EDNHAZ: a program for analyzing step drawdown tests

Computers & Geosctences Vol 15, No 6, pp 965-978. 1989

0098-3004/89 $3 00 + 0 00 Pergamon Press pie

Pnnted m Great Britain

EDNHAZ: A PROGRAM FOR ANALYZING STEP DRAWDOWN TESTS K. J. VINES* North West Water, P.O. Box 30, New Town House, Buttermarket Street, Warnngton, WAI 2QG, U.K (Recetved 22 July 1988, revised 14 October 1988)

Abstract--This paper presents a FORTRAN program for analyzing step drawdown tests which has some advantages over tradmonal graphically based methods. A more flexible approach to the planning and execution of stepped pumping tests is possible by using the method of analysis adopted in the program and outhned in the text. Particular advantages are' (I) the ability to cope with pump failures without restarting the test; (2) the easy analysis of a step test starting from a steady pumping conditmn; and (3) the easy treatment of unequal time periods Key Words: Regression analysis, Groundwater, Pumping tests

INTRODUCTION

OUTLINE OF THE EDEN-HAZEL ANALYSIS

Most hydrogeologists, at some stage in their careers, will have to analyze step drawdown tests. There are several methods available, the more important ones being summarized in Clark (1977, 1979). Almost all the methods are essentially graphically based and if carried out manually are tedious and time consuming. Microcomputers with graphics capability are widely used to reduce the tedium to little more than typing the data in, and data loggers are capable of removing even that chore. One problem remains. Whether the analysis is done manually or by micro, the answers produced by the different methods are virtually certain to be different. One would expect, logically, that one of the methods ought to be better than all the others, though it is not obvious which one it is. This paper presents a program for analyzing step tests which seems at least as good as any existing procedure and has several advantages over the graphical methods: offering a more flexible approach to the planning and analysis of step-pumping tests. Four particular advantages are: (1) The ability to cope with pump failures without restarting the test, (2) The easy analysis of a step test starting from a steady pumping condition, (3) The easy treatment of unequal time periods, and (4) The production of statistical parameters which give an idea of the "goodness-of-fit". Though the method is not new, it seems little used and deserves a wider circulation. It was introduced by Eden and Hazel (1973), although the program to which Eden and Hazel refer, exists only in ah internal report (Holloway, 1972), and therefore is not readily available. The author has since obtained a copy of Holloway's program (L. Clark, 1987, pers. comm.), but is not aware of any other, either in print, or in use.

Most hydrogeologlsts will be familiar with the Cooper-Jacob approximation to the classical Theis formula for a well pumping at a constant rate

*Present address: Geology Department, Polytechnic South West, Drake Circus, Plymouth PL4 8AA, U.K. 965

2.3Q 1-2.25Tt 1 D = 41"IT x log,0L r2 S j

(1)

where D T t S Q

= = = = =

drawdown in the aquifer at distance r, transmissivity, time, storativity, and pumping rate.

Provided that u = r2S/4Tt <~ 0.01, the approximation is satisfactory. It is convenient to write Equation (1) as D = [a + bloglo(t)]Q

(2)

where 2.3 [-2.25T 1 a = 4 n r × log,o[, r2 S ] and b =

2.3 4fiT"

The following analysis relies on the principle of superposition of solutions to linear differential equations, which states that any linear combination of two individual solutions to the governing differential equation is also a solution. This allows the variations in pumping rate during a pumping test to be treated by the imaginary introduction of extra pumps, rather than a change in the pumping rate. Suppose, for example, that a well is pumped at 1000 m3/day for 2h and the rate increased to 1500 m 3/day for another 2 h. This can be treated as one pump pumping at 1000 mJ/

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day for 4 h, joined by another pump after 2 h which pumps at 500 mS/day. Reductions in rate can be dealt with by negative pumping rates. At any stage in a pumping test the drawdown in a well is the sum of all the individual effects of the imaginary pumps. To formalize this, suppose that the initial discharge rate is Q0, which is not necessarily zero, and subsequent discharge rates are QI, Q.,, Q3, etc. starting at times tl, t.,, ta, etc. This can be treated as pump 0 pumping at Q0 for the entire test, pump I pumping at AQ~ = Qj - Qo from time to to the end, pump 2 pumping at AQ2 -- Q2 - Qi from t t to the end, and so on. In a step drawdown test, therefore, the head loss at any time t because of the ith step will be D, = AQ,[a + blogl0(t - t,)]. The total change in head at time t is determined by summing the individual steps: D, = ~ AQ,[a + blogt0(t - t,)]

VINES

If the regression analys~s leads to an exact fit, then d will be zero, though it is as well to state that the converse is not true. If water-level measurements are made from a datum other than the initial level, whether pumping or not, then the datum may be assumed to be incorporated within the constant term on the right-hand side of the equauon, as in (5). D, = (Do + d') + aQn + bH, + cQ~

(5)

so that d = Do + d'. The number of unknowns has now risen to four. The program presented in this paper (EDNHAZ, see Appendix) implements the multiple regression technique. There is nothing unusual about the program and readers who have access to other multiple regression packages might decide that it is easiest to adapt their data to fit the package. The program is merely a slightly modified least-squares version from Davis (1973). The only original part consists in organizing the data into suitable forms for the arrays.

where TESTING THE PROGRAM

tn_l < t <<.t,. So far there is no allowance for well losses, but this can easily be added. The usual assumption, and the only one considered here, is that well turbulence generates an extra drawdown which is proportional to the square of the pumping rate. This assumption has been criticized recently by Rushton and Rathod (1988), and indeed is shown here to be physically unrealistic. In spite of this, a good working tool emerges. Because the well experiences only one overall pumping rate at a time, there is only one well loss component for each step. The total drawdown now may be written as

