Conductivity model for pyrite-bearing laminated and dispersed shaly sands based on a differential equation and the generalized Archie equation

The conductance of pyrite-bearing laminated and dispersed shaly sands is not well understood and resistivity models for pyrite-bearing shaly sands are nonexistent. Thus, we first synthesize clean pyrite-matrix samples, and quartz-matrix samples with variable laminated shale, dispersed shale, and pyrite content and then perform petrophysics experiments to assess the effect of pyrite content on the conductivity of pyrite-bearing shaly sands. Second, based on the differences in conductivity and conduction pathways and geometries because of the variable composition of the pyrite-bearing laminated and dispersed shaly sands, we divide the shaly sands into their components, i.e., laminated shale, quartz grains, pyrite grains, hydrocarbon, dispersed shale, microscopic capillary water, and mobile water. A generalized resistivity model is proposed to describe the conductivity of pyrite- bearing laminated and dispersed shaly sands, based on the combined conductivity differential equation and generalized Archie equation. In the generalized resistivity model, the conductivity differential equation is used to describe the conductivity of dispersed inclusions in a host, whereas the generalized Archie equation is used to describe the conductivity of two conducting phases. Moreover, parallel conductance theory is used to describe the conductivity of dispersed shaly sands and laminated shale. Theoretical analysis suggests that the proposed model satisfies the physical constraints and the model and experimental results agree. The resistivity and resistivity index of shaly sands decrease with increasing conductivity and pyrite. Finally, the accuracy of the resistivity model is assessed based on experimental data from 46 synthetic core samples with different oil saturation. The model can describe the conductivity of clean pyrite-matrix samples, and quartz-matrix samples with different volumes of laminated shale, dispersed shale, and pyrite. An accurate saturation model of pyrite-bearing laminated and dispersed shaly sands is thus obtained and the log data interpretation in complex shaly sands can improve with the proposed model.

A quantitative interpretation of the saturation exponent in Archie’s equations

Saturation exponent is an important parameter in Archie’s equations;however,there has been no well-accepted physical interpretation for the saturation exponent.We have theoretically derived Archie’s equations from the Maxwell–Wagner theory on the assumption of homogeneous fluid distribution in the pore space of clay-free porous rocks.Further theoretical derivations showed that the saturation exponent is in essence the cementation exponent for the water–air mixture and is quantitatively and explicitly related to the aspect ratio of the air bubbles in the pores.The results have provided a theoretical backup for the empirically obtained Archie’s equations and have offered a more physical and quantitative understanding of the saturation exponent.