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yohanesnuwara committed Mar 11, 2020
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192 changes: 157 additions & 35 deletions ... 5 Finite-Difference Approximation to Linear-Flow Equations/fd_linearflow_exercises.ipynb
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"name": "fd_linearflow_exercises.ipynb",
"provenance": [],
"collapsed_sections": [],
"authorship_tag": "ABX9TyMbK4NO9NJwkoDqYQLv99Zn",
"authorship_tag": "ABX9TyNv55IEhouwh+vmA29mqwrO",
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"source": [
"!git clone https://github.com/yohanesnuwara/reservoir-engineering"
],
"execution_count": 16,
"execution_count": 2,
"outputs": [
{
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"text": [
"Cloning into 'reservoir-engineering'...\n",
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"remote: Counting objects: 100% (37/37), done.\u001b[K\n",
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"remote: Total 946 (delta 19), reused 0 (delta 0), pack-reused 909\u001b[K\n",
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"Resolving deltas: 100% (414/414), done.\n"
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"metadata": {
"id": "dDU0d5NiKC9Q",
"colab_type": "code",
"outputId": "3d72eb2b-2d08-4fbd-de1f-cab17d5b4f62",
"colab": {
"base_uri": "https://localhost:8080/",
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"outputId": "f0677107-59f9-437f-8feb-e0a14aa8e375"
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"source": [
"\"Task. determine pressure distribution during first year of production with timestep 10 days\"\n",
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"factor = (alpha * B * delta_t) / (Vb * poro * c)\n",
"factor"
],
"execution_count": 10,
"execution_count": 3,
"outputs": [
{
"output_type": "execute_result",
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"metadata": {
"tags": []
},
"execution_count": 10
"execution_count": 3
}
]
},
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"metadata": {
"id": "Jj6EB0KKOAvj",
"colab_type": "code",
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"outputId": "15ab2546-a8f8-4494-ae30-e182d6dccb2e"
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"source": [
"# calculate transmissibility of coupling cells\n",
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"T_min_half = T_plus_half\n",
"T_min_half, T_plus_half"
],
"execution_count": 11,
"execution_count": 4,
"outputs": [
{
"output_type": "execute_result",
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"metadata": {
"tags": []
},
"execution_count": 11
"execution_count": 4
}
]
},
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"metadata": {
"id": "H7aGA6dQO7P4",
"colab_type": "code",
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"outputId": "fd8750c1-f90c-4636-e526-a5ea180601a2"
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"source": [
"q = np.full(ngrid-1, T_min_half)\n",
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"print(\"At grid 3, the coupling transmissibility coeffs are:\", Ti_min_halves[2], \"for T_min_half and:\", Ti_plus_halves[2], \"for T_plus_half.\")\n",
"print(\"At grid 5, the coupling transmissibility coeffs are:\", Ti_min_halves[4], \"for T_min_half and:\", Ti_plus_halves[4], \"for T_plus_half.\")"
],
"execution_count": 12,
"execution_count": 5,
"outputs": [
{
"output_type": "stream",
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"metadata": {
"id": "RObLbHw8PYb5",
"colab_type": "code",
"outputId": "25e2d6aa-4c7e-4f0c-d95d-b90f7e1a342d",
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"height": 1000
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"outputId": "48b977b3-a6ea-426b-8cf2-49d19884b27e"
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"source": [
"pi = np.full(ngrid, p_initial) # array of pressure in each grid [6000, 6000, 6000, 6000, 6000]\n",
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"df = pd.DataFrame(pd.np.column_stack([time, df]), columns=['time', 'grid 1', 'grid 2', 'grid 3', 'grid 4', 'grid 5', 'grid6'])\n",
"df"
],
"execution_count": 14,
"execution_count": 7,
"outputs": [
{
"output_type": "execute_result",
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"metadata": {
"tags": []
},
"execution_count": 14
"execution_count": 7
}
]
},
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"metadata": {
"id": "WwMtG3j1Q5zL",
"colab_type": "code",
"outputId": "5705e4da-6c2e-4928-a00a-57686bf8d334",
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"base_uri": "https://localhost:8080/",
"height": 33
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"outputId": "82f9e5bb-1525-479f-b0b3-ceb3319c58e2"
}
},
"source": [
"# known\n",
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"factor = (Vb * poro * c) / (alpha * B * delta_t)\n",
"factor"
],
"execution_count": 15,
"execution_count": 8,
"outputs": [
{
"output_type": "execute_result",
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"metadata": {
"tags": []
},
"execution_count": 15
