05_BC2q.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 5.\n",
    "### Sensitivity of hydraulic head at a point to a <font color='blue'>Neumann BC flux</font> under steady state flow conditions\n",
    "\n",
    "## 0. Forward model\n",
    "<br>\n",
    "Governing equation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad K\\ b\\ \\frac{d^2h}{dx^2}=0\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Boundary conditions:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad q(x)=q_{\\Gamma_2}\\ ,\\quad\\quad\\quad\\quad\\quad x=0=\\Gamma_2\\cr\n",
    "\\end{align}$\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad h(x)=h_{\\Gamma_1}\\ ,\\quad\\quad\\quad\\quad\\quad x=L=\\Gamma_1\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Closed-form solution:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "Spatial derivatives from differentiation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML, display\n",
    "def set_background(color):\n",
    "    script = (\n",
    "        \"var cell = this.closest('.code_cell');\"\n",
    "        \"var editor = cell.querySelector('.input_area');\"\n",
    "        \"editor.style.background='{}';\"\n",
    "        \"this.parentNode.removeChild(this)\").format(color)\n",
    "    display(HTML('<img src onerror=\"{}\">'.format(script)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from warnings import filterwarnings\n",
    "filterwarnings(\"ignore\", category=DeprecationWarning)\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "K, b, L, BC1h, BC2q, ocol = 10., 10., 10000., 0., 0.5, 5000\n",
    "X = np.arange(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Direct sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Perturbation sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\frac{\\partial h(x')}{\\partial q_{\\Gamma_2}}\\approx\\frac{h(x,q_{\\Gamma_2}+\\Delta q_{\\Gamma_2})-h(x,q_{\\Gamma_2})}{\\Delta q_{\\Gamma_2}}\\;\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "### 2a. Analytical\n",
    "$\\begin{align}\\cr\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from os import getcwd, chdir\n",
    "cwd = getcwd()\n",
    "chdir(r'../../timml')\n",
    "import timml\n",
    "chdir(cwd)\n",
    "\n",
    "dpar = 1e-4\n",
    "M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "timml.LineSink1D(M0, xls=0, sigls=-BC2q/5000.) # <<< QUERY FUDGE FACTOR\n",
    "timml.HeadLineSink1D(M0, xls=L, hls=0.)\n",
    "M0.solve(silent=True)\n",
    "H0 = M0.headalongline(X, 0.).flatten()\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., H0, 'k--', mfc='none', label='forward solution, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M1 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "timml.LineSink1D(M1, xls=0, sigls=-BC2q/5000.-BC2q/5000.*dpar) # <<< QUERY FUDGE FACTOR\n",
    "timml.HeadLineSink1D(M1, xls=L, hls=0.)\n",
    "M1.solve(silent=True)\n",
    "H1 = M1.headalongline(X, 0.).flatten()\n",
    "dhdBC2q = (H1-H0)/(BC2q*dpar)\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdBC2q, 'k--', mfc='none', label='perturbation sensitivity, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial q_{\\Gamma_2}$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "50.010000\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f'% dhdBC2q[ocol])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2c. Numerical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import flopy\n",
    "\n",
    "nrow, ncol = 1, int(L)\n",
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=nrow, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowWel(M0, stress_period_data={0: [0, 0, 0, BC2q]})\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "success, buff = M0.run_model(silent=True)\n",
    "H0 = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., H0, 'k--', mfc='none', label='forward solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M1 = M0\n",
    "flopy.modflow.ModflowWel(M1, stress_period_data={0: [0, 0, 0, BC2q+BC2q*dpar]})\n",
    "M1.write_input()\n",
    "success, buff = M1.run_model(silent=True)\n",
    "H1 = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "dhdBC2q = (H1-H0)/(BC2q*dpar)\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdBC2q, 'k--', mfc='none', label='perturbation sensitivity, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial q_{\\Gamma_2}}$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "49.972534\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f'% dhdBC2q[ocol])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Adjoint sensitivity\n",
    "\n",
    "<br>\n",
    "    \n",
    "$\\begin{align}\n",
    "\\quad\\quad \\frac{\\partial h(x')}{\\partial q_{\\Gamma_2}}=\\oint\\limits_{\\Gamma_2}\\psi^*_1(x)\\; dx\\ =\\ \\psi^*_1(\\Gamma_2)\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Governing equation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad K\\ b\\ \\frac{d\\psi^*_1}{dx}+\\frac{1}{2\\ K\\ b}\\delta(x-x')=0\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Boundary conditions: \n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=0\\ ,\\quad\\quad\\quad\\quad\\quad x=0=\\Gamma_{1_0}\\cr\n",
    "\\end{align}$\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=0\\ ,\\quad\\quad\\quad\\quad\\quad x=L=\\Gamma_{1_L}\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Closed-form solution:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3a. Analytical\n",
    "$\\begin{align}\n",
    "\\quad\\quad\\rm{Not\\ available}\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "timml.ImpLineDoublet1D(M0, xld=0.)\n",
    "timml.HeadLineSink1D(M0, xls=L,  hls=0.)\n",
    "timml.LineSink1D(M0, xls=float(ocol), sigls=-1.)\n",
    "M0.solve(silent=True)\n",
    "A = M0.headalongline(X, 0.).flatten()\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., A, 'k--', mfc='none', label='adjoint solution, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdBC2q = np.empty(0)\n",
    "for xp in X[::1000]:\n",
    "    M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "    timml.ImpLineDoublet1D(M0, xld=0.)\n",
    "    timml.HeadLineSink1D(M0, xls=L, hls=0.)\n",
    "    timml.LineSink1D(M0, xls=xp, sigls=-1.)\n",
    "    M0.solve(silent=True)\n",
    "    A = M0.headalongline(X, 0.).flatten()  \n",
    "    dhdBC2q = np.append(dhdBC2q, A[0])\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X[::1000][1:]/1000., dhdBC2q[1:], 'k--', mfc='none', label='adjoint sensitivity, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial q_{\\Gamma_2}$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "50.000000\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f'% dhdBC2q[int(np.where(X[::1000]==float(ocol))[0])])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3c. Numerical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=1, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowWel(M0, stress_period_data={0: [0, 0, ocol, 1.]})\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "success, buff = M0.run_model(silent=True)\n",
    "A = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., A, 'k--', mfc='none', label='adjoint solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdBC2q = np.empty(0)\n",
    "for oc,xp in enumerate(X[::1000]):\n",
    "    M1 = M0\n",
    "    flopy.modflow.ModflowWel(M1, stress_period_data={0: [0, 0, oc*1000, 1.]})\n",
    "    M1.write_input()\n",
    "    success, buff = M1.run_model(silent=True)\n",
    "    A = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "    dhdBC2q = np.append(dhdBC2q, A[0])\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "f,s = plt.subplots(figsize=[8,2])\n",
    "plt.plot(X[::1000]/1000., dhdBC2q, 'k--', mfc='none', label='adjoint sensitivity, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial q_{\\Gamma_2}$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "49.990002\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f'% dhdBC2q[int(np.where(X[::1000]==float(ocol))[0])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}