INTERPRETATION OF ELECTRICAL RESISTIVITY LOGS

IN A TWO-ZONE CYLINDRICALLY SYMMETRIC GEOMETRY

by

L. J. Shamey

&

W. M. Adams

Technical Report No. 46

March 1971

This report is based on a study conducted at the Cooperative Insti­tute for Research in Environmental Sciences (CIRES)~ University of Colorado, in support of a study entitled, "Ra/liation Well Logging in Hawaii", funded by the Board of Water Supply, 0i ty and County of Honolulu and the Department of Land and Natural Resources, Division of Land and Water Development, State of Hawaii. The Cooperative Institute for Research in Environmental Sciences is a joint under­taking of the National Oceanic Atmospheric Administration and the University of Colorado, Boulder, Colorado, 80302.

ABSTRACT

A two-zone theoretical model~ consisting of a cylindrical bore hole filled with drilling mud and surrounded by homogeneous~ isotropic rock~ was studied to aid interpretation of electrical resistivity logs. Apparent resistivities are numerically calaulated as a function of the rock and the drilling mud resistivities and the separation of the electrodes on the coaxial measuring sonde. For practical use~ the inverse, interpretation problem must be solved. Therefore~ graphs for finding the true matrix resistivity--given the hole diameter~ mud resistivity, and eZectrode spacings--are presented for ranges applica­ble to Hawaiian conditions. The interpretation may be done with the interpolative digital computer program provided.

iii

CONTENTS

LIST OF FIGURES II ................................................................................................. v

LIST OF TABLES ................................................................................................. vi

FIELD CONDITIONS FOR RESISTIVITY LOGGING IN HAWAIIAN WELLS ......... l

THEORY OF FOUR-ELECTRODE RESISTIVITY ARRAY IN CYLINDRICALLY SYMMETRI C GEOMETRy ....................•.............•..........•... 1

NUMERICAL EVALUATION OF AXIAL POTENTIAL DIFFERENCES ........•....... 9

RESULTS: TABLES AND GRAPHS OF RESISTIVITIES ........•............. ll

DISCUSSION OF ERROR ................. , ............................. 12

ACKNOWLEDGEMENTS ...........................................•...... 23

REFERENCES ................... ....................................................................................... .. 24

APPENDICES ......................................................................................................... 25

LIST OF FIGURES

1. Plan view of a cylindrical well. .............................. 3 2. Sectional view of a cylindrical well .................•.•...... 3 3. Graph for determining true matrix resistivity for

normal array: L = 224 inches .............•.................. 13 4. Graph for determining true matrix resistivity for

normal array: L:::: 72 inches ....•.....................•.•.•.. 14 5. Graph for determining true matrix resistivity for

normal array: L:::: 64 inches .••.....•..................•....• 15 6. Graph for determining true matrix resistivity for

norma 1 array: L = 16 inches .•..•.........•.........•..•..... 16 7. Graph for determining true matrix resistivity for

latera 1 array: L = 224 inches ............................... 17 . 8. Graph for determining true matrix resistivity for

lateral array: L = 72 inches ..........•.......•...•......•.. 18 9. Graph for determining true matrix resistivity for

lateral array: L:::: 64 inches ....•...........•...•........... 19 10. Graph for determin"ing true matrix resistivity for

lateral array: L = 16 inches ................................ 20

v

LIST OF TABLES

1. Apparent resistivities, Pa/Po' for normal array ••...••.•••••• 21 2. Apparent resistivities, Pa/Po' for lateral array ..•••.•••••.. 21

vi

FIELD CONDITIONS FOR RESISTIVITY LOGGING IN HAWAIIAN WELLS

Resistivity, temperature, salinity, and caliper logs have been ob­

tained for a number of water wells in Hawaii for improved understanding

of the hydrology of the basaltic terrain. Analyses of these logs are

made difficult by the caving tendencies of the aa strata and by the

heterogeneity of the medium in and around the well; both conditions are

common to the layered basaltic flow structure and Hawaiian hydrology.

The purpose of this report is to provide the analytical procedures for

the practical interpretation of such resistivity logs, using the caliper

and temperature logs for the reduction procedure.

Geophysical well-logging involves the use of a caliper sonde to

measure the diameter of the well and a resistivity sonde to measure

the potential difference at points on the sonde and the electrical cur­

rent flowing between the surface and the sonde. Both of the above

measuring sondes produce a log as a function of depth into the well,

usually from the top of the casing. The current and potential dif­

ference obtained fr0m the resistivity sonde provide information about

the apparent resistivity of the material surrounding the electrodes.

This report specifically treats the problem of relating the measured

apparent resistivity, the resistivity of the drilling mud within the well,

and the diameter of the well to the true resistivity of the rock-matrix

surrounding the well. This is the two-zone case.

As interpretation of data taken in Hawaiian water wells is the im­

media:te application of these resUlts, only matrix resistivity which is

greater than mud resistivity is evaluated. The theory presented here

closely follows the work of Dakhnov (1959). Keller and Frischknecht

(1966) also provide useful background discussion on these topics.

THEORY OF FOUR-ELECTRODE RESISTIVITY ARRAY IN CYLINDRICALLY SYMMETRIC GEOMETRY

The resistivity sonde in a borehole is approximated by the following

theoretical model. A cylindrical well of diameter Do is drilled into

a rock of infinite thickness and infinite lateral extent. The well is

filled with a driliing mud having resistivity Po' The resistivity of

2

the matrix rock is Pp (Fig. 1).

The geometry of the measuring sonde, which is considered to be

on the axis of the well, is indicated in Figure 2. A current, I, flows'

from electrode A on the sonde to electrode B at the surface. Potential

differences and potential gradients are measured at electrodes M and N

on the sonde. All electrodes are considered to be points with negligible

contact resistance.

In a homogeneous isotropic medium of infinite lateral extent, with

resistivity Pa and with current I flowing from the source electrode

at A, the current density for distances very near to A (relative to

distance AB) can be apprOXimated by

I j = ---

41Tr2

where r is the radial distance from point A.

(1)

The potential U and the potential gradient E are given by Ohm's

law (neglecting the displacement current) as,

E dU . . = - dr = J Pa

Thus, the electric field is given by

E

and the potential at electrode M

M

UM = f dU = 00

where L = distance AM.

(relative to a point 00 Ip Ip

f aa -4 2 dr = 41TL

M 1Tr

(2)

(3)

at infinity) is

(4)

The resistivity can thus be obtained either by measuring a potential

gradient using Eq. (3) or by measuring a potential difference using Eq. (4).

For the practical situation o£ a heterogeneous medium, a value of

resistivity can still be calculated from the appropriate equation above.

This is defined to be the apparent resistivity. This definition of

apparent resistivity differs by a factor of 1/2 from that used for a

surface array because the medium is assumed infinite instead of semi­

infinite. If the conditions assumed in both definitions are satisfied,

Do = DIAMETER OF CYLINDRICAL WELL

Po = RESISTIVITY OF DRILLING MUD IN WELL

Pp = RESISTIVITY OF ROCK SURROUNDING WELL

FIGURE 1. PLAN VIEW OF A CYLINDRICAL WELL.

= ELECTRODES

DO = DIAMETER OF CYLINDRICAL WELL

Po = RESISTIVITY OF DRILLING MUD IN WELL

Pp = RESISTIVITY OF ROCK SURROUNDING WELL

FIGURE 2. SECTIONAL VIEW OF A CYLINDRICAL WELL.

3

6

and

1 + -

r df(r) dr = o (15)

If the real separation constant were of opposite sign, the solutions

so obtained would not satisfy the boundary conditions. Alternatively,

the case of the opposite sign can be solved by letting the separation

constant range over the complex domain and it is trivial.

The solution to Eq. (14) is of the form

z (z) C1 cos(mz) + C2

sin(mz)

where C1 and C2

are arbitrary constants. The substitution, x = mr,

is made in Eq. (15)

+ 1 df x dx - f = o

This is the modified Bessel's equation of zero order and has

solutions Io(mr) and Ko(mr).

(16)

(17)

The variables are reduced by the radius, r o ' of the well. This

scales r to r/ro and z to z/ro' Boundary condition Eq. (10) then be­

comes

Lim U(r,z) R-+O

= (18)

From the symmetry of the problem, the solution, U(r,z), must be an even

function of z, that is,

U(r,-z) = U(r,z)

This precludes sinCmx) being a part of the solution.

Thus the general solution to Eq. (11) of the form

U(r,z) = 00

f A(m)Io(mr) cos(mz)du + o

00

f B(m)Ko(mr) cos(mz)dm o

where A(m) and B(m) are continuous functions of the variable m.

(19)

(20)

In the inner zone, Io(mr) converges as r -+ 0, but Ko(mr) diverges

as r ~ O. However, the potential, Uo(r,z), in the inner zone also

diverges according to Eq. (18), hence, the term with Ko(mr) provides

just the required form. This boundary condition then requires

=

= i o

Frdm the mathematical relation (Abramowitz and Stegun, 1964,

page 486)

i cos (bt) Ko (at)dt = 7f/2

o

P I Boem)

0 =

27f2r 0

Therefore, the potential within the inner cylindrIcal zone is

00 p I 00

UoCr,z) ~ ACm)Io(mr)cosCmz)dm + 0 f KoCmr)cQs(mz)dm =

r 27f2 0 0

(21)

(22)

(23)

(24)

In the outer zone, i.e., for r > r o' the

be retained in the solution since it diverges

is used for this region of resistivity p.

function, I (mr), cannot o

=

as r ~ 00. Subscript, p,

p Hence, the potential is

JOO Bp(m)KqCmr) cosCmz)dm

o (25)

. In Uo(r,z), CoCm) is defined to be 27f2roAoCm)/(poI) and in Up(r,z),

DpCm) is defined to be 27f2roBpCm)/(ppI). The solutions then become

[~

7

8

and ::::

P I P

00

f D (m)K (mr)cos(mz)dm p a (27)

o

The two boundary conditions stated in Eqs. (9a) and (9b) are applied

at the cylindrical interface at r = 1. The two resulting equations

are:

(28)

and

(29)

These may be rewritten to emphasize the coefficients of the unknowns,

CaCm) and DpCm) , as

(30)

and,

(31)

Only the potential in the inner cylindrical zone is of practical interest

because the measuring sonde is located on the axis of the well. Thus,

CaCm) evaluated at r = ° is needed for Eq. (26). Eqs. (30) and (31) are

solved for

Co(m) =

where ~ po

As r + 0, Io(mr) + 1. Eq. (32)

(~ -l)K (m)K1 (m) po a

is substituted for CaCm) in Eq.

and the relation in Eq. (22) is used to integrate the term with

Then p I

[I + n~2 ] UoCo,z) 0

CaCm) cos(mz)dm = 2n 2r a

(32)

(26)

K (mr). a

(33)

The apparent resistivity for a normal array is, by Eq. (6), expressed

in terms of the reduced electrode spacing parameter defined by

L' = Lira

L' =

From Eq. (33), UoCo,L') is used to simplify Pa:

Pa = Po [ 1 + 2L'

Tf I CoCm) COS(mL')dm]

o

(34)

(35)

The apparent resistivity for a lateral array is, by Eq. (5), expressed

in terms of the reduced spacing, L' (the field gradient introduces

a factor llro when written in reduced coordinates)

Pa =

4Tf L' 2 [- dU I ] ro z = L' =

The derivative of the potential is evaluated by Eq. (33), so Pa

simplifies to

Pa = [ 2L' 2

Po 1 + -Tf-- fOO Co (m) sin (mL ')m dm ]

o

Eqs. (35) and (37) give the apparent resistivity, P , for a normal and lateral electrode arrays, respectively, in terms of the

(36)

(37)

9

resistivity, Po' of the inner cylindrical zone filled with drilling mud,

the resistivity, Pp ' of the surrounding rock and the reduced electrode

spacing L. The resistivity, Pp' contained in Co(m). No further

analytical reduction was found and so numerical procedures, with the

aid of a digital computer, were used to evaluate these expressions.

NUMERICAL EVALUATION OF AXIAL POTENTIAL DIFFERENCES

The' function, Co(m), was examined for small m, and the limiting

forms of Io(m), I1

(m), Ko(m), and K1(m) for m + 0 were used to find that

(38)

m « I

which is divergent. This divergence of Co(m) presents no problem for

10

the integrand of Eq. (37) because of the factor, m sin(mL), but the

integrand of Eq. (35) becomes divergent. To treat this divergent

function by numerical means

~o(m) =

::::

(lJpo - 1)

o

In(m) for m <

for m > (39)

defined. (This is equivalent to setting mo equal to 1 in the nota­

tion of Dakhov (1959) thus, there is no need to introduce the parameter

mo') Then, ~o(m) also diverges in the same manner as Co(m) as m + 0 but

with sign opposite to that of Co(m). By adding and subtracting ~oCm)

to CoCm), the integral in Eq. (35) can be developed in the following way

00

~ Co(m) cos(mL') dm " 00

~ [CoCm) + ~o{m)] cos(mL') dm

o

1

f ~o (m) cos (mL') dm

o (40)

The latter term becomes, upon substitution of ~o(m) and integration

by parts

1 1 ~ ~o(m) cos(mL') dm = (lJpo - 1) j ln(m) cos(mL') dm

o

(41)

where the integral Si(x) is defined (Abramowitz and Stegun, 1964, page

231) by

Si(x) =

Thus,

00

~ Co(m) cos(mL') dm =

o

00

f o

x

1 sin t dt o t

(42)

(lJ -1) [Co(m) + ~o(m)] cos(mL')dm + t? Si(L')

(43)

Thus, this integrand has been reduced to procedures manageable on a

digital computer. The integrand in Eq. (37) is similarly manageable

without recourse to using the ~o(m) function.

The presence of cos (mL') and sin(mL') in the integrands naturally

suggests dividing the variable of integration, m, into sections

separated by the nodes. The length of these sections will be one-half

period, i.e . ., 'IT/L'. The nodes of cos(mL') are, of course, displaced

by a quarter period from those of sinCmL').

Six-point Gaussian integration (Abramowitz and Stegun, 1964, page

11

916) was used for each such section bounded by node points of the

integrand. The integration was performed for groups of 30 nodal sections

at a time until convergence was attained. The function CoCm) decreases

slowly as m + ro and acts as a decreasing envelope for the oscillating

cosCmL') and sinCmL') functions which alternate in sign. When the con­

tributions to the integral were grouped into pairs of neighboring posi­

tive and negative terms, it was observed that the contribution of suc­

cessive pairs decreased rapidly. For most cases, convergence was

attained within the first group of 30 nodal sections. For the more

intractable cases, at most, 5 groups of 30 nodal sections were required

for convergence. Errors are discussed in the section entitled, Discussion

of Errors.

A copy of the program for calculating Pa is included as Appendix A.

RESULTS: TABLES AND GRAPHS OF RESISTIVITIES

From the theoretical point of view, Pa is calculated as a function

of Pp ' Po' and L'. From the practical point of view, Pa , Po' and L' are

measured and PP' the true rock resisfivity, is the unknown quantity to

be determined. A two-dimensional table of values of Pa/Po was calculated

as a fu~ction of pp/Po and L' = Llro ' which were allowed to vary para­

metrically. The resistivities Pp and Pa are expressed relative to Po

and, thus, are in reduced dimensionless units, as is Lt. These values

are shown in Table 1 for a normal array and in Table 2 for a lateral

array. In these tables, the electrode spacing is reduced by the diameter,

Do, of a well, rather than by its radius, roo Thus L" = LIDo is the

parameter listed. Ten values of L" and seven values of pp/po were found

12

to be sufficient to specify the variation of Pa/po '

The four-point "Lagrangian interpolation (Abramowitz and Stegun,

1964, page 878) was then used on this two-dimensional table to obtain

that value of pp/Po' which, together with a particular value of L!1,

yields a particular value of Pa/po ' The computer program for these

operations, included as Appendix B, was used to generate data to plot

pp/po as a function of Pa/po with parametric dependence on L and Do

(Figs. 3 - 10). Values of L, selected on the basis of electrode spacing

for the sondes currently in use by the Water Resources Research Center

at the University Hawaii, are 224, 72, 64, and 16 inches. Each

computer-plotted figure shows 3 curves, representing well diameters of

2, 10, and 20 inches, respectively.

By use of the interpolation program in Appendix B, similar graphs

may be generated for any given values of L and Do.

An inspection of Tables 1 and 2 shows that, for any given value

of pp/po ' as L increases, the values of Pa/Po begin near 1, rise to

a peak, and then decrease slowly to pp/po as a lower bound. These

limiting values may be seen analytically from Equations (35) and (37):

Lim L'-+O = 1 (44)

due to the presence of L' as a multiplicative factor. To investigate

the behavior of Pa/po for large L', the function ~oCm) is added to

and subtracted from CoCm) in the integrands. This procedure has already

been used for the normal array and the result is shown in Eq. (43).

Integrating by parts and using the fact that Si(X) -+ rr/2 as X -+ ~ , then,

(45)

DISCUSSION OF ERROR

The final error of any computed quantity is a combination of errors

in the input data and errors generated by the algorithm, i.e., by the

method of computation. In this calculation of the apparent resistivities,

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18

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19

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I

..,

~ pp/po

3.

10.

30.

100.

300.

1000.

3000.

~ pp/po

3.

10.

30.

100.

300.

1000.

3000.

TABLE 1. APPARENT RESISTIVITIES, P /p , FOR NORMAL ARRAY. a 0

0.3 1.0 1.7 3.0 5.7 10.0 15.7 30. 57.

1.803 2.860 3.184 3.255 3.148 3.068 3.032 3.011 3.004

3.877 8.435 10.77 12.27 12.04 11.09 10.53 10.17 10.06

7.878 20.13 28.70 37.72 42.50 39.89 35.74 31.84 30.56

15.76 46.86 71. 99 107.4 147.1 163.4 155.1 125.6 107.4

26.24 94.41 151.8 241.9 378.1 497.3 549.8 504.4 385.6

47.05 175.5 318.3 538.6 908.0 1356. 1737. 2069. 1854.

97.81 259.6 538.2 1044. 1866. 2952. 4115. 5925. 6863.

TABLE 2. APPARENT RESISTIVITIES, p/po' FOR LATERAL ARRAY.

0.3 1.0 1.7 3.0 5.7 10.0 15.7 30. 57.

