AAPG Bulletin
 Copyright ©2004. The American Association of Petroleum Geologists. All rights reserved.
Abstract
The concept of rock fabric has been shown to be very useful for characterization of carbonate reservoirs. This study shows that a Pickett crossplot of interparticle porosity vs. true resistivity (in some cases, apparent resistivity or true resistivity affected by a shale group) should result in a straight line for intervals with a constant rock fabric. The slope of the straight line is related to the porosity exponent m, the water saturation exponent n, and the size of the particles forming the interparticle porosity. Different slopes are obtained for different rock fabrics. The method helps to reconcile geology to fluid flow by illustrating the important link between geology, petrophysics, and reservoir engineering.
Lines of constant rock fabric are displayed on a Pickett plot, together with water saturation, permeability, process speed k/ϕ, capillarypressure curves, porethroat apertures r_{35} and r_{p35}, Kozeny's constant (F_{s}τ^{2}), and height above the freewater table. Pattern recognition while placing all these data in a consistent form on a Pickett plot allows determination of flow units and a more rigorous characterization of carbonate reservoirs. The method is aimed at heterogeneous carbonate reservoirs, which have a limited amount of hard data.
The use of this technique is illustrated with data from the Mission Canyon Formation in the Little Knife field of North Dakota, where a significant volume of oil in place is below the structural closure and updip wells penetrate microports that provide an effective seal in this stratigraphic trap.
Roberto Aguilera is president of Servipetrol Ltd. in Calgary, Canada and an adjunct professor in the Chemical and Petroleum Engineering Department at the University of Calgary, where he concentrates in teaching about the theoretical and practical aspects of naturally fractured reservoirs. He is a petroleum engineering graduate from the Universidad de America at Bogota, Colombia, and holds a master's degree and a Ph.D. in petroleum engineering from the Colorado School of Mines. He was an AAPG instructor on the subject of naturally fractured reservoirs from 1984 to 1996. He has presented his course on naturally fractured reservoirs and has rendered consulting services throughout the world. He is a Distinguished Author of the Journal of Canadian Petroleum Technology (1993 and 1999), a recipient of the Outstanding Service Award from the Petroleum Society of the Canadian Institute of Mining, Metallurgy, and Petroleum Engineers (CIM) in 1994, and a Society of Petroleum Engineers Distinguished Lecturer on the subject of naturally fractured reservoirs for 2000–2001.
INTRODUCTION
Pickett plots (1966, 1973) have long been recognized as very useful in log interpretation. In Pickett's method, a resistivity index, I, and water saturation, S_{w}, are calculated from loglog crossplots of porosity vs. true resistivity.
The Pickett plot has been extended throughout the years to include many situations of practical importance. For example, Aguilera (1974, 1976) demonstrated that Pickett plots could be used for evaluating naturally fractured reservoirs. In these formations, the value of the porosity exponent, m, was shown to be smaller than usual.
Sanyal and Ellithorpe (1978) and Greengold (1986) have shown that a Pickett plot should result in a straight line with a slope equal to (n − m) for intervals at irreducible water saturation.
Aguilera (1990a) extended the Pickett plot to the analysis of laminar, dispersed, and total shale models. In this approach, the resistivity included in the plot is affected by a shale group, A_{sh}, whose value depends on the type of shaly model being used. Aguilera showed that all equations for evaluation of shaly formations published in the literature, no matter how long they are, become S_{w} = I_{sh}^{−1/n}. He further showed that a Pickett plot for shaly formations should result in a straight line with a slope equal to (n − m) for intervals at irreducible water saturation.
Aguilera (1990b) demonstrated that a loglog crossplot of R_{t} vs. effective porosity as determined from neutron and density logs, minus freefluid porosity as determined from a nuclear magnetic log, should result in a straight line with a negative slope equal to the water saturation exponent, n, for intervals that are at irreducible water saturation. Extrapolation of the straight line to 100% porosity yields the product aR_{w}. Gas intervals plot above the straight line. Intervals with movable water plot below the straight line.
