WATER WELL STEP TESTING
By Darrel Dunn, Ph.D., PG , Hydrogeologist
This is a technical page on aquifer step testing. To see a non-technical page on this topic, press here.
COOPER-JACOB APPROXIMATION OF THEIS EQUATION
The purpose of this web page is to demonstrate a method for determination of aquifer transmissivity and well loss coefficients from step tests. The method is described by Birsoy and Summers (113). It involves using the Cooper and Jacob approximation of the Theis equation for drawdown in an infinite confined aquifer (66). If one starts with this equation and applies superposition to successive pumping and recovery rates the following equation is obtained via algebraic manipulation involving logarithms:
If the equation is applied to drawdown in a pumping well, r is the effective radius of the well, rw.
The equation may also be applied to injection well testing. Changes in sign and units allow drawdown to become pressure increase, and pumping rate to become injection rate.
Application of Equation 1 requires extensive repetitive calculations. I have written a Fortran program that uses this equation and the analogous equations for internal recovery and final recovery periods to process step test data for determining transmissivity and well loss constants, and another program that uses the transmissivity and well loss constants to calculate a time-drawdown plot that can be compared to the observed drawdown of a test. These programs were used to produce the step test analysis graphs and time-drawdown plots in the following presentation.
CONVENTIONAL STEP TEST
Figure 1 shows an example of an application of this equation. It shows drawdown in a hypothetical pumping well in four successive steps with rates of 200, 400, 600, and 800 gpm. Other values used are :
Aquifer transmissivity = 5000 gpd/ft
Aquifer storativity = 0.007
Effective radius = 0.75 ft
Figure 1. Hypothetical pumping well step drawdown.
It is useful to divide both sides of Equation 1 by Qn. This form of the equation shows that s/Qn (which I have called "adjusted drawdown"} plotted against the logarithm of "adjusted time" is a straight line. The slope of the line is 264/T and the y-intercept is (264/T)log[(0.3T/(1440r2S)]. Figure 2 shows the plot of adjusted drawdown versus adjusted time for the hypothetical pumping well used for Figure 1. Note that the slope is consistent with the 264/T, and the y-intercept is consistent with (264/T)log[(0.3T/(1440r2S)]. The y-intercept on the logarithmic scale is where the abscissa has the value of one. The equation for the y-intercept may be re-arranged to provide a value for r2S, once the value for T has been calculated. The equation for r2S is r2S=(2.0833E-4)xTx(10-IT/264), where I is the value of the y-intercept.
Figure 2. Hypothetical pumping well adjusted time versus adjusted drawdown.
CONVENTIONAL STEP TEST WITH WELL LOSS
Jacob (114) stated that drawdown in a pumping well has two components. One component is drawdown proportional to discharge, and the second component is drawdown proportional to approximately the square of the discharge. The second component is termed "well loss" and represents various effects on drawdown in the well, such as turbulent flow in and near the well, clogging near the well, head loss through a well screen, and losses in an artificial sand or gravel filter in the annulus around a well screen. Jacob represented "well loss" as CQ2, where C is an empirical coefficient. He recognized that CQy, where y is not equal to 2, might more accurately reflect well loss, and higher precision might warrant determination of of the exponent y by trial and error or by a graphical procedure. Rorabaugh (115) applied a graphical procedure and concluded that the exponent y "may be unity at very low rates of discharge, or it may be in excess of 2."
If a term CQ2 is added to equation 1 and C is given a hypothetical value of 0.002 (ft min2/gal2), which is a relatively high value. The extra drawdown for the hypothetical well is shown (red) in Figure 3.
Figure 3. Hypothetical pumping well step drawdown with and without well loss.
If Equation 1 with CQ2 added is divided by Qn, so that the well loss term becomes CQ, and adjusted drawdown is plotted against adjusted time, Figure 4 is the result.
Figure 4. Hypothetical pumping well adjusted time versus adjusted drawdown with well loss.
When the drawdown data is plotted this way, each step plots as a straight on a semi-log graph, and the y-intercept for each step differs from the preceding step by (Qn-Qn-1)C. Therefore, C may be determined by C = (s/Qn-s/Qn-1)/(Qn-Qn-1). That is, the difference in adjusted drawdown for successive time steps is divided by the difference in discharge for the successive time steps. In our hypothetical case shown in Figure 4, this value is 0.2/100, which is 0.002. Thus the input value used for well loss coefficient may be computed from the graph.