1.073 2.030 2.824 3.333 3.328 3.171 3.086 3.030 3.010

1.161 3.733 6.966 11.09 13.53 12.64 11.44 10.47 10.15

1.228 5.268 11.84 24.38 41.76 48.48 44.36 35.34 31.58

1. 278 6.480 16.40 40.92 96.29 160.3 192.1 168.1 122.5

1.302 7.214 19.21 52.80 150.0 321.0 504.6 672.7 544.3

1.312 7.700 21.26 61.43 194.8 495.2 952.5 1937. 2585.

1. 315 7.903 22.34 66.74 221.8 611.3 1314. 3430. 6789.

21

100.

3.001

10.02

30.20

102.4

326.3

1400.

6099.

100.

3.003

10.06

30.55

107.0

383.7

2132.

8684.

22

all input data is of parametric form and, hence, exact. Thus the

algorithm is the only source of error. The objective in these calcula­

tions is to produce results accurate to at least four significant

figures so that the maximum relative error is less in absolute value

than 0.0005.

The digital computer carried 15 significant figures and even a

most pessimistic estimate of 10 5 operations per calculation implies

negligible arithmetic rounding error. However, truncation error, caused

by approximating an infinite process by a finite one, is potentially

!,II.uch more important. The integrations in Eqs. (35) and (37) were

calculated in groups of 30 nodal sections of the integrand and the

integration process was allowed to continue for the next group of 30

nodal sections until the contribution from pairs of successive positive

and negative terms contributed only to the fifth significant figure.

Superimposed on this truncation error are those algorithm errors

caused by the method of calculating the assumed finite approximation

to the original infinite process. For these resistivity calculations,

the six-point Gaussian integration was selected after preliminary work

showed that 16- and lO-point Gaussian integration produced results which

differed only in the fifth significant figure.

The maximum total error in each part of the algorithm was thus kept

to less than 0.0005. This has been confirmed by comparing the analyti­

cally known limiting form of Eq. (45) with direct calculations for

large L. Tables 1 and 2 present these calculations with four significant

figures.

The interpolation program uses the output of the resistivity

calculations as input, and, hence, is limited to an accuracy of four

significant figures. The algorithm involves the four-point Lagrange

interpolation for both direct and inverse interpolation of Tables 1 and

2. The possible algorithm errors are that the number of points in the

table is inadequate to specify the function tabulated and that the inter­

polation itself generates errors. A thorough check of such errors was

carried out by taking the resulting interpolated rock resistivities

and putting them into the program which calculates apparent resistivities.

This iterative process produced self-consistent results accurate to three

significant figures, hence, the maximum possible relative error for inter-

23

polated resistivities is less in absolute value than 0.005.

Finally, there are errors caused by the inadequacies of the

mathematical model which was originally constructed to solve this

problem. Two possible improvements of the model have been explored.

First, preliminary work has been performed on a three-zone model, which

include the pres~nce of a concentric, cylindrical zone containing

flushed rock located between the well and the rock matrix. For a

flushed zone with a diameter of 5 to 20 times that of the well and a

resistivity intermediate between that of the rock matrix and the drill­

ing mud, preliminary work showed that the calculated apparent resistivi­

ties varied by as much as 10 to 25 percent from those of the two-zone

calculations described in this report. Secondly, situating the sonde

off-axis for the two-zone model is a logical extension because in prac­

tical work the well is not vertical and so the sonde usually rests on

the side of the well.

ACKNOWLEDGEMENTS

This work has been performed in support of logging research being

conducted by Professor Frank Peterson funded by the Department of Land

and Natural Resources, Division of Land and Water Development, State of

Hawaii and the Board of Water Supply, City and County of Honolulu.

Physical facilities have been graciously provided by the Coopera­

tive Institute for Research in Environmental Sciences (a joint effort

of the National Oceanic and Atmospheric Administration and the University

of Colorado).

Discussions with Dr. George V. Keller and Mark Matthews of the

Colorado School of Mines have also been helpful, but the authors remain

responsible for all material presented and opinions expressed.

24

REFERENCES

Abramowitz, A. and Stegun, I.A. 1964. Handbook of Mathematiaal Funations. Nat. Bu. Stan. Applied Mathematics Series No. 55. U. S. Gov. Printing Office, Washington. D. C.

Dakhov, V. N. 1959. GeophysiaaZ WelZ Logging. English translation provided by George V. Keller in Quart. Colo. School Mines 57, No.2, April 1962.

Keller, G. V. and Frischknecht, F. C. 1966. EZeatriaaZ Methods in Geophysiaal Prospeating. Pergamon Press, New York.

APPENDICES

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE

29

Program TWOZONE, which is listed below, calculates the reduced

apparent resistivities, Pa/Po ' as a function of the reduced rock resist­

ivity, pp/Po ' and the reduced electrode spacing, L/ro. The theory of

the calculation is presented in the main body of this report. The input

parameters are defined by comment cards within the program.

XG and WG are six-element arrays containing the points and weights

for the six-point Gaussian integration. XL, YlL, Y2L, and Y3L are ar­

rays containing alphanumeric information (each on a separate card) for

use as labels and titles in the plotting subroutines. NCASES is the

number of resistivity calculations to be performed. TL is another array

containing alphanumeric labeling information. RHOO is the resistivity

of drilling mud in central zone, RHOP is resistivity of surrounding rock,

L is electrode spacing, and RO is radius of the central cylindrical well.

The subroutine EZPLOT, which performs the plotting of data into

microfilm, is a custom subprogram provided by the University of Colorado

Computing Center and thus would probably not be available elsewhere. An

interested potential user of program TWOZONE could substitute a locally

available subroutine for EZPLOT and the plotting part of program TWOZONE

could be omitted entirely by removing all statements between, but not

including, statement number 101 and, but not including, statement number

500. In this case, the input cards for the plot labels, i.e.~ variables

XL, YlL, Y2L, Y3L, and TL, would be redundant. Blank cards could be

input for these parameters.

Sample input data follows the listing of program TWOZONE.

30

YES

CONSTRUCT MESH WITH

NODAL END ... POINTS

CONSTRUCT INTEGRAND

GAUSSIAN INTEGRATION

NO

CONSTRUCT MESH FOR

NEXT GROUPS OF NODES

GAUSSIAN

INTEGRATION

31

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE.

PROGRAM TWOZONE (INPUl,OUTPUT,FIlMPL,PUNCHI C TWO ZONE WELL-LOGGING CALCULATION.

ODIMFNSION CO(180) ,Xl(8),YIL(S),Y2L(81,Y3l(8),TL(7),FN(180 1 l,GN(lSO)'COSTAR(lSO"XG( 61,WG( 6"RMS(18U)tRMC(18U)'ACt31)'A~(311 2,SUM(30 ),XPLOT(1801,SUM2(3 0 1

REAL LtyO,J1,KO,K1.LP ODAlA WORD1,WORD2,WORD3,WORD4 / 10HNORMAL ARR,lOHAY 210HLATERAL AR,10HRAY I

PI = 3.141592653 LPRINT = 1 NPOINTS = ISO NNODES :: 30 READ 4,(XG(II,WG(II'I=1' 61

4 FORMAT (2F20.101 READ t;,XL READ .", YlL READ 5,V2L READ .".nL

C THESE ARE LABELS USED IN THE PLOTTING SUAROUTINE. ") F n~M AT (8 .. \1 0 I

REAl) 20,NCASES C NCASES IS THE NUMBER OF CASES TO BE RUN.

20 FORM AT (T1 0 I DO 500 NCA=l,NCASES READ 5,TL READ 21.RHOO,RHOP,L,RO

21 FORMAT (4E10.3 ) PRINT 22,NCA,RHOO,RHOP,L,RO

22 0 FORMAT (*lCASE NUMBER *12/* RESISTIVITY IN INNER ZONE = *FI2.1, 11 0X ,*RESISTIVITY IN OUTER ZONE = *F12.1/* DEPTH L TO ELECTRODES = 2 * F12.1,lOX,*RADIUS RO OF INNER ZONE = * FI2.111)

C RHOO IS THE RESISTIVITY OF THE CENTRAL WELL WIIH DRILLING MUD, C RHOP IS RESISTIVITY OF SURROUNDING ROCK' L IS ELECTRODE ~PACINGt C RO IS THE RADIUS OF WELL.

RMU :::RHOP/RHOO LP';;L/RO T '" 2.*PT/lP AC(ll=O.O $ ACI2,=T/4. $ AS(ll=O.O $ AS(2)=T/2. NNODESI = NNODES + 1 DO 10 r=3tN~ODESl I( = I-I AC(!) '" A(IK) + T / 2.

10 ASCI) '" ASIKI + T / 2. C CONSTRUCT 6 POINT MESH FOR GAUSSIAN INTEGRATION B~TWEEN NODES.

c

NSTART = -6 DO 11 T=I,NNODES I( = 1+1 NSTART = NSTART + 6 DO 12 J=I,6 JT = J + NSTART RMS(JTI=(ASIK1-ASIIII*XGIJI/2.0 +(ASIK)+AS(!)'/?O

I? RMCIJTI=IAC(K,-ACfTll*XG(J,/2.0 +(AcIKI+ACIIII/?O 11 cnNTINUE

xo = 1.00 DO 30 t=I,NPOINTS X = RMCITl $ Y = RMS(TI COllI'" CFIRMU.XI COSTAR(!) = COlli + PHIO(RMU,X,XO) GN(II = CF(RMU,Y,*Y*STNIY*LPl

32

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE (CONTID).

