{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "remove-input" ] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from myst_nb import glue\n", "%config InlineBackend.figure_format = 'svg'\n", "import sys\n", "sys.path.append(\"../py_figures\")\n", "import diagrams" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(content:chap2:steady_semi_flow)=\n", "# Steady one-dimensional semi-confined flow\n", "In semi-confined aquifers,the aquifer is capped by a leaky layer consisting of a low-permeable material such as peat, shale, or clay. This is in contrast to the confined aquifers considered in the previous chapter, which are capped by a layer that may be approximated as impermeable. The low-permeable layer is referred to as a semi-confining layer or a leaky layer. In what follows, the head above the leaky layer is fixed to a constant level $h^∗$ ({numref}`Figure %s `). This can correspond to the head in an aquifer above the semi-confining layer, or the water level of a surface water body. The fixed head $h^∗$ can also represent the mean water table in a drained area, where the water table is fixed by means of ditches or tile drains; such drained areas are sometimes referred to as polders. The fixed head $h^∗$ in the drained area can be in the low-permeable top layer ({numref}`Figure %s `) or in a permeable layer separated from the aquifer by a leaky layer ({numref}`Figure %s `)." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n", "application/papermill.record/text/plain": "

" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_solution_semi" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig_solution_semi = diagrams.solution_semi()\n", "glue(\"fig_solution_semi\", fig_solution_semi, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} fig_solution_semi\n", "---\n", "name: fig_semiconfexample2.0 \n", "---\n", "Semi-confined aquifer below surface water (left) and semi-confined aquifer below a drained area with a fixed water table in the low permeable layer (right).\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "      The specific discharge through the leaky layer is approximated as vertical and may be computed with Darcy's law (Eq. {eq}`eq_darcy0.0`) as\n", "\n", "$$\n", "q_z = -k^*\\frac{\\partial h}{\\partial z} \\approx -k^* \\frac{h^*-h}{H^*}\n", "$$(eq_qz20)\n", "\n", "where $h$ is the head in the aquifer, $h^*$ is the specified head at the top of the leaky layer, $k^*$ is the vertical hydraulic conductivity of the leaky layer, and $H^*$ is either the thickness of the leaky layer (left situation of {numref}`Figure %s `) or the distance between the drainage level and the bottom of the low permeable layer (right situation of {numref}`Figure %s `). \n", "
      The specific discharge through the leaky layer is positive for upward flow (in the positive $z$ direction) and may be written in the form \n", "\n", "$$\n", "q_z = \\frac{h-h^*}{c}\n", "$$(eq_qz2.0)\n", "\n", "where $c$ [T] is the resistance to vertical flow of the leaky layer defined as\n", "\n", "$$\n", "c=\\frac{H^*}{k^*}\n", "$$(eq_c[T]2.0)\n", "\n", "Flow in a semi-confined aquifer is governed by the Poisson equation (Eq. {eq}`eq_1dpoisson1.0`) where the infiltration rate $N$ is a function of the head difference across the semi-confining layer. Substitution of $N=-q_z$ in Eq. ({eq}`eq_1dpoisson1.0`) gives\n", "\n", "$$\n", "\\frac{\\text{d}^2h}{\\text{d}x^2} = \\frac{h-h^*}{cT}\n", "$$(eq_dh2dx2_2.0)\n", "\n", "For the case that $h^*$ is a constant, this equation can be rewritten as\n", "\n", "$$\n", "\\frac{\\text{d}^2(h-h^*)}{\\text{d}x^2} = \\frac{h-h^*}{\\lambda^2}\n", "$$(eq_1dmodhelmholz2.0)\n", "\n", "where \n", "\n", "$$\n", "\\lambda=\\sqrt{cT}\n", "$$(eq_lambda2.0)\n", "\n", "is referred to as the leakage factor [L] and is a characteristic length of the system, as will be explained later. \n", "
      Differential equation ({eq}`eq_1dmodhelmholz2.0`) is a second-order, linear, ordinary differential equation known as the modified-Helmholtz equation. The general solution is\n", "\n", "$$\n", "h-h^* = A\\text{e}^{-x/\\lambda} + B\\text{e}^{x/\\lambda}\n", "$$(eq_modhemgen2.0)\n", "\n", "where $A$ and $B$ are constants to be determined from boundary conditions. The general solution for the discharge vector is\n", "\n", "$$\n", "Q_x=\\frac{AT}{\\lambda}\\text{e}^{-x/\\lambda} - \\frac{BT}{\\lambda}\\text{e}^{x/\\lambda}\n", "$$(eq_gensolQx2.0)\n", "\n", "Alternatively, the general solution may be written as\n", "\n", "$$\n", "h-h^* = A\\sinh(x/\\lambda) + B\\cosh(x/\\lambda)\n", "$$(eq_sinhcosh2.0)\n", "\n", "with the corresponding discharge vector\n", "\n", "$$\n", "Q_x=-\\frac{AT}{\\lambda}\\cosh(x/\\lambda) - \\frac{BT}{\\lambda}\\sinh(x/\\lambda)\n", "$$(eq_Qxsinhcosh2.0)\n", "\n", "It depends on the problem whether the formulation in terms of exponential functions or the formulation in terms of hyperbolic functions is more convenient." ] } ], "metadata": { "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 }