{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "hideCode": false, "hidePrompt": false, "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\n", "import inter_fig_chap8\n", "import matplotlib as mpl\n", "mpl.rcParams['contour.negative_linestyle'] = 'solid'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(content:chap8:uniform_flow_and_wells)=\n", "# Uniform flow and wells\n", "\n", "In this section, an analytic element code is developed to simulate wells, uniform flow, and a constant. The discharge potential is written as \n", "\n", "$$\n", "\\Phi = \\sum_{m=1}^{M} \\frac{Q_m}{2\\pi}\\ln(r_m) \n", " + \\sum_{n=1}^{N} \\frac{P_n}{2\\pi}\\ln(r_n) -Q_u(x\\cos(\\theta) + y\\sin(\\theta)) + C\n", "$$(eq_phi8.1)\n", "\n", "where $M$ is the number of wells with a specified discharge $Q_m$, $N$ is the number of wells with a head-specified condition, and $P_n$ is the discharge of head-specified well $n$, which is unknown upfront; $Q_u$ is the magnitude of the uniform flow (the parameter) and $\\theta$ is the direction of uniform flow; the constant $C$ is also a priori unknown and is computed from a specified-head condition. The transmissivity is approximated as constant throughout the model.\n", "
     The analytic element code is developed using object oriented programming; an introduction to object oriented programming for Python may be found in, e.g., {cite:author}`lutz2013` ({cite:year}`lutz2013`). The analytic element code consists of a `Model` class and a number of `Element` classes. The `Model` class stores the transmissivity of the aquifer and a Python list called `elements` that contains all the elements in the model; the `element` list is initially empty. The `Model` class has a `potential` function to compute the discharge potential at a point caused by all analytic elements in the model (Eq. {eq}`eq_phi8.1`). In addition, the `Model` class has a `head` function to compute the head at a point; the `head` function internally calls the `potential` function and then converts the `potential` to a `head`. Finally, the `Model` class has a `solve` method, which computes the parameters of all head-specified elements in the model. This function is included here, but will be explained later on." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class Model:\n", " \n", " def __init__(self, T):\n", " self.T = T\n", " self.elements = [] # list of all elements\n", " \n", " def potential(self, x, y):\n", " pot = 0.0\n", " for e in self.elements:\n", " pot += e.potential(x, y)\n", " return pot\n", " \n", " def head(self, x, y):\n", " return self.potential(x, y) / self.T\n", " \n", " def solve(self):\n", " esolve = [e for e in self.elements if e.nunknowns == 1]\n", " nunknowns = len(esolve)\n", " matrix = np.zeros((nunknowns, nunknowns))\n", " rhs = np.zeros(nunknowns)\n", " for irow in range(nunknowns):\n", " matrix[irow], rhs[irow] = esolve[irow].equation()\n", " solution = np.linalg.solve(matrix, rhs)\n", " for irow in range(nunknowns):\n", " esolve[irow].parameter = solution[irow]" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "     The `Element` class is the base class of all analytic elements. It stores the model that the element is added to, the value of the parameter (e.g., the discharge of a well), the number of unknown parameters of an element, which is zero by default, and it adds the element to the specified model by adding it to the `elements` list of the model. The `Element` class includes the `potential` function, which is the same for all elements. The `potential` function computes the discharge potential of an element by returning the product of the influence function `potinf` of the element and the `parameter` of the element. Every element must implement its own potential influence function `potinf`, which is the discharge potential for the case that the parameter of the element equals one. The utility of influence functions will become apparent further on in this section, when elements are introduced that have an a priori unknown parameter, for example a well where the discharge is unknown and is computed from a specified head at the well. