{ "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", "from myst_nb import glue\n", "import sys\n", "sys.path.append(\"../py_figures\")\n", "import diagrams\n", "import diagrams_2d_basic\n", "import inter_fig_chap2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(content:chap2.4:flow_river_two_aqu)=\n", "# Flow to a river in a two-aquifer system" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Many aquifers are bounded at the bottom by a leaky layer with another aquifer below. Solutions for one-dimensional steady flow in such a two-aquifer systems can be written as the sum of the solution for confined flow ({numref}`Chapter %s `) and the solution for semi-confined flow, as will be shown in the following. \n", "
      In what follows, the tranmissivity of the top aquifer (aquifer 0) is $T_0$ and the transmissivity of the bottom aquifer (aquifer 1) is $T_1$. The resistance of the leaky layer between the two aquifers is $c$. The recharge from rainfall to the top aquifer is equal to $N$. The vertical flow $q_z$ through the leaky layer flow from aquifer 1 (the bottom aquifer) to aquifer 0 (the top aquifer) is\n", "\n", "$$\n", "q_z = \\frac{h_1 - h_0}{c}\n", "$$(eq_verticalflow2.4)\n", "\n", "where $h_0$ and $h_1$ are the head in the top and bottom aquifer, respectively. The head in the two-aquifer system is governed by a system of two differential equations\n", "\n", "\n", "$$\n", "\\begin{split}\n", "\\frac{\\text{d}^2 h_0}{\\text{d}x^2} & = \\frac{h_0-h_1}{cT_0} - \\frac{N}{T_0} \\\\\n", "\\frac{\\text{d}^2 h_1}{\\text{d}x^2} & = \\frac{h_1-h_0}{cT_1}\n", "\\end{split}\n", "$$(eq_deqsys2aq2.4)\n", "\n", "
      The solution to the system of two differential equations ({eq}`eq_deqsys2aq2.4`) is obtained as follows. Addition of the first differential equation multiplied by $T_0$ and the second differential equation multiplied by $T_1$ gives\n", "\n", "$$\n", "\\frac{\\text{d}^2(T_0h_0 + T_1h_1)}{\\text{d}x^2} = -N\n", "$$(eq_poisson2.4)\n", "\n", "This is Poisson's differential equation ({eq}`eq_1dpoisson1.0`), which has the general solution\n", "\n", "$$\n", "T_0h_0 + T_1h_1 = \\frac{-Nx^2}{2} + A'x + B'\n", "$$(eq_twoaquifer_eq1_2.4)\n", "\n", "where $A'$ and $B'$ are arbitrary constants.\n", "Subtraction of the second differential equation from the first differential equation gives, after a bit of algebra\n", "\n", "$$\n", "\\frac{\\text{d}^2(h_0 - h_1)}{\\text{d}x^2} = \\frac{h_0-h_1}{\\lambda^2} - \\frac{N}{T_0}\n", "$$(eq_const2.4)\n", "\n", "where\n", "\n", "$$\n", "\\lambda=\\sqrt{\\frac{cT_0T_1}{T_0+T_1}}\n", "$$(eq_lab2aq2.4)\n", "\n", "This is the modified Helmholtz equation ({eq}`eq_1dmodhelmholz2.0`) with an additional constant ($-N/T_0$) on the right-hand side. The general solution is\n", "\n", "$$\n", "h_0-h_1 = C'\\text{e}^{-x/\\lambda} + D'\\text{e}^{x/\\lambda} + \\frac{N\\lambda^2}{T_0}\n", "$$(eq_twoaquifer_eq2_2.4)\n", "\n", "where $C'$ and $D'$ are arbitrary constants. \n", "Expressions for $h_0$ and $h_1$ are obtained by combining Eqs. ({eq}`eq_twoaquifer_eq1_2.4`) and ({eq}`eq_twoaquifer_eq2_2.4`), which gives\n", "\n", "$$\n", "\\begin{split}\n", "h_0 &= -\\frac{N}{2T}x^2 + Ax + B + CT_1\\text{e}^{-x/\\lambda} + DT_1\\text{e}^{x/\\lambda} + \\frac{T_1N\\lambda^2}{T_0T}\n", "\\\\\n", "h_1 &= -\\frac{N}{2T}x^2 + Ax + B - CT_0\\text{e}^{-x/\\lambda} - DT_0\\text{e}^{x/\\lambda} - \\frac{N\\lambda^2}{T}\n", "\\end{split}\n", "$$(eq_gensol_2aq2.4)\n", "\n", "where $T$ = $T_0$ + $T_1$ and $A$, $B$, $C$, and $D$ are parameters that must be determined from boundary conditions. