{
"cells": [
{
"cell_type": "markdown",
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"source": [
"# 3. Slug Test - Falling Head"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Import packages"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"import timflow.transient as tft\n",
"\n",
"plt.rcParams[\"figure.figsize\"] = [5, 3]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Introduction and Conceptual Model\n",
"\n",
"A well partially penetrates a sandy unconfined aquifer that has a saturated depth of 32.57 ft. The top of the screen is located 0.47 ft below the water table and has 13.8 ft in length. The well and casing radii are 5 and 2 inches, respectively. The slug displacement is 1.48 ft. Head change has been recorded at the slug well. This slug test, taken from the AQTESOLV examples (Duffield, 2007), was reported in Batu (1998). "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"![](\"./figs/Falling_head.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Load data"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"data = np.loadtxt(\"data/falling_head.txt\", skiprows=2)\n",
"to = data[:, 0] / 60 / 60 / 24 # convert time from seconds to days\n",
"ho = (10 - data[:, 1]) * 0.3048 # convert drawdown from ft to meters"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Parameters and model"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"rw = 5 * 0.0254 # well radius, m\n",
"rc = 2 * 0.0254 # well casing radius, m\n",
"L = 13.8 * 0.3048 # screen length, m\n",
"b = 32.57 * 0.3048 # aquifer thickness, m\n",
"zt = 0 # top of aquifer, m\n",
"zst = -0.47 * 0.3048 # top of screen, m\n",
"zsb = zst - L # bottom of screen, m\n",
"zb = zt - b # bottom of aquifer, m\n",
"H0 = 1.48 * 0.3048 # initial displacement in well, m"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"convert measured displacement into volume"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"slug: 0.00366 m^3\n"
]
}
],
"source": [
"Q = np.pi * rc**2 * H0\n",
"print(f\"slug: {Q:.5f} m^3\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will create a multi-layer model. For this, we divide the second and third layers into 0.5 m thick layers. "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"# z = np.array([zt, zt - 0.1, zst, zsb, zb])\n",
"z = np.hstack(\n",
" (\n",
" zt,\n",
" np.linspace(zt - 0.01, zst, 5)[:-1],\n",
" np.linspace(zst, zsb, 5)[:-1],\n",
" np.linspace(zsb, zb, 10),\n",
" )\n",
")\n",
"nlay = len(z) - 1"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"self.neq 5\n",
"solution complete\n"
]
}
],
"source": [
"ml = tft.Model3D(\n",
" kaq=10,\n",
" z=z,\n",
" Saq=[0.1] + (nlay - 1) * [1e-4],\n",
" kzoverkh=1,\n",
" tmin=1e-5,\n",
" tmax=0.01,\n",
" topboundary=\"phreatic\",\n",
")\n",
"w = tft.Well(\n",
" ml,\n",
" xw=0,\n",
" yw=0,\n",
" rw=rw,\n",
" tsandQ=[(0, -Q)],\n",
" layers=range(6, 6 + 5),\n",
" rc=rc,\n",
" wbstype=\"slug\",\n",
")\n",
"ml.solve()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Estimate aquifer parameters"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"...................................\n",
"Fit succeeded.\n"
]
}
],
"source": [
"cal = tft.Calibrate(ml)\n",
"cal.set_parameter(name=\"kaq\", layers=list(range(1, nlay)), initial=10, pmin=0)\n",
"cal.set_parameter(name=\"Saq\", layers=list(range(1, nlay)), initial=1e-4, pmin=0)\n",
"cal.seriesinwell(name=\"obs\", element=w, t=to, h=ho)\n",
"cal.fit(report=False)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" optimal | \n",
"
\n",
" \n",
" \n",
" \n",
" | kaq_1_17 | \n",
" 0.439175 | \n",
"
\n",
" \n",
" | Saq_1_17 | \n",
" 0.000461 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"tm = np.logspace(np.log10(to[0]), np.log10(to[-1]), 100)\n",
"hm = w.headinside(tm)\n",
"plt.semilogx(to, ho / H0, \".\", label=\"obs\")\n",
"plt.semilogx(tm, hm[0] / H0, label=\"timflow\")\n",
"plt.xlabel(\"time [d]\")\n",
"plt.ylabel(\"Normalized head (h/H0)\")\n",
"plt.title(\"Model results - multi-layer model\")\n",
"plt.legend()\n",
"plt.grid()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Comparison of results\n",
"Here, the `timflow` performance in analysing slug tests is checked. The solution in `timflow` is compared with the KGS analytical model (Hyder et al. 1994) implemented in AQTESOLV (Duffield, 2007). The parameters of `timflow` and AQTESOLV are similar, even though AQTESOLV only used one layer."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
" \n",
" \n",
" | | \n",
" k [m/d] | \n",
" Ss [1/m] | \n",
" RMSE [m] | \n",
"
\n",
" \n",
" \n",
" \n",
" | timflow | \n",
" 0.44 | \n",
" 4.61e-04 | \n",
" 0.006 | \n",
"
\n",
" \n",
" | AQTESOLV | \n",
" 0.42 | \n",
" 5.70e-04 | \n",
" - | \n",
"
\n",
" \n",
"
\n"
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"text/plain": [
""
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},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
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"source": [
"t = pd.DataFrame(\n",
" columns=[\"k [m/d]\", \"Ss [1/m]\", \"RMSE [m]\"],\n",
" index=[\"timflow\", \"AQTESOLV\"],\n",
")\n",
"\n",
"t.loc[\"timflow\"] = np.append(cal.parameters[\"optimal\"].values, cal.rmse())\n",
"t.loc[\"AQTESOLV\"] = [0.4211, 5.70e-4, \"-\"]\n",
"\n",
"t_formatted = t.style.format(\n",
" {\n",
" \"k [m/d]\": \"{:.2f}\",\n",
" \"Ss [1/m]\": \"{:.2e}\",\n",
" \"RMSE [m]\": lambda x: \"-\" if x == \"-\" else f\"{float(x):.3f}\",\n",
" }\n",
")\n",
"t_formatted"
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},
{
"cell_type": "markdown",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": []
},
"source": [
"## References\n",
"\n",
"* Batu, V. (1998), Aquifer hydraulics: a comprehensive guide to hydrogeologic data analysis, John Wiley & Sons\n",
"* Duffield, G.M. (2007), AQTESOLV for Windows Version 4.5 User's Guide, HydroSOLVE, Inc., Reston, VA.\n",
"* Hyder, Z., Butler Jr, J.J., McElwee, C.D. and Liu, W. (1994), Slug tests in partially penetrating wells, Water Resources Research 30, 2945–2957."
]
}
],
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