THE a COEFFICIENT

Historically, the a coefficient always has appeared in the numerator of the modified Formation Resistivity Factor, and has been perpetuated in industry with no defined purpose. In this model there is no support for the appearance of an a with a constant value in the formation factor. The formation factor is intrinsic to the rock at 100% water saturation. The a coefficient is not an intrinsic property, but is dependent on variables not related to the electrically inert rock or the physical geometries of its pores. The a is related neither to volume nor shape imparted to the network of electrically-conductive water paths. The a coefficient contributes nothing to the efficiency of the conversion of water resistivity to resistance, or vice versa. In the model, the shift in resistivity corresponding to a always is found in the resistivity axis, and the shift varies from data to data with depth. Coefficient a emerges as an inherent and inseparable factor of the complex resistivity relationship, as in Eqs.(1a) and (1b), and is related to the proportions of all secondary electrically-conductive constituents and influences that vary the relative proportions of R_w and R_wb .

It can be seen in the graphics of Figure 2 that R_we is the product of a and R_w , therefore aR_w= R_we .   R_we varies from depth to depth, and so does coefficient a. It can be seen in this relationship that a varying a coefficient technically can be used in single-water single-porosity methods where R_we is not calculated. But, in dual-water dual-porosity methods where R_we is calculated, the a coefficient cannot be used as an independent parameter. If the a is used in single-water single-porosity methods, it must be calculated properly. And that is difficult to do when the correct value of a varies with rock constituents and depth, and only one water and one porosity is available to work with. That is one of the reasons why dual-water dual-porosity methods were conceived.

When a is associated with R_w , as in the equation developments of (1a) and (1b), and as a multiplier or reduction factor for R_w , the a coefficient more readily can be recognized as being an indispensable factor that accounts for those heterogeneities that produce additional electrical conductivity. Coefficient a is the composite factor that relates R_w to the resistivity value of the combination of R_w and R_wb , and all other natural electrically-conductive influences, in such proportions to result in R_we .

In the modified formation factor used in single-water single-porosity methods, the a coefficient appeared in the numerator. In the modified formation factor the a became a multiplier of the R_w in Archie’s saturation equation. The aR_w became a single-water single-porosity equivalent to R_we .

Secondary conductive influences, inherent to the rock, that produce coefficient a can be: clay shale, surface conductance, or solid semi- conductors such as pyrite and siderite as separate influences or in concert. One influence that is neither electrically conductive nor intrinsic is hydrocarbon saturation. These in turn, and perhaps others, can produce a change in rock resistivity.

The occurrence of one additional electrically-conductive influence would add one more term to Eqs.(1a) and (1b), or their equivalent, and if there were no change in S_wtϕ_t , the value of the compound coefficient a would be reduced. These influences, all, can change R_w to an apparent or pseudo value R_we with a corresponding change in R₀ to R_{0  corrected} at their respective porosities.

Technically, in dual-water dual-porosity methods, the coefficient a should be transposed from the modified Formation Resistivity Factor term to the water resistivity relationship. For example:

R_{0  corrected} = F_t R_we = ( a / ( ϕ_t^m ) ) R_w = ( 1.0 / ( ϕ_t^m ) ) aR_w = ( 1.0 / ( ϕ_t^m ) ) R_we

The question might be asked, "What is the difference?" There are a number of reasons, four of which are:

1.  The a coefficient is a required proportionality factor, related to all factors influencing the electrical conductivity of the rock, that convert resistivity R_w to R_we , and is calculated along with R_we at each depth increment.

2.  To prevent duplication by the user in the correction for conductive influences. Duplication occurs when both a and R_we appear in the same water-saturation equation, or when a appears as an independent variable in dual-water dual-porosity methods. Any saturation relationship involving resistivity can employ either coefficient a or R_we , but not both.

3.  To prevent the use of a constant artificial and unwarranted correction factor.

4.  Although the formation factor is meaningless at 100% porosity, the value of F_t always must be 1.0 when both porosity and water saturation in the F_t equation are 1.0.

As explained above, and in Figure 2,

                                                                           a = R_we / R_w                                                       (1c)

After substituting Eq.(1c) into Eq.(1b), a relationship is derived showing the proportionality terms in coefficient a for the usual shaly sand:

1 / a = ( S_weϕ_e ) / ( S_wtϕ_t ) + ( ϕ_ne / ( S_wtϕ_t ) ) ( R_w / R_wb )

It can be seen here that coefficient a is a function of water saturation as well as porosities. See APPENDIX (A) for further explanation. This is the evaluation for the a coefficient appearing in Figure 2.

This relationship is shown for comparison purposes or informative purposes only. It is not to be calculated and used independently in dual-water dual-porosity methods because it already has been incorporated in the calculation of R_we as can be seen in Eq.(1b).

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