Problem 53975. Compute the effective conductivity of more heterogeneous aquifers

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:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/52070\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 52070</w:t></w:r></w:hyperlink><w:r><w:t> asked for a function to compute the effective hydraulic conductivity of a heterogeneous aquifer—or the single </w:t></w:r><w:r><w:t>value of conductivity set such that the aquifer produces the same flow under the same total change in head</w:t></w:r><w:r><w:t>. In that problem, the aquifer had soil units either in series only or in parallel only. </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 the effective conductivity for two-dimensional flow in an aquifer with a more complicated distribution of conductivity. Flow is left to right, or to the east, as in the figure below. No flow occurs across the north and south boundaries. Assume the head difference is small enough that Darcy’s law applies. Use the conductivity specified on the equally-spaced grid provided. </w:t></w:r><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:r><w:t>For example, if in the aquifer below </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> = 0.1 m/d, </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> = 0.2 m/d, </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=\"K3\"/></w:customXmlPr><w:r><w:t>K_3</w:t></w:r></w:customXml><w:r><w:t> = 0.01 m/d, 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=\"K4\"/></w:customXmlPr><w:r><w:t>K_4</w:t></w:r></w:customXml><w:r><w:t> = 20 m/d, then the effective conductivity is 0.092 m/d. </w:t></w:r><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:r><w:rPr><w:b/></w:rPr><w:t>Hint</w:t></w:r><w:r><w:t>: The simple formulas that work for soil units either in series only or in parallel only will not work for these more complicated distributions because two-dimensional flow violates assumptions behind the formulas. In this problem, compute the effective conductivity directly from the definition. Darcy's law yields the specific discharges (or flow per unit cross-sectional area) of </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=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"vx = -Kdh/dx\"/></w:customXmlPr><w:r><w:t>v_x = -K\\frac{\\partial h}{\\partial 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=\"v_y = -Kdh/dy\"/></w:customXmlPr><w:r><w:t>v_y = -K\\frac{\\partial h}{\\partial y}</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>Then conservation of mass leads to </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=\"d/dx(K dh/dx) + d/dy(K dh/dy) = 0\"/></w:customXmlPr><w:r><w:t>\\frac{\\partial}{\\partial x} \\left(K \\frac{\\partial h}{\\partial x}\\right) + \\frac{\\partial}{\\partial y} \\left(K \\frac{\\partial h}{\\partial y}\\right) = 0</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>Solve this equation for the head </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>h</w:t></w:r></w:customXml><w:r><w:t>, compute the flow through the aquifer, and get the effective conductivity from the definition.</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=\"241\"/><w:attr w:name=\"width\" w:val=\"513\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" 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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: 725.117px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 362.558px; transform-origin: 407px 362.558px; vertical-align: baseline; "><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; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/52070"><span style=""><span style="text-decoration-line: underline; ">Cody Problem 52070</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: 318.25px 7.79167px; transform-origin: 318.25px 7.79167px; unicode-bidi: normal; "><span style=""> asked