Electrical resistivity is one of the most commonly collectedtypes of geophysical well log data used in groundwaterinvestigations. Because electrical currents respond to changesin a formation’s electrical properties and pore space char-acteristics, resistivity data are typically used together withgamma ray logs to qualitatively interpret changes in fluidproperties (i.e., freshwater versus contaminated ground-water) and to locate clay-rich formations. Our interest indeveloping more quantitative approaches to using electri-cal resistivity data to constrain transport properties hasresulted in development of a simple upper bound on thetotal porosity-electrical resistivity relationship.

This new upper bound on total porosity-electrical resis-tivity data is based on the theoretical Hashin-Shtrikmanbounds evaluated at the formation’s percolation thresholdφp and critical porosity φc. It is easily computed and is muchtighter than the theoretical Hashin-Shtrikman upper bound.This upper bound can be used to (1) define ranges in totalporosity-resistivity data, (2) determine a range in the totalporosity for a given electrical resistivity datum or vice versa,and (3) help constrain geologic and petrophysical interpre-tations of resistivity data.

Electrical Hashin-Shtrikman bounds. The theoreticalHashin-Shtrikman (HS) bounds for electrical conductivityprovide upper and lower limits for the total porosity-elec-trical resistivity relationship in a multiconstituent media. TheHS bounds provide the tightest constraints on the totalporosity-electrical resistivity relationship without specify-ing the geometries of each constituent in the granular sys-tem.

Data, however, do not plot anywhere near the upper HSresistivity bound since the upper HS bound immediatelyapproaches the resistivity of the mineral constituent (Figure1). The upper HS bound is therefore of limited use in con-straining total porosity given electrical resistivity data ingranular materials. The HS bounds are very wide for con-stituents with extremely different conductivities (i.e., quartzsand-conductive fluid system) and tighter for constituentswith very similar conductivities (i.e., clay-conductive fluidsystem). In the HS bound equations, the pore fluid is treatedas one of the constituents; if the pore space is fully saturatedwith a single fluid, water for example, then the ff is equal tothe porosity fraction and σf is the conductivity of the porefluid, where the subscript f denotes the fluid constituent.

The upper HS resistivity bound (Figure 1) correspondsto isolated conductive spheres (pores) covered by a resis-tive shell (mineral), a likely scenario for pumice and somebasalts. The lower HS resistivity bound corresponds to iso-lated resistive spheres (mineral) covered by a conductiveshell (water and/or clay), a more likely model for mostgranular formations.

The total porosity-formation factor relationship. Muchlike permeability, the formation resistivity factor F is a char-acteristic transport property of a material. Formation factoris a measure of the inability for an electrical current to flowthrough pore space and approximated as the measured for-

A new method for constraining total porosity: The new total porosity-electrical resistivity upper boundWENDY WEMPE, Schlumberger Water Services, Sacramento, California, USAGARY MAVKO, Stanford University, California, USA

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Box 1. Electrical Hashin-Shtrikman bounds.

The upper HS bound on resistivity RHS+ is equal to theinverse of lower conductivity HS bound σHS-

-1 and is a functionof the individual constituent fractions fi and conductivities σiand can be easily calculated by the following equation:

(1)where

and where σmin = σ1 is the minimum conductivity (σmin ≤ σi)and N is the total number of constituents.

The lower HS bound on resistivity RHS- is equal to theinverse of the upper conductivity HS bound σHS-

-1+ and is sim-

ply calculated by the following equation:

(2)where

Figure 1. The Hashin-Shtrikman resistivity bounds. Shown are boundsfor two systems, both saturated with a fluid with Rw = σ1

−1 = 1 Ω-m:(1) Rm1 = σ2

−1 = 1015 Ω-m (upper dashed line), and (2) Rm2 = σ2−1 = 104

Ω-m (lower dashed line). The lower HS bound (bold solid line) isessentially the same at φ ≥ 0.01 for both systems. The black regions inthe schematic sphere packs represent the resistive constituent and thewhite represents the conductive constituent.

mation electrical resistivity F normalized by the saturatingfluid resistivity Rw, (F ≈ R/Rw). Archie derived the empiri-cal relationship F = φ-m, which was later generalized as F =aH/φ-m. The internal geometry parameter aH and cementa-tion exponent m are computed as the y-intercept and nega-tive slope, respectively, of the least-squares fit to φ - F dataon a log-log plot. aH and m are thought to provide infor-mation about the formation’s pore space characteristics. Ithas been suggested that by subtracting percolation thresh-old φp from the total porosity, Archie’s equation could bemodified to account for the lack of the flow of electrical cur-rents through isolated pore space.

