Derivation of Well Equations

By Bill Scott©

Environmental Science
Murdoch University
Perth, Western Australia


When water is pumped from a fully penetrating well1 at a constant rate, it creates a cone of depression in the piezometric surface; the effect of this withdrawal extends outward with time. In an unconfined aquifer, the decline in the water table summed or integrated over the Radius of Influence is an indicator of the volume of the groundwater plane that is affected by the withdrawal of water. This volume, multiplied by the Storage Coefficient, must equal the volume of water withdrawn from the well. The radius of influence, however, increases with time and unsteady or transient flow exists.

Here we use the stress on the aquifer to determine properties on a scale of 100's of metres, the transmissivity $T$ and the storage coefficient $S$. These parameters identify the conveyance and storage properties of the aquifer.

Well Equations

The formulation of the problem follows an unsteady heat transfer analogy in cylindrical coordinates, assuming angular symmetry and using Darcy's Law:

The cylinder in the centre represents the bore, which penetrates the entire depth of the aquifer. At a distance $r$ from the bore, water flows through a cylindrical shell following Darcy's law \begin{equation} \text{Flow of water in (rate of input)} = \left. K A \dfrac{\partial H}{\partial r} \right|_{r+\Delta r} \end{equation}

where $A$ is the area perpendicular to the flow or the area of the cylinder wall and $K$ is the hydraulic conductivity and $H$ is the total head. The rectangle indicates that all quantities within are evaluated at the given value of the independent variable. \begin{equation} \text{Flow of water out (rate of output)} = \left. K A \dfrac{\partial H}{\partial r} \right|_{r} \end{equation}

and the Rate of Accumulation is \begin{equation} S \dfrac{\partial V}{\partial t} \end{equation}

where $S$ is the storage coefficient and $V$ is the overall volume displaced with water and $t$ is the time. The mass balance requires that the Rate of Input minus the Rate of Output equal the Rate of Accumulation; therefore

\begin{equation} \left. K A \dfrac{\partial H}{\partial r} \right|_{r+\Delta r} - \left. K A \dfrac{\partial H}{\partial r} \right|_{r} = 2 \pi\, r S \Delta r \dfrac{\partial H}{\partial t} \end{equation}

when $\Delta r$ is small. Dividing by $\Delta r$, taking the limit as $\Delta r \rightarrow 0$ and using the definition of a partial derivative,

\begin{equation} \dfrac{\partial}{\partial r} \left( K A \dfrac{\partial H}{\partial r} \right) = 2 \pi\, r S \dfrac{\partial H}{\partial t} \end{equation}

Further, if $K$ is constant and $D$, the depth of the aquifer is approximately constant, we define the transmissivity $T$ as

\begin{equation} T = K D \end{equation}

and can write

\begin{equation} \dfrac{\partial^2 H}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial H}{\partial r} = \dfrac{S}{T} \dfrac{\partial H}{\partial t} \end{equation}


\begin{equation} \dfrac{\partial^2 h}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial h}{\partial r} = \dfrac{S}{T} \dfrac{\partial h}{\partial t} \end{equation}

since the elevation of all sampling points is the bottom of the aquifer, here considered as a flat plane of impervious material. This is the basic equation for transient flow to a well.


  • one directional flow only
  • vertical flow is neglected
  • flow is uniformly distributed in depth

These are the standard Dupuit-Forchheimer assumptions, commonly used in solutions to simplified groundwater problems. In addition, note that the true three-dimensional nature of the problem limits this solution to:

  • cylindrical symmetry
  • constant Transmissivity ($T = KD$)

Of course, $D$ cannot be truly constant as the water table is lowering. This means that the water table change must be small compared to the overall depth of the aquifer or penetration of the well.

Theis’ Equation

If, in addition, we restrict the solution to specific initial and boundary conditions

\begin{align*} H &= H_{0} \text{ for all } r \text{ when } t = 0\\ & \\ H &= H_{0} \text{ when } r = \infty \text{ for all } t \end{align*}

and the well discharge rate $Q$ is a constant, the solution, in terms of the drawdown $s$: \begin{equation} s = H_{0} - H \end{equation}


\begin{equation} s = \dfrac{Q}{4 \pi T} \int_{u}^{\infty} \dfrac{\mathrm{e}^{-u}}{u} du \end{equation}

This is Theis Equation; $u = {r^2 S}/{4 T t}$ is a dimensionless group that varies with both the position $r$ of the observation well and the time $t$.