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class Element:\n", " \n", " def __init__(self, model, p):\n", " self.model = model\n", " self.parameter = p\n", " self.nunknowns = 0\n", " self.model.elements.append(self)\n", " \n", " def potential(self, x, y):\n", " return self.parameter * self.potinf(x, y)\n", " \n", " def potinf(self, x, y):\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "     Every element is derived from the `Element` base class, so that it automatically gets all attributes and functions from the `Element` class. The `Well` class, for wells with a specified discharge, takes as input arguments the model to which the well is added, the location $(x_w,y_w)$ of the well, the discharge of the well, and the radius of the well. The `Well` class is derived from the `Element` class, which stores the parameter (the discharge). The `Well` class stores the location and the radius of the well. Only one function needs to be implemented for the `Well` class: the `potinf` function, which for a well is \n", "\n", "$$\n", "\\Phi = \\frac{1}{2\\pi}\\ln(r)\n", "$$(eq_phiwell8.1)\n", "\n", "where $r=\\sqrt{(x-x_w)^2 + (y-y_w)^2}$ is the radial distance between the well and point $(x,y)$. The radial distance is \n", "set to the radius of the well for points inside the radius of the well using the `np.where` function." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class Well(Element):\n", " \n", " def __init__(self, model, xw=0, yw=0, Q=1, rw=0.3):\n", " Element.__init__(self, model, p=Q)\n", " self.xw, self.yw = xw, yw\n", " self.rw = rw\n", " \n", " def potinf(self, x, y):\n", " r = np.sqrt((x - self.xw)**2 + (y - self.yw)**2)\n", " r = np.where(r < self.rw, self.rw, r)\n", " return 1 / (2 * np.pi) * np.log(r)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "     The `UniformFlow` class is also derived from the `Element` class and takes as input arguments the model to which it is added, the absolute value of the head gradient, and the direction $\\theta$ of uniform flow (in degrees measured counter-clockwise from the positive $x$-direction). The `potinf` function is\n", "\n", "$$\n", "\\Phi = -Q_u(x\\cos(\\theta) + y\\sin(\\theta))\n", "$$(eq_phipotinf8.1)\n", "\n", "where the magnitiude $Q_u$ is computed as the product of the transmissivity and the absolute value of the gradient. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "class UniformFlow(Element):\n", " \n", " def __init__(self, model, gradient, angle):\n", " Element.__init__(self, model, p=model.T * gradient)\n", " self.theta = np.deg2rad(angle)\n", " \n", " def potinf(self, x, y):\n", " return -(x * np.cos(self.theta) + y * np.sin(self.theta))" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "      An analytic element model may now be constructed by creating a `Model` object, and optionally adding a uniform flow object and an arbitrary number of wells. The head and potential may be computed at any point. In the example below, one extraction well and one injection well are situated in an otherwise uniform flow field in the southeastern direction $(\\theta=-45^\\circ)$. Equipotentials are plotted in {numref}`Figure %s `. Note that the magnitude of the discharge of the extraction well is twice as large as for the injection well. " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "# parameters and model\n", "ml = Model(T=100) # transmissivity, m^2/d\n", "Well(ml, xw=50, yw=0, Q=200) # extraction well, Q in m^3/d\n", "Well(ml, xw=0, yw=0, Q=-100) # injection well, Q in m^3/d\n", "UniformFlow(ml, gradient=0.002, angle=-45);" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "# solution\n", "xg, yg = np.meshgrid(np.linspace(-100, 100, 100), np.linspace(-75, 75, 100))\n", "pot = ml.potential(xg, yg)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "hideCode": false, "hidePrompt": false, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-03-09T17:42:23.980826\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic equipotentials plot\n", "plt.subplot(111, aspect=1)\n", "plt.contour(xg, yg, pot, 20)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "hideCode": false, "hidePrompt": false, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-03-09T17:42:25.083545\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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_equipot" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-03-09T17:42:25.333698\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# equipotentials\n", "fig = plt.figure(figsize=diagrams_2d.figdef1new)\n", "plt.subplot(111, aspect=1)\n", "plt.contour(xg, yg, pot, 20, colors='C0')\n", "plt.plot([0, 50], [0, 0], 'k.')