These results show that real step drawdown test data from a pumping well plotted as adjusted drawdown versus adjusted time can be used to estimate the transmissivity of the aquifer near the well by using the slope of the lines on the plot, if the aquifer is confined. Furthermore if the well loss exponent is assumed to be 2, the well loss coefficient can be derived from the vertical separation of the plots for the time steps. If step drawdown data from a nearby monitoring well is available, the storativity (S) can be determined from the y-intercept. Since rw2S can be determined from the y-intercept of the adjusted pumping well data, rw can be determined when a monitoring well is available. However, rw may increase as the pumping rate increases, because turbulent flow may extend farther from the well. This is a complicating factor that should be considered when the data is interpreted.
STEP TEST WITH INTERRUPTED PUMPING
The step test analysis method described by Birsoy and Summers allows for interruptions in the pumping. When pumping stops temporarily, the drawdown begins to recover. The recovery equation is the same as the pumping equation except that the term log(0.3T/1440r2S) disappears and the last term in the adjusted time, tn becomes tn/t'n. Figure 5 shows the calculated drawdown for the hypothetical well with 60-minute interruptions between the pumping steps plus 240 minutes of post-pumping recovery.
Figure 5. Hypothetical pumping well step drawdown with interrupted pumping and recovery.
Figure 6 is a plot of the drawdown shown in Figure 5 along with a plot (red) showing drawdown with well loss, using the same well loss coefficients as above.
Figure 6. Hypothetical pumping well step drawdown with interrupted pumping and recovery showing well loss.
Figure 7 is a graph of adjusted drawdown versus adjusted time analogous to Figure 4. The slope of the pumping drawdown is the same as in Figure 4. The transmissivity may still be derived from the slope, and the well loss coefficient may still be derived from the vertical separation of the pumping plots. All of the recovery data plot as a straight line with the same slope as the pumping plots. The y-intercept of the recovery plot is zero. The recovery data includes recovery between pumping periods and recovery after the final pumping period.
Figure 7. Hypothetical pumping well adjusted time versus adjusted drawdown with well loss and recovery.
STEP TEST WITH IRREGULAR PUMPING AND RECOVERY
The step test analysis method described by Birsoy and Summers also allows for irregular changes in pumping rate and duration and for irregular interruptions in the pumping. Figure 8 shows irregular pumping using the same hypothetical aquifer and well loss parameters as above, but with an irregular pumping scenario. In this scenario, pumping begins at 500 gpm. After 30 minutes it is decreased to 200 gpm. After 120 minutes at 200 gpm, the pumping is interrupted for 20 minutes. Then the well is pumped at 400 gpm for 310 minutes, followed by 300 gpm for 180 minutes. Recovery data is graphed for 240 minutes. The resulting drawdown curve without well loss is shown in Figure 8. The drawdown curve that includes well loss using the same coefficient as above is shown in Figure 9.
Figure 8. Hypothetical pumping well drawdown with irregular pumping and recovery.
Figure 9. Hypothetical pumping well drawdown with irregular pumping and recovery showing well loss.
The plot of adjusted drawdown versus adjusted time for this irregular scenario is graphed in Figure 10. The transmissivity may still be calculated from the slope of the plots for the pumping and recovery periods, and the well loss coefficient my still be obtained from the vertical separation of the plots in the same way as for the more regular step tests described above. Consequently, it is possible to accommodate irregularities in a step test. Irregularities might be caused by such events as starting at a pumping rate that is too high and having to reduce it, or temporary pump failure.
Figure 10. Hypothetical pumping well with irregular pumping and recovery - adjusted time versus adjusted drawdown with will loss and recovery.
SINGLE PUMPING STEP (JACOB STRAIGHT LINE AND THEIS RECOVERY)
If Equation 1 and the analogous recovery equation are applied to a single pumping period and subsequent recovery, the graph of adjusted drawdown versus adjusted time shows a plot for the pumping period that is the same as that used in the Jacob straight line method of pumping test analysis, and the plot of the subsequent recovery period is the same as that used in the Theis recovery method. These two pumping test analysis methods are described in Davis and De Wiest (79). Figure 11 shows these plots for the hypothetical conditions described above for 240 minute pumping and recovery periods. The pumping rate used is 200 gpm.