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      As a first example, consider flow to a river in a two-aquifer system; areal recharge is neglected. The river is narrow and in direct contact with the the top aquifer, but not the bottom aquifer, as shown in {numref}`Figure %s `. The boundary condition at the river is\n", "\n", "$$\n", "h_0|_{x=0}=h_R\n", "$$(eq_boundconriver2.4)\n", "\n", "where $h_R$ is the river stage. Far away to the left of the river, the heads in the two aquifers are equal and the head gradient is $\\text{d}h_0/\\text{d}x=\\text{d}h_1/\\text{d}x=G_l$, so that the discharge vector $Q_{x_0}$ in the top aquifer is \n", "\n", "$$\n", "Q_{x_0}|_{x=-\\infty}=U_{l_0} = -T_0 G_l\n", "$$(eq_Qtopaq2.4)\n", "\n", "and the discharge vector $Q_{x_1}$ in the bottom aquifer is\n", "\n", "$$\n", "Q_{x_1}|_{x=-\\infty}=U_{l_1} = -T_1 G_l\n", "$$(eq_Qbotaq2.4)\n", "\n", "The head gradient far to the right is $G_r$ so that the discharge vector is \n", "\n", "$$\n", "Q_{x_0}|_{x=\\infty}=U_{r_0} = -T_0 G_r\n", "$$(eq_QGR2.4)\n", "\n", "and the discharge vector $Q_{x_1}$ in the bottom aquifer is\n", "\n", "$$\n", "Q_{x_1}|_{x=\\infty}=U_{r_1} = -T_1 G_r\n", "$$(eq_Qx1_2.4)\n", "\n", "An amount $\\sigma$ [L$^2$/T] flows from the aquifer into the river per meter river normal to the plane of flow. The discharge $\\sigma$ is obtained from continuity of flow\n", "\n", "$$\n", "\\sigma = U_{l_0} + U_{l_1} - (U_{r_0} + U_{r_1})\n", "$$(eq_sigma2.4)" ] }, { "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 2026-01-28T15:01:04.988168\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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_solution2_4" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-01-28T15:01:05.355930\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig_solution2_4 = diagrams.solution2_4()\n", "glue(\"fig_solution2_4\", fig_solution2_4, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} fig_solution2_4\n", "---\n", "name: fig_2aquifersystem2.4\n", "---\n", "Flow to a river in a two-aquifer system.\n", "```\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Application of the stated boundary conditions to the general solution {eq}`eq_gensol_2aq2.4` gives\n", "\n", "$$\n", "\\begin{split}\n", "h_0 &= -\\frac{U_{l_0}}{T_0} x - \\frac{T_1 \\lambda}{T_0 T}\\frac{\\sigma}{2} \\left(\\text{e}^{x/\\lambda} - 1\\right) + h_R\\\\ \n", "h_1 &= -\\frac{U_{l_1}}{T_1} x + \\frac{\\lambda }{T}\\frac{\\sigma}{2} \\left(\\text{e}^{x/\\lambda} + \\frac{T_1}{T_0}\\right) + h_R\n", "\\end{split} {\\hskip 2em} x \\le 0 \n", "$$(eq_h0river2.4)\n", "\n", "$$\n", "\\begin{split}\n", "h_0 &= -\\frac{U_{r_0}}{T_0} x - \\frac{T_1 \\lambda}{T_0 T}\\frac{\\sigma}{2} \\left(\\text{e}^{-x/\\lambda} - 1\\right) + h_R \\\\ \n", "h_1 &= -\\frac{U_{r_1}}{T_1} x + \\frac{\\lambda }{T}\\frac{\\sigma}{2} \\left(\\text{e}^{-x/\\lambda} + \\frac{T_1}{T_0}\\right) + h_R\n", "\\end{split} {\\hskip 2em} x \\ge 0\n", "$$(eq_h1river2.4)\n", "\n", "The corresponding expressions for the discharge vector are" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{split}\n", "Q_{x0} &= U_{l_0} + \\frac{T_1}{T}\\frac{\\sigma}{2} \\text{e}^{x/\\lambda} \\\\\n", "Q_{x1} &= U_{l_1} - \\frac{T_1}{T}\\frac{\\sigma}{2} \\text{e}^{x/\\lambda}\n", "\\end{split} {\\hskip 2em} x \\le 0\n", "$$(eq_Q0river2.4)\n", "\n", "$$\n", "\\begin{split}\n", "Q_{x0} &= U_{r_0} - \\frac{T_1}{T}\\frac{\\sigma}{2} \\text{e}^{-x/\\lambda} \\\\\n", "Q_{x1} &= U_{r_1} + \\frac{T_1}{T}\\frac{\\sigma}{2} \\text{e}^{-x/\\lambda}\n", "\\end{split} {\\hskip 2em} x \\ge 0 \n", "$$(eq_Q1river2.4)\n", "\n", "The leakage factor $\\lambda$ (Eq. {eq}`eq_lab2aq2.4`) for a two-aquifer system plays the same role as for semi-confined flow: a distance $3\\lambda$ from the river, the heads and gradients in the aquifers are approximately equal. \n", "
