True Resistivity

Key Takeaways

• Archie's equation, which uses Rt as its primary measured input, was derived empirically by Gus Archie (1942) from laboratory measurements of resistivity on water-saturated and partially water-saturated sandstone cores, establishing the power-law relationships between formation resistivity factor F = Ro/Rw (where Ro is the resistivity of the 100 percent water-saturated rock), porosity (F = a / phi^m), and water saturation (I = Rt/Ro = Sw^(-n), where I is the resistivity index); the cementation exponent m (typically 1.8 to 2.2 for clean sandstones) characterizes the effect of the pore geometry and connectivity on the formation resistivity factor, with higher m values for rocks with a higher proportion of dead-end pore space or complex pore geometry; the saturation exponent n (typically 2.0 for water-wet rocks) characterizes the effect of water saturation on the resistivity index, with n values above 2.0 for oil-wet or mixed-wet rocks where the water saturation exists as isolated droplets rather than as a continuous conducting film on the grain surfaces; the accuracy of Sw calculated from Rt using Archie's equation depends on the accuracy of all three inputs (Rt, Rw, and phi), with errors in each propagating non-linearly into the Sw result and therefore into the pore volume of hydrocarbons (HCPV = phi * (1 - Sw) * net pay volume) that determines the recoverable reserve estimate.
• Deep-reading resistivity tools designed to measure Rt include the dual induction tool (DIL, measuring the conductivity at two investigation depths using electromagnetic induction at 20 kHz, with the deep induction ILD providing the best approximation to Rt when the invasion is less than 30 inches and the contrast between Rxo and Rt is less than 10:1), the array induction tool (AIT, using 5 receiver arrays at different spacings to provide a full invasion profile and a deconvolved Rt at 1-foot resolution), the dual laterolog (DLL, using focused current injection at kilohertz frequencies to provide deep (LLD) and shallow (LLS) laterolog measurements in high-resistivity formations such as carbonates where induction tools are less accurate due to signal non-linearity at very high resistivities), and the azimuthal resistivity tools (used in LWD to provide directional formation resistivity measurements relative to the borehole wall orientation, enabling real-time geosteering and dip-corrected Rt in deviated or horizontal wells); the choice between induction and laterolog tools depends primarily on the formation resistivity range (induction preferred for Rt below 100 ohm-m, laterolog preferred for Rt above 100 ohm-m in carbonate and salt-cemented formations) and the mud system (induction tools require conductive mud, laterolog tools work in any mud but perform best in oil-based mud).
• Invasion effect on apparent deep resistivity measurements is the most significant source of error in Rt determination for permeable formations: when mud filtrate invades the near-wellbore formation during and after drilling, the flushed zone (at the borehole wall, with resistivity Rxo reflecting the filtrate saturation Sxo) and the invaded zone (at a radius of 10 to 80 inches from the borehole, with a gradual transition from Rxo to Rt) affect the deep resistivity measurement in proportion to the fraction of the tool's radial sensitivity that falls within the invaded zone; the three-resistivity separation plot (plotting Rxo, shallow resistivity, and deep resistivity at each depth) provides a visual indicator of invasion: when all three readings are equal, either there is no invasion or the invasion is both very shallow and very deep (both cases giving apparent Rt equal to the true Rt); when Rxo is lower than the deep resistivity (which occurs in an oil-bearing formation invaded by water-based mud), the deep reading may be suppressed toward Rxo and requires invasion correction to recover Rt; the invasion correction charts in the log interpretation chart books (or the software-based inversion of array tool data) use the three-resistivity readings to solve simultaneously for Rxo, Rt, and invasion radius, recovering Rt that may differ from the deep log reading by 50 percent or more in heavily invaded permeable formations.
• Non-Archie rocks (shaly sands, carbonates with complex pore systems, fractured formations) require modifications to the basic Archie Sw calculation because the simple clean sand model (where current flows only through the pore water) does not apply: in shaly sands, the clay mineral surfaces carry a negative surface charge that attracts mobile sodium cations (the diffuse double layer), creating a parallel conduction path through the clay-bound water in addition to the free water in the pore throats; this clay conductance makes the apparent Rt lower than the true non-clay-influenced resistivity at the same water saturation, causing Archie Sw to be overestimated (more water than is actually present) when applied without correction; the Waxman-Smits model and the dual water model (Clavier, Coates, Dumanoir) correct for clay conductance by adding a clay exchange cation (Qv) term to the resistivity equation, with Qv measured from cation exchange capacity (CEC) of core samples or estimated from log-based clay volume and clay type; in carbonates with vuggy or fracture-dominated porosity systems, the Archie m exponent varies significantly from the clean sandstone value (m of 2.0 to 2.5 in vugs, m of 1.5 to 1.8 in fractures versus m of 2.0 for inter-granular porosity), requiring core-calibrated m values for accurate Sw calculation from Rt.
• In core analysis, true resistivity has a slightly different definition from the log interpretation context: when a core plug is only partially saturated with water (the remainder being oil or gas), the resistivity measured in the core laboratory at the partial water saturation is referred to as Rt in the core context, as distinguished from Ro (the resistivity of the core when 100 percent saturated with the test brine); the resistivity index I = Rt/Ro at various water saturations is measured on core plugs by the air-brine centrifuge drainage capillary pressure method or by the porous plate method, producing the Sw-Rt data pairs that are used to calibrate the saturation exponent n in the Archie equation for that specific reservoir rock; n values determined from core resistivity index measurements (typically in the range of 1.5 to 3.0, with values above 2.5 indicating mixed wettability or oil-wet pore surfaces that maintain the oil phase as a continuous conductive film rather than isolating it in the pore centers) are applied to the log-measured Rt to calculate the formation Sw, with the n value being the single most uncertain parameter in the Archie equation and the largest source of Sw calculation uncertainty in formations where the rock wettability is not established by core measurements.