experimental determination of the …...stanford geothermal program interdisciplinary research in...

Stanford Geothermal Program Interdisciplinary Research in

Engineering and Earth Science STANFORD UNIVERSITY

EXPERIMENTAL DETERMINATION OF THE EFFECTIVE TAYLOR DISPER!%MTY IN A FRACTURE

BY

John R. Gilardi

A Report Submitted to The Department of Petroleum Engineering of Stanford University in Partial

Fulfillment of the Requirements for the Degree of Master of Science

June 1984

ACKNOWLEDGEMENTS

I would to thank my research advisor, Dr. R. N. Horne, for the advice, sup-

port, and encouragement he gave me throughout this work.

Many thanks are due to Jawahar Baruah for his design of the multiplexer

board; to Alee Ryals for his time discussing design ideas and his excellent

machine work; and to many fellow students and department staff who have

helped me finish this project.

Financial support was provided by the Geothermal and Hydropower Techno-

logies Division of the Department of Energy, Stanford-DOE Contract DE-AT03-

80SF11459.

1

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ......................................................................... i

ABSTRACT 11

LIST OF FIGURES ................................................................................. iv

LIST OF TABLES ................................................................................... v

INTRODUCTION .................................................................................... 1

LITERATURE SURVEY .......................................................................... 2

DESIGN ................................................................................................ 6

PROCEDURE ........................................................................................ 11

DATA PREPARATION ............................................................................ 13

RESULTS ............................................................................................. 14

CONCLUSIONS ..................................................................................... 51

REFERENCES ...................................................................................... 52

APPENDIX A: MULTIPLEXER BOARD .................................................... 53

APPENDIX B: COMPUTER SCANNING ALGORITHMS .............................. 54

APPENDIX C: COMPUTER PROGRAMS AND SUBROUTINES ................... 55

.. ...........................................................................................

... 111

LIST OF FIGURES

1. Dispersion Schematics ....... ................................... .......................... 4

2. B(t,) vs. t , (Horne & Rodriguez, 1983) ........................................... 4

3. Hele-Shaw Cell: Overall view.. ....................... ........ ....... ................ ... 7

4. Detail view ....................................................................................... 8

5. Tracer Valve ... ........ ............ .............................................. ............... 8

6. Electrode Design .................... ........... ........ ............ ... ................ ....... 10

7. Reservoir Design ............................................................................. 10

8. Comparison of Estimated and Calculated Dispersivity. Run 3. ..16

9. Comparison of Estimated and Calculated Dispersivity. Run 4. ..21

10. Comparison of Estimated and Cialculated Dispersivity. Run 5. 26

11. Comparison of Estimated and Clalculated Dispersivity. Run 7. .31

12. Comparison of Estimated and C,alculated Dispersivity. Run 8. 36

13. Comparison of Estimated and Calculated Dispersivity. Run 9. 41

14. Comparison of Estimated and Calculated Dispersivity. Run 10. 46

A. 1.1. Multiplexer Board Design ......................................................... 53

B. 1.1. Computer Scanning Algorithms ......... ............ ... ........ ............... 54

iv

1 .

2 .

3 .

4 . 5 .

6 .

7 .

8 .

9 .

LIST OF TABLFS

Fracture Dimensions ...................................................................... 14

Measured Data ................................................................................ 14

Estimated Values ............................................................................ 15

Run 3 Results .................................................................................. 16

Run 4 Results .................................................................................. 21

Run 5 Results .................................................................................. 26

Run 7 Results .................................................................................. 31

Run 8 Results .................................................................................. 36

Run 9 Results .................................................................................. 41

10 . Run 10 Results .............................................................................. 46

V

Section 1: INTRODUCTlON

Reinjection of waste hot water is commonly practiced in most geothermal

fields, primarily as a means of disposal. Surface discharge of these waste waters

is usually unacceptable due to the resulting thermal and chemical pollution.

Although reinjection can help to maintain reservoir pressure and fluid

volume, in some cases a decrease in reservoir productivity has been observed

(Horne, 1962). This is caused by rapid flow of the reinjected water through frac-

tures connecting the injector and producers. As a result, the water is not

sufficiently heated by the reservoir rock, and a reduction in enthalpy of the pro-

duced fluids is seen.

Tracer tests have proven to be valuable to reservoir engineers for the

design of a successful reinjection program. By injecting a slug of tracer and

studying the discharge of surrounding producing wells, an understanding of the

fracture network within a reservoir can be provided.

In order to quantify the results of a tracer test , a model that accurately

describes the mechanisms of tracer transport is neccessary. One such mechan-

ism, dispersion, is like a smearing out of a tracer concentration due to the velo-

city gradients over the cross section of flow. If a dispersion coefficient can be

determined from tracer test data, the fracture width can be estimated.

The purpose of this project was to design and construct an apparatus to

study the dispersion of a chemical tracer in flow through a fracture.

1

Section 2: LlTERATuRE SURVEY

The effects of water reinjection in geothermal systems worldwide is dis-

cussed in a paper by Horne(1983), which also includes a summary of tracer test-

ing procedures and results.

In order to derive a model to accurately describe the transport of tracer

through a fracture, the physics of dispersion must be understood. Taylor(1953)

presented a classic study of dispersion in flow through a capillary tube. He

showed that convective dispersion combines with transverse molecular diffusion

in what we now know as "Taylor Dispersion" (Fig. 1). He showed that the tracer

concentration is dispersed symmetrically about a plane that moves with the

mean flow velocity. Taylor presents the equation governing the effective longitu-

dinal dispersion:

where,

C = concentration

z = translated distance = x-ut

x = distance

t = time

u = mean velocity of flow

7 = net longitudinal dispersivity (derived for pipe flow in Taylor's model)

The solutions to eqn (1) for different initial and boundary conditions can be

found in Carslaw and Jaeger( 1959). For a step input,

- 3 -

found in Carslaw and Jaeger(1959). For a step input,

1 ¶u

c = C, + $c,-c,) + exp e r f c

where,

C, = base concentration

C1 = injected concentration

C = concentration a t x

erfc = complimentary error function

Horne and Rodriguez(l963) used a method similar to Taylor’s to derive an

expression for the net longitudinal dispersivity, 7, for flow in a fracture:

2 b2u2 r ) = -- 105 D

where,

b = fracture half-width

u = mean flow velocity

D = coefficient of molecular diffusion

They also showed that, due to the effects of transverse molecular diffusion, any

concentration gradients across the fracture would be equalized after a non-

dimensional time, fD = 0.5 (Fig. 2), where

I

Fossum and Horne(l962) show how the subroutine VARPRO can be used to

determine both linear and non-linear parameters from a set of experimental

data. VARPRO uses a non-linear least squares method of curve fitting.I2 Fossum

and Horne(1982) matched the calculated response to field data from tracer

4

-b

convective dispersion

- b

Fig. 1. Dispersion Schematics

F i g . 2. b ( t d ) vs. t d (Horne & Rodriguez, 1953)

- 5 -

The present study set out to examine and confirm the applicability of Equa-

tion (3), which is only approximate, and to initiate broader investigations into

dispersion in fractures. To these ends, an experimental program was undertak-

en.

The results of several experiments t o study dispersion were found in the

literature. Bear(1961) performed both one- and two-dimensional studies of

dispersion through porous media and produced results which agreed with his

theory. Hull and Koslow(l96Z) present the results of a study of dispersion in a

network of channels.

Section 3: DESIGN

The design objectives were aimed at building an apparatus capable of study-

ing dispersion through a fracture in both one- and two-dimensions. The possibil-

ity of testing both chemical and fluorescent tracers was another requirement.

1. HELE-SHAW CELL

The size of the model fracture, particularly its aperture, was constrained by

the results of Home and Rodriguez (11983). Any concentration gradient across

the width of the fracture will be equalized after a non-dimensional time, tD = 0.5

(Fig. 2). The real time it takes to become equalized is proportional to the square

of the fracture half-width:

0.5 b 2 D t = -

Using B diffusion coefficient for potassium iodide ( K I ) 2 22 /sec and

a fracture half-width of 0.25 mm, the time required is about 16 sec. By using an

aperture of 0.54 mm and flowrates of approximately 50 cc/min we could keep

the apparatus small enough to fit on a lab bench! The cell is 6 f t long by 1 f t

wide. Fig. 3 shows an overall view and Fig. 4 a detail of the design.

The lower plate is 1 in. thick cast aluminum alloy. I t is hard anodized to

prevent corrosion and provide a tough, non-conductive finish. The upper plate is

1/4 in. float glass and is separated from the aluminum by a gasket made up of

three layers of plastic electrical tape. A series of aluminum clamps holds the

cell together while four adjustable legs support it horizontally on the lab bench.

2. VALVES

The inlet and outlet ports were implemented by drilling holes 11 in. through

the width of the plate. A 0.25 in. slit was then sawed through the sur€ace (Fig. 4)

Since the pressure drop across the length of the drilled hole is negligible com-

3 u J

1 W L1

rj I

R

/ 7/32 IN. FLCAT GLASS

. 9 -

pared with that across the slit, water will flow into the cell a t uniform velocity

over i ts width.

An on-off valve (Fig. 5) was designed to allow instantaneous injection of

tracer. The valve is activated by hand and can be locked in either on or off posi-

ti on.

3. ELECTRODES

In order to continuously monitor the tracer concentration as it flows

through the cell, an array of 96 electrodes was employed. For a KI tracer the

conductivity of solution will increase linearly with the log of concentration. Thus

we are able to measure the tracer concentration at any electrode location and

any chosen time.

The coaxial electrodes were constructed using brass conductive elements

and a teAon insulator (Fig. 6). The brass surfaces were electroplated with gold

to prevent corrosion and polarization. Each electrode is press fit in to the

aluminum plate and mounted flat to within .0015 in. The central electrode is

connected to the data aquisition system, and the outer electrode is grounded to

the plate.

An instantaneous current is flowed across the electrode while the resulting

voltage is measured (requiring less than 0.1 sec.). Voltages can be measured

once each second and are stored in a Compaq personal computer. The data is

displayed on the screen so i t is possible to "watch" the tracer as it flows through

the cell. The curcuitry and electronics are described in Appendix A, and the

computer scanning algorithms are described in Appendix B.

4. CONSTANT FLOWRATE SOURCE

Two constant pressure reservoirs, one for the base concentration and one

for the tracer, were constructed (Fig. 7 ) . The flowrate can be adjusted by

10-

changing the height of the center tube.

n

-+- PLATED

INSULATOR

*, t ! ! I I I

I

I

J

- p = p.7-

Section 4: PROCEDURE

1. Solution Preparation

(a) Prepare solutions of desired concentrations using distilled water and

iodide standard. A base concentration of 175-200 ppm should be used.

Voltage readings from lower concentrations tend to be too unstable for

accurate analysis. By injecting -300 ppm the electode response

remains in the linear portion of the voltage vs. log concentration curve,

thus simplifying analysis. Using the 1000 ml volumetric flask, the con-

centration is:

(z) l i t e r of iodide standard * 12690 m g / 1 cbpm) = 1 l i ter

(b) Clean reservoirs and fill with solution.

11. Assembly

(a) Wipe clean the aluminum and glass plates with wet sponge and assem-

ble; clamps should be finger tight.

(b) Flush the Hele-Shaw cell with COz (at p <2 psi or glass may shatter).

(c) Begin flowing water slowly, making sure to clear both the inlet and

tracer valve of air, otherwise bubbles will become trapped in the cell.

(distilled water was flowed until the cell was void of all air bubbles to

save the prepared solutions for test experimental runs.) Pounding on

the glass with the but t of your hand, or tapping the glass with the

rubber mallet while flowing at high rate can prevent the water front

from fingering and forming air pockets.

(d) Now start Bowing the solution of base concentration; allow 2 pore

volumes (- 500 CC) to flow before starting to SCAN (see next step).

111. Prepare computer for run

12

(a) Plug in the multiplexor board power supply (Appendix A describes the

design and operation of the multiplexor); turn on printer, then comput-

er.

(b) When "clock" appears ('" 60 sec) hit cF10>.

(e) Type b:

(d) Remove system diskette from the A drive; replace with a blank diskette,

TV. Ready t o run

(a) To begin recording, enter SCAN. Note time. SCAN will measure and

store the voltage at each electrode, once per second, and will fill the C

(internal) diskette after about 7 min. The voltages will be plotted

against the location of each electrode on the screen (they will vary for

each electrode since each has different sensitivity). SCAN will stop au-

tomatically after 7 min., or sooner if you hit <Fl>.

(b) Allow base concentration to flow for -1 min. before injecting tracer. To

inject, open gate valve, slide tracer valve into position and shut off inlet

valve. Record time of injection.

(c) When run is over, enter FIXUP. FIXUP processes the data (-15 min) and

stores i t in a new file named 1abfix.dat on the A diskette.

(d) Enter PLOTFIX. This plots the voltage vs. time for each electrode, indivi-

dually.

( e ) Take diskette out and label it; these are your results.

(f) To start a new run, insert a blank disc in the a drive, shut off the tracer,

and flow the base concentration (-500 cc). Repeat steps 7-11.

(g) After runs are finished, disassemble and wash down thoroughly. Unplug

the multiplexor board, shut off computer and printer.