for a function to compute the effective hydraulic conductivity of a heterogeneous aquifer—or the single </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: 331.017px 7.79167px; transform-origin: 331.017px 7.79167px; unicode-bidi: normal; "><span style="">value of conductivity set such that the aquifer produces the same flow under the same total change in head</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: 25.2667px 7.79167px; transform-origin: 25.2667px 7.79167px; unicode-bidi: normal; "><span style="">. In that problem, the aquifer had soil units either in series only or in parallel only. </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: 359.933px 7.79167px; transform-origin: 359.933px 7.79167px; unicode-bidi: normal; "><span style="">Write a function to compute the effective conductivity for two-dimensional flow in an aquifer with a more complicated distribution of conductivity. Flow is left to right, or to the east, as in the figure below. No flow occurs across the north and south boundaries. Assume the head difference is small enough that Darcy’s law applies. Use the conductivity specified on the equally-spaced grid provided. </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: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; "><span style=""> </span></span></div><div style="block-size: 42.8167px; 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 21.4083px; text-align: left; transform-origin: 384px 21.4083px; 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: 110.858px 7.79167px; transform-origin: 110.858px 7.79167px; unicode-bidi: normal; "><span style="">For example, if in the aquifer below </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAoCAYAAACWwljjAAACSUlEQVRYR+1XOy8FQRS+9yd41AqPmoJIhEKDUCoQeq9OQVATNBKJV0Lr0UrIVdMIBYlE4VHQesQv4PuSmeRkMrtzdvdGrmQ3Odkx5/X5ztkzc4uFCnuKFYankAMKVSRnKCtDnQjQERPkG7pdo5+NsduH7iMEhnpNyWpgNwNZEAGfse6BvDhJhvH3odn7wnsFsqYBYm00gGhLpi5E4AYPGKq3IJMQAh6E3CUBo2WIduOQHRP8HO8+TyILhvpRiKpEbhwtQyU49hrnOacMLOkmZAji6pISpOohBv0RkbuwvjR/N+O9B2EJxyBniRE4DhqG+uFzavzYqE2mHLaM7Jf2tCVKU7IlONkv7BjraVGibaynsrIi/TUMXcOh1TgRAD/3asg8xM6gsmEKAWLDvnuyRc2hzMBCgNxB92kamImXIYuZESRsajtb6MZyHUDsgGSDs5TutM6EMcTQk2BkBOsjiJxJbHKymOSh/QPEO8XjAHHG3IpM9rhw91uigjsoefysG1YH8PbOrDhA8rhgEzeKBLKUN9hvC1DEWG+QDcN4KkAsD48DPm4D12OPQKqMPjKBA9SWOxUgeVz4AvD+s2oSaqd1akDudaMWid3TmzPqUbA0gXVoUKYGJI+LuB6RdpoxkAoQ2TkR/zkTdXu+JHntsK3CuxCvIFEXs8SAOCPqIr6YV+yz0fnE2VEvbWW4xIAisJRtOwcUojJnKI4hObciZ1botA+VQKvngWx/tVifeyyu3IH7V4C0wNU/g9QBsxrmDIUYzBn6dwz9AvmpfSn6O8EFAAAAAElFTkSuQmCC" 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: 35.1917px 7.79167px; transform-origin: 35.1917px 7.79167px; unicode-bidi: normal; "><span style=""> = 0.1 m/d, </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: 35.1917px 7.79167px; transform-origin: 35.1917px 7.79167px; unicode-bidi: normal; "><span style=""> = 0.2 m/d, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="K3" 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: 52.7px 7.79167px; transform-origin: 52.7px 7.79167px; unicode-bidi: normal; "><span style=""> = 0.01 m/d, and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="K4" 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: 88.35px 7.79167px; transform-origin: 88.35px 7.79167px; unicode-bidi: normal; "><span style=""> = 20 m/d, then the effective conductivity is 0.092 m/d. </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: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; "><span style=""> </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: 13.6083px 7.79167px; transform-origin: 13.6083px 7.79167px; unicode-bidi: normal; "><span style="font-weight: 700; ">Hint</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: 342.683px 7.79167px; transform-origin: 342.683px 7.79167px; unicode-bidi: normal; "><span style="">: The simple formulas that work for soil units either in series