      In the example below, the transmissivity (and hence the flow far from the river) in the bottom aquifer is twice the transmissivity of the top aquifer. Flow is towards the river from both sides, but the flow from the left is four times as large as from the right ({numref}`Figure %s `). " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# parameters\n", "k0 = 10 # hydraulic conductivity aquifer 0, m/d\n", "k1 = 20 # hydraulic conductivity aquifer 1, m/d\n", "H0 = 10 # thickness aquifer 0, m\n", "H1 = 10 # thickness aquifer 1, m\n", "Hstar = 2 # thickness leaky layer\n", "kstar = 0.01 # hydraulic conductivity leaky layer, m/d\n", "hR = 5 # head in river, m\n", "Gl = -0.002 # head gradient far away to the left, -\n", "Gr = 0.0005 # head gradient far away to the right, -" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "leakage factor: 115.47 m\n" ] } ], "source": [ "# solution\n", "T0 = k0 * H0\n", "T1 = k1 * H1\n", "c = Hstar / kstar\n", "T = T0 + T1\n", "Ul0 = -T0 * Gl\n", "Ul1 = -T1 * Gl\n", "Ur0 = -T0 * Gr\n", "Ur1 = -T1 * Gr\n", "σ = Ul0 + Ul1 - (Ur0 + Ur1)\n", "λ = np.sqrt(c * T0 * T1 / T)\n", "print(f'leakage factor: {λ:.2f} m')\n", "#\n", "x = np.linspace(-4 * λ, 4 * λ, 401)\n", "def hmaq(x):\n", " if x <= 0:\n", " h0 = -Ul0 / T0 * x - T1 / T0 * λ / T * σ / 2 * (np.exp(x / λ) - 1) + hR\n", " h1 = -Ul1 / T1 * x + λ / T * σ / 2 * (np.exp(x / λ) + T1 / T0) + hR\n", " else:\n", " h0 = -Ur0 / T0 * x - T1 / T0 * λ / T * σ / 2 * (np.exp(-x / λ) - 1) + hR\n", " h1 = -Ur1 / T1 * x + λ / T * σ / 2 * (np.exp(-x / λ) + T1 / T0) + hR\n", " return h0, h1\n", "\n", "hmaqvec = np.vectorize(hmaq) # vectorized version of hmaq function\n", "h0, h1 = hmaqvec(x)\n", "\n", "def disxmaq(x):\n", " if x <= 0:\n", " Qx0 = Ul0 + T1 / T * σ / 2 * np.exp(x / λ)\n", " Qx1 = Ul1 - T1 / T * σ / 2 * np.exp(x / λ)\n", " else:\n", " Qx0 = Ur0 - T1 / T * σ / 2 * np.exp(-x / λ)\n", " Qx1 = Ur1 + T1 / T * σ / 2 * np.exp(-x / λ)\n", " return Qx0, Qx1\n", "\n", "disxmaqvec = np.vectorize(disxmaq) # vectorized version of disxmaq function\n", "Qx0, Qx1 = disxmaqvec(x)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-01-28T15:01:05.906854\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic plot\n", "plt.subplot(121)\n", "plt.plot(x, h0, 'C0', label='$h_0$')\n", "plt.plot(x, h1, 'C0--', label='$h_1$')\n", "plt.legend()\n", "plt.subplot(122)\n", "plt.plot(x, Qx0, 'C1', label='$Q_{x_0}$')\n", "plt.plot(x, Qx1, 'C1--', label='$Q_{x_1}$')\n", "plt.legend();" ] }, { "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 2026-01-28T15:01:06.744273\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", "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, h0, 'C0', label='$h_0$')\n", "plt.plot(x, h1, 'C0--', label='$h_1$')\n", "diagrams.arrow((103, 5.4), (110, 5.55), 'stagnation \\n point', arrow='->')\n", "plt.xticks(np.arange(-4 * λ, 4.1 * λ, λ), [str(i) + '$\\\\lambda$' for i in range(-4, 5)])\n", "plt.xlabel('$x$')\n", "plt.ylabel('head (m)')\n", "plt.legend()\n", "plt.grid(True)\n", "plt.subplot(122)\n", "plt.title('(b)', loc='left')\n", "plt.plot(x, Qx0, 'C1', label='$Q_{x_0}$')\n", "plt.plot(x, Qx1, 'C1--', label='$Q_{x_1}$')\n", "plt.xticks(np.arange(-4 * λ, 4.1 * λ, λ), [str(i) + '$\\\\lambda$' for i in range(-4, 5)])\n", "plt.xlabel('$x$')\n", "plt.ylabel('$Q_x$ (m$^2$/d)')\n", "diagrams.arrow((103, 0), (110, 0.08), 'stagnation \\n point', arrow='->')\n", "plt.legend()\n", "plt.grid(True)\n", "plt.tight_layout()\n", "glue(\"fig_twoaquifer\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} fig_twoaquifer \n", "---\n", "name: fig_twoaquiferhQ2.4\n", "---\n", "Flow to a river in a two-aquifer system; head (left) and $Q_x$ (right).