The exponential integral, called the Well Function $W(u)$, is easily programmed on a calculator using the infinite series2:

\begin{equation} W(u) = \int_{u}^{\infty} \dfrac{\mathrm{e}^{-u}}{u} du = -\gamma - \ln u + u - \dfrac{u^2}{2 \cdot 2!} + \dfrac{u^3}{3 \cdot 3!} - \dfrac{u^4}{4 \centerdot 4!} + \ldots \end{equation}

where $\gamma$ is Euler’s constant with a value of 0.57725665.

A version of the curve, called the Reverse Type Curve, is presented on the following pages. This equation and the plotted curve are widely used in practice and are preferred over the equilibrium steady-state equation (Theim Equation, see Bouwer, pg 67) because:

  • steady state flow conditions are not required
  • only one observation well is necessary
  • $S$ can be determined
  • short pumping periods will generally suffice

It is important, however, to note the specific assumptions that apply:

  • homogeneous, isotopic, uniformly thick aquifer of infinite areal extent
  • the pumped well penetrates the entire aquifer
  • there is only horizontal flow
  • the well diameter is infinitesimal so that storage in the well can be neglected
  • Darcy’s Law is applicable through the region, including close to the turbulent withdrawal point
  • water removed from storage is discharged instantaneously with the decline of head.

Also note that it is not necessary for the piezometric surface to be horizontal before pumping; the simple use of drawdown as a measure of pumping accounts for this. In the case of a sloping piezometric surface (there is nearly always a slope), any steady, horizontal flow can be rather precisely added to the unsteady, values as calculated from Theis Equation. Nevertheless, seldom, if ever, are the above assumptions strictly satisfied; strictly speaking the Theis Equation is invalid for unconfined aquifers. The last point, as listed, reiterates that the formulation presumes that water is released instantaneously from storage when the head is lowered. In a confined aquifer this is essentially true since the released water comes chiefly from gravity drainage of the void space within the cone of depression. In an unconfined situation this draining is not instantaneous, the storage coefficient varies with time and increases at a diminishing rate; ultimately it is equivalent to Specific Yield. Hence, application of the Theis Equation to unconfined aquifers requires that

  1. the drawdown be small, and
  2. the equation only be applied after a minimum pumping time
  3. the observation well be located more than $H_0/5$ from the pumping well.

If, indeed, $s$ is large compared to $H_0$, the Boulton Equation (see Bower, 1978, page 76) should be used, this expression has a different well function. The reader should be aware that there are 100s of different forms of this curve, depending on the situation. See, for example, Fetter’s book (1994, pages 219-243). Below is presented a special, reverse Theis type curve, for both drawdown and recovery. All of the curves can use the matching method for solution, as presented below. Even these days with computers, this ‘inversion’ or ‘parameter estimation’ is easy, most informative and more satisfying when done by hand with standard logarithmic graph paper.

Theis’ Method

Normally in a drawdown test one measures the drawdown at a monitoring bore at position $r$ as a function of time. The point, of course, is to estimate the Transmissivity $T$ and Storage Coefficient $S$ of the aquifer. This means that the Theis equation is implicit and must be solved either by trial and error, clever graphical procedures, or approximate techniques. The classical procedure is to note that Theis’ well equation can be written

where the quantities in the square brackets are unknown constants. From this expression it is apparent that if $s$ is plotted as a function of $r^2/t$ on log-log paper, the shape of the curve should be the same as the $W(u)$ versus $u$ curve, with only a displacement of the two axes. This is because the argument and the dependent function are simply multiplied by constants which correspond to offsets on log-log paper. Following this, it also becomes apparent that this scheme works equally well if $s$ is plotted versus $t/r^2$; this is the basis of the Reverse Type Curve as presented earlier.

