Problem 56205. Measure the hydraulic conductivity with a falling-head permeameter

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>A falling-head permeameter is another device for measuring the hydraulic conductivity </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> of a soil sample. In this problem the sample is placed in a cylinder of length </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> and cross-sectional area </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"A_c\"/></w:customXmlPr><w:r><w:t>A_c</w:t></w:r></w:customXml><w:r><w:t>. Unlike the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/55825-measure-the-hydraulic-conductivity-with-a-constant-head-permeameter\"><w:r><w:rPr><w:u/></w:rPr><w:t>constant-head permeameter</w:t></w:r></w:hyperlink><w:r><w:t>, in which the water level in a standpipe (or tube) is kept constant, the falling-head permeameter has a water level in the tube (of cross-sectional area </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"A_t\"/></w:customXmlPr><w:r><w:t>A_t</w:t></w:r></w:customXml><w:r><w:t>) that falls. In other words, the head difference </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"delta\"/></w:customXmlPr><w:r><w:t>\\delta</w:t></w:r></w:customXml><w:r><w:t>, or the difference between the water levels in the tube and the outlet, decreases in time. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The hydraulic conductivity can be determined from a statement of conservation of mass (i.e., water volume) by equating the rate of change of volume in the tube to the flow out of the soil sample. The outflow can be computed with Darcy’s law, which states in general</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"Q = -K(dh/dx)A\"/></w:customXmlPr><w:r><w:t>Q = -K\\frac{dh}{dx}A</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Darcy’s law applies when a Reynolds number based on the specific discharge </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"v = Q/A\"/></w:customXmlPr><w:r><w:t>v = Q/A</w:t></w:r></w:customXml><w:r><w:t> and representative diameter </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t> of the soil grains is less than (approximately) 1—that is, </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"Re = vd/nu &lt; 1\"/></w:customXmlPr><w:r><w:t>Re = \\frac{vd}{\\nu} &lt; 1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"nu\"/></w:customXmlPr><w:r><w:t>\\nu</w:t></w:r></w:customXml><w:r><w:t> is the kinematic viscosity of the fluid.