\n", "```\n", " \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      A contour plot of the stream function is shown in {numref}`Figure %s `. Note that part of the water coming from the left in the bottom aquifer flows past the river before it leaks to the top aquifer and flows to the river. There is a stagnation point in the bottom aquifer ($Q_{x_1}=0$) to the right of the river as can also be seen from the right graph of {numref}`Figure %s `. " ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# solution\n", "z = [0, H1, H1 + Hstar, H1 + Hstar + H0]\n", "psi = np.zeros((len(z), len(x)))\n", "psi[1] = Qx1\n", "psi[2] = Qx1\n", "psi[3] = Qx1 + Qx0" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-01-28T15:01:07.814600\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic streamline plot\n", "plt.subplot(111, aspect=10)\n", "plt.contour(x, z, psi, np.arange(-1, 1, 0.05))\n", "plt.axhspan(H1, H1 + Hstar, color=[.8, .8, .8]);" ] }, { "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 2026-01-28T15:01:08.047318\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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_streaml" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2026-01-28T15:01:08.381747\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" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=diagrams.figdef2)\n", "plt.subplot(111, aspect=10)\n", "plt.contour(x, z, psi, np.arange(-1, 1, 0.05), colors='C1', linestyles='-')\n", "#plt.plot([0, 0], [H0 + Hstar, H0 + Hstar + H1], 'k', lw=2)\n", "plt.axhspan(H1, H1 + Hstar, color=[.8, .8, .8])\n", "plt.xticks(np.arange(-4 * λ, 4.1 * λ, λ), [str(i) + '$\\\\lambda$' for i in range(-4, 5)]);\n", "xq = 400 * np.array([[-1, 1], [-1, 1]])\n", "zq = np.array([[17], [5]]) * np.ones((2, 2))\n", "qx = np.array([[1, -1], [1, -1]])\n", "qz = np.zeros((2, 2))\n", "diagrams_2d_basic.velovecgrid(xq, zq, qx, qz, length=30)\n", "diagrams.arrow((103, 0), (110, 2.5), 'stagnation \\n point', arrow='->')\n", "plt.plot([0, 0], [H0 + Hstar, H0 + Hstar + H1], color=diagrams.C1light, lw=4)\n", "plt.xlabel('x (m)')\n", "plt.ylabel('z (m) - VE:10x');\n", "glue(\"fig_streaml\", 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", " \"
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const value = parseFloat(slider.value.toFixed(5));\\n const Gl_array = Array.from({length: h0_lines.length}, (_, i) => \\n parseFloat((slider.start + i * slider.step).toFixed(5))\\n );\\n for (let i = 0; i < h0_lines.length; i++) {\\n const is_visible = (Gl_array[i].toFixed(5) === value.toFixed(5));\\n const Gl_val = Gl_array[i];\\n h0_lines[i].visible = is_visible;\\n h1_lines[i].visible = is_visible;\\n contours[i].line_renderer.visible = is_visible;\\n \\n // Show all flow arrows for all Gl values\\n flow_arrows_left_upper_p2[i].visible = is_visible;\\n flow_arrows_left_lower_p2[i].visible = is_visible;\\n flow_arrows_right_upper_p2[i].visible = is_visible;\\n flow_arrows_right_lower_p2[i].visible = is_visible;\\n \\n // Show stagnation annotations and arrows for all non-zero Gl values\\n const show_stagnation = is_visible && (Gl_val !== 0.0);\\n stag_annotations_p1[i].visible = show_stagnation;\\n stag_arrows_p1[i].visible = show_stagnation;\\n }\\n 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