\begin{equation} \left[ \dfrac{4 \pi\, T}{Q} \right] \cdot s = W\left( \left[\dfrac{S}{4 T}\right] \cdot \dfrac{r^2}{t} \right) \end{equation}

The procedure is to plot $s$ versus $t/r^2$, as measured3, and match the $W(u)$ curve with the data. This is the solution. Use one transparent sheet and slide the two curves along, horizontally and vertically and ONLY horizontally and vertically, until the theoretical curve MATCHES or FITS the data. Select a convenient match point and drive a straight pin through the two sheets4 — Any point will do but, for convenience, say you use $1/u = 4Tt/Sr^2 = 4$ and $W(u)$ = 1.04; get the corresponding $s$ and $t/r^2$ values from the plot of the real data5 and calculate $T$ and $S$.


The above model seems restrictive but, indeed, many different variants of the matching procedure allow for almost any variation in the condition of the aquifer. Fetter (pages 219-243) presents techniques for getting parameters from unconfined situations, with leaky aquifers and with storage in the aquitard. With the superposition principle, Freeze and Cherry (pages 325-335) show how transient solutions can be obtained with a system of n wells with pumping rates $Q_1, Q_2,\ldots, Q_n$. The Theis equation is mathematically linear and variations of it can, to some degree of approximation, also be considered linear. This being the case, one simply adds up the solutions from several, simultaneous effects, noting that the boundary and initial values must be correct.

This works remarkably in the case of a well near an impermeable wall. In a vertical section, one simply inserts an artificial well into the problem at a position within the wall, directly opposite the well, the same distance from the wall:

When one considers a restrictive boundary on two sides of a pumping well, the situation is doubly reflected, as one sees in a room with mirrors on opposite sides. The reflections go off to infinity. In plan view, the situation is illustrated by a well in an alluvial valley, with impermeable boundaries on both sides:

The real well is reflected through both the left and the right boundaries, and those reflections are re-reflected through both boundaries, producing an infinite number of reflections. This means that the solution is a series if additions of well functions, each at different distances from the monitoring piezometer P:

\begin{equation} s = \dfrac{Q}{4 \pi\, T} W(u_1) + \dfrac{Q}{4 \pi\, T} W(u_2) + \dfrac{Q}{4 \pi\, T} W(u_3) + \dfrac{Q}{4 \pi\, T} W(u_4) + \ldots \quad \textrm{ where }~ u_i = \dfrac{r_i^2 S}{4 T t_i} \end{equation}

Of course, the calculations only proceed until the additions of extra image wells make no real difference on the drawdown. Note that this is simply the superposition of many wells with equal drawdown and simply related positions. In the general case of superposition, they need not be equal and, provided the initial and boundary values are correct, all sorts of different combinations can be used to suit different situations. For instance, the wells may be started at different times or the same well can be undergoing a series of stepped drawdowns, to test its efficiency (see Freeze and Cherry, 1979, page 328).


This is a special case of superposition; mathematically, instead of turning off the real pump, an artificial recharge pump is turned on. This means that there is no net flow into the well and the ‘country’ recovers from the stress of pumping. Following from above, at the time the real pump is turned off, another pump is turned on (but in recharge) and

\begin{equation} s = \dfrac{Q}{4 \pi\, T} \left[W(u_1) - W(u_2)\right] \quad \textrm{ where }~ u_1 = \dfrac{r_1^2 S}{4 T t} \textrm{ and } u_2 = \dfrac{r_2^2 S}{4 T t'} \end{equation}