Section 5: DATA PREPARATION

The processed data (1abfix.dat) can be matched to the model given by Equa-

tion (2) to provide estimates of the mean speed of flow (u), and the effective

dispersivity (q). A FORTRAN program, CURVEFIT, performs this operation, and

simultaneously calibrates the measured voltages to concentrations. The pro-

gram is run as follows:

(a) Copy 1abfix.dat from drive A to drive C (C must be erased first).

(b) Create a file P A W S on C as follows:

1.

2.

3.

4.

N = number of unknowns (12), [ =2 (u and q), can be 3 if to is unknown

as well].

Initial estimates of u, q ; one per line (F10.4).

Base concentration (C,) and injected concentration (C,) ; one per line

(F10.4).

Number of electrodes to be analyzed and electrode numbers (4012);

e.g. 080104052122232932 will analyze 8 electrodes - 1,4,5,21,22.23,29

and 32.

(c) Load a blank diskette in drive A.

(d) Type a:

(e) Type b:curvefit

(f) Type c: params <CR>, con <CR>, c:labfix.dat <CR>, prn <CR>.

(g) The program will output a summary of estimated parameters on the

printer, and will create output files (l .dat through 32.dat) on A. These may

be used for plotting. The program takes several hours to run.

(h) On completion, type b:plot, this will plot the results on the screen for a

specified range of electrodes.

14

Section 6: RESULTS

The results of seven one-dimensional experiments are presented in this sec-

tion. The results from CURVEFIT are displayed in Tables 4-10, and are compared

with the dispersivity predicted by Equation (3) in Figures 8-14. Graphs of the

data collected from each electrode and the curve that fits it are also presented.

Since it was a one-dimensional study, only the central row of 32 electrodes was

used.

The actual fracture dimensions are shown in Table 1. Since the glass plate

used for thes runs was slightly curved, the aperture was measured a t the center-

line.

width

I-

i 3un # 3 4 6 7 8 Q

10 - L

203 305 1.25 177 305 0.82 177 305 1.38 177 305 0.66 177 305 0.48 177 805 0.57

The velocities listed in Table 3 are values from CURVEFIT that best match

the data (since the fracture cross-section did not have constant aperture, the

15

flow velocity was not constant over the width of the cell, and u= q / A may not be

accurate). These best-fit values are used in Equation (3) to provide an estimate

of the dispersivity (77). The estimated times of injection, t o , used in PARAMS are

also listed in Table 3.

Table 3. ESTIMATED VALUES Run I u (cm/sec) 1 q ( c r n 2 / s e c ) to ( s e d

3 1 0.542 0.21 0 73.0 4 6 7 8 9

10

0.824 0.539 0.890 0.451 0.375 0.41 6

0.486 0.208 0.567 0.1 46 0.101 0.1 24

66.6 79.5 73.4 60.0 42.5 42.5

The dispersivities predicted by Equation (3) are compared with those es-

timated by CURVEFIT in Figures 8-14. Only selected electrodes are plotted, as

some are inconsistent due t o faulty dista collection or transmission. Electrodes

1 and 2 were usually inconsistent, probably because the tracer front had not yet

become equalized.

Electrode 1 2 3 4 6 6 7 8 8

10 11 12 13 14 15 16 18 18 20 21 22 23 24 25 26 27 28 28 30 31 32

c, ( P P d 193.6091 201.281 1 202.9373 200.0074 200.0959 202.9865 200.8080 199.9427 203.0065 203.0301 203.0107 200.091 1 203.0 1 76 203.0000 203.01 66 203.0227 203.0 147 203.031 4 203.0683 203.0394 202.991 0 202.8789 203.031 0 202.9533 203.051 7 203.0475 203.0491 203.0428 203.0459 203.0372 203.01 67 c

305.1 784 304.9338 305.0957 300,0581 300.0331 305.0680 449.3800 300.0960 305.0846 305.0726 305.1 142 300.0483 305.1 285 305.1 503 305.1 168 305.1 229 305.1 194 305.1 045 305.1 092 305.1 175 305.1 639 305.1 857 305.1 272 305.221 4 305.1 082 305.1 186 305.1325 305.1 451 305.1 377 305.1 767 305.2894

P Table 4. RUN

16

0.6992 0.61 31 0.5908 0.5835 0.6846 0.57 16 0.3034 0.5661 0.67 1 1 0.6663 0.6647 0.6638 0.5593 0.5545 0.5543 0.55 1 3 0.5498 0.5463 0.5455 0.5428 0.5442 0.64 0 1 0.54 1 0 0.5405 0.5406 0.5409 0.54 1 6 0.5429 0.5435 0.5450 0.547 1

L

7 (cm2/ sec) 1.0367 0.6220 0.41 8 1 0.3585 0.271 8 0.31 13 2.8423 0.3005 0.2633 0.2030 0.3 166 0.2449 0.21 73 0.2358 0.21 18 0.2623 0.251 1 0.2341 0.2260 0.2326 0.248 1 0.2652 0.241 1 0.2708 0.1 829 0.2078 0.1 899 0.1 909 0.1 865 0.1 787 0.2055

3 Results u (crn/sec) apparent to ( s e d

72.9238 -

72.6622 72.1 451 72.6364 72.4763 72.3291 73.101 1 73.4000 72.6024 72.89 13 72.6204 73.4000 72.4066 73.1 522 72.93 12 72.91 9 1 72.01 39 72.61 25 72.0204 72.69 15 72.8799 72.8374 72.81 48 72.2095 72.8609 72.5939 72.5580 72.8735 72.8586 72.5092 72.1 761

0*75 0.65 7 0.55

U 0045

al 0.35 t

10 20 30

Electrode #

Fig. 8. Comparison of estimated and calculated dispersivity. Run 3.

17

R u n 3 - e lecrode 1

C 0 L

0 L

Q C

V 0 F

L

so 100 150 200 250 Time (secs)

R u n 3 - elecrode 2 300

250

200 100 150 200 250

Time (secsl

R u n 3 - elecrode 3

c 0 0 c V 0

200 100 150 200 250

Time (secs)

R u n 3 - elecrode 5 - E 0 n - s

100 150 eo0 250 Time (secs)

R u n 3 - elecrode 6 300 -

s .- - 250 0 L

c 0 0

c

s V

200 100 150 200 250 330

l i m e (secsl

300 c E 8 - c 0

0 L

C Q 0 c V 0

.- - 250 c

200


E 0 0 -

1 DO 150 200 250 Time Isecsl


1 DO 150 200 250 300 Time (secs)

s f! .- - 250 c C 0 0

V 200


150 200 250 303 Time Isecsl

Run 3 - elecrode 9 Run 3 - elecrode 13 300 300 -

E n n - C 0

0 L

.- - 250

c c

f 0 C 0 W

200 I

150 200 250 300 150 200 250 300 350 T i me (sets 1 fime (SeC6)

Run 3 - elecrode 10 Run 3 - elecrode 14 300 - n A A b L &?. & A M

v V 300 - IL - - E E n n n n tt 0 E - - dPS4 + < q m < w a ~ t C . - ' ~ r , : Y

C .- .- - 250 - 0 0

- 250 - L L

C Q 0 t 0 V

t c

E 0 t 0 V

EO0 I I EO0 I I

150 200 250 300 200 250 300 350 Time Isecsl Time (secsl

Run 3 - elecrode 1 1

150 200 250 300

Run 3 - elecrode 15

200 200 250 - 300 350

In 3 - e~ecrode 12 c E n n - s

C 0 0 C 0 V

300

250

200 I I I

150 200 250 300 350

Time (secsl

19

Run 3 - elecrode 17 Run 3 - elecrode 21 300

e E n n - t 0

L 0

C Q 0 C 0 0

.- - 250 0

200 200 250 300 350

T i me (SeC6)

300 .- E n n - co -- + 250 0 L C Q 0 C

0 0

0

200


250

200 200 250 300 350 400

Time (secsl

Run 3 - elecrode 19

I 250 300 350 400

Time (secs)

Run 3 - elecrode 20 300 -

E - C 0

0 L t 0 0 C 0 u

.- - 250 - 200

250 300 350 400 Time (secsl

I I I I

250 300 350 400 Time (SeCSl

Run 3 - elecrode 22

C 0

0 L C Q 0 t 0 V

.- - 250 c

200 250 300 350 400

Time (secs)

Run 3 - elecrode 23 300 - -

E n n - 5 -- - 250 0 L t Q 0 t V 0

c

200 250 300 350 400

Time tsecs)

Run 3 - elecrode 24 300 - ’ -p-qv--v~

c E n n ’ I

C 0

t 250 0 L

.- - - 5 0 C

V 0

200 I I

E50 300 350 400 453

20

R u n 3 - elecrode 29 R u n 3 - elecrode 25 300

c

n n E

I

t 0

0 L C 0 0 t 0 0

.- - 250 c

200 300 350 400 450

300 - E n n u

t 0

0 L C 0 0

.- - 250 c

V 6

200 300 320 340 360 380 400 420 440 460

Time (secs) Time (secs)

R u n 3 - elecrode 26 R u n 3 - elecrode 30 300 300 -

c E

n

E n n

n n - u

5 C 0 _- -- - 250 * 250 -

0 L B

El c

c

E C 0 0

V

0 C 0 u

200 300 350 400 450 300 320 340 360 380 400 420 440 460


300 350 400 450

Time (cecsl

Run 3 - elecrode 28 E n n u I

Time (secsl

I I I I I I J

280 300 320 340 360 380 400 420 440 460


E n n .- 5 .- - 250 - 0 L c

5 El 0

u 200

320 340 360 380 400 420 440 460 Time Isecsl

T ime (secs )

electrode 1 2 3 4

- 6 6 7 8 8

10 11 12 13 14 16 16 17 18 18 20 21 22 23 24 26 26 27 28 28 30 31 32

U

Q> t

co ( P P d 199.2807 196.0567 200.0594 200.21 85 200.0233 200.0099 199.66 14 200.0930 199.8557 199.8767 109.8764 199.91 34 200.0458 199.3993 200.1 344 200.0068 200.0289 200.0545 199.9969 200.0546 200.37 10 200.0699 200.0753 200.1 798 200.1 206 200.0903 200.0851 200.1 322 200.1 982 200.07 16 200.1 124 200.1 661

21

able 6. RUh c1 (PPd

300.1 21 4 300.9046 300.0597 300.0896 300.0435 300.0408 289.8 199 300.0335 300.091 G 300.0824 300.0808 300.0852 300.0534 299.41 78 300.0222 300.0674 300.0669 300.0576 300.0705 300.0801 300.0504 300.0692 300.061 8 300.0337 300.0454 300.061 4 300.0662 300.041 2 300.0331 300.0772 300.0662 300.0574

1 Results u(crn/sec) 0.7671 0.6923 0.8099 0.8 56 3 0.831 8 0.8 1 48 0.8558 0.829 7 0.8 164 0.8 1 22 0.81 63 0.8305 0.8392 0.8236 0.8343 0.8255 0.8352 0.8299 0.8276 0.8263 0.8425 0.8275 0.8253 0.8296 0.8245 0.8268 0.8276 0.828 1 0.8323 0.831 6 0.8383 0.8388

0.9746 2.2386 0.5473 0.2747 0.4877 0.5452 0.3780 0.4787 0.7602 0.6532 0.7479 0.6706 0.6296 0.6332 0.4576 0.6355 0.658 1 0.5820 0.66 19 0.60 1 9 0.32 1 1 0.4723 0.4432 0.3 7 78 0.4633 0.4436 0.4348 0.41 14 0.3844 0,4726 0.44 1 7 0.43 1 9

apparent to (sec) 66.6000 66.6000 66.6000 66.6000 56.6000 66.6000 66.6000 66.6000 66.6000 56.6000 66.6000 66.6000 66.6000 66.6000 56.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000 66.6000

0.75 8

0.65

0.55

0.45

0.35

0.25 1 0.15

0 - j 0.05 1 , , , I

10 20 30

Electrode #

Fu. 9. Comparison of estimated and calculated dispersivity. Run 4.

