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    "# Test your Python skills: Darcy's law\n",
    "\n",
    "In this exercise, you will write a Python function that uses **Darcy's law** to calculate the flow rate through a porous medium.\n",
    "\n",
    "## Problem description\n",
    "\n",
    "Darcy's law states that the flow rate ($Q$) through a porous medium is given by:\n",
    "\n",
    "$$ \n",
    "Q = k \\times A \\times \\left(\\frac{\\Delta h}{L}\\right)\n",
    "$$ \n",
    "\n",
    "Where:\n",
    "- **$k$** is the hydraulic conductivity (in meters per second, $m/s$).\n",
    "- **$A$** is the cross-sectional area (in square meters, $m^2$).\n",
    "- **$\\Delta h$** is the hydraulic head difference (in meters, $m$).\n",
    "- **$L$** is the length over which the head difference occurs (in meters, $m$).\n",
    "\n",
    "**Your Task:**\n",
    "\n",
    "1. **Write a function** `darcy_flow(k, A, delta_h, L)` that:\n",
    "   - Accepts four parameters: `k`, `A`, `delta_h`, and `L`.\n",
    "   - Calculates the hydraulic gradient ($\\frac{\\Delta h}{L}$).\n",
    "   - Returns the flow rate ($Q$) using Darcy's law.\n",
    "   \n",
    "2. **Test your function** by computing flow rates for a range of hydraulic head differences (`delta_h`) while keeping `k`, `A`, and `L` constant.\n",
    "\n",
    "3. **Plot the results:** Create a plot of flow rate ($Q$) versus hydraulic head difference ($\\Delta h$).\n",
    "\n",
    "## Helpful Hints\n",
    "\n",
    "- Use the `def` keyword to define your function.\n",
    "- The hydraulic gradient is calculated as `delta_h / L`.\n",
    "- You can use the `numpy` library's `linspace` to create an array of `delta_h` values.\n",
    "\n",
    "Let's get started!\n"
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    "# Complete the function here\n",
    "def darcy_flow(k, A, ...):  # Finalize introducing ALL PARAMETERS HERE\n",
    "    \"\"\"\n",
    "    Calculate the flow rate Q using Darcy's law.\n",
    "\n",
    "    Parameters:\n",
    "        k: Hydraulic conductivity (m/s)\n",
    "        A: Cross-sectional area (m^2)\n",
    "        delta_h: Hydraulic head difference (m)\n",
    "        L: Length over which the head difference occurs (m)\n",
    "\n",
    "    Returns:\n",
    "        Q: Flow rate Q (m^3/s)\n",
    "    \"\"\"\n",
    "    \n",
    "    Q = # THE EXPRESSION HERE\n",
    "    \n",
    "    return Q\n",
    "\n",
    "# You can test your function with a simple example:\n",
    "print(\"Test Function Output:\", darcy_flow(1e-5, 10, 5, 100))\n"
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    "## Step 2: Compute flow rates for a range of hydraulic head differences\n",
    "\n",
    "Now, let’s use your function to calculate the flow rate \\( Q \\) for a range of hydraulic head differences (`delta_h`), while keeping the other parameters constant.\n",
    "\n",
    "For example, assume:\n",
    "- \\( k = 1 \\times 10^{-5} \\) m/s,\n",
    "- \\( A = 10 \\) m\\(^2\\),\n",
    "- \\( L = 100 \\) m.\n",
    "\n",
    "We'll vary `delta_h` from 0.1 m to 10 m.\n"
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    "import numpy as np\n",
    "\n",
    "# Given constant parameters\n",
    "k = 1e-5  # Hydraulic conductivity in m/s\n",
    "A = 10    # Cross-sectional area in m^2\n",
    "L = 100   # Length in m\n",
    "\n",
    "# Create an array of delta_h values from 0.1 m to 10 m (avoiding 0 to prevent any division issues)\n",
    "delta_h_values = np.linspace(0.1, 10, 50)\n",
    "\n",
    "# Calculate Q for each delta_h value using your darcy_flow function\n",
    "Q_values = []\n",
    "for dh in delta_h_values:\n",
    "    Q_values.append(darcy_flow(k, A, dh, L))\n",
    "\n",
    "# Print a few computed values to check your results\n",
    "print(\"Sample computed Q values:\", Q_values[:5])\n"
   ]
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    "## Step 3: Plot the results\n",
    "\n",
    "Finally, plot the flow rate \\( Q \\) versus the hydraulic head difference \\( \\Delta h \\) using `matplotlib`.\n",
    "\n",
    "This plot will help visualize how the flow rate increases as the hydraulic head difference increases.\n"
   ]
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    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.figure(figsize=(8, 5))\n",
    "plt.plot(delta_h_values, Q_values, marker='o', linestyle='-', color='steelblue')\n",
    "plt.xlabel('Hydraulic head difference (Δh) [m]')\n",
    "plt.ylabel('Flow rate Q [m³/s]')\n",
    "plt.title(\"Darcy's law: flow rate vs. hydraulic head difference\")\n",
    "plt.grid(True)\n",
    "plt.show()\n"
   ]
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