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AQTEBLOG (The AQTESOLV Blog)

## Thursday, August 28, 2014

###
Interview With C.V. Theis

## Tuesday, May 13, 2014

###
Classic USGS Pumping Test Video

## Saturday, May 3, 2014

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## Tuesday, February 25, 2014

###
Show/Hide Well Data on Plots

## Wednesday, February 5, 2014

###
Checking Your Cooper and Jacob Match

## Thursday, January 16, 2014

###
IARF and the Cooper and Jacob Method

## Thursday, January 9, 2014

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Infinite-Acting Radial Flow Regime

At the outset of interpreting a pumping test, an important feature to
detect in the response data is **infinite-acting radial flow**. During infinite-acting radial flow, steady pumping in a nonleaky
confined aquifer of infinite extent produces late-time drawdown that changes with the logarithm of
time since pumping began. Infinite-acting behavior occurs after wellbore storage effects have dissipated and before the influence of aquifer boundaries. One may observe this flow regime by plotting the familiar Theis (1935) solution on semilog axes (Figure 1); at late time, a graph of dimensionless drawdown, w(u), versus dimensionless time, 1/u, plots as a straight
line and the logarithmic derivative is constant.

This late-time behavior of the Theis model is well known to groundwater hydrologists. The period of infinite-acting radial flow is the basis for the tried-and-true Cooper and Jacob (1946) method which
one may use to find the transmissivity and storage coefficient of a nonleaky confined aquifer by matching a straight line to late-time drawdown data from a constant-rate pumping test plotted on semilog axes.

A derivative plot is very useful for detecting the infinite-acting radial flow regime under steady pumping conditions. One starts by looking for a**derivative plateau** (constant derivative) to tentatively identify infinite-acting radial flow. For example, the response data from a constant-rate pumping test shown in Figure 2 suggest that infinite-acting radial flow conditions are present after approximately 30 minutes of pumping once the derivative stabilizes.

Note that the derivative plateau in Figure 2 exhibits some noise; derivative fluctuation at the plateau is common in the application of derivative analysis, but the presence of a near-constant derivative in this example is clear and corresponds to the emergence of a well-defined straight line in the drawdown data.

The foregoing results suggest that the derivative plot is an indispensable tool for aquifer test interpretation; however, the mere existence of a derivative plateau does not immediately confirm infinite-acting radial flow conditions. For example, an aquifer limited by a no-flow boundary can produce a constant derivative that is not diagnostic of the infinite-acting radial flow regime. In such a situation, geologic mapping of lithologic contacts, faults and other low-permeability features would be invaluable in making a correct interpretation.

Visit Aquifer Testing 101 for a catalog of derivative plot signatures with more examples of**infinite-acting radial flow** in pumping tests.

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News and views on AQTESOLV, the world's *LEADING* software for the analysis of aquifer tests

Kick back and enjoy a lively and engaging video interview of Charles Vernon Theis who introduced the concept of transient flow to the interpretation of pumping tests. His seminal paper from 1935, "The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage," is familiar to all hydrogeologists. Learn more about his many accomplishments as well as the groundwater scientists with whom he worked during his long career at the U.S. Geological Survey.

Get more details on the Theis solution at the AQTESOLV website.

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Get more details on the Theis solution at the AQTESOLV website.

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Check out this classic video of a pumping test conducted by the U.S. Geological Survey in the Little Plover River basin in Wisconsin, USA.

The USGS study made news in a Milwaukee Sentinel article in 1961. A new study in this river basin is underway.

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The USGS study made news in a Milwaukee Sentinel article in 1961. A new study in this river basin is underway.

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Are you a member of the **LinkedIn** professional network?

If so, please visit the new AQTESOLV LinkedIn page and click the**Follow** button to receive periodic LinkedIn updates pertaining to aquifer testing and AQTESOLV.

While you are visiting LinkedIn, follow the HydroSOLVE company page and feel free to send me an invitation to connect as well!

If so, please visit the new AQTESOLV LinkedIn page and click the

While you are visiting LinkedIn, follow the HydroSOLVE company page and feel free to send me an invitation to connect as well!

When using AQTESOLV to perform visual curve matching on a data set with multiple observation wells, it's often useful to turn off the display of one or more wells and focus your attention on the remaining wells.

To hide observation data for particular wells, choose**Edit>Wells** and select the wells in the list that you'd like to hide. Right click over the selection and choose **Hide Observations**.

To turn on the display of hidden observations wells, select the wells to display, right click over the selection and choose**Show Observations**.

Find more tips on the**AQTESOLV** website!

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To hide observation data for particular wells, choose

To turn on the display of hidden observations wells, select the wells to display, right click over the selection and choose

Find more tips on the

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Groundwater hydrologists use the Cooper and Jacob (1946) solution to determine the aquifer properties of a nonleaky confined aquifer using drawdowns measured during a constant-rate pumping test. The approximate method of Cooper and Jacob derives from the Theis (1935) type-curve method when the variable u in the Theis well function, w(u), is sufficiently small (i.e., time is large or radius is small).

AQTESOLV has built-in tools to check the validity of analyses performed with the Cooper and Jacob method and to assist you with visual curve matching. First, you can superimpose on your plot the time when u is less than a critical value. Choose**View>Options>Plots tab** and check the option for **Valid time for Cooper-Jacob approximation** (Figure 1).

