Line-sinks are added to the analytic element code for wells and uniform flow developed previously. Input arguments are the model to which the element is added, the coordinates of the endpoints, and the inflow $\\sigma$. The `LineSink` class has one function: the `potinf` function, which implements Eq. ({eq}`eq_potlinesink8.2`) for $\\sigma=1$. The potential ({eq}`eq_potlinesink8.2`) for a line-sink cannot easily be computed at the end points, because either $r_0$ or $r_1$ equals zero, but the discharge potential is finite because when $r_0=0$ then also $x=-L/2$ and when $r_1=0$ then $x=L/2$. The value of $(Z-1)$ or $(Z+1)$ is set to $10^{-12}$ when their absolute value is smaller than $10^{-12}$ using the `np.where` function in the code below." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "class LineSink(Element):\n", " \n", " def __init__(self, model, x0=0, y0=0, x1=1, y1=1, sigma=1):\n", " Element.__init__(self, model, sigma)\n", " self.x0, self.y0 = x0, y0\n", " self.x1, self.y1 = x1, y1\n", " self.xc = 0.5 * (self.x0 + self.x1)\n", " self.yc = 0.5 * (self.y0 + self.y1)\n", " self.L = np.sqrt((self.x1 - self.x0) ** 2 + (self.y1 - self.y0) ** 2)\n", " self.theta = np.arctan2(self.y1 - self.y0, self.x1 - self.x0)\n", " \n", " def potinf(self, x, y):\n", " xt = (x - self.xc) * np.cos(self.theta) + (y - self.yc) * np.sin(self.theta)\n", " yt = -(x - self.xc) * np.sin(self.theta) + (y - self.yc) * np.cos(self.theta)\n", " r0 = np.sqrt((xt + self.L / 2) ** 2 + yt ** 2)\n", " r1 = np.sqrt((xt - self.L / 2) ** 2 + yt ** 2)\n", " r0 = np.where(r0 < 1e-12, 1e-12, r0)\n", " r1 = np.where(r1 < 1e-12, 1e-12, r1)\n", " theta0 = np.arctan2(yt, xt + self.L / 2)\n", " theta1 = np.arctan2(yt, xt - self.L / 2)\n", " pot = 1 / (2 * np.pi) * (\n", " (xt + self.L / 2) * np.log(r0) - (xt - self.L / 2) * np.log(r1) - yt * theta0 + yt * theta1)\n", " return pot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a first example, the equipotentials of a single line-sink are plotted in the left graph of {numref}`Figure %s
The free parameter of a line-sink is the inflow $\\sigma$ into the line-sink per unit length of line-sink. When a line-sink is used to simulate the segment of a surface water feature, the inflow (negative when it is outflow) may be computed from a specified-head condition. Here, the head is specified at the center of the line-sink (the collocation point). In the code below, the `HeadLineSink` class is derived from the `LineSink` and `HeadEquation` classes. Input arguments are the model to which the element is added, coordinates of the endpoints, and the specified head at the center of the line-sink. The location of the collocation point is already stored as `xc` and `yc` in the `LineSink` class; the potential at the collocation points is stored as `pc`. The number of unknowns is set to 1. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "class HeadLineSink(LineSink, HeadEquation):\n", " \n", " def __init__(self, model, x0=0, y0=0, x1=1, y1=1, hc=1):\n", " LineSink.__init__(self, model, x0, y0, x1, y1, 0)\n", " self.pc = self.model.T * hc\n", " self.nunknowns = 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Strings of line-sinks may be used to simulate narrow rivers or streams. A stream needs to be subdivided into line-sinks based on three criteria: the string of line-sinks must represent the geometry of the stream, the line-sinks must be short enough to simulate the variation in river stage, and the line-sinks must be short enough to simulate the variation of the inflow or outflow along the stream. Recall that the inflow is uniform for each line-sink. In the current implementation of head-specified line-sinks, the head in the aquifer is set equal to the specified head at the center of the line-sink, which means that the aquifer is in direct contact with the stream. Alternatively, the resistance of a leaky stream bed may be taken into account by specifying condition ({eq}`eq_bcleakystreambed7.6`) of {numref}`content:chap7.6:well_uni_background_near_river`, but that is beyond the scope of this book (see, e.g., {cite:alp}`haitjema1995`).\n", "
As an example of a curved stream, consider a well pumping near a meandering stream. The stream is approximated with a sine function around the $x$-axis as $y = 50\\sin(\\pi x / 400)$. A head-specified well is located at $(x,y)=(0,100)$ with a head that is 2 m below the head in the river. The head is fixed to 2 m above the river level at $(x,y)=(0,1000)$. A stretch of stream is modeled from $x=-1600$ m to $x=1600$ m, which is far enough to the left and right of the well such that a further extension does not effect the heads in the area around the well significantly. Shorter line-sinks with a length of 50 m are used to represent the stream near the well, while longer line-sinks with a length of 200 m are used farther away. All other parameters are given in the script below. A contour plot of the head for the entire model is shown in the left graph of {numref}`Figure %s
\n",
"![](\"figures_chap8/sol_ex_83.png\")
\n",
"\n",
"
\n",
"\n",
"```"
]
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"```{exercise}\n",
":label: ex_8.4\n",
"\n",
"Simulate flow to a ditch with specified head of 10 m. The ditch runs from (-200, -100) till (200, 100). The head at (0,400) is 12 m. Use enough line-sinks such that the head along the ditch is approximately uniform. The transmissivity is 200 m$^2$/d. Plot head contours and make separate plots of the head variation along the ditch and the inflow variation along the ditch. Compute the total inflow into the ditch.\n",
"\n",
"Solution
\n", "\n", "\n",
"\n",
"\n",
"![](\"figures_chap8/sol_ex_8_4a.png\")
\n",
"\n",
"![](\"figures_chap8/sol_ex_8_4b.png\")
\n",
"\n",
"
\n",
"```\n"
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"source": [
"```{exercise}\n",
":label: ex_8.5\n",
"\n",
"Consider a well pumping near a long and straight river along the $x$-axis. The transmissivity of the aquifer is 200 m$^2$/d and the head in the river is 8 m. The well is located at (0, 100), the well radius is 0.3 m, and the head in the well is 6 m. Simulate the river with head-specified line-sinks. Use enough line-sink of an appropriate length such that the analytic element solution is in reasonable agreement with the exact solution obtained with the method of images ({numref}`ex_68_discharge_well6.2`). Compute the discharge of the well. \n",
"\n",
"Solution
\n", "\n", "Using 11 line-sinks: $Q = -1855$ m$^3$/d\n", "\n", "\n",
"\n",
"\n",
"\n",
"\n",
"
\n",
"```"
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