Run 4 - elecrode 1

50 100 150 200 Time [sets)

Run 4 - elecrode 2

50 100 150 200 250 lime (recsl

Run 4 - elecrode 3

I --

- 250-

0 L

C c

! 0 u

200 50 100 . 150 200 250

T i n e (aecsl

Run 4 - elecrode 4

100 150 200 250 Time (sect)

22


250

200

300 - E n n - C 0

0 i

C 0

0 0

.- - 250 e

F 2OQ

1 300

w a

s .- - 250 0 L t *

u t 200

300

- B b .- - 250 0 i e

0 f 200

100 150 200 250 lima laecs)

Run 4 - elecrode 6

100 150 200 250

Time (aecsl

Run 4 - elecrode 7

100 150 200 250

lime Iaecs)

Run 4 - elecrode- 8

4 I

J 100 150 200 250

Time (secsl

Run 4 - etecrode 9

E n n w

C 0

100 150 200 250 T ime (secs)

Run 4 - elecrode 10 c

a n E

- C 0

c

100 150 200 25 0 Time I secs l

23

Run 4 - elecrode 13

100 150 200 250 300 Time Isecsl

Run 4 - etecrode 14 300 - -

E n n -


100 150 200 250 300

- 300 7 - E n n - I

Time (secsl

150 200 250 300 Time Isecs)

Run 4 - A elecrode l N n 15 -- h U

Time (sec t1

I

I I I

150 200 100 250 300

200 1 1 I I

150 200 250 300

Time Isecsl

24

Run 4 - elecrode 17 Run 4 - elecrode 21 300 -

E n n - C 0

0 L

Q C

0 C 0 u

-- e 250 c

200 150 200 250 300

Time lsecs)

Run 4 - elecrode 18

I

C Q 0 C 0 0

300 c E n n - C 0

0 L

C 0

U 0

-- 250

c

: 200

E n n - C 0

0 L C 0 0

0

-- - 250 c

to 200

150 200 250 300 350 Time lsecsl

Run 4 - elecrode 22 300 - -

E n n - c 0 L c C 0 0 C u 0

150 200 250 300 Time Isecsl

150 200 250 300 350 Time lsecsl

Run 4 - e ~ e c r o d e 19 A 1 Y V

I 1 I

150 200 250 300 Time Isecsl

Run 4 - elecrode 20

150 200 250 Time lsecsl

300 350

Run 4 - elecrode 23

200 250 300 350 Time I E B C S )

t 0

0 L

.- c

Run 4 - elecrode 24 -

250

- L n m CUU I 1 I

200 250 30c) 353

Time Isecsl

Run 4 - e l e c r o d e 25 300

E n n - C 0

0 L C Q 0 C 0 u

.- - 250 - 200

200 250 300 350 Time (secs)

Run 4 - e l e c r o d e 26 300 -

E

C 0 .- * 250 0 L c I

200 250 300 350 Time (secs)

Run 4 - e l e c r o d e 27 300 -

c E n n -

200 250 300 350 T ime (cecsl

Run 4 - e l e c r o d e 28

n n E - C 0 .-

200 250 300 350 Time (cecs)

25

Run 4 - e l e c r o d e 29 300

I

E n n - E .- - 250 0 c C Q

0 V

c

k! 200

s ' I - 0 1

200 250 300 350 Time (secs)

Run 4 - e l e c r o d e 30

L 0 V

200 200 250 300 350 40

Time lsecsl

Run 4 - e lecrode 31 300

c E n n - C 0

0 L

Q t 0 C 0 V

-- - 250 c

200 200 250 300 350 4 00

Time (secs)


5 -- - 250 0 L c

to E V 0

200 250 300 350 400

1 ime (secs)

1 2 9 4 6 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31

172.4794 172.5788 176.871 2 177.01 94 176.9580 176.9227 177.0075 176.9869 176.8473 176.9455 176.951 6 176.9802 177.0278 176.9525 176.9865 176.9783 176.9859 176.9790 177.0272 177.0523 177.0293 177.0075 177.321 9 177.2622 177.2921 177.8224 177.8538 177.521 3 177.7206 177.8365 I 179.41 89

c1 (PPm)- _e

305.4789 306.0025 305.1 535 304.9398 305.1213 305.1 701 305.1 239 305.1 637 305.2737 305.3065 305.2965 305.2385 305.1 734 305.2049 305.2269 305.2457 305.2278 305.2643 305.2033 305.2374 305.21 69 305.2479 305.00 1 4 305.1 656 305.1 879 305.1 204 305.1 572 305.0650 305.1 760 305.2394 305.0844

26

r Table 6. RUN 5 R e s u l t s

u (cm/sec) 0.4567

i

- 0 .?5 7

0.4574 0.501 7 0.53 1 3 0.52 1 6 0.5223 0.5370 0.536 1 0.5305 0.5356 0.5380 0.5385 0.5407 0.5385 0.5396 0.5380 0.5433 0.5404 0.5403 0.5403 0.5406 0.5399 0.5539 0.541 2 0.5403 0.5427 0.5426 0.5453 0.547 1 0.551 6 0.5596

q (cmZ/ sec 1 0.4547 0.621 6 0.2523 0.1 454 0.2398 0.2561 0.21 77 0.2446 0.31 34 0.3051 0.300 1 0.270 1 0.2603 0.2628 0.2245 0.2272 0.2246 0.2555 0.2345 0.2236 0.2265 0.2226 0.1 139 0.2284 0.2453 0.2068 0.2306 0.1 940 0.1 798 0.21 09 0.243 0

apparent to (sec) 79.4469 79.3727 78.1 31 1 79.7869 77.1 71 5 78.5297 78.91 63 79.31 20 77.8365 78.2525 79.2787 78.9465 79.0606 79.0822 78.9550 79.0627 79.2794 78.9707 79.5479 79.1 41 9 79.8696 79.1 860 79.4677 78.9000 79.9000 79.9000 79.9000 79.9000 79.9000 79.9000 79.9000

0165 1 I

0.55 1 U 01 45

c_) 0135 t i

E I ec t r o d e

F i g . 10. Comparison of estimated and calculated dispersivity. Run 5.

27

Run 5 - e lecrode 1 Run 5 - elecrode 5 - - 300 n

- n E - Ti----

.? 250 c 0 L

0 C c

; 200 0 V

100 150 200 250 Time (secsl

Run 5 - elecrode 2 - 330 n n E

.E 250 C

c 0

C a

V 0

c

; 200

100 150 200 250 lime [sees)

Run 5 - elecrode 3

250 1 d

~ 300 n - n

.? 250

E

I

C

c 0 L C Q

0 0

c

E 200

100 150 200 250 300 Time (secs)

Run 5 - elecrode 6 - 300

n a E - c

.E 250 L

c 0 L c C 0

0 V

200

100 150 200 250 330 Time (secs)

.E 250 C

100 150 200 250 lime (secs)

150 200 250 330 Time (secs)

Run 5 - elecrode 4

.E 250 C

c 0 L c C 0 : 200 0 V

100 150 200 250 lime (secsl

Run 5 - elecrode 8 1 . - 7 _ . . _ . r .L

.E 250 C

c 0 L c C 0

0 V

E 200

I I I I

150 200 253 3113 Time l s e c s l

28

150 200 250 300 Time (secsl

Run 5 - e ~ e c r o d e 10 - 300 a E Q

C

I

.! 250 c 0 L t 0

U 0

t

: 203

I I I I I 1 200 250 300 350

Time (secs)


- C

.E 250 c 0 L

Q C

0 0

c

g 200

150 200 250 300 350

Time (secs) 200 250 300 353

Time (secs)


C .E 250 c 0 L

C 01

0 U

c

20D

150 2OD 250 300 350 Time ( secs )

Run 5 - elecrode 12

0

0 L c @

0 u

0

g 200

200 250 300 350 Time lsecs)

Run 5 - elecrode 15

- E a n -

30 0

.E 250 C

c 0 L C 0

c

g 200 0 U

200 250 300 350 43:

Time lsecsl

Run 5 - elecrode 16 300 -

n n E

I

.E 250 C

c 0 L - C Q

200 U 0

~

I I I , I

200 253 300 353 4c.c

Time (secsl

Run 5 - elecrode 17

- I E n n loor! -0 250 c 0 L

C 01 0 C V 0

c

200 v 250 300 350 4 00

T ime Isecs)

Run 5 - elecrode 18 - 300 n n

-0 250

E

I

C

c 0 L

f 01

t

200 V 0

250 300 350 400 Time Isecsl

Run 5 - elecrode 19

250 300 350 400 Time (6eCS)

Run 5 - elecrode 20

29


I

C 250

c 0 L

C 01

U 0

c

E 200

- E

0

C 0

0 L C 01 V t V 0

n - - c

c

250 300 350 400 450

Time ' (secs)


250

200

300 350 400 450

Time (secs)

Run 5 - elecrode 23

.? c \ 250 c 0 L

C 01 0 C

V 0

c

Time (secs)

250 300 350 400

Time (secs)

r u n 5 - electrode 25

- 300 n n

.E 250

E

- C

c 0 L

C @

0 0

c

E 200

300 320 340 360 380 400 420 440 460 T ime Isecsl

run5 - electrode 26

I I I I I I I

300 320 340 360 380 400 420 440 460 Time (secsl

run5 - electrode 2?

.E c l - 250 c

0 L

c 0,

c

; 200 0 V V

L I I I I 1 I 1

- 300 n n

.E 250

E - t

c 0 L

C @

V 0

c

p 200

320 340 360 380 400 420 440 460 Time Isecsl


J

30


- 300 n

- n E -

.E 250 J c 0 L C @

0 u

c

E 200

run5 - electrode 30

300

250

200

340 360 380 400 420 440 460 Time Isecsl

- 300 n n

.$ 250

E

- C

c 0 L C 0

V 0

c

E 200

340 360 380 400 420 440 460 T i me (secs I . * .

run5 - electrode 31

320 340 360 380 400 420 440 460 360 380 400 420 442 462

Time (secsl T ime (secsl

31

e l e c t r o d e 1 2 3 4 6 6 7 0 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

c, (PPm) 179.2666 178.6063 7 76.7776 176.9432 177.2003 176.8742 1 76.53 1 7 177.0942 1 76.96 18 1 78.58 18 177.0820 176.9861 176.9890 1 76.98 12 177.0046 176.9860 177.01 74 177.0549 177.0296 177.0435 177.0824 176.9820 177.01 07 1 7 7.022 1 177.01 53 177.0257 177.0623 177.0006 177.0204 176.9823 177.01 60 177.0289

U

a) t

C, (ppm) 305.6806 305.421 0 305.0984 305.0747 305.0239 305.0944 304.7535 305.0427 305.1 009 308.8341 305.0543 305.1 233 305.1 339 305.1 49 1 305.1 35 1 305.1 61 4 305.1 48 1 305.1 027 305.1 550 305.2248 305.1 236 305.3 146 305.2252 305.2593 305.1 995 305.21 89 305.1 226 805.2429 305.2684 305.331 1 305.2626 305.2506

0.6423 0.7263 0.7554 0.7962 0.8025 0.8446 0.8334 0.8464 0.77 17 0.8744 0.8728 0.8793 0.8763 0.8821 0.881 2 0.8890 0.8926 0.89 14 0.8864 0.8990 0.8868 0.8908 0.8900 0.8953 0.8937 0.90 12 0.9031 0.9005 0.9062 0.9069 0.9049

7 (cmZ/ sec 1 0.4679 0.51 71 0.3560 0.4441 0.3357 0.5089 0.3708 0.4695 0.5463 2.1 440 0.4760 0.49 13 0.4733 0.561 7 0.4752 0.51 94 0.4531 0.5390 0.5489 0.4589 0.4089 0.4852 0.4432 0.3673 0.4526 0.4460 0.42 10 0.4885 0.4380 0.4202 0.4229 0.4531

apparent to (sec) 71.7837 72.7662 72.4584 71.7963 72.1 182 72.0861 73.2562 72.5221 72.3593 69.0034 72.6505 72.6992 72.4796 72.41 05 73.21 72 73.3541 72.50 10 73.22 15 73.3826 72.9996 73.0323 72.601 1 72.4894 72.66 74 72.8346 72.8886 72.51 35 73.2226 73.1 053 72.5762 73.0941 73.0907

-

0 .?5

0.45

0.35 0

",::E I , I I i 0.05

10 20 30

Electrode #


32

r u n ? - electrode 1 - 300 c t

Cl n -

1 I I I I

100 150 200 250 Time (secs)

run? - electrode 2

- 300 n n E

- L

.E 250 c 0 L

s - E 200 V 0

I I I I I

100 150 200 250

Time [secs)

- 300 a n E

- C

.E 250 0

0 L 0

E 200

0 u

run? - electrode 3 kCtCvTy+wwfv~~,*'p-wF

I I I I

100 150 200 250 Time Isecs)

7 run? - electrode 4 dtT+w++w+t-tv-*

i I I I

100 150 200 250 _ .

100 150 200 250

Time (secsl

run? - electrode 6 - 300 a a

- '.. .. -Y L

E

c I I

100 150 200 250

Time lsecsl

d l a 1' I I 1

100 150 200 250 Time I s e c s )

E P a

.E 250 C I

33

run? - electrode 9 run? - electrode 13 - 300 n E

0

C

- .E 250 c

0 L

C Q

V 0

c

E 200

100 150 200 250 Time (secsl

- 300

a E R - C

.E 250 e 0 L

C 0

c

200

u 0

100 150 200 250 Time (secsl

run? - electrode 1 1

- 300 a n

.E 250

- E

I

c - L

0 L

C c

100 150 200 250 Time [secsl

- 300 n n

-2 250

E

I

C

.- 0 L c

iil 200

0 u

run? - electrode 12 -

- I I

- 300 n E a - C .E 250 c 0 L

C Q

u 0

L

E 200

100 150 200 250 Time (secsl

run? - electrode 1 4

- E

a

C 0

0 L

C Q 0 C

u 0

n - _- e

c

100 150 200 250 Time (secsl

i 100 150 200

Time (sees)

run7 - electrode 16

I E 300 -

P a - .E 250 C

c 0 L

C Q

0 W

c

E 200

i

100 150 200 250 Time (secs) Time (secsl

run7 - electrode 17

34

run? - electrode 21

n E

a - I /

.! 250 C

c 0 L

Q C

0 W

L

E 200

100 150 200 250 Time [secsl '

100 150 200 250 Time (secsl

- 300 a a E

- .? 250 C

L

O L - C @

U 0

200

run7 - electrode 18

- 300 a E Q - C .: 250 e 0 L - c

100 150 200 250 Time [secsl

run7 - electrode 19

kl ; 200 W 0

- 300 E a a

C

- .! 250 c 0 L -

100 150 200 250 Time (secs)

run? - electrode 20

C 0

W 0

200 - I I

100 150 200 250 Time (secsl

run7 - electrode 22 - 300

C Q E 200 0 W

100 150 200 250 300 Time (secsl

run7 - electrode 23

300

.(I 250 C

c 0 L c C 0,

0 u

E 800

100 150 200 250 300 Time (secsl

run7 - electrode 2 4

- 300 n n

-0 250

- E

- - C

c 0 L

C Q

c

E 200 -

0 V

150 200 253 Time (secs)

35

run7 - electrode 25 run? - electrode 29 - 300 n n E

C .E 250 c

0 L C Q

c

; 200 u 0

.$ = c 250 c 0 L

C c

m 200

0 V

150 200 250 300 Time Isecsl

150 200 250 300 Time (secs)

run? - elecrrode 30 run? - electrode 2?