Click the **Valid Time tab** to set the **critical value of u** (Figure 2). A critical value between 0.01 and 0.05 is typical.

When you activate the valid time option, a **dashed vertical line** appears on your time-drawdown or composite plot to indicate the time when the Cooper and Jacob approximation meets the critical value of u (Figure 3). The position of the vertical line is a function of T (transmissivity) and S (storage coefficient) as well as radial distance between the pumping and observation wells..

**AQTESOLV** is through derivative analysis. After superimposing the derivative on your semilog plot of drawdown versus time, match the Cooper and Jacob straight line to drawdowns corresponding to the derivative plateau, i.e., during the period of infinite-acting radial flow (Figure 4).

Use these two techniques to obtain more reliable results when using the Cooper and Jacob method. Get AQTESOLV and start applying these powerful tools in *your* next pumping test interpretation!

Visit the AQTESOLV Knowledge Base to find more tips on the using the software. The AQTESOLV documentation also includes a number of examples illustrating the use of the Cooper and Jacob straight-line method.

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AQTESOLV has built-in tools to check the validity of analyses performed with the Cooper and Jacob method and to assist you with visual curve matching. First, you can superimpose on your plot the time when u is less than a critical value. Choose

Figure 1. Check option to display valid time for Cooper and Jacob solution. |

Figure 2. Set critical value of u used to check validity of Cooper and Jacob approximation. |

Figure 4. Cooper and Jacob method (blue line) matched to drawdown data during period of infinite-acting radial flow when derivative reaches plateau (red line). |

Visit the AQTESOLV Knowledge Base to find more tips on the using the software. The AQTESOLV documentation also includes a number of examples illustrating the use of the Cooper and Jacob straight-line method.

We invite you to follow HydroSOLVE on LinkedIn and Google+!

Groundwater hydrologists routinely use the pumping test interpretation method of Cooper and Jacob (1946), a straight-line approximation of the Theis (1935) type-curve solution, to estimate the hydraulic properties of nonleaky confined aquifers from drawdown data measured during the period of infinite-acting radial flow (IARF) when drawdown changes linearly with the logarithm of time. One expects to find IARF after the dissipation of early-time phenomena (e.g., wellbore storage) and prior to the advent of late-time effects such as aquifer boundaries.

Consider the well-known Theis solution for a constant-rate pumping test in a nonleaky confined aquifer of infinite extent:

where*s* is drawdown [L], *Q* is pumping rate [L³/T],
*T* is transmissivity
[L²/T], *t* is
time [T], *r* is radial distance [L] and *S* is storage
coefficient [-].

The Cooper and Jacob method approximates the Theis well function,*w(u)*, by truncating the infinite series in (4) after the first two terms:

The approximation in (5) assumes that*u* is small, i.e., *t* is large and *r* is small. It is precisely when (5) is valid that infinite-acting radial flow is identified from drawdown data.

How small should*u* be for (5) to be valid? The threshold for IARF (i.e., when Cooper and Jacob becomes valid) is often given as u ≤ 0.05 (Driscoll 1986) or u ≤ 0.01 (Kruseman and de Ridder 1994); however, we can see for ourselves when (5) is valid by plotting w(u) versus 1/u (dimensionless drawdown versus dimensionless time) on semilog axes and looking for a constant logarithmic derivative plateau (Figure 1).

It is evident from Figure 1 that the slope of the drawdown and the value of the derivative are each essentially constant when 1/u ≥ 50. Hence, when u ≤ 0.02, the Cooper and Jacob approximation of the Theis well function is safe to use. Figure 1 demonstrates just how useful derivative analysis can be for the detection of the IARF regime.

Visit Aquifer Testing 101 to learn more about pumping test interpretation with more examples of**infinite-acting radial flow**.

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Consider the well-known Theis solution for a constant-rate pumping test in a nonleaky confined aquifer of infinite extent:

where

The Cooper and Jacob method approximates the Theis well function,

The approximation in (5) assumes that

How small should

Figure 1. Theis (1935) solution (blue curve) plotted as w(u) versus 1/u on semilog axes. Logarithmic derivative shown by red curve. |

Visit Aquifer Testing 101 to learn more about pumping test interpretation with more examples of

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A derivative plot is very useful for detecting the infinite-acting radial flow regime under steady pumping conditions. One starts by looking for a

Figure 2. Derivative plot of drawdown (squares) and derivative (crosses) measured in an observation well during constant-rate pumping test in a nonleaky confined aquifer (Walton 1962). |

The foregoing results suggest that the derivative plot is an indispensable tool for aquifer test interpretation; however, the mere existence of a derivative plateau does not immediately confirm infinite-acting radial flow conditions. For example, an aquifer limited by a no-flow boundary can produce a constant derivative that is not diagnostic of the infinite-acting radial flow regime. In such a situation, geologic mapping of lithologic contacts, faults and other low-permeability features would be invaluable in making a correct interpretation.

Visit Aquifer Testing 101 for a catalog of derivative plot signatures with more examples of

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Labels:
derivative analysis,
flow regimes,
pumping tests

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