In the same paper, Aguilera (1990b) showed that a Pickett plot should result in a straight line for intervals of constant permeability at irreducible water saturation. The same concept has been used successfully by Doveton et al. (1996) at the Kansas Geological Survey.
Martin et al. (1997) used the Pickett plot as part of a scheme to characterize petrophysical flow units in carbonate reservoirs. They presented examples from a dolomitized grainstone in the Little Knife field of North Dakota, an oomoldic limestone in the Geneseo field of Kansas, and a dolomite in the Howard Glasscock field of the Permian basin. Their data are used for illustrating the method developed in this paper.
More recently, Aguilera (2002) presented techniques for incorporating capillary pressure, poreaperture radii, height above freewater table, and Winland r_{35} values on Pickett plots (Kolodzie, 1980). He showed that a Pickett plot should result in a straight line for intervals with constant capillary pressure at irreducible water saturation. The technique was shown to be valid for water saturations between 30 and 90%. The method was extended for determination of flow units and reservoir containers (Aguilera and Aguilera, 2002) and evaluation of matrix flow units in naturally fractured reservoirs (Aguilera, 2003a).
“Permeability and capillary properties of interparticle pore space can be related to interparticle porosity and geologic descriptions of particle size and sorting called rock fabrics” (Jennings and Lucia, 2003, p. 215). This paper shows how to construct lines of constant rock fabric on a Pickett plot. A schematic of this approach is shown in Figure 1, where P_{c1} and P_{c2} are constant capillary pressures; h_{1} and h_{2} are heights above the freewater table; r_{1} and r_{2} are constant porethroat aperture radii; (k/ϕ)_{1} and (k/ϕ)_{2} are constant process or delivery speeds; and λ_{1} and λ_{2} are constant rockfabric numbers. The rockfabric numbers are related to the size of particles in the rock.
Integrating rock fabric, poregeometry categories (intergranular, intercrystalline, vuggy, fracture) as suggested by Coalson et al. (1985), poresize classes (mega, r_{p35} > 10 μm, macro 2.5–10 μm, meso 0.5–2.5 μm, micro 0.1–0.5 μm, and nano 0.01–0.1 μm), Kozeny's constant (F_{s}τ^{2}), capillary pressures, height above freewater table, and lines of constant process speed, all on a Pickett plot, permits determination of flow units and a more rigorous characterization of fracture carbonate reservoirs.
DEFINITIONS
The attempt to integrate geology (primarily via rock fabric), petrophysics (primarily via Pickett plots), and reservoir engineering (primarily via flow or hydraulic units) on a single graph leads necessarily to terminology that is not used routinely by all disciplines. This section presents some of the key definitions I am using in this paper.
“Flow (or hydraulic) unit” is a stratigraphically continuous reservoir subdivision characterized by a similar pore type (Hartmann and Beaumont, 1999, p. 9–7). The flow unit concept was introduced originally by Ebanks (1987) as an aid to reservoir description for engineering projects in sandstone reservoirs.
“Container” is a reservoir system subdivision, consisting of a pore system, made up of one or more flow units that respond as a unit when fluid is withdrawn. Containers are defined by correlating flow units between wells (Hartmann and Beaumont, 1999, p. 9–7).
“Ports” (doors) are pore throats. The term was coined by Martin et al. (1997). A similitude is that a pore can be visualized as a room with one or more doors of different shapes in each wall. The bigger the doors, the larger the number of people that can go in and/or out of the room simultaneously. In the case of rocks, the pore throats are the doors (ports) to the pore (Hartmann and Beaumont, 1999, p. 921). Port is also a nice shortcut for “pore throat.”
“r_{35}” refers to the size of pore throats at 35% nonwetting phase saturation as determined from mercury injection capillary pressure tests. The r_{35} equation was developed originally by H. D. Winland of Amoco (Kolodzie, 1980). It is explained in detail by Hartmann and Beaumont (1999, p. 931–933). A similar technique for calculating porethroat radii at 20% nonwetting saturation in the Cardium sandstone of Canada had been developed previously by MacKenzie (1975). Pittman (1992) used the same concept for developing equations that allowed construction of a poreaperture distribution curve using porosity and permeability from core analysis of sandstone reservoirs.
“r_{p35}” refers to the size of pore throats at 35% nonwetting phase saturation. It was developed by Aguilera (2002) using data published by Kwon and Pickett (1975). For practical values of porethroat apertures, it provides similar results to r_{35}. This is significant because the r_{35} and r_{p35} equations were developed independently from different data sets.
“Process” (or) “delivery speed” is the ratio of permeability and porosity and is related directly to porethroat aperture. The term was coined by Chopra et al. (1987) and Gunter et al. (1997b) as a relative indication of how quickly fluids can move through porous media. The same ratio was used previously by Pickett and Artus (1970) for predicting recoverable hydrocarbon volumes from Ordovician carbonates in the Williston basin based on crossplots of ∑S_{o}ϕh_{n} vs. recoverable oil.
“Rock fabric” refers to the geologic description of particle size and sorting (Lucia, 1983, 1995). Poresize distribution is related to rock fabric and controls permeability and saturation.
“Pore geometry” refers to the architecture and internal anatomy of the rock. It is described preferentially by the shape of the mercury injection capillary pressure curve, scanning electron microscopy, and thinsection work.
“Kozeny's constant (F_{s}τ^{2})” refers to the combined effect of formation (shape) factor and tortuosity. Carman (1937) and several publications in the geology and petroleum engineering fields have reported values of F_{s}τ^{2} in the order of 5. However, Rose and Bruce (1949) indicated that for most reservoir rocks, the value of F_{s}τ^{2} ranges between 5 and 100. Rose and Bruce's concept is adopted in this paper.
“Height above the freewater table” refers to the height above the level at which capillary pressure is equal to zero. This is different from the production engineer's wateroil contact that can be higher than the freewater table. Wells produce 100% of water below the production engineer's wateroil contact.
“Porosity exponent m” (or cementation factor) in the Archie equation (1942) can be a good indicator of rockfabric pore geometry. When the rock has nonconnected (nontouching) vugs, the value of m is larger than the porosity exponent of the interparticle porosity. When the rock has fractures and/or connected (touching) vugs, the value of m is smaller than the porosity exponent of the interparticle porosity. When the rock has interparticle porosity plus fractures and/or connected vugs plus nonconnected vugs, the value of m can be larger or smaller than the porosity exponent of the interparticle porosity, depending on the relative volume of each pore geometry (Aguilera and Aguilera, in press).
“Partitioning coefficient v” is equal to fracture porosity (and/or connected vug porosity) divided by total porosity. The term was coined by Pirson (1957). The concept is powerful for petrophysical evaluations and for making preliminary estimates of oil and gas recovery in naturally fractured reservoirs (Aguilera, 1999).
“Vug porosity ratio v_{nc}” is equal to nonconnected vug porosity divided by total porosity. The term was coined by Lucia (1983) and is used in dual porosity models (Aguilera and Aguilera, 2003).
“Hydraulic diffusivity” is an important part of the diffusivity equation, which is at the heart of fluidflow calculations in porous media by hydrogeologists and reservoir engineers. It is equal to transmissibility (kh_{n}/μ) divided by storage (ϕc_{t}h_{n}). Because of this, the process speed (k/ϕ) is the petrophysical parameter that best and more rigorously characterizes at present a hydraulic (flow) unit, although it does not take into account viscosity (μ) and compressibility (c_{t}). Research on improvements along these lines is being conducted currently at Servipetrol in Calgary. Important support to the process speed for characterizing flow units is provided by rock fabric (Lucia, 1983, 1995), flow zone indicators (Amaefule et al., 1993), Lorenz plots (Gunter et al., 1997a), and various well logs. When using logs, it is important to perform environmental corrections (e.g., bed thickness resolution, invasion, and borehole shape), because the logs can be averaging poregeometry properties.
CARBONATE ROCK FABRIC
Lucia's (1983, 1995) work on carbonate rocks forms the basis for the incorporation of rock fabric on Pickett plots developed in this study.
Figure 2 shows a graph of porosity vs. permeability on loglog coordinates for various particlesize groups in uniformly cemented nonvuggy rocks published originally by Lucia (1983, 1995). There is a reasonable correlation for average particle sizes of greater than 100, 20–100, and less than 20 μm. An expanded version of the graph (Figure 3) shows that all the lines intersect at an interparticle porosity of 3.5% and a permeability of 0.0015 md. The same intersect also shows up in the Pickett plot that includes rock fabric developed in this study.
