C-3. Equilibrium pumping test

a. In an equilibrium type of pumping test, the well is pumped at a constant rate until the drawdown in the well and piezometers becomes stable.

b. A typical timedrawdown curve for a piezometer near a test well is plotted to an arithmetical scale in figure C-5 and to a semilog scale in figure C-6. (The computations in fig C-6 are discussed subsequently.) Generally, a time-drawdown curve plotted to a semilog scale becomes straight after the first few minutes of pumping. If true equilibrium conditions are estab­lished, the drawdown curve will become horizontal.

200 300 400

TIME AFTER STARTING TO PUMP, MIN

 

500 ‘ ‘1,000

 

(Courtesy ofVOP Johnson Division)

 

Figure C-5. Drawdown in an observation well versus pumping time (arithmetical scale).

 

C-3. Equilibrium pumping test

C-3. Equilibrium pumping test

(Courtesy of UOP Johnson Division)

 

C-3. Equilibrium pumping test

Figure C-6. Drawdown in an observation well versus pumping time (semilog scale).


Подпись: QwПодпись: (C-l)C-3. Equilibrium pumping test

The drawdown measured in the test well and adjacent observation wells or piezometers should always be plotted versus (log) time during the test to check the performance of the well and aquifer. Although the example presented in figure C-6 shows stabilization to have essentially occurred after 500 minutes, it is con­sidered good practice to pump artesian wells for 12 to 24 hours and to pump test wells where gruuity flow conditions exist for 2 or 3 days.

c. The drawdown in an artesian aquifer as measured by piezometers on a radial line from a test well is plot­ted versus (log) distance from the test well in figure C-7. In a homogeneous, isotropic aquifer with artesian flow, the drawdown (H-h) versus (log) distance from the test well will plot as a straight line when the flow in the aquifer has stabilized. The drawdown H2—h2 versus (log) distance will also plot as a straight line for gruuity flow. However, the drawdown in the well may be somewhat greater than would be indicated by a pro­jection of this straight line to the well because of well entrance losses and the effect of a “free” flow surface at gravity wells. Extension of the drawdown versus (log) distance line to zero drawdown indicates the effective source of seepage or radius of influence R, be­yond which no drawdown would be produced by pump­ing the test well (fig, C-7).

d. For flow from a circular source of seepage, the coefficient of permeability k can be computed from the formulas for fully penetrating wells.

Artesiun Flow.

2rrkD(H—h)
ln(R/r)

Gravity Flow,

_ nk(H2—h2)

4w ln(R/r)

where

Qw = flow from the well D = aquifer thickness H = initial height of groundwa­ter table (GWT) h = height of GWT at r (H-h) or (H2—h2) = drawdown at distance r from well

R = radius of influence

An example of the determination of R and k from an equilibrium pumping test is shown in figure C-7.

e. For combined artesian-gravity flow, seepage from a line source and a partially penetrating well, the coef­ficient of permeability can be computed from well-flow formulas presented in chapter 4.

C-3. Equilibrium pumping test

DISTANCE FROM PUMPED WELL, FT

NOTE: DRAWDOWNS PL DTTEDWEREM EASU RE DAFTERGROUNDWATERTABLE

HAD STABILIZED.

 

EXAMPLE: FULLY PENETRATING 12-IN. TEST WELL (FILTERED), ARTESIAN FLOW AQUIFER THICKNESS, D = SO FT PUMPING RATE, Q^= 200 GPM PUMPING PERIOD г 1,000 MIN

 

rw= 1.0 FT

 

Qw In (R/fw)^ goo In(520/1)

2ffD (h – h ) 1 7.5 (2fT> (50) (28.5) ‘ w’

 

k

 

0.0189 FPM

 

(Courtesy ofUOP Johnson Division)

 

Figure C-7. Drawdown versus distance from test well.

 

C-3. Equilibrium pumping test

C-4. Nonequilibrium pumping test.

a. Constant discharge tests. The coefficients of transmissibility T, permeability k, and storage S of a homogeneous, isotropic aquifier of infinite extent with no recharge can be determined from a nonequilibrium type of pumping test. Average values of S and T in the vicinity of a well can be obtained by measuring the drawdown with time in one or more piezometers while pumping the well at a known constant rate and analyz­ing the data according to methods described in (1), (2), and (3) below.

(1) Method 1. The formula for nonequilibrium

 

(C-3)

 

where H – h =

Q« =

W(u) =

T’ =

 

drawdown at observation peizometer, feet well discharge, gallons per minute exponential integral termed a “well func­tion” (see table C – 1)

coefficient of transmissibility, gallons per day per foot width

 

and

 

1.87r2S

T’t’

 

u

 

(C-4)

 

Подпись: TM 5-818-5/AFM 88-5, Chap 6/NAVFAC P-418
Подпись: С-8

Table C-L Values OfW(u) for values of u,

 

C-3. Equilibrium pumping test

where

r = distance from test well to observation piezometer, feet s = coefficient of storage t’ = elapsed pumping time in days The formation constants can be obtained approximate­ly from the pumping test data using a graphical meth­od of superposition, which is outlined below.

Step 1. Plot W(u) versus u on log graph paper, known as a “type-curve,” using table C-l as in figure c-8.

Step 2. Plot drawdown (H-h) versus r2/t’ on log graph paper of same size as the type-curve in figure C-8.

Step 3. Superimpose observed data curve on type-curve, keeping coordinates axes of the two curves parallel, and adjust until a position is found by trial whereby most of the plotted data fall on a segment of the type-curve as in figure C-8.

Step 4. Select an arbitrary point on coincident

segment, and record coordinates of matching point (fig. C-8). •

Step 5. With value of W(u), u, H-h, and r2/t’ thus determined, compute S and T’ from equations (C-3) and (C-4).

