Problem 57532. Compute steady drawdown in a confined aquifer

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 well extracting water from a confined aquifer will lower the piezometric head and create a cone of depression. In steady state, if the distance </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=\"r\"/></w:customXmlPr><w:r><w:t>r</w:t></w:r></w:customXml><w:r><w:t> to the point where the drawdown is wanted is smaller than the radius of influence </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=\"R\"/></w:customXmlPr><w:r><w:t>R</w:t></w:r></w:customXml><w:r><w:t>, then the drawdown </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=\"s\"/></w:customXmlPr><w:r><w:t>s</w:t></w:r></w:customXml><w:r><w:t> of a well pumping at rate </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=\"Q0\"/></w:customXmlPr><w:r><w:t>Q_0</w:t></w:r></w:customXml><w:r><w:t> in a confined aquifer of transmissivity </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=\"T\"/></w:customXmlPr><w:r><w:t>T</w:t></w:r></w:customXml><w:r><w:t> 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=\"s = (Q0/2piT) ln(R/r)\"/></w:customXmlPr><w:r><w:t>s = \\frac{Q_0}{2\\pi T} \\ln\\left(\\frac{R}{r}\\right)</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>If the distance </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=\"r\"/></w:customXmlPr><w:r><w:t>r</w:t></w:r></w:customXml><w:r><w:t> is greater than the radius of influence, then the drawdown is zero. If multiple wells are pumping, the drawdown at the requested point is the sum of the drawdowns from the individual wells. </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>Boundaries, such as no-flow and constant-head boundaries, can be modeled using the method of images, as described in </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/57497-locate-image-wells\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 57497</w:t></w:r></w:hyperlink><w:r><w:t>. Each real well will have a corresponding image, and the drawdown will be the sum of the drawdowns from all wells—real and image. Recall from the previous problem that for no-flow boundaries, the image wells pump in the same sense as the real wells, whereas for constant-head boundaries, the image wells pump in the opposite sense as the real wells. </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 to compute steady-state drawdown in a confined aquifer. Input to the function will be the </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=\"x\"/></w:customXmlPr><w:r><w:t>x</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=\"y\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t>-coordinates of the points where drawdown is requested, the </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=\"x\"/></w:customXmlPr><w:r><w:t>x</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=\"y\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t>-coordinates of the real wells, the pumping rates and radii of influence of the wells, and the transmissivity of the aquifer. If a boundary is present, it will be specified by two pairs 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=\"x\"/></w:customXmlPr><w:r><w:t>x</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=\"y\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t>-coordinates as well as a character string (‘NF’ for no-flow, ‘CH’ for constant-head). </w:t></w:r><w:r><w:t> </w:t></w:r></w:p></w:body></w:document>","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: 369.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 184.95px; transform-origin: 407px 184.95px; vertical-align: baseline; "><div style="block-size: 64px; 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 32px; text-align: left; transform-origin: 384px 32px; 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: 376.925px 8px; transform-origin: 376.925px 8px; unicode-bidi: normal; ">A well extracting water from a confined aquifer will lower the piezometric head and create a cone of depression. In steady state, if the distance r<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: 254.008px 8px; transform-origin: 254.008px 8px; unicode-bidi: normal; "> to the point where the drawdown is wanted is smaller than the radius of influence R<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: 31.1083px 8px; transform-origin: 31.1083px 8px; unicode-bidi: normal; ">, then the drawdown s<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: 80.125px 8px; transform-origin: 80.125px 8px; unicode-bidi: normal; "> of a well pumping at rate <img src="data:image/png;base64,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" alt="Q0" style="width: 19px; height: 20px;" width="19" height="20"><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: 118.633px 8px; transform-origin: 118.633px 8px; unicode-bidi: normal; "> in a confined aquifer of transmissivity T<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: 8.94167px 8px; transform-origin: 8.94167px 8px; unicode-bidi: normal; "> is </div><div style="block-size: 38.9px; 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 19.45px; text-align: left; transform-origin: 384px 19.45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="s = (Q0/2piT) ln(R/r)" style="width: 97px; height: 39px;" width="97" height="39"></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: 45.5px 8px; transform-origin: 45.5px 8px; unicode-bidi: normal; ">If the distance r<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: 310.4px 8px; transform-origin: 310.4px 8px; unicode-bidi: normal; "> is greater than the radius of influence, then the drawdown is zero. If multiple wells are pumping, the drawdown at the requested point is the sum of the drawdowns from the individual wells. </div><div style="block-size: 105px; 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 52.5px; text-align: left; transform-origin: 384px 52.5px; 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: 378.892px 8px; transform-origin: 378.892px 8px; unicode-bidi: normal; ">Boundaries, such as no-flow and constant-head boundaries, can be modeled using the method of images, as described in <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/57497-locate-image-wells">Cody Problem 57497</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: 317.433px 8px; transform-origin: 317.433px 8px; unicode-bidi: normal; ">. Each real well will have a corresponding image, and the drawdown will be the sum of the drawdowns from all wells—real and image. Recall from the previous problem that for no-flow boundaries, the image wells pump in the same sense as the real wells, whereas for constant-head boundaries, the image wells pump in the opposite sense as the real wells. </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: 323.875px 8px; transform-origin: 323.875px 8px; unicode-bidi: normal; ">Write a function to compute steady-state drawdown in a confined aquifer. Input to the function will be the x<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: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; ">- and y<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: 189.05px 8px; transform-origin: 189.05px 8px; unicode-bidi: normal; ">-coordinates of the points where drawdown is requested, the x<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: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; ">- and y<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: 150.925px 8px; transform-origin: 150.925px 8px; unicode-bidi: normal; ">-coordinates of the real wells, the pumping rates and radii of influence of the wells, and the transmissivity of the aquifer. If a boundary is present, it will be specified by two pairs of x<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: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; ">- and y<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: 257.883px 8px; transform-origin: 257.883px 8px; unicode-bidi: normal; ">-coordinates as well as a character string (‘NF’ for no-flow, ‘CH’ for constant-head). <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: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "> </div></div></div>