"execution_count": 8
}
]
},
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"metadata": {
"id": "4fS6_fx1RiUy",
"colab_type": "code",
"outputId": "301d06e4-6444-4181-81bc-f07b00d8e1d0",
"colab": {
"base_uri": "https://localhost:8080/",
"height": 1000
},
"outputId": "23945c65-3f3e-4f7a-fb84-8035277ace35"
}
},
"source": [
"import sys, os\n",
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"df = pd.DataFrame(pd.np.column_stack([time, df]), columns=['time', 'grid 1', 'grid 2', 'grid 3', 'grid 4', 'grid 5', 'grid 6'])\n",
"df"
],
"execution_count": 17,
"execution_count": 9,
"outputs": [
{
"output_type": "execute_result",
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"metadata": {
"tags": []
},
"execution_count": 17
"execution_count": 9
}
]
},
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"![Example 5 8](https://user-images.githubusercontent.com/51282928/75949840-09e36380-5eda-11ea-824c-dd5c9ac6f6d5.PNG)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "u3EvE2R_ZOV1",
"colab_type": "text"
},
"source": [
"### Explicit, timestep 10 days"
]
},
{
"cell_type": "code",
"metadata": {
"id": "9FxpnIDVXJLR",
"colab_type": "code",
"colab": {}
"colab": {
"base_uri": "https://localhost:8080/",
"height": 33
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"outputId": "94113a68-ed53-4774-bdfa-ebc3326fd2fc"
},
"source": [
"# known\n",
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"delta_z = 75\n",
"ngrid = 5\n",
"grid_loc = 4 # grid location where production well is located\n",
"B = 1 # phase FVF, assumed constant over pressure, rb/stb\n",
"\n",
"c = 3.5E-06 # phase compressibility, psi^-1\n",
"k_x = 15 # perm in x direction, md\n",
"poro = 0.18\n",
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"factor = (alpha * B * delta_t) / (Vb * poro * c)\n",
"factor"
],
"execution_count": 16,
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"1.1883597883597885"
]
},
"metadata": {
"tags": []
},
"execution_count": 16
}
]
},
{
"cell_type": "code",
"metadata": {
"id": "xl2FnDINmFyd",
"colab_type": "code",
"colab": {
"base_uri": "https://localhost:8080/",
"height": 100
},
"outputId": "d6a4d8e1-989c-44a3-e238-18cf74409a17"
},
"source": [
"# for the first timestep\n",
"\n",
"pi = np.full(ngrid, 6000) # array of pressure in each grid [6000, 6000, 6000, 6000, 6000]\n",
"# p_new = pi\n",
"p_new = [6000, 5900, 5800, 5700, 6000]\n",
"\n",
"minus_q = 0 # no flow at left boundary\n",
"plus_q = 0.15 # 0.15 psi/ft at right boundary\n",
"\n",
"B_arr = []\n",
"for j, obj in enumerate(pi):\n",
" delta_p = p_new[j] - pi[j]\n",
" B = 1 / (1 + (c * (delta_p)))\n",
" B_arr.append(float(B))\n",
"\n",
"Ti_min_halves = []\n",
"Ti_plus_halves = []\n",
"\n",
"for k, obj in enumerate(B_arr):\n",
" # calculate transmissibility of coupling cells\n",
" beta = 1.127 # transmissibility conversion factor, is a constant\n",
" if k == 0:\n",
" T_min_half = 0 \n",
" if k > 0:\n",
" minus_B = B_arr[k-1]\n",
" T_min_half = beta * ((Ax * k_x) / (mu * minus_B * delta_x))\n",
" if k < (len(pi) - 1):\n",
" plus_B = B_arr[k+1]\n",
" T_plus_half = beta * ((Ax * k_x) / (mu * plus_B * delta_x))\n",
" if k == (len(pi) - 1):\n",
" T_plus_half = 0\n",
"\n",
" Ti_min_halves.append(float(T_min_half))\n",
" Ti_plus_halves.append(float(T_plus_half))\n",
"\n",
"print(Ti_plus_halves)\n",
"print(Ti_min_halves)\n",
"print(\"At grid 1, the coupling transmissibility coeffs are:\", Ti_min_halves[0], \"for T_min_half and:\", Ti_plus_halves[0], \"for T_plus_half.\")\n",
"print(\"At grid 3, the coupling transmissibility coeffs are:\", Ti_min_halves[2], \"for T_min_half and:\", Ti_plus_halves[2], \"for T_plus_half.\")\n",
"print(\"At grid 5, the coupling transmissibility coeffs are:\", Ti_min_halves[4], \"for T_min_half and:\", Ti_plus_halves[4], \"for T_plus_half.\")"
],
"execution_count": 66,
"outputs": [
{
"output_type": "stream",
"text": [
"[0.12674312437500002, 0.12669874874999998, 0.12665437312499997, 0.1267875, 0.0]\n",
"[0.0, 0.1267875, 0.12674312437500002, 0.12669874874999998, 0.12665437312499997]\n",
"At grid 1, the coupling transmissibility coeffs are: 0.0 for T_min_half and: 0.12674312437500002 for T_plus_half.\n",
"At grid 3, the coupling transmissibility coeffs are: 0.12674312437500002 for T_min_half and: 0.12665437312499997 for T_plus_half.\n",
"At grid 5, the coupling transmissibility coeffs are: 0.12665437312499997 for T_min_half and: 0.0 for T_plus_half.\n"
],
"name": "stdout"
}
]
},
{
"cell_type": "code",
"metadata": {
"id": "DiKe4rHDrCAd",
"colab_type": "code",
"colab": {}
},
"source": [
"pi_initial = np.full(ngrid, 6000) # array of pressure in each grid [6000, 6000, 6000, 6000, 6000]\n",
"time = np.arange(10, 370, delta_t)\n",
"\n",
"pi_arr = []\n",
"min_arr = []\n",
"plus_arr = []\n",
"\n",
"delta_p = []\n",
"\n",
"for m in range(len(time)):\n",
" B_arr = []\n",
" for j, obj in enumerate(pi):\n",
" delta_p = p_new[j] - pi[j]\n",
" B = 1 / (1 + (c * (delta_p)))\n",
" B_arr.append(float(B))"
],
"execution_count": 0,
"outputs": []
}
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