Step 6. T and к from the following equations:

T’

T = ^ ^ (square feet per minute) (C-5)

Г

k= 1Q 77QD (feet per minute) (C-6)

(2) Method 2. This method can be used as an ap­proximate solution for nonequilibrium flow to a well to avoid the curve-fitting techniques of method 1 by using the techniques outlined below.

Step 1. Plot time versus drawdown on semilog graph as in figure C-9.

Step 2. Choose an arbitrary point on time-draw­down curve, and note coordinates t and H-h.

Step 3. Draw a tangent to the time-drawdown

C-3. Equilibrium pumping test

EXAMPLE: q’ z 500 GPM

w

Г= ZOO FT

115 Q‘ W(U) 115(500)12.15)

 

T’ =

 

H-h

 

■ = 103,000 GPD/ FT

 

S~ IW X 10-‘) ( 103,000) =_19B x 10,

1-97 rVf 1.87(1.95 x 107)

(From “Ground Water Hydrology" by D. K. Todd, 1959, Wiley & Sons, Inc.

Used with permission of Wiley & Sons, Inc.)

Figure C-8. Method 1 (Superposition) for solution of the nonequilibriumequation.

S 10 5 10 5 10

ELAPSED PUMPING TIME (f), DAYS

 

5 1

 

C-3. Equilibrium pumping test

EXAMPLE: Q’ = 500 GPM

DISTANCE TO OBSERVATION WELL, Г 200 FT

AT POINT A: t’ = 4.0 x 10‘3 DAY

H – h = 1.55 FT

TANGENT THROUGH A: AS = 1.26 FT/LOG CYCLE of PUMPING

H – h 1 55 TIME IN DAYS

THEN F(U) =—— = ——= 1.23 [SEE FIG. C-10 for F(uT]

AS 1.26 J

115 Q’ W(U) 115(5001(2.72)

Подпись:T’ =—– г-гг1—————— ———– = 101,000 GPD/ FT

Подпись:= 2.05 X 10

(Modifiedfrom "Ground Water Hydrology" by D. K. Todd, 1959, Wiley & Sons, Inc. Used with permission of Wiley & Sons, Inc.)

Figure C-9. Method 2 for solution of the nonequilibrium equation.

curve through the selected point, and deterine As, the drawdown in feet per log cycle of time.

Step 4. Compute F(u) = H – h/As, and deter­mine corresponding W(u) and u from figure C-10.

Step 5. Determine the formation constants by equations (C-3) and (C-4).

 

C-3. Equilibrium pumping test

(From “Ground Water Hydrology" hv D. K. Todd, І959, Wiley&Sons, Inc.

Used with permission of Wiley & Sons, Inc.)

 

(3) Method 3. This method can be used as an ap­proximate solution for nonequilibrium flow to a well if the time-drawdown curve plotted to asemilog scale be­comes a straight line (fig. C-6). The formation con­stants (T and S) can be computed from

 

2640^

As

 

T

 

(C-7)

(C-8)

 

and

 

0.3T’t„

r2

 

S

 

Figure C-10. Relation among F(u), W(u), andu,.

 

where

As = drawdown in feet per cycle of (log) time – drawdown curve

to = time at zero drawdown in days An example of the use of this method of analysis in de­termining values of T, S, and к is given in figure C-6, using the nonequilibrium portion of the time-draw­down curve.

(4) Gravity flow. Although the equations for non­equilibrium pumping tests are derived for artesian flow, they may be applied to gravity flow if the draw­down is small with respect to the saturated thickness of the aquifer and is equal to the specific yield of the dewatered portion of the aquifer plus the yield caused by compression of the saturated portion of the aquifer as a result of lowering the groundwater. The procedure for computing T’ and S for nonequilibrium gravity flow conditions is outlined below.

Step 1. Compute T’ from equation (C-3).

Step 2. Compute S from equation (C-4) for vari­ous elapsed pumping times during the test period, and plotS versus (log) t’.

Step 3. Extrapolate the S versus (log) t’ curve to an ultimate value for S’.

Step 4. Compute u from equation (C-4), using the extrapolated S’, the originally computed T’, and the original value ofr2/t’.

Step 5. Recompute T’ from equation (C-3) using

C-3. Equilibrium pumping test

(5) Recharge. Time-drawdown curves of a test well are significantly affected by recharge or depletion of the aquifer, as shown in figure C-11. Where re­charge does not occur, and all water is pumped from storage, the H’ versus (log) t curve would resemble curve a. Where the zone of influence intercepts a source of seepage, the H’ versus (log) t curve would re­semble curve b. There may be geological and recharge conditions where there is some recharge but not

C-3. Equilibrium pumping test

C и R vE a – ALL WATER FROM STORAGE-NO AQUIFER RECHARGE.

Ь – CONE OF INFLUENCE INTERCEPTS A SOURCE OF SEEPAGE AT TIME T.

C – CONE OF INFLUENCE INTERCEPTS A SOURCE OF SEEPAGE AT TIME 7 WITH SUPPLY LESS THAN RATE OF PUMPING AT TIME t. d – CONE OF INFLUENCE INTERCEPTS AN IMPERMEABLE BOUNDARY AT TIME 1.

 

U. S. Army Corps of Engineers

C-l 1

enough to equal the rate of well flow (e. g., curve c). In many areas, formation boundary conditions exist that limit the area1 extent of aquifers. The effect of such a boundary on an H’ versus (log) t graph is in reverse to the effect of recharge. Thus, when an impermeable boundary is encountered, the slope of the H’ versus (log) t curve steepens as illustrated by curve d. It should be noted that a nonequilibrium analysis of a pumping test is valid only for the first segment of a time-drawdown curve.