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Derive and solve an ordinary differential equation for the head difference </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"delta\"/></w:customXmlPr><w:r><w:t>\\delta</w:t></w:r></w:customXml><w:r><w:t>. Then write a function that takes as input measurements of head difference as a function of time, as well as the soil’s porosity, diameter of the tube, and length and diameter of the cylinder holding the soil sample. The function should compute the hydraulic conductivity by fitting the solution to the ordinary differential equation to the data and using Darcy’s law regardless of its validity. Also return a flag indicating whether Darcy’s law is valid throughout the experiment; to assess the validity, relate the hydraulic conductivity to the representative grain diameter with the Kozeny-Carman equation, as described in the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/55825-measure-the-hydraulic-conductivity-with-a-constant-head-permeameter\"><w:r><w:rPr><w:u/></w:rPr><w:t>previous problem</w:t></w:r></w:hyperlink><w:r><w:t>. Use </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"g = 981 cm/s^2\"/></w:customXmlPr><w:r><w:t>g = 981\\rm\\,cm/s^2</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"nu = 10^{-2} cm^2/s\"/></w:customXmlPr><w:r><w:t>\\nu = 10^{-2}\\rm\\,cm^2/s</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"359\"/><w:attr w:name=\"width\" w:val=\"401\"/><w:attr w:name=\"verticalAlign\" w:val=\"middle\"/><w:attr w:name=\"altText\" w:val=\"falling-head permeameter\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc 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SpU/Lcc895awAAyH40izi0Z88e0wTStGlT0wRSUFDgrQEAIHcQLhz46KOPTKjQ/hUaKgYPHuytAQAg99AskgLbr+KRRx6R7t27myYQggUAINcRLpJk+1VowNDOmi+88IK3BgCA3FZr4WLChAnenHvVue2DBw/KQw89JJs2bZJf/vKXUlxcbPpYAACA22qtz0VeXp5U166rY9s6ToX2q9Bwof0qhg8f7q0BAAB+NItUQW8nnTt3rmkC6dy5s+lXQbAAACA2wkUcGzdulFatWskHH3xgQoX2q6hXr563FgAARJPT4UI7ZUajTR96B8iPf/xjKSkpkQ0bNtCvAgCAgHI2XPzkJz+Rp556ygx8ZWm/itGjR8uIESPk2WefNX9grEePHt5aAAAQRE6GC9s5U+n//n4Vbdu2lXPnzsmoUaPMegAAkJicvFtEaye0P4Wl/Sh08Cu9C+TLX/6ytxQAACQj58KFhgoNF37an0I7bN5zzz3eEgAAkKycahbxN4f46fJ58+Z5jwAAQCpyKlxosNAgEY3eOVJaWuo9AgAAycqZZhENDnpr6f333+8tEdO/wt/HIvIxAABIHMN/AwAAp3J6EC0AAOAe4QIAADhFuAAAAE4RLgAAgFOECwAA4BThAgAAOEW4AAAAThEuAACAU4QLAADgFOECAAA4RbgAAABOES4AAIBThAsAAOAU4QIAADhFuAAAAE7llVfw5mtUXl6eRO76yWWl8ttPrnqP4nvgvgbyyowHvUfhom0bAADUjLSquRj7aAtvrmqJlAUAADUnrcJFrw5/Y2okqqJltCwAAEg/adfnIkiNBLUWqEnnzp2Txo0bS9euXeXatWve0sTF286FCxdibnv79u3medrcV7duXVm3bp23Jv46xKbnyZ43e+7GjRsnJ0+e9EoASEXahYuqai+otchc169fl/vvv1/uvvtuuXTpkrc0XJAy2Wb//v3SvHlzadKkSaVj1kAyevRouXr1qvn/iSeekN/+9rdVrkNsq1evlj179sjHH39s+maVlZVJz549TeDo3r17zvzcAdUpLe8WiVczQa0FslXbtm2lQYPwYL179265cuWKvP7667J+/XpzYZw7d26V6xDf1KlTpX79+ma+RYsWsmPHDhk6dKh5DCB1aRkuYtVeUGuBbNS7d2/zDfro0aOhC54Vr5qeKvzkFBYWyne/+105fPiwt0SkTp06smXLFvnzn/8s9957r7cUQLLSMlyoaDUU1FoASFWrVq3kpz/9qQwePFgmTZqUUj8aANGlbbiIrL2g1gJq165d5punvyPeokWLol4gtJPkiy++GNZxr1+/fmHfWP1u3bolr7zySlgHye9///umT0MiEt2O7eg5YsQIb8ntfgH63GXLlpnH2qdCH2tflAULFsRcZ/sLBD1Pdt/anKLz9jna6dTf9yCZ7Z0/f16mT58eek6nTp1innulnVO7