{"id":1884,"date":"2026-02-14T04:29:13","date_gmt":"2026-02-14T04:29:13","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch105-xhtml\/"},"modified":"2026-08-01T13:57:30","modified_gmt":"2026-08-01T13:57:30","slug":"solution-exercise-4","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/solution-exercise-4\/","title":{"raw":"Solution Exercise 4","rendered":"Solution Exercise 4"},"content":{"raw":"<div id=\"section-18\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nA well (A) pumping at 900 m<sup>3<\/sup>\/d is located near a river as shown in the image below. The river penetrates a fractured, permeable, confined limestone. A second production well (B) is located 200 m from the first well. A previous pumping test of the formation at well A yielded a <em>T<\/em> of 75 m<sup>2<\/sup>\/d and an <em>S<\/em> of 0.00003.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.15951in; height: 3.49929in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file152-6.jpg\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\r\n\r\nProduction wells located near a fully penetrating river. On average, the distance to the river in the confined aquifer is 30 m from the pumping well as shown by the red dashed line. a) Cross section showing production well A and the lithology. b) Map view of the well locations relative to the river.\r\n\r\n<\/div>\r\n<ol type=\"a\">\r\n \t<li>Compute the well interference (drawdown) that would occur at B when well A is pumped at 900 m<sup>3<\/sup>\/d for 50 days.<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>This problem deals with a linear recharge boundary where the river penetrates the confined aquifer. Using image well theory (Section 11), the river acts as an infinite source of water to the aquifer. To compute the drawdown at well location B the effect of the boundary must be represented by an image well (IA) located 30 m across the boundary from well A. As the image well represents the effects of a recharge boundary it becomes an injection well to mathematically create no drawdown at the boundary (i.e., an injection well has drawup, or negative drawdown) with an inflow of 900 m<sup>3<\/sup>\/d. The total drawdown at B from pumping A and the effects of the recharge boundary must be considered as shown in the image below.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.00752in; height: 5.79755in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file202-6.jpg\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\r\n\r\nApplication of image well theory. Well IA is an image well representing the effect of the linear recharge boundary (river) when A is pumped. The image well is represented by an injection well (\u2011900 m<sup>3<\/sup>\/d). The figure also shows the radial distance from image well IA to location B.\r\n\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]s_{T}\\ at\\ B = s_{A} + s_{IA}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Using the Theis Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X83717eb7a60cb5138762eea2a389da4682cb0dc\">(26)<\/a>, the total drawdown at B can be computed. First, <em>u<\/em> is calculated, then W(<em>u<\/em>) is read from <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> and drawdown is calculated using Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X83717eb7a60cb5138762eea2a389da4682cb0dc\">(26)<\/a>.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>First, compute <em>u<\/em> for well A.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{A} = \\frac{r^{2}S}{4Tt} = \\frac{{(200\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.00008 \\cong 8x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> to look up, or WolframAlpha to solve W(<em>u<\/em>)=\u2011Ei(\u2011<em>u<\/em>), W(<em>u<sub>A<\/sub><\/em>) is \u2245 8.8563.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]s_{A} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 8.8563 = 8.46\\ m[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Next, compute <em>u<\/em> for well IA.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{IA} = \\frac{r^{2}S}{4Tt} = \\frac{{(209\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000087 \\cong 8.7x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IA<\/sub><\/em>) is \u2245 8.7725. As the image well represents a recharge boundary effect the pumping rate is input as \u2013900 m<sup>3<\/sup>\/d.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]s_{IA} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 8.7725 = - 8.38\\ m[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>The \u201cdrawdown\u201d computed is negative because it is draw\u2011up or a rise in water level in response to injection of water.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]s_{T}\\ at\\ B = 8.46\\ m + ( - 8.38\\ m) = 0.08\\ m[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<ol start=\"2\" type=\"a\">\r\n \t<li>If during the same 50 days of pumping well A, the well at location B is pumped at 450 m<sup>3<\/sup>\/d, what would be the drawdown at unpumped observation well C at the end of the 50 days of pumping?<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>The total drawdown at C would be computed shown here.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]s_{T}\\ at\\ C = s_{A} + s_{IA} + s_{\\mathbf{B}} + s_{IB}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Drawdown at C is influenced by the two pumping wells and the two image wells that represent the recharge boundary. The image well locations and corresponding radial distances are shown in the following image.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.00142in; height: 5.47853in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file203-6.jpg\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\r\n\r\nApplication of image well theory. Well IA is an image well representing the effect of the linear recharge boundary (river) on the drawdown at A and IB is the image well representing the effect of the boundary on the drawdown at B. To calculate the drawdown at the unpumped well located at C, the effect of all four wells must be considered. The calculated radial distances from pumping and image wells to well C are shown on the figure.