THE m EXPONENTS

The porosity exponent m is an intrinsic property of the rock related to the geometry of the electrically-conductive water network imposed by the pore walls or surfaces of solid insulating materials. This has been verified by investigators of porous media in extremely meticulous and controlled laboratory investigations, particularly by Atkins and Smith (1961). All minerals have characteristic crystalline or particle shapes, whether the minerals are electrically conductive or not, and contribute to the geometry of the electrically conductive water residing in the pores through shape, physical dimensions of the pores and pore throats, tortuosity, configuration, continuity, pore isolation, orientation, irregularity, surface roughness, angularity, sphericity, and anisotropy. In addition, allogenic minerals, and authigenic mineral growths such as quartz or calcite or dolomite or clays, all, contribute to the water geometry within the pores. And, the electrical pathways through the insulating rock are configured further by secondary porosity in all its shapes and forms, such as: dissolution porosity, replacement porosity, fissures, fractures, micro-cracks, and vugs in their various orientations within the rock. Exponent m is related to all these features acting separately or in concert through the various inefficiencies (or efficiencies as the case might be) imparted to the electrically-conductive paths by the diversified void geometries, and this results in variations in electrolyte-filled path resistance and consequent rock resistivity. The greater is the inefficiency of the shape of the resulting water volume, the greater will be the value of exponent m, and the greater will be R₀ and/or R_t . This, too, can be seen in Figure 1.

Heterogeneous solid matter, if it is electrically inert, constitutes part of the framework of the rock only and has no other effect on resistivity other than by occupying a position in the rock framework and possibly displacing interstitial water and influencing the pore shape and pore geometry.

Heterogeneous electrically-conductive minerals such as pyrite and siderite present a different scenario. Their presence not only can affect the shape of the conductive water paths in the same manner as electrically inert minerals, but can provide electrical conductance in solid matter, not related to pore geometry, that can influence rock resistivity. The electrical conductivity of such minerals must be accommodated through the a coefficient in proportionality relationships in Eq.(1a) and (1b), or an equivalent.

In practice, the m exponent usually has a default value of about 2.0. In the laboratory, the minimum value for m (or m₁ ) in homogeneous granular media has been determined to be about 1.3 for spherical grains regardless of uniformity of grain size or packing (Atkins and Smith, 1961; Fricke, 1931; Pirson,1947; Wyllie and Gregory, 1952). It was shown in the development of the formation factor, above, and in Ransom (1984, 1995),that m will decrease in the presence of an open fracture, dissolution porosity, or fissure where the continuous void space is aligned favorably with the survey-current flow; and it was further demonstrated that m has an absolute minimum value of 1.0 at 100% efficiency.

In the concept in this paper, the porosity exponent m₁ can be estimated from rock containing 100% water by

                                   m = ( log₁₀ R₀ - log₁₀ R_w ) / ( log₁₀ 1 - log₁₀ϕ_e )                                   (5a)

and                             m = ( log₁₀ R_{0  corrected} - log₁₀ R_we ) / ( log₁₀ 1 - log₁₀ϕ_t )                  (5b)

The R_we in Eq.(5b) is from Eq.(1b) and reflects the change in relative proportions of R_w and R_wb as they are altered by the occupation of oil or gas. Although the slope m is intrinsic and the m from Eqs.(5a) and (5b) should be the same, it is slope m from Eq.(5b) that is seen in Figure 2.

Additionally, a value for m₁ can be determined that applies to a larger range of data. Turn to Figure 5. This figure is a plot of R_wa vs Clayiness. The assumption is made that ϕ_t , R_t , and Clayiness are reliable. The value R_wa is determined by dividing R_t by the formation factor in Eq.(3a). In interactive computer methods, the plot is entered by assuming a reasonable value for the m exponent. Select an interval that is believed to contain some wet zones, if possible. Most often an R_wz trend can be recognized in this plot, as seen in Figure 5. Both an R_w and R_wb often emerge. If an independent R_w value is known from a reliable source, convert that R_w to the in situ temperature of the logging tool environment at the depth of interest. If the R_w from the plot differs from the known value, iterate by program subroutine between the known R_w and the derived R_w by incrementally changing m in Eq.(3a) until the two values agree. The value of m determined in this manner is compatible with the log data for evaluation of R_w  and R_wb .

In the event that the m of Eq.(5a) does not agree with m in Eq.(5b), other methods for singular values of m are based on Figure 2. One method is based on the solution of m in similar triangles, and another is based on the dual salinities of R_w and R_we in a method similar to the laboratory method proposed by Worthington (2004). Both methods require considerable iteration in the evaluation of unknown variables in their respective equations.

It should be noted that on a log-log plot for either measured or corrected values of R₀ (or C₀ ) versus ϕ_t, as in Figure 2, values of m sometimes can be derived for each specific set of log data or rock sample. Individual values of R_we and R_{0 corrected} will continually change with changing clay shale content and water saturation. As a result, a reliable trend specific to comparable rock samples might not be observed for m unless the ratio of ϕ_ne / ϕ_t , or other appropriate discriminator, is held nearly constant to ensure a nearly constant relationship between the proportions of R_w and R_wb as porosity changes. And, laboratory measurements for m must show repeatability in both m and the resistivity of the influent and effluent after significant time lapses to prove that the measurement process, or sample degradation, does not influence the value of the measurement being made.