desWKt+uXbuo+0jkZyAWDRgXL140zVCNGjVK+PkAqlBeS4Lsev/xP5T/7Yz9ZtL5oGrxsBDHX/7yl/KWLVuWV3ygl1d8sHtLw8Ur89prr5n3Nj8/v7ziglo+ZcqU8ooLg1nWpUuX8oqLt1eyvPzgwYOhdX379jVlBw0aFHr+oUOHvJJ3zJgxI7TeX95OkfuIJdHtnD171rzW4cOHe0tuv359bsVF3jzHHsOzzz5bvnPnzpjrdLuJnCe7by1ny+h2/ec/le1FngN9XNW5j9zHhg0bvFKJvZaqnDp1qvyZZ54p79mzp3l+x44dy0+cOOGtBZCKtA4XaurSX5spEUG3jZplg4O+PwUFBeZDPXKyF8zIcGEvWpEXkJs3b4YuTP6L0L59+8rHjh1b6WJhL07+C7nS8tH2W1ZWFnrNQS5eyWwnWriwoh2bFW1doufJlo/2ulQq29Pj8T+nqnOvF/fIQKlByr6Hib6WWPS90LDz0ksvhbazbds28/xooRZA4mo1XOw8dbnKqXjbR2aKti7aZLed6YqKisxx2EkfW+m0LhH+cFHVFPkhH+/iYS860S7OkWKVtduPdmyxLmrRJLOdeK8/3nFHW5foebLLYl1Uk91eIkEl3j78En0t0difwWjvT6zwAyBxadvnwvrq/Z83U67RNuuK9yc0+W8xTHWd3r6YzPOirUtGxYXMtHf7t2mnig9/qfjw90repn0Yzpw5Y+a1R7/eRuif5s+fb45JxyuIbDfXfhdbt26tVNbPbj8/P1+GDBniLU2cq+0kK5XzNHDgwEp3SaSyPX0PI+98adasmTRs2NB7dJv/nLVu3dpbWlkqr8XvpZdekk8//TTq+zNmzBjzut9+++24fUMAVC3twwXc0kGqtDOc/p+JKr61yqpVq8KmNWvWeGvv0IuRdibUwamGDRsWt2w2CnqegnK9PUvfp/fff1/uuusu08kyiGRfiw0osfalt6N27tzZewQgFYSLHKN3MZSWlpr/M41+uz106FDUGg+d/ONEzJ49W5YvXy5dunSREydOhMqcPXvW3B2QzRI5T0G43p6fvaDfvHnT3GlSlVReiw0ysehradOmjfcIQCoIFzlEayv0ljul/2dK7YX90L9x44YJB1Wx31D1QlRcXCzt27f31kSX6PZjcbWdZLnef00cT9B9uHgtuo14Qcb+3HTo0MFMAJJHuMghWluh7c1K/8+k2gs7BsTixYujtqnr2AdVjVipF48VK1ZU6nOhYm1fL0I65kO050TjajvJcnGe/FxvLxob/qLt49133w1tP9XXouFi+PDhJqBov4pIBw8eNP1zZs2alXRNDIDbCBc5wl9rYWVS7YUOkT1jxgw5duyY6RCqg2FpR76JEyeagZQGDBggx48fN2X9F5E+ffqYjnpaTv8wmB14KlKPHj3M35bQ7WuziW5bR3DUToZf+tKXzP6CcLWdZCVynoJwvb1o9K+R+s+Zvl+6Dx2Mq2/fvqHtu3gtdl/z5s0zA5vZkKLBRPvmFBUVyahRo8wyACkoryW662i3kqY62W0j3A9/+ENzXiInXV5T4g2QZVVVRscjsGNh6JSfn28GkSotLfVK3LF27Vpze6Itq+WOHDlith/tdkMdL2HOnDlh216yZIlZXnFBCnQrqkp0O2fj3EYZ7/bLeOuCnqd4+/ZzsT373kY7j3puiouLw94vHQtl4cKFlcom8jMQTbR9JfJ8AFXL038qfrlqnH7TqAgD3iN3+hc0MtuupcNKS1o7ob3jbZOIX9OmTU2Vfb169bwlAACkhnCRA06fPi0bN270HompEtbqX0urgSu+JXqPAABIDeEiB3F+AADViQ6dAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAApwgXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIF8gqEyZM8OaQaXjvgOyRV17Bm69ReXl5svPUZe+RO/0LGplt19JhZYRsPj+895mL9w7IHtRcAAAApwgXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAApwgXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAApwgXAADAKcIFAABwKq+8gjdfo/Ly8mTnqcveI3f6FzQy266lw8oI2Xx+auPYnlxWKr/95Kr3KL4H7msgr8x40HsEP35vgexBzQWQorGPtvDmqpZIWQDIVIQLIEW9OvyNqZGoipbRskAyzp07J40bN5YRI0Z4S4D0RbgAHAhSI0GtBSJdv35d7r//ftMkFG26++675dKlS17pzHL+/HmZPn26CUT2eOrWrSv9+vWT999/3yuFbEW4yAGlpaVSWFgYmpT/8bZt28wyJK+q2gtqLVCVgoIC6dmzZ9j0yCOPeGszy+rVq6Vly5ayfPlyuXLlSuh4bty4IXv37pUHH3xQJk2a5JVGNiJc5AD9Rf7ss89kz549ZlJ2/vTp0+abBFIXr2aCWgvE06BBA3n33XflvffeC5t27Ngh9957r1cqM2iw0OCQn58va9euNZ107fHovH6ZadSokSk3d+5c71nINoSLHFFUVOTNhZs9e7bUq1fPe4RUxKq9oNYCuUKbeebNm2fmX3/9dRk7dqyZ9xs0aJBs2bLFzC9ZsiRjm30QH+EiRwwfPtzUYPg1bdpUnnzySe8RXIhWQ0GtReb7t8t/9ebS165du2Tw4MFhfTa0VvLkyZNeidv2799v1kXrGKq1CVWti1fbsH79eikrK5MuXbrIsGHDvKWV9ejRQ4YOHSpXr16VnTt3ekuRTQgXOSSy9oJaC/ciay+otcgO3/7BEVmy8V/TNmTohX/AgAGyfft2GT16tEyZMsUEC+3f0KFDB9m4caNXUqRbt26mP4Q2uUTWGtggogHh2rVrZt7SddrUMWTIEG9JZfb5Ghzq169v5qOpU6eO+cKjNm3aZP5HdiFc5BB/7QW1FtXHX1NBrUX22P6//q3aQsbNmzfl2WeflalTp4ZN69at80rEpjUR2sdB+22cOHHC1B6sXLlSdu/eHeqsPXny5FCQ0C8UI0eOrFRroE0ab775ppnX7ehk2XUaVHSK5tatW3LmzBkzr51TkdsIFznG1l5Qa1F9bO0FtRbZqTpCht5FsWHDBlm1alXY9NZbb3klYispKTH/v/rqq9K+fXszb2n/hhkzZlQKEracv9bgyJEjpsZCaz309bz99tvemjvrtMYjXo1EIr761a96c8hGDP+dQbSt03aWUhoUbPtnIuu+8pWvyPHjx024cLXNdFo37unZMn7abO9R7fjwd//X/P/V+z9v/kd8+45ekv3HMrNjn9ZOjRvU0nuUGK0RaNu2rfzxj380g2TFuzNE13ft2lX69u0bChRaW/DNb37TNIfs27dPHn74YbPcz9694f+dsdtq1aqVeZ4GBi331FNPmeYS7Yj5hS98IbRu5syZsmzZMhOARo0aZbYRyb6WrVu3xi1n2delNar2eJA9CBc5SD/QsrXWorp+rlD9BrRrnLa/twNnvufN3dG0cT0TLB79WhNvSeJSDRdBnh/tIh4ZSrQDpj7+5JNPTHPKt7/9bdNBVLept43qOvs43mu0IcQfZGJJpCwyD80iOYjmECB5GipmjWoj617qmlKwcEE7Rnbu3Nn02dALfzxazrIdKm3zh46mqX03tCPm5z//ebPONqXYdQMHDqxyzA3b3FLVCJz0z8h+hAsACCCdQoWlIaFNmzYmJJw9e9ZbeodexDdv3mzmIy/iOjqv1kpoM8aHH35oRtK0Zew67ZOhNRm67vHHHzfr4rHP08By+PBhb2llOriW7lc7oepdLsg+hAsAqEK6hQo/OyaF3hESOaaF1jzoRTzauBMtWrSQ3r17ywcffGAu9v4LvV2n/S/0jpWgIUD7cEyYMMGEnT59+oSCjaVhR4cE12YapZ1Qq6oNQWYiXABAFdIxVFgaAuwdIXqb6JgxY8xtrFqLoINq6dgUxcXFle7ysE0j+jy9fdXf7GFrRHSd/pmAIE0i1qJFi0zzigYMDT7aD6pXr15mfI277rrL/DEzpa9Lm3OQnQgXAJDhli5dasa00A6f9pbWAwcOmLs+jh07FvUuEmWbMZS/T4byj9IZpEnE0mCiw3trrYneCqv0tRw9etQ0uyxcuNC8Vq0NGTdunHTq1Im/kpqFuFsEWYW7RTJXOt8tAve0o6iGC/vHFLlrJLtQcwEAqHHar8OOIqo1Lsgu1Fwgq1BzkbmouQCyBzUXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAApwgXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAApwgXAADAKcIFAABwinABAACcIlwAAACnCBcAAMApwgUAAHCKcAEAAJwiXAAAAKcIFwAAwCnCBQAAcIpwAQAAnCJcAAAAp/LKK3jzNSovL092nrrsPXKnf0Ejs+1aOizUMt77zMV7B2QPwgWySoN6eXLtr94DZJT6dUWuXuf3FsgGhAtkFX3vj6z0HiCjdJsqCf/efvbZZ3LPPfd4jwCkC/pcAMhI169fl8LCQvM/gPRCuACQkV555RUpLS01/wNIL4QLABlHaysWLVpk5vV/ai+A9EK4AJBxtLbi008/NfP6P7UXQHohXADIKP5aC4vaCyC9EC4AZBR/rYVF7QWQXggXADJGtFoLi9oLIH0wzgWyCuNcZK4g41ycPn1aNm7c6D0SmTdvnhQVFXmPREaNGiUFBQXeIwC1hXCBrEK4yFzJDKLF7zqQnmgWAQAAThEuAACAU4QLAADgFOECAAA4RbgAAABOES4AAIBThAsAAOAU4QIAADhFuAAAAE4RLgAAgFOECwAA4BThAgAAOEW4AAAAThEuAACAU4QLAADgFOECAAA4RbgAAABOES4AAIBThAsAAOAU4QIAADhFuAAAAE4RLgAgi5w7d04aN24sI0aM8JYANY9wASAnXb9+Xe6//365++675dKlS95SWJwfpIJwAQAZ5NatWzJs2DCpW7euHD582FsKpBfCBcJcuHBBrl275j26jWpWAEAiCBcZRr+1vPjii+Zin5eXZ6Z+/frJ5s2bvRLJ279/vzRv3lyaNGlCNSgAIGmEiwyibaAPPPCALFiwQL74xS/KlClTZMKECbJ3715Tq7Bx40avZGratm0rDRo08B4BULt27ZLCwsJQqNdmiUWLFsWs6Zs7d66cP39epk+fHnpOp06dYjZl6BcHf9nISX/HZ86cKXfddZds3bpVbty4Id27dw+tj/b7H7nNePsHXCJcZJD169dLWVmZDB8+XE6dOiUrV66U1atXm2UjR470SiWvd+/eUl5eLkePHpX69et7SwHo79mAAQPkwIEDMnr0aBPsNYC/8MIL0qdPn0oBQ3344YfSuXNn83uq5QcNGiTHjx835SMv8PaLw/Lly6Vv376mfNeuXc26/Px883j8+PHyjW98w8w3atTIrLOv5emnn5aOHTuaZdbp06dNLWSQ/QOuES4yhH4DsU0fjz/+uPnfatGihfzsZz+TUaNGeUsQxO//6M0AcWhNxPPPPy9dunSRK1eumJCvF+yLFy/KjBkz5NixY7Jlyxav9B0bNmwwQUGfo+W3bdsmr732mqlx0NpHP/vFoaioSPbs2WPKHzx4UIYOHWrWT5o0yXyp6N+/v/zTP/2T+SKgoeO5554zZVesWCHt27c3ZS0NF0H3D7hGuMgQderUkTZt2ph5/dAISqtyBw8eHKoW1Un7aJw8edIrcUeuddz8xQGRYS+KbK34H4hFL9x6gZ41a1ZYjZ7+Tk6bNs3UImzatMlbeoeGkTfeeCPsOdqsouU1SPhrO/T3UcPCkCFDvCW3t6+BQsPA2bNnvaXBJbJ/wDXCRQaxF3391rFu3TozH4+tyt2+fXuo+lSDhfbR6NChg7M+GplMay/mvU7IQHRaY3jmzBkzr7UTU6dODZvmz59vgke0i3XLli0rNS82a9ZMGjZs6D2qXrW9f+Q2wkUG0arQOXPmmG8y48aNMx3KNGRE+waid35oVaq2C584cSJUlbt7925TPaomT57MXSEeQgaqos0cq1atCpvWrFnjrU2NNmno7/Xbb7/tLbnTFKo1Gq1bt/aWApmBcJFhXn75Zdm5c6epgbAhQ6s5I2sySkpKzP+vvvpqpbZY7dilbcVXr14120qF9oj3N7nU9qS6TQ02rfqFKR7GHzLokwFLL/CHDh0yHZ6jTal2grbNFfPmzTO/21or0qNHD3NXiDaVPPzww15JIDMQLjKQdurSGgi9Y8QfMvRCr2xVbrxvPDZwJNJ/IxrdZ7QP29qa1JGVwaYp/9kUr2RoT5FXZoo0/xtvAXKW7euUbL+HIPT39dlnnzVhX28t/81vfmNqRfTOjoULF5p+E0CmIVxksIKCAhMytAe4WrJkiWnm0A+r999/39wP36pVK7MuFi2H2zRUbHlZpGg8wQJ32L5OixcvjtoEqZ2mo3WQDsr+vur4MjpuxuXLl01Q/utf/yqzZ8+uVCNSE4EHSBXhIguMGTPGdN66efOmueNDP3z0/nr7OB4tl+sIFblNawy+/vWvS69evcIm7bOkYUL7OtlbTrUPk222mDhxommK007TWsuQrHr16pnaiY8//tiMS+Fv5tPXobWSkeHF1jzq2Bf6WvSOsOrqoF3V+QGiIVxkoaq+2diOYkprP3JV17aECtymzYM6QJZ/Ki0t9daKLF261HSE1oGt9G4rbbbQQKDjSGi5VMeY0T5SeteJDoSld3XZSWsvtD+V3lbqH/hKg43t3K2vRTtwV2enz6rODxApr9w2VNcwTeU7T132HrnTv6CR2XYtHVa10RoIbY/VQXP03ndLg4JWnS5btsx8AO3bt89Uo+qHjY7Ep9+0tCOav1On3pqq33T85ZXuQz889QPTdghVsZanI33vtT8FMo92sk309zYbftft76r+Xkf7/dJbyrWWQAfYsv2qgHRHzUUG0Y5e2v6rH6jt2rUzVZPar0KDhXbeLC4uDgUFW5WrVZo6poU2nWj1qfZK12ARWR5A7dIB7CLplwfbJJLLtYzIPISLDKEdM7XT1zPPPGNuWbPVlBoStIpUq1Qjb1fzV+Xae/T1OWPHjjXtx9zeBtS+bt26mT5TWkOhY9folwDbp0P7YNhayWHDhnnPANIfzSLIKjSLZK5cbRZRFy5cMH+07J133jFjZlg9e/aUJ598Ur71rW9Ry4iMQrhAViFcZK5cDhdAtqFZBAAAOEW4AAAAThEuAACAU4QLAADgFOECAAA4RbgAAABOES4AAIBThAsAAOAU4QIAADhFuAAAAE4RLgAAgFOECwAA4BThAgAAOEW4AAAAThEuAACAU4QLAADgFOECAAA4RbgAAABOES4AAIBThAsAAOAU4QIAADhFuAAAAE4RLgBkjNLSUiksLAxNyv9427ZtZhmA2pVXXsGbr1F5eXmy89Rl75E7/QsamW3X0mGhlul7f2Sl9wAZpdtUCfR7+9BDD5mQEalp06Zy7tw5qVevnrcEQG2h5gJARikqKvLmws2ePZtgAaQJwgWAjDJ8+HB58MEHvUe3aa3Fk08+6T0CUNsIFwAyTmTtBbUWQHohXADIOP7aC2otgPRDuACQkWztBbUWQPrhbhFklTqfy5N//3/eA2SUz1V81bn174n93j722GNSUlJCuADSDOECWaW6fq5y1Z//9H+k7OwZ+U+du3lLqs+Ado0T/r29fv06wQJIQ4QLZBXChVs3/vpXeeJvO8vlP1yUrxR0lBat20iLVm3kwYd7Vcx/VRp/oYlXMnXJhAsA6YlwgaxCuHDvX9YWy/9Y8JL3KJyGi5Ul7zoJGYQLIHvQoRNAXEMfnxgzPIz6+2ed1l4AyA6ECwBx5deta0JEpF4DviHfGveU9wgA7iBcAKhSZO2FBo4rf/g3+fSTMm8JANxBuABQJX/thc4vfX2r9BwwRJ4Z9ajsfvstsxwALMIFgEBs7cXkGf9gbk0d/ffPybwVP5VXl/2j/GjeTHNnCQAowgWAQLTGQsOEv5+FhoyVJXvl8qWLphZDx8QAAMIFgMCiDaZ1d8P/KP+4Yp08OmKUzBw/VLaXrPfWAMhVhAsATmiNxoL/+TPZ+OqP5b+/OI1mEiCHES4AOKOjeC7f+E5FsLguU7/5dZpJgBxFuADglDaTfH/pqzJq8rOmHwbNJEDuIVwAqBaDRowxtRjaTPLfZk42fwQNQG4gXACoNi1bt5GVb70r+XXrmVqMfz193FuDbHHu3Dlp3LixjBgxwltyx4ULF+TatWveo+ph99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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 902px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 451px; transform-origin: 407px 451px; vertical-align: baseline; "><div style="block-size: 107px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.5px; text-align: left; transform-origin: 384px 53.