{"id":1184,"date":"2026-05-12T21:46:50","date_gmt":"2026-05-12T21:46:50","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/simplification-of-the-theis-equation\/"},"modified":"2026-05-13T03:14:16","modified_gmt":"2026-05-13T03:14:16","slug":"simplification-of-the-theis-equation","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/simplification-of-the-theis-equation\/","title":{"raw":"8.5 Simplification of the Theis Equation","rendered":"8.5 Simplification of the Theis Equation"},"content":{"raw":"
\r\n
\r\n\r\nThe well function in the Theis equation is shown in Equation\u00a0(78)<\/span><\/a>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
[latex]W(u) = - 0.5772 - \\ln{u + u - \\frac{u^{2}}{2*2!}} + \\frac{u^{3}}{3*3!} - \\frac{u^{4}}{4*4!} + \\frac{u^{5}}{5*5!} - and\\ so\\ on\\ldots[\/latex]<\/th>\r\n(78)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]u = \\frac{r^{2}s}{4Tt}[\/latex]<\/th>\r\n(79)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nCooper and Jacob (1946) noted that, when u becomes small, everything to the right of the term [latex]ln(u)[\/latex] in the well function becomes negligible and can be omitted. Omitting this term reduced the well function to a simple term, which in turn reduces the Theis equation to that shown in Equation\u00a0(80)<\/span><\/a>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]s = \\frac{2.3Q}{4\\pi}\\log\\frac{2.25Tt}{r^{2}S}[\/latex]<\/th>\r\n(80)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nThis is called the Cooper\u2013Jacob simplification<\/em> of the Theis non\u2011equilibrium equation. This means that, in certain situations, we can use this simplified version rather than the more complex Theis equation. More specifically, the quantity [latex]u[\/latex] becomes small when [latex]t[\/latex] becomes large\u2014that is, at long times after the start of pumping\u2014or when [latex]r[\/latex] becomes small\u2014that is, very close to the pumping well. This concept can be applied to analyzing aquifer test data to estimate [latex]T[\/latex] and [latex]S[\/latex] using the methods described in the following two sections.\r\n\r\n<\/div>\r\n
\r\n

Time\u2011Drawdown Method<\/h3>\r\n
\r\n\r\nIf time\u2011drawdown data are available from a single observation well, we can use the Cooper\u2011Jacob simplification to estimate [latex]T[\/latex] and [latex]S[\/latex]. The method is as listed here.\r\n\r\n<\/div>\r\n