A well extracting water from a confined aquifer will lower the piezometric head and create a cone of depression. In steady state, if the distance  to the point where the drawdown is wanted is smaller than the radius of influence , then the drawdown  of a well pumping at rate  in a confined aquifer of transmissivity  is

If the distance  is greater than the radius of influence, then the drawdown is zero. If multiple wells are pumping, the drawdown at the requested point is the sum of the drawdowns from the individual wells. 
Boundaries, such as no-flow and constant-head boundaries, can be modeled using the method of images, as described in Cody Problem 57497. Each real well will have a corresponding image, and the drawdown will be the sum of the drawdowns from all wells—real and image. Recall from the previous problem that for no-flow boundaries, the image wells pump in the same sense as the real wells, whereas for constant-head boundaries, the image wells pump in the opposite sense as the real wells. 
Write a function to compute steady-state drawdown in a confined aquifer. Input to the function will be the - and -coordinates of the points where drawdown is requested, the - and -coordinates of the real wells, the pumping rates and radii of influence of the wells, and the transmissivity of the aquifer. If a boundary is present, it will be specified by two pairs of - and -coordinates as well as a character string (‘NF’ for no-flow, ‘CH’ for constant-head).

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Last Solution submitted on Feb 04, 2023

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William on 8 Jan 2023

My results in test 5 are s = 0.5558 0.8738 -0.2454. Also, in test 8 my results for case 3 are very slightly outside of the 1e-3 tolerance from the test values.

ChrisR on 8 Jan 2023

OK, I will check. I computed test 8 by hand, and that matched the results from my code. I might loosen the tolerance there.

Did you see that I corrected the description to mention that if r > R, the drawdown is zero? I assume you did because the other tests seem to be working for you.

William on 8 Jan 2023

No, I didn't see the correction to the description, and that was apparently the reason for the differences. My results now all agree, including the tolerance issue on the last problem. I will submet my solution after I clean it up a bit. Thanks for the tip!

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Problem 57532. Compute steady drawdown in a confined aquifer

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