{ "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.rc = {'figure.dpi': 300}\n", "%config InlineBackend.figure_format = 'svg'\n", "import sys\n", "from myst_nb import glue\n", "sys.path.append(\"../py_figures\")\n", "import diagrams\n", "import inter_fig_chap2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Flow to a river of finite width\n", " The next example considers one-dimensional steady flow to a long and straight river with width $2L$. A vertical cross section normal to the river is shown in {numref}`Figure %s `. The origin of the coordinate system is chosen in the middle of the river. The flow is confined to the left and right of the river and semi-confined below the river. The resistance of the leaky river bed is $c$, and the water level in the river is $h^∗$. The transmissivity of the aquifer is $T$. The flow is uniform and equal to $Q_x=U_l$ in the aquifer to the left of the river, and the flow is uniform and equal to $Q_x=U_r$ in the aquifer to the right of the river." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-02-25T21:04:38.283893\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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_solution12" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-02-25T21:04:38.314809\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig_solution12 = diagrams.solution12()\n", "glue(\"fig_solution12\", fig_solution12, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} fig_solution12\n", "---\n", "name: fig_longriver2.3\n", "---\n", "Flow to a long river of finite width.\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      The head and flow below the river are solved first. Boundary conditions are \n", "\n", "$$\n", "Q_x\\vert_{x=-L}=U_l {\\hskip 2em} Q_x\\vert_{x=L}=U_r\n", "$$(eq_bcQxriver2.3)\n", "\n", "The hyperbolic form (Eq. {eq}`eq_sinhcosh2.0`) is used for the head solution below the river\n", "\n", "$$\n", "h-h^* = A\\sinh(x/\\lambda) + B\\cosh(x/\\lambda)\n", "$$(eq_hyperbl2.3)\n", "\n", "with the corresponding solution for the discharge vector (Eq. {eq}`eq_Qxsinhcosh2.0`) \n", "\n", "$$\n", "Q_x=-\\frac{AT}{\\lambda}\\cosh(x/\\lambda) - \\frac{BT}{\\lambda}\\sinh(x/\\lambda)\n", "$$(eq_Qxsinhcosh2.3)\n", "\n", "
      Substitution of the boundary conditions (Eq. {eq}`eq_bcQxriver2.3`) in Eq. ({eq}`eq_Qxsinhcosh2.3`) gives \n", "\n", "$$\n", "A = \\frac{-\\lambda(U_l + U_r)}{2T\\cosh(L/\\lambda)} {\\hskip 2em}\n", "B = \\frac{\\lambda(U_l - U_r)}{2T\\sinh(L/\\lambda)}\n", "$$(eq_ABsinhcosh2.3)\n", "\n", "The solutions for the head to the left and to the right of the river are simple linear functions, because the aquifer is confined. The solution below the river is used to compute the head $h_0=h|_{x=-L}$ in the aquifer at the left river bank and the head $h_1=h|_{x=L}$ in the aquifer at the right river bank. Once $h_0$ and $h_1$ are known, the solutions in the confined parts of the aquifer to the left and right of the river can be written as\n", "\n", "$$\n", "h = -\\frac{U_l}{T}(x + L) + h_0 {\\hskip 2em} x\\le -L\n", "$$(eq_hleft2.3)\n", "\n", "$$\n", "h = -\\frac{U_r}{T}(x - L) + h_1 {\\hskip 2em} x\\ge L\n", "$$(eq_hright2.3)\n", "\n", "An example is shown for the case that the width of the river is twice the leakage factor and the inflow at the left river bank is twice the inflow at the right river bank ({numref}`Figure %s `)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "leakage factor: 100 m\n" ] } ], "source": [ "# parameters\n", "k = 5 # hydraulic conductivity, m/d\n", "H = 10 # aquifer thickness, m\n", "hstar = 10 # head in river \n", "L = 100 # half width of river\n", "c = 200 # resistance of leaky river bed, d\n", "Ul = 0.1 # Qx in left aquifer\n", "Ur = -0.05 # Qx in right aquifer\n", "T = k * H # transmissivity, m^2/d\n", "lab = np.sqrt(T * c) # leakage factor of aquifer below river\n", "print(f'leakage factor: {lab:.0f} m')" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "x = np.linspace(-300, 300, 100)\n", "A = -lab * (Ul + Ur) / (2 * T) / np.cosh(L / lab)\n", "B = lab * (Ul - Ur) / (2 * T) / np.sinh(L / lab)\n", "h0 = A * np.sinh(-L / lab) + B * np.cosh(-L / lab) + hstar\n", "h1 = A * np.sinh(L / lab) + B * np.cosh(L / lab) + hstar\n", "h = np.zeros(len(x))\n", "h[x <= -L] = -Ul / T * (x[x <= -L] + L) + h0\n", "h[x >= L] = -Ur / T * (x[x >= L] - L) + h1\n", "h[np.abs(x) < L] = A * np.sinh(x[np.abs(x) < L] / lab) + \\\n", " B * np.cosh(x[np.abs(x) < L] / lab) + hstar\n", "Qx = np.zeros(len(x))\n", "Qx[x <= 0] = Ul\n", "Qx[x >= L] = Ur\n", "Qx[np.abs(x) < L] = -A * T / lab * np.cosh(x[np.abs(x) < L] / lab) - \\\n", " B * T / lab * np.sinh(x[np.abs(x) < L] / lab)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-02-25T21:04:38.361221\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic plot\n", "plt.subplot(121)\n", "plt.plot(x, h, 'C0')\n", "plt.subplot(122)\n", "plt.plot(x, Qx, 'C1');" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-02-25T21:04:38.412281\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", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef2)\n", "plt.subplot(121)\n", "plt.title('(a)', loc='left')\n", "plt.plot(x, h)\n", "plt.axvspan(-L, L, color=[.8, .8, .8])\n", "plt.xlabel('x (m)')\n", "plt.ylabel('head (m)')\n", "plt.grid(True)\n", "plt.text(0, 10.6, 'river', ha='center')\n", "plt.subplot(122)\n", "plt.title('(b)', loc='left')\n", "plt.plot(x, Qx, 'C1')\n", "plt.axvspan(-L, L, color=[.8, .8, .8])\n", "plt.xlabel('$x$ (m)')\n", "plt.yticks(np.arange(-0.05, 0.11, 0.05))\n", "plt.ylabel('$Q_x$ (m$^2$/d)')\n", "plt.text(0, 0.08, 'river', ha='center')\n", "plt.grid(True)\n", "plt.tight_layout();\n", "glue(\"fig_flowfinriver\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} fig_flowfinriver\n", "---\n", "name: fig_flowfiniteriver2.3\n", "---\n", "Groundwater flow to a river of finite width; head (left) and $Q_x$ (right)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      Contours of the stream function are plotted below ({numref}`Figure %s `)." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# solution\n", "psi = np.zeros((2, len(x)))\n", "psi[1] = -Qx\n", "xg = x\n", "zg = [0, H]" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-02-25T21:04:38.496976\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic streamline plot\n", "plt.contour(xg, zg, psi, np.arange(-0.1, 0.05, 0.01));" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-02-25T21:04:38.530861\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", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef2)\n", "ax = plt.subplot(111)\n", "ax.set_aspect(15)\n", "cs = plt.contour(xg, zg, psi, np.arange(-0.1, 0.05, 0.01), colors='C1', linestyles='-')\n", "plt.clabel(cs, fmt='%1.2f', fontsize='smaller')\n", "plt.xlabel('$x$ (m)')\n", "plt.ylabel('$z$ (m) - VE:15x')\n", "plt.tight_layout();\n", "glue(\"fig_cont\", fig, display=False)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "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 value = parseFloat(slider.value.toFixed(3));\\n const U_array = U_values;\\n for (let i = 0; i < contour_sets.length; i++) {\\n const is_visible = (Number(U_array[i].toFixed(3)) === Number(value.toFixed(3)));\\n\\n contour_sets[i].line_renderer.visible = is_visible;\\n\\n // safe checks: items exist (null placeholders), dus test op null\\n if (flow_arrows_left[i] != null) {\\n flow_arrows_left[i].visible = is_visible;\\n }\\n if (flow_arrows_right[i] != null) {\\n flow_arrows_right[i].visible = is_visible;\\n }\\n }\\n \"}}]]]},\"width\":570,\"margin\":10,\"align\":\"center\",\"title\":\"U\\u2097 (m\\u00b2/d)\",\"format\":\"0.000\",\"start\":-0.1,\"end\":0.15,\"value\":0.1,\"step\":0.025}},{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p1372\",\"attributes\":{\"align\":\"center\",\"text\":\"
-0.10 m\\u00b2/d         |         -0.05 m\\u00b2/d         |         0.00 m\\u00b2/d         |         0.05 m\\u00b2/d         |         0.10 m\\u00b2/d         |         0.15 m\\u00b2/d
\"}}]}}]}};\n", " const render_items = [{\"docid\":\"e2aabe38-82cb-413c-904a-ce553510a591\",\"roots\":{\"p1374\":\"f5ad6077-a8b9-4cb3-b5ba-0f03906f532c\"},\"root_ids\":[\"p1374\"]}];\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": "p1374" } }, "output_type": "display_data" } ], "source": [ "inter_fig_chap2.fig_2_10()" ] }, { "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_contourplot2.3\n", "---\n", "Groundwater flow to a river of finite width; streamlines in a vertical cross-section. Discharge between two streamlines is $\\Delta\\psi=0.01$ m$^2$/d.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{exercise}\n", ":label: ex_2.4\n", "\n", "Consider the case that $U_l$ and $U_r$ are not known, but the head at $x=-300$ m is measured to be 10.4 m, and the head at $x=300$ m is measured to be 10.6 m. Solve for $U_l$ and $U_r$ using a root finding routine like `scipy.optimize.fsolve`. Plot the head as a function of $x$.\n", "\n", "
\n", " Solution \n", "\n", "$U_l=0.0574$ m$^2$/d\\\n", "$U_r=-0.0936$ m$^2$/d\n", "\n", "\n", "\n", "
\n", "```\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 }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }