Problem 52070. Compute the effective conductivity of a heterogeneous aquifer

Created by ChrisR

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{"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>Slow flow through soil or another porous medium follows Darcy’s law, which relates the flow of water </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=\"Q\"/></w:customXmlPr><w:r><w:t>Q</w:t></w:r></w:customXml><w:r><w:t> to the gradient in piezometric head </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=\"dh/dx\"/></w:customXmlPr><w:r><w:t>dh/dx</w:t></w:r></w:customXml><w:r><w:t> by</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>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=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> is the conductivity of the soil 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=\"A\"/></w:customXmlPr><w:r><w:t>A</w:t></w:r></w:customXml><w:r><w:t> is the flow area (i.e., the aquifer thickness multiplied by the aquifer width). The flow </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Q</w:t></w:r></w:customXml><w:r><w:t> (and 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>) is analogous to current in an electrical circuit, and the change in head is analogous to the voltage drop. </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>Some aquifers, or </w:t></w:r><w:r><w:t>underground water-bearing formations, consist of multiple soil units in series or parallel, as shown below. The analysis of these aquifers is often simplified by computing the effective conductivity—that is, treating the aquifer as homogeneous with the single value of </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> set such that the aquifer produces the same flow under the same total change in head. </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>For example, suppose the soil units have equal size in the two aquifers shown below and flow in both cases is from left to right. If </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=\"K1\"/></w:customXmlPr><w:r><w:t>K_1</w:t></w:r></w:customXml><w:r><w:t> is 2 m/d (meters/day) 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=\"K2\"/></w:customXmlPr><w:r><w:t>K_2</w:t></w:r></w:customXml><w:r><w:t> is 6 m/d, then the effective conductivity is 4 m/d in the parallel case and 3 m/d in the series case. </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>Write a function that computes the effective conductivity of an aquifer whose conductivity (in m/d) is specified as a matrix. Flow is always left to right, and heterogeneous cases involve either units in series or units in parallel. In the series case, the conductivity will be constant in each column, and in the parallel case, the conductivity will be constant in each row. The relative size of each unit is indicated by the number of rows or columns. </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=\"203\"/><w:attr w:name=\"width\" w:val=\"382\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"Aquifers in series and parallel\"/><w:attr w:name=\"relationshipId\" 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","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<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: 634px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 317px; transform-origin: 407px 317px; vertical-align: baseline; "><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: 311.833px 7.91667px; transform-origin: 311.833px 7.91667px; unicode-bidi: normal; "><span style="">Slow flow through soil or another porous medium follows Darcy’s law, which relates the flow of water </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);">Q</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: 56.0083px 7.91667px; transform-origin: 56.0083px 7.91667px; unicode-bidi: normal; "><span style=""> to the gradient in piezometric head </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="dh/dx" style="width: 39.5px; height: 19px;" width="39.5" height="19"></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: 9.33333px 7.91667px; transform-origin: 9.33333px 7.91667px; unicode-bidi: normal; "><span style=""> by</span></span></div><div style="block-size: 34.8333px; 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 17.4167px; text-align: left; transform-origin: 384px 17.4167px; 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: 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: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; 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);">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: 104.625px 7.91667px; transform-origin: 104.625px 7.91667px; unicode-bidi: normal; "><span style=""> is the conductivity of the soil and </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);">A</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: 244.283px 7.91667px; transform-origin: 244.283px 7.91667px; unicode-bidi: normal; "><span style=""> is the flow area (i.e., the aquifer thickness multiplied by the aquifer width). The flow </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);">Q</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: 87.1333px 7.91667px; transform-origin: 87.1333px 7.91667px; unicode-bidi: normal; "><span style=""> (and 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: 19px;" width="57.5" height="19"></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: 227.95px 7.91667px; transform-origin: 227.95px 7.91667px; unicode-bidi: normal; "><span style="">) is analogous to current in an electrical circuit, and the change in head is analogous to the voltage drop. </span></span></div><div style="block-size: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 57.175px 7.91667px; transform-origin: 57.175px 7.91667px; unicode-bidi: normal; "><span style="">Some aquifers, or </span></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: 325.6px 7.91667px; transform-origin: 325.6px 7.91667px; unicode-bidi: normal; "><span style="">underground water-bearing formations, consist of multiple soil units in series or parallel, as shown below. The analysis of these aquifers is often simplified by computing the effective conductivity—that is, treating the aquifer as homogeneous with the single value of </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: 252.433px 7.91667px; transform-origin: 252.433px 7.91667px; unicode-bidi: normal; "><span style=""> set such that the aquifer produces the same flow under the same total change in head. </span></span></div><div style="block-size: 63.25px; 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 31.625px; text-align: left; transform-origin: 384px 31.625px; 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: 376.917px 7.91667px; transform-origin: 376.917px 7.91667px; unicode-bidi: normal; "><span style="">For example, suppose the soil units have equal size in the two aquifers shown below and flow in both cases is from left to right. If </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAoCAYAAACWwljjAAABUUlEQVRYhe2WUZWDMBBFr4c4wAAGogAFOFgHdVALaEACHrCABizsfsCcTtIAIW1gP/LO4aNhTnqZmbwMFBUV3SsLPHaeHxW7F2e+CWWAJ/CrngmoArGtiplXmCyyHlAIBqDjBVzngoGlNAIzHMAMfLlEIQ0KyC+DAfqNd9mky2XVeg2MLP3SXAXT4DaqlEPKOHFBibT0CetxS9RdCSIaFVDHkpEZ14Muk8HtnyMfyi7f6Cb1+3kHkHiLlEsb5MwNWdIZadc17Un9yf2adc8ki6hx+6baWI+5IizuaU0C0tfF5L3TpRwj9hJoyW4SkHhNqIErlh46+8UfAemyhDZ44GYwxq2TgfxxI/RnBjdLMUaZDKQbcK9HdFyMDSQBWdwvnwmfJH2n6Vlp79SdBmrZnovbyDg/9iOg3CpAR/p3QDJb3Q5U824lDRePvkVFRUUx+gOBIab+pKxL6AAAAABJRU5ErkJggg==" alt="K1" style="width: 18px; height: 20px;" width="18" 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: 83.225px 7.91667px; transform-origin: 83.225px 7.91667px; unicode-bidi: normal; "><span style=""> is 2 m/d (meters/day) and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="K2" style="width: 18px; height: 20px;" width="18" 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: 251.125px 7.91667px; transform-origin: 251.125px 7.91667px; unicode-bidi: normal; "><span style=""> is 6 m/d, then the effective conductivity is 4 m/d in the parallel case and 3 m/d in the series case. </span></span></div><div style="block-size: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 376.242px 7.91667px; transform-origin: 376.242px 7.91667px; unicode-bidi: normal; "><span style="">Write a function that computes the effective conductivity of an aquifer whose conductivity (in m/d) is specified as a matrix. Flow is always left to right, and heterogeneous cases involve either units in series or units in parallel. In the series case, the conductivity will be constant in each column, and in the parallel case, the conductivity will be constant in each row. The relative size of each unit is indicated by the number of rows or columns. </span></span></div><div style="block-size: 208.917px; 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 104.458px; text-align: left; transform-origin: 384px 104.458px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: baseline;width: 382px;height: 203px" 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" alt="Aquifers in series and parallel" data-image-state="image-loaded" width="382" height="203"></div></div></div>

