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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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~ APPARENT~ISTIVITYRHOA/~O WEC 1,1970} w
L=224 IN DO= 2 IN (-), 10 IN (/j, AND 20 IN
...
NORHAL AAAA Y ( . )
FIGURE 3. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR NORMAL ARRAY: L = 224 INCHES.
...
14
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L=72 IN. DO= 2 IN (-), 10 IN (/), AND 20 IN L)
FIGURE 4. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR NORMAL ARRAY: L = 72 INCHES.
...
>t--->--lii ~ ~ C> a:::
15
[)I~Ef , I-t-+--t-t--5
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(.)
FIGURE 5. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVTY FOR NORMAL ARRAY: L = 64 INCHES.
16
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FIGURE 7. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR LATERAL ARRAY: L = 224 INCHES.
18
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FIGURE 8. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR LATERAL ARRAY: L = 72 INCHES.
W
19
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FIGURE 9. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR LATERAL ARRAY: L = 64 INCHES.
I
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II
++- :
-It
L ...
20
[ , , 5 4
5
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L = 16 IN. DO = 2 I N ( - ), 1 0 I N (J), AND 20 I N
---
:
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= r'"
..,
LA T£RAL ARRAY (.)
FIGURE 10. GRAPH FOR DETERMINING TRUE MATRIX RESISTIVITY FOR LATERAL ARRAY: L = 16 INCHES.
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