O, = aQ,, + bH, + cQ~

(3)

where

H, = ~ AQ, logl0(t - t,) and t~_l < t ~< tn. lu|

This is an equation with only three unknowns: a, b, and c. Measurements from step tests will normally provide data leading to at least 100 different equations of type (3). Though not all these equations are independent, there will be many more independent triples of equations than the one needed to solve for a, b, and c. Because the data are not perfectly consistent, the solutions for a, b, and c will differ according to the selection of the three equations. Linear regression techniques are normally used to determine a "best" solution to such overdetermined systems, and this course is adopted here. The multiple linear regression model includes a general constant term (d) which transforms (3) to D, = d + aQ~ + bH, + cQ2,,.

(4)

It is usual to test programs using theoretical data from analytical solutions. This is not possible with step-test analysis, because the form of the well-loss component is assumed rather than rigorously justified. Empirical tests are possible, however, and can operate by using the results from a short-term step test to predict the results of a long-term test. Pumping tests of new boreholes are often carried out this way: a step test to determine the yield of the hole, followed by a constant rate test for a week or more. A comparison of the step-test predictions with actual longterm data will give an idea of how well the program works. There is an obvious drawback here, in that for best results the data need to come from an aquifer as near ideal as possible. This is not always easy to arrange. If the effects of a recharge boundary become evident 3 days into a constant rate test and the step test lasted for only 1 day, there is little chance that the step test results will provide a satisfactory estimation of the longer term drawdown of the constant rate test. As an alternative to theoretical data, sets of pseudotheoretical data have been generated using the radial model program of Rushton and Redshaw (1979). This program solves numerically the onedimensional differential equation given here: t9 [ m k r ~ r ] + d"~

mkr dD r r

s dD ~t + q

(6)

where D m kr S q

= = = = =

drawdown, saturated thickness of the aquifer, radial permeability, storage coefficient (confined or unconfined), inflow per unit area, which may be a function of the radius.

Analyzing step Runnmg the program generates time: drawdown data that are essentially free from a well-loss component. The latter can easily be added at the output stage by a shght modificauon of the program, that is [drawdown + cQ~].The modified program then can be used to generate any amount of test data, with d~ffering well-loss coet~cients, well radii, transmissivities, and so on. (These data, of course, can be analyzed using the traditional graphical methods, to compare the two approaches.) This test is not entirely satisfactory because water is not conserved and the equauon of continuity is not satisfied. The term cQ~is not a function of time, and adding it to the drawdown is equivalent to removing a volume nr:cQ~ of water from the well extra to the amount pumped. The quantity of water removed from the well is therefore not equal to the quanttty pumped from it. This error can be reduced by making the well radius very small, but, as shown here, this does not always help. Even with a very small well radius, the program will generate a d~scontinuity of cQ~at the start of the test, and one of c [(Q,): - (Q,_~)2] at the start of subsequent steps. Though the mathematical analysis can cope with this, it is clear that the model used to fit real step-test data ts inadequate. No real well will experience an instantaneous drawdown. One result of this discrepancy between theory and reality is that the estimates of transmissivity from E D N H A Z tend to be too low if all the data are included. There is a way around this problem which is used in graphical methods, namely: leave out the early time data in each step. These do not normally follow the trend of the later data. This stratagem is employed in analysis by type-curve methods, where the later parts of ume: drawdown curves are used to obtain a "fit" with, say, the Theis curve. This is a perfectly rational approach. At the start of a pumping test, or of a new step in a step test, the theoretical model assumptions have little relation to reality. As pumping proceeds, however, the finite radius of the well and its storage capacity become less-and-less important, and the theoretical model becomes a better approximation. The real curve and the theoretical curve should approach one another asymptotically, until the limited areal extent of the aquifer begins to make its presence felt, though this may take a long time. Table I. Aquifer parameters for radial model runs

Permeabthty (m/day) Confined storage Unconfined storage Max,mum radms (m) Top of aqmfer (m) Bottom of aquifer (m) Water level (m) Recharge Cond,tton at boundary Well loss const [m/(m3/d) 2]

Confined model

Unconfined model

1.0 0.0001 0.1 10,000 0 500 0 800 0 00 0.0 F,xed head 10 -5

1.0 0.1 0 I 10,000.0 00 300.0 0.0 0.0 Freed head 10 -5

These parameters create aquifers which ate 300m thick, havmg a transmlsswtty of 3 0 0 m ' / d a y . The confined aquifer is at a depth of 500m

d r a w d o w n tests

967

Table 2 Analysts of results from confined model output (r ~ 0 3 m) Data set All I-9out 1-19 out 1-29 out