The symbol $t$ is the time from the original start of pumping and $t'$ is the recovery time or the time since pumping was stopped. The Reverse Type curve on this page and a following page (working size) give calculated values. Here it is expected that the time $t$ is plotted on the abscissa. The time since pumping was stopped is given in intervals of 1, 2, 5, 10,… Note that, at long times or close to the pumping well (small $r$), the recovery is abrupt and can be easily distinguished and read on the horizontal axis. Short times or measurements close to the well exhibit much less distinct turn-off times, these are distinguished by a faint symbol on the graph and the fact that values correspond to times of 1, 2, 5, 10,… It is clear that recovery effects exhibit dramatic trends on the curves and that the resolution of the Reverse Type curve is greatly enhanced by looking at recovery. Importantly, a distinct ‘knee’ in the curves allows the determination of the storage coefficient $S$. If the data are only viewed well after recovery, $S$ cannot be obtained with recovery data alone (see Bouwer, page 99 and Fetter, page 226). Recovery does allow the use of the drawdown at the pumping bore and can side-step the problem of well losses; in this case Theis has produced a logarithmic solution similar to the Jacob straight line method (see Bouwer, page 99). The Jacob method is an alternative to the use of the Theis method and is useful when $u$ is very small (?0.01, small $r$ or large times; see Fetter, pages 224-229, Groundwater Hydrology Notes, page 22, or Freeze and Cherry, pages 347-349).

Here a matching procedure is followed with recovery; the time the pump was turned off should be known; call this $t_f$. With matching of the ‘hip’ in the curve the $1/u$ value at turn off should be known; call this ($1/u_f$). Since the distance from the pumping bore is known,

\begin{equation} \dfrac{T}{S} = \dfrac{1}{u_1} \cdot \dfrac{r^2}{4 t_f} \end{equation}

It only remains to calculate $T$ from a value of the well function with a known value of the drawdown and pumping rate $Q$,

\begin{equation} T = \dfrac{Q}{4\pi\,s} W(u) \end{equation}

Of course, the ordinary matching technique will work equally as well on the drawdown/recovery data. One can even use the drawdown data only, or the recovery data only. The major point, here, is that the combined drawdown and recovery data give a VERY SPECIFIC fix to the well function curve. All that is needed is to collect recovery data for a sufficient period of time, a period that definitely establishes the convex (or concave) shape of the recovery curve.

After one has established a ‘best fit’, a ‘pinhole’ through the standard curve and reading the 4 pieces of information gives the solution. Two pieces of data come from the real data, the opaque paper sheet; two pieces of data come from the theoretical curve, the transparency sheet.


Selected from Plot of Pumping Data

$t$ = selected time
$s$ = selected drawdown


Selected from Theoretical Curve

$(1/u)$ = has a value as a symbol
$W(u)$ = has a value as a symbol

$Q$ is the average pumping rate and $r$ is the distance of the monitoring bore from the pumping bore. It remains to calculate $T$ and $S$ values:

\begin{equation} T = \dfrac{Q W(u)}{4 \pi\, s} \quad \quad S = \dfrac{4 T t}{(1/u) r^2} \end{equation}


Bouwer, H. (1978) Groundwater Hydrology, McGraw-Hill, New York, 480 pages. ISBN 0-07-006715-5

Fetter, D. W. (1994) Applied Hydrogeology, 3rd Edition, Prentice Hall, New Jersey, 691 pages. ISBN 0-02-336490-4

Freeze, R. A. and J. A. Cherry (1979) Groundwater, Prentice Hall, New Jersey, 604 pages. ISBN 0-13-365312-9

Scott, W. D. (1996) Groundwater Hydrology Notes, Murdoch Print, Perth, W. A., 290 pages. ISBN 0-86905-374-4

© Please feel free to copy and use the material. However, the author retains the copyright. He requests that he be acknowledged, should the material be passed along in any form. Permission of the author is required before copies are passed along for profit.

1 A well that fully penetrates the aquifer and draws water, more or less uniformly, from the whole depth of aquifer.

2 A program for its calculation and a high resolution graph are presented in the Groundwater Hydrology Notes (Scott, 1996, page 23a).

3 If data are acquired at different distances, simply use the appropriate distances and times. If only one collection point is used, the time t can be used directly on the plot. The logarithmic curve will, ultimately, accommodate the different arguments, provided everything is done consistently (see the abscissa of the enclosed Reverse Type curve).

4 Unfortunately, you have to do this at least once; it is not easy to understand the procedure otherwise.

5 The two curves must have precisely the same logarithmic scale. The advice here is to use one curve as a standard reference (as enclosed) and make a transparent copy along with other, opaque copies so that all the curves are identical, including any possible distortions during the copying.