Figure 2 suggests that under favorable conditions, it is possible to make an estimate of permeability based on knowledge of interparticle porosity and particle size. Recent data published by Jennings and Lucia (2003) provided a similar type of graph and led them to the generation of a rockfabric number for different rockfabric petrophysical classes. Figure 4 shows a straight line semilog correlation between particle size (d_{p}) from Figure 2 and the rockfabric number (λ). The importance of λ resides on the fact that there is “a continuum of particle size and sorting fabric” (Jennings and Lucia, 2003, p. 218) in carbonate rocks. Furthermore, the fabric and interparticle porosity is related to permeability and porethroat aperture (Jennings and Lucia, 2003). These relationships permit the generation of equations to place all of these properties on a Pickett plot.
PICKETT PLOT
The use of Pickett plots as an aid in characterizing petrophysical flow units in carbonate reservoirs has been documented by Martin et al. (1997) and Aguilera and Aguilera (2002). The basic equations in formation evaluation (Archie, 1942) are combined as proposed by Pickett (1966) with the process speed (Aguilera and Aguilera, 2002) and Jennings and Lucia (2003) rockfabric equations to obtain where a(λ) = a_{o} − a_{1}ln(λ), b(λ) = b_{o} − b_{1}ln(λ), and the constants a_{o} = 22.56, a_{1} = 12.08, b_{o} = 8.671, and b_{1} = 3.603. These constants apply strictly to the data shown in Figure 2. For better results, it is recommended to prepare the same type of plot using cores from the actual carbonate reservoir being studied to validate or change the constants as required. Equation 1 indicates that a crossplot of R_{t} vs. ϕ on loglog coordinates should result in a straight line with a slope equal to (−c_{3}n − m + b(λ)n/c_{4}) for intervals at irreducible water saturation with constant aR_{w} and constant λ. Extrapolation of the straight line to 100% porosity yields the product [aR_{w}(c_{2}^{−n})(e)^{a(λ)n/c}^{4}]. Development of equation 1 is presented in Appendix 1.
Special care must be exercised with the interpretation of resistivity data because there are many variables that affect the readings. Some of them are bed thickness, mud invasion, and borehole shape. Not making the necessary corrections via specialized charts or computer programs can lead to excessive averaging of pore geometries. A possibility is to improve the resistivity values using forward modeling simulation to generate a corrected log response.
CARMANKOZENY CONSTANT
Statistically, the flat part of a capillary pressure curve is associated with the mean port flow size and corresponds approximately to the value of r_{35} (Hartmann and Beaumont, 1999, p. 921). The porethroat aperture r_{35} provides approximately the same results as r_{p35} for most cases of practical importance (Aguilera, 2002). By assuming that r_{p35} is approximately equal to the mean hydraulic radius (r_{mh}), it is possible to calculate Kozeny's constant (Kozeny, 1927; Carman, 1937) from the equation
Equation 2 indicates that Kozeny's constant (F_{s}τ^{2}) is a function of porosity and permeability and consequently a function of r_{p35}. For unconsolidated homogenous sandstones, the value of F_{s}τ^{2} is in the order of 5.0. However, Rose and Bruce (1949) have indicated that for most reservoir rocks, the value of F_{s}τ^{2} ranges between 5 and 100. Equation 2 meets Rose and Bruce's criteria for most practical values of process speed (k/ϕ) and r_{p35}. Development of equation 2 is presented in Appendix 2.
APPLICATION
This application deals with the Mission Canyon Formation in the Little Knife field of North Dakota. The field was discovered in January 1977 by wildcat well Gulf 118.
The Mission Canyon (lower Mississippian) is part of the Madison Group and lies between the Lodgepole Limestone below and the Charles Formation above. The Mission Canyon is a shoalingupward, regressive carbonate to anhydrite sequence deposited by a shallow epeiric sea. Most of the carbonates, including the reservoir, are subtidal. The formation consists primarily of limestones interbedded with anhydrites and dolomites. The beds were deposited originally as carbonate muds in environments that ranged from open marine to coastal sabkha. The beds forming the lower part of the reservoir were deposited in a transitional open to restricted marine environment. The beds forming the upper part of the reservoir were deposited in a restricted marine environment. These beds were transformed into porous reservoir rocks by three main diagenetic changes: (1) replacement of fossil fragments by anhydrite, (2) dolomitization of the muddy limestone, and (3) complete to partial leaching of the anhydrite (Lindsay, 1982; Desch et al., 1984; Lindsay and Kendall, 1985; Heck et al., 2003). The trap is provided by structural closure in the north, west, and east sides of the reservoir. There is stratigraphic entrapment to the south provided by nanoports and microports (Martin et al., 1997). A seal is provided by anhydrite beds and nanoports and microports.