Stor x 104 Trans c x 105 d × 10 L F-stat x 10 - ° 86717 21210 I 7290 I 5823

2343 2809 287 0 289 5

09918 09987 09993 0 9996

-37651 -05974 - 0 3067 - 0 2008

0028 2251 l0 155 23 065

It is instructive to examine what effect this has on the analysis of a step drawdown test. Data for a confined aquifer have been generated from the radial model program of Rushton and Redshaw as described, simulating four steps of 3 h each, with the pumping rate increasing by 1000m~/day. A recovery step of 3 h was also simulated. Time: drawdown data were output every minute. Table l shows the aquifer parameters which were used to generate the data. Table 2 shows the results ofanalyzing these data using E D N H A Z in four different ways. At first, all the data were used to produce estimates of T and S which are shown in the appropriate columns. Subsequent analyses were performed leaving out the first I-9, 1-19, and 1-29 min of each step. The improvement in the estimates is clearly shown. F~gure 1 illustrates the closeness of the "fit" for the worst situation. For the sake of clarity, not all the data are plotted. Table 3 shows the results from analyzing a similar test, m th~s example with a well radius of 0.001 m. These estimates are s~gnificantly better, particularly those for the transmissiwty. It ~s clear that the improvement in parameter estimates is caused by the reducuon in well radius, because nothing else is changed. There may be two reasons for this. First, the effect of well storage is virtually removed; and secondly, a smaller value of r also makes u smaller. The approximation m (1) is therefore closer. However, even for a well radius of 0.3m and a time of imin (1/1440 days), u is only 0.0174. The radial model program can also be used to simulate pumping in an unconfined aquifer. First, the average saturated depth between a pair of adjacent nodes is calculated. Then, if this average depth is less than the aquifer thickness, the transmissivity between the two nodes is reduced in proportion. This is clearly not an exact method, but it is a better approximation than an unmodified confined model. Table 1 shows the parameters used to generate the data and Tables 4 and 5 show estimates for a well of 0.3m and 0.000005 m radius respectively. The results are somewhat mixed. The esumates of transmissivity for the 0.000005 m radius are better than those for the 0.3 m radius, but worse for storage. In both examples, the approximations improve if early time data in each step are left out of the analysis. The test, in fact, is a rather severe one, because the drawdown during the last step is well over 60% of the unconfined aquifer thickness (see Table l and Fig. l).

LONG-TERM TESTS Having obtained estimates for a, b, c, and d from

968

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an analysis of a step test, they can be used to estimate the drawdown at any stage of a long-term test. The drawdown data are easily generated, using the radial model. Figure 2 compares radial model data for a 20 day constant rate test (4000 m3/day, r = 0.3 m), with a step-test estimate for the confined example which uses all the data. As can be seen, the estimated transmissivity is too low and the fit is not particularly good. Figures 3 and 4 are similar comparisons using steptest estimates which leave out the first 1-9, and 119 min respectively, of each step. Leaving out the first 1-29 min results in an even better fit, but the improvement is barely detectable on a diagram. Practically speaking, the 10rain data give an excellent fit. TWO REAL STEP TESTS In 1984, the North West Water Authority carried out an investigation of the groundwater resources in the Egremont area of the West Cumbrian coast of northwestern England (see Fig. 5), by drilling and pump testing several trial boreholes. The aquifer consists largely of the Permo-Triassic St. Bees Sandstone, a well-cemented, medium-to fine-grained sandstone, with very occasional mudstone/marl bands, and locally micaccous. The sandstones lie on the eastern edge of the Irish Sea basin, and they dip and thicken to the west. Much of the aquifer is covered by glacial deposits of both boulder clay and sands and gravels, though the distribution of each is not well known. Laboratory measurements of intergranular pcrmeabilitics fall mostly in the range 10-100 mD, so

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900

fit,

almost all the useful water flow is effected through fissures. The boreholes were cored with a rotary rig, the cuttings being flushed with air. On completion, cleaning was carried out by air-lift pumping, using the borehole itself as the inductor. The testing program consisted of a I-day step test, followed, after a recovery period, by a 7-10 day constant rate test. Two of the boreholes, at Thornhill and Black Ling, have been selected for illustrative purposes, one to show how well EDNHAZ can work, the other to show how badly things can go wrong, Brief details of the boreholes are given in Table 6. At Thornhill it was possible to fit in six pumping steps plus the recovery step. At Black Ling, where the yield was lower, only four pumping steps, plus the recovery, were feasible. The steps ranged from 80 to 120 rain in length. Water levels were measured according to the following program: I rain 2rain 3 rain 10 rain

intervals intervals intervals intervals

for for for for

the the the the

first 10rain next 10rain next 40 rain remainder of the step.

Table 6 shows the results of analyzing the step tests using EDNHAZ. Data for rain 1-5 were left out for the analysis. Table 6 also shows some statistical parameters from the computer analysis which indicate the goodness-of-fits. Using the estimated values for a, b, c, and d, and the measured pumping rate, Equation (5) may be used

Table 3. Analysis of results from confined model output (r - 0.001)

Table 4. Analysis of results from unconfined model output (r = 0 3m)

Dataset

d × I0 ~ F-stat x I0 - I °

Data set

Storage Trans c × l0 s d x l02

- 9.2219 -4.6699 -2.6708 - 1.7766

All I-9 out 1-19 out 1-29 out

0 1371 0.1257 0 1222 0.1201

All I-9 out 1-19 out 1-29 out

Stor x I0 4 Trans

1.2245 1.1450 1.1070 I 9883

296.9 297.9 298.3 298 6

c x I0 s

0.9998 0.9999 0.9999 0.9999

1.337 48 089 147.878 311.276

2766 287.9 290 7 292.1

0.9991 I 0005 1.0008 1 0009

-9 -3 - I - I

2779 1336 8691 2603

F-star x 10 -9 0.764 8 189 18.467 28,369

Analyzing step drawdown tests Table 5

Analysm of results from unconfined model output (r = 0.000005 m)