- 300 a a

- E

i .E 250 1 rl c 0 L

C al

0 u

c

200

150 200 250 300

Time (secsl 150 ZOO 250 300

Time Isecs)

run? - electrode 26 run? - electrode 31

- 300 - 300 n n E R

E - n

.! c c 250 .? 250 C

c 0 L c

c 0 L c C m

200 V 0

C Q

0 V

200

1 so 200 250 300 T i me (secs 1

150 200 250 300 Time [secs)

run7 - electrode 32 - 300

r t m ? - e I ectrode 28 zaaGsza

E n n

co 250 -

200 - I

200 250 300 3' 150

L

0 L

c 0 c L

C a

0 V

E 200

c L) C

V 0

m

150 200 Time Isecs l

flectrode 2 3 4 6 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

U

0 t

c, ( P P d 170.1 399 179.8520 176.8666 176.7606 176.9526 176.9783 176.8539 176.8430 176.8034 176.841 6 176.9483 176.9379 176.8700 176.9528 1 76.9221 177.1 41 0 176.9500 176.9141 1 76.91 96 177.0144 176.8555 176.9429 176.9425 176.9841 176.9982 176.981 2 177.0255 176.981 4 177.0031 176.9886 176.9780

Table 8. RUN

306.0940 305.3446 305.51 64 306.1 938 305.4040 305.5481 305.61 09 305.71 05 305.5755 305.3648 305.4082 305.4972 305.3391 305.3843 305.1 577 305.3545 305.4277 305.5346 305.31 12 305.61 25 305.41 10 305.4284 305.4207 305.4667 305.61 10 305.61 31 306.31 42 306.69 1 6 308.1 960 31 0.5608

8 u (cm/sec) 0.551 3 0.4889 0.4808 0.4695 0.4655 0.4734 0.4669 0.4658 0.461 3 0.4622 0.4629 0.461 0 0.4588 0.4595 0.4569 0.4545 0.4557 0.4545 0.4527 0.4528 0.4489 0.4499 0.4502 0.4494 0.4490 0.4498 0.4507 0.4496 0.451 1 0.451 5 0.4541

Results - 77 ( c m Z / s e c

i

0.9233 0.4890 0.2899 0.2779 0.2536 0.1 585 0.21 08 0.2067 0.2355 0.2043 0.1 351 0.1 31 4 0.1 966 0.1 341 0.1 555 0.2325 0.1 753 0.1 594 0.1 775 0.1 384 0.1 951 0.1 390 0.1 71 2 0.1 376 0.1 397 0.1 285 0.1 225 0.1 238 0.1 401 0.1 342 0.1 365

apparent to (sec) 59.931 3 57.0895 57.8292 58.7547 57.2856 59.8243 59.7648 58.6273 59.5570 59.7669 59.01 76 59.421 7 59.5951 59.7998 59.6897 59.521 5 59.9 1 1 7 59.1 586 69.0275 59.8606 59.41 12 69.7394 69.6374 59.793 1 69.2642 59.4932 59.861 9 59.7666 59.5834 59.6492 69.51 92

0.75

0.55 1

0.05 1 , , , I 10 20 30

Electrode

1

37

I r u n 8 - electrode 1

.E 250 C

L

0 L

C a c

E 200 0 V

Time (secs)

r u n 8 - electrode 6 *-

- 300 - n E Q

rlln8 - e I ectrode 6

Time (secs)

r u n 8 - electrode 2 I

L I 1: 0 L

C Q

c

2 200 u 0 I/

L

O L C a 0 C

V 0

c

1.1 I I I I

100 150 200 250

T i me (secsl

.O_ 250 C

c 0 L c i

Time (secs)


- 300 - E - pa

Time (secsl

n

C a

0

I I I

100 150 200 250

t r u n 8 - electrode 8

r- " " L.

Time ( secs )

38

run8 - electrode 9

c

0 L

C m L

; 200 0 u

100 150 200 250 Time (secsl

rt!pFj - electrode 10

.? 250 c

c

C L

C P)

I

; 230 0 u

100 150 200 250 Time l secs l

- 3 c 0 / i " - - i J


E a a - .? 255 C

- 0 L

C m L

; 200 0 U

100 150 200 250

Time [secsl

n8 - electrode 12

- 300 a n E

- .E 250 C

c 0 L

C 0)

L

200 0 U

100 150 2.00 250

Time (secsl

Time [ s e c s )

run8 - electrode 15 - 300 a a

- E

- .: 250 c

c 0 L L I C a,

200 - u 0

I I -

150 200 250 300

Time (secsl

- E a a - C 0 .- c

0 L c C a, 0 C U 0

39

run8 - electrode 17 run8 - electrode 21

- 300 E 0. a

t

- .E 250 c 0 L

C a,

V 0

c

2 200

- 300 a a E

C .O 250 c

0 L

C (u

c

; 200 u 0

200 250 300 350 4t

Time lsecsl 150 200 250 300 350

Time Isecs)

run8 - elecrrode 22 r u n 8 - electrode 18 - 300

a E a I

C .E 250 .! 250

C

c

0 c

0 L

C 0

e

290 0 u

200 250 300 35c Time Isecs) Time I s e c s )

run8 - electrode 23 - 3oo ~ run8 - e ~ e c t r ? ~

E a a

- 300 a a E

C .E 250 L

0 L

C m L

2 200 u 0

t e

0 L

C a c

; 200 0 u

/ I I

250 300 350 420

Time (secsl 200 250 300 350

Time (secsl

r t I n R - e ~ectrode 20 - 300

- 3001 E

a

C

a - .E 250 c 0

E

a a i ~ - C .e 250 c

0 L

C m 0 C 0 0

c

I I

250 300 35c 4CC 200 250 300 350

Time [secsl

40

run8 - electrode 25 run8 - electrode 29

250 300 350 400 Time (secsl

250 300 350 400 - 450 Time (secsl

run8 - electrode 26 run8 - electrode 30 - 300 - 300 - E a n n n

t C .! 250 .E 250 -

E

I - c c

0 L

C Q

u 0

0 L

C al

c c

E 200 E 200 - u 0 +-. . W ' - ----,

250 300 350 400 450 250 300 350 400 450 Time Isecsl Time Isecsl

run8 - electrode 2?

- 300 n n

- E

- .E 250 - C

c 0 L

C al

c

E 200 - 0 V

250 300 350 400 450

- E n n I

C 0

0 L

C al

0 V

.- c

c

E


250

200

250 300 350 T i me Isecs 1 Time (secsl

run8 - electrode 28 - 300 n a E

- c .E 250 c 0 L

t a f

200 0 u

250 300 350 400 450

- e ~ectrode 32 - 300 n n E -

250 C

c 0 c C P,

f

E 0 200 0

Time I s e c s l

41

r Electrode

L

1 2 3 4 6 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 i

131 5554 176.071 9 177.1 845 1 77.1 948 176.941 7 177.1 461 176.9773 177.0871 176.8942 177.0735 177.1 774 177.2131 177.1 551 177.1 852 177.0934 177.1 607 177.1 774 177.401 5 1 77.2561 177.1 605 177.0856 177.0395 177.0251 1 77.01 09 1 77.009 1 177.0063 177.0743 177.4232 177.1 694

0 e75

01 65

0.55

304.1 395 304.6890 305.0923 305.0734 305.2280 305.0948 305.1 975 305.1 323 305.2768 305.1 51 1 305.1 143 305.1 068 305.1 408 305.1 264 305.1 743 305.1 51 6 305.1 328 305.0996 305.21 26 305.2447 305.2866 305.4663 305.6692 306.2703 306.7268 308.3059 31 4.0203 320.7780 21 0.5383

3

0.8247 0.7286 0.61 12 0.51 64 0.488 1 0.4528 0.4477 0.4303 0.4284 0.41 58 0.41 37 0.4067 0.4007 0.3955 0.3930 0.3876 0.387 1 0.3851 0.3849 0.3806 0.3809 0.3759 0.3730 0.3 728 0.3708 0.3721 0.3732 0.3721 0.3729 0.384 1

11.5627 4.8659 1.1 524 0.6782 0.3830 0.4245 0.2400 0.2683 0.2 139 0.2379 0.1 834 0.1 622 0.1 483 0.1 653 0.1 385 0.1 490 0.1 339 0.1 163 0.1 121 0.0932 0.0827 0.1 076 0.1 034 0.0868 0.0879 0.0894 0.0773 0.086 1 0.055 1

42.0653 41 5202 42.2768 40.431 6 40.2042 40.1 679 39.941 2 40.4746 42.1 549 42.3783 42.4097 42.4030 42.3131 42.1 147 41.9828 41.9846 41.6244 42.1 71 8 41.8483 4 1 5267 42.41 86 42.0458 42.021 8 42.1 476 41.6299 42.2254 41.6680 4 1.9085 42.1 72 7

0.0251 41.7788 I

U t

a, 0 * 4 5 Ob35 t 0.25

0.15

0 e 0 5

Electrode

F i g . 13. Comparison of estimated and calculated dispersivity. Run 9.

- 300 n n E

- C

.O 250 c 0 L C 01

t

200 u 0

- 300 a n E

- .? 250 C

c 0 L

C Q

c

200

u 0

run9 - electrode 1

50 100 . 150 200 250 Time lsecs)

run9 - electrode 2

42

run9 - electrode 5 - 300 n n

.E 250

E - C

e 0 L

C 0

c

200 u 0

- 300 n n

.E 250

E

- t

c 0 L C Q

u 0

c

200

50 100 150 200 250 Time [secs)

run9 - electrode 6

50 100 150 200 250 Time Isecsl

50 100 150 200 250 Time lsecs)

run9 - electrode 3

.? 250 ,( C

c

50 100 150 200 250 Time lsecsl

- '""7 run9 - electrode 4 E n n -

run9 - electrode ?

100 150 200 250

Time lsecs)

c

f 0

Time [sets)

run9 - electrode 9

.E c l 250 c 0 - L

C Q

0 u

f

200

100 150 200 250 300 Time [secs)

run9 - electrode 10

- 300 n n E

I

C .E 250 c

0 L

C m

u 0

c

p 200

150 200 250 300 Time Isecsl

J c 0 L C Q

c

p 200 0 u

150 200 250 300 Time [secs)

rung - electrode 12 E n n

.E 250

I

C

c 0

43

rung - electrode 13

E n n - C 0 .- e

150 200 250 300 350

Time (sets1

200 250 300 350 Time lsecsl

run9 - electrode 14

25 0

200 250 300 350 Time Isecs)

rung - electrode 15

- 300 E

200 250 300 350 43;

Time lsecsl

run9 - electrode 16

- C .E 250 c

0 L

C Q

0 V

e

200

250 300 353 4 3:

Time (secsl

- 300 n n E

- f

.E 250 - 0 L

C Q

u 0

c

; 200

run9 - electrode 17 # A & * I #!-

250 300 350 400 Time [sets)


E n n

.? 250

- C

c 0 L - C 0,

V 0 ; 200

250 300 350 400 Time (secsl

run9 - electrode 19

- c -0 250 e

0 L

C 0

0 u

c

200

44

run9 - electrode 21

- 300 n

- n E

- 300 E a a

C

- .E 250 c 0 L

f @

u 0

L

g 200

250 300 350 400 450 Time (secs)

ff

250 300 350 400 450 Time lsecs)

run9 - electrode 20

250 300 350 400 450 Time (secsl

run9 - electrode 22

250 300 350 403 450

1 ime [sets)

run9 - electrode 23

' 0 1 L c al

0 V

g 200 - I I

250 300 350 400 450 Time (secsl


c 0 L

t Q

0 u

c

E 200

250 300 350 432 4 5 :

T i m e ( s e c s )

45

250 300 350 400 4 50 Time (secsl


- E n n I

c 0 .- c D L

c Q V c V 0

c

run9 - electrode 25

300 ffl

250 1 200 -

I I J

250 300 350 400 450 Time lsecsl


300 -

250 -

200 -

250 300 350 .400 450

Time (secsl


run9 - electrode 29

250

200

’ 250 300 350 400 450 Time lsecs)


300 -

250 -

200 - +/ -- J - - -

1 1 11- _rc_ I

250 300 350 403 4 50 Time [eecs)

250 300 350 400 450

T ime lcecs)

Electrode

Table 10. RUN 1

46

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

U

Q)

t

c, ( P P d 155.5770 155.8977 174.2000 176.9750 177.1 250 1 76.9881 177.1 1 1 6 177.1 743 177.0269 1 76.91 13 177.0229 176.9876 177.0255 1 77.0 176 177.0756 177.0307 177.0804 177.0782 177.061 0 177.0654 177.2330 177.071 4 177.0949 177.1 479 177.0338 177.0397 177.0394 177.0389 176.9966 177.0095 176.9899 177.0404