only or in parallel only will not work for these more complicated distributions because two-dimensional flow violates assumptions behind the formulas. In this problem, compute the effective conductivity directly from the definition. Darcy's law yields the specific discharges (or flow per unit cross-sectional area) of </span></span></div><div style="block-size: 34.9167px; 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 17.4583px; text-align: left; transform-origin: 384px 17.4583px; 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="vx = -Kdh/dx" style="width: 72px; height: 35px;" width="72" height="35"></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: 23.325px 7.79167px; transform-origin: 23.325px 7.79167px; unicode-bidi: normal; "><span style="">   and   </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAABGCAYAAADM43a0AAAI80lEQVR4Xu2dOassRRTH3/sA7pqoGLgEYqC4I2ogiFsiqKiYXFDcEUR5ggaPhwu4JeIKCi9x10TcDQxcwBUFjVwQAxN3/AB6ftJnOFNTXVXTXT13uuc0HObOdFV11al/nzpb1d25wy/nQE8O7OxZ36s7B3Y4iBwEvTngIOrNQm/AQeQY6M0BB1FvFnoDDqLpYuAgGdq5Qkc0Q/xZPt8T+j0xZOocK3SG0IlCjwp9mGORgyjHofHef1y6/qfQ30L7CV0udJTQ9UJPRYZ1hfx2ttAN5h7lf8yxwEGU49B07iNl3hY6OQEkRkuZ84R+EDq6ZPgOohIuTafMkQ04kFCAKSZl/m2Ge5983lUydAdRCZemVeazBkBXyucLwdAulO9vNL+dJZ9ZfYiyDqJpAaRkNPdKoTuFXhRCD7KX3kNSHSOUUsJn9RxEJWyfVpldMpz7hT4XOiUYmkqpd+T380uH7SAq5dQ4yx0v3T5dCOtMTfyrGxAhbQ40w0Lx/q35bi046yr4Ru6/GbLCQTROcOR6faYU2CuEiY7EeVnoHCGsre+FsL5CELG0Pd80fIJ8fi1k29FnXhQCyUGUm47x3bdgCCdclytGFYIIvxI+Iv2ddu4RukXoEyOlFqw2B9H4QJLqcQpA1FN9iL9DxRoJheR6Quh1oUeEThNCubZW24JV5yCaDojQXb4TOiACEB2lBRFgubG5of4jvt4hdK3QJUIsaVxqtfG3LnUzzjmIhgcRE3SpEDpJeP0lP3wp9EzzxmvogTr2+kK+vGQmNdZrC5C2cIUuWdS3EuU6+f5k0yg61MNC1oeky2C4BP5fxUE0PIj0CfZt5zcmi3hWymvMpCEtQqdgrNe6HKXMcy1D/YMb4PI37dMXLiuh+G6ttvCeg2h1+Jk9SUMK/NDmEVZpAciuEdIlJdVdC9C2ACvm/ldNI6GP6A/5nWUwFg5J6kO055JodUiyk0FwU5VW2wMFEErvTUZS5HqJKf5BUygHTopZq83WjcXLsl5sB1Fueurdt8ppaBmxZDwrhP+mTZKkemKBsKD4SkUrqcLlzupSsbqqD2mf6SvXLCTiIKoHklxL1kdjgaIOPepvCRUFPSMP06UyBgTVeWLLVSr1w+pDKOIktQF2/EezfjqIclNf536oVOtEqxRAOqBAZxPAEt3RpTCUZNaJGKZ/WJDEljK7BGPx3S6EjjaX1BaCSNMjj5OCxFswS2ODQzSfFCKyDr8n2Yp1Aqo+pMtXcd5OhjM26QxfDxf+HnUgPhQBaS71wy51tBddakMQ8cYcKnSBEOkCXLG8E6vNF6cMFMIDhm8Vlk0Vg5Ellk2FR2WbsP4ZLCMmFmsoxttsY5kC8E/zqj+Wv3+JgEebYCkln5rrgUi7ABMfFwLllbZ2UsuZAiX2pmAuvi9EmZiV0YcRIfq7thXTDbq21bee9c/YtoYAUd++Ll0/BSJVxmJ5JzwIhQyPa5jYtHQnggoqDfu201VB7fvcsL71z3APh50mw0c9wLU7MHR7KRCpRIhlualCtk5vex9eMdEkavW5CE3EcpLDkAKJYFYysezGlpI+fVlp3RSIrDJoXeR0EICxL6m2FFrp4M3DrJ+lax9i6aa0ZUMKqhpY3qaS5rv2ZaX1UiCyjLUeTtWH2nYLrHQAlR6mVmmf5toUWBvqsHxU/wzPrGWh9el/57pdQITTbLfQQppk517MV5ySdWZfxFAtCKXfaFWDnLNR3yJ9g3RXZdF+pI6gmpJ1ZkMdsei6XeqWSo4PeHtIR153rfarrVgKIsQt23FXoQdNyTqzoY6YAh16shfylwtmmTyjywrK1SzC0n2YNlgKIrysmPo5RRoRjbf7WyE1sTUpq9VZVXN0a9SWDSnQrZLoeps7JTUsov04BFd5AaKrlgVRKoFK28KUxbMZ7mlSE3fUymOHGbIWGNVDC1ebDHWj0Zn8OUmER/pdoWVyW1SEW0URXeonoVH7QwqBhOQ9VYgxE9rQCyfjc0JWQhNiullIswopiwJO2bdM2cJHb0+xHIhgyLKRZZU89o2CoUR+1yWWNSS3czpdCKJUX9bF657kVw5EXZityqJaG5pwVbwtt8tDvc72cWAIEDEa3Pps0SXCf3cgxrdvtJv1ZHSyi4X2b4ZdsmOkE4eGApGmPhCl3hJyKdRpejpXQvqz555UEK7DhW4T0tyiqn6+oUCklgmuAbsJrjNXvGJvDtiktaopKEOBSM3WTTPre8/0wA1oJmPxUXol/RkKRARpXxUqOvOvpKNephoHNJRVfBJa7slDgYgI9Tqlp+b4sEn3NXugy9akKJ9qgYj1FuUNPxBBx4+Ehoryb9KEDzFWBVF0S3SXB9YCkVpj+IY4ksQB1GU26tZBpWAzpB4kgYnPwRHMFb/bJDqNeRK2IsiOzhRacJr9SUr0XASjFog0qWsUHta6c7V2rTEXjwkRSgEMDwoRECe8QiIh/rvw+BlAtK/QnqYMgwrzm2ya71x8rxaI1o6TG9oha8bHMgJ0Bw/siS1nNjUldl/zo+b0KQfRtNCmwe82E96m5LYp1nbLtT0YVDmFdTcnpRxE0wGRTT1pM98tiNoS4GIHgCqXoq4bB9F0QKTbkFKJbXY5a8tvssl0YW4TSnl2L/50WLpZI7EbJNtCGrZMLp9bl0ULSPQlcssWdjy7JJoG2Kzl1LZrxG4ayDkaYwdbRaUQ7HMQTQNEdrdyTBm2J8uWbN0OTwuBS3tjUshBNA0AMQqVRG0AsduwSiL44eFWpJG07jV0STQNIKm+EwOR7ljGwbhMVoU9dvhTqduaE+YgmgaIGIWa71bS4Il+TWhZANFe6oS1Oa45iKYDIs1lZ6cJkoPvhDmwxObOWCwcsi6B2S1MDqJCjo6oGMvXPkL/CPXZXaNHKmZTmx1EI0LHCrsKEJ8Wajvx35ezFU7GGB+lQdxbpfNFWRkuicY4zXX7jNThYulTACGF5o4ZTj3SQVR3QsbWWhgKAURLAYgBO4jGNu11+0s8jPgYLgDSR/gvi0tnpTqI6k7KGFvrnZXqIBrjtK9Znx1EazYhY+yOg2iMs7Zmff4Pfu7uVrpjiMAAAAAASUVORK5CYII=" alt="v_y = -Kdh/dy" style="width: 72.5px; height: 35px;" width="72.5" 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: 112.408px 7.79167px; transform-origin: 112.408px 7.79167px; unicode-bidi: normal; "><span style="">Then conservation of mass leads to </span></span></div><div style="block-size: 34.9167px; 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 17.4583px; text-align: left; transform-origin: 384px 17.4583px; 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,iVBORw0KGgoAAAANSUhEUgAAAVsAAABGCAYAAABi8DccAAAWjElEQVR4Xu2dW+h3TVXH9TYwU/MmRMQMDBXD9FXKhAJTE6RQS9GLBxRLjcA001Ii8oCml5qGwnvhqYMUUmZ54YUHPJQUJAapvIh1ox0VvLT1kf19W888M3uO+/jfG4bf8/z37Dl8Z8131qxZM3Pf+1zPhcCFwIXAhcDiCNx38RyuDC4ELgQuBC4E7nOR7SUEFwIXAhcCKyBwke0KIF9ZXAhcCFwIXGR7ycCFwIXAhcAKCJyRbJ9suD3EwodWwG8vWfyqFeSzFv5pLwW64eV4uNX/ORbeeoNweJDV9UUW3mvhP25QvYurOoJsAfmpFh465fp1+/34RoA/z/L9oIXn74xsf8HK8+gJn/+Z8PlappUYNB5l4bETib57Jj71fqeFn72hhLsnGaS9PjHJ38uKe+LyEVswYtB45CS7P2e/L8j0a2TwCRaevlH/Xx7FjhxGkC0A/5cFSOT+Fn7Fwo9a+DULcwTRUezop3slWrROOuA9U6mfa7+Pt/BHFl4fEUpI9hkWXmrhAdM3JYMH+bzZwk0k3L3I4F6JFjGqxejV9g0E+7RJBr9qv48o6LTMKOn/F+EGYI0g2xB/RtCPTYSyFuFKyCGw3y0QiK2jIPiQ6Z9YYJCIPQj7W6YXCG9OE1aHIj3IvCT+1jgslf8WMqg8qRPa3d6fUoy+N1WEvlWiqR8Nh9XaaQmypfBMPxgJ0XiX7vg07r9O+R1ByNW4X7F/QKLPtPDRSIszYKFV/H1F570E/f+BXFMGyZX2umsFeR9JDjmMmGV9csowJaex8hxN+RmJaTKtpciWDL8wCV7JFLinskxbMF38hIUjLRC90cr7OxZiGgOk+a0JlN+235qFFgT9Hy28ycIRtPyets99u5YMYsJ5l4W1ZnK5ete8n8NIMorS9GMWaha+hEkNSdeU+3BxlyRbNdTcVLkXMBae/tpCLSH15jvie5UdQX5gkKDe8eefsfCpygyFfcu3lVntOvoaMoh2yOzj8xawUx7tmcNIRPy3jXXje2ZvtUR9NAyLyrsk2crmWDMNLir0FAnt73MTUR2xMTVFi5laejQK4DmqaaWm/UviLi2DlEEzq6MObCmMemZXahvJ+DXLMkRGkS1T1ydZwBtBrl/43LHAE9PcSjpKLo6E5AhaLdoP9jzc4+T69SP2b9nDQhOINIpwViAXMtLIeXoIn6XNOLl2Wuv9FjIoMmnV/NbCRvnUYORnV14+kWVcPenr1DtnutPaQ+ki79qYrJZfL9kibHdbAEg02D+zgLsILiIsALHAswTZSnMDqD1rtQgmngfgwIIh5Ik7F54I2Gr55QmFmbg8sgH6dCQcOW1BGP3n1B767my/W8kgOIpI9q7VtmAkjxnff+VeKJfEkgVwDUil3gxnk89769NDtvJrJbHQCC7NjHdLkK3y3nMDakUWwQyJUWYCNYQnW48rf+f5sIU/tsBCmbAtMc+ow5xVu91SBrWSX+p/uhWJtGIkbxnNrrRhgcVoNjqwVhLr+7F6Kq0brd22ku1cAwK29xFdYoFMjbdXD4Q5ogUf71ITDkYiSDrxz1v4Owu3LLBI5u1oJQONPBOOMs2tIaStZVDttGcPhFaMNJDQHgzUj7EAUcon3CsLJQQqz4QjmPxqZLAqbgvZanqKxpYiUk+2JaRQU2gRyJ41Ck0vKeMTLYQuM55sw3poIAE3/IbfbkHnPHg7WmknP6NWsbUMIq+YZ+gDJWRTI9+j4vZg5EkaOfstC373mGZXpbPWo8wCRmEfTaeFbEt2NmnU18g48lAYjao5m2WswgjRUyzQ+OHzD/aHT1tgg4EOEsH+7J//tv+w731uccoTaWr67oXZD0YaSMgTMwFC7XfteI2iVKvXN6XkvKjADUp8axnUoFdiygmrnJIt4iFfX7Sgw1xS8oqs/qmFucWpHozkYaG1g2e7vPzsqmbWKoIuldtBorJYMmHb/IXl9OcWkr7ILWQrTWluaqo41PSH5wrQAIXS7lmU8IMBRUgRUWgOYRNCbhusX1hILd75/L29W9MtyhTT3CWwNVp9DzE0NM8qn2wtg2q/nmmxn6prcMUeGpMvbZlFk2TwLVFeejCS1k65QoXBz65q1gJ6lKRVhKowExQi1lDwjdc5JLSlFr9fl2qfWrL1ApIiqFA7G7mF1udfW3aPpSfR1KDh7a41nSpcWIi1oRdmPxhJo+CbcNGx1l6rfP13owe+QvkcGm1rGaQyIwZ80hGJ8u+U8iBiR4t+sYWcqxVp9WCU67+19lo1/hkGfW1gwXwUauh6R31faOGOLfi1hOWnyDnhiBFGb6+T5te74OO9JWKDhog2CdxMRdSBUgTtTQihKUTfxqanPbvKVN+ltk5qagwsNVuLW+Rhaxn0ZFTbf3x9fXumbPv+wKKX28el22V7MPKKSExeWneVnWHQ11pMin80ENF/7zj1rFZYfCPGbC9+RO0lxFhHlPC12Gtjmh5/C+uhs2HpAKlp3RxJiDBTmr+0onC/ucc29q3fVabtveCdM2tQ1hHT3rk6+91w4dbjFkItySvWdqFWt4QMarBssdf6enkNMbR9Qkzvt4B/doutvaefilBii1+eML0PeIkMUvdRM4LRMlWSXq5/ksbsek0t2ZKgyCRGtpoGlzg7l1QwjDPiYJGURoEg/aaF1OEwpeWVsMY6ibfJhlpDbkFDddeCGgTKlDK3k4xyK+0lyMcLWenqdCmWqXhbyuDcAUI19UrNrrQBgbRuWag9F0NlaMVo7khF33fwwrifhfdYKD27Vn2jxixXg+mScf06y9win/C7o6+1kG3Kv9AvDC11rKIqMmpxTBoFU0OEBgF6bSGBpRpWmk+oreR8Huc2K3iNgkUJnl+0kDoLNyybRtyahbUawV1Ts6VcW8rgCMIIF8fUef2gyEJYqcYYa6sWjDyZxkwIGmhk9kD7fkPFgKD6jXYHrZHV1rh+nWWON71b3G0L5C1k6w8dZoTieclEVID4tk4hSYHhbWU9Cz3eU0LEhWAiQK+oEJy5RpOgg8c9FnTifWqRI3foh5+ekG+Nyw3x/fctbZ4T0LXJdisZBAd1uh77tx94PXFhNugxkfl2asEodwCSBhryYRbzrMr+svQMKyenre99/8zN3jxGt2nAPR0PgdG9Y5+xf//7QiQrgEYQRqhRoOoj4Pz2ahJhQ1Len5r++M/2+28WUivJfoEJX72YRiO8wbplajk3rWwVwrBtcoLYm0/4/doySP6jZ1cMwMyoWOGucaUqxbIGI8XlMKmYe5nklIOQZn1KE4UbNShTzlulAMzEQ1ks8e7w3JOTcU+2t5kSe8h2QF2rkpA2kKvsXKJeo/DxlppeV1Vw4cgjSCJVxFGdaGEIupP3g3WPc76fXflCLUG23ZUemMAIhYni+PWNnuKVtqEvd25h1Nt2D0u2I6Yg3o+VqThnD+gEo5ZV356GXvvbi2z7ER9BFt4cRon86W89ikR/7ZZPYQR+lJJBjyNKe5/SGaIvd86E5weCG0223siNze0HLXD1OU/oitXbkHv7Xob71pVgpkepB5sWi6JgyI0FqYdtpke+qmcEWXiPFGlJXtNtbZ+9yVusPN722bPIvXZdfbvnZsFeobuxZBtqFFpk84I+anFibWEoya93Fd3vdirJLxYnpxW0prvWdyPI1ndGyZs3by3lNrkWRrl8lpxh5fJufe/NR7nZh7fZ3raIeiSbba8Zwav33u4SGr+XcltrbehR3/WSLTilnkfZi3dNL9FYUs/Si6ijsEqlM4Js/aDlO6PvpGce9I9ItsiDP59ibuPOIt4ISwt2mH4v2XoQwqmaf3d07SvVLr1kO9feN2WBrJdsw4Hd+2GG7n2lizdr98Pe/EaQ7dreCNTZ76ybU8g0U75DAw412wf3Iln5/Xct/ncKv+khW28rIrvQXhSaGFrtSWvjR12+WYjfUch2zxj2kq3fohvbzedNDD27/dbGsFQGvYbY2sdIY21vBPL0tvZU2WcPiwrJdoRdrrDvfz8aZPsDhR/0kK3fGZNaCPMuGzn3jliRQx/ewmp1Rys1BR2BbEPtrhucwgR+2eJxf17u6SVbv0U3thAWylDLxgkuEZ0z+eTq2PKehU+0vZJnhGa7tjcC9fKHsafMPN72fsfMJOyoHIy95gPZQoQlTw/Z5jQK8h8h6GvjR7k5U7Pk0fRmidXukWaELTCEbEu0sx6yzc2u1Ia9g/4fbES2ryoQQt/HejTbgqwWiSLtNnVKmxSa6HbkUq1okZJXJtpDtqUeB+F2xD3f3FsJ35CdT6k8R5Jtbb3WjN9DtuGGmtSW81C7X2JwXBMzn1cPfluVOcw3dfKg2jc5Kz4S2coUkPNzCxv31+0PHJWoBzPCCy38iwVti0UIHmIBILXJgfgA93tB3L00em05RkzfbjrZeu209O4xtLm7IrKF9vMBC3Ks19Q4Jq/E/RsXt7bt9xL/DGQLlih+r7HAZaxcY/QwC5AtNnd/jdVtuB+JbFsaas525d2Qcjauo7sseZJYYpX7pmi2dJ7aQStnXwzJdo4YS3c87YVcw3JIYcr5qu61/GG5WFjnmMlvW8iesXAksh11Jc5RGnJkOVsGqpH5nyktmaSOaHPcuh16TIFbl707/xKyRSt6joVfcrlxk+THLfSct9lS+FqtoiWP0d/sAb8WE8xoHHrSY6B9kYWfnBLhFtq/nGSw9KqYnvz9t0t6dYwqYywdprmcgfxD08uSG3pHl2fUweujy7VKeiVki6r8JAtfmkrEbqE3W+CcgddZKLnpc1RlJOhHOh1pD/gdXaNgsOAci29MgvQM++VGjZpLEEfJ4BEPwGbAZ7DieE4e1ideaQG7M/bgtc6rOOpANUR2Ssg2lpG/EFHX+Q4pUCYRrQQefYV2bfzkLH903Lx4+NXf0mtZRsio8u3ZdDCiHL1p+MPF11JedBDUjTTBtJItDb3FlOAsgr42fme1M464k66WtM60drCmeelMuNXKzPfj95CtX4Feyx9VDXaG1cy18DuzkG9lHtHgtYRnR1NH7vhorXWQMylKTXD3kC0ZLnnVSqpCZxT0JTuthPyMB+xs5XKWukyxqRNu/JHsqEsfnn8mzJqabBTZrmXz8dPvpYWjCdDKjzRYLYnfmYXck+2aaweafp9hAJvdYlopz3PRz6QkNcFSSrYI109bkOsNrl9c+PatKdeQ+DC+P9F9Q/x3uxLSSdgpgxvKCyzUuO/ohK6Ww2KaQBrwUS1+2nX0FMubf4fXReuIOVzvkjtWpnJrUaLnRuIBEHQlgTw91YLwkOsX3gkcvMKj2QF4PdLCoy3cf5LZ2MV+rDkgz3dbaPGoORqu9BsuN+WmZx5cv95rgcGYv/uBg/6J1xH4Pc4CuzZDjwXSe4sF2uLlFub68BH77ATTuJ8c2QLSeyxwog+rrwjm/1r4/akIOunHky3CzqEU91jQFef+pK1wj3iLVqeFkdItk+MQq0upBT8RMzeYIsw8fuXbH1TCuzkMzqCBUYf3WWAbNW5Kf2UBNzBwgAQkg55suZ+Km42FX0wDFVm2XoekBeIW+a2Tor7YDFTvsMCWdfD6Qwu4caLsgB0HYYNtSLZgTD8P8VVp/JGDOS8XYXWG2Whza8yRrdyTaIjwSLHw/NeUwHli5bg4rvP+sIVnW2CbGx2C0bVGs6WySjfXyM3ADPhwBH6eWNFMtbmE2QCaHpqHnzGExZbL11FdbUL5+airoO/s/Dk26Hj8wve0D7LIwyysVga18LhnFzDv3hWbCfo7+WInVfnF1dj7UhIlH561FtIHdN/xSaTIFpBpHGkTsamqP/s2tcDjGwvC5jbbV1gYsccbGxCj8twVFeMRK0txFH6ebNAKXmwBDaVk556wrzm4p6x268TyZxDHBnO/BTnlneKVgphWhTcD02RwbnlE5ksucLaUS99oBpiSAX/KXUrr1ICdwhgemKu/ZDh67GBP5Y72bYpsJURzUyxPtnP2QH+84UhNVI24x2ncKPzCM3ZrNFSVYY/4lPQTyU1Kc/RkO2e/VzoxUwJEwrbfFpstdVD77JFI/ECdkhtPtqmDyucOxNbs4BEzDSr8927yK5HJrjgxsg210dhWPq8x5Baqwqlw7XRtroI0JE/LNLALuJmPR+MnYc3h7IukMtR8sxQeLenOmQ+UnjcjzF2QqKluqDiAEUfk9crOXrXbErnxZoSUwuRPjAuVJerOaVcpU5ba8cwXWBbLd4xsvRCnRqMa47jvOKNHN2k3IzXmYvASEUfjp85cYw44uq1W5Z+rs7+va07j9+YIr73liKJUDnRdyuftA7YN7+HxylBqZuPj5OzOMkf4wTs3WIHL5yxg5rvRtloJRIxsczYavvV3KeXsVT7uEquRdBoIfS9XkI/Ez9t+wT2HNXE0AB1Zm5DGlfJj9dpWbjdhTDPLEUUtYUqhaLkzrDavkvh+wE/JjL8qKtcvY7OD3GCl3X17waQEt0XjxMhWdpyUoHtbWc6pW76M2HTQanPxWyqrFVe+fUJLAoO/GYkfAxXhpVMZc51C2gQEtObhLIMhvHdnYqq+6sjkm8PEKwfS4Gij0Pe7tw57GvSFT2og8pcX5gYrcPGzA2YRPHdbSJlgpDXv0Zbd287N389ptili9Eb1nI8nfnqsnuN3C2H4hkW74ClZWc9VUG5Wax4XlyqTNNte/BiowBetKbbIQ51x6vc2cPLG42MvWn6u3VLvpdnGiNQTRalpRZoZ8X/DAnI5emDe06AvzTZFpH6wKllA9bMD4nM8I9dFeVc8taUf8Edj3CpPu/huzmYbI4uShQsqplXKW/Zv3LzCFU1I4v0WYjt7WoHR6FsiPK15lHwnQe/BD7wQaLl5ee8GbGDaveN339GBXmNhzW2rJXi0xNGAFSNbj0XpoBJeZf+sSS5byjb3jcw+lD+3s2903j49aZYxsp3z/54rk8yBpDlnn0YZg3CPPLNapG1iZCvNgQx9x/V2oJgdRo0ByaBdIWxyqfEr9EzlyIOdaXMO+S0V1m4jn3dLOj3ftOCnTkq+rJCDn8feD1Zo77y/5QiD95DQGYgWDGSqYkFGnRZcuaZbM6RSoiW9Eg+Rnjb332rXIP1hS8LVDNQrH+D6EQuxjUq5+pcMcsRBm72INoJmys/W765hCiubKx39bRZiU3/vRhLTSEocqHMNXvKesv+4hVbfyZI8cnFq8fMrw2gOoeYV2tjC9wyEn7WQvXQuV/AdvdcAwtQfU8ldU9laTUX4hZMWA9UI09UcVLoZ4a0b4kkZmD2CG5oo/9e2+/CsjZJiyvSQ8vxRnVt2hJbkf/g4ubMRGAl5Sm6XBWxIbi4u6ZWkdXhgpwrU4KdbWL88kUuIgfBNvT8LZr4eqjN/66m3bI41m0LOgicDefENsDOV1mL3XtzbDtc+ObI9XIWuAl8IRBBgenuPhS01zSM3jEwjpVvFj1zXxcp+ke1i0F4J7wQBTCzYslvPP9hJNTYrhrwsRp1psllFts74ItutW+DKfyQCoakl9OoYmddZ00KL5cH+L6JdYjH7rPgl63WR7Y1r8lNX2O/e49+sjF9T3/ImD7fwLuU1VF6iE8W8yPZEjXlV5d4bn4ECr4XXWxh58NHZIfbbw7UBJLZx4ew4LFK/i2wXgfVKdEME0M6+bWFp964Nq7ho1jLFjDhzetGCHi3xi2yP1mJXeS8ELgQOicBFtodstqvQFwIXAkdD4CLbo7XYVd4LgQuBQyLwf6xczaF0WKndAAAAAElFTkSuQmCC" alt="d/dx(K