Figure 5 shows key characteristics of the Mission Canyon, including (1) a schematic representation of pore types using a scale of 30 μm, (2) a photomicrograph of a calcareous dolostone using a scale of 25 μm, (3) a relief pore cast using a scale of 25 μm, (4) a semilog crossplot of porosity vs. permeability from cores, (5) average capillary pressure, (6) average porethroat radii, and (7) a photograph of a slabbed core from the transitional open to restricted marine facies (Desch et al., 1984). The capillary pressure presented in Figure 5E can be matched reasonably well using Aguilera's equation (2002) as shown in Figure 6. The original equation, shown as an insert in the figure, generates the dashed line in Figure 6. Although the original equation was developed for water saturations ranging between 30 and 90%, it is interesting to observe that in this case, the equation also provides reasonable results for water saturations smaller than 30%. I have observed this on several occasions. The triangles in Figure 6 represent an adjusted curve changing constant c_{5} from 19.5 to 16. Core porosity and permeability data from Figure 5D can be used for estimating porethroat aperture r_{p35} using the template (Aguilera, 2003b) shown in Figure 7. The template is based on the equation (Aguilera and Aguilera, 2002) shown at the top of the graph. The template for calculating r_{p35} follows the same format used by Martin et al. (1997) for calculating r_{35}.
The Pickett plots for wells Leo Klatt 31948 and Kadrmas 1 discussed in this paper were published originally by Martin et al. (1997). Figures 8 and 9 show the plots incorporating lines of constant water saturation, process speed k/ϕ, capillary pressures, porethroat radii r_{p35}, and height above the freewater table. These lines were generated with equations explained by Aguilera and Aguilera (2002) using the following basic data: For the above data, the porethroat radius is r_{p} ∼ 108.1/P_{c}, and the height above freewater table is h ∼ 0.705 P_{c}.
Figures 8 and 9 also include Jennings and Lucia's (2003) rockfabric numbers (λ) ranging between 0.5 and 4.0. The corresponding particle size from Figure 4 ranges between 500 and 1.8 μm. Note that all the straight lines that represent rock fabrics intersect at a porosity of 3.5% and a permeability of 0.0015 md (same as in Figure 3).
A means of drawing a constant rockfabric line is by calculating two values of R_{t} from equation 1 at two assumed values of porosity and drawing a straight line between the two points. For example, in Figure 8, for a rockfabric number (λ) of 1.5, an assumed porosity of 1.0 calculates an R_{t} of 5.85 ohm m. A porosity of 0.03 calculates an R_{t} of 93.39. These porosity and R_{t} points are represented by triangles in Figure 8. The straight line connecting the triangles corresponds to a rockfabric number equal to 1.5. The slope of the straight line is −c_{3}n − m + b(λ) × n/c_{4} = −3 × 2 − 2 + 7.21 × 2/2 = −0.79. Based on Figure 4, this straight line also corresponds to an average particle size of 100 μm.
Well Leo Klatt 31948 tested initially 485 bbl/day of oil. The well accumulated 979,000 STBO between July 1977 and May 1993. Well Kadrmas 1 was tested on October 1987. The well died during a drillstem test while recovering only 2 L of oil and water cushion in 30 min.
The wells considered in Figures 8 and 9 are characterized by distinct Pickett plots. In the Klatt well (Figure 8), which recovered 979,000 STBO, flow (hydraulic) units 3, 4, and 6 have low water saturations, high values of process speed k/ϕ, and reasonable porethroat apertures (r_{p35}). Flow unit 4 displays the best characteristics. It shows the highest process speed (k/ϕ ∼ 7082 md, porosity is a fraction); it is composed by megaports (r_{p35} = 18.12 μm, larger than 10 μm); it has the smallest rockfabric number (λ<1.5) and the smallest Kozeny's constant (F_{s}τ^{2} = 47.0). To read the values of r_{p35} and F_{s}τ^{2} from the Pickett plot, take the zone of interest, go parallel to the corresponding line of k/ϕ until reaching S_{w} = 65%, and read the porethroat value and F_{s}τ^{2} at that point. Flow unit 6 is the second best zone. It shows a process speed, k/ϕ, of 712 md, the presence of macroports (r_{p35} = 6.45 μm, between 2.5 and 10 μm), a rockfabric number of approximately 1.8, and a Kozeny's constant of 59.17. The next best flow unit is zone 3. It shows a process speed of approximately 32 md, it is characterized by mesoports (r_{p35} = 1.6 μm, between 0.5 and 2.5 μm), and it shows a rockfabric number of about 2.0 and a Kozeny's constant of 80.66. The other zones are poor. They show the lowest process speeds, r_{p35} values that correspond mostly to microports, high water saturations, and rockfabric numbers larger than 2.5, which correspond to muddominated fabrics (packstones and wackestones). The above analysis indicates that all the methods provide essentially the same ranking. This leads to the conclusion that all the approaches are useful, support each other, and become more powerful when used jointly.