Data set

Storage

Trans

c x 105

d x 102

F-stat x 10 - ~

All I-9 out 1-19 out 1-29 out

0.2157 0 1827 0 1600 0 1419

293,2 295.3 296,9 298 4

I 0139 1.0142 1,0145 1,0146

- 1.6000 - 0 3643 0 5140 1 2058

2 0467 2 6678 3 1686 3 6221

The graphical methods usually rely on equal time periods for each step to ease the analysis. Because the program uses each data point, neither the equality of step lengths, nor the order In which the steps are carried out is important. It is feasible to start off w~th a high pumping rate, followed by lower rates during succeeding steps, or even swap increasing and decreasing steps. However, there is an advantage in using steps of increasing size. When switch-off occurs at the end of such a test, the rate of change of water level is larger than at any other stage m the test, and the well-storage effect is also at its largest. If the recovery step causes trouble in the analysis, it can be left out. If, on the other hand, the test starts off with a large pumping rate, and continues with reducing steps, the well-storage effect will probably be even larger than on recovery, and it will not be possible to leave it out of the analysis. The pumping rate is not particularly significant, although it ~s obviously important to test the borehole to its capacity at some time during the exercise. A corollary of this ~s that even pump fadures can be handled, provided water-level measurements continue to be made. The period during which the pump is inoperative is treated as a recovery step. The period of recovery can extend overmght if necessary, though corrections to water levels caused by changes in atmospheric pressure become more critical the longer the period is extended.

to calculate the drawdown at any stage m a simulated pumping test. H, in this example will consist of only one term, Qlogm(t). Thornhill was pumped for 7 days at 2220 m 3/day, the drawdown at the end of the test being 5.23 m. The simulation of the test provides an excellent match between theory and practice. After 5 rain, the difference between prediction and measurement is rarely more than 10 cm, and usually not more than 5 cm (see Fig. 6). The analytical results from the test of Black Ling seem to be better than those from Thornhill, but the simulation of the constant rate test is very poor. Figure 7, which compares the two sets of data, shows why. The measured levels fall on a smooth curve which nowhere approximates a straightline, and suggests that an element of recharge was affecting the water levels. It is possible that rainfall, present throughout almost all the test, was responsible for at least some of the recharge. A combination of permeable drift cover and groundwater movement, which in view of the low intergranular permeability, must be dominated by fissure flow, ensures that rainfall is quickly channelled, to recharge the aquifer, at least locally. At Thornhill there was little rain during either the step test or the constant rate test.

CHANGES IN STEP-PUMPING TEST PROCEDURE Using the program introduces a change in the philosophy of water-level measurement during steppumping tests. In the West Cumbrian examples described, water-level measurements were taken more frequently at the start of a step than towards the end. This scheme ensures that points plotted on logarithmic graph paper are more-or-less evenly spaced, and

ADVANTAGES OF THE EDNHAZ PROGRAM There are several advantages to the use of the program for analyzing step tests which are not immediately apparent. The major advantage is that the steps are not tied to any particular length of time. 164

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is used almost universally. This is unnecessary for the regression analysis. In fact it is better to change the relative frequency of measurement because the effects of well storage are most noticeable at the start of a step, not at the end. If data loggers are available, it is no more difficult to organize one water-level measurement every minute than any other program. There is a distinct advantage in having a large quantity of data, particularly when it is envisaged that some of the early data in a step are going to be left out of the final analysis. Not only is the statistical confidence normally improved, but the analysis is also weighted towards the later times in each step, when the effects of well storage are at a minimum. Though this worsens the prediction of the drawdown early on in a step, it is of no practical significance.

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WELL-STORAGE CORRECTION Attempts have been made to improve the fit of the regression curve at the beginning of a step by correcting for well storage. If the drawdown in a pumping well increases, some of the water comes from the well itself, not from the aqmfer. The flexibility of the program allows a well-storage correction to be made by reducing the pumping rate appropriately in each measurement period, and treating zt as a separate step. In the recovery phase, because water is added to increase well storage, water continues to be abstracted from the aquifer even though pumping has stopped, and the correction is m the opposite sense. This approach has not proved successful, partly, it is believed, because of practical difficulties in accurate measurement of water level and flow. Even when data loggers

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Analyzing step drawdown tests Table 6 Resultsfrom Thornhfll and Black Ling step Thomhdl Depth of drift (m) Depth of borehole(m) Depth to water (m)

tests

Black Ling

13 95 120.23 10 30

Borehole diameter (m)

24 00 140 45 26.30 0 20 110.0

0 20

Transmlsuvlty(mz/day) Storatlvlty(rca.ect = 0 2 m) d (estimatedmitmldrawdown) Correlation coefficient F stattstlc No of readings

918 0 0.092

0.087

- 0.03 0 9992 33,114 155

- 0 23 0 9997

971

a yield: drawdown curve needs to integrate the effect of the borehole and the aquifer• Correcting for well storage, as suggested, separates one from the other. The easiest approach, and the one which seems to give the best results, uses an average pumping rate for the whole of the step. Inaccuracies of flow measurement then are spread out throughout a longer period and are less noticeable.