dh/dx) + d/dy(K dh/dy) = 0" style="width: 173.5px; height: 35px;" width="173.5" 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: 100.358px 7.79167px; transform-origin: 100.358px 7.79167px; unicode-bidi: normal; "><span style="">Solve this equation for the head </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);">h</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: 251.9px 7.79167px; transform-origin: 251.9px 7.79167px; unicode-bidi: normal; "><span style="">, compute the flow through the aquifer, and get the effective conductivity from the definition.</span></span></div><div style="block-size: 246.467px; 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 123.233px; text-align: left; transform-origin: 384px 123.233px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: baseline;width: 513px;height: 241px" 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AFXXP2N6/QeQDSUtEnjYh7JHfY2EEqwkxuepNIpH8u7e6l4BsOh3DBluf+bUo21QXjHjh1pHzjgrhQgDy6Qp0JY7veRtgJbqZWKr5CWAnm0rG0/cJul3d9XR/8PUSQ/w3xNLfnectcw8k8CwsQxsZ5BuUtbbnxBcrJOKPMyf+Ri6V8H9df/ZkvYXb+pXvfFInMyR/P8lo8M5fM9sljlvcLqGjt2rA6MEvyCezesdsSt2n72s581o+TcloQ333zTjGLcyqitqsqd/5aEVbcS696Q5FZlCasodvSworvJkmv2335Zbo2wCkRLKG+6ch04cMCMknP/+8HFdN0bpWyPqvtR/+jRo3WQtqFXQqr8HtxQG3xyFFBM6GFFd5OLJHnqlV0C6+a51frmLQDRUtCbrnK1rJVbjf3oo/hTVpJxQ7LbYiDcG6Uk2ErrgH2EqbC/7q7bunnz5oTlrObNm2dGQPGhhxXdbff/f1k/9UpIWJXebQDRU9CWgFxVWM877zwzSn/TVXB5qgEDBphRnNsWIGHV9ru6z+WVVgZLlrdyVwzgqVCICiqsKLQ/v7Nf/duE2IoC8m81YRWIroK2BOSqwuo+616Wq0r1SDNZRsuSpzkl+++7YdR9nu+YMfFnVbvLZklYtb8naQfI1e8J8B0VVhSShNWhQy7QVX5ZBksefgEgugraEpCrCqsESLctYMiQIfpOfvv9JcDKx//ujVFuGHW5YdQN18He1GS9quPHjzcjoDjZHlZBhRWFIhdG466J3cwqKwI8WLtSjwFEVygrrEIeleqGYXlUqxzLUhESYO1H+3Ju586daf/bwTAqXxN8EECycOo+fAAoBuvWrNRL39ht3FfjK2BcOrCs/fzsqfFPIIBckrBaOXm8Xm/1wosGqk3bXuJCCUBhK6zZ6OjrJFDK41fdXtMg+bWmpqYOn0IVvHHKvcnKctdvFfL/l81SXEAx+fgI6wciPzauX6Wefy72cJjfv7474QIquN14w2j9OgDFL+8PDigUaQeQqqsIhksUHg8OQNRIgGJedp37sIqOSFHiocc3mCN0ROYoi9X7gQcHZK9oAiv8QmBF1BBY4TsCqz8IrNkr6LJWAAAAQLYIrAAAAPAagRUAAABeI7ACAADAawRWAAAAeI3ACgAAAK8RWAEAAOA1AisAAAC8RmAFAACA1wisAAAA8BqBFQAAAF4jsAIAAMBrBFYAAAB4jcAKAAAArxFYAQAA4LWStuPMGOi0hoYGtXHjRnOkVE1NjaqqqjJHSo0ePVqVl5ebo64pKSlRe/a9b44APwzodybzEl6TOcpbvh/kfYw/i+wQWJETe/fuVf379zdHJ2ppaVGlpaXmqGsIrPARgRW+I7D6g8CaPVoCMiQVw9WrV5sjBJWVlSVUVF3Tp0/PWVgFAADRQ2DNUGNjo5o4caKuIhJck7v11lvNKFFdXZ0ZAQAAZI/AmqE+ffrovXz0TXBNTqqowSqrVFcBACg0eb9G9nz9uRFYMyQ9mC6Ca3LBKuuSJUvMCACA/KutrVV9+/bl071Omj9/vpfZhsCaIVthDSK4JpIqq62qSrWV3lUAQCHIe7AE1ZkzZ55QZELmJO/4mG3yElirq6v1HXDFtC1fvtz87pIjuMbZqmqqnlYAAHLFVlTlPdgNqr179zYjZMP9GfqUbfKyrJUEVrmrPsrkrvlFixapiooKcyZaZJLLzyAf5AKC5YPgG5a1gu+KbVmr1tZWHaSophbOoEGD1Lp16/L2/p6WBNZcq6qqkr8RbMe343+4bcf/MpmfTHTk6/csP89kP2c2NjY2to63Yns/ampqaps+fXrS36tkEV/J/5+v0v08u3P+5KXCKtVVqbJGnfRwVlZWds+VSDeTq95du3blpYeVCit8RIUVvivmBwfIp3qLFy9OaN+TY3kf9pG8j/n6ZzF+/HhdRbXkZygtft19T0peelgPHTpkRtEkf7jyEYX8ZYliWJULFvnHQ/qKAADIN3mvlVUBmpqa1Lhx4/S5qGeRzrI3mbtZxocbqHk0a4akUtrRjVe+XIV0N2l+l0kuk765udmczR0qrPARFVb4rpgrrEFSNDlw4IAqLy83Z/zic4W1vr5eDRw40Lssw7JWGZIAlopvVyHdSaqr9mcl+6jffAcAKDypuPoaVn03bNgwL7MMgTVDydZhlfVGCaqJgg8K4MEBAACgqwisGXIrrLaiKv0yBNU4t7pqUWUFAABdRQ9rhqSH9ayzzqJHNQ3buxqU615WeljhI3pY4bso9bD6zuceVl8RWJETDQ0NauPGjfrJInJnplRVpRJtj3O5vBeBFT4isMJ3BFZ/EFizR2BFXuTzLyOBFT4isMJ3BFZ/EFizRw8rAAAAvEZgBQAAgNcIrAAAAPAagRUAAABeI7ACAADAawRWAAAAeI3ACgAAAK8RWAEAAOA1AisAAAC8RmAFAACA1wisAAAA8BqBFQAAAF4jsAIAAMBrBFYAAAB4jcAKAAAArxFYAQAA4DUCKwAAALxGYAUAAIDXCKwAAADwGoEVAAAAXiOwAgAAwGsEVgAAAHiNwAoAAACvEVgBAADgNQIrEDFNb+xRhw9/rP5+7Jg5A3Teb+q3qYqvD1U3jP+yerzuAXM2PZmDC+ZO0V83a9pE9cc/vG5+BQCSK2k7zoyBnCkpKVH5mlryvffse98cRY8EzXff/YsaOuQCc0aphx9dpYaPusYcpTbqixervXv36nGfPn3Uztfe0mN03YB+Z0ZyXkpQbXjpt3qcyZyS+Xv1lYPa56HIdP6ia2SO8pbvh3y+RxYrKqxAyJzUo4d65+195iimtbXFjFKbPXVMQkgYefU3zAjIjdNOP9OMUptTeW3CPJxw/RTCKoAOEViBkOnMR/nyUe22bdvMkVIXXjRQ3XHP/eYIyI0jhw+bUXLBeTho0CDmIYCMEFiBkJEK67EsQuv2LZtUzd3fNUdKnX32J9XqjfHQAHTFsSMfmZFSPXv1MqMTJZuHT6zdouczAHSEwAqEjFRYe2TwJi+vaz74N/UfM79pzsQ89vONqsdJPQgKyIkePU8zo+QV1lTzcN2metWr1ynmCADSI7ACIZNNhXXsNUPV0aNHzJFSj67coM7/3EWEVeRFsgrrsb8fO2EertqwVf3TOeeaIwDoGIEVCJlMK6xyc8vb++N3bFctvFd9cdgIcwTkXrIK6y0zr0uYhwvvuk8NvuQycwREQ0NDg6qurlY1NTV6L+xYNvl1pEdgBUImkwrrsofuT7i5Re7Cnlo5xxwB+RGssAZvspo+fbqaNOUmcwREx4ABA9oDquyFHcsmv470CKxFpLW1VfXt21ev71ZbW2vOnmjw4MH6NZWVlebMieQvkLxGvh/80lGFVRZyX1rzPXMUWxHgwdqV5gjIH7fCKvPQvcmq/NIvqHm3syIAoqm0tFRfsCUj5+XXkR6BtUht2LDBjBLJxw67du3S43Xr1ul9MnV1dXrPXyL/pKuw/vmd/WrapDHmiBUBUFi2whqch2VlZfpmPyDKlixZYkaJUp1HIgJrEZFwedVVV+mxfAwnFdegZ5991oyUamlpUfX19eYoThb1tgt7T5gwQe/hj1QVVjk/ueJqcxSzcu3TrAiAgrEVVncennxyT/XT1U+zIgAiT96jq6qqzFGMHFMYygyBtciMHTvWjJRavXq1GcW5gVVs2bLFjOLcc6NHjzYj+CJVhTX4BCF53GX/8wYQVlEwUmGdNW1i+zyUsLp+Uz0rAgDGrbfeakYxwWOkRmAtMhUVFWak1DPPPGNGca+88oreyxNmxLJly/TeZb9OngteXl6ux/BHsgrrTx99KOHmFh53ie7w0Yfvq8aGF8yR0ktZffjhB+YIgFtlpXc1OwTWImTDaDCwysf/0gYg5s2bp/dy7LYOyNj2to4bN07v4ZdkFdZ/GZx4YbH16V/qPkKgkHr06KmW/yyxV3XOjEnq8OGPzREAueFZCkL0rmaHwFqE7J2IwR5V+1H/iBEjEiqx7ooCe/bsMSOlrrvuOjOCT5JVWGVdyzsX/cgcxf7s5aNZggIK7eKBgxPuhj5w4K9q1o0TzREAuQlx8+bNVFezRGAtQqNGjTKjxH7UrVu36v2VV16p9xJcRWNjo96LjRtj1RG5+hs2bJgewy/JKqxyPHHSje3VdfH713erJfcuNEdA/tlVAmT5KllOzXr+ue16TVYAMbTbZY/AWoTk6k02sWbNGr0XdjkrG1jt3l3eyr7erjYA/6Rbh/WJtVv0xYa1ckWt2r5lkzkC8stdh/WxlRv1TVeWrMna+PKL5ggAskNgLVL2oQB2iSq7YoB7I9VNN8WfOCOtA/a1wl1tAH5JtUqAkKWDHlj2C3MUI60BzQf/poMukE/uk676nvWP6sePPGGOYuhnBdBZBNYiZaunQqqmr776qh67lVPpn7HVuCeffDKhfcDtcYVf0lVYxZDLLlc3z409q9q6cdJodezvBFbkl1thFbJShaxYYXWln3Xp9+9QA/qdqbdRX7yY4AtEDIG1SEkV1bYFyNqrtn81WDmdMWOG3m/fvr19VQHb2wo/pauwWrPmzE/oIcykn1WC8M4Xn1c3jP9yezCQ7YrLLlS/XL/KvApIza2wWnfXPKA+fe5nzFHn+ln373tTLfvJUnOk1Mm9TjMjAFFBYC1iw4cP13tZn9P2rw4ZMkTvLXuDlrQC2F7WMWPij1SEfzqqsFrSQ3jqafE39nT9rPI9rx97lZp07dWq4aXfmrMxb+9/S83/VqWaPZV5gfSCFVbryfWJDyjJtp918ffm6r1brQUQLQTWIhZclsq9GctKthIA7QB+y6TCKq+RHsKlDzxmzsSk6md9992/6IsWCQTP7fxvtWff+3p7affe9pUH5MKHSivSSVZhFfKkq0U/eNgcxcy6cYJ6771DJ8zFIKn6y9wbcP6F6pqvjzdnlX7kMIDoILAWMQmj7h3jEyZMMKNE7gMCJJywNpzfMq2wimAPoUjWzyqBYudrb+mPb+1jNOW/I6H34cfXt1dqt/36Kb0HkklVYRXjJkxKmIuyVvA3K67RF1epSJ/qLbNjX/ODHy9PmPf0ZAPRQmAtcs3NzaqtrU1vixcvNmcTPfXUU+2vcddkhZ8yqbC67rjn/oTKeqbrs9og0bt3qfp/l/yrHrf9Q/IKGiBSVVit2+5cos4++5PmKDYX5WaqVDas/YW+UUsuvM7/3EXmLIAoIrACIZNNhVVI8Pzp6qfNUUw267PK15/Ss0SPLyA0II10FVYhy6499vPER7fKzVTysX+QPFr4R/fdqcc198efxgcgmgisQMhkW2GVgJu0h3DaRN1DmIzbVyg3x0gPoeh/3gC9B5LpqMIqpFJatfBecxRz+62zzSju4QeW6LaBb1fdqT7xid7mbJz0sLrzFEBxI7ACIZRthVVID6F8tOp64tGHzCiR/RoJq/82IfY1stzZV74W73cGxM23LNDzSvrfZ3+rypxNb2rlHLXwrvtU+aVf0Ns130i80VPm3ZpfrNA3Wk2eNsucjTt6+CPdw2rnKYDiV9ImjYtAjpWUlOie2HyQ7y13sEfda7sb1dGjR/R48CWX6X0m/viH19WHH36gxwP/5ZKEN/377rpF/e6ll/VY+gstCReTpsSfjIYTyZq1zMuuk6pp5eTxer3WHz60IuEiSYLsxDEj9RrDqzdu0y0GyJzMUd7yEVZUWIGQunjgYB04ZcuGfCQrATcYVsWf/vQnHVTdsCruvn2eWrdmpTkC8mfj+lU6rKar6B89etSMAEQFgRUIMQmcnf1YNNnXPfT4hvY1WGWTNVntUkQLvjNLLZjLwu3IH1nG6r57v6vHN38ndsNVKqzDCkQLLQHIC1oCiossPWQfjfmrLb9jiaEkaAnoOmlzGffVEx9mko7cTCj92egYLQEIMyqsQMR05s7q6yZNUyef3FOPt23+L70Hcs32ZGfj1FNPNSMAxYzACkRMZ1sIzjjjdL0/0olQAWRCeqvdlpTgtmrDVv06eRDG628c0OdYuQKIBgIrEDGdqbC+8PwOvSamOP+C/6v3QHc57Yze9LACEUNgBSImWYVVVgC44rIL1a9/tU4/Ycj1eN0D+oYrIetiBtdyBQrFtqV89MEhvQ4rgOggsAIRk6rC+vb+t9Tc2VPU0CEX6Jsz7FZzd+yubfkYdv2meta+RLexPa62wsqTroDoYJUA5AWrBISPVFYfrf2xfq77nj/+3pxV+glG02bNU5cPG67DqoSEzvbBFjNWCYDvWCUAYUZgRV4QWP2VLHASQruOwArfEVgRZrQEABGTLJgSVgEAPiOwAhFD3x8AIGwIrEDEUE0FAIQNgRWIGCqsAICwIbACEZOuwkqYBQD4iMAKREy6UEq7AADARwRWIGKosAIAwobACkQMFVYAQNgQWIGIocIKAAgbAisQMVRYAQBhQ2AFIoYKKwAgbAisQMRQYQUAhA2BFYgYKqwAgLAhsAIRQ4UVABA2BFYgYqiwAgDChsAKRAwVVgBA2BBYgYihwgoACJuStuPMGOi0+vp6tWXLFnOkVE1NjaqqqjJHSo0ePVqVl5ebo64pKSlRe/a9b46QLQmlVFJzb0C/M5mX8JrMUd7yEVYEVuREa2ur6tOnjzk6UVNTkyorKzNHXUNgzR/CbOcRWOE7AivCjJYA5ERpaamaPn26OUokldZchVV0HT2sAICwocKKnJLqZ1BLS4sOtLlChTV/qLB2HhVW+I4KK8KMCityKlhllepqLsMquo4KKwAgbKiwIqeCvay5rq4KKqz5Q4W186iwwndUWBFmVFiRUxJO7eoA+aqu0g/bNVRYAQBhQ4UVOWerrLlcGcB1x5QT+2SB7rbsV0rN+Jo5ADx0508VFVaEFhVW5JxUVfMVVgFfEVYBIH8IrMgLwioA+OVTfc0ACCECKwAAEcCnAAgzAisAAAC8RmAFAACA1wisAAAA8BqBFQAAAF4jsAIAAMBrBFYAAAB4jcAKAAAArxFYAQAA4DUCKwAAALxGYAUAAIDXStqOM2MAAADAO1RYAQAA4DUCKwAAALxGYAUAAIDXCKwAAADwGoEVAAAAXiOwAgAAwGsEVgAAAHiNwIqisnfvXjMC8qO1tVVvmWJOIt+ynZNAGBFYURRqa2tV3759VV1dnTkD5Mfu3btVnz59VHV1ddqQwJxEoWQ6J4EwI7Ai1GwomDlzpmppaTFngfw5rD6h9zU1NUlDwiOPrWZOolukmpNAMSCwIpRSBdXevXubEZAfvdR7ZhRjQ8KIL31Vz8lZ0yYyJ9GtCK4oRiVtx5kx4D3pBxw8eDCVKwDI0KpVq1RFRYU5AsKJCitCpaysTP1uV5OaPHmyOZOoauG9Sq7B2Njyte3YscPMtkRS0Uqmqqoq6fdhY8vVlmpOjhgxQu3cuZOwiqJAYEXonP/PpWrFihXq7bffVtOnTzdnY076n4/MCCiM8ku/oENBc3OzrvwH5yQtASi0YcOGqc07GtXWrVtVeXm5OQuEG4EVoXXOOefoO7CbmuIV11NOOUXvgXyT6pVUtna++Hx7KCgtLW2fk9dff70+d+jQIb0H8s3OSdlGDRtkzgLFgR5WFA3pb33zzTd1dQHIl2zmmbz2wIEDVLmQV/zbhyggsAIAAMBrtAQAAADAawRWAAAAeI3ACgAAAK8RWAEAAOA1AisAAAC8RmAFAACA1wisAAAA8BqBFShyDQ0NauTIkXqTBcYBn7jzU8YAkAyBFShyu3btUtu2bdObPA0H8Ik8xtbOT5mrAJAMgRUA0G1aW1vNSKnS0lIzAoBEBFagyBEC4LNTTjnFjBLDKwC4CKxAkSMEwGcff/yxGQFAagRWAEC34RMAAJkgsAIAug09rAAyQWAFihwhAGFB+wqAVAisQJEjBCAsuLgCkAqBFShy7l3YgG/ckMrFFYBUCKxAkeMubPiMkAogEwRWoMjxMSt8xvwEkAkCK1DkqGDBZ+78JLwCSIXACgDwAhdXAFIhsAIAvECFFUAqBFagyBEC4DN3flJhBZAKgRUocoQA+Iz5CSATBFYAQLfhEwAAmShpO86MARSh2tpaNXPmTD0uKytT5513njp48KA+PuussxLGW7du1WOgUNLNz549e6ojR47osczPtWvXEnCBiCKwAkVu9erVauLEieYoPf45QKHV19erK664whyl19TUpEMtgOghsAJFTnoEpYp16NAh1bt3b/XBBx+oM844o31vz4uqqiq9Bwqpuro6YW663Hm6ePFicxZA1BBYAQAA4DVuugIAAIDXCKwAAADwGoEVAAAAXiOwAgAAwGsEVgAAAHiNwAoAAACvEVgBAADgNQIrAAAAvEZgBQAAgNcIrAAAAPAagRUAAABeI7ACAADAawRWAAAAeI3ACgAAAK8RWAEAAOA1AisAAAC8RmAFAACA1wisAAAA8BqBFQAAAF4jsAIAAMBjSv0veC+r3U695SMAAAAASUVORK5CYII=" 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Cody Problem 52070 asked for a function to compute the effective hydraulic conductivity of a heterogeneous aquifer—or the single value of conductivity set such that the aquifer produces the same flow under the same total change in head. In that problem, the aquifer had soil units either in series only or in parallel only. 
Write a function to compute the effective conductivity for two-dimensional flow in an aquifer with a more complicated distribution of conductivity. Flow is left to right, or to the east, as in the figure below. No flow occurs across the north and south boundaries. Assume the head difference is small enough that Darcy’s law applies. Use the conductivity specified on the equally-spaced grid provided.  
For example, if in the aquifer below  = 0.1 m/d,  = 0.2 m/d,  = 0.01 m/d, and  = 20 m/d, then the effective conductivity is 0.092 m/d.  
Hint: The simple formulas that work for soil units either in series only or in parallel only will not work for these more complicated distributions because two-dimensional flow violates assumptions behind the formulas. In this problem, compute the effective conductivity directly from the definition. Darcy's law yields the specific discharges (or flow per unit cross-sectional area) of 
   and   
Then conservation of mass leads to

Solve this equation for the head , compute the flow through the aquifer, and get the effective conductivity from the definition.

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Problem 53975. Compute the effective conductivity of more heterogeneous aquifers

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Problem 53975. Compute the effective conductivity of more heterogeneous aquifers

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