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('y (m)', labelpad=-6)\n", "plt.tight_layout();\n", "glue(\"fig_equipot\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "```{glue:figure} fig_equipot\n", "---\n", "name: fig_equipot8.1\n", "---\n", "Equipotentials for analytic element model of an injection well ($Q=-100$ m$^3$/d) and an extraction well ($Q=200$ m$^3$/d) in uniform flow in southeastern direction.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "```{exercise}\n", ":label: ex_8.1\n", "Create equipotentials for the example in {numref}`content:chap7:single_well` of a well in uniform flow using (like the left graph of {numref}`Figure %s `) using the analytic element code developed above. Use a transmissivity of $T=250$ m$^2$/d.\n", "\n", "
\n", " Solution \n", "\n", "\n", "\n", "
\n", "\n", "```" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "     The head must be defined at at least one point in the aquifer, otherwise the head values along the contours are meaningless (an arbitrary value can be added to all values, resulting in the same flow field). Two types of head-specified elements are implemented in this section: wells with a specified head and an additive constant which is also computed from a specified-head condition. The free parameters of head-specified elements are computed by building a system of equations for all head-specified conditions and solving for all free parameters simultaneously. This procedure is illustrated by solving a problem with uniform flow, two wells, and a constant.\n", "
     Consider an aquifer with uniform flow $U$ in the positive $x$-direction with two wells and a real constant $C$ (see {numref}`Figure %s `). Well 0 is located at $(x,y)=(0,0)$ and has a specified discharge $Q$. Well 1 is located at $(x,y)=(a,b)$ and has a discharge $P$. The radius of both wells is $r_w$. The potential for this problem is obtained from superposition of the potential of uniform flow ({eq}`eq_potuflow_7.1`), the potentials for two wells ({eq}`eq_welln6.2`), and a constant $C$:" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "$$\n", "\\Omega = -Ux + \\frac{Q}{2\\pi}\\ln(r_0) + \\frac{P}{2\\pi}\\ln(r_1) + C\n", "$$(eq_pot2wells8.1)\n", "\n", "where $r_0$ and $r_1$ are the distances from wells 0 and 1, respectively.\n", "The discharge discharge $P$ is unknown and must be computed such that the head equals $h_w$ at well 1. Furthermore, the \n", "head at $(x,y)=(a,0)$ is known to be $h_0$. This gives two equations for the two unknown parameters $C$ and $P$. \n", "
     The parameters $P$ and $C$ are obtained from the system of two linear equations\n", "\n", "$$\n", "\\begin{split}\n", "&\\Phi|_{x=a,y=0} = \\Phi_0 \\\\\n", "&\\Phi|_{x=a+r_w,y=b} = \\Phi_w\n", "\\end{split}\n", "$$(eq_sys2eq8.1)\n", "\n", "where $\\Phi_0$ and $\\Phi_w$ are the potential equivalents of $h_0$ and $h_w$, respectively. \n", "Note that the head at well 1 is specified at $(x,y)=(a+r_w, b)$ which is referred to as the collocation point; specification at another point on the well screen may result in a slightly different solution, but the difference is commonly negligible. Substitution of ({eq}`eq_pot2wells8.1`) for $\\Phi$ in ({eq}`eq_sys2eq8.1`) and rearrangement of terms gives the following system of two equations in two unknowns\n", "\n", "$$\n", "\\begin{split}\n", "\\frac{P}{2\\pi}\\ln(b) + C &= \\Phi_0 + Ua- \\frac{Q}{2\\pi}\\ln(a) \\\\\n", "\\frac{P}{2\\pi}\\ln(r_w) + C &= \\Phi_w + U(a + r_w) - \\frac{Q}{2\\pi}\\ln(d)\n", "\\end{split}\n", "$$(eq_sys2eqrearranged8.1)\n", "\n", "where $d=\\sqrt{(a + r_w)^2 + b^2}$ is the distance between the well with specified discharge and the collocation point on the well screen of the head-specified well. In matrix form, the system of equations is\n", "\n", "$$\n", "\\begin{pmatrix}\n", "\\frac{1}{2\\pi}\\ln(b) & 1 \\\\\n", "\\frac{1}{2\\pi}\\ln(r_{w}) & 1 \\\\\n", "\\end{pmatrix}\n", "\\begin{pmatrix}\n", "P \\\\\n", "C \n", "\\end{pmatrix}\n", "=\n", "\\begin{pmatrix}\n", "\\Phi_0 + U a - \\frac{Q}{2\\pi}\\ln(a)\\\\\n", "\\Phi_w + U(a + r_w) - \\frac{Q}{2\\pi}\\ln(d)\n", "\\end{pmatrix}\n", "$$(eq_matrixform8.1)\n", "\n", "The unknown free parameters $P$ and $C$ are obtained by solving this system of two linear equations.\n", "
     The `solve` method of the `Model` class, as defined above, counts the number of elements with an unknown parameter and stores them in a separate list called `esolve`, builds the matrix and the right-hand side of the system of equations, solves the system, and stores the computed parameters for the corresponding head-specified elements (in this case the values of $P$ and $C$). The `solve` method loops through all elements in the `esolve` list and calls their `equation` function, which returns one row for the matrix and one right-hand side value. The `equation` method of the `Constant` element (not yet defined) returns the first row of the matrix and the first value of the right-hand side vector, and the `equation` method of well 1 (the head-specified well) returns the second row of the matrix and the second value of the right-hand side vector. \n", "
     The `equation` function is stored in a separate mix-in class `HeadEquation`, as it is the same for all head-specified elements. Each element that uses this mix-in class must store the location of the collocation point where the head is specified as `xc` and `yc`, and the specified value of the discharge potential at the collocation point as `pc`. The `equation` function starts out with a row of the matrix that is an empty list and a right-hand side value that is equal to the specified discharge potential `pc`. The `equation` function then loops through all the elements in the model. If an element has an unknown parameter (`nunkonwns==1`), its potential influence function `potinf` is evaluated at the collocation point and is stored as the next value in the row of the matrix. If an element does not have an unknown parameter, its `potential` function is evaluated at the collocation point and the returned value is subtracted from the right-hand side value. " ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class HeadEquation:\n", "\n", " def equation(self):\n", " row = []\n", " rhs = self.pc\n", " for e in self.model.elements:\n", " if e.nunknowns == 1:\n", " row.append(e.potinf(self.xc, self.yc))\n", " else:\n", " rhs -= e.potential(self.xc, self.yc)\n", " return row, rhs" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "      The `HeadWell` class for head-specified wells can now be derived from both the `Well` class and the `HeadEquation` class. Input arguments are the model to which the element is added, the location of the well, the radius, and the specified head. The `xc` and `yc` locations of the collocation point are stored, as well as the specified potential `pc` at the collocation point. The number of unknowns parameters `nunknowns` is set to 1. All other attributes and functions are inherited from the `Well` and `HeadEquation` classes. " ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class HeadWell(Well, HeadEquation):\n", " \n", " def __init__(self, model, xw, yw, rw, hw):\n", " Well.__init__(self, model, xw, yw, 0, rw)\n", " self.xc, self.yc = xw + rw, yw\n", " self.pc = self.model.T * hw\n", " self.nunknowns = 1" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "      The `Constant` class is derived from the `Element` class and the `HeadEquation` class. Input arguments are the model to which the element is added and the location and value of the specified head. The `potinf` function of the `Constant` class is simply equal to 1. The `Constant` element is also referred to as the reference point." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "class Constant(Element, HeadEquation):\n", " \n", " def __init__(self, model, xc, yc, hc):\n", " Element.__init__(self, model, p=0)\n", " self.xc, self.yc = xc, yc\n", " self.pc = self.model.T * hc\n", " self.nunknowns = 1\n", " \n", " def potinf(self, x, y):\n", " return np.ones_like(x, dtype='float')" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "     The example problem of {numref}`Figure %s ` (as discussed above) is solved with the analytic element code developed so far. The analytic element model is created with the following five commands" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "hideCode": false, "hidePrompt": false }, "outputs": [], "source": [ "# parameters and model\n", "ml = Model(T=100) # transmissivity in m^2/d\n", "w1 = Well(ml, xw=0, yw=0, Q=100, rw=0.3) # Q in m^3/d\n", "w2 = HeadWell(ml, xw=400, yw=300, rw=0.3, hw=20)\n", "uf = UniformFlow(ml, gradient=0.002, angle=0)\n", "rf = Constant(ml, xc=400, yc=0, hc=22)" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "      In the code cell above, the model object is stored in the variable `ml`. The unknown parameters in the model may be computed by calling the `solve` method of the model. \n", "The head at the head-specified well and at the reference point are computed to verify that the solution is indeed correct. In addition, the head at the discharge-specified well and the computed discharge of the head-specified well are printed to the screen. A grid of the head is computed and the minimum and maximum values on the grid are printed to the screen, which are used to specify values for the contours to be plotted. Finally, head contours are plotted (right graph in {numref}`Figure %s `)." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "editable": true, "hideCode": false, "hidePrompt": false, "slideshow": { "slide_type": "" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "computed head at discharge-specified well: 21.81 m\n", "computed head at head-specified well: 20.00 m\n", "computed discharge of head-specified well: 185.10 m^3/d\n", "computed head at reference point: 22.00 m\n", "min and max head on grid: 20.57, 23.47\n" ] } ], "source": [ "# solution\n", "ml.solve() # compute the unkown parameters\n", "print(f'computed head at discharge-specified well: {ml.head(0, 0):.2f} m')\n", "print(f'computed head at head-specified well: {ml.head(400.3, 300):.2f} m')\n", "print(f'computed discharge of head-specified well: {w2.parameter:.2f} m^3/d')\n", "print(f'computed head at reference point: {ml.head(400, 0):.2f} m')\n", "# grid\n", "xg, yg = np.meshgrid(np.linspace(-200, 600, 100), np.linspace(-100, 500, 100))\n", "head = ml.head(xg, yg)\n", "print(f'min and max head on grid: {head.min():.2f}, {head.max():.2f}')" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "hideCode": false, "hidePrompt": false, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-03-09T17:42:25.874857\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic head contours (right graph)\n", "plt.subplot(111, aspect=1)\n", "plt.contour(xg, yg, head, np.arange(20, 24, 0.2));" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "hideCode": false, "hidePrompt": false, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2026-03-09T17:42:27.234097\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", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# head contours\n", "fig = plt.figure(figsize=diagrams_2d.figdef2)\n", "ax = plt.subplot(121, aspect=1, xlim=(-200, 600), ylim=(-100, 500))\n", "plt.plot([0, 400], [0, 300], 'k.')\n", "plt.text(0, 20, 'Well 0\\n $Q=100$ m$^3$/d', ha='center', va='bottom')\n", "plt.text(400, 320, 'Well 1\\n $h=20$ m', ha='center', va='bottom')\n", "plt.plot(400, 0, 'kx')\n", "plt.text(400, 20, 'Constant \\n $h=22$ m', ha='center', va='bottom')\n", "diagrams_2d.fancyarrow(ax, (-100, 200), (0, 200), scale=20)\n", "plt.text(-50, 220, 'UniformFlow \\n $|dh/dx|=0.002$', ha='center', va='bottom')\n", "diagrams_2d.arrow((0, -40), (400, -40), arrow='<->')\n", "plt.text(200, -50, '$a$', va='top', ha='center')\n", "diagrams_2d.arrow((520, 0), (520, 300), arrow='<->')\n", "plt.text(530, 150, '$b$', ha='left', va='center')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('y (m)', labelpad=-5)\n", "plt.title('(a)', loc='left')\n", "plt.subplot(122, aspect=1)\n", "cs = plt.contour(xg, yg, head, np.arange(20, 24, .2), colors='C0')\n", "plt.clabel(cs, fmt='%.1f', fontsize='smaller')\n", "plt.plot([0, 400], [0, 300], 'k.')