63,081

141

LIMITATIONS OF EDNHAZ

are employed for water levels, the instruments available for flow measurement are usually rather coarse in comparison. There may be scope for improvement of step-test analysis in this direction, but more precise instruments than are available to the author are needed. There is another problem with this approach, which probably has the largest effect and is insurmountable. Each new pumping rate introduces a discontinuity in the analysis because of the instantaneous change in the term cQ,~. It is these discontinuites which are hardest to reconcile with real data. Increasing their number only makes things worse. Ifstep tests are carried out to produce yield: drawdown curves, this approach might not be helpful anyway. To be useful,

E D N H A Z has its limitations, and, in common with all the other analytical methods, estimates of storativity might not be reliable. The program cannot cope with step tests where the value of the storativity changes dunng the test. This can occur either when an aquifer changes from a confined to an unconfined condition, or when the effects of delayed yield become significant. There is no way round these problems. In both situations the storativity gradually increases throughout the test, and no one value can possibly be satisfactory. However, estimates of the storativity are usually dogged by lack of knowledge of the effective radius of the well, and the test is rarely long enough for the true storativity to be developed anyway. In spite of these objections, the program has been used

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Figure 6. Thomhill borehole constant rate test: 20/7/1984. extensively for more than 2yr to construct yield: drawdown curves for use in short term predictions, normally with good results. SUGGESTED STEP-TEST PROGRAM

To make the best use of EDNHAZ, the traditional step-test program needs to be changed. Suggested changes are as follows: (I) The step length should be at least I h, preferably longer. Time available for an extra long step is best devoted to the highest pumping rate. (2) If data loggers are available, measure water levels every minute. If levels are dipped manually, once every minute can be tedious. Reducing the frequency to once every 2min after the first 30min has proved to give acceptable results. 12

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(3) The number of steps is less critical than for the graphical methods of analysis, although a s~milar principle applies: the more the better. When time is limited, it is preferable to concentrate on a few steps of adequate length, instead of cramming in several short steps. A minimum of three pumping steps is suggested. As with normal methods of analysis, it is important to test the full capacity of the borehole. CONCLUSIONS

The E D N H A Z program listed m the Appendix offers a more flexible approach to the planning and analysis of step pumping tests than traditional graphical methods permit. Four particular advantages are (I) The ability to cope with pump failures without restarting the test,

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tO000

Analyzing step drawdown tests (2) The easy analysis of a step test starting from a steady pumping condition, (3) The easy treatment of unequal time periods, and (4) The production o f statistical parameters which give an indication o f the "goodness-of-fit".

Acknowledgmems--Thls paper Is published with the perrrassion of the Chief Scientist of the North West Water Authority, although the views expressed are the author's own. J. H. Tellam and K. R. Rushton are thanked for valuable reviews of earlier drafts of the manuscript. REFERENCES Clark, L , 1977, The analysis and planning of step draw-

973

down pumping tests: Quart. Jour Engng Geol., v 10, no. 2, p. 125-143. Clark, L., 1979, The analysis and planning of step drawdown pumping tests: a clarificat,on: Quart. Jour. Engng Geol., v. 12, no. 2, p. 134. Davis, J. C., 1973, Statistics and data analym m geology: John Wiley & Sons, New York, 550 p. Eden, R. N., and Hazel, C. P., 1973, Computer and graphical analysis of variable discharge pumping tests of wells: Civil Engineering Trans. Inst. Eng. Austral,a, p. 5-10. Holloway, H. G., 1972, Analysis of pump test data: Program WU71: Systems Branch, Irngatton and Water Supply Commlsston, Queensland, 26 p. Rushton, K. R., and Rathod, K. S., 1988, Causes of nonlinear step pumping test responses' Quart. Jour. Engng. Geol., v. 21, no. 2, p. 147-158 Rushton, K. R., and Redshaw, S.C., 1979, Seepage and groundwater flow: John Wdey & Sons, New York, 339 p

APPENDIX

Program Notes The FORTRAN-77 program listed with this Appendix was developed on a Digital Equipment Corp. Professional 380 m~crocomputer. This is a desktop version of the PDP 11/73 minicomputer and memory addressing limitations restrict the size of the single precision (4 byte) arrays to about 450. Thus no more than 450 data points can be analyzed, wluch usually is more than adequate for tests completed in I day. For longer tests on a similar machine, virtual arrays will have to be used, which slows program execution down about 100 times. These figures assume the presence of a floating-point processor. On IBM PC or compatible microcomputers, there should be no problems, even with a 64 kbyte limit on array size. The tests using pseudotheoretical data were run on an ICL mainframe computer, using double precision (8 byte) variables. Input data format is as follows: TITLE (ABO) TIME, WATER-LEVEL, PUMPING RATE (free format)

-

1.0

0.0

0.0

The program expects a pumping rate for each time value, whether it is the same as the previous one or not. The rule ,s that the pumping rate ts continued fight to the end of a step. The chanB¢ is indicated in column 3 at the first measurement after a change in rate. Thus the pumping rate for time 0.0min is 0.0 m'/day if the test starts from rest. If the test starts from a condition of steady pumping, this rate needs to be known and inserted in column 3 of the data file. Water-level data can be in the measured format (i.e. meters below datum) or as drawdowns. The latter are calculated anyway. The use of a negative number to indicate the end of the data, rather than using "END ffi nnnn" in the READ statement, allows easy adjustment of the data set. Leaving out the last step, say, involves only the insertion o f " - 1.0" at the appropriate line in the file. Beware of using the program to analyze constant-rate pumping tests. Theoretically this should not work, because the bQ and cQ2 terms are perfectly correlated. However, the limited precision on computers might lead to a seemingly successful run of the program. The results are not reliable. Most of the output from the program is self-explanatory. Those readers not familiar with statistical techniques should refer to one of the many texts available on the subject (e.g. Davis, 1973). In passing, it might be useful to mention that the F-test is usually not particularly helpful in terms of model assessment. The author has not yet experienced an analysis of a step test which was not significant at the 1% level. The numbers are usually so large that they fall far outside the range of any F-test tables. The numbers are useful in the comparison of analyses of different subsets of the same data. The rule here is: the bigger the better. The program is available on disk for both the Professional 380 and the IBM PC format for an all inclusive cost of £10. A working EXE file can be provided for anyone who does not possess a FORTRAN compiler. A set of test data and its analysis by EDNHAZ will be included. The author would be interested to hear of comparisons between analyses carried out with EDNHAZ and those using graphical methods,

