{"id":1170,"date":"2026-05-12T21:46:30","date_gmt":"2026-05-12T21:46:30","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/well-hydraulics-a-quick-review\/"},"modified":"2026-05-13T16:29:16","modified_gmt":"2026-05-13T16:29:16","slug":"well-hydraulics-a-quick-review","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/well-hydraulics-a-quick-review\/","title":{"raw":"8.2 Well Hydraulics\u2014A Quick Review","rendered":"8.2 Well Hydraulics\u2014A Quick Review"},"content":{"raw":"
\r\n
\r\n\r\nFlow to a well is a special case because flow is radial<\/em>\u2014that is, converging on a specific point in the aquifer (the well). To handle this mathematically, C.V. Theis derived the equation that governs 2\u2011D radial flow to a well, as shown in Equation\u00a0(72)<\/span><\/a>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]\\frac{\\partial^{2}h}{{\\partial r}^{2}} + \\frac{1}{r}\\frac{\\partial h}{\\partial r} + \\frac{w}{T} = \\frac{S}{T}\\frac{\\partial h}{\\partial t}[\/latex]<\/th>\r\n(72)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nTheis then \u2018stole\u2019<\/em> an analytical solution for this equation from the heat flow literature to develop his eponymous equation, which is shown in Equation\u00a0(73)<\/span><\/a> and where u<\/em> is defined by Equation\u00a0(74)<\/span><\/a>, and [latex]W(u)[\/latex]\u2014the well function<\/em>\u2014is as shown in Equation\u00a0(75)<\/span><\/a>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\n[latex]S = \\frac{Q}{4\\pi T}W(u)[\/latex]<\/th>\r\n(73)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n[latex]u = \\frac{r^{2}S}{4Tt}[\/latex]<\/td>\r\n(74)<\/span><\/td>\r\n<\/tr>\r\n
<\/td>\r\n[latex]W(u) = \\int_{u}^{\\infty}\\frac{e^{- u}du}{u}[\/latex]<\/td>\r\n(75)<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nwhere (parameter dimensions are dark green font with mass as M, length as L, time as T):\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]s [\/latex]<\/th>\r\n[latex]=[\/latex]<\/th>\r\ndrawdown (L) is defined as the initial head minus the head at time [latex]t[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]Q[\/latex]<\/td>\r\n[latex]=[\/latex]<\/td>\r\npumping rate from the well (L3<\/sup>T-1<\/sup>) is assumed to be constant<\/td>\r\n<\/tr>\r\n
[latex]T[\/latex]<\/td>\r\n[latex]=[\/latex]<\/td>\r\naquifer transmissivity ([latex]T = Kb[\/latex]) (L2<\/sup>T-1<\/sup>)<\/td>\r\n<\/tr>\r\n
[latex]r[\/latex]<\/td>\r\n[latex]=[\/latex]<\/td>\r\nradius (distance from pumping well) (L)<\/td>\r\n<\/tr>\r\n
[latex]S[\/latex]<\/td>\r\n[latex]=[\/latex]<\/td>\r\nstorativity (where [latex]S\\ = \\ S_{s}b[\/latex]) (:)<\/td>\r\n<\/tr>\r\n
[latex]t[\/latex]<\/td>\r\n[latex]=[\/latex]<\/td>\r\ntime after the start of pumping ([latex]T[\/latex])<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nThe Theis equation calculates drawdown<\/em> ([latex]s[\/latex]) at some distance ([latex]r[\/latex]) from the pumping well at some time ([latex]t[\/latex]) after pumping began for a constant pumping rate ([latex]Q[\/latex]). This equation describes the cone of depression<\/em> that forms around a pumping well. If we know [latex]T[\/latex] and [latex]S[\/latex], we can use this equation to directly calculate the drawdown at any distance from the well at any time after the start of pumping.\r\n\r\n<\/div>\r\n
\r\n\r\nBut the problem is how do we measure [latex]T[\/latex] and [latex]S[\/latex] in the first place? It turns out that one of the best ways that to measure [latex]T[\/latex] and [latex]S[\/latex] in an aquifer is to pump water from a well at a constant rate, measure drawdown in the aquifer at various times after pumping starts, and then use these data and the Theis equation to calculate [latex]T[\/latex] and [latex]S[\/latex]. Doing this is called an aquifer test<\/em>, or more specifically a constant rate aquifer test<\/em>.\r\n\r\n<\/div>\r\n<\/div>","rendered":"
\n
\n