Figure 11. Jacob straight line and Theis recovery plots for the hypothetical conditions.
STEP TEST WITH NONLINEAR ADJUSTED WELL LOSS
As mentioned above, well loss might be more accurately represented by CQy rather than CQ2. The method described by Birsoy and Summers allows for estimation of y and C when y is not equal to 2. In this case, the vertical separation of plots of adjusted drawdown versus adjusted time will not be uniform. For example, assume four steps (100, 200, 300, 400 gpm) as in the first scenario examined above and transmissivity, storativity, and effective radius are still the same, but the well loss coefficient, C, is changed from 0.002 to 0.0002 and the well loss exponent, y, is changed from 2 to 2.5. The drawdown graph analogous to Figure 3 becomes as in Figure 11.
Figure 11. Hypothetical pumping well step drawdown with C=0.0002, y=2.5.
The graph of adjusted time versus adjusted drawdown analogous to Figure 4 is shown in Figure 12. The vertical separation of the plots for the pumping steps is no longer uniform. Instead, the y-intercept of each step differs from the preceding step by (Qny-1-Qn-1y-1)C. Therefore the equation for C is C=[(S/Qn)-(S/Qn-1)]/(Qny-1-Qn-1y-1). This expression gives n-1 nonlinear equations with two unknowns. The equations my be solved graphically. I use a spreadsheet. For the scenario represented in Figure 12, the graphical solution is shown in Figure 13. The lines intersect at C=0.0002 and n=2.5, as expected.
Figure 12. Hypothetical pumping well adjusted time versus adjusted drawdown, C=.0002, y=2.5.
Figure 13. Graphical solution for well loss coefficient and well loss exponent for hypothetical step test.
APPLICATION TO LEAKY AQUIFERS
Step test analysis was originally developed for application to confined aquifers, which are aquifers that are not recharged by leakage from overlying or underlying beds. Such aquifers probably do not exist in nature because all subsurface materials have a finite vertical permeability. So one needs to consider when step test analysis may be applied to leaky aquifers without unacceptable error. For small diameter wells, the drawdown curves for leaky aquifers closely follow the Theis equation until a critical time is reached. This behavior may be seen on graphs of type curves, such as those published by Walton (116). My subjective interpretation of the critical time is shown in Figure 14, where
t is time after pumping started, minutes
T is transmissivity, gpd/ft
r is radial distance, ft
S is coefficient of storativity
P is vertical hydraulic conductivity of the confining bed, gpd/ft2
m is thickness of the confining bed, ft.
Figure 14 applies when only one aquitard is leaking into the aquifer and no water is released from storage in the aquitard.
Figure 14. Estimated value of 1/u when leaky aquifer curve r/B departs from the Theis curve.
Spane and Wurstner (134) compared dimensionless drawdown and dimensionless drawdown derivative type curves for for selected values of β, where β is the second argument in the well function for leaky confined aquifers with storage in the aquitards [H(u,β)] defined as:
β = rw[(K'S'/b'TS)1/2+(K''S''/b''TS)1/2]/4, where
rw is effective radius of the well (L),
K' and K'' are vertical hydraulic conductivities of the superjacent and subjacent aquitards, respectively (L/T),
S' and S'' are storativities of the superjacent and subjacent aquitards, respectively (dimensionless),
b' and b'' are thicknesses of the superjacent and subjacent aquitards, respectively (L),
T is transmissivity of the aquifer (L2/T), and
S is storativity of the aquifer (dimensionless).
Consistent units of length and time must be used when applying this equation.
Spane and Wurstner concluded that nonleaky confined aquifer methods (Theis) can be applied to leaky aquifer test data when β is less than 0.01. Inspection of their graph suggests that a smaller value of β is preferable.
Figure 14 and β are based on a pumped well represented by a line (small diameter). The drawdown of large diameter wells does not follow the Theis curve initially due to the effect of water pumped from well storage. This effect should be considered when interpreting step test data from large diameter wells in leaky aquifers.
For most leaky aquifers, the drawdown at the effective radius of a small-diameter pumping well will remain on the Theis curve for a day or more, so the step test procedure described above may be applied. However, it is often beneficial to follow a step test with a constant discharge test including an observation well located a suitable distance from the pumped well. The computer program titled DP_LAQ may be used to analyze the data from the constant discharge test. This program includes the effect of storage depletion in a large diameter well.