30 FM!!) = COSTAR(T) * COSIX*lP) IF( lPRINT .IT. 2) GO TO 39 PRINT 31

31 FORMAT (20X, *VAlUES OF FUNcTYONS*/) DO 32 1=1,180

32 PRINT 33.I,RMC(I),(0II),(OSTAR(I),FN(I), RMS!I),GNII) 33 FORMAT (5X'I5,5X,4E12.3,lOX,2E12.3) 39 CONTINUE

C GAUSSIAN INTEGRATION. DO 14 I=l,NNODES SUM2 (!) '" 0.0

14 SUM(I) = 0.0 SUMTOT = 0.0 SUMTOT2 = 0.0 NSH.RT = -6 DO 15 I=l'~NODES K :: 1+1 NSTART = NSTART + 6 DO 16 J=I,6 JI=J + NSTMT SUM2(II = SUM2(II + WGIJ)*GN(JII

16 SUM(!) = SUM(I) + WGIJ)*FN(JII SUM2(IJ = SUM2 1 Jl * (ASIK)-ASII})/2. 0 SUM ( I) :: SU~ ( I) * ( A.C I K) -AC « r) ,/2. a SUMTOT = SUMTOT + SUM!I) SUMToT2:SUMTOT2 + SUM2(II

15 CONTINUE 795 CONTINUE

TESTI = ABS! SUM(3 0 )-SUM!29») T[5T2 = ABS! SUM I 291-SUM(28» TI='5T : TEST! IF( TESTI .lE. T[5T2) TEST = T[ST2 RATIO = AqS( TEST/SUMToTI IF( RATIO .lE. 1. 0E-04) GO TO 800 PRINT J99,RATro

79QOFORMAT !* POOR CONVERGENCE' RATro = *EIO.,,4X,*INTEGRATE FOR ANOTH 1ER SET OF 30 HALF-PERIOD5.*)

AC(1)=ACC31) $ ASCI) = AS(31) NNO~ES1 = NNODES + 1 DO 710 I=2'~NOD~Sl K= 1-1 ACCr) = ACCK) + T / 2.

710 AS(I) = A5CK) + T / 2. NSTA~T = - 6 DO 711 I=l,NNODES K = 1+1 NSTART = NSTART + 6 DO 712 J=I,6 JT = J + NSTART RMSIJI):(AS(K)-ASCI»)*XG(J)/2. 0 +(AS(K)+ASCI) )/2.0

712 RMCIJI )=(ACCK )-ACC I) I*XGIJ)/2. 0 +CAC(Kl+AC( r 1)/2.0 711 CONTlf'.lUE

DO 730 I=1tNPOrNTS x = RMC{II $ Y=RMSC II COCI) = C~(RMU,Xl cOSTAR!I) = COCT) + PH!O(RMU,X,XO, GNCYI = CF(RMU,V)*Y*STNCY*LPI

73 0 FN(I) = c05TAR(I)*CoseX*lP) C GAUSSIAN INTEGRATION FOR SECOND SET OF NODES

DO 714 1=1,NNODES

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE (CONT'D).

SU~2(Il =0.0 714 SUM(!) = 0.0

NSTART = -6 Dn 715 I=l'NNOD~S K = r +1 NST,ART = NSTART + 6 DO 716 J=1 '6 J!=J + NSTART SUM2(11 = SUM2(I) + WG(Jl*GN(JI)

716 SUM(!) = SUM(I) + WG(JI*FN(JI) SUM2(I) = SUM2(I) * (AS(K)-,AS(Il)/z.O SUM(!) = SU"1(J) * C ,AC(K)-ACCn )/Z.O SUMToT = SUMTOT + SU~(I) SUMToT2=SUMToTZ + SUM2(I)

715 CONTINUE ,. GO T" 795

800 CONTINUE XOLP = XO * LP

33

SUMToT = SUMTOT + (RMU-l.OI*<SIIXOLP)l/LP RHOA = RHOO*(l. + 2.*(LP) *SUMTOT/PII RHOA2=RHOO*(1. 0+Z.*(LP**Z.I*SUMTOTZ/PJ) PRINT 24,RHOA,SUMTOT

240FOR~AT (*OPOTENTIAL SONDE' OR NORMAL ARRAY RHOA = *E12.~'5X,

C

1* VALUE OF INTEGRAL = * E12.3/) PRINT Z5.RHOA2,SUMToT2

2S 0FORMAT C*OGRADIENT SONDE, OR LATFRAL ARRAY RHOA = *E12.3,5X, l*VALUf OF INT~GRAL = *FlZ.311)

PRHn 26 26 FORMAT C*OCONTRIBUTIONS TO INTEGRAL FROM EACH N~)DE*/)

U = 0.0 $ V = 0.0 DO 27 I=l'NNODES U = U + SUM(IJ $ V = V + SUM2(I) PRINT 28,!,SUM(!),U,SUM,2(!),V

28 FORMAT(2X,Y5,5X,2E15.5,15X'2E15.5 27 CONTINUE

PUNCH 101,RHOA ,RHOP,RHOO'L,RO,WORDl,WORD2 PUNCH 101,RHOA2,RHOP,RHOO,L,RO.WORD3,WORD4

101 FORMAT CSElO.3.l 0 X'2AlO )

C PLOT FUNCTIONS. C

N :: NPOINTS lS=O XMJN :: YMI~ :: XMAX :: YMAX :: 0.0 LT :: -2 NO=l CALL MAXMIN (COSTAR,N,Q,P, IF! Q .LE. 0.0) GO TO ry4 DO 50 !=l,N IF! cOSTAR({) .GT. 0.0) XPlOTlI) :: COSTARIJ)

50 !FICOSTAq(!1 .L~. 0.0) XPLOT(I) :: 1.0E-04*Q CALL EZPLOT(RMC,XPLOT'N'LS,XM!N,XMAX'YMIN,YMAX,XL,YlL,TL,LT,NO) !~( P .GE. 0.0) GO TO ry3

54 D0 '5 1 r :: 1 , N IF/COSTAR(I) .GE. 0.0) XPLOTII) :: 1.OE-04*(ABS(~))

t:;l IFICOSTAR(J) .IT. 0.0) XPLOTfIl = ABS!COSTARIIl) XMIN :: Y~IN :: X~AX = YMAX :: 0.0 CALL EZPLOTIRMC,XPLOT'N'LS,XMIN,XMAX,YMIN'YMAX,Xl,YIL,TL,lT,NO)

'33 CONTINUE XMIN =. YMIN :: X~AX = Y~AX :: 0.0

34

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE (CONT'D).

LS :: 0 LT 1.;' 2 L T : -1 NO :: 1 CALL EZPLOT IRMC,FN ,N'LS,XMIN.XMAX,YMIN,YMAX,XL,Y3L,TL,LT,N01 XMIN :: YMIN :: XMAX = YMAX :: 0.0 NO :: 1 LT :: -1 LS = 0 CALL EZPLOT IRMS'GN ,N'LS,XMJN,XMAX'YMIN,YMAX,XL,Y2l,TL,LT,NO)

500 CONTINUE END FUNCTION PHIO(RMU,X,XO) PHIO = (RMU - 1.0) * ALOG(X/XO) IFf X .GE. XO) PHIO = 0.0 RETURN $ END SUBROUTINE MAXMrN IF,N,XMAX,XMIN) DIMENSION FIN) XMAX :: Fll) $ XMIN :: P(l) IF ( N .EO. 1 I RETURN DO to I:2,N TP( F(l) .GE. XMAX) XMAX = F(J)

10 IF( FlU .lE. XMIN} XMtN :: F(l) ~ETURN END FUNCTION CFfX'YI REAL KO,Kl,rO'II CF :: (X-t. OJ*KOfYl*Kl(Y)/IX*KO(Y)*IlIY)+rO(Y'*KllYI) RETURN $ END REAL FUNCTION KOIX) RI:Al 10 KO .. O. JFIX.LT.2.' GO TO 120 X2 1.;' 2./X KO :: 1./SQRT1X,*EXP(-X)*Cl.25331414-.07832358*X2 +.02189568*X2**2

2 _.01062446*X2**3+. 00 587872*X2**4-.0025154*X2**5+.000 53208*X2**6) RETURN

120 IFIX.LT.O.l RETURN T = X/3.75 X2 :: X/2. 1°::1.+3. 515622('1*T**2+3. 0899424*T**4+ 1.2067492*"1 :t*6+. 26597~2*T**8

2 +.0360768*T**1 0 +.0045813*T**12 KO=_ALOGIX21*IO_.57721566+.4227842*X2**2+.2306976*X2**4+

2 .0348859*X2**6+.00262698*X2**8+.0001075*X2**10+.0000074*X2**12 RETURN END REAL FUNCTION Kl(Xl REAL J1 Kl = 0. IF(X.LT.2.) GO TO 130 X2 := 2. /X Kl :: 1./SQRTCX)*EXP(-Xl*(1.25331414+.2~49861*X2-.0~65562*X2**2 +

2 .OIS04268*X2**3-.00780353*X2**4+.00325614*X?**S-.00068245*X2**6) RETURN

130 IF(X.LT.O.l RETURN T = X/3.75 X2 := X/2. 11 = X*(.5+.8789 0594*T**2+.51498869*T**4+.15084934*T**6 +

2 .02658733*T**8+.00301532*T**10+.00032411*T**12) Kl := Il*ALOG(X2'+I.IX*(1.~.15443144*X2**2-.67278579*X2**4-

35

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE (CONTI D).

