection\",\"id\":\"p1009\",\"attributes\":{\"indices\":[],\"line_indices\":[]}},\"selection_policy\":{\"type\":\"object\",\"name\":\"UnionRenderers\",\"id\":\"p1010\"},\"data\":{\"type\":\"map\",\"entries\":[[\"t_years\",{\"type\":\"ndarray\",\"array\":{\"type\":\"bytes\",\"data\":\"H4sIAAEAAAAC/y2Te0SdYRzHE61kEymJJpFyLCmS2dK3mWhlyaaLSKREmmqmidlkjIyMGpFINvonkSUljhJHpPtVt1On605nndt77pc9z3O+zz+v9/f7Pd/f9/t5vVFRkVOvfzO6OzmIyPMX4tI/WRdiJ/n+B3kdrWUrh7Os69EpX1MX2V9CaUrDh41SA+eW8djQlWboWuH8Gnqev2w6WV3nvU20DLwuPGje4v1tVJlr47d9O9TZQ5Ec/7FPvQNkD7dPLWcdUvcIiY7ub0vzx9Q/xZffJRONOiP3GCFEhNIZ952jLVjeW68/514TGnVy4wX3X6CmWp5L+rhERa/sXNHPNV5MyBvX9HUDISKUbunvFjkxcsNf+jQjI182zPR7ByWTYqFvCx71Saf/6P8e0dOfRYJ75rBCVkWLeaz4KG6NZNiYy4b3aVLYxnx2vBNbh9vtzGmHKic6mNcB6XpoxsHcTig7DU7md6JOpP4ZrZGDhreimjSukYeGShXYRS4uvFLyLvJxQ2EZcpOTG8Ue49eEYg95efCs3/S03+QhNy8KMq8sD/u85OdF7tzN2PdcHzn6oIssJk8/1HiPn1z9UNjSA+QbQKoCGyDnIJLHtSdCiryDSFCGQuQegoo7HSL/MB4ocGF+hzD4W5T8B7qrU3cgAwAA\"},\"shape\":[100],\"dtype\":\"float64\",\"order\":\"little\"}],[\"h1\",{\"type\":\"ndarray\",\"array\":{\"type\":\"bytes\",\"data\":\"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\"},\"shape\":[100],\"dtype\":\"float64\",\"order\":\"little\"}]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1063\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1064\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1059\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"t_years\"},\"y\":{\"type\":\"field\",\"field\":\"h1\"},\"line_color\":\"#1f77b4\",\"line_width\":2}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1060\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"t_years\"},\"y\":{\"type\":\"field\",\"field\":\"h1\"},\"line_color\":\"#1f77b4\",\"line_alpha\":0.1,\"line_width\":2}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1061\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"t_years\"},\"y\":{\"type\":\"field\",\"field\":\"h1\"},\"line_color\":\"#1f77b4\",\"line_alpha\":0.2,\"line_width\":2}}}},{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1071\",\"attributes\":{\"data_source\":{\"type\":\"object\",\"name\":\"ColumnDataSource\",\"id\":\"p1017\",\"attributes\":{\"selected\":{\"type\":\"object\",\"name\":\"Selection\",\"id\":\"p1018\",\"attributes\":{\"indices\":[],\"line_indices\":[]}},\"selection_policy\":{\"type\":\"object\",\"name\":\"UnionRenderers\",\"id\":\"p1019\"},\"data\":{\"type\":\"map\",\"entries\":[[\"x\",[0.2]],[\"y\",[1.8]],[\"w\",[0.53]],[\"h\",[0.4]]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1072\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1073\"}}},\"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// Get the index directly from slider value (apply offset so 0 -> idx200)\\n const offset = idx200;\\n const idx = Math.round(slider.value) + offset;\\n const actual_r = r_vals[idx];\\n\\n // Update curves\\n source_h.data['h1'] = h_blue[idx];\\n source_h.change.emit();\\n source_Q.data['Q1'] = Q_blue[idx];\\n source_Q.change.emit();\\n\\n // Format r value for display\\n let r_display;\\n if (actual_r === 0.3) {\\n r_display = \\\"0.3\\\";\\n } else {\\n r_display = Math.round(actual_r).toString();\\n }\\n\\n // Update custom title with right alignment and black color\\n custom_title.text = \\\"
Radial distance from well: \\\" + r_display + \\\" m
\\\";\\n\\n // Update legend text for head plot\\n const r_text = \\\"r = \\\" + r_display + \\\" m\\\";\\n label_data_h.data['text'] = [r_text];\\n label_data_h.change.emit();\\n \"}}]]]},\"width\":520,\"margin\":[0,0,10,0],\"align\":\"center\",\"show_value\":false,\"format\":{\"type\":\"object\",\"name\":\"CustomJSTickFormatter\",\"id\":\"p1132\",\"attributes\":{\"args\":{\"type\":\"map\",\"entries\":[[\"r_vals\",[0.3,25.0,50.0,75.0,100.0,125.0,150.0,175.0,200.0,325.0,450.0,575.0,700.0,825.0,950.0,1075.0,1200.0]]]},\"code\":\"\\n const r_array = [0.3, 25.0, 50.0, 75.0, 100.0, 125.0, 150.0, 175.0, 200.0, 325.0, 450.0, 575.0, 700.0, 825.0, 950.0, 1075.0, 1200.0];\\n const offset = 8;\\n const idx = Math.round(tick) + offset;\\n if (idx >= 0 && idx < r_array.length) {\\n const actual_r = r_array[idx];\\n if (actual_r === 0.3) {\\n return \\\"0.3 m\\\";\\n } else {\\n return Math.round(actual_r) + \\\" m\\\";\\n }\\n }\\n return tick;\\n \"}},\"start\":-8,\"end\":8,\"value\":0}},{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p1135\",\"attributes\":{\"align\":\"center\",\"text\":\"
0.3 m              |              100 m              |              200 m              |              700 m              |              1200 m
\"}}]}}]}};\n", " const render_items = [{\"docid\":\"5ea57b65-b9a4-4352-b6ce-10931f036589\",\"roots\":{\"p1138\":\"cea2b574-45bf-4212-8058-8d92fd6af4ba\"},\"root_ids\":[\"p1138\"]}];\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": "p1138" } }, "output_type": "display_data" } ], "source": [ "inter_fig_chap9.fig_9_4()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{raw} html\n", "\n", "```\n", "\n", "```{figure} ../py_figures/1x1_pixel.png\n", "---\n", "name: fig_sine_var9.2\n", "---\n", "Sinusoidal variations vs. time at different distances for a periodic well for the head (left) and $Q_\\text{tot}$ (right).\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      As an illustration, an animation of the head surface of a periodic well is shown in {numref}`Figure %s `" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{exercise}\n", ":label: ex_97_ver_Q_r\n", "\n", "Use numerical derivatives to verify that $Q_r$ as programmed is indeed $Q_r=-T\\partial h/\\partial r$ for several values of $r$ and $t$. \n", "\n", "
\n", " Solution \n", "\n", "They are the same for several decimals\n", "\n", "
\n", "\n", "```" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "tags": [ "remove-input" ] }, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib as mpl\n", "mpl.rcParams['animation.embed_limit'] = 50 # size in MB\n", "inter_fig_chap9.fig_9_5()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{figure} ../py_figures/1x1_pixel.png\n", "---\n", "name: fig_animation_9.2\n", "---\n", "The head surface of a periodic well.\n", "```" ] } ], "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 }