APPLICATION TO WATER-TABLE AQUIFERS
The step test procedure described above is based on the Cooper Jacob approximation of the Theis equation and was originally developed for application to confined aquifers. However, the drawdown in water-table aquifers follows the Theis equation during the early part of a pumping test and during the late part of a pumping test. These Theis curves have been called Type A curves and Type Y curves, respectively. The drawdown at intermediate times is affected by delayed response of drawdown and does not follow a Theis curve. These water-table type-curves are shown in Figure 15. The curves to the left of the r/D values are Type A curves, and the curves to the right are Type Y curves. A more detailed mathematical analysis of the phenomenon of delayed response is given by Neuman (118) and he discusses the limitations of the use of the curves shown in Figure 15. Neuman concluded that the method for analyzing the results of pumping tests using such curves are limited to relatively large values of time, and the limitations become more severe as the distance from the pumping well decreases. An additional complication is that the Dupuit assumption of horizontal flow is used in the development of the water-table aquifer drawdown curves.
Figure 15. Water-Table Aquifer type curves.
As an aid for estimating the applicability of step drawdown testing to water table aquifers, I developed a logarithmic approximation of the value of 1/ua versus r/D where delayed response causes the water table curve to depart from the Type A Theis curve. This graph is shown in Figure 16. Likewise, I developed a logarithmic approximation of the value of 1/uy versus r/D where delayed response ends and the drawdown follows the Type Y Theis curve. This graph is shown in Figure 17.
Figure 16. Graph of r/D versus 1/ua where the water-table drawdown curve departs from the Type A Theis curve.
Figure 17. Graph of r/D versus 1/uy where the water table drawdown curve joins the Type Y Theis curve.
Figure 16 may be used to estimate the time after the beginning of pumping when the water table drawdown departs from the Type A Theis curve, and Figure 17 may be used to estimate the time when the water table curve joins the Type Y Theis curve. To use Figure 16 and 17, one must make preliminary estimates of values for transmissivity, specific yield (Sy), storativity (S), effective radius (rw) (or radius of a monitoring well, r), and delay index (1/α). The formula for r/D given by Prickett is:
r is distance from the center of the pumping well, ft
T is transmissivity, gpd/ft
Sy is specific yield, dimensionless
α is reciprocal of "delay index", 1/minutes
The coefficient α is an empirical constant that relates delayed response to character of the aquifer material. Prickett used values of α determined from pumping tests in glacial drift. His graph is shown in Figure 18. The delay index may be affected by influences other than the texture of the aquifer, but Prickett's results for materials in glacial drift suggests a correlation with aquifer material. Neuman (119) published a review of delayed yield and discussed the physical significance of α.
Consider a pumping test of a fully penetrating well in a 25 thick layer of silt. Assume preliminary estimates of hydraulic conductivity, specific yield, effective radius, and delay index are 0.01 gpd/ft2, 0.01, 0.0002, 0.25 ft, and 10,000 minutes, respectively. Consequently, r/D would be 0.16, which corresponds to 1/ua of about 100 on Figure 16. The formula for ua given by Prickett is ua=(2693r2S)/(Tt). Consequently, t=(1/ua)x(2693r2S/T), and t in this example is about 13 minutes. Therefore, the time on the Type A Theis curve would not be sufficient for step test analysis based on the Type A Theis curve. One might obtain an estimate of transmissivity from an initial 13 minute step if the early pumping rate could be kept sufficiently constant under field conditions and well bore storage effects were negligible. Likewise, r/D=0.16 corresponds to 1/uy of about 250. The formula for uy given by Prickett is uy=(2693r2Sy/T). Consequently t=(1/uy)x(2693r2Sy/T), and t in this example is about 18177 minutes (12.6 days). Therefore the time to reach the Type Y Theis curve would prohibit step test analysis based on the Theis curve. Since well loss is not dependent on the equation for drawdown in the aquifer, the well loss coefficient and exponent might be obtained from a step test in such an aquifer.