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 272.5px 8px; transform-origin: 272.5px 8px; unicode-bidi: normal; "><span style="">A falling-head permeameter is another device for measuring the hydraulic conductivity </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">K</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103px 8px; transform-origin: 103px 8px; unicode-bidi: normal; "><span style=""> of a soil sample. In this problem the sample is placed in a cylinder of length </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">L</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82px 8px; transform-origin: 82px 8px; unicode-bidi: normal; "><span style=""> and cross-sectional area </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="A_c" style="width: 16.5px; height: 20px;" width="16.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37px 8px; transform-origin: 37px 8px; unicode-bidi: normal; "><span style="">. Unlike the </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/55825-measure-the-hydraulic-conductivity-with-a-constant-head-permeameter"><span style=""><span style="text-decoration-line: underline; ">constant-head permeameter</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.5px 8px; transform-origin: 11.5px 8px; unicode-bidi: normal; "><span style="">, in which the water level in a standpipe (or tube) is kept constant, the falling-head permeameter has a water level in the tube (of cross-sectional area </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="A_t" style="width: 15.5px; height: 20px;" width="15.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147px 8px; transform-origin: 147px 8px; unicode-bidi: normal; "><span style="">) that falls. In other words, the head difference </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">δ</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145px 8px; transform-origin: 145px 8px; unicode-bidi: normal; "><span style="">, or the difference between the water levels in the tube and the outlet, decreases in time. </span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378px 8px; transform-origin: 378px 8px; unicode-bidi: normal; "><span style="">The hydraulic conductivity can be determined from a statement of conservation of mass (i.e., water volume) by equating the rate of change of volume in the tube to the flow out of the soil sample. The outflow can be computed with Darcy’s law, which states in general</span></span></div><div style="block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="Q = -K(dh/dx)A" style="width: 85px; height: 35px;" width="85" height="35"></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 246px 8px; transform-origin: 246px 8px; unicode-bidi: normal; "><span style="">Darcy’s law applies when a Reynolds number based on the specific discharge </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="v = Q/A" style="width: 57.5px; height: 18.5px;" width="57.