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.



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.)



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)



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)



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



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



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



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.



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.



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



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

archie parameter determination by analysis of saturation data

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