archie parameter determination by analysis of saturation data
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024223 Supplement t o SPE 19399, Improved Da ta -Ana lysi s Method
Determines Archie Parameters From Core Data
R.E. Maute, M o b i l R&D Corp.; W.D Ly le , M o b i l R&D Corp.;
E.S. Sprunt , M o b i l R&D Corp.
Copyright Society of Petroleum Engineers
This manuscript was derived directly from SPEL,hich appeared this year in a Society of Petroleum Engineers ournal. The materialin this Supplement passed SPE peer review with the published paper. Permission
to copy is restricted to an abstract of not more than 300words. Write SPE BookOrder Dept., Library Technician, P.O. Box 833836,Richardson, TX 75083-3836
U.S.A. Telex 730989 SPEDAL.
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SPE 24223
SUPPLEMENT TO SPE 19399 "Improved Data Analysis Method
of Determining Archie Parameters 'm' and 'n' from Core Data"
by
R. E. Maute, W. D. Lyle, and E.S. Sprunt
This supplement contains the detailed mathematics of CAPE with equations that will
implement the method, as well as program flow diagrams. Please note that other mathematical
methods of solving the problem could have been used.
The basic problem is to determine the Archie parameters by minimizing he mean-square enor
between the measured saturations and the saturations computed using the Archie equation. As
noted in the main text, there are two separate cases to be considered. In the first case, the
parameter a is set equal unity, and the mean-square saturation error defined by
P @ l l n 2= t t [ 5 i jl L=1
-p@brn)i j
is to be minimized,where thej ndex sums over theP cores that were measured, and the i index
sums over the number of measurements Q jmade on each core. If a is not fmed at unity but is
allowed to be a fittingparameter, the mean-squared error% to be minimized s defined by
where the indices are a s described above.
In both equations (A.1) and (A.2), SGdenotes the ith measurement of water saturation of
core j, RGdenotes the corresponding elecmcal resistivity, andR, denotes the water resistivity of
corej.
Minimum values of the and% occur when the appropriate partial derivatives are set equal
to zero. In the case, the requirements are
and in the E;! case the requirements are
(A.4.)
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Considering first the case, the two partials in equation (A.3) lead to
and
The notation for this problem as well as the problem for E;? is considerably simplified by
introduction of new functionsg and h defined by
and
Using equation (A.7) along with the observation that the 2/n term does not affect the solution
form and n, the simplified form of equations (A.5) and (A.6) becomes
whereF1 andF2 are defined in the above equations.
Equation (A.10) can be simplified by noting that
(A.9)
(A. 10)
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Consequently, the portion of the summation of equation (A.10) involving In* is zero from
equation (A.9). This leads to an equivalent expression forF1as
(A. 12)
The above expression forF1will be used in the remainder of this appendix.
Inspection of equations (A.9) and (A. 12) reveal their obvious nonlinear nature. Therefone,
an analytical solution form and n is not possible. A numerical solution can be obtained by lin-
earizingF1andF2about some point near the true solution. The linearization is accomplished by
expanding the functions in a fxst order Taylor series. ForF1the series becomes
With a similar expression for the expansion forF2 . The derivatives in equation (A.13), as
well as the corresponding derivatives in theF2expansion are evaluated at the point mk, nk. Since
the solutions of equations (A.9) and (A. 12) require that bothF1andF2be equal zero, equation
(A. 13) and the corresponding equation forF2form the basis for an iterative numerical solution by
observing that if F1andF2were approximately zero at the point mk+l, nk+l then
and
(A. 14)
(A. 15)
The above two equations can be easily solved to yield for mk+l and nk+l the expressions
(A. 16)
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SPE 2 4 2 2 - 34
Equation (A. 16) above is the basic equation to solve for them and n that minimizes of
equation (A. 1). A starting point mo, o is required along with a stopping rule of the form
and
for some selected value of 61 and &. In practice the value of m 0 and n 0= 2 works well with the
values of c$ and 4 chosen to be .001.
The partial derivatives required in equation (A. 16) can be determined from equations (A.9)
and (A.12). These derivatives have been evaluated and found to be
and
aF1- - - - Rwj- 'Z Sij 2gij ) gij
am
n'J
(A. 19)
Again, it is worth noting that in equation (A. 16) the derivatives of equations (A. 19) through
(A.22) are evaluated at the point m=mk and n = nk.
In the case of e, here are three parameters (a,m,n) that are solved for in order to minimize
equation (A.2). In exactly the same manner as was done for the first case, an iterative algorithin
can be developed by defining the new functions HI, H2, andH3 that aredetermined from the t!hree
partial derivatives of equation (A.4). These are
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The new functions H1, H2, and H3 are thus defined by
and
H3=ZZ ( S Q -h i j ) h i j = O ,
where he was defined in equation (A.8).
Again, as was the case for F 1, there is a considerable simplification possible in the above
expressions. This simplification is possible by noting that in equation (A.26) the term involving
hi can be written as
using the definition of hi from equation (A.8). Defining a new function zqby
equation (A.29)becomes-h i i= a l / n h .rl
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6 SP E 2 4 2 2 3
Inspection of equations (A.26) - (A.28) reveals that using equation (A.3 I), a constant factor of alln
can be factored out. Since the HI, H2, and H3 are each set equal to zero, the constant factor alJn
will not influence the solution. Consequently, in terms of Eij the simplified form of equations
(A.26) - (A.28) become
A three-variable iteration algorithm similar to equation (A. 16) can be developed by expanding
each of the functions H1, H2, and H3 of equations (A.32) - (A.34) in a linear series of three vari-
ables as was done for F1andF2. Canying out these operations results in the algorithm for
solution of the a,m,n that minimizes as
In equation (A.35) the partial derivatives were evaluated using the H1, H2, and H3 of
equations (A.32) - (A.34) and found to be
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SPE 2 4 2 2 3
and
Equation (A.35) along with the derivatives in equations (A.36)-(A.44)evaluated at mb nk,
and akwill solve for the minimum of E;?.A stopping rule that has been found to work in practice is
as was the case for the solution for E ~ . 1,6, nd 63 typically have values of .001.
The initial point m,, no, and a, that works in practice is to set a, equal one with m,, no being
two, or the values obtained from the solution for the E;? case.
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A final point to be noted is that for the three-parameter case there is a requirement that the
number of cores be greater than one. Otherwise, equation (A.35)will not converge since the ma-
trixof partial derivatives is singular. This can be seen from equations (A.33) and (A.34) for the
single core case. In this case, equation (A.33) becomes
Clearly, in this special case, the equations forH2 andH3 are identical since equation (A.34) forH3
becomes
which is identical to equation (A.47). Thus, in order to apply the three-parameter a,m,n algorithm,
it is necessary to have data from two or more cores.
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WE 24223FLOW CHART - m AND n DETERMINATION
START
0SET POR, SW BOUNDS ON
DATA TO BE USED
(DEFAULT, USE ALL DATA)
SET 6 621TERATION CUTOFF I
US E INITIALCUES S mo= 2, no= 2.SE T ITERATION COUNTER k = 1.I
E S
I PRINT FINAL m, n I
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FLOW CHART - m, n, AND a DETERMINATIONSFBE 24
START
0SET POR, SW 'BOUNDS ON
DATA TO BE USED
(DEFAULT, USE ALL DATA)
I SE T S 1, S2, S31TERA TION CUT OFF IU SE INITIAL GUES S (e.g., m0=2, n0=2, ao=l).
SET ITERATION COUNTER k=l.
COM PUTE m vl , n ~ ,@
II
TI PRINT FINAL m, n, a I