2 .18156897*X2**6-.0191940*XZ**8-.00110404*X2**lJ-.00004686*X2**12) RETURN FND FUNCTION ,0 tx)

REAL to DIMENSlON EY(3) CALL AESSr(X'EI) 10 = EIt1) RETURN $ EN!') FUNCTION Il (X) REAL II DIMEf\lStON EI(31 CALL RESSy(X'FI) Tl = f'J(2) RETURN $ I:f\lD SUBROUTINE RESSK (X,CKE,EI) DIMENSION FIRST(4),EI(3),COEFC4',CKE(3I,A CIO,4) DATA (A = 0.42278420, .23069756, .03488590,

1 .00262698, .00010 750, ~00000740, 3(0.0),6.0, 2 .15443144, -.67278579, -.18156897, 3 -.01919402, -.0011 0404' -.00004686' 310.0),6.0, 4 -.07832358, .02189568, -.01062446, 5 .00587872, -.00251540, .00053208. 3(0.0),6.0, 6 .23498619, -.03655620, .01504268, 7 -.00780353, .00325614, -.00068245~ 3(0.0" 6.0

CALL BESSr (X,EYI !FIX .IT. 2. 0 ) 10 ,20

10 T = X / 2. 0 Xo = ALOG CT 1 FIRST(l) = -XP * EI(l) - 0.57721566 FyRST(2) = X * XP * FI{2) + 1.0 FACTOR = T * T COEFtl) = 1. 0 COEF(2) = 1.0 / X JJ :: 1 GO TOlliO

20 T = 2.0 / X FIRST(3) = FIRST(4) = 1.25331414 JJ = 3 COEF(3) = COEF(4' = 1. 0 / (SORT (XI * ~Xp (X) ) FACTOR :: T

50 JEND ., JJ + 1 I :: 0 DO 200 J ., JJ,JEND T = I + 1 PROD = 1. 0 SUM = 0.0 KEND = AClO,J) Dn 100 K = 1-KEND PROD ., PROD * FACTOR SUM = SUM + PROD * A(K,J)

100 CONTINUE CKEel) = COFF(J} * (FYRST(J) + SUM)

200 CONTINUE CKE(3) : (2.0/XJ * CKE(2J + CKE(}) RFTURN END SUAROUTYNF REssr (X,ET! DIMENSION A(lO,4),FIRST(4J,COEF(4),E!(31 DATA(F!~ST ., 1. 0 ,O.S,2(O.3989422R) I,

36

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE {CONT'D}.

1 (A= 3.5156229. 3.0899424, 1.2067492. 2 .2659732, .0360768. .0045813, 3(0.0,. 6.0, 3 .87890 594' .51498869, .15084934, 4 .02658733, .00301532, .00032411. 3(0.0). 6.0, 5 .01328592. .00225319.-.00157565, 6 .00916281. -.02057706, .02635537, 7 -.01647633, .00392377, 0.0 , 8.0. B -.03988024. -.00362018\ .00163801. 9 -.01031555' .02282967,-.OZ89~312' 1 .01787654, -.00420059. 0.0 • 8.0

T = X I 3.75 COEF(l) = 1. 0 COFF(Z} = X COFF(31 = COEF(4) : fXP (X) I SQRT 'X) rF(X .LT. 3.75) 1°'20

10 FACTOR = T * T JJ = 1 GO TO 50

20 FACTOR = 1.0 I T JJ = 3

50 JEND = JJ + 1 I = 0 no 200 J = JJ,JFND I = I + 1 PQOI") = 1. 0 SUM = 0.0 KEN!) = A(1 0 ,J) DO 100 K = 1.KEND PROD = PROD * FACTOR SUM = SUM + PROD * AIK,J)

100 CONTINUE FytJ) = COEF(J) * ( FIRST(J) + SUM

ZOO CONTINUE Elt3) = (-Z.O/Xl * FIIz) + EI(l) RETURN Et\lD FUNCTION SIC x)

C CALCULATION OF THE SINE INTEGRAL FUNCTION. DIMFNSION XG(10),WGI10"Z(10),FII0,

C USE 10 POINT GAUSSIAN INTEGRATLON AFTWEEN NODES OF ISINCZI,/Z. XG(6,=O.1488743389816,1 $ WGI61: 0 .Z95524224714153 XG{71=O.433395394129247 $ WG(7)=O.Z69266119309996 XG(8)=O.679409568299024 $ WGI8,=O.219086362515982 XG(9)=O.865063366688985 $ WG(9)=O.149451349150581 XG(lO)=0.973906528517172 $ WG~10)=O.066671344308688 XG(S) = -XG(6) $ WGfS' = WG(6) Xr,(4) = -XG(7) $ WG(4):: WG(7, XG(3) = -XG(8) $ WG(3) = WG(8) XGIZ) = -XG(9) $ Wr,IZ, = WG(9) XG(I' - xr,cio) $ WGfl) = WGIIO, PI = 3.141592653589 T :: PI A :: -T $ 8 = 0.0 Sy = 0.0 ITFST = 0

10 A = A + T ~ ~ = p + T IFf X .IF. ~) TTFST = 1 RP :: R IF ( IT F ST. FO • 1 ) R P :: X DO 15. 1=1 9 10

APPENDIX A. COMPUTER PROGRAM FOR TWOZONE (CONTID).

ZIT) = 18P-AI*XG( 1)/2.0 + leP+AI/2.0 15 F(TI = ( S1N(Z(T» )/Z(I}

SUM = 0.0 D(') 16 T=1,10

]6 SUM = SUM + Fly) * WG(YI SUM = SUM * IBP-Al/2. 0 S I = S r + SUM IF ( ITEST .EQ. 0 I GO TO 10 RrTURN $ END

_0.932469514203152 0.171324492379170 -0.661209386466265 0.360761573048139 -O.2~8610186083197 0,46791~934~7'6Ql

0.238619186083197 0.467913934572691 0.66120938\6466265 0.36 0 761573048139 0.932469514203152 0.171324492379170 ~ (2 ZONE CASEl ~OV. 14,1970.3

FUNCTION (OSTAR(M) = COIM) + PHIO(M) FUNcTYnN COCMI*M*SYNIM*LP) ~UNrTr(,)NS COSTAP(MI*rnS(M*lP}

7 RHOP/RHOO = 3. L/RO = 114, +1.000F+00 3.000F+OO 1.140F+02 1.000E+00 RHOP/RH(,)O = 10. L/RO ='114. +1.000E+00 1.000E+01 1.140E+02 1.000E+00 RH(')P/RHnO = ,0. LIRa = 114, +1.000E+OO ~.OOOE+01 1.140F+02 1.000E+OO RH(,)P/QHnO = 100. L/PO = 114. +1.000F+OO 1.0nOf+02 1.140F+02 1.000E+00 RHOP/RHOO = 300. LIRa = 114. +1.000F+OO 3.000E+O? 1.140t+02 1.000f+OO RHOP/RHrO = 1000. L/RO = 114. +1.000E+OO 1.000E+O~ 1.140E+OZ 1.OOOE+OO RHOP/RHOO = ~OOO. L/RO = 114. +1.000E+OO 3.000E+03 1.140~+02 1.000E+OO

37

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN

Program PLOT2ZN, which is listed below, takes as input the

resistivity calculations provided by program TWOZONE. The input

information is in the form of a two-dimensional table giving Pa/po

41

and LIDo' Then, given some particular set of apparent resistivity,

electrode spacing, and well diameter, this program interpolates through

the table to find the corresponding value of rock resistivity, pp/Po'

Four-point Lagrangian interpolation is used.

The entire table of apparent resistivities is inputted. This

data was punched by program TWOZONE and is included in the sample data

following this listing. There are 140 cards, the first 70 for a normal

array and the second 70 for the lateral array, each containing apparent

resistivity, RHOA, rock resistivity, RHOP, electrode spacing, L, and

WORDI and WORD2, which contain labeling information.

Following this input data, NUMl is the number of interpolations

to be performed for a normal array and NUM2 is the number for a lat­

.eral array. For each interpolation there is a read statement for ap­

parent resistivity, XRHOA, the resistivity of drilling mud, XRHOO, the

well diameter, XDO, and the electrode spacing, XLE. The program then

interpolates the table to find the value of rock resistivity which is

implied by this input data.

The same comment made in Appendix A, concerning the plotting

routine EZPLOT, applies here. If a potential user wishes to use only

the aforementioned interpolation procedure of program PLOT2ZN, he may

terminate the program by omitting the last section of this program,

beginning with the comment cards which announce the start of the plot­

ting, i.e., two cards after statement number 100.

Sample data for this program follow the listing.

42

INTERPOLATE TABLES

TO DETERMINE Pp

INTERPOLATE TABLES

43

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN.

PROGRAM PLOT2ZN (INPUT,OUTPUT,FtLMPLI c THIS PROGRAM INTE~POLATES THE TABLE OF RHOA AS A FUNCTION OF RATIOS C RHOP/RHOO AND LIDO.