305.9257 304.6744 305.1 61 6 305.1 054 305.1 995 305.1 659 305.3780 305.1 664 305.3792 305.1 925 305.26 1 4 305.2 1 0 1 305.2232 305.1 71 8 305.2253 305.20 1 6 305.1 882 305.1 971 305.3203 305.1 388 305.23 1 9 305.1 991 305.1 808 305.41 43 305.48 1 2 305.5894 305.8246 306.9598 308.2523 31 4.8080 31 6.6703

0 Results u (cm/sec) 0.7688 0.6398 0.571 7 0.51 80 0.5005 0.4778 0.4 78 4 0.4620 0.461 0 0.4508 0.4496 0.443 1 0.4389 0.4346 0.4322 0.4272 0.4290 0.4240 0.4258 0.421 1 0.421 2 0.41 88 0.41 64 0.41 58 0.41 42 0.4 1 50 0.41 47 0.41 47 0.41 28 0.41 27 0.41 07 0.41 56

q ( c m 2 / sec 1 1.409 1 2.1 460 0.7228 0.431 7 0.3067 0.3406 0.206 1 0.2459 0.201 3 0.2708 0.1 863 0.1 885 0.1 720 0.1 776 0.1 668 0.1 629 0.1 566 0.1 542 0.1 273 0.1 299 0.1 184 0.1 301 0.1 221 0.1 174 0.1 258 0.1 262 0.1 160 0.1 092 0.0969 0.1 1 1 0 0.0983 0.083 1

apparent to (sec) 42.1 98 1 41.6750 42.0 1 1 0 40.5803 40.7347 40.2335 41.7093 40.71 68 42.4894 41.7489 42.1 240 41.6820 41.6887 42.3983 41.6534 4 1.5687 41 -601 1 42.1 21 0 42.1 639 42.4330 42.41 94 40.7729 41.9701 42.3275 41.861 4 42.2972 41.7694 41.551 4 41.61 43 42.1 783 41.9902 42.443 1

0465 i

0.05 O4I5 1 10 20 30

Electrode

1


run10 - electrode I - 300 n n E

- C .E 250 c 0 L t 0,

0 0

c

t" 200

50 100 150 200 250

Time (secs)

47

run10 - electrode 5 - 300 a n E

I

c .? 250 c

L 0

c 0,

c

t" 200 u 0

50 100 150 200 250

Time (secs)

run10 - electrode 2

n a

.E - c 250 3002 E

-

c 0 c L I /

- 300 a a E

- c

.E 250 c 0 L

c 0,

c

t" 200 u 0

- 300 n a

.? 250

E

I

C

c 0 L C c

50 100 150 200 250

Time (secs)

run10 - electrode 3

/ 50 100 150 200 250

Time [secs)

run10 - electrode 4

50 100 150 200 250

Time [secs)

- 300 a a E

- c .? 250 c 0 L

C al

0 0

c

t" 200

- 300 a n E

I

c .? 250 c

0 L c

runlo - electrode 6

150 200 250

Time Isecs)

run10 - electrode 7 - I - Y - e i i

t

I I I

50 100 150 200 250

Time lsecsl

run10 - electrode 8 - 300 a n

- E

- C 0

0 L

C al 0 C u 0

.- c

e

100 150 200 250

Time (secs )

48

run10 - electrode 9 - 300 n a E

- C .$ 250 c

L 0

C 0,

c

200 u 0

100 150 200 250

Time lsecs)

run10 - electrode 10

- 300 - E a a L

.E 250 C

c 0 L

C 0,

c

; 200 0 u

100 150 200 250 Time (secs)


E a - - a 300-

.E 250 f

c t I

run10 - electrode 13 - 300 a n E

I

.E 250 C

c 0 L

0) t

0 0

c

p 200

150 200 250 300

Time (secsl .

._ 300 n n E

- C .? 250 c 0 L

C al

c

E 200 u 0

- 300 a n E

c 0 L

a C c

p 200 0 0


.? 250 C

150 200 250 300

Time (secs)


100 150 200 250 300

Time (secsl

run10 - electrode 12 - 300 n n E

I

.$ 250 C

c 0 L

C 0,

c

p 200 0 u

150 200 250 300 Time (secs)

150 200 250 300 350

Time Isecsl

run10 - electrode 16 r

/

200 250 300 350

Time (secsl


200 250 300 350

Time (secs)


i 200 250 300 350

Time [secs)

run10 - electrode 19 - 300 n E R

C

- .? 250 c 0 L

C a

V 0

c

; 200

t t-i 200 250 300 350 400

T i me [secs 1


- 300 n n

- E

I

.E 250 C

c 0 L

t a c

200 0 V

49


- 300 a a E

- t .e 250 c 0 L

t a c

E 200 V 0

250 300 350 400 Time (secs)

run10 - electrode 22 I - 300 -

E Y V vvv

n n - .E 250 C

f

0 L

t a c

E 200 0 V

250 300 350 400

Time (sees)

run10 - electrode 23 - 300 n E 0.

C

- .E 250 c 0 L

C a +

200 u 0

250 300 350 400 450

T i me (secs 1


- 300 n n E

I

t .e 250 c 0 L

t a c

200 V 0

250 300 350 400

Time (secs)

250 300 350 400 450

Time (secs)


a E

a - .? - C 250 300/"""

200 -

I I

250 300 350 400 450

Time (secs)

run10 - electrode 26 - 300 a a E

- .? 250 t

c

0 L - C a, p 200 u 0

- 30c

50

run10 - electrode 29 - 300 a a E

I

C .? 250 c 0 L f

m C

; 200 - 0 u

I

Time (secs)


- 300

a E a - C .? 250 f

0 L

t al

0 0

c

; 200

250 300 350 400 450

Time (secs)

250 300 350 400 450

Time lsecs)

b run10 - electrode 2?

250 300 350 400 450

T i me Isecs 1


I I

250 300 350 400 450

Time (secs)

- E a a I

c 0

0 L

C al 0 C

u 0

.- c

c

300

250

200

run10 - electrode 31 L

250 300 350 400 450

Time (secs)


n E a u

.? 250 - c c P

250 300 350 400 450

Time (secs)

51

Section 7: CONCLUSIONS

The approximation for Taylor Dispersion, Equation (3), is sufficiently accu-

rate t o be used in a fracture flow model for tracer test analysis.

The slight decline of dispersivity with distance suggests that further studies

may find a stricter criterium for the non-dimensional time before tracer front

equalization.

52

Section 8: REFXRENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Bear, Jacob. Dynamics of Fluids in Porous Media , American Elsevier, New York, NY. 1972.

Bear, Jacob. "Some Experiments in Dispersion", J. Geoph. Res. ,Volume 66, no. 8 . pp. 2455-2467.

Carslaw, H.S. & Jaeger, J.C. Conduction of Feat in Solids , Oxford University Press, 2nd Ed., 1959.

Coats, K.H. "Dead End Pore Volume and Dispersion in Porous Media", Sot. Pet. Eng. Jour. , March 1964.

Davies, Cecil W. The Conductivity of Solutions , John TYiley & Sons, lnc., New York. 2nd Ed. 1933.

Fossum, M.P. Tracer Analysis in a Fractured Geothermal Reservoir , Mas- ters Report, Stanford University, June 1982.

Fossum, M.P. and Horne, R.N. "Interpretation of Tracer Return Profiles a t Wairakei Geothermal Field Using Fracture Analysis", Geoth. Res. Counc., Transaction 6 , 1982.

Horne, R.N. "Reservoir Engineering Aspects of Reinjection", Japan Geoth. Energy Assoc. Jour. , 1 9 , 23-30, (1982), (in Japanese).

Horne, R.N. and Rodriguez, F. "Dispersion in Tracer Flow in Fractured Geothermal Systems", Geoph. Res. Let. , Vol. 10, no. 4, 289-292, 1983.

Hull, L.C. and Koslow, K.N. "Dispersion in Fracture Networks", Proc. , Eighth Wkshp Geoth. Res. Engr., Stanford University, Dec. 1982.

Taylor, Sir Geoffrey, F.R.S. "Dispersion of Soluble Matter Flowing Slowly Through a Tube", Proc. Roy. SOC. , A 219, pp 186-203, 1953.

VARPRO, Computer Science Dept., Stanford University.

53

APPENDIX A: MULTIPLEXER BOARD

The multiplexer board contains circuitry which switches through the elec-

trode array, allowing instantaneous voltage measurement at each location. Its

design is described by the flow dagram of Fig. A . l . l .

The circuitry is capable of testing 128 electrodes, divided into 8 groups of

16. The computer selects a bank of 16 electrodes and the voltage across each is

measured individually using a resistance bridge. The signal is amplified, con-

verted to digital logic, and stored in 0UTPUT.DAT for later processing.

ELECTRODE’ I I h

Fig. A. 1.1. Multiplexer Board Design

54

APPENDIX B: COMPUTER SCANNING ALGORITHMS Figure B . l . l shows a flow diagram describing the computer scanning algo-

rithms used for data aquisition, transmission, curve fitting and plotting.

I

drive A I drive B

1 1-1 create

stops automatically after 7 min. I or hit <F1>

I I F D N P 1 I1abfix.dat.l creates 1

I Takes -15 min.

outputs to screen to send to

drive C

s 4ou tpu t . da t I

4- reads

I1abfix.dat.j copy to c: >[labfix.datl ----------- URVEFIT]<

change diskette params r . d a t , e . d a t , - I outputs summary to Ale 7 create using EDLIN

+ = I reads

I outputs to screen I hit shift-PrtSc to send to printer

If in trouble:

copy to A: p q i x a x ] 1 +output. dat 1

Fig. B. 1.1. Computer Scanning Algorithms

55

Appendix C: COMPUTER PROGRAMS AND SUBROUTINES

Directory of Lab Control Diskette

V o l ~ t n t e i n d r i v e B h a s D i r e c t o r y of

SCAN BAS I N I T BAS CALTABLE DAT I N I T EXE CURVEFIT EXE P L O T F I T EXE TEST BAS FLOTLAB EXE CALVOLT BAS I N I T 2 BAS SCAN2 BAS SCAN2 EXE F IXUF ' EXE

DDTEST BAS DACAL BAS F I XUF' FOR FAHAMS F L O T F I X E M S F L O T F I T BAS F L O T F I X EXE GO

SCAN EX€

C:aL DAT

no l abe l

5- 10-84 5- 10-84 5-29-84 5- 10-84 6-05-84 5-'- d l -84 3-0 1-84 5-04-84 5-09-84 5-09-84 5-0$-€34 5-09-84 5-1 1-84 5- 1 1-84 3-(52-84 5-06-84 5-1 1-84 5-31-84 5- 10-84 5-3Ct-84 5- 10-84 5-32 -84 5-28-84

1 0 : 26a i o : 15a 1 l :O lp 1 0 : 18a 3 : 1 Op 5 : 02p

l1:44a l l : 4 2 a a: 13p 1 : 36p 5 : 36p

2: 36p 10: 39a

E

5: 39p

-., 3 : 5op 2 : 2op 2 : 34p ll:21a 5 : 2 3 p J : Olp 5: 24p 9: 16a 7: 44p

c

2.3 F i l e ( = ) 4096 by tes f r e e

56

SCAN

57

47~:) VCILTS=5 48!l DEC I MAL=2(:)4.7+VOLTS 490 DECIMf iL= INT (DECIMAL) : D A H I G H = I N T ( D E C I M A L / 2 5 6 ) : DALOW=DECIMAL-256*DAHIGH c(- - -I.-)(.) FOR C Y C L E x 1 TO 2 510 REM a l t e r n a t e v o l t a g e C', d-(-) - REM I F CYCLE=1 THEN DAHIGH=hH7: DALOW=%HFF E L S E DAHIGH=&H8: DALOW=O

530 OUT &H7 1 1 , DAH I (3H 540 UUT EtH7 1 0 , DALOW &.A(:) FOR C I R C U I T = l TO 2 560 OUT E<H71D,CIRCUIT 530 P R I N T C O L = l + ( C I R C U I T- 1 ) *20 bCX) FOR ELECTRODE=(:) TO 15 6 1 0 OUT P c H 7 15, ELECTRODE 62(1 OUT &H716,(:, 639 I F ELECTRODE=(:; THEN GOTO 720 640 ADS1 G=256*ADH I GH+ADLOW 650 I F FIDS I G : :>32767 THEN ADS I G=ADS I G - 6 5 5 3 6 ! 660 ADVOLT=VOLTAGE*ADS I G/2(:)4.8 6 '70 ELECNUMz ( C I RCU I T- 1 1 * 16+ELECTRODE b80 ELECVOLT (ELECNUM) =ADVOLT

700 REM LOCATE ELECTRODE ,PRINTCOL

: ADVOLT

cc

690 F R I N T #1, U S I N G "E<- ###- ###- ##.####";OLDTIMEB,THS,ELECNUM,GDVOLT

710 F:EM F R I N T U S I N G "Ed- ###- ###- ##.####" ;OLDTIMEB,THS, ( C I R C U I T - l ) * 1 6 + E L E C T R O D

720 I F I N F ( G H 7 1 4 ) .:: 128 THEN GOTO 720 730 GDLOW=I NP (%H7 15 1 740 ADH I GH= I NP ( & H 7 16) 750 NEXT ELECTRODE 760 ADSIG=256wRDHIGH+ADLOW