The Kadrmas well (Figure 9), which recovered only 2 L of oil and water cushion during a drillstem test, shows that all the intervals have very high water saturations and lower values of process speed k/ϕ than the Klatt well. Porethroat apertures at 35% mercury saturation (65% water saturation) read from the Pickett plots are smaller than in the Klatt well. The rockfabric numbers and Kozeny's constant (F_{s}τ^{2}) are larger than in the Klatt well, indicating poorer reservoir characteristics. The rock fabric number (λ) is always larger than 2.5, indicating that the zones are composed of muddominated fabrics (packstones and wackestones) as reported by Martin et al. (1997).
Summarizing, the integrated evaluations presented in Figures 8 and 9 help to reconcile geology (primarily via rock fabric) to petrophysics (primarily via Pickett plots) to fluid flow (primarily via flow units) and are diagnostic for characterizing a carbonate reservoir and distinguishing a good well from a dry well. In this case, the production interval in the good well (Klatt) is below structural closure and forms part of a stratigraphic trap. The bad well (Kadrmas), updip from the Klatt well, penetrates microports that act as a seal.
The advantage of the pattern recognition approach presented in this paper is that we do not have to rely on just one method. The Pickett plot is an ideal tool for integrating various geological, petrophysical, and reservoir engineering methods and, as such, provides a valuable link between the various disciplines.
EFFECT OF VUGS AND FRACTURES
Vugs and/or fractures (secondary porosity) are probably more common than not in most carbonate reservoirs. If the contribution of vugs and fractures is relatively minor, the method presented in this paper can still be used (always with care). For example, some cores from the Mission Canyon application presented here have shown a minor amount of fracturing with a westnorthwest or eastnortheast orientation. The fractures propagate short distances vertically and laterally (Desch et al., 1984). In their original interpretation of well Kadrmas 1, Martin et al. (1997) indicated that some of the cores were fractured. These data calculated larger values of r_{35} and were given a different symbol in their plots. In this way, they could isolate the porethroat apertures of the interparticle porosity and still conduct a complete analysis of the matrix. Thus, one approach is to use the method by considering only those intervals without vugs and fractures.
A different approach has to be used if the vugs and/or fractures extend throughout the whole thickness being investigated by logs. In this case, the preparation of the Pickett plot requires knowledge of the isolated matrix (interparticle) porosity and isolated true resistivity of the matrix (R_{tb}). Matrix porosity can be determined from a dualporosity model (Aguilera and Aguilera, 2003). Values of R_{tb} can be calculated using the same dual porosity model based on knowledge of true resistivity of the composite system (Aguilera, 2003a).
The effect of fractures on a Pickett plot of total porosity (matrix plus fractures) vs. true resistivity of the composite system (matrix plus fractures) is to generate a curvature downward in some cases (instead of the customary straight line) for intervals of constant water saturation as shown in Figure 10. In this case, the partitioning coefficient (v) is constant. The same kind of plot using a total porosity made out of matrix plus nonconnected vugs can also result in a curvature when the vug porosity ratio (v_{nc}) is constant. If the nonconnected vug porosity is constant (instead of v_{nc} being constant), the curvature would go in an opposite direction to the one shown in Figure 10. Whenever possible, it is important to calibrate the dual porosity models with core data. A triple porosity model (Aguilera and Aguilera, in press) is required in those instances where matrix, fractures, and nonconnected vugs are present at the same level.
CONCLUSIONS