\n", "plt.plot(400, 0, 'kx')\n", "plt.xlabel('x (m)')\n", "plt.ylabel('y (m)', labelpad=-5)\n", "plt.title('(b)', loc='left')\n", "plt.tight_layout();\n", "glue(\"fig_AEM\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "```{glue:figure} fig_AEM\n", "---\n", "name: fig_layoutAEM8.1\n", "---\n", "Layout of analytic element model (left) and corresponding head contours (right).\n", "```" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false }, "source": [ "```{exercise}\n", ":label: ex_8.2\n", "Modify the `HeadWell` class to specify an optional location of the control point. Plot head contours for the previous example, but now the head for the head-specified well is 21.5 m and is specified at $(x,y)=(400,200)$. Compute the head at the head-specified well and the discharge of the head-specified well. \n", "\n", "
\n", " Solution \n", "\n", "$h = 18.8$ m, $Q = 296$ m$^3$/d\n", "\n", "\n", "\n", "
\n", "```\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An interactive figure of the head contours is added for the example of {numref}`Figure %s `, where a reference point with a head of 20 m is added at $(x,y)=(25, 70)$. " ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-03-09T17:42:28.495029\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def interactive_plot(Qextr, Qinj, theta):\n", " # parameters and model\n", " ml = Model(T=100) # transmissivity, m^2/d\n", " Well(ml, xw=50, yw=0, Q=Qextr) # extraction well, Q in m^3/d\n", " Well(ml, xw=0, yw=0, Q=Qinj) # injection well, Q in m^3/d\n", " UniformFlow(ml, gradient=0.002, angle=theta)\n", " Constant(ml, xc=25, yc=70, hc=20)\n", " ml.solve()\n", " # solution\n", " xg, yg = np.meshgrid(np.linspace(-100, 100, 100), np.linspace(-75, 75, 100))\n", " head = ml.head(xg, yg)\n", " #print(head.min(), head.max())\n", " # figure\n", " fig = plt.figure(figsize=diagrams_2d.figdef1new)\n", " plt.subplot(111, aspect=1)\n", " plt.contour(xg, yg, head, np.arange(15, 25, 0.1), colors='C0')\n", " plt.contour(xg, yg, head, [20], colors='C1')\n", " plt.plot([0, 50], [0, 0], 'k.')\n", " plt.plot(25, 70, 'kx')\n", " plt.xlabel('x (m)')\n", " plt.ylabel('y (m)', labelpad=-6)\n", " plt.tight_layout();\n", "\n", "interactive_plot(200, -100, -45)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "tags": [ "remove-cell" ] }, "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\nconst 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-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.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));", "application/vnd.bokehjs_load.v0+json": "" }, "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 = {\"3b5fe482-08f2-4361-af96-72d9f819f8c3\":{\"version\":\"3.7.3\",\"title\":\"Bokeh 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const Qextr = slider_Qextr.value;\\n const Qinj = slider_Qinj.value;\\n const theta = slider_theta.value;\\n \\n let target_idx = -1;\\n let idx = 0;\\n \\n for (let i = 0; i < Q_extr_values.length; i++) {\\n for (let j = 0; j < Q_inj_values.length; j++) {\\n for (let k = 0; k < theta_values.length; k++) {\\n if (Math.abs(Q_extr_values[i] - Qextr) < 0.1 && \\n Math.abs(Q_inj_values[j] - Qinj) < 0.1 &&\\n Math.abs(theta_values[k] - theta) < 0.1) {\\n target_idx = idx;\\n }\\n \\n contours_main[idx].line_renderer.visible = (idx === target_idx);\\n contours_highlight[idx].line_renderer.visible = (idx === target_idx);\\n \\n idx += 1;\\n }\\n }\\n }\\n \"}}]]]},\"width\":580,\"align\":\"center\",\"title\":\"Qext (m\\u00b3/d)\",\"format\":\"0\",\"start\":0,\"end\":200,\"value\":200,\"step\":50}},{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p9176\",\"attributes\":{\"align\":\"center\",\"text\":\"
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