780 ADVOLT=VOLTAGE*ADSIG/204.8 790 ELEGNUM= ( C I RCU I T- 1 1 * 16+ 16 800 ELECVOLT' (ELECNUM) =ADVOLT

770 I F ADS I G :::,32767 THEN ADS IG=ADS 16-65536 !

810 F R I N T #1, U S I N G " e ( - ###- ###- ##.####";OLDTIMES,THS,ELECNUM,ADVOLT 820 OLDT IMES=T I ME$ 830 OUT &H7 19,163 840 OIJT EA.7 13 , 17 850 THS= I NF ( &H7 18 ) 860 T H S = T H S - 6 * I N T ( T t i S / l 6 ) 870 NEXT C I R C U I T 875 OUT PtH7 1 D , ( 3

880 NUMCYCLE=NUMCYCLE+l 83C) OLDT I MEB=T I ME$ 900 KEY ( 1 ) ON 310 M I N = l : MAX=32: YBASE=130 920 FOR NUM=1 TO 32 730 SIGNAL=ELECVOLT (NUM) 940 YP=YEASE-60*SIGNAL/SIGMAX 950 XP=(NUM-MIN)*19+15 960 I F NUM.::MFIX THEN L I N E ( X P O L D ( N U M ) ,YFOLD(NUM)

LINE (XF ,YP) - (XPOLD(NUM+ l ) ,YFOLD(NUM+ l ) ) , 1

LINE (XP,YP) - (XPOLD(NUM-1) ,YPOLD(NUM- l ) ) , I 970 I F NUM>MII-4 THEN L INE (XPOLD(NUM) ,YFOLD(NUM

COMMON / A / V O L T (32,500) INTEGER ELEC,ELECN,HR,MIN,SEC,THS,RECNO D I MENS I ON T I NE (500 , CTBL ( 20 , V T B L (20 ,32 CALBF(T=(:) I F (CALBRT. EC!. 0 ) GO TO 200

W E N ( 3 , F I L E = ' c a l . d a t ' , S T A T U S = ' O L D ' , A C C E S S = ' S E ~ U E N T I A L ' ~ C N e x t sec t i on skipped if no c a l i b r a t i o n des i red

READ (3,103) NFTS 1U3 FORMAT i I 2 1

DO 1 0 E L E C = 1 , 3 2 DO 11 N = l , N F T S R E A D ( 3 , 1 0 2 ) NDUMMY,VTBL(N,ELEC)

102 FORMAT (14,11X,F6.4) CTBL (N) =NDUMMY

11 CONTINUE 1 0 CONT I NUE

,C)C) 9- - OFEN (l,FILE='c:output.dat',status='old',acce~s='direct', END FILE 3

& recl=24,form='forrnatted') R E A D ( l , l O O , R E C = l ) HR,MIN,SEC,THS,ELEC,SIG

1 C ) O FORMAT (12, l X , 12, l X , I2,2X, I2,2X, I2,2X,F6. 4 )

59

c

1 1

4 rz J

1

TZERO=3600. *HR+60. +MIN+SEC DO 1 ELEC=1,32 RECNO=ELEC 1=1 READ (1,10C),REC=RECNO,END=2) HR,MIN,SEC,THS,ELECN,SIG IF (ELEC.EL!. 1) TIME (I )=3600. *HR+6(:). *MIN+SEC-TZERO IF (CALBRT. NE. O! CONCN=CONVRT ( S 11; , NFTS, ELEC, CTBL , VTBL ( 1 , ELEC 1 1 VOLT (ELEC, I) =SIG write (*,‘(f7.l,l:.:,f10.4)’) time(i),concn I=I+1 RECNO=RECN0+32 GO ‘ro 3 CUNT I NUE WRITE (*, ’ (14) ’ ) ELEC CONT I NUE END FILE 1 OPEN (2,FILE=’a:labfi~:.dat’,status=‘new’,access=’seq~~~ntial ’ )

DO 4 ELEC=1,32

WRITE (2,101) TIME(I1) ,VOLT(ELEC,II) DO 5 II=l,I

FORMAT (F7.1,1X,F10.4) CONT I NUE CONT I NUE END FILE 2 STOP END FUNCTION CONVRT(SIG,NPTS,ELEC,CTBL,V) DIMENSION CTBL ( 2 0 ) ,V ( 2 0 ) NFTS l=NFTS- 1 IF (SIG.GT.V(l) 1 GO TO 1 DO 2 I=l,NFTSl IF (SIG.GT.V(I+l) 1 GO TO 3 CONT 1 NUE

CONVRT=CTBL (NFTS 1 +SLOFE* ( V ( NPTS 1 -S I G ) RETUF:N SLOPE=(CTFL(I+l)-CTBL(I) )/(V(I+l)-V(I)) CONVRT=CTBL(I+l)+SLOFE*~SIG-V(I+l~) RETURN S L O F E = ( C T B L ( 2 ) - C T B L ( l ) ) / ( V ( 1 ) ) CONVKT=CTFL(l)+SLOPE*(SIG-V(l) 1

SLOFE=l./(V(NFTS)-V(NFTS-l))/(CTBL(NFTS)-CTBL(NFT~-l))

RETURN END

60

PLm

62

PLm

D I M X ( lOO(1) , Y ( 1 0 0 0 ) INFUT "ymin,ymax,ydiv";YMIN,YMAX,YDIV CHOICE=!:) OPEN "a:labfix.dat" FOR INFUT AS 1 I = (j XOLDZ-1 ! IF EOF ( 1 ) THEN GOTO 120 INFUT #l,X(I),Y(I) IF X(1)c:XOLD THEN GOTO 120 XOLD=X (I 1 I=I+1 IF EOF(1) THEN GOTO 120 GOTO 50 N=I-1 CIiO I CE=CHO ICE+ 1 FT%=O XMAX=X (N)

XSCRA=9 ! YSCRA=6 ! XSCR=XSCRA-. 5 YSCR=YSCRA-.5 XDIV=60 IF XLOG% THEN XDIV=LOG(XDIV)/LOG(10! 1 IF YLOGX THEN YDIV=LOG(YDIV)/LOG(10!) DEF FNIX(F)=INT((F/XSCHA)*639!) DEF FNIY(P)=l99-INT((P/Y~C~~)*l99!) XWIDzXMAX-XMIN YWID=YMAX-YMIN SCHEEN 2 , 0 , 0 KEY OFF CLS CL-s KEY UFF LOCATE 1,40,0 : FRINT "electrode",CHOICE b:FASS= 1 LINE (FNIX(l!),FNIY(l!))-(FNIX(l!),FNIY(YSCR)) LINE (FNIX(l!),FNIY(l!))-(FNIX(XSCR),FNIY(l!)) NX=INT (XWID/XDIV) NY=INT(YWID/YDIV) FOR I=O TO NX

xrl I rd=(:)

FSET (FNIX(1!+1*(XSCR-l!)*XDIV/XWID),FNIY(l!)) LINE -STEF (0,193-FNIY (. 125) ) NEXT I FOR I=(:) TO NY FSET (FNIX(1!),FNIY(1!+I*(YSCR-1!)*~'DIV/YWID)) LINE -STEP(-FNIX(. 125) ,(I) NEXT I

64

cAlRwmT

c ~ ~ * * * * * . ) F * ~ - * * * * . ) ( ~ . ) F * Y - ~ * * * ~ K * * * * * * ~ * $ ( ~ * * ~ ~ * - * . , * ~ * $ ( . * * * * * * * * * * * * * * * * * * . ) F * * * * * * *

C C FROGRAM E{EGINS -Gilardi experiment analysis C c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C C

C C C SET D IMENSIONS FOR VARFRO. BE CAREFUL WHEN S E T T I N G THE C D IMENSIONS FOR THE INCIDENCE MATRIX INC. SEE NOTE. C

I M P L I C I T REAL*8(A-B,D-H,O-Z)

D I M E N S I O N Y (5C)C) , T (5joO) , A L F (3 ) , BETA ( 2 ) , W ( 5 0 0 ) , A (50(:) , 7) , COMMON XFOS,TZERO COMMON /DUMMY / C 5( : )0 ,6 ) EXTEFNAL ADA CHARACTER*6 C F I L E ( 3 2 )

* I N C ! i 2 , 8 ) , N E L T R Y ( 3 2 ) ,NGO(:32)

PATA C F I L E / ' 1. da t ' , '2. da t ' , '3. da t ' , '4 . d a t ' , '5. da t ' , @ ' 6 . d a t ' , '7.da.t ' , '8.dat', ' 9 . d a t ' , i ' i(S.dat ' , ' i i .dat ' , ' i 2 .da t ' , ' i Z . d a t ' , ' i 4 .da t ' , ' 15.dat ' ,

' i 6 .da t ' , ' i 7 . d a t ' , ' i 8 . d a t ' , ' 19 .da t ' ,

@ '26. da t ' , '27.dat ' , '28 .da t ' , ' 29 .da t ' , 2 '

F a- L C - ) . dat ', ' 2 i . d a t ' , '22.dat ' , '2.3.dat ' , '24.dat ' , '25.dat ' ,

7 , - - .-> .-.(-). da t ' , '31. da t ' , ' 3 2 . dat ' /. C C SET FARAMETEF:S FOR VARPRO. C C

I FLOT=O NMAX = 5r:)(3 IF 'RINT '=50

C C C READ DATA SEL!UENTIAL ORDERING AND C PROFER FORMATT I NG AKE I MPORTANT. C C C C N L I S THE NUMBER O F NONLINEAR FARAMETERS C C

READ (5,3ii) NL

WRITE (6,12) NL 311 FORMAT (11)

i2 FORMAT ( iHO, i O X , 'NUME{ER OF NONLINEAR FARAMETERS ' / / ( 13) ) C C C L I S THE NUMEiER O F L I N E A R FARAMETERS C C

C L=2

65

C c C C

ESTIMATES OF THE NONLINEAR FAHAMETERS

FOSO=9.8425 READ (5,310) VELOC,DIFFS,TZERO,DC(:),DCl FORMAT (F 10.4 ) READ (5,312) N T R I E S , (NELTRY (t:::) ,K=l , N T H I E S ) FORMAT ( 3 3 12) DO 314 t:::=l ,32 NGU (k t 1 = 1 . DO 313 K=l , N T R I E S NGO (NELTRY (I.:::) 1 =O. WRITE (6,201 VELOC,DIFFS,TZERO FORMAT ( / , ' 0 Mean V e l o c i t y ( c m / s e c ) D i f f u s i v i t y (cm2/sec) '

# ' T i m e z e r o (sec) ' , / , (5X,F9.5,18X,Fl5.3,~OX,F10.5)) C C

C C C c C c C C C

N I S THE NUMBER O F OBSERVATIONS I V I S THE NUMBER O F INDEFENDENT VAKIGBLES T

I V = 1

T I S THE INDEFENDENT V A R I A B L E Y I S THE N-VECTOR OF OBSERVATIONS

66 C

C 12

C W ( I ) AF:E THE WEIGHTING PARAMETERS

-

DO 1 I = l , N 1 W ( I ) = l . O

c IM I r \ lF '= ID INT (TZERO+XFOS/VELOC) - 1 0 0 I M I N = I D I N T (TZERO) I F ( I M I N . L T . I M I N F ) I M I N = I M I N P

N = N - I M I N 1 F (N. GT . 2(:)(:) j N=Z(>(:I DO 201 1=1 ,N T ( I ) = T ( I M I N + I ) Y ( I j = Y ( I M I N + I )

I F ( I M IN. GT . N-200 1 I M I N=N-2OO

C WRITE (6,*) T ( 1 ) , Y ( I ) 2 11) 1 CONTINUE

C A L L VARFRO(L,NL,N,NMAX,LFF2,IV,T,Y,W,ADA,A, * I P E I N T , A L F , F E T A , I E R R , 10)

L VO=BETA ( 1) V l = E E T A ( 1 1 +2. *BETA ( 2 )

AFAR=V l+EFAR*DLOGiDCl ) DO 211 I = l , N

211 Y ( I ) = D E X P ( ( A F A R - Y ! I ) ) / F P A R )

BFAR=(Vl - iVO) /DLOG!DCO/DCl )

C A L L VARFRO(L ,NL,N,NMAX,LFF2, IV ,T ,Y ,W,ADA,A, *IFRINT,ALF,BETA,IERR,10(~~)

L F l = L + l C A L L ADA (LF1,NL,N,NMAX,LFFZ,IV,A,INC,T,ALF,1) DO 8 I = l , N

DO 9 J=l,L C ( I , J ) = B E T A ( J ) * A ( I , J ) C(I,LF1)=C(I,LPl)+C(I,J)

C ( I , L F l ) = O .