ODIMENStON L(10ItLDO(10),RHOPI7I,RHOA(7I,X(200},Y(200), lXL(B)'YL(8ItTL(7)tTEMPIC10),TEMP2(7),TABLEL(7'1~)'LDOL(lO), 2RHOPL(7),TA~LF(7'lO'2),WORDl(2}tWOR02(2)

RFAL L'LOO,L!,)OL RHOO=I.O $ RO = 1.0 $ DO=2.0 DO 3 K::lt2 DO 20 I=l'lO DO 20 J=1'7 READ 15,RHOA(JI,RHOP(J),L(I),WORDl(K),WORD2(K) TA~LE(JtI.K) = RHOA(J) LOOI!) = Un/DO

15 FORMAT (2FIO.3,10X,EI0.3t20X,2AIO 20 CONTINUE

PRINT 10,WORDIIK),WORD2(K) lOOFORMAT(*I*'lOXt*TABLE OF APPARENT RESISTIVITIES RHOA/RHoa FOR *

12AI0 II) PRINT Ilt(LDO(II,Y=1,10)

11 FORMAT ( 8X,*L/DO =*,2X,(1 0 FIO.l)1 PRINT 12

12 FORMAT (/14X,*RHOP/RHOO*II) "0 13 J=I,7

13 PRINT 14,RHOP(JJ,(TABLE(J,Y,KI'Y=I'10) 14 FORMAT (1/2X'FIO.1' 5X,(IOEI0.3))

'3 CONTINUE C C CALCULATE RHOP AS A FUNCTION OF RHOA'L'OO (THESE PARAMETERS WILL C BE SPECIFIED AS A FUNCTION OF DEPTH).

KTYPE = ° REA" 16'NUMltNUM2

16 FORM A T (2 II 0) C NUMl::: ~UMBFR OF INTERPOLATIONS FOR NORMAL ARRAY' NUM2 FOR C LATERAL ARRAY.

55 KTYPE ::: KTYPE + 1 C KTYPE::: 1 FOR NORMAL ARRAY, ::: 2 FOR LATERAL ARRAY.

NUM ::: NUM1 IF(KTYPE .En. 2) NUM = NUM2 rFI NUM .~Q. 0, GO TO 100 PRINT 601,WORDl(KTYPE),WORD2(KTYPE)

601 FORMAT (*1*,2 0X ,2AlO III) DO 25 1=lt7 RHOPL(Il = ALOGIO(RHOP(Il) DO 25 J"'I'IO LDOL(J) = ALOGIO( L~O(Jl )

25 TAALEL(Y,JJ = ALOGI 0 ! TABLE(!,J,KTYPFJ Dn 100 NUMN ::: I'NUM REAO 17,XRHOA,XRHOO,XDO,XLE

17 FORMAT ( 4FI 0 .2 ) N '" 10 DO 26 1=1'7 DO 19 J=l,lO

19 TEMP1(J) = TA~LFL(J'Jl Q ::: ALOGIO(XLF/X~O ) CALL LAG!NT (F,Q.TEMPltlDOL,N)

26 TEMP2(rl = F N ::: 7 Q ::: AlnGI 0 ( XRHOA/XRHOO ) CALL LAG!NT ( P , Q ,RHOPL,TFMP2,N)

44

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN (CONT'D).

XRHOP = (1 0 • 0 ,**(PJ PRINT 22,XRHOP,XRHOA,XRHOO,XLE,XnO

220FORMAT (11* ROC~ RESISTIVyTY RHOP/RHOO = *E12.3,5X,* APPARENT RESr ISTIVITY RHOA = *EI2.3.5X, *RHOO = *E10.3/1* ELECTRODE SPACING L = 2 *F10.2* INCHES, AND WELL-DIAMETER :: *FIO.?.* tNCHES.*)

1°° CONTINUE IFI KTYPE .EO. 1) GO TO 55

C PLOT RHOP AS A FUNCTION OF RHOA WITH PARA~ETRIC DE~ENDENCE ON DO C FOR SERIES OF FIXED L.

RE/!,I) 5,Xl READ 5'YL

5 FORM AT (8 A 1 0 I DO 200 KTYPf = 1.2 DO 38 1=1'10 LDOl(Il = ALOGIO! LOO!I) DO 38 J=1t7 RHOPL(j) :: ALOGIO! RHOP(J) )

38 TABLELlj,Il = ALOGIOI TABlElj,y'KTYpEl 1)0 200 Il =1.4 IFIlL .EO. 1) XLE = 72. IFI Il .FO. 2) XLE = 64.0 IFIll .EO. 3) XLE = 16. 0 IFIll .EO. 4) XlE = 224. READ 5,TL DO 199 ID =1,3 IFI In .EO. 1) XDO:: 2. IF! Ir) .EO. 2) XOO:: 19-. 11='/ If') .FO. '::II XDO:: 2 0 • PRINT 301,WORD1(KTYPE1,WORD2(KTYPEI ,XLF,xnO

3 0 10 FQRMAT (/I/l0X'2AIQ'10X, *ELECTRODE SPACING L : *FIO.l'lOX, l*WELL-DIAMETER = * FJO.l'* INCHFS.*1 a ::: ALOGIOIXLE/XDO 1

N = 10 f')0 40 1=1,7 no 39 j=1,10

39 TF~P1(J) = TA~LI='L(I'J) CALL lAGTNT (F.O,TFMPl'LDOL,N)

40 TEMP2(11 ::: F DA :: ABSI TEMP2(7)-TEMP2(111/(1 QQ.I X(I) = TEMP2( 1 I DO 42 1=2,200 K ::: 1-1

42 XIII ::: X(I(I + nA ~ ::: 7 DO 44 1=1'2 00 o ::: Xln CALL LAGINT (F,O, RHOPL,TFMP2,NI

44 Y ( I I ::: F C FOR pURPOSE OF PLOTTING, REVERT RACK TO NON-LOG VARIABLES AND USE C LOG-LOG COORDINATES ON THE GRAPH.

DO 45 1=1.200 XITI ::: (1 0 .1**IX{Yll

4 r; Y ( I) ::: ! 1 0 • ) ** (Y ( I ) I N :: 200 IFI ID 0[0. 1) lS::: lH­IF! ID .I:Q. 2) LS :: 0 IF( ID .EO. 31 LS::: IH. XMTN ::: X~AX ::: YMIN ::: YMAX :: 0.0 NO = '":\ LT :: -4

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN (CONTI D).

~L(7) ;: WORDIIKTYPEl $ XL(8);: WORD?{KTVPF) I~ (T~ .GT. 1 1 GO TO 198 CALL EZPlOT IX,Y,N,L5,XMIN,XMAX,YMIN,VMAX,XL,Yl,TL,LT,NO) IFI ID .EO. 1) GO TO 199

198 CALL NXCURV (X,Y,N,LSl 199 CONTINUE 200 CONTINUE

FND SURROUTINE LAGtNT IFtX,FVEC,XVEC,NI

C FOUR POINT LAGRANr,~ INTERPOLATION DIMENSION FVECIN1,XVECCNl IFI XVEC(ll .GT. XVECIN) 1 GO TO 60 rSTART ;; 1 I~I N .EO. 4 1 (,0 TO 50 ySTART = 1 IFIX .GE. XVEcIll .AND. X .LE. XVEC(?) I GO TO 50 ySTART '" N-3 IFI X .GE. XVEC(N-ll .AND. X .LE. XVECINl ) GO TO 50 ISTART :: 1 IF( X .LE. XVEC( 1) GO TO 50 ISTART :: N-3 rFI X .GF. XVECINI (,0 TO 50 NLESS :: "1-1 on 10 I=l '''RESS IFi X .GE. XVEC(Il .AND. X .LE. XVECCI+ll 1 INOTE

10 CONTINUE ySTART = INOTF - 1

50 lEND;; ISTART + 3 DO 5 I = ISTART.IENO F = FVFeII) IFI X .EQ. XVFelIl I RETURN

c:; CONTINUE F = 0.0 DO 25 I=ISTARTtIEND G '" 1. 00000 DO 20 J::JSTART,JFND IF { r • EQ. J 1 GO TO 20 G :: G*IX-XVECIJ~I I (XVECII1-XVECIJ)

?O CONTINUE F :: F + G*FVEC(II

25 CONT I~IUE RFTURN

60 ISTAPT :; 1 IFIN .FO. 4} GO TO 70 ISTART "- 1 IFIX .LE. XVECl11 .AND. X.GE.XVEC ( 21) DO '0 70 I STApT :: 1\1-3 IFIX .LE. XVEC(N-il .AND. X.GE. XVECINll GO TO 70 ISTApT = 1 IF IX .GF. XVEclll } GO TO 70 rSTART :; N-3 IF IX.LE.XVECINII GO TO 70 NLFSS :; "1-1 "" 62 1=1,NLFSS

6' IF(X.LE.XVECI I I .AND. X.GE.XVEC( 1+1) I TNOTF=I rSTART = JNOT~ -1

70 IEI\IO= IST.ART + ~ DO 65 I=rSTART,IEND F=FVEC(II IF IX.EO.~VEC(Ill RETURN

45

LAG2 10 LAG2 20 LAG2 ~o LAG2 40 LAG2 50 LAG2 60 LAG2 70 LAG2 RO LAG2 90 LAG2 100 LAG2 110 LAG2 120 LAG2 130 LA.G2 140 LAG2 150 LAG2 160 LAG2 170 LAG2 lAO LAG2 1QO LAG2 200 LAG2 210 LAG2 2:'0 LAG2 2-:.,0 LAG2 240 LAG2 250 LAG2 260 LAG2 270 LAG2 280 LAG2 290 LAG? 300 L.AG 2 310 LAG2 ~?o LAG2 3':10 LAG2 340 LAG? 3"iO LAG? 360 LAGZ 370 LAG? 3~0 LAG2 390 LAG2 400 LAG? 410 LAG2 420 LAG? 430 LAG? 440 LAG2 450 LAG2 460 LAG2 470 LAG? 480 LAG? 490 LAGZ 500 LAG2 510 LAGZ 520

46

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN (CONTI D).