At the other extreme of subsurface material, consider a pumping test of a fully penetrating well in a 25 foot thick layer of gravel. Assume preliminary estimates of hydraulic conductivity, specific yield (Sy), storativity (S), effective radius, and delay index are 1,000,000 gpd/ft2, 0.25, 0.000001, 0.5 ft, and 1.0 minutes, respectively. Consequently, r/D would be 0.0052, which corresponds to 1/ua of about 25,000 on Figure 16. The time to depart from the Type A Theis curve would be less than 1E-7 minutes (~zero). The value of 1/uy would be greater than 1000 and the time to reach the Type Y Theis curve would be less than 1E-3 minutes (~zero). Consequently, all drawdown would be on a Theis curve and the step test method described above could be used. Drawdown should be adjusted according to an equation derived by Jacob and presented in many publications (for example, Walton 116).:
sa is drawdown that would occur in an equivalent nonleaky aquifer, feet
swt is observed drawdown in the water table aquifer
m is initial saturated thickness of the aquifer.
Due to the limitations of step testing applied to water table aquifers, it is often good practice to follow the step test with a constant rate pumping test. The software titled WATEQ may be used to analyze data from a constant discharge test of a water table aquifer. WATEQ also includes the effect of partially penetrating wells, without the adjustment described below.
APPLICATION TO PARTIALLY PENETRATING WELLS
The applications of step testing described above assume that the pumping well and any observation wells completely penetrate the aquifer. Often, the wells only penetrate the upper part of the aquifer or are open to some other part of the aquifer. Flow to a partially penetrating pumping well develops a vertical component as it approaches the well. Consequently, the drawdown in the pumping well is greater than the drawdown would be in a completely penetrating well that is following the Theis curve. Butler (120) tabulated values of a partial penetration constant that can be used to convert observed drawdown at the effective radius of a partially penetrating well to equivalent drawdown in a fully penetrating well. The equivalent drawdown would follow the Theis curve and be amenable to step drawdown analysis. The table is based on theory and formulas of Morris Muscat, J. Kozeny and C. E. Jacob, which involved some empirical constants and are based on confined aquifer conditions. Figure 19 is a graph based on this table. The variables in the graph are as follows:
spp is observed drawdown for partially penetrating conditions, ft,
s is equivalent drawdown for fully penetrating conditions, ft,
m is thickness of the aquifer, ft,
Kv is vertical hydraulic conductivity, gpd/ft2,
Kh is horizontal hydraulic conductivity, gpd/ft2.
Figure 19. Graph for estimating the partial penetration constant for a pumping well.
Butler also provides a similar table for adjusting drawdown in observation wells. This table applies when the distance from the pumping well to the observation well is less than 2m(Kv/Kh)1/2. Beyond this distance the effect of partial penetration is generally negligible. Figure 20 is a graph based on part of the table where the radius to negligible drawdown (R) is three times the aquifer thickness. Hantush (69) considers negligible drawdown to be of the order of 0.01 feet. Figure 20 represents the worst case part of the table. Other parts give partial penetration constants for R/m of 5, 10, and 100. This graph is for the case where the observation well is monitoring the same part of the aquifer that is open to the pumping well. Type curves for partial penetration published by Walton (116) show that the partial penetration curves based on an analytical solution are only sub-parallel to the Theis curve but approach it as distance from the pumping well increases and/or the fractional penetration increases. Consequently, there is some uncertainty in the application of step testing in partially penetrating wells based on the Theis curve, especially when the drawdown is measured in the pumping well and the fractional penetration is small.
Figure 20. Graph for estimating the partial penetration constant for an observation well.
The application of step test analysis to partial penetration under water table conditions is more complex. Hantush (69) says that the methods of analysis for tests in confined aquifers can be applied to analyze corresponding situations of complete and partial penetration in water table aquifers if the drawdown is less than 25 percent of the initial saturated thickness. I suggest that the time for the drawdown curve to merge with the Type Y Theis curve should be negligible according to Figure 17. I tentatively suggest less than 10 minutes. Also, the screened (open) part of the partially penetrating well should be near the bottom of the water table aquifer, so that the water table elevation stays above the open interval.
APPLICATION TO REAL STEP TEST DATA
Step test examples presented above are hypothetical, so that the theory could be presented in an uncomplicated manner. This section presents analysis of real step test data provided in the literature using the method of Birsoy and Summers.
Step Test Example - "Confined" Aquifer
Type of aquifer : Confined (nominal).
Lithology: Fine to medium, clayey sand, with some coarse sand and gravel.