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>The first step is to compute the radial distances from the pumping and image wells to observation well C. Then values of <em>u<\/em> are computed for each of the 4 wells and corresponding values of W(<em>u<\/em>) determined from the table shown in <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> of Section 8. <em>Q<\/em> for the recharge boundary injection wells is input as a negative rate.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{A} = \\frac{r^{2}S}{4Tt} = \\frac{{(119\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000028 \\cong 2.8x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> to look up, or WolframAlpha to solve W(<em>u<\/em>)=\u2011Ei(\u2011<em>u<\/em>), W(<em>u<sub>A<\/sub><\/em>) is \u2245 9.9061.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathbf{s}_{\\mathbf{A}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.9061 = \\mathbf{9.46\\ m}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{IA} = \\frac{r^{2}S}{4Tt} = \\frac{{(160\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000051 \\cong 5.1x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IA<\/sub><\/em>) is \u2245 9.3065.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathbf{s}_{\\mathbf{IA}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.3065 = \\mathbf{- 8.89\\ m}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{B} = \\frac{r^{2}S}{4Tt} = \\frac{{(119\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000028 \\cong 2.8x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>A<\/sub><\/em>) is \u2245 9.9061.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathbf{s}_{\\mathbf{B}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{450\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.9061 = \\mathbf{4.73\\ m}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]u_{IB} = \\frac{r^{2}S}{4Tt} = \\frac{{(160\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000051 \\cong 5.1x10^{- 5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IB<\/sub><\/em>) is \u2245 9.3065.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathbf{s}_{\\mathbf{IB}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 450\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.3065 = \\mathbf{- 4.45\\ m}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathbf{s}_{\\mathbf{T}}\\mathbf{\\ at\\ C} = s_{A} + s_{IA} + s_{\\mathbf{B}} + s_{IB} = 9.46\\ m + ( - 8.89\\ m) + 4.73\\ m + ( - 4.45\\ m) = \\mathbf{0.85\\ m}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/exercise-4\/#exercise-4\">Return to Exercise\u00a04<\/a><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/an-opportunity-to-use-well-hydraulics-to-evaluate-well-interference-in-the-presence-of-a-recharge-boundary\/#TextLinkToExer4\">Return to where text linked to Exercise\u00a04<\/a><\/blockquote>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"section-18\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<div data-custom-style=\"First Paragraph\">\n<p>A well (A) pumping at 900 m<sup>3<\/sup>\/d is located near a river as shown in the image below. The river penetrates a fractured, permeable, confined limestone. A second production well (B) is located 200 m from the first well. A previous pumping test of the formation at well A yielded a <em>T<\/em> of 75 m<sup>2<\/sup>\/d and an <em>S<\/em> of 0.00003.<\/p>\n<\/div>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.15951in; height: 3.49929in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file152-6.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\n<p>Production wells located near a fully penetrating river. On average, the distance to the river in the confined aquifer is 30 m from the pumping well as shown by the red dashed line. a) Cross section showing production well A and the lithology. b) Map view of the well locations relative to the river.<\/p>\n<\/div>\n<ol type=\"a\">\n<li>Compute the well interference (drawdown) that would occur at B when well A is pumped at 900 m<sup>3<\/sup>\/d for 50 days.<\/li>\n<\/ol>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>This problem deals with a linear recharge boundary where the river penetrates the confined aquifer. Using image well theory (Section 11), the river acts as an infinite source of water to the aquifer. To compute the drawdown at well location B the effect of the boundary must be represented by an image well (IA) located 30 m across the boundary from well A. As the image well represents the effects of a recharge boundary it becomes an injection well to mathematically create no drawdown at the boundary (i.e., an injection well has drawup, or negative drawdown) with an inflow of 900 m<sup>3<\/sup>\/d. The total drawdown at B from pumping A and the effects of the recharge boundary must be considered as shown in the image below.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.00752in; height: 5.79755in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file202-6.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\n<p>Application of image well theory. Well IA is an image well representing the effect of the linear recharge boundary (river) when A is pumped. The image well is represented by an injection well (\u2011900 m<sup>3<\/sup>\/d). The figure also shows the radial distance from image well IA to location B.<\/p>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]s_{T}\\ at\\ B = s_{A} + s_{IA}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Using the Theis Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X83717eb7a60cb5138762eea2a389da4682cb0dc\">(26)<\/a>, the total drawdown at B can be computed. First, <em>u<\/em> is calculated, then W(<em>u<\/em>) is read from <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> and drawdown is calculated using Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X83717eb7a60cb5138762eea2a389da4682cb0dc\">(26)<\/a>.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>First, compute <em>u<\/em> for well A.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{A} = \\frac{r^{2}S}{4Tt} = \\frac{{(200\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.00008 \\cong 8x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> to look up, or WolframAlpha to solve W(<em>u<\/em>)=\u2011Ei(\u2011<em>u<\/em>), W(<em>u<sub>A<\/sub><\/em>) is \u2245 8.8563.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]s_{A} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 8.8563 = 8.46\\ m[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Next, compute <em>u<\/em> for well IA.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{IA} = \\frac{r^{2}S}{4Tt} = \\frac{{(209\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000087 \\cong 8.7x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IA<\/sub><\/em>) is \u2245 8.7725. As the image well represents a recharge boundary effect the pumping rate is input as \u2013900 m<sup>3<\/sup>\/d.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]s_{IA} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 8.7725 = - 8.38\\ m[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>The \u201cdrawdown\u201d computed is negative because it is draw\u2011up or a rise in water level in response to injection of water.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]s_{T}\\ at\\ B = 8.46\\ m + ( - 8.38\\ m) = 0.08\\ m[\/latex]<\/p>\n<\/div>\n<\/div>\n<ol start=\"2\" type=\"a\">\n<li>If during the same 50 days of pumping well A, the well at location B is pumped at 450 m<sup>3<\/sup>\/d, what would be the drawdown at unpumped observation well C at the end of the 50 days of pumping?<\/li>\n<\/ol>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>The total drawdown at C would be computed shown here.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]s_{T}\\ at\\ C = s_{A} + s_{IA} + s_{\\mathbf{B}} + s_{IB}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Drawdown at C is influenced by the two pumping wells and the two image wells that represent the recharge boundary. The image well locations and corresponding radial distances are shown in the following image.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.00142in; height: 5.47853in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file203-6.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div class=\"figcaption-text\" data-custom-style=\"Block Text\">\n<p>Application of image well theory. Well IA is an image well representing the effect of the linear recharge boundary (river) on the drawdown at A and IB is the image well representing the effect of the boundary on the drawdown at B. To calculate the drawdown at the unpumped well located at C, the effect of all four wells must be considered. The calculated radial distances from pumping and image wells to well C are shown on the figure.<\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>The first step is to compute the radial distances from the pumping and image wells to observation well C. Then values of <em>u<\/em> are computed for each of the 4 wells and corresponding values of W(<em>u<\/em>) determined from the table shown in <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> of Section 8. <em>Q<\/em> for the recharge boundary injection wells is input as a negative rate.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{A} = \\frac{r^{2}S}{4Tt} = \\frac{{(119\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000028 \\cong 2.8x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a> to look up, or WolframAlpha to solve W(<em>u<\/em>)=\u2011Ei(\u2011<em>u<\/em>), W(<em>u<sub>A<\/sub><\/em>) is \u2245 9.9061.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathbf{s}_{\\mathbf{A}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.9061 = \\mathbf{9.46\\ m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{IA} = \\frac{r^{2}S}{4Tt} = \\frac{{(160\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000051 \\cong 5.1x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IA<\/sub><\/em>) is \u2245 9.3065.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathbf{s}_{\\mathbf{IA}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 900\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.3065 = \\mathbf{- 8.89\\ m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{B} = \\frac{r^{2}S}{4Tt} = \\frac{{(119\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000028 \\cong 2.8x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>A<\/sub><\/em>) is \u2245 9.9061.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathbf{s}_{\\mathbf{B}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{450\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.9061 = \\mathbf{4.73\\ m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]u_{IB} = \\frac{r^{2}S}{4Tt} = \\frac{{(160\\ m)}^{2}\\ \\ 0.00003}{4\\left( 75\\frac{m^{2}}{d} \\right)50\\ d} = 0.000051 \\cong 5.1x10^{- 5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Then using <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-the-theis-equation\/#X32bb1d0f31301257533eecaf70ec35ba2571063\">Figure 26<\/a>, W(<em>u<sub>IB<\/sub><\/em>) is \u2245 9.3065.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathbf{s}_{\\mathbf{IB}} = \\frac{Q}{4\\pi T}\\ W(u) = \\frac{- 450\\ \\frac{m^{3}}{d}}{4\\ (3.14)\\ 75\\frac{m^{2}}{d}}\\ 9.3065 = \\mathbf{- 4.45\\ m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathbf{s}_{\\mathbf{T}}\\mathbf{\\ at\\ C} = s_{A} + s_{IA} + s_{\\mathbf{B}} + s_{IB} = 9.46\\ m + ( - 8.89\\ m) + 4.73\\ m + ( - 4.45\\ m) = \\mathbf{0.85\\ m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/exercise-4\/#exercise-4\">Return to Exercise\u00a04<\/a><\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/an-opportunity-to-use-well-hydraulics-to-evaluate-well-interference-in-the-presence-of-a-recharge-boundary\/#TextLinkToExer4\">Return to where text linked to Exercise\u00a04<\/a><\/p><\/blockquote>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":97,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1884","chapter","type-chapter","status-publish","hentry"],"part":605,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1884","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":5,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1884\/revisions"}],"predecessor-version":[{"id":4145,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1884\/revisions\/4145"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/605"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1884\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1884"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapter-type?post=1884"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1884"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1884"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}