65 CI'lNTYNU"" r= = 0.0 Dn ~25 I=rSTApT,l~~n

(; = 1.00 DO 620 J=rSTART,I[N~ IF( T.F'Q. J) GO TO 620 G = G*IX-XVECIJll/{XVFC(Il-XVEC(Jll

620 CONT!NUF F = F + G*FVECIIJ

625 (ONTINUF RFTUPN nm

1.803£+00 1.000E+00 1.000£+00 6.000E-01 1. 0 00E+00 3.~77E+00 ,.000E+01 1.000E+00 A.OOOE-O, 1.000~+OO 7.87~E+OO ,.000E+01 1.000£+00 A.aOOE-Ol 1. 000 £+00 1.576E+01 1.000E+02 1.000E+00 6.000E-01 1.000E+OO 2.624£+01 3.000E+02 1.000£+00 6.000E-01 1.0001"+00 4.705£+01 1.000E+03 1.000£+00 6.000E-01 1.000E+00 9.781£+01 3.000E+03 1.000E+00 6.000E-01 1.0001='+00 2.860£+00 3.000E+00 1.000F+OO 2.000E+00 1.000F+00 8.435E+00 1.000E+01 1.000£+00 2.000E+00 1.000f+OO 2.013E+01 3.000E+01 1.000E+00 /.OOOE+OO 1. 000[+00 4.686E+01 1.000E+02 1.000E+00 2.000E+00 1.000E+00 9.441E+01 3.000E+02 1.000E+00 2.000E+00 1.000E+00 1.755E+02 1.000E+03 1.000F.+00 2.000E+00 1.000[+00 2.596E+02 3.000E+03 1.000f+00 2.000E+OO 1.000 F+00 3.184F.+00 3.000E+00 1.000E+00 ~.400E+OO 1.000F+OO 1.077E+01 1.000E+01 1.000[+00 3,400E+00 1. 0 00E+00 2.870E+01 ,.000E+01 1.000E+00 3.400E+00 1.000E+OO 7.199[+01 1.000E+02 1.000E+00 ~.400E+00 1.000E+OO 1.518£+02 3.000E+02 1.000£+00 ~.400E+00 1.000E+00 3.183£+02 1.000E+03 1.000E+00 3.400E+00 1.000E+00 5.382E+02 3.000E+03 1.008E+OO 3.400E+00 ].OOOE+OO 3.255E+00 3.000E+00 1.000E+QO 6.000E+00 ,.OOOE+OO 1.227£+01 1.000E+01 1.000E+00 6.000E+00 1.000E+00 3.772E+01 3.000E+Ol 1.000E+00 6.000E+00 1.000E+00 1.074E+02 1.000[+02 1.000F+00 6.000F+OO 1.000~+OO 2.419E+02 3.000E+02 1.000E+00 AoOOOE+OO 1. 0 00E+00 5.386E+02 1.000E+0, ,.OOOE+OO 60000E+0 0 1· 0 00E+00 1.044E+03 3.000E+03 1.000E+00 6.000E+00 1.000E+OO 3.148E+00 3.000E+00 1.000(+00 J.140£+01 1. 0001"+00 1.,04E+01 1.000E+01 ].OOOE+OO ]'J40E+01 1. 000E+00 4.250E+01 3.000E+01 1.000E+00 1.140E+01 1.000E+no 1.471f+02 1.000(+02 1.000~+OO 1.140E+01 1.000~+OO 3.781E+02 ~.OOOE+02 1.000E+00 1.140E+01 1. 000E+00 9.080E+02 1.000E+03 1.000E+00 1.140E+Ol 1.000E+00 1.866E+03 3.000E+03 1.0 00 E+00 1.140E+01 1. 0 00E+00 3.068£+00 3.000E+00 1.000E+00 2.000 E+01 1.000E+OO 1.109E+Ol 1.000E+01 1.000E+00 2.000E+01 1.000(+00 3.089E+01 3.000E+01 1.000E+00 2.000E+01 1. 000 E+00 1.634E+02 1.000E+02 1.000E+00 2.000E+01 ,.0001"+00 4.973(+02 ~.000E+02 1.000~+OO ;:>.000E+01 1.000~+00 1.356E+03 1.000E+03 1.000 E+00 2. 00 0E+01 1. 0 00E+OO 2.952E+03 ,.OOOE+O, 1.000E+00 2.000E+01 1. 0 00E+00 3.0~2E+00 3.000E+00 1.000E+00 1.140E+01 1.000[+00 1.0~3E+Ol 1.000E+01 1.000 E+00 '.140E+Ol 1.000E+00 ,.574£+01 1.000E+01 1.00 0F+00 1.140E+01 1. 000 E+00 1.5~lE+02 1.000E+O, 1.00 E+OO 1.140E+01 ~.OOOE+OO 5.498E+02 3.000E+02 1.00CE+OO 3.14CE+Cl 1. 000 E+OO

NORMAL ARRd,Y NORMAL ARRAY NORMAL ARRAY NORfv1AL ARRAY NORlv1dL ARRAY NORMAL ARRAY NOR~AL ARRAY NORMAL ARRAY NOR~AL ARRAY NORMAL ARRAY NORMAL ARRAY NORfvlAL ARRAY NORMAL ARRAY NOR~~AL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NOR"'IAL ARRAY NORMAL ARRJW NORflAL ARRAY NOR~lAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORfI.'AL ,ARRAY NOR'v1Al ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORMAL ARRAY NORt-1t>,l ARRAY NORMAL ARRAY NORMAL ARRAY NORtv'Al ARRAY NORMAL ARRAY NOR:v1Al ARRAY NORMAL ARRAY NORMAL ARRAY t-jORMAL ARRdY NOR~lAL ARRA Y NORMAL ARRAY

LAG? 5~O LAG:? 1540 LAG? 5<;0 LA(;;:> 51'0 LAG2 570 LAG2 'iRO LAG;:> ",00

LAG? 600 LAG2 610 LAC;? IS?O LAG? ~':IO

LAG2 640

47

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN (CONTID).

1.737E+O, 1.000E+03 1.OOOE+OO 3.140E+Ol 1.000E+OO NORMAL ARRAY 4'115E+03 ,.OOOE+O, l·OOOE+OO ,o14 0E + Ol 1.000 E+OO NORMAL ARRAY 3.011 E+OO ,.OOOE+OO 1.000E+OO 6.000E+Ol 1.000E+OO NORMAL ARRAY 1.017E+~1 1.OOOE+01 1.OOOE+OO 6.000E+Ol 1.000E+OO NORMAL ARRAY