(3)

Porosity domain theory. Porosity domain theory explainshow the full possible range in total porosity φ values (0 < φ< 1) in a granular material can be divided into three phys-ically, electrically, elastically, and hydraulically distinctdomains. These porosity domains are characterized by (1)isolated pores, (2) contacting grains and connected pores,and (3) suspended grains. The three total porosity domainsare separated by the granular material’s characteristic porouspercolation threshold φp and critical porosity φc.

Porosity domain 1: Total porosity is less than the mater-ial’s percolation threshold (0 < φ < φp), and all of the porespace is isolated so that no pore space is available for the flowof electrical currents and fluids.

Porosity domain 2: Total porosity values fall between thematerial’s percolation threshold and critical porosity (φc <φ < φp), and grains are in contact with each other and poresare connected; the ability for electrical currents to flowthrough the pore space depends on the constituent electri-cal properties and the pore space characteristics (i.e., sort-ing, compaction, dispersed clay volume, and cementation).

Porosity domain 3: Total porosity is greater than the mate-rial’s critical porosity (φc < φ < 1), and the grains are in sus-pension so that all of the pore space is available for the flowof electrical currents, and the suspension itself is a fluid.

Rarely in natural granular materials do we encountertotal porosities that fall within domains 1 or 3; however, thelimits of these domains (φp and φc) set limits on the totalporosity range of domain 2, the domain of most natural gran-ular materials.

Results: Defining the formation factor-porosity relation-ship in each total porosity domain. The slope of φ - R/Rwdata on a log-log plot is to some degree an indication of porespace and bulk characteristics such as shape, sorting, pack-ing, pore configuration and size, tortuosity, type of pore sys-tem (intergranular, intercrystalline, vuggy, fractured),compaction, and clay content. It is well accepted that φ -R/Rw data in cemented granular materials have higher slopesthan do data from unconsolidated materials on a log-log plot.However, regardless of the pore space characteristics, φ -R/Rw data in sands and sphere packs appear to converge near

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Figure 2. φ - R/Rw lab and borehole data. Data converges near the HSlower bound within the range of the critical porosity values for respec-tive formation types. Shown are loose clean sands, loose glass spheres,loose glass cubes, limestones and dolomites, sandstone, clay-rich sand-stone, and artificially cemented glass spheres. The lower HS bound(solid line) is plotted for reference. Data converges near the HS lowerbound at approximately 0.30 ≤ φ ≤ 0.4. See suggested reading fordata references.

the lower HS bound in some porosity range near the criti-cal porosity values of sands and sandstones (Figure 2).

We suggest that a formation’s critical porosity and per-colation threshold divide the range of total porosity valuesinto three domains within which the φ - R/Rw relationshipmay be defined (Table 1).

Porosity domain 1: Pore space is no longer connected (0<φ < φp) and the formation’s resistivity is close to the bulk

mineral resistivity; the formation factor in domain 1 may beapproximated by the upper HS bound (equation 1) nor-malized by the fluid resistivity Rw.

Porosity domain 2: Grains are in contact and pores are con-nected (φp ≤ φ ≤ φc) and the ability for electrical currents toflow through the pore space is influenced by the pore spacecharacteristics; in domain 2 formation factor and porosity arelinearly related on a log-log plot by the modified Archie’s equa-tion (equation 3).

Porosity domain 3: The grains are in suspension (φc > φ >0) so that all of the pore space is available for the flow of elec-trical currents, and the formation’s resistivity approaches the

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Box 2. New upper resistivity-porosity upper bound in porositydomain 2.

The empirical upper bound is linear on a log-log plot likethe modified Archie’s equation, and it connects the points (φp,R*

HS+ /Rw) and (φc, R*HS- /Rw). The first point corresponds to the

upper HS bound evaluated at the percolation threshold φp; thesecond point corresponds to the lower HS bound evaluated atthe critical porosity φc.

(4)where

Figure 3. Formation factor in each total porosity domain. The φ - R/Rwrelationship is defined by (1) the upper HS bound HS+ (equation 1) inporosity domain 1, where φ < φp, (2) the modified Archie’s equation(equation 3) in porosity domain 2, where φp ≤ φ ≤ φc, and (3) the lowerHS bound HS- (equation 2) in porosity domain 3, where φ > φc.

Figure 4. Example bounds for φ - R/Rw data glass sphere packs. φ -R/Rw lab data for loose glass spheres (circles) and artificially cementedglass spheres (stars) from Wyllie and Gregory (1953). Also shown arethe HS bounds (solid and dashed lines) and the new upper bound (dot-ted line), where φc = 0.40, φp = 0.035, Rm = 1014 W - m and Rw = 1 Ω- m. The data are also plotted in Figure 3.

bulk fluid resistivity; the formation factor in domain 3 maybe approximated by the lower HS bound (equation 2) nor-malized by Rw.

Results: New upper resistivity-porosity upper bound inporosity domain 2. The upper HS bound immediatelyapproaches the bulk mineral constituent resistivity at the onsetof adding resistive minerals to the system (Figure 1). This resultis not observed in data from granular systems. This discrep-ancy likely exists because the upper HS bound correspondsto isolated conductive spheres (pores) covered by a resistiveshell (mineral), an unlikely scenario in granular systems. Theupper HS bound therefore is of limited use in constrainingtotal porosity or formation factor data in sediments and mostrocks.

We develop an empirical upper resistivity bound in poros-ity domain 2 that is tighter than the upper HS bound. Theempirical upper bound on φ - R/Rw data is based on four keyobservations and theories:

• φ - R/Rw data plots along a straight line on a log-log plot(Archie’s modified equation) for a given formation whenporosity is between the percolation threshold φp and crit-ical porosity φc

• regardless of the formation pore space characteristics, φ -R/Rw data in granular systems converge toward the lowerHS bound at the formation’s characteristic φc (Figure 2)

• at total porosities greater than a formation’s φc, φ - R/Rwdata plot along the lower HS bound (Figure 2)

• at total porosities less than the formation’s φp (percolationtheory), the formation’s electrical resistivity can be approx-imated by the upper HS bound (Figure 3)

φ - R/Rw data in unconsolidated glass sphere packs fall justabove the lower HS bound for φ < φc and on the lower HSbound for φ > φc (Figure 4). Data from artificially cementedglass spheres plot just below the empirical upper φ - R/Rwbound. The empirical upper bound is easily computed andmuch tighter than the theoretical HS bound.

Conclusions. The lower Hashin-Shtrikman bound is inde-pendent of the mineral resistivities and volume fractions ifRm/Rw is greater than approximately 103, which is consistentwith most hydrogeologic systems.

φ - R/Rw data from unconsolidated glass sphere packs falljust above the lower HS bound in porosity domain 2 (φ < φp)and on the lower HS bound in porosity domain 3 (φ > φc).

The φ - R/Rw data from clean and clay-rich sands and inglass sphere packs tend to converge near the lower HS boundat porosities between approximately 0.30 and 0.40, which cor-respond to the range in accepted critical porosities of sandsand spheres.

The φ – F relationship can be defined in each porositydomain: (1) domain 1 (0 ≤ φ ≤ φp), F≈RHS+/Rw, (2) domain 2(φp < φ < φc), F = a/(φ-φp)m, and (3) domain 3 (φc ≤ φ ≥ 0),F≈RHS-/Rw.

An empirical upper bound on the R/Rw relationship canbe defined by the line F+ = a+/(φ-φp)m+, where a+ and m+ arethe y-intercept and negative slope of a line that connects thepoints (φp,/Rw) and (φc,/Rw) on a log-log plot.

Suggested reading. “The electrical resistivity log as an aid in deter-mining some reservoir characteristics” by Archie (AIMETransactions, 1942). “Avariational approach to the theory of effec-tive magnetic permeability of multiphase materials” by Hashinand Shtrikman (Journal of Applied Physics, 1962). “Three distinctporosity domains defined physically, hydraulically, electrically,

and elastically” by Wempe and Mavko (TLE, 2000). “A self-sim-ilar model for sedimentary rocks with application to the dielec-tric constant of fused glass beads” by Sen et al. (GEOPHYSICS, 1981).Data references: “Formation factors of unconsolidated porousmedia: Influence of particle shape and effect of cementation” byWyllie and Gregory (AIME Transactions, 1953). “Analogy betweendiffusion and electrical conductivity in porous rocks” byKlinkenberg (GSABulletin, 1951); “Conductivilite d’un electrolytecontentant des spheres dielectriques” by Slawinski (Journal deChimie Physique, 1926). Estudio experimental del factor de resis-tivitidad y la porosidad en medios no consolidados de muy altaporosidad by Balderas-Joers (Tesis profesional, UniversidadNacional Autonoma de Mexico, 1975). “New relationship betweenformation resistivity factor and primary and secondary porosi-ties” by Mendoza-Romero and Perez-Rosales (SPWLA, 26th

Annual Logging Symposium, 1985). “Resistivity of brine-saturatedsands in relation to pore geometry” by Winsauer et al. (AAPGBulletin, 1952). “Application of electrical resistivity measurementsto problem of fluid flow in porous media” by Wyllie and Spangler(AAPG Bulletin, 1952). “The effect of temperature on electricalresistivity of porous media” by Sanyal et al. (The Log Analyst, 1973).“Some theoretical considerations related to the quantitative eval-uation of the physical characteristics of reservoir rock from elec-trical log data” by Wyllie and Rose (AIME Transactions, 1950).“Formation factor-porosity relation derived from well log data”by Porter and Carothers (The Log Analyst, 1971). TLE

Acknowledgments: The authors thank the Stanford Rock Physics and BoreholeGeophysics (SRB) consortium for their support during Wempe’s gradu-ate research.

Corresponding author: wwempe@slb.com

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