Slow flow through soil or another porous medium follows Darcy’s law, which relates the flow of water  to the gradient in piezometric head  by

where  is the conductivity of the soil and  is the flow area (i.e., the aquifer thickness multiplied by the aquifer width). The flow  (and the specific discharge ) is analogous to current in an electrical circuit, and the change in head is analogous to the voltage drop. 
Some aquifers, or underground water-bearing formations, consist of multiple soil units in series or parallel, as shown below. The analysis of these aquifers is often simplified by computing the effective conductivity—that is, treating the aquifer as homogeneous with the single value of  set such that the aquifer produces the same flow under the same total change in head. 
For example, suppose the soil units have equal size in the two aquifers shown below and flow in both cases is from left to right. If  is 2 m/d (meters/day) and  is 6 m/d, then the effective conductivity is 4 m/d in the parallel case and 3 m/d in the series case. 
Write a function that computes the effective conductivity of an aquifer whose conductivity (in m/d) is specified as a matrix. Flow is always left to right, and heterogeneous cases involve either units in series or units in parallel. In the series case, the conductivity will be constant in each column, and in the parallel case, the conductivity will be constant in each row. The relative size of each unit is indicated by the number of rows or columns.

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Problem 52070. Compute the effective conductivity of a heterogeneous aquifer

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