A loglog crossplot of porosity vs. true resistivity (in some cases, apparent resistivity or resistivity affected by a shale group) should result in a straight line for intervals with a constant rockfabric number (λ).

In addition to rock fabric, it is possible to have, in a Pickett plot, water saturation, capillary pressure, porethroat aperture, r_{p35}, Kozeny's constant (F_{s}τ^{2}), permeability, process speed (k/ϕ), and height above the freewater table. The integration of these geologic, petrophysical, and reservoir engineering properties permits determination of flow units and a more rigorous characterization of carbonate reservoirs. The method helps to reconcile geology to fluid flow.

The method is aimed at heterogeneous carbonate reservoirs with a limited amount of hard data. However, more significant results can be obtained if the empirical equations presented in this study are calibrated with cores.

Although imperfect, the integration of geology (via rock fabrics), petrophysics (via Pickett plots), and reservoir engineering (via flow units) will help us do better work with carbonate reservoirs.
Nomenclature

a = constant in Archie formation factor equation

a_{0} = 22.56 (Jennings and Lucia, 2003)

a_{1} = 12.08 (Jennings and Lucia, 2003)

a(λ) = a_{o} − a_{1} ln (λ)

A = c_{5} S_{w}^{−c}^{6}= constant in equation for calculating capillary pressure for a given water saturation (Aguilera, 2002)

A_{sh} = shale group for miscellaneous models

b_{o} = 8.671 (Jennings and Lucia, 2003)

b_{1} = 3.603 (Jennings and Lucia, 2003)

b(λ) = b_{o} − b_{1}(λ)

c_{t} = total compressibility (psi^{−1})

c_{1} = exponent in capillary pressure equation (Aguilera, 2002)

c_{2} = constant for oil or gas in Morris and Biggs (1967) permeability equation (for example, 250 in the case of some mediumgravity oils and 79 in the case of dry gas at shallow depth)

c_{3} = exponent of porosity in equation to calculate permeability in the Morris and Biggs (1967) equation (for example, 3.0)

c_{4} = nth root of permeability in the Morris and Biggs (1967) equation (for example 2.0 or square root)

c_{5} = constant in equation to calculate A (for example, 19.5)

c_{6} = exponent of water saturation in equation to calculate A (for example, 1.7)

c_{7} = (1 − c_{3}c_{4})(c_{1})

c_{8} = (−c_{6} + c_{4}c_{1})

d_{p} = particle diameter (μm)