Program hstmg overleaf

974

K . J . V~SES

Program Listing 1 2 3 4 5 6 7 8 9 10 11 1Z 13 1l+ 15 16 17 18 19 20 21 ~

PROGRAR EDNHAZ C C C c C C C C C C C

* Written by K. J. Vines, * Program to * step test. * Data * data

calculate

s h o u d be points.

input

April/Ray

aquifer

from

• using

data

The p r o g r a m

obtained

can

handle

from

up t o

DIHENSION T ( I O O O ) , S ( I O O O ) , Q ( I O O O ) , H ( I O O O ) , D E L T Q ( I O 0 0 ) DIRENSION X ( 1 0 0 0 , 4 ) , X R ( 1 0 0 0 , 4 ) , D ( l O O O , 3 ) , A ( 4 , 4 ) , B ( 4 ) , C ( 4 ) EQUIVALENCE ( X ( 1 , 1 ) , S ( 1 ) ) , ( X ( 1 , 2 ) , q ( 1 ) ) , ( X ( 1 , 4 ) , H ( 1 ) ) CHARACTER*80 T I T L E C ND=IO00 RD=4 C C

================================

Open i n p u t

~

================================

C

25 26 27 C

and o u t p u t

C

files

========================

Read d a t a

~0

========================

C

data

OPEN(UNIT=I,STATUS='OLD',FZLE=°EDNHAZ.IN ' ) OPEN(UNIT=2,STATUS=°OLD',FZLE='EDNHAZ.OUT ' )

29 C 31 32 35 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 ?0 71 72 ?3 74 75

parameters

EDNHAZ.IH.

Z3 C

~8

1986

from

EDNHAZ.IN

READ(I,1000) TITLE 1000 FORRAT(ASO) DO 2 0 0 1 = 1 , 1 0 0 0 READCl,*,END=210)

T(I),S(I),Q(I)

C Z F ( I .EQ. 1 ) THEN DATUR=S(1) IF(q(1) .EG. 0 . 0 ) D E L T q ( 1 ) = O . O ELSE DELTQ(Z)=Q(I)-q(I-1) END I F C S(I)=S(I)-DATUR C IF(T(Z) .LT. NTIRES=I-1 GO TO 2 1 0 END I F 2 0 0 CONTINUE

0.0)

THEN

C 210

CONTINUE

C H(1)=O,O C DO 400 I = 2 , N T I R E S H(Z)=O.O DO 4 1 0 J = l , I - 1 TT=T(I)-T(J) IF(TT .LE. 1.0) TT=I.000001 H(I)=H(Z)+DELTQ(J÷I)*ALOGIO(TT) 410 CONTINUE 4 0 0 CONTINUE C DO 4 4 0 I : I , N T I M E $ X(Z,3)=Q(I)*a(Z) 440 CONTINUE ¢

C C C C

* This part * Statistical N=NTIRES M=4

C

of

t h e p r o g r a m comes f r o m D s v i s ' s book: Analysis in the Geological Sciences -

Wiley,

1973

a

* ,

1000

* ,

Analyzing step drawdown tests 7~

C

=================================

77 C

Print

78

=================================

C

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 9/+ 9S 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 116 115 116 117 118 119 120 121 122 123

title

and i n p u t

data

matrix

WRITE(2,2200)T I T L E 2200

FORMAT(1Hl,SX,86('*')/6X,'* MULTIVARIATE ANALYSIS OF PURPING ' * 'TEST DATA, THEORY GIVEN IN EDEN & HAZEL, 1973 *'1 * 6X.'*',82X,'*'/bX.'* ',ASO,' *'16X,86('*')1/I/I//)

C 130 CONTINUE C C C C

=======================================

Standardize

and p r i n t

input

data

matrix

=======================================

201

DO 201 I = I , N DO 201 J = l , R XM(I,J)=X~I,J) CONTINUE CONTINUE

C CALL S T A N D ( X R , N , R , N D , R D ) C C C C

==========================================================

Calculate

and p r i n t

matrix

of

correlations

between

columns

==========================================================

CALL RCOEF(XR,N,R,ND,RD,A,RR) CALL PRINTR(A,R,R, MR, RM) WRITE(Z,2OQT) C C C C

=======================================

Set

up and s o l v e

simultaneous

equations

=======================================

DO 100 I = Z , R C(I-1)=A(I,1) DO 100 J = 2 , M A(I-I,J-1)=A(I.J) CONTINUE 100 CONTINUE C C C C

======================

Solve

Linear

equations

======================

CALL S L E ( A . C . M - 1 . M M , I . 0 E - 0 8 ) C C C C

==========================================

Calculate

partial

regression

co-efficients

==========================================

125 127 128 129 130 131 132 133 134 135 136 137 138 139 140 161 142 163 144 165 166 167 168 169 150 151 152 153 154

975

101

DO 1 0 1 1 = I , M A(1,I)=O.O A(2,I)=O.O DO 101 J = I , N A(1,I)=A(1,I)+X(J,I) A(2.I)=A(Z.I)+X(J.I)**2 CONTINUE CONTINUE

C

AA=I.01N AB=I,0/(N-1) AC=SQRT((A(2,1)-A(1,1)*A(1,1)*AA)*AB) B(1)=A(1,1)*AA DO 102 I=~,M B(I)=C(I-1)*ACISqRT((A(2,Z)-A(1,I)*A(1,I)*AA)*AB) B(1)=B(1)-B(1)*A(1,I)*AA 102 CONTINUE C C C C