CONTINUE OFEN ( l ,F ILE=CFILE(NELEC) ,STATUS='NEW' ,ACCESS='SE~UENTIAL ' )

END F ILE 1 DCM I D=BETA ( 1 1 +BETA ( 2 1 DO 400 I 1=1 , N I F (C(II,LFl).GT.SNGL(DCMID)) GO TO 401 TMID=T (I I) CONTINUE TOCALC=TMID-XFOS/ALF (1) WRITE (7,302) N E L E C , B E T A ( l ) , B E T A ( 1 ) + 2 . * B E T A ( 2 ) , A L F ( l ) , A L F ( 2 )

FORMAT (2X,I4,5F12.4) CONT I NUE

1 , TOCALC

STOF END

67

c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C C SUBROUTINES C c Y+*******+*************~***********************~,****************** c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C c C C

C

c c: C C C C C C

C

c: c c C C C C C C C C C C r, C

SUBROUTINE ADA (LF,NL,N,NMAX,LFPZ,IV,A,INC,T,ALF,ISEL) I M P L I C I T REAL*8(A-H,O-Z) D I M E N S I O N A L F ( N L ) ,A (NPlkX,LFF2) ,T(NMAX) , I N C ( 1 2 , 8 ) COMMON XFOS,TZERO

F'1=4. *ATUN (1. )

T P I = 2 . iDSQRT ( F I 1 L = L F - 1 I F (NL. L T . 3) A L F (3) STZERO

THE I N C I D E N C E MATRIX I N C ( N L , L + l ) IS FORMED BY S E T T I N G I NC ( K , J 1 = 1 I F THE NUNL I NEAR F'ARFtMETER A L F (t:: 1 AFF'EARS I N THE J- T H FUNCTION F H I i J ) . (THE PROGRAM SETS A L L OTHER I N C ( K , J ) TO ZERO.)

I F ( I S E L . E B . 2 ) GO TO 90 I F ( I S E L . E Q . 3 ) Gi3 TO 165

THE VECTOR-SAMPLED FUNCTIONS F H I ( J ) ARE STORED IN THE F I R S T N ROWS AND F I R S T L COLUMNS O F THE MATRI X B ( 1 , J ) . B ( I , J ) CONTAINS F H I ( J , A L F ; T ( I ) , I ,... N; J=1 ,L. THE CONSTANT FUNCTIONS F H I WHICH DO NOT DEFEND UFON ANY NONLINEAR FARAMETERS A L F MUST AFPEAR F I R S T .

68

I F 1

C

C C C

81 CONT I NUE

I F ( ISEL.EB.2) GO TO 200 C C C C c c C

165 C

2 c C

170

DO 170 I = l , N

CONTINUE

69

FUNCT I ON DERFC ( Y ) DOUBLE PRECISION DIMENSION DOUBLE FRECISION

INTEGER DATA

*

* * * * DATA

st * * DATA

* * * * * * * * DATA

* * 9

* * * * DATA

* * * *. * DATA

* * * * DATA DATA DFlTA

70

X = Y ISW = 1 I F (X.GE.(:).c:)DO) GO TO 5 ISW = -1 x = - x

I F (X .LE .4 .0DO) GO TO 30 I F ( I S W .GT. (1)) GO TO 40 I F (X.LT.XLN3GE) GO TO 45 RES = 2 . ODO GO TO 70

10 I F ( X . L T . X M I N ) GO TO 110 XSQ = x * x XNUM = F ' (4) *XSL!+F'(5) XDEN = XSL!+Q(4)

5 I F ( X . L T . . 4 7 7 D O ) GO TO 10

UO 15 I = 1,s XNUM = XNUM*XS(%+P (1) XUEN = XDEN*XSB+L! ( I )

15 CONTINUE RES = X*XNUM/XDEN GO TO 25

20 RES = X*F(';) /Q( .3 ) 25 I F (ISW.EG!.- l) RES = -RES

RES = l .OD6-RES GO TO 70

XNUM = P1 (8) +X+P1 ( 9 ) XDEN = X+fi l(8) DO 35 I=1,7

3 : ) XSG = X W X

XNUM = XNUM*X+Fl (I) XDEN = XDEN*X+Ql (1)

35 CUNTINUE RES = XNUM/XDEN GO TO 60

40 I F (X .GT.XBIG) GO TO 65 45 XSQ = x * x

X I = l .ODO/XSQ XNUM= F2 (5) *X I+F I z (6 ) XDEN = XI+(%2(5) DO 50 I = 1 ,4

XNUM = XNUM*X I+F2 I I ) XDEN = X D E N * X I + Q 2 ( 1 )

5C) CONT I NUE

60 RES = HES*DEXP(-XSL!) RES = ( S B R F I + X I * X N U M / X D E N ) / X

I F ( ISW.EL! . - l ) RES =I 2.OD6-RES GO TO 70

65 RES = O.ODC) 70 DERFC = RES

RETURN END

71

SUBF:OUTINE VARPRO (L, NL, N, NMAX, L F F 2 , I V , T, Y, W , ADA, A,

DOUBLE F R E C I S I O N AI(NMAX, LF 'FZ) , B E T A ( L ) , A L F ( N L ) , T(NMAX, I V ) , X I P R I N T , ALF, BETA, IERR, ITMAX)

2 W ( N ) , Y ( N ) , ACUM,EPSl, GNSTEP, NU, FRJRES, R, RNEW, XNORM, 2 AS,BS,S

INTEGER B1, OUTPUT L O G I CAL St::: I F EXTERNAL ADA DATA E F S 1 / l . D - 6 / , OUTFUT /6 / I E H R = 1 I T E R = 0 LF1 = L + 1 B l = L + 2 LNL2 = L + NL + 2 NLFl = N L + 1 S K I P = .FALSE. MODIT = I F R I N T I F (1FF:INT .LE. 0 ) MODIT = ITMAX + 2 NU = ( 3 . NU = 1.

5 C A L L DF'A (L, NL, N, NMAX, LF'F2, I V , T, Y, W, ALF, ADA, IERR, X I F R I N T , A, BETA, A I 1 , L F l ) , R )

GNSTEF = 1 . 0 I T E R I N = 0 I F ( I T E R .GT. 0 ) GO TO 1 0

I F ( N L .Ea. 0) GO T O 90 I F ( I E R H .NE. 1) GO TO 99 I F ( I P R I N T .LE. 0) GO TO 1C) WRITE (OUTPUT, 207) I T E R I N , R WRITE (OUTFUT, 200) NU

10 C A L L O R F A C l ( N L F 1 , NMAX, N, L, I F R I N T , A ( 1 , B l ) , FRJRES, I E R R ) I F ( I E H H .LT. 0 ) GO TCI 99 I E R R = 2 I F (NU .EL!. 0 . ) GO TO 30

C A L L ORFAC2(NLP1, N M A X , NU, A ( 1 , B1)) CALL BACSUE (NMAX, NL, A ( 1 , E l ) , A ( 1 , L N L 2 ) ) DO 35 K = 1, NL

C A L L DPA (L, NL, N, NMAX, LPP2, I V , T, Y, W, A ( 1 , El), ADA, IERR, I F H I N T , A, BErTA, A ( 1 , L F l ) , RNEW) I F ( I E R R .NE. 2 ) GCI TO 99 I T E R = I T E R + 1 I T E R I N = I T E R I N + 1. % I F = MOD( ITER, MODIT) .NE. 0 I F ( S K I P ) GO TO 45

WRITE (OUTPUT, 203) I T E R WRITE (OUTPUT, 216) ( A ( t : : , Bl), K = 1, N L ) WRITE (OUTFUT, 207) I T E K I N , RNEW

A ( K , E { l ) = ALF(:t:::) + k ( K , L N L Z )

I F ( I T E R .LT . ITMAX) GO TO 50 I E R R = -1 CALL VARERR ( I P R I N T , IERR, 1 ) GO TO 95

I F (NU .NE. 0.) GO TO 60 GNSTEP = 0 . 5 G N S T E F I F (GNSTEF' .LT . EPS1) GO TO 95 DO 55 K = 1 , NL

I F (RNEW - R .LT. E P S l * ( R + 1 .DO) ) GO TO 75

72

cc d.J A ( K , B 1 ) = ALF (k::) + G N S T E P W (V i , LNL2)

6 (1 NU = 1.5*NU GO TO 40

I F ( . NOT. S K I F ) WRITE (OUTPUT, 206) NU I F (NU .LE. l(:)G. 1 GO TO 65

I E R R = -- C A L L VARERR ( I F R I N T , I E R R , 1) GO TO 95

t:::SUB = LP 1 + K DO 70 J = K , N L F 1

JSUB = L P l + J I S U B = N L F l + J

7 0 A(t::, JSUE) = f i ( ISUB, KSUE)

L

65 DO 70 t:: = 1 , NL

GO TO 25 -I= / c l R = RNEW

DO 80 t:: = 1 , NL

ACUM = GNSTEF*XNORM(NL, A ( 1 , LNL2) ) /XNURM(NL, A L F )

I F ( S K I P ) GO TO 85

8 (j A L F ( K ) = A ( K , 81)

IF (ITERIN .EQ. 1) rdu = O.S*NU

WRITE lOUTFUT, 200) NU WRITE (OUTPUT, 208) ACUM

85 I E R R = 3

Y O I E R H = I T E R 95 I F (NL .GT. ( 3 ) C A L L D F A ( L , NL, N, NMAX, L F F 2 , I V , T , Y, W, ALF,

I F (FRJRES .GT. E F S l * ( R + i . DO) ) GO TO 5

X ADA, 4, I P R I N T , A , BETA, A ( 1 , L F l ) , H)

X A ( 1 , L - F l ) , BETA, I E R R ) C A L L FOSTPR(L , NL, N, NMAX, LNL2, E F S 1 , R, I F R I N T , ALF, W , A ,

99 RETURN ' 7 - L(.)C) FORMAT ( Y H NU =, E15.7) 2OS FORMAT ( 12HO I T E R A T I O N , 14, 24H NONLINEAF: FARAMETERS) 206 FORMAT (25t.I STEP FlETRACTED, NU =, E15.7) 33'7 FORMAT ( lH0, 15, 2OH NORM OF R E S I D U A L = , E 1 5.7) 208 FORMAT (34H NORM(DELTA-ALF) / NDRM(ALF) =, E I2 .3) 216 FORMAT ( l H O , 7E15.7)

END

SUBROUTINE O R F A C l ( N L F 1 , NMAX, N, L, IF 'R INT , B, FRJF:ES, I E R R ) DOUBLE F R E C I S I O N ACUM, ALFHA, B(NMAX, NLFl ) , BETA, DSIGN, FRJRES,

C

X U, XNORM,AS,BS,S C

NL = N L F l - 1 NL2.3 = 2+PJL + 3 L F 1 = L + 1 DO 3r:) C: = 1, NL

LFt::: = L + 1:: ALPHA = D S I GN ( XNORM (N+ 1 -LF'K, B (LFt:';, 1:: 1 ) , B ( LPK , 1:::) )

U = B (LPt:::, K ) + ALFHA

BETA = ALPHA * U I F (ALPHA .NE. 0 . 0 ) GO TO 13 I E R R -8

Ec (Lpt::: , c::) = u

C A L L VARERR ( I F R I N T , IERR, LF'i + 1:::)

C SUBROUTINE DPFI (L, NL, N, NMAX, LFF2, I V , T , Y, W, ALF, FiDFi, I S E L ,

DOUBLE P R E C I S I O N A(NMAX, LFPZ) , A L F ( N L ) , T(NMAX, I V ) , W (N) 9 Y !N) 7

X I P R I N T , A , U, H, HNORM)

X ACUM,ALPHA,BETA, HNQRM, D S I G N , DSBRT, SAVE, R ( N ) , U ( L ) XNORM, x AS,BS,S

INTEGER F I R S T C , F I R S T R , I N C ( 1 2 , 8) L O G I C A L NOWATE, PH ILP l EXTERNAL ADA

74

I F ( ISEL .NE. 1) GO TO 3 L F 1 = L + 1 LNL2 = L + 2 + NL L F 2 = L + 2 L F P 1 = L F F 2 - 1 F I R S T C = 1 LASTC = L F P 1 F I R S T R = LP1 C A L L I N I T ( L , NL, N, NMAX, L F F 2 , I V , T, W, ALF, ADA, I S E L ,

I F ( I S E L .NE. 1) GO TO 99 GO TO 30

X IF 'R INT, A, I N C , NCON, NCONPI, F H I L P L , NOWATE)

3 C A L L ADA (LF' l ,NL,N,NMAX,LFF2,IV,A,INC,T,ALF,MINO(ISEL,3)) I F ( ISEL .Ea. 2) GO TO 6 F I R S T C = L F 2 LASTC = LPF1 F I R S T R = (4 - I S E L ) * L + 1 GO TO 50

LASTC = LF1 IF (NCON . EQ. 0 ) GO TO 30

6 F I R S T C = NCONF'1

IF ( ~ ( 1 , Ncord) .EO. SAVE) GO TO x )

I S E L = -7 C A L L VfiREF;R ( I F R I N T , I S E L , NCON) GO TO 99

DO 35 I = 1, N R ( I ) = Y ( 1 )

GO TO 50

30 I F I F H I L P l i GO TO 4 : )

4 J 7.c

4 0 DO 45 I = 1, N 45 H ( I ) = Y ( 1 ) - R ( I ) 50 I F (NOWATE) GO TO 58

DO 55 I = 1, N ACUM = W ( 1 ) DO 55 J = F I R S T C , LASTC

55 A ( 1 , J ) = A ( 1 , J ) * ACUM 58 I F (L .EL!. 0 ) GO TO 75

DO 70 t:: = 1 , L I.::p 1 = 1:: + 1 I F ( ISEL . GE. 3 . OR. ( I S E L . EQ. 2 . GND. 1:; . L T . NCONF1) 1 GO TO 66 ALPHA = DSIGN(XNoRM(N+l-P;, AIk;, I.:)), A(t::, 1:::)) U (1:::) = A (t::, K ) + ALPHA A (1:: , t::) = -ALPHA F I R S T C = KF1 I F (ALPHA .NE. 0.0) GO TO 66 I S E L = -8 C A L L VARERR ( I F R I N T , I S E L , K )