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93px 8px; transform-origin: 93px 8px; unicode-bidi: normal; "><span style=""> and representative diameter </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">d</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; "><span style=""> of the soil grains is less than (approximately) 1—that is, </span></span></div><div style="block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="Re = vd/nu < 1" style="width: 79px; height: 35px;" width="79" height="35"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; "><span style="">where </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">ν</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 116px 8px; transform-origin: 116px 8px; unicode-bidi: normal; "><span style=""> is the kinematic viscosity of the fluid.</span></span></div><div style="block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 231.5px 8px; transform-origin: 231.5px 8px; unicode-bidi: normal; "><span style="">Derive and solve an ordinary differential equation for the head difference </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">δ</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132px 8px; transform-origin: 132px 8px; unicode-bidi: normal; "><span style="">. Then write a function that takes as input measurements of head difference as a function of time, as well as the soil’s porosity, diameter of the tube, and length and diameter of the cylinder holding the soil sample. The function should compute the hydraulic conductivity by fitting the solution to the ordinary differential equation to the data and using Darcy’s law regardless of its validity. Also return a flag indicating whether Darcy’s law is valid throughout the experiment; to assess the validity, relate the hydraulic conductivity to the representative grain diameter with the Kozeny-Carman equation, as described in the </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/55825-measure-the-hydraulic-conductivity-with-a-constant-head-permeameter"><span style=""><span style="text-decoration-line: underline; ">previous problem</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; "><span style="">. Use </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="g = 981 cm/s^2" style="width: 90.5px; height: 19.5px;" width="90.5" height="19.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="nu = 10^{-2} cm^2/s" style="width: 94px; height: 19.5px;" width="94" height="19.