============================================================

Calculate

estimated

value

and d e v i a t i o n

for

each o b s e r v a t i o n

============================================================

DO 103 I=I,N D(I,I)=X(I,I) D(I,2)=B(1) DO 106 J=2,M

D(I,Z)=D(I,Z)+B(J)*X(Z,J) 106 103 C c c C

CONTINUE D(I,3)=D(I,1)-D(I,2) CONTINUE

Print

water

Level,

draw-down,

estimated

draw-down,

===================================================================

error,

pump r a t e

K. J. V~nEs

976 155 156 157 158 159 160 161 162 163 16~ 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 l?b 197 198 199 200 201 2O2 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 22~. 225 226 227 228 229 230 231 232 233

~RITE(2,2020) FORRAT(1H1,13X,°TIRE * ' DIFFRNCE RATE(Q)

2020

W LEVEL DRAW-DOWN EST DDOMN', DELTA Q ' I I )

DO 1 1 0 I = I , N T I R E S WRITE(2,20]O) I,T(I),S(Z)+DATUR,(D(I,J),J=I,3),~(I),H(I) 2030 FORRAT(I4,4X,FlO.2,4F10.3,2FlO.2) 110 CONTINUE C C ================================================= C Print partlal regression co-efficients C Calculate transmissivity and storage co-efficient C ================================================= WRITE(2,2009) B(1),B(Z),B(4),B(~) TRANS=2.]/4.0/3.141595/B(4) SY=B(2)/B(4) I F ( S Y .GT. 3 7 . 0 ) SY=~7.0 R2S=2.25*TRANS/1440.OI10.O**SY ~RITE(2,2010) TRANS,R2S I F ( S Y . L T . 1 5 . 0 ) THEN ~RITE(2,2011) R2SIO.O1,R2SIO.O4,R2S/O.09 ELSE WRITE(2,2012) END I F C C =================================================== C Print standardized partial regression co-efficients C WRITE(2,2013) (C(I),1=1,3) C C ======================== C CaLcuLate error measures C SY=O.O SYY=O.O SYC=O.O SYY¢=O.O DO 105 I = I , N SY=SY+D(I.1) SYY=SYY+DCI,1)**2 SYC=SYC+D(I,2) SYYC=SYYC+D(I,2)**2 105 CONTINUE C SST=SYY-SY*SY*AA SSR=SYYC-SYC*SYC*AA SSD=SST-SSR NDFI=R-1

ARSR=SSR/FLOAT(NDF1) NDF2=N-R ANSD=SSD/FLOAT(NDF2) R2=SSR/SST R=SQRT(R2) F=ARSR/ARSD NDF]=N-1 C C C C

=====~==========~===

Print

error

measures

WRITE(2,2000) WRITE(2,2001) WRITE(2,2002) WRITE(2,200]) WRITE(2,2004) C 2000

SSR,NDF1,AMSR, F SSD,NDF2,ARSD SST, NDF3 R2,R

FORRAT(///'OSOURCE OF',I]X,'SUR OF DEGRESS OF ' VARZATZON',13X,' SQUARES FREEDOR SQUARES * IX,65(1N-)) *

REAN',/ F-TEST',/

FORAAT('lREGRESSZON',TX,Flk.],ZT,2X, F11.]I46X,IPE20.9,0P)

2001 2002 2003 2004 * 2005 * * * 2006 *

FORRAT(' D E V Z A T Z O N ' , 1 1 X , F l l . 3 , Z T , 2 X , Fll.5) FORRAT('OTOTAL V A R I A T Z O N ' , Z X , F 1 4 . 3 , ZT) FORRAT('OGOODNESS OF F I T = ',F12.9/ 'OCORRELATION COEFF = ' , F 1 2 . 9 ) FORRAT('O INPUT DATA RATRIX - ' , l X , 'COLURNS = VARIABLES, ROWS = O B S E R V A T I O N S ' I / 5 X , ' C O L U R N 1 - DRAWDOMN S ( R ) , COLURN 2 - DISCHARGE Q ( R * * ] / D A Y ) ' / 5X,'COLURN ] - Q**2, COLURN 4 - H = S U R ( D Q . L O G ( T - T I ) ) ' ) FORRAT('O STANDARDIZED INPUT DATA RATRIX - ' , l X , 'COLUMNS = VARIABLES, ROMS = O B S E R V A T I O N S ' / /

