Electrical Properties of Rocks • Electrical properties of a rock depend on the pore geometry and fluid distribution • Electric current by “ionic conduction” • Consider the following tank completely filled with brine water, → apply a voltage, v → measure a current, i → calculate a resistance by Ohms Law: v = i r → Define water resistivity, Rw, as: Definitions i v A L water LAwrwR Electrical Properties of Rocks • Consider the tank completely filled with 100% brine saturated, porous sand • Resistance with respect to the water phase • Resistance with respect to fluid-filled, porous rock • Since ro  rw, Definitions i v Water+sand A L La pAaLwRwr ALoRor LaLpAAwRoRElectrical Properties of Rocks • Define the Formation Resistivity Factor, F, as: • Define tortuosity; • Define porosity, • Thus Simplified theoretical relationship between F and f does not account for heterogeneity. Definitions wRoRF 2LaLfApAfFElectrical Properties of Rocks General relationship based on both theoretical and experimental studies is given by: F = a f –m where a and m are functions of pore geometry. Methods: a. Simple theoretical models simple models designed with uniform pore geometry do not capture variation in porous media. b. Direct measurement in lab accurate but requires rock sample c. Empirical correlations based on lab data most convenient and popular, however may not be appropriate for given rock type f – F relationshipElectrical Properties of Rocks b. Direct measurement in lab “The practical application of F = f(f) for a particular rock type is best accomplished by evaluating the cementation factor using lab-measured values of F and f.” ....Helander (1983) f – F relationshipElectrical Properties of Rocks c. Empirical Correlations Archie (1942) suggested the following empirical equation based on lab measurements: F = f -m f – F relationship F dependent on degree of cementation, thus m originally defined as : “cementation exponent”.Electrical Properties of Rocks Empirical Correlations Winsauer, et al (1952) - analyzed data from 30 samples (28 ss, 1 lms, 1 unconsolidated ss) Developed correlation known as “Humble Eq.” F = 0.62f -2.15 Tixier (1979) – simplified equation using same data F = 0.81f -2 f – F relationshipElectrical Properties of Rocks f – F relationship F = f-mF = .81f-2F = .62f-2.15F = 1.45f-1.541.010.0100.01 10 100 1000 10000Formation Resistivity FactorPorosity, %2.81.61.82.02.52.21.4mfracturesvugs orspherical poresLow fnon-fracturedcarbonatesElectrical Properties of Rocks Define: m – pore geometry exponent a – pore geometry (tortuosity) factor Characteristics: • Coefficient a varies from 0.35 to 4.78 and m from 1.14 to 2.9 (higher in carbonates) • Observed variation in m-exponent, attributed to: – Degree of cementation • an increase in cementation increases m – Shape, sorting and packing of grains – Types of pores: intergranular, vuggy, fractures • fractures m ~ 1.0, vugs m > 2.0 – tortuosity – constriction in porous network – presence of conductive solids – compaction due to overburden pressure – thermal expansion f – F relationshipElectrical Properties of Rocks Consider the tank filled with a porous sand saturated with both water and hydrocarbons. Resistance with respect to the water phase only, Resistance with respect to the porous, hydrocarbon bearing rock, Since rt  rw, Resistivity-Saturation Relationship pAaLwRwrALTRtr aLaLpApAoRTR i v Water+sand +oil A L La’Electrical Properties of Rocks Define resistivity index, I as: Archie correlated experimental data and suggested: Combine, Plot, Resistivity-Saturation Relationship oRTRI nwScI nwScoRTRoRTRlogn1)clog(n1)wSlog(Electrical Properties of Rocks From plot, n=2 and c = 1, thus Resistivity-Saturation Relationship tRoRwS Electrical Properties of Rocks * Only valid when hydrocarbon and water zones are of the same porosity and salinity General form known as Archie’s Law. Fundamental relationship which the entire well logging industry is based!! Resistivity-Saturation Relationship tRoRwS tRwFRwS Electrical Properties of Rocks • Observed variation in saturation exponent, n attributed to: 1. wettability of rock surface 2. rock texture 3. presence of clays 4. measurement techniques; i.e., static vs dynamic 5. nature of displacing fluid Resistivity-Saturation Relationship Fluid distribution in the pore spaces as a function of fluid wettability. Water and oil saturations in (a) a water-wet sand and (b) an oil-wet sand. Pirson (1958)Electrical Properties of Rocks Wettability influence on rock surface Resistivity-Saturation Relationship Resistivity Ratio vs. water saturation in carbonate cores Anderson, JPT, (Dec 1986)Electrical Properties in Characterizing Porous Media • Objective – Determine tortuosity in CK equation • Methods: – Ionic transit time – Resistivity (either measured or theoretical) foTpvTkkwhereSkk2Electrical Properties in Characterizing Porous Media • Results – Ionic transit time – Theoretical a. Simple system – flow parallel to length b. Flow is at angle to length (Cornell & Katz) c. Multidimensional (Wyllie & Gardner) General Form Where 1 ≤x ≤ 2  2.1fF 2fF fF 22fFyxFfElectrical Properties in Characterizing Porous Media Archie (1942) Tixier (1979) Hutchinson (1961) General form 2fF12fF281.0fF81.02fF241fgdFgdF412fmaFfaFmfEmpirical Correlations implies a=f() Substitute for F: