{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Problem 1 (20 points)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A major fault in your reservoir has strike N35$^\\circ$W and dip 60$^\\circ$ from horizontal. $S_{hmin}$ in this area acts along 020$^\\circ$, and in situ stresses are $S_{hmin}=70$ MPa, $S_{Hmax}=80$ MPa, $S_v=65$ MPa. Assume the pore pressure $P_p =30$ MPa. Is the fault likely to slip if the respective frictional coefficient is $\\mu=0.7$?\n", "\n", "**Solution**\n", "\n", "First we'll define some functions that we will use in this problem and the next." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np\n", "\n", "def compute_Sg(S, angles=(0,0,0)):\n", " \n", " alpha, beta, gamma = np.radians(angles)\n", " \n", " Rg = np.array([[np.cos(alpha) * np.cos(beta), \n", " np.sin(alpha) * np.cos(beta), \n", " -np.sin(beta)],\n", " [np.cos(alpha) * np.sin(beta) * np.sin(gamma) - np.sin(alpha) * np.cos(gamma), \n", " np.sin(alpha) * np.sin(beta) * np.sin(gamma) + np.cos(alpha) * np.cos(gamma), \n", " np.cos(beta) * np.sin(gamma)],\n", " [np.cos(alpha) * np.sin(beta) * np.cos(gamma) + np.sin(alpha) * np.sin(gamma), \n", " np.sin(alpha) * np.sin(beta) * np.cos(gamma) - np.cos(alpha) * np.sin(gamma), \n", " np.cos(beta) * np.cos(gamma)]])\n", " \n", " return np.dot(Rg.T, np.dot(S,Rg))\n", "\n", "\n", "def compute_unit_vectors(strike, dip):\n", " \n", " strike = np.radians(strike)\n", " dip = np.radians(dip)\n", " \n", " n = np.array([-np.sin(strike) * np.sin(dip), np.cos(strike) * np.sin(dip), -np.cos(dip) ])\n", " \n", " ns = np.array([ np.cos(strike), np.sin(strike), 0 ])\n", " \n", " nd = np.array([ -np.sin(strike) * np.cos(dip), np.cos(strike) * np.cos(dip), np.sin(dip) ])\n", " \n", " return (n, ns, nd)\n", "\n", "\n", "def compute_stress_components_on_fault(S, strike, dip, angles=(0,0,0)):\n", "\n", " S_G = compute_Sg(S, angles)\n", " \n", " n, ns, nd = compute_unit_vectors(strike, dip)\n", " \n", " sigma_n = np.dot(np.dot(S_G, n), n)\n", " tau_s = np.dot(np.dot(S_G, n), ns)\n", " tau_d = np.dot(np.dot(S_G, n), nd)\n", " \n", " tau_mag = np.sqrt(tau_s ** 2. + tau_d ** 2.)\n", " \n", " return (sigma_n, tau_s, tau_d, tau_mag)\n", "\n", "\n", "def check_fault_slip(S, Pp, strike, dip, angles=(0,0,0), mu=0.6):\n", " \n", " sigma_n, tau_s, tau_d, tau_mag = compute_stress_components_on_fault(S, strike, dip, angles)\n", " \n", " return (tau_mag / (sigma_n - Pp) > mu)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can check to see if our fault is slipping under the given conditions from the funcions above." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S = np.diag([80, 70, 65])\n", "\n", "check_fault_slip(S, 30, 35, 60, angles=(-70,0,0), mu=0.7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Problem 2 (50 points)\n", "\n", "Assuming the same reservior conditions as Problem 1, write a computer code that can compute the resolved shear stress magnitude and normal effective stress on a fault characterized by the strike direction and dip angle. Now randomly generate 10000 faults (use a statistical distribution to choose random pairs of strike and dip) and plot the results on a Mohr diagram. Where do near vertical strike-slip faults appear on this diagram? Where do the oblique-slip faults fall?\n", "\n", "**Solution**\n", "\n", "The code below creates the random faults. We will assume that a near-vertical strike-slip fault is one in which the dip angle is greater than $85^\\circ$. Oblique-slip faults are those that have near-equal magnitudes of shear and dip components of shear stresses." ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/png": 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9ZYT0k6CUmuL5usRxnCXNOKfjEjpg73AI318I51B8+BcjNPcJbpvbzcCXhBlJ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\nwmnfkbcwjX4L82+mm53U8D1s2DAeWxSIiEr5moh1xqMjK7CPs7DSQqY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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "def create_plot(S, angles=(0,0,0)):\n", "\n", " strikes = np.random.rand(10000) * 360\n", " dips = np.random.rand(10000) * 90\n", "\n", " out = np.array([ compute_stress_components_on_fault(S, strikes[i], dips[i], angles) for i in range(len(strikes)) ])\n", "\n", " normal = out[:,0]\n", " shear_slip = out[:,1]\n", " shear_dip = out[:,2]\n", " shear_mag = out[:,3]\n", "\n", "\n", " vert_ss = np.logical_and(dips > 85, np.abs(shear_slip / shear_dip) > 5)\n", " oblique_slip = np.logical_and(0.75 < np.abs(shear_slip / shear_dip), np.abs(shear_slip / shear_dip) < 1.5)\n", " other = np.logical_not(np.logical_or(vert_ss, oblique_slip))\n", "\n", " plt.figure()\n", " plt.gca().set_aspect('equal')\n", " plt.plot(normal[other], shear_mag[other], 'k.',\n", " normal[oblique_slip], shear_mag[oblique_slip], 'go',\n", " normal[vert_ss], shear_mag[vert_ss], 'ro'),\n", " plt.xlabel('$\\sigma_n^{eff}$')\n", " plt.ylabel('${\\tau}$');\n", " plt.legend(['Other','Oblique-slip', 'Near vertical'],bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.);\n", " \n", " \n", "create_plot(S, angles=(-70,0,0))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The plot above sort-of speaks for itself, there is no reason to describe in words the answers to the questions with a properly made figure." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Problem 3 (10 points)\n", "\n", "Repeat Problem 2 but make $S_{hmin}=S_{Hmax}=45$ MPa (i.e. stress isotropic in the horizontal plane. How has the Mohr diagram changed? Is this still a 3D problem?\n", "\n", "**Solution**\n", "\n", "We can use our code above to quickly solve this problem by simply plugging in the new numbers." ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "S = np.diag([65, 45, 45])\n", "create_plot(S, angles=(0,-90,0))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This problem is clearly 2D as it can be fully characterized by the two principle stresses." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#Problem 4 (10 points)\n", "\n", "a. Using a constitutive model for an isotropic elastic solid that relates strains to applied effective stresses corrected by Biot's coefficient, assume you have a sample in subsurface subjected to overburden, and that confinement in both horizontal principal directions is so large that the horizontal strains are nearly zero. Derive elastic equations for horizontal stresses (use the variables $x$ and $y$ for horizontal directions) in terms of vertical loading as a function of depth (in $z$ direction).\n", "\n", "**Solution**\n", "\n", "Hooke's law\n", "\n", "\\begin{align}\n", "\\sigma_{xx} = \\lambda (\\varepsilon_{xx} + \\varepsilon_{yy} + \\varepsilon_{zz}) + 2 \\mu \\varepsilon_{xx} \\\\\n", "\\sigma_{yy} = \\lambda (\\varepsilon_{xx} + \\varepsilon_{yy} + \\varepsilon_{zz}) + 2 \\mu \\varepsilon_{yy} \\\\\n", "\\sigma_{zz} = \\lambda (\\varepsilon_{xx} + \\varepsilon_{yy} + \\varepsilon_{zz}) + 2 \\mu \\varepsilon_{zz}\n", "\\end{align}\n", "\n", "Substituting the assumptions from the problem statement\n", "\n", "\\begin{align}\n", "\\sigma_{H} = \\lambda \\varepsilon_{v} \\\\\n", "\\sigma_{H} = \\lambda \\varepsilon_{v} \\\\\n", "\\sigma_{v} = \\lambda \\varepsilon_{v} + 2 \\mu \\varepsilon_{v} = (\\lambda + 2 \\mu) \\varepsilon_v\n", "\\end{align}\n", "\n", "Solving the third equation for $\\varepsilon_{v}$\n", "\n", "$$\n", "\\varepsilon_v = \\frac{\\sigma_{v}}{(\\lambda + 2 \\mu)}\n", "$$\n", "\n", "Substituting into one of the others\n", "\n", "$$\n", "\\sigma_{H} = \\frac{\\lambda \\sigma_{v}}{(\\lambda + 2 \\mu)} \n", "$$\n", "\n", "and using the relationship \n", "\n", "$$\n", "\\lambda = \\frac{2 \\mu \\nu}{1 - 2 \\nu}\n", "$$\n", "\n", "and simplifying\n", "\n", "$$\n", "\\sigma_{H} = \\frac{\\nu \\sigma_{v}}{3\\nu - 1} = \\frac{\\nu \\bar{\\rho}g z}{3\\nu - 1}\n", "$$\n", "\n", "and using the Biot's effective stress relationship we have:\n", "\n", "$$\n", "S_{H} = \\frac{\\nu (S_{v} - \\alpha P_p)}{3\\nu - 1} + \\alpha P_p\n", "$$\n", "\n", "b. Assume you are at depth of $4$ km and the pore pressure at that depth is hydrostatic. Using the equations derived in a) above, Biot's parameter of $\\alpha=0.7$, and a Poisson’s ratio $\\nu=0.2$, estimate the horizontal stress magnitudes $S_{hmin}=S_{Hmax}$ at that depth. \n", "\n", "**Solution**\n", "\n", "\\begin{align}\n", "S_{H} &= \\frac{0.2 (92 \\mbox{ MPa} - 0.7 \\cdot 40 \\mbox{ MPa})}{3\\cdot 0.2 - 1} + 0.7 \\cdot 40 \\mbox{ MPa} \\\\\n", " &= -4 \\mbox{ MPa}\n", "\\end{align}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Problem 5 (30 points)\n", "\n", "Bernabe and Brace (1990) in their article \"Deformation and fracture of Berea Sandstone\" reported a number of triaxial test measurements for Berea sandstone, and some of them are given in the table below (all data in MPa).\n", "\n", "|$S_3$ (confining pressure) | $P_p$ (pore pressure) | $S_1-S_3$ (differential stress) |\n", "|---------------------------|-----------------------|---------------------------------|\n", "|10|0|116|\n", "|50|0|227|\n", "|20|8|119|\n", "|45|8|183|\n", "|60|8|206|\n", "|75|8|228|\n", "|50|37|120|\n", "|50|32|141|\n", "|90|64|161|\n", "|90|55|187|\n", "|130|96|186|\n", "|130|84|207|\n", "\n", "Fit this data to Mohr-Coulomb criterion to compute unconfined compressive strength $C_0$ and internal friction coefficient $\\mu_i$ for Berea sandstone.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Solution**\n", "\n", "First we create a curve fit and then use Eq. 4.5 from the book to find the internal friction and extrapolate $C_0$." ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "data = np.array([[10, 0, 116], [50, 0, 227], [20, 8, 119], [45, 8, 183], \n", " [60, 8, 206], [75, 8, 228], [50, 37, 120], [50, 32, 141],\n", " [90, 64, 161], [90, 55, 187], [130, 96, 186], [130, 84, 207]])\n", "\n", "S3_eff = data[:,0] - data[:,1]\n", "S1_eff = data[:,2] + data[:,0] - data[:,1]\n", "\n", "z = np.polyfit(S3_eff, S1_eff, 1)\n", "\n", "nfun = np.poly1d(z)\n", "\n", "line_x = np.linspace(0,70, num=100)\n", "line_y = nfun(line_x)\n", "\n", "plt.plot(S3_eff, S1_eff, 'ko', line_x, line_y, 'b-');\n", "plt.grid()\n", "plt.xlabel('$\\sigma_3^{eff}$ MPa')\n", "plt.ylabel('$\\sigma_1^{eff}$ MPa');" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "100.080890436\n", "0.614932136531\n" ] } ], "source": [ "print z[1]\n", "\n", "print (z[0] - 1) / 2 / np.sqrt(z[0])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the polynomial fit\n", "\n", "$C_0 = 100$ MPa\n", "\n", "and from the computation above\n", "\n", "$\\mu_I = 0.61$" ] } ], "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": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 1 }