Flow to a well is a special case because flow is radial<\/em>\u2014that is, converging on a specific point in the aquifer (the well). To handle this mathematically, C.V. Theis derived the equation that governs 2\u2011D radial flow to a well, as shown in Equation\u00a0(72)<\/span><\/a>.<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n
<\/th>\n[latex]\\frac{\\partial^{2}h}{{\\partial r}^{2}} + \\frac{1}{r}\\frac{\\partial h}{\\partial r} + \\frac{w}{T} = \\frac{S}{T}\\frac{\\partial h}{\\partial t}[\/latex]<\/th>\n(72)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n
\n

Theis then \u2018stole\u2019<\/em> an analytical solution for this equation from the heat flow literature to develop his eponymous equation, which is shown in Equation\u00a0(73)<\/span><\/a> and where u<\/em> is defined by Equation\u00a0(74)<\/span><\/a>, and [latex]W(u)[\/latex]\u2014the well function<\/em>\u2014is as shown in Equation\u00a0(75)<\/span><\/a>.<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n
<\/th>\n[latex]S = \\frac{Q}{4\\pi T}W(u)[\/latex]<\/th>\n(73)<\/span><\/th>\n<\/tr>\n<\/thead>\n
<\/td>\n[latex]u = \\frac{r^{2}S}{4Tt}[\/latex]<\/td>\n(74)<\/span><\/td>\n<\/tr>\n
<\/td>\n[latex]W(u) = \\int_{u}^{\\infty}\\frac{e^{- u}du}{u}[\/latex]<\/td>\n(75)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

where (parameter dimensions are dark green font with mass as M, length as L, time as T):<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n
[latex]s[\/latex]<\/th>\n[latex]=[\/latex]<\/th>\ndrawdown (L) is defined as the initial head minus the head at time [latex]t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
[latex]Q[\/latex]<\/td>\n[latex]=[\/latex]<\/td>\npumping rate from the well (L3<\/sup>T-1<\/sup>) is assumed to be constant<\/td>\n<\/tr>\n
[latex]T[\/latex]<\/td>\n[latex]=[\/latex]<\/td>\naquifer transmissivity ([latex]T = Kb[\/latex]) (L2<\/sup>T-1<\/sup>)<\/td>\n<\/tr>\n
[latex]r[\/latex]<\/td>\n[latex]=[\/latex]<\/td>\nradius (distance from pumping well) (L)<\/td>\n<\/tr>\n
[latex]S[\/latex]<\/td>\n[latex]=[\/latex]<\/td>\nstorativity (where [latex]S\\ = \\ S_{s}b[\/latex]) (:)<\/td>\n<\/tr>\n
[latex]t[\/latex]<\/td>\n[latex]=[\/latex]<\/td>\ntime after the start of pumping ([latex]T[\/latex])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

The Theis equation calculates drawdown<\/em> ([latex]s[\/latex]) at some distance ([latex]r[\/latex]) from the pumping well at some time ([latex]t[\/latex]) after pumping began for a constant pumping rate ([latex]Q[\/latex]). This equation describes the cone of depression<\/em> that forms around a pumping well. If we know [latex]T[\/latex] and [latex]S[\/latex], we can use this equation to directly calculate the drawdown at any distance from the well at any time after the start of pumping.<\/p>\n<\/div>\n

\n

But the problem is how do we measure [latex]T[\/latex] and [latex]S[\/latex] in the first place? It turns out that one of the best ways that to measure [latex]T[\/latex] and [latex]S[\/latex] in an aquifer is to pump water from a well at a constant rate, measure drawdown in the aquifer at various times after pumping starts, and then use these data and the Theis equation to calculate [latex]T[\/latex] and [latex]S[\/latex]. Doing this is called an aquifer test<\/em>, or more specifically a constant rate aquifer test<\/em>.<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":25,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1170","chapter","type-chapter","status-publish","hentry"],"part":1489,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1170","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":3,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions"}],"predecessor-version":[{"id":1859,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions\/1859"}],"part":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/parts\/1489"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/media?parent=1170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapter-type?post=1170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/contributor?post=1170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/license?post=1170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}