GO TO 99 66 BETA = -A(P:, t::) * U ( K )

DO 70 J = F I H S T C , LASTC ACUM = U(K)*A(t : : , J ) DO 68 I = Wl , N

ACUM = ACUM / BETA

DO 70 I = KFl , N

68 ACUM = ACUM + A ( 1 , K ) * A ( I , J )

A(t::: ,J) = A(t:: : ,J) - U(K)*ACUM

88

75

7 0 & ( I , J ) = A ( 1 , J ) - A ( 1 , t::).RACUM 75 I F ( ISEL .GE. 5 ) GO TO 85

RNURM = XN0F.M (N-L , R ( L F 1 ) 1 I F ( ISEL .Ea. 2) GO TO '39 I F (NCON .GT. 0 ) SAVE = A ( 1 , NCON)

DO 95 I = F I R S T R , M 85 I F (L .GT. C ) ) C A L L BACSUE ( N M A X , L, A, R )

I F (L .EQ. NCON) GO TO 95 M = LF'1 DO 90 P; = 1, NL

ACUM = 0 . DO 88 J = NCONPl , L

I F (INC(C:, J ) .Ea. ( 3 ) GO TO 88 M = M + 1 ACUM = ACUM + A ( 1 , M I * R ( J ) CONTINUE

t 3 U B = L F 1 + K I F (INC(t::, LP1) .EQ. 0 ) GO TO 90 M = M + 1 ACUM = ACUM + A I I, M )

9 0 A ( I, t:::SUB) = ACUM 95 A ! I , L N L Z ) = R ! I ) 99 RETURN

END c

SUBROUTINE I N I T ( L , NL, N, NMAX, L F F 2 , I V , T , W , ALF, ADA, I S E L ,

DOUBLE F R E C I S I O N A(NMAX, L P F 2 ) , A L F ( N L ) , T(NMAX, I V ) , W(N) ,

INTEGER OUTFUT, F, INC(l2, €3)

LOG I CAL NOWATE, PI4 I LP 1

DATA OUTFUT /6 / L F 1 = L + 1 LNL2 = L + 2 + N L I F (L . GE. 0 . AND. NL . GE. 4:) . AND. L+NL . L T . N . AND. LNL2 . LE.

X I P H I N T , A , I N C , NCON, NCONF'1, F H I L P 1 , NOWATE)

X DSQRT,AS,BS,S

EXTERNAL ADA

X L P F 2 .AND. 2*NL + 3 .LE. NMAX .AND. N .LE. NMAX .AND. x IV .GT. o .AND. .NOT. (NL .EO. o .HND. L .EQ. ( : I ) ) GO TO 1

I S E L = -4 C A L L VARERF: ( I F R I N T , I S E L , 1) GO TO 99

1 I F (L .EO. 0 .OH. NL .EO. 0 ) GO TO 3 DO 2 J = 1 , LP1

EO 2 K = 1, NL 2 I N C ( K , J ) = 0 J C A L L ADA ( L F 1 , NL, N, NMAX, L P F 2 , I V , A, I N C , T , ALF , I S E L )

NOWATE = . TRUE. D O 9 I = 1, N

NOWATE = NOWATE .FIND. ( W ( 1 ) .En. 1.0) I F ( W ( 1 ) .GE. 0.1 GO TO 9 ISEL = -6 C h L L VARERR ( I F R I N T , I S E L , I) GO TO 99

9 W ( 1 ) = D S O R T ( W ( 1 ) ) NCON = L NCONPl = L P l FHILP1 = L .EO. 0 I F ( F H I L P 1 .OR. NL .EL?. 0 ) GO TO 99

,

78

p = 0 DO 11 J = 1, LP1

I F (P .Ea. 0 ) NCONPl = J DO 11 1:: = 1, NL

I N C K J = INC(C:, J ) I F ( I N C K J .NE. 0 .AND. INCKJ .NE. 1) GO TO 15 I F ( I N C K J .EL!. 1 ) F = P + 1

11 CONTINUE NCON = NCONPl - 1 I F ( I F R I N T .GE. 0 ) WRITE (OUTPUT, 210) NCON I F (L+P+2 .Ea. LPP2) GO TO 20

15 I S E L = -5 C A L L VARERR ( I F R I N T , ISEL, 1) GO TO 99

2(1) DO 25 K = 1, NL 25 I F (INC(t:::, LF1) .Ea. 1) F H I L P l = .TRUE.

99 RETURN 215, FORMAT (33HO NUMBER OF CONSTANT FUNCTIONS =, I 4 / 1

END SUBROUTINE BACSUB (NMAX, N, A, X ) DOUBLE F R E C I S I O N A!NMAX, N ) , X ( N ) , ACUM,AS,BS,S x (N) = x (N) A (N , rd) I F (N .EQ. 1) GO TO 30 N F 1 = N + 1 DO 20 IBACC:: = 2, N

I = NF'1 - I EACK I P 1 = I + 1 ACUM = X ( 1 ) DO 10 J = IP1, N

1 0 ACUM = ACUM - A ( I , J ) * X ( J ) 2 (1) X ( 1 ) = GCUM / A ( I , I ) 30 RETURN

END SUEiROUTINE F O S T F R ( L , NL, N, NMAX, LNL2, EFS, RNORM, I P R I N T , ALF ,

DOUBLE F R E C I S I O N A(NMAX, L N L Z ) , A L F ( N L 1 , R ( N ) , U ( L ) , W ( N ) , ACUM,

INTEGER OUTPUT DATA OUTPlJT /6/ LF1 = L + 1 LFNL = L N L Z - 2 LNLl = LPNL + 1 DO 10 I = 1, N

1 (1) W ( 1 ) = W(I)**2 I F (L . EL!. ( 3 ) GO TO 30 DO 25 }::BAC}:: = 1, L

K = LF'1 - C:BACC:: p p 1 = 1::: + 1 ACUM = (1). DO 20 I = KP1, N

SAVE = R (1::) R ( K ) = ACUM / A ( K , K )

U (K 1 = SAVE

X W, A, H , U, I E R R )

X EPS, FRJKES, RNORM, SAVE, DABS,AS,BS,S

2 (1) ACUM = ACUM + & ( I , K ) * R ( I )

ACUM = -ACUM / ( U ( K ) * A ( K , k c ) )

DO a5 I = w 1 , rd

77

25 H ( 1 ) = R ( I ) - A ( 1 , K)+ACUM 3(1) ACUM = 0 .

DO 35 I = 1, N

SAVE = ALUM / N I F (NL .Ea. ( 3 ) GO TO 45 C A L L O R F A C l ( N L + l , NMAX, N, L, I F R I N T , A ( 1 , L+2), FRJRES, 4 ) 00 40 I = 1, N

TI= .-' d ACUM = ACUM + R ( I )

A ( 1 , LNL2) = R ( 1 ) DO 4 : ) C: = LP1, LNLl

4 0 A ( 1 , K ) = A ( 1 , K + l ) 45 A ( 1, LNL2) = HNOKM

ACUM = RNORM*RNORM/(N - L - NL) A ( 2 , LNL2) = ACUM C A L L COV(NMAX, LFNL, ACUM, A ) I F ( I F R I N T .LT . 0) GO TO 99 WR I T E ( OUTFUT , 209) I F (L .GT . 0) WRITE (OUTPUT, 210) ( U ( J ) , J = 1, L ) I F (NL .GT. 0 ) WHITE (OUTPUT, 211) (ALF(t : : : ) , C:: = 1, N L ) WRITE (OUTFUT, 214) RNORM, SAVE, ACUM I F (DABS(SAVE1 .GT. E F S ) WHITE (OUTPUT, 215) WR I T E (OUTFUT , 209)

99 RETlJRN 209 FOHMAT ( 1HO , 50 ( 1H ' 1 ) 210 FORMAT (20HC) L I N E A R PARAMETERS / / (7E15.7) 1 21 1 FORMAT (231-40 NONLINEAR PARAMETERS / / (7E15.7) )

214 FORMAT (21HO NORM OF R E S I D U A L =, E15.7, 33H EXPECTED ERROR O F 01 XERVATIONS =, E13.7, / 39H ESTIMATED VARIANCE OF OBSERVATIONS = X E15.7 1

X. COVARIANCE MATRIX MAY BE MEANINGLESS. 1 ) 215 FORMAT ( 9 5 H WARNING -- EXPECTED ERROR OF OBSERVATIONS I S NOT ZEI

END SUBROUTINE COV(NMAX, N, S IGMA2, A ) DOUBLE F R E C I S I O N A(NMAX, N ) , SUM, SIGMA2,AS,BS,S

C DO 1 0 J = 1, N

1 0 A ( J , J ) = 1 . / A ( J , J ) IF ( N .EQ. 1 ) GO TO 70 NM1 = N - 1 DO 60 I = 1, NM1

IF1 = I + 1 DO 60 J = IF1, N

J M 1 = J - 1 SUM = 0. DO 50 M = I , J M 1

5 (5 SUM = SUM + A ( 1 , M ) * A ( M , J ) 6 il A ( 1 , J ) = -SUM * A ( J , J ) 70 DO 90 I = 1, N

DO 90 J = I , N SUM = 0. DO 80 M = J, N

SUM = SUM * S IGMA2 A ( 1 , J ) = SUM

9 o A ( J , I) = SUM

8 0 SUM = SUlM + A ( 1 , M) * A ( J , M )

RETURN END

78

C

SUEROUTINE VARERR ( I F R I N T , I E R R , K) DOUBLE P R E C I S I O N AS,BS,S I NTEGER ERRNO, OUTPUT DATA OUTPUT /6/ I F ( I F R I N T .LT . 0 ) GO TO 99 ERRNO = I A B S ( 1 E H R ) GO TO (1, 2, 99, 4, 5 , 6 , 7, 8 1 , ERRNO

1 WHITE (OUTPUT, 101) GO TO 99

2 WRITE (OUTPUT, 102) GO TO 99

4 WRITE (OUTPUT, 104) GO TO 99

5 WHITE (OUTPUT, 105) GO TO 99

6 WRITE (OUTPUT, 106) k:: GO TO 99

7 WHITE (OUTPUT, 107) P:: GO TO 99

8 WRITE (OUTPUT, 108) b:: 99 RETURN

101 FORMAT (46HO FROBLEM TERMINATED FOR EXCESSIVE I T E R A T I O N S / / I 102 FORMAT (49HO PROBLEM TERMINATED BECAUSE OF I L L- C O N D I T I O N I N G / / ) 104 FORMAT ( 1 50H I N F U T ERROR I N FARAMETER L, NL, N, LFF2, OR NMAX. 11 105 FORMAT ( b 8 H 0 ERROR -- I N C MATRIX IMPROPERLY S F E C I F I E D , OR DISAGRE

XES W I T H L F F 2 . / )

lot, FORMAT (19HO ERROR -- WEIGHT(, 14, 1 4 ~ ) IS NEGATIVE. 1) 107 FORMAT (28HC) ERROR -- CONSTANT COLUMN , 13, 37H MUST BE COMPUTED

108 FORMAT (33HO CATASTROPHIC F A I L U R E -- COLUMN , 14, 28H IS ZERO, St XONLY WHEN I S E L = 1. / 1

XE DOCUMENTATION. / I END DOUBLE P R E C I S I O N FUNCTION XNORM (N, X ) DOUBLE P R E C I S I O N X(N), RMAX, SUM, TERM, DhBS, DSRHT

F I N D LARGEST ( I N ABSOLUTE VALUE) ELEMENT RMAX = ( 3 .

DO 10 I = 1, N IF ( D A E S ( X ( 1 ) ) .GT. RMAX) RMAX = D A B S ( X ( 1 ) )

1 0 CONTINUE SUM = 0. I F (RMAX .Ea. 0. ) GO TO 30 DO 20 I = 1, N

TERM = 0. IF (RMAX + D A B S ( X ( 1 ) ) .NE. RMAX) TERM = X ( I ) / R M A X

20 SUM = SUM + TERM+TERM 30 XNORM = RMAX*DSBRT(SUM) 99 RETURN

END

APPENDIX E: ELECTRODE MOUNTING PROCEDURE

1.

2.

3.

4.

5.

6.

7.

8.

Drill .250 in. hole in plate.

Measure electrode O.D. with micrometer.

Using hand reamer with block to keep it aligned, ream hole so that the elec-

trode can be press fit (if the fit is too tight, the electrode will deform and be

wasted).

Clean electrode and hole with acetone.

Using the silver disc on the top of the electrode and the rubber hammer,

tap the electrode part way into the plate.

Apply a bead of loctite cement around the electrode O.D.,

Tap electrode until it is flush with plate (+ .0015 in.). A dial indicator can

be used to check this tolerance.

Wipe clean with acetone.

APPENDME UTERIALS SUPPLIERS

The cast aluminum plate was purchased from Castle Metals, San Francisco,

321-2500.

- The plate was anodized a t Sanford Metal Processing, Menlo Park, 327-5172.

The float glass was purchased at Acme Glass, Palo Alto, 321-7781.

If a thicker. pyrex glass is desired, try Heussner Optics, Santa Clara, 988-

42 14.

The brass parts used to construct the electrodes were purchased at Don’s

Hobby shop, Menlo Park, 322-71’76.

The electrodes were gold plated at Hansen Labs, on campus.

experimental determination of the …...stanford geothermal program interdisciplinary research in...

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