Государственный комитет Республики Узбекистан

h(r, 0) = h₀ for all r(8.2)
where h₀ is the constant initial hydraulic head.
The boundary conditions assume no drawdown in hydraulic head at the infinite boundary:
h(∞, t) = h₀ for all t(8.3)
And a constant pumping rate Q[L³/T] at the well:
(8.4)
Condition (8.4) is the result of a straightforward application of Darcy’s law at the well face.
The solution h(r, t) describes the hydraulic head field at any radial distance r at any time after the start of pumping. For reasons that should be clear from a perusal of Figure 8.4, solutions are often presented in terms of the drawdown in head h₀ – h(r, t).
The Theis Solution
Theis (1935), in what must be considered one of the fundamental breakthroughs in the development of hydrologic methodology, utilized an analogy to heat-flow theory to arrive at an analytical solution to Eq. (8.1) subject to the initial and boundary conditions of Eqs. (8.2) through (8.4). His solution, written in terms of the drawdown, is
(8.5)
where
(8.6)
The integral in Eq. (8.5) is well known in mathematics. It is called the exponential integral and tables of values are widely available. For the specific definition of u given by Eq. (8.6), the integral is known as the well function, W(u). With this notation, Eq. (8.5) becomes
(8.7)
Table 8.1 provides values of W(u) versus u, and Figure 8.5(a) shows the relationship W(u) versus 1/u graphically. This curve is commonly called the Theis curve.
If the aquifer properties, T and S, and the pumping rate, Q, are known, it is possible to predict the drawdown in hydraulic head in a confined aquifer at any distance r from a well at any time t after the start of pumping. It is simply necessary to calculate u from Eq. (8.6), look up the value of W(u) on Table 8.1, and calculate h₀ – h from Eq. (8.7).