F = formation (shape) factor

F_{s}τ^{2} = Kozeny's constant

h = height above freewater table (ft)

h_{n} = net pay (ft)

I = resistivity index

I_{sh} = resistivity index of shaly formation

k = absolute permeability (md)

ln = logarithm base e

log = logarithm base 10

m = porosity exponent

n = water saturation exponent

p_{c} = A [k/(100ϕ)]^{−c}_{1}= capillary pressure (mercury injection) (psi)

r_{mh} = ϕ/[S_{gv}(1 − ϕ)] = mean hydraulic radius (μm)

r_{p} = porethroat aperture radius (μm)

r_{35} = 5.395 [k^{0.588}/(100ϕ)]^{0.864} = Winland porethroat aperture corresponding to a mercury saturation of 35% (μm)

r_{p35} = 2.665 [k/(100ϕ)]^{0.45} = Aguilera porethroat aperture corresponding to a mercury saturation of 35% (μm)

R_{o} = formation resistivity of zone 100% saturated with water (ohm m)

R_{t} = true formation resistivity (ohm m)

R_{tb} = true resistivity of the matrix (ohm m)

R_{w} = water resistivity (ohm m)

S_{gv} = surface area per unit grain volume (μm)

S_{wi} = irreducible water saturation, fraction (unless stated otherwise)

S_{w} = water saturation, fraction (unless stated otherwise)

S_{o} = oil saturation, fraction

v = partitioning coefficient, fraction (unless stated otherwise)

v_{nc} = vug porosity ratio, fraction (unless stated otherwise)
Subscripts

D = density

fl = free fluid

N = neutron
Greek Symbols

λ = rockfabric number

Θ = air/mercury contact angle

Θ_{h} = water/hydrocarbon contact angle

μ = viscosity (cp)

μm = micrometers

ρ_{h} = hydrocarbon density (g/cm^{3})

ρ_{o} = oil density (g/cm^{3})

ρ_{w} = water density (g/cm^{3})

σ = air/mercury interfacial tension (dynes/cm)

σ_{h} = water/hydrocarbon interfacial tension (dynes/cm)

τ = tortuosity

ϕ = effective porosity, fraction (unless stated otherwise)
APPENDIX 1: INCORPORATION OF ROCK FABRIC ON PICKETT PLOTS
Jennings and Lucia (2003) calculated permeability as a function of rock fabric from
Combining the Archie equation (1942), as proposed by Pickett (1966), and the Morris and Biggs (1967) permeability equation, Aguilera and Aguilera (2002) showed that
Inserting equation 3 into equation 4 leads to which is the same as equation 1 in the text.
APPENDIX 2: INCORPORATION OF KOZENY'S CONSTANT ON PICKETT PLOTS
The CarmanKozeny equation (Kozeny, 1927; Carman, 1937) can be written as follows: where F_{s}τ^{2} is commonly known as Kozeny's constant. The mean hydraulic radius (r_{mh}) in micrometers is given by Inserting equation 6 into 7 leads to Porethroat aperture at 35% mercury saturation is given by (Aguilera and Aguilera, 2002):
Assuming r_{mh} = r_{p35} and combining equations 8 and 9 allows calculation of Kozeny's constant from which is the same as equation 2 in the text.
Acknowledgments
I am grateful to Jack Thomas, Marc Longman, and Joseph Studlick for their detailed review, their advice, and their contribution to the manuscript.
 Manuscript receivedJuly 8, 2003.
 Revised manuscript receivedOctober 16, 2003.
 Revised manuscript receivedNovember 24, 2003.
 Final acceptanceDecember 1, 2003.