02_R.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2.\n",
    "### Sensitivity of hydraulic head at a point to <font color='blue'>spatially uniform recharge</font> under steady state flow conditions\n",
    "\n",
    "## 0. Forward model\n",
    "<br>\n",
    "Governing equation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad K\\ b\\ \\frac{d^2h}{dx^2}+R=0\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Boundary conditions:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad -K\\ b\\ \\frac{dh(x)}{dx}=0\\ ,\\quad\\quad\\;\\;\\, x=0=\\Gamma_2\\cr\n",
    "\\end{align}$\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",
    "$\\begin{align}\n",
    "\\quad\\quad h(x)=h_L+\\frac{R (L^2-x^2)}{2\\ K\\ b}\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "Spatial derivatives from differentiation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\frac{d h}{d x}=-\\frac{R\\ x}{K\\ b},\\quad\\quad \\frac{d^2 h}{d x^2}{}=-\\frac{R}{K\\ b}\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "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": 28,
   "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": [
    "from warnings import filterwarnings\n",
    "filterwarnings(\"ignore\", category=DeprecationWarning)\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "def h(x, K, R, b, L, BC1h):\n",
    "    return R/K/2./b*(L**2.-x**2.)\n",
    "\n",
    "K, R, b, L, BC1h, ocol = 10., 1e-1/1000., 10., 10000., 0., 5000\n",
    "X = np.arange(L)\n",
    "H0 = np.array([h(x, K, R, b, L, BC1h) for x in X])\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, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Direct sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\frac{\\partial h(x')}{\\partial R}=\\frac{L^2-x^2}{2\\ K\\ b}\n",
    "\\cr\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "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": [
    "dhdR = [(L**2.-x**2.)/(2.*K*b) for x in X]\n",
    "benchmark = dhdR[ocol]\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., dhdR, 'k--', mfc='none', label='direct sensitivity, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "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": [
      "375000.000000 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Perturbation sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\frac{\\partial h(x')}{\\partial R}\\approx\\frac{h(x,K+\\Delta R)-h(x,R)}{\\Delta R}\\;\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "### 2a. Analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "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": [
    "dpar = 1e-4\n",
    "H0 = np.array([h(x, K, R, b, L, BC1h) for x in X])\n",
    "H1 = np.array([h(x, K, R+R*dpar, b, L, BC1h) for x in X])\n",
    "dhdR = (H1-H0)/(R*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., dhdR, 'k--', mfc='none', label='perturbation sensitivity, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "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": [
      "374999.999999 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": false
   },
   "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": [
    "from os import getcwd, chdir\n",
    "cwd = getcwd()\n",
    "chdir(r'../../timml')\n",
    "import timml\n",
    "chdir(cwd)\n",
    "\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.StripAreaSink(M0, 0., L, R)\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(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "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.ImpLineDoublet1D(M1, xld=0.)\n",
    "timml.HeadLineSink1D(M1, xls=L, hls=0.)\n",
    "timml.StripAreaSink(M1, 0., L, R+R*dpar)\n",
    "M1.solve(silent=True)\n",
    "H1 = M1.headalongline(X, 0.).flatten()\n",
    "dhdR = (H1-H0)/(R*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., dhdR, 'k--', mfc='none', label='perturbation sensitivity, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "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": [
      "375000.000000 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2c. Numerical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "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.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\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(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "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.ModflowRch(M1, nrchop=1, rech=R+R*dpar, ipakcb=53)\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",
    "dhdR = (H1-H0)/(R*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., dhdR, 'k--', mfc='none', label='perturbation sensitivity, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "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": [
      "374984.750000 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Adjoint sensitivity\n",
    "\n",
    "<br>\n",
    "    \n",
    "$\\begin{align}\n",
    "\\quad\\quad \\frac{\\partial h(x')}{\\partial R}=\\int\\limits_X\\psi^*_1(x)\\; dx\n",
    "\\cr\\end{align}$\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",
    "$\\begin{align}\n",
    "\\quad\\quad -K\\ b\\ \\frac{d\\psi^*_1(x)}{dx}=0\\ ,\\quad\\quad\\; x=0=\\Gamma_2\\cr\n",
    "\\end{align}$\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=0\\ ,\\ \\quad\\quad\\quad\\quad\\quad x=L=\\Gamma_1\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Closed-form solution:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=\\frac{1}{2\\ K\\ b}\\left[H\\left(x'-x\\right)\\left(L-x'\\right)+H\\left(x-x'\\right)\\left(L-x\\right)\\right]\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "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": [
    "def a(x, xp, K, b, L):\n",
    "    if x>xp:\n",
    "        a = L-x\n",
    "    else:\n",
    "        a = L-xp\n",
    "    return a/K/b\n",
    "A = np.array([a(x, float(ocol), K, b, L) for x in X])\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, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3a. Analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "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": [
    "dhdR = [np.sum([a(x, xp, K, b, L) for x in X]) for xp in X[::1000]]\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]/1000., dhdR, 'k--', mfc='none', label='adjoint sensitivity, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "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": [
      "375025.000000 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "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": 43,
   "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": [
    "dhdR = 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",
    "    dhdR = np.append(dhdR, np.sum(A))\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., dhdR[1:], 'k--', mfc='none', label='adjoint sensitivity, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "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": [
      "375025.000000 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3c. Numerical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "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=np.zeros([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": 46,
   "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": [
    "dhdR = 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",
    "    dhdR = np.append(dhdR, np.sum(A))\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., dhdR, 'k--', mfc='none', label='adjoint sensitivity, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial R$')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "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": [
      "374924.968750 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdR[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3d. Numerical, discrete"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Re-run adjoint model to obtain the adjoint state vector, $\\psi^*_1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 10000)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1 = M0\n",
    "flopy.modflow.ModflowWel(M1, stress_period_data={0: [0, 0, ocol, 1.]})\n",
    "M1.write_input()\n",
    "success, buff = M1.run_model(silent=True)\n",
    "psi = np.atleast_2d(flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:])\n",
    "np.shape(psi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Re-run forward model to obtain vectors of output heads, $h$, cell-by-cell conductances, $CR$, and RHS terms, $RHS$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "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": [
    "import flopy\n",
    "from os import system\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.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\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",
    "system('mf2005_stream.exe < model.in')\n",
    "h = np.atleast_2d(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., h[0, range(500, np.shape(h)[1], 1000)], 'ko--', mfc='none', label='forward solution, numerical')\n",
    "plt.plot(X/1000., np.ravel(h), 'k--', mfc='none', label='forward solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import RHS vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1, 10000)\n"
     ]
    }
   ],
   "source": [
    "RHS = np.atleast_2d(np.loadtxt('RHS.arr').flatten())\n",
    "print(np.shape(RHS))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assemble d(RHS)/dR vector analytically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "dRHSdR = RHS/R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "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": [
      "374925.008392 (375000.000000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (np.float(-psi@dRHSdR.T), benchmark))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1., -1., -1., ..., -1., -1.,  0.]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dRHSdR\n"
   ]
  },
  {
   "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
}