Analyzing step drawdown tests 234 235 236 237 238 239 240 2&1 242

243 244 245 246 247 246 249 250 251 252 253 25~ 255 256 257 258 259 260 261 262

263

* 5X,'COLURNS ARE - DRAWDOWN, DISCHARGE, DISCHARGE**2, ' , * 'HEAD LOSS FUNCTION') 2007 FORRAT('O CORRELATION RATRIX. ' / I * SX,'COLUNN 1 - DRAWDOWN S ( R ) , COLURN 2 - DISCHARGE Q ( R * * 3 1 D A Y ) ' I * 5X,'COLURN 3 - Q * * 2 , COLURN 4 - H = S U R C D Q . L O G ( T - T I ) ) ' ) 2008 FORRAT('O COLURN I : DRAWDOWN, COLURN 2 : ESTZRATED DRA~DOWN', * ' , COLURN 3 = D E V I A T I O N ' ) 2009 FORRAT(/II'O REGRESSION C O - E F F I C I E N T S ' , I P / / * ' 1',E15.6,' (D I N I T I A L DRAWDOWN I N R E T R E S ) ' / * ' 2',E15.6,' (A : R/R**31DAY)'I * ' 3',E15.6,' (B : R I R * * 3 1 D A Y ) ' I * ' 4',E15.6,' (C : R I ( R * * 3 1 D A Y ) * * 2 ) ' I I * ' EQUATION ZS D ( T ) = D ( O ) + AQ(N) + B H ( T ) + C Q ( N ) * Q ( N ) ' ) 2010 F O R R A T ( 1 H O , 1 P , 4 X , E 1 3 , 6 , ' TRANSRZSSZVITY ( R E T R E S * * Z l D A Y ) ' , I * 5X,E13.6,' S TIRES R * * 2 ' ) 2011 FORRAT(1P, 5 X , E q 3 . 6 , ' S, I F R 1S 0 . 1 R E T R E S ' I * 5X,E13.6,' S, I F R I S 0 . 2 R E T R E S ' I * 5X,E13.6,' S, I F R I S 0 . $ R E T R E S ' / ) 2012 F O R R A T ( 1 H O , 4 X , ' * * * * UNREALISTIC VALUE FOR S R * * 2 * * * * ' ) 2013 FORRAT(IIIIP,'O 1',E15.61' 2',E15.61' 3',E15.6/ * '0 STANDARDIZED PARTIAL R E G R E S S I O N ' , I X , * 'COEFFICIENTS CONSTANT TERN = 0 . 0 ' ) C C C 9999 CONTINUE CLOSE(UNIT=l) CLOSE(UNIT=2) STOP END C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBROUTINE P R I N T R ( A , N , R , N 1 , R 1 ) DIRENSION A ( N 1 , R 1 ) DO 100 I B = l , R , l O IE=IB÷9 IF(IE .GT. R) I E : R WRITE(2,2000) (I,I=IB,IE) DO 101 J = I , N WRITE(2,2001) J,(A(J,K),K=ZB, IE) 101 CONTINUE 100 CONTINUE 2000 F O R R A T ( 1 H O , 1 X , q O I 1 5 ) 2001 FORRAT(1H , 1 5 , 1 0 F 1 5 . 3 ) RETURN END C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBROUTINE S L E ( A , B , N , N I , Z E R O ) DIMENSION A ( N 1 , N 1 ) , B ( N 1 ) DO 100 I = I , N

264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 28O 281 282 DIV:A(I,I) 283 IFCABS(DIV) .LE. ZERO) GO TO 9999 284 DO 101 J:I,N 285 A(I,J):A(I,J)IDIV 286 101 CONTINUE 287 B(I)=B(I)/DIV 288 DO 102 J:1,N 289 IF(I .NE. J) THEN 290 RATIO:A(J,I) 291 DO 103 K = I , N 292 A(J,K)=A(J,K)-RATZO*A(I,K) 293 103 CONTINUE 294 B(J)=B(J)-RATIO*B(~) 295 END I F 296 102 CONTINUE 297 100 CONTINUE 298 RETURN 299 300 C 9999 W R I T E ( * , 1 0 0 0 ) 301 1000 FORRAT(' PROGRAR ABANDONED - PIVOT ELERENT EFFECTIVELY Z E R O ' l / 3O2 * ' NO UNIQUE S O L U T I O N ' / / 303 * ' CHECK NURBER OF PURPZNG S T E P S ' l / ) 304 6 CLOSE(UNIT=2) 305 CLOSE(UNIT=l) 3O6 STOP 307 END 3O8 309 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 SUBROUTINE S T A N D ( X , N , R , N I , R I ) 311 DIMENSION X ( N I , R 1 )

977

978

K. J. VI~Es 312 DO 100 I = l , M 313 SX=O.O 314 SXX=O.O 315 DO 101 J = I , N 316 SX=SX+X(J,Z) 317 SXX=SXX+X(J,I)**2 318 101 CONTINUE 319 XMEAN=SXIFLOAT(N) 320 SD=SQRT((SXX-SX*XREAN)IFLOAT(N-1)) 321 DO 102 J = I , N 322 X(J,I)=(X(J,I)-XMEAN)ISD 323 102 CONTINUE 324 100 CONTINUE 325 RETURN 326 END 327 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 SUBROUTINE R C O E F ( X , N , R , N 1 , R 1 , A , R 2 ) 329 DIRENSION X ( N 1 , R 1 ) , A ( R Z , R 2 ) 330 AN=I.01N 331 DO 1 0 0 I = l , R 332 DO 1 0 0 J = I , R 333 SXl=O.O 334 SX2=O.O 335 SXlXl:0.O 336 SXZX2=O.O 337 SXlX2=O.O 338 DO 101 K = I , N 339 SXI=SXI+X(K,I) 340 SXZ=SXZ+X(K,J) 341 SX1XI=SX1XI+X(K,I)**2 342 SX2X2=SX2X2+X(K,J)**2 343 SXIX2=SXIX2+X(K,I)*X(K,J) 344 101 " CONTINUE 345 R=(SXlX2-SXI*SX2*AN)I 346 * SQRT((SX1XI-SXI*SXI*AN)*(SX2X2-SX2*SX2*AN)) 347 A(I,J)=R 348 A(J,I)=R 349 CONTINUE 350 100 CONTINUE 351 RETURN 352 END