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 359px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 179.5px; text-align: left; transform-origin: 384px 179.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: middle;width: 401px;height: 359px" 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" alt="falling-head permeameter" data-image-state="image-loaded" width="401" height="359"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; "><span style=""></span></span></div></div></div>

A falling-head permeameter is another device for measuring the hydraulic conductivity  of a soil sample. In this problem the sample is placed in a cylinder of length  and cross-sectional area . Unlike the constant-head permeameter, in which the water level in a standpipe (or tube) is kept constant, the falling-head permeameter has a water level in the tube (of cross-sectional area ) that falls. In other words, the head difference , or the difference between the water levels in the tube and the outlet, decreases in time. 
The hydraulic conductivity can be determined from a statement of conservation of mass (i.e., water volume) by equating the rate of change of volume in the tube to the flow out of the soil sample. The outflow can be computed with Darcy’s law, which states in general

Darcy’s law applies when a Reynolds number based on the specific discharge  and representative diameter  of the soil grains is less than (approximately) 1—that is,

where  is the kinematic viscosity of the fluid.
Derive and solve an ordinary differential equation for the head difference . Then write a function that takes as input measurements of head difference as a function of time, as well as the soil’s porosity, diameter of the tube, and length and diameter of the cylinder holding the soil sample. The function should compute the hydraulic conductivity by fitting the solution to the ordinary differential equation to the data and using Darcy’s law regardless of its validity. Also return a flag indicating whether Darcy’s law is valid throughout the experiment; to assess the validity, relate the hydraulic conductivity to the representative grain diameter with the Kozeny-Carman equation, as described in the previous problem. Use  and .

Solve

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Last Solution submitted on Sep 23, 2023

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Problem 56205. Measure the hydraulic conductivity with a falling-head permeameter

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Problem 56205. Measure the hydraulic conductivity with a falling-head permeameter

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers3

Suggested Problems

More from this Author221

Problem Tags

Community Treasure Hunt

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