3.184E+Ol 3.000E+Ol 1.000E+OO 6.000E+Ol 1.OOOE+OO NORMAL ARRAY 1.256E+02 1. 0 0 0E+02 1.000E+00 6.000E+01 1.000E+OO NORMAL ARRAY 5.044E+02 3.000E+02 1.00Of+00 6.000E+Ol 1.0001='+00 NORMAL ARRAY 2.069E+03 1.OOOE+0 l·OOOE+OO 6.000E+Ol 1.000E+OO NORMAL ARRAY 5.925E+03 3.000E+03 r.OOOE+OO 6.000E+01 1.000E+OO NORMAL ARRAY 3.004E+OO 3.000E+OO 1.000E+OO 1.140E+02 ].OOOE+OO NORMAL ARRAY 1.006E+01 1.000E+01 1·000E+OO 1·140E+02 1·000E+OO NORMAL ARRAY 3.056E+01 3.000E+01 1.000E+OO 1-140E+02 1.000E+OO NORMAL ARRAY 1.074E+02 1.000E+02 1.000E+00 le140E+02 1.000E+OO NORMAL ARRAY 3.856E+02 '3.000E+02 1.0001:+00 1.140E+02 1.000F+00 NORMAL ARRAY 1.854E+03 1.OOOE+03 1.000E+00 le140E+02 1.000E+00 NORMAL ARRAY 6.8531:+03 3.000E+03 1.0001'='+00 1.140E+02 1.000E+OO NORMAL ARRAY 3.001E+00 3.000£+00 1.000E+00 2.000E+02 1-000E+00 NORMAL ARRAY 1.002£+01 1.000E+01 1.000E+00 2.000E+02 1.000E+00 NORMAL ARRAY 3.020E+Ol 3.000E +Ol l·OOOE+OO Z·000E+02 1· 000E+00 NORMAL ARRAY 1.024E+02 1.000E+02 1.000E+00 2.000E+02 1.000E+00 NORMAL ARRAY 3.263E+02 3.000E+02 1.000E+00 2.000E+02 1.000E+00 NORMAL ARRAY 1.400£+03 1.000E+03 1·000E+00 Z·000E+02 1. 0 00E+00 NORMAL ARRAY 6.099E+03 3.000E+03 1.000F+00 2.000F+02 1.000~+OO NORMAL ARRAY 1.073£+00 3.000E+00 1.000E+00 6.000E-Ol 1.000~+OO LATFRAL ARRAY 1.161E+00 1.000E+01 1.00CE+OO 6.00CE-Ol 1.000E+00 LATERAL ARRAY 1.228E+00 3.000E+01. 1·000E+OO A.OOOE-Ol ,.OOOE+OO LATERAL ARRAY 1.278E+00 1.000E+OZ 1.000E+00 6.000E-Ol 1.000E+00 LATERAL ARRAY 1.302E+00 3.000E+02 1.000E+00 6.000E-01 1.000E+00 LATERAL ARRAY 1.312E+00 1.000E+03 1.000E+00 6.000E-01 1 .. 000E+00 LATERAL ARRAY 1.3151"+00 3.000E+03 1.000E+00 6.000E-01 1.0001"+00 LATF:RAL ARRAY 2.0301"+00 ~.OOOE+OO 1.000E+OO 2.000E+00 1.000E+OO LATERAL ARRAY 3.733E+OO 1.000E+01 1..000E+00 2.000E+0 0 1.0001"+00 LATERAL ARRAY 5.268E+00 3.000E+01 1.000E+OO Z.OOOE+OO 1.000E+00 LATERAL ARRAY 6.4801"+00 1.000£+02 1.0001"+00 2.000E+00 1.000E+00 LATERAL ARRAY 7_Z14E+00 ~.OOOE+02 1.0 00E+00 ;:>.OOOE+OO 1.000E+00 LATERAL ARRAY 7.700 E+00 1.000E+03 1·000E+00 2·000E+0 0 1.000E+(,)0 LATERAL ARRAY 7.903E+00 3.000E+03 1.000E+OO Z.OOOE+OO 1.000E+00 LATERAL ARRAY 2.824E+00 3.000E+00 1.000E+00 '.400E+00 1.0001:+00 LATERAL ARRAY 6.966E+00 1. 000E+Ol 1. 000 E+OO '.4 00E+00 1. 000 E+00 LATERAL ARRAY 1.184E.,.01 3.000E+Ol 1.000E+OO 3.400E+00 1.000~+00 LATERAL ARRAY 1.640E+Ol 1.OOOE+0;:> J.OOOE+OO 3.400E+OO 1. 00 0E+00 LATERAL ARRAY 1.921E+01 3.000E+OZ 1.000E+OO 3.400E+00 1. 000E+OO LATERAL ARRAY Z.176E+01 1.000E+O, 1·000E+00 ~'400E+OO , .OOOE+OO LATERAL ARRAY 2.2,4E+01 3.00 0E +0 , 1·000E+00 '.400E+00 ,..OOOE+OO LATERAL ARRAY 3.333E+OO 3.000E+00 1.000E+00 6.000E+00 1.000E+OO LATERAL ARRAY 1.109E+Ol 1.000E+01 1.000E+00 6.000E+00 1.000E+OO LATERAL ARRAY 2.4':!SE+01 3.000E+Ol 1.000E+00 6.000 E+OO 1.000E+(\Q LATERAL ARRAY 4.092E+01 1.000E+02 1.000E+OO 6.000E+00 1.00CE+OO LATERAL ARRAY 5.280E+Ol 3.000E+02 1.000E+OO 6.000E+00 1.000F.+OO LATERAL ARRAY 6-14,E+Ol 1.000E +0 3 1·00DE+0 0 6·000E+QO 1.000E+00 LATERAL ARRAY 6.674E+01 3.000E+03 1.000":+00 6.000E+00 1.000E+00 LATERAL ARRAY 3.328E+OO 3.000E+00 1.000E+OO 1.140E+01 1.00OE+00 LATERAL ARRAY 1.,53E+01 1. 000£+01 1.000 E+OO l'14 0E+01 1. 00 0E+00 LATERAL ARRAY 4.176E+Ol 3.000E+01 1.0 00 E+00 1.140E+01 1.000E+OO LATERAL ARRAY 9.629E+01 1. 000 E+02 1.000 E+OO 1.140E+Ol 1. 00 0E+OO LATERAL ARRAY 1.500 E+02 "3.0 00E+02 1. 000 E+00 lo140E+Ol 1. 000E+00 LATERAL ARRAY 1.948E+02 1. 000E+03 1.000 E+00 1.140E+01 1.000E+OO LATERAL ARRAY 2.21BE+02 3. 0QOE+03 1.000E+OO 1.140E+01 1.0001:+00 LATERAL ARRAY 3·1.71 E+00 ,.OOOE+OO 1·000E+00 Z·000E+01 1·000E+00 LATERAL ARRAY 1.264E+01 1. 000 E+01 1.000E+00 2.000E+01 1.000E+00 LATERAL ARRAY

48

APPENDIX B. COMPUTER PROGRAM FOR PLOT2ZN {CONTI D).

4.848E+Ol 3. 000E+01 1.603E+02 1. 000E+02 3.210E+02 3.000E+Ol 4.952E+02 1.000E+O, 6.113E+02 3.000E+O, 3.086E+OO ~.OOOE+OO 1.144E+01 1. 000 E+01 4.436E+01 3.000E+01 1.921E+02 1.000E+02 5.046E+02 3.000 E+02 9.525E+02 1. 00 0E+03 1.314E+03 3.000E+O, 3.030E+00 3.000E+00 1.047E+01 1.000E+01 3.~34E+Ol ~.OOOE+Ol 1.681E+02 1.000E +0 2 6.727E+02 3.000E+02 1.937E+03 1.000E+03 3.4~OE+03 3.00 0E +0 3 3.010E+00 3.0 00E+00 1.015E+01 1.000E+01 3.158E+Ol ,.000E+01 1.22~E+02 1.000E +0 2 5.443E+02 3.000E+02 2.585E+O, 1. 000E+03 6.789E+03 3.000E+03 3.003E+00 3.000E+OO 1.006E+01 1. 000E+01 3.055E+01 3.000E+Ol 1.070E+02 1. 000E+02 3.937E+02 ~.OOOE+O? 2.132E+03 1.000E+O~ 8.684E+03 3.000E+03

4 :3

1. 000 E+00 1.000E+OO 1.000E+00 1.000 E+OO 1·000 E+00 1. 000 E+OO 1. 000 E+00 1.000E+00 1.000E+00 1. 000 E+00 1.000E+00 1.000 E+00 1.000E+00 1.000[+00 1·000E+00 1.000 E+00 1.000E+00 1.000E+00 1·000E+00 1.0 00 E+00 1.000E+00 1.0 00 E+00 1.000E+00 1.000 E+00 1.000 E+00 1. 000 E+00 1.000E+00 1.000£+00 1.000E+OO 1. 000 E+00 1·000E+OO 1.000 E+00 1.000E+00

2.000E+01 2. 000 E+01 2.000E+01 2.000E+Ol 2.000E+01 "140E+01 3.140E+Ol 3.140E+01 3.140E+01 ,,.140E+01 "h 140E+01 3.140E+01 6.000E+01 6.000E+01 6·000E+01 6·0 0 0 E+Ol 6.000E+01 6.000E+01 6·000 E+01 1.140E+02 1.140E+02 1·140E+02 1'140E+02 1.140E+02 10140E+02 1.140E+02 2.000E+02 2.000E+02 2·000E+02 2.000E+02 ?000E+02 2.0 00 E+02 2.000E+02

10. 1.0 1.0 10. 100.0 1.0 1.0 5.0 100. 1.0 1.0 10. 100.0 1.0 1.0 30.0 10. 1.0 1. 0 10. 100. 1.0 1. 0 10.

1.000E+OO J.OOOE+OO 1.000E+00 1.000E+OO 1. 00 0E+00 1. 000E+00 1. 000 E+00 1.000E+OO 1.000E+00 1. 000E+OO 1.000E+OC 1.000E+00 1.000E+OO 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 ,.OOOE+OO 1.000E+00 1.000E+OO 1$000E+00 1. 000 E+00 1.00()EH10 1.000E+OO 1. 000E+OO 1.000~+OO ,.OOOE+OO 1. 000E+00 1.000E+OO

100. 2.0 5.0 250. APPARE~T RESISTIVITY RHOA/RHOO (O~C. 1970) T~Ut R0C~ RFSISTIVITY RHOD/pHOO ( TWO ZONE CAS~l.

L=72 T~. 00=::> IN (-)' 1 0 IN II). AND 20 I~ (.l L=64 IN. DO.? T~ (-). 10 IN II,. ~.ND 20 IN (.J L=16 IN. D0. 2 IN 1-). 10 IN (I). AND 20 IN (.1 L=??4 IN DO",? IN (-It Ii) IN 1/), AND 2u IN .,.1

L=72 IN. DO= 2 IN (-). 10 TN II). AND 20 IN (.) L=64 IN. DO. 2 IN (-), 10 IN (II. AND 20 IN (.1 L=16 IN. DO. 2 IN (-1' 10 IN (/1' ANn 20 r~ (.l L=224 IN DO=? IN (-1' 10 IN II). AND 2v IN I.)

LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERA,L ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LA H"RAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY LATERAL ARRAY