Thickness of aquifer: 70 feet.
Length of well screen: 55 feet.
Number of pumping steps: 4.
Discharge and duration of steps:
Table 1. Step test example for for confined aquifer, discharge and duration of steps.
Graph of adjusted drawdown versus adjusted time:
Figure 21. Step test example for confined aquifer, adjusted time versus adjusted drawdown.
Transmissivity = 264/(0.048-0.044) = 66,000 gpd/ft.
Graphical solution for well loss coefficient (C) and exponent (y):
Figure 22. Step test example for confined aquifer, graphical solution for well loss coefficient (C) and exponent (y).
C = 4.1133E-10
y = 3.2
Therefore the well loss contribution to the y-intercept of the first step (Q=545.4 gpm) is negligible and
rw2S = 2.08833E-(0.04*66000/264) = 1.374E-9.
Graph of calculated versus observed drawdown:
Figure 23. Step test example for confined aquifer, calculated versus observed drawdown.
Figure 23 shows that transmissivity, rw2S, and well loss parameters calculated by the method of Birsoy and Summers result in calculated drawdown that closely matches measured drawdown.
The following deficiencies were reported for this step test:
The discharge rate was allowed to decline in each step as the drawdown increased. The average discharge rate measured with a propeller-type meter was reported.
The water levels were measured by the air line method (less accurate than pressure transducer), except for the first step where a weighted steel tape was used.
Water levels in the second step appear to be less accurate than those in the other steps. They do not follow a straight-line trend as well. Consequently I did not use the third step in the analysis.
The well that was tested was gravel packed and had an eight-inch screen reported as 50 feet in one reference and 55 feet in another. I did not adjust the reported drawdown for partial penetration.
This step test data was used in seminal articles on well loss by C. E. Jacob (114) and M. I. Rorabaugh (115). Jacob used a method of analysis similar to the method described above, but it involved less computation. He assumed the well loss exponent (y) equaled 2. His resulting well loss constant (C) was 6.7E-6 ft min2/gal2. If one calculates the step test drawdown based on these well loss parameters, the result is in Figure 24. Figure 24 is analogous to Figure 23, but Jacob's well loss parameters were used instead of the parameters calculated using the Birsoy and Summers method. Figure 24 shows that Jacob's well loss parameters over-estimated the well loss.
Figure 24. Step test for confined aquifer, calculated versus observed drawdown using well loss parameters estimated by Jacob (114).
Rorabaug (115) used this same data to estimate well loss parameters, but did not assume the exponent (y) had a value of 2. He used a trial and error method and obtained y=2.64 and C-4.39E-8 ft min2.64/gal2.64. If one calculated the drawdown based on these well loss parameters, the result is in Figure 25. Figure 25 shows that Rorabaugh's well loss parameters over-estimated the well loss, but not as much as Jacob's well loss parameters.
Figure 25. Step test for confined aquifer, calculated versus observed drawdown using well loss parameters estimated by Rorabaugh (115).
The methods used by Jacob and Rorabaugh were practical when computers were not available to do rapid repetitive calculations. Now that computers are universally available, the computationally intensive method described by Birsoy and Summers may be used to get more complete and accurate results.
Step Test Example - Water Table Aquifer
Type of aquifer: Water Table.
Lithology: Sand and gravel (dominantly coarse sand and fine gravel).
Thickness of aquifer: 46 feet (initial saturated thickness was 36 feet, water table depth 10 feet).
Length of well screen: 15 feet (installed at bottom of aquifer).
Number of steps: 5
Discharge and duration of steps:
Table 2. Step test example for water table aquifer, discharge and duration of steps.
Graph of adjusted drawdown versus adjusted time:
Figure 26. Step test example for water table aquifer, adjusted time versus adjusted drawdown.
Transmissivity = 264/(0.0376-0.0337) = 67,692 gpd/ft.
rw2Sy = 3.2233E-7.
Well loss appears to be insignificant.
Graph of calculated versus observed drawdown:
Figure 27. Step test example for water table aquifer, calculated versus observed drawdown.
The discharge rate in the first step was adjusted at 14 minutes. This adjustment shows up in Figure 25, but had no significant effect on the results as may be seen by the excellent match of calculated versus observed drawdown shown in Figure 26.
Posted March 21, 2013